LMFDB - Embedded newform 507.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("507.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.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:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−2.00000	−1.00000
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	−1.73205	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−1.73205	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	2.00000	0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.46410	0.894427
$16$	4.00000	1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	6.92820	1.54919
$21$	1.73205	0.377964
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	3.46410	0.654654
$29$	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
$31$	−1.73205	−0.311086	−0.155543	−	0.987829i	$-0.549713\pi$
$31$	−1.73205	−0.311086	−0.155543	+	0.987829i	$0.549713\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	6.00000	1.01419
$36$	−2.00000	−0.333333
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−6.92820	−1.04447
$45$	−3.46410	−0.516398
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	−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
$57$	3.46410	0.458831
$58$	0	0
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	−6.92820	−0.894427
$61$	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
$63$	−1.73205	−0.218218
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	8.66025	1.05802	0.529009	−	0.848616i	$-0.322564\pi$
$67$	8.66025	1.05802	0.529009	+	0.848616i	$0.322564\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	1.73205	0.202721	0.101361	−	0.994850i	$-0.467680\pi$
$73$	1.73205	0.202721	0.101361	+	0.994850i	$0.467680\pi$
$74$	0	0
$75$	−7.00000	−0.808290
$76$	6.92820	0.794719
$77$	−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	$0.712380\pi$
$80$	−13.8564	−1.54919
$81$	1.00000	0.111111
$82$	0	0
$83$	13.8564	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$84$	−3.46410	−0.377964
$85$	0	0
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	0	0
$92$	−12.0000	−1.25109
$93$	1.73205	0.179605
$94$	0	0
$95$	12.0000	1.23117
$96$	0	0
$97$	5.19615	0.527589	0.263795	−	0.964579i	$-0.415026\pi$
$97$	5.19615	0.527589	0.263795	+	0.964579i	$0.415026\pi$
$98$	0	0
$99$	3.46410	0.348155
$100$	−14.0000	−1.40000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	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
$109$	15.5885	1.49310	0.746552	−	0.665327i	$-0.231708\pi$
$109$	15.5885	1.49310	0.746552	+	0.665327i	$0.231708\pi$
$110$	0	0
$111$	0	0
$112$	−6.92820	−0.654654
$113$	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
$115$	−20.7846	−1.93817
$116$	−12.0000	−1.11417
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.92820	0.624695
$124$	3.46410	0.311086
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	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	$0.415581\pi$
$132$	6.92820	0.603023
$133$	6.00000	0.520266
$134$	0	0
$135$	3.46410	0.298142
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	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
$141$	−3.46410	−0.291730
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	−20.7846	−1.72607
$146$	0	0
$147$	4.00000	0.329914
$148$	0	0
$149$	6.92820	0.567581	0.283790	−	0.958886i	$-0.408408\pi$
