LMFDB - Embedded newform 9300.2.g.l.3349.1

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 = [4,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0] 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:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1860)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.l.3349.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.44949i q^{7} -1.00000 q^{9} -4.44949i q^{13} -2.00000i q^{17} +4.89898 q^{19} -2.44949 q^{21} -2.89898i q^{23} +1.00000i q^{27} -0.449490 q^{29} +1.00000 q^{31} +3.55051i q^{37} -4.44949 q^{39} +7.79796 q^{41} -3.10102i q^{43} -8.89898i q^{47} +1.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} -4.89898i q^{57} +7.34847 q^{59} -6.89898 q^{61} +2.44949i q^{63} +6.44949i q^{67} -2.89898 q^{69} +1.55051 q^{71} -12.4495i q^{73} +2.89898 q^{79} +1.00000 q^{81} -1.10102i q^{83} +0.449490i q^{87} +4.44949 q^{89} -10.8990 q^{91} -1.00000i q^{93} +5.10102i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} + 8 q^{29} + 4 q^{31} - 8 q^{39} - 8 q^{41} + 4 q^{49} - 8 q^{51} - 8 q^{61} + 8 q^{69} + 16 q^{71} - 8 q^{79} + 4 q^{81} + 8 q^{89} - 24 q^{91}+O(q^{100}) $ 4 * q - 4 * q^9 + 8 * q^29 + 4 * q^31 - 8 * q^39 - 8 * q^41 + 4 * q^49 - 8 * q^51 - 8 * q^61 + 8 * q^69 + 16 * q^71 - 8 * q^79 + 4 * q^81 + 8 * q^89 - 24 * q^91

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
$7$	− 2.44949i	− 0.925820i	−0.886405	−	0.462910i	$-0.846805\pi$
$7$	− 2.44949i	− 0.925820i	0.886405	−	0.462910i	$-0.153195\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 4.44949i	− 1.23407i	−0.786937	−	0.617033i	$-0.788334\pi$
$13$	− 4.44949i	− 1.23407i	0.786937	−	0.617033i	$-0.211666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	4.89898	1.12390	0.561951	−	0.827170i	$-0.310051\pi$
$19$	4.89898	1.12390	0.561951	+	0.827170i	$0.310051\pi$
$20$	0	0
$21$	−2.44949	−0.534522
$22$	0	0
$23$	− 2.89898i	− 0.604479i	−0.953232	−	0.302240i	$-0.902266\pi$
$23$	− 2.89898i	− 0.604479i	0.953232	−	0.302240i	$-0.0977342\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−0.449490	−0.0834681	−0.0417341	−	0.999129i	$-0.513288\pi$
$29$	−0.449490	−0.0834681	−0.0417341	+	0.999129i	$0.513288\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.55051i	0.583700i	0.956464	+	0.291850i	$0.0942709\pi$
$37$	3.55051i	0.583700i	−0.956464	+	0.291850i	$0.905729\pi$
$38$	0	0
$39$	−4.44949	−0.712489
$40$	0	0
$41$	7.79796	1.21784	0.608918	−	0.793233i	$-0.291604\pi$
$41$	7.79796	1.21784	0.608918	+	0.793233i	$0.291604\pi$
$42$	0	0
$43$	− 3.10102i	− 0.472901i	−0.971644	−	0.236451i	$-0.924016\pi$
$43$	− 3.10102i	− 0.472901i	0.971644	−	0.236451i	$-0.0759841\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.89898i	− 1.29805i	−0.760767	−	0.649025i	$-0.775177\pi$
$47$	− 8.89898i	− 1.29805i	0.760767	−	0.649025i	$-0.224823\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.89898i	− 0.648886i
$58$	0	0
$59$	7.34847	0.956689	0.478345	−	0.878172i	$-0.341237\pi$
$59$	7.34847	0.956689	0.478345	+	0.878172i	$0.341237\pi$
$60$	0	0
$61$	−6.89898	−0.883324	−0.441662	−	0.897182i	$-0.645611\pi$
$61$	−6.89898	−0.883324	−0.441662	+	0.897182i	$0.645611\pi$
$62$	0	0
$63$	2.44949i	0.308607i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.44949i	0.787931i	0.919125	+	0.393965i	$0.128897\pi$
$67$	6.44949i	0.787931i	−0.919125	+	0.393965i	$0.871103\pi$
$68$	0	0
$69$	−2.89898	−0.348996
$70$	0	0
$71$	1.55051	0.184012	0.0920059	−	0.995758i	$-0.470672\pi$
$71$	1.55051	0.184012	0.0920059	+	0.995758i	$0.470672\pi$
$72$	0	0
$73$	− 12.4495i	− 1.45710i	−0.684991	−	0.728551i	$-0.740194\pi$
$73$	− 12.4495i	− 1.45710i	0.684991	−	0.728551i	$-0.259806\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.89898	0.326161	0.163080	−	0.986613i	$-0.447857\pi$
$79$	2.89898	0.326161	0.163080	+	0.986613i	$0.447857\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 1.10102i	− 0.120853i	−0.998173	−	0.0604264i	$-0.980754\pi$
$83$	− 1.10102i	− 0.120853i	0.998173	−	0.0604264i	$-0.0192460\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.449490i	0.0481904i
$88$	0	0
$89$	4.44949	0.471645	0.235822	−	0.971796i	$-0.424222\pi$
$89$	4.44949	0.471645	0.235822	+	0.971796i	$0.424222\pi$
$90$	0	0
$91$	−10.8990	−1.14252
$92$	0	0
