LMFDB - Embedded newform 1568.3.c.g.97.9

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			97.9
Root			$0.869658 + 0.314400i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.g.97.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.842794i q^{3} -1.40510i q^{5} +8.28970 q^{9} +O(q^{10})$	$q+0.842794i q^{3} -1.40510i q^{5} +8.28970 q^{9} +14.8912 q^{11} +2.67477i q^{13} +1.18421 q^{15} -13.5509i q^{17} -29.1636i q^{19} -22.8734 q^{23} +23.0257 q^{25} +14.5717i q^{27} -3.76543 q^{29} -13.2718i q^{31} +12.5503i q^{33} +2.64544 q^{37} -2.25428 q^{39} -45.2712i q^{41} -51.5858 q^{43} -11.6479i q^{45} +67.1206i q^{47} +11.4206 q^{51} +39.9231 q^{53} -20.9237i q^{55} +24.5789 q^{57} +12.7946i q^{59} +77.8365i q^{61} +3.75832 q^{65} -45.8399 q^{67} -19.2775i q^{69} +40.6812 q^{71} -64.2976i q^{73} +19.4059i q^{75} +80.5206 q^{79} +62.3264 q^{81} -121.864i q^{83} -19.0403 q^{85} -3.17349i q^{87} -45.9125i q^{89} +11.1854 q^{93} -40.9777 q^{95} -134.268i q^{97} +123.444 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.842794i	0.280931i	0.990086	+	0.140466i	$0.0448600\pi$
$3$	0.842794i	0.280931i	−0.990086	+	0.140466i	$0.955140\pi$
$4$	0	0
$5$	− 1.40510i	− 0.281020i	−0.990079	−	0.140510i	$-0.955126\pi$
$5$	− 1.40510i	− 0.281020i	0.990079	−	0.140510i	$-0.0448742\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.28970	0.921077
$10$	0	0
$11$	14.8912	1.35375	0.676874	−	0.736098i	$-0.263334\pi$
$11$	14.8912	1.35375	0.676874	+	0.736098i	$0.263334\pi$
$12$	0	0
$13$	2.67477i	0.205752i	0.994694	+	0.102876i	$0.0328045\pi$
$13$	2.67477i	0.205752i	−0.994694	+	0.102876i	$0.967196\pi$
$14$	0	0
$15$	1.18421	0.0789474
$16$	0	0
$17$	− 13.5509i	− 0.797111i	−0.917144	−	0.398555i	$-0.869512\pi$
$17$	− 13.5509i	− 0.797111i	0.917144	−	0.398555i	$-0.130488\pi$
$18$	0	0
$19$	− 29.1636i	− 1.53492i	−0.641094	−	0.767462i	$-0.721519\pi$
$19$	− 29.1636i	− 1.53492i	0.641094	−	0.767462i	$-0.278481\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−22.8734	−0.994494	−0.497247	−	0.867609i	$-0.665656\pi$
$23$	−22.8734	−0.994494	−0.497247	+	0.867609i	$0.665656\pi$
$24$	0	0
$25$	23.0257	0.921028
$26$	0	0
$27$	14.5717i	0.539691i
$28$	0	0
$29$	−3.76543	−0.129843	−0.0649213	−	0.997890i	$-0.520680\pi$
$29$	−3.76543	−0.129843	−0.0649213	+	0.997890i	$0.520680\pi$
$30$	0	0
$31$	− 13.2718i	− 0.428124i	−0.976820	−	0.214062i	$-0.931331\pi$
$31$	− 13.2718i	− 0.428124i	0.976820	−	0.214062i	$-0.0686694\pi$
$32$	0	0
$33$	12.5503i	0.380311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.64544	0.0714983	0.0357491	−	0.999361i	$-0.488618\pi$
$37$	2.64544	0.0714983	0.0357491	+	0.999361i	$0.488618\pi$
$38$	0	0
$39$	−2.25428	−0.0578022
$40$	0	0
$41$	− 45.2712i	− 1.10417i	−0.833786	−	0.552087i	$-0.813832\pi$
$41$	− 45.2712i	− 1.10417i	0.833786	−	0.552087i	$-0.186168\pi$
$42$	0	0
$43$	−51.5858	−1.19967	−0.599834	−	0.800124i	$-0.704767\pi$
$43$	−51.5858	−1.19967	−0.599834	+	0.800124i	$0.704767\pi$
$44$	0	0
$45$	− 11.6479i	− 0.258841i
$46$	0	0
$47$	67.1206i	1.42810i	0.700096	+	0.714049i	$0.253141\pi$
$47$	67.1206i	1.42810i	−0.700096	+	0.714049i	$0.746859\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	11.4206	0.223934
$52$	0	0
$53$	39.9231	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$53$	39.9231	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$54$	0	0
$55$	− 20.9237i	− 0.380431i
$56$	0	0
$57$	24.5789	0.431209
$58$	0	0
$59$	12.7946i	0.216857i	0.994104	+	0.108429i	$0.0345819\pi$
$59$	12.7946i	0.216857i	−0.994104	+	0.108429i	$0.965418\pi$
$60$	0	0
$61$	77.8365i	1.27601i	0.770033	+	0.638004i	$0.220240\pi$
$61$	77.8365i	1.27601i	−0.770033	+	0.638004i	$0.779760\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.75832	0.0578204
$66$	0	0
$67$	−45.8399	−0.684177	−0.342089	−	0.939668i	$-0.611134\pi$
$67$	−45.8399	−0.684177	−0.342089	+	0.939668i	$0.611134\pi$
$68$	0	0
$69$	− 19.2775i	− 0.279385i
$70$	0	0
$71$	40.6812	0.572975	0.286487	−	0.958084i	$-0.407512\pi$
$71$	40.6812	0.572975	0.286487	+	0.958084i	$0.407512\pi$
$72$	0	0
$73$	− 64.2976i	− 0.880789i	−0.897804	−	0.440395i	$-0.854839\pi$
$73$	− 64.2976i	− 0.880789i	0.897804	−	0.440395i	$-0.145161\pi$
$74$	0	0
$75$	19.4059i	0.258746i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	80.5206	1.01925	0.509624	−	0.860397i	$-0.329784\pi$
$79$	80.5206	1.01925	0.509624	+	0.860397i	$0.329784\pi$
$80$	0	0
$81$	62.3264	0.769461
$82$	0	0
$83$	− 121.864i	− 1.46824i	−0.679021	−	0.734118i	$-0.737596\pi$
$83$	− 121.864i	− 1.46824i	0.679021	−	0.734118i	$-0.262404\pi$
$84$	0	0
$85$	−19.0403	−0.224004
$86$	0	0
$87$	− 3.17349i	− 0.0364769i
$88$	0	0
$89$	− 45.9125i	− 0.515871i	−0.966162	−	0.257935i	$-0.916958\pi$
