LMFDB - Embedded newform 1568.3.c.g.97.3

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			97.3
Root			$2.08703 - 2.02145i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.g.97.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.98546i q^{3} -9.01776i q^{5} -6.88390 q^{9} +O(q^{10})$	$q-3.98546i q^{3} -9.01776i q^{5} -6.88390 q^{9} +16.5618 q^{11} +0.446263i q^{13} -35.9399 q^{15} -6.95177i q^{17} -12.7440i q^{19} -26.5742 q^{23} -56.3199 q^{25} -8.43362i q^{27} +26.4655 q^{29} +25.1303i q^{31} -66.0062i q^{33} -63.3984 q^{37} +1.77857 q^{39} +0.519795i q^{41} -25.5364 q^{43} +62.0774i q^{45} -68.6455i q^{47} -27.7060 q^{51} -7.17014 q^{53} -149.350i q^{55} -50.7906 q^{57} -75.4237i q^{59} +46.0559i q^{61} +4.02429 q^{65} +42.8097 q^{67} +105.911i q^{69} +60.0281 q^{71} +46.7938i q^{73} +224.461i q^{75} -54.3077 q^{79} -95.5670 q^{81} -11.4213i q^{83} -62.6894 q^{85} -105.477i q^{87} +61.4132i q^{89} +100.156 q^{93} -114.922 q^{95} +20.3570i q^{97} -114.010 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 3.98546i	− 1.32849i	−0.747516	−	0.664244i	$-0.768754\pi$
$3$	− 3.98546i	− 1.32849i	0.747516	−	0.664244i	$-0.231246\pi$
$4$	0	0
$5$	− 9.01776i	− 1.80355i	−0.432204	−	0.901776i	$-0.642264\pi$
$5$	− 9.01776i	− 1.80355i	0.432204	−	0.901776i	$-0.357736\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−6.88390	−0.764878
$10$	0	0
$11$	16.5618	1.50561	0.752807	−	0.658241i	$-0.228699\pi$
$11$	16.5618	1.50561	0.752807	+	0.658241i	$0.228699\pi$
$12$	0	0
$13$	0.446263i	0.0343279i	0.999853	+	0.0171640i	$0.00546373\pi$
$13$	0.446263i	0.0343279i	−0.999853	+	0.0171640i	$0.994536\pi$
$14$	0	0
$15$	−35.9399	−2.39599
$16$	0	0
$17$	− 6.95177i	− 0.408928i	−0.978874	−	0.204464i	$-0.934455\pi$
$17$	− 6.95177i	− 0.408928i	0.978874	−	0.204464i	$-0.0655451\pi$
$18$	0	0
$19$	− 12.7440i	− 0.670735i	−0.942087	−	0.335367i	$-0.891140\pi$
$19$	− 12.7440i	− 0.670735i	0.942087	−	0.335367i	$-0.108860\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−26.5742	−1.15540	−0.577701	−	0.816248i	$-0.696050\pi$
$23$	−26.5742	−1.15540	−0.577701	+	0.816248i	$0.696050\pi$
$24$	0	0
$25$	−56.3199	−2.25280
$26$	0	0
$27$	− 8.43362i	− 0.312356i
$28$	0	0
$29$	26.4655	0.912603	0.456301	−	0.889825i	$-0.349174\pi$
$29$	26.4655	0.912603	0.456301	+	0.889825i	$0.349174\pi$
$30$	0	0
$31$	25.1303i	0.810656i	0.914171	+	0.405328i	$0.132843\pi$
$31$	25.1303i	0.810656i	−0.914171	+	0.405328i	$0.867157\pi$
$32$	0	0
$33$	− 66.0062i	− 2.00019i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−63.3984	−1.71347	−0.856735	−	0.515757i	$-0.827511\pi$
$37$	−63.3984	−1.71347	−0.856735	+	0.515757i	$0.827511\pi$
$38$	0	0
$39$	1.77857	0.0456042
$40$	0	0
$41$	0.519795i	0.0126779i	0.999980	+	0.00633896i	$0.00201777\pi$
$41$	0.519795i	0.0126779i	−0.999980	+	0.00633896i	$0.997982\pi$
$42$	0	0
$43$	−25.5364	−0.593870	−0.296935	−	0.954898i	$-0.595964\pi$
$43$	−25.5364	−0.593870	−0.296935	+	0.954898i	$0.595964\pi$
$44$	0	0
$45$	62.0774i	1.37950i
$46$	0	0
$47$	− 68.6455i	− 1.46054i	−0.683157	−	0.730272i	$-0.739393\pi$
$47$	− 68.6455i	− 1.46054i	0.683157	−	0.730272i	$-0.260607\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−27.7060	−0.543255
$52$	0	0
$53$	−7.17014	−0.135286	−0.0676429	−	0.997710i	$-0.521548\pi$
$53$	−7.17014	−0.135286	−0.0676429	+	0.997710i	$0.521548\pi$
$54$	0	0
$55$	− 149.350i	− 2.71545i
$56$	0	0
$57$	−50.7906	−0.891062
$58$	0	0
$59$	− 75.4237i	− 1.27837i	−0.769054	−	0.639184i	$-0.779272\pi$
$59$	− 75.4237i	− 1.27837i	0.769054	−	0.639184i	$-0.220728\pi$
$60$	0	0
$61$	46.0559i	0.755014i	0.926007	+	0.377507i	$0.123219\pi$
$61$	46.0559i	0.755014i	−0.926007	+	0.377507i	$0.876781\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.02429	0.0619122
$66$	0	0
$67$	42.8097	0.638951	0.319476	−	0.947595i	$-0.396493\pi$
$67$	42.8097	0.638951	0.319476	+	0.947595i	$0.396493\pi$
$68$	0	0
$69$	105.911i	1.53494i
$70$	0	0
$71$	60.0281	0.845466	0.422733	−	0.906254i	$-0.361071\pi$
$71$	60.0281	0.845466	0.422733	+	0.906254i	$0.361071\pi$
$72$	0	0
$73$	46.7938i	0.641011i	0.947247	+	0.320505i	$0.103853\pi$
$73$	46.7938i	0.641011i	−0.947247	+	0.320505i	$0.896147\pi$
$74$	0	0
$75$	224.461i	2.99281i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−54.3077	−0.687439	−0.343720	−	0.939072i	$-0.611687\pi$
$79$	−54.3077	−0.687439	−0.343720	+	0.939072i	$0.611687\pi$
$80$	0	0
$81$	−95.5670	−1.17984
$82$	0	0
$83$	− 11.4213i	− 0.137606i	−0.997630	−	0.0688028i	$-0.978082\pi$
$83$	− 11.4213i	− 0.137606i	0.997630	−	0.0688028i	$-0.0219179\pi$
$84$	0	0
$85$	−62.6894	−0.737522
$86$	0	0
$87$	− 105.477i	− 1.21238i
$88$	0	0
$89$	61.4132i	0.690036i	0.938596	+	0.345018i	$0.112127\pi$
