LMFDB - Embedded newform 9300.2.a.ba.1.1

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 = [6,0,6,0,0,0,4,0,6,0,-3] 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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 22x^{4} + 30x^{3} + 112x^{2} - 64x - 64 $ x^6 - 2x^5 - 22x^4 + 30x^3 + 112x^2 - 64*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.20194$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.41074 q^{7} +1.00000 q^{9} +0.427911 q^{11} -3.75513 q^{13} -4.57171 q^{17} -2.72354 q^{19} -3.41074 q^{21} -7.38671 q^{23} +1.00000 q^{27} +3.93169 q^{29} -1.00000 q^{31} +0.427911 q^{33} +2.25929 q^{37} -3.75513 q^{39} +11.1659 q^{41} -6.59378 q^{43} +3.54012 q^{47} +4.63316 q^{49} -4.57171 q^{51} +6.83179 q^{53} -2.72354 q^{57} -10.2963 q^{59} +13.8146 q^{61} -3.41074 q^{63} -10.5018 q^{67} -7.38671 q^{69} +6.54698 q^{71} +2.00000 q^{73} -1.45950 q^{77} -2.33596 q^{79} +1.00000 q^{81} +3.25400 q^{83} +3.93169 q^{87} +3.11221 q^{89} +12.8078 q^{91} -1.00000 q^{93} +6.04390 q^{97} +0.427911 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 4 q^{7} + 6 q^{9} - 3 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 4 q^{21} + 5 q^{23} + 6 q^{27} - 6 q^{31} - 3 q^{33} + 8 q^{37} + 2 q^{39} + 18 q^{41} + 15 q^{43} + q^{47} + 14 q^{49} + 4 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 4 * q^7 + 6 * q^9 - 3 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 4 * q^21 + 5 * q^23 + 6 * q^27 - 6 * q^31 - 3 * q^33 + 8 * q^37 + 2 * q^39 + 18 * q^41 + 15 * q^43 + q^47 + 14 * q^49 + 4 * q^51 + 5 * q^53 - 3 * q^57 - 12 * q^59 + 28 * q^61 + 4 * q^63 + 3 * q^67 + 5 * q^69 + 25 * q^71 + 12 * q^73 + 8 * q^77 + 3 * q^79 + 6 * q^81 + 19 * q^83 + 4 * q^89 + 16 * q^91 - 6 * q^93 - 2 * q^97 - 3 * 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
$7$	−3.41074	−1.28914	−0.644570	−	0.764546i	$-0.722963\pi$
$7$	−3.41074	−1.28914	−0.644570	+	0.764546i	$0.722963\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.427911	0.129020	0.0645101	−	0.997917i	$-0.479452\pi$
$11$	0.427911	0.129020	0.0645101	+	0.997917i	$0.479452\pi$
$12$	0	0
$13$	−3.75513	−1.04148	−0.520742	−	0.853714i	$-0.674345\pi$
$13$	−3.75513	−1.04148	−0.520742	+	0.853714i	$0.674345\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.57171	−1.10880	−0.554401	−	0.832250i	$-0.687052\pi$
$17$	−4.57171	−1.10880	−0.554401	+	0.832250i	$0.687052\pi$
$18$	0	0
$19$	−2.72354	−0.624824	−0.312412	−	0.949947i	$-0.601137\pi$
$19$	−2.72354	−0.624824	−0.312412	+	0.949947i	$0.601137\pi$
$20$	0	0
$21$	−3.41074	−0.744285
$22$	0	0
$23$	−7.38671	−1.54024	−0.770118	−	0.637901i	$-0.779803\pi$
$23$	−7.38671	−1.54024	−0.770118	+	0.637901i	$0.779803\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.93169	0.730096	0.365048	−	0.930989i	$-0.381053\pi$
$29$	3.93169	0.730096	0.365048	+	0.930989i	$0.381053\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0.427911	0.0744898
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.25929	0.371425	0.185712	−	0.982604i	$-0.440541\pi$
$37$	2.25929	0.371425	0.185712	+	0.982604i	$0.440541\pi$
$38$	0	0
$39$	−3.75513	−0.601301
$40$	0	0
$41$	11.1659	1.74382	0.871908	−	0.489670i	$-0.162883\pi$
$41$	11.1659	1.74382	0.871908	+	0.489670i	$0.162883\pi$
$42$	0	0
$43$	−6.59378	−1.00554	−0.502771	−	0.864420i	$-0.667686\pi$
$43$	−6.59378	−1.00554	−0.502771	+	0.864420i	$0.667686\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.54012	0.516380	0.258190	−	0.966094i	$-0.416874\pi$
$47$	3.54012	0.516380	0.258190	+	0.966094i	$0.416874\pi$
$48$	0	0
$49$	4.63316	0.661880
$50$	0	0
$51$	−4.57171	−0.640167
$52$	0	0
$53$	6.83179	0.938419	0.469210	−	0.883087i	$-0.344539\pi$
$53$	6.83179	0.938419	0.469210	+	0.883087i	$0.344539\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.72354	−0.360742
$58$	0	0
$59$	−10.2963	−1.34047	−0.670234	−	0.742150i	$-0.733806\pi$
$59$	−10.2963	−1.34047	−0.670234	+	0.742150i	$0.733806\pi$
$60$	0	0
$61$	13.8146	1.76878	0.884391	−	0.466747i	$-0.154574\pi$
$61$	13.8146	1.76878	0.884391	+	0.466747i	$0.154574\pi$
$62$	0	0
$63$	−3.41074	−0.429713
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.5018	−1.28300	−0.641501	−	0.767122i	$-0.721688\pi$
$67$	−10.5018	−1.28300	−0.641501	+	0.767122i	$0.721688\pi$
$68$	0	0
$69$	−7.38671	−0.889256
$70$	0	0
$71$	6.54698	0.776984	0.388492	−	0.921452i	$-0.372996\pi$
$71$	6.54698	0.776984	0.388492	+	0.921452i	$0.372996\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.45950	−0.166325
$78$	0	0
$79$	−2.33596	−0.262815	−0.131408	−	0.991328i	$-0.541950\pi$
