LMFDB - Embedded newform 4788.2.a.q.1.3

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 = [4,0,0,0,0,0,-4,0,0,0,0,0,-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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 6x^{2} + 4$ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.874032$ 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)

$f(q)$	$=$	$q+0.874032 q^{5} -1.00000 q^{7} +2.82843 q^{11} +1.23607 q^{13} -3.70246 q^{17} +1.00000 q^{19} -2.82843 q^{23} -4.23607 q^{25} -7.19859 q^{29} -5.70820 q^{31} -0.874032 q^{35} -4.47214 q^{37} +8.48528 q^{41} -0.763932 q^{43} -5.45052 q^{47} +1.00000 q^{49} +4.37016 q^{53} +2.47214 q^{55} -14.8098 q^{59} -12.4721 q^{61} +1.08036 q^{65} +11.4164 q^{67} -3.03476 q^{71} +6.94427 q^{73} -2.82843 q^{77} -1.52786 q^{79} +4.78282 q^{83} -3.23607 q^{85} +8.48528 q^{89} -1.23607 q^{91} +0.874032 q^{95} -3.52786 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{7} - 4 q^{13} + 4 q^{19} - 8 q^{25} + 4 q^{31} - 12 q^{43} + 4 q^{49} - 8 q^{55} - 32 q^{61} - 8 q^{67} - 8 q^{73} - 24 q^{79} - 4 q^{85} + 4 q^{91} - 32 q^{97}+O(q^{100})$ 4 * q - 4 * q^7 - 4 * q^13 + 4 * q^19 - 8 * q^25 + 4 * q^31 - 12 * q^43 + 4 * q^49 - 8 * q^55 - 32 * q^61 - 8 * q^67 - 8 * q^73 - 24 * q^79 - 4 * q^85 + 4 * q^91 - 32 * 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
$5$	0.874032	0.390879	0.195440	−	0.980716i	$-0.437387\pi$
$5$	0.874032	0.390879	0.195440	+	0.980716i	$0.437387\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.70246	−0.897978	−0.448989	−	0.893537i	$-0.648216\pi$
$17$	−3.70246	−0.897978	−0.448989	+	0.893537i	$0.648216\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	−4.23607	−0.847214
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.19859	−1.33674	−0.668372	−	0.743827i	$-0.733009\pi$
$29$	−7.19859	−1.33674	−0.668372	+	0.743827i	$0.733009\pi$
$30$	0	0
$31$	−5.70820	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$31$	−5.70820	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.874032	−0.147738
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.48528	1.32518	0.662589	−	0.748983i	$-0.269458\pi$
$41$	8.48528	1.32518	0.662589	+	0.748983i	$0.269458\pi$
$42$	0	0
$43$	−0.763932	−0.116499	−0.0582493	−	0.998302i	$-0.518552\pi$
$43$	−0.763932	−0.116499	−0.0582493	+	0.998302i	$0.518552\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.45052	−0.795041	−0.397520	−	0.917593i	$-0.630129\pi$
$47$	−5.45052	−0.795041	−0.397520	+	0.917593i	$0.630129\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.37016	0.600288	0.300144	−	0.953894i	$-0.402965\pi$
$53$	4.37016	0.600288	0.300144	+	0.953894i	$0.402965\pi$
$54$	0	0
$55$	2.47214	0.333343
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.8098	−1.92808	−0.964038	−	0.265764i	$-0.914376\pi$
$59$	−14.8098	−1.92808	−0.964038	+	0.265764i	$0.914376\pi$
$60$	0	0
$61$	−12.4721	−1.59689	−0.798447	−	0.602066i	$-0.794345\pi$
$61$	−12.4721	−1.59689	−0.798447	+	0.602066i	$0.794345\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.08036	0.134003
$66$	0	0
$67$	11.4164	1.39474	0.697368	−	0.716713i	$-0.254354\pi$
$67$	11.4164	1.39474	0.697368	+	0.716713i	$0.254354\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.03476	−0.360160	−0.180080	−	0.983652i	$-0.557636\pi$
$71$	−3.03476	−0.360160	−0.180080	+	0.983652i	$0.557636\pi$
$72$	0	0
$73$	6.94427	0.812766	0.406383	−	0.913703i	$-0.366790\pi$
$73$	6.94427	0.812766	0.406383	+	0.913703i	$0.366790\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	−1.52786	−0.171898	−0.0859491	−	0.996300i	$-0.527392\pi$
$79$	−1.52786	−0.171898	−0.0859491	+	0.996300i	$0.527392\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.78282	0.524983	0.262491	−	0.964934i	$-0.415456\pi$
$83$	4.78282	0.524983	0.262491	+	0.964934i	$0.415456\pi$
$84$	0	0
$85$	−3.23607	−0.351001
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.48528	0.899438	0.449719	−	0.893170i	$-0.351524\pi$
$89$	8.48528	0.899438	0.449719	+	0.893170i	$0.351524\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.874032	0.0896738
