LMFDB - Embedded newform 7056.2.h.l.4607.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(4607,7056)]

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, 1, 0]))

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

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

chi := DirichletCharacter("7056.4607");

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.h (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:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4607.1
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	7056.4607
Dual form			7056.2.h.l.4607.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205i q^{5} +O(q^{10})$	$q-2.73205i q^{5} -4.24264 q^{11} +6.31319 q^{13} -0.732051i q^{17} +6.69213i q^{19} +9.14162 q^{23} -2.46410 q^{25} -2.44949i q^{29} +1.79315i q^{31} -3.46410 q^{37} +4.19615i q^{41} +9.46410i q^{43} +3.46410 q^{47} -1.41421i q^{53} +11.5911i q^{55} -10.3923 q^{59} +8.10634 q^{61} -17.2480i q^{65} +12.9282i q^{67} -4.24264 q^{71} -8.10634 q^{73} -8.53590i q^{79} -2.53590 q^{83} -2.00000 q^{85} +9.66025i q^{89} +18.2832 q^{95} -4.52004 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{25} - 48 q^{83} - 16 q^{85}+O(q^{100}) $ 8 * q + 8 * q^25 - 48 * q^83 - 16 * q^85

Display 100 coefficients 10 coefficients

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
$5$	− 2.73205i	− 1.22181i	−0.791704	−	0.610905i	$-0.790806\pi$
$5$	− 2.73205i	− 1.22181i	0.791704	−	0.610905i	$-0.209194\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	6.31319	1.75096	0.875482	−	0.483250i	$-0.160544\pi$
$13$	6.31319	1.75096	0.875482	+	0.483250i	$0.160544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.732051i	− 0.177548i	−0.996052	−	0.0887742i	$-0.971705\pi$
$17$	− 0.732051i	− 0.177548i	0.996052	−	0.0887742i	$-0.0282950\pi$
$18$	0	0
$19$	6.69213i	1.53528i	0.640881	+	0.767640i	$0.278569\pi$
$19$	6.69213i	1.53528i	−0.640881	+	0.767640i	$0.721431\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.14162	1.90616	0.953080	−	0.302719i	$-0.0978944\pi$
$23$	9.14162	1.90616	0.953080	+	0.302719i	$0.0978944\pi$
$24$	0	0
$25$	−2.46410	−0.492820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.44949i	− 0.454859i	−0.973795	−	0.227429i	$-0.926968\pi$
$29$	− 2.44949i	− 0.454859i	0.973795	−	0.227429i	$-0.0730321\pi$
$30$	0	0
$31$	1.79315i	0.322059i	0.986950	+	0.161030i	$0.0514815\pi$
$31$	1.79315i	0.322059i	−0.986950	+	0.161030i	$0.948519\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.46410	−0.569495	−0.284747	−	0.958603i	$-0.591910\pi$
$37$	−3.46410	−0.569495	−0.284747	+	0.958603i	$0.591910\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.19615i	0.655329i	0.944794	+	0.327664i	$0.106262\pi$
$41$	4.19615i	0.655329i	−0.944794	+	0.327664i	$0.893738\pi$
$42$	0	0
$43$	9.46410i	1.44326i	0.692278	+	0.721631i	$0.256607\pi$
$43$	9.46410i	1.44326i	−0.692278	+	0.721631i	$0.743393\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.41421i	− 0.194257i	−0.995272	−	0.0971286i	$-0.969034\pi$
$53$	− 1.41421i	− 0.194257i	0.995272	−	0.0971286i	$-0.0309658\pi$
$54$	0	0
$55$	11.5911i	1.56294i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.3923	−1.35296	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923	−1.35296	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	8.10634	1.03791	0.518955	−	0.854801i	$-0.326321\pi$
$61$	8.10634	1.03791	0.518955	+	0.854801i	$0.326321\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 17.2480i	− 2.13935i
$66$	0	0
$67$	12.9282i	1.57943i	0.613473	+	0.789716i	$0.289772\pi$
$67$	12.9282i	1.57943i	−0.613473	+	0.789716i	$0.710228\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.24264	−0.503509	−0.251754	−	0.967791i	$-0.581008\pi$
$71$	−4.24264	−0.503509	−0.251754	+	0.967791i	$0.581008\pi$
$72$	0	0
$73$	−8.10634	−0.948776	−0.474388	−	0.880316i	$-0.657331\pi$
