LMFDB - Embedded newform 475.2.b.b.324.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1827904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 14x^{2} + 1 $ x^6 + 9x^4 + 14x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.5
Root			$1.37720i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.b.324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.65109i q^{2} +2.37720i q^{3} -0.726109 q^{4} -3.92498 q^{6} +0.377203i q^{7} +2.10331i q^{8} -2.65109 q^{9} +O(q^{10})$	\(q+1.65109i q^{2} +2.37720i q^{3} -0.726109 q^{4} -3.92498 q^{6} +0.377203i q^{7} +2.10331i q^{8} -2.65109 q^{9} -1.37720 q^{11} -1.72611i q^{12} +2.82167i q^{13} -0.622797 q^{14} -4.92498 q^{16} -6.37720i q^{17} -4.37720i q^{18} -1.00000 q^{19} -0.896688 q^{21} -2.27389i q^{22} +6.19887i q^{23} -5.00000 q^{24} -4.65884 q^{26} +0.829422i q^{27} -0.273891i q^{28} +3.37720 q^{29} +2.48052 q^{31} -3.92498i q^{32} -3.27389i q^{33} +10.5294 q^{34} +1.92498 q^{36} -5.58383i q^{37} -1.65109i q^{38} -6.70769 q^{39} +8.50106 q^{41} -1.48052i q^{42} -12.1522i q^{43} +1.00000 q^{44} -10.2349 q^{46} +6.87826i q^{47} -11.7077i q^{48} +6.85772 q^{49} +15.1599 q^{51} -2.04884i q^{52} +11.5478i q^{53} -1.36945 q^{54} -0.793375 q^{56} -2.37720i q^{57} +5.57608i q^{58} -6.05659 q^{59} +5.02830 q^{61} +4.09556i q^{62} -1.00000i q^{63} -3.36945 q^{64} +5.40550 q^{66} -3.22717i q^{67} +4.63055i q^{68} -14.7360 q^{69} -2.30219 q^{71} -5.57608i q^{72} -3.19887i q^{73} +9.21942 q^{74} +0.726109 q^{76} -0.519485i q^{77} -11.0750i q^{78} +6.71836 q^{79} -9.92498 q^{81} +14.0360i q^{82} +18.2165i q^{83} +0.651093 q^{84} +20.0643 q^{86} +8.02830i q^{87} -2.89669i q^{88} -1.50106 q^{89} -1.06434 q^{91} -4.50106i q^{92} +5.89669i q^{93} -11.3567 q^{94} +9.33048 q^{96} +11.7827i q^{97} +11.3227i q^{98} +3.65109 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10}) $ 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9	$ 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46} + 14 q^{49} + 36 q^{51} + 10 q^{54} - 18 q^{56} + 12 q^{59} + 6 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} + 14 q^{71} + 2 q^{74} + 8 q^{76} + 36 q^{79} - 42 q^{81} - 10 q^{84} + 80 q^{86} + 40 q^{89} + 34 q^{91} - 90 q^{94} + 4 q^{96} + 8 q^{99}+O(q^{100}) $ 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9 + 2 * q^11 - 14 * q^14 - 12 * q^16 - 6 * q^19 - 12 * q^21 - 30 * q^24 - 22 * q^26 + 10 * q^29 - 2 * q^31 - 10 * q^34 - 6 * q^36 + 22 * q^39 + 2 * q^41 + 6 * q^44 - 24 * q^46 + 14 * q^49 + 36 * q^51 + 10 * q^54 - 18 * q^56 + 12 * q^59 + 6 * q^61 - 2 * q^64 - 2 * q^66 - 2 * q^69 + 14 * q^71 + 2 * q^74 + 8 * q^76 + 36 * q^79 - 42 * q^81 - 10 * q^84 + 80 * q^86 + 40 * q^89 + 34 * q^91 - 90 * q^94 + 4 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.65109i	1.16750i	0.811934	+	0.583750i	$0.198415\pi$
$2$	1.65109i	1.16750i	−0.811934	+	0.583750i	$0.801585\pi$
$3$	2.37720i	1.37248i	0.727376	+	0.686239i	$0.240740\pi$
$3$	2.37720i	1.37248i	−0.727376	+	0.686239i	$0.759260\pi$
$4$	−0.726109	−0.363055
$5$	0	0
$5$	0	0
$6$	−3.92498	−1.60237
$7$	0.377203i	0.142569i	0.997456	+	0.0712846i	$0.0227099\pi$
$7$	0.377203i	0.142569i	−0.997456	+	0.0712846i	$0.977290\pi$
$8$	2.10331i	0.743633i
$9$	−2.65109	−0.883698
$10$	0	0
$11$	−1.37720	−0.415242	−0.207621	−	0.978209i	$-0.566572\pi$
$11$	−1.37720	−0.415242	−0.207621	+	0.978209i	$0.566572\pi$
$12$	− 1.72611i	− 0.498285i
$13$	2.82167i	0.782591i	0.920265	+	0.391295i	$0.127973\pi$
$13$	2.82167i	0.782591i	−0.920265	+	0.391295i	$0.872027\pi$
$14$	−0.622797	−0.166450
$15$	0	0
$16$	−4.92498	−1.23125
$17$	− 6.37720i	− 1.54670i	−0.633980	−	0.773349i	$-0.718580\pi$
$17$	− 6.37720i	− 1.54670i	0.633980	−	0.773349i	$-0.281420\pi$
$18$	− 4.37720i	− 1.03172i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.896688	−0.195673
$22$	− 2.27389i	− 0.484795i
$23$	6.19887i	1.29255i	0.763103	+	0.646277i	$0.223675\pi$
$23$	6.19887i	1.29255i	−0.763103	+	0.646277i	$0.776325\pi$
$24$	−5.00000	−1.02062
$25$	0	0
$26$	−4.65884	−0.913674
$27$	0.829422i	0.159622i
$28$	− 0.273891i	− 0.0517604i
$29$	3.37720	0.627131	0.313565	−	0.949567i	$-0.398476\pi$
$29$	3.37720	0.627131	0.313565	+	0.949567i	$0.398476\pi$
$30$	0	0
$31$	2.48052	0.445514	0.222757	−	0.974874i	$-0.428494\pi$
$31$	2.48052	0.445514	0.222757	+	0.974874i	$0.428494\pi$
$32$	− 3.92498i	− 0.693846i
$33$	− 3.27389i	− 0.569911i
$34$	10.5294	1.80577
$35$	0	0
$36$	1.92498	0.320831
$37$	− 5.58383i	− 0.917976i	−0.888443	−	0.458988i	$-0.848212\pi$
$37$	− 5.58383i	− 0.917976i	0.888443	−	0.458988i	$-0.151788\pi$
$38$	− 1.65109i	− 0.267843i
$39$	−6.70769	−1.07409
$40$	0	0
$41$	8.50106	1.32764	0.663821	−	0.747891i	$-0.268934\pi$
$41$	8.50106	1.32764	0.663821	+	0.747891i	$0.268934\pi$
$42$	− 1.48052i	− 0.228448i
$43$	− 12.1522i	− 1.85319i	−0.376065	−	0.926593i	$-0.622723\pi$
$43$	− 12.1522i	− 1.85319i	0.376065	−	0.926593i	$-0.377277\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−10.2349	−1.50906
$47$	6.87826i	1.00330i	0.865071	+	0.501649i	$0.167273\pi$
$47$	6.87826i	1.00330i	−0.865071	+	0.501649i	$0.832727\pi$
$48$	− 11.7077i	− 1.68986i
$49$	6.85772	0.979674
$50$	0	0
$51$	15.1599	2.12281
$52$	− 2.04884i	− 0.284123i
$53$	11.5478i	1.58621i	0.609085	+	0.793105i	$0.291537\pi$
$53$	11.5478i	1.58621i	−0.609085	+	0.793105i	$0.708463\pi$
$54$	−1.36945	−0.186359
$55$	0	0
$56$	−0.793375	−0.106019
$57$	− 2.37720i	− 0.314868i
$58$	5.57608i	0.732175i
$59$	−6.05659	−0.788501	−0.394251	−	0.919003i	$-0.628996\pi$
$59$	−6.05659	−0.788501	−0.394251	+	0.919003i	$0.628996\pi$
$60$	0	0
$61$	5.02830	0.643807	0.321904	−	0.946772i	$-0.395677\pi$