$79$	−2.33596	−0.262815	−0.131408	+	0.991328i	$0.541950\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.25400	0.357173	0.178586	−	0.983924i	$-0.442848\pi$
$83$	3.25400	0.357173	0.178586	+	0.983924i	$0.442848\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.93169	0.421521
$88$	0	0
$89$	3.11221	0.329894	0.164947	−	0.986302i	$-0.447255\pi$
$89$	3.11221	0.329894	0.164947	+	0.986302i	$0.447255\pi$
$90$	0	0
$91$	12.8078	1.34262
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.04390	0.613665	0.306832	−	0.951764i	$-0.400731\pi$
$97$	6.04390	0.613665	0.306832	+	0.951764i	$0.400731\pi$
$98$	0	0
$99$	0.427911	0.0430067
$100$	0	0
$101$	13.1883	1.31229	0.656144	−	0.754636i	$-0.272187\pi$
$101$	13.1883	1.31229	0.656144	+	0.754636i	$0.272187\pi$
$102$	0	0
$103$	12.7380	1.25511	0.627554	−	0.778573i	$-0.284056\pi$
$103$	12.7380	1.25511	0.627554	+	0.778573i	$0.284056\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.4335	1.20199	0.600996	−	0.799252i	$-0.294771\pi$
$107$	12.4335	1.20199	0.600996	+	0.799252i	$0.294771\pi$
$108$	0	0
$109$	0.450120	0.0431136	0.0215568	−	0.999768i	$-0.493138\pi$
$109$	0.450120	0.0431136	0.0215568	+	0.999768i	$0.493138\pi$
$110$	0	0
$111$	2.25929	0.214442
$112$	0	0
$113$	0.640016	0.0602077	0.0301038	−	0.999547i	$-0.490416\pi$
$113$	0.640016	0.0602077	0.0301038	+	0.999547i	$0.490416\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.75513	−0.347162
$118$	0	0
$119$	15.5929	1.42940
$120$	0	0
$121$	−10.8169	−0.983354
$122$	0	0
$123$	11.1659	1.00679
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.41231	0.391529	0.195765	−	0.980651i	$-0.437281\pi$
$127$	4.41231	0.391529	0.195765	+	0.980651i	$0.437281\pi$
$128$	0	0
$129$	−6.59378	−0.580550
$130$	0	0
$131$	−5.92865	−0.517989	−0.258994	−	0.965879i	$-0.583391\pi$
$131$	−5.92865	−0.517989	−0.258994	+	0.965879i	$0.583391\pi$
$132$	0	0
$133$	9.28930	0.805484
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.37877	0.630411	0.315206	−	0.949023i	$-0.397927\pi$
$137$	7.37877	0.630411	0.315206	+	0.949023i	$0.397927\pi$
$138$	0	0
$139$	18.9265	1.60532	0.802660	−	0.596436i	$-0.203417\pi$
$139$	18.9265	1.60532	0.802660	+	0.596436i	$0.203417\pi$
$140$	0	0
$141$	3.54012	0.298132
$142$	0	0
$143$	−1.60686	−0.134373
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.63316	0.382136
$148$	0	0
$149$	−16.7853	−1.37511	−0.687553	−	0.726134i	$-0.741315\pi$
$149$	−16.7853	−1.37511	−0.687553	+	0.726134i	$0.741315\pi$
$150$	0	0
$151$	14.2757	1.16174	0.580870	−	0.813997i	$-0.302713\pi$
$151$	14.2757	1.16174	0.580870	+	0.813997i	$0.302713\pi$
$152$	0	0
$153$	−4.57171	−0.369600
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.01441	0.0809590	0.0404795	−	0.999180i	$-0.487111\pi$
$157$	1.01441	0.0809590	0.0404795	+	0.999180i	$0.487111\pi$
$158$	0	0
$159$	6.83179	0.541797
$160$	0	0
$161$	25.1942	1.98558
$162$	0	0
$163$	18.8738	1.47831	0.739156	−	0.673535i	$-0.235225\pi$
$163$	18.8738	1.47831	0.739156	+	0.673535i	$0.235225\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.93753	−0.614224	−0.307112	−	0.951673i	$-0.599363\pi$
$167$	−7.93753	−0.614224	−0.307112	+	0.951673i	$0.599363\pi$
$168$	0	0
$169$	1.10097	0.0846904
$170$	0	0
$171$	−2.72354	−0.208275
$172$	0	0
$173$	−1.07877	−0.0820171	−0.0410085	−	0.999159i	$-0.513057\pi$
$173$	−1.07877	−0.0820171	−0.0410085	+	0.999159i	$0.513057\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.2963	−0.773919
$178$	0	0
$179$	13.9285	1.04107	0.520534	−	0.853841i	$-0.325733\pi$
$179$	13.9285	1.04107	0.520534	+	0.853841i	$0.325733\pi$
$180$	0	0
$181$	−15.5354	−1.15474	−0.577369	−	0.816483i	$-0.695921\pi$
$181$	−15.5354	−1.15474	−0.577369	+	0.816483i	$0.695921\pi$
$182$	0	0
$183$	13.8146	1.02121
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.95628	−0.143058
$188$	0	0
$189$	−3.41074	−0.248095
$190$	0	0
$191$	9.76306	0.706431	0.353215	−	0.935542i	$-0.385088\pi$
$191$	9.76306	0.706431	0.353215	+	0.935542i	$0.385088\pi$
$192$	0	0
$193$	−8.36450	−0.602090	−0.301045	−	0.953610i	$-0.597335\pi$
$193$	−8.36450	−0.602090	−0.301045	+	0.953610i	$0.597335\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.13718	0.152268	0.0761340	−	0.997098i	$-0.475742\pi$
$197$	2.13718	0.152268	0.0761340	+	0.997098i	$0.475742\pi$
$198$	0	0
$199$	−22.9448	−1.62652	−0.813258	−	0.581903i	$-0.802308\pi$
$199$	−22.9448	−1.62652	−0.813258	+	0.581903i	$0.802308\pi$
$200$	0	0
$201$	−10.5018	−0.740741
$202$	0	0
$203$	−13.4100	−0.941196
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.38671	−0.513412
