LMFDB - Embedded newform 1568.3.h.a.881.25

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			881.25
Character	$\chi$	$=$	1568.881
Dual form			1568.3.h.a.881.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.89635 q^{3} -8.85969 q^{5} +6.18155 q^{9} +O(q^{10})$	$q+3.89635 q^{3} -8.85969 q^{5} +6.18155 q^{9} -3.64596i q^{11} +7.79378 q^{13} -34.5205 q^{15} -10.4749i q^{17} -10.7853 q^{19} +12.9111 q^{23} +53.4941 q^{25} -10.9817 q^{27} +17.2327i q^{29} +30.2297i q^{31} -14.2059i q^{33} +39.5843i q^{37} +30.3673 q^{39} +73.6801i q^{41} +40.8501i q^{43} -54.7667 q^{45} -41.8030i q^{47} -40.8138i q^{51} +6.41173i q^{53} +32.3020i q^{55} -42.0232 q^{57} +15.9148 q^{59} -12.1570 q^{61} -69.0505 q^{65} +7.79739i q^{67} +50.3060 q^{69} +41.3627 q^{71} +89.6091i q^{73} +208.432 q^{75} -70.7950 q^{79} -98.4224 q^{81} -60.8673 q^{83} +92.8043i q^{85} +67.1446i q^{87} +27.0204i q^{89} +117.785i q^{93} +95.5542 q^{95} -3.26608i q^{97} -22.5377i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.89635	1.29878	0.649392	−	0.760454i	$-0.275024\pi$
$3$	3.89635	1.29878	0.649392	+	0.760454i	$0.275024\pi$
$4$	0	0
$5$	−8.85969	−1.77194	−0.885969	−	0.463744i	$-0.846506\pi$
$5$	−8.85969	−1.77194	−0.885969	+	0.463744i	$0.846506\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.18155	0.686839
$10$	0	0
$11$	− 3.64596i	− 0.331451i	−0.986172	−	0.165725i	$-0.947004\pi$
$11$	− 3.64596i	− 0.331451i	0.986172	−	0.165725i	$-0.0529965\pi$
$12$	0	0
$13$	7.79378	0.599522	0.299761	−	0.954014i	$-0.403093\pi$
$13$	7.79378	0.599522	0.299761	+	0.954014i	$0.403093\pi$
$14$	0	0
$15$	−34.5205	−2.30136
$16$	0	0
$17$	− 10.4749i	− 0.616170i	−0.951359	−	0.308085i	$-0.900312\pi$
$17$	− 10.4749i	− 0.616170i	0.951359	−	0.308085i	$-0.0996881\pi$
$18$	0	0
$19$	−10.7853	−0.567646	−0.283823	−	0.958877i	$-0.591603\pi$
$19$	−10.7853	−0.567646	−0.283823	+	0.958877i	$0.591603\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	12.9111	0.561351	0.280675	−	0.959803i	$-0.409442\pi$
$23$	12.9111	0.561351	0.280675	+	0.959803i	$0.409442\pi$
$24$	0	0
$25$	53.4941	2.13977
$26$	0	0
$27$	−10.9817	−0.406728
$28$	0	0
$29$	17.2327i	0.594231i	0.954842	+	0.297115i	$0.0960246\pi$
$29$	17.2327i	0.594231i	−0.954842	+	0.297115i	$0.903975\pi$
$30$	0	0
$31$	30.2297i	0.975151i	0.873081	+	0.487575i	$0.162119\pi$
$31$	30.2297i	0.975151i	−0.873081	+	0.487575i	$0.837881\pi$
$32$	0	0
$33$	− 14.2059i	− 0.430483i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	39.5843i	1.06985i	0.844900	+	0.534924i	$0.179660\pi$
$37$	39.5843i	1.06985i	−0.844900	+	0.534924i	$0.820340\pi$
$38$	0	0
$39$	30.3673	0.778649
$40$	0	0
$41$	73.6801i	1.79707i	0.438897	+	0.898537i	$0.355369\pi$
$41$	73.6801i	1.79707i	−0.438897	+	0.898537i	$0.644631\pi$
$42$	0	0
$43$	40.8501i	0.950002i	0.879985	+	0.475001i	$0.157552\pi$
$43$	40.8501i	0.950002i	−0.879985	+	0.475001i	$0.842448\pi$
$44$	0	0
$45$	−54.7667	−1.21704
$46$	0	0
$47$	− 41.8030i	− 0.889426i	−0.895673	−	0.444713i	$-0.853306\pi$
$47$	− 41.8030i	− 0.889426i	0.895673	−	0.444713i	$-0.146694\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 40.8138i	− 0.800271i
$52$	0	0
$53$	6.41173i	0.120976i	0.998169	+	0.0604880i	$0.0192657\pi$
$53$	6.41173i	0.120976i	−0.998169	+	0.0604880i	$0.980734\pi$
$54$	0	0
$55$	32.3020i	0.587310i
$56$	0	0
$57$	−42.0232	−0.737249
$58$	0	0
$59$	15.9148	0.269743	0.134871	−	0.990863i	$-0.456938\pi$
$59$	15.9148	0.269743	0.134871	+	0.990863i	$0.456938\pi$
$60$	0	0
$61$	−12.1570	−0.199294	−0.0996472	−	0.995023i	$-0.531771\pi$
$61$	−12.1570	−0.199294	−0.0996472	+	0.995023i	$0.531771\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−69.0505	−1.06232
$66$	0	0
$67$	7.79739i	0.116379i	0.998306	+	0.0581895i	$0.0185328\pi$
$67$	7.79739i	0.116379i	−0.998306	+	0.0581895i	$0.981467\pi$
$68$	0	0
$69$	50.3060	0.729073
$70$	0	0
$71$	41.3627	0.582574	0.291287	−	0.956636i	$-0.405917\pi$
$71$	41.3627	0.582574	0.291287	+	0.956636i	$0.405917\pi$
$72$	0	0
$73$	89.6091i	1.22752i	0.789492	+	0.613761i	$0.210344\pi$
$73$	89.6091i	1.22752i	−0.789492	+	0.613761i	$0.789656\pi$
$74$	0	0
$75$	208.432	2.77909
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−70.7950	−0.896140	−0.448070	−	0.893999i	$-0.647888\pi$
$79$	−70.7950	−0.896140	−0.448070	+	0.893999i	$0.647888\pi$
$80$	0	0
$81$	−98.4224	−1.21509
$82$	0	0
$83$	−60.8673	−0.733341	−0.366671	−	0.930351i	$-0.619502\pi$
$83$	−60.8673	−0.733341	−0.366671	+	0.930351i	$0.619502\pi$
$84$	0	0
$85$	92.8043i	1.09182i
$86$	0	0
$87$	67.1446i	0.771777i
$88$	0	0
$89$	27.0204i	0.303600i	0.988411	+	0.151800i	$0.0485070\pi$
$89$	27.0204i	0.303600i	−0.988411	+	0.151800i	$0.951493\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	117.785i	1.26651i
