LMFDB - Embedded newform 1568.2.p.a.607.4

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			607.4
Root			$-0.793353 - 0.608761i$ of defining polynomial
Character	$\chi$	$=$	1568.607
Dual form			1568.2.p.a.31.4

$q$-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 $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$

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	$-0.267820\pi$
$3$	−0.541196	−	0.937379i	−0.312460	−	0.541196i	−0.978894	+	0.204367i	$0.934486\pi$
$4$		0			0
$5$	2.26303	+	1.30656i	1.01206	+	0.584313i	0.911794	−	0.410648i	$-0.134697\pi$
$5$	2.26303	+	1.30656i	1.01206	+	0.584313i	0.100265	+	0.994961i	$0.468031\pi$
$6$		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	−0.215615	−	0.976478i	$-0.569176\pi$
$11$	1.73205	−	1.00000i	0.522233	−	0.301511i	0.737848	+	0.674967i	$0.235842\pi$
$12$		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	$-0.545593\pi$
$17$	2.65131	−	1.53073i	0.643037	−	0.371257i	0.785783	+	0.618502i	$0.212260\pi$
$18$		0			0
$19$	−2.07193	+	3.58869i	−0.475333	+	0.823301i	−0.999601	−	0.0282522i	$-0.991006\pi$
$19$	−2.07193	+	3.58869i	−0.475333	+	0.823301i	0.524268	+	0.851554i	$0.324339\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.63103	−	3.82843i	−1.38267	−	0.798282i	−0.390191	−	0.920734i	$-0.627591\pi$
$23$	−6.63103	−	3.82843i	−1.38267	−	0.798282i	−0.992474	+	0.122452i	$0.960924\pi$
$24$		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	$-0.0780127\pi$
$31$	1.53073	+	2.65131i	0.274928	+	0.476189i	−0.695189	+	0.718827i	$0.744679\pi$
$32$		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	−0.358285	−	0.933612i	$-0.616638\pi$
$37$	3.82843	−	6.63103i	0.629390	−	1.09013i	0.987674	−	0.156522i	$-0.0500283\pi$
$38$		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	$-0.910053\pi$
$43$		−	3.65685i		−	0.557665i	0.960340	−	0.278833i	$-0.0899474\pi$
$44$		0			0
$45$	4.13779	−	2.38896i	0.616826	−	0.356124i
$46$		0			0
$47$	3.69552	−	6.40083i	0.539047	−	0.933656i	−0.459909	−	0.887966i	$-0.652118\pi$
$47$	3.69552	−	6.40083i	0.539047	−	0.933656i	0.998956	−	0.0456902i	$-0.0145487\pi$
$48$		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	0.926497	−	0.376303i	$-0.122805\pi$
$53$	1.00000	+	1.73205i	0.137361	+	0.237915i	−0.789136	+	0.614218i	$0.789471\pi$
$54$		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	0.998161	+	0.0606132i	$0.0193056\pi$
$59$	4.23671	+	7.33820i	0.551573	+	0.955353i	−0.446588	+	0.894740i	$0.647361\pi$
$60$		0			0
$61$	−2.26303	−	1.30656i	−0.289752	−	0.167288i	0.348078	−	0.937466i	$-0.386834\pi$
$61$	−2.26303	−	1.30656i	−0.289752	−	0.167288i	−0.637830	+	0.770177i	$0.720168\pi$
$62$		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	0.682223	−	0.731144i	$-0.261013\pi$
$67$	13.5592	−	7.82843i	1.65652	−	0.956395i	0.974301	−	0.225251i	$-0.0723201\pi$
$68$		0			0
$69$			8.28772i			0.997724i
$70$		0			0
$71$		−	8.82843i		−	1.04774i	−0.851798	−	0.523871i	$-0.824487\pi$
$71$		−	8.82843i		−	1.04774i	0.851798	−	0.523871i	$-0.175513\pi$
$72$		0			0
$73$	−10.9269	+	6.30864i	−1.27890	+	0.738371i	−0.976645	−	0.214858i	$-0.931071\pi$
$73$	−10.9269	+	6.30864i	−1.27890	+	0.738371i	−0.302251	+	0.953229i	$0.597738\pi$
$74$		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	−0.278846	−	0.960336i	$-0.589952\pi$
$79$	−11.1097	−	6.41421i	−1.24994	−	0.721655i	−0.971098	+	0.238681i	$0.923285\pi$
$80$		0			0
