LMFDB - Embedded newform 9200.2.a.cz.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 7x^{5} + 24x^{4} + x^{3} - 35x^{2} + 17x - 2 $ x^7 - 3x^6 - 7x^5 + 24x^4 + x^3 - 35x^2 + 17*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.356372$ 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)

$f(q)$	$=$	$q-0.356372 q^{3} +2.46376 q^{7} -2.87300 q^{9} +O(q^{10})$	$q-0.356372 q^{3} +2.46376 q^{7} -2.87300 q^{9} -1.61574 q^{11} +2.62553 q^{13} +2.58924 q^{17} -4.02550 q^{19} -0.878013 q^{21} +1.00000 q^{23} +2.09297 q^{27} -7.08580 q^{29} +4.58371 q^{31} +0.575803 q^{33} -2.96054 q^{37} -0.935665 q^{39} -5.71155 q^{41} -2.30448 q^{43} +6.88317 q^{47} -0.929907 q^{49} -0.922731 q^{51} +6.76467 q^{53} +1.43457 q^{57} -2.53432 q^{59} +9.25564 q^{61} -7.07837 q^{63} -15.7285 q^{67} -0.356372 q^{69} +5.25113 q^{71} -6.03858 q^{73} -3.98078 q^{77} +1.35712 q^{79} +7.87312 q^{81} -8.59500 q^{83} +2.52518 q^{87} -8.84084 q^{89} +6.46867 q^{91} -1.63350 q^{93} -8.01314 q^{97} +4.64201 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9	$ 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9} + 7 q^{11} - 7 q^{13} + 7 q^{19} - 6 q^{21} + 7 q^{23} - 11 q^{29} + 10 q^{31} - 19 q^{33} - 19 q^{37} + 24 q^{39} - 16 q^{41} - 6 q^{43} - 6 q^{47} - 17 q^{49} + 7 q^{51} - 15 q^{53} - 8 q^{57} + 11 q^{59} + 5 q^{61} - 13 q^{63} - 9 q^{67} - 3 q^{69} + 14 q^{71} - 10 q^{73} - 6 q^{77} + 32 q^{79} - 5 q^{81} - q^{83} - 10 q^{87} - 24 q^{89} + 7 q^{91} - 26 q^{93} + 7 q^{97} + 61 q^{99}+O(q^{100}) $ 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9 + 7 * q^11 - 7 * q^13 + 7 * q^19 - 6 * q^21 + 7 * q^23 - 11 * q^29 + 10 * q^31 - 19 * q^33 - 19 * q^37 + 24 * q^39 - 16 * q^41 - 6 * q^43 - 6 * q^47 - 17 * q^49 + 7 * q^51 - 15 * q^53 - 8 * q^57 + 11 * q^59 + 5 * q^61 - 13 * q^63 - 9 * q^67 - 3 * q^69 + 14 * q^71 - 10 * q^73 - 6 * q^77 + 32 * q^79 - 5 * q^81 - q^83 - 10 * q^87 - 24 * q^89 + 7 * q^91 - 26 * q^93 + 7 * q^97 + 61 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−0.356372	−0.205751	−0.102876	−	0.994694i	$-0.532804\pi$
$3$	−0.356372	−0.205751	−0.102876	+	0.994694i	$0.532804\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.46376	0.931212	0.465606	−	0.884992i	$-0.345836\pi$
$7$	2.46376	0.931212	0.465606	+	0.884992i	$0.345836\pi$
$8$	0	0
$9$	−2.87300	−0.957666
$10$	0	0
$11$	−1.61574	−0.487163	−0.243581	−	0.969880i	$-0.578322\pi$
$11$	−1.61574	−0.487163	−0.243581	+	0.969880i	$0.578322\pi$
$12$	0	0
$13$	2.62553	0.728191	0.364096	−	0.931362i	$-0.381378\pi$
$13$	2.62553	0.728191	0.364096	+	0.931362i	$0.381378\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.58924	0.627982	0.313991	−	0.949426i	$-0.398334\pi$
$17$	2.58924	0.627982	0.313991	+	0.949426i	$0.398334\pi$
$18$	0	0
$19$	−4.02550	−0.923512	−0.461756	−	0.887007i	$-0.652780\pi$
$19$	−4.02550	−0.923512	−0.461756	+	0.887007i	$0.652780\pi$
$20$	0	0
$21$	−0.878013	−0.191598
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.09297	0.402793
$28$	0	0
$29$	−7.08580	−1.31580	−0.657900	−	0.753106i	$-0.728555\pi$
$29$	−7.08580	−1.31580	−0.657900	+	0.753106i	$0.728555\pi$
$30$	0	0
$31$	4.58371	0.823258	0.411629	−	0.911352i	$-0.364960\pi$
$31$	4.58371	0.823258	0.411629	+	0.911352i	$0.364960\pi$
$32$	0	0
$33$	0.575803	0.100234
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.96054	−0.486710	−0.243355	−	0.969937i	$-0.578248\pi$
$37$	−2.96054	−0.486710	−0.243355	+	0.969937i	$0.578248\pi$
$38$	0	0
$39$	−0.935665	−0.149826
$40$	0	0
$41$	−5.71155	−0.891995	−0.445997	−	0.895034i	$-0.647151\pi$
$41$	−5.71155	−0.891995	−0.445997	+	0.895034i	$0.647151\pi$
$42$	0	0
$43$	−2.30448	−0.351430	−0.175715	−	0.984441i	$-0.556224\pi$
$43$	−2.30448	−0.351430	−0.175715	+	0.984441i	$0.556224\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.88317	1.00401	0.502007	−	0.864864i	$-0.332595\pi$
$47$	6.88317	1.00401	0.502007	+	0.864864i	$0.332595\pi$
$48$	0	0
$49$	−0.929907	−0.132844
$50$	0	0
$51$	−0.922731	−0.129208
$52$	0	0
$53$	6.76467	0.929199	0.464600	−	0.885521i	$-0.346198\pi$
$53$	6.76467	0.929199	0.464600	+	0.885521i	$0.346198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.43457	0.190014
$58$	0	0
$59$	−2.53432	−0.329941	−0.164970	−	0.986299i	$-0.552753\pi$
$59$	−2.53432	−0.329941	−0.164970	+	0.986299i	$0.552753\pi$
$60$	0	0
$61$	9.25564	1.18506	0.592532	−	0.805547i	$-0.298129\pi$
$61$	9.25564	1.18506	0.592532	+	0.805547i	$0.298129\pi$
$62$	0	0
$63$	−7.07837	−0.891791
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−15.7285	−1.92154	−0.960770	−	0.277347i	$-0.910545\pi$
$67$	−15.7285	−1.92154	−0.960770	+	0.277347i	$0.910545\pi$
$68$	0	0
$69$	−0.356372	−0.0429021
$70$	0	0