$96$	0	0
$97$	−3.52786	−0.358200	−0.179100	−	0.983831i	$-0.557319\pi$
$97$	−3.52786	−0.358200	−0.179100	+	0.983831i	$0.557319\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.5123	1.84204	0.921021	−	0.389513i	$-0.127357\pi$
$101$	18.5123	1.84204	0.921021	+	0.389513i	$0.127357\pi$
$102$	0	0
$103$	−12.9443	−1.27544	−0.637719	−	0.770270i	$-0.720122\pi$
$103$	−12.9443	−1.27544	−0.637719	+	0.770270i	$0.720122\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.6035	1.41177	0.705887	−	0.708324i	$-0.250549\pi$
$107$	14.6035	1.41177	0.705887	+	0.708324i	$0.250549\pi$
$108$	0	0
$109$	−18.9443	−1.81453	−0.907266	−	0.420557i	$-0.861835\pi$
$109$	−18.9443	−1.81453	−0.907266	+	0.420557i	$0.861835\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.54173	−0.145034	−0.0725170	−	0.997367i	$-0.523103\pi$
$113$	−1.54173	−0.145034	−0.0725170	+	0.997367i	$0.523103\pi$
$114$	0	0
$115$	−2.47214	−0.230528
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.70246	0.339404
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−8.07262	−0.722037
$126$	0	0
$127$	7.41641	0.658100	0.329050	−	0.944313i	$-0.393272\pi$
$127$	7.41641	0.658100	0.329050	+	0.944313i	$0.393272\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.03476	0.265148	0.132574	−	0.991173i	$-0.457676\pi$
$131$	3.03476	0.265148	0.132574	+	0.991173i	$0.457676\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.16073	−0.184603	−0.0923016	−	0.995731i	$-0.529422\pi$
$137$	−2.16073	−0.184603	−0.0923016	+	0.995731i	$0.529422\pi$
$138$	0	0
$139$	−4.94427	−0.419368	−0.209684	−	0.977769i	$-0.567243\pi$
$139$	−4.94427	−0.419368	−0.209684	+	0.977769i	$0.567243\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.49613	0.292361
$144$	0	0
$145$	−6.29180	−0.522505
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.24419	−0.429621	−0.214810	−	0.976656i	$-0.568913\pi$
$149$	−5.24419	−0.429621	−0.214810	+	0.976656i	$0.568913\pi$
$150$	0	0
$151$	−1.52786	−0.124336	−0.0621679	−	0.998066i	$-0.519801\pi$
$151$	−1.52786	−0.124336	−0.0621679	+	0.998066i	$0.519801\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.98915	−0.400738
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.82843	0.222911
$162$	0	0
$163$	−25.1246	−1.96791	−0.983956	−	0.178413i	$-0.942904\pi$
$163$	−25.1246	−1.96791	−0.983956	+	0.178413i	$0.942904\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.90879	−0.302471	−0.151236	−	0.988498i	$-0.548325\pi$
$167$	−3.90879	−0.302471	−0.151236	+	0.988498i	$0.548325\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−21.1344	−1.60682	−0.803409	−	0.595428i	$-0.796983\pi$
$173$	−21.1344	−1.60682	−0.803409	+	0.595428i	$0.796983\pi$
$174$	0	0
$175$	4.23607	0.320217
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.61125	−0.568891	−0.284446	−	0.958692i	$-0.591810\pi$
$179$	−7.61125	−0.568891	−0.284446	+	0.958692i	$0.591810\pi$
$180$	0	0
$181$	−14.9443	−1.11080	−0.555399	−	0.831584i	$-0.687435\pi$
$181$	−14.9443	−1.11080	−0.555399	+	0.831584i	$0.687435\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.90879	−0.287380
$186$	0	0
$187$	−10.4721	−0.765798
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.6305	1.78220	0.891101	−	0.453805i	$-0.149934\pi$
$191$	24.6305	1.78220	0.891101	+	0.453805i	$0.149934\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.81758	−0.556980	−0.278490	−	0.960439i	$-0.589834\pi$
$197$	−7.81758	−0.556980	−0.278490	+	0.960439i	$0.589834\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.19859	0.505242
$204$	0	0
$205$	7.41641	0.517984
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.82843	0.195646
$210$	0	0
$211$	1.52786	0.105182	0.0525912	−	0.998616i	$-0.483252\pi$
$211$	1.52786	0.105182	0.0525912	+	0.998616i	$0.483252\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.667701	−0.0455368
$216$	0	0
$217$	5.70820	0.387498
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.57649	−0.307848
$222$	0	0
$223$	0.763932	0.0511567	0.0255783	−	0.999673i	$-0.491857\pi$
