LMFDB - Embedded newform 1568.3.g.i.687.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(687,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("1568.687");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.g (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:	$6$
Coefficient field:	6.0.700560112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{4} - 6x^{3} - 8x^{2} + 64 $ x^6 - 2x^4 - 6x^3 - 8*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			687.1
Root			$-1.75274 + 0.963276i$ of defining polynomial
Character	$\chi$	$=$	1568.687
Dual form			1568.3.g.i.687.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.50548 q^{3} -7.01018i q^{5} +11.2994 q^{9} +O(q^{10})$	$q-4.50548 q^{3} -7.01018i q^{5} +11.2994 q^{9} -15.8048 q^{11} -2.90039i q^{13} +31.5842i q^{15} +3.31032 q^{17} +16.2171 q^{19} +19.0300i q^{23} -24.1426 q^{25} -10.3597 q^{27} +21.1392i q^{29} +27.3658i q^{31} +71.2084 q^{33} +17.6590i q^{37} +13.0676i q^{39} -1.13482 q^{41} -50.2084 q^{43} -79.2105i q^{45} +0.759384i q^{47} -14.9146 q^{51} +44.9961i q^{53} +110.795i q^{55} -73.0658 q^{57} +2.38479 q^{59} -99.4952i q^{61} -20.3322 q^{65} +66.4881 q^{67} -85.7391i q^{69} +44.4376i q^{71} -1.71941 q^{73} +108.774 q^{75} -72.6504i q^{79} -55.0188 q^{81} -102.081 q^{83} -23.2059i q^{85} -95.2421i q^{87} +61.3729 q^{89} -123.296i q^{93} -113.685i q^{95} +102.826 q^{97} -178.584 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 8 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 8 * q^9	$ 6 q - 6 q^{3} + 8 q^{9} - 14 q^{11} - 82 q^{17} + 94 q^{19} - 116 q^{25} - 30 q^{27} + 146 q^{33} - 60 q^{41} - 20 q^{43} - 106 q^{51} - 186 q^{57} - 62 q^{59} + 64 q^{65} - 178 q^{67} + 54 q^{73} - 140 q^{75} - 206 q^{81} - 196 q^{83} - 26 q^{89} + 92 q^{97} - 436 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 8 * q^9 - 14 * q^11 - 82 * q^17 + 94 * q^19 - 116 * q^25 - 30 * q^27 + 146 * q^33 - 60 * q^41 - 20 * q^43 - 106 * q^51 - 186 * q^57 - 62 * q^59 + 64 * q^65 - 178 * q^67 + 54 * q^73 - 140 * q^75 - 206 * q^81 - 196 * q^83 - 26 * q^89 + 92 * q^97 - 436 * q^99

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
$3$	−4.50548	−1.50183	−0.750913	−	0.660401i	$-0.770386\pi$
$3$	−4.50548	−1.50183	−0.750913	+	0.660401i	$0.770386\pi$
$4$	0	0
$5$	− 7.01018i	− 1.40204i	−0.713144	−	0.701018i	$-0.752729\pi$
$5$	− 7.01018i	− 1.40204i	0.713144	−	0.701018i	$-0.247271\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	11.2994	1.25548
$10$	0	0
$11$	−15.8048	−1.43680	−0.718402	−	0.695629i	$-0.755126\pi$
$11$	−15.8048	−1.43680	−0.718402	+	0.695629i	$0.755126\pi$
$12$	0	0
$13$	− 2.90039i	− 0.223107i	−0.993758	−	0.111553i	$-0.964417\pi$
$13$	− 2.90039i	− 0.223107i	0.993758	−	0.111553i	$-0.0355826\pi$
$14$	0	0
$15$	31.5842i	2.10562i
$16$	0	0
$17$	3.31032	0.194725	0.0973623	−	0.995249i	$-0.468959\pi$
$17$	3.31032	0.194725	0.0973623	+	0.995249i	$0.468959\pi$
$18$	0	0
$19$	16.2171	0.853531	0.426765	−	0.904362i	$-0.359653\pi$
$19$	16.2171	0.853531	0.426765	+	0.904362i	$0.359653\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	19.0300i	0.827390i	0.910416	+	0.413695i	$0.135762\pi$
$23$	19.0300i	0.827390i	−0.910416	+	0.413695i	$0.864238\pi$
$24$	0	0
$25$	−24.1426	−0.965705
$26$	0	0
$27$	−10.3597	−0.383693
$28$	0	0
$29$	21.1392i	0.728937i	0.931216	+	0.364468i	$0.118749\pi$
$29$	21.1392i	0.728937i	−0.931216	+	0.364468i	$0.881251\pi$
$30$	0	0
$31$	27.3658i	0.882768i	0.897318	+	0.441384i	$0.145512\pi$
$31$	27.3658i	0.882768i	−0.897318	+	0.441384i	$0.854488\pi$
$32$	0	0
$33$	71.2084	2.15783
$34$	0	0
$35$	0	0
$36$	0	0
$37$	17.6590i	0.477271i	0.971109	+	0.238635i	$0.0767001\pi$
$37$	17.6590i	0.477271i	−0.971109	+	0.238635i	$0.923300\pi$
$38$	0	0
$39$	13.0676i	0.335068i
$40$	0	0
$41$	−1.13482	−0.0276785	−0.0138392	−	0.999904i	$-0.504405\pi$
$41$	−1.13482	−0.0276785	−0.0138392	+	0.999904i	$0.504405\pi$
$42$	0	0
$43$	−50.2084	−1.16764	−0.583818	−	0.811884i	$-0.698442\pi$
$43$	−50.2084	−1.16764	−0.583818	+	0.811884i	$0.698442\pi$
$44$	0	0
$45$	− 79.2105i	− 1.76023i
$46$	0	0
$47$	0.759384i	0.0161571i	0.999967	+	0.00807855i	$0.00257151\pi$
$47$	0.759384i	0.0161571i	−0.999967	+	0.00807855i	$0.997428\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−14.9146	−0.292443
$52$	0	0
$53$	44.9961i	0.848982i	0.905432	+	0.424491i	$0.139547\pi$
$53$	44.9961i	0.848982i	−0.905432	+	0.424491i	$0.860453\pi$
$54$	0	0
$55$	110.795i	2.01445i
$56$	0	0
$57$	−73.0658	−1.28186
$58$	0	0
$59$	2.38479	0.0404201	0.0202101	−	0.999796i	$-0.493567\pi$
$59$	2.38479	0.0404201	0.0202101	+	0.999796i	$0.493567\pi$
$60$	0	0
$61$	− 99.4952i	− 1.63107i	−0.578709	−	0.815534i	$-0.696443\pi$
$61$	− 99.4952i	− 1.63107i	0.578709	−	0.815534i	$-0.303557\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−20.3322	−0.312804
$66$	0	0
$67$	66.4881	0.992359	0.496180	−	0.868220i	$-0.334736\pi$
