LMFDB - Embedded newform 4005.2.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.a (trivial)

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:	yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 128x^{6} + 14x^{5} - 358x^{4} - 59x^{3} + 344x^{2} + 71x - 21 $ x^10 - 19x^8 - x^7 + 128x^6 + 14x^5 - 358x^4 - 59x^3 + 344x^2 + 71*x - 21
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.37466$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.37466 q^{2} -0.110304 q^{4} +1.00000 q^{5} -2.28939 q^{7} +2.90096 q^{8} +O(q^{10})$	\(q-1.37466 q^{2} -0.110304 q^{4} +1.00000 q^{5} -2.28939 q^{7} +2.90096 q^{8} -1.37466 q^{10} -3.97532 q^{11} +6.54828 q^{13} +3.14713 q^{14} -3.76722 q^{16} -3.55006 q^{17} +7.14204 q^{19} -0.110304 q^{20} +5.46472 q^{22} +3.65936 q^{23} +1.00000 q^{25} -9.00166 q^{26} +0.252529 q^{28} +4.70602 q^{29} +1.10656 q^{31} -0.623251 q^{32} +4.88014 q^{34} -2.28939 q^{35} -7.55337 q^{37} -9.81789 q^{38} +2.90096 q^{40} -7.27038 q^{41} +6.83907 q^{43} +0.438496 q^{44} -5.03039 q^{46} -10.5244 q^{47} -1.75871 q^{49} -1.37466 q^{50} -0.722304 q^{52} -7.37187 q^{53} -3.97532 q^{55} -6.64140 q^{56} -6.46919 q^{58} +7.36764 q^{59} +3.90465 q^{61} -1.52115 q^{62} +8.39121 q^{64} +6.54828 q^{65} +3.25353 q^{67} +0.391588 q^{68} +3.14713 q^{70} +11.3671 q^{71} +7.36324 q^{73} +10.3833 q^{74} -0.787799 q^{76} +9.10104 q^{77} -11.2238 q^{79} -3.76722 q^{80} +9.99431 q^{82} -1.59921 q^{83} -3.55006 q^{85} -9.40141 q^{86} -11.5322 q^{88} +1.00000 q^{89} -14.9915 q^{91} -0.403644 q^{92} +14.4674 q^{94} +7.14204 q^{95} -4.76685 q^{97} +2.41764 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8	$ 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8} - 4 q^{11} + 15 q^{13} + q^{14} + 22 q^{16} - 11 q^{17} + 14 q^{19} + 18 q^{20} + 10 q^{22} + 8 q^{23} + 10 q^{25} + 14 q^{26} + 36 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 9 q^{35} + 23 q^{37} - 8 q^{38} - 3 q^{40} - 13 q^{41} + 25 q^{43} + 8 q^{44} - 14 q^{46} + q^{47} + 21 q^{49} + 51 q^{52} - 9 q^{53} - 4 q^{55} - 15 q^{56} - 8 q^{58} + 15 q^{59} + 8 q^{61} + 8 q^{62} + 9 q^{64} + 15 q^{65} + 52 q^{67} + 28 q^{68} + q^{70} + 22 q^{71} + 34 q^{73} + 18 q^{74} + 14 q^{76} - 4 q^{77} - 3 q^{79} + 22 q^{80} + 17 q^{82} + 10 q^{83} - 11 q^{85} + 6 q^{86} + 4 q^{88} + 10 q^{89} + 18 q^{91} + 14 q^{92} - 43 q^{94} + 14 q^{95} + 34 q^{97} + 38 q^{98}+O(q^{100}) $ 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8 - 4 * q^11 + 15 * q^13 + q^14 + 22 * q^16 - 11 * q^17 + 14 * q^19 + 18 * q^20 + 10 * q^22 + 8 * q^23 + 10 * q^25 + 14 * q^26 + 36 * q^28 - 5 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 9 * q^35 + 23 * q^37 - 8 * q^38 - 3 * q^40 - 13 * q^41 + 25 * q^43 + 8 * q^44 - 14 * q^46 + q^47 + 21 * q^49 + 51 * q^52 - 9 * q^53 - 4 * q^55 - 15 * q^56 - 8 * q^58 + 15 * q^59 + 8 * q^61 + 8 * q^62 + 9 * q^64 + 15 * q^65 + 52 * q^67 + 28 * q^68 + q^70 + 22 * q^71 + 34 * q^73 + 18 * q^74 + 14 * q^76 - 4 * q^77 - 3 * q^79 + 22 * q^80 + 17 * q^82 + 10 * q^83 - 11 * q^85 + 6 * q^86 + 4 * q^88 + 10 * q^89 + 18 * q^91 + 14 * q^92 - 43 * q^94 + 14 * q^95 + 34 * q^97 + 38 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.37466	−0.972033	−0.486016	−	0.873950i	$-0.661550\pi$
$2$	−1.37466	−0.972033	−0.486016	+	0.873950i	$0.661550\pi$
$3$	0	0
$3$	0	0
$4$	−0.110304	−0.0551522
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.28939	−0.865306	−0.432653	−	0.901560i	$-0.642423\pi$
$7$	−2.28939	−0.865306	−0.432653	+	0.901560i	$0.642423\pi$
$8$	2.90096	1.02564
$9$	0	0
$10$	−1.37466	−0.434706
$11$	−3.97532	−1.19860	−0.599302	−	0.800523i	$-0.704555\pi$
$11$	−3.97532	−1.19860	−0.599302	+	0.800523i	$0.704555\pi$
$12$	0	0
$13$	6.54828	1.81616	0.908082	−	0.418792i	$-0.137546\pi$
$13$	6.54828	1.81616	0.908082	+	0.418792i	$0.137546\pi$
$14$	3.14713	0.841106
$15$	0	0
$16$	−3.76722	−0.941806
$17$	−3.55006	−0.861017	−0.430509	−	0.902586i	$-0.641666\pi$
$17$	−3.55006	−0.861017	−0.430509	+	0.902586i	$0.641666\pi$
$18$	0	0
$19$	7.14204	1.63850	0.819248	−	0.573440i	$-0.194391\pi$
$19$	7.14204	1.63850	0.819248	+	0.573440i	$0.194391\pi$
$20$	−0.110304	−0.0246648
$21$	0	0
$22$	5.46472	1.16508
$23$	3.65936	0.763030	0.381515	−	0.924363i	$-0.375402\pi$
$23$	3.65936	0.763030	0.381515	+	0.924363i	$0.375402\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−9.00166	−1.76537
$27$	0	0
$28$	0.252529	0.0477236
$29$	4.70602	0.873886	0.436943	−	0.899489i	$-0.356061\pi$
$29$	4.70602	0.873886	0.436943	+	0.899489i	$0.356061\pi$
$30$	0	0
$31$	1.10656	0.198744	0.0993721	−	0.995050i	$-0.468317\pi$
$31$	1.10656	0.198744	0.0993721	+	0.995050i	$0.468317\pi$
$32$	−0.623251	−0.110176
$33$	0	0
$34$	4.88014	0.836937
$35$	−2.28939	−0.386977
$36$	0	0
$37$	−7.55337	−1.24177	−0.620884	−	0.783903i	$-0.713226\pi$
$37$	−7.55337	−1.24177	−0.620884	+	0.783903i	$0.713226\pi$
$38$	−9.81789	−1.59267
$39$	0	0
$40$	2.90096	0.458681
$41$	−7.27038	−1.13544	−0.567721	−	0.823221i	$-0.692175\pi$
$41$	−7.27038	−1.13544	−0.567721	+	0.823221i	$0.692175\pi$
$42$	0	0
$43$	6.83907	1.04295	0.521474	−	0.853267i	$-0.325382\pi$
$43$	6.83907	1.04295	0.521474	+	0.853267i	$0.325382\pi$
$44$	0.438496	0.0661057
$45$	0	0
$46$	−5.03039	−0.741690
