LMFDB - Embedded newform 2352.3.m.j.1471.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,3,Mod(1471,2352)]

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([1, 0, 0, 0]))

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

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

chi := DirichletCharacter("2352.1471");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2352.m (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:	$64.0873581775$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{37})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 10x^{2} + 9x + 81 $ x^4 - x^3 + 10x^2 + 9x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.3
Root			$1.77069 - 3.06693i$ of defining polynomial
Character	$\chi$	$=$	2352.1471
Dual form			2352.3.m.j.1471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -3.08276 q^{5} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -3.08276 q^{5} -3.00000 q^{9} -12.2677i q^{11} -5.33950i q^{15} -15.0828 q^{17} +24.5354i q^{19} -8.51691i q^{23} -15.4966 q^{25} -5.19615i q^{27} -24.4966 q^{29} +3.75080i q^{31} +21.2483 q^{33} +52.4966 q^{37} +39.0828 q^{41} +27.7128i q^{43} +9.24829 q^{45} -21.3580i q^{47} -26.1241i q^{51} +11.5034 q^{53} +37.8184i q^{55} -42.4966 q^{57} -42.1426i q^{59} +29.5034 q^{61} -73.6062i q^{67} +14.7517 q^{69} -40.5539i q^{71} -42.4966 q^{73} -26.8409i q^{75} -23.3886i q^{79} +9.00000 q^{81} +83.7118i q^{83} +46.4966 q^{85} -42.4293i q^{87} +148.076 q^{89} -6.49658 q^{93} -75.6368i q^{95} +186.497 q^{97} +36.8031i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^5 - 12 * q^9	$ 4 q + 12 q^{5} - 12 q^{9} - 36 q^{17} + 84 q^{25} + 48 q^{29} + 12 q^{33} + 64 q^{37} + 132 q^{41} - 36 q^{45} + 192 q^{53} - 24 q^{57} + 264 q^{61} + 132 q^{69} - 24 q^{73} + 36 q^{81} + 40 q^{85} + 276 q^{89} + 120 q^{93} + 600 q^{97}+O(q^{100}) $ 4 * q + 12 * q^5 - 12 * q^9 - 36 * q^17 + 84 * q^25 + 48 * q^29 + 12 * q^33 + 64 * q^37 + 132 * q^41 - 36 * q^45 + 192 * q^53 - 24 * q^57 + 264 * q^61 + 132 * q^69 - 24 * q^73 + 36 * q^81 + 40 * q^85 + 276 * q^89 + 120 * q^93 + 600 * q^97

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
$5$	−3.08276	−0.616553	−0.308276	−	0.951297i	$-0.599752\pi$
$5$	−3.08276	−0.616553	−0.308276	+	0.951297i	$0.599752\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 12.2677i	− 1.11525i	−0.830094	−	0.557623i	$-0.811714\pi$
$11$	− 12.2677i	− 1.11525i	0.830094	−	0.557623i	$-0.188286\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	− 5.33950i	− 0.355967i
$16$	0	0
$17$	−15.0828	−0.887221	−0.443611	−	0.896220i	$-0.646303\pi$
$17$	−15.0828	−0.887221	−0.443611	+	0.896220i	$0.646303\pi$
$18$	0	0
$19$	24.5354i	1.29134i	0.763618	+	0.645669i	$0.223421\pi$
$19$	24.5354i	1.29134i	−0.763618	+	0.645669i	$0.776579\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 8.51691i	− 0.370300i	−0.982710	−	0.185150i	$-0.940723\pi$
$23$	− 8.51691i	− 0.370300i	0.982710	−	0.185150i	$-0.0592771\pi$
$24$	0	0
$25$	−15.4966	−0.619863
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−24.4966	−0.844709	−0.422355	−	0.906431i	$-0.638796\pi$
$29$	−24.4966	−0.844709	−0.422355	+	0.906431i	$0.638796\pi$
$30$	0	0
$31$	3.75080i	0.120994i	0.998168	+	0.0604968i	$0.0192685\pi$
$31$	3.75080i	0.120994i	−0.998168	+	0.0604968i	$0.980732\pi$
$32$	0	0
$33$	21.2483	0.643888
$34$	0	0
$35$	0	0
$36$	0	0
$37$	52.4966	1.41883	0.709413	−	0.704793i	$-0.248960\pi$
$37$	52.4966	1.41883	0.709413	+	0.704793i	$0.248960\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	39.0828	0.953238	0.476619	−	0.879110i	$-0.341862\pi$
$41$	39.0828	0.953238	0.476619	+	0.879110i	$0.341862\pi$
$42$	0	0
$43$	27.7128i	0.644484i	0.946657	+	0.322242i	$0.104436\pi$
$43$	27.7128i	0.644484i	−0.946657	+	0.322242i	$0.895564\pi$
$44$	0	0
$45$	9.24829	0.205518
$46$	0	0
$47$	− 21.3580i	− 0.454426i	−0.973845	−	0.227213i	$-0.927039\pi$
$47$	− 21.3580i	− 0.454426i	0.973845	−	0.227213i	$-0.0729613\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 26.1241i	− 0.512237i
$52$	0	0
$53$	11.5034	0.217046	0.108523	−	0.994094i	$-0.465388\pi$
$53$	11.5034	0.217046	0.108523	+	0.994094i	$0.465388\pi$
$54$	0	0
$55$	37.8184i	0.687608i
$56$	0	0
$57$	−42.4966	−0.745554
$58$	0	0
$59$	− 42.1426i	− 0.714282i	−0.934051	−	0.357141i	$-0.883752\pi$
$59$	− 42.1426i	− 0.714282i	0.934051	−	0.357141i	$-0.116248\pi$
$60$	0	0
$61$	29.5034	0.483663	0.241831	−	0.970318i	$-0.422252\pi$
$61$	29.5034	0.483663	0.241831	+	0.970318i	$0.422252\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 73.6062i	− 1.09860i	−0.835625	−	0.549300i	$-0.814894\pi$
$67$	− 73.6062i	− 1.09860i	0.835625	−	0.549300i	$-0.185106\pi$
$68$	0	0
$69$	14.7517	0.213793
