LMFDB - Embedded newform 1875.2.b.g.1249.2

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 $ x^16 + 25x^14 + 246x^12 + 1220x^10 + 3281x^8 + 4880x^6 + 3936x^4 + 1600*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$-0.770071i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.g.1249.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59716i q^{2} +1.00000i q^{3} -4.74525 q^{4} +2.59716 q^{6} -3.28414i q^{7} +7.12986i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.59716i q^{2} +1.00000i q^{3} -4.74525 q^{4} +2.59716 q^{6} -3.28414i q^{7} +7.12986i q^{8} -1.00000 q^{9} +4.30834 q^{11} -4.74525i q^{12} -3.46120i q^{13} -8.52943 q^{14} +9.02691 q^{16} +5.44757i q^{17} +2.59716i q^{18} +7.63427 q^{19} +3.28414 q^{21} -11.1895i q^{22} -5.04986i q^{23} -7.12986 q^{24} -8.98929 q^{26} -1.00000i q^{27} +15.5841i q^{28} +3.12329 q^{29} -2.06658 q^{31} -9.18462i q^{32} +4.30834i q^{33} +14.1482 q^{34} +4.74525 q^{36} +1.89142i q^{37} -19.8274i q^{38} +3.46120 q^{39} +3.89896 q^{41} -8.52943i q^{42} -3.20815i q^{43} -20.4442 q^{44} -13.1153 q^{46} -6.28577i q^{47} +9.02691i q^{48} -3.78555 q^{49} -5.44757 q^{51} +16.4243i q^{52} -2.51480i q^{53} -2.59716 q^{54} +23.4154 q^{56} +7.63427i q^{57} -8.11170i q^{58} -7.72948 q^{59} -2.95661 q^{61} +5.36725i q^{62} +3.28414i q^{63} -5.80013 q^{64} +11.1895 q^{66} -12.9952i q^{67} -25.8501i q^{68} +5.04986 q^{69} -4.72665 q^{71} -7.12986i q^{72} -1.64933i q^{73} +4.91232 q^{74} -36.2265 q^{76} -14.1492i q^{77} -8.98929i q^{78} -13.7349 q^{79} +1.00000 q^{81} -10.1262i q^{82} -5.01687i q^{83} -15.5841 q^{84} -8.33208 q^{86} +3.12329i q^{87} +30.7179i q^{88} +9.00209 q^{89} -11.3670 q^{91} +23.9628i q^{92} -2.06658i q^{93} -16.3252 q^{94} +9.18462 q^{96} -2.57278i q^{97} +9.83169i q^{98} -4.30834 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9	$ 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9} + 24 q^{11} - 32 q^{14} + 30 q^{16} - 32 q^{19} + 24 q^{21} - 6 q^{24} - 68 q^{26} - 4 q^{29} + 26 q^{31} + 74 q^{34} + 18 q^{36} - 28 q^{39} - 24 q^{41} - 94 q^{44} + 66 q^{46} - 60 q^{49} - 2 q^{51} - 2 q^{54} + 120 q^{56} - 28 q^{59} + 20 q^{61} - 82 q^{64} + 36 q^{66} + 8 q^{69} + 42 q^{71} + 18 q^{74} - 2 q^{76} - 20 q^{79} + 16 q^{81} + 42 q^{84} + 84 q^{86} + 18 q^{89} - 24 q^{91} - 28 q^{94} - 36 q^{96} - 24 q^{99}+O(q^{100}) $ 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9 + 24 * q^11 - 32 * q^14 + 30 * q^16 - 32 * q^19 + 24 * q^21 - 6 * q^24 - 68 * q^26 - 4 * q^29 + 26 * q^31 + 74 * q^34 + 18 * q^36 - 28 * q^39 - 24 * q^41 - 94 * q^44 + 66 * q^46 - 60 * q^49 - 2 * q^51 - 2 * q^54 + 120 * q^56 - 28 * q^59 + 20 * q^61 - 82 * q^64 + 36 * q^66 + 8 * q^69 + 42 * q^71 + 18 * q^74 - 2 * q^76 - 20 * q^79 + 16 * q^81 + 42 * q^84 + 84 * q^86 + 18 * q^89 - 24 * q^91 - 28 * q^94 - 36 * q^96 - 24 * 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 $

$2$	− 2.59716i	− 1.83647i	−0.396035	−	0.918235i	$-0.629614\pi$
$2$	− 2.59716i	− 1.83647i	0.396035	−	0.918235i	$-0.370386\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−4.74525	−2.37263
$5$	0	0
$5$	0	0
$6$	2.59716	1.06029
$7$	− 3.28414i	− 1.24129i	−0.784093	−	0.620643i	$-0.786871\pi$
$7$	− 3.28414i	− 1.24129i	0.784093	−	0.620643i	$-0.213129\pi$
$8$	7.12986i	2.52079i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.30834	1.29901	0.649507	−	0.760356i	$-0.274975\pi$
$11$	4.30834	1.29901	0.649507	+	0.760356i	$0.274975\pi$
$12$	− 4.74525i	− 1.36984i
$13$	− 3.46120i	− 0.959964i	−0.877278	−	0.479982i	$-0.840643\pi$
$13$	− 3.46120i	− 0.959964i	0.877278	−	0.479982i	$-0.159357\pi$
$14$	−8.52943	−2.27959
$15$	0	0
$16$	9.02691	2.25673
$17$	5.44757i	1.32123i	0.750725	+	0.660615i	$0.229705\pi$
$17$	5.44757i	1.32123i	−0.750725	+	0.660615i	$0.770295\pi$
$18$	2.59716i	0.612157i
$19$	7.63427	1.75142	0.875711	−	0.482835i	$-0.160393\pi$
$19$	7.63427	1.75142	0.875711	+	0.482835i	$0.160393\pi$
$20$	0	0
$21$	3.28414	0.716657
$22$	− 11.1895i	− 2.38560i
$23$	− 5.04986i	− 1.05297i	−0.850185	−	0.526484i	$-0.823510\pi$
$23$	− 5.04986i	− 1.05297i	0.850185	−	0.526484i	$-0.176490\pi$
$24$	−7.12986	−1.45538
$25$	0	0
$26$	−8.98929	−1.76295
$27$	− 1.00000i	− 0.192450i
$28$	15.5841i	2.94511i
$29$	3.12329	0.579981	0.289990	−	0.957030i	$-0.406348\pi$
$29$	3.12329	0.579981	0.289990	+	0.957030i	$0.406348\pi$
$30$	0	0
$31$	−2.06658	−0.371169	−0.185584	−	0.982628i	$-0.559418\pi$
$31$	−2.06658	−0.371169	−0.185584	+	0.982628i	$0.559418\pi$
$32$	− 9.18462i	− 1.62363i
$33$	4.30834i	0.749986i
$34$	14.1482	2.42640
$35$	0	0
$36$	4.74525	0.790875
$37$	1.89142i	0.310947i	0.987840	+	0.155474i	$0.0496904\pi$
$37$	1.89142i	0.310947i	−0.987840	+	0.155474i	$0.950310\pi$
$38$	− 19.8274i	− 3.21644i
$39$	3.46120	0.554235
$40$	0	0
$41$	3.89896	0.608915	0.304457	−	0.952526i	$-0.401525\pi$
$41$	3.89896	0.608915	0.304457	+	0.952526i	$0.401525\pi$
$42$	− 8.52943i	− 1.31612i
$43$	− 3.20815i	− 0.489238i	−0.969619	−	0.244619i	$-0.921337\pi$
$43$	− 3.20815i	− 0.489238i	0.969619	−	0.244619i	$-0.0786629\pi$
$44$	−20.4442	−3.08207
$45$	0	0
$46$	−13.1153	−1.93375
$47$	− 6.28577i	− 0.916874i	−0.888727	−	0.458437i	$-0.848409\pi$
$47$	− 6.28577i	− 0.916874i	0.888727	−	0.458437i	$-0.151591\pi$
$48$	9.02691i	1.30292i
$49$	−3.78555	−0.540793
$50$	0	0
$51$	−5.44757	−0.762813
$52$	16.4243i	2.27763i
$53$	− 2.51480i	− 0.345434i	−0.984971	−	0.172717i	$-0.944745\pi$
$53$	− 2.51480i	− 0.345434i	0.984971	−	0.172717i	$-0.0552546\pi$
$54$	−2.59716	−0.353429
$55$	0	0
$56$	23.4154	3.12902
$57$	7.63427i	1.01118i
$58$	− 8.11170i	− 1.06512i
$59$	−7.72948	−1.00629	−0.503146	−	0.864201i	$-0.667824\pi$
