LMFDB - Embedded newform 1875.2.b.g.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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 $ x^16 + 25x^14 + 246x^12 + 1220x^10 + 3281x^8 + 4880x^6 + 3936x^4 + 1600*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$-0.895394i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.g.1249.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23365i q^{2} +1.00000i q^{3} -2.98921 q^{4} +2.23365 q^{6} +1.03143i q^{7} +2.20956i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.23365i q^{2} +1.00000i q^{3} -2.98921 q^{4} +2.23365 q^{6} +1.03143i q^{7} +2.20956i q^{8} -1.00000 q^{9} +6.17643 q^{11} -2.98921i q^{12} +0.937763i q^{13} +2.30385 q^{14} -1.04303 q^{16} +6.56329i q^{17} +2.23365i q^{18} -5.67453 q^{19} -1.03143 q^{21} -13.7960i q^{22} +1.64660i q^{23} -2.20956 q^{24} +2.09464 q^{26} -1.00000i q^{27} -3.08316i q^{28} -8.35819 q^{29} +5.53371 q^{31} +6.74889i q^{32} +6.17643i q^{33} +14.6601 q^{34} +2.98921 q^{36} +1.29548i q^{37} +12.6749i q^{38} -0.937763 q^{39} -4.98106 q^{41} +2.30385i q^{42} +7.75619i q^{43} -18.4627 q^{44} +3.67793 q^{46} +7.67288i q^{47} -1.04303i q^{48} +5.93616 q^{49} -6.56329 q^{51} -2.80317i q^{52} +0.500546i q^{53} -2.23365 q^{54} -2.27900 q^{56} -5.67453i q^{57} +18.6693i q^{58} +1.19340 q^{59} -12.3637 q^{61} -12.3604i q^{62} -1.03143i q^{63} +12.9886 q^{64} +13.7960 q^{66} +7.58851i q^{67} -19.6191i q^{68} -1.64660 q^{69} +10.6125 q^{71} -2.20956i q^{72} -7.98638i q^{73} +2.89365 q^{74} +16.9624 q^{76} +6.37054i q^{77} +2.09464i q^{78} +13.9213 q^{79} +1.00000 q^{81} +11.1260i q^{82} +1.46223i q^{83} +3.08316 q^{84} +17.3246 q^{86} -8.35819i q^{87} +13.6472i q^{88} +8.51161 q^{89} -0.967234 q^{91} -4.92203i q^{92} +5.53371i q^{93} +17.1386 q^{94} -6.74889 q^{96} -3.75623i q^{97} -13.2593i q^{98} -6.17643 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9	$ 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9} + 24 q^{11} - 32 q^{14} + 30 q^{16} - 32 q^{19} + 24 q^{21} - 6 q^{24} - 68 q^{26} - 4 q^{29} + 26 q^{31} + 74 q^{34} + 18 q^{36} - 28 q^{39} - 24 q^{41} - 94 q^{44} + 66 q^{46} - 60 q^{49} - 2 q^{51} - 2 q^{54} + 120 q^{56} - 28 q^{59} + 20 q^{61} - 82 q^{64} + 36 q^{66} + 8 q^{69} + 42 q^{71} + 18 q^{74} - 2 q^{76} - 20 q^{79} + 16 q^{81} + 42 q^{84} + 84 q^{86} + 18 q^{89} - 24 q^{91} - 28 q^{94} - 36 q^{96} - 24 q^{99}+O(q^{100}) $ 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9 + 24 * q^11 - 32 * q^14 + 30 * q^16 - 32 * q^19 + 24 * q^21 - 6 * q^24 - 68 * q^26 - 4 * q^29 + 26 * q^31 + 74 * q^34 + 18 * q^36 - 28 * q^39 - 24 * q^41 - 94 * q^44 + 66 * q^46 - 60 * q^49 - 2 * q^51 - 2 * q^54 + 120 * q^56 - 28 * q^59 + 20 * q^61 - 82 * q^64 + 36 * q^66 + 8 * q^69 + 42 * q^71 + 18 * q^74 - 2 * q^76 - 20 * q^79 + 16 * q^81 + 42 * q^84 + 84 * q^86 + 18 * q^89 - 24 * q^91 - 28 * q^94 - 36 * q^96 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	− 2.23365i	− 1.57943i	−0.613472	−	0.789716i	$-0.710228\pi$
$2$	− 2.23365i	− 1.57943i	0.613472	−	0.789716i	$-0.289772\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−2.98921	−1.49461
$5$	0	0
$5$	0	0
$6$	2.23365	0.911886
$7$	1.03143i	0.389843i	0.980819	+	0.194921i	$0.0624452\pi$
$7$	1.03143i	0.389843i	−0.980819	+	0.194921i	$0.937555\pi$
$8$	2.20956i	0.781198i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	6.17643	1.86227	0.931133	−	0.364681i	$-0.118822\pi$
$11$	6.17643	1.86227	0.931133	+	0.364681i	$0.118822\pi$
$12$	− 2.98921i	− 0.862912i
$13$	0.937763i	0.260089i	0.991508	+	0.130044i	$0.0415120\pi$
$13$	0.937763i	0.260089i	−0.991508	+	0.130044i	$0.958488\pi$
$14$	2.30385	0.615730
$15$	0	0
$16$	−1.04303	−0.260757
$17$	6.56329i	1.59183i	0.605407	+	0.795916i	$0.293010\pi$
$17$	6.56329i	1.59183i	−0.605407	+	0.795916i	$0.706990\pi$
$18$	2.23365i	0.526477i
$19$	−5.67453	−1.30183	−0.650913	−	0.759152i	$-0.725614\pi$
$19$	−5.67453	−1.30183	−0.650913	+	0.759152i	$0.725614\pi$
$20$	0	0
$21$	−1.03143	−0.225076
$22$	− 13.7960i	− 2.94132i
$23$	1.64660i	0.343339i	0.985155	+	0.171669i	$0.0549161\pi$
$23$	1.64660i	0.343339i	−0.985155	+	0.171669i	$0.945084\pi$
$24$	−2.20956	−0.451025
$25$	0	0
$26$	2.09464	0.410792
$27$	− 1.00000i	− 0.192450i
$28$	− 3.08316i	− 0.582662i
$29$	−8.35819	−1.55208	−0.776039	−	0.630685i	$-0.782774\pi$
$29$	−8.35819	−1.55208	−0.776039	+	0.630685i	$0.782774\pi$
$30$	0	0
$31$	5.53371	0.993883	0.496942	−	0.867784i	$-0.334456\pi$
$31$	5.53371	0.993883	0.496942	+	0.867784i	$0.334456\pi$
$32$	6.74889i	1.19305i
$33$	6.17643i	1.07518i
$34$	14.6601	2.51419
$35$	0	0
$36$	2.98921	0.498202
$37$	1.29548i	0.212976i	0.994314	+	0.106488i	$0.0339605\pi$
$37$	1.29548i	0.212976i	−0.994314	+	0.106488i	$0.966040\pi$
$38$	12.6749i	2.05615i
$39$	−0.937763	−0.150162
$40$	0	0
$41$	−4.98106	−0.777910	−0.388955	−	0.921257i	$-0.627164\pi$
$41$	−4.98106	−0.777910	−0.388955	+	0.921257i	$0.627164\pi$
$42$	2.30385i	0.355492i
$43$	7.75619i	1.18281i	0.806376	+	0.591404i	$0.201426\pi$
$43$	7.75619i	1.18281i	−0.806376	+	0.591404i	$0.798574\pi$
$44$	−18.4627	−2.78335
$45$	0	0
$46$	3.67793	0.542281
$47$	7.67288i	1.11920i	0.828761	+	0.559602i	$0.189046\pi$
$47$	7.67288i	1.11920i	−0.828761	+	0.559602i	$0.810954\pi$
$48$	− 1.04303i	− 0.150548i
$49$	5.93616	0.848023
$50$	0	0
$51$	−6.56329	−0.919044
$52$	− 2.80317i	− 0.388730i
$53$	0.500546i	0.0687553i	0.999409	+	0.0343777i	$0.0109449\pi$
$53$	0.500546i	0.0687553i	−0.999409	+	0.0343777i	$0.989055\pi$
$54$	−2.23365	−0.303962
$55$	0	0
$56$	−2.27900	−0.304544
$57$	− 5.67453i	− 0.751609i
$58$	18.6693i	2.45140i
$59$	1.19340	0.155367	0.0776837	−	0.996978i	$-0.475248\pi$
