LMFDB - Embedded newform 2352.2.b.j.1567.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(1567,2352)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2352.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.b (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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.1
Root			$-1.84776i$ of defining polynomial
Character	$\chi$	$=$	2352.1567
Dual form			2352.2.b.j.1567.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.84776i q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.84776i q^{5} +1.00000 q^{9} +5.22625i q^{11} +4.46088i q^{13} -1.84776i q^{15} +2.93015i q^{17} -5.65685 q^{19} +2.16478i q^{23} +1.58579 q^{25} +1.00000 q^{27} -5.41421 q^{29} -9.65685 q^{31} +5.22625i q^{33} +1.41421 q^{37} +4.46088i q^{39} -4.01254i q^{41} +3.06147i q^{43} -1.84776i q^{45} -1.65685 q^{47} +2.93015i q^{51} -9.65685 q^{53} +9.65685 q^{55} -5.65685 q^{57} -5.65685 q^{59} -1.39942i q^{61} +8.24264 q^{65} +13.5140i q^{67} +2.16478i q^{69} -6.49435i q^{71} +4.90923i q^{73} +1.58579 q^{75} +11.7206i q^{79} +1.00000 q^{81} +17.6569 q^{83} +5.41421 q^{85} -5.41421 q^{87} -9.05309i q^{89} -9.65685 q^{93} +10.4525i q^{95} +9.23880i q^{97} +5.22625i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} + 12 q^{25} + 4 q^{27} - 16 q^{29} - 16 q^{31} + 16 q^{47} - 16 q^{53} + 16 q^{55} + 16 q^{65} + 12 q^{75} + 4 q^{81} + 48 q^{83} + 16 q^{85} - 16 q^{87} - 16 q^{93}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 12 * q^25 + 4 * q^27 - 16 * q^29 - 16 * q^31 + 16 * q^47 - 16 * q^53 + 16 * q^55 + 16 * q^65 + 12 * q^75 + 4 * q^81 + 48 * q^83 + 16 * q^85 - 16 * q^87 - 16 * q^93

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	− 1.84776i	− 0.826343i	−0.910653	−	0.413171i	$-0.864421\pi$
$5$	− 1.84776i	− 0.826343i	0.910653	−	0.413171i	$-0.135579\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.22625i	1.57577i	0.615820	+	0.787887i	$0.288825\pi$
$11$	5.22625i	1.57577i	−0.615820	+	0.787887i	$0.711175\pi$
$12$	0	0
$13$	4.46088i	1.23723i	0.785695	+	0.618613i	$0.212305\pi$
$13$	4.46088i	1.23723i	−0.785695	+	0.618613i	$0.787695\pi$
$14$	0	0
$15$	− 1.84776i	− 0.477089i
$16$	0	0
$17$	2.93015i	0.710666i	0.934740	+	0.355333i	$0.115633\pi$
$17$	2.93015i	0.710666i	−0.934740	+	0.355333i	$0.884367\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.16478i	0.451389i	0.974198	+	0.225694i	$0.0724651\pi$
$23$	2.16478i	0.451389i	−0.974198	+	0.225694i	$0.927535\pi$
$24$	0	0
$25$	1.58579	0.317157
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.41421	−1.00539	−0.502697	−	0.864463i	$-0.667659\pi$
$29$	−5.41421	−1.00539	−0.502697	+	0.864463i	$0.667659\pi$
$30$	0	0
$31$	−9.65685	−1.73442	−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685	−1.73442	−0.867211	+	0.497941i	$0.834090\pi$
$32$	0	0
$33$	5.22625i	0.909774i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.41421	0.232495	0.116248	−	0.993220i	$-0.462913\pi$
$37$	1.41421	0.232495	0.116248	+	0.993220i	$0.462913\pi$
$38$	0	0
$39$	4.46088i	0.714313i
$40$	0	0
$41$	− 4.01254i	− 0.626654i	−0.949645	−	0.313327i	$-0.898556\pi$
$41$	− 4.01254i	− 0.626654i	0.949645	−	0.313327i	$-0.101444\pi$
$42$	0	0
$43$	3.06147i	0.466869i	0.972372	+	0.233435i	$0.0749965\pi$
$43$	3.06147i	0.466869i	−0.972372	+	0.233435i	$0.925003\pi$
$44$	0	0
$45$	− 1.84776i	− 0.275448i
$46$	0	0
$47$	−1.65685	−0.241677	−0.120839	−	0.992672i	$-0.538558\pi$
$47$	−1.65685	−0.241677	−0.120839	+	0.992672i	$0.538558\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.93015i	0.410303i
$52$	0	0
$53$	−9.65685	−1.32647	−0.663235	−	0.748411i	$-0.730817\pi$
$53$	−9.65685	−1.32647	−0.663235	+	0.748411i	$0.730817\pi$
$54$	0	0
$55$	9.65685	1.30213
$56$	0	0
$57$	−5.65685	−0.749269
$58$	0	0
$59$	−5.65685	−0.736460	−0.368230	−	0.929735i	$-0.620036\pi$
$59$	−5.65685	−0.736460	−0.368230	+	0.929735i	$0.620036\pi$
$60$	0	0
$61$	− 1.39942i	− 0.179177i	−0.995979	−	0.0895885i	$-0.971445\pi$
$61$	− 1.39942i	− 0.179177i	0.995979	−	0.0895885i	$-0.0285552\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8.24264	1.02237
$66$	0	0
$67$	13.5140i	1.65099i	0.564406	+	0.825497i	$0.309105\pi$
$67$	13.5140i	1.65099i	−0.564406	+	0.825497i	$0.690895\pi$
$68$	0	0
$69$	2.16478i	0.260609i
$70$	0	0
$71$	− 6.49435i	− 0.770738i	−0.922763	−	0.385369i	$-0.874074\pi$
$71$	− 6.49435i	− 0.770738i	0.922763	−	0.385369i	$-0.125926\pi$
$72$	0	0
$73$	4.90923i	0.574582i	0.957843	+	0.287291i	$0.0927546\pi$
$73$	4.90923i	0.574582i	−0.957843	+	0.287291i	$0.907245\pi$
$74$	0	0
$75$	1.58579	0.183111
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.7206i	1.31867i	0.751849	+	0.659336i	$0.229162\pi$
