LMFDB - Embedded newform 2352.3.f.e.97.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(97,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2352.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.f (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:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.4
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2352.97
Dual form			2352.3.f.e.97.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +8.36308i q^{5} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +8.36308i q^{5} -3.00000 q^{9} -6.00000 q^{11} +17.8639i q^{13} -14.4853 q^{15} +18.7554i q^{17} +17.0233i q^{19} -13.4558 q^{23} -44.9411 q^{25} -5.19615i q^{27} +33.9411 q^{29} +14.7479i q^{31} -10.3923i q^{33} +5.97056 q^{37} -30.9411 q^{39} -35.2354i q^{41} -15.4853 q^{43} -25.0892i q^{45} +33.2061i q^{47} -32.4853 q^{51} -34.5442 q^{53} -50.1785i q^{55} -29.4853 q^{57} +27.3647i q^{59} -40.3805i q^{61} -149.397 q^{65} +114.397 q^{67} -23.3062i q^{69} -18.6030 q^{71} +117.032i q^{73} -77.8403i q^{75} +88.3381 q^{79} +9.00000 q^{81} +75.7601i q^{83} -156.853 q^{85} +58.7878i q^{87} -20.7846i q^{89} -25.5442 q^{93} -142.368 q^{95} -30.5826i q^{97} +18.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 24 q^{11} - 24 q^{15} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 12 q^{39} - 28 q^{43} - 96 q^{51} - 240 q^{53} - 84 q^{57} - 360 q^{65} + 220 q^{67} - 312 q^{71} - 20 q^{79} + 36 q^{81} - 288 q^{85} - 204 q^{93} - 264 q^{95} + 72 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 - 24 * q^11 - 24 * q^15 + 48 * q^23 - 44 * q^25 - 44 * q^37 + 12 * q^39 - 28 * q^43 - 96 * q^51 - 240 * q^53 - 84 * q^57 - 360 * q^65 + 220 * q^67 - 312 * q^71 - 20 * q^79 + 36 * q^81 - 288 * q^85 - 204 * q^93 - 264 * q^95 + 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	8.36308i	1.67262i	0.548260	+	0.836308i	$0.315291\pi$
$5$	8.36308i	1.67262i	−0.548260	+	0.836308i	$0.684709\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	−6.00000	−0.545455	−0.272727	−	0.962091i	$-0.587926\pi$
$11$	−6.00000	−0.545455	−0.272727	+	0.962091i	$0.587926\pi$
$12$	0	0
$13$	17.8639i	1.37414i	0.726590	+	0.687072i	$0.241104\pi$
$13$	17.8639i	1.37414i	−0.726590	+	0.687072i	$0.758896\pi$
$14$	0	0
$15$	−14.4853	−0.965685
$16$	0	0
$17$	18.7554i	1.10326i	0.834090	+	0.551629i	$0.185994\pi$
$17$	18.7554i	1.10326i	−0.834090	+	0.551629i	$0.814006\pi$
$18$	0	0
$19$	17.0233i	0.895965i	0.894042	+	0.447983i	$0.147857\pi$
$19$	17.0233i	0.895965i	−0.894042	+	0.447983i	$0.852143\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−13.4558	−0.585037	−0.292518	−	0.956260i	$-0.594493\pi$
$23$	−13.4558	−0.585037	−0.292518	+	0.956260i	$0.594493\pi$
$24$	0	0
$25$	−44.9411	−1.79765
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	33.9411	1.17038	0.585192	−	0.810895i	$-0.301019\pi$
$29$	33.9411	1.17038	0.585192	+	0.810895i	$0.301019\pi$
$30$	0	0
$31$	14.7479i	0.475740i	0.971297	+	0.237870i	$0.0764491\pi$
$31$	14.7479i	0.475740i	−0.971297	+	0.237870i	$0.923551\pi$
$32$	0	0
$33$	− 10.3923i	− 0.314918i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.97056	0.161367	0.0806833	−	0.996740i	$-0.474290\pi$
$37$	5.97056	0.161367	0.0806833	+	0.996740i	$0.474290\pi$
$38$	0	0
$39$	−30.9411	−0.793362
$40$	0	0
$41$	− 35.2354i	− 0.859399i	−0.902972	−	0.429700i	$-0.858619\pi$
$41$	− 35.2354i	− 0.859399i	0.902972	−	0.429700i	$-0.141381\pi$
$42$	0	0
$43$	−15.4853	−0.360123	−0.180061	−	0.983655i	$-0.557630\pi$
$43$	−15.4853	−0.360123	−0.180061	+	0.983655i	$0.557630\pi$
$44$	0	0
$45$	− 25.0892i	− 0.557539i
$46$	0	0
$47$	33.2061i	0.706514i	0.935526	+	0.353257i	$0.114926\pi$
$47$	33.2061i	0.706514i	−0.935526	+	0.353257i	$0.885074\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−32.4853	−0.636966
$52$	0	0
$53$	−34.5442	−0.651777	−0.325888	−	0.945408i	$-0.605663\pi$
$53$	−34.5442	−0.651777	−0.325888	+	0.945408i	$0.605663\pi$
$54$	0	0
$55$	− 50.1785i	− 0.912336i
$56$	0	0
$57$	−29.4853	−0.517286
$58$	0	0
$59$	27.3647i	0.463808i	0.972739	+	0.231904i	$0.0744955\pi$
$59$	27.3647i	0.463808i	−0.972739	+	0.231904i	$0.925505\pi$
$60$	0	0
$61$	− 40.3805i	− 0.661976i	−0.943635	−	0.330988i	$-0.892618\pi$
$61$	− 40.3805i	− 0.661976i	0.943635	−	0.330988i	$-0.107382\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−149.397	−2.29841
$66$	0	0
$67$	114.397	1.70742	0.853709	−	0.520751i	$-0.174348\pi$
$67$	114.397	1.70742	0.853709	+	0.520751i	$0.174348\pi$
$68$	0	0
$69$	− 23.3062i	− 0.337771i
$70$	0	0
$71$	−18.6030	−0.262015	−0.131007	−	0.991381i	$-0.541821\pi$
$71$	−18.6030	−0.262015	−0.131007	+	0.991381i	$0.541821\pi$
$72$	0	0
$73$	117.032i	1.60318i	0.597874	+	0.801590i	$0.296012\pi$
$73$	117.032i	1.60318i	−0.597874	+	0.801590i	$0.703988\pi$
$74$	0	0
$75$	− 77.8403i	− 1.03787i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	88.3381	1.11820	0.559102	−	0.829099i	$-0.311146\pi$