$59$	−7.72948	−1.00629	−0.503146	+	0.864201i	$0.667824\pi$
$60$	0	0
$61$	−2.95661	−0.378555	−0.189278	−	0.981924i	$-0.560615\pi$
$61$	−2.95661	−0.378555	−0.189278	+	0.981924i	$0.560615\pi$
$62$	5.36725i	0.681641i
$63$	3.28414i	0.413762i
$64$	−5.80013	−0.725016
$65$	0	0
$66$	11.1895	1.37733
$67$	− 12.9952i	− 1.58762i	−0.608165	−	0.793811i	$-0.708094\pi$
$67$	− 12.9952i	− 1.58762i	0.608165	−	0.793811i	$-0.291906\pi$
$68$	− 25.8501i	− 3.13479i
$69$	5.04986	0.607931
$70$	0	0
$71$	−4.72665	−0.560950	−0.280475	−	0.959861i	$-0.590492\pi$
$71$	−4.72665	−0.560950	−0.280475	+	0.959861i	$0.590492\pi$
$72$	− 7.12986i	− 0.840262i
$73$	− 1.64933i	− 0.193040i	−0.995331	−	0.0965198i	$-0.969229\pi$
$73$	− 1.64933i	− 0.193040i	0.995331	−	0.0965198i	$-0.0307711\pi$
$74$	4.91232	0.571046
$75$	0	0
$76$	−36.2265	−4.15547
$77$	− 14.1492i	− 1.61245i
$78$	− 8.98929i	− 1.01784i
$79$	−13.7349	−1.54530	−0.772648	−	0.634835i	$-0.781068\pi$
$79$	−13.7349	−1.54530	−0.772648	+	0.634835i	$0.781068\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.1262i	− 1.11825i
$83$	− 5.01687i	− 0.550673i	−0.961348	−	0.275336i	$-0.911211\pi$
$83$	− 5.01687i	− 0.550673i	0.961348	−	0.275336i	$-0.0887893\pi$
$84$	−15.5841	−1.70036
$85$	0	0
$86$	−8.33208	−0.898472
$87$	3.12329i	0.334852i
$88$	30.7179i	3.27454i
$89$	9.00209	0.954220	0.477110	−	0.878844i	$-0.341684\pi$
$89$	9.00209	0.954220	0.477110	+	0.878844i	$0.341684\pi$
$90$	0	0
$91$	−11.3670	−1.19159
$92$	23.9628i	2.49830i
$93$	− 2.06658i	− 0.214294i
$94$	−16.3252	−1.68381
$95$	0	0
$96$	9.18462	0.937401
$97$	− 2.57278i	− 0.261226i	−0.991433	−	0.130613i	$-0.958305\pi$
$97$	− 2.57278i	− 0.261226i	0.991433	−	0.130613i	$-0.0416945\pi$
$98$	9.83169i	0.993150i
$99$	−4.30834	−0.433005
$100$	0	0
$101$	7.87863	0.783953	0.391976	−	0.919975i	$-0.371791\pi$
$101$	7.87863	0.783953	0.391976	+	0.919975i	$0.371791\pi$
$102$	14.1482i	1.40088i
$103$	12.4046i	1.22226i	0.791530	+	0.611131i	$0.209285\pi$
$103$	12.4046i	1.22226i	−0.791530	+	0.611131i	$0.790715\pi$
$104$	24.6779	2.41986
$105$	0	0
$106$	−6.53134	−0.634380
$107$	− 8.52094i	− 0.823750i	−0.911240	−	0.411875i	$-0.864874\pi$
$107$	− 8.52094i	− 0.823750i	0.911240	−	0.411875i	$-0.135126\pi$
$108$	4.74525i	0.456612i
$109$	−17.5139	−1.67752	−0.838762	−	0.544499i	$-0.816720\pi$
$109$	−17.5139	−1.67752	−0.838762	+	0.544499i	$0.816720\pi$
$110$	0	0
$111$	−1.89142	−0.179526
$112$	− 29.6456i	− 2.80125i
$113$	4.17711i	0.392949i	0.980509	+	0.196475i	$0.0629493\pi$
$113$	4.17711i	0.392949i	−0.980509	+	0.196475i	$0.937051\pi$
$114$	19.8274	1.85701
$115$	0	0
$116$	−14.8208	−1.37608
$117$	3.46120i	0.319988i
$118$	20.0747i	1.84803i
$119$	17.8906	1.64003
$120$	0	0
$121$	7.56181	0.687438
$122$	7.67880i	0.695206i
$123$	3.89896i	0.351557i
$124$	9.80645	0.880645
$125$	0	0
$126$	8.52943	0.759862
$127$	− 10.2112i	− 0.906097i	−0.891486	−	0.453049i	$-0.850336\pi$
$127$	− 10.2112i	− 0.906097i	0.891486	−	0.453049i	$-0.149664\pi$
$128$	− 3.30537i	− 0.292156i
$129$	3.20815	0.282462
$130$	0	0
$131$	−4.91686	−0.429588	−0.214794	−	0.976659i	$-0.568908\pi$
$131$	−4.91686	−0.429588	−0.214794	+	0.976659i	$0.568908\pi$
$132$	− 20.4442i	− 1.77944i
$133$	− 25.0720i	− 2.17402i
$134$	−33.7508	−2.91562
$135$	0	0
$136$	−38.8404	−3.33054
$137$	12.3817i	1.05784i	0.848672	+	0.528919i	$0.177403\pi$
$137$	12.3817i	1.05784i	−0.848672	+	0.528919i	$0.822597\pi$
$138$	− 13.1153i	− 1.11645i
$139$	−13.2420	−1.12317	−0.561584	−	0.827420i	$-0.689808\pi$
$139$	−13.2420	−1.12317	−0.561584	+	0.827420i	$0.689808\pi$
$140$	0	0
$141$	6.28577	0.529357
$142$	12.2759i	1.03017i
$143$	− 14.9120i	− 1.24701i
$144$	−9.02691	−0.752242
$145$	0	0
$146$	−4.28358	−0.354512
$147$	− 3.78555i	− 0.312227i
$148$	− 8.97526i	− 0.737762i
$149$	−5.20686	−0.426563	−0.213281	−	0.976991i	$-0.568415\pi$
$149$	−5.20686	−0.426563	−0.213281	+	0.976991i	$0.568415\pi$
$150$	0	0
$151$	7.78162	0.633259	0.316630	−	0.948549i	$-0.397449\pi$
$151$	7.78162	0.633259	0.316630	+	0.948549i	$0.397449\pi$
$152$	54.4313i	4.41496i
$153$	− 5.44757i	− 0.440410i
$154$	−36.7477	−2.96122
$155$	0	0
$156$	−16.4243	−1.31499
$157$	16.4474i	1.31264i	0.754481	+	0.656322i	$0.227889\pi$
$157$	16.4474i	1.31264i	−0.754481	+	0.656322i	$0.772111\pi$
$158$	35.6717i	2.83789i
$159$	2.51480	0.199436
$160$	0	0
$161$	−16.5844	−1.30704
$162$	− 2.59716i	− 0.204052i
$163$	7.81342i	0.611994i	0.952032	+	0.305997i	$0.0989898\pi$
$163$	7.81342i	0.611994i	−0.952032	+	0.305997i	$0.901010\pi$
$164$	−18.5015	−1.44473
$165$	0	0
$166$	−13.0296	−1.01129
$167$	2.77030i	0.214372i	0.994239	+	0.107186i	$0.0341841\pi$
$167$	2.77030i	0.214372i	−0.994239	+	0.107186i	$0.965816\pi$
$168$	23.4154i	1.80654i
$169$	1.02011	0.0784700
$170$	0	0
$171$	−7.63427	−0.583807
$172$	15.2235i	1.16078i
$173$	− 16.9573i	− 1.28924i	−0.764503	−	0.644620i	$-0.777015\pi$
$173$	− 16.9573i	− 1.28924i	0.764503	−	0.644620i	$-0.222985\pi$
$174$	8.11170	0.614946
$175$	0	0
$176$	38.8910	2.93152
$177$	− 7.72948i	− 0.580983i
$178$	− 23.3799i	− 1.75240i
$179$	−1.56173	−0.116729	−0.0583645	−	0.998295i	$-0.518589\pi$
$179$	−1.56173	−0.116729	−0.0583645	+	0.998295i	$0.518589\pi$
$180$	0	0
$181$	1.55277	0.115416	0.0577081	−	0.998333i	$-0.481621\pi$
$181$	1.55277	0.115416	0.0577081	+	0.998333i	$0.481621\pi$
$182$	29.5221i	2.18832i
$183$	− 2.95661i	− 0.218559i
$184$	36.0048	2.65431
$185$	0	0
$186$	−5.36725	−0.393546
$187$	23.4700i	1.71630i
$188$	29.8276i	2.17540i
$189$	−3.28414	−0.238886
$190$	0	0
$191$	26.5640	1.92211	0.961054	−	0.276362i	$-0.0891290\pi$
$191$	26.5640	1.92211	0.961054	+	0.276362i	$0.0891290\pi$
$192$	− 5.80013i	− 0.418588i
