LMFDB - Embedded newform 1568.3.c.h.97.4

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} + 36x^{14} + 522x^{12} + 3644x^{10} + 12219x^{8} + 15156x^{6} + 15478x^{4} - 10992x^{2} + 11025 $ x^16 + 36x^14 + 522x^12 + 3644x^10 + 12219x^8 + 15156x^6 + 15478x^4 - 10992*x^2 + 11025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.4
Root			$0.707107 - 2.60548i$ of defining polynomial
Character	$\chi$	$=$	1568.97
Dual form			1568.3.c.h.97.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.68470i q^{3} +3.04770i q^{5} -4.57700 q^{9} +O(q^{10})$	$q-3.68470i q^{3} +3.04770i q^{5} -4.57700 q^{9} +2.35033 q^{11} -25.3073i q^{13} +11.2298 q^{15} +3.56426i q^{17} +16.3704i q^{19} -17.6683 q^{23} +15.7115 q^{25} -16.2974i q^{27} +36.1220 q^{29} +7.22414i q^{31} -8.66025i q^{33} +36.8043 q^{37} -93.2498 q^{39} -53.7118i q^{41} -51.2382 q^{43} -13.9493i q^{45} +31.3627i q^{47} +13.1332 q^{51} -70.2274 q^{53} +7.16309i q^{55} +60.3201 q^{57} -94.0044i q^{59} +2.18942i q^{61} +77.1290 q^{65} +24.9619 q^{67} +65.1022i q^{69} +50.8890 q^{71} -79.5598i q^{73} -57.8923i q^{75} -115.023 q^{79} -101.244 q^{81} -154.132i q^{83} -10.8628 q^{85} -133.099i q^{87} -114.001i q^{89} +26.6188 q^{93} -49.8921 q^{95} -53.9940i q^{97} -10.7575 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^9	$ 16 q - 16 q^{9} - 32 q^{25} + 112 q^{29} - 16 q^{37} + 48 q^{53} - 528 q^{57} + 16 q^{65} - 64 q^{81} + 720 q^{85} + 464 q^{93}+O(q^{100}) $ 16 * q - 16 * q^9 - 32 * q^25 + 112 * q^29 - 16 * q^37 + 48 * q^53 - 528 * q^57 + 16 * q^65 - 64 * q^81 + 720 * q^85 + 464 * 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
$3$	− 3.68470i	− 1.22823i	−0.789215	−	0.614116i	$-0.789513\pi$
$3$	− 3.68470i	− 1.22823i	0.789215	−	0.614116i	$-0.210487\pi$
$4$	0	0
$5$	3.04770i	0.609540i	0.952426	+	0.304770i	$0.0985795\pi$
$5$	3.04770i	0.609540i	−0.952426	+	0.304770i	$0.901420\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−4.57700	−0.508556
$10$	0	0
$11$	2.35033	0.213666	0.106833	−	0.994277i	$-0.465929\pi$
$11$	2.35033	0.213666	0.106833	+	0.994277i	$0.465929\pi$
$12$	0	0
$13$	− 25.3073i	− 1.94672i	−0.229292	−	0.973358i	$-0.573641\pi$
$13$	− 25.3073i	− 1.94672i	0.229292	−	0.973358i	$-0.426359\pi$
$14$	0	0
$15$	11.2298	0.748656
$16$	0	0
$17$	3.56426i	0.209662i	0.994490	+	0.104831i	$0.0334302\pi$
$17$	3.56426i	0.209662i	−0.994490	+	0.104831i	$0.966570\pi$
$18$	0	0
$19$	16.3704i	0.861602i	0.902447	+	0.430801i	$0.141769\pi$
$19$	16.3704i	0.861602i	−0.902447	+	0.430801i	$0.858231\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−17.6683	−0.768185	−0.384093	−	0.923295i	$-0.625486\pi$
$23$	−17.6683	−0.768185	−0.384093	+	0.923295i	$0.625486\pi$
$24$	0	0
$25$	15.7115	0.628462
$26$	0	0
$27$	− 16.2974i	− 0.603608i
$28$	0	0
$29$	36.1220	1.24559	0.622793	−	0.782387i	$-0.285998\pi$
$29$	36.1220	1.24559	0.622793	+	0.782387i	$0.285998\pi$
$30$	0	0
$31$	7.22414i	0.233037i	0.993189	+	0.116518i	$0.0371734\pi$
$31$	7.22414i	0.233037i	−0.993189	+	0.116518i	$0.962827\pi$
$32$	0	0
$33$	− 8.66025i	− 0.262432i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	36.8043	0.994711	0.497355	−	0.867547i	$-0.334305\pi$
$37$	36.8043	0.994711	0.497355	+	0.867547i	$0.334305\pi$
$38$	0	0
$39$	−93.2498	−2.39102
$40$	0	0
$41$	− 53.7118i	− 1.31004i	−0.755610	−	0.655022i	$-0.772659\pi$
$41$	− 53.7118i	− 1.31004i	0.755610	−	0.655022i	$-0.227341\pi$
$42$	0	0
$43$	−51.2382	−1.19159	−0.595793	−	0.803138i	$-0.703162\pi$
$43$	−51.2382	−1.19159	−0.595793	+	0.803138i	$0.703162\pi$
$44$	0	0
$45$	− 13.9493i	− 0.309985i
$46$	0	0
$47$	31.3627i	0.667292i	0.942698	+	0.333646i	$0.108279\pi$
$47$	31.3627i	0.667292i	−0.942698	+	0.333646i	$0.891721\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	13.1332	0.257514
$52$	0	0
$53$	−70.2274	−1.32504	−0.662522	−	0.749042i	$-0.730514\pi$
$53$	−70.2274	−1.32504	−0.662522	+	0.749042i	$0.730514\pi$
$54$	0	0
$55$	7.16309i	0.130238i
$56$	0	0
$57$	60.3201	1.05825
$58$	0	0
$59$	− 94.0044i	− 1.59329i	−0.604444	−	0.796647i	$-0.706605\pi$
$59$	− 94.0044i	− 1.59329i	0.604444	−	0.796647i	$-0.293395\pi$
$60$	0	0
$61$	2.18942i	0.0358921i	0.999839	+	0.0179461i	$0.00571271\pi$
$61$	2.18942i	0.0358921i	−0.999839	+	0.0179461i	$0.994287\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	77.1290	1.18660
$66$	0	0
$67$	24.9619	0.372566	0.186283	−	0.982496i	$-0.440356\pi$
$67$	24.9619	0.372566	0.186283	+	0.982496i	$0.440356\pi$
$68$	0	0
$69$	65.1022i	0.943510i
$70$	0	0
$71$	50.8890	0.716746	0.358373	−	0.933579i	$-0.383332\pi$
$71$	50.8890	0.716746	0.358373	+	0.933579i	$0.383332\pi$
$72$	0	0
$73$	− 79.5598i	− 1.08986i	−0.838481	−	0.544930i	$-0.816556\pi$
$73$	− 79.5598i	− 1.08986i	0.838481	−	0.544930i	$-0.183444\pi$
$74$	0	0
