LMFDB - Embedded newform 975.2.c.h.274.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(274,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("975.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.c (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:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	975.274
Dual form			975.2.c.h.274.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} -2.82843i q^{7} +4.41421i q^{8} -1.00000 q^{9} -2.00000 q^{11} +3.82843i q^{12} +1.00000i q^{13} -6.82843 q^{14} +3.00000 q^{16} -3.65685i q^{17} +2.41421i q^{18} -2.82843 q^{19} -2.82843 q^{21} +4.82843i q^{22} +4.00000i q^{23} +4.41421 q^{24} +2.41421 q^{26} +1.00000i q^{27} +10.8284i q^{28} -2.00000 q^{29} -6.82843 q^{31} +1.58579i q^{32} +2.00000i q^{33} -8.82843 q^{34} +3.82843 q^{36} +3.65685i q^{37} +6.82843i q^{38} +1.00000 q^{39} +10.8284 q^{41} +6.82843i q^{42} -9.65685i q^{43} +7.65685 q^{44} +9.65685 q^{46} -0.343146i q^{47} -3.00000i q^{48} -1.00000 q^{49} -3.65685 q^{51} -3.82843i q^{52} +2.00000i q^{53} +2.41421 q^{54} +12.4853 q^{56} +2.82843i q^{57} +4.82843i q^{58} +3.65685 q^{59} -9.31371 q^{61} +16.4853i q^{62} +2.82843i q^{63} +9.82843 q^{64} +4.82843 q^{66} +1.17157i q^{67} +14.0000i q^{68} +4.00000 q^{69} +2.00000 q^{71} -4.41421i q^{72} -11.6569i q^{73} +8.82843 q^{74} +10.8284 q^{76} +5.65685i q^{77} -2.41421i q^{78} -11.3137 q^{79} +1.00000 q^{81} -26.1421i q^{82} +7.65685i q^{83} +10.8284 q^{84} -23.3137 q^{86} +2.00000i q^{87} -8.82843i q^{88} -9.17157 q^{89} +2.82843 q^{91} -15.3137i q^{92} +6.82843i q^{93} -0.828427 q^{94} +1.58579 q^{96} -7.65685i q^{97} +2.41421i q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 8 q^{11} - 16 q^{14} + 12 q^{16} + 12 q^{24} + 4 q^{26} - 8 q^{29} - 16 q^{31} - 24 q^{34} + 4 q^{36} + 4 q^{39} + 32 q^{41} + 8 q^{44} + 16 q^{46} - 4 q^{49} + 8 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 8 * q^11 - 16 * q^14 + 12 * q^16 + 12 * q^24 + 4 * q^26 - 8 * q^29 - 16 * q^31 - 24 * q^34 + 4 * q^36 + 4 * q^39 + 32 * q^41 + 8 * q^44 + 16 * q^46 - 4 * q^49 + 8 * q^51 + 4 * q^54 + 16 * q^56 - 8 * q^59 + 8 * q^61 + 28 * q^64 + 8 * q^66 + 16 * q^69 + 8 * q^71 + 24 * q^74 + 32 * q^76 + 4 * q^81 + 32 * q^84 - 48 * q^86 - 48 * q^89 + 8 * q^94 + 12 * q^96 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 2.41421i	− 1.70711i	−0.521005	−	0.853553i	$-0.674443\pi$
$2$	− 2.41421i	− 1.70711i	0.521005	−	0.853553i	$-0.325557\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	−2.41421	−0.985599
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	4.41421i	1.56066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	3.82843i	1.10517i
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	−6.82843	−1.82497
$15$	0	0
$16$	3.00000	0.750000
$17$	− 3.65685i	− 0.886917i	−0.896295	−	0.443459i	$-0.853751\pi$
$17$	− 3.65685i	− 0.886917i	0.896295	−	0.443459i	$-0.146249\pi$
$18$	2.41421i	0.569036i
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	4.82843i	1.02942i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	4.41421	0.901048
$25$	0	0
$26$	2.41421	0.473466
$27$	1.00000i	0.192450i
$28$	10.8284i	2.04638i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−6.82843	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$31$	−6.82843	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$32$	1.58579i	0.280330i
$33$	2.00000i	0.348155i
$34$	−8.82843	−1.51406
$35$	0	0
$36$	3.82843	0.638071
$37$	3.65685i	0.601183i	0.953753	+	0.300592i	$0.0971841\pi$
$37$	3.65685i	0.601183i	−0.953753	+	0.300592i	$0.902816\pi$
$38$	6.82843i	1.10772i
$39$	1.00000	0.160128
$40$	0	0
$41$	10.8284	1.69112	0.845558	−	0.533883i	$-0.179268\pi$
$41$	10.8284	1.69112	0.845558	+	0.533883i	$0.179268\pi$
$42$	6.82843i	1.05365i
$43$	− 9.65685i	− 1.47266i	−0.676625	−	0.736328i	$-0.736558\pi$
$43$	− 9.65685i	− 1.47266i	0.676625	−	0.736328i	$-0.263442\pi$
$44$	7.65685	1.15431
$45$	0	0
$46$	9.65685	1.42383
$47$	− 0.343146i	− 0.0500530i	−0.999687	−	0.0250265i	$-0.992033\pi$
$47$	− 0.343146i	− 0.0500530i	0.999687	−	0.0250265i	$-0.00796701\pi$
$48$	− 3.00000i	− 0.433013i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−3.65685	−0.512062
$52$	− 3.82843i	− 0.530907i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	2.41421	0.328533
$55$	0	0
$56$	12.4853	1.66842
$57$	2.82843i	0.374634i
$58$	4.82843i	0.634004i
$59$	3.65685	0.476082	0.238041	−	0.971255i	$-0.423495\pi$
$59$	3.65685	0.476082	0.238041	+	0.971255i	$0.423495\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	16.4853i	2.09363i
$63$	2.82843i	0.356348i
$64$	9.82843	1.22855
$65$	0	0
$66$	4.82843	0.594338
$67$	1.17157i	0.143130i	0.997436	+	0.0715652i	$0.0227994\pi$
$67$	1.17157i	0.143130i	−0.997436	+	0.0715652i	$0.977201\pi$
$68$	14.0000i	1.69775i
$69$	4.00000	0.481543
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	− 4.41421i	− 0.520220i
$73$	− 11.6569i	− 1.36433i	−0.731198	−	0.682166i	$-0.761038\pi$
$73$	− 11.6569i	− 1.36433i	0.731198	−	0.682166i	$-0.238962\pi$
$74$	8.82843	1.02628
$75$	0	0
$76$	10.8284	1.24211
$77$	5.65685i	0.644658i
$78$	− 2.41421i	− 0.273356i
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 26.1421i	− 2.88692i
$83$	7.65685i	0.840449i	0.907420	+	0.420224i	$0.138049\pi$
$83$	7.65685i	0.840449i	−0.907420	+	0.420224i	$0.861951\pi$
$84$	10.8284	1.18148
$85$	0	0
$86$	−23.3137	−2.51398
$87$	2.00000i	0.214423i
$88$	− 8.82843i	− 0.941113i