$71$	5.25113	0.623194	0.311597	−	0.950214i	$-0.399136\pi$
$71$	5.25113	0.623194	0.311597	+	0.950214i	$0.399136\pi$
$72$	0	0
$73$	−6.03858	−0.706762	−0.353381	−	0.935479i	$-0.614968\pi$
$73$	−6.03858	−0.706762	−0.353381	+	0.935479i	$0.614968\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.98078	−0.453652
$78$	0	0
$79$	1.35712	0.152688	0.0763439	−	0.997082i	$-0.475675\pi$
$79$	1.35712	0.152688	0.0763439	+	0.997082i	$0.475675\pi$
$80$	0	0
$81$	7.87312	0.874791
$82$	0	0
$83$	−8.59500	−0.943423	−0.471712	−	0.881753i	$-0.656364\pi$
$83$	−8.59500	−0.943423	−0.471712	+	0.881753i	$0.656364\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.52518	0.270727
$88$	0	0
$89$	−8.84084	−0.937127	−0.468564	−	0.883430i	$-0.655228\pi$
$89$	−8.84084	−0.937127	−0.468564	+	0.883430i	$0.655228\pi$
$90$	0	0
$91$	6.46867	0.678100
$92$	0	0
$93$	−1.63350	−0.169386
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.01314	−0.813611	−0.406806	−	0.913515i	$-0.633357\pi$
$97$	−8.01314	−0.813611	−0.406806	+	0.913515i	$0.633357\pi$
$98$	0	0
$99$	4.64201	0.466539
$100$	0	0
$101$	−6.33185	−0.630043	−0.315021	−	0.949085i	$-0.602012\pi$
$101$	−6.33185	−0.630043	−0.315021	+	0.949085i	$0.602012\pi$
$102$	0	0
$103$	13.1901	1.29966	0.649831	−	0.760078i	$-0.274839\pi$
$103$	13.1901	1.29966	0.649831	+	0.760078i	$0.274839\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.54923	0.923159	0.461580	−	0.887099i	$-0.347283\pi$
$107$	9.54923	0.923159	0.461580	+	0.887099i	$0.347283\pi$
$108$	0	0
$109$	−17.5622	−1.68215	−0.841077	−	0.540915i	$-0.818078\pi$
$109$	−17.5622	−1.68215	−0.841077	+	0.540915i	$0.818078\pi$
$110$	0	0
$111$	1.05505	0.100141
$112$	0	0
$113$	1.54267	0.145122	0.0725611	−	0.997364i	$-0.476883\pi$
$113$	1.54267	0.145122	0.0725611	+	0.997364i	$0.476883\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−7.54315	−0.697364
$118$	0	0
$119$	6.37925	0.584785
$120$	0	0
$121$	−8.38940	−0.762673
$122$	0	0
$123$	2.03544	0.183529
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.4256	−1.10260	−0.551299	−	0.834308i	$-0.685868\pi$
$127$	−12.4256	−1.10260	−0.551299	+	0.834308i	$0.685868\pi$
$128$	0	0
$129$	0.821251	0.0723071
$130$	0	0
$131$	11.3191	0.988957	0.494479	−	0.869190i	$-0.335359\pi$
$131$	11.3191	0.988957	0.494479	+	0.869190i	$0.335359\pi$
$132$	0	0
$133$	−9.91784	−0.859986
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.94093	0.422132	0.211066	−	0.977472i	$-0.432306\pi$
$137$	4.94093	0.422132	0.211066	+	0.977472i	$0.432306\pi$
$138$	0	0
$139$	2.93101	0.248605	0.124303	−	0.992244i	$-0.460331\pi$
$139$	2.93101	0.248605	0.124303	+	0.992244i	$0.460331\pi$
$140$	0	0
$141$	−2.45297	−0.206577
$142$	0	0
$143$	−4.24216	−0.354748
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0.331393	0.0273328
$148$	0	0
$149$	−5.09023	−0.417008	−0.208504	−	0.978022i	$-0.566859\pi$
$149$	−5.09023	−0.417008	−0.208504	+	0.978022i	$0.566859\pi$
$150$	0	0
$151$	14.3645	1.16897	0.584485	−	0.811405i	$-0.301297\pi$
$151$	14.3645	1.16897	0.584485	+	0.811405i	$0.301297\pi$
$152$	0	0
$153$	−7.43888	−0.601398
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.34146	−0.585912	−0.292956	−	0.956126i	$-0.594639\pi$
$157$	−7.34146	−0.585912	−0.292956	+	0.956126i	$0.594639\pi$
$158$	0	0
$159$	−2.41074	−0.191184
$160$	0	0
$161$	2.46376	0.194171
$162$	0	0
$163$	2.12762	0.166648	0.0833240	−	0.996523i	$-0.473446\pi$
$163$	2.12762	0.166648	0.0833240	+	0.996523i	$0.473446\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.16080	−0.167208	−0.0836039	−	0.996499i	$-0.526643\pi$
$167$	−2.16080	−0.167208	−0.0836039	+	0.996499i	$0.526643\pi$
$168$	0	0
$169$	−6.10659	−0.469738
$170$	0	0
$171$	11.5652	0.884417
$172$	0	0
$173$	−18.6210	−1.41573	−0.707863	−	0.706350i	$-0.750341\pi$
$173$	−18.6210	−1.41573	−0.707863	+	0.706350i	$0.750341\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.903162	0.0678858
$178$	0	0
$179$	13.4670	1.00657	0.503287	−	0.864119i	$-0.332124\pi$
$179$	13.4670	1.00657	0.503287	+	0.864119i	$0.332124\pi$
$180$	0	0
$181$	12.2927	0.913706	0.456853	−	0.889542i	$-0.348977\pi$
$181$	12.2927	0.913706	0.456853	+	0.889542i	$0.348977\pi$
$182$	0	0
$183$	−3.29845	−0.243828
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.18352	−0.305930
$188$	0	0
$189$	5.15657	0.375085
$190$	0	0
$191$	2.07099	0.149852	0.0749258	−	0.997189i	$-0.476128\pi$
$191$	2.07099	0.149852	0.0749258	+	0.997189i	$0.476128\pi$
$192$	0	0
$193$	−25.7996	−1.85710	−0.928549	−	0.371209i	$-0.878943\pi$
$193$	−25.7996	−1.85710	−0.928549	+	0.371209i	$0.878943\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.21443	0.300266	0.150133	−	0.988666i	$-0.452030\pi$
$197$	4.21443	0.300266	0.150133	+	0.988666i	$0.452030\pi$
$198$	0	0