$61$	5.02830	0.643807	0.321904	+	0.946772i	$0.395677\pi$
$62$	4.09556i	0.520137i
$63$	− 1.00000i	− 0.125988i
$64$	−3.36945	−0.421182
$65$	0	0
$66$	5.40550	0.665371
$67$	− 3.22717i	− 0.394262i	−0.980377	−	0.197131i	$-0.936838\pi$
$67$	− 3.22717i	− 0.394262i	0.980377	−	0.197131i	$-0.0631624\pi$
$68$	4.63055i	0.561536i
$69$	−14.7360	−1.77400
$70$	0	0
$71$	−2.30219	−0.273219	−0.136610	−	0.990625i	$-0.543621\pi$
$71$	−2.30219	−0.273219	−0.136610	+	0.990625i	$0.543621\pi$
$72$	− 5.57608i	− 0.657147i
$73$	− 3.19887i	− 0.374400i	−0.982322	−	0.187200i	$-0.940059\pi$
$73$	− 3.19887i	− 0.374400i	0.982322	−	0.187200i	$-0.0599412\pi$
$74$	9.21942	1.07174
$75$	0	0
$76$	0.726109	0.0832905
$77$	− 0.519485i	− 0.0592008i
$78$	− 11.0750i	− 1.25400i
$79$	6.71836	0.755874	0.377937	−	0.925831i	$-0.376634\pi$
$79$	6.71836	0.755874	0.377937	+	0.925831i	$0.376634\pi$
$80$	0	0
$81$	−9.92498	−1.10278
$82$	14.0360i	1.55002i
$83$	18.2165i	1.99952i	0.0218996	+	0.999760i	$0.493029\pi$
$83$	18.2165i	1.99952i	−0.0218996	+	0.999760i	$0.506971\pi$
$84$	0.651093	0.0710401
$85$	0	0
$86$	20.0643	2.16359
$87$	8.02830i	0.860724i
$88$	− 2.89669i	− 0.308788i
$89$	−1.50106	−0.159112	−0.0795561	−	0.996830i	$-0.525350\pi$
$89$	−1.50106	−0.159112	−0.0795561	+	0.996830i	$0.525350\pi$
$90$	0	0
$91$	−1.06434	−0.111573
$92$	− 4.50106i	− 0.469268i
$93$	5.89669i	0.611458i
$94$	−11.3567	−1.17135
$95$	0	0
$96$	9.33048	0.952288
$97$	11.7827i	1.19635i	0.801365	+	0.598176i	$0.204108\pi$
$97$	11.7827i	1.19635i	−0.801365	+	0.598176i	$0.795892\pi$
$98$	11.3227i	1.14377i
$99$	3.65109	0.366949
$100$	0	0
$101$	−9.18820	−0.914260	−0.457130	−	0.889400i	$-0.651123\pi$
$101$	−9.18820	−0.914260	−0.457130	+	0.889400i	$0.651123\pi$
$102$	25.0304i	2.47838i
$103$	− 9.47277i	− 0.933379i	−0.884421	−	0.466690i	$-0.845447\pi$
$103$	− 9.47277i	− 0.933379i	0.884421	−	0.466690i	$-0.154553\pi$
$104$	−5.93486	−0.581961
$105$	0	0
$106$	−19.0665	−1.85190
$107$	15.3510i	1.48404i	0.670378	+	0.742020i	$0.266132\pi$
$107$	15.3510i	1.48404i	−0.670378	+	0.742020i	$0.733868\pi$
$108$	− 0.602251i	− 0.0579516i
$109$	16.7643	1.60573	0.802863	−	0.596163i	$-0.203309\pi$
$109$	16.7643	1.60573	0.802863	+	0.596163i	$0.203309\pi$
$110$	0	0
$111$	13.2739	1.25990
$112$	− 1.85772i	− 0.175538i
$113$	− 13.3305i	− 1.25403i	−0.779009	−	0.627013i	$-0.784277\pi$
$113$	− 13.3305i	− 1.25403i	0.779009	−	0.627013i	$-0.215723\pi$
$114$	3.92498	0.367608
$115$	0	0
$116$	−2.45222	−0.227683
$117$	− 7.48052i	− 0.691574i
$118$	− 10.0000i	− 0.920575i
$119$	2.40550	0.220512
$120$	0	0
$121$	−9.10331	−0.827574
$122$	8.30219i	0.751645i
$123$	20.2087i	1.82216i
$124$	−1.80113	−0.161746
$125$	0	0
$126$	1.65109	0.147091
$127$	3.04672i	0.270353i	0.990822	+	0.135176i	$0.0431601\pi$
$127$	3.04672i	0.270353i	−0.990822	+	0.135176i	$0.956840\pi$
$128$	− 13.4132i	− 1.18557i
$129$	28.8881	2.54346
$130$	0	0
$131$	−8.70769	−0.760794	−0.380397	−	0.924823i	$-0.624213\pi$
$131$	−8.70769	−0.760794	−0.380397	+	0.924823i	$0.624213\pi$
$132$	2.37720i	0.206909i
$133$	− 0.377203i	− 0.0327076i
$134$	5.32836	0.460300
$135$	0	0
$136$	13.4132	1.15018
$137$	− 6.59450i	− 0.563406i	−0.959502	−	0.281703i	$-0.909101\pi$
$137$	− 6.59450i	− 0.563406i	0.959502	−	0.281703i	$-0.0908993\pi$
$138$	− 24.3305i	− 2.07115i
$139$	10.1677	0.862409	0.431205	−	0.902254i	$-0.358089\pi$
$139$	10.1677	0.862409	0.431205	+	0.902254i	$0.358089\pi$
$140$	0	0
$141$	−16.3510	−1.37701
$142$	− 3.80113i	− 0.318983i
$143$	− 3.88601i	− 0.324965i
$144$	13.0566	1.08805
$145$	0	0
$146$	5.28164	0.437112
$147$	16.3022i	1.34458i
$148$	4.05447i	0.333275i
$149$	5.54003	0.453857	0.226929	−	0.973911i	$-0.427132\pi$
$149$	5.54003	0.453857	0.226929	+	0.973911i	$0.427132\pi$
$150$	0	0
$151$	5.12386	0.416974	0.208487	−	0.978025i	$-0.433146\pi$
$151$	5.12386	0.416974	0.208487	+	0.978025i	$0.433146\pi$
$152$	− 2.10331i	− 0.170601i
$153$	16.9066i	1.36681i
$154$	0.857718	0.0691169
$155$	0	0
$156$	4.87051	0.389953
$157$	− 19.7643i	− 1.57736i	−0.614803	−	0.788681i	$-0.710765\pi$
$157$	− 19.7643i	− 1.57736i	0.614803	−	0.788681i	$-0.289235\pi$
$158$	11.0926i	0.882483i
$159$	−27.4514	−2.17704
$160$	0	0
$161$	−2.33823	−0.184279
$162$	− 16.3871i	− 1.28749i
$163$	13.5195i	1.05893i	0.848332	+	0.529464i	$0.177607\pi$
$163$	13.5195i	1.05893i	−0.848332	+	0.529464i	$0.822393\pi$
$164$	−6.17270	−0.482007
$165$	0	0
$166$	−30.0771	−2.33444
$167$	− 13.1054i	− 1.01413i	−0.861908	−	0.507065i	$-0.830731\pi$
$167$	− 13.1054i	− 1.01413i	0.861908	−	0.507065i	$-0.169269\pi$
$168$	− 1.88601i	− 0.145509i
$169$	5.03817	0.387551
$170$	0	0
$171$	2.65109	0.202734
$172$	8.82379i	0.672808i
$173$	1.48827i	0.113151i	0.998398	+	0.0565754i	$0.0180181\pi$
$173$	1.48827i	0.113151i	−0.998398	+	0.0565754i	$0.981982\pi$
$174$	−13.2555	−1.00489
$175$	0	0
$176$	6.78270	0.511265
$177$	− 14.3977i	− 1.08220i
$178$	− 2.47839i	− 0.185763i
$179$	17.1132	1.27910	0.639550	−	0.768750i	$-0.279121\pi$
$179$	17.1132	1.27910	0.639550	+	0.768750i	$0.279121\pi$
$180$	0	0
$181$	−5.73891	−0.426569	−0.213285	−	0.976990i	$-0.568416\pi$
$181$	−5.73891	−0.426569	−0.213285	+	0.976990i	$0.568416\pi$
$182$	− 1.75733i	− 0.130262i
$183$	11.9533i	0.883612i
$184$	−13.0382	−0.961187
$185$	0	0
$186$	−9.73598	−0.713877
$187$	8.78270i	0.642255i
$188$	− 4.99437i	− 0.364252i
$189$	−0.312860	−0.0227572
$190$	0	0
$191$	22.3948	1.62043	0.810216	−	0.586131i	$-0.199350\pi$
$191$	22.3948	1.62043	0.810216	+	0.586131i	$0.199350\pi$
$192$	− 8.00987i	− 0.578063i
$193$	− 21.3687i	− 1.53815i	−0.639159	−	0.769075i	$-0.720717\pi$
$193$	− 21.3687i	− 1.53815i	0.639159	−	0.769075i	$-0.279283\pi$
$194$	−19.4543	−1.39674
$195$	0	0
$196$	−4.97945	−0.355675
