LMFDB - Embedded newform 1875.2.b.g.1249.13

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			1249.13
Root			$1.31354i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.g.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.52260i q^{2} +1.00000i q^{3} -0.318310 q^{4} -1.52260 q^{6} +0.990985i q^{7} +2.56054i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.52260i q^{2} +1.00000i q^{3} -0.318310 q^{4} -1.52260 q^{6} +0.990985i q^{7} +2.56054i q^{8} -1.00000 q^{9} +5.97349 q^{11} -0.318310i q^{12} +4.02992i q^{13} -1.50887 q^{14} -4.53530 q^{16} -0.476176i q^{17} -1.52260i q^{18} -2.82322 q^{19} -0.990985 q^{21} +9.09523i q^{22} -1.74696i q^{23} -2.56054 q^{24} -6.13596 q^{26} -1.00000i q^{27} -0.315440i q^{28} +1.41641 q^{29} +8.76141 q^{31} -1.78436i q^{32} +5.97349i q^{33} +0.725025 q^{34} +0.318310 q^{36} +8.06031i q^{37} -4.29864i q^{38} -4.02992 q^{39} -5.50296 q^{41} -1.50887i q^{42} -6.82255i q^{43} -1.90142 q^{44} +2.65993 q^{46} +9.62845i q^{47} -4.53530i q^{48} +6.01795 q^{49} +0.476176 q^{51} -1.28276i q^{52} +6.57994i q^{53} +1.52260 q^{54} -2.53746 q^{56} -2.82322i q^{57} +2.15662i q^{58} -13.0760 q^{59} +12.2013 q^{61} +13.3401i q^{62} -0.990985i q^{63} -6.35373 q^{64} -9.09523 q^{66} -11.3195i q^{67} +0.151571i q^{68} +1.74696 q^{69} -5.43047 q^{71} -2.56054i q^{72} -4.08935i q^{73} -12.2726 q^{74} +0.898661 q^{76} +5.91964i q^{77} -6.13596i q^{78} -15.0070 q^{79} +1.00000 q^{81} -8.37881i q^{82} -2.29898i q^{83} +0.315440 q^{84} +10.3880 q^{86} +1.41641i q^{87} +15.2954i q^{88} -6.26920 q^{89} -3.99359 q^{91} +0.556076i q^{92} +8.76141i q^{93} -14.6603 q^{94} +1.78436 q^{96} -13.2541i q^{97} +9.16293i q^{98} -5.97349 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.52260i	1.07664i	0.842740	+	0.538320i	$0.180941\pi$
$2$	1.52260i	1.07664i	−0.842740	+	0.538320i	$0.819059\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−0.318310	−0.159155
$5$	0	0
$5$	0	0
$6$	−1.52260	−0.621599
$7$	0.990985i	0.374557i	0.982307	+	0.187279i	$0.0599667\pi$
$7$	0.990985i	0.374557i	−0.982307	+	0.187279i	$0.940033\pi$
$8$	2.56054i	0.905288i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	5.97349	1.80107	0.900537	−	0.434779i	$-0.143174\pi$
$11$	5.97349	1.80107	0.900537	+	0.434779i	$0.143174\pi$
$12$	− 0.318310i	− 0.0918882i
$13$	4.02992i	1.11770i	0.829269	+	0.558850i	$0.188757\pi$
$13$	4.02992i	1.11770i	−0.829269	+	0.558850i	$0.811243\pi$
$14$	−1.50887	−0.403263
$15$	0	0
$16$	−4.53530	−1.13382
$17$	− 0.476176i	− 0.115490i	−0.998331	−	0.0577448i	$-0.981609\pi$
$17$	− 0.476176i	− 0.115490i	0.998331	−	0.0577448i	$-0.0183910\pi$
$18$	− 1.52260i	− 0.358880i
$19$	−2.82322	−0.647692	−0.323846	−	0.946110i	$-0.604976\pi$
$19$	−2.82322	−0.647692	−0.323846	+	0.946110i	$0.604976\pi$
$20$	0	0
$21$	−0.990985	−0.216251
$22$	9.09523i	1.93911i
$23$	− 1.74696i	− 0.364267i	−0.983274	−	0.182134i	$-0.941700\pi$
$23$	− 1.74696i	− 0.364267i	0.983274	−	0.182134i	$-0.0583003\pi$
$24$	−2.56054	−0.522668
$25$	0	0
$26$	−6.13596	−1.20336
$27$	− 1.00000i	− 0.192450i
$28$	− 0.315440i	− 0.0596127i
$29$	1.41641	0.263020	0.131510	−	0.991315i	$-0.458017\pi$
$29$	1.41641	0.263020	0.131510	+	0.991315i	$0.458017\pi$
$30$	0	0
$31$	8.76141	1.57360	0.786798	−	0.617211i	$-0.211737\pi$
$31$	8.76141	1.57360	0.786798	+	0.617211i	$0.211737\pi$
$32$	− 1.78436i	− 0.315434i
$33$	5.97349i	1.03985i
$34$	0.725025	0.124341
$35$	0	0
$36$	0.318310	0.0530517
$37$	8.06031i	1.32511i	0.749015	+	0.662553i	$0.230527\pi$
$37$	8.06031i	1.32511i	−0.749015	+	0.662553i	$0.769473\pi$
$38$	− 4.29864i	− 0.697332i
$39$	−4.02992	−0.645304
$40$	0	0
$41$	−5.50296	−0.859418	−0.429709	−	0.902967i	$-0.641384\pi$
$41$	−5.50296	−0.859418	−0.429709	+	0.902967i	$0.641384\pi$
$42$	− 1.50887i	− 0.232824i
$43$	− 6.82255i	− 1.04043i	−0.854036	−	0.520214i	$-0.825852\pi$
$43$	− 6.82255i	− 1.04043i	0.854036	−	0.520214i	$-0.174148\pi$
$44$	−1.90142	−0.286650
$45$	0	0
$46$	2.65993	0.392185
$47$	9.62845i	1.40445i	0.711953	+	0.702227i	$0.247811\pi$
$47$	9.62845i	1.40445i	−0.711953	+	0.702227i	$0.752189\pi$
$48$	− 4.53530i	− 0.654614i
$49$	6.01795	0.859707
$50$	0	0
$51$	0.476176	0.0666779
$52$	− 1.28276i	− 0.177887i
$53$	6.57994i	0.903824i	0.892063	+	0.451912i	$0.149258\pi$
$53$	6.57994i	0.903824i	−0.892063	+	0.451912i	$0.850742\pi$
$54$	1.52260	0.207200
$55$	0	0
$56$	−2.53746	−0.339082
$57$	− 2.82322i	− 0.373945i
$58$	2.15662i	0.283179i
$59$	−13.0760	−1.70235	−0.851177	−	0.524879i	$-0.824111\pi$
$59$	−13.0760	−1.70235	−0.851177	+	0.524879i	$0.824111\pi$
$60$	0	0
$61$	12.2013	1.56222	0.781108	−	0.624397i	$-0.214655\pi$
$61$	12.2013	1.56222	0.781108	+	0.624397i	$0.214655\pi$
$62$	13.3401i	1.69420i
$63$	− 0.990985i	− 0.124852i
$64$	−6.35373	−0.794216
$65$	0	0
$66$	−9.09523	−1.11955
$67$	− 11.3195i	− 1.38289i	−0.722429	−	0.691445i	$-0.756974\pi$
$67$	− 11.3195i	− 1.38289i	0.722429	−	0.691445i	$-0.243026\pi$
$68$	0.151571i	0.0183807i
$69$	1.74696	0.210310
$70$	0	0
$71$	−5.43047	−0.644478	−0.322239	−	0.946658i	$-0.604436\pi$
$71$	−5.43047	−0.644478	−0.322239	+	0.946658i	$0.604436\pi$
$72$	− 2.56054i	− 0.301763i
$73$	− 4.08935i	− 0.478622i	−0.970943	−	0.239311i	$-0.923078\pi$
$73$	− 4.08935i	− 0.478622i	0.970943	−	0.239311i	$-0.0769215\pi$
$74$	−12.2726	−1.42666
$75$	0	0
$76$	0.898661	0.103083
$77$	5.91964i	0.674605i
$78$	− 6.13596i	− 0.694761i
$79$	−15.0070	−1.68842	−0.844212	−	0.536009i	$-0.819931\pi$
