LMFDB - Embedded newform 9200.2.a.cx.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9200,2,Mod(1,9200)] 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("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-4,0,0,0,-9,0,10,0,-2] 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:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.143376304.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 22x^{2} - 6x - 1 $ x^6 - 12x^4 + 22x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.420790$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73961 q^{3} -3.32224 q^{7} +0.0262434 q^{9} -5.77103 q^{11} +1.10197 q^{13} +0.893847 q^{17} -2.42839 q^{19} +5.77940 q^{21} -1.00000 q^{23} +5.17318 q^{27} +4.11268 q^{29} +9.54624 q^{31} +10.0393 q^{33} -7.69904 q^{37} -1.91700 q^{39} +0.00418347 q^{41} +9.97045 q^{43} -10.0079 q^{47} +4.03726 q^{49} -1.55495 q^{51} +6.25169 q^{53} +4.22445 q^{57} +10.7764 q^{59} +10.5929 q^{61} -0.0871868 q^{63} -10.9529 q^{67} +1.73961 q^{69} +12.9170 q^{71} +1.89943 q^{73} +19.1727 q^{77} +0.216085 q^{79} -9.07804 q^{81} +5.38967 q^{83} -7.15446 q^{87} +6.00657 q^{89} -3.66100 q^{91} -16.6067 q^{93} +2.08104 q^{97} -0.151451 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 9 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} + 5 q^{17} - 4 q^{19} - 6 q^{23} - 22 q^{27} + 5 q^{29} - 9 q^{31} - 10 q^{33} + 21 q^{37} + 8 q^{39} - q^{41} - 16 q^{43} - 16 q^{47} + 19 q^{49}+ \cdots + 16 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 9 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 + 5 * q^17 - 4 * q^19 - 6 * q^23 - 22 * q^27 + 5 * q^29 - 9 * q^31 - 10 * q^33 + 21 * q^37 + 8 * q^39 - q^41 - 16 * q^43 - 16 * q^47 + 19 * q^49 + 12 * q^51 - q^53 + 12 * q^57 + 11 * q^59 - 4 * q^61 - 19 * q^63 - 25 * q^67 + 4 * q^69 + 17 * q^71 - 14 * q^73 + 20 * q^77 - 10 * q^79 + 14 * q^81 - 21 * q^83 - 64 * q^87 - 24 * q^89 + 4 * q^91 + 4 * q^97 + 16 * 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
$3$	−1.73961	−1.00436	−0.502182	−	0.864762i	$-0.667469\pi$
$3$	−1.73961	−1.00436	−0.502182	+	0.864762i	$0.667469\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.32224	−1.25569	−0.627844	−	0.778339i	$-0.716062\pi$
$7$	−3.32224	−1.25569	−0.627844	+	0.778339i	$0.716062\pi$
$8$	0	0
$9$	0.0262434	0.00874780
$10$	0	0
$11$	−5.77103	−1.74003	−0.870016	−	0.493024i	$-0.835891\pi$
$11$	−5.77103	−1.74003	−0.870016	+	0.493024i	$0.835891\pi$
$12$	0	0
$13$	1.10197	0.305631	0.152816	−	0.988255i	$-0.451166\pi$
$13$	1.10197	0.305631	0.152816	+	0.988255i	$0.451166\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.893847	0.216790	0.108395	−	0.994108i	$-0.465429\pi$
$17$	0.893847	0.216790	0.108395	+	0.994108i	$0.465429\pi$
$18$	0	0
$19$	−2.42839	−0.557111	−0.278555	−	0.960420i	$-0.589856\pi$
$19$	−2.42839	−0.557111	−0.278555	+	0.960420i	$0.589856\pi$
$20$	0	0
$21$	5.77940	1.26117
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.17318	0.995578
$28$	0	0
$29$	4.11268	0.763705	0.381853	−	0.924223i	$-0.375286\pi$
$29$	4.11268	0.763705	0.381853	+	0.924223i	$0.375286\pi$
$30$	0	0
$31$	9.54624	1.71456	0.857278	−	0.514854i	$-0.172154\pi$
$31$	9.54624	1.71456	0.857278	+	0.514854i	$0.172154\pi$
$32$	0	0
$33$	10.0393	1.74763
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.69904	−1.26572	−0.632858	−	0.774268i	$-0.718118\pi$
$37$	−7.69904	−1.26572	−0.632858	+	0.774268i	$0.718118\pi$
$38$	0	0
$39$	−1.91700	−0.306965
$40$	0	0
$41$	0.00418347	0.000653348 0	0.000326674	−	1.00000i	$-0.499896\pi$
$41$	0.00418347	0.000653348 0	0.000326674		1.00000i	$0.499896\pi$
$42$	0	0
$43$	9.97045	1.52048	0.760240	−	0.649643i	$-0.225081\pi$
$43$	9.97045	1.52048	0.760240	+	0.649643i	$0.225081\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.0079	−1.45981	−0.729903	−	0.683551i	$-0.760435\pi$
$47$	−10.0079	−1.45981	−0.729903	+	0.683551i	$0.760435\pi$
$48$	0	0
$49$	4.03726	0.576751
$50$	0	0
$51$	−1.55495	−0.217736
$52$	0	0
$53$	6.25169	0.858735	0.429368	−	0.903130i	$-0.358736\pi$
$53$	6.25169	0.858735	0.429368	+	0.903130i	$0.358736\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.22445	0.559542
$58$	0	0
$59$	10.7764	1.40297	0.701486	−	0.712684i	$-0.252520\pi$
$59$	10.7764	1.40297	0.701486	+	0.712684i	$0.252520\pi$
$60$	0	0
$61$	10.5929	1.35628	0.678140	−	0.734932i	$-0.262786\pi$
$61$	10.5929	1.35628	0.678140	+	0.734932i	$0.262786\pi$
$62$	0	0
$63$	−0.0871868	−0.0109845
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.9529	−1.33811	−0.669055	−	0.743213i	$-0.733301\pi$
$67$	−10.9529	−1.33811	−0.669055	+	0.743213i	$0.733301\pi$
$68$	0	0
$69$	1.73961	0.209424
$70$	0	0
$71$	12.9170	1.53296	0.766481	−	0.642267i	$-0.222006\pi$
