LMFDB - Embedded newform 725.2.b.c.349.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(349,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("725.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	725.349
Dual form			725.2.b.c.349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.41421i q^{2} -2.00000i q^{3} -3.82843 q^{4} +4.82843 q^{6} -0.828427i q^{7} -4.41421i q^{8} -1.00000 q^{9} -4.82843 q^{11} +7.65685i q^{12} -2.00000i q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843i q^{17} -2.41421i q^{18} -0.828427 q^{19} -1.65685 q^{21} -11.6569i q^{22} -8.82843i q^{23} -8.82843 q^{24} +4.82843 q^{26} -4.00000i q^{27} +3.17157i q^{28} -1.00000 q^{29} -10.4853 q^{31} -1.58579i q^{32} +9.65685i q^{33} -6.82843 q^{34} +3.82843 q^{36} -8.48528i q^{37} -2.00000i q^{38} -4.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -6.00000i q^{43} +18.4853 q^{44} +21.3137 q^{46} +0.343146i q^{47} -6.00000i q^{48} +6.31371 q^{49} +5.65685 q^{51} +7.65685i q^{52} +7.65685i q^{53} +9.65685 q^{54} -3.65685 q^{56} +1.65685i q^{57} -2.41421i q^{58} +7.65685 q^{61} -25.3137i q^{62} +0.828427i q^{63} +9.82843 q^{64} -23.3137 q^{66} +10.4853i q^{67} -10.8284i q^{68} -17.6569 q^{69} +7.31371 q^{71} +4.41421i q^{72} -8.48528i q^{73} +20.4853 q^{74} +3.17157 q^{76} +4.00000i q^{77} -9.65685i q^{78} -14.4853 q^{79} -11.0000 q^{81} -14.4853i q^{82} +12.8284i q^{83} +6.34315 q^{84} +14.4853 q^{86} +2.00000i q^{87} +21.3137i q^{88} -3.65685 q^{89} -1.65685 q^{91} +33.7990i q^{92} +20.9706i q^{93} -0.828427 q^{94} -3.17157 q^{96} -4.48528i q^{97} +15.2426i q^{98} +4.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 8 q^{11} + 8 q^{14} + 12 q^{16} + 8 q^{19} + 16 q^{21} - 24 q^{24} + 8 q^{26} - 4 q^{29} - 8 q^{31} - 16 q^{34} + 4 q^{36} - 16 q^{39} - 24 q^{41} + 40 q^{44} + 40 q^{46}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^6 - 4 * q^9 - 8 * q^11 + 8 * q^14 + 12 * q^16 + 8 * q^19 + 16 * q^21 - 24 * q^24 + 8 * q^26 - 4 * q^29 - 8 * q^31 - 16 * q^34 + 4 * q^36 - 16 * q^39 - 24 * q^41 + 40 * q^44 + 40 * q^46 - 20 * q^49 + 16 * q^54 + 8 * q^56 + 8 * q^61 + 28 * q^64 - 48 * q^66 - 48 * q^69 - 16 * q^71 + 48 * q^74 + 24 * q^76 - 24 * q^79 - 44 * q^81 + 48 * q^84 + 24 * q^86 + 8 * q^89 + 16 * q^91 + 8 * q^94 - 24 * q^96 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$.

$n$	$176$	$552$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.41421i	1.70711i	0.521005	+	0.853553i	$0.325557\pi$
$2$	2.41421i	1.70711i	−0.521005	+	0.853553i	$0.674443\pi$
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	4.82843	1.97120
$7$	− 0.828427i	− 0.313116i	−0.987669	−	0.156558i	$-0.949960\pi$
$7$	− 0.828427i	− 0.313116i	0.987669	−	0.156558i	$-0.0500398\pi$
$8$	− 4.41421i	− 1.56066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	7.65685i	2.21034i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	3.00000	0.750000
$17$	2.82843i	0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$	2.82843i	0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$	− 2.41421i	− 0.569036i
$19$	−0.828427	−0.190054	−0.0950271	−	0.995475i	$-0.530294\pi$
$19$	−0.828427	−0.190054	−0.0950271	+	0.995475i	$0.530294\pi$
$20$	0	0
$21$	−1.65685	−0.361555
$22$	− 11.6569i	− 2.48525i
$23$	− 8.82843i	− 1.84085i	−0.390914	−	0.920427i	$-0.627841\pi$
$23$	− 8.82843i	− 1.84085i	0.390914	−	0.920427i	$-0.372159\pi$
$24$	−8.82843	−1.80210
$25$	0	0
$26$	4.82843	0.946932
$27$	− 4.00000i	− 0.769800i
$28$	3.17157i	0.599371i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−10.4853	−1.88321	−0.941606	−	0.336717i	$-0.890684\pi$
$31$	−10.4853	−1.88321	−0.941606	+	0.336717i	$0.890684\pi$
$32$	− 1.58579i	− 0.280330i
$33$	9.65685i	1.68104i
$34$	−6.82843	−1.17107
$35$	0	0
$36$	3.82843	0.638071
$37$	− 8.48528i	− 1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$	− 8.48528i	− 1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$	− 2.00000i	− 0.324443i
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	− 4.00000i	− 0.617213i
$43$	− 6.00000i	− 0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$	− 6.00000i	− 0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$	18.4853	2.78676
$45$	0	0
$46$	21.3137	3.14253
$47$	0.343146i	0.0500530i	0.999687	+	0.0250265i	$0.00796701\pi$
$47$	0.343146i	0.0500530i	−0.999687	+	0.0250265i	$0.992033\pi$
$48$	− 6.00000i	− 0.866025i
$49$	6.31371	0.901958
$50$	0	0
$51$	5.65685	0.792118
$52$	7.65685i	1.06181i
$53$	7.65685i	1.05175i	0.850562	+	0.525875i	$0.176262\pi$
$53$	7.65685i	1.05175i	−0.850562	+	0.525875i	$0.823738\pi$
$54$	9.65685	1.31413
$55$	0	0
$56$	−3.65685	−0.488668
$57$	1.65685i	0.219456i
$58$	− 2.41421i	− 0.317002i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	7.65685	0.980360	0.490180	−	0.871621i	$-0.336931\pi$
$61$	7.65685	0.980360	0.490180	+	0.871621i	$0.336931\pi$
$62$	− 25.3137i	− 3.21484i
$63$	0.828427i	0.104372i
$64$	9.82843	1.22855
$65$	0	0
$66$	−23.3137	−2.86972
$67$	10.4853i	1.28098i	0.767966	+	0.640490i	$0.221269\pi$
$67$	10.4853i	1.28098i	−0.767966	+	0.640490i	$0.778731\pi$
$68$	− 10.8284i	− 1.31314i
$69$	−17.6569	−2.12564
$70$	0	0
$71$	7.31371	0.867978	0.433989	−	0.900918i	$-0.357106\pi$
$71$	7.31371	0.867978	0.433989	+	0.900918i	$0.357106\pi$