$93$	− 1.00000i	− 0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.10102i	0.517930i	0.965887	+	0.258965i	$0.0833815\pi$
$97$	5.10102i	0.517930i	−0.965887	+	0.258965i	$0.916619\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	− 5.55051i	− 0.546908i	−0.961885	−	0.273454i	$-0.911834\pi$
$103$	− 5.55051i	− 0.546908i	0.961885	−	0.273454i	$-0.0881661\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.89898i	0.473602i	0.971558	+	0.236801i	$0.0760990\pi$
$107$	4.89898i	0.473602i	−0.971558	+	0.236801i	$0.923901\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	3.55051	0.337000
$112$	0	0
$113$	− 15.7980i	− 1.48615i	−0.669210	−	0.743073i	$-0.733367\pi$
$113$	− 15.7980i	− 1.48615i	0.669210	−	0.743073i	$-0.266633\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.44949i	0.411355i
$118$	0	0
$119$	−4.89898	−0.449089
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	− 7.79796i	− 0.703118i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.89898i	0.789657i	0.918755	+	0.394828i	$0.129196\pi$
$127$	8.89898i	0.789657i	−0.918755	+	0.394828i	$0.870804\pi$
$128$	0	0
$129$	−3.10102	−0.273030
$130$	0	0
$131$	12.2474	1.07006	0.535032	−	0.844832i	$-0.320299\pi$
$131$	12.2474	1.07006	0.535032	+	0.844832i	$0.320299\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.8990i	1.27291i	0.771316	+	0.636453i	$0.219599\pi$
$137$	14.8990i	1.27291i	−0.771316	+	0.636453i	$0.780401\pi$
$138$	0	0
$139$	−13.7980	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$139$	−13.7980	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$140$	0	0
$141$	−8.89898	−0.749429
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	−0.202041	−0.0165518	−0.00827592	−	0.999966i	$-0.502634\pi$
$149$	−0.202041	−0.0165518	−0.00827592	+	0.999966i	$0.502634\pi$
$150$	0	0
$151$	−10.8990	−0.886946	−0.443473	−	0.896288i	$-0.646254\pi$
$151$	−10.8990	−0.886946	−0.443473	+	0.896288i	$0.646254\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.89898i	0.550599i	0.961359	+	0.275299i	$0.0887769\pi$
$157$	6.89898i	0.550599i	−0.961359	+	0.275299i	$0.911223\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−7.10102	−0.559639
$162$	0	0
$163$	13.1464i	1.02971i	0.857278	+	0.514854i	$0.172154\pi$
$163$	13.1464i	1.02971i	−0.857278	+	0.514854i	$0.827846\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	−6.79796	−0.522920
$170$	0	0
$171$	−4.89898	−0.374634
$172$	0	0
$173$	− 0.202041i	− 0.0153609i	−0.999971	−	0.00768045i	$-0.997555\pi$
$173$	− 0.202041i	− 0.0153609i	0.999971	−	0.00768045i	$-0.00244479\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 7.34847i	− 0.552345i
$178$	0	0
$179$	15.5959	1.16569	0.582847	−	0.812582i	$-0.301939\pi$
$179$	15.5959	1.16569	0.582847	+	0.812582i	$0.301939\pi$
$180$	0	0
$181$	−2.89898	−0.215479	−0.107740	−	0.994179i	$-0.534361\pi$
$181$	−2.89898	−0.215479	−0.107740	+	0.994179i	$0.534361\pi$
$182$	0	0
$183$	6.89898i	0.509987i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.44949	0.178174
$190$	0	0
$191$	1.55051	0.112191	0.0560955	−	0.998425i	$-0.482135\pi$
$191$	1.55051	0.112191	0.0560955	+	0.998425i	$0.482135\pi$
$192$	0	0
$193$	− 8.69694i	− 0.626019i	−0.949750	−	0.313010i	$-0.898663\pi$
$193$	− 8.69694i	− 0.626019i	0.949750	−	0.313010i	$-0.101337\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.6969i	1.47460i	0.675568	+	0.737298i	$0.263899\pi$
$197$	20.6969i	1.47460i	−0.675568	+	0.737298i	$0.736101\pi$
$198$	0	0
$199$	−24.6969	−1.75072	−0.875360	−	0.483472i	$-0.839375\pi$
$199$	−24.6969	−1.75072	−0.875360	+	0.483472i	$0.839375\pi$
$200$	0	0
$201$	6.44949	0.454912
$202$	0	0
$203$	1.10102i	0.0772765i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.89898i	0.201493i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.4949	1.41093	0.705463	−	0.708746i	$-0.250739\pi$
$211$	20.4949	1.41093	0.705463	+	0.708746i	$0.250739\pi$
$212$	0	0
$213$	− 1.55051i	− 0.106239i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 2.44949i	− 0.166282i
$218$	0	0
$219$	−12.4495	−0.841259
$220$	0	0
$221$	−8.89898	−0.598610
$222$	0	0
$223$	− 9.79796i	− 0.656120i	−0.944657	−	0.328060i	$-0.893605\pi$
$223$	− 9.79796i	− 0.656120i	0.944657	−	0.328060i	$-0.106395\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.7980i	− 1.44678i	−0.690439	−	0.723391i	$-0.742583\pi$
$227$	− 21.7980i	− 1.44678i	0.690439	−	0.723391i	$-0.257417\pi$
