LMFDB - Embedded newform 8028.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1710888.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 3x^{2} + 12x + 4 $ x^5 - x^4 - 10x^3 + 3x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2676)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.42525$ of defining polynomial
Character	$\chi$	$=$	8028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40326 q^{5} -1.61825 q^{7} +O(q^{10})$	$q-1.40326 q^{5} -1.61825 q^{7} +5.53867 q^{17} +1.96865 q^{19} -5.23650 q^{23} -3.03086 q^{25} +4.23177 q^{29} +3.38127 q^{31} +2.27083 q^{35} +3.11342 q^{37} +5.61825 q^{41} -8.84460 q^{43} +1.90857 q^{47} -4.38127 q^{49} -9.89650 q^{53} +3.10551 q^{59} -15.5943 q^{61} +5.61034 q^{67} -0.420525 q^{71} -4.24490 q^{73} +11.2674 q^{79} +10.9595 q^{83} -7.77219 q^{85} -9.21258 q^{89} -2.76253 q^{95} -10.5688 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{5} + 3 q^{7}+O(q^{10}) $ 5 * q + 6 * q^5 + 3 * q^7	$ 5 q + 6 q^{5} + 3 q^{7} + 13 q^{17} - q^{19} - 4 q^{23} + 5 q^{25} - 11 q^{29} - 3 q^{31} + 8 q^{35} + 7 q^{37} + 17 q^{41} + 15 q^{43} + 25 q^{47} - 2 q^{49} + 3 q^{53} + 14 q^{59} - 18 q^{61} + 24 q^{67} + 14 q^{71} - 23 q^{73} + 14 q^{79} + 15 q^{83} + 6 q^{85} + 25 q^{89} + 26 q^{95} - 12 q^{97}+O(q^{100}) $ 5 * q + 6 * q^5 + 3 * q^7 + 13 * q^17 - q^19 - 4 * q^23 + 5 * q^25 - 11 * q^29 - 3 * q^31 + 8 * q^35 + 7 * q^37 + 17 * q^41 + 15 * q^43 + 25 * q^47 - 2 * q^49 + 3 * q^53 + 14 * q^59 - 18 * q^61 + 24 * q^67 + 14 * q^71 - 23 * q^73 + 14 * q^79 + 15 * q^83 + 6 * q^85 + 25 * q^89 + 26 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−1.40326	−0.627557	−0.313778	−	0.949496i	$-0.601595\pi$
$5$	−1.40326	−0.627557	−0.313778	+	0.949496i	$0.601595\pi$
$6$	0	0
$7$	−1.61825	−0.611641	−0.305821	−	0.952089i	$-0.598931\pi$
$7$	−1.61825	−0.611641	−0.305821	+	0.952089i	$0.598931\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.53867	1.34333	0.671663	−	0.740857i	$-0.265580\pi$
$17$	5.53867	1.34333	0.671663	+	0.740857i	$0.265580\pi$
$18$	0	0
$19$	1.96865	0.451640	0.225820	−	0.974169i	$-0.427494\pi$
$19$	1.96865	0.451640	0.225820	+	0.974169i	$0.427494\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.23650	−1.09189	−0.545943	−	0.837822i	$-0.683828\pi$
$23$	−5.23650	−1.09189	−0.545943	+	0.837822i	$0.683828\pi$
$24$	0	0
$25$	−3.03086	−0.606173
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.23177	0.785820	0.392910	−	0.919577i	$-0.371468\pi$
$29$	4.23177	0.785820	0.392910	+	0.919577i	$0.371468\pi$
$30$	0	0
$31$	3.38127	0.607293	0.303647	−	0.952785i	$-0.401796\pi$
$31$	3.38127	0.607293	0.303647	+	0.952785i	$0.401796\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.27083	0.383840
$36$	0	0
$37$	3.11342	0.511843	0.255921	−	0.966698i	$-0.417621\pi$
$37$	3.11342	0.511843	0.255921	+	0.966698i	$0.417621\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.61825	0.877423	0.438712	−	0.898628i	$-0.355435\pi$
$41$	5.61825	0.877423	0.438712	+	0.898628i	$0.355435\pi$
$42$	0	0
$43$	−8.84460	−1.34879	−0.674395	−	0.738371i	$-0.735595\pi$
$43$	−8.84460	−1.34879	−0.674395	+	0.738371i	$0.735595\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.90857	0.278394	0.139197	−	0.990265i	$-0.455548\pi$
$47$	1.90857	0.278394	0.139197	+	0.990265i	$0.455548\pi$
$48$	0	0
$49$	−4.38127	−0.625895
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.89650	−1.35939	−0.679695	−	0.733495i	$-0.737888\pi$
$53$	−9.89650	−1.35939	−0.679695	+	0.733495i	$0.737888\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.10551	0.404303	0.202151	−	0.979354i	$-0.435207\pi$
$59$	3.10551	0.404303	0.202151	+	0.979354i	$0.435207\pi$
$60$	0	0
$61$	−15.5943	−1.99665	−0.998325	−	0.0578577i	$-0.981573\pi$
$61$	−15.5943	−1.99665	−0.998325	+	0.0578577i	$0.981573\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.61034	0.685412	0.342706	−	0.939443i	$-0.388657\pi$
$67$	5.61034	0.685412	0.342706	+	0.939443i	$0.388657\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.420525	−0.0499071	−0.0249535	−	0.999689i	$-0.507944\pi$
$71$	−0.420525	−0.0499071	−0.0249535	+	0.999689i	$0.507944\pi$
$72$	0	0
$73$	−4.24490	−0.496828	−0.248414	−	0.968654i	$-0.579909\pi$
