LMFDB - Embedded newform 8028.2.h.a.4013.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(4013,8028)]

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, 1, 1]))

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

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

chi := DirichletCharacter("8028.4013");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4013.19
Character	$\chi$	$=$	8028.4013
Dual form			8028.2.h.a.4013.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.82157 q^{5} -3.59530 q^{7} +O(q^{10})$	$q-1.82157 q^{5} -3.59530 q^{7} -2.93431 q^{11} +5.99356i q^{13} +3.70505i q^{17} +7.11417 q^{19} +7.01249 q^{23} -1.68187 q^{25} +4.53327i q^{29} -6.37604 q^{31} +6.54910 q^{35} +6.61919 q^{37} +5.27708i q^{41} +7.69410 q^{43} -3.55271i q^{47} +5.92617 q^{49} -13.3505i q^{53} +5.34506 q^{55} -2.58839 q^{59} +12.8394i q^{61} -10.9177i q^{65} +0.225355i q^{67} -11.0948 q^{71} +10.0170 q^{73} +10.5497 q^{77} -6.96034i q^{79} +1.05232i q^{83} -6.74902i q^{85} +2.97199i q^{89} -21.5486i q^{91} -12.9590 q^{95} +15.0194i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q + 8 q^{7}+O(q^{10}) $ 76 * q + 8 * q^7	$ 76 q + 8 q^{7} + 16 q^{19} + 100 q^{25} - 8 q^{31} + 32 q^{37} - 24 q^{43} + 68 q^{49} + 24 q^{55} + 8 q^{73}+O(q^{100}) $ 76 * q + 8 * q^7 + 16 * q^19 + 100 * q^25 - 8 * q^31 + 32 * q^37 - 24 * q^43 + 68 * q^49 + 24 * q^55 + 8 * q^73

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8028\mathbb{Z}\right)^\times$.

$n$	$893$	$2233$	$4015$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.82157			−0.814632			−0.407316	−	0.913287i	$-0.633535\pi$
$5$	−1.82157			−0.814632			−0.407316	+	0.913287i	$0.633535\pi$
$6$		0			0
$7$	−3.59530			−1.35889			−0.679447	−	0.733724i	$-0.737780\pi$
$7$	−3.59530			−1.35889			−0.679447	+	0.733724i	$0.737780\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.93431			−0.884727			−0.442364	−	0.896836i	$-0.645860\pi$
$11$	−2.93431			−0.884727			−0.442364	+	0.896836i	$0.645860\pi$
$12$		0			0
$13$			5.99356i			1.66231i	0.556038	+	0.831157i	$0.312321\pi$
$13$			5.99356i			1.66231i	−0.556038	+	0.831157i	$0.687679\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.70505i			0.898606i	0.893379	+	0.449303i	$0.148328\pi$
$17$			3.70505i			0.898606i	−0.893379	+	0.449303i	$0.851672\pi$
$18$		0			0
$19$	7.11417			1.63210			0.816051	−	0.577980i	$-0.196159\pi$
$19$	7.11417			1.63210			0.816051	+	0.577980i	$0.196159\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.01249			1.46221			0.731103	−	0.682268i	$-0.239006\pi$
$23$	7.01249			1.46221			0.731103	+	0.682268i	$0.239006\pi$
$24$		0			0
$25$	−1.68187			−0.336374
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.53327i			0.841807i	0.907106	+	0.420903i	$0.138287\pi$
$29$			4.53327i			0.841807i	−0.907106	+	0.420903i	$0.861713\pi$
$30$		0			0
$31$	−6.37604			−1.14517			−0.572585	−	0.819845i	$-0.694059\pi$
$31$	−6.37604			−1.14517			−0.572585	+	0.819845i	$0.694059\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.54910			1.10700
$36$		0			0
$37$	6.61919			1.08819			0.544094	−	0.839024i	$-0.316873\pi$
$37$	6.61919			1.08819			0.544094	+	0.839024i	$0.316873\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.27708i			0.824142i	0.911152	+	0.412071i	$0.135194\pi$
$41$			5.27708i			0.824142i	−0.911152	+	0.412071i	$0.864806\pi$
$42$		0			0
$43$	7.69410			1.17334			0.586670	−	0.809826i	$-0.300439\pi$
$43$	7.69410			1.17334			0.586670	+	0.809826i	$0.300439\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	3.55271i		−	0.518216i	−0.965848	−	0.259108i	$-0.916571\pi$
$47$		−	3.55271i		−	0.518216i	0.965848	−	0.259108i	$-0.0834286\pi$
$48$		0			0
$49$	5.92617			0.846595
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	13.3505i		−	1.83383i	−0.399084	−	0.916914i	$-0.630672\pi$
$53$		−	13.3505i		−	1.83383i	0.399084	−	0.916914i	$-0.369328\pi$
$54$		0			0
$55$	5.34506			0.720727
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.58839			−0.336980			−0.168490	−	0.985703i	$-0.553889\pi$
$59$	−2.58839			−0.336980			−0.168490	+	0.985703i	$0.553889\pi$
$60$		0			0
$61$			12.8394i			1.64392i	0.569543	+	0.821961i	$0.307120\pi$
$61$			12.8394i			1.64392i	−0.569543	+	0.821961i	$0.692880\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	10.9177i		−	1.35418i
$66$		0			0
$67$			0.225355i			0.0275315i	0.999905	+	0.0137658i	$0.00438192\pi$
$67$			0.225355i			0.0275315i	−0.999905	+	0.0137658i	$0.995618\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−11.0948			−1.31672			−0.658358	−	0.752705i	$-0.728749\pi$
$71$	−11.0948			−1.31672			−0.658358	+	0.752705i	$0.728749\pi$
$72$		0			0
$73$	10.0170			1.17240			0.586199	−	0.810167i	$-0.300624\pi$
$73$	10.0170			1.17240			0.586199	+	0.810167i	$0.300624\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.5497			1.20225
$78$		0			0
$79$		−	6.96034i		−	0.783099i	−0.920157	−	0.391550i	$-0.871939\pi$
