LMFDB - Embedded newform 2352.2.b.j.1567.3

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			1567.3
Root			$0.765367i$ of defining polynomial
Character	$\chi$	$=$	2352.1567
Dual form			2352.2.b.j.1567.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.765367i q^{5} +1.00000 q^{9} +2.16478i q^{11} +0.317025i q^{13} +0.765367i q^{15} -3.37849i q^{17} +5.65685 q^{19} -5.22625i q^{23} +4.41421 q^{25} +1.00000 q^{27} -2.58579 q^{29} +1.65685 q^{31} +2.16478i q^{33} -1.41421 q^{37} +0.317025i q^{39} +5.99162i q^{41} +7.39104i q^{43} +0.765367i q^{45} +9.65685 q^{47} -3.37849i q^{51} +1.65685 q^{53} -1.65685 q^{55} +5.65685 q^{57} +5.65685 q^{59} +7.07401i q^{61} -0.242641 q^{65} +11.7206i q^{67} -5.22625i q^{69} +15.6788i q^{71} +6.62567i q^{73} +4.41421 q^{75} -13.5140i q^{79} +1.00000 q^{81} +6.34315 q^{83} +2.58579 q^{85} -2.58579 q^{87} -11.4036i q^{89} +1.65685 q^{93} +4.32957i q^{95} -3.82683i q^{97} +2.16478i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

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
$5$	0.765367i	0.342282i	0.985247	+	0.171141i	$0.0547454\pi$
$5$	0.765367i	0.342282i	−0.985247	+	0.171141i	$0.945255\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.16478i	0.652707i	0.945248	+	0.326354i	$0.105820\pi$
$11$	2.16478i	0.652707i	−0.945248	+	0.326354i	$0.894180\pi$
$12$	0	0
$13$	0.317025i	0.0879270i	0.999033	+	0.0439635i	$0.0139985\pi$
$13$	0.317025i	0.0879270i	−0.999033	+	0.0439635i	$0.986001\pi$
$14$	0	0
$15$	0.765367i	0.197617i
$16$	0	0
$17$	− 3.37849i	− 0.819405i	−0.912219	−	0.409702i	$-0.865633\pi$
$17$	− 3.37849i	− 0.819405i	0.912219	−	0.409702i	$-0.134367\pi$
$18$	0	0
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 5.22625i	− 1.08975i	−0.838518	−	0.544874i	$-0.816577\pi$
$23$	− 5.22625i	− 1.08975i	0.838518	−	0.544874i	$-0.183423\pi$
$24$	0	0
$25$	4.41421	0.882843
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.58579	−0.480168	−0.240084	−	0.970752i	$-0.577175\pi$
$29$	−2.58579	−0.480168	−0.240084	+	0.970752i	$0.577175\pi$
$30$	0	0
$31$	1.65685	0.297580	0.148790	−	0.988869i	$-0.452462\pi$
$31$	1.65685	0.297580	0.148790	+	0.988869i	$0.452462\pi$
$32$	0	0
$33$	2.16478i	0.376841i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.41421	−0.232495	−0.116248	−	0.993220i	$-0.537087\pi$
$37$	−1.41421	−0.232495	−0.116248	+	0.993220i	$0.537087\pi$
$38$	0	0
$39$	0.317025i	0.0507647i
$40$	0	0
$41$	5.99162i	0.935734i	0.883799	+	0.467867i	$0.154977\pi$
$41$	5.99162i	0.935734i	−0.883799	+	0.467867i	$0.845023\pi$
$42$	0	0
$43$	7.39104i	1.12712i	0.826074	+	0.563561i	$0.190569\pi$
$43$	7.39104i	1.12712i	−0.826074	+	0.563561i	$0.809431\pi$
$44$	0	0
$45$	0.765367i	0.114094i
$46$	0	0
$47$	9.65685	1.40860	0.704298	−	0.709904i	$-0.251262\pi$
$47$	9.65685	1.40860	0.704298	+	0.709904i	$0.251262\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 3.37849i	− 0.473084i
$52$	0	0
$53$	1.65685	0.227586	0.113793	−	0.993504i	$-0.463700\pi$
$53$	1.65685	0.227586	0.113793	+	0.993504i	$0.463700\pi$
$54$	0	0
$55$	−1.65685	−0.223410
$56$	0	0
$57$	5.65685	0.749269
$58$	0	0
$59$	5.65685	0.736460	0.368230	−	0.929735i	$-0.379964\pi$
$59$	5.65685	0.736460	0.368230	+	0.929735i	$0.379964\pi$
$60$	0	0
$61$	7.07401i	0.905734i	0.891578	+	0.452867i	$0.149599\pi$
$61$	7.07401i	0.905734i	−0.891578	+	0.452867i	$0.850401\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.242641	−0.0300959
$66$	0	0
$67$	11.7206i	1.43190i	0.698152	+	0.715950i	$0.254006\pi$
$67$	11.7206i	1.43190i	−0.698152	+	0.715950i	$0.745994\pi$
$68$	0	0
$69$	− 5.22625i	− 0.629167i
$70$	0	0
$71$	15.6788i	1.86073i	0.366640	+	0.930363i	$0.380508\pi$
$71$	15.6788i	1.86073i	−0.366640	+	0.930363i	$0.619492\pi$
$72$	0	0
$73$	6.62567i	0.775476i	0.921770	+	0.387738i	$0.126743\pi$
$73$	6.62567i	0.775476i	−0.921770	+	0.387738i	$0.873257\pi$
$74$	0	0
$75$	4.41421	0.509709
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 13.5140i	− 1.52044i	−0.649665	−	0.760220i	$-0.725091\pi$
$79$	− 13.5140i	− 1.52044i	0.649665	−	0.760220i	$-0.274909\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.34315	0.696251	0.348125	−	0.937448i	$-0.386818\pi$
$83$	6.34315	0.696251	0.348125	+	0.937448i	$0.386818\pi$
$84$	0	0
$85$	2.58579	0.280468
$86$	0	0
$87$	−2.58579	−0.277225
$88$	0	0
$89$	− 11.4036i	− 1.20878i	−0.796690	−	0.604389i	$-0.793417\pi$
$89$	− 11.4036i	− 1.20878i	0.796690	−	0.604389i	$-0.206583\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.65685	0.171808
$94$	0	0
$95$	4.32957i	0.444204i
$96$	0	0
