LMFDB - Embedded newform 3654.2.f.b.755.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3654,2,Mod(755,3654)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3654.755"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3654, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3654.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,-40,0,0,-4,0,0,0,0,0,0,0,0,40,0,0,0,0,0,8,0,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$29.1773368986$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			755.4
Character	$\chi$	$=$	3654.755
Dual form			3654.2.f.b.755.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +1.81839 q^{5} +(-0.812712 + 2.51784i) q^{7} -1.00000i q^{8} +1.81839i q^{10} -6.05565i q^{11} -2.51029i q^{13} +(-2.51784 - 0.812712i) q^{14} +1.00000 q^{16} +4.50010 q^{17} +4.56140i q^{19} -1.81839 q^{20} +6.05565 q^{22} -6.67026i q^{23} -1.69344 q^{25} +2.51029 q^{26} +(0.812712 - 2.51784i) q^{28} +1.00000i q^{29} -7.40750i q^{31} +1.00000i q^{32} +4.50010i q^{34} +(-1.47783 + 4.57842i) q^{35} -6.58716 q^{37} -4.56140 q^{38} -1.81839i q^{40} -8.46644 q^{41} -5.99014 q^{43} +6.05565i q^{44} +6.67026 q^{46} -5.40101 q^{47} +(-5.67900 - 4.09255i) q^{49} -1.69344i q^{50} +2.51029i q^{52} +0.309506i q^{53} -11.0115i q^{55} +(2.51784 + 0.812712i) q^{56} -1.00000 q^{58} -3.41718 q^{59} -10.0528i q^{61} +7.40750 q^{62} -1.00000 q^{64} -4.56470i q^{65} +15.2036 q^{67} -4.50010 q^{68} +(-4.57842 - 1.47783i) q^{70} +2.60992i q^{71} -6.17049i q^{73} -6.58716i q^{74} -4.56140i q^{76} +(15.2471 + 4.92149i) q^{77} -9.93366 q^{79} +1.81839 q^{80} -8.46644i q^{82} -13.0032 q^{83} +8.18295 q^{85} -5.99014i q^{86} -6.05565 q^{88} +10.4760 q^{89} +(6.32051 + 2.04015i) q^{91} +6.67026i q^{92} -5.40101i q^{94} +8.29442i q^{95} -9.92501i q^{97} +(4.09255 - 5.67900i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{4} - 4 q^{7} + 40 q^{16} + 8 q^{22} + 64 q^{25} + 4 q^{28} - 48 q^{37} - 8 q^{43} + 8 q^{46} + 36 q^{49} - 40 q^{58} - 40 q^{64} + 24 q^{67} + 28 q^{70} - 144 q^{79} - 24 q^{85} - 8 q^{88}+O(q^{100}) $ 40 * q - 40 * q^4 - 4 * q^7 + 40 * q^16 + 8 * q^22 + 64 * q^25 + 4 * q^28 - 48 * q^37 - 8 * q^43 + 8 * q^46 + 36 * q^49 - 40 * q^58 - 40 * q^64 + 24 * q^67 + 28 * q^70 - 144 * q^79 - 24 * q^85 - 8 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3654\mathbb{Z}\right)^\times$.

$n$	$379$	$407$	$2089$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.81839			0.813211			0.406605	−	0.913604i	$-0.366712\pi$
$5$	1.81839			0.813211			0.406605	+	0.913604i	$0.366712\pi$
$6$		0			0
$7$	−0.812712	+	2.51784i	−0.307176	+	0.951653i
$7$	−0.812712	+	2.51784i	−0.307176	+	0.951653i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$			1.81839i			0.575027i
$11$		−	6.05565i		−	1.82585i	−0.408132	−	0.912923i	$-0.633820\pi$
$11$		−	6.05565i		−	1.82585i	0.408132	−	0.912923i	$-0.366180\pi$
$12$		0			0
$13$		−	2.51029i		−	0.696230i	−0.937452	−	0.348115i	$-0.886822\pi$
$13$		−	2.51029i		−	0.696230i	0.937452	−	0.348115i	$-0.113178\pi$
$14$	−2.51784	−	0.812712i	−0.672920	−	0.217206i
$15$		0			0
$16$	1.00000			0.250000
$17$	4.50010			1.09143			0.545717	−	0.837970i	$-0.316257\pi$
$17$	4.50010			1.09143			0.545717	+	0.837970i	$0.316257\pi$
$18$		0			0
$19$			4.56140i			1.04646i	0.852192	+	0.523228i	$0.175273\pi$
$19$			4.56140i			1.04646i	−0.852192	+	0.523228i	$0.824727\pi$
$20$	−1.81839			−0.406605
$21$		0			0
$22$	6.05565			1.29107
$23$		−	6.67026i		−	1.39085i	−0.718601	−	0.695423i	$-0.755217\pi$
$23$		−	6.67026i		−	1.39085i	0.718601	−	0.695423i	$-0.244783\pi$
$24$		0			0
$25$	−1.69344			−0.338689
$26$	2.51029			0.492309
$27$		0			0
$28$	0.812712	−	2.51784i	0.153588	−	0.475826i
$29$			1.00000i			0.185695i
$29$			1.00000i			0.185695i
$30$		0			0
$31$		−	7.40750i		−	1.33043i	−0.746654	−	0.665213i	$-0.768341\pi$
$31$		−	7.40750i		−	1.33043i	0.746654	−	0.665213i	$-0.231659\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$			4.50010i			0.771760i
$35$	−1.47783	+	4.57842i	−0.249799	+	0.773894i
$36$		0			0
$37$	−6.58716			−1.08292			−0.541461	−	0.840726i	$-0.682129\pi$
$37$	−6.58716			−1.08292			−0.541461	+	0.840726i	$0.682129\pi$
$38$	−4.56140			−0.739957
$39$		0			0
$40$		−	1.81839i		−	0.287513i
$41$	−8.46644			−1.32224			−0.661118	−	0.750282i	$-0.729918\pi$
$41$	−8.46644			−1.32224			−0.661118	+	0.750282i	$0.729918\pi$
$42$		0			0
$43$	−5.99014			−0.913487			−0.456744	−	0.889598i	$-0.650984\pi$
$43$	−5.99014			−0.913487			−0.456744	+	0.889598i	$0.650984\pi$
$44$			6.05565i			0.912923i
$45$		0			0
$46$	6.67026			0.983476
$47$	−5.40101			−0.787818			−0.393909	−	0.919150i	$-0.628877\pi$
$47$	−5.40101			−0.787818			−0.393909	+	0.919150i	$0.628877\pi$
$48$		0			0
$49$	−5.67900	−	4.09255i	−0.811286	−	0.584650i
$50$		−	1.69344i		−	0.239489i
$51$		0			0
$52$			2.51029i			0.348115i
$53$			0.309506i			0.0425139i	0.999774	+	0.0212570i	$0.00676681\pi$
$53$			0.309506i			0.0425139i	−0.999774	+	0.0212570i	$0.993233\pi$
$54$		0			0
$55$		−	11.0115i		−	1.48480i
$56$	2.51784	+	0.812712i	0.336460	+	0.108603i
$57$		0			0
$58$	−1.00000			−0.131306
$59$	−3.41718			−0.444879			−0.222439	−	0.974947i	$-0.571402\pi$
$59$	−3.41718			−0.444879			−0.222439	+	0.974947i	$0.571402\pi$
$60$		0			0
$61$		−	10.0528i		−	1.28713i	−0.765391	−	0.643565i	$-0.777454\pi$
$61$		−	10.0528i		−	1.28713i	0.765391	−	0.643565i	$-0.222546\pi$
$62$	7.40750			0.940753
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	4.56470i		−	0.566182i
$66$		0			0
$67$	15.2036			1.85742			0.928711	−	0.370805i	$-0.120918\pi$
$67$	15.2036			1.85742			0.928711	+	0.370805i	$0.120918\pi$
$68$	−4.50010			−0.545717
$69$		0			0
$70$	−4.57842	−	1.47783i	−0.547226	−	0.176635i
$71$			2.60992i			0.309740i	0.987935	+	0.154870i	$0.0494959\pi$
$71$			2.60992i			0.309740i	−0.987935	+	0.154870i	$0.950504\pi$
$72$		0			0