$72$	4.41421i	0.520220i
$73$	− 8.48528i	− 0.993127i	−0.868000	−	0.496564i	$-0.834595\pi$
$73$	− 8.48528i	− 0.993127i	0.868000	−	0.496564i	$-0.165405\pi$
$74$	20.4853	2.38137
$75$	0	0
$76$	3.17157	0.363804
$77$	4.00000i	0.455842i
$78$	− 9.65685i	− 1.09342i
$79$	−14.4853	−1.62972	−0.814861	−	0.579657i	$-0.803187\pi$
$79$	−14.4853	−1.62972	−0.814861	+	0.579657i	$0.803187\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 14.4853i	− 1.59963i
$83$	12.8284i	1.40810i	0.710149	+	0.704051i	$0.248628\pi$
$83$	12.8284i	1.40810i	−0.710149	+	0.704051i	$0.751372\pi$
$84$	6.34315	0.692094
$85$	0	0
$86$	14.4853	1.56199
$87$	2.00000i	0.214423i
$88$	21.3137i	2.27205i
$89$	−3.65685	−0.387626	−0.193813	−	0.981039i	$-0.562085\pi$
$89$	−3.65685	−0.387626	−0.193813	+	0.981039i	$0.562085\pi$
$90$	0	0
$91$	−1.65685	−0.173686
$92$	33.7990i	3.52379i
$93$	20.9706i	2.17455i
$94$	−0.828427	−0.0854457
$95$	0	0
$96$	−3.17157	−0.323697
$97$	− 4.48528i	− 0.455411i	−0.973730	−	0.227706i	$-0.926878\pi$
$97$	− 4.48528i	− 0.455411i	0.973730	−	0.227706i	$-0.0731224\pi$
$98$	15.2426i	1.53974i
$99$	4.82843	0.485275
$100$	0	0
$101$	4.34315	0.432159	0.216080	−	0.976376i	$-0.430673\pi$
$101$	4.34315	0.432159	0.216080	+	0.976376i	$0.430673\pi$
$102$	13.6569i	1.35223i
$103$	− 12.1421i	− 1.19640i	−0.801347	−	0.598200i	$-0.795883\pi$
$103$	− 12.1421i	− 1.19640i	0.801347	−	0.598200i	$-0.204117\pi$
$104$	−8.82843	−0.865699
$105$	0	0
$106$	−18.4853	−1.79545
$107$	8.14214i	0.787130i	0.919297	+	0.393565i	$0.128758\pi$
$107$	8.14214i	0.787130i	−0.919297	+	0.393565i	$0.871242\pi$
$108$	15.3137i	1.47356i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−16.9706	−1.61077
$112$	− 2.48528i	− 0.234837i
$113$	2.82843i	0.266076i	0.991111	+	0.133038i	$0.0424732\pi$
$113$	2.82843i	0.266076i	−0.991111	+	0.133038i	$0.957527\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	3.82843	0.355461
$117$	2.00000i	0.184900i
$118$	0	0
$119$	2.34315	0.214796
$120$	0	0
$121$	12.3137	1.11943
$122$	18.4853i	1.67358i
$123$	12.0000i	1.08200i
$124$	40.1421	3.60487
$125$	0	0
$126$	−2.00000	−0.178174
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	20.5563i	1.81694i
$129$	−12.0000	−1.05654
$130$	0	0
$131$	16.1421	1.41034	0.705172	−	0.709036i	$-0.250870\pi$
$131$	16.1421	1.41034	0.705172	+	0.709036i	$0.250870\pi$
$132$	− 36.9706i	− 3.21787i
$133$	0.686292i	0.0595090i
$134$	−25.3137	−2.18677
$135$	0	0
$136$	12.4853	1.07060
$137$	10.8284i	0.925135i	0.886584	+	0.462567i	$0.153072\pi$
$137$	10.8284i	0.925135i	−0.886584	+	0.462567i	$0.846928\pi$
$138$	− 42.6274i	− 3.62869i
$139$	−10.3431	−0.877294	−0.438647	−	0.898659i	$-0.644542\pi$
$139$	−10.3431	−0.877294	−0.438647	+	0.898659i	$0.644542\pi$
$140$	0	0
$141$	0.686292	0.0577962
$142$	17.6569i	1.48173i
$143$	9.65685i	0.807547i
$144$	−3.00000	−0.250000
$145$	0	0
$146$	20.4853	1.69537
$147$	− 12.6274i	− 1.04149i
$148$	32.4853i	2.67027i
$149$	13.3137	1.09070	0.545351	−	0.838208i	$-0.316396\pi$
$149$	13.3137	1.09070	0.545351	+	0.838208i	$0.316396\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	3.65685i	0.296610i
$153$	− 2.82843i	− 0.228665i
$154$	−9.65685	−0.778171
$155$	0	0
$156$	15.3137	1.22608
$157$	16.4853i	1.31567i	0.753163	+	0.657834i	$0.228527\pi$
$157$	16.4853i	1.31567i	−0.753163	+	0.657834i	$0.771473\pi$
$158$	− 34.9706i	− 2.78211i
$159$	15.3137	1.21446
$160$	0	0
$161$	−7.31371	−0.576401
$162$	− 26.5563i	− 2.08646i
$163$	− 19.6569i	− 1.53964i	−0.638259	−	0.769822i	$-0.720345\pi$
$163$	− 19.6569i	− 1.53964i	0.638259	−	0.769822i	$-0.279655\pi$
$164$	22.9706	1.79370
$165$	0	0
$166$	−30.9706	−2.40378
$167$	− 14.4853i	− 1.12090i	−0.828187	−	0.560452i	$-0.810627\pi$
$167$	− 14.4853i	− 1.12090i	0.828187	−	0.560452i	$-0.189373\pi$
$168$	7.31371i	0.564265i
$169$	9.00000	0.692308
$170$	0	0
$171$	0.828427	0.0633514
$172$	22.9706i	1.75149i
$173$	− 5.31371i	− 0.403994i	−0.979386	−	0.201997i	$-0.935257\pi$
$173$	− 5.31371i	− 0.403994i	0.979386	−	0.201997i	$-0.0647431\pi$
$174$	−4.82843	−0.366042
$175$	0	0
$176$	−14.4853	−1.09187
$177$	0	0
$178$	− 8.82843i	− 0.661719i
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	− 4.00000i	− 0.296500i
$183$	− 15.3137i	− 1.13202i
$184$	−38.9706	−2.87295
$185$	0	0
$186$	−50.6274	−3.71218
$187$	− 13.6569i	− 0.998688i
$188$	− 1.31371i	− 0.0958120i
$189$	−3.31371	−0.241037
$190$	0	0
$191$	−15.1716	−1.09778	−0.548888	−	0.835896i	$-0.684949\pi$
$191$	−15.1716	−1.09778	−0.548888	+	0.835896i	$0.684949\pi$
$192$	− 19.6569i	− 1.41861i
$193$	− 12.4853i	− 0.898710i	−0.893353	−	0.449355i	$-0.851654\pi$
$193$	− 12.4853i	− 0.898710i	0.893353	−	0.449355i	$-0.148346\pi$
$194$	10.8284	0.777436
$195$	0	0
$196$	−24.1716	−1.72654
$197$	8.34315i	0.594425i	0.954811	+	0.297212i	$0.0960569\pi$
$197$	8.34315i	0.594425i	−0.954811	+	0.297212i	$0.903943\pi$
$198$	11.6569i	0.828417i
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	20.9706	1.47915
$202$	10.4853i	0.737742i
$203$	0.828427i	0.0581442i
$204$	−21.6569	−1.51628
$205$	0	0
$206$	29.3137	2.04238
$207$	8.82843i	0.613618i
$208$	− 6.00000i	− 0.416025i
$209$	4.00000	0.276686
$210$	0	0
$211$	4.82843	0.332403	0.166201	−	0.986092i	$-0.446850\pi$
$211$	4.82843	0.332403	0.166201	+	0.986092i	$0.446850\pi$
$212$	− 29.3137i	− 2.01327i
