LMFDB - Embedded newform 1568.3.h.a.881.8

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			881.8
Character	$\chi$	$=$	1568.881
Dual form			1568.3.h.a.881.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.40276 q^{3} -4.31716 q^{5} +2.57876 q^{9} +O(q^{10})$	$q-3.40276 q^{3} -4.31716 q^{5} +2.57876 q^{9} +17.8862i q^{11} +3.25607 q^{13} +14.6903 q^{15} -15.7343i q^{17} +1.55704 q^{19} +41.4139 q^{23} -6.36211 q^{25} +21.8499 q^{27} +3.74374i q^{29} +0.0167630i q^{31} -60.8622i q^{33} +1.34839i q^{37} -11.0796 q^{39} -70.3018i q^{41} +13.0380i q^{43} -11.1329 q^{45} +35.7723i q^{47} +53.5402i q^{51} +45.9558i q^{53} -77.2174i q^{55} -5.29824 q^{57} -68.7017 q^{59} +96.0771 q^{61} -14.0570 q^{65} -13.9497i q^{67} -140.921 q^{69} +75.7095 q^{71} -53.1487i q^{73} +21.6487 q^{75} +23.3488 q^{79} -97.5588 q^{81} -102.487 q^{83} +67.9277i q^{85} -12.7390i q^{87} +88.5638i q^{89} -0.0570404i q^{93} -6.72201 q^{95} +140.869i q^{97} +46.1241i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.40276	−1.13425	−0.567126	−	0.823631i	$-0.691945\pi$
$3$	−3.40276	−1.13425	−0.567126	+	0.823631i	$0.691945\pi$
$4$	0	0
$5$	−4.31716	−0.863433	−0.431716	−	0.902009i	$-0.642092\pi$
$5$	−4.31716	−0.863433	−0.431716	+	0.902009i	$0.642092\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.57876	0.286529
$10$	0	0
$11$	17.8862i	1.62601i	0.582254	+	0.813007i	$0.302171\pi$
$11$	17.8862i	1.62601i	−0.582254	+	0.813007i	$0.697829\pi$
$12$	0	0
$13$	3.25607	0.250467	0.125233	−	0.992127i	$-0.460032\pi$
$13$	3.25607	0.250467	0.125233	+	0.992127i	$0.460032\pi$
$14$	0	0
$15$	14.6903	0.979350
$16$	0	0
$17$	− 15.7343i	− 0.925550i	−0.886476	−	0.462775i	$-0.846854\pi$
$17$	− 15.7343i	− 0.925550i	0.886476	−	0.462775i	$-0.153146\pi$
$18$	0	0
$19$	1.55704	0.0819496	0.0409748	−	0.999160i	$-0.486954\pi$
$19$	1.55704	0.0819496	0.0409748	+	0.999160i	$0.486954\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	41.4139	1.80060	0.900301	−	0.435267i	$-0.143346\pi$
$23$	41.4139	1.80060	0.900301	+	0.435267i	$0.143346\pi$
$24$	0	0
$25$	−6.36211	−0.254484
$26$	0	0
$27$	21.8499	0.809257
$28$	0	0
$29$	3.74374i	0.129095i	0.997915	+	0.0645473i	$0.0205603\pi$
$29$	3.74374i	0.129095i	−0.997915	+	0.0645473i	$0.979440\pi$
$30$	0	0
$31$	0.0167630i	0 0.000540742i	1.00000		0.000270371i	$8.60617e-5\pi$
$31$	0.0167630i	0 0.000540742i	−1.00000		0.000270371i	$0.999914\pi$
$32$	0	0
$33$	− 60.8622i	− 1.84431i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.34839i	0.0364429i	0.999834	+	0.0182215i	$0.00580039\pi$
$37$	1.34839i	0.0364429i	−0.999834	+	0.0182215i	$0.994200\pi$
$38$	0	0
$39$	−11.0796	−0.284093
$40$	0	0
$41$	− 70.3018i	− 1.71468i	−0.514753	−	0.857339i	$-0.672116\pi$
$41$	− 70.3018i	− 1.71468i	0.514753	−	0.857339i	$-0.327884\pi$
$42$	0	0
$43$	13.0380i	0.303210i	0.988441	+	0.151605i	$0.0484442\pi$
$43$	13.0380i	0.303210i	−0.988441	+	0.151605i	$0.951556\pi$
$44$	0	0
$45$	−11.1329	−0.247398
$46$	0	0
$47$	35.7723i	0.761113i	0.924758	+	0.380557i	$0.124268\pi$
$47$	35.7723i	0.761113i	−0.924758	+	0.380557i	$0.875732\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	53.5402i	1.04981i
$52$	0	0
$53$	45.9558i	0.867091i	0.901132	+	0.433546i	$0.142738\pi$
$53$	45.9558i	0.867091i	−0.901132	+	0.433546i	$0.857262\pi$
$54$	0	0
$55$	− 77.2174i	− 1.40395i
$56$	0	0
$57$	−5.29824	−0.0929516
$58$	0	0
$59$	−68.7017	−1.16444	−0.582218	−	0.813033i	$-0.697815\pi$
$59$	−68.7017	−1.16444	−0.582218	+	0.813033i	$0.697815\pi$
$60$	0	0
$61$	96.0771	1.57503	0.787517	−	0.616292i	$-0.211366\pi$
$61$	96.0771	1.57503	0.787517	+	0.616292i	$0.211366\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.0570	−0.216261
$66$	0	0
$67$	− 13.9497i	− 0.208204i	−0.994567	−	0.104102i	$-0.966803\pi$
$67$	− 13.9497i	− 0.208204i	0.994567	−	0.104102i	$-0.0331969\pi$
$68$	0	0
$69$	−140.921	−2.04234
$70$	0	0
$71$	75.7095	1.06633	0.533166	−	0.846011i	$-0.321002\pi$
$71$	75.7095	1.06633	0.533166	+	0.846011i	$0.321002\pi$
$72$	0	0
$73$	− 53.1487i	− 0.728065i	−0.931386	−	0.364033i	$-0.881400\pi$
$73$	− 53.1487i	− 0.728065i	0.931386	−	0.364033i	$-0.118600\pi$
$74$	0	0
$75$	21.6487	0.288649
$76$	0	0
$77$	0	0
$78$	0	0
$79$	23.3488	0.295554	0.147777	−	0.989021i	$-0.452788\pi$
$79$	23.3488	0.295554	0.147777	+	0.989021i	$0.452788\pi$
$80$	0	0
$81$	−97.5588	−1.20443
$82$	0	0
$83$	−102.487	−1.23479	−0.617393	−	0.786655i	$-0.711811\pi$
$83$	−102.487	−1.23479	−0.617393	+	0.786655i	$0.711811\pi$
$84$	0	0
$85$	67.9277i	0.799150i
$86$	0	0
$87$	− 12.7390i	− 0.146426i
$88$	0	0
$89$	88.5638i	0.995099i	0.867436	+	0.497549i	$0.165767\pi$
$89$	88.5638i	0.995099i	−0.867436	+	0.497549i	$0.834233\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 0.0570404i	0 0.000613338i