$81$	0.0857864	+	0.148586i	0.00953183	+	0.0165096i
$82$		0			0
$83$	11.5349			1.26612			0.633060	−	0.774103i	$-0.281799\pi$
$83$	11.5349			1.26612			0.633060	+	0.774103i	$0.281799\pi$
$84$		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	$-0.296732\pi$
$89$	1.87476	+	1.08239i	0.198724	+	0.114733i	−0.397336	+	0.917673i	$0.630065\pi$
$90$		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	$-0.757771\pi$
$101$	−4.91434	+	2.83730i	−0.488995	+	0.282322i	0.235162	+	0.971956i	$0.424438\pi$
$102$		0			0
$103$	−8.47343	+	14.6764i	−0.834912	+	1.44611i	0.0591906	+	0.998247i	$0.481148\pi$
$103$	−8.47343	+	14.6764i	−0.834912	+	1.44611i	−0.894102	+	0.447863i	$0.852185\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.297173	+	0.171573i	0.0287288	+	0.0165866i	0.514296	−	0.857613i	$-0.328053\pi$
$107$	0.297173	+	0.171573i	0.0287288	+	0.0165866i	−0.485567	+	0.874200i	$0.661387\pi$
$108$		0			0
$109$	2.65685	+	4.60181i	0.254480	+	0.440773i	0.964754	−	0.263152i	$-0.0847622\pi$
$109$	2.65685	+	4.60181i	0.254480	+	0.440773i	−0.710274	+	0.703926i	$0.751429\pi$
$110$		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	$-0.948126\pi$
$127$		−	3.65685i		−	0.324493i	0.986750	−	0.162247i	$-0.0518740\pi$
$128$		0			0
$129$	−3.42786	+	1.97908i	−0.301806	+	0.174248i
$130$		0			0
$131$	6.84984	−	11.8643i	0.598473	−	1.03659i	−0.394573	−	0.918864i	$-0.629108\pi$
$131$	6.84984	−	11.8643i	0.598473	−	1.03659i	0.993047	−	0.117722i	$-0.0375591\pi$
$132$		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	0.820141	−	0.572161i	$-0.193895\pi$
$137$	−1.00000	−	1.73205i	−0.0854358	−	0.147979i	−0.905577	+	0.424182i	$0.860562\pi$
$138$		0			0
$139$	1.97908			0.167863			0.0839315	−	0.996472i	$-0.473252\pi$
$139$	1.97908			0.167863			0.0839315	+	0.996472i	$0.473252\pi$
$140$		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	−0.585231	−	0.810867i	$-0.698996\pi$
$149$	5.00000	−	8.66025i	0.409616	−	0.709476i	0.994847	+	0.101391i	$0.0323294\pi$
$150$		0			0
$151$	−7.22538	+	4.17157i	−0.587993	+	0.339478i	−0.764303	−	0.644857i	$-0.776917\pi$
$151$	−7.22538	+	4.17157i	−0.587993	+	0.339478i	0.176311	+	0.984335i	$0.443584\pi$
$152$		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	$-0.593958\pi$
$157$	4.91434	−	2.83730i	0.392207	−	0.226441i	0.683116	+	0.730310i	$0.260624\pi$
$158$		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	0.954719	−	0.297508i	$-0.0961555\pi$
$163$	14.9941	+	8.65685i	1.17443	+	0.678057i	0.219710	+	0.975565i	$0.429489\pi$
$164$		0			0
$165$	−2.82843	−	4.89898i	−0.220193	−	0.381385i
$166$		0			0
$167$	−6.49435			−0.502548			−0.251274	−	0.967916i	$-0.580850\pi$
$167$	−6.49435			−0.502548			−0.251274	+	0.967916i	$0.580850\pi$
$168$		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	$-0.197329\pi$
$173$	9.44041	+	5.45042i	0.717741	+	0.414388i	−0.0961797	+	0.995364i	$0.530662\pi$
$174$		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	−0.433872	−	0.900975i	$-0.642853\pi$
$179$	1.73205	−	1.00000i	0.129460	−	0.0747435i	0.563331	+	0.826231i	$0.309520\pi$
$180$		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	0.341029	−	0.940053i	$-0.389224\pi$
$191$	−4.18154	−	2.41421i	−0.302566	−	0.174686i	−0.643595	+	0.765366i	$0.722558\pi$
$192$		0			0
$193$	5.24264	+	9.08052i	0.377374	+	0.653630i	0.990679	−	0.136215i	$-0.0434940\pi$
$193$	5.24264	+	9.08052i	0.377374	+	0.653630i	−0.613306	+	0.789846i	$0.710161\pi$