$213$	− 14.6274i	− 1.00225i
$214$	−19.6569	−1.34371
$215$	0	0
$216$	−17.6569	−1.20140
$217$	8.68629i	0.589664i
$218$	− 4.82843i	− 0.327022i
$219$	−16.9706	−1.14676
$220$	0	0
$221$	5.65685	0.380521
$222$	− 40.9706i	− 2.74976i
$223$	21.7990i	1.45977i	0.683571	+	0.729884i	$0.260426\pi$
$223$	21.7990i	1.45977i	−0.683571	+	0.729884i	$0.739574\pi$
$224$	−1.31371	−0.0877758
$225$	0	0
$226$	−6.82843	−0.454220
$227$	8.14214i	0.540413i	0.962802	+	0.270206i	$0.0870919\pi$
$227$	8.14214i	0.540413i	−0.962802	+	0.270206i	$0.912908\pi$
$228$	− 6.34315i	− 0.420085i
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	4.41421i	0.289807i
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	−4.82843	−0.315644
$235$	0	0
$236$	0	0
$237$	28.9706i	1.88184i
$238$	5.65685i	0.366679i
$239$	23.3137	1.50804	0.754019	−	0.656852i	$-0.228113\pi$
$239$	23.3137	1.50804	0.754019	+	0.656852i	$0.228113\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	29.7279i	1.91098i
$243$	10.0000i	0.641500i
$244$	−29.3137	−1.87662
$245$	0	0
$246$	−28.9706	−1.84710
$247$	1.65685i	0.105423i
$248$	46.2843i	2.93905i
$249$	25.6569	1.62594
$250$	0	0
$251$	3.17157	0.200188	0.100094	−	0.994978i	$-0.468086\pi$
$251$	3.17157	0.200188	0.100094	+	0.994978i	$0.468086\pi$
$252$	− 3.17157i	− 0.199790i
$253$	42.6274i	2.67996i
$254$	14.4853	0.908887
$255$	0	0
$256$	−29.9706	−1.87316
$257$	− 29.3137i	− 1.82854i	−0.405107	−	0.914269i	$-0.632766\pi$
$257$	− 29.3137i	− 1.82854i	0.405107	−	0.914269i	$-0.367234\pi$
$258$	− 28.9706i	− 1.80363i
$259$	−7.02944	−0.436788
$260$	0	0
$261$	1.00000	0.0618984
$262$	38.9706i	2.40761i
$263$	− 8.34315i	− 0.514460i	−0.966350	−	0.257230i	$-0.917190\pi$
$263$	− 8.34315i	− 0.514460i	0.966350	−	0.257230i	$-0.0828099\pi$
$264$	42.6274	2.62354
$265$	0	0
$266$	−1.65685	−0.101588
$267$	7.31371i	0.447592i
$268$	− 40.1421i	− 2.45207i
$269$	−1.31371	−0.0800982	−0.0400491	−	0.999198i	$-0.512751\pi$
$269$	−1.31371	−0.0800982	−0.0400491	+	0.999198i	$0.512751\pi$
$270$	0	0
$271$	29.7990	1.81016	0.905080	−	0.425242i	$-0.139811\pi$
$271$	29.7990	1.81016	0.905080	+	0.425242i	$0.139811\pi$
$272$	8.48528i	0.514496i
$273$	3.31371i	0.200555i
$274$	−26.1421	−1.57930
$275$	0	0
$276$	67.5980	4.06892
$277$	− 7.65685i	− 0.460056i	−0.973184	−	0.230028i	$-0.926118\pi$
$277$	− 7.65685i	− 0.460056i	0.973184	−	0.230028i	$-0.0738817\pi$
$278$	− 24.9706i	− 1.49763i
$279$	10.4853	0.627737
$280$	0	0
$281$	−6.68629	−0.398871	−0.199435	−	0.979911i	$-0.563911\pi$
$281$	−6.68629	−0.398871	−0.199435	+	0.979911i	$0.563911\pi$
$282$	1.65685i	0.0986642i
$283$	− 0.828427i	− 0.0492449i	−0.999697	−	0.0246224i	$-0.992162\pi$
$283$	− 0.828427i	− 0.0492449i	0.999697	−	0.0246224i	$-0.00783836\pi$
$284$	−28.0000	−1.66149
$285$	0	0
$286$	−23.3137	−1.37857
$287$	4.97056i	0.293403i
$288$	1.58579i	0.0934434i
$289$	9.00000	0.529412
$290$	0	0
$291$	−8.97056	−0.525864
$292$	32.4853i	1.90106i
$293$	− 8.48528i	− 0.495715i	−0.968796	−	0.247858i	$-0.920273\pi$
$293$	− 8.48528i	− 0.495715i	0.968796	−	0.247858i	$-0.0797265\pi$
$294$	30.4853	1.77794
$295$	0	0
$296$	−37.4558	−2.17708
$297$	19.3137i	1.12070i
$298$	32.1421i	1.86194i
$299$	−17.6569	−1.02112
$300$	0	0
$301$	−4.97056	−0.286498
$302$	− 28.9706i	− 1.66707i
$303$	− 8.68629i	− 0.499014i
$304$	−2.48528	−0.142541
$305$	0	0
$306$	6.82843	0.390355
$307$	− 10.9706i	− 0.626123i	−0.949733	−	0.313062i	$-0.898645\pi$
$307$	− 10.9706i	− 0.626123i	0.949733	−	0.313062i	$-0.101355\pi$
$308$	− 15.3137i	− 0.872580i
$309$	−24.2843	−1.38148
$310$	0	0
$311$	−2.48528	−0.140927	−0.0704637	−	0.997514i	$-0.522448\pi$
$311$	−2.48528	−0.140927	−0.0704637	+	0.997514i	$0.522448\pi$
$312$	17.6569i	0.999623i
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	−39.7990	−2.24599
$315$	0	0
$316$	55.4558	3.11963
$317$	2.82843i	0.158860i	0.996840	+	0.0794301i	$0.0253101\pi$
$317$	2.82843i	0.158860i	−0.996840	+	0.0794301i	$0.974690\pi$
$318$	36.9706i	2.07321i
$319$	4.82843	0.270340
$320$	0	0
$321$	16.2843	0.908899
$322$	− 17.6569i	− 0.983978i
$323$	− 2.34315i	− 0.130376i
$324$	42.1127	2.33959
$325$	0	0
$326$	47.4558	2.62834
$327$	4.00000i	0.221201i
$328$	26.4853i	1.46241i
$329$	0.284271	0.0156724
$330$	0	0
$331$	−17.7990	−0.978321	−0.489160	−	0.872194i	$-0.662697\pi$
$331$	−17.7990	−0.978321	−0.489160	+	0.872194i	$0.662697\pi$
$332$	− 49.1127i	− 2.69541i
$333$	8.48528i	0.464991i
$334$	34.9706	1.91350
$335$	0	0
$336$	−4.97056	−0.271166
$337$	6.82843i	0.371968i	0.982553	+	0.185984i	$0.0595473\pi$
$337$	6.82843i	0.371968i	−0.982553	+	0.185984i	$0.940453\pi$
$338$	21.7279i	1.18184i
$339$	5.65685	0.307238
$340$	0	0
$341$	50.6274	2.74163
$342$	2.00000i	0.108148i
$343$	− 11.0294i	− 0.595534i
$344$	−26.4853	−1.42799
$345$	0	0
$346$	12.8284	0.689661
$347$	− 20.1421i	− 1.08129i	−0.841252	−	0.540643i	$-0.818181\pi$
$347$	− 20.1421i	− 1.08129i	0.841252	−	0.540643i	$-0.181819\pi$
$348$	− 7.65685i	− 0.410450i
$349$	24.6274	1.31828	0.659138	−	0.752022i	$-0.270921\pi$
$349$	24.6274	1.31828	0.659138	+	0.752022i	$0.270921\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	7.65685i	0.408112i
$353$	− 15.6569i	− 0.833330i	−0.909060	−	0.416665i	$-0.863199\pi$
$353$	− 15.6569i	− 0.833330i	0.909060	−	0.416665i	$-0.136801\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	− 4.68629i	− 0.248025i
