LMFDB - Embedded newform 45.4.f.a.8.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(8,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("45.8");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	45.f (of order $4$, degree $2$, 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:	$2.65508595026$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} - 14x^{8} - 512x^{6} + 3889x^{4} + 126224x^{2} + 506944 $ x^12 - 16x^10 - 14x^8 - 512x^6 + 3889x^4 + 126224*x^2 + 506944
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			8.6
Root			$3.78139 + 0.0336790i$ of defining polynomial
Character	$\chi$	$=$	45.8
Dual form			45.4.f.a.17.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.74771 + 3.74771i) q^{2} +20.0907i q^{4} +(0.572391 - 11.1657i) q^{5} +(1.80948 - 1.80948i) q^{7} +(-45.3126 + 45.3126i) q^{8} +O(q^{10})$	\(q+(3.74771 + 3.74771i) q^{2} +20.0907i q^{4} +(0.572391 - 11.1657i) q^{5} +(1.80948 - 1.80948i) q^{7} +(-45.3126 + 45.3126i) q^{8} +(43.9909 - 39.7006i) q^{10} -46.0907i q^{11} +(18.6099 + 18.6099i) q^{13} +13.5628 q^{14} -178.911 q^{16} +(-14.5793 - 14.5793i) q^{17} +74.4577i q^{19} +(224.327 + 11.4997i) q^{20} +(172.735 - 172.735i) q^{22} +(59.6003 - 59.6003i) q^{23} +(-124.345 - 12.7823i) q^{25} +139.489i q^{26} +(36.3538 + 36.3538i) q^{28} -202.168 q^{29} -49.5423 q^{31} +(-308.008 - 308.008i) q^{32} -109.278i q^{34} +(-19.1684 - 21.2398i) q^{35} +(-45.0594 + 45.0594i) q^{37} +(-279.046 + 279.046i) q^{38} +(480.009 + 531.882i) q^{40} +306.253i q^{41} +(-230.784 - 230.784i) q^{43} +925.996 q^{44} +446.730 q^{46} +(176.943 + 176.943i) q^{47} +336.452i q^{49} +(-418.104 - 513.913i) q^{50} +(-373.886 + 373.886i) q^{52} +(85.1290 - 85.1290i) q^{53} +(-514.634 - 26.3819i) q^{55} +163.985i q^{56} +(-757.669 - 757.669i) q^{58} -330.873 q^{59} +678.639 q^{61} +(-185.670 - 185.670i) q^{62} -877.361i q^{64} +(218.444 - 197.140i) q^{65} +(756.373 - 756.373i) q^{67} +(292.909 - 292.909i) q^{68} +(7.76325 - 151.438i) q^{70} +100.264i q^{71} +(586.919 + 586.919i) q^{73} -337.739 q^{74} -1495.91 q^{76} +(-83.4004 - 83.4004i) q^{77} -286.986i q^{79} +(-102.407 + 1997.67i) q^{80} +(-1147.75 + 1147.75i) q^{82} +(947.796 - 947.796i) q^{83} +(-171.133 + 154.443i) q^{85} -1729.83i q^{86} +(2088.49 + 2088.49i) q^{88} -688.442 q^{89} +67.3485 q^{91} +(1197.41 + 1197.41i) q^{92} +1326.26i q^{94} +(831.371 + 42.6189i) q^{95} +(920.676 - 920.676i) q^{97} +(-1260.92 + 1260.92i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{7}+O(q^{10}) $ 12 * q + 24 * q^7	$ 12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100}) $ 12 * q + 24 * q^7 + 192 * q^10 - 108 * q^13 - 648 * q^16 + 1056 * q^22 - 144 * q^25 - 576 * q^28 - 1248 * q^31 + 828 * q^37 + 2568 * q^40 - 96 * q^43 + 672 * q^46 - 312 * q^52 - 1512 * q^55 - 3864 * q^58 + 96 * q^61 + 1632 * q^67 - 1536 * q^70 + 3972 * q^73 - 480 * q^76 - 7848 * q^82 - 1752 * q^85 + 7968 * q^88 + 4752 * q^91 + 2772 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	3.74771	+	3.74771i	1.32502	+	1.32502i	0.909657	+	0.415360i	$0.136344\pi$
$2$	3.74771	+	3.74771i	1.32502	+	1.32502i	0.415360	+	0.909657i	$0.363656\pi$
$3$		0			0
$3$		0			0
$4$			20.0907i			2.51134i
$5$	0.572391	−	11.1657i	0.0511962	−	0.998689i
$5$	0.572391	−	11.1657i	0.0511962	−	0.998689i
$6$		0			0
$7$	1.80948	−	1.80948i	0.0977029	−	0.0977029i	−0.656566	−	0.754269i	$-0.727992\pi$
$7$	1.80948	−	1.80948i	0.0977029	−	0.0977029i	0.754269	+	0.656566i	$0.227992\pi$
$8$	−45.3126	+	45.3126i	−2.00255	+	2.00255i
$9$		0			0
$10$	43.9909	−	39.7006i	1.39112	−	1.25544i
$11$		−	46.0907i		−	1.26335i	−0.775232	−	0.631676i	$-0.782367\pi$
$11$		−	46.0907i		−	1.26335i	0.775232	−	0.631676i	$-0.217633\pi$
$12$		0			0
$13$	18.6099	+	18.6099i	0.397035	+	0.397035i	0.877186	−	0.480151i	$-0.159418\pi$
$13$	18.6099	+	18.6099i	0.397035	+	0.397035i	−0.480151	+	0.877186i	$0.659418\pi$
$14$	13.5628			0.258916
$15$		0			0
$16$	−178.911			−2.79549
$17$	−14.5793	−	14.5793i	−0.208000	−	0.208000i	0.595417	−	0.803417i	$-0.296987\pi$
$17$	−14.5793	−	14.5793i	−0.208000	−	0.208000i	−0.803417	+	0.595417i	$0.796987\pi$
$18$		0			0
$19$			74.4577i			0.899041i	0.893270	+	0.449520i	$0.148405\pi$
$19$			74.4577i			0.899041i	−0.893270	+	0.449520i	$0.851595\pi$
$20$	224.327	+	11.4997i	2.50805	+	0.128571i
$21$		0			0
$22$	172.735	−	172.735i	1.67396	−	1.67396i
$23$	59.6003	−	59.6003i	0.540327	−	0.540327i	−0.383298	−	0.923625i	$-0.625212\pi$
$23$	59.6003	−	59.6003i	0.540327	−	0.540327i	0.923625	+	0.383298i	$0.125212\pi$
$24$		0			0
$25$	−124.345	−	12.7823i	−0.994758	−	0.102258i
$26$			139.489i			1.05216i
$27$		0			0
$28$	36.3538	+	36.3538i	0.245365	+	0.245365i
$29$	−202.168			−1.29454			−0.647271	−	0.762260i	$-0.724090\pi$
$29$	−202.168			−1.29454			−0.647271	+	0.762260i	$0.724090\pi$
$30$		0			0
$31$	−49.5423			−0.287034			−0.143517	−	0.989648i	$-0.545841\pi$
$31$	−49.5423			−0.287034			−0.143517	+	0.989648i	$0.545841\pi$
$32$	−308.008	−	308.008i	−1.70152	−	1.70152i
$33$		0			0
$34$		−	109.278i		−	0.551208i
$35$	−19.1684	−	21.2398i	−0.0925727	−	0.102577i
$36$		0			0
$37$	−45.0594	+	45.0594i	−0.200209	+	0.200209i	−0.800089	−	0.599881i	$-0.795215\pi$
$37$	−45.0594	+	45.0594i	−0.200209	+	0.200209i	0.599881	+	0.800089i	$0.295215\pi$
$38$	−279.046	+	279.046i	−1.19124	+	1.19124i
$39$		0			0
$40$	480.009	+	531.882i	1.89740	+	2.10245i
$41$			306.253i			1.16655i	0.812274	+	0.583276i	$0.198229\pi$
$41$			306.253i			1.16655i	−0.812274	+	0.583276i	$0.801771\pi$
$42$		0			0
$43$	−230.784	−	230.784i	−0.818472	−	0.818472i	0.167415	−	0.985887i	$-0.446458\pi$
$43$	−230.784	−	230.784i	−0.818472	−	0.818472i	−0.985887	+	0.167415i	$0.946458\pi$
$44$	925.996			3.17271
$45$		0			0
$46$	446.730			1.43189
$47$	176.943	+	176.943i	0.549143	+	0.549143i	0.926193	−	0.377050i	$-0.123062\pi$
$47$	176.943	+	176.943i	0.549143	+	0.549143i	−0.377050	+	0.926193i	$0.623062\pi$
$48$		0			0
$49$			336.452i			0.980908i
$50$	−418.104	−	513.913i	−1.18258	−	1.45356i
$51$		0			0
$52$	−373.886	+	373.886i	−0.997090	+	0.997090i
$53$	85.1290	−	85.1290i	0.220630	−	0.220630i	−0.588134	−	0.808764i	$-0.700137\pi$
$53$	85.1290	−	85.1290i	0.220630	−	0.220630i	0.808764	+	0.588134i	$0.200137\pi$
$54$		0			0
$55$	−514.634	−	26.3819i	−1.26170	−	0.0646788i
$56$			163.985i			0.391310i
$57$		0			0
$58$	−757.669	−	757.669i	−1.71529	−	1.71529i
$59$	−330.873			−0.730103			−0.365051	−	0.930987i	$-0.618949\pi$
$59$	−330.873			−0.730103			−0.365051	+	0.930987i	$0.618949\pi$
$60$		0			0
$61$	678.639			1.42444			0.712220	−	0.701956i	$-0.247690\pi$
$61$	678.639			1.42444			0.712220	+	0.701956i	$0.247690\pi$
$62$	−185.670	−	185.670i	−0.380325	−	0.380325i
$63$		0			0
$64$		−	877.361i		−	1.71360i
$65$	218.444	−	197.140i	0.416841	−	0.376188i
$66$		0			0
$67$	756.373	−	756.373i	1.37919	−	1.37919i	0.533199	−	0.845990i	$-0.320990\pi$
$67$	756.373	−	756.373i	1.37919	−	1.37919i	0.845990	−	0.533199i	$-0.179010\pi$
$68$	292.909	−	292.909i	0.522360	−	0.522360i
$69$		0			0