$79$		−	6.96034i		−	0.783099i	0.920157	−	0.391550i	$-0.128061\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.05232i			0.115507i	0.998331	+	0.0577536i	$0.0183938\pi$
$83$			1.05232i			0.115507i	−0.998331	+	0.0577536i	$0.981606\pi$
$84$		0			0
$85$		−	6.74902i		−	0.732034i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.97199i			0.315030i	0.987517	+	0.157515i	$0.0503483\pi$
$89$			2.97199i			0.315030i	−0.987517	+	0.157515i	$0.949652\pi$
$90$		0			0
$91$		−	21.5486i		−	2.25891i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−12.9590			−1.32956
$96$		0			0
$97$			15.0194i			1.52499i	0.646993	+	0.762496i	$0.276026\pi$
$97$			15.0194i			1.52499i	−0.646993	+	0.762496i	$0.723974\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	0.940479i		−	0.0935811i	−0.998905	−	0.0467906i	$-0.985101\pi$
$101$		−	0.940479i		−	0.0935811i	0.998905	−	0.0467906i	$-0.0148993\pi$
$102$		0			0
$103$			9.99441i			0.984779i	0.870375	+	0.492389i	$0.163876\pi$
$103$			9.99441i			0.984779i	−0.870375	+	0.492389i	$0.836124\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.6740			1.41859			0.709297	−	0.704910i	$-0.249013\pi$
$107$	14.6740			1.41859			0.709297	+	0.704910i	$0.249013\pi$
$108$		0			0
$109$	3.94034			0.377416			0.188708	−	0.982033i	$-0.439570\pi$
$109$	3.94034			0.377416			0.188708	+	0.982033i	$0.439570\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−13.1919			−1.24099			−0.620496	−	0.784210i	$-0.713069\pi$
$113$	−13.1919			−1.24099			−0.620496	+	0.784210i	$0.713069\pi$
$114$		0			0
$115$	−12.7738			−1.19116
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	13.3208i		−	1.22111i
$120$		0			0
$121$	−2.38984			−0.217258
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1715			1.08865
$126$		0			0
$127$	−18.7498			−1.66377			−0.831886	−	0.554947i	$-0.812739\pi$
$127$	−18.7498			−1.66377			−0.831886	+	0.554947i	$0.812739\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			5.91915i			0.517158i	0.965990	+	0.258579i	$0.0832543\pi$
$131$			5.91915i			0.517158i	−0.965990	+	0.258579i	$0.916746\pi$
$132$		0			0
$133$	−25.5776			−2.21786
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.9698			1.36439			0.682197	−	0.731169i	$-0.261025\pi$
$137$	15.9698			1.36439			0.682197	+	0.731169i	$0.261025\pi$
$138$		0			0
$139$	9.49846			0.805649			0.402824	−	0.915277i	$-0.368029\pi$
$139$	9.49846			0.805649			0.402824	+	0.915277i	$0.368029\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	17.5869i		−	1.47069i
$144$		0			0
$145$		−	8.25768i		−	0.685763i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.397407			−0.0325568			−0.0162784	−	0.999867i	$-0.505182\pi$
$149$	−0.397407			−0.0325568			−0.0162784	+	0.999867i	$0.505182\pi$
$150$		0			0
$151$			8.17534i			0.665300i	0.943050	+	0.332650i	$0.107943\pi$
$151$			8.17534i			0.665300i	−0.943050	+	0.332650i	$0.892057\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.6144			0.932893
$156$		0			0
$157$			10.3816i			0.828545i	0.910153	+	0.414273i	$0.135964\pi$
$157$			10.3816i			0.828545i	−0.910153	+	0.414273i	$0.864036\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−25.2120			−1.98698
$162$		0			0
$163$		−	7.72001i		−	0.604678i	−0.953201	−	0.302339i	$-0.902233\pi$
$163$		−	7.72001i		−	0.604678i	0.953201	−	0.302339i	$-0.0977674\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.13593			−0.629577			−0.314788	−	0.949162i	$-0.601934\pi$
$167$	−8.13593			−0.629577			−0.314788	+	0.949162i	$0.601934\pi$
$168$		0			0
$169$	−22.9228			−1.76329
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−22.7110			−1.72668			−0.863342	−	0.504619i	$-0.831633\pi$
$173$	−22.7110			−1.72668			−0.863342	+	0.504619i	$0.831633\pi$
$174$		0			0
$175$	6.04682			0.457097
$176$		0			0
$177$		0			0
$178$		0			0
$179$			17.7330i			1.32543i	0.748874	+	0.662713i	$0.230595\pi$
$179$			17.7330i			1.32543i	−0.748874	+	0.662713i	$0.769405\pi$
$180$		0			0
$181$	−12.1268			−0.901381			−0.450690	−	0.892680i	$-0.648822\pi$
$181$	−12.1268			−0.901381			−0.450690	+	0.892680i	$0.648822\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.0573			−0.886474
$186$		0			0
$187$		−	10.8718i		−	0.795021i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.8192			0.855207			0.427604	−	0.903966i	$-0.359358\pi$
$191$	11.8192			0.855207			0.427604	+	0.903966i	$0.359358\pi$
$192$		0			0
$193$			13.6892i			0.985367i	0.870209	+	0.492683i	$0.163984\pi$
$193$			13.6892i			0.985367i	−0.870209	+	0.492683i	$0.836016\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	13.2735i		−	0.945695i	−0.881144	−	0.472848i	$-0.843226\pi$
$197$		−	13.2735i		−	0.945695i	0.881144	−	0.472848i	$-0.156774\pi$
$198$		0			0
$199$	4.07895			0.289149			0.144574	−	0.989494i	$-0.453819\pi$
$199$	4.07895			0.289149			0.144574	+	0.989494i	$0.453819\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	16.2984i		−	1.14393i