$79$	−15.0070	−1.68842	−0.844212	+	0.536009i	$0.819931\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 8.37881i	− 0.925284i
$83$	− 2.29898i	− 0.252346i	−0.992008	−	0.126173i	$-0.959731\pi$
$83$	− 2.29898i	− 0.252346i	0.992008	−	0.126173i	$-0.0402695\pi$
$84$	0.315440	0.0344174
$85$	0	0
$86$	10.3880	1.12017
$87$	1.41641i	0.151855i
$88$	15.2954i	1.63049i
$89$	−6.26920	−0.664533	−0.332267	−	0.943185i	$-0.607813\pi$
$89$	−6.26920	−0.664533	−0.332267	+	0.943185i	$0.607813\pi$
$90$	0	0
$91$	−3.99359	−0.418642
$92$	0.556076i	0.0579749i
$93$	8.76141i	0.908516i
$94$	−14.6603	−1.51209
$95$	0	0
$96$	1.78436	0.182116
$97$	− 13.2541i	− 1.34575i	−0.739756	−	0.672875i	$-0.765059\pi$
$97$	− 13.2541i	− 1.34575i	0.739756	−	0.672875i	$-0.234941\pi$
$98$	9.16293i	0.925595i
$99$	−5.97349	−0.600358
$100$	0	0
$101$	−6.41452	−0.638269	−0.319134	−	0.947709i	$-0.603392\pi$
$101$	−6.41452	−0.638269	−0.319134	+	0.947709i	$0.603392\pi$
$102$	0.725025i	0.0717882i
$103$	9.38309i	0.924543i	0.886738	+	0.462272i	$0.152966\pi$
$103$	9.38309i	0.924543i	−0.886738	+	0.462272i	$0.847034\pi$
$104$	−10.3188	−1.01184
$105$	0	0
$106$	−10.0186	−0.973094
$107$	13.4108i	1.29647i	0.761441	+	0.648234i	$0.224492\pi$
$107$	13.4108i	1.29647i	−0.761441	+	0.648234i	$0.775508\pi$
$108$	0.318310i	0.0306294i
$109$	−2.81573	−0.269698	−0.134849	−	0.990866i	$-0.543055\pi$
$109$	−2.81573	−0.269698	−0.134849	+	0.990866i	$0.543055\pi$
$110$	0	0
$111$	−8.06031	−0.765050
$112$	− 4.49441i	− 0.424682i
$113$	− 19.4252i	− 1.82737i	−0.406423	−	0.913685i	$-0.633224\pi$
$113$	− 19.4252i	− 1.82737i	0.406423	−	0.913685i	$-0.366776\pi$
$114$	4.29864	0.402605
$115$	0	0
$116$	−0.450857	−0.0418610
$117$	− 4.02992i	− 0.372566i
$118$	− 19.9096i	− 1.83282i
$119$	0.471883	0.0432574
$120$	0	0
$121$	24.6825	2.24387
$122$	18.5777i	1.68194i
$123$	− 5.50296i	− 0.496185i
$124$	−2.78884	−0.250446
$125$	0	0
$126$	1.50887	0.134421
$127$	− 6.42366i	− 0.570008i	−0.958526	−	0.285004i	$-0.908005\pi$
$127$	− 6.42366i	− 0.570008i	0.958526	−	0.285004i	$-0.0919949\pi$
$128$	− 13.2429i	− 1.17052i
$129$	6.82255	0.600692
$130$	0	0
$131$	−7.30057	−0.637853	−0.318927	−	0.947779i	$-0.603322\pi$
$131$	−7.30057	−0.637853	−0.318927	+	0.947779i	$0.603322\pi$
$132$	− 1.90142i	− 0.165497i
$133$	− 2.79777i	− 0.242598i
$134$	17.2350	1.48888
$135$	0	0
$136$	1.21927	0.104551
$137$	7.84860i	0.670552i	0.942120	+	0.335276i	$0.108829\pi$
$137$	7.84860i	0.670552i	−0.942120	+	0.335276i	$0.891171\pi$
$138$	2.65993i	0.226428i
$139$	17.0514	1.44628	0.723138	−	0.690703i	$-0.242699\pi$
$139$	17.0514	1.44628	0.723138	+	0.690703i	$0.242699\pi$
$140$	0	0
$141$	−9.62845	−0.810862
$142$	− 8.26844i	− 0.693872i
$143$	24.0727i	2.01306i
$144$	4.53530	0.377942
$145$	0	0
$146$	6.22644	0.515304
$147$	6.01795i	0.496352i
$148$	− 2.56568i	− 0.210897i
$149$	−2.84365	−0.232961	−0.116480	−	0.993193i	$-0.537161\pi$
$149$	−2.84365	−0.232961	−0.116480	+	0.993193i	$0.537161\pi$
$150$	0	0
$151$	11.5744	0.941915	0.470958	−	0.882156i	$-0.343908\pi$
$151$	11.5744	0.941915	0.470958	+	0.882156i	$0.343908\pi$
$152$	− 7.22898i	− 0.586348i
$153$	0.476176i	0.0384965i
$154$	−9.01324	−0.726307
$155$	0	0
$156$	1.28276	0.102703
$157$	− 4.77270i	− 0.380903i	−0.981697	−	0.190452i	$-0.939005\pi$
$157$	− 4.77270i	− 0.380903i	0.981697	−	0.190452i	$-0.0609952\pi$
$158$	− 22.8497i	− 1.81783i
$159$	−6.57994	−0.521823
$160$	0	0
$161$	1.73122	0.136439
$162$	1.52260i	0.119627i
$163$	2.98657i	0.233926i	0.993136	+	0.116963i	$0.0373159\pi$
$163$	2.98657i	0.233926i	−0.993136	+	0.116963i	$0.962684\pi$
$164$	1.75165	0.136781
$165$	0	0
$166$	3.50043	0.271686
$167$	− 18.6105i	− 1.44012i	−0.693912	−	0.720060i	$-0.744114\pi$
$167$	− 18.6105i	− 1.44012i	0.693912	−	0.720060i	$-0.255886\pi$
$168$	− 2.53746i	− 0.195769i
$169$	−3.24028	−0.249252
$170$	0	0
$171$	2.82322	0.215897
$172$	2.17169i	0.165589i
$173$	16.3578i	1.24366i	0.783153	+	0.621829i	$0.213610\pi$
$173$	16.3578i	1.24366i	−0.783153	+	0.621829i	$0.786390\pi$
$174$	−2.15662	−0.163493
$175$	0	0
$176$	−27.0915	−2.04210
$177$	− 13.0760i	− 0.982855i
$178$	− 9.54548i	− 0.715464i
$179$	15.2266	1.13809	0.569045	−	0.822307i	$-0.307313\pi$
$179$	15.2266	1.13809	0.569045	+	0.822307i	$0.307313\pi$
$180$	0	0
$181$	−3.44321	−0.255932	−0.127966	−	0.991779i	$-0.540845\pi$
$181$	−3.44321	−0.255932	−0.127966	+	0.991779i	$0.540845\pi$
$182$	− 6.08064i	− 0.450727i
$183$	12.2013i	0.901945i
$184$	4.47317	0.329767
$185$	0	0
$186$	−13.3401	−0.978145
$187$	− 2.84443i	− 0.208005i
$188$	− 3.06483i	− 0.223526i
$189$	0.990985	0.0720836
$190$	0	0
$191$	15.1752	1.09804	0.549020	−	0.835809i	$-0.315001\pi$
$191$	15.1752	1.09804	0.549020	+	0.835809i	$0.315001\pi$
$192$	− 6.35373i	− 0.458541i
$193$	14.0304i	1.00993i	0.863140	+	0.504965i	$0.168495\pi$
$193$	14.0304i	1.00993i	−0.863140	+	0.504965i	$0.831505\pi$
$194$	20.1807	1.44889
$195$	0	0
$196$	−1.91557	−0.136827
$197$	− 23.7871i	− 1.69476i	−0.530984	−	0.847382i	$-0.678177\pi$
$197$	− 23.7871i	− 1.69476i	0.530984	−	0.847382i	$-0.321823\pi$
$198$	− 9.09523i	− 0.646370i
$199$	15.9445	1.13028	0.565139	−	0.824996i	$-0.308823\pi$
$199$	15.9445	1.13028	0.565139	+	0.824996i	$0.308823\pi$
$200$	0	0
$201$	11.3195	0.798412
$202$	− 9.76675i	− 0.687186i
$203$	1.40364i	0.0985162i
$204$	−0.151571	−0.0106121
$205$	0	0
$206$	−14.2867	−0.995401
$207$	1.74696i	0.121422i
$208$	− 18.2769i	− 1.26728i
$209$	−16.8645	−1.16654
