LMFDB - Embedded newform 1875.2.b.f.1249.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 22x^{10} + 179x^{8} + 641x^{6} + 869x^{4} + 67x^{2} + 1 $ x^12 + 22x^10 + 179x^8 + 641x^6 + 869x^4 + 67*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.11
Root			$2.44028i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.f.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44028i q^{2} +1.00000i q^{3} -3.95498 q^{4} -2.44028 q^{6} -3.44028i q^{7} -4.77071i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.44028i q^{2} +1.00000i q^{3} -3.95498 q^{4} -2.44028 q^{6} -3.44028i q^{7} -4.77071i q^{8} -1.00000 q^{9} -3.26656 q^{11} -3.95498i q^{12} +3.23204i q^{13} +8.39527 q^{14} +3.73192 q^{16} -5.05337i q^{17} -2.44028i q^{18} +3.08119 q^{19} +3.44028 q^{21} -7.97134i q^{22} -1.54765i q^{23} +4.77071 q^{24} -7.88709 q^{26} -1.00000i q^{27} +13.6063i q^{28} +3.12218 q^{29} +7.44212 q^{31} -0.434479i q^{32} -3.26656i q^{33} +12.3317 q^{34} +3.95498 q^{36} -5.75838i q^{37} +7.51899i q^{38} -3.23204 q^{39} +5.41962 q^{41} +8.39527i q^{42} -2.53106i q^{43} +12.9192 q^{44} +3.77670 q^{46} +7.07162i q^{47} +3.73192i q^{48} -4.83555 q^{49} +5.05337 q^{51} -12.7827i q^{52} +10.1515i q^{53} +2.44028 q^{54} -16.4126 q^{56} +3.08119i q^{57} +7.61901i q^{58} -1.73056 q^{59} +7.83611 q^{61} +18.1609i q^{62} +3.44028i q^{63} +8.52409 q^{64} +7.97134 q^{66} -1.84910i q^{67} +19.9860i q^{68} +1.54765 q^{69} -0.713969 q^{71} +4.77071i q^{72} +1.88027i q^{73} +14.0521 q^{74} -12.1861 q^{76} +11.2379i q^{77} -7.88709i q^{78} +13.3332 q^{79} +1.00000 q^{81} +13.2254i q^{82} -3.95747i q^{83} -13.6063 q^{84} +6.17649 q^{86} +3.12218i q^{87} +15.5838i q^{88} -8.53392 q^{89} +11.1191 q^{91} +6.12092i q^{92} +7.44212i q^{93} -17.2567 q^{94} +0.434479 q^{96} -10.6528i q^{97} -11.8001i q^{98} +3.26656 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) $ 12 * q - 20 * q^4 - 12 * q^9	$ 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) $ 12 * q - 20 * q^4 - 12 * q^9 + 6 * q^11 + 44 * q^14 + 36 * q^16 - 22 * q^19 + 12 * q^21 - 6 * q^24 - 56 * q^26 + 6 * q^29 - 22 * q^31 - 30 * q^34 + 20 * q^36 - 12 * q^39 - 2 * q^41 - 18 * q^44 + 38 * q^46 + 28 * q^49 + 26 * q^51 - 70 * q^56 - 18 * q^59 + 22 * q^61 + 46 * q^64 + 32 * q^66 - 26 * q^69 - 16 * q^71 + 44 * q^74 - 52 * q^76 + 10 * q^79 + 12 * q^81 - 14 * q^84 - 74 * q^86 + 8 * q^89 + 68 * q^91 - 82 * q^94 + 32 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	2.44028i	1.72554i	0.505596	+	0.862770i	$0.331273\pi$
$2$	2.44028i	1.72554i	−0.505596	+	0.862770i	$0.668727\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−3.95498	−1.97749
$5$	0	0
$5$	0	0
$6$	−2.44028	−0.996241
$7$	− 3.44028i	− 1.30030i	−0.759804	−	0.650152i	$-0.774705\pi$
$7$	− 3.44028i	− 1.30030i	0.759804	−	0.650152i	$-0.225295\pi$
$8$	− 4.77071i	− 1.68670i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.26656	−0.984906	−0.492453	−	0.870339i	$-0.663900\pi$
$11$	−3.26656	−0.984906	−0.492453	+	0.870339i	$0.663900\pi$
$12$	− 3.95498i	− 1.14171i
$13$	3.23204i	0.896406i	0.893932	+	0.448203i	$0.147936\pi$
$13$	3.23204i	0.896406i	−0.893932	+	0.448203i	$0.852064\pi$
$14$	8.39527	2.24373
$15$	0	0
$16$	3.73192	0.932980
$17$	− 5.05337i	− 1.22562i	−0.790229	−	0.612811i	$-0.790039\pi$
$17$	− 5.05337i	− 1.22562i	0.790229	−	0.612811i	$-0.209961\pi$
$18$	− 2.44028i	− 0.575180i
$19$	3.08119	0.706874	0.353437	−	0.935458i	$-0.385013\pi$
$19$	3.08119	0.706874	0.353437	+	0.935458i	$0.385013\pi$
$20$	0	0
$21$	3.44028	0.750731
$22$	− 7.97134i	− 1.69950i
$23$	− 1.54765i	− 0.322707i	−0.986897	−	0.161354i	$-0.948414\pi$
$23$	− 1.54765i	− 0.322707i	0.986897	−	0.161354i	$-0.0515859\pi$
$24$	4.77071	0.973817
$25$	0	0
$26$	−7.88709	−1.54679
$27$	− 1.00000i	− 0.192450i
$28$	13.6063i	2.57134i
$29$	3.12218	0.579775	0.289887	−	0.957061i	$-0.406382\pi$
$29$	3.12218	0.579775	0.289887	+	0.957061i	$0.406382\pi$
$30$	0	0
$31$	7.44212	1.33664	0.668322	−	0.743872i	$-0.267013\pi$
$31$	7.44212	1.33664	0.668322	+	0.743872i	$0.267013\pi$
$32$	− 0.434479i	− 0.0768057i
$33$	− 3.26656i	− 0.568636i
$34$	12.3317	2.11486
$35$	0	0
$36$	3.95498	0.659164
$37$	− 5.75838i	− 0.946673i	−0.880882	−	0.473336i	$-0.843050\pi$
$37$	− 5.75838i	− 0.946673i	0.880882	−	0.473336i	$-0.156950\pi$
$38$	7.51899i	1.21974i
$39$	−3.23204	−0.517540
$40$	0	0
$41$	5.41962	0.846403	0.423201	−	0.906036i	$-0.360906\pi$
$41$	5.41962	0.846403	0.423201	+	0.906036i	$0.360906\pi$
$42$	8.39527i	1.29542i
$43$	− 2.53106i	− 0.385982i	−0.981201	−	0.192991i	$-0.938181\pi$
$43$	− 2.53106i	− 0.385982i	0.981201	−	0.192991i	$-0.0618189\pi$
$44$	12.9192	1.94764
$45$	0	0
$46$	3.77670	0.556844
$47$	7.07162i	1.03150i	0.856739	+	0.515751i	$0.172487\pi$
$47$	7.07162i	1.03150i	−0.856739	+	0.515751i	$0.827513\pi$
$48$	3.73192i	0.538656i
$49$	−4.83555	−0.690793
$50$	0	0
$51$	5.05337	0.707614
$52$	− 12.7827i	− 1.77263i
$53$	10.1515i	1.39442i	0.716866	+	0.697211i	$0.245576\pi$
$53$	10.1515i	1.39442i	−0.716866	+	0.697211i	$0.754424\pi$
$54$	2.44028	0.332080
$55$	0	0
$56$	−16.4126	−2.19323
$57$	3.08119i	0.408114i
$58$	7.61901i	1.00042i
$59$	−1.73056	−0.225300	−0.112650	−	0.993635i	$-0.535934\pi$
$59$	−1.73056	−0.225300	−0.112650	+	0.993635i	$0.535934\pi$
$60$	0	0
$61$	7.83611	1.00331	0.501656	−	0.865067i	$-0.332724\pi$
$61$	7.83611	1.00331	0.501656	+	0.865067i	$0.332724\pi$
$62$	18.1609i	2.30643i
$63$	3.44028i	0.433435i
$64$	8.52409	1.06551
$65$	0	0
$66$	7.97134	0.981204
$67$	− 1.84910i	− 0.225903i	−0.993600	−	0.112952i	$-0.963969\pi$
$67$	− 1.84910i	− 0.225903i	0.993600	−	0.112952i	$-0.0360305\pi$
$68$	19.9860i	2.42366i