$75$	− 57.8923i	− 0.771897i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−115.023	−1.45599	−0.727996	−	0.685581i	$-0.759548\pi$
$79$	−115.023	−1.45599	−0.727996	+	0.685581i	$0.759548\pi$
$80$	0	0
$81$	−101.244	−1.24993
$82$	0	0
$83$	− 154.132i	− 1.85701i	−0.371318	−	0.928506i	$-0.621094\pi$
$83$	− 154.132i	− 1.85701i	0.371318	−	0.928506i	$-0.378906\pi$
$84$	0	0
$85$	−10.8628	−0.127797
$86$	0	0
$87$	− 133.099i	− 1.52987i
$88$	0	0
$89$	− 114.001i	− 1.28091i	−0.767998	−	0.640453i	$-0.778747\pi$
$89$	− 114.001i	− 1.28091i	0.767998	−	0.640453i	$-0.221253\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	26.6188	0.286224
$94$	0	0
$95$	−49.8921	−0.525180
$96$	0	0
$97$	− 53.9940i	− 0.556640i	−0.960488	−	0.278320i	$-0.910222\pi$
$97$	− 53.9940i	− 0.556640i	0.960488	−	0.278320i	$-0.0897775\pi$
$98$	0	0
$99$	−10.7575	−0.108661
$100$	0	0
$101$	20.8198i	0.206136i	0.994674	+	0.103068i	$0.0328660\pi$
$101$	20.8198i	0.206136i	−0.994674	+	0.103068i	$0.967134\pi$
$102$	0	0
$103$	122.248i	1.18687i	0.804882	+	0.593435i	$0.202229\pi$
$103$	122.248i	1.18687i	−0.804882	+	0.593435i	$0.797771\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	114.536	1.07043	0.535216	−	0.844715i	$-0.320230\pi$
$107$	114.536	1.07043	0.535216	+	0.844715i	$0.320230\pi$
$108$	0	0
$109$	165.811	1.52121	0.760603	−	0.649218i	$-0.224904\pi$
$109$	165.811	1.52121	0.760603	+	0.649218i	$0.224904\pi$
$110$	0	0
$111$	− 135.613i	− 1.22174i
$112$	0	0
$113$	−123.071	−1.08912	−0.544560	−	0.838722i	$-0.683303\pi$
$113$	−123.071	−1.08912	−0.544560	+	0.838722i	$0.683303\pi$
$114$	0	0
$115$	− 53.8475i	− 0.468239i
$116$	0	0
$117$	115.832i	0.990014i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−115.476	−0.954347
$122$	0	0
$123$	−197.912	−1.60904
$124$	0	0
$125$	124.076i	0.992612i
$126$	0	0
$127$	−160.105	−1.26067	−0.630334	−	0.776324i	$-0.717082\pi$
$127$	−160.105	−1.26067	−0.630334	+	0.776324i	$0.717082\pi$
$128$	0	0
$129$	188.797i	1.46355i
$130$	0	0
$131$	− 61.5770i	− 0.470053i	−0.971989	−	0.235027i	$-0.924482\pi$
$131$	− 61.5770i	− 0.470053i	0.971989	−	0.235027i	$-0.0755177\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	49.6696	0.367923
$136$	0	0
$137$	95.1022	0.694176	0.347088	−	0.937832i	$-0.387170\pi$
$137$	95.1022	0.694176	0.347088	+	0.937832i	$0.387170\pi$
$138$	0	0
$139$	92.0558i	0.662272i	0.943583	+	0.331136i	$0.107432\pi$
$139$	92.0558i	0.662272i	−0.943583	+	0.331136i	$0.892568\pi$
$140$	0	0
$141$	115.562	0.819590
$142$	0	0
$143$	− 59.4805i	− 0.415947i
$144$	0	0
$145$	110.089i	0.759234i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−177.445	−1.19091	−0.595453	−	0.803390i	$-0.703027\pi$
$149$	−177.445	−1.19091	−0.595453	+	0.803390i	$0.703027\pi$
$150$	0	0
$151$	−229.788	−1.52178	−0.760888	−	0.648884i	$-0.775236\pi$
$151$	−229.788	−1.52178	−0.760888	+	0.648884i	$0.775236\pi$
$152$	0	0
$153$	− 16.3136i	− 0.106625i
$154$	0	0
$155$	−22.0170	−0.142045
$156$	0	0
$157$	− 49.6372i	− 0.316161i	−0.987426	−	0.158080i	$-0.949470\pi$
$157$	− 49.6372i	− 0.316161i	0.987426	−	0.158080i	$-0.0505305\pi$
$158$	0	0
$159$	258.767i	1.62746i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−66.5227	−0.408115	−0.204057	−	0.978959i	$-0.565413\pi$
$163$	−66.5227	−0.408115	−0.204057	+	0.978959i	$0.565413\pi$
$164$	0	0
$165$	26.3938	0.159963
$166$	0	0
$167$	− 164.292i	− 0.983786i	−0.870656	−	0.491893i	$-0.836305\pi$
$167$	− 164.292i	− 0.983786i	0.870656	−	0.491893i	$-0.163695\pi$
$168$	0	0
$169$	−471.460	−2.78970
$170$	0	0
$171$	− 74.9275i	− 0.438173i
$172$	0	0
$173$	38.6819i	0.223595i	0.993731	+	0.111797i	$0.0356607\pi$
$173$	38.6819i	0.223595i	−0.993731	+	0.111797i	$0.964339\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−346.378	−1.95694
$178$	0	0
$179$	−102.415	−0.572152	−0.286076	−	0.958207i	$-0.592351\pi$
$179$	−102.415	−0.572152	−0.286076	+	0.958207i	$0.592351\pi$
$180$	0	0
$181$	44.5843i	0.246322i	0.992387	+	0.123161i	$0.0393032\pi$
$181$	44.5843i	0.246322i	−0.992387	+	0.123161i	$0.960697\pi$
$182$	0	0
$183$	8.06735	0.0440839
$184$	0	0
$185$	112.168i	0.606315i
$186$	0	0
$187$	8.37718i	0.0447977i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−330.062	−1.72808	−0.864038	−	0.503427i	$-0.832072\pi$
$191$	−330.062	−1.72808	−0.864038	+	0.503427i	$0.832072\pi$
$192$	0	0
$193$	139.355	0.722048	0.361024	−	0.932556i	$-0.382427\pi$
$193$	139.355	0.722048	0.361024	+	0.932556i	$0.382427\pi$
$194$	0	0
$195$	− 284.197i	− 1.45742i
$196$	0	0
$197$	174.724	0.886925	0.443462	−	0.896293i	$-0.353750\pi$
$197$	174.724	0.886925	0.443462	+	0.896293i	$0.353750\pi$
$198$	0	0
$199$	− 227.487i	− 1.14315i	−0.820550	−	0.571574i	$-0.806333\pi$
$199$	− 227.487i	− 1.14315i	0.820550	−	0.571574i	$-0.193667\pi$
$200$	0	0
$201$	− 91.9772i	− 0.457598i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	163.697	0.798524
