LMFDB - Embedded newform 1875.2.b.f.1249.10

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 22x^{10} + 179x^{8} + 641x^{6} + 869x^{4} + 67x^{2} + 1 $ x^12 + 22x^10 + 179x^8 + 641x^6 + 869x^4 + 67*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.10
Root			$2.16056i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.f.1249.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.16056i q^{2} +1.00000i q^{3} -2.66802 q^{4} -2.16056 q^{6} -3.16056i q^{7} -1.44329i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.16056i q^{2} +1.00000i q^{3} -2.66802 q^{4} -2.16056 q^{6} -3.16056i q^{7} -1.44329i q^{8} -1.00000 q^{9} +1.53835 q^{11} -2.66802i q^{12} +5.24144i q^{13} +6.82858 q^{14} -2.21771 q^{16} +1.29068i q^{17} -2.16056i q^{18} -5.44587 q^{19} +3.16056 q^{21} +3.32370i q^{22} +6.44244i q^{23} +1.44329 q^{24} -11.3244 q^{26} -1.00000i q^{27} +8.43243i q^{28} +2.36361 q^{29} -4.46542 q^{31} -7.67809i q^{32} +1.53835i q^{33} -2.78860 q^{34} +2.66802 q^{36} -5.95751i q^{37} -11.7661i q^{38} -5.24144 q^{39} -8.53219 q^{41} +6.82858i q^{42} +8.48426i q^{43} -4.10435 q^{44} -13.9193 q^{46} +0.753070i q^{47} -2.21771i q^{48} -2.98914 q^{49} -1.29068 q^{51} -13.9843i q^{52} -9.74991i q^{53} +2.16056 q^{54} -4.56162 q^{56} -5.44587i q^{57} +5.10672i q^{58} -4.11270 q^{59} -10.6939 q^{61} -9.64781i q^{62} +3.16056i q^{63} +12.1535 q^{64} -3.32370 q^{66} -1.89864i q^{67} -3.44357i q^{68} -6.44244 q^{69} -0.0708774 q^{71} +1.44329i q^{72} -4.01123i q^{73} +12.8716 q^{74} +14.5297 q^{76} -4.86205i q^{77} -11.3244i q^{78} -1.61849 q^{79} +1.00000 q^{81} -18.4343i q^{82} -13.1488i q^{83} -8.43243 q^{84} -18.3308 q^{86} +2.36361i q^{87} -2.22029i q^{88} -7.27597 q^{89} +16.5659 q^{91} -17.1885i q^{92} -4.46542i q^{93} -1.62705 q^{94} +7.67809 q^{96} +10.1939i q^{97} -6.45821i q^{98} -1.53835 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) $ 12 * q - 20 * q^4 - 12 * q^9	$ 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) $ 12 * q - 20 * q^4 - 12 * q^9 + 6 * q^11 + 44 * q^14 + 36 * q^16 - 22 * q^19 + 12 * q^21 - 6 * q^24 - 56 * q^26 + 6 * q^29 - 22 * q^31 - 30 * q^34 + 20 * q^36 - 12 * q^39 - 2 * q^41 - 18 * q^44 + 38 * q^46 + 28 * q^49 + 26 * q^51 - 70 * q^56 - 18 * q^59 + 22 * q^61 + 46 * q^64 + 32 * q^66 - 26 * q^69 - 16 * q^71 + 44 * q^74 - 52 * q^76 + 10 * q^79 + 12 * q^81 - 14 * q^84 - 74 * q^86 + 8 * q^89 + 68 * q^91 - 82 * q^94 + 32 * q^96 - 6 * 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 $

$2$	2.16056i	1.52775i	0.645366	+	0.763873i	$0.276705\pi$
$2$	2.16056i	1.52775i	−0.645366	+	0.763873i	$0.723295\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−2.66802	−1.33401
$5$	0	0
$5$	0	0
$6$	−2.16056	−0.882045
$7$	− 3.16056i	− 1.19458i	−0.802026	−	0.597290i	$-0.796244\pi$
$7$	− 3.16056i	− 1.19458i	0.802026	−	0.597290i	$-0.203756\pi$
$8$	− 1.44329i	− 0.510282i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.53835	0.463830	0.231915	−	0.972736i	$-0.425501\pi$
$11$	1.53835	0.463830	0.231915	+	0.972736i	$0.425501\pi$
$12$	− 2.66802i	− 0.770191i
$13$	5.24144i	1.45371i	0.686789	+	0.726857i	$0.259019\pi$
$13$	5.24144i	1.45371i	−0.686789	+	0.726857i	$0.740981\pi$
$14$	6.82858	1.82501
$15$	0	0
$16$	−2.21771	−0.554428
$17$	1.29068i	0.313037i	0.987675	+	0.156518i	$0.0500270\pi$
$17$	1.29068i	0.313037i	−0.987675	+	0.156518i	$0.949973\pi$
$18$	− 2.16056i	− 0.509249i
$19$	−5.44587	−1.24937	−0.624685	−	0.780877i	$-0.714772\pi$
$19$	−5.44587	−1.24937	−0.624685	+	0.780877i	$0.714772\pi$
$20$	0	0
$21$	3.16056	0.689691
$22$	3.32370i	0.708615i
$23$	6.44244i	1.34334i	0.740850	+	0.671671i	$0.234423\pi$
$23$	6.44244i	1.34334i	−0.740850	+	0.671671i	$0.765577\pi$
$24$	1.44329	0.294611
$25$	0	0
$26$	−11.3244	−2.22090
$27$	− 1.00000i	− 0.192450i
$28$	8.43243i	1.59358i
$29$	2.36361	0.438912	0.219456	−	0.975622i	$-0.429572\pi$
$29$	2.36361	0.438912	0.219456	+	0.975622i	$0.429572\pi$
$30$	0	0
$31$	−4.46542	−0.802014	−0.401007	−	0.916075i	$-0.631340\pi$
$31$	−4.46542	−0.802014	−0.401007	+	0.916075i	$0.631340\pi$
$32$	− 7.67809i	− 1.35731i
$33$	1.53835i	0.267793i
$34$	−2.78860	−0.478241
$35$	0	0
$36$	2.66802	0.444670
$37$	− 5.95751i	− 0.979408i	−0.871889	−	0.489704i	$-0.837105\pi$
$37$	− 5.95751i	− 0.979408i	0.871889	−	0.489704i	$-0.162895\pi$
$38$	− 11.7661i	− 1.90872i
$39$	−5.24144	−0.839302
$40$	0	0
$41$	−8.53219	−1.33250	−0.666252	−	0.745726i	$-0.732103\pi$
$41$	−8.53219	−1.33250	−0.666252	+	0.745726i	$0.732103\pi$
$42$	6.82858i	1.05367i
$43$	8.48426i	1.29384i	0.762559	+	0.646919i	$0.223943\pi$
$43$	8.48426i	1.29384i	−0.762559	+	0.646919i	$0.776057\pi$
$44$	−4.10435	−0.618754
$45$	0	0
$46$	−13.9193	−2.05228
$47$	0.753070i	0.109847i	0.998491	+	0.0549233i	$0.0174914\pi$
$47$	0.753070i	0.109847i	−0.998491	+	0.0549233i	$0.982509\pi$
$48$	− 2.21771i	− 0.320099i
$49$	−2.98914	−0.427020
$50$	0	0
$51$	−1.29068	−0.180732
$52$	− 13.9843i	− 1.93927i
$53$	− 9.74991i	− 1.33925i	−0.742698	−	0.669626i	$-0.766454\pi$
$53$	− 9.74991i	− 1.33925i	0.742698	−	0.669626i	$-0.233546\pi$
$54$	2.16056	0.294015
$55$	0	0
$56$	−4.56162	−0.609572
$57$	− 5.44587i	− 0.721324i
$58$	5.10672i	0.670546i
$59$	−4.11270	−0.535428	−0.267714	−	0.963498i	$-0.586268\pi$
$59$	−4.11270	−0.535428	−0.267714	+	0.963498i	$0.586268\pi$
$60$	0	0
$61$	−10.6939	−1.36922	−0.684610	−	0.728910i	$-0.740027\pi$
$61$	−10.6939	−1.36922	−0.684610	+	0.728910i	$0.740027\pi$
$62$	− 9.64781i	− 1.22527i
$63$	3.16056i	0.398193i
$64$	12.1535	1.51919
$65$	0	0
$66$	−3.32370	−0.409119
$67$	− 1.89864i	− 0.231956i	−0.993252	−	0.115978i	$-0.963000\pi$
$67$	− 1.89864i	− 0.231956i	0.993252	−	0.115978i	$-0.0370003\pi$
