LMFDB - Embedded newform 378.4.f.b.127.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(127,378)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.127");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.f (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 2x^{6} + 53x^{5} + 38x^{4} - 166x^{3} + 7x^{2} + 1543x + 2707 $ x^8 - 2x^7 - 2x^6 + 53x^5 + 38x^4 - 166x^3 + 7x^2 + 1543*x + 2707
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{6} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			127.4
Root			$-2.39336 + 0.0732839i$ of defining polynomial
Character	$\chi$	$=$	378.127
Dual form			378.4.f.b.253.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.54600 + 9.60595i) q^{5} +(3.50000 - 6.06218i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.54600 + 9.60595i) q^{5} +(3.50000 - 6.06218i) q^{7} -8.00000 q^{8} +22.1840 q^{10} +(-33.4982 + 58.0206i) q^{11} +(2.43841 + 4.22346i) q^{13} +(-7.00000 - 12.1244i) q^{14} +(-8.00000 + 13.8564i) q^{16} -69.3851 q^{17} -74.9150 q^{19} +(22.1840 - 38.4238i) q^{20} +(66.9964 + 116.041i) q^{22} +(-25.4130 - 44.0166i) q^{23} +(0.983753 - 1.70391i) q^{25} +9.75366 q^{26} -28.0000 q^{28} +(-81.5021 + 141.166i) q^{29} +(57.5174 + 99.6231i) q^{31} +(16.0000 + 27.7128i) q^{32} +(-69.3851 + 120.178i) q^{34} +77.6440 q^{35} -126.697 q^{37} +(-74.9150 + 129.757i) q^{38} +(-44.3680 - 76.8476i) q^{40} +(-76.9999 - 133.368i) q^{41} +(-203.974 + 353.293i) q^{43} +267.986 q^{44} -101.652 q^{46} +(-166.808 + 288.920i) q^{47} +(-24.5000 - 42.4352i) q^{49} +(-1.96751 - 3.40782i) q^{50} +(9.75366 - 16.8938i) q^{52} +558.813 q^{53} -743.124 q^{55} +(-28.0000 + 48.4974i) q^{56} +(163.004 + 282.332i) q^{58} +(250.021 + 433.049i) q^{59} +(-136.931 + 237.172i) q^{61} +230.070 q^{62} +64.0000 q^{64} +(-27.0469 + 46.8466i) q^{65} +(203.229 + 352.004i) q^{67} +(138.770 + 240.357i) q^{68} +(77.6440 - 134.483i) q^{70} +824.686 q^{71} +141.718 q^{73} +(-126.697 + 219.446i) q^{74} +(149.830 + 259.513i) q^{76} +(234.487 + 406.144i) q^{77} +(572.244 - 991.156i) q^{79} -177.472 q^{80} -307.999 q^{82} +(234.784 - 406.657i) q^{83} +(-384.810 - 666.510i) q^{85} +(407.948 + 706.587i) q^{86} +(267.986 - 464.165i) q^{88} -1641.31 q^{89} +34.1378 q^{91} +(-101.652 + 176.067i) q^{92} +(333.616 + 577.841i) q^{94} +(-415.479 - 719.630i) q^{95} +(-607.712 + 1052.59i) q^{97} -98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8	$ 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8} - 4 q^{10} - 5 q^{11} + 21 q^{13} - 56 q^{14} - 64 q^{16} + 46 q^{17} - 188 q^{19} - 4 q^{20} + 10 q^{22} - 374 q^{23} - 41 q^{25} + 84 q^{26} - 224 q^{28} - 271 q^{29} + 243 q^{31} + 128 q^{32} + 46 q^{34} - 14 q^{35} - 362 q^{37} - 188 q^{38} + 8 q^{40} + 213 q^{41} + 238 q^{43} + 40 q^{44} - 1496 q^{46} - 675 q^{47} - 196 q^{49} + 82 q^{50} + 84 q^{52} - 108 q^{53} - 2828 q^{55} - 224 q^{56} + 542 q^{58} - 202 q^{59} + 1212 q^{61} + 972 q^{62} + 512 q^{64} - 549 q^{65} - 139 q^{67} - 92 q^{68} - 14 q^{70} + 2590 q^{71} - 4000 q^{73} - 362 q^{74} + 376 q^{76} + 35 q^{77} + 1545 q^{79} + 32 q^{80} + 852 q^{82} + 142 q^{83} + 793 q^{85} - 476 q^{86} + 40 q^{88} + 264 q^{89} + 294 q^{91} - 1496 q^{92} + 1350 q^{94} - 1244 q^{95} + 638 q^{97} - 784 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8 - 4 * q^10 - 5 * q^11 + 21 * q^13 - 56 * q^14 - 64 * q^16 + 46 * q^17 - 188 * q^19 - 4 * q^20 + 10 * q^22 - 374 * q^23 - 41 * q^25 + 84 * q^26 - 224 * q^28 - 271 * q^29 + 243 * q^31 + 128 * q^32 + 46 * q^34 - 14 * q^35 - 362 * q^37 - 188 * q^38 + 8 * q^40 + 213 * q^41 + 238 * q^43 + 40 * q^44 - 1496 * q^46 - 675 * q^47 - 196 * q^49 + 82 * q^50 + 84 * q^52 - 108 * q^53 - 2828 * q^55 - 224 * q^56 + 542 * q^58 - 202 * q^59 + 1212 * q^61 + 972 * q^62 + 512 * q^64 - 549 * q^65 - 139 * q^67 - 92 * q^68 - 14 * q^70 + 2590 * q^71 - 4000 * q^73 - 362 * q^74 + 376 * q^76 + 35 * q^77 + 1545 * q^79 + 32 * q^80 + 852 * q^82 + 142 * q^83 + 793 * q^85 - 476 * q^86 + 40 * q^88 + 264 * q^89 + 294 * q^91 - 1496 * q^92 + 1350 * q^94 - 1244 * q^95 + 638 * q^97 - 784 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	5.54600	+	9.60595i	0.496049	+	0.859183i	0.999990	−	0.00455579i	$-0.00145016\pi$
$5$	5.54600	+	9.60595i	0.496049	+	0.859183i	−0.503940	+	0.863739i	$0.668117\pi$
$6$		0			0
$7$	3.50000	−	6.06218i	0.188982	−	0.327327i
$7$	3.50000	−	6.06218i	0.188982	−	0.327327i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	22.1840			0.701520
$11$	−33.4982	+	58.0206i	−0.918190	+	1.59035i	−0.116026	+	0.993246i	$0.537016\pi$
$11$	−33.4982	+	58.0206i	−0.918190	+	1.59035i	−0.802163	+	0.597105i	$0.796318\pi$
$12$		0			0
$13$	2.43841	+	4.22346i	0.0520226	+	0.0901059i	0.890864	−	0.454270i	$-0.150100\pi$
$13$	2.43841	+	4.22346i	0.0520226	+	0.0901059i	−0.838841	+	0.544376i	$0.816767\pi$
$14$	−7.00000	−	12.1244i	−0.133631	−	0.231455i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	−69.3851			−0.989903			−0.494952	−	0.868921i	$-0.664814\pi$
$17$	−69.3851			−0.989903			−0.494952	+	0.868921i	$0.664814\pi$
$18$		0			0
$19$	−74.9150			−0.904562			−0.452281	−	0.891876i	$-0.649390\pi$
$19$	−74.9150			−0.904562			−0.452281	+	0.891876i	$0.649390\pi$
$20$	22.1840	−	38.4238i	0.248025	−	0.429591i
$21$		0			0
$22$	66.9964	+	116.041i	0.649258	+	1.12455i
$23$	−25.4130	−	44.0166i	−0.230390	−	0.399048i	0.727533	−	0.686073i	$-0.240667\pi$
$23$	−25.4130	−	44.0166i	−0.230390	−	0.399048i	−0.957923	+	0.287025i	$0.907334\pi$
$24$		0			0
$25$	0.983753	−	1.70391i	0.00787003	−	0.0136313i
$26$	9.75366			0.0735711
$27$		0			0
$28$	−28.0000			−0.188982
$29$	−81.5021	+	141.166i	−0.521881	+	0.903925i	0.477795	+	0.878472i	$0.341436\pi$
$29$	−81.5021	+	141.166i	−0.521881	+	0.903925i	−0.999676	+	0.0254535i	$0.991897\pi$
$30$		0			0
$31$	57.5174	+	99.6231i	0.333240	+	0.577188i	0.983145	−	0.182827i	$-0.0585248\pi$
$31$	57.5174	+	99.6231i	0.333240	+	0.577188i	−0.649905	+	0.760015i	$0.725191\pi$
$32$	16.0000	+	27.7128i	0.0883883	+	0.153093i
$33$		0			0
$34$	−69.3851	+	120.178i	−0.349984	+	0.606189i
$35$	77.6440			0.374978
$36$		0			0
$37$	−126.697			−0.562942			−0.281471	−	0.959570i	$-0.590822\pi$
$37$	−126.697			−0.562942			−0.281471	+	0.959570i	$0.590822\pi$
$38$	−74.9150	+	129.757i	−0.319811	+	0.553929i
$39$		0			0
$40$	−44.3680	−	76.8476i	−0.175380	−	0.303767i
$41$	−76.9999	−	133.368i	−0.293301	−	0.508013i	0.681287	−	0.732016i	$-0.261421\pi$
$41$	−76.9999	−	133.368i	−0.293301	−	0.508013i	−0.974588	+	0.224004i	$0.928087\pi$
$42$		0			0
$43$	−203.974	+	353.293i	−0.723390	+	1.25295i	0.236244	+	0.971694i	$0.424084\pi$
$43$	−203.974	+	353.293i	−0.723390	+	1.25295i	−0.959633	+	0.281254i	$0.909250\pi$
$44$	267.986			0.918190
$45$		0			0
$46$	−101.652			−0.325821
$47$	−166.808	+	288.920i	−0.517691	+	0.896667i	0.482098	+	0.876117i	$0.339875\pi$
$47$	−166.808	+	288.920i	−0.517691	+	0.896667i	−0.999789	+	0.0205498i	$0.993458\pi$
$48$		0			0
$49$	−24.5000	−	42.4352i	−0.0714286	−	0.123718i
$50$	−1.96751	−	3.40782i	−0.00556495	−	0.00963878i
$51$		0			0
$52$	9.75366	−	16.8938i	0.0260113	−	0.0450529i
$53$	558.813			1.44828			0.724140	−	0.689653i	$-0.242237\pi$
$53$	558.813			1.44828			0.724140	+	0.689653i	$0.242237\pi$
$54$		0			0
$55$	−743.124			−1.82187
$56$	−28.0000	+	48.4974i	−0.0668153	+	0.115728i
$57$		0			0
