LMFDB - Embedded newform 725.2.b.c.349.2

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			349.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	725.349
Dual form			725.2.b.c.349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.414214i q^{2} -2.00000i q^{3} +1.82843 q^{4} -0.828427 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} +0.828427 q^{11} -3.65685i q^{12} -2.00000i q^{13} +2.00000 q^{14} +3.00000 q^{16} -2.82843i q^{17} +0.414214i q^{18} +4.82843 q^{19} +9.65685 q^{21} -0.343146i q^{22} -3.17157i q^{23} -3.17157 q^{24} -0.828427 q^{26} -4.00000i q^{27} +8.82843i q^{28} -1.00000 q^{29} +6.48528 q^{31} -4.41421i q^{32} -1.65685i q^{33} -1.17157 q^{34} -1.82843 q^{36} +8.48528i q^{37} -2.00000i q^{38} -4.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -6.00000i q^{43} +1.51472 q^{44} -1.31371 q^{46} +11.6569i q^{47} -6.00000i q^{48} -16.3137 q^{49} -5.65685 q^{51} -3.65685i q^{52} -3.65685i q^{53} -1.65685 q^{54} +7.65685 q^{56} -9.65685i q^{57} +0.414214i q^{58} -3.65685 q^{61} -2.68629i q^{62} -4.82843i q^{63} +4.17157 q^{64} -0.686292 q^{66} -6.48528i q^{67} -5.17157i q^{68} -6.34315 q^{69} -15.3137 q^{71} +1.58579i q^{72} +8.48528i q^{73} +3.51472 q^{74} +8.82843 q^{76} +4.00000i q^{77} +1.65685i q^{78} +2.48528 q^{79} -11.0000 q^{81} +2.48528i q^{82} +7.17157i q^{83} +17.6569 q^{84} -2.48528 q^{86} +2.00000i q^{87} -1.31371i q^{88} +7.65685 q^{89} +9.65685 q^{91} -5.79899i q^{92} -12.9706i q^{93} +4.82843 q^{94} -8.82843 q^{96} +12.4853i q^{97} +6.75736i q^{98} -0.828427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 0.414214i	− 0.292893i	−0.989219	−	0.146447i	$-0.953216\pi$
$2$	− 0.414214i	− 0.292893i	0.989219	−	0.146447i	$-0.0467837\pi$
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	1.82843	0.914214
$5$	0	0
$5$	0	0
$6$	−0.828427	−0.338204
$7$	4.82843i	1.82497i	0.409106	+	0.912487i	$0.365841\pi$
$7$	4.82843i	1.82497i	−0.409106	+	0.912487i	$0.634159\pi$
$8$	− 1.58579i	− 0.560660i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	− 3.65685i	− 1.05564i
$13$	− 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	$-0.888558\pi$
$17$	− 2.82843i	− 0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$	0.414214i	0.0976311i
$19$	4.82843	1.10772	0.553859	−	0.832611i	$-0.313155\pi$
$19$	4.82843	1.10772	0.553859	+	0.832611i	$0.313155\pi$
$20$	0	0
$21$	9.65685	2.10730
$22$	− 0.343146i	− 0.0731589i
$23$	− 3.17157i	− 0.661319i	−0.943750	−	0.330659i	$-0.892729\pi$
$23$	− 3.17157i	− 0.661319i	0.943750	−	0.330659i	$-0.107271\pi$
$24$	−3.17157	−0.647395
$25$	0	0
$26$	−0.828427	−0.162468
$27$	− 4.00000i	− 0.769800i
$28$	8.82843i	1.66842i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.48528	1.16479	0.582395	−	0.812906i	$-0.302116\pi$
$31$	6.48528	1.16479	0.582395	+	0.812906i	$0.302116\pi$
$32$	− 4.41421i	− 0.780330i
$33$	− 1.65685i	− 0.288421i
$34$	−1.17157	−0.200923
$35$	0	0
$36$	−1.82843	−0.304738
$37$	8.48528i	1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$	8.48528i	1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$	− 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	$-0.151253\pi$
$44$	1.51472	0.228352
$45$	0	0
$46$	−1.31371	−0.193696
$47$	11.6569i	1.70033i	0.526519	+	0.850163i	$0.323497\pi$
$47$	11.6569i	1.70033i	−0.526519	+	0.850163i	$0.676503\pi$
$48$	− 6.00000i	− 0.866025i
$49$	−16.3137	−2.33053
$50$	0	0
$51$	−5.65685	−0.792118
$52$	− 3.65685i	− 0.507114i
$53$	− 3.65685i	− 0.502308i	−0.967947	−	0.251154i	$-0.919190\pi$
$53$	− 3.65685i	− 0.502308i	0.967947	−	0.251154i	$-0.0808100\pi$
$54$	−1.65685	−0.225469
$55$	0	0
$56$	7.65685	1.02319
$57$	− 9.65685i	− 1.27908i
$58$	0.414214i	0.0543889i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−3.65685	−0.468212	−0.234106	−	0.972211i	$-0.575216\pi$
$61$	−3.65685	−0.468212	−0.234106	+	0.972211i	$0.575216\pi$
$62$	− 2.68629i	− 0.341159i
$63$	− 4.82843i	− 0.608325i
$64$	4.17157	0.521447
$65$	0	0
$66$	−0.686292	−0.0844766
$67$	− 6.48528i	− 0.792303i	−0.918185	−	0.396152i	$-0.870345\pi$
$67$	− 6.48528i	− 0.792303i	0.918185	−	0.396152i	$-0.129655\pi$
$68$	− 5.17157i	− 0.627145i
$69$	−6.34315	−0.763625
$70$	0	0
$71$	−15.3137	−1.81740	−0.908701	−	0.417447i	$-0.862925\pi$
$71$	−15.3137	−1.81740	−0.908701	+	0.417447i	$0.862925\pi$
$72$	1.58579i	0.186887i
$73$	8.48528i	0.993127i	0.868000	+	0.496564i	$0.165405\pi$
$73$	8.48528i	0.993127i	−0.868000	+	0.496564i	$0.834595\pi$
$74$	3.51472	0.408578
$75$	0	0
$76$	8.82843	1.01269
$77$	4.00000i	0.455842i
$78$	1.65685i	0.187602i
$79$	2.48528	0.279616	0.139808	−	0.990179i	$-0.455351\pi$
$79$	2.48528	0.279616	0.139808	+	0.990179i	$0.455351\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	2.48528i	0.274453i
$83$	7.17157i	0.787182i	0.919286	+	0.393591i	$0.128767\pi$
$83$	7.17157i	0.787182i	−0.919286	+	0.393591i	$0.871233\pi$
$84$	17.6569	1.92652
$85$	0	0
$86$	−2.48528	−0.267995
$87$	2.00000i	0.214423i
$88$	− 1.31371i	− 0.140042i
$89$	7.65685	0.811625	0.405812	−	0.913956i	$-0.366989\pi$
$89$	7.65685	0.811625	0.405812	+	0.913956i	$0.366989\pi$
$90$	0	0
$91$	9.65685	1.01231
$92$	− 5.79899i	− 0.604586i