$89$	61.4132i	0.690036i	−0.938596	+	0.345018i	$0.887873\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	100.156	1.07695
$94$	0	0
$95$	−114.922	−1.20970
$96$	0	0
$97$	20.3570i	0.209866i	0.994479	+	0.104933i	$0.0334629\pi$
$97$	20.3570i	0.209866i	−0.994479	+	0.104933i	$0.966537\pi$
$98$	0	0
$99$	−114.010	−1.15161
$100$	0	0
$101$	179.252i	1.77477i	0.461027	+	0.887386i	$0.347481\pi$
$101$	179.252i	1.77477i	−0.461027	+	0.887386i	$0.652519\pi$
$102$	0	0
$103$	− 68.1553i	− 0.661702i	−0.943683	−	0.330851i	$-0.892664\pi$
$103$	− 68.1553i	− 0.661702i	0.943683	−	0.330851i	$-0.107336\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	127.131	1.18814	0.594072	−	0.804412i	$-0.297519\pi$
$107$	127.131	1.18814	0.594072	+	0.804412i	$0.297519\pi$
$108$	0	0
$109$	−21.5704	−0.197894	−0.0989468	−	0.995093i	$-0.531547\pi$
$109$	−21.5704	−0.197894	−0.0989468	+	0.995093i	$0.531547\pi$
$110$	0	0
$111$	252.672i	2.27632i
$112$	0	0
$113$	82.1812	0.727267	0.363634	−	0.931542i	$-0.381536\pi$
$113$	82.1812	0.727267	0.363634	+	0.931542i	$0.381536\pi$
$114$	0	0
$115$	239.640i	2.08383i
$116$	0	0
$117$	− 3.07203i	− 0.0262567i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	153.292	1.26687
$122$	0	0
$123$	2.07162	0.0168425
$124$	0	0
$125$	282.436i	2.25949i
$126$	0	0
$127$	42.9545	0.338225	0.169112	−	0.985597i	$-0.445910\pi$
$127$	42.9545	0.338225	0.169112	+	0.985597i	$0.445910\pi$
$128$	0	0
$129$	101.774i	0.788948i
$130$	0	0
$131$	− 192.812i	− 1.47185i	−0.677063	−	0.735925i	$-0.736748\pi$
$131$	− 192.812i	− 1.47185i	0.677063	−	0.735925i	$-0.263252\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−76.0523	−0.563351
$136$	0	0
$137$	146.249	1.06751	0.533754	−	0.845640i	$-0.320781\pi$
$137$	146.249	1.06751	0.533754	+	0.845640i	$0.320781\pi$
$138$	0	0
$139$	101.042i	0.726921i	0.931610	+	0.363460i	$0.118405\pi$
$139$	101.042i	0.726921i	−0.931610	+	0.363460i	$0.881595\pi$
$140$	0	0
$141$	−273.584	−1.94031
$142$	0	0
$143$	7.39090i	0.0516846i
$144$	0	0
$145$	− 238.659i	− 1.64593i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−191.108	−1.28261	−0.641303	−	0.767287i	$-0.721606\pi$
$149$	−191.108	−1.28261	−0.641303	+	0.767287i	$0.721606\pi$
$150$	0	0
$151$	−28.7170	−0.190179	−0.0950893	−	0.995469i	$-0.530314\pi$
$151$	−28.7170	−0.190179	−0.0950893	+	0.995469i	$0.530314\pi$
$152$	0	0
$153$	47.8553i	0.312780i
$154$	0	0
$155$	226.619	1.46206
$156$	0	0
$157$	− 126.133i	− 0.803397i	−0.915772	−	0.401698i	$-0.868420\pi$
$157$	− 126.133i	− 0.803397i	0.915772	−	0.401698i	$-0.131580\pi$
$158$	0	0
$159$	28.5763i	0.179725i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	254.251	1.55982	0.779911	−	0.625890i	$-0.215264\pi$
$163$	254.251	1.55982	0.779911	+	0.625890i	$0.215264\pi$
$164$	0	0
$165$	−595.228	−3.60744
$166$	0	0
$167$	1.71028i	0.0102412i	0.999987	+	0.00512059i	$0.00162994\pi$
$167$	1.71028i	0.0102412i	−0.999987	+	0.00512059i	$0.998370\pi$
$168$	0	0
$169$	168.801	0.998822
$170$	0	0
$171$	87.7282i	0.513030i
$172$	0	0
$173$	100.654i	0.581815i	0.956751	+	0.290907i	$0.0939571\pi$
$173$	100.654i	0.581815i	−0.956751	+	0.290907i	$0.906043\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−300.598	−1.69830
$178$	0	0
$179$	227.284	1.26974	0.634872	−	0.772617i	$-0.281053\pi$
$179$	227.284	1.26974	0.634872	+	0.772617i	$0.281053\pi$
$180$	0	0
$181$	− 27.1608i	− 0.150060i	−0.997181	−	0.0750298i	$-0.976095\pi$
$181$	− 27.1608i	− 0.150060i	0.997181	−	0.0750298i	$-0.0239052\pi$
$182$	0	0
$183$	183.554	1.00303
$184$	0	0
$185$	571.711i	3.09033i
$186$	0	0
$187$	− 115.133i	− 0.615687i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	55.3123	0.289593	0.144797	−	0.989461i	$-0.453747\pi$
$191$	55.3123	0.289593	0.144797	+	0.989461i	$0.453747\pi$
$192$	0	0
$193$	−222.565	−1.15319	−0.576594	−	0.817031i	$-0.695618\pi$
$193$	−222.565	−1.15319	−0.576594	+	0.817031i	$0.695618\pi$
$194$	0	0
$195$	− 16.0387i	− 0.0822496i
$196$	0	0
$197$	−15.2516	−0.0774191	−0.0387095	−	0.999251i	$-0.512325\pi$
$197$	−15.2516	−0.0774191	−0.0387095	+	0.999251i	$0.512325\pi$
$198$	0	0
$199$	44.2879i	0.222552i	0.993790	+	0.111276i	$0.0354938\pi$
$199$	44.2879i	0.222552i	−0.993790	+	0.111276i	$0.964506\pi$
$200$	0	0
$201$	− 170.617i	− 0.848838i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.68738	0.0228653
$206$	0	0
$207$	182.935	0.883742
$208$	0	0
$209$	− 211.062i	− 1.00987i
$210$	0	0
$211$	−138.721	−0.657446	−0.328723	−	0.944426i	$-0.606618\pi$
$211$	−138.721	−0.657446	−0.328723	+	0.944426i	$0.606618\pi$
$212$	0	0
$213$	− 239.240i	− 1.12319i
$214$	0	0
$215$	230.281i	1.07107i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	186.495	0.851575