$149$	6.92820	0.567581	0.283790	+	0.958886i	$0.408408\pi$
$150$	0	0
$151$	−3.46410	−0.281905	−0.140952	−	0.990016i	$-0.545016\pi$
$151$	−3.46410	−0.281905	−0.140952	+	0.990016i	$0.545016\pi$
$152$	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
$161$	−10.3923	−0.819028
$162$	0	0
$163$	−19.0526	−1.49231	−0.746156	−	0.665771i	$-0.768103\pi$
$163$	−19.0526	−1.49231	−0.746156	+	0.665771i	$0.768103\pi$
$164$	13.8564	1.08200
$165$	12.0000	0.934199
$166$	0	0
$167$	6.92820	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.46410	−0.264906
$172$	−2.00000	−0.152499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−12.1244	−0.916515
$176$	13.8564	1.04447
$177$	3.46410	0.260378
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	6.92820	0.516398
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−6.92820	−0.505291
$189$	1.73205	0.125988
$190$	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
$193$	15.5885	1.12208	0.561041	−	0.827788i	$-0.310401\pi$
$193$	15.5885	1.12208	0.561041	+	0.827788i	$0.310401\pi$
$194$	0	0
$195$	0	0
$196$	8.00000	0.571429
$197$	13.8564	0.987228	0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564	0.987228	0.493614	+	0.869681i	$0.335676\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	0	0
$201$	−8.66025	−0.610847
$202$	0	0
$203$	−10.3923	−0.729397
$204$	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
$213$	−10.3923	−0.712069
$214$	0	0
$215$	−3.46410	−0.236250
$216$	0	0
$217$	3.00000	0.203653
$218$	0	0
$219$	−1.73205	−0.117041
$220$	24.0000	1.61808
$221$	0	0
$222$	0	0
$223$	17.3205	1.15987	0.579934	−	0.814664i	$-0.303079\pi$
$223$	17.3205	1.15987	0.579934	+	0.814664i	$0.303079\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	−20.7846	−1.37952	−0.689761	−	0.724037i	$-0.742285\pi$
$227$	−20.7846	−1.37952	−0.689761	+	0.724037i	$0.742285\pi$
$228$	−6.92820	−0.458831
$229$	27.7128	1.83131	0.915657	−	0.401960i	$-0.131671\pi$
$229$	27.7128	1.83131	0.915657	+	0.401960i	$0.131671\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	6.92820	0.450988
$237$	11.0000	0.714527
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	13.8564	0.894427
$241$	20.7846	1.33885	0.669427	−	0.742878i	$-0.266540\pi$
$241$	20.7846	1.33885	0.669427	+	0.742878i	$0.266540\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	13.8564	0.885253
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−13.8564	−0.878114
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	3.46410	0.218218
$253$	20.7846	1.30672
$254$	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
$265$	−41.5692	−2.55358
$266$	0	0
$267$	6.92820	0.423999
$268$	−17.3205	−1.05802
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	5.19615	0.315644	0.157822	−	0.987468i	$-0.449553\pi$
$271$	5.19615	0.315644	0.157822	+	0.987468i	$0.449553\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.2487	1.46225
$276$	12.0000	0.722315
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	−1.73205	−0.103695
$280$	0	0
$281$	−24.2487	−1.44656	−0.723278	−	0.690557i	$-0.757366\pi$
$281$	−24.2487	−1.44656	−0.723278	+	0.690557i	$0.757366\pi$
$282$	0	0
$283$	11.0000	0.653882	0.326941	−	0.945045i	$-0.393982\pi$
$283$	11.0000	0.653882	0.326941	+	0.945045i	$0.393982\pi$
$284$	−20.7846	−1.23334
