LMFDB - Embedded newform 1568.3.c.g.97.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(97,1568)]

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, 0, 1]))

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

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

chi := DirichletCharacter("1568.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.c (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:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26 x^{14} - 16 x^{13} + 469 x^{12} + 144 x^{11} - 4526 x^{10} + 4440 x^{9} + 32608 x^{8} + \cdots + 208849 $ x^16 - 26x^14 - 16x^13 + 469x^12 + 144x^11 - 4526x^10 + 4440x^9 + 32608x^8 - 33728x^7 - 49760x^6 + 203528x^5 + 27401x^4 - 156928x^3 + 114964x^2 - 248608x + 208849
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.1
Root			$3.86852 + 1.41699i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.g.97.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.73991i q^{3} -6.30408i q^{5} -23.9466 q^{9} +O(q^{10})$	$q-5.73991i q^{3} -6.30408i q^{5} -23.9466 q^{9} -5.41114 q^{11} +15.9368i q^{13} -36.1848 q^{15} -20.4791i q^{17} +13.5873i q^{19} -4.70159 q^{23} -14.7414 q^{25} +85.7918i q^{27} +1.76543 q^{29} +13.7455i q^{31} +31.0595i q^{33} +10.4673 q^{37} +91.4760 q^{39} +11.2412i q^{41} -49.1704 q^{43} +150.961i q^{45} -10.4198i q^{47} -117.548 q^{51} +32.1013 q^{53} +34.1123i q^{55} +77.9897 q^{57} +66.1482i q^{59} +31.9626i q^{61} +100.467 q^{65} -98.5377 q^{67} +26.9867i q^{69} -61.7537 q^{71} +18.0425i q^{73} +84.6144i q^{75} -30.0036 q^{79} +276.918 q^{81} -63.4583i q^{83} -129.102 q^{85} -10.1334i q^{87} -137.558i q^{89} +78.8978 q^{93} +85.6552 q^{95} +131.075i q^{97} +129.578 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 80 q^{9}+O(q^{10}) $ 16 * q - 80 * q^9	$ 16 q - 80 q^{9} - 160 q^{25} - 16 q^{29} - 144 q^{37} - 80 q^{53} + 368 q^{57} + 336 q^{65} + 768 q^{81} - 1072 q^{85} + 336 q^{93}+O(q^{100}) $ 16 * q - 80 * q^9 - 160 * q^25 - 16 * q^29 - 144 * q^37 - 80 * q^53 + 368 * q^57 + 336 * q^65 + 768 * q^81 - 1072 * q^85 + 336 * 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
$3$	− 5.73991i	− 1.91330i	−0.291235	−	0.956651i	$-0.594066\pi$
$3$	− 5.73991i	− 1.91330i	0.291235	−	0.956651i	$-0.405934\pi$
$4$	0	0
$5$	− 6.30408i	− 1.26082i	−0.776264	−	0.630408i	$-0.782888\pi$
$5$	− 6.30408i	− 1.26082i	0.776264	−	0.630408i	$-0.217112\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−23.9466	−2.66073
$10$	0	0
$11$	−5.41114	−0.491922	−0.245961	−	0.969280i	$-0.579103\pi$
$11$	−5.41114	−0.491922	−0.245961	+	0.969280i	$0.579103\pi$
$12$	0	0
$13$	15.9368i	1.22591i	0.790118	+	0.612955i	$0.210019\pi$
$13$	15.9368i	1.22591i	−0.790118	+	0.612955i	$0.789981\pi$
$14$	0	0
$15$	−36.1848	−2.41232
$16$	0	0
$17$	− 20.4791i	− 1.20465i	−0.798250	−	0.602326i	$-0.794241\pi$
$17$	− 20.4791i	− 1.20465i	0.798250	−	0.602326i	$-0.205759\pi$
$18$	0	0
$19$	13.5873i	0.715119i	0.933890	+	0.357560i	$0.116391\pi$
$19$	13.5873i	0.715119i	−0.933890	+	0.357560i	$0.883609\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.70159	−0.204417	−0.102209	−	0.994763i	$-0.532591\pi$
$23$	−4.70159	−0.204417	−0.102209	+	0.994763i	$0.532591\pi$
$24$	0	0
$25$	−14.7414	−0.589657
$26$	0	0
$27$	85.7918i	3.17748i
$28$	0	0
$29$	1.76543	0.0608770	0.0304385	−	0.999537i	$-0.490310\pi$
$29$	1.76543	0.0608770	0.0304385	+	0.999537i	$0.490310\pi$
$30$	0	0
$31$	13.7455i	0.443403i	0.975115	+	0.221701i	$0.0711610\pi$
$31$	13.7455i	0.443403i	−0.975115	+	0.221701i	$0.928839\pi$
$32$	0	0
$33$	31.0595i	0.941196i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.4673	0.282899	0.141449	−	0.989945i	$-0.454824\pi$
$37$	10.4673	0.282899	0.141449	+	0.989945i	$0.454824\pi$
$38$	0	0
$39$	91.4760	2.34554
$40$	0	0
$41$	11.2412i	0.274175i	0.990559	+	0.137088i	$0.0437742\pi$
$41$	11.2412i	0.274175i	−0.990559	+	0.137088i	$0.956226\pi$
$42$	0	0
$43$	−49.1704	−1.14350	−0.571749	−	0.820428i	$-0.693735\pi$
$43$	−49.1704	−1.14350	−0.571749	+	0.820428i	$0.693735\pi$
$44$	0	0
$45$	150.961i	3.35469i
$46$	0	0
$47$	− 10.4198i	− 0.221698i	−0.993837	−	0.110849i	$-0.964643\pi$
$47$	− 10.4198i	− 0.221698i	0.993837	−	0.110849i	$-0.0353570\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−117.548	−2.30486
$52$	0	0
$53$	32.1013	0.605685	0.302842	−	0.953041i	$-0.402064\pi$
$53$	32.1013	0.605685	0.302842	+	0.953041i	$0.402064\pi$
$54$	0	0
$55$	34.1123i	0.620223i
$56$	0	0
$57$	77.9897	1.36824
$58$	0	0
$59$	66.1482i	1.12116i	0.828102	+	0.560578i	$0.189421\pi$
$59$	66.1482i	1.12116i	−0.828102	+	0.560578i	$0.810579\pi$
$60$	0	0
$61$	31.9626i	0.523977i	0.965071	+	0.261989i	$0.0843783\pi$
$61$	31.9626i	0.523977i	−0.965071	+	0.261989i	$0.915622\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	100.467	1.54565
$66$	0	0
$67$	−98.5377	−1.47071	−0.735356	−	0.677681i	$-0.762985\pi$