$197$	− 1.11399i	− 0.0793682i	−0.999212	−	0.0396841i	$-0.987365\pi$
$197$	− 1.11399i	− 0.0793682i	0.999212	−	0.0396841i	$-0.0126352\pi$
$198$	6.02830i	0.428412i
$199$	2.22505	0.157729	0.0788647	−	0.996885i	$-0.474870\pi$
$199$	2.22505	0.157729	0.0788647	+	0.996885i	$0.474870\pi$
$200$	0	0
$201$	7.67164	0.541116
$202$	− 15.1706i	− 1.06740i
$203$	1.27389i	0.0894096i
$204$	−11.0078	−0.770697
$205$	0	0
$206$	15.6404	1.08972
$207$	− 16.4338i	− 1.14223i
$208$	− 13.8967i	− 0.963562i
$209$	1.37720	0.0952631
$210$	0	0
$211$	−9.75441	−0.671521	−0.335760	−	0.941947i	$-0.608993\pi$
$211$	−9.75441	−0.671521	−0.335760	+	0.941947i	$0.608993\pi$
$212$	− 8.38495i	− 0.575881i
$213$	− 5.47277i	− 0.374988i
$214$	−25.3460	−1.73262
$215$	0	0
$216$	−1.74453	−0.118700
$217$	0.935657i	0.0635166i
$218$	27.6794i	1.87468i
$219$	7.60437	0.513856
$220$	0	0
$221$	17.9944	1.21043
$222$	21.9164i	1.47093i
$223$	− 18.6433i	− 1.24845i	−0.781244	−	0.624225i	$-0.785415\pi$
$223$	− 18.6433i	− 1.24845i	0.781244	−	0.624225i	$-0.214585\pi$
$224$	1.48052	0.0989211
$225$	0	0
$226$	22.0099	1.46407
$227$	8.04672i	0.534080i	0.963686	+	0.267040i	$0.0860455\pi$
$227$	8.04672i	0.534080i	−0.963686	+	0.267040i	$0.913954\pi$
$228$	1.72611i	0.114314i
$229$	−20.1415	−1.33099	−0.665493	−	0.746404i	$-0.731779\pi$
$229$	−20.1415	−1.33099	−0.665493	+	0.746404i	$0.731779\pi$
$230$	0	0
$231$	1.23492	0.0812518
$232$	7.10331i	0.466355i
$233$	− 1.73386i	− 0.113589i	−0.998386	−	0.0567945i	$-0.981912\pi$
$233$	− 1.73386i	− 0.113589i	0.998386	−	0.0567945i	$-0.0180880\pi$
$234$	12.3510	0.807412
$235$	0	0
$236$	4.39775	0.286269
$237$	15.9709i	1.03742i
$238$	3.97170i	0.257447i
$239$	18.3150	1.18470	0.592349	−	0.805682i	$-0.298201\pi$
$239$	18.3150	1.18470	0.592349	+	0.805682i	$0.298201\pi$
$240$	0	0
$241$	−2.19675	−0.141505	−0.0707526	−	0.997494i	$-0.522540\pi$
$241$	−2.19675	−0.141505	−0.0707526	+	0.997494i	$0.522540\pi$
$242$	− 15.0304i	− 0.966192i
$243$	− 21.1054i	− 1.35391i
$244$	−3.65109	−0.233737
$245$	0	0
$246$	−33.3665	−2.12737
$247$	− 2.82167i	− 0.179539i
$248$	5.21730i	0.331299i
$249$	−43.3043	−2.74430
$250$	0	0
$251$	−13.6716	−0.862946	−0.431473	−	0.902126i	$-0.642006\pi$
$251$	−13.6716	−0.862946	−0.431473	+	0.902126i	$0.642006\pi$
$252$	0.726109i	0.0457406i
$253$	− 8.53711i	− 0.536723i
$254$	−5.03042	−0.315637
$255$	0	0
$256$	15.4076	0.962976
$257$	− 28.4904i	− 1.77718i	−0.458701	−	0.888591i	$-0.651685\pi$
$257$	− 28.4904i	− 1.77718i	0.458701	−	0.888591i	$-0.348315\pi$
$258$	47.6970i	2.96949i
$259$	2.10624	0.130875
$260$	0	0
$261$	−8.95328	−0.554194
$262$	− 14.3772i	− 0.888227i
$263$	5.36945i	0.331095i	0.986202	+	0.165547i	$0.0529391\pi$
$263$	5.36945i	0.331095i	−0.986202	+	0.165547i	$0.947061\pi$
$264$	6.88601	0.423805
$265$	0	0
$266$	0.622797	0.0381861
$267$	− 3.56833i	− 0.218378i
$268$	2.34328i	0.143139i
$269$	−7.06727	−0.430899	−0.215449	−	0.976515i	$-0.569122\pi$
$269$	−7.06727	−0.430899	−0.215449	+	0.976515i	$0.569122\pi$
$270$	0	0
$271$	21.6970	1.31800	0.659000	−	0.752143i	$-0.270980\pi$
$271$	21.6970	1.31800	0.659000	+	0.752143i	$0.270980\pi$
$272$	31.4076i	1.90437i
$273$	− 2.53016i	− 0.153132i
$274$	10.8881	0.657776
$275$	0	0
$276$	10.6999	0.644060
$277$	− 25.1132i	− 1.50891i	−0.656355	−	0.754453i	$-0.727902\pi$
$277$	− 25.1132i	− 1.50891i	0.656355	−	0.754453i	$-0.272098\pi$
$278$	16.7877i	1.00686i
$279$	−6.57608	−0.393699
$280$	0	0
$281$	−10.0411	−0.599001	−0.299501	−	0.954096i	$-0.596820\pi$
$281$	−10.0411	−0.599001	−0.299501	+	0.954096i	$0.596820\pi$
$282$	− 26.9971i	− 1.60765i
$283$	− 20.9143i	− 1.24323i	−0.783324	−	0.621613i	$-0.786478\pi$
$283$	− 20.9143i	− 1.24323i	0.783324	−	0.621613i	$-0.213522\pi$
$284$	1.67164	0.0991936
$285$	0	0
$286$	6.41617	0.379396
$287$	3.20662i	0.189281i
$288$	10.4055i	0.613150i
$289$	−23.6687	−1.39228
$290$	0	0
$291$	−28.0099	−1.64197
$292$	2.32273i	0.135928i
$293$	− 0.818748i	− 0.0478318i	−0.999714	−	0.0239159i	$-0.992387\pi$
$293$	− 0.818748i	− 0.0478318i	0.999714	−	0.0239159i	$-0.00761339\pi$
$294$	−26.9164	−1.56980
$295$	0	0
$296$	11.7445	0.682637
$297$	− 1.14228i	− 0.0662819i
$298$	9.14711i	0.529878i
$299$	−17.4912	−1.01154
$300$	0	0
$301$	4.58383	0.264207
$302$	8.45997i	0.486817i
$303$	− 21.8422i	− 1.25480i
$304$	4.92498	0.282467
$305$	0	0
$306$	−27.9143	−1.59575
$307$	7.44447i	0.424878i	0.977174	+	0.212439i	$0.0681407\pi$
$307$	7.44447i	0.424878i	−0.977174	+	0.212439i	$0.931859\pi$
$308$	0.377203i	0.0214931i
$309$	22.5187	1.28104
$310$	0	0
$311$	−0.956204	−0.0542213	−0.0271107	−	0.999632i	$-0.508631\pi$
$311$	−0.956204	−0.0542213	−0.0271107	+	0.999632i	$0.508631\pi$
$312$	− 14.1084i	− 0.798729i
$313$	5.98158i	0.338099i	0.985608	+	0.169049i	$0.0540697\pi$
$313$	5.98158i	0.338099i	−0.985608	+	0.169049i	$0.945930\pi$
$314$	32.6327	1.84157
$315$	0	0
$316$	−4.87826	−0.274424
$317$	− 22.6511i	− 1.27221i	−0.771602	−	0.636106i	$-0.780544\pi$
$317$	− 22.6511i	− 1.27221i	0.771602	−	0.636106i	$-0.219456\pi$
$318$	− 45.3249i	− 2.54169i
$319$	−4.65109	−0.260411
$320$	0	0
$321$	−36.4925	−2.03681
$322$	− 3.86064i	− 0.215145i
$323$	6.37720i	0.354837i
$324$	7.20662	0.400368
$325$	0	0
$326$	−22.3219	−1.23630
$327$	39.8521i	2.20383i
$328$	17.8804i	0.987279i
$329$	−2.59450	−0.143039
$330$	0	0
$331$	4.16283	0.228810	0.114405	−	0.993434i	$-0.463504\pi$
$331$	4.16283	0.228810	0.114405	+	0.993434i	$0.463504\pi$
$332$	− 13.2272i	− 0.725935i
$333$	14.8032i	0.811213i
$334$	21.6383	1.18399
$335$	0	0
$336$	4.41617	0.240922
$337$	18.0304i	0.982180i	0.871109	+	0.491090i	$0.163401\pi$
$337$	18.0304i	0.982180i	−0.871109	+	0.491090i	$0.836599\pi$
$338$	8.31849i	0.452466i
$339$	31.6893	1.72112
