LMFDB - Embedded newform 3654.2.f.b.755.17

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)

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")

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);

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			755.17
Character	$\chi$	$=$	3654.755
Dual form			3654.2.f.b.755.18

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} -2.53682 q^{5} +(2.41720 + 1.07570i) q^{7} +1.00000i q^{8} +2.53682i q^{10} -0.368727i q^{11} -4.14980i q^{13} +(1.07570 - 2.41720i) q^{14} +1.00000 q^{16} +1.90402 q^{17} -2.46622i q^{19} +2.53682 q^{20} -0.368727 q^{22} +5.83815i q^{23} +1.43544 q^{25} -4.14980 q^{26} +(-2.41720 - 1.07570i) q^{28} -1.00000i q^{29} +7.24557i q^{31} -1.00000i q^{32} -1.90402i q^{34} +(-6.13201 - 2.72885i) q^{35} +4.79505 q^{37} -2.46622 q^{38} -2.53682i q^{40} -5.25760 q^{41} +12.0588 q^{43} +0.368727i q^{44} +5.83815 q^{46} -7.92810 q^{47} +(4.68575 + 5.20036i) q^{49} -1.43544i q^{50} +4.14980i q^{52} -2.22615i q^{53} +0.935394i q^{55} +(-1.07570 + 2.41720i) q^{56} -1.00000 q^{58} -0.966966 q^{59} -12.9452i q^{61} +7.24557 q^{62} -1.00000 q^{64} +10.5273i q^{65} -4.76675 q^{67} -1.90402 q^{68} +(-2.72885 + 6.13201i) q^{70} +6.68361i q^{71} -1.99007i q^{73} -4.79505i q^{74} +2.46622i q^{76} +(0.396639 - 0.891290i) q^{77} -0.227826 q^{79} -2.53682 q^{80} +5.25760i q^{82} +8.72527 q^{83} -4.83015 q^{85} -12.0588i q^{86} +0.368727 q^{88} +4.99081 q^{89} +(4.46393 - 10.0309i) q^{91} -5.83815i q^{92} +7.92810i q^{94} +6.25634i q^{95} -9.46666i q^{97} +(5.20036 - 4.68575i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.53682			−1.13450			−0.567250	−	0.823546i	$-0.691993\pi$
$5$	−2.53682			−1.13450			−0.567250	+	0.823546i	$0.691993\pi$
$6$		0			0
$7$	2.41720	+	1.07570i	0.913617	+	0.406575i
$7$	2.41720	+	1.07570i	0.913617	+	0.406575i
$8$			1.00000i			0.353553i
$9$		0			0
$10$			2.53682i			0.802212i
$11$		−	0.368727i		−	0.111176i	−0.998454	−	0.0555878i	$-0.982297\pi$
$11$		−	0.368727i		−	0.111176i	0.998454	−	0.0555878i	$-0.0177033\pi$
$12$		0			0
$13$		−	4.14980i		−	1.15095i	−0.817820	−	0.575474i	$-0.804818\pi$
$13$		−	4.14980i		−	1.15095i	0.817820	−	0.575474i	$-0.195182\pi$
$14$	1.07570	−	2.41720i	0.287492	−	0.646025i
$15$		0			0
$16$	1.00000			0.250000
$17$	1.90402			0.461792			0.230896	−	0.972978i	$-0.425834\pi$
$17$	1.90402			0.461792			0.230896	+	0.972978i	$0.425834\pi$
$18$		0			0
$19$		−	2.46622i		−	0.565789i	−0.959151	−	0.282894i	$-0.908705\pi$
$19$		−	2.46622i		−	0.565789i	0.959151	−	0.282894i	$-0.0912946\pi$
$20$	2.53682			0.567250
$21$		0			0
$22$	−0.368727			−0.0786130
$23$			5.83815i			1.21734i	0.793424	+	0.608669i	$0.208296\pi$
$23$			5.83815i			1.21734i	−0.793424	+	0.608669i	$0.791704\pi$
$24$		0			0
$25$	1.43544			0.287089
$26$	−4.14980			−0.813843
$27$		0			0
$28$	−2.41720	−	1.07570i	−0.456809	−	0.203288i
$29$		−	1.00000i		−	0.185695i
$29$		−	1.00000i		−	0.185695i
$30$		0			0
$31$			7.24557i			1.30134i	0.759359	+	0.650672i	$0.225513\pi$
$31$			7.24557i			1.30134i	−0.759359	+	0.650672i	$0.774487\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	1.90402i		−	0.326536i
$35$	−6.13201	−	2.72885i	−1.03650	−	0.461259i
$36$		0			0
$37$	4.79505			0.788301			0.394151	−	0.919046i	$-0.371039\pi$
$37$	4.79505			0.788301			0.394151	+	0.919046i	$0.371039\pi$
$38$	−2.46622			−0.400073
$39$		0			0
$40$		−	2.53682i		−	0.401106i
$41$	−5.25760			−0.821099			−0.410550	−	0.911838i	$-0.634663\pi$
$41$	−5.25760			−0.821099			−0.410550	+	0.911838i	$0.634663\pi$
$42$		0			0
$43$	12.0588			1.83896			0.919478	−	0.393142i	$-0.128612\pi$
$43$	12.0588			1.83896			0.919478	+	0.393142i	$0.128612\pi$
$44$			0.368727i			0.0555878i
$45$		0			0
$46$	5.83815			0.860788
$47$	−7.92810			−1.15643			−0.578216	−	0.815883i	$-0.696251\pi$
$47$	−7.92810			−1.15643			−0.578216	+	0.815883i	$0.696251\pi$
$48$		0			0
$49$	4.68575	+	5.20036i	0.669393	+	0.742909i
$50$		−	1.43544i		−	0.203002i
$51$		0			0
$52$			4.14980i			0.575474i
$53$		−	2.22615i		−	0.305785i	−0.988243	−	0.152893i	$-0.951141\pi$
$53$		−	2.22615i		−	0.305785i	0.988243	−	0.152893i	$-0.0488588\pi$
$54$		0			0
$55$			0.935394i			0.126129i
$56$	−1.07570	+	2.41720i	−0.143746	+	0.323012i
$57$		0			0
$58$	−1.00000			−0.131306
$59$	−0.966966			−0.125888			−0.0629441	−	0.998017i	$-0.520049\pi$
$59$	−0.966966			−0.125888			−0.0629441	+	0.998017i	$0.520049\pi$
$60$		0			0
$61$		−	12.9452i		−	1.65746i	−0.559645	−	0.828732i	$-0.689063\pi$
$61$		−	12.9452i		−	1.65746i	0.559645	−	0.828732i	$-0.310937\pi$
$62$	7.24557			0.920189
$63$		0			0
$64$	−1.00000			−0.125000
$65$			10.5273i			1.30575i
$66$		0			0
$67$	−4.76675			−0.582351			−0.291175	−	0.956670i	$-0.594046\pi$
$67$	−4.76675			−0.582351			−0.291175	+	0.956670i	$0.594046\pi$
$68$	−1.90402			−0.230896
$69$		0			0
$70$	−2.72885	+	6.13201i	−0.326160	+	0.732915i
$71$			6.68361i			0.793199i	0.917992	+	0.396599i	$0.129810\pi$
$71$			6.68361i			0.793199i	−0.917992	+	0.396599i	$0.870190\pi$
$72$		0			0
$73$		−	1.99007i		−	0.232920i	−0.993195	−	0.116460i	$-0.962845\pi$
$73$		−	1.99007i		−	0.232920i	0.993195	−	0.116460i	$-0.0371546\pi$
$74$		−	4.79505i		−	0.557413i
$75$		0			0
$76$			2.46622i			0.282894i
$77$	0.396639	−	0.891290i	0.0452012	−	0.101572i
$78$		0			0
$79$	−0.227826			−0.0256325			−0.0128162	−	0.999918i	$-0.504080\pi$
$79$	−0.227826			−0.0256325			−0.0128162	+	0.999918i	$0.504080\pi$
$80$	−2.53682			−0.283625
$81$		0			0
$82$			5.25760i			0.580605i
$83$	8.72527			0.957723			0.478862	−	0.877890i	$-0.341050\pi$
$83$	8.72527			0.957723			0.478862	+	0.877890i	$0.341050\pi$
$84$		0			0
$85$	−4.83015			−0.523903
$86$		−	12.0588i		−	1.30034i
$87$		0			0
$88$	0.368727			0.0393065