$228$	0	0
$229$	−19.7980	−1.30829	−0.654143	−	0.756371i	$-0.726971\pi$
$229$	−19.7980	−1.30829	−0.654143	+	0.756371i	$0.726971\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2.20204i	− 0.144261i	−0.997395	−	0.0721303i	$-0.977020\pi$
$233$	− 2.20204i	− 0.144261i	0.997395	−	0.0721303i	$-0.0229797\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2.89898i	− 0.188309i
$238$	0	0
$239$	−16.8990	−1.09310	−0.546552	−	0.837425i	$-0.684060\pi$
$239$	−16.8990	−1.09310	−0.546552	+	0.837425i	$0.684060\pi$
$240$	0	0
$241$	3.79796	0.244648	0.122324	−	0.992490i	$-0.460965\pi$
$241$	3.79796	0.244648	0.122324	+	0.992490i	$0.460965\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 21.7980i	− 1.38697i
$248$	0	0
$249$	−1.10102	−0.0697743
$250$	0	0
$251$	−9.79796	−0.618442	−0.309221	−	0.950990i	$-0.600068\pi$
$251$	−9.79796	−0.618442	−0.309221	+	0.950990i	$0.600068\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 16.0000i	− 0.998053i	−0.866587	−	0.499026i	$-0.833691\pi$
$257$	− 16.0000i	− 0.998053i	0.866587	−	0.499026i	$-0.166309\pi$
$258$	0	0
$259$	8.69694	0.540401
$260$	0	0
$261$	0.449490	0.0278227
$262$	0	0
$263$	29.3939i	1.81250i	0.422738	+	0.906252i	$0.361069\pi$
$263$	29.3939i	1.81250i	−0.422738	+	0.906252i	$0.638931\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.44949i	− 0.272304i
$268$	0	0
$269$	−12.0454	−0.734421	−0.367211	−	0.930138i	$-0.619687\pi$
$269$	−12.0454	−0.734421	−0.367211	+	0.930138i	$0.619687\pi$
$270$	0	0
$271$	−17.7980	−1.08115	−0.540575	−	0.841296i	$-0.681793\pi$
$271$	−17.7980	−1.08115	−0.540575	+	0.841296i	$0.681793\pi$
$272$	0	0
$273$	10.8990i	0.659636i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 11.1464i	− 0.669724i	−0.942267	−	0.334862i	$-0.891310\pi$
$277$	− 11.1464i	− 0.669724i	0.942267	−	0.334862i	$-0.108690\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	14.8990	0.888799	0.444399	−	0.895829i	$-0.353417\pi$
$281$	14.8990	0.888799	0.444399	+	0.895829i	$0.353417\pi$
$282$	0	0
$283$	− 9.55051i	− 0.567719i	−0.958866	−	0.283859i	$-0.908385\pi$
$283$	− 9.55051i	− 0.567719i	0.958866	−	0.283859i	$-0.0916149\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 19.1010i	− 1.12750i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	5.10102	0.299027
$292$	0	0
$293$	− 21.7980i	− 1.27345i	−0.771091	−	0.636725i	$-0.780289\pi$
$293$	− 21.7980i	− 1.27345i	0.771091	−	0.636725i	$-0.219711\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12.8990	−0.745967
$300$	0	0
$301$	−7.59592	−0.437821
$302$	0	0
$303$	− 2.00000i	− 0.114897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.4495i	1.28126i	0.767850	+	0.640630i	$0.221327\pi$
$307$	22.4495i	1.28126i	−0.767850	+	0.640630i	$0.778673\pi$
$308$	0	0
$309$	−5.55051	−0.315757
$310$	0	0
$311$	4.65153	0.263764	0.131882	−	0.991265i	$-0.457898\pi$
$311$	4.65153	0.263764	0.131882	+	0.991265i	$0.457898\pi$
$312$	0	0
$313$	− 13.3485i	− 0.754500i	−0.926111	−	0.377250i	$-0.876870\pi$
$313$	− 13.3485i	− 0.754500i	0.926111	−	0.377250i	$-0.123130\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.5959i	0.875954i	0.898986	+	0.437977i	$0.144305\pi$
$317$	15.5959i	0.875954i	−0.898986	+	0.437977i	$0.855695\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.89898	0.273434
$322$	0	0
$323$	− 9.79796i	− 0.545173i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 4.00000i	− 0.221201i
$328$	0	0
$329$	−21.7980	−1.20176
$330$	0	0
$331$	−1.10102	−0.0605176	−0.0302588	−	0.999542i	$-0.509633\pi$
$331$	−1.10102	−0.0605176	−0.0302588	+	0.999542i	$0.509633\pi$
$332$	0	0
$333$	− 3.55051i	− 0.194567i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.1464i	1.04297i	0.853260	+	0.521486i	$0.174622\pi$
$337$	19.1464i	1.04297i	−0.853260	+	0.521486i	$0.825378\pi$
$338$	0	0
$339$	−15.7980	−0.858027
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 19.5959i	− 1.05808i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	0	0
$349$	−14.2020	−0.760218	−0.380109	−	0.924942i	$-0.624113\pi$
$349$	−14.2020	−0.760218	−0.380109	+	0.924942i	$0.624113\pi$
$350$	0	0
$351$	4.44949	0.237496
$352$	0	0
$353$	− 15.7980i	− 0.840841i	−0.907329	−	0.420420i	$-0.861883\pi$
$353$	− 15.7980i	− 0.840841i	0.907329	−	0.420420i	$-0.138117\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.89898i	0.259281i
$358$	0	0
