LMFDB - Embedded newform 1568.2.f.b.1567.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(1567,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1568.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.2353561680715186176.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 $ x^16 - 4x^15 + 2x^14 + 41x^12 - 92x^11 + 66x^10 - 104x^9 + 291x^8 - 388x^7 + 366x^6 - 344x^5 + 286x^4 - 184x^3 + 84x^2 - 24x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.6
Root			$0.849168 - 0.0870829i$ of defining polynomial
Character	$\chi$	$=$	1568.1567
Dual form			1568.2.f.b.1567.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.753692 q^{3} +1.15894i q^{5} -2.43195 q^{9} +O(q^{10})$	$q-0.753692 q^{3} +1.15894i q^{5} -2.43195 q^{9} +1.70191i q^{11} -3.51537i q^{13} -0.873485i q^{15} -5.24742i q^{17} +1.32680 q^{19} +2.13386i q^{23} +3.65685 q^{25} +4.09402 q^{27} -2.43195 q^{29} +8.54366 q^{31} -1.28272i q^{33} +6.43929 q^{37} +2.64951i q^{39} -3.51537i q^{41} +12.2677i q^{43} -2.81849i q^{45} -9.68988 q^{47} +3.95494i q^{51} +12.0961 q^{53} -1.97242 q^{55} -1.00000 q^{57} +6.27705 q^{59} +3.05332i q^{61} +4.07411 q^{65} +13.2153i q^{67} -1.60827i q^{69} -10.2530i q^{71} +2.42894i q^{73} -2.75614 q^{75} -4.74473i q^{79} +4.21021 q^{81} +16.9737 q^{83} +6.08146 q^{85} +1.83294 q^{87} +7.61658i q^{89} -6.43929 q^{93} +1.53769i q^{95} -12.0646i q^{97} -4.13896i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^9	$ 16 q + 16 q^{9} - 32 q^{25} + 16 q^{29} + 16 q^{37} + 16 q^{53} - 16 q^{57} - 16 q^{65} + 96 q^{81} - 16 q^{85} - 16 q^{93}+O(q^{100}) $ 16 * q + 16 * q^9 - 32 * q^25 + 16 * q^29 + 16 * q^37 + 16 * q^53 - 16 * q^57 - 16 * q^65 + 96 * q^81 - 16 * q^85 - 16 * q^93

Display 100 coefficients 10 coefficients

Character values

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

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

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
$3$	−0.753692	−0.435144	−0.217572	−	0.976044i	$-0.569814\pi$
$3$	−0.753692	−0.435144	−0.217572	+	0.976044i	$0.569814\pi$
$4$	0	0
$5$	1.15894i	0.518294i	0.965838	+	0.259147i	$0.0834415\pi$
$5$	1.15894i	0.518294i	−0.965838	+	0.259147i	$0.916559\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.43195	−0.810649
$10$	0	0
$11$	1.70191i	0.513146i	0.966525	+	0.256573i	$0.0825934\pi$
$11$	1.70191i	0.513146i	−0.966525	+	0.256573i	$0.917407\pi$
$12$	0	0
$13$	− 3.51537i	− 0.974989i	−0.873126	−	0.487494i	$-0.837911\pi$
$13$	− 3.51537i	− 0.974989i	0.873126	−	0.487494i	$-0.162089\pi$
$14$	0	0
$15$	− 0.873485i	− 0.225533i
$16$	0	0
$17$	− 5.24742i	− 1.27269i	−0.771406	−	0.636343i	$-0.780446\pi$
$17$	− 5.24742i	− 1.27269i	0.771406	−	0.636343i	$-0.219554\pi$
$18$	0	0
$19$	1.32680	0.304389	0.152195	−	0.988351i	$-0.451366\pi$
$19$	1.32680	0.304389	0.152195	+	0.988351i	$0.451366\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.13386i	0.444941i	0.974939	+	0.222470i	$0.0714120\pi$
$23$	2.13386i	0.444941i	−0.974939	+	0.222470i	$0.928588\pi$
$24$	0	0
$25$	3.65685	0.731371
$26$	0	0
$27$	4.09402	0.787894
$28$	0	0
$29$	−2.43195	−0.451601	−0.225801	−	0.974174i	$-0.572500\pi$
$29$	−2.43195	−0.451601	−0.225801	+	0.974174i	$0.572500\pi$
$30$	0	0
$31$	8.54366	1.53449	0.767244	−	0.641356i	$-0.221628\pi$
$31$	8.54366	1.53449	0.767244	+	0.641356i	$0.221628\pi$
$32$	0	0
$33$	− 1.28272i	− 0.223293i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.43929	1.05861	0.529307	−	0.848431i	$-0.322452\pi$
$37$	6.43929	1.05861	0.529307	+	0.848431i	$0.322452\pi$
$38$	0	0
$39$	2.64951i	0.424261i
$40$	0	0
$41$	− 3.51537i	− 0.549009i	−0.961586	−	0.274504i	$-0.911486\pi$
$41$	− 3.51537i	− 0.549009i	0.961586	−	0.274504i	$-0.0885138\pi$
$42$	0	0
$43$	12.2677i	1.87081i	0.353578	+	0.935405i	$0.384965\pi$
$43$	12.2677i	1.87081i	−0.353578	+	0.935405i	$0.615035\pi$
$44$	0	0
$45$	− 2.81849i	− 0.420155i
$46$	0	0
$47$	−9.68988	−1.41341	−0.706707	−	0.707506i	$-0.749820\pi$
$47$	−9.68988	−1.41341	−0.706707	+	0.707506i	$0.749820\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.95494i	0.553803i
$52$	0	0
$53$	12.0961	1.66153	0.830767	−	0.556621i	$-0.187902\pi$
$53$	12.0961	1.66153	0.830767	+	0.556621i	$0.187902\pi$
$54$	0	0
$55$	−1.97242	−0.265961
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	6.27705	0.817202	0.408601	−	0.912713i	$-0.366017\pi$
$59$	6.27705	0.817202	0.408601	+	0.912713i	$0.366017\pi$
$60$	0	0
$61$	3.05332i	0.390937i	0.980710	+	0.195469i	$0.0626227\pi$
$61$	3.05332i	0.390937i	−0.980710	+	0.195469i	$0.937377\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.07411	0.505331
$66$	0	0
$67$	13.2153i	1.61451i	0.590204	+	0.807254i	$0.299047\pi$
$67$	13.2153i	1.61451i	−0.590204	+	0.807254i	$0.700953\pi$
$68$	0	0
$69$	− 1.60827i	− 0.193613i
$70$	0	0
$71$	− 10.2530i	− 1.21681i	−0.793626	−	0.608405i	$-0.791810\pi$
$71$	− 10.2530i	− 1.21681i	0.793626	−	0.608405i	$-0.208190\pi$
