LMFDB - Embedded newform 9300.2.a.ba.1.4

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			1.4
Root			$4.32314$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.47982 q^{7} +1.00000 q^{9} -3.58271 q^{11} -5.36030 q^{13} -3.43505 q^{17} +7.10686 q^{19} +2.47982 q^{21} +7.54340 q^{23} +1.00000 q^{27} +1.11909 q^{29} -1.00000 q^{31} -3.58271 q^{33} +10.2098 q^{37} -5.36030 q^{39} +6.88049 q^{41} +1.70222 q^{43} -9.03212 q^{47} -0.850505 q^{49} -3.43505 q^{51} -10.2290 q^{53} +7.10686 q^{57} +0.287039 q^{59} -5.12610 q^{61} +2.47982 q^{63} +4.49906 q^{67} +7.54340 q^{69} +1.13435 q^{71} +2.00000 q^{73} -8.88446 q^{77} +8.37954 q^{79} +1.00000 q^{81} +15.0965 q^{83} +1.11909 q^{87} -5.44941 q^{89} -13.2926 q^{91} -1.00000 q^{93} -5.33032 q^{97} -3.58271 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	2.47982	0.937283	0.468641	−	0.883388i	$-0.344744\pi$
$7$	2.47982	0.937283	0.468641	+	0.883388i	$0.344744\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.58271	−1.08023	−0.540113	−	0.841592i	$-0.681619\pi$
$11$	−3.58271	−1.08023	−0.540113	+	0.841592i	$0.681619\pi$
$12$	0	0
$13$	−5.36030	−1.48668	−0.743340	−	0.668913i	$-0.766760\pi$
$13$	−5.36030	−1.48668	−0.743340	+	0.668913i	$0.766760\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.43505	−0.833122	−0.416561	−	0.909108i	$-0.636765\pi$
$17$	−3.43505	−0.833122	−0.416561	+	0.909108i	$0.636765\pi$
$18$	0	0
$19$	7.10686	1.63043	0.815213	−	0.579161i	$-0.196620\pi$
$19$	7.10686	1.63043	0.815213	+	0.579161i	$0.196620\pi$
$20$	0	0
$21$	2.47982	0.541141
$22$	0	0
$23$	7.54340	1.57291	0.786454	−	0.617649i	$-0.211915\pi$
$23$	7.54340	1.57291	0.786454	+	0.617649i	$0.211915\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.11909	0.207809	0.103905	−	0.994587i	$-0.466866\pi$
$29$	1.11909	0.207809	0.103905	+	0.994587i	$0.466866\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−3.58271	−0.623669
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.2098	1.67847	0.839237	−	0.543766i	$-0.183002\pi$
$37$	10.2098	1.67847	0.839237	+	0.543766i	$0.183002\pi$
$38$	0	0
$39$	−5.36030	−0.858336
$40$	0	0
$41$	6.88049	1.07455	0.537276	−	0.843407i	$-0.319453\pi$
$41$	6.88049	1.07455	0.537276	+	0.843407i	$0.319453\pi$
$42$	0	0
$43$	1.70222	0.259586	0.129793	−	0.991541i	$-0.458569\pi$
$43$	1.70222	0.259586	0.129793	+	0.991541i	$0.458569\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.03212	−1.31747	−0.658735	−	0.752375i	$-0.728908\pi$
$47$	−9.03212	−1.31747	−0.658735	+	0.752375i	$0.728908\pi$
$48$	0	0
$49$	−0.850505	−0.121501
$50$	0	0
$51$	−3.43505	−0.481003
$52$	0	0
$53$	−10.2290	−1.40506	−0.702530	−	0.711654i	$-0.747946\pi$
$53$	−10.2290	−1.40506	−0.702530	+	0.711654i	$0.747946\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.10686	0.941327
$58$	0	0
$59$	0.287039	0.0373693	0.0186846	−	0.999825i	$-0.494052\pi$
$59$	0.287039	0.0373693	0.0186846	+	0.999825i	$0.494052\pi$
$60$	0	0
$61$	−5.12610	−0.656330	−0.328165	−	0.944620i	$-0.606430\pi$
$61$	−5.12610	−0.656330	−0.328165	+	0.944620i	$0.606430\pi$
$62$	0	0
$63$	2.47982	0.312428
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.49906	0.549647	0.274824	−	0.961495i	$-0.411380\pi$
$67$	4.49906	0.549647	0.274824	+	0.961495i	$0.411380\pi$
$68$	0	0
$69$	7.54340	0.908118
$70$	0	0
$71$	1.13435	0.134623	0.0673115	−	0.997732i	$-0.478558\pi$
$71$	1.13435	0.134623	0.0673115	+	0.997732i	$0.478558\pi$
$72$	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
$77$	−8.88446	−1.01248
$78$	0	0
$79$	8.37954	0.942772	0.471386	−	0.881927i	$-0.343754\pi$
$79$	8.37954	0.942772	0.471386	+	0.881927i	$0.343754\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0965	1.65705	0.828527	−	0.559949i	$-0.189179\pi$
$83$	15.0965	1.65705	0.828527	+	0.559949i	$0.189179\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.11909	0.119979
$88$	0	0
$89$	−5.44941	−0.577636	−0.288818	−	0.957384i	$-0.593262\pi$
$89$	−5.44941	−0.577636	−0.288818	+	0.957384i	$0.593262\pi$
$90$	0	0
$91$	−13.2926	−1.39344
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−5.33032	−0.541212	−0.270606	−	0.962690i	$-0.587224\pi$
$97$	−5.33032	−0.541212	−0.270606	+	0.962690i	$0.587224\pi$
$98$	0	0
$99$	−3.58271	−0.360076
$100$	0	0
$101$	6.89087	0.685667	0.342834	−	0.939396i	$-0.388613\pi$
$101$	6.89087	0.685667	0.342834	+	0.939396i	$0.388613\pi$
$102$	0	0