$71$	12.9170	1.53296	0.766481	+	0.642267i	$0.222006\pi$
$72$	0	0
$73$	1.89943	0.222311	0.111156	−	0.993803i	$-0.464545\pi$
$73$	1.89943	0.222311	0.111156	+	0.993803i	$0.464545\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.1727	2.18494
$78$	0	0
$79$	0.216085	0.0243114	0.0121557	−	0.999926i	$-0.496131\pi$
$79$	0.216085	0.0243114	0.0121557	+	0.999926i	$0.496131\pi$
$80$	0	0
$81$	−9.07804	−1.00867
$82$	0	0
$83$	5.38967	0.591593	0.295796	−	0.955251i	$-0.404415\pi$
$83$	5.38967	0.591593	0.295796	+	0.955251i	$0.404415\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.15446	−0.767038
$88$	0	0
$89$	6.00657	0.636696	0.318348	−	0.947974i	$-0.396872\pi$
$89$	6.00657	0.636696	0.318348	+	0.947974i	$0.396872\pi$
$90$	0	0
$91$	−3.66100	−0.383777
$92$	0	0
$93$	−16.6067	−1.72204
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.08104	0.211297	0.105649	−	0.994404i	$-0.466308\pi$
$97$	2.08104	0.211297	0.105649	+	0.994404i	$0.466308\pi$
$98$	0	0
$99$	−0.151451	−0.0152214
$100$	0	0
$101$	−13.1576	−1.30923	−0.654617	−	0.755961i	$-0.727170\pi$
$101$	−13.1576	−1.30923	−0.654617	+	0.755961i	$0.727170\pi$
$102$	0	0
$103$	−18.0956	−1.78301	−0.891507	−	0.453008i	$-0.850351\pi$
$103$	−18.0956	−1.78301	−0.891507	+	0.453008i	$0.850351\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.97263	−0.674069	−0.337035	−	0.941492i	$-0.609424\pi$
$107$	−6.97263	−0.674069	−0.337035	+	0.941492i	$0.609424\pi$
$108$	0	0
$109$	10.2873	0.985344	0.492672	−	0.870215i	$-0.336020\pi$
$109$	10.2873	0.985344	0.492672	+	0.870215i	$0.336020\pi$
$110$	0	0
$111$	13.3933	1.27124
$112$	0	0
$113$	−5.53454	−0.520646	−0.260323	−	0.965522i	$-0.583829\pi$
$113$	−5.53454	−0.520646	−0.260323	+	0.965522i	$0.583829\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.0289194	0.00267360
$118$	0	0
$119$	−2.96957	−0.272220
$120$	0	0
$121$	22.3048	2.02771
$122$	0	0
$123$	−0.00727760	−0.000656199 0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.1580	−0.990110	−0.495055	−	0.868862i	$-0.664852\pi$
$127$	−11.1580	−0.990110	−0.495055	+	0.868862i	$0.664852\pi$
$128$	0	0
$129$	−17.3447	−1.52712
$130$	0	0
$131$	15.0421	1.31423	0.657117	−	0.753789i	$-0.271776\pi$
$131$	15.0421	1.31423	0.657117	+	0.753789i	$0.271776\pi$
$132$	0	0
$133$	8.06769	0.699557
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.21231	0.189010	0.0945050	−	0.995524i	$-0.469873\pi$
$137$	2.21231	0.189010	0.0945050	+	0.995524i	$0.469873\pi$
$138$	0	0
$139$	6.53729	0.554486	0.277243	−	0.960800i	$-0.410579\pi$
$139$	6.53729	0.554486	0.277243	+	0.960800i	$0.410579\pi$
$140$	0	0
$141$	17.4099	1.46618
$142$	0	0
$143$	−6.35950	−0.531808
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.02326	−0.579269
$148$	0	0
$149$	−11.1844	−0.916263	−0.458131	−	0.888884i	$-0.651481\pi$
$149$	−11.1844	−0.916263	−0.458131	+	0.888884i	$0.651481\pi$
$150$	0	0
$151$	1.29176	0.105122	0.0525610	−	0.998618i	$-0.483262\pi$
$151$	1.29176	0.105122	0.0525610	+	0.998618i	$0.483262\pi$
$152$	0	0
$153$	0.0234576	0.00189643
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.0465	0.881609	0.440805	−	0.897603i	$-0.354693\pi$
$157$	11.0465	0.881609	0.440805	+	0.897603i	$0.354693\pi$
$158$	0	0
$159$	−10.8755	−0.862483
$160$	0	0
$161$	3.32224	0.261829
$162$	0	0
$163$	−5.04265	−0.394971	−0.197485	−	0.980306i	$-0.563277\pi$
$163$	−5.04265	−0.394971	−0.197485	+	0.980306i	$0.563277\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.20759	−0.0934465	−0.0467232	−	0.998908i	$-0.514878\pi$
$167$	−1.20759	−0.0934465	−0.0467232	+	0.998908i	$0.514878\pi$
$168$	0	0
$169$	−11.7857	−0.906590
$170$	0	0
$171$	−0.0637292	−0.00487349
$172$	0	0
$173$	−15.7392	−1.19663	−0.598313	−	0.801262i	$-0.704162\pi$
$173$	−15.7392	−1.19663	−0.598313	+	0.801262i	$0.704162\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−18.7468	−1.40909
$178$	0	0
$179$	−6.61963	−0.494774	−0.247387	−	0.968917i	$-0.579572\pi$
$179$	−6.61963	−0.494774	−0.247387	+	0.968917i	$0.579572\pi$
$180$	0	0
$181$	−16.0433	−1.19249	−0.596245	−	0.802803i	$-0.703341\pi$
$181$	−16.0433	−1.19249	−0.596245	+	0.802803i	$0.703341\pi$
$182$	0	0
$183$	−18.4275	−1.36220
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5.15842	−0.377221
$188$	0	0
$189$	−17.1865	−1.25014
$190$	0	0
$191$	−0.295590	−0.0213881	−0.0106941	−	0.999943i	$-0.503404\pi$
$191$	−0.295590	−0.0213881	−0.0106941	+	0.999943i	$0.503404\pi$
$192$	0	0
$193$	−1.42564	−0.102620	−0.0513101	−	0.998683i	$-0.516340\pi$
$193$	−1.42564	−0.102620	−0.0513101	+	0.998683i	$0.516340\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.60107	−0.612801	−0.306401	−	0.951903i	$-0.599125\pi$
$197$	−8.60107	−0.612801	−0.306401	+	0.951903i	$0.599125\pi$