$223$	0.763932	0.0511567	0.0255783	+	0.999673i	$0.491857\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.0540	1.33103	0.665516	−	0.746383i	$-0.268211\pi$
$227$	20.0540	1.33103	0.665516	+	0.746383i	$0.268211\pi$
$228$	0	0
$229$	18.3607	1.21331	0.606654	−	0.794966i	$-0.292511\pi$
$229$	18.3607	1.21331	0.606654	+	0.794966i	$0.292511\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.7186	−1.22630	−0.613149	−	0.789967i	$-0.710097\pi$
$233$	−18.7186	−1.22630	−0.613149	+	0.789967i	$0.710097\pi$
$234$	0	0
$235$	−4.76393	−0.310765
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.3863	−1.25400	−0.626999	−	0.779020i	$-0.715717\pi$
$239$	−19.3863	−1.25400	−0.626999	+	0.779020i	$0.715717\pi$
$240$	0	0
$241$	−23.8885	−1.53880	−0.769398	−	0.638769i	$-0.779444\pi$
$241$	−23.8885	−1.53880	−0.769398	+	0.638769i	$0.779444\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.874032	0.0558399
$246$	0	0
$247$	1.23607	0.0786491
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.618993	−0.0390705	−0.0195352	−	0.999809i	$-0.506219\pi$
$251$	−0.618993	−0.0390705	−0.0195352	+	0.999809i	$0.506219\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.3941	0.773121	0.386560	−	0.922264i	$-0.373663\pi$
$257$	12.3941	0.773121	0.386560	+	0.922264i	$0.373663\pi$
$258$	0	0
$259$	4.47214	0.277885
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.0509	1.11307	0.556534	−	0.830825i	$-0.312131\pi$
$263$	18.0509	1.11307	0.556534	+	0.830825i	$0.312131\pi$
$264$	0	0
$265$	3.81966	0.234640
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.73722	−0.410775	−0.205388	−	0.978681i	$-0.565846\pi$
$269$	−6.73722	−0.410775	−0.205388	+	0.978681i	$0.565846\pi$
$270$	0	0
$271$	26.8328	1.62998	0.814989	−	0.579477i	$-0.196743\pi$
$271$	26.8328	1.62998	0.814989	+	0.579477i	$0.196743\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.9814	−0.722506
$276$	0	0
$277$	−29.2361	−1.75663	−0.878313	−	0.478087i	$-0.841330\pi$
$277$	−29.2361	−1.75663	−0.878313	+	0.478087i	$0.841330\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.5927	1.16880	0.584400	−	0.811466i	$-0.301330\pi$
$281$	19.5927	1.16880	0.584400	+	0.811466i	$0.301330\pi$
$282$	0	0
$283$	−2.47214	−0.146953	−0.0734766	−	0.997297i	$-0.523409\pi$
$283$	−2.47214	−0.146953	−0.0734766	+	0.997297i	$0.523409\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.48528	−0.500870
$288$	0	0
$289$	−3.29180	−0.193635
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.1266	1.64318	0.821588	−	0.570081i	$-0.193088\pi$
$293$	28.1266	1.64318	0.821588	+	0.570081i	$0.193088\pi$
$294$	0	0
$295$	−12.9443	−0.753645
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.49613	−0.202186
$300$	0	0
$301$	0.763932	0.0440323
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−10.9010	−0.624192
$306$	0	0
$307$	9.70820	0.554076	0.277038	−	0.960859i	$-0.410647\pi$
$307$	9.70820	0.554076	0.277038	+	0.960859i	$0.410647\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.03476	0.172085	0.0860427	−	0.996291i	$-0.472578\pi$
$311$	3.03476	0.172085	0.0860427	+	0.996291i	$0.472578\pi$
$312$	0	0
$313$	−19.8885	−1.12417	−0.562083	−	0.827081i	$-0.690000\pi$
$313$	−19.8885	−1.12417	−0.562083	+	0.827081i	$0.690000\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.9867	−1.79655	−0.898277	−	0.439430i	$-0.855180\pi$
$317$	−31.9867	−1.79655	−0.898277	+	0.439430i	$0.855180\pi$
$318$	0	0
$319$	−20.3607	−1.13998
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.70246	−0.206010
$324$	0	0
$325$	−5.23607	−0.290445
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.45052	0.300497
$330$	0	0
$331$	13.8885	0.763383	0.381692	−	0.924290i	$-0.375342\pi$
$331$	13.8885	0.763383	0.381692	+	0.924290i	$0.375342\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.97831	0.545173
$336$	0	0
$337$	10.9443	0.596172	0.298086	−	0.954539i	$-0.403652\pi$
$337$	10.9443	0.596172	0.298086	+	0.954539i	$0.403652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16.1452	−0.874314
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.3786	−1.41608	−0.708038	−	0.706174i	$-0.750420\pi$
