LMFDB - Embedded newform 663.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [663,2,Mod(1,663)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(663, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("663.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 663 = 3 \cdot 13 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	663.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:	$5.29408165401$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	663.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +2.00000 q^{15} -1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} -3.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} -2.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} -4.00000 q^{33} -1.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -1.00000 q^{39} -6.00000 q^{40} +2.00000 q^{41} -4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -8.00000 q^{55} +4.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} +14.0000 q^{61} +8.00000 q^{62} +7.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} +3.00000 q^{72} -14.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +1.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} -2.00000 q^{85} +4.00000 q^{86} +2.00000 q^{87} +12.0000 q^{88} -6.00000 q^{89} +2.00000 q^{90} +8.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} +5.00000 q^{96} -6.00000 q^{97} +7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.00000	0.516398
$16$	−1.00000	−0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−3.00000	−0.612372
$25$	−1.00000	−0.200000
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−2.00000	−0.365148
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−5.00000	−0.883883
$33$	−4.00000	−0.696311
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	4.00000	0.648886
$39$	−1.00000	−0.160128
$40$	−6.00000	−0.948683
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−4.00000	−0.603023
$45$	−2.00000	−0.298142
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	−1.00000	−0.140028
$52$	−1.00000	−0.138675
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000	0.136083
$55$	−8.00000	−1.07872
$56$	0	0
$57$	4.00000	0.529813
$58$	2.00000	0.262613
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−2.00000	−0.258199
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	7.00000	0.875000
$65$	−2.00000	−0.248069
$66$	4.00000	0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000	0.353553
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	1.00000	0.113228
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	4.00000	0.431331
$87$	2.00000	0.214423
$88$	12.0000	1.27920
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	0	0
$93$	8.00000	0.829561
$94$	−8.00000	−0.825137
$95$	8.00000	0.820783
$96$	5.00000	0.510310
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	7.00000	0.707107
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	1.00000	0.0990148
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	10.0000	0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	8.00000	0.762770
$111$	2.00000	0.189832
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	1.00000	0.0924500
$118$	−4.00000	−0.368230
$119$	0	0
$120$	6.00000	0.547723
$121$	5.00000	0.454545
$122$	−14.0000	−1.26750
$123$	−2.00000	−0.180334
$124$	8.00000	0.718421
$125$	12.0000	1.07331
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	3.00000	0.265165
$129$	4.00000	0.352180
$130$	2.00000	0.175412
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	4.00000	0.345547
$135$	2.00000	0.172133
$136$	3.00000	0.257248
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	4.00000	0.334497
$144$	−1.00000	−0.0833333
$145$	4.00000	0.332182
$146$	14.0000	1.15865
$147$	7.00000	0.577350
$148$	2.00000	0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−1.00000	−0.0816497
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−12.0000	−0.973329
$153$	1.00000	0.0808452
$154$	0	0
$155$	16.0000	1.28515
$156$	1.00000	0.0800641
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	8.00000	0.636446
$159$	10.0000	0.793052
$160$	10.0000	0.790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.00000	−0.156174
$165$	8.00000	0.622799
$166$	4.00000	0.310460
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.00000	0.153393
$171$	−4.00000	−0.305888
$172$	4.00000	0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−4.00000	−0.301511
$177$	−4.00000	−0.300658
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	2.00000	0.149071
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	−14.0000	−1.03491
$184$	0	0
$185$	4.00000	0.294086
$186$	−8.00000	−0.586588
$187$	4.00000	0.292509
$188$	−8.00000	−0.583460
$189$	0	0
$190$	−8.00000	−0.580381
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	−7.00000	−0.505181
