LMFDB - Embedded newform 32.3.c.a.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,3,Mod(31,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("32.31");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	32.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$0.871936845953$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	32.31
Dual form			32.3.c.a.31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{3} +2.00000 q^{5} -8.00000i q^{7} -7.00000 q^{9} +O(q^{10})$	$q+4.00000i q^{3} +2.00000 q^{5} -8.00000i q^{7} -7.00000 q^{9} -4.00000i q^{11} -14.0000 q^{13} +8.00000i q^{15} +18.0000 q^{17} -12.0000i q^{19} +32.0000 q^{21} +40.0000i q^{23} -21.0000 q^{25} +8.00000i q^{27} -14.0000 q^{29} -32.0000i q^{31} +16.0000 q^{33} -16.0000i q^{35} -30.0000 q^{37} -56.0000i q^{39} -14.0000 q^{41} +28.0000i q^{43} -14.0000 q^{45} +16.0000i q^{47} -15.0000 q^{49} +72.0000i q^{51} +66.0000 q^{53} -8.00000i q^{55} +48.0000 q^{57} -52.0000i q^{59} +82.0000 q^{61} +56.0000i q^{63} -28.0000 q^{65} +4.00000i q^{67} -160.000 q^{69} +56.0000i q^{71} +66.0000 q^{73} -84.0000i q^{75} -32.0000 q^{77} -16.0000i q^{79} -95.0000 q^{81} -140.000i q^{83} +36.0000 q^{85} -56.0000i q^{87} -30.0000 q^{89} +112.000i q^{91} +128.000 q^{93} -24.0000i q^{95} -14.0000 q^{97} +28.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 14 q^{9}+O(q^{10}) $ 2 * q + 4 * q^5 - 14 * q^9	$ 2 q + 4 q^{5} - 14 q^{9} - 28 q^{13} + 36 q^{17} + 64 q^{21} - 42 q^{25} - 28 q^{29} + 32 q^{33} - 60 q^{37} - 28 q^{41} - 28 q^{45} - 30 q^{49} + 132 q^{53} + 96 q^{57} + 164 q^{61} - 56 q^{65} - 320 q^{69} + 132 q^{73} - 64 q^{77} - 190 q^{81} + 72 q^{85} - 60 q^{89} + 256 q^{93} - 28 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - 14 * q^9 - 28 * q^13 + 36 * q^17 + 64 * q^21 - 42 * q^25 - 28 * q^29 + 32 * q^33 - 60 * q^37 - 28 * q^41 - 28 * q^45 - 30 * q^49 + 132 * q^53 + 96 * q^57 + 164 * q^61 - 56 * q^65 - 320 * q^69 + 132 * q^73 - 64 * q^77 - 190 * q^81 + 72 * q^85 - 60 * q^89 + 256 * q^93 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$1$	$-1$

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
$2$	0	0
$3$	4.00000i	1.33333i	0.745356	+	0.666667i	$0.232280\pi$
$3$	4.00000i	1.33333i	−0.745356	+	0.666667i	$0.767720\pi$
$4$	0	0
$5$	2.00000	0.400000	0.200000	−	0.979796i	$-0.435906\pi$
$5$	2.00000	0.400000	0.200000	+	0.979796i	$0.435906\pi$
$6$	0	0
$7$	− 8.00000i	− 1.14286i	−0.820652	−	0.571429i	$-0.806389\pi$
$7$	− 8.00000i	− 1.14286i	0.820652	−	0.571429i	$-0.193611\pi$
$8$	0	0
$9$	−7.00000	−0.777778
$10$	0	0
$11$	− 4.00000i	− 0.363636i	−0.983332	−	0.181818i	$-0.941802\pi$
$11$	− 4.00000i	− 0.363636i	0.983332	−	0.181818i	$-0.0581982\pi$
$12$	0	0
$13$	−14.0000	−1.07692	−0.538462	−	0.842650i	$-0.680994\pi$
$13$	−14.0000	−1.07692	−0.538462	+	0.842650i	$0.680994\pi$
$14$	0	0
$15$	8.00000i	0.533333i
$16$	0	0
$17$	18.0000	1.05882	0.529412	−	0.848365i	$-0.322413\pi$
$17$	18.0000	1.05882	0.529412	+	0.848365i	$0.322413\pi$
$18$	0	0
$19$	− 12.0000i	− 0.631579i	−0.948829	−	0.315789i	$-0.897731\pi$
$19$	− 12.0000i	− 0.631579i	0.948829	−	0.315789i	$-0.102269\pi$
$20$	0	0
$21$	32.0000	1.52381
$22$	0	0
$23$	40.0000i	1.73913i	0.493818	+	0.869565i	$0.335601\pi$
$23$	40.0000i	1.73913i	−0.493818	+	0.869565i	$0.664399\pi$
$24$	0	0
$25$	−21.0000	−0.840000
$26$	0	0
$27$	8.00000i	0.296296i
$28$	0	0
$29$	−14.0000	−0.482759	−0.241379	−	0.970431i	$-0.577600\pi$
$29$	−14.0000	−0.482759	−0.241379	+	0.970431i	$0.577600\pi$
$30$	0	0
$31$	− 32.0000i	− 1.03226i	−0.856511	−	0.516129i	$-0.827372\pi$
$31$	− 32.0000i	− 1.03226i	0.856511	−	0.516129i	$-0.172628\pi$
$32$	0	0
$33$	16.0000	0.484848
$34$	0	0
$35$	− 16.0000i	− 0.457143i
$36$	0	0
$37$	−30.0000	−0.810811	−0.405405	−	0.914137i	$-0.632870\pi$
$37$	−30.0000	−0.810811	−0.405405	+	0.914137i	$0.632870\pi$
$38$	0	0
$39$	− 56.0000i	− 1.43590i
$40$	0	0
$41$	−14.0000	−0.341463	−0.170732	−	0.985318i	$-0.554613\pi$
$41$	−14.0000	−0.341463	−0.170732	+	0.985318i	$0.554613\pi$
$42$	0	0
$43$	28.0000i	0.651163i	0.945514	+	0.325581i	$0.105560\pi$
$43$	28.0000i	0.651163i	−0.945514	+	0.325581i	$0.894440\pi$
$44$	0	0
$45$	−14.0000	−0.311111
$46$	0	0
$47$	16.0000i	0.340426i	0.985407	+	0.170213i	$0.0544455\pi$
$47$	16.0000i	0.340426i	−0.985407	+	0.170213i	$0.945555\pi$
$48$	0	0
$49$	−15.0000	−0.306122
$50$	0	0
$51$	72.0000i	1.41176i
$52$	0	0
$53$	66.0000	1.24528	0.622642	−	0.782507i	$-0.286060\pi$
$53$	66.0000	1.24528	0.622642	+	0.782507i	$0.286060\pi$
$54$	0	0
$55$	− 8.00000i	− 0.145455i
$56$	0	0
$57$	48.0000	0.842105
$58$	0	0
$59$	− 52.0000i	− 0.881356i	−0.897665	−	0.440678i	$-0.854738\pi$
$59$	− 52.0000i	− 0.881356i	0.897665	−	0.440678i	$-0.145262\pi$
$60$	0	0
$61$	82.0000	1.34426	0.672131	−	0.740432i	$-0.265379\pi$
$61$	82.0000	1.34426	0.672131	+	0.740432i	$0.265379\pi$
$62$	0	0
$63$	56.0000i	0.888889i
$64$	0	0
$65$	−28.0000	−0.430769
$66$	0	0
$67$	4.00000i	0.0597015i	0.999554	+	0.0298507i	$0.00950320\pi$
$67$	4.00000i	0.0597015i	−0.999554	+	0.0298507i	$0.990497\pi$
$68$	0	0
