LMFDB - Embedded newform 2925.2.c.p.2224.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.c (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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.p.2224.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{2} -0.438447 q^{4} +4.56155i q^{7} +2.43845i q^{8} +O(q^{10})$	$q+1.56155i q^{2} -0.438447 q^{4} +4.56155i q^{7} +2.43845i q^{8} +2.56155 q^{11} -1.00000i q^{13} -7.12311 q^{14} -4.68466 q^{16} +2.56155i q^{17} -3.12311 q^{19} +4.00000i q^{22} +6.56155i q^{23} +1.56155 q^{26} -2.00000i q^{28} -1.12311 q^{29} +6.00000 q^{31} -2.43845i q^{32} -4.00000 q^{34} -1.68466i q^{37} -4.87689i q^{38} -0.561553 q^{41} -5.12311i q^{43} -1.12311 q^{44} -10.2462 q^{46} -2.87689i q^{47} -13.8078 q^{49} +0.438447i q^{52} +7.68466i q^{53} -11.1231 q^{56} -1.75379i q^{58} -12.0000 q^{59} +5.68466 q^{61} +9.36932i q^{62} -5.56155 q^{64} -13.3693i q^{67} -1.12311i q^{68} +14.5616 q^{71} -6.00000i q^{73} +2.63068 q^{74} +1.36932 q^{76} +11.6847i q^{77} +7.68466 q^{79} -0.876894i q^{82} +16.4924i q^{83} +8.00000 q^{86} +6.24621i q^{88} +1.68466 q^{89} +4.56155 q^{91} -2.87689i q^{92} +4.49242 q^{94} +11.9309i q^{97} -21.5616i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4}+O(q^{10}) $ 4 * q - 10 * q^4	$ 4 q - 10 q^{4} + 2 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} - 2 q^{26} + 12 q^{29} + 24 q^{31} - 16 q^{34} + 6 q^{41} + 12 q^{44} - 8 q^{46} - 14 q^{49} - 28 q^{56} - 48 q^{59} - 2 q^{61} - 14 q^{64} + 50 q^{71} + 60 q^{74} - 44 q^{76} + 6 q^{79} + 32 q^{86} - 18 q^{89} + 10 q^{91} - 48 q^{94}+O(q^{100}) $ 4 * q - 10 * q^4 + 2 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 - 2 * q^26 + 12 * q^29 + 24 * q^31 - 16 * q^34 + 6 * q^41 + 12 * q^44 - 8 * q^46 - 14 * q^49 - 28 * q^56 - 48 * q^59 - 2 * q^61 - 14 * q^64 + 50 * q^71 + 60 * q^74 - 44 * q^76 + 6 * q^79 + 32 * q^86 - 18 * q^89 + 10 * q^91 - 48 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.56155i	1.10418i	0.833783	+	0.552092i	$0.186170\pi$
$2$	1.56155i	1.10418i	−0.833783	+	0.552092i	$0.813830\pi$
$3$	0	0
$3$	0	0
$4$	−0.438447	−0.219224
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.56155i	1.72410i	0.506819	+	0.862052i	$0.330821\pi$
$7$	4.56155i	1.72410i	−0.506819	+	0.862052i	$0.669179\pi$
$8$	2.43845i	0.862121i
$9$	0	0
$10$	0	0
$11$	2.56155	0.772337	0.386169	−	0.922428i	$-0.373798\pi$
$11$	2.56155	0.772337	0.386169	+	0.922428i	$0.373798\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	−7.12311	−1.90373
$15$	0	0
$16$	−4.68466	−1.17116
$17$	2.56155i	0.621268i	0.950530	+	0.310634i	$0.100541\pi$
$17$	2.56155i	0.621268i	−0.950530	+	0.310634i	$0.899459\pi$
$18$	0	0
$19$	−3.12311	−0.716490	−0.358245	−	0.933628i	$-0.616625\pi$
$19$	−3.12311	−0.716490	−0.358245	+	0.933628i	$0.616625\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	6.56155i	1.36818i	0.729398	+	0.684089i	$0.239800\pi$
$23$	6.56155i	1.36818i	−0.729398	+	0.684089i	$0.760200\pi$
$24$	0	0
$25$	0	0
$26$	1.56155	0.306246
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	−1.12311	−0.208555	−0.104278	−	0.994548i	$-0.533253\pi$
$29$	−1.12311	−0.208555	−0.104278	+	0.994548i	$0.533253\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	− 2.43845i	− 0.431061i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	− 1.68466i	− 0.276956i	−0.990366	−	0.138478i	$-0.955779\pi$
$37$	− 1.68466i	− 0.276956i	0.990366	−	0.138478i	$-0.0442210\pi$
$38$	− 4.87689i	− 0.791137i
$39$	0	0
$40$	0	0
$41$	−0.561553	−0.0876998	−0.0438499	−	0.999038i	$-0.513962\pi$
$41$	−0.561553	−0.0876998	−0.0438499	+	0.999038i	$0.513962\pi$
$42$	0	0
$43$	− 5.12311i	− 0.781266i	−0.920546	−	0.390633i	$-0.872256\pi$
$43$	− 5.12311i	− 0.781266i	0.920546	−	0.390633i	$-0.127744\pi$
$44$	−1.12311	−0.169315
$45$	0	0
$46$	−10.2462	−1.51072
$47$	− 2.87689i	− 0.419638i	−0.977740	−	0.209819i	$-0.932712\pi$
$47$	− 2.87689i	− 0.419638i	0.977740	−	0.209819i	$-0.0672875\pi$
$48$	0	0
$49$	−13.8078	−1.97254
$50$	0	0
$51$	0	0
$52$	0.438447i	0.0608017i
$53$	7.68466i	1.05557i	0.849378	+	0.527785i	$0.176977\pi$
$53$	7.68466i	1.05557i	−0.849378	+	0.527785i	$0.823023\pi$
$54$	0	0
$55$	0	0
$56$	−11.1231	−1.48639
$57$	0	0
$58$	− 1.75379i	− 0.230284i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	5.68466	0.727846	0.363923	−	0.931429i	$-0.381437\pi$
$61$	5.68466	0.727846	0.363923	+	0.931429i	$0.381437\pi$
$62$	9.36932i	1.18990i
$63$	0	0
$64$	−5.56155	−0.695194
$65$	0	0
$66$	0	0
$67$	− 13.3693i	− 1.63332i	−0.577118	−	0.816661i	$-0.695823\pi$
$67$	− 13.3693i	− 1.63332i	0.577118	−	0.816661i	$-0.304177\pi$
$68$	− 1.12311i	− 0.136197i
$69$	0	0
$70$	0	0
$71$	14.5616	1.72814	0.864069	−	0.503373i	$-0.167908\pi$
$71$	14.5616	1.72814	0.864069	+	0.503373i	$0.167908\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	2.63068	0.305811
