LMFDB - Embedded newform 1568.2.p.a.31.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(31,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

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

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

chi := DirichletCharacter("1568.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.p (of order $6$, degree $2$, not 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:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.5
Root			$0.793353 - 0.608761i$ of defining polynomial
Character	$\chi$	$=$	1568.31
Dual form			1568.2.p.a.607.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.541196 - 0.937379i) q^{3} +(-2.26303 + 1.30656i) q^{5} +(0.914214 + 1.58346i) q^{9} +O(q^{10})$	\(q+(0.541196 - 0.937379i) q^{3} +(-2.26303 + 1.30656i) q^{5} +(0.914214 + 1.58346i) q^{9} +(1.73205 + 1.00000i) q^{11} -4.77791i q^{13} +2.82843i q^{15} +(-2.65131 - 1.53073i) q^{17} +(2.07193 + 3.58869i) q^{19} +(-6.63103 + 3.82843i) q^{23} +(0.914214 - 1.58346i) q^{25} +5.22625 q^{27} +3.65685 q^{29} +(-1.53073 + 2.65131i) q^{31} +(1.87476 - 1.08239i) q^{33} +(3.82843 + 6.63103i) q^{37} +(-4.47871 - 2.58579i) q^{39} +9.55582i q^{41} +3.65685i q^{43} +(-4.13779 - 2.38896i) q^{45} +(-3.69552 - 6.40083i) q^{47} +(-2.86976 + 1.65685i) q^{51} +(1.00000 - 1.73205i) q^{53} -5.22625 q^{55} +4.48528 q^{57} +(-4.23671 + 7.33820i) q^{59} +(2.26303 - 1.30656i) q^{61} +(6.24264 + 10.8126i) q^{65} +(13.5592 + 7.82843i) q^{67} +8.28772i q^{69} +8.82843i q^{71} +(10.9269 + 6.30864i) q^{73} +(-0.989538 - 1.71393i) q^{75} +(-11.1097 + 6.41421i) q^{79} +(0.0857864 - 0.148586i) q^{81} -11.5349 q^{83} +8.00000 q^{85} +(1.97908 - 3.42786i) q^{87} +(-1.87476 + 1.08239i) q^{89} +(1.65685 + 2.86976i) q^{93} +(-9.37769 - 5.41421i) q^{95} -13.5140i q^{97} +3.65685i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} - 8 q^{25} - 32 q^{29} + 16 q^{37} + 16 q^{53} - 64 q^{57} + 32 q^{65} + 24 q^{81} + 128 q^{85} - 64 q^{93}+O(q^{100}) $ 16 * q - 8 * q^9 - 8 * q^25 - 32 * q^29 + 16 * q^37 + 16 * q^53 - 64 * q^57 + 32 * q^65 + 24 * q^81 + 128 * q^85 - 64 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$3$	0.541196	−	0.937379i	0.312460	−	0.541196i	−0.666435	−	0.745564i	$-0.732180\pi$
$3$	0.541196	−	0.937379i	0.312460	−	0.541196i	0.978894	+	0.204367i	$0.0655137\pi$
$4$		0			0
$5$	−2.26303	+	1.30656i	−1.01206	+	0.584313i	−0.911794	−	0.410648i	$-0.865303\pi$
$5$	−2.26303	+	1.30656i	−1.01206	+	0.584313i	−0.100265	+	0.994961i	$0.531969\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	0.914214	+	1.58346i	0.304738	+	0.527821i
$10$		0			0
$11$	1.73205	+	1.00000i	0.522233	+	0.301511i	0.737848	−	0.674967i	$-0.235842\pi$
$11$	1.73205	+	1.00000i	0.522233	+	0.301511i	−0.215615	+	0.976478i	$0.569176\pi$
$12$		0			0
$13$		−	4.77791i		−	1.32515i	−0.748994	−	0.662577i	$-0.769463\pi$
$13$		−	4.77791i		−	1.32515i	0.748994	−	0.662577i	$-0.230537\pi$
$14$		0			0
$15$			2.82843i			0.730297i
$16$		0			0
$17$	−2.65131	−	1.53073i	−0.643037	−	0.371257i	0.142747	−	0.989759i	$-0.454407\pi$
$17$	−2.65131	−	1.53073i	−0.643037	−	0.371257i	−0.785783	+	0.618502i	$0.787740\pi$
$18$		0			0
$19$	2.07193	+	3.58869i	0.475333	+	0.823301i	0.999601	−	0.0282522i	$-0.00899415\pi$
$19$	2.07193	+	3.58869i	0.475333	+	0.823301i	−0.524268	+	0.851554i	$0.675661\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.63103	+	3.82843i	−1.38267	+	0.798282i	−0.992474	−	0.122452i	$-0.960924\pi$
$23$	−6.63103	+	3.82843i	−1.38267	+	0.798282i	−0.390191	+	0.920734i	$0.627591\pi$
$24$		0			0
$25$	0.914214	−	1.58346i	0.182843	−	0.316693i
$26$		0			0
$27$	5.22625			1.00579
$28$		0			0
$29$	3.65685			0.679061			0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685			0.679061			0.339530	+	0.940595i	$0.389732\pi$
$30$		0			0
$31$	−1.53073	+	2.65131i	−0.274928	+	0.476189i	−0.970117	−	0.242638i	$-0.921987\pi$
$31$	−1.53073	+	2.65131i	−0.274928	+	0.476189i	0.695189	+	0.718827i	$0.255321\pi$
$32$		0			0
$33$	1.87476	−	1.08239i	0.326354	−	0.188420i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.82843	+	6.63103i	0.629390	+	1.09013i	0.987674	+	0.156522i	$0.0500283\pi$
$37$	3.82843	+	6.63103i	0.629390	+	1.09013i	−0.358285	+	0.933612i	$0.616638\pi$
$38$		0			0
$39$	−4.47871	−	2.58579i	−0.717168	−	0.414057i
$40$		0			0
$41$			9.55582i			1.49237i	0.665740	+	0.746184i	$0.268116\pi$
$41$			9.55582i			1.49237i	−0.665740	+	0.746184i	$0.731884\pi$
$42$		0			0
$43$			3.65685i			0.557665i	0.960340	+	0.278833i	$0.0899474\pi$
$43$			3.65685i			0.557665i	−0.960340	+	0.278833i	$0.910053\pi$
$44$		0			0
$45$	−4.13779	−	2.38896i	−0.616826	−	0.356124i
$46$		0			0
$47$	−3.69552	−	6.40083i	−0.539047	−	0.933656i	−0.998956	−	0.0456902i	$-0.985451\pi$
$47$	−3.69552	−	6.40083i	−0.539047	−	0.933656i	0.459909	−	0.887966i	$-0.347882\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.86976	+	1.65685i	−0.401846	+	0.232006i
$52$		0			0
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	−0.789136	−	0.614218i	$-0.789471\pi$
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	0.926497	+	0.376303i	$0.122805\pi$
$54$		0			0
$55$	−5.22625			−0.704708
$56$		0			0
$57$	4.48528			0.594090
$58$		0			0
$59$	−4.23671	+	7.33820i	−0.551573	+	0.955353i	0.446588	+	0.894740i	$0.352639\pi$
$59$	−4.23671	+	7.33820i	−0.551573	+	0.955353i	−0.998161	+	0.0606132i	$0.980694\pi$
$60$		0			0
$61$	2.26303	−	1.30656i	0.289752	−	0.167288i	−0.348078	−	0.937466i	$-0.613166\pi$
$61$	2.26303	−	1.30656i	0.289752	−	0.167288i	0.637830	+	0.770177i	$0.279832\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.24264	+	10.8126i	0.774304	+	1.34113i
$66$		0			0
$67$	13.5592	+	7.82843i	1.65652	+	0.956395i	0.974301	+	0.225251i	$0.0723201\pi$
$67$	13.5592	+	7.82843i	1.65652	+	0.956395i	0.682223	+	0.731144i	$0.261013\pi$
$68$		0			0
$69$			8.28772i			0.997724i
$70$		0			0
$71$			8.82843i			1.04774i	0.851798	+	0.523871i	$0.175513\pi$
$71$			8.82843i			1.04774i	−0.851798	+	0.523871i	$0.824487\pi$
$72$		0			0
$73$	10.9269	+	6.30864i	1.27890	+	0.738371i	0.976645	−	0.214858i	$-0.0689287\pi$
$73$	10.9269	+	6.30864i	1.27890	+	0.738371i	0.302251	+	0.953229i	$0.402262\pi$
$74$		0			0
$75$	−0.989538	−	1.71393i	−0.114262	−	0.197908i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−11.1097	+	6.41421i	−1.24994	+	0.721655i	−0.971098	−	0.238681i	$-0.923285\pi$