$208$	0	0
$209$	−1.16543	−0.0806148
$210$	0	0
$211$	−16.8570	−1.16048	−0.580240	−	0.814445i	$-0.697041\pi$
$211$	−16.8570	−1.16048	−0.580240	+	0.814445i	$0.697041\pi$
$212$	0	0
$213$	6.54698	0.448592
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.41074	0.231536
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	17.1673	1.15480
$222$	0	0
$223$	14.4579	0.968174	0.484087	−	0.875020i	$-0.339152\pi$
$223$	14.4579	0.968174	0.484087	+	0.875020i	$0.339152\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.94130	−0.128849	−0.0644244	−	0.997923i	$-0.520521\pi$
$227$	−1.94130	−0.128849	−0.0644244	+	0.997923i	$0.520521\pi$
$228$	0	0
$229$	−18.4342	−1.21817	−0.609083	−	0.793107i	$-0.708462\pi$
$229$	−18.4342	−1.21817	−0.609083	+	0.793107i	$0.708462\pi$
$230$	0	0
$231$	−1.45950	−0.0960277
$232$	0	0
$233$	−2.13732	−0.140020	−0.0700102	−	0.997546i	$-0.522303\pi$
$233$	−2.13732	−0.140020	−0.0700102	+	0.997546i	$0.522303\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.33596	−0.151737
$238$	0	0
$239$	11.6533	0.753788	0.376894	−	0.926256i	$-0.376992\pi$
$239$	11.6533	0.753788	0.376894	+	0.926256i	$0.376992\pi$
$240$	0	0
$241$	19.8581	1.27917	0.639587	−	0.768719i	$-0.279106\pi$
$241$	19.8581	1.27917	0.639587	+	0.768719i	$0.279106\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.2272	0.650744
$248$	0	0
$249$	3.25400	0.206214
$250$	0	0
$251$	−13.4519	−0.849075	−0.424538	−	0.905410i	$-0.639563\pi$
$251$	−13.4519	−0.849075	−0.424538	+	0.905410i	$0.639563\pi$
$252$	0	0
$253$	−3.16086	−0.198721
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.8005	−0.923229	−0.461615	−	0.887081i	$-0.652730\pi$
$257$	−14.8005	−0.923229	−0.461615	+	0.887081i	$0.652730\pi$
$258$	0	0
$259$	−7.70585	−0.478818
$260$	0	0
$261$	3.93169	0.243365
$262$	0	0
$263$	−14.7766	−0.911162	−0.455581	−	0.890194i	$-0.650568\pi$
$263$	−14.7766	−0.911162	−0.455581	+	0.890194i	$0.650568\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.11221	0.190464
$268$	0	0
$269$	12.5621	0.765924	0.382962	−	0.923764i	$-0.374904\pi$
$269$	12.5621	0.765924	0.382962	+	0.923764i	$0.374904\pi$
$270$	0	0
$271$	−13.4748	−0.818533	−0.409267	−	0.912415i	$-0.634215\pi$
$271$	−13.4748	−0.818533	−0.409267	+	0.912415i	$0.634215\pi$
$272$	0	0
$273$	12.8078	0.775161
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.5508	1.05452	0.527261	−	0.849703i	$-0.323219\pi$
$277$	17.5508	1.05452	0.527261	+	0.849703i	$0.323219\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−0.150269	−0.00896431	−0.00448215	−	0.999990i	$-0.501427\pi$
$281$	−0.150269	−0.00896431	−0.00448215	+	0.999990i	$0.501427\pi$
$282$	0	0
$283$	28.8258	1.71351	0.856757	−	0.515721i	$-0.172476\pi$
$283$	28.8258	1.71351	0.856757	+	0.515721i	$0.172476\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−38.0839	−2.24802
$288$	0	0
$289$	3.90049	0.229441
$290$	0	0
$291$	6.04390	0.354300
$292$	0	0
$293$	16.3061	0.952615	0.476307	−	0.879279i	$-0.341975\pi$
$293$	16.3061	0.952615	0.476307	+	0.879279i	$0.341975\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.427911	0.0248299
$298$	0	0
$299$	27.7380	1.60413
$300$	0	0
$301$	22.4897	1.29628
$302$	0	0
$303$	13.1883	0.757650
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.1132	−1.37621	−0.688105	−	0.725611i	$-0.741557\pi$
$307$	−24.1132	−1.37621	−0.688105	+	0.725611i	$0.741557\pi$
$308$	0	0
$309$	12.7380	0.724637
$310$	0	0
$311$	30.7677	1.74468	0.872338	−	0.488903i	$-0.162603\pi$
$311$	30.7677	1.74468	0.872338	+	0.488903i	$0.162603\pi$
$312$	0	0
$313$	9.12952	0.516031	0.258016	−	0.966141i	$-0.416931\pi$
$313$	9.12952	0.516031	0.258016	+	0.966141i	$0.416931\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.38412	−0.0777399	−0.0388700	−	0.999244i	$-0.512376\pi$
$317$	−1.38412	−0.0777399	−0.0388700	+	0.999244i	$0.512376\pi$
$318$	0	0
$319$	1.68241	0.0941971
$320$	0	0
$321$	12.4335	0.693971
$322$	0	0
$323$	12.4512	0.692805
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.450120	0.0248917
$328$	0	0
$329$	−12.0744	−0.665685
$330$	0	0
$331$	−27.5834	−1.51612	−0.758060	−	0.652184i	$-0.773853\pi$
$331$	−27.5834	−1.51612	−0.758060	+	0.652184i	$0.773853\pi$
$332$	0	0
$333$	2.25929	0.123808
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.70637	0.0929520	0.0464760	−	0.998919i	$-0.485201\pi$
$337$	1.70637	0.0929520	0.0464760	+	0.998919i	$0.485201\pi$
$338$	0	0
$339$	0.640016	0.0347609
$340$	0	0
$341$	−0.427911	−0.0231727
$342$	0	0