$199$	22.0428	1.56257	0.781285	−	0.624175i	$-0.214565\pi$
$199$	22.0428	1.56257	0.781285	+	0.624175i	$0.214565\pi$
$200$	0	0
$201$	5.60519	0.395359
$202$	0	0
$203$	−17.4577	−1.22529
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.87300	−0.199687
$208$	0	0
$209$	6.50414	0.449901
$210$	0	0
$211$	−28.9217	−1.99105	−0.995524	−	0.0945049i	$-0.969873\pi$
$211$	−28.9217	−1.99105	−0.995524	+	0.0945049i	$0.969873\pi$
$212$	0	0
$213$	−1.87135	−0.128223
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.2931	0.766628
$218$	0	0
$219$	2.15198	0.145417
$220$	0	0
$221$	6.79812	0.457291
$222$	0	0
$223$	6.63731	0.444467	0.222234	−	0.974993i	$-0.428665\pi$
$223$	6.63731	0.444467	0.222234	+	0.974993i	$0.428665\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.370367	0.0245821	0.0122911	−	0.999924i	$-0.496088\pi$
$227$	0.370367	0.0245821	0.0122911	+	0.999924i	$0.496088\pi$
$228$	0	0
$229$	−11.5625	−0.764073	−0.382036	−	0.924147i	$-0.624777\pi$
$229$	−11.5625	−0.764073	−0.382036	+	0.924147i	$0.624777\pi$
$230$	0	0
$231$	1.41864	0.0933395
$232$	0	0
$233$	−16.8534	−1.10410	−0.552051	−	0.833810i	$-0.686155\pi$
$233$	−16.8534	−1.10410	−0.552051	+	0.833810i	$0.686155\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−0.483639	−0.0314157
$238$	0	0
$239$	17.7742	1.14972	0.574859	−	0.818252i	$-0.305057\pi$
$239$	17.7742	1.14972	0.574859	+	0.818252i	$0.305057\pi$
$240$	0	0
$241$	−26.6318	−1.71551	−0.857754	−	0.514061i	$-0.828141\pi$
$241$	−26.6318	−1.71551	−0.857754	+	0.514061i	$0.828141\pi$
$242$	0	0
$243$	−9.08467	−0.582782
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.5691	−0.672493
$248$	0	0
$249$	3.06301	0.194111
$250$	0	0
$251$	−16.8599	−1.06419	−0.532094	−	0.846685i	$-0.678595\pi$
$251$	−16.8599	−1.06419	−0.532094	+	0.846685i	$0.678595\pi$
$252$	0	0
$253$	−1.61574	−0.101580
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.33024	0.519626	0.259813	−	0.965659i	$-0.416339\pi$
$257$	8.33024	0.519626	0.259813	+	0.965659i	$0.416339\pi$
$258$	0	0
$259$	−7.29405	−0.453230
$260$	0	0
$261$	20.3575	1.26010
$262$	0	0
$263$	−15.0389	−0.927337	−0.463669	−	0.886009i	$-0.653467\pi$
$263$	−15.0389	−0.927337	−0.463669	+	0.886009i	$0.653467\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.15063	0.192815
$268$	0	0
$269$	−7.01922	−0.427969	−0.213985	−	0.976837i	$-0.568644\pi$
$269$	−7.01922	−0.427969	−0.213985	+	0.976837i	$0.568644\pi$
$270$	0	0
$271$	−4.40129	−0.267359	−0.133680	−	0.991025i	$-0.542679\pi$
$271$	−4.40129	−0.267359	−0.133680	+	0.991025i	$0.542679\pi$
$272$	0	0
$273$	−2.30525	−0.139520
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.41544	0.505634	0.252817	−	0.967514i	$-0.418643\pi$
$277$	8.41544	0.505634	0.252817	+	0.967514i	$0.418643\pi$
$278$	0	0
$279$	−13.1690	−0.788406
$280$	0	0
$281$	−15.0068	−0.895233	−0.447617	−	0.894226i	$-0.647727\pi$
$281$	−15.0068	−0.895233	−0.447617	+	0.894226i	$0.647727\pi$
$282$	0	0
$283$	7.09122	0.421529	0.210765	−	0.977537i	$-0.432405\pi$
$283$	7.09122	0.421529	0.210765	+	0.977537i	$0.432405\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.0719	−0.830637
$288$	0	0
$289$	−10.2958	−0.605638
$290$	0	0
$291$	2.85566	0.167402
$292$	0	0
$293$	28.0567	1.63909	0.819545	−	0.573015i	$-0.194226\pi$
$293$	28.0567	1.63909	0.819545	+	0.573015i	$0.194226\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.38169	−0.196225
$298$	0	0
$299$	2.62553	0.151838
$300$	0	0
$301$	−5.67767	−0.327256
$302$	0	0
$303$	2.25649	0.129632
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−29.2707	−1.67057	−0.835283	−	0.549820i	$-0.814696\pi$
$307$	−29.2707	−1.67057	−0.835283	+	0.549820i	$0.814696\pi$
$308$	0	0
$309$	−4.70059	−0.267407
$310$	0	0
$311$	15.8111	0.896563	0.448281	−	0.893893i	$-0.352036\pi$
$311$	15.8111	0.896563	0.448281	+	0.893893i	$0.352036\pi$
$312$	0	0
$313$	9.98602	0.564443	0.282222	−	0.959349i	$-0.408929\pi$
$313$	9.98602	0.564443	0.282222	+	0.959349i	$0.408929\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.7683	−0.773302	−0.386651	−	0.922226i	$-0.626368\pi$
$317$	−13.7683	−0.773302	−0.386651	+	0.922226i	$0.626368\pi$
$318$	0	0
$319$	11.4488	0.641008
$320$	0	0
$321$	−3.40308	−0.189941
$322$	0	0
$323$	−10.4230	−0.579950
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.25868	0.346106
$328$	0	0
$329$	16.9584	0.934950
$330$	0	0
$331$	−18.6765	−1.02656	−0.513278	−	0.858223i	$-0.671569\pi$
$331$	−18.6765	−1.02656	−0.513278	+	0.858223i	$0.671569\pi$
$332$	0	0
$333$	8.50564	0.466106
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−13.8915	−0.756716	−0.378358	−	0.925659i	$-0.623511\pi$
$337$	−13.8915	−0.756716	−0.378358	+	0.925659i	$0.623511\pi$
$338$	0	0
$339$	−0.549764	−0.0298591
$340$	0	0
$341$	−7.40606	−0.401060
