LMFDB - Embedded newform 4788.2.a.t.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4788,2,Mod(1,4788)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4788, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4788.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4788.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,-6,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.56310016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 9x^{4} + 14x^{2} - 4$ x^6 - 9x^4 + 14x^2 - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.608430$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54053 q^{5} -1.00000 q^{7} -5.35744 q^{11} +6.80536 q^{13} -4.03378 q^{17} -1.00000 q^{19} +7.79116 q^{23} +1.45427 q^{25} -2.81692 q^{29} -1.54573 q^{31} -2.54053 q^{35} +5.09146 q^{37} +9.22163 q^{41} +1.54573 q^{43} +5.46425 q^{47} +1.00000 q^{49} -9.39122 q^{53} -13.6107 q^{55} +5.09146 q^{61} +17.2892 q^{65} -3.25963 q^{67} -1.32367 q^{71} +16.5193 q^{73} +5.35744 q^{77} +11.2596 q^{79} +6.19110 q^{83} -10.2479 q^{85} +11.6554 q^{89} -6.80536 q^{91} -2.54053 q^{95} -17.9618 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 6 q^{7} + 4 q^{13} - 6 q^{19} + 14 q^{25} - 4 q^{31} + 20 q^{37} + 4 q^{43} + 6 q^{49} - 8 q^{55} + 20 q^{61} + 12 q^{67} + 36 q^{73} + 36 q^{79} + 28 q^{85} - 4 q^{91} + 8 q^{97}+O(q^{100})$ 6 * q - 6 * q^7 + 4 * q^13 - 6 * q^19 + 14 * q^25 - 4 * q^31 + 20 * q^37 + 4 * q^43 + 6 * q^49 - 8 * q^55 + 20 * q^61 + 12 * q^67 + 36 * q^73 + 36 * q^79 + 28 * q^85 - 4 * q^91 + 8 * q^97

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

)

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.54053	1.13616	0.568079	−	0.822974i	$-0.307687\pi$
$5$	2.54053	1.13616	0.568079	+	0.822974i	$0.307687\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.35744	−1.61533	−0.807665	−	0.589642i	$-0.799269\pi$
$11$	−5.35744	−1.61533	−0.807665	+	0.589642i	$0.799269\pi$
$12$	0	0
$13$	6.80536	1.88747	0.943733	−	0.330707i	$-0.107287\pi$
$13$	6.80536	1.88747	0.943733	+	0.330707i	$0.107287\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.03378	−0.978334	−0.489167	−	0.872190i	$-0.662699\pi$
$17$	−4.03378	−0.978334	−0.489167	+	0.872190i	$0.662699\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.79116	1.62457	0.812285	−	0.583261i	$-0.198224\pi$
$23$	7.79116	1.62457	0.812285	+	0.583261i	$0.198224\pi$
$24$	0	0
$25$	1.45427	0.290854
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.81692	−0.523088	−0.261544	−	0.965192i	$-0.584232\pi$
$29$	−2.81692	−0.523088	−0.261544	+	0.965192i	$0.584232\pi$
$30$	0	0
$31$	−1.54573	−0.277622	−0.138811	−	0.990319i	$-0.544328\pi$
$31$	−1.54573	−0.277622	−0.138811	+	0.990319i	$0.544328\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.54053	−0.429427
$36$	0	0
$37$	5.09146	0.837031	0.418516	−	0.908210i	$-0.362551\pi$
$37$	5.09146	0.837031	0.418516	+	0.908210i	$0.362551\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.22163	1.44018	0.720089	−	0.693882i	$-0.244101\pi$
$41$	9.22163	1.44018	0.720089	+	0.693882i	$0.244101\pi$
$42$	0	0
$43$	1.54573	0.235722	0.117861	−	0.993030i	$-0.462396\pi$
$43$	1.54573	0.235722	0.117861	+	0.993030i	$0.462396\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.46425	0.797042	0.398521	−	0.917159i	$-0.369524\pi$
$47$	5.46425	0.797042	0.398521	+	0.917159i	$0.369524\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.39122	−1.28998	−0.644991	−	0.764190i	$-0.723139\pi$
$53$	−9.39122	−1.28998	−0.644991	+	0.764190i	$0.723139\pi$
$54$	0	0
$55$	−13.6107	−1.83527
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	5.09146	0.651895	0.325947	−	0.945388i	$-0.394317\pi$
$61$	5.09146	0.651895	0.325947	+	0.945388i	$0.394317\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	17.2892	2.14446
$66$	0	0
$67$	−3.25963	−0.398227	−0.199113	−	0.979976i	$-0.563806\pi$
$67$	−3.25963	−0.398227	−0.199113	+	0.979976i	$0.563806\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.32367	−0.157090	−0.0785451	−	0.996911i	$-0.525027\pi$
$71$	−1.32367	−0.157090	−0.0785451	+	0.996911i	$0.525027\pi$
$72$	0	0
$73$	16.5193	1.93343	0.966716	−	0.255851i	$-0.0823557\pi$
$73$	16.5193	1.93343	0.966716	+	0.255851i	$0.0823557\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.35744	0.610537