$194$		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	$-0.0190851\pi$
$199$	6.30864	+	10.9269i	0.447208	+	0.774587i	−0.550995	+	0.834508i	$0.685752\pi$
$200$		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	$0.123258\pi$
$211$			10.9706i			0.755245i	−0.925960	+	0.377622i	$0.876742\pi$
$212$		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	0.990252	+	0.139288i	$0.0444815\pi$
$227$	9.27726	+	16.0687i	0.615753	+	1.06652i	−0.374499	+	0.927227i	$0.622185\pi$
$228$		0			0
$229$	−2.58469	−	1.49227i	−0.170801	−	0.0986121i	0.412162	−	0.911110i	$-0.364774\pi$
$229$	−2.58469	−	1.49227i	−0.170801	−	0.0986121i	−0.582964	+	0.812498i	$0.698107\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	0.404674	+	0.914461i	$0.367385\pi$
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	−0.994283	+	0.106773i	$0.965948\pi$
$234$		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	$-0.913030\pi$
$239$		−	8.34315i		−	0.539673i	0.962906	−	0.269837i	$-0.0869697\pi$
$240$		0			0
$241$	20.7556	−	11.9832i	1.33698	−	0.771908i	0.350625	−	0.936516i	$-0.385969\pi$
$241$	20.7556	−	11.9832i	1.33698	−	0.771908i	0.986359	+	0.164608i	$0.0526358\pi$
$242$		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	$-0.510876\pi$
$251$	−1.08239			−0.0683200			−0.0341600	+	0.999416i	$0.510876\pi$
$252$		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	$-0.394496\pi$
$257$	−5.30262	−	3.06147i	−0.330768	−	0.190969i	−0.656182	+	0.754603i	$0.727830\pi$
$258$		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	−0.606504	−	0.795081i	$-0.707429\pi$
$263$	−3.58719	+	2.07107i	−0.221196	+	0.127708i	0.385308	+	0.922788i	$0.374095\pi$
$264$		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	$-0.801011\pi$
$269$	−11.6368	+	6.71852i	−0.709510	+	0.409636i	0.101370	+	0.994849i	$0.467677\pi$
$270$		0			0
$271$	−0.896683	+	1.55310i	−0.0544696	+	0.0943441i	−0.891975	−	0.452086i	$-0.850680\pi$
$271$	−0.896683	+	1.55310i	−0.0544696	+	0.0943441i	0.837505	+	0.546430i	$0.184013\pi$
$272$		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	0.912688	+	0.408658i	$0.134003\pi$
$277$	13.4853	+	23.3572i	0.810252	+	1.40340i	−0.102436	+	0.994740i	$0.532664\pi$
$278$		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	$-0.156425\pi$
$283$	0.541196	+	0.937379i	0.0321708	+	0.0557214i	−0.849492	+	0.527602i	$0.823091\pi$
$284$		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	$-0.393233\pi$
$307$	11.5349			0.658331			0.329166	+	0.944272i	$0.393233\pi$
$308$		0			0
$309$	18.3431			1.04351
$310$		0			0
$311$	10.0042	+	17.3277i	0.567284	+	0.982565i	0.996833	+	0.0795212i	$0.0253391\pi$
$311$	10.0042	+	17.3277i	0.567284	+	0.982565i	−0.429549	+	0.903043i	$0.641328\pi$
$312$		0			0
$313$	13.5782	+	7.83938i	0.767485	+	0.443108i	0.831977	−	0.554810i	$-0.187209\pi$
$313$	13.5782	+	7.83938i	0.767485	+	0.443108i	−0.0644915	+	0.997918i	$0.520543\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.8284	+	23.9515i	−0.776682	+	1.34525i	0.157162	+	0.987573i	$0.449765\pi$
$317$	−13.8284	+	23.9515i	−0.776682	+	1.34525i	−0.933844	+	0.357680i	$0.883568\pi$
$318$		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	0.851846	−	0.523793i	$-0.175483\pi$
$331$	14.9941	+	8.65685i	0.824151	+	0.475824i	−0.0276949	+	0.999616i	$0.508817\pi$
$332$		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	−0.881671	−	0.471864i	$-0.843581\pi$