$79$	11.7206i	1.31867i	−0.751849	+	0.659336i	$0.770838\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.6569	1.93809	0.969046	−	0.246881i	$-0.0794057\pi$
$83$	17.6569	1.93809	0.969046	+	0.246881i	$0.0794057\pi$
$84$	0	0
$85$	5.41421	0.587254
$86$	0	0
$87$	−5.41421	−0.580465
$88$	0	0
$89$	− 9.05309i	− 0.959625i	−0.877371	−	0.479813i	$-0.840705\pi$
$89$	− 9.05309i	− 0.959625i	0.877371	−	0.479813i	$-0.159295\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.65685	−1.00137
$94$	0	0
$95$	10.4525i	1.07240i
$96$	0	0
$97$	9.23880i	0.938058i	0.883183	+	0.469029i	$0.155396\pi$
$97$	9.23880i	0.938058i	−0.883183	+	0.469029i	$0.844604\pi$
$98$	0	0
$99$	5.22625i	0.525258i
$100$	0	0
$101$	− 6.88830i	− 0.685412i	−0.939443	−	0.342706i	$-0.888657\pi$
$101$	− 6.88830i	− 0.685412i	0.939443	−	0.342706i	$-0.111343\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.0083i	1.93428i	0.254249	+	0.967139i	$0.418172\pi$
$107$	20.0083i	1.93428i	−0.254249	+	0.967139i	$0.581828\pi$
$108$	0	0
$109$	20.7279	1.98537	0.992687	−	0.120713i	$-0.0385181\pi$
$109$	20.7279	1.98537	0.992687	+	0.120713i	$0.0385181\pi$
$110$	0	0
$111$	1.41421	0.134231
$112$	0	0
$113$	16.9706	1.59646	0.798228	−	0.602355i	$-0.205771\pi$
$113$	16.9706	1.59646	0.798228	+	0.602355i	$0.205771\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	4.46088i	0.412409i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−16.3137	−1.48306
$122$	0	0
$123$	− 4.01254i	− 0.361799i
$124$	0	0
$125$	− 12.1689i	− 1.08842i
$126$	0	0
$127$	− 11.7206i	− 1.04004i	−0.854155	−	0.520018i	$-0.825925\pi$
$127$	− 11.7206i	− 1.04004i	0.854155	−	0.520018i	$-0.174075\pi$
$128$	0	0
$129$	3.06147i	0.269547i
$130$	0	0
$131$	−7.31371	−0.639002	−0.319501	−	0.947586i	$-0.603515\pi$
$131$	−7.31371	−0.639002	−0.319501	+	0.947586i	$0.603515\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 1.84776i	− 0.159030i
$136$	0	0
$137$	−10.5858	−0.904405	−0.452202	−	0.891915i	$-0.649362\pi$
$137$	−10.5858	−0.904405	−0.452202	+	0.891915i	$0.649362\pi$
$138$	0	0
$139$	6.34315	0.538019	0.269009	−	0.963138i	$-0.413304\pi$
$139$	6.34315	0.538019	0.269009	+	0.963138i	$0.413304\pi$
$140$	0	0
$141$	−1.65685	−0.139532
$142$	0	0
$143$	−23.3137	−1.94959
$144$	0	0
$145$	10.0042i	0.830800i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.9706	1.71798	0.858988	−	0.511996i	$-0.171094\pi$
$149$	20.9706	1.71798	0.858988	+	0.511996i	$0.171094\pi$
$150$	0	0
$151$	14.7821i	1.20295i	0.798892	+	0.601474i	$0.205420\pi$
$151$	14.7821i	1.20295i	−0.798892	+	0.601474i	$0.794580\pi$
$152$	0	0
$153$	2.93015i	0.236889i
$154$	0	0
$155$	17.8435i	1.43323i
$156$	0	0
$157$	− 4.64659i	− 0.370839i	−0.982659	−	0.185419i	$-0.940636\pi$
$157$	− 4.64659i	− 0.370839i	0.982659	−	0.185419i	$-0.0593643\pi$
$158$	0	0
$159$	−9.65685	−0.765838
$160$	0	0
$161$	0	0
$162$	0	0
$163$	14.7821i	1.15782i	0.815391	+	0.578911i	$0.196522\pi$
$163$	14.7821i	1.15782i	−0.815391	+	0.578911i	$0.803478\pi$
$164$	0	0
$165$	9.65685	0.751785
$166$	0	0
$167$	1.65685	0.128211	0.0641056	−	0.997943i	$-0.479581\pi$
$167$	1.65685	0.128211	0.0641056	+	0.997943i	$0.479581\pi$
$168$	0	0
$169$	−6.89949	−0.530730
$170$	0	0
$171$	−5.65685	−0.432590
$172$	0	0
$173$	− 19.0572i	− 1.44890i	−0.689330	−	0.724448i	$-0.742095\pi$
$173$	− 19.0572i	− 1.44890i	0.689330	−	0.724448i	$-0.257905\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.65685	−0.425195
$178$	0	0
$179$	3.95815i	0.295846i	0.988999	+	0.147923i	$0.0472588\pi$
$179$	3.95815i	0.295846i	−0.988999	+	0.147923i	$0.952741\pi$
$180$	0	0
$181$	− 13.1969i	− 0.980921i	−0.871463	−	0.490461i	$-0.836829\pi$
$181$	− 13.1969i	− 0.980921i	0.871463	−	0.490461i	$-0.163171\pi$
$182$	0	0
$183$	− 1.39942i	− 0.103448i
$184$	0	0
$185$	− 2.61313i	− 0.192121i
$186$	0	0
$187$	−15.3137	−1.11985
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 18.7402i	− 1.35599i	−0.735064	−	0.677997i	$-0.762848\pi$
$191$	− 18.7402i	− 1.35599i	0.735064	−	0.677997i	$-0.237152\pi$
$192$	0	0
$193$	−19.3137	−1.39023	−0.695116	−	0.718898i	$-0.744647\pi$
$193$	−19.3137	−1.39023	−0.695116	+	0.718898i	$0.744647\pi$
$194$	0	0
$195$	8.24264	0.590268
$196$	0	0
$197$	−2.68629	−0.191390	−0.0956952	−	0.995411i	$-0.530507\pi$
$197$	−2.68629	−0.191390	−0.0956952	+	0.995411i	$0.530507\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	13.5140i	0.953202i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−7.41421	−0.517831
$206$	0	0
$207$	2.16478i	0.150463i
$208$	0	0
$209$	− 29.5641i	− 2.04499i
$210$	0	0
$211$	− 3.06147i	− 0.210760i	−0.994432	−	0.105380i	$-0.966394\pi$
$211$	− 3.06147i	− 0.210760i	0.994432	−	0.105380i	$-0.0336059\pi$
$212$	0	0
$213$	− 6.49435i	− 0.444986i
$214$	0	0