$75$	0	0
$76$	1.36932	0.157071
$77$	11.6847i	1.33159i
$78$	0	0
$79$	7.68466	0.864592	0.432296	−	0.901732i	$-0.357704\pi$
$79$	7.68466	0.864592	0.432296	+	0.901732i	$0.357704\pi$
$80$	0	0
$81$	0	0
$82$	− 0.876894i	− 0.0968368i
$83$	16.4924i	1.81028i	0.425115	+	0.905139i	$0.360234\pi$
$83$	16.4924i	1.81028i	−0.425115	+	0.905139i	$0.639766\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	6.24621i	0.665848i
$89$	1.68466	0.178573	0.0892867	−	0.996006i	$-0.471541\pi$
$89$	1.68466	0.178573	0.0892867	+	0.996006i	$0.471541\pi$
$90$	0	0
$91$	4.56155	0.478181
$92$	− 2.87689i	− 0.299937i
$93$	0	0
$94$	4.49242	0.463358
$95$	0	0
$96$	0	0
$97$	11.9309i	1.21140i	0.795695	+	0.605698i	$0.207106\pi$
$97$	11.9309i	1.21140i	−0.795695	+	0.605698i	$0.792894\pi$
$98$	− 21.5616i	− 2.17805i
$99$	0	0
$100$	0	0
$101$	−6.24621	−0.621521	−0.310761	−	0.950488i	$-0.600584\pi$
$101$	−6.24621	−0.621521	−0.310761	+	0.950488i	$0.600584\pi$
$102$	0	0
$103$	− 6.87689i	− 0.677601i	−0.940858	−	0.338800i	$-0.889979\pi$
$103$	− 6.87689i	− 0.677601i	0.940858	−	0.338800i	$-0.110021\pi$
$104$	2.43845	0.239109
$105$	0	0
$106$	−12.0000	−1.16554
$107$	17.9309i	1.73344i	0.498793	+	0.866721i	$0.333777\pi$
$107$	17.9309i	1.73344i	−0.498793	+	0.866721i	$0.666223\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	− 21.3693i	− 2.01921i
$113$	− 13.1231i	− 1.23452i	−0.786760	−	0.617259i	$-0.788243\pi$
$113$	− 13.1231i	− 1.23452i	0.786760	−	0.617259i	$-0.211757\pi$
$114$	0	0
$115$	0	0
$116$	0.492423	0.0457203
$117$	0	0
$118$	− 18.7386i	− 1.72503i
$119$	−11.6847	−1.07113
$120$	0	0
$121$	−4.43845	−0.403495
$122$	8.87689i	0.803676i
$123$	0	0
$124$	−2.63068	−0.236242
$125$	0	0
$126$	0	0
$127$	− 18.2462i	− 1.61909i	−0.587058	−	0.809545i	$-0.699714\pi$
$127$	− 18.2462i	− 1.61909i	0.587058	−	0.809545i	$-0.300286\pi$
$128$	− 13.5616i	− 1.19868i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 14.2462i	− 1.23530i
$134$	20.8769	1.80349
$135$	0	0
$136$	−6.24621	−0.535608
$137$	7.12311i	0.608568i	0.952581	+	0.304284i	$0.0984172\pi$
$137$	7.12311i	0.608568i	−0.952581	+	0.304284i	$0.901583\pi$
$138$	0	0
$139$	−15.6847	−1.33036	−0.665178	−	0.746685i	$-0.731644\pi$
$139$	−15.6847	−1.33036	−0.665178	+	0.746685i	$0.731644\pi$
$140$	0	0
$141$	0	0
$142$	22.7386i	1.90818i
$143$	− 2.56155i	− 0.214208i
$144$	0	0
$145$	0	0
$146$	9.36932	0.775410
$147$	0	0
$148$	0.738634i	0.0607153i
$149$	−2.80776	−0.230021	−0.115010	−	0.993364i	$-0.536690\pi$
$149$	−2.80776	−0.230021	−0.115010	+	0.993364i	$0.536690\pi$
$150$	0	0
$151$	−13.3693	−1.08798	−0.543990	−	0.839092i	$-0.683087\pi$
$151$	−13.3693	−1.08798	−0.543990	+	0.839092i	$0.683087\pi$
$152$	− 7.61553i	− 0.617701i
$153$	0	0
$154$	−18.2462	−1.47032
$155$	0	0
$156$	0	0
$157$	21.3693i	1.70546i	0.522354	+	0.852729i	$0.325054\pi$
$157$	21.3693i	1.70546i	−0.522354	+	0.852729i	$0.674946\pi$
$158$	12.0000i	0.954669i
$159$	0	0
$160$	0	0
$161$	−29.9309	−2.35888
$162$	0	0
$163$	− 21.0540i	− 1.64907i	−0.565808	−	0.824537i	$-0.691436\pi$
$163$	− 21.0540i	− 1.64907i	0.565808	−	0.824537i	$-0.308564\pi$
$164$	0.246211	0.0192259
$165$	0	0
$166$	−25.7538	−1.99888
$167$	− 13.1231i	− 1.01550i	−0.861506	−	0.507748i	$-0.830478\pi$
$167$	− 13.1231i	− 1.01550i	0.861506	−	0.507748i	$-0.169522\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	2.24621i	0.171272i
$173$	21.1231i	1.60596i	0.596006	+	0.802980i	$0.296753\pi$
$173$	21.1231i	1.60596i	−0.596006	+	0.802980i	$0.703247\pi$
$174$	0	0
$175$	0	0
$176$	−12.0000	−0.904534
$177$	0	0
$178$	2.63068i	0.197178i
$179$	13.1231	0.980867	0.490433	−	0.871479i	$-0.336838\pi$
$179$	13.1231	0.980867	0.490433	+	0.871479i	$0.336838\pi$
$180$	0	0
$181$	25.6847	1.90913	0.954563	−	0.298010i	$-0.0963228\pi$
$181$	25.6847	1.90913	0.954563	+	0.298010i	$0.0963228\pi$
$182$	7.12311i	0.528000i
$183$	0	0
$184$	−16.0000	−1.17954
$185$	0	0
$186$	0	0
$187$	6.56155i	0.479828i
$188$	1.26137i	0.0919946i
$189$	0	0
$190$	0	0
$191$	−21.6155	−1.56404	−0.782022	−	0.623250i	$-0.785812\pi$
$191$	−21.6155	−1.56404	−0.782022	+	0.623250i	$0.785812\pi$
$192$	0	0
$193$	15.4384i	1.11128i	0.831422	+	0.555642i	$0.187527\pi$
$193$	15.4384i	1.11128i	−0.831422	+	0.555642i	$0.812473\pi$
$194$	−18.6307	−1.33761
$195$	0	0
$196$	6.05398	0.432427
$197$	− 21.3693i	− 1.52250i	−0.648458	−	0.761250i	$-0.724586\pi$
$197$	− 21.3693i	− 1.52250i	0.648458	−	0.761250i	$-0.275414\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	− 9.75379i	− 0.686274i
$203$	− 5.12311i	− 0.359572i
$204$	0	0
$205$	0	0
$206$	10.7386	0.748196
$207$	0	0
$208$	4.68466i	0.324823i
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−10.2462	−0.705378	−0.352689	−	0.935741i	$-0.614733\pi$
$211$	−10.2462	−0.705378	−0.352689	+	0.935741i	$0.614733\pi$
$212$	− 3.36932i	− 0.231406i
$213$	0	0
$214$	−28.0000	−1.91404