$69$	−160.000	−2.31884
$70$	0	0
$71$	56.0000i	0.788732i	0.918953	+	0.394366i	$0.129036\pi$
$71$	56.0000i	0.788732i	−0.918953	+	0.394366i	$0.870964\pi$
$72$	0	0
$73$	66.0000	0.904110	0.452055	−	0.891990i	$-0.350691\pi$
$73$	66.0000	0.904110	0.452055	+	0.891990i	$0.350691\pi$
$74$	0	0
$75$	− 84.0000i	− 1.12000i
$76$	0	0
$77$	−32.0000	−0.415584
$78$	0	0
$79$	− 16.0000i	− 0.202532i	−0.994859	−	0.101266i	$-0.967711\pi$
$79$	− 16.0000i	− 0.202532i	0.994859	−	0.101266i	$-0.0322893\pi$
$80$	0	0
$81$	−95.0000	−1.17284
$82$	0	0
$83$	− 140.000i	− 1.68675i	−0.537328	−	0.843373i	$-0.680566\pi$
$83$	− 140.000i	− 1.68675i	0.537328	−	0.843373i	$-0.319434\pi$
$84$	0	0
$85$	36.0000	0.423529
$86$	0	0
$87$	− 56.0000i	− 0.643678i
$88$	0	0
$89$	−30.0000	−0.337079	−0.168539	−	0.985695i	$-0.553905\pi$
$89$	−30.0000	−0.337079	−0.168539	+	0.985695i	$0.553905\pi$
$90$	0	0
$91$	112.000i	1.23077i
$92$	0	0
$93$	128.000	1.37634
$94$	0	0
$95$	− 24.0000i	− 0.252632i
$96$	0	0
$97$	−14.0000	−0.144330	−0.0721649	−	0.997393i	$-0.522991\pi$
$97$	−14.0000	−0.144330	−0.0721649	+	0.997393i	$0.522991\pi$
$98$	0	0
$99$	28.0000i	0.282828i
$100$	0	0
$101$	−94.0000	−0.930693	−0.465347	−	0.885129i	$-0.654070\pi$
$101$	−94.0000	−0.930693	−0.465347	+	0.885129i	$0.654070\pi$
$102$	0	0
$103$	152.000i	1.47573i	0.674949	+	0.737864i	$0.264165\pi$
$103$	152.000i	1.47573i	−0.674949	+	0.737864i	$0.735835\pi$
$104$	0	0
$105$	64.0000	0.609524
$106$	0	0
$107$	156.000i	1.45794i	0.684544	+	0.728972i	$0.260002\pi$
$107$	156.000i	1.45794i	−0.684544	+	0.728972i	$0.739998\pi$
$108$	0	0
$109$	18.0000	0.165138	0.0825688	−	0.996585i	$-0.473688\pi$
$109$	18.0000	0.165138	0.0825688	+	0.996585i	$0.473688\pi$
$110$	0	0
$111$	− 120.000i	− 1.08108i
$112$	0	0
$113$	98.0000	0.867257	0.433628	−	0.901092i	$-0.357233\pi$
$113$	98.0000	0.867257	0.433628	+	0.901092i	$0.357233\pi$
$114$	0	0
$115$	80.0000i	0.695652i
$116$	0	0
$117$	98.0000	0.837607
$118$	0	0
$119$	− 144.000i	− 1.21008i
$120$	0	0
$121$	105.000	0.867769
$122$	0	0
$123$	− 56.0000i	− 0.455285i
$124$	0	0
$125$	−92.0000	−0.736000
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	−112.000	−0.868217
$130$	0	0
$131$	68.0000i	0.519084i	0.965732	+	0.259542i	$0.0835716\pi$
$131$	68.0000i	0.519084i	−0.965732	+	0.259542i	$0.916428\pi$
$132$	0	0
$133$	−96.0000	−0.721805
$134$	0	0
$135$	16.0000i	0.118519i
$136$	0	0
$137$	−206.000	−1.50365	−0.751825	−	0.659363i	$-0.770826\pi$
$137$	−206.000	−1.50365	−0.751825	+	0.659363i	$0.770826\pi$
$138$	0	0
$139$	− 196.000i	− 1.41007i	−0.709172	−	0.705036i	$-0.750931\pi$
$139$	− 196.000i	− 1.41007i	0.709172	−	0.705036i	$-0.249069\pi$
$140$	0	0
$141$	−64.0000	−0.453901
$142$	0	0
$143$	56.0000i	0.391608i
$144$	0	0
$145$	−28.0000	−0.193103
$146$	0	0
$147$	− 60.0000i	− 0.408163i
$148$	0	0
$149$	226.000	1.51678	0.758389	−	0.651802i	$-0.225987\pi$
$149$	226.000	1.51678	0.758389	+	0.651802i	$0.225987\pi$
$150$	0	0
$151$	− 88.0000i	− 0.582781i	−0.956604	−	0.291391i	$-0.905882\pi$
$151$	− 88.0000i	− 0.582781i	0.956604	−	0.291391i	$-0.0941180\pi$
$152$	0	0
$153$	−126.000	−0.823529
$154$	0	0
$155$	− 64.0000i	− 0.412903i
$156$	0	0
$157$	−46.0000	−0.292994	−0.146497	−	0.989211i	$-0.546800\pi$
$157$	−46.0000	−0.292994	−0.146497	+	0.989211i	$0.546800\pi$
$158$	0	0
$159$	264.000i	1.66038i
$160$	0	0
$161$	320.000	1.98758
$162$	0	0
$163$	− 156.000i	− 0.957055i	−0.878073	−	0.478528i	$-0.841171\pi$
$163$	− 156.000i	− 0.957055i	0.878073	−	0.478528i	$-0.158829\pi$
$164$	0	0
$165$	32.0000	0.193939
$166$	0	0
$167$	− 40.0000i	− 0.239521i	−0.992803	−	0.119760i	$-0.961787\pi$
$167$	− 40.0000i	− 0.239521i	0.992803	−	0.119760i	$-0.0382127\pi$
$168$	0	0
$169$	27.0000	0.159763
$170$	0	0
$171$	84.0000i	0.491228i
$172$	0	0
$173$	−142.000	−0.820809	−0.410405	−	0.911904i	$-0.634613\pi$
$173$	−142.000	−0.820809	−0.410405	+	0.911904i	$0.634613\pi$
$174$	0	0
$175$	168.000i	0.960000i
$176$	0	0
$177$	208.000	1.17514
$178$	0	0
$179$	276.000i	1.54190i	0.636896	+	0.770950i	$0.280218\pi$
$179$	276.000i	1.54190i	−0.636896	+	0.770950i	$0.719782\pi$
$180$	0	0
$181$	−30.0000	−0.165746	−0.0828729	−	0.996560i	$-0.526410\pi$
$181$	−30.0000	−0.165746	−0.0828729	+	0.996560i	$0.526410\pi$
$182$	0	0
$183$	328.000i	1.79235i
$184$	0	0
$185$	−60.0000	−0.324324
$186$	0	0
$187$	− 72.0000i	− 0.385027i
$188$	0	0
$189$	64.0000	0.338624
$190$	0	0
$191$	− 320.000i	− 1.67539i	−0.546136	−	0.837696i	$-0.683902\pi$
$191$	− 320.000i	− 1.67539i	0.546136	−	0.837696i	$-0.316098\pi$
$192$	0	0
$193$	−206.000	−1.06736	−0.533679	−	0.845687i	$-0.679191\pi$
$193$	−206.000	−1.06736	−0.533679	+	0.845687i	$0.679191\pi$
$194$	0	0
$195$	− 112.000i	− 0.574359i
$196$	0	0
$197$	322.000	1.63452	0.817259	−	0.576271i	$-0.195493\pi$
$197$	322.000	1.63452	0.817259	+	0.576271i	$0.195493\pi$
$198$	0	0
$199$	− 200.000i	− 1.00503i	−0.864570	−	0.502513i	$-0.832409\pi$
$199$	− 200.000i	− 1.00503i	0.864570	−	0.502513i	$-0.167591\pi$
$200$	0	0
$201$	−16.0000	−0.0796020
$202$	0	0
$203$	112.000i	0.551724i
$204$	0	0
$205$	−28.0000	−0.136585