$103$	12.4632	1.22803	0.614017	−	0.789292i	$-0.289552\pi$
$103$	12.4632	1.22803	0.614017	+	0.789292i	$0.289552\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.37997	−0.520102	−0.260051	−	0.965595i	$-0.583739\pi$
$107$	−5.37997	−0.520102	−0.260051	+	0.965595i	$0.583739\pi$
$108$	0	0
$109$	−2.62810	−0.251726	−0.125863	−	0.992048i	$-0.540170\pi$
$109$	−2.62810	−0.251726	−0.125863	+	0.992048i	$0.540170\pi$
$110$	0	0
$111$	10.2098	0.969067
$112$	0	0
$113$	2.31596	0.217868	0.108934	−	0.994049i	$-0.465256\pi$
$113$	2.31596	0.217868	0.108934	+	0.994049i	$0.465256\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.36030	−0.495560
$118$	0	0
$119$	−8.51830	−0.780871
$120$	0	0
$121$	1.83579	0.166890
$122$	0	0
$123$	6.88049	0.620393
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.57338	0.849500	0.424750	−	0.905311i	$-0.360362\pi$
$127$	9.57338	0.849500	0.424750	+	0.905311i	$0.360362\pi$
$128$	0	0
$129$	1.70222	0.149872
$130$	0	0
$131$	13.4665	1.17658	0.588289	−	0.808651i	$-0.299802\pi$
$131$	13.4665	1.17658	0.588289	+	0.808651i	$0.299802\pi$
$132$	0	0
$133$	17.6237	1.52817
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.0946	−1.28962	−0.644811	−	0.764342i	$-0.723064\pi$
$137$	−15.0946	−1.28962	−0.644811	+	0.764342i	$0.723064\pi$
$138$	0	0
$139$	−13.7232	−1.16398	−0.581992	−	0.813194i	$-0.697727\pi$
$139$	−13.7232	−1.16398	−0.581992	+	0.813194i	$0.697727\pi$
$140$	0	0
$141$	−9.03212	−0.760641
$142$	0	0
$143$	19.2044	1.60595
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.850505	−0.0701485
$148$	0	0
$149$	7.30296	0.598282	0.299141	−	0.954209i	$-0.403300\pi$
$149$	7.30296	0.598282	0.299141	+	0.954209i	$0.403300\pi$
$150$	0	0
$151$	13.8116	1.12397	0.561984	−	0.827148i	$-0.310038\pi$
$151$	13.8116	1.12397	0.561984	+	0.827148i	$0.310038\pi$
$152$	0	0
$153$	−3.43505	−0.277707
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.5701	0.843582	0.421791	−	0.906693i	$-0.361402\pi$
$157$	10.5701	0.843582	0.421791	+	0.906693i	$0.361402\pi$
$158$	0	0
$159$	−10.2290	−0.811212
$160$	0	0
$161$	18.7062	1.47426
$162$	0	0
$163$	13.9297	1.09106	0.545529	−	0.838092i	$-0.316329\pi$
$163$	13.9297	1.09106	0.545529	+	0.838092i	$0.316329\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.6972	1.52422	0.762108	−	0.647450i	$-0.224164\pi$
$167$	19.6972	1.52422	0.762108	+	0.647450i	$0.224164\pi$
$168$	0	0
$169$	15.7329	1.21022
$170$	0	0
$171$	7.10686	0.543475
$172$	0	0
$173$	−14.7782	−1.12357	−0.561785	−	0.827283i	$-0.689885\pi$
$173$	−14.7782	−1.12357	−0.561785	+	0.827283i	$0.689885\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.287039	0.0215752
$178$	0	0
$179$	13.6444	1.01983	0.509914	−	0.860225i	$-0.329677\pi$
$179$	13.6444	1.01983	0.509914	+	0.860225i	$0.329677\pi$
$180$	0	0
$181$	5.56002	0.413273	0.206637	−	0.978418i	$-0.433748\pi$
$181$	5.56002	0.413273	0.206637	+	0.978418i	$0.433748\pi$
$182$	0	0
$183$	−5.12610	−0.378933
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.3068	0.899961
$188$	0	0
$189$	2.47982	0.180380
$190$	0	0
$191$	18.9115	1.36839	0.684196	−	0.729298i	$-0.260153\pi$
$191$	18.9115	1.36839	0.684196	+	0.729298i	$0.260153\pi$
$192$	0	0
$193$	7.49800	0.539718	0.269859	−	0.962900i	$-0.413023\pi$
$193$	7.49800	0.539718	0.269859	+	0.962900i	$0.413023\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.09293	−0.362856	−0.181428	−	0.983404i	$-0.558072\pi$
$197$	−5.09293	−0.362856	−0.181428	+	0.983404i	$0.558072\pi$
$198$	0	0
$199$	−8.72541	−0.618528	−0.309264	−	0.950976i	$-0.600083\pi$
$199$	−8.72541	−0.618528	−0.309264	+	0.950976i	$0.600083\pi$
$200$	0	0
$201$	4.49906	0.317339
$202$	0	0
$203$	2.77513	0.194776
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.54340	0.524302
$208$	0	0
$209$	−25.4618	−1.76123
$210$	0	0
$211$	24.5096	1.68731	0.843656	−	0.536884i	$-0.180399\pi$
$211$	24.5096	1.68731	0.843656	+	0.536884i	$0.180399\pi$
$212$	0	0
$213$	1.13435	0.0777246
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.47982	−0.168341
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	18.4129	1.23859
$222$	0	0
$223$	10.8313	0.725315	0.362658	−	0.931922i	$-0.381869\pi$
$223$	10.8313	0.725315	0.362658	+	0.931922i	$0.381869\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.38739	0.0920844	0.0460422	−	0.998939i	$-0.485339\pi$
$227$	1.38739	0.0920844	0.0460422	+	0.998939i	$0.485339\pi$