$198$	0	0
$199$	−3.81515	−0.270449	−0.135224	−	0.990815i	$-0.543176\pi$
$199$	−3.81515	−0.270449	−0.135224	+	0.990815i	$0.543176\pi$
$200$	0	0
$201$	19.0538	1.34395
$202$	0	0
$203$	−13.6633	−0.958975
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.0262434	−0.00182404
$208$	0	0
$209$	14.0143	0.969390
$210$	0	0
$211$	−1.32370	−0.0911273	−0.0455637	−	0.998961i	$-0.514508\pi$
$211$	−1.32370	−0.0911273	−0.0455637	+	0.998961i	$0.514508\pi$
$212$	0	0
$213$	−22.4705	−1.53965
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−31.7149	−2.15295
$218$	0	0
$219$	−3.30427	−0.223282
$220$	0	0
$221$	0.984992	0.0662577
$222$	0	0
$223$	−25.1286	−1.68274	−0.841368	−	0.540462i	$-0.818249\pi$
$223$	−25.1286	−1.68274	−0.841368	+	0.540462i	$0.818249\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.8021	1.57980	0.789900	−	0.613236i	$-0.210133\pi$
$227$	23.8021	1.57980	0.789900	+	0.613236i	$0.210133\pi$
$228$	0	0
$229$	−0.789351	−0.0521618	−0.0260809	−	0.999660i	$-0.508303\pi$
$229$	−0.789351	−0.0521618	−0.0260809	+	0.999660i	$0.508303\pi$
$230$	0	0
$231$	−33.3531	−2.19447
$232$	0	0
$233$	−20.8101	−1.36331	−0.681656	−	0.731672i	$-0.738740\pi$
$233$	−20.8101	−1.36331	−0.681656	+	0.731672i	$0.738740\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−0.375903	−0.0244175
$238$	0	0
$239$	25.2398	1.63262	0.816312	−	0.577611i	$-0.196015\pi$
$239$	25.2398	1.63262	0.816312	+	0.577611i	$0.196015\pi$
$240$	0	0
$241$	5.27710	0.339928	0.169964	−	0.985450i	$-0.445635\pi$
$241$	5.27710	0.339928	0.169964	+	0.985450i	$0.445635\pi$
$242$	0	0
$243$	0.272722	0.0174951
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.67601	−0.170270
$248$	0	0
$249$	−9.37592	−0.594175
$250$	0	0
$251$	10.8439	0.684461	0.342230	−	0.939616i	$-0.388818\pi$
$251$	10.8439	0.684461	0.342230	+	0.939616i	$0.388818\pi$
$252$	0	0
$253$	5.77103	0.362822
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.6238	1.16172	0.580859	−	0.814004i	$-0.302717\pi$
$257$	18.6238	1.16172	0.580859	+	0.814004i	$0.302717\pi$
$258$	0	0
$259$	25.5781	1.58934
$260$	0	0
$261$	0.107931	0.00668074
$262$	0	0
$263$	19.4370	1.19854	0.599269	−	0.800548i	$-0.295458\pi$
$263$	19.4370	1.19854	0.599269	+	0.800548i	$0.295458\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−10.4491	−0.639474
$268$	0	0
$269$	−3.80493	−0.231991	−0.115995	−	0.993250i	$-0.537006\pi$
$269$	−3.80493	−0.231991	−0.115995	+	0.993250i	$0.537006\pi$
$270$	0	0
$271$	18.2100	1.10618	0.553089	−	0.833122i	$-0.313449\pi$
$271$	18.2100	1.10618	0.553089	+	0.833122i	$0.313449\pi$
$272$	0	0
$273$	6.36872	0.385452
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.7624	0.886988	0.443494	−	0.896277i	$-0.353739\pi$
$277$	14.7624	0.886988	0.443494	+	0.896277i	$0.353739\pi$
$278$	0	0
$279$	0.250526	0.0149986
$280$	0	0
$281$	−6.29738	−0.375670	−0.187835	−	0.982201i	$-0.560147\pi$
$281$	−6.29738	−0.375670	−0.187835	+	0.982201i	$0.560147\pi$
$282$	0	0
$283$	−28.4443	−1.69084	−0.845418	−	0.534105i	$-0.820649\pi$
$283$	−28.4443	−1.69084	−0.845418	+	0.534105i	$0.820649\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.0138985	−0.000820401 0
$288$	0	0
$289$	−16.2010	−0.953002
$290$	0	0
$291$	−3.62020	−0.212220
$292$	0	0
$293$	14.4907	0.846556	0.423278	−	0.906000i	$-0.360879\pi$
$293$	14.4907	0.846556	0.423278	+	0.906000i	$0.360879\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−29.8546	−1.73234
$298$	0	0
$299$	−1.10197	−0.0637285
$300$	0	0
$301$	−33.1242	−1.90925
$302$	0	0
$303$	22.8892	1.31495
$304$	0	0
$305$	0	0
$306$	0	0
$307$	30.4707	1.73905	0.869527	−	0.493885i	$-0.164424\pi$
$307$	30.4707	1.73905	0.869527	+	0.493885i	$0.164424\pi$
$308$	0	0
$309$	31.4793	1.79080
$310$	0	0
$311$	−20.9031	−1.18531	−0.592654	−	0.805457i	$-0.701920\pi$
$311$	−20.9031	−1.18531	−0.592654	+	0.805457i	$0.701920\pi$
$312$	0	0
$313$	14.3614	0.811757	0.405878	−	0.913927i	$-0.366966\pi$
$313$	14.3614	0.811757	0.405878	+	0.913927i	$0.366966\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.6939	1.89244	0.946220	−	0.323525i	$-0.104868\pi$
$317$	33.6939	1.89244	0.946220	+	0.323525i	$0.104868\pi$
$318$	0	0
$319$	−23.7344	−1.32887
$320$	0	0
$321$	12.1297	0.677011
$322$	0	0
$323$	−2.17061	−0.120776
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−17.8959	−0.989644
$328$	0	0
$329$	33.2487	1.83306
$330$	0	0
$331$	−9.82292	−0.539917	−0.269958	−	0.962872i	$-0.587010\pi$
$331$	−9.82292	−0.539917	−0.269958	+	0.962872i	$0.587010\pi$
$332$	0	0
$333$	−0.202049	−0.0110722
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−13.2410	−0.721286	−0.360643	−	0.932704i	$-0.617443\pi$
$337$	−13.2410	−0.721286	−0.360643	+	0.932704i	$0.617443\pi$
$338$	0	0