$215$	0	0
$216$	0	0
$217$	27.3693i	1.85795i
$218$	3.12311i	0.211523i
$219$	0	0
$220$	0	0
$221$	2.56155	0.172309
$222$	0	0
$223$	9.36932i	0.627416i	0.949520	+	0.313708i	$0.101571\pi$
$223$	9.36932i	0.627416i	−0.949520	+	0.313708i	$0.898429\pi$
$224$	11.1231	0.743194
$225$	0	0
$226$	20.4924	1.36314
$227$	− 22.2462i	− 1.47653i	−0.674509	−	0.738266i	$-0.735645\pi$
$227$	− 22.2462i	− 1.47653i	0.674509	−	0.738266i	$-0.264355\pi$
$228$	0	0
$229$	−8.87689	−0.586602	−0.293301	−	0.956020i	$-0.594754\pi$
$229$	−8.87689	−0.586602	−0.293301	+	0.956020i	$0.594754\pi$
$230$	0	0
$231$	0	0
$232$	− 2.73863i	− 0.179800i
$233$	− 7.19224i	− 0.471179i	−0.971853	−	0.235590i	$-0.924298\pi$
$233$	− 7.19224i	− 0.471179i	0.971853	−	0.235590i	$-0.0757021\pi$
$234$	0	0
$235$	0	0
$236$	5.26137	0.342486
$237$	0	0
$238$	− 18.2462i	− 1.18273i
$239$	5.93087	0.383636	0.191818	−	0.981431i	$-0.438562\pi$
$239$	5.93087	0.383636	0.191818	+	0.981431i	$0.438562\pi$
$240$	0	0
$241$	−24.7386	−1.59356	−0.796778	−	0.604272i	$-0.793464\pi$
$241$	−24.7386	−1.59356	−0.796778	+	0.604272i	$0.793464\pi$
$242$	− 6.93087i	− 0.445533i
$243$	0	0
$244$	−2.49242	−0.159561
$245$	0	0
$246$	0	0
$247$	3.12311i	0.198718i
$248$	14.6307i	0.929049i
$249$	0	0
$250$	0	0
$251$	9.75379	0.615654	0.307827	−	0.951442i	$-0.400398\pi$
$251$	9.75379	0.615654	0.307827	+	0.951442i	$0.400398\pi$
$252$	0	0
$253$	16.8078i	1.05670i
$254$	28.4924	1.78777
$255$	0	0
$256$	10.0540	0.628373
$257$	21.1231i	1.31762i	0.752308	+	0.658812i	$0.228941\pi$
$257$	21.1231i	1.31762i	−0.752308	+	0.658812i	$0.771059\pi$
$258$	0	0
$259$	7.68466	0.477501
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.2462i	0.878459i	0.898375	+	0.439230i	$0.144749\pi$
$263$	14.2462i	0.878459i	−0.898375	+	0.439230i	$0.855251\pi$
$264$	0	0
$265$	0	0
$266$	22.2462	1.36400
$267$	0	0
$268$	5.86174i	0.358063i
$269$	9.12311	0.556246	0.278123	−	0.960546i	$-0.410288\pi$
$269$	9.12311	0.556246	0.278123	+	0.960546i	$0.410288\pi$
$270$	0	0
$271$	5.36932	0.326163	0.163081	−	0.986613i	$-0.447857\pi$
$271$	5.36932	0.326163	0.163081	+	0.986613i	$0.447857\pi$
$272$	− 12.0000i	− 0.727607i
$273$	0	0
$274$	−11.1231	−0.671971
$275$	0	0
$276$	0	0
$277$	− 4.24621i	− 0.255130i	−0.991830	−	0.127565i	$-0.959284\pi$
$277$	− 4.24621i	− 0.255130i	0.991830	−	0.127565i	$-0.0407162\pi$
$278$	− 24.4924i	− 1.46896i
$279$	0	0
$280$	0	0
$281$	−12.2462	−0.730548	−0.365274	−	0.930900i	$-0.619025\pi$
$281$	−12.2462	−0.730548	−0.365274	+	0.930900i	$0.619025\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−6.38447	−0.378849
$285$	0	0
$286$	4.00000	0.236525
$287$	− 2.56155i	− 0.151204i
$288$	0	0
$289$	10.4384	0.614026
$290$	0	0
$291$	0	0
$292$	2.63068i	0.153949i
$293$	− 3.75379i	− 0.219299i	−0.993970	−	0.109649i	$-0.965027\pi$
$293$	− 3.75379i	− 0.219299i	0.993970	−	0.109649i	$-0.0349728\pi$
$294$	0	0
$295$	0	0
$296$	4.10795	0.238770
$297$	0	0
$298$	− 4.38447i	− 0.253986i
$299$	6.56155	0.379464
$300$	0	0
$301$	23.3693	1.34699
$302$	− 20.8769i	− 1.20133i
$303$	0	0
$304$	14.6307	0.839127
$305$	0	0
$306$	0	0
$307$	− 3.43845i	− 0.196243i	−0.995174	−	0.0981213i	$-0.968717\pi$
$307$	− 3.43845i	− 0.196243i	0.995174	−	0.0981213i	$-0.0312833\pi$
$308$	− 5.12311i	− 0.291916i
$309$	0	0
$310$	0	0
$311$	27.3693	1.55197	0.775986	−	0.630750i	$-0.217253\pi$
$311$	27.3693	1.55197	0.775986	+	0.630750i	$0.217253\pi$
$312$	0	0
$313$	20.8769i	1.18003i	0.807392	+	0.590016i	$0.200879\pi$
$313$	20.8769i	1.18003i	−0.807392	+	0.590016i	$0.799121\pi$
$314$	−33.3693	−1.88314
$315$	0	0
$316$	−3.36932	−0.189539
$317$	− 34.4924i	− 1.93729i	−0.248454	−	0.968644i	$-0.579922\pi$
$317$	− 34.4924i	− 1.93729i	0.248454	−	0.968644i	$-0.420078\pi$
$318$	0	0
$319$	−2.87689	−0.161075
$320$	0	0
$321$	0	0
$322$	− 46.7386i	− 2.60464i
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	0	0
$326$	32.8769	1.82088
$327$	0	0
$328$	− 1.36932i	− 0.0756079i
$329$	13.1231	0.723500
$330$	0	0
$331$	20.8769	1.14750	0.573749	−	0.819031i	$-0.305489\pi$
$331$	20.8769	1.14750	0.573749	+	0.819031i	$0.305489\pi$
$332$	− 7.23106i	− 0.396856i
$333$	0	0
$334$	20.4924	1.12130
$335$	0	0
$336$	0	0
$337$	− 2.49242i	− 0.135771i	−0.997693	−	0.0678855i	$-0.978375\pi$
$337$	− 2.49242i	− 0.135771i	0.997693	−	0.0678855i	$-0.0216252\pi$
$338$	− 1.56155i	− 0.0849373i
$339$	0	0
$340$	0	0
$341$	15.3693	0.832295
$342$	0	0
$343$	− 31.0540i	− 1.67676i
$344$	12.4924	0.673546
$345$	0	0
$346$	−32.9848	−1.77328
$347$	− 11.0540i	− 0.593408i	−0.954969	−	0.296704i	$-0.904112\pi$
$347$	− 11.0540i	− 0.593408i	0.954969	−	0.296704i	$-0.0958876\pi$
$348$	0	0
$349$	−1.36932	−0.0732979	−0.0366489	−	0.999328i	$-0.511668\pi$
$349$	−1.36932	−0.0732979	−0.0366489	+	0.999328i	$0.511668\pi$
$350$	0	0
$351$	0	0
$352$	− 6.24621i	− 0.332924i
$353$	10.4924i	0.558455i	0.960225	+	0.279228i	$0.0900784\pi$
