LMFDB - Embedded newform 8028.2.h.a.4013.7

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			4013.7
Character	$\chi$	$=$	8028.4013
Dual form			8028.2.h.a.4013.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.80256 q^{5} +0.208948 q^{7} +O(q^{10})$	$q-3.80256 q^{5} +0.208948 q^{7} +1.33035 q^{11} +4.08205i q^{13} +0.539242i q^{17} -7.10710 q^{19} -4.33219 q^{23} +9.45948 q^{25} -5.64314i q^{29} -5.02087 q^{31} -0.794538 q^{35} +10.5243 q^{37} -9.40292i q^{41} -3.38328 q^{43} +5.78929i q^{47} -6.95634 q^{49} +6.51084i q^{53} -5.05874 q^{55} -9.05797 q^{59} +6.92448i q^{61} -15.5223i q^{65} -1.17012i q^{67} -7.05579 q^{71} +0.681020 q^{73} +0.277974 q^{77} +10.6359i q^{79} +12.9000i q^{83} -2.05050i q^{85} -11.8822i q^{89} +0.852936i q^{91} +27.0252 q^{95} +14.9736i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.80256			−1.70056			−0.850279	−	0.526333i	$-0.823567\pi$
$5$	−3.80256			−1.70056			−0.850279	+	0.526333i	$0.823567\pi$
$6$		0			0
$7$	0.208948			0.0789749			0.0394875	−	0.999220i	$-0.487427\pi$
$7$	0.208948			0.0789749			0.0394875	+	0.999220i	$0.487427\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.33035			0.401116			0.200558	−	0.979682i	$-0.435724\pi$
$11$	1.33035			0.401116			0.200558	+	0.979682i	$0.435724\pi$
$12$		0			0
$13$			4.08205i			1.13216i	0.824351	+	0.566079i	$0.191540\pi$
$13$			4.08205i			1.13216i	−0.824351	+	0.566079i	$0.808460\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.539242i			0.130785i	0.997860	+	0.0653927i	$0.0208300\pi$
$17$			0.539242i			0.130785i	−0.997860	+	0.0653927i	$0.979170\pi$
$18$		0			0
$19$	−7.10710			−1.63048			−0.815240	−	0.579124i	$-0.803395\pi$
$19$	−7.10710			−1.63048			−0.815240	+	0.579124i	$0.803395\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.33219			−0.903325			−0.451662	−	0.892189i	$-0.649169\pi$
$23$	−4.33219			−0.903325			−0.451662	+	0.892189i	$0.649169\pi$
$24$		0			0
$25$	9.45948			1.89190
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.64314i		−	1.04790i	−0.851748	−	0.523952i	$-0.824457\pi$
$29$		−	5.64314i		−	1.04790i	0.851748	−	0.523952i	$-0.175543\pi$
$30$		0			0
$31$	−5.02087			−0.901776			−0.450888	−	0.892581i	$-0.648893\pi$
$31$	−5.02087			−0.901776			−0.450888	+	0.892581i	$0.648893\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.794538			−0.134301
$36$		0			0
$37$	10.5243			1.73019			0.865094	−	0.501610i	$-0.167259\pi$
$37$	10.5243			1.73019			0.865094	+	0.501610i	$0.167259\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.40292i		−	1.46849i	−0.678885	−	0.734244i	$-0.737537\pi$
$41$		−	9.40292i		−	1.46849i	0.678885	−	0.734244i	$-0.262463\pi$
$42$		0			0
$43$	−3.38328			−0.515945			−0.257973	−	0.966152i	$-0.583054\pi$
$43$	−3.38328			−0.515945			−0.257973	+	0.966152i	$0.583054\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.78929i			0.844455i	0.906490	+	0.422228i	$0.138752\pi$
$47$			5.78929i			0.844455i	−0.906490	+	0.422228i	$0.861248\pi$
$48$		0			0
$49$	−6.95634			−0.993763
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.51084i			0.894333i	0.894451	+	0.447166i	$0.147567\pi$
$53$			6.51084i			0.894333i	−0.894451	+	0.447166i	$0.852433\pi$
$54$		0			0
$55$	−5.05874			−0.682121
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.05797			−1.17925			−0.589623	−	0.807678i	$-0.700724\pi$
$59$	−9.05797			−1.17925			−0.589623	+	0.807678i	$0.700724\pi$
$60$		0			0
$61$			6.92448i			0.886588i	0.896376	+	0.443294i	$0.146190\pi$
$61$			6.92448i			0.886588i	−0.896376	+	0.443294i	$0.853810\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	15.5223i		−	1.92530i
$66$		0			0
$67$		−	1.17012i		−	0.142952i	−0.997442	−	0.0714762i	$-0.977229\pi$
$67$		−	1.17012i		−	0.142952i	0.997442	−	0.0714762i	$-0.0227710\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.05579			−0.837368			−0.418684	−	0.908132i	$-0.637509\pi$
$71$	−7.05579			−0.837368			−0.418684	+	0.908132i	$0.637509\pi$
$72$		0			0
$73$	0.681020			0.0797074			0.0398537	−	0.999206i	$-0.487311\pi$
$73$	0.681020			0.0797074			0.0398537	+	0.999206i	$0.487311\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.277974			0.0316781
$78$		0			0
$79$			10.6359i			1.19663i	0.801260	+	0.598316i	$0.204163\pi$
$79$			10.6359i			1.19663i	−0.801260	+	0.598316i	$0.795837\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			12.9000i			1.41596i	0.706233	+	0.707979i	$0.250393\pi$
$83$			12.9000i			1.41596i	−0.706233	+	0.707979i	$0.749607\pi$
$84$		0			0
$85$		−	2.05050i		−	0.222408i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	11.8822i		−	1.25952i	−0.776791	−	0.629758i	$-0.783154\pi$
$89$		−	11.8822i		−	1.25952i	0.776791	−	0.629758i	$-0.216846\pi$
$90$		0			0
$91$			0.852936i			0.0894120i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	27.0252			2.77272
