LMFDB - Embedded newform 9300.2.g.s.3349.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(3349,9300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-8,0,4,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{6} + 167x^{4} + 389x^{2} + 196 $ x^8 + 23x^6 + 167x^4 + 389*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1860)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.7
Root			$0.829933i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.s.3349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.61894i q^{7} -1.00000 q^{9} -1.65987 q^{11} -5.27881i q^{13} +4.48128i q^{17} -1.61894 q^{21} +4.48128i q^{23} -1.00000i q^{27} +3.76009 q^{29} +1.00000 q^{31} -1.65987i q^{33} +3.61894i q^{37} +5.27881 q^{39} +3.65987 q^{41} -1.65987i q^{43} -0.756606i q^{47} +4.37903 q^{49} -4.48128 q^{51} +2.75661i q^{53} -11.4848 q^{59} +14.2823 q^{61} -1.61894i q^{63} -5.00348i q^{67} -4.48128 q^{69} -0.862336 q^{71} +2.93867i q^{73} -2.68723i q^{77} -8.41647 q^{79} +1.00000 q^{81} -0.481278i q^{83} +3.76009i q^{87} -5.13766 q^{89} +8.54608 q^{91} +1.00000i q^{93} +9.30269i q^{97} +1.65987 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} + 4 q^{11} + 8 q^{21} - 40 q^{29} + 8 q^{31} + 4 q^{39} + 12 q^{41} - 56 q^{49} - 4 q^{51} + 16 q^{61} - 4 q^{69} + 4 q^{71} - 40 q^{79} + 8 q^{81} - 52 q^{89} + 32 q^{91} - 4 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 + 4 * q^11 + 8 * q^21 - 40 * q^29 + 8 * q^31 + 4 * q^39 + 12 * q^41 - 56 * q^49 - 4 * q^51 + 16 * q^61 - 4 * q^69 + 4 * q^71 - 40 * q^79 + 8 * q^81 - 52 * q^89 + 32 * q^91 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$	$4651$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.61894i	0.611902i	0.952047	+	0.305951i	$0.0989745\pi$
$7$	1.61894i	0.611902i	−0.952047	+	0.305951i	$0.901026\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.65987	−0.500469	−0.250234	−	0.968185i	$-0.580508\pi$
$11$	−1.65987	−0.500469	−0.250234	+	0.968185i	$0.580508\pi$
$12$	0	0
$13$	− 5.27881i	− 1.46408i	−0.681263	−	0.732039i	$-0.738569\pi$
$13$	− 5.27881i	− 1.46408i	0.681263	−	0.732039i	$-0.261431\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.48128i	1.08687i	0.839451	+	0.543435i	$0.182876\pi$
$17$	4.48128i	1.08687i	−0.839451	+	0.543435i	$0.817124\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.61894	−0.353282
$22$	0	0
$23$	4.48128i	0.934411i	0.884149	+	0.467205i	$0.154739\pi$
$23$	4.48128i	0.934411i	−0.884149	+	0.467205i	$0.845261\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	3.76009	0.698230	0.349115	−	0.937080i	$-0.386482\pi$
$29$	3.76009	0.698230	0.349115	+	0.937080i	$0.386482\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	− 1.65987i	− 0.288946i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.61894i	0.594950i	0.954730	+	0.297475i	$0.0961446\pi$
$37$	3.61894i	0.594950i	−0.954730	+	0.297475i	$0.903855\pi$
$38$	0	0
$39$	5.27881	0.845286
$40$	0	0
$41$	3.65987	0.571575	0.285788	−	0.958293i	$-0.407745\pi$
$41$	3.65987	0.571575	0.285788	+	0.958293i	$0.407745\pi$
$42$	0	0
$43$	− 1.65987i	− 0.253127i	−0.991958	−	0.126564i	$-0.959605\pi$
$43$	− 1.65987i	− 0.253127i	0.991958	−	0.126564i	$-0.0403948\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.756606i	− 0.110362i	−0.998476	−	0.0551812i	$-0.982426\pi$
$47$	− 0.756606i	− 0.110362i	0.998476	−	0.0551812i	$-0.0175736\pi$
$48$	0	0
$49$	4.37903	0.625575
$50$	0	0
$51$	−4.48128	−0.627504
$52$	0	0
$53$	2.75661i	0.378649i	0.981915	+	0.189324i	$0.0606298\pi$
$53$	2.75661i	0.378649i	−0.981915	+	0.189324i	$0.939370\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.4848	−1.49519	−0.747594	−	0.664156i	$-0.768791\pi$
$59$	−11.4848	−1.49519	−0.747594	+	0.664156i	$0.768791\pi$
$60$	0	0
$61$	14.2823	1.82866	0.914330	−	0.404970i	$-0.132718\pi$
$61$	14.2823	1.82866	0.914330	+	0.404970i	$0.132718\pi$
$62$	0	0
$63$	− 1.61894i	− 0.203967i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.00348i	− 0.611272i	−0.952148	−	0.305636i	$-0.901131\pi$
$67$	− 5.00348i	− 0.611272i	0.952148	−	0.305636i	$-0.0988691\pi$
$68$	0	0
$69$	−4.48128	−0.539482
$70$	0	0
$71$	−0.862336	−0.102340	−0.0511702	−	0.998690i	$-0.516295\pi$
$71$	−0.862336	−0.102340	−0.0511702	+	0.998690i	$0.516295\pi$
$72$	0	0
$73$	2.93867i	0.343946i	0.985102	+	0.171973i	$0.0550142\pi$
$73$	2.93867i	0.343946i	−0.985102	+	0.171973i	$0.944986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.68723i	− 0.306238i
$78$	0	0
$79$	−8.41647	−0.946927	−0.473464	−	0.880813i	$-0.656996\pi$