$347$	−17.0233	+	9.82843i	−0.913861	+	0.527618i	−0.0321893	+	0.999482i	$0.510248\pi$
$348$		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	$-0.954915\pi$
$353$	−25.6033	+	14.7821i	−1.36273	+	0.786770i	−0.372740	+	0.927936i	$0.621582\pi$
$354$		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	0.801458	−	0.598051i	$-0.204058\pi$
$359$	12.9649	+	7.48528i	0.684261	+	0.395058i	−0.117198	+	0.993109i	$0.537391\pi$
$360$		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	$-0.0787142\pi$
$367$	5.22625	+	9.05213i	0.272808	+	0.472518i	−0.696772	+	0.717293i	$0.745381\pi$
$368$		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	−0.942038	−	0.335507i	$-0.891092\pi$
$373$	−3.48528	+	6.03668i	−0.180461	+	0.312568i	0.761577	+	0.648075i	$0.224426\pi$
$374$		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	$0.0435768\pi$
$379$			5.31371i			0.272947i	−0.990644	+	0.136473i	$0.956423\pi$
$380$		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	$-0.930136\pi$
$383$	−5.86030	+	10.1503i	−0.299447	+	0.518658i	0.676562	+	0.736386i	$0.263469\pi$
$384$		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	0.501677	−	0.865055i	$-0.332717\pi$
$389$	−9.82843	−	17.0233i	−0.498321	−	0.863117i	−0.999998	+	0.00193762i	$0.999383\pi$
$390$		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	$-0.658229\pi$
$397$	−29.4194	−	16.9853i	−1.47652	−	0.852469i	−0.999649	+	0.0265049i	$0.991562\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.58579	−	13.1390i	0.378816	−	0.656129i	−0.612074	−	0.790800i	$-0.709665\pi$
$401$	7.58579	−	13.1390i	0.378816	−	0.656129i	0.990890	+	0.134672i	$0.0429980\pi$
$402$		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	$-0.673724\pi$
$409$	−0.776550	+	0.448342i	−0.0383979	+	0.0221691i	0.480678	+	0.876897i	$0.340391\pi$
$410$		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	$-0.820563\pi$
$419$	−34.6047			−1.69055			−0.845275	+	0.534332i	$0.820563\pi$
$420$		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	−0.219949	−	0.975511i	$-0.570589\pi$
$431$	10.6895	−	6.17157i	0.514894	−	0.297274i	0.734843	+	0.678237i	$0.237256\pi$
$432$		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	−0.461016	−	0.887392i	$-0.652515\pi$
$439$	11.2723	−	19.5241i	0.537996	−	0.931836i	0.999012	−	0.0444443i	$-0.0141517\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−8.66025	−	5.00000i	−0.411461	−	0.237557i	0.279956	−	0.960013i	$-0.409680\pi$
$443$	−8.66025	−	5.00000i	−0.411461	−	0.237557i	−0.691417	+	0.722456i	$0.743013\pi$
$444$		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	−0.00402201	+	0.999992i	$0.501280\pi$
$457$	−18.5563	+	32.1405i	−0.868029	+	1.50347i	−0.864007	+	0.503479i	$0.832053\pi$
$458$		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	$0.122390\pi$
$463$			16.1421i			0.750189i	−0.926987	+	0.375094i	$0.877610\pi$
$464$		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	$-0.773308\pi$
$467$	4.05101	−	7.01655i	0.187458	−	0.324687i	0.944402	+	0.328793i	$0.106642\pi$
$468$		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	−0.972673	−	0.232180i	$-0.925414\pi$
$479$	−15.0447	−	26.0582i	−0.687410	−	1.19063i	0.285263	−	0.958449i	$-0.407919\pi$
$480$		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	0.230501	−	0.973072i	$-0.425964\pi$
$487$	26.2269	−	15.1421i	1.18846	−	0.686156i	0.957956	+	0.286916i	$0.0926302\pi$