$94$	0	0
$95$	95.5542	1.00583
$96$	0	0
$97$	− 3.26608i	− 0.0336710i	−0.999858	−	0.0168355i	$-0.994641\pi$
$97$	− 3.26608i	− 0.0336710i	0.999858	−	0.0168355i	$-0.00535916\pi$
$98$	0	0
$99$	− 22.5377i	− 0.227653i
$100$	0	0
$101$	−137.714	−1.36351	−0.681754	−	0.731582i	$-0.738782\pi$
$101$	−137.714	−1.36351	−0.681754	+	0.731582i	$0.738782\pi$
$102$	0	0
$103$	99.6787i	0.967754i	0.875136	+	0.483877i	$0.160772\pi$
$103$	99.6787i	0.967754i	−0.875136	+	0.483877i	$0.839228\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	94.0107i	0.878605i	0.898339	+	0.439302i	$0.144774\pi$
$107$	94.0107i	0.878605i	−0.898339	+	0.439302i	$0.855226\pi$
$108$	0	0
$109$	195.949i	1.79770i	0.438258	+	0.898849i	$0.355595\pi$
$109$	195.949i	1.79770i	−0.438258	+	0.898849i	$0.644405\pi$
$110$	0	0
$111$	154.234i	1.38950i
$112$	0	0
$113$	101.873	0.901527	0.450763	−	0.892643i	$-0.351152\pi$
$113$	101.873	0.901527	0.450763	+	0.892643i	$0.351152\pi$
$114$	0	0
$115$	−114.388	−0.994679
$116$	0	0
$117$	48.1777	0.411775
$118$	0	0
$119$	0	0
$120$	0	0
$121$	107.707	0.890141
$122$	0	0
$123$	287.083i	2.33401i
$124$	0	0
$125$	−252.449	−2.01960
$126$	0	0
$127$	139.079	1.09511	0.547554	−	0.836770i	$-0.315559\pi$
$127$	139.079	1.09511	0.547554	+	0.836770i	$0.315559\pi$
$128$	0	0
$129$	159.166i	1.23385i
$130$	0	0
$131$	−91.7051	−0.700039	−0.350020	−	0.936742i	$-0.613825\pi$
$131$	−91.7051	−0.700039	−0.350020	+	0.936742i	$0.613825\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	97.2941	0.720697
$136$	0	0
$137$	199.581	1.45679	0.728397	−	0.685155i	$-0.240266\pi$
$137$	199.581	1.45679	0.728397	+	0.685155i	$0.240266\pi$
$138$	0	0
$139$	−39.4768	−0.284006	−0.142003	−	0.989866i	$-0.545354\pi$
$139$	−39.4768	−0.284006	−0.142003	+	0.989866i	$0.545354\pi$
$140$	0	0
$141$	− 162.879i	− 1.15517i
$142$	0	0
$143$	− 28.4158i	− 0.198712i
$144$	0	0
$145$	− 152.676i	− 1.05294i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	94.7831i	0.636128i	0.948069	+	0.318064i	$0.103033\pi$
$149$	94.7831i	0.636128i	−0.948069	+	0.318064i	$0.896967\pi$
$150$	0	0
$151$	−66.5686	−0.440852	−0.220426	−	0.975404i	$-0.570745\pi$
$151$	−66.5686	−0.440852	−0.220426	+	0.975404i	$0.570745\pi$
$152$	0	0
$153$	− 64.7511i	− 0.423210i
$154$	0	0
$155$	− 267.826i	− 1.72791i
$156$	0	0
$157$	−25.5194	−0.162544	−0.0812720	−	0.996692i	$-0.525898\pi$
$157$	−25.5194	−0.162544	−0.0812720	+	0.996692i	$0.525898\pi$
$158$	0	0
$159$	24.9823i	0.157122i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 192.101i	− 1.17853i	−0.807939	−	0.589267i	$-0.799417\pi$
$163$	− 192.101i	− 1.17853i	0.807939	−	0.589267i	$-0.200583\pi$
$164$	0	0
$165$	125.860i	0.762789i
$166$	0	0
$167$	− 184.150i	− 1.10269i	−0.834276	−	0.551346i	$-0.814114\pi$
$167$	− 184.150i	− 1.10269i	0.834276	−	0.551346i	$-0.185886\pi$
$168$	0	0
$169$	−108.257	−0.640574
$170$	0	0
$171$	−66.6697	−0.389882
$172$	0	0
$173$	−69.9037	−0.404068	−0.202034	−	0.979379i	$-0.564755\pi$
$173$	−69.9037	−0.404068	−0.202034	+	0.979379i	$0.564755\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	62.0098	0.350338
$178$	0	0
$179$	− 239.313i	− 1.33695i	−0.743736	−	0.668473i	$-0.766948\pi$
$179$	− 239.313i	− 1.33695i	0.743736	−	0.668473i	$-0.233052\pi$
$180$	0	0
$181$	−36.2834	−0.200461	−0.100230	−	0.994964i	$-0.531958\pi$
$181$	−36.2834	−0.200461	−0.100230	+	0.994964i	$0.531958\pi$
$182$	0	0
$183$	−47.3678	−0.258840
$184$	0	0
$185$	− 350.705i	− 1.89570i
$186$	0	0
$187$	−38.1910	−0.204230
$188$	0	0
$189$	0	0
$190$	0	0
$191$	325.243	1.70284	0.851422	−	0.524481i	$-0.175740\pi$
$191$	325.243	1.70284	0.851422	+	0.524481i	$0.175740\pi$
$192$	0	0
$193$	199.640	1.03440	0.517201	−	0.855864i	$-0.326974\pi$
$193$	199.640	1.03440	0.517201	+	0.855864i	$0.326974\pi$
$194$	0	0
$195$	−269.045	−1.37972
$196$	0	0
$197$	15.5053i	0.0787071i	0.999225	+	0.0393536i	$0.0125299\pi$
$197$	15.5053i	0.0787071i	−0.999225	+	0.0393536i	$0.987470\pi$
$198$	0	0
$199$	56.1617i	0.282220i	0.989994	+	0.141110i	$0.0450671\pi$
$199$	56.1617i	0.282220i	−0.989994	+	0.141110i	$0.954933\pi$
$200$	0	0
$201$	30.3814i	0.151151i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 652.783i	− 3.18431i
$206$	0	0
$207$	79.8104	0.385558
$208$	0	0
$209$	39.3226i	0.188147i
$210$	0	0
$211$	370.470i	1.75578i	0.478859	+	0.877892i	$0.341051\pi$
$211$	370.470i	1.75578i	−0.478859	+	0.877892i	$0.658949\pi$
$212$	0	0
$213$	161.164	0.756638
$214$	0	0
$215$	− 361.919i	− 1.68335i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	349.149i	1.59429i
$220$	0	0
$221$	− 81.6390i	− 0.369407i
$222$	0	0
$223$	− 6.78533i	− 0.0304275i	−0.999884	−	0.0152137i	$-0.995157\pi$
$223$	− 6.78533i	− 0.0304275i	0.999884	−	0.0152137i	$-0.00484287\pi$