$78$	0	0
$79$	11.2596	1.26681	0.633403	−	0.773822i	$-0.281657\pi$
$79$	11.2596	1.26681	0.633403	+	0.773822i	$0.281657\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.19110	0.679562	0.339781	−	0.940505i	$-0.389647\pi$
$83$	6.19110	0.679562	0.339781	+	0.940505i	$0.389647\pi$
$84$	0	0
$85$	−10.2479	−1.11154
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.6554	1.23546	0.617732	−	0.786388i	$-0.288052\pi$
$89$	11.6554	1.23546	0.617732	+	0.786388i	$0.288052\pi$
$90$	0	0
$91$	−6.80536	−0.713395
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.54053	−0.260652
$96$	0	0
$97$	−17.9618	−1.82374	−0.911872	−	0.410474i	$-0.865363\pi$
$97$	−17.9618	−1.82374	−0.911872	+	0.410474i	$0.865363\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.60006	0.159212	0.0796058	−	0.996826i	$-0.474634\pi$
$101$	1.60006	0.159212	0.0796058	+	0.996826i	$0.474634\pi$
$102$	0	0
$103$	14.5193	1.43062	0.715312	−	0.698805i	$-0.246284\pi$
$103$	14.5193	1.43062	0.715312	+	0.698805i	$0.246284\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.9790	−1.25473	−0.627364	−	0.778726i	$-0.715866\pi$
$107$	−12.9790	−1.25473	−0.627364	+	0.778726i	$0.715866\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.7654	1.20087	0.600434	−	0.799674i	$-0.294994\pi$
$113$	12.7654	1.20087	0.600434	+	0.799674i	$0.294994\pi$
$114$	0	0
$115$	19.7936	1.84577
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.03378	0.369776
$120$	0	0
$121$	17.7022	1.60929
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.00802	−0.805702
$126$	0	0
$127$	−19.9618	−1.77132	−0.885662	−	0.464331i	$-0.846295\pi$
$127$	−19.9618	−1.77132	−0.885662	+	0.464331i	$0.846295\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.83844	0.772218	0.386109	−	0.922453i	$-0.373819\pi$
$131$	8.83844	0.772218	0.386109	+	0.922453i	$0.373819\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.28116	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$137$	8.28116	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$138$	0	0
$139$	12.7022	1.07738	0.538692	−	0.842503i	$-0.318919\pi$
$139$	12.7022	1.07738	0.538692	+	0.842503i	$0.318919\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−36.4593	−3.04888
$144$	0	0
$145$	−7.15645	−0.594311
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.64733	−0.216878	−0.108439	−	0.994103i	$-0.534585\pi$
$149$	−2.64733	−0.216878	−0.108439	+	0.994103i	$0.534585\pi$
$150$	0	0
$151$	−7.96180	−0.647922	−0.323961	−	0.946070i	$-0.605015\pi$
$151$	−7.96180	−0.647922	−0.323961	+	0.946070i	$0.605015\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.92697	−0.315422
$156$	0	0
$157$	16.5193	1.31838	0.659190	−	0.751976i	$-0.270899\pi$
$157$	16.5193	1.31838	0.659190	+	0.751976i	$0.270899\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.79116	−0.614029
$162$	0	0
$163$	−14.2479	−1.11598	−0.557991	−	0.829847i	$-0.688428\pi$
$163$	−14.2479	−1.11598	−0.557991	+	0.829847i	$0.688428\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.5013	−0.812613	−0.406306	−	0.913737i	$-0.633183\pi$
$167$	−10.5013	−0.812613	−0.406306	+	0.913737i	$0.633183\pi$
$168$	0	0
$169$	33.3129	2.56253
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.6554	0.886140	0.443070	−	0.896487i	$-0.353889\pi$
$173$	11.6554	0.886140	0.443070	+	0.896487i	$0.353889\pi$
$174$	0	0
$175$	−1.45427	−0.109932
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.34519	−0.549005	−0.274502	−	0.961586i	$-0.588513\pi$
$179$	−7.34519	−0.549005	−0.274502	+	0.961586i	$0.588513\pi$
$180$	0	0
$181$	25.0533	1.86219	0.931097	−	0.364771i	$-0.118853\pi$
$181$	25.0533	1.86219	0.931097	+	0.364771i	$0.118853\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.9350	0.950999
$186$	0	0
$187$	21.6107	1.58033
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.3870	−1.47515	−0.737575	−	0.675265i	$-0.764029\pi$
$191$	−20.3870	−1.47515	−0.737575	+	0.675265i	$0.764029\pi$
$192$	0	0
$193$	12.1829	0.876946	0.438473	−	0.898744i	$-0.355519\pi$
$193$	12.1829	0.876946	0.438473	+	0.898744i	$0.355519\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.4298	−1.52681	−0.763404	−	0.645922i	$-0.776473\pi$
$197$	−21.4298	−1.52681	−0.763404	+	0.645922i	$0.776473\pi$
$198$	0	0