$96$		0			0
$97$			14.9736i			1.52034i	0.649724	+	0.760170i	$0.274885\pi$
$97$			14.9736i			1.52034i	−0.649724	+	0.760170i	$0.725115\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	14.5035i		−	1.44315i	−0.692336	−	0.721575i	$-0.743418\pi$
$101$		−	14.5035i		−	1.44315i	0.692336	−	0.721575i	$-0.256582\pi$
$102$		0			0
$103$			5.54173i			0.546043i	0.962008	+	0.273022i	$0.0880231\pi$
$103$			5.54173i			0.546043i	−0.962008	+	0.273022i	$0.911977\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.21161			0.503825			0.251913	−	0.967750i	$-0.418940\pi$
$107$	5.21161			0.503825			0.251913	+	0.967750i	$0.418940\pi$
$108$		0			0
$109$	9.43175			0.903398			0.451699	−	0.892171i	$-0.350818\pi$
$109$	9.43175			0.903398			0.451699	+	0.892171i	$0.350818\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.0999			−1.60863			−0.804313	−	0.594206i	$-0.797466\pi$
$113$	−17.0999			−1.60863			−0.804313	+	0.594206i	$0.797466\pi$
$114$		0			0
$115$	16.4734			1.53616
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.112673i			0.0103288i
$120$		0			0
$121$	−9.23017			−0.839106
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−16.9574			−1.51672
$126$		0			0
$127$	19.0322			1.68883			0.844416	−	0.535688i	$-0.179948\pi$
$127$	19.0322			1.68883			0.844416	+	0.535688i	$0.179948\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	21.8866i		−	1.91224i	−0.292969	−	0.956122i	$-0.594643\pi$
$131$		−	21.8866i		−	1.91224i	0.292969	−	0.956122i	$-0.405357\pi$
$132$		0			0
$133$	−1.48501			−0.128767
$134$		0			0
$135$		0			0
$136$		0			0
$137$	17.5902			1.50283			0.751414	−	0.659831i	$-0.229372\pi$
$137$	17.5902			1.50283			0.751414	+	0.659831i	$0.229372\pi$
$138$		0			0
$139$	5.84681			0.495920			0.247960	−	0.968770i	$-0.420240\pi$
$139$	5.84681			0.495920			0.247960	+	0.968770i	$0.420240\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			5.43056i			0.454126i
$144$		0			0
$145$			21.4584i			1.78202i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	19.0913			1.56402			0.782011	−	0.623264i	$-0.214194\pi$
$149$	19.0913			1.56402			0.782011	+	0.623264i	$0.214194\pi$
$150$		0			0
$151$		−	16.3515i		−	1.33066i	−0.746548	−	0.665331i	$-0.768290\pi$
$151$		−	16.3515i		−	1.33066i	0.746548	−	0.665331i	$-0.231710\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	19.0922			1.53352
$156$		0			0
$157$			7.29739i			0.582395i	0.956663	+	0.291197i	$0.0940536\pi$
$157$			7.29739i			0.582395i	−0.956663	+	0.291197i	$0.905946\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.905203			−0.0713400
$162$		0			0
$163$			7.76339i			0.608076i	0.952660	+	0.304038i	$0.0983350\pi$
$163$			7.76339i			0.608076i	−0.952660	+	0.304038i	$0.901665\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−24.4627			−1.89298			−0.946492	−	0.322727i	$-0.895400\pi$
$167$	−24.4627			−1.89298			−0.946492	+	0.322727i	$0.895400\pi$
$168$		0			0
$169$	−3.66314			−0.281780
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.01833			0.305508			0.152754	−	0.988264i	$-0.451186\pi$
$173$	4.01833			0.305508			0.152754	+	0.988264i	$0.451186\pi$
$174$		0			0
$175$	1.97654			0.149412
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	19.4192i		−	1.45146i	−0.687980	−	0.725730i	$-0.741502\pi$
$179$		−	19.4192i		−	1.45146i	0.687980	−	0.725730i	$-0.258498\pi$
$180$		0			0
$181$	3.61933			0.269022			0.134511	−	0.990912i	$-0.457054\pi$
$181$	3.61933			0.269022			0.134511	+	0.990912i	$0.457054\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−40.0194			−2.94228
$186$		0			0
$187$			0.717381i			0.0524601i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	26.6442			1.92791			0.963954	−	0.266069i	$-0.0857249\pi$
$191$	26.6442			1.92791			0.963954	+	0.266069i	$0.0857249\pi$
$192$		0			0
$193$			0.925507i			0.0666194i	0.999445	+	0.0333097i	$0.0106048\pi$
$193$			0.925507i			0.0666194i	−0.999445	+	0.0333097i	$0.989395\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.39811i		−	0.313352i	−0.987650	−	0.156676i	$-0.949922\pi$
$197$		−	4.39811i		−	0.313352i	0.987650	−	0.156676i	$-0.0500779\pi$
$198$		0			0
$199$	−0.921762			−0.0653420			−0.0326710	−	0.999466i	$-0.510401\pi$
$199$	−0.921762			−0.0653420			−0.0326710	+	0.999466i	$0.510401\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1.17912i		−	0.0827582i
$204$		0			0
$205$			35.7552i			2.49725i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.45494			−0.654012
$210$		0			0
$211$	9.45689			0.651039			0.325520	−	0.945535i	$-0.394461\pi$
$211$	9.45689			0.651039			0.325520	+	0.945535i	$0.394461\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	12.8651			0.877394
$216$		0			0
$217$	−1.04910			−0.0712176
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.20121			−0.148070