$79$	−11.1097	+	6.41421i	−1.24994	+	0.721655i	−0.278846	+	0.960336i	$0.589952\pi$
$80$		0			0
$81$	0.0857864	−	0.148586i	0.00953183	−	0.0165096i
$82$		0			0
$83$	−11.5349			−1.26612			−0.633060	−	0.774103i	$-0.718201\pi$
$83$	−11.5349			−1.26612			−0.633060	+	0.774103i	$0.718201\pi$
$84$		0			0
$85$	8.00000			0.867722
$86$		0			0
$87$	1.97908	−	3.42786i	0.212179	−	0.367505i
$88$		0			0
$89$	−1.87476	+	1.08239i	−0.198724	+	0.114733i	−0.596060	−	0.802940i	$-0.703268\pi$
$89$	−1.87476	+	1.08239i	−0.198724	+	0.114733i	0.397336	+	0.917673i	$0.369935\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.65685	+	2.86976i	0.171808	+	0.297580i
$94$		0			0
$95$	−9.37769	−	5.41421i	−0.962131	−	0.555487i
$96$		0			0
$97$		−	13.5140i		−	1.37214i	−0.727537	−	0.686068i	$-0.759335\pi$
$97$		−	13.5140i		−	1.37214i	0.727537	−	0.686068i	$-0.240665\pi$
$98$		0			0
$99$			3.65685i			0.367528i
$100$		0			0
$101$	4.91434	+	2.83730i	0.488995	+	0.282322i	0.724158	−	0.689635i	$-0.242229\pi$
$101$	4.91434	+	2.83730i	0.488995	+	0.282322i	−0.235162	+	0.971956i	$0.575562\pi$
$102$		0			0
$103$	8.47343	+	14.6764i	0.834912	+	1.44611i	0.894102	+	0.447863i	$0.147815\pi$
$103$	8.47343	+	14.6764i	0.834912	+	1.44611i	−0.0591906	+	0.998247i	$0.518852\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.297173	−	0.171573i	0.0287288	−	0.0165866i	−0.485567	−	0.874200i	$-0.661387\pi$
$107$	0.297173	−	0.171573i	0.0287288	−	0.0165866i	0.514296	+	0.857613i	$0.328053\pi$
$108$		0			0
$109$	2.65685	−	4.60181i	0.254480	−	0.440773i	−0.710274	−	0.703926i	$-0.751429\pi$
$109$	2.65685	−	4.60181i	0.254480	−	0.440773i	0.964754	+	0.263152i	$0.0847622\pi$
$110$		0			0
$111$	8.28772			0.786636
$112$		0			0
$113$	8.82843			0.830509			0.415254	−	0.909705i	$-0.363693\pi$
$113$	8.82843			0.830509			0.415254	+	0.909705i	$0.363693\pi$
$114$		0			0
$115$	10.0042	−	17.3277i	0.932893	−	1.61582i
$116$		0			0
$117$	7.56565	−	4.36803i	0.699445	−	0.403825i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.50000	−	6.06218i	−0.318182	−	0.551107i
$122$		0			0
$123$	8.95743	+	5.17157i	0.807664	+	0.466305i
$124$		0			0
$125$		−	8.28772i		−	0.741276i
$126$		0			0
$127$			3.65685i			0.324493i	0.986750	+	0.162247i	$0.0518740\pi$
$127$			3.65685i			0.324493i	−0.986750	+	0.162247i	$0.948126\pi$
$128$		0			0
$129$	3.42786	+	1.97908i	0.301806	+	0.174248i
$130$		0			0
$131$	−6.84984	−	11.8643i	−0.598473	−	1.03659i	−0.993047	−	0.117722i	$-0.962441\pi$
$131$	−6.84984	−	11.8643i	−0.598473	−	1.03659i	0.394573	−	0.918864i	$-0.370892\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−11.8272	+	6.82843i	−1.01792	+	0.587697i
$136$		0			0
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	−0.905577	−	0.424182i	$-0.860562\pi$
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	0.820141	+	0.572161i	$0.193895\pi$
$138$		0			0
$139$	−1.97908			−0.167863			−0.0839315	−	0.996472i	$-0.526748\pi$
$139$	−1.97908			−0.167863			−0.0839315	+	0.996472i	$0.526748\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$	4.77791	−	8.27558i	0.399549	−	0.692039i
$144$		0			0
$145$	−8.27558	+	4.77791i	−0.687250	+	0.396784i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.00000	+	8.66025i	0.409616	+	0.709476i	0.994847	−	0.101391i	$-0.0323294\pi$
$149$	5.00000	+	8.66025i	0.409616	+	0.709476i	−0.585231	+	0.810867i	$0.698996\pi$
$150$		0			0
$151$	−7.22538	−	4.17157i	−0.587993	−	0.339478i	0.176311	−	0.984335i	$-0.443584\pi$
$151$	−7.22538	−	4.17157i	−0.587993	−	0.339478i	−0.764303	+	0.644857i	$0.776917\pi$
$152$		0			0
$153$		−	5.59767i		−	0.452545i
$154$		0			0
$155$		−	8.00000i		−	0.642575i
$156$		0			0
$157$	−4.91434	−	2.83730i	−0.392207	−	0.226441i	0.290909	−	0.956751i	$-0.406042\pi$
$157$	−4.91434	−	2.83730i	−0.392207	−	0.226441i	−0.683116	+	0.730310i	$0.739376\pi$
$158$		0			0
$159$	−1.08239	−	1.87476i	−0.0858393	−	0.148678i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	14.9941	−	8.65685i	1.17443	−	0.678057i	0.219710	−	0.975565i	$-0.429489\pi$
$163$	14.9941	−	8.65685i	1.17443	−	0.678057i	0.954719	+	0.297508i	$0.0961555\pi$
$164$		0			0
$165$	−2.82843	+	4.89898i	−0.220193	+	0.381385i
$166$		0			0
$167$	6.49435			0.502548			0.251274	−	0.967916i	$-0.419150\pi$
$167$	6.49435			0.502548			0.251274	+	0.967916i	$0.419150\pi$
$168$		0			0
$169$	−9.82843			−0.756033
$170$		0			0
$171$	−3.78837	+	6.56165i	−0.289704	+	0.501782i
$172$		0			0
$173$	−9.44041	+	5.45042i	−0.717741	+	0.414388i	−0.813921	−	0.580976i	$-0.802671\pi$
$173$	−9.44041	+	5.45042i	−0.717741	+	0.414388i	0.0961797	+	0.995364i	$0.469338\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.58579	+	7.94282i	0.344689	+	0.597019i
$178$		0			0
$179$	1.73205	+	1.00000i	0.129460	+	0.0747435i	0.563331	−	0.826231i	$-0.309520\pi$
$179$	1.73205	+	1.00000i	0.129460	+	0.0747435i	−0.433872	+	0.900975i	$0.642853\pi$
$180$		0			0
$181$			1.71644i			0.127582i	0.997963	+	0.0637911i	$0.0203191\pi$
$181$			1.71644i			0.127582i	−0.997963	+	0.0637911i	$0.979681\pi$
$182$		0			0
$183$		−	2.82843i		−	0.209083i
$184$		0			0
$185$	−17.3277	−	10.0042i	−1.27396	−	0.735521i
$186$		0			0
$187$	−3.06147	−	5.30262i	−0.223877	−	0.387766i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.18154	+	2.41421i	−0.302566	+	0.174686i	−0.643595	−	0.765366i	$-0.722558\pi$
$191$	−4.18154	+	2.41421i	−0.302566	+	0.174686i	0.341029	+	0.940053i	$0.389224\pi$
$192$		0			0
$193$	5.24264	−	9.08052i	0.377374	−	0.653630i	−0.613306	−	0.789846i	$-0.710161\pi$
$193$	5.24264	−	9.08052i	0.377374	−	0.653630i	0.990679	+	0.136215i	$0.0434940\pi$
$194$		0			0
$195$	13.5140			0.967756
$196$		0			0
$197$	3.65685			0.260540			0.130270	−	0.991479i	$-0.458416\pi$
$197$	3.65685			0.260540			0.130270	+	0.991479i	$0.458416\pi$
$198$		0			0
$199$	−6.30864	+	10.9269i	−0.447208	+	0.774587i	−0.998203	−	0.0599216i	$-0.980915\pi$
$199$	−6.30864	+	10.9269i	−0.447208	+	0.774587i	0.550995	+	0.834508i	$0.314248\pi$
$200$		0			0
$201$	14.6764	−	8.47343i	1.03519	−	0.597670i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−12.4853	−	21.6251i	−0.872010	−	1.51037i
$206$		0			0
$207$	−12.1244	−	7.00000i	−0.842701	−	0.486534i
$208$		0			0
$209$			8.28772i			0.573274i
$210$		0			0
$211$		−	10.9706i		−	0.755245i	−0.925960	−	0.377622i	$-0.876742\pi$
$211$		−	10.9706i		−	0.755245i	0.925960	−	0.377622i	$-0.123258\pi$
$212$		0			0