$89$	4.99081			0.529025			0.264512	−	0.964382i	$-0.414789\pi$
$89$	4.99081			0.529025			0.264512	+	0.964382i	$0.414789\pi$
$90$		0			0
$91$	4.46393	−	10.0309i	0.467947	−	1.05153i
$92$		−	5.83815i		−	0.608669i
$93$		0			0
$94$			7.92810i			0.817722i
$95$			6.25634i			0.641887i
$96$		0			0
$97$		−	9.46666i		−	0.961194i	−0.876942	−	0.480597i	$-0.840420\pi$
$97$		−	9.46666i		−	0.961194i	0.876942	−	0.480597i	$-0.159580\pi$
$98$	5.20036	−	4.68575i	0.525316	−	0.473332i
$99$		0			0
$100$	−1.43544			−0.143544
$101$	11.4978			1.14408			0.572038	−	0.820227i	$-0.306153\pi$
$101$	11.4978			1.14408			0.572038	+	0.820227i	$0.306153\pi$
$102$		0			0
$103$		−	2.65737i		−	0.261838i	−0.991393	−	0.130919i	$-0.958207\pi$
$103$		−	2.65737i		−	0.261838i	0.991393	−	0.130919i	$-0.0417928\pi$
$104$	4.14980			0.406921
$105$		0			0
$106$	−2.22615			−0.216223
$107$		−	4.77936i		−	0.462038i	−0.972949	−	0.231019i	$-0.925794\pi$
$107$		−	4.77936i		−	0.462038i	0.972949	−	0.231019i	$-0.0742060\pi$
$108$		0			0
$109$	15.9604			1.52873			0.764364	−	0.644785i	$-0.223053\pi$
$109$	15.9604			1.52873			0.764364	+	0.644785i	$0.223053\pi$
$110$	0.935394			0.0891864
$111$		0			0
$112$	2.41720	+	1.07570i	0.228404	+	0.101644i
$113$		−	14.7288i		−	1.38557i	−0.721143	−	0.692786i	$-0.756383\pi$
$113$		−	14.7288i		−	1.38557i	0.721143	−	0.692786i	$-0.243617\pi$
$114$		0			0
$115$		−	14.8103i		−	1.38107i
$116$			1.00000i			0.0928477i
$117$		0			0
$118$			0.966966i			0.0890164i
$119$	4.60240	+	2.04815i	0.421901	+	0.187753i
$120$		0			0
$121$	10.8640			0.987640
$122$	−12.9452			−1.17200
$123$		0			0
$124$		−	7.24557i		−	0.650672i
$125$	9.04263			0.808797
$126$		0			0
$127$	7.62356			0.676481			0.338241	−	0.941060i	$-0.390168\pi$
$127$	7.62356			0.676481			0.338241	+	0.941060i	$0.390168\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	10.5273			0.923304
$131$	−5.89606			−0.515141			−0.257571	−	0.966259i	$-0.582922\pi$
$131$	−5.89606			−0.515141			−0.257571	+	0.966259i	$0.582922\pi$
$132$		0			0
$133$	2.65290	−	5.96135i	0.230036	−	0.516914i
$134$			4.76675i			0.411784i
$135$		0			0
$136$			1.90402i			0.163268i
$137$		−	14.6126i		−	1.24844i	−0.781249	−	0.624219i	$-0.785417\pi$
$137$		−	14.6126i		−	1.24844i	0.781249	−	0.624219i	$-0.214583\pi$
$138$		0			0
$139$		−	4.60820i		−	0.390863i	−0.980717	−	0.195431i	$-0.937389\pi$
$139$		−	4.60820i		−	0.390863i	0.980717	−	0.195431i	$-0.0626107\pi$
$140$	6.13201	+	2.72885i	0.518249	+	0.230630i
$141$		0			0
$142$	6.68361			0.560876
$143$	−1.53015			−0.127957
$144$		0			0
$145$			2.53682i			0.210671i
$146$	−1.99007			−0.164699
$147$		0			0
$148$	−4.79505			−0.394151
$149$		−	3.31339i		−	0.271443i	−0.990747	−	0.135722i	$-0.956665\pi$
$149$		−	3.31339i		−	0.271443i	0.990747	−	0.135722i	$-0.0433353\pi$
$150$		0			0
$151$	16.7171			1.36042			0.680209	−	0.733018i	$-0.261889\pi$
$151$	16.7171			1.36042			0.680209	+	0.733018i	$0.261889\pi$
$152$	2.46622			0.200037
$153$		0			0
$154$	−0.891290	−	0.396639i	−0.0718222	−	0.0319621i
$155$		−	18.3807i		−	1.47637i
$156$		0			0
$157$			3.08156i			0.245935i	0.992411	+	0.122968i	$0.0392411\pi$
$157$			3.08156i			0.245935i	−0.992411	+	0.122968i	$0.960759\pi$
$158$			0.227826i			0.0181249i
$159$		0			0
$160$			2.53682i			0.200553i
$161$	−6.28008	+	14.1120i	−0.494940	+	1.11218i
$162$		0			0
$163$	−7.73491			−0.605845			−0.302922	−	0.953015i	$-0.597962\pi$
$163$	−7.73491			−0.605845			−0.302922	+	0.953015i	$0.597962\pi$
$164$	5.25760			0.410550
$165$		0			0
$166$		−	8.72527i		−	0.677212i
$167$	0.112969			0.00874178			0.00437089	−	0.999990i	$-0.498609\pi$
$167$	0.112969			0.00874178			0.00437089	+	0.999990i	$0.498609\pi$
$168$		0			0
$169$	−4.22084			−0.324680
$170$			4.83015i			0.370455i
$171$		0			0
$172$	−12.0588			−0.919478
$173$	15.5070			1.17897			0.589487	−	0.807778i	$-0.299330\pi$
$173$	15.5070			1.17897			0.589487	+	0.807778i	$0.299330\pi$
$174$		0			0
$175$	3.46976	+	1.54410i	0.262289	+	0.116723i
$176$		−	0.368727i		−	0.0277939i
$177$		0			0
$178$		−	4.99081i		−	0.374077i
$179$			1.04867i			0.0783814i	0.999232	+	0.0391907i	$0.0124780\pi$
$179$			1.04867i			0.0783814i	−0.999232	+	0.0391907i	$0.987522\pi$
$180$		0			0
$181$		−	13.1774i		−	0.979471i	−0.871871	−	0.489736i	$-0.837093\pi$
$181$		−	13.1774i		−	0.979471i	0.871871	−	0.489736i	$-0.162907\pi$
$182$	−10.0309	−	4.46393i	−0.743541	−	0.330888i
$183$		0			0
$184$	−5.83815			−0.430394
$185$	−12.1642			−0.894327
$186$		0			0
$187$		−	0.702064i		−	0.0513400i
$188$	7.92810			0.578216
$189$		0			0
$190$	6.25634			0.453883
$191$			5.69346i			0.411964i	0.978556	+	0.205982i	$0.0660389\pi$
$191$			5.69346i			0.411964i	−0.978556	+	0.205982i	$0.933961\pi$
$192$		0			0
$193$	9.71538			0.699328			0.349664	−	0.936875i	$-0.386296\pi$
$193$	9.71538			0.699328			0.349664	+	0.936875i	$0.386296\pi$
$194$	−9.46666			−0.679667
$195$		0			0
$196$	−4.68575	−	5.20036i	−0.334696	−	0.371454i
$197$		−	20.1969i		−	1.43897i	−0.694510	−	0.719483i	$-0.744379\pi$
$197$		−	20.1969i		−	1.43897i	0.694510	−	0.719483i	$-0.255621\pi$
$198$		0			0
$199$			4.35582i			0.308776i	0.988010	+	0.154388i	$0.0493406\pi$
$199$			4.35582i			0.308776i	−0.988010	+	0.154388i	$0.950659\pi$
$200$			1.43544i			0.101501i
$201$		0			0
$202$		−	11.4978i		−	0.808984i
$203$	1.07570	−	2.41720i	0.0754992	−	0.169654i
$204$		0			0
$205$	13.3376			0.931537
$206$	−2.65737			−0.185148
$207$		0			0
$208$		−	4.14980i		−	0.287737i
$209$	−0.909362			−0.0629019
$210$		0			0
$211$	−3.85376			−0.265304			−0.132652	−	0.991163i	$-0.542349\pi$
$211$	−3.85376			−0.265304			−0.132652	+	0.991163i	$0.542349\pi$
$212$			2.22615i			0.152893i
$213$		0			0
$214$	−4.77936			−0.326710
$215$	−30.5911			−2.08629
$216$		0			0