$222$		0			0
$223$	−14.2881	+	4.34159i	−0.956804	+	0.290735i
$223$	−14.2881	+	4.34159i	−0.956804	+	0.290735i
$224$		0			0
$225$		0			0
$226$		0			0
$227$			12.2984i			0.816271i	0.912921	+	0.408135i	$0.133821\pi$
$227$			12.2984i			0.816271i	−0.912921	+	0.408135i	$0.866179\pi$
$228$		0			0
$229$		−	26.8966i		−	1.77738i	−0.458513	−	0.888688i	$-0.651618\pi$
$229$		−	26.8966i		−	1.77738i	0.458513	−	0.888688i	$-0.348382\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.66035			−0.567358			−0.283679	−	0.958919i	$-0.591555\pi$
$233$	−8.66035			−0.567358			−0.283679	+	0.958919i	$0.591555\pi$
$234$		0			0
$235$		−	22.0141i		−	1.43604i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	21.0166i		−	1.35945i	−0.733468	−	0.679724i	$-0.762100\pi$
$239$		−	21.0166i		−	1.35945i	0.733468	−	0.679724i	$-0.237900\pi$
$240$		0			0
$241$	−14.2982			−0.921025			−0.460513	−	0.887653i	$-0.652334\pi$
$241$	−14.2982			−0.921025			−0.460513	+	0.887653i	$0.652334\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	26.4519			1.68995
$246$		0			0
$247$		−	29.0115i		−	1.84596i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			10.9771i			0.692871i	0.938074	+	0.346436i	$0.112608\pi$
$251$			10.9771i			0.692871i	−0.938074	+	0.346436i	$0.887392\pi$
$252$		0			0
$253$	−5.76334			−0.362338
$254$		0			0
$255$		0			0
$256$		0			0
$257$			9.57197i			0.597083i	0.954397	+	0.298542i	$0.0965001\pi$
$257$			9.57197i			0.597083i	−0.954397	+	0.298542i	$0.903500\pi$
$258$		0			0
$259$	2.19903			0.136641
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5.79107			−0.357092			−0.178546	−	0.983932i	$-0.557139\pi$
$263$	−5.79107			−0.357092			−0.178546	+	0.983932i	$0.557139\pi$
$264$		0			0
$265$		−	24.7579i		−	1.52086i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	24.1023			1.46954			0.734772	−	0.678314i	$-0.237289\pi$
$269$	24.1023			1.46954			0.734772	+	0.678314i	$0.237289\pi$
$270$		0			0
$271$		−	15.6218i		−	0.948955i	−0.880268	−	0.474478i	$-0.842637\pi$
$271$		−	15.6218i		−	0.948955i	0.880268	−	0.474478i	$-0.157363\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.5844			0.758870
$276$		0			0
$277$		−	17.6825i		−	1.06244i	−0.847235	−	0.531218i	$-0.821734\pi$
$277$		−	17.6825i		−	1.06244i	0.847235	−	0.531218i	$-0.178266\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			9.48574i			0.565872i	0.959139	+	0.282936i	$0.0913083\pi$
$281$			9.48574i			0.565872i	−0.959139	+	0.282936i	$0.908692\pi$
$282$		0			0
$283$	5.12935			0.304908			0.152454	−	0.988311i	$-0.451282\pi$
$283$	5.12935			0.304908			0.152454	+	0.988311i	$0.451282\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.96472i		−	0.115974i
$288$		0			0
$289$	16.7092			0.982895
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.26128			−0.541050			−0.270525	−	0.962713i	$-0.587197\pi$
$293$	−9.26128			−0.541050			−0.270525	+	0.962713i	$0.587197\pi$
$294$		0			0
$295$	34.4435			2.00538
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	17.6842i		−	1.02271i
$300$		0			0
$301$	−0.706929			−0.0407467
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	26.3308i		−	1.50769i
$306$		0			0
$307$		−	1.23488i		−	0.0704781i	−0.999379	−	0.0352391i	$-0.988781\pi$
$307$		−	1.23488i		−	0.0704781i	0.999379	−	0.0352391i	$-0.0112193\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	15.5672			0.882734			0.441367	−	0.897327i	$-0.354494\pi$
$311$	15.5672			0.882734			0.441367	+	0.897327i	$0.354494\pi$
$312$		0			0
$313$		−	34.6608i		−	1.95914i	−0.201093	−	0.979572i	$-0.564449\pi$
$313$		−	34.6608i		−	1.95914i	0.201093	−	0.979572i	$-0.435551\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	17.5693i		−	0.986792i	−0.869805	−	0.493396i	$-0.835755\pi$
$317$		−	17.5693i		−	0.986792i	0.869805	−	0.493396i	$-0.164245\pi$
$318$		0			0
$319$		−	7.50736i		−	0.420331i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	3.83244i		−	0.213243i
$324$		0			0
$325$			38.6141i			2.14192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			1.20966i			0.0666908i
$330$		0			0
$331$			32.4750i			1.78499i	0.451059	+	0.892494i	$0.351046\pi$
$331$			32.4750i			1.78499i	−0.451059	+	0.892494i	$0.648954\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			4.44944i			0.243099i
$336$		0			0
$337$			9.04909i			0.492935i	0.969151	+	0.246468i	$0.0792699\pi$
$337$			9.04909i			0.492935i	−0.969151	+	0.246468i	$0.920730\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.67953			−0.361717
$342$		0			0
$343$	−2.91615			−0.157457
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	30.5020i		−	1.63744i	−0.574196	−	0.818718i	$-0.694685\pi$
$347$		−	30.5020i		−	1.63744i	0.574196	−	0.818718i	$-0.305315\pi$
$348$		0			0
$349$	9.55081			0.511243			0.255622	−	0.966777i	$-0.417720\pi$