$213$	8.27558	+	4.77791i	0.567034	+	0.327377i
$214$		0			0
$215$	−4.77791	−	8.27558i	−0.325851	−	0.564390i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	11.8272	−	6.82843i	0.799207	−	0.461422i
$220$		0			0
$221$	−7.31371	+	12.6677i	−0.491973	+	0.852123i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	3.34315			0.222876
$226$		0			0
$227$	−9.27726	+	16.0687i	−0.615753	+	1.06652i	0.374499	+	0.927227i	$0.377815\pi$
$227$	−9.27726	+	16.0687i	−0.615753	+	1.06652i	−0.990252	+	0.139288i	$0.955519\pi$
$228$		0			0
$229$	2.58469	−	1.49227i	0.170801	−	0.0986121i	−0.412162	−	0.911110i	$-0.635226\pi$
$229$	2.58469	−	1.49227i	0.170801	−	0.0986121i	0.582964	+	0.812498i	$0.301893\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.00000	−	15.5885i	−0.589610	−	1.02123i	−0.994283	−	0.106773i	$-0.965948\pi$
$233$	−9.00000	−	15.5885i	−0.589610	−	1.02123i	0.404674	−	0.914461i	$-0.367385\pi$
$234$		0			0
$235$	16.7262	+	9.65685i	1.09109	+	0.629944i
$236$		0			0
$237$			13.8854i			0.901953i
$238$		0			0
$239$			8.34315i			0.539673i	0.962906	+	0.269837i	$0.0869697\pi$
$239$			8.34315i			0.539673i	−0.962906	+	0.269837i	$0.913030\pi$
$240$		0			0
$241$	−20.7556	−	11.9832i	−1.33698	−	0.771908i	−0.350625	−	0.936516i	$-0.614031\pi$
$241$	−20.7556	−	11.9832i	−1.33698	−	0.771908i	−0.986359	+	0.164608i	$0.947364\pi$
$242$		0			0
$243$	7.74652	+	13.4174i	0.496940	+	0.860725i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	17.1464	−	9.89949i	1.09100	−	0.629890i
$248$		0			0
$249$	−6.24264	+	10.8126i	−0.395611	+	0.685219i
$250$		0			0
$251$	1.08239			0.0683200			0.0341600	−	0.999416i	$-0.489124\pi$
$251$	1.08239			0.0683200			0.0341600	+	0.999416i	$0.489124\pi$
$252$		0			0
$253$	−15.3137			−0.962765
$254$		0			0
$255$	4.32957	−	7.49903i	0.271128	−	0.469608i
$256$		0			0
$257$	5.30262	−	3.06147i	0.330768	−	0.190969i	−0.325414	−	0.945572i	$-0.605504\pi$
$257$	5.30262	−	3.06147i	0.330768	−	0.190969i	0.656182	+	0.754603i	$0.272170\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.34315	+	5.79050i	0.206936	+	0.358423i
$262$		0			0
$263$	−3.58719	−	2.07107i	−0.221196	−	0.127708i	0.385308	−	0.922788i	$-0.374095\pi$
$263$	−3.58719	−	2.07107i	−0.221196	−	0.127708i	−0.606504	+	0.795081i	$0.707429\pi$
$264$		0			0
$265$			5.22625i			0.321046i
$266$		0			0
$267$			2.34315i			0.143398i
$268$		0			0
$269$	11.6368	+	6.71852i	0.709510	+	0.409636i	0.810879	−	0.585213i	$-0.198989\pi$
$269$	11.6368	+	6.71852i	0.709510	+	0.409636i	−0.101370	+	0.994849i	$0.532323\pi$
$270$		0			0
$271$	0.896683	+	1.55310i	0.0544696	+	0.0943441i	0.891975	−	0.452086i	$-0.149320\pi$
$271$	0.896683	+	1.55310i	0.0544696	+	0.0943441i	−0.837505	+	0.546430i	$0.815987\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.16693	−	1.82843i	0.190973	−	0.110258i
$276$		0			0
$277$	13.4853	−	23.3572i	0.810252	−	1.40340i	−0.102436	−	0.994740i	$-0.532664\pi$
$277$	13.4853	−	23.3572i	0.810252	−	1.40340i	0.912688	−	0.408658i	$-0.134003\pi$
$278$		0			0
$279$	−5.59767			−0.335124
$280$		0			0
$281$	−28.6274			−1.70777			−0.853884	−	0.520463i	$-0.825759\pi$
$281$	−28.6274			−1.70777			−0.853884	+	0.520463i	$0.825759\pi$
$282$		0			0
$283$	−0.541196	+	0.937379i	−0.0321708	+	0.0557214i	−0.881663	−	0.471880i	$-0.843575\pi$
$283$	−0.541196	+	0.937379i	−0.0321708	+	0.0557214i	0.849492	+	0.527602i	$0.176909\pi$
$284$		0			0
$285$	−10.1503	+	5.86030i	−0.601254	+	0.347134i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3.81371	−	6.60554i	−0.224336	−	0.388561i
$290$		0			0
$291$	−12.6677	−	7.31371i	−0.742595	−	0.428737i
$292$		0			0
$293$		−	5.67459i		−	0.331513i	−0.986167	−	0.165757i	$-0.946993\pi$
$293$		−	5.67459i		−	0.331513i	0.986167	−	0.165757i	$-0.0530066\pi$
$294$		0			0
$295$		−	22.1421i		−	1.28916i
$296$		0			0
$297$	9.05213	+	5.22625i	0.525258	+	0.303258i
$298$		0			0
$299$	18.2919	+	31.6825i	1.05785	+	1.83224i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	5.31925	−	3.07107i	0.305583	−	0.176428i
$304$		0			0
$305$	−3.41421	+	5.91359i	−0.195497	+	0.338611i
$306$		0			0
$307$	−11.5349			−0.658331			−0.329166	−	0.944272i	$-0.606767\pi$
$307$	−11.5349			−0.658331			−0.329166	+	0.944272i	$0.606767\pi$
$308$		0			0
$309$	18.3431			1.04351
$310$		0			0
$311$	−10.0042	+	17.3277i	−0.567284	+	0.982565i	0.429549	+	0.903043i	$0.358672\pi$
$311$	−10.0042	+	17.3277i	−0.567284	+	0.982565i	−0.996833	+	0.0795212i	$0.974661\pi$
$312$		0			0
$313$	−13.5782	+	7.83938i	−0.767485	+	0.443108i	−0.831977	−	0.554810i	$-0.812791\pi$
$313$	−13.5782	+	7.83938i	−0.767485	+	0.443108i	0.0644915	+	0.997918i	$0.479457\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.8284	−	23.9515i	−0.776682	−	1.34525i	−0.933844	−	0.357680i	$-0.883568\pi$
$317$	−13.8284	−	23.9515i	−0.776682	−	1.34525i	0.157162	−	0.987573i	$-0.449765\pi$
$318$		0			0
$319$	6.33386	+	3.65685i	0.354628	+	0.204745i
$320$		0			0
$321$		−	0.371418i		−	0.0207305i
$322$		0			0
$323$		−	12.6863i		−	0.705884i
$324$		0			0
$325$	−7.56565	−	4.36803i	−0.419667	−	0.242295i
$326$		0			0
$327$	−2.87576	−	4.98096i	−0.159030	−	0.275448i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	14.9941	−	8.65685i	0.824151	−	0.475824i	−0.0276949	−	0.999616i	$-0.508817\pi$
$331$	14.9941	−	8.65685i	0.824151	−	0.475824i	0.851846	+	0.523793i	$0.175483\pi$
$332$		0			0
$333$	−7.00000	+	12.1244i	−0.383598	+	0.664411i
$334$		0			0
$335$	−40.9133			−2.23533
$336$		0			0
$337$	8.82843			0.480915			0.240458	−	0.970660i	$-0.422703\pi$
$337$	8.82843			0.480915			0.240458	+	0.970660i	$0.422703\pi$
$338$		0			0
$339$	4.77791	−	8.27558i	0.259500	−	0.449468i
$340$		0			0
$341$	−5.30262	+	3.06147i	−0.287153	+	0.165788i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−10.8284	−	18.7554i	−0.582983	−	1.00976i
$346$		0			0
$347$	−17.0233	−	9.82843i	−0.913861	−	0.527618i	−0.0321893	−	0.999482i	$-0.510248\pi$
$347$	−17.0233	−	9.82843i	−0.913861	−	0.527618i	−0.881671	+	0.471864i	$0.843581\pi$
$348$		0			0
$349$			8.73606i			0.467631i	0.972281	+	0.233815i	$0.0751211\pi$
$349$			8.73606i			0.467631i	−0.972281	+	0.233815i	$0.924879\pi$
$350$		0			0
$351$		−	24.9706i		−	1.33283i
$352$		0			0
$353$	25.6033	+	14.7821i	1.36273	+	0.786770i	0.989986	−	0.141165i	$-0.0450849\pi$
$353$	25.6033	+	14.7821i	1.36273	+	0.786770i	0.372740	+	0.927936i	$0.378418\pi$
$354$		0			0