$68$	− 3.44357i	− 0.417594i
$69$	−6.44244	−0.775578
$70$	0	0
$71$	−0.0708774	−0.00841160	−0.00420580	−	0.999991i	$-0.501339\pi$
$71$	−0.0708774	−0.00841160	−0.00420580	+	0.999991i	$0.501339\pi$
$72$	1.44329i	0.170094i
$73$	− 4.01123i	− 0.469478i	−0.972058	−	0.234739i	$-0.924576\pi$
$73$	− 4.01123i	− 0.469478i	0.972058	−	0.234739i	$-0.0754236\pi$
$74$	12.8716	1.49629
$75$	0	0
$76$	14.5297	1.66667
$77$	− 4.86205i	− 0.554082i
$78$	− 11.3244i	− 1.28224i
$79$	−1.61849	−0.182094	−0.0910472	−	0.995847i	$-0.529021\pi$
$79$	−1.61849	−0.182094	−0.0910472	+	0.995847i	$0.529021\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 18.4343i	− 2.03573i
$83$	− 13.1488i	− 1.44327i	−0.692272	−	0.721636i	$-0.743390\pi$
$83$	− 13.1488i	− 1.44327i	0.692272	−	0.721636i	$-0.256610\pi$
$84$	−8.43243	−0.920054
$85$	0	0
$86$	−18.3308	−1.97666
$87$	2.36361i	0.253406i
$88$	− 2.22029i	− 0.236684i
$89$	−7.27597	−0.771251	−0.385626	−	0.922655i	$-0.626014\pi$
$89$	−7.27597	−0.771251	−0.385626	+	0.922655i	$0.626014\pi$
$90$	0	0
$91$	16.5659	1.73658
$92$	− 17.1885i	− 1.79203i
$93$	− 4.46542i	− 0.463043i
$94$	−1.62705	−0.167818
$95$	0	0
$96$	7.67809	0.783642
$97$	10.1939i	1.03504i	0.855673	+	0.517518i	$0.173144\pi$
$97$	10.1939i	1.03504i	−0.855673	+	0.517518i	$0.826856\pi$
$98$	− 6.45821i	− 0.652378i
$99$	−1.53835	−0.154610
$100$	0	0
$101$	1.08759	0.108219	0.0541096	−	0.998535i	$-0.482768\pi$
$101$	1.08759	0.108219	0.0541096	+	0.998535i	$0.482768\pi$
$102$	− 2.78860i	− 0.276112i
$103$	4.93756i	0.486512i	0.969962	+	0.243256i	$0.0782155\pi$
$103$	4.93756i	0.486512i	−0.969962	+	0.243256i	$0.921784\pi$
$104$	7.56494	0.741803
$105$	0	0
$106$	21.0653	2.04604
$107$	15.3059i	1.47967i	0.672786	+	0.739837i	$0.265098\pi$
$107$	15.3059i	1.47967i	−0.672786	+	0.739837i	$0.734902\pi$
$108$	2.66802i	0.256730i
$109$	−6.65900	−0.637816	−0.318908	−	0.947786i	$-0.603316\pi$
$109$	−6.65900	−0.637816	−0.318908	+	0.947786i	$0.603316\pi$
$110$	0	0
$111$	5.95751	0.565462
$112$	7.00922i	0.662309i
$113$	8.97143i	0.843961i	0.906605	+	0.421981i	$0.138665\pi$
$113$	8.97143i	0.843961i	−0.906605	+	0.421981i	$0.861335\pi$
$114$	11.7661	1.10200
$115$	0	0
$116$	−6.30616	−0.585512
$117$	− 5.24144i	− 0.484571i
$118$	− 8.88573i	− 0.817998i
$119$	4.07928	0.373947
$120$	0	0
$121$	−8.63348	−0.784861
$122$	− 23.1049i	− 2.09182i
$123$	− 8.53219i	− 0.769322i
$124$	11.9138	1.06989
$125$	0	0
$126$	−6.82858	−0.608338
$127$	10.0185i	0.888995i	0.895780	+	0.444497i	$0.146618\pi$
$127$	10.0185i	0.888995i	−0.895780	+	0.444497i	$0.853382\pi$
$128$	10.9023i	0.963635i
$129$	−8.48426	−0.746997
$130$	0	0
$131$	−6.37865	−0.557305	−0.278653	−	0.960392i	$-0.589888\pi$
$131$	−6.37865	−0.557305	−0.278653	+	0.960392i	$0.589888\pi$
$132$	− 4.10435i	− 0.357238i
$133$	17.2120i	1.49247i
$134$	4.10214	0.354371
$135$	0	0
$136$	1.86284	0.159737
$137$	− 9.17732i	− 0.784071i	−0.919950	−	0.392036i	$-0.871771\pi$
$137$	− 9.17732i	− 0.784071i	0.919950	−	0.392036i	$-0.128229\pi$
$138$	− 13.9193i	− 1.18489i
$139$	9.26539	0.785880	0.392940	−	0.919564i	$-0.371458\pi$
$139$	9.26539	0.785880	0.392940	+	0.919564i	$0.371458\pi$
$140$	0	0
$141$	−0.753070	−0.0634200
$142$	− 0.153135i	− 0.0128508i
$143$	8.06317i	0.674276i
$144$	2.21771	0.184809
$145$	0	0
$146$	8.66649	0.717244
$147$	− 2.98914i	− 0.246540i
$148$	15.8947i	1.30654i
$149$	10.0817	0.825928	0.412964	−	0.910747i	$-0.364494\pi$
$149$	10.0817	0.825928	0.412964	+	0.910747i	$0.364494\pi$
$150$	0	0
$151$	10.7375	0.873804	0.436902	−	0.899509i	$-0.356076\pi$
$151$	10.7375	0.873804	0.436902	+	0.899509i	$0.356076\pi$
$152$	7.86000i	0.637530i
$153$	− 1.29068i	− 0.104346i
$154$	10.5048	0.846497
$155$	0	0
$156$	13.9843	1.11964
$157$	17.4417i	1.39200i	0.718041	+	0.696001i	$0.245039\pi$
$157$	17.4417i	1.39200i	−0.718041	+	0.696001i	$0.754961\pi$
$158$	− 3.49684i	− 0.278194i
$159$	9.74991	0.773218
$160$	0	0
$161$	20.3617	1.60473
$162$	2.16056i	0.169750i
$163$	16.2574i	1.27338i	0.771120	+	0.636689i	$0.219697\pi$
$163$	16.2574i	1.27338i	−0.771120	+	0.636689i	$0.780303\pi$
$164$	22.7641	1.77757
$165$	0	0
$166$	28.4089	2.20496
$167$	− 11.3906i	− 0.881430i	−0.897647	−	0.440715i	$-0.854725\pi$
$167$	− 11.3906i	− 0.881430i	0.897647	−	0.440715i	$-0.145275\pi$
$168$	− 4.56162i	− 0.351936i
$169$	−14.4727	−1.11328
$170$	0	0
$171$	5.44587	0.416456
$172$	− 22.6362i	− 1.72599i
$173$	− 9.45845i	− 0.719113i	−0.933123	−	0.359556i	$-0.882928\pi$
$173$	− 9.45845i	− 0.719113i	0.933123	−	0.359556i	$-0.117072\pi$
$174$	−5.10672	−0.387140
$175$	0	0
$176$	−3.41162	−0.257161
$177$	− 4.11270i	− 0.309129i
$178$	− 15.7202i	− 1.17828i
$179$	−4.07747	−0.304764	−0.152382	−	0.988322i	$-0.548694\pi$
$179$	−4.07747	−0.304764	−0.152382	+	0.988322i	$0.548694\pi$
$180$	0	0
$181$	−11.1202	−0.826558	−0.413279	−	0.910604i	$-0.635617\pi$
$181$	−11.1202	−0.826558	−0.413279	+	0.910604i	$0.635617\pi$
$182$	35.7916i	2.65305i
$183$	− 10.6939i	− 0.790519i
$184$	9.29833	0.685482
$185$	0	0
$186$	9.64781	0.707412
$187$	1.98552i	0.145196i
$188$	− 2.00921i	− 0.146536i
$189$	−3.16056	−0.229897
$190$	0	0
$191$	3.61135	0.261308	0.130654	−	0.991428i	$-0.458292\pi$
$191$	3.61135	0.261308	0.130654	+	0.991428i	$0.458292\pi$
$192$	12.1535i	0.877107i
$193$	17.0562i	1.22774i	0.789409	+	0.613868i	$0.210387\pi$
$193$	17.0562i	1.22774i	−0.789409	+	0.613868i	$0.789613\pi$
$194$	−22.0246	−1.58127
$195$	0	0
$196$	7.97508	0.569648
$197$	− 4.40523i	− 0.313860i	−0.987610	−	0.156930i	$-0.949840\pi$
$197$	− 4.40523i	− 0.313860i	0.987610	−	0.156930i	$-0.0501597\pi$
$198$	− 3.32370i	− 0.236205i
$199$	−16.5956	−1.17643	−0.588214	−	0.808705i	$-0.700169\pi$