$79$	88.3381	1.11820	0.559102	+	0.829099i	$0.311146\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	75.7601i	0.912772i	0.889782	+	0.456386i	$0.150856\pi$
$83$	75.7601i	0.912772i	−0.889782	+	0.456386i	$0.849144\pi$
$84$	0	0
$85$	−156.853	−1.84533
$86$	0	0
$87$	58.7878i	0.675721i
$88$	0	0
$89$	− 20.7846i	− 0.233535i	−0.993159	−	0.116767i	$-0.962747\pi$
$89$	− 20.7846i	− 0.233535i	0.993159	−	0.116767i	$-0.0372532\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−25.5442	−0.274668
$94$	0	0
$95$	−142.368	−1.49861
$96$	0	0
$97$	− 30.5826i	− 0.315284i	−0.987496	−	0.157642i	$-0.949611\pi$
$97$	− 30.5826i	− 0.315284i	0.987496	−	0.157642i	$-0.0503892\pi$
$98$	0	0
$99$	18.0000	0.181818
$100$	0	0
$101$	− 127.968i	− 1.26701i	−0.773739	−	0.633504i	$-0.781616\pi$
$101$	− 127.968i	− 1.26701i	0.773739	−	0.633504i	$-0.218384\pi$
$102$	0	0
$103$	80.9563i	0.785983i	0.919542	+	0.392992i	$0.128560\pi$
$103$	80.9563i	0.785983i	−0.919542	+	0.392992i	$0.871440\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−169.456	−1.58370	−0.791850	−	0.610716i	$-0.790882\pi$
$107$	−169.456	−1.58370	−0.791850	+	0.610716i	$0.790882\pi$
$108$	0	0
$109$	178.941	1.64166	0.820831	−	0.571171i	$-0.193511\pi$
$109$	178.941	1.64166	0.820831	+	0.571171i	$0.193511\pi$
$110$	0	0
$111$	10.3413i	0.0931650i
$112$	0	0
$113$	−17.3970	−0.153955	−0.0769777	−	0.997033i	$-0.524527\pi$
$113$	−17.3970	−0.153955	−0.0769777	+	0.997033i	$0.524527\pi$
$114$	0	0
$115$	− 112.532i	− 0.978542i
$116$	0	0
$117$	− 53.5916i	− 0.458048i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−85.0000	−0.702479
$122$	0	0
$123$	61.0294	0.496174
$124$	0	0
$125$	− 166.769i	− 1.33415i
$126$	0	0
$127$	−167.426	−1.31832	−0.659159	−	0.752004i	$-0.729088\pi$
$127$	−167.426	−1.31832	−0.659159	+	0.752004i	$0.729088\pi$
$128$	0	0
$129$	− 26.8213i	− 0.207917i
$130$	0	0
$131$	1.78304i	0.0136110i	0.999977	+	0.00680549i	$0.00216627\pi$
$131$	1.78304i	0.0136110i	−0.999977	+	0.00680549i	$0.997834\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	43.4558	0.321895
$136$	0	0
$137$	100.971	0.737011	0.368506	−	0.929625i	$-0.379870\pi$
$137$	100.971	0.737011	0.368506	+	0.929625i	$0.379870\pi$
$138$	0	0
$139$	− 140.542i	− 1.01110i	−0.862799	−	0.505548i	$-0.831290\pi$
$139$	− 140.542i	− 1.01110i	0.862799	−	0.505548i	$-0.168710\pi$
$140$	0	0
$141$	−57.5147	−0.407906
$142$	0	0
$143$	− 107.183i	− 0.749533i
$144$	0	0
$145$	283.852i	1.95760i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	182.912	1.22760	0.613798	−	0.789463i	$-0.289641\pi$
$149$	182.912	1.22760	0.613798	+	0.789463i	$0.289641\pi$
$150$	0	0
$151$	288.794	1.91254	0.956271	−	0.292481i	$-0.0944808\pi$
$151$	288.794	1.91254	0.956271	+	0.292481i	$0.0944808\pi$
$152$	0	0
$153$	− 56.2662i	− 0.367753i
$154$	0	0
$155$	−123.338	−0.795730
$156$	0	0
$157$	− 187.061i	− 1.19147i	−0.803179	−	0.595737i	$-0.796860\pi$
$157$	− 187.061i	− 1.19147i	0.803179	−	0.595737i	$-0.203140\pi$
$158$	0	0
$159$	− 59.8322i	− 0.376303i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.0589	−0.0985207	−0.0492604	−	0.998786i	$-0.515686\pi$
$163$	−16.0589	−0.0985207	−0.0492604	+	0.998786i	$0.515686\pi$
$164$	0	0
$165$	86.9117	0.526738
$166$	0	0
$167$	− 176.117i	− 1.05459i	−0.849681	−	0.527297i	$-0.823206\pi$
$167$	− 176.117i	− 1.05459i	0.849681	−	0.527297i	$-0.176794\pi$
$168$	0	0
$169$	−150.118	−0.888271
$170$	0	0
$171$	− 51.0700i	− 0.298655i
$172$	0	0
$173$	− 231.152i	− 1.33614i	−0.744098	−	0.668070i	$-0.767121\pi$
$173$	− 231.152i	− 1.33614i	0.744098	−	0.668070i	$-0.232879\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−47.3970	−0.267779
$178$	0	0
$179$	85.2792	0.476420	0.238210	−	0.971214i	$-0.423439\pi$
$179$	85.2792	0.476420	0.238210	+	0.971214i	$0.423439\pi$
$180$	0	0
$181$	5.58655i	0.0308649i	0.999881	+	0.0154325i	$0.00491250\pi$
$181$	5.58655i	0.0308649i	−0.999881	+	0.0154325i	$0.995087\pi$
$182$	0	0
$183$	69.9411	0.382192
$184$	0	0
$185$	49.9323i	0.269904i
$186$	0	0
$187$	− 112.532i	− 0.601777i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−185.397	−0.970665	−0.485332	−	0.874330i	$-0.661301\pi$
$191$	−185.397	−0.970665	−0.485332	+	0.874330i	$0.661301\pi$
$192$	0	0
$193$	227.794	1.18028	0.590140	−	0.807301i	$-0.299073\pi$
$193$	227.794	1.18028	0.590140	+	0.807301i	$0.299073\pi$
$194$	0	0
$195$	− 258.763i	− 1.32699i
$196$	0	0
$197$	123.161	0.625185	0.312593	−	0.949887i	$-0.398803\pi$
$197$	123.161	0.625185	0.312593	+	0.949887i	$0.398803\pi$
$198$	0	0
$199$	6.23188i	0.0313160i	0.999877	+	0.0156580i	$0.00498430\pi$
$199$	6.23188i	0.0313160i	−0.999877	+	0.0156580i	$0.995016\pi$
$200$	0	0
$201$	198.141i	0.985778i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	294.676	1.43744
$206$	0	0
$207$	40.3675	0.195012
$208$	0	0
$209$	− 102.140i	− 0.488708i
$210$	0	0
$211$	124.912	0.591999	0.295999	−	0.955188i	$-0.404347\pi$