$89$	− 45.9125i	− 0.515871i	0.966162	−	0.257935i	$-0.0830421\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	11.1854	0.120273
$94$	0	0
$95$	−40.9777	−0.431344
$96$	0	0
$97$	− 134.268i	− 1.38421i	−0.721798	−	0.692104i	$-0.756684\pi$
$97$	− 134.268i	− 1.38421i	0.721798	−	0.692104i	$-0.243316\pi$
$98$	0	0
$99$	123.444	1.24691
$100$	0	0
$101$	− 197.234i	− 1.95281i	−0.215946	−	0.976405i	$-0.569284\pi$
$101$	− 197.234i	− 1.95281i	0.215946	−	0.976405i	$-0.430716\pi$
$102$	0	0
$103$	− 140.147i	− 1.36065i	−0.732909	−	0.680326i	$-0.761838\pi$
$103$	− 140.147i	− 1.36065i	0.732909	−	0.680326i	$-0.238162\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−106.020	−0.990840	−0.495420	−	0.868654i	$-0.664986\pi$
$107$	−106.020	−0.990840	−0.495420	+	0.868654i	$0.664986\pi$
$108$	0	0
$109$	142.752	1.30966	0.654828	−	0.755778i	$-0.272741\pi$
$109$	142.752	1.30966	0.654828	+	0.755778i	$0.272741\pi$
$110$	0	0
$111$	2.22956i	0.0200861i
$112$	0	0
$113$	108.519	0.960344	0.480172	−	0.877174i	$-0.340574\pi$
$113$	108.519	0.960344	0.480172	+	0.877174i	$0.340574\pi$
$114$	0	0
$115$	32.1394i	0.279473i
$116$	0	0
$117$	22.1731i	0.189513i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	100.749	0.832636
$122$	0	0
$123$	38.1543	0.310197
$124$	0	0
$125$	− 67.4809i	− 0.539847i
$126$	0	0
$127$	−139.789	−1.10070	−0.550351	−	0.834934i	$-0.685506\pi$
$127$	−139.789	−1.10070	−0.550351	+	0.834934i	$0.685506\pi$
$128$	0	0
$129$	− 43.4762i	− 0.337025i
$130$	0	0
$131$	95.0252i	0.725383i	0.931909	+	0.362692i	$0.118142\pi$
$131$	95.0252i	0.725383i	−0.931909	+	0.362692i	$0.881858\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	20.4746	0.151664
$136$	0	0
$137$	153.569	1.12094	0.560471	−	0.828174i	$-0.310620\pi$
$137$	153.569	1.12094	0.560471	+	0.828174i	$0.310620\pi$
$138$	0	0
$139$	217.529i	1.56496i	0.622676	+	0.782480i	$0.286046\pi$
$139$	217.529i	1.56496i	−0.622676	+	0.782480i	$0.713954\pi$
$140$	0	0
$141$	−56.5689	−0.401198
$142$	0	0
$143$	39.8307i	0.278536i
$144$	0	0
$145$	5.29081i	0.0364884i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	64.0902	0.430135	0.215068	−	0.976599i	$-0.431003\pi$
$149$	64.0902	0.430135	0.215068	+	0.976599i	$0.431003\pi$
$150$	0	0
$151$	272.046	1.80163	0.900813	−	0.434206i	$-0.142971\pi$
$151$	272.046	1.80163	0.900813	+	0.434206i	$0.142971\pi$
$152$	0	0
$153$	− 112.333i	− 0.734201i
$154$	0	0
$155$	−18.6483	−0.120311
$156$	0	0
$157$	− 216.270i	− 1.37751i	−0.724993	−	0.688757i	$-0.758157\pi$
$157$	− 216.270i	− 1.37751i	0.724993	−	0.688757i	$-0.241843\pi$
$158$	0	0
$159$	33.6470i	0.211616i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	54.6930	0.335540	0.167770	−	0.985826i	$-0.446343\pi$
$163$	54.6930	0.335540	0.167770	+	0.985826i	$0.446343\pi$
$164$	0	0
$165$	17.6344	0.106875
$166$	0	0
$167$	248.098i	1.48562i	0.669504	+	0.742809i	$0.266507\pi$
$167$	248.098i	1.48562i	−0.669504	+	0.742809i	$0.733493\pi$
$168$	0	0
$169$	161.846	0.957666
$170$	0	0
$171$	− 241.757i	− 1.41378i
$172$	0	0
$173$	259.673i	1.50100i	0.660872	+	0.750499i	$0.270187\pi$
$173$	259.673i	1.50100i	−0.660872	+	0.750499i	$0.729813\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.7832	−0.0609220
$178$	0	0
$179$	−138.177	−0.771936	−0.385968	−	0.922512i	$-0.626133\pi$
$179$	−138.177	−0.771936	−0.385968	+	0.922512i	$0.626133\pi$
$180$	0	0
$181$	114.497i	0.632579i	0.948663	+	0.316290i	$0.102437\pi$
$181$	114.497i	0.632579i	−0.948663	+	0.316290i	$0.897563\pi$
$182$	0	0
$183$	−65.6001	−0.358471
$184$	0	0
$185$	− 3.71710i	− 0.0200925i
$186$	0	0
$187$	− 201.789i	− 1.07909i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−152.429	−0.798057	−0.399029	−	0.916939i	$-0.630653\pi$
$191$	−152.429	−0.798057	−0.399029	+	0.916939i	$0.630653\pi$
$192$	0	0
$193$	242.311	1.25550	0.627749	−	0.778416i	$-0.283976\pi$
$193$	242.311	1.25550	0.627749	+	0.778416i	$0.283976\pi$
$194$	0	0
$195$	3.16750i	0.0162436i
$196$	0	0
$197$	−197.518	−1.00263	−0.501314	−	0.865265i	$-0.667150\pi$
$197$	−197.518	−1.00263	−0.501314	+	0.865265i	$0.667150\pi$
$198$	0	0
$199$	164.167i	0.824958i	0.910967	+	0.412479i	$0.135337\pi$
$199$	164.167i	0.824958i	−0.910967	+	0.412479i	$0.864663\pi$
$200$	0	0
$201$	− 38.6336i	− 0.192207i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−63.6105	−0.310295
$206$	0	0
$207$	−189.613	−0.916006
$208$	0	0
$209$	− 434.282i	− 2.07790i
$210$	0	0
$211$	53.1445	0.251870	0.125935	−	0.992039i	$-0.459807\pi$
$211$	53.1445	0.251870	0.125935	+	0.992039i	$0.459807\pi$
$212$	0	0
$213$	34.2859i	0.160967i
$214$	0	0
$215$	72.4832i	0.337131i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	54.1897	0.247441
$220$	0	0
