LMFDB - Embedded newform 7056.2.b.u.1567.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(1567,7056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7056, 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("7056.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7056.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:	$56.3424436662$
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:	no (minimal twist has level 2352)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.2
Root			$-0.765367i$ of defining polynomial
Character	$\chi$	$=$	7056.1567
Dual form			7056.2.b.u.1567.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.765367i q^{5} +2.16478i q^{11} +0.317025i q^{13} +3.37849i q^{17} -5.65685 q^{19} -5.22625i q^{23} +4.41421 q^{25} +2.58579 q^{29} -1.65685 q^{31} -1.41421 q^{37} -5.99162i q^{41} -7.39104i q^{43} +9.65685 q^{47} -1.65685 q^{53} +1.65685 q^{55} +5.65685 q^{59} +7.07401i q^{61} +0.242641 q^{65} -11.7206i q^{67} +15.6788i q^{71} +6.62567i q^{73} +13.5140i q^{79} +6.34315 q^{83} +2.58579 q^{85} +11.4036i q^{89} +4.32957i q^{95} -3.82683i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{25} + 16 q^{29} + 16 q^{31} + 16 q^{47} + 16 q^{53} - 16 q^{55} - 16 q^{65} + 48 q^{83} + 16 q^{85}+O(q^{100}) $ 4 * q + 12 * q^25 + 16 * q^29 + 16 * q^31 + 16 * q^47 + 16 * q^53 - 16 * q^55 - 16 * q^65 + 48 * q^83 + 16 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times$.

$n$	$785$	$1765$	$4609$	$6175$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 0.765367i	− 0.342282i	−0.985247	−	0.171141i	$-0.945255\pi$
$5$	− 0.765367i	− 0.342282i	0.985247	−	0.171141i	$-0.0547454\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	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	0
$16$	0	0
$17$	3.37849i	0.819405i	0.912219	+	0.409702i	$0.134367\pi$
$17$	3.37849i	0.819405i	−0.912219	+	0.409702i	$0.865633\pi$
$18$	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
$23$	− 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
$27$	0	0
$28$	0	0
$29$	2.58579	0.480168	0.240084	−	0.970752i	$-0.422825\pi$
$29$	2.58579	0.480168	0.240084	+	0.970752i	$0.422825\pi$
$30$	0	0
$31$	−1.65685	−0.297580	−0.148790	−	0.988869i	$-0.547538\pi$
$31$	−1.65685	−0.297580	−0.148790	+	0.988869i	$0.547538\pi$
$32$	0	0
$33$	0	0
$34$	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	0
$40$	0	0
$41$	− 5.99162i	− 0.935734i	−0.883799	−	0.467867i	$-0.845023\pi$
$41$	− 5.99162i	− 0.935734i	0.883799	−	0.467867i	$-0.154977\pi$
$42$	0	0
$43$	− 7.39104i	− 1.12712i	−0.826074	−	0.563561i	$-0.809431\pi$
$43$	− 7.39104i	− 1.12712i	0.826074	−	0.563561i	$-0.190569\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	0	0
$53$	−1.65685	−0.227586	−0.113793	−	0.993504i	$-0.536300\pi$
$53$	−1.65685	−0.227586	−0.113793	+	0.993504i	$0.536300\pi$
$54$	0	0
$55$	1.65685	0.223410
$56$	0	0
$57$	0	0
$58$	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	$-0.745994\pi$
$67$	− 11.7206i	− 1.43190i	0.698152	−	0.715950i	$-0.254006\pi$
$68$	0	0
$69$	0	0
$70$	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
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.5140i	1.52044i	0.649665	+	0.760220i	$0.274909\pi$
$79$	13.5140i	1.52044i	−0.649665	+	0.760220i	$0.725091\pi$
$80$	0	0
$81$	0	0
$82$	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
$87$	0	0
$88$	0	0
$89$	11.4036i	1.20878i	0.796690	+	0.604389i	$0.206583\pi$
$89$	11.4036i	1.20878i	−0.796690	+	0.604389i	$0.793417\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	16.6298i	1.65473i	0.561665	+	0.827365i	$0.310161\pi$