$89$	−9.17157	−0.972185	−0.486092	−	0.873907i	$-0.661578\pi$
$89$	−9.17157	−0.972185	−0.486092	+	0.873907i	$0.661578\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	− 15.3137i	− 1.59656i
$93$	6.82843i	0.708075i
$94$	−0.828427	−0.0854457
$95$	0	0
$96$	1.58579	0.161849
$97$	− 7.65685i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$97$	− 7.65685i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$98$	2.41421i	0.243872i
$99$	2.00000	0.201008
$100$	0	0
$101$	−3.65685	−0.363871	−0.181935	−	0.983311i	$-0.558236\pi$
$101$	−3.65685	−0.363871	−0.181935	+	0.983311i	$0.558236\pi$
$102$	8.82843i	0.874145i
$103$	− 13.6569i	− 1.34565i	−0.739802	−	0.672825i	$-0.765081\pi$
$103$	− 13.6569i	− 1.34565i	0.739802	−	0.672825i	$-0.234919\pi$
$104$	−4.41421	−0.432849
$105$	0	0
$106$	4.82843	0.468978
$107$	11.3137i	1.09374i	0.837218	+	0.546869i	$0.184180\pi$
$107$	11.3137i	1.09374i	−0.837218	+	0.546869i	$0.815820\pi$
$108$	− 3.82843i	− 0.368391i
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	3.65685	0.347093
$112$	− 8.48528i	− 0.801784i
$113$	− 17.3137i	− 1.62874i	−0.580348	−	0.814368i	$-0.697084\pi$
$113$	− 17.3137i	− 1.62874i	0.580348	−	0.814368i	$-0.302916\pi$
$114$	6.82843	0.639541
$115$	0	0
$116$	7.65685	0.710921
$117$	− 1.00000i	− 0.0924500i
$118$	− 8.82843i	− 0.812723i
$119$	−10.3431	−0.948155
$120$	0	0
$121$	−7.00000	−0.636364
$122$	22.4853i	2.03572i
$123$	− 10.8284i	− 0.976366i
$124$	26.1421	2.34763
$125$	0	0
$126$	6.82843	0.608325
$127$	− 5.65685i	− 0.501965i	−0.967992	−	0.250982i	$-0.919246\pi$
$127$	− 5.65685i	− 0.501965i	0.967992	−	0.250982i	$-0.0807536\pi$
$128$	− 20.5563i	− 1.81694i
$129$	−9.65685	−0.850239
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	− 7.65685i	− 0.666444i
$133$	8.00000i	0.693688i
$134$	2.82843	0.244339
$135$	0	0
$136$	16.1421	1.38418
$137$	− 5.17157i	− 0.441837i	−0.975292	−	0.220919i	$-0.929094\pi$
$137$	− 5.17157i	− 0.441837i	0.975292	−	0.220919i	$-0.0709055\pi$
$138$	− 9.65685i	− 0.822046i
$139$	−15.3137	−1.29889	−0.649446	−	0.760408i	$-0.724999\pi$
$139$	−15.3137	−1.29889	−0.649446	+	0.760408i	$0.724999\pi$
$140$	0	0
$141$	−0.343146	−0.0288981
$142$	− 4.82843i	− 0.405193i
$143$	− 2.00000i	− 0.167248i
$144$	−3.00000	−0.250000
$145$	0	0
$146$	−28.1421	−2.32906
$147$	1.00000i	0.0824786i
$148$	− 14.0000i	− 1.15079i
$149$	14.8284	1.21479	0.607396	−	0.794399i	$-0.292214\pi$
$149$	14.8284	1.21479	0.607396	+	0.794399i	$0.292214\pi$
$150$	0	0
$151$	−20.4853	−1.66707	−0.833534	−	0.552468i	$-0.813686\pi$
$151$	−20.4853	−1.66707	−0.833534	+	0.552468i	$0.813686\pi$
$152$	− 12.4853i	− 1.01269i
$153$	3.65685i	0.295639i
$154$	13.6569	1.10050
$155$	0	0
$156$	−3.82843	−0.306519
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	27.3137i	2.17296i
$159$	2.00000	0.158610
$160$	0	0
$161$	11.3137	0.891645
$162$	− 2.41421i	− 0.189679i
$163$	− 13.1716i	− 1.03168i	−0.856686	−	0.515839i	$-0.827480\pi$
$163$	− 13.1716i	− 1.03168i	0.856686	−	0.515839i	$-0.172520\pi$
$164$	−41.4558	−3.23716
$165$	0	0
$166$	18.4853	1.43474
$167$	7.65685i	0.592505i	0.955110	+	0.296253i	$0.0957370\pi$
$167$	7.65685i	0.592505i	−0.955110	+	0.296253i	$0.904263\pi$
$168$	− 12.4853i	− 0.963260i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	2.82843	0.216295
$172$	36.9706i	2.81898i
$173$	0.343146i	0.0260889i	0.999915	+	0.0130444i	$0.00415229\pi$
$173$	0.343146i	0.0260889i	−0.999915	+	0.0130444i	$0.995848\pi$
$174$	4.82843	0.366042
$175$	0	0
$176$	−6.00000	−0.452267
$177$	− 3.65685i	− 0.274866i
$178$	22.1421i	1.65962i
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	− 6.82843i	− 0.506157i
$183$	9.31371i	0.688489i
$184$	−17.6569	−1.30168
$185$	0	0
$186$	16.4853	1.20876
$187$	7.31371i	0.534831i
$188$	1.31371i	0.0958120i
$189$	2.82843	0.205738
$190$	0	0
$191$	−19.3137	−1.39749	−0.698745	−	0.715370i	$-0.746258\pi$
$191$	−19.3137	−1.39749	−0.698745	+	0.715370i	$0.746258\pi$
$192$	− 9.82843i	− 0.709306i
$193$	17.3137i	1.24627i	0.782115	+	0.623134i	$0.214141\pi$
$193$	17.3137i	1.24627i	−0.782115	+	0.623134i	$0.785859\pi$
$194$	−18.4853	−1.32717
$195$	0	0
$196$	3.82843	0.273459
$197$	− 16.4853i	− 1.17453i	−0.809396	−	0.587264i	$-0.800205\pi$
$197$	− 16.4853i	− 1.17453i	0.809396	−	0.587264i	$-0.199795\pi$
$198$	− 4.82843i	− 0.343141i
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	1.17157	0.0826364
$202$	8.82843i	0.621166i
$203$	5.65685i	0.397033i
$204$	14.0000	0.980196
$205$	0	0
$206$	−32.9706	−2.29717
$207$	− 4.00000i	− 0.278019i
$208$	3.00000i	0.208013i
$209$	5.65685	0.391293
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	− 7.65685i	− 0.525875i
$213$	− 2.00000i	− 0.137038i
$214$	27.3137	1.86713
$215$	0	0
$216$	−4.41421	−0.300349
$217$	19.3137i	1.31110i
$218$	− 41.7990i	− 2.83098i
$219$	−11.6569	−0.787697
$220$	0	0
$221$	3.65685	0.245987
$222$	− 8.82843i	− 0.592525i
$223$	− 4.48528i	− 0.300357i	−0.988659	−	0.150178i	$-0.952015\pi$
$223$	− 4.48528i	− 0.300357i	0.988659	−	0.150178i	$-0.0479848\pi$
$224$	4.48528	0.299685
$225$	0	0
$226$	−41.7990	−2.78043
$227$	5.31371i	0.352683i	0.984329	+	0.176342i	$0.0564263\pi$
$227$	5.31371i	0.352683i	−0.984329	+	0.176342i	$0.943574\pi$
$228$	− 10.8284i	− 0.717130i
$229$	−21.3137	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$229$	−21.3137	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	− 8.82843i	− 0.579615i
$233$	26.9706i	1.76690i	0.468525	+	0.883450i	$0.344786\pi$
$233$	26.9706i	1.76690i	−0.468525	+	0.883450i	$0.655214\pi$