$70$	7.76325	−	151.438i	0.0132555	−	0.258576i
$71$			100.264i			0.167594i	0.996483	+	0.0837972i	$0.0267048\pi$
$71$			100.264i			0.167594i	−0.996483	+	0.0837972i	$0.973295\pi$
$72$		0			0
$73$	586.919	+	586.919i	0.941010	+	0.941010i	0.998354	−	0.0573449i	$-0.0182635\pi$
$73$	586.919	+	586.919i	0.941010	+	0.941010i	−0.0573449	+	0.998354i	$0.518263\pi$
$74$	−337.739			−0.530560
$75$		0			0
$76$	−1495.91			−2.25780
$77$	−83.4004	−	83.4004i	−0.123433	−	0.123433i
$78$		0			0
$79$		−	286.986i		−	0.408714i	−0.978896	−	0.204357i	$-0.934490\pi$
$79$		−	286.986i		−	0.408714i	0.978896	−	0.204357i	$-0.0655103\pi$
$80$	−102.407	+	1997.67i	−0.143118	+	2.79182i
$81$		0			0
$82$	−1147.75	+	1147.75i	−1.54570	+	1.54570i
$83$	947.796	−	947.796i	1.25342	−	1.25342i	0.299248	−	0.954175i	$-0.403264\pi$
$83$	947.796	−	947.796i	1.25342	−	1.25342i	0.954175	−	0.299248i	$-0.0967359\pi$
$84$		0			0
$85$	−171.133	+	154.443i	−0.218376	+	0.197079i
$86$		−	1729.83i		−	2.16898i
$87$		0			0
$88$	2088.49	+	2088.49i	2.52993	+	2.52993i
$89$	−688.442			−0.819941			−0.409970	−	0.912099i	$-0.634461\pi$
$89$	−688.442			−0.819941			−0.409970	+	0.912099i	$0.634461\pi$
$90$		0			0
$91$	67.3485			0.0775829
$92$	1197.41	+	1197.41i	1.35695	+	1.35695i
$93$		0			0
$94$			1326.26i			1.45525i
$95$	831.371	+	42.6189i	0.897862	+	0.0460274i
$96$		0			0
$97$	920.676	−	920.676i	0.963716	−	0.963716i	−0.0356483	−	0.999364i	$-0.511350\pi$
$97$	920.676	−	920.676i	0.963716	−	0.963716i	0.999364	+	0.0356483i	$0.0113496\pi$
$98$	−1260.92	+	1260.92i	−1.29972	+	1.29972i
$99$		0			0
$100$	256.805	−	2498.18i	0.256805	−	2.49818i
$101$			1018.42i			1.00333i	0.865061	+	0.501667i	$0.167280\pi$
$101$			1018.42i			1.00333i	−0.865061	+	0.501667i	$0.832720\pi$
$102$		0			0
$103$	−381.856	−	381.856i	−0.365295	−	0.365295i	0.500463	−	0.865758i	$-0.333163\pi$
$103$	−381.856	−	381.856i	−0.365295	−	0.365295i	−0.865758	+	0.500463i	$0.833163\pi$
$104$	−1686.52			−1.59017
$105$		0			0
$106$	638.078			0.584676
$107$	−11.6954	−	11.6954i	−0.0105667	−	0.0105667i	0.701804	−	0.712370i	$-0.252378\pi$
$107$	−11.6954	−	11.6954i	−0.0105667	−	0.0105667i	−0.712370	+	0.701804i	$0.752378\pi$
$108$		0			0
$109$			1346.99i			1.18366i	0.806064	+	0.591828i	$0.201594\pi$
$109$			1346.99i			1.18366i	−0.806064	+	0.591828i	$0.798406\pi$
$110$	−1829.83	−	2027.57i	−1.58607	−	1.75747i
$111$		0			0
$112$	−323.737	+	323.737i	−0.273127	+	0.273127i
$113$	−677.607	+	677.607i	−0.564105	+	0.564105i	−0.930471	−	0.366366i	$-0.880602\pi$
$113$	−677.607	+	677.607i	−0.564105	+	0.564105i	0.366366	+	0.930471i	$0.380602\pi$
$114$		0			0
$115$	−631.363	−	699.593i	−0.511956	−	0.567281i
$116$		−	4061.71i		−	3.25104i
$117$		0			0
$118$	−1240.02	−	1240.02i	−0.967398	−	0.967398i
$119$	−52.7621			−0.0406445
$120$		0			0
$121$	−793.355			−0.596059
$122$	2543.35	+	2543.35i	1.88741	+	1.88741i
$123$		0			0
$124$		−	995.340i		−	0.720840i
$125$	−213.896	+	1381.08i	−0.153052	+	0.988218i
$126$		0			0
$127$	−1220.98	+	1220.98i	−0.853107	+	0.853107i	−0.990515	−	0.137408i	$-0.956123\pi$
$127$	−1220.98	+	1220.98i	−0.853107	+	0.853107i	0.137408	+	0.990515i	$0.456123\pi$
$128$	824.034	−	824.034i	0.569023	−	0.569023i
$129$		0			0
$130$	1557.49	+	79.8423i	1.05078	+	0.0538664i
$131$			215.023i			0.143409i	0.997426	+	0.0717046i	$0.0228439\pi$
$131$			215.023i			0.143409i	−0.997426	+	0.0717046i	$0.977156\pi$
$132$		0			0
$133$	134.730	+	134.730i	0.0878389	+	0.0878389i
$134$	5669.34			3.65490
$135$		0			0
$136$	1321.25			0.833063
$137$	−877.965	−	877.965i	−0.547516	−	0.547516i	0.378206	−	0.925721i	$-0.376541\pi$
$137$	−877.965	−	877.965i	−0.547516	−	0.547516i	−0.925721	+	0.378206i	$0.876541\pi$
$138$		0			0
$139$		−	2489.62i		−	1.51918i	−0.650400	−	0.759592i	$-0.725399\pi$
$139$		−	2489.62i		−	1.51918i	0.650400	−	0.759592i	$-0.274601\pi$
$140$	426.724	−	385.106i	0.257605	−	0.232482i
$141$		0			0
$142$	−375.762	+	375.762i	−0.222065	+	0.222065i
$143$	857.743	−	857.743i	0.501595	−	0.501595i
$144$		0			0
$145$	−115.719	+	2257.35i	−0.0662756	+	1.29284i
$146$			4399.21i			2.49371i
$147$		0			0
$148$	−905.276	−	905.276i	−0.502792	−	0.502792i
$149$	−1497.94			−0.823596			−0.411798	−	0.911275i	$-0.635099\pi$
$149$	−1497.94			−0.823596			−0.411798	+	0.911275i	$0.635099\pi$
$150$		0			0
$151$	−1614.69			−0.870209			−0.435104	−	0.900380i	$-0.643289\pi$
$151$	−1614.69			−0.870209			−0.435104	+	0.900380i	$0.643289\pi$
$152$	−3373.87	−	3373.87i	−1.80038	−	1.80038i
$153$		0			0
$154$		−	625.121i		−	0.327102i
$155$	−28.3575	+	553.173i	−0.0146950	+	0.286658i
$156$		0			0
$157$	−456.938	+	456.938i	−0.232278	+	0.232278i	−0.813643	−	0.581365i	$-0.802519\pi$
$157$	−456.938	+	456.938i	−0.232278	+	0.232278i	0.581365	+	0.813643i	$0.302519\pi$
$158$	1075.54	−	1075.54i	0.541553	−	0.541553i
$159$		0			0
$160$	−3615.42	+	3262.82i	−1.78640	+	1.61218i
$161$		−	215.691i		−	0.105583i
$162$		0			0
$163$	1434.13	+	1434.13i	0.689140	+	0.689140i	0.962042	−	0.272902i	$-0.0879834\pi$
$163$	1434.13	+	1434.13i	0.689140	+	0.689140i	−0.272902	+	0.962042i	$0.587983\pi$
$164$	−6152.83			−2.92961
$165$		0			0
$166$	7104.14			3.32162
$167$	−1129.18	−	1129.18i	−0.523226	−	0.523226i	0.395318	−	0.918544i	$-0.370634\pi$
$167$	−1129.18	−	1129.18i	−0.523226	−	0.523226i	−0.918544	+	0.395318i	$0.870634\pi$
$168$		0			0
$169$		−	1504.34i		−	0.684726i
$170$	−1220.17	−	62.5499i	−0.550485	−	0.0282197i
$171$		0			0
$172$	4636.62	−	4636.62i	2.05546	−	2.05546i
$173$	1656.73	−	1656.73i	0.728084	−	0.728084i	−0.242154	−	0.970238i	$-0.577854\pi$
$173$	1656.73	−	1656.73i	0.728084	−	0.728084i	0.970238	+	0.242154i	$0.0778538\pi$
$174$		0			0
$175$	−248.129	+	201.870i	−0.107182	+	0.0871998i
$176$			8246.15i			3.53169i
$177$		0			0
$178$	−2580.08	−	2580.08i	−1.08644	−	1.08644i
$179$	1449.32			0.605180			0.302590	−	0.953121i	$-0.402149\pi$
$179$	1449.32			0.605180			0.302590	+	0.953121i	$0.402149\pi$
$180$		0			0
$181$	−1120.24			−0.460037			−0.230019	−	0.973186i	$-0.573879\pi$
$181$	−1120.24			−0.460037			−0.230019	+	0.973186i	$0.573879\pi$
$182$	252.403	+	252.403i	0.102799	+	0.102799i
$183$		0			0
$184$			5401.29i			2.16407i
$185$	477.327	+	528.910i	0.189696	+	0.210196i
$186$		0			0
$187$	−671.972	+	671.972i	−0.262778	+	0.262778i
$188$	−3554.90	+	3554.90i	−1.37908	+	1.37908i
$189$		0			0
$190$	2956.02	+	3275.46i	1.12869	+	1.25067i
$191$		−	3827.59i		−	1.45002i	−0.688737	−	0.725012i	$-0.741834\pi$
$191$		−	3827.59i		−	1.45002i	0.688737	−	0.725012i	$-0.258166\pi$
$192$		0			0
$193$	1792.85	+	1792.85i	0.668665	+	0.668665i	0.957407	−	0.288742i	$-0.0932369\pi$
$193$	1792.85	+	1792.85i	0.668665	+	0.668665i	−0.288742	+	0.957407i	$0.593237\pi$
$194$	6900.86			2.55388
$195$		0			0
$196$	−6759.55			−2.46339
$197$	−2222.09	−	2222.09i	−0.803640	−	0.803640i	0.180022	−	0.983663i	$-0.442383\pi$
$197$	−2222.09	−	2222.09i	−0.803640	−	0.803640i	−0.983663	+	0.180022i	$0.942383\pi$
$198$		0			0
$199$		−	1421.45i		−	0.506352i	−0.967420	−	0.253176i	$-0.918525\pi$
$199$		−	1421.45i		−	0.506352i	0.967420	−	0.253176i	$-0.0814752\pi$