$206$	0	0
$207$	− 280.000i	− 1.35266i
$208$	0	0
$209$	−48.0000	−0.229665
$210$	0	0
$211$	− 140.000i	− 0.663507i	−0.943366	−	0.331754i	$-0.892360\pi$
$211$	− 140.000i	− 0.663507i	0.943366	−	0.331754i	$-0.107640\pi$
$212$	0	0
$213$	−224.000	−1.05164
$214$	0	0
$215$	56.0000i	0.260465i
$216$	0	0
$217$	−256.000	−1.17972
$218$	0	0
$219$	264.000i	1.20548i
$220$	0	0
$221$	−252.000	−1.14027
$222$	0	0
$223$	− 224.000i	− 1.00448i	−0.864727	−	0.502242i	$-0.832509\pi$
$223$	− 224.000i	− 1.00448i	0.864727	−	0.502242i	$-0.167491\pi$
$224$	0	0
$225$	147.000	0.653333
$226$	0	0
$227$	164.000i	0.722467i	0.932475	+	0.361233i	$0.117644\pi$
$227$	164.000i	0.722467i	−0.932475	+	0.361233i	$0.882356\pi$
$228$	0	0
$229$	2.00000	0.00873362	0.00436681	−	0.999990i	$-0.498610\pi$
$229$	2.00000	0.00873362	0.00436681	+	0.999990i	$0.498610\pi$
$230$	0	0
$231$	− 128.000i	− 0.554113i
$232$	0	0
$233$	2.00000	0.00858369	0.00429185	−	0.999991i	$-0.498634\pi$
$233$	2.00000	0.00858369	0.00429185	+	0.999991i	$0.498634\pi$
$234$	0	0
$235$	32.0000i	0.136170i
$236$	0	0
$237$	64.0000	0.270042
$238$	0	0
$239$	208.000i	0.870293i	0.900360	+	0.435146i	$0.143303\pi$
$239$	208.000i	0.870293i	−0.900360	+	0.435146i	$0.856697\pi$
$240$	0	0
$241$	−46.0000	−0.190871	−0.0954357	−	0.995436i	$-0.530424\pi$
$241$	−46.0000	−0.190871	−0.0954357	+	0.995436i	$0.530424\pi$
$242$	0	0
$243$	− 308.000i	− 1.26749i
$244$	0	0
$245$	−30.0000	−0.122449
$246$	0	0
$247$	168.000i	0.680162i
$248$	0	0
$249$	560.000	2.24900
$250$	0	0
$251$	12.0000i	0.0478088i	0.999714	+	0.0239044i	$0.00760973\pi$
$251$	12.0000i	0.0478088i	−0.999714	+	0.0239044i	$0.992390\pi$
$252$	0	0
$253$	160.000	0.632411
$254$	0	0
$255$	144.000i	0.564706i
$256$	0	0
$257$	2.00000	0.00778210	0.00389105	−	0.999992i	$-0.498761\pi$
$257$	2.00000	0.00778210	0.00389105	+	0.999992i	$0.498761\pi$
$258$	0	0
$259$	240.000i	0.926641i
$260$	0	0
$261$	98.0000	0.375479
$262$	0	0
$263$	− 136.000i	− 0.517110i	−0.965997	−	0.258555i	$-0.916754\pi$
$263$	− 136.000i	− 0.517110i	0.965997	−	0.258555i	$-0.0832464\pi$
$264$	0	0
$265$	132.000	0.498113
$266$	0	0
$267$	− 120.000i	− 0.449438i
$268$	0	0
$269$	402.000	1.49442	0.747212	−	0.664586i	$-0.231392\pi$
$269$	402.000	1.49442	0.747212	+	0.664586i	$0.231392\pi$
$270$	0	0
$271$	432.000i	1.59410i	0.603916	+	0.797048i	$0.293606\pi$
$271$	432.000i	1.59410i	−0.603916	+	0.797048i	$0.706394\pi$
$272$	0	0
$273$	−448.000	−1.64103
$274$	0	0
$275$	84.0000i	0.305455i
$276$	0	0
$277$	−382.000	−1.37906	−0.689531	−	0.724256i	$-0.742183\pi$
$277$	−382.000	−1.37906	−0.689531	+	0.724256i	$0.742183\pi$
$278$	0	0
$279$	224.000i	0.802867i
$280$	0	0
$281$	−350.000	−1.24555	−0.622776	−	0.782400i	$-0.713995\pi$
$281$	−350.000	−1.24555	−0.622776	+	0.782400i	$0.713995\pi$
$282$	0	0
$283$	− 340.000i	− 1.20141i	−0.799469	−	0.600707i	$-0.794886\pi$
$283$	− 340.000i	− 1.20141i	0.799469	−	0.600707i	$-0.205114\pi$
$284$	0	0
$285$	96.0000	0.336842
$286$	0	0
$287$	112.000i	0.390244i
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	− 56.0000i	− 0.192440i
$292$	0	0
$293$	2.00000	0.00682594	0.00341297	−	0.999994i	$-0.498914\pi$
$293$	2.00000	0.00682594	0.00341297	+	0.999994i	$0.498914\pi$
$294$	0	0
$295$	− 104.000i	− 0.352542i
$296$	0	0
$297$	32.0000	0.107744
$298$	0	0
$299$	− 560.000i	− 1.87291i
$300$	0	0
$301$	224.000	0.744186
$302$	0	0
$303$	− 376.000i	− 1.24092i
$304$	0	0
$305$	164.000	0.537705
$306$	0	0
$307$	84.0000i	0.273616i	0.990598	+	0.136808i	$0.0436843\pi$
$307$	84.0000i	0.273616i	−0.990598	+	0.136808i	$0.956316\pi$
$308$	0	0
$309$	−608.000	−1.96764
$310$	0	0
$311$	− 248.000i	− 0.797428i	−0.917075	−	0.398714i	$-0.869457\pi$
$311$	− 248.000i	− 0.797428i	0.917075	−	0.398714i	$-0.130543\pi$
$312$	0	0
$313$	530.000	1.69329	0.846645	−	0.532158i	$-0.178619\pi$
$313$	530.000	1.69329	0.846645	+	0.532158i	$0.178619\pi$
$314$	0	0
$315$	112.000i	0.355556i
$316$	0	0
$317$	−590.000	−1.86120	−0.930599	−	0.366039i	$-0.880714\pi$
$317$	−590.000	−1.86120	−0.930599	+	0.366039i	$0.880714\pi$
$318$	0	0
$319$	56.0000i	0.175549i
$320$	0	0
$321$	−624.000	−1.94393
$322$	0	0
$323$	− 216.000i	− 0.668731i
$324$	0	0
$325$	294.000	0.904615
$326$	0	0
$327$	72.0000i	0.220183i
$328$	0	0
$329$	128.000	0.389058
$330$	0	0
$331$	572.000i	1.72810i	0.503409	+	0.864048i	$0.332079\pi$
$331$	572.000i	1.72810i	−0.503409	+	0.864048i	$0.667921\pi$
$332$	0	0
$333$	210.000	0.630631
$334$	0	0
$335$	8.00000i	0.0238806i
$336$	0	0
$337$	98.0000	0.290801	0.145401	−	0.989373i	$-0.453553\pi$
$337$	98.0000	0.290801	0.145401	+	0.989373i	$0.453553\pi$
$338$	0	0
$339$	392.000i	1.15634i
$340$	0	0
$341$	−128.000	−0.375367
$342$	0	0
$343$	− 272.000i	− 0.793003i
$344$	0	0
$345$	−320.000	−0.927536
$346$	0	0
$347$	172.000i	0.495677i	0.968801	+	0.247839i	$0.0797203\pi$
$347$	172.000i	0.495677i	−0.968801	+	0.247839i	$0.920280\pi$
$348$	0	0
$349$	434.000	1.24355	0.621777	−	0.783195i	$-0.286411\pi$
$349$	434.000	1.24355	0.621777	+	0.783195i	$0.286411\pi$
$350$	0	0
$351$	− 112.000i	− 0.319088i
$352$	0	0