$358$	1.65685i	0.0875675i
$359$	32.1421	1.69640	0.848199	−	0.529678i	$-0.177687\pi$
$359$	32.1421	1.69640	0.848199	+	0.529678i	$0.177687\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	− 14.4853i	− 0.761329i
$363$	− 24.6274i	− 1.29260i
$364$	6.34315	0.332471
$365$	0	0
$366$	36.9706	1.93248
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	− 26.4853i	− 1.38064i
$369$	6.00000	0.312348
$370$	0	0
$371$	6.34315	0.329320
$372$	− 80.2843i	− 4.16255i
$373$	26.9706i	1.39648i	0.715862	+	0.698241i	$0.246034\pi$
$373$	26.9706i	1.39648i	−0.715862	+	0.698241i	$0.753966\pi$
$374$	32.9706	1.70487
$375$	0	0
$376$	1.51472	0.0781156
$377$	2.00000i	0.103005i
$378$	− 8.00000i	− 0.411476i
$379$	−5.51472	−0.283272	−0.141636	−	0.989919i	$-0.545236\pi$
$379$	−5.51472	−0.283272	−0.141636	+	0.989919i	$0.545236\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	− 36.6274i	− 1.87402i
$383$	− 14.4853i	− 0.740163i	−0.928999	−	0.370082i	$-0.879330\pi$
$383$	− 14.4853i	− 0.740163i	0.928999	−	0.370082i	$-0.120670\pi$
$384$	41.1127	2.09802
$385$	0	0
$386$	30.1421	1.53419
$387$	6.00000i	0.304997i
$388$	17.1716i	0.871755i
$389$	6.68629	0.339008	0.169504	−	0.985529i	$-0.445783\pi$
$389$	6.68629	0.339008	0.169504	+	0.985529i	$0.445783\pi$
$390$	0	0
$391$	24.9706	1.26282
$392$	− 27.8701i	− 1.40765i
$393$	− 32.2843i	− 1.62853i
$394$	−20.1421	−1.01475
$395$	0	0
$396$	−18.4853	−0.928920
$397$	8.34315i	0.418730i	0.977838	+	0.209365i	$0.0671398\pi$
$397$	8.34315i	0.418730i	−0.977838	+	0.209365i	$0.932860\pi$
$398$	− 28.9706i	− 1.45216i
$399$	1.37258	0.0687151
$400$	0	0
$401$	−29.3137	−1.46386	−0.731928	−	0.681382i	$-0.761379\pi$
$401$	−29.3137	−1.46386	−0.731928	+	0.681382i	$0.761379\pi$
$402$	50.6274i	2.52507i
$403$	20.9706i	1.04462i
$404$	−16.6274	−0.827245
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	40.9706i	2.03084i
$408$	− 24.9706i	− 1.23623i
$409$	−30.9706	−1.53140	−0.765698	−	0.643200i	$-0.777606\pi$
$409$	−30.9706	−1.53140	−0.765698	+	0.643200i	$0.777606\pi$
$410$	0	0
$411$	21.6569	1.06825
$412$	46.4853i	2.29017i
$413$	0	0
$414$	−21.3137	−1.04751
$415$	0	0
$416$	−3.17157	−0.155499
$417$	20.6863i	1.01301i
$418$	9.65685i	0.472332i
$419$	4.97056	0.242828	0.121414	−	0.992602i	$-0.461257\pi$
$419$	4.97056	0.242828	0.121414	+	0.992602i	$0.461257\pi$
$420$	0	0
$421$	−14.9706	−0.729621	−0.364810	−	0.931082i	$-0.618866\pi$
$421$	−14.9706	−0.729621	−0.364810	+	0.931082i	$0.618866\pi$
$422$	11.6569i	0.567447i
$423$	− 0.343146i	− 0.0166843i
$424$	33.7990	1.64142
$425$	0	0
$426$	35.3137	1.71095
$427$	− 6.34315i	− 0.306966i
$428$	− 31.1716i	− 1.50673i
$429$	19.3137	0.932475
$430$	0	0
$431$	−19.3137	−0.930309	−0.465154	−	0.885230i	$-0.654001\pi$
$431$	−19.3137	−0.930309	−0.465154	+	0.885230i	$0.654001\pi$
$432$	− 12.0000i	− 0.577350i
$433$	− 34.8284i	− 1.67375i	−0.547396	−	0.836874i	$-0.684381\pi$
$433$	− 34.8284i	− 1.67375i	0.547396	−	0.836874i	$-0.315619\pi$
$434$	−20.9706	−1.00662
$435$	0	0
$436$	7.65685	0.366697
$437$	7.31371i	0.349862i
$438$	− 40.9706i	− 1.95765i
$439$	21.6569	1.03363	0.516813	−	0.856099i	$-0.327118\pi$
$439$	21.6569	1.03363	0.516813	+	0.856099i	$0.327118\pi$
$440$	0	0
$441$	−6.31371	−0.300653
$442$	13.6569i	0.649590i
$443$	3.65685i	0.173742i	0.996220	+	0.0868712i	$0.0276869\pi$
$443$	3.65685i	0.173742i	−0.996220	+	0.0868712i	$0.972313\pi$
$444$	64.9706	3.08337
$445$	0	0
$446$	−52.6274	−2.49198
$447$	− 26.6274i	− 1.25943i
$448$	− 8.14214i	− 0.384680i
$449$	−0.343146	−0.0161940	−0.00809702	−	0.999967i	$-0.502577\pi$
$449$	−0.343146	−0.0161940	−0.00809702	+	0.999967i	$0.502577\pi$
$450$	0	0
$451$	28.9706	1.36417
$452$	− 10.8284i	− 0.509326i
$453$	24.0000i	1.12762i
$454$	−19.6569	−0.922542
$455$	0	0
$456$	7.31371	0.342496
$457$	− 8.34315i	− 0.390276i	−0.980776	−	0.195138i	$-0.937485\pi$
$457$	− 8.34315i	− 0.390276i	0.980776	−	0.195138i	$-0.0625155\pi$
$458$	4.82843i	0.225618i
$459$	11.3137	0.528079
$460$	0	0
$461$	−24.3431	−1.13377	−0.566887	−	0.823796i	$-0.691852\pi$
$461$	−24.3431	−1.13377	−0.566887	+	0.823796i	$0.691852\pi$
$462$	19.3137i	0.898555i
$463$	− 17.7990i	− 0.827189i	−0.910461	−	0.413595i	$-0.864273\pi$
$463$	− 17.7990i	− 0.827189i	0.910461	−	0.413595i	$-0.135727\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−43.4558	−2.01305
$467$	22.9706i	1.06295i	0.847074	+	0.531475i	$0.178362\pi$
$467$	22.9706i	1.06295i	−0.847074	+	0.531475i	$0.821638\pi$
$468$	− 7.65685i	− 0.353938i
$469$	8.68629	0.401096
$470$	0	0
$471$	32.9706	1.51920
$472$	0	0
$473$	28.9706i	1.33207i
$474$	−69.9411	−3.21250
$475$	0	0
$476$	−8.97056	−0.411165
$477$	− 7.65685i	− 0.350583i
$478$	56.2843i	2.57438i
$479$	−12.8284	−0.586146	−0.293073	−	0.956090i	$-0.594678\pi$
$479$	−12.8284	−0.586146	−0.293073	+	0.956090i	$0.594678\pi$
$480$	0	0
$481$	−16.9706	−0.773791
$482$	24.1421i	1.09964i
$483$	14.6274i	0.665571i
$484$	−47.1421	−2.14282
$485$	0	0
$486$	−24.1421	−1.09511
$487$	29.7990i	1.35032i	0.737671	+	0.675161i	$0.235926\pi$
$487$	29.7990i	1.35032i	−0.737671	+	0.675161i	$0.764074\pi$
$488$	− 33.7990i	− 1.53001i
$489$	−39.3137	−1.77783
$490$	0	0
$491$	43.4558	1.96113	0.980567	−	0.196183i	$-0.0628545\pi$
$491$	43.4558	1.96113	0.980567	+	0.196183i	$0.0628545\pi$
