LMFDB - Embedded newform 1568.3.h.a.881.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(881,1568)]

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([0, 1, 1]))

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

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

chi := DirichletCharacter("1568.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.h (of order $2$, degree $1$, 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:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.6
Character	$\chi$	$=$	1568.881
Dual form			1568.3.h.a.881.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.86988 q^{3} +4.67764 q^{5} +5.97596 q^{9} +O(q^{10})$	$q-3.86988 q^{3} +4.67764 q^{5} +5.97596 q^{9} -14.5934i q^{11} +12.7102 q^{13} -18.1019 q^{15} -19.5223i q^{17} -17.7247 q^{19} -8.86075 q^{23} -3.11968 q^{25} +11.7027 q^{27} +35.4981i q^{29} -29.0213i q^{31} +56.4747i q^{33} -12.2169i q^{37} -49.1868 q^{39} -22.0903i q^{41} +79.8001i q^{43} +27.9534 q^{45} -42.1513i q^{47} +75.5490i q^{51} -36.1532i q^{53} -68.2627i q^{55} +68.5923 q^{57} +2.40696 q^{59} +29.2449 q^{61} +59.4536 q^{65} +40.6804i q^{67} +34.2900 q^{69} +22.6174 q^{71} +76.3935i q^{73} +12.0728 q^{75} -136.803 q^{79} -99.0715 q^{81} +49.9942 q^{83} -91.3184i q^{85} -137.373i q^{87} -1.12009i q^{89} +112.309i q^{93} -82.9096 q^{95} -158.827i q^{97} -87.2097i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 64 q^{9}+O(q^{10}) $ 28 * q + 64 * q^9	$ 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) $ 28 * q + 64 * q^9 - 28 * q^15 + 60 * q^23 + 64 * q^25 - 40 * q^39 + 124 * q^57 + 104 * q^65 + 136 * q^71 - 324 * q^79 + 36 * q^81 + 580 * q^95

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
$3$	−3.86988	−1.28996	−0.644980	−	0.764200i	$-0.723134\pi$
$3$	−3.86988	−1.28996	−0.644980	+	0.764200i	$0.723134\pi$
$4$	0	0
$5$	4.67764	0.935528	0.467764	−	0.883853i	$-0.345060\pi$
$5$	4.67764	0.935528	0.467764	+	0.883853i	$0.345060\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.97596	0.663996
$10$	0	0
$11$	− 14.5934i	− 1.32667i	−0.748321	−	0.663337i	$-0.769140\pi$
$11$	− 14.5934i	− 1.32667i	0.748321	−	0.663337i	$-0.230860\pi$
$12$	0	0
$13$	12.7102	0.977705	0.488853	−	0.872366i	$-0.337416\pi$
$13$	12.7102	0.977705	0.488853	+	0.872366i	$0.337416\pi$
$14$	0	0
$15$	−18.1019	−1.20679
$16$	0	0
$17$	− 19.5223i	− 1.14837i	−0.818725	−	0.574186i	$-0.805319\pi$
$17$	− 19.5223i	− 1.14837i	0.818725	−	0.574186i	$-0.194681\pi$
$18$	0	0
$19$	−17.7247	−0.932877	−0.466439	−	0.884554i	$-0.654463\pi$
$19$	−17.7247	−0.932877	−0.466439	+	0.884554i	$0.654463\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.86075	−0.385250	−0.192625	−	0.981272i	$-0.561700\pi$
$23$	−8.86075	−0.385250	−0.192625	+	0.981272i	$0.561700\pi$
$24$	0	0
$25$	−3.11968	−0.124787
$26$	0	0
$27$	11.7027	0.433432
$28$	0	0
$29$	35.4981i	1.22407i	0.790830	+	0.612036i	$0.209649\pi$
$29$	35.4981i	1.22407i	−0.790830	+	0.612036i	$0.790351\pi$
$30$	0	0
$31$	− 29.0213i	− 0.936170i	−0.883684	−	0.468085i	$-0.844944\pi$
$31$	− 29.0213i	− 0.936170i	0.883684	−	0.468085i	$-0.155056\pi$
$32$	0	0
$33$	56.4747i	1.71136i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 12.2169i	− 0.330188i	−0.986278	−	0.165094i	$-0.947207\pi$
$37$	− 12.2169i	− 0.330188i	0.986278	−	0.165094i	$-0.0527927\pi$
$38$	0	0
$39$	−49.1868	−1.26120
$40$	0	0
$41$	− 22.0903i	− 0.538788i	−0.963030	−	0.269394i	$-0.913177\pi$
$41$	− 22.0903i	− 0.538788i	0.963030	−	0.269394i	$-0.0868233\pi$
$42$	0	0
$43$	79.8001i	1.85582i	0.372809	+	0.927908i	$0.378395\pi$
$43$	79.8001i	1.85582i	−0.372809	+	0.927908i	$0.621605\pi$
$44$	0	0
$45$	27.9534	0.621187
$46$	0	0
$47$	− 42.1513i	− 0.896836i	−0.893824	−	0.448418i	$-0.851988\pi$
$47$	− 42.1513i	− 0.896836i	0.893824	−	0.448418i	$-0.148012\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	75.5490i	1.48135i
$52$	0	0
$53$	− 36.1532i	− 0.682137i	−0.940039	−	0.341068i	$-0.889211\pi$
$53$	− 36.1532i	− 0.682137i	0.940039	−	0.341068i	$-0.110789\pi$
$54$	0	0
$55$	− 68.2627i	− 1.24114i
$56$	0	0
$57$	68.5923	1.20337
$58$	0	0
$59$	2.40696	0.0407959	0.0203979	−	0.999792i	$-0.493507\pi$
$59$	2.40696	0.0407959	0.0203979	+	0.999792i	$0.493507\pi$
$60$	0	0
$61$	29.2449	0.479424	0.239712	−	0.970844i	$-0.422947\pi$
$61$	29.2449	0.479424	0.239712	+	0.970844i	$0.422947\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	59.4536	0.914671
$66$	0	0
$67$	40.6804i	0.607170i	0.952804	+	0.303585i	$0.0981837\pi$
$67$	40.6804i	0.607170i	−0.952804	+	0.303585i	$0.901816\pi$
$68$	0	0
$69$	34.2900	0.496957
$70$	0	0
$71$	22.6174	0.318554	0.159277	−	0.987234i	$-0.449084\pi$
$71$	22.6174	0.318554	0.159277	+	0.987234i	$0.449084\pi$
$72$	0	0
$73$	76.3935i	1.04649i	0.852184	+	0.523243i	$0.175278\pi$
$73$	76.3935i	1.04649i	−0.852184	+	0.523243i	$0.824722\pi$
$74$	0	0