$59$	1.19340	0.155367	0.0776837	+	0.996978i	$0.475248\pi$
$60$	0	0
$61$	−12.3637	−1.58301	−0.791504	−	0.611164i	$-0.790701\pi$
$61$	−12.3637	−1.58301	−0.791504	+	0.611164i	$0.790701\pi$
$62$	− 12.3604i	− 1.56977i
$63$	− 1.03143i	− 0.129948i
$64$	12.9886	1.62358
$65$	0	0
$66$	13.7960	1.69817
$67$	7.58851i	0.927084i	0.886075	+	0.463542i	$0.153422\pi$
$67$	7.58851i	0.927084i	−0.886075	+	0.463542i	$0.846578\pi$
$68$	− 19.6191i	− 2.37916i
$69$	−1.64660	−0.198227
$70$	0	0
$71$	10.6125	1.25948	0.629739	−	0.776807i	$-0.283162\pi$
$71$	10.6125	1.25948	0.629739	+	0.776807i	$0.283162\pi$
$72$	− 2.20956i	− 0.260399i
$73$	− 7.98638i	− 0.934735i	−0.884063	−	0.467367i	$-0.845203\pi$
$73$	− 7.98638i	− 0.934735i	0.884063	−	0.467367i	$-0.154797\pi$
$74$	2.89365	0.336380
$75$	0	0
$76$	16.9624	1.94572
$77$	6.37054i	0.725990i
$78$	2.09464i	0.237171i
$79$	13.9213	1.56627	0.783134	−	0.621854i	$-0.213620\pi$
$79$	13.9213	1.56627	0.783134	+	0.621854i	$0.213620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	11.1260i	1.22866i
$83$	1.46223i	0.160500i	0.996775	+	0.0802500i	$0.0255719\pi$
$83$	1.46223i	0.160500i	−0.996775	+	0.0802500i	$0.974428\pi$
$84$	3.08316	0.336400
$85$	0	0
$86$	17.3246	1.86816
$87$	− 8.35819i	− 0.896092i
$88$	13.6472i	1.45480i
$89$	8.51161	0.902229	0.451115	−	0.892466i	$-0.351027\pi$
$89$	8.51161	0.902229	0.451115	+	0.892466i	$0.351027\pi$
$90$	0	0
$91$	−0.967234	−0.101394
$92$	− 4.92203i	− 0.513157i
$93$	5.53371i	0.573819i
$94$	17.1386	1.76771
$95$	0	0
$96$	−6.74889	−0.688806
$97$	− 3.75623i	− 0.381387i	−0.981650	−	0.190693i	$-0.938926\pi$
$97$	− 3.75623i	− 0.381387i	0.981650	−	0.190693i	$-0.0610737\pi$
$98$	− 13.2593i	− 1.33939i
$99$	−6.17643	−0.620755
$100$	0	0
$101$	−2.60566	−0.259273	−0.129636	−	0.991562i	$-0.541381\pi$
$101$	−2.60566	−0.259273	−0.129636	+	0.991562i	$0.541381\pi$
$102$	14.6601i	1.45157i
$103$	17.2182i	1.69656i	0.529550	+	0.848279i	$0.322361\pi$
$103$	17.2182i	1.69656i	−0.529550	+	0.848279i	$0.677639\pi$
$104$	−2.07204	−0.203181
$105$	0	0
$106$	1.11805	0.108594
$107$	− 5.54296i	− 0.535858i	−0.963439	−	0.267929i	$-0.913661\pi$
$107$	− 5.54296i	− 0.535858i	0.963439	−	0.267929i	$-0.0863393\pi$
$108$	2.98921i	0.287637i
$109$	5.98458	0.573219	0.286610	−	0.958047i	$-0.407472\pi$
$109$	5.98458	0.573219	0.286610	+	0.958047i	$0.407472\pi$
$110$	0	0
$111$	−1.29548	−0.122961
$112$	− 1.07581i	− 0.101654i
$113$	0.323636i	0.0304451i	0.999884	+	0.0152226i	$0.00484568\pi$
$113$	0.323636i	0.0304451i	−0.999884	+	0.0152226i	$0.995154\pi$
$114$	−12.6749	−1.18712
$115$	0	0
$116$	24.9844	2.31975
$117$	− 0.937763i	− 0.0866962i
$118$	− 2.66564i	− 0.245392i
$119$	−6.76955	−0.620564
$120$	0	0
$121$	27.1483	2.46803
$122$	27.6162i	2.50025i
$123$	− 4.98106i	− 0.449127i
$124$	−16.5414	−1.48546
$125$	0	0
$126$	−2.30385	−0.205243
$127$	0.0904114i	0.00802271i	0.999992	+	0.00401136i	$0.00127686\pi$
$127$	0.0904114i	0.00802271i	−0.999992	+	0.00401136i	$0.998723\pi$
$128$	− 15.5143i	− 1.37129i
$129$	−7.75619	−0.682894
$130$	0	0
$131$	−13.5700	−1.18562	−0.592809	−	0.805343i	$-0.701981\pi$
$131$	−13.5700	−1.18562	−0.592809	+	0.805343i	$0.701981\pi$
$132$	− 18.4627i	− 1.60697i
$133$	− 5.85286i	− 0.507507i
$134$	16.9501	1.46427
$135$	0	0
$136$	−14.5020	−1.24354
$137$	13.9795i	1.19435i	0.802111	+	0.597174i	$0.203710\pi$
$137$	13.9795i	1.19435i	−0.802111	+	0.597174i	$0.796290\pi$
$138$	3.67793i	0.313086i
$139$	2.21762	0.188096	0.0940481	−	0.995568i	$-0.470019\pi$
$139$	2.21762	0.188096	0.0940481	+	0.995568i	$0.470019\pi$
$140$	0	0
$141$	−7.67288	−0.646173
$142$	− 23.7048i	− 1.98926i
$143$	5.79203i	0.484354i
$144$	1.04303	0.0869191
$145$	0	0
$146$	−17.8388	−1.47635
$147$	5.93616i	0.489606i
$148$	− 3.87247i	− 0.318315i
$149$	−3.59402	−0.294433	−0.147217	−	0.989104i	$-0.547031\pi$
$149$	−3.59402	−0.294433	−0.147217	+	0.989104i	$0.547031\pi$
$150$	0	0
$151$	16.8624	1.37224	0.686121	−	0.727487i	$-0.259312\pi$
$151$	16.8624	1.37224	0.686121	+	0.727487i	$0.259312\pi$
$152$	− 12.5382i	− 1.01698i
$153$	− 6.56329i	− 0.530611i
$154$	14.2296	1.14665
$155$	0	0
$156$	2.80317	0.224433
$157$	8.42895i	0.672703i	0.941736	+	0.336352i	$0.109193\pi$
$157$	8.42895i	0.672703i	−0.941736	+	0.336352i	$0.890807\pi$
$158$	− 31.0953i	− 2.47381i
$159$	−0.500546	−0.0396959
$160$	0	0
$161$	−1.69834	−0.133848
$162$	− 2.23365i	− 0.175492i
$163$	− 11.3806i	− 0.891394i	−0.895184	−	0.445697i	$-0.852956\pi$
$163$	− 11.3806i	− 0.891394i	0.895184	−	0.445697i	$-0.147044\pi$
$164$	14.8894	1.16267
$165$	0	0
$166$	3.26611	0.253499
$167$	− 0.886875i	− 0.0686285i	−0.999411	−	0.0343142i	$-0.989075\pi$
$167$	− 0.886875i	− 0.0686285i	0.999411	−	0.0343142i	$-0.0109247\pi$
$168$	− 2.27900i	− 0.175829i
$169$	12.1206	0.932354
$170$	0	0
$171$	5.67453	0.433942
$172$	− 23.1849i	− 1.76783i
$173$	− 20.0184i	− 1.52197i	−0.648769	−	0.760986i	$-0.724716\pi$
$173$	− 20.0184i	− 1.52197i	0.648769	−	0.760986i	$-0.275284\pi$
$174$	−18.6693	−1.41532
$175$	0	0
$176$	−6.44220	−0.485599
$177$	1.19340i	0.0897014i
$178$	− 19.0120i	− 1.42501i
$179$	3.38799	0.253231	0.126615	−	0.991952i	$-0.459589\pi$
$179$	3.38799	0.253231	0.126615	+	0.991952i	$0.459589\pi$
$180$	0	0
$181$	5.41991	0.402859	0.201429	−	0.979503i	$-0.435441\pi$
$181$	5.41991	0.402859	0.201429	+	0.979503i	$0.435441\pi$
$182$	2.16047i	0.160144i
$183$	− 12.3637i	− 0.913950i
$184$	−3.63825	−0.268216
$185$	0	0
$186$	12.3604	0.906308
$187$	40.5377i	2.96441i
$188$	− 22.9359i	− 1.67277i
$189$	1.03143	0.0750253
$190$	0	0
$191$	−11.2760	−0.815901	−0.407950	−	0.913004i	$-0.633756\pi$
$191$	−11.2760	−0.815901	−0.407950	+	0.913004i	$0.633756\pi$
$192$	12.9886i	0.937374i
$193$	− 6.22383i	− 0.448001i	−0.974589	−	0.224000i	$-0.928088\pi$