$72$	0	0
$73$	2.42894i	0.284286i	0.989846	+	0.142143i	$0.0453992\pi$
$73$	2.42894i	0.284286i	−0.989846	+	0.142143i	$0.954601\pi$
$74$	0	0
$75$	−2.75614	−0.318252
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 4.74473i	− 0.533824i	−0.963721	−	0.266912i	$-0.913997\pi$
$79$	− 4.74473i	− 0.533824i	0.963721	−	0.266912i	$-0.0860032\pi$
$80$	0	0
$81$	4.21021	0.467802
$82$	0	0
$83$	16.9737	1.86311	0.931554	−	0.363604i	$-0.118454\pi$
$83$	16.9737	1.86311	0.931554	+	0.363604i	$0.118454\pi$
$84$	0	0
$85$	6.08146	0.659627
$86$	0	0
$87$	1.83294	0.196512
$88$	0	0
$89$	7.61658i	0.807355i	0.914901	+	0.403678i	$0.132268\pi$
$89$	7.61658i	0.807355i	−0.914901	+	0.403678i	$0.867732\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.43929	−0.667724
$94$	0	0
$95$	1.53769i	0.157763i
$96$	0	0
$97$	− 12.0646i	− 1.22497i	−0.790482	−	0.612486i	$-0.790170\pi$
$97$	− 12.0646i	− 1.22497i	0.790482	−	0.612486i	$-0.209830\pi$
$98$	0	0
$99$	− 4.13896i	− 0.415981i
$100$	0	0
$101$	− 16.2641i	− 1.61834i	−0.587575	−	0.809170i	$-0.699917\pi$
$101$	− 16.2641i	− 1.61834i	0.587575	−	0.809170i	$-0.300083\pi$
$102$	0	0
$103$	−2.08603	−0.205543	−0.102771	−	0.994705i	$-0.532771\pi$
$103$	−2.08603	−0.205543	−0.102771	+	0.994705i	$0.532771\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.04506i	0.391051i	0.980699	+	0.195525i	$0.0626412\pi$
$107$	4.04506i	0.391051i	−0.980699	+	0.195525i	$0.937359\pi$
$108$	0	0
$109$	−2.35049	−0.225136	−0.112568	−	0.993644i	$-0.535908\pi$
$109$	−2.35049	−0.225136	−0.112568	+	0.993644i	$0.535908\pi$
$110$	0	0
$111$	−4.85325	−0.460650
$112$	0	0
$113$	4.44664	0.418305	0.209152	−	0.977883i	$-0.432930\pi$
$113$	4.44664	0.418305	0.209152	+	0.977883i	$0.432930\pi$
$114$	0	0
$115$	−2.47302	−0.230610
$116$	0	0
$117$	8.54920i	0.790374i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	8.10349	0.736681
$122$	0	0
$123$	2.64951i	0.238898i
$124$	0	0
$125$	10.0328i	0.897360i
$126$	0	0
$127$	− 4.95401i	− 0.439598i	−0.975545	−	0.219799i	$-0.929460\pi$
$127$	− 4.95401i	− 0.439598i	0.975545	−	0.219799i	$-0.0705401\pi$
$128$	0	0
$129$	− 9.24609i	− 0.814073i
$130$	0	0
$131$	18.8736	1.64900	0.824498	−	0.565865i	$-0.191458\pi$
$131$	18.8736	1.64900	0.824498	+	0.565865i	$0.191458\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.74473i	0.408361i
$136$	0	0
$137$	−9.07411	−0.775254	−0.387627	−	0.921816i	$-0.626705\pi$
$137$	−9.07411	−0.775254	−0.387627	+	0.921816i	$0.626705\pi$
$138$	0	0
$139$	12.0904	1.02549	0.512747	−	0.858540i	$-0.328628\pi$
$139$	12.0904	1.02549	0.512747	+	0.858540i	$0.328628\pi$
$140$	0	0
$141$	7.30319	0.615040
$142$	0	0
$143$	5.98285	0.500311
$144$	0	0
$145$	− 2.81849i	− 0.234063i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.99265	−0.163245	−0.0816223	−	0.996663i	$-0.526010\pi$
$149$	−1.99265	−0.163245	−0.0816223	+	0.996663i	$0.526010\pi$
$150$	0	0
$151$	5.14121i	0.418385i	0.977874	+	0.209193i	$0.0670836\pi$
$151$	5.14121i	0.418385i	−0.977874	+	0.209193i	$0.932916\pi$
$152$	0	0
$153$	12.7615i	1.03170i
$154$	0	0
$155$	9.90161i	0.795316i
$156$	0	0
$157$	12.0518i	0.961842i	0.876764	+	0.480921i	$0.159698\pi$
$157$	12.0518i	0.961842i	−0.876764	+	0.480921i	$0.840302\pi$
$158$	0	0
$159$	−9.11677	−0.723007
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.15464i	− 0.325416i	−0.986674	−	0.162708i	$-0.947977\pi$
$163$	− 4.15464i	− 0.325416i	0.986674	−	0.162708i	$-0.0520229\pi$
$164$	0	0
$165$	1.48660	0.115731
$166$	0	0
$167$	−12.1371	−0.939195	−0.469598	−	0.882881i	$-0.655601\pi$
$167$	−12.1371	−0.939195	−0.469598	+	0.882881i	$0.655601\pi$
$168$	0	0
$169$	0.642163	0.0493971
$170$	0	0
$171$	−3.22671	−0.246753
$172$	0	0
$173$	− 15.3142i	− 1.16431i	−0.813076	−	0.582157i	$-0.802209\pi$
$173$	− 15.3142i	− 1.16431i	0.813076	−	0.582157i	$-0.197791\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.73096	−0.355601
$178$	0	0
$179$	− 10.4757i	− 0.782990i	−0.920180	−	0.391495i	$-0.871958\pi$
$179$	− 10.4757i	− 0.782990i	0.920180	−	0.391495i	$-0.128042\pi$
$180$	0	0
$181$	− 18.1199i	− 1.34684i	−0.739258	−	0.673422i	$-0.764824\pi$
$181$	− 18.1199i	− 1.34684i	0.739258	−	0.673422i	$-0.235176\pi$
$182$	0	0
$183$	− 2.30126i	− 0.170114i
$184$	0	0
$185$	7.46277i	0.548673i
$186$	0	0
$187$	8.93065	0.653074
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 9.73738i	− 0.704572i	−0.935892	−	0.352286i	$-0.885404\pi$
$191$	− 9.73738i	− 0.704572i	0.935892	−	0.352286i	$-0.114596\pi$
$192$	0	0
$193$	18.3105	1.31802	0.659011	−	0.752134i	$-0.270975\pi$
$193$	18.3105	1.31802	0.659011	+	0.752134i	$0.270975\pi$
$194$	0	0
$195$	−3.07063	−0.219892
$196$	0	0
$197$	−14.6096	−1.04089	−0.520444	−	0.853896i	$-0.674233\pi$
$197$	−14.6096	−1.04089	−0.520444	+	0.853896i	$0.674233\pi$
$198$	0	0
$199$	2.90669	0.206050	0.103025	−	0.994679i	$-0.467148\pi$
$199$	2.90669	0.206050	0.103025	+	0.994679i	$0.467148\pi$
$200$	0	0
