LMFDB - Embedded newform 1568.2.b.e.785.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,2,Mod(785,1568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1568.785"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1142512.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - x^{4} + 5x^{3} - 2x^{2} - 4x + 8 $ x^6 - x^5 - x^4 + 5x^3 - 2x^2 - 4*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			785.5
Root			$1.17445 - 0.787829i$ of defining polynomial
Character	$\chi$	$=$	1568.785
Dual form			1568.2.b.e.785.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.57566i q^{3} +0.549738i q^{5} +0.517304 q^{9} -2.39075i q^{11} -3.96641i q^{13} -0.866198 q^{15} +4.21509 q^{17} +6.64154i q^{19} +2.34889 q^{23} +4.69779 q^{25} +5.54207i q^{27} -8.21720i q^{29} +0.866198 q^{31} +3.76700 q^{33} -0.265356i q^{37} +6.24970 q^{39} +6.24970 q^{41} -5.35027i q^{43} +0.284382i q^{45} +2.59859 q^{47} +6.64154i q^{51} +10.8188i q^{53} +1.31429 q^{55} -10.4648 q^{57} +3.77461i q^{59} +7.13675i q^{61} +2.18048 q^{65} +2.67513i q^{67} +3.70105i q^{69} -8.76700 q^{71} -4.66318 q^{73} +7.40210i q^{75} -0.616498 q^{79} -7.18048 q^{81} +1.09948i q^{83} +2.31720i q^{85} +12.9475 q^{87} +6.39558 q^{89} +1.36483i q^{93} -3.65111 q^{95} -12.9475 q^{97} -1.23675i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{15} - 2 q^{17} + 2 q^{23} + 4 q^{25} - 10 q^{31} - 14 q^{33} + 4 q^{39} + 4 q^{41} - 30 q^{47} + 2 q^{55} - 2 q^{57} - 8 q^{65} - 16 q^{71} - 10 q^{73} - 22 q^{79} - 22 q^{81} + 20 q^{87} - 10 q^{89}+ \cdots - 20 q^{97}+O(q^{100}) $ 6 * q + 10 * q^15 - 2 * q^17 + 2 * q^23 + 4 * q^25 - 10 * q^31 - 14 * q^33 + 4 * q^39 + 4 * q^41 - 30 * q^47 + 2 * q^55 - 2 * q^57 - 8 * q^65 - 16 * q^71 - 10 * q^73 - 22 * q^79 - 22 * q^81 + 20 * q^87 - 10 * q^89 - 34 * q^95 - 20 * q^97

Character values

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

$n$	$197$	$1471$	$1473$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.57566i	0.909706i	0.890566	+	0.454853i	$0.150308\pi$
$3$	1.57566i	0.909706i	−0.890566	+	0.454853i	$0.849692\pi$
$4$	0	0
$5$	0.549738i	0.245850i	0.992416	+	0.122925i	$0.0392275\pi$
$5$	0.549738i	0.245850i	−0.992416	+	0.122925i	$0.960773\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.517304	0.172435
$10$	0	0
$11$	− 2.39075i	− 0.720839i	−0.932790	−	0.360419i	$-0.882634\pi$
$11$	− 2.39075i	− 0.720839i	0.932790	−	0.360419i	$-0.117366\pi$
$12$	0	0
$13$	− 3.96641i	− 1.10008i	−0.835137	−	0.550042i	$-0.814612\pi$
$13$	− 3.96641i	− 1.10008i	0.835137	−	0.550042i	$-0.185388\pi$
$14$	0	0
$15$	−0.866198	−0.223651
$16$	0	0
$17$	4.21509	1.02231	0.511155	−	0.859489i	$-0.329218\pi$
$17$	4.21509	1.02231	0.511155	+	0.859489i	$0.329218\pi$
$18$	0	0
$19$	6.64154i	1.52367i	0.647769	+	0.761837i	$0.275702\pi$
$19$	6.64154i	1.52367i	−0.647769	+	0.761837i	$0.724298\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.34889	0.489778	0.244889	−	0.969551i	$-0.421248\pi$
$23$	2.34889	0.489778	0.244889	+	0.969551i	$0.421248\pi$
$24$	0	0
$25$	4.69779	0.939558
$26$	0	0
$27$	5.54207i	1.06657i
$28$	0	0
$29$	− 8.21720i	− 1.52590i	−0.646460	−	0.762948i	$-0.723751\pi$
$29$	− 8.21720i	− 1.52590i	0.646460	−	0.762948i	$-0.276249\pi$
$30$	0	0
$31$	0.866198	0.155574	0.0777869	−	0.996970i	$-0.475215\pi$
$31$	0.866198	0.155574	0.0777869	+	0.996970i	$0.475215\pi$
$32$	0	0
$33$	3.76700	0.655751
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.265356i	− 0.0436243i	−0.999762	−	0.0218121i	$-0.993056\pi$
$37$	− 0.265356i	− 0.0436243i	0.999762	−	0.0218121i	$-0.00694357\pi$
$38$	0	0
$39$	6.24970	1.00075
$40$	0	0
$41$	6.24970	0.976039	0.488020	−	0.872833i	$-0.337719\pi$
$41$	6.24970	0.976039	0.488020	+	0.872833i	$0.337719\pi$
$42$	0	0
$43$	− 5.35027i	− 0.815908i	−0.913003	−	0.407954i	$-0.866242\pi$
$43$	− 5.35027i	− 0.815908i	0.913003	−	0.407954i	$-0.133758\pi$
$44$	0	0
$45$	0.284382i	0.0423931i
$46$	0	0
$47$	2.59859	0.379044	0.189522	−	0.981876i	$-0.439306\pi$
$47$	2.59859	0.379044	0.189522	+	0.981876i	$0.439306\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.64154i	0.930002i
$52$	0	0
$53$	10.8188i	1.48607i	0.669251	+	0.743037i	$0.266615\pi$
$53$	10.8188i	1.48607i	−0.669251	+	0.743037i	$0.733385\pi$
$54$	0	0
$55$	1.31429	0.177218
$56$	0	0
$57$	−10.4648	−1.38610
$58$	0	0
$59$	3.77461i	0.491412i	0.969344	+	0.245706i	$0.0790198\pi$
$59$	3.77461i	0.491412i	−0.969344	+	0.245706i	$0.920980\pi$
$60$	0	0
$61$	7.13675i	0.913767i	0.889527	+	0.456884i	$0.151034\pi$
$61$	7.13675i	0.913767i	−0.889527	+	0.456884i	$0.848966\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.18048	0.270456
$66$	0	0
$67$	2.67513i	0.326819i	0.986558	+	0.163410i	$0.0522493\pi$
$67$	2.67513i	0.326819i	−0.986558	+	0.163410i	$0.947751\pi$
$68$	0	0
$69$	3.70105i	0.445554i
$70$	0	0
$71$	−8.76700	−1.04045	−0.520226	−	0.854029i	$-0.674152\pi$
$71$	−8.76700	−1.04045	−0.520226	+	0.854029i	$0.674152\pi$
$72$	0	0
$73$	−4.66318	−0.545784	−0.272892	−	0.962045i	$-0.587980\pi$