$79$	−8.41647	−0.946927	−0.473464	+	0.880813i	$0.656996\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 0.481278i	− 0.0528271i	−0.999651	−	0.0264135i	$-0.991591\pi$
$83$	− 0.481278i	− 0.0528271i	0.999651	−	0.0264135i	$-0.00840867\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.76009i	0.403124i
$88$	0	0
$89$	−5.13766	−0.544591	−0.272296	−	0.962214i	$-0.587783\pi$
$89$	−5.13766	−0.544591	−0.272296	+	0.962214i	$0.587783\pi$
$90$	0	0
$91$	8.54608	0.895873
$92$	0	0
$93$	1.00000i	0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.30269i	0.944545i	0.881453	+	0.472272i	$0.156566\pi$
$97$	9.30269i	0.944545i	−0.881453	+	0.472272i	$0.843434\pi$
$98$	0	0
$99$	1.65987	0.166823
$100$	0	0
$101$	−4.62242	−0.459948	−0.229974	−	0.973197i	$-0.573864\pi$
$101$	−4.62242	−0.459948	−0.229974	+	0.973197i	$0.573864\pi$
$102$	0	0
$103$	− 0.0409247i	− 0.00403243i	−0.999998	−	0.00201621i	$-0.999358\pi$
$103$	− 0.0409247i	− 0.00403243i	0.999998	−	0.00201621i	$-0.000641781\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.48128i	− 0.239874i	−0.992781	−	0.119937i	$-0.961731\pi$
$107$	− 2.48128i	− 0.239874i	0.992781	−	0.119937i	$-0.0382693\pi$
$108$	0	0
$109$	−3.85886	−0.369611	−0.184806	−	0.982775i	$-0.559166\pi$
$109$	−3.85886	−0.369611	−0.184806	+	0.982775i	$0.559166\pi$
$110$	0	0
$111$	−3.61894	−0.343495
$112$	0	0
$113$	12.5576i	1.18132i	0.806920	+	0.590661i	$0.201133\pi$
$113$	12.5576i	1.18132i	−0.806920	+	0.590661i	$0.798867\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.27881i	0.488026i
$118$	0	0
$119$	−7.25493	−0.665058
$120$	0	0
$121$	−8.24484	−0.749531
$122$	0	0
$123$	3.65987i	0.329999i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.55762i	− 0.581894i	−0.956739	−	0.290947i	$-0.906030\pi$
$127$	− 6.55762i	− 0.581894i	0.956739	−	0.290947i	$-0.0939704\pi$
$128$	0	0
$129$	1.65987	0.146143
$130$	0	0
$131$	8.99797	0.786156	0.393078	−	0.919505i	$-0.371410\pi$
$131$	8.99797	0.786156	0.393078	+	0.919505i	$0.371410\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.7636i	1.77395i	0.461816	+	0.886976i	$0.347198\pi$
$137$	20.7636i	1.77395i	−0.461816	+	0.886976i	$0.652802\pi$
$138$	0	0
$139$	−17.0444	−1.44569	−0.722844	−	0.691011i	$-0.757165\pi$
$139$	−17.0444	−1.44569	−0.722844	+	0.691011i	$0.757165\pi$
$140$	0	0
$141$	0.756606	0.0637177
$142$	0	0
$143$	8.76212i	0.732725i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.37903i	0.361176i
$148$	0	0
$149$	−17.5850	−1.44062	−0.720309	−	0.693654i	$-0.756000\pi$
$149$	−17.5850	−1.44062	−0.720309	+	0.693654i	$0.756000\pi$
$150$	0	0
$151$	10.1411	0.825275	0.412637	−	0.910895i	$-0.364608\pi$
$151$	10.1411	0.825275	0.412637	+	0.910895i	$0.364608\pi$
$152$	0	0
$153$	− 4.48128i	− 0.362290i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.1356i	1.12815i	0.825725	+	0.564073i	$0.190766\pi$
$157$	14.1356i	1.12815i	−0.825725	+	0.564073i	$0.809234\pi$
$158$	0	0
$159$	−2.75661	−0.218613
$160$	0	0
$161$	−7.25493	−0.571768
$162$	0	0
$163$	− 3.27881i	− 0.256816i	−0.991721	−	0.128408i	$-0.959013\pi$
$163$	− 3.27881i	− 0.256816i	0.991721	−	0.128408i	$-0.0409867\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.91815i	− 0.303196i	−0.988442	−	0.151598i	$-0.951558\pi$
$167$	− 3.91815i	− 0.303196i	0.988442	−	0.151598i	$-0.0484418\pi$
$168$	0	0
$169$	−14.8658	−1.14352
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.23788i	0.398229i	0.979976	+	0.199114i	$0.0638066\pi$
$173$	5.23788i	0.398229i	−0.979976	+	0.199114i	$0.936193\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 11.4848i	− 0.863247i
$178$	0	0
$179$	8.28229	0.619047	0.309524	−	0.950892i	$-0.399830\pi$
$179$	8.28229	0.619047	0.309524	+	0.950892i	$0.399830\pi$
$180$	0	0
$181$	14.8329	1.10252	0.551262	−	0.834332i	$-0.314146\pi$
$181$	14.8329	1.10252	0.551262	+	0.834332i	$0.314146\pi$
$182$	0	0
$183$	14.2823i	1.05578i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 7.43832i	− 0.543944i
$188$	0	0
$189$	1.61894	0.117761
$190$	0	0
$191$	5.07982	0.367563	0.183781	−	0.982967i	$-0.441166\pi$
$191$	5.07982	0.367563	0.183781	+	0.982967i	$0.441166\pi$
$192$	0	0
$193$	17.4554i	1.25646i	0.778026	+	0.628232i	$0.216221\pi$
$193$	17.4554i	1.25646i	−0.778026	+	0.628232i	$0.783779\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.16154i	0.367745i	0.982950	+	0.183872i	$0.0588633\pi$
$197$	5.16154i	0.367745i	−0.982950	+	0.183872i	$0.941137\pi$
$198$	0	0
$199$	−21.4609	−1.52132	−0.760661	−	0.649150i	$-0.775125\pi$
$199$	−21.4609	−1.52132	−0.760661	+	0.649150i	$0.775125\pi$
$200$	0	0
$201$	5.00348	0.352918
$202$	0	0