$93$	− 12.9706i	− 1.34498i
$94$	4.82843	0.498014
$95$	0	0
$96$	−8.82843	−0.901048
$97$	12.4853i	1.26769i	0.773461	+	0.633844i	$0.218524\pi$
$97$	12.4853i	1.26769i	−0.773461	+	0.633844i	$0.781476\pi$
$98$	6.75736i	0.682596i
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	15.6569	1.55792	0.778958	−	0.627077i	$-0.215749\pi$
$101$	15.6569	1.55792	0.778958	+	0.627077i	$0.215749\pi$
$102$	2.34315i	0.232006i
$103$	16.1421i	1.59053i	0.606261	+	0.795266i	$0.292669\pi$
$103$	16.1421i	1.59053i	−0.606261	+	0.795266i	$0.707331\pi$
$104$	−3.17157	−0.310998
$105$	0	0
$106$	−1.51472	−0.147122
$107$	− 20.1421i	− 1.94721i	−0.228232	−	0.973607i	$-0.573294\pi$
$107$	− 20.1421i	− 1.94721i	0.228232	−	0.973607i	$-0.426706\pi$
$108$	− 7.31371i	− 0.703762i
$109$	−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
$112$	14.4853i	1.36873i
$113$	− 2.82843i	− 0.266076i	−0.991111	−	0.133038i	$-0.957527\pi$
$113$	− 2.82843i	− 0.266076i	0.991111	−	0.133038i	$-0.0424732\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−1.82843	−0.169765
$117$	2.00000i	0.184900i
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	−10.3137	−0.937610
$122$	1.51472i	0.137136i
$123$	12.0000i	1.08200i
$124$	11.8579	1.06487
$125$	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	$-0.0857705\pi$
$128$	− 10.5563i	− 0.933058i
$129$	−12.0000	−1.05654
$130$	0	0
$131$	−12.1421	−1.06086	−0.530432	−	0.847728i	$-0.677970\pi$
$131$	−12.1421	−1.06086	−0.530432	+	0.847728i	$0.677970\pi$
$132$	− 3.02944i	− 0.263679i
$133$	23.3137i	2.02155i
$134$	−2.68629	−0.232060
$135$	0	0
$136$	−4.48528	−0.384610
$137$	5.17157i	0.441837i	0.975292	+	0.220919i	$0.0709055\pi$
$137$	5.17157i	0.441837i	−0.975292	+	0.220919i	$0.929094\pi$
$138$	2.62742i	0.223661i
$139$	−21.6569	−1.83691	−0.918455	−	0.395525i	$-0.870563\pi$
$139$	−21.6569	−1.83691	−0.918455	+	0.395525i	$0.870563\pi$
$140$	0	0
$141$	23.3137	1.96337
$142$	6.34315i	0.532305i
$143$	− 1.65685i	− 0.138553i
$144$	−3.00000	−0.250000
$145$	0	0
$146$	3.51472	0.290880
$147$	32.6274i	2.69106i
$148$	15.5147i	1.27530i
$149$	−9.31371	−0.763009	−0.381504	−	0.924367i	$-0.624594\pi$
$149$	−9.31371	−0.763009	−0.381504	+	0.924367i	$0.624594\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	− 7.65685i	− 0.621053i
$153$	2.82843i	0.228665i
$154$	1.65685	0.133513
$155$	0	0
$156$	−7.31371	−0.585565
$157$	− 0.485281i	− 0.0387297i	−0.999812	−	0.0193648i	$-0.993836\pi$
$157$	− 0.485281i	− 0.0387297i	0.999812	−	0.0193648i	$-0.00616440\pi$
$158$	− 1.02944i	− 0.0818976i
$159$	−7.31371	−0.580015
$160$	0	0
$161$	15.3137	1.20689
$162$	4.55635i	0.357981i
$163$	− 8.34315i	− 0.653486i	−0.945113	−	0.326743i	$-0.894049\pi$
$163$	− 8.34315i	− 0.653486i	0.945113	−	0.326743i	$-0.105951\pi$
$164$	−10.9706	−0.856657
$165$	0	0
$166$	2.97056	0.230560
$167$	2.48528i	0.192317i	0.995366	+	0.0961584i	$0.0306555\pi$
$167$	2.48528i	0.192317i	−0.995366	+	0.0961584i	$0.969344\pi$
$168$	− 15.3137i	− 1.18148i
$169$	9.00000	0.692308
$170$	0	0
$171$	−4.82843	−0.369239
$172$	− 10.9706i	− 0.836498i
$173$	17.3137i	1.31634i	0.752871	+	0.658168i	$0.228669\pi$
$173$	17.3137i	1.31634i	−0.752871	+	0.658168i	$0.771331\pi$
$174$	0.828427	0.0628029
$175$	0	0
$176$	2.48528	0.187335
$177$	0	0
$178$	− 3.17157i	− 0.237719i
$179$	23.3137	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$179$	23.3137	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$180$	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
$183$	7.31371i	0.540645i
$184$	−5.02944	−0.370775
$185$	0	0
$186$	−5.37258	−0.393937
$187$	− 2.34315i	− 0.171348i
$188$	21.3137i	1.55446i
$189$	19.3137	1.40487
$190$	0	0
$191$	−20.8284	−1.50709	−0.753546	−	0.657395i	$-0.771658\pi$
$191$	−20.8284	−1.50709	−0.753546	+	0.657395i	$0.771658\pi$
$192$	− 8.34315i	− 0.602115i
$193$	4.48528i	0.322858i	0.986884	+	0.161429i	$0.0516102\pi$
$193$	4.48528i	0.322858i	−0.986884	+	0.161429i	$0.948390\pi$
$194$	5.17157	0.371297
$195$	0	0
$196$	−29.8284	−2.13060
$197$	19.6569i	1.40049i	0.713901	+	0.700246i	$0.246927\pi$
$197$	19.6569i	1.40049i	−0.713901	+	0.700246i	$0.753073\pi$
$198$	0.343146i	0.0243863i
$199$	−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
$201$	−12.9706	−0.914873
$202$	− 6.48528i	− 0.456303i
$203$	− 4.82843i	− 0.338889i
$204$	−10.3431	−0.724165
$205$	0	0
$206$	6.68629	0.465856
$207$	3.17157i	0.220440i
$208$	− 6.00000i	− 0.416025i
$209$	4.00000	0.276686
$210$	0	0
$211$	−0.828427	−0.0570313	−0.0285156	−	0.999593i	$-0.509078\pi$
$211$	−0.828427	−0.0570313	−0.0285156	+	0.999593i	$0.509078\pi$
$212$	− 6.68629i	− 0.459216i
$213$	30.6274i	2.09856i
$214$	−8.34315	−0.570326
$215$	0	0
$216$	−6.34315	−0.431596
$217$	31.3137i	2.12571i
$218$	0.828427i	0.0561082i
$219$	16.9706	1.14676
$220$	0	0
$221$	−5.65685	−0.380521
$222$	− 7.02944i	− 0.471785i
$223$	− 17.7990i	− 1.19191i	−0.803018	−	0.595954i	$-0.796774\pi$
$223$	− 17.7990i	− 1.19191i	0.803018	−	0.595954i	$-0.203226\pi$
$224$	21.3137	1.42408
$225$	0	0
$226$	−1.17157	−0.0779319
$227$	− 20.1421i	− 1.33688i	−0.743766	−	0.668440i	$-0.766962\pi$
$227$	− 20.1421i	− 1.33688i	0.743766	−	0.668440i	$-0.233038\pi$