$340$	0	0
$341$	−3.41617	−0.184996
$342$	4.37720i	0.236692i
$343$	5.22717i	0.282241i
$344$	25.5598	1.37809
$345$	0	0
$346$	−2.45726	−0.132103
$347$	− 7.43380i	− 0.399067i	−0.979891	−	0.199534i	$-0.936057\pi$
$347$	− 7.43380i	− 0.399067i	0.979891	−	0.199534i	$-0.0639427\pi$
$348$	− 5.82942i	− 0.312490i
$349$	−21.9194	−1.17332	−0.586658	−	0.809835i	$-0.699557\pi$
$349$	−21.9194	−1.17332	−0.586658	+	0.809835i	$0.699557\pi$
$350$	0	0
$351$	−2.34036	−0.124919
$352$	5.40550i	0.288114i
$353$	10.3695i	0.551910i	0.961171	+	0.275955i	$0.0889941\pi$
$353$	10.3695i	0.551910i	−0.961171	+	0.275955i	$0.911006\pi$
$354$	23.7720	1.26347
$355$	0	0
$356$	1.08993	0.0577664
$357$	5.71836i	0.302648i
$358$	28.2555i	1.49335i
$359$	23.9893	1.26611	0.633054	−	0.774108i	$-0.281801\pi$
$359$	23.9893	1.26611	0.633054	+	0.774108i	$0.281801\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	− 9.47547i	− 0.498020i
$363$	− 21.6404i	− 1.13583i
$364$	0.772829	0.0405073
$365$	0	0
$366$	−19.7360	−1.03162
$367$	− 35.0275i	− 1.82842i	−0.405240	−	0.914210i	$-0.632812\pi$
$367$	− 35.0275i	− 1.82842i	0.405240	−	0.914210i	$-0.367188\pi$
$368$	− 30.5294i	− 1.59145i
$369$	−22.5371	−1.17323
$370$	0	0
$371$	−4.35586	−0.226145
$372$	− 4.28164i	− 0.221993i
$373$	30.8003i	1.59478i	0.603464	+	0.797390i	$0.293787\pi$
$373$	30.8003i	1.59478i	−0.603464	+	0.797390i	$0.706213\pi$
$374$	−14.5011	−0.749832
$375$	0	0
$376$	−14.4671	−0.746086
$377$	9.52936i	0.490787i
$378$	− 0.516561i	− 0.0265691i
$379$	−0.671640	−0.0344998	−0.0172499	−	0.999851i	$-0.505491\pi$
$379$	−0.671640	−0.0344998	−0.0172499	+	0.999851i	$0.505491\pi$
$380$	0	0
$381$	−7.24267	−0.371053
$382$	36.9759i	1.89185i
$383$	− 30.1805i	− 1.54215i	−0.636745	−	0.771075i	$-0.719720\pi$
$383$	− 30.1805i	− 1.54215i	0.636745	−	0.771075i	$-0.280280\pi$
$384$	31.8860	1.62718
$385$	0	0
$386$	35.2816	1.79579
$387$	32.2165i	1.63766i
$388$	− 8.55553i	− 0.434341i
$389$	31.5109	1.59767	0.798834	−	0.601552i	$-0.205451\pi$
$389$	31.5109	1.59767	0.798834	+	0.601552i	$0.205451\pi$
$390$	0	0
$391$	39.5315	1.99919
$392$	14.4239i	0.728518i
$393$	− 20.6999i	− 1.04417i
$394$	1.83929	0.0926623
$395$	0	0
$396$	−2.65109	−0.133222
$397$	12.5761i	0.631175i	0.948896	+	0.315588i	$0.102202\pi$
$397$	12.5761i	0.631175i	−0.948896	+	0.315588i	$0.897798\pi$
$398$	3.67376i	0.184149i
$399$	0.896688	0.0448905
$400$	0	0
$401$	24.6036	1.22864	0.614322	−	0.789056i	$-0.289430\pi$
$401$	24.6036	1.22864	0.614322	+	0.789056i	$0.289430\pi$
$402$	12.6666i	0.631752i
$403$	6.99920i	0.348655i
$404$	6.67164	0.331926
$405$	0	0
$406$	−2.10331	−0.104386
$407$	7.69006i	0.381182i
$408$	31.8860i	1.57859i
$409$	−13.6666	−0.675770	−0.337885	−	0.941187i	$-0.609711\pi$
$409$	−13.6666	−0.675770	−0.337885	+	0.941187i	$0.609711\pi$
$410$	0	0
$411$	15.6765	0.773263
$412$	6.87826i	0.338868i
$413$	− 2.28456i	− 0.112416i
$414$	27.1337	1.33355
$415$	0	0
$416$	11.0750	0.542997
$417$	24.1706i	1.18364i
$418$	2.27389i	0.111220i
$419$	−12.6893	−0.619911	−0.309956	−	0.950751i	$-0.600314\pi$
$419$	−12.6893	−0.619911	−0.309956	+	0.950751i	$0.600314\pi$
$420$	0	0
$421$	29.1826	1.42227	0.711136	−	0.703055i	$-0.248181\pi$
$421$	29.1826	1.42227	0.711136	+	0.703055i	$0.248181\pi$
$422$	− 16.1054i	− 0.784000i
$423$	− 18.2349i	− 0.886612i
$424$	−24.2886	−1.17956
$425$	0	0
$426$	9.03605	0.437798
$427$	1.89669i	0.0917872i
$428$	− 11.1465i	− 0.538788i
$429$	9.23784	0.446007
$430$	0	0
$431$	−29.9816	−1.44416	−0.722081	−	0.691809i	$-0.756814\pi$
$431$	−29.9816	−1.44416	−0.722081	+	0.691809i	$0.756814\pi$
$432$	− 4.08489i	− 0.196534i
$433$	− 4.76216i	− 0.228855i	−0.993432	−	0.114427i	$-0.963497\pi$
$433$	− 4.76216i	− 0.228855i	0.993432	−	0.114427i	$-0.0365033\pi$
$434$	−1.54486	−0.0741555
$435$	0	0
$436$	−12.1727	−0.582967
$437$	− 6.19887i	− 0.296532i
$438$	12.5555i	0.599926i
$439$	−6.88601	−0.328652	−0.164326	−	0.986406i	$-0.552545\pi$
$439$	−6.88601	−0.328652	−0.164326	+	0.986406i	$0.552545\pi$
$440$	0	0
$441$	−18.1805	−0.865736
$442$	29.7104i	1.41318i
$443$	8.65109i	0.411026i	0.978654	+	0.205513i	$0.0658863\pi$
$443$	8.65109i	0.411026i	−0.978654	+	0.205513i	$0.934114\pi$
$444$	−9.63830	−0.457413
$445$	0	0
$446$	30.7819	1.45757
$447$	13.1698i	0.622909i
$448$	− 1.27097i	− 0.0600476i
$449$	−8.63055	−0.407301	−0.203650	−	0.979044i	$-0.565281\pi$
$449$	−8.63055	−0.407301	−0.203650	+	0.979044i	$0.565281\pi$
$450$	0	0
$451$	−11.7077	−0.551293
$452$	9.67939i	0.455280i
$453$	12.1805i	0.572288i
$454$	−13.2859	−0.623538
$455$	0	0
$456$	5.00000	0.234146
$457$	28.6092i	1.33828i	0.743135	+	0.669141i	$0.233338\pi$
$457$	28.6092i	1.33828i	−0.743135	+	0.669141i	$0.766662\pi$
$458$	− 33.2555i	− 1.55393i
$459$	5.28939	0.246888
$460$	0	0
$461$	−39.1025	−1.82119	−0.910593	−	0.413305i	$-0.864374\pi$
$461$	−39.1025	−1.82119	−0.910593	+	0.413305i	$0.864374\pi$
$462$	2.03897i	0.0948615i
$463$	− 8.04380i	− 0.373827i	−0.982376	−	0.186913i	$-0.940152\pi$
$463$	− 8.04380i	− 0.373827i	0.982376	−	0.186913i	$-0.0598484\pi$
$464$	−16.6327	−0.772152
$465$	0	0
$466$	2.86276	0.132615
$467$	− 17.0488i	− 0.788926i	−0.918912	−	0.394463i	$-0.870931\pi$
$467$	− 17.0488i	− 0.788926i	0.918912	−	0.394463i	$-0.129069\pi$
$468$	5.43167i	0.251079i
$469$	1.21730	0.0562096
$470$	0	0
$471$	46.9837	2.16489
$472$	− 12.7389i	− 0.586356i
$473$	16.7360i	0.769521i
$474$	−26.3695	−1.21119
$475$	0	0
$476$	−1.74666	−0.0800578
$477$	− 30.6142i	− 1.40173i
$478$	30.2397i	1.38313i
$479$	−22.5478	−1.03023	−0.515117	−	0.857120i	$-0.672252\pi$
$479$	−22.5478	−1.03023	−0.515117	+	0.857120i	$0.672252\pi$
$480$	0	0
$481$	15.7557	0.718399
$482$	− 3.62704i	− 0.165207i
$483$	− 5.55845i	− 0.252918i