$492$	− 45.9411i	− 2.07119i
$493$	− 2.82843i	− 0.127386i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−31.4558	−1.41241
$497$	− 6.05887i	− 0.271778i
$498$	61.9411i	2.77565i
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	0	0
$501$	−28.9706	−1.29431
$502$	7.65685i	0.341742i
$503$	− 30.0000i	− 1.33763i	−0.743427	−	0.668817i	$-0.766801\pi$
$503$	− 30.0000i	− 1.33763i	0.743427	−	0.668817i	$-0.233199\pi$
$504$	3.65685	0.162889
$505$	0	0
$506$	−102.912	−4.57498
$507$	− 18.0000i	− 0.799408i
$508$	22.9706i	1.01915i
$509$	44.6274	1.97808	0.989038	−	0.147663i	$-0.0471751\pi$
$509$	44.6274	1.97808	0.989038	+	0.147663i	$0.0471751\pi$
$510$	0	0
$511$	−7.02944	−0.310964
$512$	− 31.2426i	− 1.38074i
$513$	3.31371i	0.146304i
$514$	70.7696	3.12151
$515$	0	0
$516$	45.9411	2.02245
$517$	− 1.65685i	− 0.0728684i
$518$	− 16.9706i	− 0.745644i
$519$	−10.6274	−0.466492
$520$	0	0
$521$	1.31371	0.0575546	0.0287773	−	0.999586i	$-0.490839\pi$
$521$	1.31371	0.0575546	0.0287773	+	0.999586i	$0.490839\pi$
$522$	2.41421i	0.105667i
$523$	− 14.4853i	− 0.633397i	−0.948526	−	0.316699i	$-0.897426\pi$
$523$	− 14.4853i	− 0.633397i	0.948526	−	0.316699i	$-0.102574\pi$
$524$	−61.7990	−2.69970
$525$	0	0
$526$	20.1421	0.878239
$527$	− 29.6569i	− 1.29187i
$528$	28.9706i	1.26078i
$529$	−54.9411	−2.38874
$530$	0	0
$531$	0	0
$532$	− 2.62742i	− 0.113913i
$533$	12.0000i	0.519778i
$534$	−17.6569	−0.764087
$535$	0	0
$536$	46.2843	1.99918
$537$	− 1.37258i	− 0.0592313i
$538$	− 3.17157i	− 0.136736i
$539$	−30.4853	−1.31309
$540$	0	0
$541$	−38.9706	−1.67548	−0.837738	−	0.546073i	$-0.816122\pi$
$541$	−38.9706	−1.67548	−0.837738	+	0.546073i	$0.816122\pi$
$542$	71.9411i	3.09014i
$543$	12.0000i	0.514969i
$544$	4.48528	0.192305
$545$	0	0
$546$	−8.00000	−0.342368
$547$	− 14.4853i	− 0.619346i	−0.950843	−	0.309673i	$-0.899780\pi$
$547$	− 14.4853i	− 0.619346i	0.950843	−	0.309673i	$-0.100220\pi$
$548$	− 41.4558i	− 1.77091i
$549$	−7.65685	−0.326787
$550$	0	0
$551$	0.828427	0.0352922
$552$	77.9411i	3.31739i
$553$	12.0000i	0.510292i
$554$	18.4853	0.785364
$555$	0	0
$556$	39.5980	1.67933
$557$	− 39.9411i	− 1.69236i	−0.532897	−	0.846180i	$-0.678897\pi$
$557$	− 39.9411i	− 1.69236i	0.532897	−	0.846180i	$-0.321103\pi$
$558$	25.3137i	1.07161i
$559$	−12.0000	−0.507546
$560$	0	0
$561$	−27.3137	−1.15319
$562$	− 16.1421i	− 0.680915i
$563$	− 3.65685i	− 0.154118i	−0.997027	−	0.0770590i	$-0.975447\pi$
$563$	− 3.65685i	− 0.154118i	0.997027	−	0.0770590i	$-0.0245530\pi$
$564$	−2.62742	−0.110634
$565$	0	0
$566$	2.00000	0.0840663
$567$	9.11270i	0.382697i
$568$	− 32.2843i	− 1.35462i
$569$	−16.3431	−0.685140	−0.342570	−	0.939492i	$-0.611297\pi$
$569$	−16.3431	−0.685140	−0.342570	+	0.939492i	$0.611297\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	− 36.9706i	− 1.54582i
$573$	30.3431i	1.26760i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	−9.82843	−0.409518
$577$	15.7990i	0.657721i	0.944379	+	0.328860i	$0.106665\pi$
$577$	15.7990i	0.657721i	−0.944379	+	0.328860i	$0.893335\pi$
$578$	21.7279i	0.903762i
$579$	−24.9706	−1.03774
$580$	0	0
$581$	10.6274	0.440900
$582$	− 21.6569i	− 0.897705i
$583$	− 36.9706i	− 1.53116i
$584$	−37.4558	−1.54993
$585$	0	0
$586$	20.4853	0.846239
$587$	− 9.79899i	− 0.404448i	−0.979339	−	0.202224i	$-0.935183\pi$
$587$	− 9.79899i	− 0.404448i	0.979339	−	0.202224i	$-0.0648168\pi$
$588$	48.3431i	1.99364i
$589$	8.68629	0.357912
$590$	0	0
$591$	16.6863	0.686382
$592$	− 25.4558i	− 1.04623i
$593$	3.65685i	0.150169i	0.997177	+	0.0750845i	$0.0239227\pi$
$593$	3.65685i	0.150169i	−0.997177	+	0.0750845i	$0.976077\pi$
$594$	−46.6274	−1.91315
$595$	0	0
$596$	−50.9706	−2.08784
$597$	24.0000i	0.982255i
$598$	− 42.6274i	− 1.74316i
$599$	−1.79899	−0.0735047	−0.0367524	−	0.999324i	$-0.511701\pi$
$599$	−1.79899	−0.0735047	−0.0367524	+	0.999324i	$0.511701\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	− 12.0000i	− 0.489083i
$603$	− 10.4853i	− 0.426994i
$604$	45.9411	1.86932
$605$	0	0
$606$	20.9706	0.851871
$607$	− 42.9706i	− 1.74412i	−0.489398	−	0.872061i	$-0.662783\pi$
$607$	− 42.9706i	− 1.74412i	0.489398	−	0.872061i	$-0.337217\pi$
$608$	1.31371i	0.0532779i
$609$	1.65685	0.0671391
$610$	0	0
$611$	0.686292	0.0277644
$612$	10.8284i	0.437713i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	26.4853	1.06886
$615$	0	0
$616$	17.6569	0.711415
$617$	14.8284i	0.596970i	0.954414	+	0.298485i	$0.0964813\pi$
$617$	14.8284i	0.596970i	−0.954414	+	0.298485i	$0.903519\pi$
$618$	− 58.6274i	− 2.35834i
$619$	−29.7990	−1.19772	−0.598861	−	0.800853i	$-0.704380\pi$
$619$	−29.7990	−1.19772	−0.598861	+	0.800853i	$0.704380\pi$
$620$	0	0
$621$	−35.3137	−1.41709
$622$	− 6.00000i	− 0.240578i
$623$	3.02944i	0.121372i
$624$	−12.0000	−0.480384
$625$	0	0
$626$	14.4853	0.578948
$627$	− 8.00000i	− 0.319489i
$628$	− 63.1127i	− 2.51847i
$629$	24.0000	0.956943
$630$	0	0
$631$	3.02944	0.120600	0.0603000	−	0.998180i	$-0.480794\pi$
$631$	3.02944	0.120600	0.0603000	+	0.998180i	$0.480794\pi$
$632$	63.9411i	2.54344i
$633$	− 9.65685i	− 0.383825i
$634$	−6.82843	−0.271191
$635$	0	0
$636$	−58.6274	−2.32473
$637$	− 12.6274i	− 0.500316i
$638$	11.6569i	0.461499i
$639$	−7.31371	−0.289326
$640$	0	0
$641$	−44.6274	−1.76268	−0.881338	−	0.472485i	$-0.843357\pi$