$215$	5.65685	0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.90923i	0.331735i
$220$	0	0
$221$	−13.0711	−0.879255
$222$	0	0
$223$	3.31371	0.221902	0.110951	−	0.993826i	$-0.464610\pi$
$223$	3.31371	0.221902	0.110951	+	0.993826i	$0.464610\pi$
$224$	0	0
$225$	1.58579	0.105719
$226$	0	0
$227$	19.3137	1.28190	0.640948	−	0.767584i	$-0.278541\pi$
$227$	19.3137	1.28190	0.640948	+	0.767584i	$0.278541\pi$
$228$	0	0
$229$	21.8561i	1.44429i	0.691741	+	0.722145i	$0.256844\pi$
$229$	21.8561i	1.44429i	−0.691741	+	0.722145i	$0.743156\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.24264	−0.539993	−0.269997	−	0.962861i	$-0.587023\pi$
$233$	−8.24264	−0.539993	−0.269997	+	0.962861i	$0.587023\pi$
$234$	0	0
$235$	3.06147i	0.199708i
$236$	0	0
$237$	11.7206i	0.761335i
$238$	0	0
$239$	− 12.6173i	− 0.816145i	−0.912950	−	0.408072i	$-0.866201\pi$
$239$	− 12.6173i	− 0.816145i	0.912950	−	0.408072i	$-0.133799\pi$
$240$	0	0
$241$	− 18.6089i	− 1.19871i	−0.800485	−	0.599353i	$-0.795425\pi$
$241$	− 18.6089i	− 1.19871i	0.800485	−	0.599353i	$-0.204575\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 25.2346i	− 1.60564i
$248$	0	0
$249$	17.6569	1.11896
$250$	0	0
$251$	−24.9706	−1.57613	−0.788064	−	0.615593i	$-0.788916\pi$
$251$	−24.9706	−1.57613	−0.788064	+	0.615593i	$0.788916\pi$
$252$	0	0
$253$	−11.3137	−0.711287
$254$	0	0
$255$	5.41421	0.339051
$256$	0	0
$257$	− 27.3450i	− 1.70573i	−0.522130	−	0.852866i	$-0.674862\pi$
$257$	− 27.3450i	− 1.70573i	0.522130	−	0.852866i	$-0.325138\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5.41421	−0.335131
$262$	0	0
$263$	10.8239i	0.667432i	0.942674	+	0.333716i	$0.108303\pi$
$263$	10.8239i	0.667432i	−0.942674	+	0.333716i	$0.891697\pi$
$264$	0	0
$265$	17.8435i	1.09612i
$266$	0	0
$267$	− 9.05309i	− 0.554040i
$268$	0	0
$269$	24.2066i	1.47590i	0.674855	+	0.737951i	$0.264206\pi$
$269$	24.2066i	1.47590i	−0.674855	+	0.737951i	$0.735794\pi$
$270$	0	0
$271$	1.65685	0.100647	0.0503234	−	0.998733i	$-0.483975\pi$
$271$	1.65685	0.100647	0.0503234	+	0.998733i	$0.483975\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.28772i	0.499768i
$276$	0	0
$277$	13.3137	0.799943	0.399972	−	0.916528i	$-0.369020\pi$
$277$	13.3137	0.799943	0.399972	+	0.916528i	$0.369020\pi$
$278$	0	0
$279$	−9.65685	−0.578141
$280$	0	0
$281$	14.3848	0.858124	0.429062	−	0.903275i	$-0.358844\pi$
$281$	14.3848	0.858124	0.429062	+	0.903275i	$0.358844\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	10.4525i	0.619153i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.41421	0.494954
$290$	0	0
$291$	9.23880i	0.541588i
$292$	0	0
$293$	13.0112i	0.760125i	0.924961	+	0.380062i	$0.124097\pi$
$293$	13.0112i	0.760125i	−0.924961	+	0.380062i	$0.875903\pi$
$294$	0	0
$295$	10.4525i	0.608568i
$296$	0	0
$297$	5.22625i	0.303258i
$298$	0	0
$299$	−9.65685	−0.558470
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 6.88830i	− 0.395723i
$304$	0	0
$305$	−2.58579	−0.148062
$306$	0	0
$307$	−19.3137	−1.10229	−0.551146	−	0.834409i	$-0.685809\pi$
$307$	−19.3137	−1.10229	−0.551146	+	0.834409i	$0.685809\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	14.7277i	0.832458i	0.909260	+	0.416229i	$0.136648\pi$
$313$	14.7277i	0.832458i	−0.909260	+	0.416229i	$0.863352\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.3137	0.972435	0.486217	−	0.873838i	$-0.338376\pi$
$317$	17.3137	0.972435	0.486217	+	0.873838i	$0.338376\pi$
$318$	0	0
$319$	− 28.2960i	− 1.58427i
$320$	0	0
$321$	20.0083i	1.11676i
$322$	0	0
$323$	− 16.5754i	− 0.922282i
$324$	0	0
$325$	7.07401i	0.392396i
$326$	0	0
$327$	20.7279	1.14626
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.79337i	0.0985723i	0.998785	+	0.0492862i	$0.0156946\pi$
$331$	1.79337i	0.0985723i	−0.998785	+	0.0492862i	$0.984305\pi$
$332$	0	0
$333$	1.41421	0.0774984
$334$	0	0
$335$	24.9706	1.36429
$336$	0	0
$337$	−4.72792	−0.257546	−0.128773	−	0.991674i	$-0.541104\pi$
$337$	−4.72792	−0.257546	−0.128773	+	0.991674i	$0.541104\pi$
$338$	0	0
$339$	16.9706	0.921714
$340$	0	0
$341$	− 50.4692i	− 2.73306i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	− 23.0698i	− 1.23845i	−0.785213	−	0.619226i	$-0.787447\pi$
$347$	− 23.0698i	− 1.23845i	0.785213	−	0.619226i	$-0.212553\pi$
$348$	0	0
$349$	− 2.11039i	− 0.112967i	−0.998404	−	0.0564833i	$-0.982011\pi$
$349$	− 2.11039i	− 0.112967i	0.998404	−	0.0564833i	$-0.0179888\pi$
$350$	0	0
$351$	4.46088i	0.238104i
$352$	0	0
$353$	4.64659i	0.247313i	0.992325	+	0.123657i	$0.0394621\pi$
$353$	4.64659i	0.247313i	−0.992325	+	0.123657i	$0.960538\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 6.49435i	− 0.342759i	−0.985205	−	0.171379i	$-0.945178\pi$