$359$	−9.55051	−0.504057	−0.252028	−	0.967720i	$-0.581098\pi$
$359$	−9.55051	−0.504057	−0.252028	+	0.967720i	$0.581098\pi$
$360$	0	0
$361$	5.00000	0.263158
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.5959i	0.605302i	0.953101	+	0.302651i	$0.0978717\pi$
$367$	11.5959i	0.605302i	−0.953101	+	0.302651i	$0.902128\pi$
$368$	0	0
$369$	−7.79796	−0.405946
$370$	0	0
$371$	−14.6969	−0.763027
$372$	0	0
$373$	− 3.30306i	− 0.171026i	−0.996337	−	0.0855130i	$-0.972747\pi$
$373$	− 3.30306i	− 0.171026i	0.996337	−	0.0855130i	$-0.0272529\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00000i	0.103005i
$378$	0	0
$379$	−16.8990	−0.868042	−0.434021	−	0.900903i	$-0.642906\pi$
$379$	−16.8990	−0.868042	−0.434021	+	0.900903i	$0.642906\pi$
$380$	0	0
$381$	8.89898	0.455909
$382$	0	0
$383$	10.8990i	0.556912i	0.960449	+	0.278456i	$0.0898226\pi$
$383$	10.8990i	0.556912i	−0.960449	+	0.278456i	$0.910177\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.10102i	0.157634i
$388$	0	0
$389$	6.65153	0.337246	0.168623	−	0.985681i	$-0.446068\pi$
$389$	6.65153	0.337246	0.168623	+	0.985681i	$0.446068\pi$
$390$	0	0
$391$	−5.79796	−0.293215
$392$	0	0
$393$	− 12.2474i	− 0.617802i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 11.7980i	− 0.592123i	−0.955169	−	0.296061i	$-0.904327\pi$
$397$	− 11.7980i	− 0.592123i	0.955169	−	0.296061i	$-0.0956733\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	29.8434	1.49031	0.745153	−	0.666893i	$-0.232376\pi$
$401$	29.8434	1.49031	0.745153	+	0.666893i	$0.232376\pi$
$402$	0	0
$403$	− 4.44949i	− 0.221645i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0.696938	0.0344614	0.0172307	−	0.999852i	$-0.494515\pi$
$409$	0.696938	0.0344614	0.0172307	+	0.999852i	$0.494515\pi$
$410$	0	0
$411$	14.8990	0.734912
$412$	0	0
$413$	− 18.0000i	− 0.885722i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	13.7980i	0.675689i
$418$	0	0
$419$	−26.4495	−1.29214	−0.646071	−	0.763277i	$-0.723589\pi$
$419$	−26.4495	−1.29214	−0.646071	+	0.763277i	$0.723589\pi$
$420$	0	0
$421$	5.79796	0.282575	0.141288	−	0.989969i	$-0.454876\pi$
$421$	5.79796	0.282575	0.141288	+	0.989969i	$0.454876\pi$
$422$	0	0
$423$	8.89898i	0.432683i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.8990i	0.817799i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.5505	−0.845378	−0.422689	−	0.906275i	$-0.638914\pi$
$431$	−17.5505	−0.845378	−0.422689	+	0.906275i	$0.638914\pi$
$432$	0	0
$433$	31.1464i	1.49680i	0.663247	+	0.748401i	$0.269178\pi$
$433$	31.1464i	1.49680i	−0.663247	+	0.748401i	$0.730822\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 14.2020i	− 0.679376i
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.202041i	0.00955621i
$448$	0	0
$449$	3.14643	0.148489	0.0742446	−	0.997240i	$-0.476345\pi$
$449$	3.14643	0.148489	0.0742446	+	0.997240i	$0.476345\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	10.8990i	0.512079i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 5.34847i	− 0.250191i	−0.992145	−	0.125095i	$-0.960076\pi$
$457$	− 5.34847i	− 0.250191i	0.992145	−	0.125095i	$-0.0399237\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	3.55051	0.165364	0.0826819	−	0.996576i	$-0.473651\pi$
$461$	3.55051	0.165364	0.0826819	+	0.996576i	$0.473651\pi$
$462$	0	0
$463$	1.30306i	0.0605584i	0.999541	+	0.0302792i	$0.00963964\pi$
$463$	1.30306i	0.0605584i	−0.999541	+	0.0302792i	$0.990360\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 19.1010i	− 0.883890i	−0.897042	−	0.441945i	$-0.854289\pi$
$467$	− 19.1010i	− 0.883890i	0.897042	−	0.441945i	$-0.145711\pi$
$468$	0	0
$469$	15.7980	0.729482
$470$	0	0
$471$	6.89898	0.317888
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	−19.3485	−0.884054	−0.442027	−	0.897002i	$-0.645740\pi$
$479$	−19.3485	−0.884054	−0.442027	+	0.897002i	$0.645740\pi$
$480$	0	0
$481$	15.7980	0.720325
$482$	0	0
$483$	7.10102i	0.323108i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.8990i	1.49080i	0.666620	+	0.745398i	$0.267740\pi$
$487$	32.8990i	1.49080i	−0.666620	+	0.745398i	$0.732260\pi$
$488$	0	0
$489$	13.1464	0.594502
$490$	0	0
$491$	−11.1010	−0.500982	−0.250491	−	0.968119i	$-0.580592\pi$
$491$	−11.1010	−0.500982	−0.250491	+	0.968119i	$0.580592\pi$
$492$	0	0
$493$	0.898979i	0.0404880i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.79796i	− 0.170362i