$342$	0	0
$343$	−19.5374	−1.05492
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.706854	−0.0379459	−0.0189730	−	0.999820i	$-0.506040\pi$
$347$	−0.706854	−0.0379459	−0.0189730	+	0.999820i	$0.506040\pi$
$348$	0	0
$349$	6.93817	0.371392	0.185696	−	0.982607i	$-0.440546\pi$
$349$	6.93817	0.371392	0.185696	+	0.982607i	$0.440546\pi$
$350$	0	0
$351$	5.49516	0.293310
$352$	0	0
$353$	0.413170	0.0219908	0.0109954	−	0.999940i	$-0.496500\pi$
$353$	0.413170	0.0219908	0.0109954	+	0.999940i	$0.496500\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−2.27338	−0.120320
$358$	0	0
$359$	32.6209	1.72167	0.860834	−	0.508886i	$-0.169943\pi$
$359$	32.6209	1.72167	0.860834	+	0.508886i	$0.169943\pi$
$360$	0	0
$361$	−2.79537	−0.147125
$362$	0	0
$363$	2.98974	0.156921
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.5293	−0.706223	−0.353112	−	0.935581i	$-0.614876\pi$
$367$	−13.5293	−0.706223	−0.353112	+	0.935581i	$0.614876\pi$
$368$	0	0
$369$	16.4093	0.854234
$370$	0	0
$371$	16.6665	0.865282
$372$	0	0
$373$	−20.7690	−1.07538	−0.537689	−	0.843143i	$-0.680703\pi$
$373$	−20.7690	−1.07538	−0.537689	+	0.843143i	$0.680703\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−18.6040	−0.958153
$378$	0	0
$379$	25.0528	1.28688	0.643438	−	0.765498i	$-0.277507\pi$
$379$	25.0528	1.28688	0.643438	+	0.765498i	$0.277507\pi$
$380$	0	0
$381$	4.42815	0.226861
$382$	0	0
$383$	−4.51677	−0.230796	−0.115398	−	0.993319i	$-0.536814\pi$
$383$	−4.51677	−0.230796	−0.115398	+	0.993319i	$0.536814\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.62076	0.336552
$388$	0	0
$389$	0.455188	0.0230789	0.0115395	−	0.999933i	$-0.496327\pi$
$389$	0.455188	0.0230789	0.0115395	+	0.999933i	$0.496327\pi$
$390$	0	0
$391$	2.58924	0.130943
$392$	0	0
$393$	−4.03382	−0.203479
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.0798	0.756832	0.378416	−	0.925636i	$-0.376469\pi$
$397$	15.0798	0.756832	0.378416	+	0.925636i	$0.376469\pi$
$398$	0	0
$399$	3.53444	0.176943
$400$	0	0
$401$	−38.0845	−1.90185	−0.950924	−	0.309424i	$-0.899864\pi$
$401$	−38.0845	−1.90185	−0.950924	+	0.309424i	$0.899864\pi$
$402$	0	0
$403$	12.0347	0.599489
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.78345	0.237107
$408$	0	0
$409$	16.3672	0.809305	0.404653	−	0.914470i	$-0.367392\pi$
$409$	16.3672	0.809305	0.404653	+	0.914470i	$0.367392\pi$
$410$	0	0
$411$	−1.76081	−0.0868543
$412$	0	0
$413$	−6.24396	−0.307245
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.04453	−0.0511509
$418$	0	0
$419$	5.32621	0.260202	0.130101	−	0.991501i	$-0.458470\pi$
$419$	5.32621	0.260202	0.130101	+	0.991501i	$0.458470\pi$
$420$	0	0
$421$	−11.3000	−0.550730	−0.275365	−	0.961340i	$-0.588799\pi$
$421$	−11.3000	−0.550730	−0.275365	+	0.961340i	$0.588799\pi$
$422$	0	0
$423$	−19.7753	−0.961510
$424$	0	0
$425$	0	0
$426$	0	0
$427$	22.8036	1.10355
$428$	0	0
$429$	1.51179	0.0729898
$430$	0	0
$431$	−33.6364	−1.62021	−0.810105	−	0.586285i	$-0.800590\pi$
$431$	−33.6364	−1.62021	−0.810105	+	0.586285i	$0.800590\pi$
$432$	0	0
$433$	18.3932	0.883920	0.441960	−	0.897035i	$-0.354283\pi$
$433$	18.3932	0.883920	0.441960	+	0.897035i	$0.354283\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.02550	−0.192566
$438$	0	0
$439$	−28.7628	−1.37277	−0.686387	−	0.727236i	$-0.740804\pi$
$439$	−28.7628	−1.37277	−0.686387	+	0.727236i	$0.740804\pi$
$440$	0	0
$441$	2.67162	0.127220
$442$	0	0
$443$	−22.2359	−1.05646	−0.528229	−	0.849102i	$-0.677144\pi$
$443$	−22.2359	−1.05646	−0.528229	+	0.849102i	$0.677144\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.81401	0.0857999
$448$	0	0
$449$	28.4538	1.34282	0.671409	−	0.741087i	$-0.265689\pi$
$449$	28.4538	1.34282	0.671409	+	0.741087i	$0.265689\pi$
$450$	0	0
$451$	9.22836	0.434547
$452$	0	0
$453$	−5.11912	−0.240517
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.58512	0.354817	0.177408	−	0.984137i	$-0.443229\pi$
$457$	7.58512	0.354817	0.177408	+	0.984137i	$0.443229\pi$
$458$	0	0
$459$	5.41920	0.252947
$460$	0	0
$461$	6.02951	0.280822	0.140411	−	0.990093i	$-0.455158\pi$
$461$	6.02951	0.280822	0.140411	+	0.990093i	$0.455158\pi$
$462$	0	0
$463$	−24.2763	−1.12821	−0.564107	−	0.825701i	$-0.690780\pi$
$463$	−24.2763	−1.12821	−0.564107	+	0.825701i	$0.690780\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.99614	−0.277468	−0.138734	−	0.990330i	$-0.544303\pi$
$467$	−5.99614	−0.277468	−0.138734	+	0.990330i	$0.544303\pi$
$468$	0	0
$469$	−38.7511	−1.78936
$470$	0	0
$471$	2.61629	0.120552
$472$	0	0
$473$	3.72343	0.171203
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−19.4349	−0.889863
$478$	0	0
$479$	17.5065	0.799892	0.399946	−	0.916539i	$-0.369029\pi$
$479$	17.5065	0.799892	0.399946	+	0.916539i	$0.369029\pi$