$69$	1.54765	0.186315
$70$	0	0
$71$	−0.713969	−0.0847325	−0.0423663	−	0.999102i	$-0.513490\pi$
$71$	−0.713969	−0.0847325	−0.0423663	+	0.999102i	$0.513490\pi$
$72$	4.77071i	0.562234i
$73$	1.88027i	0.220069i	0.993928	+	0.110035i	$0.0350962\pi$
$73$	1.88027i	0.220069i	−0.993928	+	0.110035i	$0.964904\pi$
$74$	14.0521	1.63352
$75$	0	0
$76$	−12.1861	−1.39784
$77$	11.2379i	1.28068i
$78$	− 7.88709i	− 0.893037i
$79$	13.3332	1.50011	0.750053	−	0.661378i	$-0.230028\pi$
$79$	13.3332	1.50011	0.750053	+	0.661378i	$0.230028\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	13.2254i	1.46050i
$83$	− 3.95747i	− 0.434389i	−0.976128	−	0.217195i	$-0.930309\pi$
$83$	− 3.95747i	− 0.434389i	0.976128	−	0.217195i	$-0.0696906\pi$
$84$	−13.6063	−1.48456
$85$	0	0
$86$	6.17649	0.666028
$87$	3.12218i	0.334733i
$88$	15.5838i	1.66124i
$89$	−8.53392	−0.904593	−0.452297	−	0.891868i	$-0.649395\pi$
$89$	−8.53392	−0.904593	−0.452297	+	0.891868i	$0.649395\pi$
$90$	0	0
$91$	11.1191	1.16560
$92$	6.12092i	0.638150i
$93$	7.44212i	0.771712i
$94$	−17.2567	−1.77990
$95$	0	0
$96$	0.434479	0.0443438
$97$	− 10.6528i	− 1.08163i	−0.841142	−	0.540815i	$-0.818116\pi$
$97$	− 10.6528i	− 1.08163i	0.841142	−	0.540815i	$-0.181884\pi$
$98$	− 11.8001i	− 1.19199i
$99$	3.26656	0.328302
$100$	0	0
$101$	1.76173	0.175299	0.0876496	−	0.996151i	$-0.472064\pi$
$101$	1.76173	0.175299	0.0876496	+	0.996151i	$0.472064\pi$
$102$	12.3317i	1.22102i
$103$	− 15.9143i	− 1.56808i	−0.620712	−	0.784039i	$-0.713156\pi$
$103$	− 15.9143i	− 1.56808i	0.620712	−	0.784039i	$-0.286844\pi$
$104$	15.4191	1.51197
$105$	0	0
$106$	−24.7726	−2.40613
$107$	− 15.7807i	− 1.52558i	−0.646649	−	0.762788i	$-0.723830\pi$
$107$	− 15.7807i	− 1.52558i	0.646649	−	0.762788i	$-0.276170\pi$
$108$	3.95498i	0.380568i
$109$	9.91980	0.950144	0.475072	−	0.879947i	$-0.342422\pi$
$109$	9.91980	0.950144	0.475072	+	0.879947i	$0.342422\pi$
$110$	0	0
$111$	5.75838	0.546562
$112$	− 12.8389i	− 1.21316i
$113$	− 5.51386i	− 0.518700i	−0.965783	−	0.259350i	$-0.916492\pi$
$113$	− 5.51386i	− 0.518700i	0.965783	−	0.259350i	$-0.0835084\pi$
$114$	−7.51899	−0.704218
$115$	0	0
$116$	−12.3482	−1.14650
$117$	− 3.23204i	− 0.298802i
$118$	− 4.22306i	− 0.388764i
$119$	−17.3850	−1.59368
$120$	0	0
$121$	−0.329568	−0.0299607
$122$	19.1223i	1.73126i
$123$	5.41962i	0.488671i
$124$	−29.4334	−2.64320
$125$	0	0
$126$	−8.39527	−0.747910
$127$	− 0.992089i	− 0.0880336i	−0.999031	−	0.0440168i	$-0.985984\pi$
$127$	− 0.992089i	− 0.0880336i	0.999031	−	0.0440168i	$-0.0140155\pi$
$128$	19.9322i	1.76178i
$129$	2.53106	0.222847
$130$	0	0
$131$	−12.6115	−1.10187	−0.550937	−	0.834547i	$-0.685729\pi$
$131$	−12.6115	−1.10187	−0.550937	+	0.834547i	$0.685729\pi$
$132$	12.9192i	1.12447i
$133$	− 10.6002i	− 0.919152i
$134$	4.51232	0.389805
$135$	0	0
$136$	−24.1082	−2.06726
$137$	8.24065i	0.704046i	0.935991	+	0.352023i	$0.114506\pi$
$137$	8.24065i	0.704046i	−0.935991	+	0.352023i	$0.885494\pi$
$138$	3.77670i	0.321494i
$139$	−8.06371	−0.683955	−0.341977	−	0.939708i	$-0.611097\pi$
$139$	−8.06371	−0.683955	−0.341977	+	0.939708i	$0.611097\pi$
$140$	0	0
$141$	−7.07162	−0.595538
$142$	− 1.74229i	− 0.146209i
$143$	− 10.5577i	− 0.882875i
$144$	−3.73192	−0.310993
$145$	0	0
$146$	−4.58840	−0.379739
$147$	− 4.83555i	− 0.398829i
$148$	22.7743i	1.87204i
$149$	19.1101	1.56556	0.782781	−	0.622298i	$-0.213801\pi$
$149$	19.1101	1.56556	0.782781	+	0.622298i	$0.213801\pi$
$150$	0	0
$151$	1.58550	0.129026	0.0645132	−	0.997917i	$-0.479451\pi$
$151$	1.58550	0.129026	0.0645132	+	0.997917i	$0.479451\pi$
$152$	− 14.6995i	− 1.19229i
$153$	5.05337i	0.408541i
$154$	−27.4237	−2.20986
$155$	0	0
$156$	12.7827	1.02343
$157$	− 21.8510i	− 1.74390i	−0.489599	−	0.871948i	$-0.662857\pi$
$157$	− 21.8510i	− 1.74390i	0.489599	−	0.871948i	$-0.337143\pi$
$158$	32.5369i	2.58849i
$159$	−10.1515	−0.805070
$160$	0	0
$161$	−5.32435	−0.419617
$162$	2.44028i	0.191727i
$163$	− 10.1338i	− 0.793744i	−0.917874	−	0.396872i	$-0.870096\pi$
$163$	− 10.1338i	− 0.793744i	0.917874	−	0.396872i	$-0.129904\pi$
$164$	−21.4345	−1.67375
$165$	0	0
$166$	9.65736	0.749556
$167$	− 1.51109i	− 0.116931i	−0.998289	−	0.0584657i	$-0.981379\pi$
$167$	− 1.51109i	− 0.116931i	0.998289	−	0.0584657i	$-0.0186208\pi$
$168$	− 16.4126i	− 1.26626i
$169$	2.55393	0.196456
$170$	0	0
$171$	−3.08119	−0.235625
$172$	10.0103i	0.763277i
$173$	13.8553i	1.05340i	0.850051	+	0.526700i	$0.176571\pi$
$173$	13.8553i	1.05340i	−0.850051	+	0.526700i	$0.823429\pi$
$174$	−7.61901	−0.577595
$175$	0	0
$176$	−12.1906	−0.918897
$177$	− 1.73056i	− 0.130077i
$178$	− 20.8252i	− 1.56091i
$179$	−14.4557	−1.08047	−0.540235	−	0.841514i	$-0.681665\pi$
$179$	−14.4557	−1.08047	−0.540235	+	0.841514i	$0.681665\pi$
$180$	0	0
$181$	17.1898	1.27771	0.638854	−	0.769328i	$-0.279409\pi$
$181$	17.1898	1.27771	0.638854	+	0.769328i	$0.279409\pi$
$182$	27.1338i	2.01129i
$183$	7.83611i	0.579262i
$184$	−7.38338	−0.544310
$185$	0	0
$186$	−18.1609	−1.33162
$187$	16.5072i	1.20712i
$188$	− 27.9681i	− 2.03978i
$189$	−3.44028	−0.250244
$190$	0	0
$191$	10.5424	0.762821	0.381410	−	0.924406i	$-0.375438\pi$
$191$	10.5424	0.762821	0.381410	+	0.924406i	$0.375438\pi$
$192$	8.52409i	0.615173i
$193$	0.682908i	0.0491568i	0.999698	+	0.0245784i	$0.00782434\pi$
$193$	0.682908i	0.0491568i	−0.999698	+	0.0245784i	$0.992176\pi$
$194$	25.9959	1.86640
$195$	0	0
$196$	19.1245	1.36604
$197$	− 23.2972i	− 1.65986i	−0.557870	−	0.829928i	$-0.688381\pi$
$197$	− 23.2972i	− 1.65986i	0.557870	−	0.829928i	$-0.311619\pi$
$198$	7.97134i	0.566498i
$199$	−10.1946	−0.722679	−0.361339	−	0.932434i	$-0.617680\pi$
$199$	−10.1946	−0.722679	−0.361339	+	0.932434i	$0.617680\pi$