$73$	−8.10634	−0.948776	−0.474388	+	0.880316i	$0.657331\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 8.53590i	− 0.960364i	−0.877169	−	0.480182i	$-0.840571\pi$
$79$	− 8.53590i	− 0.960364i	0.877169	−	0.480182i	$-0.159429\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.53590	−0.278351	−0.139176	−	0.990268i	$-0.544445\pi$
$83$	−2.53590	−0.278351	−0.139176	+	0.990268i	$0.544445\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.66025i	1.02398i	0.858990	+	0.511992i	$0.171092\pi$
$89$	9.66025i	1.02398i	−0.858990	+	0.511992i	$0.828908\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	18.2832	1.87582
$96$	0	0
$97$	−4.52004	−0.458941	−0.229470	−	0.973316i	$-0.573699\pi$
$97$	−4.52004	−0.458941	−0.229470	+	0.973316i	$0.573699\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.1962i	1.81058i	0.424789	+	0.905292i	$0.360348\pi$
$101$	18.1962i	1.81058i	−0.424789	+	0.905292i	$0.639652\pi$
$102$	0	0
$103$	3.10583i	0.306026i	0.988224	+	0.153013i	$0.0488977\pi$
$103$	3.10583i	0.306026i	−0.988224	+	0.153013i	$0.951102\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.4543	1.01066	0.505328	−	0.862928i	$-0.331372\pi$
$107$	10.4543	1.01066	0.505328	+	0.862928i	$0.331372\pi$
$108$	0	0
$109$	19.8564	1.90190	0.950949	−	0.309346i	$-0.100110\pi$
$109$	19.8564	1.90190	0.950949	+	0.309346i	$0.100110\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13.2827i	1.24953i	0.780811	+	0.624767i	$0.214806\pi$
$113$	13.2827i	1.24953i	−0.780811	+	0.624767i	$0.785194\pi$
$114$	0	0
$115$	− 24.9754i	− 2.32897i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.07180	−0.443125	−0.221562	−	0.975146i	$-0.571116\pi$
$131$	−5.07180	−0.443125	−0.221562	+	0.975146i	$0.571116\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.03579i	− 0.515672i	−0.966189	−	0.257836i	$-0.916990\pi$
$137$	− 6.03579i	− 0.515672i	0.966189	−	0.257836i	$-0.0830095\pi$
$138$	0	0
$139$	13.3843i	1.13524i	0.823291	+	0.567619i	$0.192135\pi$
$139$	13.3843i	1.13524i	−0.823291	+	0.567619i	$0.807865\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−26.7846	−2.23984
$144$	0	0
$145$	−6.69213	−0.555751
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.2827i	1.08816i	0.839032	+	0.544082i	$0.183122\pi$
$149$	13.2827i	1.08816i	−0.839032	+	0.544082i	$0.816878\pi$
$150$	0	0
$151$	4.39230i	0.357441i	0.983900	+	0.178720i	$0.0571957\pi$
$151$	4.39230i	0.357441i	−0.983900	+	0.178720i	$0.942804\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.89898	0.393496
$156$	0	0
$157$	6.59059	0.525987	0.262993	−	0.964798i	$-0.415290\pi$
$157$	6.59059	0.525987	0.262993	+	0.964798i	$0.415290\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.8564i	1.55527i	0.628714	+	0.777637i	$0.283582\pi$
$163$	19.8564i	1.55527i	−0.628714	+	0.777637i	$0.716418\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.3923	0.804181	0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923	0.804181	0.402090	+	0.915600i	$0.368284\pi$
$168$	0	0
$169$	26.8564	2.06588
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 20.7321i	− 1.57623i	−0.615529	−	0.788114i	$-0.711058\pi$
$173$	− 20.7321i	− 1.57623i	0.615529	−	0.788114i	$-0.288942\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.9396	1.41561	0.707805	−	0.706408i	$-0.249685\pi$
$179$	18.9396	1.41561	0.707805	+	0.706408i	$0.249685\pi$
$180$	0	0
$181$	7.07107	0.525588	0.262794	−	0.964852i	$-0.415356\pi$
$181$	7.07107	0.525588	0.262794	+	0.964852i	$0.415356\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.46410i	0.695815i
$186$	0	0
$187$	3.10583i	0.227121i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.8386	1.72490	0.862449	−	0.506144i	$-0.168930\pi$
$191$	23.8386	1.72490	0.862449	+	0.506144i	$0.168930\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.2827i	− 0.946355i	−0.880967	−	0.473177i	$-0.843107\pi$
$197$	− 13.2827i	− 0.946355i	0.880967	−	0.473177i	$-0.156893\pi$