$73$	−4.66318	−0.545784	−0.272892	+	0.962045i	$0.587980\pi$
$74$	0	0
$75$	7.40210i	0.854721i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.616498	−0.0693614	−0.0346807	−	0.999398i	$-0.511041\pi$
$79$	−0.616498	−0.0693614	−0.0346807	+	0.999398i	$0.511041\pi$
$80$	0	0
$81$	−7.18048	−0.797832
$82$	0	0
$83$	1.09948i	0.120683i	0.998178	+	0.0603416i	$0.0192190\pi$
$83$	1.09948i	0.120683i	−0.998178	+	0.0603416i	$0.980781\pi$
$84$	0	0
$85$	2.31720i	0.251335i
$86$	0	0
$87$	12.9475	1.38812
$88$	0	0
$89$	6.39558	0.677930	0.338965	−	0.940799i	$-0.389923\pi$
$89$	6.39558	0.677930	0.338965	+	0.940799i	$0.389923\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.36483i	0.141526i
$94$	0	0
$95$	−3.65111	−0.374596
$96$	0	0
$97$	−12.9475	−1.31462	−0.657309	−	0.753621i	$-0.728305\pi$
$97$	−12.9475	−1.31462	−0.657309	+	0.753621i	$0.728305\pi$
$98$	0	0
$99$	− 1.23675i	− 0.124298i
$100$	0	0
$101$	15.8847i	1.58058i	0.612731	+	0.790291i	$0.290071\pi$
$101$	15.8847i	1.58058i	−0.612731	+	0.790291i	$0.709929\pi$
$102$	0	0
$103$	18.7791	1.85036	0.925179	−	0.379531i	$-0.123915\pi$
$103$	18.7791	1.85036	0.925179	+	0.379531i	$0.123915\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.338912i	− 0.0327639i	−0.999866	−	0.0163819i	$-0.994785\pi$
$107$	− 0.338912i	− 0.0327639i	0.999866	−	0.0163819i	$-0.00521476\pi$
$108$	0	0
$109$	− 7.80473i	− 0.747558i	−0.927518	−	0.373779i	$-0.878062\pi$
$109$	− 7.80473i	− 0.747558i	0.927518	−	0.373779i	$-0.121938\pi$
$110$	0	0
$111$	0.418110	0.0396853
$112$	0	0
$113$	2.51730	0.236808	0.118404	−	0.992966i	$-0.462222\pi$
$113$	2.51730	0.236808	0.118404	+	0.992966i	$0.462222\pi$
$114$	0	0
$115$	1.29128i	0.120412i
$116$	0	0
$117$	− 2.05184i	− 0.189693i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.28431	0.480392
$122$	0	0
$123$	9.84739i	0.887909i
$124$	0	0
$125$	5.33124i	0.476841i
$126$	0	0
$127$	5.30221	0.470495	0.235248	−	0.971935i	$-0.424410\pi$
$127$	5.30221	0.470495	0.235248	+	0.971935i	$0.424410\pi$
$128$	0	0
$129$	8.43018	0.742236
$130$	0	0
$131$	− 10.0392i	− 0.877128i	−0.898700	−	0.438564i	$-0.855487\pi$
$131$	− 10.0392i	− 0.877128i	0.898700	−	0.438564i	$-0.144513\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.04668	−0.262217
$136$	0	0
$137$	3.24970	0.277641	0.138820	−	0.990318i	$-0.455669\pi$
$137$	3.24970	0.277641	0.138820	+	0.990318i	$0.455669\pi$
$138$	0	0
$139$	− 15.3349i	− 1.30069i	−0.759639	−	0.650346i	$-0.774624\pi$
$139$	− 15.3349i	− 1.30069i	0.759639	−	0.650346i	$-0.225376\pi$
$140$	0	0
$141$	4.09449i	0.344819i
$142$	0	0
$143$	−9.48270	−0.792983
$144$	0	0
$145$	4.51730	0.375142
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.38308i	0.604845i	0.953174	+	0.302423i	$0.0977954\pi$
$149$	7.38308i	0.604845i	−0.953174	+	0.302423i	$0.902205\pi$
$150$	0	0
$151$	−8.33099	−0.677966	−0.338983	−	0.940792i	$-0.610083\pi$
$151$	−8.33099	−0.677966	−0.338983	+	0.940792i	$0.610083\pi$
$152$	0	0
$153$	2.18048	0.176282
$154$	0	0
$155$	0.476182i	0.0382478i
$156$	0	0
$157$	− 7.13675i	− 0.569575i	−0.958591	−	0.284787i	$-0.908077\pi$
$157$	− 7.13675i	− 0.569575i	0.958591	−	0.284787i	$-0.0919230\pi$
$158$	0	0
$159$	−17.0467	−1.35189
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 7.30952i	− 0.572526i	−0.958151	−	0.286263i	$-0.907587\pi$
$163$	− 7.30952i	− 0.572526i	0.958151	−	0.286263i	$-0.0924131\pi$
$164$	0	0
$165$	2.07086i	0.161217i
$166$	0	0
$167$	1.88873	0.146154	0.0730772	−	0.997326i	$-0.476718\pi$
$167$	1.88873	0.146154	0.0730772	+	0.997326i	$0.476718\pi$
$168$	0	0
$169$	−2.73240	−0.210184
$170$	0	0
$171$	3.43570i	0.262734i
$172$	0	0
$173$	16.5526i	1.25847i	0.777213	+	0.629237i	$0.216633\pi$
$173$	16.5526i	1.25847i	−0.777213	+	0.629237i	$0.783367\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.94749	−0.447041
$178$	0	0
$179$	5.54207i	0.414233i	0.978316	+	0.207117i	$0.0664080\pi$
$179$	5.54207i	0.414233i	−0.978316	+	0.207117i	$0.933592\pi$
$180$	0	0
$181$	9.98466i	0.742154i	0.928602	+	0.371077i	$0.121011\pi$
$181$	9.98466i	0.742154i	−0.928602	+	0.371077i	$0.878989\pi$
$182$	0	0
$183$	−11.2451	−0.831259
$184$	0	0
$185$	0.145876	0.0107250
$186$	0	0
$187$	− 10.0772i	− 0.736921i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.168410	−0.0121857	−0.00609285	−	0.999981i	$-0.501939\pi$
$191$	−0.168410	−0.0121857	−0.00609285	+	0.999981i	$0.501939\pi$
$192$	0	0
$193$	7.51730	0.541107	0.270554	−	0.962705i	$-0.412793\pi$
$193$	7.51730	0.541107	0.270554	+	0.962705i	$0.412793\pi$
$194$	0	0
$195$	3.43570i	0.246035i
$196$	0	0
$197$	− 1.34581i	− 0.0958847i	−0.998850	−	0.0479424i	$-0.984734\pi$
$197$	− 1.34581i	− 0.0958847i	0.998850	−	0.0479424i	$-0.0152664\pi$
$198$	0	0
$199$	−12.7612	−0.904616	−0.452308	−	0.891862i	$-0.649399\pi$
$199$	−12.7612	−0.904616	−0.452308	+	0.891862i	$0.649399\pi$
$200$	0	0
$201$	−4.21509	−0.297310
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.43570i	0.239959i
$206$	0	0
$207$	1.21509	0.0844548
$208$	0	0
$209$	15.8783	1.09832
$210$	0	0