$228$	− 17.6569i	− 1.16935i
$229$	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
$232$	1.58579i	0.104112i
$233$	18.0000i	1.17922i	0.807688	+	0.589610i	$0.200718\pi$
$233$	18.0000i	1.17922i	−0.807688	+	0.589610i	$0.799282\pi$
$234$	0.828427	0.0541560
$235$	0	0
$236$	0	0
$237$	− 4.97056i	− 0.322873i
$238$	− 5.65685i	− 0.366679i
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	4.27208i	0.274620i
$243$	10.0000i	0.641500i
$244$	−6.68629	−0.428046
$245$	0	0
$246$	4.97056	0.316912
$247$	− 9.65685i	− 0.614451i
$248$	− 10.2843i	− 0.653052i
$249$	14.3431	0.908960
$250$	0	0
$251$	8.82843	0.557245	0.278623	−	0.960401i	$-0.410122\pi$
$251$	8.82843	0.557245	0.278623	+	0.960401i	$0.410122\pi$
$252$	− 8.82843i	− 0.556139i
$253$	− 2.62742i	− 0.165184i
$254$	−2.48528	−0.155940
$255$	0	0
$256$	3.97056	0.248160
$257$	− 6.68629i	− 0.417079i	−0.978014	−	0.208540i	$-0.933129\pi$
$257$	− 6.68629i	− 0.417079i	0.978014	−	0.208540i	$-0.0668710\pi$
$258$	4.97056i	0.309454i
$259$	−40.9706	−2.54579
$260$	0	0
$261$	1.00000	0.0618984
$262$	5.02944i	0.310720i
$263$	− 19.6569i	− 1.21209i	−0.795429	−	0.606047i	$-0.792754\pi$
$263$	− 19.6569i	− 1.21209i	0.795429	−	0.606047i	$-0.207246\pi$
$264$	−2.62742	−0.161706
$265$	0	0
$266$	9.65685	0.592100
$267$	− 15.3137i	− 0.937184i
$268$	− 11.8579i	− 0.724334i
$269$	21.3137	1.29952	0.649760	−	0.760140i	$-0.274869\pi$
$269$	21.3137	1.29952	0.649760	+	0.760140i	$0.274869\pi$
$270$	0	0
$271$	−9.79899	−0.595246	−0.297623	−	0.954683i	$-0.596194\pi$
$271$	−9.79899	−0.595246	−0.297623	+	0.954683i	$0.596194\pi$
$272$	− 8.48528i	− 0.514496i
$273$	− 19.3137i	− 1.16892i
$274$	2.14214	0.129411
$275$	0	0
$276$	−11.5980	−0.698116
$277$	3.65685i	0.219719i	0.993947	+	0.109860i	$0.0350401\pi$
$277$	3.65685i	0.219719i	−0.993947	+	0.109860i	$0.964960\pi$
$278$	8.97056i	0.538019i
$279$	−6.48528	−0.388264
$280$	0	0
$281$	−29.3137	−1.74871	−0.874355	−	0.485288i	$-0.838715\pi$
$281$	−29.3137	−1.74871	−0.874355	+	0.485288i	$0.838715\pi$
$282$	− 9.65685i	− 0.575057i
$283$	4.82843i	0.287020i	0.989649	+	0.143510i	$0.0458390\pi$
$283$	4.82843i	0.287020i	−0.989649	+	0.143510i	$0.954161\pi$
$284$	−28.0000	−1.66149
$285$	0	0
$286$	−0.686292	−0.0405813
$287$	− 28.9706i	− 1.71008i
$288$	4.41421i	0.260110i
$289$	9.00000	0.529412
$290$	0	0
$291$	24.9706	1.46380
$292$	15.5147i	0.907930i
$293$	8.48528i	0.495715i	0.968796	+	0.247858i	$0.0797265\pi$
$293$	8.48528i	0.495715i	−0.968796	+	0.247858i	$0.920273\pi$
$294$	13.5147	0.788194
$295$	0	0
$296$	13.4558	0.782105
$297$	− 3.31371i	− 0.192281i
$298$	3.85786i	0.223480i
$299$	−6.34315	−0.366834
$300$	0	0
$301$	28.9706	1.66984
$302$	4.97056i	0.286024i
$303$	− 31.3137i	− 1.79893i
$304$	14.4853	0.830788
$305$	0	0
$306$	1.17157	0.0669744
$307$	22.9706i	1.31100i	0.755195	+	0.655500i	$0.227542\pi$
$307$	22.9706i	1.31100i	−0.755195	+	0.655500i	$0.772458\pi$
$308$	7.31371i	0.416737i
$309$	32.2843	1.83659
$310$	0	0
$311$	14.4853	0.821385	0.410692	−	0.911774i	$-0.365287\pi$
$311$	14.4853	0.821385	0.410692	+	0.911774i	$0.365287\pi$
$312$	6.34315i	0.359110i
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	−0.201010	−0.0113437
$315$	0	0
$316$	4.54416	0.255629
$317$	− 2.82843i	− 0.158860i	−0.996840	−	0.0794301i	$-0.974690\pi$
$317$	− 2.82843i	− 0.158860i	0.996840	−	0.0794301i	$-0.0253101\pi$
$318$	3.02944i	0.169882i
$319$	−0.828427	−0.0463830
$320$	0	0
$321$	−40.2843	−2.24845
$322$	− 6.34315i	− 0.353490i
$323$	− 13.6569i	− 0.759888i
$324$	−20.1127	−1.11737
$325$	0	0
$326$	−3.45584	−0.191402
$327$	4.00000i	0.221201i
$328$	9.51472i	0.525362i
$329$	−56.2843	−3.10305
$330$	0	0
$331$	21.7990	1.19818	0.599090	−	0.800681i	$-0.295529\pi$
$331$	21.7990	1.19818	0.599090	+	0.800681i	$0.295529\pi$
$332$	13.1127i	0.719653i
$333$	− 8.48528i	− 0.464991i
$334$	1.02944	0.0563283
$335$	0	0
$336$	28.9706	1.58047
$337$	1.17157i	0.0638196i	0.999491	+	0.0319098i	$0.0101589\pi$
$337$	1.17157i	0.0638196i	−0.999491	+	0.0319098i	$0.989841\pi$
$338$	− 3.72792i	− 0.202772i
$339$	−5.65685	−0.307238
$340$	0	0
$341$	5.37258	0.290942
$342$	2.00000i	0.108148i
$343$	− 44.9706i	− 2.42818i
$344$	−9.51472	−0.512999
$345$	0	0
$346$	7.17157	0.385546
$347$	8.14214i	0.437093i	0.975827	+	0.218546i	$0.0701315\pi$
$347$	8.14214i	0.437093i	−0.975827	+	0.218546i	$0.929869\pi$
$348$	3.65685i	0.196028i
$349$	−20.6274	−1.10416	−0.552080	−	0.833791i	$-0.686166\pi$
$349$	−20.6274	−1.10416	−0.552080	+	0.833791i	$0.686166\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	− 3.65685i	− 0.194911i
$353$	− 4.34315i	− 0.231162i	−0.993298	−	0.115581i	$-0.963127\pi$
$353$	− 4.34315i	− 0.231162i	0.993298	−	0.115581i	$-0.0368730\pi$
$354$	0	0
$355$	0	0
$356$	14.0000	0.741999
$357$	− 27.3137i	− 1.44559i
$358$	− 9.65685i	− 0.510381i
$359$	3.85786	0.203610	0.101805	−	0.994804i	$-0.467538\pi$
$359$	3.85786	0.203610	0.101805	+	0.994804i	$0.467538\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	2.48528i	0.130623i
$363$	20.6274i	1.08266i
$364$	17.6569	0.925471
$365$	0	0
$366$	3.02944	0.158351
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	− 9.51472i	− 0.495989i