$199$	6.18292	0.438296	0.219148	−	0.975692i	$-0.429672\pi$
$199$	6.18292	0.438296	0.219148	+	0.975692i	$0.429672\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.81692	0.197709
$204$	0	0
$205$	23.4278	1.63627
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.35744	0.370582
$210$	0	0
$211$	15.9618	1.09886	0.549428	−	0.835541i	$-0.314846\pi$
$211$	15.9618	1.09886	0.549428	+	0.835541i	$0.314846\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.92697	0.267817
$216$	0	0
$217$	1.54573	0.104931
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−27.4513	−1.84657
$222$	0	0
$223$	3.36281	0.225190	0.112595	−	0.993641i	$-0.464084\pi$
$223$	3.36281	0.225190	0.112595	+	0.993641i	$0.464084\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.0771	−1.59805	−0.799027	−	0.601296i	$-0.794651\pi$
$227$	−24.0771	−1.59805	−0.799027	+	0.601296i	$0.794651\pi$
$228$	0	0
$229$	0.389285	0.0257247	0.0128623	−	0.999917i	$-0.495906\pi$
$229$	0.389285	0.0257247	0.0128623	+	0.999917i	$0.495906\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.52827	−0.296657	−0.148328	−	0.988938i	$-0.547389\pi$
$233$	−4.52827	−0.296657	−0.148328	+	0.988938i	$0.547389\pi$
$234$	0	0
$235$	13.8821	0.905566
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.0208	−1.68315	−0.841573	−	0.540143i	$-0.818370\pi$
$239$	−26.0208	−1.68315	−0.841573	+	0.540143i	$0.818370\pi$
$240$	0	0
$241$	−18.8703	−1.21555	−0.607773	−	0.794111i	$-0.707937\pi$
$241$	−18.8703	−1.21555	−0.607773	+	0.794111i	$0.707937\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.54053	0.162308
$246$	0	0
$247$	−6.80536	−0.433015
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.4803	1.48206	0.741031	−	0.671470i	$-0.234337\pi$
$251$	23.4803	1.48206	0.741031	+	0.671470i	$0.234337\pi$
$252$	0	0
$253$	−41.7407	−2.62421
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.8420	−1.11295	−0.556476	−	0.830864i	$-0.687847\pi$
$257$	−17.8420	−1.11295	−0.556476	+	0.830864i	$0.687847\pi$
$258$	0	0
$259$	−5.09146	−0.316368
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.45199	−0.459509	−0.229755	−	0.973249i	$-0.573792\pi$
$263$	−7.45199	−0.459509	−0.229755	+	0.973249i	$0.573792\pi$
$264$	0	0
$265$	−23.8586	−1.46562
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.70686	−0.104069	−0.0520346	−	0.998645i	$-0.516571\pi$
$269$	−1.70686	−0.104069	−0.0520346	+	0.998645i	$0.516571\pi$
$270$	0	0
$271$	16.7022	1.01459	0.507293	−	0.861774i	$-0.330646\pi$
$271$	16.7022	1.01459	0.507293	+	0.861774i	$0.330646\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.79116	−0.469825
$276$	0	0
$277$	29.1564	1.75184	0.875921	−	0.482455i	$-0.160255\pi$
$277$	29.1564	1.75184	0.875921	+	0.482455i	$0.160255\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.0005	1.13348	0.566738	−	0.823898i	$-0.308205\pi$
$281$	19.0005	1.13348	0.566738	+	0.823898i	$0.308205\pi$
$282$	0	0
$283$	−15.0915	−0.897094	−0.448547	−	0.893759i	$-0.648058\pi$
$283$	−15.0915	−0.897094	−0.448547	+	0.893759i	$0.648058\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.22163	−0.544336
$288$	0	0
$289$	−0.728656	−0.0428621
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.2081	0.713207	0.356603	−	0.934256i	$-0.383935\pi$
$293$	12.2081	0.713207	0.356603	+	0.934256i	$0.383935\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	53.0216	3.06632
$300$	0	0
$301$	−1.54573	−0.0890945
$302$	0	0
$303$	0	0
$304$	0	0
$305$	12.9350	0.740655
$306$	0	0
$307$	18.5842	1.06066	0.530329	−	0.847792i	$-0.322068\pi$
$307$	18.5842	1.06066	0.530329	+	0.847792i	$0.322068\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.75738	−0.213062	−0.106531	−	0.994309i	$-0.533974\pi$
$311$	−3.75738	−0.213062	−0.106531	+	0.994309i	$0.533974\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.23714	−0.462644	−0.231322	−	0.972877i	$-0.574305\pi$
$317$	−8.23714	−0.462644	−0.231322	+	0.972877i	$0.574305\pi$
$318$	0	0
$319$	15.0915	0.844960
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.03378	0.224445
$324$	0	0
$325$	9.89682	0.548977
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.46425	−0.301254
$330$	0	0
$331$	−4.16816	−0.229103	−0.114552	−	0.993417i	$-0.536543\pi$
$331$	−4.16816	−0.229103	−0.114552	+	0.993417i	$0.536543\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.28116	−0.452448