$234$	−2.41421	−0.157822
$235$	0	0
$236$	−14.0000	−0.911322
$237$	11.3137i	0.734904i
$238$	24.9706i	1.61860i
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	11.6569	0.750884	0.375442	−	0.926846i	$-0.377491\pi$
$241$	11.6569	0.750884	0.375442	+	0.926846i	$0.377491\pi$
$242$	16.8995i	1.08634i
$243$	− 1.00000i	− 0.0641500i
$244$	35.6569	2.28270
$245$	0	0
$246$	−26.1421	−1.66676
$247$	− 2.82843i	− 0.179969i
$248$	− 30.1421i	− 1.91403i
$249$	7.65685	0.485233
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	− 10.8284i	− 0.682127i
$253$	− 8.00000i	− 0.502956i
$254$	−13.6569	−0.856907
$255$	0	0
$256$	−29.9706	−1.87316
$257$	− 15.6569i	− 0.976648i	−0.872662	−	0.488324i	$-0.837608\pi$
$257$	− 15.6569i	− 0.976648i	0.872662	−	0.488324i	$-0.162392\pi$
$258$	23.3137i	1.45145i
$259$	10.3431	0.642692
$260$	0	0
$261$	2.00000	0.123797
$262$	19.3137i	1.19320i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	−8.82843	−0.543352
$265$	0	0
$266$	19.3137	1.18420
$267$	9.17157i	0.561291i
$268$	− 4.48528i	− 0.273982i
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	11.7990	0.716738	0.358369	−	0.933580i	$-0.383333\pi$
$271$	11.7990	0.716738	0.358369	+	0.933580i	$0.383333\pi$
$272$	− 10.9706i	− 0.665188i
$273$	− 2.82843i	− 0.171184i
$274$	−12.4853	−0.754263
$275$	0	0
$276$	−15.3137	−0.921777
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	36.9706i	2.21735i
$279$	6.82843	0.408807
$280$	0	0
$281$	26.8284	1.60045	0.800225	−	0.599700i	$-0.204713\pi$
$281$	26.8284	1.60045	0.800225	+	0.599700i	$0.204713\pi$
$282$	0.828427i	0.0493321i
$283$	4.97056i	0.295469i	0.989027	+	0.147735i	$0.0471982\pi$
$283$	4.97056i	0.295469i	−0.989027	+	0.147735i	$0.952802\pi$
$284$	−7.65685	−0.454351
$285$	0	0
$286$	−4.82843	−0.285511
$287$	− 30.6274i	− 1.80788i
$288$	− 1.58579i	− 0.0934434i
$289$	3.62742	0.213377
$290$	0	0
$291$	−7.65685	−0.448853
$292$	44.6274i	2.61162i
$293$	26.1421i	1.52724i	0.645666	+	0.763620i	$0.276580\pi$
$293$	26.1421i	1.52724i	−0.645666	+	0.763620i	$0.723420\pi$
$294$	2.41421	0.140800
$295$	0	0
$296$	−16.1421	−0.938243
$297$	− 2.00000i	− 0.116052i
$298$	− 35.7990i	− 2.07378i
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−27.3137	−1.57434
$302$	49.4558i	2.84586i
$303$	3.65685i	0.210081i
$304$	−8.48528	−0.486664
$305$	0	0
$306$	8.82843	0.504688
$307$	− 17.1716i	− 0.980033i	−0.871713	−	0.490017i	$-0.836991\pi$
$307$	− 17.1716i	− 0.980033i	0.871713	−	0.490017i	$-0.163009\pi$
$308$	− 21.6569i	− 1.23401i
$309$	−13.6569	−0.776911
$310$	0	0
$311$	34.6274	1.96354	0.981770	−	0.190071i	$-0.0608718\pi$
$311$	34.6274	1.96354	0.981770	+	0.190071i	$0.0608718\pi$
$312$	4.41421i	0.249906i
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	−24.1421	−1.36242
$315$	0	0
$316$	43.3137	2.43659
$317$	− 8.48528i	− 0.476581i	−0.971194	−	0.238290i	$-0.923413\pi$
$317$	− 8.48528i	− 0.476581i	0.971194	−	0.238290i	$-0.0765870\pi$
$318$	− 4.82843i	− 0.270765i
$319$	4.00000	0.223957
$320$	0	0
$321$	11.3137	0.631470
$322$	− 27.3137i	− 1.52213i
$323$	10.3431i	0.575508i
$324$	−3.82843	−0.212690
$325$	0	0
$326$	−31.7990	−1.76118
$327$	− 17.3137i	− 0.957450i
$328$	47.7990i	2.63926i
$329$	−0.970563	−0.0535089
$330$	0	0
$331$	−2.14214	−0.117742	−0.0588712	−	0.998266i	$-0.518750\pi$
$331$	−2.14214	−0.117742	−0.0588712	+	0.998266i	$0.518750\pi$
$332$	− 29.3137i	− 1.60880i
$333$	− 3.65685i	− 0.200394i
$334$	18.4853	1.01147
$335$	0	0
$336$	−8.48528	−0.462910
$337$	− 13.3137i	− 0.725244i	−0.931936	−	0.362622i	$-0.881882\pi$
$337$	− 13.3137i	− 0.725244i	0.931936	−	0.362622i	$-0.118118\pi$
$338$	2.41421i	0.131316i
$339$	−17.3137	−0.940352
$340$	0	0
$341$	13.6569	0.739560
$342$	− 6.82843i	− 0.369239i
$343$	− 16.9706i	− 0.916324i
$344$	42.6274	2.29832
$345$	0	0
$346$	0.828427	0.0445365
$347$	− 31.3137i	− 1.68101i	−0.541805	−	0.840504i	$-0.682259\pi$
$347$	− 31.3137i	− 1.68101i	0.541805	−	0.840504i	$-0.317741\pi$
$348$	− 7.65685i	− 0.410450i
$349$	7.65685	0.409862	0.204931	−	0.978776i	$-0.434303\pi$
$349$	7.65685	0.409862	0.204931	+	0.978776i	$0.434303\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	− 3.17157i	− 0.169045i
$353$	− 17.4558i	− 0.929081i	−0.885552	−	0.464540i	$-0.846220\pi$
$353$	− 17.4558i	− 0.929081i	0.885552	−	0.464540i	$-0.153780\pi$
$354$	−8.82843	−0.469226
$355$	0	0
$356$	35.1127	1.86097
$357$	10.3431i	0.547417i
$358$	− 1.65685i	− 0.0875675i
$359$	−1.02944	−0.0543316	−0.0271658	−	0.999631i	$-0.508648\pi$
$359$	−1.02944	−0.0543316	−0.0271658	+	0.999631i	$0.508648\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	− 33.7990i	− 1.77644i
$363$	7.00000i	0.367405i
$364$	−10.8284	−0.567564
$365$	0	0
$366$	22.4853	1.17532
$367$	− 24.0000i	− 1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$	− 24.0000i	− 1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$	12.0000i	0.625543i
$369$	−10.8284	−0.563705
$370$	0	0
$371$	5.65685	0.293689
$372$	− 26.1421i	− 1.35541i
$373$	− 10.0000i	− 0.517780i	−0.965907	−	0.258890i	$-0.916643\pi$
$373$	− 10.0000i	− 0.517780i	0.965907	−	0.258890i	$-0.0833568\pi$
$374$	17.6569	0.913014
$375$	0	0
$376$	1.51472	0.0781156
$377$	− 2.00000i	− 0.103005i
$378$	− 6.82843i	− 0.351216i
$379$	16.4853	0.846792	0.423396	−	0.905945i	$-0.360838\pi$
$379$	16.4853	0.846792	0.423396	+	0.905945i	$0.360838\pi$
$380$	0	0
$381$	−5.65685	−0.289809
$382$	46.6274i	2.38567i
$383$	2.97056i	0.151789i	0.997116	+	0.0758943i	$0.0241812\pi$
$383$	2.97056i	0.151789i	−0.997116	+	0.0758943i	$0.975819\pi$