$199$	−16.5956	−1.17643	−0.588214	+	0.808705i	$0.700169\pi$
$200$	0	0
$201$	1.89864	0.133920
$202$	2.34980i	0.165332i
$203$	− 7.47034i	− 0.524315i
$204$	3.44357	0.241098
$205$	0	0
$206$	−10.6679	−0.743267
$207$	− 6.44244i	− 0.447780i
$208$	− 11.6240i	− 0.805980i
$209$	−8.37767	−0.579495
$210$	0	0
$211$	18.5454	1.27671	0.638357	−	0.769740i	$-0.279614\pi$
$211$	18.5454	1.27671	0.638357	+	0.769740i	$0.279614\pi$
$212$	26.0129i	1.78658i
$213$	− 0.0708774i	− 0.00485644i
$214$	−33.0693	−2.26057
$215$	0	0
$216$	−1.44329	−0.0982037
$217$	14.1132i	0.958069i
$218$	− 14.3872i	− 0.974422i
$219$	4.01123	0.271053
$220$	0	0
$221$	−6.76504	−0.455066
$222$	12.8716i	0.863882i
$223$	0.771557i	0.0516673i	0.999666	+	0.0258336i	$0.00822402\pi$
$223$	0.771557i	0.0516673i	−0.999666	+	0.0258336i	$0.991776\pi$
$224$	−24.2671	−1.62141
$225$	0	0
$226$	−19.3833	−1.28936
$227$	11.4314i	0.758727i	0.925248	+	0.379363i	$0.123857\pi$
$227$	11.4314i	0.758727i	−0.925248	+	0.379363i	$0.876143\pi$
$228$	14.5297i	0.962253i
$229$	−10.2248	−0.675671	−0.337835	−	0.941205i	$-0.609695\pi$
$229$	−10.2248	−0.675671	−0.337835	+	0.941205i	$0.609695\pi$
$230$	0	0
$231$	4.86205	0.319899
$232$	− 3.41139i	− 0.223969i
$233$	22.5229i	1.47553i	0.675059	+	0.737763i	$0.264118\pi$
$233$	22.5229i	1.47553i	−0.675059	+	0.737763i	$0.735882\pi$
$234$	11.3244	0.740302
$235$	0	0
$236$	10.9728	0.714266
$237$	− 1.61849i	− 0.105132i
$238$	8.81353i	0.571297i
$239$	0.589074	0.0381040	0.0190520	−	0.999818i	$-0.493935\pi$
$239$	0.589074	0.0381040	0.0190520	+	0.999818i	$0.493935\pi$
$240$	0	0
$241$	1.50937	0.0972270	0.0486135	−	0.998818i	$-0.484520\pi$
$241$	1.50937	0.0972270	0.0486135	+	0.998818i	$0.484520\pi$
$242$	− 18.6531i	− 1.19907i
$243$	1.00000i	0.0641500i
$244$	28.5317	1.82655
$245$	0	0
$246$	18.4343	1.17533
$247$	− 28.5442i	− 1.81622i
$248$	6.44492i	0.409253i
$249$	13.1488	0.833274
$250$	0	0
$251$	11.8953	0.750823	0.375412	−	0.926858i	$-0.377501\pi$
$251$	11.8953	0.750823	0.375412	+	0.926858i	$0.377501\pi$
$252$	− 8.43243i	− 0.531193i
$253$	9.91073i	0.623082i
$254$	−21.6455	−1.35816
$255$	0	0
$256$	0.752056	0.0470035
$257$	− 18.0492i	− 1.12588i	−0.826498	−	0.562939i	$-0.809671\pi$
$257$	− 18.0492i	− 1.12588i	0.826498	−	0.562939i	$-0.190329\pi$
$258$	− 18.3308i	− 1.14122i
$259$	−18.8291	−1.16998
$260$	0	0
$261$	−2.36361	−0.146304
$262$	− 13.7815i	− 0.851421i
$263$	19.1066i	1.17817i	0.808072	+	0.589083i	$0.200511\pi$
$263$	19.1066i	1.17817i	−0.808072	+	0.589083i	$0.799489\pi$
$264$	2.22029	0.136650
$265$	0	0
$266$	−37.1876	−2.28012
$267$	− 7.27597i	− 0.445282i
$268$	5.06562i	0.309432i
$269$	22.2250	1.35508	0.677542	−	0.735484i	$-0.263045\pi$
$269$	22.2250	1.35508	0.677542	+	0.735484i	$0.263045\pi$
$270$	0	0
$271$	20.2107	1.22772	0.613858	−	0.789417i	$-0.289617\pi$
$271$	20.2107	1.22772	0.613858	+	0.789417i	$0.289617\pi$
$272$	− 2.86237i	− 0.173556i
$273$	16.5659i	1.00261i
$274$	19.8281	1.19786
$275$	0	0
$276$	17.1885	1.03463
$277$	− 16.0413i	− 0.963826i	−0.876219	−	0.481913i	$-0.839942\pi$
$277$	− 16.0413i	− 0.963826i	0.876219	−	0.481913i	$-0.160058\pi$
$278$	20.0184i	1.20063i
$279$	4.46542	0.267338
$280$	0	0
$281$	8.37308	0.499496	0.249748	−	0.968311i	$-0.419652\pi$
$281$	8.37308	0.499496	0.249748	+	0.968311i	$0.419652\pi$
$282$	− 1.62705i	− 0.0968896i
$283$	1.12995i	0.0671685i	0.999436	+	0.0335843i	$0.0106922\pi$
$283$	1.12995i	0.0671685i	−0.999436	+	0.0335843i	$0.989308\pi$
$284$	0.189102	0.0112211
$285$	0	0
$286$	−17.4210	−1.03012
$287$	26.9665i	1.59178i
$288$	7.67809i	0.452436i
$289$	15.3341	0.902008
$290$	0	0
$291$	−10.1939	−0.597578
$292$	10.7020i	0.626289i
$293$	2.37857i	0.138958i	0.997583	+	0.0694789i	$0.0221337\pi$
$293$	2.37857i	0.138958i	−0.997583	+	0.0694789i	$0.977866\pi$
$294$	6.45821	0.376651
$295$	0	0
$296$	−8.59844	−0.499774
$297$	− 1.53835i	− 0.0892642i
$298$	21.7822i	1.26181i
$299$	−33.7676	−1.95283
$300$	0	0
$301$	26.8150	1.54559
$302$	23.1990i	1.33495i
$303$	1.08759i	0.0624804i
$304$	12.0774	0.692686
$305$	0	0
$306$	2.78860	0.159414
$307$	2.89366i	0.165150i	0.996585	+	0.0825748i	$0.0263143\pi$
$307$	2.89366i	0.165150i	−0.996585	+	0.0825748i	$0.973686\pi$
$308$	12.9720i	0.739151i
$309$	−4.93756	−0.280888
$310$	0	0
$311$	−7.14545	−0.405181	−0.202591	−	0.979264i	$-0.564936\pi$
$311$	−7.14545	−0.405181	−0.202591	+	0.979264i	$0.564936\pi$
$312$	7.56494i	0.428280i
$313$	− 23.0318i	− 1.30184i	−0.759148	−	0.650918i	$-0.774384\pi$
$313$	− 23.0318i	− 1.30184i	0.759148	−	0.650918i	$-0.225616\pi$
$314$	−37.6839	−2.12663
$315$	0	0
$316$	4.31816	0.242916
$317$	− 20.7750i	− 1.16684i	−0.812171	−	0.583419i	$-0.801715\pi$
$317$	− 20.7750i	− 1.16684i	0.812171	−	0.583419i	$-0.198285\pi$
$318$	21.0653i	1.18128i
$319$	3.63606	0.203581
$320$	0	0
$321$	−15.3059	−0.854291
$322$	43.9927i	2.45162i
$323$	− 7.02890i	− 0.391098i
$324$	−2.66802	−0.148223
$325$	0	0
$326$	−35.1251	−1.94540
$327$	− 6.65900i	− 0.368243i
$328$	12.3145i	0.679953i
$329$	2.38012	0.131220
$330$	0	0
$331$	−19.2504	−1.05810	−0.529048	−	0.848592i	$-0.677451\pi$
$331$	−19.2504	−1.05810	−0.529048	+	0.848592i	$0.677451\pi$
$332$	35.0814i	1.92534i
$333$	5.95751i	0.326469i
$334$	24.6100	1.34660
$335$	0	0
$336$	−7.00922	−0.382384
$337$	21.7064i	1.18243i	0.806516	+	0.591213i	$0.201351\pi$
$337$	21.7064i	1.18243i	−0.806516	+	0.591213i	$0.798649\pi$
$338$	− 31.2690i	− 1.70081i
$339$	−8.97143	−0.487261
$340$	0	0
$341$	−6.86939	−0.371998
$342$	11.7661i	0.636240i
$343$	− 12.6766i	− 0.684470i
$344$	12.2453	0.660221
$345$	0	0
$346$	20.4356	1.09862
$347$	− 1.71494i	− 0.0920626i	−0.998940	−	0.0460313i	$-0.985343\pi$