$73$		−	6.17049i		−	0.722201i	−0.932527	−	0.361101i	$-0.882401\pi$
$73$		−	6.17049i		−	0.722201i	0.932527	−	0.361101i	$-0.117599\pi$
$74$		−	6.58716i		−	0.765742i
$75$		0			0
$76$		−	4.56140i		−	0.523228i
$77$	15.2471	+	4.92149i	1.73757	+	0.560856i
$78$		0			0
$79$	−9.93366			−1.11762			−0.558812	−	0.829294i	$-0.688743\pi$
$79$	−9.93366			−1.11762			−0.558812	+	0.829294i	$0.688743\pi$
$80$	1.81839			0.203303
$81$		0			0
$82$		−	8.46644i		−	0.934962i
$83$	−13.0032			−1.42728			−0.713642	−	0.700510i	$-0.752956\pi$
$83$	−13.0032			−1.42728			−0.713642	+	0.700510i	$0.752956\pi$
$84$		0			0
$85$	8.18295			0.887566
$86$		−	5.99014i		−	0.645933i
$87$		0			0
$88$	−6.05565			−0.645534
$89$	10.4760			1.11045			0.555226	−	0.831700i	$-0.312632\pi$
$89$	10.4760			1.11045			0.555226	+	0.831700i	$0.312632\pi$
$90$		0			0
$91$	6.32051	+	2.04015i	0.662570	+	0.213865i
$92$			6.67026i			0.695423i
$93$		0			0
$94$		−	5.40101i		−	0.557071i
$95$			8.29442i			0.850990i
$96$		0			0
$97$		−	9.92501i		−	1.00773i	−0.863782	−	0.503866i	$-0.831911\pi$
$97$		−	9.92501i		−	1.00773i	0.863782	−	0.503866i	$-0.168089\pi$
$98$	4.09255	−	5.67900i	0.413410	−	0.573666i
$99$		0			0
$100$	1.69344			0.169344
$101$	16.0644			1.59847			0.799235	−	0.601019i	$-0.205239\pi$
$101$	16.0644			1.59847			0.799235	+	0.601019i	$0.205239\pi$
$102$		0			0
$103$			0.332202i			0.0327328i	0.999866	+	0.0163664i	$0.00520982\pi$
$103$			0.332202i			0.0327328i	−0.999866	+	0.0163664i	$0.994790\pi$
$104$	−2.51029			−0.246155
$105$		0			0
$106$	−0.309506			−0.0300619
$107$		−	15.3799i		−	1.48683i	−0.668831	−	0.743415i	$-0.733205\pi$
$107$		−	15.3799i		−	1.48683i	0.668831	−	0.743415i	$-0.266795\pi$
$108$		0			0
$109$	5.70082			0.546039			0.273020	−	0.962008i	$-0.411978\pi$
$109$	5.70082			0.546039			0.273020	+	0.962008i	$0.411978\pi$
$110$	11.0115			1.04991
$111$		0			0
$112$	−0.812712	+	2.51784i	−0.0767940	+	0.237913i
$113$		−	1.14651i		−	0.107855i	−0.998545	−	0.0539274i	$-0.982826\pi$
$113$		−	1.14651i		−	0.107855i	0.998545	−	0.0539274i	$-0.0171740\pi$
$114$		0			0
$115$		−	12.1292i		−	1.13105i
$116$		−	1.00000i		−	0.0928477i
$117$		0			0
$118$		−	3.41718i		−	0.314577i
$119$	−3.65728	+	11.3305i	−0.335262	+	1.03867i
$120$		0			0
$121$	−25.6708			−2.33371
$122$	10.0528			0.910139
$123$		0			0
$124$			7.40750i			0.665213i
$125$	−12.1713			−1.08864
$126$		0			0
$127$	−9.94342			−0.882336			−0.441168	−	0.897425i	$-0.645436\pi$
$127$	−9.94342			−0.882336			−0.441168	+	0.897425i	$0.645436\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	4.56470			0.400351
$131$	3.35409			0.293048			0.146524	−	0.989207i	$-0.453191\pi$
$131$	3.35409			0.293048			0.146524	+	0.989207i	$0.453191\pi$
$132$		0			0
$133$	−11.4849	−	3.70710i	−0.995864	−	0.321447i
$134$			15.2036i			1.31340i
$135$		0			0
$136$		−	4.50010i		−	0.385880i
$137$			14.5958i			1.24700i	0.781822	+	0.623501i	$0.214290\pi$
$137$			14.5958i			1.24700i	−0.781822	+	0.623501i	$0.785710\pi$
$138$		0			0
$139$		−	5.95257i		−	0.504891i	−0.967611	−	0.252445i	$-0.918765\pi$
$139$		−	5.95257i		−	0.504891i	0.967611	−	0.252445i	$-0.0812348\pi$
$140$	1.47783	−	4.57842i	0.124899	−	0.386947i
$141$		0			0
$142$	−2.60992			−0.219019
$143$	−15.2015			−1.27121
$144$		0			0
$145$			1.81839i			0.151009i
$146$	6.17049			0.510674
$147$		0			0
$148$	6.58716			0.541461
$149$			15.2344i			1.24805i	0.781404	+	0.624025i	$0.214504\pi$
$149$			15.2344i			1.24805i	−0.781404	+	0.624025i	$0.785496\pi$
$150$		0			0
$151$	−5.39307			−0.438882			−0.219441	−	0.975626i	$-0.570423\pi$
$151$	−5.39307			−0.438882			−0.219441	+	0.975626i	$0.570423\pi$
$152$	4.56140			0.369978
$153$		0			0
$154$	−4.92149	+	15.2471i	−0.396585	+	1.22865i
$155$		−	13.4698i		−	1.08192i
$156$		0			0
$157$			5.90694i			0.471425i	0.971823	+	0.235712i	$0.0757424\pi$
$157$			5.90694i			0.471425i	−0.971823	+	0.235712i	$0.924258\pi$
$158$		−	9.93366i		−	0.790280i
$159$		0			0
$160$			1.81839i			0.143757i
$161$	16.7946	+	5.42100i	1.32360	+	0.427234i
$162$		0			0
$163$	10.6581			0.834805			0.417402	−	0.908722i	$-0.362941\pi$
$163$	10.6581			0.834805			0.417402	+	0.908722i	$0.362941\pi$
$164$	8.46644			0.661118
$165$		0			0
$166$		−	13.0032i		−	1.00924i
$167$	10.7289			0.830223			0.415112	−	0.909770i	$-0.363742\pi$
$167$	10.7289			0.830223			0.415112	+	0.909770i	$0.363742\pi$
$168$		0			0
$169$	6.69842			0.515263
$170$			8.18295i			0.627604i
$171$		0			0
$172$	5.99014			0.456744
$173$	23.0797			1.75472			0.877360	−	0.479833i	$-0.159303\pi$
$173$	23.0797			1.75472			0.877360	+	0.479833i	$0.159303\pi$
$174$		0			0
$175$	1.37628	−	4.26381i	0.104037	−	0.322314i
$176$		−	6.05565i		−	0.456461i
$177$		0			0
$178$			10.4760i			0.785207i
$179$		−	16.6281i		−	1.24284i	−0.783477	−	0.621421i	$-0.786556\pi$
$179$		−	16.6281i		−	1.24284i	0.783477	−	0.621421i	$-0.213444\pi$
$180$		0			0
$181$		−	5.27347i		−	0.391974i	−0.980607	−	0.195987i	$-0.937209\pi$
$181$		−	5.27347i		−	0.391974i	0.980607	−	0.195987i	$-0.0627910\pi$
$182$	−2.04015	+	6.32051i	−0.151226	+	0.468507i
$183$		0			0
$184$	−6.67026			−0.491738
$185$	−11.9781			−0.880644
$186$		0			0
$187$		−	27.2510i		−	1.99279i
$188$	5.40101			0.393909
$189$		0			0
$190$	−8.29442			−0.601741
$191$		−	10.2997i		−	0.745257i	−0.927981	−	0.372629i	$-0.878457\pi$
$191$		−	10.2997i		−	0.745257i	0.927981	−	0.372629i	$-0.121543\pi$
$192$		0			0
$193$	13.3100			0.958078			0.479039	−	0.877794i	$-0.340985\pi$
$193$	13.3100			0.958078			0.479039	+	0.877794i	$0.340985\pi$
$194$	9.92501			0.712574
$195$		0			0
$196$	5.67900	+	4.09255i	0.405643	+	0.292325i
$197$			18.2881i			1.30297i	0.758660	+	0.651487i	$0.225855\pi$
$197$			18.2881i			1.30297i	−0.758660	+	0.651487i	$0.774145\pi$
$198$		0			0
$199$			8.30887i			0.589000i	0.955651	+	0.294500i	$0.0951531\pi$
$199$			8.30887i			0.589000i	−0.955651	+	0.294500i	$0.904847\pi$