$347$	−26.3786	−1.41608	−0.708038	+	0.706174i	$0.750420\pi$
$348$	0	0
$349$	9.41641	0.504049	0.252024	−	0.967721i	$-0.418904\pi$
$349$	9.41641	0.504049	0.252024	+	0.967721i	$0.418904\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.0270	−0.533684	−0.266842	−	0.963740i	$-0.585980\pi$
$353$	−10.0270	−0.533684	−0.266842	+	0.963740i	$0.585980\pi$
$354$	0	0
$355$	−2.65248	−0.140779
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.9442	1.89706	0.948532	−	0.316682i	$-0.102569\pi$
$359$	35.9442	1.89706	0.948532	+	0.316682i	$0.102569\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.06952	0.317693
$366$	0	0
$367$	3.41641	0.178335	0.0891675	−	0.996017i	$-0.471579\pi$
$367$	3.41641	0.178335	0.0891675	+	0.996017i	$0.471579\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.37016	−0.226887
$372$	0	0
$373$	−22.9443	−1.18801	−0.594005	−	0.804462i	$-0.702454\pi$
$373$	−22.9443	−1.18801	−0.594005	+	0.804462i	$0.702454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.89794	−0.458267
$378$	0	0
$379$	0.583592	0.0299771	0.0149886	−	0.999888i	$-0.495229\pi$
$379$	0.583592	0.0299771	0.0149886	+	0.999888i	$0.495229\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.08347	0.157558	0.0787789	−	0.996892i	$-0.474898\pi$
$383$	3.08347	0.157558	0.0787789	+	0.996892i	$0.474898\pi$
$384$	0	0
$385$	−2.47214	−0.125992
$386$	0	0
$387$	0	0
$388$	0	0
$389$	17.8933	0.907226	0.453613	−	0.891199i	$-0.350135\pi$
$389$	17.8933	0.907226	0.453613	+	0.891199i	$0.350135\pi$
$390$	0	0
$391$	10.4721	0.529599
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.33540	−0.0671914
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.8369	1.24029	0.620147	−	0.784486i	$-0.287073\pi$
$401$	24.8369	1.24029	0.620147	+	0.784486i	$0.287073\pi$
$402$	0	0
$403$	−7.05573	−0.351471
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.6491	−0.626993
$408$	0	0
$409$	15.1246	0.747864	0.373932	−	0.927456i	$-0.378009\pi$
$409$	15.1246	0.747864	0.373932	+	0.927456i	$0.378009\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	14.8098	0.728744
$414$	0	0
$415$	4.18034	0.205205
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.6870	−0.864065	−0.432033	−	0.901858i	$-0.642203\pi$
$419$	−17.6870	−0.864065	−0.432033	+	0.901858i	$0.642203\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	15.6839	0.760779
$426$	0	0
$427$	12.4721	0.603569
$428$	0	0
$429$	0	0
$430$	0	0
$431$	23.7565	1.14431	0.572155	−	0.820146i	$-0.306108\pi$
$431$	23.7565	1.14431	0.572155	+	0.820146i	$0.306108\pi$
$432$	0	0
$433$	23.8885	1.14801	0.574005	−	0.818852i	$-0.305389\pi$
$433$	23.8885	1.14801	0.574005	+	0.818852i	$0.305389\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.82843	−0.135302
$438$	0	0
$439$	−25.1246	−1.19913	−0.599566	−	0.800325i	$-0.704660\pi$
$439$	−25.1246	−1.19913	−0.599566	+	0.800325i	$0.704660\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−28.5393	−1.35594	−0.677972	−	0.735088i	$-0.737141\pi$
$443$	−28.5393	−1.35594	−0.677972	+	0.735088i	$0.737141\pi$
$444$	0	0
$445$	7.41641	0.351571
$446$	0	0
$447$	0	0
$448$	0	0
$449$	31.3190	1.47804	0.739018	−	0.673685i	$-0.235290\pi$
$449$	31.3190	1.47804	0.739018	+	0.673685i	$0.235290\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.08036	−0.0506482
$456$	0	0
$457$	−17.5967	−0.823141	−0.411571	−	0.911378i	$-0.635020\pi$
$457$	−17.5967	−0.823141	−0.411571	+	0.911378i	$0.635020\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−39.9017	−1.85841	−0.929204	−	0.369566i	$-0.879506\pi$
$461$	−39.9017	−1.85841	−0.929204	+	0.369566i	$0.879506\pi$
$462$	0	0
$463$	13.8885	0.645455	0.322728	−	0.946492i	$-0.395400\pi$
$463$	13.8885	0.645455	0.322728	+	0.946492i	$0.395400\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.5014	1.08752	0.543759	−	0.839242i	$-0.317001\pi$
$467$	23.5014	1.08752	0.543759	+	0.839242i	$0.317001\pi$
$468$	0	0
$469$	−11.4164	−0.527161
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.16073	−0.0993503
$474$	0	0
$475$	−4.23607	−0.194364
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15.9389	0.728267	0.364134	−	0.931347i	$-0.381365\pi$