$58$	163.004	+	282.332i	0.369026	+	0.639172i
$59$	250.021	+	433.049i	0.551694	+	0.955562i	0.998153	+	0.0607578i	$0.0193517\pi$
$59$	250.021	+	433.049i	0.551694	+	0.955562i	−0.446459	+	0.894804i	$0.647315\pi$
$60$		0			0
$61$	−136.931	+	237.172i	−0.287414	+	0.497816i	−0.973192	−	0.229995i	$-0.926129\pi$
$61$	−136.931	+	237.172i	−0.287414	+	0.497816i	0.685777	+	0.727811i	$0.259462\pi$
$62$	230.070			0.471272
$63$		0			0
$64$	64.0000			0.125000
$65$	−27.0469	+	46.8466i	−0.0516116	+	0.0893939i
$66$		0			0
$67$	203.229	+	352.004i	0.370574	+	0.641852i	0.989654	−	0.143475i	$-0.0458278\pi$
$67$	203.229	+	352.004i	0.370574	+	0.641852i	−0.619080	+	0.785328i	$0.712494\pi$
$68$	138.770	+	240.357i	0.247476	+	0.428641i
$69$		0			0
$70$	77.6440	−	134.483i	0.132575	−	0.229626i
$71$	824.686			1.37848			0.689241	−	0.724532i	$-0.257944\pi$
$71$	824.686			1.37848			0.689241	+	0.724532i	$0.257944\pi$
$72$		0			0
$73$	141.718			0.227216			0.113608	−	0.993526i	$-0.463759\pi$
$73$	141.718			0.227216			0.113608	+	0.993526i	$0.463759\pi$
$74$	−126.697	+	219.446i	−0.199030	+	0.344730i
$75$		0			0
$76$	149.830	+	259.513i	0.226140	+	0.391687i
$77$	234.487	+	406.144i	0.347043	+	0.601096i
$78$		0			0
$79$	572.244	−	991.156i	0.814968	−	1.41157i	−0.0943829	−	0.995536i	$-0.530088\pi$
$79$	572.244	−	991.156i	0.814968	−	1.41157i	0.909351	−	0.416030i	$-0.136579\pi$
$80$	−177.472			−0.248025
$81$		0			0
$82$	−307.999			−0.414791
$83$	234.784	−	406.657i	0.310492	−	0.537789i	−0.667977	−	0.744182i	$-0.732839\pi$
$83$	234.784	−	406.657i	0.310492	−	0.537789i	0.978469	+	0.206394i	$0.0661727\pi$
$84$		0			0
$85$	−384.810	−	666.510i	−0.491041	−	0.850508i
$86$	407.948	+	706.587i	0.511514	+	0.885968i
$87$		0			0
$88$	267.986	−	464.165i	0.324629	−	0.562274i
$89$	−1641.31			−1.95482			−0.977409	−	0.211355i	$-0.932212\pi$
$89$	−1641.31			−1.95482			−0.977409	+	0.211355i	$0.932212\pi$
$90$		0			0
$91$	34.1378			0.0393254
$92$	−101.652	+	176.067i	−0.115195	+	0.199524i
$93$		0			0
$94$	333.616	+	577.841i	0.366063	+	0.634039i
$95$	−415.479	−	719.630i	−0.448707	−	0.777184i
$96$		0			0
$97$	−607.712	+	1052.59i	−0.636122	+	1.10180i	0.350154	+	0.936692i	$0.386129\pi$
$97$	−607.712	+	1052.59i	−0.636122	+	1.10180i	−0.986276	+	0.165104i	$0.947204\pi$
$98$	−98.0000			−0.101015
$99$		0			0
$100$	−7.87003			−0.00787003
$101$	873.265	−	1512.54i	0.860328	−	1.49013i	−0.0112851	−	0.999936i	$-0.503592\pi$
$101$	873.265	−	1512.54i	0.860328	−	1.49013i	0.871613	−	0.490195i	$-0.163074\pi$
$102$		0			0
$103$	779.171	+	1349.56i	0.745379	+	1.29103i	0.950018	+	0.312196i	$0.101065\pi$
$103$	779.171	+	1349.56i	0.745379	+	1.29103i	−0.204639	+	0.978837i	$0.565602\pi$
$104$	−19.5073	−	33.7877i	−0.0183928	−	0.0318572i
$105$		0			0
$106$	558.813	−	967.892i	0.512044	−	0.886886i
$107$	−1034.24			−0.934432			−0.467216	−	0.884143i	$-0.654743\pi$
$107$	−1034.24			−0.934432			−0.467216	+	0.884143i	$0.654743\pi$
$108$		0			0
$109$	−1240.98			−1.09049			−0.545247	−	0.838275i	$-0.683564\pi$
$109$	−1240.98			−1.09049			−0.545247	+	0.838275i	$0.683564\pi$
$110$	−743.124	+	1287.13i	−0.644128	+	1.11566i
$111$		0			0
$112$	56.0000	+	96.9948i	0.0472456	+	0.0818317i
$113$	−729.856	−	1264.15i	−0.607602	−	1.05240i	−0.991634	−	0.129078i	$-0.958798\pi$
$113$	−729.856	−	1264.15i	−0.607602	−	1.05240i	0.384032	−	0.923320i	$-0.374535\pi$
$114$		0			0
$115$	281.881	−	488.233i	0.228570	−	0.395895i
$116$	652.017			0.521881
$117$		0			0
$118$	1000.08			0.780213
$119$	−242.848	+	420.625i	−0.187074	+	0.324022i
$120$		0			0
$121$	−1578.76	−	2734.49i	−1.18614	−	2.05446i
$122$	273.863	+	474.344i	0.203233	+	0.352009i
$123$		0			0
$124$	230.070	−	398.492i	0.166620	−	0.288594i
$125$	1408.32			1.00771
$126$		0			0
$127$	750.041			0.524058			0.262029	−	0.965060i	$-0.415608\pi$
$127$	750.041			0.524058			0.262029	+	0.965060i	$0.415608\pi$
$128$	64.0000	−	110.851i	0.0441942	−	0.0765466i
$129$		0			0
$130$	54.0938	+	93.6932i	0.0364949	+	0.0632110i
$131$	−538.519	−	932.742i	−0.359165	−	0.622092i	0.628657	−	0.777683i	$-0.283605\pi$
$131$	−538.519	−	932.742i	−0.359165	−	0.622092i	−0.987822	+	0.155591i	$0.950272\pi$
$132$		0			0
$133$	−262.202	+	454.148i	−0.170946	+	0.296087i
$134$	812.918			0.524070
$135$		0			0
$136$	555.081			0.349984
$137$	−66.7606	+	115.633i	−0.0416331	+	0.0721107i	−0.886091	−	0.463511i	$-0.846589\pi$
$137$	−66.7606	+	115.633i	−0.0416331	+	0.0721107i	0.844458	+	0.535622i	$0.179923\pi$
$138$		0			0
$139$	1078.68	+	1868.33i	0.658220	+	1.14007i	0.981076	+	0.193623i	$0.0620238\pi$
$139$	1078.68	+	1868.33i	0.658220	+	1.14007i	−0.322856	+	0.946448i	$0.604643\pi$
$140$	−155.288	−	268.967i	−0.0937445	−	0.162370i
$141$		0			0
$142$	824.686	−	1428.40i	0.487367	−	0.844144i
$143$	−326.730			−0.191067
$144$		0			0
$145$	−1808.04			−1.03552
$146$	141.718	−	245.462i	0.0803331	−	0.139141i
$147$		0			0
$148$	253.394	+	438.891i	0.140735	+	0.243761i
$149$	479.377	+	830.305i	0.263571	+	0.456518i	0.967188	−	0.254061i	$-0.0817663\pi$
$149$	479.377	+	830.305i	0.263571	+	0.456518i	−0.703617	+	0.710579i	$0.748433\pi$
$150$		0			0
$151$	279.084	−	483.388i	0.150408	−	0.260514i	−0.780970	−	0.624569i	$-0.785275\pi$
$151$	279.084	−	483.388i	0.150408	−	0.260514i	0.931377	+	0.364055i	$0.118608\pi$
$152$	599.320			0.319811
$153$		0			0
$154$	937.949			0.490793
$155$	−637.983	+	1105.02i	−0.330607	+	0.572628i
$156$		0			0
$157$	1157.19	+	2004.32i	0.588243	+	1.01887i	0.994463	+	0.105091i	$0.0335135\pi$
$157$	1157.19	+	2004.32i	0.588243	+	1.01887i	−0.406219	+	0.913776i	$0.633153\pi$
$158$	−1144.49	−	1982.31i	−0.576269	−	0.998128i
$159$		0			0
$160$	−177.472	+	307.391i	−0.0876900	+	0.151883i
$161$	−355.782			−0.174159
$162$		0			0
$163$	−1110.71			−0.533727			−0.266864	−	0.963734i	$-0.585987\pi$
$163$	−1110.71			−0.533727			−0.266864	+	0.963734i	$0.585987\pi$
$164$	−307.999	+	533.471i	−0.146651	+	0.254006i
$165$		0			0
$166$	−469.568	−	813.315i	−0.219551	−	0.380274i
$167$	−974.863	−	1688.51i	−0.451719	−	0.782401i	0.546774	−	0.837281i	$-0.315856\pi$
$167$	−974.863	−	1688.51i	−0.451719	−	0.782401i	−0.998493	+	0.0548795i	$0.982523\pi$
$168$		0			0
$169$	1086.61	−	1882.06i	0.494587	−	0.856650i
$170$	−1539.24			−0.694437
$171$		0			0
$172$	1631.79			0.723390
$173$	630.605	−	1092.24i	0.277133	−	0.480008i	−0.693538	−	0.720420i	$-0.743949\pi$
$173$	630.605	−	1092.24i	0.277133	−	0.480008i	0.970671	+	0.240412i	$0.0772825\pi$
$174$		0			0
$175$	−6.88627	−	11.9274i	−0.00297459	−	0.00515214i
$176$	−535.971	−	928.329i	−0.229547	−	0.397588i
$177$		0			0
$178$	−1641.31	+	2842.84i	−0.691133	+	1.19708i
$179$	−4177.67			−1.74444			−0.872218	−	0.489118i	$-0.837319\pi$
$179$	−4177.67			−1.74444			−0.872218	+	0.489118i	$0.837319\pi$
$180$		0			0
$181$	−1703.41			−0.699520			−0.349760	−	0.936839i	$-0.613737\pi$
$181$	−1703.41			−0.699520			−0.349760	+	0.936839i	$0.613737\pi$
$182$	34.1378	−	59.1284i	0.0139036	−	0.0240818i
$183$		0			0
$184$	203.304	+	352.133i	0.0814553	+	0.141085i
$185$	−702.661	−	1217.04i	−0.279247	−	0.483670i
$186$		0			0
$187$	2324.27	−	4025.76i	0.908919	−	1.57429i
$188$	1334.47			0.517691
$189$		0			0
$190$	−1661.91			−0.634568
$191$	2230.14	−	3862.71i	0.844854	−	1.46333i	−0.0408931	−	0.999164i	$-0.513020\pi$
$191$	2230.14	−	3862.71i	0.844854	−	1.46333i	0.885748	−	0.464167i	$-0.153646\pi$
$192$		0			0