$347$	− 1.71494i	− 0.0920626i	0.998940	−	0.0460313i	$-0.0146574\pi$
$348$	− 6.30616i	− 0.338046i
$349$	15.5553	0.832654	0.416327	−	0.909215i	$-0.363317\pi$
$349$	15.5553	0.832654	0.416327	+	0.909215i	$0.363317\pi$
$350$	0	0
$351$	5.24144	0.279767
$352$	− 11.8116i	− 0.629560i
$353$	− 15.5536i	− 0.827833i	−0.910315	−	0.413916i	$-0.864161\pi$
$353$	− 15.5536i	− 0.827833i	0.910315	−	0.413916i	$-0.135839\pi$
$354$	8.88573	0.472271
$355$	0	0
$356$	19.4124	1.02886
$357$	4.07928i	0.215899i
$358$	− 8.80961i	− 0.465602i
$359$	24.5371	1.29502	0.647508	−	0.762059i	$-0.275811\pi$
$359$	24.5371	1.29502	0.647508	+	0.762059i	$0.275811\pi$
$360$	0	0
$361$	10.6575	0.560924
$362$	− 24.0259i	− 1.26277i
$363$	− 8.63348i	− 0.453140i
$364$	−44.1981	−2.31661
$365$	0	0
$366$	23.1049	1.20771
$367$	28.1068i	1.46716i	0.679602	+	0.733581i	$0.262153\pi$
$367$	28.1068i	1.46716i	−0.679602	+	0.733581i	$0.737847\pi$
$368$	− 14.2875i	− 0.744786i
$369$	8.53219	0.444168
$370$	0	0
$371$	−30.8152	−1.59984
$372$	11.9138i	0.617703i
$373$	9.70825i	0.502674i	0.967900	+	0.251337i	$0.0808703\pi$
$373$	9.70825i	0.502674i	−0.967900	+	0.251337i	$0.919130\pi$
$374$	−4.28984	−0.221823
$375$	0	0
$376$	1.08690	0.0560527
$377$	12.3887i	0.638052i
$378$	− 6.82858i	− 0.351224i
$379$	22.2665	1.14375	0.571875	−	0.820340i	$-0.306216\pi$
$379$	22.2665	1.14375	0.571875	+	0.820340i	$0.306216\pi$
$380$	0	0
$381$	−10.0185	−0.513261
$382$	7.80253i	0.399212i
$383$	13.6299i	0.696458i	0.937410	+	0.348229i	$0.113217\pi$
$383$	13.6299i	0.696458i	−0.937410	+	0.348229i	$0.886783\pi$
$384$	−10.9023	−0.556355
$385$	0	0
$386$	−36.8510	−1.87567
$387$	− 8.48426i	− 0.431279i
$388$	− 27.1976i	− 1.38075i
$389$	−19.2304	−0.975021	−0.487510	−	0.873117i	$-0.662095\pi$
$389$	−19.2304	−0.975021	−0.487510	+	0.873117i	$0.662095\pi$
$390$	0	0
$391$	−8.31515	−0.420515
$392$	4.31421i	0.217900i
$393$	− 6.37865i	− 0.321760i
$394$	9.51777	0.479498
$395$	0	0
$396$	4.10435	0.206251
$397$	22.2281i	1.11560i	0.829976	+	0.557799i	$0.188354\pi$
$397$	22.2281i	1.11560i	−0.829976	+	0.557799i	$0.811646\pi$
$398$	− 35.8557i	− 1.79729i
$399$	−17.2120	−0.861678
$400$	0	0
$401$	4.71728	0.235570	0.117785	−	0.993039i	$-0.462421\pi$
$401$	4.71728	0.235570	0.117785	+	0.993039i	$0.462421\pi$
$402$	4.10214i	0.204596i
$403$	− 23.4052i	− 1.16590i
$404$	−2.90171	−0.144365
$405$	0	0
$406$	16.1401	0.801020
$407$	− 9.16474i	− 0.454279i
$408$	1.86284i	0.0922241i
$409$	12.4807	0.617130	0.308565	−	0.951203i	$-0.400151\pi$
$409$	12.4807	0.617130	0.308565	+	0.951203i	$0.400151\pi$
$410$	0	0
$411$	9.17732	0.452684
$412$	− 13.1735i	− 0.649012i
$413$	12.9984i	0.639611i
$414$	13.9193	0.684095
$415$	0	0
$416$	40.2442	1.97314
$417$	9.26539i	0.453728i
$418$	− 18.1005i	− 0.885322i
$419$	−34.9484	−1.70734	−0.853669	−	0.520815i	$-0.825628\pi$
$419$	−34.9484	−1.70734	−0.853669	+	0.520815i	$0.825628\pi$
$420$	0	0
$421$	−39.5601	−1.92804	−0.964020	−	0.265829i	$-0.914354\pi$
$421$	−39.5601	−1.92804	−0.964020	+	0.265829i	$0.914354\pi$
$422$	40.0683i	1.95050i
$423$	− 0.753070i	− 0.0366155i
$424$	−14.0720	−0.683396
$425$	0	0
$426$	0.153135	0.00741940
$427$	33.7989i	1.63564i
$428$	− 40.8364i	− 1.97390i
$429$	−8.06317	−0.389294
$430$	0	0
$431$	−14.0584	−0.677170	−0.338585	−	0.940936i	$-0.609948\pi$
$431$	−14.0584	−0.677170	−0.338585	+	0.940936i	$0.609948\pi$
$432$	2.21771i	0.106700i
$433$	0.549678i	0.0264158i	0.999913	+	0.0132079i	$0.00420433\pi$
$433$	0.549678i	0.0264158i	−0.999913	+	0.0132079i	$0.995796\pi$
$434$	−30.4925	−1.46369
$435$	0	0
$436$	17.7663	0.850853
$437$	− 35.0847i	− 1.67833i
$438$	8.66649i	0.414101i
$439$	2.97377	0.141930	0.0709651	−	0.997479i	$-0.477392\pi$
$439$	2.97377	0.141930	0.0709651	+	0.997479i	$0.477392\pi$
$440$	0	0
$441$	2.98914	0.142340
$442$	− 14.6163i	− 0.695225i
$443$	36.6893i	1.74316i	0.490253	+	0.871580i	$0.336905\pi$
$443$	36.6893i	1.74316i	−0.490253	+	0.871580i	$0.663095\pi$
$444$	−15.8947	−0.754331
$445$	0	0
$446$	−1.66700	−0.0789345
$447$	10.0817i	0.476850i
$448$	− 38.4120i	− 1.81480i
$449$	17.8432	0.842071	0.421036	−	0.907044i	$-0.361667\pi$
$449$	17.8432	0.842071	0.421036	+	0.907044i	$0.361667\pi$
$450$	0	0
$451$	−13.1255	−0.618056
$452$	− 23.9359i	− 1.12585i
$453$	10.7375i	0.504491i
$454$	−24.6982	−1.15914
$455$	0	0
$456$	−7.86000	−0.368078
$457$	− 24.1784i	− 1.13102i	−0.824742	−	0.565509i	$-0.808680\pi$
$457$	− 24.1784i	− 1.13102i	0.824742	−	0.565509i	$-0.191320\pi$
$458$	− 22.0912i	− 1.03225i
$459$	1.29068	0.0602439
$460$	0	0
$461$	−37.3874	−1.74130	−0.870651	−	0.491901i	$-0.836302\pi$
$461$	−37.3874	−1.74130	−0.870651	+	0.491901i	$0.836302\pi$
$462$	10.5048i	0.488725i
$463$	− 20.4314i	− 0.949530i	−0.880113	−	0.474765i	$-0.842533\pi$
$463$	− 20.4314i	− 0.949530i	0.880113	−	0.474765i	$-0.157467\pi$
$464$	−5.24181	−0.243345
$465$	0	0
$466$	−48.6622	−2.25423
$467$	31.2124i	1.44434i	0.691716	+	0.722169i	$0.256855\pi$
$467$	31.2124i	1.44434i	−0.691716	+	0.722169i	$0.743145\pi$
$468$	13.9843i	0.646422i
$469$	−6.00078	−0.277090
$470$	0	0
$471$	−17.4417	−0.803673
$472$	5.93584i	0.273219i
$473$	13.0518i	0.600121i
$474$	3.49684	0.160615
$475$	0	0
$476$	−10.8836	−0.498849
$477$	9.74991i	0.446418i
$478$	1.27273i	0.0582133i
$479$	−11.0970	−0.507033	−0.253516	−	0.967331i	$-0.581587\pi$
$479$	−11.0970	−0.507033	−0.253516	+	0.967331i	$0.581587\pi$
$480$	0	0
$481$	31.2259	1.42378
$482$	3.26108i	0.148538i
$483$	20.3617i	0.926490i
$484$	23.0343	1.04701
$485$	0	0
$486$	−2.16056	−0.0980050
$487$	7.13702i	0.323409i	0.986839	+	0.161705i	$0.0516992\pi$
$487$	7.13702i	0.323409i	−0.986839	+	0.161705i	$0.948301\pi$
