LMFDB - Embedded newform 4788.2.a.r.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,-12] 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:	$\Z[a_1, \ldots, a_{11}]$
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)

$f(q)$	$=$	$q+0.540182 q^{5} +1.00000 q^{7} +1.74806 q^{11} -0.763932 q^{13} +2.28825 q^{17} -1.00000 q^{19} -7.40492 q^{23} -4.70820 q^{25} -7.94510 q^{29} -2.76393 q^{31} +0.540182 q^{35} -6.00000 q^{37} +1.74806 q^{41} -10.1803 q^{43} -5.78437 q^{47} +1.00000 q^{49} -6.19704 q^{53} +0.944272 q^{55} +12.6491 q^{59} +6.94427 q^{61} -0.412662 q^{65} -2.47214 q^{67} +9.69316 q^{71} -3.52786 q^{73} +1.74806 q^{77} +4.94427 q^{79} +7.27740 q^{83} +1.23607 q^{85} +1.74806 q^{89} -0.763932 q^{91} -0.540182 q^{95} -17.4164 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{7} - 12 q^{13} - 4 q^{19} + 8 q^{25} - 20 q^{31} - 24 q^{37} + 4 q^{43} + 4 q^{49} - 32 q^{55} - 8 q^{61} + 8 q^{67} - 32 q^{73} - 16 q^{79} - 4 q^{85} - 12 q^{91} - 16 q^{97}+O(q^{100})$ 4 * q + 4 * q^7 - 12 * q^13 - 4 * q^19 + 8 * q^25 - 20 * q^31 - 24 * q^37 + 4 * q^43 + 4 * q^49 - 32 * q^55 - 8 * q^61 + 8 * q^67 - 32 * q^73 - 16 * q^79 - 4 * q^85 - 12 * q^91 - 16 * 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
$5$	0.540182	0.241577	0.120788	−	0.992678i	$-0.461458\pi$
$5$	0.540182	0.241577	0.120788	+	0.992678i	$0.461458\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.74806	0.527061	0.263531	−	0.964651i	$-0.415113\pi$
$11$	1.74806	0.527061	0.263531	+	0.964651i	$0.415113\pi$
$12$	0	0
$13$	−0.763932	−0.211877	−0.105938	−	0.994373i	$-0.533785\pi$
$13$	−0.763932	−0.211877	−0.105938	+	0.994373i	$0.533785\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.28825	0.554981	0.277491	−	0.960728i	$-0.410497\pi$
$17$	2.28825	0.554981	0.277491	+	0.960728i	$0.410497\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.40492	−1.54403	−0.772016	−	0.635603i	$-0.780752\pi$
$23$	−7.40492	−1.54403	−0.772016	+	0.635603i	$0.780752\pi$
$24$	0	0
$25$	−4.70820	−0.941641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.94510	−1.47537	−0.737684	−	0.675146i	$-0.764081\pi$
$29$	−7.94510	−1.47537	−0.737684	+	0.675146i	$0.764081\pi$
$30$	0	0
$31$	−2.76393	−0.496417	−0.248208	−	0.968707i	$-0.579842\pi$
$31$	−2.76393	−0.496417	−0.248208	+	0.968707i	$0.579842\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.540182	0.0913073
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.74806	0.273002	0.136501	−	0.990640i	$-0.456414\pi$
$41$	1.74806	0.273002	0.136501	+	0.990640i	$0.456414\pi$
$42$	0	0
$43$	−10.1803	−1.55249	−0.776244	−	0.630433i	$-0.782877\pi$
$43$	−10.1803	−1.55249	−0.776244	+	0.630433i	$0.782877\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.78437	−0.843738	−0.421869	−	0.906657i	$-0.638626\pi$
$47$	−5.78437	−0.843738	−0.421869	+	0.906657i	$0.638626\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.19704	−0.851228	−0.425614	−	0.904905i	$-0.639942\pi$
$53$	−6.19704	−0.851228	−0.425614	+	0.904905i	$0.639942\pi$
$54$	0	0
$55$	0.944272	0.127326
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.6491	1.64677	0.823387	−	0.567480i	$-0.192082\pi$
$59$	12.6491	1.64677	0.823387	+	0.567480i	$0.192082\pi$
$60$	0	0
$61$	6.94427	0.889123	0.444561	−	0.895748i	$-0.353360\pi$
$61$	6.94427	0.889123	0.444561	+	0.895748i	$0.353360\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.412662	−0.0511844
$66$	0	0
$67$	−2.47214	−0.302019	−0.151010	−	0.988532i	$-0.548252\pi$
$67$	−2.47214	−0.302019	−0.151010	+	0.988532i	$0.548252\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	9.69316	1.15037	0.575183	−	0.818024i	$-0.304931\pi$
$71$	9.69316	1.15037	0.575183	+	0.818024i	$0.304931\pi$
$72$	0	0
$73$	−3.52786	−0.412905	−0.206453	−	0.978457i	$-0.566192\pi$
$73$	−3.52786	−0.412905	−0.206453	+	0.978457i	$0.566192\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.74806	0.199210
$78$	0	0
$79$	4.94427	0.556274	0.278137	−	0.960541i	$-0.410283\pi$
$79$	4.94427	0.556274	0.278137	+	0.960541i	$0.410283\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.27740	0.798798	0.399399	−	0.916777i	$-0.369219\pi$
$83$	7.27740	0.798798	0.399399	+	0.916777i	$0.369219\pi$