$211$	− 8.46353i	− 0.582653i	−0.956624	−	0.291327i	$-0.905903\pi$
$211$	− 8.46353i	− 0.582653i	0.956624	−	0.291327i	$-0.0940967\pi$
$212$	0	0
$213$	− 13.8138i	− 0.946506i
$214$	0	0
$215$	2.94124	0.200591
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 7.34757i	− 0.496503i
$220$	0	0
$221$	− 16.7188i	− 1.12463i
$222$	0	0
$223$	−5.80161	−0.388505	−0.194252	−	0.980952i	$-0.562228\pi$
$223$	−5.80161	−0.388505	−0.194252	+	0.980952i	$0.562228\pi$
$224$	0	0
$225$	2.43018	0.162012
$226$	0	0
$227$	− 11.4230i	− 0.758174i	−0.925361	−	0.379087i	$-0.876238\pi$
$227$	− 11.4230i	− 0.758174i	0.925361	−	0.379087i	$-0.123762\pi$
$228$	0	0
$229$	− 18.5053i	− 1.22286i	−0.791298	−	0.611431i	$-0.790594\pi$
$229$	− 18.5053i	− 1.22286i	0.791298	−	0.611431i	$-0.209406\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.0513	−0.723996	−0.361998	−	0.932179i	$-0.617905\pi$
$233$	−11.0513	−0.723996	−0.361998	+	0.932179i	$0.617905\pi$
$234$	0	0
$235$	1.42855i	0.0931880i
$236$	0	0
$237$	− 0.971389i	− 0.0630985i
$238$	0	0
$239$	22.6107	1.46256	0.731281	−	0.682076i	$-0.238923\pi$
$239$	22.6107	1.46256	0.731281	+	0.682076i	$0.238923\pi$
$240$	0	0
$241$	−13.9296	−0.897284	−0.448642	−	0.893712i	$-0.648092\pi$
$241$	−13.9296	−0.897284	−0.448642	+	0.893712i	$0.648092\pi$
$242$	0	0
$243$	5.31221i	0.340779i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.3431	1.67617
$248$	0	0
$249$	−1.73240	−0.109786
$250$	0	0
$251$	0.706033i	0.0445644i	0.999752	+	0.0222822i	$0.00709323\pi$
$251$	0.706033i	0.0445644i	−0.999752	+	0.0222822i	$0.992907\pi$
$252$	0	0
$253$	− 5.61562i	− 0.353051i
$254$	0	0
$255$	−3.65111	−0.228641
$256$	0	0
$257$	−20.7837	−1.29645	−0.648226	−	0.761448i	$-0.724489\pi$
$257$	−20.7837	−1.29645	−0.648226	+	0.761448i	$0.724489\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 4.25079i	− 0.263117i
$262$	0	0
$263$	22.7791	1.40462	0.702309	−	0.711872i	$-0.252152\pi$
$263$	22.7791	1.40462	0.702309	+	0.711872i	$0.252152\pi$
$264$	0	0
$265$	−5.94749	−0.365351
$266$	0	0
$267$	10.0772i	0.616717i
$268$	0	0
$269$	2.99502i	0.182610i	0.995823	+	0.0913048i	$0.0291037\pi$
$269$	2.99502i	0.182610i	−0.995823	+	0.0913048i	$0.970896\pi$
$270$	0	0
$271$	−25.1851	−1.52989	−0.764943	−	0.644098i	$-0.777233\pi$
$271$	−25.1851	−1.52989	−0.764943	+	0.644098i	$0.777233\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 11.2312i	− 0.677269i
$276$	0	0
$277$	− 9.44476i	− 0.567481i	−0.958901	−	0.283740i	$-0.908425\pi$
$277$	− 9.44476i	− 0.567481i	0.958901	−	0.283740i	$-0.0915754\pi$
$278$	0	0
$279$	0.448088	0.0268263
$280$	0	0
$281$	26.8425	1.60129	0.800644	−	0.599141i	$-0.204491\pi$
$281$	26.8425	1.60129	0.800644	+	0.599141i	$0.204491\pi$
$282$	0	0
$283$	− 12.9822i	− 0.771713i	−0.922559	−	0.385856i	$-0.873906\pi$
$283$	− 12.9822i	− 0.771713i	0.922559	−	0.385856i	$-0.126094\pi$
$284$	0	0
$285$	− 5.75289i	− 0.340772i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0.767005	0.0451179
$290$	0	0
$291$	− 20.4008i	− 1.19592i
$292$	0	0
$293$	− 9.56300i	− 0.558677i	−0.960193	−	0.279338i	$-0.909885\pi$
$293$	− 9.56300i	− 0.558677i	0.960193	−	0.279338i	$-0.0901151\pi$
$294$	0	0
$295$	−2.07504	−0.120814
$296$	0	0
$297$	13.2497	0.768826
$298$	0	0
$299$	− 9.31667i	− 0.538797i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−25.0288	−1.43787
$304$	0	0
$305$	−3.92334	−0.224650
$306$	0	0
$307$	− 12.2217i	− 0.697527i	−0.937211	−	0.348763i	$-0.886602\pi$
$307$	− 12.2217i	− 0.697527i	0.937211	−	0.348763i	$-0.113398\pi$
$308$	0	0
$309$	29.5894i	1.68328i
$310$	0	0
$311$	31.8829	1.80791	0.903957	−	0.427624i	$-0.140649\pi$
$311$	31.8829	1.80791	0.903957	+	0.427624i	$0.140649\pi$
$312$	0	0
$313$	−21.5236	−1.21658	−0.608291	−	0.793714i	$-0.708145\pi$
$313$	−21.5236	−1.21658	−0.608291	+	0.793714i	$0.708145\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.1495i	− 1.30021i	−0.759846	−	0.650103i	$-0.774726\pi$
$317$	− 23.1495i	− 1.30021i	0.759846	−	0.650103i	$-0.225274\pi$
$318$	0	0
$319$	−19.6453	−1.09992
$320$	0	0
$321$	0.534009	0.0298055
$322$	0	0
$323$	27.9947i	1.55767i
$324$	0	0
$325$	− 18.6333i	− 1.03359i
$326$	0	0
$327$	12.2976	0.680058
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.5158i	1.62234i	0.584812	+	0.811169i	$0.301168\pi$
$331$	29.5158i	1.62234i	−0.584812	+	0.811169i	$0.698832\pi$
$332$	0	0
$333$	− 0.137270i	− 0.00752234i
$334$	0	0
$335$	−1.47062	−0.0803486
$336$	0	0
$337$	−3.28431	−0.178908	−0.0894538	−	0.995991i	$-0.528512\pi$
$337$	−3.28431	−0.178908	−0.0894538	+	0.995991i	$0.528512\pi$
$338$	0	0
$339$	3.96641i	0.215426i
$340$	0	0
$341$	− 2.07086i	− 0.112144i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.03461	−0.109540
$346$	0	0
$347$	− 31.6767i	− 1.70050i	−0.526382	−	0.850248i	$-0.676452\pi$
$347$	− 31.6767i	− 1.70050i	0.526382	−	0.850248i	$-0.323548\pi$
$348$	0	0
$349$	− 28.4807i	− 1.52454i	−0.647260	−	0.762269i	$-0.724085\pi$
$349$	− 28.4807i	− 1.52454i	0.647260	−	0.762269i	$-0.275915\pi$
$350$	0	0
$351$	21.9821	1.17332
$352$	0	0