$220$	0	0
$221$	3.10232	0.0140376
$222$	0	0
$223$	53.1564i	0.238370i	0.992872	+	0.119185i	$0.0380281\pi$
$223$	53.1564i	0.238370i	−0.992872	+	0.119185i	$0.961972\pi$
$224$	0	0
$225$	387.701	1.72312
$226$	0	0
$227$	− 400.609i	− 1.76480i	−0.470501	−	0.882399i	$-0.655927\pi$
$227$	− 400.609i	− 1.76480i	0.470501	−	0.882399i	$-0.344073\pi$
$228$	0	0
$229$	− 380.550i	− 1.66179i	−0.556430	−	0.830895i	$-0.687829\pi$
$229$	− 380.550i	− 1.66179i	0.556430	−	0.830895i	$-0.312171\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−333.003	−1.42920	−0.714598	−	0.699535i	$-0.753391\pi$
$233$	−333.003	−1.42920	−0.714598	+	0.699535i	$0.753391\pi$
$234$	0	0
$235$	−619.029	−2.63417
$236$	0	0
$237$	216.441i	0.913255i
$238$	0	0
$239$	123.781	0.517912	0.258956	−	0.965889i	$-0.416622\pi$
$239$	123.781	0.517912	0.258956	+	0.965889i	$0.416622\pi$
$240$	0	0
$241$	83.2837i	0.345575i	0.984959	+	0.172788i	$0.0552774\pi$
$241$	83.2837i	0.345575i	−0.984959	+	0.172788i	$0.944723\pi$
$242$	0	0
$243$	304.976i	1.25505i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.68716	0.0230249
$248$	0	0
$249$	−45.5190	−0.182807
$250$	0	0
$251$	403.749i	1.60856i	0.594250	+	0.804281i	$0.297449\pi$
$251$	403.749i	1.60856i	−0.594250	+	0.804281i	$0.702551\pi$
$252$	0	0
$253$	−440.116	−1.73959
$254$	0	0
$255$	249.846i	0.979788i
$256$	0	0
$257$	− 258.268i	− 1.00493i	−0.864596	−	0.502467i	$-0.832426\pi$
$257$	− 258.268i	− 1.00493i	0.864596	−	0.502467i	$-0.167574\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−182.186	−0.698030
$262$	0	0
$263$	−393.857	−1.49756	−0.748778	−	0.662821i	$-0.769359\pi$
$263$	−393.857	−1.49756	−0.748778	+	0.662821i	$0.769359\pi$
$264$	0	0
$265$	64.6586i	0.243995i
$266$	0	0
$267$	244.760	0.916704
$268$	0	0
$269$	− 53.8363i	− 0.200135i	−0.994981	−	0.100067i	$-0.968094\pi$
$269$	− 53.8363i	− 0.200135i	0.994981	−	0.100067i	$-0.0319059\pi$
$270$	0	0
$271$	− 47.5133i	− 0.175326i	−0.996150	−	0.0876629i	$-0.972060\pi$
$271$	− 47.5133i	− 0.175326i	0.996150	−	0.0876629i	$-0.0279398\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−932.757	−3.39184
$276$	0	0
$277$	−243.918	−0.880569	−0.440285	−	0.897858i	$-0.645122\pi$
$277$	−243.918	−0.880569	−0.440285	+	0.897858i	$0.645122\pi$
$278$	0	0
$279$	− 172.995i	− 0.620053i
$280$	0	0
$281$	173.857	0.618709	0.309355	−	0.950947i	$-0.399887\pi$
$281$	173.857	0.618709	0.309355	+	0.950947i	$0.399887\pi$
$282$	0	0
$283$	52.5435i	0.185666i	0.995682	+	0.0928330i	$0.0295923\pi$
$283$	52.5435i	0.185666i	−0.995682	+	0.0928330i	$0.970408\pi$
$284$	0	0
$285$	458.017i	1.60708i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	240.673	0.832778
$290$	0	0
$291$	81.1322	0.278805
$292$	0	0
$293$	321.060i	1.09577i	0.836554	+	0.547885i	$0.184567\pi$
$293$	321.060i	1.09577i	−0.836554	+	0.547885i	$0.815433\pi$
$294$	0	0
$295$	−680.153	−2.30560
$296$	0	0
$297$	− 139.676i	− 0.470288i
$298$	0	0
$299$	− 11.8591i	− 0.0396626i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	714.402	2.35776
$304$	0	0
$305$	415.321	1.36171
$306$	0	0
$307$	− 568.177i	− 1.85074i	−0.379066	−	0.925370i	$-0.623755\pi$
$307$	− 568.177i	− 1.85074i	0.379066	−	0.925370i	$-0.376245\pi$
$308$	0	0
$309$	−271.630	−0.879063
$310$	0	0
$311$	− 48.1978i	− 0.154977i	−0.996993	−	0.0774884i	$-0.975310\pi$
$311$	− 48.1978i	− 0.154977i	0.996993	−	0.0774884i	$-0.0246901\pi$
$312$	0	0
$313$	81.4683i	0.260282i	0.991495	+	0.130141i	$0.0415430\pi$
$313$	81.4683i	0.260282i	−0.991495	+	0.130141i	$0.958457\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	396.581	1.25104	0.625522	−	0.780206i	$-0.284886\pi$
$317$	396.581	1.25104	0.625522	+	0.780206i	$0.284886\pi$
$318$	0	0
$319$	438.315	1.37403
$320$	0	0
$321$	− 506.678i	− 1.57844i
$322$	0	0
$323$	−88.5930	−0.274282
$324$	0	0
$325$	− 25.1335i	− 0.0773339i
$326$	0	0
$327$	85.9680i	0.262899i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−474.236	−1.43274	−0.716369	−	0.697722i	$-0.754197\pi$
$331$	−474.236	−1.43274	−0.716369	+	0.697722i	$0.754197\pi$
$332$	0	0
$333$	436.428	1.31060
$334$	0	0
$335$	− 386.048i	− 1.15238i
$336$	0	0
$337$	234.392	0.695526	0.347763	−	0.937583i	$-0.386941\pi$
$337$	234.392	0.695526	0.347763	+	0.937583i	$0.386941\pi$
$338$	0	0
$339$	− 327.530i	− 0.966165i
$340$	0	0
$341$	416.203i	1.22054i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	955.076	2.76834
$346$	0	0
$347$	224.989	0.648384	0.324192	−	0.945991i	$-0.394908\pi$
$347$	224.989	0.648384	0.324192	+	0.945991i	$0.394908\pi$
$348$	0	0
$349$	− 414.621i	− 1.18802i	−0.804456	−	0.594012i	$-0.797543\pi$
$349$	− 414.621i	− 1.18802i	0.804456	−	0.594012i	$-0.202457\pi$
$350$	0	0
$351$	3.76362	0.0107226
$352$	0	0