$75$	12.0728	0.160970
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−136.803	−1.73168	−0.865840	−	0.500321i	$-0.833215\pi$
$79$	−136.803	−1.73168	−0.865840	+	0.500321i	$0.833215\pi$
$80$	0	0
$81$	−99.0715	−1.22311
$82$	0	0
$83$	49.9942	0.602340	0.301170	−	0.953571i	$-0.402623\pi$
$83$	49.9942	0.602340	0.301170	+	0.953571i	$0.402623\pi$
$84$	0	0
$85$	− 91.3184i	− 1.07433i
$86$	0	0
$87$	− 137.373i	− 1.57900i
$88$	0	0
$89$	− 1.12009i	− 0.0125852i	−0.999980	−	0.00629262i	$-0.997997\pi$
$89$	− 1.12009i	− 0.0125852i	0.999980	−	0.00629262i	$-0.00200302\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	112.309i	1.20762i
$94$	0	0
$95$	−82.9096	−0.872733
$96$	0	0
$97$	− 158.827i	− 1.63740i	−0.574225	−	0.818698i	$-0.694696\pi$
$97$	− 158.827i	− 1.63740i	0.574225	−	0.818698i	$-0.305304\pi$
$98$	0	0
$99$	− 87.2097i	− 0.880906i
$100$	0	0
$101$	−69.6244	−0.689350	−0.344675	−	0.938722i	$-0.612011\pi$
$101$	−69.6244	−0.689350	−0.344675	+	0.938722i	$0.612011\pi$
$102$	0	0
$103$	19.6197i	0.190482i	0.995454	+	0.0952411i	$0.0303622\pi$
$103$	19.6197i	0.190482i	−0.995454	+	0.0952411i	$0.969638\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 44.7493i	− 0.418218i	−0.977892	−	0.209109i	$-0.932944\pi$
$107$	− 44.7493i	− 0.418218i	0.977892	−	0.209109i	$-0.0670563\pi$
$108$	0	0
$109$	56.6045i	0.519308i	0.965702	+	0.259654i	$0.0836084\pi$
$109$	56.6045i	0.519308i	−0.965702	+	0.259654i	$0.916392\pi$
$110$	0	0
$111$	47.2781i	0.425929i
$112$	0	0
$113$	−188.632	−1.66931	−0.834657	−	0.550770i	$-0.814334\pi$
$113$	−188.632	−1.66931	−0.834657	+	0.550770i	$0.814334\pi$
$114$	0	0
$115$	−41.4474	−0.360412
$116$	0	0
$117$	75.9555	0.649192
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−91.9677	−0.760063
$122$	0	0
$123$	85.4868i	0.695014i
$124$	0	0
$125$	−131.534	−1.05227
$126$	0	0
$127$	−45.8547	−0.361060	−0.180530	−	0.983569i	$-0.557781\pi$
$127$	−45.8547	−0.361060	−0.180530	+	0.983569i	$0.557781\pi$
$128$	0	0
$129$	− 308.817i	− 2.39393i
$130$	0	0
$131$	120.996	0.923637	0.461818	−	0.886975i	$-0.347197\pi$
$131$	120.996	0.923637	0.461818	+	0.886975i	$0.347197\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	54.7408	0.405488
$136$	0	0
$137$	−145.241	−1.06016	−0.530078	−	0.847949i	$-0.677837\pi$
$137$	−145.241	−1.06016	−0.530078	+	0.847949i	$0.677837\pi$
$138$	0	0
$139$	−86.3503	−0.621225	−0.310613	−	0.950537i	$-0.600534\pi$
$139$	−86.3503	−0.621225	−0.310613	+	0.950537i	$0.600534\pi$
$140$	0	0
$141$	163.120i	1.15688i
$142$	0	0
$143$	− 185.485i	− 1.29710i
$144$	0	0
$145$	166.047i	1.14515i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 218.857i	− 1.46884i	−0.678695	−	0.734421i	$-0.737454\pi$
$149$	− 218.857i	− 1.46884i	0.678695	−	0.734421i	$-0.262546\pi$
$150$	0	0
$151$	−83.3104	−0.551725	−0.275862	−	0.961197i	$-0.588963\pi$
$151$	−83.3104	−0.551725	−0.275862	+	0.961197i	$0.588963\pi$
$152$	0	0
$153$	− 116.665i	− 0.762514i
$154$	0	0
$155$	− 135.751i	− 0.875813i
$156$	0	0
$157$	35.4415	0.225742	0.112871	−	0.993610i	$-0.463995\pi$
$157$	35.4415	0.225742	0.112871	+	0.993610i	$0.463995\pi$
$158$	0	0
$159$	139.909i	0.879929i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 10.4548i	− 0.0641399i	−0.999486	−	0.0320699i	$-0.989790\pi$
$163$	− 10.4548i	− 0.0641399i	0.999486	−	0.0320699i	$-0.0102099\pi$
$164$	0	0
$165$	264.169i	1.60102i
$166$	0	0
$167$	− 78.8843i	− 0.472361i	−0.971709	−	0.236181i	$-0.924104\pi$
$167$	− 78.8843i	− 0.472361i	0.971709	−	0.236181i	$-0.0758957\pi$
$168$	0	0
$169$	−7.45163	−0.0440925
$170$	0	0
$171$	−105.922	−0.619427
$172$	0	0
$173$	−191.842	−1.10891	−0.554456	−	0.832213i	$-0.687073\pi$
$173$	−191.842	−1.10891	−0.554456	+	0.832213i	$0.687073\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.31463	−0.0526250
$178$	0	0
$179$	− 69.9086i	− 0.390551i	−0.980748	−	0.195275i	$-0.937440\pi$
$179$	− 69.9086i	− 0.390551i	0.980748	−	0.195275i	$-0.0625601\pi$
$180$	0	0
$181$	−343.635	−1.89853	−0.949267	−	0.314472i	$-0.898173\pi$
$181$	−343.635	−1.89853	−0.949267	+	0.314472i	$0.898173\pi$
$182$	0	0
$183$	−113.174	−0.618438
$184$	0	0
$185$	− 57.1465i	− 0.308900i
$186$	0	0
$187$	−284.897	−1.52351
$188$	0	0
$189$	0	0
$190$	0	0
$191$	139.779	0.731827	0.365913	−	0.930649i	$-0.380757\pi$
$191$	139.779	0.731827	0.365913	+	0.930649i	$0.380757\pi$
$192$	0	0
$193$	22.9233	0.118773	0.0593867	−	0.998235i	$-0.481086\pi$
$193$	22.9233	0.118773	0.0593867	+	0.998235i	$0.481086\pi$
$194$	0	0
$195$	−230.078	−1.17989
$196$	0	0
$197$	− 287.788i	− 1.46085i	−0.682992	−	0.730426i	$-0.739322\pi$
$197$	− 287.788i	− 1.46085i	0.682992	−	0.730426i	$-0.260678\pi$
$198$	0	0
$199$	65.4821i	0.329056i	0.986372	+	0.164528i	$0.0526100\pi$
$199$	65.4821i	0.329056i	−0.986372	+	0.164528i	$0.947390\pi$
$200$	0	0
$201$	− 157.428i	− 0.783225i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 103.330i	− 0.504051i