$70$	0	0
$71$	− 40.5539i	− 0.571182i	−0.958352	−	0.285591i	$-0.907810\pi$
$71$	− 40.5539i	− 0.571182i	0.958352	−	0.285591i	$-0.0921899\pi$
$72$	0	0
$73$	−42.4966	−0.582145	−0.291072	−	0.956701i	$-0.594012\pi$
$73$	−42.4966	−0.582145	−0.291072	+	0.956701i	$0.594012\pi$
$74$	0	0
$75$	− 26.8409i	− 0.357878i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 23.3886i	− 0.296058i	−0.988983	−	0.148029i	$-0.952707\pi$
$79$	− 23.3886i	− 0.296058i	0.988983	−	0.148029i	$-0.0472930\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	83.7118i	1.00858i	0.863535	+	0.504288i	$0.168245\pi$
$83$	83.7118i	1.00858i	−0.863535	+	0.504288i	$0.831755\pi$
$84$	0	0
$85$	46.4966	0.547019
$86$	0	0
$87$	− 42.4293i	− 0.487693i
$88$	0	0
$89$	148.076	1.66377	0.831887	−	0.554945i	$-0.187261\pi$
$89$	148.076	1.66377	0.831887	+	0.554945i	$0.187261\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.49658	−0.0698556
$94$	0	0
$95$	− 75.6368i	− 0.796177i
$96$	0	0
$97$	186.497	1.92265	0.961323	−	0.275425i	$-0.0888186\pi$
$97$	186.497	1.92265	0.961323	+	0.275425i	$0.0888186\pi$
$98$	0	0
$99$	36.8031i	0.371749i
$100$	0	0
$101$	16.0759	0.159167	0.0795837	−	0.996828i	$-0.474641\pi$
$101$	16.0759	0.159167	0.0795837	+	0.996828i	$0.474641\pi$
$102$	0	0
$103$	109.394i	1.06208i	0.847347	+	0.531039i	$0.178198\pi$
$103$	109.394i	1.06208i	−0.847347	+	0.531039i	$0.821802\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 68.8401i	− 0.643366i	−0.946847	−	0.321683i	$-0.895752\pi$
$107$	− 68.8401i	− 0.643366i	0.946847	−	0.321683i	$-0.104248\pi$
$108$	0	0
$109$	2.99315	0.0274601	0.0137300	−	0.999906i	$-0.495629\pi$
$109$	2.99315	0.0274601	0.0137300	+	0.999906i	$0.495629\pi$
$110$	0	0
$111$	90.9267i	0.819160i
$112$	0	0
$113$	30.9932	0.274276	0.137138	−	0.990552i	$-0.456210\pi$
$113$	30.9932	0.274276	0.137138	+	0.990552i	$0.456210\pi$
$114$	0	0
$115$	26.2556i	0.228310i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−29.4966	−0.243773
$122$	0	0
$123$	67.6933i	0.550352i
$124$	0	0
$125$	124.841	0.998731
$126$	0	0
$127$	87.4626i	0.688682i	0.938845	+	0.344341i	$0.111898\pi$
$127$	87.4626i	0.688682i	−0.938845	+	0.344341i	$0.888102\pi$
$128$	0	0
$129$	−48.0000	−0.372093
$130$	0	0
$131$	187.061i	1.42795i	0.700171	+	0.713975i	$0.253107\pi$
$131$	187.061i	1.42795i	−0.700171	+	0.713975i	$0.746893\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	16.0185i	0.118656i
$136$	0	0
$137$	151.986	1.10939	0.554695	−	0.832054i	$-0.312835\pi$
$137$	151.986	1.10939	0.554695	+	0.832054i	$0.312835\pi$
$138$	0	0
$139$	− 43.2894i	− 0.311435i	−0.987802	−	0.155717i	$-0.950231\pi$
$139$	− 43.2894i	− 0.311435i	0.987802	−	0.155717i	$-0.0497689\pi$
$140$	0	0
$141$	36.9932	0.262363
$142$	0	0
$143$	0	0
$144$	0	0
$145$	75.5171	0.520808
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.5034	0.0772042	0.0386021	−	0.999255i	$-0.487710\pi$
$149$	11.5034	0.0772042	0.0386021	+	0.999255i	$0.487710\pi$
$150$	0	0
$151$	266.712i	1.76631i	0.469086	+	0.883153i	$0.344584\pi$
$151$	266.712i	1.76631i	−0.469086	+	0.883153i	$0.655416\pi$
$152$	0	0
$153$	45.2483	0.295740
$154$	0	0
$155$	− 11.5628i	− 0.0745989i
$156$	0	0
$157$	−284.483	−1.81199	−0.905996	−	0.423285i	$-0.860877\pi$
$157$	−284.483	−1.81199	−0.905996	+	0.423285i	$0.860877\pi$
$158$	0	0
$159$	19.9245i	0.125311i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 82.2546i	− 0.504630i	−0.967645	−	0.252315i	$-0.918808\pi$
$163$	− 82.2546i	− 0.504630i	0.967645	−	0.252315i	$-0.0811918\pi$
$164$	0	0
$165$	−65.5034	−0.396990
$166$	0	0
$167$	144.919i	0.867778i	0.900966	+	0.433889i	$0.142859\pi$
$167$	144.919i	0.867778i	−0.900966	+	0.433889i	$0.857141\pi$
$168$	0	0
$169$	−169.000	−1.00000
$170$	0	0
$171$	− 73.6062i	− 0.430446i
$172$	0	0
$173$	219.083	1.26637	0.633187	−	0.773999i	$-0.281746\pi$
$173$	219.083	1.26637	0.633187	+	0.773999i	$0.281746\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	72.9932	0.412391
$178$	0	0
$179$	− 119.942i	− 0.670065i	−0.942207	−	0.335032i	$-0.891253\pi$
$179$	− 119.942i	− 0.670065i	0.942207	−	0.335032i	$-0.108747\pi$
$180$	0	0
$181$	313.986	1.73473	0.867365	−	0.497672i	$-0.165812\pi$
$181$	313.986	1.73473	0.867365	+	0.497672i	$0.165812\pi$
$182$	0	0
$183$	51.1014i	0.279243i
$184$	0	0
$185$	−161.834	−0.874781
$186$	0	0
$187$	185.031i	0.989470i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	344.511i	1.80372i	0.432026	+	0.901861i	$0.357799\pi$
$191$	344.511i	1.80372i	−0.432026	+	0.901861i	$0.642201\pi$
$192$	0	0
$193$	69.4897	0.360050	0.180025	−	0.983662i	$-0.442382\pi$
$193$	69.4897	0.360050	0.180025	+	0.983662i	$0.442382\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	325.490	1.65223	0.826116	−	0.563500i	$-0.190546\pi$