$228$	0	0
$229$	14.3487	0.948186	0.474093	−	0.880475i	$-0.342776\pi$
$229$	14.3487	0.948186	0.474093	+	0.880475i	$0.342776\pi$
$230$	0	0
$231$	−8.88446	−0.584555
$232$	0	0
$233$	−2.99894	−0.196467	−0.0982337	−	0.995163i	$-0.531319\pi$
$233$	−2.99894	−0.196467	−0.0982337	+	0.995163i	$0.531319\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.37954	0.544310
$238$	0	0
$239$	−17.1886	−1.11184	−0.555920	−	0.831236i	$-0.687634\pi$
$239$	−17.1886	−1.11184	−0.555920	+	0.831236i	$0.687634\pi$
$240$	0	0
$241$	18.5650	1.19588	0.597939	−	0.801541i	$-0.295986\pi$
$241$	18.5650	1.19588	0.597939	+	0.801541i	$0.295986\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−38.0949	−2.42392
$248$	0	0
$249$	15.0965	0.956701
$250$	0	0
$251$	−3.86281	−0.243818	−0.121909	−	0.992541i	$-0.538902\pi$
$251$	−3.86281	−0.243818	−0.121909	+	0.992541i	$0.538902\pi$
$252$	0	0
$253$	−27.0258	−1.69910
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.5624	−0.658865	−0.329433	−	0.944179i	$-0.606857\pi$
$257$	−10.5624	−0.658865	−0.329433	+	0.944179i	$0.606857\pi$
$258$	0	0
$259$	25.3183	1.57320
$260$	0	0
$261$	1.11909	0.0692697
$262$	0	0
$263$	−7.01960	−0.432847	−0.216423	−	0.976300i	$-0.569439\pi$
$263$	−7.01960	−0.432847	−0.216423	+	0.976300i	$0.569439\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.44941	−0.333498
$268$	0	0
$269$	11.9415	0.728088	0.364044	−	0.931382i	$-0.381396\pi$
$269$	11.9415	0.728088	0.364044	+	0.931382i	$0.381396\pi$
$270$	0	0
$271$	27.1483	1.64914	0.824570	−	0.565760i	$-0.191417\pi$
$271$	27.1483	1.64914	0.824570	+	0.565760i	$0.191417\pi$
$272$	0	0
$273$	−13.2926	−0.804503
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−9.77189	−0.587136	−0.293568	−	0.955938i	$-0.594843\pi$
$277$	−9.77189	−0.587136	−0.293568	+	0.955938i	$0.594843\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−5.03657	−0.300457	−0.150228	−	0.988651i	$-0.548001\pi$
$281$	−5.03657	−0.300457	−0.150228	+	0.988651i	$0.548001\pi$
$282$	0	0
$283$	5.80255	0.344926	0.172463	−	0.985016i	$-0.444828\pi$
$283$	5.80255	0.344926	0.172463	+	0.985016i	$0.444828\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	17.0623	1.00716
$288$	0	0
$289$	−5.20042	−0.305907
$290$	0	0
$291$	−5.33032	−0.312469
$292$	0	0
$293$	−12.3558	−0.721834	−0.360917	−	0.932598i	$-0.617536\pi$
$293$	−12.3558	−0.721834	−0.360917	+	0.932598i	$0.617536\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.58271	−0.207890
$298$	0	0
$299$	−40.4349	−2.33841
$300$	0	0
$301$	4.22120	0.243306
$302$	0	0
$303$	6.89087	0.395870
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−32.4752	−1.85346	−0.926728	−	0.375733i	$-0.877391\pi$
$307$	−32.4752	−1.85346	−0.926728	+	0.375733i	$0.877391\pi$
$308$	0	0
$309$	12.4632	0.709006
$310$	0	0
$311$	31.2503	1.77204	0.886020	−	0.463647i	$-0.153459\pi$
$311$	31.2503	1.77204	0.886020	+	0.463647i	$0.153459\pi$
$312$	0	0
$313$	18.6144	1.05215	0.526074	−	0.850439i	$-0.323664\pi$
$313$	18.6144	1.05215	0.526074	+	0.850439i	$0.323664\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.2684	−0.801392	−0.400696	−	0.916211i	$-0.631232\pi$
$317$	−14.2684	−0.801392	−0.400696	+	0.916211i	$0.631232\pi$
$318$	0	0
$319$	−4.00936	−0.224481
$320$	0	0
$321$	−5.37997	−0.300281
$322$	0	0
$323$	−24.4124	−1.35834
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.62810	−0.145334
$328$	0	0
$329$	−22.3980	−1.23484
$330$	0	0
$331$	−9.50894	−0.522659	−0.261329	−	0.965250i	$-0.584161\pi$
$331$	−9.50894	−0.522659	−0.261329	+	0.965250i	$0.584161\pi$
$332$	0	0
$333$	10.2098	0.559491
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10.0040	−0.544951	−0.272476	−	0.962163i	$-0.587842\pi$
$337$	−10.0040	−0.544951	−0.272476	+	0.962163i	$0.587842\pi$
$338$	0	0
$339$	2.31596	0.125786
$340$	0	0
$341$	3.58271	0.194014
$342$	0	0
$343$	−19.4678	−1.05116
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.12454	−0.221417	−0.110709	−	0.993853i	$-0.535312\pi$
$347$	−4.12454	−0.221417	−0.110709	+	0.993853i	$0.535312\pi$
$348$	0	0
$349$	12.7217	0.680975	0.340487	−	0.940249i	$-0.389408\pi$
$349$	12.7217	0.680975	0.340487	+	0.940249i	$0.389408\pi$
$350$	0	0
$351$	−5.36030	−0.286112
$352$	0	0
$353$	22.3828	1.19131	0.595657	−	0.803239i	$-0.296892\pi$
$353$	22.3828	1.19131	0.595657	+	0.803239i	$0.296892\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−8.51830	−0.450836