$97$	− 3.82683i	− 0.388556i	−0.980946	−	0.194278i	$-0.937764\pi$
$97$	− 3.82683i	− 0.388556i	0.980946	−	0.194278i	$-0.0622364\pi$
$98$	0	0
$99$	2.16478i	0.217569i
$100$	0	0
$101$	− 16.6298i	− 1.65473i	−0.561665	−	0.827365i	$-0.689839\pi$
$101$	− 16.6298i	− 1.65473i	0.561665	−	0.827365i	$-0.310161\pi$
$102$	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
$107$	− 3.95815i	− 0.382649i	−0.981527	−	0.191324i	$-0.938722\pi$
$107$	− 3.95815i	− 0.382649i	0.981527	−	0.191324i	$-0.0612783\pi$
$108$	0	0
$109$	−4.72792	−0.452853	−0.226426	−	0.974028i	$-0.572704\pi$
$109$	−4.72792	−0.452853	−0.226426	+	0.974028i	$0.572704\pi$
$110$	0	0
$111$	−1.41421	−0.134231
$112$	0	0
$113$	−16.9706	−1.59646	−0.798228	−	0.602355i	$-0.794229\pi$
$113$	−16.9706	−1.59646	−0.798228	+	0.602355i	$0.794229\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	0.317025i	0.0293090i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.31371	0.573973
$122$	0	0
$123$	5.99162i	0.540246i
$124$	0	0
$125$	7.20533i	0.644464i
$126$	0	0
$127$	13.5140i	1.19917i	0.800311	+	0.599586i	$0.204668\pi$
$127$	13.5140i	1.19917i	−0.800311	+	0.599586i	$0.795332\pi$
$128$	0	0
$129$	7.39104i	0.650744i
$130$	0	0
$131$	15.3137	1.33796	0.668982	−	0.743278i	$-0.266730\pi$
$131$	15.3137	1.33796	0.668982	+	0.743278i	$0.266730\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.765367i	0.0658723i
$136$	0	0
$137$	−13.4142	−1.14605	−0.573027	−	0.819537i	$-0.694231\pi$
$137$	−13.4142	−1.14605	−0.573027	+	0.819537i	$0.694231\pi$
$138$	0	0
$139$	17.6569	1.49763	0.748817	−	0.662776i	$-0.230622\pi$
$139$	17.6569	1.49763	0.748817	+	0.662776i	$0.230622\pi$
$140$	0	0
$141$	9.65685	0.813254
$142$	0	0
$143$	−0.686292	−0.0573906
$144$	0	0
$145$	− 1.97908i	− 0.164353i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.9706	−1.06259	−0.531295	−	0.847187i	$-0.678294\pi$
$149$	−12.9706	−1.06259	−0.531295	+	0.847187i	$0.678294\pi$
$150$	0	0
$151$	− 6.12293i	− 0.498277i	−0.968468	−	0.249139i	$-0.919852\pi$
$151$	− 6.12293i	− 0.498277i	0.968468	−	0.249139i	$-0.0801475\pi$
$152$	0	0
$153$	− 3.37849i	− 0.273135i
$154$	0	0
$155$	1.26810i	0.101856i
$156$	0	0
$157$	14.9134i	1.19022i	0.803645	+	0.595109i	$0.202891\pi$
$157$	14.9134i	1.19022i	−0.803645	+	0.595109i	$0.797109\pi$
$158$	0	0
$159$	1.65685	0.131397
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 6.12293i	− 0.479585i	−0.970824	−	0.239793i	$-0.922921\pi$
$163$	− 6.12293i	− 0.479585i	0.970824	−	0.239793i	$-0.0770795\pi$
$164$	0	0
$165$	−1.65685	−0.128986
$166$	0	0
$167$	−9.65685	−0.747270	−0.373635	−	0.927576i	$-0.621889\pi$
$167$	−9.65685	−0.747270	−0.373635	+	0.927576i	$0.621889\pi$
$168$	0	0
$169$	12.8995	0.992269
$170$	0	0
$171$	5.65685	0.432590
$172$	0	0
$173$	− 9.42450i	− 0.716532i	−0.933620	−	0.358266i	$-0.883368\pi$
$173$	− 9.42450i	− 0.716532i	0.933620	−	0.358266i	$-0.116632\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.65685	0.425195
$178$	0	0
$179$	20.0083i	1.49549i	0.663985	+	0.747746i	$0.268864\pi$
$179$	20.0083i	1.49549i	−0.663985	+	0.747746i	$0.731136\pi$
$180$	0	0
$181$	− 16.1815i	− 1.20276i	−0.798963	−	0.601380i	$-0.794618\pi$
$181$	− 16.1815i	− 1.20276i	0.798963	−	0.601380i	$-0.205382\pi$
$182$	0	0
$183$	7.07401i	0.522926i
$184$	0	0
$185$	− 1.08239i	− 0.0795791i
$186$	0	0
$187$	7.31371	0.534831
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 13.8854i	− 1.00471i	−0.864661	−	0.502356i	$-0.832467\pi$
$191$	− 13.8854i	− 1.00471i	0.864661	−	0.502356i	$-0.167533\pi$
$192$	0	0
$193$	3.31371	0.238526	0.119263	−	0.992863i	$-0.461947\pi$
$193$	3.31371	0.238526	0.119263	+	0.992863i	$0.461947\pi$
$194$	0	0
$195$	−0.242641	−0.0173759
$196$	0	0
$197$	−25.3137	−1.80353	−0.901764	−	0.432230i	$-0.857727\pi$
$197$	−25.3137	−1.80353	−0.901764	+	0.432230i	$0.857727\pi$
$198$	0	0
$199$	5.65685	0.401004	0.200502	−	0.979693i	$-0.435743\pi$
$199$	5.65685	0.401004	0.200502	+	0.979693i	$0.435743\pi$
$200$	0	0
$201$	11.7206i	0.826708i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.58579	−0.320285
$206$	0	0
$207$	− 5.22625i	− 0.363250i
$208$	0	0
$209$	12.2459i	0.847065i
$210$	0	0
$211$	− 7.39104i	− 0.508820i	−0.967096	−	0.254410i	$-0.918119\pi$
$211$	− 7.39104i	− 0.508820i	0.967096	−	0.254410i	$-0.0818813\pi$
$212$	0	0
$213$	15.6788i	1.07429i
$214$	0	0
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.62567i	0.447721i
$220$	0	0
$221$	1.07107	0.0720478
$222$	0	0
$223$	−19.3137	−1.29334	−0.646671	−	0.762769i	$-0.723839\pi$
$223$	−19.3137	−1.29334	−0.646671	+	0.762769i	$0.723839\pi$
$224$	0	0
$225$	4.41421	0.294281
$226$	0	0