$211$	124.912	0.591999	0.295999	+	0.955188i	$0.404347\pi$
$212$	0	0
$213$	− 32.2214i	− 0.151274i
$214$	0	0
$215$	− 129.505i	− 0.602347i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−202.706	−0.925596
$220$	0	0
$221$	−335.044	−1.51603
$222$	0	0
$223$	− 228.631i	− 1.02525i	−0.858613	−	0.512625i	$-0.828673\pi$
$223$	− 228.631i	− 1.02525i	0.858613	−	0.512625i	$-0.171327\pi$
$224$	0	0
$225$	134.823	0.599215
$226$	0	0
$227$	− 169.537i	− 0.746859i	−0.927659	−	0.373430i	$-0.878182\pi$
$227$	− 169.537i	− 0.746859i	0.927659	−	0.373430i	$-0.121818\pi$
$228$	0	0
$229$	34.6920i	0.151493i	0.997127	+	0.0757467i	$0.0241340\pi$
$229$	34.6920i	0.151493i	−0.997127	+	0.0757467i	$0.975866\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−254.485	−1.09221	−0.546106	−	0.837716i	$-0.683890\pi$
$233$	−254.485	−1.09221	−0.546106	+	0.837716i	$0.683890\pi$
$234$	0	0
$235$	−277.706	−1.18173
$236$	0	0
$237$	153.006i	0.645595i
$238$	0	0
$239$	−197.147	−0.824884	−0.412442	−	0.910984i	$-0.635324\pi$
$239$	−197.147	−0.824884	−0.412442	+	0.910984i	$0.635324\pi$
$240$	0	0
$241$	− 88.4701i	− 0.367096i	−0.983011	−	0.183548i	$-0.941242\pi$
$241$	− 88.4701i	− 0.367096i	0.983011	−	0.183548i	$-0.0587582\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−304.103	−1.23118
$248$	0	0
$249$	−131.220	−0.526989
$250$	0	0
$251$	− 215.903i	− 0.860172i	−0.902788	−	0.430086i	$-0.858483\pi$
$251$	− 215.903i	− 0.860172i	0.902788	−	0.430086i	$-0.141517\pi$
$252$	0	0
$253$	80.7351	0.319111
$254$	0	0
$255$	− 271.677i	− 1.06540i
$256$	0	0
$257$	− 4.30463i	− 0.0167495i	−0.999965	−	0.00837477i	$-0.997334\pi$
$257$	− 4.30463i	− 0.0167495i	0.999965	−	0.00837477i	$-0.00266580\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−101.823	−0.390128
$262$	0	0
$263$	−282.676	−1.07481	−0.537407	−	0.843323i	$-0.680596\pi$
$263$	−282.676	−1.07481	−0.537407	+	0.843323i	$0.680596\pi$
$264$	0	0
$265$	− 288.896i	− 1.09017i
$266$	0	0
$267$	36.0000	0.134831
$268$	0	0
$269$	381.934i	1.41983i	0.704288	+	0.709914i	$0.251266\pi$
$269$	381.934i	1.41983i	−0.704288	+	0.709914i	$0.748734\pi$
$270$	0	0
$271$	84.3271i	0.311170i	0.987822	+	0.155585i	$0.0497263\pi$
$271$	84.3271i	0.311170i	−0.987822	+	0.155585i	$0.950274\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	269.647	0.980534
$276$	0	0
$277$	−137.118	−0.495010	−0.247505	−	0.968887i	$-0.579611\pi$
$277$	−137.118	−0.495010	−0.247505	+	0.968887i	$0.579611\pi$
$278$	0	0
$279$	− 44.2438i	− 0.158580i
$280$	0	0
$281$	−325.103	−1.15695	−0.578474	−	0.815701i	$-0.696352\pi$
$281$	−325.103	−1.15695	−0.578474	+	0.815701i	$0.696352\pi$
$282$	0	0
$283$	194.575i	0.687545i	0.939053	+	0.343773i	$0.111705\pi$
$283$	194.575i	0.687545i	−0.939053	+	0.343773i	$0.888295\pi$
$284$	0	0
$285$	− 246.588i	− 0.865220i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−62.7645	−0.217178
$290$	0	0
$291$	52.9706	0.182029
$292$	0	0
$293$	239.702i	0.818095i	0.912513	+	0.409048i	$0.134139\pi$
$293$	239.702i	0.818095i	−0.912513	+	0.409048i	$0.865861\pi$
$294$	0	0
$295$	−228.853	−0.775772
$296$	0	0
$297$	31.1769i	0.104973i
$298$	0	0
$299$	− 240.373i	− 0.803924i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	221.647	0.731507
$304$	0	0
$305$	337.706	1.10723
$306$	0	0
$307$	540.272i	1.75984i	0.475120	+	0.879921i	$0.342405\pi$
$307$	540.272i	1.75984i	−0.475120	+	0.879921i	$0.657595\pi$
$308$	0	0
$309$	−140.220	−0.453788
$310$	0	0
$311$	404.196i	1.29966i	0.760078	+	0.649832i	$0.225161\pi$
$311$	404.196i	1.29966i	−0.760078	+	0.649832i	$0.774839\pi$
$312$	0	0
$313$	− 131.296i	− 0.419477i	−0.977757	−	0.209739i	$-0.932739\pi$
$313$	− 131.296i	− 0.419477i	0.977757	−	0.209739i	$-0.0672613\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−93.9411	−0.296344	−0.148172	−	0.988962i	$-0.547339\pi$
$317$	−93.9411	−0.296344	−0.148172	+	0.988962i	$0.547339\pi$
$318$	0	0
$319$	−203.647	−0.638391
$320$	0	0
$321$	− 293.506i	− 0.914349i
$322$	0	0
$323$	−319.279	−0.988481
$324$	0	0
$325$	− 802.822i	− 2.47022i
$326$	0	0
$327$	309.935i	0.947814i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	261.368	0.789630	0.394815	−	0.918761i	$-0.370809\pi$
$331$	261.368	0.789630	0.394815	+	0.918761i	$0.370809\pi$
$332$	0	0
$333$	−17.9117	−0.0537889
$334$	0	0
$335$	956.711i	2.85585i
$336$	0	0
$337$	136.265	0.404347	0.202173	−	0.979350i	$-0.435200\pi$
$337$	136.265	0.404347	0.202173	+	0.979350i	$0.435200\pi$
$338$	0	0
$339$	− 30.1324i	− 0.0888862i
$340$	0	0
$341$	− 88.4876i	− 0.259494i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	194.912	0.564961
$346$	0	0
$347$	322.191	0.928504	0.464252	−	0.885703i	$-0.346323\pi$
$347$	322.191	0.928504	0.464252	+	0.885703i	$0.346323\pi$
$348$	0	0
$349$	− 346.495i	− 0.992821i	−0.868088	−	0.496411i	$-0.834651\pi$
$349$	− 346.495i	− 0.992821i	0.868088	−	0.496411i	$-0.165349\pi$
$350$	0	0
$351$	92.8234	0.264454
$352$	0	0