$193$	−995.410	−	1724.10i	−0.371250	−	0.643023i	0.618509	−	0.785778i	$-0.287737\pi$
$193$	−995.410	−	1724.10i	−0.371250	−	0.643023i	−0.989758	+	0.142755i	$0.954404\pi$
$194$	1215.42	+	2105.18i	0.449806	+	0.779087i
$195$		0			0
$196$	−98.0000	+	169.741i	−0.0357143	+	0.0618590i
$197$	1215.19			0.439486			0.219743	−	0.975558i	$-0.429478\pi$
$197$	1215.19			0.439486			0.219743	+	0.975558i	$0.429478\pi$
$198$		0			0
$199$	−1676.82			−0.597320			−0.298660	−	0.954360i	$-0.596540\pi$
$199$	−1676.82			−0.597320			−0.298660	+	0.954360i	$0.596540\pi$
$200$	−7.87003	+	13.6313i	−0.00278247	+	0.00481939i
$201$		0			0
$202$	−1746.53	−	3025.08i	−0.608344	−	1.05368i
$203$	570.515	+	988.160i	0.197253	+	0.341652i
$204$		0			0
$205$	854.082	−	1479.31i	0.290984	−	0.503999i
$206$	3116.68			1.05412
$207$		0			0
$208$	−78.0292			−0.0260113
$209$	2509.52	−	4346.61i	0.830559	−	1.43857i
$210$		0			0
$211$	1901.37	+	3293.26i	0.620358	+	1.07449i	0.989419	+	0.145086i	$0.0463459\pi$
$211$	1901.37	+	3293.26i	0.620358	+	1.07449i	−0.369061	+	0.929405i	$0.620321\pi$
$212$	−1117.63	−	1935.78i	−0.362070	−	0.627123i
$213$		0			0
$214$	−1034.24	+	1791.36i	−0.330372	+	0.572220i
$215$	−4524.96			−1.43535
$216$		0			0
$217$	805.244			0.251906
$218$	−1240.98	+	2149.43i	−0.385548	+	0.667789i
$219$		0			0
$220$	1486.25	+	2574.26i	0.455467	+	0.788893i
$221$	−169.190	−	293.045i	−0.0514974	−	0.0891961i
$222$		0			0
$223$	1332.86	−	2308.58i	0.400247	−	0.693248i	−0.593509	−	0.804828i	$-0.702258\pi$
$223$	1332.86	−	2308.58i	0.400247	−	0.693248i	0.993755	+	0.111580i	$0.0355911\pi$
$224$	224.000			0.0668153
$225$		0			0
$226$	−2919.42			−0.859279
$227$	−2076.53	+	3596.66i	−0.607155	+	1.05162i	0.384552	+	0.923103i	$0.374356\pi$
$227$	−2076.53	+	3596.66i	−0.607155	+	1.05162i	−0.991707	+	0.128520i	$0.958977\pi$
$228$		0			0
$229$	−509.127	−	881.834i	−0.146917	−	0.254468i	0.783169	−	0.621809i	$-0.213602\pi$
$229$	−509.127	−	881.834i	−0.146917	−	0.254468i	−0.930087	+	0.367340i	$0.880268\pi$
$230$	−563.762	−	976.465i	−0.161623	−	0.279940i
$231$		0			0
$232$	652.017	−	1129.33i	0.184513	−	0.319586i
$233$	649.020			0.182484			0.0912418	−	0.995829i	$-0.470916\pi$
$233$	649.020			0.182484			0.0912418	+	0.995829i	$0.470916\pi$
$234$		0			0
$235$	−3700.47			−1.02720
$236$	1000.08	−	1732.20i	0.275847	−	0.477781i
$237$		0			0
$238$	485.696	+	841.249i	0.132281	+	0.229118i
$239$	2411.87	+	4177.48i	0.652765	+	1.13062i	0.982449	+	0.186531i	$0.0597245\pi$
$239$	2411.87	+	4177.48i	0.652765	+	1.13062i	−0.329684	+	0.944091i	$0.606942\pi$
$240$		0			0
$241$	1096.08	−	1898.47i	0.292966	−	0.507431i	−0.681544	−	0.731777i	$-0.738691\pi$
$241$	1096.08	−	1898.47i	0.292966	−	0.507431i	0.974510	+	0.224346i	$0.0720245\pi$
$242$	−6315.03			−1.67746
$243$		0			0
$244$	1095.45			0.287414
$245$	271.754	−	470.692i	0.0708642	−	0.122740i
$246$		0			0
$247$	−182.674	−	316.400i	−0.0470577	−	0.0815063i
$248$	−460.139	−	796.985i	−0.117818	−	0.204067i
$249$		0			0
$250$	1408.32	−	2439.29i	0.356281	−	0.617097i
$251$	1244.15			0.312869			0.156434	−	0.987688i	$-0.450000\pi$
$251$	1244.15			0.312869			0.156434	+	0.987688i	$0.450000\pi$
$252$		0			0
$253$	3405.16			0.846168
$254$	750.041	−	1299.11i	0.185283	−	0.320919i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	1648.28	+	2854.90i	0.400065	+	0.692933i	0.993733	−	0.111777i	$-0.0356542\pi$
$257$	1648.28	+	2854.90i	0.400065	+	0.692933i	−0.593668	+	0.804710i	$0.702321\pi$
$258$		0			0
$259$	−443.439	+	768.059i	−0.106386	+	0.184266i
$260$	216.375			0.0516116
$261$		0			0
$262$	−2154.07			−0.507936
$263$	−3542.87	+	6136.42i	−0.830656	+	1.43874i	0.0668633	+	0.997762i	$0.478701\pi$
$263$	−3542.87	+	6136.42i	−0.830656	+	1.43874i	−0.897519	+	0.440976i	$0.854632\pi$
$264$		0			0
$265$	3099.18	+	5367.93i	0.718418	+	1.24434i
$266$	524.405	+	908.296i	0.120877	+	0.209365i
$267$		0			0
$268$	812.918	−	1408.01i	0.185287	−	0.320926i
$269$	3955.50			0.896547			0.448273	−	0.893896i	$-0.352039\pi$
$269$	3955.50			0.896547			0.448273	+	0.893896i	$0.352039\pi$
$270$		0			0
$271$	4100.35			0.919110			0.459555	−	0.888149i	$-0.348009\pi$
$271$	4100.35			0.919110			0.459555	+	0.888149i	$0.348009\pi$
$272$	555.081	−	961.428i	0.123738	−	0.214320i
$273$		0			0
$274$	133.521	+	231.265i	0.0294391	+	0.0509900i
$275$	65.9079	+	114.156i	0.0144524	+	0.0250322i
$276$		0			0
$277$	3010.29	−	5213.98i	0.652964	−	1.13097i	−0.329436	−	0.944178i	$-0.606859\pi$
$277$	3010.29	−	5213.98i	0.652964	−	1.13097i	0.982400	−	0.186789i	$-0.0598082\pi$
$278$	4314.73			0.930864
$279$		0			0
$280$	−621.152			−0.132575
$281$	2637.03	−	4567.46i	0.559829	−	0.969652i	−0.437682	−	0.899130i	$-0.644200\pi$
$281$	2637.03	−	4567.46i	0.559829	−	0.969652i	0.997510	−	0.0705215i	$-0.0224663\pi$
$282$		0			0
$283$	2166.69	+	3752.82i	0.455111	+	0.788275i	0.998695	−	0.0510797i	$-0.0162662\pi$
$283$	2166.69	+	3752.82i	0.455111	+	0.788275i	−0.543584	+	0.839355i	$0.682933\pi$
$284$	−1649.37	−	2856.80i	−0.344620	−	0.596900i
$285$		0			0
$286$	−326.730	+	565.913i	−0.0675522	+	0.117004i
$287$	−1078.00			−0.221715
$288$		0			0
$289$	−98.7112			−0.0200918
$290$	−1808.04	+	3131.62i	−0.366110	+	0.634121i
$291$		0			0
$292$	−283.435	−	490.924i	−0.0568041	−	0.0983875i
$293$	4042.89	+	7002.49i	0.806103	+	1.39621i	0.915544	+	0.402218i	$0.131760\pi$
$293$	4042.89	+	7002.49i	0.806103	+	1.39621i	−0.109441	+	0.993993i	$0.534906\pi$
$294$		0			0
$295$	−2773.23	+	4803.38i	−0.547335	+	0.948012i
$296$	1013.58			0.199030
$297$		0			0
$298$	1917.51			0.372746
$299$	123.935	−	214.662i	0.0239710	−	0.0415191i
$300$		0			0
$301$	1427.82	+	2473.05i	0.273416	+	0.473570i
$302$	−558.168	−	966.776i	−0.106354	−	0.184211i
$303$		0			0
$304$	599.320	−	1038.05i	0.113070	−	0.195843i
$305$	−3037.69			−0.570287
$306$		0			0
$307$	3828.04			0.711655			0.355827	−	0.934552i	$-0.384199\pi$
$307$	3828.04			0.711655			0.355827	+	0.934552i	$0.384199\pi$
$308$	937.949	−	1624.58i	0.173522	−	0.300548i
$309$		0			0
$310$	1275.97	+	2210.04i	0.233774	+	0.404909i
$311$	3651.59	+	6324.73i	0.665796	+	1.15319i	0.979069	+	0.203529i	$0.0652412\pi$
$311$	3651.59	+	6324.73i	0.665796	+	1.15319i	−0.313273	+	0.949663i	$0.601426\pi$
$312$		0			0
$313$	−2434.26	+	4216.27i	−0.439593	+	0.761398i	−0.997658	−	0.0683995i	$-0.978211\pi$
$313$	−2434.26	+	4216.27i	−0.439593	+	0.761398i	0.558065	+	0.829797i	$0.311544\pi$
$314$	4628.78			0.831901
$315$		0			0
$316$	−4577.95			−0.814968
$317$	467.364	−	809.499i	0.0828069	−	0.143426i	−0.821648	−	0.569996i	$-0.806945\pi$
$317$	467.364	−	809.499i	0.0828069	−	0.143426i	0.904455	+	0.426570i	$0.140278\pi$
$318$		0			0
$319$	−5460.35	−	9457.60i	−0.958372	−	1.65995i
$320$	354.944	+	614.781i	0.0620062	+	0.107398i
$321$		0			0
$322$	−355.782	+	616.233i	−0.0615744	+	0.106650i
$323$	5197.98			0.895429
$324$		0			0
$325$	9.59519			0.00163768
$326$	−1110.71	+	1923.81i	−0.188701	+	0.326840i
$327$		0			0
$328$	615.999	+	1066.94i	0.103698	+	0.179610i
$329$	1167.66	+	2022.44i	0.195669	+	0.338908i
$330$		0			0
$331$	−4798.45	+	8311.16i	−0.796818	+	1.38013i	0.124860	+	0.992174i	$0.460152\pi$
$331$	−4798.45	+	8311.16i	−0.796818	+	1.38013i	−0.921678	+	0.387955i	$0.873182\pi$
$332$	−1878.27			−0.310492
$333$		0			0
$334$	−3899.45			−0.638828
$335$	−2254.22	+	3904.43i	−0.367646	+	0.636781i
$336$		0			0