$358$	0	0
$359$	10.9766	0.579323	0.289662	−	0.957129i	$-0.406457\pi$
$359$	10.9766	0.579323	0.289662	+	0.957129i	$0.406457\pi$
$360$	0	0
$361$	31.5075	1.65829
$362$	0	0
$363$	1.83579	0.0963541
$364$	0	0
$365$	0	0
$366$	0	0
$367$	13.4003	0.699490	0.349745	−	0.936845i	$-0.386268\pi$
$367$	13.4003	0.699490	0.349745	+	0.936845i	$0.386268\pi$
$368$	0	0
$369$	6.88049	0.358184
$370$	0	0
$371$	−25.3660	−1.31694
$372$	0	0
$373$	−1.36752	−0.0708076	−0.0354038	−	0.999373i	$-0.511272\pi$
$373$	−1.36752	−0.0708076	−0.0354038	+	0.999373i	$0.511272\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.99864	−0.308946
$378$	0	0
$379$	29.5243	1.51656	0.758282	−	0.651927i	$-0.226039\pi$
$379$	29.5243	1.51656	0.758282	+	0.651927i	$0.226039\pi$
$380$	0	0
$381$	9.57338	0.490459
$382$	0	0
$383$	−18.3471	−0.937494	−0.468747	−	0.883332i	$-0.655294\pi$
$383$	−18.3471	−0.937494	−0.468747	+	0.883332i	$0.655294\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.70222	0.0865288
$388$	0	0
$389$	5.16835	0.262046	0.131023	−	0.991379i	$-0.458174\pi$
$389$	5.16835	0.262046	0.131023	+	0.991379i	$0.458174\pi$
$390$	0	0
$391$	−25.9120	−1.31042
$392$	0	0
$393$	13.4665	0.679297
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−34.9095	−1.75206	−0.876030	−	0.482257i	$-0.839817\pi$
$397$	−34.9095	−1.75206	−0.876030	+	0.482257i	$0.839817\pi$
$398$	0	0
$399$	17.6237	0.882290
$400$	0	0
$401$	−6.66739	−0.332954	−0.166477	−	0.986045i	$-0.553239\pi$
$401$	−6.66739	−0.332954	−0.166477	+	0.986045i	$0.553239\pi$
$402$	0	0
$403$	5.36030	0.267016
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−36.5786	−1.81313
$408$	0	0
$409$	−6.58965	−0.325837	−0.162919	−	0.986639i	$-0.552091\pi$
$409$	−6.58965	−0.325837	−0.162919	+	0.986639i	$0.552091\pi$
$410$	0	0
$411$	−15.0946	−0.744564
$412$	0	0
$413$	0.711804	0.0350256
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−13.7232	−0.672027
$418$	0	0
$419$	18.5844	0.907906	0.453953	−	0.891026i	$-0.350013\pi$
$419$	18.5844	0.907906	0.453953	+	0.891026i	$0.350013\pi$
$420$	0	0
$421$	25.3577	1.23586	0.617930	−	0.786233i	$-0.287971\pi$
$421$	25.3577	1.23586	0.617930	+	0.786233i	$0.287971\pi$
$422$	0	0
$423$	−9.03212	−0.439157
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−12.7118	−0.615167
$428$	0	0
$429$	19.2044	0.927197
$430$	0	0
$431$	−27.4346	−1.32148	−0.660739	−	0.750616i	$-0.729757\pi$
$431$	−27.4346	−1.32148	−0.660739	+	0.750616i	$0.729757\pi$
$432$	0	0
$433$	17.6388	0.847668	0.423834	−	0.905740i	$-0.360684\pi$
$433$	17.6388	0.847668	0.423834	+	0.905740i	$0.360684\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	53.6099	2.56451
$438$	0	0
$439$	−26.2548	−1.25307	−0.626537	−	0.779391i	$-0.715528\pi$
$439$	−26.2548	−1.25307	−0.626537	+	0.779391i	$0.715528\pi$
$440$	0	0
$441$	−0.850505	−0.0405002
$442$	0	0
$443$	33.3813	1.58599	0.792996	−	0.609227i	$-0.208520\pi$
$443$	33.3813	1.58599	0.792996	+	0.609227i	$0.208520\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.30296	0.345418
$448$	0	0
$449$	−8.51712	−0.401948	−0.200974	−	0.979597i	$-0.564411\pi$
$449$	−8.51712	−0.401948	−0.200974	+	0.979597i	$0.564411\pi$
$450$	0	0
$451$	−24.6508	−1.16076
$452$	0	0
$453$	13.8116	0.648923
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.78489	0.364162	0.182081	−	0.983284i	$-0.441717\pi$
$457$	7.78489	0.364162	0.182081	+	0.983284i	$0.441717\pi$
$458$	0	0
$459$	−3.43505	−0.160334
$460$	0	0
$461$	−7.10254	−0.330798	−0.165399	−	0.986227i	$-0.552891\pi$
$461$	−7.10254	−0.330798	−0.165399	+	0.986227i	$0.552891\pi$
$462$	0	0
$463$	−2.46131	−0.114387	−0.0571934	−	0.998363i	$-0.518215\pi$
$463$	−2.46131	−0.114387	−0.0571934	+	0.998363i	$0.518215\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.6917	−1.51279	−0.756396	−	0.654114i	$-0.773042\pi$
$467$	−32.6917	−1.51279	−0.756396	+	0.654114i	$0.773042\pi$
$468$	0	0
$469$	11.1568	0.515175
$470$	0	0
$471$	10.5701	0.487043
$472$	0	0
$473$	−6.09856	−0.280412
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.2290	−0.468353
$478$	0	0
$479$	17.9416	0.819771	0.409885	−	0.912137i	$-0.365569\pi$
$479$	17.9416	0.819771	0.409885	+	0.912137i	$0.365569\pi$
$480$	0	0
$481$	−54.7274	−2.49535
$482$	0	0
$483$	18.7062	0.851164
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−32.3158	−1.46437	−0.732184	−	0.681107i	$-0.761499\pi$