$224$	0	0
$225$	330.677	1.46968
$226$	0	0
$227$	−296.618	−1.30669	−0.653344	−	0.757061i	$-0.726635\pi$
$227$	−296.618	−1.30669	−0.653344	+	0.757061i	$0.726635\pi$
$228$	0	0
$229$	178.193	0.778135	0.389067	−	0.921209i	$-0.372797\pi$
$229$	178.193	0.778135	0.389067	+	0.921209i	$0.372797\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	117.802	0.505589	0.252795	−	0.967520i	$-0.418650\pi$
$233$	117.802	0.505589	0.252795	+	0.967520i	$0.418650\pi$
$234$	0	0
$235$	370.362i	1.57601i
$236$	0	0
$237$	−275.842	−1.16389
$238$	0	0
$239$	−46.3543	−0.193951	−0.0969755	−	0.995287i	$-0.530917\pi$
$239$	−46.3543	−0.193951	−0.0969755	+	0.995287i	$0.530917\pi$
$240$	0	0
$241$	366.618i	1.52124i	0.649199	+	0.760619i	$0.275104\pi$
$241$	366.618i	1.52124i	−0.649199	+	0.760619i	$0.724896\pi$
$242$	0	0
$243$	−284.653	−1.17141
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−84.0581	−0.340316
$248$	0	0
$249$	−237.160	−0.952452
$250$	0	0
$251$	129.896	0.517513	0.258756	−	0.965943i	$-0.416687\pi$
$251$	129.896	0.517513	0.258756	+	0.965943i	$0.416687\pi$
$252$	0	0
$253$	− 47.0732i	− 0.186060i
$254$	0	0
$255$	361.598i	1.41803i
$256$	0	0
$257$	− 268.345i	− 1.04415i	−0.852901	−	0.522073i	$-0.825159\pi$
$257$	− 268.345i	− 1.04415i	0.852901	−	0.522073i	$-0.174841\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	106.525i	0.408141i
$262$	0	0
$263$	235.383	0.894991	0.447495	−	0.894286i	$-0.352316\pi$
$263$	235.383	0.894991	0.447495	+	0.894286i	$0.352316\pi$
$264$	0	0
$265$	− 56.8059i	− 0.214362i
$266$	0	0
$267$	105.281i	0.394311i
$268$	0	0
$269$	−354.695	−1.31857	−0.659285	−	0.751893i	$-0.729141\pi$
$269$	−354.695	−1.31857	−0.659285	+	0.751893i	$0.729141\pi$
$270$	0	0
$271$	− 421.870i	− 1.55672i	−0.627820	−	0.778358i	$-0.716053\pi$
$271$	− 421.870i	− 1.55672i	0.627820	−	0.778358i	$-0.283947\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 195.037i	− 0.709226i
$276$	0	0
$277$	− 368.529i	− 1.33043i	−0.746653	−	0.665214i	$-0.768340\pi$
$277$	− 368.529i	− 1.33043i	0.746653	−	0.665214i	$-0.231660\pi$
$278$	0	0
$279$	186.866i	0.669772i
$280$	0	0
$281$	−35.2868	−0.125576	−0.0627879	−	0.998027i	$-0.519999\pi$
$281$	−35.2868	−0.125576	−0.0627879	+	0.998027i	$0.519999\pi$
$282$	0	0
$283$	−196.617	−0.694761	−0.347380	−	0.937724i	$-0.612929\pi$
$283$	−196.617	−0.694761	−0.347380	+	0.937724i	$0.612929\pi$
$284$	0	0
$285$	372.313	1.30636
$286$	0	0
$287$	0	0
$288$	0	0
$289$	179.277	0.620335
$290$	0	0
$291$	− 12.7258i	− 0.0437313i
$292$	0	0
$293$	317.573	1.08387	0.541933	−	0.840421i	$-0.317693\pi$
$293$	317.573	1.08387	0.541933	+	0.840421i	$0.317693\pi$
$294$	0	0
$295$	−141.001	−0.477968
$296$	0	0
$297$	40.0386i	0.134810i
$298$	0	0
$299$	100.626	0.336542
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−536.583	−1.77090
$304$	0	0
$305$	107.707	0.353137
$306$	0	0
$307$	132.193	0.430596	0.215298	−	0.976548i	$-0.430928\pi$
$307$	132.193	0.430596	0.215298	+	0.976548i	$0.430928\pi$
$308$	0	0
$309$	388.383i	1.25690i
$310$	0	0
$311$	− 462.404i	− 1.48683i	−0.668831	−	0.743414i	$-0.733205\pi$
$311$	− 462.404i	− 1.48683i	0.668831	−	0.743414i	$-0.266795\pi$
$312$	0	0
$313$	566.042i	1.80844i	0.427067	+	0.904220i	$0.359547\pi$
$313$	566.042i	1.80844i	−0.427067	+	0.904220i	$0.640453\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	177.033i	0.558463i	0.960224	+	0.279231i	$0.0900796\pi$
$317$	177.033i	0.558463i	−0.960224	+	0.279231i	$0.909920\pi$
$318$	0	0
$319$	62.8296	0.196958
$320$	0	0
$321$	366.299i	1.14112i
$322$	0	0
$323$	112.975i	0.349766i
$324$	0	0
$325$	416.922	1.28284
$326$	0	0
$327$	763.486i	2.33482i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	496.160i	1.49897i	0.662019	+	0.749487i	$0.269700\pi$
$331$	496.160i	1.49897i	−0.662019	+	0.749487i	$0.730300\pi$
$332$	0	0
$333$	244.693i	0.734813i
$334$	0	0
$335$	− 69.0825i	− 0.206216i
$336$	0	0
$337$	−206.191	−0.611843	−0.305922	−	0.952057i	$-0.598965\pi$
$337$	−206.191	−0.611843	−0.305922	+	0.952057i	$0.598965\pi$
$338$	0	0
$339$	396.931	1.17089
$340$	0	0
$341$	110.216	0.323214
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−445.696	−1.29187
$346$	0	0
$347$	− 606.190i	− 1.74694i	−0.486874	−	0.873472i	$-0.661863\pi$
$347$	− 606.190i	− 1.74694i	0.486874	−	0.873472i	$-0.338137\pi$
$348$	0	0
$349$	136.343	0.390669	0.195335	−	0.980737i	$-0.437421\pi$
$349$	136.343	0.390669	0.195335	+	0.980737i	$0.437421\pi$
$350$	0	0
$351$	−85.5887	−0.243842
$352$	0	0
$353$	− 10.0743i	− 0.0285390i	−0.999898	−	0.0142695i	$-0.995458\pi$
$353$	− 10.0743i	− 0.0285390i	0.999898	−	0.0142695i	$-0.00454228\pi$
$354$	0	0
$355$	−366.461	−1.03229
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−395.616	−1.10199	−0.550997	−	0.834507i	$-0.685752\pi$