$67$	−98.5377	−1.47071	−0.735356	+	0.677681i	$0.762985\pi$
$68$	0	0
$69$	26.9867i	0.391112i
$70$	0	0
$71$	−61.7537	−0.869770	−0.434885	−	0.900486i	$-0.643211\pi$
$71$	−61.7537	−0.869770	−0.434885	+	0.900486i	$0.643211\pi$
$72$	0	0
$73$	18.0425i	0.247158i	0.992335	+	0.123579i	$0.0394372\pi$
$73$	18.0425i	0.247158i	−0.992335	+	0.123579i	$0.960563\pi$
$74$	0	0
$75$	84.6144i	1.12819i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−30.0036	−0.379793	−0.189896	−	0.981804i	$-0.560815\pi$
$79$	−30.0036	−0.379793	−0.189896	+	0.981804i	$0.560815\pi$
$80$	0	0
$81$	276.918	3.41875
$82$	0	0
$83$	− 63.4583i	− 0.764558i	−0.924047	−	0.382279i	$-0.875139\pi$
$83$	− 63.4583i	− 0.764558i	0.924047	−	0.382279i	$-0.124861\pi$
$84$	0	0
$85$	−129.102	−1.51884
$86$	0	0
$87$	− 10.1334i	− 0.116476i
$88$	0	0
$89$	− 137.558i	− 1.54560i	−0.634651	−	0.772799i	$-0.718856\pi$
$89$	− 137.558i	− 1.54560i	0.634651	−	0.772799i	$-0.281144\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	78.8978	0.848364
$94$	0	0
$95$	85.6552	0.901634
$96$	0	0
$97$	131.075i	1.35129i	0.737228	+	0.675644i	$0.236134\pi$
$97$	131.075i	1.35129i	−0.737228	+	0.675644i	$0.763866\pi$
$98$	0	0
$99$	129.578	1.30887
$100$	0	0
$101$	− 114.648i	− 1.13512i	−0.823331	−	0.567562i	$-0.807887\pi$
$101$	− 114.648i	− 1.13512i	0.823331	−	0.567562i	$-0.192113\pi$
$102$	0	0
$103$	− 200.613i	− 1.94770i	−0.227187	−	0.973851i	$-0.572953\pi$
$103$	− 200.613i	− 1.94770i	0.227187	−	0.973851i	$-0.427047\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−99.4109	−0.929074	−0.464537	−	0.885554i	$-0.653779\pi$
$107$	−99.4109	−0.929074	−0.464537	+	0.885554i	$0.653779\pi$
$108$	0	0
$109$	−15.1889	−0.139348	−0.0696740	−	0.997570i	$-0.522196\pi$
$109$	−15.1889	−0.139348	−0.0696740	+	0.997570i	$0.522196\pi$
$110$	0	0
$111$	− 60.0811i	− 0.541271i
$112$	0	0
$113$	114.050	1.00929	0.504645	−	0.863327i	$-0.331623\pi$
$113$	114.050	1.00929	0.504645	+	0.863327i	$0.331623\pi$
$114$	0	0
$115$	29.6392i	0.257732i
$116$	0	0
$117$	− 381.632i	− 3.26181i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−91.7196	−0.758013
$122$	0	0
$123$	64.5234	0.524580
$124$	0	0
$125$	− 64.6709i	− 0.517367i
$126$	0	0
$127$	27.7466	0.218477	0.109239	−	0.994016i	$-0.465159\pi$
$127$	27.7466	0.218477	0.109239	+	0.994016i	$0.465159\pi$
$128$	0	0
$129$	282.234i	2.18786i
$130$	0	0
$131$	182.750i	1.39504i	0.716566	+	0.697519i	$0.245713\pi$
$131$	182.750i	1.39504i	−0.716566	+	0.697519i	$0.754287\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	540.839	4.00621
$136$	0	0
$137$	101.107	0.738009	0.369004	−	0.929428i	$-0.379699\pi$
$137$	101.107	0.738009	0.369004	+	0.929428i	$0.379699\pi$
$138$	0	0
$139$	88.5595i	0.637119i	0.947903	+	0.318559i	$0.103199\pi$
$139$	88.5595i	0.637119i	−0.947903	+	0.318559i	$0.896801\pi$
$140$	0	0
$141$	−59.8088	−0.424176
$142$	0	0
$143$	− 86.2365i	− 0.603052i
$144$	0	0
$145$	− 11.1294i	− 0.0767547i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−127.036	−0.852593	−0.426296	−	0.904583i	$-0.640182\pi$
$149$	−127.036	−0.852593	−0.426296	+	0.904583i	$0.640182\pi$
$150$	0	0
$151$	−291.385	−1.92970	−0.964852	−	0.262793i	$-0.915356\pi$
$151$	−291.385	−1.92970	−0.964852	+	0.262793i	$0.915356\pi$
$152$	0	0
$153$	490.403i	3.20525i
$154$	0	0
$155$	86.6526	0.559049
$156$	0	0
$157$	215.495i	1.37258i	0.727329	+	0.686289i	$0.240761\pi$
$157$	215.495i	1.37258i	−0.727329	+	0.686289i	$0.759239\pi$
$158$	0	0
$159$	− 184.258i	− 1.15886i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	174.017	1.06759	0.533794	−	0.845615i	$-0.320766\pi$
$163$	174.017	1.06759	0.533794	+	0.845615i	$0.320766\pi$
$164$	0	0
$165$	195.801	1.18667
$166$	0	0
$167$	− 241.170i	− 1.44413i	−0.691823	−	0.722067i	$-0.743192\pi$
$167$	− 241.170i	− 1.44413i	0.691823	−	0.722067i	$-0.256808\pi$
$168$	0	0
$169$	−84.9827	−0.502856
$170$	0	0
$171$	− 325.368i	− 1.90274i
$172$	0	0
$173$	215.972i	1.24839i	0.781267	+	0.624197i	$0.214574\pi$
$173$	215.972i	1.24839i	−0.781267	+	0.624197i	$0.785426\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	379.685	2.14511
$178$	0	0
$179$	−178.074	−0.994829	−0.497414	−	0.867513i	$-0.665717\pi$
$179$	−178.074	−0.994829	−0.497414	+	0.867513i	$0.665717\pi$
$180$	0	0
$181$	− 172.429i	− 0.952648i	−0.879270	−	0.476324i	$-0.841969\pi$
$181$	− 172.429i	− 0.952648i	0.879270	−	0.476324i	$-0.158031\pi$
$182$	0	0
$183$	183.462	1.00253
$184$	0	0
$185$	− 65.9865i	− 0.356684i
$186$	0	0
$187$	110.815i	0.592595i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−139.246	−0.729036	−0.364518	−	0.931196i	$-0.618766\pi$
$191$	−139.246	−0.729036	−0.364518	+	0.931196i	$0.618766\pi$
$192$	0	0
$193$	−43.2331	−0.224005	−0.112003	−	0.993708i	$-0.535727\pi$
$193$	−43.2331	−0.224005	−0.112003	+	0.993708i	$0.535727\pi$
$194$	0	0
$195$	− 576.672i	− 2.95729i
$196$	0	0