$200$	6213.58	−	5055.18i	2.19683	−	1.78728i
$201$		0			0
$202$	−3816.75	+	3816.75i	−1.32943	+	1.32943i
$203$	−365.820	+	365.820i	−0.126481	+	0.126481i
$204$		0			0
$205$	3419.52	+	175.296i	1.16502	+	0.0597230i
$206$		−	2862.17i		−	0.968043i
$207$		0			0
$208$	−3329.52	−	3329.52i	−1.10991	−	1.10991i
$209$	3431.81			1.13580
$210$		0			0
$211$	1133.67			0.369883			0.184942	−	0.982749i	$-0.440790\pi$
$211$	1133.67			0.369883			0.184942	+	0.982749i	$0.440790\pi$
$212$	1710.30	+	1710.30i	0.554076	+	0.554076i
$213$		0			0
$214$		−	87.6623i		−	0.0280022i
$215$	−2708.96	+	2444.76i	−0.859301	+	0.775496i
$216$		0			0
$217$	−89.6459	+	89.6459i	−0.0280441	+	0.0280441i
$218$	−5048.15	+	5048.15i	−1.56837	+	1.56837i
$219$		0			0
$220$	530.031	−	10339.4i	0.162430	−	3.16855i
$221$		−	542.639i		−	0.165167i
$222$		0			0
$223$	666.458	+	666.458i	0.200131	+	0.200131i	0.800056	−	0.599925i	$-0.204803\pi$
$223$	666.458	+	666.458i	0.200131	+	0.200131i	−0.599925	+	0.800056i	$0.704803\pi$
$224$	−1114.67			−0.332487
$225$		0			0
$226$	−5078.95			−1.49490
$227$	2292.99	+	2292.99i	0.670446	+	0.670446i	0.957819	−	0.287373i	$-0.0927819\pi$
$227$	2292.99	+	2292.99i	0.670446	+	0.670446i	−0.287373	+	0.957819i	$0.592782\pi$
$228$		0			0
$229$		−	2535.40i		−	0.731634i	−0.930687	−	0.365817i	$-0.880790\pi$
$229$		−	2535.40i		−	0.731634i	0.930687	−	0.365817i	$-0.119210\pi$
$230$	255.704	−	4988.04i	0.0733071	−	1.43001i
$231$		0			0
$232$	9160.77	−	9160.77i	2.59239	−	2.59239i
$233$	−1699.70	+	1699.70i	−0.477902	+	0.477902i	−0.904460	−	0.426558i	$-0.859726\pi$
$233$	−1699.70	+	1699.70i	−0.477902	+	0.477902i	0.426558	+	0.904460i	$0.359726\pi$
$234$		0			0
$235$	2076.96	−	1874.40i	0.576537	−	0.520309i
$236$		−	6647.49i		−	1.83354i
$237$		0			0
$238$	−197.737	−	197.737i	−0.0538546	−	0.0538546i
$239$	−3948.41			−1.06863			−0.534313	−	0.845287i	$-0.679430\pi$
$239$	−3948.41			−1.06863			−0.534313	+	0.845287i	$0.679430\pi$
$240$		0			0
$241$	−2447.20			−0.654101			−0.327050	−	0.945007i	$-0.606055\pi$
$241$	−2447.20			−0.654101			−0.327050	+	0.945007i	$0.606055\pi$
$242$	−2973.27	−	2973.27i	−0.789788	−	0.789788i
$243$		0			0
$244$			13634.3i			3.57725i
$245$	3756.71	+	192.582i	0.979622	+	0.0502188i
$246$		0			0
$247$	−1385.65	+	1385.65i	−0.356951	+	0.356951i
$248$	2244.89	−	2244.89i	0.574801	−	0.574801i
$249$		0			0
$250$	−5977.50	+	4374.26i	−1.51220	+	1.10661i
$251$			4755.75i			1.19594i	0.801519	+	0.597969i	$0.204025\pi$
$251$			4755.75i			1.19594i	−0.801519	+	0.597969i	$0.795975\pi$
$252$		0			0
$253$	−2747.02	−	2747.02i	−0.682624	−	0.682624i
$254$	−9151.78			−2.26076
$255$		0			0
$256$	−842.400			−0.205664
$257$	1127.34	+	1127.34i	0.273624	+	0.273624i	0.830557	−	0.556933i	$-0.188022\pi$
$257$	1127.34	+	1127.34i	0.273624	+	0.273624i	−0.556933	+	0.830557i	$0.688022\pi$
$258$		0			0
$259$			163.068i			0.0391219i
$260$	3960.68	+	4388.70i	0.944735	+	1.04683i
$261$		0			0
$262$	−805.843	+	805.843i	−0.190020	+	0.190020i
$263$	358.621	−	358.621i	0.0840817	−	0.0840817i	−0.663815	−	0.747897i	$-0.731064\pi$
$263$	358.621	−	358.621i	0.0840817	−	0.0840817i	0.747897	+	0.663815i	$0.231064\pi$
$264$		0			0
$265$	−901.796	−	999.250i	−0.209045	−	0.231636i
$266$			1009.86i			0.232776i
$267$		0			0
$268$	15196.1	+	15196.1i	3.46361	+	3.46361i
$269$	3824.41			0.866834			0.433417	−	0.901193i	$-0.357308\pi$
$269$	3824.41			0.866834			0.433417	+	0.901193i	$0.357308\pi$
$270$		0			0
$271$	535.546			0.120045			0.0600223	−	0.998197i	$-0.480883\pi$
$271$	535.546			0.120045			0.0600223	+	0.998197i	$0.480883\pi$
$272$	2608.41	+	2608.41i	0.581463	+	0.581463i
$273$		0			0
$274$		−	6580.72i		−	1.45093i
$275$	−589.144	+	5731.14i	−0.129188	+	1.25673i
$276$		0			0
$277$	1108.08	−	1108.08i	0.240354	−	0.240354i	−0.576643	−	0.816996i	$-0.695638\pi$
$277$	1108.08	−	1108.08i	0.240354	−	0.240354i	0.816996	+	0.576643i	$0.195638\pi$
$278$	9330.37	−	9330.37i	2.01294	−	2.01294i
$279$		0			0
$280$	1831.00	+	93.8633i	0.390797	+	0.0200336i
$281$			3439.91i			0.730277i	0.930953	+	0.365139i	$0.118978\pi$
$281$			3439.91i			0.730277i	−0.930953	+	0.365139i	$0.881022\pi$
$282$		0			0
$283$	2726.08	+	2726.08i	0.572610	+	0.572610i	0.932857	−	0.360247i	$-0.117307\pi$
$283$	2726.08	+	2726.08i	0.572610	+	0.572610i	−0.360247	+	0.932857i	$0.617307\pi$
$284$	−2014.38			−0.420886
$285$		0			0
$286$	6429.15			1.32924
$287$	554.159	+	554.159i	0.113975	+	0.113975i
$288$		0			0
$289$		−	4487.89i		−	0.913472i
$290$	−8893.57	+	8026.21i	−1.80086	+	1.62522i
$291$		0			0
$292$	−11791.6	+	11791.6i	−2.36320	+	2.36320i
$293$	−4347.55	+	4347.55i	−0.866849	+	0.866849i	−0.992122	−	0.125273i	$-0.960019\pi$
$293$	−4347.55	+	4347.55i	−0.866849	+	0.866849i	0.125273	+	0.992122i	$0.460019\pi$
$294$		0			0
$295$	−189.389	+	3694.43i	−0.0373785	+	0.729145i
$296$		−	4083.51i		−	0.801856i
$297$		0			0
$298$	−5613.84	−	5613.84i	−1.09128	−	1.09128i
$299$	2218.31			0.429058
$300$		0			0
$301$	−835.200			−0.159934
$302$	−6051.39	−	6051.39i	−1.15304	−	1.15304i
$303$		0			0
$304$		−	13321.3i		−	2.51326i
$305$	388.447	−	7577.47i	0.0729259	−	1.42257i
$306$		0			0
$307$	3137.18	−	3137.18i	0.583220	−	0.583220i	−0.352567	−	0.935787i	$-0.614691\pi$
$307$	3137.18	−	3137.18i	0.583220	−	0.583220i	0.935787	+	0.352567i	$0.114691\pi$
$308$	1675.57	−	1675.57i	0.309983	−	0.309983i
$309$		0			0
$310$	−2179.41	+	1966.86i	−0.399298	+	0.360355i
$311$		−	8268.00i		−	1.50751i	−0.657157	−	0.753754i	$-0.728241\pi$
$311$		−	8268.00i		−	1.50751i	0.657157	−	0.753754i	$-0.271759\pi$
$312$		0			0
$313$	−4673.46	−	4673.46i	−0.843959	−	0.843959i	0.145412	−	0.989371i	$-0.453549\pi$
$313$	−4673.46	−	4673.46i	−0.843959	−	0.843959i	−0.989371	+	0.145412i	$0.953549\pi$
$314$	−3424.95			−0.615544
$315$		0			0
$316$	5765.75			1.02642
$317$	−3271.78	−	3271.78i	−0.579690	−	0.579690i	0.355128	−	0.934818i	$-0.384437\pi$
$317$	−3271.78	−	3271.78i	−0.579690	−	0.579690i	−0.934818	+	0.355128i	$0.884437\pi$
$318$		0			0
$319$			9318.09i			1.63546i
$320$	−9796.33	−	502.193i	−1.71135	−	0.0877295i
$321$		0			0
$322$	808.350	−	808.350i	0.139899	−	0.139899i
$323$	1085.54	−	1085.54i	0.187001	−	0.187001i
$324$		0			0
$325$	−2076.17	−	2551.92i	−0.354354	−	0.435554i
$326$			10749.4i			1.82624i
$327$		0			0
$328$	−13877.1	−	13877.1i	−2.33608	−	2.33608i
$329$	640.349			0.107306
$330$		0			0
$331$	9665.06			1.60495			0.802477	−	0.596684i	$-0.203515\pi$
$331$	9665.06			1.60495			0.802477	+	0.596684i	$0.203515\pi$
$332$	19041.9	+	19041.9i	3.14777	+	3.14777i
$333$		0			0
$334$		−	8463.70i		−	1.38657i
$335$	−8012.47	−	8878.36i	−1.30677	−	1.44799i
$336$		0			0
$337$	1911.95	−	1911.95i	0.309053	−	0.309053i	−0.535489	−	0.844542i	$-0.679873\pi$
$337$	1911.95	−	1911.95i	0.309053	−	0.309053i	0.844542	+	0.535489i	$0.179873\pi$
$338$	5637.85	−	5637.85i	0.907274	−	0.907274i
$339$		0			0
$340$	−3102.87	−	3438.19i	−0.494932	−	0.548418i
$341$			2283.44i			0.362625i
$342$		0			0
$343$	1229.46	+	1229.46i	0.193540	+	0.193540i
$344$	20914.9			3.27806
$345$		0			0
