LMFDB - Embedded newform 507.2.b.c.337.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [507,2,Mod(337,507)] 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("507.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,4,0,0,0,0,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.2.b.c.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.00000 q^{4} -3.46410i q^{5} +1.73205i q^{7} +1.00000 q^{9} -3.46410i q^{11} -2.00000 q^{12} +3.46410i q^{15} +4.00000 q^{16} -3.46410i q^{19} -6.92820i q^{20} -1.73205i q^{21} -6.00000 q^{23} -7.00000 q^{25} -1.00000 q^{27} +3.46410i q^{28} +6.00000 q^{29} -1.73205i q^{31} +3.46410i q^{33} +6.00000 q^{35} +2.00000 q^{36} -6.92820i q^{41} -1.00000 q^{43} -6.92820i q^{44} -3.46410i q^{45} -3.46410i q^{47} -4.00000 q^{48} +4.00000 q^{49} +12.0000 q^{53} -12.0000 q^{55} +3.46410i q^{57} +3.46410i q^{59} +6.92820i q^{60} +1.00000 q^{61} +1.73205i q^{63} +8.00000 q^{64} +8.66025i q^{67} +6.00000 q^{69} +10.3923i q^{71} -1.73205i q^{73} +7.00000 q^{75} -6.92820i q^{76} +6.00000 q^{77} -11.0000 q^{79} -13.8564i q^{80} +1.00000 q^{81} +13.8564i q^{83} -3.46410i q^{84} -6.00000 q^{87} +6.92820i q^{89} -12.0000 q^{92} +1.73205i q^{93} -12.0000 q^{95} +5.19615i q^{97} -3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{4} + 2 q^{9} - 4 q^{12} + 8 q^{16} - 12 q^{23} - 14 q^{25} - 2 q^{27} + 12 q^{29} + 12 q^{35} + 4 q^{36} - 2 q^{43} - 8 q^{48} + 8 q^{49} + 24 q^{53} - 24 q^{55} + 2 q^{61} + 16 q^{64}+ \cdots - 24 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^4 + 2 * q^9 - 4 * q^12 + 8 * q^16 - 12 * q^23 - 14 * q^25 - 2 * q^27 + 12 * q^29 + 12 * q^35 + 4 * q^36 - 2 * q^43 - 8 * q^48 + 8 * q^49 + 24 * q^53 - 24 * q^55 + 2 * q^61 + 16 * q^64 + 12 * q^69 + 14 * q^75 + 12 * q^77 - 22 * q^79 + 2 * q^81 - 12 * q^87 - 24 * q^92 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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			$\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	− 3.46410i	− 1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	− 3.46410i	− 1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$	0	0
$7$	1.73205i	0.654654i	0.944911	+	0.327327i	$0.106148\pi$
$7$	1.73205i	0.654654i	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 3.46410i	− 1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	− 3.46410i	− 1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$	−2.00000	−0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.46410i	0.894427i
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	− 6.92820i	− 1.54919i
$21$	− 1.73205i	− 0.377964i
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	3.46410i	0.654654i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	− 1.73205i	− 0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	− 1.73205i	− 0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$	0	0
$33$	3.46410i	0.603023i
$34$	0	0
$35$	6.00000	1.01419
$36$	2.00000	0.333333
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.92820i	− 1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$	− 6.92820i	− 1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	− 6.92820i	− 1.04447i
$45$	− 3.46410i	− 0.516398i
$46$	0	0
$47$	− 3.46410i	− 0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$	− 3.46410i	− 0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$	−4.00000	−0.577350
$49$	4.00000	0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	0	0
$57$	3.46410i	0.458831i
$58$	0	0
$59$	3.46410i	0.450988i	0.974245	+	0.225494i	$0.0723995\pi$
$59$	3.46410i	0.450988i	−0.974245	+	0.225494i	$0.927600\pi$
$60$	6.92820i	0.894427i
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	1.73205i	0.218218i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	8.66025i	1.05802i	0.848616	+	0.529009i	$0.177436\pi$
$67$	8.66025i	1.05802i	−0.848616	+	0.529009i	$0.822564\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	10.3923i	1.23334i	0.787222	+	0.616670i	$0.211519\pi$
$71$	10.3923i	1.23334i	−0.787222	+	0.616670i	$0.788481\pi$
$72$	0	0
$73$	− 1.73205i	− 0.202721i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	− 1.73205i	− 0.202721i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	− 6.92820i	− 0.794719i