$498$	0	0
$499$	−13.7980	−0.617681	−0.308841	−	0.951114i	$-0.599941\pi$
$499$	−13.7980	−0.617681	−0.308841	+	0.951114i	$0.599941\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	27.1010i	1.20837i	0.796842	+	0.604187i	$0.206502\pi$
$503$	27.1010i	1.20837i	−0.796842	+	0.604187i	$0.793498\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.79796i	0.301908i
$508$	0	0
$509$	−6.24745	−0.276913	−0.138457	−	0.990368i	$-0.544214\pi$
$509$	−6.24745	−0.276913	−0.138457	+	0.990368i	$0.544214\pi$
$510$	0	0
$511$	−30.4949	−1.34901
$512$	0	0
$513$	4.89898i	0.216295i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−0.202041	−0.00886862
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	− 11.1010i	− 0.485414i	−0.970100	−	0.242707i	$-0.921965\pi$
$523$	− 11.1010i	− 0.485414i	0.970100	−	0.242707i	$-0.0780353\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.00000i	− 0.0871214i
$528$	0	0
$529$	14.5959	0.634605
$530$	0	0
$531$	−7.34847	−0.318896
$532$	0	0
$533$	− 34.6969i	− 1.50289i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.5959i	− 0.673014i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.404082	−0.0173728	−0.00868642	−	0.999962i	$-0.502765\pi$
$541$	−0.404082	−0.0173728	−0.00868642	+	0.999962i	$0.502765\pi$
$542$	0	0
$543$	2.89898i	0.124407i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 30.9444i	− 1.32309i	−0.749907	−	0.661543i	$-0.769902\pi$
$547$	− 30.9444i	− 1.32309i	0.749907	−	0.661543i	$-0.230098\pi$
$548$	0	0
$549$	6.89898	0.294441
$550$	0	0
$551$	−2.20204	−0.0938101
$552$	0	0
$553$	− 7.10102i	− 0.301966i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21.5959i	− 0.915048i	−0.889197	−	0.457524i	$-0.848736\pi$
$557$	− 21.5959i	− 0.915048i	0.889197	−	0.457524i	$-0.151264\pi$
$558$	0	0
$559$	−13.7980	−0.583591
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.898979i	0.0378875i	0.999821	+	0.0189437i	$0.00603034\pi$
$563$	0.898979i	0.0378875i	−0.999821	+	0.0189437i	$0.993970\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.44949i	− 0.102869i
$568$	0	0
$569$	40.4495	1.69573	0.847865	−	0.530212i	$-0.177887\pi$
$569$	40.4495	1.69573	0.847865	+	0.530212i	$0.177887\pi$
$570$	0	0
$571$	−10.2020	−0.426942	−0.213471	−	0.976949i	$-0.568477\pi$
$571$	−10.2020	−0.426942	−0.213471	+	0.976949i	$0.568477\pi$
$572$	0	0
$573$	− 1.55051i	− 0.0647735i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.40408i	− 0.100083i	−0.998747	−	0.0500416i	$-0.984065\pi$
$577$	− 2.40408i	− 0.100083i	0.998747	−	0.0500416i	$-0.0159354\pi$
$578$	0	0
$579$	−8.69694	−0.361432
$580$	0	0
$581$	−2.69694	−0.111888
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.5959i	− 0.643712i	−0.946789	−	0.321856i	$-0.895693\pi$
$587$	− 15.5959i	− 0.643712i	0.946789	−	0.321856i	$-0.104307\pi$
$588$	0	0
$589$	4.89898	0.201859
$590$	0	0
$591$	20.6969	0.851358
$592$	0	0
$593$	− 33.5959i	− 1.37962i	−0.723991	−	0.689809i	$-0.757694\pi$
$593$	− 33.5959i	− 1.37962i	0.723991	−	0.689809i	$-0.242306\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	24.6969i	1.01078i
$598$	0	0
$599$	14.9444	0.610611	0.305306	−	0.952254i	$-0.401241\pi$
$599$	14.9444	0.610611	0.305306	+	0.952254i	$0.401241\pi$
$600$	0	0
$601$	3.30306	0.134735	0.0673673	−	0.997728i	$-0.478540\pi$
$601$	3.30306	0.134735	0.0673673	+	0.997728i	$0.478540\pi$
$602$	0	0
$603$	− 6.44949i	− 0.262644i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	42.9444i	1.74306i	0.490343	+	0.871529i	$0.336871\pi$
$607$	42.9444i	1.74306i	−0.490343	+	0.871529i	$0.663129\pi$
$608$	0	0
$609$	1.10102	0.0446156
$610$	0	0
$611$	−39.5959	−1.60188
$612$	0	0
$613$	36.0454i	1.45586i	0.685651	+	0.727930i	$0.259518\pi$
$613$	36.0454i	1.45586i	−0.685651	+	0.727930i	$0.740482\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	−46.4949	−1.86879	−0.934394	−	0.356242i	$-0.884058\pi$
$619$	−46.4949	−1.86879	−0.934394	+	0.356242i	$0.884058\pi$
$620$	0	0
$621$	2.89898	0.116332
$622$	0	0
$623$	− 10.8990i	− 0.436658i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.10102	0.283136
$630$	0	0
$631$	3.59592	0.143151	0.0715756	−	0.997435i	$-0.477197\pi$
$631$	3.59592	0.143151	0.0715756	+	0.997435i	$0.477197\pi$
$632$	0	0
$633$	− 20.4949i	− 0.814599i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.44949i	− 0.176295i
$638$	0	0
$639$	−1.55051	−0.0613372