$73$	−4.24490	−0.496828	−0.248414	+	0.968654i	$0.579909\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.2674	1.26768	0.633839	−	0.773465i	$-0.281478\pi$
$79$	11.2674	1.26768	0.633839	+	0.773465i	$0.281478\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.9595	1.20296	0.601481	−	0.798887i	$-0.294578\pi$
$83$	10.9595	1.20296	0.601481	+	0.798887i	$0.294578\pi$
$84$	0	0
$85$	−7.77219	−0.843013
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.21258	−0.976532	−0.488266	−	0.872695i	$-0.662370\pi$
$89$	−9.21258	−0.976532	−0.488266	+	0.872695i	$0.662370\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.76253	−0.283430
$96$	0	0
$97$	−10.5688	−1.07310	−0.536552	−	0.843867i	$-0.680273\pi$
$97$	−10.5688	−1.07310	−0.536552	+	0.843867i	$0.680273\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.51274	0.847050	0.423525	−	0.905884i	$-0.360793\pi$
$101$	8.51274	0.847050	0.423525	+	0.905884i	$0.360793\pi$
$102$	0	0
$103$	9.56347	0.942317	0.471158	−	0.882049i	$-0.343836\pi$
$103$	9.56347	0.942317	0.471158	+	0.882049i	$0.343836\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.20834	−0.696856	−0.348428	−	0.937335i	$-0.613284\pi$
$107$	−7.20834	−0.696856	−0.348428	+	0.937335i	$0.613284\pi$
$108$	0	0
$109$	−8.92716	−0.855067	−0.427533	−	0.904000i	$-0.640617\pi$
$109$	−8.92716	−0.855067	−0.427533	+	0.904000i	$0.640617\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.1831	1.42830	0.714151	−	0.699992i	$-0.246813\pi$
$113$	15.1831	1.42830	0.714151	+	0.699992i	$0.246813\pi$
$114$	0	0
$115$	7.34817	0.685220
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.96296	−0.821633
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.2694	1.00796
$126$	0	0
$127$	22.3255	1.98107	0.990535	−	0.137263i	$-0.0438304\pi$
$127$	22.3255	1.98107	0.990535	+	0.137263i	$0.0438304\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	20.9933	1.83420	0.917099	−	0.398660i	$-0.130525\pi$
$131$	20.9933	1.83420	0.917099	+	0.398660i	$0.130525\pi$
$132$	0	0
$133$	−3.18577	−0.276242
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.65558	−0.739497	−0.369748	−	0.929132i	$-0.620556\pi$
$137$	−8.65558	−0.739497	−0.369748	+	0.929132i	$0.620556\pi$
$138$	0	0
$139$	−11.0816	−0.939928	−0.469964	−	0.882686i	$-0.655733\pi$
$139$	−11.0816	−0.939928	−0.469964	+	0.882686i	$0.655733\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.93827	−0.493147
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.9707	1.71798	0.858992	−	0.511989i	$-0.171091\pi$
$149$	20.9707	1.71798	0.858992	+	0.511989i	$0.171091\pi$
$150$	0	0
$151$	−17.4473	−1.41984	−0.709922	−	0.704280i	$-0.751270\pi$
$151$	−17.4473	−1.41984	−0.709922	+	0.704280i	$0.751270\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.74479	−0.381111
$156$	0	0
$157$	21.1728	1.68978	0.844888	−	0.534943i	$-0.179667\pi$
$157$	21.1728	1.68978	0.844888	+	0.534943i	$0.179667\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.47397	0.667842
$162$	0	0
$163$	−5.43248	−0.425504	−0.212752	−	0.977106i	$-0.568243\pi$
$163$	−5.43248	−0.425504	−0.212752	+	0.977106i	$0.568243\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.2512	1.56708	0.783541	−	0.621341i	$-0.213412\pi$
$167$	20.2512	1.56708	0.783541	+	0.621341i	$0.213412\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.4451	1.70647	0.853235	−	0.521527i	$-0.174637\pi$
$173$	22.4451	1.70647	0.853235	+	0.521527i	$0.174637\pi$
$174$	0	0
$175$	4.90469	0.370760
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.1857	−0.985542	−0.492771	−	0.870159i	$-0.664016\pi$
$179$	−13.1857	−0.985542	−0.492771	+	0.870159i	$0.664016\pi$
$180$	0	0
$181$	22.1278	1.64475	0.822373	−	0.568949i	$-0.192650\pi$
$181$	22.1278	1.64475	0.822373	+	0.568949i	$0.192650\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.36893	−0.321210
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.82502	0.276769	0.138384	−	0.990379i	$-0.455809\pi$
$191$	3.82502	0.276769	0.138384	+	0.990379i	$0.455809\pi$
$192$	0	0
$193$	−14.1468	−1.01831	−0.509155	−	0.860675i	$-0.670042\pi$
$193$	−14.1468	−1.01831	−0.509155	+	0.860675i	$0.670042\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.16956	0.510810	0.255405	−	0.966834i	$-0.417791\pi$
$197$	7.16956	0.510810	0.255405	+	0.966834i	$0.417791\pi$
$198$	0	0
$199$	−7.28542	−0.516450	−0.258225	−	0.966085i	$-0.583138\pi$