$94$	0	0
$95$	−6.72201	−0.0707580
$96$	0	0
$97$	140.869i	1.45226i	0.687558	+	0.726130i	$0.258683\pi$
$97$	140.869i	1.45226i	−0.687558	+	0.726130i	$0.741317\pi$
$98$	0	0
$99$	46.1241i	0.465900i
$100$	0	0
$101$	−35.3977	−0.350472	−0.175236	−	0.984526i	$-0.556069\pi$
$101$	−35.3977	−0.350472	−0.175236	+	0.984526i	$0.556069\pi$
$102$	0	0
$103$	100.650i	0.977181i	0.872513	+	0.488590i	$0.162489\pi$
$103$	100.650i	0.977181i	−0.872513	+	0.488590i	$0.837511\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	106.950i	0.999534i	0.866160	+	0.499767i	$0.166581\pi$
$107$	106.950i	0.999534i	−0.866160	+	0.499767i	$0.833419\pi$
$108$	0	0
$109$	52.6312i	0.482855i	0.970419	+	0.241427i	$0.0776156\pi$
$109$	52.6312i	0.482855i	−0.970419	+	0.241427i	$0.922384\pi$
$110$	0	0
$111$	− 4.58824i	− 0.0413355i
$112$	0	0
$113$	45.4346	0.402076	0.201038	−	0.979583i	$-0.435568\pi$
$113$	45.4346	0.402076	0.201038	+	0.979583i	$0.435568\pi$
$114$	0	0
$115$	−178.790	−1.55470
$116$	0	0
$117$	8.39662	0.0717659
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−198.914	−1.64392
$122$	0	0
$123$	239.220i	1.94488i
$124$	0	0
$125$	135.395	1.08316
$126$	0	0
$127$	−125.695	−0.989723	−0.494861	−	0.868972i	$-0.664781\pi$
$127$	−125.695	−0.989723	−0.494861	+	0.868972i	$0.664781\pi$
$128$	0	0
$129$	− 44.3653i	− 0.343917i
$130$	0	0
$131$	−113.301	−0.864891	−0.432446	−	0.901660i	$-0.642349\pi$
$131$	−113.301	−0.864891	−0.432446	+	0.901660i	$0.642349\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−94.3297	−0.698738
$136$	0	0
$137$	78.3358	0.571794	0.285897	−	0.958260i	$-0.407708\pi$
$137$	78.3358	0.571794	0.285897	+	0.958260i	$0.407708\pi$
$138$	0	0
$139$	−149.038	−1.07222	−0.536109	−	0.844149i	$-0.680106\pi$
$139$	−149.038	−1.07222	−0.536109	+	0.844149i	$0.680106\pi$
$140$	0	0
$141$	− 121.725i	− 0.863295i
$142$	0	0
$143$	58.2385i	0.407263i
$144$	0	0
$145$	− 16.1623i	− 0.111464i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 85.2595i	− 0.572212i	−0.958198	−	0.286106i	$-0.907639\pi$
$149$	− 85.2595i	− 0.572212i	0.958198	−	0.286106i	$-0.0923609\pi$
$150$	0	0
$151$	−131.802	−0.872864	−0.436432	−	0.899737i	$-0.643758\pi$
$151$	−131.802	−0.872864	−0.436432	+	0.899737i	$0.643758\pi$
$152$	0	0
$153$	− 40.5751i	− 0.265197i
$154$	0	0
$155$	− 0.0723686i	0 0.000466894i
$156$	0	0
$157$	−245.105	−1.56118	−0.780589	−	0.625045i	$-0.785081\pi$
$157$	−245.105	−1.56118	−0.780589	+	0.625045i	$0.785081\pi$
$158$	0	0
$159$	− 156.377i	− 0.983501i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 240.281i	− 1.47411i	−0.675830	−	0.737057i	$-0.736215\pi$
$163$	− 240.281i	− 1.47411i	0.675830	−	0.737057i	$-0.263785\pi$
$164$	0	0
$165$	262.752i	1.59244i
$166$	0	0
$167$	73.1965i	0.438302i	0.975691	+	0.219151i	$0.0703288\pi$
$167$	73.1965i	0.438302i	−0.975691	+	0.219151i	$0.929671\pi$
$168$	0	0
$169$	−158.398	−0.937266
$170$	0	0
$171$	4.01524	0.0234809
$172$	0	0
$173$	37.0491	0.214157	0.107078	−	0.994251i	$-0.465850\pi$
$173$	37.0491	0.214157	0.107078	+	0.994251i	$0.465850\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	233.775	1.32076
$178$	0	0
$179$	237.848i	1.32876i	0.747395	+	0.664379i	$0.231304\pi$
$179$	237.848i	1.32876i	−0.747395	+	0.664379i	$0.768696\pi$
$180$	0	0
$181$	−292.553	−1.61631	−0.808157	−	0.588966i	$-0.799535\pi$
$181$	−292.553	−1.61631	−0.808157	+	0.588966i	$0.799535\pi$
$182$	0	0
$183$	−326.927	−1.78649
$184$	0	0
$185$	− 5.82121i	− 0.0314660i
$186$	0	0
$187$	281.427	1.50496
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−141.227	−0.739408	−0.369704	−	0.929150i	$-0.620541\pi$
$191$	−141.227	−0.739408	−0.369704	+	0.929150i	$0.620541\pi$
$192$	0	0
$193$	−65.9598	−0.341761	−0.170880	−	0.985292i	$-0.554661\pi$
$193$	−65.9598	−0.341761	−0.170880	+	0.985292i	$0.554661\pi$
$194$	0	0
$195$	47.8325	0.245295
$196$	0	0
$197$	− 199.421i	− 1.01229i	−0.862448	−	0.506145i	$-0.831070\pi$
$197$	− 199.421i	− 1.01229i	0.862448	−	0.506145i	$-0.168930\pi$
$198$	0	0
$199$	− 67.6920i	− 0.340161i	−0.985430	−	0.170080i	$-0.945597\pi$
$199$	− 67.6920i	− 0.340161i	0.985430	−	0.170080i	$-0.0544028\pi$
$200$	0	0
$201$	47.4674i	0.236156i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	303.504i	1.48051i
$206$	0	0
$207$	106.796	0.515924
$208$	0	0
$209$	27.8495i	0.133251i
$210$	0	0
$211$	62.1464i	0.294533i	0.989097	+	0.147266i	$0.0470475\pi$
$211$	62.1464i	0.294533i	−0.989097	+	0.147266i	$0.952953\pi$
$212$	0	0
$213$	−257.621	−1.20949
$214$	0	0
$215$	− 56.2874i	− 0.261802i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	180.852i	0.825810i
$220$	0	0
$221$	− 51.2321i	− 0.231820i
$222$	0	0
$223$	115.525i	0.518050i	0.965871	+	0.259025i	$0.0834012\pi$
$223$	115.525i	0.518050i	−0.965871	+	0.259025i	$0.916599\pi$