$206$	0	0
$207$	−52.9515	−0.255805
$208$	0	0
$209$	258.663i	1.23762i
$210$	0	0
$211$	17.8985i	0.0848270i	0.999100	+	0.0424135i	$0.0135047\pi$
$211$	17.8985i	0.0848270i	−0.999100	+	0.0424135i	$0.986495\pi$
$212$	0	0
$213$	−87.5265	−0.410922
$214$	0	0
$215$	373.276i	1.73617i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 295.633i	− 1.34992i
$220$	0	0
$221$	− 248.132i	− 1.12277i
$222$	0	0
$223$	− 258.973i	− 1.16132i	−0.814148	−	0.580658i	$-0.802795\pi$
$223$	− 258.973i	− 1.16132i	0.814148	−	0.580658i	$-0.197205\pi$
$224$	0	0
$225$	−18.6431	−0.0828581
$226$	0	0
$227$	−117.344	−0.516935	−0.258468	−	0.966020i	$-0.583218\pi$
$227$	−117.344	−0.516935	−0.258468	+	0.966020i	$0.583218\pi$
$228$	0	0
$229$	−87.0949	−0.380327	−0.190164	−	0.981752i	$-0.560902\pi$
$229$	−87.0949	−0.380327	−0.190164	+	0.981752i	$0.560902\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−311.651	−1.33756	−0.668778	−	0.743462i	$-0.733183\pi$
$233$	−311.651	−1.33756	−0.668778	+	0.743462i	$0.733183\pi$
$234$	0	0
$235$	− 197.169i	− 0.839015i
$236$	0	0
$237$	529.410	2.23380
$238$	0	0
$239$	140.823	0.589218	0.294609	−	0.955618i	$-0.404811\pi$
$239$	140.823	0.589218	0.294609	+	0.955618i	$0.404811\pi$
$240$	0	0
$241$	− 48.0871i	− 0.199532i	−0.995011	−	0.0997658i	$-0.968191\pi$
$241$	− 48.0871i	− 0.199532i	0.995011	−	0.0997658i	$-0.0318094\pi$
$242$	0	0
$243$	278.071	1.14432
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−225.283	−0.912079
$248$	0	0
$249$	−193.471	−0.776994
$250$	0	0
$251$	−152.080	−0.605895	−0.302947	−	0.953007i	$-0.597971\pi$
$251$	−152.080	−0.605895	−0.302947	+	0.953007i	$0.597971\pi$
$252$	0	0
$253$	129.309i	0.511101i
$254$	0	0
$255$	353.391i	1.38585i
$256$	0	0
$257$	− 145.669i	− 0.566807i	−0.959001	−	0.283403i	$-0.908536\pi$
$257$	− 145.669i	− 0.566807i	0.959001	−	0.283403i	$-0.0914635\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	212.135i	0.812779i
$262$	0	0
$263$	−353.393	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$263$	−353.393	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$264$	0	0
$265$	− 169.112i	− 0.638158i
$266$	0	0
$267$	4.33460i	0.0162345i
$268$	0	0
$269$	−133.298	−0.495532	−0.247766	−	0.968820i	$-0.579696\pi$
$269$	−133.298	−0.495532	−0.247766	+	0.968820i	$0.579696\pi$
$270$	0	0
$271$	− 376.827i	− 1.39051i	−0.718765	−	0.695253i	$-0.755292\pi$
$271$	− 376.827i	− 1.39051i	0.718765	−	0.695253i	$-0.244708\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	45.5267i	0.165552i
$276$	0	0
$277$	− 77.9178i	− 0.281292i	−0.990060	−	0.140646i	$-0.955082\pi$
$277$	− 77.9178i	− 0.281292i	0.990060	−	0.140646i	$-0.0449179\pi$
$278$	0	0
$279$	− 173.430i	− 0.621613i
$280$	0	0
$281$	324.564	1.15503	0.577516	−	0.816380i	$-0.304022\pi$
$281$	324.564	1.15503	0.577516	+	0.816380i	$0.304022\pi$
$282$	0	0
$283$	299.482	1.05824	0.529119	−	0.848547i	$-0.322522\pi$
$283$	299.482	1.05824	0.529119	+	0.848547i	$0.322522\pi$
$284$	0	0
$285$	320.850	1.12579
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−92.1208	−0.318757
$290$	0	0
$291$	614.643i	2.11217i
$292$	0	0
$293$	−81.7250	−0.278925	−0.139463	−	0.990227i	$-0.544537\pi$
$293$	−81.7250	−0.278925	−0.139463	+	0.990227i	$0.544537\pi$
$294$	0	0
$295$	11.2589	0.0381657
$296$	0	0
$297$	− 170.782i	− 0.575022i
$298$	0	0
$299$	−112.622	−0.376661
$300$	0	0
$301$	0	0
$302$	0	0
$303$	269.438	0.889234
$304$	0	0
$305$	136.797	0.448515
$306$	0	0
$307$	361.930	1.17892	0.589462	−	0.807796i	$-0.299340\pi$
$307$	361.930	1.17892	0.589462	+	0.807796i	$0.299340\pi$
$308$	0	0
$309$	− 75.9257i	− 0.245714i
$310$	0	0
$311$	− 188.803i	− 0.607085i	−0.952818	−	0.303542i	$-0.901831\pi$
$311$	− 188.803i	− 0.607085i	0.952818	−	0.303542i	$-0.0981694\pi$
$312$	0	0
$313$	25.6878i	0.0820697i	0.999158	+	0.0410349i	$0.0130655\pi$
$313$	25.6878i	0.0820697i	−0.999158	+	0.0410349i	$0.986935\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	499.128i	1.57454i	0.616611	+	0.787268i	$0.288505\pi$
$317$	499.128i	1.57454i	−0.616611	+	0.787268i	$0.711495\pi$
$318$	0	0
$319$	518.038	1.62394
$320$	0	0
$321$	173.174i	0.539484i
$322$	0	0
$323$	346.027i	1.07129i
$324$	0	0
$325$	−39.6516	−0.122005
$326$	0	0
$327$	− 219.053i	− 0.669886i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	249.859i	0.754861i	0.926038	+	0.377431i	$0.123192\pi$
$331$	249.859i	0.754861i	−0.926038	+	0.377431i	$0.876808\pi$
$332$	0	0
$333$	− 73.0080i	− 0.219243i
$334$	0	0
$335$	190.288i	0.568025i
$336$	0	0
$337$	−84.4039	−0.250457	−0.125228	−	0.992128i	$-0.539966\pi$
$337$	−84.4039	−0.250457	−0.125228	+	0.992128i	$0.539966\pi$
$338$	0	0
$339$	729.985	2.15335
$340$	0	0
$341$	−423.519	−1.24199
$342$	0	0
$343$	0	0
$344$	0	0
$345$	160.396	0.464917
$346$	0	0
$347$	− 377.249i	− 1.08717i	−0.839354	−	0.543586i	$-0.817066\pi$
$347$	− 377.249i	− 1.08717i	0.839354	−	0.543586i	$-0.182934\pi$