$384$	−20.5563	−1.04901
$385$	0	0
$386$	41.7990	2.12751
$387$	9.65685i	0.490885i
$388$	29.3137i	1.48818i
$389$	−6.97056	−0.353422	−0.176711	−	0.984263i	$-0.556546\pi$
$389$	−6.97056	−0.353422	−0.176711	+	0.984263i	$0.556546\pi$
$390$	0	0
$391$	14.6274	0.739740
$392$	− 4.41421i	− 0.222951i
$393$	8.00000i	0.403547i
$394$	−39.7990	−2.00504
$395$	0	0
$396$	−7.65685	−0.384771
$397$	− 2.97056i	− 0.149088i	−0.997218	−	0.0745441i	$-0.976250\pi$
$397$	− 2.97056i	− 0.149088i	0.997218	−	0.0745441i	$-0.0237502\pi$
$398$	24.9706i	1.25166i
$399$	8.00000	0.400501
$400$	0	0
$401$	−2.14214	−0.106973	−0.0534866	−	0.998569i	$-0.517033\pi$
$401$	−2.14214	−0.106973	−0.0534866	+	0.998569i	$0.517033\pi$
$402$	− 2.82843i	− 0.141069i
$403$	− 6.82843i	− 0.340148i
$404$	14.0000	0.696526
$405$	0	0
$406$	13.6569	0.677778
$407$	− 7.31371i	− 0.362527i
$408$	− 16.1421i	− 0.799155i
$409$	1.02944	0.0509024	0.0254512	−	0.999676i	$-0.491898\pi$
$409$	1.02944	0.0509024	0.0254512	+	0.999676i	$0.491898\pi$
$410$	0	0
$411$	−5.17157	−0.255095
$412$	52.2843i	2.57586i
$413$	− 10.3431i	− 0.508953i
$414$	−9.65685	−0.474608
$415$	0	0
$416$	−1.58579	−0.0777496
$417$	15.3137i	0.749916i
$418$	− 13.6569i	− 0.667979i
$419$	30.6274	1.49625	0.748124	−	0.663559i	$-0.230955\pi$
$419$	30.6274	1.49625	0.748124	+	0.663559i	$0.230955\pi$
$420$	0	0
$421$	14.6863	0.715766	0.357883	−	0.933766i	$-0.383499\pi$
$421$	14.6863	0.715766	0.357883	+	0.933766i	$0.383499\pi$
$422$	28.9706i	1.41026i
$423$	0.343146i	0.0166843i
$424$	−8.82843	−0.428746
$425$	0	0
$426$	−4.82843	−0.233938
$427$	26.3431i	1.27483i
$428$	− 43.3137i	− 2.09365i
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	19.6569	0.946837	0.473419	−	0.880838i	$-0.343020\pi$
$431$	19.6569	0.946837	0.473419	+	0.880838i	$0.343020\pi$
$432$	3.00000i	0.144338i
$433$	− 1.31371i	− 0.0631328i	−0.999502	−	0.0315664i	$-0.989950\pi$
$433$	− 1.31371i	− 0.0631328i	0.999502	−	0.0315664i	$-0.0100496\pi$
$434$	46.6274	2.23819
$435$	0	0
$436$	−66.2843	−3.17444
$437$	− 11.3137i	− 0.541208i
$438$	28.1421i	1.34468i
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	− 8.82843i	− 0.419925i
$443$	− 41.9411i	− 1.99268i	−0.0854611	−	0.996342i	$-0.527236\pi$
$443$	− 41.9411i	− 1.99268i	0.0854611	−	0.996342i	$-0.472764\pi$
$444$	−14.0000	−0.664411
$445$	0	0
$446$	−10.8284	−0.512741
$447$	− 14.8284i	− 0.701361i
$448$	− 27.7990i	− 1.31338i
$449$	−7.79899	−0.368057	−0.184029	−	0.982921i	$-0.558914\pi$
$449$	−7.79899	−0.368057	−0.184029	+	0.982921i	$0.558914\pi$
$450$	0	0
$451$	−21.6569	−1.01978
$452$	66.2843i	3.11775i
$453$	20.4853i	0.962482i
$454$	12.8284	0.602068
$455$	0	0
$456$	−12.4853	−0.584677
$457$	3.65685i	0.171060i	0.996336	+	0.0855302i	$0.0272584\pi$
$457$	3.65685i	0.171060i	−0.996336	+	0.0855302i	$0.972742\pi$
$458$	51.4558i	2.40437i
$459$	3.65685	0.170687
$460$	0	0
$461$	10.8284	0.504330	0.252165	−	0.967684i	$-0.418857\pi$
$461$	10.8284	0.504330	0.252165	+	0.967684i	$0.418857\pi$
$462$	− 13.6569i	− 0.635374i
$463$	7.51472i	0.349239i	0.984636	+	0.174619i	$0.0558695\pi$
$463$	7.51472i	0.349239i	−0.984636	+	0.174619i	$0.944131\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	65.1127	3.01629
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	3.82843i	0.176969i
$469$	3.31371	0.153013
$470$	0	0
$471$	−10.0000	−0.460776
$472$	16.1421i	0.743002i
$473$	19.3137i	0.888045i
$474$	27.3137	1.25456
$475$	0	0
$476$	39.5980	1.81497
$477$	− 2.00000i	− 0.0915737i
$478$	4.82843i	0.220847i
$479$	−2.68629	−0.122740	−0.0613699	−	0.998115i	$-0.519547\pi$
$479$	−2.68629	−0.122740	−0.0613699	+	0.998115i	$0.519547\pi$
$480$	0	0
$481$	−3.65685	−0.166738
$482$	− 28.1421i	− 1.28184i
$483$	− 11.3137i	− 0.514792i
$484$	26.7990	1.21814
$485$	0	0
$486$	−2.41421	−0.109511
$487$	31.7990i	1.44095i	0.693481	+	0.720475i	$0.256076\pi$
$487$	31.7990i	1.44095i	−0.693481	+	0.720475i	$0.743924\pi$
$488$	− 41.1127i	− 1.86108i
$489$	−13.1716	−0.595639
$490$	0	0
$491$	−14.6274	−0.660126	−0.330063	−	0.943959i	$-0.607070\pi$
$491$	−14.6274	−0.660126	−0.330063	+	0.943959i	$0.607070\pi$
$492$	41.4558i	1.86897i
$493$	7.31371i	0.329393i
$494$	−6.82843	−0.307225
$495$	0	0
$496$	−20.4853	−0.919816
$497$	− 5.65685i	− 0.253745i
$498$	− 18.4853i	− 0.828345i
$499$	2.14214	0.0958952	0.0479476	−	0.998850i	$-0.484732\pi$
$499$	2.14214	0.0958952	0.0479476	+	0.998850i	$0.484732\pi$
$500$	0	0
$501$	7.65685	0.342083
$502$	0	0
$503$	− 15.3137i	− 0.682805i	−0.939917	−	0.341402i	$-0.889098\pi$
$503$	− 15.3137i	− 0.682805i	0.939917	−	0.341402i	$-0.110902\pi$
$504$	−12.4853	−0.556139
$505$	0	0
$506$	−19.3137	−0.858599
$507$	1.00000i	0.0444116i
$508$	21.6569i	0.960868i
$509$	27.7990	1.23217	0.616084	−	0.787680i	$-0.288718\pi$
$509$	27.7990	1.23217	0.616084	+	0.787680i	$0.288718\pi$
$510$	0	0
$511$	−32.9706	−1.45853
$512$	31.2426i	1.38074i
$513$	− 2.82843i	− 0.124878i
$514$	−37.7990	−1.66724
$515$	0	0
$516$	36.9706	1.62754
$517$	0.686292i	0.0301831i
$518$	− 24.9706i	− 1.09714i
$519$	0.343146	0.0150624
$520$	0	0
$521$	2.68629	0.117689	0.0588443	−	0.998267i	$-0.481258\pi$
$521$	2.68629	0.117689	0.0588443	+	0.998267i	$0.481258\pi$
$522$	− 4.82843i	− 0.211335i
$523$	− 7.31371i	− 0.319806i	−0.987133	−	0.159903i	$-0.948882\pi$
$523$	− 7.31371i	− 0.319806i	0.987133	−	0.159903i	$-0.0511182\pi$
$524$	30.6274	1.33796
$525$	0	0
$526$	−28.9706	−1.26318
$527$	24.9706i	1.08773i
$528$	6.00000i	0.261116i
$529$	7.00000	0.304348