$200$			1.69344i			0.119744i
$201$		0			0
$202$			16.0644i			1.13029i
$203$	−2.51784	−	0.812712i	−0.176717	−	0.0570412i
$204$		0			0
$205$	−15.3953			−1.07526
$206$	−0.332202			−0.0231456
$207$		0			0
$208$		−	2.51029i		−	0.174058i
$209$	27.6222			1.91067
$210$		0			0
$211$	−23.4970			−1.61760			−0.808799	−	0.588085i	$-0.799882\pi$
$211$	−23.4970			−1.61760			−0.808799	+	0.588085i	$0.799882\pi$
$212$		−	0.309506i		−	0.0212570i
$213$		0			0
$214$	15.3799			1.05135
$215$	−10.8924			−0.742857
$216$		0			0
$217$	18.6509	+	6.02016i	1.26610	+	0.408675i
$218$			5.70082i			0.386108i
$219$		0			0
$220$			11.0115i			0.742399i
$221$		−	11.2966i		−	0.759889i
$222$		0			0
$223$		−	8.34460i		−	0.558796i	−0.960175	−	0.279398i	$-0.909865\pi$
$223$		−	8.34460i		−	0.558796i	0.960175	−	0.279398i	$-0.0901348\pi$
$224$	−2.51784	−	0.812712i	−0.168230	−	0.0543016i
$225$		0			0
$226$	1.14651			0.0762649
$227$	13.0567			0.866600			0.433300	−	0.901250i	$-0.357349\pi$
$227$	13.0567			0.866600			0.433300	+	0.901250i	$0.357349\pi$
$228$		0			0
$229$		−	10.5442i		−	0.696779i	−0.937350	−	0.348390i	$-0.886729\pi$
$229$		−	10.5442i		−	0.696779i	0.937350	−	0.348390i	$-0.113271\pi$
$230$	12.1292			0.799773
$231$		0			0
$232$	1.00000			0.0656532
$233$		−	15.6046i		−	1.02229i	−0.859493	−	0.511147i	$-0.829221\pi$
$233$		−	15.6046i		−	1.02229i	0.859493	−	0.511147i	$-0.170779\pi$
$234$		0			0
$235$	−9.82116			−0.640662
$236$	3.41718			0.222439
$237$		0			0
$238$	−11.3305	−	3.65728i	−0.734448	−	0.237066i
$239$		−	9.94549i		−	0.643320i	−0.946855	−	0.321660i	$-0.895759\pi$
$239$		−	9.94549i		−	0.643320i	0.946855	−	0.321660i	$-0.104241\pi$
$240$		0			0
$241$			25.6474i			1.65210i	0.563599	+	0.826049i	$0.309416\pi$
$241$			25.6474i			1.65210i	−0.563599	+	0.826049i	$0.690584\pi$
$242$		−	25.6708i		−	1.65018i
$243$		0			0
$244$			10.0528i			0.643565i
$245$	−10.3267	−	7.44187i	−0.659746	−	0.475444i
$246$		0			0
$247$	11.4505			0.728575
$248$	−7.40750			−0.470377
$249$		0			0
$250$		−	12.1713i		−	0.769782i
$251$	0.195266			0.0123251			0.00616254	−	0.999981i	$-0.498038\pi$
$251$	0.195266			0.0123251			0.00616254	+	0.999981i	$0.498038\pi$
$252$		0			0
$253$	−40.3927			−2.53947
$254$		−	9.94342i		−	0.623906i
$255$		0			0
$256$	1.00000			0.0625000
$257$	3.37691			0.210646			0.105323	−	0.994438i	$-0.466412\pi$
$257$	3.37691			0.210646			0.105323	+	0.994438i	$0.466412\pi$
$258$		0			0
$259$	5.35346	−	16.5854i	0.332648	−	1.03057i
$260$			4.56470i			0.283091i
$261$		0			0
$262$			3.35409i			0.207216i
$263$		−	2.02058i		−	0.124595i	−0.998058	−	0.0622973i	$-0.980157\pi$
$263$		−	2.02058i		−	0.124595i	0.998058	−	0.0622973i	$-0.0198427\pi$
$264$		0			0
$265$			0.562804i			0.0345728i
$266$	3.70710	−	11.4849i	0.227297	−	0.704182i
$267$		0			0
$268$	−15.2036			−0.928711
$269$	18.4193			1.12304			0.561522	−	0.827462i	$-0.310216\pi$
$269$	18.4193			1.12304			0.561522	+	0.827462i	$0.310216\pi$
$270$		0			0
$271$			24.5042i			1.48853i	0.667887	+	0.744263i	$0.267199\pi$
$271$			24.5042i			1.48853i	−0.667887	+	0.744263i	$0.732801\pi$
$272$	4.50010			0.272858
$273$		0			0
$274$	−14.5958			−0.881764
$275$			10.2549i			0.618393i
$276$		0			0
$277$	12.1782			0.731717			0.365859	−	0.930670i	$-0.380775\pi$
$277$	12.1782			0.731717			0.365859	+	0.930670i	$0.380775\pi$
$278$	5.95257			0.357011
$279$		0			0
$280$	4.57842	+	1.47783i	0.273613	+	0.0883173i
$281$			11.5814i			0.690888i	0.938439	+	0.345444i	$0.112272\pi$
$281$			11.5814i			0.690888i	−0.938439	+	0.345444i	$0.887728\pi$
$282$		0			0
$283$		−	27.1540i		−	1.61414i	−0.590458	−	0.807068i	$-0.701053\pi$
$283$		−	27.1540i		−	1.61414i	0.590458	−	0.807068i	$-0.298947\pi$
$284$		−	2.60992i		−	0.154870i
$285$		0			0
$286$		−	15.2015i		−	0.898881i
$287$	6.88078	−	21.3171i	0.406159	−	1.25831i
$288$		0			0
$289$	3.25087			0.191228
$290$	−1.81839			−0.106780
$291$		0			0
$292$			6.17049i			0.361101i
$293$	−31.7235			−1.85331			−0.926655	−	0.375914i	$-0.877329\pi$
$293$	−31.7235			−1.85331			−0.926655	+	0.375914i	$0.877329\pi$
$294$		0			0
$295$	−6.21377			−0.361780
$296$			6.58716i			0.382871i
$297$		0			0
$298$	−15.2344			−0.882505
$299$	−16.7443			−0.968349
$300$		0			0
$301$	4.86825	−	15.0822i	0.280601	−	0.869323i
$302$		−	5.39307i		−	0.310336i
$303$		0			0
$304$			4.56140i			0.261614i
$305$		−	18.2800i		−	1.04671i
$306$		0			0
$307$		−	2.72493i		−	0.155520i	−0.996972	−	0.0777601i	$-0.975223\pi$
$307$		−	2.72493i		−	0.155520i	0.996972	−	0.0777601i	$-0.0247768\pi$
$308$	−15.2471	−	4.92149i	−0.868785	−	0.280428i
$309$		0			0
$310$	13.4698			0.765030
$311$	−7.74611			−0.439241			−0.219621	−	0.975585i	$-0.570482\pi$
$311$	−7.74611			−0.439241			−0.219621	+	0.975585i	$0.570482\pi$
$312$		0			0
$313$		−	26.8167i		−	1.51577i	−0.652390	−	0.757884i	$-0.726233\pi$
$313$		−	26.8167i		−	1.51577i	0.652390	−	0.757884i	$-0.273767\pi$
$314$	−5.90694			−0.333348
$315$		0			0
$316$	9.93366			0.558812
$317$		−	15.0707i		−	0.846458i	−0.906023	−	0.423229i	$-0.860897\pi$
$317$		−	15.0707i		−	0.846458i	0.906023	−	0.423229i	$-0.139103\pi$
$318$		0			0
$319$	6.05565			0.339051
$320$	−1.81839			−0.101651
$321$		0			0
$322$	−5.42100	+	16.7946i	−0.302100	+	0.935928i
$323$			20.5267i			1.14214i
$324$		0			0
$325$			4.25104i			0.235805i
$326$			10.6581i			0.590296i
$327$		0			0
$328$			8.46644i			0.467481i
$329$	4.38946	−	13.5988i	0.241999	−	0.749729i
$330$		0			0
$331$	1.24808			0.0686004			0.0343002	−	0.999412i	$-0.489080\pi$
$331$	1.24808			0.0686004			0.0343002	+	0.999412i	$0.489080\pi$
$332$	13.0032			0.713642
$333$		0			0
$334$			10.7289i			0.587057i
$335$	27.6462			1.51047
$336$		0			0
$337$	13.1359			0.715560			0.357780	−	0.933806i	$-0.383534\pi$
$337$	13.1359			0.715560			0.357780	+	0.933806i	$0.383534\pi$
$338$			6.69842i			0.364346i
$339$		0			0
$340$	−8.18295			−0.443783
$341$	−44.8572			−2.42915