$47$	−10.5244	−1.53514	−0.767568	−	0.640967i	$-0.778533\pi$
$47$	−10.5244	−1.53514	−0.767568	+	0.640967i	$0.778533\pi$
$48$	0	0
$49$	−1.75871	−0.251245
$50$	−1.37466	−0.194407
$51$	0	0
$52$	−0.722304	−0.100166
$53$	−7.37187	−1.01260	−0.506302	−	0.862356i	$-0.668988\pi$
$53$	−7.37187	−1.01260	−0.506302	+	0.862356i	$0.668988\pi$
$54$	0	0
$55$	−3.97532	−0.536032
$56$	−6.64140	−0.887495
$57$	0	0
$58$	−6.46919	−0.849446
$59$	7.36764	0.959185	0.479593	−	0.877491i	$-0.340784\pi$
$59$	7.36764	0.959185	0.479593	+	0.877491i	$0.340784\pi$
$60$	0	0
$61$	3.90465	0.499939	0.249970	−	0.968254i	$-0.419579\pi$
$61$	3.90465	0.499939	0.249970	+	0.968254i	$0.419579\pi$
$62$	−1.52115	−0.193186
$63$	0	0
$64$	8.39121	1.04890
$65$	6.54828	0.812214
$66$	0	0
$67$	3.25353	0.397482	0.198741	−	0.980052i	$-0.436315\pi$
$67$	3.25353	0.397482	0.198741	+	0.980052i	$0.436315\pi$
$68$	0.391588	0.0474870
$69$	0	0
$70$	3.14713	0.376154
$71$	11.3671	1.34903	0.674516	−	0.738260i	$-0.264352\pi$
$71$	11.3671	1.34903	0.674516	+	0.738260i	$0.264352\pi$
$72$	0	0
$73$	7.36324	0.861802	0.430901	−	0.902399i	$-0.358196\pi$
$73$	7.36324	0.861802	0.430901	+	0.902399i	$0.358196\pi$
$74$	10.3833	1.20704
$75$	0	0
$76$	−0.787799	−0.0903667
$77$	9.10104	1.03716
$78$	0	0
$79$	−11.2238	−1.26278	−0.631390	−	0.775465i	$-0.717515\pi$
$79$	−11.2238	−1.26278	−0.631390	+	0.775465i	$0.717515\pi$
$80$	−3.76722	−0.421188
$81$	0	0
$82$	9.99431	1.10369
$83$	−1.59921	−0.175536	−0.0877680	−	0.996141i	$-0.527973\pi$
$83$	−1.59921	−0.175536	−0.0877680	+	0.996141i	$0.527973\pi$
$84$	0	0
$85$	−3.55006	−0.385059
$86$	−9.40141	−1.01378
$87$	0	0
$88$	−11.5322	−1.22934
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−14.9915	−1.57154
$92$	−0.403644	−0.0420828
$93$	0	0
$94$	14.4674	1.49220
$95$	7.14204	0.732757
$96$	0	0
$97$	−4.76685	−0.484000	−0.242000	−	0.970276i	$-0.577803\pi$
$97$	−4.76685	−0.484000	−0.242000	+	0.970276i	$0.577803\pi$
$98$	2.41764	0.244218
$99$	0	0
$100$	−0.110304	−0.0110304
$101$	−3.40420	−0.338731	−0.169365	−	0.985553i	$-0.554172\pi$
$101$	−3.40420	−0.338731	−0.169365	+	0.985553i	$0.554172\pi$
$102$	0	0
$103$	12.7412	1.25543	0.627715	−	0.778443i	$-0.283990\pi$
$103$	12.7412	1.25543	0.627715	+	0.778443i	$0.283990\pi$
$104$	18.9963	1.86274
$105$	0	0
$106$	10.1338	0.984284
$107$	12.4923	1.20768	0.603840	−	0.797106i	$-0.293637\pi$
$107$	12.4923	1.20768	0.603840	+	0.797106i	$0.293637\pi$
$108$	0	0
$109$	−11.9503	−1.14463	−0.572314	−	0.820034i	$-0.693954\pi$
$109$	−11.9503	−1.14463	−0.572314	+	0.820034i	$0.693954\pi$
$110$	5.46472	0.521041
$111$	0	0
$112$	8.62463	0.814951
$113$	10.3102	0.969906	0.484953	−	0.874540i	$-0.338837\pi$
$113$	10.3102	0.969906	0.484953	+	0.874540i	$0.338837\pi$
$114$	0	0
$115$	3.65936	0.341237
$116$	−0.519095	−0.0481968
$117$	0	0
$118$	−10.1280	−0.932359
$119$	8.12747	0.745044
$120$	0	0
$121$	4.80318	0.436653
$122$	−5.36758	−0.485957
$123$	0	0
$124$	−0.122059	−0.0109612
$125$	1.00000	0.0894427
$126$	0	0
$127$	−21.5528	−1.91250	−0.956252	−	0.292546i	$-0.905498\pi$
$127$	−21.5528	−1.91250	−0.956252	+	0.292546i	$0.905498\pi$
$128$	−10.2886	−0.909390
$129$	0	0
$130$	−9.00166	−0.789498
$131$	9.41146	0.822283	0.411142	−	0.911571i	$-0.365130\pi$
$131$	9.41146	0.822283	0.411142	+	0.911571i	$0.365130\pi$
$132$	0	0
$133$	−16.3509	−1.41780
$134$	−4.47251	−0.386366
$135$	0	0
$136$	−10.2986	−0.883096
$137$	8.45290	0.722180	0.361090	−	0.932531i	$-0.382405\pi$
$137$	8.45290	0.722180	0.361090	+	0.932531i	$0.382405\pi$
$138$	0	0
$139$	−4.04600	−0.343177	−0.171589	−	0.985169i	$-0.554890\pi$
$139$	−4.04600	−0.343177	−0.171589	+	0.985169i	$0.554890\pi$
$140$	0.252529	0.0213426
$141$	0	0
$142$	−15.6260	−1.31130
$143$	−26.0315	−2.17686
$144$	0	0
$145$	4.70602	0.390814
$146$	−10.1220	−0.837699
$147$	0	0
$148$	0.833171	0.0684862
$149$	−8.90940	−0.729886	−0.364943	−	0.931030i	$-0.618912\pi$
$149$	−8.90940	−0.729886	−0.364943	+	0.931030i	$0.618912\pi$
$150$	0	0
$151$	−3.30603	−0.269041	−0.134520	−	0.990911i	$-0.542949\pi$
$151$	−3.30603	−0.269041	−0.134520	+	0.990911i	$0.542949\pi$
$152$	20.7187	1.68051
$153$	0	0
$154$	−12.5109	−1.00815
$155$	1.10656	0.0888811
$156$	0	0
$157$	16.2493	1.29684	0.648420	−	0.761283i	$-0.275430\pi$
$157$	16.2493	1.29684	0.648420	+	0.761283i	$0.275430\pi$
$158$	15.4290	1.22746
$159$	0	0
$160$	−0.623251	−0.0492723
$161$	−8.37769	−0.660255
$162$	0	0
$163$	18.0532	1.41404	0.707019	−	0.707194i	$-0.250039\pi$
$163$	18.0532	1.41404	0.707019	+	0.707194i	$0.250039\pi$
$164$	0.801955	0.0626222
$165$	0	0
$166$	2.19837	0.170627
$167$	22.3280	1.72779	0.863897	−	0.503668i	$-0.168017\pi$
$167$	22.3280	1.72779	0.863897	+	0.503668i	$0.168017\pi$
$168$	0	0
$169$	29.8799	2.29845
$170$	4.88014	0.374290
$171$	0	0
$172$	−0.754380	−0.0575210
$173$	−6.98409	−0.530990	−0.265495	−	0.964112i	$-0.585535\pi$
$173$	−6.98409	−0.530990	−0.265495	+	0.964112i	$0.585535\pi$
$174$	0	0
$175$	−2.28939	−0.173061
$176$	14.9759	1.12885
$177$	0	0
$178$	−1.37466	−0.103035
$179$	−21.3067	−1.59254	−0.796268	−	0.604944i	$-0.793196\pi$
$179$	−21.3067	−1.59254	−0.796268	+	0.604944i	$0.793196\pi$
$180$	0	0
$181$	8.45589	0.628521	0.314261	−	0.949337i	$-0.398243\pi$
$181$	8.45589	0.628521	0.314261	+	0.949337i	$0.398243\pi$
$182$	20.6083	1.52759