$641$	−44.6274	−1.76268	−0.881338	+	0.472485i	$0.843357\pi$
$642$	39.3137i	1.55159i
$643$	− 31.4558i	− 1.24050i	−0.784405	−	0.620249i	$-0.787032\pi$
$643$	− 31.4558i	− 1.24050i	0.784405	−	0.620249i	$-0.212968\pi$
$644$	28.0000	1.10335
$645$	0	0
$646$	5.65685	0.222566
$647$	− 21.1127i	− 0.830026i	−0.909816	−	0.415013i	$-0.863777\pi$
$647$	− 21.1127i	− 0.830026i	0.909816	−	0.415013i	$-0.136223\pi$
$648$	48.5563i	1.90747i
$649$	0	0
$650$	0	0
$651$	17.3726	0.680885
$652$	75.2548i	2.94721i
$653$	− 22.8284i	− 0.893345i	−0.894697	−	0.446673i	$-0.852609\pi$
$653$	− 22.8284i	− 0.893345i	0.894697	−	0.446673i	$-0.147391\pi$
$654$	−9.65685	−0.377613
$655$	0	0
$656$	−18.0000	−0.702782
$657$	8.48528i	0.331042i
$658$	0.686292i	0.0267544i
$659$	37.7990	1.47244	0.736220	−	0.676743i	$-0.236609\pi$
$659$	37.7990	1.47244	0.736220	+	0.676743i	$0.236609\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	− 42.9706i	− 1.67010i
$663$	− 11.3137i	− 0.439388i
$664$	56.6274	2.19757
$665$	0	0
$666$	−20.4853	−0.793789
$667$	8.82843i	0.341838i
$668$	55.4558i	2.14565i
$669$	43.5980	1.68560
$670$	0	0
$671$	−36.9706	−1.42723
$672$	2.62742i	0.101355i
$673$	− 10.9706i	− 0.422884i	−0.977391	−	0.211442i	$-0.932184\pi$
$673$	− 10.9706i	− 0.422884i	0.977391	−	0.211442i	$-0.0678160\pi$
$674$	−16.4853	−0.634989
$675$	0	0
$676$	−34.4558	−1.32522
$677$	− 36.7696i	− 1.41317i	−0.707629	−	0.706584i	$-0.750235\pi$
$677$	− 36.7696i	− 1.41317i	0.707629	−	0.706584i	$-0.249765\pi$
$678$	13.6569i	0.524488i
$679$	−3.71573	−0.142597
$680$	0	0
$681$	16.2843	0.624015
$682$	122.225i	4.68025i
$683$	40.1421i	1.53600i	0.640452	+	0.767998i	$0.278747\pi$
$683$	40.1421i	1.53600i	−0.640452	+	0.767998i	$0.721253\pi$
$684$	−3.17157	−0.121268
$685$	0	0
$686$	26.6274	1.01664
$687$	− 4.00000i	− 0.152610i
$688$	− 18.0000i	− 0.686244i
$689$	15.3137	0.583406
$690$	0	0
$691$	−11.0294	−0.419580	−0.209790	−	0.977747i	$-0.567278\pi$
$691$	−11.0294	−0.419580	−0.209790	+	0.977747i	$0.567278\pi$
$692$	20.3431i	0.773330i
$693$	− 4.00000i	− 0.151947i
$694$	48.6274	1.84587
$695$	0	0
$696$	8.82843	0.334641
$697$	− 16.9706i	− 0.642806i
$698$	59.4558i	2.25044i
$699$	36.0000	1.36165
$700$	0	0
$701$	29.3137	1.10716	0.553582	−	0.832795i	$-0.313261\pi$
$701$	29.3137	1.10716	0.553582	+	0.832795i	$0.313261\pi$
$702$	− 19.3137i	− 0.728949i
$703$	7.02944i	0.265120i
$704$	−47.4558	−1.78856
$705$	0	0
$706$	37.7990	1.42258
$707$	− 3.59798i	− 0.135316i
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	14.4853	0.543240
$712$	16.1421i	0.604952i
$713$	92.5685i	3.46672i
$714$	11.3137	0.423405
$715$	0	0
$716$	−2.62742	−0.0981912
$717$	− 46.6274i	− 1.74133i
$718$	77.5980i	2.89593i
$719$	−10.6274	−0.396336	−0.198168	−	0.980168i	$-0.563499\pi$
$719$	−10.6274	−0.396336	−0.198168	+	0.980168i	$0.563499\pi$
$720$	0	0
$721$	−10.0589	−0.374612
$722$	− 44.2132i	− 1.64545i
$723$	− 20.0000i	− 0.743808i
$724$	22.9706	0.853694
$725$	0	0
$726$	59.4558	2.20661
$727$	− 43.9411i	− 1.62969i	−0.579682	−	0.814843i	$-0.696823\pi$
$727$	− 43.9411i	− 1.62969i	0.579682	−	0.814843i	$-0.303177\pi$
$728$	7.31371i	0.271064i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	16.9706	0.627679
$732$	58.6274i	2.16693i
$733$	− 17.1716i	− 0.634247i	−0.948384	−	0.317123i	$-0.897283\pi$
$733$	− 17.1716i	− 0.634247i	0.948384	−	0.317123i	$-0.102717\pi$
$734$	43.4558	1.60398
$735$	0	0
$736$	−14.0000	−0.516047
$737$	− 50.6274i	− 1.86488i
$738$	14.4853i	0.533211i
$739$	2.48528	0.0914226	0.0457113	−	0.998955i	$-0.485445\pi$
$739$	2.48528	0.0914226	0.0457113	+	0.998955i	$0.485445\pi$
$740$	0	0
$741$	3.31371	0.121732
$742$	15.3137i	0.562184i
$743$	− 7.37258i	− 0.270474i	−0.990813	−	0.135237i	$-0.956820\pi$
$743$	− 7.37258i	− 0.270474i	0.990813	−	0.135237i	$-0.0431796\pi$
$744$	92.5685	3.39373
$745$	0	0
$746$	−65.1127	−2.38395
$747$	− 12.8284i	− 0.469368i
$748$	52.2843i	1.91170i
$749$	6.74517	0.246463
$750$	0	0
$751$	−12.1421	−0.443073	−0.221536	−	0.975152i	$-0.571107\pi$
$751$	−12.1421	−0.443073	−0.221536	+	0.975152i	$0.571107\pi$
$752$	1.02944i	0.0375397i
$753$	− 6.34315i	− 0.231157i
$754$	−4.82843	−0.175841
$755$	0	0
$756$	12.6863	0.461396
$757$	− 36.4853i	− 1.32608i	−0.748584	−	0.663040i	$-0.769266\pi$
$757$	− 36.4853i	− 1.32608i	0.748584	−	0.663040i	$-0.230734\pi$
$758$	− 13.3137i	− 0.483576i
$759$	85.2548	3.09455
$760$	0	0
$761$	−36.6274	−1.32774	−0.663871	−	0.747847i	$-0.731088\pi$
$761$	−36.6274	−1.32774	−0.663871	+	0.747847i	$0.731088\pi$
$762$	− 28.9706i	− 1.04949i
$763$	1.65685i	0.0599822i
$764$	58.0833	2.10138
$765$	0	0
$766$	34.9706	1.26354
$767$	0	0
$768$	59.9411i	2.16294i
$769$	4.34315	0.156618	0.0783089	−	0.996929i	$-0.475048\pi$
$769$	4.34315	0.156618	0.0783089	+	0.996929i	$0.475048\pi$
$770$	0	0
$771$	−58.6274	−2.11141
$772$	47.7990i	1.72032i
$773$	8.48528i	0.305194i	0.988288	+	0.152597i	$0.0487637\pi$
$773$	8.48528i	0.305194i	−0.988288	+	0.152597i	$0.951236\pi$
$774$	−14.4853	−0.520663
$775$	0	0
$776$	−19.7990	−0.710742
$777$	14.0589i	0.504359i
$778$	16.1421i	0.578724i
$779$	4.97056	0.178089
$780$	0	0
$781$	−35.3137	−1.26362
$782$	60.2843i	2.15576i
$783$	4.00000i	0.142948i
$784$	18.9411	0.676469
$785$	0	0
$786$	77.9411	2.78007
$787$	21.7990i	0.777050i	0.921438	+	0.388525i	$0.127015\pi$