$355$	−11.5349	−	19.9790i	−0.612209	−	1.06038i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.9649	−	7.48528i	0.684261	−	0.395058i	−0.117198	−	0.993109i	$-0.537391\pi$
$359$	12.9649	−	7.48528i	0.684261	−	0.395058i	0.801458	+	0.598051i	$0.204058\pi$
$360$		0			0
$361$	0.914214	−	1.58346i	0.0481165	−	0.0833402i
$362$		0			0
$363$	−7.57675			−0.397676
$364$		0			0
$365$	−32.9706			−1.72576
$366$		0			0
$367$	−5.22625	+	9.05213i	−0.272808	+	0.472518i	−0.969580	−	0.244775i	$-0.921286\pi$
$367$	−5.22625	+	9.05213i	−0.272808	+	0.472518i	0.696772	+	0.717293i	$0.254619\pi$
$368$		0			0
$369$	−15.1313	+	8.73606i	−0.787704	+	0.454781i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−3.48528	−	6.03668i	−0.180461	−	0.312568i	0.761577	−	0.648075i	$-0.224426\pi$
$373$	−3.48528	−	6.03668i	−0.180461	−	0.312568i	−0.942038	+	0.335507i	$0.891092\pi$
$374$		0			0
$375$	−7.76874	−	4.48528i	−0.401176	−	0.231619i
$376$		0			0
$377$		−	17.4721i		−	0.899860i
$378$		0			0
$379$		−	5.31371i		−	0.272947i	−0.990644	−	0.136473i	$-0.956423\pi$
$379$		−	5.31371i		−	0.272947i	0.990644	−	0.136473i	$-0.0435768\pi$
$380$		0			0
$381$	3.42786	+	1.97908i	0.175615	+	0.101391i
$382$		0			0
$383$	5.86030	+	10.1503i	0.299447	+	0.518658i	0.976010	−	0.217727i	$-0.0698643\pi$
$383$	5.86030	+	10.1503i	0.299447	+	0.518658i	−0.676562	+	0.736386i	$0.736531\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.79050	+	3.34315i	−0.294348	+	0.169942i
$388$		0			0
$389$	−9.82843	+	17.0233i	−0.498321	+	0.863117i	−0.999998	−	0.00193762i	$-0.999383\pi$
$389$	−9.82843	+	17.0233i	−0.498321	+	0.863117i	0.501677	+	0.865055i	$0.332717\pi$
$390$		0			0
$391$	23.4412			1.18547
$392$		0			0
$393$	−14.8284			−0.747995
$394$		0			0
$395$	16.7611	−	29.0312i	0.843345	−	1.46072i
$396$		0			0
$397$	29.4194	−	16.9853i	1.47652	−	0.852469i	0.476870	−	0.878974i	$-0.341771\pi$
$397$	29.4194	−	16.9853i	1.47652	−	0.852469i	0.999649	+	0.0265049i	$0.00843776\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.58579	+	13.1390i	0.378816	+	0.656129i	0.990890	−	0.134672i	$-0.0429980\pi$
$401$	7.58579	+	13.1390i	0.378816	+	0.656129i	−0.612074	+	0.790800i	$0.709665\pi$
$402$		0			0
$403$	12.6677	+	7.31371i	0.631024	+	0.364322i
$404$		0			0
$405$			0.448342i			0.0222783i
$406$		0			0
$407$			15.3137i			0.759072i
$408$		0			0
$409$	0.776550	+	0.448342i	0.0383979	+	0.0221691i	0.519076	−	0.854728i	$-0.326276\pi$
$409$	0.776550	+	0.448342i	0.0383979	+	0.0221691i	−0.480678	+	0.876897i	$0.659609\pi$
$410$		0			0
$411$	1.08239	+	1.87476i	0.0533905	+	0.0924750i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	26.1039	−	15.0711i	1.28139	−	0.739810i
$416$		0			0
$417$	−1.07107	+	1.85514i	−0.0524504	+	0.0908468i
$418$		0			0
$419$	34.6047			1.69055			0.845275	−	0.534332i	$-0.179437\pi$
$419$	34.6047			1.69055			0.845275	+	0.534332i	$0.179437\pi$
$420$		0			0
$421$	−21.3137			−1.03877			−0.519383	−	0.854541i	$-0.673838\pi$
$421$	−21.3137			−1.03877			−0.519383	+	0.854541i	$0.673838\pi$
$422$		0			0
$423$	6.75699	−	11.7034i	0.328536	−	0.569041i
$424$		0			0
$425$	−4.84772	+	2.79884i	−0.235149	+	0.135763i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−5.17157	−	8.95743i	−0.249686	−	0.432469i
$430$		0			0
$431$	10.6895	+	6.17157i	0.514894	+	0.297274i	0.734843	−	0.678237i	$-0.237256\pi$
$431$	10.6895	+	6.17157i	0.514894	+	0.297274i	−0.219949	+	0.975511i	$0.570589\pi$
$432$		0			0
$433$			5.59767i			0.269007i	0.990913	+	0.134503i	$0.0429439\pi$
$433$			5.59767i			0.269007i	−0.990913	+	0.134503i	$0.957056\pi$
$434$		0			0
$435$			10.3431i			0.495916i
$436$		0			0
$437$	−27.4781	−	15.8645i	−1.31445	−	0.758900i
$438$		0			0
$439$	−11.2723	−	19.5241i	−0.537996	−	0.931836i	−0.999012	−	0.0444443i	$-0.985848\pi$
$439$	−11.2723	−	19.5241i	−0.537996	−	0.931836i	0.461016	−	0.887392i	$-0.347485\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.66025	+	5.00000i	−0.411461	+	0.237557i	−0.691417	−	0.722456i	$-0.743013\pi$
$443$	−8.66025	+	5.00000i	−0.411461	+	0.237557i	0.279956	+	0.960013i	$0.409680\pi$
$444$		0			0
$445$	2.82843	−	4.89898i	0.134080	−	0.232234i
$446$		0			0
$447$	10.8239			0.511954
$448$		0			0
$449$	21.3137			1.00586			0.502928	−	0.864328i	$-0.332256\pi$
$449$	21.3137			1.00586			0.502928	+	0.864328i	$0.332256\pi$
$450$		0			0
$451$	−9.55582	+	16.5512i	−0.449966	+	0.779364i
$452$		0			0
$453$	−7.82069	+	4.51528i	−0.367448	+	0.212146i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−18.5563	−	32.1405i	−0.868029	−	1.50347i	−0.864007	−	0.503479i	$-0.832053\pi$
$457$	−18.5563	−	32.1405i	−0.868029	−	1.50347i	−0.00402201	−	0.999992i	$-0.501280\pi$
$458$		0			0
$459$	−13.8564	−	8.00000i	−0.646762	−	0.373408i
$460$		0			0
$461$		−	10.3756i		−	0.483239i	−0.970371	−	0.241619i	$-0.922321\pi$
$461$		−	10.3756i		−	0.483239i	0.970371	−	0.241619i	$-0.0776786\pi$
$462$		0			0
$463$		−	16.1421i		−	0.750189i	−0.926987	−	0.375094i	$-0.877610\pi$
$463$		−	16.1421i		−	0.750189i	0.926987	−	0.375094i	$-0.122390\pi$
$464$		0			0
$465$	−7.49903	−	4.32957i	−0.347759	−	0.200779i
$466$		0			0
$467$	−4.05101	−	7.01655i	−0.187458	−	0.324687i	0.756944	−	0.653480i	$-0.226692\pi$
$467$	−4.05101	−	7.01655i	−0.187458	−	0.324687i	−0.944402	+	0.328793i	$0.893358\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−5.31925	+	3.07107i	−0.245098	+	0.141507i
$472$		0			0
$473$	−3.65685	+	6.33386i	−0.168142	+	0.291231i
$474$		0			0
$475$	7.57675			0.347645
$476$		0			0
$477$	3.65685			0.167436
$478$		0			0
$479$	15.0447	−	26.0582i	0.687410	−	1.19063i	−0.285263	−	0.958449i	$-0.592081\pi$
$479$	15.0447	−	26.0582i	0.687410	−	1.19063i	0.972673	−	0.232180i	$-0.0745858\pi$
$480$		0			0
$481$	31.6825	−	18.2919i	1.44460	−	0.834038i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	17.6569	+	30.5826i	0.801756	+	1.38868i
$486$		0			0
$487$	26.2269	+	15.1421i	1.18846	+	0.686156i	0.957956	−	0.286916i	$-0.0926302\pi$
$487$	26.2269	+	15.1421i	1.18846	+	0.686156i	0.230501	+	0.973072i	$0.425964\pi$
$488$		0			0
$489$		−	18.7402i		−	0.847462i
$490$		0			0
$491$		−	30.2843i		−	1.36671i	−0.730086	−	0.683355i	$-0.760520\pi$
$491$		−	30.2843i		−	1.36671i	0.730086	−	0.683355i	$-0.239480\pi$
$492$		0			0
$493$	−9.69545	−	5.59767i	−0.436661	−	0.252106i
$494$		0			0
$495$	−4.77791	−	8.27558i	−0.214751	−	0.371960i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	14.9941	−	8.65685i	0.671229	−	0.387534i	−0.125313	−	0.992117i	$-0.539994\pi$