$353$	620.591i	1.75805i	0.476777	+	0.879025i	$0.341805\pi$
$353$	620.591i	1.75805i	−0.476777	+	0.879025i	$0.658195\pi$
$354$	0	0
$355$	− 155.579i	− 0.438250i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.2355	−0.0563663	−0.0281831	−	0.999603i	$-0.508972\pi$
$359$	−20.2355	−0.0563663	−0.0281831	+	0.999603i	$0.508972\pi$
$360$	0	0
$361$	71.2061	0.197247
$362$	0	0
$363$	− 147.224i	− 0.405577i
$364$	0	0
$365$	−978.749	−2.68151
$366$	0	0
$367$	− 311.574i	− 0.848975i	−0.905434	−	0.424488i	$-0.860454\pi$
$367$	− 311.574i	− 0.848975i	0.905434	−	0.424488i	$-0.139546\pi$
$368$	0	0
$369$	105.706i	0.286466i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−681.382	−1.82676	−0.913380	−	0.407107i	$-0.866538\pi$
$373$	−681.382	−1.82676	−0.913380	+	0.407107i	$0.866538\pi$
$374$	0	0
$375$	288.853	0.770274
$376$	0	0
$377$	606.320i	1.60828i
$378$	0	0
$379$	624.779	1.64849	0.824246	−	0.566231i	$-0.191599\pi$
$379$	624.779	1.64849	0.824246	+	0.566231i	$0.191599\pi$
$380$	0	0
$381$	− 289.991i	− 0.761131i
$382$	0	0
$383$	− 138.300i	− 0.361098i	−0.983566	−	0.180549i	$-0.942213\pi$
$383$	− 138.300i	− 0.361098i	0.983566	−	0.180549i	$-0.0577874\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	46.4558	0.120041
$388$	0	0
$389$	−563.574	−1.44878	−0.724388	−	0.689393i	$-0.757877\pi$
$389$	−563.574	−1.44878	−0.724388	+	0.689393i	$0.757877\pi$
$390$	0	0
$391$	− 252.370i	− 0.645446i
$392$	0	0
$393$	−3.08831	−0.00785830
$394$	0	0
$395$	738.779i	1.87033i
$396$	0	0
$397$	− 453.338i	− 1.14191i	−0.820981	−	0.570955i	$-0.806573\pi$
$397$	− 453.338i	− 1.14191i	0.820981	−	0.570955i	$-0.193427\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−275.750	−0.687656	−0.343828	−	0.939033i	$-0.611724\pi$
$401$	−275.750	−0.687656	−0.343828	+	0.939033i	$0.611724\pi$
$402$	0	0
$403$	−263.455	−0.653734
$404$	0	0
$405$	75.2677i	0.185846i
$406$	0	0
$407$	−35.8234	−0.0880181
$408$	0	0
$409$	435.831i	1.06560i	0.846240	+	0.532801i	$0.178861\pi$
$409$	435.831i	1.06560i	−0.846240	+	0.532801i	$0.821139\pi$
$410$	0	0
$411$	174.886i	0.425514i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−633.588	−1.52672
$416$	0	0
$417$	243.426	0.583756
$418$	0	0
$419$	− 301.257i	− 0.718991i	−0.933147	−	0.359496i	$-0.882949\pi$
$419$	− 301.257i	− 0.718991i	0.933147	−	0.359496i	$-0.117051\pi$
$420$	0	0
$421$	−203.794	−0.484071	−0.242036	−	0.970267i	$-0.577815\pi$
$421$	−203.794	−0.484071	−0.242036	+	0.970267i	$0.577815\pi$
$422$	0	0
$423$	− 99.6184i	− 0.235505i
$424$	0	0
$425$	− 842.888i	− 1.98327i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	185.647	0.432743
$430$	0	0
$431$	395.720	0.918144	0.459072	−	0.888399i	$-0.348182\pi$
$431$	395.720	0.918144	0.459072	+	0.888399i	$0.348182\pi$
$432$	0	0
$433$	44.2685i	0.102237i	0.998693	+	0.0511184i	$0.0162786\pi$
$433$	44.2685i	0.102237i	−0.998693	+	0.0511184i	$0.983721\pi$
$434$	0	0
$435$	−491.647	−1.13022
$436$	0	0
$437$	− 229.063i	− 0.524172i
$438$	0	0
$439$	− 397.862i	− 0.906291i	−0.891437	−	0.453146i	$-0.850302\pi$
$439$	− 397.862i	− 0.906291i	0.891437	−	0.453146i	$-0.149698\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	118.544	0.267594	0.133797	−	0.991009i	$-0.457283\pi$
$443$	118.544	0.267594	0.133797	+	0.991009i	$0.457283\pi$
$444$	0	0
$445$	173.823	0.390614
$446$	0	0
$447$	316.812i	0.708752i
$448$	0	0
$449$	713.897	1.58997	0.794985	−	0.606629i	$-0.207479\pi$
$449$	713.897	1.58997	0.794985	+	0.606629i	$0.207479\pi$
$450$	0	0
$451$	211.412i	0.468763i
$452$	0	0
$453$	500.206i	1.10421i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−125.177	−0.273909	−0.136955	−	0.990577i	$-0.543731\pi$
$457$	−125.177	−0.273909	−0.136955	+	0.990577i	$0.543731\pi$
$458$	0	0
$459$	97.4558	0.212322
$460$	0	0
$461$	− 655.767i	− 1.42249i	−0.702945	−	0.711244i	$-0.748132\pi$
$461$	− 655.767i	− 1.42249i	0.702945	−	0.711244i	$-0.251868\pi$
$462$	0	0
$463$	−869.396	−1.87775	−0.938873	−	0.344265i	$-0.888128\pi$
$463$	−869.396	−1.87775	−0.938873	+	0.344265i	$0.888128\pi$
$464$	0	0
$465$	− 213.628i	− 0.459415i
$466$	0	0
$467$	267.372i	0.572532i	0.958150	+	0.286266i	$0.0924141\pi$
$467$	267.372i	0.572532i	−0.958150	+	0.286266i	$0.907586\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	324.000	0.687898
$472$	0	0
$473$	92.9117	0.196431
$474$	0	0
$475$	− 765.048i	− 1.61063i
$476$	0	0
$477$	103.632	0.217259
$478$	0	0
$479$	271.737i	0.567300i	0.958928	+	0.283650i	$0.0915454\pi$
$479$	271.737i	0.567300i	−0.958928	+	0.283650i	$0.908455\pi$
$480$	0	0
$481$	106.657i	0.221741i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	255.765	0.527349
$486$	0	0
$487$	561.514	1.15301	0.576503	−	0.817095i	$-0.304417\pi$
$487$	561.514	1.15301	0.576503	+	0.817095i	$0.304417\pi$
$488$	0	0
$489$	− 27.8148i	− 0.0568810i
$490$	0	0
$491$	406.441	0.827781	0.413891	−	0.910327i	$-0.364170\pi$
$491$	406.441	0.827781	0.413891	+	0.910327i	$0.364170\pi$
$492$	0	0
$493$	636.579i	1.29124i