$101$	16.6298i	1.65473i	−0.561665	+	0.827365i	$0.689839\pi$
$102$	0	0
$103$	−12.0000	−1.18240	−0.591198	−	0.806527i	$-0.701345\pi$
$103$	−12.0000	−1.18240	−0.591198	+	0.806527i	$0.701345\pi$
$104$	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
$111$	0	0
$112$	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
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.31371	0.573973
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 7.20533i	− 0.644464i
$126$	0	0
$127$	− 13.5140i	− 1.19917i	−0.800311	−	0.599586i	$-0.795332\pi$
$127$	− 13.5140i	− 1.19917i	0.800311	−	0.599586i	$-0.204668\pi$
$128$	0	0
$129$	0	0
$130$	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	0
$136$	0	0
$137$	13.4142	1.14605	0.573027	−	0.819537i	$-0.305769\pi$
$137$	13.4142	1.14605	0.573027	+	0.819537i	$0.305769\pi$
$138$	0	0
$139$	−17.6569	−1.49763	−0.748817	−	0.662776i	$-0.769378\pi$
$139$	−17.6569	−1.49763	−0.748817	+	0.662776i	$0.769378\pi$
$140$	0	0
$141$	0	0
$142$	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	$-0.321706\pi$
$149$	12.9706	1.06259	0.531295	+	0.847187i	$0.321706\pi$
$150$	0	0
$151$	6.12293i	0.498277i	0.968468	+	0.249139i	$0.0801475\pi$
$151$	6.12293i	0.498277i	−0.968468	+	0.249139i	$0.919852\pi$
$152$	0	0
$153$	0	0
$154$	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
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.12293i	0.479585i	0.970824	+	0.239793i	$0.0770795\pi$
$163$	6.12293i	0.479585i	−0.970824	+	0.239793i	$0.922921\pi$
$164$	0	0
$165$	0	0
$166$	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
$171$	0	0
$172$	0	0
$173$	9.42450i	0.716532i	0.933620	+	0.358266i	$0.116632\pi$
$173$	9.42450i	0.716532i	−0.933620	+	0.358266i	$0.883368\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	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
$183$	0	0
$184$	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	0
$196$	0	0
$197$	25.3137	1.80353	0.901764	−	0.432230i	$-0.142273\pi$
$197$	25.3137	1.80353	0.901764	+	0.432230i	$0.142273\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−4.58579	−0.320285
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 12.2459i	− 0.847065i
$210$	0	0
$211$	7.39104i	0.508820i	0.967096	+	0.254410i	$0.0818813\pi$
$211$	7.39104i	0.508820i	−0.967096	+	0.254410i	$0.918119\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.07107	−0.0720478
$222$	0	0
$223$	19.3137	1.29334	0.646671	−	0.762769i	$-0.276161\pi$
$223$	19.3137	1.29334	0.646671	+	0.762769i	$0.276161\pi$
$224$	0	0
$225$	0	0
$226$	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	$-0.502530\pi$
$233$	−0.242641	−0.0158959	−0.00794796	+	0.999968i	$0.502530\pi$
$234$	0	0
$235$	− 7.39104i	− 0.482138i
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.79337i	− 0.114109i
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	18.9803i	1.18396i	0.805953	+	0.591980i	$0.201654\pi$
$257$	18.9803i	1.18396i	−0.805953	+	0.591980i	$0.798346\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	0	0
$269$	25.1802i	1.53526i	0.640892	+	0.767631i	$0.278565\pi$
$269$	25.1802i	1.53526i	−0.640892	+	0.767631i	$0.721435\pi$
$270$	0	0
$271$	9.65685	0.586612	0.293306	−	0.956019i	$-0.405245\pi$
$271$	9.65685	0.586612	0.293306	+	0.956019i	$0.405245\pi$
$272$	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
$279$	0	0
$280$	0	0
$281$	22.3848	1.33536	0.667682	−	0.744447i	$-0.267287\pi$
$281$	22.3848	1.33536	0.667682	+	0.744447i	$0.267287\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	5.58579	0.328576