$285$	−12.0000	−0.710819
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−5.19615	−0.304604
$292$	−3.46410	−0.202721
$293$	−17.3205	−1.01187	−0.505937	−	0.862570i	$-0.668853\pi$
$293$	−17.3205	−1.01187	−0.505937	+	0.862570i	$0.668853\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	−3.46410	−0.201008
$298$	0	0
$299$	0	0
$300$	14.0000	0.808290
$301$	−1.73205	−0.0998337
$302$	0	0
$303$	−18.0000	−1.03407
$304$	−13.8564	−0.794719
$305$	−3.46410	−0.198354
$306$	0	0
$307$	−1.73205	−0.0988534	−0.0494267	−	0.998778i	$-0.515739\pi$
$307$	−1.73205	−0.0988534	−0.0494267	+	0.998778i	$0.515739\pi$
$308$	12.0000	0.683763
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	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
$317$	−6.92820	−0.389127	−0.194563	−	0.980890i	$-0.562329\pi$
$317$	−6.92820	−0.389127	−0.194563	+	0.980890i	$0.562329\pi$
$318$	0	0
$319$	20.7846	1.16371
$320$	27.7128	1.54919
$321$	6.00000	0.334887
$322$	0	0
$323$	0	0
$324$	−2.00000	−0.111111
$325$	0	0
$326$	0	0
$327$	−15.5885	−0.862044
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−5.19615	−0.285606	−0.142803	−	0.989751i	$-0.545612\pi$
$331$	−5.19615	−0.285606	−0.142803	+	0.989751i	$0.545612\pi$
$332$	−27.7128	−1.52094
$333$	0	0
$334$	0	0
$335$	−30.0000	−1.63908
$336$	6.92820	0.377964
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	19.0526	1.02874
$344$	0	0
$345$	20.7846	1.11901
$346$	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
$349$	19.0526	1.01986	0.509930	−	0.860216i	$-0.329671\pi$
$349$	19.0526	1.01986	0.509930	+	0.860216i	$0.329671\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.3923	0.553127	0.276563	−	0.960996i	$-0.410804\pi$
$353$	10.3923	0.553127	0.276563	+	0.960996i	$0.410804\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	13.8564	0.734388
$357$	0	0
$358$	0	0
$359$	6.92820	0.365657	0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820	0.365657	0.182828	+	0.983145i	$0.441475\pi$
$360$	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
$369$	−6.92820	−0.360668
$370$	0	0
$371$	−20.7846	−1.07908
$372$	−3.46410	−0.179605
$373$	−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
$375$	6.92820	0.357771
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−22.5167	−1.15660	−0.578302	−	0.815823i	$-0.696284\pi$
$379$	−22.5167	−1.15660	−0.578302	+	0.815823i	$0.696284\pi$
$380$	−24.0000	−1.23117
$381$	13.0000	0.666010
$382$	0	0
$383$	−27.7128	−1.41606	−0.708029	−	0.706183i	$-0.750416\pi$
$383$	−27.7128	−1.41606	−0.708029	+	0.706183i	$0.750416\pi$
$384$	0	0
$385$	20.7846	1.05928
$386$	0	0
$387$	1.00000	0.0508329
$388$	−10.3923	−0.527589
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	38.1051	1.91728
$396$	−6.92820	−0.348155
$397$	15.5885	0.782362	0.391181	−	0.920314i	$-0.372067\pi$
$397$	15.5885	0.782362	0.391181	+	0.920314i	$0.372067\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	28.0000	1.40000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	−36.0000	−1.79107
$405$	−3.46410	−0.172133
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8.66025	−0.428222	−0.214111	−	0.976809i	$-0.568685\pi$
$409$	−8.66025	−0.428222	−0.214111	+	0.976809i	$0.568685\pi$
$410$	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
$421$	12.1244	0.590905	0.295452	−	0.955357i	$-0.404530\pi$