$84$	0	0
$85$	1.23607	0.134070
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.74806	0.185294	0.0926472	−	0.995699i	$-0.470467\pi$
$89$	1.74806	0.185294	0.0926472	+	0.995699i	$0.470467\pi$
$90$	0	0
$91$	−0.763932	−0.0800818
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.540182	−0.0554215
$96$	0	0
$97$	−17.4164	−1.76837	−0.884184	−	0.467139i	$-0.845285\pi$
$97$	−17.4164	−1.76837	−0.884184	+	0.467139i	$0.845285\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.44897	−0.442689	−0.221345	−	0.975196i	$-0.571045\pi$
$101$	−4.44897	−0.442689	−0.221345	+	0.975196i	$0.571045\pi$
$102$	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
$107$	−14.9374	−1.44405	−0.722024	−	0.691868i	$-0.756788\pi$
$107$	−14.9374	−1.44405	−0.722024	+	0.691868i	$0.756788\pi$
$108$	0	0
$109$	6.94427	0.665141	0.332570	−	0.943078i	$-0.392084\pi$
$109$	6.94427	0.665141	0.332570	+	0.943078i	$0.392084\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.1784	1.71008	0.855042	−	0.518559i	$-0.173531\pi$
$113$	18.1784	1.71008	0.855042	+	0.518559i	$0.173531\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.28825	0.209763
$120$	0	0
$121$	−7.94427	−0.722207
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.24419	−0.469055
$126$	0	0
$127$	−9.52786	−0.845461	−0.422731	−	0.906255i	$-0.638928\pi$
$127$	−9.52786	−0.845461	−0.422731	+	0.906255i	$0.638928\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.19704	0.541438	0.270719	−	0.962658i	$-0.412739\pi$
$131$	6.19704	0.541438	0.270719	+	0.962658i	$0.412739\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.99226	0.597389	0.298694	−	0.954349i	$-0.403449\pi$
$137$	6.99226	0.597389	0.298694	+	0.954349i	$0.403449\pi$
$138$	0	0
$139$	10.4721	0.888235	0.444117	−	0.895969i	$-0.353517\pi$
$139$	10.4721	0.888235	0.444117	+	0.895969i	$0.353517\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.33540	−0.111672
$144$	0	0
$145$	−4.29180	−0.356414
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.08036	0.0885068	0.0442534	−	0.999020i	$-0.485909\pi$
$149$	1.08036	0.0885068	0.0442534	+	0.999020i	$0.485909\pi$
$150$	0	0
$151$	4.94427	0.402359	0.201180	−	0.979554i	$-0.435523\pi$
$151$	4.94427	0.402359	0.201180	+	0.979554i	$0.435523\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.49302	−0.119923
$156$	0	0
$157$	−8.47214	−0.676150	−0.338075	−	0.941119i	$-0.609776\pi$
$157$	−8.47214	−0.676150	−0.338075	+	0.941119i	$0.609776\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.40492	−0.583589
$162$	0	0
$163$	5.23607	0.410120	0.205060	−	0.978749i	$-0.434261\pi$
$163$	5.23607	0.410120	0.205060	+	0.978749i	$0.434261\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−25.0432	−1.93790	−0.968950	−	0.247257i	$-0.920471\pi$
$167$	−25.0432	−1.93790	−0.968950	+	0.247257i	$0.920471\pi$
$168$	0	0
$169$	−12.4164	−0.955108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.56564	0.727262	0.363631	−	0.931543i	$-0.381537\pi$
$173$	9.56564	0.727262	0.363631	+	0.931543i	$0.381537\pi$
$174$	0	0
$175$	−4.70820	−0.355907
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.952843	0.0712189	0.0356094	−	0.999366i	$-0.488663\pi$
$179$	0.952843	0.0712189	0.0356094	+	0.999366i	$0.488663\pi$
$180$	0	0
$181$	−19.8885	−1.47830	−0.739152	−	0.673539i	$-0.764773\pi$
$181$	−19.8885	−1.47830	−0.739152	+	0.673539i	$0.764773\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.24109	−0.238290
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.74806	−0.126485	−0.0632427	−	0.997998i	$-0.520144\pi$
$191$	−1.74806	−0.126485	−0.0632427	+	0.997998i	$0.520144\pi$
$192$	0	0
$193$	21.4164	1.54159	0.770793	−	0.637085i	$-0.219860\pi$
$193$	21.4164	1.54159	0.770793	+	0.637085i	$0.219860\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.3060	−1.30425	−0.652123	−	0.758113i	$-0.726121\pi$
$197$	−18.3060	−1.30425	−0.652123	+	0.758113i	$0.726121\pi$
$198$	0	0
$199$	−20.9443	−1.48470	−0.742350	−	0.670012i	$-0.766289\pi$
$199$	−20.9443	−1.48470	−0.742350	+	0.670012i	$0.766289\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.94510	−0.557637
$204$	0	0
$205$	0.944272	0.0659508
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.74806	−0.120916