$499$	14.9941	−	8.65685i	0.671229	−	0.387534i	0.796542	+	0.604583i	$0.206660\pi$
$500$		0			0
$501$	3.51472	−	6.08767i	0.157026	−	0.271977i
$502$		0			0
$503$	17.4721			0.779043			0.389522	−	0.921017i	$-0.372640\pi$
$503$	17.4721			0.779043			0.389522	+	0.921017i	$0.372640\pi$
$504$		0			0
$505$	−14.8284			−0.659856
$506$		0			0
$507$	−5.31911	+	9.21296i	−0.236230	+	0.409162i
$508$		0			0
$509$	22.5637	−	13.0272i	1.00012	−	0.577419i	0.0918356	−	0.995774i	$-0.470727\pi$
$509$	22.5637	−	13.0272i	1.00012	−	0.577419i	0.908284	+	0.418355i	$0.137393\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	10.8284	+	18.7554i	0.478087	+	0.828071i
$514$		0			0
$515$	−38.3513	−	22.1421i	−1.68996	−	0.975699i
$516$		0			0
$517$		−	14.7821i		−	0.650115i
$518$		0			0
$519$			11.7990i			0.517918i
$520$		0			0
$521$	−0.776550	−	0.448342i	−0.0340213	−	0.0196422i	0.482893	−	0.875679i	$-0.339586\pi$
$521$	−0.776550	−	0.448342i	−0.0340213	−	0.0196422i	−0.516914	+	0.856037i	$0.672919\pi$
$522$		0			0
$523$	−9.46297	−	16.3903i	−0.413787	−	0.716699i	0.581514	−	0.813537i	$-0.302461\pi$
$523$	−9.46297	−	16.3903i	−0.413787	−	0.716699i	−0.995300	+	0.0968372i	$0.969127\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.11689	−	4.68629i	0.353577	−	0.204138i
$528$		0			0
$529$	17.8137	−	30.8542i	0.774509	−	1.34149i
$530$		0			0
$531$	−15.4930			−0.672341
$532$		0			0
$533$	45.6569			1.97762
$534$		0			0
$535$	−0.448342	+	0.776550i	−0.0193835	+	0.0335732i
$536$		0			0
$537$	1.87476	−	1.08239i	0.0809018	−	0.0467087i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−12.6569	−	21.9223i	−0.544161	−	0.942514i	−0.998659	−	0.0517664i	$-0.983515\pi$
$541$	−12.6569	−	21.9223i	−0.544161	−	0.942514i	0.454499	−	0.890747i	$-0.349818\pi$
$542$		0			0
$543$	1.60896	+	0.928932i	0.0690470	+	0.0398643i
$544$		0			0
$545$			13.8854i			0.594785i
$546$		0			0
$547$			33.3137i			1.42439i	0.701981	+	0.712196i	$0.252299\pi$
$547$			33.3137i			1.42439i	−0.701981	+	0.712196i	$0.747701\pi$
$548$		0			0
$549$	4.13779	+	2.38896i	0.176597	+	0.101958i
$550$		0			0
$551$	7.57675	+	13.1233i	0.322780	+	0.559072i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−18.7554	+	10.8284i	−0.796122	+	0.459641i
$556$		0			0
$557$	2.65685	−	4.60181i	0.112575	−	0.194985i	−0.804233	−	0.594314i	$-0.797424\pi$
$557$	2.65685	−	4.60181i	0.112575	−	0.194985i	0.916808	+	0.399329i	$0.130757\pi$
$558$		0			0
$559$	17.4721			0.738992
$560$		0			0
$561$	−6.62742			−0.279810
$562$		0			0
$563$	−2.25764	+	3.91035i	−0.0951481	+	0.164801i	−0.909670	−	0.415331i	$-0.863666\pi$
$563$	−2.25764	+	3.91035i	−0.0951481	+	0.164801i	0.814522	+	0.580132i	$0.196999\pi$
$564$		0			0
$565$	−19.9790	+	11.5349i	−0.840524	+	0.485277i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.92893	−	6.80511i	−0.164709	−	0.285285i	0.771843	−	0.635814i	$-0.219335\pi$
$569$	−3.92893	−	6.80511i	−0.164709	−	0.285285i	−0.936552	+	0.350529i	$0.886002\pi$
$570$		0			0
$571$	−7.22538	−	4.17157i	−0.302373	−	0.174575i	0.341136	−	0.940014i	$-0.389188\pi$
$571$	−7.22538	−	4.17157i	−0.302373	−	0.174575i	−0.643508	+	0.765439i	$0.722522\pi$
$572$		0			0
$573$			5.22625i			0.218330i
$574$		0			0
$575$			14.0000i			0.583840i
$576$		0			0
$577$	30.9059	+	17.8435i	1.28663	+	0.742836i	0.978052	−	0.208363i	$-0.0668134\pi$
$577$	30.9059	+	17.8435i	1.28663	+	0.742836i	0.308579	+	0.951199i	$0.400147\pi$
$578$		0			0
$579$	−5.67459	−	9.82868i	−0.235828	−	0.408466i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3.46410	−	2.00000i	0.143468	−	0.0828315i
$584$		0			0
$585$	−11.4142	+	19.7700i	−0.471920	+	0.817389i
$586$		0			0
$587$	−29.0070			−1.19725			−0.598624	−	0.801030i	$-0.704286\pi$
$587$	−29.0070			−1.19725			−0.598624	+	0.801030i	$0.704286\pi$
$588$		0			0
$589$	−12.6863			−0.522730
$590$		0			0
$591$	1.97908	−	3.42786i	0.0814083	−	0.141003i
$592$		0			0
$593$	−1.55310	+	0.896683i	−0.0637782	+	0.0368224i	−0.531550	−	0.847027i	$-0.678390\pi$
$593$	−1.55310	+	0.896683i	−0.0637782	+	0.0368224i	0.467772	+	0.883849i	$0.345057\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.82843	+	11.8272i	0.279469	+	0.484054i
$598$		0			0
$599$	26.9954	+	15.5858i	1.10300	+	0.636818i	0.937008	−	0.349309i	$-0.113584\pi$
$599$	26.9954	+	15.5858i	1.10300	+	0.636818i	0.165993	+	0.986127i	$0.446917\pi$
$600$		0			0
$601$			42.1814i			1.72062i	0.509774	+	0.860308i	$0.329729\pi$
$601$			42.1814i			1.72062i	−0.509774	+	0.860308i	$0.670271\pi$
$602$		0			0
$603$			28.6274i			1.16580i
$604$		0			0
$605$	15.8412	+	9.14594i	0.644038	+	0.371835i
$606$		0			0
$607$	8.65914	+	14.9981i	0.351464	+	0.608753i	0.986506	−	0.163724i	$-0.0523508\pi$
$607$	8.65914	+	14.9981i	0.351464	+	0.608753i	−0.635043	+	0.772477i	$0.719017\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−30.5826	+	17.6569i	−1.23724	+	0.714320i
$612$		0			0
$613$	2.65685	−	4.60181i	0.107309	−	0.185865i	−0.807370	−	0.590045i	$-0.799110\pi$
$613$	2.65685	−	4.60181i	0.107309	−	0.185865i	0.914679	+	0.404180i	$0.132443\pi$
$614$		0			0
$615$	−27.0279			−1.08987
$616$		0			0
$617$	13.1127			0.527897			0.263949	−	0.964537i	$-0.414975\pi$
$617$	13.1127			0.527897			0.263949	+	0.964537i	$0.414975\pi$
$618$		0			0
$619$	−1.43788	+	2.49048i	−0.0577932	+	0.100101i	−0.893475	−	0.449114i	$-0.851740\pi$
$619$	−1.43788	+	2.49048i	−0.0577932	+	0.100101i	0.835681	+	0.549215i	$0.185073\pi$
$620$		0			0
$621$	−34.6554	+	20.0083i	−1.39067	+	0.802906i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15.3995	+	26.6727i	0.615980	+	1.06691i
$626$		0			0
$627$	7.76874	+	4.48528i	0.310253	+	0.179125i
$628$		0			0
$629$		−	23.4412i		−	0.934662i
$630$		0			0
$631$		−	26.4853i		−	1.05436i	−0.849753	−	0.527181i	$-0.823249\pi$
$631$		−	26.4853i		−	1.05436i	0.849753	−	0.527181i	$-0.176751\pi$
$632$		0			0
$633$	−10.2836	−	5.93723i	−0.408735	−	0.235984i
$634$		0			0
$635$	−4.77791	−	8.27558i	−0.189606	−	0.328407i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−13.9795	+	8.07107i	−0.553020	+	0.319287i
$640$		0			0
$641$	−14.0711	+	24.3718i	−0.555774	+	0.962628i	0.442069	+	0.896981i	$0.354245\pi$
$641$	−14.0711	+	24.3718i	−0.555774	+	0.962628i	−0.997843	+	0.0656474i	$0.979089\pi$
$642$		0			0
$643$	−17.1326			−0.675642			−0.337821	−	0.941210i	$-0.609690\pi$
$643$	−17.1326			−0.675642			−0.337821	+	0.941210i	$0.609690\pi$