$359$	− 6.49435i	− 0.342759i	0.985205	−	0.171379i	$-0.0548224\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	−16.3137	−0.856248
$364$	0	0
$365$	9.07107	0.474801
$366$	0	0
$367$	−10.3431	−0.539908	−0.269954	−	0.962873i	$-0.587008\pi$
$367$	−10.3431	−0.539908	−0.269954	+	0.962873i	$0.587008\pi$
$368$	0	0
$369$	− 4.01254i	− 0.208885i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.6274	−0.550267	−0.275133	−	0.961406i	$-0.588722\pi$
$373$	−10.6274	−0.550267	−0.275133	+	0.961406i	$0.588722\pi$
$374$	0	0
$375$	− 12.1689i	− 0.628402i
$376$	0	0
$377$	− 24.1522i	− 1.24390i
$378$	0	0
$379$	38.2233i	1.96340i	0.190439	+	0.981699i	$0.439009\pi$
$379$	38.2233i	1.96340i	−0.190439	+	0.981699i	$0.560991\pi$
$380$	0	0
$381$	− 11.7206i	− 0.600465i
$382$	0	0
$383$	20.2843	1.03648	0.518239	−	0.855236i	$-0.326588\pi$
$383$	20.2843	1.03648	0.518239	+	0.855236i	$0.326588\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.06147i	0.155623i
$388$	0	0
$389$	−28.0416	−1.42177	−0.710884	−	0.703310i	$-0.751705\pi$
$389$	−28.0416	−1.42177	−0.710884	+	0.703310i	$0.751705\pi$
$390$	0	0
$391$	−6.34315	−0.320787
$392$	0	0
$393$	−7.31371	−0.368928
$394$	0	0
$395$	21.6569	1.08967
$396$	0	0
$397$	21.0363i	1.05578i	0.849312	+	0.527891i	$0.177017\pi$
$397$	21.0363i	1.05578i	−0.849312	+	0.527891i	$0.822983\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.8701	1.54158	0.770789	−	0.637091i	$-0.219862\pi$
$401$	30.8701	1.54158	0.770789	+	0.637091i	$0.219862\pi$
$402$	0	0
$403$	− 43.0781i	− 2.14587i
$404$	0	0
$405$	− 1.84776i	− 0.0918159i
$406$	0	0
$407$	7.39104i	0.366360i
$408$	0	0
$409$	22.7528i	1.12505i	0.826780	+	0.562526i	$0.190170\pi$
$409$	22.7528i	1.12505i	−0.826780	+	0.562526i	$0.809830\pi$
$410$	0	0
$411$	−10.5858	−0.522158
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 32.6256i	− 1.60153i
$416$	0	0
$417$	6.34315	0.310625
$418$	0	0
$419$	−22.6274	−1.10542	−0.552711	−	0.833373i	$-0.686407\pi$
$419$	−22.6274	−1.10542	−0.552711	+	0.833373i	$0.686407\pi$
$420$	0	0
$421$	−29.3137	−1.42866	−0.714331	−	0.699808i	$-0.753269\pi$
$421$	−29.3137	−1.42866	−0.714331	+	0.699808i	$0.753269\pi$
$422$	0	0
$423$	−1.65685	−0.0805590
$424$	0	0
$425$	4.64659i	0.225393i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−23.3137	−1.12560
$430$	0	0
$431$	− 36.5838i	− 1.76218i	−0.472951	−	0.881089i	$-0.656811\pi$
$431$	− 36.5838i	− 1.76218i	0.472951	−	0.881089i	$-0.343189\pi$
$432$	0	0
$433$	12.7486i	0.612659i	0.951926	+	0.306329i	$0.0991009\pi$
$433$	12.7486i	0.612659i	−0.951926	+	0.306329i	$0.900899\pi$
$434$	0	0
$435$	10.0042i	0.479663i
$436$	0	0
$437$	− 12.2459i	− 0.585799i
$438$	0	0
$439$	22.6274	1.07995	0.539974	−	0.841682i	$-0.318434\pi$
$439$	22.6274	1.07995	0.539974	+	0.841682i	$0.318434\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 10.8239i	− 0.514260i	−0.966377	−	0.257130i	$-0.917223\pi$
$443$	− 10.8239i	− 0.514260i	0.966377	−	0.257130i	$-0.0827769\pi$
$444$	0	0
$445$	−16.7279	−0.792980
$446$	0	0
$447$	20.9706	0.991874
$448$	0	0
$449$	−18.3431	−0.865667	−0.432833	−	0.901474i	$-0.642486\pi$
$449$	−18.3431	−0.865667	−0.432833	+	0.901474i	$0.642486\pi$
$450$	0	0
$451$	20.9706	0.987465
$452$	0	0
$453$	14.7821i	0.694522i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.31371	0.155009	0.0775044	−	0.996992i	$-0.475305\pi$
$457$	3.31371	0.155009	0.0775044	+	0.996992i	$0.475305\pi$
$458$	0	0
$459$	2.93015i	0.136768i
$460$	0	0
$461$	25.1802i	1.17276i	0.810037	+	0.586379i	$0.199447\pi$
$461$	25.1802i	1.17276i	−0.810037	+	0.586379i	$0.800553\pi$
$462$	0	0
$463$	− 25.2346i	− 1.17275i	−0.810040	−	0.586375i	$-0.800554\pi$
$463$	− 25.2346i	− 1.17275i	0.810040	−	0.586375i	$-0.199446\pi$
$464$	0	0
$465$	17.8435i	0.827474i
$466$	0	0
$467$	3.31371	0.153340	0.0766701	−	0.997057i	$-0.475571\pi$
$467$	3.31371	0.153340	0.0766701	+	0.997057i	$0.475571\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 4.64659i	− 0.214104i
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	−8.97056	−0.411598
$476$	0	0
$477$	−9.65685	−0.442157
$478$	0	0
$479$	18.6274	0.851108	0.425554	−	0.904933i	$-0.360079\pi$
$479$	18.6274	0.851108	0.425554	+	0.904933i	$0.360079\pi$
$480$	0	0
$481$	6.30864i	0.287649i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.0711	0.775157
$486$	0	0
$487$	2.53620i	0.114926i	0.998348	+	0.0574632i	$0.0183012\pi$
$487$	2.53620i	0.114926i	−0.998348	+	0.0574632i	$0.981699\pi$
$488$	0	0
$489$	14.7821i	0.668468i
$490$	0	0
$491$	24.3379i	1.09835i	0.835706	+	0.549177i	$0.185059\pi$
$491$	24.3379i	1.09835i	−0.835706	+	0.549177i	$0.814941\pi$
$492$	0	0
$493$	− 15.8645i	− 0.714500i
$494$	0	0
$495$	9.65685	0.434043
$496$	0	0
$497$	0	0
$498$	0	0