$221$	36.2455	0.164007
$222$	0	0
$223$	− 310.755i	− 1.39352i	−0.717303	−	0.696761i	$-0.754624\pi$
$223$	− 310.755i	− 1.39352i	0.717303	−	0.696761i	$-0.245376\pi$
$224$	0	0
$225$	190.876	0.848338
$226$	0	0
$227$	49.8982i	0.219816i	0.993942	+	0.109908i	$0.0350556\pi$
$227$	49.8982i	0.219816i	−0.993942	+	0.109908i	$0.964944\pi$
$228$	0	0
$229$	211.118i	0.921911i	0.887423	+	0.460955i	$0.152493\pi$
$229$	211.118i	0.921911i	−0.887423	+	0.460955i	$0.847507\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	75.4547	0.323840	0.161920	−	0.986804i	$-0.448231\pi$
$233$	75.4547	0.323840	0.161920	+	0.986804i	$0.448231\pi$
$234$	0	0
$235$	94.3112	0.401324
$236$	0	0
$237$	67.8623i	0.286339i
$238$	0	0
$239$	310.396	1.29873	0.649365	−	0.760477i	$-0.275035\pi$
$239$	310.396	1.29873	0.649365	+	0.760477i	$0.275035\pi$
$240$	0	0
$241$	− 108.219i	− 0.449039i	−0.974470	−	0.224520i	$-0.927919\pi$
$241$	− 108.219i	− 0.449039i	0.974470	−	0.224520i	$-0.0720813\pi$
$242$	0	0
$243$	183.673i	0.755857i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	78.0059	0.315813
$248$	0	0
$249$	102.706	0.412474
$250$	0	0
$251$	− 302.023i	− 1.20328i	−0.798768	−	0.601639i	$-0.794515\pi$
$251$	− 302.023i	− 1.20328i	0.798768	−	0.601639i	$-0.205485\pi$
$252$	0	0
$253$	−340.613	−1.34629
$254$	0	0
$255$	− 16.0471i	− 0.0629298i
$256$	0	0
$257$	− 103.700i	− 0.403502i	−0.979437	−	0.201751i	$-0.935337\pi$
$257$	− 103.700i	− 0.403502i	0.979437	−	0.201751i	$-0.0646631\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−31.2143	−0.119595
$262$	0	0
$263$	−257.943	−0.980773	−0.490387	−	0.871505i	$-0.663144\pi$
$263$	−257.943	−0.980773	−0.490387	+	0.871505i	$0.663144\pi$
$264$	0	0
$265$	− 56.0960i	− 0.211683i
$266$	0	0
$267$	38.6948	0.144924
$268$	0	0
$269$	− 162.262i	− 0.603204i	−0.953434	−	0.301602i	$-0.902479\pi$
$269$	− 162.262i	− 0.603204i	0.953434	−	0.301602i	$-0.0975214\pi$
$270$	0	0
$271$	62.7498i	0.231549i	0.993276	+	0.115775i	$0.0369350\pi$
$271$	62.7498i	0.231549i	−0.993276	+	0.115775i	$0.963065\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	342.881	1.24684
$276$	0	0
$277$	−257.153	−0.928349	−0.464175	−	0.885744i	$-0.653649\pi$
$277$	−257.153	−0.928349	−0.464175	+	0.885744i	$0.653649\pi$
$278$	0	0
$279$	− 110.019i	− 0.394335i
$280$	0	0
$281$	230.243	0.819369	0.409685	−	0.912227i	$-0.365639\pi$
$281$	230.243	0.819369	0.409685	+	0.912227i	$0.365639\pi$
$282$	0	0
$283$	409.499i	1.44699i	0.690328	+	0.723497i	$0.257466\pi$
$283$	409.499i	1.44699i	−0.690328	+	0.723497i	$0.742534\pi$
$284$	0	0
$285$	− 34.5358i	− 0.121178i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	105.374	0.364615
$290$	0	0
$291$	113.161	0.388868
$292$	0	0
$293$	438.106i	1.49524i	0.664126	+	0.747621i	$0.268804\pi$
$293$	438.106i	1.49524i	−0.664126	+	0.747621i	$0.731196\pi$
$294$	0	0
$295$	17.9776	0.0609412
$296$	0	0
$297$	216.990i	0.730606i
$298$	0	0
$299$	− 61.1810i	− 0.204619i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	166.228	0.548606
$304$	0	0
$305$	109.368	0.358584
$306$	0	0
$307$	136.514i	0.444672i	0.974970	+	0.222336i	$0.0713682\pi$
$307$	136.514i	0.444672i	−0.974970	+	0.222336i	$0.928632\pi$
$308$	0	0
$309$	118.115	0.382250
$310$	0	0
$311$	− 580.921i	− 1.86791i	−0.357386	−	0.933957i	$-0.616332\pi$
$311$	− 580.921i	− 1.86791i	0.357386	−	0.933957i	$-0.383668\pi$
$312$	0	0
$313$	− 187.197i	− 0.598075i	−0.954241	−	0.299037i	$-0.903334\pi$
$313$	− 187.197i	− 0.598075i	0.954241	−	0.299037i	$-0.0966655\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	165.888	0.523306	0.261653	−	0.965162i	$-0.415732\pi$
$317$	165.888	0.523306	0.261653	+	0.965162i	$0.415732\pi$
$318$	0	0
$319$	−56.0720	−0.175774
$320$	0	0
$321$	− 89.3530i	− 0.278358i
$322$	0	0
$323$	−395.192	−1.22350
$324$	0	0
$325$	61.5885i	0.189503i
$326$	0	0
$327$	120.311i	0.367923i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−418.198	−1.26344	−0.631719	−	0.775198i	$-0.717650\pi$
$331$	−418.198	−1.26344	−0.631719	+	0.775198i	$0.717650\pi$
$332$	0	0
$333$	21.9299	0.0658555
$334$	0	0
$335$	64.4096i	0.192268i
$336$	0	0
$337$	104.000	0.308607	0.154303	−	0.988024i	$-0.450687\pi$
$337$	104.000	0.308607	0.154303	+	0.988024i	$0.450687\pi$
$338$	0	0
$339$	91.4591i	0.269791i
$340$	0	0
$341$	− 197.634i	− 0.579572i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−27.0869	−0.0785127
$346$	0	0
$347$	97.9592	0.282303	0.141152	−	0.989988i	$-0.454919\pi$
$347$	97.9592	0.282303	0.141152	+	0.989988i	$0.454919\pi$
$348$	0	0
$349$	248.606i	0.712339i	0.934421	+	0.356169i	$0.115917\pi$
$349$	248.606i	0.712339i	−0.934421	+	0.356169i	$0.884083\pi$
$350$	0	0
$351$	−38.9759	−0.111042
$352$	0	0