$67$	66.4881	0.992359	0.496180	+	0.868220i	$0.334736\pi$
$68$	0	0
$69$	− 85.7391i	− 1.24260i
$70$	0	0
$71$	44.4376i	0.625882i	0.949773	+	0.312941i	$0.101314\pi$
$71$	44.4376i	0.625882i	−0.949773	+	0.312941i	$0.898686\pi$
$72$	0	0
$73$	−1.71941	−0.0235535	−0.0117768	−	0.999931i	$-0.503749\pi$
$73$	−1.71941	−0.0235535	−0.0117768	+	0.999931i	$0.503749\pi$
$74$	0	0
$75$	108.774	1.45032
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 72.6504i	− 0.919625i	−0.888016	−	0.459812i	$-0.847917\pi$
$79$	− 72.6504i	− 0.919625i	0.888016	−	0.459812i	$-0.152083\pi$
$80$	0	0
$81$	−55.0188	−0.679244
$82$	0	0
$83$	−102.081	−1.22990	−0.614948	−	0.788568i	$-0.710823\pi$
$83$	−102.081	−1.22990	−0.614948	+	0.788568i	$0.710823\pi$
$84$	0	0
$85$	− 23.2059i	− 0.273011i
$86$	0	0
$87$	− 95.2421i	− 1.09474i
$88$	0	0
$89$	61.3729	0.689583	0.344792	−	0.938679i	$-0.387950\pi$
$89$	61.3729	0.689583	0.344792	+	0.938679i	$0.387950\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 123.296i	− 1.32576i
$94$	0	0
$95$	− 113.685i	− 1.19668i
$96$	0	0
$97$	102.826	1.06006	0.530030	−	0.847979i	$-0.322181\pi$
$97$	102.826	1.06006	0.530030	+	0.847979i	$0.322181\pi$
$98$	0	0
$99$	−178.584	−1.80388
$100$	0	0
$101$	182.039i	1.80237i	0.433435	+	0.901185i	$0.357301\pi$
$101$	182.039i	1.80237i	−0.433435	+	0.901185i	$0.642699\pi$
$102$	0	0
$103$	− 45.8536i	− 0.445181i	−0.974912	−	0.222591i	$-0.928549\pi$
$103$	− 45.8536i	− 0.445181i	0.974912	−	0.222591i	$-0.0714513\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	73.8925	0.690584	0.345292	−	0.938495i	$-0.387780\pi$
$107$	73.8925	0.690584	0.345292	+	0.938495i	$0.387780\pi$
$108$	0	0
$109$	120.924i	1.10940i	0.832051	+	0.554699i	$0.187167\pi$
$109$	120.924i	1.10940i	−0.832051	+	0.554699i	$0.812833\pi$
$110$	0	0
$111$	− 79.5623i	− 0.716778i
$112$	0	0
$113$	−97.7663	−0.865188	−0.432594	−	0.901589i	$-0.642402\pi$
$113$	−97.7663	−0.865188	−0.432594	+	0.901589i	$0.642402\pi$
$114$	0	0
$115$	133.403	1.16003
$116$	0	0
$117$	− 32.7725i	− 0.280107i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	128.793	1.06440
$122$	0	0
$123$	5.11289	0.0415682
$124$	0	0
$125$	− 6.01041i	− 0.0480833i
$126$	0	0
$127$	− 175.050i	− 1.37835i	−0.724595	−	0.689175i	$-0.757973\pi$
$127$	− 175.050i	− 1.37835i	0.724595	−	0.689175i	$-0.242027\pi$
$128$	0	0
$129$	226.213	1.75359
$130$	0	0
$131$	−8.61985	−0.0658004	−0.0329002	−	0.999459i	$-0.510474\pi$
$131$	−8.61985	−0.0658004	−0.0329002	+	0.999459i	$0.510474\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	72.6234i	0.537951i
$136$	0	0
$137$	256.084	1.86923	0.934615	−	0.355662i	$-0.115745\pi$
$137$	256.084	1.86923	0.934615	+	0.355662i	$0.115745\pi$
$138$	0	0
$139$	218.834	1.57434	0.787171	−	0.616735i	$-0.211545\pi$
$139$	218.834	1.57434	0.787171	+	0.616735i	$0.211545\pi$
$140$	0	0
$141$	− 3.42139i	− 0.0242652i
$142$	0	0
$143$	45.8402i	0.320561i
$144$	0	0
$145$	148.189	1.02200
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 23.8782i	− 0.160257i	−0.996785	−	0.0801284i	$-0.974467\pi$
$149$	− 23.8782i	− 0.160257i	0.996785	−	0.0801284i	$-0.0255330\pi$
$150$	0	0
$151$	− 183.501i	− 1.21524i	−0.794229	−	0.607619i	$-0.792125\pi$
$151$	− 183.501i	− 1.21524i	0.794229	−	0.607619i	$-0.207875\pi$
$152$	0	0
$153$	37.4044	0.244474
$154$	0	0
$155$	191.839	1.23767
$156$	0	0
$157$	− 165.934i	− 1.05691i	−0.848963	−	0.528453i	$-0.822772\pi$
$157$	− 165.934i	− 1.05691i	0.848963	−	0.528453i	$-0.177228\pi$
$158$	0	0
$159$	− 202.729i	− 1.27502i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.1043	0.0619896	0.0309948	−	0.999520i	$-0.490132\pi$
$163$	10.1043	0.0619896	0.0309948	+	0.999520i	$0.490132\pi$
$164$	0	0
$165$	− 499.184i	− 3.02535i
$166$	0	0
$167$	− 285.049i	− 1.70688i	−0.521190	−	0.853441i	$-0.674512\pi$
$167$	− 285.049i	− 1.70688i	0.521190	−	0.853441i	$-0.325488\pi$
$168$	0	0
$169$	160.588	0.950223
$170$	0	0
$171$	183.243	1.07159
$172$	0	0
$173$	97.3720i	0.562844i	0.959584	+	0.281422i	$0.0908060\pi$
$173$	97.3720i	0.562844i	−0.959584	+	0.281422i	$0.909194\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.7446	−0.0607040
$178$	0	0
$179$	301.816	1.68612	0.843061	−	0.537819i	$-0.180751\pi$
$179$	301.816	1.68612	0.843061	+	0.537819i	$0.180751\pi$
$180$	0	0
$181$	60.4535i	0.333997i	0.985957	+	0.166999i	$0.0534076\pi$
$181$	60.4535i	0.333997i	−0.985957	+	0.166999i	$0.946592\pi$
$182$	0	0
$183$	448.273i	2.44958i
$184$	0	0
$185$	123.793	0.669151
$186$	0	0
$187$	−52.3190	−0.279781
$188$	0	0
$189$	0	0
$190$	0	0
$191$	73.4123i	0.384358i	0.981360	+	0.192179i	$0.0615554\pi$
$191$	73.4123i	0.384358i	−0.981360	+	0.192179i	$0.938445\pi$
$192$	0	0
$193$	65.0893	0.337250	0.168625	−	0.985680i	$-0.446067\pi$
$193$	65.0893	0.337250	0.168625	+	0.985680i	$0.446067\pi$
$194$	0	0
$195$	91.6065	0.469777
$196$	0	0
$197$	348.324i	1.76814i	0.467355	+	0.884070i	$0.345207\pi$
$197$	348.324i	1.76814i	−0.467355	+	0.884070i	$0.654793\pi$