$197$	325.490	1.65223	0.826116	+	0.563500i	$0.190546\pi$
$198$	0	0
$199$	56.5724i	0.284284i	0.989846	+	0.142142i	$0.0453989\pi$
$199$	56.5724i	0.284284i	−0.989846	+	0.142142i	$0.954601\pi$
$200$	0	0
$201$	127.490	0.634277
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−120.483	−0.587721
$206$	0	0
$207$	25.5507i	0.123433i
$208$	0	0
$209$	300.993	1.44016
$210$	0	0
$211$	− 147.212i	− 0.697689i	−0.937181	−	0.348845i	$-0.886574\pi$
$211$	− 147.212i	− 0.697689i	0.937181	−	0.348845i	$-0.113426\pi$
$212$	0	0
$213$	70.2414	0.329772
$214$	0	0
$215$	− 85.4320i	− 0.397358i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 73.6062i	− 0.336101i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	340.055i	1.52491i	0.647040	+	0.762456i	$0.276007\pi$
$223$	340.055i	1.52491i	−0.647040	+	0.762456i	$0.723993\pi$
$224$	0	0
$225$	46.4897	0.206621
$226$	0	0
$227$	− 334.274i	− 1.47257i	−0.676670	−	0.736286i	$-0.736578\pi$
$227$	− 334.274i	− 1.47257i	0.676670	−	0.736286i	$-0.263422\pi$
$228$	0	0
$229$	398.979	1.74227	0.871134	−	0.491045i	$-0.163385\pi$
$229$	398.979	1.74227	0.871134	+	0.491045i	$0.163385\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−306.000	−1.31330	−0.656652	−	0.754193i	$-0.728028\pi$
$233$	−306.000	−1.31330	−0.656652	+	0.754193i	$0.728028\pi$
$234$	0	0
$235$	65.8417i	0.280177i
$236$	0	0
$237$	40.5103	0.170929
$238$	0	0
$239$	− 307.003i	− 1.28453i	−0.766482	−	0.642266i	$-0.777995\pi$
$239$	− 307.003i	− 1.28453i	0.766482	−	0.642266i	$-0.222005\pi$
$240$	0	0
$241$	153.476	0.636830	0.318415	−	0.947951i	$-0.396849\pi$
$241$	153.476	0.636830	0.318415	+	0.947951i	$0.396849\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−144.993	−0.582302
$250$	0	0
$251$	147.786i	0.588788i	0.955684	+	0.294394i	$0.0951178\pi$
$251$	147.786i	0.588788i	−0.955684	+	0.294394i	$0.904882\pi$
$252$	0	0
$253$	−104.483	−0.412976
$254$	0	0
$255$	80.5344i	0.315821i
$256$	0	0
$257$	74.0896	0.288286	0.144143	−	0.989557i	$-0.453957\pi$
$257$	74.0896	0.288286	0.144143	+	0.989557i	$0.453957\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	73.4897	0.281570
$262$	0	0
$263$	− 216.363i	− 0.822673i	−0.911484	−	0.411337i	$-0.865062\pi$
$263$	− 216.363i	− 0.822673i	0.911484	−	0.411337i	$-0.134938\pi$
$264$	0	0
$265$	−35.4623	−0.133820
$266$	0	0
$267$	256.475i	0.960581i
$268$	0	0
$269$	−221.069	−0.821818	−0.410909	−	0.911676i	$-0.634789\pi$
$269$	−221.069	−0.821818	−0.410909	+	0.911676i	$0.634789\pi$
$270$	0	0
$271$	113.455i	0.418654i	0.977846	+	0.209327i	$0.0671273\pi$
$271$	113.455i	0.418654i	−0.977846	+	0.209327i	$0.932873\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	190.107i	0.691300i
$276$	0	0
$277$	−286.483	−1.03423	−0.517117	−	0.855915i	$-0.672995\pi$
$277$	−286.483	−1.03423	−0.517117	+	0.855915i	$0.672995\pi$
$278$	0	0
$279$	− 11.2524i	− 0.0403312i
$280$	0	0
$281$	−56.9795	−0.202774	−0.101387	−	0.994847i	$-0.532328\pi$
$281$	−56.9795	−0.202774	−0.101387	+	0.994847i	$0.532328\pi$
$282$	0	0
$283$	− 428.665i	− 1.51472i	−0.653000	−	0.757358i	$-0.726490\pi$
$283$	− 428.665i	− 1.51472i	0.653000	−	0.757358i	$-0.273510\pi$
$284$	0	0
$285$	131.007	0.459673
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−61.5103	−0.212838
$290$	0	0
$291$	323.022i	1.11004i
$292$	0	0
$293$	376.076	1.28354	0.641768	−	0.766899i	$-0.278201\pi$
$293$	376.076	1.28354	0.641768	+	0.766899i	$0.278201\pi$
$294$	0	0
$295$	129.916i	0.440392i
$296$	0	0
$297$	−63.7449	−0.214629
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	27.8443i	0.0918954i
$304$	0	0
$305$	−90.9521	−0.298203
$306$	0	0
$307$	130.179i	0.424035i	0.977266	+	0.212017i	$0.0680033\pi$
$307$	130.179i	0.424035i	−0.977266	+	0.212017i	$0.931997\pi$
$308$	0	0
$309$	−189.476	−0.613191
$310$	0	0
$311$	269.627i	0.866966i	0.901162	+	0.433483i	$0.142716\pi$
$311$	269.627i	0.866966i	−0.901162	+	0.433483i	$0.857284\pi$
$312$	0	0
$313$	−25.9863	−0.0830233	−0.0415117	−	0.999138i	$-0.513217\pi$
$313$	−25.9863	−0.0830233	−0.0415117	+	0.999138i	$0.513217\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	364.469	1.14975	0.574873	−	0.818243i	$-0.305052\pi$
$317$	364.469	1.14975	0.574873	+	0.818243i	$0.305052\pi$
$318$	0	0
$319$	300.517i	0.942059i
$320$	0	0
$321$	119.235	0.371447
$322$	0	0
$323$	− 370.062i	− 1.14570i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.18429i	0.0158541i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	219.935i	0.664456i	0.943199	+	0.332228i	$0.107800\pi$
$331$	219.935i	0.664456i	−0.943199	+	0.332228i	$0.892200\pi$
$332$	0	0
$333$	−157.490	−0.472942
$334$	0	0
$335$	226.911i	0.677345i
$336$	0	0
$337$	−155.986	−0.462867	−0.231434	−	0.972851i	$-0.574342\pi$