$530$	0	0
$531$	−3.65685	−0.158694
$532$	− 30.6274i	− 1.32787i
$533$	10.8284i	0.469031i
$534$	22.1421	0.958184
$535$	0	0
$536$	−5.17157	−0.223378
$537$	− 0.686292i	− 0.0296157i
$538$	43.4558i	1.87351i
$539$	2.00000	0.0861461
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	− 28.4853i	− 1.22355i
$543$	− 14.0000i	− 0.600798i
$544$	5.79899	0.248630
$545$	0	0
$546$	−6.82843	−0.292230
$547$	0.686292i	0.0293437i	0.999892	+	0.0146719i	$0.00467036\pi$
$547$	0.686292i	0.0293437i	−0.999892	+	0.0146719i	$0.995330\pi$
$548$	19.7990i	0.845771i
$549$	9.31371	0.397499
$550$	0	0
$551$	5.65685	0.240990
$552$	17.6569i	0.751526i
$553$	32.0000i	1.36078i
$554$	−4.82843	−0.205140
$555$	0	0
$556$	58.6274	2.48636
$557$	31.7990i	1.34737i	0.739020	+	0.673683i	$0.235289\pi$
$557$	31.7990i	1.34737i	−0.739020	+	0.673683i	$0.764711\pi$
$558$	− 16.4853i	− 0.697878i
$559$	9.65685	0.408441
$560$	0	0
$561$	7.31371	0.308785
$562$	− 64.7696i	− 2.73214i
$563$	− 4.00000i	− 0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$	− 4.00000i	− 0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$	1.31371	0.0553171
$565$	0	0
$566$	12.0000	0.504398
$567$	− 2.82843i	− 0.118783i
$568$	8.82843i	0.370433i
$569$	9.02944	0.378534	0.189267	−	0.981926i	$-0.439389\pi$
$569$	9.02944	0.378534	0.189267	+	0.981926i	$0.439389\pi$
$570$	0	0
$571$	20.9706	0.877591	0.438795	−	0.898587i	$-0.355405\pi$
$571$	20.9706	0.877591	0.438795	+	0.898587i	$0.355405\pi$
$572$	7.65685i	0.320149i
$573$	19.3137i	0.806842i
$574$	−73.9411	−3.08624
$575$	0	0
$576$	−9.82843	−0.409518
$577$	35.9411i	1.49625i	0.663559	+	0.748124i	$0.269045\pi$
$577$	35.9411i	1.49625i	−0.663559	+	0.748124i	$0.730955\pi$
$578$	− 8.75736i	− 0.364258i
$579$	17.3137	0.719533
$580$	0	0
$581$	21.6569	0.898478
$582$	18.4853i	0.766240i
$583$	− 4.00000i	− 0.165663i
$584$	51.4558	2.12926
$585$	0	0
$586$	63.1127	2.60716
$587$	22.9706i	0.948097i	0.880499	+	0.474048i	$0.157208\pi$
$587$	22.9706i	0.948097i	−0.880499	+	0.474048i	$0.842792\pi$
$588$	− 3.82843i	− 0.157882i
$589$	19.3137	0.795807
$590$	0	0
$591$	−16.4853	−0.678114
$592$	10.9706i	0.450887i
$593$	3.51472i	0.144332i	0.997393	+	0.0721661i	$0.0229912\pi$
$593$	3.51472i	0.144332i	−0.997393	+	0.0721661i	$0.977009\pi$
$594$	−4.82843	−0.198113
$595$	0	0
$596$	−56.7696	−2.32537
$597$	10.3431i	0.423317i
$598$	9.65685i	0.394898i
$599$	0.686292	0.0280411	0.0140206	−	0.999902i	$-0.495537\pi$
$599$	0.686292	0.0280411	0.0140206	+	0.999902i	$0.495537\pi$
$600$	0	0
$601$	44.6274	1.82039	0.910195	−	0.414180i	$-0.135931\pi$
$601$	44.6274	1.82039	0.910195	+	0.414180i	$0.135931\pi$
$602$	65.9411i	2.68756i
$603$	− 1.17157i	− 0.0477101i
$604$	78.4264	3.19113
$605$	0	0
$606$	8.82843	0.358630
$607$	− 25.9411i	− 1.05292i	−0.850201	−	0.526459i	$-0.823519\pi$
$607$	− 25.9411i	− 1.05292i	0.850201	−	0.526459i	$-0.176481\pi$
$608$	− 4.48528i	− 0.181902i
$609$	5.65685	0.229227
$610$	0	0
$611$	0.343146	0.0138822
$612$	− 14.0000i	− 0.565916i
$613$	36.3431i	1.46789i	0.679211	+	0.733943i	$0.262322\pi$
$613$	36.3431i	1.46789i	−0.679211	+	0.733943i	$0.737678\pi$
$614$	−41.4558	−1.67302
$615$	0	0
$616$	−24.9706	−1.00609
$617$	− 29.1716i	− 1.17440i	−0.809441	−	0.587202i	$-0.800230\pi$
$617$	− 29.1716i	− 1.17440i	0.809441	−	0.587202i	$-0.199770\pi$
$618$	32.9706i	1.32627i
$619$	15.7990	0.635015	0.317508	−	0.948256i	$-0.397154\pi$
$619$	15.7990	0.635015	0.317508	+	0.948256i	$0.397154\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	− 83.5980i	− 3.35197i
$623$	25.9411i	1.03931i
$624$	3.00000	0.120096
$625$	0	0
$626$	−14.4853	−0.578948
$627$	− 5.65685i	− 0.225913i
$628$	38.2843i	1.52771i
$629$	13.3726	0.533200
$630$	0	0
$631$	−19.1127	−0.760865	−0.380432	−	0.924809i	$-0.624225\pi$
$631$	−19.1127	−0.760865	−0.380432	+	0.924809i	$0.624225\pi$
$632$	− 49.9411i	− 1.98655i
$633$	12.0000i	0.476957i
$634$	−20.4853	−0.813574
$635$	0	0
$636$	−7.65685	−0.303614
$637$	− 1.00000i	− 0.0396214i
$638$	− 9.65685i	− 0.382319i
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−26.2843	−1.03817	−0.519083	−	0.854724i	$-0.673727\pi$
$641$	−26.2843	−1.03817	−0.519083	+	0.854724i	$0.673727\pi$
$642$	− 27.3137i	− 1.07799i
$643$	− 17.1716i	− 0.677181i	−0.940934	−	0.338590i	$-0.890050\pi$
$643$	− 17.1716i	− 0.677181i	0.940934	−	0.338590i	$-0.109950\pi$
$644$	−43.3137	−1.70680
$645$	0	0
$646$	24.9706	0.982454
$647$	− 11.3137i	− 0.444788i	−0.974957	−	0.222394i	$-0.928613\pi$
$647$	− 11.3137i	− 0.444788i	0.974957	−	0.222394i	$-0.0713871\pi$
$648$	4.41421i	0.173407i
$649$	−7.31371	−0.287088
$650$	0	0
$651$	19.3137	0.756964
$652$	50.4264i	1.97485i
$653$	2.68629i	0.105123i	0.998618	+	0.0525614i	$0.0167385\pi$
$653$	2.68629i	0.105123i	−0.998618	+	0.0525614i	$0.983261\pi$
$654$	−41.7990	−1.63447
$655$	0	0
$656$	32.4853	1.26834
$657$	11.6569i	0.454777i
$658$	2.34315i	0.0913453i
$659$	24.6863	0.961641	0.480821	−	0.876819i	$-0.340339\pi$
$659$	24.6863	0.961641	0.480821	+	0.876819i	$0.340339\pi$
$660$	0	0
$661$	−1.02944	−0.0400405	−0.0200202	−	0.999800i	$-0.506373\pi$
$661$	−1.02944	−0.0400405	−0.0200202	+	0.999800i	$0.506373\pi$
$662$	5.17157i	0.200999i
$663$	− 3.65685i	− 0.142020i
$664$	−33.7990	−1.31166
$665$	0	0
$666$	−8.82843	−0.342095
$667$	− 8.00000i	− 0.309761i
$668$	− 29.3137i	− 1.13418i
$669$	−4.48528	−0.173411
$670$	0	0
$671$	18.6274	0.719103
$672$	− 4.48528i	− 0.173023i
$673$	28.6274i	1.10351i	0.834008	+	0.551753i	$0.186041\pi$