$193$	− 6.22383i	− 0.448001i	0.974589	−	0.224000i	$-0.0719117\pi$
$194$	−8.39011	−0.602375
$195$	0	0
$196$	−17.7444	−1.26746
$197$	19.7296i	1.40568i	0.711349	+	0.702839i	$0.248084\pi$
$197$	19.7296i	1.40568i	−0.711349	+	0.702839i	$0.751916\pi$
$198$	13.7960i	0.980441i
$199$	−18.5731	−1.31661	−0.658307	−	0.752750i	$-0.728727\pi$
$199$	−18.5731	−1.31661	−0.658307	+	0.752750i	$0.728727\pi$
$200$	0	0
$201$	−7.58851	−0.535252
$202$	5.82014i	0.409504i
$203$	− 8.62087i	− 0.605066i
$204$	19.6191	1.37361
$205$	0	0
$206$	38.4595	2.67960
$207$	− 1.64660i	− 0.114446i
$208$	− 0.978114i	− 0.0678200i
$209$	−35.0483	−2.42434
$210$	0	0
$211$	2.71998	0.187251	0.0936255	−	0.995607i	$-0.470154\pi$
$211$	2.71998	0.187251	0.0936255	+	0.995607i	$0.470154\pi$
$212$	− 1.49624i	− 0.102762i
$213$	10.6125i	0.727160i
$214$	−12.3811	−0.846352
$215$	0	0
$216$	2.20956	0.150342
$217$	5.70762i	0.387458i
$218$	− 13.3675i	− 0.905361i
$219$	7.98638	0.539669
$220$	0	0
$221$	−6.15481	−0.414017
$222$	2.89365i	0.194209i
$223$	7.04569i	0.471814i	0.971776	+	0.235907i	$0.0758061\pi$
$223$	7.04569i	0.471814i	−0.971776	+	0.235907i	$0.924194\pi$
$224$	−6.96099	−0.465101
$225$	0	0
$226$	0.722892	0.0480860
$227$	− 16.9671i	− 1.12615i	−0.826407	−	0.563073i	$-0.809619\pi$
$227$	− 16.9671i	− 1.12615i	0.826407	−	0.563073i	$-0.190381\pi$
$228$	16.9624i	1.12336i
$229$	−5.14606	−0.340061	−0.170030	−	0.985439i	$-0.554387\pi$
$229$	−5.14606	−0.340061	−0.170030	+	0.985439i	$0.554387\pi$
$230$	0	0
$231$	−6.37054	−0.419151
$232$	− 18.4679i	− 1.21248i
$233$	27.3476i	1.79160i	0.444460	+	0.895799i	$0.353396\pi$
$233$	27.3476i	1.79160i	−0.444460	+	0.895799i	$0.646604\pi$
$234$	−2.09464	−0.136931
$235$	0	0
$236$	−3.56733	−0.232213
$237$	13.9213i	0.904285i
$238$	15.1208i	0.980139i
$239$	−19.6994	−1.27425	−0.637123	−	0.770762i	$-0.719876\pi$
$239$	−19.6994	−1.27425	−0.637123	+	0.770762i	$0.719876\pi$
$240$	0	0
$241$	−9.57114	−0.616532	−0.308266	−	0.951300i	$-0.599749\pi$
$241$	−9.57114	−0.616532	−0.308266	+	0.951300i	$0.599749\pi$
$242$	− 60.6400i	− 3.89809i
$243$	1.00000i	0.0641500i
$244$	36.9577	2.36597
$245$	0	0
$246$	−11.1260	−0.709365
$247$	− 5.32136i	− 0.338590i
$248$	12.2271i	0.776420i
$249$	−1.46223	−0.0926648
$250$	0	0
$251$	−5.82514	−0.367680	−0.183840	−	0.982956i	$-0.558853\pi$
$251$	−5.82514	−0.367680	−0.183840	+	0.982956i	$0.558853\pi$
$252$	3.08316i	0.194221i
$253$	10.1701i	0.639388i
$254$	0.201948	0.0126713
$255$	0	0
$256$	−8.67641	−0.542276
$257$	3.47558i	0.216801i	0.994107	+	0.108400i	$0.0345729\pi$
$257$	3.47558i	0.216801i	−0.994107	+	0.108400i	$0.965427\pi$
$258$	17.3246i	1.07859i
$259$	−1.33619	−0.0830270
$260$	0	0
$261$	8.35819	0.517359
$262$	30.3107i	1.87260i
$263$	− 22.7227i	− 1.40114i	−0.713584	−	0.700570i	$-0.752929\pi$
$263$	− 22.7227i	− 1.40114i	0.713584	−	0.700570i	$-0.247071\pi$
$264$	−13.6472	−0.839928
$265$	0	0
$266$	−13.0733	−0.801573
$267$	8.51161i	0.520902i
$268$	− 22.6837i	− 1.38563i
$269$	−1.63546	−0.0997160	−0.0498580	−	0.998756i	$-0.515877\pi$
$269$	−1.63546	−0.0997160	−0.0498580	+	0.998756i	$0.515877\pi$
$270$	0	0
$271$	12.4498	0.756275	0.378137	−	0.925750i	$-0.376565\pi$
$271$	12.4498	0.756275	0.378137	+	0.925750i	$0.376565\pi$
$272$	− 6.84571i	− 0.415082i
$273$	− 0.967234i	− 0.0585397i
$274$	31.2254	1.88639
$275$	0	0
$276$	4.92203	0.296271
$277$	− 2.16353i	− 0.129994i	−0.997885	−	0.0649971i	$-0.979296\pi$
$277$	− 2.16353i	− 0.129994i	0.997885	−	0.0649971i	$-0.0207038\pi$
$278$	− 4.95340i	− 0.297085i
$279$	−5.53371	−0.331294
$280$	0	0
$281$	4.97817	0.296973	0.148486	−	0.988914i	$-0.452560\pi$
$281$	4.97817	0.296973	0.148486	+	0.988914i	$0.452560\pi$
$282$	17.1386i	1.02059i
$283$	− 17.3357i	− 1.03050i	−0.857041	−	0.515249i	$-0.827699\pi$
$283$	− 17.3357i	− 1.03050i	0.857041	−	0.515249i	$-0.172301\pi$
$284$	−31.7232	−1.88242
$285$	0	0
$286$	12.9374	0.765004
$287$	− 5.13759i	− 0.303263i
$288$	− 6.74889i	− 0.397682i
$289$	−26.0768	−1.53393
$290$	0	0
$291$	3.75623	0.220194
$292$	23.8730i	1.39706i
$293$	20.1863i	1.17930i	0.807661	+	0.589648i	$0.200733\pi$
$293$	20.1863i	1.17930i	−0.807661	+	0.589648i	$0.799267\pi$
$294$	13.2593	0.773300
$295$	0	0
$296$	−2.86244	−0.166376
$297$	− 6.17643i	− 0.358393i
$298$	8.02779i	0.465038i
$299$	−1.54412	−0.0892986
$300$	0	0
$301$	−7.99994	−0.461109
$302$	− 37.6648i	− 2.16736i
$303$	− 2.60566i	− 0.149691i
$304$	5.91870	0.339461
$305$	0	0
$306$	−14.6601	−0.838064
$307$	− 24.7882i	− 1.41474i	−0.706844	−	0.707369i	$-0.749882\pi$
$307$	− 24.7882i	− 1.41474i	0.706844	−	0.707369i	$-0.250118\pi$
$308$	− 19.0429i	− 1.08507i
$309$	−17.2182	−0.979508
$310$	0	0
$311$	−28.1046	−1.59367	−0.796834	−	0.604198i	$-0.793493\pi$
$311$	−28.1046	−1.59367	−0.796834	+	0.604198i	$0.793493\pi$
$312$	− 2.07204i	− 0.117306i
$313$	− 27.1858i	− 1.53663i	−0.640070	−	0.768317i	$-0.721095\pi$
$313$	− 27.1858i	− 1.53663i	0.640070	−	0.768317i	$-0.278905\pi$
$314$	18.8274	1.06249
$315$	0	0
$316$	−41.6137	−2.34095
$317$	14.9749i	0.841073i	0.907276	+	0.420536i	$0.138158\pi$
$317$	14.9749i	0.841073i	−0.907276	+	0.420536i	$0.861842\pi$
$318$	1.11805i	0.0626970i
$319$	−51.6238	−2.89038
$320$	0	0
$321$	5.54296	0.309378
$322$	3.79351i	0.211404i
$323$	− 37.2436i	− 2.07229i
$324$	−2.98921	−0.166067
$325$	0	0
$326$	−25.4202	−1.40790
$327$	5.98458i	0.330948i
$328$	− 11.0059i	− 0.607702i
$329$	−7.91401	−0.436314
$330$	0	0
$331$	−13.9393	−0.766173	−0.383086	−	0.923713i	$-0.625139\pi$
$331$	−13.9393	−0.766173	−0.383086	+	0.923713i	$0.625139\pi$
$332$	− 4.37090i	− 0.239885i
$333$	− 1.29548i	− 0.0709918i
$334$	−1.98097	−0.108394
$335$	0	0
$336$	1.07581	0.0586902
$337$	− 35.3670i	− 1.92656i	−0.268492	−	0.963282i	$-0.586525\pi$
