LMFDB - Embedded newform 1875.2.b.d.1249.3

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:	$8$
Coefficient field:	8.0.324000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9x^6 + 26x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$1.82709i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.d.1249.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.33826i q^{2} -1.00000i q^{3} +0.209057 q^{4} -1.33826 q^{6} -1.27977i q^{7} -2.95630i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.33826i q^{2} -1.00000i q^{3} +0.209057 q^{4} -1.33826 q^{6} -1.27977i q^{7} -2.95630i q^{8} -1.00000 q^{9} +1.16535 q^{11} -0.209057i q^{12} -3.61048i q^{13} -1.71267 q^{14} -3.53818 q^{16} -5.35772i q^{17} +1.33826i q^{18} +7.61048 q^{19} -1.27977 q^{21} -1.55955i q^{22} +3.41811i q^{23} -2.95630 q^{24} -4.83176 q^{26} +1.00000i q^{27} -0.267545i q^{28} +3.21661 q^{29} -1.09306 q^{31} -1.17758i q^{32} -1.16535i q^{33} -7.17002 q^{34} -0.209057 q^{36} +7.80126i q^{37} -10.1848i q^{38} -3.61048 q^{39} -3.00565 q^{41} +1.71267i q^{42} -3.42278i q^{43} +0.243625 q^{44} +4.57433 q^{46} +9.41462i q^{47} +3.53818i q^{48} +5.36218 q^{49} -5.35772 q^{51} -0.754795i q^{52} -7.64760i q^{53} +1.33826 q^{54} -3.78339 q^{56} -7.61048i q^{57} -4.30467i q^{58} -12.7500 q^{59} -10.1506 q^{61} +1.46280i q^{62} +1.27977i q^{63} -8.65227 q^{64} -1.55955 q^{66} -8.78903i q^{67} -1.12007i q^{68} +3.41811 q^{69} -15.0647 q^{71} +2.95630i q^{72} +13.1533i q^{73} +10.4401 q^{74} +1.59102 q^{76} -1.49139i q^{77} +4.83176i q^{78} +11.5155 q^{79} +1.00000 q^{81} +4.02234i q^{82} -10.7189i q^{83} -0.267545 q^{84} -4.58058 q^{86} -3.21661i q^{87} -3.44512i q^{88} +4.51584 q^{89} -4.62059 q^{91} +0.714580i q^{92} +1.09306i q^{93} +12.5992 q^{94} -1.17758 q^{96} -18.0135i q^{97} -7.17600i q^{98} -1.16535 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9	$ 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9} - 12 q^{11} + 20 q^{14} - 18 q^{16} + 18 q^{19} - 10 q^{21} - 6 q^{24} - 4 q^{26} + 56 q^{29} - 20 q^{31} - 14 q^{34} + 2 q^{36} + 14 q^{39} + 18 q^{44} + 10 q^{46} + 6 q^{49} + 14 q^{51} + 2 q^{54} - 8 q^{59} - 86 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} - 54 q^{71} - 10 q^{74} + 18 q^{76} - 20 q^{79} + 8 q^{81} + 10 q^{84} + 48 q^{86} + 18 q^{89} + 10 q^{91} + 44 q^{94} + 12 q^{99}+O(q^{100}) $ 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9 - 12 * q^11 + 20 * q^14 - 18 * q^16 + 18 * q^19 - 10 * q^21 - 6 * q^24 - 4 * q^26 + 56 * q^29 - 20 * q^31 - 14 * q^34 + 2 * q^36 + 14 * q^39 + 18 * q^44 + 10 * q^46 + 6 * q^49 + 14 * q^51 + 2 * q^54 - 8 * q^59 - 86 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 - 54 * q^71 - 10 * q^74 + 18 * q^76 - 20 * q^79 + 8 * q^81 + 10 * q^84 + 48 * q^86 + 18 * q^89 + 10 * q^91 + 44 * q^94 + 12 * 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 $

$2$	− 1.33826i	− 0.946294i	−0.880984	−	0.473147i	$-0.843118\pi$
$2$	− 1.33826i	− 0.946294i	0.880984	−	0.473147i	$-0.156882\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0.209057	0.104528
$5$	0	0
$5$	0	0
$6$	−1.33826	−0.546343
$7$	− 1.27977i	− 0.483709i	−0.970313	−	0.241854i	$-0.922244\pi$
$7$	− 1.27977i	− 0.483709i	0.970313	−	0.241854i	$-0.0777556\pi$
$8$	− 2.95630i	− 1.04521i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.16535	0.351367	0.175683	−	0.984447i	$-0.443786\pi$
$11$	1.16535	0.351367	0.175683	+	0.984447i	$0.443786\pi$
$12$	− 0.209057i	− 0.0603495i
$13$	− 3.61048i	− 1.00137i	−0.865631	−	0.500683i	$-0.833082\pi$
$13$	− 3.61048i	− 1.00137i	0.865631	−	0.500683i	$-0.166918\pi$
$14$	−1.71267	−0.457730
$15$	0	0
$16$	−3.53818	−0.884545
$17$	− 5.35772i	− 1.29944i	−0.760175	−	0.649718i	$-0.774887\pi$
$17$	− 5.35772i	− 1.29944i	0.760175	−	0.649718i	$-0.225113\pi$
$18$	1.33826i	0.315431i
$19$	7.61048	1.74596	0.872982	−	0.487753i	$-0.162183\pi$
$19$	7.61048	1.74596	0.872982	+	0.487753i	$0.162183\pi$
$20$	0	0
$21$	−1.27977	−0.279269
$22$	− 1.55955i	− 0.332496i
$23$	3.41811i	0.712726i	0.934348	+	0.356363i	$0.115983\pi$
$23$	3.41811i	0.712726i	−0.934348	+	0.356363i	$0.884017\pi$
$24$	−2.95630	−0.603451
$25$	0	0
$26$	−4.83176	−0.947586
$27$	1.00000i	0.192450i
$28$	− 0.267545i	− 0.0505613i
$29$	3.21661	0.597310	0.298655	−	0.954361i	$-0.403462\pi$
$29$	3.21661	0.597310	0.298655	+	0.954361i	$0.403462\pi$
$30$	0	0
$31$	−1.09306	−0.196319	−0.0981594	−	0.995171i	$-0.531295\pi$
$31$	−1.09306	−0.196319	−0.0981594	+	0.995171i	$0.531295\pi$
$32$	− 1.17758i	− 0.208169i
$33$	− 1.16535i	− 0.202862i
$34$	−7.17002	−1.22965
$35$	0	0
$36$	−0.209057	−0.0348428
$37$	7.80126i	1.28252i	0.767324	+	0.641260i	$0.221588\pi$
$37$	7.80126i	1.28252i	−0.767324	+	0.641260i	$0.778412\pi$
$38$	− 10.1848i	− 1.65219i
$39$	−3.61048	−0.578139
$40$	0	0
$41$	−3.00565	−0.469403	−0.234702	−	0.972067i	$-0.575411\pi$
$41$	−3.00565	−0.469403	−0.234702	+	0.972067i	$0.575411\pi$
$42$	1.71267i	0.264271i
$43$	− 3.42278i	− 0.521970i	−0.965343	−	0.260985i	$-0.915953\pi$
$43$	− 3.42278i	− 0.521970i	0.965343	−	0.260985i	$-0.0840473\pi$
$44$	0.243625	0.0367278
$45$	0	0
$46$	4.57433	0.674448
$47$	9.41462i	1.37326i	0.727005	+	0.686632i	$0.240912\pi$
$47$	9.41462i	1.37326i	−0.727005	+	0.686632i	$0.759088\pi$
$48$	3.53818i	0.510692i
$49$	5.36218	0.766026
$50$	0	0
$51$	−5.35772	−0.750230
$52$	− 0.754795i	− 0.104671i
$53$	− 7.64760i	− 1.05048i	−0.850954	−	0.525239i	$-0.823976\pi$
$53$	− 7.64760i	− 1.05048i	0.850954	−	0.525239i	$-0.176024\pi$
$54$	1.33826	0.182114
$55$	0	0
$56$	−3.78339	−0.505576
$57$	− 7.61048i	− 1.00803i
$58$	− 4.30467i	− 0.565231i
$59$	−12.7500	−1.65991	−0.829954	−	0.557832i	$-0.811634\pi$