$421$	12.1244	0.590905	0.295452	+	0.955357i	$0.404530\pi$
$422$	0	0
$423$	3.46410	0.168430
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.73205	−0.0838198
$428$	12.0000	0.580042
$429$	0	0
$430$	0	0
$431$	20.7846	1.00116	0.500580	−	0.865690i	$-0.333120\pi$
$431$	20.7846	1.00116	0.500580	+	0.865690i	$0.333120\pi$
$432$	−4.00000	−0.192450
$433$	23.0000	1.10531	0.552655	−	0.833410i	$-0.313615\pi$
$433$	23.0000	1.10531	0.552655	+	0.833410i	$0.313615\pi$
$434$	0	0
$435$	20.7846	0.996546
$436$	−31.1769	−1.49310
$437$	−20.7846	−0.994263
$438$	0	0
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	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
$447$	−6.92820	−0.327693
$448$	13.8564	0.654654
$449$	38.1051	1.79829	0.899146	−	0.437649i	$-0.144189\pi$
$449$	38.1051	1.79829	0.899146	+	0.437649i	$0.144189\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	−12.0000	−0.564433
$453$	3.46410	0.162758
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−36.3731	−1.70146	−0.850730	−	0.525603i	$-0.823840\pi$
$457$	−36.3731	−1.70146	−0.850730	+	0.525603i	$0.823840\pi$
$458$	0	0
$459$	0	0
$460$	41.5692	1.93817
$461$	41.5692	1.93607	0.968036	−	0.250812i	$-0.0806976\pi$
$461$	41.5692	1.93607	0.968036	+	0.250812i	$0.0806976\pi$
$462$	0	0
$463$	−36.3731	−1.69040	−0.845200	−	0.534450i	$-0.820519\pi$
$463$	−36.3731	−1.69040	−0.845200	+	0.534450i	$0.820519\pi$
$464$	24.0000	1.11417
$465$	−6.00000	−0.278243
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	−15.0000	−0.692636
$470$	0	0
$471$	−11.0000	−0.506853
$472$	0	0
$473$	3.46410	0.159280
$474$	0	0
$475$	−24.2487	−1.11261
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	−34.6410	−1.58279	−0.791394	−	0.611306i	$-0.790644\pi$
$479$	−34.6410	−1.58279	−0.791394	+	0.611306i	$0.790644\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	10.3923	0.472866
$484$	−2.00000	−0.0909091
$485$	−18.0000	−0.817338
$486$	0	0
$487$	24.2487	1.09881	0.549407	−	0.835555i	$-0.314854\pi$
$487$	24.2487	1.09881	0.549407	+	0.835555i	$0.314854\pi$
$488$	0	0
$489$	19.0526	0.861586
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	−13.8564	−0.624695
$493$	0	0
$494$	0	0
$495$	−12.0000	−0.539360
$496$	−6.92820	−0.311086
$497$	−18.0000	−0.807410
$498$	0	0
$499$	−31.1769	−1.39567	−0.697835	−	0.716258i	$-0.745853\pi$
$499$	−31.1769	−1.39567	−0.697835	+	0.716258i	$0.745853\pi$
$500$	13.8564	0.619677
$501$	−6.92820	−0.309529
$502$	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
$505$	−62.3538	−2.77471
$506$	0	0
$507$	0	0
$508$	26.0000	1.15356
$509$	−17.3205	−0.767718	−0.383859	−	0.923392i	$-0.625405\pi$
$509$	−17.3205	−0.767718	−0.383859	+	0.923392i	$0.625405\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	0	0
$513$	3.46410	0.152944
$514$	0	0
$515$	−3.46410	−0.152647
$516$	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
$525$	12.1244	0.529150
$526$	0	0
$527$	0	0
$528$	−13.8564	−0.603023
$529$	13.0000	0.565217
$530$	0	0
$531$	−3.46410	−0.150329
$532$	−12.0000	−0.520266
$533$	0	0
$534$	0	0
$535$	20.7846	0.898597
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	−13.8564	−0.596838
$540$	−6.92820	−0.298142
$541$	−29.4449	−1.26593	−0.632967	−	0.774179i	$-0.718163\pi$
$541$	−29.4449	−1.26593	−0.632967	+	0.774179i	$0.718163\pi$