$183$	0	0
$184$	10.6157	0.782596
$185$	−7.55337	−0.555335
$186$	0	0
$187$	14.1126	1.03202
$188$	1.16088	0.0846662
$189$	0	0
$190$	−9.81789	−0.712264
$191$	19.9797	1.44568	0.722842	−	0.691014i	$-0.242836\pi$
$191$	19.9797	1.44568	0.722842	+	0.691014i	$0.242836\pi$
$192$	0	0
$193$	5.04259	0.362974	0.181487	−	0.983393i	$-0.441909\pi$
$193$	5.04259	0.362974	0.181487	+	0.983393i	$0.441909\pi$
$194$	6.55281	0.470464
$195$	0	0
$196$	0.193994	0.0138567
$197$	9.78601	0.697224	0.348612	−	0.937267i	$-0.386653\pi$
$197$	9.78601	0.697224	0.348612	+	0.937267i	$0.386653\pi$
$198$	0	0
$199$	−5.98296	−0.424121	−0.212060	−	0.977257i	$-0.568017\pi$
$199$	−5.98296	−0.424121	−0.212060	+	0.977257i	$0.568017\pi$
$200$	2.90096	0.205129
$201$	0	0
$202$	4.67963	0.329257
$203$	−10.7739	−0.756179
$204$	0	0
$205$	−7.27038	−0.507785
$206$	−17.5149	−1.22032
$207$	0	0
$208$	−24.6688	−1.71047
$209$	−28.3919	−1.96391
$210$	0	0
$211$	3.42611	0.235863	0.117931	−	0.993022i	$-0.462374\pi$
$211$	3.42611	0.235863	0.117931	+	0.993022i	$0.462374\pi$
$212$	0.813150	0.0558474
$213$	0	0
$214$	−17.1727	−1.17390
$215$	6.83907	0.466421
$216$	0	0
$217$	−2.53334	−0.171975
$218$	16.4276	1.11262
$219$	0	0
$220$	0.438496	0.0295634
$221$	−23.2468	−1.56375
$222$	0	0
$223$	3.72595	0.249508	0.124754	−	0.992188i	$-0.460186\pi$
$223$	3.72595	0.249508	0.124754	+	0.992188i	$0.460186\pi$
$224$	1.42686	0.0953362
$225$	0	0
$226$	−14.1731	−0.942780
$227$	3.75690	0.249354	0.124677	−	0.992197i	$-0.460210\pi$
$227$	3.75690	0.249354	0.124677	+	0.992197i	$0.460210\pi$
$228$	0	0
$229$	−12.7045	−0.839535	−0.419767	−	0.907632i	$-0.637888\pi$
$229$	−12.7045	−0.839535	−0.419767	+	0.907632i	$0.637888\pi$
$230$	−5.03039	−0.331694
$231$	0	0
$232$	13.6520	0.896295
$233$	13.2602	0.868706	0.434353	−	0.900743i	$-0.356977\pi$
$233$	13.2602	0.868706	0.434353	+	0.900743i	$0.356977\pi$
$234$	0	0
$235$	−10.5244	−0.686534
$236$	−0.812684	−0.0529012
$237$	0	0
$238$	−11.1725	−0.724207
$239$	5.04324	0.326220	0.163110	−	0.986608i	$-0.447847\pi$
$239$	5.04324	0.326220	0.163110	+	0.986608i	$0.447847\pi$
$240$	0	0
$241$	−14.9409	−0.962431	−0.481216	−	0.876602i	$-0.659805\pi$
$241$	−14.9409	−0.962431	−0.481216	+	0.876602i	$0.659805\pi$
$242$	−6.60275	−0.424441
$243$	0	0
$244$	−0.430701	−0.0275728
$245$	−1.75871	−0.112360
$246$	0	0
$247$	46.7680	2.97578
$248$	3.21008	0.203841
$249$	0	0
$250$	−1.37466	−0.0869413
$251$	−19.9025	−1.25623	−0.628116	−	0.778120i	$-0.716174\pi$
$251$	−19.9025	−1.25623	−0.628116	+	0.778120i	$0.716174\pi$
$252$	0	0
$253$	−14.5471	−0.914571
$254$	29.6278	1.85902
$255$	0	0
$256$	−2.63911	−0.164944
$257$	10.8233	0.675136	0.337568	−	0.941301i	$-0.390396\pi$
$257$	10.8233	0.675136	0.337568	+	0.941301i	$0.390396\pi$
$258$	0	0
$259$	17.2926	1.07451
$260$	−0.722304	−0.0447954
$261$	0	0
$262$	−12.9376	−0.799286
$263$	26.1606	1.61313	0.806566	−	0.591144i	$-0.201324\pi$
$263$	26.1606	1.61313	0.806566	+	0.591144i	$0.201324\pi$
$264$	0	0
$265$	−7.37187	−0.452850
$266$	22.4769	1.37815
$267$	0	0
$268$	−0.358879	−0.0219220
$269$	−2.66751	−0.162641	−0.0813205	−	0.996688i	$-0.525914\pi$
$269$	−2.66751	−0.162641	−0.0813205	+	0.996688i	$0.525914\pi$
$270$	0	0
$271$	23.2561	1.41271	0.706353	−	0.707860i	$-0.250339\pi$
$271$	23.2561	1.41271	0.706353	+	0.707860i	$0.250339\pi$
$272$	13.3739	0.810911
$273$	0	0
$274$	−11.6199	−0.701983
$275$	−3.97532	−0.239721
$276$	0	0
$277$	9.24683	0.555588	0.277794	−	0.960641i	$-0.410397\pi$
$277$	9.24683	0.555588	0.277794	+	0.960641i	$0.410397\pi$
$278$	5.56188	0.333579
$279$	0	0
$280$	−6.64140	−0.396900
$281$	−0.733660	−0.0437665	−0.0218832	−	0.999761i	$-0.506966\pi$
$281$	−0.733660	−0.0437665	−0.0218832	+	0.999761i	$0.506966\pi$
$282$	0	0
$283$	2.08489	0.123934	0.0619668	−	0.998078i	$-0.480263\pi$
$283$	2.08489	0.123934	0.0619668	+	0.998078i	$0.480263\pi$
$284$	−1.25385	−0.0744021
$285$	0	0
$286$	35.7845	2.11598
$287$	16.6447	0.982506
$288$	0	0
$289$	−4.39704	−0.258649
$290$	−6.46919	−0.379884
$291$	0	0
$292$	−0.812198	−0.0475303
$293$	3.64176	0.212754	0.106377	−	0.994326i	$-0.466075\pi$
$293$	3.64176	0.212754	0.106377	+	0.994326i	$0.466075\pi$
$294$	0	0
$295$	7.36764	0.428961
$296$	−21.9120	−1.27361
$297$	0	0
$298$	12.2474	0.709473
$299$	23.9625	1.38579
$300$	0	0
$301$	−15.6573	−0.902470
$302$	4.54467	0.261516
$303$	0	0
$304$	−26.9057	−1.54314
$305$	3.90465	0.223580
$306$	0	0
$307$	26.8470	1.53224	0.766118	−	0.642700i	$-0.222186\pi$
$307$	26.8470	1.53224	0.766118	+	0.642700i	$0.222186\pi$
$308$	−1.00389	−0.0572017
$309$	0	0
$310$	−1.52115	−0.0863954
$311$	28.7155	1.62830	0.814152	−	0.580651i	$-0.197202\pi$
$311$	28.7155	1.62830	0.814152	+	0.580651i	$0.197202\pi$
$312$	0	0
$313$	33.0233	1.86659	0.933293	−	0.359116i	$-0.116922\pi$
$313$	33.0233	1.86659	0.933293	+	0.359116i	$0.116922\pi$
$314$	−22.3374	−1.26057
$315$	0	0
$316$	1.23804	0.0696452
$317$	−11.9134	−0.669125	−0.334563	−	0.942374i	$-0.608589\pi$
$317$	−11.9134	−0.669125	−0.334563	+	0.942374i	$0.608589\pi$
$318$	0	0
$319$	−18.7079	−1.04744
$320$	8.39121	0.469083
$321$	0	0
$322$	11.5165	0.641789
$323$	−25.3547	−1.41077
$324$	0	0
$325$	6.54828	0.363233
$326$	−24.8171	−1.37449
$327$	0	0
$328$	−21.0910	−1.16456
$329$	24.0943	1.32836
$330$	0	0