$227$	−3.31371	−0.219939	−0.109969	−	0.993935i	$-0.535075\pi$
$227$	−3.31371	−0.219939	−0.109969	+	0.993935i	$0.535075\pi$
$228$	0	0
$229$	− 4.72352i	− 0.312139i	−0.987746	−	0.156069i	$-0.950118\pi$
$229$	− 4.72352i	− 0.312139i	0.987746	−	0.156069i	$-0.0498824\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.242641	0.0158959	0.00794796	−	0.999968i	$-0.497470\pi$
$233$	0.242641	0.0158959	0.00794796	+	0.999968i	$0.497470\pi$
$234$	0	0
$235$	7.39104i	0.482138i
$236$	0	0
$237$	− 13.5140i	− 0.877827i
$238$	0	0
$239$	0.896683i	0.0580016i	0.999579	+	0.0290008i	$0.00923254\pi$
$239$	0.896683i	0.0580016i	−0.999579	+	0.0290008i	$0.990767\pi$
$240$	0	0
$241$	− 3.11586i	− 0.200710i	−0.994952	−	0.100355i	$-0.968002\pi$
$241$	− 3.11586i	− 0.200710i	0.994952	−	0.100355i	$-0.0319979\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.79337i	0.114109i
$248$	0	0
$249$	6.34315	0.401981
$250$	0	0
$251$	8.97056	0.566217	0.283108	−	0.959088i	$-0.408634\pi$
$251$	8.97056	0.566217	0.283108	+	0.959088i	$0.408634\pi$
$252$	0	0
$253$	11.3137	0.711287
$254$	0	0
$255$	2.58579	0.161928
$256$	0	0
$257$	− 18.9803i	− 1.18396i	−0.805953	−	0.591980i	$-0.798346\pi$
$257$	− 18.9803i	− 1.18396i	0.805953	−	0.591980i	$-0.201654\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.58579	−0.160056
$262$	0	0
$263$	− 26.1313i	− 1.61132i	−0.592377	−	0.805661i	$-0.701810\pi$
$263$	− 26.1313i	− 1.61132i	0.592377	−	0.805661i	$-0.298190\pi$
$264$	0	0
$265$	1.26810i	0.0778988i
$266$	0	0
$267$	− 11.4036i	− 0.697888i
$268$	0	0
$269$	− 25.1802i	− 1.53526i	−0.640892	−	0.767631i	$-0.721435\pi$
$269$	− 25.1802i	− 1.53526i	0.640892	−	0.767631i	$-0.278565\pi$
$270$	0	0
$271$	−9.65685	−0.586612	−0.293306	−	0.956019i	$-0.594755\pi$
$271$	−9.65685	−0.586612	−0.293306	+	0.956019i	$0.594755\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.55582i	0.576238i
$276$	0	0
$277$	−9.31371	−0.559607	−0.279803	−	0.960057i	$-0.590269\pi$
$277$	−9.31371	−0.559607	−0.279803	+	0.960057i	$0.590269\pi$
$278$	0	0
$279$	1.65685	0.0991933
$280$	0	0
$281$	−22.3848	−1.33536	−0.667682	−	0.744447i	$-0.732713\pi$
$281$	−22.3848	−1.33536	−0.667682	+	0.744447i	$0.732713\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	4.32957i	0.256462i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	5.58579	0.328576
$290$	0	0
$291$	− 3.82683i	− 0.224333i
$292$	0	0
$293$	31.4119i	1.83510i	0.397617	+	0.917552i	$0.369837\pi$
$293$	31.4119i	1.83510i	−0.397617	+	0.917552i	$0.630163\pi$
$294$	0	0
$295$	4.32957i	0.252077i
$296$	0	0
$297$	2.16478i	0.125614i
$298$	0	0
$299$	1.65685	0.0958184
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 16.6298i	− 0.955359i
$304$	0	0
$305$	−5.41421	−0.310017
$306$	0	0
$307$	3.31371	0.189123	0.0945617	−	0.995519i	$-0.469855\pi$
$307$	3.31371	0.189123	0.0945617	+	0.995519i	$0.469855\pi$
$308$	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
$313$	19.8770i	1.12351i	0.827302	+	0.561757i	$0.189875\pi$
$313$	19.8770i	1.12351i	−0.827302	+	0.561757i	$0.810125\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.31371	−0.298448	−0.149224	−	0.988803i	$-0.547677\pi$
$317$	−5.31371	−0.298448	−0.149224	+	0.988803i	$0.547677\pi$
$318$	0	0
$319$	− 5.59767i	− 0.313409i
$320$	0	0
$321$	− 3.95815i	− 0.220922i
$322$	0	0
$323$	− 19.1116i	− 1.06340i
$324$	0	0
$325$	1.39942i	0.0776257i
$326$	0	0
$327$	−4.72792	−0.261455
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.2346i	1.38702i	0.720448	+	0.693509i	$0.243936\pi$
$331$	25.2346i	1.38702i	−0.720448	+	0.693509i	$0.756064\pi$
$332$	0	0
$333$	−1.41421	−0.0774984
$334$	0	0
$335$	−8.97056	−0.490114
$336$	0	0
$337$	20.7279	1.12912	0.564561	−	0.825391i	$-0.309046\pi$
$337$	20.7279	1.12912	0.564561	+	0.825391i	$0.309046\pi$
$338$	0	0
$339$	−16.9706	−0.921714
$340$	0	0
$341$	3.58673i	0.194232i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	− 3.43289i	− 0.184287i	−0.995746	−	0.0921435i	$-0.970628\pi$
$347$	− 3.43289i	− 0.184287i	0.995746	−	0.0921435i	$-0.0293718\pi$
$348$	0	0
$349$	− 20.7737i	− 1.11199i	−0.831186	−	0.555995i	$-0.812337\pi$
$349$	− 20.7737i	− 1.11199i	0.831186	−	0.555995i	$-0.187663\pi$
$350$	0	0
$351$	0.317025i	0.0169216i
$352$	0	0
$353$	− 14.9134i	− 0.793760i	−0.917871	−	0.396880i	$-0.870093\pi$
$353$	− 14.9134i	− 0.793760i	0.917871	−	0.396880i	$-0.129907\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.6788i	0.827493i	0.910392	+	0.413747i	$0.135780\pi$
$359$	15.6788i	0.827493i	−0.910392	+	0.413747i	$0.864220\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	6.31371	0.331384
$364$	0	0