$488$	15.4345i	0.698688i
$489$	−16.2574	−0.735185
$490$	0	0
$491$	7.23348	0.326442	0.163221	−	0.986590i	$-0.447812\pi$
$491$	7.23348	0.326442	0.163221	+	0.986590i	$0.447812\pi$
$492$	22.7641i	1.02628i
$493$	3.05067i	0.137395i
$494$	61.6715	2.77473
$495$	0	0
$496$	9.90303	0.444659
$497$	0.224012i	0.0100483i
$498$	28.4089i	1.27303i
$499$	−16.0169	−0.717014	−0.358507	−	0.933527i	$-0.616714\pi$
$499$	−16.0169	−0.717014	−0.358507	+	0.933527i	$0.616714\pi$
$500$	0	0
$501$	11.3906	0.508894
$502$	25.7005i	1.14707i
$503$	33.7963i	1.50690i	0.657503	+	0.753452i	$0.271613\pi$
$503$	33.7963i	1.50690i	−0.657503	+	0.753452i	$0.728387\pi$
$504$	4.56162	0.203191
$505$	0	0
$506$	−21.4127	−0.951912
$507$	− 14.4727i	− 0.642753i
$508$	− 26.7294i	− 1.18593i
$509$	36.4270	1.61460	0.807300	−	0.590142i	$-0.200928\pi$
$509$	36.4270	1.61460	0.807300	+	0.590142i	$0.200928\pi$
$510$	0	0
$511$	−12.6777	−0.560829
$512$	23.4294i	1.03544i
$513$	5.44587i	0.240441i
$514$	38.9964	1.72006
$515$	0	0
$516$	22.6362	0.996502
$517$	1.15849i	0.0509502i
$518$	− 40.6813i	− 1.78743i
$519$	9.45845	0.415180
$520$	0	0
$521$	25.4993	1.11714	0.558572	−	0.829456i	$-0.311349\pi$
$521$	25.4993	1.11714	0.558572	+	0.829456i	$0.311349\pi$
$522$	− 5.10672i	− 0.223515i
$523$	7.03174i	0.307477i	0.988112	+	0.153738i	$0.0491313\pi$
$523$	7.03174i	0.307477i	−0.988112	+	0.153738i	$0.950869\pi$
$524$	17.0184	0.743450
$525$	0	0
$526$	−41.2811	−1.79994
$527$	− 5.76345i	− 0.251060i
$528$	− 3.41162i	− 0.148472i
$529$	−18.5050	−0.804565
$530$	0	0
$531$	4.11270	0.178476
$532$	− 45.9220i	− 1.99097i
$533$	− 44.7210i	− 1.93708i
$534$	15.7202	0.680278
$535$	0	0
$536$	−2.74030	−0.118363
$537$	− 4.07747i	− 0.175956i
$538$	48.0185i	2.07022i
$539$	−4.59834	−0.198065
$540$	0	0
$541$	−27.1476	−1.16717	−0.583584	−	0.812053i	$-0.698350\pi$
$541$	−27.1476	−1.16717	−0.583584	+	0.812053i	$0.698350\pi$
$542$	43.6665i	1.87564i
$543$	− 11.1202i	− 0.477214i
$544$	9.90999	0.424887
$545$	0	0
$546$	−35.7916	−1.53174
$547$	44.0769i	1.88459i	0.334782	+	0.942296i	$0.391337\pi$
$547$	44.0769i	1.88459i	−0.334782	+	0.942296i	$0.608663\pi$
$548$	24.4853i	1.04596i
$549$	10.6939	0.456407
$550$	0	0
$551$	−12.8719	−0.548363
$552$	9.29833i	0.395763i
$553$	5.11533i	0.217526i
$554$	34.6581	1.47248
$555$	0	0
$556$	−24.7202	−1.04837
$557$	− 9.85667i	− 0.417641i	−0.977954	−	0.208820i	$-0.933038\pi$
$557$	− 9.85667i	− 0.417641i	0.977954	−	0.208820i	$-0.0669624\pi$
$558$	9.64781i	0.408425i
$559$	−44.4697	−1.88087
$560$	0	0
$561$	−1.98552	−0.0838289
$562$	18.0905i	0.763103i
$563$	12.3917i	0.522249i	0.965305	+	0.261125i	$0.0840933\pi$
$563$	12.3917i	0.522249i	−0.965305	+	0.261125i	$0.915907\pi$
$564$	2.00921	0.0846028
$565$	0	0
$566$	−2.44132	−0.102616
$567$	− 3.16056i	− 0.132731i
$568$	0.102297i	0.00429228i
$569$	11.7158	0.491150	0.245575	−	0.969378i	$-0.421023\pi$
$569$	11.7158	0.491150	0.245575	+	0.969378i	$0.421023\pi$
$570$	0	0
$571$	28.9797	1.21276	0.606382	−	0.795173i	$-0.292620\pi$
$571$	28.9797	1.21276	0.606382	+	0.795173i	$0.292620\pi$
$572$	− 21.5127i	− 0.899491i
$573$	3.61135i	0.150866i
$574$	−58.2628	−2.43184
$575$	0	0
$576$	−12.1535	−0.506398
$577$	− 32.8024i	− 1.36558i	−0.730614	−	0.682790i	$-0.760766\pi$
$577$	− 32.8024i	− 1.36558i	0.730614	−	0.682790i	$-0.239234\pi$
$578$	33.1303i	1.37804i
$579$	−17.0562	−0.708833
$580$	0	0
$581$	−41.5577	−1.72410
$582$	− 22.0246i	− 0.912947i
$583$	− 14.9988i	− 0.621186i
$584$	−5.78938	−0.239566
$585$	0	0
$586$	−5.13905	−0.212292
$587$	18.6395i	0.769334i	0.923055	+	0.384667i	$0.125684\pi$
$587$	18.6395i	0.769334i	−0.923055	+	0.384667i	$0.874316\pi$
$588$	7.97508i	0.328887i
$589$	24.3181	1.00201
$590$	0	0
$591$	4.40523	0.181207
$592$	13.2120i	0.543012i
$593$	− 1.55882i	− 0.0640129i	−0.999488	−	0.0320064i	$-0.989810\pi$
$593$	− 1.55882i	− 0.0640129i	0.999488	−	0.0320064i	$-0.0101897\pi$
$594$	3.32370	0.136373
$595$	0	0
$596$	−26.8983	−1.10180
$597$	− 16.5956i	− 0.679212i
$598$	− 72.9570i	− 2.98343i
$599$	31.5470	1.28898	0.644489	−	0.764614i	$-0.277070\pi$
$599$	31.5470	1.28898	0.644489	+	0.764614i	$0.277070\pi$
$600$	0	0
$601$	−32.3999	−1.32162	−0.660809	−	0.750554i	$-0.729787\pi$
$601$	−32.3999	−1.32162	−0.660809	+	0.750554i	$0.729787\pi$
$602$	57.9354i	2.36127i
$603$	1.89864i	0.0773188i
$604$	−28.6478	−1.16566
$605$	0	0
$606$	−2.34980	−0.0954542
$607$	24.8410i	1.00827i	0.863626	+	0.504133i	$0.168188\pi$
$607$	24.8410i	1.00827i	−0.863626	+	0.504133i	$0.831812\pi$
$608$	41.8139i	1.69578i
$609$	7.47034	0.302713
$610$	0	0
$611$	−3.94717	−0.159685
$612$	3.44357i	0.139198i
$613$	3.94558i	0.159360i	0.996820	+	0.0796802i	$0.0253899\pi$
$613$	3.94558i	0.159360i	−0.996820	+	0.0796802i	$0.974610\pi$
$614$	−6.25192	−0.252307
$615$	0	0
$616$	−7.01737	−0.282738
$617$	− 17.4821i	− 0.703801i	−0.936037	−	0.351900i	$-0.885536\pi$
$617$	− 17.4821i	− 0.703801i	0.936037	−	0.351900i	$-0.114464\pi$
$618$	− 10.6679i	− 0.429126i
$619$	14.3869	0.578260	0.289130	−	0.957290i	$-0.406634\pi$
$619$	14.3869	0.578260	0.289130	+	0.957290i	$0.406634\pi$
$620$	0	0
$621$	6.44244	0.258526
$622$	− 15.4382i	− 0.619014i
$623$	22.9961i	0.921321i
$624$	11.6240	0.465333
$625$	0	0
$626$	49.7617	1.98888
$627$	− 8.37767i	− 0.334572i
$628$	− 46.5349i	− 1.85694i
$629$	7.68926	0.306591
$630$	0	0
$631$	−17.6758	−0.703664	−0.351832	−	0.936063i	$-0.614441\pi$
$631$	−17.6758	−0.703664	−0.351832	+	0.936063i	$0.614441\pi$
$632$	2.33596i	0.0929194i
$633$	18.5454i	0.737112i
$634$	44.8855	1.78263
$635$	0	0
$636$	−26.0129	−1.03148
$637$	− 15.6674i	− 0.620764i
$638$	7.85594i	0.311019i
$639$	0.0708774	0.00280387
$640$	0	0