$337$	−155.986	−0.462867	−0.231434	+	0.972851i	$0.574342\pi$
$338$	0	0
$339$	53.6817i	0.158353i
$340$	0	0
$341$	46.0137	0.134938
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−45.4760	−0.131815
$346$	0	0
$347$	− 459.997i	− 1.32564i	−0.748779	−	0.662820i	$-0.769360\pi$
$347$	− 459.997i	− 1.32564i	0.748779	−	0.662820i	$-0.230640\pi$
$348$	0	0
$349$	395.462	1.13313	0.566565	−	0.824017i	$-0.308272\pi$
$349$	395.462	1.13313	0.566565	+	0.824017i	$0.308272\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.07591	−0.0115465	−0.00577325	−	0.999983i	$-0.501838\pi$
$353$	−4.07591	−0.0115465	−0.00577325	+	0.999983i	$0.501838\pi$
$354$	0	0
$355$	125.018i	0.352164i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 28.9911i	− 0.0807551i	−0.999184	−	0.0403776i	$-0.987144\pi$
$359$	− 28.9911i	− 0.0807551i	0.999184	−	0.0403776i	$-0.0128561\pi$
$360$	0	0
$361$	−240.986	−0.667552
$362$	0	0
$363$	− 51.0896i	− 0.140743i
$364$	0	0
$365$	131.007	0.358923
$366$	0	0
$367$	302.547i	0.824380i	0.911098	+	0.412190i	$0.135236\pi$
$367$	302.547i	0.824380i	−0.911098	+	0.412190i	$0.864764\pi$
$368$	0	0
$369$	−117.248	−0.317746
$370$	0	0
$371$	0	0
$372$	0	0
$373$	317.973	0.852473	0.426237	−	0.904612i	$-0.359839\pi$
$373$	317.973	0.852473	0.426237	+	0.904612i	$0.359839\pi$
$374$	0	0
$375$	216.232i	0.576617i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	285.777i	0.754028i	0.926208	+	0.377014i	$0.123049\pi$
$379$	285.777i	0.754028i	−0.926208	+	0.377014i	$0.876951\pi$
$380$	0	0
$381$	−151.490	−0.397611
$382$	0	0
$383$	506.858i	1.32339i	0.749773	+	0.661695i	$0.230163\pi$
$383$	506.858i	1.32339i	−0.749773	+	0.661695i	$0.769837\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 83.1384i	− 0.214828i
$388$	0	0
$389$	747.476	1.92153	0.960766	−	0.277360i	$-0.0894594\pi$
$389$	747.476	1.92153	0.960766	+	0.277360i	$0.0894594\pi$
$390$	0	0
$391$	128.458i	0.328538i
$392$	0	0
$393$	−324.000	−0.824427
$394$	0	0
$395$	72.1016i	0.182536i
$396$	0	0
$397$	107.462	0.270686	0.135343	−	0.990799i	$-0.456786\pi$
$397$	107.462	0.270686	0.135343	+	0.990799i	$0.456786\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−475.986	−1.18700	−0.593499	−	0.804835i	$-0.702254\pi$
$401$	−475.986	−1.18700	−0.593499	+	0.804835i	$0.702254\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−27.7449	−0.0685058
$406$	0	0
$407$	− 644.012i	− 1.58234i
$408$	0	0
$409$	−330.497	−0.808060	−0.404030	−	0.914746i	$-0.632391\pi$
$409$	−330.497	−0.808060	−0.404030	+	0.914746i	$0.632391\pi$
$410$	0	0
$411$	263.248i	0.640506i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 258.064i	− 0.621840i
$416$	0	0
$417$	74.9795	0.179807
$418$	0	0
$419$	− 818.627i	− 1.95376i	−0.213778	−	0.976882i	$-0.568577\pi$
$419$	− 818.627i	− 1.95376i	0.213778	−	0.976882i	$-0.431423\pi$
$420$	0	0
$421$	319.476	0.758850	0.379425	−	0.925222i	$-0.376122\pi$
$421$	319.476	0.758850	0.379425	+	0.925222i	$0.376122\pi$
$422$	0	0
$423$	64.0740i	0.151475i
$424$	0	0
$425$	233.731	0.549956
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 826.308i	− 1.91719i	−0.284777	−	0.958594i	$-0.591920\pi$
$431$	− 826.308i	− 1.91719i	0.284777	−	0.958594i	$-0.408080\pi$
$432$	0	0
$433$	−280.966	−0.648882	−0.324441	−	0.945906i	$-0.605176\pi$
$433$	−280.966	−0.648882	−0.324441	+	0.945906i	$0.605176\pi$
$434$	0	0
$435$	130.799i	0.300688i
$436$	0	0
$437$	208.966	0.478182
$438$	0	0
$439$	− 777.631i	− 1.77137i	−0.464286	−	0.885685i	$-0.653689\pi$
$439$	− 777.631i	− 1.77137i	0.464286	−	0.885685i	$-0.346311\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	412.336i	0.930781i	0.885106	+	0.465390i	$0.154086\pi$
$443$	412.336i	0.930781i	−0.885106	+	0.465390i	$0.845914\pi$
$444$	0	0
$445$	−456.483	−1.02580
$446$	0	0
$447$	19.9245i	0.0445739i
$448$	0	0
$449$	306.000	0.681514	0.340757	−	0.940151i	$-0.389317\pi$
$449$	306.000	0.681514	0.340757	+	0.940151i	$0.389317\pi$
$450$	0	0
$451$	− 479.456i	− 1.06309i
$452$	0	0
$453$	−461.959	−1.01978
$454$	0	0
$455$	0	0
$456$	0	0
$457$	443.986	0.971524	0.485762	−	0.874091i	$-0.338542\pi$
$457$	443.986	0.971524	0.485762	+	0.874091i	$0.338542\pi$
$458$	0	0
$459$	78.3723i	0.170746i
$460$	0	0
$461$	−613.883	−1.33163	−0.665817	−	0.746115i	$-0.731917\pi$
$461$	−613.883	−1.33163	−0.665817	+	0.746115i	$0.731917\pi$
$462$	0	0
$463$	695.377i	1.50189i	0.660363	+	0.750947i	$0.270403\pi$
$463$	695.377i	1.50189i	−0.660363	+	0.750947i	$0.729597\pi$
$464$	0	0
$465$	20.0274	0.0430697
$466$	0	0
$467$	187.061i	0.400560i	0.979739	+	0.200280i	$0.0641852\pi$
$467$	187.061i	0.400560i	−0.979739	+	0.200280i	$0.935815\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 492.739i	− 1.04615i
$472$	0	0