$353$	130.000	0.368272	0.184136	−	0.982901i	$-0.441051\pi$
$353$	130.000	0.368272	0.184136	+	0.982901i	$0.441051\pi$
$354$	0	0
$355$	112.000i	0.315493i
$356$	0	0
$357$	576.000	1.61345
$358$	0	0
$359$	408.000i	1.13649i	0.822859	+	0.568245i	$0.192377\pi$
$359$	408.000i	1.13649i	−0.822859	+	0.568245i	$0.807623\pi$
$360$	0	0
$361$	217.000	0.601108
$362$	0	0
$363$	420.000i	1.15702i
$364$	0	0
$365$	132.000	0.361644
$366$	0	0
$367$	− 304.000i	− 0.828338i	−0.910200	−	0.414169i	$-0.864072\pi$
$367$	− 304.000i	− 0.828338i	0.910200	−	0.414169i	$-0.135928\pi$
$368$	0	0
$369$	98.0000	0.265583
$370$	0	0
$371$	− 528.000i	− 1.42318i
$372$	0	0
$373$	−254.000	−0.680965	−0.340483	−	0.940251i	$-0.610590\pi$
$373$	−254.000	−0.680965	−0.340483	+	0.940251i	$0.610590\pi$
$374$	0	0
$375$	− 368.000i	− 0.981333i
$376$	0	0
$377$	196.000	0.519894
$378$	0	0
$379$	268.000i	0.707124i	0.935411	+	0.353562i	$0.115030\pi$
$379$	268.000i	0.707124i	−0.935411	+	0.353562i	$0.884970\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	128.000i	0.334204i	0.985940	+	0.167102i	$0.0534409\pi$
$383$	128.000i	0.334204i	−0.985940	+	0.167102i	$0.946559\pi$
$384$	0	0
$385$	−64.0000	−0.166234
$386$	0	0
$387$	− 196.000i	− 0.506460i
$388$	0	0
$389$	−94.0000	−0.241645	−0.120823	−	0.992674i	$-0.538553\pi$
$389$	−94.0000	−0.241645	−0.120823	+	0.992674i	$0.538553\pi$
$390$	0	0
$391$	720.000i	1.84143i
$392$	0	0
$393$	−272.000	−0.692112
$394$	0	0
$395$	− 32.0000i	− 0.0810127i
$396$	0	0
$397$	−558.000	−1.40554	−0.702771	−	0.711416i	$-0.748054\pi$
$397$	−558.000	−1.40554	−0.702771	+	0.711416i	$0.748054\pi$
$398$	0	0
$399$	− 384.000i	− 0.962406i
$400$	0	0
$401$	−110.000	−0.274314	−0.137157	−	0.990549i	$-0.543797\pi$
$401$	−110.000	−0.274314	−0.137157	+	0.990549i	$0.543797\pi$
$402$	0	0
$403$	448.000i	1.11166i
$404$	0	0
$405$	−190.000	−0.469136
$406$	0	0
$407$	120.000i	0.294840i
$408$	0	0
$409$	−430.000	−1.05134	−0.525672	−	0.850687i	$-0.676186\pi$
$409$	−430.000	−1.05134	−0.525672	+	0.850687i	$0.676186\pi$
$410$	0	0
$411$	− 824.000i	− 2.00487i
$412$	0	0
$413$	−416.000	−1.00726
$414$	0	0
$415$	− 280.000i	− 0.674699i
$416$	0	0
$417$	784.000	1.88010
$418$	0	0
$419$	− 476.000i	− 1.13604i	−0.823015	−	0.568019i	$-0.807710\pi$
$419$	− 476.000i	− 1.13604i	0.823015	−	0.568019i	$-0.192290\pi$
$420$	0	0
$421$	322.000	0.764846	0.382423	−	0.923987i	$-0.375090\pi$
$421$	322.000	0.764846	0.382423	+	0.923987i	$0.375090\pi$
$422$	0	0
$423$	− 112.000i	− 0.264775i
$424$	0	0
$425$	−378.000	−0.889412
$426$	0	0
$427$	− 656.000i	− 1.53630i
$428$	0	0
$429$	−224.000	−0.522145
$430$	0	0
$431$	528.000i	1.22506i	0.790448	+	0.612529i	$0.209848\pi$
$431$	528.000i	1.22506i	−0.790448	+	0.612529i	$0.790152\pi$
$432$	0	0
$433$	210.000	0.484988	0.242494	−	0.970153i	$-0.422034\pi$
$433$	210.000	0.484988	0.242494	+	0.970153i	$0.422034\pi$
$434$	0	0
$435$	− 112.000i	− 0.257471i
$436$	0	0
$437$	480.000	1.09840
$438$	0	0
$439$	− 376.000i	− 0.856492i	−0.903662	−	0.428246i	$-0.859132\pi$
$439$	− 376.000i	− 0.856492i	0.903662	−	0.428246i	$-0.140868\pi$
$440$	0	0
$441$	105.000	0.238095
$442$	0	0
$443$	268.000i	0.604966i	0.953155	+	0.302483i	$0.0978156\pi$
$443$	268.000i	0.604966i	−0.953155	+	0.302483i	$0.902184\pi$
$444$	0	0
$445$	−60.0000	−0.134831
$446$	0	0
$447$	904.000i	2.02237i
$448$	0	0
$449$	−462.000	−1.02895	−0.514477	−	0.857504i	$-0.672014\pi$
$449$	−462.000	−1.02895	−0.514477	+	0.857504i	$0.672014\pi$
$450$	0	0
$451$	56.0000i	0.124169i
$452$	0	0
$453$	352.000	0.777042
$454$	0	0
$455$	224.000i	0.492308i
$456$	0	0
$457$	−590.000	−1.29103	−0.645514	−	0.763748i	$-0.723357\pi$
$457$	−590.000	−1.29103	−0.645514	+	0.763748i	$0.723357\pi$
$458$	0	0
$459$	144.000i	0.313725i
$460$	0	0
$461$	658.000	1.42733	0.713666	−	0.700486i	$-0.247033\pi$
$461$	658.000	1.42733	0.713666	+	0.700486i	$0.247033\pi$
$462$	0	0
$463$	112.000i	0.241901i	0.992659	+	0.120950i	$0.0385942\pi$
$463$	112.000i	0.241901i	−0.992659	+	0.120950i	$0.961406\pi$
$464$	0	0
$465$	256.000	0.550538
$466$	0	0
$467$	180.000i	0.385439i	0.981254	+	0.192719i	$0.0617307\pi$
$467$	180.000i	0.385439i	−0.981254	+	0.192719i	$0.938269\pi$
$468$	0	0
$469$	32.0000	0.0682303
$470$	0	0
$471$	− 184.000i	− 0.390658i
$472$	0	0
$473$	112.000	0.236786
$474$	0	0
$475$	252.000i	0.530526i
$476$	0	0
$477$	−462.000	−0.968553
$478$	0	0
$479$	32.0000i	0.0668058i	0.999442	+	0.0334029i	$0.0106345\pi$
$479$	32.0000i	0.0668058i	−0.999442	+	0.0334029i	$0.989366\pi$
$480$	0	0
$481$	420.000	0.873181
$482$	0	0
$483$	1280.00i	2.65010i
$484$	0	0
$485$	−28.0000	−0.0577320
$486$	0	0
$487$	− 360.000i	− 0.739220i	−0.929187	−	0.369610i	$-0.879491\pi$
$487$	− 360.000i	− 0.739220i	0.929187	−	0.369610i	$-0.120509\pi$
$488$	0	0
$489$	624.000	1.27607
$490$	0	0
$491$	− 292.000i	− 0.594705i	−0.954768	−	0.297352i	$-0.903896\pi$
$491$	− 292.000i	− 0.594705i	0.954768	−	0.297352i	$-0.0961036\pi$
$492$	0	0
$493$	−252.000	−0.511156
$494$	0	0
$495$	56.0000i	0.113131i
$496$	0	0
$497$	448.000	0.901408
$498$	0	0
$499$	− 940.000i	− 1.88377i	−0.335939	−	0.941884i	$-0.609054\pi$
