LMFDB - Embedded newform 245.2.b.d.99.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [245,2,Mod(99,245)] 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("245.99"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	245.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$1.95633484952$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			99.1
Root			$-2.23607i$ of defining polynomial
Character	$\chi$	$=$	245.99
Dual form			245.2.b.d.99.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.00000 q^{9} -3.00000 q^{11} -4.47214i q^{12} +6.70820i q^{13} -5.00000 q^{15} +4.00000 q^{16} -2.23607i q^{17} -4.47214i q^{20} -5.00000 q^{25} -2.23607i q^{27} +9.00000 q^{29} +6.70820i q^{33} -4.00000 q^{36} +15.0000 q^{39} -6.00000 q^{44} +4.47214i q^{45} +11.1803i q^{47} -8.94427i q^{48} -5.00000 q^{51} +13.4164i q^{52} +6.70820i q^{55} -10.0000 q^{60} +8.00000 q^{64} +15.0000 q^{65} -4.47214i q^{68} -12.0000 q^{71} +13.4164i q^{73} +11.1803i q^{75} +1.00000 q^{79} -8.94427i q^{80} -11.0000 q^{81} -8.94427i q^{83} -5.00000 q^{85} -20.1246i q^{87} +6.70820i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} - 4 q^{9} - 6 q^{11} - 10 q^{15} + 8 q^{16} - 10 q^{25} + 18 q^{29} - 8 q^{36} + 30 q^{39} - 12 q^{44} - 10 q^{51} - 20 q^{60} + 16 q^{64} + 30 q^{65} - 24 q^{71} + 2 q^{79} - 22 q^{81} - 10 q^{85}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q + 4 * q^4 - 4 * q^9 - 6 * q^11 - 10 * q^15 + 8 * q^16 - 10 * q^25 + 18 * q^29 - 8 * q^36 + 30 * q^39 - 12 * q^44 - 10 * q^51 - 20 * q^60 + 16 * q^64 + 30 * q^65 - 24 * q^71 + 2 * q^79 - 22 * q^81 - 10 * q^85 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$1$	$-1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	− 2.23607i	− 1.29099i	−0.763763	−	0.645497i	$-0.776650\pi$
$3$	− 2.23607i	− 1.29099i	0.763763	−	0.645497i	$-0.223350\pi$
$4$	2.00000	1.00000
$5$	− 2.23607i	− 1.00000i
$5$	− 2.23607i	− 1.00000i
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	− 4.47214i	− 1.29099i
$13$	6.70820i	1.86052i	0.366900	+	0.930261i	$0.380419\pi$
$13$	6.70820i	1.86052i	−0.366900	+	0.930261i	$0.619581\pi$
$14$	0	0
$15$	−5.00000	−1.29099
$16$	4.00000	1.00000
$17$	− 2.23607i	− 0.542326i	−0.962533	−	0.271163i	$-0.912592\pi$
$17$	− 2.23607i	− 0.542326i	0.962533	−	0.271163i	$-0.0874083\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	− 4.47214i	− 1.00000i
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	− 2.23607i	− 0.430331i
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	6.70820i	1.16775i
$34$	0	0
$35$	0	0
$36$	−4.00000	−0.666667
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	15.0000	2.40192
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−6.00000	−0.904534
$45$	4.47214i	0.666667i
$46$	0	0
$47$	11.1803i	1.63082i	0.578884	+	0.815410i	$0.303489\pi$
$47$	11.1803i	1.63082i	−0.578884	+	0.815410i	$0.696511\pi$
$48$	− 8.94427i	− 1.29099i
$49$	0	0
$50$	0	0
$51$	−5.00000	−0.700140
$52$	13.4164i	1.86052i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	6.70820i	0.904534i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−10.0000	−1.29099
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	8.00000	1.00000
$65$	15.0000	1.86052
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	− 4.47214i	− 0.542326i