$198$	0	0
$199$	− 11.1106i	− 0.787612i	−0.919194	−	0.393806i	$-0.871158\pi$
$199$	− 11.1106i	− 0.787612i	0.919194	−	0.393806i	$-0.128842\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	11.4641	0.800688
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 28.3923i	− 1.96394i
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	25.8564	1.76339
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4.62158i	− 0.310881i
$222$	0	0
$223$	− 21.8695i	− 1.46449i	−0.681040	−	0.732246i	$-0.738472\pi$
$223$	− 21.8695i	− 1.46449i	0.681040	−	0.732246i	$-0.261528\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.32051	−0.353135	−0.176567	−	0.984289i	$-0.556499\pi$
$227$	−5.32051	−0.353135	−0.176567	+	0.984289i	$0.556499\pi$
$228$	0	0
$229$	20.4553	1.35173	0.675863	−	0.737027i	$-0.263771\pi$
$229$	20.4553	1.35173	0.675863	+	0.737027i	$0.263771\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 19.4201i	− 1.27225i	−0.771586	−	0.636125i	$-0.780536\pi$
$233$	− 19.4201i	− 1.27225i	0.771586	−	0.636125i	$-0.219464\pi$
$234$	0	0
$235$	− 9.46410i	− 0.617370i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	17.6269	1.14019	0.570095	−	0.821579i	$-0.306906\pi$
$239$	17.6269	1.14019	0.570095	+	0.821579i	$0.306906\pi$
$240$	0	0
$241$	−23.5612	−1.51771	−0.758854	−	0.651261i	$-0.774240\pi$
$241$	−23.5612	−1.51771	−0.758854	+	0.651261i	$0.774240\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	42.2487i	2.68822i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.8564	1.25333	0.626663	−	0.779291i	$-0.284420\pi$
$251$	19.8564	1.25333	0.626663	+	0.779291i	$0.284420\pi$
$252$	0	0
$253$	−38.7846	−2.43837
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.0526i	− 0.876575i	−0.898835	−	0.438287i	$-0.855585\pi$
$257$	− 14.0526i	− 0.876575i	0.898835	−	0.438287i	$-0.144415\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.3142	−1.00598	−0.502989	−	0.864293i	$-0.667766\pi$
$263$	−16.3142	−1.00598	−0.502989	+	0.864293i	$0.667766\pi$
$264$	0	0
$265$	−3.86370	−0.237345
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.3397i	0.630425i	0.949021	+	0.315213i	$0.102076\pi$
$269$	10.3397i	0.630425i	−0.949021	+	0.315213i	$0.897924\pi$
$270$	0	0
$271$	6.69213i	0.406518i	0.979125	+	0.203259i	$0.0651533\pi$
$271$	6.69213i	0.406518i	−0.979125	+	0.203259i	$0.934847\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.4543	0.630418
$276$	0	0
$277$	−18.7846	−1.12866	−0.564329	−	0.825550i	$-0.690865\pi$
$277$	−18.7846	−1.12866	−0.564329	+	0.825550i	$0.690865\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 29.2180i	− 1.74300i	−0.490395	−	0.871500i	$-0.663148\pi$
$281$	− 29.2180i	− 1.74300i	0.490395	−	0.871500i	$-0.336852\pi$
$282$	0	0
$283$	0.480473i	0.0285612i	0.999898	+	0.0142806i	$0.00454581\pi$
$283$	0.480473i	0.0285612i	−0.999898	+	0.0142806i	$0.995454\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.4641	0.968477
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.0526i	1.17148i	0.810498	+	0.585741i	$0.199197\pi$
$293$	20.0526i	1.17148i	−0.810498	+	0.585741i	$0.800803\pi$
$294$	0	0
$295$	28.3923i	1.65306i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	57.7128	3.33762
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 22.1469i	− 1.26813i
$306$	0	0
$307$	4.41851i	0.252177i	0.992019	+	0.126089i	$0.0402424\pi$
$307$	4.41851i	0.252177i	−0.992019	+	0.126089i	$0.959758\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.2487	−1.37502	−0.687509	−	0.726176i	$-0.741296\pi$
$311$	−24.2487	−1.37502	−0.687509	+	0.726176i	$0.741296\pi$
$312$	0	0
$313$	−6.31319	−0.356843	−0.178421	−	0.983954i	$-0.557099\pi$
$313$	−6.31319	−0.356843	−0.178421	+	0.983954i	$0.557099\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 21.7680i	− 1.22261i	−0.791394	−	0.611307i	$-0.790644\pi$
$317$	− 21.7680i	− 1.22261i	0.791394	−	0.611307i	$-0.209356\pi$
