LMFDB - Embedded newform 725.2.b.a.349.1

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	725.349
Dual form			725.2.b.a.349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} -2.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} -6.00000 q^{11} -2.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} -3.00000i q^{18} +2.00000 q^{19} +6.00000i q^{22} -2.00000i q^{23} -2.00000 q^{26} -2.00000i q^{28} +1.00000 q^{29} +2.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} +3.00000 q^{36} +10.0000i q^{37} -2.00000i q^{38} +2.00000 q^{41} -8.00000i q^{43} -6.00000 q^{44} -2.00000 q^{46} -12.0000i q^{47} +3.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} -6.00000 q^{56} -1.00000i q^{58} +8.00000 q^{59} -6.00000 q^{61} -2.00000i q^{62} -6.00000i q^{63} -7.00000 q^{64} +2.00000i q^{67} -2.00000i q^{68} -12.0000 q^{71} -9.00000i q^{72} +6.00000i q^{73} +10.0000 q^{74} +2.00000 q^{76} +12.0000i q^{77} +10.0000 q^{79} +9.00000 q^{81} -2.00000i q^{82} +14.0000i q^{83} -8.00000 q^{86} +18.0000i q^{88} -18.0000 q^{89} -4.00000 q^{91} -2.00000i q^{92} -12.0000 q^{94} +2.00000i q^{97} -3.00000i q^{98} -18.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 6 q^{9} - 12 q^{11} - 4 q^{14} - 2 q^{16} + 4 q^{19} - 4 q^{26} + 2 q^{29} + 4 q^{31} - 4 q^{34} + 6 q^{36} + 4 q^{41} - 12 q^{44} - 4 q^{46} + 6 q^{49} - 12 q^{56} + 16 q^{59} - 12 q^{61}+ \cdots - 36 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 6 * q^9 - 12 * q^11 - 4 * q^14 - 2 * q^16 + 4 * q^19 - 4 * q^26 + 2 * q^29 + 4 * q^31 - 4 * q^34 + 6 * q^36 + 4 * q^41 - 12 * q^44 - 4 * q^46 + 6 * q^49 - 12 * q^56 + 16 * q^59 - 12 * q^61 - 14 * q^64 - 24 * q^71 + 20 * q^74 + 4 * q^76 + 20 * q^79 + 18 * q^81 - 16 * q^86 - 36 * q^89 - 8 * q^91 - 24 * q^94 - 36 * 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 $

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	− 3.00000i	− 1.06066i
$9$	3.00000	1.00000
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	− 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
$16$	−1.00000	−0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	− 3.00000i	− 0.707107i
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	6.00000i	1.27920i
$23$	− 2.00000i	− 0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$	− 2.00000i	− 0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 5.00000i	− 0.883883i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	3.00000	0.500000
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	− 2.00000i	− 0.324443i
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	−2.00000	−0.294884
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	−6.00000	−0.801784
$57$	0	0
$58$	− 1.00000i	− 0.131306i
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	− 2.00000i	− 0.254000i
$63$	− 6.00000i	− 0.755929i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	− 9.00000i	− 1.06066i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	2.00000	0.229416
$77$	12.0000i	1.36753i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	− 2.00000i	− 0.220863i
$83$	14.0000i	1.53670i	0.640030	+	0.768350i	$0.278922\pi$
$83$	14.0000i	1.53670i	−0.640030	+	0.768350i	$0.721078\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	18.0000i	1.91881i
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	− 2.00000i	− 0.208514i
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 3.00000i	− 0.303046i
$99$	−18.0000	−1.80907
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	6.00000i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$107$	6.00000i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	2.00000i	0.188982i
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	0	0
$115$	0	0
$116$	1.00000	0.0928477
$117$	− 6.00000i	− 0.554700i
$118$	− 8.00000i	− 0.736460i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	25.0000	2.27273
$122$	6.00000i	0.543214i
$123$	0	0
$124$	2.00000	0.179605
$125$	0	0
$126$	−6.00000	−0.534522