$217$	−7.79405	+	17.5140i	−0.529094	+	1.18893i
$218$		−	15.9604i		−	1.08097i
$219$		0			0
$220$		−	0.935394i		−	0.0630643i
$221$		−	7.90129i		−	0.531499i
$222$		0			0
$223$		−	20.3202i		−	1.36074i	−0.732869	−	0.680370i	$-0.761819\pi$
$223$		−	20.3202i		−	1.36074i	0.732869	−	0.680370i	$-0.238181\pi$
$224$	1.07570	−	2.41720i	0.0718731	−	0.161506i
$225$		0			0
$226$	−14.7288			−0.979748
$227$	−10.4603			−0.694275			−0.347138	−	0.937814i	$-0.612846\pi$
$227$	−10.4603			−0.694275			−0.347138	+	0.937814i	$0.612846\pi$
$228$		0			0
$229$			12.9271i			0.854248i	0.904193	+	0.427124i	$0.140473\pi$
$229$			12.9271i			0.854248i	−0.904193	+	0.427124i	$0.859527\pi$
$230$	−14.8103			−0.976563
$231$		0			0
$232$	1.00000			0.0656532
$233$			10.8665i			0.711885i	0.934508	+	0.355943i	$0.115840\pi$
$233$			10.8665i			0.711885i	−0.934508	+	0.355943i	$0.884160\pi$
$234$		0			0
$235$	20.1122			1.31197
$236$	0.966966			0.0629441
$237$		0			0
$238$	2.04815	−	4.60240i	0.132762	−	0.298329i
$239$		−	10.7624i		−	0.696163i	−0.937464	−	0.348082i	$-0.886833\pi$
$239$		−	10.7624i		−	0.696163i	0.937464	−	0.348082i	$-0.113167\pi$
$240$		0			0
$241$		−	14.8239i		−	0.954889i	−0.878662	−	0.477445i	$-0.841563\pi$
$241$		−	14.8239i		−	0.954889i	0.878662	−	0.477445i	$-0.158437\pi$
$242$		−	10.8640i		−	0.698367i
$243$		0			0
$244$			12.9452i			0.828732i
$245$	−11.8869	−	13.1924i	−0.759426	−	0.842829i
$246$		0			0
$247$	−10.2343			−0.651193
$248$	−7.24557			−0.460094
$249$		0			0
$250$		−	9.04263i		−	0.571906i
$251$	−27.9058			−1.76140			−0.880698	−	0.473678i	$-0.842926\pi$
$251$	−27.9058			−1.76140			−0.880698	+	0.473678i	$0.842926\pi$
$252$		0			0
$253$	2.15269			0.135338
$254$		−	7.62356i		−	0.478345i
$255$		0			0
$256$	1.00000			0.0625000
$257$	17.7089			1.10465			0.552326	−	0.833628i	$-0.313740\pi$
$257$	17.7089			1.10465			0.552326	+	0.833628i	$0.313740\pi$
$258$		0			0
$259$	11.5906	+	5.15802i	0.720206	+	0.320504i
$260$		−	10.5273i		−	0.652875i
$261$		0			0
$262$			5.89606i			0.364260i
$263$		−	19.9029i		−	1.22726i	−0.789592	−	0.613632i	$-0.789708\pi$
$263$		−	19.9029i		−	1.22726i	0.789592	−	0.613632i	$-0.210292\pi$
$264$		0			0
$265$			5.64733i			0.346913i
$266$	−5.96135	−	2.65290i	−0.365514	−	0.162660i
$267$		0			0
$268$	4.76675			0.291175
$269$	16.4333			1.00196			0.500979	−	0.865459i	$-0.332973\pi$
$269$	16.4333			1.00196			0.500979	+	0.865459i	$0.332973\pi$
$270$		0			0
$271$		−	3.56501i		−	0.216559i	−0.994120	−	0.108279i	$-0.965466\pi$
$271$		−	3.56501i		−	0.216559i	0.994120	−	0.108279i	$-0.0345341\pi$
$272$	1.90402			0.115448
$273$		0			0
$274$	−14.6126			−0.882780
$275$		−	0.529287i		−	0.0319172i
$276$		0			0
$277$	−17.0617			−1.02514			−0.512569	−	0.858646i	$-0.671306\pi$
$277$	−17.0617			−1.02514			−0.512569	+	0.858646i	$0.671306\pi$
$278$	−4.60820			−0.276382
$279$		0			0
$280$	2.72885	−	6.13201i	0.163080	−	0.366457i
$281$		−	18.6570i		−	1.11298i	−0.830854	−	0.556491i	$-0.812147\pi$
$281$		−	18.6570i		−	1.11298i	0.830854	−	0.556491i	$-0.187853\pi$
$282$		0			0
$283$			26.6294i			1.58295i	0.611201	+	0.791475i	$0.290687\pi$
$283$			26.6294i			1.58295i	−0.611201	+	0.791475i	$0.709313\pi$
$284$		−	6.68361i		−	0.396599i
$285$		0			0
$286$			1.53015i			0.0904794i
$287$	−12.7087	−	5.65559i	−0.750171	−	0.333839i
$288$		0			0
$289$	−13.3747			−0.786748
$290$	2.53682			0.148967
$291$		0			0
$292$			1.99007i			0.116460i
$293$	12.0659			0.704899			0.352450	−	0.935831i	$-0.385349\pi$
$293$	12.0659			0.704899			0.352450	+	0.935831i	$0.385349\pi$
$294$		0			0
$295$	2.45302			0.142820
$296$			4.79505i			0.278707i
$297$		0			0
$298$	−3.31339			−0.191940
$299$	24.2271			1.40109
$300$		0			0
$301$	29.1487	+	12.9717i	1.68010	+	0.747674i
$302$		−	16.7171i		−	0.961961i
$303$		0			0
$304$		−	2.46622i		−	0.141447i
$305$			32.8396i			1.88039i
$306$		0			0
$307$		−	21.2649i		−	1.21365i	−0.794835	−	0.606825i	$-0.792443\pi$
$307$		−	21.2649i		−	1.21365i	0.794835	−	0.606825i	$-0.207557\pi$
$308$	−0.396639	+	0.891290i	−0.0226006	+	0.0507859i
$309$		0			0
$310$	−18.3807			−1.04395
$311$	−6.27453			−0.355796			−0.177898	−	0.984049i	$-0.556930\pi$
$311$	−6.27453			−0.355796			−0.177898	+	0.984049i	$0.556930\pi$
$312$		0			0
$313$			5.57141i			0.314915i	0.987526	+	0.157457i	$0.0503297\pi$
$313$			5.57141i			0.314915i	−0.987526	+	0.157457i	$0.949670\pi$
$314$	3.08156			0.173903
$315$		0			0
$316$	0.227826			0.0128162
$317$			2.91135i			0.163518i	0.996652	+	0.0817588i	$0.0260537\pi$
$317$			2.91135i			0.163518i	−0.996652	+	0.0817588i	$0.973946\pi$
$318$		0			0
$319$	−0.368727			−0.0206448
$320$	2.53682			0.141812
$321$		0			0
$322$	14.1120	+	6.28008i	0.786431	+	0.349975i
$323$		−	4.69572i		−	0.261277i
$324$		0			0
$325$		−	5.95680i		−	0.330424i
$326$			7.73491i			0.428397i
$327$		0			0
$328$		−	5.25760i		−	0.290302i
$329$	−19.1638	−	8.52824i	−1.05654	−	0.470177i
$330$		0			0
$331$	12.0652			0.663166			0.331583	−	0.943426i	$-0.392417\pi$
$331$	12.0652			0.663166			0.331583	+	0.943426i	$0.392417\pi$
$332$	−8.72527			−0.478862
$333$		0			0
$334$		−	0.112969i		−	0.00618137i
$335$	12.0924			0.660676
$336$		0			0
$337$	6.12184			0.333478			0.166739	−	0.986001i	$-0.446676\pi$
$337$	6.12184			0.333478			0.166739	+	0.986001i	$0.446676\pi$
$338$			4.22084i			0.229583i
$339$		0			0
$340$	4.83015			0.261951
$341$	2.67164			0.144678
$342$		0			0
$343$	5.73240	+	17.6108i	0.309521	+	0.950893i
$344$			12.0588i			0.650169i
$345$		0			0
$346$		−	15.5070i		−	0.833660i
$347$			9.47108i			0.508434i	0.967147	+	0.254217i	$0.0818178\pi$
$347$			9.47108i			0.508434i	−0.967147	+	0.254217i	$0.918182\pi$
$348$		0			0
$349$		−	0.612881i		−	0.0328068i	−0.999865	−	0.0164034i	$-0.994778\pi$