$499$	41.8100i	1.87167i	0.352434	+	0.935837i	$0.385354\pi$
$499$	41.8100i	1.87167i	−0.352434	+	0.935837i	$0.614646\pi$
$500$	0	0
$501$	1.65685	0.0740228
$502$	0	0
$503$	−26.3431	−1.17458	−0.587291	−	0.809376i	$-0.699806\pi$
$503$	−26.3431	−1.17458	−0.587291	+	0.809376i	$0.699806\pi$
$504$	0	0
$505$	−12.7279	−0.566385
$506$	0	0
$507$	−6.89949	−0.306417
$508$	0	0
$509$	21.6704i	0.960522i	0.877126	+	0.480261i	$0.159458\pi$
$509$	21.6704i	0.960522i	−0.877126	+	0.480261i	$0.840542\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.65685	−0.249756
$514$	0	0
$515$	− 22.1731i	− 0.977064i
$516$	0	0
$517$	− 8.65914i	− 0.380828i
$518$	0	0
$519$	− 19.0572i	− 0.836520i
$520$	0	0
$521$	− 14.4650i	− 0.633725i	−0.948471	−	0.316863i	$-0.897371\pi$
$521$	− 14.4650i	− 0.633725i	0.948471	−	0.316863i	$-0.102629\pi$
$522$	0	0
$523$	34.6274	1.51415	0.757076	−	0.653327i	$-0.226627\pi$
$523$	34.6274	1.51415	0.757076	+	0.653327i	$0.226627\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 28.2960i	− 1.23260i
$528$	0	0
$529$	18.3137	0.796248
$530$	0	0
$531$	−5.65685	−0.245487
$532$	0	0
$533$	17.8995	0.775313
$534$	0	0
$535$	36.9706	1.59838
$536$	0	0
$537$	3.95815i	0.170807i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−12.0000	−0.515920	−0.257960	−	0.966156i	$-0.583050\pi$
$541$	−12.0000	−0.515920	−0.257960	+	0.966156i	$0.583050\pi$
$542$	0	0
$543$	− 13.1969i	− 0.566335i
$544$	0	0
$545$	− 38.3002i	− 1.64060i
$546$	0	0
$547$	− 30.0894i	− 1.28653i	−0.765644	−	0.643265i	$-0.777579\pi$
$547$	− 30.0894i	− 1.28653i	0.765644	−	0.643265i	$-0.222421\pi$
$548$	0	0
$549$	− 1.39942i	− 0.0597257i
$550$	0	0
$551$	30.6274	1.30477
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 2.61313i	− 0.110921i
$556$	0	0
$557$	−1.31371	−0.0556636	−0.0278318	−	0.999613i	$-0.508860\pi$
$557$	−1.31371	−0.0556636	−0.0278318	+	0.999613i	$0.508860\pi$
$558$	0	0
$559$	−13.6569	−0.577623
$560$	0	0
$561$	−15.3137	−0.646545
$562$	0	0
$563$	0.970563	0.0409043	0.0204522	−	0.999791i	$-0.493489\pi$
$563$	0.970563	0.0409043	0.0204522	+	0.999791i	$0.493489\pi$
$564$	0	0
$565$	− 31.3575i	− 1.31922i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.41421	0.226976	0.113488	−	0.993539i	$-0.463798\pi$
$569$	5.41421	0.226976	0.113488	+	0.993539i	$0.463798\pi$
$570$	0	0
$571$	− 9.18440i	− 0.384355i	−0.981360	−	0.192178i	$-0.938445\pi$
$571$	− 9.18440i	− 0.384355i	0.981360	−	0.192178i	$-0.0615550\pi$
$572$	0	0
$573$	− 18.7402i	− 0.782884i
$574$	0	0
$575$	3.43289i	0.143161i
$576$	0	0
$577$	− 5.35757i	− 0.223038i	−0.993762	−	0.111519i	$-0.964428\pi$
$577$	− 5.35757i	− 0.223038i	0.993762	−	0.111519i	$-0.0355717\pi$
$578$	0	0
$579$	−19.3137	−0.802650
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 50.4692i	− 2.09022i
$584$	0	0
$585$	8.24264	0.340791
$586$	0	0
$587$	−24.9706	−1.03065	−0.515323	−	0.856996i	$-0.672328\pi$
$587$	−24.9706	−1.03065	−0.515323	+	0.856996i	$0.672328\pi$
$588$	0	0
$589$	54.6274	2.25088
$590$	0	0
$591$	−2.68629	−0.110499
$592$	0	0
$593$	− 22.5671i	− 0.926718i	−0.886171	−	0.463359i	$-0.846644\pi$
$593$	− 22.5671i	− 0.926718i	0.886171	−	0.463359i	$-0.153356\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.65685	−0.231520
$598$	0	0
$599$	− 8.28772i	− 0.338627i	−0.985562	−	0.169314i	$-0.945845\pi$
$599$	− 8.28772i	− 0.338627i	0.985562	−	0.169314i	$-0.0541550\pi$
$600$	0	0
$601$	22.8616i	0.932542i	0.884642	+	0.466271i	$0.154403\pi$
$601$	22.8616i	0.932542i	−0.884642	+	0.466271i	$0.845597\pi$
$602$	0	0
$603$	13.5140i	0.550331i
$604$	0	0
$605$	30.1438i	1.22552i
$606$	0	0
$607$	−20.6863	−0.839631	−0.419815	−	0.907610i	$-0.637905\pi$
$607$	−20.6863	−0.839631	−0.419815	+	0.907610i	$0.637905\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 7.39104i	− 0.299009i
$612$	0	0
$613$	4.24264	0.171359	0.0856793	−	0.996323i	$-0.472694\pi$
$613$	4.24264	0.171359	0.0856793	+	0.996323i	$0.472694\pi$
$614$	0	0
$615$	−7.41421	−0.298970
$616$	0	0
$617$	27.5563	1.10938	0.554688	−	0.832058i	$-0.312837\pi$
$617$	27.5563	1.10938	0.554688	+	0.832058i	$0.312837\pi$
$618$	0	0
$619$	41.6569	1.67433	0.837165	−	0.546950i	$-0.184211\pi$
$619$	41.6569	1.67433	0.837165	+	0.546950i	$0.184211\pi$
$620$	0	0
$621$	2.16478i	0.0868698i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−14.5563	−0.582254
$626$	0	0
$627$	− 29.5641i	− 1.18068i
$628$	0	0
$629$	4.14386i	0.165227i
$630$	0	0
$631$	− 14.7821i	− 0.588465i	−0.955734	−	0.294233i	$-0.904936\pi$
$631$	− 14.7821i	− 0.588465i	0.955734	−	0.294233i	$-0.0950640\pi$
$632$	0	0
$633$	− 3.06147i	− 0.121682i
$634$	0	0
$635$	−21.6569	−0.859426
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 6.49435i	− 0.256913i
$640$	0	0
