LMFDB - Embedded newform 9300.2.a.be.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(1,9300)] 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("9300.1"); 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, 0, 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.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,7,0,0,0,-4,0,7,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.2608738798$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 21x^{5} + 31x^{4} + 113x^{3} - 187x^{2} - 161x + 281 $ x^7 - x^6 - 21x^5 + 31x^4 + 113x^3 - 187x^2 - 161*x + 281
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.61743$ of defining polynomial
Character	$\chi$	$=$	9300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.73400 q^{7} +1.00000 q^{9} -3.98164 q^{11} -1.30377 q^{13} -0.895853 q^{17} -5.46229 q^{19} +1.73400 q^{21} +1.06044 q^{23} +1.00000 q^{27} +3.43448 q^{29} +1.00000 q^{31} -3.98164 q^{33} -6.25745 q^{37} -1.30377 q^{39} +4.68395 q^{41} +12.1207 q^{43} -1.61346 q^{47} -3.99326 q^{49} -0.895853 q^{51} +10.5029 q^{53} -5.46229 q^{57} -9.43866 q^{59} -0.198690 q^{61} +1.73400 q^{63} +1.79907 q^{67} +1.06044 q^{69} -7.78632 q^{71} -12.5095 q^{73} -6.90415 q^{77} +4.70850 q^{79} +1.00000 q^{81} +1.54706 q^{83} +3.43448 q^{87} -7.00766 q^{89} -2.26073 q^{91} +1.00000 q^{93} -5.55763 q^{97} -3.98164 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} - 4 q^{7} + 7 q^{9} + 4 q^{11} - 10 q^{13} - 14 q^{17} - 6 q^{19} - 4 q^{21} - 8 q^{23} + 7 q^{27} - 2 q^{29} + 7 q^{31} + 4 q^{33} - 12 q^{37} - 10 q^{39} + 10 q^{41} - 8 q^{43} - 16 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 - 4 * q^7 + 7 * q^9 + 4 * q^11 - 10 * q^13 - 14 * q^17 - 6 * q^19 - 4 * q^21 - 8 * q^23 + 7 * q^27 - 2 * q^29 + 7 * q^31 + 4 * q^33 - 12 * q^37 - 10 * q^39 + 10 * q^41 - 8 * q^43 - 16 * q^47 + q^49 - 14 * q^51 - 12 * q^53 - 6 * q^57 - 2 * q^59 + 2 * q^61 - 4 * q^63 - 18 * q^67 - 8 * q^69 + 6 * q^71 - 22 * q^73 - 22 * q^77 - 24 * q^79 + 7 * q^81 - 12 * q^83 - 2 * q^87 - 12 * q^89 - 28 * q^91 + 7 * q^93 - 36 * q^97 + 4 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.73400	0.655389	0.327694	−	0.944784i	$-0.393728\pi$
$7$	1.73400	0.655389	0.327694	+	0.944784i	$0.393728\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.98164	−1.20051	−0.600255	−	0.799808i	$-0.704934\pi$
$11$	−3.98164	−1.20051	−0.600255	+	0.799808i	$0.704934\pi$
$12$	0	0
$13$	−1.30377	−0.361601	−0.180801	−	0.983520i	$-0.557869\pi$
$13$	−1.30377	−0.361601	−0.180801	+	0.983520i	$0.557869\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.895853	−0.217276	−0.108638	−	0.994081i	$-0.534649\pi$
$17$	−0.895853	−0.217276	−0.108638	+	0.994081i	$0.534649\pi$
$18$	0	0
$19$	−5.46229	−1.25314	−0.626568	−	0.779367i	$-0.715541\pi$
$19$	−5.46229	−1.25314	−0.626568	+	0.779367i	$0.715541\pi$
$20$	0	0
$21$	1.73400	0.378389
$22$	0	0
$23$	1.06044	0.221118	0.110559	−	0.993870i	$-0.464736\pi$
$23$	1.06044	0.221118	0.110559	+	0.993870i	$0.464736\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.43448	0.637767	0.318884	−	0.947794i	$-0.396692\pi$
$29$	3.43448	0.637767	0.318884	+	0.947794i	$0.396692\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−3.98164	−0.693115
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.25745	−1.02872	−0.514359	−	0.857575i	$-0.671970\pi$
$37$	−6.25745	−1.02872	−0.514359	+	0.857575i	$0.671970\pi$
$38$	0	0
$39$	−1.30377	−0.208771
$40$	0	0
$41$	4.68395	0.731509	0.365755	−	0.930711i	$-0.380811\pi$
$41$	4.68395	0.731509	0.365755	+	0.930711i	$0.380811\pi$
$42$	0	0
$43$	12.1207	1.84838	0.924191	−	0.381930i	$-0.124741\pi$
$43$	12.1207	1.84838	0.924191	+	0.381930i	$0.124741\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.61346	−0.235347	−0.117674	−	0.993052i	$-0.537544\pi$
$47$	−1.61346	−0.235347	−0.117674	+	0.993052i	$0.537544\pi$
$48$	0	0
$49$	−3.99326	−0.570466
$50$	0	0
$51$	−0.895853	−0.125444
$52$	0	0
$53$	10.5029	1.44268	0.721339	−	0.692582i	$-0.243527\pi$
$53$	10.5029	1.44268	0.721339	+	0.692582i	$0.243527\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.46229	−0.723498
$58$	0	0
$59$	−9.43866	−1.22881	−0.614404	−	0.788991i	$-0.710604\pi$
$59$	−9.43866	−1.22881	−0.614404	+	0.788991i	$0.710604\pi$
$60$	0	0
$61$	−0.198690	−0.0254397	−0.0127198	−	0.999919i	$-0.504049\pi$
$61$	−0.198690	−0.0254397	−0.0127198	+	0.999919i	$0.504049\pi$
$62$	0	0
$63$	1.73400	0.218463
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.79907	0.219791	0.109896	−	0.993943i	$-0.464948\pi$
$67$	1.79907	0.219791	0.109896	+	0.993943i	$0.464948\pi$
$68$	0	0
$69$	1.06044	0.127662
$70$	0	0