$77$	6.00000	0.683763
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	− 13.8564i	− 1.54919i
$81$	1.00000	0.111111
$82$	0	0
$83$	13.8564i	1.52094i	0.649374	+	0.760469i	$0.275031\pi$
$83$	13.8564i	1.52094i	−0.649374	+	0.760469i	$0.724969\pi$
$84$	− 3.46410i	− 0.377964i
$85$	0	0
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	6.92820i	0.734388i	0.930144	+	0.367194i	$0.119682\pi$
$89$	6.92820i	0.734388i	−0.930144	+	0.367194i	$0.880318\pi$
$90$	0	0
$91$	0	0
$92$	−12.0000	−1.25109
$93$	1.73205i	0.179605i
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	5.19615i	0.527589i	0.964579	+	0.263795i	$0.0849741\pi$
$97$	5.19615i	0.527589i	−0.964579	+	0.263795i	$0.915026\pi$
$98$	0	0
$99$	− 3.46410i	− 0.348155i
$100$	−14.0000	−1.40000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−2.00000	−0.192450
$109$	15.5885i	1.49310i	0.665327	+	0.746552i	$0.268292\pi$
$109$	15.5885i	1.49310i	−0.665327	+	0.746552i	$0.731708\pi$
$110$	0	0
$111$	0	0
$112$	6.92820i	0.654654i
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	20.7846i	1.93817i
$116$	12.0000	1.11417
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	6.92820i	0.624695i
$124$	− 3.46410i	− 0.311086i
$125$	6.92820i	0.619677i
$126$	0	0
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	6.92820i	0.603023i
$133$	6.00000	0.520266
$134$	0	0
$135$	3.46410i	0.298142i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	12.0000	1.01419
$141$	3.46410i	0.291730i
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	− 20.7846i	− 1.72607i
$146$	0	0
$147$	−4.00000	−0.329914
$148$	0	0
$149$	6.92820i	0.567581i	0.958886	+	0.283790i	$0.0915919\pi$
$149$	6.92820i	0.567581i	−0.958886	+	0.283790i	$0.908408\pi$
$150$	0	0
$151$	3.46410i	0.281905i	0.990016	+	0.140952i	$0.0450164\pi$
$151$	3.46410i	0.281905i	−0.990016	+	0.140952i	$0.954984\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	− 10.3923i	− 0.819028i
$162$	0	0
$163$	19.0526i	1.49231i	0.665771	+	0.746156i	$0.268103\pi$
$163$	19.0526i	1.49231i	−0.665771	+	0.746156i	$0.731897\pi$
$164$	− 13.8564i	− 1.08200i
$165$	12.0000	0.934199
$166$	0	0
$167$	− 6.92820i	− 0.536120i	−0.963402	−	0.268060i	$-0.913617\pi$
$167$	− 6.92820i	− 0.536120i	0.963402	−	0.268060i	$-0.0863826\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 3.46410i	− 0.264906i
$172$	−2.00000	−0.152499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	− 12.1244i	− 0.916515i
$176$	− 13.8564i	− 1.04447i
$177$	− 3.46410i	− 0.260378i
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	− 6.92820i	− 0.516398i
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	− 6.92820i	− 0.505291i
$189$	− 1.73205i	− 0.125988i
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	−8.00000	−0.577350
$193$	− 15.5885i	− 1.12208i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	− 15.5885i	− 1.12208i	0.827788	−	0.561041i	$-0.189599\pi$
$194$	0	0
$195$	0	0
$196$	8.00000	0.571429
$197$	13.8564i	0.987228i	0.869681	+	0.493614i	$0.164324\pi$
$197$	13.8564i	0.987228i	−0.869681	+	0.493614i	$0.835676\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	− 8.66025i	− 0.610847i
$202$	0	0
$203$	10.3923i	0.729397i
$204$	0	0
$205$	−24.0000	−1.67623
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	24.0000	1.64833
$213$	− 10.3923i	− 0.712069i
$214$	0	0
$215$	3.46410i	0.236250i
$216$	0	0
$217$	3.00000	0.203653
$218$	0	0
$219$	1.73205i	0.117041i
$220$	−24.0000	−1.61808
$221$	0	0
$222$	0	0
$223$	17.3205i	1.15987i	0.814664	+	0.579934i	$0.196921\pi$
$223$	17.3205i	1.15987i	−0.814664	+	0.579934i	$0.803079\pi$
$224$	0	0
$225$	−7.00000	−0.466667
$226$	0	0
$227$	− 20.7846i	− 1.37952i	−0.724037	−	0.689761i	$-0.757715\pi$
$227$	− 20.7846i	− 1.37952i	0.724037	−	0.689761i	$-0.242285\pi$
$228$	6.92820i	0.458831i
$229$	− 27.7128i	− 1.83131i	−0.401960	−	0.915657i	$-0.631671\pi$
$229$	− 27.7128i	− 1.83131i	0.401960	−	0.915657i	$-0.368329\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	6.92820i	0.450988i
$237$	11.0000	0.714527
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	13.8564i	0.894427i