$204$		0			0
$205$		−	9.61259i		−	0.671373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−20.8752			−1.44396
$210$		0			0
$211$	−20.1497			−1.38716			−0.693582	−	0.720377i	$-0.743969\pi$
$211$	−20.1497			−1.38716			−0.693582	+	0.720377i	$0.743969\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−14.0154			−0.955840
$216$		0			0
$217$	22.9238			1.55617
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−22.2064			−1.49377
$222$		0			0
$223$	8.24812	+	12.4486i	0.552335	+	0.833622i
$223$	8.24812	+	12.4486i	0.552335	+	0.833622i
$224$		0			0
$225$		0			0
$226$		0			0
$227$			6.25858i			0.415396i	0.978193	+	0.207698i	$0.0665972\pi$
$227$			6.25858i			0.415396i	−0.978193	+	0.207698i	$0.933403\pi$
$228$		0			0
$229$			20.6548i			1.36491i	0.730929	+	0.682454i	$0.239087\pi$
$229$			20.6548i			1.36491i	−0.730929	+	0.682454i	$0.760913\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.5569			−1.41224			−0.706118	−	0.708094i	$-0.749555\pi$
$233$	−21.5569			−1.41224			−0.706118	+	0.708094i	$0.749555\pi$
$234$		0			0
$235$			6.47153i			0.422156i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2.29312i		−	0.148330i	−0.997246	−	0.0741648i	$-0.976371\pi$
$239$		−	2.29312i		−	0.148330i	0.997246	−	0.0741648i	$-0.0236291\pi$
$240$		0			0
$241$	0.507591			0.0326968			0.0163484	−	0.999866i	$-0.494796\pi$
$241$	0.507591			0.0326968			0.0163484	+	0.999866i	$0.494796\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.7950			−0.689664
$246$		0			0
$247$			42.6392i			2.71307i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	31.4966i		−	1.98805i	−0.109160	−	0.994024i	$-0.534816\pi$
$251$		−	31.4966i		−	1.98805i	0.109160	−	0.994024i	$-0.465184\pi$
$252$		0			0
$253$	−20.5768			−1.29365
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	2.41507i		−	0.150648i	−0.997159	−	0.0753238i	$-0.976001\pi$
$257$		−	2.41507i		−	0.150648i	0.997159	−	0.0753238i	$-0.0239991\pi$
$258$		0			0
$259$	−23.7980			−1.47873
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−17.6419			−1.08784			−0.543922	−	0.839136i	$-0.683061\pi$
$263$	−17.6419			−1.08784			−0.543922	+	0.839136i	$0.683061\pi$
$264$		0			0
$265$			24.3189i			1.49390i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−20.7552			−1.26547			−0.632735	−	0.774369i	$-0.718068\pi$
$269$	−20.7552			−1.26547			−0.632735	+	0.774369i	$0.718068\pi$
$270$		0			0
$271$			6.69737i			0.406837i	0.979092	+	0.203418i	$0.0652051\pi$
$271$			6.69737i			0.406837i	−0.979092	+	0.203418i	$0.934795\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.93512			0.297599
$276$		0			0
$277$		−	22.4023i		−	1.34602i	−0.739632	−	0.673011i	$-0.765001\pi$
$277$		−	22.4023i		−	1.34602i	0.739632	−	0.673011i	$-0.234999\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	24.5352i		−	1.46365i	−0.681493	−	0.731825i	$-0.738669\pi$
$281$		−	24.5352i		−	1.46365i	0.681493	−	0.731825i	$-0.261331\pi$
$282$		0			0
$283$	−21.1749			−1.25872			−0.629359	−	0.777114i	$-0.716683\pi$
$283$	−21.1749			−1.25872			−0.629359	+	0.777114i	$0.716683\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	18.9727i		−	1.11992i
$288$		0			0
$289$	3.27261			0.192507
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−13.8435			−0.808748			−0.404374	−	0.914594i	$-0.632511\pi$
$293$	−13.8435			−0.808748			−0.404374	+	0.914594i	$0.632511\pi$
$294$		0			0
$295$	4.71495			0.274515
$296$		0			0
$297$		0			0
$298$		0			0
$299$			42.0298i			2.43064i
$300$		0			0
$301$	−27.6626			−1.59445
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	23.3880i		−	1.33919i
$306$		0			0
$307$			10.2816i			0.586800i	0.955990	+	0.293400i	$0.0947868\pi$
$307$			10.2816i			0.586800i	−0.955990	+	0.293400i	$0.905213\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.07671			−0.174464			−0.0872320	−	0.996188i	$-0.527802\pi$
$311$	−3.07671			−0.174464			−0.0872320	+	0.996188i	$0.527802\pi$
$312$		0			0
$313$			17.3876i			0.982804i	0.870933	+	0.491402i	$0.163515\pi$
$313$			17.3876i			0.982804i	−0.870933	+	0.491402i	$0.836485\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6.70109i		−	0.376371i	−0.982134	−	0.188185i	$-0.939739\pi$
$317$		−	6.70109i		−	0.376371i	0.982134	−	0.188185i	$-0.0602606\pi$
$318$		0			0
$319$		−	13.3020i		−	0.744769i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			26.3583i			1.46662i
$324$		0			0
$325$		−	10.0804i		−	0.559159i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			12.7731i			0.704202i
$330$		0			0
$331$			14.6195i			0.803559i	0.915736	+	0.401780i	$0.131608\pi$
$331$			14.6195i			0.803559i	−0.915736	+	0.401780i	$0.868392\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	0.410501i		−	0.0224281i
$336$		0			0
$337$			13.1687i			0.717343i	0.933464	+	0.358671i	$0.116770\pi$
$337$			13.1687i			0.717343i	−0.933464	+	0.358671i	$0.883230\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.7093			1.01316