$197$	205.910	1.04523	0.522614	−	0.852570i	$-0.324957\pi$
$197$	205.910	1.04523	0.522614	+	0.852570i	$0.324957\pi$
$198$	0	0
$199$	− 301.292i	− 1.51403i	−0.653397	−	0.757016i	$-0.726657\pi$
$199$	− 301.292i	− 1.51403i	0.653397	−	0.757016i	$-0.273343\pi$
$200$	0	0
$201$	565.597i	2.81392i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	70.8653	0.345685
$206$	0	0
$207$	112.587	0.543898
$208$	0	0
$209$	− 73.5226i	− 0.351783i
$210$	0	0
$211$	310.102	1.46968	0.734839	−	0.678241i	$-0.237258\pi$
$211$	310.102	1.46968	0.734839	+	0.678241i	$0.237258\pi$
$212$	0	0
$213$	354.460i	1.66413i
$214$	0	0
$215$	309.974i	1.44174i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	103.562	0.472888
$220$	0	0
$221$	326.372	1.47680
$222$	0	0
$223$	266.090i	1.19323i	0.802528	+	0.596614i	$0.203488\pi$
$223$	266.090i	1.19323i	−0.802528	+	0.596614i	$0.796512\pi$
$224$	0	0
$225$	353.006	1.56892
$226$	0	0
$227$	305.058i	1.34387i	0.740612	+	0.671933i	$0.234536\pi$
$227$	305.058i	1.34387i	−0.740612	+	0.671933i	$0.765464\pi$
$228$	0	0
$229$	− 198.571i	− 0.867124i	−0.901124	−	0.433562i	$-0.857256\pi$
$229$	− 198.571i	− 0.867124i	0.901124	−	0.433562i	$-0.142744\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−247.857	−1.06376	−0.531881	−	0.846819i	$-0.678515\pi$
$233$	−247.857	−1.06376	−0.531881	+	0.846819i	$0.678515\pi$
$234$	0	0
$235$	−65.6873	−0.279521
$236$	0	0
$237$	172.218i	0.726658i
$238$	0	0
$239$	120.884	0.505790	0.252895	−	0.967494i	$-0.418617\pi$
$239$	120.884	0.505790	0.252895	+	0.967494i	$0.418617\pi$
$240$	0	0
$241$	145.442i	0.603493i	0.953388	+	0.301746i	$0.0975696\pi$
$241$	145.442i	0.603493i	−0.953388	+	0.301746i	$0.902430\pi$
$242$	0	0
$243$	− 817.360i	− 3.36362i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−216.538	−0.876672
$248$	0	0
$249$	−364.245	−1.46283
$250$	0	0
$251$	212.061i	0.844865i	0.906394	+	0.422432i	$0.138824\pi$
$251$	212.061i	0.844865i	−0.906394	+	0.422432i	$0.861176\pi$
$252$	0	0
$253$	25.4410	0.100557
$254$	0	0
$255$	741.032i	2.90601i
$256$	0	0
$257$	− 10.0672i	− 0.0391720i	−0.999808	−	0.0195860i	$-0.993765\pi$
$257$	− 10.0672i	− 0.0391720i	0.999808	−	0.0195860i	$-0.00623482\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−42.2760	−0.161977
$262$	0	0
$263$	−0.0835284	−0.000317599 0	−0.000158799	−	1.00000i	$-0.500051\pi$
$263$	−0.0835284	−0.000317599 0	−0.000158799		1.00000i	$0.500051\pi$
$264$	0	0
$265$	− 202.369i	− 0.763657i
$266$	0	0
$267$	−789.572	−2.95720
$268$	0	0
$269$	− 1.27445i	− 0.00473774i	−0.999997	−	0.00236887i	$-0.999246\pi$
$269$	− 1.27445i	− 0.00473774i	0.999997	−	0.00236887i	$-0.000754036\pi$
$270$	0	0
$271$	357.667i	1.31981i	0.751351	+	0.659903i	$0.229402\pi$
$271$	357.667i	1.31981i	−0.751351	+	0.659903i	$0.770598\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	79.7679	0.290065
$276$	0	0
$277$	−169.215	−0.610884	−0.305442	−	0.952211i	$-0.598804\pi$
$277$	−169.215	−0.610884	−0.305442	+	0.952211i	$0.598804\pi$
$278$	0	0
$279$	− 329.157i	− 1.17977i
$280$	0	0
$281$	246.835	0.878418	0.439209	−	0.898385i	$-0.355259\pi$
$281$	246.835	0.878418	0.439209	+	0.898385i	$0.355259\pi$
$282$	0	0
$283$	229.656i	0.811504i	0.913983	+	0.405752i	$0.132990\pi$
$283$	229.656i	0.811504i	−0.913983	+	0.405752i	$0.867010\pi$
$284$	0	0
$285$	− 491.653i	− 1.72510i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−130.393	−0.451187
$290$	0	0
$291$	752.358	2.58542
$292$	0	0
$293$	228.171i	0.778740i	0.921081	+	0.389370i	$0.127307\pi$
$293$	228.171i	0.778740i	−0.921081	+	0.389370i	$0.872693\pi$
$294$	0	0
$295$	417.004	1.41357
$296$	0	0
$297$	− 464.232i	− 1.56307i
$298$	0	0
$299$	− 74.9285i	− 0.250597i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−658.066	−2.17184
$304$	0	0
$305$	201.495	0.660639
$306$	0	0
$307$	333.745i	1.08712i	0.839371	+	0.543559i	$0.182924\pi$
$307$	333.745i	1.08712i	−0.839371	+	0.543559i	$0.817076\pi$
$308$	0	0
$309$	−1151.50	−3.72654
$310$	0	0
$311$	− 249.016i	− 0.800693i	−0.916364	−	0.400347i	$-0.868890\pi$
$311$	− 249.016i	− 0.800693i	0.916364	−	0.400347i	$-0.131110\pi$
$312$	0	0
$313$	39.8542i	0.127330i	0.997971	+	0.0636648i	$0.0202789\pi$
$313$	39.8542i	0.127330i	−0.997971	+	0.0636648i	$0.979721\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−442.412	−1.39562	−0.697811	−	0.716282i	$-0.745842\pi$
$317$	−442.412	−1.39562	−0.697811	+	0.716282i	$0.745842\pi$
$318$	0	0
$319$	−9.55301	−0.0299467
$320$	0	0
$321$	570.609i	1.77760i
$322$	0	0
$323$	278.255	0.861470
$324$	0	0
$325$	− 234.932i	− 0.722866i
$326$	0	0
$327$	87.1830i	0.266615i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−143.928	−0.434829	−0.217414	−	0.976079i	$-0.569762\pi$
$331$	−143.928	−0.434829	−0.217414	+	0.976079i	$0.569762\pi$
$332$	0	0
$333$	−250.655	−0.752717
$334$	0	0
$335$	621.189i	1.85430i
$336$	0	0
$337$	−428.372	−1.27113	−0.635567	−	0.772046i	$-0.719234\pi$
