LMFDB - Embedded newform 1875.2.b.h.1249.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 $ x^16 + 20x^14 + 156x^12 + 610x^10 + 1286x^8 + 1440x^6 + 761x^4 + 130*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.13
Root			$-0.536547i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.h.1249.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53655i q^{2} +1.00000i q^{3} -0.360976 q^{4} -1.53655 q^{6} +1.49550i q^{7} +2.51844i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.53655i q^{2} +1.00000i q^{3} -0.360976 q^{4} -1.53655 q^{6} +1.49550i q^{7} +2.51844i q^{8} -1.00000 q^{9} -2.35626 q^{11} -0.360976i q^{12} -1.34951i q^{13} -2.29790 q^{14} -4.59165 q^{16} -2.19405i q^{17} -1.53655i q^{18} -5.71069 q^{19} -1.49550 q^{21} -3.62050i q^{22} -8.79501i q^{23} -2.51844 q^{24} +2.07358 q^{26} -1.00000i q^{27} -0.539839i q^{28} -7.90017 q^{29} -3.69717 q^{31} -2.01841i q^{32} -2.35626i q^{33} +3.37126 q^{34} +0.360976 q^{36} +9.75097i q^{37} -8.77474i q^{38} +1.34951 q^{39} +1.85550 q^{41} -2.29790i q^{42} +8.01874i q^{43} +0.850553 q^{44} +13.5139 q^{46} +6.66298i q^{47} -4.59165i q^{48} +4.76349 q^{49} +2.19405 q^{51} +0.487140i q^{52} -4.17153i q^{53} +1.53655 q^{54} -3.76631 q^{56} -5.71069i q^{57} -12.1390i q^{58} -11.0647 q^{59} -12.2372 q^{61} -5.68088i q^{62} -1.49550i q^{63} -6.08192 q^{64} +3.62050 q^{66} +4.31358i q^{67} +0.792000i q^{68} +8.79501 q^{69} +5.77750 q^{71} -2.51844i q^{72} -6.92684i q^{73} -14.9828 q^{74} +2.06142 q^{76} -3.52377i q^{77} +2.07358i q^{78} +10.6687 q^{79} +1.00000 q^{81} +2.85106i q^{82} -0.224003i q^{83} +0.539839 q^{84} -12.3212 q^{86} -7.90017i q^{87} -5.93408i q^{88} -0.429167 q^{89} +2.01818 q^{91} +3.17479i q^{92} -3.69717i q^{93} -10.2380 q^{94} +2.01841 q^{96} +12.4945i q^{97} +7.31933i q^{98} +2.35626 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	1.53655i	1.08650i	0.839570	+	0.543251i	$0.182807\pi$
$2$	1.53655i	1.08650i	−0.839570	+	0.543251i	$0.817193\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−0.360976	−0.180488
$5$	0	0
$5$	0	0
$6$	−1.53655	−0.627293
$7$	1.49550i	0.565244i	0.959231	+	0.282622i	$0.0912043\pi$
$7$	1.49550i	0.565244i	−0.959231	+	0.282622i	$0.908796\pi$
$8$	2.51844i	0.890402i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.35626	−0.710438	−0.355219	−	0.934783i	$-0.615594\pi$
$11$	−2.35626	−0.710438	−0.355219	+	0.934783i	$0.615594\pi$
$12$	− 0.360976i	− 0.104205i
$13$	− 1.34951i	− 0.374286i	−0.982333	−	0.187143i	$-0.940077\pi$
$13$	− 1.34951i	− 0.374286i	0.982333	−	0.187143i	$-0.0599227\pi$
$14$	−2.29790	−0.614140
$15$	0	0
$16$	−4.59165	−1.14791
$17$	− 2.19405i	− 0.532135i	−0.963954	−	0.266068i	$-0.914276\pi$
$17$	− 2.19405i	− 0.532135i	0.963954	−	0.266068i	$-0.0857245\pi$
$18$	− 1.53655i	− 0.362168i
$19$	−5.71069	−1.31012	−0.655061	−	0.755576i	$-0.727357\pi$
$19$	−5.71069	−1.31012	−0.655061	+	0.755576i	$0.727357\pi$
$20$	0	0
$21$	−1.49550	−0.326344
$22$	− 3.62050i	− 0.771892i
$23$	− 8.79501i	− 1.83389i	−0.399018	−	0.916943i	$-0.630649\pi$
$23$	− 8.79501i	− 1.83389i	0.399018	−	0.916943i	$-0.369351\pi$
$24$	−2.51844	−0.514074
$25$	0	0
$26$	2.07358	0.406662
$27$	− 1.00000i	− 0.192450i
$28$	− 0.539839i	− 0.102020i
$29$	−7.90017	−1.46702	−0.733512	−	0.679676i	$-0.762120\pi$
$29$	−7.90017	−1.46702	−0.733512	+	0.679676i	$0.762120\pi$
$30$	0	0
$31$	−3.69717	−0.664031	−0.332016	−	0.943274i	$-0.607729\pi$
$31$	−3.69717	−0.664031	−0.332016	+	0.943274i	$0.607729\pi$
$32$	− 2.01841i	− 0.356808i
$33$	− 2.35626i	− 0.410171i
$34$	3.37126	0.578167
$35$	0	0
$36$	0.360976	0.0601627
$37$	9.75097i	1.60305i	0.597962	+	0.801525i	$0.295977\pi$
$37$	9.75097i	1.60305i	−0.597962	+	0.801525i	$0.704023\pi$
$38$	− 8.77474i	− 1.42345i
$39$	1.34951	0.216094
$40$	0	0
$41$	1.85550	0.289780	0.144890	−	0.989448i	$-0.453717\pi$
$41$	1.85550	0.289780	0.144890	+	0.989448i	$0.453717\pi$
$42$	− 2.29790i	− 0.354574i
$43$	8.01874i	1.22285i	0.791304	+	0.611423i	$0.209403\pi$
$43$	8.01874i	1.22285i	−0.791304	+	0.611423i	$0.790597\pi$
$44$	0.850553	0.128226
$45$	0	0
$46$	13.5139	1.99252
$47$	6.66298i	0.971895i	0.873988	+	0.485948i	$0.161525\pi$
$47$	6.66298i	0.971895i	−0.873988	+	0.485948i	$0.838475\pi$
$48$	− 4.59165i	− 0.662747i
$49$	4.76349	0.680499
$50$	0	0
$51$	2.19405	0.307228
$52$	0.487140i	0.0675542i
$53$	− 4.17153i	− 0.573003i	−0.958080	−	0.286502i	$-0.907508\pi$
$53$	− 4.17153i	− 0.573003i	0.958080	−	0.286502i	$-0.0924924\pi$
$54$	1.53655	0.209098
$55$	0	0
$56$	−3.76631	−0.503295
$57$	− 5.71069i	− 0.756399i
$58$	− 12.1390i	− 1.59393i
$59$	−11.0647	−1.44050	−0.720248	−	0.693716i	$-0.755972\pi$
$59$	−11.0647	−1.44050	−0.720248	+	0.693716i	$0.755972\pi$
$60$	0	0
$61$	−12.2372	−1.56682	−0.783408	−	0.621508i	$-0.786520\pi$
$61$	−12.2372	−1.56682	−0.783408	+	0.621508i	$0.786520\pi$
$62$	− 5.68088i	− 0.721472i
$63$	− 1.49550i	− 0.188415i
$64$	−6.08192	−0.760239
$65$	0	0
$66$	3.62050	0.445652
$67$	4.31358i	0.526988i	0.964661	+	0.263494i	$0.0848749\pi$
$67$	4.31358i	0.526988i	−0.964661	+	0.263494i	$0.915125\pi$
$68$	0.792000i	0.0960442i
$69$	8.79501	1.05879
$70$	0	0
$71$	5.77750	0.685663	0.342832	−	0.939397i	$-0.388614\pi$
$71$	5.77750	0.685663	0.342832	+	0.939397i	$0.388614\pi$
$72$	− 2.51844i	− 0.296801i
$73$	− 6.92684i	− 0.810725i	−0.914156	−	0.405362i	$-0.867145\pi$
$73$	− 6.92684i	− 0.810725i	0.914156	−	0.405362i	$-0.132855\pi$
$74$	−14.9828	−1.74172
$75$	0	0
$76$	2.06142	0.236461
$77$	− 3.52377i	− 0.401571i
$78$	2.07358i	0.234787i