$203$	6.08736i	0.427249i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.48128i	− 0.311470i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−21.9959	−1.51426	−0.757131	−	0.653263i	$-0.773400\pi$
$211$	−21.9959	−1.51426	−0.757131	+	0.653263i	$0.773400\pi$
$212$	0	0
$213$	− 0.862336i	− 0.0590863i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.61894i	0.109901i
$218$	0	0
$219$	−2.93867	−0.198577
$220$	0	0
$221$	23.6558	1.59126
$222$	0	0
$223$	− 0.897750i	− 0.0601178i	−0.999548	−	0.0300589i	$-0.990431\pi$
$223$	− 0.897750i	− 0.0601178i	0.999548	−	0.0300589i	$-0.00956948\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.99449i	− 0.265124i	−0.991175	−	0.132562i	$-0.957680\pi$
$227$	− 3.99449i	− 0.265124i	0.991175	−	0.132562i	$-0.0423203\pi$
$228$	0	0
$229$	−18.0708	−1.19415	−0.597077	−	0.802184i	$-0.703671\pi$
$229$	−18.0708	−1.19415	−0.597077	+	0.802184i	$0.703671\pi$
$230$	0	0
$231$	2.68723	0.176807
$232$	0	0
$233$	26.0014i	1.70341i	0.524020	+	0.851706i	$0.324432\pi$
$233$	26.0014i	1.70341i	−0.524020	+	0.851706i	$0.675568\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.41647i	− 0.546709i
$238$	0	0
$239$	21.8603	1.41403	0.707013	−	0.707201i	$-0.250042\pi$
$239$	21.8603	1.41403	0.707013	+	0.707201i	$0.250042\pi$
$240$	0	0
$241$	11.8773	0.765087	0.382544	−	0.923937i	$-0.375048\pi$
$241$	11.8773	0.765087	0.382544	+	0.923937i	$0.375048\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.481278	0.0304997
$250$	0	0
$251$	−27.6020	−1.74222	−0.871112	−	0.491084i	$-0.836601\pi$
$251$	−27.6020	−1.74222	−0.871112	+	0.491084i	$0.836601\pi$
$252$	0	0
$253$	− 7.43832i	− 0.467643i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.71916i	− 0.356751i	−0.983962	−	0.178376i	$-0.942916\pi$
$257$	− 5.71916i	− 0.356751i	0.983962	−	0.178376i	$-0.0570843\pi$
$258$	0	0
$259$	−5.85886	−0.364052
$260$	0	0
$261$	−3.76009	−0.232743
$262$	0	0
$263$	20.0708i	1.23762i	0.785541	+	0.618810i	$0.212385\pi$
$263$	20.0708i	1.23762i	−0.785541	+	0.618810i	$0.787615\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 5.13766i	− 0.314420i
$268$	0	0
$269$	4.60405	0.280714	0.140357	−	0.990101i	$-0.455175\pi$
$269$	4.60405	0.280714	0.140357	+	0.990101i	$0.455175\pi$
$270$	0	0
$271$	−22.8399	−1.38743	−0.693713	−	0.720252i	$-0.744026\pi$
$271$	−22.8399	−1.38743	−0.693713	+	0.720252i	$0.744026\pi$
$272$	0	0
$273$	8.54608i	0.517232i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 13.4144i	− 0.805996i	−0.915201	−	0.402998i	$-0.867968\pi$
$277$	− 13.4144i	− 0.805996i	0.915201	−	0.402998i	$-0.132032\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−2.08185	−0.124193	−0.0620964	−	0.998070i	$-0.519779\pi$
$281$	−2.08185	−0.124193	−0.0620964	+	0.998070i	$0.519779\pi$
$282$	0	0
$283$	0.0238807i	0.00141956i	1.00000		0.000709780i	$0.000225930\pi$
$283$	0.0238807i	0.00141956i	−1.00000		0.000709780i	$0.999774\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.92511i	0.349748i
$288$	0	0
$289$	−3.08185	−0.181285
$290$	0	0
$291$	−9.30269	−0.545333
$292$	0	0
$293$	7.92366i	0.462905i	0.972846	+	0.231453i	$0.0743478\pi$
$293$	7.92366i	0.462905i	−0.972846	+	0.231453i	$0.925652\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.65987i	0.0963152i
$298$	0	0
$299$	23.6558	1.36805
$300$	0	0
$301$	2.68723	0.154889
$302$	0	0
$303$	− 4.62242i	− 0.265551i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.30617i	0.474058i	0.971503	+	0.237029i	$0.0761737\pi$
$307$	8.30617i	0.474058i	−0.971503	+	0.237029i	$0.923826\pi$
$308$	0	0
$309$	0.0409247	0.00232812
$310$	0	0
$311$	−28.1002	−1.59342	−0.796709	−	0.604364i	$-0.793427\pi$
$311$	−28.1002	−1.59342	−0.796709	+	0.604364i	$0.793427\pi$
$312$	0	0
$313$	− 11.8364i	− 0.669034i	−0.942390	−	0.334517i	$-0.891427\pi$
$313$	− 11.8364i	− 0.669034i	0.942390	−	0.334517i	$-0.108573\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.23237i	− 0.181548i	−0.995872	−	0.0907741i	$-0.971066\pi$
$317$	− 3.23237i	− 0.181548i	0.995872	−	0.0907741i	$-0.0289341\pi$
$318$	0	0
$319$	−6.24124	−0.349442
$320$	0	0
$321$	2.48128	0.138491
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 3.85886i	− 0.213395i
$328$	0	0
$329$	1.22490	0.0675310
$330$	0	0
$331$	−14.3416	−0.788285	−0.394142	−	0.919049i	$-0.628958\pi$
$331$	−14.3416	−0.788285	−0.394142	+	0.919049i	$0.628958\pi$
$332$	0	0
$333$	− 3.61894i	− 0.198317i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.4144i	0.730731i	0.930864	+	0.365366i	$0.119056\pi$
$337$	13.4144i	0.730731i	−0.930864	+	0.365366i	$0.880944\pi$
$338$	0	0
$339$	−12.5576	−0.682036
$340$	0	0
$341$	−1.65987	−0.0898868
$342$	0	0
$343$	18.4220i	0.994694i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 27.8025i	− 1.49251i	−0.665658	−	0.746257i	$-0.731849\pi$