$499$	− 940.000i	− 1.88377i	0.335939	−	0.941884i	$-0.390946\pi$
$500$	0	0
$501$	160.000	0.319361
$502$	0	0
$503$	− 56.0000i	− 0.111332i	−0.998449	−	0.0556660i	$-0.982272\pi$
$503$	− 56.0000i	− 0.111332i	0.998449	−	0.0556660i	$-0.0177282\pi$
$504$	0	0
$505$	−188.000	−0.372277
$506$	0	0
$507$	108.000i	0.213018i
$508$	0	0
$509$	−110.000	−0.216110	−0.108055	−	0.994145i	$-0.534462\pi$
$509$	−110.000	−0.216110	−0.108055	+	0.994145i	$0.534462\pi$
$510$	0	0
$511$	− 528.000i	− 1.03327i
$512$	0	0
$513$	96.0000	0.187135
$514$	0	0
$515$	304.000i	0.590291i
$516$	0	0
$517$	64.0000	0.123791
$518$	0	0
$519$	− 568.000i	− 1.09441i
$520$	0	0
$521$	178.000	0.341651	0.170825	−	0.985301i	$-0.445357\pi$
$521$	178.000	0.341651	0.170825	+	0.985301i	$0.445357\pi$
$522$	0	0
$523$	380.000i	0.726577i	0.931677	+	0.363289i	$0.118346\pi$
$523$	380.000i	0.726577i	−0.931677	+	0.363289i	$0.881654\pi$
$524$	0	0
$525$	−672.000	−1.28000
$526$	0	0
$527$	− 576.000i	− 1.09298i
$528$	0	0
$529$	−1071.00	−2.02457
$530$	0	0
$531$	364.000i	0.685499i
$532$	0	0
$533$	196.000	0.367730
$534$	0	0
$535$	312.000i	0.583178i
$536$	0	0
$537$	−1104.00	−2.05587
$538$	0	0
$539$	60.0000i	0.111317i
$540$	0	0
$541$	−110.000	−0.203327	−0.101664	−	0.994819i	$-0.532417\pi$
$541$	−110.000	−0.203327	−0.101664	+	0.994819i	$0.532417\pi$
$542$	0	0
$543$	− 120.000i	− 0.220994i
$544$	0	0
$545$	36.0000	0.0660550
$546$	0	0
$547$	− 604.000i	− 1.10420i	−0.833776	−	0.552102i	$-0.813826\pi$
$547$	− 604.000i	− 1.10420i	0.833776	−	0.552102i	$-0.186174\pi$
$548$	0	0
$549$	−574.000	−1.04554
$550$	0	0
$551$	168.000i	0.304900i
$552$	0	0
$553$	−128.000	−0.231465
$554$	0	0
$555$	− 240.000i	− 0.432432i
$556$	0	0
$557$	690.000	1.23878	0.619390	−	0.785084i	$-0.287380\pi$
$557$	690.000	1.23878	0.619390	+	0.785084i	$0.287380\pi$
$558$	0	0
$559$	− 392.000i	− 0.701252i
$560$	0	0
$561$	288.000	0.513369
$562$	0	0
$563$	340.000i	0.603908i	0.953323	+	0.301954i	$0.0976388\pi$
$563$	340.000i	0.603908i	−0.953323	+	0.301954i	$0.902361\pi$
$564$	0	0
$565$	196.000	0.346903
$566$	0	0
$567$	760.000i	1.34039i
$568$	0	0
$569$	530.000	0.931459	0.465729	−	0.884927i	$-0.345792\pi$
$569$	530.000	0.931459	0.465729	+	0.884927i	$0.345792\pi$
$570$	0	0
$571$	− 116.000i	− 0.203152i	−0.994828	−	0.101576i	$-0.967611\pi$
$571$	− 116.000i	− 0.203152i	0.994828	−	0.101576i	$-0.0323886\pi$
$572$	0	0
$573$	1280.00	2.23386
$574$	0	0
$575$	− 840.000i	− 1.46087i
$576$	0	0
$577$	66.0000	0.114385	0.0571924	−	0.998363i	$-0.481785\pi$
$577$	66.0000	0.114385	0.0571924	+	0.998363i	$0.481785\pi$
$578$	0	0
$579$	− 824.000i	− 1.42314i
$580$	0	0
$581$	−1120.00	−1.92771
$582$	0	0
$583$	− 264.000i	− 0.452830i
$584$	0	0
$585$	196.000	0.335043
$586$	0	0
$587$	700.000i	1.19250i	0.802797	+	0.596252i	$0.203344\pi$
$587$	700.000i	1.19250i	−0.802797	+	0.596252i	$0.796656\pi$
$588$	0	0
$589$	−384.000	−0.651952
$590$	0	0
$591$	1288.00i	2.17936i
$592$	0	0
$593$	−222.000	−0.374368	−0.187184	−	0.982325i	$-0.559936\pi$
$593$	−222.000	−0.374368	−0.187184	+	0.982325i	$0.559936\pi$
$594$	0	0
$595$	− 288.000i	− 0.484034i
$596$	0	0
$597$	800.000	1.34003
$598$	0	0
$599$	872.000i	1.45576i	0.685705	+	0.727880i	$0.259494\pi$
$599$	872.000i	1.45576i	−0.685705	+	0.727880i	$0.740506\pi$
$600$	0	0
$601$	994.000	1.65391	0.826955	−	0.562268i	$-0.190071\pi$
$601$	994.000	1.65391	0.826955	+	0.562268i	$0.190071\pi$
$602$	0	0
$603$	− 28.0000i	− 0.0464345i
$604$	0	0
$605$	210.000	0.347107
$606$	0	0
$607$	800.000i	1.31796i	0.752162	+	0.658979i	$0.229011\pi$
$607$	800.000i	1.31796i	−0.752162	+	0.658979i	$0.770989\pi$
$608$	0	0
$609$	−448.000	−0.735632
$610$	0	0
$611$	− 224.000i	− 0.366612i
$612$	0	0
$613$	−318.000	−0.518760	−0.259380	−	0.965775i	$-0.583518\pi$
$613$	−318.000	−0.518760	−0.259380	+	0.965775i	$0.583518\pi$
$614$	0	0
$615$	− 112.000i	− 0.182114i
$616$	0	0
$617$	322.000	0.521880	0.260940	−	0.965355i	$-0.415968\pi$
$617$	322.000	0.521880	0.260940	+	0.965355i	$0.415968\pi$
$618$	0	0
$619$	284.000i	0.458805i	0.973332	+	0.229402i	$0.0736771\pi$
$619$	284.000i	0.458805i	−0.973332	+	0.229402i	$0.926323\pi$
$620$	0	0
$621$	−320.000	−0.515298
$622$	0	0
$623$	240.000i	0.385233i
$624$	0	0
$625$	341.000	0.545600
$626$	0	0
$627$	− 192.000i	− 0.306220i
$628$	0	0
$629$	−540.000	−0.858506
$630$	0	0
$631$	840.000i	1.33122i	0.746300	+	0.665610i	$0.231829\pi$
$631$	840.000i	1.33122i	−0.746300	+	0.665610i	$0.768171\pi$
$632$	0	0
$633$	560.000	0.884676
$634$	0	0
$635$	0	0
$636$	0	0
$637$	210.000	0.329670
$638$	0	0
$639$	− 392.000i	− 0.613459i
$640$	0	0
$641$	−718.000	−1.12012	−0.560062	−	0.828450i	$-0.689223\pi$
$641$	−718.000	−1.12012	−0.560062	+	0.828450i	$0.689223\pi$
$642$	0	0
$643$	196.000i	0.304821i	0.988317	+	0.152411i	$0.0487036\pi$
$643$	196.000i	0.304821i	−0.988317	+	0.152411i	$0.951296\pi$
$644$	0	0
$645$	−224.000	−0.347287
$646$	0	0
$647$	− 1032.00i	− 1.59505i	−0.603283	−	0.797527i	$-0.706141\pi$
$647$	− 1032.00i	− 1.59505i	0.603283	−	0.797527i	$-0.293859\pi$
$648$	0	0
$649$	−208.000	−0.320493