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	− 3.00000i	− 0.265165i
$129$	0	0
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	2.00000	0.172774
$135$	0	0
$136$	−6.00000	−0.514496
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	12.0000i	1.00702i
$143$	12.0000i	1.00349i
$144$	−3.00000	−0.250000
$145$	0	0
$146$	6.00000	0.496564
$147$	0	0
$148$	10.0000i	0.821995i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	− 6.00000i	− 0.486664i
$153$	− 6.00000i	− 0.485071i
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	22.0000i	1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$	22.0000i	1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$	− 10.0000i	− 0.795557i
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	− 9.00000i	− 0.707107i
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	14.0000	1.08661
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	− 8.00000i	− 0.609994i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	0	0
$175$	0	0
$176$	6.00000	0.452267
$177$	0	0
$178$	18.0000i	1.34916i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	4.00000i	0.296500i
$183$	0	0
$184$	−6.00000	−0.442326
$185$	0	0
$186$	0	0
$187$	12.0000i	0.877527i
$188$	− 12.0000i	− 0.875190i
$189$	0	0
$190$	0	0
$191$	−22.0000	−1.59186	−0.795932	−	0.605386i	$-0.793019\pi$
$191$	−22.0000	−1.59186	−0.795932	+	0.605386i	$0.793019\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	3.00000	0.214286
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	18.0000i	1.27920i
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	− 10.0000i	− 0.703598i
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	6.00000	0.418040
$207$	− 6.00000i	− 0.417029i
$208$	2.00000i	0.138675i
$209$	−12.0000	−0.830057
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	6.00000	0.410152
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	− 14.0000i	− 0.948200i
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	−10.0000	−0.668153
$225$	0	0
$226$	−2.00000	−0.133038
$227$	22.0000i	1.46019i	0.683345	+	0.730096i	$0.260525\pi$
$227$	22.0000i	1.46019i	−0.683345	+	0.730096i	$0.739475\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	− 3.00000i	− 0.196960i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	8.00000	0.520756
$237$	0	0
$238$	4.00000i	0.259281i
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	− 25.0000i	− 1.60706i
$243$	0	0
$244$	−6.00000	−0.384111
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	− 6.00000i	− 0.381000i
$249$	0	0
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	− 6.00000i	− 0.377964i
$253$	12.0000i	0.754434i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 30.0000i	− 1.87135i	−0.352865	−	0.935674i	$-0.614792\pi$
$257$	− 30.0000i	− 1.87135i	0.352865	−	0.935674i	$-0.385208\pi$
$258$	0	0
$259$	20.0000	1.24274
$260$	0	0
$261$	3.00000	0.185695
$262$	− 14.0000i	− 0.864923i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	0	0
$268$	2.00000i	0.122169i
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	18.0000i	1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$	18.0000i	1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	6.00000	0.359211
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	12.0000	0.709575
$287$	− 4.00000i	− 0.236113i
$288$	− 15.0000i	− 0.883883i
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	6.00000i	0.351123i
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	0	0
$296$	30.0000	1.74371
$297$	0	0
$298$	− 10.0000i	− 0.579284i
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−16.0000	−0.922225
$302$	4.00000i	0.230174i
$303$	0	0
$304$	−2.00000	−0.114708
$305$	0	0
$306$	−6.00000	−0.342997
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	12.0000i	0.683763i
$309$	0	0
$310$	0	0
$311$	−22.0000	−1.24751	−0.623753	−	0.781622i	$-0.714393\pi$
$311$	−22.0000	−1.24751	−0.623753	+	0.781622i	$0.714393\pi$
$312$	0	0
$313$	− 2.00000i	− 0.113047i	−0.998401	−	0.0565233i	$-0.981998\pi$
$313$	− 2.00000i	− 0.113047i	0.998401	−	0.0565233i	$-0.0180015\pi$