$210$	0	0
$211$	−6.49016	−0.446801	−0.223401	−	0.974727i	$-0.571716\pi$
$211$	−6.49016	−0.446801	−0.223401	+	0.974727i	$0.571716\pi$
$212$	− 2.09446i	− 0.143848i
$213$	− 5.43047i	− 0.372090i
$214$	−20.4192	−1.39583
$215$	0	0
$216$	2.56054	0.174223
$217$	8.68243i	0.589402i
$218$	− 4.28723i	− 0.290368i
$219$	4.08935	0.276332
$220$	0	0
$221$	1.91895	0.129083
$222$	− 12.2726i	− 0.823684i
$223$	10.7454i	0.719562i	0.933037	+	0.359781i	$0.117149\pi$
$223$	10.7454i	0.719562i	−0.933037	+	0.359781i	$0.882851\pi$
$224$	1.76828	0.118148
$225$	0	0
$226$	29.5768	1.96742
$227$	6.84922i	0.454599i	0.973825	+	0.227299i	$0.0729896\pi$
$227$	6.84922i	0.454599i	−0.973825	+	0.227299i	$0.927010\pi$
$228$	0.898661i	0.0595153i
$229$	−2.21379	−0.146291	−0.0731457	−	0.997321i	$-0.523304\pi$
$229$	−2.21379	−0.146291	−0.0731457	+	0.997321i	$0.523304\pi$
$230$	0	0
$231$	−5.91964	−0.389484
$232$	3.62677i	0.238109i
$233$	− 9.39682i	− 0.615606i	−0.951450	−	0.307803i	$-0.900406\pi$
$233$	− 9.39682i	− 0.615606i	0.951450	−	0.307803i	$-0.0995938\pi$
$234$	6.13596	0.401120
$235$	0	0
$236$	4.16223	0.270938
$237$	− 15.0070i	− 0.974812i
$238$	0.718489i	0.0465727i
$239$	0.737315	0.0476930	0.0238465	−	0.999716i	$-0.492409\pi$
$239$	0.737315	0.0476930	0.0238465	+	0.999716i	$0.492409\pi$
$240$	0	0
$241$	−4.21895	−0.271767	−0.135883	−	0.990725i	$-0.543387\pi$
$241$	−4.21895	−0.271767	−0.135883	+	0.990725i	$0.543387\pi$
$242$	37.5816i	2.41584i
$243$	1.00000i	0.0641500i
$244$	−3.88379	−0.248634
$245$	0	0
$246$	8.37881	0.534213
$247$	− 11.3774i	− 0.723925i
$248$	22.4339i	1.42456i
$249$	2.29898	0.145692
$250$	0	0
$251$	−9.82121	−0.619909	−0.309955	−	0.950751i	$-0.600314\pi$
$251$	−9.82121	−0.619909	−0.309955	+	0.950751i	$0.600314\pi$
$252$	0.315440i	0.0198709i
$253$	− 10.4355i	− 0.656072i
$254$	9.78067	0.613694
$255$	0	0
$256$	7.45620	0.466012
$257$	− 0.0383881i	− 0.00239458i	−0.999999	−	0.00119729i	$-0.999619\pi$
$257$	− 0.0383881i	− 0.00239458i	0.999999	−	0.00119729i	$-0.000381110\pi$
$258$	10.3880i	0.646729i
$259$	−7.98764	−0.496328
$260$	0	0
$261$	−1.41641	−0.0876735
$262$	− 11.1158i	− 0.686739i
$263$	− 14.2706i	− 0.879965i	−0.898006	−	0.439982i	$-0.854985\pi$
$263$	− 14.2706i	− 0.879965i	0.898006	−	0.439982i	$-0.145015\pi$
$264$	−15.2954	−0.941364
$265$	0	0
$266$	4.25989	0.261191
$267$	− 6.26920i	− 0.383669i
$268$	3.60309i	0.220094i
$269$	−21.4090	−1.30533	−0.652666	−	0.757646i	$-0.726349\pi$
$269$	−21.4090	−1.30533	−0.652666	+	0.757646i	$0.726349\pi$
$270$	0	0
$271$	6.22804	0.378327	0.189163	−	0.981946i	$-0.439422\pi$
$271$	6.22804	0.378327	0.189163	+	0.981946i	$0.439422\pi$
$272$	2.15960i	0.130945i
$273$	− 3.99359i	− 0.241703i
$274$	−11.9503	−0.721943
$275$	0	0
$276$	−0.556076	−0.0334718
$277$	14.0431i	0.843768i	0.906650	+	0.421884i	$0.138631\pi$
$277$	14.0431i	0.843768i	−0.906650	+	0.421884i	$0.861369\pi$
$278$	25.9624i	1.55712i
$279$	−8.76141	−0.524532
$280$	0	0
$281$	−0.516993	−0.0308412	−0.0154206	−	0.999881i	$-0.504909\pi$
$281$	−0.516993	−0.0308412	−0.0154206	+	0.999881i	$0.504909\pi$
$282$	− 14.6603i	− 0.873007i
$283$	0.977154i	0.0580858i	0.999578	+	0.0290429i	$0.00924594\pi$
$283$	0.977154i	0.0580858i	−0.999578	+	0.0290429i	$0.990754\pi$
$284$	1.72857	0.102572
$285$	0	0
$286$	−36.6531	−2.16734
$287$	− 5.45335i	− 0.321901i
$288$	1.78436i	0.105145i
$289$	16.7733	0.986662
$290$	0	0
$291$	13.2541	0.776969
$292$	1.30168i	0.0761751i
$293$	2.75561i	0.160984i	0.996755	+	0.0804922i	$0.0256492\pi$
$293$	2.75561i	0.160984i	−0.996755	+	0.0804922i	$0.974351\pi$
$294$	−9.16293	−0.534393
$295$	0	0
$296$	−20.6387	−1.19960
$297$	− 5.97349i	− 0.346617i
$298$	− 4.32974i	− 0.250815i
$299$	7.04013	0.407141
$300$	0	0
$301$	6.76104	0.389700
$302$	17.6233i	1.01410i
$303$	− 6.41452i	− 0.368505i
$304$	12.8042	0.734369
$305$	0	0
$306$	−0.725025	−0.0414469
$307$	− 16.1940i	− 0.924241i	−0.886817	−	0.462120i	$-0.847089\pi$
$307$	− 16.1940i	− 0.924241i	0.886817	−	0.462120i	$-0.152911\pi$
$308$	− 1.88428i	− 0.107367i
$309$	−9.38309	−0.533785
$310$	0	0
$311$	8.20094	0.465033	0.232516	−	0.972593i	$-0.425304\pi$
$311$	8.20094	0.465033	0.232516	+	0.972593i	$0.425304\pi$
$312$	− 10.3188i	− 0.584186i
$313$	20.0065i	1.13083i	0.824806	+	0.565416i	$0.191284\pi$
$313$	20.0065i	1.13083i	−0.824806	+	0.565416i	$0.808716\pi$
$314$	7.26691	0.410096
$315$	0	0
$316$	4.77689	0.268721
$317$	6.52708i	0.366597i	0.983057	+	0.183299i	$0.0586775\pi$
$317$	6.52708i	0.366597i	−0.983057	+	0.183299i	$0.941322\pi$
$318$	− 10.0186i	− 0.561816i
$319$	8.46090	0.473719
$320$	0	0
$321$	−13.4108	−0.748516
$322$	2.63595i	0.146896i
$323$	1.34435i	0.0748017i
$324$	−0.318310	−0.0176839
$325$	0	0
$326$	−4.54735	−0.251854
$327$	− 2.81573i	− 0.155710i
$328$	− 14.0906i	− 0.778021i
$329$	−9.54165	−0.526048
$330$	0	0
$331$	33.2523	1.82771	0.913857	−	0.406037i	$-0.133090\pi$
$331$	33.2523	1.82771	0.913857	+	0.406037i	$0.133090\pi$
$332$	0.731789i	0.0401621i
$333$	− 8.06031i	− 0.441702i
$334$	28.3363	1.55049
$335$	0	0
$336$	4.49441	0.245190
$337$	− 16.9653i	− 0.924160i	−0.886838	−	0.462080i	$-0.847103\pi$
$337$	− 16.9653i	− 0.924160i	0.886838	−	0.462080i	$-0.152897\pi$
$338$	− 4.93365i	− 0.268355i
$339$	19.4252	1.05503
$340$	0	0
$341$	52.3362	2.83416
$342$	4.29864i	0.232444i
$343$	12.9006i	0.696567i
$344$	17.4694	0.941888
$345$	0	0
$346$	−24.9063	−1.33897
$347$	20.1449i	1.08144i	0.841203	+	0.540719i	$0.181848\pi$