$71$	−7.78632	−0.924066	−0.462033	−	0.886863i	$-0.652880\pi$
$71$	−7.78632	−0.924066	−0.462033	+	0.886863i	$0.652880\pi$
$72$	0	0
$73$	−12.5095	−1.46413	−0.732065	−	0.681235i	$-0.761443\pi$
$73$	−12.5095	−1.46413	−0.732065	+	0.681235i	$0.761443\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.90415	−0.786801
$78$	0	0
$79$	4.70850	0.529747	0.264874	−	0.964283i	$-0.414670\pi$
$79$	4.70850	0.529747	0.264874	+	0.964283i	$0.414670\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.54706	0.169812	0.0849058	−	0.996389i	$-0.472941\pi$
$83$	1.54706	0.169812	0.0849058	+	0.996389i	$0.472941\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.43448	0.368215
$88$	0	0
$89$	−7.00766	−0.742810	−0.371405	−	0.928471i	$-0.621124\pi$
$89$	−7.00766	−0.742810	−0.371405	+	0.928471i	$0.621124\pi$
$90$	0	0
$91$	−2.26073	−0.236989
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.55763	−0.564292	−0.282146	−	0.959372i	$-0.591046\pi$
$97$	−5.55763	−0.564292	−0.282146	+	0.959372i	$0.591046\pi$
$98$	0	0
$99$	−3.98164	−0.400170
$100$	0	0
$101$	−0.894108	−0.0889671	−0.0444836	−	0.999010i	$-0.514164\pi$
$101$	−0.894108	−0.0889671	−0.0444836	+	0.999010i	$0.514164\pi$
$102$	0	0
$103$	5.75934	0.567485	0.283742	−	0.958901i	$-0.408424\pi$
$103$	5.75934	0.567485	0.283742	+	0.958901i	$0.408424\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.7961	−1.43039	−0.715195	−	0.698925i	$-0.753662\pi$
$107$	−14.7961	−1.43039	−0.715195	+	0.698925i	$0.753662\pi$
$108$	0	0
$109$	−13.6183	−1.30440	−0.652201	−	0.758046i	$-0.726154\pi$
$109$	−13.6183	−1.30440	−0.652201	+	0.758046i	$0.726154\pi$
$110$	0	0
$111$	−6.25745	−0.593931
$112$	0	0
$113$	−4.20432	−0.395509	−0.197754	−	0.980252i	$-0.563365\pi$
$113$	−4.20432	−0.395509	−0.197754	+	0.980252i	$0.563365\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.30377	−0.120534
$118$	0	0
$119$	−1.55340	−0.142400
$120$	0	0
$121$	4.85349	0.441226
$122$	0	0
$123$	4.68395	0.422337
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−17.2549	−1.53112	−0.765562	−	0.643362i	$-0.777539\pi$
$127$	−17.2549	−1.53112	−0.765562	+	0.643362i	$0.777539\pi$
$128$	0	0
$129$	12.1207	1.06716
$130$	0	0
$131$	7.01254	0.612688	0.306344	−	0.951921i	$-0.400894\pi$
$131$	7.01254	0.612688	0.306344	+	0.951921i	$0.400894\pi$
$132$	0	0
$133$	−9.47158	−0.821290
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.2396	−1.21657	−0.608284	−	0.793719i	$-0.708142\pi$
$137$	−14.2396	−1.21657	−0.608284	+	0.793719i	$0.708142\pi$
$138$	0	0
$139$	−14.9797	−1.27056	−0.635280	−	0.772282i	$-0.719115\pi$
$139$	−14.9797	−1.27056	−0.635280	+	0.772282i	$0.719115\pi$
$140$	0	0
$141$	−1.61346	−0.135878
$142$	0	0
$143$	5.19115	0.434106
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.99326	−0.329359
$148$	0	0
$149$	6.18919	0.507039	0.253519	−	0.967330i	$-0.418412\pi$
$149$	6.18919	0.507039	0.253519	+	0.967330i	$0.418412\pi$
$150$	0	0
$151$	−11.5380	−0.938946	−0.469473	−	0.882947i	$-0.655556\pi$
$151$	−11.5380	−0.938946	−0.469473	+	0.882947i	$0.655556\pi$
$152$	0	0
$153$	−0.895853	−0.0724254
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.5823	−0.844561	−0.422281	−	0.906465i	$-0.638770\pi$
$157$	−10.5823	−0.844561	−0.422281	+	0.906465i	$0.638770\pi$
$158$	0	0
$159$	10.5029	0.832930
$160$	0	0
$161$	1.83880	0.144918
$162$	0	0
$163$	−10.7180	−0.839500	−0.419750	−	0.907640i	$-0.637882\pi$
$163$	−10.7180	−0.839500	−0.419750	+	0.907640i	$0.637882\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.55536	−0.197740	−0.0988700	−	0.995100i	$-0.531523\pi$
$167$	−2.55536	−0.197740	−0.0988700	+	0.995100i	$0.531523\pi$
$168$	0	0
$169$	−11.3002	−0.869245
$170$	0	0
$171$	−5.46229	−0.417712
$172$	0	0
$173$	2.90517	0.220876	0.110438	−	0.993883i	$-0.464775\pi$
$173$	2.90517	0.220876	0.110438	+	0.993883i	$0.464775\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.43866	−0.709453
$178$	0	0
$179$	19.0291	1.42231	0.711153	−	0.703038i	$-0.248174\pi$
$179$	19.0291	1.42231	0.711153	+	0.703038i	$0.248174\pi$
$180$	0	0
$181$	−15.9144	−1.18290	−0.591452	−	0.806340i	$-0.701445\pi$
$181$	−15.9144	−1.18290	−0.591452	+	0.806340i	$0.701445\pi$
$182$	0	0
$183$	−0.198690	−0.0146876
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.56697	0.260842
$188$	0	0
$189$	1.73400	0.126130
$190$	0	0
$191$	13.2916	0.961747	0.480874	−	0.876790i	$-0.340320\pi$
$191$	13.2916	0.961747	0.480874	+	0.876790i	$0.340320\pi$
$192$	0	0
$193$	−19.9489	−1.43595	−0.717975	−	0.696068i	$-0.754931\pi$
$193$	−19.9489	−1.43595	−0.717975	+	0.696068i	$0.754931\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.81774	0.272003	0.136001	−	0.990709i	$-0.456575\pi$