$349$	9.55081			0.511243			0.255622	+	0.966777i	$0.417720\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			28.9863i			1.54279i	0.636359	+	0.771393i	$0.280440\pi$
$353$			28.9863i			1.54279i	−0.636359	+	0.771393i	$0.719560\pi$
$354$		0			0
$355$	26.8301			1.42399
$356$		0			0
$357$		0			0
$358$		0			0
$359$			21.5630i			1.13805i	0.822320	+	0.569026i	$0.192680\pi$
$359$			21.5630i			1.13805i	−0.822320	+	0.569026i	$0.807320\pi$
$360$		0			0
$361$	31.5108			1.65846
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.58962			−0.135547
$366$		0			0
$367$	26.3072			1.37322			0.686612	−	0.727024i	$-0.259097\pi$
$367$	26.3072			1.37322			0.686612	+	0.727024i	$0.259097\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1.36043i			0.0706298i
$372$		0			0
$373$		−	25.2124i		−	1.30545i	−0.757594	−	0.652726i	$-0.773626\pi$
$373$		−	25.2124i		−	1.30545i	0.757594	−	0.652726i	$-0.226374\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	23.0356			1.18639
$378$		0			0
$379$	15.4051			0.791309			0.395655	−	0.918399i	$-0.370518\pi$
$379$	15.4051			0.791309			0.395655	+	0.918399i	$0.370518\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.86900			0.504282			0.252141	−	0.967690i	$-0.418865\pi$
$383$	9.86900			0.504282			0.252141	+	0.967690i	$0.418865\pi$
$384$		0			0
$385$	−1.05701			−0.0538704
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	34.9090i		−	1.76996i	−0.465632	−	0.884978i	$-0.654173\pi$
$389$		−	34.9090i		−	1.76996i	0.465632	−	0.884978i	$-0.345827\pi$
$390$		0			0
$391$		−	2.33610i		−	0.118142i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	40.4437i		−	2.03494i
$396$		0			0
$397$		−	35.1142i		−	1.76233i	−0.472810	−	0.881165i	$-0.656760\pi$
$397$		−	35.1142i		−	1.76233i	0.472810	−	0.881165i	$-0.343240\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	11.6067i		−	0.579611i	−0.957086	−	0.289805i	$-0.906409\pi$
$401$		−	11.6067i		−	0.579611i	0.957086	−	0.289805i	$-0.0935906\pi$
$402$		0			0
$403$		−	20.4955i		−	1.02095i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	14.0010			0.694006
$408$		0			0
$409$			12.5856i			0.622318i	0.950358	+	0.311159i	$0.100717\pi$
$409$			12.5856i			0.622318i	−0.950358	+	0.311159i	$0.899283\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.89264			−0.0931309
$414$		0			0
$415$		−	49.0530i		−	2.40792i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			11.4094i			0.557384i	0.960381	+	0.278692i	$0.0899009\pi$
$419$			11.4094i			0.557384i	−0.960381	+	0.278692i	$0.910099\pi$
$420$		0			0
$421$		−	4.85755i		−	0.236743i	−0.992969	−	0.118371i	$-0.962233\pi$
$421$		−	4.85755i		−	0.236743i	0.992969	−	0.118371i	$-0.0377673\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			5.10095i			0.247432i
$426$		0			0
$427$			1.44686i			0.0700182i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.723967			−0.0348723			−0.0174361	−	0.999848i	$-0.505550\pi$
$431$	−0.723967			−0.0348723			−0.0174361	+	0.999848i	$0.505550\pi$
$432$		0			0
$433$	22.1978			1.06676			0.533378	−	0.845877i	$-0.320922\pi$
$433$	22.1978			1.06676			0.533378	+	0.845877i	$0.320922\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	30.7893			1.47285
$438$		0			0
$439$		−	32.5907i		−	1.55547i	−0.628592	−	0.777735i	$-0.716368\pi$
$439$		−	32.5907i		−	1.55547i	0.628592	−	0.777735i	$-0.283632\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			29.6712i			1.40972i	0.709346	+	0.704861i	$0.248991\pi$
$443$			29.6712i			1.40972i	−0.709346	+	0.704861i	$0.751009\pi$
$444$		0			0
$445$			45.1830i			2.14188i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.9981			−0.896574			−0.448287	−	0.893890i	$-0.647966\pi$
$449$	−18.9981			−0.896574			−0.448287	+	0.893890i	$0.647966\pi$
$450$		0			0
$451$		−	12.5092i		−	0.589035i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	3.24334i		−	0.152050i
$456$		0			0
$457$			32.5387i			1.52210i	0.648695	+	0.761049i	$0.275315\pi$
$457$			32.5387i			1.52210i	−0.648695	+	0.761049i	$0.724685\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			1.34281i			0.0625408i	0.999511	+	0.0312704i	$0.00995531\pi$
$461$			1.34281i			0.0625408i	−0.999511	+	0.0312704i	$0.990045\pi$
$462$		0			0
$463$	36.8958			1.71469			0.857347	−	0.514739i	$-0.172111\pi$
$463$	36.8958			1.71469			0.857347	+	0.514739i	$0.172111\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.0610			1.39106			0.695528	−	0.718499i	$-0.255170\pi$
$467$	30.0610			1.39106			0.695528	+	0.718499i	$0.255170\pi$
$468$		0			0
$469$		−	0.244493i		−	0.0112897i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.50095			−0.206954
$474$		0			0
$475$	−67.2294			−3.08470
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	13.9562i		−	0.637675i	−0.947809	−	0.318838i	$-0.896708\pi$
$479$		−	13.9562i		−	0.637675i	0.947809	−	0.318838i	$-0.103292\pi$