$359$	−395.616	−1.10199	−0.550997	+	0.834507i	$0.685752\pi$
$360$	0	0
$361$	−244.678	−0.677778
$362$	0	0
$363$	419.664	1.15610
$364$	0	0
$365$	− 793.909i	− 2.17509i
$366$	0	0
$367$	− 189.932i	− 0.517526i	−0.965941	−	0.258763i	$-0.916685\pi$
$367$	− 189.932i	− 0.517526i	0.965941	−	0.258763i	$-0.0833149\pi$
$368$	0	0
$369$	455.457i	1.23430i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	360.104i	0.965425i	0.875779	+	0.482713i	$0.160348\pi$
$373$	360.104i	0.965425i	−0.875779	+	0.482713i	$0.839652\pi$
$374$	0	0
$375$	−983.631	−2.62302
$376$	0	0
$377$	134.308i	0.356254i
$378$	0	0
$379$	− 11.2929i	− 0.0297966i	−0.999889	−	0.0148983i	$-0.995258\pi$
$379$	− 11.2929i	− 0.0297966i	0.999889	−	0.0148983i	$-0.00474246\pi$
$380$	0	0
$381$	541.900	1.42231
$382$	0	0
$383$	434.253i	1.13382i	0.823779	+	0.566910i	$0.191861\pi$
$383$	434.253i	1.13382i	−0.823779	+	0.566910i	$0.808139\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	252.517i	0.652499i
$388$	0	0
$389$	43.1631i	0.110959i	0.998460	+	0.0554796i	$0.0176688\pi$
$389$	43.1631i	0.110959i	−0.998460	+	0.0554796i	$0.982331\pi$
$390$	0	0
$391$	− 135.242i	− 0.345887i
$392$	0	0
$393$	−357.315	−0.909199
$394$	0	0
$395$	627.222	1.58790
$396$	0	0
$397$	−486.790	−1.22617	−0.613086	−	0.790016i	$-0.710072\pi$
$397$	−486.790	−1.22617	−0.613086	+	0.790016i	$0.710072\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−546.914	−1.36388	−0.681938	−	0.731410i	$-0.738862\pi$
$401$	−546.914	−1.36388	−0.681938	+	0.731410i	$0.738862\pi$
$402$	0	0
$403$	235.603i	0.584624i
$404$	0	0
$405$	871.992	2.15307
$406$	0	0
$407$	144.323	0.354601
$408$	0	0
$409$	66.7668i	0.163244i	0.996663	+	0.0816220i	$0.0260100\pi$
$409$	66.7668i	0.163244i	−0.996663	+	0.0816220i	$0.973990\pi$
$410$	0	0
$411$	777.637	1.89206
$412$	0	0
$413$	0	0
$414$	0	0
$415$	539.266	1.29944
$416$	0	0
$417$	−153.815	−0.368862
$418$	0	0
$419$	550.169	1.31305	0.656527	−	0.754303i	$-0.272025\pi$
$419$	550.169	1.31305	0.656527	+	0.754303i	$0.272025\pi$
$420$	0	0
$421$	579.599i	1.37672i	0.725369	+	0.688360i	$0.241669\pi$
$421$	579.599i	1.37672i	−0.725369	+	0.688360i	$0.758331\pi$
$422$	0	0
$423$	− 258.408i	− 0.610893i
$424$	0	0
$425$	− 560.345i	− 1.31846i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 110.718i	− 0.258084i
$430$	0	0
$431$	431.870	1.00202	0.501009	−	0.865442i	$-0.332962\pi$
$431$	431.870	1.00202	0.501009	+	0.865442i	$0.332962\pi$
$432$	0	0
$433$	0.143463i	0 0.000331322i	1.00000		0.000165661i	$5.27316e-5\pi$
$433$	0.143463i	0 0.000331322i	−1.00000		0.000165661i	$0.999947\pi$
$434$	0	0
$435$	− 594.881i	− 1.36754i
$436$	0	0
$437$	−139.249	−0.318648
$438$	0	0
$439$	− 191.349i	− 0.435874i	−0.975963	−	0.217937i	$-0.930067\pi$
$439$	− 191.349i	− 0.435874i	0.975963	−	0.217937i	$-0.0699328\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 393.501i	− 0.888265i	−0.895961	−	0.444133i	$-0.853512\pi$
$443$	− 393.501i	− 0.888265i	0.895961	−	0.444133i	$-0.146488\pi$
$444$	0	0
$445$	− 239.393i	− 0.537961i
$446$	0	0
$447$	369.308i	0.826193i
$448$	0	0
$449$	−725.831	−1.61655	−0.808275	−	0.588805i	$-0.799598\pi$
$449$	−725.831	−1.61655	−0.808275	+	0.588805i	$0.799598\pi$
$450$	0	0
$451$	268.634	0.595641
$452$	0	0
$453$	−259.375	−0.572571
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−69.3427	−0.151735	−0.0758673	−	0.997118i	$-0.524173\pi$
$457$	−69.3427	−0.151735	−0.0758673	+	0.997118i	$0.524173\pi$
$458$	0	0
$459$	115.032i	0.250614i
$460$	0	0
$461$	−768.006	−1.66596	−0.832978	−	0.553306i	$-0.813366\pi$
$461$	−768.006	−1.66596	−0.832978	+	0.553306i	$0.813366\pi$
$462$	0	0
$463$	−215.717	−0.465911	−0.232956	−	0.972487i	$-0.574840\pi$
$463$	−215.717	−0.465911	−0.232956	+	0.972487i	$0.574840\pi$
$464$	0	0
$465$	− 1043.54i	− 2.24418i
$466$	0	0
$467$	−28.9376	−0.0619648	−0.0309824	−	0.999520i	$-0.509864\pi$
$467$	−28.9376	−0.0619648	−0.0309824	+	0.999520i	$0.509864\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−99.4325	−0.211109
$472$	0	0
$473$	148.938	0.314879
$474$	0	0
$475$	−576.949	−1.21463
$476$	0	0
$477$	39.6344i	0.0830911i
$478$	0	0
$479$	802.952i	1.67631i	0.545433	+	0.838154i	$0.316365\pi$
$479$	802.952i	1.67631i	−0.545433	+	0.838154i	$0.683635\pi$
$480$	0	0
$481$	308.512i	0.641396i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.9365i	0.0596629i
$486$	0	0
$487$	−19.9239	−0.0409116	−0.0204558	−	0.999791i	$-0.506512\pi$
$487$	−19.9239	−0.0409116	−0.0204558	+	0.999791i	$0.506512\pi$
$488$	0	0
$489$	− 748.493i	− 1.53066i
$490$	0	0
$491$	− 76.2017i	− 0.155197i	−0.996985	−	0.0775985i	$-0.975275\pi$
$491$	− 76.2017i	− 0.155197i	0.996985	−	0.0775985i	$-0.0247252\pi$
$492$	0	0
$493$	180.510	0.366147
$494$	0	0