$197$	3.81774	0.272003	0.136001	+	0.990709i	$0.456575\pi$
$198$	0	0
$199$	12.1645	0.862321	0.431160	−	0.902275i	$-0.358104\pi$
$199$	12.1645	0.862321	0.431160	+	0.902275i	$0.358104\pi$
$200$	0	0
$201$	1.79907	0.126897
$202$	0	0
$203$	5.95537	0.417985
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.06044	0.0737059
$208$	0	0
$209$	21.7489	1.50440
$210$	0	0
$211$	4.44272	0.305849	0.152925	−	0.988238i	$-0.451131\pi$
$211$	4.44272	0.305849	0.152925	+	0.988238i	$0.451131\pi$
$212$	0	0
$213$	−7.78632	−0.533510
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.73400	0.117711
$218$	0	0
$219$	−12.5095	−0.845316
$220$	0	0
$221$	1.16799	0.0785673
$222$	0	0
$223$	14.4309	0.966364	0.483182	−	0.875520i	$-0.339481\pi$
$223$	14.4309	0.966364	0.483182	+	0.875520i	$0.339481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.7550	−1.70942	−0.854708	−	0.519108i	$-0.826264\pi$
$227$	−25.7550	−1.70942	−0.854708	+	0.519108i	$0.826264\pi$
$228$	0	0
$229$	−19.6828	−1.30068	−0.650339	−	0.759644i	$-0.725373\pi$
$229$	−19.6828	−1.30068	−0.650339	+	0.759644i	$0.725373\pi$
$230$	0	0
$231$	−6.90415	−0.454260
$232$	0	0
$233$	−16.1012	−1.05482	−0.527411	−	0.849610i	$-0.676837\pi$
$233$	−16.1012	−1.05482	−0.527411	+	0.849610i	$0.676837\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.70850	0.305850
$238$	0	0
$239$	−25.6408	−1.65857	−0.829283	−	0.558828i	$-0.811251\pi$
$239$	−25.6408	−1.65857	−0.829283	+	0.558828i	$0.811251\pi$
$240$	0	0
$241$	14.2483	0.917815	0.458908	−	0.888484i	$-0.348241\pi$
$241$	14.2483	0.917815	0.458908	+	0.888484i	$0.348241\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.12158	0.453135
$248$	0	0
$249$	1.54706	0.0980408
$250$	0	0
$251$	2.24657	0.141802	0.0709011	−	0.997483i	$-0.477413\pi$
$251$	2.24657	0.141802	0.0709011	+	0.997483i	$0.477413\pi$
$252$	0	0
$253$	−4.22231	−0.265454
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.5388	0.906907	0.453453	−	0.891280i	$-0.350192\pi$
$257$	14.5388	0.906907	0.453453	+	0.891280i	$0.350192\pi$
$258$	0	0
$259$	−10.8504	−0.674210
$260$	0	0
$261$	3.43448	0.212589
$262$	0	0
$263$	−2.24388	−0.138363	−0.0691817	−	0.997604i	$-0.522039\pi$
$263$	−2.24388	−0.138363	−0.0691817	+	0.997604i	$0.522039\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−7.00766	−0.428862
$268$	0	0
$269$	10.2405	0.624373	0.312187	−	0.950021i	$-0.398939\pi$
$269$	10.2405	0.624373	0.312187	+	0.950021i	$0.398939\pi$
$270$	0	0
$271$	−9.66554	−0.587140	−0.293570	−	0.955938i	$-0.594843\pi$
$271$	−9.66554	−0.587140	−0.293570	+	0.955938i	$0.594843\pi$
$272$	0	0
$273$	−2.26073	−0.136826
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.2885	1.03876	0.519382	−	0.854542i	$-0.326162\pi$
$277$	17.2885	1.03876	0.519382	+	0.854542i	$0.326162\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−4.78330	−0.285348	−0.142674	−	0.989770i	$-0.545570\pi$
$281$	−4.78330	−0.285348	−0.142674	+	0.989770i	$0.545570\pi$
$282$	0	0
$283$	23.8427	1.41730	0.708651	−	0.705560i	$-0.249304\pi$
$283$	23.8427	1.41730	0.708651	+	0.705560i	$0.249304\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.12194	0.479423
$288$	0	0
$289$	−16.1974	−0.952791
$290$	0	0
$291$	−5.55763	−0.325794
$292$	0	0
$293$	7.75172	0.452860	0.226430	−	0.974027i	$-0.427294\pi$
$293$	7.75172	0.452860	0.226430	+	0.974027i	$0.427294\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.98164	−0.231038
$298$	0	0
$299$	−1.38258	−0.0799564
$300$	0	0
$301$	21.0172	1.21141
$302$	0	0
$303$	−0.894108	−0.0513652
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−30.3970	−1.73485	−0.867425	−	0.497567i	$-0.834227\pi$
$307$	−30.3970	−1.73485	−0.867425	+	0.497567i	$0.834227\pi$
$308$	0	0
$309$	5.75934	0.327638
$310$	0	0
$311$	17.9492	1.01781	0.508903	−	0.860824i	$-0.330051\pi$
$311$	17.9492	1.01781	0.508903	+	0.860824i	$0.330051\pi$
$312$	0	0
$313$	−20.3920	−1.15262	−0.576312	−	0.817230i	$-0.695509\pi$
$313$	−20.3920	−1.15262	−0.576312	+	0.817230i	$0.695509\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.98326	0.504550	0.252275	−	0.967656i	$-0.418821\pi$
$317$	8.98326	0.504550	0.252275	+	0.967656i	$0.418821\pi$
$318$	0	0
$319$	−13.6749	−0.765646
$320$	0	0
$321$	−14.7961	−0.825836
$322$	0	0
$323$	4.89341	0.272276
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−13.6183	−0.753096
$328$	0	0
$329$	−2.79773	−0.154244
$330$	0	0
$331$	19.5465	1.07437	0.537187	−	0.843463i	$-0.319487\pi$
$331$	19.5465	1.07437	0.537187	+	0.843463i	$0.319487\pi$
$332$	0	0
$333$	−6.25745	−0.342906
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.91986	0.159055	0.0795276	−	0.996833i	$-0.474659\pi$