$346$	12417.9			1.92945
$347$	7023.71	+	7023.71i	1.08661	+	1.08661i	0.995875	+	0.0907318i	$0.0289206\pi$
$347$	7023.71	+	7023.71i	1.08661	+	1.08661i	0.0907318	+	0.995875i	$0.471079\pi$
$348$		0			0
$349$			9259.70i			1.42023i	0.704085	+	0.710115i	$0.251357\pi$
$349$			9259.70i			1.42023i	−0.704085	+	0.710115i	$0.748643\pi$
$350$	−1686.47	−	173.364i	−0.257559	−	0.0264762i
$351$		0			0
$352$	−14196.3	+	14196.3i	−2.14962	+	2.14962i
$353$	4338.33	−	4338.33i	0.654125	−	0.654125i	−0.299859	−	0.953984i	$-0.596940\pi$
$353$	4338.33	−	4338.33i	0.654125	−	0.654125i	0.953984	+	0.299859i	$0.0969396\pi$
$354$		0			0
$355$	1119.52	+	57.3904i	0.167375	+	0.00858019i
$356$		−	13831.3i		−	2.05915i
$357$		0			0
$358$	5431.64	+	5431.64i	0.801874	+	0.801874i
$359$	−5949.44			−0.874650			−0.437325	−	0.899304i	$-0.644074\pi$
$359$	−5949.44			−0.874650			−0.437325	+	0.899304i	$0.644074\pi$
$360$		0			0
$361$	1315.05			0.191726
$362$	−4198.34	−	4198.34i	−0.609557	−	0.609557i
$363$		0			0
$364$			1353.08i			0.194837i
$365$	6889.30	−	6217.40i	0.987952	−	0.891599i
$366$		0			0
$367$	−9153.79	+	9153.79i	−1.30197	+	1.30197i	−0.374912	+	0.927061i	$0.622327\pi$
$367$	−9153.79	+	9153.79i	−1.30197	+	1.30197i	−0.927061	+	0.374912i	$0.877673\pi$
$368$	−10663.2	+	10663.2i	−1.51048	+	1.51048i
$369$		0			0
$370$	−193.319	+	3771.09i	−0.0271626	+	0.529864i
$371$		−	308.079i		−	0.0431123i
$372$		0			0
$373$	−6473.68	−	6473.68i	−0.898644	−	0.898644i	0.0966723	−	0.995316i	$-0.469180\pi$
$373$	−6473.68	−	6473.68i	−0.898644	−	0.898644i	−0.995316	+	0.0966723i	$0.969180\pi$
$374$	−5036.72			−0.696370
$375$		0			0
$376$	−16035.4			−2.19937
$377$	−3762.33	−	3762.33i	−0.513979	−	0.513979i
$378$		0			0
$379$		−	245.031i		−	0.0332094i	−0.999862	−	0.0166047i	$-0.994714\pi$
$379$		−	245.031i		−	0.0332094i	0.999862	−	0.0166047i	$-0.00528569\pi$
$380$	−856.244	+	16702.8i	−0.115591	+	2.25484i
$381$		0			0
$382$	14344.7	−	14344.7i	1.92131	−	1.92131i
$383$	−4476.48	+	4476.48i	−0.597225	+	0.597225i	−0.939573	−	0.342348i	$-0.888778\pi$
$383$	−4476.48	+	4476.48i	−0.597225	+	0.597225i	0.342348	+	0.939573i	$0.388778\pi$
$384$		0			0
$385$	−978.959	+	883.484i	−0.129591	+	0.116952i
$386$			13438.2i			1.77199i
$387$		0			0
$388$	18497.0	+	18497.0i	2.42022	+	2.42022i
$389$	4429.26			0.577307			0.288654	−	0.957434i	$-0.406792\pi$
$389$	4429.26			0.577307			0.288654	+	0.957434i	$0.406792\pi$
$390$		0			0
$391$	−1737.87			−0.224777
$392$	−15245.5	−	15245.5i	−1.96432	−	1.96432i
$393$		0			0
$394$		−	16655.5i		−	2.12967i
$395$	−3204.39	−	164.268i	−0.408178	−	0.0209246i
$396$		0			0
$397$	4786.16	−	4786.16i	0.605065	−	0.605065i	−0.336588	−	0.941652i	$-0.609273\pi$
$397$	4786.16	−	4786.16i	0.605065	−	0.605065i	0.941652	+	0.336588i	$0.109273\pi$
$398$	5327.19	−	5327.19i	0.670925	−	0.670925i
$399$		0			0
$400$	22246.7	+	2286.89i	2.78084	+	0.285861i
$401$		−	4521.85i		−	0.563119i	−0.959544	−	0.281559i	$-0.909148\pi$
$401$		−	4521.85i		−	0.563119i	0.959544	−	0.281559i	$-0.0908516\pi$
$402$		0			0
$403$	−921.977	−	921.977i	−0.113963	−	0.113963i
$404$	−20460.8			−2.51971
$405$		0			0
$406$	−2741.98			−0.335178
$407$	2076.82	+	2076.82i	0.252934	+	0.252934i
$408$		0			0
$409$			10162.9i			1.22866i	0.789048	+	0.614331i	$0.210574\pi$
$409$			10162.9i			1.22866i	−0.789048	+	0.614331i	$0.789426\pi$
$410$	12158.4	+	13472.3i	1.46454	+	1.62281i
$411$		0			0
$412$	7671.76	−	7671.76i	0.917379	−	0.917379i
$413$	−598.710	+	598.710i	−0.0713331	+	0.0713331i
$414$		0			0
$415$	−10040.3	−	11125.3i	−1.18761	−	1.31595i
$416$		−	11464.0i		−	1.35113i
$417$		0			0
$418$	12861.4	+	12861.4i	1.50496	+	1.50496i
$419$	−13243.2			−1.54409			−0.772044	−	0.635569i	$-0.780766\pi$
$419$	−13243.2			−1.54409			−0.772044	+	0.635569i	$0.780766\pi$
$420$		0			0
$421$	−7488.37			−0.866891			−0.433445	−	0.901180i	$-0.642702\pi$
$421$	−7488.37			−0.866891			−0.433445	+	0.901180i	$0.642702\pi$
$422$	4248.69	+	4248.69i	0.490102	+	0.490102i
$423$		0			0
$424$			7714.83i			0.883644i
$425$	1626.51	+	1999.22i	0.185640	+	0.228180i
$426$		0			0
$427$	1227.99	−	1227.99i	0.139172	−	0.139172i
$428$	234.970	−	234.970i	0.0265367	−	0.0265367i
$429$		0			0
$430$	−19314.7	−	990.137i	−2.16613	−	0.111043i
$431$			5842.63i			0.652969i	0.945202	+	0.326485i	$0.105864\pi$
$431$			5842.63i			0.652969i	−0.945202	+	0.326485i	$0.894136\pi$
$432$		0			0
$433$	−5310.04	−	5310.04i	−0.589340	−	0.589340i	0.348112	−	0.937453i	$-0.386823\pi$
$433$	−5310.04	−	5310.04i	−0.589340	−	0.589340i	−0.937453	+	0.348112i	$0.886823\pi$
$434$	−671.934			−0.0743177
$435$		0			0
$436$	−27062.1			−2.97257
$437$	4437.70	+	4437.70i	0.485776	+	0.485776i
$438$		0			0
$439$			10440.4i			1.13506i	0.823352	+	0.567530i	$0.192101\pi$
$439$			10440.4i			1.13506i	−0.823352	+	0.567530i	$0.807899\pi$
$440$	24514.8	−	22124.0i	2.65613	−	2.39709i
$441$		0			0
$442$	2033.66	−	2033.66i	0.218849	−	0.218849i
$443$	−2729.46	+	2729.46i	−0.292732	+	0.292732i	−0.838159	−	0.545426i	$-0.816368\pi$
$443$	−2729.46	+	2729.46i	−0.292732	+	0.292732i	0.545426	+	0.838159i	$0.316368\pi$
$444$		0			0
$445$	−394.058	+	7686.92i	−0.0419778	+	0.818865i
$446$			4995.38i			0.530355i
$447$		0			0
$448$	−1587.57	−	1587.57i	−0.167423	−	0.167423i
$449$	2227.63			0.234139			0.117070	−	0.993124i	$-0.462650\pi$
$449$	2227.63			0.234139			0.117070	+	0.993124i	$0.462650\pi$
$450$		0			0
$451$	14115.4			1.47377
$452$	−13613.6	−	13613.6i	−1.41666	−	1.41666i
$453$		0			0
$454$			17187.0i			1.77670i
$455$	38.5497	−	751.992i	0.00397195	−	0.0774812i
$456$		0			0
$457$	4328.64	−	4328.64i	0.443075	−	0.443075i	−0.449969	−	0.893044i	$-0.648565\pi$
$457$	4328.64	−	4328.64i	0.443075	−	0.443075i	0.893044	+	0.449969i	$0.148565\pi$
$458$	9501.96	−	9501.96i	0.969427	−	0.969427i
$459$		0			0
$460$	14055.3	−	12684.5i	1.42464	−	1.28570i
$461$			8223.98i			0.830865i	0.909624	+	0.415433i	$0.136370\pi$
$461$			8223.98i			0.830865i	−0.909624	+	0.415433i	$0.863630\pi$
$462$		0			0
$463$	−2611.01	−	2611.01i	−0.262082	−	0.262082i	0.563817	−	0.825899i	$-0.309332\pi$
$463$	−2611.01	−	2611.01i	−0.262082	−	0.262082i	−0.825899	+	0.563817i	$0.809332\pi$
$464$	36170.2			3.61888
$465$		0			0
$466$	−12740.0			−1.26646
$467$	12010.6	+	12010.6i	1.19012	+	1.19012i	0.977034	+	0.213085i	$0.0683511\pi$
$467$	12010.6	+	12010.6i	1.19012	+	1.19012i	0.213085	+	0.977034i	$0.431649\pi$
$468$		0			0
$469$		−	2737.29i		−	0.269501i
$470$	14808.6	+	759.139i	1.45334	+	0.0745031i
$471$		0			0
$472$	14992.7	−	14992.7i	1.46207	−	1.46207i
$473$	−10637.0	+	10637.0i	−1.03402	+	1.03402i
$474$		0			0
$475$	951.738	−	9258.42i	0.0919342	−	0.894328i
$476$		−	1060.03i		−	0.102072i
$477$		0			0
$478$	−14797.5	−	14797.5i	−1.41595	−	1.41595i
$479$	6186.30			0.590103			0.295051	−	0.955481i	$-0.404663\pi$
$479$	6186.30			0.590103			0.295051	+	0.955481i	$0.404663\pi$
$480$		0			0
$481$	−1677.10			−0.158980
$482$	−9171.42	−	9171.42i	−0.866694	−	0.866694i
$483$		0			0
$484$		−	15939.1i		−	1.49691i
$485$	−9752.98	−	10807.0i	−0.913114	−	1.01179i
$486$		0			0