$479$	15.9389	0.728267	0.364134	+	0.931347i	$0.381365\pi$
$480$	0	0
$481$	−5.52786	−0.252049
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.08347	−0.140013
$486$	0	0
$487$	−37.8885	−1.71689	−0.858447	−	0.512902i	$-0.828570\pi$
$487$	−37.8885	−1.71689	−0.858447	+	0.512902i	$0.828570\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	32.8608	1.48299	0.741493	−	0.670961i	$-0.234118\pi$
$491$	32.8608	1.48299	0.741493	+	0.670961i	$0.234118\pi$
$492$	0	0
$493$	26.6525	1.20037
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.03476	0.136128
$498$	0	0
$499$	−17.8885	−0.800801	−0.400401	−	0.916340i	$-0.631129\pi$
$499$	−17.8885	−0.800801	−0.400401	+	0.916340i	$0.631129\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.7456	−1.28170	−0.640852	−	0.767664i	$-0.721419\pi$
$503$	−28.7456	−1.28170	−0.640852	+	0.767664i	$0.721419\pi$
$504$	0	0
$505$	16.1803	0.720016
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.2117	−0.895866	−0.447933	−	0.894067i	$-0.647840\pi$
$509$	−20.2117	−0.895866	−0.447933	+	0.894067i	$0.647840\pi$
$510$	0	0
$511$	−6.94427	−0.307197
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.3137	−0.498542
$516$	0	0
$517$	−15.4164	−0.678013
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.2333	−0.448331	−0.224166	−	0.974551i	$-0.571966\pi$
$521$	−10.2333	−0.448331	−0.224166	+	0.974551i	$0.571966\pi$
$522$	0	0
$523$	−16.1803	−0.707517	−0.353758	−	0.935337i	$-0.615097\pi$
$523$	−16.1803	−0.707517	−0.353758	+	0.935337i	$0.615097\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.1344	0.920629
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.4884	0.454302
$534$	0	0
$535$	12.7639	0.551833
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.82843	0.121829
$540$	0	0
$541$	15.5279	0.667595	0.333798	−	0.942645i	$-0.391670\pi$
$541$	15.5279	0.667595	0.333798	+	0.942645i	$0.391670\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.5579	−0.709263
$546$	0	0
$547$	16.5836	0.709063	0.354532	−	0.935044i	$-0.384640\pi$
$547$	16.5836	0.709063	0.354532	+	0.935044i	$0.384640\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.19859	−0.306670
$552$	0	0
$553$	1.52786	0.0649714
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.5579	0.701581	0.350790	−	0.936454i	$-0.385913\pi$
$557$	16.5579	0.701581	0.350790	+	0.936454i	$0.385913\pi$
$558$	0	0
$559$	−0.944272	−0.0399384
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−34.8639	−1.46934	−0.734668	−	0.678426i	$-0.762662\pi$
$563$	−34.8639	−1.46934	−0.734668	+	0.678426i	$0.762662\pi$
$564$	0	0
$565$	−1.34752	−0.0566908
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.03476	−0.127224	−0.0636118	−	0.997975i	$-0.520262\pi$
$569$	−3.03476	−0.127224	−0.0636118	+	0.997975i	$0.520262\pi$
$570$	0	0
$571$	37.8885	1.58559	0.792793	−	0.609491i	$-0.208626\pi$
$571$	37.8885	1.58559	0.792793	+	0.609491i	$0.208626\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	11.9814	0.499659
$576$	0	0
$577$	−24.8328	−1.03380	−0.516902	−	0.856045i	$-0.672915\pi$
$577$	−24.8328	−1.03380	−0.516902	+	0.856045i	$0.672915\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.78282	−0.198425
$582$	0	0
$583$	12.3607	0.511927
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.1074	−0.458451	−0.229225	−	0.973373i	$-0.573619\pi$
$587$	−11.1074	−0.458451	−0.229225	+	0.973373i	$0.573619\pi$
$588$	0	0
$589$	−5.70820	−0.235202
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.7456	−1.18044	−0.590221	−	0.807242i	$-0.700959\pi$
$593$	−28.7456	−1.18044	−0.590221	+	0.807242i	$0.700959\pi$
$594$	0	0
$595$	3.23607	0.132666
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.1908	0.579822	0.289911	−	0.957054i	$-0.406374\pi$
$599$	14.1908	0.579822	0.289911	+	0.957054i	$0.406374\pi$
$600$	0	0
$601$	29.4164	1.19992	0.599960	−	0.800030i	$-0.295183\pi$
$601$	29.4164	1.19992	0.599960	+	0.800030i	$0.295183\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.62210	−0.106603
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.73722	−0.272559