$200$	0	0
$201$	1.84910	0.130425
$202$	4.29913i	0.302486i
$203$	− 10.7412i	− 0.753884i
$204$	−19.9860	−1.39930
$205$	0	0
$206$	38.8353	2.70578
$207$	1.54765i	0.107569i
$208$	12.0617i	0.836329i
$209$	−10.0649	−0.696205
$210$	0	0
$211$	18.2570	1.25686	0.628431	−	0.777865i	$-0.283697\pi$
$211$	18.2570	1.25686	0.628431	+	0.777865i	$0.283697\pi$
$212$	− 40.1492i	− 2.75746i
$213$	− 0.713969i	− 0.0489203i
$214$	38.5093	2.63244
$215$	0	0
$216$	−4.77071	−0.324606
$217$	− 25.6030i	− 1.73804i
$218$	24.2071i	1.63951i
$219$	−1.88027	−0.127057
$220$	0	0
$221$	16.3327	1.09866
$222$	14.0521i	0.943115i
$223$	− 17.2244i	− 1.15343i	−0.816946	−	0.576714i	$-0.804335\pi$
$223$	− 17.2244i	− 1.15343i	0.816946	−	0.576714i	$-0.195665\pi$
$224$	−1.49473	−0.0998708
$225$	0	0
$226$	13.4554	0.895039
$227$	− 4.32879i	− 0.287312i	−0.989628	−	0.143656i	$-0.954114\pi$
$227$	− 4.32879i	− 0.287312i	0.989628	−	0.143656i	$-0.0458858\pi$
$228$	− 12.1861i	− 0.807042i
$229$	−20.3147	−1.34243	−0.671216	−	0.741262i	$-0.734228\pi$
$229$	−20.3147	−1.34243	−0.671216	+	0.741262i	$0.734228\pi$
$230$	0	0
$231$	−11.2379	−0.739400
$232$	− 14.8950i	− 0.977906i
$233$	0.340956i	0.0223368i	0.999938	+	0.0111684i	$0.00355508\pi$
$233$	0.340956i	0.0223368i	−0.999938	+	0.0111684i	$0.996445\pi$
$234$	7.88709	0.515595
$235$	0	0
$236$	6.84434	0.445529
$237$	13.3332i	0.866087i
$238$	− 42.4244i	− 2.74997i
$239$	7.16488	0.463458	0.231729	−	0.972780i	$-0.425562\pi$
$239$	7.16488	0.463458	0.231729	+	0.972780i	$0.425562\pi$
$240$	0	0
$241$	−12.5622	−0.809204	−0.404602	−	0.914493i	$-0.632590\pi$
$241$	−12.5622	−0.809204	−0.404602	+	0.914493i	$0.632590\pi$
$242$	− 0.804239i	− 0.0516985i
$243$	1.00000i	0.0641500i
$244$	−30.9917	−1.98404
$245$	0	0
$246$	−13.2254	−0.843222
$247$	9.95854i	0.633646i
$248$	− 35.5042i	− 2.25452i
$249$	3.95747	0.250795
$250$	0	0
$251$	17.0160	1.07404	0.537022	−	0.843568i	$-0.319549\pi$
$251$	17.0160	1.07404	0.537022	+	0.843568i	$0.319549\pi$
$252$	− 13.6063i	− 0.857114i
$253$	5.05549i	0.317836i
$254$	2.42098	0.151906
$255$	0	0
$256$	−31.5921	−1.97451
$257$	4.13200i	0.257747i	0.991661	+	0.128874i	$0.0411361\pi$
$257$	4.13200i	0.257747i	−0.991661	+	0.128874i	$0.958864\pi$
$258$	6.17649i	0.384532i
$259$	−19.8105	−1.23096
$260$	0	0
$261$	−3.12218	−0.193258
$262$	− 30.7757i	− 1.90133i
$263$	1.03773i	0.0639892i	0.999488	+	0.0319946i	$0.0101859\pi$
$263$	1.03773i	0.0639892i	−0.999488	+	0.0319946i	$0.989814\pi$
$264$	−15.5838	−0.959118
$265$	0	0
$266$	25.8674	1.58603
$267$	− 8.53392i	− 0.522267i
$268$	7.31315i	0.446722i
$269$	15.7897	0.962713	0.481357	−	0.876525i	$-0.340144\pi$
$269$	15.7897	0.962713	0.481357	+	0.876525i	$0.340144\pi$
$270$	0	0
$271$	−13.7003	−0.832237	−0.416118	−	0.909310i	$-0.636610\pi$
$271$	−13.7003	−0.832237	−0.416118	+	0.909310i	$0.636610\pi$
$272$	− 18.8588i	− 1.14348i
$273$	11.1191i	0.672960i
$274$	−20.1095	−1.21486
$275$	0	0
$276$	−6.12092	−0.368436
$277$	6.97989i	0.419381i	0.977768	+	0.209691i	$0.0672457\pi$
$277$	6.97989i	0.419381i	−0.977768	+	0.209691i	$0.932754\pi$
$278$	− 19.6777i	− 1.18019i
$279$	−7.44212	−0.445548
$280$	0	0
$281$	7.95823	0.474748	0.237374	−	0.971418i	$-0.423713\pi$
$281$	7.95823	0.474748	0.237374	+	0.971418i	$0.423713\pi$
$282$	− 17.2567i	− 1.02762i
$283$	27.7952i	1.65225i	0.563486	+	0.826126i	$0.309460\pi$
$283$	27.7952i	1.65225i	−0.563486	+	0.826126i	$0.690540\pi$
$284$	2.82373	0.167558
$285$	0	0
$286$	25.7637	1.52344
$287$	− 18.6450i	− 1.10058i
$288$	0.434479i	0.0256019i
$289$	−8.53657	−0.502151
$290$	0	0
$291$	10.6528	0.624479
$292$	− 7.43645i	− 0.435185i
$293$	14.2098i	0.830146i	0.909788	+	0.415073i	$0.136244\pi$
$293$	14.2098i	0.830146i	−0.909788	+	0.415073i	$0.863756\pi$
$294$	11.8001	0.688196
$295$	0	0
$296$	−27.4716	−1.59675
$297$	3.26656i	0.189545i
$298$	46.6341i	2.70144i
$299$	5.00206	0.289276
$300$	0	0
$301$	−8.70755	−0.501895
$302$	3.86908i	0.222640i
$303$	1.76173i	0.101209i
$304$	11.4988	0.659500
$305$	0	0
$306$	−12.3317	−0.704954
$307$	− 23.2911i	− 1.32930i	−0.747157	−	0.664648i	$-0.768582\pi$
$307$	− 23.2911i	− 1.32930i	0.747157	−	0.664648i	$-0.231418\pi$
$308$	− 44.4457i	− 2.53253i
$309$	15.9143	0.905330
$310$	0	0
$311$	−7.54924	−0.428078	−0.214039	−	0.976825i	$-0.568662\pi$
$311$	−7.54924	−0.428078	−0.214039	+	0.976825i	$0.568662\pi$
$312$	15.4191i	0.872936i
$313$	31.6565i	1.78933i	0.446734	+	0.894667i	$0.352587\pi$
$313$	31.6565i	1.78933i	−0.446734	+	0.894667i	$0.647413\pi$
$314$	53.3225	3.00916
$315$	0	0
$316$	−52.7327	−2.96645
$317$	25.8362i	1.45110i	0.688167	+	0.725552i	$0.258416\pi$
$317$	25.8362i	1.45110i	−0.688167	+	0.725552i	$0.741584\pi$
$318$	− 24.7726i	− 1.38918i
$319$	−10.1988	−0.571023
$320$	0	0
$321$	15.7807	0.880792
$322$	− 12.9929i	− 0.724067i
$323$	− 15.5704i	− 0.866361i
$324$	−3.95498	−0.219721
$325$	0	0
$326$	24.7295	1.36964
$327$	9.91980i	0.548566i
$328$	− 25.8555i	− 1.42763i
$329$	24.3284	1.34127
$330$	0	0
$331$	20.1083	1.10525	0.552625	−	0.833430i	$-0.313626\pi$
$331$	20.1083	1.10525	0.552625	+	0.833430i	$0.313626\pi$
$332$	15.6517i	0.859001i
$333$	5.75838i	0.315558i
$334$	3.68748	0.201770
$335$	0	0
$336$	12.8389	0.700417
$337$	3.38378i	0.184326i	0.995744	+	0.0921630i	$0.0293781\pi$
$337$	3.38378i	0.184326i	−0.995744	+	0.0921630i	$0.970622\pi$
$338$	6.23232i	0.338993i
$339$	5.51386	0.299472
$340$	0	0
$341$	−24.3101	−1.31647
$342$	− 7.51899i	− 0.406580i
$343$	− 7.44633i	− 0.402064i
$344$	−12.0749	−0.651037
$345$	0	0
$346$	−33.8109	−1.81768
$347$	− 29.1964i	− 1.56734i	−0.621175	−	0.783672i	$-0.713344\pi$
$347$	− 29.1964i	− 1.56734i	0.621175	−	0.783672i	$-0.286656\pi$