$339$	9.62795	0.522918
$340$	0	0
$341$	−55.0917	−2.98338
$342$	0	0
$343$	9.84292	0.531468
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.15627	0.491534	0.245767	−	0.969329i	$-0.420960\pi$
$347$	9.15627	0.491534	0.245767	+	0.969329i	$0.420960\pi$
$348$	0	0
$349$	25.7744	1.37967	0.689836	−	0.723966i	$-0.257683\pi$
$349$	25.7744	1.37967	0.689836	+	0.723966i	$0.257683\pi$
$350$	0	0
$351$	5.70068	0.304280
$352$	0	0
$353$	−2.40654	−0.128087	−0.0640436	−	0.997947i	$-0.520400\pi$
$353$	−2.40654	−0.128087	−0.0640436	+	0.997947i	$0.520400\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.16590	0.273408
$358$	0	0
$359$	−21.5236	−1.13597	−0.567985	−	0.823039i	$-0.692277\pi$
$359$	−21.5236	−1.13597	−0.567985	+	0.823039i	$0.692277\pi$
$360$	0	0
$361$	−13.1029	−0.689628
$362$	0	0
$363$	−38.8016	−2.03656
$364$	0	0
$365$	0	0
$366$	0	0
$367$	6.47203	0.337837	0.168919	−	0.985630i	$-0.445972\pi$
$367$	6.47203	0.337837	0.168919	+	0.985630i	$0.445972\pi$
$368$	0	0
$369$	0.000109788 0	5.71536e−6 0
$370$	0	0
$371$	−20.7696	−1.07830
$372$	0	0
$373$	−27.7866	−1.43874	−0.719368	−	0.694629i	$-0.755569\pi$
$373$	−27.7866	−1.43874	−0.719368	+	0.694629i	$0.755569\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.53204	0.233412
$378$	0	0
$379$	−7.63904	−0.392391	−0.196196	−	0.980565i	$-0.562859\pi$
$379$	−7.63904	−0.392391	−0.196196	+	0.980565i	$0.562859\pi$
$380$	0	0
$381$	19.4105	0.994432
$382$	0	0
$383$	19.4206	0.992348	0.496174	−	0.868223i	$-0.334738\pi$
$383$	19.4206	0.992348	0.496174	+	0.868223i	$0.334738\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.261659	0.0133009
$388$	0	0
$389$	−26.1255	−1.32461	−0.662306	−	0.749233i	$-0.730422\pi$
$389$	−26.1255	−1.32461	−0.662306	+	0.749233i	$0.730422\pi$
$390$	0	0
$391$	−0.893847	−0.0452038
$392$	0	0
$393$	−26.1674	−1.31997
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.5388	−0.579117	−0.289558	−	0.957160i	$-0.593508\pi$
$397$	−11.5388	−0.579117	−0.289558	+	0.957160i	$0.593508\pi$
$398$	0	0
$399$	−14.0346	−0.702610
$400$	0	0
$401$	−0.188549	−0.00941568	−0.00470784	−	0.999989i	$-0.501499\pi$
$401$	−0.188549	−0.00941568	−0.00470784	+	0.999989i	$0.501499\pi$
$402$	0	0
$403$	10.5197	0.524022
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44.4314	2.20238
$408$	0	0
$409$	−8.48327	−0.419471	−0.209735	−	0.977758i	$-0.567260\pi$
$409$	−8.48327	−0.419471	−0.209735	+	0.977758i	$0.567260\pi$
$410$	0	0
$411$	−3.84855	−0.189835
$412$	0	0
$413$	−35.8018	−1.76169
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−11.3723	−0.556906
$418$	0	0
$419$	2.15604	0.105329	0.0526647	−	0.998612i	$-0.483229\pi$
$419$	2.15604	0.105329	0.0526647	+	0.998612i	$0.483229\pi$
$420$	0	0
$421$	8.01842	0.390794	0.195397	−	0.980724i	$-0.437400\pi$
$421$	8.01842	0.390794	0.195397	+	0.980724i	$0.437400\pi$
$422$	0	0
$423$	−0.262642	−0.0127701
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−35.1921	−1.70307
$428$	0	0
$429$	11.0630	0.534129
$430$	0	0
$431$	15.3343	0.738629	0.369315	−	0.929304i	$-0.379592\pi$
$431$	15.3343	0.738629	0.369315	+	0.929304i	$0.379592\pi$
$432$	0	0
$433$	16.3093	0.783777	0.391888	−	0.920013i	$-0.371822\pi$
$433$	16.3093	0.783777	0.391888	+	0.920013i	$0.371822\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.42839	0.116166
$438$	0	0
$439$	23.9534	1.14323	0.571617	−	0.820520i	$-0.306316\pi$
$439$	23.9534	1.14323	0.571617	+	0.820520i	$0.306316\pi$
$440$	0	0
$441$	0.105951	0.00504531
$442$	0	0
$443$	29.2360	1.38905	0.694523	−	0.719471i	$-0.255616\pi$
$443$	29.2360	1.38905	0.694523	+	0.719471i	$0.255616\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	19.4565	0.920262
$448$	0	0
$449$	−21.2244	−1.00164	−0.500822	−	0.865550i	$-0.666969\pi$
$449$	−21.2244	−1.00164	−0.500822	+	0.865550i	$0.666969\pi$
$450$	0	0
$451$	−0.0241429	−0.00113685
$452$	0	0
$453$	−2.24716	−0.105581
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.5959	−1.47799	−0.738996	−	0.673710i	$-0.764700\pi$
$457$	−31.5959	−1.47799	−0.738996	+	0.673710i	$0.764700\pi$
$458$	0	0
$459$	4.62403	0.215831
$460$	0	0
$461$	16.4858	0.767820	0.383910	−	0.923370i	$-0.374577\pi$
$461$	16.4858	0.767820	0.383910	+	0.923370i	$0.374577\pi$
$462$	0	0
$463$	−30.1555	−1.40144	−0.700722	−	0.713435i	$-0.747139\pi$
$463$	−30.1555	−1.40144	−0.700722	+	0.713435i	$0.747139\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.89645	0.365404	0.182702	−	0.983168i	$-0.441516\pi$
$467$	7.89645	0.365404	0.182702	+	0.983168i	$0.441516\pi$
$468$	0	0
$469$	36.3882	1.68025
$470$	0	0
$471$	−19.2167	−0.885457
$472$	0	0
$473$	−57.5398	−2.64568
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.164066	0.00751205
$478$	0	0