$487$	−12791.1	+	12791.1i	−1.19019	+	1.19019i	−0.213173	+	0.977015i	$0.568380\pi$
$487$	−12791.1	+	12791.1i	−1.19019	+	1.19019i	−0.977015	+	0.213173i	$0.931620\pi$
$488$	−30750.9	+	30750.9i	−2.85251	+	2.85251i
$489$		0			0
$490$	13357.3	+	14800.8i	1.23148	+	1.36456i
$491$		−	13622.3i		−	1.25207i	−0.779796	−	0.626034i	$-0.784677\pi$
$491$		−	13622.3i		−	1.25207i	0.779796	−	0.626034i	$-0.215323\pi$
$492$		0			0
$493$	2947.48	+	2947.48i	0.269265	+	0.269265i
$494$	−10386.0			−0.945931
$495$		0			0
$496$	8863.68			0.802401
$497$	181.427	+	181.427i	0.0163744	+	0.0163744i
$498$		0			0
$499$		−	15358.5i		−	1.37784i	−0.724837	−	0.688920i	$-0.758085\pi$
$499$		−	15358.5i		−	1.37784i	0.724837	−	0.688920i	$-0.241915\pi$
$500$	−27746.8	−	4297.33i	−2.48175	−	0.384365i
$501$		0			0
$502$	−17823.2	+	17823.2i	−1.58464	+	1.58464i
$503$	9784.44	−	9784.44i	0.867329	−	0.867329i	−0.124847	−	0.992176i	$-0.539844\pi$
$503$	9784.44	−	9784.44i	0.867329	−	0.867329i	0.992176	+	0.124847i	$0.0398440\pi$
$504$		0			0
$505$	11371.4	+	582.935i	1.00202	+	0.0513669i
$506$		−	20590.1i		−	1.80898i
$507$		0			0
$508$	−24530.4	−	24530.4i	−2.14244	−	2.14244i
$509$	5300.61			0.461583			0.230791	−	0.973003i	$-0.425869\pi$
$509$	5300.61			0.461583			0.230791	+	0.973003i	$0.425869\pi$
$510$		0			0
$511$	2124.04			0.183879
$512$	−9749.34	−	9749.34i	−0.841532	−	0.841532i
$513$		0			0
$514$			8449.89i			0.725114i
$515$	−4482.25	+	4045.11i	−0.383517	+	0.346114i
$516$		0			0
$517$	8155.41	−	8155.41i	0.693761	−	0.693761i
$518$	−611.134	+	611.134i	−0.0518372	+	0.0518372i
$519$		0			0
$520$	−965.351	+	18831.2i	−0.0814104	+	1.58808i
$521$			8512.67i			0.715829i	0.933754	+	0.357915i	$0.116512\pi$
$521$			8512.67i			0.715829i	−0.933754	+	0.357915i	$0.883488\pi$
$522$		0			0
$523$	7880.88	+	7880.88i	0.658904	+	0.658904i	0.955121	−	0.296217i	$-0.0957251\pi$
$523$	7880.88	+	7880.88i	0.658904	+	0.658904i	−0.296217	+	0.955121i	$0.595725\pi$
$524$	−4319.96			−0.360149
$525$		0			0
$526$	2688.02			0.222819
$527$	722.293	+	722.293i	0.0597032	+	0.0597032i
$528$		0			0
$529$			5062.61i			0.416093i
$530$	365.230	−	7124.58i	0.0299332	−	0.583909i
$531$		0			0
$532$	−2706.82	+	2706.82i	−0.220593	+	0.220593i
$533$	−5699.33	+	5699.33i	−0.463162	+	0.463162i
$534$		0			0
$535$	−137.282	+	123.893i	−0.0110938	+	0.0100119i
$536$			68546.4i			5.52379i
$537$		0			0
$538$	14332.8	+	14332.8i	1.14857	+	1.14857i
$539$	15507.3			1.23923
$540$		0			0
$541$	8961.43			0.712166			0.356083	−	0.934454i	$-0.384112\pi$
$541$	8961.43			0.712166			0.356083	+	0.934454i	$0.384112\pi$
$542$	2007.07	+	2007.07i	0.159061	+	0.159061i
$543$		0			0
$544$			8981.10i			0.707833i
$545$	15040.1	+	771.007i	1.18210	+	0.0605987i
$546$		0			0
$547$	13458.8	−	13458.8i	1.05203	−	1.05203i	0.0534553	−	0.998570i	$-0.482977\pi$
$547$	13458.8	−	13458.8i	1.05203	−	1.05203i	0.998570	−	0.0534553i	$-0.0170235\pi$
$548$	17639.0	−	17639.0i	1.37500	−	1.37500i
$549$		0			0
$550$	−23686.6	+	19270.7i	−1.83636	+	1.49401i
$551$		−	15053.0i		−	1.16385i
$552$		0			0
$553$	−519.296	−	519.296i	−0.0399325	−	0.0399325i
$554$	8305.52			0.636946
$555$		0			0
$556$	50018.2			3.81519
$557$	−12265.3	−	12265.3i	−0.933031	−	0.933031i	0.0648635	−	0.997894i	$-0.479339\pi$
$557$	−12265.3	−	12265.3i	−0.933031	−	0.933031i	−0.997894	+	0.0648635i	$0.979339\pi$
$558$		0			0
$559$		−	8589.74i		−	0.649924i
$560$	3429.44	+	3800.05i	0.258786	+	0.286752i
$561$		0			0
$562$	−12891.8	+	12891.8i	−0.967630	+	0.967630i
$563$	10874.1	−	10874.1i	0.814011	−	0.814011i	−0.171222	−	0.985233i	$-0.554771\pi$
$563$	10874.1	−	10874.1i	0.814011	−	0.814011i	0.985233	+	0.171222i	$0.0547714\pi$
$564$		0			0
$565$	7178.08	+	7953.79i	0.534485	+	0.592245i
$566$			20433.2i			1.51744i
$567$		0			0
$568$	−4543.24	−	4543.24i	−0.335616	−	0.335616i
$569$	−5162.54			−0.380360			−0.190180	−	0.981749i	$-0.560907\pi$
$569$	−5162.54			−0.380360			−0.190180	+	0.981749i	$0.560907\pi$
$570$		0			0
$571$	6356.12			0.465842			0.232921	−	0.972496i	$-0.425172\pi$
$571$	6356.12			0.465842			0.232921	+	0.972496i	$0.425172\pi$
$572$	17232.7	+	17232.7i	1.25968	+	1.25968i
$573$		0			0
$574$			4153.66i			0.302039i
$575$	−8172.81	+	6649.16i	−0.592748	+	0.482242i
$576$		0			0
$577$	−7183.63	+	7183.63i	−0.518299	+	0.518299i	−0.917056	−	0.398758i	$-0.869441\pi$
$577$	−7183.63	+	7183.63i	−0.518299	+	0.518299i	0.398758	+	0.917056i	$0.369441\pi$
$578$	16819.3	−	16819.3i	1.21037	−	1.21037i
$579$		0			0
$580$	−45351.7	−	2324.88i	−3.24677	−	0.166441i
$581$		−	3430.04i		−	0.244926i
$582$		0			0
$583$	−3923.66	−	3923.66i	−0.278733	−	0.278733i
$584$	−53189.6			−3.76884
$585$		0			0
$586$	−32586.8			−2.29718
$587$	−6007.01	−	6007.01i	−0.422378	−	0.422378i	0.463644	−	0.886022i	$-0.346542\pi$
$587$	−6007.01	−	6007.01i	−0.422378	−	0.422378i	−0.886022	+	0.463644i	$0.846542\pi$
$588$		0			0
$589$		−	3688.81i		−	0.258055i
$590$	−14555.4	+	13135.9i	−1.01566	+	0.916603i
$591$		0			0
$592$	8061.64	−	8061.64i	0.559681	−	0.559681i
$593$	−0.276387	+	0.276387i	−1.91397e−5	+	1.91397e-5i	−0.707116	−	0.707097i	$-0.750004\pi$
$593$	−0.276387	+	0.276387i	−1.91397e−5	+	1.91397e-5i	0.707097	+	0.707116i	$0.250004\pi$
$594$		0			0
$595$	−30.2005	+	589.124i	−0.00208084	+	0.0405912i
$596$		−	30094.7i		−	2.06833i
$597$		0			0
$598$	8313.59	+	8313.59i	0.568509	+	0.568509i
$599$	792.766			0.0540761			0.0270380	−	0.999634i	$-0.491392\pi$
$599$	792.766			0.0540761			0.0270380	+	0.999634i	$0.491392\pi$
$600$		0			0
$601$	−6779.60			−0.460142			−0.230071	−	0.973174i	$-0.573896\pi$
$601$	−6779.60			−0.460142			−0.230071	+	0.973174i	$0.573896\pi$
$602$	−3130.09	−	3130.09i	−0.211915	−	0.211915i
$603$		0			0
$604$		−	32440.3i		−	2.18539i
$605$	−454.109	+	8858.34i	−0.0305159	+	0.595277i
$606$		0			0
$607$	−14598.9	+	14598.9i	−0.976194	+	0.976194i	−0.999723	−	0.0235289i	$-0.992510\pi$
$607$	−14598.9	+	14598.9i	−0.976194	+	0.976194i	0.0235289	+	0.999723i	$0.492510\pi$
$608$	22933.6	−	22933.6i	1.52974	−	1.52974i
$609$		0			0
$610$	29854.0	−	26942.4i	1.98156	−	1.78830i
$611$			6585.76i			0.436058i
$612$		0			0
$613$	9862.97	+	9862.97i	0.649856	+	0.649856i	0.952958	−	0.303102i	$-0.0980224\pi$
$613$	9862.97	+	9862.97i	0.649856	+	0.649856i	−0.303102	+	0.952958i	$0.598022\pi$
$614$	23514.5			1.54555
$615$		0			0
$616$	7558.17			0.494363
$617$	−10135.5	−	10135.5i	−0.661329	−	0.661329i	0.294364	−	0.955693i	$-0.404892\pi$
$617$	−10135.5	−	10135.5i	−0.661329	−	0.661329i	−0.955693	+	0.294364i	$0.904892\pi$
$618$		0			0
$619$		−	27282.6i		−	1.77153i	−0.464132	−	0.885766i	$-0.653634\pi$
$619$		−	27282.6i		−	1.77153i	0.464132	−	0.885766i	$-0.346366\pi$
$620$	−11113.6	−	569.724i	−0.719895	−	0.0369043i
$621$		0			0
$622$	30986.1	−	30986.1i	1.99747	−	1.99747i
$623$	−1245.72	+	1245.72i	−0.0801106	+	0.0801106i
$624$		0			0
$625$	15298.2	+	3178.81i	0.979087	+	0.203444i
$626$		−	35029.5i		−	2.23652i
$627$		0			0
$628$	−9180.22	−	9180.22i	−0.583329	−	0.583329i
$629$	1313.87			0.0832869
$630$		0			0
$631$	−28414.9			−1.79268			−0.896340	−	0.443368i	$-0.853783\pi$