$542$	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
$551$	−20.7846	−0.885454
$552$	0	0
$553$	19.0526	0.810197
$554$	0	0
$555$	0	0
$556$	−10.0000	−0.424094
$557$	27.7128	1.17423	0.587115	−	0.809504i	$-0.300264\pi$
$557$	27.7128	1.17423	0.587115	+	0.809504i	$0.300264\pi$
$558$	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	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	6.92820	0.291730
$565$	−20.7846	−0.874415
$566$	0	0
$567$	−1.73205	−0.0727393
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	42.0000	1.75152
$576$	−8.00000	−0.333333
$577$	34.6410	1.44212	0.721062	−	0.692870i	$-0.243654\pi$
$577$	34.6410	1.44212	0.721062	+	0.692870i	$0.243654\pi$
$578$	0	0
$579$	−15.5885	−0.647834
$580$	41.5692	1.72607
$581$	−24.0000	−0.995688
$582$	0	0
$583$	41.5692	1.72162
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−31.1769	−1.28681	−0.643404	−	0.765526i	$-0.722479\pi$
$587$	−31.1769	−1.28681	−0.643404	+	0.765526i	$0.722479\pi$
$588$	−8.00000	−0.329914
$589$	6.00000	0.247226
$590$	0	0
$591$	−13.8564	−0.569976
$592$	0	0
$593$	3.46410	0.142254	0.0711268	−	0.997467i	$-0.477341\pi$
$593$	3.46410	0.142254	0.0711268	+	0.997467i	$0.477341\pi$
$594$	0	0
$595$	0	0
$596$	−13.8564	−0.567581
$597$	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
$603$	8.66025	0.352673
$604$	6.92820	0.281905
$605$	−3.46410	−0.140836
$606$	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
$609$	10.3923	0.421117
$610$	0	0
$611$	0	0
$612$	0	0
$613$	8.66025	0.349784	0.174892	−	0.984588i	$-0.444042\pi$
$613$	8.66025	0.349784	0.174892	+	0.984588i	$0.444042\pi$
$614$	0	0
$615$	−24.0000	−0.967773
$616$	0	0
$617$	10.3923	0.418378	0.209189	−	0.977875i	$-0.432918\pi$
$617$	10.3923	0.418378	0.209189	+	0.977875i	$0.432918\pi$
$618$	0	0
$619$	25.9808	1.04425	0.522127	−	0.852867i	$-0.325139\pi$
$619$	25.9808	1.04425	0.522127	+	0.852867i	$0.325139\pi$
$620$	−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
$631$	1.73205	0.0689519	0.0344759	−	0.999406i	$-0.489024\pi$
$631$	1.73205	0.0689519	0.0344759	+	0.999406i	$0.489024\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	0	0
$635$	45.0333	1.78709
$636$	24.0000	0.951662
$637$	0	0
$638$	0	0
$639$	10.3923	0.411113
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	−19.0526	−0.751360	−0.375680	−	0.926750i	$-0.622591\pi$
$643$	−19.0526	−0.751360	−0.375680	+	0.926750i	$0.622591\pi$
$644$	20.7846	0.819028
$645$	3.46410	0.136399
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−3.00000	−0.117579
$652$	38.1051	1.49231
$653$	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
$655$	−20.7846	−0.812122
$656$	−27.7128	−1.08200
$657$	1.73205	0.0675737
$658$	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
$661$	25.9808	1.01053	0.505267	−	0.862963i	$-0.331394\pi$
$661$	25.9808	1.01053	0.505267	+	0.862963i	$0.331394\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−20.7846	−0.805993
$666$	0	0
$667$	36.0000	1.39393
$668$	−13.8564	−0.536120
$669$	−17.3205	−0.669650
$670$	0	0
$671$	3.46410	0.133730
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	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
$681$	20.7846	0.796468
$682$	0	0
$683$	24.2487	0.927851	0.463926	−	0.885874i	$-0.346441\pi$