$495$	199.677i	0.403387i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 522.871i	− 1.04784i	−0.851768	−	0.523919i	$-0.824470\pi$
$499$	− 522.871i	− 1.04784i	0.851768	−	0.523919i	$-0.175530\pi$
$500$	0	0
$501$	− 717.512i	− 1.43216i
$502$	0	0
$503$	− 132.060i	− 0.262545i	−0.991346	−	0.131273i	$-0.958094\pi$
$503$	− 132.060i	− 0.262545i	0.991346	−	0.131273i	$-0.0419064\pi$
$504$	0	0
$505$	1220.11	2.41605
$506$	0	0
$507$	−421.807	−0.831967
$508$	0	0
$509$	−310.157	−0.609346	−0.304673	−	0.952457i	$-0.598547\pi$
$509$	−310.157	−0.609346	−0.304673	+	0.952457i	$0.598547\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	118.440	0.230878
$514$	0	0
$515$	− 883.122i	− 1.71480i
$516$	0	0
$517$	−152.412	−0.294801
$518$	0	0
$519$	−272.369	−0.524797
$520$	0	0
$521$	61.0976i	0.117270i	0.998279	+	0.0586349i	$0.0186748\pi$
$521$	61.0976i	0.117270i	−0.998279	+	0.0586349i	$0.981325\pi$
$522$	0	0
$523$	−513.846	−0.982497	−0.491249	−	0.871019i	$-0.663459\pi$
$523$	−513.846	−0.982497	−0.491249	+	0.871019i	$0.663459\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	316.652	0.600858
$528$	0	0
$529$	−362.304	−0.684886
$530$	0	0
$531$	98.3784	0.185270
$532$	0	0
$533$	574.246i	1.07739i
$534$	0	0
$535$	− 832.906i	− 1.55683i
$536$	0	0
$537$	− 932.449i	− 1.73640i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 107.078i	− 0.197926i	−0.995091	−	0.0989630i	$-0.968447\pi$
$541$	− 107.078i	− 0.197926i	0.995091	−	0.0989630i	$-0.0315525\pi$
$542$	0	0
$543$	−141.373	−0.260355
$544$	0	0
$545$	− 1736.05i	− 3.18541i
$546$	0	0
$547$	43.3240i	0.0792030i	0.999216	+	0.0396015i	$0.0126088\pi$
$547$	43.3240i	0.0792030i	−0.999216	+	0.0396015i	$0.987391\pi$
$548$	0	0
$549$	−75.1489	−0.136883
$550$	0	0
$551$	− 185.859i	− 0.337313i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 1366.47i	− 2.46211i
$556$	0	0
$557$	72.5011i	0.130164i	0.997880	+	0.0650818i	$0.0207308\pi$
$557$	72.5011i	0.130164i	−0.997880	+	0.0650818i	$0.979269\pi$
$558$	0	0
$559$	318.377i	0.569547i
$560$	0	0
$561$	−148.805	−0.265250
$562$	0	0
$563$	584.943	1.03897	0.519487	−	0.854478i	$-0.326123\pi$
$563$	584.943	1.03897	0.519487	+	0.854478i	$0.326123\pi$
$564$	0	0
$565$	−902.559	−1.59745
$566$	0	0
$567$	0	0
$568$	0	0
$569$	743.529	1.30673	0.653365	−	0.757043i	$-0.273357\pi$
$569$	743.529	1.30673	0.653365	+	0.757043i	$0.273357\pi$
$570$	0	0
$571$	− 1032.06i	− 1.80747i	−0.428097	−	0.903733i	$-0.640816\pi$
$571$	− 1032.06i	− 1.80747i	0.428097	−	0.903733i	$-0.359184\pi$
$572$	0	0
$573$	1267.26	2.21163
$574$	0	0
$575$	690.666	1.20116
$576$	0	0
$577$	− 953.094i	− 1.65181i	−0.563810	−	0.825905i	$-0.690665\pi$
$577$	− 953.094i	− 1.65181i	0.563810	−	0.825905i	$-0.309335\pi$
$578$	0	0
$579$	777.866	1.34346
$580$	0	0
$581$	0	0
$582$	0	0
$583$	23.3769	0.0400976
$584$	0	0
$585$	−426.839	−0.729640
$586$	0	0
$587$	96.2876	0.164033	0.0820167	−	0.996631i	$-0.473864\pi$
$587$	96.2876	0.164033	0.0820167	+	0.996631i	$0.473864\pi$
$588$	0	0
$589$	− 326.035i	− 0.553540i
$590$	0	0
$591$	60.4141i	0.102224i
$592$	0	0
$593$	− 51.0193i	− 0.0860358i	−0.999074	−	0.0430179i	$-0.986303\pi$
$593$	− 51.0193i	− 0.0860358i	0.999074	−	0.0430179i	$-0.0136973\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	218.826i	0.366542i
$598$	0	0
$599$	−902.236	−1.50624	−0.753119	−	0.657885i	$-0.771451\pi$
$599$	−902.236	−1.50624	−0.753119	+	0.657885i	$0.771451\pi$
$600$	0	0
$601$	903.595i	1.50349i	0.659456	+	0.751743i	$0.270787\pi$
$601$	903.595i	1.50349i	−0.659456	+	0.751743i	$0.729213\pi$
$602$	0	0
$603$	48.2000i	0.0799337i
$604$	0	0
$605$	−954.251	−1.57727
$606$	0	0
$607$	− 354.410i	− 0.583872i	−0.956438	−	0.291936i	$-0.905701\pi$
$607$	− 354.410i	− 0.583872i	0.956438	−	0.291936i	$-0.0942994\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 325.804i	− 0.533230i
$612$	0	0
$613$	− 335.999i	− 0.548123i	−0.961712	−	0.274061i	$-0.911633\pi$
$613$	− 335.999i	− 0.548123i	0.961712	−	0.274061i	$-0.0883671\pi$
$614$	0	0
$615$	− 2543.47i	− 4.13573i
$616$	0	0
$617$	223.359	0.362008	0.181004	−	0.983482i	$-0.442065\pi$
$617$	223.359	0.362008	0.181004	+	0.983482i	$0.442065\pi$
$618$	0	0
$619$	726.053	1.17294	0.586472	−	0.809969i	$-0.300516\pi$
$619$	726.053	1.17294	0.586472	+	0.809969i	$0.300516\pi$
$620$	0	0
$621$	−141.785	−0.228317
$622$	0	0
$623$	0	0
$624$	0	0
$625$	899.270	1.43883
$626$	0	0
$627$	153.215i	0.244362i
$628$	0	0
$629$	414.642	0.659208
$630$	0	0
$631$	326.157	0.516888	0.258444	−	0.966026i	$-0.416790\pi$
$631$	326.157	0.516888	0.258444	+	0.966026i	$0.416790\pi$
$632$	0	0
$633$	1443.48i	2.28038i
$634$	0	0
$635$	−1232.19	−1.94046
$636$	0	0
$637$	0	0
$638$	0	0
$639$	255.686	0.400135
$640$	0	0
$641$	−598.375	−0.933502	−0.466751	−	0.884389i	$-0.654576\pi$