$369$	6.00000	0.312348
$370$	0	0
$371$	17.6569	0.916698
$372$	− 23.7157i	− 1.22960i
$373$	− 6.97056i	− 0.360922i	−0.983582	−	0.180461i	$-0.942241\pi$
$373$	− 6.97056i	− 0.360922i	0.983582	−	0.180461i	$-0.0577590\pi$
$374$	−0.970563	−0.0501866
$375$	0	0
$376$	18.4853	0.953306
$377$	2.00000i	0.103005i
$378$	− 8.00000i	− 0.411476i
$379$	−22.4853	−1.15499	−0.577496	−	0.816394i	$-0.695970\pi$
$379$	−22.4853	−1.15499	−0.577496	+	0.816394i	$0.695970\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	8.62742i	0.441417i
$383$	2.48528i	0.126992i	0.997982	+	0.0634960i	$0.0202250\pi$
$383$	2.48528i	0.126992i	−0.997982	+	0.0634960i	$0.979775\pi$
$384$	−21.1127	−1.07740
$385$	0	0
$386$	1.85786	0.0945628
$387$	6.00000i	0.304997i
$388$	22.8284i	1.15894i
$389$	29.3137	1.48626	0.743132	−	0.669145i	$-0.233339\pi$
$389$	29.3137	1.48626	0.743132	+	0.669145i	$0.233339\pi$
$390$	0	0
$391$	−8.97056	−0.453661
$392$	25.8701i	1.30664i
$393$	24.2843i	1.22498i
$394$	8.14214	0.410195
$395$	0	0
$396$	−1.51472	−0.0761175
$397$	19.6569i	0.986549i	0.869874	+	0.493275i	$0.164200\pi$
$397$	19.6569i	0.986549i	−0.869874	+	0.493275i	$0.835800\pi$
$398$	4.97056i	0.249152i
$399$	46.6274	2.33429
$400$	0	0
$401$	−6.68629	−0.333897	−0.166949	−	0.985966i	$-0.553391\pi$
$401$	−6.68629	−0.333897	−0.166949	+	0.985966i	$0.553391\pi$
$402$	5.37258i	0.267960i
$403$	− 12.9706i	− 0.646110i
$404$	28.6274	1.42427
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	7.02944i	0.348436i
$408$	8.97056i	0.444109i
$409$	2.97056	0.146885	0.0734424	−	0.997299i	$-0.476601\pi$
$409$	2.97056	0.146885	0.0734424	+	0.997299i	$0.476601\pi$
$410$	0	0
$411$	10.3431	0.510190
$412$	29.5147i	1.45409i
$413$	0	0
$414$	1.31371	0.0645653
$415$	0	0
$416$	−8.82843	−0.432849
$417$	43.3137i	2.12108i
$418$	− 1.65685i	− 0.0810394i
$419$	−28.9706	−1.41530	−0.707652	−	0.706561i	$-0.750246\pi$
$419$	−28.9706	−1.41530	−0.707652	+	0.706561i	$0.750246\pi$
$420$	0	0
$421$	18.9706	0.924569	0.462284	−	0.886732i	$-0.347030\pi$
$421$	18.9706	0.924569	0.462284	+	0.886732i	$0.347030\pi$
$422$	0.343146i	0.0167041i
$423$	− 11.6569i	− 0.566776i
$424$	−5.79899	−0.281624
$425$	0	0
$426$	12.6863	0.614653
$427$	− 17.6569i	− 0.854475i
$428$	− 36.8284i	− 1.78017i
$429$	−3.31371	−0.159987
$430$	0	0
$431$	3.31371	0.159616	0.0798079	−	0.996810i	$-0.474569\pi$
$431$	3.31371	0.159616	0.0798079	+	0.996810i	$0.474569\pi$
$432$	− 12.0000i	− 0.577350i
$433$	− 29.1716i	− 1.40190i	−0.713212	−	0.700948i	$-0.752760\pi$
$433$	− 29.1716i	− 1.40190i	0.713212	−	0.700948i	$-0.247240\pi$
$434$	12.9706	0.622607
$435$	0	0
$436$	−3.65685	−0.175132
$437$	− 15.3137i	− 0.732554i
$438$	− 7.02944i	− 0.335880i
$439$	10.3431	0.493651	0.246826	−	0.969060i	$-0.420612\pi$
$439$	10.3431	0.493651	0.246826	+	0.969060i	$0.420612\pi$
$440$	0	0
$441$	16.3137	0.776843
$442$	2.34315i	0.111452i
$443$	− 7.65685i	− 0.363788i	−0.983318	−	0.181894i	$-0.941777\pi$
$443$	− 7.65685i	− 0.363788i	0.983318	−	0.181894i	$-0.0582228\pi$
$444$	31.0294	1.47259
$445$	0	0
$446$	−7.37258	−0.349102
$447$	18.6274i	0.881047i
$448$	20.1421i	0.951626i
$449$	−11.6569	−0.550121	−0.275060	−	0.961427i	$-0.588698\pi$
$449$	−11.6569	−0.550121	−0.275060	+	0.961427i	$0.588698\pi$
$450$	0	0
$451$	−4.97056	−0.234055
$452$	− 5.17157i	− 0.243250i
$453$	24.0000i	1.12762i
$454$	−8.34315	−0.391563
$455$	0	0
$456$	−15.3137	−0.717130
$457$	− 19.6569i	− 0.919509i	−0.888046	−	0.459754i	$-0.847937\pi$
$457$	− 19.6569i	− 0.919509i	0.888046	−	0.459754i	$-0.152063\pi$
$458$	− 0.828427i	− 0.0387099i
$459$	−11.3137	−0.528079
$460$	0	0
$461$	−35.6569	−1.66071	−0.830353	−	0.557238i	$-0.811861\pi$
$461$	−35.6569	−1.66071	−0.830353	+	0.557238i	$0.811861\pi$
$462$	− 3.31371i	− 0.154168i
$463$	21.7990i	1.01308i	0.862215	+	0.506542i	$0.169077\pi$
$463$	21.7990i	1.01308i	−0.862215	+	0.506542i	$0.830923\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	7.45584	0.345385
$467$	− 10.9706i	− 0.507657i	−0.967249	−	0.253829i	$-0.918310\pi$
$467$	− 10.9706i	− 0.507657i	0.967249	−	0.253829i	$-0.0816899\pi$
$468$	3.65685i	0.169038i
$469$	31.3137	1.44593
$470$	0	0
$471$	−0.970563	−0.0447212
$472$	0	0
$473$	− 4.97056i	− 0.228547i
$474$	−2.05887	−0.0945672
$475$	0	0
$476$	24.9706	1.14452
$477$	3.65685i	0.167436i
$478$	− 0.284271i	− 0.0130023i
$479$	−7.17157	−0.327678	−0.163839	−	0.986487i	$-0.552388\pi$
$479$	−7.17157	−0.327678	−0.163839	+	0.986487i	$0.552388\pi$
$480$	0	0
$481$	16.9706	0.773791
$482$	− 4.14214i	− 0.188669i
$483$	− 30.6274i	− 1.39360i
$484$	−18.8579	−0.857176
$485$	0	0
$486$	4.14214	0.187891
$487$	− 9.79899i	− 0.444035i	−0.975043	−	0.222017i	$-0.928736\pi$
$487$	− 9.79899i	− 0.444035i	0.975043	−	0.222017i	$-0.0712641\pi$
$488$	5.79899i	0.262508i
$489$	−16.6863	−0.754580
$490$	0	0
$491$	−7.45584	−0.336478	−0.168239	−	0.985746i	$-0.553808\pi$
$491$	−7.45584	−0.336478	−0.168239	+	0.985746i	$0.553808\pi$
$492$	21.9411i	0.989182i
$493$	2.82843i	0.127386i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	19.4558	0.873593
$497$	− 73.9411i	− 3.31671i
$498$	− 5.94113i	− 0.266228i
$499$	−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