$342$		0			0
$343$	14.9198	−	10.9727i	0.805591	−	0.592472i
$344$			5.99014i			0.322966i
$345$		0			0
$346$			23.0797i			1.24077i
$347$			29.5484i			1.58624i	0.609065	+	0.793120i	$0.291545\pi$
$347$			29.5484i			1.58624i	−0.609065	+	0.793120i	$0.708455\pi$
$348$		0			0
$349$		−	19.2752i		−	1.03178i	−0.856655	−	0.515889i	$-0.827462\pi$
$349$		−	19.2752i		−	1.03178i	0.856655	−	0.515889i	$-0.172538\pi$
$350$	4.26381	+	1.37628i	0.227910	+	0.0735653i
$351$		0			0
$352$	6.05565			0.322767
$353$	−8.50869			−0.452872			−0.226436	−	0.974026i	$-0.572707\pi$
$353$	−8.50869			−0.452872			−0.226436	+	0.974026i	$0.572707\pi$
$354$		0			0
$355$			4.74586i			0.251884i
$356$	−10.4760			−0.555226
$357$		0			0
$358$	16.6281			0.878822
$359$		−	7.94081i		−	0.419100i	−0.977798	−	0.209550i	$-0.932800\pi$
$359$		−	7.94081i		−	0.419100i	0.977798	−	0.209550i	$-0.0671999\pi$
$360$		0			0
$361$	−1.80637			−0.0950722
$362$	5.27347			0.277167
$363$		0			0
$364$	−6.32051	−	2.04015i	−0.331285	−	0.106933i
$365$		−	11.2204i		−	0.587302i
$366$		0			0
$367$		−	36.3595i		−	1.89795i	−0.315348	−	0.948976i	$-0.602121\pi$
$367$		−	36.3595i		−	1.89795i	0.315348	−	0.948976i	$-0.397879\pi$
$368$		−	6.67026i		−	0.347711i
$369$		0			0
$370$		−	11.9781i		−	0.622709i
$371$	−0.779285	−	0.251539i	−0.0404585	−	0.0130593i
$372$		0			0
$373$	31.4672			1.62931			0.814656	−	0.579944i	$-0.196926\pi$
$373$	31.4672			1.62931			0.814656	+	0.579944i	$0.196926\pi$
$374$	27.2510			1.40912
$375$		0			0
$376$			5.40101i			0.278536i
$377$	2.51029			0.129287
$378$		0			0
$379$	16.8128			0.863617			0.431809	−	0.901965i	$-0.357876\pi$
$379$	16.8128			0.863617			0.431809	+	0.901965i	$0.357876\pi$
$380$		−	8.29442i		−	0.425495i
$381$		0			0
$382$	10.2997			0.526976
$383$	20.1586			1.03006			0.515029	−	0.857173i	$-0.327781\pi$
$383$	20.1586			1.03006			0.515029	+	0.857173i	$0.327781\pi$
$384$		0			0
$385$	27.7253	+	8.94922i	1.41301	+	0.456094i
$386$			13.3100i			0.677464i
$387$		0			0
$388$			9.92501i			0.503866i
$389$			34.6042i			1.75450i	0.480033	+	0.877251i	$0.340625\pi$
$389$			34.6042i			1.75450i	−0.480033	+	0.877251i	$0.659375\pi$
$390$		0			0
$391$		−	30.0168i		−	1.51802i
$392$	−4.09255	+	5.67900i	−0.206705	+	0.286833i
$393$		0			0
$394$	−18.2881			−0.921342
$395$	−18.0633			−0.908864
$396$		0			0
$397$			28.8882i			1.44986i	0.688823	+	0.724929i	$0.258128\pi$
$397$			28.8882i			1.44986i	−0.688823	+	0.724929i	$0.741872\pi$
$398$	−8.30887			−0.416486
$399$		0			0
$400$	−1.69344			−0.0846721
$401$		−	18.5186i		−	0.924774i	−0.886678	−	0.462387i	$-0.846993\pi$
$401$		−	18.5186i		−	0.924774i	0.886678	−	0.462387i	$-0.153007\pi$
$402$		0			0
$403$	−18.5950			−0.926283
$404$	−16.0644			−0.799235
$405$		0			0
$406$	0.812712	−	2.51784i	0.0403342	−	0.124958i
$407$			39.8895i			1.97725i
$408$		0			0
$409$			22.0226i			1.08895i	0.838777	+	0.544475i	$0.183271\pi$
$409$			22.0226i			1.08895i	−0.838777	+	0.544475i	$0.816729\pi$
$410$		−	15.3953i		−	0.760321i
$411$		0			0
$412$		−	0.332202i		−	0.0163664i
$413$	2.77718	−	8.60389i	0.136656	−	0.423370i
$414$		0			0
$415$	−23.6449			−1.16068
$416$	2.51029			0.123077
$417$		0			0
$418$			27.6222i			1.35105i
$419$	35.3086			1.72494			0.862468	−	0.506111i	$-0.168917\pi$
$419$	35.3086			1.72494			0.862468	+	0.506111i	$0.168917\pi$
$420$		0			0
$421$	5.21246			0.254040			0.127020	−	0.991900i	$-0.459459\pi$
$421$	5.21246			0.254040			0.127020	+	0.991900i	$0.459459\pi$
$422$		−	23.4970i		−	1.14382i
$423$		0			0
$424$	0.309506			0.0150309
$425$	−7.62066			−0.369656
$426$		0			0
$427$	25.3113	+	8.17004i	1.22490	+	0.395376i
$428$			15.3799i			0.743415i
$429$		0			0
$430$		−	10.8924i		−	0.525280i
$431$		−	4.01661i		−	0.193474i	−0.995310	−	0.0967368i	$-0.969160\pi$
$431$		−	4.01661i		−	0.193474i	0.995310	−	0.0967368i	$-0.0308405\pi$
$432$		0			0
$433$		−	2.36693i		−	0.113748i	−0.998381	−	0.0568738i	$-0.981887\pi$
$433$		−	2.36693i		−	0.113748i	0.998381	−	0.0568738i	$-0.0181133\pi$
$434$	−6.02016	+	18.6509i	−0.288977	+	0.895270i
$435$		0			0
$436$	−5.70082			−0.273020
$437$	30.4257			1.45546
$438$		0			0
$439$		−	14.4641i		−	0.690334i	−0.938541	−	0.345167i	$-0.887822\pi$
$439$		−	14.4641i		−	0.690334i	0.938541	−	0.345167i	$-0.112178\pi$
$440$	−11.0115			−0.524955
$441$		0			0
$442$	11.2966			0.537323
$443$			7.09574i			0.337129i	0.985691	+	0.168564i	$0.0539131\pi$
$443$			7.09574i			0.337129i	−0.985691	+	0.168564i	$0.946087\pi$
$444$		0			0
$445$	19.0495			0.903031
$446$	8.34460			0.395128
$447$		0			0
$448$	0.812712	−	2.51784i	0.0383970	−	0.118957i
$449$			22.9034i			1.08088i	0.841383	+	0.540440i	$0.181742\pi$
$449$			22.9034i			1.08088i	−0.841383	+	0.540440i	$0.818258\pi$
$450$		0			0
$451$			51.2698i			2.41420i
$452$			1.14651i			0.0539274i
$453$		0			0
$454$			13.0567i			0.612779i
$455$	11.4932	+	3.70979i	0.538809	+	0.173918i
$456$		0			0
$457$	−8.71177			−0.407519			−0.203760	−	0.979021i	$-0.565316\pi$
$457$	−8.71177			−0.407519			−0.203760	+	0.979021i	$0.565316\pi$
$458$	10.5442			0.492697
$459$		0			0
$460$			12.1292i			0.565525i
$461$	25.0888			1.16850			0.584252	−	0.811572i	$-0.301388\pi$
$461$	25.0888			1.16850			0.584252	+	0.811572i	$0.301388\pi$
$462$		0			0
$463$	−18.0716			−0.839860			−0.419930	−	0.907557i	$-0.637945\pi$
$463$	−18.0716			−0.839860			−0.419930	+	0.907557i	$0.637945\pi$
$464$			1.00000i			0.0464238i
$465$		0			0
$466$	15.6046			0.722871
$467$	−26.3517			−1.21941			−0.609706	−	0.792627i	$-0.708713\pi$
$467$	−26.3517			−1.21941			−0.609706	+	0.792627i	$0.708713\pi$
$468$		0			0
$469$	−12.3562	+	38.2803i	−0.570556	+	1.76762i
$470$		−	9.82116i		−	0.453016i
$471$		0			0
$472$			3.41718i			0.157288i
$473$			36.2741i			1.66789i
$474$		0			0
$475$		−	7.72447i		−	0.354423i
$476$	3.65728	−	11.3305i	0.167631	−	0.519333i
$477$		0			0
$478$	9.94549			0.454896