$331$	5.66736	0.311506	0.155753	−	0.987796i	$-0.450220\pi$
$331$	5.66736	0.311506	0.155753	+	0.987796i	$0.450220\pi$
$332$	0.176400	0.00968121
$333$	0	0
$334$	−30.6935	−1.67947
$335$	3.25353	0.177760
$336$	0	0
$337$	−18.8504	−1.02685	−0.513424	−	0.858135i	$-0.671623\pi$
$337$	−18.8504	−1.02685	−0.513424	+	0.858135i	$0.671623\pi$
$338$	−41.0748	−2.23417
$339$	0	0
$340$	0.391588	0.0212368
$341$	−4.39894	−0.238216
$342$	0	0
$343$	20.0521	1.08271
$344$	19.8398	1.06969
$345$	0	0
$346$	9.60076	0.516140
$347$	−19.0792	−1.02422	−0.512112	−	0.858919i	$-0.671137\pi$
$347$	−19.0792	−1.02422	−0.512112	+	0.858919i	$0.671137\pi$
$348$	0	0
$349$	10.5867	0.566696	0.283348	−	0.959017i	$-0.408555\pi$
$349$	10.5867	0.566696	0.283348	+	0.959017i	$0.408555\pi$
$350$	3.14713	0.168221
$351$	0	0
$352$	2.47762	0.132058
$353$	−23.6295	−1.25767	−0.628837	−	0.777537i	$-0.716469\pi$
$353$	−23.6295	−1.25767	−0.628837	+	0.777537i	$0.716469\pi$
$354$	0	0
$355$	11.3671	0.603305
$356$	−0.110304	−0.00584613
$357$	0	0
$358$	29.2895	1.54800
$359$	−17.7877	−0.938798	−0.469399	−	0.882986i	$-0.655529\pi$
$359$	−17.7877	−0.938798	−0.469399	+	0.882986i	$0.655529\pi$
$360$	0	0
$361$	32.0087	1.68467
$362$	−11.6240	−0.610943
$363$	0	0
$364$	1.65363	0.0866739
$365$	7.36324	0.385409
$366$	0	0
$367$	22.5633	1.17779	0.588897	−	0.808208i	$-0.299562\pi$
$367$	22.5633	1.17779	0.588897	+	0.808208i	$0.299562\pi$
$368$	−13.7856	−0.718626
$369$	0	0
$370$	10.3833	0.539804
$371$	16.8770	0.876212
$372$	0	0
$373$	26.2815	1.36080	0.680402	−	0.732839i	$-0.261805\pi$
$373$	26.2815	1.36080	0.680402	+	0.732839i	$0.261805\pi$
$374$	−19.4001	−1.00316
$375$	0	0
$376$	−30.5307	−1.57450
$377$	30.8163	1.58712
$378$	0	0
$379$	−3.96279	−0.203555	−0.101777	−	0.994807i	$-0.532453\pi$
$379$	−3.96279	−0.203555	−0.101777	+	0.994807i	$0.532453\pi$
$380$	−0.787799	−0.0404132
$381$	0	0
$382$	−27.4654	−1.40525
$383$	7.80078	0.398601	0.199301	−	0.979938i	$-0.436133\pi$
$383$	7.80078	0.398601	0.199301	+	0.979938i	$0.436133\pi$
$384$	0	0
$385$	9.10104	0.463832
$386$	−6.93186	−0.352822
$387$	0	0
$388$	0.525805	0.0266937
$389$	5.71843	0.289936	0.144968	−	0.989436i	$-0.453692\pi$
$389$	5.71843	0.289936	0.144968	+	0.989436i	$0.453692\pi$
$390$	0	0
$391$	−12.9910	−0.656982
$392$	−5.10195	−0.257688
$393$	0	0
$394$	−13.4524	−0.677725
$395$	−11.2238	−0.564733
$396$	0	0
$397$	3.95725	0.198608	0.0993042	−	0.995057i	$-0.468338\pi$
$397$	3.95725	0.198608	0.0993042	+	0.995057i	$0.468338\pi$
$398$	8.22455	0.412259
$399$	0	0
$400$	−3.76722	−0.188361
$401$	31.2629	1.56119	0.780597	−	0.625034i	$-0.214915\pi$
$401$	31.2629	1.56119	0.780597	+	0.625034i	$0.214915\pi$
$402$	0	0
$403$	7.24607	0.360952
$404$	0.375499	0.0186818
$405$	0	0
$406$	14.8105	0.735031
$407$	30.0271	1.48839
$408$	0	0
$409$	−19.0376	−0.941346	−0.470673	−	0.882308i	$-0.655989\pi$
$409$	−19.0376	−0.941346	−0.470673	+	0.882308i	$0.655989\pi$
$410$	9.99431	0.493584
$411$	0	0
$412$	−1.40541	−0.0692397
$413$	−16.8674	−0.829989
$414$	0	0
$415$	−1.59921	−0.0785021
$416$	−4.08122	−0.200098
$417$	0	0
$418$	39.0292	1.90898
$419$	34.4946	1.68517	0.842585	−	0.538563i	$-0.181033\pi$
$419$	34.4946	1.68517	0.842585	+	0.538563i	$0.181033\pi$
$420$	0	0
$421$	5.77627	0.281518	0.140759	−	0.990044i	$-0.455046\pi$
$421$	5.77627	0.281518	0.140759	+	0.990044i	$0.455046\pi$
$422$	−4.70974	−0.229266
$423$	0	0
$424$	−21.3855	−1.03857
$425$	−3.55006	−0.172203
$426$	0	0
$427$	−8.93925	−0.432601
$428$	−1.37796	−0.0666062
$429$	0	0
$430$	−9.40141	−0.453376
$431$	14.5633	0.701491	0.350746	−	0.936471i	$-0.385928\pi$
$431$	14.5633	0.701491	0.350746	+	0.936471i	$0.385928\pi$
$432$	0	0
$433$	21.6954	1.04261	0.521306	−	0.853370i	$-0.325445\pi$
$433$	21.6954	1.04261	0.521306	+	0.853370i	$0.325445\pi$
$434$	3.48249	0.167165
$435$	0	0
$436$	1.31817	0.0631288
$437$	26.1353	1.25022
$438$	0	0
$439$	−25.0947	−1.19771	−0.598853	−	0.800859i	$-0.704377\pi$
$439$	−25.0947	−1.19771	−0.598853	+	0.800859i	$0.704377\pi$
$440$	−11.5322	−0.549777
$441$	0	0
$442$	31.9565	1.52002
$443$	17.5944	0.835935	0.417967	−	0.908462i	$-0.362743\pi$
$443$	17.5944	0.835935	0.417967	+	0.908462i	$0.362743\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−5.12193	−0.242530
$447$	0	0
$448$	−19.2107	−0.907621
$449$	5.73457	0.270631	0.135316	−	0.990803i	$-0.456795\pi$
$449$	5.73457	0.270631	0.135316	+	0.990803i	$0.456795\pi$
$450$	0	0
$451$	28.9021	1.36095
$452$	−1.13727	−0.0534925
$453$	0	0
$454$	−5.16447	−0.242381
$455$	−14.9915	−0.702814
$456$	0	0
$457$	−40.2313	−1.88194	−0.940970	−	0.338490i	$-0.890084\pi$
$457$	−40.2313	−1.88194	−0.940970	+	0.338490i	$0.890084\pi$
$458$	17.4643	0.816055
$459$	0	0
$460$	−0.403644	−0.0188200
$461$	9.68017	0.450851	0.225425	−	0.974260i	$-0.427623\pi$
$461$	9.68017	0.450851	0.225425	+	0.974260i	$0.427623\pi$
$462$	0	0
$463$	−4.87346	−0.226489	−0.113244	−	0.993567i	$-0.536124\pi$
$463$	−4.87346	−0.226489	−0.113244	+	0.993567i	$0.536124\pi$
$464$	−17.7286	−0.823031
$465$	0	0
$466$	−18.2283	−0.844410
$467$	5.15143	0.238380	0.119190	−	0.992871i	$-0.461970\pi$
$467$	5.15143	0.238380	0.119190	+	0.992871i	$0.461970\pi$
$468$	0	0
$469$	−7.44859	−0.343944
$470$	14.4674	0.667333
$471$	0	0
$472$	21.3732	0.983781
$473$	−27.1875	−1.25008
$474$	0	0