$487$	−32.3158	−1.46437	−0.732184	+	0.681107i	$0.761499\pi$
$488$	0	0
$489$	13.9297	0.629923
$490$	0	0
$491$	−2.68303	−0.121084	−0.0605418	−	0.998166i	$-0.519283\pi$
$491$	−2.68303	−0.121084	−0.0605418	+	0.998166i	$0.519283\pi$
$492$	0	0
$493$	−3.84412	−0.173130
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.81299	0.126180
$498$	0	0
$499$	−41.3317	−1.85026	−0.925131	−	0.379649i	$-0.876045\pi$
$499$	−41.3317	−1.85026	−0.925131	+	0.379649i	$0.876045\pi$
$500$	0	0
$501$	19.6972	0.880007
$502$	0	0
$503$	−34.6398	−1.54451	−0.772256	−	0.635311i	$-0.780872\pi$
$503$	−34.6398	−1.54451	−0.772256	+	0.635311i	$0.780872\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	15.7329	0.698721
$508$	0	0
$509$	−16.3452	−0.724490	−0.362245	−	0.932083i	$-0.617990\pi$
$509$	−16.3452	−0.724490	−0.362245	+	0.932083i	$0.617990\pi$
$510$	0	0
$511$	4.95963	0.219401
$512$	0	0
$513$	7.10686	0.313776
$514$	0	0
$515$	0	0
$516$	0	0
$517$	32.3594	1.42317
$518$	0	0
$519$	−14.7782	−0.648693
$520$	0	0
$521$	16.3951	0.718282	0.359141	−	0.933283i	$-0.383070\pi$
$521$	16.3951	0.718282	0.359141	+	0.933283i	$0.383070\pi$
$522$	0	0
$523$	−20.8314	−0.910896	−0.455448	−	0.890262i	$-0.650521\pi$
$523$	−20.8314	−0.910896	−0.455448	+	0.890262i	$0.650521\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.43505	0.149633
$528$	0	0
$529$	33.9028	1.47404
$530$	0	0
$531$	0.287039	0.0124564
$532$	0	0
$533$	−36.8815	−1.59751
$534$	0	0
$535$	0	0
$536$	0	0
$537$	13.6444	0.588798
$538$	0	0
$539$	3.04711	0.131248
$540$	0	0
$541$	−15.7734	−0.678153	−0.339077	−	0.940759i	$-0.610115\pi$
$541$	−15.7734	−0.678153	−0.339077	+	0.940759i	$0.610115\pi$
$542$	0	0
$543$	5.56002	0.238603
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.43593	−0.0613960	−0.0306980	−	0.999529i	$-0.509773\pi$
$547$	−1.43593	−0.0613960	−0.0306980	+	0.999529i	$0.509773\pi$
$548$	0	0
$549$	−5.12610	−0.218777
$550$	0	0
$551$	7.95319	0.338817
$552$	0	0
$553$	20.7797	0.883644
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.9536	−1.56577	−0.782887	−	0.622164i	$-0.786254\pi$
$557$	−36.9536	−1.56577	−0.782887	+	0.622164i	$0.786254\pi$
$558$	0	0
$559$	−9.12442	−0.385922
$560$	0	0
$561$	12.3068	0.519593
$562$	0	0
$563$	15.9105	0.670548	0.335274	−	0.942121i	$-0.391171\pi$
$563$	15.9105	0.670548	0.335274	+	0.942121i	$0.391171\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.47982	0.104143
$568$	0	0
$569$	14.5503	0.609980	0.304990	−	0.952356i	$-0.401347\pi$
$569$	14.5503	0.609980	0.304990	+	0.952356i	$0.401347\pi$
$570$	0	0
$571$	0.560199	0.0234436	0.0117218	−	0.999931i	$-0.496269\pi$
$571$	0.560199	0.0234436	0.0117218	+	0.999931i	$0.496269\pi$
$572$	0	0
$573$	18.9115	0.790041
$574$	0	0
$575$	0	0
$576$	0	0
$577$	32.1923	1.34018	0.670091	−	0.742279i	$-0.266255\pi$
$577$	32.1923	1.34018	0.670091	+	0.742279i	$0.266255\pi$
$578$	0	0
$579$	7.49800	0.311606
$580$	0	0
$581$	37.4365	1.55313
$582$	0	0
$583$	36.6475	1.51778
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.1990	0.586054	0.293027	−	0.956104i	$-0.405337\pi$
$587$	14.1990	0.586054	0.293027	+	0.956104i	$0.405337\pi$
$588$	0	0
$589$	−7.10686	−0.292833
$590$	0	0
$591$	−5.09293	−0.209495
$592$	0	0
$593$	33.4034	1.37172	0.685858	−	0.727736i	$-0.259427\pi$
$593$	33.4034	1.37172	0.685858	+	0.727736i	$0.259427\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.72541	−0.357107
$598$	0	0
$599$	40.4896	1.65436	0.827179	−	0.561938i	$-0.189944\pi$
$599$	40.4896	1.65436	0.827179	+	0.561938i	$0.189944\pi$
$600$	0	0
$601$	20.3045	0.828239	0.414119	−	0.910223i	$-0.364090\pi$
$601$	20.3045	0.828239	0.414119	+	0.910223i	$0.364090\pi$
$602$	0	0
$603$	4.49906	0.183216
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.5277	0.589659	0.294830	−	0.955550i	$-0.404737\pi$
$607$	14.5277	0.589659	0.294830	+	0.955550i	$0.404737\pi$
$608$	0	0
$609$	2.77513	0.112454
$610$	0	0
$611$	48.4149	1.95866
$612$	0	0
$613$	−25.7081	−1.03834	−0.519171	−	0.854671i	$-0.673759\pi$
$613$	−25.7081	−1.03834	−0.519171	+	0.854671i	$0.673759\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.56578	−0.103294	−0.0516472	−	0.998665i	$-0.516447\pi$
$617$	−2.56578	−0.103294	−0.0516472	+	0.998665i	$0.516447\pi$
$618$	0	0
$619$	8.92971	0.358915	0.179458	−	0.983766i	$-0.442566\pi$
$619$	8.92971	0.358915	0.179458	+	0.983766i	$0.442566\pi$
$620$	0	0
$621$	7.54340	0.302706
$622$	0	0