$198$	0	0
$199$	− 260.113i	− 1.30710i	−0.756883	−	0.653551i	$-0.773279\pi$
$199$	− 260.113i	− 1.30710i	0.756883	−	0.653551i	$-0.226721\pi$
$200$	0	0
$201$	−299.561	−1.49035
$202$	0	0
$203$	0	0
$204$	0	0
$205$	7.95527i	0.0388062i
$206$	0	0
$207$	215.026i	1.03877i
$208$	0	0
$209$	−256.308	−1.22636
$210$	0	0
$211$	−4.73025	−0.0224182	−0.0112091	−	0.999937i	$-0.503568\pi$
$211$	−4.73025	−0.0224182	−0.0112091	+	0.999937i	$0.503568\pi$
$212$	0	0
$213$	− 200.213i	− 0.939967i
$214$	0	0
$215$	351.970i	1.63707i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.74675	0.0353733
$220$	0	0
$221$	− 9.60120i	− 0.0434444i
$222$	0	0
$223$	109.315i	0.490200i	0.969498	+	0.245100i	$0.0788209\pi$
$223$	109.315i	0.490200i	−0.969498	+	0.245100i	$0.921179\pi$
$224$	0	0
$225$	−272.796	−1.21243
$226$	0	0
$227$	−121.441	−0.534982	−0.267491	−	0.963560i	$-0.586195\pi$
$227$	−121.441	−0.534982	−0.267491	+	0.963560i	$0.586195\pi$
$228$	0	0
$229$	− 144.197i	− 0.629680i	−0.949145	−	0.314840i	$-0.898049\pi$
$229$	− 144.197i	− 0.629680i	0.949145	−	0.314840i	$-0.101951\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	194.262	0.833742	0.416871	−	0.908966i	$-0.363127\pi$
$233$	194.262	0.833742	0.416871	+	0.908966i	$0.363127\pi$
$234$	0	0
$235$	5.32342	0.0226528
$236$	0	0
$237$	327.325i	1.38112i
$238$	0	0
$239$	− 315.567i	− 1.32036i	−0.751106	−	0.660181i	$-0.770479\pi$
$239$	− 315.567i	− 1.32036i	0.751106	−	0.660181i	$-0.229521\pi$
$240$	0	0
$241$	−362.711	−1.50503	−0.752513	−	0.658578i	$-0.771158\pi$
$241$	−362.711	−1.50503	−0.752513	+	0.658578i	$0.771158\pi$
$242$	0	0
$243$	341.123	1.40380
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 47.0358i	− 0.190429i
$248$	0	0
$249$	459.926	1.84709
$250$	0	0
$251$	−9.04237	−0.0360254	−0.0180127	−	0.999838i	$-0.505734\pi$
$251$	−9.04237	−0.0360254	−0.0180127	+	0.999838i	$0.505734\pi$
$252$	0	0
$253$	− 300.765i	− 1.18880i
$254$	0	0
$255$	104.554i	0.410015i
$256$	0	0
$257$	371.078	1.44388	0.721942	−	0.691954i	$-0.243250\pi$
$257$	371.078	1.44388	0.721942	+	0.691954i	$0.243250\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	238.859i	0.915169i
$262$	0	0
$263$	473.880i	1.80182i	0.434002	+	0.900912i	$0.357101\pi$
$263$	473.880i	1.80182i	−0.434002	+	0.900912i	$0.642899\pi$
$264$	0	0
$265$	315.431	1.19030
$266$	0	0
$267$	−276.515	−1.03563
$268$	0	0
$269$	− 94.1111i	− 0.349855i	−0.984581	−	0.174928i	$-0.944031\pi$
$269$	− 94.1111i	− 0.349855i	0.984581	−	0.174928i	$-0.0559692\pi$
$270$	0	0
$271$	− 158.219i	− 0.583832i	−0.956444	−	0.291916i	$-0.905707\pi$
$271$	− 158.219i	− 0.583832i	0.956444	−	0.291916i	$-0.0942928\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	381.570	1.38753
$276$	0	0
$277$	236.740i	0.854656i	0.904097	+	0.427328i	$0.140545\pi$
$277$	236.740i	0.854656i	−0.904097	+	0.427328i	$0.859455\pi$
$278$	0	0
$279$	309.216i	1.10830i
$280$	0	0
$281$	241.948	0.861025	0.430512	−	0.902585i	$-0.358333\pi$
$281$	241.948	0.861025	0.430512	+	0.902585i	$0.358333\pi$
$282$	0	0
$283$	−233.474	−0.824996	−0.412498	−	0.910958i	$-0.635344\pi$
$283$	−233.474	−0.824996	−0.412498	+	0.910958i	$0.635344\pi$
$284$	0	0
$285$	512.204i	1.79721i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−278.042	−0.962082
$290$	0	0
$291$	−463.280	−1.59203
$292$	0	0
$293$	− 216.492i	− 0.738882i	−0.929254	−	0.369441i	$-0.879549\pi$
$293$	− 216.492i	− 0.738882i	0.929254	−	0.369441i	$-0.120451\pi$
$294$	0	0
$295$	− 16.7178i	− 0.0566704i
$296$	0	0
$297$	163.733	0.551291
$298$	0	0
$299$	55.1943	0.184596
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 820.175i	− 2.70685i
$304$	0	0
$305$	−697.479	−2.28682
$306$	0	0
$307$	173.487	0.565106	0.282553	−	0.959252i	$-0.408819\pi$
$307$	173.487	0.565106	0.282553	+	0.959252i	$0.408819\pi$
$308$	0	0
$309$	206.593i	0.668585i
$310$	0	0
$311$	115.707i	0.372049i	0.982545	+	0.186024i	$0.0595604\pi$
$311$	115.707i	0.372049i	−0.982545	+	0.186024i	$0.940440\pi$
$312$	0	0
$313$	−210.522	−0.672593	−0.336297	−	0.941756i	$-0.609174\pi$
$313$	−210.522	−0.672593	−0.336297	+	0.941756i	$0.609174\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	349.871i	1.10369i	0.833945	+	0.551847i	$0.186077\pi$
$317$	349.871i	1.10369i	−0.833945	+	0.551847i	$0.813923\pi$
$318$	0	0
$319$	− 334.101i	− 1.04734i
$320$	0	0
$321$	−332.921	−1.03714
$322$	0	0
$323$	53.6837	0.166203
$324$	0	0
$325$	70.0230i	0.215455i
$326$	0	0
$327$	− 544.822i	− 1.66612i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−214.110	−0.646859	−0.323430	−	0.946252i	$-0.604836\pi$
$331$	−214.110	−0.646859	−0.323430	+	0.946252i	$0.604836\pi$
$332$	0	0
$333$	199.535i	0.599206i
$334$	0	0
$335$	− 466.093i	− 1.39132i
$336$	0	0
$337$	−437.275	−1.29755	−0.648777	−	0.760979i	$-0.724719\pi$
$337$	−437.275	−1.29755	−0.648777	+	0.760979i	$0.724719\pi$
$338$	0	0
$339$	440.484	1.29936
$340$	0	0
$341$	− 432.512i	− 1.26836i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−601.047	−1.74216