$353$	−20.2392	−1.07723	−0.538613	−	0.842553i	$-0.681052\pi$
$353$	−20.2392	−1.07723	−0.538613	+	0.842553i	$0.681052\pi$
$354$	0	0
$355$	− 4.81955i	− 0.255795i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−25.1221	−1.32590	−0.662948	−	0.748665i	$-0.730695\pi$
$359$	−25.1221	−1.32590	−0.662948	+	0.748665i	$0.730695\pi$
$360$	0	0
$361$	−25.1101	−1.32158
$362$	0	0
$363$	8.32626i	0.437015i
$364$	0	0
$365$	− 2.56353i	− 0.134181i
$366$	0	0
$367$	−30.3823	−1.58594	−0.792972	−	0.609258i	$-0.791467\pi$
$367$	−30.3823	−1.58594	−0.792972	+	0.609258i	$0.791467\pi$
$368$	0	0
$369$	3.23300	0.168303
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5.61562i	0.290766i	0.989375	+	0.145383i	$0.0464414\pi$
$373$	5.61562i	0.290766i	−0.989375	+	0.145383i	$0.953559\pi$
$374$	0	0
$375$	−8.40021	−0.433785
$376$	0	0
$377$	−32.5928	−1.67861
$378$	0	0
$379$	− 8.07009i	− 0.414533i	−0.978285	−	0.207266i	$-0.933543\pi$
$379$	− 8.07009i	− 0.414533i	0.978285	−	0.207266i	$-0.0664566\pi$
$380$	0	0
$381$	8.35447i	0.428012i
$382$	0	0
$383$	−25.6332	−1.30980	−0.654898	−	0.755718i	$-0.727288\pi$
$383$	−25.6332	−1.30980	−0.654898	+	0.755718i	$0.727288\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.76771i	− 0.140691i
$388$	0	0
$389$	26.5471i	1.34599i	0.739645	+	0.672997i	$0.234993\pi$
$389$	26.5471i	1.34599i	−0.739645	+	0.672997i	$0.765007\pi$
$390$	0	0
$391$	9.90081	0.500705
$392$	0	0
$393$	15.8183	0.797929
$394$	0	0
$395$	− 0.338912i	− 0.0170525i
$396$	0	0
$397$	− 34.0964i	− 1.71125i	−0.517599	−	0.855624i	$-0.673174\pi$
$397$	− 34.0964i	− 1.71125i	0.517599	−	0.855624i	$-0.326826\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	18.3535	0.916531	0.458266	−	0.888815i	$-0.348471\pi$
$401$	18.3535	0.916531	0.458266	+	0.888815i	$0.348471\pi$
$402$	0	0
$403$	− 3.43570i	− 0.171144i
$404$	0	0
$405$	− 3.94738i	− 0.196147i
$406$	0	0
$407$	−0.634401	−0.0314461
$408$	0	0
$409$	−11.7491	−0.580956	−0.290478	−	0.956882i	$-0.593814\pi$
$409$	−11.7491	−0.580956	−0.290478	+	0.956882i	$0.593814\pi$
$410$	0	0
$411$	5.12041i	0.252571i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.604423	−0.0296700
$416$	0	0
$417$	24.1626	1.18325
$418$	0	0
$419$	11.0841i	0.541495i	0.962650	+	0.270748i	$0.0872709\pi$
$419$	11.0841i	0.541495i	−0.962650	+	0.270748i	$0.912729\pi$
$420$	0	0
$421$	− 0.137270i	− 0.00669012i	−0.999994	−	0.00334506i	$-0.998935\pi$
$421$	− 0.137270i	− 0.00669012i	0.999994	−	0.00334506i	$-0.00106477\pi$
$422$	0	0
$423$	1.34426	0.0653603
$424$	0	0
$425$	19.8016	0.960519
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 14.9415i	− 0.721381i
$430$	0	0
$431$	−12.4602	−0.600185	−0.300092	−	0.953910i	$-0.597018\pi$
$431$	−12.4602	−0.600185	−0.300092	+	0.953910i	$0.597018\pi$
$432$	0	0
$433$	14.1563	0.680310	0.340155	−	0.940369i	$-0.389520\pi$
$433$	14.1563	0.680310	0.340155	+	0.940369i	$0.389520\pi$
$434$	0	0
$435$	7.11772i	0.341269i
$436$	0	0
$437$	15.6003i	0.746262i
$438$	0	0
$439$	−5.45896	−0.260542	−0.130271	−	0.991478i	$-0.541585\pi$
$439$	−5.45896	−0.260542	−0.130271	+	0.991478i	$0.541585\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 33.3351i	− 1.58380i	−0.610651	−	0.791900i	$-0.709092\pi$
$443$	− 33.3351i	− 1.58380i	0.610651	−	0.791900i	$-0.290908\pi$
$444$	0	0
$445$	3.51589i	0.166669i
$446$	0	0
$447$	−11.6332	−0.550232
$448$	0	0
$449$	−26.9716	−1.27287	−0.636435	−	0.771330i	$-0.719592\pi$
$449$	−26.9716	−1.27287	−0.636435	+	0.771330i	$0.719592\pi$
$450$	0	0
$451$	− 14.9415i	− 0.703567i
$452$	0	0
$453$	− 13.1268i	− 0.616750i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.0934	0.893150	0.446575	−	0.894746i	$-0.352644\pi$
$457$	19.0934	0.893150	0.446575	+	0.894746i	$0.352644\pi$
$458$	0	0
$459$	23.3603i	1.09037i
$460$	0	0
$461$	18.9177i	0.881087i	0.897731	+	0.440543i	$0.145214\pi$
$461$	18.9177i	0.881087i	−0.897731	+	0.440543i	$0.854786\pi$
$462$	0	0
$463$	−0.860370	−0.0399848	−0.0199924	−	0.999800i	$-0.506364\pi$
$463$	−0.860370	−0.0399848	−0.0199924	+	0.999800i	$0.506364\pi$
$464$	0	0
$465$	−0.750299	−0.0347943
$466$	0	0
$467$	− 18.6879i	− 0.864772i	−0.901689	−	0.432386i	$-0.857672\pi$
$467$	− 18.6879i	− 0.864772i	0.901689	−	0.432386i	$-0.142328\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	11.2451	0.518145
$472$	0	0
$473$	−12.7912	−0.588138
$474$	0	0
$475$	31.2006i	1.43158i
$476$	0	0
$477$	5.59660i	0.256251i
$478$	0	0
$479$	−18.5473	−0.847447	−0.423723	−	0.905792i	$-0.639277\pi$
$479$	−18.5473	−0.847447	−0.423723	+	0.905792i	$0.639277\pi$
$480$	0	0
$481$	−1.05251	−0.0479904
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 7.11772i	− 0.323199i
$486$	0	0
$487$	22.9175	1.03849	0.519246	−	0.854625i	$-0.326213\pi$
$487$	22.9175	1.03849	0.519246	+	0.854625i	$0.326213\pi$
$488$	0	0
$489$	11.5173	0.520830
$490$	0	0
$491$	24.7987i	1.11915i	0.828780	+	0.559575i	$0.189036\pi$
$491$	24.7987i	1.11915i	−0.828780	+	0.559575i	$0.810964\pi$
$492$	0	0
$493$	− 34.6363i	− 1.55994i
$494$	0	0
$495$	0.679886	0.0305586
$496$	0	0