$501$	4.97056	0.222068
$502$	− 3.65685i	− 0.163213i
$503$	− 30.0000i	− 1.33763i	−0.743427	−	0.668817i	$-0.766801\pi$
$503$	− 30.0000i	− 1.33763i	0.743427	−	0.668817i	$-0.233199\pi$
$504$	−7.65685	−0.341063
$505$	0	0
$506$	−1.08831	−0.0483814
$507$	− 18.0000i	− 0.799408i
$508$	− 10.9706i	− 0.486740i
$509$	−0.627417	−0.0278098	−0.0139049	−	0.999903i	$-0.504426\pi$
$509$	−0.627417	−0.0278098	−0.0139049	+	0.999903i	$0.504426\pi$
$510$	0	0
$511$	−40.9706	−1.81243
$512$	− 22.7574i	− 1.00574i
$513$	− 19.3137i	− 0.852721i
$514$	−2.76955	−0.122160
$515$	0	0
$516$	−21.9411	−0.965904
$517$	9.65685i	0.424708i
$518$	16.9706i	0.745644i
$519$	34.6274	1.51997
$520$	0	0
$521$	−21.3137	−0.933771	−0.466885	−	0.884318i	$-0.654624\pi$
$521$	−21.3137	−0.933771	−0.466885	+	0.884318i	$0.654624\pi$
$522$	− 0.414214i	− 0.0181296i
$523$	2.48528i	0.108674i	0.998523	+	0.0543369i	$0.0173045\pi$
$523$	2.48528i	0.108674i	−0.998523	+	0.0543369i	$0.982696\pi$
$524$	−22.2010	−0.969856
$525$	0	0
$526$	−8.14214	−0.355014
$527$	− 18.3431i	− 0.799040i
$528$	− 4.97056i	− 0.216316i
$529$	12.9411	0.562658
$530$	0	0
$531$	0	0
$532$	42.6274i	1.84813i
$533$	12.0000i	0.519778i
$534$	−6.34315	−0.274495
$535$	0	0
$536$	−10.2843	−0.444213
$537$	− 46.6274i	− 2.01212i
$538$	− 8.82843i	− 0.380621i
$539$	−13.5147	−0.582120
$540$	0	0
$541$	−5.02944	−0.216232	−0.108116	−	0.994138i	$-0.534482\pi$
$541$	−5.02944	−0.216232	−0.108116	+	0.994138i	$0.534482\pi$
$542$	4.05887i	0.174344i
$543$	12.0000i	0.514969i
$544$	−12.4853	−0.535302
$545$	0	0
$546$	−8.00000	−0.342368
$547$	2.48528i	0.106263i	0.998588	+	0.0531315i	$0.0169202\pi$
$547$	2.48528i	0.106263i	−0.998588	+	0.0531315i	$0.983080\pi$
$548$	9.45584i	0.403934i
$549$	3.65685	0.156071
$550$	0	0
$551$	−4.82843	−0.205698
$552$	10.0589i	0.428134i
$553$	12.0000i	0.510292i
$554$	1.51472	0.0643542
$555$	0	0
$556$	−39.5980	−1.67933
$557$	27.9411i	1.18390i	0.805973	+	0.591952i	$0.201642\pi$
$557$	27.9411i	1.18390i	−0.805973	+	0.591952i	$0.798358\pi$
$558$	2.68629i	0.113720i
$559$	−12.0000	−0.507546
$560$	0	0
$561$	−4.68629	−0.197855
$562$	12.1421i	0.512185i
$563$	7.65685i	0.322698i	0.986897	+	0.161349i	$0.0515845\pi$
$563$	7.65685i	0.322698i	−0.986897	+	0.161349i	$0.948416\pi$
$564$	42.6274	1.79494
$565$	0	0
$566$	2.00000	0.0840663
$567$	− 53.1127i	− 2.23052i
$568$	24.2843i	1.01895i
$569$	−27.6569	−1.15944	−0.579718	−	0.814817i	$-0.696837\pi$
$569$	−27.6569	−1.15944	−0.579718	+	0.814817i	$0.696837\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	− 3.02944i	− 0.126667i
$573$	41.6569i	1.74024i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	−4.17157	−0.173816
$577$	− 23.7990i	− 0.990765i	−0.868675	−	0.495382i	$-0.835028\pi$
$577$	− 23.7990i	− 0.990765i	0.868675	−	0.495382i	$-0.164972\pi$
$578$	− 3.72792i	− 0.155061i
$579$	8.97056	0.372804
$580$	0	0
$581$	−34.6274	−1.43659
$582$	− 10.3431i	− 0.428737i
$583$	− 3.02944i	− 0.125466i
$584$	13.4558	0.556807
$585$	0	0
$586$	3.51472	0.145192
$587$	29.7990i	1.22994i	0.788552	+	0.614968i	$0.210831\pi$
$587$	29.7990i	1.22994i	−0.788552	+	0.614968i	$0.789169\pi$
$588$	59.6569i	2.46021i
$589$	31.3137	1.29026
$590$	0	0
$591$	39.3137	1.61715
$592$	25.4558i	1.04623i
$593$	− 7.65685i	− 0.314429i	−0.987564	−	0.157215i	$-0.949749\pi$
$593$	− 7.65685i	− 0.314429i	0.987564	−	0.157215i	$-0.0502515\pi$
$594$	−1.37258	−0.0563178
$595$	0	0
$596$	−17.0294	−0.697553
$597$	24.0000i	0.982255i
$598$	2.62742i	0.107443i
$599$	37.7990	1.54442	0.772212	−	0.635364i	$-0.219150\pi$
$599$	37.7990	1.54442	0.772212	+	0.635364i	$0.219150\pi$
$600$	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
$603$	6.48528i	0.264101i
$604$	−21.9411	−0.892772
$605$	0	0
$606$	−12.9706	−0.526893
$607$	− 9.02944i	− 0.366494i	−0.983067	−	0.183247i	$-0.941339\pi$
$607$	− 9.02944i	− 0.366494i	0.983067	−	0.183247i	$-0.0586607\pi$
$608$	− 21.3137i	− 0.864385i
$609$	−9.65685	−0.391315
$610$	0	0
$611$	23.3137	0.943172
$612$	5.17157i	0.209048i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	9.51472	0.383983
$615$	0	0
$616$	6.34315	0.255573
$617$	9.17157i	0.369234i	0.982811	+	0.184617i	$0.0591044\pi$
$617$	9.17157i	0.369234i	−0.982811	+	0.184617i	$0.940896\pi$
$618$	− 13.3726i	− 0.537924i
$619$	9.79899	0.393855	0.196927	−	0.980418i	$-0.436904\pi$
$619$	9.79899	0.393855	0.196927	+	0.980418i	$0.436904\pi$
$620$	0	0
$621$	−12.6863	−0.509083
$622$	− 6.00000i	− 0.240578i
$623$	36.9706i	1.48119i
$624$	−12.0000	−0.480384
$625$	0	0
$626$	−2.48528	−0.0993318
$627$	− 8.00000i	− 0.319489i
$628$	− 0.887302i	− 0.0354072i
$629$	24.0000	0.956943
$630$	0	0
$631$	36.9706	1.47177	0.735887	−	0.677104i	$-0.236765\pi$
$631$	36.9706	1.47177	0.735887	+	0.677104i	$0.236765\pi$
$632$	− 3.94113i	− 0.156770i
$633$	1.65685i	0.0658540i
$634$	−1.17157	−0.0465291
$635$	0	0
$636$	−13.3726	−0.530257
$637$	32.6274i	1.29275i
$638$	0.343146i	0.0135853i
$639$	15.3137	0.605801
$640$	0	0
$641$	0.627417	0.0247815	0.0123907	−	0.999923i	$-0.496056\pi$
$641$	0.627417	0.0247815	0.0123907	+	0.999923i	$0.496056\pi$
$642$	16.6863i	0.658555i