$193$	− 13.7642i	− 0.990772i	−0.868673	−	0.495386i	$-0.835027\pi$
$193$	− 13.7642i	− 0.990772i	0.868673	−	0.495386i	$-0.164973\pi$
$194$	−6.68192	−0.479734
$195$	0	0
$196$	17.9634	1.28310
$197$	− 6.25851i	− 0.445901i	−0.974830	−	0.222950i	$-0.928431\pi$
$197$	− 6.25851i	− 0.445901i	0.974830	−	0.222950i	$-0.0715688\pi$
$198$	11.1895i	0.795201i
$199$	−3.35009	−0.237482	−0.118741	−	0.992925i	$-0.537886\pi$
$199$	−3.35009	−0.237482	−0.118741	+	0.992925i	$0.537886\pi$
$200$	0	0
$201$	12.9952	0.916614
$202$	− 20.4621i	− 1.43971i
$203$	− 10.2573i	− 0.719923i
$204$	25.8501	1.80987
$205$	0	0
$206$	32.2168	2.24465
$207$	5.04986i	0.350989i
$208$	− 31.2439i	− 2.16638i
$209$	32.8911	2.27512
$210$	0	0
$211$	24.0984	1.65900	0.829502	−	0.558504i	$-0.188624\pi$
$211$	24.0984	1.65900	0.829502	+	0.558504i	$0.188624\pi$
$212$	11.9333i	0.819586i
$213$	− 4.72665i	− 0.323865i
$214$	−22.1303	−1.51279
$215$	0	0
$216$	7.12986	0.485126
$217$	6.78693i	0.460727i
$218$	45.4863i	3.08072i
$219$	1.64933	0.111452
$220$	0	0
$221$	18.8551	1.26833
$222$	4.91232i	0.329693i
$223$	− 18.2697i	− 1.22343i	−0.791079	−	0.611714i	$-0.790480\pi$
$223$	− 18.2697i	− 1.22343i	0.791079	−	0.611714i	$-0.209520\pi$
$224$	−30.1635	−2.01539
$225$	0	0
$226$	10.8486	0.721640
$227$	17.9602i	1.19206i	0.802962	+	0.596031i	$0.203256\pi$
$227$	17.9602i	1.19206i	−0.802962	+	0.596031i	$0.796744\pi$
$228$	− 36.2265i	− 2.39916i
$229$	−6.00658	−0.396926	−0.198463	−	0.980108i	$-0.563595\pi$
$229$	−6.00658	−0.396926	−0.198463	+	0.980108i	$0.563595\pi$
$230$	0	0
$231$	14.1492	0.930948
$232$	22.2687i	1.46201i
$233$	− 20.8526i	− 1.36610i	−0.730373	−	0.683049i	$-0.760654\pi$
$233$	− 20.8526i	− 1.36610i	0.730373	−	0.683049i	$-0.239346\pi$
$234$	8.98929	0.587648
$235$	0	0
$236$	36.6783	2.38755
$237$	− 13.7349i	− 0.892177i
$238$	− 46.4647i	− 3.01186i
$239$	27.8722	1.80290	0.901452	−	0.432880i	$-0.142503\pi$
$239$	27.8722	1.80290	0.901452	+	0.432880i	$0.142503\pi$
$240$	0	0
$241$	−18.7159	−1.20560	−0.602798	−	0.797894i	$-0.705947\pi$
$241$	−18.7159	−1.20560	−0.602798	+	0.797894i	$0.705947\pi$
$242$	− 19.6393i	− 1.26246i
$243$	1.00000i	0.0641500i
$244$	14.0299	0.898170
$245$	0	0
$246$	10.1262	0.645624
$247$	− 26.4237i	− 1.68130i
$248$	− 14.7344i	− 0.935638i
$249$	5.01687	0.317931
$250$	0	0
$251$	11.0995	0.700594	0.350297	−	0.936639i	$-0.386081\pi$
$251$	11.0995	0.700594	0.350297	+	0.936639i	$0.386081\pi$
$252$	− 15.5841i	− 0.981703i
$253$	− 21.7565i	− 1.36782i
$254$	−26.5201	−1.66402
$255$	0	0
$256$	−20.1848	−1.26155
$257$	17.8844i	1.11560i	0.829975	+	0.557800i	$0.188354\pi$
$257$	17.8844i	1.11560i	−0.829975	+	0.557800i	$0.811646\pi$
$258$	− 8.33208i	− 0.518733i
$259$	6.21168	0.385975
$260$	0	0
$261$	−3.12329	−0.193327
$262$	12.7699i	0.788927i
$263$	− 18.1341i	− 1.11819i	−0.829102	−	0.559097i	$-0.811148\pi$
$263$	− 18.1341i	− 1.11819i	0.829102	−	0.559097i	$-0.188852\pi$
$264$	−30.7179	−1.89056
$265$	0	0
$266$	−65.1160	−3.99252
$267$	9.00209i	0.550919i
$268$	61.6657i	3.76683i
$269$	22.2963	1.35943	0.679714	−	0.733477i	$-0.262104\pi$
$269$	22.2963	1.35943	0.679714	+	0.733477i	$0.262104\pi$
$270$	0	0
$271$	16.8621	1.02430	0.512149	−	0.858896i	$-0.328849\pi$
$271$	16.8621	1.02430	0.512149	+	0.858896i	$0.328849\pi$
$272$	49.1747i	2.98166i
$273$	− 11.3670i	− 0.687965i
$274$	32.1572	1.94269
$275$	0	0
$276$	−23.9628	−1.44239
$277$	18.6929i	1.12315i	0.827426	+	0.561575i	$0.189804\pi$
$277$	18.6929i	1.12315i	−0.827426	+	0.561575i	$0.810196\pi$
$278$	34.3915i	2.06266i
$279$	2.06658	0.123723
$280$	0	0
$281$	25.9852	1.55015	0.775075	−	0.631870i	$-0.217712\pi$
$281$	25.9852	1.55015	0.775075	+	0.631870i	$0.217712\pi$
$282$	− 16.3252i	− 0.972150i
$283$	16.9219i	1.00591i	0.864314	+	0.502953i	$0.167753\pi$
$283$	16.9219i	1.00591i	−0.864314	+	0.502953i	$0.832247\pi$
$284$	22.4291	1.33092
$285$	0	0
$286$	−38.7289	−2.29009
$287$	− 12.8047i	− 0.755838i
$288$	9.18462i	0.541209i
$289$	−12.6760	−0.745650
$290$	0	0
$291$	2.57278	0.150819
$292$	7.82649i	0.458011i
$293$	1.05979i	0.0619134i	0.999521	+	0.0309567i	$0.00985540\pi$
$293$	1.05979i	0.0619134i	−0.999521	+	0.0309567i	$0.990145\pi$
$294$	−9.83169	−0.573396
$295$	0	0
$296$	−13.4856	−0.783832
$297$	− 4.30834i	− 0.249995i
$298$	13.5231i	0.783370i
$299$	−17.4786	−1.01081
$300$	0	0
$301$	−10.5360	−0.607285
$302$	− 20.2101i	− 1.16296i
$303$	7.87863i	0.452615i
$304$	68.9139	3.95248
$305$	0	0
$306$	−14.1482	−0.808800
$307$	− 29.7239i	− 1.69643i	−0.529652	−	0.848215i	$-0.677677\pi$
$307$	− 29.7239i	− 1.69643i	0.529652	−	0.848215i	$-0.322323\pi$
$308$	67.1414i	3.82574i
$309$	−12.4046	−0.705673
$310$	0	0
$311$	5.08513	0.288351	0.144176	−	0.989552i	$-0.453947\pi$
$311$	5.08513	0.288351	0.144176	+	0.989552i	$0.453947\pi$
$312$	24.6779i	1.39711i
$313$	25.9388i	1.46615i	0.680149	+	0.733074i	$0.261915\pi$
$313$	25.9388i	1.46615i	−0.680149	+	0.733074i	$0.738085\pi$
$314$	42.7165	2.41063
$315$	0	0
$316$	65.1755	3.66641
$317$	− 19.7727i	− 1.11054i	−0.831669	−	0.555271i	$-0.812614\pi$
$317$	− 19.7727i	− 1.11054i	0.831669	−	0.555271i	$-0.187386\pi$
$318$	− 6.53134i	− 0.366259i
$319$	13.4562	0.753403
$320$	0	0
$321$	8.52094	0.475592
$322$	43.0724i	2.40033i
$323$	41.5883i	2.31403i
$324$	−4.74525	−0.263625
$325$	0	0
$326$	20.2927	1.12391
$327$	− 17.5139i	− 0.968519i
$328$	27.7990i	1.53494i
$329$	−20.6433	−1.13810
$330$	0	0
$331$	−23.9589	−1.31690	−0.658449	−	0.752625i	$-0.728787\pi$
$331$	−23.9589	−1.31690	−0.658449	+	0.752625i	$0.728787\pi$
$332$	23.8063i	1.30654i
$333$	− 1.89142i	− 0.103649i
$334$	7.19492	0.393689
$335$	0	0
$336$	29.6456	1.61730
$337$	27.6779i	1.50771i	0.657039	+	0.753857i	$0.271809\pi$