$497$	0	0
$498$	0	0
$499$	39.2260i	1.75599i	0.478665	+	0.877997i	$0.341121\pi$
$499$	39.2260i	1.75599i	−0.478665	+	0.877997i	$0.658879\pi$
$500$	0	0
$501$	2.97599i	0.132958i
$502$	0	0
$503$	7.59396	0.338598	0.169299	−	0.985565i	$-0.445850\pi$
$503$	7.59396	0.338598	0.169299	+	0.985565i	$0.445850\pi$
$504$	0	0
$505$	−8.73240	−0.388587
$506$	0	0
$507$	− 4.30532i	− 0.191206i
$508$	0	0
$509$	4.37888i	0.194090i	0.995280	+	0.0970451i	$0.0309391\pi$
$509$	4.37888i	0.194090i	−0.995280	+	0.0970451i	$0.969061\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−36.8079	−1.62511
$514$	0	0
$515$	10.3236i	0.454911i
$516$	0	0
$517$	− 6.21259i	− 0.273230i
$518$	0	0
$519$	−26.0813	−1.14484
$520$	0	0
$521$	10.7565	0.471253	0.235626	−	0.971844i	$-0.424286\pi$
$521$	10.7565	0.471253	0.235626	+	0.971844i	$0.424286\pi$
$522$	0	0
$523$	2.81240i	0.122978i	0.998108	+	0.0614889i	$0.0195849\pi$
$523$	2.81240i	0.122978i	−0.998108	+	0.0614889i	$0.980415\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.65111	0.159045
$528$	0	0
$529$	−17.4827	−0.760117
$530$	0	0
$531$	1.95262i	0.0847365i
$532$	0	0
$533$	− 24.7889i	− 1.07372i
$534$	0	0
$535$	0.186313	0.00805500
$536$	0	0
$537$	−8.73240	−0.376831
$538$	0	0
$539$	0	0
$540$	0	0
$541$	31.2576i	1.34387i	0.740610	+	0.671935i	$0.234537\pi$
$541$	31.2576i	1.34387i	−0.740610	+	0.671935i	$0.765463\pi$
$542$	0	0
$543$	−15.7324	−0.675142
$544$	0	0
$545$	4.29055	0.183787
$546$	0	0
$547$	− 29.4711i	− 1.26010i	−0.776556	−	0.630048i	$-0.783035\pi$
$547$	− 29.4711i	− 1.26010i	0.776556	−	0.630048i	$-0.216965\pi$
$548$	0	0
$549$	3.69187i	0.157565i
$550$	0	0
$551$	54.5749	2.32497
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.229851i	0.00975663i
$556$	0	0
$557$	22.3725i	0.947951i	0.880538	+	0.473976i	$0.157182\pi$
$557$	22.3725i	0.947951i	−0.880538	+	0.473976i	$0.842818\pi$
$558$	0	0
$559$	−21.2213	−0.897567
$560$	0	0
$561$	15.8783	0.670381
$562$	0	0
$563$	32.3926i	1.36519i	0.730799	+	0.682593i	$0.239148\pi$
$563$	32.3926i	1.36519i	−0.730799	+	0.682593i	$0.760852\pi$
$564$	0	0
$565$	1.38386i	0.0582193i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−37.0576	−1.55353	−0.776767	−	0.629788i	$-0.783142\pi$
$569$	−37.0576	−1.55353	−0.776767	+	0.629788i	$0.783142\pi$
$570$	0	0
$571$	− 22.4362i	− 0.938924i	−0.882953	−	0.469462i	$-0.844448\pi$
$571$	− 22.4362i	− 0.938924i	0.882953	−	0.469462i	$-0.155552\pi$
$572$	0	0
$573$	− 0.265356i	− 0.0110854i
$574$	0	0
$575$	11.0346	0.460175
$576$	0	0
$577$	−9.56862	−0.398347	−0.199173	−	0.979964i	$-0.563826\pi$
$577$	−9.56862	−0.398347	−0.199173	+	0.979964i	$0.563826\pi$
$578$	0	0
$579$	11.8447i	0.492249i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	25.8650	1.07122
$584$	0	0
$585$	1.12797	0.0466360
$586$	0	0
$587$	− 23.6894i	− 0.977766i	−0.872349	−	0.488883i	$-0.837405\pi$
$587$	− 23.6894i	− 0.977766i	0.872349	−	0.488883i	$-0.162595\pi$
$588$	0	0
$589$	5.75289i	0.237044i
$590$	0	0
$591$	2.12053	0.0872269
$592$	0	0
$593$	7.44809	0.305856	0.152928	−	0.988237i	$-0.451130\pi$
$593$	7.44809	0.305856	0.152928	+	0.988237i	$0.451130\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 20.1072i	− 0.822935i
$598$	0	0
$599$	−1.67525	−0.0684491	−0.0342245	−	0.999414i	$-0.510896\pi$
$599$	−1.67525	−0.0684491	−0.0342245	+	0.999414i	$0.510896\pi$
$600$	0	0
$601$	8.27385	0.337497	0.168749	−	0.985659i	$-0.446027\pi$
$601$	8.27385	0.337497	0.168749	+	0.985659i	$0.446027\pi$
$602$	0	0
$603$	1.38386i	0.0563550i
$604$	0	0
$605$	2.90498i	0.118104i
$606$	0	0
$607$	−26.5294	−1.07679	−0.538397	−	0.842691i	$-0.680970\pi$
$607$	−26.5294	−1.07679	−0.538397	+	0.842691i	$0.680970\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 10.3071i	− 0.416980i
$612$	0	0
$613$	− 31.6032i	− 1.27644i	−0.769854	−	0.638220i	$-0.779671\pi$
$613$	− 31.6032i	− 1.27644i	0.769854	−	0.638220i	$-0.220329\pi$
$614$	0	0
$615$	−5.41348	−0.218293
$616$	0	0
$617$	10.1113	0.407064	0.203532	−	0.979068i	$-0.434758\pi$
$617$	10.1113	0.407064	0.203532	+	0.979068i	$0.434758\pi$
$618$	0	0
$619$	5.53222i	0.222359i	0.993800	+	0.111179i	$0.0354628\pi$
$619$	5.53222i	0.222359i	−0.993800	+	0.111179i	$0.964537\pi$
$620$	0	0
$621$	13.0177i	0.522383i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	20.5582	0.822326
$626$	0	0
$627$	25.0187i	0.999151i
$628$	0	0
$629$	− 1.11850i	− 0.0445975i
$630$	0	0
$631$	−35.5582	−1.41555	−0.707774	−	0.706439i	$-0.750300\pi$
$631$	−35.5582	−1.41555	−0.707774	+	0.706439i	$0.750300\pi$
$632$	0	0
$633$	13.3356	0.530043
$634$	0	0
$635$	2.91483i	0.115671i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.53521	−0.179410
$640$	0	0
$641$	9.46599	0.373884	0.186942	−	0.982371i	$-0.440142\pi$
$641$	9.46599	0.373884	0.186942	+	0.982371i	$0.440142\pi$
$642$	0	0
$643$	− 13.0085i	− 0.513007i	−0.966543	−	0.256503i	$-0.917430\pi$
$643$	− 13.0085i	− 0.513007i	0.966543	−	0.256503i	$-0.0825705\pi$
$644$	0	0
$645$	4.63439i	0.182479i
$646$	0	0
$647$	27.5219	1.08200	0.540999	−	0.841023i	$-0.318046\pi$