$365$	−5.07107	−0.265432
$366$	0	0
$367$	−21.6569	−1.13048	−0.565239	−	0.824927i	$-0.691216\pi$
$367$	−21.6569	−1.13048	−0.565239	+	0.824927i	$0.691216\pi$
$368$	0	0
$369$	5.99162i	0.311911i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	34.6274	1.79294	0.896470	−	0.443105i	$-0.146123\pi$
$373$	34.6274	1.79294	0.896470	+	0.443105i	$0.146123\pi$
$374$	0	0
$375$	7.20533i	0.372081i
$376$	0	0
$377$	− 0.819760i	− 0.0422198i
$378$	0	0
$379$	− 33.1509i	− 1.70285i	−0.524479	−	0.851423i	$-0.675740\pi$
$379$	− 33.1509i	− 1.70285i	0.524479	−	0.851423i	$-0.324260\pi$
$380$	0	0
$381$	13.5140i	0.692342i
$382$	0	0
$383$	−36.2843	−1.85404	−0.927020	−	0.375012i	$-0.877638\pi$
$383$	−36.2843	−1.85404	−0.927020	+	0.375012i	$0.877638\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.39104i	0.375707i
$388$	0	0
$389$	20.0416	1.01615	0.508076	−	0.861313i	$-0.330357\pi$
$389$	20.0416	1.01615	0.508076	+	0.861313i	$0.330357\pi$
$390$	0	0
$391$	−17.6569	−0.892946
$392$	0	0
$393$	15.3137	0.772474
$394$	0	0
$395$	10.3431	0.520420
$396$	0	0
$397$	19.4287i	0.975097i	0.873096	+	0.487548i	$0.162109\pi$
$397$	19.4287i	0.975097i	−0.873096	+	0.487548i	$0.837891\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.8701	−1.14208	−0.571038	−	0.820924i	$-0.693459\pi$
$401$	−22.8701	−1.14208	−0.571038	+	0.820924i	$0.693459\pi$
$402$	0	0
$403$	0.525265i	0.0261653i
$404$	0	0
$405$	0.765367i	0.0380314i
$406$	0	0
$407$	− 3.06147i	− 0.151751i
$408$	0	0
$409$	7.89377i	0.390322i	0.980771	+	0.195161i	$0.0625229\pi$
$409$	7.89377i	0.390322i	−0.980771	+	0.195161i	$0.937477\pi$
$410$	0	0
$411$	−13.4142	−0.661674
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.85483i	0.238314i
$416$	0	0
$417$	17.6569	0.864660
$418$	0	0
$419$	22.6274	1.10542	0.552711	−	0.833373i	$-0.313593\pi$
$419$	22.6274	1.10542	0.552711	+	0.833373i	$0.313593\pi$
$420$	0	0
$421$	−6.68629	−0.325870	−0.162935	−	0.986637i	$-0.552096\pi$
$421$	−6.68629	−0.325870	−0.162935	+	0.986637i	$0.552096\pi$
$422$	0	0
$423$	9.65685	0.469532
$424$	0	0
$425$	− 14.9134i	− 0.723406i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.686292	−0.0331345
$430$	0	0
$431$	− 15.1535i	− 0.729918i	−0.931024	−	0.364959i	$-0.881083\pi$
$431$	− 15.1535i	− 0.729918i	0.931024	−	0.364959i	$-0.118917\pi$
$432$	0	0
$433$	9.87285i	0.474459i	0.971454	+	0.237229i	$0.0762393\pi$
$433$	9.87285i	0.474459i	−0.971454	+	0.237229i	$0.923761\pi$
$434$	0	0
$435$	− 1.97908i	− 0.0948894i
$436$	0	0
$437$	− 29.5641i	− 1.41424i
$438$	0	0
$439$	−22.6274	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$439$	−22.6274	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.1313i	1.24153i	0.783995	+	0.620767i	$0.213179\pi$
$443$	26.1313i	1.24153i	−0.783995	+	0.620767i	$0.786821\pi$
$444$	0	0
$445$	8.72792	0.413743
$446$	0	0
$447$	−12.9706	−0.613487
$448$	0	0
$449$	−29.6569	−1.39959	−0.699797	−	0.714342i	$-0.746726\pi$
$449$	−29.6569	−1.39959	−0.699797	+	0.714342i	$0.746726\pi$
$450$	0	0
$451$	−12.9706	−0.610760
$452$	0	0
$453$	− 6.12293i	− 0.287681i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.3137	−0.903457	−0.451729	−	0.892155i	$-0.649192\pi$
$457$	−19.3137	−0.903457	−0.451729	+	0.892155i	$0.649192\pi$
$458$	0	0
$459$	− 3.37849i	− 0.157695i
$460$	0	0
$461$	24.2066i	1.12741i	0.825975	+	0.563706i	$0.190625\pi$
$461$	24.2066i	1.12741i	−0.825975	+	0.563706i	$0.809375\pi$
$462$	0	0
$463$	1.79337i	0.0833448i	0.999131	+	0.0416724i	$0.0132686\pi$
$463$	1.79337i	0.0833448i	−0.999131	+	0.0416724i	$0.986731\pi$
$464$	0	0
$465$	1.26810i	0.0588068i
$466$	0	0
$467$	−19.3137	−0.893732	−0.446866	−	0.894601i	$-0.647460\pi$
$467$	−19.3137	−0.893732	−0.446866	+	0.894601i	$0.647460\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.9134i	0.687173i
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	24.9706	1.14573
$476$	0	0
$477$	1.65685	0.0758621
$478$	0	0
$479$	−26.6274	−1.21664	−0.608319	−	0.793693i	$-0.708156\pi$
$479$	−26.6274	−1.21664	−0.608319	+	0.793693i	$0.708156\pi$
$480$	0	0
$481$	− 0.448342i	− 0.0204426i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.92893	0.132996
$486$	0	0
$487$	− 35.6871i	− 1.61714i	−0.588403	−	0.808568i	$-0.700243\pi$
$487$	− 35.6871i	− 1.61714i	0.588403	−	0.808568i	$-0.299757\pi$
$488$	0	0
$489$	− 6.12293i	− 0.276889i
$490$	0	0
$491$	− 14.4107i	− 0.650344i	−0.945655	−	0.325172i	$-0.894578\pi$
$491$	− 14.4107i	− 0.650344i	0.945655	−	0.325172i	$-0.105422\pi$
$492$	0	0
$493$	8.73606i	0.393452i
$494$	0	0
$495$	−1.65685	−0.0744701
$496$	0	0
$497$	0	0
$498$	0	0