$484$	6.61000	0.300455
$485$	0	0
$486$	34.8470	1.58069
$487$	24.5577i	1.11281i	0.830910	+	0.556407i	$0.187820\pi$
$487$	24.5577i	1.11281i	−0.830910	+	0.556407i	$0.812180\pi$
$488$	10.5761i	0.478757i
$489$	−32.1386	−1.45336
$490$	0	0
$491$	−16.9298	−0.764032	−0.382016	−	0.924156i	$-0.624770\pi$
$491$	−16.9298	−0.764032	−0.382016	+	0.924156i	$0.624770\pi$
$492$	− 14.6738i	− 0.661544i
$493$	− 21.5371i	− 0.969983i
$494$	4.65884	0.209611
$495$	0	0
$496$	−12.2165	−0.548537
$497$	− 0.868391i	− 0.0389527i
$498$	− 71.4995i	− 3.20397i
$499$	−0.418295	−0.0187255	−0.00936273	−	0.999956i	$-0.502980\pi$
$499$	−0.418295	−0.0187255	−0.00936273	+	0.999956i	$0.502980\pi$
$500$	0	0
$501$	31.1543	1.39187
$502$	− 22.5732i	− 1.00749i
$503$	− 12.5993i	− 0.561776i	−0.959741	−	0.280888i	$-0.909371\pi$
$503$	− 12.5993i	− 0.561776i	0.959741	−	0.280888i	$-0.0906290\pi$
$504$	2.10331	0.0936890
$505$	0	0
$506$	14.0956	0.626624
$507$	11.9767i	0.531906i
$508$	− 2.21225i	− 0.0981528i
$509$	−12.8209	−0.568275	−0.284138	−	0.958784i	$-0.591707\pi$
$509$	−12.8209	−0.568275	−0.284138	+	0.958784i	$0.591707\pi$
$510$	0	0
$511$	1.20662	0.0533779
$512$	− 1.38708i	− 0.0613007i
$513$	− 0.829422i	− 0.0366199i
$514$	47.0403	2.07486
$515$	0	0
$516$	−20.9759	−0.923415
$517$	− 9.47277i	− 0.416612i
$518$	3.47759i	0.152797i
$519$	−3.53791	−0.155297
$520$	0	0
$521$	11.3743	0.498316	0.249158	−	0.968463i	$-0.419846\pi$
$521$	11.3743	0.498316	0.249158	+	0.968463i	$0.419846\pi$
$522$	− 14.7827i	− 0.647021i
$523$	33.0948i	1.44713i	0.690255	+	0.723566i	$0.257498\pi$
$523$	33.0948i	1.44713i	−0.690255	+	0.723566i	$0.742502\pi$
$524$	6.32273	0.276210
$525$	0	0
$526$	−8.86547	−0.386553
$527$	− 15.8187i	− 0.689075i
$528$	16.1239i	0.701701i
$529$	−15.4260	−0.670698
$530$	0	0
$531$	16.0566	0.696797
$532$	0.273891i	0.0118747i
$533$	23.9872i	1.03900i
$534$	5.89164	0.254956
$535$	0	0
$536$	6.78775	0.293186
$537$	40.6815i	1.75554i
$538$	− 11.6687i	− 0.503074i
$539$	−9.44447	−0.406802
$540$	0	0
$541$	31.4338	1.35144	0.675722	−	0.737156i	$-0.263832\pi$
$541$	31.4338	1.35144	0.675722	+	0.737156i	$0.263832\pi$
$542$	35.8238i	1.53876i
$543$	− 13.6425i	− 0.585458i
$544$	−25.0304	−1.07317
$545$	0	0
$546$	4.17753	0.178782
$547$	23.3460i	0.998202i	0.866544	+	0.499101i	$0.166336\pi$
$547$	23.3460i	0.998202i	−0.866544	+	0.499101i	$0.833664\pi$
$548$	4.78833i	0.204547i
$549$	−13.3305	−0.568931
$550$	0	0
$551$	−3.37720	−0.143874
$552$	− 30.9944i	− 1.31921i
$553$	2.53418i	0.107764i
$554$	41.4642	1.76165
$555$	0	0
$556$	−7.38283	−0.313102
$557$	− 15.9229i	− 0.674673i	−0.941384	−	0.337337i	$-0.890474\pi$
$557$	− 15.9229i	− 0.674673i	0.941384	−	0.337337i	$-0.109526\pi$
$558$	− 10.8577i	− 0.459644i
$559$	34.2894	1.45029
$560$	0	0
$561$	−20.8783	−0.881481
$562$	− 16.5788i	− 0.699334i
$563$	− 29.6404i	− 1.24919i	−0.780947	−	0.624597i	$-0.785263\pi$
$563$	− 29.6404i	− 1.24919i	0.780947	−	0.624597i	$-0.214737\pi$
$564$	11.8726	0.499928
$565$	0	0
$566$	34.5315	1.45147
$567$	− 3.74373i	− 0.157222i
$568$	− 4.84222i	− 0.203175i
$569$	7.64334	0.320426	0.160213	−	0.987082i	$-0.448782\pi$
$569$	7.64334	0.320426	0.160213	+	0.987082i	$0.448782\pi$
$570$	0	0
$571$	−10.5577	−0.441824	−0.220912	−	0.975294i	$-0.570903\pi$
$571$	−10.5577	−0.441824	−0.220912	+	0.975294i	$0.570903\pi$
$572$	2.82167i	0.117980i
$573$	53.2370i	2.22401i
$574$	−5.29444	−0.220986
$575$	0	0
$576$	8.93273	0.372197
$577$	− 38.9447i	− 1.62129i	−0.585538	−	0.810645i	$-0.699117\pi$
$577$	− 38.9447i	− 1.62129i	0.585538	−	0.810645i	$-0.300883\pi$
$578$	− 39.0793i	− 1.62548i
$579$	50.7976	2.11108
$580$	0	0
$581$	−6.87131	−0.285070
$582$	− 46.2469i	− 1.91700i
$583$	− 15.9036i	− 0.658661i
$584$	6.72823	0.278416
$585$	0	0
$586$	1.35183	0.0558436
$587$	10.9992i	0.453986i	0.973896	+	0.226993i	$0.0728894\pi$
$587$	10.9992i	0.453986i	−0.973896	+	0.226993i	$0.927111\pi$
$588$	− 11.8372i	− 0.488157i
$589$	−2.48052	−0.102208
$590$	0	0
$591$	2.64817	0.108931
$592$	27.5003i	1.13025i
$593$	26.7848i	1.09992i	0.835191	+	0.549960i	$0.185357\pi$
$593$	26.7848i	1.09992i	−0.835191	+	0.549960i	$0.814643\pi$
$594$	1.88601	0.0773841
$595$	0	0
$596$	−4.02267	−0.164775
$597$	5.28939i	0.216480i
$598$	− 28.8796i	− 1.18097i
$599$	3.44930	0.140934	0.0704672	−	0.997514i	$-0.477551\pi$
$599$	3.44930	0.140934	0.0704672	+	0.997514i	$0.477551\pi$
$600$	0	0
$601$	−37.0686	−1.51206	−0.756030	−	0.654537i	$-0.772863\pi$
$601$	−37.0686	−1.51206	−0.756030	+	0.654537i	$0.772863\pi$
$602$	7.56833i	0.308462i
$603$	8.55553i	0.348408i
$604$	−3.72048	−0.151384
$605$	0	0
$606$	36.0635	1.46498
$607$	14.8732i	0.603685i	0.953358	+	0.301843i	$0.0976017\pi$
$607$	14.8732i	0.603685i	−0.953358	+	0.301843i	$0.902398\pi$
$608$	3.92498i	0.159179i
$609$	−3.02830	−0.122713
$610$	0	0
$611$	−19.4082	−0.785172
$612$	− 12.2760i	− 0.496228i
$613$	− 40.4671i	− 1.63445i	−0.576317	−	0.817226i	$-0.695511\pi$
$613$	− 40.4671i	− 1.63445i	0.576317	−	0.817226i	$-0.304489\pi$
$614$	−12.2915	−0.496045
$615$	0	0
$616$	1.09264	0.0440237
$617$	− 8.76991i	− 0.353063i	−0.984295	−	0.176532i	$-0.943512\pi$
$617$	− 8.76991i	− 0.353063i	0.984295	−	0.176532i	$-0.0564878\pi$
$618$	37.1805i	1.49562i
$619$	−32.0510	−1.28824	−0.644119	−	0.764926i	$-0.722776\pi$
$619$	−32.0510	−1.28824	−0.644119	+	0.764926i	$0.722776\pi$
$620$	0	0
$621$	−5.14148	−0.206321
$622$	− 1.57878i	− 0.0633034i
$623$	− 0.566205i	− 0.0226845i
$624$	33.0352	1.32247
$625$	0	0
$626$	−9.87614	−0.394730
$627$	3.27389i	0.130747i
$628$	14.3510i	0.572668i
$629$	−35.6092	−1.41983
$630$	0	0
$631$	−18.9709	−0.755220	−0.377610	−	0.925965i	$-0.623254\pi$
$631$	−18.9709	−0.755220	−0.377610	+	0.925965i	$0.623254\pi$