$206$	0	0
$207$	80.8677	0.390665
$208$	0	0
$209$	38.4759i	0.184095i
$210$	0	0
$211$	251.350	1.19123	0.595617	−	0.803269i	$-0.296908\pi$
$211$	251.350	1.19123	0.595617	+	0.803269i	$0.296908\pi$
$212$	0	0
$213$	− 187.510i	− 0.880331i
$214$	0	0
$215$	− 156.159i	− 0.726319i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−293.154	−1.33860
$220$	0	0
$221$	90.2017	0.408153
$222$	0	0
$223$	108.297i	0.485636i	0.970072	+	0.242818i	$0.0780718\pi$
$223$	108.297i	0.485636i	−0.970072	+	0.242818i	$0.921928\pi$
$224$	0	0
$225$	−71.9118	−0.319608
$226$	0	0
$227$	− 282.953i	− 1.24649i	−0.782027	−	0.623245i	$-0.785814\pi$
$227$	− 282.953i	− 1.24649i	0.782027	−	0.623245i	$-0.214186\pi$
$228$	0	0
$229$	176.682i	0.771539i	0.922595	+	0.385769i	$0.126064\pi$
$229$	176.682i	0.771539i	−0.922595	+	0.385769i	$0.873936\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	355.385	1.52526	0.762630	−	0.646835i	$-0.223908\pi$
$233$	355.385	1.52526	0.762630	+	0.646835i	$0.223908\pi$
$234$	0	0
$235$	−95.5841	−0.406741
$236$	0	0
$237$	423.826i	1.78830i
$238$	0	0
$239$	17.5451	0.0734104	0.0367052	−	0.999326i	$-0.488314\pi$
$239$	17.5451	0.0734104	0.0367052	+	0.999326i	$0.488314\pi$
$240$	0	0
$241$	121.139i	0.502650i	0.967903	+	0.251325i	$0.0808663\pi$
$241$	121.139i	0.502650i	−0.967903	+	0.251325i	$0.919134\pi$
$242$	0	0
$243$	226.377i	0.931593i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	414.292	1.67729
$248$	0	0
$249$	−567.930	−2.28084
$250$	0	0
$251$	219.342i	0.873874i	0.899492	+	0.436937i	$0.143937\pi$
$251$	219.342i	0.873874i	−0.899492	+	0.436937i	$0.856063\pi$
$252$	0	0
$253$	−41.5262	−0.164135
$254$	0	0
$255$	40.0261i	0.156965i
$256$	0	0
$257$	− 482.026i	− 1.87559i	−0.347192	−	0.937794i	$-0.612865\pi$
$257$	− 482.026i	− 1.87559i	0.347192	−	0.937794i	$-0.387135\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−165.330	−0.633450
$262$	0	0
$263$	−89.6879	−0.341019	−0.170509	−	0.985356i	$-0.554541\pi$
$263$	−89.6879	−0.341019	−0.170509	+	0.985356i	$0.554541\pi$
$264$	0	0
$265$	− 214.032i	− 0.807667i
$266$	0	0
$267$	−420.058	−1.57325
$268$	0	0
$269$	− 38.0783i	− 0.141555i	−0.997492	−	0.0707776i	$-0.977452\pi$
$269$	− 38.0783i	− 0.141555i	0.997492	−	0.0707776i	$-0.0225481\pi$
$270$	0	0
$271$	− 84.9632i	− 0.313517i	−0.987637	−	0.156759i	$-0.949896\pi$
$271$	− 84.9632i	− 0.313517i	0.987637	−	0.156759i	$-0.0501045\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	36.9273	0.134281
$276$	0	0
$277$	62.5046	0.225648	0.112824	−	0.993615i	$-0.464010\pi$
$277$	62.5046	0.225648	0.112824	+	0.993615i	$0.464010\pi$
$278$	0	0
$279$	− 33.0649i	− 0.118512i
$280$	0	0
$281$	58.6599	0.208754	0.104377	−	0.994538i	$-0.466715\pi$
$281$	58.6599	0.208754	0.104377	+	0.994538i	$0.466715\pi$
$282$	0	0
$283$	239.556i	0.846486i	0.906016	+	0.423243i	$0.139108\pi$
$283$	239.556i	0.846486i	−0.906016	+	0.423243i	$0.860892\pi$
$284$	0	0
$285$	183.837i	0.645044i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	276.296	0.956042
$290$	0	0
$291$	−198.952	−0.683683
$292$	0	0
$293$	196.503i	0.670658i	0.942101	+	0.335329i	$0.108848\pi$
$293$	196.503i	0.670658i	−0.942101	+	0.335329i	$0.891152\pi$
$294$	0	0
$295$	286.497	0.971176
$296$	0	0
$297$	− 38.3043i	− 0.128971i
$298$	0	0
$299$	447.136i	1.49544i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	76.7146	0.253183
$304$	0	0
$305$	−6.67268	−0.0218777
$306$	0	0
$307$	− 246.955i	− 0.804415i	−0.915549	−	0.402208i	$-0.868243\pi$
$307$	− 246.955i	− 0.804415i	0.915549	−	0.402208i	$-0.131757\pi$
$308$	0	0
$309$	450.446	1.45775
$310$	0	0
$311$	340.045i	1.09339i	0.837331	+	0.546696i	$0.184115\pi$
$311$	340.045i	1.09339i	−0.837331	+	0.546696i	$0.815885\pi$
$312$	0	0
$313$	− 113.397i	− 0.362292i	−0.983456	−	0.181146i	$-0.942019\pi$
$313$	− 113.397i	− 0.362292i	0.983456	−	0.181146i	$-0.0579806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	242.310	0.764385	0.382192	−	0.924083i	$-0.375169\pi$
$317$	242.310	0.764385	0.382192	+	0.924083i	$0.375169\pi$
$318$	0	0
$319$	84.8986	0.266140
$320$	0	0
$321$	− 422.031i	− 1.31474i
$322$	0	0
$323$	−58.3484	−0.180645
$324$	0	0
$325$	− 397.617i	− 1.22344i
$326$	0	0
$327$	− 610.965i	− 1.86839i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−69.7089	−0.210601	−0.105300	−	0.994440i	$-0.533580\pi$
$331$	−69.7089	−0.210601	−0.105300	+	0.994440i	$0.533580\pi$
$332$	0	0
$333$	−168.453	−0.505866
$334$	0	0
$335$	76.0764i	0.227094i
$336$	0	0
$337$	165.816	0.492037	0.246019	−	0.969265i	$-0.420878\pi$
$337$	165.816	0.492037	0.246019	+	0.969265i	$0.420878\pi$
$338$	0	0
$339$	453.478i	1.33769i
$340$	0	0
$341$	16.9791i	0.0497921i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−198.412	−0.575107
$346$	0	0
$347$	566.904	1.63373	0.816864	−	0.576830i	$-0.195711\pi$