$353$	131.409i	0.372264i	0.982525	+	0.186132i	$0.0595953\pi$
$353$	131.409i	0.372264i	−0.982525	+	0.186132i	$0.940405\pi$
$354$	0	0
$355$	− 57.1612i	− 0.161017i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−353.267	−0.984032	−0.492016	−	0.870586i	$-0.663740\pi$
$359$	−353.267	−0.984032	−0.492016	+	0.870586i	$0.663740\pi$
$360$	0	0
$361$	−489.513	−1.35599
$362$	0	0
$363$	84.9107i	0.233914i
$364$	0	0
$365$	−90.3446	−0.247519
$366$	0	0
$367$	− 121.597i	− 0.331326i	−0.986182	−	0.165663i	$-0.947024\pi$
$367$	− 121.597i	− 0.331326i	0.986182	−	0.165663i	$-0.0529764\pi$
$368$	0	0
$369$	− 375.284i	− 1.01703i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−490.915	−1.31613	−0.658063	−	0.752963i	$-0.728624\pi$
$373$	−490.915	−1.31613	−0.658063	+	0.752963i	$0.728624\pi$
$374$	0	0
$375$	56.8725	0.151660
$376$	0	0
$377$	− 10.0717i	− 0.0267153i
$378$	0	0
$379$	−325.094	−0.857768	−0.428884	−	0.903360i	$-0.641093\pi$
$379$	−325.094	−0.857768	−0.428884	+	0.903360i	$0.641093\pi$
$380$	0	0
$381$	− 117.813i	− 0.309222i
$382$	0	0
$383$	− 444.592i	− 1.16081i	−0.814327	−	0.580407i	$-0.802894\pi$
$383$	− 444.592i	− 1.16081i	0.814327	−	0.580407i	$-0.197106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−427.630	−1.10499
$388$	0	0
$389$	−418.439	−1.07568	−0.537839	−	0.843048i	$-0.680759\pi$
$389$	−418.439	−1.07568	−0.537839	+	0.843048i	$0.680759\pi$
$390$	0	0
$391$	309.954i	0.792721i
$392$	0	0
$393$	−80.0867	−0.203783
$394$	0	0
$395$	− 113.140i	− 0.286429i
$396$	0	0
$397$	437.417i	1.10181i	0.834569	+	0.550903i	$0.185717\pi$
$397$	437.417i	1.10181i	−0.834569	+	0.550903i	$0.814283\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−90.4723	−0.225617	−0.112808	−	0.993617i	$-0.535985\pi$
$401$	−90.4723	−0.225617	−0.112808	+	0.993617i	$0.535985\pi$
$402$	0	0
$403$	35.4991	0.0880872
$404$	0	0
$405$	− 87.5748i	− 0.216234i
$406$	0	0
$407$	39.3938	0.0967907
$408$	0	0
$409$	701.049i	1.71406i	0.515270	+	0.857028i	$0.327692\pi$
$409$	701.049i	1.71406i	−0.515270	+	0.857028i	$0.672308\pi$
$410$	0	0
$411$	129.427i	0.314908i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−171.231	−0.412604
$416$	0	0
$417$	−183.333	−0.439646
$418$	0	0
$419$	374.252i	0.893202i	0.894733	+	0.446601i	$0.147366\pi$
$419$	374.252i	0.893202i	−0.894733	+	0.446601i	$0.852634\pi$
$420$	0	0
$421$	−733.801	−1.74299	−0.871497	−	0.490400i	$-0.836851\pi$
$421$	−733.801	−1.74299	−0.871497	+	0.490400i	$0.836851\pi$
$422$	0	0
$423$	556.409i	1.31539i
$424$	0	0
$425$	− 312.018i	− 0.734161i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−33.5691	−0.0782496
$430$	0	0
$431$	833.006	1.93273	0.966365	−	0.257176i	$-0.0827919\pi$
$431$	833.006	1.93273	0.966365	+	0.257176i	$0.0827919\pi$
$432$	0	0
$433$	414.085i	0.956316i	0.878274	+	0.478158i	$0.158695\pi$
$433$	414.085i	0.956316i	−0.878274	+	0.478158i	$0.841305\pi$
$434$	0	0
$435$	−4.45907	−0.0102507
$436$	0	0
$437$	667.068i	1.52647i
$438$	0	0
$439$	71.5686i	0.163026i	0.996672	+	0.0815132i	$0.0259753\pi$
$439$	71.5686i	0.163026i	−0.996672	+	0.0815132i	$0.974025\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	829.257	1.87191	0.935956	−	0.352118i	$-0.114538\pi$
$443$	829.257	1.87191	0.935956	+	0.352118i	$0.114538\pi$
$444$	0	0
$445$	−64.5116	−0.144970
$446$	0	0
$447$	54.0149i	0.120839i
$448$	0	0
$449$	247.153	0.550452	0.275226	−	0.961380i	$-0.411247\pi$
$449$	247.153	0.550452	0.275226	+	0.961380i	$0.411247\pi$
$450$	0	0
$451$	− 674.144i	− 1.49478i
$452$	0	0
$453$	229.279i	0.506134i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−481.491	−1.05359	−0.526796	−	0.849992i	$-0.676607\pi$
$457$	−481.491	−1.05359	−0.526796	+	0.849992i	$0.676607\pi$
$458$	0	0
$459$	197.459	0.430194
$460$	0	0
$461$	256.190i	0.555727i	0.960621	+	0.277863i	$0.0896263\pi$
$461$	256.190i	0.555727i	−0.960621	+	0.277863i	$0.910374\pi$
$462$	0	0
$463$	−142.004	−0.306705	−0.153353	−	0.988172i	$-0.549007\pi$
$463$	−142.004	−0.306705	−0.153353	+	0.988172i	$0.549007\pi$
$464$	0	0
$465$	− 15.7166i	− 0.0337992i
$466$	0	0
$467$	− 433.189i	− 0.927599i	−0.885940	−	0.463800i	$-0.846486\pi$
$467$	− 433.189i	− 0.927599i	0.885940	−	0.463800i	$-0.153514\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	182.271	0.386987
$472$	0	0
$473$	−768.176	−1.62405
$474$	0	0
$475$	− 671.511i	− 1.41371i
$476$	0	0
$477$	330.950	0.693816
$478$	0	0
$479$	263.149i	0.549372i	0.961534	+	0.274686i	$0.0885740\pi$
$479$	263.149i	0.549372i	−0.961534	+	0.274686i	$0.911426\pi$
$480$	0	0
$481$	7.07594i	0.0147109i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−188.660	−0.388990
$486$	0	0
$487$	−677.009	−1.39016	−0.695081	−	0.718931i	$-0.744632\pi$
$487$	−677.009	−1.39016	−0.695081	+	0.718931i	$0.744632\pi$