$479$	−39.5234	−1.80587	−0.902935	−	0.429778i	$-0.858592\pi$
$479$	−39.5234	−1.80587	−0.902935	+	0.429778i	$0.858592\pi$
$480$	0	0
$481$	−8.48411	−0.386842
$482$	0	0
$483$	−5.77940	−0.262972
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.20444	−0.145207	−0.0726035	−	0.997361i	$-0.523131\pi$
$487$	−3.20444	−0.145207	−0.0726035	+	0.997361i	$0.523131\pi$
$488$	0	0
$489$	8.77224	0.396695
$490$	0	0
$491$	−34.8865	−1.57441	−0.787203	−	0.616694i	$-0.788472\pi$
$491$	−34.8865	−1.57441	−0.787203	+	0.616694i	$0.788472\pi$
$492$	0	0
$493$	3.67611	0.165564
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−42.9132	−1.92492
$498$	0	0
$499$	−33.1594	−1.48442	−0.742210	−	0.670167i	$-0.766222\pi$
$499$	−33.1594	−1.48442	−0.742210	+	0.670167i	$0.766222\pi$
$500$	0	0
$501$	2.10074	0.0938543
$502$	0	0
$503$	−7.05945	−0.314765	−0.157383	−	0.987538i	$-0.550306\pi$
$503$	−7.05945	−0.314765	−0.157383	+	0.987538i	$0.550306\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	20.5025	0.910546
$508$	0	0
$509$	16.6812	0.739382	0.369691	−	0.929155i	$-0.379464\pi$
$509$	16.6812	0.739382	0.369691	+	0.929155i	$0.379464\pi$
$510$	0	0
$511$	−6.31035	−0.279154
$512$	0	0
$513$	−12.5625	−0.554648
$514$	0	0
$515$	0	0
$516$	0	0
$517$	57.7560	2.54011
$518$	0	0
$519$	27.3800	1.20185
$520$	0	0
$521$	−24.9941	−1.09501	−0.547507	−	0.836801i	$-0.684423\pi$
$521$	−24.9941	−1.09501	−0.547507	+	0.836801i	$0.684423\pi$
$522$	0	0
$523$	14.3283	0.626535	0.313267	−	0.949665i	$-0.398576\pi$
$523$	14.3283	0.626535	0.313267	+	0.949665i	$0.398576\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.53289	0.371698
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.282810	0.0122729
$532$	0	0
$533$	0.00461005	0.000199683 0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.5156	0.496934
$538$	0	0
$539$	−23.2992	−1.00357
$540$	0	0
$541$	22.0323	0.947244	0.473622	−	0.880728i	$-0.342946\pi$
$541$	22.0323	0.947244	0.473622	+	0.880728i	$0.342946\pi$
$542$	0	0
$543$	27.9091	1.19769
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−21.5194	−0.920104	−0.460052	−	0.887892i	$-0.652169\pi$
$547$	−21.5194	−0.920104	−0.460052	+	0.887892i	$0.652169\pi$
$548$	0	0
$549$	0.277993	0.0118645
$550$	0	0
$551$	−9.98719	−0.425468
$552$	0	0
$553$	−0.717884	−0.0305276
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.1923	−0.897945	−0.448972	−	0.893546i	$-0.648210\pi$
$557$	−21.1923	−0.897945	−0.448972	+	0.893546i	$0.648210\pi$
$558$	0	0
$559$	10.9871	0.464706
$560$	0	0
$561$	8.97364	0.378867
$562$	0	0
$563$	−23.6292	−0.995851	−0.497925	−	0.867220i	$-0.665905\pi$
$563$	−23.6292	−0.995851	−0.497925	+	0.867220i	$0.665905\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	30.1594	1.26658
$568$	0	0
$569$	−33.7490	−1.41483	−0.707415	−	0.706798i	$-0.750139\pi$
$569$	−33.7490	−1.41483	−0.707415	+	0.706798i	$0.750139\pi$
$570$	0	0
$571$	25.4667	1.06575	0.532874	−	0.846194i	$-0.321112\pi$
$571$	25.4667	1.06575	0.532874	+	0.846194i	$0.321112\pi$
$572$	0	0
$573$	0.514211	0.0214815
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−22.7293	−0.946234	−0.473117	−	0.881000i	$-0.656871\pi$
$577$	−22.7293	−0.946234	−0.473117	+	0.881000i	$0.656871\pi$
$578$	0	0
$579$	2.48006	0.103068
$580$	0	0
$581$	−17.9057	−0.742856
$582$	0	0
$583$	−36.0787	−1.49423
$584$	0	0
$585$	0	0
$586$	0	0
$587$	35.6609	1.47188	0.735941	−	0.677045i	$-0.236740\pi$
$587$	35.6609	1.47188	0.735941	+	0.677045i	$0.236740\pi$
$588$	0	0
$589$	−23.1820	−0.955198
$590$	0	0
$591$	14.9625	0.615476
$592$	0	0
$593$	20.8412	0.855845	0.427922	−	0.903815i	$-0.359246\pi$
$593$	20.8412	0.855845	0.427922	+	0.903815i	$0.359246\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.63687	0.271629
$598$	0	0
$599$	10.9952	0.449252	0.224626	−	0.974445i	$-0.427884\pi$
$599$	10.9952	0.449252	0.224626	+	0.974445i	$0.427884\pi$
$600$	0	0
$601$	2.35032	0.0958714	0.0479357	−	0.998850i	$-0.484736\pi$
$601$	2.35032	0.0958714	0.0479357	+	0.998850i	$0.484736\pi$
$602$	0	0
$603$	−0.287442	−0.0117055
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−31.8412	−1.29239	−0.646197	−	0.763170i	$-0.723642\pi$
$607$	−31.8412	−1.29239	−0.646197	+	0.763170i	$0.723642\pi$
$608$	0	0
$609$	23.7688	0.963160
$610$	0	0
$611$	−11.0284	−0.446162
$612$	0	0
$613$	29.0597	1.17371	0.586856	−	0.809692i	$-0.300366\pi$
$613$	29.0597	1.17371	0.586856	+	0.809692i	$0.300366\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.9851	−1.20715	−0.603577	−	0.797304i	$-0.706259\pi$
$617$	−29.9851	−1.20715	−0.603577	+	0.797304i	$0.706259\pi$
$618$	0	0
$619$	16.9204	0.680089	0.340045	−	0.940409i	$-0.389558\pi$
$619$	16.9204	0.680089	0.340045	+	0.940409i	$0.389558\pi$
$620$	0	0
$621$	−5.17318	−0.207592
$622$	0	0