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	13.4164i	1.57027i	0.619324	+	0.785136i	$0.287407\pi$
$73$	13.4164i	1.57027i	−0.619324	+	0.785136i	$0.712593\pi$
$74$	0	0
$75$	11.1803i	1.29099i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	− 8.94427i	− 1.00000i
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 8.94427i	− 0.981761i	−0.871227	−	0.490881i	$-0.836675\pi$
$83$	− 8.94427i	− 0.981761i	0.871227	−	0.490881i	$-0.163325\pi$
$84$	0	0
$85$	−5.00000	−0.542326
$86$	0	0
$87$	− 20.1246i	− 2.15758i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.70820i	0.681115i	0.940224	+	0.340557i	$0.110616\pi$
$97$	6.70820i	0.681115i	−0.940224	+	0.340557i	$0.889384\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	−10.0000	−1.00000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 20.1246i	− 1.98294i	−0.130347	−	0.991468i	$-0.541609\pi$
$103$	− 20.1246i	− 1.98294i	0.130347	−	0.991468i	$-0.458391\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	− 4.47214i	− 0.430331i
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	18.0000	1.67126
$117$	− 13.4164i	− 1.24035i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803i	1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	13.4164i	1.16775i
$133$	0	0
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	25.0000	2.10538
$142$	0	0
$143$	− 20.1246i	− 1.68290i
$144$	−8.00000	−0.666667
$145$	− 20.1246i	− 1.67126i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	0	0
$153$	4.47214i	0.361551i
$154$	0	0
$155$	0	0
$156$	30.0000	2.40192
$157$	13.4164i	1.07075i	0.844616	+	0.535373i	$0.179829\pi$
$157$	13.4164i	1.07075i	−0.844616	+	0.535373i	$0.820171\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	15.0000	1.16775
$166$	0	0
$167$	− 24.5967i	− 1.90335i	−0.307102	−	0.951677i	$-0.599359\pi$
$167$	− 24.5967i	− 1.90335i	0.307102	−	0.951677i	$-0.400641\pi$
$168$	0	0
$169$	−32.0000	−2.46154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.1803i	0.850026i	0.905187	+	0.425013i	$0.139730\pi$
$173$	11.1803i	0.850026i	−0.905187	+	0.425013i	$0.860270\pi$
$174$	0	0
$175$	0	0
$176$	−12.0000	−0.904534
$177$	0	0
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	8.94427i	0.666667i
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.70820i	0.490552i
$188$	22.3607i	1.63082i
$189$	0	0
$190$	0	0
$191$	−27.0000	−1.95365	−0.976826	−	0.214036i	$-0.931339\pi$
$191$	−27.0000	−1.95365	−0.976826	+	0.214036i	$0.931339\pi$
$192$	− 17.8885i	− 1.29099i
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	− 33.5410i	− 2.40192i
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	−10.0000	−0.700140
$205$	0	0
$206$	0	0
$207$	0	0
$208$	26.8328i	1.86052i
$209$	0	0
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	0	0
$213$	26.8328i	1.83855i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	30.0000	2.02721
$220$	13.4164i	0.904534i
$221$	15.0000	1.00901
$222$	0	0
$223$	6.70820i	0.449215i	0.974449	+	0.224607i	$0.0721099\pi$
$223$	6.70820i	0.449215i	−0.974449	+	0.224607i	$0.927890\pi$
$224$	0	0
$225$	10.0000	0.666667
$226$	0	0
$227$	29.0689i	1.92937i	0.263407	+	0.964685i	$0.415154\pi$
$227$	29.0689i	1.92937i	−0.263407	+	0.964685i	$0.584846\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	25.0000	1.63082
$236$	0	0
$237$	− 2.23607i	− 0.145248i
$238$	0	0
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	−20.0000	−1.29099