$346$	0	0
$347$	−377.388	−1.08757	−0.543786	−	0.839224i	$-0.683010\pi$
$347$	−377.388	−1.08757	−0.543786	+	0.839224i	$0.683010\pi$
$348$	0	0
$349$	− 211.146i	− 0.605003i	−0.953149	−	0.302502i	$-0.902178\pi$
$349$	− 211.146i	− 0.605003i	0.953149	−	0.302502i	$-0.0978218\pi$
$350$	0	0
$351$	30.0472i	0.0856044i
$352$	0	0
$353$	−298.750	−0.846319	−0.423159	−	0.906055i	$-0.639079\pi$
$353$	−298.750	−0.846319	−0.423159	+	0.906055i	$0.639079\pi$
$354$	0	0
$355$	311.516	0.877510
$356$	0	0
$357$	0	0
$358$	0	0
$359$	220.845i	0.615168i	0.951521	+	0.307584i	$0.0995205\pi$
$359$	220.845i	0.615168i	−0.951521	+	0.307584i	$0.900479\pi$
$360$	0	0
$361$	−98.0061	−0.271485
$362$	0	0
$363$	−580.274	−1.59855
$364$	0	0
$365$	12.0533i	0.0330229i
$366$	0	0
$367$	215.348i	0.586778i	0.955993	+	0.293389i	$0.0947831\pi$
$367$	215.348i	0.586778i	−0.955993	+	0.293389i	$0.905217\pi$
$368$	0	0
$369$	−12.8227	−0.0347499
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 223.878i	− 0.600209i	−0.953906	−	0.300104i	$-0.902978\pi$
$373$	− 223.878i	− 0.600209i	0.953906	−	0.300104i	$-0.0970215\pi$
$374$	0	0
$375$	27.0798i	0.0722128i
$376$	0	0
$377$	61.3118	0.162631
$378$	0	0
$379$	507.051	1.33787	0.668933	−	0.743323i	$-0.266751\pi$
$379$	507.051	1.33787	0.668933	+	0.743323i	$0.266751\pi$
$380$	0	0
$381$	788.687i	2.07004i
$382$	0	0
$383$	− 148.588i	− 0.387958i	−0.981006	−	0.193979i	$-0.937861\pi$
$383$	− 148.588i	− 0.387958i	0.981006	−	0.193979i	$-0.0621393\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−567.322	−1.46595
$388$	0	0
$389$	− 158.317i	− 0.406985i	−0.979076	−	0.203493i	$-0.934771\pi$
$389$	− 158.317i	− 0.406985i	0.979076	−	0.203493i	$-0.0652293\pi$
$390$	0	0
$391$	62.9952i	0.161113i
$392$	0	0
$393$	38.8366	0.0988208
$394$	0	0
$395$	−509.292	−1.28935
$396$	0	0
$397$	− 277.742i	− 0.699601i	−0.936824	−	0.349801i	$-0.886249\pi$
$397$	− 277.742i	− 0.699601i	0.936824	−	0.349801i	$-0.113751\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	623.798	1.55561	0.777803	−	0.628508i	$-0.216334\pi$
$401$	623.798	1.55561	0.777803	+	0.628508i	$0.216334\pi$
$402$	0	0
$403$	79.3714	0.196951
$404$	0	0
$405$	385.691i	0.952324i
$406$	0	0
$407$	− 279.098i	− 0.685744i
$408$	0	0
$409$	495.137	1.21060	0.605302	−	0.795996i	$-0.293052\pi$
$409$	495.137	1.21060	0.605302	+	0.795996i	$0.293052\pi$
$410$	0	0
$411$	−1153.78	−2.80726
$412$	0	0
$413$	0	0
$414$	0	0
$415$	715.609i	1.72436i
$416$	0	0
$417$	−985.950	−2.36439
$418$	0	0
$419$	112.631	0.268810	0.134405	−	0.990927i	$-0.457088\pi$
$419$	112.631	0.268810	0.134405	+	0.990927i	$0.457088\pi$
$420$	0	0
$421$	− 821.936i	− 1.95234i	−0.217003	−	0.976171i	$-0.569628\pi$
$421$	− 821.936i	− 1.95234i	0.217003	−	0.976171i	$-0.430372\pi$
$422$	0	0
$423$	8.58055i	0.0202850i
$424$	0	0
$425$	−79.9197	−0.188046
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 206.532i	− 0.481426i
$430$	0	0
$431$	333.809i	0.774498i	0.921975	+	0.387249i	$0.126575\pi$
$431$	333.809i	0.774498i	−0.921975	+	0.387249i	$0.873425\pi$
$432$	0	0
$433$	−604.681	−1.39649	−0.698246	−	0.715858i	$-0.746036\pi$
$433$	−604.681	−1.39649	−0.698246	+	0.715858i	$0.746036\pi$
$434$	0	0
$435$	−667.664	−1.53486
$436$	0	0
$437$	308.610i	0.706202i
$438$	0	0
$439$	481.114i	1.09593i	0.836501	+	0.547966i	$0.184598\pi$
$439$	481.114i	1.09593i	−0.836501	+	0.547966i	$0.815402\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−184.720	−0.416975	−0.208487	−	0.978025i	$-0.566854\pi$
$443$	−184.720	−0.416975	−0.208487	+	0.978025i	$0.566854\pi$
$444$	0	0
$445$	− 430.235i	− 0.966821i
$446$	0	0
$447$	107.583i	0.240678i
$448$	0	0
$449$	115.894	0.258115	0.129057	−	0.991637i	$-0.458805\pi$
$449$	115.894	0.258115	0.129057	+	0.991637i	$0.458805\pi$
$450$	0	0
$451$	17.9356	0.0397685
$452$	0	0
$453$	826.759i	1.82508i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−158.462	−0.346745	−0.173372	−	0.984856i	$-0.555466\pi$
$457$	−158.462	−0.346745	−0.173372	+	0.984856i	$0.555466\pi$
$458$	0	0
$459$	−34.2939	−0.0747144
$460$	0	0
$461$	42.3829i	0.0919368i	0.998943	+	0.0459684i	$0.0146374\pi$
$461$	42.3829i	0.0919368i	−0.998943	+	0.0459684i	$0.985363\pi$
$462$	0	0
$463$	390.038i	0.842416i	0.906964	+	0.421208i	$0.138394\pi$
$463$	390.038i	0.842416i	−0.906964	+	0.421208i	$0.861606\pi$
$464$	0	0
$465$	−864.328	−1.85877
$466$	0	0
$467$	38.3445	0.0821082	0.0410541	−	0.999157i	$-0.486928\pi$
$467$	38.3445	0.0821082	0.0410541	+	0.999157i	$0.486928\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	747.613i	1.58729i
$472$	0	0
$473$	793.535	1.67766
$474$	0	0
$475$	−391.523	−0.824259
$476$	0	0
$477$	508.427i	1.06588i
$478$	0	0
$479$	648.164i	1.35316i	0.736369	+	0.676580i	$0.236539\pi$
$479$	648.164i	1.35316i	−0.736369	+	0.676580i	$0.763461\pi$
$480$	0	0
$481$	51.2180	0.106482
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 720.828i	− 1.48624i
$486$	0	0