$473$	339.973	0.718758
$474$	0	0
$475$	− 380.215i	− 0.800452i
$476$	0	0
$477$	−34.5103	−0.0723486
$478$	0	0
$479$	248.269i	0.518306i	0.965836	+	0.259153i	$0.0834434\pi$
$479$	248.269i	0.518306i	−0.965836	+	0.259153i	$0.916557\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−574.925	−1.18541
$486$	0	0
$487$	147.212i	0.302284i	0.988512	+	0.151142i	$0.0482951\pi$
$487$	147.212i	0.302284i	−0.988512	+	0.151142i	$0.951705\pi$
$488$	0	0
$489$	142.469	0.291348
$490$	0	0
$491$	− 309.344i	− 0.630029i	−0.949087	−	0.315014i	$-0.897991\pi$
$491$	− 309.344i	− 0.630029i	0.949087	−	0.315014i	$-0.102009\pi$
$492$	0	0
$493$	369.476	0.749444
$494$	0	0
$495$	− 113.455i	− 0.229203i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 590.617i	− 1.18360i	−0.806084	−	0.591801i	$-0.798417\pi$
$499$	− 590.617i	− 1.18360i	0.806084	−	0.591801i	$-0.201583\pi$
$500$	0	0
$501$	−251.007	−0.501012
$502$	0	0
$503$	820.348i	1.63091i	0.578821	+	0.815455i	$0.303513\pi$
$503$	820.348i	1.63091i	−0.578821	+	0.815455i	$0.696487\pi$
$504$	0	0
$505$	−49.5582	−0.0981351
$506$	0	0
$507$	− 292.717i	− 0.577350i
$508$	0	0
$509$	−415.055	−0.815433	−0.407716	−	0.913109i	$-0.633675\pi$
$509$	−415.055	−0.815433	−0.407716	+	0.913109i	$0.633675\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	127.490	0.248518
$514$	0	0
$515$	− 337.236i	− 0.654827i
$516$	0	0
$517$	−262.014	−0.506796
$518$	0	0
$519$	379.462i	0.731142i
$520$	0	0
$521$	135.951	0.260943	0.130472	−	0.991452i	$-0.458351\pi$
$521$	135.951	0.260943	0.130472	+	0.991452i	$0.458351\pi$
$522$	0	0
$523$	− 386.785i	− 0.739551i	−0.929121	−	0.369775i	$-0.879435\pi$
$523$	− 386.785i	− 0.739551i	0.929121	−	0.369775i	$-0.120565\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 56.5724i	− 0.107348i
$528$	0	0
$529$	456.462	0.862878
$530$	0	0
$531$	126.428i	0.238094i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	212.218i	0.396669i
$536$	0	0
$537$	207.745	0.386862
$538$	0	0
$539$	0	0
$540$	0	0
$541$	193.517	0.357703	0.178851	−	0.983876i	$-0.442762\pi$
$541$	193.517	0.357703	0.178851	+	0.983876i	$0.442762\pi$
$542$	0	0
$543$	543.840i	1.00155i
$544$	0	0
$545$	−9.22717	−0.0169306
$546$	0	0
$547$	452.937i	0.828039i	0.910268	+	0.414019i	$0.135875\pi$
$547$	452.937i	0.828039i	−0.910268	+	0.414019i	$0.864125\pi$
$548$	0	0
$549$	−88.5103	−0.161221
$550$	0	0
$551$	− 601.033i	− 1.09080i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 280.306i	− 0.505055i
$556$	0	0
$557$	495.476	0.889544	0.444772	−	0.895644i	$-0.353285\pi$
$557$	495.476	0.889544	0.444772	+	0.895644i	$0.353285\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−320.483	−0.571271
$562$	0	0
$563$	− 549.001i	− 0.975135i	−0.873085	−	0.487567i	$-0.837884\pi$
$563$	− 549.001i	− 0.975135i	0.873085	−	0.487567i	$-0.162116\pi$
$564$	0	0
$565$	−95.5445	−0.169105
$566$	0	0
$567$	0	0
$568$	0	0
$569$	360.952	0.634362	0.317181	−	0.948365i	$-0.397264\pi$
$569$	360.952	0.634362	0.317181	+	0.948365i	$0.397264\pi$
$570$	0	0
$571$	856.446i	1.49990i	0.661492	+	0.749952i	$0.269924\pi$
$571$	856.446i	1.49990i	−0.661492	+	0.749952i	$0.730076\pi$
$572$	0	0
$573$	−596.711	−1.04138
$574$	0	0
$575$	131.983i	0.229535i
$576$	0	0
$577$	450.952	0.781546	0.390773	−	0.920487i	$-0.372208\pi$
$577$	450.952	0.781546	0.390773	+	0.920487i	$0.372208\pi$
$578$	0	0
$579$	120.360i	0.207875i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 141.121i	− 0.242059i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 708.397i	− 1.20681i	−0.797435	−	0.603405i	$-0.793810\pi$
$587$	− 708.397i	− 1.20681i	0.797435	−	0.603405i	$-0.206190\pi$
$588$	0	0
$589$	−92.0274	−0.156243
$590$	0	0
$591$	563.765i	0.953917i
$592$	0	0
$593$	−467.110	−0.787707	−0.393853	−	0.919173i	$-0.628858\pi$
$593$	−467.110	−0.787707	−0.393853	+	0.919173i	$0.628858\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−97.9863	−0.164131
$598$	0	0
$599$	− 906.006i	− 1.51253i	−0.654265	−	0.756265i	$-0.727022\pi$
$599$	− 906.006i	− 1.51253i	0.654265	−	0.756265i	$-0.272978\pi$
$600$	0	0
$601$	−738.952	−1.22954	−0.614769	−	0.788707i	$-0.710751\pi$
$601$	−738.952	−1.22954	−0.614769	+	0.788707i	$0.710751\pi$
$602$	0	0
$603$	220.819i	0.366200i
$604$	0	0
$605$	90.9309	0.150299
$606$	0	0
$607$	− 438.197i	− 0.721906i	−0.932584	−	0.360953i	$-0.882451\pi$
$607$	− 438.197i	− 0.721906i	0.932584	−	0.360953i	$-0.117549\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	317.945	0.518671	0.259335	−	0.965787i	$-0.416496\pi$
$613$	317.945	0.518671	0.259335	+	0.965787i	$0.416496\pi$
$614$	0	0
$615$	− 208.682i	− 0.339321i
$616$	0	0
$617$	766.966	1.24306	0.621528	−	0.783392i	$-0.286512\pi$
$617$	766.966	1.24306	0.621528	+	0.783392i	$0.286512\pi$