$650$	0	0
$651$	− 1024.00i	− 1.57296i
$652$	0	0
$653$	530.000	0.811639	0.405819	−	0.913953i	$-0.366986\pi$
$653$	530.000	0.811639	0.405819	+	0.913953i	$0.366986\pi$
$654$	0	0
$655$	136.000i	0.207634i
$656$	0	0
$657$	−462.000	−0.703196
$658$	0	0
$659$	308.000i	0.467375i	0.972312	+	0.233687i	$0.0750792\pi$
$659$	308.000i	0.467375i	−0.972312	+	0.233687i	$0.924921\pi$
$660$	0	0
$661$	290.000	0.438729	0.219365	−	0.975643i	$-0.429602\pi$
$661$	290.000	0.438729	0.219365	+	0.975643i	$0.429602\pi$
$662$	0	0
$663$	− 1008.00i	− 1.52036i
$664$	0	0
$665$	−192.000	−0.288722
$666$	0	0
$667$	− 560.000i	− 0.839580i
$668$	0	0
$669$	896.000	1.33931
$670$	0	0
$671$	− 328.000i	− 0.488823i
$672$	0	0
$673$	−14.0000	−0.0208024	−0.0104012	−	0.999946i	$-0.503311\pi$
$673$	−14.0000	−0.0208024	−0.0104012	+	0.999946i	$0.503311\pi$
$674$	0	0
$675$	− 168.000i	− 0.248889i
$676$	0	0
$677$	−670.000	−0.989660	−0.494830	−	0.868990i	$-0.664770\pi$
$677$	−670.000	−0.989660	−0.494830	+	0.868990i	$0.664770\pi$
$678$	0	0
$679$	112.000i	0.164948i
$680$	0	0
$681$	−656.000	−0.963289
$682$	0	0
$683$	− 228.000i	− 0.333821i	−0.985972	−	0.166911i	$-0.946621\pi$
$683$	− 228.000i	− 0.333821i	0.985972	−	0.166911i	$-0.0533792\pi$
$684$	0	0
$685$	−412.000	−0.601460
$686$	0	0
$687$	8.00000i	0.0116448i
$688$	0	0
$689$	−924.000	−1.34107
$690$	0	0
$691$	1300.00i	1.88133i	0.339335	+	0.940666i	$0.389798\pi$
$691$	1300.00i	1.88133i	−0.339335	+	0.940666i	$0.610202\pi$
$692$	0	0
$693$	224.000	0.323232
$694$	0	0
$695$	− 392.000i	− 0.564029i
$696$	0	0
$697$	−252.000	−0.361549
$698$	0	0
$699$	8.00000i	0.0114449i
$700$	0	0
$701$	−558.000	−0.796006	−0.398003	−	0.917384i	$-0.630297\pi$
$701$	−558.000	−0.796006	−0.398003	+	0.917384i	$0.630297\pi$
$702$	0	0
$703$	360.000i	0.512091i
$704$	0	0
$705$	−128.000	−0.181560
$706$	0	0
$707$	752.000i	1.06365i
$708$	0	0
$709$	−478.000	−0.674189	−0.337094	−	0.941471i	$-0.609444\pi$
$709$	−478.000	−0.674189	−0.337094	+	0.941471i	$0.609444\pi$
$710$	0	0
$711$	112.000i	0.157525i
$712$	0	0
$713$	1280.00	1.79523
$714$	0	0
$715$	112.000i	0.156643i
$716$	0	0
$717$	−832.000	−1.16039
$718$	0	0
$719$	− 912.000i	− 1.26843i	−0.773157	−	0.634214i	$-0.781324\pi$
$719$	− 912.000i	− 1.26843i	0.773157	−	0.634214i	$-0.218676\pi$
$720$	0	0
$721$	1216.00	1.68655
$722$	0	0
$723$	− 184.000i	− 0.254495i
$724$	0	0
$725$	294.000	0.405517
$726$	0	0
$727$	− 280.000i	− 0.385144i	−0.981283	−	0.192572i	$-0.938317\pi$
$727$	− 280.000i	− 0.385144i	0.981283	−	0.192572i	$-0.0616830\pi$
$728$	0	0
$729$	377.000	0.517147
$730$	0	0
$731$	504.000i	0.689466i
$732$	0	0
$733$	754.000	1.02865	0.514325	−	0.857596i	$-0.328043\pi$
$733$	754.000	1.02865	0.514325	+	0.857596i	$0.328043\pi$
$734$	0	0
$735$	− 120.000i	− 0.163265i
$736$	0	0
$737$	16.0000	0.0217096
$738$	0	0
$739$	164.000i	0.221922i	0.993825	+	0.110961i	$0.0353928\pi$
$739$	164.000i	0.221922i	−0.993825	+	0.110961i	$0.964607\pi$
$740$	0	0
$741$	−672.000	−0.906883
$742$	0	0
$743$	− 616.000i	− 0.829071i	−0.910033	−	0.414536i	$-0.863944\pi$
$743$	− 616.000i	− 0.829071i	0.910033	−	0.414536i	$-0.136056\pi$
$744$	0	0
$745$	452.000	0.606711
$746$	0	0
$747$	980.000i	1.31191i
$748$	0	0
$749$	1248.00	1.66622
$750$	0	0
$751$	464.000i	0.617843i	0.951088	+	0.308921i	$0.0999680\pi$
$751$	464.000i	0.617843i	−0.951088	+	0.308921i	$0.900032\pi$
$752$	0	0
$753$	−48.0000	−0.0637450
$754$	0	0
$755$	− 176.000i	− 0.233113i
$756$	0	0
$757$	1442.00	1.90489	0.952444	−	0.304714i	$-0.0985609\pi$
$757$	1442.00	1.90489	0.952444	+	0.304714i	$0.0985609\pi$
$758$	0	0
$759$	640.000i	0.843215i
$760$	0	0
$761$	−110.000	−0.144547	−0.0722733	−	0.997385i	$-0.523025\pi$
$761$	−110.000	−0.144547	−0.0722733	+	0.997385i	$0.523025\pi$
$762$	0	0
$763$	− 144.000i	− 0.188729i
$764$	0	0
$765$	−252.000	−0.329412
$766$	0	0
$767$	728.000i	0.949153i
$768$	0	0
$769$	434.000	0.564369	0.282185	−	0.959360i	$-0.408941\pi$
$769$	434.000	0.564369	0.282185	+	0.959360i	$0.408941\pi$
$770$	0	0
$771$	8.00000i	0.0103761i
$772$	0	0
$773$	−990.000	−1.28072	−0.640362	−	0.768073i	$-0.721216\pi$
$773$	−990.000	−1.28072	−0.640362	+	0.768073i	$0.721216\pi$
$774$	0	0
$775$	672.000i	0.867097i
$776$	0	0
$777$	−960.000	−1.23552
$778$	0	0
$779$	168.000i	0.215661i
$780$	0	0
$781$	224.000	0.286812
$782$	0	0
$783$	− 112.000i	− 0.143040i
$784$	0	0
$785$	−92.0000	−0.117197
$786$	0	0
$787$	− 1228.00i	− 1.56036i	−0.625558	−	0.780178i	$-0.715129\pi$
$787$	− 1228.00i	− 1.56036i	0.625558	−	0.780178i	$-0.284871\pi$
$788$	0	0
$789$	544.000	0.689480
$790$	0	0
$791$	− 784.000i	− 0.991150i
$792$	0	0
$793$	−1148.00	−1.44767
$794$	0	0
$795$	528.000i	0.664151i
$796$	0	0
$797$	−686.000	−0.860728	−0.430364	−	0.902655i	$-0.641615\pi$
$797$	−686.000	−0.860728	−0.430364	+	0.902655i	$0.641615\pi$
$798$	0	0
$799$	288.000i	0.360451i
$800$	0	0
$801$	210.000	0.262172
$802$	0	0
$803$	− 264.000i	− 0.328767i
$804$	0	0
$805$	640.000	0.795031
$806$	0	0
$807$	1608.00i	1.99257i
$808$	0	0
$809$	−270.000	−0.333745	−0.166873	−	0.985978i	$-0.553367\pi$