$201$	− 9.96028i	− 0.702544i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.07411	0.284548
$206$	0	0
$207$	− 5.18944i	− 0.360691i
$208$	0	0
$209$	2.25810i	0.156196i
$210$	0	0
$211$	19.5814i	1.34804i	0.738713	+	0.674021i	$0.235434\pi$
$211$	19.5814i	1.34804i	−0.738713	+	0.674021i	$0.764566\pi$
$212$	0	0
$213$	7.72763i	0.529488i
$214$	0	0
$215$	−14.2176	−0.969630
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 1.83067i	− 0.123705i
$220$	0	0
$221$	−18.4466	−1.24086
$222$	0	0
$223$	3.48955	0.233677	0.116839	−	0.993151i	$-0.462724\pi$
$223$	3.48955	0.233677	0.116839	+	0.993151i	$0.462724\pi$
$224$	0	0
$225$	−8.89328	−0.592885
$226$	0	0
$227$	−3.05034	−0.202458	−0.101229	−	0.994863i	$-0.532277\pi$
$227$	−3.05034	−0.202458	−0.101229	+	0.994863i	$0.532277\pi$
$228$	0	0
$229$	− 7.39026i	− 0.488362i	−0.969730	−	0.244181i	$-0.921481\pi$
$229$	− 7.39026i	− 0.488362i	0.969730	−	0.244181i	$-0.0785191\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.88176	−0.516351	−0.258176	−	0.966098i	$-0.583121\pi$
$233$	−7.88176	−0.516351	−0.258176	+	0.966098i	$0.583121\pi$
$234$	0	0
$235$	− 11.2300i	− 0.732565i
$236$	0	0
$237$	3.57606i	0.232290i
$238$	0	0
$239$	− 13.2604i	− 0.857742i	−0.903366	−	0.428871i	$-0.858911\pi$
$239$	− 13.2604i	− 0.857742i	0.903366	−	0.428871i	$-0.141089\pi$
$240$	0	0
$241$	3.00626i	0.193650i	0.995301	+	0.0968250i	$0.0308687\pi$
$241$	3.00626i	0.193650i	−0.995301	+	0.0968250i	$0.969131\pi$
$242$	0	0
$243$	−15.4553	−0.991455
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.66420i	− 0.296776i
$248$	0	0
$249$	−12.7930	−0.810721
$250$	0	0
$251$	−20.4166	−1.28868	−0.644342	−	0.764738i	$-0.722868\pi$
$251$	−20.4166	−1.28868	−0.644342	+	0.764738i	$0.722868\pi$
$252$	0	0
$253$	−3.63164	−0.228319
$254$	0	0
$255$	−4.58355	−0.287033
$256$	0	0
$257$	20.6001i	1.28500i	0.766285	+	0.642501i	$0.222103\pi$
$257$	20.6001i	1.28500i	−0.766285	+	0.642501i	$0.777897\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.91437	0.366090
$262$	0	0
$263$	− 26.9370i	− 1.66101i	−0.557013	−	0.830504i	$-0.688053\pi$
$263$	− 26.9370i	− 1.66101i	0.557013	−	0.830504i	$-0.311947\pi$
$264$	0	0
$265$	14.0187i	0.861164i
$266$	0	0
$267$	− 5.74055i	− 0.351316i
$268$	0	0
$269$	19.0826i	1.16349i	0.813373	+	0.581743i	$0.197629\pi$
$269$	19.0826i	1.16349i	−0.813373	+	0.581743i	$0.802371\pi$
$270$	0	0
$271$	22.0431	1.33902	0.669512	−	0.742801i	$-0.266503\pi$
$271$	22.0431	1.33902	0.669512	+	0.742801i	$0.266503\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.22365i	0.375300i
$276$	0	0
$277$	−4.11084	−0.246996	−0.123498	−	0.992345i	$-0.539411\pi$
$277$	−4.11084	−0.246996	−0.123498	+	0.992345i	$0.539411\pi$
$278$	0	0
$279$	−20.7777	−1.24393
$280$	0	0
$281$	−7.73096	−0.461191	−0.230595	−	0.973050i	$-0.574067\pi$
$281$	−7.73096	−0.461191	−0.230595	+	0.973050i	$0.574067\pi$
$282$	0	0
$283$	−28.5733	−1.69850	−0.849252	−	0.527988i	$-0.822947\pi$
$283$	−28.5733	−1.69850	−0.849252	+	0.527988i	$0.822947\pi$
$284$	0	0
$285$	− 1.15894i	− 0.0686498i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.5354	−0.619732
$290$	0	0
$291$	9.09298i	0.533040i
$292$	0	0
$293$	6.78319i	0.396278i	0.980174	+	0.198139i	$0.0634898\pi$
$293$	6.78319i	0.396278i	−0.980174	+	0.198139i	$0.936510\pi$
$294$	0	0
$295$	7.27474i	0.423551i
$296$	0	0
$297$	6.96766i	0.404305i
$298$	0	0
$299$	7.50131	0.433812
$300$	0	0
$301$	0	0
$302$	0	0
$303$	12.2581i	0.704211i
$304$	0	0
$305$	−3.53861	−0.202620
$306$	0	0
$307$	13.4842	0.769582	0.384791	−	0.923004i	$-0.374274\pi$
$307$	13.4842	0.769582	0.384791	+	0.923004i	$0.374274\pi$
$308$	0	0
$309$	1.57222	0.0894407
$310$	0	0
$311$	−13.1641	−0.746470	−0.373235	−	0.927737i	$-0.621752\pi$
$311$	−13.1641	−0.746470	−0.373235	+	0.927737i	$0.621752\pi$
$312$	0	0
$313$	− 14.9941i	− 0.847517i	−0.905775	−	0.423759i	$-0.860710\pi$
$313$	− 14.9941i	− 0.847517i	0.905775	−	0.423759i	$-0.139290\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.2176	−0.630041	−0.315020	−	0.949085i	$-0.602011\pi$
$317$	−11.2176	−0.630041	−0.315020	+	0.949085i	$0.602011\pi$
$318$	0	0
$319$	− 4.13896i	− 0.231737i
$320$	0	0
$321$	− 3.04873i	− 0.170163i
$322$	0	0
$323$	− 6.96229i	− 0.387392i
$324$	0	0
$325$	− 12.8552i	− 0.713078i
$326$	0	0
$327$	1.77155	0.0979668
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.4371i	0.573672i	0.957980	+	0.286836i	$0.0926035\pi$
$331$	10.4371i	0.573672i	−0.957980	+	0.286836i	$0.907396\pi$
$332$	0	0
$333$	−15.6600	−0.858164
$334$	0	0
$335$	−15.3158	−0.836791
$336$	0	0
$337$	19.2543	1.04885	0.524424	−	0.851457i	$-0.324281\pi$
$337$	19.2543	1.04885	0.524424	+	0.851457i	$0.324281\pi$
$338$	0	0
$339$	−3.35140	−0.182023
$340$	0	0
$341$	14.5406i	0.787416i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.86390	0.100349
$346$	0	0
$347$	12.7802i	0.686077i	0.939321	+	0.343038i	$0.111456\pi$
$347$	12.7802i	0.686077i	−0.939321	+	0.343038i	$0.888544\pi$