$348$	− 12.3482i	− 0.661932i
$349$	20.3979	1.09187	0.545937	−	0.837826i	$-0.316174\pi$
$349$	20.3979	1.09187	0.545937	+	0.837826i	$0.316174\pi$
$350$	0	0
$351$	3.23204	0.172513
$352$	1.41925i	0.0756464i
$353$	1.86346i	0.0991821i	0.998770	+	0.0495910i	$0.0157918\pi$
$353$	1.86346i	0.0991821i	−0.998770	+	0.0495910i	$0.984208\pi$
$354$	4.22306	0.224453
$355$	0	0
$356$	33.7515	1.78883
$357$	− 17.3850i	− 0.920113i
$358$	− 35.2760i	− 1.86440i
$359$	−3.83795	−0.202559	−0.101280	−	0.994858i	$-0.532294\pi$
$359$	−3.83795	−0.202559	−0.101280	+	0.994858i	$0.532294\pi$
$360$	0	0
$361$	−9.50624	−0.500329
$362$	41.9480i	2.20474i
$363$	− 0.329568i	− 0.0172978i
$364$	−43.9759	−2.30497
$365$	0	0
$366$	−19.1223	−0.999541
$367$	25.6281i	1.33777i	0.743364	+	0.668887i	$0.233229\pi$
$367$	25.6281i	1.33777i	−0.743364	+	0.668887i	$0.766771\pi$
$368$	− 5.77570i	− 0.301079i
$369$	−5.41962	−0.282134
$370$	0	0
$371$	34.9242	1.81317
$372$	− 29.4334i	− 1.52605i
$373$	− 21.4369i	− 1.10996i	−0.831863	−	0.554980i	$-0.812726\pi$
$373$	− 21.4369i	− 1.10996i	0.831863	−	0.554980i	$-0.187274\pi$
$374$	−40.2821	−2.08294
$375$	0	0
$376$	33.7366	1.73983
$377$	10.0910i	0.519713i
$378$	− 8.39527i	− 0.431806i
$379$	25.8713	1.32892	0.664461	−	0.747323i	$-0.268661\pi$
$379$	25.8713	1.32892	0.664461	+	0.747323i	$0.268661\pi$
$380$	0	0
$381$	0.992089	0.0508262
$382$	25.7264i	1.31628i
$383$	14.5336i	0.742632i	0.928507	+	0.371316i	$0.121093\pi$
$383$	14.5336i	0.742632i	−0.928507	+	0.371316i	$0.878907\pi$
$384$	−19.9322	−1.01716
$385$	0	0
$386$	−1.66649	−0.0848221
$387$	2.53106i	0.128661i
$388$	42.1317i	2.13891i
$389$	−23.1047	−1.17145	−0.585726	−	0.810509i	$-0.699191\pi$
$389$	−23.1047	−1.17145	−0.585726	+	0.810509i	$0.699191\pi$
$390$	0	0
$391$	−7.82084	−0.395517
$392$	23.0690i	1.16516i
$393$	− 12.6115i	− 0.636167i
$394$	56.8517	2.86415
$395$	0	0
$396$	−12.9192	−0.649214
$397$	13.9345i	0.699355i	0.936870	+	0.349677i	$0.113709\pi$
$397$	13.9345i	0.699355i	−0.936870	+	0.349677i	$0.886291\pi$
$398$	− 24.8778i	− 1.24701i
$399$	10.6002	0.530673
$400$	0	0
$401$	−27.5822	−1.37739	−0.688694	−	0.725052i	$-0.741816\pi$
$401$	−27.5822	−1.37739	−0.688694	+	0.725052i	$0.741816\pi$
$402$	4.51232i	0.225054i
$403$	24.0532i	1.19818i
$404$	−6.96763	−0.346653
$405$	0	0
$406$	26.2115	1.30086
$407$	18.8101i	0.932383i
$408$	− 24.1082i	− 1.19353i
$409$	−13.2125	−0.653315	−0.326658	−	0.945143i	$-0.605922\pi$
$409$	−13.2125	−0.653315	−0.326658	+	0.945143i	$0.605922\pi$
$410$	0	0
$411$	−8.24065	−0.406481
$412$	62.9406i	3.10086i
$413$	5.95362i	0.292959i
$414$	−3.77670	−0.185615
$415$	0	0
$416$	1.40425	0.0688491
$417$	− 8.06371i	− 0.394881i
$418$	− 24.5612i	− 1.20133i
$419$	8.84774	0.432240	0.216120	−	0.976367i	$-0.430660\pi$
$419$	8.84774	0.432240	0.216120	+	0.976367i	$0.430660\pi$
$420$	0	0
$421$	−37.4766	−1.82650	−0.913248	−	0.407404i	$-0.866434\pi$
$421$	−37.4766	−1.82650	−0.913248	+	0.407404i	$0.866434\pi$
$422$	44.5522i	2.16877i
$423$	− 7.07162i	− 0.343834i
$424$	48.4301	2.35197
$425$	0	0
$426$	1.74229	0.0844140
$427$	− 26.9585i	− 1.30461i
$428$	62.4123i	3.01681i
$429$	10.5577	0.509728
$430$	0	0
$431$	−26.6245	−1.28246	−0.641228	−	0.767350i	$-0.721575\pi$
$431$	−26.6245	−1.28246	−0.641228	+	0.767350i	$0.721575\pi$
$432$	− 3.73192i	− 0.179552i
$433$	− 34.9972i	− 1.68186i	−0.541146	−	0.840928i	$-0.682010\pi$
$433$	− 34.9972i	− 1.68186i	0.541146	−	0.840928i	$-0.317990\pi$
$434$	62.4785	2.99907
$435$	0	0
$436$	−39.2326	−1.87890
$437$	− 4.76861i	− 0.228113i
$438$	− 4.58840i	− 0.219242i
$439$	−20.0539	−0.957122	−0.478561	−	0.878054i	$-0.658841\pi$
$439$	−20.0539	−0.957122	−0.478561	+	0.878054i	$0.658841\pi$
$440$	0	0
$441$	4.83555	0.230264
$442$	39.8564i	1.89577i
$443$	− 4.14871i	− 0.197111i	−0.995132	−	0.0985556i	$-0.968578\pi$
$443$	− 4.14871i	− 0.197111i	0.995132	−	0.0985556i	$-0.0314222\pi$
$444$	−22.7743	−1.08082
$445$	0	0
$446$	42.0323	1.99029
$447$	19.1101i	0.903877i
$448$	− 29.3253i	− 1.38549i
$449$	8.34804	0.393969	0.196984	−	0.980407i	$-0.436885\pi$
$449$	8.34804	0.393969	0.196984	+	0.980407i	$0.436885\pi$
$450$	0	0
$451$	−17.7035	−0.833627
$452$	21.8072i	1.02573i
$453$	1.58550i	0.0744935i
$454$	10.5635	0.495768
$455$	0	0
$456$	14.6995	0.688366
$457$	20.5774i	0.962571i	0.876564	+	0.481285i	$0.159830\pi$
$457$	20.5774i	0.962571i	−0.876564	+	0.481285i	$0.840170\pi$
$458$	− 49.5736i	− 2.31642i
$459$	−5.05337	−0.235871
$460$	0	0
$461$	28.6891	1.33618	0.668092	−	0.744079i	$-0.267111\pi$
$461$	28.6891	1.33618	0.668092	+	0.744079i	$0.267111\pi$
$462$	− 27.4237i	− 1.27586i
$463$	− 40.0797i	− 1.86266i	−0.364173	−	0.931331i	$-0.618648\pi$
$463$	− 40.0797i	− 1.86266i	0.364173	−	0.931331i	$-0.381352\pi$
$464$	11.6517	0.540918
$465$	0	0
$466$	−0.832030	−0.0385430
$467$	− 8.14191i	− 0.376763i	−0.982096	−	0.188381i	$-0.939676\pi$
$467$	− 8.14191i	− 0.376763i	0.982096	−	0.188381i	$-0.0603241\pi$
$468$	12.7827i	0.590878i
$469$	−6.36142	−0.293743
$470$	0	0
$471$	21.8510	1.00684
$472$	8.25601i	0.380014i
$473$	8.26785i	0.380156i
$474$	−32.5369	−1.49447
$475$	0	0
$476$	68.7575	3.15149
$477$	− 10.1515i	− 0.464807i
$478$	17.4843i	0.799715i
$479$	28.0621	1.28219	0.641094	−	0.767462i	$-0.278481\pi$
$479$	28.0621	1.28219	0.641094	+	0.767462i	$0.278481\pi$
$480$	0	0
$481$	18.6113	0.848603
$482$	− 30.6554i	− 1.39631i
$483$	− 5.32435i	− 0.242266i
$484$	1.30344	0.0592471
$485$	0	0
$486$	−2.44028	−0.110693
$487$	− 15.2173i	− 0.689560i	−0.938684	−	0.344780i	$-0.887954\pi$
$487$	− 15.2173i	− 0.689560i	0.938684	−	0.344780i	$-0.112046\pi$
$488$	− 37.3838i	− 1.69229i
$489$	10.1338	0.458269
$490$	0	0