$210$	0	0
$211$	15.4164	1.06131	0.530655	−	0.847588i	$-0.321946\pi$
$211$	15.4164	1.06131	0.530655	+	0.847588i	$0.321946\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.49923	−0.375044
$216$	0	0
$217$	−2.76393	−0.187628
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.74806	−0.117588
$222$	0	0
$223$	−5.23607	−0.350633	−0.175317	−	0.984512i	$-0.556095\pi$
$223$	−5.23607	−0.350633	−0.175317	+	0.984512i	$0.556095\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.41577	−0.160340	−0.0801700	−	0.996781i	$-0.525546\pi$
$227$	−2.41577	−0.160340	−0.0801700	+	0.996781i	$0.525546\pi$
$228$	0	0
$229$	−18.3607	−1.21331	−0.606654	−	0.794966i	$-0.707489\pi$
$229$	−18.3607	−1.21331	−0.606654	+	0.794966i	$0.707489\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.2039	1.78219	0.891094	−	0.453819i	$-0.149939\pi$
$233$	27.2039	1.78219	0.891094	+	0.453819i	$0.149939\pi$
$234$	0	0
$235$	−3.12461	−0.203827
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.6460	0.688633	0.344316	−	0.938854i	$-0.388111\pi$
$239$	10.6460	0.688633	0.344316	+	0.938854i	$0.388111\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.540182	0.0345109
$246$	0	0
$247$	0.763932	0.0486078
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.1707	1.58876	0.794380	−	0.607421i	$-0.207796\pi$
$251$	25.1707	1.58876	0.794380	+	0.607421i	$0.207796\pi$
$252$	0	0
$253$	−12.9443	−0.813799
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.16383	−0.259733	−0.129866	−	0.991532i	$-0.541455\pi$
$257$	−4.16383	−0.259733	−0.129866	+	0.991532i	$0.541455\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.4775	0.954386	0.477193	−	0.878799i	$-0.341654\pi$
$263$	15.4775	0.954386	0.477193	+	0.878799i	$0.341654\pi$
$264$	0	0
$265$	−3.34752	−0.205637
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−24.6305	−1.50175	−0.750875	−	0.660445i	$-0.770368\pi$
$269$	−24.6305	−1.50175	−0.750875	+	0.660445i	$0.770368\pi$
$270$	0	0
$271$	11.4164	0.693497	0.346749	−	0.937958i	$-0.387286\pi$
$271$	11.4164	0.693497	0.346749	+	0.937958i	$0.387286\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.23024	−0.496302
$276$	0	0
$277$	−8.76393	−0.526574	−0.263287	−	0.964718i	$-0.584807\pi$
$277$	−8.76393	−0.526574	−0.263287	+	0.964718i	$0.584807\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.70091	0.161123	0.0805613	−	0.996750i	$-0.474329\pi$
$281$	2.70091	0.161123	0.0805613	+	0.996750i	$0.474329\pi$
$282$	0	0
$283$	6.47214	0.384729	0.192364	−	0.981324i	$-0.438384\pi$
$283$	6.47214	0.384729	0.192364	+	0.981324i	$0.438384\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.74806	0.103185
$288$	0	0
$289$	−11.7639	−0.691996
$290$	0	0
$291$	0	0
$292$	0	0
$293$	16.5579	0.967323	0.483661	−	0.875255i	$-0.339307\pi$
$293$	16.5579	0.967323	0.483661	+	0.875255i	$0.339307\pi$
$294$	0	0
$295$	6.83282	0.397822
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	−10.1803	−0.586785
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.75117	0.214791
$306$	0	0
$307$	−14.1803	−0.809315	−0.404657	−	0.914468i	$-0.632609\pi$
$307$	−14.1803	−0.809315	−0.404657	+	0.914468i	$0.632609\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.1815	1.14439	0.572195	−	0.820117i	$-0.306092\pi$
$311$	20.1815	1.14439	0.572195	+	0.820117i	$0.306092\pi$
$312$	0	0
$313$	10.3607	0.585620	0.292810	−	0.956171i	$-0.405410\pi$
$313$	10.3607	0.585620	0.292810	+	0.956171i	$0.405410\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.0100	1.29237	0.646184	−	0.763181i	$-0.276364\pi$
$317$	23.0100	1.29237	0.646184	+	0.763181i	$0.276364\pi$
$318$	0	0
$319$	−13.8885	−0.777609
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.28825	−0.127321
$324$	0	0
$325$	3.59675	0.199512
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.78437	−0.318903
$330$	0	0
$331$	−11.4164	−0.627503	−0.313751	−	0.949505i	$-0.601586\pi$
$331$	−11.4164	−0.627503	−0.313751	+	0.949505i	$0.601586\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.33540	−0.0729608
$336$	0	0
$337$	−0.111456	−0.00607140	−0.00303570	−	0.999995i	$-0.500966\pi$
$337$	−0.111456	−0.00607140	−0.00303570	+	0.999995i	$0.500966\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.83153	−0.261642