$347$	566.904	1.63373	0.816864	+	0.576830i	$0.195711\pi$
$348$	0	0
$349$	− 245.773i	− 0.704219i	−0.935959	−	0.352110i	$-0.885464\pi$
$349$	− 245.773i	− 0.704219i	0.935959	−	0.352110i	$-0.114536\pi$
$350$	0	0
$351$	−412.443	−1.17505
$352$	0	0
$353$	− 159.160i	− 0.450879i	−0.974257	−	0.225439i	$-0.927618\pi$
$353$	− 159.160i	− 0.450879i	0.974257	−	0.225439i	$-0.0723817\pi$
$354$	0	0
$355$	155.094i	0.436885i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	205.025	0.571101	0.285550	−	0.958364i	$-0.407824\pi$
$359$	205.025	0.571101	0.285550	+	0.958364i	$0.407824\pi$
$360$	0	0
$361$	93.0089	0.257642
$362$	0	0
$363$	425.494i	1.17216i
$364$	0	0
$365$	242.474	0.664313
$366$	0	0
$367$	1.68715i	0.00459715i	0.999997	+	0.00229858i	$0.000731660\pi$
$367$	1.68715i	0.00459715i	−0.999997	+	0.00229858i	$0.999268\pi$
$368$	0	0
$369$	245.839i	0.666231i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	463.405	1.24237	0.621186	−	0.783663i	$-0.286651\pi$
$373$	463.405	1.24237	0.621186	+	0.783663i	$0.286651\pi$
$374$	0	0
$375$	457.184	1.21916
$376$	0	0
$377$	− 914.150i	− 2.42480i
$378$	0	0
$379$	493.215	1.30136	0.650680	−	0.759352i	$-0.274484\pi$
$379$	493.215	1.30136	0.650680	+	0.759352i	$0.274484\pi$
$380$	0	0
$381$	589.938i	1.54839i
$382$	0	0
$383$	− 573.038i	− 1.49618i	−0.663595	−	0.748092i	$-0.730970\pi$
$383$	− 573.038i	− 1.49618i	0.663595	−	0.748092i	$-0.269030\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	234.518	0.605988
$388$	0	0
$389$	−46.5513	−0.119669	−0.0598346	−	0.998208i	$-0.519057\pi$
$389$	−46.5513	−0.119669	−0.0598346	+	0.998208i	$0.519057\pi$
$390$	0	0
$391$	− 62.9742i	− 0.161059i
$392$	0	0
$393$	−226.893	−0.577335
$394$	0	0
$395$	− 350.556i	− 0.887485i
$396$	0	0
$397$	− 208.704i	− 0.525704i	−0.964836	−	0.262852i	$-0.915337\pi$
$397$	− 208.704i	− 0.525704i	0.964836	−	0.262852i	$-0.0846631\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−221.190	−0.551596	−0.275798	−	0.961216i	$-0.588942\pi$
$401$	−221.190	−0.551596	−0.275798	+	0.961216i	$0.588942\pi$
$402$	0	0
$403$	182.824	0.453657
$404$	0	0
$405$	− 308.561i	− 0.761880i
$406$	0	0
$407$	86.5022	0.212536
$408$	0	0
$409$	96.4289i	0.235767i	0.993027	+	0.117884i	$0.0376110\pi$
$409$	96.4289i	0.235767i	−0.993027	+	0.117884i	$0.962389\pi$
$410$	0	0
$411$	− 350.423i	− 0.852610i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	469.748	1.13192
$416$	0	0
$417$	339.198	0.813424
$418$	0	0
$419$	239.093i	0.570627i	0.958434	+	0.285313i	$0.0920977\pi$
$419$	239.093i	0.570627i	−0.958434	+	0.285313i	$0.907902\pi$
$420$	0	0
$421$	508.228	1.20719	0.603596	−	0.797290i	$-0.293734\pi$
$421$	508.228	1.20719	0.603596	+	0.797290i	$0.293734\pi$
$422$	0	0
$423$	− 143.547i	− 0.339355i
$424$	0	0
$425$	56.0000i	0.131765i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−219.168	−0.510880
$430$	0	0
$431$	598.349	1.38828	0.694140	−	0.719840i	$-0.255785\pi$
$431$	598.349	1.38828	0.694140	+	0.719840i	$0.255785\pi$
$432$	0	0
$433$	283.405i	0.654516i	0.944935	+	0.327258i	$0.106125\pi$
$433$	283.405i	0.654516i	−0.944935	+	0.327258i	$0.893875\pi$
$434$	0	0
$435$	405.644	0.932516
$436$	0	0
$437$	− 289.237i	− 0.661870i
$438$	0	0
$439$	342.480i	0.780137i	0.920786	+	0.390068i	$0.127549\pi$
$439$	342.480i	0.780137i	−0.920786	+	0.390068i	$0.872451\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−587.817	−1.32690	−0.663451	−	0.748220i	$-0.730909\pi$
$443$	−587.817	−1.32690	−0.663451	+	0.748220i	$0.730909\pi$
$444$	0	0
$445$	347.439	0.780762
$446$	0	0
$447$	653.831i	1.46271i
$448$	0	0
$449$	−265.522	−0.591364	−0.295682	−	0.955286i	$-0.595547\pi$
$449$	−265.522	−0.591364	−0.295682	+	0.955286i	$0.595547\pi$
$450$	0	0
$451$	− 126.240i	− 0.279912i
$452$	0	0
$453$	846.700i	1.86909i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−396.591	−0.867815	−0.433907	−	0.900957i	$-0.642866\pi$
$457$	−396.591	−0.867815	−0.433907	+	0.900957i	$0.642866\pi$
$458$	0	0
$459$	58.0882	0.126554
$460$	0	0
$461$	143.388i	0.311037i	0.987833	+	0.155519i	$0.0497049\pi$
$461$	143.388i	0.311037i	−0.987833	+	0.155519i	$0.950295\pi$
$462$	0	0
$463$	−72.3400	−0.156242	−0.0781210	−	0.996944i	$-0.524892\pi$
$463$	−72.3400	−0.156242	−0.0781210	+	0.996944i	$0.524892\pi$
$464$	0	0
$465$	81.1260i	0.174465i
$466$	0	0
$467$	− 523.869i	− 1.12178i	−0.827892	−	0.560888i	$-0.810460\pi$
$467$	− 523.869i	− 1.12178i	0.827892	−	0.560888i	$-0.189540\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−182.898	−0.388319
$472$	0	0
$473$	−120.427	−0.254602
$474$	0	0
$475$	257.205i	0.541484i
$476$	0	0
$477$	321.431	0.673859
$478$	0	0
$479$	849.580i	1.77365i	0.462103	+	0.886826i	$0.347095\pi$
$479$	849.580i	1.77365i	−0.462103	+	0.886826i	$0.652905\pi$
$480$	0	0