$641$	−2.74127	−0.108274	−0.0541369	−	0.998534i	$-0.517241\pi$
$641$	−2.74127	−0.108274	−0.0541369	+	0.998534i	$0.517241\pi$
$642$	− 33.0693i	− 1.30514i
$643$	− 20.9276i	− 0.825303i	−0.910889	−	0.412651i	$-0.864603\pi$
$643$	− 20.9276i	− 0.825303i	0.910889	−	0.412651i	$-0.135397\pi$
$644$	−54.3254	−2.14072
$645$	0	0
$646$	15.1864	0.597499
$647$	− 1.84667i	− 0.0726002i	−0.999341	−	0.0363001i	$-0.988443\pi$
$647$	− 1.84667i	− 0.0726002i	0.999341	−	0.0363001i	$-0.0115572\pi$
$648$	− 1.44329i	− 0.0566980i
$649$	−6.32678	−0.248348
$650$	0	0
$651$	−14.1132	−0.553141
$652$	− 43.3751i	− 1.69870i
$653$	18.4133i	0.720567i	0.932843	+	0.360284i	$0.117320\pi$
$653$	18.4133i	0.720567i	−0.932843	+	0.360284i	$0.882680\pi$
$654$	14.3872	0.562583
$655$	0	0
$656$	18.9220	0.738778
$657$	4.01123i	0.156493i
$658$	5.14240i	0.200472i
$659$	13.3800	0.521210	0.260605	−	0.965446i	$-0.416078\pi$
$659$	13.3800	0.521210	0.260605	+	0.965446i	$0.416078\pi$
$660$	0	0
$661$	39.1834	1.52406	0.762030	−	0.647542i	$-0.224203\pi$
$661$	39.1834	1.52406	0.762030	+	0.647542i	$0.224203\pi$
$662$	− 41.5915i	− 1.61650i
$663$	− 6.76504i	− 0.262732i
$664$	−18.9776	−0.736476
$665$	0	0
$666$	−12.8716	−0.498763
$667$	15.2274i	0.589608i
$668$	30.3903i	1.17584i
$669$	−0.771557	−0.0298301
$670$	0	0
$671$	−16.4510	−0.635086
$672$	− 24.2671i	− 0.936122i
$673$	− 19.3911i	− 0.747473i	−0.927535	−	0.373736i	$-0.878077\pi$
$673$	− 19.3911i	− 0.747473i	0.927535	−	0.373736i	$-0.121923\pi$
$674$	−46.8981	−1.80645
$675$	0	0
$676$	38.6133	1.48513
$677$	− 45.6836i	− 1.75576i	−0.478876	−	0.877882i	$-0.658956\pi$
$677$	− 45.6836i	− 1.75576i	0.478876	−	0.877882i	$-0.341044\pi$
$678$	− 19.3833i	− 0.744412i
$679$	32.2185	1.23643
$680$	0	0
$681$	−11.4314	−0.438051
$682$	− 14.8417i	− 0.568319i
$683$	− 11.5893i	− 0.443451i	−0.975109	−	0.221725i	$-0.928831\pi$
$683$	− 11.5893i	− 0.443451i	0.975109	−	0.221725i	$-0.0711688\pi$
$684$	−14.5297	−0.555557
$685$	0	0
$686$	27.3885	1.04570
$687$	− 10.2248i	− 0.390099i
$688$	− 18.8157i	− 0.717340i
$689$	51.1035	1.94689
$690$	0	0
$691$	−21.6424	−0.823317	−0.411658	−	0.911338i	$-0.635050\pi$
$691$	−21.6424	−0.823317	−0.411658	+	0.911338i	$0.635050\pi$
$692$	25.2353i	0.959303i
$693$	4.86205i	0.184694i
$694$	3.70522	0.140648
$695$	0	0
$696$	3.41139	0.129308
$697$	− 11.0124i	− 0.417123i
$698$	33.6081i	1.27208i
$699$	−22.5229	−0.851896
$700$	0	0
$701$	−8.61904	−0.325537	−0.162768	−	0.986664i	$-0.552042\pi$
$701$	−8.61904	−0.325537	−0.162768	+	0.986664i	$0.552042\pi$
$702$	11.3244i	0.427413i
$703$	32.4438i	1.22364i
$704$	18.6964	0.704648
$705$	0	0
$706$	33.6044	1.26472
$707$	− 3.43739i	− 0.129276i
$708$	10.9728i	0.412381i
$709$	−16.0001	−0.600895	−0.300447	−	0.953798i	$-0.597136\pi$
$709$	−16.0001	−0.600895	−0.300447	+	0.953798i	$0.597136\pi$
$710$	0	0
$711$	1.61849	0.0606981
$712$	10.5014i	0.393555i
$713$	− 28.7682i	− 1.07738i
$714$	−8.81353	−0.329838
$715$	0	0
$716$	10.8788	0.406558
$717$	0.589074i	0.0219994i
$718$	53.0138i	1.97846i
$719$	48.5838	1.81187	0.905935	−	0.423416i	$-0.139169\pi$
$719$	48.5838	1.81187	0.905935	+	0.423416i	$0.139169\pi$
$720$	0	0
$721$	15.6055	0.581177
$722$	23.0263i	0.856949i
$723$	1.50937i	0.0561340i
$724$	29.6689	1.10264
$725$	0	0
$726$	18.6531	0.692283
$727$	25.4328i	0.943251i	0.881799	+	0.471625i	$0.156333\pi$
$727$	25.4328i	0.943251i	−0.881799	+	0.471625i	$0.843667\pi$
$728$	− 23.9094i	− 0.886143i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−10.9505	−0.405019
$732$	28.5317i	1.05456i
$733$	18.6190i	0.687708i	0.939023	+	0.343854i	$0.111733\pi$
$733$	18.6190i	0.687708i	−0.939023	+	0.343854i	$0.888267\pi$
$734$	−60.7264	−2.24145
$735$	0	0
$736$	49.4656	1.82333
$737$	− 2.92078i	− 0.107588i
$738$	18.4343i	0.678576i
$739$	−46.8872	−1.72478	−0.862388	−	0.506249i	$-0.831032\pi$
$739$	−46.8872	−1.72478	−0.862388	+	0.506249i	$0.831032\pi$
$740$	0	0
$741$	28.5442	1.04860
$742$	− 66.5780i	− 2.44416i
$743$	42.7061i	1.56674i	0.621558	+	0.783368i	$0.286500\pi$
$743$	42.7061i	1.56674i	−0.621558	+	0.783368i	$0.713500\pi$
$744$	−6.44492	−0.236282
$745$	0	0
$746$	−20.9753	−0.767959
$747$	13.1488i	0.481091i
$748$	− 5.29742i	− 0.193693i
$749$	48.3751	1.76759
$750$	0	0
$751$	−2.64893	−0.0966608	−0.0483304	−	0.998831i	$-0.515390\pi$
$751$	−2.64893	−0.0966608	−0.0483304	+	0.998831i	$0.515390\pi$
$752$	− 1.67009i	− 0.0609021i
$753$	11.8953i	0.433488i
$754$	−26.7666	−0.974781
$755$	0	0
$756$	8.43243	0.306685
$757$	52.9153i	1.92324i	0.274383	+	0.961620i	$0.411526\pi$
$757$	52.9153i	1.92324i	−0.274383	+	0.961620i	$0.588474\pi$
$758$	48.1080i	1.74736i
$759$	−9.91073	−0.359737
$760$	0	0
$761$	50.7787	1.84073	0.920363	−	0.391064i	$-0.127893\pi$
$761$	50.7787	1.84073	0.920363	+	0.391064i	$0.127893\pi$
$762$	− 21.6455i	− 0.784133i
$763$	21.0462i	0.761922i
$764$	−9.63514	−0.348587
$765$	0	0
$766$	−29.4483	−1.06401
$767$	− 21.5565i	− 0.778358i
$768$	0.752056i	0.0271375i
$769$	−24.5805	−0.886396	−0.443198	−	0.896424i	$-0.646156\pi$
$769$	−24.5805	−0.886396	−0.443198	+	0.896424i	$0.646156\pi$
$770$	0	0
$771$	18.0492	0.650026
$772$	− 45.5064i	− 1.63781i
$773$	20.0112i	0.719754i	0.933000	+	0.359877i	$0.117181\pi$
$773$	20.0112i	0.719754i	−0.933000	+	0.359877i	$0.882819\pi$
$774$	18.3308	0.658885
$775$	0	0
$776$	14.7128	0.528159
$777$	− 18.8291i	− 0.675489i
$778$	− 41.5485i	− 1.48958i
$779$	46.4653	1.66479
$780$	0	0
$781$	−0.109034	−0.00390155
$782$	− 17.9654i	− 0.642441i
$783$	− 2.36361i	− 0.0844686i
$784$	6.62905	0.236752
$785$	0	0
$786$	13.7815	0.491568
$787$	8.99272i	0.320556i	0.987072	+	0.160278i	$0.0512391\pi$
$787$	8.99272i	0.320556i	−0.987072	+	0.160278i	$0.948761\pi$