$224$	0	0
$225$	−16.4063	−0.0729171
$226$	0	0
$227$	56.5064	0.248927	0.124463	−	0.992224i	$-0.460279\pi$
$227$	56.5064	0.248927	0.124463	+	0.992224i	$0.460279\pi$
$228$	0	0
$229$	−118.339	−0.516765	−0.258383	−	0.966043i	$-0.583190\pi$
$229$	−118.339	−0.516765	−0.258383	+	0.966043i	$0.583190\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.6805	−0.105925	−0.0529625	−	0.998597i	$-0.516866\pi$
$233$	−24.6805	−0.105925	−0.0529625	+	0.998597i	$0.516866\pi$
$234$	0	0
$235$	− 154.435i	− 0.657170i
$236$	0	0
$237$	−79.4503	−0.335233
$238$	0	0
$239$	251.189	1.05100	0.525499	−	0.850794i	$-0.323879\pi$
$239$	251.189	1.05100	0.525499	+	0.850794i	$0.323879\pi$
$240$	0	0
$241$	112.443i	0.466567i	0.972409	+	0.233283i	$0.0749470\pi$
$241$	112.443i	0.466567i	−0.972409	+	0.233283i	$0.925053\pi$
$242$	0	0
$243$	135.320	0.556871
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.06984	0.0205257
$248$	0	0
$249$	348.739	1.40056
$250$	0	0
$251$	−121.248	−0.483059	−0.241529	−	0.970394i	$-0.577649\pi$
$251$	−121.248	−0.483059	−0.241529	+	0.970394i	$0.577649\pi$
$252$	0	0
$253$	740.735i	2.92781i
$254$	0	0
$255$	− 231.142i	− 0.906438i
$256$	0	0
$257$	104.775i	0.407684i	0.979004	+	0.203842i	$0.0653430\pi$
$257$	104.775i	0.407684i	−0.979004	+	0.203842i	$0.934657\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	9.65421i	0.0369893i
$262$	0	0
$263$	104.678	0.398017	0.199008	−	0.979998i	$-0.436228\pi$
$263$	104.678	0.398017	0.199008	+	0.979998i	$0.436228\pi$
$264$	0	0
$265$	− 198.399i	− 0.748675i
$266$	0	0
$267$	− 301.361i	− 1.12869i
$268$	0	0
$269$	304.932	1.13358	0.566789	−	0.823863i	$-0.308185\pi$
$269$	304.932	1.13358	0.566789	+	0.823863i	$0.308185\pi$
$270$	0	0
$271$	102.646i	0.378768i	0.981903	+	0.189384i	$0.0606491\pi$
$271$	102.646i	0.378768i	−0.981903	+	0.189384i	$0.939351\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 113.794i	− 0.413795i
$276$	0	0
$277$	16.6548i	0.0601256i	0.999548	+	0.0300628i	$0.00957073\pi$
$277$	16.6548i	0.0601256i	−0.999548	+	0.0300628i	$0.990429\pi$
$278$	0	0
$279$	0.0432277i	0 0.000154938i
$280$	0	0
$281$	75.8291	0.269855	0.134927	−	0.990856i	$-0.456920\pi$
$281$	75.8291	0.269855	0.134927	+	0.990856i	$0.456920\pi$
$282$	0	0
$283$	87.3313	0.308591	0.154296	−	0.988025i	$-0.450689\pi$
$283$	87.3313	0.308591	0.154296	+	0.988025i	$0.450689\pi$
$284$	0	0
$285$	22.8734	0.0802574
$286$	0	0
$287$	0	0
$288$	0	0
$289$	41.4304	0.143358
$290$	0	0
$291$	− 479.344i	− 1.64723i
$292$	0	0
$293$	27.5057	0.0938760	0.0469380	−	0.998898i	$-0.485054\pi$
$293$	27.5057	0.0938760	0.0469380	+	0.998898i	$0.485054\pi$
$294$	0	0
$295$	296.597	1.00541
$296$	0	0
$297$	390.811i	1.31586i
$298$	0	0
$299$	134.846	0.450991
$300$	0	0
$301$	0	0
$302$	0	0
$303$	120.450	0.397524
$304$	0	0
$305$	−414.781	−1.35994
$306$	0	0
$307$	−247.996	−0.807805	−0.403902	−	0.914802i	$-0.632346\pi$
$307$	−247.996	−0.807805	−0.403902	+	0.914802i	$0.632346\pi$
$308$	0	0
$309$	− 342.486i	− 1.10837i
$310$	0	0
$311$	− 437.036i	− 1.40526i	−0.711556	−	0.702630i	$-0.752009\pi$
$311$	− 437.036i	− 1.40526i	0.711556	−	0.702630i	$-0.247991\pi$
$312$	0	0
$313$	82.8301i	0.264633i	0.991208	+	0.132317i	$0.0422415\pi$
$313$	82.8301i	0.264633i	−0.991208	+	0.132317i	$0.957758\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	244.536i	0.771408i	0.922623	+	0.385704i	$0.126041\pi$
$317$	244.536i	0.771408i	−0.922623	+	0.385704i	$0.873959\pi$
$318$	0	0
$319$	−66.9611	−0.209910
$320$	0	0
$321$	− 363.925i	− 1.13372i
$322$	0	0
$323$	− 24.4991i	− 0.0758485i
$324$	0	0
$325$	−20.7155	−0.0637399
$326$	0	0
$327$	− 179.091i	− 0.547679i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 76.5472i	− 0.231261i	−0.993292	−	0.115630i	$-0.963111\pi$
$331$	− 76.5472i	− 0.231261i	0.993292	−	0.115630i	$-0.0368888\pi$
$332$	0	0
$333$	3.47717i	0.0104419i
$334$	0	0
$335$	60.2230i	0.179770i
$336$	0	0
$337$	−38.2520	−0.113507	−0.0567537	−	0.998388i	$-0.518075\pi$
$337$	−38.2520	−0.113507	−0.0567537	+	0.998388i	$0.518075\pi$
$338$	0	0
$339$	−154.603	−0.456056
$340$	0	0
$341$	−0.299825	−0.000879253 0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	608.380	1.76342
$346$	0	0
$347$	240.633i	0.693468i	0.937964	+	0.346734i	$0.112709\pi$
$347$	240.633i	0.693468i	−0.937964	+	0.346734i	$0.887291\pi$
$348$	0	0
$349$	430.367	1.23314	0.616572	−	0.787298i	$-0.288521\pi$
$349$	430.367	1.23314	0.616572	+	0.787298i	$0.288521\pi$
$350$	0	0
$351$	71.1449	0.202692
$352$	0	0
$353$	− 307.093i	− 0.869951i	−0.900442	−	0.434975i	$-0.856757\pi$
$353$	− 307.093i	− 0.869951i	0.900442	−	0.434975i	$-0.143243\pi$
$354$	0	0
$355$	−326.850	−0.920705
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−461.760	−1.28624	−0.643120	−	0.765766i	$-0.722360\pi$