$641$	14.8701	0.587332	0.293666	−	0.955908i	$-0.405125\pi$
$641$	14.8701	0.587332	0.293666	+	0.955908i	$0.405125\pi$
$642$	0	0
$643$	−10.3431	−0.407894	−0.203947	−	0.978982i	$-0.565377\pi$
$643$	−10.3431	−0.407894	−0.203947	+	0.978982i	$0.565377\pi$
$644$	0	0
$645$	5.65685	0.222738
$646$	0	0
$647$	−20.9706	−0.824438	−0.412219	−	0.911085i	$-0.635246\pi$
$647$	−20.9706	−0.824438	−0.412219	+	0.911085i	$0.635246\pi$
$648$	0	0
$649$	− 29.5641i	− 1.16049i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20.9289	0.819012	0.409506	−	0.912307i	$-0.365701\pi$
$653$	20.9289	0.819012	0.409506	+	0.912307i	$0.365701\pi$
$654$	0	0
$655$	13.5140i	0.528035i
$656$	0	0
$657$	4.90923i	0.191527i
$658$	0	0
$659$	7.76245i	0.302382i	0.988505	+	0.151191i	$0.0483109\pi$
$659$	7.76245i	0.302382i	−0.988505	+	0.151191i	$0.951689\pi$
$660$	0	0
$661$	33.4679i	1.30175i	0.759185	+	0.650875i	$0.225598\pi$
$661$	33.4679i	1.30175i	−0.759185	+	0.650875i	$0.774402\pi$
$662$	0	0
$663$	−13.0711	−0.507638
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 11.7206i	− 0.453824i
$668$	0	0
$669$	3.31371	0.128115
$670$	0	0
$671$	7.31371	0.282343
$672$	0	0
$673$	7.07107	0.272570	0.136285	−	0.990670i	$-0.456484\pi$
$673$	7.07107	0.272570	0.136285	+	0.990670i	$0.456484\pi$
$674$	0	0
$675$	1.58579	0.0610369
$676$	0	0
$677$	− 9.23880i	− 0.355076i	−0.984114	−	0.177538i	$-0.943187\pi$
$677$	− 9.23880i	− 0.355076i	0.984114	−	0.177538i	$-0.0568132\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	19.3137	0.740103
$682$	0	0
$683$	21.2764i	0.814120i	0.913401	+	0.407060i	$0.133446\pi$
$683$	21.2764i	0.814120i	−0.913401	+	0.407060i	$0.866554\pi$
$684$	0	0
$685$	19.5600i	0.747349i
$686$	0	0
$687$	21.8561i	0.833862i
$688$	0	0
$689$	− 43.0781i	− 1.64115i
$690$	0	0
$691$	9.65685	0.367364	0.183682	−	0.982986i	$-0.441198\pi$
$691$	9.65685	0.367364	0.183682	+	0.982986i	$0.441198\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 11.7206i	− 0.444588i
$696$	0	0
$697$	11.7574	0.445342
$698$	0	0
$699$	−8.24264	−0.311765
$700$	0	0
$701$	−30.3848	−1.14762	−0.573809	−	0.818989i	$-0.694535\pi$
$701$	−30.3848	−1.14762	−0.573809	+	0.818989i	$0.694535\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	3.06147i	0.115302i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−14.1005	−0.529556	−0.264778	−	0.964309i	$-0.585299\pi$
$709$	−14.1005	−0.529556	−0.264778	+	0.964309i	$0.585299\pi$
$710$	0	0
$711$	11.7206i	0.439557i
$712$	0	0
$713$	− 20.9050i	− 0.782899i
$714$	0	0
$715$	43.0781i	1.61103i
$716$	0	0
$717$	− 12.6173i	− 0.471201i
$718$	0	0
$719$	−36.2843	−1.35317	−0.676587	−	0.736362i	$-0.736542\pi$
$719$	−36.2843	−1.35317	−0.676587	+	0.736362i	$0.736542\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 18.6089i	− 0.692073i
$724$	0	0
$725$	−8.58579	−0.318868
$726$	0	0
$727$	−20.9706	−0.777755	−0.388878	−	0.921289i	$-0.627137\pi$
$727$	−20.9706	−0.777755	−0.388878	+	0.921289i	$0.627137\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.97056	−0.331788
$732$	0	0
$733$	50.2609i	1.85643i	0.372045	+	0.928215i	$0.378657\pi$
$733$	50.2609i	1.85643i	−0.372045	+	0.928215i	$0.621343\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−70.6274	−2.60159
$738$	0	0
$739$	− 17.3183i	− 0.637063i	−0.947912	−	0.318532i	$-0.896810\pi$
$739$	− 17.3183i	− 0.637063i	0.947912	−	0.318532i	$-0.103190\pi$
$740$	0	0
$741$	− 25.2346i	− 0.927015i
$742$	0	0
$743$	− 18.2150i	− 0.668242i	−0.942530	−	0.334121i	$-0.891561\pi$
$743$	− 18.2150i	− 0.668242i	0.942530	−	0.334121i	$-0.108439\pi$
$744$	0	0
$745$	− 38.7485i	− 1.41964i
$746$	0	0
$747$	17.6569	0.646031
$748$	0	0
$749$	0	0
$750$	0	0
$751$	5.59767i	0.204262i	0.994771	+	0.102131i	$0.0325661\pi$
$751$	5.59767i	0.204262i	−0.994771	+	0.102131i	$0.967434\pi$
$752$	0	0
$753$	−24.9706	−0.909978
$754$	0	0
$755$	27.3137	0.994048
$756$	0	0
$757$	26.3848	0.958971	0.479486	−	0.877550i	$-0.340823\pi$
$757$	26.3848	0.958971	0.479486	+	0.877550i	$0.340823\pi$
$758$	0	0
$759$	−11.3137	−0.410662
$760$	0	0
$761$	25.1033i	0.909992i	0.890493	+	0.454996i	$0.150359\pi$
$761$	25.1033i	0.909992i	−0.890493	+	0.454996i	$0.849641\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	5.41421	0.195751
$766$	0	0
$767$	− 25.2346i	− 0.911168i
$768$	0	0
$769$	− 9.16187i	− 0.330386i	−0.986261	−	0.165193i	$-0.947175\pi$
$769$	− 9.16187i	− 0.330386i	0.986261	−	0.165193i	$-0.0528246\pi$
$770$	0	0
$771$	− 27.3450i	− 0.984805i
$772$	0	0
$773$	41.8644i	1.50576i	0.658159	+	0.752879i	$0.271335\pi$
$773$	41.8644i	1.50576i	−0.658159	+	0.752879i	$0.728665\pi$
$774$	0	0
$775$	−15.3137	−0.550085
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22.6984i	0.813254i
$780$	0	0
$781$	33.9411	1.21451
$782$	0	0