$643$	19.4558i	0.767264i	0.923486	+	0.383632i	$0.125327\pi$
$643$	19.4558i	0.767264i	−0.923486	+	0.383632i	$0.874673\pi$
$644$	28.0000	1.10335
$645$	0	0
$646$	−5.65685	−0.222566
$647$	41.1127i	1.61631i	0.588972	+	0.808153i	$0.299533\pi$
$647$	41.1127i	1.61631i	−0.588972	+	0.808153i	$0.700467\pi$
$648$	17.4437i	0.685251i
$649$	0	0
$650$	0	0
$651$	62.6274	2.45456
$652$	− 15.2548i	− 0.597425i
$653$	− 17.1716i	− 0.671976i	−0.941866	−	0.335988i	$-0.890930\pi$
$653$	− 17.1716i	− 0.671976i	0.941866	−	0.335988i	$-0.109070\pi$
$654$	1.65685	0.0647881
$655$	0	0
$656$	−18.0000	−0.702782
$657$	− 8.48528i	− 0.331042i
$658$	23.3137i	0.908863i
$659$	−1.79899	−0.0700787	−0.0350393	−	0.999386i	$-0.511156\pi$
$659$	−1.79899	−0.0700787	−0.0350393	+	0.999386i	$0.511156\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	− 9.02944i	− 0.350939i
$663$	11.3137i	0.439388i
$664$	11.3726	0.441342
$665$	0	0
$666$	−3.51472	−0.136193
$667$	3.17157i	0.122804i
$668$	4.54416i	0.175819i
$669$	−35.5980	−1.37630
$670$	0	0
$671$	−3.02944	−0.116950
$672$	− 42.6274i	− 1.64439i
$673$	22.9706i	0.885450i	0.896657	+	0.442725i	$0.145988\pi$
$673$	22.9706i	0.885450i	−0.896657	+	0.442725i	$0.854012\pi$
$674$	0.485281	0.0186923
$675$	0	0
$676$	16.4558	0.632917
$677$	36.7696i	1.41317i	0.707629	+	0.706584i	$0.249765\pi$
$677$	36.7696i	1.41317i	−0.707629	+	0.706584i	$0.750235\pi$
$678$	2.34315i	0.0899880i
$679$	−60.2843	−2.31350
$680$	0	0
$681$	−40.2843	−1.54370
$682$	− 2.22540i	− 0.0852148i
$683$	11.8579i	0.453729i	0.973926	+	0.226864i	$0.0728474\pi$
$683$	11.8579i	0.453729i	−0.973926	+	0.226864i	$0.927153\pi$
$684$	−8.82843	−0.337563
$685$	0	0
$686$	−18.6274	−0.711198
$687$	− 4.00000i	− 0.152610i
$688$	− 18.0000i	− 0.686244i
$689$	−7.31371	−0.278630
$690$	0	0
$691$	−44.9706	−1.71076	−0.855380	−	0.518000i	$-0.826677\pi$
$691$	−44.9706	−1.71076	−0.855380	+	0.518000i	$0.826677\pi$
$692$	31.6569i	1.20341i
$693$	− 4.00000i	− 0.151947i
$694$	3.37258	0.128022
$695$	0	0
$696$	3.17157	0.120218
$697$	16.9706i	0.642806i
$698$	8.54416i	0.323401i
$699$	36.0000	1.36165
$700$	0	0
$701$	6.68629	0.252538	0.126269	−	0.991996i	$-0.459700\pi$
$701$	6.68629	0.252538	0.126269	+	0.991996i	$0.459700\pi$
$702$	3.31371i	0.125068i
$703$	40.9706i	1.54523i
$704$	3.45584	0.130247
$705$	0	0
$706$	−1.79899	−0.0677059
$707$	75.5980i	2.84315i
$708$	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
$711$	−2.48528	−0.0932053
$712$	− 12.1421i	− 0.455046i
$713$	− 20.5685i	− 0.770298i
$714$	−11.3137	−0.423405
$715$	0	0
$716$	42.6274	1.59306
$717$	− 1.37258i	− 0.0512601i
$718$	− 1.59798i	− 0.0596361i
$719$	34.6274	1.29138	0.645692	−	0.763598i	$-0.276569\pi$
$719$	34.6274	1.29138	0.645692	+	0.763598i	$0.276569\pi$
$720$	0	0
$721$	−77.9411	−2.90268
$722$	− 1.78680i	− 0.0664977i
$723$	− 20.0000i	− 0.743808i
$724$	−10.9706	−0.407718
$725$	0	0
$726$	8.54416	0.317103
$727$	23.9411i	0.887927i	0.896045	+	0.443964i	$0.146428\pi$
$727$	23.9411i	0.887927i	−0.896045	+	0.443964i	$0.853572\pi$
$728$	− 15.3137i	− 0.567564i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−16.9706	−0.627679
$732$	13.3726i	0.494265i
$733$	− 22.8284i	− 0.843187i	−0.906785	−	0.421594i	$-0.861471\pi$
$733$	− 22.8284i	− 0.843187i	0.906785	−	0.421594i	$-0.138529\pi$
$734$	−7.45584	−0.275200
$735$	0	0
$736$	−14.0000	−0.516047
$737$	− 5.37258i	− 0.197902i
$738$	− 2.48528i	− 0.0914845i
$739$	−14.4853	−0.532850	−0.266425	−	0.963856i	$-0.585842\pi$
$739$	−14.4853	−0.532850	−0.266425	+	0.963856i	$0.585842\pi$
$740$	0	0
$741$	−19.3137	−0.709507
$742$	− 7.31371i	− 0.268495i
$743$	− 52.6274i	− 1.93071i	−0.260935	−	0.965356i	$-0.584031\pi$
$743$	− 52.6274i	− 1.93071i	0.260935	−	0.965356i	$-0.415969\pi$
$744$	−20.5685	−0.754079
$745$	0	0
$746$	−2.88730	−0.105712
$747$	− 7.17157i	− 0.262394i
$748$	− 4.28427i	− 0.156648i
$749$	97.2548	3.55361
$750$	0	0
$751$	16.1421	0.589035	0.294517	−	0.955646i	$-0.404841\pi$
$751$	16.1421	0.589035	0.294517	+	0.955646i	$0.404841\pi$
$752$	34.9706i	1.27525i
$753$	− 17.6569i	− 0.643452i
$754$	0.828427	0.0301695
$755$	0	0
$756$	35.3137	1.28435
$757$	− 19.5147i	− 0.709275i	−0.935004	−	0.354637i	$-0.884604\pi$
$757$	− 19.5147i	− 0.709275i	0.935004	−	0.354637i	$-0.115396\pi$
$758$	9.31371i	0.338289i
$759$	−5.25483	−0.190738
$760$	0	0
$761$	8.62742	0.312744	0.156372	−	0.987698i	$-0.450020\pi$
$761$	8.62742	0.312744	0.156372	+	0.987698i	$0.450020\pi$
$762$	4.97056i	0.180064i
$763$	− 9.65685i	− 0.349602i
$764$	−38.0833	−1.37780
$765$	0	0
$766$	1.02944	0.0371951
$767$	0	0
$768$	− 7.94113i	− 0.286551i
$769$	15.6569	0.564601	0.282300	−	0.959326i	$-0.408903\pi$
$769$	15.6569	0.564601	0.282300	+	0.959326i	$0.408903\pi$
$770$	0	0
$771$	−13.3726	−0.481602
$772$	8.20101i	0.295161i
$773$	− 8.48528i	− 0.305194i	−0.988288	−	0.152597i	$-0.951236\pi$
$773$	− 8.48528i	− 0.305194i	0.988288	−	0.152597i	$-0.0487637\pi$
$774$	2.48528	0.0893316
$775$	0	0
$776$	19.7990	0.710742
$777$	81.9411i	2.93962i
$778$	− 12.1421i	− 0.435317i
$779$	−28.9706	−1.03798
$780$	0	0
$781$	−12.6863	−0.453951
$782$	3.71573i	0.132874i
$783$	4.00000i	0.142948i
$784$	−48.9411	−1.74790
$785$	0	0
$786$	10.0589	0.358788