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	17.8885i	1.14755i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−20.0000	−1.26745
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	11.1803i	0.700140i
$256$	16.0000	1.00000
$257$	− 4.47214i	− 0.278964i	−0.990225	−	0.139482i	$-0.955456\pi$
$257$	− 4.47214i	− 0.278964i	0.990225	−	0.139482i	$-0.0445438\pi$
$258$	0	0
$259$	0	0
$260$	30.0000	1.86052
$261$	−18.0000	−1.11417
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	− 8.94427i	− 0.542326i
$273$	0	0
$274$	0	0
$275$	15.0000	0.904534
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−33.0000	−1.96861	−0.984307	−	0.176462i	$-0.943535\pi$
$281$	−33.0000	−1.96861	−0.984307	+	0.176462i	$0.943535\pi$
$282$	0	0
$283$	− 33.5410i	− 1.99381i	−0.0786368	−	0.996903i	$-0.525057\pi$
$283$	− 33.5410i	− 1.99381i	0.0786368	−	0.996903i	$-0.474943\pi$
$284$	−24.0000	−1.42414
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12.0000	0.705882
$290$	0	0
$291$	15.0000	0.879316
$292$	26.8328i	1.57027i
$293$	− 24.5967i	− 1.43696i	−0.695549	−	0.718479i	$-0.744839\pi$
$293$	− 24.5967i	− 1.43696i	0.695549	−	0.718479i	$-0.255161\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.70820i	0.389249i
$298$	0	0
$299$	0	0
$300$	22.3607i	1.29099i
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.70820i	0.382857i	0.981507	+	0.191429i	$0.0613121\pi$
$307$	6.70820i	0.382857i	−0.981507	+	0.191429i	$0.938688\pi$
$308$	0	0
$309$	−45.0000	−2.55996
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 20.1246i	− 1.13751i	−0.822507	−	0.568755i	$-0.807425\pi$
$313$	− 20.1246i	− 1.13751i	0.822507	−	0.568755i	$-0.192575\pi$
$314$	0	0
$315$	0	0
$316$	2.00000	0.112509
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	−27.0000	−1.51171
$320$	− 17.8885i	− 1.00000i
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−22.0000	−1.22222
$325$	− 33.5410i	− 1.86052i
$326$	0	0
$327$	− 24.5967i	− 1.36020i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	− 17.8885i	− 0.981761i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	−10.0000	−0.542326
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	− 40.2492i	− 2.15758i
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	15.0000	0.800641
$352$	0	0
$353$	29.0689i	1.54718i	0.633686	+	0.773590i	$0.281541\pi$
$353$	29.0689i	1.54718i	−0.633686	+	0.773590i	$0.718459\pi$
$354$	0	0
$355$	26.8328i	1.42414i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	4.47214i	0.234726i
$364$	0	0
$365$	30.0000	1.57027
$366$	0	0
$367$	− 33.5410i	− 1.75083i	−0.483375	−	0.875413i	$-0.660589\pi$
$367$	− 33.5410i	− 1.75083i	0.483375	−	0.875413i	$-0.339411\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	25.0000	1.29099
$376$	0	0
$377$	60.3738i	3.10941i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 35.7771i	− 1.82812i	−0.405575	−	0.914062i	$-0.632929\pi$
$383$	− 35.7771i	− 1.82812i	0.405575	−	0.914062i	$-0.367071\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	13.4164i	0.681115i
$389$	39.0000	1.97738	0.988689	−	0.149979i	$-0.0479205\pi$
$389$	39.0000	1.97738	0.988689	+	0.149979i	$0.0479205\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 2.23607i	− 0.112509i
$396$	12.0000	0.603023
$397$	− 20.1246i	− 1.01003i	−0.863112	−	0.505013i	$-0.831488\pi$