$612$	0	0
$613$	−30.1803	−1.21897	−0.609486	−	0.792797i	$-0.708624\pi$
$613$	−30.1803	−1.21897	−0.609486	+	0.792797i	$0.708624\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−40.1081	−1.61469	−0.807345	−	0.590080i	$-0.799096\pi$
$617$	−40.1081	−1.61469	−0.807345	+	0.590080i	$0.799096\pi$
$618$	0	0
$619$	35.7771	1.43800	0.719001	−	0.695009i	$-0.244600\pi$
$619$	35.7771	1.43800	0.719001	+	0.695009i	$0.244600\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.48528	−0.339956
$624$	0	0
$625$	14.1246	0.564984
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.5579	0.660207
$630$	0	0
$631$	−11.5967	−0.461659	−0.230830	−	0.972994i	$-0.574144\pi$
$631$	−11.5967	−0.461659	−0.230830	+	0.972994i	$0.574144\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6.48218	0.257237
$636$	0	0
$637$	1.23607	0.0489748
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−32.9095	−1.29985	−0.649923	−	0.760000i	$-0.725199\pi$
$641$	−32.9095	−1.29985	−0.649923	+	0.760000i	$0.725199\pi$
$642$	0	0
$643$	37.5279	1.47995	0.739977	−	0.672632i	$-0.234836\pi$
$643$	37.5279	1.47995	0.739977	+	0.672632i	$0.234836\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.86319	0.230506	0.115253	−	0.993336i	$-0.463232\pi$
$647$	5.86319	0.230506	0.115253	+	0.993336i	$0.463232\pi$
$648$	0	0
$649$	−41.8885	−1.64427
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.2920	0.833221	0.416610	−	0.909085i	$-0.363218\pi$
$653$	21.2920	0.833221	0.416610	+	0.909085i	$0.363218\pi$
$654$	0	0
$655$	2.65248	0.103641
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16.0965	−0.627032	−0.313516	−	0.949583i	$-0.601507\pi$
$659$	−16.0965	−0.627032	−0.313516	+	0.949583i	$0.601507\pi$
$660$	0	0
$661$	14.7639	0.574250	0.287125	−	0.957893i	$-0.407300\pi$
$661$	14.7639	0.574250	0.287125	+	0.957893i	$0.407300\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.874032	−0.0338935
$666$	0	0
$667$	20.3607	0.788369
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−35.2765	−1.36184
$672$	0	0
$673$	−24.8328	−0.957235	−0.478617	−	0.878024i	$-0.658862\pi$
$673$	−24.8328	−0.957235	−0.478617	+	0.878024i	$0.658862\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.5610	−0.713357	−0.356679	−	0.934227i	$-0.616091\pi$
$677$	−18.5610	−0.713357	−0.356679	+	0.934227i	$0.616091\pi$
$678$	0	0
$679$	3.52786	0.135387
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.3299	1.00748	0.503742	−	0.863854i	$-0.331956\pi$
$683$	26.3299	1.00748	0.503742	+	0.863854i	$0.331956\pi$
$684$	0	0
$685$	−1.88854	−0.0721576
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.40182	0.205793
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.32145	−0.163922
$696$	0	0
$697$	−31.4164	−1.18998
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19.6414	0.741844	0.370922	−	0.928664i	$-0.379042\pi$
$701$	19.6414	0.741844	0.370922	+	0.928664i	$0.379042\pi$
$702$	0	0
$703$	−4.47214	−0.168670
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.5123	−0.696227
$708$	0	0
$709$	−27.7082	−1.04060	−0.520302	−	0.853983i	$-0.674181\pi$
$709$	−27.7082	−1.04060	−0.520302	+	0.853983i	$0.674181\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.1452	0.604644
$714$	0	0
$715$	3.05573	0.114278
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.8229	−1.03762	−0.518810	−	0.854890i	$-0.673625\pi$
$719$	−27.8229	−1.03762	−0.518810	+	0.854890i	$0.673625\pi$
$720$	0	0
$721$	12.9443	0.482070
$722$	0	0
$723$	0	0
$724$	0	0
$725$	30.4937	1.13251
$726$	0	0
$727$	−5.52786	−0.205017	−0.102509	−	0.994732i	$-0.532687\pi$
$727$	−5.52786	−0.205017	−0.102509	+	0.994732i	$0.532687\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.82843	0.104613
$732$	0	0
$733$	12.8328	0.473991	0.236995	−	0.971511i	$-0.423837\pi$
$733$	12.8328	0.473991	0.236995	+	0.971511i	$0.423837\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.2905	1.18944
$738$	0	0
$739$	−25.1246	−0.924224	−0.462112	−	0.886822i	$-0.652908\pi$
$739$	−25.1246	−0.924224	−0.462112	+	0.886822i	$0.652908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.2681	0.486760	0.243380	−	0.969931i	$-0.421744\pi$