$480$	0	0
$481$	−7.77299	−0.354418
$482$	0	0
$483$	−0.878013	−0.0399510
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.6269	1.43315	0.716576	−	0.697509i	$-0.245708\pi$
$487$	31.6269	1.43315	0.716576	+	0.697509i	$0.245708\pi$
$488$	0	0
$489$	−0.758224	−0.0342881
$490$	0	0
$491$	−24.7642	−1.11759	−0.558797	−	0.829305i	$-0.688737\pi$
$491$	−24.7642	−1.11759	−0.558797	+	0.829305i	$0.688737\pi$
$492$	0	0
$493$	−18.3468	−0.826299
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.9375	0.580326
$498$	0	0
$499$	−37.7906	−1.69174	−0.845871	−	0.533388i	$-0.820919\pi$
$499$	−37.7906	−1.69174	−0.845871	+	0.533388i	$0.820919\pi$
$500$	0	0
$501$	0.770049	0.0344032
$502$	0	0
$503$	−1.40481	−0.0626376	−0.0313188	−	0.999509i	$-0.509971\pi$
$503$	−1.40481	−0.0626376	−0.0313188	+	0.999509i	$0.509971\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.17622	0.0966492
$508$	0	0
$509$	30.4044	1.34765	0.673825	−	0.738891i	$-0.264650\pi$
$509$	30.4044	1.34765	0.673825	+	0.738891i	$0.264650\pi$
$510$	0	0
$511$	−14.8776	−0.658146
$512$	0	0
$513$	−8.42525	−0.371984
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−11.1214	−0.489118
$518$	0	0
$519$	6.63599	0.291288
$520$	0	0
$521$	−13.3500	−0.584873	−0.292437	−	0.956285i	$-0.594466\pi$
$521$	−13.3500	−0.584873	−0.292437	+	0.956285i	$0.594466\pi$
$522$	0	0
$523$	−8.71026	−0.380873	−0.190437	−	0.981699i	$-0.560990\pi$
$523$	−8.71026	−0.380873	−0.190437	+	0.981699i	$0.560990\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.8683	0.516991
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	7.28111	0.315973
$532$	0	0
$533$	−14.9959	−0.649543
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.79927	−0.207104
$538$	0	0
$539$	1.50248	0.0647166
$540$	0	0
$541$	21.2922	0.915423	0.457711	−	0.889101i	$-0.348669\pi$
$541$	21.2922	0.915423	0.457711	+	0.889101i	$0.348669\pi$
$542$	0	0
$543$	−4.38075	−0.187996
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.27222	−0.310937	−0.155469	−	0.987841i	$-0.549689\pi$
$547$	−7.27222	−0.310937	−0.155469	+	0.987841i	$0.549689\pi$
$548$	0	0
$549$	−26.5914	−1.13490
$550$	0	0
$551$	28.5238	1.21516
$552$	0	0
$553$	3.34361	0.142185
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−27.0665	−1.14684	−0.573422	−	0.819260i	$-0.694384\pi$
$557$	−27.0665	−1.14684	−0.573422	+	0.819260i	$0.694384\pi$
$558$	0	0
$559$	−6.05048	−0.255908
$560$	0	0
$561$	1.49089	0.0629454
$562$	0	0
$563$	−12.7921	−0.539124	−0.269562	−	0.962983i	$-0.586879\pi$
$563$	−12.7921	−0.539124	−0.269562	+	0.962983i	$0.586879\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	19.3974	0.814616
$568$	0	0
$569$	−13.3013	−0.557621	−0.278810	−	0.960346i	$-0.589940\pi$
$569$	−13.3013	−0.557621	−0.278810	+	0.960346i	$0.589940\pi$
$570$	0	0
$571$	−31.8239	−1.33179	−0.665894	−	0.746046i	$-0.731950\pi$
$571$	−31.8239	−1.33179	−0.665894	+	0.746046i	$0.731950\pi$
$572$	0	0
$573$	−0.738043	−0.0308322
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.0091	1.08277	0.541386	−	0.840774i	$-0.317900\pi$
$577$	26.0091	1.08277	0.541386	+	0.840774i	$0.317900\pi$
$578$	0	0
$579$	9.19426	0.382101
$580$	0	0
$581$	−21.1760	−0.878527
$582$	0	0
$583$	−10.9299	−0.452671
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.4259	1.62728	0.813641	−	0.581367i	$-0.197482\pi$
$587$	39.4259	1.62728	0.813641	+	0.581367i	$0.197482\pi$
$588$	0	0
$589$	−18.4517	−0.760289
$590$	0	0
$591$	−1.50191	−0.0617801
$592$	0	0
$593$	20.8918	0.857922	0.428961	−	0.903323i	$-0.358880\pi$
$593$	20.8918	0.857922	0.428961	+	0.903323i	$0.358880\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.85542	−0.321501
$598$	0	0
$599$	−12.7986	−0.522938	−0.261469	−	0.965212i	$-0.584207\pi$
$599$	−12.7986	−0.522938	−0.261469	+	0.965212i	$0.584207\pi$
$600$	0	0
$601$	−18.7566	−0.765099	−0.382550	−	0.923935i	$-0.624954\pi$
$601$	−18.7566	−0.765099	−0.382550	+	0.923935i	$0.624954\pi$
$602$	0	0
$603$	45.1879	1.84019
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.51153	0.0613512	0.0306756	−	0.999529i	$-0.490234\pi$
$607$	1.51153	0.0613512	0.0306756	+	0.999529i	$0.490234\pi$
$608$	0	0
$609$	6.22142	0.252105
$610$	0	0
$611$	18.0720	0.731114
$612$	0	0
$613$	−39.5673	−1.59811	−0.799054	−	0.601259i	$-0.794666\pi$
$613$	−39.5673	−1.59811	−0.799054	+	0.601259i	$0.794666\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	41.0023	1.65069	0.825345	−	0.564629i	$-0.190981\pi$
$617$	41.0023	1.65069	0.825345	+	0.564629i	$0.190981\pi$
$618$	0	0
$619$	−10.9089	−0.438464	−0.219232	−	0.975673i	$-0.570355\pi$
$619$	−10.9089	−0.438464	−0.219232	+	0.975673i	$0.570355\pi$
$620$	0	0
$621$	2.09297	0.0839880
$622$	0	0
$623$	−21.7817	−0.872664