$336$	0	0
$337$	−8.51925	−0.464073	−0.232037	−	0.972707i	$-0.574539\pi$
$337$	−8.51925	−0.464073	−0.232037	+	0.972707i	$0.574539\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.28116	0.448450
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.2724	1.03460	0.517299	−	0.855805i	$-0.326937\pi$
$347$	19.2724	1.03460	0.517299	+	0.855805i	$0.326937\pi$
$348$	0	0
$349$	−11.6107	−0.621507	−0.310754	−	0.950490i	$-0.600581\pi$
$349$	−11.6107	−0.621507	−0.310754	+	0.950490i	$0.600581\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.4420	1.03479	0.517397	−	0.855745i	$-0.326901\pi$
$353$	19.4420	1.03479	0.517397	+	0.855745i	$0.326901\pi$
$354$	0	0
$355$	−3.36281	−0.178479
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.34394	0.440376	0.220188	−	0.975457i	$-0.429333\pi$
$359$	8.34394	0.440376	0.220188	+	0.975457i	$0.429333\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	41.9676	2.19668
$366$	0	0
$367$	−30.3129	−1.58232	−0.791160	−	0.611609i	$-0.790523\pi$
$367$	−30.3129	−1.58232	−0.791160	+	0.611609i	$0.790523\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.39122	0.487568
$372$	0	0
$373$	−10.3363	−0.535195	−0.267597	−	0.963531i	$-0.586230\pi$
$373$	−10.3363	−0.535195	−0.267597	+	0.963531i	$0.586230\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.1701	−0.987311
$378$	0	0
$379$	19.9618	1.02537	0.512685	−	0.858577i	$-0.328651\pi$
$379$	19.9618	1.02537	0.512685	+	0.858577i	$0.328651\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	14.6904	0.750643	0.375321	−	0.926895i	$-0.377532\pi$
$383$	14.6904	0.750643	0.375321	+	0.926895i	$0.377532\pi$
$384$	0	0
$385$	13.6107	0.693666
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−34.7920	−1.76402	−0.882012	−	0.471228i	$-0.843811\pi$
$389$	−34.7920	−1.76402	−0.882012	+	0.471228i	$0.843811\pi$
$390$	0	0
$391$	−31.4278	−1.58937
$392$	0	0
$393$	0	0
$394$	0	0
$395$	28.6054	1.43929
$396$	0	0
$397$	−33.9236	−1.70258	−0.851289	−	0.524698i	$-0.824178\pi$
$397$	−33.9236	−1.70258	−0.851289	+	0.524698i	$0.824178\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.11005	−0.0554334	−0.0277167	−	0.999616i	$-0.508824\pi$
$401$	−1.11005	−0.0554334	−0.0277167	+	0.999616i	$0.508824\pi$
$402$	0	0
$403$	−10.5193	−0.524001
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−27.2772	−1.35208
$408$	0	0
$409$	−17.8968	−0.884941	−0.442470	−	0.896783i	$-0.645898\pi$
$409$	−17.8968	−0.884941	−0.442470	+	0.896783i	$0.645898\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	15.7287	0.772089
$416$	0	0
$417$	0	0
$418$	0	0
$419$	23.1411	1.13052	0.565259	−	0.824914i	$-0.308776\pi$
$419$	23.1411	1.13052	0.565259	+	0.824914i	$0.308776\pi$
$420$	0	0
$421$	6.33633	0.308814	0.154407	−	0.988007i	$-0.450653\pi$
$421$	6.33633	0.308814	0.154407	+	0.988007i	$0.450653\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.86619	−0.284552
$426$	0	0
$427$	−5.09146	−0.246393
$428$	0	0
$429$	0	0
$430$	0	0
$431$	34.1952	1.64712	0.823562	−	0.567227i	$-0.191984\pi$
$431$	34.1952	1.64712	0.823562	+	0.567227i	$0.191984\pi$
$432$	0	0
$433$	31.2362	1.50112	0.750558	−	0.660805i	$-0.229785\pi$
$433$	31.2362	1.50112	0.750558	+	0.660805i	$0.229785\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.79116	−0.372702
$438$	0	0
$439$	−9.33937	−0.445744	−0.222872	−	0.974848i	$-0.571543\pi$
$439$	−9.33937	−0.445744	−0.222872	+	0.974848i	$0.571543\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.3020	−1.62974	−0.814868	−	0.579647i	$-0.803191\pi$
$443$	−34.3020	−1.62974	−0.814868	+	0.579647i	$0.803191\pi$
$444$	0	0
$445$	29.6107	1.40368
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.8545	−1.26734	−0.633671	−	0.773603i	$-0.718452\pi$
$449$	−26.8545	−1.26734	−0.633671	+	0.773603i	$0.718452\pi$
$450$	0	0
$451$	−49.4044	−2.32636
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−17.2892	−0.810529
$456$	0	0
$457$	9.36281	0.437974	0.218987	−	0.975728i	$-0.429725\pi$
$457$	9.36281	0.437974	0.218987	+	0.975728i	$0.429725\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−25.1244	−1.17016	−0.585079	−	0.810976i	$-0.698937\pi$
$461$	−25.1244	−1.17016	−0.585079	+	0.810976i	$0.698937\pi$
$462$	0	0
$463$	42.4429	1.97249	0.986244	−	0.165299i	$-0.0528589\pi$