$337$	− 35.3670i	− 1.92656i	0.268492	−	0.963282i	$-0.413475\pi$
$338$	− 27.0732i	− 1.47259i
$339$	−0.323636	−0.0175775
$340$	0	0
$341$	34.1786	1.85087
$342$	− 12.6749i	− 0.685382i
$343$	13.3427i	0.720438i
$344$	−17.1378	−0.924007
$345$	0	0
$346$	−44.7142	−2.40385
$347$	− 9.34346i	− 0.501583i	−0.968041	−	0.250792i	$-0.919309\pi$
$347$	− 9.34346i	− 0.501583i	0.968041	−	0.250792i	$-0.0806909\pi$
$348$	24.9844i	1.33931i
$349$	−6.92379	−0.370622	−0.185311	−	0.982680i	$-0.559329\pi$
$349$	−6.92379	−0.370622	−0.185311	+	0.982680i	$0.559329\pi$
$350$	0	0
$351$	0.937763	0.0500541
$352$	41.6841i	2.22177i
$353$	5.13902i	0.273522i	0.990604	+	0.136761i	$0.0436693\pi$
$353$	5.13902i	0.273522i	−0.990604	+	0.136761i	$0.956331\pi$
$354$	2.66564	0.141677
$355$	0	0
$356$	−25.4430	−1.34848
$357$	− 6.76955i	− 0.358283i
$358$	− 7.56761i	− 0.399961i
$359$	19.0429	1.00504	0.502522	−	0.864564i	$-0.332406\pi$
$359$	19.0429	1.00504	0.502522	+	0.864564i	$0.332406\pi$
$360$	0	0
$361$	13.2003	0.694750
$362$	− 12.1062i	− 0.636288i
$363$	27.1483i	1.42492i
$364$	2.89127	0.151544
$365$	0	0
$366$	−27.6162	−1.44352
$367$	− 14.1916i	− 0.740796i	−0.928873	−	0.370398i	$-0.879221\pi$
$367$	− 14.1916i	− 0.740796i	0.928873	−	0.370398i	$-0.120779\pi$
$368$	− 1.71745i	− 0.0895282i
$369$	4.98106	0.259303
$370$	0	0
$371$	−0.516277	−0.0268038
$372$	− 16.5414i	− 0.857633i
$373$	35.4663i	1.83637i	0.396148	+	0.918187i	$0.370347\pi$
$373$	35.4663i	1.83637i	−0.396148	+	0.918187i	$0.629653\pi$
$374$	90.5473	4.68209
$375$	0	0
$376$	−16.9537	−0.874320
$377$	− 7.83800i	− 0.403678i
$378$	− 2.30385i	− 0.118497i
$379$	1.99692	0.102575	0.0512874	−	0.998684i	$-0.483668\pi$
$379$	1.99692	0.102575	0.0512874	+	0.998684i	$0.483668\pi$
$380$	0	0
$381$	−0.0904114	−0.00463191
$382$	25.1866i	1.28866i
$383$	− 9.43941i	− 0.482331i	−0.970484	−	0.241166i	$-0.922470\pi$
$383$	− 9.43941i	− 0.482331i	0.970484	−	0.241166i	$-0.0775297\pi$
$384$	15.5143	0.791713
$385$	0	0
$386$	−13.9019	−0.707587
$387$	− 7.75619i	− 0.394269i
$388$	11.2282i	0.570024i
$389$	1.31535	0.0666908	0.0333454	−	0.999444i	$-0.489384\pi$
$389$	1.31535	0.0666908	0.0333454	+	0.999444i	$0.489384\pi$
$390$	0	0
$391$	−10.8071	−0.546538
$392$	13.1163i	0.662474i
$393$	− 13.5700i	− 0.684517i
$394$	44.0692	2.22017
$395$	0	0
$396$	18.4627	0.927785
$397$	25.3086i	1.27020i	0.772430	+	0.635100i	$0.219041\pi$
$397$	25.3086i	1.27020i	−0.772430	+	0.635100i	$0.780959\pi$
$398$	41.4860i	2.07950i
$399$	5.85286	0.293009
$400$	0	0
$401$	13.4276	0.670543	0.335271	−	0.942122i	$-0.391172\pi$
$401$	13.4276	0.670543	0.335271	+	0.942122i	$0.391172\pi$
$402$	16.9501i	0.845395i
$403$	5.18931i	0.258498i
$404$	7.78887	0.387511
$405$	0	0
$406$	−19.2560	−0.955661
$407$	8.00144i	0.396617i
$408$	− 14.5020i	− 0.717956i
$409$	18.8088	0.930035	0.465018	−	0.885301i	$-0.346048\pi$
$409$	18.8088	0.930035	0.465018	+	0.885301i	$0.346048\pi$
$410$	0	0
$411$	−13.9795	−0.689558
$412$	− 51.4688i	− 2.53569i
$413$	1.23090i	0.0605688i
$414$	−3.67793	−0.180760
$415$	0	0
$416$	−6.32886	−0.310298
$417$	2.21762i	0.108597i
$418$	78.2859i	3.82909i
$419$	21.7812	1.06408	0.532042	−	0.846718i	$-0.321425\pi$
$419$	21.7812	1.06408	0.532042	+	0.846718i	$0.321425\pi$
$420$	0	0
$421$	28.3679	1.38257	0.691283	−	0.722584i	$-0.257046\pi$
$421$	28.3679	1.38257	0.691283	+	0.722584i	$0.257046\pi$
$422$	− 6.07549i	− 0.295750i
$423$	− 7.67288i	− 0.373068i
$424$	−1.10599	−0.0537115
$425$	0	0
$426$	23.7048	1.14850
$427$	− 12.7522i	− 0.617124i
$428$	16.5691i	0.800898i
$429$	−5.79203	−0.279642
$430$	0	0
$431$	−23.7069	−1.14192	−0.570960	−	0.820978i	$-0.693429\pi$
$431$	−23.7069	−1.14192	−0.570960	+	0.820978i	$0.693429\pi$
$432$	1.04303i	0.0501828i
$433$	7.11846i	0.342091i	0.985263	+	0.171046i	$0.0547146\pi$
$433$	7.11846i	0.342091i	−0.985263	+	0.171046i	$0.945285\pi$
$434$	12.7488	0.611964
$435$	0	0
$436$	−17.8892	−0.856737
$437$	− 9.34365i	− 0.446967i
$438$	− 17.8388i	− 0.852371i
$439$	−6.68142	−0.318887	−0.159443	−	0.987207i	$-0.550970\pi$
$439$	−6.68142	−0.318887	−0.159443	+	0.987207i	$0.550970\pi$
$440$	0	0
$441$	−5.93616	−0.282674
$442$	13.7477i	0.653912i
$443$	20.7065i	0.983797i	0.870653	+	0.491898i	$0.163697\pi$
$443$	20.7065i	0.983797i	−0.870653	+	0.491898i	$0.836303\pi$
$444$	3.87247	0.183779
$445$	0	0
$446$	15.7376	0.745199
$447$	− 3.59402i	− 0.169991i
$448$	13.3968i	0.632940i
$449$	0.218310	0.0103027	0.00515134	−	0.999987i	$-0.498360\pi$
$449$	0.218310	0.0103027	0.00515134	+	0.999987i	$0.498360\pi$
$450$	0	0
$451$	−30.7652	−1.44867
$452$	− 0.967418i	− 0.0455035i
$453$	16.8624i	0.792264i
$454$	−37.8987	−1.77867
$455$	0	0
$456$	12.5382	0.587156
$457$	− 5.96061i	− 0.278826i	−0.990234	−	0.139413i	$-0.955479\pi$
$457$	− 5.96061i	− 0.278826i	0.990234	−	0.139413i	$-0.0445215\pi$
$458$	11.4945i	0.537103i
$459$	6.56329	0.306348
$460$	0	0
$461$	−12.8249	−0.597314	−0.298657	−	0.954360i	$-0.596539\pi$
$461$	−12.8249	−0.597314	−0.298657	+	0.954360i	$0.596539\pi$
$462$	14.2296i	0.662020i
$463$	26.9181i	1.25099i	0.780227	+	0.625496i	$0.215103\pi$
$463$	26.9181i	1.25099i	−0.780227	+	0.625496i	$0.784897\pi$
$464$	8.71784	0.404716
$465$	0	0
$466$	61.0850	2.82971
$467$	26.8600i	1.24293i	0.783441	+	0.621466i	$0.213463\pi$
$467$	26.8600i	1.24293i	−0.783441	+	0.621466i	$0.786537\pi$
$468$	2.80317i	0.129577i
$469$	−7.82699	−0.361417
$470$	0	0
$471$	−8.42895	−0.388386
$472$	2.63689i	0.121373i
$473$	47.9056i	2.20270i
$474$	31.0953	1.42826
$475$	0	0
$476$	20.2356	0.927499
$477$	− 0.500546i	− 0.0229184i
$478$	44.0016i	2.01258i
$479$	−39.2952	−1.79544	−0.897722	−	0.440562i	$-0.854779\pi$
$479$	−39.2952	−1.79544	−0.897722	+	0.440562i	$0.854779\pi$