$683$	24.2487	0.927851	0.463926	+	0.885874i	$0.346441\pi$
$684$	6.92820	0.264906
$685$	0	0
$686$	0	0
$687$	−27.7128	−1.05731
$688$	4.00000	0.152499
$689$	0	0
$690$	0	0
$691$	43.3013	1.64726	0.823629	−	0.567129i	$-0.191946\pi$
$691$	43.3013	1.64726	0.823629	+	0.567129i	$0.191946\pi$
$692$	−12.0000	−0.456172
$693$	−6.00000	−0.227921
$694$	0	0
$695$	−17.3205	−0.657004
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−18.0000	−0.680823
$700$	24.2487	0.916515
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	0	0
$704$	−27.7128	−1.04447
$705$	12.0000	0.451946
$706$	0	0
$707$	−31.1769	−1.17253
$708$	−6.92820	−0.260378
$709$	−19.0526	−0.715534	−0.357767	−	0.933811i	$-0.616462\pi$
$709$	−19.0526	−0.715534	−0.357767	+	0.933811i	$0.616462\pi$
$710$	0	0
$711$	−11.0000	−0.412532
$712$	0	0
$713$	−10.3923	−0.389195
$714$	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	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−13.8564	−0.516398
$721$	−1.73205	−0.0645049
$722$	0	0
$723$	−20.7846	−0.772988
$724$	28.0000	1.04061
$725$	42.0000	1.55984
$726$	0	0
$727$	35.0000	1.29808	0.649039	−	0.760755i	$-0.275171\pi$
$727$	35.0000	1.29808	0.649039	+	0.760755i	$0.275171\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	2.00000	0.0739221
$733$	−39.8372	−1.47142	−0.735710	−	0.677297i	$-0.763151\pi$
$733$	−39.8372	−1.47142	−0.735710	+	0.677297i	$0.763151\pi$
$734$	0	0
$735$	−13.8564	−0.511101
$736$	0	0
$737$	30.0000	1.10506
$738$	0	0
$739$	−45.0333	−1.65658	−0.828289	−	0.560301i	$-0.810685\pi$
$739$	−45.0333	−1.65658	−0.828289	+	0.560301i	$0.810685\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.3923	−0.381257	−0.190628	−	0.981662i	$-0.561053\pi$
$743$	−10.3923	−0.381257	−0.190628	+	0.981662i	$0.561053\pi$
$744$	0	0
$745$	−24.0000	−0.879292
$746$	0	0
$747$	13.8564	0.506979
$748$	0	0
$749$	10.3923	0.379727
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	13.8564	0.505291
$753$	12.0000	0.437304
$754$	0	0
$755$	12.0000	0.436725
$756$	−3.46410	−0.125988
$757$	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
$759$	−20.7846	−0.754434
$760$	0	0
$761$	−20.7846	−0.753442	−0.376721	−	0.926327i	$-0.622948\pi$
$761$	−20.7846	−0.753442	−0.376721	+	0.926327i	$0.622948\pi$
$762$	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
$769$	−6.92820	−0.249837	−0.124919	−	0.992167i	$-0.539867\pi$
$769$	−6.92820	−0.249837	−0.124919	+	0.992167i	$0.539867\pi$
$770$	0	0
$771$	0	0
$772$	−31.1769	−1.12208
$773$	51.9615	1.86893	0.934463	−	0.356060i	$-0.115880\pi$
$773$	51.9615	1.86893	0.934463	+	0.356060i	$0.115880\pi$
$774$	0	0
$775$	−12.1244	−0.435520
$776$	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
$785$	−38.1051	−1.36003
$786$	0	0
$787$	32.9090	1.17308	0.586539	−	0.809921i	$-0.300490\pi$
$787$	32.9090	1.17308	0.586539	+	0.809921i	$0.300490\pi$
$788$	−27.7128	−0.987228
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	0	0
$794$	0	0
$795$	41.5692	1.47431
$796$	14.0000	0.496217
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−6.92820	−0.244796
$802$	0	0
$803$	6.00000	0.211735
$804$	17.3205	0.610847
$805$	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
$811$	25.9808	0.912308	0.456154	−	0.889901i	$-0.349227\pi$
$811$	25.9808	0.912308	0.456154	+	0.889901i	$0.349227\pi$
$812$	20.7846	0.729397