$314$	22.0000	1.24153
$315$	0	0
$316$	10.0000	0.562544
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	0	0
$322$	4.00000i	0.222911i
$323$	− 4.00000i	− 0.222566i
$324$	9.00000	0.500000
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	− 6.00000i	− 0.331295i
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	14.0000i	0.768350i
$333$	30.0000i	1.64399i
$334$	18.0000	0.984916
$335$	0	0
$336$	0	0
$337$	2.00000i	0.108947i	0.998515	+	0.0544735i	$0.0173480\pi$
$337$	2.00000i	0.108947i	−0.998515	+	0.0544735i	$0.982652\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	−12.0000	−0.649836
$342$	− 6.00000i	− 0.324443i
$343$	− 20.0000i	− 1.07990i
$344$	−24.0000	−1.29399
$345$	0	0
$346$	14.0000	0.752645
$347$	− 6.00000i	− 0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	− 6.00000i	− 0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$	0	0
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	0	0
$352$	30.0000i	1.59901i
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	0	0
$355$	0	0
$356$	−18.0000	−0.953998
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	−22.0000	−1.16112	−0.580558	−	0.814219i	$-0.697165\pi$
$359$	−22.0000	−1.16112	−0.580558	+	0.814219i	$0.697165\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	− 6.00000i	− 0.315353i
$363$	0	0
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	24.0000i	1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$	24.0000i	1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$	2.00000i	0.104257i
$369$	6.00000	0.312348
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	− 18.0000i	− 0.932005i	−0.884783	−	0.466002i	$-0.845694\pi$
$373$	− 18.0000i	− 0.932005i	0.884783	−	0.466002i	$-0.154306\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	−36.0000	−1.85656
$377$	− 2.00000i	− 0.103005i
$378$	0	0
$379$	−6.00000	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$380$	0	0
$381$	0	0
$382$	22.0000i	1.12562i
$383$	− 14.0000i	− 0.715367i	−0.933843	−	0.357683i	$-0.883567\pi$
$383$	− 14.0000i	− 0.715367i	0.933843	−	0.357683i	$-0.116433\pi$
$384$	0	0
$385$	0	0
$386$	10.0000	0.508987
$387$	− 24.0000i	− 1.21999i
$388$	2.00000i	0.101535i
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	2.00000	0.100759
$395$	0	0
$396$	−18.0000	−0.904534
$397$	− 30.0000i	− 1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$	− 30.0000i	− 1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$	− 4.00000i	− 0.200502i
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	10.0000	0.497519
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	− 60.0000i	− 2.97409i
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	6.00000i	0.295599i
$413$	− 16.0000i	− 0.787309i
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−10.0000	−0.490290
$417$	0	0
$418$	12.0000i	0.586939i
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	− 14.0000i	− 0.681509i
$423$	− 36.0000i	− 1.75038i
$424$	18.0000	0.874157
$425$	0	0
$426$	0	0
$427$	12.0000i	0.580721i
$428$	6.00000i	0.290021i
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	14.0000	0.670478
$437$	− 4.00000i	− 0.191346i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	4.00000i	0.190261i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	14.0000	0.662919
$447$	0	0
$448$	14.0000i	0.661438i
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	− 2.00000i	− 0.0940721i
$453$	0	0
$454$	22.0000	1.03251
$455$	0	0
$456$	0	0
$457$	− 38.0000i	− 1.77757i	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	− 38.0000i	− 1.77757i	0.458329	−	0.888783i	$-0.348448\pi$
$458$	− 6.00000i	− 0.280362i
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	− 22.0000i	− 1.02243i	−0.859454	−	0.511213i	$-0.829196\pi$
$463$	− 22.0000i	− 1.02243i	0.859454	−	0.511213i	$-0.170804\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−18.0000	−0.833834
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	− 6.00000i	− 0.277350i