$348$	0	0
$349$	400.193	1.14668	0.573342	−	0.819316i	$-0.305647\pi$
$349$	400.193	1.14668	0.573342	+	0.819316i	$0.305647\pi$
$350$	0	0
$351$	148.743	0.423768
$352$	0	0
$353$	295.942i	0.838363i	0.907903	+	0.419181i	$0.137683\pi$
$353$	295.942i	0.838363i	−0.907903	+	0.419181i	$0.862317\pi$
$354$	0	0
$355$	105.796	0.298017
$356$	0	0
$357$	0	0
$358$	0	0
$359$	205.174	0.571515	0.285757	−	0.958302i	$-0.407755\pi$
$359$	205.174	0.571515	0.285757	+	0.958302i	$0.407755\pi$
$360$	0	0
$361$	−46.8362	−0.129740
$362$	0	0
$363$	355.904	0.980451
$364$	0	0
$365$	357.341i	0.979017i
$366$	0	0
$367$	353.588i	0.963456i	0.876321	+	0.481728i	$0.159991\pi$
$367$	353.588i	0.963456i	−0.876321	+	0.481728i	$0.840009\pi$
$368$	0	0
$369$	− 132.011i	− 0.357753i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 358.028i	− 0.959862i	−0.877306	−	0.479931i	$-0.840662\pi$
$373$	− 358.028i	− 0.959862i	0.877306	−	0.479931i	$-0.159338\pi$
$374$	0	0
$375$	509.020	1.35739
$376$	0	0
$377$	451.187i	1.19678i
$378$	0	0
$379$	514.679i	1.35799i	0.734142	+	0.678996i	$0.237585\pi$
$379$	514.679i	1.35799i	−0.734142	+	0.678996i	$0.762415\pi$
$380$	0	0
$381$	177.452	0.465753
$382$	0	0
$383$	519.838i	1.35728i	0.734472	+	0.678639i	$0.237430\pi$
$383$	519.838i	1.35728i	−0.734472	+	0.678639i	$0.762570\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	476.882i	1.23225i
$388$	0	0
$389$	105.456i	0.271096i	0.990771	+	0.135548i	$0.0432795\pi$
$389$	105.456i	0.271096i	−0.990771	+	0.135548i	$0.956721\pi$
$390$	0	0
$391$	172.982i	0.442410i
$392$	0	0
$393$	−468.241	−1.19145
$394$	0	0
$395$	−639.914	−1.62004
$396$	0	0
$397$	371.524	0.935828	0.467914	−	0.883774i	$-0.345006\pi$
$397$	371.524	0.935828	0.467914	+	0.883774i	$0.345006\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	299.272	0.746314	0.373157	−	0.927768i	$-0.378275\pi$
$401$	299.272	0.746314	0.373157	+	0.927768i	$0.378275\pi$
$402$	0	0
$403$	− 368.865i	− 0.915298i
$404$	0	0
$405$	−463.421	−1.14425
$406$	0	0
$407$	−178.287	−0.438052
$408$	0	0
$409$	− 20.3701i	− 0.0498046i	−0.999690	−	0.0249023i	$-0.992073\pi$
$409$	− 20.3701i	− 0.0498046i	0.999690	−	0.0249023i	$-0.00792747\pi$
$410$	0	0
$411$	562.067	1.36756
$412$	0	0
$413$	0	0
$414$	0	0
$415$	233.855	0.563506
$416$	0	0
$417$	334.165	0.801355
$418$	0	0
$419$	183.085	0.436956	0.218478	−	0.975842i	$-0.429891\pi$
$419$	183.085	0.436956	0.218478	+	0.975842i	$0.429891\pi$
$420$	0	0
$421$	293.022i	0.696014i	0.937492	+	0.348007i	$0.113141\pi$
$421$	293.022i	0.696014i	−0.937492	+	0.348007i	$0.886859\pi$
$422$	0	0
$423$	− 251.894i	− 0.595495i
$424$	0	0
$425$	60.9033i	0.143302i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	717.803i	1.67320i
$430$	0	0
$431$	75.2217	0.174528	0.0872641	−	0.996185i	$-0.472188\pi$
$431$	75.2217	0.174528	0.0872641	+	0.996185i	$0.472188\pi$
$432$	0	0
$433$	− 506.209i	− 1.16907i	−0.811367	−	0.584536i	$-0.801276\pi$
$433$	− 506.209i	− 1.16907i	0.811367	−	0.584536i	$-0.198724\pi$
$434$	0	0
$435$	− 642.583i	− 1.47720i
$436$	0	0
$437$	157.054	0.359391
$438$	0	0
$439$	− 163.226i	− 0.371813i	−0.982567	−	0.185907i	$-0.940478\pi$
$439$	− 163.226i	− 0.371813i	0.982567	−	0.185907i	$-0.0595222\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 113.707i	− 0.256675i	−0.991731	−	0.128338i	$-0.959036\pi$
$443$	− 113.707i	− 0.256675i	0.991731	−	0.128338i	$-0.0409641\pi$
$444$	0	0
$445$	− 5.23936i	− 0.0117738i
$446$	0	0
$447$	846.952i	1.89475i
$448$	0	0
$449$	391.120	0.871091	0.435546	−	0.900167i	$-0.356555\pi$
$449$	391.120	0.871091	0.435546	+	0.900167i	$0.356555\pi$
$450$	0	0
$451$	−322.373	−0.714796
$452$	0	0
$453$	322.401	0.711703
$454$	0	0
$455$	0	0
$456$	0	0
$457$	573.786	1.25555	0.627774	−	0.778396i	$-0.283966\pi$
$457$	573.786	1.25555	0.627774	+	0.778396i	$0.283966\pi$
$458$	0	0
$459$	− 228.463i	− 0.497741i
$460$	0	0
$461$	847.131	1.83759	0.918797	−	0.394729i	$-0.129162\pi$
$461$	847.131	1.83759	0.918797	+	0.394729i	$0.129162\pi$
$462$	0	0
$463$	−109.055	−0.235539	−0.117770	−	0.993041i	$-0.537574\pi$
$463$	−109.055	−0.235539	−0.117770	+	0.993041i	$0.537574\pi$
$464$	0	0
$465$	525.340i	1.12976i
$466$	0	0
$467$	−642.410	−1.37561	−0.687805	−	0.725895i	$-0.741426\pi$
$467$	−642.410	−1.37561	−0.687805	+	0.725895i	$0.741426\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−137.154	−0.291198
$472$	0	0
$473$	1164.56	2.46206
$474$	0	0
$475$	55.2952	0.116411
$476$	0	0
$477$	− 216.050i	− 0.452936i
$478$	0	0
$479$	424.825i	0.886899i	0.896299	+	0.443449i	$0.146245\pi$
$479$	424.825i	0.886899i	−0.896299	+	0.443449i	$0.853755\pi$
$480$	0	0
$481$	− 155.279i	− 0.322826i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 742.937i	− 1.53183i
$486$	0	0
$487$	−777.233	−1.59596	−0.797980	−	0.602684i	$-0.794098\pi$
$487$	−777.233	−1.59596	−0.797980	+	0.602684i	$0.794098\pi$
$488$	0	0