$491$	8.29230	0.374226	0.187113	−	0.982338i	$-0.440087\pi$
$491$	8.29230	0.374226	0.187113	+	0.982338i	$0.440087\pi$
$492$	− 21.4345i	− 0.966343i
$493$	− 15.7775i	− 0.710585i
$494$	−24.3016	−1.09338
$495$	0	0
$496$	27.7734	1.24706
$497$	2.45625i	0.110178i
$498$	9.65736i	0.432756i
$499$	24.4006	1.09232	0.546160	−	0.837681i	$-0.316089\pi$
$499$	24.4006	1.09232	0.546160	+	0.837681i	$0.316089\pi$
$500$	0	0
$501$	1.51109	0.0675104
$502$	41.5240i	1.85331i
$503$	− 10.4731i	− 0.466974i	−0.972360	−	0.233487i	$-0.924986\pi$
$503$	− 10.4731i	− 0.466974i	0.972360	−	0.233487i	$-0.0750137\pi$
$504$	16.4126	0.731075
$505$	0	0
$506$	−12.3368	−0.548439
$507$	2.55393i	0.113424i
$508$	3.92369i	0.174086i
$509$	11.0599	0.490220	0.245110	−	0.969495i	$-0.421176\pi$
$509$	11.0599	0.490220	0.245110	+	0.969495i	$0.421176\pi$
$510$	0	0
$511$	6.46867	0.286157
$512$	− 37.2293i	− 1.64532i
$513$	− 3.08119i	− 0.136038i
$514$	−10.0832	−0.444753
$515$	0	0
$516$	−10.0103	−0.440678
$517$	− 23.0999i	− 1.01593i
$518$	− 48.3432i	− 2.12408i
$519$	−13.8553	−0.608181
$520$	0	0
$521$	−41.4212	−1.81470	−0.907348	−	0.420380i	$-0.861897\pi$
$521$	−41.4212	−1.81470	−0.907348	+	0.420380i	$0.861897\pi$
$522$	− 7.61901i	− 0.333475i
$523$	− 13.6425i	− 0.596544i	−0.954481	−	0.298272i	$-0.903590\pi$
$523$	− 13.6425i	− 0.596544i	0.954481	−	0.298272i	$-0.0964102\pi$
$524$	49.8783	2.17894
$525$	0	0
$526$	−2.53235	−0.110416
$527$	− 37.6078i	− 1.63822i
$528$	− 12.1906i	− 0.530526i
$529$	20.6048	0.895860
$530$	0	0
$531$	1.73056	0.0751000
$532$	41.9235i	1.81762i
$533$	17.5164i	0.758721i
$534$	20.8252	0.901194
$535$	0	0
$536$	−8.82151	−0.381031
$537$	− 14.4557i	− 0.623810i
$538$	38.5313i	1.66120i
$539$	15.7956	0.680366
$540$	0	0
$541$	14.6598	0.630272	0.315136	−	0.949046i	$-0.397950\pi$
$541$	14.6598	0.630272	0.315136	+	0.949046i	$0.397950\pi$
$542$	− 33.4327i	− 1.43606i
$543$	17.1898i	0.737685i
$544$	−2.19558	−0.0941348
$545$	0	0
$546$	−27.1338	−1.16122
$547$	26.6123i	1.13786i	0.822385	+	0.568931i	$0.192643\pi$
$547$	26.6123i	1.13786i	−0.822385	+	0.568931i	$0.807357\pi$
$548$	− 32.5916i	− 1.39225i
$549$	−7.83611	−0.334437
$550$	0	0
$551$	9.62005	0.409828
$552$	− 7.38338i	− 0.314258i
$553$	− 45.8701i	− 1.95060i
$554$	−17.0329	−0.723659
$555$	0	0
$556$	31.8918	1.35251
$557$	− 10.3141i	− 0.437020i	−0.975835	−	0.218510i	$-0.929880\pi$
$557$	− 10.3141i	− 0.437020i	0.975835	−	0.218510i	$-0.0701197\pi$
$558$	− 18.1609i	− 0.768811i
$559$	8.18047	0.345997
$560$	0	0
$561$	−16.5072	−0.696933
$562$	19.4203i	0.819197i
$563$	34.4031i	1.44992i	0.688793	+	0.724958i	$0.258141\pi$
$563$	34.4031i	1.44992i	−0.688793	+	0.724958i	$0.741859\pi$
$564$	27.9681	1.17767
$565$	0	0
$566$	−67.8281	−2.85103
$567$	− 3.44028i	− 0.144478i
$568$	3.40614i	0.142918i
$569$	12.2048	0.511651	0.255825	−	0.966723i	$-0.417653\pi$
$569$	12.2048	0.511651	0.255825	+	0.966723i	$0.417653\pi$
$570$	0	0
$571$	42.0644	1.76034	0.880170	−	0.474658i	$-0.157428\pi$
$571$	42.0644	1.76034	0.880170	+	0.474658i	$0.157428\pi$
$572$	41.7553i	1.74588i
$573$	10.5424i	0.440415i
$574$	45.4992	1.89910
$575$	0	0
$576$	−8.52409	−0.355170
$577$	17.5351i	0.729994i	0.931009	+	0.364997i	$0.118930\pi$
$577$	17.5351i	0.729994i	−0.931009	+	0.364997i	$0.881070\pi$
$578$	− 20.8316i	− 0.866482i
$579$	−0.682908	−0.0283807
$580$	0	0
$581$	−13.6148	−0.564838
$582$	25.9959i	1.07756i
$583$	− 33.1607i	− 1.37337i
$584$	8.97024	0.371191
$585$	0	0
$586$	−34.6759	−1.43245
$587$	− 21.9119i	− 0.904402i	−0.891916	−	0.452201i	$-0.850639\pi$
$587$	− 21.9119i	− 0.904402i	0.891916	−	0.452201i	$-0.149361\pi$
$588$	19.1245i	0.788681i
$589$	22.9306	0.944839
$590$	0	0
$591$	23.2972	0.958318
$592$	− 21.4898i	− 0.883227i
$593$	38.0061i	1.56072i	0.625330	+	0.780361i	$0.284965\pi$
$593$	38.0061i	1.56072i	−0.625330	+	0.780361i	$0.715035\pi$
$594$	−7.97134	−0.327068
$595$	0	0
$596$	−75.5802	−3.09588
$597$	− 10.1946i	− 0.417239i
$598$	12.2064i	0.499158i
$599$	−16.3154	−0.666629	−0.333314	−	0.942816i	$-0.608167\pi$
$599$	−16.3154	−0.666629	−0.333314	+	0.942816i	$0.608167\pi$
$600$	0	0
$601$	2.31871	0.0945822	0.0472911	−	0.998881i	$-0.484941\pi$
$601$	2.31871	0.0945822	0.0472911	+	0.998881i	$0.484941\pi$
$602$	− 21.2489i	− 0.866040i
$603$	1.84910i	0.0753011i
$604$	−6.27064	−0.255149
$605$	0	0
$606$	−4.29913	−0.174640
$607$	32.2134i	1.30750i	0.756709	+	0.653752i	$0.226806\pi$
$607$	32.2134i	1.30750i	−0.756709	+	0.653752i	$0.773194\pi$
$608$	− 1.33871i	− 0.0542920i
$609$	10.7412	0.435255
$610$	0	0
$611$	−22.8557	−0.924644
$612$	− 19.9860i	− 0.807886i
$613$	− 25.9482i	− 1.04804i	−0.851707	−	0.524018i	$-0.824432\pi$
$613$	− 25.9482i	− 1.04804i	0.851707	−	0.524018i	$-0.175568\pi$
$614$	56.8370	2.29375
$615$	0	0
$616$	53.6128	2.16012
$617$	1.90374i	0.0766416i	0.999265	+	0.0383208i	$0.0122009\pi$
$617$	1.90374i	0.0766416i	−0.999265	+	0.0383208i	$0.987799\pi$
$618$	38.8353i	1.56218i
$619$	19.8795	0.799025	0.399513	−	0.916728i	$-0.369179\pi$
$619$	19.8795	0.799025	0.399513	+	0.916728i	$0.369179\pi$
$620$	0	0
$621$	−1.54765	−0.0621050
$622$	− 18.4223i	− 0.738666i
$623$	29.3591i	1.17625i
$624$	−12.0617	−0.482855
$625$	0	0
$626$	−77.2509	−3.08757
$627$	− 10.0649i	− 0.401954i
$628$	86.4201i	3.44854i
$629$	−29.0993	−1.16026
$630$	0	0
$631$	−32.8982	−1.30966	−0.654828	−	0.755778i	$-0.727259\pi$
$631$	−32.8982	−1.30966	−0.654828	+	0.755778i	$0.727259\pi$
$632$	− 63.6090i	− 2.53023i
$633$	18.2570i	0.725650i
$634$	−63.0476	−2.50394
$635$	0	0
$636$	40.1492	1.59202
$637$	− 15.6287i	− 0.619231i
$638$	− 24.8880i	− 0.985324i
$639$	0.713969	0.0282442
$640$	0	0
$641$	−40.7624	−1.61002	−0.805009	−	0.593263i	$-0.797839\pi$