$809$	−270.000	−0.333745	−0.166873	+	0.985978i	$0.553367\pi$
$810$	0	0
$811$	− 420.000i	− 0.517879i	−0.965894	−	0.258940i	$-0.916627\pi$
$811$	− 420.000i	− 0.517879i	0.965894	−	0.258940i	$-0.0833731\pi$
$812$	0	0
$813$	−1728.00	−2.12546
$814$	0	0
$815$	− 312.000i	− 0.382822i
$816$	0	0
$817$	336.000	0.411261
$818$	0	0
$819$	− 784.000i	− 0.957265i
$820$	0	0
$821$	898.000	1.09379	0.546894	−	0.837202i	$-0.315810\pi$
$821$	898.000	1.09379	0.546894	+	0.837202i	$0.315810\pi$
$822$	0	0
$823$	− 248.000i	− 0.301337i	−0.988584	−	0.150668i	$-0.951857\pi$
$823$	− 248.000i	− 0.301337i	0.988584	−	0.150668i	$-0.0481425\pi$
$824$	0	0
$825$	−336.000	−0.407273
$826$	0	0
$827$	− 756.000i	− 0.914148i	−0.889429	−	0.457074i	$-0.848898\pi$
$827$	− 756.000i	− 0.914148i	0.889429	−	0.457074i	$-0.151102\pi$
$828$	0	0
$829$	18.0000	0.0217129	0.0108565	−	0.999941i	$-0.496544\pi$
$829$	18.0000	0.0217129	0.0108565	+	0.999941i	$0.496544\pi$
$830$	0	0
$831$	− 1528.00i	− 1.83875i
$832$	0	0
$833$	−270.000	−0.324130
$834$	0	0
$835$	− 80.0000i	− 0.0958084i
$836$	0	0
$837$	256.000	0.305854
$838$	0	0
$839$	952.000i	1.13468i	0.823482	+	0.567342i	$0.192028\pi$
$839$	952.000i	1.13468i	−0.823482	+	0.567342i	$0.807972\pi$
$840$	0	0
$841$	−645.000	−0.766944
$842$	0	0
$843$	− 1400.00i	− 1.66074i
$844$	0	0
$845$	54.0000	0.0639053
$846$	0	0
$847$	− 840.000i	− 0.991736i
$848$	0	0
$849$	1360.00	1.60188
$850$	0	0
$851$	− 1200.00i	− 1.41011i
$852$	0	0
$853$	−1022.00	−1.19812	−0.599062	−	0.800703i	$-0.704460\pi$
$853$	−1022.00	−1.19812	−0.599062	+	0.800703i	$0.704460\pi$
$854$	0	0
$855$	168.000i	0.196491i
$856$	0	0
$857$	850.000	0.991832	0.495916	−	0.868371i	$-0.334832\pi$
$857$	850.000	0.991832	0.495916	+	0.868371i	$0.334832\pi$
$858$	0	0
$859$	940.000i	1.09430i	0.837036	+	0.547148i	$0.184286\pi$
$859$	940.000i	1.09430i	−0.837036	+	0.547148i	$0.815714\pi$
$860$	0	0
$861$	−448.000	−0.520325
$862$	0	0
$863$	1440.00i	1.66860i	0.551312	+	0.834299i	$0.314127\pi$
$863$	1440.00i	1.66860i	−0.551312	+	0.834299i	$0.685873\pi$
$864$	0	0
$865$	−284.000	−0.328324
$866$	0	0
$867$	140.000i	0.161476i
$868$	0	0
$869$	−64.0000	−0.0736479
$870$	0	0
$871$	− 56.0000i	− 0.0642939i
$872$	0	0
$873$	98.0000	0.112257
$874$	0	0
$875$	736.000i	0.841143i
$876$	0	0
$877$	−1454.00	−1.65792	−0.828962	−	0.559304i	$-0.811068\pi$
$877$	−1454.00	−1.65792	−0.828962	+	0.559304i	$0.811068\pi$
$878$	0	0
$879$	8.00000i	0.00910125i
$880$	0	0
$881$	−1374.00	−1.55959	−0.779796	−	0.626034i	$-0.784677\pi$
$881$	−1374.00	−1.55959	−0.779796	+	0.626034i	$0.784677\pi$
$882$	0	0
$883$	1428.00i	1.61721i	0.588349	+	0.808607i	$0.299778\pi$
$883$	1428.00i	1.61721i	−0.588349	+	0.808607i	$0.700222\pi$
$884$	0	0
$885$	416.000	0.470056
$886$	0	0
$887$	− 824.000i	− 0.928974i	−0.885580	−	0.464487i	$-0.846239\pi$
$887$	− 824.000i	− 0.928974i	0.885580	−	0.464487i	$-0.153761\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	380.000i	0.426487i
$892$	0	0
$893$	192.000	0.215006
$894$	0	0
$895$	552.000i	0.616760i
$896$	0	0
$897$	2240.00	2.49721
$898$	0	0
$899$	448.000i	0.498331i
$900$	0	0
$901$	1188.00	1.31853
$902$	0	0
$903$	896.000i	0.992248i
$904$	0	0
$905$	−60.0000	−0.0662983
$906$	0	0
$907$	− 836.000i	− 0.921720i	−0.887473	−	0.460860i	$-0.847541\pi$
$907$	− 836.000i	− 0.921720i	0.887473	−	0.460860i	$-0.152459\pi$
$908$	0	0
$909$	658.000	0.723872
$910$	0	0
$911$	560.000i	0.614709i	0.951595	+	0.307355i	$0.0994438\pi$
$911$	560.000i	0.614709i	−0.951595	+	0.307355i	$0.900556\pi$
$912$	0	0
$913$	−560.000	−0.613363
$914$	0	0
$915$	656.000i	0.716940i
$916$	0	0
$917$	544.000	0.593239
$918$	0	0
$919$	− 600.000i	− 0.652884i	−0.945217	−	0.326442i	$-0.894150\pi$
$919$	− 600.000i	− 0.652884i	0.945217	−	0.326442i	$-0.105850\pi$
$920$	0	0
$921$	−336.000	−0.364821
$922$	0	0
$923$	− 784.000i	− 0.849404i
$924$	0	0
$925$	630.000	0.681081
$926$	0	0
$927$	− 1064.00i	− 1.14779i
$928$	0	0
$929$	754.000	0.811625	0.405813	−	0.913956i	$-0.366989\pi$
$929$	754.000	0.811625	0.405813	+	0.913956i	$0.366989\pi$
$930$	0	0
$931$	180.000i	0.193340i
$932$	0	0
$933$	992.000	1.06324
$934$	0	0
$935$	− 144.000i	− 0.154011i
$936$	0	0
$937$	−894.000	−0.954109	−0.477054	−	0.878874i	$-0.658296\pi$
$937$	−894.000	−0.954109	−0.477054	+	0.878874i	$0.658296\pi$
$938$	0	0
$939$	2120.00i	2.25772i
$940$	0	0
$941$	1298.00	1.37938	0.689692	−	0.724103i	$-0.257746\pi$
$941$	1298.00	1.37938	0.689692	+	0.724103i	$0.257746\pi$
$942$	0	0
$943$	− 560.000i	− 0.593849i
$944$	0	0
$945$	128.000	0.135450
$946$	0	0
$947$	1748.00i	1.84583i	0.385005	+	0.922914i	$0.374200\pi$
$947$	1748.00i	1.84583i	−0.385005	+	0.922914i	$0.625800\pi$
$948$	0	0
$949$	−924.000	−0.973656
$950$	0	0
$951$	− 2360.00i	− 2.48160i
$952$	0	0
$953$	−1134.00	−1.18993	−0.594963	−	0.803753i	$-0.702833\pi$
$953$	−1134.00	−1.18993	−0.594963	+	0.803753i	$0.702833\pi$
$954$	0	0
$955$	− 640.000i	− 0.670157i
$956$	0	0
$957$	−224.000	−0.234065
$958$	0	0
$959$	1648.00i	1.71846i
$960$	0	0
$961$	−63.0000	−0.0655567
$962$	0	0
$963$	− 1092.00i	− 1.13396i
$964$	0	0
$965$	−412.000	−0.426943