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.31789	−0.0925677
$628$	0	0
$629$	−7.66555	−0.305645
$630$	0	0
$631$	24.7516	0.985347	0.492674	−	0.870214i	$-0.336020\pi$
$631$	24.7516	0.985347	0.492674	+	0.870214i	$0.336020\pi$
$632$	0	0
$633$	10.3069	0.409661
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.44150	−0.0967357
$638$	0	0
$639$	−15.0865	−0.596812
$640$	0	0
$641$	33.5110	1.32360	0.661802	−	0.749679i	$-0.269792\pi$
$641$	33.5110	1.32360	0.661802	+	0.749679i	$0.269792\pi$
$642$	0	0
$643$	−9.15767	−0.361143	−0.180572	−	0.983562i	$-0.557795\pi$
$643$	−9.15767	−0.361143	−0.180572	+	0.983562i	$0.557795\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.8011	−1.21092	−0.605458	−	0.795877i	$-0.707010\pi$
$647$	−30.8011	−1.21092	−0.605458	+	0.795877i	$0.707010\pi$
$648$	0	0
$649$	4.09480	0.160735
$650$	0	0
$651$	−4.02455	−0.157735
$652$	0	0
$653$	25.4779	0.997026	0.498513	−	0.866882i	$-0.333880\pi$
$653$	25.4779	0.997026	0.498513	+	0.866882i	$0.333880\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	17.3488	0.676843
$658$	0	0
$659$	−2.34417	−0.0913159	−0.0456580	−	0.998957i	$-0.514538\pi$
$659$	−2.34417	−0.0913159	−0.0456580	+	0.998957i	$0.514538\pi$
$660$	0	0
$661$	43.4164	1.68870	0.844351	−	0.535791i	$-0.179987\pi$
$661$	43.4164	1.68870	0.844351	+	0.535791i	$0.179987\pi$
$662$	0	0
$663$	−2.42266	−0.0940883
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.08580	−0.274363
$668$	0	0
$669$	−2.36535	−0.0914498
$670$	0	0
$671$	−14.9547	−0.577319
$672$	0	0
$673$	−48.4902	−1.86916	−0.934580	−	0.355754i	$-0.884224\pi$
$673$	−48.4902	−1.86916	−0.934580	+	0.355754i	$0.884224\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−46.9213	−1.80333	−0.901665	−	0.432435i	$-0.857655\pi$
$677$	−46.9213	−1.80333	−0.901665	+	0.432435i	$0.857655\pi$
$678$	0	0
$679$	−19.7424	−0.757645
$680$	0	0
$681$	−0.131988	−0.00505780
$682$	0	0
$683$	15.3660	0.587965	0.293983	−	0.955811i	$-0.405019\pi$
$683$	15.3660	0.587965	0.293983	+	0.955811i	$0.405019\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	4.12056	0.157209
$688$	0	0
$689$	17.7609	0.676635
$690$	0	0
$691$	40.0235	1.52257	0.761283	−	0.648419i	$-0.224570\pi$
$691$	40.0235	1.52257	0.761283	+	0.648419i	$0.224570\pi$
$692$	0	0
$693$	11.4368	0.434447
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−14.7886	−0.560157
$698$	0	0
$699$	6.00607	0.227171
$700$	0	0
$701$	−28.2943	−1.06866	−0.534331	−	0.845276i	$-0.679436\pi$
$701$	−28.2943	−1.06866	−0.534331	+	0.845276i	$0.679436\pi$
$702$	0	0
$703$	11.9177	0.449483
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.6001	−0.586703
$708$	0	0
$709$	26.8823	1.00959	0.504793	−	0.863240i	$-0.331569\pi$
$709$	26.8823	1.00959	0.504793	+	0.863240i	$0.331569\pi$
$710$	0	0
$711$	−3.89900	−0.146224
$712$	0	0
$713$	4.58371	0.171661
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.33423	−0.236556
$718$	0	0
$719$	−11.2624	−0.420015	−0.210008	−	0.977700i	$-0.567349\pi$
$719$	−11.2624	−0.420015	−0.210008	+	0.977700i	$0.567349\pi$
$720$	0	0
$721$	32.4973	1.21026
$722$	0	0
$723$	9.49084	0.352968
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.07629	0.0770054	0.0385027	−	0.999258i	$-0.487741\pi$
$727$	2.07629	0.0770054	0.0385027	+	0.999258i	$0.487741\pi$
$728$	0	0
$729$	−20.3818	−0.754883
$730$	0	0
$731$	−5.96684	−0.220692
$732$	0	0
$733$	−16.6294	−0.614221	−0.307111	−	0.951674i	$-0.599362\pi$
$733$	−16.6294	−0.614221	−0.307111	+	0.951674i	$0.599362\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.4131	0.936103
$738$	0	0
$739$	−50.8052	−1.86890	−0.934450	−	0.356095i	$-0.884108\pi$
$739$	−50.8052	−1.86890	−0.934450	+	0.356095i	$0.884108\pi$
$740$	0	0
$741$	3.76652	0.138366
$742$	0	0
$743$	31.3667	1.15073	0.575366	−	0.817896i	$-0.304859\pi$
$743$	31.3667	1.15073	0.575366	+	0.817896i	$0.304859\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	24.6934	0.903485
$748$	0	0
$749$	23.5270	0.859657
$750$	0	0
$751$	−41.2551	−1.50542	−0.752711	−	0.658351i	$-0.771254\pi$
$751$	−41.2551	−1.50542	−0.752711	+	0.658351i	$0.771254\pi$
$752$	0	0
$753$	6.00839	0.218958
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−11.3081	−0.411000	−0.205500	−	0.978657i	$-0.565882\pi$
$757$	−11.3081	−0.411000	−0.205500	+	0.978657i	$0.565882\pi$
$758$	0	0
$759$	0.575803	0.0209003
$760$	0	0
$761$	−4.32880	−0.156919	−0.0784594	−	0.996917i	$-0.525000\pi$
$761$	−4.32880	−0.156919	−0.0784594	+	0.996917i	$0.525000\pi$
$762$	0	0
$763$	−43.2690	−1.56644
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.65395	−0.240260
$768$	0	0
$769$	40.6517	1.46594	0.732968	−	0.680263i	$-0.238134\pi$
$769$	40.6517	1.46594	0.732968	+	0.680263i	$0.238134\pi$
$770$	0	0
$771$	−2.96866	−0.106914
$772$	0	0