$641$	−40.7624	−1.61002	−0.805009	+	0.593263i	$0.797839\pi$
$642$	38.5093i	1.51984i
$643$	− 24.9947i	− 0.985695i	−0.870116	−	0.492847i	$-0.835956\pi$
$643$	− 24.9947i	− 0.985695i	0.870116	−	0.492847i	$-0.164044\pi$
$644$	21.0577	0.829790
$645$	0	0
$646$	37.9962	1.49494
$647$	− 32.0232i	− 1.25896i	−0.777016	−	0.629481i	$-0.783268\pi$
$647$	− 32.0232i	− 1.25896i	0.777016	−	0.629481i	$-0.216732\pi$
$648$	− 4.77071i	− 0.187411i
$649$	5.65299	0.221899
$650$	0	0
$651$	25.6030	1.00346
$652$	40.0792i	1.56962i
$653$	44.6772i	1.74835i	0.485607	+	0.874177i	$0.338599\pi$
$653$	44.6772i	1.74835i	−0.485607	+	0.874177i	$0.661401\pi$
$654$	−24.2071	−0.946573
$655$	0	0
$656$	20.2256	0.789677
$657$	− 1.88027i	− 0.0733564i
$658$	59.3681i	2.31441i
$659$	24.0215	0.935747	0.467873	−	0.883796i	$-0.345020\pi$
$659$	24.0215	0.935747	0.467873	+	0.883796i	$0.345020\pi$
$660$	0	0
$661$	37.8928	1.47386	0.736929	−	0.675970i	$-0.236275\pi$
$661$	37.8928	1.47386	0.736929	+	0.675970i	$0.236275\pi$
$662$	49.0699i	1.90715i
$663$	16.3327i	0.634309i
$664$	−18.8800	−0.732684
$665$	0	0
$666$	−14.0521	−0.544507
$667$	− 4.83204i	− 0.187097i
$668$	5.97633i	0.231231i
$669$	17.2244	0.665932
$670$	0	0
$671$	−25.5972	−0.988167
$672$	− 1.49473i	− 0.0576604i
$673$	18.4267i	0.710299i	0.934810	+	0.355149i	$0.115570\pi$
$673$	18.4267i	0.710299i	−0.934810	+	0.355149i	$0.884430\pi$
$674$	−8.25737	−0.318062
$675$	0	0
$676$	−10.1008	−0.388491
$677$	− 22.8164i	− 0.876904i	−0.898755	−	0.438452i	$-0.855527\pi$
$677$	− 22.8164i	− 0.876904i	0.898755	−	0.438452i	$-0.144473\pi$
$678$	13.4554i	0.516751i
$679$	−36.6487	−1.40645
$680$	0	0
$681$	4.32879	0.165879
$682$	− 59.3236i	− 2.27162i
$683$	− 32.2064i	− 1.23234i	−0.787612	−	0.616172i	$-0.788683\pi$
$683$	− 32.2064i	− 1.23234i	0.787612	−	0.616172i	$-0.211317\pi$
$684$	12.1861	0.465946
$685$	0	0
$686$	18.1711	0.693777
$687$	− 20.3147i	− 0.775053i
$688$	− 9.44570i	− 0.360114i
$689$	−32.8102	−1.24997
$690$	0	0
$691$	24.5769	0.934947	0.467474	−	0.884007i	$-0.345164\pi$
$691$	24.5769	0.934947	0.467474	+	0.884007i	$0.345164\pi$
$692$	− 54.7975i	− 2.08309i
$693$	− 11.2379i	− 0.426893i
$694$	71.2475	2.70452
$695$	0	0
$696$	14.8950	0.564594
$697$	− 27.3874i	− 1.03737i
$698$	49.7766i	1.88407i
$699$	−0.340956	−0.0128961
$700$	0	0
$701$	−3.66355	−0.138370	−0.0691852	−	0.997604i	$-0.522040\pi$
$701$	−3.66355	−0.138370	−0.0691852	+	0.997604i	$0.522040\pi$
$702$	7.88709i	0.297679i
$703$	− 17.7427i	− 0.669179i
$704$	−27.8445	−1.04943
$705$	0	0
$706$	−4.54737	−0.171143
$707$	− 6.06087i	− 0.227942i
$708$	6.84434i	0.257226i
$709$	26.6969	1.00262	0.501312	−	0.865267i	$-0.332851\pi$
$709$	26.6969	1.00262	0.501312	+	0.865267i	$0.332851\pi$
$710$	0	0
$711$	−13.3332	−0.500035
$712$	40.7129i	1.52578i
$713$	− 11.5178i	− 0.431344i
$714$	42.4244	1.58769
$715$	0	0
$716$	57.1721	2.13662
$717$	7.16488i	0.267577i
$718$	− 9.36569i	− 0.349524i
$719$	−17.8699	−0.666433	−0.333217	−	0.942850i	$-0.608134\pi$
$719$	−17.8699	−0.666433	−0.333217	+	0.942850i	$0.608134\pi$
$720$	0	0
$721$	−54.7495	−2.03898
$722$	− 23.1979i	− 0.863337i
$723$	− 12.5622i	− 0.467194i
$724$	−67.9853	−2.52665
$725$	0	0
$726$	0.804239	0.0298481
$727$	− 34.0727i	− 1.26369i	−0.775097	−	0.631843i	$-0.782299\pi$
$727$	− 34.0727i	− 1.26369i	0.775097	−	0.631843i	$-0.217701\pi$
$728$	− 53.0461i	− 1.96602i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.7904	−0.473069
$732$	− 30.9917i	− 1.14549i
$733$	40.1806i	1.48410i	0.670342	+	0.742052i	$0.266147\pi$
$733$	40.1806i	1.48410i	−0.670342	+	0.742052i	$0.733853\pi$
$734$	−62.5398	−2.30838
$735$	0	0
$736$	−0.672420	−0.0247857
$737$	6.04019i	0.222493i
$738$	− 13.2254i	− 0.486834i
$739$	−15.4750	−0.569256	−0.284628	−	0.958638i	$-0.591870\pi$
$739$	−15.4750	−0.569256	−0.284628	+	0.958638i	$0.591870\pi$
$740$	0	0
$741$	−9.95854	−0.365836
$742$	85.2249i	3.12870i
$743$	17.1140i	0.627853i	0.949447	+	0.313926i	$0.101645\pi$
$743$	17.1140i	0.627853i	−0.949447	+	0.313926i	$0.898355\pi$
$744$	35.5042	1.30165
$745$	0	0
$746$	52.3121	1.91528
$747$	3.95747i	0.144796i
$748$	− 65.2855i	− 2.38707i
$749$	−54.2900	−1.98371
$750$	0	0
$751$	−11.1559	−0.407086	−0.203543	−	0.979066i	$-0.565246\pi$
$751$	−11.1559	−0.407086	−0.203543	+	0.979066i	$0.565246\pi$
$752$	26.3907i	0.962370i
$753$	17.0160i	0.620099i
$754$	−24.6249	−0.896787
$755$	0	0
$756$	13.6063	0.494855
$757$	24.6773i	0.896911i	0.893805	+	0.448456i	$0.148026\pi$
$757$	24.6773i	0.896911i	−0.893805	+	0.448456i	$0.851974\pi$
$758$	63.1334i	2.29311i
$759$	−5.05549	−0.183503
$760$	0	0
$761$	−40.2478	−1.45898	−0.729490	−	0.683991i	$-0.760243\pi$
$761$	−40.2478	−1.45898	−0.729490	+	0.683991i	$0.760243\pi$
$762$	2.42098i	0.0877028i
$763$	− 34.1269i	− 1.23548i
$764$	−41.6950	−1.50847
$765$	0	0
$766$	−35.4661	−1.28144
$767$	− 5.59324i	− 0.201960i
$768$	− 31.5921i	− 1.13998i
$769$	−22.1010	−0.796983	−0.398492	−	0.917172i	$-0.630466\pi$
$769$	−22.1010	−0.796983	−0.398492	+	0.917172i	$0.630466\pi$
$770$	0	0
$771$	−4.13200	−0.148810
$772$	− 2.70089i	− 0.0972072i
$773$	− 4.90033i	− 0.176253i	−0.996109	−	0.0881264i	$-0.971912\pi$
$773$	− 4.90033i	− 0.176253i	0.996109	−	0.0881264i	$-0.0280879\pi$
$774$	−6.17649	−0.222009
$775$	0	0
$776$	−50.8215	−1.82439
$777$	− 19.8105i	− 0.710697i
$778$	− 56.3819i	− 2.02139i
$779$	16.6989	0.598301
$780$	0	0
$781$	2.33222	0.0834535
$782$	− 19.0851i	− 0.682481i
$783$	− 3.12218i	− 0.111578i
$784$	−18.0459	−0.644496
$785$	0	0
$786$	30.7757	1.09773
$787$	− 3.73477i	− 0.133130i	−0.997782	−	0.0665652i	$-0.978796\pi$
$787$	− 3.73477i	− 0.133130i	0.997782	−	0.0665652i	$-0.0212040\pi$
$788$	92.1400i	3.28235i