$499$	17.3183i	0.775272i	0.921812	+	0.387636i	$0.126708\pi$
$499$	17.3183i	0.775272i	−0.921812	+	0.387636i	$0.873292\pi$
$500$	0	0
$501$	−9.65685	−0.431436
$502$	0	0
$503$	−37.6569	−1.67904	−0.839518	−	0.543332i	$-0.817163\pi$
$503$	−37.6569	−1.67904	−0.839518	+	0.543332i	$0.817163\pi$
$504$	0	0
$505$	12.7279	0.566385
$506$	0	0
$507$	12.8995	0.572887
$508$	0	0
$509$	10.5069i	0.465710i	0.972511	+	0.232855i	$0.0748068\pi$
$509$	10.5069i	0.465710i	−0.972511	+	0.232855i	$0.925193\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.65685	0.249756
$514$	0	0
$515$	9.18440i	0.404713i
$516$	0	0
$517$	20.9050i	0.919401i
$518$	0	0
$519$	− 9.42450i	− 0.413690i
$520$	0	0
$521$	1.66205i	0.0728157i	0.999337	+	0.0364079i	$0.0115915\pi$
$521$	1.66205i	0.0728157i	−0.999337	+	0.0364079i	$0.988408\pi$
$522$	0	0
$523$	−10.6274	−0.464704	−0.232352	−	0.972632i	$-0.574642\pi$
$523$	−10.6274	−0.464704	−0.232352	+	0.972632i	$0.574642\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5.59767i	− 0.243838i
$528$	0	0
$529$	−4.31371	−0.187553
$530$	0	0
$531$	5.65685	0.245487
$532$	0	0
$533$	−1.89949	−0.0822763
$534$	0	0
$535$	3.02944	0.130974
$536$	0	0
$537$	20.0083i	0.863423i
$538$	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
$543$	− 16.1815i	− 0.694414i
$544$	0	0
$545$	− 3.61859i	− 0.155004i
$546$	0	0
$547$	− 30.8322i	− 1.31829i	−0.752015	−	0.659146i	$-0.770918\pi$
$547$	− 30.8322i	− 1.31829i	0.752015	−	0.659146i	$-0.229082\pi$
$548$	0	0
$549$	7.07401i	0.301911i
$550$	0	0
$551$	−14.6274	−0.623149
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 1.08239i	− 0.0459450i
$556$	0	0
$557$	21.3137	0.903091	0.451545	−	0.892248i	$-0.350873\pi$
$557$	21.3137	0.903091	0.451545	+	0.892248i	$0.350873\pi$
$558$	0	0
$559$	−2.34315	−0.0991045
$560$	0	0
$561$	7.31371	0.308785
$562$	0	0
$563$	−32.9706	−1.38954	−0.694772	−	0.719230i	$-0.744495\pi$
$563$	−32.9706	−1.38954	−0.694772	+	0.719230i	$0.744495\pi$
$564$	0	0
$565$	− 12.9887i	− 0.546439i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.58579	0.108402	0.0542009	−	0.998530i	$-0.482739\pi$
$569$	2.58579	0.108402	0.0542009	+	0.998530i	$0.482739\pi$
$570$	0	0
$571$	− 22.1731i	− 0.927916i	−0.885857	−	0.463958i	$-0.846429\pi$
$571$	− 22.1731i	− 0.927916i	0.885857	−	0.463958i	$-0.153571\pi$
$572$	0	0
$573$	− 13.8854i	− 0.580070i
$574$	0	0
$575$	− 23.0698i	− 0.962077i
$576$	0	0
$577$	− 12.9343i	− 0.538463i	−0.963076	−	0.269231i	$-0.913230\pi$
$577$	− 12.9343i	− 0.538463i	0.963076	−	0.269231i	$-0.0867696\pi$
$578$	0	0
$579$	3.31371	0.137713
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.58673i	0.148547i
$584$	0	0
$585$	−0.242641	−0.0100320
$586$	0	0
$587$	8.97056	0.370255	0.185127	−	0.982715i	$-0.440730\pi$
$587$	8.97056	0.370255	0.185127	+	0.982715i	$0.440730\pi$
$588$	0	0
$589$	9.37258	0.386191
$590$	0	0
$591$	−25.3137	−1.04127
$592$	0	0
$593$	− 23.1242i	− 0.949596i	−0.880095	−	0.474798i	$-0.842521\pi$
$593$	− 23.1242i	− 0.949596i	0.880095	−	0.474798i	$-0.157479\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.65685	0.231520
$598$	0	0
$599$	− 9.55582i	− 0.390440i	−0.980759	−	0.195220i	$-0.937458\pi$
$599$	− 9.55582i	− 0.390440i	0.980759	−	0.195220i	$-0.0625421\pi$
$600$	0	0
$601$	− 44.1061i	− 1.79913i	−0.436791	−	0.899563i	$-0.643885\pi$
$601$	− 44.1061i	− 1.79913i	0.436791	−	0.899563i	$-0.356115\pi$
$602$	0	0
$603$	11.7206i	0.477300i
$604$	0	0
$605$	4.83230i	0.196461i
$606$	0	0
$607$	−43.3137	−1.75805	−0.879025	−	0.476776i	$-0.841805\pi$
$607$	−43.3137	−1.75805	−0.879025	+	0.476776i	$0.841805\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.06147i	0.123854i
$612$	0	0
$613$	−4.24264	−0.171359	−0.0856793	−	0.996323i	$-0.527306\pi$
$613$	−4.24264	−0.171359	−0.0856793	+	0.996323i	$0.527306\pi$
$614$	0	0
$615$	−4.58579	−0.184917
$616$	0	0
$617$	−3.55635	−0.143173	−0.0715866	−	0.997434i	$-0.522806\pi$
$617$	−3.55635	−0.143173	−0.0715866	+	0.997434i	$0.522806\pi$
$618$	0	0
$619$	30.3431	1.21959	0.609797	−	0.792558i	$-0.291251\pi$
$619$	30.3431	1.21959	0.609797	+	0.792558i	$0.291251\pi$
$620$	0	0
$621$	− 5.22625i	− 0.209722i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	16.5563	0.662254
$626$	0	0
$627$	12.2459i	0.489053i
$628$	0	0
$629$	4.77791i	0.190508i
$630$	0	0
$631$	6.12293i	0.243750i	0.992545	+	0.121875i	$0.0388907\pi$
$631$	6.12293i	0.243750i	−0.992545	+	0.121875i	$0.961109\pi$
$632$	0	0
$633$	− 7.39104i	− 0.293767i
$634$	0	0
$635$	−10.3431	−0.410455
$636$	0	0
$637$	0	0
$638$	0	0
$639$	15.6788i	0.620242i
$640$	0	0
$641$	−38.8701	−1.53527	−0.767637	−	0.640884i	$-0.778568\pi$