$632$	14.1308i	0.562093i
$633$	− 23.1882i	− 0.921648i
$634$	37.3991	1.48531
$635$	0	0
$636$	19.9327	0.790384
$637$	19.3502i	0.766684i
$638$	− 7.67939i	− 0.304030i
$639$	6.10331	0.241443
$640$	0	0
$641$	−44.3433	−1.75145	−0.875727	−	0.482806i	$-0.839617\pi$
$641$	−44.3433	−1.75145	−0.875727	+	0.482806i	$0.839617\pi$
$642$	− 60.2525i	− 2.37798i
$643$	42.9397i	1.69338i	0.532090	+	0.846688i	$0.321407\pi$
$643$	42.9397i	1.69338i	−0.532090	+	0.846688i	$0.678593\pi$
$644$	1.69781	0.0669032
$645$	0	0
$646$	−10.5294	−0.414272
$647$	− 17.9864i	− 0.707118i	−0.935412	−	0.353559i	$-0.884971\pi$
$647$	− 17.9864i	− 0.707118i	0.935412	−	0.353559i	$-0.115029\pi$
$648$	− 20.8753i	− 0.820061i
$649$	8.34116	0.327419
$650$	0	0
$651$	−2.22425	−0.0871751
$652$	− 9.81663i	− 0.384449i
$653$	− 21.7274i	− 0.850260i	−0.905132	−	0.425130i	$-0.860228\pi$
$653$	− 21.7274i	− 0.850260i	0.905132	−	0.425130i	$-0.139772\pi$
$654$	−65.7995	−2.57297
$655$	0	0
$656$	−41.8676	−1.63465
$657$	8.48052i	0.330856i
$658$	− 4.28376i	− 0.166998i
$659$	40.8590	1.59164	0.795821	−	0.605532i	$-0.207040\pi$
$659$	40.8590	1.59164	0.795821	+	0.605532i	$0.207040\pi$
$660$	0	0
$661$	33.7282	1.31188	0.655938	−	0.754815i	$-0.272273\pi$
$661$	33.7282	1.31188	0.655938	+	0.754815i	$0.272273\pi$
$662$	6.87322i	0.267135i
$663$	42.7763i	1.66129i
$664$	−38.3150	−1.48691
$665$	0	0
$666$	−24.4415	−0.947091
$667$	20.9349i	0.810601i
$668$	9.51598i	0.368184i
$669$	44.3190	1.71347
$670$	0	0
$671$	−6.92498	−0.267336
$672$	3.51948i	0.135767i
$673$	− 11.0622i	− 0.426417i	−0.977007	−	0.213209i	$-0.931609\pi$
$673$	− 11.0622i	− 0.426417i	0.977007	−	0.213209i	$-0.0683914\pi$
$674$	−29.7699	−1.14669
$675$	0	0
$676$	−3.65826	−0.140702
$677$	6.14711i	0.236253i	0.992999	+	0.118126i	$0.0376888\pi$
$677$	6.14711i	0.236253i	−0.992999	+	0.118126i	$0.962311\pi$
$678$	52.3219i	2.00941i
$679$	−4.44447	−0.170563
$680$	0	0
$681$	−19.1287	−0.733013
$682$	− 5.64042i	− 0.215983i
$683$	− 13.6073i	− 0.520669i	−0.965519	−	0.260334i	$-0.916167\pi$
$683$	− 13.6073i	− 0.520669i	0.965519	−	0.260334i	$-0.0838328\pi$
$684$	−1.92498	−0.0736036
$685$	0	0
$686$	−8.63055	−0.329516
$687$	− 47.8804i	− 1.82675i
$688$	59.8492i	2.28173i
$689$	−32.5840	−1.24135
$690$	0	0
$691$	−40.3043	−1.53325	−0.766624	−	0.642096i	$-0.778065\pi$
$691$	−40.3043	−1.53325	−0.766624	+	0.642096i	$0.778065\pi$
$692$	− 1.08064i	− 0.0410799i
$693$	1.37720i	0.0523156i
$694$	12.2739	0.465911
$695$	0	0
$696$	−16.8860	−0.640063
$697$	− 54.2130i	− 2.05346i
$698$	− 36.1909i	− 1.36985i
$699$	4.12174	0.155898
$700$	0	0
$701$	−3.23704	−0.122261	−0.0611307	−	0.998130i	$-0.519471\pi$
$701$	−3.23704	−0.122261	−0.0611307	+	0.998130i	$0.519471\pi$
$702$	− 3.86415i	− 0.145843i
$703$	5.58383i	0.210598i
$704$	4.64042	0.174892
$705$	0	0
$706$	−17.1209	−0.644355
$707$	− 3.46582i	− 0.130345i
$708$	10.4543i	0.392898i
$709$	−20.2370	−0.760018	−0.380009	−	0.924983i	$-0.624079\pi$
$709$	−20.2370	−0.760018	−0.380009	+	0.924983i	$0.624079\pi$
$710$	0	0
$711$	−17.8110	−0.667965
$712$	− 3.15720i	− 0.118321i
$713$	15.3764i	0.575851i
$714$	−9.44155	−0.353341
$715$	0	0
$716$	−12.4260	−0.464383
$717$	43.5384i	1.62597i
$718$	39.6086i	1.47818i
$719$	−47.8804	−1.78564	−0.892819	−	0.450416i	$-0.851276\pi$
$719$	−47.8804	−1.78564	−0.892819	+	0.450416i	$0.851276\pi$
$720$	0	0
$721$	3.57315	0.133071
$722$	1.65109i	0.0614473i
$723$	− 5.22212i	− 0.194213i
$724$	4.16707	0.154868
$725$	0	0
$726$	35.7304	1.32608
$727$	− 17.7926i	− 0.659890i	−0.944000	−	0.329945i	$-0.892970\pi$
$727$	− 17.7926i	− 0.659890i	0.944000	−	0.329945i	$-0.107030\pi$
$728$	− 2.23864i	− 0.0829697i
$729$	20.3969	0.755443
$730$	0	0
$731$	−77.4968	−2.86632
$732$	− 8.67939i	− 0.320799i
$733$	− 5.66097i	− 0.209093i	−0.994520	−	0.104546i	$-0.966661\pi$
$733$	− 5.66097i	− 0.209093i	0.994520	−	0.104546i	$-0.0333390\pi$
$734$	57.8337	2.13468
$735$	0	0
$736$	24.3305	0.896834
$737$	4.44447i	0.163714i
$738$	− 37.2109i	− 1.36975i
$739$	5.59955	0.205983	0.102991	−	0.994682i	$-0.467159\pi$
$739$	5.59955	0.205983	0.102991	+	0.994682i	$0.467159\pi$
$740$	0	0
$741$	6.70769	0.246413
$742$	− 7.19193i	− 0.264024i
$743$	5.81663i	0.213391i	0.994292	+	0.106696i	$0.0340270\pi$
$743$	5.81663i	0.213391i	−0.994292	+	0.106696i	$0.965973\pi$
$744$	−12.4026	−0.454700
$745$	0	0
$746$	−50.8542	−1.86191
$747$	− 48.2936i	− 1.76697i
$748$	− 6.37720i	− 0.233174i
$749$	−5.79045	−0.211579
$750$	0	0
$751$	−41.1797	−1.50267	−0.751333	−	0.659923i	$-0.770589\pi$
$751$	−41.1797	−1.50267	−0.751333	+	0.659923i	$0.770589\pi$
$752$	− 33.8753i	− 1.23531i
$753$	− 32.5003i	− 1.18438i
$754$	−15.7339	−0.572993
$755$	0	0
$756$	0.227171	0.00826212
$757$	33.5208i	1.21833i	0.793042	+	0.609167i	$0.208496\pi$
$757$	33.5208i	1.21833i	−0.793042	+	0.609167i	$0.791504\pi$
$758$	− 1.10894i	− 0.0402785i
$759$	20.2944	0.736641
$760$	0	0
$761$	31.5032	1.14199	0.570995	−	0.820954i	$-0.306558\pi$
$761$	31.5032	1.14199	0.570995	+	0.820954i	$0.306558\pi$
$762$	− 11.9583i	− 0.433204i
$763$	6.32353i	0.228927i
$764$	−16.2611	−0.588306
$765$	0	0
$766$	49.8307	1.80046
$767$	− 17.0897i	− 0.617074i
$768$	36.6270i	1.32166i
$769$	1.22425	0.0441475	0.0220737	−	0.999756i	$-0.492973\pi$
$769$	1.22425	0.0441475	0.0220737	+	0.999756i	$0.492973\pi$
$770$	0	0
$771$	67.7274	2.43914
$772$	15.5160i	0.558432i
$773$	40.6687i	1.46275i	0.681974	+	0.731376i	$0.261122\pi$
$773$	40.6687i	1.46275i	−0.681974	+	0.731376i	$0.738878\pi$
$774$	−53.1924	−1.91196
$775$	0	0
$776$	−24.7827	−0.889647
$777$	5.00695i	0.179623i
$778$	52.0275i	1.86528i
$779$	−8.50106	−0.304582
$780$	0	0
$781$	3.17058	0.113452
$782$	65.2702i	2.33406i