$788$	11.7532i	0.418692i
$789$	−19.1066	−0.680215
$790$	0	0
$791$	28.3547	1.00818
$792$	2.22029i	0.0788947i
$793$	− 56.0517i	− 1.99045i
$794$	−48.0252	−1.70435
$795$	0	0
$796$	44.2773	1.56937
$797$	− 37.7564i	− 1.33740i	−0.743533	−	0.668700i	$-0.766851\pi$
$797$	− 37.7564i	− 1.33740i	0.743533	−	0.668700i	$-0.233149\pi$
$798$	− 37.1876i	− 1.31643i
$799$	−0.971976	−0.0343860
$800$	0	0
$801$	7.27597	0.257084
$802$	10.1920i	0.359891i
$803$	− 6.17067i	− 0.217758i
$804$	−5.06562	−0.178651
$805$	0	0
$806$	50.5684	1.78120
$807$	22.2250i	0.782358i
$808$	− 1.56971i	− 0.0552223i
$809$	48.9001	1.71924	0.859618	−	0.510937i	$-0.170702\pi$
$809$	48.9001	1.71924	0.859618	+	0.510937i	$0.170702\pi$
$810$	0	0
$811$	−18.0551	−0.633999	−0.317000	−	0.948426i	$-0.602675\pi$
$811$	−18.0551	−0.633999	−0.317000	+	0.948426i	$0.602675\pi$
$812$	19.9310i	0.699441i
$813$	20.2107i	0.708822i
$814$	19.8010	0.694024
$815$	0	0
$816$	2.86237	0.100203
$817$	− 46.2042i	− 1.61648i
$818$	26.9653i	0.942819i
$819$	−16.5659	−0.578859
$820$	0	0
$821$	50.6097	1.76629	0.883146	−	0.469099i	$-0.155421\pi$
$821$	50.6097	1.76629	0.883146	+	0.469099i	$0.155421\pi$
$822$	19.8281i	0.691586i
$823$	26.8070i	0.934433i	0.884143	+	0.467217i	$0.154743\pi$
$823$	26.8070i	0.934433i	−0.884143	+	0.467217i	$0.845257\pi$
$824$	7.12635	0.248258
$825$	0	0
$826$	−28.0839	−0.977163
$827$	− 21.3649i	− 0.742929i	−0.928447	−	0.371465i	$-0.878856\pi$
$827$	− 21.3649i	− 0.742929i	0.928447	−	0.371465i	$-0.121144\pi$
$828$	17.1885i	0.597343i
$829$	48.1123	1.67101	0.835504	−	0.549484i	$-0.185176\pi$
$829$	48.1123	1.67101	0.835504	+	0.549484i	$0.185176\pi$
$830$	0	0
$831$	16.0413	0.556465
$832$	63.7021i	2.20847i
$833$	− 3.85803i	− 0.133673i
$834$	−20.0184	−0.693182
$835$	0	0
$836$	22.3518	0.773052
$837$	4.46542i	0.154348i
$838$	− 75.5080i	− 2.60838i
$839$	41.8540	1.44496	0.722481	−	0.691391i	$-0.243002\pi$
$839$	41.8540	1.44496	0.722481	+	0.691391i	$0.243002\pi$
$840$	0	0
$841$	−23.4133	−0.807357
$842$	− 85.4719i	− 2.94556i
$843$	8.37308i	0.288384i
$844$	−49.4794	−1.70315
$845$	0	0
$846$	1.62705	0.0559393
$847$	27.2866i	0.937579i
$848$	21.6225i	0.742520i
$849$	−1.12995	−0.0387798
$850$	0	0
$851$	38.3809	1.31568
$852$	0.189102i	0.00647853i
$853$	58.2023i	1.99281i	0.0847260	+	0.996404i	$0.472998\pi$
$853$	58.2023i	1.99281i	−0.0847260	+	0.996404i	$0.527002\pi$
$854$	−73.0245	−2.49885
$855$	0	0
$856$	22.0909	0.755051
$857$	− 3.22045i	− 0.110008i	−0.998486	−	0.0550042i	$-0.982483\pi$
$857$	− 3.22045i	− 0.110008i	0.998486	−	0.0550042i	$-0.0175172\pi$
$858$	− 17.4210i	− 0.594742i
$859$	−49.4431	−1.68698	−0.843488	−	0.537147i	$-0.819502\pi$
$859$	−49.4431	−1.68698	−0.843488	+	0.537147i	$0.819502\pi$
$860$	0	0
$861$	−26.9665	−0.919016
$862$	− 30.3740i	− 1.03454i
$863$	− 21.4924i	− 0.731611i	−0.930691	−	0.365806i	$-0.880793\pi$
$863$	− 21.4924i	− 0.731611i	0.930691	−	0.365806i	$-0.119207\pi$
$864$	−7.67809	−0.261214
$865$	0	0
$866$	−1.18761	−0.0403567
$867$	15.3341i	0.520775i
$868$	− 37.6544i	− 1.27807i
$869$	−2.48981	−0.0844609
$870$	0	0
$871$	9.95163	0.337198
$872$	9.61090i	0.325466i
$873$	− 10.1939i	− 0.345012i
$874$	75.8026	2.56406
$875$	0	0
$876$	−10.7020	−0.361588
$877$	15.3793i	0.519321i	0.965700	+	0.259661i	$0.0836107\pi$
$877$	15.3793i	0.519321i	−0.965700	+	0.259661i	$0.916389\pi$
$878$	6.42501i	0.216833i
$879$	−2.37857	−0.0802273
$880$	0	0
$881$	18.9275	0.637682	0.318841	−	0.947808i	$-0.396706\pi$
$881$	18.9275	0.637682	0.318841	+	0.947808i	$0.396706\pi$
$882$	6.45821i	0.217459i
$883$	− 6.31708i	− 0.212587i	−0.994335	−	0.106293i	$-0.966102\pi$
$883$	− 6.31708i	− 0.212587i	0.994335	−	0.106293i	$-0.0338982\pi$
$884$	18.0492	0.607062
$885$	0	0
$886$	−79.2694	−2.66311
$887$	0.0721586i	0.00242285i	0.999999	+	0.00121142i	$0.000385609\pi$
$887$	0.0721586i	0.00242285i	−0.999999	+	0.00121142i	$0.999614\pi$
$888$	− 8.59844i	− 0.288545i
$889$	31.6639	1.06197
$890$	0	0
$891$	1.53835	0.0515367
$892$	− 2.05853i	− 0.0689246i
$893$	− 4.10113i	− 0.137239i
$894$	−21.7822	−0.728505
$895$	0	0
$896$	34.4573	1.15114
$897$	− 33.7676i	− 1.12747i
$898$	38.5512i	1.28647i
$899$	−10.5545	−0.352013
$900$	0	0
$901$	12.5840	0.419235
$902$	− 28.3584i	− 0.944233i
$903$	26.8150i	0.892348i
$904$	12.9484	0.430658
$905$	0	0
$906$	−23.1990	−0.770735
$907$	− 25.4951i	− 0.846550i	−0.906001	−	0.423275i	$-0.860880\pi$
$907$	− 25.4951i	− 0.846550i	0.906001	−	0.423275i	$-0.139120\pi$
$908$	− 30.4991i	− 1.01215i
$909$	−1.08759	−0.0360731
$910$	0	0
$911$	3.55803	0.117883	0.0589414	−	0.998261i	$-0.481227\pi$
$911$	3.55803	0.117883	0.0589414	+	0.998261i	$0.481227\pi$
$912$	12.0774i	0.399922i
$913$	− 20.2275i	− 0.669434i
$914$	52.2389	1.72791
$915$	0	0
$916$	27.2798	0.901351
$917$	20.1601i	0.665745i
$918$	2.78860i	0.0920375i
$919$	−7.96968	−0.262896	−0.131448	−	0.991323i	$-0.541963\pi$
$919$	−7.96968	−0.262896	−0.131448	+	0.991323i	$0.541963\pi$
$920$	0	0
$921$	−2.89366	−0.0953492
$922$	− 80.7776i	− 2.66027i
$923$	− 0.371499i	− 0.0122280i
$924$	−12.9720	−0.426749
$925$	0	0
$926$	44.1434	1.45064
$927$	− 4.93756i	− 0.162171i
$928$	− 18.1480i	− 0.595738i
$929$	−48.5424	−1.59263	−0.796313	−	0.604885i	$-0.793219\pi$
$929$	−48.5424	−1.59263	−0.796313	+	0.604885i	$0.793219\pi$
$930$	0	0
$931$	16.2785	0.533505
$932$	− 60.0916i	− 1.96837i
$933$	− 7.14545i	− 0.233932i
$934$	−67.4363	−2.20658
$935$	0	0
$936$	−7.56494	−0.247268
$937$	31.7296i	1.03656i	0.855211	+	0.518280i	$0.173427\pi$
$937$	31.7296i	1.03656i	−0.855211	+	0.518280i	$0.826573\pi$
$938$	− 12.9650i	− 0.423324i
$939$	23.0318	0.751616
$940$	0	0