$397$	− 20.1246i	− 1.01003i	0.863112	−	0.505013i	$-0.168512\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	24.5967i	1.22222i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	− 40.2492i	− 1.98294i
$413$	0	0
$414$	0	0
$415$	−20.0000	−0.981761
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	37.0000	1.80327	0.901635	−	0.432498i	$-0.142368\pi$
$421$	37.0000	1.80327	0.901635	+	0.432498i	$0.142368\pi$
$422$	0	0
$423$	− 22.3607i	− 1.08721i
$424$	0	0
$425$	11.1803i	0.542326i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−45.0000	−2.17262
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	− 8.94427i	− 0.430331i
$433$	− 40.2492i	− 1.93425i	−0.254293	−	0.967127i	$-0.581843\pi$
$433$	− 40.2492i	− 1.93425i	0.254293	−	0.967127i	$-0.418157\pi$
$434$	0	0
$435$	−45.0000	−2.15758
$436$	22.0000	1.05361
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.4164i	0.634574i
$448$	0	0
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	38.0132i	1.78601i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	36.0000	1.67126
$465$	0	0
$466$	0	0
$467$	42.4853i	1.96598i	0.183646	+	0.982992i	$0.441210\pi$
$467$	42.4853i	1.96598i	−0.183646	+	0.982992i	$0.558790\pi$
$468$	− 26.8328i	− 1.24035i
$469$	0	0
$470$	0	0
$471$	30.0000	1.38233
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−4.00000	−0.181818
$485$	15.0000	0.681115
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	0	0
$493$	− 20.1246i	− 0.906367i
$494$	0	0
$495$	− 13.4164i	− 0.603023i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−41.0000	−1.83541	−0.917706	−	0.397260i	$-0.869961\pi$
$499$	−41.0000	−1.83541	−0.917706	+	0.397260i	$0.869961\pi$
$500$	22.3607i	1.00000i
$501$	−55.0000	−2.45722
$502$	0	0
$503$	38.0132i	1.69492i	0.530857	+	0.847461i	$0.321870\pi$
$503$	38.0132i	1.69492i	−0.530857	+	0.847461i	$0.678130\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	71.5542i	3.17783i
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−45.0000	−1.98294
$516$	0	0
$517$	− 33.5410i	− 1.47513i
$518$	0	0
$519$	25.0000	1.09738
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	26.8328i	1.17332i	0.809834	+	0.586659i	$0.199557\pi$
$523$	26.8328i	1.17332i	−0.809834	+	0.586659i	$0.800443\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	26.8328i	1.16775i
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	53.6656i	2.31584i
$538$	0	0
$539$	0	0
$540$	−10.0000	−0.430331
$541$	43.0000	1.84871	0.924357	−	0.381528i	$-0.124602\pi$
$541$	43.0000	1.84871	0.924357	+	0.381528i	$0.124602\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 24.5967i	− 1.05361i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	15.0000	0.633300
$562$	0	0
$563$	44.7214i	1.88478i	0.334515	+	0.942390i	$0.391427\pi$
$563$	44.7214i	1.88478i	−0.334515	+	0.942390i	$0.608573\pi$
$564$	50.0000	2.10538
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	− 40.2492i	− 1.68290i
$573$	60.3738i	2.52215i
$574$	0	0
$575$	0	0
$576$	−16.0000	−0.666667
$577$	− 33.5410i	− 1.39633i	−0.715936	−	0.698165i	$-0.754000\pi$
$577$	− 33.5410i	− 1.39633i	0.715936	−	0.698165i	$-0.246000\pi$
$578$	0	0
$579$	0	0
$580$	− 40.2492i	− 1.67126i
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−30.0000	−1.24035
$586$	0	0
$587$	− 8.94427i	− 0.369170i	−0.982817	−	0.184585i	$-0.940906\pi$