$342$		0			0
$343$	3.86074			0.208461
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	17.4430i		−	0.936387i	−0.883626	−	0.468193i	$-0.844905\pi$
$347$		−	17.4430i		−	0.936387i	0.883626	−	0.468193i	$-0.155095\pi$
$348$		0			0
$349$	2.23039			0.119390			0.0596949	−	0.998217i	$-0.480987\pi$
$349$	2.23039			0.119390			0.0596949	+	0.998217i	$0.480987\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			23.0713i			1.22796i	0.789320	+	0.613982i	$0.210433\pi$
$353$			23.0713i			1.22796i	−0.789320	+	0.613982i	$0.789567\pi$
$354$		0			0
$355$	20.2101			1.07264
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.30173i		−	0.438148i	−0.975708	−	0.219074i	$-0.929696\pi$
$359$		−	8.30173i		−	0.438148i	0.975708	−	0.219074i	$-0.0703037\pi$
$360$		0			0
$361$	31.6114			1.66376
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−18.2467			−0.955074
$366$		0			0
$367$	15.0548			0.785854			0.392927	−	0.919570i	$-0.371462\pi$
$367$	15.0548			0.785854			0.392927	+	0.919570i	$0.371462\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			47.9989i			2.49198i
$372$		0			0
$373$			29.9699i			1.55178i	0.630866	+	0.775892i	$0.282700\pi$
$373$			29.9699i			1.55178i	−0.630866	+	0.775892i	$0.717300\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−27.1704			−1.39935
$378$		0			0
$379$	30.1720			1.54983			0.774915	−	0.632065i	$-0.217792\pi$
$379$	30.1720			1.54983			0.774915	+	0.632065i	$0.217792\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.99612			0.408583			0.204291	−	0.978910i	$-0.434511\pi$
$383$	7.99612			0.408583			0.204291	+	0.978910i	$0.434511\pi$
$384$		0			0
$385$	−19.2171			−0.979393
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	31.9560i		−	1.62023i	−0.586269	−	0.810117i	$-0.699404\pi$
$389$		−	31.9560i		−	1.62023i	0.586269	−	0.810117i	$-0.300596\pi$
$390$		0			0
$391$			25.9816i			1.31395i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			12.6788i			0.637938i
$396$		0			0
$397$			6.05135i			0.303708i	0.988403	+	0.151854i	$0.0485244\pi$
$397$			6.05135i			0.303708i	−0.988403	+	0.151854i	$0.951476\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	31.5173i		−	1.57390i	−0.617018	−	0.786949i	$-0.711660\pi$
$401$		−	31.5173i		−	1.57390i	0.617018	−	0.786949i	$-0.288340\pi$
$402$		0			0
$403$		−	38.2152i		−	1.90363i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−19.4227			−0.962750
$408$		0			0
$409$		−	24.0389i		−	1.18865i	−0.804226	−	0.594324i	$-0.797420\pi$
$409$		−	24.0389i		−	1.18865i	0.804226	−	0.594324i	$-0.202580\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.30604			0.457921
$414$		0			0
$415$		−	1.91688i		−	0.0940959i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			29.4207i			1.43730i	0.695373	+	0.718649i	$0.255239\pi$
$419$			29.4207i			1.43730i	−0.695373	+	0.718649i	$0.744761\pi$
$420$		0			0
$421$			17.4455i			0.850240i	0.905137	+	0.425120i	$0.139768\pi$
$421$			17.4455i			0.850240i	−0.905137	+	0.425120i	$0.860232\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	6.23141i		−	0.302268i
$426$		0			0
$427$		−	46.1616i		−	2.23392i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.25866			−0.253301			−0.126650	−	0.991947i	$-0.540423\pi$
$431$	−5.25866			−0.253301			−0.126650	+	0.991947i	$0.540423\pi$
$432$		0			0
$433$	−12.5845			−0.604771			−0.302385	−	0.953186i	$-0.597783\pi$
$433$	−12.5845			−0.604771			−0.302385	+	0.953186i	$0.597783\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	49.8880			2.38647
$438$		0			0
$439$		−	32.3809i		−	1.54546i	−0.634737	−	0.772728i	$-0.718892\pi$
$439$		−	32.3809i		−	1.54546i	0.634737	−	0.772728i	$-0.281108\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			1.88216i			0.0894241i	0.999000	+	0.0447120i	$0.0142370\pi$
$443$			1.88216i			0.0894241i	−0.999000	+	0.0447120i	$0.985763\pi$
$444$		0			0
$445$		−	5.41370i		−	0.256634i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−19.7727			−0.933133			−0.466567	−	0.884486i	$-0.654509\pi$
$449$	−19.7727			−0.933133			−0.466567	+	0.884486i	$0.654509\pi$
$450$		0			0
$451$		−	15.4846i		−	0.729140i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			39.2524i			1.84018i
$456$		0			0
$457$		−	1.80062i		−	0.0842295i	−0.999113	−	0.0421148i	$-0.986590\pi$
$457$		−	1.80062i		−	0.0842295i	0.999113	−	0.0421148i	$-0.0134095\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	7.74553i		−	0.360745i	−0.983598	−	0.180373i	$-0.942270\pi$
$461$		−	7.74553i		−	0.360745i	0.983598	−	0.180373i	$-0.0577304\pi$
$462$		0			0
$463$	6.37315			0.296185			0.148093	−	0.988973i	$-0.452687\pi$
$463$	6.37315			0.296185			0.148093	+	0.988973i	$0.452687\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	38.0782			1.76205			0.881026	−	0.473069i	$-0.156854\pi$
$467$	38.0782			1.76205			0.881026	+	0.473069i	$0.156854\pi$
$468$		0			0
$469$		−	0.810220i		−	0.0374125i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−22.5769			−1.03809