$641$	−598.375	−0.933502	−0.466751	+	0.884389i	$0.654576\pi$
$642$	0	0
$643$	−1008.20	−1.56796	−0.783979	−	0.620787i	$-0.786813\pi$
$643$	−1008.20	−1.56796	−0.783979	+	0.620787i	$0.786813\pi$
$644$	0	0
$645$	− 1410.16i	− 2.18630i
$646$	0	0
$647$	− 663.235i	− 1.02509i	−0.858660	−	0.512546i	$-0.828702\pi$
$647$	− 663.235i	− 1.02509i	0.858660	−	0.512546i	$-0.171298\pi$
$648$	0	0
$649$	− 58.0248i	− 0.0894064i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 990.608i	− 1.51701i	−0.651666	−	0.758506i	$-0.725930\pi$
$653$	− 990.608i	− 1.51701i	0.651666	−	0.758506i	$-0.274070\pi$
$654$	0	0
$655$	812.479	1.24043
$656$	0	0
$657$	553.924i	0.843110i
$658$	0	0
$659$	− 82.2318i	− 0.124783i	−0.998052	−	0.0623914i	$-0.980127\pi$
$659$	− 82.2318i	− 0.124783i	0.998052	−	0.0623914i	$-0.0198727\pi$
$660$	0	0
$661$	626.220	0.947382	0.473691	−	0.880691i	$-0.342921\pi$
$661$	626.220	0.947382	0.473691	+	0.880691i	$0.342921\pi$
$662$	0	0
$663$	− 318.094i	− 0.479780i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	222.492i	0.333572i
$668$	0	0
$669$	− 26.4380i	− 0.0395187i
$670$	0	0
$671$	44.3237i	0.0660562i
$672$	0	0
$673$	−150.211	−0.223196	−0.111598	−	0.993753i	$-0.535597\pi$
$673$	−150.211	−0.223196	−0.111598	+	0.993753i	$0.535597\pi$
$674$	0	0
$675$	−587.455	−0.870303
$676$	0	0
$677$	556.414	0.821882	0.410941	−	0.911662i	$-0.365200\pi$
$677$	556.414	0.821882	0.410941	+	0.911662i	$0.365200\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1155.73	−1.69711
$682$	0	0
$683$	791.356i	1.15865i	0.815098	+	0.579323i	$0.196683\pi$
$683$	791.356i	1.15865i	−0.815098	+	0.579323i	$0.803317\pi$
$684$	0	0
$685$	−1768.22	−2.58135
$686$	0	0
$687$	694.302	1.01063
$688$	0	0
$689$	49.9716i	0.0725277i
$690$	0	0
$691$	976.534	1.41322	0.706609	−	0.707604i	$-0.250224\pi$
$691$	976.534	1.41322	0.706609	+	0.707604i	$0.250224\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	349.752	0.503241
$696$	0	0
$697$	771.790	1.10730
$698$	0	0
$699$	458.999	0.656651
$700$	0	0
$701$	855.098i	1.21983i	0.792468	+	0.609913i	$0.208796\pi$
$701$	855.098i	1.21983i	−0.792468	+	0.609913i	$0.791204\pi$
$702$	0	0
$703$	− 426.928i	− 0.607294i
$704$	0	0
$705$	1443.06i	2.04689i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	333.471i	0.470340i	0.971954	+	0.235170i	$0.0755647\pi$
$709$	333.471i	0.470340i	−0.971954	+	0.235170i	$0.924435\pi$
$710$	0	0
$711$	−437.623	−0.615504
$712$	0	0
$713$	390.297i	0.547401i
$714$	0	0
$715$	251.755i	0.352105i
$716$	0	0
$717$	−180.613	−0.251900
$718$	0	0
$719$	− 39.8230i	− 0.0553866i	−0.999616	−	0.0276933i	$-0.991184\pi$
$719$	− 39.8230i	− 0.0553866i	0.999616	−	0.0276933i	$-0.00881618\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1428.47i	1.97576i
$724$	0	0
$725$	921.848i	1.27151i
$726$	0	0
$727$	489.402i	0.673180i	0.941651	+	0.336590i	$0.109274\pi$
$727$	489.402i	0.673180i	−0.941651	+	0.336590i	$0.890726\pi$
$728$	0	0
$729$	−223.307	−0.306320
$730$	0	0
$731$	427.900	0.585363
$732$	0	0
$733$	−178.318	−0.243272	−0.121636	−	0.992575i	$-0.538814\pi$
$733$	−178.318	−0.243272	−0.121636	+	0.992575i	$0.538814\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28.4289	0.0385739
$738$	0	0
$739$	882.401i	1.19405i	0.802224	+	0.597024i	$0.203650\pi$
$739$	882.401i	1.19405i	−0.802224	+	0.597024i	$0.796350\pi$
$740$	0	0
$741$	−327.520	−0.441997
$742$	0	0
$743$	−1404.00	−1.88964	−0.944819	−	0.327591i	$-0.893763\pi$
$743$	−1404.00	−1.88964	−0.944819	+	0.327591i	$0.893763\pi$
$744$	0	0
$745$	− 839.749i	− 1.12718i
$746$	0	0
$747$	−376.255	−0.503687
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−205.680	−0.273875	−0.136938	−	0.990580i	$-0.543726\pi$
$751$	−205.680	−0.273875	−0.136938	+	0.990580i	$0.543726\pi$
$752$	0	0
$753$	506.119	0.672137
$754$	0	0
$755$	589.777	0.781162
$756$	0	0
$757$	15.0345i	0.0198606i	0.999951	+	0.00993032i	$0.00316097\pi$
$757$	15.0345i	0.0198606i	−0.999951	+	0.00993032i	$0.996839\pi$
$758$	0	0
$759$	− 183.414i	− 0.241652i
$760$	0	0
$761$	628.491i	0.825876i	0.910759	+	0.412938i	$0.135497\pi$
$761$	628.491i	0.825876i	−0.910759	+	0.412938i	$0.864503\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	573.675i	0.749901i
$766$	0	0
$767$	124.037	0.161717
$768$	0	0
$769$	442.918i	0.575967i	0.957635	+	0.287983i	$0.0929848\pi$
$769$	442.918i	0.575967i	−0.957635	+	0.287983i	$0.907015\pi$
$770$	0	0
$771$	− 1045.57i	− 1.35612i
$772$	0	0
$773$	169.198	0.218885	0.109442	−	0.993993i	$-0.465093\pi$
$773$	169.198	0.218885	0.109442	+	0.993993i	$0.465093\pi$
$774$	0	0
$775$	1617.11i	2.08659i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 794.660i	− 1.02010i
$780$	0	0
$781$	− 150.807i	− 0.193094i
$782$	0	0
$783$	− 189.244i	− 0.241690i