$475$	7.14204	0.327699
$476$	−0.896496	−0.0410908
$477$	0	0
$478$	−6.93275	−0.317097
$479$	41.2039	1.88265	0.941327	−	0.337496i	$-0.109580\pi$
$479$	41.2039	1.88265	0.941327	+	0.337496i	$0.109580\pi$
$480$	0	0
$481$	−49.4616	−2.25525
$482$	20.5388	0.935515
$483$	0	0
$484$	−0.529812	−0.0240824
$485$	−4.76685	−0.216451
$486$	0	0
$487$	33.6081	1.52293	0.761464	−	0.648207i	$-0.224481\pi$
$487$	33.6081	1.52293	0.761464	+	0.648207i	$0.224481\pi$
$488$	11.3272	0.512759
$489$	0	0
$490$	2.41764	0.109218
$491$	10.1569	0.458373	0.229186	−	0.973383i	$-0.426393\pi$
$491$	10.1569	0.458373	0.229186	+	0.973383i	$0.426393\pi$
$492$	0	0
$493$	−16.7067	−0.752431
$494$	−64.2902	−2.89255
$495$	0	0
$496$	−4.16866	−0.187179
$497$	−26.0238	−1.16733
$498$	0	0
$499$	−14.7342	−0.659593	−0.329796	−	0.944052i	$-0.606980\pi$
$499$	−14.7342	−0.659593	−0.329796	+	0.944052i	$0.606980\pi$
$500$	−0.110304	−0.00493297
$501$	0	0
$502$	27.3591	1.22110
$503$	38.5524	1.71897	0.859483	−	0.511164i	$-0.170786\pi$
$503$	38.5524	1.71897	0.859483	+	0.511164i	$0.170786\pi$
$504$	0	0
$505$	−3.40420	−0.151485
$506$	19.9974	0.888993
$507$	0	0
$508$	2.37737	0.105479
$509$	−4.26005	−0.188823	−0.0944117	−	0.995533i	$-0.530097\pi$
$509$	−4.26005	−0.188823	−0.0944117	+	0.995533i	$0.530097\pi$
$510$	0	0
$511$	−16.8573	−0.745722
$512$	24.2050	1.06972
$513$	0	0
$514$	−14.8783	−0.656254
$515$	12.7412	0.561445
$516$	0	0
$517$	41.8377	1.84002
$518$	−23.7715	−1.04446
$519$	0	0
$520$	18.9963	0.833041
$521$	−34.9585	−1.53156	−0.765779	−	0.643104i	$-0.777646\pi$
$521$	−34.9585	−1.53156	−0.765779	+	0.643104i	$0.777646\pi$
$522$	0	0
$523$	−10.4394	−0.456483	−0.228242	−	0.973605i	$-0.573298\pi$
$523$	−10.4394	−0.456483	−0.228242	+	0.973605i	$0.573298\pi$
$524$	−1.03813	−0.0453508
$525$	0	0
$526$	−35.9620	−1.56802
$527$	−3.92836	−0.171122
$528$	0	0
$529$	−9.60906	−0.417785
$530$	10.1338	0.440185
$531$	0	0
$532$	1.80357	0.0781949
$533$	−47.6084	−2.06215
$534$	0	0
$535$	12.4923	0.540091
$536$	9.43835	0.407675
$537$	0	0
$538$	3.66692	0.158092
$539$	6.99146	0.301143
$540$	0	0
$541$	15.5770	0.669705	0.334853	−	0.942270i	$-0.391313\pi$
$541$	15.5770	0.669705	0.334853	+	0.942270i	$0.391313\pi$
$542$	−31.9692	−1.37320
$543$	0	0
$544$	2.21258	0.0948637
$545$	−11.9503	−0.511894
$546$	0	0
$547$	15.1133	0.646199	0.323100	−	0.946365i	$-0.395275\pi$
$547$	15.1133	0.646199	0.323100	+	0.946365i	$0.395275\pi$
$548$	−0.932393	−0.0398299
$549$	0	0
$550$	5.46472	0.233017
$551$	33.6106	1.43186
$552$	0	0
$553$	25.6957	1.09269
$554$	−12.7113	−0.540050
$555$	0	0
$556$	0.446292	0.0189270
$557$	−44.6980	−1.89391	−0.946957	−	0.321359i	$-0.895860\pi$
$557$	−44.6980	−1.89391	−0.946957	+	0.321359i	$0.895860\pi$
$558$	0	0
$559$	44.7841	1.89417
$560$	8.62463	0.364457
$561$	0	0
$562$	1.00853	0.0425424
$563$	10.5847	0.446092	0.223046	−	0.974808i	$-0.428400\pi$
$563$	10.5847	0.446092	0.223046	+	0.974808i	$0.428400\pi$
$564$	0	0
$565$	10.3102	0.433755
$566$	−2.86601	−0.120467
$567$	0	0
$568$	32.9756	1.38362
$569$	−7.56471	−0.317129	−0.158565	−	0.987349i	$-0.550687\pi$
$569$	−7.56471	−0.317129	−0.158565	+	0.987349i	$0.550687\pi$
$570$	0	0
$571$	−31.9429	−1.33677	−0.668384	−	0.743816i	$-0.733014\pi$
$571$	−31.9429	−1.33677	−0.668384	+	0.743816i	$0.733014\pi$
$572$	2.87139	0.120059
$573$	0	0
$574$	−22.8808	−0.955028
$575$	3.65936	0.152606
$576$	0	0
$577$	−29.9958	−1.24874	−0.624372	−	0.781127i	$-0.714645\pi$
$577$	−29.9958	−1.24874	−0.624372	+	0.781127i	$0.714645\pi$
$578$	6.04444	0.251416
$579$	0	0
$580$	−0.519095	−0.0215542
$581$	3.66121	0.151892
$582$	0	0
$583$	29.3055	1.21371
$584$	21.3604	0.883900
$585$	0	0
$586$	−5.00618	−0.206804
$587$	−8.98349	−0.370788	−0.185394	−	0.982664i	$-0.559356\pi$
$587$	−8.98349	−0.370788	−0.185394	+	0.982664i	$0.559356\pi$
$588$	0	0
$589$	7.90310	0.325642
$590$	−10.1280	−0.416964
$591$	0	0
$592$	28.4553	1.16950
$593$	25.6784	1.05449	0.527244	−	0.849714i	$-0.323226\pi$
$593$	25.6784	1.05449	0.527244	+	0.849714i	$0.323226\pi$
$594$	0	0
$595$	8.12747	0.333194
$596$	0.982746	0.0402549
$597$	0	0
$598$	−32.9404	−1.34703
$599$	19.7209	0.805776	0.402888	−	0.915249i	$-0.368006\pi$
$599$	19.7209	0.805776	0.402888	+	0.915249i	$0.368006\pi$
$600$	0	0
$601$	5.22658	0.213197	0.106598	−	0.994302i	$-0.466004\pi$
$601$	5.22658	0.213197	0.106598	+	0.994302i	$0.466004\pi$
$602$	21.5235	0.877231
$603$	0	0
$604$	0.364670	0.0148382
$605$	4.80318	0.195277
$606$	0	0
$607$	−36.8250	−1.49468	−0.747339	−	0.664443i	$-0.768669\pi$
$607$	−36.8250	−1.49468	−0.747339	+	0.664443i	$0.768669\pi$
$608$	−4.45128	−0.180523
$609$	0	0
$610$	−5.36758	−0.217327
$611$	−68.9164	−2.78806
$612$	0	0
$613$	−31.9964	−1.29232	−0.646161	−	0.763201i	$-0.723626\pi$
$613$	−31.9964	−1.29232	−0.646161	+	0.763201i	$0.723626\pi$
$614$	−36.9055	−1.48938
$615$	0	0
$616$	26.4017	1.06376
$617$	42.2042	1.69908	0.849540	−	0.527525i	$-0.176880\pi$
$617$	42.2042	1.69908	0.849540	+	0.527525i	$0.176880\pi$
$618$	0	0
$619$	12.3991	0.498361	0.249181	−	0.968457i	$-0.419839\pi$
$619$	12.3991	0.498361	0.249181	+	0.968457i	$0.419839\pi$
$620$	−0.122059	−0.00490199
$621$	0	0
$622$	−39.4741	−1.58277
$623$	−2.28939	−0.0917223
$624$	0	0
$625$	1.00000	0.0400000
$626$	−45.3958	−1.81438
$627$	0	0
$628$	−1.79238	−0.0715236