$359$	−461.760	−1.28624	−0.643120	+	0.765766i	$0.722360\pi$
$360$	0	0
$361$	−358.576	−0.993284
$362$	0	0
$363$	676.858	1.86462
$364$	0	0
$365$	229.452i	0.628635i
$366$	0	0
$367$	626.943i	1.70829i	0.520034	+	0.854146i	$0.325919\pi$
$367$	626.943i	1.70829i	−0.520034	+	0.854146i	$0.674081\pi$
$368$	0	0
$369$	− 181.291i	− 0.491304i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 412.594i	− 1.10615i	−0.833132	−	0.553075i	$-0.813454\pi$
$373$	− 412.594i	− 1.10615i	0.833132	−	0.553075i	$-0.186546\pi$
$374$	0	0
$375$	−460.717	−1.22858
$376$	0	0
$377$	12.1899i	0.0323339i
$378$	0	0
$379$	327.118i	0.863107i	0.902087	+	0.431554i	$0.142034\pi$
$379$	327.118i	0.863107i	−0.902087	+	0.431554i	$0.857966\pi$
$380$	0	0
$381$	427.709	1.12260
$382$	0	0
$383$	− 248.865i	− 0.649777i	−0.945752	−	0.324889i	$-0.894673\pi$
$383$	− 248.865i	− 0.649777i	0.945752	−	0.324889i	$-0.105327\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	33.6220i	0.0868785i
$388$	0	0
$389$	− 377.273i	− 0.969854i	−0.874555	−	0.484927i	$-0.838846\pi$
$389$	− 377.273i	− 0.969854i	0.874555	−	0.484927i	$-0.161154\pi$
$390$	0	0
$391$	− 651.620i	− 1.66655i
$392$	0	0
$393$	385.535	0.981005
$394$	0	0
$395$	−100.801	−0.255191
$396$	0	0
$397$	−671.749	−1.69206	−0.846031	−	0.533133i	$-0.821014\pi$
$397$	−671.749	−1.69206	−0.846031	+	0.533133i	$0.821014\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−470.400	−1.17307	−0.586534	−	0.809925i	$-0.699508\pi$
$401$	−470.400	−1.17307	−0.586534	+	0.809925i	$0.699508\pi$
$402$	0	0
$403$	0.0545815i	0 0.000135438i
$404$	0	0
$405$	421.177	1.03994
$406$	0	0
$407$	−24.1175	−0.0592567
$408$	0	0
$409$	− 66.6725i	− 0.163013i	−0.996673	−	0.0815067i	$-0.974027\pi$
$409$	− 66.6725i	− 0.163013i	0.996673	−	0.0815067i	$-0.0259732\pi$
$410$	0	0
$411$	−266.558	−0.648559
$412$	0	0
$413$	0	0
$414$	0	0
$415$	442.454	1.06615
$416$	0	0
$417$	507.141	1.21617
$418$	0	0
$419$	437.380	1.04387	0.521933	−	0.852986i	$-0.325211\pi$
$419$	437.380	1.04387	0.521933	+	0.852986i	$0.325211\pi$
$420$	0	0
$421$	− 703.800i	− 1.67173i	−0.548933	−	0.835867i	$-0.684966\pi$
$421$	− 703.800i	− 1.67173i	0.548933	−	0.835867i	$-0.315034\pi$
$422$	0	0
$423$	92.2482i	0.218081i
$424$	0	0
$425$	100.104i	0.235538i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 198.172i	− 0.461939i
$430$	0	0
$431$	549.738	1.27549	0.637747	−	0.770246i	$-0.279866\pi$
$431$	549.738	1.27549	0.637747	+	0.770246i	$0.279866\pi$
$432$	0	0
$433$	355.012i	0.819890i	0.912110	+	0.409945i	$0.134452\pi$
$433$	355.012i	0.819890i	−0.912110	+	0.409945i	$0.865548\pi$
$434$	0	0
$435$	54.9965i	0.126429i
$436$	0	0
$437$	64.4832	0.147559
$438$	0	0
$439$	550.830i	1.25474i	0.778722	+	0.627369i	$0.215868\pi$
$439$	550.830i	1.25474i	−0.778722	+	0.627369i	$0.784132\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 270.231i	− 0.610002i	−0.952352	−	0.305001i	$-0.901343\pi$
$443$	− 270.231i	− 0.610002i	0.952352	−	0.305001i	$-0.0986569\pi$
$444$	0	0
$445$	− 382.344i	− 0.859200i
$446$	0	0
$447$	290.117i	0.649032i
$448$	0	0
$449$	455.397	1.01425	0.507124	−	0.861873i	$-0.330709\pi$
$449$	455.397	1.01425	0.507124	+	0.861873i	$0.330709\pi$
$450$	0	0
$451$	1257.43	2.78809
$452$	0	0
$453$	448.492	0.990048
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−168.634	−0.369003	−0.184501	−	0.982832i	$-0.559067\pi$
$457$	−168.634	−0.369003	−0.184501	+	0.982832i	$0.559067\pi$
$458$	0	0
$459$	− 343.794i	− 0.749007i
$460$	0	0
$461$	265.062	0.574971	0.287485	−	0.957785i	$-0.407181\pi$
$461$	265.062	0.574971	0.287485	+	0.957785i	$0.407181\pi$
$462$	0	0
$463$	−97.4735	−0.210526	−0.105263	−	0.994444i	$-0.533568\pi$
$463$	−97.4735	−0.210526	−0.105263	+	0.994444i	$0.533568\pi$
$464$	0	0
$465$	0.246253i	0 0.000529576i
$466$	0	0
$467$	−74.1994	−0.158885	−0.0794427	−	0.996839i	$-0.525314\pi$
$467$	−74.1994	−0.158885	−0.0794427	+	0.996839i	$0.525314\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	834.032	1.77077
$472$	0	0
$473$	−233.201	−0.493024
$474$	0	0
$475$	−9.90608	−0.0208549
$476$	0	0
$477$	118.509i	0.248447i
$478$	0	0
$479$	548.737i	1.14559i	0.819699	+	0.572794i	$0.194140\pi$
$479$	548.737i	1.14559i	−0.819699	+	0.572794i	$0.805860\pi$
$480$	0	0
$481$	4.39045i	0.00912775i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 608.155i	− 1.25393i
$486$	0	0
$487$	−567.876	−1.16607	−0.583034	−	0.812447i	$-0.698135\pi$
$487$	−567.876	−1.16607	−0.583034	+	0.812447i	$0.698135\pi$
$488$	0	0
$489$	817.617i	1.67202i
$490$	0	0
$491$	− 78.8005i	− 0.160490i	−0.996775	−	0.0802449i	$-0.974430\pi$
$491$	− 78.8005i	− 0.160490i	0.996775	−	0.0802449i	$-0.0255702\pi$
$492$	0	0
$493$	58.9053	0.119483
$494$	0	0