$787$	− 17.7990i	− 0.634465i	−0.948348	−	0.317233i	$-0.897246\pi$
$787$	− 17.7990i	− 0.634465i	0.948348	−	0.317233i	$-0.102754\pi$
$788$	35.9411i	1.28035i
$789$	−39.3137	−1.39961
$790$	0	0
$791$	13.6569	0.485582
$792$	1.31371i	0.0466806i
$793$	7.31371i	0.259717i
$794$	8.14214	0.288954
$795$	0	0
$796$	−21.9411	−0.777683
$797$	5.85786i	0.207496i	0.994604	+	0.103748i	$0.0330836\pi$
$797$	5.85786i	0.207496i	−0.994604	+	0.103748i	$0.966916\pi$
$798$	− 19.3137i	− 0.683698i
$799$	32.9706	1.16641
$800$	0	0
$801$	−7.65685	−0.270542
$802$	2.76955i	0.0977963i
$803$	7.02944i	0.248063i
$804$	−23.7157	−0.836389
$805$	0	0
$806$	−5.37258	−0.189241
$807$	− 42.6274i	− 1.50056i
$808$	− 24.8284i	− 0.873461i
$809$	−42.2843	−1.48664	−0.743318	−	0.668938i	$-0.766749\pi$
$809$	−42.2843	−1.48664	−0.743318	+	0.668938i	$0.766749\pi$
$810$	0	0
$811$	37.6569	1.32231	0.661155	−	0.750249i	$-0.270066\pi$
$811$	37.6569	1.32231	0.661155	+	0.750249i	$0.270066\pi$
$812$	− 8.82843i	− 0.309817i
$813$	19.5980i	0.687331i
$814$	2.91169	0.102055
$815$	0	0
$816$	−16.9706	−0.594089
$817$	− 28.9706i	− 1.01355i
$818$	− 1.23045i	− 0.0430216i
$819$	−9.65685	−0.337438
$820$	0	0
$821$	−22.6863	−0.791757	−0.395879	−	0.918303i	$-0.629560\pi$
$821$	−22.6863	−0.791757	−0.395879	+	0.918303i	$0.629560\pi$
$822$	− 4.28427i	− 0.149431i
$823$	− 30.9706i	− 1.07957i	−0.841804	−	0.539783i	$-0.818506\pi$
$823$	− 30.9706i	− 1.07957i	0.841804	−	0.539783i	$-0.181494\pi$
$824$	25.5980	0.891748
$825$	0	0
$826$	0	0
$827$	− 17.3137i	− 0.602057i	−0.953615	−	0.301028i	$-0.902670\pi$
$827$	− 17.3137i	− 0.602057i	0.953615	−	0.301028i	$-0.0973299\pi$
$828$	5.79899i	0.201529i
$829$	−20.6274	−0.716420	−0.358210	−	0.933641i	$-0.616613\pi$
$829$	−20.6274	−0.716420	−0.358210	+	0.933641i	$0.616613\pi$
$830$	0	0
$831$	7.31371	0.253710
$832$	− 8.34315i	− 0.289247i
$833$	46.1421i	1.59873i
$834$	17.9411	0.621250
$835$	0	0
$836$	7.31371	0.252950
$837$	− 25.9411i	− 0.896656i
$838$	12.0000i	0.414533i
$839$	−2.48528	−0.0858014	−0.0429007	−	0.999079i	$-0.513660\pi$
$839$	−2.48528	−0.0858014	−0.0429007	+	0.999079i	$0.513660\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 7.85786i	− 0.270800i
$843$	58.6274i	2.01924i
$844$	−1.51472	−0.0521388
$845$	0	0
$846$	−4.82843	−0.166005
$847$	− 49.7990i	− 1.71111i
$848$	− 10.9706i	− 0.376731i
$849$	9.65685	0.331422
$850$	0	0
$851$	26.9117	0.922521
$852$	56.0000i	1.91853i
$853$	51.1127i	1.75007i	0.484064	+	0.875033i	$0.339160\pi$
$853$	51.1127i	1.75007i	−0.484064	+	0.875033i	$0.660840\pi$
$854$	−7.31371	−0.250270
$855$	0	0
$856$	−31.9411	−1.09173
$857$	− 3.37258i	− 0.115205i	−0.998340	−	0.0576026i	$-0.981654\pi$
$857$	− 3.37258i	− 0.115205i	0.998340	−	0.0576026i	$-0.0183456\pi$
$858$	1.37258i	0.0468592i
$859$	56.4264	1.92524	0.962622	−	0.270848i	$-0.0873041\pi$
$859$	56.4264	1.92524	0.962622	+	0.270848i	$0.0873041\pi$
$860$	0	0
$861$	−57.9411	−1.97463
$862$	− 1.37258i	− 0.0467504i
$863$	− 36.1421i	− 1.23029i	−0.788413	−	0.615146i	$-0.789097\pi$
$863$	− 36.1421i	− 1.23029i	0.788413	−	0.615146i	$-0.210903\pi$
$864$	−17.6569	−0.600698
$865$	0	0
$866$	−12.0833	−0.410606
$867$	− 18.0000i	− 0.611312i
$868$	57.2548i	1.94336i
$869$	2.05887	0.0698425
$870$	0	0
$871$	−12.9706	−0.439491
$872$	3.17157i	0.107403i
$873$	− 12.4853i	− 0.422563i
$874$	−6.34315	−0.214560
$875$	0	0
$876$	31.0294	1.04839
$877$	− 38.2843i	− 1.29277i	−0.763012	−	0.646384i	$-0.776280\pi$
$877$	− 38.2843i	− 1.29277i	0.763012	−	0.646384i	$-0.223720\pi$
$878$	− 4.28427i	− 0.144587i
$879$	16.9706	0.572403
$880$	0	0
$881$	29.3137	0.987604	0.493802	−	0.869574i	$-0.335607\pi$
$881$	29.3137	0.987604	0.493802	+	0.869574i	$0.335607\pi$
$882$	− 6.75736i	− 0.227532i
$883$	− 14.4853i	− 0.487469i	−0.969842	−	0.243734i	$-0.921628\pi$
$883$	− 14.4853i	− 0.487469i	0.969842	−	0.243734i	$-0.0783725\pi$
$884$	−10.3431	−0.347878
$885$	0	0
$886$	−3.17157	−0.106551
$887$	6.68629i	0.224504i	0.993680	+	0.112252i	$0.0358063\pi$
$887$	6.68629i	0.224504i	−0.993680	+	0.112252i	$0.964194\pi$
$888$	− 26.9117i	− 0.903097i
$889$	28.9706	0.971641
$890$	0	0
$891$	−9.11270	−0.305287
$892$	− 32.5442i	− 1.08966i
$893$	56.2843i	1.88348i
$894$	7.71573	0.258053
$895$	0	0
$896$	50.9706	1.70281
$897$	12.6863i	0.423583i
$898$	4.82843i	0.161127i
$899$	−6.48528	−0.216296
$900$	0	0
$901$	−10.3431	−0.344580
$902$	2.05887i	0.0685530i
$903$	− 57.9411i	− 1.92816i
$904$	−4.48528	−0.149178
$905$	0	0
$906$	9.94113	0.330272
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	− 36.8284i	− 1.22219i
$909$	−15.6569	−0.519305
$910$	0	0
$911$	32.1421	1.06492	0.532458	−	0.846456i	$-0.321268\pi$
$911$	32.1421	1.06492	0.532458	+	0.846456i	$0.321268\pi$
$912$	− 28.9706i	− 0.959311i
$913$	5.94113i	0.196623i
$914$	−8.14214	−0.269318
$915$	0	0
$916$	3.65685	0.120826
$917$	− 58.6274i	− 1.93605i
$918$	4.68629i	0.154671i
$919$	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
$921$	45.9411	1.51381
$922$	14.7696i	0.486409i
$923$	30.6274i	1.00811i
$924$	14.6274	0.481207
$925$	0	0
$926$	9.02944	0.296726
$927$	− 16.1421i	− 0.530177i
$928$	4.41421i	0.144904i