$463$	42.4429	1.97249	0.986244	+	0.165299i	$0.0528589\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.2587	0.659812	0.329906	−	0.944014i	$-0.392983\pi$
$467$	14.2587	0.659812	0.329906	+	0.944014i	$0.392983\pi$
$468$	0	0
$469$	3.25963	0.150516
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.28116	−0.380768
$474$	0	0
$475$	−1.45427	−0.0667264
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.2602	0.971402	0.485701	−	0.874125i	$-0.338564\pi$
$479$	21.2602	0.971402	0.485701	+	0.874125i	$0.338564\pi$
$480$	0	0
$481$	34.6492	1.57987
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−45.6324	−2.07206
$486$	0	0
$487$	10.3511	0.469053	0.234526	−	0.972110i	$-0.424646\pi$
$487$	10.3511	0.469053	0.234526	+	0.972110i	$0.424646\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.2994	0.600195	0.300098	−	0.953909i	$-0.402981\pi$
$491$	13.2994	0.600195	0.300098	+	0.953909i	$0.402981\pi$
$492$	0	0
$493$	11.3628	0.511755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.32367	0.0593745
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.2252	0.812620	0.406310	−	0.913735i	$-0.366815\pi$
$503$	18.2252	0.812620	0.406310	+	0.913735i	$0.366815\pi$
$504$	0	0
$505$	4.06498	0.180889
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.03502	0.134525	0.0672624	−	0.997735i	$-0.478574\pi$
$509$	3.03502	0.134525	0.0672624	+	0.997735i	$0.478574\pi$
$510$	0	0
$511$	−16.5193	−0.730769
$512$	0	0
$513$	0	0
$514$	0	0
$515$	36.8865	1.62541
$516$	0	0
$517$	−29.2744	−1.28749
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.9906	1.35772	0.678861	−	0.734267i	$-0.262474\pi$
$521$	30.9906	1.35772	0.678861	+	0.734267i	$0.262474\pi$
$522$	0	0
$523$	5.88206	0.257205	0.128602	−	0.991696i	$-0.458951\pi$
$523$	5.88206	0.257205	0.128602	+	0.991696i	$0.458951\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.23513	0.271607
$528$	0	0
$529$	37.7022	1.63923
$530$	0	0
$531$	0	0
$532$	0	0
$533$	62.7565	2.71829
$534$	0	0
$535$	−32.9735	−1.42557
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.35744	−0.230761
$540$	0	0
$541$	−10.1300	−0.435521	−0.217761	−	0.976002i	$-0.569875\pi$
$541$	−10.1300	−0.435521	−0.217761	+	0.976002i	$0.569875\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	25.4053	1.08824
$546$	0	0
$547$	−5.77888	−0.247087	−0.123544	−	0.992339i	$-0.539426\pi$
$547$	−5.77888	−0.247087	−0.123544	+	0.992339i	$0.539426\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.81692	0.120005
$552$	0	0
$553$	−11.2596	−0.478808
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.6729	−1.55388	−0.776941	−	0.629574i	$-0.783230\pi$
$557$	−36.6729	−1.55388	−0.776941	+	0.629574i	$0.783230\pi$
$558$	0	0
$559$	10.5193	0.444917
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.3822	0.521848	0.260924	−	0.965359i	$-0.415973\pi$
$563$	12.3822	0.521848	0.260924	+	0.965359i	$0.415973\pi$
$564$	0	0
$565$	32.4308	1.36438
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.4818	−1.27786	−0.638932	−	0.769263i	$-0.720624\pi$
$569$	−30.4818	−1.27786	−0.638932	+	0.769263i	$0.720624\pi$
$570$	0	0
$571$	12.3363	0.516259	0.258130	−	0.966110i	$-0.416894\pi$
$571$	12.3363	0.516259	0.258130	+	0.966110i	$0.416894\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.3304	0.472512
$576$	0	0
$577$	10.3658	0.431536	0.215768	−	0.976445i	$-0.430775\pi$
$577$	10.3658	0.431536	0.215768	+	0.976445i	$0.430775\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.19110	−0.256850
$582$	0	0
$583$	50.3129	2.08375
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.2126	0.504069	0.252034	−	0.967718i	$-0.418900\pi$
$587$	12.2126	0.504069	0.252034	+	0.967718i	$0.418900\pi$
$588$	0	0
$589$	1.54573	0.0636907
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.29342	−0.258440	−0.129220	−	0.991616i	$-0.541247\pi$
$593$	−6.29342	−0.258440	−0.129220	+	0.991616i	$0.541247\pi$
$594$	0	0
$595$	10.2479	0.420123
$596$	0	0
$597$	0	0
$598$	0	0
$599$	33.9816	1.38845	0.694224	−	0.719759i	$-0.255748\pi$
$599$	33.9816	1.38845	0.694224	+	0.719759i	$0.255748\pi$
$600$	0	0
$601$	−10.7404	−0.438109	−0.219054	−	0.975713i	$-0.570297\pi$
$601$	−10.7404	−0.438109	−0.219054	+	0.975713i	$0.570297\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	44.9728	1.82841