$488$		0			0
$489$		−	18.7402i		−	0.847462i
$490$		0			0
$491$			30.2843i			1.36671i	0.730086	+	0.683355i	$0.239480\pi$
$491$			30.2843i			1.36671i	−0.730086	+	0.683355i	$0.760520\pi$
$492$		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	0.796542	−	0.604583i	$-0.206660\pi$
$499$	14.9941	+	8.65685i	0.671229	+	0.387534i	−0.125313	+	0.992117i	$0.539994\pi$
$500$		0			0
$501$	3.51472	+	6.08767i	0.157026	+	0.271977i
$502$		0			0
$503$	−17.4721			−0.779043			−0.389522	−	0.921017i	$-0.627360\pi$
$503$	−17.4721			−0.779043			−0.389522	+	0.921017i	$0.627360\pi$
$504$		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	$-0.529273\pi$
$509$	−22.5637	−	13.0272i	−1.00012	−	0.577419i	−0.908284	+	0.418355i	$0.862607\pi$
$510$		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	$-0.660414\pi$
$521$	0.776550	−	0.448342i	0.0340213	−	0.0196422i	0.516914	+	0.856037i	$0.327081\pi$
$522$		0			0
$523$	9.46297	−	16.3903i	0.413787	−	0.716699i	−0.581514	−	0.813537i	$-0.697539\pi$
$523$	9.46297	−	16.3903i	0.413787	−	0.716699i	0.995300	+	0.0968372i	$0.0308726\pi$
$524$		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	0.454499	+	0.890747i	$0.349818\pi$
$541$	−12.6569	+	21.9223i	−0.544161	+	0.942514i	−0.998659	+	0.0517664i	$0.983515\pi$
$542$		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	$-0.747701\pi$
$547$		−	33.3137i		−	1.42439i	0.701981	−	0.712196i	$-0.252299\pi$
$548$		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	0.916808	−	0.399329i	$-0.130757\pi$
$557$	2.65685	+	4.60181i	0.112575	+	0.194985i	−0.804233	+	0.594314i	$0.797424\pi$
$558$		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	$-0.136334\pi$
$563$	2.25764	+	3.91035i	0.0951481	+	0.164801i	−0.814522	+	0.580132i	$0.803001\pi$
$564$		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	−0.936552	−	0.350529i	$-0.886002\pi$
$569$	−3.92893	+	6.80511i	−0.164709	+	0.285285i	0.771843	+	0.635814i	$0.219335\pi$
$570$		0			0
$571$	−7.22538	+	4.17157i	−0.302373	+	0.174575i	−0.643508	−	0.765439i	$-0.722522\pi$
$571$	−7.22538	+	4.17157i	−0.302373	+	0.174575i	0.341136	+	0.940014i	$0.389188\pi$
$572$		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	$-0.933187\pi$
$577$	−30.9059	+	17.8435i	−1.28663	+	0.742836i	−0.308579	+	0.951199i	$0.599853\pi$
$578$		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	$-0.295714\pi$
$587$	29.0070			1.19725			0.598624	+	0.801030i	$0.295714\pi$
$588$		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	$-0.321610\pi$
$593$	1.55310	+	0.896683i	0.0637782	+	0.0368224i	−0.467772	+	0.883849i	$0.654943\pi$
$594$		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	0.165993	−	0.986127i	$-0.446917\pi$
$599$	26.9954	−	15.5858i	1.10300	−	0.636818i	0.937008	+	0.349309i	$0.113584\pi$
$600$		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	$-0.947649\pi$
$607$	−8.65914	+	14.9981i	−0.351464	+	0.608753i	0.635043	+	0.772477i	$0.280983\pi$
$608$		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	0.914679	−	0.404180i	$-0.132443\pi$
$613$	2.65685	+	4.60181i	0.107309	+	0.185865i	−0.807370	+	0.590045i	$0.799110\pi$
$614$		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	$-0.148260\pi$
$619$	1.43788	+	2.49048i	0.0577932	+	0.100101i	−0.835681	+	0.549215i	$0.814927\pi$
$620$		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	$0.176751\pi$
$631$			26.4853i			1.05436i	−0.849753	+	0.527181i	$0.823249\pi$