$59$	−12.7500	−1.65991	−0.829954	+	0.557832i	$0.811634\pi$
$60$	0	0
$61$	−10.1506	−1.29965	−0.649824	−	0.760085i	$-0.725157\pi$
$61$	−10.1506	−1.29965	−0.649824	+	0.760085i	$0.725157\pi$
$62$	1.46280i	0.185775i
$63$	1.27977i	0.161236i
$64$	−8.65227	−1.08153
$65$	0	0
$66$	−1.55955	−0.191967
$67$	− 8.78903i	− 1.07375i	−0.843661	−	0.536876i	$-0.819604\pi$
$67$	− 8.78903i	− 1.07375i	0.843661	−	0.536876i	$-0.180396\pi$
$68$	− 1.12007i	− 0.135828i
$69$	3.41811	0.411493
$70$	0	0
$71$	−15.0647	−1.78786	−0.893928	−	0.448211i	$-0.852061\pi$
$71$	−15.0647	−1.78786	−0.893928	+	0.448211i	$0.852061\pi$
$72$	2.95630i	0.348403i
$73$	13.1533i	1.53948i	0.638357	+	0.769740i	$0.279614\pi$
$73$	13.1533i	1.53948i	−0.638357	+	0.769740i	$0.720386\pi$
$74$	10.4401	1.21364
$75$	0	0
$76$	1.59102	0.182503
$77$	− 1.49139i	− 0.169959i
$78$	4.83176i	0.547089i
$79$	11.5155	1.29560	0.647798	−	0.761812i	$-0.275690\pi$
$79$	11.5155	1.29560	0.647798	+	0.761812i	$0.275690\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.02234i	0.444193i
$83$	− 10.7189i	− 1.17655i	−0.808659	−	0.588277i	$-0.799806\pi$
$83$	− 10.7189i	− 1.17655i	0.808659	−	0.588277i	$-0.200194\pi$
$84$	−0.267545	−0.0291916
$85$	0	0
$86$	−4.58058	−0.493937
$87$	− 3.21661i	− 0.344857i
$88$	− 3.44512i	− 0.367252i
$89$	4.51584	0.478678	0.239339	−	0.970936i	$-0.423069\pi$
$89$	4.51584	0.478678	0.239339	+	0.970936i	$0.423069\pi$
$90$	0	0
$91$	−4.62059	−0.484369
$92$	0.714580i	0.0745002i
$93$	1.09306i	0.113345i
$94$	12.5992	1.29951
$95$	0	0
$96$	−1.17758	−0.120186
$97$	− 18.0135i	− 1.82899i	−0.404596	−	0.914496i	$-0.632588\pi$
$97$	− 18.0135i	− 1.82899i	0.404596	−	0.914496i	$-0.367412\pi$
$98$	− 7.17600i	− 0.724885i
$99$	−1.16535	−0.117122
$100$	0	0
$101$	−6.54634	−0.651385	−0.325693	−	0.945476i	$-0.605597\pi$
$101$	−6.54634	−0.651385	−0.325693	+	0.945476i	$0.605597\pi$
$102$	7.17002i	0.709938i
$103$	11.1588i	1.09951i	0.835327	+	0.549753i	$0.185278\pi$
$103$	11.1588i	1.09951i	−0.835327	+	0.549753i	$0.814722\pi$
$104$	−10.6736	−1.04664
$105$	0	0
$106$	−10.2345	−0.994061
$107$	3.72681i	0.360284i	0.983641	+	0.180142i	$0.0576557\pi$
$107$	3.72681i	0.360284i	−0.983641	+	0.180142i	$0.942344\pi$
$108$	0.209057i	0.0201165i
$109$	6.99533	0.670031	0.335016	−	0.942213i	$-0.391258\pi$
$109$	6.99533	0.670031	0.335016	+	0.942213i	$0.391258\pi$
$110$	0	0
$111$	7.80126	0.740463
$112$	4.52807i	0.427862i
$113$	− 5.59858i	− 0.526670i	−0.964704	−	0.263335i	$-0.915178\pi$
$113$	− 5.59858i	− 0.526670i	0.964704	−	0.263335i	$-0.0848225\pi$
$114$	−10.1848	−0.953895
$115$	0	0
$116$	0.672455	0.0624359
$117$	3.61048i	0.333789i
$118$	17.0628i	1.57076i
$119$	−6.85666	−0.628549
$120$	0	0
$121$	−9.64195	−0.876541
$122$	13.5841i	1.22985i
$123$	3.00565i	0.271010i
$124$	−0.228511	−0.0205209
$125$	0	0
$126$	1.71267	0.152577
$127$	4.38919i	0.389478i	0.980855	+	0.194739i	$0.0623860\pi$
$127$	4.38919i	0.389478i	−0.980855	+	0.194739i	$0.937614\pi$
$128$	9.22384i	0.815280i
$129$	−3.42278	−0.301359
$130$	0	0
$131$	−4.99244	−0.436192	−0.218096	−	0.975927i	$-0.569985\pi$
$131$	−4.99244	−0.436192	−0.218096	+	0.975927i	$0.569985\pi$
$132$	− 0.243625i	− 0.0212048i
$133$	− 9.73968i	− 0.844537i
$134$	−11.7620	−1.01608
$135$	0	0
$136$	−15.8390	−1.35818
$137$	− 12.2056i	− 1.04279i	−0.853315	−	0.521396i	$-0.825411\pi$
$137$	− 12.2056i	− 1.04279i	0.853315	−	0.521396i	$-0.174589\pi$
$138$	− 4.57433i	− 0.389393i
$139$	5.20617	0.441582	0.220791	−	0.975321i	$-0.429136\pi$
$139$	5.20617	0.441582	0.220791	+	0.975321i	$0.429136\pi$
$140$	0	0
$141$	9.41462	0.792854
$142$	20.1606i	1.69184i
$143$	− 4.20748i	− 0.351847i
$144$	3.53818	0.294848
$145$	0	0
$146$	17.6026	1.45680
$147$	− 5.36218i	− 0.442265i
$148$	1.63091i	0.134060i
$149$	21.6208	1.77124	0.885622	−	0.464406i	$-0.153732\pi$
$149$	21.6208	1.77124	0.885622	+	0.464406i	$0.153732\pi$
$150$	0	0
$151$	11.8918	0.967743	0.483872	−	0.875139i	$-0.339230\pi$
$151$	11.8918	0.967743	0.483872	+	0.875139i	$0.339230\pi$
$152$	− 22.4988i	− 1.82490i
$153$	5.35772i	0.433146i
$154$	−1.99586	−0.160831
$155$	0	0
$156$	−0.754795	−0.0604320
$157$	12.1112i	0.966579i	0.875461	+	0.483290i	$0.160558\pi$
$157$	12.1112i	0.966579i	−0.875461	+	0.483290i	$0.839442\pi$
$158$	− 15.4108i	− 1.22601i
$159$	−7.64760	−0.606494
$160$	0	0
$161$	4.37441	0.344752
$162$	− 1.33826i	− 0.105144i
$163$	8.26913i	0.647688i	0.946111	+	0.323844i	$0.104975\pi$
$163$	8.26913i	0.647688i	−0.946111	+	0.323844i	$0.895025\pi$
$164$	−0.628351	−0.0490660
$165$	0	0
$166$	−14.3447	−1.11337
$167$	− 6.26564i	− 0.484849i	−0.970170	−	0.242425i	$-0.922057\pi$
$167$	− 6.26564i	− 0.484849i	0.970170	−	0.242425i	$-0.0779427\pi$
$168$	3.78339i	0.291895i
$169$	−0.0355444	−0.00273418
$170$	0	0
$171$	−7.61048	−0.581988
$172$	− 0.715557i	− 0.0545607i
$173$	6.65609i	0.506053i	0.967459	+	0.253027i	$0.0814260\pi$
$173$	6.65609i	0.506053i	−0.967459	+	0.253027i	$0.918574\pi$
$174$	−4.30467	−0.326336
$175$	0	0
$176$	−4.12323	−0.310800
$177$	12.7500i	0.958349i
$178$	− 6.04337i	− 0.452970i
$179$	−21.7154	−1.62309	−0.811544	−	0.584292i	$-0.801372\pi$
$179$	−21.7154	−1.62309	−0.811544	+	0.584292i	$0.801372\pi$
$180$	0	0
$181$	12.9474	0.962375	0.481188	−	0.876618i	$-0.340206\pi$
$181$	12.9474	0.962375	0.481188	+	0.876618i	$0.340206\pi$
$182$	6.18356i	0.458356i
$183$	10.1506i	0.750352i
$184$	10.1050	0.744947
$185$	0	0
$186$	1.46280	0.107257
$187$	− 6.24362i	− 0.456579i
$188$	1.96819i	0.143545i
$189$	1.27977	0.0930898
$190$	0	0
$191$	15.7683	1.14095	0.570476	−	0.821314i	$-0.306759\pi$
$191$	15.7683	1.14095	0.570476	+	0.821314i	$0.306759\pi$
$192$	8.65227i	0.624424i
$193$	6.93514i	0.499202i	0.968349	+	0.249601i	$0.0802995\pi$