$353$	10.4924i	0.558455i	−0.960225	+	0.279228i	$0.909922\pi$
$354$	0	0
$355$	0	0
$356$	−0.738634	−0.0391475
$357$	0	0
$358$	20.4924i	1.08306i
$359$	−2.24621	−0.118550	−0.0592752	−	0.998242i	$-0.518879\pi$
$359$	−2.24621	−0.118550	−0.0592752	+	0.998242i	$0.518879\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	40.1080i	2.10803i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	18.2462i	0.952444i	0.879325	+	0.476222i	$0.157994\pi$
$367$	18.2462i	0.952444i	−0.879325	+	0.476222i	$0.842006\pi$
$368$	− 30.7386i	− 1.60236i
$369$	0	0
$370$	0	0
$371$	−35.0540	−1.81991
$372$	0	0
$373$	− 4.24621i	− 0.219860i	−0.993939	−	0.109930i	$-0.964937\pi$
$373$	− 4.24621i	− 0.219860i	0.993939	−	0.109930i	$-0.0350627\pi$
$374$	−10.2462	−0.529819
$375$	0	0
$376$	7.01515	0.361779
$377$	1.12311i	0.0578429i
$378$	0	0
$379$	−2.49242	−0.128027	−0.0640136	−	0.997949i	$-0.520390\pi$
$379$	−2.49242	−0.128027	−0.0640136	+	0.997949i	$0.520390\pi$
$380$	0	0
$381$	0	0
$382$	− 33.7538i	− 1.72699i
$383$	36.4924i	1.86468i	0.361588	+	0.932338i	$0.382235\pi$
$383$	36.4924i	1.86468i	−0.361588	+	0.932338i	$0.617765\pi$
$384$	0	0
$385$	0	0
$386$	−24.1080	−1.22706
$387$	0	0
$388$	− 5.23106i	− 0.265567i
$389$	19.8617	1.00703	0.503515	−	0.863986i	$-0.332040\pi$
$389$	19.8617	1.00703	0.503515	+	0.863986i	$0.332040\pi$
$390$	0	0
$391$	−16.8078	−0.850005
$392$	− 33.6695i	− 1.70057i
$393$	0	0
$394$	33.3693	1.68112
$395$	0	0
$396$	0	0
$397$	8.56155i	0.429692i	0.976648	+	0.214846i	$0.0689250\pi$
$397$	8.56155i	0.429692i	−0.976648	+	0.214846i	$0.931075\pi$
$398$	12.4924i	0.626189i
$399$	0	0
$400$	0	0
$401$	20.2462	1.01105	0.505524	−	0.862813i	$-0.331299\pi$
$401$	20.2462	1.01105	0.505524	+	0.862813i	$0.331299\pi$
$402$	0	0
$403$	− 6.00000i	− 0.298881i
$404$	2.73863	0.136252
$405$	0	0
$406$	8.00000	0.397033
$407$	− 4.31534i	− 0.213904i
$408$	0	0
$409$	15.1231	0.747789	0.373895	−	0.927471i	$-0.378022\pi$
$409$	15.1231	0.747789	0.373895	+	0.927471i	$0.378022\pi$
$410$	0	0
$411$	0	0
$412$	3.01515i	0.148546i
$413$	− 54.7386i	− 2.69351i
$414$	0	0
$415$	0	0
$416$	−2.43845	−0.119555
$417$	0	0
$418$	− 12.4924i	− 0.611024i
$419$	33.6155	1.64223	0.821113	−	0.570766i	$-0.193354\pi$
$419$	33.6155	1.64223	0.821113	+	0.570766i	$0.193354\pi$
$420$	0	0
$421$	14.6307	0.713056	0.356528	−	0.934285i	$-0.383960\pi$
$421$	14.6307	0.713056	0.356528	+	0.934285i	$0.383960\pi$
$422$	− 16.0000i	− 0.778868i
$423$	0	0
$424$	−18.7386	−0.910029
$425$	0	0
$426$	0	0
$427$	25.9309i	1.25488i
$428$	− 7.86174i	− 0.380012i
$429$	0	0
$430$	0	0
$431$	−36.4924	−1.75778	−0.878889	−	0.477026i	$-0.841715\pi$
$431$	−36.4924	−1.75778	−0.878889	+	0.477026i	$0.841715\pi$
$432$	0	0
$433$	31.6155i	1.51935i	0.650306	+	0.759673i	$0.274641\pi$
$433$	31.6155i	1.51935i	−0.650306	+	0.759673i	$0.725359\pi$
$434$	−42.7386	−2.05152
$435$	0	0
$436$	−0.876894	−0.0419956
$437$	− 20.4924i	− 0.980286i
$438$	0	0
$439$	15.0540	0.718487	0.359244	−	0.933244i	$-0.383035\pi$
$439$	15.0540	0.718487	0.359244	+	0.933244i	$0.383035\pi$
$440$	0	0
$441$	0	0
$442$	4.00000i	0.190261i
$443$	12.8078i	0.608515i	0.952590	+	0.304258i	$0.0984084\pi$
$443$	12.8078i	0.608515i	−0.952590	+	0.304258i	$0.901592\pi$
$444$	0	0
$445$	0	0
$446$	−14.6307	−0.692783
$447$	0	0
$448$	− 25.3693i	− 1.19859i
$449$	−10.1771	−0.480286	−0.240143	−	0.970738i	$-0.577194\pi$
$449$	−10.1771	−0.480286	−0.240143	+	0.970738i	$0.577194\pi$
$450$	0	0
$451$	−1.43845	−0.0677338
$452$	5.75379i	0.270635i
$453$	0	0
$454$	34.7386	1.63036
$455$	0	0
$456$	0	0
$457$	25.6847i	1.20148i	0.799445	+	0.600739i	$0.205127\pi$
$457$	25.6847i	1.20148i	−0.799445	+	0.600739i	$0.794873\pi$
$458$	− 13.8617i	− 0.647717i
$459$	0	0
$460$	0	0
$461$	−29.0540	−1.35318	−0.676589	−	0.736361i	$-0.736543\pi$
$461$	−29.0540	−1.35318	−0.676589	+	0.736361i	$0.736543\pi$
$462$	0	0
$463$	− 0.0691303i	− 0.00321276i	−0.999999	−	0.00160638i	$-0.999489\pi$
$463$	− 0.0691303i	− 0.00321276i	0.999999	−	0.00160638i	$-0.000511327\pi$
$464$	5.26137	0.244253
$465$	0	0
$466$	11.2311	0.520269
$467$	21.9309i	1.01484i	0.861699	+	0.507420i	$0.169401\pi$
$467$	21.9309i	1.01484i	−0.861699	+	0.507420i	$0.830599\pi$
$468$	0	0
$469$	60.9848	2.81602
$470$	0	0
$471$	0	0
$472$	− 29.2614i	− 1.34686i
$473$	− 13.1231i	− 0.603401i
$474$	0	0
$475$	0	0
$476$	5.12311	0.234817
$477$	0	0
$478$	9.26137i	0.423605i
$479$	11.0540	0.505069	0.252535	−	0.967588i	$-0.418736\pi$
$479$	11.0540	0.505069	0.252535	+	0.967588i	$0.418736\pi$
$480$	0	0
$481$	−1.68466	−0.0768138
$482$	− 38.6307i	− 1.75958i
$483$	0	0
$484$	1.94602	0.0884557
$485$	0	0
$486$	0	0
$487$	− 5.05398i	− 0.229017i	−0.993422	−	0.114509i	$-0.963471\pi$
$487$	− 5.05398i	− 0.229017i	0.993422	−	0.114509i	$-0.0365294\pi$
$488$	13.8617i	0.627491i
$489$	0	0
$490$	0	0