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	− 24.0000i	− 1.10469i
$473$	48.0000i	2.20704i
$474$	0	0
$475$	0	0
$476$	−4.00000	−0.183340
$477$	18.0000i	0.824163i
$478$	− 12.0000i	− 0.548867i
$479$	14.0000	0.639676	0.319838	−	0.947472i	$-0.396371\pi$
$479$	14.0000	0.639676	0.319838	+	0.947472i	$0.396371\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	26.0000i	1.18427i
$483$	0	0
$484$	25.0000	1.13636
$485$	0	0
$486$	0	0
$487$	− 34.0000i	− 1.54069i	−0.637629	−	0.770344i	$-0.720085\pi$
$487$	− 34.0000i	− 1.54069i	0.637629	−	0.770344i	$-0.279915\pi$
$488$	18.0000i	0.814822i
$489$	0	0
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	− 2.00000i	− 0.0900755i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	24.0000i	1.07655i
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	0	0
$502$	6.00000i	0.267793i
$503$	− 16.0000i	− 0.713405i	−0.934218	−	0.356702i	$-0.883901\pi$
$503$	− 16.0000i	− 0.713405i	0.934218	−	0.356702i	$-0.116099\pi$
$504$	−18.0000	−0.801784
$505$	0	0
$506$	12.0000	0.533465
$507$	0	0
$508$	− 16.0000i	− 0.709885i
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	11.0000i	0.486136i
$513$	0	0
$514$	−30.0000	−1.32324
$515$	0	0
$516$	0	0
$517$	72.0000i	3.16656i
$518$	− 20.0000i	− 0.878750i
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	− 3.00000i	− 0.131306i
$523$	− 42.0000i	− 1.83653i	−0.395964	−	0.918266i	$-0.629590\pi$
$523$	− 42.0000i	− 1.83653i	0.395964	−	0.918266i	$-0.370410\pi$
$524$	14.0000	0.611593
$525$	0	0
$526$	−12.0000	−0.523225
$527$	− 4.00000i	− 0.174243i
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	24.0000	1.04151
$532$	− 4.00000i	− 0.173422i
$533$	− 4.00000i	− 0.173259i
$534$	0	0
$535$	0	0
$536$	6.00000	0.259161
$537$	0	0
$538$	26.0000i	1.12094i
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	2.00000i	0.0859074i
$543$	0	0
$544$	−10.0000	−0.428746
$545$	0	0
$546$	0	0
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	6.00000i	0.256307i
$549$	−18.0000	−0.768221
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	− 20.0000i	− 0.850487i
$554$	18.0000	0.764747
$555$	0	0
$556$	0	0
$557$	− 22.0000i	− 0.932170i	−0.884740	−	0.466085i	$-0.845664\pi$
$557$	− 22.0000i	− 0.932170i	0.884740	−	0.466085i	$-0.154336\pi$
$558$	− 6.00000i	− 0.254000i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	− 22.0000i	− 0.928014i
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	0	0
$565$	0	0
$566$	−22.0000	−0.924729
$567$	− 18.0000i	− 0.755929i
$568$	36.0000i	1.51053i
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	12.0000i	0.501745i
$573$	0	0
$574$	−4.00000	−0.166957
$575$	0	0
$576$	−21.0000	−0.875000
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	− 13.0000i	− 0.540729i
$579$	0	0
$580$	0	0
$581$	28.0000	1.16164
$582$	0	0
$583$	− 36.0000i	− 1.49097i
$584$	18.0000	0.744845
$585$	0	0
$586$	2.00000	0.0826192
$587$	− 18.0000i	− 0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$	− 18.0000i	− 0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	− 10.0000i	− 0.410997i
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	4.00000i	0.163572i
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	16.0000i	0.652111i
$603$	6.00000i	0.244339i
$604$	−4.00000	−0.162758
$605$	0	0
$606$	0	0
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	− 10.0000i	− 0.405554i
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	− 6.00000i	− 0.242536i
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	36.0000	1.45048
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	0	0
$621$	0	0
$622$	22.0000i	0.882120i
$623$	36.0000i	1.44231i
$624$	0	0
$625$	0	0
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	22.0000i	0.877896i
$629$	20.0000	0.797452
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	− 30.0000i	− 1.19334i
$633$	0	0
$634$	−14.0000	−0.556011
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	6.00000i	0.237542i
$639$	−36.0000	−1.42414