$487$	− 350.225i	− 0.719148i	−0.933117	−	0.359574i	$-0.882922\pi$
$487$	− 350.225i	− 0.719148i	0.933117	−	0.359574i	$-0.117078\pi$
$488$	0	0
$489$	−45.5248	−0.0930977
$490$	0	0
$491$	453.547	0.923722	0.461861	−	0.886952i	$-0.347182\pi$
$491$	453.547	0.923722	0.461861	+	0.886952i	$0.347182\pi$
$492$	0	0
$493$	69.9774i	0.141942i
$494$	0	0
$495$	1251.91i	2.52911i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	529.565	1.06125	0.530626	−	0.847606i	$-0.321957\pi$
$499$	529.565	1.06125	0.530626	+	0.847606i	$0.321957\pi$
$500$	0	0
$501$	1284.28i	2.56344i
$502$	0	0
$503$	321.597i	0.639358i	0.947526	+	0.319679i	$0.103575\pi$
$503$	321.597i	0.639358i	−0.947526	+	0.319679i	$0.896425\pi$
$504$	0	0
$505$	1276.13	2.52699
$506$	0	0
$507$	−723.525	−1.42707
$508$	0	0
$509$	− 719.914i	− 1.41437i	−0.707029	−	0.707185i	$-0.749965\pi$
$509$	− 719.914i	− 1.41437i	0.707029	−	0.707185i	$-0.250035\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−168.004	−0.327494
$514$	0	0
$515$	−321.442	−0.624160
$516$	0	0
$517$	− 12.0019i	− 0.0232146i
$518$	0	0
$519$	− 438.708i	− 0.845294i
$520$	0	0
$521$	−413.980	−0.794588	−0.397294	−	0.917691i	$-0.630051\pi$
$521$	−413.980	−0.794588	−0.397294	+	0.917691i	$0.630051\pi$
$522$	0	0
$523$	−348.649	−0.666632	−0.333316	−	0.942815i	$-0.608168\pi$
$523$	−348.649	−0.666632	−0.333316	+	0.942815i	$0.608168\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	90.5895i	0.171897i
$528$	0	0
$529$	166.861	0.315427
$530$	0	0
$531$	26.9466	0.0507468
$532$	0	0
$533$	3.29141i	0.00617525i
$534$	0	0
$535$	− 518.000i	− 0.968224i
$536$	0	0
$537$	−1359.82	−2.53226
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 234.220i	− 0.432939i	−0.976289	−	0.216469i	$-0.930546\pi$
$541$	− 234.220i	− 0.432939i	0.976289	−	0.216469i	$-0.0694541\pi$
$542$	0	0
$543$	− 272.372i	− 0.501606i
$544$	0	0
$545$	847.701	1.55542
$546$	0	0
$547$	577.082	1.05499	0.527497	−	0.849557i	$-0.323130\pi$
$547$	577.082	1.05499	0.527497	+	0.849557i	$0.323130\pi$
$548$	0	0
$549$	− 1124.23i	− 2.04778i
$550$	0	0
$551$	342.816i	0.622170i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−557.746	−1.00495
$556$	0	0
$557$	− 42.4066i	− 0.0761339i	−0.999275	−	0.0380670i	$-0.987880\pi$
$557$	− 42.4066i	− 0.0761339i	0.999275	−	0.0380670i	$-0.0121200\pi$
$558$	0	0
$559$	145.624i	0.260508i
$560$	0	0
$561$	235.722	0.420182
$562$	0	0
$563$	−247.296	−0.439247	−0.219624	−	0.975585i	$-0.570483\pi$
$563$	−247.296	−0.439247	−0.219624	+	0.975585i	$0.570483\pi$
$564$	0	0
$565$	685.359i	1.21303i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	625.474	1.09925	0.549626	−	0.835411i	$-0.314770\pi$
$569$	625.474	1.09925	0.549626	+	0.835411i	$0.314770\pi$
$570$	0	0
$571$	582.574	1.02027	0.510135	−	0.860095i	$-0.329596\pi$
$571$	582.574	1.02027	0.510135	+	0.860095i	$0.329596\pi$
$572$	0	0
$573$	− 330.758i	− 0.577239i
$574$	0	0
$575$	− 459.433i	− 0.799014i
$576$	0	0
$577$	396.761	0.687627	0.343814	−	0.939038i	$-0.388281\pi$
$577$	396.761	0.687627	0.343814	+	0.939038i	$0.388281\pi$
$578$	0	0
$579$	−293.259	−0.506491
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 711.156i	− 1.21982i
$584$	0	0
$585$	−229.741	−0.392720
$586$	0	0
$587$	355.763	0.606070	0.303035	−	0.952979i	$-0.402000\pi$
$587$	355.763	0.606070	0.303035	+	0.952979i	$0.402000\pi$
$588$	0	0
$589$	443.794i	0.753470i
$590$	0	0
$591$	− 1569.36i	− 2.65544i
$592$	0	0
$593$	179.722	0.303073	0.151536	−	0.988452i	$-0.451578\pi$
$593$	179.722	0.303073	0.151536	+	0.988452i	$0.451578\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1171.93i	1.96304i
$598$	0	0
$599$	40.6812i	0.0679152i	0.999423	+	0.0339576i	$0.0108111\pi$
$599$	40.6812i	0.0679152i	−0.999423	+	0.0339576i	$0.989189\pi$
$600$	0	0
$601$	1020.79	1.69849	0.849245	−	0.527998i	$-0.177057\pi$
$601$	1020.79	1.69849	0.849245	+	0.527998i	$0.177057\pi$
$602$	0	0
$603$	751.272	1.24589
$604$	0	0
$605$	− 902.861i	− 1.49233i
$606$	0	0
$607$	− 315.487i	− 0.519749i	−0.965642	−	0.259874i	$-0.916319\pi$
$607$	− 315.487i	− 0.519749i	0.965642	−	0.259874i	$-0.0836811\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.20251	0.00360476
$612$	0	0
$613$	413.864i	0.675145i	0.941300	+	0.337572i	$0.109606\pi$
$613$	413.864i	0.675145i	−0.941300	+	0.337572i	$0.890394\pi$
$614$	0	0
$615$	− 35.8423i	− 0.0582802i
$616$	0	0
$617$	139.167	0.225554	0.112777	−	0.993620i	$-0.464025\pi$
$617$	139.167	0.225554	0.112777	+	0.993620i	$0.464025\pi$
$618$	0	0
$619$	334.013	0.539601	0.269800	−	0.962916i	$-0.413042\pi$
$619$	334.013	0.539601	0.269800	+	0.962916i	$0.413042\pi$
$620$	0	0
$621$	− 197.145i	− 0.317463i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−645.699	−1.03312
$626$	0	0
$627$	1154.79	1.84177
$628$	0	0
$629$	58.4569i	0.0929363i
$630$	0	0
$631$	284.343i	0.450623i	0.974287	+	0.225311i	$0.0723399\pi$
$631$	284.343i	0.450623i	−0.974287	+	0.225311i	$0.927660\pi$
$632$	0	0
$633$	21.3120	0.0336683