$337$	2.91986	0.159055	0.0795276	+	0.996833i	$0.474659\pi$
$338$	0	0
$339$	−4.20432	−0.228347
$340$	0	0
$341$	−3.98164	−0.215618
$342$	0	0
$343$	−19.0623	−1.02927
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.412028	−0.0221188	−0.0110594	−	0.999939i	$-0.503520\pi$
$347$	−0.412028	−0.0221188	−0.0110594	+	0.999939i	$0.503520\pi$
$348$	0	0
$349$	13.2711	0.710385	0.355192	−	0.934793i	$-0.384415\pi$
$349$	13.2711	0.710385	0.355192	+	0.934793i	$0.384415\pi$
$350$	0	0
$351$	−1.30377	−0.0695902
$352$	0	0
$353$	−20.9774	−1.11651	−0.558257	−	0.829668i	$-0.688530\pi$
$353$	−20.9774	−1.11651	−0.558257	+	0.829668i	$0.688530\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.55340	−0.0822149
$358$	0	0
$359$	14.4040	0.760213	0.380106	−	0.924943i	$-0.375887\pi$
$359$	14.4040	0.760213	0.380106	+	0.924943i	$0.375887\pi$
$360$	0	0
$361$	10.8366	0.570348
$362$	0	0
$363$	4.85349	0.254742
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−27.7612	−1.44912	−0.724562	−	0.689210i	$-0.757958\pi$
$367$	−27.7612	−1.44912	−0.724562	+	0.689210i	$0.757958\pi$
$368$	0	0
$369$	4.68395	0.243836
$370$	0	0
$371$	18.2119	0.945514
$372$	0	0
$373$	−8.98754	−0.465357	−0.232679	−	0.972554i	$-0.574749\pi$
$373$	−8.98754	−0.465357	−0.232679	+	0.972554i	$0.574749\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.47778	−0.230617
$378$	0	0
$379$	11.4182	0.586515	0.293257	−	0.956034i	$-0.405261\pi$
$379$	11.4182	0.586515	0.293257	+	0.956034i	$0.405261\pi$
$380$	0	0
$381$	−17.2549	−0.883995
$382$	0	0
$383$	−5.62537	−0.287443	−0.143722	−	0.989618i	$-0.545907\pi$
$383$	−5.62537	−0.287443	−0.143722	+	0.989618i	$0.545907\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.1207	0.616127
$388$	0	0
$389$	30.4230	1.54251	0.771253	−	0.636528i	$-0.219630\pi$
$389$	30.4230	1.54251	0.771253	+	0.636528i	$0.219630\pi$
$390$	0	0
$391$	−0.950001	−0.0480436
$392$	0	0
$393$	7.01254	0.353736
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.11796	0.256863	0.128432	−	0.991718i	$-0.459006\pi$
$397$	5.11796	0.256863	0.128432	+	0.991718i	$0.459006\pi$
$398$	0	0
$399$	−9.47158	−0.474172
$400$	0	0
$401$	7.27462	0.363277	0.181639	−	0.983365i	$-0.441860\pi$
$401$	7.27462	0.363277	0.181639	+	0.983365i	$0.441860\pi$
$402$	0	0
$403$	−1.30377	−0.0649455
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.9149	1.23499
$408$	0	0
$409$	23.3718	1.15566	0.577831	−	0.816156i	$-0.303899\pi$
$409$	23.3718	1.15566	0.577831	+	0.816156i	$0.303899\pi$
$410$	0	0
$411$	−14.2396	−0.702386
$412$	0	0
$413$	−16.3666	−0.805347
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−14.9797	−0.733558
$418$	0	0
$419$	12.4740	0.609397	0.304698	−	0.952449i	$-0.401444\pi$
$419$	12.4740	0.609397	0.304698	+	0.952449i	$0.401444\pi$
$420$	0	0
$421$	22.8399	1.11315	0.556575	−	0.830797i	$-0.312115\pi$
$421$	22.8399	1.11315	0.556575	+	0.830797i	$0.312115\pi$
$422$	0	0
$423$	−1.61346	−0.0784491
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.344528	−0.0166729
$428$	0	0
$429$	5.19115	0.250631
$430$	0	0
$431$	20.8737	1.00545	0.502726	−	0.864446i	$-0.332330\pi$
$431$	20.8737	1.00545	0.502726	+	0.864446i	$0.332330\pi$
$432$	0	0
$433$	−7.46460	−0.358726	−0.179363	−	0.983783i	$-0.557404\pi$
$433$	−7.46460	−0.358726	−0.179363	+	0.983783i	$0.557404\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.79245	−0.277090
$438$	0	0
$439$	4.14459	0.197810	0.0989051	−	0.995097i	$-0.468466\pi$
$439$	4.14459	0.197810	0.0989051	+	0.995097i	$0.468466\pi$
$440$	0	0
$441$	−3.99326	−0.190155
$442$	0	0
$443$	36.6456	1.74108	0.870542	−	0.492094i	$-0.163768\pi$
$443$	36.6456	1.74108	0.870542	+	0.492094i	$0.163768\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.18919	0.292739
$448$	0	0
$449$	3.71731	0.175431	0.0877154	−	0.996146i	$-0.472043\pi$
$449$	3.71731	0.175431	0.0877154	+	0.996146i	$0.472043\pi$
$450$	0	0
$451$	−18.6498	−0.878185
$452$	0	0
$453$	−11.5380	−0.542101
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.1919	1.22521	0.612603	−	0.790391i	$-0.290123\pi$
$457$	26.1919	1.22521	0.612603	+	0.790391i	$0.290123\pi$
$458$	0	0
$459$	−0.895853	−0.0418148
$460$	0	0
$461$	−39.3759	−1.83392	−0.916958	−	0.398983i	$-0.869363\pi$
$461$	−39.3759	−1.83392	−0.916958	+	0.398983i	$0.869363\pi$
$462$	0	0
$463$	−10.4890	−0.487464	−0.243732	−	0.969843i	$-0.578372\pi$
$463$	−10.4890	−0.487464	−0.243732	+	0.969843i	$0.578372\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.02916	−0.0938983	−0.0469491	−	0.998897i	$-0.514950\pi$
$467$	−2.02916	−0.0938983	−0.0469491	+	0.998897i	$0.514950\pi$
$468$	0	0
$469$	3.11958	0.144049
$470$	0	0
$471$	−10.5823	−0.487608
$472$	0	0