$491$	13.6155	0.614460	0.307230	−	0.951635i	$-0.400598\pi$
$491$	13.6155	0.614460	0.307230	+	0.951635i	$0.400598\pi$
$492$	0	0
$493$	− 2.87689i	− 0.129569i
$494$	−4.87689	−0.219422
$495$	0	0
$496$	−28.1080	−1.26208
$497$	66.4233i	2.97949i
$498$	0	0
$499$	30.9848	1.38707	0.693536	−	0.720422i	$-0.256052\pi$
$499$	30.9848	1.38707	0.693536	+	0.720422i	$0.256052\pi$
$500$	0	0
$501$	0	0
$502$	15.2311i	0.679795i
$503$	30.2462i	1.34861i	0.738452	+	0.674306i	$0.235557\pi$
$503$	30.2462i	1.34861i	−0.738452	+	0.674306i	$0.764443\pi$
$504$	0	0
$505$	0	0
$506$	−26.2462	−1.16679
$507$	0	0
$508$	8.00000i	0.354943i
$509$	19.4384	0.861594	0.430797	−	0.902449i	$-0.358232\pi$
$509$	19.4384	0.861594	0.430797	+	0.902449i	$0.358232\pi$
$510$	0	0
$511$	27.3693	1.21075
$512$	− 11.4233i	− 0.504843i
$513$	0	0
$514$	−32.9848	−1.45490
$515$	0	0
$516$	0	0
$517$	− 7.36932i	− 0.324102i
$518$	12.0000i	0.527250i
$519$	0	0
$520$	0	0
$521$	31.8617	1.39589	0.697944	−	0.716152i	$-0.254098\pi$
$521$	31.8617	1.39589	0.697944	+	0.716152i	$0.254098\pi$
$522$	0	0
$523$	18.7386i	0.819383i	0.912224	+	0.409692i	$0.134364\pi$
$523$	18.7386i	0.819383i	−0.912224	+	0.409692i	$0.865636\pi$
$524$	0	0
$525$	0	0
$526$	−22.2462	−0.969981
$527$	15.3693i	0.669498i
$528$	0	0
$529$	−20.0540	−0.871912
$530$	0	0
$531$	0	0
$532$	6.24621i	0.270808i
$533$	0.561553i	0.0243236i
$534$	0	0
$535$	0	0
$536$	32.6004	1.40812
$537$	0	0
$538$	14.2462i	0.614198i
$539$	−35.3693	−1.52346
$540$	0	0
$541$	27.1231	1.16611	0.583057	−	0.812431i	$-0.301857\pi$
$541$	27.1231	1.16611	0.583057	+	0.812431i	$0.301857\pi$
$542$	8.38447i	0.360144i
$543$	0	0
$544$	6.24621	0.267804
$545$	0	0
$546$	0	0
$547$	− 19.3693i	− 0.828172i	−0.910238	−	0.414086i	$-0.864101\pi$
$547$	− 19.3693i	− 0.828172i	0.910238	−	0.414086i	$-0.135899\pi$
$548$	− 3.12311i	− 0.133412i
$549$	0	0
$550$	0	0
$551$	3.50758	0.149428
$552$	0	0
$553$	35.0540i	1.49065i
$554$	6.63068	0.281711
$555$	0	0
$556$	6.87689	0.291645
$557$	26.4924i	1.12252i	0.827640	+	0.561260i	$0.189683\pi$
$557$	26.4924i	1.12252i	−0.827640	+	0.561260i	$0.810317\pi$
$558$	0	0
$559$	−5.12311	−0.216684
$560$	0	0
$561$	0	0
$562$	− 19.1231i	− 0.806660i
$563$	− 31.6847i	− 1.33535i	−0.744453	−	0.667675i	$-0.767290\pi$
$563$	− 31.6847i	− 1.33535i	0.744453	−	0.667675i	$-0.232710\pi$
$564$	0	0
$565$	0	0
$566$	6.24621	0.262548
$567$	0	0
$568$	35.5076i	1.48986i
$569$	41.1231	1.72397	0.861985	−	0.506934i	$-0.169221\pi$
$569$	41.1231	1.72397	0.861985	+	0.506934i	$0.169221\pi$
$570$	0	0
$571$	15.0540	0.629989	0.314995	−	0.949093i	$-0.397997\pi$
$571$	15.0540	0.629989	0.314995	+	0.949093i	$0.397997\pi$
$572$	1.12311i	0.0469594i
$573$	0	0
$574$	4.00000	0.166957
$575$	0	0
$576$	0	0
$577$	− 28.5616i	− 1.18903i	−0.804083	−	0.594517i	$-0.797343\pi$
$577$	− 28.5616i	− 1.18903i	0.804083	−	0.594517i	$-0.202657\pi$
$578$	16.3002i	0.677998i
$579$	0	0
$580$	0	0
$581$	−75.2311	−3.12111
$582$	0	0
$583$	19.6847i	0.815255i
$584$	14.6307	0.605422
$585$	0	0
$586$	5.86174	0.242146
$587$	− 0.492423i	− 0.0203245i	−0.999948	−	0.0101622i	$-0.996765\pi$
$587$	− 0.492423i	− 0.0203245i	0.999948	−	0.0101622i	$-0.00323479\pi$
$588$	0	0
$589$	−18.7386	−0.772112
$590$	0	0
$591$	0	0
$592$	7.89205i	0.324361i
$593$	7.75379i	0.318410i	0.987246	+	0.159205i	$0.0508931\pi$
$593$	7.75379i	0.318410i	−0.987246	+	0.159205i	$0.949107\pi$
$594$	0	0
$595$	0	0
$596$	1.23106	0.0504260
$597$	0	0
$598$	10.2462i	0.418999i
$599$	−15.3693	−0.627973	−0.313987	−	0.949427i	$-0.601665\pi$
$599$	−15.3693	−0.627973	−0.313987	+	0.949427i	$0.601665\pi$
$600$	0	0
$601$	41.5464	1.69471	0.847356	−	0.531025i	$-0.178193\pi$
$601$	41.5464	1.69471	0.847356	+	0.531025i	$0.178193\pi$
$602$	36.4924i	1.48732i
$603$	0	0
$604$	5.86174	0.238511
$605$	0	0
$606$	0	0
$607$	24.0000i	0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$	24.0000i	0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	7.61553i	0.308850i
$609$	0	0
$610$	0	0
$611$	−2.87689	−0.116387
$612$	0	0
$613$	37.5464i	1.51648i	0.651973	+	0.758242i	$0.273942\pi$
$613$	37.5464i	1.51648i	−0.651973	+	0.758242i	$0.726058\pi$
$614$	5.36932	0.216688
$615$	0	0
$616$	−28.4924	−1.14799
$617$	48.7386i	1.96214i	0.193645	+	0.981072i	$0.437969\pi$
$617$	48.7386i	1.96214i	−0.193645	+	0.981072i	$0.562031\pi$
$618$	0	0
$619$	33.3693	1.34123	0.670613	−	0.741807i	$-0.266031\pi$
$619$	33.3693	1.34123	0.670613	+	0.741807i	$0.266031\pi$
$620$	0	0
$621$	0	0
$622$	42.7386i	1.71366i
$623$	7.68466i	0.307879i
$624$	0	0
$625$	0	0
$626$	−32.6004	−1.30297
$627$	0	0
$628$	− 9.36932i	− 0.373876i
$629$	4.31534	0.172064
$630$	0	0
$631$	16.7386	0.666354	0.333177	−	0.942864i	$-0.391879\pi$
$631$	16.7386	0.666354	0.333177	+	0.942864i	$0.391879\pi$
$632$	18.7386i	0.745383i
$633$	0	0
$634$	53.8617	2.13912
$635$	0	0
$636$	0	0
$637$	13.8078i	0.547084i
$638$	− 4.49242i	− 0.177857i