$673$	28.6274i	1.10351i	−0.834008	+	0.551753i	$0.813959\pi$
$674$	−32.1421	−1.23807
$675$	0	0
$676$	3.82843	0.147247
$677$	49.3137i	1.89528i	0.319341	+	0.947640i	$0.396538\pi$
$677$	49.3137i	1.89528i	−0.319341	+	0.947640i	$0.603462\pi$
$678$	41.7990i	1.60528i
$679$	−21.6569	−0.831114
$680$	0	0
$681$	5.31371	0.203622
$682$	− 32.9706i	− 1.26251i
$683$	19.9411i	0.763026i	0.924363	+	0.381513i	$0.124597\pi$
$683$	19.9411i	0.763026i	−0.924363	+	0.381513i	$0.875403\pi$
$684$	−10.8284	−0.414035
$685$	0	0
$686$	−40.9706	−1.56426
$687$	21.3137i	0.813169i
$688$	− 28.9706i	− 1.10449i
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−34.1421	−1.29883	−0.649414	−	0.760435i	$-0.724986\pi$
$691$	−34.1421	−1.29883	−0.649414	+	0.760435i	$0.724986\pi$
$692$	− 1.31371i	− 0.0499397i
$693$	− 5.65685i	− 0.214886i
$694$	−75.5980	−2.86966
$695$	0	0
$696$	−8.82843	−0.334641
$697$	− 39.5980i	− 1.49988i
$698$	− 18.4853i	− 0.699678i
$699$	26.9706	1.02012
$700$	0	0
$701$	38.9706	1.47190	0.735949	−	0.677037i	$-0.236736\pi$
$701$	38.9706	1.47190	0.735949	+	0.677037i	$0.236736\pi$
$702$	2.41421i	0.0911186i
$703$	− 10.3431i	− 0.390099i
$704$	−19.6569	−0.740846
$705$	0	0
$706$	−42.1421	−1.58604
$707$	10.3431i	0.388994i
$708$	14.0000i	0.526152i
$709$	−40.6274	−1.52579	−0.762897	−	0.646520i	$-0.776224\pi$
$709$	−40.6274	−1.52579	−0.762897	+	0.646520i	$0.776224\pi$
$710$	0	0
$711$	11.3137	0.424297
$712$	− 40.4853i	− 1.51725i
$713$	− 27.3137i	− 1.02291i
$714$	24.9706	0.934500
$715$	0	0
$716$	−2.62742	−0.0981912
$717$	2.00000i	0.0746914i
$718$	2.48528i	0.0927499i
$719$	−37.9411	−1.41497	−0.707483	−	0.706731i	$-0.750169\pi$
$719$	−37.9411	−1.41497	−0.707483	+	0.706731i	$0.750169\pi$
$720$	0	0
$721$	−38.6274	−1.43856
$722$	26.5563i	0.988325i
$723$	− 11.6569i	− 0.433523i
$724$	−53.5980	−1.99195
$725$	0	0
$726$	16.8995	0.627199
$727$	− 21.6569i	− 0.803208i	−0.915813	−	0.401604i	$-0.868453\pi$
$727$	− 21.6569i	− 0.803208i	0.915813	−	0.401604i	$-0.131547\pi$
$728$	12.4853i	0.462735i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−35.3137	−1.30612
$732$	− 35.6569i	− 1.31792i
$733$	− 8.62742i	− 0.318661i	−0.987225	−	0.159330i	$-0.949066\pi$
$733$	− 8.62742i	− 0.318661i	0.987225	−	0.159330i	$-0.0509335\pi$
$734$	−57.9411	−2.13865
$735$	0	0
$736$	−6.34315	−0.233811
$737$	− 2.34315i	− 0.0863109i
$738$	26.1421i	0.962305i
$739$	−10.1421	−0.373084	−0.186542	−	0.982447i	$-0.559728\pi$
$739$	−10.1421	−0.373084	−0.186542	+	0.982447i	$0.559728\pi$
$740$	0	0
$741$	−2.82843	−0.103905
$742$	− 13.6569i	− 0.501359i
$743$	− 2.00000i	− 0.0733729i	−0.999327	−	0.0366864i	$-0.988320\pi$
$743$	− 2.00000i	− 0.0733729i	0.999327	−	0.0366864i	$-0.0116803\pi$
$744$	−30.1421	−1.10506
$745$	0	0
$746$	−24.1421	−0.883906
$747$	− 7.65685i	− 0.280150i
$748$	− 28.0000i	− 1.02378i
$749$	32.0000	1.16925
$750$	0	0
$751$	32.9706	1.20311	0.601556	−	0.798830i	$-0.294547\pi$
$751$	32.9706	1.20311	0.601556	+	0.798830i	$0.294547\pi$
$752$	− 1.02944i	− 0.0375397i
$753$	0	0
$754$	−4.82843	−0.175841
$755$	0	0
$756$	−10.8284	−0.393826
$757$	− 15.9411i	− 0.579390i	−0.957119	−	0.289695i	$-0.906446\pi$
$757$	− 15.9411i	− 0.579390i	0.957119	−	0.289695i	$-0.0935539\pi$
$758$	− 39.7990i	− 1.44556i
$759$	−8.00000	−0.290382
$760$	0	0
$761$	15.5147	0.562408	0.281204	−	0.959648i	$-0.409266\pi$
$761$	15.5147	0.562408	0.281204	+	0.959648i	$0.409266\pi$
$762$	13.6569i	0.494736i
$763$	− 48.9706i	− 1.77285i
$764$	73.9411	2.67510
$765$	0	0
$766$	7.17157	0.259119
$767$	3.65685i	0.132041i
$768$	29.9706i	1.08147i
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	−15.6569	−0.563868
$772$	− 66.2843i	− 2.38562i
$773$	− 5.85786i	− 0.210693i	−0.994436	−	0.105346i	$-0.966405\pi$
$773$	− 5.85786i	− 0.210693i	0.994436	−	0.105346i	$-0.0335951\pi$
$774$	23.3137	0.837994
$775$	0	0
$776$	33.7990	1.21331
$777$	− 10.3431i	− 0.371058i
$778$	16.8284i	0.603328i
$779$	−30.6274	−1.09734
$780$	0	0
$781$	−4.00000	−0.143131
$782$	− 35.3137i	− 1.26282i
$783$	− 2.00000i	− 0.0714742i
$784$	−3.00000	−0.107143
$785$	0	0
$786$	19.3137	0.688897
$787$	32.7696i	1.16811i	0.811715	+	0.584054i	$0.198534\pi$
$787$	32.7696i	1.16811i	−0.811715	+	0.584054i	$0.801466\pi$
$788$	63.1127i	2.24830i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−48.9706	−1.74119
$792$	8.82843i	0.313704i
$793$	− 9.31371i	− 0.330739i
$794$	−7.17157	−0.254510
$795$	0	0
$796$	39.5980	1.40351
$797$	35.6569i	1.26303i	0.775363	+	0.631515i	$0.217567\pi$
$797$	35.6569i	1.26303i	−0.775363	+	0.631515i	$0.782433\pi$
$798$	− 19.3137i	− 0.683698i
$799$	−1.25483	−0.0443928
$800$	0	0
$801$	9.17157	0.324062
$802$	5.17157i	0.182615i
$803$	23.3137i	0.822723i
$804$	−4.48528	−0.158184
$805$	0	0
$806$	−16.4853	−0.580669
$807$	18.0000i	0.633630i
$808$	− 16.1421i	− 0.567878i
$809$	−41.3137	−1.45251	−0.726256	−	0.687424i	$-0.758741\pi$
$809$	−41.3137	−1.45251	−0.726256	+	0.687424i	$0.758741\pi$
$810$	0	0
$811$	−1.85786	−0.0652384	−0.0326192	−	0.999468i	$-0.510385\pi$
$811$	−1.85786	−0.0652384	−0.0326192	+	0.999468i	$0.510385\pi$
$812$	− 21.6569i	− 0.760007i
$813$	− 11.7990i	− 0.413809i
$814$	−17.6569	−0.618872
$815$	0	0
$816$	−10.9706	−0.384047
$817$	27.3137i	0.955586i
$818$	− 2.48528i	− 0.0868958i
$819$	−2.82843	−0.0988332
$820$	0	0
$821$	15.7990	0.551389	0.275694	−	0.961245i	$-0.411092\pi$
$821$	15.7990	0.551389	0.275694	+	0.961245i	$0.411092\pi$
$822$	12.4853i	0.435474i
$823$	− 48.9706i	− 1.70701i	−0.521088	−	0.853503i	$-0.674473\pi$