$337$	−428.372	−1.27113	−0.635567	+	0.772046i	$0.719234\pi$
$338$	0	0
$339$	− 654.635i	− 1.93108i
$340$	0	0
$341$	− 74.3788i	− 0.218120i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	170.126	0.493120
$346$	0	0
$347$	−245.605	−0.707795	−0.353898	−	0.935284i	$-0.615144\pi$
$347$	−245.605	−0.707795	−0.353898	+	0.935284i	$0.615144\pi$
$348$	0	0
$349$	− 675.578i	− 1.93575i	−0.251428	−	0.967876i	$-0.580900\pi$
$349$	− 675.578i	− 1.93575i	0.251428	−	0.967876i	$-0.419100\pi$
$350$	0	0
$351$	−1367.25	−3.89530
$352$	0	0
$353$	− 568.508i	− 1.61050i	−0.592932	−	0.805252i	$-0.702030\pi$
$353$	− 568.508i	− 1.61050i	0.592932	−	0.805252i	$-0.297970\pi$
$354$	0	0
$355$	389.300i	1.09662i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−673.912	−1.87719	−0.938596	−	0.345017i	$-0.887873\pi$
$359$	−673.912	−1.87719	−0.938596	+	0.345017i	$0.887873\pi$
$360$	0	0
$361$	176.386	0.488605
$362$	0	0
$363$	526.462i	1.45031i
$364$	0	0
$365$	113.742	0.311621
$366$	0	0
$367$	43.1126i	0.117473i	0.998274	+	0.0587366i	$0.0187072\pi$
$367$	43.1126i	0.117473i	−0.998274	+	0.0587366i	$0.981293\pi$
$368$	0	0
$369$	− 269.188i	− 0.729506i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	633.165	1.69749	0.848746	−	0.528800i	$-0.177358\pi$
$373$	633.165	1.69749	0.848746	+	0.528800i	$0.177358\pi$
$374$	0	0
$375$	−371.205	−0.989880
$376$	0	0
$377$	28.1354i	0.0746297i
$378$	0	0
$379$	−611.641	−1.61383	−0.806914	−	0.590669i	$-0.798864\pi$
$379$	−611.641	−1.61383	−0.806914	+	0.590669i	$0.798864\pi$
$380$	0	0
$381$	− 159.263i	− 0.418014i
$382$	0	0
$383$	641.087i	1.67386i	0.547312	+	0.836929i	$0.315651\pi$
$383$	641.087i	1.67386i	−0.547312	+	0.836929i	$0.684349\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1177.46	3.04254
$388$	0	0
$389$	−354.359	−0.910949	−0.455475	−	0.890249i	$-0.650530\pi$
$389$	−354.359	−0.910949	−0.455475	+	0.890249i	$0.650530\pi$
$390$	0	0
$391$	96.2843i	0.246251i
$392$	0	0
$393$	1048.97	2.66913
$394$	0	0
$395$	189.145i	0.478849i
$396$	0	0
$397$	− 313.843i	− 0.790537i	−0.918566	−	0.395268i	$-0.870652\pi$
$397$	− 313.843i	− 0.790537i	0.918566	−	0.395268i	$-0.129348\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	322.119	0.803289	0.401645	−	0.915796i	$-0.368439\pi$
$401$	322.119	0.803289	0.401645	+	0.915796i	$0.368439\pi$
$402$	0	0
$403$	−219.060	−0.543572
$404$	0	0
$405$	− 1745.72i	− 4.31041i
$406$	0	0
$407$	−56.6398	−0.139164
$408$	0	0
$409$	477.869i	1.16838i	0.811615	+	0.584192i	$0.198588\pi$
$409$	477.869i	1.16838i	−0.811615	+	0.584192i	$0.801412\pi$
$410$	0	0
$411$	− 580.346i	− 1.41203i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−400.046	−0.963967
$416$	0	0
$417$	508.323	1.21900
$418$	0	0
$419$	194.885i	0.465119i	0.972582	+	0.232560i	$0.0747101\pi$
$419$	194.885i	0.465119i	−0.972582	+	0.232560i	$0.925290\pi$
$420$	0	0
$421$	290.331	0.689621	0.344811	−	0.938672i	$-0.387943\pi$
$421$	290.331	0.689621	0.344811	+	0.938672i	$0.387943\pi$
$422$	0	0
$423$	249.519i	0.589878i
$424$	0	0
$425$	301.891i	0.710331i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−494.989	−1.15382
$430$	0	0
$431$	382.339	0.887097	0.443549	−	0.896250i	$-0.353719\pi$
$431$	382.339	0.887097	0.443549	+	0.896250i	$0.353719\pi$
$432$	0	0
$433$	295.254i	0.681879i	0.940085	+	0.340940i	$0.110745\pi$
$433$	295.254i	0.681879i	−0.940085	+	0.340940i	$0.889255\pi$
$434$	0	0
$435$	−63.8819	−0.146855
$436$	0	0
$437$	− 63.8818i	− 0.146183i
$438$	0	0
$439$	49.0800i	0.111799i	0.998436	+	0.0558997i	$0.0178027\pi$
$439$	49.0800i	0.111799i	−0.998436	+	0.0558997i	$0.982197\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−633.724	−1.43053	−0.715264	−	0.698854i	$-0.753694\pi$
$443$	−633.724	−1.43053	−0.715264	+	0.698854i	$0.753694\pi$
$444$	0	0
$445$	−867.178	−1.94871
$446$	0	0
$447$	729.177i	1.63127i
$448$	0	0
$449$	−27.3296	−0.0608677	−0.0304339	−	0.999537i	$-0.509689\pi$
$449$	−27.3296	−0.0608677	−0.0304339	+	0.999537i	$0.509689\pi$
$450$	0	0
$451$	− 60.8277i	− 0.134873i
$452$	0	0
$453$	1672.53i	3.69211i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	339.110	0.742034	0.371017	−	0.928626i	$-0.379009\pi$
$457$	339.110	0.742034	0.371017	+	0.928626i	$0.379009\pi$
$458$	0	0
$459$	1756.94	3.82775
$460$	0	0
$461$	580.237i	1.25865i	0.777143	+	0.629324i	$0.216668\pi$
$461$	580.237i	1.25865i	−0.777143	+	0.629324i	$0.783332\pi$
$462$	0	0
$463$	433.685	0.936684	0.468342	−	0.883547i	$-0.344852\pi$
$463$	433.685	0.936684	0.468342	+	0.883547i	$0.344852\pi$
$464$	0	0
$465$	− 497.378i	− 1.06963i
$466$	0	0
$467$	− 100.062i	− 0.214264i	−0.994245	−	0.107132i	$-0.965833\pi$
$467$	− 100.062i	− 0.214264i	0.994245	−	0.107132i	$-0.0341668\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1236.92	2.62616
$472$	0	0
$473$	266.068	0.562512
$474$	0	0
$475$	− 200.296i	− 0.421675i
$476$	0	0
$477$	−768.715	−1.61156