$618$	0	0
$619$	− 470.544i	− 0.760169i	−0.924952	−	0.380084i	$-0.875895\pi$
$619$	− 470.544i	− 0.760169i	0.924952	−	0.380084i	$-0.124105\pi$
$620$	0	0
$621$	−44.2551	−0.0712643
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.55822	0.00409316
$626$	0	0
$627$	521.335i	0.831476i
$628$	0	0
$629$	−791.793	−1.25881
$630$	0	0
$631$	− 251.972i	− 0.399322i	−0.979865	−	0.199661i	$-0.936016\pi$
$631$	− 251.972i	− 0.399322i	0.979865	−	0.199661i	$-0.0639840\pi$
$632$	0	0
$633$	254.979	0.402811
$634$	0	0
$635$	− 269.627i	− 0.424609i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	121.662i	0.190394i
$640$	0	0
$641$	792.952	1.23705	0.618527	−	0.785763i	$-0.287730\pi$
$641$	792.952	1.23705	0.618527	+	0.785763i	$0.287730\pi$
$642$	0	0
$643$	− 682.141i	− 1.06087i	−0.847725	−	0.530436i	$-0.822028\pi$
$643$	− 682.141i	− 1.06087i	0.847725	−	0.530436i	$-0.177972\pi$
$644$	0	0
$645$	147.973	0.229415
$646$	0	0
$647$	1030.49i	1.59272i	0.604826	+	0.796358i	$0.293243\pi$
$647$	1030.49i	1.59272i	−0.604826	+	0.796358i	$0.706757\pi$
$648$	0	0
$649$	−516.993	−0.796600
$650$	0	0
$651$	0	0
$652$	0	0
$653$	315.476	0.483118	0.241559	−	0.970386i	$-0.422341\pi$
$653$	315.476	0.483118	0.241559	+	0.970386i	$0.422341\pi$
$654$	0	0
$655$	− 576.666i	− 0.880406i
$656$	0	0
$657$	127.490	0.194048
$658$	0	0
$659$	− 1084.32i	− 1.64541i	−0.568470	−	0.822704i	$-0.692464\pi$
$659$	− 1084.32i	− 1.64541i	0.568470	−	0.822704i	$-0.307536\pi$
$660$	0	0
$661$	553.531	0.837414	0.418707	−	0.908121i	$-0.362483\pi$
$661$	553.531	0.837414	0.418707	+	0.908121i	$0.362483\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	208.635i	0.312796i
$668$	0	0
$669$	−588.993	−0.880408
$670$	0	0
$671$	− 361.939i	− 0.539403i
$672$	0	0
$673$	−579.449	−0.860994	−0.430497	−	0.902592i	$-0.641662\pi$
$673$	−579.449	−0.860994	−0.430497	+	0.902592i	$0.641662\pi$
$674$	0	0
$675$	80.5226i	0.119293i
$676$	0	0
$677$	−1121.86	−1.65710	−0.828549	−	0.559916i	$-0.810833\pi$
$677$	−1121.86	−1.65710	−0.828549	+	0.559916i	$0.810833\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	578.979	0.850190
$682$	0	0
$683$	− 1238.25i	− 1.81296i	−0.422253	−	0.906478i	$-0.638760\pi$
$683$	− 1238.25i	− 1.81296i	0.422253	−	0.906478i	$-0.361240\pi$
$684$	0	0
$685$	−468.538	−0.683997
$686$	0	0
$687$	691.053i	1.00590i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	775.911i	1.12288i	0.827517	+	0.561441i	$0.189753\pi$
$691$	775.911i	1.12288i	−0.827517	+	0.561441i	$0.810247\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	133.451i	0.192016i
$696$	0	0
$697$	−589.476	−0.845733
$698$	0	0
$699$	− 530.008i	− 0.758237i
$700$	0	0
$701$	315.476	0.450037	0.225019	−	0.974354i	$-0.427756\pi$
$701$	315.476	0.450037	0.225019	+	0.974354i	$0.427756\pi$
$702$	0	0
$703$	1288.02i	1.83218i
$704$	0	0
$705$	−114.041	−0.161760
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−813.007	−1.14670	−0.573348	−	0.819312i	$-0.694356\pi$
$709$	−813.007	−1.14670	−0.573348	+	0.819312i	$0.694356\pi$
$710$	0	0
$711$	70.1659i	0.0986862i
$712$	0	0
$713$	31.9452	0.0448039
$714$	0	0
$715$	0	0
$716$	0	0
$717$	531.745	0.741625
$718$	0	0
$719$	− 798.989i	− 1.11125i	−0.831433	−	0.555626i	$-0.812479\pi$
$719$	− 798.989i	− 1.11125i	0.831433	−	0.555626i	$-0.187521\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	265.828i	0.367674i
$724$	0	0
$725$	379.613	0.523604
$726$	0	0
$727$	381.935i	0.525358i	0.964883	+	0.262679i	$0.0846060\pi$
$727$	381.935i	0.525358i	−0.964883	+	0.262679i	$0.915394\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 417.986i	− 0.571800i
$732$	0	0
$733$	−1059.97	−1.44607	−0.723037	−	0.690809i	$-0.757255\pi$
$733$	−1059.97	−1.44607	−0.723037	+	0.690809i	$0.757255\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−902.979	−1.22521
$738$	0	0
$739$	956.881i	1.29483i	0.762137	+	0.647416i	$0.224150\pi$
$739$	956.881i	1.29483i	−0.762137	+	0.647416i	$0.775850\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 440.622i	− 0.593031i	−0.955028	−	0.296516i	$-0.904175\pi$
$743$	− 440.622i	− 0.593031i	0.955028	−	0.296516i	$-0.0958246\pi$
$744$	0	0
$745$	−35.4623	−0.0476004
$746$	0	0
$747$	− 251.136i	− 0.336192i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 748.246i	− 0.996333i	−0.867081	−	0.498166i	$-0.834007\pi$
$751$	− 748.246i	− 0.996333i	0.867081	−	0.498166i	$-0.165993\pi$
$752$	0	0
$753$	−255.973	−0.339937
$754$	0	0
$755$	− 822.210i	− 1.08902i
$756$	0	0
$757$	−924.897	−1.22179	−0.610896	−	0.791710i	$-0.709191\pi$
$757$	−924.897	−1.22179	−0.610896	+	0.791710i	$0.709191\pi$
$758$	0	0
$759$	− 180.970i	− 0.238432i
$760$	0	0
$761$	−745.883	−0.980135	−0.490068	−	0.871684i	$-0.663028\pi$