$480$	0	0
$481$	−1.21485	−0.0553925
$482$	21.3786i	0.973770i
$483$	− 1.69834i	− 0.0772773i
$484$	−81.1522	−3.68874
$485$	0	0
$486$	2.23365	0.101321
$487$	− 37.3269i	− 1.69144i	−0.533626	−	0.845721i	$-0.679171\pi$
$487$	− 37.3269i	− 1.69144i	0.533626	−	0.845721i	$-0.320829\pi$
$488$	− 27.3183i	− 1.23664i
$489$	11.3806	0.514647
$490$	0	0
$491$	−16.8123	−0.758729	−0.379364	−	0.925247i	$-0.623857\pi$
$491$	−16.8123	−0.758729	−0.379364	+	0.925247i	$0.623857\pi$
$492$	14.8894i	0.671268i
$493$	− 54.8573i	− 2.47065i
$494$	−11.8861	−0.534780
$495$	0	0
$496$	−5.77182	−0.259162
$497$	10.9461i	0.490998i
$498$	3.26611i	0.146358i
$499$	20.6970	0.926526	0.463263	−	0.886221i	$-0.346678\pi$
$499$	20.6970	0.926526	0.463263	+	0.886221i	$0.346678\pi$
$500$	0	0
$501$	0.886875	0.0396227
$502$	13.0114i	0.580725i
$503$	− 38.8631i	− 1.73282i	−0.499333	−	0.866410i	$-0.666421\pi$
$503$	− 38.8631i	− 1.73282i	0.499333	−	0.866410i	$-0.333579\pi$
$504$	2.27900	0.101515
$505$	0	0
$506$	22.7165	1.00987
$507$	12.1206i	0.538295i
$508$	− 0.270259i	− 0.0119908i
$509$	−20.8260	−0.923096	−0.461548	−	0.887115i	$-0.652706\pi$
$509$	−20.8260	−0.923096	−0.461548	+	0.887115i	$0.652706\pi$
$510$	0	0
$511$	8.23736	0.364400
$512$	− 11.6486i	− 0.514799i
$513$	5.67453i	0.250536i
$514$	7.76325	0.342422
$515$	0	0
$516$	23.1849	1.02066
$517$	47.3910i	2.08425i
$518$	2.98459i	0.131135i
$519$	20.0184	0.878710
$520$	0	0
$521$	10.0302	0.439430	0.219715	−	0.975564i	$-0.429487\pi$
$521$	10.0302	0.439430	0.219715	+	0.975564i	$0.429487\pi$
$522$	− 18.6693i	− 0.817134i
$523$	21.8640i	0.956046i	0.878347	+	0.478023i	$0.158646\pi$
$523$	21.8640i	0.956046i	−0.878347	+	0.478023i	$0.841354\pi$
$524$	40.5637	1.77203
$525$	0	0
$526$	−50.7546	−2.21300
$527$	36.3193i	1.58210i
$528$	− 6.44220i	− 0.280361i
$529$	20.2887	0.882118
$530$	0	0
$531$	−1.19340	−0.0517891
$532$	17.4954i	0.758524i
$533$	− 4.67105i	− 0.202326i
$534$	19.0120	0.822730
$535$	0	0
$536$	−16.7673	−0.724236
$537$	3.38799i	0.146203i
$538$	3.65306i	0.157495i
$539$	36.6643	1.57924
$540$	0	0
$541$	39.2383	1.68699	0.843493	−	0.537140i	$-0.180495\pi$
$541$	39.2383	1.68699	0.843493	+	0.537140i	$0.180495\pi$
$542$	− 27.8087i	− 1.19448i
$543$	5.41991i	0.232591i
$544$	−44.2949	−1.89913
$545$	0	0
$546$	−2.16047	−0.0924594
$547$	− 2.82535i	− 0.120803i	−0.998174	−	0.0604016i	$-0.980762\pi$
$547$	− 2.82535i	− 0.120803i	0.998174	−	0.0604016i	$-0.0192381\pi$
$548$	− 41.7877i	− 1.78508i
$549$	12.3637	0.527669
$550$	0	0
$551$	47.4288	2.02053
$552$	− 3.63825i	− 0.154854i
$553$	14.3588i	0.610598i
$554$	−4.83259	−0.205317
$555$	0	0
$556$	−6.62895	−0.281130
$557$	10.4495i	0.442759i	0.975188	+	0.221379i	$0.0710559\pi$
$557$	10.4495i	0.442759i	−0.975188	+	0.221379i	$0.928944\pi$
$558$	12.3604i	0.523257i
$559$	−7.27347	−0.307635
$560$	0	0
$561$	−40.5377	−1.71150
$562$	− 11.1195i	− 0.469048i
$563$	5.74113i	0.241960i	0.992655	+	0.120980i	$0.0386037\pi$
$563$	5.74113i	0.241960i	−0.992655	+	0.120980i	$0.961396\pi$
$564$	22.9359	0.965774
$565$	0	0
$566$	−38.7219	−1.62760
$567$	1.03143i	0.0433159i
$568$	23.4491i	0.983901i
$569$	−5.50079	−0.230605	−0.115302	−	0.993330i	$-0.536784\pi$
$569$	−5.50079	−0.230605	−0.115302	+	0.993330i	$0.536784\pi$
$570$	0	0
$571$	−9.36436	−0.391886	−0.195943	−	0.980615i	$-0.562777\pi$
$571$	−9.36436	−0.391886	−0.195943	+	0.980615i	$0.562777\pi$
$572$	− 17.3136i	− 0.723919i
$573$	− 11.2760i	− 0.471061i
$574$	−11.4756	−0.478983
$575$	0	0
$576$	−12.9886	−0.541193
$577$	− 23.9595i	− 0.997448i	−0.866761	−	0.498724i	$-0.833802\pi$
$577$	− 23.9595i	− 0.997448i	0.866761	−	0.498724i	$-0.166198\pi$
$578$	58.2465i	2.42274i
$579$	6.22383	0.258653
$580$	0	0
$581$	−1.50818	−0.0625698
$582$	− 8.39011i	− 0.347781i
$583$	3.09159i	0.128041i
$584$	17.6464	0.730213
$585$	0	0
$586$	45.0892	1.86262
$587$	22.4831i	0.927978i	0.885841	+	0.463989i	$0.153582\pi$
$587$	22.4831i	0.927978i	−0.885841	+	0.463989i	$0.846418\pi$
$588$	− 17.7444i	− 0.731769i
$589$	−31.4012	−1.29386
$590$	0	0
$591$	−19.7296	−0.811568
$592$	− 1.35122i	− 0.0555349i
$593$	− 31.0854i	− 1.27652i	−0.769819	−	0.638262i	$-0.779654\pi$
$593$	− 31.0854i	− 1.27652i	0.769819	−	0.638262i	$-0.220346\pi$
$594$	−13.7960	−0.566058
$595$	0	0
$596$	10.7433	0.440062
$597$	− 18.5731i	− 0.760148i
$598$	3.44902i	0.141041i
$599$	9.30580	0.380225	0.190112	−	0.981762i	$-0.439115\pi$
$599$	9.30580	0.380225	0.190112	+	0.981762i	$0.439115\pi$
$600$	0	0
$601$	39.0074	1.59114	0.795572	−	0.605859i	$-0.207170\pi$
$601$	39.0074	1.59114	0.795572	+	0.605859i	$0.207170\pi$
$602$	17.8691i	0.728290i
$603$	− 7.58851i	− 0.309028i
$604$	−50.4053	−2.05096
$605$	0	0
$606$	−5.82014	−0.236427
$607$	− 18.6229i	− 0.755880i	−0.925830	−	0.377940i	$-0.876632\pi$
$607$	− 18.6229i	− 0.755880i	0.925830	−	0.377940i	$-0.123368\pi$
$608$	− 38.2968i	− 1.55314i
$609$	8.62087	0.349335
$610$	0	0
$611$	−7.19534	−0.291092
$612$	19.6191i	0.793054i
$613$	− 36.3935i	− 1.46992i	−0.678111	−	0.734960i	$-0.737201\pi$
$613$	− 36.3935i	− 1.46992i	0.678111	−	0.734960i	$-0.262799\pi$
$614$	−55.3683	−2.23448
$615$	0	0
$616$	−14.0761	−0.567142
$617$	− 0.139173i	− 0.00560289i	−0.999996	−	0.00280145i	$-0.999108\pi$
$617$	− 0.139173i	− 0.00560289i	0.999996	−	0.00280145i	$-0.000891729\pi$
$618$	38.4595i	1.54707i
$619$	−1.04686	−0.0420769	−0.0210384	−	0.999779i	$-0.506697\pi$
$619$	−1.04686	−0.0420769	−0.0210384	+	0.999779i	$0.506697\pi$
$620$	0	0
$621$	1.64660	0.0660756
$622$	62.7761i	2.51709i
$623$	8.77910i	0.351727i
$624$	0.978114	0.0391559
$625$	0	0
$626$	−60.7237	−2.42701
$627$	− 35.0483i	− 1.39970i
$628$	− 25.1959i	− 1.00543i
$629$	−8.50261	−0.339021
$630$	0	0