$347$	20.1449i	1.08144i	−0.841203	+	0.540719i	$0.818152\pi$
$348$	− 0.450857i	− 0.0241685i
$349$	29.8420	1.59741	0.798703	−	0.601725i	$-0.205520\pi$
$349$	29.8420	1.59741	0.798703	+	0.601725i	$0.205520\pi$
$350$	0	0
$351$	4.02992	0.215101
$352$	− 10.6589i	− 0.568120i
$353$	0.982924i	0.0523157i	0.999658	+	0.0261579i	$0.00832726\pi$
$353$	0.982924i	0.0523157i	−0.999658	+	0.0261579i	$0.991673\pi$
$354$	19.9096	1.05818
$355$	0	0
$356$	1.99555	0.105764
$357$	0.471883i	0.0249747i
$358$	23.1840i	1.22531i
$359$	33.4999	1.76806	0.884030	−	0.467431i	$-0.154820\pi$
$359$	33.4999	1.76806	0.884030	+	0.467431i	$0.154820\pi$
$360$	0	0
$361$	−11.0294	−0.580495
$362$	− 5.24263i	− 0.275546i
$363$	24.6825i	1.29550i
$364$	1.27120	0.0666290
$365$	0	0
$366$	−18.5777	−0.971071
$367$	− 0.352799i	− 0.0184160i	−0.999958	−	0.00920798i	$-0.997069\pi$
$367$	− 0.352799i	− 0.0184160i	0.999958	−	0.00920798i	$-0.00293103\pi$
$368$	7.92300i	0.413015i
$369$	5.50296	0.286473
$370$	0	0
$371$	−6.52062	−0.338534
$372$	− 2.78884i	− 0.144595i
$373$	− 24.2811i	− 1.25723i	−0.777718	−	0.628614i	$-0.783623\pi$
$373$	− 24.2811i	− 1.25723i	0.777718	−	0.628614i	$-0.216377\pi$
$374$	4.33093	0.223947
$375$	0	0
$376$	−24.6540	−1.27144
$377$	5.70802i	0.293978i
$378$	1.50887i	0.0776081i
$379$	36.4446	1.87203	0.936017	−	0.351956i	$-0.114483\pi$
$379$	36.4446	1.87203	0.936017	+	0.351956i	$0.114483\pi$
$380$	0	0
$381$	6.42366	0.329094
$382$	23.1058i	1.18219i
$383$	6.63250i	0.338905i	0.985538	+	0.169453i	$0.0542000\pi$
$383$	6.63250i	0.338905i	−0.985538	+	0.169453i	$0.945800\pi$
$384$	13.2429	0.675799
$385$	0	0
$386$	−21.3627	−1.08733
$387$	6.82255i	0.346810i
$388$	4.21891i	0.214183i
$389$	−2.94097	−0.149113	−0.0745566	−	0.997217i	$-0.523754\pi$
$389$	−2.94097	−0.149113	−0.0745566	+	0.997217i	$0.523754\pi$
$390$	0	0
$391$	−0.831862	−0.0420690
$392$	15.4092i	0.778282i
$393$	− 7.30057i	− 0.368265i
$394$	36.2183	1.82465
$395$	0	0
$396$	1.90142	0.0955500
$397$	9.44435i	0.473998i	0.971510	+	0.236999i	$0.0761639\pi$
$397$	9.44435i	0.473998i	−0.971510	+	0.236999i	$0.923836\pi$
$398$	24.2771i	1.21690i
$399$	2.79777	0.140064
$400$	0	0
$401$	24.3968	1.21832	0.609159	−	0.793048i	$-0.291507\pi$
$401$	24.3968	1.21832	0.609159	+	0.793048i	$0.291507\pi$
$402$	17.2350i	0.859603i
$403$	35.3078i	1.75881i
$404$	2.04181	0.101584
$405$	0	0
$406$	−2.13718	−0.106067
$407$	48.1481i	2.38661i
$408$	1.21927i	0.0603627i
$409$	−13.6073	−0.672836	−0.336418	−	0.941713i	$-0.609216\pi$
$409$	−13.6073	−0.672836	−0.336418	+	0.941713i	$0.609216\pi$
$410$	0	0
$411$	−7.84860	−0.387143
$412$	− 2.98673i	− 0.147146i
$413$	− 12.9582i	− 0.637629i
$414$	−2.65993	−0.130728
$415$	0	0
$416$	7.19085	0.352560
$417$	17.0514i	0.835008i
$418$	− 25.6779i	− 1.25595i
$419$	−25.0654	−1.22452	−0.612262	−	0.790655i	$-0.709740\pi$
$419$	−25.0654	−1.22452	−0.612262	+	0.790655i	$0.709740\pi$
$420$	0	0
$421$	−5.56078	−0.271016	−0.135508	−	0.990776i	$-0.543267\pi$
$421$	−5.56078	−0.271016	−0.135508	+	0.990776i	$0.543267\pi$
$422$	− 9.88192i	− 0.481044i
$423$	− 9.62845i	− 0.468151i
$424$	−16.8482	−0.818221
$425$	0	0
$426$	8.26844	0.400607
$427$	12.0913i	0.585139i
$428$	− 4.26878i	− 0.206339i
$429$	−24.0727	−1.16224
$430$	0	0
$431$	24.2599	1.16856	0.584279	−	0.811553i	$-0.301377\pi$
$431$	24.2599	1.16856	0.584279	+	0.811553i	$0.301377\pi$
$432$	4.53530i	0.218205i
$433$	38.8459i	1.86681i	0.358821	+	0.933407i	$0.383179\pi$
$433$	38.8459i	1.86681i	−0.358821	+	0.933407i	$0.616821\pi$
$434$	−13.2199	−0.634574
$435$	0	0
$436$	0.896274	0.0429238
$437$	4.93207i	0.235933i
$438$	6.22644i	0.297511i
$439$	9.22879	0.440466	0.220233	−	0.975447i	$-0.429318\pi$
$439$	9.22879	0.440466	0.220233	+	0.975447i	$0.429318\pi$
$440$	0	0
$441$	−6.01795	−0.286569
$442$	2.92179i	0.138976i
$443$	− 5.30591i	− 0.252091i	−0.992024	−	0.126046i	$-0.959771\pi$
$443$	− 5.30591i	− 0.252091i	0.992024	−	0.126046i	$-0.0402286\pi$
$444$	2.56568	0.121762
$445$	0	0
$446$	−16.3609	−0.774710
$447$	− 2.84365i	− 0.134500i
$448$	− 6.29645i	− 0.297479i
$449$	22.2199	1.04862	0.524311	−	0.851527i	$-0.324323\pi$
$449$	22.2199	1.04862	0.524311	+	0.851527i	$0.324323\pi$
$450$	0	0
$451$	−32.8719	−1.54788
$452$	6.18324i	0.290835i
$453$	11.5744i	0.543815i
$454$	−10.4286	−0.489440
$455$	0	0
$456$	7.22898	0.338528
$457$	6.77157i	0.316761i	0.987378	+	0.158380i	$0.0506272\pi$
$457$	6.77157i	0.316761i	−0.987378	+	0.158380i	$0.949373\pi$
$458$	− 3.37072i	− 0.157503i
$459$	−0.476176	−0.0222260
$460$	0	0
$461$	33.9340	1.58046	0.790231	−	0.612809i	$-0.209960\pi$
$461$	33.9340	1.58046	0.790231	+	0.612809i	$0.209960\pi$
$462$	− 9.01324i	− 0.419334i
$463$	− 28.8896i	− 1.34262i	−0.741179	−	0.671308i	$-0.765733\pi$
$463$	− 28.8896i	− 1.34262i	0.741179	−	0.671308i	$-0.234267\pi$
$464$	−6.42384	−0.298219
$465$	0	0
$466$	14.3076	0.662786
$467$	31.2100i	1.44423i	0.691776	+	0.722113i	$0.256829\pi$
$467$	31.2100i	1.44423i	−0.691776	+	0.722113i	$0.743171\pi$
$468$	1.28276i	0.0592958i
$469$	11.2174	0.517972
$470$	0	0
$471$	4.77270	0.219914
$472$	− 33.4817i	− 1.54112i
$473$	− 40.7544i	− 1.87389i
$474$	22.8497	1.04952
$475$	0	0
$476$	−0.150205	−0.00688464
$477$	− 6.57994i	− 0.301275i
$478$	1.12264i	0.0513482i
$479$	14.6580	0.669741	0.334870	−	0.942264i	$-0.391307\pi$
$479$	14.6580	0.669741	0.334870	+	0.942264i	$0.391307\pi$
$480$	0	0
$481$	−32.4824	−1.48107
$482$	− 6.42377i	− 0.292595i
$483$	1.73122i	0.0787730i