$241$	− 20.7846i	− 1.33885i	−0.742878	−	0.669427i	$-0.766540\pi$
$241$	− 20.7846i	− 1.33885i	0.742878	−	0.669427i	$-0.233460\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	− 13.8564i	− 0.885253i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 13.8564i	− 0.878114i
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	3.46410i	0.218218i
$253$	20.7846i	1.30672i
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	− 41.5692i	− 2.55358i
$266$	0	0
$267$	− 6.92820i	− 0.423999i
$268$	17.3205i	1.05802i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	− 5.19615i	− 0.315644i	−0.987468	−	0.157822i	$-0.949553\pi$
$271$	− 5.19615i	− 0.315644i	0.987468	−	0.157822i	$-0.0504472\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.2487i	1.46225i
$276$	12.0000	0.722315
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	− 1.73205i	− 0.103695i
$280$	0	0
$281$	24.2487i	1.44656i	0.690557	+	0.723278i	$0.257366\pi$
$281$	24.2487i	1.44656i	−0.690557	+	0.723278i	$0.742634\pi$
$282$	0	0
$283$	−11.0000	−0.653882	−0.326941	−	0.945045i	$-0.606018\pi$
$283$	−11.0000	−0.653882	−0.326941	+	0.945045i	$0.606018\pi$
$284$	20.7846i	1.23334i
$285$	12.0000	0.710819
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	− 5.19615i	− 0.304604i
$292$	− 3.46410i	− 0.202721i
$293$	17.3205i	1.01187i	0.862570	+	0.505937i	$0.168853\pi$
$293$	17.3205i	1.01187i	−0.862570	+	0.505937i	$0.831147\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	3.46410i	0.201008i
$298$	0	0
$299$	0	0
$300$	14.0000	0.808290
$301$	− 1.73205i	− 0.0998337i
$302$	0	0
$303$	18.0000	1.03407
$304$	− 13.8564i	− 0.794719i
$305$	− 3.46410i	− 0.198354i
$306$	0	0
$307$	1.73205i	0.0988534i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$	1.73205i	0.0988534i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$	12.0000	0.683763
$309$	1.00000	0.0568880
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	13.0000	0.734803	0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000	0.734803	0.367402	+	0.930062i	$0.380247\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	−22.0000	−1.23760
$317$	− 6.92820i	− 0.389127i	−0.980890	−	0.194563i	$-0.937671\pi$
$317$	− 6.92820i	− 0.389127i	0.980890	−	0.194563i	$-0.0623290\pi$
$318$	0	0
$319$	− 20.7846i	− 1.16371i
$320$	− 27.7128i	− 1.54919i
$321$	6.00000	0.334887
$322$	0	0
$323$	0	0
$324$	2.00000	0.111111
$325$	0	0
$326$	0	0
$327$	− 15.5885i	− 0.862044i
$328$	0	0
$329$	6.00000	0.330791
$330$	0	0
$331$	− 5.19615i	− 0.285606i	−0.989751	−	0.142803i	$-0.954388\pi$
$331$	− 5.19615i	− 0.285606i	0.989751	−	0.142803i	$-0.0456116\pi$
$332$	27.7128i	1.52094i
$333$	0	0
$334$	0	0
$335$	30.0000	1.63908
$336$	− 6.92820i	− 0.377964i
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	19.0526i	1.02874i
$344$	0	0
$345$	− 20.7846i	− 1.11901i
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	−12.0000	−0.643268
$349$	− 19.0526i	− 1.01986i	−0.860216	−	0.509930i	$-0.829671\pi$
$349$	− 19.0526i	− 1.01986i	0.860216	−	0.509930i	$-0.170329\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.3923i	0.553127i	0.960996	+	0.276563i	$0.0891955\pi$
$353$	10.3923i	0.553127i	−0.960996	+	0.276563i	$0.910804\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	13.8564i	0.734388i
$357$	0	0
$358$	0	0
$359$	− 6.92820i	− 0.365657i	−0.983145	−	0.182828i	$-0.941475\pi$
$359$	− 6.92820i	− 0.365657i	0.983145	−	0.182828i	$-0.0585252\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	−24.0000	−1.25109
$369$	− 6.92820i	− 0.360668i
$370$	0	0
$371$	20.7846i	1.07908i
$372$	3.46410i	0.179605i
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	0	0
$375$	− 6.92820i	− 0.357771i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 22.5167i	− 1.15660i	−0.815823	−	0.578302i	$-0.803716\pi$
$379$	− 22.5167i	− 1.15660i	0.815823	−	0.578302i	$-0.196284\pi$
$380$	−24.0000	−1.23117
$381$	−13.0000	−0.666010
$382$	0	0
$383$	− 27.7128i	− 1.41606i	−0.706183	−	0.708029i	$-0.749584\pi$
$383$	− 27.7128i	− 1.41606i	0.706183	−	0.708029i	$-0.250416\pi$
$384$	0	0
$385$	− 20.7846i	− 1.05928i
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	10.3923i	0.527589i