$481$	− 931.417i	− 1.93642i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	164.558	0.339294
$486$	0	0
$487$	405.963	0.833600	0.416800	−	0.908998i	$-0.363152\pi$
$487$	405.963	0.833600	0.416800	+	0.908998i	$0.363152\pi$
$488$	0	0
$489$	245.116i	0.501260i
$490$	0	0
$491$	−661.455	−1.34716	−0.673580	−	0.739115i	$-0.735244\pi$
$491$	−661.455	−1.34716	−0.673580	+	0.739115i	$0.735244\pi$
$492$	0	0
$493$	128.748i	0.261152i
$494$	0	0
$495$	− 32.7855i	− 0.0662333i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	239.568	0.480096	0.240048	−	0.970761i	$-0.422837\pi$
$499$	239.568	0.480096	0.240048	+	0.970761i	$0.422837\pi$
$500$	0	0
$501$	−605.367	−1.20832
$502$	0	0
$503$	− 578.149i	− 1.14940i	−0.818364	−	0.574701i	$-0.805118\pi$
$503$	− 578.149i	− 1.14940i	0.818364	−	0.574701i	$-0.194882\pi$
$504$	0	0
$505$	−63.4524	−0.125648
$506$	0	0
$507$	1737.19i	3.42640i
$508$	0	0
$509$	− 301.289i	− 0.591923i	−0.955200	−	0.295962i	$-0.904360\pi$
$509$	− 301.289i	− 0.591923i	0.955200	−	0.295962i	$-0.0956401\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	266.796	0.520070
$514$	0	0
$515$	−372.574	−0.723445
$516$	0	0
$517$	73.7127i	0.142578i
$518$	0	0
$519$	142.531	0.274626
$520$	0	0
$521$	− 132.574i	− 0.254460i	−0.991873	−	0.127230i	$-0.959391\pi$
$521$	− 132.574i	− 0.254460i	0.991873	−	0.127230i	$-0.0406086\pi$
$522$	0	0
$523$	111.000i	0.212238i	0.994353	+	0.106119i	$0.0338424\pi$
$523$	111.000i	0.212238i	−0.994353	+	0.106119i	$0.966158\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.7487	−0.0488590
$528$	0	0
$529$	−216.833	−0.409891
$530$	0	0
$531$	430.258i	0.810279i
$532$	0	0
$533$	−1359.30	−2.55028
$534$	0	0
$535$	349.071i	0.652470i
$536$	0	0
$537$	377.369i	0.702736i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	85.7043	0.158418	0.0792092	−	0.996858i	$-0.474761\pi$
$541$	85.7043	0.158418	0.0792092	+	0.996858i	$0.474761\pi$
$542$	0	0
$543$	164.280	0.302541
$544$	0	0
$545$	505.343i	0.927235i
$546$	0	0
$547$	674.162	1.23247	0.616236	−	0.787562i	$-0.288657\pi$
$547$	674.162	1.23247	0.616236	+	0.787562i	$0.288657\pi$
$548$	0	0
$549$	− 10.0210i	− 0.0182531i
$550$	0	0
$551$	591.333i	1.07320i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	413.307	0.744696
$556$	0	0
$557$	763.490	1.37072	0.685359	−	0.728205i	$-0.259645\pi$
$557$	763.490	1.37072	0.685359	+	0.728205i	$0.259645\pi$
$558$	0	0
$559$	1296.70i	2.31968i
$560$	0	0
$561$	30.8674	0.0550221
$562$	0	0
$563$	355.022i	0.630590i	0.948994	+	0.315295i	$0.102103\pi$
$563$	355.022i	0.630590i	−0.948994	+	0.315295i	$0.897897\pi$
$564$	0	0
$565$	− 375.082i	− 0.663861i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	527.069	0.926308	0.463154	−	0.886278i	$-0.346718\pi$
$569$	527.069	0.926308	0.463154	+	0.886278i	$0.346718\pi$
$570$	0	0
$571$	515.772	0.903278	0.451639	−	0.892201i	$-0.350839\pi$
$571$	515.772	0.903278	0.451639	+	0.892201i	$0.350839\pi$
$572$	0	0
$573$	1216.18i	2.12248i
$574$	0	0
$575$	−277.596	−0.482775
$576$	0	0
$577$	210.190i	0.364280i	0.983273	+	0.182140i	$0.0583024\pi$
$577$	210.190i	0.364280i	−0.983273	+	0.182140i	$0.941698\pi$
$578$	0	0
$579$	− 513.482i	− 0.886844i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−165.057	−0.283117
$584$	0	0
$585$	−353.020	−0.603452
$586$	0	0
$587$	− 91.8797i	− 0.156524i	−0.996933	−	0.0782621i	$-0.975063\pi$
$587$	− 91.8797i	− 0.156524i	0.996933	−	0.0782621i	$-0.0249371\pi$
$588$	0	0
$589$	−118.262	−0.200785
$590$	0	0
$591$	− 643.806i	− 1.08935i
$592$	0	0
$593$	448.871i	0.756950i	0.925612	+	0.378475i	$0.123551\pi$
$593$	448.871i	0.756950i	−0.925612	+	0.378475i	$0.876449\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−838.219	−1.40405
$598$	0	0
$599$	−197.705	−0.330058	−0.165029	−	0.986289i	$-0.552772\pi$
$599$	−197.705	−0.330058	−0.165029	+	0.986289i	$0.552772\pi$
$600$	0	0
$601$	232.075i	0.386148i	0.981184	+	0.193074i	$0.0618458\pi$
$601$	232.075i	0.386148i	−0.981184	+	0.193074i	$0.938154\pi$
$602$	0	0
$603$	−114.251	−0.189471
$604$	0	0
$605$	− 351.936i	− 0.581712i
$606$	0	0
$607$	819.633i	1.35030i	0.737680	+	0.675151i	$0.235921\pi$
$607$	819.633i	1.35030i	−0.737680	+	0.675151i	$0.764079\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	793.706	1.29903
$612$	0	0
$613$	−811.390	−1.32364	−0.661819	−	0.749664i	$-0.730215\pi$
$613$	−811.390	−1.32364	−0.661819	+	0.749664i	$0.730215\pi$
$614$	0	0
$615$	− 603.175i	− 0.980773i
$616$	0	0
$617$	248.808	0.403255	0.201627	−	0.979462i	$-0.435377\pi$
$617$	248.808	0.403255	0.201627	+	0.979462i	$0.435377\pi$
$618$	0	0
$619$	− 544.744i	− 0.880039i	−0.897988	−	0.440020i	$-0.854971\pi$
$619$	− 544.744i	− 0.880039i	0.897988	−	0.440020i	$-0.145029\pi$
$620$	0	0
$621$	287.947i	0.463683i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14.6410	0.0234256
$626$	0	0
$627$	141.772	0.226112