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.9737	−1.01856	−0.509280	−	0.860601i	$-0.670088\pi$
$347$	−18.9737	−1.01856	−0.509280	+	0.860601i	$0.670088\pi$
$348$	0	0
$349$	13.4164	0.718164	0.359082	−	0.933306i	$-0.383090\pi$
$349$	13.4164	0.718164	0.359082	+	0.933306i	$0.383090\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−35.8167	−1.90633	−0.953166	−	0.302449i	$-0.902196\pi$
$353$	−35.8167	−1.90633	−0.953166	+	0.302449i	$0.902196\pi$
$354$	0	0
$355$	5.23607	0.277902
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.24419	0.276778	0.138389	−	0.990378i	$-0.455808\pi$
$359$	5.24419	0.276778	0.138389	+	0.990378i	$0.455808\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.90569	−0.0997482
$366$	0	0
$367$	20.3607	1.06282	0.531409	−	0.847115i	$-0.321663\pi$
$367$	20.3607	1.06282	0.531409	+	0.847115i	$0.321663\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.19704	−0.321734
$372$	0	0
$373$	−5.05573	−0.261776	−0.130888	−	0.991397i	$-0.541783\pi$
$373$	−5.05573	−0.261776	−0.130888	+	0.991397i	$0.541783\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.06952	0.312596
$378$	0	0
$379$	−24.9443	−1.28130	−0.640651	−	0.767833i	$-0.721335\pi$
$379$	−24.9443	−1.28130	−0.640651	+	0.767833i	$0.721335\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.2333	−0.522900	−0.261450	−	0.965217i	$-0.584201\pi$
$383$	−10.2333	−0.522900	−0.261450	+	0.965217i	$0.584201\pi$
$384$	0	0
$385$	0.944272	0.0481246
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.3863	0.982926	0.491463	−	0.870898i	$-0.336462\pi$
$389$	19.3863	0.982926	0.491463	+	0.870898i	$0.336462\pi$
$390$	0	0
$391$	−16.9443	−0.856909
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.67080	0.134383
$396$	0	0
$397$	−3.88854	−0.195160	−0.0975802	−	0.995228i	$-0.531110\pi$
$397$	−3.88854	−0.195160	−0.0975802	+	0.995228i	$0.531110\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.8461	−0.941132	−0.470566	−	0.882365i	$-0.655950\pi$
$401$	−18.8461	−0.941132	−0.470566	+	0.882365i	$0.655950\pi$
$402$	0	0
$403$	2.11146	0.105179
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.4884	−0.519890
$408$	0	0
$409$	−1.70820	−0.0844652	−0.0422326	−	0.999108i	$-0.513447\pi$
$409$	−1.70820	−0.0844652	−0.0422326	+	0.999108i	$0.513447\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.6491	0.622422
$414$	0	0
$415$	3.93112	0.192971
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.4844	−1.78238	−0.891190	−	0.453630i	$-0.850129\pi$
$419$	−36.4844	−1.78238	−0.891190	+	0.453630i	$0.850129\pi$
$420$	0	0
$421$	−14.3607	−0.699897	−0.349948	−	0.936769i	$-0.613801\pi$
$421$	−14.3607	−0.699897	−0.349948	+	0.936769i	$0.613801\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−10.7735	−0.522593
$426$	0	0
$427$	6.94427	0.336057
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.9157	−1.20015	−0.600073	−	0.799946i	$-0.704862\pi$
$431$	−24.9157	−1.20015	−0.600073	+	0.799946i	$0.704862\pi$
$432$	0	0
$433$	−35.8885	−1.72469	−0.862347	−	0.506318i	$-0.831006\pi$
$433$	−35.8885	−1.72469	−0.862347	+	0.506318i	$0.831006\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.40492	0.354225
$438$	0	0
$439$	−20.0689	−0.957836	−0.478918	−	0.877860i	$-0.658971\pi$
$439$	−20.0689	−0.957836	−0.478918	+	0.877860i	$0.658971\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	40.7758	1.93731	0.968657	−	0.248401i	$-0.0799049\pi$
$443$	40.7758	1.93731	0.968657	+	0.248401i	$0.0799049\pi$
$444$	0	0
$445$	0.944272	0.0447628
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.43812	0.445413	0.222706	−	0.974886i	$-0.428511\pi$
$449$	9.43812	0.445413	0.222706	+	0.974886i	$0.428511\pi$
$450$	0	0
$451$	3.05573	0.143889
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.412662	−0.0193459
$456$	0	0
$457$	−31.2361	−1.46116	−0.730581	−	0.682826i	$-0.760751\pi$
$457$	−31.2361	−1.46116	−0.730581	+	0.682826i	$0.760751\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17.9234	−0.834776	−0.417388	−	0.908728i	$-0.637054\pi$
$461$	−17.9234	−0.834776	−0.417388	+	0.908728i	$0.637054\pi$
$462$	0	0
$463$	24.9443	1.15926	0.579629	−	0.814880i	$-0.303197\pi$