$623$	−19.9553	−0.799491
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−24.3794	−0.973621
$628$	0	0
$629$	−6.88177	−0.274394
$630$	0	0
$631$	20.4522	0.814188	0.407094	−	0.913386i	$-0.366542\pi$
$631$	20.4522	0.814188	0.407094	+	0.913386i	$0.366542\pi$
$632$	0	0
$633$	2.30272	0.0915250
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.44894	0.176273
$638$	0	0
$639$	0.338985	0.0134100
$640$	0	0
$641$	−8.27686	−0.326916	−0.163458	−	0.986550i	$-0.552265\pi$
$641$	−8.27686	−0.326916	−0.163458	+	0.986550i	$0.552265\pi$
$642$	0	0
$643$	−0.627685	−0.0247535	−0.0123767	−	0.999923i	$-0.503940\pi$
$643$	−0.627685	−0.0247535	−0.0123767	+	0.999923i	$0.503940\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.3508	−0.485561	−0.242781	−	0.970081i	$-0.578060\pi$
$647$	−12.3508	−0.485561	−0.242781	+	0.970081i	$0.578060\pi$
$648$	0	0
$649$	−62.1911	−2.44121
$650$	0	0
$651$	55.1715	2.16234
$652$	0	0
$653$	−18.3211	−0.716959	−0.358480	−	0.933538i	$-0.616705\pi$
$653$	−18.3211	−0.716959	−0.358480	+	0.933538i	$0.616705\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.0498475	0.00194473
$658$	0	0
$659$	39.1896	1.52661	0.763306	−	0.646038i	$-0.223575\pi$
$659$	39.1896	1.52661	0.763306	+	0.646038i	$0.223575\pi$
$660$	0	0
$661$	9.38565	0.365060	0.182530	−	0.983200i	$-0.441571\pi$
$661$	9.38565	0.365060	0.182530	+	0.983200i	$0.441571\pi$
$662$	0	0
$663$	−1.71350	−0.0665469
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.11268	−0.159244
$668$	0	0
$669$	43.7140	1.69008
$670$	0	0
$671$	−61.1319	−2.35997
$672$	0	0
$673$	8.05600	0.310536	0.155268	−	0.987872i	$-0.450376\pi$
$673$	8.05600	0.310536	0.155268	+	0.987872i	$0.450376\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.9461	−0.651293	−0.325647	−	0.945492i	$-0.605582\pi$
$677$	−16.9461	−0.651293	−0.325647	+	0.945492i	$0.605582\pi$
$678$	0	0
$679$	−6.91370	−0.265324
$680$	0	0
$681$	−41.4063	−1.58669
$682$	0	0
$683$	0.111363	0.00426119	0.00213059	−	0.999998i	$-0.499322\pi$
$683$	0.111363	0.00426119	0.00213059	+	0.999998i	$0.499322\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.37316	0.0523894
$688$	0	0
$689$	6.88917	0.262456
$690$	0	0
$691$	38.1718	1.45212	0.726062	−	0.687629i	$-0.241348\pi$
$691$	38.1718	1.45212	0.726062	+	0.687629i	$0.241348\pi$
$692$	0	0
$693$	0.503158	0.0191134
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.00373938	0.000141639 0
$698$	0	0
$699$	36.2014	1.36926
$700$	0	0
$701$	38.7698	1.46431	0.732157	−	0.681136i	$-0.238514\pi$
$701$	38.7698	1.46431	0.732157	+	0.681136i	$0.238514\pi$
$702$	0	0
$703$	18.6963	0.705144
$704$	0	0
$705$	0	0
$706$	0	0
$707$	43.7128	1.64399
$708$	0	0
$709$	−47.4028	−1.78025	−0.890126	−	0.455715i	$-0.849383\pi$
$709$	−47.4028	−1.78025	−0.890126	+	0.455715i	$0.849383\pi$
$710$	0	0
$711$	0.00567080	0.000212671 0
$712$	0	0
$713$	−9.54624	−0.357510
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−43.9073	−1.63975
$718$	0	0
$719$	17.7370	0.661479	0.330740	−	0.943722i	$-0.392702\pi$
$719$	17.7370	0.661479	0.330740	+	0.943722i	$0.392702\pi$
$720$	0	0
$721$	60.1179	2.23891
$722$	0	0
$723$	−9.18010	−0.341412
$724$	0	0
$725$	0	0
$726$	0	0
$727$	50.7341	1.88163	0.940813	−	0.338927i	$-0.110064\pi$
$727$	50.7341	1.88163	0.940813	+	0.338927i	$0.110064\pi$
$728$	0	0
$729$	26.7597	0.991100
$730$	0	0
$731$	8.91206	0.329625
$732$	0	0
$733$	−18.9970	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$733$	−18.9970	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	63.2096	2.32835
$738$	0	0
$739$	44.6999	1.64431	0.822156	−	0.569263i	$-0.192771\pi$
$739$	44.6999	1.64431	0.822156	+	0.569263i	$0.192771\pi$
$740$	0	0
$741$	4.65522	0.171014
$742$	0	0
$743$	−38.9841	−1.43019	−0.715094	−	0.699029i	$-0.753616\pi$
$743$	−38.9841	−1.43019	−0.715094	+	0.699029i	$0.753616\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.141443	0.00517513
$748$	0	0
$749$	23.1647	0.846421
$750$	0	0
$751$	−3.97027	−0.144877	−0.0724386	−	0.997373i	$-0.523078\pi$
$751$	−3.97027	−0.144877	−0.0724386	+	0.997373i	$0.523078\pi$
$752$	0	0
$753$	−18.8641	−0.687448
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−23.2885	−0.846436	−0.423218	−	0.906028i	$-0.639100\pi$
$757$	−23.2885	−0.846436	−0.423218	+	0.906028i	$0.639100\pi$
$758$	0	0
$759$	−10.0393	−0.364405
$760$	0	0
$761$	5.62214	0.203802	0.101901	−	0.994795i	$-0.467507\pi$
$761$	5.62214	0.203802	0.101901	+	0.994795i	$0.467507\pi$
$762$	0	0
$763$	−34.1768	−1.23728
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.8753	0.428792
$768$	0	0
$769$	−33.0457	−1.19166	−0.595828	−	0.803112i	$-0.703176\pi$
$769$	−33.0457	−1.19166	−0.595828	+	0.803112i	$0.703176\pi$
$770$	0	0
$771$	−32.3981	−1.16679