$639$	0	0
$640$	0	0
$641$	32.9848	1.30282	0.651412	−	0.758725i	$-0.274177\pi$
$641$	32.9848	1.30282	0.651412	+	0.758725i	$0.274177\pi$
$642$	0	0
$643$	1.68466i	0.0664364i	0.999448	+	0.0332182i	$0.0105756\pi$
$643$	1.68466i	0.0664364i	−0.999448	+	0.0332182i	$0.989424\pi$
$644$	13.1231	0.517123
$645$	0	0
$646$	12.4924	0.491508
$647$	− 11.1922i	− 0.440012i	−0.975498	−	0.220006i	$-0.929392\pi$
$647$	− 11.1922i	− 0.440012i	0.975498	−	0.220006i	$-0.0706077\pi$
$648$	0	0
$649$	−30.7386	−1.20660
$650$	0	0
$651$	0	0
$652$	9.23106i	0.361516i
$653$	− 8.63068i	− 0.337745i	−0.985638	−	0.168872i	$-0.945987\pi$
$653$	− 8.63068i	− 0.337745i	0.985638	−	0.168872i	$-0.0540126\pi$
$654$	0	0
$655$	0	0
$656$	2.63068	0.102711
$657$	0	0
$658$	20.4924i	0.798878i
$659$	−9.12311	−0.355386	−0.177693	−	0.984086i	$-0.556863\pi$
$659$	−9.12311	−0.355386	−0.177693	+	0.984086i	$0.556863\pi$
$660$	0	0
$661$	−26.4924	−1.03044	−0.515218	−	0.857059i	$-0.672289\pi$
$661$	−26.4924	−1.03044	−0.515218	+	0.857059i	$0.672289\pi$
$662$	32.6004i	1.26705i
$663$	0	0
$664$	−40.2159	−1.56068
$665$	0	0
$666$	0	0
$667$	− 7.36932i	− 0.285341i
$668$	5.75379i	0.222621i
$669$	0	0
$670$	0	0
$671$	14.5616	0.562143
$672$	0	0
$673$	− 32.7386i	− 1.26198i	−0.775790	−	0.630991i	$-0.782649\pi$
$673$	− 32.7386i	− 1.26198i	0.775790	−	0.630991i	$-0.217351\pi$
$674$	3.89205	0.149916
$675$	0	0
$676$	0.438447	0.0168634
$677$	18.5616i	0.713378i	0.934223	+	0.356689i	$0.116094\pi$
$677$	18.5616i	0.713378i	−0.934223	+	0.356689i	$0.883906\pi$
$678$	0	0
$679$	−54.4233	−2.08857
$680$	0	0
$681$	0	0
$682$	24.0000i	0.919007i
$683$	− 0.492423i	− 0.0188420i	−0.999956	−	0.00942101i	$-0.997001\pi$
$683$	− 0.492423i	− 0.0188420i	0.999956	−	0.00942101i	$-0.00299885\pi$
$684$	0	0
$685$	0	0
$686$	48.4924	1.85145
$687$	0	0
$688$	24.0000i	0.914991i
$689$	7.68466	0.292762
$690$	0	0
$691$	19.6155	0.746210	0.373105	−	0.927789i	$-0.378293\pi$
$691$	19.6155	0.746210	0.373105	+	0.927789i	$0.378293\pi$
$692$	− 9.26137i	− 0.352064i
$693$	0	0
$694$	17.2614	0.655233
$695$	0	0
$696$	0	0
$697$	− 1.43845i	− 0.0544851i
$698$	− 2.13826i	− 0.0809344i
$699$	0	0
$700$	0	0
$701$	16.9848	0.641509	0.320754	−	0.947162i	$-0.396064\pi$
$701$	16.9848	0.641509	0.320754	+	0.947162i	$0.396064\pi$
$702$	0	0
$703$	5.26137i	0.198436i
$704$	−14.2462	−0.536924
$705$	0	0
$706$	−16.3845	−0.616638
$707$	− 28.4924i	− 1.07157i
$708$	0	0
$709$	−0.876894	−0.0329325	−0.0164662	−	0.999864i	$-0.505242\pi$
$709$	−0.876894	−0.0329325	−0.0164662	+	0.999864i	$0.505242\pi$
$710$	0	0
$711$	0	0
$712$	4.10795i	0.153952i
$713$	39.3693i	1.47439i
$714$	0	0
$715$	0	0
$716$	−5.75379	−0.215029
$717$	0	0
$718$	− 3.50758i	− 0.130902i
$719$	−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
$721$	31.3693	1.16825
$722$	− 14.4384i	− 0.537343i
$723$	0	0
$724$	−11.2614	−0.418525
$725$	0	0
$726$	0	0
$727$	− 21.1231i	− 0.783413i	−0.920090	−	0.391706i	$-0.871885\pi$
$727$	− 21.1231i	− 0.783413i	0.920090	−	0.391706i	$-0.128115\pi$
$728$	11.1231i	0.412250i
$729$	0	0
$730$	0	0
$731$	13.1231	0.485376
$732$	0	0
$733$	− 20.5616i	− 0.759458i	−0.925098	−	0.379729i	$-0.876017\pi$
$733$	− 20.5616i	− 0.759458i	0.925098	−	0.379729i	$-0.123983\pi$
$734$	−28.4924	−1.05167
$735$	0	0
$736$	16.0000	0.589768
$737$	− 34.2462i	− 1.26148i
$738$	0	0
$739$	0.738634	0.0271711	0.0135855	−	0.999908i	$-0.495675\pi$
$739$	0.738634	0.0271711	0.0135855	+	0.999908i	$0.495675\pi$
$740$	0	0
$741$	0	0
$742$	− 54.7386i	− 2.00952i
$743$	− 30.7386i	− 1.12769i	−0.825880	−	0.563846i	$-0.809321\pi$
$743$	− 30.7386i	− 1.12769i	0.825880	−	0.563846i	$-0.190679\pi$
$744$	0	0
$745$	0	0
$746$	6.63068	0.242767
$747$	0	0
$748$	− 2.87689i	− 0.105190i
$749$	−81.7926	−2.98864
$750$	0	0
$751$	−23.0540	−0.841252	−0.420626	−	0.907234i	$-0.638189\pi$
$751$	−23.0540	−0.841252	−0.420626	+	0.907234i	$0.638189\pi$
$752$	13.4773i	0.491465i
$753$	0	0
$754$	−1.75379	−0.0638692
$755$	0	0
$756$	0	0
$757$	− 18.4924i	− 0.672119i	−0.941841	−	0.336059i	$-0.890906\pi$
$757$	− 18.4924i	− 0.672119i	0.941841	−	0.336059i	$-0.109094\pi$
$758$	− 3.89205i	− 0.141366i
$759$	0	0
$760$	0	0
$761$	−37.2311	−1.34962	−0.674812	−	0.737989i	$-0.735775\pi$
$761$	−37.2311	−1.34962	−0.674812	+	0.737989i	$0.735775\pi$
$762$	0	0
$763$	9.12311i	0.330279i
$764$	9.47727	0.342876
$765$	0	0
$766$	−56.9848	−2.05895
$767$	12.0000i	0.433295i
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	0	0
$772$	− 6.76894i	− 0.243620i
$773$	− 0.876894i	− 0.0315397i	−0.999876	−	0.0157698i	$-0.994980\pi$
$773$	− 0.876894i	− 0.0315397i	0.999876	−	0.0157698i	$-0.00501991\pi$
$774$	0	0
$775$	0	0
$776$	−29.0928	−1.04437
$777$	0	0
$778$	31.0152i	1.11195i
$779$	1.75379	0.0628360
$780$	0	0
$781$	37.3002	1.33471
$782$	− 26.2462i	− 0.938563i
$783$	0	0
$784$	64.6847	2.31017
$785$	0	0
$786$	0	0