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	6.00000	0.430775
$195$	2.00000	0.143223
$196$	7.00000	0.500000
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−4.00000	−0.284268
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−3.00000	−0.212132
$201$	4.00000	0.282138
$202$	−6.00000	−0.422159
$203$	0	0
$204$	1.00000	0.0700140
$205$	−4.00000	−0.279372
$206$	8.00000	0.557386
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−16.0000	−1.10674
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	10.0000	0.686803
$213$	0	0
$214$	12.0000	0.820303
$215$	8.00000	0.545595
$216$	−3.00000	−0.204124
$217$	0	0
$218$	10.0000	0.677285
$219$	14.0000	0.946032
$220$	8.00000	0.539360
$221$	1.00000	0.0672673
$222$	−2.00000	−0.134231
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−2.00000	−0.133038
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	−4.00000	−0.264906
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−1.00000	−0.0653720
$235$	−16.0000	−1.04372
$236$	−4.00000	−0.260378
$237$	8.00000	0.519656
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−2.00000	−0.129099
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	14.0000	0.894427
$246$	2.00000	0.127515
$247$	−4.00000	−0.254514
$248$	−24.0000	−1.52400
$249$	4.00000	0.253490
$250$	−12.0000	−0.758947
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2.00000	0.125245
$256$	−17.0000	−1.06250
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	2.00000	0.124035
$261$	−2.00000	−0.123797
$262$	4.00000	0.247121
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	−12.0000	−0.738549
$265$	20.0000	1.22859
$266$	0	0
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−2.00000	−0.121716
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−1.00000	−0.0606339
$273$	0	0
$274$	22.0000	1.32907
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−20.0000	−1.19952
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	8.00000	0.476393
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	−4.00000	−0.236525
$287$	0	0
$288$	−5.00000	−0.294628
$289$	1.00000	0.0588235
$290$	−4.00000	−0.234888
$291$	6.00000	0.351726
$292$	14.0000	0.819288
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	−7.00000	−0.408248
$295$	−8.00000	−0.465778
$296$	−6.00000	−0.348743
$297$	−4.00000	−0.232104
$298$	10.0000	0.579284
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	16.0000	0.920697
$303$	−6.00000	−0.344691
$304$	4.00000	0.229416
$305$	−28.0000	−1.60328
$306$	−1.00000	−0.0571662
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	−16.0000	−0.908739
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−3.00000	−0.169842
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	−10.0000	−0.560772
$319$	−8.00000	−0.447914
$320$	−14.0000	−0.782624
$321$	12.0000	0.669775
$322$	0	0
$323$	−4.00000	−0.222566
$324$	−1.00000	−0.0555556
$325$	−1.00000	−0.0554700
$326$	4.00000	0.221540
$327$	10.0000	0.553001
$328$	6.00000	0.331295
$329$	0	0
$330$	−8.00000	−0.440386
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	4.00000	0.219529
$333$	−2.00000	−0.109599
$334$	−16.0000	−0.875481
$335$	8.00000	0.437087
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−1.00000	−0.0543928
$339$	−2.00000	−0.108625
$340$	2.00000	0.108465
$341$	−32.0000	−1.73290
$342$	4.00000	0.216295
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−14.0000	−0.752645
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−2.00000	−0.107211
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	−20.0000	−1.06600
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	4.00000	0.212598
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−6.00000	−0.316228
$361$	−3.00000	−0.157895
$362$	−6.00000	−0.315353
$363$	−5.00000	−0.262432
$364$	0	0
$365$	28.0000	1.46559
$366$	14.0000	0.731792
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	−4.00000	−0.207950
$371$	0	0
$372$	−8.00000	−0.414781
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−4.00000	−0.206835
$375$	−12.0000	−0.619677
$376$	24.0000	1.23771
$377$	−2.00000	−0.103005
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	−16.0000	−0.818631
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	22.0000	1.11977
$387$	−4.00000	−0.203331
$388$	6.00000	0.304604
$389$	38.0000	1.92668	0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000	1.92668	0.963338	+	0.268290i	$0.0864585\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	−21.0000	−1.06066
$393$	4.00000	0.201773
$394$	2.00000	0.100759
$395$	16.0000	0.805047
$396$	−4.00000	−0.201008
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−4.00000	−0.199502
$403$	−8.00000	−0.398508
$404$	−6.00000	−0.298511
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−8.00000	−0.396545
$408$	−3.00000	−0.148522
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	4.00000	0.197546