$644$		0			0
$645$	−10.3431			−0.407261
$646$		0			0
$647$	2.35049	−	4.07117i	0.0924074	−	0.160054i	−0.816116	−	0.577888i	$-0.803877\pi$
$647$	2.35049	−	4.07117i	0.0924074	−	0.160054i	0.908524	+	0.417833i	$0.137210\pi$
$648$		0			0
$649$	−14.6764	+	8.47343i	−0.576099	+	0.332611i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.34315	−	2.32640i	−0.0525614	−	0.0910389i	0.838548	−	0.544828i	$-0.183405\pi$
$653$	−1.34315	−	2.32640i	−0.0525614	−	0.0910389i	−0.891109	+	0.453789i	$0.850072\pi$
$654$		0			0
$655$	31.0028	+	17.8995i	1.21138	+	0.699391i
$656$		0			0
$657$			23.0698i			0.900038i
$658$		0			0
$659$		−	17.0294i		−	0.663373i	−0.943390	−	0.331686i	$-0.892382\pi$
$659$		−	17.0294i		−	0.663373i	0.943390	−	0.331686i	$-0.107618\pi$
$660$		0			0
$661$	26.7681	+	15.4546i	1.04116	+	0.601114i	0.920161	−	0.391539i	$-0.128057\pi$
$661$	26.7681	+	15.4546i	1.04116	+	0.601114i	0.120998	+	0.992653i	$0.461391\pi$
$662$		0			0
$663$	7.91630	+	13.7114i	0.307444	+	0.532508i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−24.2487	+	14.0000i	−0.938914	+	0.542082i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.22625			0.201757
$672$		0			0
$673$	6.68629			0.257738			0.128869	−	0.991662i	$-0.458865\pi$
$673$	6.68629			0.257738			0.128869	+	0.991662i	$0.458865\pi$
$674$		0			0
$675$	4.77791	−	8.27558i	0.183902	−	0.318527i
$676$		0			0
$677$	21.1439	−	12.2074i	0.812624	−	0.469169i	−0.0352421	−	0.999379i	$-0.511220\pi$
$677$	21.1439	−	12.2074i	0.812624	−	0.469169i	0.847866	+	0.530210i	$0.177887\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	10.0416	+	17.3926i	0.384796	+	0.666486i
$682$		0			0
$683$	−37.8079	−	21.8284i	−1.44668	−	0.835242i	−0.448399	−	0.893834i	$-0.648006\pi$
$683$	−37.8079	−	21.8284i	−1.44668	−	0.835242i	−0.998282	+	0.0585921i	$0.981339\pi$
$684$		0			0
$685$		−	5.22625i		−	0.199685i
$686$		0			0
$687$		−	3.23045i		−	0.123249i
$688$		0			0
$689$	−8.27558	−	4.77791i	−0.315275	−	0.182024i
$690$		0			0
$691$	16.2200	+	28.0938i	0.617036	+	1.06874i	0.990024	+	0.140901i	$0.0450001\pi$
$691$	16.2200	+	28.0938i	0.617036	+	1.06874i	−0.372988	+	0.927836i	$0.621667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.47871	−	2.58579i	0.169887	−	0.0980845i
$696$		0			0
$697$	14.6274	−	25.3354i	0.554053	−	0.959648i
$698$		0			0
$699$	−19.4831			−0.736917
$700$		0			0
$701$	9.31371			0.351774			0.175887	−	0.984410i	$-0.443721\pi$
$701$	9.31371			0.351774			0.175887	+	0.984410i	$0.443721\pi$
$702$		0			0
$703$	−15.8645	+	27.4781i	−0.598340	+	1.03635i
$704$		0			0
$705$	18.1043	−	10.4525i	0.681846	−	0.393664i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9.00000	+	15.5885i	0.338002	+	0.585437i	0.984057	−	0.177854i	$-0.0569156\pi$
$709$	9.00000	+	15.5885i	0.338002	+	0.585437i	−0.646055	+	0.763291i	$0.723582\pi$
$710$		0			0
$711$	−20.3134	−	11.7279i	−0.761810	−	0.439831i
$712$		0			0
$713$		−	23.4412i		−	0.877880i
$714$		0			0
$715$			24.9706i			0.933846i
$716$		0			0
$717$	7.82069	+	4.51528i	0.292069	+	0.168626i
$718$		0			0
$719$	17.2095	+	29.8077i	0.641806	+	1.11164i	0.985029	+	0.172386i	$0.0551477\pi$
$719$	17.2095	+	29.8077i	0.641806	+	1.11164i	−0.343224	+	0.939254i	$0.611519\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−22.4657	+	12.9706i	−0.835507	+	0.482380i
$724$		0			0
$725$	3.34315	−	5.79050i	0.124161	−	0.215054i
$726$		0			0
$727$	−42.1814			−1.56442			−0.782211	−	0.623013i	$-0.785908\pi$
$727$	−42.1814			−1.56442			−0.782211	+	0.623013i	$0.785908\pi$
$728$		0			0
$729$	17.2843			0.640158
$730$		0			0
$731$	5.59767	−	9.69545i	0.207037	−	0.358599i
$732$		0			0
$733$	−21.1439	+	12.2074i	−0.780966	+	0.450891i	−0.836773	−	0.547551i	$-0.815560\pi$
$733$	−21.1439	+	12.2074i	−0.780966	+	0.450891i	0.0558065	+	0.998442i	$0.482227\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15.6569	+	27.1185i	0.576728	+	0.998922i
$738$		0			0
$739$	14.3998	+	8.31371i	0.529704	+	0.305825i	0.740896	−	0.671620i	$-0.234401\pi$
$739$	14.3998	+	8.31371i	0.529704	+	0.305825i	−0.211192	+	0.977445i	$0.567735\pi$
$740$		0			0
$741$		−	21.4303i		−	0.787261i
$742$		0			0
$743$			24.3431i			0.893063i	0.894768	+	0.446532i	$0.147341\pi$
$743$			24.3431i			0.893063i	−0.894768	+	0.446532i	$0.852659\pi$
$744$		0			0
$745$	−22.6303	−	13.0656i	−0.829111	−	0.478688i
$746$		0			0
$747$	−10.5454	−	18.2651i	−0.385834	−	0.668285i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	12.9649	−	7.48528i	0.473095	−	0.273142i	−0.244439	−	0.969665i	$-0.578604\pi$
$751$	12.9649	−	7.48528i	0.473095	−	0.273142i	0.717535	+	0.696523i	$0.245271\pi$
$752$		0			0
$753$	0.585786	−	1.01461i	0.0213472	−	0.0369745i
$754$		0			0
$755$	21.8017			0.793445
$756$		0			0
$757$	49.3137			1.79234			0.896169	−	0.443714i	$-0.146339\pi$
$757$	49.3137			1.79234			0.896169	+	0.443714i	$0.146339\pi$
$758$		0			0
$759$	−8.28772	+	14.3548i	−0.300825	+	0.521044i
$760$		0			0
$761$	6.72248	−	3.88123i	0.243690	−	0.140694i	−0.373182	−	0.927758i	$-0.621733\pi$
$761$	6.72248	−	3.88123i	0.243690	−	0.140694i	0.616871	+	0.787064i	$0.288400\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	7.31371	+	12.6677i	0.264428	+	0.458002i
$766$		0			0
$767$	35.0613	+	20.2426i	1.26599	+	0.730919i
$768$		0			0
$769$			13.5140i			0.487326i	0.969860	+	0.243663i	$0.0783491\pi$
$769$			13.5140i			0.487326i	−0.969860	+	0.243663i	$0.921651\pi$
$770$		0			0
$771$		−	6.62742i		−	0.238681i
$772$		0			0
$773$	−23.3403	−	13.4755i	−0.839491	−	0.484680i	0.0176001	−	0.999845i	$-0.494397\pi$
$773$	−23.3403	−	13.4755i	−0.839491	−	0.484680i	−0.857091	+	0.515165i	$0.827731\pi$
$774$		0			0
$775$	2.79884	+	4.84772i	0.100537	+	0.174135i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−34.2929	+	19.7990i	−1.22867	+	0.709372i
$780$		0			0
$781$	−8.82843	+	15.2913i	−0.315906	+	0.547165i
$782$		0			0
$783$	19.1116			0.682994
$784$		0			0
$785$	14.8284			0.529249
$786$		0			0
$787$	−2.25764	+	3.91035i	−0.0804761	+	0.139389i	−0.903455	−	0.428684i	$-0.858977\pi$
$787$	−2.25764	+	3.91035i	−0.0804761	+	0.139389i	0.822978	+	0.568073i	$0.192311\pi$
$788$		0			0
$789$	−3.88275	+	2.24171i	−0.138230	+	0.0798069i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−6.24264	−	10.8126i	−0.221683	−	0.383966i
$794$		0			0
$795$	4.89898	+	2.82843i	0.173749	+	0.100314i
$796$		0			0