$348$	0	0
$349$	14.8521i	0.795016i	0.917599	+	0.397508i	$0.130125\pi$
$349$	14.8521i	0.795016i	−0.917599	+	0.397508i	$0.869875\pi$
$350$	0	0
$351$	− 14.3920i	− 0.768188i
$352$	0	0
$353$	0.264863i	0.0140972i	0.999975	+	0.00704862i	$0.00224367\pi$
$353$	0.264863i	0.0140972i	−0.999975	+	0.00704862i	$0.997756\pi$
$354$	0	0
$355$	11.8827	0.630666
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 10.1725i	− 0.536884i	−0.963296	−	0.268442i	$-0.913491\pi$
$359$	− 10.1725i	− 0.536884i	0.963296	−	0.268442i	$-0.0865088\pi$
$360$	0	0
$361$	−17.2396	−0.907347
$362$	0	0
$363$	−6.10754	−0.320563
$364$	0	0
$365$	−2.81500	−0.147344
$366$	0	0
$367$	−32.8626	−1.71541	−0.857707	−	0.514139i	$-0.828112\pi$
$367$	−32.8626	−1.71541	−0.857707	+	0.514139i	$0.828112\pi$
$368$	0	0
$369$	8.54920i	0.445054i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−18.0961	−0.936983	−0.468492	−	0.883468i	$-0.655202\pi$
$373$	−18.0961	−0.936983	−0.468492	+	0.883468i	$0.655202\pi$
$374$	0	0
$375$	− 7.56164i	− 0.390481i
$376$	0	0
$377$	8.54920i	0.440306i
$378$	0	0
$379$	− 12.1096i	− 0.622027i	−0.950405	−	0.311014i	$-0.899332\pi$
$379$	− 12.1096i	− 0.622027i	0.950405	−	0.311014i	$-0.100668\pi$
$380$	0	0
$381$	3.73380i	0.191288i
$382$	0	0
$383$	5.76538	0.294597	0.147298	−	0.989092i	$-0.452942\pi$
$383$	5.76538	0.294597	0.147298	+	0.989092i	$0.452942\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 29.8345i	− 1.51657i
$388$	0	0
$389$	12.8566	0.651853	0.325926	−	0.945395i	$-0.394324\pi$
$389$	12.8566	0.651853	0.325926	+	0.945395i	$0.394324\pi$
$390$	0	0
$391$	11.1973	0.566270
$392$	0	0
$393$	−14.2249	−0.717552
$394$	0	0
$395$	5.49886	0.276678
$396$	0	0
$397$	− 19.3560i	− 0.971448i	−0.874112	−	0.485724i	$-0.838556\pi$
$397$	− 19.3560i	− 0.971448i	0.874112	−	0.485724i	$-0.161444\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.4319	1.41982	0.709911	−	0.704292i	$-0.248735\pi$
$401$	28.4319	1.41982	0.709911	+	0.704292i	$0.248735\pi$
$402$	0	0
$403$	− 30.0342i	− 1.49611i
$404$	0	0
$405$	4.87939i	0.242459i
$406$	0	0
$407$	10.9591i	0.543223i
$408$	0	0
$409$	30.8474i	1.52531i	0.646807	+	0.762654i	$0.276104\pi$
$409$	30.8474i	1.52531i	−0.646807	+	0.762654i	$0.723896\pi$
$410$	0	0
$411$	6.83909	0.337347
$412$	0	0
$413$	0	0
$414$	0	0
$415$	19.6715i	0.965638i
$416$	0	0
$417$	−9.11244	−0.446238
$418$	0	0
$419$	−30.5826	−1.49406	−0.747028	−	0.664792i	$-0.768520\pi$
$419$	−30.5826	−1.49406	−0.747028	+	0.664792i	$0.768520\pi$
$420$	0	0
$421$	32.7015	1.59377	0.796887	−	0.604128i	$-0.206478\pi$
$421$	32.7015	1.59377	0.796887	+	0.604128i	$0.206478\pi$
$422$	0	0
$423$	23.5653	1.14578
$424$	0	0
$425$	− 19.1891i	− 0.930806i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.50923	−0.217708
$430$	0	0
$431$	0.676702i	0.0325956i	0.999867	+	0.0162978i	$0.00518798\pi$
$431$	0.676702i	0.0325956i	−0.999867	+	0.0162978i	$0.994812\pi$
$432$	0	0
$433$	− 12.1162i	− 0.582268i	−0.956682	−	0.291134i	$-0.905967\pi$
$433$	− 12.1162i	− 0.582268i	0.956682	−	0.291134i	$-0.0940326\pi$
$434$	0	0
$435$	2.12427i	0.101851i
$436$	0	0
$437$	2.83121i	0.135435i
$438$	0	0
$439$	10.9497	0.522601	0.261301	−	0.965257i	$-0.415849\pi$
$439$	10.9497	0.522601	0.261301	+	0.965257i	$0.415849\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3.66804i	− 0.174274i	−0.996196	−	0.0871370i	$-0.972228\pi$
$443$	− 3.66804i	− 0.174274i	0.996196	−	0.0871370i	$-0.0277718\pi$
$444$	0	0
$445$	−8.82717	−0.418448
$446$	0	0
$447$	1.50185	0.0710350
$448$	0	0
$449$	−14.3904	−0.679125	−0.339562	−	0.940584i	$-0.610279\pi$
$449$	−14.3904	−0.679125	−0.339562	+	0.940584i	$0.610279\pi$
$450$	0	0
$451$	5.98285	0.281722
$452$	0	0
$453$	− 3.87489i	− 0.182058i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11.3725	−0.531984	−0.265992	−	0.963975i	$-0.585700\pi$
$457$	−11.3725	−0.531984	−0.265992	+	0.963975i	$0.585700\pi$
$458$	0	0
$459$	− 21.4830i	− 1.00274i
$460$	0	0
$461$	26.1261i	1.21681i	0.793626	+	0.608406i	$0.208191\pi$
$461$	26.1261i	1.21681i	−0.793626	+	0.608406i	$0.791809\pi$
$462$	0	0
$463$	17.7470i	0.824772i	0.911009	+	0.412386i	$0.135304\pi$
$463$	17.7470i	0.824772i	−0.911009	+	0.412386i	$0.864696\pi$
$464$	0	0
$465$	− 7.46277i	− 0.346077i
$466$	0	0
$467$	−27.7415	−1.28372	−0.641862	−	0.766820i	$-0.721838\pi$
$467$	−27.7415	−1.28372	−0.641862	+	0.766820i	$0.721838\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 9.08339i	− 0.418540i
$472$	0	0
$473$	−20.8786	−0.959998
$474$	0	0
$475$	4.85192	0.222621
$476$	0	0
$477$	−29.4172	−1.34692
$478$	0	0
$479$	−3.34586	−0.152876	−0.0764382	−	0.997074i	$-0.524355\pi$
$479$	−3.34586	−0.152876	−0.0764382	+	0.997074i	$0.524355\pi$
$480$	0	0
$481$	− 22.6365i	− 1.03214i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.9821	0.634896
$486$	0	0
$487$	1.44757i	0.0655956i	0.999462	+	0.0327978i	$0.0104417\pi$