$411$	22.0000	1.08518
$412$	8.00000	0.394132
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	−5.00000	−0.245145
$417$	−20.0000	−0.979404
$418$	16.0000	0.782586
$419$	28.0000	1.36789	0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000	1.36789	0.683945	+	0.729534i	$0.260263\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−12.0000	−0.584151
$423$	8.00000	0.388973
$424$	−30.0000	−1.45693
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	−4.00000	−0.193122
$430$	−8.00000	−0.385794
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000	0.0481125
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	10.0000	0.478913
$437$	0	0
$438$	−14.0000	−0.668946
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	−24.0000	−1.14416
$441$	−7.00000	−0.333333
$442$	−1.00000	−0.0475651
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	−2.00000	−0.0949158
$445$	12.0000	0.568855
$446$	−8.00000	−0.378811
$447$	10.0000	0.472984
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	1.00000	0.0471405
$451$	8.00000	0.376705
$452$	−2.00000	−0.0940721
$453$	16.0000	0.751746
$454$	−12.0000	−0.563188
$455$	0	0
$456$	12.0000	0.561951
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	10.0000	0.467269
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	2.00000	0.0928477
$465$	−16.0000	−0.741982
$466$	6.00000	0.277945
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	−1.00000	−0.0462250
$469$	0	0
$470$	16.0000	0.738025
$471$	2.00000	0.0921551
$472$	12.0000	0.552345
$473$	−16.0000	−0.735681
$474$	−8.00000	−0.367452
$475$	4.00000	0.183533
$476$	0	0
$477$	−10.0000	−0.457869
$478$	24.0000	1.09773
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−10.0000	−0.456435
$481$	−2.00000	−0.0911922
$482$	−10.0000	−0.455488
$483$	0	0
$484$	−5.00000	−0.227273
$485$	12.0000	0.544892
$486$	1.00000	0.0453609
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	42.0000	1.90125
$489$	4.00000	0.180886
$490$	−14.0000	−0.632456
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	2.00000	0.0901670
$493$	−2.00000	−0.0900755
$494$	4.00000	0.179969
$495$	−8.00000	−0.359573
$496$	8.00000	0.359211
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−12.0000	−0.536656
$501$	−16.0000	−0.714827
$502$	−12.0000	−0.535586
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	11.0000	0.486136
$513$	4.00000	0.176604
$514$	30.0000	1.32324
$515$	16.0000	0.705044
$516$	−4.00000	−0.176090
$517$	32.0000	1.40736
$518$	0	0
$519$	−14.0000	−0.614532
$520$	−6.00000	−0.263117
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	2.00000	0.0875376
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−8.00000	−0.348485
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	−20.0000	−0.868744
$531$	4.00000	0.173585
$532$	0	0
$533$	2.00000	0.0866296
$534$	−6.00000	−0.259645
$535$	24.0000	1.03761
$536$	−12.0000	−0.518321
$537$	12.0000	0.517838
$538$	18.0000	0.776035
$539$	−28.0000	−1.20605
$540$	−2.00000	−0.0860663
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	−8.00000	−0.343629
$543$	−6.00000	−0.257485
$544$	−5.00000	−0.214373
$545$	20.0000	0.856706
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	22.0000	0.939793
$549$	14.0000	0.597505
$550$	4.00000	0.170561
$551$	8.00000	0.340811
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	−4.00000	−0.169791
$556$	−20.0000	−0.848189
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	8.00000	0.338667
$559$	−4.00000	−0.169182
$560$	0	0
$561$	−4.00000	−0.168880
$562$	6.00000	0.253095
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	8.00000	0.336861
$565$	−4.00000	−0.168281
$566$	−20.0000	−0.840663
$567$	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	$0.540139\pi$
$570$	8.00000	0.335083
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	−4.00000	−0.167248
$573$	−16.0000	−0.668410
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−1.00000	−0.0415945
$579$	22.0000	0.914289
$580$	−4.00000	−0.166091
$581$	0	0
$582$	−6.00000	−0.248708
$583$	−40.0000	−1.65663
$584$	−42.0000	−1.73797
$585$	−2.00000	−0.0826898
$586$	26.0000	1.07405
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	−7.00000	−0.288675
$589$	32.0000	1.31854
$590$	8.00000	0.329355
$591$	2.00000	0.0822690
$592$	2.00000	0.0821995
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	10.0000	0.409616
$597$	−16.0000	−0.654836
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	3.00000	0.122474
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	16.0000	0.651031
$605$	−10.0000	−0.406558
$606$	6.00000	0.243733
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	20.0000	0.811107
$609$	0	0
$610$	28.0000	1.13369
$611$	8.00000	0.323645
$612$	−1.00000	−0.0404226
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	20.0000	0.807134
$615$	4.00000	0.161296