$347$	− 27.8025i	− 1.49251i	0.665658	−	0.746257i	$-0.268151\pi$
$348$	0	0
$349$	16.4234	0.879126	0.439563	−	0.898212i	$-0.355133\pi$
$349$	16.4234	0.879126	0.439563	+	0.898212i	$0.355133\pi$
$350$	0	0
$351$	−5.27881	−0.281762
$352$	0	0
$353$	8.19899i	0.436388i	0.975905	+	0.218194i	$0.0700166\pi$
$353$	8.19899i	0.436388i	−0.975905	+	0.218194i	$0.929983\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 7.25493i	− 0.383971i
$358$	0	0
$359$	−10.7226	−0.565919	−0.282960	−	0.959132i	$-0.591316\pi$
$359$	−10.7226	−0.565919	−0.282960	+	0.959132i	$0.591316\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 8.24484i	− 0.432742i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−3.65987	−0.190525
$370$	0	0
$371$	−4.46278	−0.231696
$372$	0	0
$373$	− 3.72467i	− 0.192856i	−0.995340	−	0.0964281i	$-0.969258\pi$
$373$	− 3.72467i	− 0.192856i	0.995340	−	0.0964281i	$-0.0307418\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 19.8488i	− 1.02226i
$378$	0	0
$379$	−7.15603	−0.367581	−0.183790	−	0.982965i	$-0.558837\pi$
$379$	−7.15603	−0.367581	−0.183790	+	0.982965i	$0.558837\pi$
$380$	0	0
$381$	6.55762	0.335957
$382$	0	0
$383$	− 2.95704i	− 0.151098i	−0.997142	−	0.0755490i	$-0.975929\pi$
$383$	− 2.95704i	− 0.151098i	0.997142	−	0.0755490i	$-0.0240709\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.65987i	0.0843758i
$388$	0	0
$389$	1.89978	0.0963227	0.0481613	−	0.998840i	$-0.484664\pi$
$389$	1.89978	0.0963227	0.0481613	+	0.998840i	$0.484664\pi$
$390$	0	0
$391$	−20.0818	−1.01558
$392$	0	0
$393$	8.99797i	0.453888i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 12.3231i	− 0.618478i	−0.950984	−	0.309239i	$-0.899926\pi$
$397$	− 12.3231i	− 0.618478i	0.950984	−	0.309239i	$-0.100074\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.13766	−0.256563	−0.128281	−	0.991738i	$-0.540946\pi$
$401$	−5.13766	−0.256563	−0.128281	+	0.991738i	$0.540946\pi$
$402$	0	0
$403$	− 5.27881i	− 0.262956i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.00696i	− 0.297754i
$408$	0	0
$409$	−25.4383	−1.25784	−0.628922	−	0.777468i	$-0.716504\pi$
$409$	−25.4383	−1.25784	−0.628922	+	0.777468i	$0.716504\pi$
$410$	0	0
$411$	−20.7636	−1.02419
$412$	0	0
$413$	− 18.5932i	− 0.914909i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 17.0444i	− 0.834668i
$418$	0	0
$419$	8.78049	0.428955	0.214477	−	0.976729i	$-0.431195\pi$
$419$	8.78049	0.428955	0.214477	+	0.976729i	$0.431195\pi$
$420$	0	0
$421$	−7.37903	−0.359632	−0.179816	−	0.983700i	$-0.557550\pi$
$421$	−7.37903	−0.359632	−0.179816	+	0.983700i	$0.557550\pi$
$422$	0	0
$423$	0.756606i	0.0367874i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	23.1222i	1.11896i
$428$	0	0
$429$	−8.76212	−0.423039
$430$	0	0
$431$	22.9401	1.10499	0.552493	−	0.833517i	$-0.313676\pi$
$431$	22.9401	1.10499	0.552493	+	0.833517i	$0.313676\pi$
$432$	0	0
$433$	21.5441i	1.03534i	0.855580	+	0.517671i	$0.173201\pi$
$433$	21.5441i	1.03534i	−0.855580	+	0.517671i	$0.826799\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−17.9251	−0.855519	−0.427759	−	0.903893i	$-0.640697\pi$
$439$	−17.9251	−0.855519	−0.427759	+	0.903893i	$0.640697\pi$
$440$	0	0
$441$	−4.37903	−0.208525
$442$	0	0
$443$	− 32.5591i	− 1.54693i	−0.633840	−	0.773464i	$-0.718522\pi$
$443$	− 32.5591i	− 1.54693i	0.633840	−	0.773464i	$-0.281478\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 17.5850i	− 0.831741i
$448$	0	0
$449$	3.61343	0.170528	0.0852642	−	0.996358i	$-0.472827\pi$
$449$	3.61343	0.170528	0.0852642	+	0.996358i	$0.472827\pi$
$450$	0	0
$451$	−6.07489	−0.286055
$452$	0	0
$453$	10.1411i	0.476473i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 34.5407i	− 1.61575i	−0.589357	−	0.807873i	$-0.700619\pi$
$457$	− 34.5407i	− 1.61575i	0.589357	−	0.807873i	$-0.299381\pi$
$458$	0	0
$459$	4.48128	0.209168
$460$	0	0
$461$	−11.2803	−0.525374	−0.262687	−	0.964881i	$-0.584609\pi$
$461$	−11.2803	−0.525374	−0.262687	+	0.964881i	$0.584609\pi$
$462$	0	0
$463$	19.8673i	0.923310i	0.887060	+	0.461655i	$0.152744\pi$
$463$	19.8673i	0.923310i	−0.887060	+	0.461655i	$0.847256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.4772i	1.13267i	0.824175	+	0.566335i	$0.191639\pi$
$467$	24.4772i	1.13267i	−0.824175	+	0.566335i	$0.808361\pi$
$468$	0	0
$469$	8.10034	0.374039
$470$	0	0
$471$	−14.1356	−0.651336
$472$	0	0
$473$	2.75516i	0.126682i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.75661i	− 0.126216i
$478$	0	0
$479$	22.9401	1.04816	0.524081	−	0.851669i	$-0.324409\pi$
$479$	22.9401	1.04816	0.524081	+	0.851669i	$0.324409\pi$