$487$	1.44757i	0.0655956i	−0.999462	+	0.0327978i	$0.989558\pi$
$488$	0	0
$489$	3.13132i	0.141603i
$490$	0	0
$491$	− 16.1390i	− 0.728341i	−0.931332	−	0.364171i	$-0.881353\pi$
$491$	− 16.1390i	− 0.728341i	0.931332	−	0.364171i	$-0.118647\pi$
$492$	0	0
$493$	12.7615i	0.574747i
$494$	0	0
$495$	4.79682	0.215601
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 4.15464i	− 0.185987i	−0.995667	−	0.0929936i	$-0.970356\pi$
$499$	− 4.15464i	− 0.185987i	0.995667	−	0.0929936i	$-0.0296436\pi$
$500$	0	0
$501$	9.14762	0.408686
$502$	0	0
$503$	30.4579	1.35805	0.679025	−	0.734115i	$-0.262403\pi$
$503$	30.4579	1.35805	0.679025	+	0.734115i	$0.262403\pi$
$504$	0	0
$505$	18.8492	0.838776
$506$	0	0
$507$	−0.483993	−0.0214949
$508$	0	0
$509$	13.0730i	0.579451i	0.957110	+	0.289725i	$0.0935640\pi$
$509$	13.0730i	0.579451i	−0.957110	+	0.289725i	$0.906436\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.43195	0.239826
$514$	0	0
$515$	− 2.41759i	− 0.106532i
$516$	0	0
$517$	− 16.4913i	− 0.725288i
$518$	0	0
$519$	11.5422i	0.506645i
$520$	0	0
$521$	20.7964i	0.911108i	0.890208	+	0.455554i	$0.150559\pi$
$521$	20.7964i	0.911108i	−0.890208	+	0.455554i	$0.849441\pi$
$522$	0	0
$523$	−11.5284	−0.504102	−0.252051	−	0.967714i	$-0.581105\pi$
$523$	−11.5284	−0.504102	−0.252051	+	0.967714i	$0.581105\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 44.8322i	− 1.95292i
$528$	0	0
$529$	18.4466	0.802028
$530$	0	0
$531$	−15.2655	−0.662465
$532$	0	0
$533$	−12.3578	−0.535277
$534$	0	0
$535$	−4.68799	−0.202679
$536$	0	0
$537$	7.89545i	0.340714i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.33575	0.315388	0.157694	−	0.987488i	$-0.449594\pi$
$541$	7.33575	0.315388	0.157694	+	0.987488i	$0.449594\pi$
$542$	0	0
$543$	13.6569i	0.586072i
$544$	0	0
$545$	− 2.72408i	− 0.116687i
$546$	0	0
$547$	− 3.09746i	− 0.132438i	−0.997805	−	0.0662190i	$-0.978906\pi$
$547$	− 3.09746i	− 0.132438i	0.997805	−	0.0662190i	$-0.0210936\pi$
$548$	0	0
$549$	− 7.42550i	− 0.316913i
$550$	0	0
$551$	−3.22671	−0.137463
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 5.62463i	− 0.238752i
$556$	0	0
$557$	−14.5281	−0.615575	−0.307788	−	0.951455i	$-0.599589\pi$
$557$	−14.5281	−0.615575	−0.307788	+	0.951455i	$0.599589\pi$
$558$	0	0
$559$	43.1256	1.82402
$560$	0	0
$561$	−6.73096	−0.284182
$562$	0	0
$563$	19.8343	0.835914	0.417957	−	0.908467i	$-0.362746\pi$
$563$	19.8343	0.835914	0.417957	+	0.908467i	$0.362746\pi$
$564$	0	0
$565$	5.15340i	0.216805i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−33.2779	−1.39508	−0.697542	−	0.716544i	$-0.745723\pi$
$569$	−33.2779	−1.39508	−0.697542	+	0.716544i	$0.745723\pi$
$570$	0	0
$571$	7.00093i	0.292980i	0.989212	+	0.146490i	$0.0467976\pi$
$571$	7.00093i	0.292980i	−0.989212	+	0.146490i	$0.953202\pi$
$572$	0	0
$573$	7.33899i	0.306591i
$574$	0	0
$575$	7.80322i	0.325417i
$576$	0	0
$577$	− 27.9043i	− 1.16167i	−0.814021	−	0.580835i	$-0.802726\pi$
$577$	− 27.9043i	− 1.16167i	0.814021	−	0.580835i	$-0.197274\pi$
$578$	0	0
$579$	−13.8005	−0.573530
$580$	0	0
$581$	0	0
$582$	0	0
$583$	20.5866i	0.852609i
$584$	0	0
$585$	−9.90802	−0.409646
$586$	0	0
$587$	−13.4842	−0.556551	−0.278276	−	0.960501i	$-0.589763\pi$
$587$	−13.4842	−0.556551	−0.278276	+	0.960501i	$0.589763\pi$
$588$	0	0
$589$	11.3357	0.467081
$590$	0	0
$591$	11.0111	0.452936
$592$	0	0
$593$	19.3311i	0.793831i	0.917855	+	0.396916i	$0.129919\pi$
$593$	19.3311i	0.793831i	−0.917855	+	0.396916i	$0.870081\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.19075	−0.0896615
$598$	0	0
$599$	28.5773i	1.16764i	0.811884	+	0.583819i	$0.198442\pi$
$599$	28.5773i	1.16764i	−0.811884	+	0.583819i	$0.801558\pi$
$600$	0	0
$601$	7.82137i	0.319040i	0.987195	+	0.159520i	$0.0509947\pi$
$601$	7.82137i	0.319040i	−0.987195	+	0.159520i	$0.949005\pi$
$602$	0	0
$603$	− 32.1390i	− 1.30880i
$604$	0	0
$605$	9.39148i	0.381818i
$606$	0	0
$607$	23.8049	0.966212	0.483106	−	0.875562i	$-0.339509\pi$
$607$	23.8049	0.966212	0.483106	+	0.875562i	$0.339509\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	34.0635i	1.37806i
$612$	0	0
$613$	35.8124	1.44645	0.723225	−	0.690612i	$-0.242659\pi$
$613$	35.8124	1.44645	0.723225	+	0.690612i	$0.242659\pi$
$614$	0	0
$615$	−3.07063	−0.123820
$616$	0	0
$617$	12.7751	0.514306	0.257153	−	0.966371i	$-0.417216\pi$
$617$	12.7751	0.514306	0.257153	+	0.966371i	$0.417216\pi$
$618$	0	0
$619$	−7.33180	−0.294690	−0.147345	−	0.989085i	$-0.547073\pi$
$619$	−7.33180	−0.294690	−0.147345	+	0.989085i	$0.547073\pi$
$620$	0	0
$621$	8.73606i	0.350566i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.65685	0.266274
$626$	0	0
$627$	− 1.70191i	− 0.0679678i
$628$	0	0
$629$	− 33.7897i	− 1.34728i
$630$	0	0
$631$	34.3891i	1.36901i	0.729009	+	0.684504i	$0.239981\pi$
$631$	34.3891i	1.36901i	−0.729009	+	0.684504i	$0.760019\pi$
$632$	0	0
$633$	− 14.7584i	− 0.586593i
$634$	0	0
$635$	5.74141	0.227841
$636$	0	0
$637$	0	0
$638$	0	0