$463$	24.9443	1.15926	0.579629	+	0.814880i	$0.303197\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.8384	1.19566	0.597829	−	0.801623i	$-0.296030\pi$
$467$	25.8384	1.19566	0.597829	+	0.801623i	$0.296030\pi$
$468$	0	0
$469$	−2.47214	−0.114153
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−17.7959	−0.818256
$474$	0	0
$475$	4.70820	0.216027
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.62365	0.165569	0.0827843	−	0.996567i	$-0.473619\pi$
$479$	3.62365	0.165569	0.0827843	+	0.996567i	$0.473619\pi$
$480$	0	0
$481$	4.58359	0.208994
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.40802	−0.427196
$486$	0	0
$487$	42.8328	1.94094	0.970470	−	0.241222i	$-0.0775480\pi$
$487$	42.8328	1.94094	0.970470	+	0.241222i	$0.0775480\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−0.667701	−0.0301329	−0.0150665	−	0.999886i	$-0.504796\pi$
$491$	−0.667701	−0.0301329	−0.0150665	+	0.999886i	$0.504796\pi$
$492$	0	0
$493$	−18.1803	−0.818801
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.69316	0.434798
$498$	0	0
$499$	28.9443	1.29572	0.647862	−	0.761758i	$-0.275663\pi$
$499$	28.9443	1.29572	0.647862	+	0.761758i	$0.275663\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	41.4736	1.84921	0.924607	−	0.380921i	$-0.124393\pi$
$503$	41.4736	1.84921	0.924607	+	0.380921i	$0.124393\pi$
$504$	0	0
$505$	−2.40325	−0.106943
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.16383	0.184558	0.0922792	−	0.995733i	$-0.470585\pi$
$509$	4.16383	0.184558	0.0922792	+	0.995733i	$0.470585\pi$
$510$	0	0
$511$	−3.52786	−0.156064
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.99226	−0.308116
$516$	0	0
$517$	−10.1115	−0.444701
$518$	0	0
$519$	0	0
$520$	0	0
$521$	28.9520	1.26841	0.634204	−	0.773165i	$-0.281328\pi$
$521$	28.9520	1.26841	0.634204	+	0.773165i	$0.281328\pi$
$522$	0	0
$523$	27.7082	1.21160	0.605798	−	0.795619i	$-0.292854\pi$
$523$	27.7082	1.21160	0.605798	+	0.795619i	$0.292854\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.32456	−0.275502
$528$	0	0
$529$	31.8328	1.38404
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.33540	−0.0578427
$534$	0	0
$535$	−8.06888	−0.348848
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.74806	0.0752945
$540$	0	0
$541$	13.4164	0.576816	0.288408	−	0.957508i	$-0.406874\pi$
$541$	13.4164	0.576816	0.288408	+	0.957508i	$0.406874\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.75117	0.160682
$546$	0	0
$547$	−7.05573	−0.301681	−0.150841	−	0.988558i	$-0.548198\pi$
$547$	−7.05573	−0.301681	−0.150841	+	0.988558i	$0.548198\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.94510	0.338473
$552$	0	0
$553$	4.94427	0.210252
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−32.0354	−1.35739	−0.678693	−	0.734423i	$-0.737453\pi$
$557$	−32.0354	−1.35739	−0.678693	+	0.734423i	$0.737453\pi$
$558$	0	0
$559$	7.77709	0.328936
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−26.3786	−1.11172	−0.555862	−	0.831274i	$-0.687612\pi$
$563$	−26.3786	−1.11172	−0.555862	+	0.831274i	$0.687612\pi$
$564$	0	0
$565$	9.81966	0.413116
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.2402	−1.30966	−0.654829	−	0.755777i	$-0.727259\pi$
$569$	−31.2402	−1.30966	−0.654829	+	0.755777i	$0.727259\pi$
$570$	0	0
$571$	32.9443	1.37867	0.689337	−	0.724440i	$-0.257902\pi$
$571$	32.9443	1.37867	0.689337	+	0.724440i	$0.257902\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	34.8639	1.45392
$576$	0	0
$577$	−30.3607	−1.26393	−0.631966	−	0.774996i	$-0.717752\pi$
$577$	−30.3607	−1.26393	−0.631966	+	0.774996i	$0.717752\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.27740	0.301917
$582$	0	0
$583$	−10.8328	−0.448649
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.9079	−1.31698	−0.658490	−	0.752589i	$-0.728805\pi$
$587$	−31.9079	−1.31698	−0.658490	+	0.752589i	$0.728805\pi$
$588$	0	0
$589$	2.76393	0.113886
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.78127	−0.155278	−0.0776391	−	0.996982i	$-0.524738\pi$
$593$	−3.78127	−0.155278	−0.0776391	+	0.996982i	$0.524738\pi$
$594$	0	0
$595$	1.23607	0.0506738
$596$	0	0
$597$	0	0
$598$	0	0
$599$	10.1058	0.412913	0.206456	−	0.978456i	$-0.433807\pi$