$489$	40.4588i	0.0827379i
$490$	0	0
$491$	− 476.370i	− 0.970203i	−0.874458	−	0.485102i	$-0.838783\pi$
$491$	− 476.370i	− 0.970203i	0.874458	−	0.485102i	$-0.161217\pi$
$492$	0	0
$493$	693.005	1.40569
$494$	0	0
$495$	− 407.936i	− 0.824112i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 435.397i	− 0.872539i	−0.899816	−	0.436269i	$-0.856299\pi$
$499$	− 435.397i	− 0.872539i	0.899816	−	0.436269i	$-0.143701\pi$
$500$	0	0
$501$	305.273i	0.609327i
$502$	0	0
$503$	− 375.404i	− 0.746329i	−0.927765	−	0.373165i	$-0.878273\pi$
$503$	− 375.404i	− 0.746329i	0.927765	−	0.373165i	$-0.121727\pi$
$504$	0	0
$505$	−325.678	−0.644907
$506$	0	0
$507$	28.8369	0.0568776
$508$	0	0
$509$	−147.823	−0.290419	−0.145210	−	0.989401i	$-0.546386\pi$
$509$	−147.823	−0.290419	−0.145210	+	0.989401i	$0.546386\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−207.426	−0.404339
$514$	0	0
$515$	91.7737i	0.178201i
$516$	0	0
$517$	−615.131	−1.18981
$518$	0	0
$519$	742.404	1.43045
$520$	0	0
$521$	872.148i	1.67399i	0.547212	+	0.836994i	$0.315689\pi$
$521$	872.148i	1.67399i	−0.547212	+	0.836994i	$0.684311\pi$
$522$	0	0
$523$	−357.200	−0.682982	−0.341491	−	0.939885i	$-0.610932\pi$
$523$	−357.200	−0.682982	−0.341491	+	0.939885i	$0.610932\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−566.562	−1.07507
$528$	0	0
$529$	−450.487	−0.851582
$530$	0	0
$531$	14.3839	0.0270883
$532$	0	0
$533$	− 280.771i	− 0.526776i
$534$	0	0
$535$	− 209.321i	− 0.391255i
$536$	0	0
$537$	270.538i	0.503795i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 932.072i	− 1.72287i	−0.507869	−	0.861434i	$-0.669567\pi$
$541$	− 932.072i	− 1.72287i	0.507869	−	0.861434i	$-0.330433\pi$
$542$	0	0
$543$	1329.82	2.44903
$544$	0	0
$545$	264.776i	0.485827i
$546$	0	0
$547$	− 151.397i	− 0.276778i	−0.990378	−	0.138389i	$-0.955808\pi$
$547$	− 151.397i	− 0.276778i	0.990378	−	0.138389i	$-0.0441924\pi$
$548$	0	0
$549$	174.766	0.318336
$550$	0	0
$551$	− 629.192i	− 1.14191i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	221.150i	0.398469i
$556$	0	0
$557$	− 895.577i	− 1.60786i	−0.594725	−	0.803929i	$-0.702739\pi$
$557$	− 895.577i	− 1.60786i	0.594725	−	0.803929i	$-0.297261\pi$
$558$	0	0
$559$	1014.27i	1.81444i
$560$	0	0
$561$	1102.52	1.96527
$562$	0	0
$563$	−279.915	−0.497185	−0.248592	−	0.968608i	$-0.579968\pi$
$563$	−279.915	−0.497185	−0.248592	+	0.968608i	$0.579968\pi$
$564$	0	0
$565$	−882.355	−1.56169
$566$	0	0
$567$	0	0
$568$	0	0
$569$	321.926	0.565775	0.282887	−	0.959153i	$-0.408708\pi$
$569$	321.926	0.565775	0.282887	+	0.959153i	$0.408708\pi$
$570$	0	0
$571$	822.820i	1.44102i	0.693446	+	0.720508i	$0.256091\pi$
$571$	822.820i	1.44102i	−0.693446	+	0.720508i	$0.743909\pi$
$572$	0	0
$573$	−540.928	−0.944027
$574$	0	0
$575$	27.6427	0.0480742
$576$	0	0
$577$	962.053i	1.66734i	0.552266	+	0.833668i	$0.313763\pi$
$577$	962.053i	1.66734i	−0.552266	+	0.833668i	$0.686237\pi$
$578$	0	0
$579$	−88.7102	−0.153213
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−527.599	−0.904973
$584$	0	0
$585$	355.293	0.607338
$586$	0	0
$587$	−885.638	−1.50875	−0.754377	−	0.656442i	$-0.772061\pi$
$587$	−885.638	−1.50875	−0.754377	+	0.656442i	$0.772061\pi$
$588$	0	0
$589$	514.392i	0.873331i
$590$	0	0
$591$	1113.70i	1.88444i
$592$	0	0
$593$	− 335.808i	− 0.566286i	−0.959078	−	0.283143i	$-0.908623\pi$
$593$	− 335.808i	− 0.566286i	0.959078	−	0.283143i	$-0.0913772\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 253.408i	− 0.424468i
$598$	0	0
$599$	60.3064	0.100679	0.0503393	−	0.998732i	$-0.483970\pi$
$599$	60.3064	0.100679	0.0503393	+	0.998732i	$0.483970\pi$
$600$	0	0
$601$	509.153i	0.847176i	0.905855	+	0.423588i	$0.139230\pi$
$601$	509.153i	0.847176i	−0.905855	+	0.423588i	$0.860770\pi$
$602$	0	0
$603$	243.105i	0.403159i
$604$	0	0
$605$	−430.192	−0.711061
$606$	0	0
$607$	− 1039.65i	− 1.71277i	−0.516340	−	0.856384i	$-0.672706\pi$
$607$	− 1039.65i	− 1.71277i	0.516340	−	0.856384i	$-0.327294\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 535.750i	− 0.876841i
$612$	0	0
$613$	− 198.928i	− 0.324516i	−0.986748	−	0.162258i	$-0.948122\pi$
$613$	− 198.928i	− 0.324516i	0.986748	−	0.162258i	$-0.0518777\pi$
$614$	0	0
$615$	399.876i	0.650206i
$616$	0	0
$617$	136.689	0.221538	0.110769	−	0.993846i	$-0.464669\pi$
$617$	136.689	0.221538	0.110769	+	0.993846i	$0.464669\pi$
$618$	0	0
$619$	461.877	0.746166	0.373083	−	0.927798i	$-0.378301\pi$
$619$	461.877	0.746166	0.373083	+	0.927798i	$0.378301\pi$
$620$	0	0
$621$	−103.694	−0.166980
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−537.276	−0.859641
$626$	0	0
$627$	− 1001.00i	− 1.59648i
$628$	0	0
$629$	−238.503	−0.379178
$630$	0	0
$631$	−1042.33	−1.65187	−0.825933	−	0.563768i	$-0.809351\pi$
$631$	−1042.33	−1.65187	−0.825933	+	0.563768i	$0.809351\pi$
$632$	0	0
$633$	− 69.2650i	− 0.109423i
$634$	0	0
$635$	−214.492	−0.337782