$632$		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	−0.997843	−	0.0656474i	$-0.979089\pi$
$641$	−14.0711	−	24.3718i	−0.555774	−	0.962628i	0.442069	−	0.896981i	$-0.354245\pi$
$642$		0			0
$643$	17.1326			0.675642			0.337821	−	0.941210i	$-0.390310\pi$
$643$	17.1326			0.675642			0.337821	+	0.941210i	$0.390310\pi$
$644$		0			0
$645$	−10.3431			−0.407261
$646$		0			0
$647$	−2.35049	−	4.07117i	−0.0924074	−	0.160054i	0.816116	−	0.577888i	$-0.196123\pi$
$647$	−2.35049	−	4.07117i	−0.0924074	−	0.160054i	−0.908524	+	0.417833i	$0.862790\pi$
$648$		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	−0.891109	−	0.453789i	$-0.850072\pi$
$653$	−1.34315	+	2.32640i	−0.0525614	+	0.0910389i	0.838548	+	0.544828i	$0.183405\pi$
$654$		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	$0.107618\pi$
$659$			17.0294i			0.663373i	−0.943390	+	0.331686i	$0.892382\pi$
$660$		0			0
$661$	−26.7681	+	15.4546i	−1.04116	+	0.601114i	−0.920161	−	0.391539i	$-0.871943\pi$
$661$	−26.7681	+	15.4546i	−1.04116	+	0.601114i	−0.120998	+	0.992653i	$0.538609\pi$
$662$		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	$-0.488780\pi$
$677$	−21.1439	−	12.2074i	−0.812624	−	0.469169i	−0.847866	+	0.530210i	$0.822113\pi$
$678$		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	−0.998282	−	0.0585921i	$-0.981339\pi$
$683$	−37.8079	+	21.8284i	−1.44668	+	0.835242i	−0.448399	+	0.893834i	$0.648006\pi$
$684$		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	0.372988	+	0.927836i	$0.378333\pi$
$691$	−16.2200	+	28.0938i	−0.617036	+	1.06874i	−0.990024	+	0.140901i	$0.955000\pi$
$692$		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	−0.646055	−	0.763291i	$-0.723582\pi$
$709$	9.00000	−	15.5885i	0.338002	−	0.585437i	0.984057	+	0.177854i	$0.0569156\pi$
$710$		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	0.343224	+	0.939254i	$0.388481\pi$
$719$	−17.2095	+	29.8077i	−0.641806	+	1.11164i	−0.985029	+	0.172386i	$0.944852\pi$
$720$		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	$-0.214092\pi$
$727$	42.1814			1.56442			0.782211	+	0.623013i	$0.214092\pi$
$728$		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	$-0.184440\pi$
$733$	21.1439	+	12.2074i	0.780966	+	0.450891i	−0.0558065	+	0.998442i	$0.517773\pi$
$734$		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	−0.211192	−	0.977445i	$-0.567735\pi$
$739$	14.3998	−	8.31371i	0.529704	−	0.305825i	0.740896	+	0.671620i	$0.234401\pi$
$740$		0			0
$741$		−	21.4303i		−	0.787261i
$742$		0			0
$743$		−	24.3431i		−	0.893063i	−0.894768	−	0.446532i	$-0.852659\pi$
$743$		−	24.3431i		−	0.893063i	0.894768	−	0.446532i	$-0.147341\pi$
$744$		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	0.717535	−	0.696523i	$-0.245271\pi$
$751$	12.9649	+	7.48528i	0.473095	+	0.273142i	−0.244439	+	0.969665i	$0.578604\pi$
$752$		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	$-0.378267\pi$
$761$	−6.72248	−	3.88123i	−0.243690	−	0.140694i	−0.616871	+	0.787064i	$0.711600\pi$
$762$		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	$-0.505603\pi$
$773$	23.3403	−	13.4755i	0.839491	−	0.484680i	0.857091	+	0.515165i	$0.172269\pi$
$774$		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	$-0.141023\pi$
$787$	2.25764	+	3.91035i	0.0804761	+	0.139389i	−0.822978	+	0.568073i	$0.807689\pi$
$788$		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	0.977901	−	0.209068i	$-0.0670430\pi$