$599$	10.1058	0.412913	0.206456	+	0.978456i	$0.433807\pi$
$600$	0	0
$601$	−13.4164	−0.547267	−0.273633	−	0.961834i	$-0.588225\pi$
$601$	−13.4164	−0.547267	−0.273633	+	0.961834i	$0.588225\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.29135	−0.174468
$606$	0	0
$607$	5.88854	0.239009	0.119504	−	0.992834i	$-0.461869\pi$
$607$	5.88854	0.239009	0.119504	+	0.992834i	$0.461869\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.41887	0.178768
$612$	0	0
$613$	−11.5967	−0.468388	−0.234194	−	0.972190i	$-0.575245\pi$
$613$	−11.5967	−0.468388	−0.234194	+	0.972190i	$0.575245\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.32766	−0.335259	−0.167629	−	0.985850i	$-0.553611\pi$
$617$	−8.32766	−0.335259	−0.167629	+	0.985850i	$0.553611\pi$
$618$	0	0
$619$	−10.4721	−0.420911	−0.210455	−	0.977603i	$-0.567495\pi$
$619$	−10.4721	−0.420911	−0.210455	+	0.977603i	$0.567495\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.74806	0.0700347
$624$	0	0
$625$	20.7082	0.828328
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−13.7295	−0.547430
$630$	0	0
$631$	1.81966	0.0724395	0.0362198	−	0.999344i	$-0.488468\pi$
$631$	1.81966	0.0724395	0.0362198	+	0.999344i	$0.488468\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−5.14678	−0.204244
$636$	0	0
$637$	−0.763932	−0.0302681
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12.5216	−0.494573	−0.247287	−	0.968942i	$-0.579539\pi$
$641$	−12.5216	−0.494573	−0.247287	+	0.968942i	$0.579539\pi$
$642$	0	0
$643$	22.4721	0.886215	0.443107	−	0.896469i	$-0.353876\pi$
$643$	22.4721	0.886215	0.443107	+	0.896469i	$0.353876\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	46.4627	1.82664	0.913319	−	0.407245i	$-0.133510\pi$
$647$	46.4627	1.82664	0.913319	+	0.407245i	$0.133510\pi$
$648$	0	0
$649$	22.1115	0.867951
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.6491	−0.494998	−0.247499	−	0.968888i	$-0.579609\pi$
$653$	−12.6491	−0.494998	−0.247499	+	0.968888i	$0.579609\pi$
$654$	0	0
$655$	3.34752	0.130799
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.45207	0.251337	0.125669	−	0.992072i	$-0.459892\pi$
$659$	6.45207	0.251337	0.125669	+	0.992072i	$0.459892\pi$
$660$	0	0
$661$	11.8197	0.459731	0.229866	−	0.973222i	$-0.426171\pi$
$661$	11.8197	0.459731	0.229866	+	0.973222i	$0.426171\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.540182	−0.0209473
$666$	0	0
$667$	58.8328	2.27802
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.1390	0.468622
$672$	0	0
$673$	−43.3050	−1.66928	−0.834642	−	0.550793i	$-0.814325\pi$
$673$	−43.3050	−1.66928	−0.834642	+	0.550793i	$0.814325\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.9225	1.76495	0.882473	−	0.470363i	$-0.155877\pi$
$677$	45.9225	1.76495	0.882473	+	0.470363i	$0.155877\pi$
$678$	0	0
$679$	−17.4164	−0.668380
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21.6746	−0.829355	−0.414677	−	0.909969i	$-0.636106\pi$
$683$	−21.6746	−0.829355	−0.414677	+	0.909969i	$0.636106\pi$
$684$	0	0
$685$	3.77709	0.144315
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.73411	0.180355
$690$	0	0
$691$	30.2492	1.15073	0.575367	−	0.817895i	$-0.304859\pi$
$691$	30.2492	1.15073	0.575367	+	0.817895i	$0.304859\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.65685	0.214577
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.3060	−0.691407	−0.345703	−	0.938344i	$-0.612360\pi$
$701$	−18.3060	−0.691407	−0.345703	+	0.938344i	$0.612360\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.44897	−0.167321
$708$	0	0
$709$	17.7082	0.665046	0.332523	−	0.943095i	$-0.392100\pi$
$709$	17.7082	0.665046	0.332523	+	0.943095i	$0.392100\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.4667	0.766484
$714$	0	0
$715$	−0.721360	−0.0269773
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−48.4658	−1.80747	−0.903735	−	0.428092i	$-0.859186\pi$
$719$	−48.4658	−1.80747	−0.903735	+	0.428092i	$0.859186\pi$
$720$	0	0
$721$	−12.9443	−0.482070
$722$	0	0
$723$	0	0
$724$	0	0
$725$	37.4072	1.38927
$726$	0	0
$727$	−37.8885	−1.40521	−0.702604	−	0.711581i	$-0.747980\pi$
$727$	−37.8885	−1.40521	−0.702604	+	0.711581i	$0.747980\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−23.2951	−0.861601