$79$	10.6687	1.20033	0.600163	−	0.799878i	$-0.295102\pi$
$79$	10.6687	1.20033	0.600163	+	0.799878i	$0.295102\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.85106i	0.314846i
$83$	− 0.224003i	− 0.0245875i	−0.999924	−	0.0122938i	$-0.996087\pi$
$83$	− 0.224003i	− 0.0245875i	0.999924	−	0.0122938i	$-0.00391332\pi$
$84$	0.539839	0.0589012
$85$	0	0
$86$	−12.3212	−1.32863
$87$	− 7.90017i	− 0.846987i
$88$	− 5.93408i	− 0.632575i
$89$	−0.429167	−0.0454916	−0.0227458	−	0.999741i	$-0.507241\pi$
$89$	−0.429167	−0.0454916	−0.0227458	+	0.999741i	$0.507241\pi$
$90$	0	0
$91$	2.01818	0.211563
$92$	3.17479i	0.330995i
$93$	− 3.69717i	− 0.383379i
$94$	−10.2380	−1.05597
$95$	0	0
$96$	2.01841	0.206003
$97$	12.4945i	1.26863i	0.773075	+	0.634315i	$0.218718\pi$
$97$	12.4945i	1.26863i	−0.773075	+	0.634315i	$0.781282\pi$
$98$	7.31933i	0.739364i
$99$	2.35626	0.236813
$100$	0	0
$101$	8.19767	0.815698	0.407849	−	0.913049i	$-0.366279\pi$
$101$	8.19767	0.815698	0.407849	+	0.913049i	$0.366279\pi$
$102$	3.37126i	0.333805i
$103$	− 2.50005i	− 0.246337i	−0.992386	−	0.123169i	$-0.960694\pi$
$103$	− 2.50005i	− 0.246337i	0.992386	−	0.123169i	$-0.0393056\pi$
$104$	3.39865	0.333265
$105$	0	0
$106$	6.40975	0.622570
$107$	− 1.81004i	− 0.174983i	−0.996165	−	0.0874914i	$-0.972115\pi$
$107$	− 1.81004i	− 0.174983i	0.996165	−	0.0874914i	$-0.0278850\pi$
$108$	0.360976i	0.0347350i
$109$	2.94778	0.282346	0.141173	−	0.989985i	$-0.454913\pi$
$109$	2.94778	0.282346	0.141173	+	0.989985i	$0.454913\pi$
$110$	0	0
$111$	−9.75097	−0.925521
$112$	− 6.86679i	− 0.648851i
$113$	− 13.8365i	− 1.30163i	−0.759238	−	0.650813i	$-0.774428\pi$
$113$	− 13.8365i	− 1.30163i	0.759238	−	0.650813i	$-0.225572\pi$
$114$	8.77474	0.821829
$115$	0	0
$116$	2.85178	0.264781
$117$	1.34951i	0.124762i
$118$	− 17.0014i	− 1.56510i
$119$	3.28119	0.300787
$120$	0	0
$121$	−5.44806	−0.495278
$122$	− 18.8031i	− 1.70235i
$123$	1.85550i	0.167304i
$124$	1.33459	0.119850
$125$	0	0
$126$	2.29790	0.204713
$127$	5.73995i	0.509338i	0.967028	+	0.254669i	$0.0819666\pi$
$127$	5.73995i	0.509338i	−0.967028	+	0.254669i	$0.918033\pi$
$128$	− 13.3820i	− 1.18281i
$129$	−8.01874	−0.706011
$130$	0	0
$131$	−4.35756	−0.380722	−0.190361	−	0.981714i	$-0.560966\pi$
$131$	−4.35756	−0.380722	−0.190361	+	0.981714i	$0.560966\pi$
$132$	0.850553i	0.0740311i
$133$	− 8.54031i	− 0.740539i
$134$	−6.62802	−0.572574
$135$	0	0
$136$	5.52558	0.473814
$137$	1.97461i	0.168702i	0.996436	+	0.0843510i	$0.0268817\pi$
$137$	1.97461i	0.168702i	−0.996436	+	0.0843510i	$0.973118\pi$
$138$	13.5139i	1.15038i
$139$	−1.67910	−0.142419	−0.0712095	−	0.997461i	$-0.522686\pi$
$139$	−1.67910	−0.142419	−0.0712095	+	0.997461i	$0.522686\pi$
$140$	0	0
$141$	−6.66298	−0.561124
$142$	8.87740i	0.744975i
$143$	3.17978i	0.265907i
$144$	4.59165	0.382637
$145$	0	0
$146$	10.6434	0.880855
$147$	4.76349i	0.392886i
$148$	− 3.51987i	− 0.289331i
$149$	7.38524	0.605023	0.302511	−	0.953146i	$-0.402175\pi$
$149$	7.38524	0.605023	0.302511	+	0.953146i	$0.402175\pi$
$150$	0	0
$151$	−4.26137	−0.346785	−0.173393	−	0.984853i	$-0.555473\pi$
$151$	−4.26137	−0.346785	−0.173393	+	0.984853i	$0.555473\pi$
$152$	− 14.3820i	− 1.16653i
$153$	2.19405i	0.177378i
$154$	5.41444	0.436308
$155$	0	0
$156$	−0.487140	−0.0390024
$157$	16.0573i	1.28152i	0.767743	+	0.640758i	$0.221380\pi$
$157$	16.0573i	1.28152i	−0.767743	+	0.640758i	$0.778620\pi$
$158$	16.3930i	1.30416i
$159$	4.17153	0.330824
$160$	0	0
$161$	13.1529	1.03659
$162$	1.53655i	0.120723i
$163$	22.2938i	1.74618i	0.487556	+	0.873092i	$0.337889\pi$
$163$	22.2938i	1.74618i	−0.487556	+	0.873092i	$0.662111\pi$
$164$	−0.669790	−0.0523018
$165$	0	0
$166$	0.344191	0.0267144
$167$	− 6.46601i	− 0.500355i	−0.968200	−	0.250177i	$-0.919511\pi$
$167$	− 6.46601i	− 0.500355i	0.968200	−	0.250177i	$-0.0804889\pi$
$168$	− 3.76631i	− 0.290577i
$169$	11.1788	0.859910
$170$	0	0
$171$	5.71069	0.436707
$172$	− 2.89458i	− 0.220709i
$173$	11.8180i	0.898509i	0.893404	+	0.449255i	$0.148310\pi$
$173$	11.8180i	0.898509i	−0.893404	+	0.449255i	$0.851690\pi$
$174$	12.1390	0.920254
$175$	0	0
$176$	10.8191	0.815520
$177$	− 11.0647i	− 0.831671i
$178$	− 0.659435i	− 0.0494268i
$179$	−15.5746	−1.16410	−0.582049	−	0.813154i	$-0.697749\pi$
$179$	−15.5746	−1.16410	−0.582049	+	0.813154i	$0.697749\pi$
$180$	0	0
$181$	−14.5797	−1.08370	−0.541851	−	0.840475i	$-0.682276\pi$
$181$	−14.5797	−1.08370	−0.541851	+	0.840475i	$0.682276\pi$
$182$	3.10103i	0.229864i
$183$	− 12.2372i	− 0.904602i
$184$	22.1497	1.63290
$185$	0	0
$186$	5.68088	0.416542
$187$	5.16974i	0.378049i
$188$	− 2.40518i	− 0.175416i
$189$	1.49550	0.108781
$190$	0	0
$191$	−20.9884	−1.51867	−0.759333	−	0.650702i	$-0.774475\pi$
$191$	−20.9884	−1.51867	−0.759333	+	0.650702i	$0.774475\pi$
$192$	− 6.08192i	− 0.438924i
$193$	− 22.7094i	− 1.63466i	−0.576173	−	0.817328i	$-0.695454\pi$
$193$	− 22.7094i	− 1.63466i	0.576173	−	0.817328i	$-0.304546\pi$
$194$	−19.1985	−1.37837
$195$	0	0
$196$	−1.71951	−0.122822
$197$	1.35341i	0.0964268i	0.998837	+	0.0482134i	$0.0153528\pi$
$197$	1.35341i	0.0964268i	−0.998837	+	0.0482134i	$0.984647\pi$
$198$	3.62050i	0.257297i
$199$	8.96061	0.635201	0.317600	−	0.948225i	$-0.397123\pi$
$199$	8.96061	0.635201	0.317600	+	0.948225i	$0.397123\pi$
$200$	0	0
$201$	−4.31358	−0.304257
$202$	12.5961i	0.886259i
$203$	− 11.8147i	− 0.829228i
$204$	−0.792000	−0.0554511
$205$	0	0
$206$	3.84145	0.267646
$207$	8.79501i	0.611295i
$208$	6.19646i	0.429647i
$209$	13.4558	0.930760
$210$	0	0