$349$		−	0.612881i		−	0.0328068i	0.999865	−	0.0164034i	$-0.00522159\pi$
$350$	1.54410	−	3.46976i	0.0825358	−	0.185466i
$351$		0			0
$352$	−0.368727			−0.0196532
$353$	−10.6174			−0.565108			−0.282554	−	0.959251i	$-0.591182\pi$
$353$	−10.6174			−0.565108			−0.282554	+	0.959251i	$0.591182\pi$
$354$		0			0
$355$		−	16.9551i		−	0.899883i
$356$	−4.99081			−0.264512
$357$		0			0
$358$	1.04867			0.0554240
$359$			29.5978i			1.56211i	0.624460	+	0.781057i	$0.285319\pi$
$359$			29.5978i			1.56211i	−0.624460	+	0.781057i	$0.714681\pi$
$360$		0			0
$361$	12.9178			0.679883
$362$	−13.1774			−0.692591
$363$		0			0
$364$	−4.46393	+	10.0309i	−0.233973	+	0.525763i
$365$			5.04843i			0.264247i
$366$		0			0
$367$			1.88880i			0.0985946i	0.998784	+	0.0492973i	$0.0156982\pi$
$367$			1.88880i			0.0985946i	−0.998784	+	0.0492973i	$0.984302\pi$
$368$			5.83815i			0.304334i
$369$		0			0
$370$			12.1642i			0.632385i
$371$	2.39466	−	5.38106i	0.124325	−	0.279371i
$372$		0			0
$373$	−25.8181			−1.33681			−0.668406	−	0.743797i	$-0.733023\pi$
$373$	−25.8181			−1.33681			−0.668406	+	0.743797i	$0.733023\pi$
$374$	−0.702064			−0.0363029
$375$		0			0
$376$		−	7.92810i		−	0.408861i
$377$	−4.14980			−0.213726
$378$		0			0
$379$	−18.2508			−0.937483			−0.468741	−	0.883335i	$-0.655292\pi$
$379$	−18.2508			−0.937483			−0.468741	+	0.883335i	$0.655292\pi$
$380$		−	6.25634i		−	0.320944i
$381$		0			0
$382$	5.69346			0.291303
$383$	15.3875			0.786265			0.393132	−	0.919482i	$-0.371391\pi$
$383$	15.3875			0.786265			0.393132	+	0.919482i	$0.371391\pi$
$384$		0			0
$385$	−1.00620	+	2.26104i	−0.0512808	+	0.115233i
$386$		−	9.71538i		−	0.494500i
$387$		0			0
$388$			9.46666i			0.480597i
$389$			12.4856i			0.633045i	0.948585	+	0.316522i	$0.102515\pi$
$389$			12.4856i			0.633045i	−0.948585	+	0.316522i	$0.897485\pi$
$390$		0			0
$391$			11.1159i			0.562157i
$392$	−5.20036	+	4.68575i	−0.262658	+	0.236666i
$393$		0			0
$394$	−20.1969			−1.01750
$395$	0.577954			0.0290800
$396$		0			0
$397$			34.1206i			1.71246i	0.516593	+	0.856231i	$0.327200\pi$
$397$			34.1206i			1.71246i	−0.516593	+	0.856231i	$0.672800\pi$
$398$	4.35582			0.218337
$399$		0			0
$400$	1.43544			0.0717722
$401$		−	21.8765i		−	1.09246i	−0.837636	−	0.546229i	$-0.816063\pi$
$401$		−	21.8765i		−	1.09246i	0.837636	−	0.546229i	$-0.183937\pi$
$402$		0			0
$403$	30.0677			1.49778
$404$	−11.4978			−0.572038
$405$		0			0
$406$	−2.41720	−	1.07570i	−0.119964	−	0.0533860i
$407$		−	1.76807i		−	0.0876398i
$408$		0			0
$409$			13.8313i			0.683916i	0.939715	+	0.341958i	$0.111090\pi$
$409$			13.8313i			0.683916i	−0.939715	+	0.341958i	$0.888910\pi$
$410$		−	13.3376i		−	0.658696i
$411$		0			0
$412$			2.65737i			0.130919i
$413$	−2.33735	−	1.04016i	−0.115014	−	0.0511830i
$414$		0			0
$415$	−22.1344			−1.08654
$416$	−4.14980			−0.203461
$417$		0			0
$418$			0.909362i			0.0444783i
$419$	20.4414			0.998626			0.499313	−	0.866422i	$-0.333586\pi$
$419$	20.4414			0.998626			0.499313	+	0.866422i	$0.333586\pi$
$420$		0			0
$421$	17.6489			0.860154			0.430077	−	0.902792i	$-0.358486\pi$
$421$	17.6489			0.860154			0.430077	+	0.902792i	$0.358486\pi$
$422$			3.85376i			0.187598i
$423$		0			0
$424$	2.22615			0.108111
$425$	2.73311			0.132575
$426$		0			0
$427$	13.9251	−	31.2912i	0.673884	−	1.51429i
$428$			4.77936i			0.231019i
$429$		0			0
$430$			30.5911i			1.47523i
$431$		−	6.07161i		−	0.292459i	−0.989251	−	0.146230i	$-0.953286\pi$
$431$		−	6.07161i		−	0.292459i	0.989251	−	0.146230i	$-0.0467139\pi$
$432$		0			0
$433$			26.3646i			1.26700i	0.773742	+	0.633501i	$0.218383\pi$
$433$			26.3646i			1.26700i	−0.773742	+	0.633501i	$0.781617\pi$
$434$	17.5140	+	7.79405i	0.840700	+	0.374126i
$435$		0			0
$436$	−15.9604			−0.764364
$437$	14.3981			0.688756
$438$		0			0
$439$			27.3138i			1.30362i	0.758383	+	0.651809i	$0.225990\pi$
$439$			27.3138i			1.30362i	−0.758383	+	0.651809i	$0.774010\pi$
$440$	−0.935394			−0.0445932
$441$		0			0
$442$	−7.90129			−0.375826
$443$			23.8216i			1.13180i	0.824474	+	0.565899i	$0.191471\pi$
$443$			23.8216i			1.13180i	−0.824474	+	0.565899i	$0.808529\pi$
$444$		0			0
$445$	−12.6608			−0.600178
$446$	−20.3202			−0.962189
$447$		0			0
$448$	−2.41720	−	1.07570i	−0.114202	−	0.0508219i
$449$		−	29.4708i		−	1.39081i	−0.718616	−	0.695407i	$-0.755224\pi$
$449$		−	29.4708i		−	1.39081i	0.718616	−	0.695407i	$-0.244776\pi$
$450$		0			0
$451$			1.93862i			0.0912862i
$452$			14.7288i			0.692786i
$453$		0			0
$454$			10.4603i			0.490927i
$455$	−11.3242	+	25.4466i	−0.530885	+	1.19295i
$456$		0			0
$457$	9.59622			0.448892			0.224446	−	0.974487i	$-0.427943\pi$
$457$	9.59622			0.448892			0.224446	+	0.974487i	$0.427943\pi$
$458$	12.9271			0.604044
$459$		0			0
$460$			14.8103i			0.690534i
$461$	23.0246			1.07236			0.536181	−	0.844103i	$-0.319867\pi$
$461$	23.0246			1.07236			0.536181	+	0.844103i	$0.319867\pi$
$462$		0			0
$463$	2.99885			0.139368			0.0696842	−	0.997569i	$-0.477801\pi$
$463$	2.99885			0.139368			0.0696842	+	0.997569i	$0.477801\pi$
$464$		−	1.00000i		−	0.0464238i
$465$		0			0
$466$	10.8665			0.503379
$467$	42.1930			1.95246			0.976230	−	0.216738i	$-0.0695417\pi$
$467$	42.1930			1.95246			0.976230	+	0.216738i	$0.0695417\pi$
$468$		0			0
$469$	−11.5222	−	5.12758i	−0.532046	−	0.236769i
$470$		−	20.1122i		−	0.927705i
$471$		0			0
$472$		−	0.966966i		−	0.0445082i
$473$		−	4.44643i		−	0.204447i
$474$		0			0
$475$		−	3.54011i		−	0.162432i
$476$	−4.60240	−	2.04815i	−0.210951	−	0.0938767i
$477$		0			0
$478$	−10.7624			−0.492262
$479$	0.981120			0.0448285			0.0224143	−	0.999749i	$-0.492865\pi$
$479$	0.981120			0.0448285			0.0224143	+	0.999749i	$0.492865\pi$
$480$		0			0
$481$		−	19.8985i		−	0.907293i
$482$	−14.8239			−0.675209
$483$		0			0
$484$	−10.8640			−0.493820