$606$	0	0
$607$	−17.0385	−0.691572	−0.345786	−	0.938313i	$-0.612388\pi$
$607$	−17.0385	−0.691572	−0.345786	+	0.938313i	$0.612388\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	37.1862	1.50439
$612$	0	0
$613$	34.9735	1.41257	0.706284	−	0.707929i	$-0.250370\pi$
$613$	34.9735	1.41257	0.706284	+	0.707929i	$0.250370\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.2677	0.453619	0.226810	−	0.973939i	$-0.427171\pi$
$617$	11.2677	0.453619	0.226810	+	0.973939i	$0.427171\pi$
$618$	0	0
$619$	−9.48075	−0.381063	−0.190532	−	0.981681i	$-0.561021\pi$
$619$	−9.48075	−0.381063	−0.190532	+	0.981681i	$0.561021\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.6554	−0.466962
$624$	0	0
$625$	−30.1564	−1.20626
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.5378	−0.818896
$630$	0	0
$631$	−11.9350	−0.475125	−0.237563	−	0.971372i	$-0.576349\pi$
$631$	−11.9350	−0.475125	−0.237563	+	0.971372i	$0.576349\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−50.7135	−2.01250
$636$	0	0
$637$	6.80536	0.269638
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−30.7814	−1.21579	−0.607897	−	0.794016i	$-0.707987\pi$
$641$	−30.7814	−1.21579	−0.607897	+	0.794016i	$0.707987\pi$
$642$	0	0
$643$	−15.0915	−0.595149	−0.297575	−	0.954699i	$-0.596178\pi$
$643$	−15.0915	−0.595149	−0.297575	+	0.954699i	$0.596178\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−47.1302	−1.85288	−0.926439	−	0.376445i	$-0.877146\pi$
$647$	−47.1302	−1.85288	−0.926439	+	0.376445i	$0.877146\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	34.5784	1.35316	0.676578	−	0.736371i	$-0.263462\pi$
$653$	34.5784	1.35316	0.676578	+	0.736371i	$0.263462\pi$
$654$	0	0
$655$	22.4543	0.877361
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.3532	0.637030	0.318515	−	0.947918i	$-0.396816\pi$
$659$	16.3532	0.637030	0.318515	+	0.947918i	$0.396816\pi$
$660$	0	0
$661$	−19.1712	−0.745674	−0.372837	−	0.927897i	$-0.621615\pi$
$661$	−19.1712	−0.745674	−0.372837	+	0.927897i	$0.621615\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.54053	0.0985173
$666$	0	0
$667$	−21.9470	−0.849793
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−27.2772	−1.05303
$672$	0	0
$673$	−0.519253	−0.0200157	−0.0100079	−	0.999950i	$-0.503186\pi$
$673$	−0.519253	−0.0200157	−0.0100079	+	0.999950i	$0.503186\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−51.7420	−1.98861	−0.994303	−	0.106589i	$-0.966007\pi$
$677$	−51.7420	−1.98861	−0.994303	+	0.106589i	$0.966007\pi$
$678$	0	0
$679$	17.9618	0.689311
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23.0531	−0.882101	−0.441050	−	0.897482i	$-0.645394\pi$
$683$	−23.0531	−0.882101	−0.441050	+	0.897482i	$0.645394\pi$
$684$	0	0
$685$	21.0385	0.803840
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−63.9106	−2.43480
$690$	0	0
$691$	−24.3363	−0.925798	−0.462899	−	0.886411i	$-0.653191\pi$
$691$	−24.3363	−0.925798	−0.462899	+	0.886411i	$0.653191\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	32.2702	1.22408
$696$	0	0
$697$	−37.1980	−1.40897
$698$	0	0
$699$	0	0
$700$	0	0
$701$	34.5784	1.30601	0.653003	−	0.757355i	$-0.273509\pi$
$701$	34.5784	1.30601	0.653003	+	0.757355i	$0.273509\pi$
$702$	0	0
$703$	−5.09146	−0.192028
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.60006	−0.0601763
$708$	0	0
$709$	14.7672	0.554592	0.277296	−	0.960784i	$-0.410562\pi$
$709$	14.7672	0.554592	0.277296	+	0.960784i	$0.410562\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.0430	−0.451015
$714$	0	0
$715$	−92.6258	−3.46401
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.5764	−1.21489	−0.607447	−	0.794360i	$-0.707806\pi$
$719$	−32.5764	−1.21489	−0.607447	+	0.794360i	$0.707806\pi$
$720$	0	0
$721$	−14.5193	−0.540725
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.09655	−0.152142
$726$	0	0
$727$	−23.4278	−0.868889	−0.434444	−	0.900699i	$-0.643055\pi$
$727$	−23.4278	−0.868889	−0.434444	+	0.900699i	$0.643055\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.23513	−0.230615
$732$	0	0
$733$	−23.0385	−0.850947	−0.425473	−	0.904971i	$-0.639892\pi$
$733$	−23.0385	−0.850947	−0.425473	+	0.904971i	$0.639892\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	17.4633	0.643267
$738$	0	0