$587$	− 8.94427i	− 0.369170i	0.982817	−	0.184585i	$-0.0590940\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	42.4853i	1.74466i	0.488916	+	0.872331i	$0.337392\pi$
$593$	42.4853i	1.74466i	−0.488916	+	0.872331i	$0.662608\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	0	0
$598$	0	0
$599$	39.0000	1.59350	0.796748	−	0.604311i	$-0.206552\pi$
$599$	39.0000	1.59350	0.796748	+	0.604311i	$0.206552\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	−34.0000	−1.38344
$605$	4.47214i	0.181818i
$606$	0	0
$607$	− 20.1246i	− 0.816833i	−0.912796	−	0.408416i	$-0.866081\pi$
$607$	− 20.1246i	− 0.816833i	0.912796	−	0.408416i	$-0.133919\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−75.0000	−3.03418
$612$	8.94427i	0.361551i
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	60.0000	2.40192
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	26.8328i	1.07075i
$629$	0	0
$630$	0	0
$631$	−47.0000	−1.87104	−0.935520	−	0.353273i	$-0.885069\pi$
$631$	−47.0000	−1.87104	−0.935520	+	0.353273i	$0.885069\pi$
$632$	0	0
$633$	− 51.4296i	− 2.04414i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	0	0
$643$	6.70820i	0.264546i	0.991213	+	0.132273i	$0.0422275\pi$
$643$	6.70820i	0.264546i	−0.991213	+	0.132273i	$0.957772\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 17.8885i	− 0.703271i	−0.936137	−	0.351636i	$-0.885626\pi$
$647$	− 17.8885i	− 0.703271i	0.936137	−	0.351636i	$-0.114374\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 26.8328i	− 1.04685i
$658$	0	0
$659$	51.0000	1.98668	0.993339	−	0.115229i	$-0.0367601\pi$
$659$	51.0000	1.98668	0.993339	+	0.115229i	$0.0367601\pi$
$660$	30.0000	1.16775
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	− 33.5410i	− 1.30263i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 49.1935i	− 1.90335i
$669$	15.0000	0.579934
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	11.1803i	0.430331i
$676$	−64.0000	−2.46154
$677$	− 51.4296i	− 1.97660i	−0.152527	−	0.988299i	$-0.548741\pi$
$677$	− 51.4296i	− 1.97660i	0.152527	−	0.988299i	$-0.451259\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	65.0000	2.49081
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	22.3607i	0.850026i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	0	0
$703$	0	0
$704$	−24.0000	−0.904534
$705$	− 55.9017i	− 2.10538i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	−2.00000	−0.0750059
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−45.0000	−1.68290
$716$	−48.0000	−1.79384
$717$	− 20.1246i	− 0.751567i
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	17.8885i	0.666667i
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−45.0000	−1.67126
$726$	0	0
$727$	53.6656i	1.99035i	0.0981255	+	0.995174i	$0.468715\pi$
$727$	53.6656i	1.99035i	−0.0981255	+	0.995174i	$0.531285\pi$
$728$	0	0
$729$	7.00000	0.259259
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 20.1246i	− 0.743319i	−0.928369	−	0.371660i	$-0.878789\pi$
$733$	− 20.1246i	− 0.743319i	0.928369	−	0.371660i	$-0.121211\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	13.4164i	0.491539i
$746$	0	0
$747$	17.8885i	0.654508i
$748$	13.4164i	0.490552i
$749$	0	0
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	44.7214i	1.63082i
$753$	0	0
$754$	0	0
$755$	38.0132i	1.38344i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	−54.0000	−1.95365
$765$	10.0000	0.361551
$766$	0	0
$767$	0	0
$768$	− 35.7771i	− 1.29099i