$636$	0	0
$637$	0	0
$638$	0	0
$639$	135.161	0.211519
$640$	0	0
$641$	122.667	0.191368	0.0956840	−	0.995412i	$-0.469496\pi$
$641$	122.667	0.191368	0.0956840	+	0.995412i	$0.469496\pi$
$642$	0	0
$643$	−720.813	−1.12102	−0.560508	−	0.828149i	$-0.689394\pi$
$643$	−720.813	−1.12102	−0.560508	+	0.828149i	$0.689394\pi$
$644$	0	0
$645$	− 1444.53i	− 2.23959i
$646$	0	0
$647$	− 589.850i	− 0.911669i	−0.890065	−	0.455834i	$-0.849341\pi$
$647$	− 589.850i	− 0.911669i	0.890065	−	0.455834i	$-0.150659\pi$
$648$	0	0
$649$	− 35.1257i	− 0.0541228i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	248.600i	0.380705i	0.981716	+	0.190352i	$0.0609631\pi$
$653$	248.600i	0.380705i	−0.981716	+	0.190352i	$0.939037\pi$
$654$	0	0
$655$	565.978	0.864088
$656$	0	0
$657$	456.524i	0.694862i
$658$	0	0
$659$	640.918i	0.972562i	0.873802	+	0.486281i	$0.161647\pi$
$659$	640.918i	0.972562i	−0.873802	+	0.486281i	$0.838353\pi$
$660$	0	0
$661$	−966.642	−1.46239	−0.731197	−	0.682167i	$-0.761038\pi$
$661$	−966.642	−1.46239	−0.731197	+	0.682167i	$0.761038\pi$
$662$	0	0
$663$	960.240i	1.44833i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 314.540i	− 0.471574i
$668$	0	0
$669$	1002.20i	1.49805i
$670$	0	0
$671$	− 426.783i	− 0.636040i
$672$	0	0
$673$	−754.537	−1.12115	−0.560577	−	0.828102i	$-0.689421\pi$
$673$	−754.537	−1.12115	−0.560577	+	0.828102i	$0.689421\pi$
$674$	0	0
$675$	−36.5085	−0.0540867
$676$	0	0
$677$	761.957	1.12549	0.562745	−	0.826631i	$-0.309745\pi$
$677$	761.957	1.12549	0.562745	+	0.826631i	$0.309745\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	454.108	0.666825
$682$	0	0
$683$	759.336i	1.11177i	0.831261	+	0.555883i	$0.187620\pi$
$683$	759.336i	1.11177i	−0.831261	+	0.555883i	$0.812380\pi$
$684$	0	0
$685$	−679.387	−0.991806
$686$	0	0
$687$	337.047	0.490607
$688$	0	0
$689$	− 459.514i	− 0.666928i
$690$	0	0
$691$	−490.152	−0.709337	−0.354669	−	0.934992i	$-0.615406\pi$
$691$	−490.152	−0.709337	−0.354669	+	0.934992i	$0.615406\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−403.916	−0.581174
$696$	0	0
$697$	−431.254	−0.618728
$698$	0	0
$699$	1206.05	1.72539
$700$	0	0
$701$	− 577.533i	− 0.823870i	−0.911213	−	0.411935i	$-0.864853\pi$
$701$	− 577.533i	− 0.823870i	0.911213	−	0.411935i	$-0.135147\pi$
$702$	0	0
$703$	216.541i	0.308025i
$704$	0	0
$705$	763.018i	1.08230i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	711.080i	1.00293i	0.865177	+	0.501467i	$0.167206\pi$
$709$	711.080i	1.00293i	−0.865177	+	0.501467i	$0.832794\pi$
$710$	0	0
$711$	−817.528	−1.14983
$712$	0	0
$713$	257.150i	0.360659i
$714$	0	0
$715$	− 867.631i	− 1.21347i
$716$	0	0
$717$	−544.968	−0.760068
$718$	0	0
$719$	− 173.713i	− 0.241603i	−0.992677	−	0.120802i	$-0.961453\pi$
$719$	− 173.713i	− 0.241603i	0.992677	−	0.120802i	$-0.0385465\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	186.091i	0.257388i
$724$	0	0
$725$	− 110.743i	− 0.152748i
$726$	0	0
$727$	− 1056.57i	− 1.45333i	−0.686993	−	0.726664i	$-0.741070\pi$
$727$	− 1056.57i	− 1.45333i	0.686993	−	0.726664i	$-0.258930\pi$
$728$	0	0
$729$	−184.457	−0.253028
$730$	0	0
$731$	1557.88	2.13117
$732$	0	0
$733$	−230.597	−0.314594	−0.157297	−	0.987551i	$-0.550278\pi$
$733$	−230.597	−0.314594	−0.157297	+	0.987551i	$0.550278\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	593.666	0.805517
$738$	0	0
$739$	− 89.4860i	− 0.121091i	−0.998165	−	0.0605453i	$-0.980716\pi$
$739$	− 89.4860i	− 0.121091i	0.998165	−	0.0605453i	$-0.0192840\pi$
$740$	0	0
$741$	871.820	1.17654
$742$	0	0
$743$	236.120	0.317793	0.158897	−	0.987295i	$-0.449206\pi$
$743$	236.120	0.317793	0.158897	+	0.987295i	$0.449206\pi$
$744$	0	0
$745$	− 1023.74i	− 1.37414i
$746$	0	0
$747$	298.763	0.399951
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1435.38	1.91130	0.955649	−	0.294510i	$-0.0951563\pi$
$751$	1435.38	1.91130	0.955649	+	0.294510i	$0.0951563\pi$
$752$	0	0
$753$	588.530	0.781580
$754$	0	0
$755$	−389.696	−0.516154
$756$	0	0
$757$	− 692.645i	− 0.914987i	−0.889213	−	0.457494i	$-0.848747\pi$
$757$	− 692.645i	− 0.914987i	0.889213	−	0.457494i	$-0.151253\pi$
$758$	0	0
$759$	− 500.409i	− 0.659300i
$760$	0	0
$761$	1154.48i	1.51706i	0.651639	+	0.758529i	$0.274082\pi$
$761$	1154.48i	1.51706i	−0.651639	+	0.758529i	$0.725918\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 545.715i	− 0.713353i
$766$	0	0
$767$	30.5928	0.0398863
$768$	0	0
$769$	− 894.095i	− 1.16267i	−0.813663	−	0.581336i	$-0.802530\pi$
$769$	− 894.095i	− 1.16267i	0.813663	−	0.581336i	$-0.197470\pi$
$770$	0	0
$771$	563.723i	0.731158i
$772$	0	0
$773$	929.076	1.20191	0.600954	−	0.799283i	$-0.294787\pi$
$773$	929.076	1.20191	0.600954	+	0.799283i	$0.294787\pi$
$774$	0	0
$775$	90.5370i	0.116822i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	391.543i	0.502623i
$780$	0	0
$781$	− 330.065i	− 0.422618i
$782$	0	0
$783$	415.422i	0.530552i