$623$	−13.5135	−0.541409
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−25.4618	−1.01685
$628$	0	0
$629$	−35.0710	−1.39837
$630$	0	0
$631$	39.9960	1.59222	0.796108	−	0.605154i	$-0.206888\pi$
$631$	39.9960	1.59222	0.796108	+	0.605154i	$0.206888\pi$
$632$	0	0
$633$	24.5096	0.974170
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.55897	0.180633
$638$	0	0
$639$	1.13435	0.0448743
$640$	0	0
$641$	16.5176	0.652407	0.326203	−	0.945300i	$-0.394231\pi$
$641$	16.5176	0.652407	0.326203	+	0.945300i	$0.394231\pi$
$642$	0	0
$643$	1.45524	0.0573891	0.0286946	−	0.999588i	$-0.490865\pi$
$643$	1.45524	0.0573891	0.0286946	+	0.999588i	$0.490865\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.6023	−1.12447	−0.562236	−	0.826977i	$-0.690059\pi$
$647$	−28.6023	−1.12447	−0.562236	+	0.826977i	$0.690059\pi$
$648$	0	0
$649$	−1.02838	−0.0403673
$650$	0	0
$651$	−2.47982	−0.0971917
$652$	0	0
$653$	−42.3069	−1.65560	−0.827799	−	0.561025i	$-0.810407\pi$
$653$	−42.3069	−1.65560	−0.827799	+	0.561025i	$0.810407\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	−23.1362	−0.901257	−0.450628	−	0.892712i	$-0.648800\pi$
$659$	−23.1362	−0.901257	−0.450628	+	0.892712i	$0.648800\pi$
$660$	0	0
$661$	31.9179	1.24146	0.620731	−	0.784024i	$-0.286836\pi$
$661$	31.9179	1.24146	0.620731	+	0.784024i	$0.286836\pi$
$662$	0	0
$663$	18.4129	0.715098
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.44171	0.326864
$668$	0	0
$669$	10.8313	0.418761
$670$	0	0
$671$	18.3653	0.708986
$672$	0	0
$673$	8.15065	0.314184	0.157092	−	0.987584i	$-0.449788\pi$
$673$	8.15065	0.314184	0.157092	+	0.987584i	$0.449788\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.8875	1.84047	0.920233	−	0.391372i	$-0.127999\pi$
$677$	47.8875	1.84047	0.920233	+	0.391372i	$0.127999\pi$
$678$	0	0
$679$	−13.2182	−0.507269
$680$	0	0
$681$	1.38739	0.0531649
$682$	0	0
$683$	32.3127	1.23641	0.618206	−	0.786016i	$-0.287860\pi$
$683$	32.3127	1.23641	0.618206	+	0.786016i	$0.287860\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14.3487	0.547435
$688$	0	0
$689$	54.8305	2.08888
$690$	0	0
$691$	−12.9290	−0.491841	−0.245921	−	0.969290i	$-0.579090\pi$
$691$	−12.9290	−0.491841	−0.245921	+	0.969290i	$0.579090\pi$
$692$	0	0
$693$	−8.88446	−0.337493
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−23.6348	−0.895233
$698$	0	0
$699$	−2.99894	−0.113431
$700$	0	0
$701$	−19.9800	−0.754636	−0.377318	−	0.926084i	$-0.623154\pi$
$701$	−19.9800	−0.754636	−0.377318	+	0.926084i	$0.623154\pi$
$702$	0	0
$703$	72.5593	2.73663
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.0881	0.642664
$708$	0	0
$709$	17.5867	0.660484	0.330242	−	0.943896i	$-0.392870\pi$
$709$	17.5867	0.660484	0.330242	+	0.943896i	$0.392870\pi$
$710$	0	0
$711$	8.37954	0.314257
$712$	0	0
$713$	−7.54340	−0.282502
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−17.1886	−0.641921
$718$	0	0
$719$	5.59060	0.208494	0.104247	−	0.994551i	$-0.466757\pi$
$719$	5.59060	0.208494	0.104247	+	0.994551i	$0.466757\pi$
$720$	0	0
$721$	30.9064	1.15102
$722$	0	0
$723$	18.5650	0.690441
$724$	0	0
$725$	0	0
$726$	0	0
$727$	34.2933	1.27187	0.635934	−	0.771744i	$-0.280615\pi$
$727$	34.2933	1.27187	0.635934	+	0.771744i	$0.280615\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.84722	−0.216267
$732$	0	0
$733$	25.3863	0.937663	0.468831	−	0.883288i	$-0.344675\pi$
$733$	25.3863	0.937663	0.468831	+	0.883288i	$0.344675\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.1188	−0.593744
$738$	0	0
$739$	26.0370	0.957787	0.478894	−	0.877873i	$-0.341038\pi$
$739$	26.0370	0.957787	0.478894	+	0.877873i	$0.341038\pi$
$740$	0	0
$741$	−38.0949	−1.39945
$742$	0	0
$743$	−19.0925	−0.700437	−0.350218	−	0.936668i	$-0.613893\pi$
$743$	−19.0925	−0.700437	−0.350218	+	0.936668i	$0.613893\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.0965	0.552351
$748$	0	0
$749$	−13.3413	−0.487482
$750$	0	0
$751$	−29.6559	−1.08216	−0.541079	−	0.840972i	$-0.681984\pi$
$751$	−29.6559	−1.08216	−0.541079	+	0.840972i	$0.681984\pi$
$752$	0	0
$753$	−3.86281	−0.140769
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13.5948	−0.494112	−0.247056	−	0.969001i	$-0.579463\pi$
$757$	−13.5948	−0.494112	−0.247056	+	0.969001i	$0.579463\pi$
$758$	0	0
$759$	−27.0258	−0.980974
$760$	0	0
$761$	16.9854	0.615719	0.307860	−	0.951432i	$-0.400387\pi$
$761$	16.9854	0.615719	0.307860	+	0.951432i	$0.400387\pi$