$318$	0	0
$319$	10.3923i	0.581857i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.89898	0.272587
$324$	0	0
$325$	−15.5563	−0.862911
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	32.7846i	1.80201i	0.433814	+	0.901003i	$0.357168\pi$
$331$	32.7846i	1.80201i	−0.433814	+	0.901003i	$0.642832\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	35.3205	1.92977
$336$	0	0
$337$	1.60770	0.0875767	0.0437884	−	0.999041i	$-0.486057\pi$
$337$	1.60770	0.0875767	0.0437884	+	0.999041i	$0.486057\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 7.60770i	− 0.411980i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	23.8386	1.27972	0.639860	−	0.768491i	$-0.278992\pi$
$347$	23.8386	1.27972	0.639860	+	0.768491i	$0.278992\pi$
$348$	0	0
$349$	−6.79367	−0.363657	−0.181828	−	0.983330i	$-0.558202\pi$
$349$	−6.79367	−0.363657	−0.181828	+	0.983330i	$0.558202\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 26.9808i	− 1.43604i	−0.696022	−	0.718021i	$-0.745048\pi$
$353$	− 26.9808i	− 1.43604i	0.696022	−	0.718021i	$-0.254952\pi$
$354$	0	0
$355$	11.5911i	0.615192i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.4152	−0.602474	−0.301237	−	0.953549i	$-0.597399\pi$
$359$	−11.4152	−0.602474	−0.301237	+	0.953549i	$0.597399\pi$
$360$	0	0
$361$	−25.7846	−1.35708
$362$	0	0
$363$	0	0
$364$	0	0
$365$	22.1469i	1.15922i
$366$	0	0
$367$	− 26.7685i	− 1.39731i	−0.715461	−	0.698653i	$-0.753783\pi$
$367$	− 26.7685i	− 1.39731i	0.715461	−	0.698653i	$-0.246217\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	30.7846	1.59397	0.796983	−	0.604001i	$-0.206428\pi$
$373$	30.7846	1.59397	0.796983	+	0.604001i	$0.206428\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.4641i	− 0.796442i
$378$	0	0
$379$	− 13.8564i	− 0.711756i	−0.934532	−	0.355878i	$-0.884182\pi$
$379$	− 13.8564i	− 0.711756i	0.934532	−	0.355878i	$-0.115818\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−17.0718	−0.872328	−0.436164	−	0.899867i	$-0.643663\pi$
$383$	−17.0718	−0.872328	−0.436164	+	0.899867i	$0.643663\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.8338i	0.802803i	0.915902	+	0.401402i	$0.131477\pi$
$389$	15.8338i	0.802803i	−0.915902	+	0.401402i	$0.868523\pi$
$390$	0	0
$391$	− 6.69213i	− 0.338436i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−23.3205	−1.17338
$396$	0	0
$397$	−11.4896	−0.576645	−0.288323	−	0.957533i	$-0.593098\pi$
$397$	−11.4896	−0.576645	−0.288323	+	0.957533i	$0.593098\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.72311i	− 0.235861i	−0.993022	−	0.117931i	$-0.962374\pi$
$401$	− 4.72311i	− 0.235861i	0.993022	−	0.117931i	$-0.0376260\pi$
$402$	0	0
$403$	11.3205i	0.563915i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.6969	0.728500
$408$	0	0
$409$	−9.41902	−0.465741	−0.232870	−	0.972508i	$-0.574812\pi$
$409$	−9.41902	−0.465741	−0.232870	+	0.972508i	$0.574812\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6.92820i	0.340092i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.9282	1.21782	0.608911	−	0.793238i	$-0.291607\pi$
$419$	24.9282	1.21782	0.608911	+	0.793238i	$0.291607\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.80385i	0.0874995i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	27.4249	1.32101	0.660505	−	0.750822i	$-0.270342\pi$
$431$	27.4249	1.32101	0.660505	+	0.750822i	$0.270342\pi$
$432$	0	0
$433$	31.7690	1.52672	0.763361	−	0.645972i	$-0.223548\pi$
$433$	31.7690	1.52672	0.763361	+	0.645972i	$0.223548\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	61.1769i	2.92649i
$438$	0	0
$439$	6.21166i	0.296466i	0.988952	+	0.148233i	$0.0473586\pi$
$439$	6.21166i	0.296466i	−0.988952	+	0.148233i	$0.952641\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.3238	1.53575	0.767876	−	0.640599i	$-0.221314\pi$
$443$	32.3238	1.53575	0.767876	+	0.640599i	$0.221314\pi$
$444$	0	0
$445$	26.3923	1.25112