$337$	−3357.80	−	5815.89i	−0.542763	−	0.940094i	−0.998744	−	0.0501041i	$-0.984045\pi$
$337$	−3357.80	−	5815.89i	−0.542763	−	0.940094i	0.455981	−	0.889990i	$-0.349289\pi$
$338$	−2173.22	−	3764.12i	−0.349726	−	0.605743i
$339$		0			0
$340$	−1539.24	+	2666.04i	−0.245520	+	0.425254i
$341$	−7706.92			−1.22391
$342$		0			0
$343$	−343.000			−0.0539949
$344$	1631.79	−	2826.35i	0.255757	−	0.442984i
$345$		0			0
$346$	−1261.21	−	2184.48i	−0.195963	−	0.339417i
$347$	3992.85	+	6915.82i	0.617716	+	1.06992i	0.989901	+	0.141757i	$0.0452752\pi$
$347$	3992.85	+	6915.82i	0.617716	+	1.06992i	−0.372185	+	0.928158i	$0.621391\pi$
$348$		0			0
$349$	−3007.64	+	5209.39i	−0.461305	+	0.799003i	−0.999026	−	0.0441194i	$-0.985952\pi$
$349$	−3007.64	+	5209.39i	−0.461305	+	0.799003i	0.537722	+	0.843122i	$0.319285\pi$
$350$	−27.5451			−0.00420671
$351$		0			0
$352$	−2143.88			−0.324629
$353$	641.078	−	1110.38i	0.0966604	−	0.167421i	−0.813640	−	0.581369i	$-0.802517\pi$
$353$	641.078	−	1110.38i	0.0966604	−	0.167421i	0.910300	+	0.413948i	$0.135851\pi$
$354$		0			0
$355$	4573.71	+	7921.89i	0.683795	+	1.18437i
$356$	3282.63	+	5685.68i	0.488705	+	0.846461i
$357$		0			0
$358$	−4177.67	+	7235.94i	−0.616751	+	1.06824i
$359$	−6459.33			−0.949612			−0.474806	−	0.880091i	$-0.657482\pi$
$359$	−6459.33			−0.949612			−0.474806	+	0.880091i	$0.657482\pi$
$360$		0			0
$361$	−1246.75			−0.181768
$362$	−1703.41	+	2950.38i	−0.247318	+	0.428367i
$363$		0			0
$364$	−68.2756	−	118.257i	−0.00983135	−	0.0170284i
$365$	785.966	+	1361.33i	0.112710	+	0.195220i
$366$		0			0
$367$	−2098.34	+	3634.44i	−0.298454	+	0.516938i	−0.975783	−	0.218743i	$-0.929804\pi$
$367$	−2098.34	+	3634.44i	−0.298454	+	0.516938i	0.677328	+	0.735681i	$0.263138\pi$
$368$	813.217			0.115195
$369$		0			0
$370$	−2810.64			−0.394915
$371$	1955.84	−	3387.62i	0.273699	−	0.474061i
$372$		0			0
$373$	−4117.90	−	7132.41i	−0.571627	−	0.990087i	−0.996399	−	0.0847864i	$-0.972979\pi$
$373$	−4117.90	−	7132.41i	−0.571627	−	0.990087i	0.424772	−	0.905300i	$-0.360354\pi$
$374$	−4648.55	−	8051.52i	−0.642703	−	1.11319i
$375$		0			0
$376$	1334.47	−	2311.36i	0.183031	−	0.317020i
$377$	−794.943			−0.108599
$378$		0			0
$379$	−3135.68			−0.424985			−0.212492	−	0.977163i	$-0.568158\pi$
$379$	−3135.68			−0.424985			−0.212492	+	0.977163i	$0.568158\pi$
$380$	−1661.91	+	2878.52i	−0.224354	+	0.388592i
$381$		0			0
$382$	−4460.28	−	7725.43i	−0.597402	−	1.03473i
$383$	−4083.64	−	7073.06i	−0.544815	−	0.943647i	−0.998619	−	0.0525451i	$-0.983267\pi$
$383$	−4083.64	−	7073.06i	−0.544815	−	0.943647i	0.453804	−	0.891102i	$-0.350067\pi$
$384$		0			0
$385$	−2600.93	+	4504.95i	−0.344301	+	0.596347i
$386$	−3981.64			−0.525026
$387$		0			0
$388$	4861.70			0.636122
$389$	1247.06	−	2159.97i	0.162541	−	0.281530i	−0.773238	−	0.634116i	$-0.781364\pi$
$389$	1247.06	−	2159.97i	0.162541	−	0.281530i	0.935779	+	0.352586i	$0.114698\pi$
$390$		0			0
$391$	1763.28	+	3054.10i	0.228064	+	0.395019i
$392$	196.000	+	339.482i	0.0252538	+	0.0437409i
$393$		0			0
$394$	1215.19	−	2104.77i	0.155382	−	0.269129i
$395$	12694.7			1.61706
$396$		0			0
$397$	−8156.76			−1.03117			−0.515587	−	0.856837i	$-0.672426\pi$
$397$	−8156.76			−1.03117			−0.515587	+	0.856837i	$0.672426\pi$
$398$	−1676.82	+	2904.34i	−0.211184	+	0.365782i
$399$		0			0
$400$	15.7401	+	27.2626i	0.00196751	+	0.00340782i
$401$	3643.93	+	6311.48i	0.453789	+	0.785985i	0.998618	−	0.0525619i	$-0.0167387\pi$
$401$	3643.93	+	6311.48i	0.453789	+	0.785985i	−0.544829	+	0.838547i	$0.683405\pi$
$402$		0			0
$403$	−280.503	+	485.845i	−0.0346720	+	0.0600537i
$404$	−6986.12			−0.860328
$405$		0			0
$406$	2282.06			0.278957
$407$	4244.12	−	7351.03i	0.516887	−	0.895275i
$408$		0			0
$409$	4856.20	+	8411.18i	0.587099	+	1.01689i	0.994610	+	0.103685i	$0.0330635\pi$
$409$	4856.20	+	8411.18i	0.587099	+	1.01689i	−0.407511	+	0.913200i	$0.633603\pi$
$410$	−1708.16	−	2958.63i	−0.205757	−	0.356381i
$411$		0			0
$412$	3116.68	−	5398.25i	0.372689	−	0.645517i
$413$	3500.29			0.417041
$414$		0			0
$415$	5208.44			0.616078
$416$	−78.0292	+	135.151i	−0.00919639	+	0.0159286i
$417$		0			0
$418$	−5019.03	−	8693.22i	−0.587294	−	1.01722i
$419$	446.035	+	772.556i	0.0520054	+	0.0900760i	0.890856	−	0.454285i	$-0.150105\pi$
$419$	446.035	+	772.556i	0.0520054	+	0.0900760i	−0.838851	+	0.544361i	$0.816772\pi$
$420$		0			0
$421$	410.936	−	711.763i	0.0475720	−	0.0823971i	−0.841259	−	0.540632i	$-0.818185\pi$
$421$	410.936	−	711.763i	0.0475720	−	0.0823971i	0.888831	+	0.458235i	$0.151518\pi$
$422$	7605.47			0.877318
$423$		0			0
$424$	−4470.50			−0.512044
$425$	−68.2578	+	118.226i	−0.00779056	+	0.0134937i
$426$		0			0
$427$	958.520	+	1660.21i	0.108632	+	0.188157i
$428$	2068.49	+	3582.73i	0.233608	+	0.404621i
$429$		0			0
$430$	−4524.96	+	7837.46i	−0.507472	+	0.878967i
$431$	−4506.54			−0.503648			−0.251824	−	0.967773i	$-0.581030\pi$
$431$	−4506.54			−0.503648			−0.251824	+	0.967773i	$0.581030\pi$
$432$		0			0
$433$	9316.05			1.03395			0.516975	−	0.856000i	$-0.327058\pi$
$433$	9316.05			1.03395			0.516975	+	0.856000i	$0.327058\pi$
$434$	805.244	−	1394.72i	0.0890621	−	0.154260i
$435$		0			0
$436$	2481.95	+	4298.87i	0.272624	+	0.472198i
$437$	1903.82	+	3297.51i	0.208402	+	0.360964i
$438$		0			0
$439$	−1574.49	+	2727.10i	−0.171176	+	0.296486i	−0.938831	−	0.344377i	$-0.888090\pi$
$439$	−1574.49	+	2727.10i	−0.171176	+	0.296486i	0.767655	+	0.640863i	$0.221423\pi$
$440$	5944.99			0.644128
$441$		0			0
$442$	−676.758			−0.0728283
$443$	−3221.18	+	5579.25i	−0.345470	+	0.598371i	−0.985439	−	0.170029i	$-0.945614\pi$
$443$	−3221.18	+	5579.25i	−0.345470	+	0.598371i	0.639969	+	0.768400i	$0.278947\pi$
$444$		0			0
$445$	−9102.73	−	15766.4i	−0.969687	−	1.67955i
$446$	−2665.72	−	4617.17i	−0.283017	−	0.490200i
$447$		0			0
$448$	224.000	−	387.979i	0.0236228	−	0.0409159i
$449$	1059.39			0.111349			0.0556746	−	0.998449i	$-0.482269\pi$
$449$	1059.39			0.111349			0.0556746	+	0.998449i	$0.482269\pi$
$450$		0			0
$451$	10317.4			1.07723
$452$	−2919.42	+	5056.59i	−0.303801	+	0.526199i
$453$		0			0
$454$	4153.06	+	7193.31i	0.429324	+	0.743610i
$455$	189.328	+	327.926i	0.0195073	+	0.0337877i
$456$		0			0
$457$	2334.53	−	4043.53i	0.238960	−	0.413892i	−0.721456	−	0.692460i	$-0.756527\pi$
$457$	2334.53	−	4043.53i	0.238960	−	0.413892i	0.960416	+	0.278569i	$0.0898600\pi$
$458$	−2036.51			−0.207772
$459$		0			0
$460$	−2255.05			−0.228570
$461$	−3825.66	+	6626.23i	−0.386504	+	0.669445i	−0.991977	−	0.126421i	$-0.959651\pi$
$461$	−3825.66	+	6626.23i	−0.386504	+	0.669445i	0.605472	+	0.795866i	$0.292984\pi$
$462$		0			0
$463$	9196.75	+	15929.2i	0.923130	+	1.59891i	0.794542	+	0.607210i	$0.207711\pi$
$463$	9196.75	+	15929.2i	0.923130	+	1.59891i	0.128588	+	0.991698i	$0.458955\pi$
$464$	−1304.03	−	2258.65i	−0.130470	−	0.225981i
$465$		0			0
$466$	649.020	−	1124.14i	0.0645177	−	0.111748i
$467$	−6141.08			−0.608512			−0.304256	−	0.952590i	$-0.598408\pi$
$467$	−6141.08			−0.608512			−0.304256	+	0.952590i	$0.598408\pi$
$468$		0			0
$469$	2845.21			0.280127
$470$	−3700.47	+	6409.41i	−0.363170	+	0.629030i
$471$		0			0
$472$	−2000.17	−	3464.39i	−0.195053	−	0.337842i
$473$	−13665.5	−	23669.4i	−1.32842	−	2.30089i
$474$		0			0
$475$	−73.6979	+	127.648i	−0.00711893	+	0.0123303i
$476$	1942.78			0.187074
$477$		0			0
$478$	9647.48			0.923149
$479$	−2636.85	+	4567.16i	−0.251526	+	0.435655i	−0.963946	−	0.266098i	$-0.914266\pi$