$628$	0	0
$629$	131.180i	0.208553i
$630$	0	0
$631$	407.805	0.646284	0.323142	−	0.946350i	$-0.395261\pi$
$631$	407.805	0.646284	0.323142	+	0.946350i	$0.395261\pi$
$632$	0	0
$633$	− 926.150i	− 1.46311i
$634$	0	0
$635$	− 487.951i	− 0.768427i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−232.919	−0.364505
$640$	0	0
$641$	645.679	1.00730	0.503650	−	0.863908i	$-0.331990\pi$
$641$	645.679	1.00730	0.503650	+	0.863908i	$0.331990\pi$
$642$	0	0
$643$	932.869i	1.45081i	0.688324	+	0.725403i	$0.258347\pi$
$643$	932.869i	1.45081i	−0.688324	+	0.725403i	$0.741653\pi$
$644$	0	0
$645$	−575.397	−0.892089
$646$	0	0
$647$	485.306i	0.750086i	0.927007	+	0.375043i	$0.122372\pi$
$647$	485.306i	0.750086i	−0.927007	+	0.375043i	$0.877628\pi$
$648$	0	0
$649$	− 220.941i	− 0.340433i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−361.174	−0.553099	−0.276550	−	0.961000i	$-0.589191\pi$
$653$	−361.174	−0.553099	−0.276550	+	0.961000i	$0.589191\pi$
$654$	0	0
$655$	187.668	0.286516
$656$	0	0
$657$	364.146i	0.554255i
$658$	0	0
$659$	180.761	0.274296	0.137148	−	0.990551i	$-0.456206\pi$
$659$	180.761	0.274296	0.137148	+	0.990551i	$0.456206\pi$
$660$	0	0
$661$	589.821i	0.892316i	0.894954	+	0.446158i	$0.147208\pi$
$661$	589.821i	0.892316i	−0.894954	+	0.446158i	$0.852792\pi$
$662$	0	0
$663$	− 332.366i	− 0.501307i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−638.213	−0.956841
$668$	0	0
$669$	399.041	0.596474
$670$	0	0
$671$	5.14585i	0.00766893i
$672$	0	0
$673$	−416.772	−0.619276	−0.309638	−	0.950855i	$-0.600208\pi$
$673$	−416.772	−0.619276	−0.309638	+	0.950855i	$0.600208\pi$
$674$	0	0
$675$	− 256.057i	− 0.379344i
$676$	0	0
$677$	− 459.702i	− 0.679028i	−0.940601	−	0.339514i	$-0.889737\pi$
$677$	− 459.702i	− 0.679028i	0.940601	−	0.339514i	$-0.110263\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1042.60	−1.53098
$682$	0	0
$683$	1120.19	1.64010	0.820048	−	0.572295i	$-0.193947\pi$
$683$	1120.19	1.64010	0.820048	+	0.572295i	$0.193947\pi$
$684$	0	0
$685$	289.843i	0.423128i
$686$	0	0
$687$	651.021	0.947629
$688$	0	0
$689$	1777.27i	2.57949i
$690$	0	0
$691$	− 427.332i	− 0.618425i	−0.950993	−	0.309213i	$-0.899934\pi$
$691$	− 427.332i	− 0.618425i	0.950993	−	0.309213i	$-0.100066\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−280.558	−0.403681
$696$	0	0
$697$	191.443	0.274667
$698$	0	0
$699$	− 1309.49i	− 1.87337i
$700$	0	0
$701$	−99.9460	−0.142576	−0.0712882	−	0.997456i	$-0.522711\pi$
$701$	−99.9460	−0.142576	−0.0712882	+	0.997456i	$0.522711\pi$
$702$	0	0
$703$	602.502i	0.857044i
$704$	0	0
$705$	352.199i	0.499573i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	39.1725	0.0552504	0.0276252	−	0.999618i	$-0.491206\pi$
$709$	39.1725	0.0552504	0.0276252	+	0.999618i	$0.491206\pi$
$710$	0	0
$711$	526.462	0.740453
$712$	0	0
$713$	− 127.638i	− 0.179015i
$714$	0	0
$715$	181.279	0.253536
$716$	0	0
$717$	− 64.6484i	− 0.0901651i
$718$	0	0
$719$	− 771.897i	− 1.07357i	−0.843719	−	0.536785i	$-0.819639\pi$
$719$	− 771.897i	− 1.07357i	0.843719	−	0.536785i	$-0.180361\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	446.359	0.617371
$724$	0	0
$725$	567.532	0.782803
$726$	0	0
$727$	963.864i	1.32581i	0.748703	+	0.662905i	$0.230677\pi$
$727$	963.864i	1.32581i	−0.748703	+	0.662905i	$0.769323\pi$
$728$	0	0
$729$	−77.0650	−0.105713
$730$	0	0
$731$	− 182.626i	− 0.249831i
$732$	0	0
$733$	95.0862i	0.129722i	0.997894	+	0.0648610i	$0.0206604\pi$
$733$	95.0862i	0.129722i	−0.997894	+	0.0648610i	$0.979340\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	58.6687	0.0796048
$738$	0	0
$739$	1148.80	1.55454	0.777268	−	0.629169i	$-0.216605\pi$
$739$	1148.80	1.55454	0.777268	+	0.629169i	$0.216605\pi$
$740$	0	0
$741$	− 1526.54i	− 2.06011i
$742$	0	0
$743$	−232.652	−0.313126	−0.156563	−	0.987668i	$-0.550041\pi$
$743$	−232.652	−0.313126	−0.156563	+	0.987668i	$0.550041\pi$
$744$	0	0
$745$	− 540.799i	− 0.725904i
$746$	0	0
$747$	705.463i	0.944394i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−686.158	−0.913659	−0.456829	−	0.889554i	$-0.651015\pi$
$751$	−686.158	−0.913659	−0.456829	+	0.889554i	$0.651015\pi$
$752$	0	0
$753$	808.210	1.07332
$754$	0	0
$755$	− 700.325i	− 0.927582i
$756$	0	0
$757$	657.058	0.867976	0.433988	−	0.900919i	$-0.357106\pi$
$757$	657.058	0.867976	0.433988	+	0.900919i	$0.357106\pi$
$758$	0	0
$759$	153.012i	0.201596i
$760$	0	0
$761$	1228.17i	1.61389i	0.590626	+	0.806946i	$0.298881\pi$
$761$	1228.17i	1.61389i	−0.590626	+	0.806946i	$0.701119\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	49.7190	0.0649921
$766$	0	0
$767$	−2379.00	−3.10169
$768$	0	0
$769$	499.279i	0.649257i	0.945842	+	0.324629i	$0.105239\pi$
$769$	499.279i	0.649257i	−0.945842	+	0.324629i	$0.894761\pi$
$770$	0	0
$771$	−1776.12	−2.30366
$772$	0	0