$639$	24.9348i	0.986407i
$640$	0	0
$641$	48.2191	1.90454	0.952270	−	0.305257i	$-0.0987423\pi$
$641$	48.2191	1.90454	0.952270	+	0.305257i	$0.0987423\pi$
$642$	0	0
$643$	−4.30600	−0.169812	−0.0849060	−	0.996389i	$-0.527059\pi$
$643$	−4.30600	−0.169812	−0.0849060	+	0.996389i	$0.527059\pi$
$644$	0	0
$645$	10.7157	0.421929
$646$	0	0
$647$	−37.3449	−1.46818	−0.734091	−	0.679052i	$-0.762391\pi$
$647$	−37.3449	−1.46818	−0.734091	+	0.679052i	$0.762391\pi$
$648$	0	0
$649$	10.6830i	0.419344i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.5460	1.03882	0.519412	−	0.854524i	$-0.326151\pi$
$653$	26.5460	1.03882	0.519412	+	0.854524i	$0.326151\pi$
$654$	0	0
$655$	21.8734i	0.854666i
$656$	0	0
$657$	− 5.90705i	− 0.230456i
$658$	0	0
$659$	− 30.3597i	− 1.18265i	−0.806435	−	0.591323i	$-0.798606\pi$
$659$	− 30.3597i	− 1.18265i	0.806435	−	0.591323i	$-0.201394\pi$
$660$	0	0
$661$	− 31.0221i	− 1.20662i	−0.797507	−	0.603310i	$-0.793848\pi$
$661$	− 31.0221i	− 1.20662i	0.797507	−	0.603310i	$-0.206152\pi$
$662$	0	0
$663$	13.9031	0.539951
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5.18944i	− 0.200936i
$668$	0	0
$669$	−2.63005	−0.101683
$670$	0	0
$671$	−5.19648	−0.200608
$672$	0	0
$673$	24.9527	0.961856	0.480928	−	0.876760i	$-0.340300\pi$
$673$	24.9527	0.961856	0.480928	+	0.876760i	$0.340300\pi$
$674$	0	0
$675$	14.9712	0.576243
$676$	0	0
$677$	3.88368i	0.149262i	0.997211	+	0.0746311i	$0.0237779\pi$
$677$	3.88368i	0.149262i	−0.997211	+	0.0746311i	$0.976222\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	2.29902	0.0880985
$682$	0	0
$683$	18.7434i	0.717197i	0.933492	+	0.358598i	$0.116745\pi$
$683$	18.7434i	0.717197i	−0.933492	+	0.358598i	$0.883255\pi$
$684$	0	0
$685$	− 10.5164i	− 0.401810i
$686$	0	0
$687$	5.56998i	0.212508i
$688$	0	0
$689$	− 42.5225i	− 1.61998i
$690$	0	0
$691$	1.08346	0.0412167	0.0206083	−	0.999788i	$-0.493440\pi$
$691$	1.08346	0.0412167	0.0206083	+	0.999788i	$0.493440\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	14.0121i	0.531508i
$696$	0	0
$697$	−18.4466	−0.698716
$698$	0	0
$699$	5.94042	0.224687
$700$	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
$703$	8.54366	0.322230
$704$	0	0
$705$	8.46397i	0.318772i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	39.3032	1.47606	0.738031	−	0.674767i	$-0.235756\pi$
$709$	39.3032	1.47606	0.738031	+	0.674767i	$0.235756\pi$
$710$	0	0
$711$	11.5389i	0.432744i
$712$	0	0
$713$	18.2310i	0.682756i
$714$	0	0
$715$	6.93378i	0.259309i
$716$	0	0
$717$	9.99424i	0.373242i
$718$	0	0
$719$	−33.2976	−1.24179	−0.620895	−	0.783894i	$-0.713231\pi$
$719$	−33.2976	−1.24179	−0.620895	+	0.783894i	$0.713231\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 2.26579i	− 0.0842657i
$724$	0	0
$725$	−8.89328	−0.330288
$726$	0	0
$727$	15.7046	0.582452	0.291226	−	0.956654i	$-0.405937\pi$
$727$	15.7046	0.582452	0.291226	+	0.956654i	$0.405937\pi$
$728$	0	0
$729$	−0.982135	−0.0363754
$730$	0	0
$731$	64.3739	2.38096
$732$	0	0
$733$	4.57177i	0.168862i	0.996429	+	0.0844311i	$0.0269073\pi$
$733$	4.57177i	0.168862i	−0.996429	+	0.0844311i	$0.973093\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22.4913	−0.828478
$738$	0	0
$739$	21.1932i	0.779605i	0.920898	+	0.389803i	$0.127457\pi$
$739$	21.1932i	0.779605i	−0.920898	+	0.389803i	$0.872543\pi$
$740$	0	0
$741$	3.51537i	0.129140i
$742$	0	0
$743$	− 18.7112i	− 0.686447i	−0.939254	−	0.343223i	$-0.888481\pi$
$743$	− 18.7112i	− 0.686447i	0.939254	−	0.343223i	$-0.111519\pi$
$744$	0	0
$745$	− 2.30937i	− 0.0846088i
$746$	0	0
$747$	−41.2792	−1.51033
$748$	0	0
$749$	0	0
$750$	0	0
$751$	49.9315i	1.82203i	0.412376	+	0.911014i	$0.364699\pi$
$751$	49.9315i	1.82203i	−0.412376	+	0.911014i	$0.635301\pi$
$752$	0	0
$753$	15.3878	0.560763
$754$	0	0
$755$	−5.95836	−0.216847
$756$	0	0
$757$	−24.3373	−0.884556	−0.442278	−	0.896878i	$-0.645830\pi$
$757$	−24.3373	−0.884556	−0.442278	+	0.896878i	$0.645830\pi$
$758$	0	0
$759$	2.73714	0.0993520
$760$	0	0
$761$	− 29.5562i	− 1.07141i	−0.844405	−	0.535706i	$-0.820046\pi$
$761$	− 29.5562i	− 1.07141i	0.844405	−	0.535706i	$-0.179954\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−14.7898	−0.534726
$766$	0	0
$767$	− 22.0662i	− 0.796763i
$768$	0	0
$769$	− 51.5243i	− 1.85801i	−0.370063	−	0.929007i	$-0.620664\pi$
$769$	− 51.5243i	− 1.85801i	0.370063	−	0.929007i	$-0.379336\pi$
$770$	0	0
$771$	− 15.5262i	− 0.559161i
$772$	0	0
$773$	− 43.1808i	− 1.55311i	−0.630052	−	0.776553i	$-0.716967\pi$
$773$	− 43.1808i	− 1.55311i	0.630052	−	0.776553i	$-0.283033\pi$
$774$	0	0
$775$	31.2429	1.12228
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 4.66420i	− 0.167112i
$780$	0	0
$781$	17.4498	0.624401
$782$	0	0
$783$	−9.95644	−0.355814
$784$	0	0
$785$	−13.9674	−0.498518
$786$	0	0
$787$	0.187446	0.00668173	0.00334087	−	0.999994i	$-0.498937\pi$
$787$	0.187446	0.00668173	0.00334087	+	0.999994i	$0.498937\pi$
$788$	0	0