$353$	− 200.599i	− 0.568268i	−0.958785	−	0.284134i	$-0.908294\pi$
$353$	− 200.599i	− 0.568268i	0.958785	−	0.284134i	$-0.0917060\pi$
$354$	0	0
$355$	− 541.319i	− 1.52484i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−149.929	−0.417630	−0.208815	−	0.977955i	$-0.566961\pi$
$359$	−149.929	−0.417630	−0.208815	+	0.977955i	$0.566961\pi$
$360$	0	0
$361$	198.591	0.550115
$362$	0	0
$363$	− 610.939i	− 1.68303i
$364$	0	0
$365$	421.975	1.15610
$366$	0	0
$367$	− 485.835i	− 1.32380i	−0.749592	−	0.661900i	$-0.769750\pi$
$367$	− 485.835i	− 1.32380i	0.749592	−	0.661900i	$-0.230250\pi$
$368$	0	0
$369$	− 3.57822i	− 0.00969706i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−162.047	−0.434442	−0.217221	−	0.976122i	$-0.569699\pi$
$373$	−162.047	−0.434442	−0.217221	+	0.976122i	$0.569699\pi$
$374$	0	0
$375$	1125.64	3.00170
$376$	0	0
$377$	11.8106i	0.0313278i
$378$	0	0
$379$	388.817	1.02590	0.512951	−	0.858418i	$-0.328552\pi$
$379$	388.817	1.02590	0.512951	+	0.858418i	$0.328552\pi$
$380$	0	0
$381$	− 171.194i	− 0.449327i
$382$	0	0
$383$	− 28.2666i	− 0.0738031i	−0.999319	−	0.0369015i	$-0.988251\pi$
$383$	− 28.2666i	− 0.0738031i	0.999319	−	0.0369015i	$-0.0117488\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	175.790	0.454238
$388$	0	0
$389$	606.292	1.55859	0.779296	−	0.626656i	$-0.215577\pi$
$389$	606.292	1.55859	0.779296	+	0.626656i	$0.215577\pi$
$390$	0	0
$391$	184.738i	0.472476i
$392$	0	0
$393$	−768.446	−1.95533
$394$	0	0
$395$	489.734i	1.23983i
$396$	0	0
$397$	610.769i	1.53846i	0.638971	+	0.769231i	$0.279360\pi$
$397$	610.769i	1.53846i	−0.638971	+	0.769231i	$0.720640\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−214.109	−0.533938	−0.266969	−	0.963705i	$-0.586022\pi$
$401$	−214.109	−0.533938	−0.266969	+	0.963705i	$0.586022\pi$
$402$	0	0
$403$	−11.2148	−0.0278282
$404$	0	0
$405$	861.800i	2.12790i
$406$	0	0
$407$	−1049.99	−2.57983
$408$	0	0
$409$	− 238.496i	− 0.583120i	−0.956553	−	0.291560i	$-0.905826\pi$
$409$	− 238.496i	− 0.583120i	0.956553	−	0.291560i	$-0.0941742\pi$
$410$	0	0
$411$	− 582.869i	− 1.41817i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−102.994	−0.248179
$416$	0	0
$417$	402.699	0.965705
$418$	0	0
$419$	− 126.446i	− 0.301779i	−0.988551	−	0.150890i	$-0.951786\pi$
$419$	− 126.446i	− 0.301779i	0.988551	−	0.150890i	$-0.0482138\pi$
$420$	0	0
$421$	−113.097	−0.268639	−0.134319	−	0.990938i	$-0.542885\pi$
$421$	−113.097	−0.268639	−0.134319	+	0.990938i	$0.542885\pi$
$422$	0	0
$423$	472.549i	1.11714i
$424$	0	0
$425$	391.523i	0.921231i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	29.4562	0.0686624
$430$	0	0
$431$	−689.184	−1.59903	−0.799517	−	0.600643i	$-0.794911\pi$
$431$	−689.184	−1.59903	−0.799517	+	0.600643i	$0.794911\pi$
$432$	0	0
$433$	− 600.031i	− 1.38575i	−0.721057	−	0.692876i	$-0.756343\pi$
$433$	− 600.031i	− 1.38575i	0.721057	−	0.692876i	$-0.243657\pi$
$434$	0	0
$435$	−951.167	−2.18659
$436$	0	0
$437$	338.661i	0.774968i
$438$	0	0
$439$	− 165.206i	− 0.376324i	−0.982138	−	0.188162i	$-0.939747\pi$
$439$	− 165.206i	− 0.376324i	0.982138	−	0.188162i	$-0.0602530\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−204.833	−0.462377	−0.231189	−	0.972909i	$-0.574261\pi$
$443$	−204.833	−0.462377	−0.231189	+	0.972909i	$0.574261\pi$
$444$	0	0
$445$	553.810	1.24452
$446$	0	0
$447$	761.655i	1.70393i
$448$	0	0
$449$	−112.007	−0.249458	−0.124729	−	0.992191i	$-0.539806\pi$
$449$	−112.007	−0.249458	−0.124729	+	0.992191i	$0.539806\pi$
$450$	0	0
$451$	8.60871i	0.0190881i
$452$	0	0
$453$	114.450i	0.252650i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	165.235	0.361564	0.180782	−	0.983523i	$-0.442137\pi$
$457$	165.235	0.361564	0.180782	+	0.983523i	$0.442137\pi$
$458$	0	0
$459$	−58.6286	−0.127731
$460$	0	0
$461$	− 187.790i	− 0.407353i	−0.979038	−	0.203677i	$-0.934711\pi$
$461$	− 187.790i	− 0.407353i	0.979038	−	0.203677i	$-0.0652891\pi$
$462$	0	0
$463$	678.682	1.46584	0.732918	−	0.680317i	$-0.238158\pi$
$463$	678.682	1.46584	0.732918	+	0.680317i	$0.238158\pi$
$464$	0	0
$465$	− 903.183i	− 1.94233i
$466$	0	0
$467$	− 15.9116i	− 0.0340720i	−0.999855	−	0.0170360i	$-0.994577\pi$
$467$	− 15.9116i	− 0.0340720i	0.999855	−	0.0170360i	$-0.00542299\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−502.699	−1.06730
$472$	0	0
$473$	−422.928	−0.894139
$474$	0	0
$475$	717.739i	1.51103i
$476$	0	0
$477$	49.3586	0.103477
$478$	0	0
$479$	553.743i	1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$479$	553.743i	1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$480$	0	0
$481$	− 28.2924i	− 0.0588199i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	183.575	0.378505
$486$	0	0