$484$	−7.85670	−0.357123
$485$	0	0
$486$	−1.52260	−0.0690665
$487$	− 5.55868i	− 0.251888i	−0.992037	−	0.125944i	$-0.959804\pi$
$487$	− 5.55868i	− 0.251888i	0.992037	−	0.125944i	$-0.0401959\pi$
$488$	31.2419i	1.41425i
$489$	−2.98657	−0.135057
$490$	0	0
$491$	−17.4593	−0.787925	−0.393962	−	0.919127i	$-0.628896\pi$
$491$	−17.4593	−0.787925	−0.393962	+	0.919127i	$0.628896\pi$
$492$	1.75165i	0.0789704i
$493$	− 0.674459i	− 0.0303761i
$494$	17.3232	0.779407
$495$	0	0
$496$	−39.7356	−1.78418
$497$	− 5.38152i	− 0.241394i
$498$	3.50043i	0.156858i
$499$	−16.3254	−0.730825	−0.365412	−	0.930846i	$-0.619072\pi$
$499$	−16.3254	−0.730825	−0.365412	+	0.930846i	$0.619072\pi$
$500$	0	0
$501$	18.6105	0.831454
$502$	− 14.9538i	− 0.667420i
$503$	− 16.7612i	− 0.747345i	−0.927561	−	0.373672i	$-0.878098\pi$
$503$	− 16.7612i	− 0.747345i	0.927561	−	0.373672i	$-0.121902\pi$
$504$	2.53746	0.113027
$505$	0	0
$506$	15.8890	0.706354
$507$	− 3.24028i	− 0.143906i
$508$	2.04472i	0.0907196i
$509$	−8.35851	−0.370485	−0.185242	−	0.982693i	$-0.559307\pi$
$509$	−8.35851	−0.370485	−0.185242	+	0.982693i	$0.559307\pi$
$510$	0	0
$511$	4.05248	0.179271
$512$	− 15.1330i	− 0.668791i
$513$	2.82322i	0.124648i
$514$	0.0584497	0.00257811
$515$	0	0
$516$	−2.17169	−0.0956031
$517$	57.5154i	2.52953i
$518$	− 12.1620i	− 0.534367i
$519$	−16.3578	−0.718026
$520$	0	0
$521$	−45.6262	−1.99892	−0.999460	−	0.0328599i	$-0.989538\pi$
$521$	−45.6262	−1.99892	−0.999460	+	0.0328599i	$0.989538\pi$
$522$	− 2.15662i	− 0.0943928i
$523$	− 26.6701i	− 1.16620i	−0.812399	−	0.583102i	$-0.801839\pi$
$523$	− 26.6701i	− 1.16620i	0.812399	−	0.583102i	$-0.198161\pi$
$524$	2.32384	0.101518
$525$	0	0
$526$	21.7285	0.947406
$527$	− 4.17197i	− 0.181734i
$528$	− 27.0915i	− 1.17901i
$529$	19.9481	0.867309
$530$	0	0
$531$	13.0760	0.567451
$532$	0.890559i	0.0386106i
$533$	− 22.1765i	− 0.960571i
$534$	9.54548	0.413073
$535$	0	0
$536$	28.9839	1.25191
$537$	15.2266i	0.657076i
$538$	− 32.5974i	− 1.40537i
$539$	35.9481	1.54840
$540$	0	0
$541$	−0.225005	−0.00967370	−0.00483685	−	0.999988i	$-0.501540\pi$
$541$	−0.225005	−0.00967370	−0.00483685	+	0.999988i	$0.501540\pi$
$542$	9.48282i	0.407322i
$543$	− 3.44321i	− 0.147762i
$544$	−0.849670	−0.0364293
$545$	0	0
$546$	6.08064	0.260228
$547$	− 8.58474i	− 0.367057i	−0.983014	−	0.183529i	$-0.941248\pi$
$547$	− 8.58474i	− 0.367057i	0.983014	−	0.183529i	$-0.0587520\pi$
$548$	− 2.49829i	− 0.106722i
$549$	−12.2013	−0.520738
$550$	0	0
$551$	−3.99884	−0.170356
$552$	4.47317i	0.190391i
$553$	− 14.8718i	− 0.632411i
$554$	−21.3820	−0.908435
$555$	0	0
$556$	−5.42762	−0.230182
$557$	20.6736i	0.875968i	0.898983	+	0.437984i	$0.144307\pi$
$557$	20.6736i	0.875968i	−0.898983	+	0.437984i	$0.855693\pi$
$558$	− 13.3401i	− 0.564732i
$559$	27.4943	1.16289
$560$	0	0
$561$	2.84443	0.120092
$562$	− 0.787174i	− 0.0332049i
$563$	12.6162i	0.531708i	0.964013	+	0.265854i	$0.0856540\pi$
$563$	12.6162i	0.531708i	−0.964013	+	0.265854i	$0.914346\pi$
$564$	3.06483	0.129053
$565$	0	0
$566$	−1.48781	−0.0625375
$567$	0.990985i	0.0416175i
$568$	− 13.9049i	− 0.583439i
$569$	−10.9897	−0.460714	−0.230357	−	0.973106i	$-0.573989\pi$
$569$	−10.9897	−0.460714	−0.230357	+	0.973106i	$0.573989\pi$
$570$	0	0
$571$	0.276517	0.0115719	0.00578593	−	0.999983i	$-0.498158\pi$
$571$	0.276517	0.0115719	0.00578593	+	0.999983i	$0.498158\pi$
$572$	− 7.66258i	− 0.320389i
$573$	15.1752i	0.633953i
$574$	8.30327	0.346572
$575$	0	0
$576$	6.35373	0.264739
$577$	15.7460i	0.655515i	0.944762	+	0.327757i	$0.106293\pi$
$577$	15.7460i	0.655515i	−0.944762	+	0.327757i	$0.893707\pi$
$578$	25.5390i	1.06228i
$579$	−14.0304	−0.583083
$580$	0	0
$581$	2.27826	0.0945180
$582$	20.1807i	0.836517i
$583$	39.3052i	1.62785i
$584$	10.4709	0.433291
$585$	0	0
$586$	−4.19569	−0.173322
$587$	11.3751i	0.469501i	0.972056	+	0.234751i	$0.0754273\pi$
$587$	11.3751i	0.469501i	−0.972056	+	0.234751i	$0.924573\pi$
$588$	− 1.91557i	− 0.0789969i
$589$	−24.7354	−1.01921
$590$	0	0
$591$	23.7871	0.978473
$592$	− 36.5559i	− 1.50244i
$593$	− 30.4617i	− 1.25091i	−0.780259	−	0.625457i	$-0.784913\pi$
$593$	− 30.4617i	− 1.25091i	0.780259	−	0.625457i	$-0.215087\pi$
$594$	9.09523	0.373182
$595$	0	0
$596$	0.905161	0.0370769
$597$	15.9445i	0.652566i
$598$	10.7193i	0.438345i
$599$	−3.09493	−0.126455	−0.0632277	−	0.997999i	$-0.520139\pi$
$599$	−3.09493	−0.126455	−0.0632277	+	0.997999i	$0.520139\pi$
$600$	0	0
$601$	9.19481	0.375064	0.187532	−	0.982258i	$-0.439951\pi$
$601$	9.19481	0.375064	0.187532	+	0.982258i	$0.439951\pi$
$602$	10.2944i	0.419567i
$603$	11.3195i	0.460964i
$604$	−3.68426	−0.149911
$605$	0	0
$606$	9.76675	0.396747
$607$	− 37.9938i	− 1.54212i	−0.636762	−	0.771060i	$-0.719727\pi$
$607$	− 37.9938i	− 1.54212i	0.636762	−	0.771060i	$-0.280273\pi$
$608$	5.03766i	0.204304i
$609$	−1.40364	−0.0568784
$610$	0	0
$611$	−38.8019	−1.56976
$612$	− 0.151571i	− 0.00612691i
$613$	− 24.7188i	− 0.998381i	−0.866492	−	0.499191i	$-0.833631\pi$
$613$	− 24.7188i	− 0.998381i	0.866492	−	0.499191i	$-0.166369\pi$
$614$	24.6570	0.995075
$615$	0	0
$616$	−15.1575	−0.610712
$617$	− 29.8302i	− 1.20092i	−0.799655	−	0.600460i	$-0.794984\pi$
$617$	− 29.8302i	− 1.20092i	0.799655	−	0.600460i	$-0.205016\pi$
$618$	− 14.2867i	− 0.574695i
$619$	20.9427	0.841757	0.420878	−	0.907117i	$-0.361722\pi$
$619$	20.9427	0.841757	0.420878	+	0.907117i	$0.361722\pi$
$620$	0	0
$621$	−1.74696	−0.0701032
$622$	12.4867i	0.500673i