$199$	−7.28542	−0.516450	−0.258225	+	0.966085i	$0.583138\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.84807	−0.480640
$204$	0	0
$205$	−7.88386	−0.550633
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.96769	−0.135461	−0.0677306	−	0.997704i	$-0.521576\pi$
$211$	−1.96769	−0.135461	−0.0677306	+	0.997704i	$0.521576\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	12.4113	0.846442
$216$	0	0
$217$	−5.47173	−0.371446
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.7520	−1.24462	−0.622309	−	0.782772i	$-0.713805\pi$
$227$	−18.7520	−1.24462	−0.622309	+	0.782772i	$0.713805\pi$
$228$	0	0
$229$	19.9190	1.31629	0.658143	−	0.752893i	$-0.271342\pi$
$229$	19.9190	1.31629	0.658143	+	0.752893i	$0.271342\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.64619	−0.631943	−0.315971	−	0.948769i	$-0.602330\pi$
$233$	−9.64619	−0.631943	−0.315971	+	0.948769i	$0.602330\pi$
$234$	0	0
$235$	−2.67823	−0.174708
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25.7083	1.66293	0.831465	−	0.555577i	$-0.187503\pi$
$239$	25.7083	1.66293	0.831465	+	0.555577i	$0.187503\pi$
$240$	0	0
$241$	20.6000	1.32696	0.663482	−	0.748193i	$-0.269078\pi$
$241$	20.6000	1.32696	0.663482	+	0.748193i	$0.269078\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.14805	0.392785
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.9205	1.06801	0.534007	−	0.845480i	$-0.320686\pi$
$251$	16.9205	1.06801	0.534007	+	0.845480i	$0.320686\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.1440	1.44368	0.721842	−	0.692058i	$-0.243296\pi$
$257$	23.1440	1.44368	0.721842	+	0.692058i	$0.243296\pi$
$258$	0	0
$259$	−5.03829	−0.313064
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.60379	0.345544	0.172772	−	0.984962i	$-0.444728\pi$
$263$	5.60379	0.345544	0.172772	+	0.984962i	$0.444728\pi$
$264$	0	0
$265$	13.8874	0.853094
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.22443	−0.0746548	−0.0373274	−	0.999303i	$-0.511884\pi$
$269$	−1.22443	−0.0746548	−0.0373274	+	0.999303i	$0.511884\pi$
$270$	0	0
$271$	−19.2162	−1.16730	−0.583651	−	0.812005i	$-0.698376\pi$
$271$	−19.2162	−1.16730	−0.583651	+	0.812005i	$0.698376\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.99034	0.299840	0.149920	−	0.988698i	$-0.452098\pi$
$277$	4.99034	0.299840	0.149920	+	0.988698i	$0.452098\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.9032	−1.06802	−0.534009	−	0.845479i	$-0.679315\pi$
$281$	−17.9032	−1.06802	−0.534009	+	0.845479i	$0.679315\pi$
$282$	0	0
$283$	10.4435	0.620800	0.310400	−	0.950606i	$-0.399537\pi$
$283$	10.4435	0.620800	0.310400	+	0.950606i	$0.399537\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.09174	−0.536668
$288$	0	0
$289$	13.6769	0.804523
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−21.4545	−1.25339	−0.626693	−	0.779267i	$-0.715592\pi$
$293$	−21.4545	−1.25339	−0.626693	+	0.779267i	$0.715592\pi$
$294$	0	0
$295$	−4.35783	−0.253723
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	14.3128	0.824975
$302$	0	0
$303$	0	0
$304$	0	0
$305$	21.8829	1.25301
$306$	0	0
$307$	24.7307	1.41146	0.705728	−	0.708483i	$-0.250620\pi$
$307$	24.7307	1.41146	0.705728	+	0.708483i	$0.250620\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.1904	1.82535	0.912674	−	0.408688i	$-0.134013\pi$
$311$	32.1904	1.82535	0.912674	+	0.408688i	$0.134013\pi$
$312$	0	0
$313$	9.12036	0.515513	0.257757	−	0.966210i	$-0.417017\pi$
$313$	9.12036	0.515513	0.257757	+	0.966210i	$0.417017\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.3003	1.30867	0.654337	−	0.756203i	$-0.272948\pi$
$317$	23.3003	1.30867	0.654337	+	0.756203i	$0.272948\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.9037	0.606700
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.08855	−0.170277
$330$	0	0
$331$	30.6795	1.68630	0.843150	−	0.537678i	$-0.180698\pi$
$331$	30.6795	1.68630	0.843150	+	0.537678i	$0.180698\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.87276	−0.430135
$336$	0	0
$337$	29.3997	1.60150	0.800751	−	0.598997i	$-0.204434\pi$
$337$	29.3997	1.60150	0.800751	+	0.598997i	$0.204434\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	18.4177	0.994464
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.3745	1.73795	0.868977	−	0.494853i	$-0.164778\pi$