$797$			41.3617i			1.46511i	0.680710	+	0.732553i	$0.261671\pi$
$797$			41.3617i			1.46511i	−0.680710	+	0.732553i	$0.738329\pi$
$798$		0			0
$799$			22.6274i			0.800500i
$800$		0			0
$801$	−3.42786	−	1.97908i	−0.121117	−	0.0699272i
$802$		0			0
$803$	12.6173	+	21.8538i	0.445254	+	0.771203i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	12.5956	−	7.27208i	0.443386	−	0.255989i
$808$		0			0
$809$	8.75736	−	15.1682i	0.307892	−	0.533285i	−0.670009	−	0.742353i	$-0.733710\pi$
$809$	8.75736	−	15.1682i	0.307892	−	0.533285i	0.977901	+	0.209068i	$0.0670430\pi$
$810$		0			0
$811$	38.5628			1.35412			0.677062	−	0.735926i	$-0.263253\pi$
$811$	38.5628			1.35412			0.677062	+	0.735926i	$0.263253\pi$
$812$		0			0
$813$	1.94113			0.0680782
$814$		0			0
$815$	−22.6215	+	39.1815i	−0.792395	+	1.37247i
$816$		0			0
$817$	−13.1233	+	7.57675i	−0.459126	+	0.265077i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	24.7990	+	42.9531i	0.865491	+	1.49907i	0.866559	+	0.499074i	$0.166327\pi$
$821$	24.7990	+	42.9531i	0.865491	+	1.49907i	−0.00106839	+	0.999999i	$0.500340\pi$
$822$		0			0
$823$	21.7482	+	12.5563i	0.758096	+	0.437687i	0.828612	−	0.559824i	$-0.189131\pi$
$823$	21.7482	+	12.5563i	0.758096	+	0.437687i	−0.0705158	+	0.997511i	$0.522465\pi$
$824$		0			0
$825$		−	3.95815i		−	0.137805i
$826$		0			0
$827$			43.2548i			1.50412i	0.659096	+	0.752059i	$0.270939\pi$
$827$			43.2548i			1.50412i	−0.659096	+	0.752059i	$0.729061\pi$
$828$		0			0
$829$	−5.36923	−	3.09993i	−0.186481	−	0.107665i	0.403853	−	0.914824i	$-0.367671\pi$
$829$	−5.36923	−	3.09993i	−0.186481	−	0.107665i	−0.590334	+	0.807159i	$0.701004\pi$
$830$		0			0
$831$	−14.5964	−	25.2816i	−0.506342	−	0.877010i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−14.6969	+	8.48528i	−0.508609	+	0.293645i
$836$		0			0
$837$	−8.00000	+	13.8564i	−0.276520	+	0.478947i
$838$		0			0
$839$	42.1814			1.45626			0.728132	−	0.685437i	$-0.240389\pi$
$839$	42.1814			1.45626			0.728132	+	0.685437i	$0.240389\pi$
$840$		0			0
$841$	−15.6274			−0.538876
$842$		0			0
$843$	−15.4930	+	26.8347i	−0.533609	+	0.924238i
$844$		0			0
$845$	22.2421	−	12.8415i	0.765150	−	0.441760i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	0.585786	+	1.01461i	0.0201041	+	0.0348214i
$850$		0			0
$851$	−50.7728	−	29.3137i	−1.74047	−	1.00486i
$852$		0			0
$853$			31.8059i			1.08901i	0.838757	+	0.544506i	$0.183283\pi$
$853$			31.8059i			1.08901i	−0.838757	+	0.544506i	$0.816717\pi$
$854$		0			0
$855$		−	19.7990i		−	0.677111i
$856$		0			0
$857$	6.07917	+	3.50981i	0.207660	+	0.119893i	0.600224	−	0.799832i	$-0.295078\pi$
$857$	6.07917	+	3.50981i	0.207660	+	0.119893i	−0.392563	+	0.919725i	$0.628412\pi$
$858$		0			0
$859$	1.25217	+	2.16882i	0.0427235	+	0.0739993i	0.886596	−	0.462544i	$-0.153063\pi$
$859$	1.25217	+	2.16882i	0.0427235	+	0.0739993i	−0.843873	+	0.536543i	$0.819730\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−23.7775	+	13.7279i	−0.809394	+	0.467304i	−0.846745	−	0.531998i	$-0.821441\pi$
$863$	−23.7775	+	13.7279i	−0.809394	+	0.467304i	0.0373513	+	0.999302i	$0.488108\pi$
$864$		0			0
$865$	14.2426	−	24.6690i	0.484264	−	0.838770i
$866$		0			0
$867$	−8.25586			−0.280384
$868$		0			0
$869$	−25.6569			−0.870349
$870$		0			0
$871$	37.4035	−	64.7848i	1.26737	−	2.19515i
$872$		0			0
$873$	21.3989	−	12.3547i	0.724243	−	0.418142i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.171573	−	0.297173i	−0.00579360	−	0.0100348i	0.863114	−	0.505009i	$-0.168511\pi$
$877$	−0.171573	−	0.297173i	−0.00579360	−	0.0100348i	−0.868908	+	0.494974i	$0.835178\pi$
$878$		0			0
$879$	−5.31925	−	3.07107i	−0.179414	−	0.103585i
$880$		0			0
$881$		−	27.7708i		−	0.935621i	−0.883829	−	0.467811i	$-0.845043\pi$
$881$		−	27.7708i		−	0.935621i	0.883829	−	0.467811i	$-0.154957\pi$
$882$		0			0
$883$			24.3431i			0.819212i	0.912263	+	0.409606i	$0.134334\pi$
$883$			24.3431i			0.819212i	−0.912263	+	0.409606i	$0.865666\pi$
$884$		0			0
$885$	−20.7556	−	11.9832i	−0.697691	−	0.402812i
$886$		0			0
$887$	−27.5851	−	47.7787i	−0.926216	−	1.60425i	−0.789594	−	0.613630i	$-0.789709\pi$
$887$	−27.5851	−	47.7787i	−0.926216	−	1.60425i	−0.136622	−	0.990623i	$-0.543625\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.297173	−	0.171573i	0.00995567	−	0.00574791i
$892$		0			0
$893$	15.3137	−	26.5241i	0.512454	−	0.887596i
$894$		0			0
$895$	−5.22625			−0.174694
$896$		0			0
$897$	39.5980			1.32214
$898$		0			0
$899$	−5.59767	+	9.69545i	−0.186693	+	0.323361i
$900$		0			0
$901$	−5.30262	+	3.06147i	−0.176656	+	0.101992i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.24264	−	3.88437i	−0.0745479	−	0.129121i
$906$		0			0
$907$	22.5167	+	13.0000i	0.747653	+	0.431658i	0.824845	−	0.565358i	$-0.191262\pi$
$907$	22.5167	+	13.0000i	0.747653	+	0.431658i	−0.0771920	+	0.997016i	$0.524595\pi$
$908$		0			0
$909$			10.3756i			0.344136i
$910$		0			0
$911$			24.3431i			0.806524i	0.915084	+	0.403262i	$0.132124\pi$
$911$			24.3431i			0.806524i	−0.915084	+	0.403262i	$0.867876\pi$
$912$		0			0
$913$	−19.9790	−	11.5349i	−0.661209	−	0.381749i
$914$		0			0
$915$	3.69552	+	6.40083i	0.122170	+	0.211605i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−25.8067	+	14.8995i	−0.851284	+	0.491489i	−0.861084	−	0.508463i	$-0.830214\pi$
$919$	−25.8067	+	14.8995i	−0.851284	+	0.491489i	0.00979995	+	0.999952i	$0.496881\pi$
$920$		0			0
$921$	−6.24264	+	10.8126i	−0.205702	+	0.356286i
$922$		0			0
$923$	42.1814			1.38842
$924$		0			0
$925$	14.0000			0.460317
$926$		0			0
$927$	−15.4930	+	26.8347i	−0.508858	+	0.881369i
$928$		0			0
$929$	−50.1084	+	28.9301i	−1.64400	+	0.949166i	−0.664613	+	0.747188i	$0.731404\pi$
$929$	−50.1084	+	28.9301i	−1.64400	+	0.949166i	−0.979390	+	0.201978i	$0.935263\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	10.8284	+	18.7554i	0.354507	+	0.614024i
$934$		0			0
$935$	13.8564	+	8.00000i	0.453153	+	0.261628i
$936$		0			0
$937$		−	44.7176i		−	1.46086i	−0.682987	−	0.730431i	$-0.739319\pi$
$937$		−	44.7176i		−	1.46086i	0.682987	−	0.730431i	$-0.260681\pi$
$938$		0			0
$939$			16.9706i			0.553813i
$940$		0			0
$941$	−21.4655	−	12.3931i	−0.699756	−	0.404004i	0.107501	−	0.994205i	$-0.465715\pi$
$941$	−21.4655	−	12.3931i	−0.699756	−	0.404004i	−0.807256	+	0.590201i	$0.799049\pi$
$942$		0			0
$943$	−36.5838	−	63.3649i	−1.19133	−	2.06345i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.5459	+	14.1716i	−0.797634	+	0.460514i	−0.842643	−	0.538472i	$-0.819002\pi$