$487$	−393.559	−0.808129	−0.404065	−	0.914730i	$-0.632403\pi$
$487$	−393.559	−0.808129	−0.404065	+	0.914730i	$0.632403\pi$
$488$	0	0
$489$	− 1013.31i	− 2.07220i
$490$	0	0
$491$	−96.8828	−0.197317	−0.0986586	−	0.995121i	$-0.531455\pi$
$491$	−96.8828	−0.197317	−0.0986586	+	0.995121i	$0.531455\pi$
$492$	0	0
$493$	− 183.982i	− 0.373188i
$494$	0	0
$495$	1028.11i	2.07699i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−154.872	−0.310366	−0.155183	−	0.987886i	$-0.549597\pi$
$499$	−154.872	−0.310366	−0.155183	+	0.987886i	$0.549597\pi$
$500$	0	0
$501$	6.81624	0.0136053
$502$	0	0
$503$	− 710.432i	− 1.41239i	−0.708018	−	0.706194i	$-0.750410\pi$
$503$	− 710.432i	− 1.41239i	0.708018	−	0.706194i	$-0.249590\pi$
$504$	0	0
$505$	1616.45	3.20089
$506$	0	0
$507$	− 672.749i	− 1.32692i
$508$	0	0
$509$	− 85.0058i	− 0.167005i	−0.996508	−	0.0835027i	$-0.973389\pi$
$509$	− 85.0058i	− 0.167005i	0.996508	−	0.0835027i	$-0.0266107\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−107.478	−0.209508
$514$	0	0
$515$	−614.608	−1.19341
$516$	0	0
$517$	− 1136.89i	− 2.19902i
$518$	0	0
$519$	401.152	0.772933
$520$	0	0
$521$	− 480.680i	− 0.922610i	−0.887242	−	0.461305i	$-0.847381\pi$
$521$	− 480.680i	− 0.922610i	0.887242	−	0.461305i	$-0.152619\pi$
$522$	0	0
$523$	532.458i	1.01808i	0.860742	+	0.509042i	$0.170000\pi$
$523$	532.458i	1.01808i	−0.860742	+	0.509042i	$0.830000\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	174.700	0.331500
$528$	0	0
$529$	177.190	0.334954
$530$	0	0
$531$	519.210i	0.977796i
$532$	0	0
$533$	−0.231965	−0.000435207 0
$534$	0	0
$535$	− 1146.44i	− 2.14288i
$536$	0	0
$537$	− 905.832i	− 1.68684i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−827.485	−1.52955	−0.764774	−	0.644299i	$-0.777149\pi$
$541$	−827.485	−1.52955	−0.764774	+	0.644299i	$0.777149\pi$
$542$	0	0
$543$	−108.248	−0.199352
$544$	0	0
$545$	194.517i	0.356911i
$546$	0	0
$547$	665.687	1.21698	0.608489	−	0.793562i	$-0.291776\pi$
$547$	665.687	1.21698	0.608489	+	0.793562i	$0.291776\pi$
$548$	0	0
$549$	− 317.044i	− 0.577494i
$550$	0	0
$551$	− 337.275i	− 0.612114i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	2278.53	4.10547
$556$	0	0
$557$	−18.8463	−0.0338353	−0.0169177	−	0.999857i	$-0.505385\pi$
$557$	−18.8463	−0.0338353	−0.0169177	+	0.999857i	$0.505385\pi$
$558$	0	0
$559$	− 11.3960i	− 0.0203863i
$560$	0	0
$561$	−458.860	−0.817932
$562$	0	0
$563$	512.440i	0.910196i	0.890441	+	0.455098i	$0.150396\pi$
$563$	512.440i	0.910196i	−0.890441	+	0.455098i	$0.849604\pi$
$564$	0	0
$565$	− 741.090i	− 1.31166i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−631.779	−1.11033	−0.555166	−	0.831740i	$-0.687345\pi$
$569$	−631.779	−1.11033	−0.555166	+	0.831740i	$0.687345\pi$
$570$	0	0
$571$	916.717	1.60546	0.802729	−	0.596344i	$-0.203380\pi$
$571$	916.717	1.60546	0.802729	+	0.596344i	$0.203380\pi$
$572$	0	0
$573$	− 220.445i	− 0.384721i
$574$	0	0
$575$	1496.66	2.60289
$576$	0	0
$577$	− 184.769i	− 0.320223i	−0.987099	−	0.160111i	$-0.948815\pi$
$577$	− 184.769i	− 0.320223i	0.987099	−	0.160111i	$-0.0511854\pi$
$578$	0	0
$579$	887.025i	1.53199i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−118.750	−0.203688
$584$	0	0
$585$	−27.7029	−0.0473553
$586$	0	0
$587$	− 141.805i	− 0.241575i	−0.992678	−	0.120788i	$-0.961458\pi$
$587$	− 141.805i	− 0.241575i	0.992678	−	0.120788i	$-0.0385420\pi$
$588$	0	0
$589$	320.260	0.543735
$590$	0	0
$591$	60.7845i	0.102850i
$592$	0	0
$593$	567.422i	0.956867i	0.878124	+	0.478433i	$0.158795\pi$
$593$	567.422i	0.956867i	−0.878124	+	0.478433i	$0.841205\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	176.508	0.295658
$598$	0	0
$599$	448.906	0.749426	0.374713	−	0.927141i	$-0.377741\pi$
$599$	448.906	0.749426	0.374713	+	0.927141i	$0.377741\pi$
$600$	0	0
$601$	939.633i	1.56345i	0.623623	+	0.781725i	$0.285660\pi$
$601$	939.633i	1.56345i	−0.623623	+	0.781725i	$0.714340\pi$
$602$	0	0
$603$	−294.698	−0.488720
$604$	0	0
$605$	− 1382.35i	− 2.28487i
$606$	0	0
$607$	− 218.704i	− 0.360303i	−0.983639	−	0.180151i	$-0.942341\pi$
$607$	− 218.704i	− 0.360303i	0.983639	−	0.180151i	$-0.0576588\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.6340	0.0501375
$612$	0	0
$613$	283.223	0.462028	0.231014	−	0.972950i	$-0.425796\pi$
$613$	283.223	0.462028	0.231014	+	0.972950i	$0.425796\pi$
$614$	0	0
$615$	− 18.6814i	− 0.0303762i
$616$	0	0
$617$	−1099.79	−1.78247	−0.891236	−	0.453539i	$-0.850161\pi$
$617$	−1099.79	−1.78247	−0.891236	+	0.453539i	$0.850161\pi$
$618$	0	0
$619$	598.401i	0.966722i	0.875421	+	0.483361i	$0.160584\pi$
$619$	598.401i	0.966722i	−0.875421	+	0.483361i	$0.839416\pi$
$620$	0	0
$621$	224.117i	0.360897i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1138.94	1.82230