$494$	0	0
$495$	150.535i	0.304112i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	371.426	0.744341	0.372171	−	0.928164i	$-0.378614\pi$
$499$	371.426	0.744341	0.372171	+	0.928164i	$0.378614\pi$
$500$	0	0
$501$	305.044	0.608870
$502$	0	0
$503$	64.6292i	0.128488i	0.997934	+	0.0642438i	$0.0204635\pi$
$503$	64.6292i	0.128488i	−0.997934	+	0.0642438i	$0.979536\pi$
$504$	0	0
$505$	1070.21	2.11922
$506$	0	0
$507$	− 260.012i	− 0.512843i
$508$	0	0
$509$	1006.77i	1.97794i	0.148125	+	0.988969i	$0.452676\pi$
$509$	1006.77i	1.97794i	−0.148125	+	0.988969i	$0.547324\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	88.4558	0.172429
$514$	0	0
$515$	−677.044	−1.31465
$516$	0	0
$517$	− 199.237i	− 0.385371i
$518$	0	0
$519$	400.368	0.771421
$520$	0	0
$521$	372.153i	0.714306i	0.934046	+	0.357153i	$0.116253\pi$
$521$	372.153i	0.714306i	−0.934046	+	0.357153i	$0.883747\pi$
$522$	0	0
$523$	637.284i	1.21852i	0.792972	+	0.609258i	$0.208533\pi$
$523$	637.284i	1.21852i	−0.792972	+	0.609258i	$0.791467\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−276.603	−0.524863
$528$	0	0
$529$	−347.940	−0.657732
$530$	0	0
$531$	− 82.0940i	− 0.154603i
$532$	0	0
$533$	629.440	1.18094
$534$	0	0
$535$	− 1417.17i	− 2.64892i
$536$	0	0
$537$	147.708i	0.275061i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	220.823	0.408176	0.204088	−	0.978953i	$-0.434577\pi$
$541$	220.823	0.408176	0.204088	+	0.978953i	$0.434577\pi$
$542$	0	0
$543$	−9.67619	−0.0178199
$544$	0	0
$545$	1496.50i	2.74587i
$546$	0	0
$547$	160.676	0.293741	0.146870	−	0.989156i	$-0.453080\pi$
$547$	160.676	0.293741	0.146870	+	0.989156i	$0.453080\pi$
$548$	0	0
$549$	121.142i	0.220659i
$550$	0	0
$551$	577.791i	1.04862i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−86.4853	−0.155829
$556$	0	0
$557$	−474.353	−0.851622	−0.425811	−	0.904812i	$-0.640011\pi$
$557$	−474.353	−0.851622	−0.425811	+	0.904812i	$0.640011\pi$
$558$	0	0
$559$	− 276.627i	− 0.494860i
$560$	0	0
$561$	194.912	0.347436
$562$	0	0
$563$	− 496.868i	− 0.882537i	−0.897375	−	0.441269i	$-0.854529\pi$
$563$	− 496.868i	− 0.882537i	0.897375	−	0.441269i	$-0.145471\pi$
$564$	0	0
$565$	− 145.492i	− 0.257508i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	785.294	1.38013	0.690065	−	0.723748i	$-0.257582\pi$
$569$	785.294	1.38013	0.690065	+	0.723748i	$0.257582\pi$
$570$	0	0
$571$	715.043	1.25226	0.626132	−	0.779717i	$-0.284637\pi$
$571$	715.043	1.25226	0.626132	+	0.779717i	$0.284637\pi$
$572$	0	0
$573$	− 321.117i	− 0.560414i
$574$	0	0
$575$	604.721	1.05169
$576$	0	0
$577$	− 772.630i	− 1.33905i	−0.742791	−	0.669524i	$-0.766498\pi$
$577$	− 772.630i	− 1.33905i	0.742791	−	0.669524i	$-0.233502\pi$
$578$	0	0
$579$	394.551i	0.681435i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	207.265	0.355514
$584$	0	0
$585$	448.191	0.766138
$586$	0	0
$587$	− 436.477i	− 0.743572i	−0.928318	−	0.371786i	$-0.878746\pi$
$587$	− 436.477i	− 0.743572i	0.928318	−	0.371786i	$-0.121254\pi$
$588$	0	0
$589$	−251.059	−0.426246
$590$	0	0
$591$	213.322i	0.360951i
$592$	0	0
$593$	834.152i	1.40666i	0.710861	+	0.703332i	$0.248305\pi$
$593$	834.152i	1.40666i	−0.710861	+	0.703332i	$0.751695\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−10.7939	−0.0180803
$598$	0	0
$599$	−873.588	−1.45841	−0.729205	−	0.684295i	$-0.760110\pi$
$599$	−873.588	−1.45841	−0.729205	+	0.684295i	$0.760110\pi$
$600$	0	0
$601$	− 198.982i	− 0.331085i	−0.986203	−	0.165542i	$-0.947063\pi$
$601$	− 198.982i	− 0.331085i	0.986203	−	0.165542i	$-0.0529375\pi$
$602$	0	0
$603$	−343.191	−0.569139
$604$	0	0
$605$	− 710.862i	− 1.17498i
$606$	0	0
$607$	− 158.950i	− 0.261861i	−0.991392	−	0.130930i	$-0.958204\pi$
$607$	− 158.950i	− 0.261861i	0.991392	−	0.130930i	$-0.0417964\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−593.190	−0.970851
$612$	0	0
$613$	−714.735	−1.16596	−0.582981	−	0.812486i	$-0.698114\pi$
$613$	−714.735	−1.16596	−0.582981	+	0.812486i	$0.698114\pi$
$614$	0	0
$615$	510.394i	0.829909i
$616$	0	0
$617$	−639.381	−1.03627	−0.518137	−	0.855298i	$-0.673374\pi$
$617$	−639.381	−1.03627	−0.518137	+	0.855298i	$0.673374\pi$
$618$	0	0
$619$	172.025i	0.277908i	0.990299	+	0.138954i	$0.0443740\pi$
$619$	172.025i	0.277908i	−0.990299	+	0.138954i	$0.955626\pi$
$620$	0	0
$621$	69.9186i	0.112590i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	271.177	0.433883
$626$	0	0
$627$	176.912	0.282156
$628$	0	0
$629$	111.980i	0.178029i
$630$	0	0
$631$	1141.06	1.80833	0.904166	−	0.427180i	$-0.140493\pi$
$631$	1141.06	1.80833	0.904166	+	0.427180i	$0.140493\pi$
$632$	0	0
$633$	216.353i	0.341791i
$634$	0	0
$635$	− 1400.20i	− 2.20504i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	55.8091	0.0873382
$640$	0	0
$641$	229.103	0.357414	0.178707	−	0.983902i	$-0.442809\pi$
$641$	229.103	0.357414	0.178707	+	0.983902i	$0.442809\pi$
$642$	0	0
$643$	707.670i	1.10058i	0.834975	+	0.550288i	$0.185482\pi$