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 31.4119i	− 1.83510i	−0.397617	−	0.917552i	$-0.630163\pi$
$293$	− 31.4119i	− 1.83510i	0.397617	−	0.917552i	$-0.369837\pi$
$294$	0	0
$295$	− 4.32957i	− 0.252077i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.65685	0.0958184
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.41421	0.310017
$306$	0	0
$307$	−3.31371	−0.189123	−0.0945617	−	0.995519i	$-0.530145\pi$
$307$	−3.31371	−0.189123	−0.0945617	+	0.995519i	$0.530145\pi$
$308$	0	0
$309$	0	0
$310$	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	$-0.452323\pi$
$317$	5.31371	0.298448	0.149224	+	0.988803i	$0.452323\pi$
$318$	0	0
$319$	5.59767i	0.313409i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 19.1116i	− 1.06340i
$324$	0	0
$325$	1.39942i	0.0776257i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 25.2346i	− 1.38702i	−0.720448	−	0.693509i	$-0.756064\pi$
$331$	− 25.2346i	− 1.38702i	0.720448	−	0.693509i	$-0.243936\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	− 3.58673i	− 0.194232i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	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	0
$352$	0	0
$353$	14.9134i	0.793760i	0.917871	+	0.396880i	$0.129907\pi$
$353$	14.9134i	0.793760i	−0.917871	+	0.396880i	$0.870093\pi$
$354$	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
$363$	0	0
$364$	0	0
$365$	5.07107	0.265432
$366$	0	0
$367$	21.6569	1.13048	0.565239	−	0.824927i	$-0.308784\pi$
$367$	21.6569	1.13048	0.565239	+	0.824927i	$0.308784\pi$
$368$	0	0
$369$	0	0
$370$	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
$375$	0	0
$376$	0	0
$377$	0.819760i	0.0422198i
$378$	0	0
$379$	33.1509i	1.70285i	0.524479	+	0.851423i	$0.324260\pi$
$379$	33.1509i	1.70285i	−0.524479	+	0.851423i	$0.675740\pi$
$380$	0	0
$381$	0	0
$382$	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
$387$	0	0
$388$	0	0
$389$	−20.0416	−1.01615	−0.508076	−	0.861313i	$-0.669643\pi$
$389$	−20.0416	−1.01615	−0.508076	+	0.861313i	$0.669643\pi$
$390$	0	0
$391$	17.6569	0.892946
$392$	0	0
$393$	0	0
$394$	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	$-0.306541\pi$
$401$	22.8701	1.14208	0.571038	+	0.820924i	$0.306541\pi$
$402$	0	0
$403$	− 0.525265i	− 0.0261653i
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 4.85483i	− 0.238314i
$416$	0	0
$417$	0	0
$418$	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
$423$	0	0
$424$	0	0
$425$	14.9134i	0.723406i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	0	0
$437$	29.5641i	1.41424i
$438$	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
$443$	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
$447$	0	0
$448$	0	0
$449$	29.6569	1.39959	0.699797	−	0.714342i	$-0.253274\pi$
$449$	29.6569	1.39959	0.699797	+	0.714342i	$0.253274\pi$
$450$	0	0
$451$	12.9706	0.610760
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	− 24.2066i	− 1.12741i	−0.825975	−	0.563706i	$-0.809375\pi$
$461$	− 24.2066i	− 1.12741i	0.825975	−	0.563706i	$-0.190625\pi$
$462$	0	0
$463$	− 1.79337i	− 0.0833448i	−0.999131	−	0.0416724i	$-0.986731\pi$
$463$	− 1.79337i	− 0.0833448i	0.999131	−	0.0416724i	$-0.0132686\pi$
$464$	0	0
$465$	0	0
$466$	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
$471$	0	0
$472$	0	0
$473$	16.0000	0.735681
$474$	0	0
$475$	−24.9706	−1.14573
$476$	0	0
$477$	0	0
$478$	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	$0.299757\pi$
$487$	35.6871i	1.61714i	−0.588403	+	0.808568i	$0.700243\pi$
$488$	0	0
$489$	0	0
$490$	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
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 17.3183i	− 0.775272i	−0.921812	−	0.387636i	$-0.873292\pi$