$743$	13.2681	0.486760	0.243380	+	0.969931i	$0.421744\pi$
$744$	0	0
$745$	−4.58359	−0.167930
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−14.6035	−0.533600
$750$	0	0
$751$	−16.3607	−0.597010	−0.298505	−	0.954408i	$-0.596488\pi$
$751$	−16.3607	−0.597010	−0.298505	+	0.954408i	$0.596488\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.33540	−0.0486003
$756$	0	0
$757$	43.3050	1.57395	0.786973	−	0.616988i	$-0.211647\pi$
$757$	43.3050	1.57395	0.786973	+	0.616988i	$0.211647\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.60815	−0.203295	−0.101648	−	0.994820i	$-0.532411\pi$
$761$	−5.60815	−0.203295	−0.101648	+	0.994820i	$0.532411\pi$
$762$	0	0
$763$	18.9443	0.685829
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18.3060	−0.660990
$768$	0	0
$769$	39.8885	1.43842	0.719209	−	0.694794i	$-0.244504\pi$
$769$	39.8885	1.43842	0.719209	+	0.694794i	$0.244504\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.2719	1.59235	0.796175	−	0.605067i	$-0.206854\pi$
$773$	44.2719	1.59235	0.796175	+	0.605067i	$0.206854\pi$
$774$	0	0
$775$	24.1803	0.868583
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.48528	0.304017
$780$	0	0
$781$	−8.58359	−0.307145
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.24419	−0.187173
$786$	0	0
$787$	53.1246	1.89369	0.946844	−	0.321693i	$-0.104252\pi$
$787$	53.1246	1.89369	0.946844	+	0.321693i	$0.104252\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.54173	0.0548177
$792$	0	0
$793$	−15.4164	−0.547453
$794$	0	0
$795$	0	0
$796$	0	0
$797$	48.5933	1.72126	0.860632	−	0.509227i	$-0.170069\pi$
$797$	48.5933	1.72126	0.860632	+	0.509227i	$0.170069\pi$
$798$	0	0
$799$	20.1803	0.713929
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19.6414	0.693129
$804$	0	0
$805$	2.47214	0.0871313
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.83153	0.169868	0.0849338	−	0.996387i	$-0.472932\pi$
$809$	4.83153	0.169868	0.0849338	+	0.996387i	$0.472932\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−21.9597	−0.769215
$816$	0	0
$817$	−0.763932	−0.0267266
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.0618	−0.455859	−0.227930	−	0.973678i	$-0.573196\pi$
$821$	−13.0618	−0.455859	−0.227930	+	0.973678i	$0.573196\pi$
$822$	0	0
$823$	0.180340	0.00628625	0.00314313	−	0.999995i	$-0.499000\pi$
$823$	0.180340	0.00628625	0.00314313	+	0.999995i	$0.499000\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22.6761	−0.788526	−0.394263	−	0.918998i	$-0.629000\pi$
$827$	−22.6761	−0.788526	−0.394263	+	0.918998i	$0.629000\pi$
$828$	0	0
$829$	19.5279	0.678231	0.339115	−	0.940745i	$-0.389872\pi$
$829$	19.5279	0.678231	0.339115	+	0.940745i	$0.389872\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.70246	−0.128283
$834$	0	0
$835$	−3.41641	−0.118230
$836$	0	0
$837$	0	0
$838$	0	0
$839$	16.1452	0.557396	0.278698	−	0.960379i	$-0.410097\pi$
$839$	16.1452	0.557396	0.278698	+	0.960379i	$0.410097\pi$
$840$	0	0
$841$	22.8197	0.786885
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.0270	−0.344940
$846$	0	0
$847$	3.00000	0.103081
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12.6491	0.433606
$852$	0	0
$853$	−18.3607	−0.628658	−0.314329	−	0.949314i	$-0.601779\pi$
$853$	−18.3607	−0.628658	−0.314329	+	0.949314i	$0.601779\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.7618	1.49487	0.747437	−	0.664332i	$-0.231284\pi$
$857$	43.7618	1.49487	0.747437	+	0.664332i	$0.231284\pi$
$858$	0	0
$859$	41.3050	1.40931	0.704653	−	0.709552i	$-0.251103\pi$
$859$	41.3050	1.40931	0.704653	+	0.709552i	$0.251103\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−52.5508	−1.78885	−0.894426	−	0.447217i	$-0.852415\pi$
$863$	−52.5508	−1.78885	−0.894426	+	0.447217i	$0.852415\pi$
$864$	0	0
$865$	−18.4721	−0.628071
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.32145	−0.146595
$870$	0	0
$871$	14.1115	0.478148
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.07262	0.272904
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	44.0656	1.48461	0.742303	−	0.670064i	$-0.233733\pi$