$629$	26.8150	1.06918
$630$	0	0
$631$	−11.9796	−0.476899	−0.238449	−	0.971155i	$-0.576639\pi$
$631$	−11.9796	−0.476899	−0.238449	+	0.971155i	$0.576639\pi$
$632$	−32.5599	−1.29516
$633$	0	0
$634$	16.3769	0.650411
$635$	−21.5528	−0.855297
$636$	0	0
$637$	−11.5165	−0.456302
$638$	25.7171	1.01815
$639$	0	0
$640$	−10.2886	−0.406691
$641$	−17.8411	−0.704681	−0.352341	−	0.935872i	$-0.614614\pi$
$641$	−17.8411	−0.704681	−0.352341	+	0.935872i	$0.614614\pi$
$642$	0	0
$643$	−36.2388	−1.42912	−0.714559	−	0.699575i	$-0.753373\pi$
$643$	−36.2388	−1.42912	−0.714559	+	0.699575i	$0.753373\pi$
$644$	0.924097	0.0364145
$645$	0	0
$646$	34.8541	1.37132
$647$	14.7562	0.580126	0.290063	−	0.957007i	$-0.406324\pi$
$647$	14.7562	0.580126	0.290063	+	0.957007i	$0.406324\pi$
$648$	0	0
$649$	−29.2887	−1.14968
$650$	−9.00166	−0.353074
$651$	0	0
$652$	−1.99135	−0.0779874
$653$	37.7281	1.47642	0.738208	−	0.674573i	$-0.235672\pi$
$653$	37.7281	1.47642	0.738208	+	0.674573i	$0.235672\pi$
$654$	0	0
$655$	9.41146	0.367736
$656$	27.3891	1.06937
$657$	0	0
$658$	−33.1216	−1.29121
$659$	37.4534	1.45898	0.729489	−	0.683993i	$-0.239758\pi$
$659$	37.4534	1.45898	0.729489	+	0.683993i	$0.239758\pi$
$660$	0	0
$661$	17.5093	0.681033	0.340516	−	0.940239i	$-0.389398\pi$
$661$	17.5093	0.681033	0.340516	+	0.940239i	$0.389398\pi$
$662$	−7.79070	−0.302794
$663$	0	0
$664$	−4.63924	−0.180037
$665$	−16.3509	−0.634060
$666$	0	0
$667$	17.2210	0.666801
$668$	−2.46288	−0.0952917
$669$	0	0
$670$	−4.47251	−0.172788
$671$	−15.5222	−0.599230
$672$	0	0
$673$	45.1306	1.73965	0.869827	−	0.493356i	$-0.164230\pi$
$673$	45.1306	1.73965	0.869827	+	0.493356i	$0.164230\pi$
$674$	25.9130	0.998131
$675$	0	0
$676$	−3.29589	−0.126765
$677$	−39.3012	−1.51047	−0.755233	−	0.655456i	$-0.772476\pi$
$677$	−39.3012	−1.51047	−0.755233	+	0.655456i	$0.772476\pi$
$678$	0	0
$679$	10.9132	0.418808
$680$	−10.2986	−0.394932
$681$	0	0
$682$	6.04705	0.231553
$683$	−7.13615	−0.273057	−0.136529	−	0.990636i	$-0.543595\pi$
$683$	−7.13615	−0.273057	−0.136529	+	0.990636i	$0.543595\pi$
$684$	0	0
$685$	8.45290	0.322969
$686$	−27.5648	−1.05243
$687$	0	0
$688$	−25.7643	−0.982256
$689$	−48.2730	−1.83906
$690$	0	0
$691$	29.3985	1.11837	0.559186	−	0.829043i	$-0.311114\pi$
$691$	29.3985	1.11837	0.559186	+	0.829043i	$0.311114\pi$
$692$	0.770376	0.0292853
$693$	0	0
$694$	26.2274	0.995579
$695$	−4.04600	−0.153473
$696$	0	0
$697$	25.8103	0.977635
$698$	−14.5532	−0.550847
$699$	0	0
$700$	0.252529	0.00954472
$701$	4.39112	0.165850	0.0829252	−	0.996556i	$-0.473574\pi$
$701$	4.39112	0.165850	0.0829252	+	0.996556i	$0.473574\pi$
$702$	0	0
$703$	−53.9465	−2.03463
$704$	−33.3577	−1.25722
$705$	0	0
$706$	32.4826	1.22250
$707$	7.79353	0.293106
$708$	0	0
$709$	35.5086	1.33355	0.666777	−	0.745257i	$-0.267673\pi$
$709$	35.5086	1.33355	0.666777	+	0.745257i	$0.267673\pi$
$710$	−15.6260	−0.586433
$711$	0	0
$712$	2.90096	0.108718
$713$	4.04931	0.151648
$714$	0	0
$715$	−26.0315	−0.973523
$716$	2.35022	0.0878320
$717$	0	0
$718$	24.4521	0.912543
$719$	21.0712	0.785823	0.392911	−	0.919576i	$-0.371468\pi$
$719$	21.0712	0.785823	0.392911	+	0.919576i	$0.371468\pi$
$720$	0	0
$721$	−29.1696	−1.08633
$722$	−44.0011	−1.63755
$723$	0	0
$724$	−0.932723	−0.0346644
$725$	4.70602	0.174777
$726$	0	0
$727$	14.4153	0.534633	0.267317	−	0.963609i	$-0.413863\pi$
$727$	14.4153	0.534633	0.267317	+	0.963609i	$0.413863\pi$
$728$	−43.4897	−1.61184
$729$	0	0
$730$	−10.1220	−0.374631
$731$	−24.2792	−0.897997
$732$	0	0
$733$	−26.4716	−0.977751	−0.488875	−	0.872354i	$-0.662593\pi$
$733$	−26.4716	−0.977751	−0.488875	+	0.872354i	$0.662593\pi$
$734$	−31.0169	−1.14485
$735$	0	0
$736$	−2.28070	−0.0840678
$737$	−12.9338	−0.476424
$738$	0	0
$739$	−21.4584	−0.789361	−0.394681	−	0.918818i	$-0.629145\pi$
$739$	−21.4584	−0.789361	−0.394681	+	0.918818i	$0.629145\pi$
$740$	0.833171	0.0306280
$741$	0	0
$742$	−23.2002	−0.851707
$743$	−38.1613	−1.40000	−0.700002	−	0.714141i	$-0.746818\pi$
$743$	−38.1613	−1.40000	−0.700002	+	0.714141i	$0.746818\pi$
$744$	0	0
$745$	−8.90940	−0.326415
$746$	−36.1282	−1.32275
$747$	0	0
$748$	−1.55669	−0.0569182
$749$	−28.5998	−1.04501
$750$	0	0
$751$	25.6588	0.936302	0.468151	−	0.883648i	$-0.344920\pi$
$751$	25.6588	0.936302	0.468151	+	0.883648i	$0.344920\pi$
$752$	39.6476	1.44580
$753$	0	0
$754$	−42.3620	−1.54273
$755$	−3.30603	−0.120319
$756$	0	0
$757$	36.8881	1.34072	0.670361	−	0.742035i	$-0.266139\pi$
$757$	36.8881	1.34072	0.670361	+	0.742035i	$0.266139\pi$
$758$	5.44749	0.197862
$759$	0	0
$760$	20.7187	0.751547
$761$	−22.2746	−0.807453	−0.403727	−	0.914880i	$-0.632285\pi$
$761$	−22.2746	−0.807453	−0.403727	+	0.914880i	$0.632285\pi$
$762$	0	0
$763$	27.3588	0.990455
$764$	−2.20386	−0.0797327
$765$	0	0
$766$	−10.7234	−0.387453
$767$	48.2453	1.74204
$768$	0	0
$769$	−40.7859	−1.47078	−0.735389	−	0.677645i	$-0.763001\pi$
$769$	−40.7859	−1.47078	−0.735389	+	0.677645i	$0.763001\pi$
$770$	−12.5109	−0.450860
$771$	0	0
$772$	−0.556220	−0.0200188
$773$	7.31064	0.262946	0.131473	−	0.991320i	$-0.458029\pi$
$773$	7.31064	0.262946	0.131473	+	0.991320i	$0.458029\pi$
$774$	0	0
$775$	1.10656	0.0397488
$776$	−13.8284	−0.496411
$777$	0	0
$778$	−7.86091	−0.281827
$779$	−51.9253	−1.86042
$780$	0	0
$781$	−45.1880	−1.61696