$480$	0	0
$481$	19.1037	0.871054
$482$	0	0
$483$	− 7.25493i	− 0.330111i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 13.7247i	− 0.621924i	−0.950422	−	0.310962i	$-0.899349\pi$
$487$	− 13.7247i	− 0.621924i	0.950422	−	0.310962i	$-0.100651\pi$
$488$	0	0
$489$	3.27881	0.148273
$490$	0	0
$491$	−0.751095	−0.0338965	−0.0169482	−	0.999856i	$-0.505395\pi$
$491$	−0.751095	−0.0338965	−0.0169482	+	0.999856i	$0.505395\pi$
$492$	0	0
$493$	16.8500i	0.758885i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.39607i	− 0.0626224i
$498$	0	0
$499$	−33.1152	−1.48244	−0.741221	−	0.671261i	$-0.765753\pi$
$499$	−33.1152	−1.48244	−0.741221	+	0.671261i	$0.765753\pi$
$500$	0	0
$501$	3.91815	0.175050
$502$	0	0
$503$	30.6817i	1.36803i	0.729468	+	0.684015i	$0.239768\pi$
$503$	30.6817i	1.36803i	−0.729468	+	0.684015i	$0.760232\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 14.8658i	− 0.660214i
$508$	0	0
$509$	−5.05581	−0.224095	−0.112048	−	0.993703i	$-0.535741\pi$
$509$	−5.05581	−0.224095	−0.112048	+	0.993703i	$0.535741\pi$
$510$	0	0
$511$	−4.75754	−0.210461
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.25586i	0.0552329i
$518$	0	0
$519$	−5.23788	−0.229918
$520$	0	0
$521$	−9.45537	−0.414247	−0.207124	−	0.978315i	$-0.566410\pi$
$521$	−9.45537	−0.414247	−0.207124	+	0.978315i	$0.566410\pi$
$522$	0	0
$523$	− 1.16010i	− 0.0507274i	−0.999678	−	0.0253637i	$-0.991926\pi$
$523$	− 1.16010i	− 0.0507274i	0.999678	−	0.0253637i	$-0.00807439\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.48128i	0.195208i
$528$	0	0
$529$	2.91815	0.126876
$530$	0	0
$531$	11.4848	0.498396
$532$	0	0
$533$	− 19.3197i	− 0.836831i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.28229i	0.357407i
$538$	0	0
$539$	−7.26860	−0.313081
$540$	0	0
$541$	−7.50864	−0.322822	−0.161411	−	0.986887i	$-0.551604\pi$
$541$	−7.50864	−0.322822	−0.161411	+	0.986887i	$0.551604\pi$
$542$	0	0
$543$	14.8329i	0.636543i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	5.75458i	0.246048i	0.992404	+	0.123024i	$0.0392592\pi$
$547$	5.75458i	0.246048i	−0.992404	+	0.123024i	$0.960741\pi$
$548$	0	0
$549$	−14.2823	−0.609553
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 13.6258i	− 0.579427i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.7636i	0.879781i	0.898052	+	0.439890i	$0.144983\pi$
$557$	20.7636i	0.879781i	−0.898052	+	0.439890i	$0.855017\pi$
$558$	0	0
$559$	−8.76212	−0.370598
$560$	0	0
$561$	7.43832	0.314046
$562$	0	0
$563$	− 1.80101i	− 0.0759035i	−0.999280	−	0.0379518i	$-0.987917\pi$
$563$	− 1.80101i	− 0.0759035i	0.999280	−	0.0379518i	$-0.0120833\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.61894i	0.0679892i
$568$	0	0
$569$	−10.2469	−0.429571	−0.214786	−	0.976661i	$-0.568905\pi$
$569$	−10.2469	−0.429571	−0.214786	+	0.976661i	$0.568905\pi$
$570$	0	0
$571$	7.24890	0.303357	0.151679	−	0.988430i	$-0.451532\pi$
$571$	7.24890	0.303357	0.151679	+	0.988430i	$0.451532\pi$
$572$	0	0
$573$	5.07982i	0.212212i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22.6155i	− 0.941494i	−0.882268	−	0.470747i	$-0.843984\pi$
$577$	− 22.6155i	− 0.941494i	0.882268	−	0.470747i	$-0.156016\pi$
$578$	0	0
$579$	−17.4554	−0.725420
$580$	0	0
$581$	0.779160	0.0323250
$582$	0	0
$583$	− 4.57560i	− 0.189502i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.52017i	− 0.145293i	−0.997358	−	0.0726465i	$-0.976856\pi$
$587$	− 3.52017i	− 0.145293i	0.997358	−	0.0726465i	$-0.0231445\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−5.16154	−0.212318
$592$	0	0
$593$	10.8807i	0.446817i	0.974725	+	0.223409i	$0.0717184\pi$
$593$	10.8807i	0.446817i	−0.974725	+	0.223409i	$0.928282\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 21.4609i	− 0.878335i
$598$	0	0
$599$	43.9735	1.79671	0.898354	−	0.439271i	$-0.144763\pi$
$599$	43.9735	1.79671	0.898354	+	0.439271i	$0.144763\pi$
$600$	0	0
$601$	28.2782	1.15349	0.576746	−	0.816923i	$-0.304322\pi$
$601$	28.2782	1.15349	0.576746	+	0.816923i	$0.304322\pi$
$602$	0	0
$603$	5.00348i	0.203757i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.72119i	0.191627i	0.995399	+	0.0958136i	$0.0305453\pi$
$607$	4.72119i	0.191627i	−0.995399	+	0.0958136i	$0.969455\pi$
$608$	0	0
$609$	−6.08736	−0.246672
$610$	0	0
$611$	−3.99398	−0.161579
$612$	0	0
$613$	19.8842i	0.803115i	0.915834	+	0.401557i	$0.131531\pi$
$613$	19.8842i	0.803115i	−0.915834	+	0.401557i	$0.868469\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.4350i	1.38630i	0.720794	+	0.693150i	$0.243778\pi$
$617$	34.4350i	1.38630i	−0.720794	+	0.693150i	$0.756222\pi$
$618$	0	0