$473$	−48.2601	−2.21900
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.5029	0.480893
$478$	0	0
$479$	−15.4839	−0.707475	−0.353738	−	0.935345i	$-0.615089\pi$
$479$	−15.4839	−0.707475	−0.353738	+	0.935345i	$0.615089\pi$
$480$	0	0
$481$	8.15829	0.371986
$482$	0	0
$483$	1.83880	0.0836685
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−8.74808	−0.396413	−0.198207	−	0.980160i	$-0.563512\pi$
$487$	−8.74808	−0.396413	−0.198207	+	0.980160i	$0.563512\pi$
$488$	0	0
$489$	−10.7180	−0.484686
$490$	0	0
$491$	17.9939	0.812053	0.406027	−	0.913861i	$-0.366914\pi$
$491$	17.9939	0.812053	0.406027	+	0.913861i	$0.366914\pi$
$492$	0	0
$493$	−3.07679	−0.138572
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.5014	−0.605622
$498$	0	0
$499$	16.9146	0.757201	0.378601	−	0.925560i	$-0.376405\pi$
$499$	16.9146	0.757201	0.378601	+	0.925560i	$0.376405\pi$
$500$	0	0
$501$	−2.55536	−0.114165
$502$	0	0
$503$	−32.4466	−1.44672	−0.723360	−	0.690471i	$-0.757404\pi$
$503$	−32.4466	−1.44672	−0.723360	+	0.690471i	$0.757404\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−11.3002	−0.501859
$508$	0	0
$509$	9.73565	0.431525	0.215763	−	0.976446i	$-0.430776\pi$
$509$	9.73565	0.431525	0.215763	+	0.976446i	$0.430776\pi$
$510$	0	0
$511$	−21.6915	−0.959574
$512$	0	0
$513$	−5.46229	−0.241166
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.42422	0.282537
$518$	0	0
$519$	2.90517	0.127523
$520$	0	0
$521$	−11.2149	−0.491333	−0.245667	−	0.969354i	$-0.579007\pi$
$521$	−11.2149	−0.491333	−0.245667	+	0.969354i	$0.579007\pi$
$522$	0	0
$523$	6.52014	0.285106	0.142553	−	0.989787i	$-0.454469\pi$
$523$	6.52014	0.285106	0.142553	+	0.989787i	$0.454469\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.895853	−0.0390240
$528$	0	0
$529$	−21.8755	−0.951107
$530$	0	0
$531$	−9.43866	−0.409603
$532$	0	0
$533$	−6.10679	−0.264515
$534$	0	0
$535$	0	0
$536$	0	0
$537$	19.0291	0.821168
$538$	0	0
$539$	15.8997	0.684850
$540$	0	0
$541$	17.3045	0.743979	0.371989	−	0.928237i	$-0.378676\pi$
$541$	17.3045	0.743979	0.371989	+	0.928237i	$0.378676\pi$
$542$	0	0
$543$	−15.9144	−0.682950
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.0891	−0.516893	−0.258446	−	0.966026i	$-0.583211\pi$
$547$	−12.0891	−0.516893	−0.258446	+	0.966026i	$0.583211\pi$
$548$	0	0
$549$	−0.198690	−0.00847989
$550$	0	0
$551$	−18.7601	−0.799209
$552$	0	0
$553$	8.16451	0.347190
$554$	0	0
$555$	0	0
$556$	0	0
$557$	35.8051	1.51711	0.758556	−	0.651608i	$-0.225905\pi$
$557$	35.8051	1.51711	0.758556	+	0.651608i	$0.225905\pi$
$558$	0	0
$559$	−15.8026	−0.668377
$560$	0	0
$561$	3.56697	0.150597
$562$	0	0
$563$	−25.8203	−1.08820	−0.544099	−	0.839021i	$-0.683128\pi$
$563$	−25.8203	−1.08820	−0.544099	+	0.839021i	$0.683128\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.73400	0.0728209
$568$	0	0
$569$	14.1633	0.593758	0.296879	−	0.954915i	$-0.404054\pi$
$569$	14.1633	0.593758	0.296879	+	0.954915i	$0.404054\pi$
$570$	0	0
$571$	−9.63733	−0.403310	−0.201655	−	0.979457i	$-0.564632\pi$
$571$	−9.63733	−0.403310	−0.201655	+	0.979457i	$0.564632\pi$
$572$	0	0
$573$	13.2916	0.555265
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−37.1419	−1.54624	−0.773120	−	0.634260i	$-0.781305\pi$
$577$	−37.1419	−1.54624	−0.773120	+	0.634260i	$0.781305\pi$
$578$	0	0
$579$	−19.9489	−0.829047
$580$	0	0
$581$	2.68259	0.111293
$582$	0	0
$583$	−41.8186	−1.73195
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−43.7218	−1.80459	−0.902295	−	0.431119i	$-0.858119\pi$
$587$	−43.7218	−1.80459	−0.902295	+	0.431119i	$0.858119\pi$
$588$	0	0
$589$	−5.46229	−0.225070
$590$	0	0
$591$	3.81774	0.157041
$592$	0	0
$593$	37.9873	1.55995	0.779975	−	0.625811i	$-0.215232\pi$
$593$	37.9873	1.55995	0.779975	+	0.625811i	$0.215232\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.1645	0.497861
$598$	0	0
$599$	−4.09227	−0.167205	−0.0836027	−	0.996499i	$-0.526643\pi$
$599$	−4.09227	−0.167205	−0.0836027	+	0.996499i	$0.526643\pi$
$600$	0	0
$601$	−24.8217	−1.01250	−0.506249	−	0.862387i	$-0.668968\pi$
$601$	−24.8217	−1.01250	−0.506249	+	0.862387i	$0.668968\pi$
$602$	0	0
$603$	1.79907	0.0732638
$604$	0	0
$605$	0	0
$606$	0	0
$607$	40.8891	1.65964	0.829818	−	0.558034i	$-0.188444\pi$
$607$	40.8891	1.65964	0.829818	+	0.558034i	$0.188444\pi$
$608$	0	0
$609$	5.95537	0.241324
$610$	0	0
$611$	2.10358	0.0851018
$612$	0	0
$613$	−18.6052	−0.751457	−0.375728	−	0.926730i	$-0.622607\pi$
$613$	−18.6052	−0.751457	−0.375728	+	0.926730i	$0.622607\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−45.4410	−1.82939	−0.914693	−	0.404149i	$-0.867568\pi$
$617$	−45.4410	−1.82939	−0.914693	+	0.404149i	$0.867568\pi$