$784$	0	0
$785$	165.782	0.211188
$786$	0	0
$787$	−1386.03	−1.76115	−0.880576	−	0.473905i	$-0.842844\pi$
$787$	−1386.03	−1.76115	−0.880576	+	0.473905i	$0.842844\pi$
$788$	0	0
$789$	1367.59	1.73332
$790$	0	0
$791$	0	0
$792$	0	0
$793$	371.708	0.468736
$794$	0	0
$795$	654.442i	0.823198i
$796$	0	0
$797$	388.524	0.487484	0.243742	−	0.969840i	$-0.421625\pi$
$797$	388.524	0.487484	0.243742	+	0.969840i	$0.421625\pi$
$798$	0	0
$799$	−822.890	−1.02990
$800$	0	0
$801$	− 6.69359i	− 0.00835655i
$802$	0	0
$803$	1114.84	1.38835
$804$	0	0
$805$	0	0
$806$	0	0
$807$	515.847	0.639216
$808$	0	0
$809$	510.224	0.630685	0.315342	−	0.948978i	$-0.397881\pi$
$809$	510.224	0.630685	0.315342	+	0.948978i	$0.397881\pi$
$810$	0	0
$811$	−54.8689	−0.0676558	−0.0338279	−	0.999428i	$-0.510770\pi$
$811$	−54.8689	−0.0676558	−0.0338279	+	0.999428i	$0.510770\pi$
$812$	0	0
$813$	1458.28i	1.79370i
$814$	0	0
$815$	− 48.9038i	− 0.0600047i
$816$	0	0
$817$	− 1414.43i	− 1.73125i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 192.888i	− 0.234942i	−0.993076	−	0.117471i	$-0.962521\pi$
$821$	− 192.888i	− 0.234942i	0.993076	−	0.117471i	$-0.0374787\pi$
$822$	0	0
$823$	−142.648	−0.173327	−0.0866636	−	0.996238i	$-0.527621\pi$
$823$	−142.648	−0.173327	−0.0866636	+	0.996238i	$0.527621\pi$
$824$	0	0
$825$	− 176.183i	− 0.213555i
$826$	0	0
$827$	91.3639i	0.110476i	0.998473	+	0.0552382i	$0.0175918\pi$
$827$	91.3639i	0.110476i	−0.998473	+	0.0552382i	$0.982408\pi$
$828$	0	0
$829$	465.999	0.562122	0.281061	−	0.959690i	$-0.409314\pi$
$829$	465.999	0.562122	0.281061	+	0.959690i	$0.409314\pi$
$830$	0	0
$831$	301.533i	0.362855i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 368.993i	− 0.441907i
$836$	0	0
$837$	− 339.626i	− 0.405766i
$838$	0	0
$839$	− 24.9426i	− 0.0297289i	−0.999890	−	0.0148645i	$-0.995268\pi$
$839$	− 24.9426i	− 0.0297289i	0.999890	−	0.0148645i	$-0.00473168\pi$
$840$	0	0
$841$	−419.114	−0.498352
$842$	0	0
$843$	−1256.02	−1.48994
$844$	0	0
$845$	−34.8561	−0.0412498
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1158.96	−1.36509
$850$	0	0
$851$	108.251i	0.127205i
$852$	0	0
$853$	1642.90	1.92603	0.963016	−	0.269445i	$-0.0868403\pi$
$853$	1642.90	1.92603	0.963016	+	0.269445i	$0.0868403\pi$
$854$	0	0
$855$	−495.465	−0.579491
$856$	0	0
$857$	− 151.136i	− 0.176355i	−0.996105	−	0.0881775i	$-0.971896\pi$
$857$	− 151.136i	− 0.176355i	0.996105	−	0.0881775i	$-0.0281043\pi$
$858$	0	0
$859$	900.014	1.04775	0.523873	−	0.851796i	$-0.324487\pi$
$859$	900.014	1.04775	0.523873	+	0.851796i	$0.324487\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−415.864	−0.481882	−0.240941	−	0.970540i	$-0.577456\pi$
$863$	−415.864	−0.481882	−0.240941	+	0.970540i	$0.577456\pi$
$864$	0	0
$865$	−897.366	−1.03742
$866$	0	0
$867$	356.496	0.411184
$868$	0	0
$869$	1996.42i	2.29737i
$870$	0	0
$871$	517.055i	0.593634i
$872$	0	0
$873$	− 949.146i	− 1.08722i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1718.84i	1.95991i	0.199226	+	0.979954i	$0.436157\pi$
$877$	1718.84i	1.95991i	−0.199226	+	0.979954i	$0.563843\pi$
$878$	0	0
$879$	316.266	0.359802
$880$	0	0
$881$	1094.47i	1.24231i	0.783689	+	0.621153i	$0.213336\pi$
$881$	1094.47i	1.24231i	−0.783689	+	0.621153i	$0.786664\pi$
$882$	0	0
$883$	− 527.301i	− 0.597170i	−0.954383	−	0.298585i	$-0.903485\pi$
$883$	− 527.301i	− 0.597170i	0.954383	−	0.298585i	$-0.0965146\pi$
$884$	0	0
$885$	−43.5705	−0.0492322
$886$	0	0
$887$	1685.85i	1.90062i	0.311304	+	0.950310i	$0.399234\pi$
$887$	1685.85i	1.90062i	−0.311304	+	0.950310i	$0.600766\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1445.79i	1.62266i
$892$	0	0
$893$	747.117i	0.836637i
$894$	0	0
$895$	− 327.007i	− 0.365371i
$896$	0	0
$897$	435.832	0.485878
$898$	0	0
$899$	1030.20	1.14594
$900$	0	0
$901$	−705.795	−0.783346
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1607.40	−1.77613
$906$	0	0
$907$	− 1662.05i	− 1.83247i	−0.400644	−	0.916234i	$-0.631214\pi$
$907$	− 1662.05i	− 1.83247i	0.400644	−	0.916234i	$-0.368786\pi$
$908$	0	0
$909$	−416.073	−0.457726
$910$	0	0
$911$	1220.19	1.33940	0.669698	−	0.742633i	$-0.266423\pi$
$911$	1220.19	1.33940	0.669698	+	0.742633i	$0.266423\pi$
$912$	0	0
$913$	− 729.586i	− 0.799108i
$914$	0	0
$915$	−529.388	−0.578566
$916$	0	0
$917$	0	0
$918$	0	0
$919$	76.2434	0.0829635	0.0414817	−	0.999139i	$-0.486792\pi$
$919$	76.2434	0.0829635	0.0414817	+	0.999139i	$0.486792\pi$
$920$	0	0
$921$	−1400.62	−1.52076
$922$	0	0
$923$	287.471	0.311452
$924$	0	0
$925$	38.1129i	0.0412032i
$926$	0	0
$927$	117.246i	0.126479i
$928$	0	0
$929$	− 231.821i	− 0.249538i	−0.992186	−	0.124769i	$-0.960181\pi$
$929$	− 231.821i	− 0.249538i	0.992186	−	0.124769i	$-0.0398189\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	730.646i	0.783115i
$934$	0	0
$935$	−1332.65	−1.42529
$936$	0	0
$937$	− 985.061i	− 1.05129i	−0.850703	−	0.525646i	$-0.823824\pi$