$211$	5.83983	0.402031	0.201015	−	0.979588i	$-0.435576\pi$
$211$	5.83983	0.402031	0.201015	+	0.979588i	$0.435576\pi$
$212$	1.50582i	0.103420i
$213$	5.77750i	0.395868i
$214$	2.78121	0.190119
$215$	0	0
$216$	2.51844	0.171358
$217$	− 5.52910i	− 0.375340i
$218$	4.52939i	0.306769i
$219$	6.92684	0.468072
$220$	0	0
$221$	−2.96088	−0.199171
$222$	− 14.9828i	− 1.00558i
$223$	7.82097i	0.523731i	0.965104	+	0.261865i	$0.0843377\pi$
$223$	7.82097i	0.523731i	−0.965104	+	0.261865i	$0.915662\pi$
$224$	3.01853	0.201684
$225$	0	0
$226$	21.2604	1.41422
$227$	16.3090i	1.08246i	0.840873	+	0.541232i	$0.182042\pi$
$227$	16.3090i	1.08246i	−0.840873	+	0.541232i	$0.817958\pi$
$228$	2.06142i	0.136521i
$229$	21.7088	1.43456	0.717278	−	0.696787i	$-0.245388\pi$
$229$	21.7088	1.43456	0.717278	+	0.696787i	$0.245388\pi$
$230$	0	0
$231$	3.52377	0.231847
$232$	− 19.8961i	− 1.30624i
$233$	13.5341i	0.886646i	0.896362	+	0.443323i	$0.146201\pi$
$233$	13.5341i	0.886646i	−0.896362	+	0.443323i	$0.853799\pi$
$234$	−2.07358	−0.135554
$235$	0	0
$236$	3.99408	0.259993
$237$	10.6687i	0.693009i
$238$	5.04171i	0.326805i
$239$	10.5338	0.681377	0.340689	−	0.940176i	$-0.389340\pi$
$239$	10.5338	0.681377	0.340689	+	0.940176i	$0.389340\pi$
$240$	0	0
$241$	19.4838	1.25506	0.627530	−	0.778592i	$-0.284066\pi$
$241$	19.4838	1.25506	0.627530	+	0.778592i	$0.284066\pi$
$242$	− 8.37120i	− 0.538121i
$243$	1.00000i	0.0641500i
$244$	4.41735	0.282792
$245$	0	0
$246$	−2.85106	−0.181777
$247$	7.70661i	0.490360i
$248$	− 9.31109i	− 0.591255i
$249$	0.224003	0.0141956
$250$	0	0
$251$	−20.9446	−1.32201	−0.661007	−	0.750380i	$-0.729871\pi$
$251$	−20.9446	−1.32201	−0.661007	+	0.750380i	$0.729871\pi$
$252$	0.539839i	0.0340066i
$253$	20.7233i	1.30286i
$254$	−8.81971	−0.553398
$255$	0	0
$256$	8.39819	0.524887
$257$	− 1.67121i	− 0.104247i	−0.998641	−	0.0521237i	$-0.983401\pi$
$257$	− 1.67121i	− 0.104247i	0.998641	−	0.0521237i	$-0.0165990\pi$
$258$	− 12.3212i	− 0.767083i
$259$	−14.5825	−0.906115
$260$	0	0
$261$	7.90017	0.489008
$262$	− 6.69559i	− 0.413655i
$263$	− 7.55667i	− 0.465964i	−0.972481	−	0.232982i	$-0.925152\pi$
$263$	− 7.55667i	− 0.465964i	0.972481	−	0.232982i	$-0.0748483\pi$
$264$	5.93408	0.365217
$265$	0	0
$266$	13.1226	0.804597
$267$	− 0.429167i	− 0.0262646i
$268$	− 1.55710i	− 0.0951151i
$269$	11.2841	0.688005	0.344002	−	0.938969i	$-0.388217\pi$
$269$	11.2841	0.688005	0.344002	+	0.938969i	$0.388217\pi$
$270$	0	0
$271$	−10.9752	−0.666696	−0.333348	−	0.942804i	$-0.608178\pi$
$271$	−10.9752	−0.666696	−0.333348	+	0.942804i	$0.608178\pi$
$272$	10.0743i	0.610845i
$273$	2.01818i	0.122146i
$274$	−3.03407	−0.183295
$275$	0	0
$276$	−3.17479	−0.191100
$277$	− 5.75112i	− 0.345551i	−0.984961	−	0.172776i	$-0.944726\pi$
$277$	− 5.75112i	− 0.345551i	0.984961	−	0.172776i	$-0.0552735\pi$
$278$	− 2.58001i	− 0.154739i
$279$	3.69717	0.221344
$280$	0	0
$281$	8.49962	0.507045	0.253522	−	0.967330i	$-0.418411\pi$
$281$	8.49962	0.507045	0.253522	+	0.967330i	$0.418411\pi$
$282$	− 10.2380i	− 0.609663i
$283$	17.2248i	1.02391i	0.859013	+	0.511955i	$0.171078\pi$
$283$	17.2248i	1.02391i	−0.859013	+	0.511955i	$0.828922\pi$
$284$	−2.08554	−0.123754
$285$	0	0
$286$	−4.88588	−0.288908
$287$	2.77489i	0.163796i
$288$	2.01841i	0.118936i
$289$	12.1861	0.716832
$290$	0	0
$291$	−12.4945	−0.732443
$292$	2.50042i	0.146326i
$293$	− 9.38764i	− 0.548432i	−0.961668	−	0.274216i	$-0.911582\pi$
$293$	− 9.38764i	− 0.548432i	0.961668	−	0.274216i	$-0.0884183\pi$
$294$	−7.31933	−0.426872
$295$	0	0
$296$	−24.5572	−1.42736
$297$	2.35626i	0.136724i
$298$	11.3478i	0.657359i
$299$	−11.8689	−0.686397
$300$	0	0
$301$	−11.9920	−0.691207
$302$	− 6.54779i	− 0.376783i
$303$	8.19767i	0.470944i
$304$	26.2215	1.50390
$305$	0	0
$306$	−3.37126	−0.192722
$307$	10.6465i	0.607627i	0.952731	+	0.303814i	$0.0982600\pi$
$307$	10.6465i	0.607627i	−0.952731	+	0.303814i	$0.901740\pi$
$308$	1.27200i	0.0724788i
$309$	2.50005	0.142223
$310$	0	0
$311$	−27.3572	−1.55128	−0.775641	−	0.631174i	$-0.782573\pi$
$311$	−27.3572	−1.55128	−0.775641	+	0.631174i	$0.782573\pi$
$312$	3.39865i	0.192410i
$313$	1.99509i	0.112769i	0.998409	+	0.0563846i	$0.0179573\pi$
$313$	1.99509i	0.112769i	−0.998409	+	0.0563846i	$0.982043\pi$
$314$	−24.6729	−1.39237
$315$	0	0
$316$	−3.85116	−0.216645
$317$	8.64876i	0.485763i	0.970056	+	0.242881i	$0.0780925\pi$
$317$	8.64876i	0.485763i	−0.970056	+	0.242881i	$0.921907\pi$
$318$	6.40975i	0.359441i
$319$	18.6148	1.04223
$320$	0	0
$321$	1.81004	0.101026
$322$	20.2101i	1.12626i
$323$	12.5295i	0.697162i
$324$	−0.360976	−0.0200542
$325$	0	0
$326$	−34.2554	−1.89723
$327$	2.94778i	0.163012i
$328$	4.67295i	0.258020i
$329$	−9.96446	−0.549358
$330$	0	0
$331$	−3.94331	−0.216744	−0.108372	−	0.994110i	$-0.534564\pi$
$331$	−3.94331	−0.216744	−0.108372	+	0.994110i	$0.534564\pi$
$332$	0.0808598i	0.00443776i
$333$	− 9.75097i	− 0.534350i
$334$	9.93532	0.543637
$335$	0	0
$336$	6.86679	0.374614
$337$	3.95471i	0.215427i	0.994182	+	0.107713i	$0.0343529\pi$
$337$	3.95471i	0.215427i	−0.994182	+	0.107713i	$0.965647\pi$
$338$	17.1768i	0.934295i
$339$	13.8365	0.751494
$340$	0	0
$341$	8.71148	0.471753
$342$	8.77474i	0.474483i
$343$	17.5923i	0.949893i
$344$	−20.1947	−1.08882
$345$	0	0
$346$	−18.1590	−0.976233
$347$	− 9.99596i	− 0.536611i	−0.963334	−	0.268306i	$-0.913536\pi$
$347$	− 9.99596i	− 0.536611i	0.963334	−	0.268306i	$-0.0864637\pi$
$348$	2.85178i	0.152871i
$349$	−18.4534	−0.987789	−0.493895	−	0.869522i	$-0.664427\pi$
$349$	−18.4534	−0.987789	−0.493895	+	0.869522i	$0.664427\pi$