$641$	−38.8701	−1.53527	−0.767637	+	0.640884i	$0.778568\pi$
$642$	0	0
$643$	−21.6569	−0.854063	−0.427031	−	0.904237i	$-0.640441\pi$
$643$	−21.6569	−0.854063	−0.427031	+	0.904237i	$0.640441\pi$
$644$	0	0
$645$	−5.65685	−0.222738
$646$	0	0
$647$	12.9706	0.509925	0.254963	−	0.966951i	$-0.417937\pi$
$647$	12.9706	0.509925	0.254963	+	0.966951i	$0.417937\pi$
$648$	0	0
$649$	12.2459i	0.480692i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.0711	1.37244	0.686218	−	0.727395i	$-0.259269\pi$
$653$	35.0711	1.37244	0.686218	+	0.727395i	$0.259269\pi$
$654$	0	0
$655$	11.7206i	0.457962i
$656$	0	0
$657$	6.62567i	0.258492i
$658$	0	0
$659$	− 33.5223i	− 1.30584i	−0.757425	−	0.652922i	$-0.773543\pi$
$659$	− 33.5223i	− 1.30584i	0.757425	−	0.652922i	$-0.226457\pi$
$660$	0	0
$661$	33.7624i	1.31321i	0.754237	+	0.656603i	$0.228007\pi$
$661$	33.7624i	1.31321i	−0.754237	+	0.656603i	$0.771993\pi$
$662$	0	0
$663$	1.07107	0.0415968
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.5140i	0.523263i
$668$	0	0
$669$	−19.3137	−0.746711
$670$	0	0
$671$	−15.3137	−0.591179
$672$	0	0
$673$	−7.07107	−0.272570	−0.136285	−	0.990670i	$-0.543516\pi$
$673$	−7.07107	−0.272570	−0.136285	+	0.990670i	$0.543516\pi$
$674$	0	0
$675$	4.41421	0.169903
$676$	0	0
$677$	3.82683i	0.147077i	0.997292	+	0.0735386i	$0.0234292\pi$
$677$	3.82683i	0.147077i	−0.997292	+	0.0735386i	$0.976571\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.31371	−0.126982
$682$	0	0
$683$	− 21.8017i	− 0.834219i	−0.908856	−	0.417109i	$-0.863043\pi$
$683$	− 21.8017i	− 0.834219i	0.908856	−	0.417109i	$-0.136957\pi$
$684$	0	0
$685$	− 10.2668i	− 0.392274i
$686$	0	0
$687$	− 4.72352i	− 0.180213i
$688$	0	0
$689$	0.525265i	0.0200110i
$690$	0	0
$691$	−1.65685	−0.0630297	−0.0315149	−	0.999503i	$-0.510033\pi$
$691$	−1.65685	−0.0630297	−0.0315149	+	0.999503i	$0.510033\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.5140i	0.512614i
$696$	0	0
$697$	20.2426	0.766745
$698$	0	0
$699$	0.242641	0.00917751
$700$	0	0
$701$	6.38478	0.241150	0.120575	−	0.992704i	$-0.461526\pi$
$701$	6.38478	0.241150	0.120575	+	0.992704i	$0.461526\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	7.39104i	0.278363i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−33.8995	−1.27312	−0.636561	−	0.771226i	$-0.719644\pi$
$709$	−33.8995	−1.27312	−0.636561	+	0.771226i	$0.719644\pi$
$710$	0	0
$711$	− 13.5140i	− 0.506814i
$712$	0	0
$713$	− 8.65914i	− 0.324287i
$714$	0	0
$715$	− 0.525265i	− 0.0196438i
$716$	0	0
$717$	0.896683i	0.0334872i
$718$	0	0
$719$	20.2843	0.756476	0.378238	−	0.925708i	$-0.376530\pi$
$719$	20.2843	0.756476	0.378238	+	0.925708i	$0.376530\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 3.11586i	− 0.115880i
$724$	0	0
$725$	−11.4142	−0.423913
$726$	0	0
$727$	12.9706	0.481052	0.240526	−	0.970643i	$-0.422680\pi$
$727$	12.9706	0.481052	0.240526	+	0.970643i	$0.422680\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.9706	0.923570
$732$	0	0
$733$	− 51.1257i	− 1.88837i	−0.329413	−	0.944186i	$-0.606851\pi$
$733$	− 51.1257i	− 1.88837i	0.329413	−	0.944186i	$-0.393149\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.3726	−0.934611
$738$	0	0
$739$	41.8100i	1.53801i	0.639245	+	0.769003i	$0.279247\pi$
$739$	41.8100i	1.53801i	−0.639245	+	0.769003i	$0.720753\pi$
$740$	0	0
$741$	1.79337i	0.0658810i
$742$	0	0
$743$	29.1927i	1.07098i	0.844542	+	0.535489i	$0.179873\pi$
$743$	29.1927i	1.07098i	−0.844542	+	0.535489i	$0.820127\pi$
$744$	0	0
$745$	− 9.92724i	− 0.363706i
$746$	0	0
$747$	6.34315	0.232084
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 28.2960i	− 1.03254i	−0.856427	−	0.516269i	$-0.827321\pi$
$751$	− 28.2960i	− 1.03254i	0.856427	−	0.516269i	$-0.172679\pi$
$752$	0	0
$753$	8.97056	0.326905
$754$	0	0
$755$	4.68629	0.170552
$756$	0	0
$757$	−10.3848	−0.377441	−0.188721	−	0.982031i	$-0.560434\pi$
$757$	−10.3848	−0.377441	−0.188721	+	0.982031i	$0.560434\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	− 12.5629i	− 0.455405i	−0.973731	−	0.227702i	$-0.926879\pi$
$761$	− 12.5629i	− 0.455405i	0.973731	−	0.227702i	$-0.0731213\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.58579	0.0934893
$766$	0	0
$767$	1.79337i	0.0647547i
$768$	0	0
$769$	40.5963i	1.46394i	0.681337	+	0.731970i	$0.261399\pi$
$769$	40.5963i	1.46394i	−0.681337	+	0.731970i	$0.738601\pi$
$770$	0	0
$771$	− 18.9803i	− 0.683560i
$772$	0	0
$773$	− 8.68167i	− 0.312258i	−0.987737	−	0.156129i	$-0.950098\pi$
$773$	− 8.68167i	− 0.312258i	0.987737	−	0.156129i	$-0.0499015\pi$
$774$	0	0
$775$	7.31371	0.262716