$343$	8.07269	0.435884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.0096	0.591024	0.295512	−	0.955339i	$-0.404510\pi$
$347$	11.0096	0.591024	0.295512	+	0.955339i	$0.404510\pi$
$348$	0	0
$349$	10.3729	0.555250	0.277625	−	0.960689i	$-0.410453\pi$
$349$	10.3729	0.555250	0.277625	+	0.960689i	$0.410453\pi$
$350$	0	0
$351$	−3.75513	−0.200434
$352$	0	0
$353$	−33.8769	−1.80309	−0.901544	−	0.432688i	$-0.857565\pi$
$353$	−33.8769	−1.80309	−0.901544	+	0.432688i	$0.857565\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	15.5929	0.825264
$358$	0	0
$359$	−13.4478	−0.709747	−0.354873	−	0.934914i	$-0.615476\pi$
$359$	−13.4478	−0.709747	−0.354873	+	0.934914i	$0.615476\pi$
$360$	0	0
$361$	−11.5823	−0.609596
$362$	0	0
$363$	−10.8169	−0.567740
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−36.7264	−1.91710	−0.958552	−	0.284918i	$-0.908034\pi$
$367$	−36.7264	−1.91710	−0.958552	+	0.284918i	$0.908034\pi$
$368$	0	0
$369$	11.1659	0.581272
$370$	0	0
$371$	−23.3015	−1.20975
$372$	0	0
$373$	0.810461	0.0419641	0.0209820	−	0.999780i	$-0.493321\pi$
$373$	0.810461	0.0419641	0.0209820	+	0.999780i	$0.493321\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.7640	−0.760384
$378$	0	0
$379$	−4.67322	−0.240047	−0.120024	−	0.992771i	$-0.538297\pi$
$379$	−4.67322	−0.240047	−0.120024	+	0.992771i	$0.538297\pi$
$380$	0	0
$381$	4.41231	0.226050
$382$	0	0
$383$	27.5702	1.40877	0.704385	−	0.709818i	$-0.251223\pi$
$383$	27.5702	1.40877	0.704385	+	0.709818i	$0.251223\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.59378	−0.335181
$388$	0	0
$389$	28.0105	1.42019	0.710094	−	0.704106i	$-0.248652\pi$
$389$	28.0105	1.42019	0.710094	+	0.704106i	$0.248652\pi$
$390$	0	0
$391$	33.7699	1.70782
$392$	0	0
$393$	−5.92865	−0.299061
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.76970	0.339761	0.169881	−	0.985465i	$-0.445662\pi$
$397$	6.76970	0.339761	0.169881	+	0.985465i	$0.445662\pi$
$398$	0	0
$399$	9.28930	0.465047
$400$	0	0
$401$	28.9267	1.44453	0.722265	−	0.691616i	$-0.243101\pi$
$401$	28.9267	1.44453	0.722265	+	0.691616i	$0.243101\pi$
$402$	0	0
$403$	3.75513	0.187056
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.966775	0.0479212
$408$	0	0
$409$	−4.79098	−0.236899	−0.118449	−	0.992960i	$-0.537792\pi$
$409$	−4.79098	−0.236899	−0.118449	+	0.992960i	$0.537792\pi$
$410$	0	0
$411$	7.37877	0.363968
$412$	0	0
$413$	35.1181	1.72805
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.9265	0.926832
$418$	0	0
$419$	−20.0404	−0.979036	−0.489518	−	0.871993i	$-0.662827\pi$
$419$	−20.0404	−0.979036	−0.489518	+	0.871993i	$0.662827\pi$
$420$	0	0
$421$	29.8466	1.45463	0.727317	−	0.686302i	$-0.240767\pi$
$421$	29.8466	1.45463	0.727317	+	0.686302i	$0.240767\pi$
$422$	0	0
$423$	3.54012	0.172127
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−47.1181	−2.28021
$428$	0	0
$429$	−1.60686	−0.0775800
$430$	0	0
$431$	31.1114	1.49858	0.749291	−	0.662241i	$-0.230394\pi$
$431$	31.1114	1.49858	0.749291	+	0.662241i	$0.230394\pi$
$432$	0	0
$433$	−9.99539	−0.480348	−0.240174	−	0.970730i	$-0.577205\pi$
$433$	−9.99539	−0.480348	−0.240174	+	0.970730i	$0.577205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.1180	0.962376
$438$	0	0
$439$	14.5716	0.695466	0.347733	−	0.937594i	$-0.386952\pi$
$439$	14.5716	0.695466	0.347733	+	0.937594i	$0.386952\pi$
$440$	0	0
$441$	4.63316	0.220627
$442$	0	0
$443$	−16.6421	−0.790688	−0.395344	−	0.918533i	$-0.629375\pi$
$443$	−16.6421	−0.790688	−0.395344	+	0.918533i	$0.629375\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.7853	−0.793918
$448$	0	0
$449$	18.9847	0.895943	0.447972	−	0.894048i	$-0.352147\pi$
$449$	18.9847	0.895943	0.447972	+	0.894048i	$0.352147\pi$
$450$	0	0
$451$	4.77800	0.224987
$452$	0	0
$453$	14.2757	0.670730
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.4707	−0.957578	−0.478789	−	0.877930i	$-0.658924\pi$
$457$	−20.4707	−0.957578	−0.478789	+	0.877930i	$0.658924\pi$
$458$	0	0
$459$	−4.57171	−0.213389
$460$	0	0
$461$	−18.5102	−0.862106	−0.431053	−	0.902327i	$-0.641858\pi$
$461$	−18.5102	−0.862106	−0.431053	+	0.902327i	$0.641858\pi$
$462$	0	0
$463$	27.2657	1.26714	0.633572	−	0.773684i	$-0.281588\pi$
$463$	27.2657	1.26714	0.633572	+	0.773684i	$0.281588\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.49060	0.161526	0.0807629	−	0.996733i	$-0.474264\pi$
$467$	3.49060	0.161526	0.0807629	+	0.996733i	$0.474264\pi$
$468$	0	0
$469$	35.8190	1.65397
$470$	0	0
$471$	1.01441	0.0467417
$472$	0	0
$473$	−2.82155	−0.129735
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.83179	0.312806
$478$	0	0