$479$	15.0834			0.689177			0.344588	−	0.938754i	$-0.388018\pi$
$479$	15.0834			0.689177			0.344588	+	0.938754i	$0.388018\pi$
$480$		0			0
$481$			16.5357i			0.753963i
$482$	−25.6474			−1.16821
$483$		0			0
$484$	25.6708			1.16686
$485$		−	18.0476i		−	0.819498i
$486$		0			0
$487$	−13.8291			−0.626658			−0.313329	−	0.949645i	$-0.601444\pi$
$487$	−13.8291			−0.626658			−0.313329	+	0.949645i	$0.601444\pi$
$488$	−10.0528			−0.455069
$489$		0			0
$490$	7.44187	−	10.3267i	0.336189	−	0.466511i
$491$		−	6.77918i		−	0.305940i	−0.988231	−	0.152970i	$-0.951116\pi$
$491$		−	6.77918i		−	0.305940i	0.988231	−	0.152970i	$-0.0488838\pi$
$492$		0			0
$493$			4.50010i			0.202674i
$494$			11.4505i			0.515180i
$495$		0			0
$496$		−	7.40750i		−	0.332606i
$497$	−6.57134	−	2.12111i	−0.294765	−	0.0951448i
$498$		0			0
$499$	−20.2540			−0.906695			−0.453347	−	0.891334i	$-0.649770\pi$
$499$	−20.2540			−0.906695			−0.453347	+	0.891334i	$0.649770\pi$
$500$	12.1713			0.544318
$501$		0			0
$502$			0.195266i			0.00871515i
$503$	−18.4959			−0.824693			−0.412346	−	0.911027i	$-0.635291\pi$
$503$	−18.4959			−0.824693			−0.412346	+	0.911027i	$0.635291\pi$
$504$		0			0
$505$	29.2114			1.29989
$506$		−	40.3927i		−	1.79568i
$507$		0			0
$508$	9.94342			0.441168
$509$	−8.26194			−0.366204			−0.183102	−	0.983094i	$-0.558614\pi$
$509$	−8.26194			−0.366204			−0.183102	+	0.983094i	$0.558614\pi$
$510$		0			0
$511$	15.5363	+	5.01483i	0.687285	+	0.221843i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			3.37691i			0.148949i
$515$			0.604073i			0.0266187i
$516$		0			0
$517$			32.7066i			1.43843i
$518$	16.5854	+	5.35346i	0.728720	+	0.235218i
$519$		0			0
$520$	−4.56470			−0.200176
$521$	9.44983			0.414005			0.207002	−	0.978340i	$-0.433629\pi$
$521$	9.44983			0.414005			0.207002	+	0.978340i	$0.433629\pi$
$522$		0			0
$523$			27.7171i			1.21198i	0.795470	+	0.605992i	$0.207224\pi$
$523$			27.7171i			1.21198i	−0.795470	+	0.605992i	$0.792776\pi$
$524$	−3.35409			−0.146524
$525$		0			0
$526$	2.02058			0.0881016
$527$		−	33.3345i		−	1.45207i
$528$		0			0
$529$	−21.4924			−0.934450
$530$	−0.562804			−0.0244466
$531$		0			0
$532$	11.4849	+	3.70710i	0.497932	+	0.160723i
$533$			21.2533i			0.920581i
$534$		0			0
$535$		−	27.9667i		−	1.20911i
$536$		−	15.2036i		−	0.656698i
$537$		0			0
$538$			18.4193i			0.794113i
$539$	−24.7830	+	34.3900i	−1.06748	+	1.48128i
$540$		0			0
$541$	−6.11681			−0.262982			−0.131491	−	0.991317i	$-0.541977\pi$
$541$	−6.11681			−0.262982			−0.131491	+	0.991317i	$0.541977\pi$
$542$	−24.5042			−1.05255
$543$		0			0
$544$			4.50010i			0.192940i
$545$	10.3663			0.444045
$546$		0			0
$547$	−45.0722			−1.92715			−0.963573	−	0.267445i	$-0.913821\pi$
$547$	−45.0722			−1.92715			−0.963573	+	0.267445i	$0.913821\pi$
$548$		−	14.5958i		−	0.623501i
$549$		0			0
$550$	−10.2549			−0.437270
$551$	−4.56140			−0.194322
$552$		0			0
$553$	8.07321	−	25.0113i	0.343308	−	1.06359i
$554$			12.1782i			0.517402i
$555$		0			0
$556$			5.95257i			0.252445i
$557$			3.74211i			0.158558i	0.996852	+	0.0792792i	$0.0252619\pi$
$557$			3.74211i			0.158558i	−0.996852	+	0.0792792i	$0.974738\pi$
$558$		0			0
$559$			15.0370i			0.635998i
$560$	−1.47783	+	4.57842i	−0.0624497	+	0.193474i
$561$		0			0
$562$	−11.5814			−0.488532
$563$	−13.8494			−0.583683			−0.291841	−	0.956467i	$-0.594268\pi$
$563$	−13.8494			−0.583683			−0.291841	+	0.956467i	$0.594268\pi$
$564$		0			0
$565$		−	2.08481i		−	0.0877087i
$566$	27.1540			1.14137
$567$		0			0
$568$	2.60992			0.109510
$569$			0.00674482i			0.000282758i	1.00000		0.000141379i	$4.50023e-5\pi$
$569$			0.00674482i			0.000282758i	−1.00000		0.000141379i	$0.999955\pi$
$570$		0			0
$571$	−36.7110			−1.53631			−0.768155	−	0.640264i	$-0.778825\pi$
$571$	−36.7110			−1.53631			−0.768155	+	0.640264i	$0.778825\pi$
$572$	15.2015			0.635605
$573$		0			0
$574$	21.3171	+	6.88078i	0.889759	+	0.287198i
$575$			11.2957i			0.471063i
$576$		0			0
$577$		−	21.5738i		−	0.898127i	−0.893500	−	0.449064i	$-0.851758\pi$
$577$		−	21.5738i		−	0.898127i	0.893500	−	0.449064i	$-0.148242\pi$
$578$			3.25087i			0.135218i
$579$		0			0
$580$		−	1.81839i		−	0.0755047i
$581$	10.5678	−	32.7399i	0.438428	−	1.35828i
$582$		0			0
$583$	1.87426			0.0776239
$584$	−6.17049			−0.255337
$585$		0			0
$586$		−	31.7235i		−	1.31049i
$587$	9.51169			0.392590			0.196295	−	0.980545i	$-0.437109\pi$
$587$	9.51169			0.392590			0.196295	+	0.980545i	$0.437109\pi$
$588$		0			0
$589$	33.7886			1.39223
$590$		−	6.21377i		−	0.255817i
$591$		0			0
$592$	−6.58716			−0.270731
$593$	−28.5199			−1.17117			−0.585585	−	0.810611i	$-0.699135\pi$
$593$	−28.5199			−1.17117			−0.585585	+	0.810611i	$0.699135\pi$
$594$		0			0
$595$	−6.65038	+	20.6033i	−0.272639	+	0.844654i
$596$		−	15.2344i		−	0.624025i
$597$		0			0
$598$		−	16.7443i		−	0.684726i
$599$		−	41.3490i		−	1.68948i	−0.535180	−	0.844738i	$-0.679756\pi$
$599$		−	41.3490i		−	1.68948i	0.535180	−	0.844738i	$-0.320244\pi$
$600$		0			0
$601$			35.2793i			1.43907i	0.694455	+	0.719536i	$0.255646\pi$
$601$			35.2793i			1.43907i	−0.694455	+	0.719536i	$0.744354\pi$
$602$	15.0822	+	4.86825i	0.614704	+	0.198415i
$603$		0			0
$604$	5.39307			0.219441
$605$	−46.6797			−1.89780
$606$		0			0
$607$		−	20.6794i		−	0.839353i	−0.907674	−	0.419676i	$-0.862144\pi$
$607$		−	20.6794i		−	0.839353i	0.907674	−	0.419676i	$-0.137856\pi$
$608$	−4.56140			−0.184989
$609$		0			0
$610$	18.2800			0.740135
$611$			13.5581i			0.548503i
$612$		0			0
$613$	−24.9741			−1.00869			−0.504347	−	0.863501i	$-0.668267\pi$
$613$	−24.9741			−1.00869			−0.504347	+	0.863501i	$0.668267\pi$
$614$	2.72493			0.109969
$615$		0			0
$616$	4.92149	−	15.2471i	0.198293	−	0.614324i
$617$		−	9.90881i		−	0.398914i	−0.979907	−	0.199457i	$-0.936082\pi$
$617$		−	9.90881i		−	0.398914i	0.979907	−	0.199457i	$-0.0639178\pi$
$618$		0			0