$623$	− 6.21268i	− 0.248906i
$624$	18.2769	0.731662
$625$	0	0
$626$	−30.4618	−1.21750
$627$	− 16.8645i	− 0.673503i
$628$	1.51920i	0.0606226i
$629$	3.83812	0.153036
$630$	0	0
$631$	18.1871	0.724017	0.362009	−	0.932175i	$-0.382091\pi$
$631$	18.1871	0.724017	0.362009	+	0.932175i	$0.382091\pi$
$632$	− 38.4261i	− 1.52851i
$633$	− 6.49016i	− 0.257961i
$634$	−9.93813	−0.394693
$635$	0	0
$636$	2.09446	0.0830507
$637$	24.2519i	0.960894i
$638$	12.8826i	0.510025i
$639$	5.43047	0.214826
$640$	0	0
$641$	3.94506	0.155821	0.0779103	−	0.996960i	$-0.475175\pi$
$641$	3.94506	0.155821	0.0779103	+	0.996960i	$0.475175\pi$
$642$	− 20.4192i	− 0.805883i
$643$	23.1026i	0.911077i	0.890216	+	0.455538i	$0.150553\pi$
$643$	23.1026i	0.911077i	−0.890216	+	0.455538i	$0.849447\pi$
$644$	−0.551063	−0.0217149
$645$	0	0
$646$	−2.04691	−0.0805345
$647$	9.35622i	0.367831i	0.982942	+	0.183916i	$0.0588773\pi$
$647$	9.35622i	0.367831i	−0.982942	+	0.183916i	$0.941123\pi$
$648$	2.56054i	0.100588i
$649$	−78.1095	−3.06607
$650$	0	0
$651$	−8.68243	−0.340291
$652$	− 0.950655i	− 0.0372305i
$653$	2.29702i	0.0898892i	0.998989	+	0.0449446i	$0.0143111\pi$
$653$	2.29702i	0.0898892i	−0.998989	+	0.0449446i	$0.985689\pi$
$654$	4.28723	0.167644
$655$	0	0
$656$	24.9576	0.974429
$657$	4.08935i	0.159541i
$658$	− 14.5281i	− 0.566365i
$659$	17.5619	0.684116	0.342058	−	0.939679i	$-0.388876\pi$
$659$	17.5619	0.684116	0.342058	+	0.939679i	$0.388876\pi$
$660$	0	0
$661$	−6.88170	−0.267667	−0.133834	−	0.991004i	$-0.542729\pi$
$661$	−6.88170	−0.267667	−0.133834	+	0.991004i	$0.542729\pi$
$662$	50.6300i	1.96779i
$663$	1.91895i	0.0745259i
$664$	5.88664	0.228446
$665$	0	0
$666$	12.2726	0.475554
$667$	− 2.47441i	− 0.0958097i
$668$	5.92390i	0.229202i
$669$	−10.7454	−0.415440
$670$	0	0
$671$	72.8842	2.81367
$672$	1.76828i	0.0682128i
$673$	− 12.4307i	− 0.479168i	−0.970876	−	0.239584i	$-0.922989\pi$
$673$	− 12.4307i	− 0.479168i	0.970876	−	0.239584i	$-0.0770110\pi$
$674$	25.8314	0.994989
$675$	0	0
$676$	1.03141	0.0396697
$677$	6.61696i	0.254310i	0.991883	+	0.127155i	$0.0405846\pi$
$677$	6.61696i	0.254310i	−0.991883	+	0.127155i	$0.959415\pi$
$678$	29.5768i	1.13589i
$679$	13.1346	0.504060
$680$	0	0
$681$	−6.84922	−0.262463
$682$	79.6870i	3.05137i
$683$	− 18.6885i	− 0.715097i	−0.933895	−	0.357549i	$-0.883613\pi$
$683$	− 18.6885i	− 0.715097i	0.933895	−	0.357549i	$-0.116387\pi$
$684$	−0.898661	−0.0343612
$685$	0	0
$686$	−19.6424	−0.749952
$687$	− 2.21379i	− 0.0844614i
$688$	30.9423i	1.17966i
$689$	−26.5166	−1.01020
$690$	0	0
$691$	−18.4352	−0.701306	−0.350653	−	0.936505i	$-0.614040\pi$
$691$	−18.4352	−0.701306	−0.350653	+	0.936505i	$0.614040\pi$
$692$	− 5.20684i	− 0.197934i
$693$	− 5.91964i	− 0.224868i
$694$	−30.6727	−1.16432
$695$	0	0
$696$	−3.62677	−0.137472
$697$	2.62038i	0.0992538i
$698$	45.4375i	1.71983i
$699$	9.39682	0.355420
$700$	0	0
$701$	14.9443	0.564437	0.282219	−	0.959350i	$-0.408930\pi$
$701$	14.9443	0.564437	0.282219	+	0.959350i	$0.408930\pi$
$702$	6.13596i	0.231587i
$703$	− 22.7561i	− 0.858261i
$704$	−37.9539	−1.43044
$705$	0	0
$706$	−1.49660	−0.0563253
$707$	− 6.35670i	− 0.239068i
$708$	4.16223i	0.156426i
$709$	−44.4198	−1.66822	−0.834110	−	0.551598i	$-0.814018\pi$
$709$	−44.4198	−1.66822	−0.834110	+	0.551598i	$0.814018\pi$
$710$	0	0
$711$	15.0070	0.562808
$712$	− 16.0525i	− 0.601594i
$713$	− 15.3059i	− 0.573209i
$714$	−0.718489	−0.0268888
$715$	0	0
$716$	−4.84678	−0.181133
$717$	0.737315i	0.0275355i
$718$	51.0070i	1.90356i
$719$	38.9409	1.45225	0.726125	−	0.687563i	$-0.241319\pi$
$719$	38.9409	1.45225	0.726125	+	0.687563i	$0.241319\pi$
$720$	0	0
$721$	−9.29850	−0.346294
$722$	− 16.7934i	− 0.624984i
$723$	− 4.21895i	− 0.156905i
$724$	1.09601	0.0407328
$725$	0	0
$726$	−37.5816	−1.39479
$727$	− 2.48415i	− 0.0921320i	−0.998938	−	0.0460660i	$-0.985332\pi$
$727$	− 2.48415i	− 0.0921320i	0.998938	−	0.0460660i	$-0.0146685\pi$
$728$	− 10.2258i	− 0.378992i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−3.24873	−0.120159
$732$	− 3.88379i	− 0.143549i
$733$	− 31.7905i	− 1.17421i	−0.809511	−	0.587105i	$-0.800267\pi$
$733$	− 31.7905i	− 1.17421i	0.809511	−	0.587105i	$-0.199733\pi$
$734$	0.537172	0.0198274
$735$	0	0
$736$	−3.11722	−0.114902
$737$	− 67.6166i	− 2.49069i
$738$	8.37881i	0.308428i
$739$	32.4392	1.19329	0.596647	−	0.802504i	$-0.296499\pi$
$739$	32.4392	1.19329	0.596647	+	0.802504i	$0.296499\pi$
$740$	0	0
$741$	11.3774	0.417958
$742$	− 9.92829i	− 0.364479i
$743$	− 2.02921i	− 0.0744447i	−0.999307	−	0.0372223i	$-0.988149\pi$
$743$	− 2.02921i	− 0.0744447i	0.999307	−	0.0372223i	$-0.0118510\pi$
$744$	−22.4339	−0.822468
$745$	0	0
$746$	36.9704	1.35358
$747$	2.29898i	0.0841154i
$748$	0.905410i	0.0331051i
$749$	−13.2899	−0.485602
$750$	0	0
$751$	−31.8934	−1.16381	−0.581904	−	0.813257i	$-0.697692\pi$
$751$	−31.8934	−1.16381	−0.581904	+	0.813257i	$0.697692\pi$
$752$	− 43.6679i	− 1.59241i
$753$	− 9.82121i	− 0.357905i
$754$	−8.69102	−0.316508
$755$	0	0
$756$	−0.315440	−0.0114725
$757$	− 39.2724i	− 1.42738i	−0.700463	−	0.713689i	$-0.747023\pi$
$757$	− 39.2724i	− 1.42738i	0.700463	−	0.713689i	$-0.252977\pi$
$758$	55.4905i	2.01551i
$759$	10.4355	0.378783
$760$	0	0
$761$	14.2422	0.516278	0.258139	−	0.966108i	$-0.416891\pi$
$761$	14.2422	0.516278	0.258139	+	0.966108i	$0.416891\pi$
$762$	9.78067i	0.354316i
$763$	− 2.79034i	− 0.101017i
$764$	−4.83042	−0.174758
$765$	0	0
$766$	−10.0987	−0.364879