$634$	0	0
$635$	−1227.14	−1.93250
$636$	0	0
$637$	0	0
$638$	0	0
$639$	502.117i	0.785785i
$640$	0	0
$641$	64.9089	0.101262	0.0506310	−	0.998717i	$-0.483877\pi$
$641$	64.9089	0.101262	0.0506310	+	0.998717i	$0.483877\pi$
$642$	0	0
$643$	683.365	1.06278	0.531388	−	0.847128i	$-0.321671\pi$
$643$	683.365	1.06278	0.531388	+	0.847128i	$0.321671\pi$
$644$	0	0
$645$	− 1585.79i	− 2.45859i
$646$	0	0
$647$	1274.34i	1.96962i	0.173644	+	0.984808i	$0.444446\pi$
$647$	1274.34i	1.96962i	−0.173644	+	0.984808i	$0.555554\pi$
$648$	0	0
$649$	−37.6912	−0.0580757
$650$	0	0
$651$	0	0
$652$	0	0
$653$	546.047i	0.836214i	0.908398	+	0.418107i	$0.137306\pi$
$653$	546.047i	0.836214i	−0.908398	+	0.418107i	$0.862694\pi$
$654$	0	0
$655$	60.4267i	0.0922545i
$656$	0	0
$657$	−19.4282	−0.0295710
$658$	0	0
$659$	398.883	0.605285	0.302642	−	0.953104i	$-0.402131\pi$
$659$	398.883	0.605285	0.302642	+	0.953104i	$0.402131\pi$
$660$	0	0
$661$	829.243i	1.25453i	0.778806	+	0.627264i	$0.215825\pi$
$661$	829.243i	1.25453i	−0.778806	+	0.627264i	$0.784175\pi$
$662$	0	0
$663$	43.2580i	0.0652459i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−402.278	−0.603115
$668$	0	0
$669$	− 492.515i	− 0.736196i
$670$	0	0
$671$	1572.50i	2.34352i
$672$	0	0
$673$	946.218	1.40597	0.702985	−	0.711205i	$-0.251850\pi$
$673$	946.218	1.40597	0.702985	+	0.711205i	$0.251850\pi$
$674$	0	0
$675$	250.110	0.370534
$676$	0	0
$677$	1140.61i	1.68479i	0.538858	+	0.842397i	$0.318856\pi$
$677$	1140.61i	1.68479i	−0.538858	+	0.842397i	$0.681144\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	547.150	0.803451
$682$	0	0
$683$	−117.408	−0.171901	−0.0859504	−	0.996299i	$-0.527393\pi$
$683$	−117.408	−0.171901	−0.0859504	+	0.996299i	$0.527393\pi$
$684$	0	0
$685$	− 1795.20i	− 2.62073i
$686$	0	0
$687$	649.675i	0.945670i
$688$	0	0
$689$	130.506	0.189414
$690$	0	0
$691$	955.922	1.38339	0.691695	−	0.722190i	$-0.256864\pi$
$691$	955.922	1.38339	0.691695	+	0.722190i	$0.256864\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1534.06i	− 2.20728i
$696$	0	0
$697$	−3.75660	−0.00538967
$698$	0	0
$699$	−875.243	−1.25214
$700$	0	0
$701$	− 824.950i	− 1.17682i	−0.808563	−	0.588410i	$-0.799754\pi$
$701$	− 824.950i	− 1.17682i	0.808563	−	0.588410i	$-0.200246\pi$
$702$	0	0
$703$	286.378i	0.407365i
$704$	0	0
$705$	−23.9846	−0.0340206
$706$	0	0
$707$	0	0
$708$	0	0
$709$	129.992i	0.183345i	0.995789	+	0.0916726i	$0.0292213\pi$
$709$	129.992i	0.183345i	−0.995789	+	0.0916726i	$0.970779\pi$
$710$	0	0
$711$	− 820.902i	− 1.15457i
$712$	0	0
$713$	−520.770	−0.730393
$714$	0	0
$715$	321.348	0.449437
$716$	0	0
$717$	1421.78i	1.98296i
$718$	0	0
$719$	506.479i	0.704421i	0.935921	+	0.352210i	$0.114570\pi$
$719$	506.479i	0.704421i	−0.935921	+	0.352210i	$0.885430\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1634.19	2.26029
$724$	0	0
$725$	− 510.355i	− 0.703938i
$726$	0	0
$727$	− 1272.41i	− 1.75022i	−0.483924	−	0.875110i	$-0.660789\pi$
$727$	− 1272.41i	− 1.75022i	0.483924	−	0.875110i	$-0.339211\pi$
$728$	0	0
$729$	−1041.76	−1.42902
$730$	0	0
$731$	−166.206	−0.227368
$732$	0	0
$733$	710.689i	0.969563i	0.874635	+	0.484781i	$0.161101\pi$
$733$	710.689i	0.969563i	−0.874635	+	0.484781i	$0.838899\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1050.83	−1.42583
$738$	0	0
$739$	211.019	0.285547	0.142773	−	0.989755i	$-0.454398\pi$
$739$	211.019	0.285547	0.142773	+	0.989755i	$0.454398\pi$
$740$	0	0
$741$	211.919i	0.285991i
$742$	0	0
$743$	− 244.355i	− 0.328876i	−0.986387	−	0.164438i	$-0.947419\pi$
$743$	− 244.355i	− 0.328876i	0.986387	−	0.164438i	$-0.0525811\pi$
$744$	0	0
$745$	−167.391	−0.224686
$746$	0	0
$747$	−1153.45	−1.54411
$748$	0	0
$749$	0	0
$750$	0	0
$751$	303.383i	0.403972i	0.979388	+	0.201986i	$0.0647396\pi$
$751$	303.383i	0.403972i	−0.979388	+	0.201986i	$0.935260\pi$
$752$	0	0
$753$	40.7402	0.0541039
$754$	0	0
$755$	−1286.37	−1.70381
$756$	0	0
$757$	668.225i	0.882728i	0.897328	+	0.441364i	$0.145505\pi$
$757$	668.225i	0.882728i	−0.897328	+	0.441364i	$0.854495\pi$
$758$	0	0
$759$	1355.09i	1.78537i
$760$	0	0
$761$	−90.8305	−0.119357	−0.0596784	−	0.998218i	$-0.519008\pi$
$761$	−90.8305	−0.119357	−0.0596784	+	0.998218i	$0.519008\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 262.212i	− 0.342761i
$766$	0	0
$767$	− 6.91681i	− 0.00901800i
$768$	0	0
$769$	−421.091	−0.547582	−0.273791	−	0.961789i	$-0.588278\pi$
$769$	−421.091	−0.547582	−0.273791	+	0.961789i	$0.588278\pi$
$770$	0	0
$771$	−1671.88	−2.16846
$772$	0	0
$773$	− 586.703i	− 0.758995i	−0.925193	−	0.379498i	$-0.876097\pi$
$773$	− 586.703i	− 0.758995i	0.925193	−	0.379498i	$-0.123903\pi$
$774$	0	0
$775$	− 660.682i	− 0.852493i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−18.4034	−0.0236244
$780$	0	0
$781$	− 702.330i	− 0.899270i
$782$	0	0
$783$	− 218.996i	− 0.279688i
$784$	0	0
$785$	−1163.23	−1.48182
$786$	0	0
$787$	1362.27	1.73097	0.865485	−	0.500935i	$-0.167010\pi$