$783$	2.80113i	0.100104i
$784$	−33.7742	−1.20622
$785$	0	0
$786$	34.1775	1.21907
$787$	39.9525i	1.42415i	0.702102	+	0.712076i	$0.252245\pi$
$787$	39.9525i	1.42415i	−0.702102	+	0.712076i	$0.747755\pi$
$788$	0.808876i	0.0288150i
$789$	−12.7643	−0.454420
$790$	0	0
$791$	5.02830	0.178786
$792$	7.67939i	0.272875i
$793$	14.1882i	0.503838i
$794$	−20.7643	−0.736897
$795$	0	0
$796$	−1.61563	−0.0572644
$797$	4.53791i	0.160741i	0.996765	+	0.0803705i	$0.0256103\pi$
$797$	4.53791i	0.160741i	−0.996765	+	0.0803705i	$0.974390\pi$
$798$	1.48052i	0.0524097i
$799$	43.8641	1.55180
$800$	0	0
$801$	3.97945	0.140607
$802$	40.6228i	1.43444i
$803$	4.40550i	0.155467i
$804$	−5.57045	−0.196455
$805$	0	0
$806$	−11.5563	−0.407054
$807$	− 16.8003i	− 0.591399i
$808$	− 19.3257i	− 0.679874i
$809$	−55.5958	−1.95465	−0.977323	−	0.211756i	$-0.932082\pi$
$809$	−55.5958	−1.95465	−0.977323	+	0.211756i	$0.932082\pi$
$810$	0	0
$811$	42.3326	1.48650	0.743249	−	0.669014i	$-0.233284\pi$
$811$	42.3326	1.48650	0.743249	+	0.669014i	$0.233284\pi$
$812$	− 0.924984i	− 0.0324606i
$813$	51.5782i	1.80893i
$814$	−12.6970	−0.445030
$815$	0	0
$816$	−74.6623	−2.61370
$817$	12.1522i	0.425150i
$818$	− 22.5648i	− 0.788961i
$819$	2.82167	0.0985972
$820$	0	0
$821$	−3.10543	−0.108380	−0.0541902	−	0.998531i	$-0.517258\pi$
$821$	−3.10543	−0.108380	−0.0541902	+	0.998531i	$0.517258\pi$
$822$	25.8833i	0.902784i
$823$	− 45.2002i	− 1.57558i	−0.615944	−	0.787790i	$-0.711225\pi$
$823$	− 45.2002i	− 1.57558i	0.615944	−	0.787790i	$-0.288775\pi$
$824$	19.9242	0.694092
$825$	0	0
$826$	3.77203	0.131246
$827$	35.6503i	1.23968i	0.784727	+	0.619841i	$0.212803\pi$
$827$	35.6503i	1.23968i	−0.784727	+	0.619841i	$0.787197\pi$
$828$	11.9327i	0.414691i
$829$	18.4330	0.640204	0.320102	−	0.947383i	$-0.396283\pi$
$829$	18.4330	0.640204	0.320102	+	0.947383i	$0.396283\pi$
$830$	0	0
$831$	59.6991	2.07094
$832$	− 9.50749i	− 0.329613i
$833$	− 43.7331i	− 1.51526i
$834$	−39.9079	−1.38190
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	2.05739i	0.0711139i
$838$	− 20.9512i	− 0.723746i
$839$	14.1805	0.489564	0.244782	−	0.969578i	$-0.421284\pi$
$839$	14.1805	0.489564	0.244782	+	0.969578i	$0.421284\pi$
$840$	0	0
$841$	−17.5945	−0.606707
$842$	48.1832i	1.66050i
$843$	− 23.8697i	− 0.822117i
$844$	7.08277	0.243799
$845$	0	0
$846$	30.1076	1.03512
$847$	− 3.43380i	− 0.117987i
$848$	− 56.8726i	− 1.95301i
$849$	49.7176	1.70630
$850$	0	0
$851$	34.6134	1.18653
$852$	3.97383i	0.136141i
$853$	42.6639i	1.46078i	0.683028	+	0.730392i	$0.260663\pi$
$853$	42.6639i	1.46078i	−0.683028	+	0.730392i	$0.739337\pi$
$854$	−3.13161	−0.107161
$855$	0	0
$856$	−32.2880	−1.10358
$857$	1.42392i	0.0486403i	0.999704	+	0.0243201i	$0.00774210\pi$
$857$	1.42392i	0.0486403i	−0.999704	+	0.0243201i	$0.992258\pi$
$858$	15.2525i	0.520713i
$859$	8.27894	0.282474	0.141237	−	0.989976i	$-0.454892\pi$
$859$	8.27894	0.282474	0.141237	+	0.989976i	$0.454892\pi$
$860$	0	0
$861$	−7.62280	−0.259784
$862$	− 49.5024i	− 1.68606i
$863$	30.4154i	1.03535i	0.855577	+	0.517676i	$0.173203\pi$
$863$	30.4154i	1.03535i	−0.855577	+	0.517676i	$0.826797\pi$
$864$	3.25547	0.110753
$865$	0	0
$866$	7.86276	0.267188
$867$	− 56.2653i	− 1.91087i
$868$	− 0.679390i	− 0.0230600i
$869$	−9.25254	−0.313871
$870$	0	0
$871$	9.10602	0.308546
$872$	35.2605i	1.19407i
$873$	− 31.2370i	− 1.05721i
$874$	10.2349	0.346201
$875$	0	0
$876$	−5.52161	−0.186558
$877$	− 57.8484i	− 1.95340i	−0.214608	−	0.976700i	$-0.568847\pi$
$877$	− 57.8484i	− 1.95340i	0.214608	−	0.976700i	$-0.431153\pi$
$878$	− 11.3695i	− 0.383700i
$879$	1.94633	0.0656481
$880$	0	0
$881$	18.4055	0.620097	0.310049	−	0.950721i	$-0.399655\pi$
$881$	18.4055	0.620097	0.310049	+	0.950721i	$0.399655\pi$
$882$	− 30.0176i	− 1.01075i
$883$	− 18.7947i	− 0.632492i	−0.948677	−	0.316246i	$-0.897578\pi$
$883$	− 18.7947i	− 0.632492i	0.948677	−	0.316246i	$-0.102422\pi$
$884$	−13.0659	−0.439453
$885$	0	0
$886$	−14.2838	−0.479872
$887$	11.1471i	0.374283i	0.982333	+	0.187142i	$0.0599223\pi$
$887$	11.1471i	0.374283i	−0.982333	+	0.187142i	$0.940078\pi$
$888$	27.9191i	0.936905i
$889$	−1.14923	−0.0385440
$890$	0	0
$891$	13.6687	0.457919
$892$	13.5371i	0.453256i
$893$	− 6.87826i	− 0.230172i
$894$	−21.7445	−0.727246
$895$	0	0
$896$	5.05952	0.169027
$897$	− 41.5801i	− 1.38832i
$898$	− 14.2498i	− 0.475523i
$899$	8.37720	0.279395
$900$	0	0
$901$	73.6425	2.45339
$902$	− 19.3305i	− 0.643635i
$903$	10.8967i	0.362619i
$904$	28.0382	0.932536
$905$	0	0
$906$	−20.1111	−0.668145
$907$	− 38.5598i	− 1.28036i	−0.768226	−	0.640178i	$-0.778861\pi$
$907$	− 38.5598i	− 1.28036i	0.768226	−	0.640178i	$-0.221139\pi$
$908$	− 5.84280i	− 0.193900i
$909$	24.3588	0.807930
$910$	0	0
$911$	−1.79820	−0.0595771	−0.0297885	−	0.999556i	$-0.509483\pi$
$911$	−1.79820	−0.0595771	−0.0297885	+	0.999556i	$0.509483\pi$
$912$	11.7077i	0.387680i
$913$	− 25.0878i	− 0.830285i
$914$	−47.2365	−1.56244
$915$	0	0
$916$	14.6249	0.483221
$917$	− 3.28456i	− 0.108466i
$918$	8.73328i	0.288241i
$919$	32.4458	1.07029	0.535144	−	0.844761i	$-0.320257\pi$
$919$	32.4458	1.07029	0.535144	+	0.844761i	$0.320257\pi$
$920$	0	0
$921$	−17.6970	−0.583136
$922$	− 64.5619i	− 2.12623i
$923$	− 6.49602i	− 0.213819i
$924$	−0.896688	−0.0294989
$925$	0	0
$926$	13.2811	0.436443
$927$	25.1132i	0.824825i
$928$	− 13.2555i	− 0.435132i
$929$	18.4231	0.604443	0.302222	−	0.953238i	$-0.402272\pi$
$929$	18.4231	0.604443	0.302222	+	0.953238i	$0.402272\pi$
$930$	0	0
$931$	−6.85772	−0.224753
$932$	1.25897i	0.0412390i
$933$	− 2.27309i	− 0.0744176i
$934$	28.1492	0.921071
$935$	0	0
$936$	15.7339	0.514277
$937$	− 8.29926i	− 0.271125i	−0.990769	−	0.135563i	$-0.956716\pi$