$647$	27.5219	1.08200	0.540999	+	0.841023i	$0.318046\pi$
$648$	0	0
$649$	9.02415	0.354229
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.1479i	1.37545i	0.725974	+	0.687723i	$0.241389\pi$
$653$	35.1479i	1.37545i	−0.725974	+	0.687723i	$0.758611\pi$
$654$	0	0
$655$	5.51892	0.215642
$656$	0	0
$657$	−2.41228	−0.0941121
$658$	0	0
$659$	− 3.86719i	− 0.150644i	−0.997159	−	0.0753222i	$-0.976001\pi$
$659$	− 3.86719i	− 0.150644i	0.997159	−	0.0753222i	$-0.0239985\pi$
$660$	0	0
$661$	16.6617i	0.648065i	0.946046	+	0.324033i	$0.105039\pi$
$661$	16.6617i	0.648065i	−0.946046	+	0.324033i	$0.894961\pi$
$662$	0	0
$663$	26.3431	1.02308
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 19.3013i	− 0.747350i
$668$	0	0
$669$	− 9.14135i	− 0.353425i
$670$	0	0
$671$	17.0622	0.658679
$672$	0	0
$673$	−3.95795	−0.152568	−0.0762838	−	0.997086i	$-0.524306\pi$
$673$	−3.95795	−0.152568	−0.0762838	+	0.997086i	$0.524306\pi$
$674$	0	0
$675$	26.0355i	1.00211i
$676$	0	0
$677$	41.2042i	1.58361i	0.610776	+	0.791804i	$0.290858\pi$
$677$	41.2042i	1.58361i	−0.610776	+	0.791804i	$0.709142\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	17.9988	0.689716
$682$	0	0
$683$	33.0986i	1.26648i	0.773954	+	0.633242i	$0.218276\pi$
$683$	33.0986i	1.26648i	−0.773954	+	0.633242i	$0.781724\pi$
$684$	0	0
$685$	1.78648i	0.0682580i
$686$	0	0
$687$	29.1580	1.11245
$688$	0	0
$689$	42.9117	1.63480
$690$	0	0
$691$	14.8968i	0.566701i	0.959016	+	0.283350i	$0.0914459\pi$
$691$	14.8968i	0.566701i	−0.959016	+	0.283350i	$0.908554\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.43018	0.319775
$696$	0	0
$697$	26.3431	0.997815
$698$	0	0
$699$	− 17.4131i	− 0.658623i
$700$	0	0
$701$	32.5746i	1.23032i	0.788401	+	0.615162i	$0.210910\pi$
$701$	32.5746i	1.23032i	−0.788401	+	0.615162i	$0.789090\pi$
$702$	0	0
$703$	1.76237	0.0664692
$704$	0	0
$705$	−2.25090	−0.0847737
$706$	0	0
$707$	0	0
$708$	0	0
$709$	11.4966i	0.431764i	0.976419	+	0.215882i	$0.0692626\pi$
$709$	11.4966i	0.431764i	−0.976419	+	0.215882i	$0.930737\pi$
$710$	0	0
$711$	−0.318917	−0.0119603
$712$	0	0
$713$	2.03461	0.0761967
$714$	0	0
$715$	− 5.21300i	− 0.194955i
$716$	0	0
$717$	35.6267i	1.33050i
$718$	0	0
$719$	−26.9775	−1.00609	−0.503045	−	0.864260i	$-0.667787\pi$
$719$	−26.9775	−1.00609	−0.503045	+	0.864260i	$0.667787\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 21.9483i	− 0.816264i
$724$	0	0
$725$	− 38.6027i	− 1.43367i
$726$	0	0
$727$	−27.0230	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$727$	−27.0230	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$728$	0	0
$729$	−29.9117	−1.10784
$730$	0	0
$731$	− 22.5519i	− 0.834111i
$732$	0	0
$733$	20.6760i	0.763686i	0.924227	+	0.381843i	$0.124710\pi$
$733$	20.6760i	0.763686i	−0.924227	+	0.381843i	$0.875290\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.39558	0.235584
$738$	0	0
$739$	− 10.7452i	− 0.395269i	−0.980276	−	0.197635i	$-0.936674\pi$
$739$	− 10.7452i	− 0.395269i	0.980276	−	0.197635i	$-0.0633260\pi$
$740$	0	0
$741$	41.5076i	1.52482i
$742$	0	0
$743$	−11.8708	−0.435498	−0.217749	−	0.976005i	$-0.569871\pi$
$743$	−11.8708	−0.435498	−0.217749	+	0.976005i	$0.569871\pi$
$744$	0	0
$745$	−4.05876	−0.148701
$746$	0	0
$747$	0.568763i	0.0208100i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−17.0646	−0.622696	−0.311348	−	0.950296i	$-0.600780\pi$
$751$	−17.0646	−0.622696	−0.311348	+	0.950296i	$0.600780\pi$
$752$	0	0
$753$	−1.11247	−0.0405405
$754$	0	0
$755$	− 4.57986i	− 0.166678i
$756$	0	0
$757$	− 46.3272i	− 1.68379i	−0.539641	−	0.841895i	$-0.681440\pi$
$757$	− 46.3272i	− 1.68379i	0.539641	−	0.841895i	$-0.318560\pi$
$758$	0	0
$759$	8.84830	0.321173
$760$	0	0
$761$	14.6095	0.529593	0.264796	−	0.964304i	$-0.414695\pi$
$761$	14.6095	0.529593	0.264796	+	0.964304i	$0.414695\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.19869i	0.0433389i
$766$	0	0
$767$	14.9716	0.540595
$768$	0	0
$769$	−49.3177	−1.77844	−0.889221	−	0.457477i	$-0.848753\pi$
$769$	−49.3177	−1.77844	−0.889221	+	0.457477i	$0.848753\pi$
$770$	0	0
$771$	− 32.7480i	− 1.17939i
$772$	0	0
$773$	− 1.04240i	− 0.0374925i	−0.999824	−	0.0187462i	$-0.994033\pi$
$773$	− 1.04240i	− 0.0374925i	0.999824	−	0.0187462i	$-0.00596747\pi$
$774$	0	0
$775$	4.06922	0.146171
$776$	0	0
$777$	0	0
$778$	0	0
$779$	41.5076i	1.48717i
$780$	0	0
$781$	20.9597i	0.749998i
$782$	0	0
$783$	45.5403	1.62748
$784$	0	0
$785$	3.92334	0.140030
$786$	0	0
$787$	− 40.2874i	− 1.43609i	−0.695997	−	0.718045i	$-0.745037\pi$
$787$	− 40.2874i	− 1.43609i	0.695997	−	0.718045i	$-0.254963\pi$
$788$	0	0
$789$	35.8920i	1.27779i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	28.3073	1.00522
$794$	0	0
$795$	− 9.37120i	− 0.332362i
$796$	0	0
$797$	32.2902i	1.14378i	0.820331	+	0.571889i	$0.193789\pi$
$797$	32.2902i	1.14378i	−0.820331	+	0.571889i	$0.806211\pi$
$798$	0	0
$799$	10.9533	0.387501
$800$	0	0
$801$	3.30846	0.116899
$802$	0	0
$803$	11.1485i	0.393422i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.71912	−0.166121