$618$	0	0
$619$	0.478372	0.0192274	0.00961369	−	0.999954i	$-0.496940\pi$
$619$	0.478372	0.0192274	0.00961369	+	0.999954i	$0.496940\pi$
$620$	0	0
$621$	1.06044	0.0425541
$622$	0	0
$623$	−12.1512	−0.486829
$624$	0	0
$625$	0	0
$626$	0	0
$627$	21.7489	0.868567
$628$	0	0
$629$	5.60575	0.223516
$630$	0	0
$631$	−24.6954	−0.983109	−0.491555	−	0.870847i	$-0.663571\pi$
$631$	−24.6954	−0.983109	−0.491555	+	0.870847i	$0.663571\pi$
$632$	0	0
$633$	4.44272	0.176582
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.20630	0.206281
$638$	0	0
$639$	−7.78632	−0.308022
$640$	0	0
$641$	19.9875	0.789460	0.394730	−	0.918797i	$-0.370838\pi$
$641$	19.9875	0.789460	0.394730	+	0.918797i	$0.370838\pi$
$642$	0	0
$643$	0.105515	0.00416109	0.00208054	−	0.999998i	$-0.499338\pi$
$643$	0.105515	0.00416109	0.00208054	+	0.999998i	$0.499338\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.6968	−1.01025	−0.505123	−	0.863048i	$-0.668553\pi$
$647$	−25.6968	−1.01025	−0.505123	+	0.863048i	$0.668553\pi$
$648$	0	0
$649$	37.5814	1.47520
$650$	0	0
$651$	1.73400	0.0679606
$652$	0	0
$653$	8.65871	0.338841	0.169421	−	0.985544i	$-0.445810\pi$
$653$	8.65871	0.338841	0.169421	+	0.985544i	$0.445810\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−12.5095	−0.488043
$658$	0	0
$659$	−4.56827	−0.177954	−0.0889772	−	0.996034i	$-0.528360\pi$
$659$	−4.56827	−0.177954	−0.0889772	+	0.996034i	$0.528360\pi$
$660$	0	0
$661$	−30.9424	−1.20352	−0.601759	−	0.798678i	$-0.705533\pi$
$661$	−30.9424	−1.20352	−0.601759	+	0.798678i	$0.705533\pi$
$662$	0	0
$663$	1.16799	0.0453609
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.64207	0.141022
$668$	0	0
$669$	14.4309	0.557930
$670$	0	0
$671$	0.791113	0.0305406
$672$	0	0
$673$	22.2863	0.859075	0.429537	−	0.903049i	$-0.358677\pi$
$673$	22.2863	0.859075	0.429537	+	0.903049i	$0.358677\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26.3885	−1.01419	−0.507097	−	0.861889i	$-0.669281\pi$
$677$	−26.3885	−1.01419	−0.507097	+	0.861889i	$0.669281\pi$
$678$	0	0
$679$	−9.63690	−0.369830
$680$	0	0
$681$	−25.7550	−0.986932
$682$	0	0
$683$	24.2104	0.926385	0.463192	−	0.886258i	$-0.346704\pi$
$683$	24.2104	0.926385	0.463192	+	0.886258i	$0.346704\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−19.6828	−0.750946
$688$	0	0
$689$	−13.6933	−0.521674
$690$	0	0
$691$	19.2885	0.733768	0.366884	−	0.930267i	$-0.380425\pi$
$691$	19.2885	0.733768	0.366884	+	0.930267i	$0.380425\pi$
$692$	0	0
$693$	−6.90415	−0.262267
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.19612	−0.158940
$698$	0	0
$699$	−16.1012	−0.609001
$700$	0	0
$701$	37.2625	1.40739	0.703693	−	0.710504i	$-0.251533\pi$
$701$	37.2625	1.40739	0.703693	+	0.710504i	$0.251533\pi$
$702$	0	0
$703$	34.1800	1.28912
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.55038	−0.0583080
$708$	0	0
$709$	8.82346	0.331372	0.165686	−	0.986179i	$-0.447016\pi$
$709$	8.82346	0.331372	0.165686	+	0.986179i	$0.447016\pi$
$710$	0	0
$711$	4.70850	0.176582
$712$	0	0
$713$	1.06044	0.0397139
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−25.6408	−0.957574
$718$	0	0
$719$	17.4383	0.650338	0.325169	−	0.945656i	$-0.394579\pi$
$719$	17.4383	0.650338	0.325169	+	0.945656i	$0.394579\pi$
$720$	0	0
$721$	9.98667	0.371923
$722$	0	0
$723$	14.2483	0.529901
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−16.4239	−0.609129	−0.304565	−	0.952492i	$-0.598511\pi$
$727$	−16.4239	−0.609129	−0.304565	+	0.952492i	$0.598511\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.8583	−0.401609
$732$	0	0
$733$	36.8924	1.36265	0.681326	−	0.731980i	$-0.261403\pi$
$733$	36.8924	1.36265	0.681326	+	0.731980i	$0.261403\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.16325	−0.263862
$738$	0	0
$739$	37.0574	1.36318	0.681589	−	0.731735i	$-0.261289\pi$
$739$	37.0574	1.36318	0.681589	+	0.731735i	$0.261289\pi$
$740$	0	0
$741$	7.12158	0.261618
$742$	0	0
$743$	23.8375	0.874514	0.437257	−	0.899337i	$-0.355950\pi$
$743$	23.8375	0.874514	0.437257	+	0.899337i	$0.355950\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.54706	0.0566039
$748$	0	0
$749$	−25.6563	−0.937461
$750$	0	0
$751$	10.4071	0.379760	0.189880	−	0.981807i	$-0.439190\pi$
$751$	10.4071	0.379760	0.189880	+	0.981807i	$0.439190\pi$
$752$	0	0
$753$	2.24657	0.0818695
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−38.4950	−1.39913	−0.699563	−	0.714571i	$-0.746622\pi$
$757$	−38.4950	−1.39913	−0.699563	+	0.714571i	$0.746622\pi$
$758$	0	0
$759$	−4.22231	−0.153260
$760$	0	0
$761$	−8.81181	−0.319428	−0.159714	−	0.987163i	$-0.551057\pi$
$761$	−8.81181	−0.319428	−0.159714	+	0.987163i	$0.551057\pi$