$488$	0	0
$489$	46.0950i	0.0942637i
$490$	0	0
$491$	807.748	1.64511	0.822554	−	0.568686i	$-0.192548\pi$
$491$	807.748	1.64511	0.822554	+	0.568686i	$0.192548\pi$
$492$	0	0
$493$	51.0249i	0.103499i
$494$	0	0
$495$	− 173.451i	− 0.350406i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−156.504	−0.313635	−0.156817	−	0.987628i	$-0.550123\pi$
$499$	−156.504	−0.313635	−0.156817	+	0.987628i	$0.550123\pi$
$500$	0	0
$501$	−209.096	−0.417357
$502$	0	0
$503$	− 499.753i	− 0.993545i	−0.867881	−	0.496773i	$-0.834518\pi$
$503$	− 499.753i	− 0.993545i	0.867881	−	0.496773i	$-0.165482\pi$
$504$	0	0
$505$	−277.133	−0.548779
$506$	0	0
$507$	136.403i	0.269039i
$508$	0	0
$509$	− 534.024i	− 1.04916i	−0.851360	−	0.524582i	$-0.824222\pi$
$509$	− 534.024i	− 1.04916i	0.851360	−	0.524582i	$-0.175778\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	424.962	0.828385
$514$	0	0
$515$	−196.921	−0.382371
$516$	0	0
$517$	999.509i	1.93329i
$518$	0	0
$519$	−218.851	−0.421677
$520$	0	0
$521$	720.367i	1.38266i	0.722538	+	0.691331i	$0.242976\pi$
$521$	720.367i	1.38266i	−0.722538	+	0.691331i	$0.757024\pi$
$522$	0	0
$523$	604.485i	1.15580i	0.816106	+	0.577902i	$0.196128\pi$
$523$	604.485i	1.15580i	−0.816106	+	0.577902i	$0.803872\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−179.845	−0.341262
$528$	0	0
$529$	−5.80974	−0.0109825
$530$	0	0
$531$	106.063i	0.199742i
$532$	0	0
$533$	121.090	0.227186
$534$	0	0
$535$	148.969i	0.278446i
$536$	0	0
$537$	− 116.454i	− 0.216861i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	354.688	0.655615	0.327807	−	0.944745i	$-0.393690\pi$
$541$	354.688	0.655615	0.327807	+	0.944745i	$0.393690\pi$
$542$	0	0
$543$	−96.4973	−0.177711
$544$	0	0
$545$	− 200.581i	− 0.368039i
$546$	0	0
$547$	867.407	1.58575	0.792877	−	0.609382i	$-0.208583\pi$
$547$	867.407	1.58575	0.792877	+	0.609382i	$0.208583\pi$
$548$	0	0
$549$	645.241i	1.17530i
$550$	0	0
$551$	109.813i	0.199298i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.13275	0.00564460
$556$	0	0
$557$	−134.735	−0.241894	−0.120947	−	0.992659i	$-0.538593\pi$
$557$	−134.735	−0.241894	−0.120947	+	0.992659i	$0.538593\pi$
$558$	0	0
$559$	− 137.980i	− 0.246834i
$560$	0	0
$561$	170.067	0.303150
$562$	0	0
$563$	477.596i	0.848306i	0.905591	+	0.424153i	$0.139428\pi$
$563$	477.596i	0.848306i	−0.905591	+	0.424153i	$0.860572\pi$
$564$	0	0
$565$	− 152.480i	− 0.269876i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−332.681	−0.584677	−0.292338	−	0.956315i	$-0.594433\pi$
$569$	−332.681	−0.584677	−0.292338	+	0.956315i	$0.594433\pi$
$570$	0	0
$571$	−1051.13	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$571$	−1051.13	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$572$	0	0
$573$	− 128.466i	− 0.224199i
$574$	0	0
$575$	−526.675	−0.915956
$576$	0	0
$577$	− 404.568i	− 0.701158i	−0.936533	−	0.350579i	$-0.885985\pi$
$577$	− 404.568i	− 0.701158i	0.936533	−	0.350579i	$-0.114015\pi$
$578$	0	0
$579$	204.219i	0.352709i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	594.505	1.01973
$584$	0	0
$585$	31.1554	0.0532571
$586$	0	0
$587$	64.6639i	0.110160i	0.998482	+	0.0550800i	$0.0175414\pi$
$587$	64.6639i	0.110160i	−0.998482	+	0.0550800i	$0.982459\pi$
$588$	0	0
$589$	−387.054	−0.657137
$590$	0	0
$591$	− 166.467i	− 0.281670i
$592$	0	0
$593$	41.6665i	0.0702640i	0.999383	+	0.0351320i	$0.0111852\pi$
$593$	41.6665i	0.0702640i	−0.999383	+	0.0351320i	$0.988815\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−138.359	−0.231757
$598$	0	0
$599$	−120.005	−0.200342	−0.100171	−	0.994970i	$-0.531939\pi$
$599$	−120.005	−0.200342	−0.100171	+	0.994970i	$0.531939\pi$
$600$	0	0
$601$	1120.08i	1.86369i	0.362858	+	0.931844i	$0.381801\pi$
$601$	1120.08i	1.86369i	−0.362858	+	0.931844i	$0.618199\pi$
$602$	0	0
$603$	−379.999	−0.630180
$604$	0	0
$605$	− 141.562i	− 0.233988i
$606$	0	0
$607$	443.210i	0.730165i	0.930975	+	0.365083i	$0.118959\pi$
$607$	443.210i	0.730165i	−0.930975	+	0.365083i	$0.881041\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−179.532	−0.293834
$612$	0	0
$613$	−1137.28	−1.85527	−0.927637	−	0.373482i	$-0.878164\pi$
$613$	−1137.28	−1.85527	−0.927637	+	0.373482i	$0.878164\pi$
$614$	0	0
$615$	− 53.6106i	− 0.0871717i
$616$	0	0
$617$	−357.454	−0.579343	−0.289671	−	0.957126i	$-0.593546\pi$
$617$	−357.454	−0.579343	−0.289671	+	0.957126i	$0.593546\pi$
$618$	0	0
$619$	− 792.836i	− 1.28083i	−0.768028	−	0.640416i	$-0.778762\pi$
$619$	− 792.836i	− 1.28083i	0.768028	−	0.640416i	$-0.221238\pi$
$620$	0	0
$621$	− 333.303i	− 0.536719i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	480.825	0.769320
$626$	0	0
$627$	366.010	0.583748
$628$	0	0