$787$	1362.27	1.73097	0.865485	+	0.500935i	$0.167010\pi$
$788$	0	0
$789$	− 2135.06i	− 2.70603i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−288.575	−0.363902
$794$	0	0
$795$	−1421.17	−1.78763
$796$	0	0
$797$	874.598i	1.09736i	0.836032	+	0.548681i	$0.184870\pi$
$797$	874.598i	1.09736i	−0.836032	+	0.548681i	$0.815130\pi$
$798$	0	0
$799$	2.51380i	0.00314618i
$800$	0	0
$801$	693.474	0.865761
$802$	0	0
$803$	27.1749	0.0338417
$804$	0	0
$805$	0	0
$806$	0	0
$807$	424.016i	0.525422i
$808$	0	0
$809$	−55.8830	−0.0690767	−0.0345383	−	0.999403i	$-0.510996\pi$
$809$	−55.8830	−0.0690767	−0.0345383	+	0.999403i	$0.510996\pi$
$810$	0	0
$811$	917.924	1.13184	0.565921	−	0.824459i	$-0.308521\pi$
$811$	917.924	1.13184	0.565921	+	0.824459i	$0.308521\pi$
$812$	0	0
$813$	712.850i	0.876815i
$814$	0	0
$815$	− 70.8330i	− 0.0869117i
$816$	0	0
$817$	−814.234	−0.996614
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1360.35i	− 1.65694i	−0.560030	−	0.828472i	$-0.689211\pi$
$821$	− 1360.35i	− 1.65694i	0.560030	−	0.828472i	$-0.310789\pi$
$822$	0	0
$823$	563.560i	0.684763i	0.939561	+	0.342382i	$0.111234\pi$
$823$	563.560i	0.684763i	−0.939561	+	0.342382i	$0.888766\pi$
$824$	0	0
$825$	−1719.16	−2.08383
$826$	0	0
$827$	−63.4180	−0.0766844	−0.0383422	−	0.999265i	$-0.512208\pi$
$827$	−63.4180	−0.0766844	−0.0383422	+	0.999265i	$0.512208\pi$
$828$	0	0
$829$	− 291.700i	− 0.351870i	−0.984402	−	0.175935i	$-0.943705\pi$
$829$	− 291.700i	− 0.351870i	0.984402	−	0.175935i	$-0.0562949\pi$
$830$	0	0
$831$	− 1066.63i	− 1.28354i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1998.25	−2.39311
$836$	0	0
$837$	− 283.502i	− 0.338712i
$838$	0	0
$839$	463.917i	0.552940i	0.961023	+	0.276470i	$0.0891646\pi$
$839$	463.917i	0.552940i	−0.961023	+	0.276470i	$0.910835\pi$
$840$	0	0
$841$	394.135	0.468651
$842$	0	0
$843$	−1090.09	−1.29311
$844$	0	0
$845$	− 1125.75i	− 1.33225i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1051.91	1.23900
$850$	0	0
$851$	−336.050	−0.394889
$852$	0	0
$853$	649.909i	0.761909i	0.924594	+	0.380955i	$0.124405\pi$
$853$	649.909i	0.761909i	−0.924594	+	0.380955i	$0.875595\pi$
$854$	0	0
$855$	− 1284.56i	− 1.50241i
$856$	0	0
$857$	−1168.28	−1.36322	−0.681610	−	0.731716i	$-0.738720\pi$
$857$	−1168.28	−1.36322	−0.681610	+	0.731716i	$0.738720\pi$
$858$	0	0
$859$	1466.42	1.70712	0.853560	−	0.520994i	$-0.174439\pi$
$859$	1466.42	1.70712	0.853560	+	0.520994i	$0.174439\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42.7148i	0.0494957i	0.999694	+	0.0247479i	$0.00787830\pi$
$863$	42.7148i	0.0494957i	−0.999694	+	0.0247479i	$0.992122\pi$
$864$	0	0
$865$	682.595	0.789127
$866$	0	0
$867$	1252.71	1.44488
$868$	0	0
$869$	1148.23i	1.32132i
$870$	0	0
$871$	− 192.841i	− 0.221402i
$872$	0	0
$873$	1161.87	1.33089
$874$	0	0
$875$	0	0
$876$	0	0
$877$	972.508i	1.10890i	0.832216	+	0.554452i	$0.187072\pi$
$877$	972.508i	1.10890i	−0.832216	+	0.554452i	$0.812928\pi$
$878$	0	0
$879$	975.402i	1.10967i
$880$	0	0
$881$	110.722	0.125678	0.0628390	−	0.998024i	$-0.479985\pi$
$881$	110.722	0.125678	0.0628390	+	0.998024i	$0.479985\pi$
$882$	0	0
$883$	−1283.66	−1.45375	−0.726874	−	0.686770i	$-0.759028\pi$
$883$	−1283.66	−1.45375	−0.726874	+	0.686770i	$0.759028\pi$
$884$	0	0
$885$	75.3216i	0.0851092i
$886$	0	0
$887$	− 496.238i	− 0.559457i	−0.960079	−	0.279729i	$-0.909756\pi$
$887$	− 496.238i	− 0.559457i	0.960079	−	0.279729i	$-0.0902445\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	869.563	0.975940
$892$	0	0
$893$	12.3150i	0.0137906i
$894$	0	0
$895$	− 2115.78i	− 2.36400i
$896$	0	0
$897$	−248.677	−0.277232
$898$	0	0
$899$	−578.490	−0.643482
$900$	0	0
$901$	148.951i	0.165318i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	423.790	0.468276
$906$	0	0
$907$	−1592.26	−1.75553	−0.877763	−	0.479096i	$-0.840965\pi$
$907$	−1592.26	−1.75553	−0.877763	+	0.479096i	$0.840965\pi$
$908$	0	0
$909$	2056.93i	2.26285i
$910$	0	0
$911$	663.197i	0.727988i	0.931401	+	0.363994i	$0.118587\pi$
$911$	663.197i	0.727988i	−0.931401	+	0.363994i	$0.881413\pi$
$912$	0	0
$913$	1613.38	1.76712
$914$	0	0
$915$	3142.48	3.43440
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 509.651i	− 0.554571i	−0.960788	−	0.277286i	$-0.910565\pi$
$919$	− 509.651i	− 0.554571i	0.960788	−	0.277286i	$-0.0894349\pi$
$920$	0	0
$921$	−781.644	−0.848691
$922$	0	0
$923$	128.886	0.139639
$924$	0	0
$925$	− 426.335i	− 0.460902i
$926$	0	0
$927$	− 518.117i	− 0.558918i
$928$	0	0
$929$	322.286	0.346917	0.173459	−	0.984841i	$-0.444506\pi$
$929$	322.286	0.346917	0.173459	+	0.984841i	$0.444506\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 521.317i	− 0.558753i
$934$	0	0
$935$	366.766i	0.392263i
$936$	0	0
$937$	1077.79	1.15026	0.575130	−	0.818062i	$-0.304952\pi$
$937$	1077.79	1.15026	0.575130	+	0.818062i	$0.304952\pi$
$938$	0	0
$939$	948.502	1.01012
$940$	0	0
$941$	73.4336i	0.0780378i	0.999238	+	0.0390189i	$0.0124233\pi$