$479$	−24.7740	−1.13195	−0.565976	−	0.824422i	$-0.691501\pi$
$479$	−24.7740	−1.13195	−0.565976	+	0.824422i	$0.691501\pi$
$480$	0	0
$481$	−8.48391	−0.386833
$482$	0	0
$483$	25.1942	1.14637
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.6052	0.661826	0.330913	−	0.943661i	$-0.392643\pi$
$487$	14.6052	0.661826	0.330913	+	0.943661i	$0.392643\pi$
$488$	0	0
$489$	18.8738	0.853503
$490$	0	0
$491$	−29.9916	−1.35350	−0.676751	−	0.736212i	$-0.736613\pi$
$491$	−29.9916	−1.35350	−0.676751	+	0.736212i	$0.736613\pi$
$492$	0	0
$493$	−17.9745	−0.809532
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−22.3301	−1.00164
$498$	0	0
$499$	19.0942	0.854774	0.427387	−	0.904069i	$-0.359434\pi$
$499$	19.0942	0.854774	0.427387	+	0.904069i	$0.359434\pi$
$500$	0	0
$501$	−7.93753	−0.354623
$502$	0	0
$503$	−8.48724	−0.378427	−0.189214	−	0.981936i	$-0.560594\pi$
$503$	−8.48724	−0.378427	−0.189214	+	0.981936i	$0.560594\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.10097	0.0488960
$508$	0	0
$509$	−17.0988	−0.757889	−0.378945	−	0.925419i	$-0.623713\pi$
$509$	−17.0988	−0.757889	−0.378945	+	0.925419i	$0.623713\pi$
$510$	0	0
$511$	−6.82148	−0.301765
$512$	0	0
$513$	−2.72354	−0.120247
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.51486	0.0666234
$518$	0	0
$519$	−1.07877	−0.0473526
$520$	0	0
$521$	41.9887	1.83956	0.919780	−	0.392434i	$-0.128367\pi$
$521$	41.9887	1.83956	0.919780	+	0.392434i	$0.128367\pi$
$522$	0	0
$523$	14.0185	0.612988	0.306494	−	0.951873i	$-0.400844\pi$
$523$	14.0185	0.612988	0.306494	+	0.951873i	$0.400844\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.57171	0.199147
$528$	0	0
$529$	31.5635	1.37233
$530$	0	0
$531$	−10.2963	−0.446823
$532$	0	0
$533$	−41.9292	−1.81616
$534$	0	0
$535$	0	0
$536$	0	0
$537$	13.9285	0.601061
$538$	0	0
$539$	1.98258	0.0853958
$540$	0	0
$541$	15.3558	0.660199	0.330099	−	0.943946i	$-0.392918\pi$
$541$	15.3558	0.660199	0.330099	+	0.943946i	$0.392918\pi$
$542$	0	0
$543$	−15.5354	−0.666688
$544$	0	0
$545$	0	0
$546$	0	0
$547$	35.6879	1.52591	0.762953	−	0.646454i	$-0.223749\pi$
$547$	35.6879	1.52591	0.762953	+	0.646454i	$0.223749\pi$
$548$	0	0
$549$	13.8146	0.589594
$550$	0	0
$551$	−10.7081	−0.456181
$552$	0	0
$553$	7.96734	0.338806
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.0924	0.808972	0.404486	−	0.914544i	$-0.367450\pi$
$557$	19.0924	0.808972	0.404486	+	0.914544i	$0.367450\pi$
$558$	0	0
$559$	24.7605	1.04726
$560$	0	0
$561$	−1.95628	−0.0825944
$562$	0	0
$563$	−37.4357	−1.57773	−0.788863	−	0.614569i	$-0.789330\pi$
$563$	−37.4357	−1.57773	−0.788863	+	0.614569i	$0.789330\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.41074	−0.143238
$568$	0	0
$569$	−24.6621	−1.03389	−0.516945	−	0.856018i	$-0.672931\pi$
$569$	−24.6621	−1.03389	−0.516945	+	0.856018i	$0.672931\pi$
$570$	0	0
$571$	27.7596	1.16170	0.580851	−	0.814010i	$-0.302720\pi$
$571$	27.7596	1.16170	0.580851	+	0.814010i	$0.302720\pi$
$572$	0	0
$573$	9.76306	0.407858
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.16772	0.0486127	0.0243064	−	0.999705i	$-0.492262\pi$
$577$	1.16772	0.0486127	0.0243064	+	0.999705i	$0.492262\pi$
$578$	0	0
$579$	−8.36450	−0.347617
$580$	0	0
$581$	−11.0986	−0.460446
$582$	0	0
$583$	2.92340	0.121075
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.7301	−0.525428	−0.262714	−	0.964874i	$-0.584618\pi$
$587$	−12.7301	−0.525428	−0.262714	+	0.964874i	$0.584618\pi$
$588$	0	0
$589$	2.72354	0.112222
$590$	0	0
$591$	2.13718	0.0879120
$592$	0	0
$593$	−31.1787	−1.28036	−0.640179	−	0.768226i	$-0.721140\pi$
$593$	−31.1787	−1.28036	−0.640179	+	0.768226i	$0.721140\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−22.9448	−0.939070
$598$	0	0
$599$	−1.63527	−0.0668152	−0.0334076	−	0.999442i	$-0.510636\pi$
$599$	−1.63527	−0.0668152	−0.0334076	+	0.999442i	$0.510636\pi$
$600$	0	0
$601$	−7.59037	−0.309618	−0.154809	−	0.987944i	$-0.549476\pi$
$601$	−7.59037	−0.309618	−0.154809	+	0.987944i	$0.549476\pi$
$602$	0	0
$603$	−10.5018	−0.427667
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−15.2753	−0.620006	−0.310003	−	0.950736i	$-0.600330\pi$
$607$	−15.2753	−0.620006	−0.310003	+	0.950736i	$0.600330\pi$
$608$	0	0
$609$	−13.4100	−0.543400
$610$	0	0
$611$	−13.2936	−0.537802
$612$	0	0
$613$	37.8200	1.52754	0.763768	−	0.645491i	$-0.223347\pi$
$613$	37.8200	1.52754	0.763768	+	0.645491i	$0.223347\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−24.4463	−0.984172	−0.492086	−	0.870547i	$-0.663765\pi$
$617$	−24.4463	−0.984172	−0.492086	+	0.870547i	$0.663765\pi$
$618$	0	0