$762$	0	0
$763$	−23.6141	−0.854890
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.3059	0.444339
$768$	0	0
$769$	−42.8973	−1.54692	−0.773458	−	0.633848i	$-0.781475\pi$
$769$	−42.8973	−1.54692	−0.773458	+	0.633848i	$0.781475\pi$
$770$	0	0
$771$	14.5388	0.523603
$772$	0	0
$773$	−24.9276	−0.896582	−0.448291	−	0.893888i	$-0.647967\pi$
$773$	−24.9276	−0.896582	−0.448291	+	0.893888i	$0.647967\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−10.8504	−0.389256
$778$	0	0
$779$	−25.5851	−0.916680
$780$	0	0
$781$	31.0024	1.10935
$782$	0	0
$783$	3.43448	0.122738
$784$	0	0
$785$	0	0
$786$	0	0
$787$	17.6213	0.628131	0.314066	−	0.949401i	$-0.398309\pi$
$787$	17.6213	0.628131	0.314066	+	0.949401i	$0.398309\pi$
$788$	0	0
$789$	−2.24388	−0.0798841
$790$	0	0
$791$	−7.29027	−0.259212
$792$	0	0
$793$	0.259046	0.00919901
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.57890	−0.162193	−0.0810964	−	0.996706i	$-0.525842\pi$
$797$	−4.57890	−0.162193	−0.0810964	+	0.996706i	$0.525842\pi$
$798$	0	0
$799$	1.44542	0.0511354
$800$	0	0
$801$	−7.00766	−0.247603
$802$	0	0
$803$	49.8085	1.75770
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.2405	0.360482
$808$	0	0
$809$	32.1427	1.13008	0.565038	−	0.825065i	$-0.308861\pi$
$809$	32.1427	1.13008	0.565038	+	0.825065i	$0.308861\pi$
$810$	0	0
$811$	−25.3997	−0.891902	−0.445951	−	0.895057i	$-0.647135\pi$
$811$	−25.3997	−0.891902	−0.445951	+	0.895057i	$0.647135\pi$
$812$	0	0
$813$	−9.66554	−0.338985
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−66.2065	−2.31627
$818$	0	0
$819$	−2.26073	−0.0789964
$820$	0	0
$821$	34.4795	1.20334	0.601671	−	0.798744i	$-0.294502\pi$
$821$	34.4795	1.20334	0.601671	+	0.798744i	$0.294502\pi$
$822$	0	0
$823$	4.43720	0.154671	0.0773356	−	0.997005i	$-0.475359\pi$
$823$	4.43720	0.154671	0.0773356	+	0.997005i	$0.475359\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−53.5566	−1.86234	−0.931172	−	0.364580i	$-0.881213\pi$
$827$	−53.5566	−1.86234	−0.931172	+	0.364580i	$0.881213\pi$
$828$	0	0
$829$	11.7932	0.409595	0.204797	−	0.978804i	$-0.434346\pi$
$829$	11.7932	0.409595	0.204797	+	0.978804i	$0.434346\pi$
$830$	0	0
$831$	17.2885	0.599730
$832$	0	0
$833$	3.57737	0.123949
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	35.0907	1.21146	0.605732	−	0.795669i	$-0.292880\pi$
$839$	35.0907	1.21146	0.605732	+	0.795669i	$0.292880\pi$
$840$	0	0
$841$	−17.2043	−0.593253
$842$	0	0
$843$	−4.78330	−0.164746
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.41593	0.289175
$848$	0	0
$849$	23.8427	0.818279
$850$	0	0
$851$	−6.63567	−0.227468
$852$	0	0
$853$	−14.9627	−0.512312	−0.256156	−	0.966635i	$-0.582456\pi$
$853$	−14.9627	−0.512312	−0.256156	+	0.966635i	$0.582456\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.3843	0.901270	0.450635	−	0.892708i	$-0.351198\pi$
$857$	26.3843	0.901270	0.450635	+	0.892708i	$0.351198\pi$
$858$	0	0
$859$	10.9447	0.373427	0.186714	−	0.982414i	$-0.440216\pi$
$859$	10.9447	0.373427	0.186714	+	0.982414i	$0.440216\pi$
$860$	0	0
$861$	8.12194	0.276795
$862$	0	0
$863$	−35.4748	−1.20758	−0.603788	−	0.797145i	$-0.706343\pi$
$863$	−35.4748	−1.20758	−0.603788	+	0.797145i	$0.706343\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−16.1974	−0.550094
$868$	0	0
$869$	−18.7476	−0.635967
$870$	0	0
$871$	−2.34558	−0.0794768
$872$	0	0
$873$	−5.55763	−0.188097
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.79049	0.127996	0.0639979	−	0.997950i	$-0.479615\pi$
$877$	3.79049	0.127996	0.0639979	+	0.997950i	$0.479615\pi$
$878$	0	0
$879$	7.75172	0.261459
$880$	0	0
$881$	19.2724	0.649304	0.324652	−	0.945833i	$-0.394753\pi$
$881$	19.2724	0.649304	0.324652	+	0.945833i	$0.394753\pi$
$882$	0	0
$883$	52.3337	1.76117	0.880585	−	0.473889i	$-0.157150\pi$
$883$	52.3337	1.76117	0.880585	+	0.473889i	$0.157150\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.1396	0.877680	0.438840	−	0.898565i	$-0.355389\pi$
$887$	26.1396	0.877680	0.438840	+	0.898565i	$0.355389\pi$
$888$	0	0
$889$	−29.9199	−1.00348
$890$	0	0
$891$	−3.98164	−0.133390
$892$	0	0
$893$	8.81318	0.294922
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.38258	−0.0461629
$898$	0	0
$899$	3.43448	0.114546
$900$	0	0
$901$	−9.40901	−0.313459
$902$	0	0
$903$	21.0172	0.699407
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.7957	0.657307	0.328653	−	0.944451i	$-0.393405\pi$
$907$	19.7957	0.657307	0.328653	+	0.944451i	$0.393405\pi$
$908$	0	0
$909$	−0.894108	−0.0296557
$910$	0	0
$911$	22.5317	0.746510	0.373255	−	0.927729i	$-0.378242\pi$
$911$	22.5317	0.746510	0.373255	+	0.927729i	$0.378242\pi$
$912$	0	0
$913$	−6.15983	−0.203861