$347$	32.3745	1.73795	0.868977	+	0.494853i	$0.164778\pi$
$348$	0	0
$349$	−34.6193	−1.85313	−0.926565	−	0.376134i	$-0.877253\pi$
$349$	−34.6193	−1.85313	−0.926565	+	0.376134i	$0.877253\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.5794	1.20178	0.600890	−	0.799332i	$-0.294813\pi$
$353$	22.5794	1.20178	0.600890	+	0.799332i	$0.294813\pi$
$354$	0	0
$355$	0.590105	0.0313195
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.4079	0.549307	0.274654	−	0.961543i	$-0.411437\pi$
$359$	10.4079	0.549307	0.274654	+	0.961543i	$0.411437\pi$
$360$	0	0
$361$	−15.1244	−0.796021
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.95669	0.311787
$366$	0	0
$367$	21.2606	1.10979	0.554897	−	0.831919i	$-0.312758\pi$
$367$	21.2606	1.10979	0.554897	+	0.831919i	$0.312758\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	16.0150	0.831458
$372$	0	0
$373$	−12.5847	−0.651610	−0.325805	−	0.945437i	$-0.605635\pi$
$373$	−12.5847	−0.651610	−0.325805	+	0.945437i	$0.605635\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−11.1600	−0.573253	−0.286626	−	0.958042i	$-0.592534\pi$
$379$	−11.1600	−0.573253	−0.286626	+	0.958042i	$0.592534\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.7245	−0.650192	−0.325096	−	0.945681i	$-0.605397\pi$
$383$	−12.7245	−0.650192	−0.325096	+	0.945681i	$0.605397\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8.63290	−0.437705	−0.218853	−	0.975758i	$-0.570231\pi$
$389$	−8.63290	−0.437705	−0.218853	+	0.975758i	$0.570231\pi$
$390$	0	0
$391$	−29.0033	−1.46676
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.8110	−0.795540
$396$	0	0
$397$	−3.36749	−0.169010	−0.0845049	−	0.996423i	$-0.526931\pi$
$397$	−3.36749	−0.169010	−0.0845049	+	0.996423i	$0.526931\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.5377	0.676041	0.338020	−	0.941139i	$-0.390243\pi$
$401$	13.5377	0.676041	0.338020	+	0.941139i	$0.390243\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−22.0841	−1.09199	−0.545995	−	0.837789i	$-0.683848\pi$
$409$	−22.0841	−1.09199	−0.545995	+	0.837789i	$0.683848\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−5.02549	−0.247288
$414$	0	0
$415$	−15.3790	−0.754927
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−27.7920	−1.35773	−0.678865	−	0.734263i	$-0.737528\pi$
$419$	−27.7920	−1.35773	−0.678865	+	0.734263i	$0.737528\pi$
$420$	0	0
$421$	6.43635	0.313688	0.156844	−	0.987623i	$-0.449868\pi$
$421$	6.43635	0.313688	0.156844	+	0.987623i	$0.449868\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−16.7870	−0.814287
$426$	0	0
$427$	25.2355	1.22123
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.3169	−0.834127	−0.417064	−	0.908877i	$-0.636941\pi$
$431$	−17.3169	−0.834127	−0.417064	+	0.908877i	$0.636941\pi$
$432$	0	0
$433$	21.5064	1.03353	0.516766	−	0.856127i	$-0.327136\pi$
$433$	21.5064	1.03353	0.516766	+	0.856127i	$0.327136\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.3089	−0.493139
$438$	0	0
$439$	19.8989	0.949723	0.474861	−	0.880061i	$-0.342498\pi$
$439$	19.8989	0.949723	0.474861	+	0.880061i	$0.342498\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.26186	−0.202487	−0.101244	−	0.994862i	$-0.532282\pi$
$443$	−4.26186	−0.202487	−0.101244	+	0.994862i	$0.532282\pi$
$444$	0	0
$445$	12.9276	0.612829
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.3100	−1.14726	−0.573630	−	0.819115i	$-0.694465\pi$
$449$	−24.3100	−1.14726	−0.573630	+	0.819115i	$0.694465\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0924	1.03344	0.516720	−	0.856155i	$-0.327153\pi$
$457$	22.0924	1.03344	0.516720	+	0.856155i	$0.327153\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−31.3448	−1.45987	−0.729936	−	0.683515i	$-0.760450\pi$
$461$	−31.3448	−1.45987	−0.729936	+	0.683515i	$0.760450\pi$
$462$	0	0
$463$	23.2468	1.08037	0.540186	−	0.841546i	$-0.318354\pi$
$463$	23.2468	1.08037	0.540186	+	0.841546i	$0.318354\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.08202	0.0963443	0.0481721	−	0.998839i	$-0.484660\pi$
$467$	2.08202	0.0963443	0.0481721	+	0.998839i	$0.484660\pi$
$468$	0	0
$469$	−9.07893	−0.419226
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−5.96672	−0.273772
$476$	0	0
$477$	0	0
$478$	0	0
$479$	26.3044	1.20188	0.600939	−	0.799295i	$-0.294793\pi$