$480$		0			0
$481$			42.9608i			1.95884i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	56.9381i		−	2.58543i
$486$		0			0
$487$	−42.4738			−1.92467			−0.962337	−	0.271860i	$-0.912361\pi$
$487$	−42.4738			−1.92467			−0.962337	+	0.271860i	$0.912361\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13.8349			0.624362			0.312181	−	0.950023i	$-0.398941\pi$
$491$	13.8349			0.624362			0.312181	+	0.950023i	$0.398941\pi$
$492$		0			0
$493$	3.04302			0.137051
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.47429			−0.0661311
$498$		0			0
$499$	25.6015			1.14608			0.573040	−	0.819527i	$-0.305764\pi$
$499$	25.6015			1.14608			0.573040	+	0.819527i	$0.305764\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.01682			−0.0453377			−0.0226689	−	0.999743i	$-0.507216\pi$
$503$	−1.01682			−0.0453377			−0.0226689	+	0.999743i	$0.507216\pi$
$504$		0			0
$505$			55.1504i			2.45416i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	31.8353i		−	1.41107i	−0.708673	−	0.705537i	$-0.750706\pi$
$509$		−	31.8353i		−	1.41107i	0.708673	−	0.705537i	$-0.249294\pi$
$510$		0			0
$511$	0.142298			0.00629488
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	21.0728i		−	0.928578i
$516$		0			0
$517$			7.70179i			0.338725i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.4550			−0.720906			−0.360453	−	0.932777i	$-0.617378\pi$
$521$	−16.4550			−0.720906			−0.360453	+	0.932777i	$0.617378\pi$
$522$		0			0
$523$		−	21.8645i		−	0.956068i	−0.878341	−	0.478034i	$-0.841350\pi$
$523$		−	21.8645i		−	0.956068i	0.878341	−	0.478034i	$-0.158650\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	2.70747i		−	0.117939i
$528$		0			0
$529$	−4.23211			−0.184005
$530$		0			0
$531$		0			0
$532$		0			0
$533$	38.3832			1.66256
$534$		0			0
$535$	−19.8175			−0.856784
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−9.25438			−0.398614
$540$		0			0
$541$		−	41.1404i		−	1.76877i	−0.466762	−	0.884383i	$-0.654580\pi$
$541$		−	41.1404i		−	1.76877i	0.466762	−	0.884383i	$-0.345420\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−35.8648			−1.53628
$546$		0			0
$547$	−20.4174			−0.872986			−0.436493	−	0.899708i	$-0.643780\pi$
$547$	−20.4174			−0.872986			−0.436493	+	0.899708i	$0.643780\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			40.1063i			1.70859i
$552$		0			0
$553$			2.22235i			0.0945040i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	30.3141			1.28445			0.642224	−	0.766517i	$-0.278012\pi$
$557$	30.3141			1.28445			0.642224	+	0.766517i	$0.278012\pi$
$558$		0			0
$559$		−	13.8107i		−	0.584131i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17.7620			−0.748579			−0.374290	−	0.927312i	$-0.622113\pi$
$563$	−17.7620			−0.748579			−0.374290	+	0.927312i	$0.622113\pi$
$564$		0			0
$565$	65.0235			2.73556
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.7176			0.784682			0.392341	−	0.919820i	$-0.371665\pi$
$569$	18.7176			0.784682			0.392341	+	0.919820i	$0.371665\pi$
$570$		0			0
$571$			41.6857i			1.74449i	0.489068	+	0.872246i	$0.337337\pi$
$571$			41.6857i			1.74449i	−0.489068	+	0.872246i	$0.662663\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−40.9803			−1.70900
$576$		0			0
$577$	13.9161			0.579337			0.289668	−	0.957127i	$-0.406455\pi$
$577$	13.9161			0.579337			0.289668	+	0.957127i	$0.406455\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			2.69543i			0.111825i
$582$		0			0
$583$			8.66170i			0.358731i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	33.7363			1.39245			0.696223	−	0.717826i	$-0.254862\pi$
$587$	33.7363			1.39245			0.696223	+	0.717826i	$0.254862\pi$
$588$		0			0
$589$	35.6838			1.47033
$590$		0			0
$591$		0			0
$592$		0			0
$593$	27.3831			1.12449			0.562244	−	0.826971i	$-0.309938\pi$
$593$	27.3831			1.12449			0.562244	+	0.826971i	$0.309938\pi$
$594$		0			0
$595$		−	0.428448i		−	0.0175646i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			20.4076i			0.833832i	0.908945	+	0.416916i	$0.136889\pi$
$599$			20.4076i			0.833832i	−0.908945	+	0.416916i	$0.863111\pi$
$600$		0			0
$601$		−	2.78817i		−	0.113732i	−0.998382	−	0.0568660i	$-0.981889\pi$
$601$		−	2.78817i		−	0.113732i	0.998382	−	0.0568660i	$-0.0181108\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	35.0983			1.42695
$606$		0			0
$607$		−	3.29179i		−	0.133610i	−0.997766	−	0.0668048i	$-0.978720\pi$
$607$		−	3.29179i		−	0.133610i	0.997766	−	0.0668048i	$-0.0212805\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−23.6322			−0.956056
$612$		0			0
$613$			41.6117i			1.68068i	0.542058	+	0.840341i	$0.317645\pi$
$613$			41.6117i			1.68068i	−0.542058	+	0.840341i	$0.682355\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	16.9244i		−	0.681350i	−0.940181	−	0.340675i	$-0.889344\pi$