$193$	6.93514i	0.499202i	−0.968349	+	0.249601i	$0.919701\pi$
$194$	−24.1067	−1.73076
$195$	0	0
$196$	1.12100	0.0800715
$197$	7.43948i	0.530041i	0.964243	+	0.265020i	$0.0853787\pi$
$197$	7.43948i	0.530041i	−0.964243	+	0.265020i	$0.914621\pi$
$198$	1.55955i	0.110832i
$199$	14.2398	1.00943	0.504716	−	0.863285i	$-0.331597\pi$
$199$	14.2398	1.00943	0.504716	+	0.863285i	$0.331597\pi$
$200$	0	0
$201$	−8.78903	−0.619931
$202$	8.76072i	0.616402i
$203$	− 4.11653i	− 0.288924i
$204$	−1.12007	−0.0784204
$205$	0	0
$206$	14.9334	1.04046
$207$	− 3.41811i	− 0.237575i
$208$	12.7745i	0.885754i
$209$	8.86889	0.613474
$210$	0	0
$211$	6.89569	0.474719	0.237360	−	0.971422i	$-0.423718\pi$
$211$	6.89569	0.474719	0.237360	+	0.971422i	$0.423718\pi$
$212$	− 1.59878i	− 0.109805i
$213$	15.0647i	1.03222i
$214$	4.98744	0.340935
$215$	0	0
$216$	2.95630	0.201150
$217$	1.39886i	0.0949611i
$218$	− 9.36158i	− 0.634046i
$219$	13.1533	0.888820
$220$	0	0
$221$	−19.3439	−1.30121
$222$	− 10.4401i	− 0.700695i
$223$	13.3983i	0.897219i	0.893728	+	0.448609i	$0.148080\pi$
$223$	13.3983i	0.897219i	−0.893728	+	0.448609i	$0.851920\pi$
$224$	−1.50703	−0.100693
$225$	0	0
$226$	−7.49236	−0.498385
$227$	0.849831i	0.0564053i	0.999602	+	0.0282026i	$0.00897837\pi$
$227$	0.849831i	0.0564053i	−0.999602	+	0.0282026i	$0.991022\pi$
$228$	− 1.59102i	− 0.105368i
$229$	8.79561	0.581231	0.290615	−	0.956840i	$-0.406140\pi$
$229$	8.79561	0.581231	0.290615	+	0.956840i	$0.406140\pi$
$230$	0	0
$231$	−1.49139	−0.0981260
$232$	− 9.50926i	− 0.624314i
$233$	21.2093i	1.38947i	0.719267	+	0.694734i	$0.244478\pi$
$233$	21.2093i	1.38947i	−0.719267	+	0.694734i	$0.755522\pi$
$234$	4.83176	0.315862
$235$	0	0
$236$	−2.66548	−0.173508
$237$	− 11.5155i	− 0.748013i
$238$	9.17600i	0.594792i
$239$	16.7972	1.08652	0.543261	−	0.839564i	$-0.317189\pi$
$239$	16.7972	1.08652	0.543261	+	0.839564i	$0.317189\pi$
$240$	0	0
$241$	9.51860	0.613147	0.306573	−	0.951847i	$-0.400817\pi$
$241$	9.51860	0.613147	0.306573	+	0.951847i	$0.400817\pi$
$242$	12.9035i	0.829465i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.12205	−0.135850
$245$	0	0
$246$	4.02234	0.256455
$247$	− 27.4775i	− 1.74835i
$248$	3.23140i	0.205194i
$249$	−10.7189	−0.679284
$250$	0	0
$251$	12.9486	0.817309	0.408655	−	0.912689i	$-0.365998\pi$
$251$	12.9486	0.817309	0.408655	+	0.912689i	$0.365998\pi$
$252$	0.267545i	0.0168538i
$253$	3.98331i	0.250428i
$254$	5.87389	0.368560
$255$	0	0
$256$	−4.96064	−0.310040
$257$	16.6837i	1.04070i	0.853952	+	0.520352i	$0.174199\pi$
$257$	16.6837i	1.04070i	−0.853952	+	0.520352i	$0.825801\pi$
$258$	4.58058i	0.285174i
$259$	9.98384	0.620366
$260$	0	0
$261$	−3.21661	−0.199103
$262$	6.68119i	0.412765i
$263$	− 9.45426i	− 0.582975i	−0.956575	−	0.291487i	$-0.905850\pi$
$263$	− 9.45426i	− 0.582975i	0.956575	−	0.291487i	$-0.0941501\pi$
$264$	−3.44512	−0.212033
$265$	0	0
$266$	−13.0342	−0.799180
$267$	− 4.51584i	− 0.276365i
$268$	− 1.83741i	− 0.112238i
$269$	−31.7384	−1.93513	−0.967563	−	0.252631i	$-0.918704\pi$
$269$	−31.7384	−1.93513	−0.967563	+	0.252631i	$0.918704\pi$
$270$	0	0
$271$	19.0556	1.15755	0.578773	−	0.815489i	$-0.303532\pi$
$271$	19.0556	1.15755	0.578773	+	0.815489i	$0.303532\pi$
$272$	18.9566i	1.14941i
$273$	4.62059i	0.279651i
$274$	−16.3342	−0.986787
$275$	0	0
$276$	0.714580	0.0430127
$277$	2.71929i	0.163387i	0.996658	+	0.0816933i	$0.0260328\pi$
$277$	2.71929i	0.163387i	−0.996658	+	0.0816933i	$0.973967\pi$
$278$	− 6.96722i	− 0.417866i
$279$	1.09306	0.0654396
$280$	0	0
$281$	−14.6076	−0.871416	−0.435708	−	0.900088i	$-0.643502\pi$
$281$	−14.6076	−0.871416	−0.435708	+	0.900088i	$0.643502\pi$
$282$	− 12.5992i	− 0.750273i
$283$	18.0890i	1.07528i	0.843175	+	0.537639i	$0.180684\pi$
$283$	18.0890i	1.07528i	−0.843175	+	0.537639i	$0.819316\pi$
$284$	−3.14939	−0.186882
$285$	0	0
$286$	−5.63070	−0.332950
$287$	3.84655i	0.227054i
$288$	1.17758i	0.0693895i
$289$	−11.7051	−0.688536
$290$	0	0
$291$	−18.0135	−1.05597
$292$	2.74979i	0.160920i
$293$	− 0.293910i	− 0.0171704i	−0.999963	−	0.00858521i	$-0.997267\pi$
$293$	− 0.293910i	− 0.0171704i	0.999963	−	0.00858521i	$-0.00273279\pi$
$294$	−7.17600	−0.418513
$295$	0	0
$296$	23.0628	1.34050
$297$	1.16535i	0.0676206i
$298$	− 28.9343i	− 1.67612i
$299$	12.3410	0.713700
$300$	0	0
$301$	−4.38039	−0.252481
$302$	− 15.9144i	− 0.915769i
$303$	6.54634i	0.376078i
$304$	−26.9272	−1.54438
$305$	0	0
$306$	7.17002	0.409883
$307$	1.34411i	0.0767125i	0.999264	+	0.0383563i	$0.0122122\pi$
$307$	1.34411i	0.0767125i	−0.999264	+	0.0383563i	$0.987788\pi$
$308$	− 0.311785i	− 0.0177656i
$309$	11.1588	0.634800
$310$	0	0
$311$	19.9569	1.13165	0.565826	−	0.824525i	$-0.308557\pi$
$311$	19.9569	1.13165	0.565826	+	0.824525i	$0.308557\pi$
$312$	10.6736i	0.604276i
$313$	14.1446i	0.799500i	0.916624	+	0.399750i	$0.130903\pi$
$313$	14.1446i	0.799500i	−0.916624	+	0.399750i	$0.869097\pi$
$314$	16.2080	0.914668
$315$	0	0
$316$	2.40740	0.135427
$317$	− 17.5897i	− 0.987937i	−0.869480	−	0.493968i	$-0.835546\pi$
$317$	− 17.5897i	− 0.987937i	0.869480	−	0.493968i	$-0.164454\pi$
$318$	10.2345i	0.573922i
$319$	3.74849	0.209875
$320$	0	0
$321$	3.72681	0.208010
$322$	− 5.85410i	− 0.326236i
$323$	− 40.7748i	− 2.26877i
$324$	0.209057	0.0116143
$325$	0	0
$326$	11.0662	0.612903
$327$	− 6.99533i	− 0.386843i
$328$	8.88558i	0.490624i
$329$	12.0486	0.664260
$330$	0	0
$331$	8.00683	0.440095	0.220048	−	0.975489i	$-0.429379\pi$
$331$	8.00683	0.440095	0.220048	+	0.975489i	$0.429379\pi$
$332$	− 2.24086i	− 0.122983i
$333$	− 7.80126i	− 0.427506i
$334$	−8.38506	−0.458810
$335$	0	0
$336$	4.52807	0.247026
$337$	− 10.8474i	− 0.590895i	−0.955359	−	0.295448i	$-0.904531\pi$