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	−10.0000	−0.360141
$772$	0	0
$773$	− 2.23607i	− 0.0804258i	−0.999191	−	0.0402129i	$-0.987196\pi$
$773$	− 2.23607i	− 0.0804258i	0.999191	−	0.0402129i	$-0.0128036\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	− 67.0820i	− 2.40192i
$781$	36.0000	1.28818
$782$	0	0
$783$	− 20.1246i	− 0.719195i
$784$	0	0
$785$	30.0000	1.07075
$786$	0	0
$787$	− 33.5410i	− 1.19561i	−0.801642	−	0.597804i	$-0.796040\pi$
$787$	− 33.5410i	− 1.19561i	0.801642	−	0.597804i	$-0.203960\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 55.9017i	− 1.98014i	−0.140576	−	0.990070i	$-0.544895\pi$
$797$	− 55.9017i	− 1.98014i	0.140576	−	0.990070i	$-0.455105\pi$
$798$	0	0
$799$	25.0000	0.884436
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 40.2492i	− 1.42036i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−20.0000	−0.700140
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	57.0000	1.98931	0.994657	−	0.103236i	$-0.0329198\pi$
$821$	57.0000	1.98931	0.994657	+	0.103236i	$0.0329198\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	− 33.5410i	− 1.16775i
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	53.6656i	1.86052i
$833$	0	0
$834$	0	0
$835$	−55.0000	−1.90335
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	73.7902i	2.54147i
$844$	46.0000	1.58339
$845$	71.5542i	2.46154i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−75.0000	−2.57399
$850$	0	0
$851$	0	0
$852$	53.6656i	1.83855i
$853$	− 40.2492i	− 1.37811i	−0.724710	−	0.689054i	$-0.758026\pi$
$853$	− 40.2492i	− 1.37811i	0.724710	−	0.689054i	$-0.241974\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 49.1935i	− 1.68042i	−0.542263	−	0.840209i	$-0.682432\pi$
$857$	− 49.1935i	− 1.68042i	0.542263	−	0.840209i	$-0.317568\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	25.0000	0.850026
$866$	0	0
$867$	− 26.8328i	− 0.911290i
$868$	0	0
$869$	−3.00000	−0.101768
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 13.4164i	− 0.454077i
$874$	0	0
$875$	0	0
$876$	60.0000	2.02721
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	−55.0000	−1.85510
$880$	26.8328i	0.904534i
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	30.0000	1.00901
$885$	0	0
$886$	0	0
$887$	− 35.7771i	− 1.20128i	−0.799521	−	0.600639i	$-0.794913\pi$
$887$	− 35.7771i	− 1.20128i	0.799521	−	0.600639i	$-0.205087\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	33.0000	1.10554
$892$	13.4164i	0.449215i
$893$	0	0
$894$	0	0
$895$	53.6656i	1.79384i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	20.0000	0.666667
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	58.1378i	1.92937i
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	26.8328i	0.888037i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	15.0000	0.494267
$922$	0	0
$923$	− 80.4984i	− 2.64964i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	40.2492i	1.32196i
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	15.0000	0.490552
$936$	0	0
$937$	6.70820i	0.219147i	0.993979	+	0.109574i	$0.0349486\pi$
$937$	6.70820i	0.219147i	−0.993979	+	0.109574i	$0.965051\pi$
$938$	0	0
$939$	−45.0000	−1.46852
$940$	50.0000	1.63082
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	− 4.47214i	− 0.145248i
$949$	−90.0000	−2.92152
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	60.3738i	1.95365i