$499$	− 17.3183i	− 0.775272i	0.921812	−	0.387636i	$-0.126708\pi$
$500$	0	0
$501$	0	0
$502$	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
$507$	0	0
$508$	0	0
$509$	− 10.5069i	− 0.465710i	−0.972511	−	0.232855i	$-0.925193\pi$
$509$	− 10.5069i	− 0.465710i	0.972511	−	0.232855i	$-0.0748068\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.18440i	0.404713i
$516$	0	0
$517$	20.9050i	0.919401i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.66205i	− 0.0728157i	−0.999337	−	0.0364079i	$-0.988408\pi$
$521$	− 1.66205i	− 0.0728157i	0.999337	−	0.0364079i	$-0.0115915\pi$
$522$	0	0
$523$	10.6274	0.464704	0.232352	−	0.972632i	$-0.425358\pi$
$523$	10.6274	0.464704	0.232352	+	0.972632i	$0.425358\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5.59767i	− 0.243838i
$528$	0	0
$529$	−4.31371	−0.187553
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.89949	0.0822763
$534$	0	0
$535$	−3.02944	−0.130974
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	0	0
$545$	3.61859i	0.155004i
$546$	0	0
$547$	30.8322i	1.31829i	0.752015	+	0.659146i	$0.229082\pi$
$547$	30.8322i	1.31829i	−0.752015	+	0.659146i	$0.770918\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−14.6274	−0.623149
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.3137	−0.903091	−0.451545	−	0.892248i	$-0.649127\pi$
$557$	−21.3137	−0.903091	−0.451545	+	0.892248i	$0.649127\pi$
$558$	0	0
$559$	2.34315	0.0991045
$560$	0	0
$561$	0	0
$562$	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	$-0.517261\pi$
$569$	−2.58579	−0.108402	−0.0542009	+	0.998530i	$0.517261\pi$
$570$	0	0
$571$	22.1731i	0.927916i	0.885857	+	0.463958i	$0.153571\pi$
$571$	22.1731i	0.927916i	−0.885857	+	0.463958i	$0.846429\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 3.58673i	− 0.148547i
$584$	0	0
$585$	0	0
$586$	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
$591$	0	0
$592$	0	0
$593$	23.1242i	0.949596i	0.880095	+	0.474798i	$0.157479\pi$
$593$	23.1242i	0.949596i	−0.880095	+	0.474798i	$0.842521\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	− 4.83230i	− 0.196461i
$606$	0	0
$607$	43.3137	1.75805	0.879025	−	0.476776i	$-0.158195\pi$
$607$	43.3137	1.75805	0.879025	+	0.476776i	$0.158195\pi$
$608$	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
$615$	0	0
$616$	0	0
$617$	3.55635	0.143173	0.0715866	−	0.997434i	$-0.477194\pi$
$617$	3.55635	0.143173	0.0715866	+	0.997434i	$0.477194\pi$
$618$	0	0
$619$	−30.3431	−1.21959	−0.609797	−	0.792558i	$-0.708749\pi$
$619$	−30.3431	−1.21959	−0.609797	+	0.792558i	$0.708749\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	16.5563	0.662254
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 4.77791i	− 0.190508i
$630$	0	0
$631$	− 6.12293i	− 0.243750i	−0.992545	−	0.121875i	$-0.961109\pi$
$631$	− 6.12293i	− 0.243750i	0.992545	−	0.121875i	$-0.0388907\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10.3431	−0.410455
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	38.8701	1.53527	0.767637	−	0.640884i	$-0.221432\pi$
$641$	38.8701	1.53527	0.767637	+	0.640884i	$0.221432\pi$
$642$	0	0
$643$	21.6569	0.854063	0.427031	−	0.904237i	$-0.359559\pi$
$643$	21.6569	0.854063	0.427031	+	0.904237i	$0.359559\pi$
$644$	0	0
$645$	0	0
$646$	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	$-0.740731\pi$
$653$	−35.0711	−1.37244	−0.686218	+	0.727395i	$0.740731\pi$
$654$	0	0
$655$	− 11.7206i	− 0.457962i
$656$	0	0
$657$	0	0
$658$	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
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 13.5140i	− 0.523263i