$337$	− 10.8474i	− 0.590895i	0.955359	−	0.295448i	$-0.0954688\pi$
$338$	0.0475677i	0.00258734i
$339$	−5.59858	−0.304073
$340$	0	0
$341$	−1.27380	−0.0689799
$342$	10.1848i	0.550731i
$343$	− 15.8208i	− 0.854242i
$344$	−10.1188	−0.545567
$345$	0	0
$346$	8.90759	0.478875
$347$	− 1.86153i	− 0.0999323i	−0.998751	−	0.0499662i	$-0.984089\pi$
$347$	− 1.86153i	− 0.0999323i	0.998751	−	0.0499662i	$-0.0159113\pi$
$348$	− 0.672455i	− 0.0360474i
$349$	−24.4623	−1.30944	−0.654719	−	0.755872i	$-0.727213\pi$
$349$	−24.4623	−1.30944	−0.654719	+	0.755872i	$0.727213\pi$
$350$	0	0
$351$	3.61048	0.192713
$352$	− 1.37229i	− 0.0731436i
$353$	− 37.3621i	− 1.98858i	−0.106704	−	0.994291i	$-0.534030\pi$
$353$	− 37.3621i	− 1.98858i	0.106704	−	0.994291i	$-0.465970\pi$
$354$	17.0628	0.906879
$355$	0	0
$356$	0.944068	0.0500355
$357$	6.85666i	0.362893i
$358$	29.0609i	1.53592i
$359$	25.3360	1.33718	0.668592	−	0.743629i	$-0.266897\pi$
$359$	25.3360	1.33718	0.668592	+	0.743629i	$0.266897\pi$
$360$	0	0
$361$	38.9194	2.04839
$362$	− 17.3270i	− 0.910689i
$363$	9.64195i	0.506071i
$364$	−0.965966	−0.0506304
$365$	0	0
$366$	13.5841	0.710053
$367$	− 23.7057i	− 1.23743i	−0.785616	−	0.618714i	$-0.787654\pi$
$367$	− 23.7057i	− 1.23743i	0.785616	−	0.618714i	$-0.212346\pi$
$368$	− 12.0939i	− 0.630438i
$369$	3.00565	0.156468
$370$	0	0
$371$	−9.78719	−0.508126
$372$	0.228511i	0.0118477i
$373$	− 2.93244i	− 0.151836i	−0.997114	−	0.0759181i	$-0.975811\pi$
$373$	− 2.93244i	− 0.151836i	0.997114	−	0.0759181i	$-0.0241888\pi$
$374$	−8.35560	−0.432058
$375$	0	0
$376$	27.8324	1.43535
$377$	− 11.6135i	− 0.598126i
$378$	− 1.71267i	− 0.0880903i
$379$	17.7349	0.910980	0.455490	−	0.890241i	$-0.349464\pi$
$379$	17.7349	0.910980	0.455490	+	0.890241i	$0.349464\pi$
$380$	0	0
$381$	4.38919	0.224865
$382$	− 21.1021i	− 1.07968i
$383$	− 4.40524i	− 0.225097i	−0.993646	−	0.112549i	$-0.964099\pi$
$383$	− 4.40524i	− 0.225097i	0.993646	−	0.112549i	$-0.0359014\pi$
$384$	9.22384	0.470702
$385$	0	0
$386$	9.28102	0.472392
$387$	3.42278i	0.173990i
$388$	− 3.76584i	− 0.191182i
$389$	31.3215	1.58806	0.794031	−	0.607877i	$-0.207979\pi$
$389$	31.3215	1.58806	0.794031	+	0.607877i	$0.207979\pi$
$390$	0	0
$391$	18.3133	0.926142
$392$	− 15.8522i	− 0.800657i
$393$	4.99244i	0.251835i
$394$	9.95596	0.501574
$395$	0	0
$396$	−0.243625	−0.0122426
$397$	16.4281i	0.824501i	0.911071	+	0.412250i	$0.135257\pi$
$397$	16.4281i	0.824501i	−0.911071	+	0.412250i	$0.864743\pi$
$398$	− 19.0566i	− 0.955220i
$399$	−9.73968	−0.487594
$400$	0	0
$401$	−15.1813	−0.758120	−0.379060	−	0.925372i	$-0.623753\pi$
$401$	−15.1813	−0.758120	−0.379060	+	0.925372i	$0.623753\pi$
$402$	11.7620i	0.586636i
$403$	3.94646i	0.196587i
$404$	−1.36856	−0.0680883
$405$	0	0
$406$	−5.50900	−0.273407
$407$	9.09122i	0.450635i
$408$	15.8390i	0.784147i
$409$	14.8421	0.733897	0.366948	−	0.930241i	$-0.380403\pi$
$409$	14.8421	0.733897	0.366948	+	0.930241i	$0.380403\pi$
$410$	0	0
$411$	−12.2056	−0.602056
$412$	2.33282i	0.114930i
$413$	16.3171i	0.802912i
$414$	−4.57433	−0.224816
$415$	0	0
$416$	−4.25162	−0.208453
$417$	− 5.20617i	− 0.254947i
$418$	− 11.8689i	− 0.580526i
$419$	35.2636	1.72274	0.861370	−	0.507978i	$-0.169607\pi$
$419$	35.2636	1.72274	0.861370	+	0.507978i	$0.169607\pi$
$420$	0	0
$421$	−11.2827	−0.549887	−0.274943	−	0.961460i	$-0.588659\pi$
$421$	−11.2827	−0.549887	−0.274943	+	0.961460i	$0.588659\pi$
$422$	− 9.22824i	− 0.449224i
$423$	− 9.41462i	− 0.457755i
$424$	−22.6086	−1.09797
$425$	0	0
$426$	20.1606	0.976782
$427$	12.9904i	0.628651i
$428$	0.779115i	0.0376599i
$429$	−4.20748	−0.203139
$430$	0	0
$431$	25.8249	1.24394	0.621971	−	0.783041i	$-0.286332\pi$
$431$	25.8249	1.24394	0.621971	+	0.783041i	$0.286332\pi$
$432$	− 3.53818i	− 0.170231i
$433$	− 9.99902i	− 0.480522i	−0.970708	−	0.240261i	$-0.922767\pi$
$433$	− 9.99902i	− 0.480522i	0.970708	−	0.240261i	$-0.0772331\pi$
$434$	1.87205	0.0898610
$435$	0	0
$436$	1.46242	0.0700373
$437$	26.0135i	1.24439i
$438$	− 17.6026i	− 0.841084i
$439$	−9.20385	−0.439276	−0.219638	−	0.975581i	$-0.570488\pi$
$439$	−9.20385	−0.439276	−0.219638	+	0.975581i	$0.570488\pi$
$440$	0	0
$441$	−5.36218	−0.255342
$442$	25.8872i	1.23133i
$443$	− 9.12790i	− 0.433680i	−0.976207	−	0.216840i	$-0.930425\pi$
$443$	− 9.12790i	− 0.433680i	0.976207	−	0.216840i	$-0.0695749\pi$
$444$	1.63091	0.0773994
$445$	0	0
$446$	17.9305	0.849032
$447$	− 21.6208i	− 1.02263i
$448$	11.0729i	0.523147i
$449$	−35.4831	−1.67455	−0.837275	−	0.546781i	$-0.815853\pi$
$449$	−35.4831	−1.67455	−0.837275	+	0.546781i	$0.815853\pi$
$450$	0	0
$451$	−3.50264	−0.164933
$452$	− 1.17042i	− 0.0550520i
$453$	− 11.8918i	− 0.558727i
$454$	1.13730	0.0533759
$455$	0	0
$456$	−22.4988	−1.05360
$457$	− 18.7516i	− 0.877162i	−0.898692	−	0.438581i	$-0.855481\pi$
$457$	− 18.7516i	− 0.877162i	0.898692	−	0.438581i	$-0.144519\pi$
$458$	− 11.7708i	− 0.550015i
$459$	5.35772	0.250077
$460$	0	0
$461$	−16.0877	−0.749280	−0.374640	−	0.927170i	$-0.622234\pi$
$461$	−16.0877	−0.749280	−0.374640	+	0.927170i	$0.622234\pi$
$462$	1.99586i	0.0928560i
$463$	− 21.1040i	− 0.980787i	−0.871501	−	0.490393i	$-0.836853\pi$
$463$	− 21.1040i	− 0.980787i	0.871501	−	0.490393i	$-0.163147\pi$
$464$	−11.3810	−0.528348
$465$	0	0
$466$	28.3836	1.31484
$467$	− 18.0122i	− 0.833504i	−0.909020	−	0.416752i	$-0.863168\pi$
$467$	− 18.0122i	− 0.833504i	0.909020	−	0.416752i	$-0.136832\pi$
$468$	0.754795i	0.0348904i
$469$	−11.2480	−0.519383
$470$	0	0
$471$	12.1112	0.558055
$472$	37.6928i	1.73495i
$473$	− 3.98875i	− 0.183403i
$474$	−15.4108	−0.707840
$475$	0	0
$476$	−1.43343	−0.0657012
$477$	7.64760i	0.350160i
$478$	− 22.4791i	− 1.02817i
$479$	−7.31414	−0.334191	−0.167096	−	0.985941i	$-0.553439\pi$