$956$	18.0000	0.582162
$957$	60.3738i	1.95161i
$958$	0	0
$959$	0	0
$960$	−40.0000	−1.29099
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	35.7771i	1.14755i
$973$	0	0
$974$	0	0
$975$	−75.0000	−2.40192
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	29.0689i	0.927153i	0.886057	+	0.463577i	$0.153434\pi$
$983$	29.0689i	0.927153i	−0.886057	+	0.463577i	$0.846566\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	− 17.8885i	− 0.567676i
$994$	0	0
$995$	0	0
$996$	−40.0000	−1.26745
$997$	60.3738i	1.91206i	0.293271	+	0.956029i	$0.405256\pi$
$997$	60.3738i	1.91206i	−0.293271	+	0.956029i	$0.594744\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.2.b.d.99.1	✓	2
3.2	odd	2		2205.2.d.h.1324.2		2
5.2	odd	4		1225.2.a.q.1.1		2
5.3	odd	4		1225.2.a.q.1.2		2
5.4	even	2	inner	245.2.b.d.99.2	yes	2
7.2	even	3		245.2.j.b.214.2		4
7.3	odd	6		245.2.j.b.79.2		4
7.4	even	3		245.2.j.b.79.1		4
7.5	odd	6		245.2.j.b.214.1		4
7.6	odd	2	inner	245.2.b.d.99.2	yes	2
15.14	odd	2		2205.2.d.h.1324.1		2
21.20	even	2		2205.2.d.h.1324.1		2
35.4	even	6		245.2.j.b.79.2		4
35.9	even	6		245.2.j.b.214.1		4
35.13	even	4		1225.2.a.q.1.1		2
35.19	odd	6		245.2.j.b.214.2		4
35.24	odd	6		245.2.j.b.79.1		4
35.27	even	4		1225.2.a.q.1.2		2
35.34	odd	2	CM	245.2.b.d.99.1	✓	2
105.104	even	2		2205.2.d.h.1324.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.2.b.d.99.1	✓	2	1.1	even	1	trivial
245.2.b.d.99.1	✓	2	35.34	odd	2	CM
245.2.b.d.99.2	yes	2	5.4	even	2	inner
245.2.b.d.99.2	yes	2	7.6	odd	2	inner
245.2.j.b.79.1		4	7.4	even	3
245.2.j.b.79.1		4	35.24	odd	6
245.2.j.b.79.2		4	7.3	odd	6
245.2.j.b.79.2		4	35.4	even	6
245.2.j.b.214.1		4	7.5	odd	6
245.2.j.b.214.1		4	35.9	even	6
245.2.j.b.214.2		4	7.2	even	3
245.2.j.b.214.2		4	35.19	odd	6
1225.2.a.q.1.1		2	5.2	odd	4
1225.2.a.q.1.1		2	35.13	even	4
1225.2.a.q.1.2		2	5.3	odd	4
1225.2.a.q.1.2		2	35.27	even	4
2205.2.d.h.1324.1		2	15.14	odd	2
2205.2.d.h.1324.1		2	21.20	even	2
2205.2.d.h.1324.2		2	3.2	odd	2
2205.2.d.h.1324.2		2	105.104	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:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.00000 q^{9} -3.00000 q^{11} -4.47214i q^{12} +6.70820i q^{13} -5.00000 q^{15} +4.00000 q^{16} -2.23607i q^{17} -4.47214i q^{20} -5.00000 q^{25} -2.23607i q^{27} +9.00000 q^{29} +6.70820i q^{33} -4.00000 q^{36} +15.0000 q^{39} -6.00000 q^{44} +4.47214i q^{45} +11.1803i q^{47} -8.94427i q^{48} -5.00000 q^{51} +13.4164i q^{52} +6.70820i q^{55} -10.0000 q^{60} +8.00000 q^{64} +15.0000 q^{65} -4.47214i q^{68} -12.0000 q^{71} +13.4164i q^{73} +11.1803i q^{75} +1.00000 q^{79} -8.94427i q^{80} -11.0000 q^{81} -8.94427i q^{83} -5.00000 q^{85} -20.1246i q^{87} +6.70820i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} - 4 q^{9} - 6 q^{11} - 10 q^{15} + 8 q^{16} - 10 q^{25} + 18 q^{29} - 8 q^{36} + 30 q^{39} - 12 q^{44} - 10 q^{51} - 20 q^{60} + 16 q^{64} + 30 q^{65} - 24 q^{71} + 2 q^{79} - 22 q^{81} - 10 q^{85}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q + 4 * q^4 - 4 * q^9 - 6 * q^11 - 10 * q^15 + 8 * q^16 - 10 * q^25 + 18 * q^29 - 8 * q^36 + 30 * q^39 - 12 * q^44 - 10 * q^51 - 20 * q^60 + 16 * q^64 + 30 * q^65 - 24 * q^71 + 2 * q^79 - 22 * q^81 - 10 * q^85 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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