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	38.1051i	1.91728i
$396$	− 6.92820i	− 0.348155i
$397$	− 15.5885i	− 0.782362i	−0.920314	−	0.391181i	$-0.872067\pi$
$397$	− 15.5885i	− 0.782362i	0.920314	−	0.391181i	$-0.127933\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	−28.0000	−1.40000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	−36.0000	−1.79107
$405$	− 3.46410i	− 0.172133i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 8.66025i	− 0.428222i	−0.976809	−	0.214111i	$-0.931315\pi$
$409$	− 8.66025i	− 0.428222i	0.976809	−	0.214111i	$-0.0686854\pi$
$410$	0	0
$411$	0	0
$412$	−2.00000	−0.0985329
$413$	−6.00000	−0.295241
$414$	0	0
$415$	48.0000	2.35623
$416$	0	0
$417$	−5.00000	−0.244851
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	−12.0000	−0.585540
$421$	12.1244i	0.590905i	0.955357	+	0.295452i	$0.0954704\pi$
$421$	12.1244i	0.590905i	−0.955357	+	0.295452i	$0.904530\pi$
$422$	0	0
$423$	− 3.46410i	− 0.168430i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.73205i	0.0838198i
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	20.7846i	1.00116i	0.865690	+	0.500580i	$0.166880\pi$
$431$	20.7846i	1.00116i	−0.865690	+	0.500580i	$0.833120\pi$
$432$	−4.00000	−0.192450
$433$	−23.0000	−1.10531	−0.552655	−	0.833410i	$-0.686385\pi$
$433$	−23.0000	−1.10531	−0.552655	+	0.833410i	$0.686385\pi$
$434$	0	0
$435$	20.7846i	0.996546i
$436$	31.1769i	1.49310i
$437$	20.7846i	0.994263i
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	4.00000	0.190476
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	− 6.92820i	− 0.327693i
$448$	13.8564i	0.654654i
$449$	− 38.1051i	− 1.79829i	−0.437649	−	0.899146i	$-0.644189\pi$
$449$	− 38.1051i	− 1.79829i	0.437649	−	0.899146i	$-0.355811\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	12.0000	0.564433
$453$	− 3.46410i	− 0.162758i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 36.3731i	− 1.70146i	−0.525603	−	0.850730i	$-0.676160\pi$
$457$	− 36.3731i	− 1.70146i	0.525603	−	0.850730i	$-0.323840\pi$
$458$	0	0
$459$	0	0
$460$	41.5692i	1.93817i
$461$	41.5692i	1.93607i	0.250812	+	0.968036i	$0.419302\pi$
$461$	41.5692i	1.93607i	−0.250812	+	0.968036i	$0.580698\pi$
$462$	0	0
$463$	36.3731i	1.69040i	0.534450	+	0.845200i	$0.320519\pi$
$463$	36.3731i	1.69040i	−0.534450	+	0.845200i	$0.679481\pi$
$464$	24.0000	1.11417
$465$	6.00000	0.278243
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−15.0000	−0.692636
$470$	0	0
$471$	−11.0000	−0.506853
$472$	0	0
$473$	3.46410i	0.159280i
$474$	0	0
$475$	24.2487i	1.11261i
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	34.6410i	1.58279i	0.611306	+	0.791394i	$0.290644\pi$
$479$	34.6410i	1.58279i	−0.611306	+	0.791394i	$0.709356\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	10.3923i	0.472866i
$484$	−2.00000	−0.0909091
$485$	18.0000	0.817338
$486$	0	0
$487$	24.2487i	1.09881i	0.835555	+	0.549407i	$0.185146\pi$
$487$	24.2487i	1.09881i	−0.835555	+	0.549407i	$0.814854\pi$
$488$	0	0
$489$	− 19.0526i	− 0.861586i
$490$	0	0
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	13.8564i	0.624695i
$493$	0	0
$494$	0	0
$495$	−12.0000	−0.539360
$496$	− 6.92820i	− 0.311086i
$497$	−18.0000	−0.807410
$498$	0	0
$499$	− 31.1769i	− 1.39567i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	− 31.1769i	− 1.39567i	0.716258	−	0.697835i	$-0.245853\pi$
$500$	13.8564i	0.619677i
$501$	6.92820i	0.309529i
$502$	0	0
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	62.3538i	2.77471i
$506$	0	0
$507$	0	0
$508$	26.0000	1.15356
$509$	− 17.3205i	− 0.767718i	−0.923392	−	0.383859i	$-0.874595\pi$
$509$	− 17.3205i	− 0.767718i	0.923392	−	0.383859i	$-0.125405\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	0	0
$513$	3.46410i	0.152944i
$514$	0	0
$515$	3.46410i	0.152647i
$516$	2.00000	0.0880451
$517$	−12.0000	−0.527759
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	12.0000	0.524222
$525$	12.1244i	0.529150i
$526$	0	0
$527$	0	0
$528$	13.8564i	0.603023i
$529$	13.0000	0.565217
$530$	0	0
$531$	3.46410i	0.150329i
$532$	12.0000	0.520266
$533$	0	0
$534$	0	0
$535$	20.7846i	0.898597i