$813$	−5.19615	−0.182237
$814$	0	0
$815$	66.0000	2.31188
$816$	0	0
$817$	−3.46410	−0.121194
$818$	0	0
$819$	0	0
$820$	−48.0000	−1.67623
$821$	24.2487	0.846286	0.423143	−	0.906063i	$-0.360927\pi$
$821$	24.2487	0.846286	0.423143	+	0.906063i	$0.360927\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	−24.2487	−0.844232
$826$	0	0
$827$	48.4974	1.68642	0.843210	−	0.537584i	$-0.180663\pi$
$827$	48.4974	1.68642	0.843210	+	0.537584i	$0.180663\pi$
$828$	−12.0000	−0.417029
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	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
$837$	1.73205	0.0598684
$838$	0	0
$839$	31.1769	1.07635	0.538173	−	0.842834i	$-0.319115\pi$
$839$	31.1769	1.07635	0.538173	+	0.842834i	$0.319115\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	24.2487	0.835170
$844$	−26.0000	−0.894957
$845$	0	0
$846$	0	0
$847$	−1.73205	−0.0595140
$848$	48.0000	1.64833
$849$	−11.0000	−0.377519
$850$	0	0
$851$	0	0
$852$	20.7846	0.712069
$853$	−25.9808	−0.889564	−0.444782	−	0.895639i	$-0.646719\pi$
$853$	−25.9808	−0.889564	−0.444782	+	0.895639i	$0.646719\pi$
$854$	0	0
$855$	12.0000	0.410391
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	6.92820	0.236250
$861$	−12.0000	−0.408959
$862$	0	0
$863$	17.3205	0.589597	0.294798	−	0.955559i	$-0.404747\pi$
$863$	17.3205	0.589597	0.294798	+	0.955559i	$0.404747\pi$
$864$	0	0
$865$	−20.7846	−0.706698
$866$	0	0
$867$	17.0000	0.577350
$868$	−6.00000	−0.203653
$869$	−38.1051	−1.29263
$870$	0	0
$871$	0	0
$872$	0	0
$873$	5.19615	0.175863
$874$	0	0
$875$	12.0000	0.405674
$876$	3.46410	0.117041
$877$	−41.5692	−1.40369	−0.701846	−	0.712328i	$-0.747641\pi$
$877$	−41.5692	−1.40369	−0.701846	+	0.712328i	$0.747641\pi$
$878$	0	0
$879$	17.3205	0.584206
$880$	−48.0000	−1.61808
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	0	0
$883$	5.00000	0.168263	0.0841317	−	0.996455i	$-0.473188\pi$
$883$	5.00000	0.168263	0.0841317	+	0.996455i	$0.473188\pi$
$884$	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
$889$	22.5167	0.755185
$890$	0	0
$891$	3.46410	0.116052
$892$	−34.6410	−1.15987
$893$	−12.0000	−0.401565
$894$	0	0
$895$	41.5692	1.38951
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.3923	−0.346603
$900$	−14.0000	−0.466667
$901$	0	0
$902$	0	0
$903$	1.73205	0.0576390
$904$	0	0
$905$	48.4974	1.61211
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	41.5692	1.37952
$909$	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	$0.563701\pi$
$912$	13.8564	0.458831
$913$	48.0000	1.58857
$914$	0	0
$915$	3.46410	0.114520
$916$	−55.4256	−1.83131
$917$	−10.3923	−0.343184
$918$	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
$921$	1.73205	0.0570730
$922$	0	0
$923$	0	0
$924$	−12.0000	−0.394771
$925$	0	0
$926$	0	0
$927$	1.00000	0.0328443
$928$	0	0
$929$	20.7846	0.681921	0.340960	−	0.940078i	$-0.389248\pi$
$929$	20.7846	0.681921	0.340960	+	0.940078i	$0.389248\pi$
$930$	0	0
$931$	13.8564	0.454125
$932$	−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
$941$	10.3923	0.338779	0.169390	−	0.985549i	$-0.445820\pi$
$941$	10.3923	0.338779	0.169390	+	0.985549i	$0.445820\pi$
$942$	0	0
$943$	−41.5692	−1.35368
$944$	−13.8564	−0.450988
$945$	−6.00000	−0.195180
$946$	0	0
$947$	48.4974	1.57595	0.787977	−	0.615704i	$-0.211128\pi$