$485$			24.0152i			1.09047i
$486$		0			0
$487$	−35.9998			−1.63131			−0.815654	−	0.578540i	$-0.803623\pi$
$487$	−35.9998			−1.63131			−0.815654	+	0.578540i	$0.803623\pi$
$488$	12.9452			0.586002
$489$		0			0
$490$	−13.1924	+	11.8869i	−0.595970	+	0.536995i
$491$			21.5000i			0.970281i	0.874436	+	0.485141i	$0.161232\pi$
$491$			21.5000i			0.970281i	−0.874436	+	0.485141i	$0.838768\pi$
$492$		0			0
$493$		−	1.90402i		−	0.0857527i
$494$			10.2343i			0.460463i
$495$		0			0
$496$			7.24557i			0.325336i
$497$	−7.18954	+	16.1556i	−0.322495	+	0.724680i
$498$		0			0
$499$	31.8156			1.42426			0.712131	−	0.702046i	$-0.247730\pi$
$499$	31.8156			1.42426			0.712131	+	0.702046i	$0.247730\pi$
$500$	−9.04263			−0.404399
$501$		0			0
$502$			27.9058i			1.24550i
$503$	−35.5044			−1.58306			−0.791532	−	0.611128i	$-0.790716\pi$
$503$	−35.5044			−1.58306			−0.791532	+	0.611128i	$0.790716\pi$
$504$		0			0
$505$	−29.1679			−1.29795
$506$		−	2.15269i		−	0.0956985i
$507$		0			0
$508$	−7.62356			−0.338241
$509$	5.35311			0.237273			0.118636	−	0.992938i	$-0.462148\pi$
$509$	5.35311			0.237273			0.118636	+	0.992938i	$0.462148\pi$
$510$		0			0
$511$	2.14071	−	4.81040i	0.0946994	−	0.212799i
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	17.7089i		−	0.781108i
$515$			6.74126i			0.297055i
$516$		0			0
$517$			2.92331i			0.128567i
$518$	5.15802	−	11.5906i	0.226630	−	0.509262i
$519$		0			0
$520$	−10.5273			−0.461652
$521$	18.8059			0.823902			0.411951	−	0.911206i	$-0.364847\pi$
$521$	18.8059			0.823902			0.411951	+	0.911206i	$0.364847\pi$
$522$		0			0
$523$		−	14.7727i		−	0.645966i	−0.946405	−	0.322983i	$-0.895314\pi$
$523$		−	14.7727i		−	0.645966i	0.946405	−	0.322983i	$-0.104686\pi$
$524$	5.89606			0.257571
$525$		0			0
$526$	−19.9029			−0.867807
$527$			13.7957i			0.600950i
$528$		0			0
$529$	−11.0840			−0.481911
$530$	5.64733			0.245305
$531$		0			0
$532$	−2.65290	+	5.96135i	−0.115018	+	0.258457i
$533$			21.8180i			0.945042i
$534$		0			0
$535$			12.1244i			0.524182i
$536$		−	4.76675i		−	0.205892i
$537$		0			0
$538$		−	16.4333i		−	0.708491i
$539$	1.91752	−	1.72777i	0.0825932	−	0.0744201i
$540$		0			0
$541$	−6.23509			−0.268067			−0.134034	−	0.990977i	$-0.542793\pi$
$541$	−6.23509			−0.268067			−0.134034	+	0.990977i	$0.542793\pi$
$542$	−3.56501			−0.153130
$543$		0			0
$544$		−	1.90402i		−	0.0816341i
$545$	−40.4886			−1.73434
$546$		0			0
$547$	8.72732			0.373153			0.186577	−	0.982440i	$-0.440261\pi$
$547$	8.72732			0.373153			0.186577	+	0.982440i	$0.440261\pi$
$548$			14.6126i			0.624219i
$549$		0			0
$550$	−0.529287			−0.0225689
$551$	−2.46622			−0.105064
$552$		0			0
$553$	−0.550703	−	0.245072i	−0.0234183	−	0.0104215i
$554$			17.0617i			0.724882i
$555$		0			0
$556$			4.60820i			0.195431i
$557$		−	8.50314i		−	0.360290i	−0.983640	−	0.180145i	$-0.942343\pi$
$557$		−	8.50314i		−	0.360290i	0.983640	−	0.180145i	$-0.0576567\pi$
$558$		0			0
$559$		−	50.0418i		−	2.11654i
$560$	−6.13201	−	2.72885i	−0.259125	−	0.115315i
$561$		0			0
$562$	−18.6570			−0.786997
$563$	−29.9941			−1.26410			−0.632050	−	0.774928i	$-0.717786\pi$
$563$	−29.9941			−1.26410			−0.632050	+	0.774928i	$0.717786\pi$
$564$		0			0
$565$			37.3644i			1.57193i
$566$	26.6294			1.11932
$567$		0			0
$568$	−6.68361			−0.280438
$569$			23.8331i			0.999134i	0.866275	+	0.499567i	$0.166508\pi$
$569$			23.8331i			0.999134i	−0.866275	+	0.499567i	$0.833492\pi$
$570$		0			0
$571$	−18.8833			−0.790243			−0.395121	−	0.918629i	$-0.629298\pi$
$571$	−18.8833			−0.790243			−0.395121	+	0.918629i	$0.629298\pi$
$572$	1.53015			0.0639786
$573$		0			0
$574$	−5.65559	+	12.7087i	−0.236060	+	0.530451i
$575$			8.38033i			0.349484i
$576$		0			0
$577$		−	27.8156i		−	1.15798i	−0.815335	−	0.578990i	$-0.803447\pi$
$577$		−	27.8156i		−	1.15798i	0.815335	−	0.578990i	$-0.196553\pi$
$578$			13.3747i			0.556315i
$579$		0			0
$580$		−	2.53682i		−	0.105336i
$581$	21.0908	+	9.38575i	0.874992	+	0.389387i
$582$		0			0
$583$	−0.820842			−0.0339958
$584$	1.99007			0.0823495
$585$		0			0
$586$		−	12.0659i		−	0.498439i
$587$	37.9404			1.56597			0.782985	−	0.622041i	$-0.213696\pi$
$587$	37.9404			1.56597			0.782985	+	0.622041i	$0.213696\pi$
$588$		0			0
$589$	17.8692			0.736286
$590$		−	2.45302i		−	0.100989i
$591$		0			0
$592$	4.79505			0.197075
$593$	14.4804			0.594640			0.297320	−	0.954778i	$-0.403907\pi$
$593$	14.4804			0.594640			0.297320	+	0.954778i	$0.403907\pi$
$594$		0			0
$595$	−11.6755	−	5.19578i	−0.478647	−	0.213006i
$596$			3.31339i			0.135722i
$597$		0			0
$598$		−	24.2271i		−	0.990722i
$599$		−	14.9799i		−	0.612062i	−0.952021	−	0.306031i	$-0.900999\pi$
$599$		−	14.9799i		−	0.612062i	0.952021	−	0.306031i	$-0.0990012\pi$
$600$		0			0
$601$		−	34.9192i		−	1.42438i	−0.701984	−	0.712192i	$-0.747702\pi$
$601$		−	34.9192i		−	1.42438i	0.701984	−	0.712192i	$-0.252298\pi$
$602$	12.9717	−	29.1487i	0.528685	−	1.18801i
$603$		0			0
$604$	−16.7171			−0.680209
$605$	−27.5601			−1.12048
$606$		0			0
$607$			17.8793i			0.725699i	0.931848	+	0.362849i	$0.118196\pi$
$607$			17.8793i			0.725699i	−0.931848	+	0.362849i	$0.881804\pi$
$608$	−2.46622			−0.100018
$609$		0			0
$610$	32.8396			1.32964
$611$			32.9000i			1.33099i
$612$		0			0
$613$	−39.5347			−1.59679			−0.798396	−	0.602133i	$-0.794318\pi$
$613$	−39.5347			−1.59679			−0.798396	+	0.602133i	$0.794318\pi$
$614$	−21.2649			−0.858180
$615$		0			0
$616$	0.891290	+	0.396639i	0.0359111	+	0.0159810i
$617$		−	37.6172i		−	1.51441i	−0.653176	−	0.757206i	$-0.726564\pi$
$617$		−	37.6172i		−	1.51441i	0.653176	−	0.757206i	$-0.273436\pi$
$618$		0			0
$619$			15.6572i			0.629314i	0.949205	+	0.314657i	$0.101890\pi$
$619$			15.6572i			0.629314i	−0.949205	+	0.314657i	$0.898110\pi$