$776$	0	0
$777$	0	0
$778$	0	0
$779$	33.8937i	1.21437i
$780$	0	0
$781$	−33.9411	−1.21451
$782$	0	0
$783$	−2.58579	−0.0924085
$784$	0	0
$785$	−11.4142	−0.407391
$786$	0	0
$787$	−23.3137	−0.831044	−0.415522	−	0.909583i	$-0.636401\pi$
$787$	−23.3137	−0.831044	−0.415522	+	0.909583i	$0.636401\pi$
$788$	0	0
$789$	− 26.1313i	− 0.930297i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.24264	−0.0796385
$794$	0	0
$795$	1.26810i	0.0449749i
$796$	0	0
$797$	− 2.48181i	− 0.0879102i	−0.999034	−	0.0439551i	$-0.986004\pi$
$797$	− 2.48181i	− 0.0879102i	0.999034	−	0.0439551i	$-0.0139959\pi$
$798$	0	0
$799$	− 32.6256i	− 1.15421i
$800$	0	0
$801$	− 11.4036i	− 0.402926i
$802$	0	0
$803$	−14.3431	−0.506159
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 25.1802i	− 0.886384i
$808$	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
$811$	−1.65685	−0.0581800	−0.0290900	−	0.999577i	$-0.509261\pi$
$811$	−1.65685	−0.0581800	−0.0290900	+	0.999577i	$0.509261\pi$
$812$	0	0
$813$	−9.65685	−0.338681
$814$	0	0
$815$	4.68629	0.164154
$816$	0	0
$817$	41.8100i	1.46275i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.3137	−0.883455	−0.441727	−	0.897149i	$-0.645634\pi$
$821$	−25.3137	−0.883455	−0.441727	+	0.897149i	$0.645634\pi$
$822$	0	0
$823$	− 34.4190i	− 1.19977i	−0.800086	−	0.599885i	$-0.795213\pi$
$823$	− 34.4190i	− 1.19977i	0.800086	−	0.599885i	$-0.204787\pi$
$824$	0	0
$825$	9.55582i	0.332691i
$826$	0	0
$827$	12.6173i	0.438746i	0.975641	+	0.219373i	$0.0704012\pi$
$827$	12.6173i	0.438746i	−0.975641	+	0.219373i	$0.929599\pi$
$828$	0	0
$829$	21.7473i	0.755315i	0.925945	+	0.377657i	$0.123270\pi$
$829$	21.7473i	0.755315i	−0.925945	+	0.377657i	$0.876730\pi$
$830$	0	0
$831$	−9.31371	−0.323089
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 7.39104i	− 0.255777i
$836$	0	0
$837$	1.65685	0.0572693
$838$	0	0
$839$	29.9411	1.03368	0.516841	−	0.856081i	$-0.327108\pi$
$839$	29.9411	1.03368	0.516841	+	0.856081i	$0.327108\pi$
$840$	0	0
$841$	−22.3137	−0.769438
$842$	0	0
$843$	−22.3848	−0.770973
$844$	0	0
$845$	9.87285i	0.339636i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	7.39104i	0.253361i
$852$	0	0
$853$	− 5.46635i	− 0.187164i	−0.995612	−	0.0935822i	$-0.970168\pi$
$853$	− 5.46635i	− 0.187164i	0.995612	−	0.0935822i	$-0.0298318\pi$
$854$	0	0
$855$	4.32957i	0.148068i
$856$	0	0
$857$	47.6159i	1.62653i	0.581894	+	0.813264i	$0.302312\pi$
$857$	47.6159i	1.62653i	−0.581894	+	0.813264i	$0.697688\pi$
$858$	0	0
$859$	38.6274	1.31795	0.658975	−	0.752165i	$-0.270990\pi$
$859$	38.6274	1.31795	0.658975	+	0.752165i	$0.270990\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 31.7289i	− 1.08007i	−0.841644	−	0.540033i	$-0.818412\pi$
$863$	− 31.7289i	− 1.08007i	0.841644	−	0.540033i	$-0.181588\pi$
$864$	0	0
$865$	7.21320	0.245256
$866$	0	0
$867$	5.58579	0.189703
$868$	0	0
$869$	29.2548	0.992402
$870$	0	0
$871$	−3.71573	−0.125903
$872$	0	0
$873$	− 3.82683i	− 0.129519i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.2426	1.22383	0.611914	−	0.790925i	$-0.290400\pi$
$877$	36.2426	1.22383	0.611914	+	0.790925i	$0.290400\pi$
$878$	0	0
$879$	31.4119i	1.05950i
$880$	0	0
$881$	− 50.7862i	− 1.71103i	−0.517778	−	0.855515i	$-0.673241\pi$
$881$	− 50.7862i	− 1.71103i	0.517778	−	0.855515i	$-0.326759\pi$
$882$	0	0
$883$	16.5754i	0.557808i	0.960319	+	0.278904i	$0.0899711\pi$
$883$	16.5754i	0.557808i	−0.960319	+	0.278904i	$0.910029\pi$
$884$	0	0
$885$	4.32957i	0.145537i
$886$	0	0
$887$	24.6863	0.828885	0.414442	−	0.910076i	$-0.363977\pi$
$887$	24.6863	0.828885	0.414442	+	0.910076i	$0.363977\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.16478i	0.0725230i
$892$	0	0
$893$	54.6274	1.82804
$894$	0	0
$895$	−15.3137	−0.511881
$896$	0	0
$897$	1.65685	0.0553208
$898$	0	0
$899$	−4.28427	−0.142888
$900$	0	0
$901$	− 5.59767i	− 0.186485i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.3848	0.411684
$906$	0	0
$907$	− 1.79337i	− 0.0595477i	−0.999557	−	0.0297739i	$-0.990521\pi$
$907$	− 1.79337i	− 0.0595477i	0.999557	−	0.0297739i	$-0.00947872\pi$
$908$	0	0
$909$	− 16.6298i	− 0.551577i
$910$	0	0
$911$	26.6565i	0.883170i	0.897219	+	0.441585i	$0.145584\pi$
$911$	26.6565i	0.883170i	−0.897219	+	0.441585i	$0.854416\pi$
$912$	0	0
$913$	13.7315i	0.454448i
$914$	0	0
$915$	−5.41421	−0.178988
$916$	0	0
$917$	0	0
$918$	0	0
$919$	29.0389i	0.957904i	0.877841	+	0.478952i	$0.158983\pi$
$919$	29.0389i	0.957904i	−0.877841	+	0.478952i	$0.841017\pi$
$920$	0	0
$921$	3.31371	0.109190
$922$	0	0
$923$	−4.97056	−0.163608
$924$	0	0
$925$	−6.24264	−0.205257
$926$	0	0