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	26.0000i	1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$	26.0000i	1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−4.00000	−0.157378
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	− 27.0000i	− 1.06066i
$649$	−48.0000	−1.88416
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	34.0000i	1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$	34.0000i	1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$	0	0
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	18.0000i	0.702247i
$658$	24.0000i	0.935617i
$659$	26.0000	1.01282	0.506408	−	0.862294i	$-0.330973\pi$
$659$	26.0000	1.01282	0.506408	+	0.862294i	$0.330973\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	18.0000i	0.699590i
$663$	0	0
$664$	42.0000	1.62992
$665$	0	0
$666$	30.0000	1.16248
$667$	− 2.00000i	− 0.0774403i
$668$	18.0000i	0.696441i
$669$	0	0
$670$	0	0
$671$	36.0000	1.38976
$672$	0	0
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	9.00000	0.346154
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	0	0
$682$	12.0000i	0.459504i
$683$	− 18.0000i	− 0.688751i	−0.938832	−	0.344375i	$-0.888091\pi$
$683$	− 18.0000i	− 0.688751i	0.938832	−	0.344375i	$-0.111909\pi$
$684$	6.00000	0.229416
$685$	0	0
$686$	−20.0000	−0.763604
$687$	0	0
$688$	8.00000i	0.304997i
$689$	12.0000	0.457164
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	14.0000i	0.532200i
$693$	36.0000i	1.36753i
$694$	−6.00000	−0.227757
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	34.0000i	1.28692i
$699$	0	0
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	20.0000i	0.754314i
$704$	42.0000	1.58293
$705$	0	0
$706$	14.0000	0.526897
$707$	− 20.0000i	− 0.752177i
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	30.0000	1.12509
$712$	54.0000i	2.02374i
$713$	− 4.00000i	− 0.149801i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	22.0000i	0.821033i
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	15.0000i	0.558242i
$723$	0	0
$724$	6.00000	0.222988
$725$	0	0
$726$	0	0
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	12.0000i	0.444750i
$729$	27.0000	1.00000
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	24.0000	0.885856
$735$	0	0
$736$	−10.0000	−0.368605
$737$	− 12.0000i	− 0.442026i
$738$	− 6.00000i	− 0.220863i
$739$	−46.0000	−1.69214	−0.846069	−	0.533074i	$-0.821037\pi$
$739$	−46.0000	−1.69214	−0.846069	+	0.533074i	$0.821037\pi$
$740$	0	0
$741$	0	0
$742$	− 12.0000i	− 0.440534i
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	−18.0000	−0.659027
$747$	42.0000i	1.53670i
$748$	12.0000i	0.438763i
$749$	12.0000	0.438470
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	12.0000i	0.437595i
$753$	0	0
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	0	0
$757$	− 26.0000i	− 0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	− 26.0000i	− 0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$	6.00000i	0.217930i
$759$	0	0
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	− 28.0000i	− 1.01367i
$764$	−22.0000	−0.795932
$765$	0	0
$766$	−14.0000	−0.505841
$767$	− 16.0000i	− 0.577727i
$768$	0	0
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	0	0
$772$	10.0000i	0.359908i
$773$	− 10.0000i	− 0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$	− 10.0000i	− 0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$	−24.0000	−0.862662
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	− 14.0000i	− 0.501924i
$779$	4.00000	0.143315
$780$	0	0
$781$	72.0000	2.57636
$782$	4.00000i	0.143040i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	18.0000i	0.641631i	0.947142	+	0.320815i	$0.103957\pi$
$787$	18.0000i	0.641631i	−0.947142	+	0.320815i	$0.896043\pi$
$788$	2.00000i	0.0712470i
$789$	0	0
$790$	0	0
$791$	−4.00000	−0.142224
$792$	54.0000i	1.91881i
$793$	12.0000i	0.426132i
$794$	−30.0000	−1.06466
$795$	0	0
$796$	4.00000	0.141776
$797$	− 34.0000i	− 1.20434i	−0.798367	−	0.602171i	$-0.794303\pi$
$797$	− 34.0000i	− 1.20434i	0.798367	−	0.602171i	$-0.205697\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	0	0