$479$	−2636.85	+	4567.16i	−0.251526	+	0.435655i	0.712421	+	0.701753i	$0.247599\pi$
$480$		0			0
$481$	−308.940	−	535.099i	−0.0292857	−	0.0507244i
$482$	−2192.16	−	3796.93i	−0.207158	−	0.358808i
$483$		0			0
$484$	−6315.03	+	10938.0i	−0.593072	+	1.02723i
$485$	−13481.5			−1.26219
$486$		0			0
$487$	18946.8			1.76296			0.881480	−	0.472220i	$-0.156547\pi$
$487$	18946.8			1.76296			0.881480	+	0.472220i	$0.156547\pi$
$488$	1095.45	−	1897.38i	0.101616	−	0.176005i
$489$		0			0
$490$	−543.508	−	941.384i	−0.0501086	−	0.0867906i
$491$	2321.05	+	4020.17i	0.213335	+	0.369507i	0.952756	−	0.303736i	$-0.0982341\pi$
$491$	2321.05	+	4020.17i	0.213335	+	0.369507i	−0.739421	+	0.673243i	$0.764901\pi$
$492$		0			0
$493$	5655.03	−	9794.80i	0.516612	−	0.894798i
$494$	−730.695			−0.0665496
$495$		0			0
$496$	−1840.56			−0.166620
$497$	2886.40	−	4999.39i	0.260509	−	0.451214i
$498$		0			0
$499$	−4145.44	−	7180.12i	−0.371895	−	0.644141i	0.617962	−	0.786208i	$-0.287958\pi$
$499$	−4145.44	−	7180.12i	−0.371895	−	0.644141i	−0.989857	+	0.142067i	$0.954625\pi$
$500$	−2816.65	−	4878.58i	−0.251929	−	0.436353i
$501$		0			0
$502$	1244.15	−	2154.93i	0.110616	−	0.191592i
$503$	8418.29			0.746228			0.373114	−	0.927785i	$-0.378290\pi$
$503$	8418.29			0.746228			0.373114	+	0.927785i	$0.378290\pi$
$504$		0			0
$505$	19372.5			1.70706
$506$	3405.16	−	5897.91i	0.299166	−	0.518170i
$507$		0			0
$508$	−1500.08	−	2598.22i	−0.131015	−	0.226924i
$509$	−3677.55	−	6369.71i	−0.320245	−	0.554681i	0.660293	−	0.751008i	$-0.270432\pi$
$509$	−3677.55	−	6369.71i	−0.320245	−	0.554681i	−0.980538	+	0.196327i	$0.937099\pi$
$510$		0			0
$511$	496.011	−	859.117i	0.0429398	−	0.0743740i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	6593.11			0.565777
$515$	−8642.57	+	14969.4i	−0.739489	+	1.28083i
$516$		0			0
$517$	−11175.5	−	19356.6i	−0.950677	−	1.64662i
$518$	886.878	+	1536.12i	0.0752263	+	0.130296i
$519$		0			0
$520$	216.375	−	374.773i	0.0182475	−	0.0316055i
$521$	13601.1			1.14372			0.571858	−	0.820353i	$-0.306223\pi$
$521$	13601.1			1.14372			0.571858	+	0.820353i	$0.306223\pi$
$522$		0			0
$523$	−16439.2			−1.37445			−0.687223	−	0.726447i	$-0.741170\pi$
$523$	−16439.2			−1.37445			−0.687223	+	0.726447i	$0.741170\pi$
$524$	−2154.07	+	3730.97i	−0.179582	+	0.311046i
$525$		0			0
$526$	7085.73	+	12272.8i	0.587362	+	1.01734i
$527$	−3990.85	−	6912.36i	−0.329875	−	0.571361i
$528$		0			0
$529$	4791.86	−	8299.74i	0.393840	−	0.682152i
$530$	12396.7			1.01600
$531$		0			0
$532$	2097.62			0.170946
$533$	375.515	−	650.411i	0.0305166	−	0.0528563i
$534$		0			0
$535$	−5735.92	−	9934.91i	−0.463524	−	0.802848i
$536$	−1625.84	−	2816.03i	−0.131018	−	0.226929i
$537$		0			0
$538$	3955.50	−	6851.13i	0.316977	−	0.549021i
$539$	3282.82			0.262340
$540$		0			0
$541$	6388.56			0.507700			0.253850	−	0.967244i	$-0.418303\pi$
$541$	6388.56			0.507700			0.253850	+	0.967244i	$0.418303\pi$
$542$	4100.35	−	7102.02i	0.324954	−	0.562837i
$543$		0			0
$544$	−1110.16	−	1922.86i	−0.0874959	−	0.151547i
$545$	−6882.45	−	11920.8i	−0.540939	−	0.936934i
$546$		0			0
$547$	8765.70	−	15182.6i	0.685182	−	1.18677i	−0.288198	−	0.957571i	$-0.593056\pi$
$547$	8765.70	−	15182.6i	0.685182	−	1.18677i	0.973380	−	0.229198i	$-0.0736105\pi$
$548$	534.085			0.0416331
$549$		0			0
$550$	263.632			0.0204387
$551$	6105.73	−	10575.4i	0.472074	−	0.817656i
$552$		0			0
$553$	−4005.71	−	6938.09i	−0.308029	−	0.533522i
$554$	−6020.59	−	10428.0i	−0.461715	−	0.799715i
$555$		0			0
$556$	4314.73	−	7473.33i	0.329110	−	0.570035i
$557$	−757.119			−0.0575945			−0.0287973	−	0.999585i	$-0.509168\pi$
$557$	−757.119			−0.0575945			−0.0287973	+	0.999585i	$0.509168\pi$
$558$		0			0
$559$	−1989.49			−0.150531
$560$	−621.152	+	1075.87i	−0.0468723	+	0.0811851i
$561$		0			0
$562$	−5274.05	−	9134.93i	−0.395859	−	0.685647i
$563$	5365.80	+	9293.84i	0.401673	+	0.695717i	0.993928	−	0.110033i	$-0.0350957\pi$
$563$	5365.80	+	9293.84i	0.401673	+	0.695717i	−0.592255	+	0.805750i	$0.701762\pi$
$564$		0			0
$565$	8095.56	−	14021.9i	0.602801	−	1.04408i
$566$	8666.76			0.643624
$567$		0			0
$568$	−6597.49			−0.487367
$569$	6334.14	−	10971.1i	0.466680	−	0.808314i	−0.532595	−	0.846370i	$-0.678783\pi$
$569$	6334.14	−	10971.1i	0.466680	−	0.808314i	0.999276	+	0.0380561i	$0.0121165\pi$
$570$		0			0
$571$	−10071.8	−	17444.8i	−0.738161	−	1.27853i	−0.953322	−	0.301954i	$-0.902361\pi$
$571$	−10071.8	−	17444.8i	−0.738161	−	1.27853i	0.215161	−	0.976579i	$-0.430972\pi$
$572$	653.460	+	1131.83i	0.0477666	+	0.0827343i
$573$		0			0
$574$	−1078.00	+	1867.15i	−0.0783881	+	0.135772i
$575$	−100.001			−0.00725272
$576$		0			0
$577$	−7062.25			−0.509541			−0.254771	−	0.967002i	$-0.582000\pi$
$577$	−7062.25			−0.509541			−0.254771	+	0.967002i	$0.582000\pi$
$578$	−98.7112	+	170.973i	−0.00710353	+	0.0123037i
$579$		0			0
$580$	3616.09	+	6263.24i	0.258879	+	0.448391i
$581$	−1643.49	−	2846.60i	−0.117355	−	0.203265i
$582$		0			0
$583$	−18719.2	+	32422.6i	−1.32980	+	2.30327i
$584$	−1133.74			−0.0803331
$585$		0			0
$586$	16171.6			1.14000
$587$	−9186.35	+	15911.2i	−0.645931	+	1.11878i	0.338155	+	0.941090i	$0.390197\pi$
$587$	−9186.35	+	15911.2i	−0.645931	+	1.11878i	−0.984086	+	0.177694i	$0.943136\pi$
$588$		0			0
$589$	−4308.92	−	7463.26i	−0.301436	−	0.522103i
$590$	5546.46	+	9606.76i	0.387024	+	0.670346i
$591$		0			0
$592$	1013.58	−	1755.56i	0.0703677	−	0.121880i
$593$	−20506.5			−1.42007			−0.710033	−	0.704168i	$-0.751320\pi$
$593$	−20506.5			−1.42007			−0.710033	+	0.704168i	$0.751320\pi$
$594$		0			0
$595$	−5387.34			−0.371192
$596$	1917.51	−	3321.22i	0.131786	−	0.228259i
$597$		0			0
$598$	−247.870	−	429.323i	−0.0169501	−	0.0293584i
$599$	−5810.14	−	10063.5i	−0.396320	−	0.686447i	0.596949	−	0.802280i	$-0.296380\pi$
$599$	−5810.14	−	10063.5i	−0.396320	−	0.686447i	−0.993269	+	0.115833i	$0.963046\pi$
$600$		0			0
$601$	−10119.0	+	17526.7i	−0.686795	+	1.18956i	0.286074	+	0.958207i	$0.407650\pi$
$601$	−10119.0	+	17526.7i	−0.686795	+	1.18956i	−0.972869	+	0.231356i	$0.925684\pi$
$602$	5711.27			0.386668
$603$		0			0
$604$	−2232.67			−0.150408
$605$	17511.6	−	30331.0i	1.17677	−	2.03823i
$606$		0			0
$607$	−3877.84	−	6716.61i	−0.259303	−	0.449125i	0.706753	−	0.707461i	$-0.250159\pi$
$607$	−3877.84	−	6716.61i	−0.259303	−	0.449125i	−0.966055	+	0.258336i	$0.916826\pi$
$608$	−1198.64	−	2076.10i	−0.0799527	−	0.138482i
$609$		0			0
$610$	−3037.69	+	5261.43i	−0.201627	+	0.349228i
$611$	−1626.99			−0.107727
$612$		0			0
$613$	2832.37			0.186621			0.0933103	−	0.995637i	$-0.470255\pi$
$613$	2832.37			0.186621			0.0933103	+	0.995637i	$0.470255\pi$
$614$	3828.04	−	6630.37i	0.251608	−	0.435798i
$615$		0			0
$616$	−1875.90	−	3249.15i	−0.122698	−	0.212520i
$617$	7266.09	+	12585.2i	0.474103	+	0.821171i	0.999560	−	0.0296494i	$-0.00943908\pi$
$617$	7266.09	+	12585.2i	0.474103	+	0.821171i	−0.525457	+	0.850820i	$0.676106\pi$
$618$		0			0
$619$	−8107.78	+	14043.1i	−0.526461	+	0.911856i	0.473064	+	0.881028i	$0.343148\pi$
$619$	−8107.78	+	14043.1i	−0.526461	+	0.911856i	−0.999525	+	0.0308284i	$0.990185\pi$
$620$	5103.87			0.330607
$621$		0			0
$622$	14606.3			0.941578
$623$	−5744.60	+	9949.93i	−0.369426	+	0.639865i
$624$		0			0
$625$	7687.60	+	13315.3i	0.492006	+	0.852180i
$626$	4868.53	+	8432.54i	0.310839	+	0.538390i
$627$		0			0