$782$	17.8582	0.638608
$783$	0	0
$784$	6.62547	0.236624
$785$	16.2493	0.579964
$786$	0	0
$787$	19.6892	0.701843	0.350922	−	0.936405i	$-0.385868\pi$
$787$	19.6892	0.701843	0.350922	+	0.936405i	$0.385868\pi$
$788$	−1.07944	−0.0384535
$789$	0	0
$790$	15.4290	0.548939
$791$	−23.6041	−0.839265
$792$	0	0
$793$	25.5687	0.907972
$794$	−5.43987	−0.193054
$795$	0	0
$796$	0.659947	0.0233912
$797$	−43.7692	−1.55039	−0.775193	−	0.631725i	$-0.782347\pi$
$797$	−43.7692	−1.55039	−0.775193	+	0.631725i	$0.782347\pi$
$798$	0	0
$799$	37.3622	1.32178
$800$	−0.623251	−0.0220353
$801$	0	0
$802$	−42.9759	−1.51753
$803$	−29.2712	−1.03296
$804$	0	0
$805$	−8.37769	−0.295275
$806$	−9.96089	−0.350857
$807$	0	0
$808$	−9.87544	−0.347417
$809$	19.4362	0.683339	0.341669	−	0.939820i	$-0.389008\pi$
$809$	19.4362	0.683339	0.341669	+	0.939820i	$0.389008\pi$
$810$	0	0
$811$	−49.1245	−1.72499	−0.862497	−	0.506062i	$-0.831101\pi$
$811$	−49.1245	−1.72499	−0.862497	+	0.506062i	$0.831101\pi$
$812$	1.18841	0.0417050
$813$	0	0
$814$	−41.2771	−1.44676
$815$	18.0532	0.632377
$816$	0	0
$817$	48.8449	1.70887
$818$	26.1702	0.915020
$819$	0	0
$820$	0.801955	0.0280055
$821$	−8.18538	−0.285672	−0.142836	−	0.989746i	$-0.545622\pi$
$821$	−8.18538	−0.285672	−0.142836	+	0.989746i	$0.545622\pi$
$822$	0	0
$823$	−15.1316	−0.527453	−0.263727	−	0.964597i	$-0.584952\pi$
$823$	−15.1316	−0.527453	−0.263727	+	0.964597i	$0.584952\pi$
$824$	36.9617	1.28762
$825$	0	0
$826$	23.1869	0.806777
$827$	27.4856	0.955770	0.477885	−	0.878423i	$-0.341404\pi$
$827$	27.4856	0.955770	0.477885	+	0.878423i	$0.341404\pi$
$828$	0	0
$829$	−37.4398	−1.30034	−0.650169	−	0.759790i	$-0.725302\pi$
$829$	−37.4398	−1.30034	−0.650169	+	0.759790i	$0.725302\pi$
$830$	2.19837	0.0763066
$831$	0	0
$832$	54.9479	1.90498
$833$	6.24355	0.216326
$834$	0	0
$835$	22.3280	0.772693
$836$	3.13175	0.108314
$837$	0	0
$838$	−47.4184	−1.63804
$839$	−43.7261	−1.50959	−0.754797	−	0.655959i	$-0.772265\pi$
$839$	−43.7261	−1.50959	−0.754797	+	0.655959i	$0.772265\pi$
$840$	0	0
$841$	−6.85338	−0.236323
$842$	−7.94041	−0.273645
$843$	0	0
$844$	−0.377915	−0.0130084
$845$	29.8799	1.02790
$846$	0	0
$847$	−10.9963	−0.377838
$848$	27.7715	0.953676
$849$	0	0
$850$	4.88014	0.167387
$851$	−27.6405	−0.947506
$852$	0	0
$853$	2.44945	0.0838676	0.0419338	−	0.999120i	$-0.486648\pi$
$853$	2.44945	0.0838676	0.0419338	+	0.999120i	$0.486648\pi$
$854$	12.2884	0.420502
$855$	0	0
$856$	36.2397	1.23865
$857$	22.2184	0.758967	0.379483	−	0.925199i	$-0.376102\pi$
$857$	22.2184	0.758967	0.379483	+	0.925199i	$0.376102\pi$
$858$	0	0
$859$	−41.5511	−1.41771	−0.708853	−	0.705356i	$-0.750787\pi$
$859$	−41.5511	−1.41771	−0.708853	+	0.705356i	$0.750787\pi$
$860$	−0.754380	−0.0257242
$861$	0	0
$862$	−20.0197	−0.681873
$863$	−1.91313	−0.0651236	−0.0325618	−	0.999470i	$-0.510367\pi$
$863$	−1.91313	−0.0651236	−0.0325618	+	0.999470i	$0.510367\pi$
$864$	0	0
$865$	−6.98409	−0.237466
$866$	−29.8238	−1.01345
$867$	0	0
$868$	0.279439	0.00948479
$869$	44.6184	1.51357
$870$	0	0
$871$	21.3050	0.721893
$872$	−34.6672	−1.17398
$873$	0	0
$874$	−35.9272	−1.21526
$875$	−2.28939	−0.0773953
$876$	0	0
$877$	1.54459	0.0521571	0.0260785	−	0.999660i	$-0.491698\pi$
$877$	1.54459	0.0521571	0.0260785	+	0.999660i	$0.491698\pi$
$878$	34.4968	1.16421
$879$	0	0
$880$	14.9759	0.504838
$881$	−2.22951	−0.0751140	−0.0375570	−	0.999294i	$-0.511958\pi$
$881$	−2.22951	−0.0751140	−0.0375570	+	0.999294i	$0.511958\pi$
$882$	0	0
$883$	−33.5580	−1.12932	−0.564658	−	0.825325i	$-0.690992\pi$
$883$	−33.5580	−1.12932	−0.564658	+	0.825325i	$0.690992\pi$
$884$	2.56423	0.0862443
$885$	0	0
$886$	−24.1863	−0.812556
$887$	−19.5292	−0.655726	−0.327863	−	0.944725i	$-0.606328\pi$
$887$	−19.5292	−0.655726	−0.327863	+	0.944725i	$0.606328\pi$
$888$	0	0
$889$	49.3427	1.65490
$890$	−1.37466	−0.0460788
$891$	0	0
$892$	−0.410989	−0.0137609
$893$	−75.1654	−2.51531
$894$	0	0
$895$	−21.3067	−0.712204
$896$	23.5545	0.786901
$897$	0	0
$898$	−7.88310	−0.263062
$899$	5.20750	0.173680
$900$	0	0
$901$	26.1706	0.871869
$902$	−39.7306	−1.32288
$903$	0	0
$904$	29.9095	0.994776
$905$	8.45589	0.281083
$906$	0	0
$907$	12.4602	0.413735	0.206868	−	0.978369i	$-0.433673\pi$
$907$	12.4602	0.413735	0.206868	+	0.978369i	$0.433673\pi$
$908$	−0.414403	−0.0137525
$909$	0	0
$910$	20.6083	0.683158
$911$	42.6518	1.41312	0.706559	−	0.707654i	$-0.250246\pi$
$911$	42.6518	1.41312	0.706559	+	0.707654i	$0.250246\pi$
$912$	0	0
$913$	6.35737	0.210398
$914$	55.3044	1.82931
$915$	0	0
$916$	1.40136	0.0463022
$917$	−21.5465	−0.711527
$918$	0	0
$919$	43.6574	1.44013	0.720063	−	0.693909i	$-0.244113\pi$
$919$	43.6574	1.44013	0.720063	+	0.693909i	$0.244113\pi$
$920$	10.6157	0.349988
$921$	0	0
$922$	−13.3070	−0.438241
$923$	74.4352	2.45006
$924$	0	0
$925$	−7.55337	−0.248353
$926$	6.69936	0.220155
$927$	0	0
$928$	−2.93303	−0.0962815
$929$	2.54088	0.0833636	0.0416818	−	0.999131i	$-0.486728\pi$
$929$	2.54088	0.0833636	0.0416818	+	0.999131i	$0.486728\pi$
$930$	0	0
$931$	−12.5608	−0.411664
$932$	−1.46266	−0.0479111
$933$	0	0
$934$	−7.08148	−0.231713
$935$	14.1126	0.461533
$936$	0	0
$937$	24.4505	0.798762	0.399381	−	0.916785i	$-0.369225\pi$
$937$	24.4505	0.798762	0.399381	+	0.916785i	$0.369225\pi$