$937$	− 985.061i	− 1.05129i	0.850703	−	0.525646i	$-0.176176\pi$
$938$	0	0
$939$	− 99.4088i	− 0.105867i
$940$	0	0
$941$	1408.48	1.49679	0.748394	−	0.663254i	$-0.230825\pi$
$941$	1408.48	1.49679	0.748394	+	0.663254i	$0.230825\pi$
$942$	0	0
$943$	195.737i	0.207568i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	166.417i	0.175731i	0.996132	+	0.0878654i	$0.0280045\pi$
$947$	166.417i	0.175731i	−0.996132	+	0.0878654i	$0.971995\pi$
$948$	0	0
$949$	970.974i	1.02315i
$950$	0	0
$951$	− 1931.56i	− 2.03109i
$952$	0	0
$953$	815.618	0.855843	0.427921	−	0.903816i	$-0.359246\pi$
$953$	815.618	0.855843	0.427921	+	0.903816i	$0.359246\pi$
$954$	0	0
$955$	653.836	0.684645
$956$	0	0
$957$	−2004.74	−2.09482
$958$	0	0
$959$	0	0
$960$	0	0
$961$	118.767	0.123587
$962$	0	0
$963$	− 267.420i	− 0.277695i
$964$	0	0
$965$	107.227	0.111116
$966$	0	0
$967$	86.3395	0.0892860	0.0446430	−	0.999003i	$-0.485785\pi$
$967$	86.3395	0.0892860	0.0446430	+	0.999003i	$0.485785\pi$
$968$	0	0
$969$	− 1339.08i	− 1.38192i
$970$	0	0
$971$	679.301	0.699589	0.349795	−	0.936826i	$-0.386251\pi$
$971$	679.301	0.699589	0.349795	+	0.936826i	$0.386251\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	153.447	0.157382
$976$	0	0
$977$	−214.057	−0.219096	−0.109548	−	0.993981i	$-0.534940\pi$
$977$	−214.057	−0.219096	−0.109548	+	0.993981i	$0.534940\pi$
$978$	0	0
$979$	−16.3459	−0.0166965
$980$	0	0
$981$	338.267i	0.344818i
$982$	0	0
$983$	− 1800.29i	− 1.83143i	−0.401833	−	0.915713i	$-0.631627\pi$
$983$	− 1800.29i	− 1.83143i	0.401833	−	0.915713i	$-0.368373\pi$
$984$	0	0
$985$	− 1346.17i	− 1.36667i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 707.089i	− 0.714953i
$990$	0	0
$991$	−411.427	−0.415163	−0.207582	−	0.978218i	$-0.566559\pi$
$991$	−411.427	−0.415163	−0.207582	+	0.978218i	$0.566559\pi$
$992$	0	0
$993$	− 966.925i	− 0.973741i
$994$	0	0
$995$	306.302i	0.307841i
$996$	0	0
$997$	−1612.00	−1.61685	−0.808425	−	0.588599i	$-0.799680\pi$
$997$	−1612.00	−1.61685	−0.808425	+	0.588599i	$0.799680\pi$
$998$	0	0
$999$	− 142.971i	− 0.143114i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.h.a.881.6		28
4.3	odd	2		392.3.h.a.293.2		28
7.4	even	3		224.3.n.a.145.12		28
7.5	odd	6		224.3.n.a.17.3		28
7.6	odd	2	inner	1568.3.h.a.881.24		28
8.3	odd	2		392.3.h.a.293.3		28
8.5	even	2	inner	1568.3.h.a.881.23		28
28.3	even	6		392.3.j.e.117.11		28
28.11	odd	6		56.3.j.a.5.11	yes	28
28.19	even	6		56.3.j.a.45.9	yes	28
28.23	odd	6		392.3.j.e.325.9		28
28.27	even	2		392.3.h.a.293.1		28
56.3	even	6		392.3.j.e.117.9		28
56.5	odd	6		224.3.n.a.17.12		28
56.11	odd	6		56.3.j.a.5.9	✓	28
56.13	odd	2	inner	1568.3.h.a.881.5		28
56.19	even	6		56.3.j.a.45.11	yes	28
56.27	even	2		392.3.h.a.293.4		28
56.51	odd	6		392.3.j.e.325.11		28
56.53	even	6		224.3.n.a.145.3		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.j.a.5.9	✓	28	56.11	odd	6
56.3.j.a.5.11	yes	28	28.11	odd	6
56.3.j.a.45.9	yes	28	28.19	even	6
56.3.j.a.45.11	yes	28	56.19	even	6
224.3.n.a.17.3		28	7.5	odd	6
224.3.n.a.17.12		28	56.5	odd	6
224.3.n.a.145.3		28	56.53	even	6
224.3.n.a.145.12		28	7.4	even	3
392.3.h.a.293.1		28	28.27	even	2
392.3.h.a.293.2		28	4.3	odd	2
392.3.h.a.293.3		28	8.3	odd	2
392.3.h.a.293.4		28	56.27	even	2
392.3.j.e.117.9		28	56.3	even	6
392.3.j.e.117.11		28	28.3	even	6
392.3.j.e.325.9		28	28.23	odd	6
392.3.j.e.325.11		28	56.51	odd	6
1568.3.h.a.881.5		28	56.13	odd	2	inner
1568.3.h.a.881.6		28	1.1	even	1	trivial
1568.3.h.a.881.23		28	8.5	even	2	inner
1568.3.h.a.881.24		28	7.6	odd	2	inner

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.86988 q^{3} +4.67764 q^{5} +5.97596 q^{9} +O(q^{10})\)	\(q-3.86988 q^{3} +4.67764 q^{5} +5.97596 q^{9} -14.5934i q^{11} +12.7102 q^{13} -18.1019 q^{15} -19.5223i q^{17} -17.7247 q^{19} -8.86075 q^{23} -3.11968 q^{25} +11.7027 q^{27} +35.4981i q^{29} -29.0213i q^{31} +56.4747i q^{33} -12.2169i q^{37} -49.1868 q^{39} -22.0903i q^{41} +79.8001i q^{43} +27.9534 q^{45} -42.1513i q^{47} +75.5490i q^{51} -36.1532i q^{53} -68.2627i q^{55} +68.5923 q^{57} +2.40696 q^{59} +29.2449 q^{61} +59.4536 q^{65} +40.6804i q^{67} +34.2900 q^{69} +22.6174 q^{71} +76.3935i q^{73} +12.0728 q^{75} -136.803 q^{79} -99.0715 q^{81} +49.9942 q^{83} -91.3184i q^{85} -137.373i q^{87} -1.12009i q^{89} +112.309i q^{93} -82.9096 q^{95} -158.827i q^{97} -87.2097i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 64 q^{9}+O(q^{10}) \) 28 * q + 64 * q^9	\( 28 q + 64 q^{9} - 28 q^{15} + 60 q^{23} + 64 q^{25} - 40 q^{39} + 124 q^{57} + 104 q^{65} + 136 q^{71} - 324 q^{79} + 36 q^{81} + 580 q^{95}+O(q^{100}) \) 28 * q + 64 * q^9 - 28 * q^15 + 60 * q^23 + 64 * q^25 - 40 * q^39 + 124 * q^57 + 104 * q^65 + 136 * q^71 - 324 * q^79 + 36 * q^81 + 580 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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