$619$	13.8738	0.557636	0.278818	−	0.960344i	$-0.410057\pi$
$619$	13.8738	0.557636	0.278818	+	0.960344i	$0.410057\pi$
$620$	0	0
$621$	−7.38671	−0.296419
$622$	0	0
$623$	−10.6149	−0.425279
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.16543	−0.0465430
$628$	0	0
$629$	−10.3288	−0.411836
$630$	0	0
$631$	−21.9165	−0.872480	−0.436240	−	0.899830i	$-0.643690\pi$
$631$	−21.9165	−0.872480	−0.436240	+	0.899830i	$0.643690\pi$
$632$	0	0
$633$	−16.8570	−0.670004
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−17.3981	−0.689338
$638$	0	0
$639$	6.54698	0.258995
$640$	0	0
$641$	28.5022	1.12577	0.562885	−	0.826536i	$-0.309692\pi$
$641$	28.5022	1.12577	0.562885	+	0.826536i	$0.309692\pi$
$642$	0	0
$643$	35.2496	1.39011	0.695054	−	0.718958i	$-0.255381\pi$
$643$	35.2496	1.39011	0.695054	+	0.718958i	$0.255381\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.3487	−1.27176	−0.635878	−	0.771789i	$-0.719362\pi$
$647$	−32.3487	−1.27176	−0.635878	+	0.771789i	$0.719362\pi$
$648$	0	0
$649$	−4.40592	−0.172947
$650$	0	0
$651$	3.41074	0.133677
$652$	0	0
$653$	36.9271	1.44507	0.722534	−	0.691336i	$-0.242977\pi$
$653$	36.9271	1.44507	0.722534	+	0.691336i	$0.242977\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	17.5020	0.681781	0.340891	−	0.940103i	$-0.389271\pi$
$659$	17.5020	0.681781	0.340891	+	0.940103i	$0.389271\pi$
$660$	0	0
$661$	17.0217	0.662068	0.331034	−	0.943619i	$-0.392602\pi$
$661$	17.0217	0.662068	0.331034	+	0.943619i	$0.392602\pi$
$662$	0	0
$663$	17.1673	0.666724
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−29.0423	−1.12452
$668$	0	0
$669$	14.4579	0.558976
$670$	0	0
$671$	5.91144	0.228208
$672$	0	0
$673$	−33.5339	−1.29264	−0.646318	−	0.763068i	$-0.723692\pi$
$673$	−33.5339	−1.29264	−0.646318	+	0.763068i	$0.723692\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0.132810	0.00510432	0.00255216	−	0.999997i	$-0.499188\pi$
$677$	0.132810	0.00510432	0.00255216	+	0.999997i	$0.499188\pi$
$678$	0	0
$679$	−20.6142	−0.791100
$680$	0	0
$681$	−1.94130	−0.0743909
$682$	0	0
$683$	−48.0007	−1.83669	−0.918347	−	0.395776i	$-0.870476\pi$
$683$	−48.0007	−1.83669	−0.918347	+	0.395776i	$0.870476\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−18.4342	−0.703308
$688$	0	0
$689$	−25.6543	−0.977349
$690$	0	0
$691$	49.1980	1.87158	0.935790	−	0.352558i	$-0.114688\pi$
$691$	49.1980	1.87158	0.935790	+	0.352558i	$0.114688\pi$
$692$	0	0
$693$	−1.45950	−0.0554416
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−51.0471	−1.93355
$698$	0	0
$699$	−2.13732	−0.0808408
$700$	0	0
$701$	21.6846	0.819015	0.409508	−	0.912307i	$-0.365700\pi$
$701$	21.6846	0.819015	0.409508	+	0.912307i	$0.365700\pi$
$702$	0	0
$703$	−6.15327	−0.232075
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−44.9820	−1.69172
$708$	0	0
$709$	28.3600	1.06508	0.532542	−	0.846404i	$-0.321237\pi$
$709$	28.3600	1.06508	0.532542	+	0.846404i	$0.321237\pi$
$710$	0	0
$711$	−2.33596	−0.0876052
$712$	0	0
$713$	7.38671	0.276635
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.6533	0.435200
$718$	0	0
$719$	−48.3350	−1.80259	−0.901295	−	0.433205i	$-0.857382\pi$
$719$	−48.3350	−1.80259	−0.901295	+	0.433205i	$0.857382\pi$
$720$	0	0
$721$	−43.4459	−1.61801
$722$	0	0
$723$	19.8581	0.738531
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.5823	0.689179	0.344589	−	0.938754i	$-0.388018\pi$
$727$	18.5823	0.689179	0.344589	+	0.938754i	$0.388018\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.1448	1.11495
$732$	0	0
$733$	21.8160	0.805791	0.402896	−	0.915246i	$-0.368004\pi$
$733$	21.8160	0.805791	0.402896	+	0.915246i	$0.368004\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.49385	−0.165533
$738$	0	0
$739$	−10.0118	−0.368290	−0.184145	−	0.982899i	$-0.558952\pi$
$739$	−10.0118	−0.368290	−0.184145	+	0.982899i	$0.558952\pi$
$740$	0	0
$741$	10.2272	0.375707
$742$	0	0
$743$	14.2768	0.523766	0.261883	−	0.965100i	$-0.415656\pi$
$743$	14.2768	0.523766	0.261883	+	0.965100i	$0.415656\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.25400	0.119058
$748$	0	0
$749$	−42.4075	−1.54954
$750$	0	0
$751$	17.3646	0.633643	0.316821	−	0.948485i	$-0.397384\pi$
$751$	17.3646	0.633643	0.316821	+	0.948485i	$0.397384\pi$
$752$	0	0
$753$	−13.4519	−0.490214
$754$	0	0
$755$	0	0
$756$	0	0
$757$	46.9831	1.70763	0.853815	−	0.520576i	$-0.174283\pi$
$757$	46.9831	1.70763	0.853815	+	0.520576i	$0.174283\pi$
$758$	0	0
$759$	−3.16086	−0.114732
$760$	0	0
$761$	12.2461	0.443920	0.221960	−	0.975056i	$-0.428755\pi$
$761$	12.2461	0.443920	0.221960	+	0.975056i	$0.428755\pi$