$640$	0	0
$641$	−44.0454	−1.73969	−0.869845	−	0.493326i	$-0.835781\pi$
$641$	−44.0454	−1.73969	−0.869845	+	0.493326i	$0.835781\pi$
$642$	0	0
$643$	− 20.4949i	− 0.808240i	−0.914706	−	0.404120i	$-0.867578\pi$
$643$	− 20.4949i	− 0.808240i	0.914706	−	0.404120i	$-0.132422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 19.5959i	− 0.770395i	−0.922834	−	0.385198i	$-0.874133\pi$
$647$	− 19.5959i	− 0.770395i	0.922834	−	0.385198i	$-0.125867\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.44949	−0.0960031
$652$	0	0
$653$	4.00000i	0.156532i	0.996933	+	0.0782660i	$0.0249384\pi$
$653$	4.00000i	0.156532i	−0.996933	+	0.0782660i	$0.975062\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.4495i	0.485701i
$658$	0	0
$659$	−0.651531	−0.0253800	−0.0126900	−	0.999919i	$-0.504039\pi$
$659$	−0.651531	−0.0253800	−0.0126900	+	0.999919i	$0.504039\pi$
$660$	0	0
$661$	39.3939	1.53224	0.766122	−	0.642695i	$-0.222184\pi$
$661$	39.3939	1.53224	0.766122	+	0.642695i	$0.222184\pi$
$662$	0	0
$663$	8.89898i	0.345608i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.30306i	0.0504547i
$668$	0	0
$669$	−9.79796	−0.378811
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.9444i	1.11572i	0.829934	+	0.557862i	$0.188378\pi$
$673$	28.9444i	1.11572i	−0.829934	+	0.557862i	$0.811622\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.20204i	0.315230i	0.987501	+	0.157615i	$0.0503805\pi$
$677$	8.20204i	0.315230i	−0.987501	+	0.157615i	$0.949619\pi$
$678$	0	0
$679$	12.4949	0.479510
$680$	0	0
$681$	−21.7980	−0.835300
$682$	0	0
$683$	3.59592i	0.137594i	0.997631	+	0.0687970i	$0.0219161\pi$
$683$	3.59592i	0.137594i	−0.997631	+	0.0687970i	$0.978084\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	19.7980i	0.755339i
$688$	0	0
$689$	−26.6969	−1.01707
$690$	0	0
$691$	−45.3939	−1.72686	−0.863432	−	0.504465i	$-0.831690\pi$
$691$	−45.3939	−1.72686	−0.863432	+	0.504465i	$0.831690\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 15.5959i	− 0.590738i
$698$	0	0
$699$	−2.20204	−0.0832888
$700$	0	0
$701$	44.6969	1.68818	0.844090	−	0.536201i	$-0.180141\pi$
$701$	44.6969	1.68818	0.844090	+	0.536201i	$0.180141\pi$
$702$	0	0
$703$	17.3939i	0.656022i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.89898i	− 0.184245i
$708$	0	0
$709$	7.79796	0.292858	0.146429	−	0.989221i	$-0.453222\pi$
$709$	7.79796	0.292858	0.146429	+	0.989221i	$0.453222\pi$
$710$	0	0
$711$	−2.89898	−0.108720
$712$	0	0
$713$	− 2.89898i	− 0.108568i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.8990i	0.631104i
$718$	0	0
$719$	−17.7980	−0.663752	−0.331876	−	0.943323i	$-0.607682\pi$
$719$	−17.7980	−0.663752	−0.331876	+	0.943323i	$0.607682\pi$
$720$	0	0
$721$	−13.5959	−0.506338
$722$	0	0
$723$	− 3.79796i	− 0.141248i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 17.1464i	− 0.635926i	−0.948103	−	0.317963i	$-0.897001\pi$
$727$	− 17.1464i	− 0.635926i	0.948103	−	0.317963i	$-0.102999\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−6.20204	−0.229391
$732$	0	0
$733$	10.4041i	0.384284i	0.981367	+	0.192142i	$0.0615433\pi$
$733$	10.4041i	0.384284i	−0.981367	+	0.192142i	$0.938457\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	40.6969	1.49706	0.748531	−	0.663100i	$-0.230760\pi$
$739$	40.6969	1.49706	0.748531	+	0.663100i	$0.230760\pi$
$740$	0	0
$741$	−21.7980	−0.800768
$742$	0	0
$743$	− 1.79796i	− 0.0659607i	−0.999456	−	0.0329804i	$-0.989500\pi$
$743$	− 1.79796i	− 0.0659607i	0.999456	−	0.0329804i	$-0.0104999\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.10102i	0.0402842i
$748$	0	0
$749$	12.0000	0.438470
$750$	0	0
$751$	−5.79796	−0.211571	−0.105785	−	0.994389i	$-0.533736\pi$
$751$	−5.79796	−0.211571	−0.105785	+	0.994389i	$0.533736\pi$
$752$	0	0
$753$	9.79796i	0.357057i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.4495i	0.452484i	0.974071	+	0.226242i	$0.0726441\pi$
$757$	12.4495i	0.452484i	−0.974071	+	0.226242i	$0.927356\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−37.8434	−1.37182	−0.685910	−	0.727686i	$-0.740596\pi$
$761$	−37.8434	−1.37182	−0.685910	+	0.727686i	$0.740596\pi$
$762$	0	0
$763$	− 9.79796i	− 0.354710i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 32.6969i	− 1.18062i
$768$	0	0
$769$	−33.7980	−1.21879	−0.609393	−	0.792868i	$-0.708587\pi$