$739$	23.8586	0.877654	0.438827	−	0.898572i	$-0.355394\pi$
$739$	23.8586	0.877654	0.438827	+	0.898572i	$0.355394\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	19.1656	0.703119	0.351559	−	0.936166i	$-0.385652\pi$
$743$	19.1656	0.703119	0.351559	+	0.936166i	$0.385652\pi$
$744$	0	0
$745$	−6.72561	−0.246407
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12.9790	0.474243
$750$	0	0
$751$	−31.6255	−1.15403	−0.577015	−	0.816734i	$-0.695783\pi$
$751$	−31.6255	−1.15403	−0.577015	+	0.816734i	$0.695783\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−20.2272	−0.736142
$756$	0	0
$757$	41.0151	1.49072	0.745359	−	0.666663i	$-0.232278\pi$
$757$	41.0151	1.49072	0.745359	+	0.666663i	$0.232278\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.77413	−0.0643122	−0.0321561	−	0.999483i	$-0.510237\pi$
$761$	−1.77413	−0.0643122	−0.0321561	+	0.999483i	$0.510237\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−11.8171	−0.426135	−0.213067	−	0.977038i	$-0.568345\pi$
$769$	−11.8171	−0.426135	−0.213067	+	0.977038i	$0.568345\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16.9500	0.609650	0.304825	−	0.952408i	$-0.401402\pi$
$773$	16.9500	0.609650	0.304825	+	0.952408i	$0.401402\pi$
$774$	0	0
$775$	−2.24791	−0.0807473
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.22163	−0.330399
$780$	0	0
$781$	7.09146	0.253752
$782$	0	0
$783$	0	0
$784$	0	0
$785$	41.9676	1.49789
$786$	0	0
$787$	−36.0650	−1.28558	−0.642789	−	0.766043i	$-0.722223\pi$
$787$	−36.0650	−1.28558	−0.642789	+	0.766043i	$0.722223\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.7654	−0.453886
$792$	0	0
$793$	34.6492	1.23043
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.03253	0.178261	0.0891307	−	0.996020i	$-0.471591\pi$
$797$	5.03253	0.178261	0.0891307	+	0.996020i	$0.471591\pi$
$798$	0	0
$799$	−22.0415	−0.779774
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−88.5009	−3.12313
$804$	0	0
$805$	−19.7936	−0.697634
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.1716	−0.920146	−0.460073	−	0.887881i	$-0.652177\pi$
$809$	−26.1716	−0.920146	−0.460073	+	0.887881i	$0.652177\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−36.1972	−1.26793
$816$	0	0
$817$	−1.54573	−0.0540783
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.213613	0.00745514	0.00372757	−	0.999993i	$-0.498813\pi$
$821$	0.213613	0.00745514	0.00372757	+	0.999993i	$0.498813\pi$
$822$	0	0
$823$	−29.6757	−1.03443	−0.517215	−	0.855856i	$-0.673031\pi$
$823$	−29.6757	−1.03443	−0.517215	+	0.855856i	$0.673031\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.06555	−0.210920	−0.105460	−	0.994424i	$-0.533631\pi$
$827$	−6.06555	−0.210920	−0.105460	+	0.994424i	$0.533631\pi$
$828$	0	0
$829$	−11.0767	−0.384710	−0.192355	−	0.981325i	$-0.561612\pi$
$829$	−11.0767	−0.384710	−0.192355	+	0.981325i	$0.561612\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.03378	−0.139762
$834$	0	0
$835$	−26.6787	−0.923256
$836$	0	0
$837$	0	0
$838$	0	0
$839$	23.7379	0.819524	0.409762	−	0.912192i	$-0.365612\pi$
$839$	23.7379	0.819524	0.409762	+	0.912192i	$0.365612\pi$
$840$	0	0
$841$	−21.0650	−0.726379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	84.6322	2.91144
$846$	0	0
$847$	−17.7022	−0.608254
$848$	0	0
$849$	0	0
$850$	0	0
$851$	39.6684	1.35982
$852$	0	0
$853$	−6.90854	−0.236544	−0.118272	−	0.992981i	$-0.537735\pi$
$853$	−6.90854	−0.236544	−0.118272	+	0.992981i	$0.537735\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.8470	−1.25867	−0.629335	−	0.777134i	$-0.716672\pi$
$857$	−36.8470	−1.25867	−0.629335	+	0.777134i	$0.716672\pi$
$858$	0	0
$859$	−24.9085	−0.849868	−0.424934	−	0.905224i	$-0.639703\pi$
$859$	−24.9085	−0.849868	−0.424934	+	0.905224i	$0.639703\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.7949	−0.946151	−0.473075	−	0.881022i	$-0.656856\pi$
$863$	−27.7949	−0.946151	−0.473075	+	0.881022i	$0.656856\pi$
$864$	0	0
$865$	29.6107	1.00679
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−60.3228	−2.04631
$870$	0	0
$871$	−22.1829	−0.751640
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.00802	0.304527
$876$	0	0
$877$	−19.8171	−0.669175	−0.334588	−	0.942365i	$-0.608597\pi$
$877$	−19.8171	−0.669175	−0.334588	+	0.942365i	$0.608597\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18.1228	−0.610574	−0.305287	−	0.952260i	$-0.598752\pi$