$929$	4.62742	0.151821	0.0759103	−	0.997115i	$-0.475814\pi$
$929$	4.62742	0.151821	0.0759103	+	0.997115i	$0.475814\pi$
$930$	0	0
$931$	−78.7696	−2.58157
$932$	32.9117i	1.07806i
$933$	− 28.9706i	− 0.948454i
$934$	−4.54416	−0.148689
$935$	0	0
$936$	3.17157	0.103666
$937$	− 19.6569i	− 0.642161i	−0.947052	−	0.321081i	$-0.895954\pi$
$937$	− 19.6569i	− 0.642161i	0.947052	−	0.321081i	$-0.104046\pi$
$938$	− 12.9706i	− 0.423504i
$939$	−12.0000	−0.391605
$940$	0	0
$941$	−27.9411	−0.910855	−0.455427	−	0.890273i	$-0.650514\pi$
$941$	−27.9411	−0.910855	−0.455427	+	0.890273i	$0.650514\pi$
$942$	0.402020i	0.0130985i
$943$	19.0294i	0.619684i
$944$	0	0
$945$	0	0
$946$	−2.05887	−0.0669398
$947$	− 44.9117i	− 1.45943i	−0.683749	−	0.729717i	$-0.739652\pi$
$947$	− 44.9117i	− 1.45943i	0.683749	−	0.729717i	$-0.260348\pi$
$948$	− 9.08831i	− 0.295175i
$949$	16.9706	0.550888
$950$	0	0
$951$	−5.65685	−0.183436
$952$	− 21.6569i	− 0.701903i
$953$	29.3137i	0.949564i	0.880103	+	0.474782i	$0.157473\pi$
$953$	29.3137i	0.949564i	−0.880103	+	0.474782i	$0.842527\pi$
$954$	1.51472	0.0490408
$955$	0	0
$956$	1.25483	0.0405842
$957$	1.65685i	0.0535585i
$958$	2.97056i	0.0959745i
$959$	−24.9706	−0.806342
$960$	0	0
$961$	11.0589	0.356738
$962$	− 7.02944i	− 0.226638i
$963$	20.1421i	0.649071i
$964$	18.2843	0.588897
$965$	0	0
$966$	−12.6863	−0.408175
$967$	− 14.9706i	− 0.481421i	−0.970597	−	0.240710i	$-0.922620\pi$
$967$	− 14.9706i	− 0.481421i	0.970597	−	0.240710i	$-0.0773804\pi$
$968$	16.3553i	0.525681i
$969$	−27.3137	−0.877443
$970$	0	0
$971$	28.1421	0.903124	0.451562	−	0.892240i	$-0.350867\pi$
$971$	28.1421	0.903124	0.451562	+	0.892240i	$0.350867\pi$
$972$	18.2843i	0.586468i
$973$	− 104.569i	− 3.35231i
$974$	−4.05887	−0.130055
$975$	0	0
$976$	−10.9706	−0.351159
$977$	2.68629i	0.0859421i	0.999076	+	0.0429710i	$0.0136823\pi$
$977$	2.68629i	0.0859421i	−0.999076	+	0.0429710i	$0.986318\pi$
$978$	6.91169i	0.221011i
$979$	6.34315	0.202728
$980$	0	0
$981$	2.00000	0.0638551
$982$	3.08831i	0.0985520i
$983$	− 9.31371i	− 0.297061i	−0.988908	−	0.148531i	$-0.952546\pi$
$983$	− 9.31371i	− 0.297061i	0.988908	−	0.148531i	$-0.0474543\pi$
$984$	19.0294	0.606636
$985$	0	0
$986$	1.17157	0.0373105
$987$	112.569i	3.58310i
$988$	− 17.6569i	− 0.561739i
$989$	−19.0294	−0.605101
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	− 28.6274i	− 0.908921i
$993$	− 43.5980i	− 1.38354i
$994$	−30.6274	−0.971443
$995$	0	0
$996$	26.2254	0.830983
$997$	− 6.82843i	− 0.216258i	−0.994137	−	0.108129i	$-0.965514\pi$
$997$	− 6.82843i	− 0.216258i	0.994137	−	0.108129i	$-0.0344860\pi$
$998$	14.9117i	0.472021i
$999$	33.9411	1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.b.c.349.2		4
5.2	odd	4		145.2.a.b.1.2	✓	2
5.3	odd	4		725.2.a.c.1.1		2
5.4	even	2	inner	725.2.b.c.349.3		4
15.2	even	4		1305.2.a.n.1.1		2
15.8	even	4		6525.2.a.p.1.2		2
20.7	even	4		2320.2.a.k.1.2		2
35.27	even	4		7105.2.a.e.1.2		2
40.27	even	4		9280.2.a.w.1.2		2
40.37	odd	4		9280.2.a.be.1.1		2
145.57	odd	4		4205.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.2	✓	2	5.2	odd	4
725.2.a.c.1.1		2	5.3	odd	4
725.2.b.c.349.2		4	1.1	even	1	trivial
725.2.b.c.349.3		4	5.4	even	2	inner
1305.2.a.n.1.1		2	15.2	even	4
2320.2.a.k.1.2		2	20.7	even	4
4205.2.a.d.1.1		2	145.57	odd	4
6525.2.a.p.1.2		2	15.8	even	4
7105.2.a.e.1.2		2	35.27	even	4
9280.2.a.w.1.2		2	40.27	even	4
9280.2.a.be.1.1		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)

\(f(q)\)	\(=\)	\(q-0.414214i q^{2} -2.00000i q^{3} +1.82843 q^{4} -0.828427 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} +0.828427 q^{11} -3.65685i q^{12} -2.00000i q^{13} +2.00000 q^{14} +3.00000 q^{16} -2.82843i q^{17} +0.414214i q^{18} +4.82843 q^{19} +9.65685 q^{21} -0.343146i q^{22} -3.17157i q^{23} -3.17157 q^{24} -0.828427 q^{26} -4.00000i q^{27} +8.82843i q^{28} -1.00000 q^{29} +6.48528 q^{31} -4.41421i q^{32} -1.65685i q^{33} -1.17157 q^{34} -1.82843 q^{36} +8.48528i q^{37} -2.00000i q^{38} -4.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} -6.00000i q^{43} +1.51472 q^{44} -1.31371 q^{46} +11.6569i q^{47} -6.00000i q^{48} -16.3137 q^{49} -5.65685 q^{51} -3.65685i q^{52} -3.65685i q^{53} -1.65685 q^{54} +7.65685 q^{56} -9.65685i q^{57} +0.414214i q^{58} -3.65685 q^{61} -2.68629i q^{62} -4.82843i q^{63} +4.17157 q^{64} -0.686292 q^{66} -6.48528i q^{67} -5.17157i q^{68} -6.34315 q^{69} -15.3137 q^{71} +1.58579i q^{72} +8.48528i q^{73} +3.51472 q^{74} +8.82843 q^{76} +4.00000i q^{77} +1.65685i q^{78} +2.48528 q^{79} -11.0000 q^{81} +2.48528i q^{82} +7.17157i q^{83} +17.6569 q^{84} -2.48528 q^{86} +2.00000i q^{87} -1.31371i q^{88} +7.65685 q^{89} +9.65685 q^{91} -5.79899i q^{92} -12.9706i q^{93} +4.82843 q^{94} -8.82843 q^{96} +12.4853i q^{97} +6.75736i q^{98} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 8 q^{11} + 8 q^{14} + 12 q^{16} + 8 q^{19} + 16 q^{21} - 24 q^{24} + 8 q^{26} - 4 q^{29} - 8 q^{31} - 16 q^{34} + 4 q^{36} - 16 q^{39} - 24 q^{41} + 40 q^{44} + 40 q^{46}+ \cdots + 8 q^{99}+O(q^{100}) \) 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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