$337$	27.6779i	1.50771i	−0.657039	+	0.753857i	$0.728191\pi$
$338$	− 2.64939i	− 0.144108i
$339$	−4.17711	−0.226869
$340$	0	0
$341$	−8.90354	−0.482154
$342$	19.8274i	1.07215i
$343$	− 10.5567i	− 0.570008i
$344$	22.8737	1.23327
$345$	0	0
$346$	−44.0409	−2.36765
$347$	13.4880i	0.724075i	0.932164	+	0.362037i	$0.117919\pi$
$347$	13.4880i	0.724075i	−0.932164	+	0.362037i	$0.882081\pi$
$348$	− 14.8208i	− 0.794479i
$349$	−19.1678	−1.02603	−0.513015	−	0.858379i	$-0.671472\pi$
$349$	−19.1678	−1.02603	−0.513015	+	0.858379i	$0.671472\pi$
$350$	0	0
$351$	−3.46120	−0.184745
$352$	− 39.5705i	− 2.10911i
$353$	7.61950i	0.405545i	0.979226	+	0.202773i	$0.0649952\pi$
$353$	7.61950i	0.405545i	−0.979226	+	0.202773i	$0.935005\pi$
$354$	−20.0747	−1.06696
$355$	0	0
$356$	−42.7172	−2.26401
$357$	17.8906i	0.946869i
$358$	4.05606i	0.214369i
$359$	−31.2943	−1.65165	−0.825826	−	0.563925i	$-0.809291\pi$
$359$	−31.2943	−1.65165	−0.825826	+	0.563925i	$0.809291\pi$
$360$	0	0
$361$	39.2821	2.06748
$362$	− 4.03279i	− 0.211959i
$363$	7.56181i	0.396892i
$364$	53.9395	2.82720
$365$	0	0
$366$	−7.67880	−0.401377
$367$	6.35079i	0.331508i	0.986167	+	0.165754i	$0.0530058\pi$
$367$	6.35079i	0.331508i	−0.986167	+	0.165754i	$0.946994\pi$
$368$	− 45.5846i	− 2.37626i
$369$	−3.89896	−0.202972
$370$	0	0
$371$	−8.25894	−0.428783
$372$	9.80645i	0.508441i
$373$	11.1710i	0.578410i	0.957267	+	0.289205i	$0.0933909\pi$
$373$	11.1710i	0.578410i	−0.957267	+	0.289205i	$0.906609\pi$
$374$	60.9554	3.15193
$375$	0	0
$376$	44.8167	2.31124
$377$	− 10.8103i	− 0.556761i
$378$	8.52943i	0.438707i
$379$	20.7113	1.06387	0.531933	−	0.846787i	$-0.321466\pi$
$379$	20.7113	1.06387	0.531933	+	0.846787i	$0.321466\pi$
$380$	0	0
$381$	10.2112	0.523135
$382$	− 68.9911i	− 3.52989i
$383$	− 12.5869i	− 0.643158i	−0.946883	−	0.321579i	$-0.895786\pi$
$383$	− 12.5869i	− 0.643158i	0.946883	−	0.321579i	$-0.104214\pi$
$384$	3.30537	0.168676
$385$	0	0
$386$	−35.7480	−1.81952
$387$	3.20815i	0.163079i
$388$	12.2085i	0.619792i
$389$	−10.8301	−0.549110	−0.274555	−	0.961571i	$-0.588531\pi$
$389$	−10.8301	−0.549110	−0.274555	+	0.961571i	$0.588531\pi$
$390$	0	0
$391$	27.5095	1.39121
$392$	− 26.9905i	− 1.36322i
$393$	− 4.91686i	− 0.248023i
$394$	−16.2544	−0.818884
$395$	0	0
$396$	20.4442	1.02736
$397$	4.90704i	0.246277i	0.992389	+	0.123139i	$0.0392960\pi$
$397$	4.90704i	0.246277i	−0.992389	+	0.123139i	$0.960704\pi$
$398$	8.70072i	0.436128i
$399$	25.0720	1.25517
$400$	0	0
$401$	−27.5223	−1.37440	−0.687198	−	0.726470i	$-0.741160\pi$
$401$	−27.5223	−1.37440	−0.687198	+	0.726470i	$0.741160\pi$
$402$	− 33.7508i	− 1.68333i
$403$	7.15285i	0.356309i
$404$	−37.3861	−1.86003
$405$	0	0
$406$	−26.6399	−1.32212
$407$	8.14888i	0.403925i
$408$	− 38.8404i	− 1.92289i
$409$	−19.7665	−0.977390	−0.488695	−	0.872455i	$-0.662527\pi$
$409$	−19.7665	−0.977390	−0.488695	+	0.872455i	$0.662527\pi$
$410$	0	0
$411$	−12.3817	−0.610743
$412$	− 58.8630i	− 2.89997i
$413$	25.3847i	1.24910i
$414$	13.1153	0.644582
$415$	0	0
$416$	−31.7898	−1.55862
$417$	− 13.2420i	− 0.648461i
$418$	− 85.4234i	− 4.17820i
$419$	6.67948	0.326314	0.163157	−	0.986600i	$-0.447832\pi$
$419$	6.67948	0.326314	0.163157	+	0.986600i	$0.447832\pi$
$420$	0	0
$421$	5.65996	0.275849	0.137925	−	0.990443i	$-0.455957\pi$
$421$	5.65996	0.275849	0.137925	+	0.990443i	$0.455957\pi$
$422$	− 62.5875i	− 3.04671i
$423$	6.28577i	0.305625i
$424$	17.9302	0.870766
$425$	0	0
$426$	−12.2759	−0.594768
$427$	9.70991i	0.469896i
$428$	40.4340i	1.95445i
$429$	14.9120	0.719959
$430$	0	0
$431$	6.41717	0.309104	0.154552	−	0.987985i	$-0.450607\pi$
$431$	6.41717	0.309104	0.154552	+	0.987985i	$0.450607\pi$
$432$	− 9.02691i	− 0.434307i
$433$	− 1.10506i	− 0.0531059i	−0.999647	−	0.0265530i	$-0.991547\pi$
$433$	− 1.10506i	− 0.0531059i	0.999647	−	0.0265530i	$-0.00845307\pi$
$434$	17.6268	0.846112
$435$	0	0
$436$	83.1077	3.98014
$437$	− 38.5520i	− 1.84419i
$438$	− 4.28358i	− 0.204677i
$439$	5.11313	0.244036	0.122018	−	0.992528i	$-0.461063\pi$
$439$	5.11313	0.244036	0.122018	+	0.992528i	$0.461063\pi$
$440$	0	0
$441$	3.78555	0.180264
$442$	− 48.9698i	− 2.32926i
$443$	− 17.9079i	− 0.850829i	−0.904999	−	0.425414i	$-0.860128\pi$
$443$	− 17.9079i	− 0.850829i	0.904999	−	0.425414i	$-0.139872\pi$
$444$	8.97526	0.425947
$445$	0	0
$446$	−47.4493	−2.24679
$447$	− 5.20686i	− 0.246276i
$448$	19.0484i	0.899953i
$449$	−27.3710	−1.29172	−0.645859	−	0.763457i	$-0.723501\pi$
$449$	−27.3710	−1.29172	−0.645859	+	0.763457i	$0.723501\pi$
$450$	0	0
$451$	16.7980	0.790989
$452$	− 19.8214i	− 0.932322i
$453$	7.78162i	0.365612i
$454$	46.6456	2.18919
$455$	0	0
$456$	−54.4313	−2.54898
$457$	− 8.92806i	− 0.417637i	−0.977954	−	0.208819i	$-0.933038\pi$
$457$	− 8.92806i	− 0.417637i	0.977954	−	0.208819i	$-0.0669618\pi$
$458$	15.6001i	0.728943i
$459$	5.44757	0.254271
$460$	0	0
$461$	1.81204	0.0843952	0.0421976	−	0.999109i	$-0.486564\pi$
$461$	1.81204	0.0843952	0.0421976	+	0.999109i	$0.486564\pi$
$462$	− 36.7477i	− 1.70966i
$463$	2.72192i	0.126498i	0.997998	+	0.0632492i	$0.0201463\pi$
$463$	2.72192i	0.126498i	−0.997998	+	0.0632492i	$0.979854\pi$
$464$	28.1937	1.30886
$465$	0	0
$466$	−54.1575	−2.50880
$467$	40.2261i	1.86144i	0.365729	+	0.930721i	$0.380820\pi$
$467$	40.2261i	1.86144i	−0.365729	+	0.930721i	$0.619180\pi$
$468$	− 16.4243i	− 0.759211i
$469$	−42.6781	−1.97069
$470$	0	0
$471$	−16.4474	−0.757855
$472$	− 55.1101i	− 2.53665i
$473$	− 13.8218i	− 0.635527i
$474$	−35.6717	−1.63846
$475$	0	0
$476$	−84.8953	−3.89117
$477$	2.51480i	0.115145i
$478$	− 72.3886i	− 3.31098i
$479$	27.9457	1.27687	0.638435	−	0.769676i	$-0.279582\pi$