$787$	21.7990i	0.777050i	−0.921438	+	0.388525i	$0.872985\pi$
$788$	− 31.9411i	− 1.13786i
$789$	−16.6863	−0.594048
$790$	0	0
$791$	2.34315	0.0833127
$792$	− 21.3137i	− 0.757350i
$793$	− 15.3137i	− 0.543806i
$794$	−20.1421	−0.714818
$795$	0	0
$796$	45.9411	1.62834
$797$	34.1421i	1.20938i	0.796462	+	0.604688i	$0.206702\pi$
$797$	34.1421i	1.20938i	−0.796462	+	0.604688i	$0.793298\pi$
$798$	3.31371i	0.117304i
$799$	−0.970563	−0.0343360
$800$	0	0
$801$	3.65685	0.129209
$802$	− 70.7696i	− 2.49896i
$803$	40.9706i	1.44582i
$804$	−80.2843	−2.83141
$805$	0	0
$806$	−50.6274	−1.78327
$807$	2.62742i	0.0924895i
$808$	− 19.1716i	− 0.674454i
$809$	14.2843	0.502208	0.251104	−	0.967960i	$-0.419206\pi$
$809$	14.2843	0.502208	0.251104	+	0.967960i	$0.419206\pi$
$810$	0	0
$811$	26.3431	0.925033	0.462516	−	0.886611i	$-0.346947\pi$
$811$	26.3431	0.925033	0.462516	+	0.886611i	$0.346947\pi$
$812$	− 3.17157i	− 0.111300i
$813$	− 59.5980i	− 2.09019i
$814$	−98.9117	−3.46685
$815$	0	0
$816$	16.9706	0.594089
$817$	4.97056i	0.173898i
$818$	− 74.7696i	− 2.61426i
$819$	1.65685	0.0578952
$820$	0	0
$821$	−45.3137	−1.58146	−0.790730	−	0.612165i	$-0.790299\pi$
$821$	−45.3137	−1.58146	−0.790730	+	0.612165i	$0.790299\pi$
$822$	52.2843i	1.82362i
$823$	2.97056i	0.103547i	0.998659	+	0.0517737i	$0.0164874\pi$
$823$	2.97056i	0.103547i	−0.998659	+	0.0517737i	$0.983513\pi$
$824$	−53.5980	−1.86717
$825$	0	0
$826$	0	0
$827$	5.31371i	0.184776i	0.995723	+	0.0923879i	$0.0294500\pi$
$827$	5.31371i	0.184776i	−0.995723	+	0.0923879i	$0.970550\pi$
$828$	− 33.7990i	− 1.17460i
$829$	24.6274	0.855346	0.427673	−	0.903934i	$-0.359334\pi$
$829$	24.6274	0.855346	0.427673	+	0.903934i	$0.359334\pi$
$830$	0	0
$831$	−15.3137	−0.531227
$832$	− 19.6569i	− 0.681479i
$833$	17.8579i	0.618738i
$834$	−49.9411	−1.72932
$835$	0	0
$836$	−15.3137	−0.529636
$837$	41.9411i	1.44970i
$838$	12.0000i	0.414533i
$839$	14.4853	0.500087	0.250044	−	0.968235i	$-0.419555\pi$
$839$	14.4853	0.500087	0.250044	+	0.968235i	$0.419555\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 36.1421i	− 1.24554i
$843$	13.3726i	0.460576i
$844$	−18.4853	−0.636290
$845$	0	0
$846$	0.828427	0.0284819
$847$	− 10.2010i	− 0.350511i
$848$	22.9706i	0.788812i
$849$	−1.65685	−0.0568631
$850$	0	0
$851$	−74.9117	−2.56794
$852$	56.0000i	1.91853i
$853$	− 11.1127i	− 0.380492i	−0.981736	−	0.190246i	$-0.939072\pi$
$853$	− 11.1127i	− 0.380492i	0.981736	−	0.190246i	$-0.0609285\pi$
$854$	15.3137	0.524024
$855$	0	0
$856$	35.9411	1.22844
$857$	− 48.6274i	− 1.66108i	−0.556958	−	0.830540i	$-0.688032\pi$
$857$	− 48.6274i	− 1.66108i	0.556958	−	0.830540i	$-0.311968\pi$
$858$	46.6274i	1.59183i
$859$	−28.4264	−0.969896	−0.484948	−	0.874543i	$-0.661162\pi$
$859$	−28.4264	−0.969896	−0.484948	+	0.874543i	$0.661162\pi$
$860$	0	0
$861$	9.94113	0.338793
$862$	− 46.6274i	− 1.58814i
$863$	− 7.85786i	− 0.267485i	−0.991016	−	0.133742i	$-0.957301\pi$
$863$	− 7.85786i	− 0.267485i	0.991016	−	0.133742i	$-0.0426995\pi$
$864$	−6.34315	−0.215798
$865$	0	0
$866$	84.0833	2.85727
$867$	− 18.0000i	− 0.611312i
$868$	− 33.2548i	− 1.12874i
$869$	69.9411	2.37259
$870$	0	0
$871$	20.9706	0.710560
$872$	8.82843i	0.298968i
$873$	4.48528i	0.151804i
$874$	−17.6569	−0.597252
$875$	0	0
$876$	64.9706	2.19515
$877$	18.2843i	0.617416i	0.951157	+	0.308708i	$0.0998966\pi$
$877$	18.2843i	0.617416i	−0.951157	+	0.308708i	$0.900103\pi$
$878$	52.2843i	1.76451i
$879$	−16.9706	−0.572403
$880$	0	0
$881$	6.68629	0.225267	0.112633	−	0.993637i	$-0.464071\pi$
$881$	6.68629	0.225267	0.112633	+	0.993637i	$0.464071\pi$
$882$	− 15.2426i	− 0.513246i
$883$	2.48528i	0.0836364i	0.999125	+	0.0418182i	$0.0133150\pi$
$883$	2.48528i	0.0836364i	−0.999125	+	0.0418182i	$0.986685\pi$
$884$	−21.6569	−0.728399
$885$	0	0
$886$	−8.82843	−0.296597
$887$	29.3137i	0.984258i	0.870522	+	0.492129i	$0.163781\pi$
$887$	29.3137i	0.984258i	−0.870522	+	0.492129i	$0.836219\pi$
$888$	74.9117i	2.51387i
$889$	−4.97056	−0.166707
$890$	0	0
$891$	53.1127	1.77934
$892$	− 83.4558i	− 2.79431i
$893$	− 0.284271i	− 0.00951277i
$894$	64.2843	2.14999
$895$	0	0
$896$	17.0294	0.568914
$897$	35.3137i	1.17909i
$898$	− 0.828427i	− 0.0276450i
$899$	10.4853	0.349704
$900$	0	0
$901$	−21.6569	−0.721494
$902$	69.9411i	2.32878i
$903$	9.94113i	0.330820i
$904$	12.4853	0.415254
$905$	0	0
$906$	−57.9411	−1.92496
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	− 31.1716i	− 1.03446i
$909$	−4.34315	−0.144053
$910$	0	0
$911$	3.85786	0.127817	0.0639084	−	0.997956i	$-0.479643\pi$
$911$	3.85786	0.127817	0.0639084	+	0.997956i	$0.479643\pi$
$912$	4.97056i	0.164592i
$913$	− 61.9411i	− 2.04995i
$914$	20.1421	0.666243
$915$	0	0
$916$	−7.65685	−0.252990
$917$	− 13.3726i	− 0.441602i
$918$	27.3137i	0.901487i
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	−21.9411	−0.722985
$922$	− 58.7696i	− 1.93547i
$923$	− 14.6274i	− 0.481467i
$924$	−30.6274	−1.00757
$925$	0	0
$926$	42.9706	1.41210
$927$	12.1421i	0.398800i
$928$	1.58579i	0.0520560i
$929$	−40.6274	−1.33294	−0.666471	−	0.745531i	$-0.732196\pi$
$929$	−40.6274	−1.33294	−0.666471	+	0.745531i	$0.732196\pi$
$930$	0	0
$931$	−5.23045	−0.171421
$932$	− 68.9117i	− 2.25728i
$933$	4.97056i	0.162729i
$934$	−55.4558	−1.81457
$935$	0	0