$478$	0	0
$479$	− 483.815i	− 1.01005i	−0.863104	−	0.505026i	$-0.831483\pi$
$479$	− 483.815i	− 1.01005i	0.863104	−	0.505026i	$-0.168517\pi$
$480$	0	0
$481$	166.815i	0.346809i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	826.307	1.70373
$486$	0	0
$487$	−229.527	−0.471307	−0.235654	−	0.971837i	$-0.575723\pi$
$487$	−229.527	−0.471307	−0.235654	+	0.971837i	$0.575723\pi$
$488$	0	0
$489$	− 998.840i	− 2.04262i
$490$	0	0
$491$	−221.445	−0.451008	−0.225504	−	0.974242i	$-0.572403\pi$
$491$	−221.445	−0.451008	−0.225504	+	0.974242i	$0.572403\pi$
$492$	0	0
$493$	− 36.1545i	− 0.0733356i
$494$	0	0
$495$	− 816.871i	− 1.65024i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−513.406	−1.02887	−0.514435	−	0.857530i	$-0.671998\pi$
$499$	−513.406	−1.02887	−0.514435	+	0.857530i	$0.671998\pi$
$500$	0	0
$501$	−1384.30	−2.76306
$502$	0	0
$503$	− 360.553i	− 0.716806i	−0.933567	−	0.358403i	$-0.883321\pi$
$503$	− 360.553i	− 0.716806i	0.933567	−	0.358403i	$-0.116679\pi$
$504$	0	0
$505$	−722.747	−1.43118
$506$	0	0
$507$	487.793i	0.962116i
$508$	0	0
$509$	− 188.862i	− 0.371046i	−0.982640	−	0.185523i	$-0.940602\pi$
$509$	− 188.862i	− 0.371046i	0.982640	−	0.185523i	$-0.0593979\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1165.68	−2.27227
$514$	0	0
$515$	−1264.68	−2.45569
$516$	0	0
$517$	56.3831i	0.109058i
$518$	0	0
$519$	1239.66	2.38856
$520$	0	0
$521$	− 473.032i	− 0.907930i	−0.891019	−	0.453965i	$-0.850009\pi$
$521$	− 473.032i	− 0.907930i	0.891019	−	0.453965i	$-0.149991\pi$
$522$	0	0
$523$	419.149i	0.801432i	0.916202	+	0.400716i	$0.131239\pi$
$523$	419.149i	0.801432i	−0.916202	+	0.400716i	$0.868761\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	281.495	0.534146
$528$	0	0
$529$	−506.895	−0.958214
$530$	0	0
$531$	− 1584.02i	− 2.98309i
$532$	0	0
$533$	−179.149	−0.336114
$534$	0	0
$535$	626.694i	1.17139i
$536$	0	0
$537$	1022.13i	1.90341i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−618.487	−1.14323	−0.571614	−	0.820522i	$-0.693683\pi$
$541$	−618.487	−1.14323	−0.571614	+	0.820522i	$0.693683\pi$
$542$	0	0
$543$	−989.728	−1.82270
$544$	0	0
$545$	95.7522i	0.175692i
$546$	0	0
$547$	−255.031	−0.466236	−0.233118	−	0.972448i	$-0.574893\pi$
$547$	−255.031	−0.466236	−0.233118	+	0.972448i	$0.574893\pi$
$548$	0	0
$549$	− 765.394i	− 1.39416i
$550$	0	0
$551$	23.9874i	0.0435343i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−378.756	−0.682444
$556$	0	0
$557$	488.260	0.876588	0.438294	−	0.898832i	$-0.355583\pi$
$557$	488.260	0.876588	0.438294	+	0.898832i	$0.355583\pi$
$558$	0	0
$559$	− 783.621i	− 1.40183i
$560$	0	0
$561$	636.069	1.13381
$562$	0	0
$563$	− 49.0923i	− 0.0871977i	−0.999049	−	0.0435989i	$-0.986118\pi$
$563$	− 49.0923i	− 0.0871977i	0.999049	−	0.0435989i	$-0.0138824\pi$
$564$	0	0
$565$	− 718.978i	− 1.27253i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−608.113	−1.06874	−0.534370	−	0.845251i	$-0.679451\pi$
$569$	−608.113	−1.06874	−0.534370	+	0.845251i	$0.679451\pi$
$570$	0	0
$571$	−300.793	−0.526782	−0.263391	−	0.964689i	$-0.584841\pi$
$571$	−300.793	−0.526782	−0.263391	+	0.964689i	$0.584841\pi$
$572$	0	0
$573$	799.259i	1.39487i
$574$	0	0
$575$	69.3082	0.120536
$576$	0	0
$577$	− 174.035i	− 0.301621i	−0.988563	−	0.150810i	$-0.951812\pi$
$577$	− 174.035i	− 0.301621i	0.988563	−	0.150810i	$-0.0481883\pi$
$578$	0	0
$579$	248.154i	0.428590i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−173.705	−0.297949
$584$	0	0
$585$	−2405.84	−4.11255
$586$	0	0
$587$	− 477.592i	− 0.813614i	−0.913514	−	0.406807i	$-0.866642\pi$
$587$	− 477.592i	− 0.813614i	0.913514	−	0.406807i	$-0.133358\pi$
$588$	0	0
$589$	−186.764	−0.317086
$590$	0	0
$591$	− 1181.90i	− 1.99984i
$592$	0	0
$593$	− 441.542i	− 0.744590i	−0.928114	−	0.372295i	$-0.878571\pi$
$593$	− 441.542i	− 0.744590i	0.928114	−	0.372295i	$-0.121429\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1729.39	−2.89680
$598$	0	0
$599$	292.092	0.487633	0.243816	−	0.969821i	$-0.421601\pi$
$599$	292.092	0.487633	0.243816	+	0.969821i	$0.421601\pi$
$600$	0	0
$601$	− 614.010i	− 1.02165i	−0.859685	−	0.510824i	$-0.829340\pi$
$601$	− 614.010i	− 1.02165i	0.859685	−	0.510824i	$-0.170660\pi$
$602$	0	0
$603$	2359.64	3.91316
$604$	0	0
$605$	578.207i	0.955715i
$606$	0	0
$607$	325.232i	0.535802i	0.963446	+	0.267901i	$0.0863301\pi$
$607$	325.232i	0.535802i	−0.963446	+	0.267901i	$0.913670\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	166.059	0.271782
$612$	0	0
$613$	556.936	0.908542	0.454271	−	0.890864i	$-0.349900\pi$
$613$	556.936	0.908542	0.454271	+	0.890864i	$0.349900\pi$
$614$	0	0
$615$	− 406.761i	− 0.661399i
$616$	0	0
$617$	277.944	0.450476	0.225238	−	0.974304i	$-0.427684\pi$
$617$	277.944	0.450476	0.225238	+	0.974304i	$0.427684\pi$
$618$	0	0
$619$	128.939i	0.208302i	0.994561	+	0.104151i	$0.0332126\pi$