$350$	0	0
$351$	−1.34951	−0.0720313
$352$	4.75589i	0.253490i
$353$	− 9.32398i	− 0.496265i	−0.968726	−	0.248133i	$-0.920183\pi$
$353$	− 9.32398i	− 0.496265i	0.968726	−	0.248133i	$-0.0798169\pi$
$354$	17.0014	0.903613
$355$	0	0
$356$	0.154919	0.00821070
$357$	3.28119i	0.173659i
$358$	− 23.9311i	− 1.26480i
$359$	−28.9438	−1.52759	−0.763797	−	0.645457i	$-0.776667\pi$
$359$	−28.9438	−1.52759	−0.763797	+	0.645457i	$0.776667\pi$
$360$	0	0
$361$	13.6119	0.716418
$362$	− 22.4024i	− 1.17744i
$363$	− 5.44806i	− 0.285949i
$364$	−0.728516	−0.0381846
$365$	0	0
$366$	18.8031	0.982852
$367$	− 3.99869i	− 0.208730i	−0.994539	−	0.104365i	$-0.966719\pi$
$367$	− 3.99869i	− 0.208730i	0.994539	−	0.104365i	$-0.0332810\pi$
$368$	40.3836i	2.10514i
$369$	−1.85550	−0.0965932
$370$	0	0
$371$	6.23850	0.323887
$372$	1.33459i	0.0691953i
$373$	3.17260i	0.164271i	0.996621	+	0.0821354i	$0.0261740\pi$
$373$	3.17260i	0.164271i	−0.996621	+	0.0821354i	$0.973826\pi$
$374$	−7.94355	−0.410751
$375$	0	0
$376$	−16.7803	−0.865377
$377$	10.6613i	0.549086i
$378$	2.29790i	0.118191i
$379$	−28.5206	−1.46500	−0.732501	−	0.680766i	$-0.761647\pi$
$379$	−28.5206	−1.46500	−0.732501	+	0.680766i	$0.761647\pi$
$380$	0	0
$381$	−5.73995	−0.294067
$382$	− 32.2497i	− 1.65004i
$383$	11.4611i	0.585636i	0.956168	+	0.292818i	$0.0945929\pi$
$383$	11.4611i	0.585636i	−0.956168	+	0.292818i	$0.905407\pi$
$384$	13.3820	0.682896
$385$	0	0
$386$	34.8940	1.77606
$387$	− 8.01874i	− 0.407616i
$388$	− 4.51024i	− 0.228973i
$389$	−34.2463	−1.73636	−0.868179	−	0.496252i	$-0.834709\pi$
$389$	−34.2463	−1.73636	−0.868179	+	0.496252i	$0.834709\pi$
$390$	0	0
$391$	−19.2967	−0.975876
$392$	11.9966i	0.605917i
$393$	− 4.35756i	− 0.219810i
$394$	−2.07958	−0.104768
$395$	0	0
$396$	−0.850553	−0.0427419
$397$	− 20.0333i	− 1.00545i	−0.864448	−	0.502723i	$-0.832332\pi$
$397$	− 20.0333i	− 1.00545i	0.864448	−	0.502723i	$-0.167668\pi$
$398$	13.7684i	0.690148i
$399$	8.54031	0.427550
$400$	0	0
$401$	4.98200	0.248789	0.124395	−	0.992233i	$-0.460301\pi$
$401$	4.98200	0.248789	0.124395	+	0.992233i	$0.460301\pi$
$402$	− 6.62802i	− 0.330576i
$403$	4.98935i	0.248537i
$404$	−2.95916	−0.147224
$405$	0	0
$406$	18.1538	0.900958
$407$	− 22.9758i	− 1.13887i
$408$	5.52558i	0.273557i
$409$	−23.8591	−1.17976	−0.589878	−	0.807493i	$-0.700824\pi$
$409$	−23.8591	−1.17976	−0.589878	+	0.807493i	$0.700824\pi$
$410$	0	0
$411$	−1.97461	−0.0974001
$412$	0.902460i	0.0444610i
$413$	− 16.5472i	− 0.814233i
$414$	−13.5139	−0.664174
$415$	0	0
$416$	−2.72386	−0.133548
$417$	− 1.67910i	− 0.0822257i
$418$	20.6755i	1.01127i
$419$	−0.482550	−0.0235741	−0.0117871	−	0.999931i	$-0.503752\pi$
$419$	−0.482550	−0.0235741	−0.0117871	+	0.999931i	$0.503752\pi$
$420$	0	0
$421$	−17.7183	−0.863537	−0.431769	−	0.901984i	$-0.642110\pi$
$421$	−17.7183	−0.863537	−0.431769	+	0.901984i	$0.642110\pi$
$422$	8.97317i	0.436807i
$423$	− 6.66298i	− 0.323965i
$424$	10.5057	0.510203
$425$	0	0
$426$	−8.87740	−0.430112
$427$	− 18.3007i	− 0.885634i
$428$	0.653380i	0.0315823i
$429$	−3.17978	−0.153521
$430$	0	0
$431$	26.8061	1.29121	0.645603	−	0.763673i	$-0.276606\pi$
$431$	26.8061	1.29121	0.645603	+	0.763673i	$0.276606\pi$
$432$	4.59165i	0.220916i
$433$	9.23115i	0.443621i	0.975090	+	0.221810i	$0.0711966\pi$
$433$	9.23115i	0.443621i	−0.975090	+	0.221810i	$0.928803\pi$
$434$	8.49573	0.407808
$435$	0	0
$436$	−1.06408	−0.0509601
$437$	50.2255i	2.40261i
$438$	10.6434i	0.508562i
$439$	−1.00240	−0.0478421	−0.0239211	−	0.999714i	$-0.507615\pi$
$439$	−1.00240	−0.0478421	−0.0239211	+	0.999714i	$0.507615\pi$
$440$	0	0
$441$	−4.76349	−0.226833
$442$	− 4.54954i	− 0.216399i
$443$	26.2872i	1.24894i	0.781048	+	0.624471i	$0.214685\pi$
$443$	26.2872i	1.24894i	−0.781048	+	0.624471i	$0.785315\pi$
$444$	3.51987	0.167046
$445$	0	0
$446$	−12.0173	−0.569035
$447$	7.38524i	0.349310i
$448$	− 9.09548i	− 0.429721i
$449$	−4.75449	−0.224378	−0.112189	−	0.993687i	$-0.535786\pi$
$449$	−4.75449	−0.224378	−0.112189	+	0.993687i	$0.535786\pi$
$450$	0	0
$451$	−4.37202	−0.205870
$452$	4.99464i	0.234928i
$453$	− 4.26137i	− 0.200216i
$454$	−25.0595	−1.17610
$455$	0	0
$456$	14.3820	0.673499
$457$	15.9703i	0.747059i	0.927618	+	0.373529i	$0.121852\pi$
$457$	15.9703i	0.747059i	−0.927618	+	0.373529i	$0.878148\pi$
$458$	33.3566i	1.55865i
$459$	−2.19405	−0.102409
$460$	0	0
$461$	39.7558	1.85161	0.925806	−	0.377998i	$-0.123387\pi$
$461$	39.7558	1.85161	0.925806	+	0.377998i	$0.123387\pi$
$462$	5.41444i	0.251902i
$463$	26.1209i	1.21394i	0.794724	+	0.606971i	$0.207616\pi$
$463$	26.1209i	1.21394i	−0.794724	+	0.606971i	$0.792384\pi$
$464$	36.2748	1.68402
$465$	0	0
$466$	−20.7957	−0.963343
$467$	3.85204i	0.178251i	0.996020	+	0.0891256i	$0.0284073\pi$
$467$	3.85204i	0.178251i	−0.996020	+	0.0891256i	$0.971593\pi$
$468$	− 0.487140i	− 0.0225181i
$469$	−6.45095	−0.297877
$470$	0	0
$471$	−16.0573	−0.739883
$472$	− 27.8657i	− 1.28262i
$473$	− 18.8942i	− 0.868756i
$474$	−16.3930	−0.752956
$475$	0	0
$476$	−1.18443	−0.0542884
$477$	4.17153i	0.191001i
$478$	16.1857i	0.740318i
$479$	14.2698	0.652004	0.326002	−	0.945369i	$-0.394298\pi$
$479$	14.2698	0.652004	0.326002	+	0.945369i	$0.394298\pi$
$480$	0	0
$481$	13.1590	0.599998
$482$	29.9377i	1.36363i
$483$	13.1529i	0.598478i
$484$	1.96662	0.0893919
$485$	0	0
$486$	−1.53655	−0.0696992
$487$	− 24.8222i	− 1.12480i	−0.826865	−	0.562401i	$-0.809878\pi$
$487$	− 24.8222i	− 1.12480i	0.826865	−	0.562401i	$-0.190122\pi$
$488$	− 30.8187i	− 1.39510i
$489$	−22.2938	−1.00816