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	− 13.8564i	− 0.596838i
$540$	6.92820i	0.298142i
$541$	29.4449i	1.26593i	0.774179	+	0.632967i	$0.218163\pi$
$541$	29.4449i	1.26593i	−0.774179	+	0.632967i	$0.781837\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	54.0000	2.31311
$546$	0	0
$547$	−19.0000	−0.812381	−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000	−0.812381	−0.406191	+	0.913788i	$0.633143\pi$
$548$	0	0
$549$	1.00000	0.0426790
$550$	0	0
$551$	− 20.7846i	− 0.885454i
$552$	0	0
$553$	− 19.0526i	− 0.810197i
$554$	0	0
$555$	0	0
$556$	10.0000	0.424094
$557$	− 27.7128i	− 1.17423i	−0.809504	−	0.587115i	$-0.800264\pi$
$557$	− 27.7128i	− 1.17423i	0.809504	−	0.587115i	$-0.199736\pi$
$558$	0	0
$559$	0	0
$560$	24.0000	1.01419
$561$	0	0
$562$	0	0
$563$	−42.0000	−1.77009	−0.885044	−	0.465506i	$-0.845872\pi$
$563$	−42.0000	−1.77009	−0.885044	+	0.465506i	$0.845872\pi$
$564$	6.92820i	0.291730i
$565$	− 20.7846i	− 0.874415i
$566$	0	0
$567$	1.73205i	0.0727393i
$568$	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
$571$	−44.0000	−1.84134	−0.920671	−	0.390339i	$-0.872358\pi$
$571$	−44.0000	−1.84134	−0.920671	+	0.390339i	$0.872358\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	42.0000	1.75152
$576$	8.00000	0.333333
$577$	34.6410i	1.44212i	0.692870	+	0.721062i	$0.256346\pi$
$577$	34.6410i	1.44212i	−0.692870	+	0.721062i	$0.743654\pi$
$578$	0	0
$579$	15.5885i	0.647834i
$580$	− 41.5692i	− 1.72607i
$581$	−24.0000	−0.995688
$582$	0	0
$583$	− 41.5692i	− 1.72162i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 31.1769i	− 1.28681i	−0.765526	−	0.643404i	$-0.777521\pi$
$587$	− 31.1769i	− 1.28681i	0.765526	−	0.643404i	$-0.222479\pi$
$588$	−8.00000	−0.329914
$589$	−6.00000	−0.247226
$590$	0	0
$591$	− 13.8564i	− 0.569976i
$592$	0	0
$593$	− 3.46410i	− 0.142254i	−0.997467	−	0.0711268i	$-0.977341\pi$
$593$	− 3.46410i	− 0.142254i	0.997467	−	0.0711268i	$-0.0226595\pi$
$594$	0	0
$595$	0	0
$596$	13.8564i	0.567581i
$597$	−7.00000	−0.286491
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	8.66025i	0.352673i
$604$	6.92820i	0.281905i
$605$	3.46410i	0.140836i
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	− 10.3923i	− 0.421117i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.66025i	0.349784i	0.984588	+	0.174892i	$0.0559577\pi$
$613$	8.66025i	0.349784i	−0.984588	+	0.174892i	$0.944042\pi$
$614$	0	0
$615$	24.0000	0.967773
$616$	0	0
$617$	10.3923i	0.418378i	0.977875	+	0.209189i	$0.0670825\pi$
$617$	10.3923i	0.418378i	−0.977875	+	0.209189i	$0.932918\pi$
$618$	0	0
$619$	− 25.9808i	− 1.04425i	−0.852867	−	0.522127i	$-0.825139\pi$
$619$	− 25.9808i	− 1.04425i	0.852867	−	0.522127i	$-0.174861\pi$
$620$	−12.0000	−0.481932
$621$	6.00000	0.240772
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	12.0000	0.479234
$628$	22.0000	0.877896
$629$	0	0
$630$	0	0
$631$	− 1.73205i	− 0.0689519i	−0.999406	−	0.0344759i	$-0.989024\pi$
$631$	− 1.73205i	− 0.0689519i	0.999406	−	0.0344759i	$-0.0109762\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	0	0
$635$	− 45.0333i	− 1.78709i
$636$	−24.0000	−0.951662
$637$	0	0
$638$	0	0
$639$	10.3923i	0.411113i
$640$	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
$643$	− 19.0526i	− 0.751360i	−0.926750	−	0.375680i	$-0.877409\pi$
$643$	− 19.0526i	− 0.751360i	0.926750	−	0.375680i	$-0.122591\pi$
$644$	− 20.7846i	− 0.819028i
$645$	− 3.46410i	− 0.136399i
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−3.00000	−0.117579
$652$	38.1051i	1.49231i
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	− 20.7846i	− 0.812122i
$656$	− 27.7128i	− 1.08200i
$657$	− 1.73205i	− 0.0675737i
$658$	0	0
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	24.0000	0.934199
$661$	− 25.9808i	− 1.01053i	−0.862963	−	0.505267i	$-0.831394\pi$
$661$	− 25.9808i	− 1.01053i	0.862963	−	0.505267i	$-0.168606\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 20.7846i	− 0.805993i
$666$	0	0
$667$	−36.0000	−1.39393
$668$	− 13.8564i	− 0.536120i
$669$	− 17.3205i	− 0.669650i
$670$	0	0
$671$	− 3.46410i	− 0.133730i
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	0	0
$675$	7.00000	0.269430