$787$	13.3693i	0.476565i	0.971196	+	0.238282i	$0.0765844\pi$
$787$	13.3693i	0.476565i	−0.971196	+	0.238282i	$0.923416\pi$
$788$	9.36932i	0.333768i
$789$	0	0
$790$	0	0
$791$	59.8617	2.12844
$792$	0	0
$793$	− 5.68466i	− 0.201868i
$794$	−13.3693	−0.474459
$795$	0	0
$796$	−3.50758	−0.124323
$797$	7.68466i	0.272205i	0.990695	+	0.136102i	$0.0434576\pi$
$797$	7.68466i	0.272205i	−0.990695	+	0.136102i	$0.956542\pi$
$798$	0	0
$799$	7.36932	0.260708
$800$	0	0
$801$	0	0
$802$	31.6155i	1.11638i
$803$	− 15.3693i	− 0.542371i
$804$	0	0
$805$	0	0
$806$	9.36932	0.330020
$807$	0	0
$808$	− 15.2311i	− 0.535827i
$809$	−40.4924	−1.42364	−0.711819	−	0.702363i	$-0.752128\pi$
$809$	−40.4924	−1.42364	−0.711819	+	0.702363i	$0.752128\pi$
$810$	0	0
$811$	−12.2462	−0.430023	−0.215011	−	0.976612i	$-0.568979\pi$
$811$	−12.2462	−0.430023	−0.215011	+	0.976612i	$0.568979\pi$
$812$	2.24621i	0.0788266i
$813$	0	0
$814$	6.73863	0.236189
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	23.6155i	0.825698i
$819$	0	0
$820$	0	0
$821$	−21.5464	−0.751974	−0.375987	−	0.926625i	$-0.622696\pi$
$821$	−21.5464	−0.751974	−0.375987	+	0.926625i	$0.622696\pi$
$822$	0	0
$823$	− 46.2462i	− 1.61204i	−0.591887	−	0.806021i	$-0.701617\pi$
$823$	− 46.2462i	− 1.61204i	0.591887	−	0.806021i	$-0.298383\pi$
$824$	16.7689	0.584174
$825$	0	0
$826$	85.4773	2.97413
$827$	1.12311i	0.0390542i	0.999809	+	0.0195271i	$0.00621607\pi$
$827$	1.12311i	0.0390542i	−0.999809	+	0.0195271i	$0.993784\pi$
$828$	0	0
$829$	48.7386	1.69276	0.846381	−	0.532577i	$-0.178776\pi$
$829$	48.7386	1.69276	0.846381	+	0.532577i	$0.178776\pi$
$830$	0	0
$831$	0	0
$832$	5.56155i	0.192812i
$833$	− 35.3693i	− 1.22547i
$834$	0	0
$835$	0	0
$836$	3.50758	0.121312
$837$	0	0
$838$	52.4924i	1.81332i
$839$	−41.7926	−1.44284	−0.721421	−	0.692497i	$-0.756510\pi$
$839$	−41.7926	−1.44284	−0.721421	+	0.692497i	$0.756510\pi$
$840$	0	0
$841$	−27.7386	−0.956505
$842$	22.8466i	0.787345i
$843$	0	0
$844$	4.49242	0.154636
$845$	0	0
$846$	0	0
$847$	− 20.2462i	− 0.695668i
$848$	− 36.0000i	− 1.23625i
$849$	0	0
$850$	0	0
$851$	11.0540	0.378925
$852$	0	0
$853$	− 20.4233i	− 0.699280i	−0.936884	−	0.349640i	$-0.886304\pi$
$853$	− 20.4233i	− 0.699280i	0.936884	−	0.349640i	$-0.113696\pi$
$854$	−40.4924	−1.38562
$855$	0	0
$856$	−43.7235	−1.49444
$857$	− 6.56155i	− 0.224138i	−0.993700	−	0.112069i	$-0.964252\pi$
$857$	− 6.56155i	− 0.224138i	0.993700	−	0.112069i	$-0.0357478\pi$
$858$	0	0
$859$	16.8078	0.573474	0.286737	−	0.958009i	$-0.407429\pi$
$859$	16.8078	0.573474	0.286737	+	0.958009i	$0.407429\pi$
$860$	0	0
$861$	0	0
$862$	− 56.9848i	− 1.94091i
$863$	15.3693i	0.523178i	0.965179	+	0.261589i	$0.0842464\pi$
$863$	15.3693i	0.523178i	−0.965179	+	0.261589i	$0.915754\pi$
$864$	0	0
$865$	0	0
$866$	−49.3693	−1.67764
$867$	0	0
$868$	− 12.0000i	− 0.407307i
$869$	19.6847	0.667756
$870$	0	0
$871$	−13.3693	−0.453002
$872$	4.87689i	0.165152i
$873$	0	0
$874$	32.0000	1.08242
$875$	0	0
$876$	0	0
$877$	18.9848i	0.641073i	0.947236	+	0.320536i	$0.103863\pi$
$877$	18.9848i	0.641073i	−0.947236	+	0.320536i	$0.896137\pi$
$878$	23.5076i	0.793342i
$879$	0	0
$880$	0	0
$881$	3.36932	0.113515	0.0567576	−	0.998388i	$-0.481924\pi$
$881$	3.36932	0.113515	0.0567576	+	0.998388i	$0.481924\pi$
$882$	0	0
$883$	18.1080i	0.609381i	0.952451	+	0.304691i	$0.0985530\pi$
$883$	18.1080i	0.609381i	−0.952451	+	0.304691i	$0.901447\pi$
$884$	−1.12311	−0.0377741
$885$	0	0
$886$	−20.0000	−0.671913
$887$	− 1.43845i	− 0.0482983i	−0.999708	−	0.0241492i	$-0.992312\pi$
$887$	− 1.43845i	− 0.0482983i	0.999708	−	0.0241492i	$-0.00768767\pi$
$888$	0	0
$889$	83.2311	2.79148
$890$	0	0
$891$	0	0
$892$	− 4.10795i	− 0.137544i
$893$	8.98485i	0.300666i
$894$	0	0
$895$	0	0
$896$	61.8617	2.06666
$897$	0	0
$898$	− 15.8920i	− 0.530325i
$899$	−6.73863	−0.224746
$900$	0	0
$901$	−19.6847	−0.655791
$902$	− 2.24621i	− 0.0747907i
$903$	0	0
$904$	32.0000	1.06430
$905$	0	0
$906$	0	0
$907$	27.3693i	0.908783i	0.890802	+	0.454392i	$0.150143\pi$
$907$	27.3693i	0.908783i	−0.890802	+	0.454392i	$0.849857\pi$
$908$	9.75379i	0.323691i
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	42.2462i	1.39815i
$914$	−40.1080	−1.32665
$915$	0	0
$916$	3.89205	0.128597
$917$	0	0
$918$	0	0
$919$	−39.0540	−1.28827	−0.644136	−	0.764911i	$-0.722783\pi$
$919$	−39.0540	−1.28827	−0.644136	+	0.764911i	$0.722783\pi$
$920$	0	0
$921$	0	0
$922$	− 45.3693i	− 1.49416i
$923$	− 14.5616i	− 0.479299i
$924$	0	0
$925$	0	0
$926$	0.107951	0.00354748
$927$	0	0
$928$	2.73863i	0.0899001i
$929$	38.6695	1.26871	0.634353	−	0.773044i	$-0.281267\pi$
$929$	38.6695	1.26871	0.634353	+	0.773044i	$0.281267\pi$
$930$	0	0
$931$	43.1231	1.41330
$932$	3.15342i	0.103294i
$933$	0	0
$934$	−34.2462	−1.12057
$935$	0	0
$936$	0	0
$937$	2.63068i	0.0859407i	0.999076	+	0.0429703i	$0.0136821\pi$