$628$	4628.78	−	8017.28i	0.294122	−	0.509433i
$629$	8790.88			0.557258
$630$		0			0
$631$	−1472.45			−0.0928959			−0.0464479	−	0.998921i	$-0.514790\pi$
$631$	−1472.45			−0.0928959			−0.0464479	+	0.998921i	$0.514790\pi$
$632$	−4577.95	+	7929.24i	−0.288135	+	0.499064i
$633$		0			0
$634$	−934.729	−	1619.00i	−0.0585533	−	0.101417i
$635$	4159.73	+	7204.86i	0.259959	+	0.450262i
$636$		0			0
$637$	119.482	−	206.949i	0.00743181	−	0.0128723i
$638$	−21841.4			−1.35534
$639$		0			0
$640$	1419.78			0.0876900
$641$	9521.13	−	16491.1i	0.586680	−	1.01616i	−0.407984	−	0.912989i	$-0.633768\pi$
$641$	9521.13	−	16491.1i	0.586680	−	1.01616i	0.994664	−	0.103171i	$-0.0328987\pi$
$642$		0			0
$643$	15927.9	+	27588.0i	0.976883	+	1.69201i	0.673575	+	0.739119i	$0.264758\pi$
$643$	15927.9	+	27588.0i	0.976883	+	1.69201i	0.303309	+	0.952892i	$0.401909\pi$
$644$	711.564	+	1232.47i	0.0435397	+	0.0754130i
$645$		0			0
$646$	5197.98	−	9003.17i	0.316582	−	0.548336i
$647$	−18434.8			−1.12016			−0.560082	−	0.828437i	$-0.689230\pi$
$647$	−18434.8			−1.12016			−0.560082	+	0.828437i	$0.689230\pi$
$648$		0			0
$649$	−33501.0			−2.02624
$650$	9.59519	−	16.6194i	0.000579007	−	0.00100287i
$651$		0			0
$652$	2221.42	+	3847.61i	0.133432	+	0.231111i
$653$	4246.16	+	7354.56i	0.254464	+	0.440745i	0.964750	−	0.263169i	$-0.0847676\pi$
$653$	4246.16	+	7354.56i	0.254464	+	0.440745i	−0.710286	+	0.703913i	$0.751434\pi$
$654$		0			0
$655$	5973.25	−	10346.0i	0.356327	−	0.617176i
$656$	2464.00			0.146651
$657$		0			0
$658$	4670.63			0.276717
$659$	14434.5	−	25001.2i	0.853243	−	1.47786i	−0.0250231	−	0.999687i	$-0.507966\pi$
$659$	14434.5	−	25001.2i	0.853243	−	1.47786i	0.878266	−	0.478173i	$-0.158701\pi$
$660$		0			0
$661$	−7722.84	−	13376.3i	−0.454438	−	0.787110i	0.544218	−	0.838944i	$-0.316827\pi$
$661$	−7722.84	−	13376.3i	−0.454438	−	0.787110i	−0.998656	+	0.0518343i	$0.983493\pi$
$662$	9596.90	+	16622.3i	0.563435	+	0.975899i
$663$		0			0
$664$	−1878.27	+	3253.26i	−0.109776	+	0.190137i
$665$	−5816.70			−0.339191
$666$		0			0
$667$	8284.86			0.480946
$668$	−3899.45	+	6754.05i	−0.225860	+	0.391201i
$669$		0			0
$670$	4508.44	+	7808.85i	0.259965	+	0.450272i
$671$	−9173.91	−	15889.7i	−0.527802	−	0.914179i
$672$		0			0
$673$	11259.1	−	19501.3i	0.644884	−	1.11697i	−0.339445	−	0.940626i	$-0.610239\pi$
$673$	11259.1	−	19501.3i	0.644884	−	1.11697i	0.984328	−	0.176345i	$-0.0564275\pi$
$674$	−13431.2			−0.767583
$675$		0			0
$676$	−8692.87			−0.494587
$677$	−11463.7	+	19855.7i	−0.650791	+	1.12720i	0.332140	+	0.943230i	$0.392229\pi$
$677$	−11463.7	+	19855.7i	−0.650791	+	1.12720i	−0.982931	+	0.183973i	$0.941104\pi$
$678$		0			0
$679$	4253.99	+	7368.12i	0.240432	+	0.416440i
$680$	3078.48	+	5332.08i	0.173609	+	0.300700i
$681$		0			0
$682$	−7706.92	+	13348.8i	−0.432717	+	0.749488i
$683$	−1565.19			−0.0876869			−0.0438434	−	0.999038i	$-0.513960\pi$
$683$	−1565.19			−0.0876869			−0.0438434	+	0.999038i	$0.513960\pi$
$684$		0			0
$685$	−1481.02			−0.0826084
$686$	−343.000	+	594.093i	−0.0190901	+	0.0330650i
$687$		0			0
$688$	−3263.59	−	5652.70i	−0.180847	−	0.313237i
$689$	1362.62	+	2360.12i	0.0753433	+	0.130498i
$690$		0			0
$691$	7176.64	−	12430.3i	0.395097	−	0.684329i	−0.598016	−	0.801484i	$-0.704044\pi$
$691$	7176.64	−	12430.3i	0.395097	−	0.684329i	0.993114	+	0.117155i	$0.0373775\pi$
$692$	−5044.84			−0.277133
$693$		0			0
$694$	15971.4			0.873582
$695$	−11964.7	+	20723.5i	−0.653019	+	1.13106i
$696$		0			0
$697$	5342.64	+	9253.73i	0.290340	+	0.502884i
$698$	6015.28	+	10418.8i	0.326192	+	0.564980i
$699$		0			0
$700$	−27.5451	+	47.7095i	−0.00148730	+	0.00257607i
$701$	24786.3			1.33547			0.667736	−	0.744398i	$-0.267263\pi$
$701$	24786.3			1.33547			0.667736	+	0.744398i	$0.267263\pi$
$702$		0			0
$703$	9491.50			0.509216
$704$	−2143.88	+	3713.32i	−0.114774	+	0.198794i
$705$		0			0
$706$	−1282.16	−	2220.76i	−0.0683493	−	0.118384i
$707$	−6112.85	−	10587.8i	−0.325173	−	0.563217i
$708$		0			0
$709$	−5161.03	+	8939.17i	−0.273380	+	0.473509i	−0.969725	−	0.244199i	$-0.921475\pi$
$709$	−5161.03	+	8939.17i	−0.273380	+	0.473509i	0.696345	+	0.717707i	$0.254808\pi$
$710$	18294.8			0.967032
$711$		0			0
$712$	13130.5			0.691133
$713$	2923.38	−	5063.45i	0.153551	−	0.265957i
$714$		0			0
$715$	−1812.04	−	3138.55i	−0.0947785	−	0.164161i
$716$	8355.35	+	14471.9i	0.436109	+	0.755363i
$717$		0			0
$718$	−6459.33	+	11187.9i	−0.335738	+	0.581516i
$719$	11243.4			0.583183			0.291592	−	0.956543i	$-0.405815\pi$
$719$	11243.4			0.583183			0.291592	+	0.956543i	$0.405815\pi$
$720$		0			0
$721$	10908.4			0.563453
$722$	−1246.75	+	2159.43i	−0.0642646	+	0.111310i
$723$		0			0
$724$	3406.81	+	5900.77i	0.174880	+	0.302901i
$725$	160.356	+	277.745i	0.00821444	+	0.0142278i
$726$		0			0
$727$	−16858.1	+	29199.1i	−0.860017	+	1.48959i	0.0118942	+	0.999929i	$0.496214\pi$
$727$	−16858.1	+	29199.1i	−0.860017	+	1.48959i	−0.871911	+	0.489664i	$0.837119\pi$
$728$	−273.102			−0.0139036
$729$		0			0
$730$	3143.86			0.159397
$731$	14152.8	−	24513.3i	0.716086	−	1.24030i
$732$		0			0
$733$	982.732	+	1702.14i	0.0495198	+	0.0857708i	0.889723	−	0.456501i	$-0.150898\pi$
$733$	982.732	+	1702.14i	0.0495198	+	0.0857708i	−0.840203	+	0.542272i	$0.817564\pi$
$734$	4196.69	+	7268.88i	0.211039	+	0.365530i
$735$		0			0
$736$	813.217	−	1408.53i	0.0407277	−	0.0705424i
$737$	−27231.3			−1.36103
$738$		0			0
$739$	−25596.3			−1.27412			−0.637060	−	0.770814i	$-0.719850\pi$
$739$	−25596.3			−1.27412			−0.637060	+	0.770814i	$0.719850\pi$
$740$	−2810.64	+	4868.18i	−0.139623	+	0.241835i
$741$		0			0
$742$	−3911.69	−	6775.24i	−0.193534	−	0.335212i
$743$	−3522.27	−	6100.75i	−0.173916	−	0.301231i	0.765870	−	0.642996i	$-0.222309\pi$
$743$	−3522.27	−	6100.75i	−0.173916	−	0.301231i	−0.939786	+	0.341765i	$0.888975\pi$
$744$		0			0
$745$	−5317.25	+	9209.75i	−0.261489	+	0.452911i
$746$	−16471.6			−0.808402
$747$		0			0
$748$	−18594.2			−0.908919
$749$	−3619.86	+	6269.78i	−0.176591	+	0.305865i
$750$		0			0
$751$	6973.75	+	12078.9i	0.338849	+	0.586904i	0.984217	−	0.176968i	$-0.0566290\pi$
$751$	6973.75	+	12078.9i	0.338849	+	0.586904i	−0.645367	+	0.763872i	$0.723296\pi$
$752$	−2668.93	−	4622.72i	−0.129423	−	0.224167i
$753$		0			0
$754$	−794.943	+	1376.88i	−0.0383954	+	0.0665028i
$755$	6191.21			0.298438
$756$		0			0
$757$	−35518.4			−1.70533			−0.852667	−	0.522455i	$-0.825016\pi$
$757$	−35518.4			−1.70533			−0.852667	+	0.522455i	$0.825016\pi$
$758$	−3135.68	+	5431.16i	−0.150255	+	0.260249i
$759$		0			0
$760$	3323.83	+	5757.04i	0.158642	+	0.274776i
$761$	6006.72	+	10403.9i	0.286128	+	0.495588i	0.972882	−	0.231302i	$-0.0742984\pi$
$761$	6006.72	+	10403.9i	0.286128	+	0.495588i	−0.686754	+	0.726890i	$0.740965\pi$
$762$		0			0
$763$	−4343.41	+	7523.01i	−0.206084	+	0.356948i
$764$	−17841.1			−0.844854
$765$		0			0
$766$	−16334.5			−0.770484
$767$	−1219.31	+	2111.90i	−0.0574012	+	0.0994217i
$768$		0			0
$769$	9848.80	+	17058.6i	0.461842	+	0.799934i	0.999053	−	0.0435138i	$-0.0138552\pi$
$769$	9848.80	+	17058.6i	0.461842	+	0.799934i	−0.537210	+	0.843448i	$0.680522\pi$
$770$	5201.87	+	9009.90i	0.243458	+	0.421681i
$771$		0			0
$772$	−3981.64	+	6896.40i	−0.185625	+	0.321512i
$773$	−7910.32			−0.368065			−0.184033	−	0.982920i	$-0.558915\pi$
$773$	−7910.32			−0.368065			−0.184033	+	0.982920i	$0.558915\pi$
$774$		0			0
$775$	226.332			0.0104904