$941$	73.4336i	0.0780378i	−0.999238	+	0.0390189i	$0.987577\pi$
$942$	0	0
$943$	− 21.5955i	− 0.0229009i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−76.8154	−0.0811145	−0.0405572	−	0.999177i	$-0.512913\pi$
$947$	−76.8154	−0.0811145	−0.0405572	+	0.999177i	$0.512913\pi$
$948$	0	0
$949$	4.98694i	0.00525495i
$950$	0	0
$951$	− 1576.34i	− 1.65756i
$952$	0	0
$953$	−1329.39	−1.39495	−0.697477	−	0.716607i	$-0.745694\pi$
$953$	−1329.39	−1.39495	−0.697477	+	0.716607i	$0.745694\pi$
$954$	0	0
$955$	514.634	0.538883
$956$	0	0
$957$	1505.29i	1.57292i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	212.113	0.220721
$962$	0	0
$963$	834.938	0.867018
$964$	0	0
$965$	− 456.288i	− 0.472837i
$966$	0	0
$967$	− 601.743i	− 0.622279i	−0.950364	−	0.311139i	$-0.899289\pi$
$967$	− 601.743i	− 0.622279i	0.950364	−	0.311139i	$-0.100711\pi$
$968$	0	0
$969$	−241.871	−0.249609
$970$	0	0
$971$	426.278	0.439009	0.219505	−	0.975611i	$-0.429556\pi$
$971$	426.278	0.439009	0.219505	+	0.975611i	$0.429556\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 315.487i	− 0.323576i
$976$	0	0
$977$	219.857	0.225032	0.112516	−	0.993650i	$-0.464109\pi$
$977$	219.857	0.225032	0.112516	+	0.993650i	$0.464109\pi$
$978$	0	0
$979$	−969.989	−0.990796
$980$	0	0
$981$	1366.37i	1.39283i
$982$	0	0
$983$	− 1668.22i	− 1.69707i	−0.529136	−	0.848537i	$-0.677484\pi$
$983$	− 1668.22i	− 1.69707i	0.529136	−	0.848537i	$-0.322516\pi$
$984$	0	0
$985$	2441.81	2.47900
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 955.463i	− 0.966090i
$990$	0	0
$991$	1367.39i	1.37981i	0.723899	+	0.689906i	$0.242348\pi$
$991$	1367.39i	1.37981i	−0.723899	+	0.689906i	$0.757652\pi$
$992$	0	0
$993$	964.670	0.971471
$994$	0	0
$995$	−1823.44	−1.83260
$996$	0	0
$997$	− 927.580i	− 0.930371i	−0.885213	−	0.465186i	$-0.845988\pi$
$997$	− 927.580i	− 0.930371i	0.885213	−	0.465186i	$-0.154012\pi$
$998$	0	0
$999$	− 182.942i	− 0.183125i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.g.i.687.1		6
4.3	odd	2		392.3.g.l.99.1		6
7.3	odd	6		224.3.o.c.79.1		12
7.5	odd	6		224.3.o.c.207.2		12
7.6	odd	2		1568.3.g.k.687.6		6
8.3	odd	2	inner	1568.3.g.i.687.2		6
8.5	even	2		392.3.g.l.99.2		6
28.3	even	6		56.3.k.c.51.5	yes	12
28.11	odd	6		392.3.k.k.275.5		12
28.19	even	6		56.3.k.c.11.4	✓	12
28.23	odd	6		392.3.k.k.67.4		12
28.27	even	2		392.3.g.k.99.1		6
56.3	even	6		224.3.o.c.79.2		12
56.5	odd	6		56.3.k.c.11.5	yes	12
56.13	odd	2		392.3.g.k.99.2		6
56.19	even	6		224.3.o.c.207.1		12
56.27	even	2		1568.3.g.k.687.5		6
56.37	even	6		392.3.k.k.67.5		12
56.45	odd	6		56.3.k.c.51.4	yes	12
56.53	even	6		392.3.k.k.275.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.k.c.11.4	✓	12	28.19	even	6
56.3.k.c.11.5	yes	12	56.5	odd	6
56.3.k.c.51.4	yes	12	56.45	odd	6
56.3.k.c.51.5	yes	12	28.3	even	6
224.3.o.c.79.1		12	7.3	odd	6
224.3.o.c.79.2		12	56.3	even	6
224.3.o.c.207.1		12	56.19	even	6
224.3.o.c.207.2		12	7.5	odd	6
392.3.g.k.99.1		6	28.27	even	2
392.3.g.k.99.2		6	56.13	odd	2
392.3.g.l.99.1		6	4.3	odd	2
392.3.g.l.99.2		6	8.5	even	2
392.3.k.k.67.4		12	28.23	odd	6
392.3.k.k.67.5		12	56.37	even	6
392.3.k.k.275.4		12	56.53	even	6
392.3.k.k.275.5		12	28.11	odd	6
1568.3.g.i.687.1		6	1.1	even	1	trivial
1568.3.g.i.687.2		6	8.3	odd	2	inner
1568.3.g.k.687.5		6	56.27	even	2
1568.3.g.k.687.6		6	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.50548 q^{3} -7.01018i q^{5} +11.2994 q^{9} +O(q^{10})\)	\(q-4.50548 q^{3} -7.01018i q^{5} +11.2994 q^{9} -15.8048 q^{11} -2.90039i q^{13} +31.5842i q^{15} +3.31032 q^{17} +16.2171 q^{19} +19.0300i q^{23} -24.1426 q^{25} -10.3597 q^{27} +21.1392i q^{29} +27.3658i q^{31} +71.2084 q^{33} +17.6590i q^{37} +13.0676i q^{39} -1.13482 q^{41} -50.2084 q^{43} -79.2105i q^{45} +0.759384i q^{47} -14.9146 q^{51} +44.9961i q^{53} +110.795i q^{55} -73.0658 q^{57} +2.38479 q^{59} -99.4952i q^{61} -20.3322 q^{65} +66.4881 q^{67} -85.7391i q^{69} +44.4376i q^{71} -1.71941 q^{73} +108.774 q^{75} -72.6504i q^{79} -55.0188 q^{81} -102.081 q^{83} -23.2059i q^{85} -95.2421i q^{87} +61.3729 q^{89} -123.296i q^{93} -113.685i q^{95} +102.826 q^{97} -178.584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 8 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 8 * q^9	\( 6 q - 6 q^{3} + 8 q^{9} - 14 q^{11} - 82 q^{17} + 94 q^{19} - 116 q^{25} - 30 q^{27} + 146 q^{33} - 60 q^{41} - 20 q^{43} - 106 q^{51} - 186 q^{57} - 62 q^{59} + 64 q^{65} - 178 q^{67} + 54 q^{73} - 140 q^{75} - 206 q^{81} - 196 q^{83} - 26 q^{89} + 92 q^{97} - 436 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 8 * q^9 - 14 * q^11 - 82 * q^17 + 94 * q^19 - 116 * q^25 - 30 * q^27 + 146 * q^33 - 60 * q^41 - 20 * q^43 - 106 * q^51 - 186 * q^57 - 62 * q^59 + 64 * q^65 - 178 * q^67 + 54 * q^73 - 140 * q^75 - 206 * q^81 - 196 * q^83 - 26 * q^89 + 92 * q^97 - 436 * q^99

Introduction

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