$479$	−7.31414	−0.334191	−0.167096	+	0.985941i	$0.553439\pi$
$480$	0	0
$481$	28.1663	1.28427
$482$	− 12.7384i	− 0.580217i
$483$	− 4.37441i	− 0.199043i
$484$	−2.01572	−0.0916235
$485$	0	0
$486$	−1.33826	−0.0607048
$487$	− 11.9433i	− 0.541202i	−0.962692	−	0.270601i	$-0.912778\pi$
$487$	− 11.9433i	− 0.541202i	0.962692	−	0.270601i	$-0.0872225\pi$
$488$	30.0081i	1.35840i
$489$	8.26913	0.373943
$490$	0	0
$491$	11.1658	0.503904	0.251952	−	0.967740i	$-0.418928\pi$
$491$	11.1658	0.503904	0.251952	+	0.967740i	$0.418928\pi$
$492$	0.628351i	0.0283283i
$493$	− 17.2337i	− 0.776167i
$494$	−36.7720	−1.65445
$495$	0	0
$496$	3.86743	0.173653
$497$	19.2794i	0.864801i
$498$	14.3447i	0.642802i
$499$	−30.6045	−1.37005	−0.685023	−	0.728522i	$-0.740208\pi$
$499$	−30.6045	−1.37005	−0.685023	+	0.728522i	$0.740208\pi$
$500$	0	0
$501$	−6.26564	−0.279928
$502$	− 17.3286i	− 0.773414i
$503$	− 1.90950i	− 0.0851404i	−0.999093	−	0.0425702i	$-0.986445\pi$
$503$	− 1.90950i	− 0.0851404i	0.999093	−	0.0425702i	$-0.0135546\pi$
$504$	3.78339	0.168525
$505$	0	0
$506$	5.33070	0.236979
$507$	0.0355444i	0.00157858i
$508$	0.917591i	0.0407115i
$509$	−2.52518	−0.111927	−0.0559634	−	0.998433i	$-0.517823\pi$
$509$	−2.52518	−0.111927	−0.0559634	+	0.998433i	$0.517823\pi$
$510$	0	0
$511$	16.8333	0.744660
$512$	25.0863i	1.10867i
$513$	7.61048i	0.336011i
$514$	22.3272	0.984811
$515$	0	0
$516$	−0.715557	−0.0315006
$517$	10.9714i	0.482520i
$518$	− 13.3610i	− 0.587048i
$519$	6.65609	0.292170
$520$	0	0
$521$	3.65479	0.160119	0.0800595	−	0.996790i	$-0.474489\pi$
$521$	3.65479	0.160119	0.0800595	+	0.996790i	$0.474489\pi$
$522$	4.30467i	0.188410i
$523$	− 36.3215i	− 1.58823i	−0.607768	−	0.794114i	$-0.707935\pi$
$523$	− 36.3215i	− 1.58823i	0.607768	−	0.794114i	$-0.292065\pi$
$524$	−1.04370	−0.0455945
$525$	0	0
$526$	−12.6523	−0.551665
$527$	5.85629i	0.255104i
$528$	4.12323i	0.179440i
$529$	11.3165	0.492022
$530$	0	0
$531$	12.7500	0.553303
$532$	− 2.03615i	− 0.0882782i
$533$	10.8518i	0.470044i
$534$	−6.04337	−0.261522
$535$	0	0
$536$	−25.9830	−1.12229
$537$	21.7154i	0.937090i
$538$	42.4743i	1.83120i
$539$	6.24883	0.269156
$540$	0	0
$541$	−13.3159	−0.572496	−0.286248	−	0.958156i	$-0.592408\pi$
$541$	−13.3159	−0.572496	−0.286248	+	0.958156i	$0.592408\pi$
$542$	− 25.5014i	− 1.09538i
$543$	− 12.9474i	− 0.555627i
$544$	−6.30914	−0.270502
$545$	0	0
$546$	6.18356	0.264632
$547$	29.6657i	1.26841i	0.773163	+	0.634207i	$0.218673\pi$
$547$	29.6657i	1.26841i	−0.773163	+	0.634207i	$0.781327\pi$
$548$	− 2.55166i	− 0.109001i
$549$	10.1506	0.433216
$550$	0	0
$551$	24.4800	1.04288
$552$	− 10.1050i	− 0.430095i
$553$	− 14.7372i	− 0.626691i
$554$	3.63913	0.154612
$555$	0	0
$556$	1.08839	0.0461578
$557$	42.4585i	1.79902i	0.436897	+	0.899512i	$0.356078\pi$
$557$	42.4585i	1.79902i	−0.436897	+	0.899512i	$0.643922\pi$
$558$	− 1.46280i	− 0.0619251i
$559$	−12.3579	−0.522683
$560$	0	0
$561$	−6.24362	−0.263606
$562$	19.5488i	0.824615i
$563$	− 41.0132i	− 1.72850i	−0.503063	−	0.864250i	$-0.667794\pi$
$563$	− 41.0132i	− 1.72850i	0.503063	−	0.864250i	$-0.332206\pi$
$564$	1.96819	0.0828758
$565$	0	0
$566$	24.2078	1.01753
$567$	− 1.27977i	− 0.0537454i
$568$	44.5358i	1.86868i
$569$	29.0619	1.21834	0.609169	−	0.793040i	$-0.291503\pi$
$569$	29.0619	1.21834	0.609169	+	0.793040i	$0.291503\pi$
$570$	0	0
$571$	−0.230751	−0.00965665	−0.00482832	−	0.999988i	$-0.501537\pi$
$571$	−0.230751	−0.00965665	−0.00482832	+	0.999988i	$0.501537\pi$
$572$	− 0.879602i	− 0.0367780i
$573$	− 15.7683i	− 0.658729i
$574$	5.14768	0.214860
$575$	0	0
$576$	8.65227	0.360511
$577$	37.4944i	1.56091i	0.625211	+	0.780456i	$0.285013\pi$
$577$	37.4944i	1.56091i	−0.625211	+	0.780456i	$0.714987\pi$
$578$	15.6645i	0.651557i
$579$	6.93514	0.288214
$580$	0	0
$581$	−13.7178	−0.569110
$582$	24.1067i	0.999256i
$583$	− 8.91215i	− 0.369103i
$584$	38.8851	1.60908
$585$	0	0
$586$	−0.393329	−0.0162483
$587$	15.3719i	0.634464i	0.948348	+	0.317232i	$0.102753\pi$
$587$	15.3719i	0.634464i	−0.948348	+	0.317232i	$0.897247\pi$
$588$	− 1.12100i	− 0.0462293i
$589$	−8.31868	−0.342765
$590$	0	0
$591$	7.43948	0.306019
$592$	− 27.6023i	− 1.13445i
$593$	8.30710i	0.341132i	0.985346	+	0.170566i	$0.0545596\pi$
$593$	8.30710i	0.341132i	−0.985346	+	0.170566i	$0.945440\pi$
$594$	1.55955	0.0639889
$595$	0	0
$596$	4.51998	0.185145
$597$	− 14.2398i	− 0.582796i
$598$	− 16.5155i	− 0.675369i
$599$	29.0293	1.18610	0.593052	−	0.805164i	$-0.297923\pi$
$599$	29.0293	1.18610	0.593052	+	0.805164i	$0.297923\pi$
$600$	0	0
$601$	−43.2988	−1.76620	−0.883098	−	0.469188i	$-0.844547\pi$
$601$	−43.2988	−1.76620	−0.883098	+	0.469188i	$0.844547\pi$
$602$	5.86210i	0.238921i
$603$	8.78903i	0.357917i
$604$	2.48607	0.101157
$605$	0	0
$606$	8.76072	0.355880
$607$	− 14.4484i	− 0.586443i	−0.956044	−	0.293222i	$-0.905273\pi$
$607$	− 14.4484i	− 0.586443i	0.956044	−	0.293222i	$-0.0947274\pi$
$608$	− 8.96194i	− 0.363455i
$609$	−4.11653	−0.166810
$610$	0	0
$611$	33.9913	1.37514
$612$	1.12007i	0.0452760i
$613$	− 8.63973i	− 0.348955i	−0.984661	−	0.174478i	$-0.944176\pi$
$613$	− 8.63973i	− 0.348955i	0.984661	−	0.174478i	$-0.0558237\pi$
$614$	1.79877	0.0725926
$615$	0	0
$616$	−4.40898	−0.177643
$617$	32.9268i	1.32559i	0.748803	+	0.662793i	$0.230629\pi$
$617$	32.9268i	1.32559i	−0.748803	+	0.662793i	$0.769371\pi$
$618$	− 14.9334i	− 0.600707i
$619$	−9.82425	−0.394870	−0.197435	−	0.980316i	$-0.563261\pi$
$619$	−9.82425	−0.394870	−0.197435	+	0.980316i	$0.563261\pi$
$620$	0	0
$621$	−3.41811	−0.137164
$622$	− 26.7075i	− 1.07087i
$623$	− 5.77925i	− 0.231541i
$624$	12.7745	0.511390
$625$	0	0
$626$	18.9292	0.756561
$627$	− 8.86889i	− 0.354189i
$628$	2.53193i	0.101035i