$479$	26.3044	1.20188	0.600939	+	0.799295i	$0.294793\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.8308	0.673433
$486$	0	0
$487$	20.7378	0.939718	0.469859	−	0.882742i	$-0.344305\pi$
$487$	20.7378	0.939718	0.469859	+	0.882742i	$0.344305\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−27.6215	−1.24654	−0.623271	−	0.782006i	$-0.714197\pi$
$491$	−27.6215	−1.24654	−0.623271	+	0.782006i	$0.714197\pi$
$492$	0	0
$493$	23.4384	1.05561
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.680514	0.0305252
$498$	0	0
$499$	−13.5546	−0.606787	−0.303393	−	0.952865i	$-0.598120\pi$
$499$	−13.5546	−0.606787	−0.303393	+	0.952865i	$0.598120\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−30.1809	−1.34570	−0.672850	−	0.739779i	$-0.734930\pi$
$503$	−30.1809	−1.34570	−0.672850	+	0.739779i	$0.734930\pi$
$504$	0	0
$505$	−11.9456	−0.531572
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14.0784	−0.624014	−0.312007	−	0.950080i	$-0.601001\pi$
$509$	−14.0784	−0.624014	−0.312007	+	0.950080i	$0.601001\pi$
$510$	0	0
$511$	6.86931	0.303880
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.4200	−0.591357
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−22.1554	−0.970644	−0.485322	−	0.874335i	$-0.661298\pi$
$521$	−22.1554	−0.970644	−0.485322	+	0.874335i	$0.661298\pi$
$522$	0	0
$523$	4.62326	0.202161	0.101080	−	0.994878i	$-0.467770\pi$
$523$	4.62326	0.202161	0.101080	+	0.994878i	$0.467770\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.7277	0.815792
$528$	0	0
$529$	4.42094	0.192215
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	10.1152	0.437317
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	6.80940	0.292759	0.146379	−	0.989229i	$-0.453238\pi$
$541$	6.80940	0.292759	0.146379	+	0.989229i	$0.453238\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.5271	0.536603
$546$	0	0
$547$	−5.04185	−0.215574	−0.107787	−	0.994174i	$-0.534376\pi$
$547$	−5.04185	−0.215574	−0.107787	+	0.994174i	$0.534376\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.33089	0.354908
$552$	0	0
$553$	−18.2334	−0.775364
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−40.3438	−1.70942	−0.854711	−	0.519103i	$-0.826266\pi$
$557$	−40.3438	−1.70942	−0.854711	+	0.519103i	$0.826266\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.34899	−0.267578	−0.133789	−	0.991010i	$-0.542714\pi$
$563$	−6.34899	−0.267578	−0.133789	+	0.991010i	$0.542714\pi$
$564$	0	0
$565$	−21.3058	−0.896340
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.1226	1.05319	0.526596	−	0.850115i	$-0.323468\pi$
$569$	25.1226	1.05319	0.526596	+	0.850115i	$0.323468\pi$
$570$	0	0
$571$	27.9840	1.17109	0.585547	−	0.810639i	$-0.300880\pi$
$571$	27.9840	1.17109	0.585547	+	0.810639i	$0.300880\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15.8711	0.661871
$576$	0	0
$577$	−1.73656	−0.0722940	−0.0361470	−	0.999346i	$-0.511508\pi$
$577$	−1.73656	−0.0722940	−0.0361470	+	0.999346i	$0.511508\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.7352	−0.735781
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.2415	−0.463986	−0.231993	−	0.972717i	$-0.574525\pi$
$587$	−11.2415	−0.463986	−0.231993	+	0.972717i	$0.574525\pi$
$588$	0	0
$589$	6.65654	0.274278
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.71924	−0.358056	−0.179028	−	0.983844i	$-0.557295\pi$
$593$	−8.71924	−0.358056	−0.179028	+	0.983844i	$0.557295\pi$
$594$	0	0
$595$	12.5774	0.515621
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.79611	−0.0733872	−0.0366936	−	0.999327i	$-0.511683\pi$
$599$	−1.79611	−0.0733872	−0.0366936	+	0.999327i	$0.511683\pi$
$600$	0	0
$601$	−28.5664	−1.16525	−0.582623	−	0.812742i	$-0.697974\pi$
$601$	−28.5664	−1.16525	−0.582623	+	0.812742i	$0.697974\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.4359	0.627557
$606$	0	0
$607$	28.1950	1.14440	0.572200	−	0.820114i	$-0.306090\pi$
$607$	28.1950	1.14440	0.572200	+	0.820114i	$0.306090\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	23.9384	0.966865	0.483432	−	0.875382i	$-0.339390\pi$
$613$	23.9384	0.966865	0.483432	+	0.875382i	$0.339390\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.05111	−0.324126	−0.162063	−	0.986780i	$-0.551815\pi$
$617$	−8.05111	−0.324126	−0.162063	+	0.986780i	$0.551815\pi$