$620$			18.3807i			0.738187i
$621$		0			0
$622$			6.27453i			0.251586i
$623$	12.0638	+	5.36860i	0.483326	+	0.215088i
$624$		0			0
$625$	−30.1167			−1.20467
$626$	5.57141			0.222678
$627$		0			0
$628$		−	3.08156i		−	0.122968i
$629$	9.12986			0.364031
$630$		0			0
$631$	−8.43047			−0.335612			−0.167806	−	0.985820i	$-0.553668\pi$
$631$	−8.43047			−0.335612			−0.167806	+	0.985820i	$0.553668\pi$
$632$		−	0.227826i		−	0.00906245i
$633$		0			0
$634$	2.91135			0.115624
$635$	−19.3396			−0.767468
$636$		0			0
$637$	21.5805	−	19.4449i	0.855049	−	0.770436i
$638$			0.368727i			0.0145981i
$639$		0			0
$640$		−	2.53682i		−	0.100277i
$641$			4.26393i			0.168415i	0.996448	+	0.0842075i	$0.0268359\pi$
$641$			4.26393i			0.168415i	−0.996448	+	0.0842075i	$0.973164\pi$
$642$		0			0
$643$		−	44.3604i		−	1.74941i	−0.484660	−	0.874703i	$-0.661057\pi$
$643$		−	44.3604i		−	1.74941i	0.484660	−	0.874703i	$-0.338943\pi$
$644$	6.28008	−	14.1120i	0.247470	−	0.556090i
$645$		0			0
$646$	−4.69572			−0.184751
$647$	44.7238			1.75827			0.879137	−	0.476568i	$-0.158119\pi$
$647$	44.7238			1.75827			0.879137	+	0.476568i	$0.158119\pi$
$648$		0			0
$649$			0.356547i			0.0139957i
$650$	−5.95680			−0.233645
$651$		0			0
$652$	7.73491			0.302922
$653$		−	18.3560i		−	0.718326i	−0.933275	−	0.359163i	$-0.883062\pi$
$653$		−	18.3560i		−	0.718326i	0.933275	−	0.359163i	$-0.116938\pi$
$654$		0			0
$655$	14.9572			0.584427
$656$	−5.25760			−0.205275
$657$		0			0
$658$	−8.52824	+	19.1638i	−0.332465	+	0.747085i
$659$			36.7210i			1.43045i	0.698897	+	0.715223i	$0.253675\pi$
$659$			36.7210i			1.43045i	−0.698897	+	0.715223i	$0.746325\pi$
$660$		0			0
$661$			25.3874i			0.987457i	0.869616	+	0.493728i	$0.164366\pi$
$661$			25.3874i			0.987457i	−0.869616	+	0.493728i	$0.835634\pi$
$662$		−	12.0652i		−	0.468929i
$663$		0			0
$664$			8.72527i			0.338606i
$665$	−6.72993	+	15.1229i	−0.260975	+	0.586439i
$666$		0			0
$667$	5.83815			0.226054
$668$	−0.112969			−0.00437089
$669$		0			0
$670$		−	12.0924i		−	0.467169i
$671$	−4.77326			−0.184270
$672$		0			0
$673$	−23.5006			−0.905881			−0.452940	−	0.891541i	$-0.649625\pi$
$673$	−23.5006			−0.905881			−0.452940	+	0.891541i	$0.649625\pi$
$674$		−	6.12184i		−	0.235805i
$675$		0			0
$676$	4.22084			0.162340
$677$	21.0630			0.809515			0.404757	−	0.914424i	$-0.367356\pi$
$677$	21.0630			0.809515			0.404757	+	0.914424i	$0.367356\pi$
$678$		0			0
$679$	10.1833	−	22.8829i	0.390798	−	0.878164i
$680$		−	4.83015i		−	0.185228i
$681$		0			0
$682$		−	2.67164i		−	0.102302i
$683$			33.9689i			1.29978i	0.760027	+	0.649891i	$0.225186\pi$
$683$			33.9689i			1.29978i	−0.760027	+	0.649891i	$0.774814\pi$
$684$		0			0
$685$			37.0695i			1.41635i
$686$	17.6108	−	5.73240i	0.672383	−	0.218864i
$687$		0			0
$688$	12.0588			0.459739
$689$	−9.23807			−0.351943
$690$		0			0
$691$			9.61070i			0.365608i	0.983149	+	0.182804i	$0.0585174\pi$
$691$			9.61070i			0.365608i	−0.983149	+	0.182804i	$0.941483\pi$
$692$	−15.5070			−0.589487
$693$		0			0
$694$	9.47108			0.359517
$695$			11.6902i			0.443433i
$696$		0			0
$697$	−10.0106			−0.379177
$698$	−0.612881			−0.0231979
$699$		0			0
$700$	−3.46976	−	1.54410i	−0.131145	−	0.0583616i
$701$		−	34.6335i		−	1.30809i	−0.756456	−	0.654044i	$-0.773071\pi$
$701$		−	34.6335i		−	1.30809i	0.756456	−	0.654044i	$-0.226929\pi$
$702$		0			0
$703$		−	11.8256i		−	0.446012i
$704$			0.368727i			0.0138969i
$705$		0			0
$706$			10.6174i			0.399592i
$707$	27.7926	+	12.3682i	1.04525	+	0.465153i
$708$		0			0
$709$	−51.5502			−1.93601			−0.968004	−	0.250935i	$-0.919262\pi$
$709$	−51.5502			−1.93601			−0.968004	+	0.250935i	$0.919262\pi$
$710$	−16.9551			−0.636314
$711$		0			0
$712$			4.99081i			0.187038i
$713$	−42.3007			−1.58417
$714$		0			0
$715$	3.88170			0.145167
$716$		−	1.04867i		−	0.0391907i
$717$		0			0
$718$	29.5978			1.10458
$719$	40.1463			1.49721			0.748603	−	0.663018i	$-0.230725\pi$
$719$	40.1463			1.49721			0.748603	+	0.663018i	$0.230725\pi$
$720$		0			0
$721$	2.85852	−	6.42340i	0.106457	−	0.239220i
$722$		−	12.9178i		−	0.480750i
$723$		0			0
$724$			13.1774i			0.489736i
$725$		−	1.43544i		−	0.0533110i
$726$		0			0
$727$		−	31.0243i		−	1.15063i	−0.817933	−	0.575314i	$-0.804880\pi$
$727$		−	31.0243i		−	1.15063i	0.817933	−	0.575314i	$-0.195120\pi$
$728$	10.0309	+	4.46393i	0.371770	+	0.165444i
$729$		0			0
$730$	5.04843			0.186851
$731$	22.9603			0.849216
$732$		0			0
$733$		−	29.3018i		−	1.08229i	−0.840930	−	0.541144i	$-0.817991\pi$
$733$		−	29.3018i		−	1.08229i	0.840930	−	0.541144i	$-0.182009\pi$
$734$	1.88880			0.0697169
$735$		0			0
$736$	5.83815			0.215197
$737$			1.75763i			0.0647431i
$738$		0			0
$739$	−21.9736			−0.808313			−0.404157	−	0.914690i	$-0.632435\pi$
$739$	−21.9736			−0.808313			−0.404157	+	0.914690i	$0.632435\pi$
$740$	12.1642			0.447164
$741$		0			0
$742$	−5.38106	−	2.39466i	−0.197545	−	0.0879108i
$743$			22.2285i			0.815484i	0.913097	+	0.407742i	$0.133684\pi$
$743$			22.2285i			0.815484i	−0.913097	+	0.407742i	$0.866316\pi$
$744$		0			0
$745$			8.40546i			0.307952i
$746$			25.8181i			0.945268i
$747$		0			0
$748$			0.702064i			0.0256700i
$749$	5.14114	−	11.5527i	0.187853	−	0.422126i
$750$		0			0
$751$	12.0845			0.440971			0.220485	−	0.975390i	$-0.429236\pi$
$751$	12.0845			0.440971			0.220485	+	0.975390i	$0.429236\pi$
$752$	−7.92810			−0.289108
$753$		0			0
$754$			4.14980i			0.151127i
$755$	−42.4082			−1.54339
$756$		0			0
$757$	29.4694			1.07108			0.535541	−	0.844509i	$-0.320108\pi$
$757$	29.4694			1.07108			0.535541	+	0.844509i	$0.320108\pi$
$758$			18.2508i			0.662900i
$759$		0			0
$760$	−6.25634			−0.226941
$761$	−29.8713			−1.08283			−0.541417	−	0.840754i	$-0.682112\pi$
$761$	−29.8713			−1.08283			−0.541417	+	0.840754i	$0.682112\pi$