$762$	0	0
$763$	−6.51721	−0.235939
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.53862	−0.0555562
$768$	0	0
$769$	36.9376	1.33200	0.666002	−	0.745950i	$-0.268004\pi$
$769$	36.9376	1.33200	0.666002	+	0.745950i	$0.268004\pi$
$770$	0	0
$771$	−10.5624	−0.380396
$772$	0	0
$773$	−5.34165	−0.192126	−0.0960628	−	0.995375i	$-0.530625\pi$
$773$	−5.34165	−0.192126	−0.0960628	+	0.995375i	$0.530625\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	25.3183	0.908290
$778$	0	0
$779$	48.8987	1.75198
$780$	0	0
$781$	−4.06406	−0.145423
$782$	0	0
$783$	1.11909	0.0399929
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−51.6039	−1.83948	−0.919740	−	0.392528i	$-0.871601\pi$
$787$	−51.6039	−1.83948	−0.919740	+	0.392528i	$0.871601\pi$
$788$	0	0
$789$	−7.01960	−0.249904
$790$	0	0
$791$	5.74317	0.204204
$792$	0	0
$793$	27.4775	0.975754
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.6631	−1.01530	−0.507650	−	0.861563i	$-0.669486\pi$
$797$	−28.6631	−1.01530	−0.507650	+	0.861563i	$0.669486\pi$
$798$	0	0
$799$	31.0258	1.09761
$800$	0	0
$801$	−5.44941	−0.192545
$802$	0	0
$803$	−7.16541	−0.252862
$804$	0	0
$805$	0	0
$806$	0	0
$807$	11.9415	0.420362
$808$	0	0
$809$	23.6636	0.831969	0.415985	−	0.909372i	$-0.363437\pi$
$809$	23.6636	0.831969	0.415985	+	0.909372i	$0.363437\pi$
$810$	0	0
$811$	3.46617	0.121714	0.0608568	−	0.998147i	$-0.480617\pi$
$811$	3.46617	0.121714	0.0608568	+	0.998147i	$0.480617\pi$
$812$	0	0
$813$	27.1483	0.952131
$814$	0	0
$815$	0	0
$816$	0	0
$817$	12.0975	0.423236
$818$	0	0
$819$	−13.2926	−0.464480
$820$	0	0
$821$	−49.8638	−1.74026	−0.870129	−	0.492824i	$-0.835965\pi$
$821$	−49.8638	−1.74026	−0.870129	+	0.492824i	$0.835965\pi$
$822$	0	0
$823$	17.6381	0.614825	0.307412	−	0.951576i	$-0.400537\pi$
$823$	17.6381	0.614825	0.307412	+	0.951576i	$0.400537\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−56.5863	−1.96770	−0.983849	−	0.179001i	$-0.942714\pi$
$827$	−56.5863	−1.96770	−0.983849	+	0.179001i	$0.942714\pi$
$828$	0	0
$829$	−30.6884	−1.06585	−0.532926	−	0.846162i	$-0.678908\pi$
$829$	−30.6884	−1.06585	−0.532926	+	0.846162i	$0.678908\pi$
$830$	0	0
$831$	−9.77189	−0.338983
$832$	0	0
$833$	2.92153	0.101225
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	45.2923	1.56366	0.781831	−	0.623490i	$-0.214286\pi$
$839$	45.2923	1.56366	0.781831	+	0.623490i	$0.214286\pi$
$840$	0	0
$841$	−27.7476	−0.956815
$842$	0	0
$843$	−5.03657	−0.173469
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.55243	0.156423
$848$	0	0
$849$	5.80255	0.199143
$850$	0	0
$851$	77.0162	2.64008
$852$	0	0
$853$	−4.25135	−0.145563	−0.0727817	−	0.997348i	$-0.523188\pi$
$853$	−4.25135	−0.145563	−0.0727817	+	0.997348i	$0.523188\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.252234	0.00861615	0.00430807	−	0.999991i	$-0.498629\pi$
$857$	0.252234	0.00861615	0.00430807	+	0.999991i	$0.498629\pi$
$858$	0	0
$859$	−41.7841	−1.42566	−0.712828	−	0.701339i	$-0.752586\pi$
$859$	−41.7841	−1.42566	−0.712828	+	0.701339i	$0.752586\pi$
$860$	0	0
$861$	17.0623	0.581483
$862$	0	0
$863$	−32.9413	−1.12133	−0.560667	−	0.828042i	$-0.689455\pi$
$863$	−32.9413	−1.12133	−0.560667	+	0.828042i	$0.689455\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−5.20042	−0.176616
$868$	0	0
$869$	−30.0215	−1.01841
$870$	0	0
$871$	−24.1163	−0.817150
$872$	0	0
$873$	−5.33032	−0.180404
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.2855	−1.25904	−0.629520	−	0.776984i	$-0.716749\pi$
$877$	−37.2855	−1.25904	−0.629520	+	0.776984i	$0.716749\pi$
$878$	0	0
$879$	−12.3558	−0.416751
$880$	0	0
$881$	−35.4123	−1.19307	−0.596536	−	0.802587i	$-0.703457\pi$
$881$	−35.4123	−1.19307	−0.596536	+	0.802587i	$0.703457\pi$
$882$	0	0
$883$	−58.5860	−1.97158	−0.985789	−	0.167990i	$-0.946272\pi$
$883$	−58.5860	−1.97158	−0.985789	+	0.167990i	$0.946272\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.7748	−0.932588	−0.466294	−	0.884630i	$-0.654411\pi$
$887$	−27.7748	−0.932588	−0.466294	+	0.884630i	$0.654411\pi$
$888$	0	0
$889$	23.7402	0.796222
$890$	0	0
$891$	−3.58271	−0.120025
$892$	0	0
$893$	−64.1900	−2.14804
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−40.4349	−1.35008
$898$	0	0
$899$	−1.11909	−0.0373236
$900$	0	0
$901$	35.1371	1.17059
$902$	0	0
$903$	4.22120	0.140473
$904$	0	0
$905$	0	0
$906$	0	0
$907$	36.6213	1.21599	0.607996	−	0.793940i	$-0.291974\pi$