$479$	27.9457	1.27687	0.638435	+	0.769676i	$0.279582\pi$
$480$	0	0
$481$	6.54657	0.298498
$482$	48.6082i	2.21404i
$483$	− 16.5844i	− 0.754617i
$484$	−35.8827	−1.63103
$485$	0	0
$486$	2.59716	0.117810
$487$	25.8089i	1.16951i	0.811209	+	0.584756i	$0.198810\pi$
$487$	25.8089i	1.16951i	−0.811209	+	0.584756i	$0.801190\pi$
$488$	− 21.0802i	− 0.954257i
$489$	−7.81342	−0.353335
$490$	0	0
$491$	23.7044	1.06976	0.534882	−	0.844927i	$-0.320356\pi$
$491$	23.7044	1.06976	0.534882	+	0.844927i	$0.320356\pi$
$492$	− 18.5015i	− 0.834113i
$493$	17.0144i	0.766288i
$494$	−68.6267	−3.08766
$495$	0	0
$496$	−18.6548	−0.837627
$497$	15.5230i	0.696300i
$498$	− 13.0296i	− 0.583871i
$499$	−8.95752	−0.400994	−0.200497	−	0.979694i	$-0.564256\pi$
$499$	−8.95752	−0.400994	−0.200497	+	0.979694i	$0.564256\pi$
$500$	0	0
$501$	−2.77030	−0.123768
$502$	− 28.8272i	− 1.28662i
$503$	10.1655i	0.453256i	0.973981	+	0.226628i	$0.0727701\pi$
$503$	10.1655i	0.453256i	−0.973981	+	0.226628i	$0.927230\pi$
$504$	−23.4154	−1.04301
$505$	0	0
$506$	−56.5052	−2.51196
$507$	1.02011i	0.0453047i
$508$	48.4547i	2.14983i
$509$	6.54473	0.290090	0.145045	−	0.989425i	$-0.453667\pi$
$509$	6.54473	0.290090	0.145045	+	0.989425i	$0.453667\pi$
$510$	0	0
$511$	−5.41663	−0.239618
$512$	45.8125i	2.02465i
$513$	− 7.63427i	− 0.337061i
$514$	46.4488	2.04877
$515$	0	0
$516$	−15.2235	−0.670176
$517$	− 27.0813i	− 1.19103i
$518$	− 16.1327i	− 0.708832i
$519$	16.9573	0.744343
$520$	0	0
$521$	−25.6022	−1.12165	−0.560827	−	0.827933i	$-0.689517\pi$
$521$	−25.6022	−1.12165	−0.560827	+	0.827933i	$0.689517\pi$
$522$	8.11170i	0.355039i
$523$	5.02210i	0.219601i	0.993954	+	0.109801i	$0.0350212\pi$
$523$	5.02210i	0.219601i	−0.993954	+	0.109801i	$0.964979\pi$
$524$	23.3318	1.01925
$525$	0	0
$526$	−47.0971	−2.05353
$527$	− 11.2579i	− 0.490400i
$528$	38.8910i	1.69251i
$529$	−2.50105	−0.108741
$530$	0	0
$531$	7.72948	0.335431
$532$	118.973i	5.15813i
$533$	− 13.4951i	− 0.584536i
$534$	23.3799	1.01175
$535$	0	0
$536$	92.6543	4.00206
$537$	− 1.56173i	− 0.0673935i
$538$	− 57.9071i	− 2.49655i
$539$	−16.3094	−0.702498
$540$	0	0
$541$	−7.99099	−0.343560	−0.171780	−	0.985135i	$-0.554952\pi$
$541$	−7.99099	−0.343560	−0.171780	+	0.985135i	$0.554952\pi$
$542$	− 43.7936i	− 1.88109i
$543$	1.55277i	0.0666356i
$544$	50.0339	2.14518
$545$	0	0
$546$	−29.5221	−1.26343
$547$	21.9399i	0.938084i	0.883176	+	0.469042i	$0.155401\pi$
$547$	21.9399i	0.938084i	−0.883176	+	0.469042i	$0.844599\pi$
$548$	− 58.7542i	− 2.50986i
$549$	2.95661	0.126185
$550$	0	0
$551$	23.8441	1.01579
$552$	36.0048i	1.53247i
$553$	45.1072i	1.91815i
$554$	48.5486	2.06263
$555$	0	0
$556$	62.8364	2.66486
$557$	− 32.6366i	− 1.38286i	−0.722444	−	0.691429i	$-0.756981\pi$
$557$	− 32.6366i	− 1.38286i	0.722444	−	0.691429i	$-0.243019\pi$
$558$	− 5.36725i	− 0.227214i
$559$	−11.1040	−0.469651
$560$	0	0
$561$	−23.4700	−0.990905
$562$	− 67.4879i	− 2.84680i
$563$	− 5.25771i	− 0.221586i	−0.993843	−	0.110793i	$-0.964661\pi$
$563$	− 5.25771i	− 0.221586i	0.993843	−	0.110793i	$-0.0353391\pi$
$564$	−29.8276	−1.25597
$565$	0	0
$566$	43.9490	1.84732
$567$	− 3.28414i	− 0.137921i
$568$	− 33.7004i	− 1.41404i
$569$	−15.1931	−0.636926	−0.318463	−	0.947935i	$-0.603167\pi$
$569$	−15.1931	−0.636926	−0.318463	+	0.947935i	$0.603167\pi$
$570$	0	0
$571$	−8.29434	−0.347107	−0.173554	−	0.984824i	$-0.555525\pi$
$571$	−8.29434	−0.347107	−0.173554	+	0.984824i	$0.555525\pi$
$572$	70.7613i	2.95868i
$573$	26.5640i	1.10973i
$574$	−33.2559	−1.38807
$575$	0	0
$576$	5.80013	0.241672
$577$	− 2.69155i	− 0.112051i	−0.998429	−	0.0560253i	$-0.982157\pi$
$577$	− 2.69155i	− 0.112051i	0.998429	−	0.0560253i	$-0.0178427\pi$
$578$	32.9218i	1.36936i
$579$	13.7642	0.572022
$580$	0	0
$581$	−16.4761	−0.683543
$582$	− 6.68192i	− 0.276975i
$583$	− 10.8346i	− 0.448724i
$584$	11.7595	0.486612
$585$	0	0
$586$	2.75244	0.113702
$587$	− 27.2841i	− 1.12613i	−0.826411	−	0.563067i	$-0.809621\pi$
$587$	− 27.2841i	− 1.12613i	0.826411	−	0.563067i	$-0.190379\pi$
$588$	17.9634i	0.740798i
$589$	−15.7768	−0.650074
$590$	0	0
$591$	6.25851	0.257441
$592$	17.0737i	0.701723i
$593$	− 3.04738i	− 0.125141i	−0.998041	−	0.0625704i	$-0.980070\pi$
$593$	− 3.04738i	− 0.125141i	0.998041	−	0.0625704i	$-0.0199298\pi$
$594$	−11.1895	−0.459109
$595$	0	0
$596$	24.7079	1.01207
$597$	− 3.35009i	− 0.137110i
$598$	45.3946i	1.85632i
$599$	21.1104	0.862549	0.431274	−	0.902221i	$-0.358064\pi$
$599$	21.1104	0.862549	0.431274	+	0.902221i	$0.358064\pi$
$600$	0	0
$601$	1.78443	0.0727884	0.0363942	−	0.999338i	$-0.488413\pi$
$601$	1.78443	0.0727884	0.0363942	+	0.999338i	$0.488413\pi$
$602$	27.3637i	1.11526i
$603$	12.9952i	0.529207i
$604$	−36.9258	−1.50249
$605$	0	0
$606$	20.4621	0.831215
$607$	− 43.9927i	− 1.78561i	−0.450444	−	0.892805i	$-0.648734\pi$
$607$	− 43.9927i	− 1.78561i	0.450444	−	0.892805i	$-0.351266\pi$
$608$	− 70.1179i	− 2.84366i
$609$	10.2573	0.415648
$610$	0	0
$611$	−21.7563	−0.880166
$612$	25.8501i	1.04493i
$613$	− 37.9419i	− 1.53246i	−0.642567	−	0.766230i	$-0.722130\pi$
$613$	− 37.9419i	− 1.53246i	0.642567	−	0.766230i	$-0.277870\pi$
$614$	−77.1977	−3.11544
$615$	0	0
$616$	100.882	4.06464
$617$	7.92668i	0.319116i	0.987189	+	0.159558i	$0.0510069\pi$
$617$	7.92668i	0.319116i	−0.987189	+	0.159558i	$0.948993\pi$
$618$	32.2168i	1.29595i
$619$	7.83411	0.314880	0.157440	−	0.987529i	$-0.449676\pi$
$619$	7.83411	0.314880	0.157440	+	0.987529i	$0.449676\pi$
$620$	0	0
$621$	−5.04986	−0.202644
$622$	− 13.2069i	− 0.529548i
$623$	− 29.5641i	− 1.18446i
$624$	31.2439	1.25076
$625$	0	0
$626$	67.3673	2.69254
$627$	32.8911i	1.31354i
$628$	− 78.0469i	− 3.11441i