$495$	− 199.125i	− 0.402273i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	335.939i	0.673225i	0.941643	+	0.336612i	$0.109281\pi$
$499$	335.939i	0.673225i	−0.941643	+	0.336612i	$0.890719\pi$
$500$	0	0
$501$	− 249.070i	− 0.497146i
$502$	0	0
$503$	274.052i	0.544836i	0.962179	+	0.272418i	$0.0878233\pi$
$503$	274.052i	0.544836i	−0.962179	+	0.272418i	$0.912177\pi$
$504$	0	0
$505$	152.817	0.302609
$506$	0	0
$507$	538.990	1.06310
$508$	0	0
$509$	336.017	0.660152	0.330076	−	0.943954i	$-0.392926\pi$
$509$	336.017	0.660152	0.330076	+	0.943954i	$0.392926\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	34.0213	0.0663183
$514$	0	0
$515$	− 434.521i	− 0.843730i
$516$	0	0
$517$	−639.829	−1.23758
$518$	0	0
$519$	−126.069	−0.242908
$520$	0	0
$521$	− 632.282i	− 1.21359i	−0.794857	−	0.606796i	$-0.792454\pi$
$521$	− 632.282i	− 1.21359i	0.794857	−	0.606796i	$-0.207546\pi$
$522$	0	0
$523$	−779.246	−1.48995	−0.744977	−	0.667090i	$-0.767539\pi$
$523$	−779.246	−1.48995	−0.744977	+	0.667090i	$0.767539\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.263755	0.000500483 0
$528$	0	0
$529$	1186.11	2.24217
$530$	0	0
$531$	−177.165	−0.333644
$532$	0	0
$533$	− 228.907i	− 0.429470i
$534$	0	0
$535$	− 461.721i	− 0.863030i
$536$	0	0
$537$	− 809.338i	− 1.50715i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	673.903i	1.24566i	0.782356	+	0.622831i	$0.214018\pi$
$541$	673.903i	1.24566i	−0.782356	+	0.622831i	$0.785982\pi$
$542$	0	0
$543$	995.487	1.83331
$544$	0	0
$545$	− 227.217i	− 0.416913i
$546$	0	0
$547$	52.5329i	0.0960382i	0.998846	+	0.0480191i	$0.0152908\pi$
$547$	52.5329i	0.0960382i	−0.998846	+	0.0480191i	$0.984709\pi$
$548$	0	0
$549$	247.760	0.451293
$550$	0	0
$551$	5.82917i	0.0105793i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	19.8082i	0.0356904i
$556$	0	0
$557$	− 783.029i	− 1.40580i	−0.711290	−	0.702899i	$-0.751889\pi$
$557$	− 783.029i	− 1.40580i	0.711290	−	0.702899i	$-0.248111\pi$
$558$	0	0
$559$	42.4528i	0.0759441i
$560$	0	0
$561$	−957.628	−1.70700
$562$	0	0
$563$	−892.403	−1.58509	−0.792543	−	0.609816i	$-0.791243\pi$
$563$	−892.403	−1.58509	−0.792543	+	0.609816i	$0.791243\pi$
$564$	0	0
$565$	−196.149	−0.347166
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−297.443	−0.522747	−0.261373	−	0.965238i	$-0.584175\pi$
$569$	−297.443	−0.522747	−0.261373	+	0.965238i	$0.584175\pi$
$570$	0	0
$571$	− 252.746i	− 0.442638i	−0.975202	−	0.221319i	$-0.928964\pi$
$571$	− 252.746i	− 0.442638i	0.975202	−	0.221319i	$-0.0710362\pi$
$572$	0	0
$573$	480.561	0.838676
$574$	0	0
$575$	−263.479	−0.458225
$576$	0	0
$577$	882.716i	1.52984i	0.644127	+	0.764918i	$0.277221\pi$
$577$	882.716i	1.52984i	−0.644127	+	0.764918i	$0.722779\pi$
$578$	0	0
$579$	224.445	0.387643
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−821.973	−1.40990
$584$	0	0
$585$	−36.2496	−0.0619650
$586$	0	0
$587$	−66.7814	−0.113767	−0.0568836	−	0.998381i	$-0.518116\pi$
$587$	−66.7814	−0.113767	−0.0568836	+	0.998381i	$0.518116\pi$
$588$	0	0
$589$	0.0261007i	0 4.43136e-5i
$590$	0	0
$591$	678.582i	1.14819i
$592$	0	0
$593$	360.164i	0.607360i	0.952774	+	0.303680i	$0.0982153\pi$
$593$	360.164i	0.607360i	−0.952774	+	0.303680i	$0.901785\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	230.340i	0.385828i
$598$	0	0
$599$	198.044	0.330624	0.165312	−	0.986241i	$-0.447137\pi$
$599$	198.044	0.330624	0.165312	+	0.986241i	$0.447137\pi$
$600$	0	0
$601$	373.907i	0.622141i	0.950387	+	0.311071i	$0.100688\pi$
$601$	373.907i	0.622141i	−0.950387	+	0.311071i	$0.899312\pi$
$602$	0	0
$603$	− 35.9728i	− 0.0596565i
$604$	0	0
$605$	858.746	1.41942
$606$	0	0
$607$	231.130i	0.380774i	0.981709	+	0.190387i	$0.0609743\pi$
$607$	231.130i	0.380774i	−0.981709	+	0.190387i	$0.939026\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	116.477i	0.190634i
$612$	0	0
$613$	513.516i	0.837710i	0.908053	+	0.418855i	$0.137569\pi$
$613$	513.516i	0.837710i	−0.908053	+	0.418855i	$0.862431\pi$
$614$	0	0
$615$	− 1032.75i	− 1.67927i
$616$	0	0
$617$	−1119.01	−1.81363	−0.906815	−	0.421529i	$-0.861493\pi$
$617$	−1119.01	−1.81363	−0.906815	+	0.421529i	$0.861493\pi$
$618$	0	0
$619$	−128.204	−0.207114	−0.103557	−	0.994624i	$-0.533022\pi$
$619$	−128.204	−0.207114	−0.103557	+	0.994624i	$0.533022\pi$
$620$	0	0
$621$	904.890	1.45715
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−425.471	−0.680753
$626$	0	0
$627$	− 94.7652i	− 0.151141i
$628$	0	0
$629$	21.2160	0.0337297
$630$	0	0
$631$	−313.995	−0.497615	−0.248808	−	0.968553i	$-0.580039\pi$
$631$	−313.995	−0.497615	−0.248808	+	0.968553i	$0.580039\pi$
$632$	0	0
$633$	− 211.469i	− 0.334075i
$634$	0	0
$635$	542.645	0.854559
$636$	0	0
$637$	0	0
$638$	0	0
$639$	195.237	0.305535
$640$	0	0
$641$	−231.188	−0.360668	−0.180334	−	0.983605i	$-0.557718\pi$