$629$	− 35.8480i	− 0.0569920i
$630$	0	0
$631$	587.326	0.930786	0.465393	−	0.885104i	$-0.345913\pi$
$631$	587.326	0.930786	0.465393	+	0.885104i	$0.345913\pi$
$632$	0	0
$633$	44.7899i	0.0707581i
$634$	0	0
$635$	196.418i	0.309319i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	337.235	0.527754
$640$	0	0
$641$	458.698	0.715597	0.357798	−	0.933799i	$-0.383527\pi$
$641$	458.698	0.715597	0.357798	+	0.933799i	$0.383527\pi$
$642$	0	0
$643$	242.009i	0.376374i	0.982133	+	0.188187i	$0.0602611\pi$
$643$	242.009i	0.376374i	−0.982133	+	0.188187i	$0.939739\pi$
$644$	0	0
$645$	−61.0884	−0.0947107
$646$	0	0
$647$	− 79.3244i	− 0.122603i	−0.998119	−	0.0613017i	$-0.980475\pi$
$647$	− 79.3244i	− 0.122603i	0.998119	−	0.0613017i	$-0.0195252\pi$
$648$	0	0
$649$	190.527i	0.293570i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−112.216	−0.171846	−0.0859232	−	0.996302i	$-0.527384\pi$
$653$	−112.216	−0.171846	−0.0859232	+	0.996302i	$0.527384\pi$
$654$	0	0
$655$	133.520	0.203847
$656$	0	0
$657$	− 533.008i	− 0.811275i
$658$	0	0
$659$	−811.930	−1.23206	−0.616032	−	0.787721i	$-0.711261\pi$
$659$	−811.930	−1.23206	−0.616032	+	0.787721i	$0.711261\pi$
$660$	0	0
$661$	367.012i	0.555237i	0.960691	+	0.277619i	$0.0895452\pi$
$661$	367.012i	0.555237i	−0.960691	+	0.277619i	$0.910455\pi$
$662$	0	0
$663$	30.5475i	0.0460747i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	86.1281	0.129128
$668$	0	0
$669$	261.903	0.391484
$670$	0	0
$671$	1159.08i	1.72739i
$672$	0	0
$673$	−504.526	−0.749668	−0.374834	−	0.927092i	$-0.622300\pi$
$673$	−504.526	−0.749668	−0.374834	+	0.927092i	$0.622300\pi$
$674$	0	0
$675$	335.523i	0.497071i
$676$	0	0
$677$	− 600.144i	− 0.886476i	−0.896404	−	0.443238i	$-0.853830\pi$
$677$	− 600.144i	− 0.886476i	0.896404	−	0.443238i	$-0.146170\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−42.0539	−0.0617532
$682$	0	0
$683$	−211.434	−0.309567	−0.154783	−	0.987948i	$-0.549468\pi$
$683$	−211.434	−0.309567	−0.154783	+	0.987948i	$0.549468\pi$
$684$	0	0
$685$	− 215.780i	− 0.315007i
$686$	0	0
$687$	−177.929	−0.258994
$688$	0	0
$689$	106.785i	0.154986i
$690$	0	0
$691$	19.2464i	0.0278529i	0.999903	+	0.0139265i	$0.00443307\pi$
$691$	19.2464i	0.0278529i	−0.999903	+	0.0139265i	$0.995567\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	305.651	0.439785
$696$	0	0
$697$	−613.464	−0.880149
$698$	0	0
$699$	63.5928i	0.0909768i
$700$	0	0
$701$	1073.36	1.53119	0.765595	−	0.643323i	$-0.222445\pi$
$701$	1073.36	1.53119	0.765595	+	0.643323i	$0.222445\pi$
$702$	0	0
$703$	− 77.1503i	− 0.109744i
$704$	0	0
$705$	79.4849i	0.112745i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−257.598	−0.363325	−0.181663	−	0.983361i	$-0.558148\pi$
$709$	−257.598	−0.363325	−0.181663	+	0.983361i	$0.558148\pi$
$710$	0	0
$711$	667.491	0.938807
$712$	0	0
$713$	303.571i	0.425766i
$714$	0	0
$715$	55.9661	0.0782743
$716$	0	0
$717$	261.600i	0.364854i
$718$	0	0
$719$	− 321.622i	− 0.447318i	−0.974667	−	0.223659i	$-0.928200\pi$
$719$	− 321.622i	− 0.447318i	0.974667	−	0.223659i	$-0.0718002\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	91.2060	0.126149
$724$	0	0
$725$	−86.7017	−0.119589
$726$	0	0
$727$	− 19.9115i	− 0.0273886i	−0.999906	−	0.0136943i	$-0.995641\pi$
$727$	− 19.9115i	− 0.0273886i	0.999906	−	0.0136943i	$-0.00435917\pi$
$728$	0	0
$729$	406.138	0.557117
$730$	0	0
$731$	699.033i	0.956269i
$732$	0	0
$733$	593.609i	0.809834i	0.914353	+	0.404917i	$0.132700\pi$
$733$	593.609i	0.809834i	−0.914353	+	0.404917i	$0.867300\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−682.612	−0.926204
$738$	0	0
$739$	1011.00	1.36806	0.684031	−	0.729453i	$-0.260225\pi$
$739$	1011.00	1.36806	0.684031	+	0.729453i	$0.260225\pi$
$740$	0	0
$741$	65.7430i	0.0887219i
$742$	0	0
$743$	581.910	0.783189	0.391595	−	0.920138i	$-0.371924\pi$
$743$	581.910	0.783189	0.391595	+	0.920138i	$0.371924\pi$
$744$	0	0
$745$	− 90.0531i	− 0.120877i
$746$	0	0
$747$	− 1010.21i	− 1.35236i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1210.55	−1.61192	−0.805962	−	0.591968i	$-0.798351\pi$
$751$	−1210.55	−1.61192	−0.805962	+	0.591968i	$0.798351\pi$
$752$	0	0
$753$	254.543	0.338039
$754$	0	0
$755$	− 382.251i	− 0.506293i
$756$	0	0
$757$	−245.833	−0.324746	−0.162373	−	0.986729i	$-0.551915\pi$
$757$	−245.833	−0.324746	−0.162373	+	0.986729i	$0.551915\pi$
$758$	0	0
$759$	− 287.066i	− 0.378217i
$760$	0	0
$761$	68.6017i	0.0901468i	0.998984	+	0.0450734i	$0.0143522\pi$
$761$	68.6017i	0.0901468i	−0.998984	+	0.0450734i	$0.985648\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−157.839	−0.206325
$766$	0	0
$767$	−34.2226	−0.0446187
$768$	0	0
$769$	540.953i	0.703450i	0.936103	+	0.351725i	$0.114405\pi$