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	− 3.82683i	− 0.147077i	−0.997292	−	0.0735386i	$-0.976571\pi$
$677$	− 3.82683i	− 0.147077i	0.997292	−	0.0735386i	$-0.0234292\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	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
$687$	0	0
$688$	0	0
$689$	− 0.525265i	− 0.0200110i
$690$	0	0
$691$	1.65685	0.0630297	0.0315149	−	0.999503i	$-0.489967\pi$
$691$	1.65685	0.0630297	0.0315149	+	0.999503i	$0.489967\pi$
$692$	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	0
$700$	0	0
$701$	−6.38478	−0.241150	−0.120575	−	0.992704i	$-0.538474\pi$
$701$	−6.38478	−0.241150	−0.120575	+	0.992704i	$0.538474\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	8.65914i	0.324287i
$714$	0	0
$715$	0.525265i	0.0196438i
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	11.4142	0.423913
$726$	0	0
$727$	−12.9706	−0.481052	−0.240526	−	0.970643i	$-0.577320\pi$
$727$	−12.9706	−0.481052	−0.240526	+	0.970643i	$0.577320\pi$
$728$	0	0
$729$	0	0
$730$	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	$-0.720753\pi$
$739$	− 41.8100i	− 1.53801i	0.639245	−	0.769003i	$-0.279247\pi$
$740$	0	0
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	28.2960i	1.03254i	0.856427	+	0.516269i	$0.172679\pi$
$751$	28.2960i	1.03254i	−0.856427	+	0.516269i	$0.827321\pi$
$752$	0	0
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	12.5629i	0.455405i	0.973731	+	0.227702i	$0.0731213\pi$
$761$	12.5629i	0.455405i	−0.973731	+	0.227702i	$0.926879\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	8.68167i	0.312258i	0.987737	+	0.156129i	$0.0499015\pi$
$773$	8.68167i	0.312258i	−0.987737	+	0.156129i	$0.950098\pi$
$774$	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
$783$	0	0
$784$	0	0
$785$	11.4142	0.407391
$786$	0	0
$787$	23.3137	0.831044	0.415522	−	0.909583i	$-0.363599\pi$
$787$	23.3137	0.831044	0.415522	+	0.909583i	$0.363599\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.24264	−0.0796385
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.48181i	0.0879102i	0.999034	+	0.0439551i	$0.0139959\pi$
$797$	2.48181i	0.0879102i	−0.999034	+	0.0439551i	$0.986004\pi$
$798$	0	0
$799$	32.6256i	1.15421i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−14.3431	−0.506159
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	1.65685	0.0581800	0.0290900	−	0.999577i	$-0.490739\pi$
$811$	1.65685	0.0581800	0.0290900	+	0.999577i	$0.490739\pi$
$812$	0	0
$813$	0	0
$814$	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	$-0.354366\pi$
$821$	25.3137	0.883455	0.441727	+	0.897149i	$0.354366\pi$
$822$	0	0
$823$	34.4190i	1.19977i	0.800086	+	0.599885i	$0.204787\pi$
$823$	34.4190i	1.19977i	−0.800086	+	0.599885i	$0.795213\pi$
$824$	0	0
$825$	0	0
$826$	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
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.39104i	0.255777i
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	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
$855$	0	0
$856$	0	0
$857$	− 47.6159i	− 1.62653i	−0.581894	−	0.813264i	$-0.697688\pi$
$857$	− 47.6159i	− 1.62653i	0.581894	−	0.813264i	$-0.302312\pi$
$858$	0	0
$859$	−38.6274	−1.31795	−0.658975	−	0.752165i	$-0.729010\pi$
$859$	−38.6274	−1.31795	−0.658975	+	0.752165i	$0.729010\pi$
$860$	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
$867$	0	0
$868$	0	0
$869$	−29.2548	−0.992402
$870$	0	0
$871$	3.71573	0.125903
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	50.7862i	1.71103i	0.517778	+	0.855515i	$0.326759\pi$