$761$	−745.883	−0.980135	−0.490068	+	0.871684i	$0.663028\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−139.490	−0.182340
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−897.021	−1.16648	−0.583238	−	0.812301i	$-0.698215\pi$
$769$	−897.021	−1.16648	−0.583238	+	0.812301i	$0.698215\pi$
$770$	0	0
$771$	128.327i	0.166442i
$772$	0	0
$773$	−39.8693	−0.0515773	−0.0257887	−	0.999667i	$-0.508210\pi$
$773$	−39.8693	−0.0515773	−0.0257887	+	0.999667i	$0.508210\pi$
$774$	0	0
$775$	− 58.1245i	− 0.0749994i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	958.912i	1.23095i
$780$	0	0
$781$	−497.503	−0.637008
$782$	0	0
$783$	127.288i	0.162564i
$784$	0	0
$785$	876.993	1.11719
$786$	0	0
$787$	216.447i	0.275028i	0.990500	+	0.137514i	$0.0439112\pi$
$787$	216.447i	0.275028i	−0.990500	+	0.137514i	$0.956089\pi$
$788$	0	0
$789$	374.752	0.474970
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 61.4226i	− 0.0772611i
$796$	0	0
$797$	−681.910	−0.855596	−0.427798	−	0.903874i	$-0.640711\pi$
$797$	−681.910	−0.855596	−0.427798	+	0.903874i	$0.640711\pi$
$798$	0	0
$799$	322.138i	0.403176i
$800$	0	0
$801$	−444.228	−0.554591
$802$	0	0
$803$	521.335i	0.649235i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 382.903i	− 0.474477i
$808$	0	0
$809$	−1224.95	−1.51416	−0.757078	−	0.653325i	$-0.773374\pi$
$809$	−1224.95	−1.51416	−0.757078	+	0.653325i	$0.773374\pi$
$810$	0	0
$811$	− 953.751i	− 1.17602i	−0.808854	−	0.588009i	$-0.799912\pi$
$811$	− 953.751i	− 1.17602i	0.808854	−	0.588009i	$-0.200088\pi$
$812$	0	0
$813$	−196.510	−0.241710
$814$	0	0
$815$	253.571i	0.311131i
$816$	0	0
$817$	−679.945	−0.832246
$818$	0	0
$819$	0	0
$820$	0	0
$821$	769.408	0.937159	0.468579	−	0.883421i	$-0.344766\pi$
$821$	769.408	0.937159	0.468579	+	0.883421i	$0.344766\pi$
$822$	0	0
$823$	605.358i	0.735550i	0.929915	+	0.367775i	$0.119880\pi$
$823$	605.358i	0.735550i	−0.929915	+	0.367775i	$0.880120\pi$
$824$	0	0
$825$	−329.276	−0.399122
$826$	0	0
$827$	− 764.575i	− 0.924516i	−0.886745	−	0.462258i	$-0.847039\pi$
$827$	− 764.575i	− 0.924516i	0.886745	−	0.462258i	$-0.152961\pi$
$828$	0	0
$829$	−313.986	−0.378753	−0.189377	−	0.981905i	$-0.560647\pi$
$829$	−313.986	−0.378753	−0.189377	+	0.981905i	$0.560647\pi$
$830$	0	0
$831$	− 496.203i	− 0.597115i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 446.750i	− 0.535031i
$836$	0	0
$837$	19.4897	0.0232852
$838$	0	0
$839$	− 43.8628i	− 0.0522799i	−0.999658	−	0.0261399i	$-0.991678\pi$
$839$	− 43.8628i	− 0.0522799i	0.999658	−	0.0261399i	$-0.00832155\pi$
$840$	0	0
$841$	−240.918	−0.286466
$842$	0	0
$843$	− 98.6913i	− 0.117072i
$844$	0	0
$845$	520.987	0.616553
$846$	0	0
$847$	0	0
$848$	0	0
$849$	742.469	0.874522
$850$	0	0
$851$	− 447.108i	− 0.525392i
$852$	0	0
$853$	−860.483	−1.00877	−0.504386	−	0.863478i	$-0.668281\pi$
$853$	−860.483	−1.00877	−0.504386	+	0.863478i	$0.668281\pi$
$854$	0	0
$855$	226.911i	0.265392i
$856$	0	0
$857$	879.994	1.02683	0.513415	−	0.858140i	$-0.328380\pi$
$857$	879.994	1.02683	0.513415	+	0.858140i	$0.328380\pi$
$858$	0	0
$859$	− 221.440i	− 0.257788i	−0.991658	−	0.128894i	$-0.958857\pi$
$859$	− 221.440i	− 0.257788i	0.991658	−	0.128894i	$-0.0411426\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1282.95i	1.48661i	0.668950	+	0.743307i	$0.266744\pi$
$863$	1282.95i	1.48661i	−0.668950	+	0.743307i	$0.733256\pi$
$864$	0	0
$865$	−675.380	−0.780786
$866$	0	0
$867$	− 106.539i	− 0.122882i
$868$	0	0
$869$	−286.925	−0.330178
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−559.490	−0.640882
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44.0411	−0.0502179	−0.0251089	−	0.999685i	$-0.507993\pi$
$877$	−44.0411	−0.0502179	−0.0251089	+	0.999685i	$0.507993\pi$
$878$	0	0
$879$	651.383i	0.741050i
$880$	0	0
$881$	−1605.95	−1.82287	−0.911437	−	0.411439i	$-0.865026\pi$
$881$	−1605.95	−1.82287	−0.911437	+	0.411439i	$0.865026\pi$
$882$	0	0
$883$	694.588i	0.786623i	0.919405	+	0.393311i	$0.128671\pi$
$883$	694.588i	0.786623i	−0.919405	+	0.393311i	$0.871329\pi$
$884$	0	0
$885$	−225.021	−0.254261
$886$	0	0
$887$	− 1102.16i	− 1.24257i	−0.783586	−	0.621284i	$-0.786611\pi$
$887$	− 1102.16i	− 1.24257i	0.783586	−	0.621284i	$-0.213389\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 110.409i	− 0.123916i
$892$	0	0
$893$	524.027	0.586817
$894$	0	0
$895$	369.751i	0.413130i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 91.8817i	− 0.102204i
$900$	0	0
$901$	−173.503	−0.192568
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−967.945	−1.06955
$906$	0	0
$907$	720.533i	0.794414i	0.917729	+	0.397207i	$0.130020\pi$
$907$	720.533i	0.794414i	−0.917729	+	0.397207i	$0.869980\pi$
$908$	0	0
$909$	−48.2277	−0.0530558
$910$	0	0