$762$	0	0
$763$	−1.53524	−0.0555795
$764$	0	0
$765$	0	0
$766$	0	0
$767$	38.6640	1.39608
$768$	0	0
$769$	25.2040	0.908878	0.454439	−	0.890778i	$-0.349840\pi$
$769$	25.2040	0.908878	0.454439	+	0.890778i	$0.349840\pi$
$770$	0	0
$771$	−14.8005	−0.533027
$772$	0	0
$773$	−7.55851	−0.271861	−0.135930	−	0.990718i	$-0.543402\pi$
$773$	−7.55851	−0.271861	−0.135930	+	0.990718i	$0.543402\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.70585	−0.276446
$778$	0	0
$779$	−30.4107	−1.08958
$780$	0	0
$781$	2.80153	0.100247
$782$	0	0
$783$	3.93169	0.140507
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−21.3562	−0.761265	−0.380632	−	0.924726i	$-0.624294\pi$
$787$	−21.3562	−0.761265	−0.380632	+	0.924726i	$0.624294\pi$
$788$	0	0
$789$	−14.7766	−0.526060
$790$	0	0
$791$	−2.18293	−0.0776161
$792$	0	0
$793$	−51.8757	−1.84216
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−50.9002	−1.80298	−0.901489	−	0.432801i	$-0.857525\pi$
$797$	−50.9002	−1.80298	−0.901489	+	0.432801i	$0.857525\pi$
$798$	0	0
$799$	−16.1844	−0.572563
$800$	0	0
$801$	3.11221	0.109965
$802$	0	0
$803$	0.855823	0.0302013
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.5621	0.442207
$808$	0	0
$809$	−9.22704	−0.324406	−0.162203	−	0.986757i	$-0.551860\pi$
$809$	−9.22704	−0.324406	−0.162203	+	0.986757i	$0.551860\pi$
$810$	0	0
$811$	17.6632	0.620240	0.310120	−	0.950697i	$-0.399631\pi$
$811$	17.6632	0.620240	0.310120	+	0.950697i	$0.399631\pi$
$812$	0	0
$813$	−13.4748	−0.472580
$814$	0	0
$815$	0	0
$816$	0	0
$817$	17.9584	0.628286
$818$	0	0
$819$	12.8078	0.447540
$820$	0	0
$821$	6.56537	0.229133	0.114566	−	0.993416i	$-0.463452\pi$
$821$	6.56537	0.229133	0.114566	+	0.993416i	$0.463452\pi$
$822$	0	0
$823$	−0.394615	−0.0137554	−0.00687771	−	0.999976i	$-0.502189\pi$
$823$	−0.394615	−0.0137554	−0.00687771	+	0.999976i	$0.502189\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.2938	−1.74889	−0.874443	−	0.485128i	$-0.838773\pi$
$827$	−50.2938	−1.74889	−0.874443	+	0.485128i	$0.838773\pi$
$828$	0	0
$829$	2.76505	0.0960342	0.0480171	−	0.998847i	$-0.484710\pi$
$829$	2.76505	0.0960342	0.0480171	+	0.998847i	$0.484710\pi$
$830$	0	0
$831$	17.5508	0.608829
$832$	0	0
$833$	−21.1814	−0.733893
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−28.5821	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$839$	−28.5821	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$840$	0	0
$841$	−13.5418	−0.466959
$842$	0	0
$843$	−0.150269	−0.00517554
$844$	0	0
$845$	0	0
$846$	0	0
$847$	36.8936	1.26768
$848$	0	0
$849$	28.8258	0.989297
$850$	0	0
$851$	−16.6887	−0.572082
$852$	0	0
$853$	45.8292	1.56916	0.784580	−	0.620027i	$-0.212879\pi$
$853$	45.8292	1.56916	0.784580	+	0.620027i	$0.212879\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29.1514	0.995792	0.497896	−	0.867237i	$-0.334106\pi$
$857$	29.1514	0.995792	0.497896	+	0.867237i	$0.334106\pi$
$858$	0	0
$859$	−29.4350	−1.00431	−0.502154	−	0.864778i	$-0.667459\pi$
$859$	−29.4350	−1.00431	−0.502154	+	0.864778i	$0.667459\pi$
$860$	0	0
$861$	−38.0839	−1.29790
$862$	0	0
$863$	−19.1243	−0.650997	−0.325499	−	0.945543i	$-0.605532\pi$
$863$	−19.1243	−0.650997	−0.325499	+	0.945543i	$0.605532\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.90049	0.132468
$868$	0	0
$869$	−0.999582	−0.0339085
$870$	0	0
$871$	39.4357	1.33623
$872$	0	0
$873$	6.04390	0.204555
$874$	0	0
$875$	0	0
$876$	0	0
$877$	9.87688	0.333518	0.166759	−	0.985998i	$-0.446670\pi$
$877$	9.87688	0.333518	0.166759	+	0.985998i	$0.446670\pi$
$878$	0	0
$879$	16.3061	0.549992
$880$	0	0
$881$	3.98035	0.134101	0.0670507	−	0.997750i	$-0.478641\pi$
$881$	3.98035	0.134101	0.0670507	+	0.997750i	$0.478641\pi$
$882$	0	0
$883$	−11.7422	−0.395156	−0.197578	−	0.980287i	$-0.563308\pi$
$883$	−11.7422	−0.395156	−0.197578	+	0.980287i	$0.563308\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.25106	−0.243467	−0.121733	−	0.992563i	$-0.538845\pi$
$887$	−7.25106	−0.243467	−0.121733	+	0.992563i	$0.538845\pi$
$888$	0	0
$889$	−15.0493	−0.504736
$890$	0	0
$891$	0.427911	0.0143356
$892$	0	0
$893$	−9.64167	−0.322646
$894$	0	0
$895$	0	0
$896$	0	0
$897$	27.7380	0.926146
$898$	0	0
$899$	−3.93169	−0.131129
$900$	0	0
$901$	−31.2329	−1.04052
$902$	0	0
$903$	22.4897	0.748410
$904$	0	0
$905$	0	0
$906$	0	0
$907$	52.6937	1.74966	0.874832	−	0.484426i	$-0.160972\pi$
$907$	52.6937	1.74966	0.874832	+	0.484426i	$0.160972\pi$
$908$	0	0
$909$	13.1883	0.437429
$910$	0	0
$911$	9.08965	0.301153	0.150577	−	0.988598i	$-0.451887\pi$