$776$	4861.70	−	8420.71i	0.224903	−	0.389544i
$777$		0			0
$778$	−2494.12	−	4319.95i	−0.114934	−	0.199072i
$779$	5768.44	+	9991.24i	0.265309	+	0.459529i
$780$		0			0
$781$	−27625.5	+	47848.7i	−1.26571	+	2.19227i
$782$	7053.14			0.322532
$783$		0			0
$784$	784.000			0.0357143
$785$	−12835.6	+	22231.9i	−0.583595	+	1.01082i
$786$		0			0
$787$	−12714.6	−	22022.3i	−0.575890	−	0.997470i	−0.995944	−	0.0899715i	$-0.971322\pi$
$787$	−12714.6	−	22022.3i	−0.575890	−	0.997470i	0.420055	−	0.907499i	$-0.362011\pi$
$788$	−2430.38	−	4209.55i	−0.109872	−	0.190303i
$789$		0			0
$790$	12694.7	−	21987.8i	0.571716	−	0.990241i
$791$	−10218.0			−0.459304
$792$		0			0
$793$	−1335.58			−0.0598082
$794$	−8156.76	+	14127.9i	−0.364575	+	0.631463i
$795$		0			0
$796$	3353.64	+	5808.68i	0.149330	+	0.258647i
$797$	−16031.2	−	27766.9i	−0.712490	−	1.23407i	−0.963920	−	0.266193i	$-0.914234\pi$
$797$	−16031.2	−	27766.9i	−0.712490	−	1.23407i	0.251430	−	0.967875i	$-0.419099\pi$
$798$		0			0
$799$	11574.0	−	20046.8i	0.512464	−	0.887614i
$800$	62.9602			0.00278247
$801$		0			0
$802$	14575.7			0.641754
$803$	−4747.28	+	8222.53i	−0.208628	+	0.361354i
$804$		0			0
$805$	−1973.17	−	3417.63i	−0.0863914	−	0.149634i
$806$	561.005	+	971.689i	0.0245168	+	0.0424644i
$807$		0			0
$808$	−6986.12	+	12100.3i	−0.304172	+	0.526841i
$809$	3239.35			0.140778			0.0703891	−	0.997520i	$-0.477576\pi$
$809$	3239.35			0.140778			0.0703891	+	0.997520i	$0.477576\pi$
$810$		0			0
$811$	−4404.15			−0.190691			−0.0953457	−	0.995444i	$-0.530396\pi$
$811$	−4404.15			−0.190691			−0.0953457	+	0.995444i	$0.530396\pi$
$812$	2282.06	−	3952.64i	0.0986263	−	0.170826i
$813$		0			0
$814$	−8488.24	−	14702.1i	−0.365494	−	0.633055i
$815$	−6160.00	−	10669.4i	−0.264755	−	0.458569i
$816$		0			0
$817$	15280.7	−	26467.0i	0.654351	−	1.13337i
$818$	19424.8			0.830284
$819$		0			0
$820$	−6832.66			−0.290984
$821$	−532.421	+	922.180i	−0.0226329	+	0.0392013i	−0.877120	−	0.480271i	$-0.840538\pi$
$821$	−532.421	+	922.180i	−0.0226329	+	0.0392013i	0.854487	+	0.519473i	$0.173872\pi$
$822$		0			0
$823$	14620.2	+	25322.9i	0.619230	+	1.07254i	0.989626	+	0.143665i	$0.0458887\pi$
$823$	14620.2	+	25322.9i	0.619230	+	1.07254i	−0.370396	+	0.928874i	$0.620778\pi$
$824$	−6233.37	−	10796.5i	−0.263531	−	0.456449i
$825$		0			0
$826$	3500.29	−	6062.68i	0.147446	−	0.255385i
$827$	11840.8			0.497880			0.248940	−	0.968519i	$-0.419918\pi$
$827$	11840.8			0.497880			0.248940	+	0.968519i	$0.419918\pi$
$828$		0			0
$829$	12288.7			0.514844			0.257422	−	0.966299i	$-0.417127\pi$
$829$	12288.7			0.514844			0.257422	+	0.966299i	$0.417127\pi$
$830$	5208.44	−	9021.29i	0.217817	−	0.377269i
$831$		0			0
$832$	156.058	+	270.301i	0.00650283	+	0.0112632i
$833$	1699.93	+	2944.37i	0.0707074	+	0.122469i
$834$		0			0
$835$	10813.2	−	18729.0i	0.448150	−	0.776219i
$836$	−20076.1			−0.830559
$837$		0			0
$838$	1784.14			0.0735467
$839$	−16277.9	+	28194.1i	−0.669815	+	1.16015i	0.308141	+	0.951341i	$0.400293\pi$
$839$	−16277.9	+	28194.1i	−0.669815	+	1.16015i	−0.977956	+	0.208813i	$0.933040\pi$
$840$		0			0
$841$	−1090.68	−	1889.12i	−0.0447203	−	0.0774579i
$842$	−821.873	−	1423.53i	−0.0336385	−	0.0582636i
$843$		0			0
$844$	7605.47	−	13173.1i	0.310179	−	0.537245i
$845$	24105.3			0.981359
$846$		0			0
$847$	−22102.6			−0.896641
$848$	−4470.50	+	7743.13i	−0.181035	+	0.313562i
$849$		0			0
$850$	136.516	+	236.452i	0.00550876	+	0.00954145i
$851$	3219.75	+	5576.77i	0.129696	+	0.224641i
$852$		0			0
$853$	−9329.27	+	16158.8i	−0.374476	+	0.648611i	−0.990248	−	0.139313i	$-0.955511\pi$
$853$	−9329.27	+	16158.8i	−0.374476	+	0.648611i	0.615773	+	0.787924i	$0.288844\pi$
$854$	3834.08			0.153629
$855$		0			0
$856$	8273.96			0.330372
$857$	−6546.04	+	11338.1i	−0.260920	+	0.451927i	−0.966487	−	0.256717i	$-0.917359\pi$
$857$	−6546.04	+	11338.1i	−0.260920	+	0.451927i	0.705567	+	0.708644i	$0.250693\pi$
$858$		0			0
$859$	17527.3	+	30358.1i	0.696185	+	1.20583i	0.969780	+	0.243982i	$0.0784539\pi$
$859$	17527.3	+	30358.1i	0.696185	+	1.20583i	−0.273595	+	0.961845i	$0.588213\pi$
$860$	9049.92	+	15674.9i	0.358837	+	0.621524i
$861$		0			0
$862$	−4506.54	+	7805.56i	−0.178067	+	0.308420i
$863$	−6050.35			−0.238652			−0.119326	−	0.992855i	$-0.538073\pi$
$863$	−6050.35			−0.238652			−0.119326	+	0.992855i	$0.538073\pi$
$864$		0			0
$865$	13989.3			0.549886
$866$	9316.05	−	16135.9i	0.365557	−	0.633163i
$867$		0			0
$868$	−1610.49	−	2789.45i	−0.0629764	−	0.109078i
$869$	38338.3	+	66403.8i	1.49659	+	2.59217i
$870$		0			0
$871$	−991.115	+	1716.66i	−0.0385564	+	0.0667817i
$872$	9927.80			0.385548
$873$		0			0
$874$	7615.26			0.294726
$875$	4929.13	−	8537.51i	0.190440	−	0.329852i
$876$		0			0
$877$	−8661.08	−	15001.4i	−0.333482	−	0.577608i	0.649710	−	0.760182i	$-0.274890\pi$
$877$	−8661.08	−	15001.4i	−0.333482	−	0.577608i	−0.983192	+	0.182574i	$0.941557\pi$
$878$	3148.98	+	5454.20i	0.121040	+	0.209647i
$879$		0			0
$880$	5944.99	−	10297.0i	0.227734	−	0.394446i
$881$	33935.3			1.29774			0.648871	−	0.760898i	$-0.275241\pi$
$881$	33935.3			1.29774			0.648871	+	0.760898i	$0.275241\pi$
$882$		0			0
$883$	29815.2			1.13631			0.568154	−	0.822922i	$-0.307658\pi$
$883$	29815.2			1.13631			0.568154	+	0.822922i	$0.307658\pi$
$884$	−676.758	+	1172.18i	−0.0257487	+	0.0445980i
$885$		0			0
$886$	6442.37	+	11158.5i	0.244284	+	0.423112i
$887$	−41.9482	−	72.6564i	−0.00158792	−	0.00275035i	0.865230	−	0.501375i	$-0.167172\pi$
$887$	−41.9482	−	72.6564i	−0.00158792	−	0.00275035i	−0.866818	+	0.498624i	$0.833839\pi$
$888$		0			0
$889$	2625.14	−	4546.88i	0.0990377	−	0.171538i
$890$	−36410.9			−1.37134
$891$		0			0
$892$	−10662.9			−0.400247
$893$	12496.4	−	21644.5i	0.468284	−	0.811091i
$894$		0			0
$895$	−23169.4	−	40130.5i	−0.865326	−	1.49879i
$896$	−448.000	−	775.959i	−0.0167038	−	0.0289319i
$897$		0			0
$898$	1059.39	−	1834.92i	0.0393679	−	0.0681872i
$899$	−18751.2			−0.695647
$900$		0			0
$901$	−38773.3			−1.43366
$902$	10317.4	−	17870.3i	0.380857	−	0.659663i
$903$		0			0
$904$	5838.84	+	10113.2i	0.214820	+	0.372079i
$905$	−9447.09	−	16362.8i	−0.346996	−	0.601016i
$906$		0			0
$907$	−7174.85	+	12427.2i	−0.262665	+	0.454949i	−0.966949	−	0.254969i	$-0.917935\pi$
$907$	−7174.85	+	12427.2i	−0.262665	+	0.454949i	0.704284	+	0.709918i	$0.251268\pi$
$908$	16612.2			0.607155
$909$		0			0
$910$	757.313			0.0275876
$911$	−11421.7	+	19783.0i	−0.415388	+	0.719473i	−0.995469	−	0.0950858i	$-0.969687\pi$
$911$	−11421.7	+	19783.0i	−0.415388	+	0.719473i	0.580081	+	0.814559i	$0.303021\pi$
$912$		0			0
$913$	15729.7	+	27244.6i	0.570182	+	0.987584i
$914$	−4669.07	−	8087.06i	−0.168971	−	0.292666i
$915$		0			0
$916$	−2036.51	+	3527.34i	−0.0734587	+	0.127234i
$917$	−7539.26			−0.271503
$918$		0			0
$919$	−9755.75			−0.350177			−0.175088	−	0.984553i	$-0.556021\pi$
$919$	−9755.75			−0.350177			−0.175088	+	0.984553i	$0.556021\pi$
$920$	−2255.05	+	3905.86i	−0.0808117	+	0.139970i
$921$		0			0
$922$	7651.31	+	13252.5i	0.273300	+	0.473369i
$923$	2010.93	+	3483.02i	0.0717122	+	0.124209i
$924$		0			0
$925$	−124.639	+	215.880i	−0.00443037	+	0.00767362i
$926$	36787.0			1.30550
$927$		0			0
$928$	−5216.13			−0.184513
$929$	16206.2	−	28070.0i	0.572346	−	0.991332i	−0.423979	−	0.905672i	$-0.639367\pi$
$929$	16206.2	−	28070.0i	0.572346	−	0.991332i	0.996324	−	0.0856599i	$-0.0272998\pi$