$643$	707.670i	1.10058i	−0.834975	+	0.550288i	$0.814518\pi$
$644$	0	0
$645$	224.309	0.347765
$646$	0	0
$647$	1179.37i	1.82283i	0.411484	+	0.911417i	$0.365011\pi$
$647$	1179.37i	1.82283i	−0.411484	+	0.911417i	$0.634989\pi$
$648$	0	0
$649$	− 164.188i	− 0.252986i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−154.764	−0.237004	−0.118502	−	0.992954i	$-0.537809\pi$
$653$	−154.764	−0.237004	−0.118502	+	0.992954i	$0.537809\pi$
$654$	0	0
$655$	−14.9117	−0.0227659
$656$	0	0
$657$	− 351.096i	− 0.534393i
$658$	0	0
$659$	−591.308	−0.897280	−0.448640	−	0.893712i	$-0.648092\pi$
$659$	−591.308	−0.897280	−0.448640	+	0.893712i	$0.648092\pi$
$660$	0	0
$661$	− 162.167i	− 0.245337i	−0.992448	−	0.122668i	$-0.960855\pi$
$661$	− 162.167i	− 0.245337i	0.992448	−	0.122668i	$-0.0391451\pi$
$662$	0	0
$663$	− 580.313i	− 0.875283i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−456.706	−0.684717
$668$	0	0
$669$	396.000	0.591928
$670$	0	0
$671$	242.283i	0.361078i
$672$	0	0
$673$	42.3238	0.0628883	0.0314441	−	0.999506i	$-0.489989\pi$
$673$	42.3238	0.0628883	0.0314441	+	0.999506i	$0.489989\pi$
$674$	0	0
$675$	233.521i	0.345957i
$676$	0	0
$677$	497.354i	0.734643i	0.930094	+	0.367322i	$0.119725\pi$
$677$	497.354i	0.734643i	−0.930094	+	0.367322i	$0.880275\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	293.647	0.431199
$682$	0	0
$683$	−1216.16	−1.78062	−0.890308	−	0.455359i	$-0.849511\pi$
$683$	−1216.16	−1.78062	−0.890308	+	0.455359i	$0.849511\pi$
$684$	0	0
$685$	844.425i	1.23274i
$686$	0	0
$687$	−60.0883	−0.0874648
$688$	0	0
$689$	− 617.092i	− 0.895635i
$690$	0	0
$691$	1076.39i	1.55773i	0.627191	+	0.778865i	$0.284204\pi$
$691$	1076.39i	1.55773i	−0.627191	+	0.778865i	$0.715796\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1175.37	1.69118
$696$	0	0
$697$	660.853	0.948139
$698$	0	0
$699$	− 440.781i	− 0.630589i
$700$	0	0
$701$	−695.897	−0.992720	−0.496360	−	0.868117i	$-0.665330\pi$
$701$	−695.897	−0.992720	−0.496360	+	0.868117i	$0.665330\pi$
$702$	0	0
$703$	101.639i	0.144579i
$704$	0	0
$705$	− 481.000i	− 0.682270i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	254.824	0.359414	0.179707	−	0.983720i	$-0.442485\pi$
$709$	254.824	0.359414	0.179707	+	0.983720i	$0.442485\pi$
$710$	0	0
$711$	−265.014	−0.372735
$712$	0	0
$713$	− 198.446i	− 0.278325i
$714$	0	0
$715$	896.382	1.25368
$716$	0	0
$717$	− 341.469i	− 0.476247i
$718$	0	0
$719$	1114.20i	1.54965i	0.632175	+	0.774826i	$0.282162\pi$
$719$	1114.20i	1.54965i	−0.632175	+	0.774826i	$0.717838\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	153.235	0.211943
$724$	0	0
$725$	−1525.35	−2.10393
$726$	0	0
$727$	− 398.345i	− 0.547930i	−0.961740	−	0.273965i	$-0.911665\pi$
$727$	− 398.345i	− 0.547930i	0.961740	−	0.273965i	$-0.0883353\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 290.432i	− 0.397308i
$732$	0	0
$733$	945.139i	1.28941i	0.764431	+	0.644706i	$0.223020\pi$
$733$	945.139i	1.28941i	−0.764431	+	0.644706i	$0.776980\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−686.382	−0.931319
$738$	0	0
$739$	−192.632	−0.260666	−0.130333	−	0.991470i	$-0.541605\pi$
$739$	−192.632	−0.260666	−0.130333	+	0.991470i	$0.541605\pi$
$740$	0	0
$741$	− 526.721i	− 0.710825i
$742$	0	0
$743$	911.616	1.22694	0.613470	−	0.789718i	$-0.289773\pi$
$743$	911.616	1.22694	0.613470	+	0.789718i	$0.289773\pi$
$744$	0	0
$745$	1529.71i	2.05330i
$746$	0	0
$747$	− 227.280i	− 0.304257i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	391.662	0.521521	0.260760	−	0.965404i	$-0.416027\pi$
$751$	391.662	0.521521	0.260760	+	0.965404i	$0.416027\pi$
$752$	0	0
$753$	373.955	0.496621
$754$	0	0
$755$	2415.21i	3.19895i
$756$	0	0
$757$	−152.823	−0.201879	−0.100940	−	0.994893i	$-0.532185\pi$
$757$	−152.823	−0.201879	−0.100940	+	0.994893i	$0.532185\pi$
$758$	0	0
$759$	139.837i	0.184239i
$760$	0	0
$761$	− 126.245i	− 0.165893i	−0.996554	−	0.0829465i	$-0.973567\pi$
$761$	− 126.245i	− 0.165893i	0.996554	−	0.0829465i	$-0.0264330\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	470.558	0.615109
$766$	0	0
$767$	−488.839	−0.637338
$768$	0	0
$769$	369.148i	0.480037i	0.970768	+	0.240018i	$0.0771535\pi$
$769$	369.148i	0.480037i	−0.970768	+	0.240018i	$0.922847\pi$
$770$	0	0
$771$	7.45584	0.00967036
$772$	0	0
$773$	1403.71i	1.81592i	0.419056	+	0.907961i	$0.362361\pi$
$773$	1403.71i	1.81592i	−0.419056	+	0.907961i	$0.637639\pi$
$774$	0	0
$775$	− 662.788i	− 0.855211i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	599.823	0.769991
$780$	0	0
$781$	111.618	0.142917
$782$	0	0
$783$	− 176.363i	− 0.225240i
$784$	0	0
$785$	1564.41	1.99288
$786$	0	0
$787$	226.507i	0.287810i	0.989591	+	0.143905i	$0.0459660\pi$
$787$	226.507i	0.287810i	−0.989591	+	0.143905i	$0.954034\pi$
$788$	0	0
$789$	− 489.610i	− 0.620544i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	721.352	0.909650
$794$	0	0
$795$	500.382	0.629411
$796$	0	0