$773$	− 320.220i	− 0.414256i	−0.978314	−	0.207128i	$-0.933588\pi$
$773$	− 320.220i	− 0.414256i	0.978314	−	0.207128i	$-0.0664116\pi$
$774$	0	0
$775$	113.502i	0.146455i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	879.286	1.12874
$780$	0	0
$781$	119.606	0.153144
$782$	0	0
$783$	− 588.695i	− 0.751845i
$784$	0	0
$785$	151.279	0.192712
$786$	0	0
$787$	− 566.896i	− 0.720325i	−0.932890	−	0.360162i	$-0.882721\pi$
$787$	− 566.896i	− 0.720325i	0.932890	−	0.360162i	$-0.117279\pi$
$788$	0	0
$789$	330.473i	0.418850i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	55.4083	0.0698717
$794$	0	0
$795$	−788.643	−0.992003
$796$	0	0
$797$	− 1416.99i	− 1.77791i	−0.457996	−	0.888954i	$-0.651433\pi$
$797$	− 1416.99i	− 1.77791i	0.457996	−	0.888954i	$-0.348567\pi$
$798$	0	0
$799$	−111.785	−0.139906
$800$	0	0
$801$	521.781i	0.651412i
$802$	0	0
$803$	− 186.992i	− 0.232867i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−140.307	−0.173863
$808$	0	0
$809$	−997.261	−1.23271	−0.616354	−	0.787469i	$-0.711391\pi$
$809$	−997.261	−1.23271	−0.616354	+	0.787469i	$0.711391\pi$
$810$	0	0
$811$	1201.56i	1.48158i	0.671735	+	0.740792i	$0.265550\pi$
$811$	1201.56i	1.48158i	−0.671735	+	0.740792i	$0.734450\pi$
$812$	0	0
$813$	−313.064	−0.385072
$814$	0	0
$815$	− 202.741i	− 0.248762i
$816$	0	0
$817$	− 838.792i	− 1.02667i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−185.790	−0.226298	−0.113149	−	0.993578i	$-0.536094\pi$
$821$	−185.790	−0.226298	−0.113149	+	0.993578i	$0.536094\pi$
$822$	0	0
$823$	−858.902	−1.04362	−0.521812	−	0.853061i	$-0.674744\pi$
$823$	−858.902	−1.04362	−0.521812	+	0.853061i	$0.674744\pi$
$824$	0	0
$825$	− 136.066i	− 0.164928i
$826$	0	0
$827$	−217.847	−0.263418	−0.131709	−	0.991288i	$-0.542046\pi$
$827$	−217.847	−0.263418	−0.131709	+	0.991288i	$0.542046\pi$
$828$	0	0
$829$	991.084i	1.19552i	0.801676	+	0.597759i	$0.203942\pi$
$829$	991.084i	1.19552i	−0.801676	+	0.597759i	$0.796058\pi$
$830$	0	0
$831$	− 230.311i	− 0.277149i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	500.713	0.599656
$836$	0	0
$837$	117.735	0.140663
$838$	0	0
$839$	415.268i	0.494956i	0.968894	+	0.247478i	$0.0796017\pi$
$839$	415.268i	0.494956i	−0.968894	+	0.247478i	$0.920398\pi$
$840$	0	0
$841$	463.798	0.551484
$842$	0	0
$843$	− 216.144i	− 0.256398i
$844$	0	0
$845$	− 1436.87i	− 1.70043i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	882.690	1.03968
$850$	0	0
$851$	−650.268	−0.764122
$852$	0	0
$853$	− 1335.36i	− 1.56549i	−0.622343	−	0.782745i	$-0.713819\pi$
$853$	− 1335.36i	− 1.56549i	0.622343	−	0.782745i	$-0.286181\pi$
$854$	0	0
$855$	228.356	0.267084
$856$	0	0
$857$	− 1607.75i	− 1.87602i	−0.346611	−	0.938009i	$-0.612668\pi$
$857$	− 1607.75i	− 1.87602i	0.346611	−	0.938009i	$-0.387332\pi$
$858$	0	0
$859$	152.282i	0.177278i	0.996064	+	0.0886391i	$0.0282518\pi$
$859$	152.282i	0.177278i	−0.996064	+	0.0886391i	$0.971748\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	960.275	1.11272	0.556358	−	0.830942i	$-0.312198\pi$
$863$	960.275	1.11272	0.556358	+	0.830942i	$0.312198\pi$
$864$	0	0
$865$	−117.891	−0.136290
$866$	0	0
$867$	− 1018.07i	− 1.17424i
$868$	0	0
$869$	−270.343	−0.311096
$870$	0	0
$871$	− 631.719i	− 0.725280i
$872$	0	0
$873$	247.131i	0.283082i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.9410	0.0136158	0.00680790	−	0.999977i	$-0.497833\pi$
$877$	11.9410	0.0136158	0.00680790	+	0.999977i	$0.497833\pi$
$878$	0	0
$879$	724.054	0.823725
$880$	0	0
$881$	625.457i	0.709940i	0.934878	+	0.354970i	$0.115509\pi$
$881$	625.457i	0.709940i	−0.934878	+	0.354970i	$0.884491\pi$
$882$	0	0
$883$	−78.6167	−0.0890337	−0.0445168	−	0.999009i	$-0.514175\pi$
$883$	−78.6167	−0.0890337	−0.0445168	+	0.999009i	$0.514175\pi$
$884$	0	0
$885$	− 1055.65i	− 1.19283i
$886$	0	0
$887$	731.689i	0.824903i	0.910980	+	0.412451i	$0.135327\pi$
$887$	731.689i	0.824903i	−0.910980	+	0.412451i	$0.864673\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−237.957	−0.267067
$892$	0	0
$893$	−513.422	−0.574940
$894$	0	0
$895$	− 312.131i	− 0.348749i
$896$	0	0
$897$	1647.56	1.83675
$898$	0	0
$899$	260.950i	0.290267i
$900$	0	0
$901$	− 250.308i	− 0.277812i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−135.880	−0.150143
$906$	0	0
$907$	287.701	0.317201	0.158601	−	0.987343i	$-0.449302\pi$
$907$	287.701	0.317201	0.158601	+	0.987343i	$0.449302\pi$
$908$	0	0
$909$	− 95.2922i	− 0.104832i
$910$	0	0
$911$	249.315	0.273672	0.136836	−	0.990594i	$-0.456307\pi$
$911$	249.315	0.273672	0.136836	+	0.990594i	$0.456307\pi$
$912$	0	0
$913$	− 362.261i	− 0.396781i
$914$	0	0
$915$	24.5868i	0.0268709i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	20.1150	0.0218879	0.0109440	−	0.999940i	$-0.496516\pi$
$919$	20.1150	0.0218879	0.0109440	+	0.999940i	$0.496516\pi$
$920$	0	0
$921$	−909.956	−0.988009