$629$	−10.3036	−0.410833
$630$	0	0
$631$	19.0248	0.757366	0.378683	−	0.925526i	$-0.376377\pi$
$631$	19.0248	0.757366	0.378683	+	0.925526i	$0.376377\pi$
$632$	− 97.9279i	− 3.89536i
$633$	24.0984i	0.957826i
$634$	−51.3528	−2.03948
$635$	0	0
$636$	−11.9333	−0.473188
$637$	13.1025i	0.519141i
$638$	− 34.9480i	− 1.38360i
$639$	4.72665	0.186983
$640$	0	0
$641$	−32.5783	−1.28677	−0.643383	−	0.765545i	$-0.722470\pi$
$641$	−32.5783	−1.28677	−0.643383	+	0.765545i	$0.722470\pi$
$642$	− 22.1303i	− 0.873411i
$643$	40.0025i	1.57755i	0.614684	+	0.788773i	$0.289283\pi$
$643$	40.0025i	1.57755i	−0.614684	+	0.788773i	$0.710717\pi$
$644$	78.6972	3.10111
$645$	0	0
$646$	108.011	4.24965
$647$	20.0199i	0.787063i	0.919311	+	0.393532i	$0.128747\pi$
$647$	20.0199i	0.787063i	−0.919311	+	0.393532i	$0.871253\pi$
$648$	7.12986i	0.280087i
$649$	−33.3012	−1.30719
$650$	0	0
$651$	−6.78693	−0.266001
$652$	− 37.0766i	− 1.45203i
$653$	− 10.3332i	− 0.404369i	−0.979347	−	0.202185i	$-0.935196\pi$
$653$	− 10.3332i	− 0.404369i	0.979347	−	0.202185i	$-0.0648041\pi$
$654$	−45.4863	−1.77866
$655$	0	0
$656$	35.1955	1.37415
$657$	1.64933i	0.0643466i
$658$	53.6141i	2.09009i
$659$	47.2403	1.84022	0.920111	−	0.391657i	$-0.128098\pi$
$659$	47.2403	1.84022	0.920111	+	0.391657i	$0.128098\pi$
$660$	0	0
$661$	51.3528	1.99739	0.998696	−	0.0510604i	$-0.0162601\pi$
$661$	51.3528	1.99739	0.998696	+	0.0510604i	$0.0162601\pi$
$662$	62.2250i	2.41844i
$663$	18.8551i	0.732272i
$664$	35.7696	1.38813
$665$	0	0
$666$	−4.91232	−0.190349
$667$	− 15.7722i	− 0.610701i
$668$	− 13.1458i	− 0.508625i
$669$	18.2697	0.706346
$670$	0	0
$671$	−12.7381	−0.491749
$672$	− 30.1635i	− 1.16358i
$673$	40.5790i	1.56420i	0.623150	+	0.782102i	$0.285853\pi$
$673$	40.5790i	1.56420i	−0.623150	+	0.782102i	$0.714147\pi$
$674$	71.8841	2.76887
$675$	0	0
$676$	−4.84068	−0.186180
$677$	− 9.21482i	− 0.354154i	−0.984197	−	0.177077i	$-0.943336\pi$
$677$	− 9.21482i	− 0.354154i	0.984197	−	0.177077i	$-0.0566642\pi$
$678$	10.8486i	0.416639i
$679$	−8.44935	−0.324256
$680$	0	0
$681$	−17.9602	−0.688237
$682$	23.1239i	0.885461i
$683$	13.3281i	0.509987i	0.966943	+	0.254994i	$0.0820734\pi$
$683$	13.3281i	0.509987i	−0.966943	+	0.254994i	$0.917927\pi$
$684$	36.2265	1.38516
$685$	0	0
$686$	−27.4174	−1.04680
$687$	− 6.00658i	− 0.229165i
$688$	− 28.9597i	− 1.10408i
$689$	−8.70421	−0.331604
$690$	0	0
$691$	−32.5862	−1.23964	−0.619819	−	0.784745i	$-0.712794\pi$
$691$	−32.5862	−1.23964	−0.619819	+	0.784745i	$0.712794\pi$
$692$	80.4667i	3.05889i
$693$	14.1492i	0.537483i
$694$	35.0306	1.32974
$695$	0	0
$696$	−22.2687	−0.844091
$697$	21.2398i	0.804516i
$698$	49.7820i	1.88428i
$699$	20.8526	0.788717
$700$	0	0
$701$	−33.5547	−1.26734	−0.633672	−	0.773602i	$-0.718453\pi$
$701$	−33.5547	−1.26734	−0.633672	+	0.773602i	$0.718453\pi$
$702$	8.98929i	0.339279i
$703$	14.4396i	0.544600i
$704$	−24.9889	−0.941806
$705$	0	0
$706$	19.7891	0.744772
$707$	− 25.8745i	− 0.973110i
$708$	36.6783i	1.37846i
$709$	−12.4731	−0.468436	−0.234218	−	0.972184i	$-0.575253\pi$
$709$	−12.4731	−0.468436	−0.234218	+	0.972184i	$0.575253\pi$
$710$	0	0
$711$	13.7349	0.515098
$712$	64.1837i	2.40539i
$713$	10.4359i	0.390829i
$714$	46.4647	1.73890
$715$	0	0
$716$	7.41079	0.276954
$717$	27.8722i	1.04091i
$718$	81.2765i	3.03321i
$719$	7.53586	0.281040	0.140520	−	0.990078i	$-0.455123\pi$
$719$	7.53586	0.281040	0.140520	+	0.990078i	$0.455123\pi$
$720$	0	0
$721$	40.7384	1.51718
$722$	− 102.022i	− 3.79687i
$723$	− 18.7159i	− 0.696051i
$724$	−7.36827	−0.273839
$725$	0	0
$726$	19.6393	0.728881
$727$	42.8265i	1.58835i	0.607691	+	0.794173i	$0.292096\pi$
$727$	42.8265i	1.58835i	−0.607691	+	0.794173i	$0.707904\pi$
$728$	− 81.0455i	− 3.00375i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	17.4766	0.646396
$732$	14.0299i	0.518559i
$733$	48.4170i	1.78832i	0.447745	+	0.894161i	$0.352227\pi$
$733$	48.4170i	1.78832i	−0.447745	+	0.894161i	$0.647773\pi$
$734$	16.4940	0.608806
$735$	0	0
$736$	−46.3810	−1.70963
$737$	− 55.9880i	− 2.06234i
$738$	10.1262i	0.372751i
$739$	−31.1926	−1.14744	−0.573720	−	0.819052i	$-0.694500\pi$
$739$	−31.1926	−1.14744	−0.573720	+	0.819052i	$0.694500\pi$
$740$	0	0
$741$	26.4237	0.970700
$742$	21.4498i	0.787447i
$743$	45.9840i	1.68699i	0.537137	+	0.843495i	$0.319506\pi$
$743$	45.9840i	1.68699i	−0.537137	+	0.843495i	$0.680494\pi$
$744$	14.7344	0.540191
$745$	0	0
$746$	29.0128	1.06223
$747$	5.01687i	0.183558i
$748$	− 111.371i	− 4.07213i
$749$	−27.9839	−1.02251
$750$	0	0
$751$	2.51488	0.0917695	0.0458847	−	0.998947i	$-0.485389\pi$
$751$	2.51488	0.0917695	0.0458847	+	0.998947i	$0.485389\pi$
$752$	− 56.7411i	− 2.06913i
$753$	11.0995i	0.404488i
$754$	−28.0762	−1.02247
$755$	0	0
$756$	15.5841	0.566786
$757$	39.6597i	1.44146i	0.693218	+	0.720728i	$0.256192\pi$
$757$	39.6597i	1.44146i	−0.693218	+	0.720728i	$0.743808\pi$
$758$	− 53.7905i	− 1.95376i
$759$	21.7565	0.789711
$760$	0	0
$761$	17.2752	0.626225	0.313113	−	0.949716i	$-0.398628\pi$
$761$	17.2752	0.626225	0.313113	+	0.949716i	$0.398628\pi$
$762$	− 26.5201i	− 0.960723i
$763$	57.5179i	2.08229i
$764$	−126.053	−4.56044
$765$	0	0
$766$	−32.6901	−1.18114
$767$	26.7533i	0.966004i
$768$	− 20.1848i	− 0.728357i
$769$	4.50943	0.162614	0.0813071	−	0.996689i	$-0.474091\pi$
$769$	4.50943	0.162614	0.0813071	+	0.996689i	$0.474091\pi$
$770$	0	0
$771$	−17.8844	−0.644092
$772$	65.3148i	2.35073i
$773$	− 33.6503i	− 1.21032i	−0.796105	−	0.605159i	$-0.793110\pi$
$773$	− 33.6503i	− 1.21032i	0.796105	−	0.605159i	$-0.206890\pi$
$774$	8.33208	0.299491
$775$	0	0
$776$	18.3436	0.658495
$777$	6.21168i	0.222843i
$778$	28.1276i	1.00842i
$779$	29.7657	1.06647