$966$	0	0
$967$	− 1736.00i	− 1.79524i	−0.440767	−	0.897622i	$-0.645294\pi$
$967$	− 1736.00i	− 1.79524i	0.440767	−	0.897622i	$-0.354706\pi$
$968$	0	0
$969$	864.000	0.891641
$970$	0	0
$971$	124.000i	0.127703i	0.997959	+	0.0638517i	$0.0203385\pi$
$971$	124.000i	0.127703i	−0.997959	+	0.0638517i	$0.979662\pi$
$972$	0	0
$973$	−1568.00	−1.61151
$974$	0	0
$975$	1176.00i	1.20615i
$976$	0	0
$977$	914.000	0.935517	0.467758	−	0.883856i	$-0.345062\pi$
$977$	914.000	0.935517	0.467758	+	0.883856i	$0.345062\pi$
$978$	0	0
$979$	120.000i	0.122574i
$980$	0	0
$981$	−126.000	−0.128440
$982$	0	0
$983$	1256.00i	1.27772i	0.769322	+	0.638861i	$0.220594\pi$
$983$	1256.00i	1.27772i	−0.769322	+	0.638861i	$0.779406\pi$
$984$	0	0
$985$	644.000	0.653807
$986$	0	0
$987$	512.000i	0.518744i
$988$	0	0
$989$	−1120.00	−1.13246
$990$	0	0
$991$	− 864.000i	− 0.871847i	−0.899984	−	0.435923i	$-0.856422\pi$
$991$	− 864.000i	− 0.871847i	0.899984	−	0.435923i	$-0.143578\pi$
$992$	0	0
$993$	−2288.00	−2.30413
$994$	0	0
$995$	− 400.000i	− 0.402010i
$996$	0	0
$997$	−1054.00	−1.05717	−0.528586	−	0.848880i	$-0.677277\pi$
$997$	−1054.00	−1.05717	−0.528586	+	0.848880i	$0.677277\pi$
$998$	0	0
$999$	− 240.000i	− 0.240240i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.3.c.a.31.2	yes	2
3.2	odd	2		288.3.g.b.127.1		2
4.3	odd	2	inner	32.3.c.a.31.1	✓	2
5.2	odd	4		800.3.h.b.799.1		2
5.3	odd	4		800.3.h.a.799.1		2
5.4	even	2		800.3.b.a.351.1		2
7.6	odd	2		1568.3.d.b.1471.1		2
8.3	odd	2		64.3.c.b.63.2		2
8.5	even	2		64.3.c.b.63.1		2
12.11	even	2		288.3.g.b.127.2		2
16.3	odd	4		256.3.d.c.127.1		2
16.5	even	4		256.3.d.c.127.2		2
16.11	odd	4		256.3.d.a.127.2		2
16.13	even	4		256.3.d.a.127.1		2
20.3	even	4		800.3.h.b.799.2		2
20.7	even	4		800.3.h.a.799.2		2
20.19	odd	2		800.3.b.a.351.2		2
24.5	odd	2		576.3.g.g.127.1		2
24.11	even	2		576.3.g.g.127.2		2
28.27	even	2		1568.3.d.b.1471.2		2
40.3	even	4		1600.3.h.a.1599.1		2
40.13	odd	4		1600.3.h.c.1599.2		2
40.19	odd	2		1600.3.b.e.1151.1		2
40.27	even	4		1600.3.h.c.1599.1		2
40.29	even	2		1600.3.b.e.1151.2		2
40.37	odd	4		1600.3.h.a.1599.2		2
48.5	odd	4		2304.3.b.c.127.1		2
48.11	even	4		2304.3.b.g.127.1		2
48.29	odd	4		2304.3.b.g.127.2		2
48.35	even	4		2304.3.b.c.127.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.3.c.a.31.1	✓	2	4.3	odd	2	inner
32.3.c.a.31.2	yes	2	1.1	even	1	trivial
64.3.c.b.63.1		2	8.5	even	2
64.3.c.b.63.2		2	8.3	odd	2
256.3.d.a.127.1		2	16.13	even	4
256.3.d.a.127.2		2	16.11	odd	4
256.3.d.c.127.1		2	16.3	odd	4
256.3.d.c.127.2		2	16.5	even	4
288.3.g.b.127.1		2	3.2	odd	2
288.3.g.b.127.2		2	12.11	even	2
576.3.g.g.127.1		2	24.5	odd	2
576.3.g.g.127.2		2	24.11	even	2
800.3.b.a.351.1		2	5.4	even	2
800.3.b.a.351.2		2	20.19	odd	2
800.3.h.a.799.1		2	5.3	odd	4
800.3.h.a.799.2		2	20.7	even	4
800.3.h.b.799.1		2	5.2	odd	4
800.3.h.b.799.2		2	20.3	even	4
1568.3.d.b.1471.1		2	7.6	odd	2
1568.3.d.b.1471.2		2	28.27	even	2
1600.3.b.e.1151.1		2	40.19	odd	2
1600.3.b.e.1151.2		2	40.29	even	2
1600.3.h.a.1599.1		2	40.3	even	4
1600.3.h.a.1599.2		2	40.37	odd	4
1600.3.h.c.1599.1		2	40.27	even	4
1600.3.h.c.1599.2		2	40.13	odd	4
2304.3.b.c.127.1		2	48.5	odd	4
2304.3.b.c.127.2		2	48.35	even	4
2304.3.b.g.127.1		2	48.11	even	4
2304.3.b.g.127.2		2	48.29	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	32.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{3} +2.00000 q^{5} -8.00000i q^{7} -7.00000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{3} +2.00000 q^{5} -8.00000i q^{7} -7.00000 q^{9} -4.00000i q^{11} -14.0000 q^{13} +8.00000i q^{15} +18.0000 q^{17} -12.0000i q^{19} +32.0000 q^{21} +40.0000i q^{23} -21.0000 q^{25} +8.00000i q^{27} -14.0000 q^{29} -32.0000i q^{31} +16.0000 q^{33} -16.0000i q^{35} -30.0000 q^{37} -56.0000i q^{39} -14.0000 q^{41} +28.0000i q^{43} -14.0000 q^{45} +16.0000i q^{47} -15.0000 q^{49} +72.0000i q^{51} +66.0000 q^{53} -8.00000i q^{55} +48.0000 q^{57} -52.0000i q^{59} +82.0000 q^{61} +56.0000i q^{63} -28.0000 q^{65} +4.00000i q^{67} -160.000 q^{69} +56.0000i q^{71} +66.0000 q^{73} -84.0000i q^{75} -32.0000 q^{77} -16.0000i q^{79} -95.0000 q^{81} -140.000i q^{83} +36.0000 q^{85} -56.0000i q^{87} -30.0000 q^{89} +112.000i q^{91} +128.000 q^{93} -24.0000i q^{95} -14.0000 q^{97} +28.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 14 q^{9}+O(q^{10}) \) 2 * q + 4 * q^5 - 14 * q^9	\( 2 q + 4 q^{5} - 14 q^{9} - 28 q^{13} + 36 q^{17} + 64 q^{21} - 42 q^{25} - 28 q^{29} + 32 q^{33} - 60 q^{37} - 28 q^{41} - 28 q^{45} - 30 q^{49} + 132 q^{53} + 96 q^{57} + 164 q^{61} - 56 q^{65} - 320 q^{69} + 132 q^{73} - 64 q^{77} - 190 q^{81} + 72 q^{85} - 60 q^{89} + 256 q^{93} - 28 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - 14 * q^9 - 28 * q^13 + 36 * q^17 + 64 * q^21 - 42 * q^25 - 28 * q^29 + 32 * q^33 - 60 * q^37 - 28 * q^41 - 28 * q^45 - 30 * q^49 + 132 * q^53 + 96 * q^57 + 164 * q^61 - 56 * q^65 - 320 * q^69 + 132 * q^73 - 64 * q^77 - 190 * q^81 + 72 * q^85 - 60 * q^89 + 256 * q^93 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more