$808$	0	0
$809$	41.6262	1.46350	0.731749	−	0.681574i	$-0.238704\pi$
$809$	41.6262	1.46350	0.731749	+	0.681574i	$0.238704\pi$
$810$	0	0
$811$	− 18.7227i	− 0.657444i	−0.944427	−	0.328722i	$-0.893382\pi$
$811$	− 18.7227i	− 0.657444i	0.944427	−	0.328722i	$-0.106618\pi$
$812$	0	0
$813$	− 39.6831i	− 1.39175i
$814$	0	0
$815$	4.01832	0.140756
$816$	0	0
$817$	35.5340	1.24318
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 18.7418i	− 0.654092i	−0.945008	−	0.327046i	$-0.893947\pi$
$821$	− 18.7418i	− 0.654092i	0.945008	−	0.327046i	$-0.106053\pi$
$822$	0	0
$823$	20.4423	0.712572	0.356286	−	0.934377i	$-0.384043\pi$
$823$	20.4423	0.712572	0.356286	+	0.934377i	$0.384043\pi$
$824$	0	0
$825$	17.6966	0.616116
$826$	0	0
$827$	− 48.6254i	− 1.69087i	−0.534079	−	0.845435i	$-0.679341\pi$
$827$	− 48.6254i	− 1.69087i	0.534079	−	0.845435i	$-0.320659\pi$
$828$	0	0
$829$	− 6.99948i	− 0.243102i	−0.992585	−	0.121551i	$-0.961213\pi$
$829$	− 6.99948i	− 0.243102i	0.992585	−	0.121551i	$-0.0387868\pi$
$830$	0	0
$831$	14.8817	0.516241
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.03831i	0.0359321i
$836$	0	0
$837$	4.80053i	0.165931i
$838$	0	0
$839$	40.1867	1.38740	0.693700	−	0.720264i	$-0.255979\pi$
$839$	40.1867	1.38740	0.693700	+	0.720264i	$0.255979\pi$
$840$	0	0
$841$	−38.5224	−1.32836
$842$	0	0
$843$	42.2945i	1.45670i
$844$	0	0
$845$	− 1.50210i	− 0.0516739i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	20.4555	0.702032
$850$	0	0
$851$	− 0.623294i	− 0.0213662i
$852$	0	0
$853$	− 30.8071i	− 1.05482i	−0.849612	−	0.527408i	$-0.823164\pi$
$853$	− 30.8071i	− 1.05482i	0.849612	−	0.527408i	$-0.176836\pi$
$854$	0	0
$855$	−1.88873	−0.0645933
$856$	0	0
$857$	−13.6978	−0.467908	−0.233954	−	0.972248i	$-0.575166\pi$
$857$	−13.6978	−0.467908	−0.233954	+	0.972248i	$0.575166\pi$
$858$	0	0
$859$	8.69338i	0.296614i	0.988941	+	0.148307i	$0.0473824\pi$
$859$	8.69338i	0.296614i	−0.988941	+	0.148307i	$0.952618\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.592349	0.0201638	0.0100819	−	0.999949i	$-0.496791\pi$
$863$	0.592349	0.0201638	0.0100819	+	0.999949i	$0.496791\pi$
$864$	0	0
$865$	−9.09961	−0.309396
$866$	0	0
$867$	1.20854i	0.0410440i
$868$	0	0
$869$	1.47389i	0.0499984i
$870$	0	0
$871$	10.6107	0.359529
$872$	0	0
$873$	−6.69779	−0.226686
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 15.2778i	− 0.515896i	−0.966159	−	0.257948i	$-0.916954\pi$
$877$	− 15.2778i	− 0.515896i	0.966159	−	0.257948i	$-0.0830464\pi$
$878$	0	0
$879$	15.0680	0.508232
$880$	0	0
$881$	43.1280	1.45302	0.726509	−	0.687157i	$-0.241141\pi$
$881$	43.1280	1.45302	0.726509	+	0.687157i	$0.241141\pi$
$882$	0	0
$883$	20.2255i	0.680642i	0.940309	+	0.340321i	$0.110536\pi$
$883$	20.2255i	0.680642i	−0.940309	+	0.340321i	$0.889464\pi$
$884$	0	0
$885$	− 3.26956i	− 0.109905i
$886$	0	0
$887$	21.5640	0.724048	0.362024	−	0.932169i	$-0.382086\pi$
$887$	21.5640	0.724048	0.362024	+	0.932169i	$0.382086\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	17.1668i	0.575108i
$892$	0	0
$893$	17.2587i	0.577540i
$894$	0	0
$895$	−3.04668	−0.101839
$896$	0	0
$897$	14.6799	0.490147
$898$	0	0
$899$	− 7.11772i	− 0.237389i
$900$	0	0
$901$	45.6021i	1.51923i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.48894	−0.182459
$906$	0	0
$907$	31.5677i	1.04819i	0.851661	+	0.524094i	$0.175596\pi$
$907$	31.5677i	1.04819i	−0.851661	+	0.524094i	$0.824404\pi$
$908$	0	0
$909$	8.21720i	0.272547i
$910$	0	0
$911$	−15.6873	−0.519744	−0.259872	−	0.965643i	$-0.583680\pi$
$911$	−15.6873	−0.519744	−0.259872	+	0.965643i	$0.583680\pi$
$912$	0	0
$913$	2.62857	0.0869930
$914$	0	0
$915$	− 6.18184i	− 0.204365i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−15.5881	−0.514205	−0.257103	−	0.966384i	$-0.582768\pi$
$919$	−15.5881	−0.514205	−0.257103	+	0.966384i	$0.582768\pi$
$920$	0	0
$921$	19.2571	0.634544
$922$	0	0
$923$	34.7735i	1.14458i
$924$	0	0
$925$	− 1.24659i	− 0.0409875i
$926$	0	0
$927$	9.71449	0.319066
$928$	0	0
$929$	−41.3851	−1.35780	−0.678901	−	0.734230i	$-0.737543\pi$
$929$	−41.3851	−1.35780	−0.678901	+	0.734230i	$0.737543\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	50.2365i	1.64467i
$934$	0	0
$935$	5.53984	0.181172
$936$	0	0
$937$	23.9308	0.781785	0.390892	−	0.920436i	$-0.372166\pi$
$937$	23.9308	0.781785	0.390892	+	0.920436i	$0.372166\pi$
$938$	0	0
$939$	− 33.9137i	− 1.10673i
$940$	0	0
$941$	38.4844i	1.25456i	0.778795	+	0.627278i	$0.215831\pi$
$941$	38.4844i	1.25456i	−0.778795	+	0.627278i	$0.784169\pi$
$942$	0	0
$943$	14.6799	0.478043
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9.65559i	− 0.313764i	−0.987617	−	0.156882i	$-0.949856\pi$
$947$	− 9.65559i	− 0.313764i	0.987617	−	0.156882i	$-0.0501443\pi$
$948$	0	0
$949$	18.4961i	0.600408i
$950$	0	0
$951$	36.4757	1.18280
$952$	0	0
$953$	−19.3777	−0.627704	−0.313852	−	0.949472i	$-0.601620\pi$
$953$	−19.3777	−0.627704	−0.313852	+	0.949472i	$0.601620\pi$
$954$	0	0
$955$	− 0.0925812i	− 0.00299586i
$956$	0	0