$676$	0	0
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	0	0
$679$	−9.00000	−0.345388
$680$	0	0
$681$	20.7846i	0.796468i
$682$	0	0
$683$	− 24.2487i	− 0.927851i	−0.885874	−	0.463926i	$-0.846441\pi$
$683$	− 24.2487i	− 0.927851i	0.885874	−	0.463926i	$-0.153559\pi$
$684$	− 6.92820i	− 0.264906i
$685$	0	0
$686$	0	0
$687$	27.7128i	1.05731i
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	43.3013i	1.64726i	0.567129	+	0.823629i	$0.308054\pi$
$691$	43.3013i	1.64726i	−0.567129	+	0.823629i	$0.691946\pi$
$692$	−12.0000	−0.456172
$693$	6.00000	0.227921
$694$	0	0
$695$	− 17.3205i	− 0.657004i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	18.0000	0.680823
$700$	− 24.2487i	− 0.916515i
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	0	0
$704$	− 27.7128i	− 1.04447i
$705$	12.0000	0.451946
$706$	0	0
$707$	− 31.1769i	− 1.17253i
$708$	− 6.92820i	− 0.260378i
$709$	19.0526i	0.715534i	0.933811	+	0.357767i	$0.116462\pi$
$709$	19.0526i	0.715534i	−0.933811	+	0.357767i	$0.883538\pi$
$710$	0	0
$711$	−11.0000	−0.412532
$712$	0	0
$713$	10.3923i	0.389195i
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	0	0
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	− 13.8564i	− 0.516398i
$721$	− 1.73205i	− 0.0645049i
$722$	0	0
$723$	20.7846i	0.772988i
$724$	28.0000	1.04061
$725$	−42.0000	−1.55984
$726$	0	0
$727$	−35.0000	−1.29808	−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000	−1.29808	−0.649039	+	0.760755i	$0.724829\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−2.00000	−0.0739221
$733$	− 39.8372i	− 1.47142i	−0.677297	−	0.735710i	$-0.736849\pi$
$733$	− 39.8372i	− 1.47142i	0.677297	−	0.735710i	$-0.263151\pi$
$734$	0	0
$735$	13.8564i	0.511101i
$736$	0	0
$737$	30.0000	1.10506
$738$	0	0
$739$	45.0333i	1.65658i	0.560301	+	0.828289i	$0.310685\pi$
$739$	45.0333i	1.65658i	−0.560301	+	0.828289i	$0.689315\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 10.3923i	− 0.381257i	−0.981662	−	0.190628i	$-0.938947\pi$
$743$	− 10.3923i	− 0.381257i	0.981662	−	0.190628i	$-0.0610525\pi$
$744$	0	0
$745$	24.0000	0.879292
$746$	0	0
$747$	13.8564i	0.506979i
$748$	0	0
$749$	− 10.3923i	− 0.379727i
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	− 13.8564i	− 0.505291i
$753$	−12.0000	−0.437304
$754$	0	0
$755$	12.0000	0.436725
$756$	− 3.46410i	− 0.125988i
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	0	0
$759$	− 20.7846i	− 0.754434i
$760$	0	0
$761$	20.7846i	0.753442i	0.926327	+	0.376721i	$0.122948\pi$
$761$	20.7846i	0.753442i	−0.926327	+	0.376721i	$0.877052\pi$
$762$	0	0
$763$	−27.0000	−0.977466
$764$	−36.0000	−1.30243
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−16.0000	−0.577350
$769$	− 6.92820i	− 0.249837i	−0.992167	−	0.124919i	$-0.960133\pi$
$769$	− 6.92820i	− 0.249837i	0.992167	−	0.124919i	$-0.0398670\pi$
$770$	0	0
$771$	0	0
$772$	− 31.1769i	− 1.12208i
$773$	51.9615i	1.86893i	0.356060	+	0.934463i	$0.384120\pi$
$773$	51.9615i	1.86893i	−0.356060	+	0.934463i	$0.615880\pi$
$774$	0	0
$775$	12.1244i	0.435520i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	−6.00000	−0.214423
$784$	16.0000	0.571429
$785$	− 38.1051i	− 1.36003i
$786$	0	0
$787$	− 32.9090i	− 1.17308i	−0.809921	−	0.586539i	$-0.800490\pi$
$787$	− 32.9090i	− 1.17308i	0.809921	−	0.586539i	$-0.199510\pi$
$788$	27.7128i	0.987228i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	10.3923i	0.369508i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	41.5692i	1.47431i
$796$	14.0000	0.496217
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.92820i	0.244796i
$802$	0	0
$803$	−6.00000	−0.211735
$804$	− 17.3205i	− 0.610847i
$805$	−36.0000	−1.26883
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	25.9808i	0.912308i	0.889901	+	0.456154i	$0.150773\pi$
$811$	25.9808i	0.912308i	−0.889901	+	0.456154i	$0.849227\pi$
$812$	20.7846i	0.729397i
$813$	5.19615i	0.182237i
$814$	0	0
$815$	66.0000	2.31188
$816$	0	0
$817$	3.46410i	0.121194i
$818$	0	0
$819$	0	0
$820$	−48.0000	−1.67623
$821$	24.2487i	0.846286i	0.906063	+	0.423143i	$0.139073\pi$
$821$	24.2487i	0.846286i	−0.906063	+	0.423143i	$0.860927\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	− 24.2487i	− 0.844232i