$789$	−1.03773	−0.0369442
$790$	0	0
$791$	−18.9692	−0.674469
$792$	− 15.5838i	− 0.553747i
$793$	25.3266i	0.899375i
$794$	−34.0042	−1.20677
$795$	0	0
$796$	40.3196	1.42909
$797$	3.34404i	0.118452i	0.998245	+	0.0592259i	$0.0188632\pi$
$797$	3.34404i	0.118452i	−0.998245	+	0.0592259i	$0.981137\pi$
$798$	25.8674i	0.915698i
$799$	35.7355	1.26423
$800$	0	0
$801$	8.53392	0.301531
$802$	− 67.3083i	− 2.37674i
$803$	− 6.14203i	− 0.216748i
$804$	−7.31315	−0.257915
$805$	0	0
$806$	−58.6966	−2.06750
$807$	15.7897i	0.555823i
$808$	− 8.40473i	− 0.295677i
$809$	−34.2875	−1.20549	−0.602743	−	0.797935i	$-0.705926\pi$
$809$	−34.2875	−1.20549	−0.602743	+	0.797935i	$0.705926\pi$
$810$	0	0
$811$	−3.46750	−0.121760	−0.0608801	−	0.998145i	$-0.519391\pi$
$811$	−3.46750	−0.121760	−0.0608801	+	0.998145i	$0.519391\pi$
$812$	42.4812i	1.49080i
$813$	− 13.7003i	− 0.480492i
$814$	−45.9020	−1.60887
$815$	0	0
$816$	18.8588	0.660189
$817$	− 7.79867i	− 0.272841i
$818$	− 32.2422i	− 1.12732i
$819$	−11.1191	−0.388534
$820$	0	0
$821$	−14.6809	−0.512367	−0.256184	−	0.966628i	$-0.582465\pi$
$821$	−14.6809	−0.512367	−0.256184	+	0.966628i	$0.582465\pi$
$822$	− 20.1095i	− 0.701400i
$823$	20.8698i	0.727475i	0.931502	+	0.363737i	$0.118499\pi$
$823$	20.8698i	0.727475i	−0.931502	+	0.363737i	$0.881501\pi$
$824$	−75.9223	−2.64488
$825$	0	0
$826$	−14.5285	−0.505512
$827$	45.2876i	1.57480i	0.616441	+	0.787401i	$0.288574\pi$
$827$	45.2876i	1.57480i	−0.616441	+	0.787401i	$0.711426\pi$
$828$	− 6.12092i	− 0.212717i
$829$	9.65729	0.335411	0.167706	−	0.985837i	$-0.446364\pi$
$829$	9.65729	0.335411	0.167706	+	0.985837i	$0.446364\pi$
$830$	0	0
$831$	−6.97989	−0.242130
$832$	27.5502i	0.955131i
$833$	24.4358i	0.846651i
$834$	19.6777	0.681384
$835$	0	0
$836$	39.8066	1.37674
$837$	− 7.44212i	− 0.257237i
$838$	21.5910i	0.745848i
$839$	25.1717	0.869023	0.434511	−	0.900666i	$-0.356921\pi$
$839$	25.1717	0.869023	0.434511	+	0.900666i	$0.356921\pi$
$840$	0	0
$841$	−19.2520	−0.663861
$842$	− 91.4534i	− 3.15169i
$843$	7.95823i	0.274096i
$844$	−72.2060	−2.48543
$845$	0	0
$846$	17.2567	0.593299
$847$	1.13381i	0.0389581i
$848$	37.8848i	1.30097i
$849$	−27.7952	−0.953928
$850$	0	0
$851$	−8.91196	−0.305498
$852$	2.82373i	0.0967395i
$853$	3.20847i	0.109856i	0.998490	+	0.0549280i	$0.0174929\pi$
$853$	3.20847i	0.109856i	−0.998490	+	0.0549280i	$0.982507\pi$
$854$	65.7863	2.25116
$855$	0	0
$856$	−75.2851	−2.57319
$857$	19.4569i	0.664634i	0.943168	+	0.332317i	$0.107830\pi$
$857$	19.4569i	0.664634i	−0.943168	+	0.332317i	$0.892170\pi$
$858$	25.7637i	0.879557i
$859$	−16.8332	−0.574340	−0.287170	−	0.957880i	$-0.592714\pi$
$859$	−16.8332	−0.574340	−0.287170	+	0.957880i	$0.592714\pi$
$860$	0	0
$861$	18.6450	0.635421
$862$	− 64.9713i	− 2.21293i
$863$	− 3.81218i	− 0.129768i	−0.997893	−	0.0648841i	$-0.979332\pi$
$863$	− 3.81218i	− 0.129768i	0.997893	−	0.0648841i	$-0.0206678\pi$
$864$	−0.434479	−0.0147813
$865$	0	0
$866$	85.4030	2.90211
$867$	− 8.53657i	− 0.289917i
$868$	101.259i	3.43697i
$869$	−43.5538	−1.47746
$870$	0	0
$871$	5.97635	0.202501
$872$	− 47.3245i	− 1.60261i
$873$	10.6528i	0.360543i
$874$	11.6367	0.393619
$875$	0	0
$876$	7.43645	0.251254
$877$	− 25.7451i	− 0.869352i	−0.900587	−	0.434676i	$-0.856863\pi$
$877$	− 25.7451i	− 0.869352i	0.900587	−	0.434676i	$-0.143137\pi$
$878$	− 48.9373i	− 1.65155i
$879$	−14.2098	−0.479285
$880$	0	0
$881$	45.2537	1.52464	0.762318	−	0.647202i	$-0.224061\pi$
$881$	45.2537	1.52464	0.762318	+	0.647202i	$0.224061\pi$
$882$	11.8001i	0.397330i
$883$	− 44.3978i	− 1.49410i	−0.664765	−	0.747052i	$-0.731469\pi$
$883$	− 44.3978i	− 1.49410i	0.664765	−	0.747052i	$-0.268531\pi$
$884$	−64.5955	−2.17258
$885$	0	0
$886$	10.1240	0.340124
$887$	− 38.9928i	− 1.30925i	−0.755953	−	0.654626i	$-0.772826\pi$
$887$	− 38.9928i	− 1.30925i	0.755953	−	0.654626i	$-0.227174\pi$
$888$	− 27.4716i	− 0.921886i
$889$	−3.41307	−0.114471
$890$	0	0
$891$	−3.26656	−0.109434
$892$	68.1220i	2.28089i
$893$	21.7890i	0.729142i
$894$	−46.6341	−1.55968
$895$	0	0
$896$	68.5726	2.29085
$897$	5.00206i	0.167014i
$898$	20.3716i	0.679809i
$899$	23.2356	0.774952
$900$	0	0
$901$	51.2995	1.70903
$902$	− 43.2017i	− 1.43846i
$903$	− 8.70755i	− 0.289769i
$904$	−26.3050	−0.874892
$905$	0	0
$906$	−3.86908	−0.128541
$907$	11.0201i	0.365915i	0.983121	+	0.182958i	$0.0585671\pi$
$907$	11.0201i	0.365915i	−0.983121	+	0.182958i	$0.941433\pi$
$908$	17.1203i	0.568156i
$909$	−1.76173	−0.0584331
$910$	0	0
$911$	−52.9227	−1.75341	−0.876704	−	0.481031i	$-0.840263\pi$
$911$	−52.9227	−1.75341	−0.876704	+	0.481031i	$0.840263\pi$
$912$	11.4988i	0.380762i
$913$	12.9273i	0.427832i
$914$	−50.2147	−1.66096
$915$	0	0
$916$	80.3442	2.65465
$917$	43.3872i	1.43277i
$918$	− 12.3317i	− 0.407005i
$919$	47.6045	1.57033	0.785164	−	0.619288i	$-0.212579\pi$
$919$	47.6045	1.57033	0.785164	+	0.619288i	$0.212579\pi$
$920$	0	0
$921$	23.2911	0.767469
$922$	70.0095i	2.30564i
$923$	− 2.30757i	− 0.0759547i
$924$	44.4457	1.46216
$925$	0	0
$926$	97.8059	3.21410
$927$	15.9143i	0.522693i
$928$	− 1.35652i	− 0.0445300i
$929$	−21.6887	−0.711582	−0.355791	−	0.934566i	$-0.615789\pi$
$929$	−21.6887	−0.711582	−0.355791	+	0.934566i	$0.615789\pi$
$930$	0	0
$931$	−14.8993	−0.488304
$932$	− 1.34848i	− 0.0441708i
$933$	− 7.54924i	− 0.247151i
$934$	19.8686	0.650119
$935$	0	0
$936$	−15.4191	−0.503990
$937$	2.20089i	0.0718999i	0.999354	+	0.0359500i	$0.0114457\pi$
$937$	2.20089i	0.0718999i	−0.999354	+	0.0359500i	$0.988554\pi$
$938$	− 15.5237i	− 0.506866i
$939$	−31.6565	−1.03307
$940$	0	0
$941$	−16.7898	−0.547331	−0.273666	−	0.961825i	$-0.588236\pi$
$941$	−16.7898	−0.547331	−0.273666	+	0.961825i	$0.588236\pi$