$626$	0	0
$627$	−841.181	−1.34160
$628$	0	0
$629$	440.731i	0.700685i
$630$	0	0
$631$	1056.45	1.67425	0.837127	−	0.547008i	$-0.184233\pi$
$631$	1056.45	1.67425	0.837127	+	0.547008i	$0.184233\pi$
$632$	0	0
$633$	552.868i	0.873409i
$634$	0	0
$635$	− 387.353i	− 0.610005i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−413.228	−0.646679
$640$	0	0
$641$	598.958	0.934412	0.467206	−	0.884149i	$-0.345261\pi$
$641$	598.958	0.934412	0.467206	+	0.884149i	$0.345261\pi$
$642$	0	0
$643$	− 707.781i	− 1.10075i	−0.834918	−	0.550374i	$-0.814485\pi$
$643$	− 707.781i	− 1.10075i	0.834918	−	0.550374i	$-0.185515\pi$
$644$	0	0
$645$	917.776	1.42291
$646$	0	0
$647$	− 488.067i	− 0.754353i	−0.926141	−	0.377177i	$-0.876895\pi$
$647$	− 488.067i	− 0.754353i	0.926141	−	0.377177i	$-0.123105\pi$
$648$	0	0
$649$	− 1249.15i	− 1.92473i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	355.995	0.545168	0.272584	−	0.962132i	$-0.412122\pi$
$653$	355.995	0.545168	0.272584	+	0.962132i	$0.412122\pi$
$654$	0	0
$655$	−1738.74	−2.65456
$656$	0	0
$657$	− 322.124i	− 0.490295i
$658$	0	0
$659$	1182.36	1.79418	0.897088	−	0.441852i	$-0.145678\pi$
$659$	1182.36	1.79418	0.897088	+	0.441852i	$0.145678\pi$
$660$	0	0
$661$	87.8710i	0.132936i	0.997789	+	0.0664682i	$0.0211731\pi$
$661$	87.8710i	0.132936i	−0.997789	+	0.0664682i	$0.978827\pi$
$662$	0	0
$663$	− 12.3642i	− 0.0186488i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−703.300	−1.05442
$668$	0	0
$669$	211.853	0.316671
$670$	0	0
$671$	762.766i	1.13676i
$672$	0	0
$673$	939.720	1.39631	0.698157	−	0.715944i	$-0.254004\pi$
$673$	939.720	1.39631	0.698157	+	0.715944i	$0.254004\pi$
$674$	0	0
$675$	474.981i	0.703676i
$676$	0	0
$677$	84.8451i	0.125325i	0.998035	+	0.0626626i	$0.0199592\pi$
$677$	84.8451i	0.125325i	−0.998035	+	0.0626626i	$0.980041\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1596.61	−2.34451
$682$	0	0
$683$	303.080	0.443748	0.221874	−	0.975075i	$-0.428783\pi$
$683$	303.080	0.443748	0.221874	+	0.975075i	$0.428783\pi$
$684$	0	0
$685$	− 1318.84i	− 1.92531i
$686$	0	0
$687$	−1516.67	−2.20767
$688$	0	0
$689$	− 3.19977i	− 0.00464408i
$690$	0	0
$691$	255.450i	0.369682i	0.982768	+	0.184841i	$0.0591770\pi$
$691$	255.450i	0.369682i	−0.982768	+	0.184841i	$0.940823\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	911.172	1.31104
$696$	0	0
$697$	3.61349	0.00518435
$698$	0	0
$699$	1327.17i	1.89867i
$700$	0	0
$701$	−560.333	−0.799333	−0.399667	−	0.916661i	$-0.630874\pi$
$701$	−560.333	−0.799333	−0.399667	+	0.916661i	$0.630874\pi$
$702$	0	0
$703$	807.947i	1.14928i
$704$	0	0
$705$	2467.12i	3.49945i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	822.738	1.16042	0.580210	−	0.814467i	$-0.302970\pi$
$709$	822.738	1.16042	0.580210	+	0.814467i	$0.302970\pi$
$710$	0	0
$711$	373.849	0.525807
$712$	0	0
$713$	− 667.820i	− 0.936634i
$714$	0	0
$715$	66.6494	0.0932159
$716$	0	0
$717$	− 493.324i	− 0.688039i
$718$	0	0
$719$	− 1242.96i	− 1.72874i	−0.502859	−	0.864368i	$-0.667719\pi$
$719$	− 1242.96i	− 1.72874i	0.502859	−	0.864368i	$-0.332281\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	331.924	0.459092
$724$	0	0
$725$	−1490.53	−2.05591
$726$	0	0
$727$	1025.14i	1.41010i	0.709156	+	0.705052i	$0.249076\pi$
$727$	1025.14i	1.41010i	−0.709156	+	0.705052i	$0.750924\pi$
$728$	0	0
$729$	355.367	0.487472
$730$	0	0
$731$	177.523i	0.242850i
$732$	0	0
$733$	− 768.426i	− 1.04833i	−0.851617	−	0.524165i	$-0.824378\pi$
$733$	− 768.426i	− 1.04833i	0.851617	−	0.524165i	$-0.175622\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	709.004	0.962014
$738$	0	0
$739$	880.474	1.19144	0.595720	−	0.803192i	$-0.296867\pi$
$739$	880.474	1.19144	0.595720	+	0.803192i	$0.296867\pi$
$740$	0	0
$741$	− 22.6660i	− 0.0305883i
$742$	0	0
$743$	−984.949	−1.32564	−0.662819	−	0.748780i	$-0.730640\pi$
$743$	−984.949	−1.32564	−0.662819	+	0.748780i	$0.730640\pi$
$744$	0	0
$745$	1723.37i	2.31325i
$746$	0	0
$747$	78.6228i	0.105251i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−333.968	−0.444697	−0.222349	−	0.974967i	$-0.571372\pi$
$751$	−333.968	−0.444697	−0.222349	+	0.974967i	$0.571372\pi$
$752$	0	0
$753$	1609.13	2.13695
$754$	0	0
$755$	258.963i	0.342997i
$756$	0	0
$757$	−964.869	−1.27460	−0.637298	−	0.770617i	$-0.719948\pi$
$757$	−964.869	−1.27460	−0.637298	+	0.770617i	$0.719948\pi$
$758$	0	0
$759$	1754.07i	2.31102i
$760$	0	0
$761$	885.964i	1.16421i	0.813113	+	0.582105i	$0.197771\pi$
$761$	885.964i	1.16421i	−0.813113	+	0.582105i	$0.802229\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	431.547	0.564114
$766$	0	0
$767$	33.6588	0.0438838
$768$	0	0
$769$	416.779i	0.541976i	0.962583	+	0.270988i	$0.0873503\pi$