$631$	25.6813	1.02236	0.511178	−	0.859475i	$-0.329209\pi$
$631$	25.6813	1.02236	0.511178	+	0.859475i	$0.329209\pi$
$632$	30.7599i	1.22356i
$633$	2.71998i	0.108109i
$634$	33.4487	1.32842
$635$	0	0
$636$	1.49624	0.0593298
$637$	5.56671i	0.220561i
$638$	115.310i	4.56516i
$639$	−10.6125	−0.419826
$640$	0	0
$641$	−12.7784	−0.504718	−0.252359	−	0.967634i	$-0.581206\pi$
$641$	−12.7784	−0.504718	−0.252359	+	0.967634i	$0.581206\pi$
$642$	− 12.3811i	− 0.488642i
$643$	14.8374i	0.585131i	0.956246	+	0.292565i	$0.0945089\pi$
$643$	14.8374i	0.585131i	−0.956246	+	0.292565i	$0.905491\pi$
$644$	5.07671	0.200050
$645$	0	0
$646$	−83.1893	−3.27304
$647$	− 17.9733i	− 0.706602i	−0.935510	−	0.353301i	$-0.885059\pi$
$647$	− 17.9733i	− 0.706602i	0.935510	−	0.353301i	$-0.114941\pi$
$648$	2.20956i	0.0867998i
$649$	7.37095	0.289335
$650$	0	0
$651$	−5.70762	−0.223699
$652$	34.0189i	1.33228i
$653$	3.36997i	0.131877i	0.997824	+	0.0659385i	$0.0210041\pi$
$653$	3.36997i	0.131877i	−0.997824	+	0.0659385i	$0.978996\pi$
$654$	13.3675	0.522710
$655$	0	0
$656$	5.19539	0.202846
$657$	7.98638i	0.311578i
$658$	17.6772i	0.689128i
$659$	28.9767	1.12877	0.564386	−	0.825511i	$-0.309113\pi$
$659$	28.9767	1.12877	0.564386	+	0.825511i	$0.309113\pi$
$660$	0	0
$661$	18.3608	0.714153	0.357076	−	0.934075i	$-0.383774\pi$
$661$	18.3608	0.714153	0.357076	+	0.934075i	$0.383774\pi$
$662$	31.1356i	1.21012i
$663$	− 6.15481i	− 0.239033i
$664$	−3.23088	−0.125382
$665$	0	0
$666$	−2.89365	−0.112127
$667$	− 13.7626i	− 0.532889i
$668$	2.65106i	0.102573i
$669$	−7.04569	−0.272402
$670$	0	0
$671$	−76.3635	−2.94798
$672$	− 6.96099i	− 0.268526i
$673$	− 15.6585i	− 0.603591i	−0.953373	−	0.301796i	$-0.902414\pi$
$673$	− 15.6585i	− 0.603591i	0.953373	−	0.301796i	$-0.0975860\pi$
$674$	−78.9977	−3.04288
$675$	0	0
$676$	−36.2311	−1.39350
$677$	− 32.5070i	− 1.24934i	−0.780887	−	0.624672i	$-0.785233\pi$
$677$	− 32.5070i	− 1.24934i	0.780887	−	0.624672i	$-0.214767\pi$
$678$	0.722892i	0.0277625i
$679$	3.87427	0.148681
$680$	0	0
$681$	16.9671	0.650181
$682$	− 76.3432i	− 2.92333i
$683$	− 0.00991954i	0 0.000379561i	−1.00000		0.000189780i	$-0.999940\pi$
$683$	− 0.00991954i	0 0.000379561i	1.00000		0.000189780i	$-6.04090e-5\pi$
$684$	−16.9624	−0.648572
$685$	0	0
$686$	29.8030	1.13788
$687$	− 5.14606i	− 0.196334i
$688$	− 8.08993i	− 0.308426i
$689$	−0.469394	−0.0178825
$690$	0	0
$691$	15.1856	0.577688	0.288844	−	0.957376i	$-0.406729\pi$
$691$	15.1856	0.577688	0.288844	+	0.957376i	$0.406729\pi$
$692$	59.8393i	2.27475i
$693$	− 6.37054i	− 0.241997i
$694$	−20.8701	−0.792217
$695$	0	0
$696$	18.4679	0.700026
$697$	− 32.6921i	− 1.23830i
$698$	15.4654i	0.585373i
$699$	−27.3476	−1.03438
$700$	0	0
$701$	38.0017	1.43530	0.717652	−	0.696402i	$-0.245217\pi$
$701$	38.0017	1.43530	0.717652	+	0.696402i	$0.245217\pi$
$702$	− 2.09464i	− 0.0790570i
$703$	− 7.35123i	− 0.277257i
$704$	80.2234	3.02353
$705$	0	0
$706$	11.4788	0.432010
$707$	− 2.68754i	− 0.101076i
$708$	− 3.56733i	− 0.134068i
$709$	−23.2204	−0.872062	−0.436031	−	0.899932i	$-0.643616\pi$
$709$	−23.2204	−0.872062	−0.436031	+	0.899932i	$0.643616\pi$
$710$	0	0
$711$	−13.9213	−0.522089
$712$	18.8069i	0.704819i
$713$	9.11178i	0.341239i
$714$	−15.1208	−0.565883
$715$	0	0
$716$	−10.1274	−0.378480
$717$	− 19.6994i	− 0.735686i
$718$	− 42.5352i	− 1.58740i
$719$	−23.3002	−0.868951	−0.434476	−	0.900684i	$-0.643066\pi$
$719$	−23.3002	−0.868951	−0.434476	+	0.900684i	$0.643066\pi$
$720$	0	0
$721$	−17.7593	−0.661390
$722$	− 29.4848i	− 1.09731i
$723$	− 9.57114i	− 0.355955i
$724$	−16.2013	−0.602116
$725$	0	0
$726$	60.6400	2.25056
$727$	− 23.3640i	− 0.866523i	−0.901268	−	0.433261i	$-0.857363\pi$
$727$	− 23.3640i	− 0.866523i	0.901268	−	0.433261i	$-0.142637\pi$
$728$	− 2.13716i	− 0.0792085i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−50.9061	−1.88283
$732$	36.9577i	1.36600i
$733$	10.8277i	0.399931i	0.979803	+	0.199966i	$0.0640831\pi$
$733$	10.8277i	0.399931i	−0.979803	+	0.199966i	$0.935917\pi$
$734$	−31.6992	−1.17004
$735$	0	0
$736$	−11.1127	−0.409619
$737$	46.8699i	1.72648i
$738$	− 11.1260i	− 0.409552i
$739$	−24.5505	−0.903105	−0.451553	−	0.892244i	$-0.649130\pi$
$739$	−24.5505	−0.903105	−0.451553	+	0.892244i	$0.649130\pi$
$740$	0	0
$741$	5.32136	0.195485
$742$	1.15318i	0.0423347i
$743$	0.745387i	0.0273456i	0.999907	+	0.0136728i	$0.00435232\pi$
$743$	0.745387i	0.0273456i	−0.999907	+	0.0136728i	$0.995648\pi$
$744$	−12.2271	−0.448266
$745$	0	0
$746$	79.2194	2.90043
$747$	− 1.46223i	− 0.0535000i
$748$	− 121.176i	− 4.43063i
$749$	5.71716	0.208900
$750$	0	0
$751$	1.84004	0.0671440	0.0335720	−	0.999436i	$-0.489312\pi$
$751$	1.84004	0.0671440	0.0335720	+	0.999436i	$0.489312\pi$
$752$	− 8.00304i	− 0.291841i
$753$	− 5.82514i	− 0.212280i
$754$	−17.5074	−0.637582
$755$	0	0
$756$	−3.08316	−0.112133
$757$	− 24.4525i	− 0.888742i	−0.895843	−	0.444371i	$-0.853427\pi$
$757$	− 24.4525i	− 0.888742i	0.895843	−	0.444371i	$-0.146573\pi$
$758$	− 4.46043i	− 0.162010i
$759$	−10.1701	−0.369151
$760$	0	0
$761$	15.6021	0.565574	0.282787	−	0.959183i	$-0.408741\pi$
$761$	15.6021	0.565574	0.282787	+	0.959183i	$0.408741\pi$
$762$	0.201948i	0.00731579i
$763$	6.17266i	0.223465i
$764$	33.7063	1.21945
$765$	0	0
$766$	−21.0844	−0.761810
$767$	1.11913i	0.0404093i
$768$	− 8.67641i	− 0.313083i
$769$	9.63438	0.347425	0.173712	−	0.984796i	$-0.444424\pi$
$769$	9.63438	0.347425	0.173712	+	0.984796i	$0.444424\pi$
$770$	0	0
$771$	−3.47558	−0.125170
$772$	18.6043i	0.669585i
$773$	− 5.00666i	− 0.180077i	−0.995938	−	0.0900386i	$-0.971301\pi$
$773$	− 5.00666i	− 0.180077i	0.995938	−	0.0900386i	$-0.0286990\pi$
$774$	−17.3246	−0.622722
$775$	0	0
$776$	8.29961	0.297939
$777$	− 1.33619i	− 0.0479356i