$617$		−	16.9244i		−	0.681350i	0.940181	−	0.340675i	$-0.110656\pi$
$618$		0			0
$619$			0.647181i			0.0260124i	0.999915	+	0.0130062i	$0.00414012\pi$
$619$			0.647181i			0.0260124i	−0.999915	+	0.0130062i	$0.995860\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	2.48277i		−	0.0994701i
$624$		0			0
$625$	17.1843			0.687373
$626$		0			0
$627$		0			0
$628$		0			0
$629$			5.67515i			0.226283i
$630$		0			0
$631$			25.6818i			1.02238i	0.859468	+	0.511189i	$0.170795\pi$
$631$			25.6818i			1.02238i	−0.859468	+	0.511189i	$0.829205\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−72.3710			−2.87195
$636$		0			0
$637$		−	28.3961i		−	1.12510i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.6616			−0.421107			−0.210553	−	0.977582i	$-0.567527\pi$
$641$	−10.6616			−0.421107			−0.210553	+	0.977582i	$0.567527\pi$
$642$		0			0
$643$	−7.92399			−0.312492			−0.156246	−	0.987718i	$-0.549939\pi$
$643$	−7.92399			−0.312492			−0.156246	+	0.987718i	$0.549939\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			39.2006i			1.54114i	0.637358	+	0.770568i	$0.280027\pi$
$647$			39.2006i			1.54114i	−0.637358	+	0.770568i	$0.719973\pi$
$648$		0			0
$649$	−12.0503			−0.473015
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.23977			0.283314			0.141657	−	0.989916i	$-0.454757\pi$
$653$	7.23977			0.283314			0.141657	+	0.989916i	$0.454757\pi$
$654$		0			0
$655$			83.2253i			3.25188i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	22.1480i		−	0.862763i	−0.902170	−	0.431382i	$-0.858026\pi$
$659$		−	22.1480i		−	0.862763i	0.902170	−	0.431382i	$-0.141974\pi$
$660$		0			0
$661$		−	37.0371i		−	1.44057i	−0.693676	−	0.720287i	$-0.744010\pi$
$661$		−	37.0371i		−	1.44057i	0.693676	−	0.720287i	$-0.255990\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.64685			0.218976
$666$		0			0
$667$			24.4472i			0.946598i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			9.21199i			0.355625i
$672$		0			0
$673$	21.3359			0.822438			0.411219	−	0.911537i	$-0.365103\pi$
$673$	21.3359			0.822438			0.411219	+	0.911537i	$0.365103\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			31.9732i			1.22883i	0.788983	+	0.614415i	$0.210608\pi$
$677$			31.9732i			1.22883i	−0.788983	+	0.614415i	$0.789392\pi$
$678$		0			0
$679$			3.12871i			0.120069i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			15.0872i			0.577297i	0.957435	+	0.288648i	$0.0932059\pi$
$683$			15.0872i			0.577297i	−0.957435	+	0.288648i	$0.906794\pi$
$684$		0			0
$685$	−66.8877			−2.55565
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−26.5776			−1.01253
$690$		0			0
$691$		−	28.1405i		−	1.07051i	−0.844689	−	0.535257i	$-0.820215\pi$
$691$		−	28.1405i		−	1.07051i	0.844689	−	0.535257i	$-0.179785\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−22.2329			−0.843341
$696$		0			0
$697$	5.07045			0.192057
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	34.1011i		−	1.28798i	−0.765033	−	0.643991i	$-0.777278\pi$
$701$		−	34.1011i		−	1.28798i	0.765033	−	0.643991i	$-0.222722\pi$
$702$		0			0
$703$	−74.7973			−2.82104
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	3.03047i		−	0.113973i
$708$		0			0
$709$			9.47213i			0.355733i	0.984055	+	0.177867i	$0.0569196\pi$
$709$			9.47213i			0.355733i	−0.984055	+	0.177867i	$0.943080\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.7514			0.814596
$714$		0			0
$715$		−	20.6501i		−	0.772268i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	34.6715i		−	1.29303i	−0.762901	−	0.646515i	$-0.776226\pi$
$719$		−	34.6715i		−	1.29303i	0.762901	−	0.646515i	$-0.223774\pi$
$720$		0			0
$721$			1.15793i			0.0431237i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	53.3812i		−	1.98253i
$726$		0			0
$727$	9.84554			0.365151			0.182576	−	0.983192i	$-0.441557\pi$
$727$	9.84554			0.365151			0.182576	+	0.983192i	$0.441557\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1.82440i		−	0.0674781i
$732$		0			0
$733$	28.8377			1.06515			0.532573	−	0.846384i	$-0.321225\pi$
$733$	28.8377			1.06515			0.532573	+	0.846384i	$0.321225\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1.55667i		−	0.0573405i
$738$		0			0
$739$			49.5666i			1.82334i	0.410926	+	0.911669i	$0.365205\pi$
$739$			49.5666i			1.82334i	−0.410926	+	0.911669i	$0.634795\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	0.639292i		−	0.0234533i	−0.999931	−	0.0117267i	$-0.996267\pi$
$743$		−	0.639292i		−	0.0234533i	0.999931	−	0.0117267i	$-0.00373280\pi$
$744$		0			0
$745$	−72.5959			−2.65971
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.08896			0.0397896
$750$		0			0
$751$	−17.3853			−0.634400			−0.317200	−	0.948359i	$-0.602743\pi$
$751$	−17.3853			−0.634400			−0.317200	+	0.948359i	$0.602743\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			62.1774i			2.26287i