$809$	8.75736	+	15.1682i	0.307892	+	0.533285i	−0.670009	+	0.742353i	$0.733710\pi$
$810$		0			0
$811$	−38.5628			−1.35412			−0.677062	−	0.735926i	$-0.736747\pi$
$811$	−38.5628			−1.35412			−0.677062	+	0.735926i	$0.736747\pi$
$812$		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	−0.00106839	−	0.999999i	$-0.500340\pi$
$821$	24.7990	−	42.9531i	0.865491	−	1.49907i	0.866559	−	0.499074i	$-0.166327\pi$
$822$		0			0
$823$	21.7482	−	12.5563i	0.758096	−	0.437687i	−0.0705158	−	0.997511i	$-0.522465\pi$
$823$	21.7482	−	12.5563i	0.758096	−	0.437687i	0.828612	+	0.559824i	$0.189131\pi$
$824$		0			0
$825$		−	3.95815i		−	0.137805i
$826$		0			0
$827$		−	43.2548i		−	1.50412i	−0.659096	−	0.752059i	$-0.729061\pi$
$827$		−	43.2548i		−	1.50412i	0.659096	−	0.752059i	$-0.270939\pi$
$828$		0			0
$829$	5.36923	−	3.09993i	0.186481	−	0.107665i	−0.403853	−	0.914824i	$-0.632329\pi$
$829$	5.36923	−	3.09993i	0.186481	−	0.107665i	0.590334	+	0.807159i	$0.298996\pi$
$830$		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	$-0.759611\pi$
$839$	−42.1814			−1.45626			−0.728132	+	0.685437i	$0.759611\pi$
$840$		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	$-0.704922\pi$
$857$	−6.07917	+	3.50981i	−0.207660	+	0.119893i	0.392563	+	0.919725i	$0.371588\pi$
$858$		0			0
$859$	−1.25217	+	2.16882i	−0.0427235	+	0.0739993i	−0.886596	−	0.462544i	$-0.846937\pi$
$859$	−1.25217	+	2.16882i	−0.0427235	+	0.0739993i	0.843873	+	0.536543i	$0.180270\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−23.7775	−	13.7279i	−0.809394	−	0.467304i	0.0373513	−	0.999302i	$-0.488108\pi$
$863$	−23.7775	−	13.7279i	−0.809394	−	0.467304i	−0.846745	+	0.531998i	$0.821441\pi$
$864$		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	−0.868908	−	0.494974i	$-0.835178\pi$
$877$	−0.171573	+	0.297173i	−0.00579360	+	0.0100348i	0.863114	+	0.505009i	$0.168511\pi$
$878$		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	$-0.865666\pi$
$883$		−	24.3431i		−	0.819212i	0.912263	−	0.409606i	$-0.134334\pi$
$884$		0			0
$885$	20.7556	−	11.9832i	0.697691	−	0.402812i
$886$		0			0
$887$	27.5851	−	47.7787i	0.926216	−	1.60425i	0.136622	−	0.990623i	$-0.456375\pi$
$887$	27.5851	−	47.7787i	0.926216	−	1.60425i	0.789594	−	0.613630i	$-0.210291\pi$
$888$		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	−0.0771920	−	0.997016i	$-0.524595\pi$
$907$	22.5167	−	13.0000i	0.747653	−	0.431658i	0.824845	+	0.565358i	$0.191262\pi$
$908$		0			0
$909$			10.3756i			0.344136i
$910$		0			0
$911$		−	24.3431i		−	0.806524i	−0.915084	−	0.403262i	$-0.867876\pi$
$911$		−	24.3431i		−	0.806524i	0.915084	−	0.403262i	$-0.132124\pi$
$912$		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	0.00979995	−	0.999952i	$-0.496881\pi$
$919$	−25.8067	−	14.8995i	−0.851284	−	0.491489i	−0.861084	+	0.508463i	$0.830214\pi$
$920$		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	0.979390	+	0.201978i	$0.0647369\pi$
$929$	50.1084	+	28.9301i	1.64400	+	0.949166i	0.664613	+	0.747188i	$0.268596\pi$
$930$		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	$-0.534285\pi$
$941$	21.4655	−	12.3931i	0.699756	−	0.404004i	0.807256	+	0.590201i	$0.200951\pi$
$942$		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	0.0450091	−	0.998987i	$-0.485668\pi$
$947$	−24.5459	−	14.1716i	−0.797634	−	0.460514i	−0.842643	+	0.538472i	$0.819002\pi$
$948$		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	$0.0882856\pi$