$778$	− 2.93803i	− 0.105334i
$779$	28.2651	1.01270
$780$	0	0
$781$	65.5477	2.34548
$782$	24.1393i	0.863220i
$783$	8.35819i	0.298697i
$784$	−6.19159	−0.221128
$785$	0	0
$786$	−30.3107	−1.08115
$787$	− 45.0282i	− 1.60508i	−0.596597	−	0.802541i	$-0.703481\pi$
$787$	− 45.0282i	− 1.60508i	0.596597	−	0.802541i	$-0.296519\pi$
$788$	− 58.9761i	− 2.10093i
$789$	22.7227	0.808948
$790$	0	0
$791$	−0.333807	−0.0118688
$792$	− 13.6472i	− 0.484933i
$793$	− 11.5942i	− 0.411722i
$794$	56.5306	2.00620
$795$	0	0
$796$	55.5190	1.96782
$797$	3.26600i	0.115688i	0.998326	+	0.0578438i	$0.0184225\pi$
$797$	3.26600i	0.115688i	−0.998326	+	0.0578438i	$0.981577\pi$
$798$	− 13.0733i	− 0.462789i
$799$	−50.3593	−1.78158
$800$	0	0
$801$	−8.51161	−0.300743
$802$	− 29.9926i	− 1.05908i
$803$	− 49.3273i	− 1.74072i
$804$	22.6837	0.799992
$805$	0	0
$806$	11.5911	0.408280
$807$	− 1.63546i	− 0.0575710i
$808$	− 5.75736i	− 0.202543i
$809$	24.5785	0.864135	0.432067	−	0.901841i	$-0.357784\pi$
$809$	24.5785	0.864135	0.432067	+	0.901841i	$0.357784\pi$
$810$	0	0
$811$	−17.0396	−0.598340	−0.299170	−	0.954200i	$-0.596710\pi$
$811$	−17.0396	−0.598340	−0.299170	+	0.954200i	$0.596710\pi$
$812$	25.7696i	0.904336i
$813$	12.4498i	0.436635i
$814$	17.8725	0.626430
$815$	0	0
$816$	6.84571	0.239648
$817$	− 44.0127i	− 1.53981i
$818$	− 42.0124i	− 1.46893i
$819$	0.967234	0.0337979
$820$	0	0
$821$	−3.30808	−0.115453	−0.0577264	−	0.998332i	$-0.518385\pi$
$821$	−3.30808	−0.115453	−0.0577264	+	0.998332i	$0.518385\pi$
$822$	31.2254i	1.08911i
$823$	14.5583i	0.507469i	0.967274	+	0.253735i	$0.0816590\pi$
$823$	14.5583i	0.507469i	−0.967274	+	0.253735i	$0.918341\pi$
$824$	−38.0446	−1.32535
$825$	0	0
$826$	2.74941	0.0956644
$827$	− 8.34632i	− 0.290230i	−0.989415	−	0.145115i	$-0.953645\pi$
$827$	− 8.34632i	− 0.290230i	0.989415	−	0.145115i	$-0.0463552\pi$
$828$	4.92203i	0.171052i
$829$	−11.1514	−0.387303	−0.193652	−	0.981070i	$-0.562033\pi$
$829$	−11.1514	−0.387303	−0.193652	+	0.981070i	$0.562033\pi$
$830$	0	0
$831$	2.16353	0.0750522
$832$	12.1803i	0.422274i
$833$	38.9607i	1.34991i
$834$	4.95340	0.171522
$835$	0	0
$836$	104.767	3.62344
$837$	− 5.53371i	− 0.191273i
$838$	− 48.6518i	− 1.68065i
$839$	−10.3600	−0.357665	−0.178833	−	0.983879i	$-0.557232\pi$
$839$	−10.3600	−0.357665	−0.178833	+	0.983879i	$0.557232\pi$
$840$	0	0
$841$	40.8594	1.40894
$842$	− 63.3641i	− 2.18367i
$843$	4.97817i	0.171457i
$844$	−8.13060	−0.279867
$845$	0	0
$846$	−17.1386	−0.589236
$847$	28.0015i	0.962144i
$848$	− 0.522085i	− 0.0179285i
$849$	17.3357	0.594958
$850$	0	0
$851$	−2.13313	−0.0731228
$852$	− 31.7232i	− 1.08682i
$853$	− 16.3865i	− 0.561064i	−0.959845	−	0.280532i	$-0.909489\pi$
$853$	− 16.3865i	− 0.561064i	0.959845	−	0.280532i	$-0.0905108\pi$
$854$	−28.4841	−0.974706
$855$	0	0
$856$	12.2475	0.418611
$857$	− 0.794707i	− 0.0271467i	−0.999908	−	0.0135733i	$-0.995679\pi$
$857$	− 0.794707i	− 0.0271467i	0.999908	−	0.0135733i	$-0.00432066\pi$
$858$	12.9374i	0.441675i
$859$	1.06774	0.0364307	0.0182154	−	0.999834i	$-0.494202\pi$
$859$	1.06774	0.0364307	0.0182154	+	0.999834i	$0.494202\pi$
$860$	0	0
$861$	5.13759	0.175089
$862$	52.9530i	1.80359i
$863$	49.3832i	1.68102i	0.541794	+	0.840511i	$0.317745\pi$
$863$	49.3832i	1.68102i	−0.541794	+	0.840511i	$0.682255\pi$
$864$	6.74889	0.229602
$865$	0	0
$866$	15.9002	0.540310
$867$	− 26.0768i	− 0.885614i
$868$	− 17.0613i	− 0.579098i
$869$	85.9839	2.91680
$870$	0	0
$871$	−7.11622	−0.241124
$872$	13.2233i	0.447798i
$873$	3.75623i	0.127129i
$874$	−20.8705	−0.705955
$875$	0	0
$876$	−23.8730	−0.806594
$877$	46.1096i	1.55701i	0.627638	+	0.778506i	$0.284022\pi$
$877$	46.1096i	1.55701i	−0.627638	+	0.778506i	$0.715978\pi$
$878$	14.9240i	0.503660i
$879$	−20.1863	−0.680866
$880$	0	0
$881$	−14.0984	−0.474988	−0.237494	−	0.971389i	$-0.576326\pi$
$881$	−14.0984	−0.474988	−0.237494	+	0.971389i	$0.576326\pi$
$882$	13.2593i	0.446465i
$883$	− 54.9493i	− 1.84919i	−0.380951	−	0.924595i	$-0.624403\pi$
$883$	− 54.9493i	− 1.84919i	0.380951	−	0.924595i	$-0.375597\pi$
$884$	18.3980	0.618793
$885$	0	0
$886$	46.2512	1.55384
$887$	− 0.338258i	− 0.0113576i	−0.999984	−	0.00567879i	$-0.998192\pi$
$887$	− 0.338258i	− 0.0113576i	0.999984	−	0.00567879i	$-0.00180763\pi$
$888$	− 2.86244i	− 0.0960573i
$889$	−0.0932527	−0.00312759
$890$	0	0
$891$	6.17643	0.206918
$892$	− 21.0611i	− 0.705177i
$893$	− 43.5399i	− 1.45701i
$894$	−8.02779	−0.268490
$895$	0	0
$896$	16.0019	0.534586
$897$	− 1.54412i	− 0.0515565i
$898$	− 0.487629i	− 0.0162724i
$899$	−46.2518	−1.54258
$900$	0	0
$901$	−3.28523	−0.109447
$902$	68.7188i	2.28808i
$903$	− 7.99994i	− 0.266221i
$904$	−0.715094	−0.0237837
$905$	0	0
$906$	37.6648	1.25133
$907$	− 25.1534i	− 0.835204i	−0.908630	−	0.417602i	$-0.862871\pi$
$907$	− 25.1534i	− 0.835204i	0.908630	−	0.417602i	$-0.137129\pi$
$908$	50.7183i	1.68315i
$909$	2.60566	0.0864242
$910$	0	0
$911$	−16.2238	−0.537519	−0.268759	−	0.963207i	$-0.586614\pi$
$911$	−16.2238	−0.537519	−0.268759	+	0.963207i	$0.586614\pi$
$912$	5.91870i	0.195988i
$913$	9.03134i	0.298894i
$914$	−13.3139	−0.440386
$915$	0	0
$916$	15.3827	0.508257
$917$	− 13.9965i	− 0.462205i
$918$	− 14.6601i	− 0.483856i
$919$	−21.5696	−0.711515	−0.355758	−	0.934578i	$-0.615777\pi$
$919$	−21.5696	−0.711515	−0.355758	+	0.934578i	$0.615777\pi$
$920$	0	0
$921$	24.7882	0.816799
$922$	28.6464i	0.943417i
$923$	9.95205i	0.327576i
$924$	19.0429	0.626466
$925$	0	0
$926$	60.1258	1.97586
$927$	− 17.2182i	− 0.565519i
$928$	− 56.4085i	− 1.85170i
$929$	37.6727	1.23600	0.618001	−	0.786177i	$-0.287943\pi$
$929$	37.6727	1.23600	0.618001	+	0.786177i	$0.287943\pi$
$930$	0	0
$931$	−33.6849	−1.10398
$932$	− 81.7477i	− 2.67773i