$629$	41.7969	1.66655
$630$	0	0
$631$	2.88051	0.114671	0.0573356	−	0.998355i	$-0.481739\pi$
$631$	2.88051	0.114671	0.0573356	+	0.998355i	$0.481739\pi$
$632$	− 34.0432i	− 1.35417i
$633$	− 6.89569i	− 0.274079i
$634$	−23.5396	−0.934878
$635$	0	0
$636$	−1.59878	−0.0633959
$637$	− 19.3600i	− 0.767072i
$638$	− 5.01646i	− 0.198603i
$639$	15.0647	0.595952
$640$	0	0
$641$	−18.6464	−0.736490	−0.368245	−	0.929729i	$-0.620041\pi$
$641$	−18.6464	−0.736490	−0.368245	+	0.929729i	$0.620041\pi$
$642$	− 4.98744i	− 0.196839i
$643$	1.96215i	0.0773795i	0.999251	+	0.0386897i	$0.0123184\pi$
$643$	1.96215i	0.0773795i	−0.999251	+	0.0386897i	$0.987682\pi$
$644$	0.914501	0.0360364
$645$	0	0
$646$	−54.5673	−2.14692
$647$	1.72726i	0.0679055i	0.999423	+	0.0339528i	$0.0108096\pi$
$647$	1.72726i	0.0679055i	−0.999423	+	0.0339528i	$0.989190\pi$
$648$	− 2.95630i	− 0.116134i
$649$	−14.8582	−0.583237
$650$	0	0
$651$	1.39886	0.0548258
$652$	1.72872i	0.0677018i
$653$	8.37928i	0.327907i	0.986468	+	0.163953i	$0.0524247\pi$
$653$	8.37928i	0.327907i	−0.986468	+	0.163953i	$0.947575\pi$
$654$	−9.36158	−0.366067
$655$	0	0
$656$	10.6345	0.415208
$657$	− 13.1533i	− 0.513160i
$658$	− 16.1241i	− 0.628585i
$659$	−31.2624	−1.21781	−0.608905	−	0.793243i	$-0.708391\pi$
$659$	−31.2624	−1.21781	−0.608905	+	0.793243i	$0.708391\pi$
$660$	0	0
$661$	−10.8798	−0.423175	−0.211588	−	0.977359i	$-0.567863\pi$
$661$	−10.8798	−0.423175	−0.211588	+	0.977359i	$0.567863\pi$
$662$	− 10.7152i	− 0.416459i
$663$	19.3439i	0.751255i
$664$	−31.6883	−1.22974
$665$	0	0
$666$	−10.4401	−0.404547
$667$	10.9948i	0.425719i
$668$	− 1.30987i	− 0.0506806i
$669$	13.3983	0.518009
$670$	0	0
$671$	−11.8290	−0.456653
$672$	1.50703i	0.0581351i
$673$	28.9803i	1.11711i	0.829468	+	0.558554i	$0.188644\pi$
$673$	28.9803i	1.11711i	−0.829468	+	0.558554i	$0.811356\pi$
$674$	−14.5166	−0.559160
$675$	0	0
$676$	−0.00743080	−0.000285800 0
$677$	5.89919i	0.226724i	0.993554	+	0.113362i	$0.0361620\pi$
$677$	5.89919i	0.226724i	−0.993554	+	0.113362i	$0.963838\pi$
$678$	7.49236i	0.287742i
$679$	−23.0532	−0.884699
$680$	0	0
$681$	0.849831	0.0325656
$682$	1.70467i	0.0652752i
$683$	45.9083i	1.75663i	0.478081	+	0.878316i	$0.341332\pi$
$683$	45.9083i	1.75663i	−0.478081	+	0.878316i	$0.658668\pi$
$684$	−1.59102	−0.0608343
$685$	0	0
$686$	−21.1723	−0.808364
$687$	− 8.79561i	− 0.335574i
$688$	12.1104i	0.461706i
$689$	−27.6115	−1.05191
$690$	0	0
$691$	20.9475	0.796880	0.398440	−	0.917194i	$-0.369552\pi$
$691$	20.9475	0.796880	0.398440	+	0.917194i	$0.369552\pi$
$692$	1.39150i	0.0528970i
$693$	1.49139i	0.0566531i
$694$	−2.49122	−0.0945653
$695$	0	0
$696$	−9.50926	−0.360448
$697$	16.1034i	0.609960i
$698$	32.7370i	1.23911i
$699$	21.2093	0.802210
$700$	0	0
$701$	52.2223	1.97241	0.986204	−	0.165533i	$-0.0529346\pi$
$701$	52.2223	1.97241	0.986204	+	0.165533i	$0.0529346\pi$
$702$	− 4.83176i	− 0.182363i
$703$	59.3713i	2.23923i
$704$	−10.0829	−0.380015
$705$	0	0
$706$	−50.0002	−1.88178
$707$	8.37783i	0.315081i
$708$	2.66548i	0.100175i
$709$	46.9974	1.76503	0.882513	−	0.470288i	$-0.155850\pi$
$709$	46.9974	1.76503	0.882513	+	0.470288i	$0.155850\pi$
$710$	0	0
$711$	−11.5155	−0.431865
$712$	− 13.3502i	− 0.500318i
$713$	− 3.73619i	− 0.139921i
$714$	9.17600	0.343403
$715$	0	0
$716$	−4.53976	−0.169659
$717$	− 16.7972i	− 0.627304i
$718$	− 33.9062i	− 1.26537i
$719$	4.69037	0.174921	0.0874607	−	0.996168i	$-0.472125\pi$
$719$	4.69037	0.174921	0.0874607	+	0.996168i	$0.472125\pi$
$720$	0	0
$721$	14.2807	0.531841
$722$	− 52.0843i	− 1.93838i
$723$	− 9.51860i	− 0.354001i
$724$	2.70675	0.100596
$725$	0	0
$726$	12.9035	0.478892
$727$	− 27.7594i	− 1.02954i	−0.857329	−	0.514769i	$-0.827878\pi$
$727$	− 27.7594i	− 1.02954i	0.857329	−	0.514769i	$-0.172122\pi$
$728$	13.6598i	0.506267i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−18.3383	−0.678267
$732$	2.12205i	0.0784331i
$733$	− 30.6040i	− 1.13038i	−0.824960	−	0.565192i	$-0.808802\pi$
$733$	− 30.6040i	− 1.13038i	0.824960	−	0.565192i	$-0.191198\pi$
$734$	−31.7244	−1.17097
$735$	0	0
$736$	4.02510	0.148367
$737$	− 10.2423i	− 0.377281i
$738$	− 4.02234i	− 0.148064i
$739$	−4.93383	−0.181494	−0.0907469	−	0.995874i	$-0.528925\pi$
$739$	−4.93383	−0.181494	−0.0907469	+	0.995874i	$0.528925\pi$
$740$	0	0
$741$	−27.4775	−1.00941
$742$	13.0978i	0.480836i
$743$	21.7109i	0.796496i	0.917278	+	0.398248i	$0.130382\pi$
$743$	21.7109i	0.796496i	−0.917278	+	0.398248i	$0.869618\pi$
$744$	3.23140	0.118469
$745$	0	0
$746$	−3.92438	−0.143682
$747$	10.7189i	0.392185i
$748$	− 1.30527i	− 0.0477255i
$749$	4.76947	0.174273
$750$	0	0
$751$	−33.6080	−1.22637	−0.613186	−	0.789939i	$-0.710112\pi$
$751$	−33.6080	−1.22637	−0.613186	+	0.789939i	$0.710112\pi$
$752$	− 33.3106i	− 1.21471i
$753$	− 12.9486i	− 0.471874i
$754$	−15.5419	−0.566003
$755$	0	0
$756$	0.267545	0.00973053
$757$	4.88834i	0.177670i	0.996046	+	0.0888349i	$0.0283143\pi$
$757$	4.88834i	0.177670i	−0.996046	+	0.0888349i	$0.971686\pi$
$758$	− 23.7339i	− 0.862054i
$759$	3.98331	0.144585
$760$	0	0
$761$	−30.2366	−1.09608	−0.548038	−	0.836453i	$-0.684625\pi$
$761$	−30.2366	−1.09608	−0.548038	+	0.836453i	$0.684625\pi$
$762$	− 5.87389i	− 0.212788i
$763$	− 8.95243i	− 0.324100i
$764$	3.29647	0.119262
$765$	0	0
$766$	−5.89536	−0.213008
$767$	46.0336i	1.66218i
$768$	4.96064i	0.179002i
$769$	−51.0523	−1.84099	−0.920497	−	0.390751i	$-0.872215\pi$
$769$	−51.0523	−1.84099	−0.920497	+	0.390751i	$0.872215\pi$
$770$	0	0
$771$	16.6837	0.600851
$772$	1.44984i	0.0521808i
$773$	− 6.54233i	− 0.235311i	−0.993054	−	0.117656i	$-0.962462\pi$
$773$	− 6.54233i	− 0.235311i	0.993054	−	0.117656i	$-0.0375379\pi$
$774$	4.58058	0.164646
$775$	0	0