$881$	−18.1228	−0.610574	−0.305287	+	0.952260i	$0.598752\pi$
$882$	0	0
$883$	11.2214	0.377631	0.188816	−	0.982013i	$-0.439535\pi$
$883$	11.2214	0.377631	0.188816	+	0.982013i	$0.439535\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−34.2392	−1.14964	−0.574820	−	0.818280i	$-0.694928\pi$
$887$	−34.2392	−1.14964	−0.574820	+	0.818280i	$0.694928\pi$
$888$	0	0
$889$	19.9618	0.669497
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.46425	−0.182854
$894$	0	0
$895$	−18.6606	−0.623756
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.35419	0.145221
$900$	0	0
$901$	37.8821	1.26203
$902$	0	0
$903$	0	0
$904$	0	0
$905$	63.6485	2.11575
$906$	0	0
$907$	1.31258	0.0435836	0.0217918	−	0.999763i	$-0.493063\pi$
$907$	1.31258	0.0435836	0.0217918	+	0.999763i	$0.493063\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.4208	−0.809096	−0.404548	−	0.914517i	$-0.632571\pi$
$911$	−24.4208	−0.809096	−0.404548	+	0.914517i	$0.632571\pi$
$912$	0	0
$913$	−33.1685	−1.09772
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.83844	−0.291871
$918$	0	0
$919$	−52.6258	−1.73596	−0.867982	−	0.496595i	$-0.834583\pi$
$919$	−52.6258	−1.73596	−0.867982	+	0.496595i	$0.834583\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.00802	−0.296503
$924$	0	0
$925$	7.40436	0.243454
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22.0498	−0.723431	−0.361715	−	0.932289i	$-0.617809\pi$
$929$	−22.0498	−0.723431	−0.361715	+	0.932289i	$0.617809\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	54.9026	1.79551
$936$	0	0
$937$	−40.8851	−1.33566	−0.667829	−	0.744315i	$-0.732776\pi$
$937$	−40.8851	−1.33566	−0.667829	+	0.744315i	$0.732776\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.83242	0.0597351	0.0298676	−	0.999554i	$-0.490491\pi$
$941$	1.83242	0.0597351	0.0298676	+	0.999554i	$0.490491\pi$
$942$	0	0
$943$	71.8472	2.33967
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−60.8128	−1.97615	−0.988075	−	0.153973i	$-0.950793\pi$
$947$	−60.8128	−1.97615	−0.988075	+	0.153973i	$0.950793\pi$
$948$	0	0
$949$	112.419	3.64929
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.1196	−0.554558	−0.277279	−	0.960789i	$-0.589433\pi$
$953$	−17.1196	−0.554558	−0.277279	+	0.960789i	$0.589433\pi$
$954$	0	0
$955$	−51.7936	−1.67600
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.28116	−0.267413
$960$	0	0
$961$	−28.6107	−0.922926
$962$	0	0
$963$	0	0
$964$	0	0
$965$	30.9510	0.996349
$966$	0	0
$967$	20.1950	0.649426	0.324713	−	0.945813i	$-0.394732\pi$
$967$	20.1950	0.649426	0.324713	+	0.945813i	$0.394732\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.40023	0.205393	0.102697	−	0.994713i	$-0.467253\pi$
$971$	6.40023	0.205393	0.102697	+	0.994713i	$0.467253\pi$
$972$	0	0
$973$	−12.7022	−0.407213
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.7225	1.52678	0.763389	−	0.645939i	$-0.223534\pi$
$977$	47.7225	1.52678	0.763389	+	0.645939i	$0.223534\pi$
$978$	0	0
$979$	−62.4429	−1.99568
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4.44021	0.141621	0.0708104	−	0.997490i	$-0.477441\pi$
$983$	4.44021	0.141621	0.0708104	+	0.997490i	$0.477441\pi$
$984$	0	0
$985$	−54.4429	−1.73469
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.0430	0.382946
$990$	0	0
$991$	−20.1682	−0.640663	−0.320331	−	0.947306i	$-0.603794\pi$
$991$	−20.1682	−0.640663	−0.320331	+	0.947306i	$0.603794\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.7079	0.497973
$996$	0	0
$997$	5.29782	0.167784	0.0838919	−	0.996475i	$-0.473265\pi$
$997$	5.29782	0.167784	0.0838919	+	0.996475i	$0.473265\pi$
$998$	0	0
$999$	0	0

Display

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a_n

instead) (See

a_n

instead) Display

a_n

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up to: 50 250 1000 (See only

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.t.1.5	yes	6
3.2	odd	2	inner	4788.2.a.t.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.a.t.1.2	✓	6	3.2	odd	2	inner
4788.2.a.t.1.5	yes	6	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4788.2.a.t.1.5

Level	$4788$
Weight	$2$
Character	4788.1
Self dual	yes
Analytic conductor	$38.232$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$