$641$	−231.188	−0.360668	−0.180334	+	0.983605i	$0.557718\pi$
$642$	0	0
$643$	637.869	0.992020	0.496010	−	0.868317i	$-0.334798\pi$
$643$	637.869	0.992020	0.496010	+	0.868317i	$0.334798\pi$
$644$	0	0
$645$	191.532i	0.296949i
$646$	0	0
$647$	677.187i	1.04666i	0.852131	+	0.523329i	$0.175310\pi$
$647$	677.187i	1.04666i	−0.852131	+	0.523329i	$0.824690\pi$
$648$	0	0
$649$	− 1228.81i	− 1.89339i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1057.73i	− 1.61980i	−0.586567	−	0.809901i	$-0.699521\pi$
$653$	− 1057.73i	− 1.61980i	0.586567	−	0.809901i	$-0.300479\pi$
$654$	0	0
$655$	489.138	0.746775
$656$	0	0
$657$	− 137.058i	− 0.208612i
$658$	0	0
$659$	− 644.502i	− 0.978000i	−0.872284	−	0.489000i	$-0.837362\pi$
$659$	− 644.502i	− 0.978000i	0.872284	−	0.489000i	$-0.162638\pi$
$660$	0	0
$661$	1120.14	1.69461	0.847306	−	0.531105i	$-0.178223\pi$
$661$	1120.14	1.69461	0.847306	+	0.531105i	$0.178223\pi$
$662$	0	0
$663$	174.330i	0.262942i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	155.043i	0.232448i
$668$	0	0
$669$	− 393.104i	− 0.587600i
$670$	0	0
$671$	1718.45i	2.56103i
$672$	0	0
$673$	−307.811	−0.457371	−0.228686	−	0.973500i	$-0.573443\pi$
$673$	−307.811	−0.457371	−0.228686	+	0.973500i	$0.573443\pi$
$674$	0	0
$675$	−139.012	−0.205943
$676$	0	0
$677$	−1015.55	−1.50007	−0.750033	−	0.661400i	$-0.769963\pi$
$677$	−1015.55	−1.50007	−0.750033	+	0.661400i	$0.769963\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−192.278	−0.282346
$682$	0	0
$683$	− 970.203i	− 1.42050i	−0.703948	−	0.710251i	$-0.748581\pi$
$683$	− 970.203i	− 1.42050i	0.703948	−	0.710251i	$-0.251419\pi$
$684$	0	0
$685$	−338.188	−0.493706
$686$	0	0
$687$	402.680	0.586143
$688$	0	0
$689$	149.635i	0.217178i
$690$	0	0
$691$	−549.161	−0.794734	−0.397367	−	0.917660i	$-0.630076\pi$
$691$	−549.161	−0.794734	−0.397367	+	0.917660i	$0.630076\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	643.423	0.925788
$696$	0	0
$697$	−1106.15	−1.58702
$698$	0	0
$699$	83.9818	0.120146
$700$	0	0
$701$	− 452.665i	− 0.645742i	−0.946443	−	0.322871i	$-0.895352\pi$
$701$	− 452.665i	− 0.645742i	0.946443	−	0.322871i	$-0.104648\pi$
$702$	0	0
$703$	2.09950i	0.00298649i
$704$	0	0
$705$	525.505i	0.745397i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 703.414i	− 0.992121i	−0.868288	−	0.496060i	$-0.834779\pi$
$709$	− 703.414i	− 0.992121i	0.868288	−	0.496060i	$-0.165221\pi$
$710$	0	0
$711$	60.2109	0.0846848
$712$	0	0
$713$	0.694220i	0 0.000973661i
$714$	0	0
$715$	− 251.425i	− 0.351644i
$716$	0	0
$717$	−854.734	−1.19210
$718$	0	0
$719$	62.4878i	0.0869093i	0.999055	+	0.0434546i	$0.0138364\pi$
$719$	62.4878i	0.0869093i	−0.999055	+	0.0434546i	$0.986164\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 382.615i	− 0.529205i
$724$	0	0
$725$	− 23.8181i	− 0.0328525i
$726$	0	0
$727$	889.995i	1.22420i	0.790779	+	0.612101i	$0.209676\pi$
$727$	889.995i	1.22420i	−0.790779	+	0.612101i	$0.790324\pi$
$728$	0	0
$729$	417.569	0.572797
$730$	0	0
$731$	205.145	0.280636
$732$	0	0
$733$	−912.254	−1.24455	−0.622274	−	0.782800i	$-0.713791\pi$
$733$	−912.254	−1.24455	−0.622274	+	0.782800i	$0.713791\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	249.506	0.338543
$738$	0	0
$739$	1248.83i	1.68989i	0.534852	+	0.844946i	$0.320367\pi$
$739$	1248.83i	1.68989i	−0.534852	+	0.844946i	$0.679633\pi$
$740$	0	0
$741$	−17.2514	−0.0232813
$742$	0	0
$743$	305.880	0.411682	0.205841	−	0.978585i	$-0.434007\pi$
$743$	305.880	0.411682	0.205841	+	0.978585i	$0.434007\pi$
$744$	0	0
$745$	368.079i	0.494066i
$746$	0	0
$747$	−264.290	−0.353802
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−517.791	−0.689468	−0.344734	−	0.938700i	$-0.612031\pi$
$751$	−517.791	−0.689468	−0.344734	+	0.938700i	$0.612031\pi$
$752$	0	0
$753$	412.577	0.547911
$754$	0	0
$755$	569.012	0.753659
$756$	0	0
$757$	939.898i	1.24161i	0.783965	+	0.620804i	$0.213194\pi$
$757$	939.898i	1.24161i	−0.783965	+	0.620804i	$0.786806\pi$
$758$	0	0
$759$	− 2520.54i	− 3.32087i
$760$	0	0
$761$	1127.86i	1.48208i	0.671461	+	0.741039i	$0.265667\pi$
$761$	1127.86i	1.48208i	−0.671461	+	0.741039i	$0.734333\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	175.169i	0.228979i
$766$	0	0
$767$	−223.698	−0.291653
$768$	0	0
$769$	− 300.115i	− 0.390267i	−0.980777	−	0.195133i	$-0.937486\pi$
$769$	− 300.115i	− 0.390267i	0.980777	−	0.195133i	$-0.0625139\pi$
$770$	0	0
$771$	− 356.524i	− 0.462417i
$772$	0	0
$773$	−750.240	−0.970557	−0.485278	−	0.874360i	$-0.661282\pi$
$773$	−750.240	−0.970557	−0.485278	+	0.874360i	$0.661282\pi$
$774$	0	0
$775$	− 0.106648i	0 0.000137610i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 109.463i	− 0.140517i
$780$	0	0
$781$	1354.15i	1.73387i
$782$	0	0