$631$	−28414.9			−1.79268			−0.896340	+	0.443368i	$0.853783\pi$
$632$	13004.1	+	13004.1i	0.818471	+	0.818471i
$633$		0			0
$634$		−	24523.4i		−	1.53620i
$635$	12934.2	+	14332.0i	0.808312	+	0.895664i
$636$		0			0
$637$	−6261.33	+	6261.33i	−0.389455	+	0.389455i
$638$	−34921.5	+	34921.5i	−2.16702	+	2.16702i
$639$		0			0
$640$	−8729.23	−	9672.57i	−0.539145	−	0.597409i
$641$			26402.3i			1.62688i	0.581650	+	0.813439i	$0.302407\pi$
$641$			26402.3i			1.62688i	−0.581650	+	0.813439i	$0.697593\pi$
$642$		0			0
$643$	−2361.97	−	2361.97i	−0.144863	−	0.144863i	0.630956	−	0.775819i	$-0.282663\pi$
$643$	−2361.97	−	2361.97i	−0.144863	−	0.144863i	−0.775819	+	0.630956i	$0.782663\pi$
$644$	4333.40			0.265155
$645$		0			0
$646$	8136.61			0.495558
$647$	11204.5	+	11204.5i	0.680826	+	0.680826i	0.960186	−	0.279360i	$-0.0901224\pi$
$647$	11204.5	+	11204.5i	0.680826	+	0.680826i	−0.279360	+	0.960186i	$0.590122\pi$
$648$		0			0
$649$			15250.2i			0.922377i
$650$	1782.99	−	17344.7i	0.107591	−	1.04664i
$651$		0			0
$652$	−28812.7	+	28812.7i	−1.73066	+	1.73066i
$653$	−18407.7	+	18407.7i	−1.10314	+	1.10314i	−0.109108	+	0.994030i	$0.534799\pi$
$653$	−18407.7	+	18407.7i	−1.10314	+	1.10314i	−0.994030	+	0.109108i	$0.965201\pi$
$654$		0			0
$655$	2400.87	+	123.077i	0.143221	+	0.00734201i
$656$		−	54792.0i		−	3.26108i
$657$		0			0
$658$	2399.84	+	2399.84i	0.142182	+	0.142182i
$659$	−8165.28			−0.482662			−0.241331	−	0.970443i	$-0.577584\pi$
$659$	−8165.28			−0.482662			−0.241331	+	0.970443i	$0.577584\pi$
$660$		0			0
$661$	−16725.1			−0.984163			−0.492082	−	0.870549i	$-0.663764\pi$
$661$	−16725.1			−0.984163			−0.492082	+	0.870549i	$0.663764\pi$
$662$	36221.9	+	36221.9i	2.12659	+	2.12659i
$663$		0			0
$664$			85894.2i			5.02009i
$665$	1581.47	−	1427.23i	0.0922207	−	0.0832266i
$666$		0			0
$667$	−12049.3	+	12049.3i	−0.699476	+	0.699476i
$668$	22686.1	−	22686.1i	1.31400	−	1.31400i
$669$		0			0
$670$	3245.08	−	63302.0i	0.187117	−	3.65010i
$671$		−	31279.0i		−	1.79957i
$672$		0			0
$673$	321.610	+	321.610i	0.0184208	+	0.0184208i	0.716257	−	0.697836i	$-0.245854\pi$
$673$	321.610	+	321.610i	0.0184208	+	0.0184208i	−0.697836	+	0.716257i	$0.745854\pi$
$674$	14330.9			0.819000
$675$		0			0
$676$	30223.4			1.71958
$677$	−4497.58	−	4497.58i	−0.255326	−	0.255326i	0.567824	−	0.823150i	$-0.307785\pi$
$677$	−4497.58	−	4497.58i	−0.255326	−	0.255326i	−0.823150	+	0.567824i	$0.807785\pi$
$678$		0			0
$679$		−	3331.89i		−	0.188316i
$680$	756.273	−	14752.7i	0.0426496	−	0.831971i
$681$		0			0
$682$	−8557.68	+	8557.68i	−0.480485	+	0.480485i
$683$	7479.00	−	7479.00i	0.418999	−	0.418999i	−0.465860	−	0.884859i	$-0.654255\pi$
$683$	7479.00	−	7479.00i	0.418999	−	0.418999i	0.884859	+	0.465860i	$0.154255\pi$
$684$		0			0
$685$	−10305.6	+	9300.54i	−0.574828	+	0.518767i
$686$			9215.30i			0.512889i
$687$		0			0
$688$	41289.9	+	41289.9i	2.28803	+	2.28803i
$689$	3168.48			0.175195
$690$		0			0
$691$	−22650.6			−1.24699			−0.623496	−	0.781827i	$-0.714288\pi$
$691$	−22650.6			−1.24699			−0.623496	+	0.781827i	$0.714288\pi$
$692$	33284.8	+	33284.8i	1.82847	+	1.82847i
$693$		0			0
$694$			52645.7i			2.87955i
$695$	−27798.3	−	1425.03i	−1.51719	−	0.0777764i
$696$		0			0
$697$	4464.96	−	4464.96i	0.242643	−	0.242643i
$698$	−34702.7	+	34702.7i	−1.88183	+	1.88183i
$699$		0			0
$700$	−4055.72	−	4985.09i	−0.218988	−	0.269170i
$701$		−	1858.50i		−	0.100135i	−0.998746	−	0.0500676i	$-0.984056\pi$
$701$		−	1858.50i		−	0.100135i	0.998746	−	0.0500676i	$-0.0159437\pi$
$702$		0			0
$703$	−3355.02	−	3355.02i	−0.179996	−	0.179996i
$704$	−40438.2			−2.16487
$705$		0			0
$706$	32517.6			1.73345
$707$	1842.82	+	1842.82i	0.0980286	+	0.0980286i
$708$		0			0
$709$		−	14344.6i		−	0.759836i	−0.925020	−	0.379918i	$-0.875952\pi$
$709$		−	14344.6i		−	0.759836i	0.925020	−	0.379918i	$-0.124048\pi$
$710$	3980.56	+	4410.72i	0.210405	+	0.233143i
$711$		0			0
$712$	31195.1	−	31195.1i	1.64197	−	1.64197i
$713$	−2952.74	+	2952.74i	−0.155092	+	0.155092i
$714$		0			0
$715$	−9086.32	−	10068.3i	−0.475258	−	0.526617i
$716$			29117.9i			1.51981i
$717$		0			0
$718$	−22296.8	−	22296.8i	−1.15893	−	1.15893i
$719$	10128.2			0.525337			0.262669	−	0.964886i	$-0.415397\pi$
$719$	10128.2			0.525337			0.262669	+	0.964886i	$0.415397\pi$
$720$		0			0
$721$	−1381.92			−0.0713807
$722$	4928.43	+	4928.43i	0.254040	+	0.254040i
$723$		0			0
$724$		−	22506.4i		−	1.15531i
$725$	25138.6	+	2584.17i	1.28776	+	0.132377i
$726$		0			0
$727$	15769.7	−	15769.7i	0.804490	−	0.804490i	−0.179304	−	0.983794i	$-0.557385\pi$
$727$	15769.7	−	15769.7i	0.804490	−	0.804490i	0.983794	+	0.179304i	$0.0573846\pi$
$728$	−3051.74	+	3051.74i	−0.155364	+	0.155364i
$729$		0			0
$730$	49120.2	+	2518.07i	2.49044	+	0.127668i
$731$			6729.36i			0.340485i
$732$		0			0
$733$	22982.0	+	22982.0i	1.15806	+	1.15806i	0.984892	+	0.173169i	$0.0554007\pi$
$733$	22982.0	+	22982.0i	1.15806	+	1.15806i	0.173169	+	0.984892i	$0.444599\pi$
$734$	−68611.5			−3.45027
$735$		0			0
$736$	−36714.7			−1.83875
$737$	−34861.8	−	34861.8i	−1.74240	−	1.74240i
$738$		0			0
$739$			1989.79i			0.0990467i	0.998773	+	0.0495234i	$0.0157702\pi$
$739$			1989.79i			0.0990467i	−0.998773	+	0.0495234i	$0.984230\pi$
$740$	−10626.2	+	9589.85i	−0.527874	+	0.476392i
$741$		0			0
$742$	1154.59	−	1154.59i	0.0571245	−	0.0571245i
$743$	4460.57	−	4460.57i	0.220246	−	0.220246i	−0.588356	−	0.808602i	$-0.700225\pi$
$743$	4460.57	−	4460.57i	0.220246	−	0.220246i	0.808602	+	0.588356i	$0.200225\pi$
$744$		0			0
$745$	−857.406	+	16725.5i	−0.0421650	+	0.822516i
$746$		−	48523.0i		−	2.38144i
$747$		0			0
$748$	−13500.4	−	13500.4i	−0.659924	−	0.659924i
$749$	−42.3254			−0.00206480
$750$		0			0
$751$	7038.34			0.341988			0.170994	−	0.985272i	$-0.445302\pi$
$751$	7038.34			0.341988			0.170994	+	0.985272i	$0.445302\pi$
$752$	−31657.0	−	31657.0i	−1.53512	−	1.53512i
$753$		0			0
$754$		−	28200.3i		−	1.36206i
$755$	−924.233	+	18029.1i	−0.0445514	+	0.869068i
$756$		0			0
$757$	−8036.02	+	8036.02i	−0.385831	+	0.385831i	−0.873198	−	0.487366i	$-0.837958\pi$
$757$	−8036.02	+	8036.02i	−0.385831	+	0.385831i	0.487366	+	0.873198i	$0.337958\pi$
$758$	918.304	−	918.304i	0.0440031	−	0.0440031i
$759$		0			0
$760$	−39602.7	+	35740.4i	−1.89019	+	1.70584i
$761$		−	11486.5i		−	0.547154i	−0.961850	−	0.273577i	$-0.911793\pi$
$761$		−	11486.5i		−	0.547154i	0.961850	−	0.273577i	$-0.0882068\pi$
$762$		0			0
$763$	2437.36	+	2437.36i	0.115647	+	0.115647i
$764$	76899.0			3.64150
$765$		0			0
$766$	−33553.1			−1.58267
$767$	−6157.52	−	6157.52i	−0.289876	−	0.289876i
$768$		0			0
$769$		−	11244.4i		−	0.527289i	−0.964620	−	0.263644i	$-0.915075\pi$
$769$		−	11244.4i		−	0.527289i	0.964620	−	0.263644i	$-0.0849245\pi$
$770$	−6979.90	−	357.814i	−0.326673	−	0.0167464i
$771$		0			0
$772$	−36019.7	+	36019.7i	−1.67925	+	1.67925i
$773$	21176.6	−	21176.6i	0.985341	−	0.985341i	−0.0145529	−	0.999894i	$-0.504633\pi$
$773$	21176.6	−	21176.6i	0.985341	−	0.985341i	0.999894	+	0.0145529i	$0.00463251\pi$
$774$		0			0