$826$	0	0
$827$	− 48.4974i	− 1.68642i	−0.537584	−	0.843210i	$-0.680663\pi$
$827$	− 48.4974i	− 1.68642i	0.537584	−	0.843210i	$-0.319337\pi$
$828$	−12.0000	−0.417029
$829$	31.0000	1.07667	0.538337	−	0.842729i	$-0.319053\pi$
$829$	31.0000	1.07667	0.538337	+	0.842729i	$0.319053\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−24.0000	−0.830554
$836$	−24.0000	−0.830057
$837$	1.73205i	0.0598684i
$838$	0	0
$839$	− 31.1769i	− 1.07635i	−0.842834	−	0.538173i	$-0.819115\pi$
$839$	− 31.1769i	− 1.07635i	0.842834	−	0.538173i	$-0.180885\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	− 24.2487i	− 0.835170i
$844$	26.0000	0.894957
$845$	0	0
$846$	0	0
$847$	− 1.73205i	− 0.0595140i
$848$	48.0000	1.64833
$849$	11.0000	0.377519
$850$	0	0
$851$	0	0
$852$	− 20.7846i	− 0.712069i
$853$	25.9808i	0.889564i	0.895639	+	0.444782i	$0.146719\pi$
$853$	25.9808i	0.889564i	−0.895639	+	0.444782i	$0.853281\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	6.92820i	0.236250i
$861$	−12.0000	−0.408959
$862$	0	0
$863$	17.3205i	0.589597i	0.955559	+	0.294798i	$0.0952525\pi$
$863$	17.3205i	0.589597i	−0.955559	+	0.294798i	$0.904747\pi$
$864$	0	0
$865$	20.7846i	0.706698i
$866$	0	0
$867$	17.0000	0.577350
$868$	6.00000	0.203653
$869$	38.1051i	1.29263i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	5.19615i	0.175863i
$874$	0	0
$875$	−12.0000	−0.405674
$876$	3.46410i	0.117041i
$877$	− 41.5692i	− 1.40369i	−0.712328	−	0.701846i	$-0.752359\pi$
$877$	− 41.5692i	− 1.40369i	0.712328	−	0.701846i	$-0.247641\pi$
$878$	0	0
$879$	− 17.3205i	− 0.584206i
$880$	−48.0000	−1.61808
$881$	−12.0000	−0.404290	−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000	−0.404290	−0.202145	+	0.979356i	$0.564791\pi$
$882$	0	0
$883$	−5.00000	−0.168263	−0.0841317	−	0.996455i	$-0.526812\pi$
$883$	−5.00000	−0.168263	−0.0841317	+	0.996455i	$0.526812\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	0	0
$889$	22.5167i	0.755185i
$890$	0	0
$891$	− 3.46410i	− 0.116052i
$892$	34.6410i	1.15987i
$893$	−12.0000	−0.401565
$894$	0	0
$895$	− 41.5692i	− 1.38951i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 10.3923i	− 0.346603i
$900$	−14.0000	−0.466667
$901$	0	0
$902$	0	0
$903$	1.73205i	0.0576390i
$904$	0	0
$905$	− 48.4974i	− 1.61211i
$906$	0	0
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	− 41.5692i	− 1.37952i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	13.8564i	0.458831i
$913$	48.0000	1.58857
$914$	0	0
$915$	3.46410i	0.114520i
$916$	− 55.4256i	− 1.83131i
$917$	10.3923i	0.343184i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	− 1.73205i	− 0.0570730i
$922$	0	0
$923$	0	0
$924$	−12.0000	−0.394771
$925$	0	0
$926$	0	0
$927$	−1.00000	−0.0328443
$928$	0	0
$929$	20.7846i	0.681921i	0.940078	+	0.340960i	$0.110752\pi$
$929$	20.7846i	0.681921i	−0.940078	+	0.340960i	$0.889248\pi$
$930$	0	0
$931$	− 13.8564i	− 0.454125i
$932$	−36.0000	−1.17922
$933$	24.0000	0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	−13.0000	−0.424239
$940$	−24.0000	−0.782794
$941$	10.3923i	0.338779i	0.985549	+	0.169390i	$0.0541797\pi$
$941$	10.3923i	0.338779i	−0.985549	+	0.169390i	$0.945820\pi$
$942$	0	0
$943$	41.5692i	1.35368i
$944$	13.8564i	0.450988i
$945$	−6.00000	−0.195180
$946$	0	0
$947$	− 48.4974i	− 1.57595i	−0.615704	−	0.787977i	$-0.711128\pi$
$947$	− 48.4974i	− 1.57595i	0.615704	−	0.787977i	$-0.288872\pi$
$948$	22.0000	0.714527
$949$	0	0
$950$	0	0
$951$	6.92820i	0.224662i
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	62.3538i	2.01772i
$956$	0	0
$957$	20.7846i	0.671871i
$958$	0	0
$959$	0	0
$960$	27.7128i	0.894427i
$961$	28.0000	0.903226
$962$	0	0
$963$	−6.00000	−0.193347
$964$	− 41.5692i	− 1.33885i
$965$	−54.0000	−1.73832
$966$	0	0
$967$	24.2487i	0.779786i	0.920860	+	0.389893i	$0.127488\pi$
$967$	24.2487i	0.779786i	−0.920860	+	0.389893i	$0.872512\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	−2.00000	−0.0641500
$973$	8.66025i	0.277635i
$974$	0	0
$975$	0	0
$976$	4.00000	0.128037
$977$	− 27.7128i	− 0.886611i	−0.896370	−	0.443306i	$-0.853806\pi$