$789$	20.3022i	0.722778i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.7335	0.381159
$794$	0	0
$795$	− 10.5658i	− 0.374731i
$796$	0	0
$797$	− 28.9136i	− 1.02417i	−0.858934	−	0.512086i	$-0.828873\pi$
$797$	− 28.9136i	− 1.02417i	0.858934	−	0.512086i	$-0.171127\pi$
$798$	0	0
$799$	50.8469i	1.79883i
$800$	0	0
$801$	− 18.5231i	− 0.654482i
$802$	0	0
$803$	−4.13384	−0.145880
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 14.3824i	− 0.506285i
$808$	0	0
$809$	34.5916	1.21618	0.608089	−	0.793869i	$-0.291937\pi$
$809$	34.5916	1.21618	0.608089	+	0.793869i	$0.291937\pi$
$810$	0	0
$811$	15.5799	0.547086	0.273543	−	0.961860i	$-0.411804\pi$
$811$	15.5799	0.547086	0.273543	+	0.961860i	$0.411804\pi$
$812$	0	0
$813$	−16.6137	−0.582669
$814$	0	0
$815$	4.81498	0.168661
$816$	0	0
$817$	16.2768i	0.569454i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.1077	−0.980964	−0.490482	−	0.871451i	$-0.663179\pi$
$821$	−28.1077	−0.980964	−0.490482	+	0.871451i	$0.663179\pi$
$822$	0	0
$823$	24.5479i	0.855688i	0.903853	+	0.427844i	$0.140727\pi$
$823$	24.5479i	0.855688i	−0.903853	+	0.427844i	$0.859273\pi$
$824$	0	0
$825$	− 4.69071i	− 0.163310i
$826$	0	0
$827$	27.2945i	0.949124i	0.880222	+	0.474562i	$0.157393\pi$
$827$	27.2945i	0.949124i	−0.880222	+	0.474562i	$0.842607\pi$
$828$	0	0
$829$	38.1466i	1.32489i	0.749112	+	0.662443i	$0.230480\pi$
$829$	38.1466i	1.32489i	−0.749112	+	0.662443i	$0.769520\pi$
$830$	0	0
$831$	3.09831	0.107479
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 14.0662i	− 0.486780i
$836$	0	0
$837$	34.9779	1.20901
$838$	0	0
$839$	55.9292	1.93089	0.965444	−	0.260609i	$-0.0839234\pi$
$839$	55.9292	1.93089	0.965444	+	0.260609i	$0.0839234\pi$
$840$	0	0
$841$	−23.0856	−0.796056
$842$	0	0
$843$	5.82677	0.200685
$844$	0	0
$845$	0.744229i	0.0256023i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	21.5354	0.739094
$850$	0	0
$851$	13.7406i	0.471020i
$852$	0	0
$853$	− 40.4997i	− 1.38668i	−0.720609	−	0.693342i	$-0.756138\pi$
$853$	− 40.4997i	− 1.38668i	0.720609	−	0.693342i	$-0.243862\pi$
$854$	0	0
$855$	− 3.73957i	− 0.127891i
$856$	0	0
$857$	15.2841i	0.522096i	0.965326	+	0.261048i	$0.0840681\pi$
$857$	15.2841i	0.522096i	−0.965326	+	0.261048i	$0.915932\pi$
$858$	0	0
$859$	−43.1409	−1.47195	−0.735975	−	0.677009i	$-0.763276\pi$
$859$	−43.1409	−1.47195	−0.735975	+	0.677009i	$0.763276\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 45.3575i	− 1.54399i	−0.635631	−	0.771993i	$-0.719260\pi$
$863$	− 45.3575i	− 1.54399i	0.635631	−	0.771993i	$-0.280740\pi$
$864$	0	0
$865$	17.7482	0.603458
$866$	0	0
$867$	7.94048	0.269673
$868$	0	0
$869$	8.07511	0.273929
$870$	0	0
$871$	46.4567	1.57413
$872$	0	0
$873$	29.3404i	0.993022i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33.2265	−1.12198	−0.560989	−	0.827823i	$-0.689579\pi$
$877$	−33.2265	−1.12198	−0.560989	+	0.827823i	$0.689579\pi$
$878$	0	0
$879$	− 5.11244i	− 0.172438i
$880$	0	0
$881$	22.4056i	0.754864i	0.926037	+	0.377432i	$0.123193\pi$
$881$	22.4056i	0.754864i	−0.926037	+	0.377432i	$0.876807\pi$
$882$	0	0
$883$	4.55462i	0.153275i	0.997059	+	0.0766376i	$0.0244184\pi$
$883$	4.55462i	0.153275i	−0.997059	+	0.0766376i	$0.975582\pi$
$884$	0	0
$885$	− 5.48291i	− 0.184306i
$886$	0	0
$887$	−23.6997	−0.795759	−0.397879	−	0.917438i	$-0.630254\pi$
$887$	−23.6997	−0.795759	−0.397879	+	0.917438i	$0.630254\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.16542i	0.240051i
$892$	0	0
$893$	−12.8566	−0.430228
$894$	0	0
$895$	12.1407	0.405819
$896$	0	0
$897$	−5.65368	−0.188771
$898$	0	0
$899$	−20.7777	−0.692977
$900$	0	0
$901$	− 63.4736i	− 2.11461i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.9999	0.698062
$906$	0	0
$907$	26.1380i	0.867897i	0.900938	+	0.433949i	$0.142880\pi$
$907$	26.1380i	0.867897i	−0.900938	+	0.433949i	$0.857120\pi$
$908$	0	0
$909$	39.5535i	1.31191i
$910$	0	0
$911$	42.5016i	1.40814i	0.710131	+	0.704070i	$0.248636\pi$
$911$	42.5016i	1.40814i	−0.710131	+	0.704070i	$0.751364\pi$
$912$	0	0
$913$	28.8878i	0.956046i
$914$	0	0
$915$	2.66703	0.0881692
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 19.9003i	− 0.656450i	−0.944600	−	0.328225i	$-0.893550\pi$
$919$	− 19.9003i	− 0.656450i	0.944600	−	0.328225i	$-0.106450\pi$
$920$	0	0
$921$	−10.1629	−0.334879
$922$	0	0
$923$	−36.0432	−1.18638
$924$	0	0
$925$	23.5476	0.774239
$926$	0	0
$927$	5.07312	0.166623
$928$	0	0
$929$	− 2.77903i	− 0.0911770i	−0.998960	−	0.0455885i	$-0.985484\pi$
$929$	− 2.77903i	− 0.0911770i	0.998960	−	0.0455885i	$-0.0145163\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.92172	0.324822
$934$	0	0
$935$	10.3501i	0.338485i
$936$	0	0
$937$	− 41.9794i	− 1.37141i	−0.727881	−	0.685703i	$-0.759495\pi$
$937$	− 41.9794i	− 1.37141i	0.727881	−	0.685703i	$-0.240505\pi$
$938$	0	0
$939$	11.3009i	0.368792i
$940$	0	0
$941$	− 41.7187i	− 1.35999i	−0.733216	−	0.679996i	$-0.761982\pi$