$937$	2.63068i	0.0859407i	−0.999076	+	0.0429703i	$0.986318\pi$
$938$	95.2311i	3.10940i
$939$	0	0
$940$	0	0
$941$	−21.1922	−0.690847	−0.345424	−	0.938447i	$-0.612265\pi$
$941$	−21.1922	−0.690847	−0.345424	+	0.938447i	$0.612265\pi$
$942$	0	0
$943$	− 3.68466i	− 0.119989i
$944$	56.2159	1.82967
$945$	0	0
$946$	20.4924	0.666266
$947$	46.6004i	1.51431i	0.653236	+	0.757154i	$0.273411\pi$
$947$	46.6004i	1.51431i	−0.653236	+	0.757154i	$0.726589\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	− 28.4924i	− 0.923445i
$953$	25.4384i	0.824032i	0.911177	+	0.412016i	$0.135175\pi$
$953$	25.4384i	0.824032i	−0.911177	+	0.412016i	$0.864825\pi$
$954$	0	0
$955$	0	0
$956$	−2.60037	−0.0841021
$957$	0	0
$958$	17.2614i	0.557689i
$959$	−32.4924	−1.04924
$960$	0	0
$961$	5.00000	0.161290
$962$	− 2.63068i	− 0.0848166i
$963$	0	0
$964$	10.8466	0.349345
$965$	0	0
$966$	0	0
$967$	− 17.3693i	− 0.558560i	−0.960210	−	0.279280i	$-0.909904\pi$
$967$	− 17.3693i	− 0.558560i	0.960210	−	0.279280i	$-0.0900957\pi$
$968$	− 10.8229i	− 0.347862i
$969$	0	0
$970$	0	0
$971$	8.49242	0.272535	0.136267	−	0.990672i	$-0.456489\pi$
$971$	8.49242	0.272535	0.136267	+	0.990672i	$0.456489\pi$
$972$	0	0
$973$	− 71.5464i	− 2.29367i
$974$	7.89205	0.252878
$975$	0	0
$976$	−26.6307	−0.852427
$977$	− 55.4773i	− 1.77488i	−0.460928	−	0.887438i	$-0.652483\pi$
$977$	− 55.4773i	− 1.77488i	0.460928	−	0.887438i	$-0.347517\pi$
$978$	0	0
$979$	4.31534	0.137919
$980$	0	0
$981$	0	0
$982$	21.2614i	0.678477i
$983$	− 6.73863i	− 0.214929i	−0.994209	−	0.107465i	$-0.965727\pi$
$983$	− 6.73863i	− 0.214929i	0.994209	−	0.107465i	$-0.0342732\pi$
$984$	0	0
$985$	0	0
$986$	4.49242	0.143068
$987$	0	0
$988$	− 1.36932i	− 0.0435638i
$989$	33.6155	1.06891
$990$	0	0
$991$	−27.0540	−0.859398	−0.429699	−	0.902972i	$-0.641380\pi$
$991$	−27.0540	−0.859398	−0.429699	+	0.902972i	$0.641380\pi$
$992$	− 14.6307i	− 0.464525i
$993$	0	0
$994$	−103.723	−3.28991
$995$	0	0
$996$	0	0
$997$	− 11.7538i	− 0.372246i	−0.982526	−	0.186123i	$-0.940408\pi$
$997$	− 11.7538i	− 0.372246i	0.982526	−	0.186123i	$-0.0595923\pi$
$998$	48.3845i	1.53158i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.p.2224.3		4
3.2	odd	2		2925.2.c.o.2224.2		4
5.2	odd	4		585.2.a.l.1.1	yes	2
5.3	odd	4		2925.2.a.x.1.2		2
5.4	even	2	inner	2925.2.c.p.2224.2		4
15.2	even	4		585.2.a.j.1.2	✓	2
15.8	even	4		2925.2.a.bc.1.1		2
15.14	odd	2		2925.2.c.o.2224.3		4
20.7	even	4		9360.2.a.cw.1.2		2
60.47	odd	4		9360.2.a.cl.1.2		2
65.12	odd	4		7605.2.a.bd.1.2		2
195.77	even	4		7605.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.a.j.1.2	✓	2	15.2	even	4
585.2.a.l.1.1	yes	2	5.2	odd	4
2925.2.a.x.1.2		2	5.3	odd	4
2925.2.a.bc.1.1		2	15.8	even	4
2925.2.c.o.2224.2		4	3.2	odd	2
2925.2.c.o.2224.3		4	15.14	odd	2
2925.2.c.p.2224.2		4	5.4	even	2	inner
2925.2.c.p.2224.3		4	1.1	even	1	trivial
7605.2.a.bd.1.2		2	65.12	odd	4
7605.2.a.bi.1.1		2	195.77	even	4
9360.2.a.cl.1.2		2	60.47	odd	4
9360.2.a.cw.1.2		2	20.7	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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{2} -0.438447 q^{4} +4.56155i q^{7} +2.43845i q^{8} +O(q^{10})\)	\(q+1.56155i q^{2} -0.438447 q^{4} +4.56155i q^{7} +2.43845i q^{8} +2.56155 q^{11} -1.00000i q^{13} -7.12311 q^{14} -4.68466 q^{16} +2.56155i q^{17} -3.12311 q^{19} +4.00000i q^{22} +6.56155i q^{23} +1.56155 q^{26} -2.00000i q^{28} -1.12311 q^{29} +6.00000 q^{31} -2.43845i q^{32} -4.00000 q^{34} -1.68466i q^{37} -4.87689i q^{38} -0.561553 q^{41} -5.12311i q^{43} -1.12311 q^{44} -10.2462 q^{46} -2.87689i q^{47} -13.8078 q^{49} +0.438447i q^{52} +7.68466i q^{53} -11.1231 q^{56} -1.75379i q^{58} -12.0000 q^{59} +5.68466 q^{61} +9.36932i q^{62} -5.56155 q^{64} -13.3693i q^{67} -1.12311i q^{68} +14.5616 q^{71} -6.00000i q^{73} +2.63068 q^{74} +1.36932 q^{76} +11.6847i q^{77} +7.68466 q^{79} -0.876894i q^{82} +16.4924i q^{83} +8.00000 q^{86} +6.24621i q^{88} +1.68466 q^{89} +4.56155 q^{91} -2.87689i q^{92} +4.49242 q^{94} +11.9309i q^{97} -21.5616i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4}+O(q^{10}) \) 4 * q - 10 * q^4	\( 4 q - 10 q^{4} + 2 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} - 2 q^{26} + 12 q^{29} + 24 q^{31} - 16 q^{34} + 6 q^{41} + 12 q^{44} - 8 q^{46} - 14 q^{49} - 28 q^{56} - 48 q^{59} - 2 q^{61} - 14 q^{64} + 50 q^{71} + 60 q^{74} - 44 q^{76} + 6 q^{79} + 32 q^{86} - 18 q^{89} + 10 q^{91} - 48 q^{94}+O(q^{100}) \) 4 * q - 10 * q^4 + 2 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 - 2 * q^26 + 12 * q^29 + 24 * q^31 - 16 * q^34 + 6 * q^41 + 12 * q^44 - 8 * q^46 - 14 * q^49 - 28 * q^56 - 48 * q^59 - 2 * q^61 - 14 * q^64 + 50 * q^71 + 60 * q^74 - 44 * q^76 + 6 * q^79 + 32 * q^86 - 18 * q^89 + 10 * q^91 - 48 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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