$957$	− 30.9542i	− 1.00061i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.2497	−0.975797
$962$	0	0
$963$	− 0.175321i	− 0.00564963i
$964$	0	0
$965$	4.13255i	0.133031i
$966$	0	0
$967$	−34.8845	−1.12181	−0.560905	−	0.827880i	$-0.689547\pi$
$967$	−34.8845	−1.12181	−0.560905	+	0.827880i	$0.689547\pi$
$968$	0	0
$969$	−44.1101	−1.41702
$970$	0	0
$971$	44.4291i	1.42580i	0.701267	+	0.712899i	$0.252618\pi$
$971$	44.4291i	1.42580i	−0.701267	+	0.712899i	$0.747382\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	29.3598	0.940265
$976$	0	0
$977$	−27.0871	−0.866594	−0.433297	−	0.901251i	$-0.642650\pi$
$977$	−27.0871	−0.866594	−0.433297	+	0.901251i	$0.642650\pi$
$978$	0	0
$979$	− 15.2902i	− 0.488678i
$980$	0	0
$981$	− 4.03742i	− 0.128905i
$982$	0	0
$983$	25.0887	0.800206	0.400103	−	0.916470i	$-0.368974\pi$
$983$	25.0887	0.800206	0.400103	+	0.916470i	$0.368974\pi$
$984$	0	0
$985$	0.739840	0.0235733
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 12.5672i	− 0.399614i
$990$	0	0
$991$	−17.6394	−0.560336	−0.280168	−	0.959951i	$-0.590390\pi$
$991$	−17.6394	−0.560336	−0.280168	+	0.959951i	$0.590390\pi$
$992$	0	0
$993$	−46.5068	−1.47585
$994$	0	0
$995$	− 7.01530i	− 0.222400i
$996$	0	0
$997$	− 30.6607i	− 0.971033i	−0.874228	−	0.485516i	$-0.838632\pi$
$997$	− 30.6607i	− 0.971033i	0.874228	−	0.485516i	$-0.161368\pi$
$998$	0	0
$999$	1.47062	0.0465284

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.b.e.785.5		6
4.3	odd	2		392.2.b.f.197.2		6
7.2	even	3		1568.2.t.g.753.2		12
7.3	odd	6		224.2.t.a.177.2		12
7.4	even	3		1568.2.t.g.177.5		12
7.5	odd	6		224.2.t.a.81.5		12
7.6	odd	2		1568.2.b.f.785.2		6
8.3	odd	2		392.2.b.f.197.1		6
8.5	even	2	inner	1568.2.b.e.785.2		6
21.5	even	6		2016.2.cr.c.1873.4		12
21.17	even	6		2016.2.cr.c.1297.3		12
28.3	even	6		56.2.p.a.37.3	✓	12
28.11	odd	6		392.2.p.g.373.3		12
28.19	even	6		56.2.p.a.53.6	yes	12
28.23	odd	6		392.2.p.g.165.6		12
28.27	even	2		392.2.b.e.197.2		6
56.3	even	6		56.2.p.a.37.6	yes	12
56.5	odd	6		224.2.t.a.81.2		12
56.11	odd	6		392.2.p.g.373.6		12
56.13	odd	2		1568.2.b.f.785.5		6
56.19	even	6		56.2.p.a.53.3	yes	12
56.27	even	2		392.2.b.e.197.1		6
56.37	even	6		1568.2.t.g.753.5		12
56.45	odd	6		224.2.t.a.177.5		12
56.51	odd	6		392.2.p.g.165.3		12
56.53	even	6		1568.2.t.g.177.2		12
84.47	odd	6		504.2.cj.c.109.1		12
84.59	odd	6		504.2.cj.c.37.4		12
168.5	even	6		2016.2.cr.c.1873.3		12
168.59	odd	6		504.2.cj.c.37.1		12
168.101	even	6		2016.2.cr.c.1297.4		12
168.131	odd	6		504.2.cj.c.109.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.p.a.37.3	✓	12	28.3	even	6
56.2.p.a.37.6	yes	12	56.3	even	6
56.2.p.a.53.3	yes	12	56.19	even	6
56.2.p.a.53.6	yes	12	28.19	even	6
224.2.t.a.81.2		12	56.5	odd	6
224.2.t.a.81.5		12	7.5	odd	6
224.2.t.a.177.2		12	7.3	odd	6
224.2.t.a.177.5		12	56.45	odd	6
392.2.b.e.197.1		6	56.27	even	2
392.2.b.e.197.2		6	28.27	even	2
392.2.b.f.197.1		6	8.3	odd	2
392.2.b.f.197.2		6	4.3	odd	2
392.2.p.g.165.3		12	56.51	odd	6
392.2.p.g.165.6		12	28.23	odd	6
392.2.p.g.373.3		12	28.11	odd	6
392.2.p.g.373.6		12	56.11	odd	6
504.2.cj.c.37.1		12	168.59	odd	6
504.2.cj.c.37.4		12	84.59	odd	6
504.2.cj.c.109.1		12	84.47	odd	6
504.2.cj.c.109.4		12	168.131	odd	6
1568.2.b.e.785.2		6	8.5	even	2	inner
1568.2.b.e.785.5		6	1.1	even	1	trivial
1568.2.b.f.785.2		6	7.6	odd	2
1568.2.b.f.785.5		6	56.13	odd	2
1568.2.t.g.177.2		12	56.53	even	6
1568.2.t.g.177.5		12	7.4	even	3
1568.2.t.g.753.2		12	7.2	even	3
1568.2.t.g.753.5		12	56.37	even	6
2016.2.cr.c.1297.3		12	21.17	even	6
2016.2.cr.c.1297.4		12	168.101	even	6
2016.2.cr.c.1873.3		12	168.5	even	6
2016.2.cr.c.1873.4		12	21.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.57566i q^{3} +0.549738i q^{5} +0.517304 q^{9} -2.39075i q^{11} -3.96641i q^{13} -0.866198 q^{15} +4.21509 q^{17} +6.64154i q^{19} +2.34889 q^{23} +4.69779 q^{25} +5.54207i q^{27} -8.21720i q^{29} +0.866198 q^{31} +3.76700 q^{33} -0.265356i q^{37} +6.24970 q^{39} +6.24970 q^{41} -5.35027i q^{43} +0.284382i q^{45} +2.59859 q^{47} +6.64154i q^{51} +10.8188i q^{53} +1.31429 q^{55} -10.4648 q^{57} +3.77461i q^{59} +7.13675i q^{61} +2.18048 q^{65} +2.67513i q^{67} +3.70105i q^{69} -8.76700 q^{71} -4.66318 q^{73} +7.40210i q^{75} -0.616498 q^{79} -7.18048 q^{81} +1.09948i q^{83} +2.31720i q^{85} +12.9475 q^{87} +6.39558 q^{89} +1.36483i q^{93} -3.65111 q^{95} -12.9475 q^{97} -1.23675i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{15} - 2 q^{17} + 2 q^{23} + 4 q^{25} - 10 q^{31} - 14 q^{33} + 4 q^{39} + 4 q^{41} - 30 q^{47} + 2 q^{55} - 2 q^{57} - 8 q^{65} - 16 q^{71} - 10 q^{73} - 22 q^{79} - 22 q^{81} + 20 q^{87} - 10 q^{89}+ \cdots - 20 q^{97}+O(q^{100}) \) 6 * q + 10 * q^15 - 2 * q^17 + 2 * q^23 + 4 * q^25 - 10 * q^31 - 14 * q^33 + 4 * q^39 + 4 * q^41 - 30 * q^47 + 2 * q^55 - 2 * q^57 - 8 * q^65 - 16 * q^71 - 10 * q^73 - 22 * q^79 - 22 * q^81 + 20 * q^87 - 10 * q^89 - 34 * q^95 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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