$801$	−54.0000	−1.90800
$802$	14.0000i	0.494357i
$803$	− 36.0000i	− 1.27041i
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	− 30.0000i	− 1.05540i
$809$	14.0000	0.492214	0.246107	−	0.969243i	$-0.420849\pi$
$809$	14.0000	0.492214	0.246107	+	0.969243i	$0.420849\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	− 2.00000i	− 0.0701862i
$813$	0	0
$814$	−60.0000	−2.10300
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	− 22.0000i	− 0.769212i
$819$	−12.0000	−0.419314
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	18.0000	0.627060
$825$	0	0
$826$	−16.0000	−0.556711
$827$	− 48.0000i	− 1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$	− 48.0000i	− 1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$	− 6.00000i	− 0.208514i
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	0	0
$832$	14.0000i	0.485363i
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	4.00000i	0.138178i
$839$	14.0000	0.483334	0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000	0.483334	0.241667	+	0.970359i	$0.422306\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 2.00000i	− 0.0689246i
$843$	0	0
$844$	14.0000	0.481900
$845$	0	0
$846$	−36.0000	−1.23771
$847$	− 50.0000i	− 1.71802i
$848$	− 6.00000i	− 0.206041i
$849$	0	0
$850$	0	0
$851$	20.0000	0.685591
$852$	0	0
$853$	22.0000i	0.753266i	0.926363	+	0.376633i	$0.122918\pi$
$853$	22.0000i	0.753266i	−0.926363	+	0.376633i	$0.877082\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	18.0000	0.615227
$857$	58.0000i	1.98124i	0.136637	+	0.990621i	$0.456370\pi$
$857$	58.0000i	1.98124i	−0.136637	+	0.990621i	$0.543630\pi$
$858$	0	0
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	0	0
$861$	0	0
$862$	24.0000i	0.817443i
$863$	42.0000i	1.42970i	0.699280	+	0.714848i	$0.253504\pi$
$863$	42.0000i	1.42970i	−0.699280	+	0.714848i	$0.746496\pi$
$864$	0	0
$865$	0	0
$866$	34.0000	1.15537
$867$	0	0
$868$	− 4.00000i	− 0.135769i
$869$	−60.0000	−2.03536
$870$	0	0
$871$	4.00000	0.135535
$872$	− 42.0000i	− 1.42230i
$873$	6.00000i	0.203069i
$874$	−4.00000	−0.135302
$875$	0	0
$876$	0	0
$877$	− 38.0000i	− 1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$	− 38.0000i	− 1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	− 9.00000i	− 0.303046i
$883$	50.0000i	1.68263i	0.540542	+	0.841317i	$0.318219\pi$
$883$	50.0000i	1.68263i	−0.540542	+	0.841317i	$0.681781\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	12.0000	0.403148
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	0	0
$891$	−54.0000	−1.80907
$892$	14.0000i	0.468755i
$893$	− 24.0000i	− 0.803129i
$894$	0	0
$895$	0	0
$896$	−6.00000	−0.200446
$897$	0	0
$898$	10.0000i	0.333704i
$899$	2.00000	0.0667037
$900$	0	0
$901$	12.0000	0.399778
$902$	12.0000i	0.399556i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	24.0000i	0.796907i	0.917189	+	0.398453i	$0.130453\pi$
$907$	24.0000i	0.796907i	−0.917189	+	0.398453i	$0.869547\pi$
$908$	22.0000i	0.730096i
$909$	30.0000	0.995037
$910$	0	0
$911$	14.0000	0.463841	0.231920	−	0.972735i	$-0.425499\pi$
$911$	14.0000	0.463841	0.231920	+	0.972735i	$0.425499\pi$
$912$	0	0
$913$	− 84.0000i	− 2.77999i
$914$	−38.0000	−1.25693
$915$	0	0
$916$	6.00000	0.198246
$917$	− 28.0000i	− 0.924641i
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	0	0
$922$	− 18.0000i	− 0.592798i
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	−22.0000	−0.722965
$927$	18.0000i	0.591198i
$928$	− 5.00000i	− 0.164133i
$929$	−2.00000	−0.0656179	−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000	−0.0656179	−0.0328089	+	0.999462i	$0.510445\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	− 18.0000i	− 0.589610i
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	−18.0000	−0.588348
$937$	18.0000i	0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$	18.0000i	0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$	− 4.00000i	− 0.130605i
$939$	0	0
$940$	0	0
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	0	0