$618$	0	0
$619$	−15.7404	−0.632659	−0.316329	−	0.948649i	$-0.602450\pi$
$619$	−15.7404	−0.632659	−0.316329	+	0.948649i	$0.602450\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.9083	0.597287
$624$	0	0
$625$	−0.659558	−0.0263823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.2442	0.687572
$630$	0	0
$631$	41.4647	1.65068	0.825341	−	0.564635i	$-0.190983\pi$
$631$	41.4647	1.65068	0.825341	+	0.564635i	$0.190983\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−31.3285	−1.24323
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−25.1833	−0.994681	−0.497341	−	0.867555i	$-0.665690\pi$
$641$	−25.1833	−0.994681	−0.497341	+	0.867555i	$0.665690\pi$
$642$	0	0
$643$	−19.1666	−0.755856	−0.377928	−	0.925835i	$-0.623363\pi$
$643$	−19.1666	−0.755856	−0.377928	+	0.925835i	$0.623363\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9.83625	−0.386703	−0.193351	−	0.981130i	$-0.561936\pi$
$647$	−9.83625	−0.386703	−0.193351	+	0.981130i	$0.561936\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.57989	0.0618257	0.0309129	−	0.999522i	$-0.490159\pi$
$653$	1.57989	0.0618257	0.0309129	+	0.999522i	$0.490159\pi$
$654$	0	0
$655$	−29.4591	−1.15106
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.92291	0.230724	0.115362	−	0.993324i	$-0.463197\pi$
$659$	5.92291	0.230724	0.115362	+	0.993324i	$0.463197\pi$
$660$	0	0
$661$	37.5740	1.46146	0.730730	−	0.682666i	$-0.239180\pi$
$661$	37.5740	1.46146	0.730730	+	0.682666i	$0.239180\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.47047	0.173357
$666$	0	0
$667$	−22.1597	−0.858026
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−44.3968	−1.71137	−0.855685	−	0.517497i	$-0.826864\pi$
$673$	−44.3968	−1.71137	−0.855685	+	0.517497i	$0.826864\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.4640	1.47829	0.739145	−	0.673546i	$-0.235230\pi$
$677$	38.4640	1.47829	0.739145	+	0.673546i	$0.235230\pi$
$678$	0	0
$679$	17.1030	0.656354
$680$	0	0
$681$	0	0
$682$	0	0
$683$	31.3478	1.19949	0.599745	−	0.800191i	$-0.295269\pi$
$683$	31.3478	1.19949	0.599745	+	0.800191i	$0.295269\pi$
$684$	0	0
$685$	12.1460	0.464076
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	40.0142	1.52221	0.761107	−	0.648627i	$-0.224656\pi$
$691$	40.0142	1.52221	0.761107	+	0.648627i	$0.224656\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.5503	0.589858
$696$	0	0
$697$	31.1176	1.17866
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31.6851	1.19673	0.598364	−	0.801224i	$-0.295818\pi$
$701$	31.6851	1.19673	0.598364	+	0.801224i	$0.295818\pi$
$702$	0	0
$703$	6.12924	0.231169
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13.7757	−0.518090
$708$	0	0
$709$	36.1918	1.35921	0.679605	−	0.733578i	$-0.262151\pi$
$709$	36.1918	1.35921	0.679605	+	0.733578i	$0.262151\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−17.7060	−0.663095
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	36.5160	1.36182	0.680908	−	0.732369i	$-0.261585\pi$
$719$	36.5160	1.36182	0.680908	+	0.732369i	$0.261585\pi$
$720$	0	0
$721$	−15.4761	−0.576360
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.8259	−0.476343
$726$	0	0
$727$	−35.1730	−1.30450	−0.652248	−	0.758006i	$-0.726174\pi$
$727$	−35.1730	−1.30450	−0.652248	+	0.758006i	$0.726174\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−48.9874	−1.81186
$732$	0	0
$733$	13.9797	0.516350	0.258175	−	0.966098i	$-0.416879\pi$
$733$	13.9797	0.516350	0.258175	+	0.966098i	$0.416879\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	18.6304	0.685330	0.342665	−	0.939458i	$-0.388670\pi$
$739$	18.6304	0.685330	0.342665	+	0.939458i	$0.388670\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.2286	−1.51253	−0.756265	−	0.654266i	$-0.772978\pi$
$743$	−41.2286	−1.51253	−0.756265	+	0.654266i	$0.772978\pi$
$744$	0	0
$745$	−29.4273	−1.07813
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11.6649	0.426226
$750$	0	0
$751$	−19.7567	−0.720931	−0.360466	−	0.932773i	$-0.617382\pi$
$751$	−19.7567	−0.720931	−0.360466	+	0.932773i	$0.617382\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24.4831	0.891033
$756$	0	0
$757$	−9.48574	−0.344765	−0.172383	−	0.985030i	$-0.555147\pi$
$757$	−9.48574	−0.344765	−0.172383	+	0.985030i	$0.555147\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	43.8699	1.59028	0.795141	−	0.606425i	$-0.207397\pi$