$967$			17.0294i			0.547630i	−0.961782	+	0.273815i	$0.911714\pi$
$968$		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	$-0.812153\pi$
$971$	2.07193	−	3.58869i	0.0664914	−	0.115166i	0.897355	+	0.441310i	$0.145486\pi$
$972$		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	0.701841	−	0.712334i	$-0.252362\pi$
$977$	−8.31371	−	14.3998i	−0.265979	−	0.460689i	−0.967820	+	0.251645i	$0.919029\pi$
$978$		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	−0.999183	−	0.0404217i	$-0.987130\pi$
$983$	−16.7611	−	29.0312i	−0.534598	−	0.925950i	0.464585	−	0.885528i	$-0.346203\pi$
$984$		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	−0.685834	−	0.727758i	$-0.740562\pi$
$991$	−12.5446	+	7.24264i	−0.398493	+	0.230070i	0.287341	+	0.957828i	$0.407229\pi$
$992$		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	$-0.755495\pi$
$997$	−15.0647	+	8.69760i	−0.477103	+	0.275456i	0.242105	+	0.970250i	$0.422162\pi$
$998$		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.607.4		16
4.3	odd	2	inner	1568.2.p.a.607.6		16
7.2	even	3		224.2.f.a.223.5	yes	8
7.3	odd	6	inner	1568.2.p.a.31.6		16
7.4	even	3	inner	1568.2.p.a.31.3		16
7.5	odd	6		224.2.f.a.223.4	yes	8
7.6	odd	2	inner	1568.2.p.a.607.5		16
21.2	odd	6		2016.2.b.b.1567.8		8
21.5	even	6		2016.2.b.b.1567.1		8
28.3	even	6	inner	1568.2.p.a.31.4		16
28.11	odd	6	inner	1568.2.p.a.31.5		16
28.19	even	6		224.2.f.a.223.6	yes	8
28.23	odd	6		224.2.f.a.223.3	✓	8
28.27	even	2	inner	1568.2.p.a.607.3		16
56.5	odd	6		448.2.f.d.447.5		8
56.19	even	6		448.2.f.d.447.3		8
56.37	even	6		448.2.f.d.447.4		8
56.51	odd	6		448.2.f.d.447.6		8
84.23	even	6		2016.2.b.b.1567.7		8
84.47	odd	6		2016.2.b.b.1567.2		8
112.5	odd	12		1792.2.e.f.895.5		8
112.19	even	12		1792.2.e.f.895.6		8
112.37	even	12		1792.2.e.f.895.4		8
112.51	odd	12		1792.2.e.f.895.3		8
112.61	odd	12		1792.2.e.g.895.4		8
112.75	even	12		1792.2.e.g.895.3		8
112.93	even	12		1792.2.e.g.895.5		8
112.107	odd	12		1792.2.e.g.895.6		8
168.5	even	6		4032.2.b.p.3583.7		8
168.107	even	6		4032.2.b.p.3583.1		8
168.131	odd	6		4032.2.b.p.3583.8		8
168.149	odd	6		4032.2.b.p.3583.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.f.a.223.3	✓	8	28.23	odd	6
224.2.f.a.223.4	yes	8	7.5	odd	6
224.2.f.a.223.5	yes	8	7.2	even	3
224.2.f.a.223.6	yes	8	28.19	even	6
448.2.f.d.447.3		8	56.19	even	6
448.2.f.d.447.4		8	56.37	even	6
448.2.f.d.447.5		8	56.5	odd	6
448.2.f.d.447.6		8	56.51	odd	6
1568.2.p.a.31.3		16	7.4	even	3	inner
1568.2.p.a.31.4		16	28.3	even	6	inner
1568.2.p.a.31.5		16	28.11	odd	6	inner
1568.2.p.a.31.6		16	7.3	odd	6	inner
1568.2.p.a.607.3		16	28.27	even	2	inner
1568.2.p.a.607.4		16	1.1	even	1	trivial
1568.2.p.a.607.5		16	7.6	odd	2	inner
1568.2.p.a.607.6		16	4.3	odd	2	inner
1792.2.e.f.895.3		8	112.51	odd	12
1792.2.e.f.895.4		8	112.37	even	12
1792.2.e.f.895.5		8	112.5	odd	12
1792.2.e.f.895.6		8	112.19	even	12
1792.2.e.g.895.3		8	112.75	even	12
1792.2.e.g.895.4		8	112.61	odd	12
1792.2.e.g.895.5		8	112.93	even	12
1792.2.e.g.895.6		8	112.107	odd	12
2016.2.b.b.1567.1		8	21.5	even	6
2016.2.b.b.1567.2		8	84.47	odd	6
2016.2.b.b.1567.7		8	84.23	even	6
2016.2.b.b.1567.8		8	21.2	odd	6
4032.2.b.p.3583.1		8	168.107	even	6
4032.2.b.p.3583.2		8	168.149	odd	6
4032.2.b.p.3583.7		8	168.5	even	6
4032.2.b.p.3583.8		8	168.131	odd	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

Related objects

Downloads

Learn more