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	−8.00000	−0.321807
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	−16.0000	−0.642575
$621$	0	0
$622$	0	0
$623$	0	0
$624$	1.00000	0.0400320
$625$	−19.0000	−0.760000
$626$	22.0000	0.879297
$627$	16.0000	0.638978
$628$	2.00000	0.0798087
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	−24.0000	−0.954669
$633$	−12.0000	−0.476957
$634$	10.0000	0.397151
$635$	0	0
$636$	−10.0000	−0.396526
$637$	−7.00000	−0.277350
$638$	8.00000	0.316723
$639$	0	0
$640$	−6.00000	−0.237171
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	−12.0000	−0.473602
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	4.00000	0.157378
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	3.00000	0.117851
$649$	16.0000	0.628055
$650$	1.00000	0.0392232
$651$	0	0
$652$	4.00000	0.156652
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−10.0000	−0.391031
$655$	8.00000	0.312586
$656$	−2.00000	−0.0780869
$657$	−14.0000	−0.546192
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	−8.00000	−0.311400
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−4.00000	−0.155464
$663$	−1.00000	−0.0388368
$664$	−12.0000	−0.465690
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	−16.0000	−0.619059
$669$	−8.00000	−0.309298
$670$	−8.00000	−0.309067
$671$	56.0000	2.16186
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	−18.0000	−0.693334
$675$	1.00000	0.0384900
$676$	−1.00000	−0.0384615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	2.00000	0.0768095
$679$	0	0
$680$	−6.00000	−0.230089
$681$	−12.0000	−0.459841
$682$	32.0000	1.22534
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	4.00000	0.152944
$685$	44.0000	1.68115
$686$	0	0
$687$	10.0000	0.381524
$688$	4.00000	0.152499
$689$	−10.0000	−0.380970
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	12.0000	0.455514
$695$	−40.0000	−1.51729
$696$	6.00000	0.227429
$697$	2.00000	0.0757554
$698$	−30.0000	−1.13552
$699$	6.00000	0.226941
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	1.00000	0.0377426
$703$	8.00000	0.301726
$704$	28.0000	1.05529
$705$	16.0000	0.602595
$706$	14.0000	0.526897
$707$	0	0
$708$	4.00000	0.150329
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	−18.0000	−0.674579
$713$	0	0
$714$	0	0
$715$	−8.00000	−0.299183
$716$	12.0000	0.448461
$717$	24.0000	0.896296
$718$	0	0
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	3.00000	0.111648
$723$	−10.0000	−0.371904
$724$	−6.00000	−0.222988
$725$	2.00000	0.0742781
$726$	5.00000	0.185567
$727$	−40.0000	−1.48352	−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000	−1.48352	−0.741759	+	0.670667i	$0.766008\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−28.0000	−1.03633
$731$	−4.00000	−0.147945
$732$	14.0000	0.517455
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−8.00000	−0.295285
$735$	−14.0000	−0.516398
$736$	0	0
$737$	−16.0000	−0.589368
$738$	−2.00000	−0.0736210
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	−4.00000	−0.147043
$741$	4.00000	0.146944
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	24.0000	0.879883
$745$	20.0000	0.732743
$746$	−22.0000	−0.805477
$747$	−4.00000	−0.146352
$748$	−4.00000	−0.146254
$749$	0	0
$750$	12.0000	0.438178
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	−8.00000	−0.291730
$753$	−12.0000	−0.437304
$754$	2.00000	0.0728357
$755$	32.0000	1.16460
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	24.0000	0.870572
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	0	0
$763$	0	0
$764$	−16.0000	−0.578860
$765$	−2.00000	−0.0723102
$766$	24.0000	0.867155
$767$	4.00000	0.144432
$768$	17.0000	0.613435
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	30.0000	1.08042
$772$	22.0000	0.791797
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	4.00000	0.143777
$775$	8.00000	0.287368
$776$	−18.0000	−0.646162
$777$	0	0
$778$	−38.0000	−1.36237
$779$	−8.00000	−0.286630
$780$	−2.00000	−0.0716115
$781$	0	0
$782$	0	0
$783$	2.00000	0.0714742
$784$	7.00000	0.250000
$785$	4.00000	0.142766
$786$	−4.00000	−0.142675
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	2.00000	0.0712470
$789$	−24.0000	−0.854423
$790$	−16.0000	−0.569254
$791$	0	0
$792$	12.0000	0.426401
$793$	14.0000	0.497155
$794$	26.0000	0.922705
$795$	−20.0000	−0.709327
$796$	−16.0000	−0.567105
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	5.00000	0.176777
$801$	−6.00000	−0.212000
$802$	6.00000	0.211867
$803$	−56.0000	−1.97620
$804$	−4.00000	−0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	18.0000	0.633630
$808$	18.0000	0.633238
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	2.00000	0.0702728
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	8.00000	0.280400
$815$	8.00000	0.280228
$816$	1.00000	0.0350070
$817$	16.0000	0.559769
$818$	−26.0000	−0.909069
$819$	0	0
$820$	4.00000	0.139686