$783$	−5.41421	−0.193488
$784$	0	0
$785$	−8.58579	−0.306440
$786$	0	0
$787$	−0.686292	−0.0244636	−0.0122318	−	0.999925i	$-0.503894\pi$
$787$	−0.686292	−0.0244636	−0.0122318	+	0.999925i	$0.503894\pi$
$788$	0	0
$789$	10.8239i	0.385342i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.24264	0.221683
$794$	0	0
$795$	17.8435i	0.632845i
$796$	0	0
$797$	− 9.68714i	− 0.343136i	−0.985172	−	0.171568i	$-0.945117\pi$
$797$	− 9.68714i	− 0.343136i	0.985172	−	0.171568i	$-0.0548833\pi$
$798$	0	0
$799$	− 4.85483i	− 0.171752i
$800$	0	0
$801$	− 9.05309i	− 0.319875i
$802$	0	0
$803$	−25.6569	−0.905411
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.2066i	0.852112i
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	9.65685	0.339098	0.169549	−	0.985522i	$-0.445769\pi$
$811$	9.65685	0.339098	0.169549	+	0.985522i	$0.445769\pi$
$812$	0	0
$813$	1.65685	0.0581084
$814$	0	0
$815$	27.3137	0.956757
$816$	0	0
$817$	− 17.3183i	− 0.605890i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.68629	−0.0937522	−0.0468761	−	0.998901i	$-0.514927\pi$
$821$	−2.68629	−0.0937522	−0.0468761	+	0.998901i	$0.514927\pi$
$822$	0	0
$823$	20.3797i	0.710393i	0.934792	+	0.355197i	$0.115586\pi$
$823$	20.3797i	0.710393i	−0.934792	+	0.355197i	$0.884414\pi$
$824$	0	0
$825$	8.28772i	0.288541i
$826$	0	0
$827$	0.896683i	0.0311807i	0.999878	+	0.0155904i	$0.00496277\pi$
$827$	0.896683i	0.0311807i	−0.999878	+	0.0155904i	$0.995037\pi$
$828$	0	0
$829$	− 47.2764i	− 1.64198i	−0.570945	−	0.820988i	$-0.693423\pi$
$829$	− 47.2764i	− 1.64198i	0.570945	−	0.820988i	$-0.306577\pi$
$830$	0	0
$831$	13.3137	0.461847
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 3.06147i	− 0.105946i
$836$	0	0
$837$	−9.65685	−0.333790
$838$	0	0
$839$	−37.9411	−1.30987	−0.654937	−	0.755684i	$-0.727305\pi$
$839$	−37.9411	−1.30987	−0.654937	+	0.755684i	$0.727305\pi$
$840$	0	0
$841$	0.313708	0.0108175
$842$	0	0
$843$	14.3848	0.495438
$844$	0	0
$845$	12.7486i	0.438565i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.06147i	0.104946i
$852$	0	0
$853$	− 39.0656i	− 1.33758i	−0.743451	−	0.668790i	$-0.766813\pi$
$853$	− 39.0656i	− 1.33758i	0.743451	−	0.668790i	$-0.233187\pi$
$854$	0	0
$855$	10.4525i	0.357468i
$856$	0	0
$857$	− 36.5612i	− 1.24891i	−0.781062	−	0.624454i	$-0.785322\pi$
$857$	− 36.5612i	− 1.24891i	0.781062	−	0.624454i	$-0.214678\pi$
$858$	0	0
$859$	−6.62742	−0.226125	−0.113062	−	0.993588i	$-0.536066\pi$
$859$	−6.62742	−0.226125	−0.113062	+	0.993588i	$0.536066\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 17.4721i	− 0.594758i	−0.954760	−	0.297379i	$-0.903888\pi$
$863$	− 17.4721i	− 0.594758i	0.954760	−	0.297379i	$-0.0961125\pi$
$864$	0	0
$865$	−35.2132	−1.19728
$866$	0	0
$867$	8.41421	0.285762
$868$	0	0
$869$	−61.2548	−2.07793
$870$	0	0
$871$	−60.2843	−2.04265
$872$	0	0
$873$	9.23880i	0.312686i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	27.7574	0.937299	0.468650	−	0.883384i	$-0.344741\pi$
$877$	27.7574	0.937299	0.468650	+	0.883384i	$0.344741\pi$
$878$	0	0
$879$	13.0112i	0.438858i
$880$	0	0
$881$	− 8.04762i	− 0.271131i	−0.990768	−	0.135566i	$-0.956715\pi$
$881$	− 8.04762i	− 0.271131i	0.990768	−	0.135566i	$-0.0432851\pi$
$882$	0	0
$883$	− 19.1116i	− 0.643158i	−0.946883	−	0.321579i	$-0.895786\pi$
$883$	− 19.1116i	− 0.643158i	0.946883	−	0.321579i	$-0.104214\pi$
$884$	0	0
$885$	10.4525i	0.351357i
$886$	0	0
$887$	47.3137	1.58864	0.794319	−	0.607500i	$-0.207828\pi$
$887$	47.3137	1.58864	0.794319	+	0.607500i	$0.207828\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.22625i	0.175086i
$892$	0	0
$893$	9.37258	0.313642
$894$	0	0
$895$	7.31371	0.244470
$896$	0	0
$897$	−9.65685	−0.322433
$898$	0	0
$899$	52.2843	1.74378
$900$	0	0
$901$	− 28.2960i	− 0.942678i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.3848	−0.810577
$906$	0	0
$907$	25.2346i	0.837900i	0.908009	+	0.418950i	$0.137602\pi$
$907$	25.2346i	0.837900i	−0.908009	+	0.418950i	$0.862398\pi$
$908$	0	0
$909$	− 6.88830i	− 0.228471i
$910$	0	0
$911$	− 53.9020i	− 1.78585i	−0.450201	−	0.892927i	$-0.648648\pi$
$911$	− 53.9020i	− 1.78585i	0.450201	−	0.892927i	$-0.351352\pi$
$912$	0	0
$913$	92.2792i	3.05399i
$914$	0	0
$915$	−2.58579	−0.0854835
$916$	0	0
$917$	0	0
$918$	0	0
$919$	55.3240i	1.82497i	0.409110	+	0.912485i	$0.365839\pi$
$919$	55.3240i	1.82497i	−0.409110	+	0.912485i	$0.634161\pi$
$920$	0	0
$921$	−19.3137	−0.636408
$922$	0	0
$923$	28.9706	0.953578
$924$	0	0
$925$	2.24264	0.0737376
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	− 7.25972i	− 0.238184i	−0.992883	−	0.119092i	$-0.962002\pi$
$929$	− 7.25972i	− 0.238184i	0.992883	−	0.119092i	$-0.0379983\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.00000	0.130954