$762$		0			0
$763$	38.5795	+	17.1685i	1.39667	+	0.621543i
$764$		−	5.69346i		−	0.205982i
$765$		0			0
$766$		−	15.3875i		−	0.555973i
$767$			4.01271i			0.144891i
$768$		0			0
$769$			49.3711i			1.78037i	0.455602	+	0.890183i	$0.349424\pi$
$769$			49.3711i			1.78037i	−0.455602	+	0.890183i	$0.650576\pi$
$770$	2.26104	+	1.00620i	0.0814822	+	0.0362610i
$771$		0			0
$772$	−9.71538			−0.349664
$773$	−32.2350			−1.15941			−0.579706	−	0.814826i	$-0.696832\pi$
$773$	−32.2350			−1.15941			−0.579706	+	0.814826i	$0.696832\pi$
$774$		0			0
$775$			10.4006i			0.373601i
$776$	9.46666			0.339833
$777$		0			0
$778$	12.4856			0.447630
$779$			12.9664i			0.464569i
$780$		0			0
$781$	2.46443			0.0881843
$782$	11.1159			0.397505
$783$		0			0
$784$	4.68575	+	5.20036i	0.167348	+	0.185727i
$785$		−	7.81736i		−	0.279013i
$786$		0			0
$787$		−	39.9126i		−	1.42273i	−0.702823	−	0.711365i	$-0.748077\pi$
$787$		−	39.9126i		−	1.42273i	0.702823	−	0.711365i	$-0.251923\pi$
$788$			20.1969i			0.719483i
$789$		0			0
$790$		−	0.577954i		−	0.0205627i
$791$	15.8438	−	35.6026i	0.563340	−	1.26588i
$792$		0			0
$793$	−53.7201			−1.90765
$794$	34.1206			1.21089
$795$		0			0
$796$		−	4.35582i		−	0.154388i
$797$	28.2498			1.00066			0.500330	−	0.865835i	$-0.333212\pi$
$797$	28.2498			1.00066			0.500330	+	0.865835i	$0.333212\pi$
$798$		0			0
$799$	−15.0953			−0.534032
$800$		−	1.43544i		−	0.0507506i
$801$		0			0
$802$	−21.8765			−0.772485
$803$	−0.733792			−0.0258950
$804$		0			0
$805$	15.9314	−	35.7995i	0.561509	−	1.26177i
$806$		−	30.0677i		−	1.05909i
$807$		0			0
$808$			11.4978i			0.404492i
$809$		−	2.08396i		−	0.0732682i	−0.999329	−	0.0366341i	$-0.988336\pi$
$809$		−	2.08396i		−	0.0732682i	0.999329	−	0.0366341i	$-0.0116636\pi$
$810$		0			0
$811$			24.3914i			0.856497i	0.903661	+	0.428249i	$0.140869\pi$
$811$			24.3914i			0.856497i	−0.903661	+	0.428249i	$0.859131\pi$
$812$	−1.07570	+	2.41720i	−0.0377496	+	0.0848272i
$813$		0			0
$814$	−1.76807			−0.0619707
$815$	19.6220			0.687330
$816$		0			0
$817$		−	29.7397i		−	1.04046i
$818$	13.8313			0.483602
$819$		0			0
$820$	−13.3376			−0.465768
$821$		−	18.3927i		−	0.641911i	−0.947094	−	0.320956i	$-0.895996\pi$
$821$		−	18.3927i		−	0.641911i	0.947094	−	0.320956i	$-0.104004\pi$
$822$		0			0
$823$	51.0597			1.77983			0.889914	−	0.456127i	$-0.150764\pi$
$823$	51.0597			1.77983			0.889914	+	0.456127i	$0.150764\pi$
$824$	2.65737			0.0925738
$825$		0			0
$826$	−1.04016	+	2.33735i	−0.0361919	+	0.0813269i
$827$		−	36.8496i		−	1.28139i	−0.767797	−	0.640693i	$-0.778647\pi$
$827$		−	36.8496i		−	1.28139i	0.767797	−	0.640693i	$-0.221353\pi$
$828$		0			0
$829$		−	21.2708i		−	0.738765i	−0.929277	−	0.369383i	$-0.879569\pi$
$829$		−	21.2708i		−	0.738765i	0.929277	−	0.369383i	$-0.120431\pi$
$830$			22.1344i			0.768297i
$831$		0			0
$832$			4.14980i			0.143868i
$833$	8.92175	+	9.90158i	0.309120	+	0.343069i
$834$		0			0
$835$	−0.286581			−0.00991754
$836$	0.909362			0.0314509
$837$		0			0
$838$		−	20.4414i		−	0.706135i
$839$	14.0472			0.484964			0.242482	−	0.970156i	$-0.422038\pi$
$839$	14.0472			0.484964			0.242482	+	0.970156i	$0.422038\pi$
$840$		0			0
$841$	−1.00000			−0.0344828
$842$		−	17.6489i		−	0.608221i
$843$		0			0
$844$	3.85376			0.132652
$845$	10.7075			0.368349
$846$		0			0
$847$	26.2606	+	11.6864i	0.902325	+	0.401550i
$848$		−	2.22615i		−	0.0764463i
$849$		0			0
$850$		−	2.73311i		−	0.0937449i
$851$			27.9942i			0.959629i
$852$		0			0
$853$			54.8330i			1.87745i	0.344671	+	0.938724i	$0.387991\pi$
$853$			54.8330i			1.87745i	−0.344671	+	0.938724i	$0.612009\pi$
$854$	−31.2912	−	13.9251i	−1.07076	−	0.476508i
$855$		0			0
$856$	4.77936			0.163355
$857$	−44.4145			−1.51717			−0.758585	−	0.651575i	$-0.774109\pi$
$857$	−44.4145			−1.51717			−0.758585	+	0.651575i	$0.774109\pi$
$858$		0			0
$859$			7.16208i			0.244367i	0.992508	+	0.122184i	$0.0389897\pi$
$859$			7.16208i			0.244367i	−0.992508	+	0.122184i	$0.961010\pi$
$860$	30.5911			1.04315
$861$		0			0
$862$	−6.07161			−0.206800
$863$			33.9163i			1.15453i	0.816558	+	0.577263i	$0.195879\pi$
$863$			33.9163i			1.15453i	−0.816558	+	0.577263i	$0.804121\pi$
$864$		0			0
$865$	−39.3384			−1.33755
$866$	26.3646			0.895905
$867$		0			0
$868$	7.79405	−	17.5140i	0.264547	−	0.594465i
$869$			0.0840059i			0.00284970i
$870$		0			0
$871$			19.7810i			0.670255i
$872$			15.9604i			0.540487i
$873$		0			0
$874$		−	14.3981i		−	0.487024i
$875$	21.8579	+	9.72713i	0.738931	+	0.328837i
$876$		0			0
$877$	37.1782			1.25542			0.627710	−	0.778448i	$-0.283993\pi$
$877$	37.1782			1.25542			0.627710	+	0.778448i	$0.283993\pi$
$878$	27.3138			0.921797
$879$		0			0
$880$			0.935394i			0.0315321i
$881$	6.81184			0.229497			0.114748	−	0.993395i	$-0.463394\pi$
$881$	6.81184			0.229497			0.114748	+	0.993395i	$0.463394\pi$
$882$		0			0
$883$	−23.4622			−0.789565			−0.394783	−	0.918775i	$-0.629180\pi$
$883$	−23.4622			−0.789565			−0.394783	+	0.918775i	$0.629180\pi$
$884$			7.90129i			0.265749i
$885$		0			0
$886$	23.8216			0.800302
$887$	−45.7869			−1.53738			−0.768688	−	0.639624i	$-0.779090\pi$
$887$	−45.7869			−1.53738			−0.768688	+	0.639624i	$0.779090\pi$
$888$		0			0
$889$	18.4277	+	8.20064i	0.618045	+	0.275041i
$890$			12.6608i			0.424390i
$891$		0			0
$892$			20.3202i			0.680370i
$893$			19.5524i			0.654297i
$894$		0			0
$895$		−	2.66029i		−	0.0889237i
$896$	−1.07570	+	2.41720i	−0.0359365	+	0.0807531i
$897$		0			0
$898$	−29.4708			−0.983454
$899$	7.24557			0.241653
$900$		0			0
$901$		−	4.23863i		−	0.141209i
$902$	1.93862			0.0645491
$903$		0			0
$904$	14.7288			0.489874
$905$			33.4288i			1.11121i
$906$		0			0
$907$	47.4624			1.57596			0.787982	−	0.615698i	$-0.211126\pi$