$619$	128.939i	0.208302i	−0.994561	+	0.104151i	$0.966787\pi$
$620$	0	0
$621$	− 403.358i	− 0.649530i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−776.226	−1.24196
$626$	0	0
$627$	−422.013	−0.673067
$628$	0	0
$629$	− 214.360i	− 0.340795i
$630$	0	0
$631$	279.339	0.442692	0.221346	−	0.975195i	$-0.428955\pi$
$631$	279.339	0.442692	0.221346	+	0.975195i	$0.428955\pi$
$632$	0	0
$633$	− 1779.96i	− 2.81194i
$634$	0	0
$635$	− 174.917i	− 0.275460i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1478.79	2.31422
$640$	0	0
$641$	−882.265	−1.37639	−0.688194	−	0.725526i	$-0.741596\pi$
$641$	−882.265	−1.37639	−0.688194	+	0.725526i	$0.741596\pi$
$642$	0	0
$643$	− 575.210i	− 0.894572i	−0.894391	−	0.447286i	$-0.852391\pi$
$643$	− 575.210i	− 0.894572i	0.894391	−	0.447286i	$-0.147609\pi$
$644$	0	0
$645$	1779.22	2.75849
$646$	0	0
$647$	469.048i	0.724958i	0.931992	+	0.362479i	$0.118070\pi$
$647$	469.048i	0.724958i	−0.931992	+	0.362479i	$0.881930\pi$
$648$	0	0
$649$	− 357.937i	− 0.551521i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	824.926	1.26329	0.631644	−	0.775259i	$-0.282381\pi$
$653$	824.926	1.26329	0.631644	+	0.775259i	$0.282381\pi$
$654$	0	0
$655$	1152.07	1.75889
$656$	0	0
$657$	− 432.056i	− 0.657620i
$658$	0	0
$659$	888.926	1.34890	0.674451	−	0.738320i	$-0.264381\pi$
$659$	888.926	1.34890	0.674451	+	0.738320i	$0.264381\pi$
$660$	0	0
$661$	− 151.288i	− 0.228877i	−0.993430	−	0.114439i	$-0.963493\pi$
$661$	− 151.288i	− 0.228877i	0.993430	−	0.114439i	$-0.0365069\pi$
$662$	0	0
$663$	− 1873.34i	− 2.82556i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.30035	−0.0124443
$668$	0	0
$669$	1527.33	2.28301
$670$	0	0
$671$	− 172.954i	− 0.257756i
$672$	0	0
$673$	−904.158	−1.34347	−0.671737	−	0.740790i	$-0.734452\pi$
$673$	−904.158	−1.34347	−0.671737	+	0.740790i	$0.734452\pi$
$674$	0	0
$675$	− 1264.69i	− 1.87362i
$676$	0	0
$677$	880.547i	1.30066i	0.759652	+	0.650330i	$0.225369\pi$
$677$	880.547i	1.30066i	−0.759652	+	0.650330i	$0.774631\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1751.00	2.57122
$682$	0	0
$683$	−545.551	−0.798757	−0.399379	−	0.916786i	$-0.630774\pi$
$683$	−545.551	−0.798757	−0.399379	+	0.916786i	$0.630774\pi$
$684$	0	0
$685$	− 637.388i	− 0.930493i
$686$	0	0
$687$	−1139.78	−1.65907
$688$	0	0
$689$	511.593i	0.742515i
$690$	0	0
$691$	941.927i	1.36314i	0.731755	+	0.681568i	$0.238701\pi$
$691$	941.927i	1.36314i	−0.731755	+	0.681568i	$0.761299\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	558.286	0.803289
$696$	0	0
$697$	230.209	0.330286
$698$	0	0
$699$	1422.67i	2.03530i
$700$	0	0
$701$	−304.580	−0.434494	−0.217247	−	0.976117i	$-0.569708\pi$
$701$	−304.580	−0.434494	−0.217247	+	0.976117i	$0.569708\pi$
$702$	0	0
$703$	142.221i	0.202306i
$704$	0	0
$705$	377.039i	0.534807i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−643.877	−0.908148	−0.454074	−	0.890964i	$-0.650030\pi$
$709$	−643.877	−0.908148	−0.454074	+	0.890964i	$0.650030\pi$
$710$	0	0
$711$	718.483	1.01052
$712$	0	0
$713$	− 64.6257i	− 0.0906391i
$714$	0	0
$715$	−543.641	−0.760338
$716$	0	0
$717$	− 693.862i	− 0.967730i
$718$	0	0
$719$	− 725.206i	− 1.00863i	−0.863519	−	0.504316i	$-0.831745\pi$
$719$	− 725.206i	− 1.00863i	0.863519	−	0.504316i	$-0.168255\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	834.822	1.15466
$724$	0	0
$725$	−26.0250	−0.0358965
$726$	0	0
$727$	− 1090.68i	− 1.50025i	−0.661295	−	0.750126i	$-0.729993\pi$
$727$	− 1090.68i	− 1.50025i	0.661295	−	0.750126i	$-0.270007\pi$
$728$	0	0
$729$	−2199.30	−3.01688
$730$	0	0
$731$	1006.97i	1.37752i
$732$	0	0
$733$	236.205i	0.322244i	0.986935	+	0.161122i	$0.0515112\pi$
$733$	236.205i	0.322244i	−0.986935	+	0.161122i	$0.948489\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	533.201	0.723475
$738$	0	0
$739$	−0.281069	−0.000380336 0	−0.000190168	−	1.00000i	$-0.500061\pi$
$739$	−0.281069	−0.000380336 0	−0.000190168		1.00000i	$0.500061\pi$
$740$	0	0
$741$	1242.91i	1.67734i
$742$	0	0
$743$	−172.285	−0.231878	−0.115939	−	0.993256i	$-0.536988\pi$
$743$	−172.285	−0.231878	−0.115939	+	0.993256i	$0.536988\pi$
$744$	0	0
$745$	800.847i	1.07496i
$746$	0	0
$747$	1519.61i	2.03428i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−50.9471	−0.0678390	−0.0339195	−	0.999425i	$-0.510799\pi$
$751$	−50.9471	−0.0678390	−0.0339195	+	0.999425i	$0.510799\pi$
$752$	0	0
$753$	1217.21	1.61648
$754$	0	0
$755$	1836.92i	2.43300i
$756$	0	0
$757$	−472.598	−0.624304	−0.312152	−	0.950032i	$-0.601050\pi$
$757$	−472.598	−0.624304	−0.312152	+	0.950032i	$0.601050\pi$
$758$	0	0
$759$	− 146.029i	− 0.192396i
$760$	0	0
$761$	427.525i	0.561794i	0.959738	+	0.280897i	$0.0906319\pi$
$761$	427.525i	0.561794i	−0.959738	+	0.280897i	$0.909368\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3091.54	4.04123
$766$	0	0
$767$	−1054.19	−1.37444
$768$	0	0
$769$	− 199.651i	− 0.259625i	−0.991539	−	0.129812i	$-0.958563\pi$