$922$	0	0
$923$	− 1287.86i	− 1.39530i
$924$	0	0
$925$	578.252	0.625137
$926$	0	0
$927$	− 559.528i	− 0.603590i
$928$	0	0
$929$	1232.64i	1.32685i	0.748244	+	0.663424i	$0.230897\pi$
$929$	1232.64i	1.32685i	−0.748244	+	0.663424i	$0.769103\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1252.96	1.34294
$934$	0	0
$935$	−25.5311	−0.0273060
$936$	0	0
$937$	283.840i	0.302924i	0.988463	+	0.151462i	$0.0483981\pi$
$937$	283.840i	0.302924i	−0.988463	+	0.151462i	$0.951602\pi$
$938$	0	0
$939$	−417.835	−0.444979
$940$	0	0
$941$	648.090i	0.688725i	0.938837	+	0.344362i	$0.111905\pi$
$941$	648.090i	0.688725i	−0.938837	+	0.344362i	$0.888095\pi$
$942$	0	0
$943$	948.994i	1.00636i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1718.20	−1.81436	−0.907181	−	0.420740i	$-0.861771\pi$
$947$	−1718.20	−1.81436	−0.907181	+	0.420740i	$0.861771\pi$
$948$	0	0
$949$	−2013.44	−2.12165
$950$	0	0
$951$	− 892.839i	− 0.938842i
$952$	0	0
$953$	1085.95	1.13950	0.569752	−	0.821817i	$-0.307039\pi$
$953$	1085.95	1.13950	0.569752	+	0.821817i	$0.307039\pi$
$954$	0	0
$955$	− 1005.93i	− 1.05333i
$956$	0	0
$957$	− 312.826i	− 0.326882i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	908.812	0.945694
$962$	0	0
$963$	−524.232	−0.544374
$964$	0	0
$965$	424.713i	0.440117i
$966$	0	0
$967$	969.512	1.00260	0.501299	−	0.865274i	$-0.332856\pi$
$967$	969.512	1.00260	0.501299	+	0.865274i	$0.332856\pi$
$968$	0	0
$969$	214.996i	0.221875i
$970$	0	0
$971$	− 837.111i	− 0.862112i	−0.902325	−	0.431056i	$-0.858141\pi$
$971$	− 837.111i	− 0.862112i	0.902325	−	0.431056i	$-0.141859\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1465.10	−1.50266
$976$	0	0
$977$	1360.23	1.39225	0.696125	−	0.717921i	$-0.254906\pi$
$977$	1360.23	1.39225	0.696125	+	0.717921i	$0.254906\pi$
$978$	0	0
$979$	− 267.939i	− 0.273686i
$980$	0	0
$981$	−758.919	−0.773618
$982$	0	0
$983$	− 397.275i	− 0.404145i	−0.979371	−	0.202073i	$-0.935232\pi$
$983$	− 397.275i	− 0.404145i	0.979371	−	0.202073i	$-0.0647677\pi$
$984$	0	0
$985$	532.506i	0.540616i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	905.290	0.915359
$990$	0	0
$991$	−278.500	−0.281029	−0.140514	−	0.990079i	$-0.544876\pi$
$991$	−278.500	−0.281029	−0.140514	+	0.990079i	$0.544876\pi$
$992$	0	0
$993$	256.856i	0.258667i
$994$	0	0
$995$	693.310	0.696794
$996$	0	0
$997$	− 827.397i	− 0.829887i	−0.909847	−	0.414943i	$-0.863801\pi$
$997$	− 827.397i	− 0.829887i	0.909847	−	0.414943i	$-0.136199\pi$
$998$	0	0
$999$	− 599.815i	− 0.600415i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.c.h.97.4		16
4.3	odd	2	inner	1568.3.c.h.97.14		16
7.4	even	3		224.3.s.a.33.2	✓	16
7.5	odd	6		224.3.s.a.129.2	yes	16
7.6	odd	2	inner	1568.3.c.h.97.13		16
28.11	odd	6		224.3.s.a.33.7	yes	16
28.19	even	6		224.3.s.a.129.7	yes	16
28.27	even	2	inner	1568.3.c.h.97.3		16
56.5	odd	6		448.3.s.g.129.7		16
56.11	odd	6		448.3.s.g.257.2		16
56.19	even	6		448.3.s.g.129.2		16
56.53	even	6		448.3.s.g.257.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.s.a.33.2	✓	16	7.4	even	3
224.3.s.a.33.7	yes	16	28.11	odd	6
224.3.s.a.129.2	yes	16	7.5	odd	6
224.3.s.a.129.7	yes	16	28.19	even	6
448.3.s.g.129.2		16	56.19	even	6
448.3.s.g.129.7		16	56.5	odd	6
448.3.s.g.257.2		16	56.11	odd	6
448.3.s.g.257.7		16	56.53	even	6
1568.3.c.h.97.3		16	28.27	even	2	inner
1568.3.c.h.97.4		16	1.1	even	1	trivial
1568.3.c.h.97.13		16	7.6	odd	2	inner
1568.3.c.h.97.14		16	4.3	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)

\(f(q)\)	\(=\)	\(q-3.68470i q^{3} +3.04770i q^{5} -4.57700 q^{9} +O(q^{10})\)	\(q-3.68470i q^{3} +3.04770i q^{5} -4.57700 q^{9} +2.35033 q^{11} -25.3073i q^{13} +11.2298 q^{15} +3.56426i q^{17} +16.3704i q^{19} -17.6683 q^{23} +15.7115 q^{25} -16.2974i q^{27} +36.1220 q^{29} +7.22414i q^{31} -8.66025i q^{33} +36.8043 q^{37} -93.2498 q^{39} -53.7118i q^{41} -51.2382 q^{43} -13.9493i q^{45} +31.3627i q^{47} +13.1332 q^{51} -70.2274 q^{53} +7.16309i q^{55} +60.3201 q^{57} -94.0044i q^{59} +2.18942i q^{61} +77.1290 q^{65} +24.9619 q^{67} +65.1022i q^{69} +50.8890 q^{71} -79.5598i q^{73} -57.8923i q^{75} -115.023 q^{79} -101.244 q^{81} -154.132i q^{83} -10.8628 q^{85} -133.099i q^{87} -114.001i q^{89} +26.6188 q^{93} -49.8921 q^{95} -53.9940i q^{97} -10.7575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^9	\( 16 q - 16 q^{9} - 32 q^{25} + 112 q^{29} - 16 q^{37} + 48 q^{53} - 528 q^{57} + 16 q^{65} - 64 q^{81} + 720 q^{85} + 464 q^{93}+O(q^{100}) \) 16 * q - 16 * q^9 - 32 * q^25 + 112 * q^29 - 16 * q^37 + 48 * q^53 - 528 * q^57 + 16 * q^65 - 64 * q^81 + 720 * q^85 + 464 * q^93

Introduction

L-functions

Modular forms

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Representations

Groups

Database

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