$769$	−33.7980	−1.21879	−0.609393	+	0.792868i	$0.708587\pi$
$770$	0	0
$771$	−16.0000	−0.576226
$772$	0	0
$773$	− 10.0000i	− 0.359675i	−0.983696	−	0.179838i	$-0.942443\pi$
$773$	− 10.0000i	− 0.359675i	0.983696	−	0.179838i	$-0.0575572\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.69694i	− 0.312001i
$778$	0	0
$779$	38.2020	1.36873
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 0.449490i	− 0.0160635i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	47.1918i	1.68221i	0.540874	+	0.841104i	$0.318094\pi$
$787$	47.1918i	1.68221i	−0.540874	+	0.841104i	$0.681906\pi$
$788$	0	0
$789$	29.3939	1.04645
$790$	0	0
$791$	−38.6969	−1.37590
$792$	0	0
$793$	30.6969i	1.09008i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 26.4949i	− 0.938497i	−0.883066	−	0.469249i	$-0.844525\pi$
$797$	− 26.4949i	− 0.938497i	0.883066	−	0.469249i	$-0.155475\pi$
$798$	0	0
$799$	−17.7980	−0.629647
$800$	0	0
$801$	−4.44949	−0.157215
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.0454i	0.424018i
$808$	0	0
$809$	17.3485	0.609940	0.304970	−	0.952362i	$-0.401354\pi$
$809$	17.3485	0.609940	0.304970	+	0.952362i	$0.401354\pi$
$810$	0	0
$811$	46.2929	1.62556	0.812781	−	0.582569i	$-0.197953\pi$
$811$	46.2929	1.62556	0.812781	+	0.582569i	$0.197953\pi$
$812$	0	0
$813$	17.7980i	0.624202i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 15.1918i	− 0.531495i
$818$	0	0
$819$	10.8990	0.380841
$820$	0	0
$821$	−55.6413	−1.94190	−0.970948	−	0.239291i	$-0.923085\pi$
$821$	−55.6413	−1.94190	−0.970948	+	0.239291i	$0.923085\pi$
$822$	0	0
$823$	24.4949i	0.853838i	0.904290	+	0.426919i	$0.140401\pi$
$823$	24.4949i	0.853838i	−0.904290	+	0.426919i	$0.859599\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42.0908i	1.46364i	0.681497	+	0.731821i	$0.261329\pi$
$827$	42.0908i	1.46364i	−0.681497	+	0.731821i	$0.738671\pi$
$828$	0	0
$829$	−2.89898	−0.100686	−0.0503429	−	0.998732i	$-0.516031\pi$
$829$	−2.89898	−0.100686	−0.0503429	+	0.998732i	$0.516031\pi$
$830$	0	0
$831$	−11.1464	−0.386665
$832$	0	0
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000i	0.0345651i
$838$	0	0
$839$	14.4495	0.498852	0.249426	−	0.968394i	$-0.419758\pi$
$839$	14.4495	0.498852	0.249426	+	0.968394i	$0.419758\pi$
$840$	0	0
$841$	−28.7980	−0.993033
$842$	0	0
$843$	− 14.8990i	− 0.513148i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.9444i	0.925820i
$848$	0	0
$849$	−9.55051	−0.327773
$850$	0	0
$851$	10.2929	0.352835
$852$	0	0
$853$	− 19.3939i	− 0.664034i	−0.943273	−	0.332017i	$-0.892271\pi$
$853$	− 19.3939i	− 0.664034i	0.943273	−	0.332017i	$-0.107729\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 33.1918i	− 1.13381i	−0.823783	−	0.566906i	$-0.808140\pi$
$857$	− 33.1918i	− 1.13381i	0.823783	−	0.566906i	$-0.191860\pi$
$858$	0	0
$859$	47.5959	1.62395	0.811976	−	0.583691i	$-0.198392\pi$
$859$	47.5959	1.62395	0.811976	+	0.583691i	$0.198392\pi$
$860$	0	0
$861$	−19.1010	−0.650961
$862$	0	0
$863$	− 56.0000i	− 1.90626i	−0.302558	−	0.953131i	$-0.597840\pi$
$863$	− 56.0000i	− 1.90626i	0.302558	−	0.953131i	$-0.402160\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	28.6969	0.972359
$872$	0	0
$873$	− 5.10102i	− 0.172643i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16.2020i	0.547104i	0.961857	+	0.273552i	$0.0881986\pi$
$877$	16.2020i	0.547104i	−0.961857	+	0.273552i	$0.911801\pi$
$878$	0	0
$879$	−21.7980	−0.735227
$880$	0	0
$881$	1.75255	0.0590450	0.0295225	−	0.999564i	$-0.490601\pi$
$881$	1.75255	0.0590450	0.0295225	+	0.999564i	$0.490601\pi$
$882$	0	0
$883$	− 21.3939i	− 0.719961i	−0.932960	−	0.359981i	$-0.882783\pi$
$883$	− 21.3939i	− 0.719961i	0.932960	−	0.359981i	$-0.117217\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 20.0000i	− 0.671534i	−0.941945	−	0.335767i	$-0.891004\pi$
$887$	− 20.0000i	− 0.671534i	0.941945	−	0.335767i	$-0.108996\pi$
$888$	0	0
$889$	21.7980	0.731080
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 43.5959i	− 1.45888i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.8990i	0.430684i
$898$	0	0
$899$	−0.449490	−0.0149913
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	7.59592i	0.252776i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 37.1464i	− 1.23343i	−0.787188	−	0.616713i	$-0.788464\pi$
$907$	− 37.1464i	− 1.23343i	0.787188	−	0.616713i	$-0.211536\pi$
$908$	0	0