$756$		0			0
$757$		−	9.66493i		−	0.351278i	−0.984455	−	0.175639i	$-0.943801\pi$
$757$		−	9.66493i		−	0.351278i	0.984455	−	0.175639i	$-0.0561991\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.12471			0.185771			0.0928854	−	0.995677i	$-0.470391\pi$
$761$	5.12471			0.185771			0.0928854	+	0.995677i	$0.470391\pi$
$762$		0			0
$763$	1.97074			0.0713457
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	36.9751i		−	1.33509i
$768$		0			0
$769$	15.7622			0.568399			0.284200	−	0.958765i	$-0.408272\pi$
$769$	15.7622			0.568399			0.284200	+	0.958765i	$0.408272\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−35.9742			−1.29390			−0.646950	−	0.762532i	$-0.723956\pi$
$773$	−35.9742			−1.29390			−0.646950	+	0.762532i	$0.723956\pi$
$774$		0			0
$775$	−47.4949			−1.70607
$776$		0			0
$777$		0			0
$778$		0			0
$779$			66.8274i			2.39434i
$780$		0			0
$781$	−9.38668			−0.335882
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	27.7488i		−	0.990396i
$786$		0			0
$787$			10.3048i			0.367325i	0.982989	+	0.183662i	$0.0587953\pi$
$787$			10.3048i			0.367325i	−0.982989	+	0.183662i	$0.941205\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.57299			−0.127041
$792$		0			0
$793$	−28.2661			−1.00376
$794$		0			0
$795$		0			0
$796$		0			0
$797$			30.5205i			1.08109i	0.841314	+	0.540547i	$0.181783\pi$
$797$			30.5205i			1.08109i	−0.841314	+	0.540547i	$0.818217\pi$
$798$		0			0
$799$	−3.12183			−0.110442
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.905996			0.0319719
$804$		0			0
$805$	3.44209			0.121318
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−21.1702			−0.744304			−0.372152	−	0.928172i	$-0.621380\pi$
$809$	−21.1702			−0.744304			−0.372152	+	0.928172i	$0.621380\pi$
$810$		0			0
$811$		−	11.4567i		−	0.402298i	−0.979561	−	0.201149i	$-0.935532\pi$
$811$		−	11.4567i		−	0.402298i	0.979561	−	0.201149i	$-0.0644676\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	29.5208i		−	1.03407i
$816$		0			0
$817$	24.0453			0.841238
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	3.84440i		−	0.134170i	−0.997747	−	0.0670852i	$-0.978630\pi$
$821$		−	3.84440i		−	0.134170i	0.997747	−	0.0670852i	$-0.0213699\pi$
$822$		0			0
$823$		−	43.7338i		−	1.52447i	−0.647303	−	0.762233i	$-0.724103\pi$
$823$		−	43.7338i		−	1.52447i	0.647303	−	0.762233i	$-0.275897\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.19473			−0.319732			−0.159866	−	0.987139i	$-0.551106\pi$
$827$	−9.19473			−0.319732			−0.159866	+	0.987139i	$0.551106\pi$
$828$		0			0
$829$			6.04115i			0.209818i	0.994482	+	0.104909i	$0.0334551\pi$
$829$			6.04115i			0.209818i	−0.994482	+	0.104909i	$0.966545\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	3.75115i		−	0.129970i
$834$		0			0
$835$	93.0211			3.21913
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.66335			−0.126473			−0.0632365	−	0.997999i	$-0.520142\pi$
$839$	−3.66335			−0.126473			−0.0632365	+	0.997999i	$0.520142\pi$
$840$		0			0
$841$	−2.84503			−0.0981044
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.9293			0.479183
$846$		0			0
$847$	−1.92862			−0.0662683
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−45.5934			−1.56292
$852$		0			0
$853$			37.5694i			1.28635i	0.765718	+	0.643176i	$0.222384\pi$
$853$			37.5694i			1.28635i	−0.765718	+	0.643176i	$0.777616\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.90581i		−	0.0651011i	−0.999470	−	0.0325506i	$-0.989637\pi$
$857$		−	1.90581i		−	0.0651011i	0.999470	−	0.0325506i	$-0.0103630\pi$
$858$		0			0
$859$			31.2647i			1.06674i	0.845883	+	0.533369i	$0.179074\pi$
$859$			31.2647i			1.06674i	−0.845883	+	0.533369i	$0.820926\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−10.5088			−0.357725			−0.178863	−	0.983874i	$-0.557242\pi$
$863$	−10.5088			−0.357725			−0.178863	+	0.983874i	$0.557242\pi$
$864$		0			0
$865$	−15.2799			−0.519533
$866$		0			0
$867$		0			0
$868$		0			0
$869$			14.1495i			0.479989i
$870$		0			0
$871$	4.77647			0.161845
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3.54322			−0.119783
$876$		0			0
$877$		−	14.1636i		−	0.478269i	−0.970986	−	0.239135i	$-0.923136\pi$
$877$		−	14.1636i		−	0.478269i	0.970986	−	0.239135i	$-0.0768637\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			24.0117i			0.808976i	0.914543	+	0.404488i	$0.132550\pi$
$881$			24.0117i			0.808976i	−0.914543	+	0.404488i	$0.867450\pi$
$882$		0			0
$883$		−	39.6345i		−	1.33381i	−0.745145	−	0.666903i	$-0.767620\pi$
$883$		−	39.6345i		−	1.33381i	0.745145	−	0.666903i	$-0.232380\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			18.8831i			0.634032i	0.948420	+	0.317016i	$0.102681\pi$
$887$			18.8831i			0.634032i	−0.948420	+	0.317016i	$0.897319\pi$
$888$		0			0
$889$	3.97673			0.133375