$769$	540.953i	0.703450i	−0.936103	+	0.351725i	$0.885595\pi$
$770$	0	0
$771$	87.3977	0.113356
$772$	0	0
$773$	258.032i	0.333806i	0.985973	+	0.166903i	$0.0533767\pi$
$773$	258.032i	0.333806i	−0.985973	+	0.166903i	$0.946623\pi$
$774$	0	0
$775$	− 305.593i	− 0.394314i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1320.27	−1.69482
$780$	0	0
$781$	605.793	0.775664
$782$	0	0
$783$	− 54.8686i	− 0.0700749i
$784$	0	0
$785$	−303.880	−0.387109
$786$	0	0
$787$	− 180.488i	− 0.229336i	−0.993404	−	0.114668i	$-0.963420\pi$
$787$	− 180.488i	− 0.229336i	0.993404	−	0.114668i	$-0.0365804\pi$
$788$	0	0
$789$	− 217.393i	− 0.275530i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−208.195	−0.262541
$794$	0	0
$795$	47.2774	0.0594684
$796$	0	0
$797$	− 1387.83i	− 1.74132i	−0.491886	−	0.870660i	$-0.663692\pi$
$797$	− 1387.83i	− 1.74132i	0.491886	−	0.870660i	$-0.336308\pi$
$798$	0	0
$799$	909.543	1.13835
$800$	0	0
$801$	− 380.601i	− 0.475157i
$802$	0	0
$803$	− 957.471i	− 1.19237i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	136.753	0.169459
$808$	0	0
$809$	−708.894	−0.876260	−0.438130	−	0.898912i	$-0.644359\pi$
$809$	−708.894	−0.876260	−0.438130	+	0.898912i	$0.644359\pi$
$810$	0	0
$811$	281.652i	0.347289i	0.984808	+	0.173645i	$0.0555544\pi$
$811$	281.652i	0.347289i	−0.984808	+	0.173645i	$0.944446\pi$
$812$	0	0
$813$	−52.8852	−0.0650495
$814$	0	0
$815$	− 76.8492i	− 0.0942934i
$816$	0	0
$817$	1504.42i	1.84140i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1221.44	1.48775	0.743876	−	0.668318i	$-0.232985\pi$
$821$	1221.44	1.48775	0.743876	+	0.668318i	$0.232985\pi$
$822$	0	0
$823$	−83.4003	−0.101337	−0.0506685	−	0.998716i	$-0.516135\pi$
$823$	−83.4003	−0.101337	−0.0506685	+	0.998716i	$0.516135\pi$
$824$	0	0
$825$	288.978i	0.350277i
$826$	0	0
$827$	−804.602	−0.972916	−0.486458	−	0.873704i	$-0.661711\pi$
$827$	−804.602	−0.972916	−0.486458	+	0.873704i	$0.661711\pi$
$828$	0	0
$829$	− 1029.19i	− 1.24148i	−0.784017	−	0.620740i	$-0.786832\pi$
$829$	− 1029.19i	− 1.24148i	0.784017	−	0.620740i	$-0.213168\pi$
$830$	0	0
$831$	− 216.727i	− 0.260803i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	348.603	0.417488
$836$	0	0
$837$	193.393	0.231055
$838$	0	0
$839$	− 864.563i	− 1.03047i	−0.857049	−	0.515234i	$-0.827705\pi$
$839$	− 864.563i	− 1.03047i	0.857049	−	0.515234i	$-0.172295\pi$
$840$	0	0
$841$	−826.822	−0.983141
$842$	0	0
$843$	194.047i	0.230187i
$844$	0	0
$845$	− 227.409i	− 0.269123i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−345.124	−0.406506
$850$	0	0
$851$	−60.5100	−0.0711046
$852$	0	0
$853$	− 285.568i	− 0.334781i	−0.985891	−	0.167391i	$-0.946466\pi$
$853$	− 285.568i	− 0.334781i	0.985891	−	0.167391i	$-0.0535341\pi$
$854$	0	0
$855$	−339.693	−0.397302
$856$	0	0
$857$	652.995i	0.761954i	0.924584	+	0.380977i	$0.124412\pi$
$857$	652.995i	0.761954i	−0.924584	+	0.380977i	$0.875588\pi$
$858$	0	0
$859$	87.3539i	0.101693i	0.998706	+	0.0508463i	$0.0161919\pi$
$859$	87.3539i	0.101693i	−0.998706	+	0.0508463i	$0.983808\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	427.278	0.495108	0.247554	−	0.968874i	$-0.420373\pi$
$863$	427.278	0.495108	0.247554	+	0.968874i	$0.420373\pi$
$864$	0	0
$865$	364.866	0.421810
$866$	0	0
$867$	88.8083i	0.102432i
$868$	0	0
$869$	1199.05	1.37981
$870$	0	0
$871$	− 122.611i	− 0.140771i
$872$	0	0
$873$	− 1113.04i	− 1.27496i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−581.302	−0.662830	−0.331415	−	0.943485i	$-0.607526\pi$
$877$	−581.302	−0.662830	−0.331415	+	0.943485i	$0.607526\pi$
$878$	0	0
$879$	−369.233	−0.420061
$880$	0	0
$881$	− 617.976i	− 0.701448i	−0.936479	−	0.350724i	$-0.885936\pi$
$881$	− 617.976i	− 0.701448i	0.936479	−	0.350724i	$-0.114064\pi$
$882$	0	0
$883$	150.782	0.170761	0.0853807	−	0.996348i	$-0.472789\pi$
$883$	150.782	0.170761	0.0853807	+	0.996348i	$0.472789\pi$
$884$	0	0
$885$	15.1515i	0.0171203i
$886$	0	0
$887$	1337.14i	1.50748i	0.657171	+	0.753742i	$0.271753\pi$
$887$	1337.14i	1.50748i	−0.657171	+	0.753742i	$0.728247\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	928.117	1.04166
$892$	0	0
$893$	1957.48	2.19202
$894$	0	0
$895$	194.152i	0.216930i
$896$	0	0
$897$	51.5630	0.0574839
$898$	0	0
$899$	49.9742i	0.0555887i
$900$	0	0
$901$	− 540.993i	− 0.600437i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	160.880	0.177768
$906$	0	0
$907$	645.355	0.711527	0.355763	−	0.934576i	$-0.384221\pi$
$907$	645.355	0.711527	0.355763	+	0.934576i	$0.384221\pi$
$908$	0	0
$909$	− 1635.01i	− 1.79869i
$910$	0	0
$911$	−865.195	−0.949720	−0.474860	−	0.880061i	$-0.657501\pi$
$911$	−865.195	−0.949720	−0.474860	+	0.880061i	$0.657501\pi$
$912$	0	0
$913$	− 1814.70i	− 1.98762i