$732$	0	0
$733$	37.4164	1.38201	0.691003	−	0.722852i	$-0.257169\pi$
$733$	37.4164	1.38201	0.691003	+	0.722852i	$0.257169\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.32145	−0.159183
$738$	0	0
$739$	36.0689	1.32682	0.663408	−	0.748258i	$-0.269110\pi$
$739$	36.0689	1.32682	0.663408	+	0.748258i	$0.269110\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15.7627	−0.578277	−0.289138	−	0.957287i	$-0.593369\pi$
$743$	−15.7627	−0.578277	−0.289138	+	0.957287i	$0.593369\pi$
$744$	0	0
$745$	0.583592	0.0213812
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−14.9374	−0.545799
$750$	0	0
$751$	20.9443	0.764267	0.382134	−	0.924107i	$-0.375189\pi$
$751$	20.9443	0.764267	0.382134	+	0.924107i	$0.375189\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.67080	0.0972005
$756$	0	0
$757$	−41.4164	−1.50530	−0.752652	−	0.658418i	$-0.771226\pi$
$757$	−41.4164	−1.50530	−0.752652	+	0.658418i	$0.771226\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31.4953	1.14170	0.570851	−	0.821054i	$-0.306613\pi$
$761$	31.4953	1.14170	0.570851	+	0.821054i	$0.306613\pi$
$762$	0	0
$763$	6.94427	0.251400
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.66306	−0.348913
$768$	0	0
$769$	−43.3050	−1.56162	−0.780808	−	0.624771i	$-0.785192\pi$
$769$	−43.3050	−1.56162	−0.780808	+	0.624771i	$0.785192\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−17.8933	−0.643577	−0.321789	−	0.946812i	$-0.604284\pi$
$773$	−17.8933	−0.643577	−0.321789	+	0.946812i	$0.604284\pi$
$774$	0	0
$775$	13.0132	0.467446
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.74806	−0.0626309
$780$	0	0
$781$	16.9443	0.606314
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.57649	−0.163342
$786$	0	0
$787$	−14.7639	−0.526277	−0.263139	−	0.964758i	$-0.584758\pi$
$787$	−14.7639	−0.526277	−0.263139	+	0.964758i	$0.584758\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.1784	0.646351
$792$	0	0
$793$	−5.30495	−0.188384
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.0246	1.31148	0.655739	−	0.754988i	$-0.272357\pi$
$797$	37.0246	1.31148	0.655739	+	0.754988i	$0.272357\pi$
$798$	0	0
$799$	−13.2361	−0.468258
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.16693	−0.217626
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.7881	0.871505	0.435752	−	0.900067i	$-0.356482\pi$
$809$	24.7881	0.871505	0.435752	+	0.900067i	$0.356482\pi$
$810$	0	0
$811$	16.9443	0.594994	0.297497	−	0.954723i	$-0.403848\pi$
$811$	16.9443	0.594994	0.297497	+	0.954723i	$0.403848\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.82843	0.0990755
$816$	0	0
$817$	10.1803	0.356165
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.3491	−1.51290	−0.756448	−	0.654054i	$-0.773067\pi$
$821$	−43.3491	−1.51290	−0.756448	+	0.654054i	$0.773067\pi$
$822$	0	0
$823$	56.6525	1.97478	0.987391	−	0.158303i	$-0.0506023\pi$
$823$	56.6525	1.97478	0.987391	+	0.158303i	$0.0506023\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.7363	−1.20790	−0.603951	−	0.797022i	$-0.706408\pi$
$827$	−34.7363	−1.20790	−0.603951	+	0.797022i	$0.706408\pi$
$828$	0	0
$829$	−29.4164	−1.02167	−0.510837	−	0.859678i	$-0.670664\pi$
$829$	−29.4164	−1.02167	−0.510837	+	0.859678i	$0.670664\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.28825	0.0792830
$834$	0	0
$835$	−13.5279	−0.468151
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.9845	0.482799	0.241399	−	0.970426i	$-0.422394\pi$
$839$	13.9845	0.482799	0.241399	+	0.970426i	$0.422394\pi$
$840$	0	0
$841$	34.1246	1.17671
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6.70711	−0.230732
$846$	0	0
$847$	−7.94427	−0.272968
$848$	0	0
$849$	0	0
$850$	0	0
$851$	44.4295	1.52302
$852$	0	0
$853$	24.4721	0.837910	0.418955	−	0.908007i	$-0.362397\pi$
$853$	24.4721	0.837910	0.418955	+	0.908007i	$0.362397\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.0757	0.344180	0.172090	−	0.985081i	$-0.444948\pi$
$857$	10.0757	0.344180	0.172090	+	0.985081i	$0.444948\pi$
$858$	0	0
$859$	35.4164	1.20839	0.604196	−	0.796836i	$-0.293494\pi$
$859$	35.4164	1.20839	0.604196	+	0.796836i	$0.293494\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−39.1552	−1.33286	−0.666430	−	0.745568i	$-0.732178\pi$