$943$	− 4.00000i	− 0.130258i
$944$	−8.00000	−0.260378
$945$	0	0
$946$	48.0000	1.56061
$947$	60.0000i	1.94974i	0.222779	+	0.974869i	$0.428487\pi$
$947$	60.0000i	1.94974i	−0.222779	+	0.974869i	$0.571513\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	0	0
$952$	12.0000i	0.388922i
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	18.0000	0.582772
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 14.0000i	− 0.452319i
$959$	12.0000	0.387500
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 20.0000i	− 0.644826i
$963$	18.0000i	0.580042i
$964$	−26.0000	−0.837404
$965$	0	0
$966$	0	0
$967$	− 20.0000i	− 0.643157i	−0.946883	−	0.321578i	$-0.895787\pi$
$967$	− 20.0000i	− 0.643157i	0.946883	−	0.321578i	$-0.104213\pi$
$968$	− 75.0000i	− 2.41059i
$969$	0	0
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	0	0
$974$	−34.0000	−1.08943
$975$	0	0
$976$	6.00000	0.192055
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	0	0
$979$	108.000	3.45169
$980$	0	0
$981$	42.0000	1.34096
$982$	2.00000i	0.0638226i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	−2.00000	−0.0636930
$987$	0	0
$988$	− 4.00000i	− 0.127257i
$989$	−16.0000	−0.508770
$990$	0	0
$991$	28.0000	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$992$	− 10.0000i	− 0.317500i
$993$	0	0
$994$	24.0000	0.761234
$995$	0	0
$996$	0	0
$997$	− 58.0000i	− 1.83688i	−0.395562	−	0.918439i	$-0.629450\pi$
$997$	− 58.0000i	− 1.83688i	0.395562	−	0.918439i	$-0.370550\pi$
$998$	− 12.0000i	− 0.379853i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.b.a.349.1		2
5.2	odd	4		725.2.a.a.1.1		1
5.3	odd	4		145.2.a.a.1.1	✓	1
5.4	even	2	inner	725.2.b.a.349.2		2
15.2	even	4		6525.2.a.d.1.1		1
15.8	even	4		1305.2.a.f.1.1		1
20.3	even	4		2320.2.a.e.1.1		1
35.13	even	4		7105.2.a.b.1.1		1
40.3	even	4		9280.2.a.o.1.1		1
40.13	odd	4		9280.2.a.l.1.1		1
145.28	odd	4		4205.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.a.1.1	✓	1	5.3	odd	4
725.2.a.a.1.1		1	5.2	odd	4
725.2.b.a.349.1		2	1.1	even	1	trivial
725.2.b.a.349.2		2	5.4	even	2	inner
1305.2.a.f.1.1		1	15.8	even	4
2320.2.a.e.1.1		1	20.3	even	4
4205.2.a.a.1.1		1	145.28	odd	4
6525.2.a.d.1.1		1	15.2	even	4
7105.2.a.b.1.1		1	35.13	even	4
9280.2.a.l.1.1		1	40.13	odd	4
9280.2.a.o.1.1		1	40.3	even	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} -2.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} -6.00000 q^{11} -2.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} -3.00000i q^{18} +2.00000 q^{19} +6.00000i q^{22} -2.00000i q^{23} -2.00000 q^{26} -2.00000i q^{28} +1.00000 q^{29} +2.00000 q^{31} -5.00000i q^{32} -2.00000 q^{34} +3.00000 q^{36} +10.0000i q^{37} -2.00000i q^{38} +2.00000 q^{41} -8.00000i q^{43} -6.00000 q^{44} -2.00000 q^{46} -12.0000i q^{47} +3.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} -6.00000 q^{56} -1.00000i q^{58} +8.00000 q^{59} -6.00000 q^{61} -2.00000i q^{62} -6.00000i q^{63} -7.00000 q^{64} +2.00000i q^{67} -2.00000i q^{68} -12.0000 q^{71} -9.00000i q^{72} +6.00000i q^{73} +10.0000 q^{74} +2.00000 q^{76} +12.0000i q^{77} +10.0000 q^{79} +9.00000 q^{81} -2.00000i q^{82} +14.0000i q^{83} -8.00000 q^{86} +18.0000i q^{88} -18.0000 q^{89} -4.00000 q^{91} -2.00000i q^{92} -12.0000 q^{94} +2.00000i q^{97} -3.00000i q^{98} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 6 q^{9} - 12 q^{11} - 4 q^{14} - 2 q^{16} + 4 q^{19} - 4 q^{26} + 2 q^{29} + 4 q^{31} - 4 q^{34} + 6 q^{36} + 4 q^{41} - 12 q^{44} - 4 q^{46} + 6 q^{49} - 12 q^{56} + 16 q^{59} - 12 q^{61}+ \cdots - 36 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 6 * q^9 - 12 * q^11 - 4 * q^14 - 2 * q^16 + 4 * q^19 - 4 * q^26 + 2 * q^29 + 4 * q^31 - 4 * q^34 + 6 * q^36 + 4 * q^41 - 12 * q^44 - 4 * q^46 + 6 * q^49 - 12 * q^56 + 16 * q^59 - 12 * q^61 - 14 * q^64 - 24 * q^71 + 20 * q^74 + 4 * q^76 + 20 * q^79 + 18 * q^81 - 16 * q^86 - 36 * q^89 - 8 * q^91 - 24 * q^94 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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