$767$	− 52.6954i	− 1.90272i
$768$	7.45620i	0.269052i
$769$	6.86409	0.247526	0.123763	−	0.992312i	$-0.460504\pi$
$769$	6.86409	0.247526	0.123763	+	0.992312i	$0.460504\pi$
$770$	0	0
$771$	0.0383881	0.00138251
$772$	− 4.46601i	− 0.160735i
$773$	− 8.46681i	− 0.304530i	−0.988340	−	0.152265i	$-0.951343\pi$
$773$	− 8.46681i	− 0.304530i	0.988340	−	0.152265i	$-0.0486567\pi$
$774$	−10.3880	−0.373389
$775$	0	0
$776$	33.9377	1.21829
$777$	− 7.98764i	− 0.286555i
$778$	− 4.47792i	− 0.160541i
$779$	15.5361	0.556638
$780$	0	0
$781$	−32.4389	−1.16075
$782$	− 1.26659i	− 0.0452932i
$783$	− 1.41641i	− 0.0506183i
$784$	−27.2932	−0.974757
$785$	0	0
$786$	11.1158	0.396489
$787$	14.9960i	0.534551i	0.963620	+	0.267275i	$0.0861233\pi$
$787$	14.9960i	0.534551i	−0.963620	+	0.267275i	$0.913877\pi$
$788$	7.57169i	0.269730i
$789$	14.2706	0.508048
$790$	0	0
$791$	19.2501	0.684455
$792$	− 15.2954i	− 0.543497i
$793$	49.1703i	1.74609i
$794$	−14.3800	−0.510326
$795$	0	0
$796$	−5.07530	−0.179889
$797$	10.9127i	0.386549i	0.981145	+	0.193274i	$0.0619107\pi$
$797$	10.9127i	0.386549i	−0.981145	+	0.193274i	$0.938089\pi$
$798$	4.25989i	0.150798i
$799$	4.58483	0.162200
$800$	0	0
$801$	6.26920	0.221511
$802$	37.1465i	1.31169i
$803$	− 24.4277i	− 0.862033i
$804$	−3.60309	−0.127071
$805$	0	0
$806$	−53.7597	−1.89360
$807$	− 21.4090i	− 0.753633i
$808$	− 16.4246i	− 0.577817i
$809$	−13.8729	−0.487746	−0.243873	−	0.969807i	$-0.578418\pi$
$809$	−13.8729	−0.487746	−0.243873	+	0.969807i	$0.578418\pi$
$810$	0	0
$811$	13.1036	0.460129	0.230065	−	0.973175i	$-0.426106\pi$
$811$	13.1036	0.460129	0.230065	+	0.973175i	$0.426106\pi$
$812$	− 0.446793i	− 0.0156793i
$813$	6.22804i	0.218427i
$814$	−73.3103	−2.56953
$815$	0	0
$816$	−2.15960	−0.0756011
$817$	19.2616i	0.673878i
$818$	− 20.7184i	− 0.724403i
$819$	3.99359	0.139547
$820$	0	0
$821$	15.5589	0.543011	0.271505	−	0.962437i	$-0.412479\pi$
$821$	15.5589	0.543011	0.271505	+	0.962437i	$0.412479\pi$
$822$	− 11.9503i	− 0.416814i
$823$	45.2205i	1.57629i	0.615491	+	0.788144i	$0.288958\pi$
$823$	45.2205i	1.57629i	−0.615491	+	0.788144i	$0.711042\pi$
$824$	−24.0258	−0.836978
$825$	0	0
$826$	19.7301	0.686497
$827$	25.6349i	0.891411i	0.895180	+	0.445706i	$0.147047\pi$
$827$	25.6349i	0.891411i	−0.895180	+	0.445706i	$0.852953\pi$
$828$	− 0.556076i	− 0.0193250i
$829$	−14.6302	−0.508129	−0.254064	−	0.967187i	$-0.581768\pi$
$829$	−14.6302	−0.508129	−0.254064	+	0.967187i	$0.581768\pi$
$830$	0	0
$831$	−14.0431	−0.487150
$832$	− 25.6050i	− 0.887695i
$833$	− 2.86560i	− 0.0992872i
$834$	−25.9624	−0.899004
$835$	0	0
$836$	5.36814	0.185661
$837$	− 8.76141i	− 0.302839i
$838$	− 38.1645i	− 1.31837i
$839$	−37.4349	−1.29240	−0.646199	−	0.763169i	$-0.723642\pi$
$839$	−37.4349	−1.29240	−0.646199	+	0.763169i	$0.723642\pi$
$840$	0	0
$841$	−26.9938	−0.930820
$842$	− 8.46684i	− 0.291787i
$843$	− 0.516993i	− 0.0178062i
$844$	2.06588	0.0711106
$845$	0	0
$846$	14.6603	0.504031
$847$	24.4600i	0.840457i
$848$	− 29.8420i	− 1.02478i
$849$	−0.977154	−0.0335358
$850$	0	0
$851$	14.0811	0.482693
$852$	1.72857i	0.0592200i
$853$	− 4.45661i	− 0.152591i	−0.997085	−	0.0762956i	$-0.975691\pi$
$853$	− 4.45661i	− 0.152591i	0.997085	−	0.0762956i	$-0.0243093\pi$
$854$	−18.4102	−0.629984
$855$	0	0
$856$	−34.3388	−1.17368
$857$	− 48.0738i	− 1.64217i	−0.570807	−	0.821084i	$-0.693370\pi$
$857$	− 48.0738i	− 1.64217i	0.570807	−	0.821084i	$-0.306630\pi$
$858$	− 36.6531i	− 1.25132i
$859$	40.5317	1.38292	0.691461	−	0.722414i	$-0.256967\pi$
$859$	40.5317	1.38292	0.691461	+	0.722414i	$0.256967\pi$
$860$	0	0
$861$	5.45335	0.185850
$862$	36.9381i	1.25812i
$863$	7.99001i	0.271983i	0.990710	+	0.135992i	$0.0434220\pi$
$863$	7.99001i	0.271983i	−0.990710	+	0.135992i	$0.956578\pi$
$864$	−1.78436	−0.0607053
$865$	0	0
$866$	−59.1467	−2.00989
$867$	16.7733i	0.569650i
$868$	− 2.76370i	− 0.0938062i
$869$	−89.6444	−3.04098
$870$	0	0
$871$	45.6165	1.54566
$872$	− 7.20979i	− 0.244154i
$873$	13.2541i	0.448583i
$874$	−7.50957	−0.254015
$875$	0	0
$876$	−1.30168	−0.0439797
$877$	− 30.7432i	− 1.03812i	−0.854737	−	0.519062i	$-0.826281\pi$
$877$	− 30.7432i	− 1.03812i	0.854737	−	0.519062i	$-0.173719\pi$
$878$	14.0518i	0.474224i
$879$	−2.75561	−0.0929444
$880$	0	0
$881$	−35.1508	−1.18426	−0.592130	−	0.805843i	$-0.701713\pi$
$881$	−35.1508	−1.18426	−0.592130	+	0.805843i	$0.701713\pi$
$882$	− 9.16293i	− 0.308532i
$883$	− 35.8185i	− 1.20539i	−0.797972	−	0.602694i	$-0.794094\pi$
$883$	− 35.8185i	− 1.20539i	0.797972	−	0.602694i	$-0.205906\pi$
$884$	−0.610821	−0.0205441
$885$	0	0
$886$	8.07877	0.271412
$887$	30.3154i	1.01789i	0.860798	+	0.508946i	$0.169965\pi$
$887$	30.3154i	1.01789i	−0.860798	+	0.508946i	$0.830035\pi$
$888$	− 20.6387i	− 0.692591i
$889$	6.36575	0.213501
$890$	0	0
$891$	5.97349	0.200119
$892$	− 3.42035i	− 0.114522i
$893$	− 27.1833i	− 0.909654i
$894$	4.32974	0.144808
$895$	0	0
$896$	13.1235	0.438426
$897$	7.04013i	0.235063i
$898$	33.8320i	1.12899i
$899$	12.4097	0.413888
$900$	0	0
$901$	3.13321	0.104382
$902$	− 50.0507i	− 1.66651i
$903$	6.76104i	0.224993i
$904$	49.7390	1.65430
$905$	0	0
$906$	−17.6233	−0.585493
$907$	6.79755i	0.225709i	0.993612	+	0.112854i	$0.0359994\pi$
$907$	6.79755i	0.225709i	−0.993612	+	0.112854i	$0.964001\pi$
$908$	− 2.18018i	− 0.0723517i
$909$	6.41452	0.212756
$910$	0	0
$911$	−34.3310	−1.13744	−0.568719	−	0.822532i	$-0.692561\pi$
$911$	−34.3310	−1.13744	−0.568719	+	0.822532i	$0.692561\pi$