$881$	44.0656	1.48461	0.742303	+	0.670064i	$0.233733\pi$
$882$	0	0
$883$	52.9443	1.78172	0.890858	−	0.454281i	$-0.150104\pi$
$883$	52.9443	1.78172	0.890858	+	0.454281i	$0.150104\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	53.9952	1.81298	0.906490	−	0.422227i	$-0.138752\pi$
$887$	53.9952	1.81298	0.906490	+	0.422227i	$0.138752\pi$
$888$	0	0
$889$	−7.41641	−0.248738
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.45052	−0.182395
$894$	0	0
$895$	−6.65248	−0.222368
$896$	0	0
$897$	0	0
$898$	0	0
$899$	41.0910	1.37046
$900$	0	0
$901$	−16.1803	−0.539045
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.0618	−0.434188
$906$	0	0
$907$	−20.9443	−0.695443	−0.347722	−	0.937598i	$-0.613045\pi$
$907$	−20.9443	−0.695443	−0.347722	+	0.937598i	$0.613045\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.1877	0.403798	0.201899	−	0.979406i	$-0.435289\pi$
$911$	12.1877	0.403798	0.201899	+	0.979406i	$0.435289\pi$
$912$	0	0
$913$	13.5279	0.447707
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.03476	−0.100217
$918$	0	0
$919$	−11.0557	−0.364695	−0.182347	−	0.983234i	$-0.558370\pi$
$919$	−11.0557	−0.364695	−0.182347	+	0.983234i	$0.558370\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.75117	−0.123471
$924$	0	0
$925$	18.9443	0.622884
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12.4428	0.408234	0.204117	−	0.978946i	$-0.434568\pi$
$929$	12.4428	0.408234	0.204117	+	0.978946i	$0.434568\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.15298	−0.299335
$936$	0	0
$937$	6.94427	0.226859	0.113430	−	0.993546i	$-0.463816\pi$
$937$	6.94427	0.226859	0.113430	+	0.993546i	$0.463816\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.90569	0.0621236	0.0310618	−	0.999517i	$-0.490111\pi$
$941$	1.90569	0.0621236	0.0310618	+	0.999517i	$0.490111\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.6011	−1.35185	−0.675927	−	0.736969i	$-0.736256\pi$
$947$	−41.6011	−1.35185	−0.675927	+	0.736969i	$0.736256\pi$
$948$	0	0
$949$	8.58359	0.278635
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−20.9281	−0.677926	−0.338963	−	0.940800i	$-0.610076\pi$
$953$	−20.9281	−0.677926	−0.338963	+	0.940800i	$0.610076\pi$
$954$	0	0
$955$	21.5279	0.696625
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.16073	0.0697735
$960$	0	0
$961$	1.58359	0.0510836
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.7326	−0.506450
$966$	0	0
$967$	16.5410	0.531923	0.265962	−	0.963984i	$-0.414311\pi$
$967$	16.5410	0.531923	0.265962	+	0.963984i	$0.414311\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11.3137	−0.363074	−0.181537	−	0.983384i	$-0.558107\pi$
$971$	−11.3137	−0.363074	−0.181537	+	0.983384i	$0.558107\pi$
$972$	0	0
$973$	4.94427	0.158506
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.0888	−0.738676	−0.369338	−	0.929295i	$-0.620415\pi$
$977$	−23.0888	−0.738676	−0.369338	+	0.929295i	$0.620415\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.32145	−0.137833	−0.0689165	−	0.997622i	$-0.521954\pi$
$983$	−4.32145	−0.137833	−0.0689165	+	0.997622i	$0.521954\pi$
$984$	0	0
$985$	−6.83282	−0.217712
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.16073	0.0687071
$990$	0	0
$991$	5.88854	0.187056	0.0935279	−	0.995617i	$-0.470186\pi$
$991$	5.88854	0.187056	0.0935279	+	0.995617i	$0.470186\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.99226	0.221669
$996$	0	0
$997$	−17.7771	−0.563006	−0.281503	−	0.959560i	$-0.590833\pi$
$997$	−17.7771	−0.563006	−0.281503	+	0.959560i	$0.590833\pi$
$998$	0	0
$999$	0	0

Display

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a_n

instead) (See

a_n

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.q.1.3	yes	4
3.2	odd	2	inner	4788.2.a.q.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.a.q.1.2	✓	4	3.2	odd	2	inner
4788.2.a.q.1.3	yes	4	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.q.1.3

Level	$4788$
Weight	$2$
Character	4788.1
Self dual	yes
Analytic conductor	$38.232$
Analytic rank	$1$
Dimension	$4$
Inner twists	$2$