$947$	48.4974	1.57595	0.787977	+	0.615704i	$0.211128\pi$
$948$	−22.0000	−0.714527
$949$	0	0
$950$	0	0
$951$	6.92820	0.224662
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	62.3538	2.01772
$956$	0	0
$957$	−20.7846	−0.671871
$958$	0	0
$959$	0	0
$960$	−27.7128	−0.894427
$961$	−28.0000	−0.903226
$962$	0	0
$963$	−6.00000	−0.193347
$964$	−41.5692	−1.33885
$965$	−54.0000	−1.73832
$966$	0	0
$967$	24.2487	0.779786	0.389893	−	0.920860i	$-0.372512\pi$
$967$	24.2487	0.779786	0.389893	+	0.920860i	$0.372512\pi$
$968$	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
$973$	−8.66025	−0.277635
$974$	0	0
$975$	0	0
$976$	4.00000	0.128037
$977$	−27.7128	−0.886611	−0.443306	−	0.896370i	$-0.646194\pi$
$977$	−27.7128	−0.886611	−0.443306	+	0.896370i	$0.646194\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	−27.7128	−0.885253
$981$	15.5885	0.497701
$982$	0	0
$983$	−10.3923	−0.331463	−0.165732	−	0.986171i	$-0.552999\pi$
$983$	−10.3923	−0.331463	−0.165732	+	0.986171i	$0.552999\pi$
$984$	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
$993$	5.19615	0.164895
$994$	0	0
$995$	24.2487	0.768736
$996$	27.7128	0.878114
$997$	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.a.e.1.1		2
3.2	odd	2		1521.2.a.h.1.2		2
4.3	odd	2		8112.2.a.bu.1.1		2
13.2	odd	12		39.2.j.a.4.1	✓	2
13.3	even	3		507.2.e.f.22.1		4
13.4	even	6		507.2.e.f.484.2		4
13.5	odd	4		507.2.b.c.337.2		2
13.6	odd	12		507.2.j.b.361.1		2
13.7	odd	12		39.2.j.a.10.1	yes	2
13.8	odd	4		507.2.b.c.337.1		2
13.9	even	3		507.2.e.f.484.1		4
13.10	even	6		507.2.e.f.22.2		4
13.11	odd	12		507.2.j.b.316.1		2
13.12	even	2	inner	507.2.a.e.1.2		2
39.2	even	12		117.2.q.a.82.1		2
39.5	even	4		1521.2.b.f.1351.1		2
39.8	even	4		1521.2.b.f.1351.2		2
39.20	even	12		117.2.q.a.10.1		2
39.38	odd	2		1521.2.a.h.1.1		2
52.7	even	12		624.2.bv.b.49.1		2
52.15	even	12		624.2.bv.b.433.1		2
52.51	odd	2		8112.2.a.bu.1.2		2
65.2	even	12		975.2.w.d.199.1		4
65.7	even	12		975.2.w.d.49.2		4
65.28	even	12		975.2.w.d.199.2		4
65.33	even	12		975.2.w.d.49.1		4
65.54	odd	12		975.2.bc.c.901.1		2
65.59	odd	12		975.2.bc.c.751.1		2
156.59	odd	12		1872.2.by.f.1297.1		2
156.119	odd	12		1872.2.by.f.433.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.j.a.4.1	✓	2	13.2	odd	12
39.2.j.a.10.1	yes	2	13.7	odd	12
117.2.q.a.10.1		2	39.20	even	12
117.2.q.a.82.1		2	39.2	even	12
507.2.a.e.1.1		2	1.1	even	1	trivial
507.2.a.e.1.2		2	13.12	even	2	inner
507.2.b.c.337.1		2	13.8	odd	4
507.2.b.c.337.2		2	13.5	odd	4
507.2.e.f.22.1		4	13.3	even	3
507.2.e.f.22.2		4	13.10	even	6
507.2.e.f.484.1		4	13.9	even	3
507.2.e.f.484.2		4	13.4	even	6
507.2.j.b.316.1		2	13.11	odd	12
507.2.j.b.361.1		2	13.6	odd	12
624.2.bv.b.49.1		2	52.7	even	12
624.2.bv.b.433.1		2	52.15	even	12
975.2.w.d.49.1		4	65.33	even	12
975.2.w.d.49.2		4	65.7	even	12
975.2.w.d.199.1		4	65.2	even	12
975.2.w.d.199.2		4	65.28	even	12
975.2.bc.c.751.1		2	65.59	odd	12
975.2.bc.c.901.1		2	65.54	odd	12
1521.2.a.h.1.1		2	39.38	odd	2
1521.2.a.h.1.2		2	3.2	odd	2
1521.2.b.f.1351.1		2	39.5	even	4
1521.2.b.f.1351.2		2	39.8	even	4
1872.2.by.f.433.1		2	156.119	odd	12
1872.2.by.f.1297.1		2	156.59	odd	12
8112.2.a.bu.1.1		2	4.3	odd	2
8112.2.a.bu.1.2		2	52.51	odd	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

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