$619$	28.4194	1.14227	0.571135	−	0.820856i	$-0.306503\pi$
$619$	28.4194	1.14227	0.571135	+	0.820856i	$0.306503\pi$
$620$	0	0
$621$	4.48128	0.179827
$622$	0	0
$623$	− 8.31758i	− 0.333237i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.2175	−0.646633
$630$	0	0
$631$	27.5910	1.09838	0.549190	−	0.835697i	$-0.314936\pi$
$631$	27.5910	1.09838	0.549190	+	0.835697i	$0.314936\pi$
$632$	0	0
$633$	− 21.9959i	− 0.874260i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 23.1160i	− 0.915891i
$638$	0	0
$639$	0.862336	0.0341135
$640$	0	0
$641$	−5.47374	−0.216200	−0.108100	−	0.994140i	$-0.534477\pi$
$641$	−5.47374	−0.216200	−0.108100	+	0.994140i	$0.534477\pi$
$642$	0	0
$643$	− 34.2892i	− 1.35224i	−0.736793	−	0.676118i	$-0.763661\pi$
$643$	− 34.2892i	− 1.35224i	0.736793	−	0.676118i	$-0.236339\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.51321i	0.374003i	0.982360	+	0.187001i	$0.0598769\pi$
$647$	9.51321i	0.374003i	−0.982360	+	0.187001i	$0.940123\pi$
$648$	0	0
$649$	19.0632	0.748295
$650$	0	0
$651$	−1.61894	−0.0634513
$652$	0	0
$653$	45.7561i	1.79058i	0.445487	+	0.895288i	$0.353030\pi$
$653$	45.7561i	1.79058i	−0.445487	+	0.895288i	$0.646970\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.93867i	− 0.114649i
$658$	0	0
$659$	−10.8215	−0.421547	−0.210774	−	0.977535i	$-0.567598\pi$
$659$	−10.8215	−0.421547	−0.210774	+	0.977535i	$0.567598\pi$
$660$	0	0
$661$	−12.4280	−0.483393	−0.241697	−	0.970352i	$-0.577704\pi$
$661$	−12.4280	−0.483393	−0.241697	+	0.970352i	$0.577704\pi$
$662$	0	0
$663$	23.6558i	0.918715i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.8500i	0.652434i
$668$	0	0
$669$	0.897750	0.0347090
$670$	0	0
$671$	−23.7067	−0.915187
$672$	0	0
$673$	43.1591i	1.66366i	0.555031	+	0.831830i	$0.312706\pi$
$673$	43.1591i	1.66366i	−0.555031	+	0.831830i	$0.687294\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.7192i	1.83400i	0.398891	+	0.916998i	$0.369395\pi$
$677$	47.7192i	1.83400i	−0.398891	+	0.916998i	$0.630605\pi$
$678$	0	0
$679$	−15.0605	−0.577969
$680$	0	0
$681$	3.99449	0.153069
$682$	0	0
$683$	− 29.1097i	− 1.11385i	−0.830562	−	0.556926i	$-0.811981\pi$
$683$	− 29.1097i	− 1.11385i	0.830562	−	0.556926i	$-0.188019\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 18.0708i	− 0.689445i
$688$	0	0
$689$	14.5516	0.554372
$690$	0	0
$691$	20.8807	0.794339	0.397170	−	0.917745i	$-0.369992\pi$
$691$	20.8807	0.794339	0.397170	+	0.917745i	$0.369992\pi$
$692$	0	0
$693$	2.68723i	0.102079i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	16.4009i	0.621228i
$698$	0	0
$699$	−26.0014	−0.983465
$700$	0	0
$701$	−14.2175	−0.536987	−0.268493	−	0.963282i	$-0.586526\pi$
$701$	−14.2175	−0.536987	−0.268493	+	0.963282i	$0.586526\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.48343i	− 0.281443i
$708$	0	0
$709$	−47.6798	−1.79065	−0.895326	−	0.445411i	$-0.853058\pi$
$709$	−47.6798	−1.79065	−0.895326	+	0.445411i	$0.853058\pi$
$710$	0	0
$711$	8.41647	0.315642
$712$	0	0
$713$	4.48128i	0.167825i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.8603i	0.816388i
$718$	0	0
$719$	−38.2244	−1.42553	−0.712766	−	0.701402i	$-0.752558\pi$
$719$	−38.2244	−1.42553	−0.712766	+	0.701402i	$0.752558\pi$
$720$	0	0
$721$	0.0662547	0.00246745
$722$	0	0
$723$	11.8773i	0.441723i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	15.0104i	0.556706i	0.960479	+	0.278353i	$0.0897886\pi$
$727$	15.0104i	0.556706i	−0.960479	+	0.278353i	$0.910211\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	7.43832	0.275116
$732$	0	0
$733$	− 6.43497i	− 0.237681i	−0.992913	−	0.118840i	$-0.962082\pi$
$733$	− 6.43497i	− 0.237681i	0.992913	−	0.118840i	$-0.0379177\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.30511i	0.305923i
$738$	0	0
$739$	−12.3456	−0.454142	−0.227071	−	0.973878i	$-0.572915\pi$
$739$	−12.3456	−0.454142	−0.227071	+	0.973878i	$0.572915\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	53.9959i	1.98092i	0.137805	+	0.990459i	$0.455995\pi$
$743$	53.9959i	1.98092i	−0.137805	+	0.990459i	$0.544005\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.481278i	0.0176090i
$748$	0	0
$749$	4.01704	0.146780
$750$	0	0
$751$	−0.197540	−0.00720834	−0.00360417	−	0.999994i	$-0.501147\pi$
$751$	−0.197540	−0.00720834	−0.00360417	+	0.999994i	$0.501147\pi$
$752$	0	0
$753$	− 27.6020i	− 1.00587i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 23.2747i	− 0.845935i	−0.906145	−	0.422968i	$-0.860988\pi$
$757$	− 23.2747i	− 0.845935i	0.906145	−	0.422968i	$-0.139012\pi$
$758$	0	0
$759$	7.43832	0.269994
$760$	0	0
$761$	−7.00899	−0.254076	−0.127038	−	0.991898i	$-0.540547\pi$