$933$	− 28.1046i	− 0.920105i
$934$	59.9960	1.96313
$935$	0	0
$936$	2.07204	0.0677269
$937$	42.7073i	1.39519i	0.716495	+	0.697593i	$0.245745\pi$
$937$	42.7073i	1.39519i	−0.716495	+	0.697593i	$0.754255\pi$
$938$	17.4828i	0.570834i
$939$	27.1858	0.887176
$940$	0	0
$941$	−4.27023	−0.139206	−0.0696028	−	0.997575i	$-0.522173\pi$
$941$	−4.27023	−0.139206	−0.0696028	+	0.997575i	$0.522173\pi$
$942$	18.8274i	0.613429i
$943$	− 8.20179i	− 0.267087i
$944$	−1.24475	−0.0405132
$945$	0	0
$946$	107.005	3.47902
$947$	26.4988i	0.861097i	0.902567	+	0.430548i	$0.141680\pi$
$947$	26.4988i	0.861097i	−0.902567	+	0.430548i	$0.858320\pi$
$948$	− 41.6137i	− 1.35155i
$949$	7.48933	0.243114
$950$	0	0
$951$	−14.9749	−0.485594
$952$	− 14.9577i	− 0.484783i
$953$	− 16.9907i	− 0.550383i	−0.961389	−	0.275191i	$-0.911259\pi$
$953$	− 16.9907i	− 0.550383i	0.961389	−	0.275191i	$-0.0887412\pi$
$954$	−1.11805	−0.0361981
$955$	0	0
$956$	58.8856	1.90450
$957$	− 51.6238i	− 1.66876i
$958$	87.7719i	2.83578i
$959$	−14.4188	−0.465608
$960$	0	0
$961$	−0.378072	−0.0121959
$962$	2.71356i	0.0874887i
$963$	5.54296i	0.178619i
$964$	28.6102	0.921472
$965$	0	0
$966$	−3.79351	−0.122054
$967$	− 2.79897i	− 0.0900089i	−0.998987	−	0.0450045i	$-0.985670\pi$
$967$	− 2.79897i	− 0.0900089i	0.998987	−	0.0450045i	$-0.0143302\pi$
$968$	59.9859i	1.92802i
$969$	37.2436	1.19644
$970$	0	0
$971$	−5.63046	−0.180690	−0.0903450	−	0.995911i	$-0.528797\pi$
$971$	−5.63046	−0.180690	−0.0903450	+	0.995911i	$0.528797\pi$
$972$	− 2.98921i	− 0.0958791i
$973$	2.28731i	0.0733279i
$974$	−83.3753	−2.67152
$975$	0	0
$976$	12.8957	0.412781
$977$	40.5445i	1.29713i	0.761157	+	0.648567i	$0.224632\pi$
$977$	40.5445i	1.29713i	−0.761157	+	0.648567i	$0.775368\pi$
$978$	− 25.4202i	− 0.812850i
$979$	52.5714	1.68019
$980$	0	0
$981$	−5.98458	−0.191073
$982$	37.5529i	1.19836i
$983$	− 16.6381i	− 0.530674i	−0.964156	−	0.265337i	$-0.914517\pi$
$983$	− 16.6381i	− 0.530674i	0.964156	−	0.265337i	$-0.0854832\pi$
$984$	11.0059	0.350857
$985$	0	0
$986$	−122.532	−3.90222
$987$	− 7.91401i	− 0.251906i
$988$	15.9067i	0.506059i
$989$	−12.7713	−0.406104
$990$	0	0
$991$	1.05069	0.0333764	0.0166882	−	0.999861i	$-0.494688\pi$
$991$	1.05069	0.0333764	0.0166882	+	0.999861i	$0.494688\pi$
$992$	37.3464i	1.18575i
$993$	− 13.9393i	− 0.442350i
$994$	24.4497	0.775498
$995$	0	0
$996$	4.37090	0.138497
$997$	0.318717i	0.0100939i	0.999987	+	0.00504693i	$0.00160650\pi$
$997$	0.318717i	0.0100939i	−0.999987	+	0.00504693i	$0.998394\pi$
$998$	− 46.2300i	− 1.46339i
$999$	1.29548	0.0409872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.g.1249.3		16
5.2	odd	4		1875.2.a.o.1.7	yes	8
5.3	odd	4		1875.2.a.n.1.2	✓	8
5.4	even	2	inner	1875.2.b.g.1249.14		16
15.2	even	4		5625.2.a.u.1.2		8
15.8	even	4		5625.2.a.bc.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.2	✓	8	5.3	odd	4
1875.2.a.o.1.7	yes	8	5.2	odd	4
1875.2.b.g.1249.3		16	1.1	even	1	trivial
1875.2.b.g.1249.14		16	5.4	even	2	inner
5625.2.a.u.1.2		8	15.2	even	4
5625.2.a.bc.1.7		8	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23365i q^{2} +1.00000i q^{3} -2.98921 q^{4} +2.23365 q^{6} +1.03143i q^{7} +2.20956i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.23365i q^{2} +1.00000i q^{3} -2.98921 q^{4} +2.23365 q^{6} +1.03143i q^{7} +2.20956i q^{8} -1.00000 q^{9} +6.17643 q^{11} -2.98921i q^{12} +0.937763i q^{13} +2.30385 q^{14} -1.04303 q^{16} +6.56329i q^{17} +2.23365i q^{18} -5.67453 q^{19} -1.03143 q^{21} -13.7960i q^{22} +1.64660i q^{23} -2.20956 q^{24} +2.09464 q^{26} -1.00000i q^{27} -3.08316i q^{28} -8.35819 q^{29} +5.53371 q^{31} +6.74889i q^{32} +6.17643i q^{33} +14.6601 q^{34} +2.98921 q^{36} +1.29548i q^{37} +12.6749i q^{38} -0.937763 q^{39} -4.98106 q^{41} +2.30385i q^{42} +7.75619i q^{43} -18.4627 q^{44} +3.67793 q^{46} +7.67288i q^{47} -1.04303i q^{48} +5.93616 q^{49} -6.56329 q^{51} -2.80317i q^{52} +0.500546i q^{53} -2.23365 q^{54} -2.27900 q^{56} -5.67453i q^{57} +18.6693i q^{58} +1.19340 q^{59} -12.3637 q^{61} -12.3604i q^{62} -1.03143i q^{63} +12.9886 q^{64} +13.7960 q^{66} +7.58851i q^{67} -19.6191i q^{68} -1.64660 q^{69} +10.6125 q^{71} -2.20956i q^{72} -7.98638i q^{73} +2.89365 q^{74} +16.9624 q^{76} +6.37054i q^{77} +2.09464i q^{78} +13.9213 q^{79} +1.00000 q^{81} +11.1260i q^{82} +1.46223i q^{83} +3.08316 q^{84} +17.3246 q^{86} -8.35819i q^{87} +13.6472i q^{88} +8.51161 q^{89} -0.967234 q^{91} -4.92203i q^{92} +5.53371i q^{93} +17.1386 q^{94} -6.74889 q^{96} -3.75623i q^{97} -13.2593i q^{98} -6.17643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9	\( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9} + 24 q^{11} - 32 q^{14} + 30 q^{16} - 32 q^{19} + 24 q^{21} - 6 q^{24} - 68 q^{26} - 4 q^{29} + 26 q^{31} + 74 q^{34} + 18 q^{36} - 28 q^{39} - 24 q^{41} - 94 q^{44} + 66 q^{46} - 60 q^{49} - 2 q^{51} - 2 q^{54} + 120 q^{56} - 28 q^{59} + 20 q^{61} - 82 q^{64} + 36 q^{66} + 8 q^{69} + 42 q^{71} + 18 q^{74} - 2 q^{76} - 20 q^{79} + 16 q^{81} + 42 q^{84} + 84 q^{86} + 18 q^{89} - 24 q^{91} - 28 q^{94} - 36 q^{96} - 24 q^{99}+O(q^{100}) \) 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9 + 24 * q^11 - 32 * q^14 + 30 * q^16 - 32 * q^19 + 24 * q^21 - 6 * q^24 - 68 * q^26 - 4 * q^29 + 26 * q^31 + 74 * q^34 + 18 * q^36 - 28 * q^39 - 24 * q^41 - 94 * q^44 + 66 * q^46 - 60 * q^49 - 2 * q^51 - 2 * q^54 + 120 * q^56 - 28 * q^59 + 20 * q^61 - 82 * q^64 + 36 * q^66 + 8 * q^69 + 42 * q^71 + 18 * q^74 - 2 * q^76 - 20 * q^79 + 16 * q^81 + 42 * q^84 + 84 * q^86 + 18 * q^89 - 24 * q^91 - 28 * q^94 - 36 * q^96 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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