$911$	− 751.292i	− 0.824689i	−0.911028	−	0.412345i	$-0.864710\pi$
$911$	− 751.292i	− 0.824689i	0.911028	−	0.412345i	$-0.135290\pi$
$912$	0	0
$913$	1026.95	1.12481
$914$	0	0
$915$	− 157.534i	− 0.172168i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	874.626i	0.951715i	0.879522	+	0.475858i	$0.157862\pi$
$919$	874.626i	0.951715i	−0.879522	+	0.475858i	$0.842138\pi$
$920$	0	0
$921$	−225.476	−0.244817
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−813.517	−0.879478
$926$	0	0
$927$	− 328.182i	− 0.354026i
$928$	0	0
$929$	350.007	0.376757	0.188379	−	0.982096i	$-0.439677\pi$
$929$	350.007	0.376757	0.188379	+	0.982096i	$0.439677\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−467.007	−0.500543
$934$	0	0
$935$	− 570.406i	− 0.610060i
$936$	0	0
$937$	−169.986	−0.181415	−0.0907077	−	0.995878i	$-0.528913\pi$
$937$	−169.986	−0.181415	−0.0907077	+	0.995878i	$0.528913\pi$
$938$	0	0
$939$	− 45.0096i	− 0.0479335i
$940$	0	0
$941$	1454.79	1.54601	0.773004	−	0.634401i	$-0.218753\pi$
$941$	1454.79	1.54601	0.773004	+	0.634401i	$0.218753\pi$
$942$	0	0
$943$	− 332.864i	− 0.352984i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	712.853i	0.752748i	0.926468	+	0.376374i	$0.122829\pi$
$947$	712.853i	0.752748i	−0.926468	+	0.376374i	$0.877171\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	631.279i	0.663806i
$952$	0	0
$953$	84.0411	0.0881858	0.0440929	−	0.999027i	$-0.485960\pi$
$953$	84.0411	0.0881858	0.0440929	+	0.999027i	$0.485960\pi$
$954$	0	0
$955$	− 1062.05i	− 1.11209i
$956$	0	0
$957$	−520.510	−0.543898
$958$	0	0
$959$	0	0
$960$	0	0
$961$	946.932	0.985361
$962$	0	0
$963$	206.520i	0.214455i
$964$	0	0
$965$	−214.220	−0.221990
$966$	0	0
$967$	− 1772.64i	− 1.83313i	−0.399880	−	0.916567i	$-0.630948\pi$
$967$	− 1772.64i	− 1.83313i	0.399880	−	0.916567i	$-0.369052\pi$
$968$	0	0
$969$	640.966	0.661471
$970$	0	0
$971$	− 383.871i	− 0.395335i	−0.980269	−	0.197668i	$-0.936663\pi$
$971$	− 383.871i	− 0.395335i	0.980269	−	0.197668i	$-0.0633367\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	373.945	0.382748	0.191374	−	0.981517i	$-0.438706\pi$
$977$	373.945	0.382748	0.191374	+	0.981517i	$0.438706\pi$
$978$	0	0
$979$	− 1816.55i	− 1.85552i
$980$	0	0
$981$	−8.97945	−0.00915337
$982$	0	0
$983$	681.878i	0.693671i	0.937926	+	0.346835i	$0.112744\pi$
$983$	681.878i	0.693671i	−0.937926	+	0.346835i	$0.887256\pi$
$984$	0	0
$985$	−1003.41	−1.01869
$986$	0	0
$987$	0	0
$988$	0	0
$989$	236.027	0.238653
$990$	0	0
$991$	− 81.3708i	− 0.0821098i	−0.999157	−	0.0410549i	$-0.986928\pi$
$991$	− 81.3708i	− 0.0821098i	0.999157	−	0.0410549i	$-0.0130719\pi$
$992$	0	0
$993$	−380.938	−0.383624
$994$	0	0
$995$	− 174.399i	− 0.175276i
$996$	0	0
$997$	−1408.35	−1.41258	−0.706292	−	0.707921i	$-0.749633\pi$
$997$	−1408.35	−1.41258	−0.706292	+	0.707921i	$0.749633\pi$
$998$	0	0
$999$	− 272.780i	− 0.273053i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.3.m.j.1471.3	yes	4
4.3	odd	2	inner	2352.3.m.j.1471.1	yes	4
7.6	odd	2		2352.3.m.e.1471.2	✓	4
28.27	even	2		2352.3.m.e.1471.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.3.m.e.1471.2	✓	4	7.6	odd	2
2352.3.m.e.1471.4	yes	4	28.27	even	2
2352.3.m.j.1471.1	yes	4	4.3	odd	2	inner
2352.3.m.j.1471.3	yes	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.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -3.08276 q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -3.08276 q^{5} -3.00000 q^{9} -12.2677i q^{11} -5.33950i q^{15} -15.0828 q^{17} +24.5354i q^{19} -8.51691i q^{23} -15.4966 q^{25} -5.19615i q^{27} -24.4966 q^{29} +3.75080i q^{31} +21.2483 q^{33} +52.4966 q^{37} +39.0828 q^{41} +27.7128i q^{43} +9.24829 q^{45} -21.3580i q^{47} -26.1241i q^{51} +11.5034 q^{53} +37.8184i q^{55} -42.4966 q^{57} -42.1426i q^{59} +29.5034 q^{61} -73.6062i q^{67} +14.7517 q^{69} -40.5539i q^{71} -42.4966 q^{73} -26.8409i q^{75} -23.3886i q^{79} +9.00000 q^{81} +83.7118i q^{83} +46.4966 q^{85} -42.4293i q^{87} +148.076 q^{89} -6.49658 q^{93} -75.6368i q^{95} +186.497 q^{97} +36.8031i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^5 - 12 * q^9	\( 4 q + 12 q^{5} - 12 q^{9} - 36 q^{17} + 84 q^{25} + 48 q^{29} + 12 q^{33} + 64 q^{37} + 132 q^{41} - 36 q^{45} + 192 q^{53} - 24 q^{57} + 264 q^{61} + 132 q^{69} - 24 q^{73} + 36 q^{81} + 40 q^{85} + 276 q^{89} + 120 q^{93} + 600 q^{97}+O(q^{100}) \) 4 * q + 12 * q^5 - 12 * q^9 - 36 * q^17 + 84 * q^25 + 48 * q^29 + 12 * q^33 + 64 * q^37 + 132 * q^41 - 36 * q^45 + 192 * q^53 - 24 * q^57 + 264 * q^61 + 132 * q^69 - 24 * q^73 + 36 * q^81 + 40 * q^85 + 276 * q^89 + 120 * q^93 + 600 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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