$474$		0			0
$475$	−11.9651			−0.548997
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	26.7222i		−	1.22097i	−0.792028	−	0.610485i	$-0.790975\pi$
$479$		−	26.7222i		−	1.22097i	0.792028	−	0.610485i	$-0.209025\pi$
$480$		0			0
$481$			39.6725i			1.80891i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	27.3590i		−	1.24231i
$486$		0			0
$487$	−1.34182			−0.0608037			−0.0304018	−	0.999538i	$-0.509679\pi$
$487$	−1.34182			−0.0608037			−0.0304018	+	0.999538i	$0.509679\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	11.5719			0.522233			0.261117	−	0.965307i	$-0.415909\pi$
$491$	11.5719			0.522233			0.261117	+	0.965307i	$0.415909\pi$
$492$		0			0
$493$	−16.7960			−0.756453
$494$		0			0
$495$		0			0
$496$		0			0
$497$	39.8893			1.78928
$498$		0			0
$499$	−5.20106			−0.232831			−0.116416	−	0.993201i	$-0.537140\pi$
$499$	−5.20106			−0.232831			−0.116416	+	0.993201i	$0.537140\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.90687			0.263374			0.131687	−	0.991291i	$-0.457961\pi$
$503$	5.90687			0.263374			0.131687	+	0.991291i	$0.457961\pi$
$504$		0			0
$505$			1.71315i			0.0762342i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	28.3989i		−	1.25876i	−0.777098	−	0.629379i	$-0.783309\pi$
$509$		−	28.3989i		−	1.25876i	0.777098	−	0.629379i	$-0.216691\pi$
$510$		0			0
$511$	−36.0140			−1.59317
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	18.2056i		−	0.802233i
$516$		0			0
$517$			10.4248i			0.458480i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.0181			−1.18369			−0.591843	−	0.806053i	$-0.701599\pi$
$521$	−27.0181			−1.18369			−0.591843	+	0.806053i	$0.701599\pi$
$522$		0			0
$523$		−	45.3648i		−	1.98366i	−0.127551	−	0.991832i	$-0.540712\pi$
$523$		−	45.3648i		−	1.98366i	0.127551	−	0.991832i	$-0.459288\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	23.6235i		−	1.02906i
$528$		0			0
$529$	26.1750			1.13804
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−31.6285			−1.36998
$534$		0			0
$535$	−26.7299			−1.15563
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−17.3892			−0.749006
$540$		0			0
$541$		−	38.6289i		−	1.66079i	−0.557177	−	0.830394i	$-0.688115\pi$
$541$		−	38.6289i		−	1.66079i	0.557177	−	0.830394i	$-0.311885\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7.17762			−0.307456
$546$		0			0
$547$	−0.210736			−0.00901042			−0.00450521	−	0.999990i	$-0.501434\pi$
$547$	−0.210736			−0.00901042			−0.00450521	+	0.999990i	$0.501434\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			32.2504i			1.37391i
$552$		0			0
$553$			25.0245i			1.06415i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.8249			−0.967121			−0.483560	−	0.875311i	$-0.660657\pi$
$557$	−22.8249			−0.967121			−0.483560	+	0.875311i	$0.660657\pi$
$558$		0			0
$559$			46.1151i			1.95046i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.3756			−0.900876			−0.450438	−	0.892808i	$-0.648732\pi$
$563$	−21.3756			−0.900876			−0.450438	+	0.892808i	$0.648732\pi$
$564$		0			0
$565$	24.0301			1.01095
$566$		0			0
$567$		0			0
$568$		0			0
$569$	38.1375			1.59881			0.799404	−	0.600794i	$-0.205149\pi$
$569$	38.1375			1.59881			0.799404	+	0.600794i	$0.205149\pi$
$570$		0			0
$571$		−	23.7573i		−	0.994210i	−0.867690	−	0.497105i	$-0.834396\pi$
$571$		−	23.7573i		−	0.994210i	0.867690	−	0.497105i	$-0.165604\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.7941			−0.491848
$576$		0			0
$577$	−29.4946			−1.22788			−0.613938	−	0.789354i	$-0.710416\pi$
$577$	−29.4946			−1.22788			−0.613938	+	0.789354i	$0.710416\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	3.78341i		−	0.156962i
$582$		0			0
$583$			39.1744i			1.62244i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−44.5425			−1.83847			−0.919233	−	0.393714i	$-0.871190\pi$
$587$	−44.5425			−1.83847			−0.919233	+	0.393714i	$0.871190\pi$
$588$		0			0
$589$	−45.3602			−1.86903
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.32982			−0.136739			−0.0683697	−	0.997660i	$-0.521780\pi$
$593$	−3.32982			−0.136739			−0.0683697	+	0.997660i	$0.521780\pi$
$594$		0			0
$595$			24.2647i			0.994757i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			11.1459i			0.455407i	0.973731	+	0.227704i	$0.0731217\pi$
$599$			11.1459i			0.455407i	−0.973731	+	0.227704i	$0.926878\pi$
$600$		0			0
$601$		−	10.2968i		−	0.420015i	−0.977700	−	0.210007i	$-0.932651\pi$
$601$		−	10.2968i		−	0.420015i	0.977700	−	0.210007i	$-0.0673488\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.35326			0.176985
$606$		0			0
$607$		−	8.77381i		−	0.356118i	−0.984020	−	0.178059i	$-0.943018\pi$
$607$		−	8.77381i		−	0.356118i	0.984020	−	0.178059i	$-0.0569818\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	21.2934			0.861439
$612$		0			0
$613$		−	10.4228i		−	0.420974i	−0.977597	−	0.210487i	$-0.932495\pi$