$907$	47.4624			1.57596			0.787982	+	0.615698i	$0.211126\pi$
$908$	10.4603			0.347138
$909$		0			0
$910$	25.4466	+	11.3242i	0.843546	+	0.375393i
$911$		−	38.7767i		−	1.28473i	−0.766399	−	0.642365i	$-0.777953\pi$
$911$		−	38.7767i		−	1.28473i	0.766399	−	0.642365i	$-0.222047\pi$
$912$		0			0
$913$		−	3.21725i		−	0.106475i
$914$		−	9.59622i		−	0.317415i
$915$		0			0
$916$		−	12.9271i		−	0.427124i
$917$	−14.2520	−	6.34238i	−0.470642	−	0.209444i
$918$		0			0
$919$	−19.6619			−0.648586			−0.324293	−	0.945957i	$-0.605126\pi$
$919$	−19.6619			−0.648586			−0.324293	+	0.945957i	$0.605126\pi$
$920$	14.8103			0.488282
$921$		0			0
$922$		−	23.0246i		−	0.758274i
$923$	27.7356			0.912930
$924$		0			0
$925$	6.88302			0.226312
$926$		−	2.99885i		−	0.0985484i
$927$		0			0
$928$	−1.00000			−0.0328266
$929$	−22.4413			−0.736275			−0.368137	−	0.929771i	$-0.620004\pi$
$929$	−22.4413			−0.736275			−0.368137	+	0.929771i	$0.620004\pi$
$930$		0			0
$931$	12.8252	−	11.5561i	0.420329	−	0.378735i
$932$		−	10.8665i		−	0.355943i
$933$		0			0
$934$		−	42.1930i		−	1.38060i
$935$			1.78101i			0.0582452i
$936$		0			0
$937$			51.7289i			1.68991i	0.534838	+	0.844955i	$0.320373\pi$
$937$			51.7289i			1.68991i	−0.534838	+	0.844955i	$0.679627\pi$
$938$	−5.12758	+	11.5222i	−0.167421	+	0.376213i
$939$		0			0
$940$	−20.1122			−0.655986
$941$	3.78935			0.123529			0.0617646	−	0.998091i	$-0.480327\pi$
$941$	3.78935			0.123529			0.0617646	+	0.998091i	$0.480327\pi$
$942$		0			0
$943$		−	30.6947i		−	0.999555i
$944$	−0.966966			−0.0314721
$945$		0			0
$946$	−4.44643			−0.144566
$947$			46.8050i			1.52096i	0.649363	+	0.760478i	$0.275036\pi$
$947$			46.8050i			1.52096i	−0.649363	+	0.760478i	$0.724964\pi$
$948$		0			0
$949$	−8.25838			−0.268078
$950$	−3.54011			−0.114856
$951$		0			0
$952$	−2.04815	+	4.60240i	−0.0663808	+	0.149165i
$953$			36.5161i			1.18287i	0.806352	+	0.591436i	$0.201439\pi$
$953$			36.5161i			1.18287i	−0.806352	+	0.591436i	$0.798561\pi$
$954$		0			0
$955$		−	14.4433i		−	0.467373i
$956$			10.7624i			0.348082i
$957$		0			0
$958$		−	0.981120i		−	0.0316986i
$959$	15.7187	−	35.3216i	0.507585	−	1.14060i
$960$		0			0
$961$	−21.4984			−0.693495
$962$	−19.8985			−0.641553
$963$		0			0
$964$			14.8239i			0.477445i
$965$	−24.6461			−0.793387
$966$		0			0
$967$	−36.9577			−1.18848			−0.594239	−	0.804288i	$-0.702547\pi$
$967$	−36.9577			−1.18848			−0.594239	+	0.804288i	$0.702547\pi$
$968$			10.8640i			0.349183i
$969$		0			0
$970$	24.0152			0.771082
$971$	−16.9611			−0.544308			−0.272154	−	0.962254i	$-0.587736\pi$
$971$	−16.9611			−0.544308			−0.272154	+	0.962254i	$0.587736\pi$
$972$		0			0
$973$	4.95703	−	11.1390i	0.158915	−	0.357099i
$974$			35.9998i			1.15351i
$975$		0			0
$976$		−	12.9452i		−	0.414366i
$977$			17.2686i			0.552472i	0.961090	+	0.276236i	$0.0890871\pi$
$977$			17.2686i			0.552472i	−0.961090	+	0.276236i	$0.910913\pi$
$978$		0			0
$979$		−	1.84025i		−	0.0588146i
$980$	11.8869	+	13.1924i	0.379713	+	0.421415i
$981$		0			0
$982$	21.5000			0.686093
$983$	36.6197			1.16799			0.583994	−	0.811758i	$-0.301489\pi$
$983$	36.6197			1.16799			0.583994	+	0.811758i	$0.301489\pi$
$984$		0			0
$985$			51.2357i			1.63251i
$986$	−1.90402			−0.0606363
$987$		0			0
$988$	10.2343			0.325597
$989$			70.4013i			2.23863i
$990$		0			0
$991$	−38.5856			−1.22571			−0.612857	−	0.790194i	$-0.709980\pi$
$991$	−38.5856			−1.22571			−0.612857	+	0.790194i	$0.709980\pi$
$992$	7.24557			0.230047
$993$		0			0
$994$	16.1556	+	7.18954i	0.512426	+	0.228038i
$995$		−	11.0499i		−	0.350306i
$996$		0			0
$997$		−	19.1516i		−	0.606537i	−0.952905	−	0.303268i	$-0.901922\pi$
$997$		−	19.1516i		−	0.606537i	0.952905	−	0.303268i	$-0.0980779\pi$
$998$		−	31.8156i		−	1.00711i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3654.2.f.b.755.17	✓	40
3.2	odd	2	inner	3654.2.f.b.755.24	yes	40
7.6	odd	2	inner	3654.2.f.b.755.23	yes	40
21.20	even	2	inner	3654.2.f.b.755.18	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3654.2.f.b.755.17	✓	40	1.1	even	1	trivial
3654.2.f.b.755.18	yes	40	21.20	even	2	inner
3654.2.f.b.755.23	yes	40	7.6	odd	2	inner
3654.2.f.b.755.24	yes	40	3.2	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -2.53682 q^{5} +(2.41720 + 1.07570i) q^{7} +1.00000i q^{8} +2.53682i q^{10} -0.368727i q^{11} -4.14980i q^{13} +(1.07570 - 2.41720i) q^{14} +1.00000 q^{16} +1.90402 q^{17} -2.46622i q^{19} +2.53682 q^{20} -0.368727 q^{22} +5.83815i q^{23} +1.43544 q^{25} -4.14980 q^{26} +(-2.41720 - 1.07570i) q^{28} -1.00000i q^{29} +7.24557i q^{31} -1.00000i q^{32} -1.90402i q^{34} +(-6.13201 - 2.72885i) q^{35} +4.79505 q^{37} -2.46622 q^{38} -2.53682i q^{40} -5.25760 q^{41} +12.0588 q^{43} +0.368727i q^{44} +5.83815 q^{46} -7.92810 q^{47} +(4.68575 + 5.20036i) q^{49} -1.43544i q^{50} +4.14980i q^{52} -2.22615i q^{53} +0.935394i q^{55} +(-1.07570 + 2.41720i) q^{56} -1.00000 q^{58} -0.966966 q^{59} -12.9452i q^{61} +7.24557 q^{62} -1.00000 q^{64} +10.5273i q^{65} -4.76675 q^{67} -1.90402 q^{68} +(-2.72885 + 6.13201i) q^{70} +6.68361i q^{71} -1.99007i q^{73} -4.79505i q^{74} +2.46622i q^{76} +(0.396639 - 0.891290i) q^{77} -0.227826 q^{79} -2.53682 q^{80} +5.25760i q^{82} +8.72527 q^{83} -4.83015 q^{85} -12.0588i q^{86} +0.368727 q^{88} +4.99081 q^{89} +(4.46393 - 10.0309i) q^{91} -5.83815i q^{92} +7.92810i q^{94} +6.25634i q^{95} -9.46666i q^{97} +(5.20036 - 4.68575i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{4} - 4 q^{7} + 40 q^{16} + 8 q^{22} + 64 q^{25} + 4 q^{28} - 48 q^{37} - 8 q^{43} + 8 q^{46} + 36 q^{49} - 40 q^{58} - 40 q^{64} + 24 q^{67} + 28 q^{70} - 144 q^{79} - 24 q^{85} - 8 q^{88}+O(q^{100}) \) 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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