$912$	12.8042i	0.423988i
$913$	− 13.7329i	− 0.454494i
$914$	−10.3104	−0.341037
$915$	0	0
$916$	0.704672	0.0232830
$917$	− 7.23475i	− 0.238913i
$918$	− 0.725025i	− 0.0239294i
$919$	−6.72902	−0.221970	−0.110985	−	0.993822i	$-0.535401\pi$
$919$	−6.72902	−0.221970	−0.110985	+	0.993822i	$0.535401\pi$
$920$	0	0
$921$	16.1940	0.533611
$922$	51.6679i	1.70159i
$923$	− 21.8844i	− 0.720333i
$924$	1.88428	0.0619883
$925$	0	0
$926$	43.9874	1.44551
$927$	− 9.38309i	− 0.308181i
$928$	− 2.52739i	− 0.0829656i
$929$	4.70020	0.154209	0.0771043	−	0.997023i	$-0.475433\pi$
$929$	4.70020	0.154209	0.0771043	+	0.997023i	$0.475433\pi$
$930$	0	0
$931$	−16.9900	−0.556825
$932$	2.99110i	0.0979768i
$933$	8.20094i	0.268487i
$934$	−47.5203	−1.55491
$935$	0	0
$936$	10.3188	0.337280
$937$	− 1.41847i	− 0.0463395i	−0.999732	−	0.0231698i	$-0.992624\pi$
$937$	− 1.41847i	− 0.0463395i	0.999732	−	0.0231698i	$-0.00737582\pi$
$938$	17.0796i	0.557669i
$939$	−20.0065	−0.652886
$940$	0	0
$941$	−48.7929	−1.59060	−0.795302	−	0.606213i	$-0.792688\pi$
$941$	−48.7929	−1.59060	−0.795302	+	0.606213i	$0.792688\pi$
$942$	7.26691i	0.236769i
$943$	9.61347i	0.313058i
$944$	59.3037	1.93017
$945$	0	0
$946$	62.0526	2.01751
$947$	38.1885i	1.24096i	0.784222	+	0.620480i	$0.213062\pi$
$947$	38.1885i	1.24096i	−0.784222	+	0.620480i	$0.786938\pi$
$948$	4.77689i	0.155146i
$949$	16.4798	0.534955
$950$	0	0
$951$	−6.52708	−0.211655
$952$	1.20828i	0.0391604i
$953$	− 8.25677i	− 0.267463i	−0.991018	−	0.133731i	$-0.957304\pi$
$953$	− 8.25677i	− 0.267463i	0.991018	−	0.133731i	$-0.0426960\pi$
$954$	10.0186	0.324365
$955$	0	0
$956$	−0.234695	−0.00759057
$957$	8.46090i	0.273502i
$958$	22.3183i	0.721070i
$959$	−7.77785	−0.251160
$960$	0	0
$961$	45.7623	1.47620
$962$	− 49.4577i	− 1.59458i
$963$	− 13.4108i	− 0.432156i
$964$	1.34293	0.0432530
$965$	0	0
$966$	−2.63595	−0.0848102
$967$	− 43.3033i	− 1.39254i	−0.717780	−	0.696270i	$-0.754842\pi$
$967$	− 43.3033i	− 1.39254i	0.717780	−	0.696270i	$-0.245158\pi$
$968$	63.2007i	2.03135i
$969$	−1.34435	−0.0431868
$970$	0	0
$971$	−35.1452	−1.12786	−0.563932	−	0.825821i	$-0.690712\pi$
$971$	−35.1452	−1.12786	−0.563932	+	0.825821i	$0.690712\pi$
$972$	− 0.318310i	− 0.0102098i
$973$	16.8976i	0.541713i
$974$	8.46364	0.271192
$975$	0	0
$976$	−55.3365	−1.77128
$977$	− 40.4411i	− 1.29383i	−0.762563	−	0.646913i	$-0.776059\pi$
$977$	− 40.4411i	− 1.29383i	0.762563	−	0.646913i	$-0.223941\pi$
$978$	− 4.54735i	− 0.145408i
$979$	−37.4490	−1.19687
$980$	0	0
$981$	2.81573	0.0898993
$982$	− 26.5835i	− 0.848312i
$983$	− 58.9585i	− 1.88048i	−0.340509	−	0.940241i	$-0.610599\pi$
$983$	− 58.9585i	− 1.88048i	0.340509	−	0.940241i	$-0.389401\pi$
$984$	14.0906	0.449190
$985$	0	0
$986$	1.02693	0.0327042
$987$	− 9.54165i	− 0.303714i
$988$	3.62153i	0.115216i
$989$	−11.9187	−0.378994
$990$	0	0
$991$	11.0145	0.349888	0.174944	−	0.984578i	$-0.444026\pi$
$991$	11.0145	0.349888	0.174944	+	0.984578i	$0.444026\pi$
$992$	− 15.6335i	− 0.496365i
$993$	33.2523i	1.05523i
$994$	8.19390	0.259895
$995$	0	0
$996$	−0.731789	−0.0231876
$997$	− 52.8873i	− 1.67496i	−0.546469	−	0.837479i	$-0.684029\pi$
$997$	− 52.8873i	− 1.67496i	0.546469	−	0.837479i	$-0.315971\pi$
$998$	− 24.8570i	− 0.786836i
$999$	8.06031	0.255017

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.g.1249.13		16
5.2	odd	4		1875.2.a.o.1.2	yes	8
5.3	odd	4		1875.2.a.n.1.7	✓	8
5.4	even	2	inner	1875.2.b.g.1249.4		16
15.2	even	4		5625.2.a.u.1.7		8
15.8	even	4		5625.2.a.bc.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.7	✓	8	5.3	odd	4
1875.2.a.o.1.2	yes	8	5.2	odd	4
1875.2.b.g.1249.4		16	5.4	even	2	inner
1875.2.b.g.1249.13		16	1.1	even	1	trivial
5625.2.a.u.1.7		8	15.2	even	4
5625.2.a.bc.1.2		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)

\(f(q)\)	\(=\)	\(q+1.52260i q^{2} +1.00000i q^{3} -0.318310 q^{4} -1.52260 q^{6} +0.990985i q^{7} +2.56054i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.52260i q^{2} +1.00000i q^{3} -0.318310 q^{4} -1.52260 q^{6} +0.990985i q^{7} +2.56054i q^{8} -1.00000 q^{9} +5.97349 q^{11} -0.318310i q^{12} +4.02992i q^{13} -1.50887 q^{14} -4.53530 q^{16} -0.476176i q^{17} -1.52260i q^{18} -2.82322 q^{19} -0.990985 q^{21} +9.09523i q^{22} -1.74696i q^{23} -2.56054 q^{24} -6.13596 q^{26} -1.00000i q^{27} -0.315440i q^{28} +1.41641 q^{29} +8.76141 q^{31} -1.78436i q^{32} +5.97349i q^{33} +0.725025 q^{34} +0.318310 q^{36} +8.06031i q^{37} -4.29864i q^{38} -4.02992 q^{39} -5.50296 q^{41} -1.50887i q^{42} -6.82255i q^{43} -1.90142 q^{44} +2.65993 q^{46} +9.62845i q^{47} -4.53530i q^{48} +6.01795 q^{49} +0.476176 q^{51} -1.28276i q^{52} +6.57994i q^{53} +1.52260 q^{54} -2.53746 q^{56} -2.82322i q^{57} +2.15662i q^{58} -13.0760 q^{59} +12.2013 q^{61} +13.3401i q^{62} -0.990985i q^{63} -6.35373 q^{64} -9.09523 q^{66} -11.3195i q^{67} +0.151571i q^{68} +1.74696 q^{69} -5.43047 q^{71} -2.56054i q^{72} -4.08935i q^{73} -12.2726 q^{74} +0.898661 q^{76} +5.91964i q^{77} -6.13596i q^{78} -15.0070 q^{79} +1.00000 q^{81} -8.37881i q^{82} -2.29898i q^{83} +0.315440 q^{84} +10.3880 q^{86} +1.41641i q^{87} +15.2954i q^{88} -6.26920 q^{89} -3.99359 q^{91} +0.556076i q^{92} +8.76141i q^{93} -14.6603 q^{94} +1.78436 q^{96} -13.2541i q^{97} +9.16293i q^{98} -5.97349 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) 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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