$863$	−39.1552	−1.33286	−0.666430	+	0.745568i	$0.732178\pi$
$864$	0	0
$865$	5.16718	0.175690
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.64290	0.293190
$870$	0	0
$871$	1.88854	0.0639909
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.24419	−0.177286
$876$	0	0
$877$	28.8328	0.973615	0.486808	−	0.873509i	$-0.338161\pi$
$877$	28.8328	0.973615	0.486808	+	0.873509i	$0.338161\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42.2989	−1.42509	−0.712543	−	0.701629i	$-0.752457\pi$
$881$	−42.2989	−1.42509	−0.712543	+	0.701629i	$0.752457\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.5687	0.388441	0.194220	−	0.980958i	$-0.437782\pi$
$887$	11.5687	0.388441	0.194220	+	0.980958i	$0.437782\pi$
$888$	0	0
$889$	−9.52786	−0.319554
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.78437	0.193567
$894$	0	0
$895$	0.514708	0.0172048
$896$	0	0
$897$	0	0
$898$	0	0
$899$	21.9597	0.732398
$900$	0	0
$901$	−14.1803	−0.472416
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.7434	−0.357123
$906$	0	0
$907$	20.9443	0.695443	0.347722	−	0.937598i	$-0.386955\pi$
$907$	20.9443	0.695443	0.347722	+	0.937598i	$0.386955\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−51.9619	−1.72158	−0.860788	−	0.508964i	$-0.830029\pi$
$911$	−51.9619	−1.72158	−0.860788	+	0.508964i	$0.830029\pi$
$912$	0	0
$913$	12.7214	0.421016
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.19704	0.204644
$918$	0	0
$919$	3.05573	0.100799	0.0503996	−	0.998729i	$-0.483951\pi$
$919$	3.05573	0.100799	0.0503996	+	0.998729i	$0.483951\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.40492	−0.243736
$924$	0	0
$925$	28.2492	0.928829
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28.9219	0.948896	0.474448	−	0.880284i	$-0.342648\pi$
$929$	28.9219	0.948896	0.474448	+	0.880284i	$0.342648\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.16073	0.0706633
$936$	0	0
$937$	16.8328	0.549904	0.274952	−	0.961458i	$-0.411338\pi$
$937$	16.8328	0.549904	0.274952	+	0.961458i	$0.411338\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	41.6011	1.35616	0.678078	−	0.734990i	$-0.262813\pi$
$941$	41.6011	1.35616	0.678078	+	0.734990i	$0.262813\pi$
$942$	0	0
$943$	−12.9443	−0.421523
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.7186	−0.608274	−0.304137	−	0.952628i	$-0.598368\pi$
$947$	−18.7186	−0.608274	−0.304137	+	0.952628i	$0.598368\pi$
$948$	0	0
$949$	2.69505	0.0874850
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.7441	0.898719	0.449360	−	0.893351i	$-0.351652\pi$
$953$	27.7441	0.898719	0.449360	+	0.893351i	$0.351652\pi$
$954$	0	0
$955$	−0.944272	−0.0305559
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.99226	0.225792
$960$	0	0
$961$	−23.3607	−0.753570
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.5687	0.372411
$966$	0	0
$967$	−14.7639	−0.474776	−0.237388	−	0.971415i	$-0.576291\pi$
$967$	−14.7639	−0.474776	−0.237388	+	0.971415i	$0.576291\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.65685	0.181537	0.0907685	−	0.995872i	$-0.471068\pi$
$971$	5.65685	0.181537	0.0907685	+	0.995872i	$0.471068\pi$
$972$	0	0
$973$	10.4721	0.335721
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.7735	0.344676	0.172338	−	0.985038i	$-0.444868\pi$
$977$	10.7735	0.344676	0.172338	+	0.985038i	$0.444868\pi$
$978$	0	0
$979$	3.05573	0.0976615
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57.9039	−1.84685	−0.923425	−	0.383780i	$-0.874622\pi$
$983$	−57.9039	−1.84685	−0.923425	+	0.383780i	$0.874622\pi$
$984$	0	0
$985$	−9.88854	−0.315075
$986$	0	0
$987$	0	0
$988$	0	0
$989$	75.3846	2.39709
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.3137	−0.358669
$996$	0	0
$997$	−27.5279	−0.871816	−0.435908	−	0.899991i	$-0.643573\pi$
$997$	−27.5279	−0.871816	−0.435908	+	0.899991i	$0.643573\pi$
$998$	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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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.r.1.3	yes	4
3.2	odd	2	inner	4788.2.a.r.1.2	✓	4

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

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