$761$	43.8699	1.59028	0.795141	+	0.606425i	$0.207397\pi$
$762$	0	0
$763$	14.4464	0.522994
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	10.9663	0.395457	0.197728	−	0.980257i	$-0.436644\pi$
$769$	10.9663	0.395457	0.197728	+	0.980257i	$0.436644\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.9950	0.970942	0.485471	−	0.874253i	$-0.338648\pi$
$773$	26.9950	0.970942	0.485471	+	0.874253i	$0.338648\pi$
$774$	0	0
$775$	−10.2482	−0.368125
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.0604	0.396280
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−29.7110	−1.06043
$786$	0	0
$787$	−9.80517	−0.349516	−0.174758	−	0.984611i	$-0.555914\pi$
$787$	−9.80517	−0.349516	−0.174758	+	0.984611i	$0.555914\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−24.5700	−0.873608
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.2265	0.751880	0.375940	−	0.926644i	$-0.377320\pi$
$797$	21.2265	0.751880	0.375940	+	0.926644i	$0.377320\pi$
$798$	0	0
$799$	10.5710	0.373974
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−11.8912	−0.419109
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.52618	0.299764	0.149882	−	0.988704i	$-0.452111\pi$
$809$	8.52618	0.299764	0.149882	+	0.988704i	$0.452111\pi$
$810$	0	0
$811$	48.1939	1.69232	0.846158	−	0.532933i	$-0.178910\pi$
$811$	48.1939	1.69232	0.846158	+	0.532933i	$0.178910\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.62317	0.267028
$816$	0	0
$817$	−17.4120	−0.609167
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−55.1829	−1.92590	−0.962949	−	0.269685i	$-0.913081\pi$
$821$	−55.1829	−1.92590	−0.962949	+	0.269685i	$0.913081\pi$
$822$	0	0
$823$	−24.8480	−0.866146	−0.433073	−	0.901359i	$-0.642571\pi$
$823$	−24.8480	−0.866146	−0.433073	+	0.901359i	$0.642571\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.4243	−0.884091	−0.442045	−	0.896993i	$-0.645747\pi$
$827$	−25.4243	−0.884091	−0.442045	+	0.896993i	$0.645747\pi$
$828$	0	0
$829$	12.6733	0.440161	0.220080	−	0.975482i	$-0.429368\pi$
$829$	12.6733	0.440161	0.220080	+	0.975482i	$0.429368\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−24.2664	−0.840781
$834$	0	0
$835$	−28.4176	−0.983432
$836$	0	0
$837$	0	0
$838$	0	0
$839$	49.4966	1.70881	0.854405	−	0.519607i	$-0.173922\pi$
$839$	49.4966	1.70881	0.854405	+	0.519607i	$0.173922\pi$
$840$	0	0
$841$	−11.0921	−0.382486
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.2424	0.627557
$846$	0	0
$847$	17.8008	0.611641
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−16.3034	−0.558874
$852$	0	0
$853$	−2.22570	−0.0762064	−0.0381032	−	0.999274i	$-0.512132\pi$
$853$	−2.22570	−0.0762064	−0.0381032	+	0.999274i	$0.512132\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.5170	1.21324	0.606618	−	0.794993i	$-0.292526\pi$
$857$	35.5170	1.21324	0.606618	+	0.794993i	$0.292526\pi$
$858$	0	0
$859$	41.3643	1.41133	0.705667	−	0.708544i	$-0.250648\pi$
$859$	41.3643	1.41133	0.705667	+	0.708544i	$0.250648\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.9377	1.01909	0.509546	−	0.860443i	$-0.329813\pi$
$863$	29.9377	1.01909	0.509546	+	0.860443i	$0.329813\pi$
$864$	0	0
$865$	−31.4963	−1.07091
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−18.2367	−0.616512
$876$	0	0
$877$	8.97301	0.302997	0.151499	−	0.988457i	$-0.451590\pi$
$877$	8.97301	0.302997	0.151499	+	0.988457i	$0.451590\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47.5950	1.60352	0.801758	−	0.597649i	$-0.203898\pi$
$881$	47.5950	1.60352	0.801758	+	0.597649i	$0.203898\pi$
$882$	0	0
$883$	−0.336218	−0.0113146	−0.00565732	−	0.999984i	$-0.501801\pi$
$883$	−0.336218	−0.0113146	−0.00565732	+	0.999984i	$0.501801\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.07987	−0.204142	−0.102071	−	0.994777i	$-0.532547\pi$
$887$	−6.07987	−0.204142	−0.102071	+	0.994777i	$0.532547\pi$
$888$	0	0
$889$	−36.1283	−1.21170
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.75732	0.125734
$894$	0	0
$895$	18.5029	0.618484
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.3087	0.477223
$900$	0	0
$901$	−54.8135	−1.82610
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−31.0510	−1.03217
$906$	0	0