$927$	12.0000	0.394132
$928$	0	0
$929$	13.8310i	0.453780i	0.973920	+	0.226890i	$0.0728558\pi$
$929$	13.8310i	0.453780i	−0.973920	+	0.226890i	$0.927144\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.00000	0.130954
$934$	0	0
$935$	5.59767i	0.183063i
$936$	0	0
$937$	− 3.82683i	− 0.125017i	−0.998044	−	0.0625086i	$-0.980090\pi$
$937$	− 3.82683i	− 0.125017i	0.998044	−	0.0625086i	$-0.0199101\pi$
$938$	0	0
$939$	19.8770i	0.648662i
$940$	0	0
$941$	− 11.0322i	− 0.359638i	−0.983700	−	0.179819i	$-0.942449\pi$
$941$	− 11.0322i	− 0.359638i	0.983700	−	0.179819i	$-0.0575512\pi$
$942$	0	0
$943$	31.3137	1.01971
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 38.3771i	− 1.24709i	−0.781788	−	0.623545i	$-0.785692\pi$
$947$	− 38.3771i	− 1.24709i	0.781788	−	0.623545i	$-0.214308\pi$
$948$	0	0
$949$	−2.10051	−0.0681853
$950$	0	0
$951$	−5.31371	−0.172309
$952$	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
$955$	10.6274	0.343895
$956$	0	0
$957$	− 5.59767i	− 0.180947i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.2548	−0.911446
$962$	0	0
$963$	− 3.95815i	− 0.127550i
$964$	0	0
$965$	2.53620i	0.0816433i
$966$	0	0
$967$	49.9439i	1.60609i	0.595920	+	0.803044i	$0.296787\pi$
$967$	49.9439i	1.60609i	−0.595920	+	0.803044i	$0.703213\pi$
$968$	0	0
$969$	− 19.1116i	− 0.613954i
$970$	0	0
$971$	11.0294	0.353951	0.176976	−	0.984215i	$-0.443369\pi$
$971$	11.0294	0.353951	0.176976	+	0.984215i	$0.443369\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.39942i	0.0448172i
$976$	0	0
$977$	−42.1838	−1.34958	−0.674789	−	0.738011i	$-0.735765\pi$
$977$	−42.1838	−1.34958	−0.674789	+	0.738011i	$0.735765\pi$
$978$	0	0
$979$	24.6863	0.788977
$980$	0	0
$981$	−4.72792	−0.150951
$982$	0	0
$983$	23.5980	0.752659	0.376329	−	0.926486i	$-0.377186\pi$
$983$	23.5980	0.752659	0.376329	+	0.926486i	$0.377186\pi$
$984$	0	0
$985$	− 19.3743i	− 0.617316i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	38.6274	1.22828
$990$	0	0
$991$	15.3073i	0.486254i	0.969995	+	0.243127i	$0.0781731\pi$
$991$	15.3073i	0.486254i	−0.969995	+	0.243127i	$0.921827\pi$
$992$	0	0
$993$	25.2346i	0.800795i
$994$	0	0
$995$	4.32957i	0.137257i
$996$	0	0
$997$	− 48.8071i	− 1.54574i	−0.634567	−	0.772868i	$-0.718821\pi$
$997$	− 48.8071i	− 1.54574i	0.634567	−	0.772868i	$-0.281179\pi$
$998$	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.1567.3	yes	4
3.2	odd	2		7056.2.b.t.1567.2		4
4.3	odd	2		2352.2.b.i.1567.3	yes	4
7.2	even	3		2352.2.bl.p.31.3		8
7.3	odd	6		2352.2.bl.s.607.3		8
7.4	even	3		2352.2.bl.p.607.2		8
7.5	odd	6		2352.2.bl.s.31.2		8
7.6	odd	2		2352.2.b.i.1567.2	✓	4
12.11	even	2		7056.2.b.u.1567.2		4
21.20	even	2		7056.2.b.u.1567.3		4
28.3	even	6		2352.2.bl.p.607.3		8
28.11	odd	6		2352.2.bl.s.607.2		8
28.19	even	6		2352.2.bl.p.31.2		8
28.23	odd	6		2352.2.bl.s.31.3		8
28.27	even	2	inner	2352.2.b.j.1567.2	yes	4
84.83	odd	2		7056.2.b.t.1567.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.b.i.1567.2	✓	4	7.6	odd	2
2352.2.b.i.1567.3	yes	4	4.3	odd	2
2352.2.b.j.1567.2	yes	4	28.27	even	2	inner
2352.2.b.j.1567.3	yes	4	1.1	even	1	trivial
2352.2.bl.p.31.2		8	28.19	even	6
2352.2.bl.p.31.3		8	7.2	even	3
2352.2.bl.p.607.2		8	7.4	even	3
2352.2.bl.p.607.3		8	28.3	even	6
2352.2.bl.s.31.2		8	7.5	odd	6
2352.2.bl.s.31.3		8	28.23	odd	6
2352.2.bl.s.607.2		8	28.11	odd	6
2352.2.bl.s.607.3		8	7.3	odd	6
7056.2.b.t.1567.2		4	3.2	odd	2
7056.2.b.t.1567.3		4	84.83	odd	2
7056.2.b.u.1567.2		4	12.11	even	2
7056.2.b.u.1567.3		4	21.20	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.765367i q^{5} +1.00000 q^{9} +2.16478i q^{11} +0.317025i q^{13} +0.765367i q^{15} -3.37849i q^{17} +5.65685 q^{19} -5.22625i q^{23} +4.41421 q^{25} +1.00000 q^{27} -2.58579 q^{29} +1.65685 q^{31} +2.16478i q^{33} -1.41421 q^{37} +0.317025i q^{39} +5.99162i q^{41} +7.39104i q^{43} +0.765367i q^{45} +9.65685 q^{47} -3.37849i q^{51} +1.65685 q^{53} -1.65685 q^{55} +5.65685 q^{57} +5.65685 q^{59} +7.07401i q^{61} -0.242641 q^{65} +11.7206i q^{67} -5.22625i q^{69} +15.6788i q^{71} +6.62567i q^{73} +4.41421 q^{75} -13.5140i q^{79} +1.00000 q^{81} +6.34315 q^{83} +2.58579 q^{85} -2.58579 q^{87} -11.4036i q^{89} +1.65685 q^{93} +4.32957i q^{95} -3.82683i q^{97} +2.16478i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9} + 12 q^{25} + 4 q^{27} - 16 q^{29} - 16 q^{31} + 16 q^{47} - 16 q^{53} + 16 q^{55} + 16 q^{65} + 12 q^{75} + 4 q^{81} + 48 q^{83} + 16 q^{85} - 16 q^{87} - 16 q^{93}+O(q^{100}) \) 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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