$769$	416.779i	0.541976i	−0.962583	+	0.270988i	$0.912650\pi$
$770$	0	0
$771$	−1029.32	−1.33504
$772$	0	0
$773$	66.9101i	0.0865590i	0.999063	+	0.0432795i	$0.0137806\pi$
$773$	66.9101i	0.0865590i	−0.999063	+	0.0432795i	$0.986219\pi$
$774$	0	0
$775$	− 1415.34i	− 1.82624i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.62424	0.00850352
$780$	0	0
$781$	994.171	1.27295
$782$	0	0
$783$	− 223.200i	− 0.285057i
$784$	0	0
$785$	−1137.44	−1.44897
$786$	0	0
$787$	− 1223.31i	− 1.55439i	−0.629258	−	0.777196i	$-0.716641\pi$
$787$	− 1223.31i	− 1.55439i	0.629258	−	0.777196i	$-0.283359\pi$
$788$	0	0
$789$	1569.70i	1.98948i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−20.5530	−0.0259181
$794$	0	0
$795$	257.694	0.324144
$796$	0	0
$797$	578.768i	0.726184i	0.931753	+	0.363092i	$0.118279\pi$
$797$	578.768i	0.726184i	−0.931753	+	0.363092i	$0.881721\pi$
$798$	0	0
$799$	−477.208	−0.597256
$800$	0	0
$801$	− 422.763i	− 0.527794i
$802$	0	0
$803$	774.987i	0.965115i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−214.562	−0.265877
$808$	0	0
$809$	−892.128	−1.10275	−0.551377	−	0.834256i	$-0.685897\pi$
$809$	−892.128	−1.10275	−0.551377	+	0.834256i	$0.685897\pi$
$810$	0	0
$811$	− 200.724i	− 0.247502i	−0.992313	−	0.123751i	$-0.960508\pi$
$811$	− 200.724i	− 0.247502i	0.992313	−	0.123751i	$-0.0394925\pi$
$812$	0	0
$813$	−189.362	−0.232918
$814$	0	0
$815$	− 2292.77i	− 2.81322i
$816$	0	0
$817$	325.435i	0.398329i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	632.629	0.770559	0.385280	−	0.922800i	$-0.374105\pi$
$821$	632.629	0.770559	0.385280	+	0.922800i	$0.374105\pi$
$822$	0	0
$823$	−3.81646	−0.00463726	−0.00231863	−	0.999997i	$-0.500738\pi$
$823$	−3.81646	−0.00463726	−0.00231863	+	0.999997i	$0.500738\pi$
$824$	0	0
$825$	3717.47i	4.50602i
$826$	0	0
$827$	−1174.97	−1.42076	−0.710382	−	0.703817i	$-0.751478\pi$
$827$	−1174.97	−1.42076	−0.710382	+	0.703817i	$0.751478\pi$
$828$	0	0
$829$	744.824i	0.898461i	0.893416	+	0.449230i	$0.148302\pi$
$829$	744.824i	0.898461i	−0.893416	+	0.449230i	$0.851698\pi$
$830$	0	0
$831$	972.125i	1.16983i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	15.4229	0.0184705
$836$	0	0
$837$	211.940	0.253214
$838$	0	0
$839$	− 1124.03i	− 1.33973i	−0.742485	−	0.669863i	$-0.766353\pi$
$839$	− 1124.03i	− 1.33973i	0.742485	−	0.669863i	$-0.233647\pi$
$840$	0	0
$841$	−140.579	−0.167156
$842$	0	0
$843$	− 692.902i	− 0.821947i
$844$	0	0
$845$	− 1522.21i	− 1.80143i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	209.410	0.246655
$850$	0	0
$851$	1684.76	1.97975
$852$	0	0
$853$	− 96.2021i	− 0.112781i	−0.998409	−	0.0563905i	$-0.982041\pi$
$853$	− 96.2021i	− 0.112781i	0.998409	−	0.0563905i	$-0.0179592\pi$
$854$	0	0
$855$	791.112	0.925277
$856$	0	0
$857$	− 856.417i	− 0.999319i	−0.866222	−	0.499660i	$-0.833458\pi$
$857$	− 856.417i	− 0.999319i	0.866222	−	0.499660i	$-0.166542\pi$
$858$	0	0
$859$	1676.12i	1.95125i	0.219443	+	0.975625i	$0.429576\pi$
$859$	1676.12i	1.95125i	−0.219443	+	0.975625i	$0.570424\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−283.249	−0.328215	−0.164107	−	0.986442i	$-0.552474\pi$
$863$	−283.249	−0.328215	−0.164107	+	0.986442i	$0.552474\pi$
$864$	0	0
$865$	907.672	1.04933
$866$	0	0
$867$	− 959.193i	− 1.10634i
$868$	0	0
$869$	−899.431	−1.03502
$870$	0	0
$871$	19.1044i	0.0219339i
$872$	0	0
$873$	− 140.136i	− 0.160522i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−832.492	−0.949250	−0.474625	−	0.880188i	$-0.657416\pi$
$877$	−832.492	−0.949250	−0.474625	+	0.880188i	$0.657416\pi$
$878$	0	0
$879$	1279.57	1.45572
$880$	0	0
$881$	− 890.282i	− 1.01054i	−0.862962	−	0.505268i	$-0.831394\pi$
$881$	− 890.282i	− 1.01054i	0.862962	−	0.505268i	$-0.168606\pi$
$882$	0	0
$883$	322.429	0.365152	0.182576	−	0.983192i	$-0.441556\pi$
$883$	322.429	0.365152	0.182576	+	0.983192i	$0.441556\pi$
$884$	0	0
$885$	2710.72i	3.06296i
$886$	0	0
$887$	− 542.546i	− 0.611665i	−0.952085	−	0.305832i	$-0.901065\pi$
$887$	− 542.546i	− 0.611665i	0.952085	−	0.305832i	$-0.0989347\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1582.76	−1.77638
$892$	0	0
$893$	−874.816	−0.979637
$894$	0	0
$895$	− 2049.59i	− 2.29005i
$896$	0	0
$897$	−47.2640	−0.0526912
$898$	0	0
$899$	665.087i	0.739807i
$900$	0	0
$901$	49.8452i	0.0553221i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−244.929	−0.270640
$906$	0	0
$907$	−1455.95	−1.60524	−0.802619	−	0.596493i	$-0.796560\pi$
$907$	−1455.95	−1.60524	−0.802619	+	0.596493i	$0.796560\pi$
$908$	0	0
$909$	− 1233.95i	− 1.35748i
$910$	0	0
$911$	−336.756	−0.369655	−0.184828	−	0.982771i	$-0.559173\pi$
$911$	−336.756	−0.369655	−0.184828	+	0.982771i	$0.559173\pi$
$912$	0	0