$930$		0			0
$931$	1835.42	+	3179.04i	0.0646116	+	0.111911i
$932$	−1298.04	−	2248.27i	−0.0456209	−	0.0790177i
$933$		0			0
$934$	−6141.08	+	10636.7i	−0.215142	+	0.372636i
$935$	51561.7			1.80347
$936$		0			0
$937$	−45686.5			−1.59286			−0.796431	−	0.604729i	$-0.793281\pi$
$937$	−45686.5			−1.59286			−0.796431	+	0.604729i	$0.793281\pi$
$938$	2845.21	−	4928.05i	0.0990400	−	0.171542i
$939$		0			0
$940$	7400.95	+	12818.8i	0.256800	+	0.444791i
$941$	9728.77	+	16850.7i	0.337034	+	0.583760i	0.983873	−	0.178867i	$-0.0572430\pi$
$941$	9728.77	+	16850.7i	0.337034	+	0.583760i	−0.646840	+	0.762626i	$0.723910\pi$
$942$		0			0
$943$	−3913.60	+	6778.55i	−0.135148	+	0.234083i
$944$	−8000.67			−0.275847
$945$		0			0
$946$	−54662.1			−1.87867
$947$	256.737	−	444.681i	0.00880973	−	0.0152589i	−0.861587	−	0.507610i	$-0.830529\pi$
$947$	256.737	−	444.681i	0.00880973	−	0.0152589i	0.870397	+	0.492351i	$0.163862\pi$
$948$		0			0
$949$	345.566	+	598.538i	0.0118204	+	0.0204735i
$950$	147.396	+	255.297i	0.00503384	+	0.00871887i
$951$		0			0
$952$	1942.78	−	3365.00i	0.0661407	−	0.114559i
$953$	−24933.2			−0.847499			−0.423749	−	0.905779i	$-0.639286\pi$
$953$	−24933.2			−0.847499			−0.423749	+	0.905779i	$0.639286\pi$
$954$		0			0
$955$	49473.4			1.67636
$956$	9647.48	−	16709.9i	0.326383	−	0.565311i
$957$		0			0
$958$	5273.70	+	9134.31i	0.177855	+	0.308055i
$959$	467.324	+	809.429i	0.0157358	+	0.0272553i
$960$		0			0
$961$	8278.99	−	14339.6i	0.277902	−	0.481341i
$962$	−1235.76			−0.0414163
$963$		0			0
$964$	−8768.64			−0.292966
$965$	11041.1	−	19123.7i	0.368316	−	0.637942i
$966$		0			0
$967$	−7942.88	−	13757.5i	−0.264142	−	0.457508i	0.703196	−	0.710996i	$-0.251756\pi$
$967$	−7942.88	−	13757.5i	−0.264142	−	0.457508i	−0.967339	+	0.253488i	$0.918422\pi$
$968$	12630.1	+	21875.9i	0.419365	+	0.726362i
$969$		0			0
$970$	−13481.5	+	23350.6i	−0.446252	+	0.772932i
$971$	26642.9			0.880547			0.440274	−	0.897864i	$-0.354881\pi$
$971$	26642.9			0.880547			0.440274	+	0.897864i	$0.354881\pi$
$972$		0			0
$973$	15101.5			0.497568
$974$	18946.8	−	32816.8i	0.623301	−	1.07959i
$975$		0			0
$976$	−2190.90	−	3794.76i	−0.0718536	−	0.124454i
$977$	18841.5	+	32634.5i	0.616984	+	1.06865i	0.990033	+	0.140836i	$0.0449791\pi$
$977$	18841.5	+	32634.5i	0.616984	+	1.06865i	−0.373049	+	0.927812i	$0.621688\pi$
$978$		0			0
$979$	54981.0	−	95229.9i	1.79489	−	3.10885i
$980$	−2174.03			−0.0708642
$981$		0			0
$982$	9284.19			0.301701
$983$	−16441.9	+	28478.2i	−0.533483	+	0.924020i	0.465752	+	0.884915i	$0.345784\pi$
$983$	−16441.9	+	28478.2i	−0.533483	+	0.924020i	−0.999235	+	0.0391050i	$0.987549\pi$
$984$		0			0
$985$	6739.45	+	11673.1i	0.218007	+	0.377599i
$986$	−11310.1	−	19589.6i	−0.365300	−	0.632718i
$987$		0			0
$988$	−730.695	+	1265.60i	−0.0235288	+	0.0407532i
$989$	20734.4			0.666648
$990$		0			0
$991$	37238.4			1.19366			0.596831	−	0.802367i	$-0.296426\pi$
$991$	37238.4			1.19366			0.596831	+	0.802367i	$0.296426\pi$
$992$	−1840.56	+	3187.94i	−0.0589090	+	0.102033i
$993$		0			0
$994$	−5772.80	−	9998.78i	−0.184207	−	0.319056i
$995$	−9299.65	−	16107.5i	−0.296300	−	0.513207i
$996$		0			0
$997$	12388.4	−	21457.4i	0.393526	−	0.681607i	−0.599386	−	0.800460i	$-0.704589\pi$
$997$	12388.4	−	21457.4i	0.393526	−	0.681607i	0.992912	+	0.118853i	$0.0379219\pi$
$998$	−16581.8			−0.525939
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.f.b.127.4		8
3.2	odd	2		126.4.f.b.43.4	✓	8
9.2	odd	6		1134.4.a.o.1.4		4
9.4	even	3	inner	378.4.f.b.253.4		8
9.5	odd	6		126.4.f.b.85.4	yes	8
9.7	even	3		1134.4.a.l.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.f.b.43.4	✓	8	3.2	odd	2
126.4.f.b.85.4	yes	8	9.5	odd	6
378.4.f.b.127.4		8	1.1	even	1	trivial
378.4.f.b.253.4		8	9.4	even	3	inner
1134.4.a.l.1.1		4	9.7	even	3
1134.4.a.o.1.4		4	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.54600 + 9.60595i) q^{5} +(3.50000 - 6.06218i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(5.54600 + 9.60595i) q^{5} +(3.50000 - 6.06218i) q^{7} -8.00000 q^{8} +22.1840 q^{10} +(-33.4982 + 58.0206i) q^{11} +(2.43841 + 4.22346i) q^{13} +(-7.00000 - 12.1244i) q^{14} +(-8.00000 + 13.8564i) q^{16} -69.3851 q^{17} -74.9150 q^{19} +(22.1840 - 38.4238i) q^{20} +(66.9964 + 116.041i) q^{22} +(-25.4130 - 44.0166i) q^{23} +(0.983753 - 1.70391i) q^{25} +9.75366 q^{26} -28.0000 q^{28} +(-81.5021 + 141.166i) q^{29} +(57.5174 + 99.6231i) q^{31} +(16.0000 + 27.7128i) q^{32} +(-69.3851 + 120.178i) q^{34} +77.6440 q^{35} -126.697 q^{37} +(-74.9150 + 129.757i) q^{38} +(-44.3680 - 76.8476i) q^{40} +(-76.9999 - 133.368i) q^{41} +(-203.974 + 353.293i) q^{43} +267.986 q^{44} -101.652 q^{46} +(-166.808 + 288.920i) q^{47} +(-24.5000 - 42.4352i) q^{49} +(-1.96751 - 3.40782i) q^{50} +(9.75366 - 16.8938i) q^{52} +558.813 q^{53} -743.124 q^{55} +(-28.0000 + 48.4974i) q^{56} +(163.004 + 282.332i) q^{58} +(250.021 + 433.049i) q^{59} +(-136.931 + 237.172i) q^{61} +230.070 q^{62} +64.0000 q^{64} +(-27.0469 + 46.8466i) q^{65} +(203.229 + 352.004i) q^{67} +(138.770 + 240.357i) q^{68} +(77.6440 - 134.483i) q^{70} +824.686 q^{71} +141.718 q^{73} +(-126.697 + 219.446i) q^{74} +(149.830 + 259.513i) q^{76} +(234.487 + 406.144i) q^{77} +(572.244 - 991.156i) q^{79} -177.472 q^{80} -307.999 q^{82} +(234.784 - 406.657i) q^{83} +(-384.810 - 666.510i) q^{85} +(407.948 + 706.587i) q^{86} +(267.986 - 464.165i) q^{88} -1641.31 q^{89} +34.1378 q^{91} +(-101.652 + 176.067i) q^{92} +(333.616 + 577.841i) q^{94} +(-415.479 - 719.630i) q^{95} +(-607.712 + 1052.59i) q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8	\( 8 q + 8 q^{2} - 16 q^{4} - q^{5} + 28 q^{7} - 64 q^{8} - 4 q^{10} - 5 q^{11} + 21 q^{13} - 56 q^{14} - 64 q^{16} + 46 q^{17} - 188 q^{19} - 4 q^{20} + 10 q^{22} - 374 q^{23} - 41 q^{25} + 84 q^{26} - 224 q^{28} - 271 q^{29} + 243 q^{31} + 128 q^{32} + 46 q^{34} - 14 q^{35} - 362 q^{37} - 188 q^{38} + 8 q^{40} + 213 q^{41} + 238 q^{43} + 40 q^{44} - 1496 q^{46} - 675 q^{47} - 196 q^{49} + 82 q^{50} + 84 q^{52} - 108 q^{53} - 2828 q^{55} - 224 q^{56} + 542 q^{58} - 202 q^{59} + 1212 q^{61} + 972 q^{62} + 512 q^{64} - 549 q^{65} - 139 q^{67} - 92 q^{68} - 14 q^{70} + 2590 q^{71} - 4000 q^{73} - 362 q^{74} + 376 q^{76} + 35 q^{77} + 1545 q^{79} + 32 q^{80} + 852 q^{82} + 142 q^{83} + 793 q^{85} - 476 q^{86} + 40 q^{88} + 264 q^{89} + 294 q^{91} - 1496 q^{92} + 1350 q^{94} - 1244 q^{95} + 638 q^{97} - 784 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^4 - q^5 + 28 * q^7 - 64 * q^8 - 4 * q^10 - 5 * q^11 + 21 * q^13 - 56 * q^14 - 64 * q^16 + 46 * q^17 - 188 * q^19 - 4 * q^20 + 10 * q^22 - 374 * q^23 - 41 * q^25 + 84 * q^26 - 224 * q^28 - 271 * q^29 + 243 * q^31 + 128 * q^32 + 46 * q^34 - 14 * q^35 - 362 * q^37 - 188 * q^38 + 8 * q^40 + 213 * q^41 + 238 * q^43 + 40 * q^44 - 1496 * q^46 - 675 * q^47 - 196 * q^49 + 82 * q^50 + 84 * q^52 - 108 * q^53 - 2828 * q^55 - 224 * q^56 + 542 * q^58 - 202 * q^59 + 1212 * q^61 + 972 * q^62 + 512 * q^64 - 549 * q^65 - 139 * q^67 - 92 * q^68 - 14 * q^70 + 2590 * q^71 - 4000 * q^73 - 362 * q^74 + 376 * q^76 + 35 * q^77 + 1545 * q^79 + 32 * q^80 + 852 * q^82 + 142 * q^83 + 793 * q^85 - 476 * q^86 + 40 * q^88 + 264 * q^89 + 294 * q^91 - 1496 * q^92 + 1350 * q^94 - 1244 * q^95 + 638 * q^97 - 784 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more