$780$	0	0
$781$	−20.3640	−0.728682
$782$	− 71.4465i	− 2.55492i
$783$	− 3.12329i	− 0.111617i
$784$	−34.1718	−1.22042
$785$	0	0
$786$	−12.7699	−0.455487
$787$	− 36.8366i	− 1.31308i	−0.754289	−	0.656542i	$-0.772018\pi$
$787$	− 36.8366i	− 1.31308i	0.754289	−	0.656542i	$-0.227982\pi$
$788$	29.6982i	1.05796i
$789$	18.1341	0.645589
$790$	0	0
$791$	13.7182	0.487763
$792$	− 30.7179i	− 1.09151i
$793$	10.2334i	0.363399i
$794$	12.7444	0.452281
$795$	0	0
$796$	15.8970	0.563455
$797$	40.4962i	1.43445i	0.696843	+	0.717224i	$0.254588\pi$
$797$	40.4962i	1.43445i	−0.696843	+	0.717224i	$0.745412\pi$
$798$	− 65.1160i	− 2.30508i
$799$	34.2422	1.21140
$800$	0	0
$801$	−9.00209	−0.318073
$802$	71.4798i	2.52404i
$803$	− 7.10589i	− 0.250761i
$804$	−61.6657	−2.17478
$805$	0	0
$806$	18.5771	0.654350
$807$	22.2963i	0.784866i
$808$	56.1735i	1.97618i
$809$	11.9110	0.418767	0.209384	−	0.977834i	$-0.432854\pi$
$809$	11.9110	0.418767	0.209384	+	0.977834i	$0.432854\pi$
$810$	0	0
$811$	−42.2433	−1.48336	−0.741681	−	0.670752i	$-0.765971\pi$
$811$	−42.2433	−1.48336	−0.741681	+	0.670752i	$0.765971\pi$
$812$	48.6736i	1.70811i
$813$	16.8621i	0.591379i
$814$	21.1640	0.741796
$815$	0	0
$816$	−49.1747	−1.72146
$817$	− 24.4919i	− 0.856863i
$818$	51.3368i	1.79495i
$819$	11.3670	0.397197
$820$	0	0
$821$	44.5710	1.55554	0.777769	−	0.628551i	$-0.216352\pi$
$821$	44.5710	1.55554	0.777769	+	0.628551i	$0.216352\pi$
$822$	32.1572i	1.12161i
$823$	12.3125i	0.429186i	0.976704	+	0.214593i	$0.0688425\pi$
$823$	12.3125i	0.429186i	−0.976704	+	0.214593i	$0.931157\pi$
$824$	−88.4431	−3.08106
$825$	0	0
$826$	65.9281	2.29393
$827$	24.7421i	0.860367i	0.902742	+	0.430183i	$0.141551\pi$
$827$	24.7421i	0.860367i	−0.902742	+	0.430183i	$0.858449\pi$
$828$	− 23.9628i	− 0.832766i
$829$	53.7221	1.86584	0.932922	−	0.360077i	$-0.117250\pi$
$829$	53.7221	1.86584	0.932922	+	0.360077i	$0.117250\pi$
$830$	0	0
$831$	−18.6929	−0.648451
$832$	20.0754i	0.695989i
$833$	− 20.6221i	− 0.714512i
$834$	−34.3915	−1.19088
$835$	0	0
$836$	−156.076	−5.39801
$837$	2.06658i	0.0714315i
$838$	− 17.3477i	− 0.599266i
$839$	45.7865	1.58073	0.790363	−	0.612639i	$-0.209892\pi$
$839$	45.7865	1.58073	0.790363	+	0.612639i	$0.209892\pi$
$840$	0	0
$841$	−19.2450	−0.663622
$842$	− 14.6998i	− 0.506589i
$843$	25.9852i	0.894979i
$844$	−114.353	−3.93620
$845$	0	0
$846$	16.3252	0.561271
$847$	− 24.8340i	− 0.853307i
$848$	− 22.7009i	− 0.779550i
$849$	−16.9219	−0.580760
$850$	0	0
$851$	9.55139	0.327418
$852$	22.4291i	0.768410i
$853$	− 20.3074i	− 0.695311i	−0.937622	−	0.347655i	$-0.886978\pi$
$853$	− 20.3074i	− 0.695311i	0.937622	−	0.347655i	$-0.113022\pi$
$854$	25.2182	0.862950
$855$	0	0
$856$	60.7531	2.07650
$857$	47.4031i	1.61926i	0.586941	+	0.809630i	$0.300332\pi$
$857$	47.4031i	1.61926i	−0.586941	+	0.809630i	$0.699668\pi$
$858$	− 38.7289i	− 1.32218i
$859$	5.37538	0.183406	0.0917028	−	0.995786i	$-0.470769\pi$
$859$	5.37538	0.183406	0.0917028	+	0.995786i	$0.470769\pi$
$860$	0	0
$861$	12.8047	0.436383
$862$	− 16.6664i	− 0.567661i
$863$	− 15.0679i	− 0.512916i	−0.966555	−	0.256458i	$-0.917444\pi$
$863$	− 15.0679i	− 0.512916i	0.966555	−	0.256458i	$-0.0825556\pi$
$864$	−9.18462	−0.312467
$865$	0	0
$866$	−2.87003	−0.0975275
$867$	− 12.6760i	− 0.430501i
$868$	− 32.2057i	− 1.09313i
$869$	−59.1746	−2.00736
$870$	0	0
$871$	−44.9791	−1.52406
$872$	− 124.871i	− 4.22868i
$873$	2.57278i	0.0870753i
$874$	−100.126	−3.38680
$875$	0	0
$876$	−7.82649	−0.264433
$877$	− 6.43572i	− 0.217319i	−0.994079	−	0.108659i	$-0.965344\pi$
$877$	− 6.43572i	− 0.217319i	0.994079	−	0.108659i	$-0.0346558\pi$
$878$	− 13.2796i	− 0.448165i
$879$	−1.05979	−0.0357457
$880$	0	0
$881$	46.2411	1.55790	0.778950	−	0.627085i	$-0.215752\pi$
$881$	46.2411	1.55790	0.778950	+	0.627085i	$0.215752\pi$
$882$	− 9.83169i	− 0.331050i
$883$	− 12.8545i	− 0.432590i	−0.976328	−	0.216295i	$-0.930603\pi$
$883$	− 12.8545i	− 0.432590i	0.976328	−	0.216295i	$-0.0693972\pi$
$884$	−89.4723	−3.00928
$885$	0	0
$886$	−46.5097	−1.56252
$887$	33.5617i	1.12689i	0.826153	+	0.563446i	$0.190525\pi$
$887$	33.5617i	1.12689i	−0.826153	+	0.563446i	$0.809475\pi$
$888$	− 13.4856i	− 0.452546i
$889$	−33.5350	−1.12473
$890$	0	0
$891$	4.30834	0.144335
$892$	86.6942i	2.90274i
$893$	− 47.9873i	− 1.60583i
$894$	−13.5231	−0.452279
$895$	0	0
$896$	−10.8553	−0.362649
$897$	− 17.4786i	− 0.583592i
$898$	71.0870i	2.37220i
$899$	−6.45454	−0.215271
$900$	0	0
$901$	13.6995	0.456398
$902$	− 43.6272i	− 1.45263i
$903$	− 10.5360i	− 0.350616i
$904$	−29.7822	−0.990542
$905$	0	0
$906$	20.2101	0.671437
$907$	− 5.73678i	− 0.190487i	−0.995454	−	0.0952433i	$-0.969637\pi$
$907$	− 5.73678i	− 0.190487i	0.995454	−	0.0952433i	$-0.0303629\pi$
$908$	− 85.2257i	− 2.82831i
$909$	−7.87863	−0.261318
$910$	0	0
$911$	4.08321	0.135283	0.0676413	−	0.997710i	$-0.478453\pi$
$911$	4.08321	0.135283	0.0676413	+	0.997710i	$0.478453\pi$
$912$	68.9139i	2.28197i
$913$	− 21.6144i	− 0.715332i
$914$	−23.1876	−0.766978
$915$	0	0
$916$	28.5027	0.941757
$917$	16.1476i	0.533242i
$918$	− 14.1482i	− 0.466961i
$919$	−34.1894	−1.12780	−0.563902	−	0.825842i	$-0.690700\pi$
$919$	−34.1894	−1.12780	−0.563902	+	0.825842i	$0.690700\pi$
$920$	0	0
$921$	29.7239	0.979434
$922$	− 4.70617i	− 0.154989i
$923$	16.3599i	0.538492i
$924$	−67.1414	−2.20879
$925$	0	0
$926$	7.06927	0.232311
$927$	− 12.4046i	− 0.407421i
$928$	− 28.6863i	− 0.941672i
$929$	23.9398	0.785438	0.392719	−	0.919659i	$-0.371535\pi$
$929$	23.9398	0.785438	0.392719	+	0.919659i	$0.371535\pi$
$930$	0	0
$931$	−28.8999	−0.947157
$932$	98.9507i	3.24124i
$933$	5.08513i	0.166480i
$934$	104.474	3.41849
$935$	0	0