$936$	8.82843	0.288566
$937$	− 8.34315i	− 0.272559i	−0.990670	−	0.136279i	$-0.956486\pi$
$937$	− 8.34315i	− 0.272559i	0.990670	−	0.136279i	$-0.0435145\pi$
$938$	20.9706i	0.684713i
$939$	−12.0000	−0.391605
$940$	0	0
$941$	39.9411	1.30204	0.651022	−	0.759059i	$-0.274341\pi$
$941$	39.9411	1.30204	0.651022	+	0.759059i	$0.274341\pi$
$942$	79.5980i	2.59344i
$943$	52.9706i	1.72496i
$944$	0	0
$945$	0	0
$946$	−69.9411	−2.27398
$947$	56.9117i	1.84938i	0.380719	+	0.924691i	$0.375676\pi$
$947$	56.9117i	1.84938i	−0.380719	+	0.924691i	$0.624324\pi$
$948$	− 110.912i	− 3.60224i
$949$	−16.9706	−0.550888
$950$	0	0
$951$	5.65685	0.183436
$952$	− 10.3431i	− 0.335223i
$953$	6.68629i	0.216590i	0.994119	+	0.108295i	$0.0345391\pi$
$953$	6.68629i	0.216590i	−0.994119	+	0.108295i	$0.965461\pi$
$954$	18.4853	0.598483
$955$	0	0
$956$	−89.2548	−2.88671
$957$	− 9.65685i	− 0.312162i
$958$	− 30.9706i	− 1.00061i
$959$	8.97056	0.289675
$960$	0	0
$961$	78.9411	2.54649
$962$	− 40.9706i	− 1.32094i
$963$	− 8.14214i	− 0.262377i
$964$	−38.2843	−1.23305
$965$	0	0
$966$	−35.3137	−1.13620
$967$	18.9706i	0.610052i	0.952344	+	0.305026i	$0.0986652\pi$
$967$	18.9706i	0.610052i	−0.952344	+	0.305026i	$0.901335\pi$
$968$	− 54.3553i	− 1.74705i
$969$	−4.68629	−0.150545
$970$	0	0
$971$	−0.142136	−0.00456135	−0.00228067	−	0.999997i	$-0.500726\pi$
$971$	−0.142136	−0.00456135	−0.00228067	+	0.999997i	$0.500726\pi$
$972$	− 38.2843i	− 1.22797i
$973$	8.56854i	0.274695i
$974$	−71.9411	−2.30514
$975$	0	0
$976$	22.9706	0.735270
$977$	25.3137i	0.809857i	0.914348	+	0.404929i	$0.132704\pi$
$977$	25.3137i	0.809857i	−0.914348	+	0.404929i	$0.867296\pi$
$978$	− 94.9117i	− 3.03494i
$979$	17.6569	0.564316
$980$	0	0
$981$	2.00000	0.0638551
$982$	104.912i	3.34787i
$983$	13.3137i	0.424641i	0.977200	+	0.212321i	$0.0681021\pi$
$983$	13.3137i	0.424641i	−0.977200	+	0.212321i	$0.931898\pi$
$984$	52.9706	1.68864
$985$	0	0
$986$	6.82843	0.217461
$987$	− 0.568542i	− 0.0180969i
$988$	− 6.34315i	− 0.201802i
$989$	−52.9706	−1.68437
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	16.6274i	0.527921i
$993$	35.5980i	1.12967i
$994$	14.6274	0.463953
$995$	0	0
$996$	−98.2254	−3.11239
$997$	− 1.17157i	− 0.0371041i	−0.999828	−	0.0185520i	$-0.994094\pi$
$997$	− 1.17157i	− 0.0371041i	0.999828	−	0.0185520i	$-0.00590564\pi$
$998$	− 86.9117i	− 2.75114i
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.b.c.349.4		4
5.2	odd	4		145.2.a.b.1.1	✓	2
5.3	odd	4		725.2.a.c.1.2		2
5.4	even	2	inner	725.2.b.c.349.1		4
15.2	even	4		1305.2.a.n.1.2		2
15.8	even	4		6525.2.a.p.1.1		2
20.7	even	4		2320.2.a.k.1.1		2
35.27	even	4		7105.2.a.e.1.1		2
40.27	even	4		9280.2.a.w.1.1		2
40.37	odd	4		9280.2.a.be.1.2		2
145.57	odd	4		4205.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.1	✓	2	5.2	odd	4
725.2.a.c.1.2		2	5.3	odd	4
725.2.b.c.349.1		4	5.4	even	2	inner
725.2.b.c.349.4		4	1.1	even	1	trivial
1305.2.a.n.1.2		2	15.2	even	4
2320.2.a.k.1.1		2	20.7	even	4
4205.2.a.d.1.2		2	145.57	odd	4
6525.2.a.p.1.1		2	15.8	even	4
7105.2.a.e.1.1		2	35.27	even	4
9280.2.a.w.1.1		2	40.27	even	4
9280.2.a.be.1.2		2	40.37	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.41421i q^{2} -2.00000i q^{3} -3.82843 q^{4} +4.82843 q^{6} -0.828427i q^{7} -4.41421i q^{8} -1.00000 q^{9} -4.82843 q^{11} +7.65685i q^{12} -2.00000i q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843i q^{17} -2.41421i q^{18} -0.828427 q^{19} -1.65685 q^{21} -11.6569i q^{22} -8.82843i q^{23} -8.82843 q^{24} +4.82843 q^{26} -4.00000i q^{27} +3.17157i q^{28} -1.00000 q^{29} -10.4853 q^{31} -1.58579i q^{32} +9.65685i q^{33} -6.82843 q^{34} +3.82843 q^{36} -8.48528i q^{37} -2.00000i q^{38} -4.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -6.00000i q^{43} +18.4853 q^{44} +21.3137 q^{46} +0.343146i q^{47} -6.00000i q^{48} +6.31371 q^{49} +5.65685 q^{51} +7.65685i q^{52} +7.65685i q^{53} +9.65685 q^{54} -3.65685 q^{56} +1.65685i q^{57} -2.41421i q^{58} +7.65685 q^{61} -25.3137i q^{62} +0.828427i q^{63} +9.82843 q^{64} -23.3137 q^{66} +10.4853i q^{67} -10.8284i q^{68} -17.6569 q^{69} +7.31371 q^{71} +4.41421i q^{72} -8.48528i q^{73} +20.4853 q^{74} +3.17157 q^{76} +4.00000i q^{77} -9.65685i q^{78} -14.4853 q^{79} -11.0000 q^{81} -14.4853i q^{82} +12.8284i q^{83} +6.34315 q^{84} +14.4853 q^{86} +2.00000i q^{87} +21.3137i q^{88} -3.65685 q^{89} -1.65685 q^{91} +33.7990i q^{92} +20.9706i q^{93} -0.828427 q^{94} -3.17157 q^{96} -4.48528i q^{97} +15.2426i q^{98} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 8 q^{11} + 8 q^{14} + 12 q^{16} + 8 q^{19} + 16 q^{21} - 24 q^{24} + 8 q^{26} - 4 q^{29} - 8 q^{31} - 16 q^{34} + 4 q^{36} - 16 q^{39} - 24 q^{41} + 40 q^{44} + 40 q^{46}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^6 - 4 * q^9 - 8 * q^11 + 8 * q^14 + 12 * q^16 + 8 * q^19 + 16 * q^21 - 24 * q^24 + 8 * q^26 - 4 * q^29 - 8 * q^31 - 16 * q^34 + 4 * q^36 - 16 * q^39 - 24 * q^41 + 40 * q^44 + 40 * q^46 - 20 * q^49 + 16 * q^54 + 8 * q^56 + 8 * q^61 + 28 * q^64 - 48 * q^66 - 48 * q^69 - 16 * q^71 + 48 * q^74 + 24 * q^76 - 24 * q^79 - 44 * q^81 + 48 * q^84 + 24 * q^86 + 8 * q^89 + 16 * q^91 + 8 * q^94 - 24 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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