$775$	6160.32	+	633.262i	0.285529	+	0.0293516i
$776$			83436.4i			3.85978i
$777$		0			0
$778$	16599.6	+	16599.6i	0.764942	+	0.764942i
$779$	−22802.9			−1.04878
$780$		0			0
$781$	4621.26			0.211731
$782$	−6513.02	−	6513.02i	−0.297833	−	0.297833i
$783$		0			0
$784$		−	60195.0i		−	2.74212i
$785$	4840.48	+	5363.57i	0.220082	+	0.243865i
$786$		0			0
$787$	−1608.25	+	1608.25i	−0.0728437	+	0.0728437i	−0.742590	−	0.669746i	$-0.766403\pi$
$787$	−1608.25	+	1608.25i	−0.0728437	+	0.0728437i	0.669746	+	0.742590i	$0.266403\pi$
$788$	44643.3	−	44643.3i	2.01821	−	2.01821i
$789$		0			0
$790$	−11393.5	−	12624.8i	−0.513117	−	0.568568i
$791$			2452.24i			0.110229i
$792$		0			0
$793$	12629.4	+	12629.4i	0.565553	+	0.565553i
$794$	35874.3			1.60344
$795$		0			0
$796$	28558.0			1.27162
$797$	7418.34	+	7418.34i	0.329700	+	0.329700i	0.852472	−	0.522772i	$-0.175102\pi$
$797$	7418.34	+	7418.34i	0.329700	+	0.329700i	−0.522772	+	0.852472i	$0.675102\pi$
$798$		0			0
$799$		−	5159.41i		−	0.228444i
$800$	34362.1	+	42236.2i	1.51861	+	1.86659i
$801$		0			0
$802$	16946.6	−	16946.6i	0.746142	−	0.746142i
$803$	27051.5	−	27051.5i	1.18883	−	1.18883i
$804$		0			0
$805$	−2408.34	−	123.460i	−0.105445	−	0.00540545i
$806$		−	6910.61i		−	0.302005i
$807$		0			0
$808$	−46147.3	−	46147.3i	−2.00923	−	2.00923i
$809$	1490.14			0.0647595			0.0323797	−	0.999476i	$-0.489691\pi$
$809$	1490.14			0.0647595			0.0323797	+	0.999476i	$0.489691\pi$
$810$		0			0
$811$	26161.5			1.13274			0.566371	−	0.824150i	$-0.308347\pi$
$811$	26161.5			1.13274			0.566371	+	0.824150i	$0.308347\pi$
$812$	−7349.59	−	7349.59i	−0.317636	−	0.317636i
$813$		0			0
$814$			15566.7i			0.670284i
$815$	16833.9	−	15192.2i	0.723518	−	0.652955i
$816$		0			0
$817$	17183.7	−	17183.7i	0.735839	−	0.735839i
$818$	−38087.7	+	38087.7i	−1.62800	+	1.62800i
$819$		0			0
$820$	−3521.82	+	68700.6i	−0.149985	+	2.92577i
$821$			21056.4i			0.895095i	0.894260	+	0.447548i	$0.147702\pi$
$821$			21056.4i			0.895095i	−0.894260	+	0.447548i	$0.852298\pi$
$822$		0			0
$823$	187.614	+	187.614i	0.00794631	+	0.00794631i	0.711069	−	0.703122i	$-0.248211\pi$
$823$	187.614	+	187.614i	0.00794631	+	0.00794631i	−0.703122	+	0.711069i	$0.748211\pi$
$824$	34605.7			1.46304
$825$		0			0
$826$	−4487.59			−0.189035
$827$	10888.3	+	10888.3i	0.457827	+	0.457827i	0.897942	−	0.440114i	$-0.145062\pi$
$827$	10888.3	+	10888.3i	0.457827	+	0.457827i	−0.440114	+	0.897942i	$0.645062\pi$
$828$		0			0
$829$			9441.85i			0.395572i	0.980245	+	0.197786i	$0.0633751\pi$
$829$			9441.85i			0.395572i	−0.980245	+	0.197786i	$0.936625\pi$
$830$	4066.34	−	79322.5i	0.170054	−	3.31726i
$831$		0			0
$832$	16327.6	−	16327.6i	0.680357	−	0.680357i
$833$	4905.24	−	4905.24i	0.204029	−	0.204029i
$834$		0			0
$835$	−13254.4	+	11961.7i	−0.549327	+	0.495753i
$836$			68947.5i			2.85239i
$837$		0			0
$838$	−49631.8	−	49631.8i	−2.04594	−	2.04594i
$839$	−25744.3			−1.05935			−0.529673	−	0.848202i	$-0.677685\pi$
$839$	−25744.3			−1.05935			−0.529673	+	0.848202i	$0.677685\pi$
$840$		0			0
$841$	16483.1			0.675840
$842$	−28064.3	−	28064.3i	−1.14865	−	1.14865i
$843$		0			0
$844$			22776.3i			0.928903i
$845$	−16797.0	−	861.072i	−0.683828	−	0.0350554i
$846$		0			0
$847$	−1435.56	+	1435.56i	−0.0582367	+	0.0582367i
$848$	−15230.5	+	15230.5i	−0.616768	+	0.616768i
$849$		0			0
$850$	−1396.82	+	13588.2i	−0.0563655	+	0.548319i
$851$			5371.11i			0.216356i
$852$		0			0
$853$	−32181.0	−	32181.0i	−1.29174	−	1.29174i	−0.933709	−	0.358033i	$-0.883448\pi$
$853$	−32181.0	−	32181.0i	−1.29174	−	1.29174i	−0.358033	−	0.933709i	$-0.616552\pi$
$854$	9204.28			0.368810
$855$		0			0
$856$	1059.90			0.0423208
$857$	15272.0	+	15272.0i	0.608729	+	0.608729i	0.942614	−	0.333885i	$-0.108360\pi$
$857$	15272.0	+	15272.0i	0.608729	+	0.608729i	−0.333885	+	0.942614i	$0.608360\pi$
$858$		0			0
$859$			11140.0i			0.442484i	0.975219	+	0.221242i	$0.0710110\pi$
$859$			11140.0i			0.442484i	−0.975219	+	0.221242i	$0.928989\pi$
$860$	−49117.1	−	54425.0i	−1.94753	−	2.15800i
$861$		0			0
$862$	−21896.5	+	21896.5i	−0.865196	+	0.865196i
$863$	−2425.86	+	2425.86i	−0.0956863	+	0.0956863i	−0.753330	−	0.657643i	$-0.771553\pi$
$863$	−2425.86	+	2425.86i	−0.0956863	+	0.0956863i	0.657643	+	0.753330i	$0.271553\pi$
$864$		0			0
$865$	−17550.2	−	19446.8i	−0.689854	−	0.764404i
$866$		−	39801.0i		−	1.56177i
$867$		0			0
$868$	−1801.05	−	1801.05i	−0.0704282	−	0.0704282i
$869$	−13227.4			−0.516350
$870$		0			0
$871$	28152.0			1.09517
$872$	−61035.7	−	61035.7i	−2.37033	−	2.37033i
$873$		0			0
$874$			33262.5i			1.28732i
$875$	2111.99	+	2886.08i	0.0815982	+	0.111505i
$876$		0			0
$877$	28777.5	−	28777.5i	1.10803	−	1.10803i	0.114625	−	0.993409i	$-0.463433\pi$
$877$	28777.5	−	28777.5i	1.10803	−	1.10803i	0.993409	−	0.114625i	$-0.0365668\pi$
$878$	−39127.5	+	39127.5i	−1.50397	+	1.50397i
$879$		0			0
$880$	92073.9	+	4720.02i	3.52706	+	0.180809i
$881$			3701.33i			0.141545i	0.997492	+	0.0707724i	$0.0225464\pi$
$881$			3701.33i			0.141545i	−0.997492	+	0.0707724i	$0.977454\pi$
$882$		0			0
$883$	30772.1	+	30772.1i	1.17278	+	1.17278i	0.981544	+	0.191236i	$0.0612494\pi$
$883$	30772.1	+	30772.1i	1.17278	+	1.17278i	0.191236	+	0.981544i	$0.438751\pi$
$884$	10902.0			0.414790
$885$		0			0
$886$	−20458.4			−0.775750
$887$	12139.6	+	12139.6i	0.459536	+	0.459536i	0.898503	−	0.438967i	$-0.144656\pi$
$887$	12139.6	+	12139.6i	0.459536	+	0.459536i	−0.438967	+	0.898503i	$0.644656\pi$
$888$		0			0
$889$			4418.69i			0.166702i
$890$	−30285.2	+	27331.6i	−1.14063	+	1.02939i
$891$		0			0
$892$	−13389.6	+	13389.6i	−0.502598	+	0.502598i
$893$	−13174.7	+	13174.7i	−0.493702	+	0.493702i
$894$		0			0
$895$	829.578	−	16182.6i	0.0309829	−	0.604387i
$896$		−	2982.15i		−	0.111190i
$897$		0			0
$898$	8348.53	+	8348.53i	0.310238	+	0.310238i
$899$	10015.9			0.371578
$900$		0			0
$901$	−2482.25			−0.0917821
$902$	52900.5	+	52900.5i	1.95276	+	1.95276i
$903$		0			0
$904$		−	61408.2i		−	2.25930i
$905$	−641.215	+	12508.2i	−0.0235521	+	0.459434i
$906$		0			0
$907$	−316.892	+	316.892i	−0.0116011	+	0.0116011i	−0.712884	−	0.701282i	$-0.752611\pi$
$907$	−316.892	+	316.892i	−0.0116011	+	0.0116011i	0.701282	+	0.712884i	$0.252611\pi$
$908$	−46067.8	+	46067.8i	−1.68372	+	1.68372i
$909$		0			0
$910$	2962.73	−	2673.78i	0.107927	−	0.0974010i
$911$			15635.7i			0.568643i	0.958729	+	0.284321i	$0.0917683\pi$
$911$			15635.7i			0.568643i	−0.958729	+	0.284321i	$0.908232\pi$
$912$		0			0
$913$	−43684.6	−	43684.6i	−1.58352	−	1.58352i
$914$	32445.0			1.17416
$915$		0			0
$916$	50938.1			1.83738
$917$	389.080	+	389.080i	0.0140115	+	0.0140115i
$918$		0			0
$919$		−	7430.80i		−	0.266724i	−0.991067	−	0.133362i	$-0.957423\pi$
$919$		−	7430.80i		−	0.266724i	0.991067	−	0.133362i	$-0.0425773\pi$
$920$	60309.0	+	3091.65i	2.16123	+	0.110792i
$921$		0			0
$922$	−30821.1	+	30821.1i	−1.10091	+	1.10091i
$923$	−1865.91	+	1865.91i	−0.0665408	+	0.0665408i
$924$		0			0
$925$	6178.86	−	5026.94i	0.219632	−	0.178686i
$926$		−	19570.6i		−	0.694526i
$927$		0			0
$928$	62269.5	+	62269.5i	2.20269	+	2.20269i