$784$	0	0
$785$	226.094	0.288018
$786$	0	0
$787$	−46.4874	−0.0590691	−0.0295346	−	0.999564i	$-0.509403\pi$
$787$	−46.4874	−0.0590691	−0.0295346	+	0.999564i	$0.509403\pi$
$788$	0	0
$789$	917.133	1.16240
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−94.7487	−0.119481
$794$	0	0
$795$	− 221.336i	− 0.278410i
$796$	0	0
$797$	1351.86	1.69618	0.848092	−	0.529850i	$-0.177752\pi$
$797$	1351.86	1.69618	0.848092	+	0.529850i	$0.177752\pi$
$798$	0	0
$799$	−437.882	−0.548038
$800$	0	0
$801$	167.028i	0.208525i
$802$	0	0
$803$	326.711	0.406863
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1382.02	−1.71254
$808$	0	0
$809$	−1403.13	−1.73440	−0.867198	−	0.497963i	$-0.834081\pi$
$809$	−1403.13	−1.73440	−0.867198	+	0.497963i	$0.834081\pi$
$810$	0	0
$811$	−689.037	−0.849614	−0.424807	−	0.905284i	$-0.639658\pi$
$811$	−689.037	−0.849614	−0.424807	+	0.905284i	$0.639658\pi$
$812$	0	0
$813$	− 1643.75i	− 2.02184i
$814$	0	0
$815$	1701.95i	2.08829i
$816$	0	0
$817$	− 440.579i	− 0.539265i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.3423i	0.0272135i	0.999907	+	0.0136067i	$0.00433129\pi$
$821$	22.3423i	0.0272135i	−0.999907	+	0.0136067i	$0.995669\pi$
$822$	0	0
$823$	−1024.22	−1.24450	−0.622249	−	0.782819i	$-0.713781\pi$
$823$	−1024.22	−1.24450	−0.622249	+	0.782819i	$0.713781\pi$
$824$	0	0
$825$	− 759.934i	− 0.921132i
$826$	0	0
$827$	− 466.377i	− 0.563938i	−0.959424	−	0.281969i	$-0.909012\pi$
$827$	− 466.377i	− 0.563938i	0.959424	−	0.281969i	$-0.0909875\pi$
$828$	0	0
$829$	1500.70	1.81025	0.905126	−	0.425142i	$-0.139776\pi$
$829$	1500.70	1.81025	0.905126	+	0.425142i	$0.139776\pi$
$830$	0	0
$831$	− 1435.92i	− 1.72794i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1631.51i	1.95390i
$836$	0	0
$837$	− 331.972i	− 0.396621i
$838$	0	0
$839$	1068.18i	1.27316i	0.771212	+	0.636579i	$0.219651\pi$
$839$	1068.18i	1.27316i	−0.771212	+	0.636579i	$0.780349\pi$
$840$	0	0
$841$	544.034	0.646890
$842$	0	0
$843$	−137.490	−0.163096
$844$	0	0
$845$	959.123	1.13506
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−766.090	−0.902344
$850$	0	0
$851$	511.076i	0.600559i
$852$	0	0
$853$	918.640	1.07695	0.538476	−	0.842641i	$-0.319000\pi$
$853$	918.640	1.07695	0.538476	+	0.842641i	$0.319000\pi$
$854$	0	0
$855$	590.673	0.690846
$856$	0	0
$857$	− 505.952i	− 0.590375i	−0.955439	−	0.295188i	$-0.904618\pi$
$857$	− 505.952i	− 0.590375i	0.955439	−	0.295188i	$-0.0953822\pi$
$858$	0	0
$859$	−1377.03	−1.60306	−0.801532	−	0.597952i	$-0.795981\pi$
$859$	−1377.03	−1.60306	−0.801532	+	0.597952i	$0.795981\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−917.635	−1.06331	−0.531654	−	0.846962i	$-0.678429\pi$
$863$	−917.635	−1.06331	−0.531654	+	0.846962i	$0.678429\pi$
$864$	0	0
$865$	619.325	0.715983
$866$	0	0
$867$	698.525	0.805681
$868$	0	0
$869$	258.116i	0.297026i
$870$	0	0
$871$	60.7712i	0.0697717i
$872$	0	0
$873$	− 20.1895i	− 0.0231265i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	699.948i	0.798116i	0.916926	+	0.399058i	$0.130663\pi$
$877$	699.948i	0.798116i	−0.916926	+	0.399058i	$0.869337\pi$
$878$	0	0
$879$	1237.38	1.40771
$880$	0	0
$881$	− 6.37652i	− 0.00723783i	−0.999993	−	0.00361891i	$-0.998848\pi$
$881$	− 6.37652i	− 0.00723783i	0.999993	−	0.00361891i	$-0.00115194\pi$
$882$	0	0
$883$	1548.35i	1.75351i	0.480935	+	0.876756i	$0.340297\pi$
$883$	1548.35i	1.75351i	−0.480935	+	0.876756i	$0.659703\pi$
$884$	0	0
$885$	−549.388	−0.620777
$886$	0	0
$887$	732.369i	0.825670i	0.910806	+	0.412835i	$0.135461\pi$
$887$	732.369i	0.825670i	−0.910806	+	0.412835i	$0.864539\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	358.844i	0.402743i
$892$	0	0
$893$	450.857i	0.504879i
$894$	0	0
$895$	2120.24i	2.36899i
$896$	0	0
$897$	392.074	0.437095
$898$	0	0
$899$	−520.938	−0.579464
$900$	0	0
$901$	67.1621	0.0745418
$902$	0	0
$903$	0	0
$904$	0	0
$905$	321.459	0.355204
$906$	0	0
$907$	− 723.838i	− 0.798057i	−0.916939	−	0.399029i	$-0.869347\pi$
$907$	− 723.838i	− 0.798057i	0.916939	−	0.399029i	$-0.130653\pi$
$908$	0	0
$909$	−851.288	−0.936511
$910$	0	0
$911$	−1600.04	−1.75636	−0.878179	−	0.478332i	$-0.841242\pi$
$911$	−1600.04	−1.75636	−0.878179	+	0.478332i	$0.841242\pi$
$912$	0	0
$913$	221.920i	0.243066i
$914$	0	0
$915$	419.664	0.458649
$916$	0	0
$917$	0	0
$918$	0	0
$919$	524.085	0.570277	0.285139	−	0.958486i	$-0.407960\pi$
$919$	524.085	0.570277	0.285139	+	0.958486i	$0.407960\pi$
$920$	0	0
$921$	515.070	0.559251
$922$	0	0
$923$	322.372	0.349266
$924$	0	0
$925$	2117.53i	2.28922i
$926$	0	0
$927$	616.169i	0.664691i
$928$	0	0
$929$	− 637.154i	− 0.685849i	−0.939363	−	0.342924i	$-0.888583\pi$
$929$	− 637.154i	− 0.685849i	0.939363	−	0.342924i	$-0.111417\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 1801.69i	− 1.93107i