$977$	− 27.7128i	− 0.886611i	0.896370	−	0.443306i	$-0.146194\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	− 27.7128i	− 0.885253i
$981$	15.5885i	0.497701i
$982$	0	0
$983$	10.3923i	0.331463i	0.986171	+	0.165732i	$0.0529985\pi$
$983$	10.3923i	0.331463i	−0.986171	+	0.165732i	$0.947001\pi$
$984$	0	0
$985$	48.0000	1.52941
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	6.00000	0.190789
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	5.19615i	0.164895i
$994$	0	0
$995$	− 24.2487i	− 0.768736i
$996$	− 27.7128i	− 0.878114i
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.b.c.337.1		2
3.2	odd	2		1521.2.b.f.1351.2		2
13.2	odd	12		507.2.e.f.22.1		4
13.3	even	3		507.2.j.b.316.1		2
13.4	even	6		507.2.j.b.361.1		2
13.5	odd	4		507.2.a.e.1.1		2
13.6	odd	12		507.2.e.f.484.1		4
13.7	odd	12		507.2.e.f.484.2		4
13.8	odd	4		507.2.a.e.1.2		2
13.9	even	3		39.2.j.a.10.1	yes	2
13.10	even	6		39.2.j.a.4.1	✓	2
13.11	odd	12		507.2.e.f.22.2		4
13.12	even	2	inner	507.2.b.c.337.2		2
39.5	even	4		1521.2.a.h.1.2		2
39.8	even	4		1521.2.a.h.1.1		2
39.23	odd	6		117.2.q.a.82.1		2
39.35	odd	6		117.2.q.a.10.1		2
39.38	odd	2		1521.2.b.f.1351.1		2
52.23	odd	6		624.2.bv.b.433.1		2
52.31	even	4		8112.2.a.bu.1.1		2
52.35	odd	6		624.2.bv.b.49.1		2
52.47	even	4		8112.2.a.bu.1.2		2
65.9	even	6		975.2.bc.c.751.1		2
65.22	odd	12		975.2.w.d.49.2		4
65.23	odd	12		975.2.w.d.199.2		4
65.48	odd	12		975.2.w.d.49.1		4
65.49	even	6		975.2.bc.c.901.1		2
65.62	odd	12		975.2.w.d.199.1		4
156.23	even	6		1872.2.by.f.433.1		2
156.35	even	6		1872.2.by.f.1297.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.j.a.4.1	✓	2	13.10	even	6
39.2.j.a.10.1	yes	2	13.9	even	3
117.2.q.a.10.1		2	39.35	odd	6
117.2.q.a.82.1		2	39.23	odd	6
507.2.a.e.1.1		2	13.5	odd	4
507.2.a.e.1.2		2	13.8	odd	4
507.2.b.c.337.1		2	1.1	even	1	trivial
507.2.b.c.337.2		2	13.12	even	2	inner
507.2.e.f.22.1		4	13.2	odd	12
507.2.e.f.22.2		4	13.11	odd	12
507.2.e.f.484.1		4	13.6	odd	12
507.2.e.f.484.2		4	13.7	odd	12
507.2.j.b.316.1		2	13.3	even	3
507.2.j.b.361.1		2	13.4	even	6
624.2.bv.b.49.1		2	52.35	odd	6
624.2.bv.b.433.1		2	52.23	odd	6
975.2.w.d.49.1		4	65.48	odd	12
975.2.w.d.49.2		4	65.22	odd	12
975.2.w.d.199.1		4	65.62	odd	12
975.2.w.d.199.2		4	65.23	odd	12
975.2.bc.c.751.1		2	65.9	even	6
975.2.bc.c.901.1		2	65.49	even	6
1521.2.a.h.1.1		2	39.8	even	4
1521.2.a.h.1.2		2	39.5	even	4
1521.2.b.f.1351.1		2	39.38	odd	2
1521.2.b.f.1351.2		2	3.2	odd	2
1872.2.by.f.433.1		2	156.23	even	6
1872.2.by.f.1297.1		2	156.35	even	6
8112.2.a.bu.1.1		2	52.31	even	4
8112.2.a.bu.1.2		2	52.47	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.00000 q^{4} -3.46410i q^{5} +1.73205i q^{7} +1.00000 q^{9} -3.46410i q^{11} -2.00000 q^{12} +3.46410i q^{15} +4.00000 q^{16} -3.46410i q^{19} -6.92820i q^{20} -1.73205i q^{21} -6.00000 q^{23} -7.00000 q^{25} -1.00000 q^{27} +3.46410i q^{28} +6.00000 q^{29} -1.73205i q^{31} +3.46410i q^{33} +6.00000 q^{35} +2.00000 q^{36} -6.92820i q^{41} -1.00000 q^{43} -6.92820i q^{44} -3.46410i q^{45} -3.46410i q^{47} -4.00000 q^{48} +4.00000 q^{49} +12.0000 q^{53} -12.0000 q^{55} +3.46410i q^{57} +3.46410i q^{59} +6.92820i q^{60} +1.00000 q^{61} +1.73205i q^{63} +8.00000 q^{64} +8.66025i q^{67} +6.00000 q^{69} +10.3923i q^{71} -1.73205i q^{73} +7.00000 q^{75} -6.92820i q^{76} +6.00000 q^{77} -11.0000 q^{79} -13.8564i q^{80} +1.00000 q^{81} +13.8564i q^{83} -3.46410i q^{84} -6.00000 q^{87} +6.92820i q^{89} -12.0000 q^{92} +1.73205i q^{93} -12.0000 q^{95} +5.19615i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{4} + 2 q^{9} - 4 q^{12} + 8 q^{16} - 12 q^{23} - 14 q^{25} - 2 q^{27} + 12 q^{29} + 12 q^{35} + 4 q^{36} - 2 q^{43} - 8 q^{48} + 8 q^{49} + 24 q^{53} - 24 q^{55} + 2 q^{61} + 16 q^{64}+ \cdots - 24 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^4 + 2 * q^9 - 4 * q^12 + 8 * q^16 - 12 * q^23 - 14 * q^25 - 2 * q^27 + 12 * q^29 + 12 * q^35 + 4 * q^36 - 2 * q^43 - 8 * q^48 + 8 * q^49 + 24 * q^53 - 24 * q^55 + 2 * q^61 + 16 * q^64 + 12 * q^69 + 14 * q^75 + 12 * q^77 - 22 * q^79 + 2 * q^81 - 12 * q^87 - 24 * q^92 - 24 * q^95

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