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−22.0000	−0.767338
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−24.0000	−0.836080
$825$	4.00000	0.139262
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−8.00000	−0.277684
$831$	10.0000	0.346896
$832$	7.00000	0.242681
$833$	−7.00000	−0.242536
$834$	20.0000	0.692543
$835$	−32.0000	−1.10741
$836$	16.0000	0.553372
$837$	8.00000	0.276520
$838$	−28.0000	−0.967244
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	26.0000	0.896019
$843$	6.00000	0.206651
$844$	−12.0000	−0.413057
$845$	−2.00000	−0.0688021
$846$	−8.00000	−0.275046
$847$	0	0
$848$	10.0000	0.343401
$849$	−20.0000	−0.686398
$850$	1.00000	0.0342997
$851$	0	0
$852$	0	0
$853$	−50.0000	−1.71197	−0.855984	−	0.517003i	$-0.827048\pi$
$853$	−50.0000	−1.71197	−0.855984	+	0.517003i	$0.827048\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	−36.0000	−1.23045
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	4.00000	0.136558
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	−24.0000	−0.817443
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	5.00000	0.170103
$865$	−28.0000	−0.952029
$866$	14.0000	0.475739
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−32.0000	−1.08553
$870$	4.00000	0.135613
$871$	−4.00000	−0.135535
$872$	−30.0000	−1.01593
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	−14.0000	−0.473016
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−16.0000	−0.539974
$879$	26.0000	0.876958
$880$	8.00000	0.269680
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	7.00000	0.235702
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−1.00000	−0.0336336
$885$	8.00000	0.268917
$886$	20.0000	0.671913
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	−12.0000	−0.402241
$891$	4.00000	0.134005
$892$	−8.00000	−0.267860
$893$	−32.0000	−1.07084
$894$	−10.0000	−0.334450
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	22.0000	0.734150
$899$	16.0000	0.533630
$900$	1.00000	0.0333333
$901$	−10.0000	−0.333148
$902$	−8.00000	−0.266371
$903$	0	0
$904$	6.00000	0.199557
$905$	−12.0000	−0.398893
$906$	−16.0000	−0.531564
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−12.0000	−0.398234
$909$	6.00000	0.199007
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	−4.00000	−0.132453
$913$	−16.0000	−0.529523
$914$	22.0000	0.727695
$915$	28.0000	0.925651
$916$	10.0000	0.330409
$917$	0	0
$918$	1.00000	0.0330049
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	−24.0000	−0.788689
$927$	−8.00000	−0.262754
$928$	10.0000	0.328266
$929$	−22.0000	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$929$	−22.0000	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$930$	16.0000	0.524661
$931$	28.0000	0.917663
$932$	6.00000	0.196537
$933$	0	0
$934$	−4.00000	−0.130884
$935$	−8.00000	−0.261628
$936$	3.00000	0.0980581
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	22.0000	0.717943
$940$	16.0000	0.521862
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−2.00000	−0.0651635
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−8.00000	−0.259828
$949$	−14.0000	−0.454459
$950$	−4.00000	−0.129777
$951$	10.0000	0.324272
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	10.0000	0.323762
$955$	−32.0000	−1.03550
$956$	24.0000	0.776215
$957$	8.00000	0.258603
$958$	24.0000	0.775405
$959$	0	0
$960$	14.0000	0.451848
$961$	33.0000	1.06452
$962$	2.00000	0.0644826
$963$	−12.0000	−0.386695
$964$	−10.0000	−0.322078
$965$	44.0000	1.41641
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	15.0000	0.482118
$969$	4.00000	0.128499
$970$	−12.0000	−0.385297
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−32.0000	−1.02535
$975$	1.00000	0.0320256
$976$	−14.0000	−0.448129
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−4.00000	−0.127906
$979$	−24.0000	−0.767043
$980$	−14.0000	−0.447214
$981$	−10.0000	−0.319275
$982$	−28.0000	−0.893516
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	−6.00000	−0.191273
$985$	4.00000	0.127451
$986$	2.00000	0.0636930
$987$	0	0
$988$	4.00000	0.127257
$989$	0	0
$990$	8.00000	0.254257
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	40.0000	1.27000
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−32.0000	−1.01447
$996$	−4.00000	−0.126745
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	4.00000	0.126618
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	663.2.a.a.1.1	✓	1
3.2	odd	2		1989.2.a.e.1.1		1
13.12	even	2		8619.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
663.2.a.a.1.1	✓	1	1.1	even	1	trivial
1989.2.a.e.1.1		1	3.2	odd	2
8619.2.a.i.1.1		1	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 663 = 3 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	663.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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