$941$	41.0068	1.33678	0.668392	−	0.743809i	$-0.266983\pi$
$941$	41.0068	1.33678	0.668392	+	0.743809i	$0.266983\pi$
$942$	− 37.6839i	− 1.22781i
$943$	− 54.9681i	− 1.79001i
$944$	9.12079	0.296856
$945$	0	0
$946$	−28.1991	−0.916833
$947$	− 9.81789i	− 0.319038i	−0.987195	−	0.159519i	$-0.949006\pi$
$947$	− 9.81789i	− 0.319038i	0.987195	−	0.159519i	$-0.0509944\pi$
$948$	4.31816i	0.140247i
$949$	21.0246	0.682487
$950$	0	0
$951$	20.7750	0.673674
$952$	− 5.88761i	− 0.190818i
$953$	− 53.1118i	− 1.72046i	−0.509906	−	0.860230i	$-0.670320\pi$
$953$	− 53.1118i	− 1.72046i	0.509906	−	0.860230i	$-0.329680\pi$
$954$	−21.0653	−0.682013
$955$	0	0
$956$	−1.57166	−0.0508311
$957$	3.63606i	0.117537i
$958$	− 23.9756i	− 0.774618i
$959$	−29.0055	−0.936635
$960$	0	0
$961$	−11.0600	−0.356774
$962$	67.4654i	2.17517i
$963$	− 15.3059i	− 0.493225i
$964$	−4.02702	−0.129702
$965$	0	0
$966$	−43.9927	−1.41544
$967$	− 2.44732i	− 0.0787004i	−0.999225	−	0.0393502i	$-0.987471\pi$
$967$	− 2.44732i	− 0.0787004i	0.999225	−	0.0393502i	$-0.0125288\pi$
$968$	12.4606i	0.400500i
$969$	7.02890	0.225801
$970$	0	0
$971$	−47.9683	−1.53938	−0.769688	−	0.638421i	$-0.779588\pi$
$971$	−47.9683	−1.53938	−0.769688	+	0.638421i	$0.779588\pi$
$972$	− 2.66802i	− 0.0855767i
$973$	− 29.2838i	− 0.938796i
$974$	−15.4200	−0.494088
$975$	0	0
$976$	23.7161	0.759134
$977$	23.6950i	0.758070i	0.925382	+	0.379035i	$0.123744\pi$
$977$	23.6950i	0.758070i	−0.925382	+	0.379035i	$0.876256\pi$
$978$	− 35.1251i	− 1.12318i
$979$	−11.1930	−0.357730
$980$	0	0
$981$	6.65900	0.212605
$982$	15.6284i	0.498721i
$983$	− 12.9505i	− 0.413057i	−0.978441	−	0.206529i	$-0.933783\pi$
$983$	− 12.9505i	− 0.413057i	0.978441	−	0.206529i	$-0.0662167\pi$
$984$	−12.3145	−0.392571
$985$	0	0
$986$	−6.59116	−0.209905
$987$	2.38012i	0.0757602i
$988$	76.1565i	2.42286i
$989$	−54.6593	−1.73807
$990$	0	0
$991$	44.4967	1.41348	0.706742	−	0.707471i	$-0.250164\pi$
$991$	44.4967	1.41348	0.706742	+	0.707471i	$0.250164\pi$
$992$	34.2859i	1.08858i
$993$	− 19.2504i	− 0.610891i
$994$	−0.483992	−0.0153513
$995$	0	0
$996$	−35.0814	−1.11160
$997$	− 55.5513i	− 1.75933i	−0.475597	−	0.879663i	$-0.657768\pi$
$997$	− 55.5513i	− 1.75933i	0.475597	−	0.879663i	$-0.342232\pi$
$998$	− 34.6054i	− 1.09542i
$999$	−5.95751	−0.188487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.f.1249.10		12
5.2	odd	4		1875.2.a.k.1.2		6
5.3	odd	4		1875.2.a.j.1.5		6
5.4	even	2	inner	1875.2.b.f.1249.3		12
15.2	even	4		5625.2.a.q.1.5		6
15.8	even	4		5625.2.a.p.1.2		6
25.3	odd	20		75.2.g.c.16.1	✓	12
25.4	even	10		375.2.i.d.49.1		24
25.6	even	5		375.2.i.d.199.1		24
25.8	odd	20		75.2.g.c.61.1	yes	12
25.17	odd	20		375.2.g.c.301.3		12
25.19	even	10		375.2.i.d.199.6		24
25.21	even	5		375.2.i.d.49.6		24
25.22	odd	20		375.2.g.c.76.3		12
75.8	even	20		225.2.h.d.136.3		12
75.53	even	20		225.2.h.d.91.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.c.16.1	✓	12	25.3	odd	20
75.2.g.c.61.1	yes	12	25.8	odd	20
225.2.h.d.91.3		12	75.53	even	20
225.2.h.d.136.3		12	75.8	even	20
375.2.g.c.76.3		12	25.22	odd	20
375.2.g.c.301.3		12	25.17	odd	20
375.2.i.d.49.1		24	25.4	even	10
375.2.i.d.49.6		24	25.21	even	5
375.2.i.d.199.1		24	25.6	even	5
375.2.i.d.199.6		24	25.19	even	10
1875.2.a.j.1.5		6	5.3	odd	4
1875.2.a.k.1.2		6	5.2	odd	4
1875.2.b.f.1249.3		12	5.4	even	2	inner
1875.2.b.f.1249.10		12	1.1	even	1	trivial
5625.2.a.p.1.2		6	15.8	even	4
5625.2.a.q.1.5		6	15.2	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)

\(f(q)\)	\(=\)	\(q+2.16056i q^{2} +1.00000i q^{3} -2.66802 q^{4} -2.16056 q^{6} -3.16056i q^{7} -1.44329i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.16056i q^{2} +1.00000i q^{3} -2.66802 q^{4} -2.16056 q^{6} -3.16056i q^{7} -1.44329i q^{8} -1.00000 q^{9} +1.53835 q^{11} -2.66802i q^{12} +5.24144i q^{13} +6.82858 q^{14} -2.21771 q^{16} +1.29068i q^{17} -2.16056i q^{18} -5.44587 q^{19} +3.16056 q^{21} +3.32370i q^{22} +6.44244i q^{23} +1.44329 q^{24} -11.3244 q^{26} -1.00000i q^{27} +8.43243i q^{28} +2.36361 q^{29} -4.46542 q^{31} -7.67809i q^{32} +1.53835i q^{33} -2.78860 q^{34} +2.66802 q^{36} -5.95751i q^{37} -11.7661i q^{38} -5.24144 q^{39} -8.53219 q^{41} +6.82858i q^{42} +8.48426i q^{43} -4.10435 q^{44} -13.9193 q^{46} +0.753070i q^{47} -2.21771i q^{48} -2.98914 q^{49} -1.29068 q^{51} -13.9843i q^{52} -9.74991i q^{53} +2.16056 q^{54} -4.56162 q^{56} -5.44587i q^{57} +5.10672i q^{58} -4.11270 q^{59} -10.6939 q^{61} -9.64781i q^{62} +3.16056i q^{63} +12.1535 q^{64} -3.32370 q^{66} -1.89864i q^{67} -3.44357i q^{68} -6.44244 q^{69} -0.0708774 q^{71} +1.44329i q^{72} -4.01123i q^{73} +12.8716 q^{74} +14.5297 q^{76} -4.86205i q^{77} -11.3244i q^{78} -1.61849 q^{79} +1.00000 q^{81} -18.4343i q^{82} -13.1488i q^{83} -8.43243 q^{84} -18.3308 q^{86} +2.36361i q^{87} -2.22029i q^{88} -7.27597 q^{89} +16.5659 q^{91} -17.1885i q^{92} -4.46542i q^{93} -1.62705 q^{94} +7.67809 q^{96} +10.1939i q^{97} -6.45821i q^{98} -1.53835 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) \) 12 * q - 20 * q^4 - 12 * q^9	\( 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) \) 12 * q - 20 * q^4 - 12 * q^9 + 6 * q^11 + 44 * q^14 + 36 * q^16 - 22 * q^19 + 12 * q^21 - 6 * q^24 - 56 * q^26 + 6 * q^29 - 22 * q^31 - 30 * q^34 + 20 * q^36 - 12 * q^39 - 2 * q^41 - 18 * q^44 + 38 * q^46 + 28 * q^49 + 26 * q^51 - 70 * q^56 - 18 * q^59 + 22 * q^61 + 46 * q^64 + 32 * q^66 - 26 * q^69 - 16 * q^71 + 44 * q^74 - 52 * q^76 + 10 * q^79 + 12 * q^81 - 14 * q^84 - 74 * q^86 + 8 * q^89 + 68 * q^91 - 82 * q^94 + 32 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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