$776$	−53.2532	−1.91168
$777$	− 9.98384i	− 0.358168i
$778$	− 41.9163i	− 1.50277i
$779$	−22.8744	−0.819561
$780$	0	0
$781$	−17.5557	−0.628193
$782$	− 24.5080i	− 0.876403i
$783$	3.21661i	0.114952i
$784$	−18.9724	−0.677585
$785$	0	0
$786$	6.68119	0.238310
$787$	− 10.9765i	− 0.391269i	−0.980677	−	0.195635i	$-0.937323\pi$
$787$	− 10.9765i	− 0.391269i	0.980677	−	0.195635i	$-0.0626767\pi$
$788$	1.55527i	0.0554044i
$789$	−9.45426	−0.336581
$790$	0	0
$791$	−7.16491	−0.254755
$792$	3.44512i	0.122417i
$793$	36.6484i	1.30142i
$794$	21.9850	0.780220
$795$	0	0
$796$	2.97693	0.105514
$797$	− 12.6891i	− 0.449473i	−0.974420	−	0.224736i	$-0.927848\pi$
$797$	− 12.6891i	− 0.449473i	0.974420	−	0.224736i	$-0.0721521\pi$
$798$	13.0342i	0.461407i
$799$	50.4409	1.78447
$800$	0	0
$801$	−4.51584	−0.159559
$802$	20.3166i	0.717404i
$803$	15.3283i	0.540923i
$804$	−1.83741	−0.0648004
$805$	0	0
$806$	5.28139	0.186029
$807$	31.7384i	1.11725i
$808$	19.3529i	0.680833i
$809$	17.7818	0.625174	0.312587	−	0.949889i	$-0.398805\pi$
$809$	17.7818	0.625174	0.312587	+	0.949889i	$0.398805\pi$
$810$	0	0
$811$	37.0811	1.30210	0.651048	−	0.759037i	$-0.274330\pi$
$811$	37.0811	1.30210	0.651048	+	0.759037i	$0.274330\pi$
$812$	− 0.860590i	− 0.0302008i
$813$	− 19.0556i	− 0.668309i
$814$	12.1664	0.426433
$815$	0	0
$816$	18.9566	0.663613
$817$	− 26.0490i	− 0.911340i
$818$	− 19.8627i	− 0.694482i
$819$	4.62059	0.161456
$820$	0	0
$821$	−0.337853	−0.0117911	−0.00589557	−	0.999983i	$-0.501877\pi$
$821$	−0.337853	−0.0117911	−0.00589557	+	0.999983i	$0.501877\pi$
$822$	16.3342i	0.569722i
$823$	26.6629i	0.929409i	0.885466	+	0.464704i	$0.153839\pi$
$823$	26.6629i	0.929409i	−0.885466	+	0.464704i	$0.846161\pi$
$824$	32.9886	1.14921
$825$	0	0
$826$	21.8365	0.759791
$827$	43.4655i	1.51144i	0.654893	+	0.755722i	$0.272714\pi$
$827$	43.4655i	1.51144i	−0.654893	+	0.755722i	$0.727286\pi$
$828$	− 0.714580i	− 0.0248334i
$829$	−18.9493	−0.658137	−0.329068	−	0.944306i	$-0.606735\pi$
$829$	−18.9493	−0.658137	−0.329068	+	0.944306i	$0.606735\pi$
$830$	0	0
$831$	2.71929	0.0943312
$832$	31.2388i	1.08301i
$833$	− 28.7290i	− 0.995402i
$834$	−6.96722	−0.241255
$835$	0	0
$836$	1.85410	0.0641255
$837$	− 1.09306i	− 0.0377816i
$838$	− 47.1919i	− 1.63022i
$839$	−22.7357	−0.784922	−0.392461	−	0.919769i	$-0.628376\pi$
$839$	−22.7357	−0.784922	−0.392461	+	0.919769i	$0.628376\pi$
$840$	0	0
$841$	−18.6534	−0.643221
$842$	15.0992i	0.520355i
$843$	14.6076i	0.503112i
$844$	1.44159	0.0496217
$845$	0	0
$846$	−12.5992	−0.433170
$847$	12.3395i	0.423991i
$848$	27.0586i	0.929196i
$849$	18.0890	0.620813
$850$	0	0
$851$	−26.6656	−0.914085
$852$	3.14939i	0.107896i
$853$	− 35.9922i	− 1.23235i	−0.787610	−	0.616175i	$-0.788682\pi$
$853$	− 35.9922i	− 1.23235i	0.787610	−	0.616175i	$-0.211318\pi$
$854$	17.3846	0.594888
$855$	0	0
$856$	11.0175	0.376572
$857$	20.1724i	0.689076i	0.938772	+	0.344538i	$0.111964\pi$
$857$	20.1724i	0.689076i	−0.938772	+	0.344538i	$0.888036\pi$
$858$	5.63070i	0.192229i
$859$	−39.0077	−1.33093	−0.665464	−	0.746430i	$-0.731766\pi$
$859$	−39.0077	−1.33093	−0.665464	+	0.746430i	$0.731766\pi$
$860$	0	0
$861$	3.84655	0.131090
$862$	− 34.5605i	− 1.17713i
$863$	24.4962i	0.833861i	0.908938	+	0.416930i	$0.136894\pi$
$863$	24.4962i	0.833861i	−0.908938	+	0.416930i	$0.863106\pi$
$864$	1.17758	0.0400621
$865$	0	0
$866$	−13.3813	−0.454715
$867$	11.7051i	0.397526i
$868$	0.292442i	0.00992613i
$869$	13.4196	0.455230
$870$	0	0
$871$	−31.7326	−1.07522
$872$	− 20.6803i	− 0.700322i
$873$	18.0135i	0.609664i
$874$	34.8128	1.17756
$875$	0	0
$876$	2.74979	0.0929069
$877$	− 29.2278i	− 0.986952i	−0.869759	−	0.493476i	$-0.835726\pi$
$877$	− 29.2278i	− 0.986952i	0.869759	−	0.493476i	$-0.164274\pi$
$878$	12.3172i	0.415684i
$879$	−0.293910	−0.00991335
$880$	0	0
$881$	−3.72194	−0.125395	−0.0626977	−	0.998033i	$-0.519970\pi$
$881$	−3.72194	−0.125395	−0.0626977	+	0.998033i	$0.519970\pi$
$882$	7.17600i	0.241628i
$883$	− 7.01518i	− 0.236080i	−0.993009	−	0.118040i	$-0.962339\pi$
$883$	− 7.01518i	− 0.236080i	0.993009	−	0.118040i	$-0.0376611\pi$
$884$	−4.04398	−0.136014
$885$	0	0
$886$	−12.2155	−0.410388
$887$	− 27.4537i	− 0.921806i	−0.887451	−	0.460903i	$-0.847526\pi$
$887$	− 27.4537i	− 0.921806i	0.887451	−	0.460903i	$-0.152474\pi$
$888$	− 23.0628i	− 0.773938i
$889$	5.61717	0.188394
$890$	0	0
$891$	1.16535	0.0390408
$892$	2.80101i	0.0937849i
$893$	71.6498i	2.39767i
$894$	−28.9343	−0.967707
$895$	0	0
$896$	11.8044	0.394358
$897$	− 12.3410i	− 0.412055i
$898$	47.4857i	1.58462i
$899$	−3.51594	−0.117263
$900$	0	0
$901$	−40.9737	−1.36503
$902$	4.68744i	0.156075i
$903$	4.38039i	0.145770i
$904$	−16.5511	−0.550480
$905$	0	0
$906$	−15.9144	−0.528720
$907$	6.09167i	0.202271i	0.994873	+	0.101135i	$0.0322475\pi$
$907$	6.09167i	0.202271i	−0.994873	+	0.101135i	$0.967752\pi$
$908$	0.177663i	0.00589595i
$909$	6.54634	0.217128
$910$	0	0
$911$	−2.77586	−0.0919682	−0.0459841	−	0.998942i	$-0.514642\pi$
$911$	−2.77586	−0.0919682	−0.0459841	+	0.998942i	$0.514642\pi$
$912$	26.9272i	0.891650i
$913$	− 12.4913i	− 0.413402i
$914$	−25.0945	−0.830053
$915$	0	0
$916$	1.83878	0.0607551
$917$	6.38919i	0.210990i
$918$	− 7.17002i	− 0.236646i
$919$	23.7566	0.783659	0.391829	−	0.920038i	$-0.371842\pi$
$919$	23.7566	0.783659	0.391829	+	0.920038i	$0.371842\pi$
$920$	0	0
$921$	1.34411	0.0442900
$922$	21.5296i	0.709039i
$923$	54.3909i	1.79030i
$924$	−0.311785	−0.0102570
$925$	0	0
$926$	−28.2427	−0.928112
$927$	− 11.1588i	− 0.366502i
$928$	− 3.78782i	− 0.124341i
$929$	−28.7158	−0.942136	−0.471068	−	0.882097i	$-0.656131\pi$
$929$	−28.7158	−0.942136	−0.471068	+	0.882097i	$0.656131\pi$
$930$	0	0
$931$	40.8088	1.33745