$619$			23.4963i			0.944396i	0.881492	+	0.472198i	$0.156539\pi$
$619$			23.4963i			0.944396i	−0.881492	+	0.472198i	$0.843461\pi$
$620$			13.4698i			0.540958i
$621$		0			0
$622$		−	7.74611i		−	0.310591i
$623$	−8.51395	+	26.3768i	−0.341104	+	1.05676i
$624$		0			0
$625$	−13.6650			−0.546602
$626$	26.8167			1.07181
$627$		0			0
$628$		−	5.90694i		−	0.235712i
$629$	−29.6429			−1.18194
$630$		0			0
$631$	26.8846			1.07026			0.535129	−	0.844770i	$-0.320263\pi$
$631$	26.8846			1.07026			0.535129	+	0.844770i	$0.320263\pi$
$632$			9.93366i			0.395140i
$633$		0			0
$634$	15.0707			0.598536
$635$	−18.0811			−0.717525
$636$		0			0
$637$	−10.2735	+	14.2560i	−0.407051	+	0.564842i
$638$			6.05565i			0.239745i
$639$		0			0
$640$		−	1.81839i		−	0.0718783i
$641$			31.7461i			1.25389i	0.779062	+	0.626947i	$0.215696\pi$
$641$			31.7461i			1.25389i	−0.779062	+	0.626947i	$0.784304\pi$
$642$		0			0
$643$		−	13.1293i		−	0.517768i	−0.965908	−	0.258884i	$-0.916645\pi$
$643$		−	13.1293i		−	0.517768i	0.965908	−	0.258884i	$-0.0833547\pi$
$644$	−16.7946	−	5.42100i	−0.661801	−	0.213617i
$645$		0			0
$646$	−20.5267			−0.807614
$647$	−1.96318			−0.0771805			−0.0385902	−	0.999255i	$-0.512287\pi$
$647$	−1.96318			−0.0771805			−0.0385902	+	0.999255i	$0.512287\pi$
$648$		0			0
$649$			20.6932i			0.812280i
$650$	−4.25104			−0.166739
$651$		0			0
$652$	−10.6581			−0.417402
$653$			2.19506i			0.0858993i	0.999077	+	0.0429497i	$0.0136755\pi$
$653$			2.19506i			0.0858993i	−0.999077	+	0.0429497i	$0.986324\pi$
$654$		0			0
$655$	6.09906			0.238310
$656$	−8.46644			−0.330559
$657$		0			0
$658$	13.5988	+	4.38946i	0.530138	+	0.171119i
$659$			9.97577i			0.388601i	0.980942	+	0.194300i	$0.0622436\pi$
$659$			9.97577i			0.388601i	−0.980942	+	0.194300i	$0.937756\pi$
$660$		0			0
$661$		−	11.9875i		−	0.466259i	−0.972446	−	0.233130i	$-0.925103\pi$
$661$		−	11.9875i		−	0.466259i	0.972446	−	0.233130i	$-0.0748966\pi$
$662$			1.24808i			0.0485078i
$663$		0			0
$664$			13.0032i			0.504621i
$665$	−20.8840	−	6.74098i	−0.809847	−	0.261404i
$666$		0			0
$667$	6.67026			0.258273
$668$	−10.7289			−0.415112
$669$		0			0
$670$			27.6462i			1.06807i
$671$	−60.8763			−2.35010
$672$		0			0
$673$	29.2508			1.12754			0.563769	−	0.825933i	$-0.309351\pi$
$673$	29.2508			1.12754			0.563769	+	0.825933i	$0.309351\pi$
$674$			13.1359i			0.505977i
$675$		0			0
$676$	−6.69842			−0.257632
$677$	−22.2483			−0.855071			−0.427535	−	0.903999i	$-0.640618\pi$
$677$	−22.2483			−0.855071			−0.427535	+	0.903999i	$0.640618\pi$
$678$		0			0
$679$	24.9896	+	8.06617i	0.959011	+	0.309551i
$680$		−	8.18295i		−	0.313802i
$681$		0			0
$682$		−	44.8572i		−	1.71767i
$683$		−	29.1501i		−	1.11540i	−0.830043	−	0.557700i	$-0.811684\pi$
$683$		−	29.1501i		−	1.11540i	0.830043	−	0.557700i	$-0.188316\pi$
$684$		0			0
$685$			26.5409i			1.01408i
$686$	10.9727	+	14.9198i	0.418941	+	0.569639i
$687$		0			0
$688$	−5.99014			−0.228372
$689$	0.776951			0.0295995
$690$		0			0
$691$			12.4910i			0.475181i	0.971365	+	0.237591i	$0.0763577\pi$
$691$			12.4910i			0.475181i	−0.971365	+	0.237591i	$0.923642\pi$
$692$	−23.0797			−0.877360
$693$		0			0
$694$	−29.5484			−1.12164
$695$		−	10.8241i		−	0.410582i
$696$		0			0
$697$	−38.0998			−1.44313
$698$	19.2752			0.729577
$699$		0			0
$700$	−1.37628	+	4.26381i	−0.0520185	+	0.161157i
$701$			39.2074i			1.48084i	0.672142	+	0.740422i	$0.265374\pi$
$701$			39.2074i			1.48084i	−0.672142	+	0.740422i	$0.734626\pi$
$702$		0			0
$703$		−	30.0467i		−	1.13323i
$704$			6.05565i			0.228231i
$705$		0			0
$706$		−	8.50869i		−	0.320229i
$707$	−13.0557	+	40.4476i	−0.491012	+	1.52119i
$708$		0			0
$709$	21.8254			0.819672			0.409836	−	0.912159i	$-0.365586\pi$
$709$	21.8254			0.819672			0.409836	+	0.912159i	$0.365586\pi$
$710$	−4.74586			−0.178109
$711$		0			0
$712$		−	10.4760i		−	0.392604i
$713$	−49.4099			−1.85042
$714$		0			0
$715$	−27.6422			−1.03376
$716$			16.6281i			0.621421i
$717$		0			0
$718$	7.94081			0.296348
$719$	23.6604			0.882385			0.441193	−	0.897412i	$-0.354556\pi$
$719$	23.6604			0.882385			0.441193	+	0.897412i	$0.354556\pi$
$720$		0			0
$721$	−0.836429	−	0.269984i	−0.0311502	−	0.0100547i
$722$		−	1.80637i		−	0.0672262i
$723$		0			0
$724$			5.27347i			0.195987i
$725$		−	1.69344i		−	0.0628929i
$726$		0			0
$727$		−	31.4492i		−	1.16638i	−0.812334	−	0.583192i	$-0.801803\pi$
$727$		−	31.4492i		−	1.16638i	0.812334	−	0.583192i	$-0.198197\pi$
$728$	2.04015	−	6.32051i	0.0756128	−	0.234254i
$729$		0			0
$730$	11.2204			0.415285
$731$	−26.9562			−0.997011
$732$		0			0
$733$		−	24.5013i		−	0.904976i	−0.891770	−	0.452488i	$-0.850537\pi$
$733$		−	24.5013i		−	0.904976i	0.891770	−	0.452488i	$-0.149463\pi$
$734$	36.3595			1.34205
$735$		0			0
$736$	6.67026			0.245869
$737$		−	92.0679i		−	3.39136i
$738$		0			0
$739$	10.2134			0.375707			0.187854	−	0.982197i	$-0.439847\pi$
$739$	10.2134			0.375707			0.187854	+	0.982197i	$0.439847\pi$
$740$	11.9781			0.440322
$741$		0			0
$742$	0.251539	−	0.779285i	0.00923429	−	0.0286085i
$743$		−	5.32104i		−	0.195210i	−0.995225	−	0.0976050i	$-0.968882\pi$
$743$		−	5.32104i		−	0.195210i	0.995225	−	0.0976050i	$-0.0311182\pi$
$744$		0			0
$745$			27.7021i			1.01493i
$746$			31.4672i			1.15210i
$747$		0			0
$748$			27.2510i			0.996395i
$749$	38.7240	+	12.4994i	1.41495	+	0.456719i
$750$		0			0
$751$	−37.2309			−1.35857			−0.679287	−	0.733873i	$-0.737711\pi$
$751$	−37.2309			−1.35857			−0.679287	+	0.733873i	$0.737711\pi$
$752$	−5.40101			−0.196954
$753$		0			0
$754$			2.51029i			0.0914195i
$755$	−9.80672			−0.356903
$756$		0			0
$757$	−41.0652			−1.49254			−0.746270	−	0.665644i	$-0.768157\pi$
$757$	−41.0652			−1.49254			−0.746270	+	0.665644i	$0.768157\pi$
$758$			16.8128i			0.610670i
$759$		0			0
$760$	8.29442			0.300870
$761$	39.4965			1.43175			0.715874	−	0.698230i	$-0.246029\pi$
$761$	39.4965			1.43175			0.715874	+	0.698230i	$0.246029\pi$