$773$	−37.9716	−1.36574	−0.682872	−	0.730538i	$-0.739269\pi$
$773$	−37.9716	−1.36574	−0.682872	+	0.730538i	$0.739269\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.59940	0.0932528
$778$	0	0
$779$	22.9918	0.823768
$780$	0	0
$781$	−8.48443	−0.303597
$782$	0	0
$783$	−14.8304	−0.529994
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−55.1654	−1.96643	−0.983216	−	0.182443i	$-0.941599\pi$
$787$	−55.1654	−1.96643	−0.983216	+	0.182443i	$0.941599\pi$
$788$	0	0
$789$	5.35944	0.190801
$790$	0	0
$791$	3.80076	0.135140
$792$	0	0
$793$	24.3010	0.862952
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33.0601	−1.17105	−0.585524	−	0.810655i	$-0.699111\pi$
$797$	−33.0601	−1.17105	−0.585524	+	0.810655i	$0.699111\pi$
$798$	0	0
$799$	17.8222	0.630503
$800$	0	0
$801$	25.3997	0.897455
$802$	0	0
$803$	9.75675	0.344308
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.50145	0.0880553
$808$	0	0
$809$	−26.4318	−0.929291	−0.464646	−	0.885497i	$-0.653818\pi$
$809$	−26.4318	−0.929291	−0.464646	+	0.885497i	$0.653818\pi$
$810$	0	0
$811$	−5.49057	−0.192800	−0.0964000	−	0.995343i	$-0.530733\pi$
$811$	−5.49057	−0.192800	−0.0964000	+	0.995343i	$0.530733\pi$
$812$	0	0
$813$	1.56850	0.0550096
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.27667	0.324550
$818$	0	0
$819$	−18.5845	−0.649394
$820$	0	0
$821$	−42.2732	−1.47534	−0.737672	−	0.675160i	$-0.764075\pi$
$821$	−42.2732	−1.47534	−0.737672	+	0.675160i	$0.764075\pi$
$822$	0	0
$823$	52.2599	1.82167	0.910833	−	0.412774i	$-0.135440\pi$
$823$	52.2599	1.82167	0.910833	+	0.412774i	$0.135440\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−46.7667	−1.62624	−0.813119	−	0.582097i	$-0.802232\pi$
$827$	−46.7667	−1.62624	−0.813119	+	0.582097i	$0.802232\pi$
$828$	0	0
$829$	15.2175	0.528525	0.264262	−	0.964451i	$-0.414871\pi$
$829$	15.2175	0.528525	0.264262	+	0.964451i	$0.414871\pi$
$830$	0	0
$831$	−2.99902	−0.104035
$832$	0	0
$833$	−2.40775	−0.0834236
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.59356	0.331602
$838$	0	0
$839$	−54.7388	−1.88979	−0.944897	−	0.327369i	$-0.893838\pi$
$839$	−54.7388	−1.88979	−0.944897	+	0.327369i	$0.893838\pi$
$840$	0	0
$841$	21.2085	0.731328
$842$	0	0
$843$	5.34802	0.184195
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−20.6694	−0.710210
$848$	0	0
$849$	−2.52711	−0.0867302
$850$	0	0
$851$	−2.96054	−0.101486
$852$	0	0
$853$	−16.8683	−0.577559	−0.288779	−	0.957396i	$-0.593249\pi$
$853$	−16.8683	−0.577559	−0.288779	+	0.957396i	$0.593249\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.9513	1.36471	0.682355	−	0.731021i	$-0.260956\pi$
$857$	39.9513	1.36471	0.682355	+	0.731021i	$0.260956\pi$
$858$	0	0
$859$	−21.5728	−0.736053	−0.368027	−	0.929815i	$-0.619966\pi$
$859$	−21.5728	−0.736053	−0.368027	+	0.929815i	$0.619966\pi$
$860$	0	0
$861$	5.01482	0.170905
$862$	0	0
$863$	−2.71384	−0.0923803	−0.0461902	−	0.998933i	$-0.514708\pi$
$863$	−2.71384	−0.0923803	−0.0461902	+	0.998933i	$0.514708\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.66915	0.124611
$868$	0	0
$869$	−2.19275	−0.0743838
$870$	0	0
$871$	−41.2956	−1.39925
$872$	0	0
$873$	23.0217	0.779168
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−40.8294	−1.37871	−0.689355	−	0.724423i	$-0.742106\pi$
$877$	−40.8294	−1.37871	−0.689355	+	0.724423i	$0.742106\pi$
$878$	0	0
$879$	−9.99862	−0.337245
$880$	0	0
$881$	36.3676	1.22525	0.612627	−	0.790372i	$-0.290113\pi$
$881$	36.3676	1.22525	0.612627	+	0.790372i	$0.290113\pi$
$882$	0	0
$883$	−18.2495	−0.614146	−0.307073	−	0.951686i	$-0.599350\pi$
$883$	−18.2495	−0.614146	−0.307073	+	0.951686i	$0.599350\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.33546	−0.111994	−0.0559968	−	0.998431i	$-0.517834\pi$
$887$	−3.33546	−0.111994	−0.0559968	+	0.998431i	$0.517834\pi$
$888$	0	0
$889$	−30.6138	−1.02675
$890$	0	0
$891$	−12.7209	−0.426166
$892$	0	0
$893$	−27.7082	−0.927219
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.935665	−0.0312409
$898$	0	0
$899$	−32.4792	−1.08324
$900$	0	0
$901$	17.5153	0.583521
$902$	0	0
$903$	2.02336	0.0673333
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−5.50057	−0.182643	−0.0913217	−	0.995821i	$-0.529109\pi$
$907$	−5.50057	−0.182643	−0.0913217	+	0.995821i	$0.529109\pi$
$908$	0	0
$909$	18.1914	0.603371
$910$	0	0
$911$	−11.9531	−0.396024	−0.198012	−	0.980200i	$-0.563449\pi$
$911$	−11.9531	−0.396024	−0.198012	+	0.980200i	$0.563449\pi$
$912$	0	0
$913$	13.8872	0.459601
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.8876	0.920929
$918$	0	0
$919$	−22.3756	−0.738102	−0.369051	−	0.929409i	$-0.620317\pi$
$919$	−22.3756	−0.738102	−0.369051	+	0.929409i	$0.620317\pi$
$920$	0	0
$921$	10.4312	0.343721
$922$	0	0
$923$	13.7870	0.453804