$783$	81.8005i	0.104471i
$784$	0	0
$785$	1058.16	1.34797
$786$	0	0
$787$	289.552	0.367919	0.183960	−	0.982934i	$-0.441108\pi$
$787$	289.552	0.367919	0.183960	+	0.982934i	$0.441108\pi$
$788$	0	0
$789$	−356.195	−0.451452
$790$	0	0
$791$	0	0
$792$	0	0
$793$	312.834	0.394494
$794$	0	0
$795$	675.103i	0.849186i
$796$	0	0
$797$	1086.57	1.36332	0.681659	−	0.731670i	$-0.261259\pi$
$797$	1086.57	1.36332	0.681659	+	0.731670i	$0.261259\pi$
$798$	0	0
$799$	562.854	0.704448
$800$	0	0
$801$	228.385i	0.285124i
$802$	0	0
$803$	950.627	1.18384
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1037.61	−1.28576
$808$	0	0
$809$	181.949	0.224906	0.112453	−	0.993657i	$-0.464129\pi$
$809$	181.949	0.224906	0.112453	+	0.993657i	$0.464129\pi$
$810$	0	0
$811$	−1005.31	−1.23960	−0.619799	−	0.784760i	$-0.712786\pi$
$811$	−1005.31	−1.23960	−0.619799	+	0.784760i	$0.712786\pi$
$812$	0	0
$813$	− 349.280i	− 0.429619i
$814$	0	0
$815$	1037.33i	1.27280i
$816$	0	0
$817$	20.3008i	0.0248480i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	982.661i	1.19691i	0.801158	+	0.598453i	$0.204218\pi$
$821$	982.661i	1.19691i	−0.801158	+	0.598453i	$0.795782\pi$
$822$	0	0
$823$	−1485.01	−1.80439	−0.902194	−	0.431331i	$-0.858044\pi$
$823$	−1485.01	−1.80439	−0.902194	+	0.431331i	$0.858044\pi$
$824$	0	0
$825$	387.212i	0.469348i
$826$	0	0
$827$	708.113i	0.856243i	0.903721	+	0.428121i	$0.140824\pi$
$827$	708.113i	0.856243i	−0.903721	+	0.428121i	$0.859176\pi$
$828$	0	0
$829$	−150.033	−0.180980	−0.0904902	−	0.995897i	$-0.528843\pi$
$829$	−150.033	−0.180980	−0.0904902	+	0.995897i	$0.528843\pi$
$830$	0	0
$831$	− 56.6722i	− 0.0681976i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 316.001i	− 0.378445i
$836$	0	0
$837$	0.366270i	0 0.000437599i
$838$	0	0
$839$	− 1106.41i	− 1.31873i	−0.751824	−	0.659364i	$-0.770826\pi$
$839$	− 1106.41i	− 1.31873i	0.751824	−	0.659364i	$-0.229174\pi$
$840$	0	0
$841$	826.984	0.983335
$842$	0	0
$843$	−258.028	−0.306083
$844$	0	0
$845$	683.830	0.809266
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−297.167	−0.350020
$850$	0	0
$851$	55.8420i	0.0656192i
$852$	0	0
$853$	−1243.82	−1.45817	−0.729086	−	0.684423i	$-0.760054\pi$
$853$	−1243.82	−1.45817	−0.729086	+	0.684423i	$0.760054\pi$
$854$	0	0
$855$	−17.3344	−0.0202742
$856$	0	0
$857$	283.652i	0.330982i	0.986211	+	0.165491i	$0.0529209\pi$
$857$	283.652i	0.330982i	−0.986211	+	0.165491i	$0.947079\pi$
$858$	0	0
$859$	−911.799	−1.06147	−0.530733	−	0.847539i	$-0.678083\pi$
$859$	−911.799	−1.06147	−0.530733	+	0.847539i	$0.678083\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	873.816	1.01253	0.506266	−	0.862377i	$-0.331025\pi$
$863$	873.816	1.01253	0.506266	+	0.862377i	$0.331025\pi$
$864$	0	0
$865$	−159.947	−0.184910
$866$	0	0
$867$	−140.978	−0.162604
$868$	0	0
$869$	417.620i	0.480575i
$870$	0	0
$871$	− 45.4211i	− 0.0521482i
$872$	0	0
$873$	363.268i	0.416114i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	634.480i	0.723467i	0.932282	+	0.361733i	$0.117815\pi$
$877$	634.480i	0.723467i	−0.932282	+	0.361733i	$0.882185\pi$
$878$	0	0
$879$	−93.5951	−0.106479
$880$	0	0
$881$	− 670.044i	− 0.760549i	−0.924874	−	0.380274i	$-0.875830\pi$
$881$	− 670.044i	− 0.760549i	0.924874	−	0.380274i	$-0.124170\pi$
$882$	0	0
$883$	875.514i	0.991522i	0.868459	+	0.495761i	$0.165111\pi$
$883$	875.514i	0.991522i	−0.868459	+	0.495761i	$0.834889\pi$
$884$	0	0
$885$	−1009.25	−1.14039
$886$	0	0
$887$	986.289i	1.11194i	0.831203	+	0.555969i	$0.187653\pi$
$887$	986.289i	1.11194i	−0.831203	+	0.555969i	$0.812347\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 1744.95i	− 1.95842i
$892$	0	0
$893$	55.6991i	0.0623730i
$894$	0	0
$895$	− 1026.83i	− 1.14729i
$896$	0	0
$897$	−458.850	−0.511538
$898$	0	0
$899$	−0.0627563	−6.98068e−5 0
$900$	0	0
$901$	723.085	0.802536
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1263.00	1.39558
$906$	0	0
$907$	− 866.709i	− 0.955578i	−0.878475	−	0.477789i	$-0.841438\pi$
$907$	− 866.709i	− 0.955578i	0.878475	−	0.477789i	$-0.158562\pi$
$908$	0	0
$909$	−91.2820	−0.100420
$910$	0	0
$911$	128.713	0.141288	0.0706438	−	0.997502i	$-0.477495\pi$
$911$	128.713	0.141288	0.0706438	+	0.997502i	$0.477495\pi$
$912$	0	0
$913$	− 1833.10i	− 2.00778i
$914$	0	0
$915$	1411.40	1.54251
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−860.175	−0.935990	−0.467995	−	0.883731i	$-0.655023\pi$
$919$	−860.175	−0.935990	−0.467995	+	0.883731i	$0.655023\pi$
$920$	0	0
$921$	843.871	0.916255
$922$	0	0
$923$	246.515	0.267081
$924$	0	0
$925$	− 8.57859i	− 0.00927415i
$926$	0	0
$927$	259.551i	0.279990i
$928$	0	0
$929$	− 233.279i	− 0.251107i	−0.992087	−	0.125554i	$-0.959929\pi$