$947$	−24.5459	+	14.1716i	−0.797634	+	0.460514i	0.0450091	+	0.998987i	$0.485668\pi$
$948$		0			0
$949$	30.1421	−	52.2077i	0.978455	−	1.69473i
$950$		0			0
$951$	−29.9356			−0.970727
$952$		0			0
$953$	0.627417			0.0203240			0.0101620	−	0.999948i	$-0.496765\pi$
$953$	0.627417			0.0203240			0.0101620	+	0.999948i	$0.496765\pi$
$954$		0			0
$955$	6.30864	−	10.9269i	0.204143	−	0.353586i
$956$		0			0
$957$	6.85572	−	3.95815i	0.221614	−	0.127949i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	10.8137	+	18.7299i	0.348829	+	0.604190i
$962$		0			0
$963$	0.543359	+	0.313708i	0.0175095	+	0.0101091i
$964$		0			0
$965$			27.3994i			0.882017i
$966$		0			0
$967$		−	17.0294i		−	0.547630i	−0.961782	−	0.273815i	$-0.911714\pi$
$967$		−	17.0294i		−	0.547630i	0.961782	−	0.273815i	$-0.0882856\pi$
$968$		0			0
$969$	−11.8919	−	6.86577i	−0.382022	−	0.220560i
$970$		0			0
$971$	−2.07193	−	3.58869i	−0.0664914	−	0.115166i	0.830863	−	0.556477i	$-0.187847\pi$
$971$	−2.07193	−	3.58869i	−0.0664914	−	0.115166i	−0.897355	+	0.441310i	$0.854514\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−8.18900	+	4.72792i	−0.262258	+	0.151415i
$976$		0			0
$977$	−8.31371	+	14.3998i	−0.265979	+	0.460689i	−0.967820	−	0.251645i	$-0.919029\pi$
$977$	−8.31371	+	14.3998i	−0.265979	+	0.460689i	0.701841	+	0.712334i	$0.252362\pi$
$978$		0			0
$979$	−4.32957			−0.138374
$980$		0			0
$981$	9.71573			0.310199
$982$		0			0
$983$	16.7611	−	29.0312i	0.534598	−	0.925950i	−0.464585	−	0.885528i	$-0.653797\pi$
$983$	16.7611	−	29.0312i	0.534598	−	0.925950i	0.999183	−	0.0404217i	$-0.0128701\pi$
$984$		0			0
$985$	−8.27558	+	4.77791i	−0.263682	+	0.152237i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14.0000	−	24.2487i	−0.445174	−	0.771064i
$990$		0			0
$991$	−12.5446	−	7.24264i	−0.398493	−	0.230070i	0.287341	−	0.957828i	$-0.407229\pi$
$991$	−12.5446	−	7.24264i	−0.398493	−	0.230070i	−0.685834	+	0.727758i	$0.740562\pi$
$992$		0			0
$993$		−	18.7402i		−	0.594703i
$994$		0			0
$995$		−	32.9706i		−	1.04524i
$996$		0			0
$997$	15.0647	+	8.69760i	0.477103	+	0.275456i	0.719209	−	0.694794i	$-0.244505\pi$
$997$	15.0647	+	8.69760i	0.477103	+	0.275456i	−0.242105	+	0.970250i	$0.577838\pi$
$998$		0			0
$999$	20.0083	+	34.6554i	0.633035	+	1.09645i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.p.a.31.5		16
4.3	odd	2	inner	1568.2.p.a.31.3		16
7.2	even	3	inner	1568.2.p.a.607.6		16
7.3	odd	6		224.2.f.a.223.6	yes	8
7.4	even	3		224.2.f.a.223.3	✓	8
7.5	odd	6	inner	1568.2.p.a.607.3		16
7.6	odd	2	inner	1568.2.p.a.31.4		16
21.11	odd	6		2016.2.b.b.1567.7		8
21.17	even	6		2016.2.b.b.1567.2		8
28.3	even	6		224.2.f.a.223.4	yes	8
28.11	odd	6		224.2.f.a.223.5	yes	8
28.19	even	6	inner	1568.2.p.a.607.5		16
28.23	odd	6	inner	1568.2.p.a.607.4		16
28.27	even	2	inner	1568.2.p.a.31.6		16
56.3	even	6		448.2.f.d.447.5		8
56.11	odd	6		448.2.f.d.447.4		8
56.45	odd	6		448.2.f.d.447.3		8
56.53	even	6		448.2.f.d.447.6		8
84.11	even	6		2016.2.b.b.1567.8		8
84.59	odd	6		2016.2.b.b.1567.1		8
112.3	even	12		1792.2.e.g.895.4		8
112.11	odd	12		1792.2.e.f.895.4		8
112.45	odd	12		1792.2.e.f.895.6		8
112.53	even	12		1792.2.e.g.895.6		8
112.59	even	12		1792.2.e.f.895.5		8
112.67	odd	12		1792.2.e.g.895.5		8
112.101	odd	12		1792.2.e.g.895.3		8
112.109	even	12		1792.2.e.f.895.3		8
168.11	even	6		4032.2.b.p.3583.2		8
168.53	odd	6		4032.2.b.p.3583.1		8
168.59	odd	6		4032.2.b.p.3583.7		8
168.101	even	6		4032.2.b.p.3583.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.f.a.223.3	✓	8	7.4	even	3
224.2.f.a.223.4	yes	8	28.3	even	6
224.2.f.a.223.5	yes	8	28.11	odd	6
224.2.f.a.223.6	yes	8	7.3	odd	6
448.2.f.d.447.3		8	56.45	odd	6
448.2.f.d.447.4		8	56.11	odd	6
448.2.f.d.447.5		8	56.3	even	6
448.2.f.d.447.6		8	56.53	even	6
1568.2.p.a.31.3		16	4.3	odd	2	inner
1568.2.p.a.31.4		16	7.6	odd	2	inner
1568.2.p.a.31.5		16	1.1	even	1	trivial
1568.2.p.a.31.6		16	28.27	even	2	inner
1568.2.p.a.607.3		16	7.5	odd	6	inner
1568.2.p.a.607.4		16	28.23	odd	6	inner
1568.2.p.a.607.5		16	28.19	even	6	inner
1568.2.p.a.607.6		16	7.2	even	3	inner
1792.2.e.f.895.3		8	112.109	even	12
1792.2.e.f.895.4		8	112.11	odd	12
1792.2.e.f.895.5		8	112.59	even	12
1792.2.e.f.895.6		8	112.45	odd	12
1792.2.e.g.895.3		8	112.101	odd	12
1792.2.e.g.895.4		8	112.3	even	12
1792.2.e.g.895.5		8	112.67	odd	12
1792.2.e.g.895.6		8	112.53	even	12
2016.2.b.b.1567.1		8	84.59	odd	6
2016.2.b.b.1567.2		8	21.17	even	6
2016.2.b.b.1567.7		8	21.11	odd	6
2016.2.b.b.1567.8		8	84.11	even	6
4032.2.b.p.3583.1		8	168.53	odd	6
4032.2.b.p.3583.2		8	168.11	even	6
4032.2.b.p.3583.7		8	168.59	odd	6
4032.2.b.p.3583.8		8	168.101	even	6

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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.541196 - 0.937379i) q^{3} +(-2.26303 + 1.30656i) q^{5} +(0.914214 + 1.58346i) q^{9} +O(q^{10})\)	\(q+(0.541196 - 0.937379i) q^{3} +(-2.26303 + 1.30656i) q^{5} +(0.914214 + 1.58346i) q^{9} +(1.73205 + 1.00000i) q^{11} -4.77791i q^{13} +2.82843i q^{15} +(-2.65131 - 1.53073i) q^{17} +(2.07193 + 3.58869i) q^{19} +(-6.63103 + 3.82843i) q^{23} +(0.914214 - 1.58346i) q^{25} +5.22625 q^{27} +3.65685 q^{29} +(-1.53073 + 2.65131i) q^{31} +(1.87476 - 1.08239i) q^{33} +(3.82843 + 6.63103i) q^{37} +(-4.47871 - 2.58579i) q^{39} +9.55582i q^{41} +3.65685i q^{43} +(-4.13779 - 2.38896i) q^{45} +(-3.69552 - 6.40083i) q^{47} +(-2.86976 + 1.65685i) q^{51} +(1.00000 - 1.73205i) q^{53} -5.22625 q^{55} +4.48528 q^{57} +(-4.23671 + 7.33820i) q^{59} +(2.26303 - 1.30656i) q^{61} +(6.24264 + 10.8126i) q^{65} +(13.5592 + 7.82843i) q^{67} +8.28772i q^{69} +8.82843i q^{71} +(10.9269 + 6.30864i) q^{73} +(-0.989538 - 1.71393i) q^{75} +(-11.1097 + 6.41421i) q^{79} +(0.0857864 - 0.148586i) q^{81} -11.5349 q^{83} +8.00000 q^{85} +(1.97908 - 3.42786i) q^{87} +(-1.87476 + 1.08239i) q^{89} +(1.65685 + 2.86976i) q^{93} +(-9.37769 - 5.41421i) q^{95} -13.5140i q^{97} +3.65685i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} - 8 q^{25} - 32 q^{29} + 16 q^{37} + 16 q^{53} - 64 q^{57} + 32 q^{65} + 24 q^{81} + 128 q^{85} - 64 q^{93}+O(q^{100}) \) 16 * q - 8 * q^9 - 8 * q^25 - 32 * q^29 + 16 * q^37 + 16 * q^53 - 64 * q^57 + 32 * q^65 + 24 * q^81 + 128 * q^85 - 64 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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