$613$		−	10.4228i		−	0.420974i	0.977597	−	0.210487i	$-0.0675050\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	32.8525i		−	1.32259i	−0.750126	−	0.661295i	$-0.770007\pi$
$617$		−	32.8525i		−	1.32259i	0.750126	−	0.661295i	$-0.229993\pi$
$618$		0			0
$619$		−	4.92094i		−	0.197789i	−0.995098	−	0.0988947i	$-0.968469\pi$
$619$		−	4.92094i		−	0.197789i	0.995098	−	0.0988947i	$-0.0315307\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	10.6852i		−	0.428093i
$624$		0			0
$625$	−13.7620			−0.550479
$626$		0			0
$627$		0			0
$628$		0			0
$629$			24.5244i			0.977853i
$630$		0			0
$631$		−	8.81382i		−	0.350873i	−0.984491	−	0.175436i	$-0.943866\pi$
$631$		−	8.81382i		−	0.350873i	0.984491	−	0.175436i	$-0.0561336\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	34.1541			1.35536
$636$		0			0
$637$			35.5188i			1.40731i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.0382			0.435983			0.217991	−	0.975951i	$-0.430050\pi$
$641$	11.0382			0.435983			0.217991	+	0.975951i	$0.430050\pi$
$642$		0			0
$643$	13.9369			0.549618			0.274809	−	0.961499i	$-0.411385\pi$
$643$	13.9369			0.549618			0.274809	+	0.961499i	$0.411385\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			15.6147i			0.613876i	0.951730	+	0.306938i	$0.0993044\pi$
$647$			15.6147i			0.613876i	−0.951730	+	0.306938i	$0.900696\pi$
$648$		0			0
$649$	7.59514			0.298135
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−27.7720			−1.08680			−0.543402	−	0.839473i	$-0.682864\pi$
$653$	−27.7720			−1.08680			−0.543402	+	0.839473i	$0.682864\pi$
$654$		0			0
$655$		−	10.7822i		−	0.421294i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			30.3202i			1.18111i	0.806998	+	0.590554i	$0.201091\pi$
$659$			30.3202i			1.18111i	−0.806998	+	0.590554i	$0.798909\pi$
$660$		0			0
$661$			35.5499i			1.38273i	0.722505	+	0.691366i	$0.242991\pi$
$661$			35.5499i			1.38273i	−0.722505	+	0.691366i	$0.757009\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	46.5914			1.80674
$666$		0			0
$667$			31.7895i			1.23089i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	37.6749i		−	1.45442i
$672$		0			0
$673$	33.9354			1.30811			0.654057	−	0.756445i	$-0.273066\pi$
$673$	33.9354			1.30811			0.654057	+	0.756445i	$0.273066\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	39.6185i		−	1.52266i	−0.648362	−	0.761332i	$-0.724546\pi$
$677$		−	39.6185i		−	1.52266i	0.648362	−	0.761332i	$-0.275454\pi$
$678$		0			0
$679$		−	53.9993i		−	2.07230i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	37.5635i		−	1.43733i	−0.695357	−	0.718664i	$-0.744754\pi$
$683$		−	37.5635i		−	1.43733i	0.695357	−	0.718664i	$-0.255246\pi$
$684$		0			0
$685$	−29.0902			−1.11148
$686$		0			0
$687$		0			0
$688$		0			0
$689$	80.0169			3.04840
$690$		0			0
$691$		−	31.3005i		−	1.19073i	−0.803456	−	0.595363i	$-0.797008\pi$
$691$		−	31.3005i		−	1.19073i	0.803456	−	0.595363i	$-0.202992\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−17.3021			−0.656308
$696$		0			0
$697$	−19.5518			−0.740579
$698$		0			0
$699$		0			0
$700$		0			0
$701$			42.2384i			1.59532i	0.603106	+	0.797661i	$0.293930\pi$
$701$			42.2384i			1.59532i	−0.603106	+	0.797661i	$0.706070\pi$
$702$		0			0
$703$	47.0900			1.77603
$704$		0			0
$705$		0			0
$706$		0			0
$707$			3.38130i			0.127167i
$708$		0			0
$709$		−	14.1995i		−	0.533274i	−0.963797	−	0.266637i	$-0.914087\pi$
$709$		−	14.1995i		−	0.533274i	0.963797	−	0.266637i	$-0.0859125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−44.7119			−1.67447
$714$		0			0
$715$			32.0359i			1.19808i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			16.0836i			0.599816i	0.953968	+	0.299908i	$0.0969560\pi$
$719$			16.0836i			0.599816i	−0.953968	+	0.299908i	$0.903044\pi$
$720$		0			0
$721$		−	35.9329i		−	1.33821i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	7.62436i		−	0.283162i
$726$		0			0
$727$	−7.42296			−0.275302			−0.137651	−	0.990481i	$-0.543955\pi$
$727$	−7.42296			−0.275302			−0.137651	+	0.990481i	$0.543955\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			28.5070i			1.05437i
$732$		0			0
$733$	−8.63629			−0.318989			−0.159494	−	0.987199i	$-0.550986\pi$
$733$	−8.63629			−0.318989			−0.159494	+	0.987199i	$0.550986\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	0.661262i		−	0.0243579i
$738$		0			0
$739$			35.9787i			1.32350i	0.749725	+	0.661750i	$0.230186\pi$
$739$			35.9787i			1.32350i	−0.749725	+	0.661750i	$0.769814\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	12.0775i		−	0.443080i	−0.975151	−	0.221540i	$-0.928892\pi$
$743$		−	12.0775i		−	0.443080i	0.975151	−	0.221540i	$-0.0711084\pi$
$744$		0			0
$745$	0.723906			0.0265219
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−52.7576			−1.92772
$750$		0			0
$751$	−31.5918			−1.15280			−0.576401	−	0.817167i	$-0.695543\pi$
$751$	−31.5918			−1.15280			−0.576401	+	0.817167i	$0.695543\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	14.8920i		−	0.541975i