$881$	50.7862i	1.71103i	−0.517778	+	0.855515i	$0.673241\pi$
$882$	0	0
$883$	− 16.5754i	− 0.557808i	−0.960319	−	0.278904i	$-0.910029\pi$
$883$	− 16.5754i	− 0.557808i	0.960319	−	0.278904i	$-0.0899711\pi$
$884$	0	0
$885$	0	0
$886$	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
$891$	0	0
$892$	0	0
$893$	−54.6274	−1.82804
$894$	0	0
$895$	15.3137	0.511881
$896$	0	0
$897$	0	0
$898$	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	$0.00947872\pi$
$907$	1.79337i	0.0595477i	−0.999557	+	0.0297739i	$0.990521\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 29.0389i	− 0.957904i	−0.877841	−	0.478952i	$-0.841017\pi$
$919$	− 29.0389i	− 0.957904i	0.877841	−	0.478952i	$-0.158983\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.97056	−0.163608
$924$	0	0
$925$	−6.24264	−0.205257
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 13.8310i	− 0.453780i	−0.973920	−	0.226890i	$-0.927144\pi$
$929$	− 13.8310i	− 0.453780i	0.973920	−	0.226890i	$-0.0728558\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	11.0322i	0.359638i	0.983700	+	0.179819i	$0.0575512\pi$
$941$	11.0322i	0.359638i	−0.983700	+	0.179819i	$0.942449\pi$
$942$	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
$951$	0	0
$952$	0	0
$953$	−38.0000	−1.23094	−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000	−1.23094	−0.615470	+	0.788160i	$0.711034\pi$
$954$	0	0
$955$	−10.6274	−0.343895
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28.2548	−0.911446
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 2.53620i	− 0.0816433i
$966$	0	0
$967$	− 49.9439i	− 1.60609i	−0.595920	−	0.803044i	$-0.703213\pi$
$967$	− 49.9439i	− 1.60609i	0.595920	−	0.803044i	$-0.296787\pi$
$968$	0	0
$969$	0	0
$970$	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
$975$	0	0
$976$	0	0
$977$	42.1838	1.34958	0.674789	−	0.738011i	$-0.264235\pi$
$977$	42.1838	1.34958	0.674789	+	0.738011i	$0.264235\pi$
$978$	0	0
$979$	−24.6863	−0.788977
$980$	0	0
$981$	0	0
$982$	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	$-0.921827\pi$
$991$	− 15.3073i	− 0.486254i	0.969995	−	0.243127i	$-0.0781731\pi$
$992$	0	0
$993$	0	0
$994$	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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.b.u.1567.2		4
3.2	odd	2		2352.2.b.i.1567.3	yes	4
4.3	odd	2		7056.2.b.t.1567.2		4
7.6	odd	2		7056.2.b.t.1567.3		4
12.11	even	2		2352.2.b.j.1567.3	yes	4
21.2	odd	6		2352.2.bl.s.31.3		8
21.5	even	6		2352.2.bl.p.31.2		8
21.11	odd	6		2352.2.bl.s.607.2		8
21.17	even	6		2352.2.bl.p.607.3		8
21.20	even	2		2352.2.b.j.1567.2	yes	4
28.27	even	2	inner	7056.2.b.u.1567.3		4
84.11	even	6		2352.2.bl.p.607.2		8
84.23	even	6		2352.2.bl.p.31.3		8
84.47	odd	6		2352.2.bl.s.31.2		8
84.59	odd	6		2352.2.bl.s.607.3		8
84.83	odd	2		2352.2.b.i.1567.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2352.2.b.i.1567.2	✓	4	84.83	odd	2
2352.2.b.i.1567.3	yes	4	3.2	odd	2
2352.2.b.j.1567.2	yes	4	21.20	even	2
2352.2.b.j.1567.3	yes	4	12.11	even	2
2352.2.bl.p.31.2		8	21.5	even	6
2352.2.bl.p.31.3		8	84.23	even	6
2352.2.bl.p.607.2		8	84.11	even	6
2352.2.bl.p.607.3		8	21.17	even	6
2352.2.bl.s.31.2		8	84.47	odd	6
2352.2.bl.s.31.3		8	21.2	odd	6
2352.2.bl.s.607.2		8	21.11	odd	6
2352.2.bl.s.607.3		8	84.59	odd	6
7056.2.b.t.1567.2		4	4.3	odd	2
7056.2.b.t.1567.3		4	7.6	odd	2
7056.2.b.u.1567.2		4	1.1	even	1	trivial
7056.2.b.u.1567.3		4	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.b (of order \(2\), degree \(1\), minimal)

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

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