$761$	−7.00899	−0.254076	−0.127038	+	0.991898i	$0.540547\pi$
$762$	0	0
$763$	− 6.24726i	− 0.226166i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	60.6258i	2.18907i
$768$	0	0
$769$	39.1815	1.41292	0.706460	−	0.707753i	$-0.250291\pi$
$769$	39.1815	1.41292	0.706460	+	0.707753i	$0.250291\pi$
$770$	0	0
$771$	5.71916	0.205971
$772$	0	0
$773$	16.0833i	0.578476i	0.957257	+	0.289238i	$0.0934019\pi$
$773$	16.0833i	0.578476i	−0.957257	+	0.289238i	$0.906598\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 5.85886i	− 0.210185i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	1.43136	0.0512182
$782$	0	0
$783$	− 3.76009i	− 0.134375i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 48.3500i	− 1.72349i	−0.507341	−	0.861746i	$-0.669371\pi$
$787$	− 48.3500i	− 1.72349i	0.507341	−	0.861746i	$-0.330629\pi$
$788$	0	0
$789$	−20.0708	−0.714540
$790$	0	0
$791$	−20.3300	−0.722853
$792$	0	0
$793$	− 75.3935i	− 2.67730i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.23933i	0.256430i	0.991746	+	0.128215i	$0.0409248\pi$
$797$	7.23933i	0.256430i	−0.991746	+	0.128215i	$0.959075\pi$
$798$	0	0
$799$	3.39056	0.119949
$800$	0	0
$801$	5.13766	0.181530
$802$	0	0
$803$	− 4.87781i	− 0.172134i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.60405i	0.162070i
$808$	0	0
$809$	6.91010	0.242946	0.121473	−	0.992595i	$-0.461238\pi$
$809$	6.91010	0.242946	0.121473	+	0.992595i	$0.461238\pi$
$810$	0	0
$811$	−28.4827	−1.00016	−0.500082	−	0.865978i	$-0.666697\pi$
$811$	−28.4827	−1.00016	−0.500082	+	0.865978i	$0.666697\pi$
$812$	0	0
$813$	− 22.8399i	− 0.801030i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−8.54608	−0.298624
$820$	0	0
$821$	44.1890	1.54221	0.771104	−	0.636709i	$-0.219705\pi$
$821$	44.1890	1.54221	0.771104	+	0.636709i	$0.219705\pi$
$822$	0	0
$823$	40.2883i	1.40436i	0.711998	+	0.702181i	$0.247790\pi$
$823$	40.2883i	1.40436i	−0.711998	+	0.702181i	$0.752210\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.2504i	1.36487i	0.730947	+	0.682434i	$0.239079\pi$
$827$	39.2504i	1.36487i	−0.730947	+	0.682434i	$0.760921\pi$
$828$	0	0
$829$	40.5466	1.40824	0.704121	−	0.710080i	$-0.251341\pi$
$829$	40.5466	1.40824	0.704121	+	0.710080i	$0.251341\pi$
$830$	0	0
$831$	13.4144	0.465342
$832$	0	0
$833$	19.6236i	0.679919i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	0	0
$839$	26.3585	0.909997	0.454998	−	0.890492i	$-0.349640\pi$
$839$	26.3585	0.909997	0.454998	+	0.890492i	$0.349640\pi$
$840$	0	0
$841$	−14.8618	−0.512474
$842$	0	0
$843$	− 2.08185i	− 0.0717027i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 13.3479i	− 0.458640i
$848$	0	0
$849$	−0.0238807	−0.000819583 0
$850$	0	0
$851$	−16.2175	−0.555928
$852$	0	0
$853$	16.4220i	0.562278i	0.959667	+	0.281139i	$0.0907122\pi$
$853$	16.4220i	0.562278i	−0.959667	+	0.281139i	$0.909288\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.39392i	0.286731i	0.989670	+	0.143365i	$0.0457924\pi$
$857$	8.39392i	0.286731i	−0.989670	+	0.143365i	$0.954208\pi$
$858$	0	0
$859$	−23.4493	−0.800081	−0.400041	−	0.916497i	$-0.631004\pi$
$859$	−23.4493	−0.800081	−0.400041	+	0.916497i	$0.631004\pi$
$860$	0	0
$861$	−5.92511	−0.201927
$862$	0	0
$863$	8.43497i	0.287130i	0.989641	+	0.143565i	$0.0458565\pi$
$863$	8.43497i	0.287130i	−0.989641	+	0.143565i	$0.954143\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 3.08185i	− 0.104665i
$868$	0	0
$869$	13.9702	0.473907
$870$	0	0
$871$	−26.4124	−0.894950
$872$	0	0
$873$	− 9.30269i	− 0.314848i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 27.0163i	− 0.912277i	−0.889909	−	0.456138i	$-0.849232\pi$
$877$	− 27.0163i	− 0.912277i	0.889909	−	0.456138i	$-0.150768\pi$
$878$	0	0
$879$	−7.92366	−0.267258
$880$	0	0
$881$	21.9365	0.739060	0.369530	−	0.929219i	$-0.379519\pi$
$881$	21.9365	0.739060	0.369530	+	0.929219i	$0.379519\pi$
$882$	0	0
$883$	− 4.19442i	− 0.141153i	−0.997506	−	0.0705767i	$-0.977516\pi$
$883$	− 4.19442i	− 0.141153i	0.997506	−	0.0705767i	$-0.0224839\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 27.3851i	− 0.919500i	−0.888048	−	0.459750i	$-0.847939\pi$
$887$	− 27.3851i	− 0.919500i	0.888048	−	0.459750i	$-0.152061\pi$
$888$	0	0
$889$	10.6164	0.356063
$890$	0	0
$891$	−1.65987	−0.0556076
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	23.6558i	0.789844i
$898$	0	0
$899$	3.76009	0.125406
$900$	0	0
$901$	−12.3531	−0.411542
$902$	0	0
$903$	2.68723i	0.0894253i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	21.9831i	0.729936i	0.931020	+	0.364968i	$0.118920\pi$
$907$	21.9831i	0.729936i	−0.931020	+	0.364968i	$0.881080\pi$
$908$	0	0
$909$	4.62242	0.153316