$929$	889.629			0.0314185			0.0157092	−	0.999877i	$-0.494999\pi$
$929$	889.629			0.0314185			0.0157092	+	0.999877i	$0.494999\pi$
$930$		0			0
$931$	−25051.4			−0.881876
$932$	−34148.2	−	34148.2i	−1.20017	−	1.20017i
$933$		0			0
$934$			90024.8i			3.15386i
$935$	7118.39	+	7887.65i	0.248980	+	0.275886i
$936$		0			0
$937$	−4028.44	+	4028.44i	−0.140452	+	0.140452i	−0.773837	−	0.633385i	$-0.781665\pi$
$937$	−4028.44	+	4028.44i	−0.140452	+	0.140452i	0.633385	+	0.773837i	$0.281665\pi$
$938$	10258.6	−	10258.6i	0.357094	−	0.357094i
$939$		0			0
$940$	37658.1	+	41727.7i	1.30667	+	1.44788i
$941$			14232.9i			0.493072i	0.969134	+	0.246536i	$0.0792924\pi$
$941$			14232.9i			0.493072i	−0.969134	+	0.246536i	$0.920708\pi$
$942$		0			0
$943$	18252.7	+	18252.7i	0.630319	+	0.630319i
$944$	59197.0			2.04099
$945$		0			0
$946$	−79729.0			−2.74018
$947$	−13637.1	−	13637.1i	−0.467947	−	0.467947i	0.433302	−	0.901249i	$-0.357348\pi$
$947$	−13637.1	−	13637.1i	−0.467947	−	0.467947i	−0.901249	+	0.433302i	$0.857348\pi$
$948$		0			0
$949$			21845.0i			0.747227i
$950$	38264.8	−	31131.1i	1.30681	−	1.06319i
$951$		0			0
$952$	2390.79	−	2390.79i	0.0813927	−	0.0813927i
$953$	10234.2	−	10234.2i	0.347869	−	0.347869i	−0.511446	−	0.859315i	$-0.670890\pi$
$953$	10234.2	−	10234.2i	0.347869	−	0.347869i	0.859315	+	0.511446i	$0.170890\pi$
$954$		0			0
$955$	−42737.6	−	2190.88i	−1.44812	−	0.0742357i
$956$		−	79326.5i		−	2.68368i
$957$		0			0
$958$	23184.5	+	23184.5i	0.781896	+	0.781896i
$959$	−3177.33			−0.106988
$960$		0			0
$961$	−27336.6			−0.917611
$962$	−6285.29	−	6285.29i	−0.210651	−	0.210651i
$963$		0			0
$964$		−	49166.1i		−	1.64267i
$965$	21044.6	−	18992.2i	0.702021	−	0.633555i
$966$		0			0
$967$	7178.73	−	7178.73i	0.238731	−	0.238731i	−0.577594	−	0.816324i	$-0.696008\pi$
$967$	7178.73	−	7178.73i	0.238731	−	0.238731i	0.816324	+	0.577594i	$0.196008\pi$
$968$	35948.9	−	35948.9i	1.19364	−	1.19364i
$969$		0			0
$970$	3949.99	−	77052.8i	0.130749	−	2.55053i
$971$			43718.0i			1.44488i	0.691434	+	0.722440i	$0.256979\pi$
$971$			43718.0i			1.44488i	−0.691434	+	0.722440i	$0.743021\pi$
$972$		0			0
$973$	−4504.92	−	4504.92i	−0.148429	−	0.148429i
$974$	−95875.0			−3.15404
$975$		0			0
$976$	−121416.			−3.98201
$977$	36446.0	+	36446.0i	1.19346	+	1.19346i	0.976089	+	0.217371i	$0.0697482\pi$
$977$	36446.0	+	36446.0i	1.19346	+	1.19346i	0.217371	+	0.976089i	$0.430252\pi$
$978$		0			0
$979$			31730.8i			1.03587i
$980$	−3869.11	+	75475.0i	−0.126116	+	2.46016i
$981$		0			0
$982$	51052.5	−	51052.5i	1.65901	−	1.65901i
$983$	−28612.6	+	28612.6i	−0.928382	+	0.928382i	−0.997601	−	0.0692194i	$-0.977949\pi$
$983$	−28612.6	+	28612.6i	−0.928382	+	0.928382i	0.0692194	+	0.997601i	$0.477949\pi$
$984$		0			0
$985$	−26083.0	+	23539.2i	−0.843730	+	0.761443i
$986$			22092.6i			0.713562i
$987$		0			0
$988$	−27838.7	−	27838.7i	−0.896424	−	0.896424i
$989$	−27509.6			−0.884485
$990$		0			0
$991$	56452.4			1.80956			0.904778	−	0.425884i	$-0.140037\pi$
$991$	56452.4			1.80956			0.904778	+	0.425884i	$0.140037\pi$
$992$	15259.4	+	15259.4i	0.488394	+	0.488394i
$993$		0			0
$994$			1359.87i			0.0433928i
$995$	−15871.5	−	813.626i	−0.505688	−	0.0259233i
$996$		0			0
$997$	19121.4	−	19121.4i	0.607402	−	0.607402i	−0.334864	−	0.942266i	$-0.608690\pi$
$997$	19121.4	−	19121.4i	0.607402	−	0.607402i	0.942266	+	0.334864i	$0.108690\pi$
$998$	57559.4	−	57559.4i	1.82566	−	1.82566i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.f.a.8.6	yes	12
3.2	odd	2	inner	45.4.f.a.8.1	✓	12
4.3	odd	2		720.4.w.d.593.4		12
5.2	odd	4	inner	45.4.f.a.17.1	yes	12
5.3	odd	4		225.4.f.c.107.6		12
5.4	even	2		225.4.f.c.143.1		12
12.11	even	2		720.4.w.d.593.3		12
15.2	even	4	inner	45.4.f.a.17.6	yes	12
15.8	even	4		225.4.f.c.107.1		12
15.14	odd	2		225.4.f.c.143.6		12
20.7	even	4		720.4.w.d.17.3		12
60.47	odd	4		720.4.w.d.17.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.f.a.8.1	✓	12	3.2	odd	2	inner
45.4.f.a.8.6	yes	12	1.1	even	1	trivial
45.4.f.a.17.1	yes	12	5.2	odd	4	inner
45.4.f.a.17.6	yes	12	15.2	even	4	inner
225.4.f.c.107.1		12	15.8	even	4
225.4.f.c.107.6		12	5.3	odd	4
225.4.f.c.143.1		12	5.4	even	2
225.4.f.c.143.6		12	15.14	odd	2
720.4.w.d.17.3		12	20.7	even	4
720.4.w.d.17.4		12	60.47	odd	4
720.4.w.d.593.3		12	12.11	even	2
720.4.w.d.593.4		12	4.3	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.74771 + 3.74771i) q^{2} +20.0907i q^{4} +(0.572391 - 11.1657i) q^{5} +(1.80948 - 1.80948i) q^{7} +(-45.3126 + 45.3126i) q^{8} +O(q^{10})\)	\(q+(3.74771 + 3.74771i) q^{2} +20.0907i q^{4} +(0.572391 - 11.1657i) q^{5} +(1.80948 - 1.80948i) q^{7} +(-45.3126 + 45.3126i) q^{8} +(43.9909 - 39.7006i) q^{10} -46.0907i q^{11} +(18.6099 + 18.6099i) q^{13} +13.5628 q^{14} -178.911 q^{16} +(-14.5793 - 14.5793i) q^{17} +74.4577i q^{19} +(224.327 + 11.4997i) q^{20} +(172.735 - 172.735i) q^{22} +(59.6003 - 59.6003i) q^{23} +(-124.345 - 12.7823i) q^{25} +139.489i q^{26} +(36.3538 + 36.3538i) q^{28} -202.168 q^{29} -49.5423 q^{31} +(-308.008 - 308.008i) q^{32} -109.278i q^{34} +(-19.1684 - 21.2398i) q^{35} +(-45.0594 + 45.0594i) q^{37} +(-279.046 + 279.046i) q^{38} +(480.009 + 531.882i) q^{40} +306.253i q^{41} +(-230.784 - 230.784i) q^{43} +925.996 q^{44} +446.730 q^{46} +(176.943 + 176.943i) q^{47} +336.452i q^{49} +(-418.104 - 513.913i) q^{50} +(-373.886 + 373.886i) q^{52} +(85.1290 - 85.1290i) q^{53} +(-514.634 - 26.3819i) q^{55} +163.985i q^{56} +(-757.669 - 757.669i) q^{58} -330.873 q^{59} +678.639 q^{61} +(-185.670 - 185.670i) q^{62} -877.361i q^{64} +(218.444 - 197.140i) q^{65} +(756.373 - 756.373i) q^{67} +(292.909 - 292.909i) q^{68} +(7.76325 - 151.438i) q^{70} +100.264i q^{71} +(586.919 + 586.919i) q^{73} -337.739 q^{74} -1495.91 q^{76} +(-83.4004 - 83.4004i) q^{77} -286.986i q^{79} +(-102.407 + 1997.67i) q^{80} +(-1147.75 + 1147.75i) q^{82} +(947.796 - 947.796i) q^{83} +(-171.133 + 154.443i) q^{85} -1729.83i q^{86} +(2088.49 + 2088.49i) q^{88} -688.442 q^{89} +67.3485 q^{91} +(1197.41 + 1197.41i) q^{92} +1326.26i q^{94} +(831.371 + 42.6189i) q^{95} +(920.676 - 920.676i) q^{97} +(-1260.92 + 1260.92i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{7}+O(q^{10}) \) 12 * q + 24 * q^7	\( 12 q + 24 q^{7} + 192 q^{10} - 108 q^{13} - 648 q^{16} + 1056 q^{22} - 144 q^{25} - 576 q^{28} - 1248 q^{31} + 828 q^{37} + 2568 q^{40} - 96 q^{43} + 672 q^{46} - 312 q^{52} - 1512 q^{55} - 3864 q^{58} + 96 q^{61} + 1632 q^{67} - 1536 q^{70} + 3972 q^{73} - 480 q^{76} - 7848 q^{82} - 1752 q^{85} + 7968 q^{88} + 4752 q^{91} + 2772 q^{97}+O(q^{100}) \) 12 * q + 24 * q^7 + 192 * q^10 - 108 * q^13 - 648 * q^16 + 1056 * q^22 - 144 * q^25 - 576 * q^28 - 1248 * q^31 + 828 * q^37 + 2568 * q^40 - 96 * q^43 + 672 * q^46 - 312 * q^52 - 1512 * q^55 - 3864 * q^58 + 96 * q^61 + 1632 * q^67 - 1536 * q^70 + 3972 * q^73 - 480 * q^76 - 7848 * q^82 - 1752 * q^85 + 7968 * q^88 + 4752 * q^91 + 2772 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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