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.1597	0.401549
$918$	0	0
$919$	27.9476	0.921905	0.460952	−	0.887425i	$-0.347508\pi$
$919$	27.9476	0.921905	0.460952	+	0.887425i	$0.347508\pi$
$920$	0	0
$921$	−30.3970	−1.00162
$922$	0	0
$923$	10.1516	0.334143
$924$	0	0
$925$	0	0
$926$	0	0
$927$	5.75934	0.189162
$928$	0	0
$929$	−53.0243	−1.73967	−0.869835	−	0.493343i	$-0.835775\pi$
$929$	−53.0243	−1.73967	−0.869835	+	0.493343i	$0.835775\pi$
$930$	0	0
$931$	21.8123	0.714871
$932$	0	0
$933$	17.9492	0.587631
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13.0948	0.427788	0.213894	−	0.976857i	$-0.431385\pi$
$937$	13.0948	0.427788	0.213894	+	0.976857i	$0.431385\pi$
$938$	0	0
$939$	−20.3920	−0.665468
$940$	0	0
$941$	51.1404	1.66713	0.833564	−	0.552423i	$-0.186297\pi$
$941$	51.1404	1.66713	0.833564	+	0.552423i	$0.186297\pi$
$942$	0	0
$943$	4.96706	0.161750
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.6221	1.22255	0.611277	−	0.791417i	$-0.290656\pi$
$947$	37.6221	1.22255	0.611277	+	0.791417i	$0.290656\pi$
$948$	0	0
$949$	16.3096	0.529431
$950$	0	0
$951$	8.98326	0.291302
$952$	0	0
$953$	−16.7017	−0.541022	−0.270511	−	0.962717i	$-0.587193\pi$
$953$	−16.7017	−0.541022	−0.270511	+	0.962717i	$0.587193\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−13.6749	−0.442046
$958$	0	0
$959$	−24.6913	−0.797325
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−14.7961	−0.476796
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−54.3428	−1.74755	−0.873773	−	0.486334i	$-0.838334\pi$
$967$	−54.3428	−1.74755	−0.873773	+	0.486334i	$0.838334\pi$
$968$	0	0
$969$	4.89341	0.157199
$970$	0	0
$971$	54.5310	1.74998	0.874992	−	0.484137i	$-0.160866\pi$
$971$	54.5310	1.74998	0.874992	+	0.484137i	$0.160866\pi$
$972$	0	0
$973$	−25.9747	−0.832711
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.7885	−1.40092	−0.700459	−	0.713693i	$-0.747021\pi$
$977$	−43.7885	−1.40092	−0.700459	+	0.713693i	$0.747021\pi$
$978$	0	0
$979$	27.9020	0.891752
$980$	0	0
$981$	−13.6183	−0.434800
$982$	0	0
$983$	−29.4010	−0.937746	−0.468873	−	0.883266i	$-0.655340\pi$
$983$	−29.4010	−0.937746	−0.468873	+	0.883266i	$0.655340\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.79773	−0.0890528
$988$	0	0
$989$	12.8533	0.408710
$990$	0	0
$991$	−53.1958	−1.68982	−0.844910	−	0.534908i	$-0.820346\pi$
$991$	−53.1958	−1.68982	−0.844910	+	0.534908i	$0.820346\pi$
$992$	0	0
$993$	19.5465	0.620290
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.96230	−0.0621467	−0.0310733	−	0.999517i	$-0.509893\pi$
$997$	−1.96230	−0.0621467	−0.0310733	+	0.999517i	$0.509893\pi$
$998$	0	0
$999$	−6.25745	−0.197977

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.a.be.1.6		7
5.2	odd	4		1860.2.g.b.1489.2	✓	14
5.3	odd	4		1860.2.g.b.1489.9	yes	14
5.4	even	2		9300.2.a.bd.1.2		7
15.2	even	4		5580.2.g.e.3349.11		14
15.8	even	4		5580.2.g.e.3349.12		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.g.b.1489.2	✓	14	5.2	odd	4
1860.2.g.b.1489.9	yes	14	5.3	odd	4
5580.2.g.e.3349.11		14	15.2	even	4
5580.2.g.e.3349.12		14	15.8	even	4
9300.2.a.bd.1.2		7	5.4	even	2
9300.2.a.be.1.6		7	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.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.73400 q^{7} +1.00000 q^{9} -3.98164 q^{11} -1.30377 q^{13} -0.895853 q^{17} -5.46229 q^{19} +1.73400 q^{21} +1.06044 q^{23} +1.00000 q^{27} +3.43448 q^{29} +1.00000 q^{31} -3.98164 q^{33} -6.25745 q^{37} -1.30377 q^{39} +4.68395 q^{41} +12.1207 q^{43} -1.61346 q^{47} -3.99326 q^{49} -0.895853 q^{51} +10.5029 q^{53} -5.46229 q^{57} -9.43866 q^{59} -0.198690 q^{61} +1.73400 q^{63} +1.79907 q^{67} +1.06044 q^{69} -7.78632 q^{71} -12.5095 q^{73} -6.90415 q^{77} +4.70850 q^{79} +1.00000 q^{81} +1.54706 q^{83} +3.43448 q^{87} -7.00766 q^{89} -2.26073 q^{91} +1.00000 q^{93} -5.55763 q^{97} -3.98164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} - 4 q^{7} + 7 q^{9} + 4 q^{11} - 10 q^{13} - 14 q^{17} - 6 q^{19} - 4 q^{21} - 8 q^{23} + 7 q^{27} - 2 q^{29} + 7 q^{31} + 4 q^{33} - 12 q^{37} - 10 q^{39} + 10 q^{41} - 8 q^{43} - 16 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 - 4 * q^7 + 7 * q^9 + 4 * q^11 - 10 * q^13 - 14 * q^17 - 6 * q^19 - 4 * q^21 - 8 * q^23 + 7 * q^27 - 2 * q^29 + 7 * q^31 + 4 * q^33 - 12 * q^37 - 10 * q^39 + 10 * q^41 - 8 * q^43 - 16 * q^47 + q^49 - 14 * q^51 - 12 * q^53 - 6 * q^57 - 2 * q^59 + 2 * q^61 - 4 * q^63 - 18 * q^67 - 8 * q^69 + 6 * q^71 - 22 * q^73 - 22 * q^77 - 24 * q^79 + 7 * q^81 - 12 * q^83 - 2 * q^87 - 12 * q^89 - 28 * q^91 + 7 * q^93 - 36 * q^97 + 4 * q^99

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