$910$	0	0
$911$	−31.4554	−1.04216	−0.521081	−	0.853507i	$-0.674471\pi$
$911$	−31.4554	−1.04216	−0.521081	+	0.853507i	$0.674471\pi$
$912$	0	0
$913$	0.798857i	0.0264383i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.5672i	0.481051i
$918$	0	0
$919$	−1.51321	−0.0499162	−0.0249581	−	0.999688i	$-0.507945\pi$
$919$	−1.51321	−0.0499162	−0.0249581	+	0.999688i	$0.507945\pi$
$920$	0	0
$921$	−8.30617	−0.273698
$922$	0	0
$923$	4.55211i	0.149834i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.0409247i	0.00134414i
$928$	0	0
$929$	−33.6264	−1.10325	−0.551623	−	0.834093i	$-0.685991\pi$
$929$	−33.6264	−1.10325	−0.551623	+	0.834093i	$0.685991\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 28.1002i	− 0.919960i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 37.3027i	− 1.21863i	−0.792930	−	0.609313i	$-0.791445\pi$
$937$	− 37.3027i	− 1.21863i	0.792930	−	0.609313i	$-0.208555\pi$
$938$	0	0
$939$	11.8364	0.386267
$940$	0	0
$941$	21.7022	0.707473	0.353736	−	0.935345i	$-0.384911\pi$
$941$	21.7022	0.707473	0.353736	+	0.935345i	$0.384911\pi$
$942$	0	0
$943$	16.4009i	0.534086i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.91624i	− 0.159756i	−0.996805	−	0.0798782i	$-0.974547\pi$
$947$	− 4.91624i	− 0.159756i	0.996805	−	0.0798782i	$-0.0254531\pi$
$948$	0	0
$949$	15.5127	0.503564
$950$	0	0
$951$	3.23237	0.104817
$952$	0	0
$953$	− 55.6853i	− 1.80382i	−0.431919	−	0.901912i	$-0.642164\pi$
$953$	− 55.6853i	− 1.80382i	0.431919	−	0.901912i	$-0.357836\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 6.24124i	− 0.201751i
$958$	0	0
$959$	−33.6150	−1.08549
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	2.48128i	0.0799581i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	22.1767i	0.713154i	0.934266	+	0.356577i	$0.116056\pi$
$967$	22.1767i	0.713154i	−0.934266	+	0.356577i	$0.883944\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33.4977	−1.07499	−0.537497	−	0.843266i	$-0.680630\pi$
$971$	−33.4977	−1.07499	−0.537497	+	0.843266i	$0.680630\pi$
$972$	0	0
$973$	− 27.5939i	− 0.884620i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.0403i	0.801112i	0.916272	+	0.400556i	$0.131183\pi$
$977$	25.0403i	0.801112i	−0.916272	+	0.400556i	$0.868817\pi$
$978$	0	0
$979$	8.52784	0.272551
$980$	0	0
$981$	3.85886	0.123204
$982$	0	0
$983$	− 14.7581i	− 0.470709i	−0.971910	−	0.235354i	$-0.924375\pi$
$983$	− 14.7581i	− 0.470709i	0.971910	−	0.235354i	$-0.0756251\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.22490i	0.0389890i
$988$	0	0
$989$	7.43832	0.236525
$990$	0	0
$991$	6.83990	0.217277	0.108638	−	0.994081i	$-0.465351\pi$
$991$	6.83990	0.217277	0.108638	+	0.994081i	$0.465351\pi$
$992$	0	0
$993$	− 14.3416i	− 0.455116i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 55.8663i	− 1.76930i	−0.466252	−	0.884652i	$-0.654396\pi$
$997$	− 55.8663i	− 1.76930i	0.466252	−	0.884652i	$-0.345604\pi$
$998$	0	0
$999$	3.61894	0.114498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.s.3349.7		8
5.2	odd	4		9300.2.a.x.1.2		4
5.3	odd	4		1860.2.a.i.1.3	✓	4
5.4	even	2	inner	9300.2.g.s.3349.2		8
15.8	even	4		5580.2.a.m.1.3		4
20.3	even	4		7440.2.a.cb.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.i.1.3	✓	4	5.3	odd	4
5580.2.a.m.1.3		4	15.8	even	4
7440.2.a.cb.1.2		4	20.3	even	4
9300.2.a.x.1.2		4	5.2	odd	4
9300.2.g.s.3349.2		8	5.4	even	2	inner
9300.2.g.s.3349.7		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.61894i q^{7} -1.00000 q^{9} -1.65987 q^{11} -5.27881i q^{13} +4.48128i q^{17} -1.61894 q^{21} +4.48128i q^{23} -1.00000i q^{27} +3.76009 q^{29} +1.00000 q^{31} -1.65987i q^{33} +3.61894i q^{37} +5.27881 q^{39} +3.65987 q^{41} -1.65987i q^{43} -0.756606i q^{47} +4.37903 q^{49} -4.48128 q^{51} +2.75661i q^{53} -11.4848 q^{59} +14.2823 q^{61} -1.61894i q^{63} -5.00348i q^{67} -4.48128 q^{69} -0.862336 q^{71} +2.93867i q^{73} -2.68723i q^{77} -8.41647 q^{79} +1.00000 q^{81} -0.481278i q^{83} +3.76009i q^{87} -5.13766 q^{89} +8.54608 q^{91} +1.00000i q^{93} +9.30269i q^{97} +1.65987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} + 4 q^{11} + 8 q^{21} - 40 q^{29} + 8 q^{31} + 4 q^{39} + 12 q^{41} - 56 q^{49} - 4 q^{51} + 16 q^{61} - 4 q^{69} + 4 q^{71} - 40 q^{79} + 8 q^{81} - 52 q^{89} + 32 q^{91} - 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 + 4 * q^11 + 8 * q^21 - 40 * q^29 + 8 * q^31 + 4 * q^39 + 12 * q^41 - 56 * q^49 - 4 * q^51 + 16 * q^61 - 4 * q^69 + 4 * q^71 - 40 * q^79 + 8 * q^81 - 52 * q^89 + 32 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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