$756$		0			0
$757$		−	43.0921i		−	1.56621i	−0.621889	−	0.783105i	$-0.713635\pi$
$757$		−	43.0921i		−	1.56621i	0.621889	−	0.783105i	$-0.286365\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.22720			−0.261986			−0.130993	−	0.991383i	$-0.541817\pi$
$761$	−7.22720			−0.261986			−0.130993	+	0.991383i	$0.541817\pi$
$762$		0			0
$763$	−14.1667			−0.512869
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	15.5137i		−	0.560167i
$768$		0			0
$769$	17.2510			0.622088			0.311044	−	0.950395i	$-0.399321\pi$
$769$	17.2510			0.622088			0.311044	+	0.950395i	$0.399321\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	10.3480			0.372193			0.186096	−	0.982532i	$-0.440416\pi$
$773$	10.3480			0.372193			0.186096	+	0.982532i	$0.440416\pi$
$774$		0			0
$775$	10.7237			0.385205
$776$		0			0
$777$		0			0
$778$		0			0
$779$			37.5420i			1.34508i
$780$		0			0
$781$	32.5557			1.16493
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	18.9109i		−	0.674960i
$786$		0			0
$787$			48.9133i			1.74357i	0.489888	+	0.871785i	$0.337038\pi$
$787$			48.9133i			1.74357i	−0.489888	+	0.871785i	$0.662962\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	47.4289			1.68638
$792$		0			0
$793$	−76.9540			−2.73272
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	13.4894i		−	0.477818i	−0.971042	−	0.238909i	$-0.923210\pi$
$797$		−	13.4894i		−	0.477818i	0.971042	−	0.238909i	$-0.0767898\pi$
$798$		0			0
$799$	13.1630			0.465673
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−29.3929			−1.03725
$804$		0			0
$805$	45.9255			1.61866
$806$		0			0
$807$		0			0
$808$		0			0
$809$	12.5609			0.441618			0.220809	−	0.975317i	$-0.429130\pi$
$809$	12.5609			0.441618			0.220809	+	0.975317i	$0.429130\pi$
$810$		0			0
$811$		−	3.27161i		−	0.114882i	−0.998349	−	0.0574408i	$-0.981706\pi$
$811$		−	3.27161i		−	0.114882i	0.998349	−	0.0574408i	$-0.0182941\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			14.0626i			0.492590i
$816$		0			0
$817$	54.7371			1.91501
$818$		0			0
$819$		0			0
$820$		0			0
$821$			11.8605i			0.413934i	0.978348	+	0.206967i	$0.0663593\pi$
$821$			11.8605i			0.413934i	−0.978348	+	0.206967i	$0.933641\pi$
$822$		0			0
$823$		−	4.83048i		−	0.168380i	−0.996450	−	0.0841900i	$-0.973170\pi$
$823$		−	4.83048i		−	0.168380i	0.996450	−	0.0841900i	$-0.0268303\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.45844			−0.120262			−0.0601308	−	0.998191i	$-0.519152\pi$
$827$	−3.45844			−0.120262			−0.0601308	+	0.998191i	$0.519152\pi$
$828$		0			0
$829$			38.4503i			1.33544i	0.744415	+	0.667718i	$0.232729\pi$
$829$			38.4503i			1.33544i	−0.744415	+	0.667718i	$0.767271\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			21.9567i			0.760756i
$834$		0			0
$835$	14.8202			0.512874
$836$		0			0
$837$		0			0
$838$		0			0
$839$	27.3832			0.945373			0.472686	−	0.881231i	$-0.343284\pi$
$839$	27.3832			0.945373			0.472686	+	0.881231i	$0.343284\pi$
$840$		0			0
$841$	8.44949			0.291362
$842$		0			0
$843$		0			0
$844$		0			0
$845$	41.7555			1.43643
$846$		0			0
$847$	8.59218			0.295231
$848$		0			0
$849$		0			0
$850$		0			0
$851$	46.4170			1.59115
$852$		0			0
$853$			35.8903i			1.22886i	0.788971	+	0.614431i	$0.210614\pi$
$853$			35.8903i			1.22886i	−0.788971	+	0.614431i	$0.789386\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			4.40626i			0.150515i	0.997164	+	0.0752575i	$0.0239779\pi$
$857$			4.40626i			0.150515i	−0.997164	+	0.0752575i	$0.976022\pi$
$858$		0			0
$859$			40.3941i			1.37823i	0.724653	+	0.689114i	$0.242000\pi$
$859$			40.3941i			1.37823i	−0.724653	+	0.689114i	$0.758000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−5.44915			−0.185491			−0.0927456	−	0.995690i	$-0.529564\pi$
$863$	−5.44915			−0.185491			−0.0927456	+	0.995690i	$0.529564\pi$
$864$		0			0
$865$	41.3697			1.40661
$866$		0			0
$867$		0			0
$868$		0			0
$869$			20.4238i			0.692829i
$870$		0			0
$871$	−1.35068			−0.0457661
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−43.7602			−1.47937
$876$		0			0
$877$		−	50.6924i		−	1.71176i	−0.517174	−	0.855880i	$-0.673016\pi$
$877$		−	50.6924i		−	1.71176i	0.517174	−	0.855880i	$-0.326984\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			31.2034i			1.05127i	0.850711	+	0.525634i	$0.176172\pi$
$881$			31.2034i			1.05127i	−0.850711	+	0.525634i	$0.823828\pi$
$882$		0			0
$883$			5.45363i			0.183529i	0.995781	+	0.0917646i	$0.0292507\pi$
$883$			5.45363i			0.183529i	−0.995781	+	0.0917646i	$0.970749\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	21.8482i		−	0.733591i	−0.930302	−	0.366795i	$-0.880455\pi$
$887$		−	21.8482i		−	0.733591i	0.930302	−	0.366795i	$-0.119545\pi$
$888$		0			0
$889$	67.4110			2.26089
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	25.2746i		−	0.845782i
$894$		0			0
$895$		−	32.3019i		−	1.07973i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	28.9043i		−	0.964012i
$900$		0			0