$934$	0	0
$935$	338.360	0.361883
$936$	0	0
$937$	− 383.587i	− 0.409378i	−0.978827	−	0.204689i	$-0.934382\pi$
$937$	− 383.587i	− 0.409378i	0.978827	−	0.204689i	$-0.0656182\pi$
$938$	0	0
$939$	2205.50i	2.34877i
$940$	0	0
$941$	260.591	0.276930	0.138465	−	0.990367i	$-0.455783\pi$
$941$	260.591	0.276930	0.138465	+	0.990367i	$0.455783\pi$
$942$	0	0
$943$	951.288i	1.00879i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	983.163i	1.03819i	0.854717	+	0.519093i	$0.173730\pi$
$947$	983.163i	1.03819i	−0.854717	+	0.519093i	$0.826270\pi$
$948$	0	0
$949$	698.394i	0.735926i
$950$	0	0
$951$	689.781i	0.725322i
$952$	0	0
$953$	−1137.60	−1.19370	−0.596851	−	0.802352i	$-0.703582\pi$
$953$	−1137.60	−1.19370	−0.596851	+	0.802352i	$0.703582\pi$
$954$	0	0
$955$	−2881.56	−3.01734
$956$	0	0
$957$	244.806	0.255806
$958$	0	0
$959$	0	0
$960$	0	0
$961$	47.1670	0.0490812
$962$	0	0
$963$	581.132i	0.603460i
$964$	0	0
$965$	−1768.75	−1.83290
$966$	0	0
$967$	1296.35	1.34059	0.670297	−	0.742093i	$-0.266167\pi$
$967$	1296.35	1.34059	0.670297	+	0.742093i	$0.266167\pi$
$968$	0	0
$969$	440.188i	0.454271i
$970$	0	0
$971$	1330.47	1.37021	0.685105	−	0.728445i	$-0.259756\pi$
$971$	1330.47	1.37021	0.685105	+	0.728445i	$0.259756\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1624.47	1.66613
$976$	0	0
$977$	1386.16	1.41879	0.709397	−	0.704809i	$-0.248967\pi$
$977$	1386.16	1.41879	0.709397	+	0.704809i	$0.248967\pi$
$978$	0	0
$979$	98.5153	0.100629
$980$	0	0
$981$	1211.27i	1.23473i
$982$	0	0
$983$	694.161i	0.706166i	0.935592	+	0.353083i	$0.114867\pi$
$983$	694.161i	0.706166i	−0.935592	+	0.353083i	$0.885133\pi$
$984$	0	0
$985$	− 137.372i	− 0.139464i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	527.418i	0.533284i
$990$	0	0
$991$	934.514	0.943001	0.471500	−	0.881866i	$-0.343713\pi$
$991$	934.514	0.943001	0.471500	+	0.881866i	$0.343713\pi$
$992$	0	0
$993$	1933.22i	1.94684i
$994$	0	0
$995$	− 497.576i	− 0.500076i
$996$	0	0
$997$	−1538.21	−1.54283	−0.771417	−	0.636330i	$-0.780452\pi$
$997$	−1538.21	−1.54283	−0.771417	+	0.636330i	$0.780452\pi$
$998$	0	0
$999$	− 434.702i	− 0.435137i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.h.a.881.25		28
4.3	odd	2		392.3.h.a.293.17		28
7.4	even	3		224.3.n.a.145.2		28
7.5	odd	6		224.3.n.a.17.13		28
7.6	odd	2	inner	1568.3.h.a.881.3		28
8.3	odd	2		392.3.h.a.293.20		28
8.5	even	2	inner	1568.3.h.a.881.4		28
28.3	even	6		392.3.j.e.117.10		28
28.11	odd	6		56.3.j.a.5.10	yes	28
28.19	even	6		56.3.j.a.45.1	yes	28
28.23	odd	6		392.3.j.e.325.1		28
28.27	even	2		392.3.h.a.293.18		28
56.3	even	6		392.3.j.e.117.1		28
56.5	odd	6		224.3.n.a.17.2		28
56.11	odd	6		56.3.j.a.5.1	✓	28
56.13	odd	2	inner	1568.3.h.a.881.26		28
56.19	even	6		56.3.j.a.45.10	yes	28
56.27	even	2		392.3.h.a.293.19		28
56.51	odd	6		392.3.j.e.325.10		28
56.53	even	6		224.3.n.a.145.13		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.j.a.5.1	✓	28	56.11	odd	6
56.3.j.a.5.10	yes	28	28.11	odd	6
56.3.j.a.45.1	yes	28	28.19	even	6
56.3.j.a.45.10	yes	28	56.19	even	6
224.3.n.a.17.2		28	56.5	odd	6
224.3.n.a.17.13		28	7.5	odd	6
224.3.n.a.145.2		28	7.4	even	3
224.3.n.a.145.13		28	56.53	even	6
392.3.h.a.293.17		28	4.3	odd	2
392.3.h.a.293.18		28	28.27	even	2
392.3.h.a.293.19		28	56.27	even	2
392.3.h.a.293.20		28	8.3	odd	2
392.3.j.e.117.1		28	56.3	even	6
392.3.j.e.117.10		28	28.3	even	6
392.3.j.e.325.1		28	28.23	odd	6
392.3.j.e.325.10		28	56.51	odd	6
1568.3.h.a.881.3		28	7.6	odd	2	inner
1568.3.h.a.881.4		28	8.5	even	2	inner
1568.3.h.a.881.25		28	1.1	even	1	trivial
1568.3.h.a.881.26		28	56.13	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)

\(f(q)\)	\(=\)	\(q+3.89635 q^{3} -8.85969 q^{5} +6.18155 q^{9} +O(q^{10})\)	\(q+3.89635 q^{3} -8.85969 q^{5} +6.18155 q^{9} -3.64596i q^{11} +7.79378 q^{13} -34.5205 q^{15} -10.4749i q^{17} -10.7853 q^{19} +12.9111 q^{23} +53.4941 q^{25} -10.9817 q^{27} +17.2327i q^{29} +30.2297i q^{31} -14.2059i q^{33} +39.5843i q^{37} +30.3673 q^{39} +73.6801i q^{41} +40.8501i q^{43} -54.7667 q^{45} -41.8030i q^{47} -40.8138i q^{51} +6.41173i q^{53} +32.3020i q^{55} -42.0232 q^{57} +15.9148 q^{59} -12.1570 q^{61} -69.0505 q^{65} +7.79739i q^{67} +50.3060 q^{69} +41.3627 q^{71} +89.6091i q^{73} +208.432 q^{75} -70.7950 q^{79} -98.4224 q^{81} -60.8673 q^{83} +92.8043i q^{85} +67.1446i q^{87} +27.0204i q^{89} +117.785i q^{93} +95.5542 q^{95} -3.26608i q^{97} -22.5377i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 64 q^{9}+O(q^{10}) \) 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

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Database

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