$938$	10.2393	0.334325
$939$	0	0
$940$	1.16088	0.0378639
$941$	−10.4644	−0.341131	−0.170566	−	0.985346i	$-0.554559\pi$
$941$	−10.4644	−0.341131	−0.170566	+	0.985346i	$0.554559\pi$
$942$	0	0
$943$	−26.6050	−0.866377
$944$	−27.7556	−0.903366
$945$	0	0
$946$	37.3736	1.21512
$947$	−5.04812	−0.164042	−0.0820209	−	0.996631i	$-0.526137\pi$
$947$	−5.04812	−0.164042	−0.0820209	+	0.996631i	$0.526137\pi$
$948$	0	0
$949$	48.2165	1.56517
$950$	−9.81789	−0.318534
$951$	0	0
$952$	23.5774	0.764148
$953$	13.0322	0.422154	0.211077	−	0.977469i	$-0.432303\pi$
$953$	13.0322	0.422154	0.211077	+	0.977469i	$0.432303\pi$
$954$	0	0
$955$	19.9797	0.646529
$956$	−0.556292	−0.0179918
$957$	0	0
$958$	−56.6414	−1.83000
$959$	−19.3520	−0.624907
$960$	0	0
$961$	−29.7755	−0.960501
$962$	67.9929	2.19218
$963$	0	0
$964$	1.64805	0.0530802
$965$	5.04259	0.162327
$966$	0	0
$967$	38.0202	1.22265	0.611323	−	0.791381i	$-0.290637\pi$
$967$	38.0202	1.22265	0.611323	+	0.791381i	$0.290637\pi$
$968$	13.9338	0.447849
$969$	0	0
$970$	6.55281	0.210398
$971$	25.4068	0.815342	0.407671	−	0.913129i	$-0.366341\pi$
$971$	25.4068	0.815342	0.407671	+	0.913129i	$0.366341\pi$
$972$	0	0
$973$	9.26285	0.296953
$974$	−46.1998	−1.48034
$975$	0	0
$976$	−14.7097	−0.470846
$977$	−30.8516	−0.987030	−0.493515	−	0.869737i	$-0.664288\pi$
$977$	−30.8516	−0.987030	−0.493515	+	0.869737i	$0.664288\pi$
$978$	0	0
$979$	−3.97532	−0.127052
$980$	0.193994	0.00619692
$981$	0	0
$982$	−13.9623	−0.445553
$983$	−44.8009	−1.42893	−0.714464	−	0.699672i	$-0.753329\pi$
$983$	−44.8009	−1.42893	−0.714464	+	0.699672i	$0.753329\pi$
$984$	0	0
$985$	9.78601	0.311808
$986$	22.9660	0.731387
$987$	0	0
$988$	−5.15872	−0.164121
$989$	25.0267	0.795801
$990$	0	0
$991$	−49.7959	−1.58182	−0.790909	−	0.611933i	$-0.790392\pi$
$991$	−49.7959	−1.58182	−0.790909	+	0.611933i	$0.790392\pi$
$992$	−0.689665	−0.0218969
$993$	0	0
$994$	35.7739	1.13468
$995$	−5.98296	−0.189673
$996$	0	0
$997$	29.5896	0.937111	0.468555	−	0.883434i	$-0.344775\pi$
$997$	29.5896	0.937111	0.468555	+	0.883434i	$0.344775\pi$
$998$	20.2545	0.641146
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.t.1.4		10
3.2	odd	2		1335.2.a.i.1.7	✓	10
15.14	odd	2		6675.2.a.ba.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.i.1.7	✓	10	3.2	odd	2
4005.2.a.t.1.4		10	1.1	even	1	trivial
6675.2.a.ba.1.4		10	15.14	odd	2

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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-1.37466 q^{2} -0.110304 q^{4} +1.00000 q^{5} -2.28939 q^{7} +2.90096 q^{8} +O(q^{10})\)	\(q-1.37466 q^{2} -0.110304 q^{4} +1.00000 q^{5} -2.28939 q^{7} +2.90096 q^{8} -1.37466 q^{10} -3.97532 q^{11} +6.54828 q^{13} +3.14713 q^{14} -3.76722 q^{16} -3.55006 q^{17} +7.14204 q^{19} -0.110304 q^{20} +5.46472 q^{22} +3.65936 q^{23} +1.00000 q^{25} -9.00166 q^{26} +0.252529 q^{28} +4.70602 q^{29} +1.10656 q^{31} -0.623251 q^{32} +4.88014 q^{34} -2.28939 q^{35} -7.55337 q^{37} -9.81789 q^{38} +2.90096 q^{40} -7.27038 q^{41} +6.83907 q^{43} +0.438496 q^{44} -5.03039 q^{46} -10.5244 q^{47} -1.75871 q^{49} -1.37466 q^{50} -0.722304 q^{52} -7.37187 q^{53} -3.97532 q^{55} -6.64140 q^{56} -6.46919 q^{58} +7.36764 q^{59} +3.90465 q^{61} -1.52115 q^{62} +8.39121 q^{64} +6.54828 q^{65} +3.25353 q^{67} +0.391588 q^{68} +3.14713 q^{70} +11.3671 q^{71} +7.36324 q^{73} +10.3833 q^{74} -0.787799 q^{76} +9.10104 q^{77} -11.2238 q^{79} -3.76722 q^{80} +9.99431 q^{82} -1.59921 q^{83} -3.55006 q^{85} -9.40141 q^{86} -11.5322 q^{88} +1.00000 q^{89} -14.9915 q^{91} -0.403644 q^{92} +14.4674 q^{94} +7.14204 q^{95} -4.76685 q^{97} +2.41764 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8	\( 10 q + 18 q^{4} + 10 q^{5} + 9 q^{7} - 3 q^{8} - 4 q^{11} + 15 q^{13} + q^{14} + 22 q^{16} - 11 q^{17} + 14 q^{19} + 18 q^{20} + 10 q^{22} + 8 q^{23} + 10 q^{25} + 14 q^{26} + 36 q^{28} - 5 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 9 q^{35} + 23 q^{37} - 8 q^{38} - 3 q^{40} - 13 q^{41} + 25 q^{43} + 8 q^{44} - 14 q^{46} + q^{47} + 21 q^{49} + 51 q^{52} - 9 q^{53} - 4 q^{55} - 15 q^{56} - 8 q^{58} + 15 q^{59} + 8 q^{61} + 8 q^{62} + 9 q^{64} + 15 q^{65} + 52 q^{67} + 28 q^{68} + q^{70} + 22 q^{71} + 34 q^{73} + 18 q^{74} + 14 q^{76} - 4 q^{77} - 3 q^{79} + 22 q^{80} + 17 q^{82} + 10 q^{83} - 11 q^{85} + 6 q^{86} + 4 q^{88} + 10 q^{89} + 18 q^{91} + 14 q^{92} - 43 q^{94} + 14 q^{95} + 34 q^{97} + 38 q^{98}+O(q^{100}) \) 10 * q + 18 * q^4 + 10 * q^5 + 9 * q^7 - 3 * q^8 - 4 * q^11 + 15 * q^13 + q^14 + 22 * q^16 - 11 * q^17 + 14 * q^19 + 18 * q^20 + 10 * q^22 + 8 * q^23 + 10 * q^25 + 14 * q^26 + 36 * q^28 - 5 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 9 * q^35 + 23 * q^37 - 8 * q^38 - 3 * q^40 - 13 * q^41 + 25 * q^43 + 8 * q^44 - 14 * q^46 + q^47 + 21 * q^49 + 51 * q^52 - 9 * q^53 - 4 * q^55 - 15 * q^56 - 8 * q^58 + 15 * q^59 + 8 * q^61 + 8 * q^62 + 9 * q^64 + 15 * q^65 + 52 * q^67 + 28 * q^68 + q^70 + 22 * q^71 + 34 * q^73 + 18 * q^74 + 14 * q^76 - 4 * q^77 - 3 * q^79 + 22 * q^80 + 17 * q^82 + 10 * q^83 - 11 * q^85 + 6 * q^86 + 4 * q^88 + 10 * q^89 + 18 * q^91 + 14 * q^92 - 43 * q^94 + 14 * q^95 + 34 * q^97 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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