$769$	− 199.651i	− 0.259625i	0.991539	−	0.129812i	$-0.0414375\pi$
$770$	0	0
$771$	−57.7849	−0.0749480
$772$	0	0
$773$	− 42.4021i	− 0.0548540i	−0.999624	−	0.0274270i	$-0.991269\pi$
$773$	− 42.4021i	− 0.0548540i	0.999624	−	0.0274270i	$-0.00873137\pi$
$774$	0	0
$775$	− 202.628i	− 0.261455i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−152.737	−0.196068
$780$	0	0
$781$	334.158	0.427859
$782$	0	0
$783$	151.460i	0.193435i
$784$	0	0
$785$	1358.50	1.73057
$786$	0	0
$787$	605.183i	0.768974i	0.923130	+	0.384487i	$0.125622\pi$
$787$	605.183i	0.768974i	−0.923130	+	0.384487i	$0.874378\pi$
$788$	0	0
$789$	0.479445i	0 0.000607662i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−509.383	−0.642349
$794$	0	0
$795$	−1161.58	−1.46111
$796$	0	0
$797$	− 349.434i	− 0.438437i	−0.975676	−	0.219219i	$-0.929649\pi$
$797$	− 349.434i	− 0.438437i	0.975676	−	0.219219i	$-0.0703508\pi$
$798$	0	0
$799$	−213.388	−0.267069
$800$	0	0
$801$	3294.04i	4.11242i
$802$	0	0
$803$	− 97.6307i	− 0.121582i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.31524	−0.00906474
$808$	0	0
$809$	−59.5871	−0.0736553	−0.0368276	−	0.999322i	$-0.511725\pi$
$809$	−59.5871	−0.0736553	−0.0368276	+	0.999322i	$0.511725\pi$
$810$	0	0
$811$	− 157.670i	− 0.194414i	−0.995264	−	0.0972069i	$-0.969009\pi$
$811$	− 157.670i	− 0.194414i	0.995264	−	0.0972069i	$-0.0309909\pi$
$812$	0	0
$813$	2052.98	2.52519
$814$	0	0
$815$	− 1097.02i	− 1.34603i
$816$	0	0
$817$	− 668.092i	− 0.817738i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−338.137	−0.411859	−0.205930	−	0.978567i	$-0.566022\pi$
$821$	−338.137	−0.411859	−0.205930	+	0.978567i	$0.566022\pi$
$822$	0	0
$823$	1115.34	1.35521	0.677605	−	0.735426i	$-0.263018\pi$
$823$	1115.34	1.35521	0.677605	+	0.735426i	$0.263018\pi$
$824$	0	0
$825$	− 457.861i	− 0.554982i
$826$	0	0
$827$	−1368.45	−1.65472	−0.827359	−	0.561673i	$-0.810158\pi$
$827$	−1368.45	−1.65472	−0.827359	+	0.561673i	$0.810158\pi$
$828$	0	0
$829$	339.141i	0.409096i	0.978857	+	0.204548i	$0.0655725\pi$
$829$	339.141i	0.409096i	−0.978857	+	0.204548i	$0.934428\pi$
$830$	0	0
$831$	971.277i	1.16881i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1520.36	−1.82079
$836$	0	0
$837$	−1179.25	−1.40890
$838$	0	0
$839$	− 402.959i	− 0.480284i	−0.970738	−	0.240142i	$-0.922806\pi$
$839$	− 402.959i	− 0.480284i	0.970738	−	0.240142i	$-0.0771941\pi$
$840$	0	0
$841$	−837.883	−0.996294
$842$	0	0
$843$	− 1416.81i	− 1.68068i
$844$	0	0
$845$	535.738i	0.634009i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1318.20	1.55265
$850$	0	0
$851$	−49.2128	−0.0578294
$852$	0	0
$853$	− 1179.46i	− 1.38272i	−0.722512	−	0.691359i	$-0.757012\pi$
$853$	− 1179.46i	− 1.38272i	0.722512	−	0.691359i	$-0.242988\pi$
$854$	0	0
$855$	−2051.15	−2.39900
$856$	0	0
$857$	669.619i	0.781352i	0.920528	+	0.390676i	$0.127759\pi$
$857$	669.619i	0.781352i	−0.920528	+	0.390676i	$0.872241\pi$
$858$	0	0
$859$	− 869.756i	− 1.01252i	−0.862380	−	0.506261i	$-0.831027\pi$
$859$	− 869.756i	− 1.01252i	0.862380	−	0.506261i	$-0.168973\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−91.9408	−0.106536	−0.0532681	−	0.998580i	$-0.516964\pi$
$863$	−91.9408	−0.106536	−0.0532681	+	0.998580i	$0.516964\pi$
$864$	0	0
$865$	1361.51	1.57400
$866$	0	0
$867$	748.444i	0.863257i
$868$	0	0
$869$	162.354	0.186828
$870$	0	0
$871$	− 1570.38i	− 1.80296i
$872$	0	0
$873$	− 3138.79i	− 3.59541i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−436.320	−0.497515	−0.248757	−	0.968566i	$-0.580022\pi$
$877$	−436.320	−0.497515	−0.248757	+	0.968566i	$0.580022\pi$
$878$	0	0
$879$	1309.68	1.48997
$880$	0	0
$881$	884.017i	1.00342i	0.865035	+	0.501712i	$0.167296\pi$
$881$	884.017i	1.00342i	−0.865035	+	0.501712i	$0.832704\pi$
$882$	0	0
$883$	−62.8499	−0.0711776	−0.0355888	−	0.999367i	$-0.511331\pi$
$883$	−62.8499	−0.0711776	−0.0355888	+	0.999367i	$0.511331\pi$
$884$	0	0
$885$	− 2393.56i	− 2.70459i
$886$	0	0
$887$	1040.67i	1.17324i	0.809862	+	0.586621i	$0.199542\pi$
$887$	1040.67i	1.17324i	−0.809862	+	0.586621i	$0.800458\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1498.44	−1.68176
$892$	0	0
$893$	141.577	0.158541
$894$	0	0
$895$	1122.60i	1.25430i
$896$	0	0
$897$	−430.083	−0.479468
$898$	0	0
$899$	24.2667i	0.0269930i
$900$	0	0
$901$	− 657.405i	− 0.729639i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1087.01	−1.20111
$906$	0	0
$907$	−1199.14	−1.32209	−0.661045	−	0.750346i	$-0.729887\pi$
$907$	−1199.14	−1.32209	−0.661045	+	0.750346i	$0.729887\pi$
$908$	0	0
$909$	2745.41i	3.02026i
$910$	0	0
$911$	1588.16	1.74331	0.871657	−	0.490116i	$-0.163046\pi$
$911$	1588.16	1.74331	0.871657	+	0.490116i	$0.163046\pi$
$912$	0	0
$913$	343.382i	0.376103i
$914$	0	0
$915$	− 1156.56i	− 1.26400i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1146.15	−1.24717	−0.623584	−	0.781757i	$-0.714324\pi$