$934$	0	0
$935$	28.2960i	0.925380i
$936$	0	0
$937$	9.23880i	0.301818i	0.988548	+	0.150909i	$0.0482201\pi$
$937$	9.23880i	0.301818i	−0.988548	+	0.150909i	$0.951780\pi$
$938$	0	0
$939$	14.7277i	0.480620i
$940$	0	0
$941$	21.4077i	0.697872i	0.937146	+	0.348936i	$0.113457\pi$
$941$	21.4077i	0.697872i	−0.937146	+	0.348936i	$0.886543\pi$
$942$	0	0
$943$	8.68629	0.282865
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.3881i	1.31244i	0.754571	+	0.656218i	$0.227845\pi$
$947$	40.3881i	1.31244i	−0.754571	+	0.656218i	$0.772155\pi$
$948$	0	0
$949$	−21.8995	−0.710888
$950$	0	0
$951$	17.3137	0.561435
$952$	0	0
$953$	38.0000	1.23094	0.615470	−	0.788160i	$-0.288966\pi$
$953$	38.0000	1.23094	0.615470	+	0.788160i	$0.288966\pi$
$954$	0	0
$955$	−34.6274	−1.12052
$956$	0	0
$957$	− 28.2960i	− 0.914681i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	62.2548	2.00822
$962$	0	0
$963$	20.0083i	0.644759i
$964$	0	0
$965$	35.6871i	1.14881i
$966$	0	0
$967$	46.6648i	1.50064i	0.661075	+	0.750320i	$0.270101\pi$
$967$	46.6648i	1.50064i	−0.661075	+	0.750320i	$0.729899\pi$
$968$	0	0
$969$	− 16.5754i	− 0.532480i
$970$	0	0
$971$	44.9706	1.44317	0.721587	−	0.692324i	$-0.243413\pi$
$971$	44.9706	1.44317	0.721587	+	0.692324i	$0.243413\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	7.07401i	0.226550i
$976$	0	0
$977$	34.1838	1.09364	0.546818	−	0.837252i	$-0.315839\pi$
$977$	34.1838	1.09364	0.546818	+	0.837252i	$0.315839\pi$
$978$	0	0
$979$	47.3137	1.51215
$980$	0	0
$981$	20.7279	0.661792
$982$	0	0
$983$	−55.5980	−1.77330	−0.886650	−	0.462441i	$-0.846974\pi$
$983$	−55.5980	−1.77330	−0.886650	+	0.462441i	$0.846974\pi$
$984$	0	0
$985$	4.96362i	0.158154i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.62742	−0.210740
$990$	0	0
$991$	− 36.9552i	− 1.17392i	−0.809616	−	0.586960i	$-0.800325\pi$
$991$	− 36.9552i	− 1.17392i	0.809616	−	0.586960i	$-0.199675\pi$
$992$	0	0
$993$	1.79337i	0.0569108i
$994$	0	0
$995$	10.4525i	0.331367i
$996$	0	0
$997$	− 18.0518i	− 0.571706i	−0.958274	−	0.285853i	$-0.907723\pi$
$997$	− 18.0518i	− 0.571706i	0.958274	−	0.285853i	$-0.0922768\pi$
$998$	0	0
$999$	1.41421	0.0447437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.b.j.1567.1	yes	4
3.2	odd	2		7056.2.b.t.1567.4		4
4.3	odd	2		2352.2.b.i.1567.1	✓	4
7.2	even	3		2352.2.bl.p.31.1		8
7.3	odd	6		2352.2.bl.s.607.1		8
7.4	even	3		2352.2.bl.p.607.4		8
7.5	odd	6		2352.2.bl.s.31.4		8
7.6	odd	2		2352.2.b.i.1567.4	yes	4
12.11	even	2		7056.2.b.u.1567.4		4
21.20	even	2		7056.2.b.u.1567.1		4
28.3	even	6		2352.2.bl.p.607.1		8
28.11	odd	6		2352.2.bl.s.607.4		8
28.19	even	6		2352.2.bl.p.31.4		8
28.23	odd	6		2352.2.bl.s.31.1		8
28.27	even	2	inner	2352.2.b.j.1567.4	yes	4
84.83	odd	2		7056.2.b.t.1567.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.b.i.1567.1	✓	4	4.3	odd	2
2352.2.b.i.1567.4	yes	4	7.6	odd	2
2352.2.b.j.1567.1	yes	4	1.1	even	1	trivial
2352.2.b.j.1567.4	yes	4	28.27	even	2	inner
2352.2.bl.p.31.1		8	7.2	even	3
2352.2.bl.p.31.4		8	28.19	even	6
2352.2.bl.p.607.1		8	28.3	even	6
2352.2.bl.p.607.4		8	7.4	even	3
2352.2.bl.s.31.1		8	28.23	odd	6
2352.2.bl.s.31.4		8	7.5	odd	6
2352.2.bl.s.607.1		8	7.3	odd	6
2352.2.bl.s.607.4		8	28.11	odd	6
7056.2.b.t.1567.1		4	84.83	odd	2
7056.2.b.t.1567.4		4	3.2	odd	2
7056.2.b.u.1567.1		4	21.20	even	2
7056.2.b.u.1567.4		4	12.11	even	2

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.84776i q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.84776i q^{5} +1.00000 q^{9} +5.22625i q^{11} +4.46088i q^{13} -1.84776i q^{15} +2.93015i q^{17} -5.65685 q^{19} +2.16478i q^{23} +1.58579 q^{25} +1.00000 q^{27} -5.41421 q^{29} -9.65685 q^{31} +5.22625i q^{33} +1.41421 q^{37} +4.46088i q^{39} -4.01254i q^{41} +3.06147i q^{43} -1.84776i q^{45} -1.65685 q^{47} +2.93015i q^{51} -9.65685 q^{53} +9.65685 q^{55} -5.65685 q^{57} -5.65685 q^{59} -1.39942i q^{61} +8.24264 q^{65} +13.5140i q^{67} +2.16478i q^{69} -6.49435i q^{71} +4.90923i q^{73} +1.58579 q^{75} +11.7206i q^{79} +1.00000 q^{81} +17.6569 q^{83} +5.41421 q^{85} -5.41421 q^{87} -9.05309i q^{89} -9.65685 q^{93} +10.4525i q^{95} +9.23880i q^{97} +5.22625i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 12 q^{25} + 4 q^{27} - 16 q^{29} - 16 q^{31} + 16 q^{47} - 16 q^{53} + 16 q^{55} + 16 q^{65} + 12 q^{75} + 4 q^{81} + 48 q^{83} + 16 q^{85} - 16 q^{87} - 16 q^{93}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 12 * q^25 + 4 * q^27 - 16 * q^29 - 16 * q^31 + 16 * q^47 - 16 * q^53 + 16 * q^55 + 16 * q^65 + 12 * q^75 + 4 * q^81 + 48 * q^83 + 16 * q^85 - 16 * q^87 - 16 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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