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−6.20204	−0.205483	−0.102741	−	0.994708i	$-0.532761\pi$
$911$	−6.20204	−0.205483	−0.102741	+	0.994708i	$0.532761\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 30.0000i	− 0.990687i
$918$	0	0
$919$	−25.3031	−0.834671	−0.417335	−	0.908752i	$-0.637036\pi$
$919$	−25.3031	−0.834671	−0.417335	+	0.908752i	$0.637036\pi$
$920$	0	0
$921$	22.4495	0.739736
$922$	0	0
$923$	− 6.89898i	− 0.227083i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.55051i	0.182303i
$928$	0	0
$929$	40.9444	1.34334	0.671671	−	0.740850i	$-0.265577\pi$
$929$	40.9444	1.34334	0.671671	+	0.740850i	$0.265577\pi$
$930$	0	0
$931$	4.89898	0.160558
$932$	0	0
$933$	− 4.65153i	− 0.152284i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	22.4949i	0.734876i	0.930048	+	0.367438i	$0.119765\pi$
$937$	22.4949i	0.734876i	−0.930048	+	0.367438i	$0.880235\pi$
$938$	0	0
$939$	−13.3485	−0.435611
$940$	0	0
$941$	−2.24745	−0.0732647	−0.0366324	−	0.999329i	$-0.511663\pi$
$941$	−2.24745	−0.0732647	−0.0366324	+	0.999329i	$0.511663\pi$
$942$	0	0
$943$	− 22.6061i	− 0.736157i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.49490i	0.211056i	0.994416	+	0.105528i	$0.0336532\pi$
$947$	6.49490i	0.211056i	−0.994416	+	0.105528i	$0.966347\pi$
$948$	0	0
$949$	−55.3939	−1.79816
$950$	0	0
$951$	15.5959	0.505732
$952$	0	0
$953$	40.6969i	1.31830i	0.752010	+	0.659152i	$0.229084\pi$
$953$	40.6969i	1.31830i	−0.752010	+	0.659152i	$0.770916\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.4949	1.17848
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	− 4.89898i	− 0.157867i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 29.3939i	− 0.945243i	−0.881265	−	0.472622i	$-0.843308\pi$
$967$	− 29.3939i	− 0.945243i	0.881265	−	0.472622i	$-0.156692\pi$
$968$	0	0
$969$	−9.79796	−0.314756
$970$	0	0
$971$	−21.5505	−0.691589	−0.345794	−	0.938310i	$-0.612391\pi$
$971$	−21.5505	−0.691589	−0.345794	+	0.938310i	$0.612391\pi$
$972$	0	0
$973$	33.7980i	1.08351i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.1918i	0.805958i	0.915209	+	0.402979i	$0.132025\pi$
$977$	25.1918i	0.805958i	−0.915209	+	0.402979i	$0.867975\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	6.20204i	0.197814i	0.995097	+	0.0989072i	$0.0315347\pi$
$983$	6.20204i	0.197814i	−0.995097	+	0.0989072i	$0.968465\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	21.7980i	0.693837i
$988$	0	0
$989$	−8.98979	−0.285859
$990$	0	0
$991$	37.3939	1.18786	0.593928	−	0.804518i	$-0.297576\pi$
$991$	37.3939	1.18786	0.593928	+	0.804518i	$0.297576\pi$
$992$	0	0
$993$	1.10102i	0.0349398i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.2929i	0.896044i	0.894023	+	0.448022i	$0.147871\pi$
$997$	28.2929i	0.896044i	−0.894023	+	0.448022i	$0.852129\pi$
$998$	0	0
$999$	−3.55051	−0.112333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.l.3349.1		4
5.2	odd	4		1860.2.a.d.1.2	✓	2
5.3	odd	4		9300.2.a.p.1.1		2
5.4	even	2	inner	9300.2.g.l.3349.4		4
15.2	even	4		5580.2.a.g.1.2		2
20.7	even	4		7440.2.a.bj.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.d.1.2	✓	2	5.2	odd	4
5580.2.a.g.1.2		2	15.2	even	4
7440.2.a.bj.1.1		2	20.7	even	4
9300.2.a.p.1.1		2	5.3	odd	4
9300.2.g.l.3349.1		4	1.1	even	1	trivial
9300.2.g.l.3349.4		4	5.4	even	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.44949i q^{7} -1.00000 q^{9} -4.44949i q^{13} -2.00000i q^{17} +4.89898 q^{19} -2.44949 q^{21} -2.89898i q^{23} +1.00000i q^{27} -0.449490 q^{29} +1.00000 q^{31} +3.55051i q^{37} -4.44949 q^{39} +7.79796 q^{41} -3.10102i q^{43} -8.89898i q^{47} +1.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} -4.89898i q^{57} +7.34847 q^{59} -6.89898 q^{61} +2.44949i q^{63} +6.44949i q^{67} -2.89898 q^{69} +1.55051 q^{71} -12.4495i q^{73} +2.89898 q^{79} +1.00000 q^{81} -1.10102i q^{83} +0.449490i q^{87} +4.44949 q^{89} -10.8990 q^{91} -1.00000i q^{93} +5.10102i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} + 8 q^{29} + 4 q^{31} - 8 q^{39} - 8 q^{41} + 4 q^{49} - 8 q^{51} - 8 q^{61} + 8 q^{69} + 16 q^{71} - 8 q^{79} + 4 q^{81} + 8 q^{89} - 24 q^{91}+O(q^{100}) \) 4 * q - 4 * q^9 + 8 * q^29 + 4 * q^31 - 8 * q^39 - 8 * q^41 + 4 * q^49 - 8 * q^51 - 8 * q^61 + 8 * q^69 + 16 * q^71 - 8 * q^79 + 4 * q^81 + 8 * q^89 - 24 * q^91

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