$772$	0	0
$773$	−46.8196	−1.68398	−0.841991	−	0.539491i	$-0.818617\pi$
$773$	−46.8196	−1.68398	−0.841991	+	0.539491i	$0.818617\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−44.4958	−1.59628
$778$	0	0
$779$	−0.0101591	−0.000363987 0
$780$	0	0
$781$	−74.5442	−2.66740
$782$	0	0
$783$	21.2756	0.760328
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−35.1470	−1.25286	−0.626428	−	0.779479i	$-0.715484\pi$
$787$	−35.1470	−1.25286	−0.626428	+	0.779479i	$0.715484\pi$
$788$	0	0
$789$	−33.8128	−1.20377
$790$	0	0
$791$	18.3871	0.653769
$792$	0	0
$793$	11.6730	0.414522
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.8566	0.774200	0.387100	−	0.922038i	$-0.373477\pi$
$797$	21.8566	0.774200	0.387100	+	0.922038i	$0.373477\pi$
$798$	0	0
$799$	−8.94556	−0.316471
$800$	0	0
$801$	0.157633	0.00556969
$802$	0	0
$803$	−10.9617	−0.386829
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.61909	0.233003
$808$	0	0
$809$	−2.24801	−0.0790357	−0.0395178	−	0.999219i	$-0.512582\pi$
$809$	−2.24801	−0.0790357	−0.0395178	+	0.999219i	$0.512582\pi$
$810$	0	0
$811$	−29.5155	−1.03643	−0.518215	−	0.855250i	$-0.673403\pi$
$811$	−29.5155	−1.03643	−0.518215	+	0.855250i	$0.673403\pi$
$812$	0	0
$813$	−31.6783	−1.11101
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.2121	−0.847076
$818$	0	0
$819$	−0.0960772	−0.00335721
$820$	0	0
$821$	−29.1470	−1.01724	−0.508618	−	0.860992i	$-0.669843\pi$
$821$	−29.1470	−1.01724	−0.508618	+	0.860992i	$0.669843\pi$
$822$	0	0
$823$	13.1354	0.457873	0.228936	−	0.973441i	$-0.426475\pi$
$823$	13.1354	0.457873	0.228936	+	0.973441i	$0.426475\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−54.7619	−1.90426	−0.952129	−	0.305698i	$-0.901110\pi$
$827$	−54.7619	−1.90426	−0.952129	+	0.305698i	$0.901110\pi$
$828$	0	0
$829$	−41.3545	−1.43630	−0.718151	−	0.695887i	$-0.755011\pi$
$829$	−41.3545	−1.43630	−0.718151	+	0.695887i	$0.755011\pi$
$830$	0	0
$831$	−25.6809	−0.890859
$832$	0	0
$833$	3.60869	0.125034
$834$	0	0
$835$	0	0
$836$	0	0
$837$	49.3844	1.70698
$838$	0	0
$839$	−8.60909	−0.297219	−0.148609	−	0.988896i	$-0.547480\pi$
$839$	−8.60909	−0.297219	−0.148609	+	0.988896i	$0.547480\pi$
$840$	0	0
$841$	−12.0859	−0.416755
$842$	0	0
$843$	10.9550	0.377310
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−74.1018	−2.54617
$848$	0	0
$849$	49.4819	1.69822
$850$	0	0
$851$	7.69904	0.263920
$852$	0	0
$853$	53.2697	1.82392	0.911960	−	0.410280i	$-0.134569\pi$
$853$	53.2697	1.82392	0.911960	+	0.410280i	$0.134569\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.3201	0.455005	0.227502	−	0.973778i	$-0.426944\pi$
$857$	13.3201	0.455005	0.227502	+	0.973778i	$0.426944\pi$
$858$	0	0
$859$	−23.3425	−0.796434	−0.398217	−	0.917291i	$-0.630371\pi$
$859$	−23.3425	−0.796434	−0.398217	+	0.917291i	$0.630371\pi$
$860$	0	0
$861$	0.0241779	0.000823981 0
$862$	0	0
$863$	−21.8395	−0.743426	−0.371713	−	0.928348i	$-0.621229\pi$
$863$	−21.8395	−0.743426	−0.371713	+	0.928348i	$0.621229\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	28.1835	0.957161
$868$	0	0
$869$	−1.24703	−0.0423026
$870$	0	0
$871$	−12.0698	−0.408968
$872$	0	0
$873$	0.0546135	0.00184839
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−29.0903	−0.982309	−0.491155	−	0.871072i	$-0.663425\pi$
$877$	−29.0903	−0.982309	−0.491155	+	0.871072i	$0.663425\pi$
$878$	0	0
$879$	−25.2082	−0.850251
$880$	0	0
$881$	−44.7215	−1.50671	−0.753353	−	0.657616i	$-0.771565\pi$
$881$	−44.7215	−1.50671	−0.753353	+	0.657616i	$0.771565\pi$
$882$	0	0
$883$	−19.4889	−0.655855	−0.327927	−	0.944703i	$-0.606350\pi$
$883$	−19.4889	−0.655855	−0.327927	+	0.944703i	$0.606350\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.0317	0.806906	0.403453	−	0.915000i	$-0.367810\pi$
$887$	24.0317	0.806906	0.403453	+	0.915000i	$0.367810\pi$
$888$	0	0
$889$	37.0694	1.24327
$890$	0	0
$891$	52.3897	1.75512
$892$	0	0
$893$	24.3031	0.813274
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.91700	0.0640067
$898$	0	0
$899$	39.2606	1.30942
$900$	0	0
$901$	5.58806	0.186165
$902$	0	0
$903$	57.6232	1.91758
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16.2777	−0.540491	−0.270245	−	0.962792i	$-0.587105\pi$
$907$	−16.2777	−0.540491	−0.270245	+	0.962792i	$0.587105\pi$
$908$	0	0
$909$	−0.345301	−0.0114529
$910$	0	0
$911$	−23.8128	−0.788952	−0.394476	−	0.918906i	$-0.629074\pi$
$911$	−23.8128	−0.788952	−0.394476	+	0.918906i	$0.629074\pi$
$912$	0	0
$913$	−31.1039	−1.02939
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−49.9734	−1.65027
$918$	0	0
$919$	−22.0433	−0.727141	−0.363571	−	0.931567i	$-0.618442\pi$
$919$	−22.0433	−0.727141	−0.363571	+	0.931567i	$0.618442\pi$
$920$	0	0
$921$	−53.0071	−1.74664
$922$	0	0