$913$	− 189.156i	− 0.207181i
$914$	0	0
$915$	− 1655.24i	− 1.80901i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−980.740	−1.06718	−0.533591	−	0.845743i	$-0.679158\pi$
$919$	−980.740	−1.06718	−0.533591	+	0.845743i	$0.679158\pi$
$920$	0	0
$921$	−2264.45	−2.45868
$922$	0	0
$923$	26.7883i	0.0290231i
$924$	0	0
$925$	3570.59	3.86010
$926$	0	0
$927$	469.175i	0.506121i
$928$	0	0
$929$	− 414.270i	− 0.445931i	−0.974826	−	0.222965i	$-0.928426\pi$
$929$	− 414.270i	− 0.445931i	0.974826	−	0.222965i	$-0.0715737\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−192.090	−0.205885
$934$	0	0
$935$	−1038.25	−1.11042
$936$	0	0
$937$	− 1323.89i	− 1.41290i	−0.707764	−	0.706449i	$-0.750296\pi$
$937$	− 1323.89i	− 1.41290i	0.707764	−	0.706449i	$-0.249704\pi$
$938$	0	0
$939$	324.689	0.345781
$940$	0	0
$941$	− 950.630i	− 1.01023i	−0.863051	−	0.505117i	$-0.831449\pi$
$941$	− 950.630i	− 1.01023i	0.863051	−	0.505117i	$-0.168551\pi$
$942$	0	0
$943$	− 13.8132i	− 0.0146481i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1160.49	1.22543	0.612717	−	0.790302i	$-0.290077\pi$
$947$	1160.49	1.22543	0.612717	+	0.790302i	$0.290077\pi$
$948$	0	0
$949$	−20.8823	−0.0220046
$950$	0	0
$951$	− 1580.56i	− 1.66200i
$952$	0	0
$953$	−1025.81	−1.07640	−0.538201	−	0.842816i	$-0.680896\pi$
$953$	−1025.81	−1.07640	−0.538201	+	0.842816i	$0.680896\pi$
$954$	0	0
$955$	− 498.793i	− 0.522297i
$956$	0	0
$957$	− 1746.89i	− 1.82538i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	329.466	0.342836
$962$	0	0
$963$	−875.161	−0.908786
$964$	0	0
$965$	2007.04i	2.07983i
$966$	0	0
$967$	−648.379	−0.670505	−0.335253	−	0.942128i	$-0.608822\pi$
$967$	−648.379	−0.670505	−0.335253	+	0.942128i	$0.608822\pi$
$968$	0	0
$969$	353.084i	0.364380i
$970$	0	0
$971$	− 731.698i	− 0.753551i	−0.926305	−	0.376776i	$-0.877033\pi$
$971$	− 731.698i	− 0.753551i	0.926305	−	0.376776i	$-0.122967\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−100.169	−0.102737
$976$	0	0
$977$	−777.446	−0.795749	−0.397874	−	0.917440i	$-0.630252\pi$
$977$	−777.446	−0.795749	−0.397874	+	0.917440i	$0.630252\pi$
$978$	0	0
$979$	1017.11i	1.03893i
$980$	0	0
$981$	148.489	0.151364
$982$	0	0
$983$	1359.09i	1.38260i	0.722570	+	0.691298i	$0.242961\pi$
$983$	1359.09i	1.38260i	−0.722570	+	0.691298i	$0.757039\pi$
$984$	0	0
$985$	137.535i	0.139629i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	678.611	0.686158
$990$	0	0
$991$	705.097	0.711500	0.355750	−	0.934581i	$-0.384225\pi$
$991$	705.097	0.711500	0.355750	+	0.934581i	$0.384225\pi$
$992$	0	0
$993$	1890.05i	1.90337i
$994$	0	0
$995$	399.377	0.401384
$996$	0	0
$997$	1245.74i	1.24949i	0.780829	+	0.624745i	$0.214797\pi$
$997$	1245.74i	1.24949i	−0.780829	+	0.624745i	$0.785203\pi$
$998$	0	0
$999$	534.678i	0.535213i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.g.97.3		16
4.3	odd	2	inner	1568.3.c.g.97.13		16
7.2	even	3		224.3.s.b.129.7	yes	16
7.3	odd	6		224.3.s.b.33.7	yes	16
7.6	odd	2	inner	1568.3.c.g.97.14		16
28.3	even	6		224.3.s.b.33.2	✓	16
28.23	odd	6		224.3.s.b.129.2	yes	16
28.27	even	2	inner	1568.3.c.g.97.4		16
56.3	even	6		448.3.s.h.257.7		16
56.37	even	6		448.3.s.h.129.2		16
56.45	odd	6		448.3.s.h.257.2		16
56.51	odd	6		448.3.s.h.129.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.s.b.33.2	✓	16	28.3	even	6
224.3.s.b.33.7	yes	16	7.3	odd	6
224.3.s.b.129.2	yes	16	28.23	odd	6
224.3.s.b.129.7	yes	16	7.2	even	3
448.3.s.h.129.2		16	56.37	even	6
448.3.s.h.129.7		16	56.51	odd	6
448.3.s.h.257.2		16	56.45	odd	6
448.3.s.h.257.7		16	56.3	even	6
1568.3.c.g.97.3		16	1.1	even	1	trivial
1568.3.c.g.97.4		16	28.27	even	2	inner
1568.3.c.g.97.13		16	4.3	odd	2	inner
1568.3.c.g.97.14		16	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.98546i q^{3} -9.01776i q^{5} -6.88390 q^{9} +O(q^{10})\)	\(q-3.98546i q^{3} -9.01776i q^{5} -6.88390 q^{9} +16.5618 q^{11} +0.446263i q^{13} -35.9399 q^{15} -6.95177i q^{17} -12.7440i q^{19} -26.5742 q^{23} -56.3199 q^{25} -8.43362i q^{27} +26.4655 q^{29} +25.1303i q^{31} -66.0062i q^{33} -63.3984 q^{37} +1.77857 q^{39} +0.519795i q^{41} -25.5364 q^{43} +62.0774i q^{45} -68.6455i q^{47} -27.7060 q^{51} -7.17014 q^{53} -149.350i q^{55} -50.7906 q^{57} -75.4237i q^{59} +46.0559i q^{61} +4.02429 q^{65} +42.8097 q^{67} +105.911i q^{69} +60.0281 q^{71} +46.7938i q^{73} +224.461i q^{75} -54.3077 q^{79} -95.5670 q^{81} -11.4213i q^{83} -62.6894 q^{85} -105.477i q^{87} +61.4132i q^{89} +100.156 q^{93} -114.922 q^{95} +20.3570i q^{97} -114.010 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 80 q^{9}+O(q^{10}) \) 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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