$823$	− 48.9706i	− 1.70701i	0.521088	−	0.853503i	$-0.325527\pi$
$824$	60.2843	2.10010
$825$	0	0
$826$	−24.9706	−0.868837
$827$	− 26.0000i	− 0.904109i	−0.891990	−	0.452054i	$-0.850691\pi$
$827$	− 26.0000i	− 0.904109i	0.891990	−	0.452054i	$-0.149309\pi$
$828$	15.3137i	0.532188i
$829$	−5.31371	−0.184553	−0.0922764	−	0.995733i	$-0.529414\pi$
$829$	−5.31371	−0.184553	−0.0922764	+	0.995733i	$0.529414\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	9.82843i	0.340739i
$833$	3.65685i	0.126702i
$834$	36.9706	1.28019
$835$	0	0
$836$	−21.6569	−0.749018
$837$	− 6.82843i	− 0.236025i
$838$	− 73.9411i	− 2.55425i
$839$	47.2548	1.63142	0.815709	−	0.578462i	$-0.196347\pi$
$839$	47.2548	1.63142	0.815709	+	0.578462i	$0.196347\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 35.4558i	− 1.22189i
$843$	− 26.8284i	− 0.924020i
$844$	45.9411	1.58136
$845$	0	0
$846$	0.828427	0.0284819
$847$	19.7990i	0.680301i
$848$	6.00000i	0.206041i
$849$	4.97056	0.170589
$850$	0	0
$851$	−14.6274	−0.501421
$852$	7.65685i	0.262320i
$853$	7.65685i	0.262166i	0.991371	+	0.131083i	$0.0418454\pi$
$853$	7.65685i	0.262166i	−0.991371	+	0.131083i	$0.958155\pi$
$854$	63.5980	2.17628
$855$	0	0
$856$	−49.9411	−1.70695
$857$	29.5980i	1.01105i	0.862813	+	0.505524i	$0.168701\pi$
$857$	29.5980i	1.01105i	−0.862813	+	0.505524i	$0.831299\pi$
$858$	4.82843i	0.164840i
$859$	23.3137	0.795453	0.397727	−	0.917504i	$-0.369799\pi$
$859$	23.3137	0.795453	0.397727	+	0.917504i	$0.369799\pi$
$860$	0	0
$861$	−30.6274	−1.04378
$862$	− 47.4558i	− 1.61635i
$863$	39.6569i	1.34994i	0.737847	+	0.674968i	$0.235842\pi$
$863$	39.6569i	1.34994i	−0.737847	+	0.674968i	$0.764158\pi$
$864$	−1.58579	−0.0539496
$865$	0	0
$866$	−3.17157	−0.107774
$867$	− 3.62742i	− 0.123194i
$868$	− 73.9411i	− 2.50973i
$869$	22.6274	0.767583
$870$	0	0
$871$	−1.17157	−0.0396972
$872$	76.4264i	2.58812i
$873$	7.65685i	0.259145i
$874$	−27.3137	−0.923900
$875$	0	0
$876$	44.6274	1.50782
$877$	− 14.2843i	− 0.482346i	−0.970482	−	0.241173i	$-0.922468\pi$
$877$	− 14.2843i	− 0.482346i	0.970482	−	0.241173i	$-0.0775321\pi$
$878$	− 40.9706i	− 1.38269i
$879$	26.1421	0.881752
$880$	0	0
$881$	53.5980	1.80576	0.902881	−	0.429891i	$-0.141448\pi$
$881$	53.5980	1.80576	0.902881	+	0.429891i	$0.141448\pi$
$882$	− 2.41421i	− 0.0812908i
$883$	51.5980i	1.73641i	0.496205	+	0.868205i	$0.334726\pi$
$883$	51.5980i	1.73641i	−0.496205	+	0.868205i	$0.665274\pi$
$884$	−14.0000	−0.470871
$885$	0	0
$886$	−101.255	−3.40172
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	16.1421i	0.541695i
$889$	−16.0000	−0.536623
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	17.1716i	0.574947i
$893$	0.970563i	0.0324786i
$894$	−35.7990	−1.19730
$895$	0	0
$896$	−58.1421	−1.94239
$897$	4.00000i	0.133556i
$898$	18.8284i	0.628313i
$899$	13.6569	0.455482
$900$	0	0
$901$	7.31371	0.243655
$902$	52.2843i	1.74088i
$903$	27.3137i	0.908943i
$904$	76.4264	2.54190
$905$	0	0
$906$	49.4558	1.64306
$907$	20.9706i	0.696316i	0.937436	+	0.348158i	$0.113193\pi$
$907$	20.9706i	0.696316i	−0.937436	+	0.348158i	$0.886807\pi$
$908$	− 20.3431i	− 0.675111i
$909$	3.65685	0.121290
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	8.48528i	0.280976i
$913$	− 15.3137i	− 0.506810i
$914$	8.82843	0.292018
$915$	0	0
$916$	81.5980	2.69607
$917$	22.6274i	0.747223i
$918$	− 8.82843i	− 0.291382i
$919$	19.3137	0.637100	0.318550	−	0.947906i	$-0.396804\pi$
$919$	19.3137	0.637100	0.318550	+	0.947906i	$0.396804\pi$
$920$	0	0
$921$	−17.1716	−0.565823
$922$	− 26.1421i	− 0.860945i
$923$	2.00000i	0.0658308i
$924$	−21.6569	−0.712458
$925$	0	0
$926$	18.1421	0.596188
$927$	13.6569i	0.448550i
$928$	− 3.17157i	− 0.104112i
$929$	27.7990	0.912055	0.456028	−	0.889966i	$-0.349272\pi$
$929$	27.7990	0.912055	0.456028	+	0.889966i	$0.349272\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	− 103.255i	− 3.38222i
$933$	− 34.6274i	− 1.13365i
$934$	−19.3137	−0.631964
$935$	0	0
$936$	4.41421	0.144283
$937$	1.31371i	0.0429170i	0.999770	+	0.0214585i	$0.00683097\pi$
$937$	1.31371i	0.0429170i	−0.999770	+	0.0214585i	$0.993169\pi$
$938$	− 8.00000i	− 0.261209i
$939$	−6.00000	−0.195803
$940$	0	0
$941$	5.85786	0.190961	0.0954805	−	0.995431i	$-0.469561\pi$
$941$	5.85786	0.190961	0.0954805	+	0.995431i	$0.469561\pi$
$942$	24.1421i	0.786593i
$943$	43.3137i	1.41049i
$944$	10.9706	0.357061
$945$	0	0
$946$	46.6274	1.51599
$947$	− 54.9706i	− 1.78630i	−0.449756	−	0.893152i	$-0.648489\pi$
$947$	− 54.9706i	− 1.78630i	0.449756	−	0.893152i	$-0.351511\pi$
$948$	− 43.3137i	− 1.40676i
$949$	11.6569	0.378398
$950$	0	0
$951$	−8.48528	−0.275154
$952$	− 45.6569i	− 1.47975i
$953$	− 51.6569i	− 1.67333i	−0.547715	−	0.836665i	$-0.684502\pi$
$953$	− 51.6569i	− 1.67333i	0.547715	−	0.836665i	$-0.315498\pi$
$954$	−4.82843	−0.156326
$955$	0	0
$956$	7.65685	0.247640
$957$	− 4.00000i	− 0.129302i
$958$	6.48528i	0.209530i
$959$	−14.6274	−0.472344
$960$	0	0
$961$	15.6274	0.504110
$962$	8.82843i	0.284640i
$963$	− 11.3137i	− 0.364579i
$964$	−44.6274	−1.43735
$965$	0	0
$966$	−27.3137	−0.878804
$967$	− 10.1421i	− 0.326149i	−0.986614	−	0.163075i	$-0.947859\pi$
$967$	− 10.1421i	− 0.326149i	0.986614	−	0.163075i	$-0.0521411\pi$
$968$	− 30.8995i	− 0.993147i
$969$	10.3431	0.332270
$970$	0	0
$971$	7.31371	0.234708	0.117354	−	0.993090i	$-0.462559\pi$
$971$	7.31371	0.234708	0.117354	+	0.993090i	$0.462559\pi$
$972$	3.82843i	0.122797i
$973$	43.3137i	1.38857i
$974$	76.7696	2.45986
$975$	0	0
$976$	−27.9411	−0.894374