$490$	0	0
$491$	11.4893	0.518507	0.259253	−	0.965809i	$-0.416524\pi$
$491$	11.4893	0.518507	0.259253	+	0.965809i	$0.416524\pi$
$492$	− 0.669790i	− 0.0301965i
$493$	17.3334i	0.780656i
$494$	−11.8416	−0.532777
$495$	0	0
$496$	16.9761	0.762250
$497$	8.64023i	0.387567i
$498$	0.344191i	0.0154236i
$499$	4.17487	0.186893	0.0934465	−	0.995624i	$-0.470212\pi$
$499$	4.17487	0.186893	0.0934465	+	0.995624i	$0.470212\pi$
$500$	0	0
$501$	6.46601	0.288880
$502$	− 32.1824i	− 1.43637i
$503$	38.7163i	1.72627i	0.504970	+	0.863137i	$0.331503\pi$
$503$	38.7163i	1.72627i	−0.504970	+	0.863137i	$0.668497\pi$
$504$	3.76631	0.167765
$505$	0	0
$506$	−31.8423	−1.41556
$507$	11.1788i	0.496469i
$508$	− 2.07199i	− 0.0919296i
$509$	−30.1797	−1.33769	−0.668845	−	0.743402i	$-0.733211\pi$
$509$	−30.1797	−1.33769	−0.668845	+	0.743402i	$0.733211\pi$
$510$	0	0
$511$	10.3591	0.458258
$512$	− 13.8597i	− 0.612519i
$513$	5.71069i	0.252133i
$514$	2.56790	0.113265
$515$	0	0
$516$	2.89458	0.127427
$517$	− 15.6997i	− 0.690471i
$518$	− 22.4067i	− 0.984496i
$519$	−11.8180	−0.518755
$520$	0	0
$521$	−25.4856	−1.11654	−0.558272	−	0.829658i	$-0.688536\pi$
$521$	−25.4856	−1.11654	−0.558272	+	0.829658i	$0.688536\pi$
$522$	12.1390i	0.531309i
$523$	− 3.89180i	− 0.170176i	−0.996373	−	0.0850882i	$-0.972883\pi$
$523$	− 3.89180i	− 0.170176i	0.996373	−	0.0850882i	$-0.0271172\pi$
$524$	1.57298	0.0687158
$525$	0	0
$526$	11.6112	0.506271
$527$	8.11178i	0.353355i
$528$	10.8191i	0.470841i
$529$	−54.3522	−2.36314
$530$	0	0
$531$	11.0647	0.480166
$532$	3.08285i	0.133659i
$533$	− 2.50400i	− 0.108460i
$534$	0.659435	0.0285366
$535$	0	0
$536$	−10.8635	−0.469231
$537$	− 15.5746i	− 0.672092i
$538$	17.3386i	0.747519i
$539$	−11.2240	−0.483452
$540$	0	0
$541$	40.2148	1.72897	0.864484	−	0.502661i	$-0.167646\pi$
$541$	40.2148	1.72897	0.864484	+	0.502661i	$0.167646\pi$
$542$	− 16.8639i	− 0.724367i
$543$	− 14.5797i	− 0.625675i
$544$	−4.42849	−0.189870
$545$	0	0
$546$	−3.10103	−0.132712
$547$	7.37923i	0.315513i	0.987478	+	0.157757i	$0.0504261\pi$
$547$	7.37923i	0.315513i	−0.987478	+	0.157757i	$0.949574\pi$
$548$	− 0.712786i	− 0.0304487i
$549$	12.2372	0.522272
$550$	0	0
$551$	45.1154	1.92198
$552$	22.1497i	0.942753i
$553$	15.9550i	0.678478i
$554$	8.83686	0.375442
$555$	0	0
$556$	0.606114	0.0257050
$557$	1.52499i	0.0646160i	0.999478	+	0.0323080i	$0.0102857\pi$
$557$	1.52499i	0.0646160i	−0.999478	+	0.0323080i	$0.989714\pi$
$558$	5.68088i	0.240491i
$559$	10.8213	0.457694
$560$	0	0
$561$	−5.16974	−0.218267
$562$	13.0601i	0.550905i
$563$	− 13.7955i	− 0.581411i	−0.956813	−	0.290706i	$-0.906110\pi$
$563$	− 13.7955i	− 0.581411i	0.956813	−	0.290706i	$-0.0938901\pi$
$564$	2.40518	0.101276
$565$	0	0
$566$	−26.4667	−1.11248
$567$	1.49550i	0.0628049i
$568$	14.5503i	0.610516i
$569$	8.49904	0.356298	0.178149	−	0.984004i	$-0.442989\pi$
$569$	8.49904	0.356298	0.178149	+	0.984004i	$0.442989\pi$
$570$	0	0
$571$	39.3230	1.64562	0.822809	−	0.568319i	$-0.192406\pi$
$571$	39.3230	1.64562	0.822809	+	0.568319i	$0.192406\pi$
$572$	− 1.14783i	− 0.0479930i
$573$	− 20.9884i	− 0.876803i
$574$	−4.26374	−0.177965
$575$	0	0
$576$	6.08192	0.253413
$577$	24.5832i	1.02341i	0.859160	+	0.511707i	$0.170987\pi$
$577$	24.5832i	1.02341i	−0.859160	+	0.511707i	$0.829013\pi$
$578$	18.7246i	0.778840i
$579$	22.7094	0.943769
$580$	0	0
$581$	0.334995	0.0138980
$582$	− 19.1985i	− 0.795802i
$583$	9.82918i	0.407083i
$584$	17.4448	0.721871
$585$	0	0
$586$	14.4245	0.595872
$587$	− 9.24270i	− 0.381487i	−0.981640	−	0.190744i	$-0.938910\pi$
$587$	− 9.24270i	− 0.381487i	0.981640	−	0.190744i	$-0.0610899\pi$
$588$	− 1.71951i	− 0.0709113i
$589$	21.1134	0.869962
$590$	0	0
$591$	−1.35341	−0.0556720
$592$	− 44.7730i	− 1.84016i
$593$	6.07888i	0.249630i	0.992180	+	0.124815i	$0.0398337\pi$
$593$	6.07888i	0.249630i	−0.992180	+	0.124815i	$0.960166\pi$
$594$	−3.62050	−0.148551
$595$	0	0
$596$	−2.66590	−0.109199
$597$	8.96061i	0.366733i
$598$	− 18.2372i	− 0.745773i
$599$	−6.40129	−0.261550	−0.130775	−	0.991412i	$-0.541746\pi$
$599$	−6.40129	−0.261550	−0.130775	+	0.991412i	$0.541746\pi$
$600$	0	0
$601$	−38.4675	−1.56912	−0.784560	−	0.620052i	$-0.787111\pi$
$601$	−38.4675	−1.56912	−0.784560	+	0.620052i	$0.787111\pi$
$602$	− 18.4263i	− 0.750999i
$603$	− 4.31358i	− 0.175663i
$604$	1.53825	0.0625906
$605$	0	0
$606$	−12.5961	−0.511682
$607$	5.22464i	0.212062i	0.994363	+	0.106031i	$0.0338142\pi$
$607$	5.22464i	0.212062i	−0.994363	+	0.106031i	$0.966186\pi$
$608$	11.5265i	0.467462i
$609$	11.8147	0.478755
$610$	0	0
$611$	8.99173	0.363766
$612$	− 0.792000i	− 0.0320147i
$613$	− 29.8153i	− 1.20423i	−0.798410	−	0.602114i	$-0.794325\pi$
$613$	− 29.8153i	− 1.20423i	0.798410	−	0.602114i	$-0.205675\pi$
$614$	−16.3588	−0.660189
$615$	0	0
$616$	8.87439	0.357559
$617$	23.3025i	0.938124i	0.883165	+	0.469062i	$0.155408\pi$
$617$	23.3025i	0.938124i	−0.883165	+	0.469062i	$0.844592\pi$
$618$	3.84145i	0.154526i
$619$	0.236492	0.00950543	0.00475272	−	0.999989i	$-0.498487\pi$
$619$	0.236492	0.00950543	0.00475272	+	0.999989i	$0.498487\pi$
$620$	0	0
$621$	−8.79501	−0.352932
$622$	− 42.0356i	− 1.68547i
$623$	− 0.641818i	− 0.0257139i
$624$	−6.19646	−0.248057
$625$	0	0
$626$	−3.06555	−0.122524
$627$	13.4558i	0.537374i
$628$	− 5.79632i	− 0.231298i
$629$	21.3941	0.853039
$630$	0	0
$631$	−17.8789	−0.711748	−0.355874	−	0.934534i	$-0.615817\pi$
$631$	−17.8789	−0.711748	−0.355874	+	0.934534i	$0.615817\pi$
$632$	26.8685i	1.06877i
$633$	5.83983i	0.232112i
$634$	−13.2892	−0.527782