$919$	−1146.15	−1.24717	−0.623584	+	0.781757i	$0.714324\pi$
$920$	0	0
$921$	1915.67	2.07999
$922$	0	0
$923$	− 984.158i	− 1.06626i
$924$	0	0
$925$	−154.302	−0.166813
$926$	0	0
$927$	4804.00i	5.18231i
$928$	0	0
$929$	342.137i	0.368285i	0.982900	+	0.184142i	$0.0589508\pi$
$929$	342.137i	0.368285i	−0.982900	+	0.184142i	$0.941049\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1429.33	−1.53197
$934$	0	0
$935$	698.588	0.747153
$936$	0	0
$937$	77.2601i	0.0824548i	0.999150	+	0.0412274i	$0.0131268\pi$
$937$	77.2601i	0.0824548i	−0.999150	+	0.0412274i	$0.986873\pi$
$938$	0	0
$939$	228.759	0.243620
$940$	0	0
$941$	480.342i	0.510459i	0.966881	+	0.255229i	$0.0821510\pi$
$941$	480.342i	0.510459i	−0.966881	+	0.255229i	$0.917849\pi$
$942$	0	0
$943$	− 52.8515i	− 0.0560461i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	390.489	0.412343	0.206171	−	0.978516i	$-0.433900\pi$
$947$	390.489	0.412343	0.206171	+	0.978516i	$0.433900\pi$
$948$	0	0
$949$	−287.541	−0.302994
$950$	0	0
$951$	2539.40i	2.67025i
$952$	0	0
$953$	−1234.80	−1.29570	−0.647851	−	0.761767i	$-0.724332\pi$
$953$	−1234.80	−1.29570	−0.647851	+	0.761767i	$0.724332\pi$
$954$	0	0
$955$	877.817i	0.919181i
$956$	0	0
$957$	54.8334i	0.0572972i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	772.062	0.803394
$962$	0	0
$963$	2380.55	2.47201
$964$	0	0
$965$	272.545i	0.282430i
$966$	0	0
$967$	−719.055	−0.743593	−0.371797	−	0.928314i	$-0.621258\pi$
$967$	−719.055	−0.743593	−0.371797	+	0.928314i	$0.621258\pi$
$968$	0	0
$969$	− 1597.16i	− 1.64825i
$970$	0	0
$971$	967.347i	0.996238i	0.867109	+	0.498119i	$0.165976\pi$
$971$	967.347i	0.996238i	−0.867109	+	0.498119i	$0.834024\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1348.49	−1.38306
$976$	0	0
$977$	−1336.91	−1.36838	−0.684192	−	0.729302i	$-0.739845\pi$
$977$	−1336.91	−1.36838	−0.684192	+	0.729302i	$0.739845\pi$
$978$	0	0
$979$	744.347i	0.760313i
$980$	0	0
$981$	363.722	0.370767
$982$	0	0
$983$	537.591i	0.546888i	0.961888	+	0.273444i	$0.0881628\pi$
$983$	537.591i	0.546888i	−0.961888	+	0.273444i	$0.911837\pi$
$984$	0	0
$985$	− 1298.07i	− 1.31784i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	231.179	0.233751
$990$	0	0
$991$	371.174	0.374545	0.187273	−	0.982308i	$-0.440035\pi$
$991$	371.174	0.374545	0.187273	+	0.982308i	$0.440035\pi$
$992$	0	0
$993$	826.135i	0.831959i
$994$	0	0
$995$	−1899.37	−1.90891
$996$	0	0
$997$	− 810.625i	− 0.813064i	−0.913636	−	0.406532i	$-0.866738\pi$
$997$	− 810.625i	− 0.813064i	0.913636	−	0.406532i	$-0.133262\pi$
$998$	0	0
$999$	898.006i	0.898905i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.g.97.1		16
4.3	odd	2	inner	1568.3.c.g.97.15		16
7.4	even	3		224.3.s.b.33.1	✓	16
7.5	odd	6		224.3.s.b.129.1	yes	16
7.6	odd	2	inner	1568.3.c.g.97.16		16
28.11	odd	6		224.3.s.b.33.8	yes	16
28.19	even	6		224.3.s.b.129.8	yes	16
28.27	even	2	inner	1568.3.c.g.97.2		16
56.5	odd	6		448.3.s.h.129.8		16
56.11	odd	6		448.3.s.h.257.1		16
56.19	even	6		448.3.s.h.129.1		16
56.53	even	6		448.3.s.h.257.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.s.b.33.1	✓	16	7.4	even	3
224.3.s.b.33.8	yes	16	28.11	odd	6
224.3.s.b.129.1	yes	16	7.5	odd	6
224.3.s.b.129.8	yes	16	28.19	even	6
448.3.s.h.129.1		16	56.19	even	6
448.3.s.h.129.8		16	56.5	odd	6
448.3.s.h.257.1		16	56.11	odd	6
448.3.s.h.257.8		16	56.53	even	6
1568.3.c.g.97.1		16	1.1	even	1	trivial
1568.3.c.g.97.2		16	28.27	even	2	inner
1568.3.c.g.97.15		16	4.3	odd	2	inner
1568.3.c.g.97.16		16	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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.73991i q^{3} -6.30408i q^{5} -23.9466 q^{9} +O(q^{10})\)	\(q-5.73991i q^{3} -6.30408i q^{5} -23.9466 q^{9} -5.41114 q^{11} +15.9368i q^{13} -36.1848 q^{15} -20.4791i q^{17} +13.5873i q^{19} -4.70159 q^{23} -14.7414 q^{25} +85.7918i q^{27} +1.76543 q^{29} +13.7455i q^{31} +31.0595i q^{33} +10.4673 q^{37} +91.4760 q^{39} +11.2412i q^{41} -49.1704 q^{43} +150.961i q^{45} -10.4198i q^{47} -117.548 q^{51} +32.1013 q^{53} +34.1123i q^{55} +77.9897 q^{57} +66.1482i q^{59} +31.9626i q^{61} +100.467 q^{65} -98.5377 q^{67} +26.9867i q^{69} -61.7537 q^{71} +18.0425i q^{73} +84.6144i q^{75} -30.0036 q^{79} +276.918 q^{81} -63.4583i q^{83} -129.102 q^{85} -10.1334i q^{87} -137.558i q^{89} +78.8978 q^{93} +85.6552 q^{95} +131.075i q^{97} +129.578 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 80 q^{9}+O(q^{10}) \) 16 * q - 80 * q^9	\( 16 q - 80 q^{9} - 160 q^{25} - 16 q^{29} - 144 q^{37} - 80 q^{53} + 368 q^{57} + 336 q^{65} + 768 q^{81} - 1072 q^{85} + 336 q^{93}+O(q^{100}) \) 16 * q - 80 * q^9 - 160 * q^25 - 16 * q^29 - 144 * q^37 - 80 * q^53 + 368 * q^57 + 336 * q^65 + 768 * q^81 - 1072 * q^85 + 336 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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