$762$		0			0
$763$	−4.63312	+	14.3537i	−0.167730	+	0.519640i
$764$			10.2997i			0.372629i
$765$		0			0
$766$			20.1586i			0.728361i
$767$			8.57812i			0.309738i
$768$		0			0
$769$			30.2630i			1.09131i	0.838009	+	0.545656i	$0.183719\pi$
$769$			30.2630i			1.09131i	−0.838009	+	0.545656i	$0.816281\pi$
$770$	−8.94922	+	27.7253i	−0.322507	+	0.999150i
$771$		0			0
$772$	−13.3100			−0.479039
$773$	−54.9880			−1.97778			−0.988890	−	0.148647i	$-0.952508\pi$
$773$	−54.9880			−1.97778			−0.988890	+	0.148647i	$0.952508\pi$
$774$		0			0
$775$			12.5442i			0.450600i
$776$	−9.92501			−0.356287
$777$		0			0
$778$	−34.6042			−1.24062
$779$		−	38.6188i		−	1.38366i
$780$		0			0
$781$	15.8047			0.565538
$782$	30.0168			1.07340
$783$		0			0
$784$	−5.67900	−	4.09255i	−0.202821	−	0.146163i
$785$			10.7411i			0.383368i
$786$		0			0
$787$			21.1533i			0.754034i	0.926206	+	0.377017i	$0.123050\pi$
$787$			21.1533i			0.754034i	−0.926206	+	0.377017i	$0.876950\pi$
$788$		−	18.2881i		−	0.651487i
$789$		0			0
$790$		−	18.0633i		−	0.642664i
$791$	2.88673	+	0.931784i	0.102640	+	0.0331304i
$792$		0			0
$793$	−25.2355			−0.896140
$794$	−28.8882			−1.02520
$795$		0			0
$796$		−	8.30887i		−	0.294500i
$797$	42.6655			1.51129			0.755645	−	0.654981i	$-0.227324\pi$
$797$	42.6655			1.51129			0.755645	+	0.654981i	$0.227324\pi$
$798$		0			0
$799$	−24.3051			−0.859851
$800$		−	1.69344i		−	0.0598722i
$801$		0			0
$802$	18.5186			0.653914
$803$	−37.3663			−1.31863
$804$		0			0
$805$	30.5392	+	9.85751i	1.07637	+	0.347432i
$806$		−	18.5950i		−	0.654981i
$807$		0			0
$808$		−	16.0644i		−	0.565144i
$809$			10.9426i			0.384722i	0.981324	+	0.192361i	$0.0616145\pi$
$809$			10.9426i			0.384722i	−0.981324	+	0.192361i	$0.938386\pi$
$810$		0			0
$811$			38.1748i			1.34050i	0.742137	+	0.670248i	$0.233812\pi$
$811$			38.1748i			1.34050i	−0.742137	+	0.670248i	$0.766188\pi$
$812$	2.51784	+	0.812712i	0.0883587	+	0.0285206i
$813$		0			0
$814$	−39.8895			−1.39813
$815$	19.3806			0.678872
$816$		0			0
$817$		−	27.3234i		−	0.955925i
$818$	−22.0226			−0.770004
$819$		0			0
$820$	15.3953			0.537628
$821$			3.40618i			0.118876i	0.998232	+	0.0594382i	$0.0189309\pi$
$821$			3.40618i			0.118876i	−0.998232	+	0.0594382i	$0.981069\pi$
$822$		0			0
$823$	−1.26763			−0.0441867			−0.0220934	−	0.999756i	$-0.507033\pi$
$823$	−1.26763			−0.0441867			−0.0220934	+	0.999756i	$0.507033\pi$
$824$	0.332202			0.0115728
$825$		0			0
$826$	8.60389	+	2.77718i	0.299368	+	0.0966304i
$827$			35.1059i			1.22075i	0.792112	+	0.610376i	$0.208982\pi$
$827$			35.1059i			1.22075i	−0.792112	+	0.610376i	$0.791018\pi$
$828$		0			0
$829$		−	43.1802i		−	1.49971i	−0.661601	−	0.749856i	$-0.730123\pi$
$829$		−	43.1802i		−	1.49971i	0.661601	−	0.749856i	$-0.269877\pi$
$830$		−	23.6449i		−	0.820727i
$831$		0			0
$832$			2.51029i			0.0870288i
$833$	−25.5560	−	18.4169i	−0.885465	−	0.638107i
$834$		0			0
$835$	19.5093			0.675146
$836$	−27.6222			−0.955334
$837$		0			0
$838$			35.3086i			1.21971i
$839$	−21.1455			−0.730023			−0.365012	−	0.931003i	$-0.618935\pi$
$839$	−21.1455			−0.730023			−0.365012	+	0.931003i	$0.618935\pi$
$840$		0			0
$841$	−1.00000			−0.0344828
$842$			5.21246i			0.179633i
$843$		0			0
$844$	23.4970			0.808799
$845$	12.1804			0.419017
$846$		0			0
$847$	20.8630	−	64.6350i	0.716861	−	2.22088i
$848$			0.309506i			0.0106285i
$849$		0			0
$850$		−	7.62066i		−	0.261386i
$851$			43.9381i			1.50618i
$852$		0			0
$853$		−	23.7703i		−	0.813880i	−0.913455	−	0.406940i	$-0.866596\pi$
$853$		−	23.7703i		−	0.813880i	0.913455	−	0.406940i	$-0.133404\pi$
$854$	−8.17004	+	25.3113i	−0.279573	+	0.866136i
$855$		0			0
$856$	−15.3799			−0.525674
$857$	6.61694			0.226030			0.113015	−	0.993593i	$-0.463949\pi$
$857$	6.61694			0.226030			0.113015	+	0.993593i	$0.463949\pi$
$858$		0			0
$859$			7.17360i			0.244760i	0.992483	+	0.122380i	$0.0390527\pi$
$859$			7.17360i			0.244760i	−0.992483	+	0.122380i	$0.960947\pi$
$860$	10.8924			0.371429
$861$		0			0
$862$	4.01661			0.136806
$863$			47.7103i			1.62408i	0.583604	+	0.812038i	$0.301642\pi$
$863$			47.7103i			1.62408i	−0.583604	+	0.812038i	$0.698358\pi$
$864$		0			0
$865$	41.9681			1.42696
$866$	2.36693			0.0804317
$867$		0			0
$868$	−18.6509	−	6.02016i	−0.633052	−	0.204338i
$869$			60.1547i			2.04061i
$870$		0			0
$871$		−	38.1656i		−	1.29319i
$872$		−	5.70082i		−	0.193054i
$873$		0			0
$874$			30.4257i			1.02917i
$875$	9.89177	−	30.6454i	0.334403	−	1.03600i
$876$		0			0
$877$	−30.7609			−1.03872			−0.519361	−	0.854555i	$-0.673830\pi$
$877$	−30.7609			−1.03872			−0.519361	+	0.854555i	$0.673830\pi$
$878$	14.4641			0.488140
$879$		0			0
$880$		−	11.0115i		−	0.371199i
$881$	5.28813			0.178162			0.0890808	−	0.996024i	$-0.471607\pi$
$881$	5.28813			0.178162			0.0890808	+	0.996024i	$0.471607\pi$
$882$		0			0
$883$	25.7628			0.866988			0.433494	−	0.901156i	$-0.357281\pi$
$883$	25.7628			0.866988			0.433494	+	0.901156i	$0.357281\pi$
$884$			11.2966i			0.379945i
$885$		0			0
$886$	−7.09574			−0.238386
$887$	19.9871			0.671101			0.335551	−	0.942022i	$-0.391078\pi$
$887$	19.9871			0.671101			0.335551	+	0.942022i	$0.391078\pi$
$888$		0			0
$889$	8.08113	−	25.0359i	0.271033	−	0.839677i
$890$			19.0495i			0.638539i
$891$		0			0
$892$			8.34460i			0.279398i
$893$		−	24.6361i		−	0.824417i
$894$		0			0
$895$		−	30.2364i		−	1.01069i
$896$	2.51784	+	0.812712i	0.0841150	+	0.0271508i
$897$		0			0
$898$	−22.9034			−0.764297
$899$	7.40750			0.247054
$900$		0			0
$901$			1.39281i			0.0464011i
$902$	−51.2698			−1.70710
$903$		0			0
$904$	−1.14651			−0.0381324
$905$		−	9.58924i		−	0.318757i
$906$		0			0
$907$	−46.1269			−1.53162			−0.765809	−	0.643068i	$-0.777661\pi$
$907$	−46.1269			−1.53162			−0.765809	+	0.643068i	$0.777661\pi$
$908$	−13.0567			−0.433300
$909$		0			0
$910$	−3.70979	+	11.4932i	−0.122978	+	0.380995i