$635$	0	0
$636$	−1.50582	−0.0597098
$637$	− 6.42836i	− 0.254701i
$638$	28.6025i	1.13239i
$639$	−5.77750	−0.228554
$640$	0	0
$641$	10.8680	0.429259	0.214630	−	0.976695i	$-0.431146\pi$
$641$	10.8680	0.429259	0.214630	+	0.976695i	$0.431146\pi$
$642$	2.78121i	0.109765i
$643$	− 3.09039i	− 0.121873i	−0.998142	−	0.0609366i	$-0.980591\pi$
$643$	− 3.09039i	− 0.121873i	0.998142	−	0.0609366i	$-0.0194088\pi$
$644$	−4.74789	−0.187093
$645$	0	0
$646$	−19.2522	−0.757468
$647$	− 5.53705i	− 0.217684i	−0.994059	−	0.108842i	$-0.965286\pi$
$647$	− 5.53705i	− 0.217684i	0.994059	−	0.108842i	$-0.0347142\pi$
$648$	2.51844i	0.0989335i
$649$	26.0712	1.02338
$650$	0	0
$651$	5.52910	0.216703
$652$	− 8.04753i	− 0.315166i
$653$	− 38.8754i	− 1.52131i	−0.649154	−	0.760657i	$-0.724877\pi$
$653$	− 38.8754i	− 1.52131i	0.649154	−	0.760657i	$-0.275123\pi$
$654$	−4.52939	−0.177113
$655$	0	0
$656$	−8.51978	−0.332642
$657$	6.92684i	0.270242i
$658$	− 15.3109i	− 0.596879i
$659$	20.5389	0.800082	0.400041	−	0.916497i	$-0.368996\pi$
$659$	20.5389	0.800082	0.400041	+	0.916497i	$0.368996\pi$
$660$	0	0
$661$	38.4254	1.49457	0.747287	−	0.664502i	$-0.231356\pi$
$661$	38.4254	1.49457	0.747287	+	0.664502i	$0.231356\pi$
$662$	− 6.05907i	− 0.235493i
$663$	− 2.96088i	− 0.114991i
$664$	0.564137	0.0218928
$665$	0	0
$666$	14.9828	0.580572
$667$	69.4821i	2.69036i
$668$	2.33408i	0.0903081i
$669$	−7.82097	−0.302376
$670$	0	0
$671$	28.8340	1.11313
$672$	3.01853i	0.116442i
$673$	12.9819i	0.500417i	0.968192	+	0.250208i	$0.0804991\pi$
$673$	12.9819i	0.500417i	−0.968192	+	0.250208i	$0.919501\pi$
$674$	−6.07660	−0.234062
$675$	0	0
$676$	−4.03529	−0.155204
$677$	7.23957i	0.278239i	0.990276	+	0.139120i	$0.0444273\pi$
$677$	7.23957i	0.278239i	−0.990276	+	0.139120i	$0.955573\pi$
$678$	21.2604i	0.816500i
$679$	−18.6855	−0.717086
$680$	0	0
$681$	−16.3090	−0.624961
$682$	13.3856i	0.512561i
$683$	− 24.8800i	− 0.952005i	−0.879444	−	0.476003i	$-0.842085\pi$
$683$	− 24.8800i	− 0.952005i	0.879444	−	0.476003i	$-0.157915\pi$
$684$	−2.06142	−0.0788205
$685$	0	0
$686$	−27.0313	−1.03206
$687$	21.7088i	0.828241i
$688$	− 36.8192i	− 1.40372i
$689$	−5.62950	−0.214467
$690$	0	0
$691$	−33.2706	−1.26567	−0.632836	−	0.774286i	$-0.718109\pi$
$691$	−33.2706	−1.26567	−0.632836	+	0.774286i	$0.718109\pi$
$692$	− 4.26604i	− 0.162170i
$693$	3.52377i	0.133857i
$694$	15.3593	0.583029
$695$	0	0
$696$	19.8961	0.754159
$697$	− 4.07105i	− 0.154202i
$698$	− 28.3546i	− 1.07324i
$699$	−13.5341	−0.511905
$700$	0	0
$701$	−13.2163	−0.499173	−0.249586	−	0.968353i	$-0.580295\pi$
$701$	−13.2163	−0.499173	−0.249586	+	0.968353i	$0.580295\pi$
$702$	− 2.07358i	− 0.0782622i
$703$	− 55.6847i	− 2.10019i
$704$	14.3305	0.540103
$705$	0	0
$706$	14.3267	0.539194
$707$	12.2596i	0.461069i
$708$	3.99408i	0.150107i
$709$	−6.25068	−0.234749	−0.117375	−	0.993088i	$-0.537448\pi$
$709$	−6.25068	−0.234749	−0.117375	+	0.993088i	$0.537448\pi$
$710$	0	0
$711$	−10.6687	−0.400109
$712$	− 1.08083i	− 0.0405058i
$713$	32.5167i	1.21776i
$714$	−5.04171	−0.188681
$715$	0	0
$716$	5.62205	0.210106
$717$	10.5338i	0.393393i
$718$	− 44.4735i	− 1.65973i
$719$	−27.5016	−1.02564	−0.512818	−	0.858497i	$-0.671398\pi$
$719$	−27.5016	−1.02564	−0.512818	+	0.858497i	$0.671398\pi$
$720$	0	0
$721$	3.73882	0.139241
$722$	20.9154i	0.778390i
$723$	19.4838i	0.724610i
$724$	5.26293	0.195595
$725$	0	0
$726$	8.37120	0.310684
$727$	− 22.0397i	− 0.817406i	−0.912668	−	0.408703i	$-0.865981\pi$
$727$	− 22.0397i	− 0.817406i	0.912668	−	0.408703i	$-0.134019\pi$
$728$	5.08266i	0.188376i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	17.5935	0.650720
$732$	4.41735i	0.163270i
$733$	− 34.7134i	− 1.28217i	−0.767470	−	0.641085i	$-0.778485\pi$
$733$	− 34.7134i	− 1.28217i	0.767470	−	0.641085i	$-0.221515\pi$
$734$	6.14417	0.226785
$735$	0	0
$736$	−17.7519	−0.654345
$737$	− 10.1639i	− 0.374392i
$738$	− 2.85106i	− 0.104949i
$739$	7.58593	0.279053	0.139526	−	0.990218i	$-0.455442\pi$
$739$	7.58593	0.279053	0.139526	+	0.990218i	$0.455442\pi$
$740$	0	0
$741$	−7.70661	−0.283109
$742$	9.58575i	0.351904i
$743$	− 27.5328i	− 1.01008i	−0.863096	−	0.505040i	$-0.831478\pi$
$743$	− 27.5328i	− 1.01008i	0.863096	−	0.505040i	$-0.168522\pi$
$744$	9.31109	0.341361
$745$	0	0
$746$	−4.87484	−0.178481
$747$	0.224003i	0.00819584i
$748$	− 1.86615i	− 0.0682334i
$749$	2.70690	0.0989081
$750$	0	0
$751$	4.24930	0.155059	0.0775296	−	0.996990i	$-0.475297\pi$
$751$	4.24930	0.155059	0.0775296	+	0.996990i	$0.475297\pi$
$752$	− 30.5940i	− 1.11565i
$753$	− 20.9446i	− 0.763265i
$754$	−16.3816	−0.596584
$755$	0	0
$756$	−0.539839	−0.0196337
$757$	− 45.6609i	− 1.65957i	−0.558081	−	0.829787i	$-0.688462\pi$
$757$	− 45.6609i	− 1.65957i	0.558081	−	0.829787i	$-0.311538\pi$
$758$	− 43.8232i	− 1.59173i
$759$	−20.7233	−0.752208
$760$	0	0
$761$	−40.1268	−1.45460	−0.727298	−	0.686322i	$-0.759224\pi$
$761$	−40.1268	−1.45460	−0.727298	+	0.686322i	$0.759224\pi$
$762$	− 8.81971i	− 0.319504i
$763$	4.40839i	0.159594i
$764$	7.57631	0.274101
$765$	0	0
$766$	−17.6105	−0.636295
$767$	14.9318i	0.539157i
$768$	8.39819i	0.303044i
$769$	37.8350	1.36437	0.682183	−	0.731181i	$-0.261031\pi$
$769$	37.8350	1.36437	0.682183	+	0.731181i	$0.261031\pi$
$770$	0	0
$771$	1.67121	0.0601872
$772$	8.19755i	0.295036i
$773$	26.2148i	0.942881i	0.881898	+	0.471441i	$0.156266\pi$
$773$	26.2148i	0.942881i	−0.881898	+	0.471441i	$0.843734\pi$
$774$	12.3212	0.442875
$775$	0	0
$776$	−31.4667	−1.12959
$777$	− 14.5825i	− 0.523146i
$778$	− 52.6211i	− 1.88656i