$797$	688.414i	0.863756i	0.901932	+	0.431878i	$0.142149\pi$
$797$	688.414i	0.863756i	−0.901932	+	0.431878i	$0.857851\pi$
$798$	0	0
$799$	−622.794	−0.779467
$800$	0	0
$801$	62.3538i	0.0778450i
$802$	0	0
$803$	− 702.193i	− 0.874462i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−661.529	−0.819739
$808$	0	0
$809$	25.2792	0.0312475	0.0156237	−	0.999878i	$-0.495027\pi$
$809$	25.2792	0.0312475	0.0156237	+	0.999878i	$0.495027\pi$
$810$	0	0
$811$	1527.62i	1.88362i	0.336145	+	0.941810i	$0.390877\pi$
$811$	1527.62i	1.88362i	−0.336145	+	0.941810i	$0.609123\pi$
$812$	0	0
$813$	−146.059	−0.179654
$814$	0	0
$815$	− 134.302i	− 0.164787i
$816$	0	0
$817$	− 263.611i	− 0.322657i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−116.662	−0.142097	−0.0710487	−	0.997473i	$-0.522635\pi$
$821$	−116.662	−0.142097	−0.0710487	+	0.997473i	$0.522635\pi$
$822$	0	0
$823$	−125.911	−0.152990	−0.0764950	−	0.997070i	$-0.524373\pi$
$823$	−125.911	−0.152990	−0.0764950	+	0.997070i	$0.524373\pi$
$824$	0	0
$825$	467.042i	0.566111i
$826$	0	0
$827$	1434.40	1.73446	0.867229	−	0.497910i	$-0.165899\pi$
$827$	1434.40	1.73446	0.867229	+	0.497910i	$0.165899\pi$
$828$	0	0
$829$	37.3228i	0.0450215i	0.999747	+	0.0225107i	$0.00716600\pi$
$829$	37.3228i	0.0450215i	−0.999747	+	0.0225107i	$0.992834\pi$
$830$	0	0
$831$	− 237.495i	− 0.285794i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1472.88	1.76393
$836$	0	0
$837$	76.6325	0.0915561
$838$	0	0
$839$	3.07370i	0.00366353i	0.999998	+	0.00183177i	$0.000583069\pi$
$839$	3.07370i	0.00366353i	−0.999998	+	0.00183177i	$0.999417\pi$
$840$	0	0
$841$	311.000	0.369798
$842$	0	0
$843$	− 563.094i	− 0.667965i
$844$	0	0
$845$	− 1255.45i	− 1.48574i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−337.014	−0.396954
$850$	0	0
$851$	−80.3390	−0.0944054
$852$	0	0
$853$	155.257i	0.182013i	0.995850	+	0.0910063i	$0.0290083\pi$
$853$	155.257i	0.182013i	−0.995850	+	0.0910063i	$0.970992\pi$
$854$	0	0
$855$	427.103	0.499535
$856$	0	0
$857$	1603.86i	1.87148i	0.352686	+	0.935742i	$0.385269\pi$
$857$	1603.86i	1.87148i	−0.352686	+	0.935742i	$0.614731\pi$
$858$	0	0
$859$	− 629.735i	− 0.733103i	−0.930398	−	0.366551i	$-0.880538\pi$
$859$	− 629.735i	− 0.733103i	0.930398	−	0.366551i	$-0.119462\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1029.41	−1.19283	−0.596414	−	0.802677i	$-0.703408\pi$
$863$	−1029.41	−1.19283	−0.596414	+	0.802677i	$0.703408\pi$
$864$	0	0
$865$	1933.15	2.23485
$866$	0	0
$867$	− 108.711i	− 0.125388i
$868$	0	0
$869$	−530.029	−0.609929
$870$	0	0
$871$	2043.57i	2.34624i
$872$	0	0
$873$	91.7477i	0.105095i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	648.441	0.739385	0.369693	−	0.929154i	$-0.379463\pi$
$877$	648.441	0.739385	0.369693	+	0.929154i	$0.379463\pi$
$878$	0	0
$879$	−415.176	−0.472327
$880$	0	0
$881$	363.857i	0.413005i	0.978446	+	0.206502i	$0.0662082\pi$
$881$	363.857i	0.413005i	−0.978446	+	0.206502i	$0.933792\pi$
$882$	0	0
$883$	−1536.16	−1.73971	−0.869853	−	0.493312i	$-0.835786\pi$
$883$	−1536.16	−1.73971	−0.869853	+	0.493312i	$0.835786\pi$
$884$	0	0
$885$	− 396.385i	− 0.447892i
$886$	0	0
$887$	− 1125.51i	− 1.26889i	−0.772966	−	0.634447i	$-0.781228\pi$
$887$	− 1125.51i	− 1.26889i	0.772966	−	0.634447i	$-0.218772\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−54.0000	−0.0606061
$892$	0	0
$893$	−565.279	−0.633011
$894$	0	0
$895$	713.197i	0.796868i
$896$	0	0
$897$	416.339	0.464146
$898$	0	0
$899$	500.561i	0.556798i
$900$	0	0
$901$	− 647.889i	− 0.719078i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−46.7208	−0.0516252
$906$	0	0
$907$	−234.897	−0.258982	−0.129491	−	0.991581i	$-0.541334\pi$
$907$	−234.897	−0.258982	−0.129491	+	0.991581i	$0.541334\pi$
$908$	0	0
$909$	383.903i	0.422336i
$910$	0	0
$911$	−224.278	−0.246189	−0.123095	−	0.992395i	$-0.539282\pi$
$911$	−224.278	−0.246189	−0.123095	+	0.992395i	$0.539282\pi$
$912$	0	0
$913$	− 454.561i	− 0.497876i
$914$	0	0
$915$	584.923i	0.639260i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	932.161	1.01432	0.507161	−	0.861851i	$-0.330695\pi$
$919$	932.161	1.01432	0.507161	+	0.861851i	$0.330695\pi$
$920$	0	0
$921$	−935.778	−1.01605
$922$	0	0
$923$	− 332.322i	− 0.360046i
$924$	0	0
$925$	−268.324	−0.290080
$926$	0	0
$927$	− 242.869i	− 0.261994i
$928$	0	0
$929$	− 714.055i	− 0.768628i	−0.923203	−	0.384314i	$-0.874438\pi$
$929$	− 714.055i	− 0.768628i	0.923203	−	0.384314i	$-0.125562\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−700.087	−0.750362
$934$	0	0
$935$	941.117	1.00654
$936$	0	0
$937$	1723.25i	1.83912i	0.392952	+	0.919559i	$0.371454\pi$
$937$	1723.25i	1.83912i	−0.392952	+	0.919559i	$0.628546\pi$
$938$	0	0
$939$	227.412	0.242185
$940$	0	0
$941$	− 964.761i	− 1.02525i	−0.858612	−	0.512625i	$-0.828673\pi$
$941$	− 964.761i	− 1.02525i	0.858612	−	0.512625i	$-0.171327\pi$
$942$	0	0