$911$		−	35.7417i		−	1.18418i	−0.805873	−	0.592088i	$-0.798304\pi$
$911$		−	35.7417i		−	1.18418i	0.805873	−	0.592088i	$-0.201696\pi$
$912$		0			0
$913$			78.7427i			2.60600i
$914$		−	8.71177i		−	0.288160i
$915$		0			0
$916$			10.5442i			0.348390i
$917$	−2.72591	+	8.44505i	−0.0900174	+	0.278880i
$918$		0			0
$919$	36.5080			1.20429			0.602145	−	0.798387i	$-0.294313\pi$
$919$	36.5080			1.20429			0.602145	+	0.798387i	$0.294313\pi$
$920$	−12.1292			−0.399887
$921$		0			0
$922$			25.0888i			0.826257i
$923$	6.55166			0.215650
$924$		0			0
$925$	11.1550			0.366773
$926$		−	18.0716i		−	0.593870i
$927$		0			0
$928$	−1.00000			−0.0328266
$929$	18.8445			0.618269			0.309135	−	0.951018i	$-0.399961\pi$
$929$	18.8445			0.618269			0.309135	+	0.951018i	$0.399961\pi$
$930$		0			0
$931$	18.6678	−	25.9042i	0.611811	−	0.848975i
$932$			15.6046i			0.511147i
$933$		0			0
$934$		−	26.3517i		−	0.862255i
$935$		−	49.5530i		−	1.62056i
$936$		0			0
$937$			37.0437i			1.21017i	0.796162	+	0.605083i	$0.206860\pi$
$937$			37.0437i			1.21017i	−0.796162	+	0.605083i	$0.793140\pi$
$938$	−38.2803	−	12.3562i	−1.24990	−	0.403444i
$939$		0			0
$940$	9.82116			0.320331
$941$	31.4026			1.02369			0.511847	−	0.859077i	$-0.328961\pi$
$941$	31.4026			1.02369			0.511847	+	0.859077i	$0.328961\pi$
$942$		0			0
$943$			56.4733i			1.83903i
$944$	−3.41718			−0.111220
$945$		0			0
$946$	−36.2741			−1.17937
$947$		−	13.9045i		−	0.451837i	−0.974146	−	0.225919i	$-0.927462\pi$
$947$		−	13.9045i		−	0.451837i	0.974146	−	0.225919i	$-0.0725383\pi$
$948$		0			0
$949$	−15.4898			−0.502819
$950$	7.72447			0.250615
$951$		0			0
$952$	11.3305	+	3.65728i	0.367224	+	0.118533i
$953$			51.0716i			1.65437i	0.561928	+	0.827186i	$0.310060\pi$
$953$			51.0716i			1.65437i	−0.561928	+	0.827186i	$0.689940\pi$
$954$		0			0
$955$		−	18.7288i		−	0.606051i
$956$			9.94549i			0.321660i
$957$		0			0
$958$			15.0834i			0.487322i
$959$	−36.7498	−	11.8622i	−1.18671	−	0.383050i
$960$		0			0
$961$	−23.8710			−0.770033
$962$	−16.5357			−0.533133
$963$		0			0
$964$		−	25.6474i		−	0.826049i
$965$	24.2029			0.779119
$966$		0			0
$967$	−18.5529			−0.596620			−0.298310	−	0.954469i	$-0.596423\pi$
$967$	−18.5529			−0.596620			−0.298310	+	0.954469i	$0.596423\pi$
$968$			25.6708i			0.825092i
$969$		0			0
$970$	18.0476			0.579473
$971$	−23.6943			−0.760388			−0.380194	−	0.924907i	$-0.624143\pi$
$971$	−23.6943			−0.760388			−0.380194	+	0.924907i	$0.624143\pi$
$972$		0			0
$973$	14.9876	+	4.83772i	0.480480	+	0.155090i
$974$		−	13.8291i		−	0.443114i
$975$		0			0
$976$		−	10.0528i		−	0.321783i
$977$		−	46.6304i		−	1.49184i	−0.666036	−	0.745920i	$-0.732010\pi$
$977$		−	46.6304i		−	1.49184i	0.666036	−	0.745920i	$-0.267990\pi$
$978$		0			0
$979$		−	63.4388i		−	2.02751i
$980$	10.3267	+	7.44187i	0.329873	+	0.237722i
$981$		0			0
$982$	6.77918			0.216332
$983$	−1.23148			−0.0392781			−0.0196390	−	0.999807i	$-0.506252\pi$
$983$	−1.23148			−0.0392781			−0.0196390	+	0.999807i	$0.506252\pi$
$984$		0			0
$985$			33.2550i			1.05959i
$986$	−4.50010			−0.143312
$987$		0			0
$988$	−11.4505			−0.364288
$989$			39.9558i			1.27052i
$990$		0			0
$991$	2.29700			0.0729666			0.0364833	−	0.999334i	$-0.488384\pi$
$991$	2.29700			0.0729666			0.0364833	+	0.999334i	$0.488384\pi$
$992$	7.40750			0.235188
$993$		0			0
$994$	2.12111	−	6.57134i	0.0672775	−	0.208430i
$995$			15.1088i			0.478981i
$996$		0			0
$997$			11.4488i			0.362588i	0.983429	+	0.181294i	$0.0580286\pi$
$997$			11.4488i			0.362588i	−0.983429	+	0.181294i	$0.941971\pi$
$998$		−	20.2540i		−	0.641130i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3654.2.f.b.755.4	yes	40
3.2	odd	2	inner	3654.2.f.b.755.37	yes	40
7.6	odd	2	inner	3654.2.f.b.755.38	yes	40
21.20	even	2	inner	3654.2.f.b.755.3	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3654.2.f.b.755.3	✓	40	21.20	even	2	inner
3654.2.f.b.755.4	yes	40	1.1	even	1	trivial
3654.2.f.b.755.37	yes	40	3.2	odd	2	inner
3654.2.f.b.755.38	yes	40	7.6	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3654 = 2 \cdot 3^{2} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3654.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.81839 q^{5} +(-0.812712 + 2.51784i) q^{7} -1.00000i q^{8} +1.81839i q^{10} -6.05565i q^{11} -2.51029i q^{13} +(-2.51784 - 0.812712i) q^{14} +1.00000 q^{16} +4.50010 q^{17} +4.56140i q^{19} -1.81839 q^{20} +6.05565 q^{22} -6.67026i q^{23} -1.69344 q^{25} +2.51029 q^{26} +(0.812712 - 2.51784i) q^{28} +1.00000i q^{29} -7.40750i q^{31} +1.00000i q^{32} +4.50010i q^{34} +(-1.47783 + 4.57842i) q^{35} -6.58716 q^{37} -4.56140 q^{38} -1.81839i q^{40} -8.46644 q^{41} -5.99014 q^{43} +6.05565i q^{44} +6.67026 q^{46} -5.40101 q^{47} +(-5.67900 - 4.09255i) q^{49} -1.69344i q^{50} +2.51029i q^{52} +0.309506i q^{53} -11.0115i q^{55} +(2.51784 + 0.812712i) q^{56} -1.00000 q^{58} -3.41718 q^{59} -10.0528i q^{61} +7.40750 q^{62} -1.00000 q^{64} -4.56470i q^{65} +15.2036 q^{67} -4.50010 q^{68} +(-4.57842 - 1.47783i) q^{70} +2.60992i q^{71} -6.17049i q^{73} -6.58716i q^{74} -4.56140i q^{76} +(15.2471 + 4.92149i) q^{77} -9.93366 q^{79} +1.81839 q^{80} -8.46644i q^{82} -13.0032 q^{83} +8.18295 q^{85} -5.99014i q^{86} -6.05565 q^{88} +10.4760 q^{89} +(6.32051 + 2.04015i) q^{91} +6.67026i q^{92} -5.40101i q^{94} +8.29442i q^{95} -9.92501i q^{97} +(4.09255 - 5.67900i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{4} - 4 q^{7} + 40 q^{16} + 8 q^{22} + 64 q^{25} + 4 q^{28} - 48 q^{37} - 8 q^{43} + 8 q^{46} + 36 q^{49} - 40 q^{58} - 40 q^{64} + 24 q^{67} + 28 q^{70} - 144 q^{79} - 24 q^{85} - 8 q^{88}+O(q^{100}) \) 40 * q - 40 * q^4 - 4 * q^7 + 40 * q^16 + 8 * q^22 + 64 * q^25 + 4 * q^28 - 48 * q^37 - 8 * q^43 + 8 * q^46 + 36 * q^49 - 40 * q^58 - 40 * q^64 + 24 * q^67 + 28 * q^70 - 144 * q^79 - 24 * q^85 - 8 * q^88

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