$779$	−10.5962	−0.379646
$780$	0	0
$781$	−13.6133	−0.487121
$782$	− 29.6503i	− 1.06029i
$783$	7.90017i	0.282329i
$784$	−21.8723	−0.781153
$785$	0	0
$786$	6.69559	0.238824
$787$	− 2.41254i	− 0.0859977i	−0.999075	−	0.0429988i	$-0.986309\pi$
$787$	− 2.41254i	− 0.0859977i	0.999075	−	0.0429988i	$-0.0136912\pi$
$788$	− 0.488551i	− 0.0174039i
$789$	7.55667	0.269025
$790$	0	0
$791$	20.6924	0.735737
$792$	5.93408i	0.210858i
$793$	16.5142i	0.586437i
$794$	30.7822	1.09242
$795$	0	0
$796$	−3.23457	−0.114646
$797$	8.11230i	0.287353i	0.989625	+	0.143676i	$0.0458924\pi$
$797$	8.11230i	0.287353i	−0.989625	+	0.143676i	$0.954108\pi$
$798$	13.1226i	0.464534i
$799$	14.6189	0.517180
$800$	0	0
$801$	0.429167	0.0151639
$802$	7.65507i	0.270310i
$803$	16.3214i	0.575970i
$804$	1.55710	0.0549148
$805$	0	0
$806$	−7.66638	−0.270037
$807$	11.2841i	0.397220i
$808$	20.6453i	0.726299i
$809$	40.7091	1.43125	0.715627	−	0.698482i	$-0.246141\pi$
$809$	40.7091	1.43125	0.715627	+	0.698482i	$0.246141\pi$
$810$	0	0
$811$	−49.4990	−1.73814	−0.869072	−	0.494685i	$-0.835283\pi$
$811$	−49.4990	−1.73814	−0.869072	+	0.494685i	$0.835283\pi$
$812$	4.26482i	0.149666i
$813$	− 10.9752i	− 0.384917i
$814$	35.3033	1.23738
$815$	0	0
$816$	−10.0743	−0.352671
$817$	− 45.7925i	− 1.60208i
$818$	− 36.6606i	− 1.28181i
$819$	−2.01818	−0.0705210
$820$	0	0
$821$	42.2114	1.47319	0.736594	−	0.676335i	$-0.236433\pi$
$821$	42.2114	1.47319	0.736594	+	0.676335i	$0.236433\pi$
$822$	− 3.03407i	− 0.105825i
$823$	49.2349i	1.71622i	0.513464	+	0.858111i	$0.328362\pi$
$823$	49.2349i	1.71622i	−0.513464	+	0.858111i	$0.671638\pi$
$824$	6.29622	0.219339
$825$	0	0
$826$	25.4255	0.884666
$827$	51.0011i	1.77348i	0.462266	+	0.886742i	$0.347037\pi$
$827$	51.0011i	1.77348i	−0.462266	+	0.886742i	$0.652963\pi$
$828$	− 3.17479i	− 0.110332i
$829$	38.2342	1.32793	0.663964	−	0.747764i	$-0.268873\pi$
$829$	38.2342	1.32793	0.663964	+	0.747764i	$0.268873\pi$
$830$	0	0
$831$	5.75112	0.199504
$832$	8.20758i	0.284547i
$833$	− 10.4513i	− 0.362117i
$834$	2.58001	0.0893384
$835$	0	0
$836$	−4.85724	−0.167991
$837$	3.69717i	0.127793i
$838$	− 0.741461i	− 0.0256133i
$839$	16.7086	0.576845	0.288422	−	0.957503i	$-0.406869\pi$
$839$	16.7086	0.576845	0.288422	+	0.957503i	$0.406869\pi$
$840$	0	0
$841$	33.4127	1.15216
$842$	− 27.2250i	− 0.938236i
$843$	8.49962i	0.292742i
$844$	−2.10804	−0.0725618
$845$	0	0
$846$	10.2380	0.351989
$847$	− 8.14755i	− 0.279953i
$848$	19.1542i	0.657757i
$849$	−17.2248	−0.591154
$850$	0	0
$851$	85.7599	2.93981
$852$	− 2.08554i	− 0.0714495i
$853$	18.2644i	0.625361i	0.949858	+	0.312681i	$0.101227\pi$
$853$	18.2644i	0.625361i	−0.949858	+	0.312681i	$0.898773\pi$
$854$	28.1199	0.962244
$855$	0	0
$856$	4.55846	0.155805
$857$	− 53.4773i	− 1.82675i	−0.407119	−	0.913375i	$-0.633466\pi$
$857$	− 53.4773i	− 1.82675i	0.407119	−	0.913375i	$-0.366534\pi$
$858$	− 4.88588i	− 0.166801i
$859$	18.7575	0.639998	0.319999	−	0.947418i	$-0.396317\pi$
$859$	18.7575	0.639998	0.319999	+	0.947418i	$0.396317\pi$
$860$	0	0
$861$	−2.77489	−0.0945678
$862$	41.1889i	1.40290i
$863$	51.2363i	1.74410i	0.489414	+	0.872051i	$0.337211\pi$
$863$	51.2363i	1.74410i	−0.489414	+	0.872051i	$0.662789\pi$
$864$	−2.01841	−0.0686677
$865$	0	0
$866$	−14.1841	−0.481995
$867$	12.1861i	0.413863i
$868$	1.99588i	0.0677444i
$869$	−25.1383	−0.852757
$870$	0	0
$871$	5.82121	0.197244
$872$	7.42378i	0.251401i
$873$	− 12.4945i	− 0.422876i
$874$	−77.1739	−2.61045
$875$	0	0
$876$	−2.50042	−0.0844815
$877$	− 6.06306i	− 0.204735i	−0.994747	−	0.102368i	$-0.967358\pi$
$877$	− 6.06306i	− 0.204735i	0.994747	−	0.102368i	$-0.0326418\pi$
$878$	− 1.54024i	− 0.0519806i
$879$	9.38764	0.316637
$880$	0	0
$881$	22.6698	0.763765	0.381883	−	0.924211i	$-0.375276\pi$
$881$	22.6698	0.763765	0.381883	+	0.924211i	$0.375276\pi$
$882$	− 7.31933i	− 0.246455i
$883$	5.53899i	0.186402i	0.995647	+	0.0932009i	$0.0297099\pi$
$883$	5.53899i	0.186402i	−0.995647	+	0.0932009i	$0.970290\pi$
$884$	1.06881	0.0359480
$885$	0	0
$886$	−40.3915	−1.35698
$887$	− 10.9716i	− 0.368390i	−0.982890	−	0.184195i	$-0.941032\pi$
$887$	− 10.9716i	− 0.368390i	0.982890	−	0.184195i	$-0.0589678\pi$
$888$	− 24.5572i	− 0.824086i
$889$	−8.58408	−0.287901
$890$	0	0
$891$	−2.35626	−0.0789375
$892$	− 2.82319i	− 0.0945272i
$893$	− 38.0502i	− 1.27330i
$894$	−11.3478	−0.379526
$895$	0	0
$896$	20.0127	0.668577
$897$	− 11.8689i	− 0.396292i
$898$	− 7.30550i	− 0.243788i
$899$	29.2083	0.974151
$900$	0	0
$901$	−9.15254	−0.304915
$902$	− 6.71781i	− 0.223679i
$903$	− 11.9920i	− 0.399069i
$904$	34.8463	1.15897
$905$	0	0
$906$	6.54779	0.217536
$907$	− 19.3907i	− 0.643856i	−0.946764	−	0.321928i	$-0.895669\pi$
$907$	− 19.3907i	− 0.643856i	0.946764	−	0.321928i	$-0.104331\pi$
$908$	− 5.88715i	− 0.195372i
$909$	−8.19767	−0.271899
$910$	0	0
$911$	−13.7190	−0.454530	−0.227265	−	0.973833i	$-0.572978\pi$
$911$	−13.7190	−0.454530	−0.227265	+	0.973833i	$0.572978\pi$
$912$	26.2215i	0.868280i
$913$	0.527808i	0.0174679i
$914$	−24.5391	−0.811681
$915$	0	0
$916$	−7.83636	−0.258920
$917$	− 6.51671i	− 0.215201i
$918$	− 3.37126i	− 0.111268i
$919$	32.4108	1.06913	0.534567	−	0.845126i	$-0.320475\pi$
$919$	32.4108	1.06913	0.534567	+	0.845126i	$0.320475\pi$
$920$	0	0
$921$	−10.6465	−0.350814
$922$	61.0867i	2.01178i
$923$	− 7.79678i	− 0.256634i
$924$	−1.27200	−0.0418457
$925$	0	0
$926$	−40.1360	−1.31895
$927$	2.50005i	0.0821125i
$928$	15.9458i	0.523446i
$929$	4.33444	0.142208	0.0711042	−	0.997469i	$-0.477348\pi$