$937$	− 8.29926i	− 0.271125i	0.990769	−	0.135563i	$-0.0432842\pi$
$938$	2.00987i	0.0656247i
$939$	−14.2194	−0.464033
$940$	0	0
$941$	33.6687	1.09757	0.548784	−	0.835964i	$-0.315091\pi$
$941$	33.6687	1.09757	0.548784	+	0.835964i	$0.315091\pi$
$942$	77.5745i	2.52751i
$943$	52.6970i	1.71605i
$944$	29.8286	0.970839
$945$	0	0
$946$	−27.6327	−0.898416
$947$	− 1.35103i	− 0.0439026i	−0.999759	−	0.0219513i	$-0.993012\pi$
$947$	− 1.35103i	− 0.0439026i	0.999759	−	0.0219513i	$-0.00698787\pi$
$948$	− 11.5966i	− 0.376641i
$949$	9.02617	0.293002
$950$	0	0
$951$	53.8462	1.74608
$952$	5.05952i	0.163980i
$953$	− 6.70769i	− 0.217283i	−0.994081	−	0.108642i	$-0.965350\pi$
$953$	− 6.70769i	− 0.217283i	0.994081	−	0.108642i	$-0.0346501\pi$
$954$	50.5470	1.63652
$955$	0	0
$956$	−13.2987	−0.430110
$957$	− 11.0566i	− 0.357409i
$958$	− 37.2285i	− 1.20280i
$959$	2.48746	0.0803244
$960$	0	0
$961$	−24.8470	−0.801518
$962$	26.0142i	0.838731i
$963$	− 40.6970i	− 1.31144i
$964$	1.59508	0.0513741
$965$	0	0
$966$	9.17753	0.295282
$967$	− 19.5174i	− 0.627636i	−0.949483	−	0.313818i	$-0.898392\pi$
$967$	− 19.5174i	− 0.627636i	0.949483	−	0.313818i	$-0.101608\pi$
$968$	− 19.1471i	− 0.615411i
$969$	−15.1599	−0.487006
$970$	0	0
$971$	−8.50669	−0.272993	−0.136496	−	0.990641i	$-0.543584\pi$
$971$	−8.50669	−0.272993	−0.136496	+	0.990641i	$0.543584\pi$
$972$	15.3249i	0.491545i
$973$	3.83527i	0.122953i
$974$	−40.5470	−1.29921
$975$	0	0
$976$	−24.7643	−0.792685
$977$	− 19.7048i	− 0.630411i	−0.949023	−	0.315206i	$-0.897927\pi$
$977$	− 19.7048i	− 0.630411i	0.949023	−	0.315206i	$-0.102073\pi$
$978$	− 53.0638i	− 1.69679i
$979$	2.06727	0.0660701
$980$	0	0
$981$	−44.4437	−1.41898
$982$	− 27.9527i	− 0.892006i
$983$	− 12.4677i	− 0.397658i	−0.980034	−	0.198829i	$-0.936286\pi$
$983$	− 12.4677i	− 0.397658i	0.980034	−	0.198829i	$-0.0637139\pi$
$984$	−42.5053	−1.35502
$985$	0	0
$986$	35.5598	1.13245
$987$	− 6.16765i	− 0.196319i
$988$	2.04884i	0.0651824i
$989$	75.3297	2.39534
$990$	0	0
$991$	25.0841	0.796822	0.398411	−	0.917207i	$-0.369562\pi$
$991$	25.0841	0.796822	0.398411	+	0.917207i	$0.369562\pi$
$992$	− 9.73598i	− 0.309118i
$993$	9.89589i	0.314036i
$994$	1.43380	0.0454772
$995$	0	0
$996$	31.4437	0.996331
$997$	− 11.8775i	− 0.376163i	−0.982153	−	0.188082i	$-0.939773\pi$
$997$	− 11.8775i	− 0.376163i	0.982153	−	0.188082i	$-0.0602269\pi$
$998$	− 0.690644i	− 0.0218620i
$999$	4.63135	0.146529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.b.324.5		6
5.2	odd	4		475.2.a.g.1.1	yes	3
5.3	odd	4		475.2.a.e.1.3	✓	3
5.4	even	2	inner	475.2.b.b.324.2		6
15.2	even	4		4275.2.a.ba.1.3		3
15.8	even	4		4275.2.a.bm.1.1		3
20.3	even	4		7600.2.a.cc.1.3		3
20.7	even	4		7600.2.a.bh.1.1		3
95.18	even	4		9025.2.a.bc.1.1		3
95.37	even	4		9025.2.a.y.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.3	✓	3	5.3	odd	4
475.2.a.g.1.1	yes	3	5.2	odd	4
475.2.b.b.324.2		6	5.4	even	2	inner
475.2.b.b.324.5		6	1.1	even	1	trivial
4275.2.a.ba.1.3		3	15.2	even	4
4275.2.a.bm.1.1		3	15.8	even	4
7600.2.a.bh.1.1		3	20.7	even	4
7600.2.a.cc.1.3		3	20.3	even	4
9025.2.a.y.1.3		3	95.37	even	4
9025.2.a.bc.1.1		3	95.18	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.65109i q^{2} +2.37720i q^{3} -0.726109 q^{4} -3.92498 q^{6} +0.377203i q^{7} +2.10331i q^{8} -2.65109 q^{9} +O(q^{10})\)	\(q+1.65109i q^{2} +2.37720i q^{3} -0.726109 q^{4} -3.92498 q^{6} +0.377203i q^{7} +2.10331i q^{8} -2.65109 q^{9} -1.37720 q^{11} -1.72611i q^{12} +2.82167i q^{13} -0.622797 q^{14} -4.92498 q^{16} -6.37720i q^{17} -4.37720i q^{18} -1.00000 q^{19} -0.896688 q^{21} -2.27389i q^{22} +6.19887i q^{23} -5.00000 q^{24} -4.65884 q^{26} +0.829422i q^{27} -0.273891i q^{28} +3.37720 q^{29} +2.48052 q^{31} -3.92498i q^{32} -3.27389i q^{33} +10.5294 q^{34} +1.92498 q^{36} -5.58383i q^{37} -1.65109i q^{38} -6.70769 q^{39} +8.50106 q^{41} -1.48052i q^{42} -12.1522i q^{43} +1.00000 q^{44} -10.2349 q^{46} +6.87826i q^{47} -11.7077i q^{48} +6.85772 q^{49} +15.1599 q^{51} -2.04884i q^{52} +11.5478i q^{53} -1.36945 q^{54} -0.793375 q^{56} -2.37720i q^{57} +5.57608i q^{58} -6.05659 q^{59} +5.02830 q^{61} +4.09556i q^{62} -1.00000i q^{63} -3.36945 q^{64} +5.40550 q^{66} -3.22717i q^{67} +4.63055i q^{68} -14.7360 q^{69} -2.30219 q^{71} -5.57608i q^{72} -3.19887i q^{73} +9.21942 q^{74} +0.726109 q^{76} -0.519485i q^{77} -11.0750i q^{78} +6.71836 q^{79} -9.92498 q^{81} +14.0360i q^{82} +18.2165i q^{83} +0.651093 q^{84} +20.0643 q^{86} +8.02830i q^{87} -2.89669i q^{88} -1.50106 q^{89} -1.06434 q^{91} -4.50106i q^{92} +5.89669i q^{93} -11.3567 q^{94} +9.33048 q^{96} +11.7827i q^{97} +11.3227i q^{98} +3.65109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10}) \) 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9	\( 6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46} + 14 q^{49} + 36 q^{51} + 10 q^{54} - 18 q^{56} + 12 q^{59} + 6 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} + 14 q^{71} + 2 q^{74} + 8 q^{76} + 36 q^{79} - 42 q^{81} - 10 q^{84} + 80 q^{86} + 40 q^{89} + 34 q^{91} - 90 q^{94} + 4 q^{96} + 8 q^{99}+O(q^{100}) \) 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9 + 2 * q^11 - 14 * q^14 - 12 * q^16 - 6 * q^19 - 12 * q^21 - 30 * q^24 - 22 * q^26 + 10 * q^29 - 2 * q^31 - 10 * q^34 - 6 * q^36 + 22 * q^39 + 2 * q^41 + 6 * q^44 - 24 * q^46 + 14 * q^49 + 36 * q^51 + 10 * q^54 - 18 * q^56 + 12 * q^59 + 6 * q^61 - 2 * q^64 - 2 * q^66 - 2 * q^69 + 14 * q^71 + 2 * q^74 + 8 * q^76 + 36 * q^79 - 42 * q^81 - 10 * q^84 + 80 * q^86 + 40 * q^89 + 34 * q^91 - 90 * q^94 + 4 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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