$929$	− 233.279i	− 0.251107i	0.992087	−	0.125554i	$-0.0400707\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1487.13i	1.59392i
$934$	0	0
$935$	−1214.97	−1.29943
$936$	0	0
$937$	− 1426.29i	− 1.52219i	−0.648641	−	0.761095i	$-0.724662\pi$
$937$	− 1426.29i	− 1.52219i	0.648641	−	0.761095i	$-0.275338\pi$
$938$	0	0
$939$	− 281.851i	− 0.300161i
$940$	0	0
$941$	−1270.85	−1.35053	−0.675265	−	0.737575i	$-0.735971\pi$
$941$	−1270.85	−1.35053	−0.675265	+	0.737575i	$0.735971\pi$
$942$	0	0
$943$	− 2911.47i	− 3.08745i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1632.17i	1.72352i	0.507318	+	0.861759i	$0.330637\pi$
$947$	1632.17i	1.72352i	−0.507318	+	0.861759i	$0.669363\pi$
$948$	0	0
$949$	− 173.056i	− 0.182356i
$950$	0	0
$951$	− 832.098i	− 0.874972i
$952$	0	0
$953$	95.9158	0.100646	0.0503231	−	0.998733i	$-0.483975\pi$
$953$	95.9158	0.100646	0.0503231	+	0.998733i	$0.483975\pi$
$954$	0	0
$955$	609.700	0.638429
$956$	0	0
$957$	227.853	0.238090
$958$	0	0
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	275.799i	0.286395i
$964$	0	0
$965$	284.759	0.295087
$966$	0	0
$967$	−1419.97	−1.46843	−0.734216	−	0.678916i	$-0.762450\pi$
$967$	−1419.97	−1.46843	−0.734216	+	0.678916i	$0.762450\pi$
$968$	0	0
$969$	83.3644i	0.0860313i
$970$	0	0
$971$	659.635	0.679336	0.339668	−	0.940545i	$-0.389685\pi$
$971$	659.635	0.679336	0.339668	+	0.940545i	$0.389685\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	70.4897	0.0722971
$976$	0	0
$977$	−1914.30	−1.95937	−0.979683	−	0.200550i	$-0.935727\pi$
$977$	−1914.30	−1.95937	−0.979683	+	0.200550i	$0.935727\pi$
$978$	0	0
$979$	−1584.07	−1.61804
$980$	0	0
$981$	135.723i	0.138352i
$982$	0	0
$983$	− 223.613i	− 0.227480i	−0.993511	−	0.113740i	$-0.963717\pi$
$983$	− 223.613i	− 0.227480i	0.993511	−	0.113740i	$-0.0362831\pi$
$984$	0	0
$985$	860.934i	0.874045i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	539.956i	0.545961i
$990$	0	0
$991$	1472.74	1.48612	0.743058	−	0.669227i	$-0.233374\pi$
$991$	1472.74	1.48612	0.743058	+	0.669227i	$0.233374\pi$
$992$	0	0
$993$	260.472i	0.262308i
$994$	0	0
$995$	292.237i	0.293706i
$996$	0	0
$997$	−106.896	−0.107218	−0.0536088	−	0.998562i	$-0.517072\pi$
$997$	−106.896	−0.107218	−0.0536088	+	0.998562i	$0.517072\pi$
$998$	0	0
$999$	29.4622i	0.0294917i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.h.a.881.8		28
4.3	odd	2		392.3.h.a.293.10		28
7.4	even	3		224.3.n.a.145.11		28
7.5	odd	6		224.3.n.a.17.4		28
7.6	odd	2	inner	1568.3.h.a.881.22		28
8.3	odd	2		392.3.h.a.293.11		28
8.5	even	2	inner	1568.3.h.a.881.21		28
28.3	even	6		392.3.j.e.117.14		28
28.11	odd	6		56.3.j.a.5.14	yes	28
28.19	even	6		56.3.j.a.45.5	yes	28
28.23	odd	6		392.3.j.e.325.5		28
28.27	even	2		392.3.h.a.293.9		28
56.3	even	6		392.3.j.e.117.5		28
56.5	odd	6		224.3.n.a.17.11		28
56.11	odd	6		56.3.j.a.5.5	✓	28
56.13	odd	2	inner	1568.3.h.a.881.7		28
56.19	even	6		56.3.j.a.45.14	yes	28
56.27	even	2		392.3.h.a.293.12		28
56.51	odd	6		392.3.j.e.325.14		28
56.53	even	6		224.3.n.a.145.4		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.j.a.5.5	✓	28	56.11	odd	6
56.3.j.a.5.14	yes	28	28.11	odd	6
56.3.j.a.45.5	yes	28	28.19	even	6
56.3.j.a.45.14	yes	28	56.19	even	6
224.3.n.a.17.4		28	7.5	odd	6
224.3.n.a.17.11		28	56.5	odd	6
224.3.n.a.145.4		28	56.53	even	6
224.3.n.a.145.11		28	7.4	even	3
392.3.h.a.293.9		28	28.27	even	2
392.3.h.a.293.10		28	4.3	odd	2
392.3.h.a.293.11		28	8.3	odd	2
392.3.h.a.293.12		28	56.27	even	2
392.3.j.e.117.5		28	56.3	even	6
392.3.j.e.117.14		28	28.3	even	6
392.3.j.e.325.5		28	28.23	odd	6
392.3.j.e.325.14		28	56.51	odd	6
1568.3.h.a.881.7		28	56.13	odd	2	inner
1568.3.h.a.881.8		28	1.1	even	1	trivial
1568.3.h.a.881.21		28	8.5	even	2	inner
1568.3.h.a.881.22		28	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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.40276 q^{3} -4.31716 q^{5} +2.57876 q^{9} +O(q^{10})\)	\(q-3.40276 q^{3} -4.31716 q^{5} +2.57876 q^{9} +17.8862i q^{11} +3.25607 q^{13} +14.6903 q^{15} -15.7343i q^{17} +1.55704 q^{19} +41.4139 q^{23} -6.36211 q^{25} +21.8499 q^{27} +3.74374i q^{29} +0.0167630i q^{31} -60.8622i q^{33} +1.34839i q^{37} -11.0796 q^{39} -70.3018i q^{41} +13.0380i q^{43} -11.1329 q^{45} +35.7723i q^{47} +53.5402i q^{51} +45.9558i q^{53} -77.2174i q^{55} -5.29824 q^{57} -68.7017 q^{59} +96.0771 q^{61} -14.0570 q^{65} -13.9497i q^{67} -140.921 q^{69} +75.7095 q^{71} -53.1487i q^{73} +21.6487 q^{75} +23.3488 q^{79} -97.5588 q^{81} -102.487 q^{83} +67.9277i q^{85} -12.7390i q^{87} +88.5638i q^{89} -0.0570404i q^{93} -6.72201 q^{95} +140.869i q^{97} +46.1241i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 64 q^{9}+O(q^{10}) \) 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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