$932$	4.43395i	0.145239i
$933$	− 19.9569i	− 0.653360i
$934$	−24.1050	−0.788739
$935$	0	0
$936$	10.6736	0.348879
$937$	13.1368i	0.429161i	0.976706	+	0.214580i	$0.0688384\pi$
$937$	13.1368i	0.429161i	−0.976706	+	0.214580i	$0.931162\pi$
$938$	15.0527i	0.491489i
$939$	14.1446	0.461591
$940$	0	0
$941$	−39.4446	−1.28586	−0.642929	−	0.765926i	$-0.722281\pi$
$941$	−39.4446	−1.28586	−0.642929	+	0.765926i	$0.722281\pi$
$942$	− 16.2080i	− 0.528084i
$943$	− 10.2736i	− 0.334556i
$944$	45.1118	1.46826
$945$	0	0
$946$	−5.33799	−0.173553
$947$	− 25.1266i	− 0.816504i	−0.912869	−	0.408252i	$-0.866138\pi$
$947$	− 25.1266i	− 0.816504i	0.912869	−	0.408252i	$-0.133862\pi$
$948$	− 2.40740i	− 0.0781886i
$949$	47.4898	1.54158
$950$	0	0
$951$	−17.5897	−0.570386
$952$	20.2703i	0.656964i
$953$	− 6.14850i	− 0.199169i	−0.995029	−	0.0995847i	$-0.968249\pi$
$953$	− 6.14850i	− 0.199169i	0.995029	−	0.0995847i	$-0.0317514\pi$
$954$	10.2345	0.331354
$955$	0	0
$956$	3.51158	0.113573
$957$	− 3.74849i	− 0.121171i
$958$	9.78823i	0.316243i
$959$	−15.6204	−0.504407
$960$	0	0
$961$	−29.8052	−0.961459
$962$	− 37.6938i	− 1.21530i
$963$	− 3.72681i	− 0.120095i
$964$	1.98993	0.0640913
$965$	0	0
$966$	−5.85410	−0.188353
$967$	49.4461i	1.59008i	0.606557	+	0.795040i	$0.292550\pi$
$967$	49.4461i	1.59008i	−0.606557	+	0.795040i	$0.707450\pi$
$968$	28.5045i	0.916168i
$969$	−40.7748	−1.30987
$970$	0	0
$971$	51.7332	1.66020	0.830099	−	0.557616i	$-0.188284\pi$
$971$	51.7332	1.66020	0.830099	+	0.557616i	$0.188284\pi$
$972$	− 0.209057i	− 0.00670550i
$973$	− 6.66272i	− 0.213597i
$974$	−15.9832	−0.512136
$975$	0	0
$976$	35.9146	1.14960
$977$	36.3167i	1.16187i	0.813948	+	0.580937i	$0.197314\pi$
$977$	36.3167i	1.16187i	−0.813948	+	0.580937i	$0.802686\pi$
$978$	− 11.0662i	− 0.353860i
$979$	5.26254	0.168192
$980$	0	0
$981$	−6.99533	−0.223344
$982$	− 14.9427i	− 0.476841i
$983$	52.0481i	1.66008i	0.557706	+	0.830039i	$0.311682\pi$
$983$	52.0481i	1.66008i	−0.557706	+	0.830039i	$0.688318\pi$
$984$	8.88558	0.283262
$985$	0	0
$986$	−23.0632	−0.734482
$987$	− 12.0486i	− 0.383511i
$988$	− 5.74435i	− 0.182752i
$989$	11.6995	0.372021
$990$	0	0
$991$	−29.4843	−0.936599	−0.468300	−	0.883570i	$-0.655133\pi$
$991$	−29.4843	−0.936599	−0.468300	+	0.883570i	$0.655133\pi$
$992$	1.28716i	0.0408674i
$993$	− 8.00683i	− 0.254089i
$994$	25.8009	0.818356
$995$	0	0
$996$	−2.24086	−0.0710045
$997$	− 42.4390i	− 1.34406i	−0.740525	−	0.672029i	$-0.765423\pi$
$997$	− 42.4390i	− 1.34406i	0.740525	−	0.672029i	$-0.234577\pi$
$998$	40.9568i	1.29647i
$999$	−7.80126	−0.246821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.d.1249.3		8
5.2	odd	4		1875.2.a.g.1.3	yes	4
5.3	odd	4		1875.2.a.f.1.2	✓	4
5.4	even	2	inner	1875.2.b.d.1249.6		8
15.2	even	4		5625.2.a.j.1.2		4
15.8	even	4		5625.2.a.m.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.f.1.2	✓	4	5.3	odd	4
1875.2.a.g.1.3	yes	4	5.2	odd	4
1875.2.b.d.1249.3		8	1.1	even	1	trivial
1875.2.b.d.1249.6		8	5.4	even	2	inner
5625.2.a.j.1.2		4	15.2	even	4
5625.2.a.m.1.3		4	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)

\(f(q)\)	\(=\)	\(q-1.33826i q^{2} -1.00000i q^{3} +0.209057 q^{4} -1.33826 q^{6} -1.27977i q^{7} -2.95630i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.33826i q^{2} -1.00000i q^{3} +0.209057 q^{4} -1.33826 q^{6} -1.27977i q^{7} -2.95630i q^{8} -1.00000 q^{9} +1.16535 q^{11} -0.209057i q^{12} -3.61048i q^{13} -1.71267 q^{14} -3.53818 q^{16} -5.35772i q^{17} +1.33826i q^{18} +7.61048 q^{19} -1.27977 q^{21} -1.55955i q^{22} +3.41811i q^{23} -2.95630 q^{24} -4.83176 q^{26} +1.00000i q^{27} -0.267545i q^{28} +3.21661 q^{29} -1.09306 q^{31} -1.17758i q^{32} -1.16535i q^{33} -7.17002 q^{34} -0.209057 q^{36} +7.80126i q^{37} -10.1848i q^{38} -3.61048 q^{39} -3.00565 q^{41} +1.71267i q^{42} -3.42278i q^{43} +0.243625 q^{44} +4.57433 q^{46} +9.41462i q^{47} +3.53818i q^{48} +5.36218 q^{49} -5.35772 q^{51} -0.754795i q^{52} -7.64760i q^{53} +1.33826 q^{54} -3.78339 q^{56} -7.61048i q^{57} -4.30467i q^{58} -12.7500 q^{59} -10.1506 q^{61} +1.46280i q^{62} +1.27977i q^{63} -8.65227 q^{64} -1.55955 q^{66} -8.78903i q^{67} -1.12007i q^{68} +3.41811 q^{69} -15.0647 q^{71} +2.95630i q^{72} +13.1533i q^{73} +10.4401 q^{74} +1.59102 q^{76} -1.49139i q^{77} +4.83176i q^{78} +11.5155 q^{79} +1.00000 q^{81} +4.02234i q^{82} -10.7189i q^{83} -0.267545 q^{84} -4.58058 q^{86} -3.21661i q^{87} -3.44512i q^{88} +4.51584 q^{89} -4.62059 q^{91} +0.714580i q^{92} +1.09306i q^{93} +12.5992 q^{94} -1.17758 q^{96} -18.0135i q^{97} -7.17600i q^{98} -1.16535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9	\( 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9} - 12 q^{11} + 20 q^{14} - 18 q^{16} + 18 q^{19} - 10 q^{21} - 6 q^{24} - 4 q^{26} + 56 q^{29} - 20 q^{31} - 14 q^{34} + 2 q^{36} + 14 q^{39} + 18 q^{44} + 10 q^{46} + 6 q^{49} + 14 q^{51} + 2 q^{54} - 8 q^{59} - 86 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} - 54 q^{71} - 10 q^{74} + 18 q^{76} - 20 q^{79} + 8 q^{81} + 10 q^{84} + 48 q^{86} + 18 q^{89} + 10 q^{91} + 44 q^{94} + 12 q^{99}+O(q^{100}) \) 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9 - 12 * q^11 + 20 * q^14 - 18 * q^16 + 18 * q^19 - 10 * q^21 - 6 * q^24 - 4 * q^26 + 56 * q^29 - 20 * q^31 - 14 * q^34 + 2 * q^36 + 14 * q^39 + 18 * q^44 + 10 * q^46 + 6 * q^49 + 14 * q^51 + 2 * q^54 - 8 * q^59 - 86 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 - 54 * q^71 - 10 * q^74 + 18 * q^76 - 20 * q^79 + 8 * q^81 + 10 * q^84 + 48 * q^86 + 18 * q^89 + 10 * q^91 + 44 * q^94 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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