$923$	14.2341	0.468521
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.474890	−0.0155974
$928$	0	0
$929$	−5.19696	−0.170507	−0.0852534	−	0.996359i	$-0.527170\pi$
$929$	−5.19696	−0.170507	−0.0852534	+	0.996359i	$0.527170\pi$
$930$	0	0
$931$	−9.80404	−0.321315
$932$	0	0
$933$	36.3633	1.19048
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−29.0029	−0.947482	−0.473741	−	0.880664i	$-0.657097\pi$
$937$	−29.0029	−0.947482	−0.473741	+	0.880664i	$0.657097\pi$
$938$	0	0
$939$	−24.9833	−0.815300
$940$	0	0
$941$	50.2138	1.63692	0.818461	−	0.574562i	$-0.194827\pi$
$941$	50.2138	1.63692	0.818461	+	0.574562i	$0.194827\pi$
$942$	0	0
$943$	−0.00418347	−0.000136232 0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−18.3661	−0.596817	−0.298408	−	0.954438i	$-0.596456\pi$
$947$	−18.3661	−0.596817	−0.298408	+	0.954438i	$0.596456\pi$
$948$	0	0
$949$	2.09311	0.0679453
$950$	0	0
$951$	−58.6143	−1.90070
$952$	0	0
$953$	34.9542	1.13228	0.566139	−	0.824310i	$-0.308437\pi$
$953$	34.9542	1.13228	0.566139	+	0.824310i	$0.308437\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	41.2886	1.33467
$958$	0	0
$959$	−7.34980	−0.237338
$960$	0	0
$961$	60.1308	1.93970
$962$	0	0
$963$	−0.182985	−0.00589662
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−48.3107	−1.55357	−0.776784	−	0.629767i	$-0.783150\pi$
$967$	−48.3107	−1.55357	−0.776784	+	0.629767i	$0.783150\pi$
$968$	0	0
$969$	3.77602	0.121303
$970$	0	0
$971$	14.8815	0.477570	0.238785	−	0.971072i	$-0.423251\pi$
$971$	14.8815	0.477570	0.238785	+	0.971072i	$0.423251\pi$
$972$	0	0
$973$	−21.7184	−0.696261
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.6833	0.597730	0.298865	−	0.954295i	$-0.403392\pi$
$977$	18.6833	0.597730	0.298865	+	0.954295i	$0.403392\pi$
$978$	0	0
$979$	−34.6641	−1.10787
$980$	0	0
$981$	0.269974	0.00861959
$982$	0	0
$983$	−28.4191	−0.906428	−0.453214	−	0.891402i	$-0.649723\pi$
$983$	−28.4191	−0.906428	−0.453214	+	0.891402i	$0.649723\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−57.8398	−1.84106
$988$	0	0
$989$	−9.97045	−0.317042
$990$	0	0
$991$	28.4942	0.905148	0.452574	−	0.891727i	$-0.350506\pi$
$991$	28.4942	0.905148	0.452574	+	0.891727i	$0.350506\pi$
$992$	0	0
$993$	17.0881	0.542273
$994$	0	0
$995$	0	0
$996$	0	0
$997$	26.5435	0.840642	0.420321	−	0.907375i	$-0.361917\pi$
$997$	26.5435	0.840642	0.420321	+	0.907375i	$0.361917\pi$
$998$	0	0
$999$	−39.8285	−1.26012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cx.1.3		6
4.3	odd	2		2300.2.a.o.1.4		6
5.2	odd	4		1840.2.e.f.369.9		12
5.3	odd	4		1840.2.e.f.369.4		12
5.4	even	2		9200.2.a.cy.1.4		6
20.3	even	4		460.2.c.a.369.9	yes	12
20.7	even	4		460.2.c.a.369.4	✓	12
20.19	odd	2		2300.2.a.n.1.3		6
60.23	odd	4		4140.2.f.b.829.1		12
60.47	odd	4		4140.2.f.b.829.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.4	✓	12	20.7	even	4
460.2.c.a.369.9	yes	12	20.3	even	4
1840.2.e.f.369.4		12	5.3	odd	4
1840.2.e.f.369.9		12	5.2	odd	4
2300.2.a.n.1.3		6	20.19	odd	2
2300.2.a.o.1.4		6	4.3	odd	2
4140.2.f.b.829.1		12	60.23	odd	4
4140.2.f.b.829.2		12	60.47	odd	4
9200.2.a.cx.1.3		6	1.1	even	1	trivial
9200.2.a.cy.1.4		6	5.4	even	2

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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73961 q^{3} -3.32224 q^{7} +0.0262434 q^{9} -5.77103 q^{11} +1.10197 q^{13} +0.893847 q^{17} -2.42839 q^{19} +5.77940 q^{21} -1.00000 q^{23} +5.17318 q^{27} +4.11268 q^{29} +9.54624 q^{31} +10.0393 q^{33} -7.69904 q^{37} -1.91700 q^{39} +0.00418347 q^{41} +9.97045 q^{43} -10.0079 q^{47} +4.03726 q^{49} -1.55495 q^{51} +6.25169 q^{53} +4.22445 q^{57} +10.7764 q^{59} +10.5929 q^{61} -0.0871868 q^{63} -10.9529 q^{67} +1.73961 q^{69} +12.9170 q^{71} +1.89943 q^{73} +19.1727 q^{77} +0.216085 q^{79} -9.07804 q^{81} +5.38967 q^{83} -7.15446 q^{87} +6.00657 q^{89} -3.66100 q^{91} -16.6067 q^{93} +2.08104 q^{97} -0.151451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 9 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} + 5 q^{17} - 4 q^{19} - 6 q^{23} - 22 q^{27} + 5 q^{29} - 9 q^{31} - 10 q^{33} + 21 q^{37} + 8 q^{39} - q^{41} - 16 q^{43} - 16 q^{47} + 19 q^{49}+ \cdots + 16 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 9 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^13 + 5 * q^17 - 4 * q^19 - 6 * q^23 - 22 * q^27 + 5 * q^29 - 9 * q^31 - 10 * q^33 + 21 * q^37 + 8 * q^39 - q^41 - 16 * q^43 - 16 * q^47 + 19 * q^49 + 12 * q^51 - q^53 + 12 * q^57 + 11 * q^59 - 4 * q^61 - 19 * q^63 - 25 * q^67 + 4 * q^69 + 17 * q^71 - 14 * q^73 + 20 * q^77 - 10 * q^79 + 14 * q^81 - 21 * q^83 - 64 * q^87 - 24 * q^89 + 4 * q^91 + 4 * q^97 + 16 * q^99

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