$907$	36.6213	1.21599	0.607996	+	0.793940i	$0.291974\pi$
$908$	0	0
$909$	6.89087	0.228556
$910$	0	0
$911$	21.6511	0.717334	0.358667	−	0.933466i	$-0.383231\pi$
$911$	21.6511	0.717334	0.358667	+	0.933466i	$0.383231\pi$
$912$	0	0
$913$	−54.0863	−1.78999
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.3946	1.10279
$918$	0	0
$919$	25.0948	0.827800	0.413900	−	0.910322i	$-0.364166\pi$
$919$	25.0948	0.827800	0.413900	+	0.910322i	$0.364166\pi$
$920$	0	0
$921$	−32.4752	−1.07009
$922$	0	0
$923$	−6.08048	−0.200141
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.4632	0.409345
$928$	0	0
$929$	12.1541	0.398762	0.199381	−	0.979922i	$-0.436107\pi$
$929$	12.1541	0.398762	0.199381	+	0.979922i	$0.436107\pi$
$930$	0	0
$931$	−6.04442	−0.198098
$932$	0	0
$933$	31.2503	1.02309
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−53.4724	−1.74687	−0.873433	−	0.486944i	$-0.838112\pi$
$937$	−53.4724	−1.74687	−0.873433	+	0.486944i	$0.838112\pi$
$938$	0	0
$939$	18.6144	0.607458
$940$	0	0
$941$	−9.01864	−0.293999	−0.147000	−	0.989137i	$-0.546962\pi$
$941$	−9.01864	−0.293999	−0.147000	+	0.989137i	$0.546962\pi$
$942$	0	0
$943$	51.9022	1.69017
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.4863	−0.763201	−0.381600	−	0.924327i	$-0.624627\pi$
$947$	−23.4863	−0.763201	−0.381600	+	0.924327i	$0.624627\pi$
$948$	0	0
$949$	−10.7206	−0.348006
$950$	0	0
$951$	−14.2684	−0.462684
$952$	0	0
$953$	−20.6115	−0.667671	−0.333836	−	0.942631i	$-0.608343\pi$
$953$	−20.6115	−0.667671	−0.333836	+	0.942631i	$0.608343\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.00936	−0.129604
$958$	0	0
$959$	−37.4320	−1.20874
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−5.37997	−0.173367
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.4748	0.915686	0.457843	−	0.889033i	$-0.348622\pi$
$967$	28.4748	0.915686	0.457843	+	0.889033i	$0.348622\pi$
$968$	0	0
$969$	−24.4124	−0.784241
$970$	0	0
$971$	8.62046	0.276644	0.138322	−	0.990387i	$-0.455829\pi$
$971$	8.62046	0.276644	0.138322	+	0.990387i	$0.455829\pi$
$972$	0	0
$973$	−34.0310	−1.09098
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40.7433	−1.30349	−0.651747	−	0.758436i	$-0.725964\pi$
$977$	−40.7433	−1.30349	−0.651747	+	0.758436i	$0.725964\pi$
$978$	0	0
$979$	19.5236	0.623978
$980$	0	0
$981$	−2.62810	−0.0839088
$982$	0	0
$983$	−35.8588	−1.14372	−0.571859	−	0.820352i	$-0.693778\pi$
$983$	−35.8588	−1.14372	−0.571859	+	0.820352i	$0.693778\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−22.3980	−0.712936
$988$	0	0
$989$	12.8405	0.408305
$990$	0	0
$991$	51.4647	1.63483	0.817415	−	0.576049i	$-0.195406\pi$
$991$	51.4647	1.63483	0.817415	+	0.576049i	$0.195406\pi$
$992$	0	0
$993$	−9.50894	−0.301757
$994$	0	0
$995$	0	0
$996$	0	0
$997$	58.5164	1.85323	0.926616	−	0.376008i	$-0.122703\pi$
$997$	58.5164	1.85323	0.926616	+	0.376008i	$0.122703\pi$
$998$	0	0
$999$	10.2098	0.323022

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.a.ba.1.4	yes	6
5.2	odd	4		9300.2.g.t.3349.4		12
5.3	odd	4		9300.2.g.t.3349.9		12
5.4	even	2		9300.2.a.y.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9300.2.a.y.1.3	✓	6	5.4	even	2
9300.2.a.ba.1.4	yes	6	1.1	even	1	trivial
9300.2.g.t.3349.4		12	5.2	odd	4
9300.2.g.t.3349.9		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.47982 q^{7} +1.00000 q^{9} -3.58271 q^{11} -5.36030 q^{13} -3.43505 q^{17} +7.10686 q^{19} +2.47982 q^{21} +7.54340 q^{23} +1.00000 q^{27} +1.11909 q^{29} -1.00000 q^{31} -3.58271 q^{33} +10.2098 q^{37} -5.36030 q^{39} +6.88049 q^{41} +1.70222 q^{43} -9.03212 q^{47} -0.850505 q^{49} -3.43505 q^{51} -10.2290 q^{53} +7.10686 q^{57} +0.287039 q^{59} -5.12610 q^{61} +2.47982 q^{63} +4.49906 q^{67} +7.54340 q^{69} +1.13435 q^{71} +2.00000 q^{73} -8.88446 q^{77} +8.37954 q^{79} +1.00000 q^{81} +15.0965 q^{83} +1.11909 q^{87} -5.44941 q^{89} -13.2926 q^{91} -1.00000 q^{93} -5.33032 q^{97} -3.58271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 4 q^{7} + 6 q^{9} - 3 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 4 q^{21} + 5 q^{23} + 6 q^{27} - 6 q^{31} - 3 q^{33} + 8 q^{37} + 2 q^{39} + 18 q^{41} + 15 q^{43} + q^{47} + 14 q^{49} + 4 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) \) 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

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