$936$	−24.6779	−0.806621
$937$	− 21.4640i	− 0.701199i	−0.936525	−	0.350600i	$-0.885978\pi$
$937$	− 21.4640i	− 0.701199i	0.936525	−	0.350600i	$-0.114022\pi$
$938$	110.842i	3.61912i
$939$	−25.9388	−0.846481
$940$	0	0
$941$	−1.78646	−0.0582371	−0.0291185	−	0.999576i	$-0.509270\pi$
$941$	−1.78646	−0.0582371	−0.0291185	+	0.999576i	$0.509270\pi$
$942$	42.7165i	1.39178i
$943$	− 19.6892i	− 0.641167i
$944$	−69.7733	−2.27093
$945$	0	0
$946$	−35.8975	−1.16713
$947$	45.4097i	1.47562i	0.675009	+	0.737809i	$0.264140\pi$
$947$	45.4097i	1.47562i	−0.675009	+	0.737809i	$0.735860\pi$
$948$	65.1755i	2.11680i
$949$	−5.70866	−0.185311
$950$	0	0
$951$	19.7727	0.641172
$952$	127.557i	4.13416i
$953$	1.00850i	0.0326686i	0.999867	+	0.0163343i	$0.00519960\pi$
$953$	1.00850i	0.0326686i	−0.999867	+	0.0163343i	$0.994800\pi$
$954$	6.53134	0.211460
$955$	0	0
$956$	−132.261	−4.27761
$957$	13.4562i	0.434978i
$958$	− 72.5794i	− 2.34493i
$959$	40.6631	1.31308
$960$	0	0
$961$	−26.7292	−0.862234
$962$	− 17.0025i	− 0.548183i
$963$	8.52094i	0.274583i
$964$	88.8115	2.86043
$965$	0	0
$966$	−43.0724	−1.38583
$967$	− 21.4625i	− 0.690188i	−0.938568	−	0.345094i	$-0.887847\pi$
$967$	− 21.4625i	− 0.690188i	0.938568	−	0.345094i	$-0.112153\pi$
$968$	53.9147i	1.73288i
$969$	−41.5883	−1.33601
$970$	0	0
$971$	47.3619	1.51992	0.759958	−	0.649972i	$-0.225219\pi$
$971$	47.3619	1.51992	0.759958	+	0.649972i	$0.225219\pi$
$972$	− 4.74525i	− 0.152204i
$973$	43.4884i	1.39417i
$974$	67.0298	2.14777
$975$	0	0
$976$	−26.6891	−0.854296
$977$	− 6.29459i	− 0.201382i	−0.994918	−	0.100691i	$-0.967895\pi$
$977$	− 6.29459i	− 0.201382i	0.994918	−	0.100691i	$-0.0321053\pi$
$978$	20.2927i	0.648889i
$979$	38.7841	1.23955
$980$	0	0
$981$	17.5139	0.559174
$982$	− 61.5641i	− 1.96459i
$983$	− 7.80790i	− 0.249033i	−0.992218	−	0.124517i	$-0.960262\pi$
$983$	− 7.80790i	− 0.249033i	0.992218	−	0.124517i	$-0.0397381\pi$
$984$	−27.7990	−0.886200
$985$	0	0
$986$	44.1891	1.40727
$987$	− 20.6433i	− 0.657084i
$988$	125.387i	3.98910i
$989$	−16.2007	−0.515152
$990$	0	0
$991$	−21.0569	−0.668896	−0.334448	−	0.942414i	$-0.608550\pi$
$991$	−21.0569	−0.668896	−0.334448	+	0.942414i	$0.608550\pi$
$992$	18.9808i	0.602640i
$993$	− 23.9589i	− 0.760311i
$994$	40.3156	1.27873
$995$	0	0
$996$	−23.8063	−0.754332
$997$	− 29.3042i	− 0.928072i	−0.885816	−	0.464036i	$-0.846401\pi$
$997$	− 29.3042i	− 0.928072i	0.885816	−	0.464036i	$-0.153599\pi$
$998$	23.2641i	0.736413i
$999$	1.89142	0.0598418

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.g.1249.2		16
5.2	odd	4		1875.2.a.o.1.8	yes	8
5.3	odd	4		1875.2.a.n.1.1	✓	8
5.4	even	2	inner	1875.2.b.g.1249.15		16
15.2	even	4		5625.2.a.u.1.1		8
15.8	even	4		5625.2.a.bc.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.1	✓	8	5.3	odd	4
1875.2.a.o.1.8	yes	8	5.2	odd	4
1875.2.b.g.1249.2		16	1.1	even	1	trivial
1875.2.b.g.1249.15		16	5.4	even	2	inner
5625.2.a.u.1.1		8	15.2	even	4
5625.2.a.bc.1.8		8	15.8	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)

\(f(q)\)	\(=\)	\(q-2.59716i q^{2} +1.00000i q^{3} -4.74525 q^{4} +2.59716 q^{6} -3.28414i q^{7} +7.12986i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.59716i q^{2} +1.00000i q^{3} -4.74525 q^{4} +2.59716 q^{6} -3.28414i q^{7} +7.12986i q^{8} -1.00000 q^{9} +4.30834 q^{11} -4.74525i q^{12} -3.46120i q^{13} -8.52943 q^{14} +9.02691 q^{16} +5.44757i q^{17} +2.59716i q^{18} +7.63427 q^{19} +3.28414 q^{21} -11.1895i q^{22} -5.04986i q^{23} -7.12986 q^{24} -8.98929 q^{26} -1.00000i q^{27} +15.5841i q^{28} +3.12329 q^{29} -2.06658 q^{31} -9.18462i q^{32} +4.30834i q^{33} +14.1482 q^{34} +4.74525 q^{36} +1.89142i q^{37} -19.8274i q^{38} +3.46120 q^{39} +3.89896 q^{41} -8.52943i q^{42} -3.20815i q^{43} -20.4442 q^{44} -13.1153 q^{46} -6.28577i q^{47} +9.02691i q^{48} -3.78555 q^{49} -5.44757 q^{51} +16.4243i q^{52} -2.51480i q^{53} -2.59716 q^{54} +23.4154 q^{56} +7.63427i q^{57} -8.11170i q^{58} -7.72948 q^{59} -2.95661 q^{61} +5.36725i q^{62} +3.28414i q^{63} -5.80013 q^{64} +11.1895 q^{66} -12.9952i q^{67} -25.8501i q^{68} +5.04986 q^{69} -4.72665 q^{71} -7.12986i q^{72} -1.64933i q^{73} +4.91232 q^{74} -36.2265 q^{76} -14.1492i q^{77} -8.98929i q^{78} -13.7349 q^{79} +1.00000 q^{81} -10.1262i q^{82} -5.01687i q^{83} -15.5841 q^{84} -8.33208 q^{86} +3.12329i q^{87} +30.7179i q^{88} +9.00209 q^{89} -11.3670 q^{91} +23.9628i q^{92} -2.06658i q^{93} -16.3252 q^{94} +9.18462 q^{96} -2.57278i q^{97} +9.83169i q^{98} -4.30834 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9	\( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9} + 24 q^{11} - 32 q^{14} + 30 q^{16} - 32 q^{19} + 24 q^{21} - 6 q^{24} - 68 q^{26} - 4 q^{29} + 26 q^{31} + 74 q^{34} + 18 q^{36} - 28 q^{39} - 24 q^{41} - 94 q^{44} + 66 q^{46} - 60 q^{49} - 2 q^{51} - 2 q^{54} + 120 q^{56} - 28 q^{59} + 20 q^{61} - 82 q^{64} + 36 q^{66} + 8 q^{69} + 42 q^{71} + 18 q^{74} - 2 q^{76} - 20 q^{79} + 16 q^{81} + 42 q^{84} + 84 q^{86} + 18 q^{89} - 24 q^{91} - 28 q^{94} - 36 q^{96} - 24 q^{99}+O(q^{100}) \) 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9 + 24 * q^11 - 32 * q^14 + 30 * q^16 - 32 * q^19 + 24 * q^21 - 6 * q^24 - 68 * q^26 - 4 * q^29 + 26 * q^31 + 74 * q^34 + 18 * q^36 - 28 * q^39 - 24 * q^41 - 94 * q^44 + 66 * q^46 - 60 * q^49 - 2 * q^51 - 2 * q^54 + 120 * q^56 - 28 * q^59 + 20 * q^61 - 82 * q^64 + 36 * q^66 + 8 * q^69 + 42 * q^71 + 18 * q^74 - 2 * q^76 - 20 * q^79 + 16 * q^81 + 42 * q^84 + 84 * q^86 + 18 * q^89 - 24 * q^91 - 28 * q^94 - 36 * q^96 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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