$929$	4.33444	0.142208	0.0711042	+	0.997469i	$0.477348\pi$
$930$	0	0
$931$	−27.2028	−0.891536
$932$	− 4.88548i	− 0.160029i
$933$	− 27.3572i	− 0.895633i
$934$	−5.91884	−0.193670
$935$	0	0
$936$	−3.39865	−0.111088
$937$	− 54.5925i	− 1.78346i	−0.452567	−	0.891730i	$-0.649492\pi$
$937$	− 54.5925i	− 1.78346i	0.452567	−	0.891730i	$-0.350508\pi$
$938$	− 9.91218i	− 0.323644i
$939$	−1.99509	−0.0651073
$940$	0	0
$941$	4.23853	0.138172	0.0690861	−	0.997611i	$-0.477992\pi$
$941$	4.23853	0.138172	0.0690861	+	0.997611i	$0.477992\pi$
$942$	− 24.6729i	− 0.803885i
$943$	− 16.3191i	− 0.531423i
$944$	50.8051	1.65356
$945$	0	0
$946$	29.0318	0.943906
$947$	− 12.6069i	− 0.409671i	−0.978796	−	0.204835i	$-0.934334\pi$
$947$	− 12.6069i	− 0.409671i	0.978796	−	0.204835i	$-0.0656659\pi$
$948$	− 3.85116i	− 0.125080i
$949$	−9.34781	−0.303443
$950$	0	0
$951$	−8.64876	−0.280455
$952$	8.26348i	0.267821i
$953$	− 31.1635i	− 1.00948i	−0.863270	−	0.504742i	$-0.831588\pi$
$953$	− 31.1635i	− 1.00948i	0.863270	−	0.504742i	$-0.168412\pi$
$954$	−6.40975	−0.207523
$955$	0	0
$956$	−3.80247	−0.122981
$957$	18.6148i	0.601732i
$958$	21.9262i	0.708405i
$959$	−2.95301	−0.0953578
$960$	0	0
$961$	−17.3309	−0.559062
$962$	20.2194i	0.651900i
$963$	1.81004i	0.0583276i
$964$	−7.03319	−0.226524
$965$	0	0
$966$	−20.2101	−0.650248
$967$	− 1.55308i	− 0.0499438i	−0.999688	−	0.0249719i	$-0.992050\pi$
$967$	− 1.55308i	− 0.0499438i	0.999688	−	0.0249719i	$-0.00794963\pi$
$968$	− 13.7206i	− 0.440997i
$969$	−12.5295	−0.402507
$970$	0	0
$971$	−38.2623	−1.22790	−0.613948	−	0.789347i	$-0.710419\pi$
$971$	−38.2623	−1.22790	−0.613948	+	0.789347i	$0.710419\pi$
$972$	− 0.360976i	− 0.0115783i
$973$	− 2.51108i	− 0.0805016i
$974$	38.1405	1.22210
$975$	0	0
$976$	56.1890	1.79857
$977$	20.3661i	0.651570i	0.945444	+	0.325785i	$0.105629\pi$
$977$	20.3661i	0.651570i	−0.945444	+	0.325785i	$0.894371\pi$
$978$	− 34.2554i	− 1.09537i
$979$	1.01123	0.0323190
$980$	0	0
$981$	−2.94778	−0.0941152
$982$	17.6539i	0.563359i
$983$	− 13.2557i	− 0.422791i	−0.977401	−	0.211395i	$-0.932199\pi$
$983$	− 13.2557i	− 0.422791i	0.977401	−	0.211395i	$-0.0678008\pi$
$984$	−4.67295	−0.148968
$985$	0	0
$986$	−26.6335	−0.848185
$987$	− 9.96446i	− 0.317172i
$988$	− 2.78190i	− 0.0885041i
$989$	70.5249	2.24256
$990$	0	0
$991$	−2.24081	−0.0711816	−0.0355908	−	0.999366i	$-0.511331\pi$
$991$	−2.24081	−0.0711816	−0.0355908	+	0.999366i	$0.511331\pi$
$992$	7.46241i	0.236932i
$993$	− 3.94331i	− 0.125137i
$994$	−13.2761	−0.421093
$995$	0	0
$996$	−0.0808598	−0.00256214
$997$	14.0404i	0.444663i	0.974971	+	0.222331i	$0.0713667\pi$
$997$	14.0404i	0.444663i	−0.974971	+	0.222331i	$0.928633\pi$
$998$	6.41489i	0.203060i
$999$	9.75097	0.308507

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.h.1249.13		16
5.2	odd	4		1875.2.a.m.1.3		8
5.3	odd	4		1875.2.a.p.1.6		8
5.4	even	2	inner	1875.2.b.h.1249.4		16
15.2	even	4		5625.2.a.bd.1.6		8
15.8	even	4		5625.2.a.t.1.3		8
25.2	odd	20		375.2.g.e.226.2		16
25.9	even	10		75.2.i.a.19.4	yes	16
25.11	even	5		75.2.i.a.4.4	✓	16
25.12	odd	20		375.2.g.e.151.2		16
25.13	odd	20		375.2.g.d.151.3		16
25.14	even	10		375.2.i.c.274.1		16
25.16	even	5		375.2.i.c.349.1		16
25.23	odd	20		375.2.g.d.226.3		16
75.11	odd	10		225.2.m.b.154.1		16
75.59	odd	10		225.2.m.b.19.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.4.4	✓	16	25.11	even	5
75.2.i.a.19.4	yes	16	25.9	even	10
225.2.m.b.19.1		16	75.59	odd	10
225.2.m.b.154.1		16	75.11	odd	10
375.2.g.d.151.3		16	25.13	odd	20
375.2.g.d.226.3		16	25.23	odd	20
375.2.g.e.151.2		16	25.12	odd	20
375.2.g.e.226.2		16	25.2	odd	20
375.2.i.c.274.1		16	25.14	even	10
375.2.i.c.349.1		16	25.16	even	5
1875.2.a.m.1.3		8	5.2	odd	4
1875.2.a.p.1.6		8	5.3	odd	4
1875.2.b.h.1249.4		16	5.4	even	2	inner
1875.2.b.h.1249.13		16	1.1	even	1	trivial
5625.2.a.t.1.3		8	15.8	even	4
5625.2.a.bd.1.6		8	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)

\(f(q)\)	\(=\)	\(q+1.53655i q^{2} +1.00000i q^{3} -0.360976 q^{4} -1.53655 q^{6} +1.49550i q^{7} +2.51844i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.53655i q^{2} +1.00000i q^{3} -0.360976 q^{4} -1.53655 q^{6} +1.49550i q^{7} +2.51844i q^{8} -1.00000 q^{9} -2.35626 q^{11} -0.360976i q^{12} -1.34951i q^{13} -2.29790 q^{14} -4.59165 q^{16} -2.19405i q^{17} -1.53655i q^{18} -5.71069 q^{19} -1.49550 q^{21} -3.62050i q^{22} -8.79501i q^{23} -2.51844 q^{24} +2.07358 q^{26} -1.00000i q^{27} -0.539839i q^{28} -7.90017 q^{29} -3.69717 q^{31} -2.01841i q^{32} -2.35626i q^{33} +3.37126 q^{34} +0.360976 q^{36} +9.75097i q^{37} -8.77474i q^{38} +1.34951 q^{39} +1.85550 q^{41} -2.29790i q^{42} +8.01874i q^{43} +0.850553 q^{44} +13.5139 q^{46} +6.66298i q^{47} -4.59165i q^{48} +4.76349 q^{49} +2.19405 q^{51} +0.487140i q^{52} -4.17153i q^{53} +1.53655 q^{54} -3.76631 q^{56} -5.71069i q^{57} -12.1390i q^{58} -11.0647 q^{59} -12.2372 q^{61} -5.68088i q^{62} -1.49550i q^{63} -6.08192 q^{64} +3.62050 q^{66} +4.31358i q^{67} +0.792000i q^{68} +8.79501 q^{69} +5.77750 q^{71} -2.51844i q^{72} -6.92684i q^{73} -14.9828 q^{74} +2.06142 q^{76} -3.52377i q^{77} +2.07358i q^{78} +10.6687 q^{79} +1.00000 q^{81} +2.85106i q^{82} -0.224003i q^{83} +0.539839 q^{84} -12.3212 q^{86} -7.90017i q^{87} -5.93408i q^{88} -0.429167 q^{89} +2.01818 q^{91} +3.17479i q^{92} -3.69717i q^{93} -10.2380 q^{94} +2.01841 q^{96} +12.4945i q^{97} +7.31933i q^{98} +2.35626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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