$901$	49.4642			1.64789
$902$		0			0
$903$		0			0
$904$		0			0
$905$	22.0899			0.734294
$906$		0			0
$907$	−56.5692			−1.87835			−0.939175	−	0.343440i	$-0.888408\pi$
$907$	−56.5692			−1.87835			−0.939175	+	0.343440i	$0.888408\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			53.2039i			1.76272i	0.472441	+	0.881362i	$0.343373\pi$
$911$			53.2039i			1.76272i	−0.472441	+	0.881362i	$0.656627\pi$
$912$		0			0
$913$		−	3.08783i		−	0.102192i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	21.2811i		−	0.702764i
$918$		0			0
$919$		−	29.5879i		−	0.976016i	−0.872839	−	0.488008i	$-0.837724\pi$
$919$		−	29.5879i		−	0.976016i	0.872839	−	0.488008i	$-0.162276\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	66.4976i		−	2.18880i
$924$		0			0
$925$	−11.1326			−0.366038
$926$		0			0
$927$		0			0
$928$		0			0
$929$			38.1910i			1.25301i	0.779419	+	0.626503i	$0.215514\pi$
$929$			38.1910i			1.25301i	−0.779419	+	0.626503i	$0.784486\pi$
$930$		0			0
$931$	42.1598			1.38173
$932$		0			0
$933$		0			0
$934$		0			0
$935$			19.8037i			0.647650i
$936$		0			0
$937$		−	7.78924i		−	0.254463i	−0.991873	−	0.127232i	$-0.959391\pi$
$937$		−	7.78924i		−	0.254463i	0.991873	−	0.127232i	$-0.0406092\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	10.8396i		−	0.353360i	−0.984268	−	0.176680i	$-0.943464\pi$
$941$		−	10.8396i		−	0.353360i	0.984268	−	0.176680i	$-0.0565357\pi$
$942$		0			0
$943$			37.0055i			1.20506i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			53.5127i			1.73893i	0.493996	+	0.869464i	$0.335536\pi$
$947$			53.5127i			1.73893i	−0.493996	+	0.869464i	$0.664464\pi$
$948$		0			0
$949$			60.0373i			1.94889i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.34920			0.238064			0.119032	−	0.992890i	$-0.462021\pi$
$953$	7.34920			0.238064			0.119032	+	0.992890i	$0.462021\pi$
$954$		0			0
$955$	−21.5295			−0.696680
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−57.4162			−1.85407
$960$		0			0
$961$	9.65388			0.311415
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	24.9358i		−	0.802712i
$966$		0			0
$967$		−	2.53611i		−	0.0815560i	−0.999168	−	0.0407780i	$-0.987016\pi$
$967$		−	2.53611i		−	0.0815560i	0.999168	−	0.0407780i	$-0.0129836\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.30595			−0.0740016			−0.0370008	−	0.999315i	$-0.511780\pi$
$971$	−2.30595			−0.0740016			−0.0370008	+	0.999315i	$0.511780\pi$
$972$		0			0
$973$	−34.1498			−1.09479
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.416021			−0.0133097			−0.00665484	−	0.999978i	$-0.502118\pi$
$977$	−0.416021			−0.0133097			−0.00665484	+	0.999978i	$0.502118\pi$
$978$		0			0
$979$		−	8.72074i		−	0.278716i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−43.0879			−1.37429			−0.687146	−	0.726520i	$-0.741137\pi$
$983$	−43.0879			−1.37429			−0.687146	+	0.726520i	$0.741137\pi$
$984$		0			0
$985$			24.1786i			0.770394i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	53.9548			1.71566
$990$		0			0
$991$			9.21313i			0.292665i	0.989235	+	0.146332i	$0.0467469\pi$
$991$			9.21313i			0.292665i	−0.989235	+	0.146332i	$0.953253\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7.43010			−0.235550
$996$		0			0
$997$	−38.3510			−1.21459			−0.607295	−	0.794477i	$-0.707745\pi$
$997$	−38.3510			−1.21459			−0.607295	+	0.794477i	$0.707745\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.h.a.4013.19	✓	76
3.2	odd	2	inner	8028.2.h.a.4013.57	yes	76
223.222	odd	2	inner	8028.2.h.a.4013.58	yes	76
669.668	even	2	inner	8028.2.h.a.4013.20	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8028.2.h.a.4013.19	✓	76	1.1	even	1	trivial
8028.2.h.a.4013.20	yes	76	669.668	even	2	inner
8028.2.h.a.4013.57	yes	76	3.2	odd	2	inner
8028.2.h.a.4013.58	yes	76	223.222	odd	2	inner

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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.82157 q^{5} -3.59530 q^{7} +O(q^{10})\)	\(q-1.82157 q^{5} -3.59530 q^{7} -2.93431 q^{11} +5.99356i q^{13} +3.70505i q^{17} +7.11417 q^{19} +7.01249 q^{23} -1.68187 q^{25} +4.53327i q^{29} -6.37604 q^{31} +6.54910 q^{35} +6.61919 q^{37} +5.27708i q^{41} +7.69410 q^{43} -3.55271i q^{47} +5.92617 q^{49} -13.3505i q^{53} +5.34506 q^{55} -2.58839 q^{59} +12.8394i q^{61} -10.9177i q^{65} +0.225355i q^{67} -11.0948 q^{71} +10.0170 q^{73} +10.5497 q^{77} -6.96034i q^{79} +1.05232i q^{83} -6.74902i q^{85} +2.97199i q^{89} -21.5486i q^{91} -12.9590 q^{95} +15.0194i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 8 q^{7}+O(q^{10}) \) 76 * q + 8 * q^7	\( 76 q + 8 q^{7} + 16 q^{19} + 100 q^{25} - 8 q^{31} + 32 q^{37} - 24 q^{43} + 68 q^{49} + 24 q^{55} + 8 q^{73}+O(q^{100}) \) 76 * q + 8 * q^7 + 16 * q^19 + 100 * q^25 - 8 * q^31 + 32 * q^37 - 24 * q^43 + 68 * q^49 + 24 * q^55 + 8 * q^73

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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