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	41.1451i		−	1.37687i
$894$		0			0
$895$			73.8427i			2.46829i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			28.3335i			0.944975i
$900$		0			0
$901$	−3.51092			−0.116966
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−13.7627			−0.457488
$906$		0			0
$907$	36.9432			1.22668			0.613339	−	0.789819i	$-0.289826\pi$
$907$	36.9432			1.22668			0.613339	+	0.789819i	$0.289826\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			27.7706i			0.920082i	0.887898	+	0.460041i	$0.152165\pi$
$911$			27.7706i			0.920082i	−0.887898	+	0.460041i	$0.847835\pi$
$912$		0			0
$913$			17.1615i			0.567964i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	4.57317i		−	0.151019i
$918$		0			0
$919$		−	39.6093i		−	1.30659i	−0.757103	−	0.653295i	$-0.773386\pi$
$919$		−	39.6093i		−	1.30659i	0.757103	−	0.653295i	$-0.226614\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	28.8021i		−	0.948032i
$924$		0			0
$925$	99.5546			3.27333
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	36.6313i		−	1.20183i	−0.799311	−	0.600917i	$-0.794802\pi$
$929$		−	36.6313i		−	1.20183i	0.799311	−	0.600917i	$-0.205198\pi$
$930$		0			0
$931$	49.4394			1.62031
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	2.72789i		−	0.0892114i
$936$		0			0
$937$		−	17.6302i		−	0.575952i	−0.957638	−	0.287976i	$-0.907018\pi$
$937$		−	17.6302i		−	0.575952i	0.957638	−	0.287976i	$-0.0929824\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	30.8495i		−	1.00566i	−0.864384	−	0.502832i	$-0.832291\pi$
$941$		−	30.8495i		−	1.00566i	0.864384	−	0.502832i	$-0.167709\pi$
$942$		0			0
$943$			40.7352i			1.32652i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1.26037i			0.0409564i	0.999790	+	0.0204782i	$0.00651887\pi$
$947$			1.26037i			0.0409564i	−0.999790	+	0.0204782i	$0.993481\pi$
$948$		0			0
$949$			2.77996i			0.0902413i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−25.3553			−0.821337			−0.410669	−	0.911785i	$-0.634705\pi$
$953$	−25.3553			−0.821337			−0.410669	+	0.911785i	$0.634705\pi$
$954$		0			0
$955$	−101.316			−3.27852
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.67543			0.118686
$960$		0			0
$961$	−5.79082			−0.186801
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	3.51930i		−	0.113290i
$966$		0			0
$967$			45.3716i			1.45905i	0.683953	+	0.729526i	$0.260259\pi$
$967$			45.3716i			1.45905i	−0.683953	+	0.729526i	$0.739741\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−23.9067			−0.767202			−0.383601	−	0.923499i	$-0.625316\pi$
$971$	−23.9067			−0.767202			−0.383601	+	0.923499i	$0.625316\pi$
$972$		0			0
$973$	1.22168			0.0391652
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.6411			0.596381			0.298191	−	0.954506i	$-0.403617\pi$
$977$	18.6411			0.596381			0.298191	+	0.954506i	$0.403617\pi$
$978$		0			0
$979$		−	15.8076i		−	0.505212i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20.3477			0.648990			0.324495	−	0.945887i	$-0.394806\pi$
$983$	20.3477			0.648990			0.324495	+	0.945887i	$0.394806\pi$
$984$		0			0
$985$			16.7241i			0.532873i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.6570			0.466066
$990$		0			0
$991$			30.9044i			0.981711i	0.871241	+	0.490855i	$0.163316\pi$
$991$			30.9044i			0.981711i	−0.871241	+	0.490855i	$0.836684\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.50506			0.111118
$996$		0			0
$997$	30.4878			0.965559			0.482780	−	0.875742i	$-0.339627\pi$
$997$	30.4878			0.965559			0.482780	+	0.875742i	$0.339627\pi$
$998$		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.4013.7	✓	76
3.2	odd	2	inner	8028.2.h.a.4013.70	yes	76
223.222	odd	2	inner	8028.2.h.a.4013.69	yes	76
669.668	even	2	inner	8028.2.h.a.4013.8	yes	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8028.2.h.a.4013.7	✓	76	1.1	even	1	trivial
8028.2.h.a.4013.8	yes	76	669.668	even	2	inner
8028.2.h.a.4013.69	yes	76	223.222	odd	2	inner
8028.2.h.a.4013.70	yes	76	3.2	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)

\(f(q)\)	\(=\)	\(q-3.80256 q^{5} +0.208948 q^{7} +O(q^{10})\)	\(q-3.80256 q^{5} +0.208948 q^{7} +1.33035 q^{11} +4.08205i q^{13} +0.539242i q^{17} -7.10710 q^{19} -4.33219 q^{23} +9.45948 q^{25} -5.64314i q^{29} -5.02087 q^{31} -0.794538 q^{35} +10.5243 q^{37} -9.40292i q^{41} -3.38328 q^{43} +5.78929i q^{47} -6.95634 q^{49} +6.51084i q^{53} -5.05874 q^{55} -9.05797 q^{59} +6.92448i q^{61} -15.5223i q^{65} -1.17012i q^{67} -7.05579 q^{71} +0.681020 q^{73} +0.277974 q^{77} +10.6359i q^{79} +12.9000i q^{83} -2.05050i q^{85} -11.8822i q^{89} +0.852936i q^{91} +27.0252 q^{95} +14.9736i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q + 8 q^{7}+O(q^{10}) \) 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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