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−37.8953	−1.24464
$928$	0	0
$929$	−42.9751	−1.40997	−0.704983	−	0.709224i	$-0.749045\pi$
$929$	−42.9751	−1.40997	−0.704983	+	0.709224i	$0.749045\pi$
$930$	0	0
$931$	3.74334	0.122683
$932$	0	0
$933$	−5.63461	−0.184469
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.76556	0.123015	0.0615077	−	0.998107i	$-0.480409\pi$
$937$	3.76556	0.123015	0.0615077	+	0.998107i	$0.480409\pi$
$938$	0	0
$939$	−3.55874	−0.116135
$940$	0	0
$941$	−15.9193	−0.518953	−0.259477	−	0.965749i	$-0.583550\pi$
$941$	−15.9193	−0.518953	−0.259477	+	0.965749i	$0.583550\pi$
$942$	0	0
$943$	−5.71155	−0.185994
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.8553	1.68507	0.842535	−	0.538642i	$-0.181062\pi$
$947$	51.8553	1.68507	0.842535	+	0.538642i	$0.181062\pi$
$948$	0	0
$949$	−15.8545	−0.514658
$950$	0	0
$951$	4.90662	0.159108
$952$	0	0
$953$	−44.6848	−1.44748	−0.723741	−	0.690072i	$-0.757579\pi$
$953$	−44.6848	−1.44748	−0.723741	+	0.690072i	$0.757579\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.08002	−0.131888
$958$	0	0
$959$	12.1733	0.393095
$960$	0	0
$961$	−9.98964	−0.322247
$962$	0	0
$963$	−27.4349	−0.884079
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−31.5594	−1.01488	−0.507440	−	0.861687i	$-0.669408\pi$
$967$	−31.5594	−1.01488	−0.507440	+	0.861687i	$0.669408\pi$
$968$	0	0
$969$	3.71445	0.119325
$970$	0	0
$971$	20.6291	0.662018	0.331009	−	0.943628i	$-0.392611\pi$
$971$	20.6291	0.662018	0.331009	+	0.943628i	$0.392611\pi$
$972$	0	0
$973$	7.22130	0.231504
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.76933	0.0566058	0.0283029	−	0.999599i	$-0.490990\pi$
$977$	1.76933	0.0566058	0.0283029	+	0.999599i	$0.490990\pi$
$978$	0	0
$979$	14.2845	0.456533
$980$	0	0
$981$	50.4562	1.61094
$982$	0	0
$983$	43.4073	1.38448	0.692239	−	0.721668i	$-0.256624\pi$
$983$	43.4073	1.38448	0.692239	+	0.721668i	$0.256624\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−6.04351	−0.192367
$988$	0	0
$989$	−2.30448	−0.0732781
$990$	0	0
$991$	−16.2222	−0.515317	−0.257658	−	0.966236i	$-0.582951\pi$
$991$	−16.2222	−0.515317	−0.257658	+	0.966236i	$0.582951\pi$
$992$	0	0
$993$	6.65579	0.211215
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.61347	−0.0827695	−0.0413847	−	0.999143i	$-0.513177\pi$
$997$	−2.61347	−0.0827695	−0.0413847	+	0.999143i	$0.513177\pi$
$998$	0	0
$999$	−6.19633	−0.196043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.cz.1.4		7
4.3	odd	2		4600.2.a.bi.1.4		7
5.2	odd	4		1840.2.e.g.369.9		14
5.3	odd	4		1840.2.e.g.369.6		14
5.4	even	2		9200.2.a.dc.1.4		7
20.3	even	4		920.2.e.b.369.9	yes	14
20.7	even	4		920.2.e.b.369.6	✓	14
20.19	odd	2		4600.2.a.bh.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.b.369.6	✓	14	20.7	even	4
920.2.e.b.369.9	yes	14	20.3	even	4
1840.2.e.g.369.6		14	5.3	odd	4
1840.2.e.g.369.9		14	5.2	odd	4
4600.2.a.bh.1.4		7	20.19	odd	2
4600.2.a.bi.1.4		7	4.3	odd	2
9200.2.a.cz.1.4		7	1.1	even	1	trivial
9200.2.a.dc.1.4		7	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.356372 q^{3} +2.46376 q^{7} -2.87300 q^{9} +O(q^{10})\)	\(q-0.356372 q^{3} +2.46376 q^{7} -2.87300 q^{9} -1.61574 q^{11} +2.62553 q^{13} +2.58924 q^{17} -4.02550 q^{19} -0.878013 q^{21} +1.00000 q^{23} +2.09297 q^{27} -7.08580 q^{29} +4.58371 q^{31} +0.575803 q^{33} -2.96054 q^{37} -0.935665 q^{39} -5.71155 q^{41} -2.30448 q^{43} +6.88317 q^{47} -0.929907 q^{49} -0.922731 q^{51} +6.76467 q^{53} +1.43457 q^{57} -2.53432 q^{59} +9.25564 q^{61} -7.07837 q^{63} -15.7285 q^{67} -0.356372 q^{69} +5.25113 q^{71} -6.03858 q^{73} -3.98078 q^{77} +1.35712 q^{79} +7.87312 q^{81} -8.59500 q^{83} +2.52518 q^{87} -8.84084 q^{89} +6.46867 q^{91} -1.63350 q^{93} -8.01314 q^{97} +4.64201 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9	\( 7 q - 3 q^{3} - 4 q^{7} + 2 q^{9} + 7 q^{11} - 7 q^{13} + 7 q^{19} - 6 q^{21} + 7 q^{23} - 11 q^{29} + 10 q^{31} - 19 q^{33} - 19 q^{37} + 24 q^{39} - 16 q^{41} - 6 q^{43} - 6 q^{47} - 17 q^{49} + 7 q^{51} - 15 q^{53} - 8 q^{57} + 11 q^{59} + 5 q^{61} - 13 q^{63} - 9 q^{67} - 3 q^{69} + 14 q^{71} - 10 q^{73} - 6 q^{77} + 32 q^{79} - 5 q^{81} - q^{83} - 10 q^{87} - 24 q^{89} + 7 q^{91} - 26 q^{93} + 7 q^{97} + 61 q^{99}+O(q^{100}) \) 7 * q - 3 * q^3 - 4 * q^7 + 2 * q^9 + 7 * q^11 - 7 * q^13 + 7 * q^19 - 6 * q^21 + 7 * q^23 - 11 * q^29 + 10 * q^31 - 19 * q^33 - 19 * q^37 + 24 * q^39 - 16 * q^41 - 6 * q^43 - 6 * q^47 - 17 * q^49 + 7 * q^51 - 15 * q^53 - 8 * q^57 + 11 * q^59 + 5 * q^61 - 13 * q^63 - 9 * q^67 - 3 * q^69 + 14 * q^71 - 10 * q^73 - 6 * q^77 + 32 * q^79 - 5 * q^81 - q^83 - 10 * q^87 - 24 * q^89 + 7 * q^91 - 26 * q^93 + 7 * q^97 + 61 * q^99

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