$977$	13.8579i	0.443352i	0.975120	+	0.221676i	$0.0711528\pi$
$977$	13.8579i	0.443352i	−0.975120	+	0.221676i	$0.928847\pi$
$978$	31.7990i	1.01682i
$979$	18.3431	0.586249
$980$	0	0
$981$	−17.3137	−0.552784
$982$	35.3137i	1.12691i
$983$	− 2.68629i	− 0.0856794i	−0.999082	−	0.0428397i	$-0.986360\pi$
$983$	− 2.68629i	− 0.0856794i	0.999082	−	0.0428397i	$-0.0136405\pi$
$984$	47.7990	1.52378
$985$	0	0
$986$	17.6569	0.562309
$987$	0.970563i	0.0308934i
$988$	10.8284i	0.344498i
$989$	38.6274	1.22828
$990$	0	0
$991$	27.3137	0.867649	0.433824	−	0.900998i	$-0.357164\pi$
$991$	27.3137	0.867649	0.433824	+	0.900998i	$0.357164\pi$
$992$	− 10.8284i	− 0.343803i
$993$	2.14214i	0.0679786i
$994$	−13.6569	−0.433169
$995$	0	0
$996$	−29.3137	−0.928840
$997$	51.2548i	1.62326i	0.584174	+	0.811628i	$0.301419\pi$
$997$	51.2548i	1.62326i	−0.584174	+	0.811628i	$0.698581\pi$
$998$	− 5.17157i	− 0.163703i
$999$	−3.65685	−0.115698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.c.h.274.1		4
3.2	odd	2		2925.2.c.u.2224.4		4
5.2	odd	4		975.2.a.l.1.2		2
5.3	odd	4		39.2.a.b.1.1	✓	2
5.4	even	2	inner	975.2.c.h.274.4		4
15.2	even	4		2925.2.a.v.1.1		2
15.8	even	4		117.2.a.c.1.2		2
15.14	odd	2		2925.2.c.u.2224.1		4
20.3	even	4		624.2.a.k.1.2		2
35.13	even	4		1911.2.a.h.1.1		2
40.3	even	4		2496.2.a.bi.1.1		2
40.13	odd	4		2496.2.a.bf.1.1		2
45.13	odd	12		1053.2.e.m.703.2		4
45.23	even	12		1053.2.e.e.703.1		4
45.38	even	12		1053.2.e.e.352.1		4
45.43	odd	12		1053.2.e.m.352.2		4
55.43	even	4		4719.2.a.p.1.2		2
60.23	odd	4		1872.2.a.w.1.1		2
65.3	odd	12		507.2.e.h.22.2		4
65.8	even	4		507.2.b.e.337.1		4
65.18	even	4		507.2.b.e.337.4		4
65.23	odd	12		507.2.e.d.22.1		4
65.28	even	12		507.2.j.f.316.1		8
65.33	even	12		507.2.j.f.361.1		8
65.38	odd	4		507.2.a.h.1.2		2
65.43	odd	12		507.2.e.d.484.1		4
65.48	odd	12		507.2.e.h.484.2		4
65.58	even	12		507.2.j.f.361.4		8
65.63	even	12		507.2.j.f.316.4		8
105.83	odd	4		5733.2.a.u.1.2		2
120.53	even	4		7488.2.a.cl.1.2		2
120.83	odd	4		7488.2.a.co.1.2		2
195.8	odd	4		1521.2.b.j.1351.4		4
195.38	even	4		1521.2.a.f.1.1		2
195.83	odd	4		1521.2.b.j.1351.1		4
260.103	even	4		8112.2.a.bm.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.1	✓	2	5.3	odd	4
117.2.a.c.1.2		2	15.8	even	4
507.2.a.h.1.2		2	65.38	odd	4
507.2.b.e.337.1		4	65.8	even	4
507.2.b.e.337.4		4	65.18	even	4
507.2.e.d.22.1		4	65.23	odd	12
507.2.e.d.484.1		4	65.43	odd	12
507.2.e.h.22.2		4	65.3	odd	12
507.2.e.h.484.2		4	65.48	odd	12
507.2.j.f.316.1		8	65.28	even	12
507.2.j.f.316.4		8	65.63	even	12
507.2.j.f.361.1		8	65.33	even	12
507.2.j.f.361.4		8	65.58	even	12
624.2.a.k.1.2		2	20.3	even	4
975.2.a.l.1.2		2	5.2	odd	4
975.2.c.h.274.1		4	1.1	even	1	trivial
975.2.c.h.274.4		4	5.4	even	2	inner
1053.2.e.e.352.1		4	45.38	even	12
1053.2.e.e.703.1		4	45.23	even	12
1053.2.e.m.352.2		4	45.43	odd	12
1053.2.e.m.703.2		4	45.13	odd	12
1521.2.a.f.1.1		2	195.38	even	4
1521.2.b.j.1351.1		4	195.83	odd	4
1521.2.b.j.1351.4		4	195.8	odd	4
1872.2.a.w.1.1		2	60.23	odd	4
1911.2.a.h.1.1		2	35.13	even	4
2496.2.a.bf.1.1		2	40.13	odd	4
2496.2.a.bi.1.1		2	40.3	even	4
2925.2.a.v.1.1		2	15.2	even	4
2925.2.c.u.2224.1		4	15.14	odd	2
2925.2.c.u.2224.4		4	3.2	odd	2
4719.2.a.p.1.2		2	55.43	even	4
5733.2.a.u.1.2		2	105.83	odd	4
7488.2.a.cl.1.2		2	120.53	even	4
7488.2.a.co.1.2		2	120.83	odd	4
8112.2.a.bm.1.1		2	260.103	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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} -2.41421 q^{6} -2.82843i q^{7} +4.41421i q^{8} -1.00000 q^{9} -2.00000 q^{11} +3.82843i q^{12} +1.00000i q^{13} -6.82843 q^{14} +3.00000 q^{16} -3.65685i q^{17} +2.41421i q^{18} -2.82843 q^{19} -2.82843 q^{21} +4.82843i q^{22} +4.00000i q^{23} +4.41421 q^{24} +2.41421 q^{26} +1.00000i q^{27} +10.8284i q^{28} -2.00000 q^{29} -6.82843 q^{31} +1.58579i q^{32} +2.00000i q^{33} -8.82843 q^{34} +3.82843 q^{36} +3.65685i q^{37} +6.82843i q^{38} +1.00000 q^{39} +10.8284 q^{41} +6.82843i q^{42} -9.65685i q^{43} +7.65685 q^{44} +9.65685 q^{46} -0.343146i q^{47} -3.00000i q^{48} -1.00000 q^{49} -3.65685 q^{51} -3.82843i q^{52} +2.00000i q^{53} +2.41421 q^{54} +12.4853 q^{56} +2.82843i q^{57} +4.82843i q^{58} +3.65685 q^{59} -9.31371 q^{61} +16.4853i q^{62} +2.82843i q^{63} +9.82843 q^{64} +4.82843 q^{66} +1.17157i q^{67} +14.0000i q^{68} +4.00000 q^{69} +2.00000 q^{71} -4.41421i q^{72} -11.6569i q^{73} +8.82843 q^{74} +10.8284 q^{76} +5.65685i q^{77} -2.41421i q^{78} -11.3137 q^{79} +1.00000 q^{81} -26.1421i q^{82} +7.65685i q^{83} +10.8284 q^{84} -23.3137 q^{86} +2.00000i q^{87} -8.82843i q^{88} -9.17157 q^{89} +2.82843 q^{91} -15.3137i q^{92} +6.82843i q^{93} -0.828427 q^{94} +1.58579 q^{96} -7.65685i q^{97} +2.41421i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 8 q^{11} - 16 q^{14} + 12 q^{16} + 12 q^{24} + 4 q^{26} - 8 q^{29} - 16 q^{31} - 24 q^{34} + 4 q^{36} + 4 q^{39} + 32 q^{41} + 8 q^{44} + 16 q^{46} - 4 q^{49} + 8 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 8 * q^11 - 16 * q^14 + 12 * q^16 + 12 * q^24 + 4 * q^26 - 8 * q^29 - 16 * q^31 - 24 * q^34 + 4 * q^36 + 4 * q^39 + 32 * q^41 + 8 * q^44 + 16 * q^46 - 4 * q^49 + 8 * q^51 + 4 * q^54 + 16 * q^56 - 8 * q^59 + 8 * q^61 + 28 * q^64 + 8 * q^66 + 16 * q^69 + 8 * q^71 + 24 * q^74 + 32 * q^76 + 4 * q^81 + 32 * q^84 - 48 * q^86 - 48 * q^89 + 8 * q^94 + 12 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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