$911$	9.08965	0.301153	0.150577	+	0.988598i	$0.451887\pi$
$912$	0	0
$913$	1.39242	0.0460825
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20.2211	0.667760
$918$	0	0
$919$	31.5454	1.04059	0.520293	−	0.853988i	$-0.325823\pi$
$919$	31.5454	1.04059	0.520293	+	0.853988i	$0.325823\pi$
$920$	0	0
$921$	−24.1132	−0.794556
$922$	0	0
$923$	−24.5847	−0.809217
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.7380	0.418369
$928$	0	0
$929$	−5.48385	−0.179919	−0.0899596	−	0.995945i	$-0.528674\pi$
$929$	−5.48385	−0.179919	−0.0899596	+	0.995945i	$0.528674\pi$
$930$	0	0
$931$	−12.6186	−0.413558
$932$	0	0
$933$	30.7677	1.00729
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.28857	0.238107	0.119054	−	0.992888i	$-0.462014\pi$
$937$	7.28857	0.238107	0.119054	+	0.992888i	$0.462014\pi$
$938$	0	0
$939$	9.12952	0.297931
$940$	0	0
$941$	38.2388	1.24655	0.623274	−	0.782004i	$-0.285802\pi$
$941$	38.2388	1.24655	0.623274	+	0.782004i	$0.285802\pi$
$942$	0	0
$943$	−82.4791	−2.68589
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.4759	−0.795360	−0.397680	−	0.917524i	$-0.630185\pi$
$947$	−24.4759	−0.795360	−0.397680	+	0.917524i	$0.630185\pi$
$948$	0	0
$949$	−7.51025	−0.243793
$950$	0	0
$951$	−1.38412	−0.0448832
$952$	0	0
$953$	52.9740	1.71600	0.857998	−	0.513653i	$-0.171708\pi$
$953$	52.9740	1.71600	0.857998	+	0.513653i	$0.171708\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.68241	0.0543847
$958$	0	0
$959$	−25.1671	−0.812688
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	12.4335	0.400664
$964$	0	0
$965$	0	0
$966$	0	0
$967$	53.4238	1.71799	0.858996	−	0.511982i	$-0.171088\pi$
$967$	53.4238	1.71799	0.858996	+	0.511982i	$0.171088\pi$
$968$	0	0
$969$	12.4512	0.399991
$970$	0	0
$971$	13.2130	0.424026	0.212013	−	0.977267i	$-0.431998\pi$
$971$	13.2130	0.424026	0.212013	+	0.977267i	$0.431998\pi$
$972$	0	0
$973$	−64.5532	−2.06948
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.4303	−0.781595	−0.390797	−	0.920477i	$-0.627801\pi$
$977$	−24.4303	−0.781595	−0.390797	+	0.920477i	$0.627801\pi$
$978$	0	0
$979$	1.33175	0.0425629
$980$	0	0
$981$	0.450120	0.0143712
$982$	0	0
$983$	19.6136	0.625578	0.312789	−	0.949823i	$-0.398737\pi$
$983$	19.6136	0.625578	0.312789	+	0.949823i	$0.398737\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−12.0744	−0.384334
$988$	0	0
$989$	48.7064	1.54877
$990$	0	0
$991$	−52.1843	−1.65769	−0.828844	−	0.559480i	$-0.811001\pi$
$991$	−52.1843	−1.65769	−0.828844	+	0.559480i	$0.811001\pi$
$992$	0	0
$993$	−27.5834	−0.875333
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−25.8833	−0.819732	−0.409866	−	0.912146i	$-0.634425\pi$
$997$	−25.8833	−0.819732	−0.409866	+	0.912146i	$0.634425\pi$
$998$	0	0
$999$	2.25929	0.0714807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.a.ba.1.1	yes	6
5.2	odd	4		9300.2.g.t.3349.1		12
5.3	odd	4		9300.2.g.t.3349.12		12
5.4	even	2		9300.2.a.y.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9300.2.a.y.1.6	✓	6	5.4	even	2
9300.2.a.ba.1.1	yes	6	1.1	even	1	trivial
9300.2.g.t.3349.1		12	5.2	odd	4
9300.2.g.t.3349.12		12	5.3	odd	4

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.41074 q^{7} +1.00000 q^{9} +0.427911 q^{11} -3.75513 q^{13} -4.57171 q^{17} -2.72354 q^{19} -3.41074 q^{21} -7.38671 q^{23} +1.00000 q^{27} +3.93169 q^{29} -1.00000 q^{31} +0.427911 q^{33} +2.25929 q^{37} -3.75513 q^{39} +11.1659 q^{41} -6.59378 q^{43} +3.54012 q^{47} +4.63316 q^{49} -4.57171 q^{51} +6.83179 q^{53} -2.72354 q^{57} -10.2963 q^{59} +13.8146 q^{61} -3.41074 q^{63} -10.5018 q^{67} -7.38671 q^{69} +6.54698 q^{71} +2.00000 q^{73} -1.45950 q^{77} -2.33596 q^{79} +1.00000 q^{81} +3.25400 q^{83} +3.93169 q^{87} +3.11221 q^{89} +12.8078 q^{91} -1.00000 q^{93} +6.04390 q^{97} +0.427911 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 4 q^{7} + 6 q^{9} - 3 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 4 q^{21} + 5 q^{23} + 6 q^{27} - 6 q^{31} - 3 q^{33} + 8 q^{37} + 2 q^{39} + 18 q^{41} + 15 q^{43} + q^{47} + 14 q^{49} + 4 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 4 * q^7 + 6 * q^9 - 3 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 4 * q^21 + 5 * q^23 + 6 * q^27 - 6 * q^31 - 3 * q^33 + 8 * q^37 + 2 * q^39 + 18 * q^41 + 15 * q^43 + q^47 + 14 * q^49 + 4 * q^51 + 5 * q^53 - 3 * q^57 - 12 * q^59 + 28 * q^61 + 4 * q^63 + 3 * q^67 + 5 * q^69 + 25 * q^71 + 12 * q^73 + 8 * q^77 + 3 * q^79 + 6 * q^81 + 19 * q^83 + 4 * q^89 + 16 * q^91 - 6 * q^93 - 2 * q^97 - 3 * q^99

Introduction

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