$907$	−0.847074	−0.0281266	−0.0140633	−	0.999901i	$-0.504477\pi$
$907$	−0.847074	−0.0281266	−0.0140633	+	0.999901i	$0.504477\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	19.7682	0.654950	0.327475	−	0.944860i	$-0.393802\pi$
$911$	19.7682	0.654950	0.327475	+	0.944860i	$0.393802\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−33.9725	−1.12187
$918$	0	0
$919$	−51.1093	−1.68594	−0.842970	−	0.537960i	$-0.819195\pi$
$919$	−51.1093	−1.68594	−0.842970	+	0.537960i	$0.819195\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−9.43635	−0.310265
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−44.7628	−1.46862	−0.734310	−	0.678814i	$-0.762494\pi$
$929$	−44.7628	−1.46862	−0.734310	+	0.678814i	$0.762494\pi$
$930$	0	0
$931$	−8.62520	−0.282679
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−41.4763	−1.35497	−0.677485	−	0.735536i	$-0.736930\pi$
$937$	−41.4763	−1.35497	−0.677485	+	0.735536i	$0.736930\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−33.3121	−1.08594	−0.542972	−	0.839751i	$-0.682701\pi$
$941$	−33.3121	−1.08594	−0.542972	+	0.839751i	$0.682701\pi$
$942$	0	0
$943$	−29.4200	−0.958046
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.49201	−0.0484839	−0.0242419	−	0.999706i	$-0.507717\pi$
$947$	−1.49201	−0.0484839	−0.0242419	+	0.999706i	$0.507717\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.08591	−0.164749	−0.0823745	−	0.996601i	$-0.526250\pi$
$953$	−5.08591	−0.164749	−0.0823745	+	0.996601i	$0.526250\pi$
$954$	0	0
$955$	−5.36749	−0.173688
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.0069	0.452306
$960$	0	0
$961$	−19.5670	−0.631195
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.8517	0.639047
$966$	0	0
$967$	−10.7554	−0.345870	−0.172935	−	0.984933i	$-0.555325\pi$
$967$	−10.7554	−0.345870	−0.172935	+	0.984933i	$0.555325\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−20.5808	−0.660470	−0.330235	−	0.943899i	$-0.607128\pi$
$971$	−20.5808	−0.660470	−0.330235	+	0.943899i	$0.607128\pi$
$972$	0	0
$973$	17.9328	0.574899
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31.1926	−0.997939	−0.498969	−	0.866620i	$-0.666288\pi$
$977$	−31.1926	−0.997939	−0.498969	+	0.866620i	$0.666288\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.66724	0.308337	0.154169	−	0.988045i	$-0.450730\pi$
$983$	9.66724	0.308337	0.154169	+	0.988045i	$0.450730\pi$
$984$	0	0
$985$	−10.0608	−0.320562
$986$	0	0
$987$	0	0
$988$	0	0
$989$	46.3148	1.47272
$990$	0	0
$991$	−26.8843	−0.854007	−0.427004	−	0.904250i	$-0.640431\pi$
$991$	−26.8843	−0.854007	−0.427004	+	0.904250i	$0.640431\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10.2233	0.324102
$996$	0	0
$997$	0.507419	0.0160701	0.00803506	−	0.999968i	$-0.497442\pi$
$997$	0.507419	0.0160701	0.00803506	+	0.999968i	$0.497442\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.a.i.1.1		5
3.2	odd	2		2676.2.a.c.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2676.2.a.c.1.5	✓	5	3.2	odd	2
8028.2.a.i.1.1		5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\(q-1.40326 q^{5} -1.61825 q^{7} +O(q^{10})\)	\(q-1.40326 q^{5} -1.61825 q^{7} +5.53867 q^{17} +1.96865 q^{19} -5.23650 q^{23} -3.03086 q^{25} +4.23177 q^{29} +3.38127 q^{31} +2.27083 q^{35} +3.11342 q^{37} +5.61825 q^{41} -8.84460 q^{43} +1.90857 q^{47} -4.38127 q^{49} -9.89650 q^{53} +3.10551 q^{59} -15.5943 q^{61} +5.61034 q^{67} -0.420525 q^{71} -4.24490 q^{73} +11.2674 q^{79} +10.9595 q^{83} -7.77219 q^{85} -9.21258 q^{89} -2.76253 q^{95} -10.5688 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{5} + 3 q^{7}+O(q^{10}) \) 5 * q + 6 * q^5 + 3 * q^7	\( 5 q + 6 q^{5} + 3 q^{7} + 13 q^{17} - q^{19} - 4 q^{23} + 5 q^{25} - 11 q^{29} - 3 q^{31} + 8 q^{35} + 7 q^{37} + 17 q^{41} + 15 q^{43} + 25 q^{47} - 2 q^{49} + 3 q^{53} + 14 q^{59} - 18 q^{61} + 24 q^{67} + 14 q^{71} - 23 q^{73} + 14 q^{79} + 15 q^{83} + 6 q^{85} + 25 q^{89} + 26 q^{95} - 12 q^{97}+O(q^{100}) \) 5 * q + 6 * q^5 + 3 * q^7 + 13 * q^17 - q^19 - 4 * q^23 + 5 * q^25 - 11 * q^29 - 3 * q^31 + 8 * q^35 + 7 * q^37 + 17 * q^41 + 15 * q^43 + 25 * q^47 - 2 * q^49 + 3 * q^53 + 14 * q^59 - 18 * q^61 + 24 * q^67 + 14 * q^71 - 23 * q^73 + 14 * q^79 + 15 * q^83 + 6 * q^85 + 25 * q^89 + 26 * q^95 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more