$942$	53.3225i	1.73734i
$943$	− 8.38767i	− 0.273140i
$944$	−6.45832	−0.210200
$945$	0	0
$946$	−20.1759	−0.655975
$947$	36.6132i	1.18977i	0.803811	+	0.594885i	$0.202802\pi$
$947$	36.6132i	1.18977i	−0.803811	+	0.594885i	$0.797198\pi$
$948$	− 52.7327i	− 1.71268i
$949$	−6.07711	−0.197271
$950$	0	0
$951$	−25.8362	−0.837796
$952$	82.9389i	2.68807i
$953$	− 43.9277i	− 1.42296i	−0.702707	−	0.711479i	$-0.748026\pi$
$953$	− 43.9277i	− 1.42296i	0.702707	−	0.711479i	$-0.251974\pi$
$954$	24.7726	0.802044
$955$	0	0
$956$	−28.3370	−0.916483
$957$	− 10.1988i	− 0.329680i
$958$	68.4794i	2.21247i
$959$	28.3502	0.915475
$960$	0	0
$961$	24.3851	0.786616
$962$	45.4169i	1.46430i
$963$	15.7807i	0.508525i
$964$	49.6834	1.60019
$965$	0	0
$966$	12.9929	0.418040
$967$	24.4815i	0.787272i	0.919266	+	0.393636i	$0.128783\pi$
$967$	24.4815i	0.787272i	−0.919266	+	0.393636i	$0.871217\pi$
$968$	1.57227i	0.0505348i
$969$	15.5704	0.500194
$970$	0	0
$971$	−19.6254	−0.629808	−0.314904	−	0.949123i	$-0.601972\pi$
$971$	−19.6254	−0.629808	−0.314904	+	0.949123i	$0.601972\pi$
$972$	− 3.95498i	− 0.126856i
$973$	27.7414i	0.889349i
$974$	37.1344	1.18986
$975$	0	0
$976$	29.2438	0.936070
$977$	30.5238i	0.976541i	0.872692	+	0.488271i	$0.162372\pi$
$977$	30.5238i	0.976541i	−0.872692	+	0.488271i	$0.837628\pi$
$978$	24.7295i	0.790761i
$979$	27.8766	0.890939
$980$	0	0
$981$	−9.91980	−0.316715
$982$	20.2356i	0.645743i
$983$	− 10.3655i	− 0.330609i	−0.986243	−	0.165305i	$-0.947139\pi$
$983$	− 10.3655i	− 0.330609i	0.986243	−	0.165305i	$-0.0528608\pi$
$984$	25.8555	0.824242
$985$	0	0
$986$	38.5017	1.22614
$987$	24.3284i	0.774380i
$988$	− 39.3858i	− 1.25303i
$989$	−3.91718	−0.124559
$990$	0	0
$991$	−10.1087	−0.321113	−0.160557	−	0.987027i	$-0.551329\pi$
$991$	−10.1087	−0.321113	−0.160557	+	0.987027i	$0.551329\pi$
$992$	− 3.23344i	− 0.102662i
$993$	20.1083i	0.638116i
$994$	−5.99396	−0.190117
$995$	0	0
$996$	−15.6517	−0.495944
$997$	21.3639i	0.676602i	0.941038	+	0.338301i	$0.109852\pi$
$997$	21.3639i	0.676602i	−0.941038	+	0.338301i	$0.890148\pi$
$998$	59.5443i	1.88484i
$999$	−5.75838	−0.182187

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.f.1249.11		12
5.2	odd	4		1875.2.a.k.1.1		6
5.3	odd	4		1875.2.a.j.1.6		6
5.4	even	2	inner	1875.2.b.f.1249.2		12
15.2	even	4		5625.2.a.q.1.6		6
15.8	even	4		5625.2.a.p.1.1		6
25.2	odd	20		375.2.g.c.226.1		12
25.9	even	10		375.2.i.d.349.5		24
25.11	even	5		375.2.i.d.274.5		24
25.12	odd	20		375.2.g.c.151.1		12
25.13	odd	20		75.2.g.c.31.3	✓	12
25.14	even	10		375.2.i.d.274.2		24
25.16	even	5		375.2.i.d.349.2		24
25.23	odd	20		75.2.g.c.46.3	yes	12
75.23	even	20		225.2.h.d.46.1		12
75.38	even	20		225.2.h.d.181.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.c.31.3	✓	12	25.13	odd	20
75.2.g.c.46.3	yes	12	25.23	odd	20
225.2.h.d.46.1		12	75.23	even	20
225.2.h.d.181.1		12	75.38	even	20
375.2.g.c.151.1		12	25.12	odd	20
375.2.g.c.226.1		12	25.2	odd	20
375.2.i.d.274.2		24	25.14	even	10
375.2.i.d.274.5		24	25.11	even	5
375.2.i.d.349.2		24	25.16	even	5
375.2.i.d.349.5		24	25.9	even	10
1875.2.a.j.1.6		6	5.3	odd	4
1875.2.a.k.1.1		6	5.2	odd	4
1875.2.b.f.1249.2		12	5.4	even	2	inner
1875.2.b.f.1249.11		12	1.1	even	1	trivial
5625.2.a.p.1.1		6	15.8	even	4
5625.2.a.q.1.6		6	15.2	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44028i q^{2} +1.00000i q^{3} -3.95498 q^{4} -2.44028 q^{6} -3.44028i q^{7} -4.77071i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.44028i q^{2} +1.00000i q^{3} -3.95498 q^{4} -2.44028 q^{6} -3.44028i q^{7} -4.77071i q^{8} -1.00000 q^{9} -3.26656 q^{11} -3.95498i q^{12} +3.23204i q^{13} +8.39527 q^{14} +3.73192 q^{16} -5.05337i q^{17} -2.44028i q^{18} +3.08119 q^{19} +3.44028 q^{21} -7.97134i q^{22} -1.54765i q^{23} +4.77071 q^{24} -7.88709 q^{26} -1.00000i q^{27} +13.6063i q^{28} +3.12218 q^{29} +7.44212 q^{31} -0.434479i q^{32} -3.26656i q^{33} +12.3317 q^{34} +3.95498 q^{36} -5.75838i q^{37} +7.51899i q^{38} -3.23204 q^{39} +5.41962 q^{41} +8.39527i q^{42} -2.53106i q^{43} +12.9192 q^{44} +3.77670 q^{46} +7.07162i q^{47} +3.73192i q^{48} -4.83555 q^{49} +5.05337 q^{51} -12.7827i q^{52} +10.1515i q^{53} +2.44028 q^{54} -16.4126 q^{56} +3.08119i q^{57} +7.61901i q^{58} -1.73056 q^{59} +7.83611 q^{61} +18.1609i q^{62} +3.44028i q^{63} +8.52409 q^{64} +7.97134 q^{66} -1.84910i q^{67} +19.9860i q^{68} +1.54765 q^{69} -0.713969 q^{71} +4.77071i q^{72} +1.88027i q^{73} +14.0521 q^{74} -12.1861 q^{76} +11.2379i q^{77} -7.88709i q^{78} +13.3332 q^{79} +1.00000 q^{81} +13.2254i q^{82} -3.95747i q^{83} -13.6063 q^{84} +6.17649 q^{86} +3.12218i q^{87} +15.5838i q^{88} -8.53392 q^{89} +11.1191 q^{91} +6.12092i q^{92} +7.44212i q^{93} -17.2567 q^{94} +0.434479 q^{96} -10.6528i q^{97} -11.8001i q^{98} +3.26656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) \) 12 * q - 20 * q^4 - 12 * q^9	\( 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) \) 12 * q - 20 * q^4 - 12 * q^9 + 6 * q^11 + 44 * q^14 + 36 * q^16 - 22 * q^19 + 12 * q^21 - 6 * q^24 - 56 * q^26 + 6 * q^29 - 22 * q^31 - 30 * q^34 + 20 * q^36 - 12 * q^39 - 2 * q^41 - 18 * q^44 + 38 * q^46 + 28 * q^49 + 26 * q^51 - 70 * q^56 - 18 * q^59 + 22 * q^61 + 46 * q^64 + 32 * q^66 - 26 * q^69 - 16 * q^71 + 44 * q^74 - 52 * q^76 + 10 * q^79 + 12 * q^81 - 14 * q^84 - 74 * q^86 + 8 * q^89 + 68 * q^91 - 82 * q^94 + 32 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

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Representations

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Database

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