$941$	− 41.7187i	− 1.35999i	0.733216	−	0.679996i	$-0.238018\pi$
$942$	0	0
$943$	7.50131	0.244276
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.6025i	1.22192i	0.791662	+	0.610959i	$0.209216\pi$
$947$	37.6025i	1.22192i	−0.791662	+	0.610959i	$0.790784\pi$
$948$	0	0
$949$	8.53861	0.277175
$950$	0	0
$951$	8.45459	0.274159
$952$	0	0
$953$	−10.1808	−0.329788	−0.164894	−	0.986311i	$-0.552728\pi$
$953$	−10.1808	−0.329788	−0.164894	+	0.986311i	$0.552728\pi$
$954$	0	0
$955$	11.2851	0.365176
$956$	0	0
$957$	3.11950i	0.100839i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	41.9942	1.35465
$962$	0	0
$963$	− 9.83737i	− 0.317005i
$964$	0	0
$965$	21.2208i	0.683123i
$966$	0	0
$967$	− 24.0102i	− 0.772116i	−0.922474	−	0.386058i	$-0.873836\pi$
$967$	− 24.0102i	− 0.772116i	0.922474	−	0.386058i	$-0.126164\pi$
$968$	0	0
$969$	5.24742i	0.168571i
$970$	0	0
$971$	21.0210	0.674598	0.337299	−	0.941398i	$-0.390487\pi$
$971$	21.0210	0.674598	0.337299	+	0.941398i	$0.390487\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	9.68887i	0.310292i
$976$	0	0
$977$	4.55336	0.145675	0.0728375	−	0.997344i	$-0.476795\pi$
$977$	4.55336	0.145675	0.0728375	+	0.997344i	$0.476795\pi$
$978$	0	0
$979$	−12.9627	−0.414291
$980$	0	0
$981$	5.71627	0.182507
$982$	0	0
$983$	−18.3730	−0.586008	−0.293004	−	0.956111i	$-0.594655\pi$
$983$	−18.3730	−0.586008	−0.293004	+	0.956111i	$0.594655\pi$
$984$	0	0
$985$	− 16.9316i	− 0.539486i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26.1776	−0.832399
$990$	0	0
$991$	− 7.04632i	− 0.223834i	−0.993718	−	0.111917i	$-0.964301\pi$
$991$	− 7.04632i	− 0.223834i	0.993718	−	0.111917i	$-0.0356990\pi$
$992$	0	0
$993$	− 7.86632i	− 0.249630i
$994$	0	0
$995$	3.36869i	0.106795i
$996$	0	0
$997$	− 22.8367i	− 0.723246i	−0.932324	−	0.361623i	$-0.882223\pi$
$997$	− 22.8367i	− 0.723246i	0.932324	−	0.361623i	$-0.117777\pi$
$998$	0	0
$999$	26.3626	0.834075

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.f.b.1567.6		16
4.3	odd	2	inner	1568.2.f.b.1567.12		16
7.2	even	3		224.2.p.a.31.6	yes	16
7.3	odd	6		224.2.p.a.159.3	yes	16
7.4	even	3		1568.2.p.b.607.6		16
7.5	odd	6		1568.2.p.b.31.3		16
7.6	odd	2	inner	1568.2.f.b.1567.11		16
8.3	odd	2		3136.2.f.j.3135.5		16
8.5	even	2		3136.2.f.j.3135.11		16
21.2	odd	6		2016.2.cs.b.703.4		16
21.17	even	6		2016.2.cs.b.1279.3		16
28.3	even	6		224.2.p.a.159.6	yes	16
28.11	odd	6		1568.2.p.b.607.3		16
28.19	even	6		1568.2.p.b.31.6		16
28.23	odd	6		224.2.p.a.31.3	✓	16
28.27	even	2	inner	1568.2.f.b.1567.5		16
56.3	even	6		448.2.p.e.383.3		16
56.13	odd	2		3136.2.f.j.3135.6		16
56.27	even	2		3136.2.f.j.3135.12		16
56.37	even	6		448.2.p.e.255.3		16
56.45	odd	6		448.2.p.e.383.6		16
56.51	odd	6		448.2.p.e.255.6		16
84.23	even	6		2016.2.cs.b.703.3		16
84.59	odd	6		2016.2.cs.b.1279.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.p.a.31.3	✓	16	28.23	odd	6
224.2.p.a.31.6	yes	16	7.2	even	3
224.2.p.a.159.3	yes	16	7.3	odd	6
224.2.p.a.159.6	yes	16	28.3	even	6
448.2.p.e.255.3		16	56.37	even	6
448.2.p.e.255.6		16	56.51	odd	6
448.2.p.e.383.3		16	56.3	even	6
448.2.p.e.383.6		16	56.45	odd	6
1568.2.f.b.1567.5		16	28.27	even	2	inner
1568.2.f.b.1567.6		16	1.1	even	1	trivial
1568.2.f.b.1567.11		16	7.6	odd	2	inner
1568.2.f.b.1567.12		16	4.3	odd	2	inner
1568.2.p.b.31.3		16	7.5	odd	6
1568.2.p.b.31.6		16	28.19	even	6
1568.2.p.b.607.3		16	28.11	odd	6
1568.2.p.b.607.6		16	7.4	even	3
2016.2.cs.b.703.3		16	84.23	even	6
2016.2.cs.b.703.4		16	21.2	odd	6
2016.2.cs.b.1279.3		16	21.17	even	6
2016.2.cs.b.1279.4		16	84.59	odd	6
3136.2.f.j.3135.5		16	8.3	odd	2
3136.2.f.j.3135.6		16	56.13	odd	2
3136.2.f.j.3135.11		16	8.5	even	2
3136.2.f.j.3135.12		16	56.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.753692 q^{3} +1.15894i q^{5} -2.43195 q^{9} +O(q^{10})\)	\(q-0.753692 q^{3} +1.15894i q^{5} -2.43195 q^{9} +1.70191i q^{11} -3.51537i q^{13} -0.873485i q^{15} -5.24742i q^{17} +1.32680 q^{19} +2.13386i q^{23} +3.65685 q^{25} +4.09402 q^{27} -2.43195 q^{29} +8.54366 q^{31} -1.28272i q^{33} +6.43929 q^{37} +2.64951i q^{39} -3.51537i q^{41} +12.2677i q^{43} -2.81849i q^{45} -9.68988 q^{47} +3.95494i q^{51} +12.0961 q^{53} -1.97242 q^{55} -1.00000 q^{57} +6.27705 q^{59} +3.05332i q^{61} +4.07411 q^{65} +13.2153i q^{67} -1.60827i q^{69} -10.2530i q^{71} +2.42894i q^{73} -2.75614 q^{75} -4.74473i q^{79} +4.21021 q^{81} +16.9737 q^{83} +6.08146 q^{85} +1.83294 q^{87} +7.61658i q^{89} -6.43929 q^{93} +1.53769i q^{95} -12.0646i q^{97} -4.13896i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^9	\( 16 q + 16 q^{9} - 32 q^{25} + 16 q^{29} + 16 q^{37} + 16 q^{53} - 16 q^{57} - 16 q^{65} + 96 q^{81} - 16 q^{85} - 16 q^{93}+O(q^{100}) \) 16 * q + 16 * q^9 - 32 * q^25 + 16 * q^29 + 16 * q^37 + 16 * q^53 - 16 * q^57 - 16 * q^65 + 96 * q^81 - 16 * q^85 - 16 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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