$943$	474.122i	0.502780i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1451.76	−1.53301	−0.766506	−	0.642237i	$-0.778007\pi$
$947$	−1451.76	−1.53301	−0.766506	+	0.642237i	$0.778007\pi$
$948$	0	0
$949$	−2090.65	−2.20300
$950$	0	0
$951$	− 162.711i	− 0.171094i
$952$	0	0
$953$	1147.43	1.20401	0.602007	−	0.798491i	$-0.294368\pi$
$953$	1147.43	1.20401	0.602007	+	0.798491i	$0.294368\pi$
$954$	0	0
$955$	− 1550.49i	− 1.62355i
$956$	0	0
$957$	− 352.727i	− 0.368575i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	743.499	0.773672
$962$	0	0
$963$	508.368	0.527900
$964$	0	0
$965$	1905.06i	1.97415i
$966$	0	0
$967$	−412.190	−0.426257	−0.213128	−	0.977024i	$-0.568365\pi$
$967$	−412.190	−0.426257	−0.213128	+	0.977024i	$0.568365\pi$
$968$	0	0
$969$	− 553.008i	− 0.570700i
$970$	0	0
$971$	− 1004.12i	− 1.03411i	−0.855952	−	0.517056i	$-0.827028\pi$
$971$	− 1004.12i	− 1.03411i	0.855952	−	0.517056i	$-0.172972\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1390.53	1.42618
$976$	0	0
$977$	−1588.23	−1.62562	−0.812812	−	0.582527i	$-0.802064\pi$
$977$	−1588.23	−1.62562	−0.812812	+	0.582527i	$0.802064\pi$
$978$	0	0
$979$	124.708i	0.127383i
$980$	0	0
$981$	−536.823	−0.547221
$982$	0	0
$983$	833.533i	0.847948i	0.905674	+	0.423974i	$0.139365\pi$
$983$	833.533i	0.847948i	−0.905674	+	0.423974i	$0.860635\pi$
$984$	0	0
$985$	1030.01i	1.04569i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	208.368	0.210685
$990$	0	0
$991$	−66.8965	−0.0675041	−0.0337520	−	0.999430i	$-0.510746\pi$
$991$	−66.8965	−0.0675041	−0.0337520	+	0.999430i	$0.510746\pi$
$992$	0	0
$993$	452.702i	0.455893i
$994$	0	0
$995$	−52.1177	−0.0523796
$996$	0	0
$997$	− 1464.91i	− 1.46931i	−0.678439	−	0.734657i	$-0.737343\pi$
$997$	− 1464.91i	− 1.46931i	0.678439	−	0.734657i	$-0.262657\pi$
$998$	0	0
$999$	− 31.0240i	− 0.0310550i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.f.e.97.4		4
4.3	odd	2		294.3.c.a.97.3		4
7.2	even	3		336.3.bh.e.241.2		4
7.3	odd	6		336.3.bh.e.145.2		4
7.6	odd	2	inner	2352.3.f.e.97.1		4
12.11	even	2		882.3.c.b.685.1		4
21.2	odd	6		1008.3.cg.h.577.1		4
21.17	even	6		1008.3.cg.h.145.1		4
28.3	even	6		42.3.g.a.19.1	✓	4
28.11	odd	6		294.3.g.a.19.1		4
28.19	even	6		294.3.g.a.31.1		4
28.23	odd	6		42.3.g.a.31.1	yes	4
28.27	even	2		294.3.c.a.97.4		4
84.11	even	6		882.3.n.e.19.2		4
84.23	even	6		126.3.n.a.73.2		4
84.47	odd	6		882.3.n.e.325.2		4
84.59	odd	6		126.3.n.a.19.2		4
84.83	odd	2		882.3.c.b.685.2		4
140.3	odd	12		1050.3.q.a.649.4		8
140.23	even	12		1050.3.q.a.199.1		8
140.59	even	6		1050.3.p.a.901.2		4
140.79	odd	6		1050.3.p.a.451.2		4
140.87	odd	12		1050.3.q.a.649.1		8
140.107	even	12		1050.3.q.a.199.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.g.a.19.1	✓	4	28.3	even	6
42.3.g.a.31.1	yes	4	28.23	odd	6
126.3.n.a.19.2		4	84.59	odd	6
126.3.n.a.73.2		4	84.23	even	6
294.3.c.a.97.3		4	4.3	odd	2
294.3.c.a.97.4		4	28.27	even	2
294.3.g.a.19.1		4	28.11	odd	6
294.3.g.a.31.1		4	28.19	even	6
336.3.bh.e.145.2		4	7.3	odd	6
336.3.bh.e.241.2		4	7.2	even	3
882.3.c.b.685.1		4	12.11	even	2
882.3.c.b.685.2		4	84.83	odd	2
882.3.n.e.19.2		4	84.11	even	6
882.3.n.e.325.2		4	84.47	odd	6
1008.3.cg.h.145.1		4	21.17	even	6
1008.3.cg.h.577.1		4	21.2	odd	6
1050.3.p.a.451.2		4	140.79	odd	6
1050.3.p.a.901.2		4	140.59	even	6
1050.3.q.a.199.1		8	140.23	even	12
1050.3.q.a.199.4		8	140.107	even	12
1050.3.q.a.649.1		8	140.87	odd	12
1050.3.q.a.649.4		8	140.3	odd	12
2352.3.f.e.97.1		4	7.6	odd	2	inner
2352.3.f.e.97.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +8.36308i q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +8.36308i q^{5} -3.00000 q^{9} -6.00000 q^{11} +17.8639i q^{13} -14.4853 q^{15} +18.7554i q^{17} +17.0233i q^{19} -13.4558 q^{23} -44.9411 q^{25} -5.19615i q^{27} +33.9411 q^{29} +14.7479i q^{31} -10.3923i q^{33} +5.97056 q^{37} -30.9411 q^{39} -35.2354i q^{41} -15.4853 q^{43} -25.0892i q^{45} +33.2061i q^{47} -32.4853 q^{51} -34.5442 q^{53} -50.1785i q^{55} -29.4853 q^{57} +27.3647i q^{59} -40.3805i q^{61} -149.397 q^{65} +114.397 q^{67} -23.3062i q^{69} -18.6030 q^{71} +117.032i q^{73} -77.8403i q^{75} +88.3381 q^{79} +9.00000 q^{81} +75.7601i q^{83} -156.853 q^{85} +58.7878i q^{87} -20.7846i q^{89} -25.5442 q^{93} -142.368 q^{95} -30.5826i q^{97} +18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 24 q^{11} - 24 q^{15} + 48 q^{23} - 44 q^{25} - 44 q^{37} + 12 q^{39} - 28 q^{43} - 96 q^{51} - 240 q^{53} - 84 q^{57} - 360 q^{65} + 220 q^{67} - 312 q^{71} - 20 q^{79} + 36 q^{81} - 288 q^{85} - 204 q^{93} - 264 q^{95} + 72 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 - 24 * q^11 - 24 * q^15 + 48 * q^23 - 44 * q^25 - 44 * q^37 + 12 * q^39 - 28 * q^43 - 96 * q^51 - 240 * q^53 - 84 * q^57 - 360 * q^65 + 220 * q^67 - 312 * q^71 - 20 * q^79 + 36 * q^81 - 288 * q^85 - 204 * q^93 - 264 * q^95 + 72 * q^99

Introduction

L-functions

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