$914$	0	0
$915$	92.1748i	0.100737i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−63.2186	−0.0687907	−0.0343953	−	0.999408i	$-0.510951\pi$
$919$	−63.2186	−0.0687907	−0.0343953	+	0.999408i	$0.510951\pi$
$920$	0	0
$921$	−115.054	−0.124922
$922$	0	0
$923$	108.813i	0.117891i
$924$	0	0
$925$	60.9130	0.0658519
$926$	0	0
$927$	− 1161.78i	− 1.25327i
$928$	0	0
$929$	1280.91i	1.37880i	0.724379	+	0.689402i	$0.242126\pi$
$929$	1280.91i	1.37880i	−0.724379	+	0.689402i	$0.757874\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	489.597	0.524756
$934$	0	0
$935$	−283.534	−0.303245
$936$	0	0
$937$	1176.84i	1.25597i	0.778227	+	0.627983i	$0.216119\pi$
$937$	1176.84i	1.25597i	−0.778227	+	0.627983i	$0.783881\pi$
$938$	0	0
$939$	157.769	0.168018
$940$	0	0
$941$	− 1367.90i	− 1.45367i	−0.686815	−	0.726833i	$-0.740992\pi$
$941$	− 1367.90i	− 1.45367i	0.686815	−	0.726833i	$-0.259008\pi$
$942$	0	0
$943$	1035.50i	1.09809i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	175.210	0.185016	0.0925078	−	0.995712i	$-0.470512\pi$
$947$	175.210	0.185016	0.0925078	+	0.995712i	$0.470512\pi$
$948$	0	0
$949$	171.982	0.181224
$950$	0	0
$951$	139.809i	0.147013i
$952$	0	0
$953$	89.5669	0.0939842	0.0469921	−	0.998895i	$-0.485036\pi$
$953$	89.5669	0.0939842	0.0469921	+	0.998895i	$0.485036\pi$
$954$	0	0
$955$	214.178i	0.224270i
$956$	0	0
$957$	− 47.2571i	− 0.0493805i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	784.858	0.816710
$962$	0	0
$963$	−878.873	−0.912640
$964$	0	0
$965$	− 340.472i	− 0.352820i
$966$	0	0
$967$	−640.201	−0.662049	−0.331024	−	0.943622i	$-0.607394\pi$
$967$	−640.201	−0.662049	−0.331024	+	0.943622i	$0.607394\pi$
$968$	0	0
$969$	− 333.066i	− 0.343721i
$970$	0	0
$971$	1143.41i	1.17756i	0.808294	+	0.588779i	$0.200391\pi$
$971$	1143.41i	1.17756i	−0.808294	+	0.588779i	$0.799609\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−51.9065	−0.0532374
$976$	0	0
$977$	1603.07	1.64081	0.820403	−	0.571786i	$-0.193749\pi$
$977$	1603.07	1.64081	0.820403	+	0.571786i	$0.193749\pi$
$978$	0	0
$979$	− 683.694i	− 0.698359i
$980$	0	0
$981$	1183.37	1.20629
$982$	0	0
$983$	981.769i	0.998748i	0.866387	+	0.499374i	$0.166437\pi$
$983$	981.769i	0.998748i	−0.866387	+	0.499374i	$0.833563\pi$
$984$	0	0
$985$	277.532i	0.281759i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1179.94	1.19306
$990$	0	0
$991$	−827.709	−0.835226	−0.417613	−	0.908625i	$-0.637133\pi$
$991$	−827.709	−0.835226	−0.417613	+	0.908625i	$0.637133\pi$
$992$	0	0
$993$	− 352.455i	− 0.354939i
$994$	0	0
$995$	230.671	0.231830
$996$	0	0
$997$	180.797i	0.181341i	0.995881	+	0.0906705i	$0.0289010\pi$
$997$	180.797i	0.181341i	−0.995881	+	0.0906705i	$0.971099\pi$
$998$	0	0
$999$	38.5484i	0.0385870i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.g.97.9		16
4.3	odd	2	inner	1568.3.c.g.97.7		16
7.2	even	3		224.3.s.b.129.4	yes	16
7.3	odd	6		224.3.s.b.33.4	✓	16
7.6	odd	2	inner	1568.3.c.g.97.8		16
28.3	even	6		224.3.s.b.33.5	yes	16
28.23	odd	6		224.3.s.b.129.5	yes	16
28.27	even	2	inner	1568.3.c.g.97.10		16
56.3	even	6		448.3.s.h.257.4		16
56.37	even	6		448.3.s.h.129.5		16
56.45	odd	6		448.3.s.h.257.5		16
56.51	odd	6		448.3.s.h.129.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.s.b.33.4	✓	16	7.3	odd	6
224.3.s.b.33.5	yes	16	28.3	even	6
224.3.s.b.129.4	yes	16	7.2	even	3
224.3.s.b.129.5	yes	16	28.23	odd	6
448.3.s.h.129.4		16	56.51	odd	6
448.3.s.h.129.5		16	56.37	even	6
448.3.s.h.257.4		16	56.3	even	6
448.3.s.h.257.5		16	56.45	odd	6
1568.3.c.g.97.7		16	4.3	odd	2	inner
1568.3.c.g.97.8		16	7.6	odd	2	inner
1568.3.c.g.97.9		16	1.1	even	1	trivial
1568.3.c.g.97.10		16	28.27	even	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)

\(f(q)\)	\(=\)	\(q+0.842794i q^{3} -1.40510i q^{5} +8.28970 q^{9} +O(q^{10})\)	\(q+0.842794i q^{3} -1.40510i q^{5} +8.28970 q^{9} +14.8912 q^{11} +2.67477i q^{13} +1.18421 q^{15} -13.5509i q^{17} -29.1636i q^{19} -22.8734 q^{23} +23.0257 q^{25} +14.5717i q^{27} -3.76543 q^{29} -13.2718i q^{31} +12.5503i q^{33} +2.64544 q^{37} -2.25428 q^{39} -45.2712i q^{41} -51.5858 q^{43} -11.6479i q^{45} +67.1206i q^{47} +11.4206 q^{51} +39.9231 q^{53} -20.9237i q^{55} +24.5789 q^{57} +12.7946i q^{59} +77.8365i q^{61} +3.75832 q^{65} -45.8399 q^{67} -19.2775i q^{69} +40.6812 q^{71} -64.2976i q^{73} +19.4059i q^{75} +80.5206 q^{79} +62.3264 q^{81} -121.864i q^{83} -19.0403 q^{85} -3.17349i q^{87} -45.9125i q^{89} +11.1854 q^{93} -40.9777 q^{95} -134.268i q^{97} +123.444 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 80 q^{9}+O(q^{10}) \) 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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