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 9.89949i	− 0.467186i	−0.972334	−	0.233593i	$-0.924952\pi$
$449$	− 9.89949i	− 0.467186i	0.972334	−	0.233593i	$-0.0750483\pi$
$450$	0	0
$451$	− 17.8028i	− 0.838300i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.78461	−0.223815	−0.111907	−	0.993719i	$-0.535696\pi$
$457$	−4.78461	−0.223815	−0.111907	+	0.993719i	$0.535696\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.1962i	1.03378i	0.856052	+	0.516889i	$0.172910\pi$
$461$	22.1962i	1.03378i	−0.856052	+	0.516889i	$0.827090\pi$
$462$	0	0
$463$	− 18.0000i	− 0.836531i	−0.908325	−	0.418265i	$-0.862638\pi$
$463$	− 18.0000i	− 0.836531i	0.908325	−	0.418265i	$-0.137362\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.9282	−1.15354	−0.576770	−	0.816907i	$-0.695687\pi$
$467$	−24.9282	−1.15354	−0.576770	+	0.816907i	$0.695687\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 40.1528i	− 1.84623i
$474$	0	0
$475$	− 16.4901i	− 0.756617i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.85641	−0.358968	−0.179484	−	0.983761i	$-0.557443\pi$
$479$	−7.85641	−0.358968	−0.179484	+	0.983761i	$0.557443\pi$
$480$	0	0
$481$	−21.8695	−0.997165
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.3490i	0.560739i
$486$	0	0
$487$	7.60770i	0.344738i	0.985032	+	0.172369i	$0.0551421\pi$
$487$	7.60770i	0.344738i	−0.985032	+	0.172369i	$0.944858\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−17.6269	−0.795491	−0.397745	−	0.917496i	$-0.630207\pi$
$491$	−17.6269	−0.795491	−0.397745	+	0.917496i	$0.630207\pi$
$492$	0	0
$493$	−1.79315	−0.0807595
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 26.5359i	− 1.18791i	−0.804498	−	0.593955i	$-0.797566\pi$
$499$	− 26.5359i	− 1.18791i	0.804498	−	0.593955i	$-0.202434\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−33.4641	−1.49209	−0.746045	−	0.665895i	$-0.768050\pi$
$503$	−33.4641	−1.49209	−0.746045	+	0.665895i	$0.768050\pi$
$504$	0	0
$505$	49.7128	2.21219
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 23.8038i	− 1.05509i	−0.849528	−	0.527543i	$-0.823113\pi$
$509$	− 23.8038i	− 1.05509i	0.849528	−	0.527543i	$-0.176887\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.48528	0.373906
$516$	0	0
$517$	−14.6969	−0.646371
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.5885i	0.639132i	0.947564	+	0.319566i	$0.103537\pi$
$521$	14.5885i	0.639132i	−0.947564	+	0.319566i	$0.896463\pi$
$522$	0	0
$523$	− 30.3548i	− 1.32732i	−0.748033	−	0.663662i	$-0.769001\pi$
$523$	− 30.3548i	− 1.32732i	0.748033	−	0.663662i	$-0.230999\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.31268	0.0571811
$528$	0	0
$529$	60.5692	2.63344
$530$	0	0
$531$	0	0
$532$	0	0
$533$	26.4911i	1.14746i
$534$	0	0
$535$	− 28.5617i	− 1.23483i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 54.2487i	− 2.32376i
$546$	0	0
$547$	5.32051i	0.227488i	0.993510	+	0.113744i	$0.0362844\pi$
$547$	5.32051i	0.227488i	−0.993510	+	0.113744i	$0.963716\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	16.3923	0.698336
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.07107i	− 0.299611i	−0.988716	−	0.149805i	$-0.952135\pi$
$557$	− 7.07107i	− 0.299611i	0.988716	−	0.149805i	$-0.0478647\pi$
$558$	0	0
$559$	59.7487i	2.52710i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−41.3205	−1.74145	−0.870726	−	0.491769i	$-0.836351\pi$
$563$	−41.3205	−1.74145	−0.870726	+	0.491769i	$0.836351\pi$
$564$	0	0
$565$	36.2891	1.52669
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.4591i	0.773846i	0.922112	+	0.386923i	$0.126462\pi$
$569$	18.4591i	0.773846i	−0.922112	+	0.386923i	$0.873538\pi$
$570$	0	0
$571$	17.3205i	0.724841i	0.932015	+	0.362420i	$0.118050\pi$
$571$	17.3205i	0.724841i	−0.932015	+	0.362420i	$0.881950\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−22.5259	−0.939394
$576$	0	0
$577$	17.4238	0.725364	0.362682	−	0.931913i	$-0.381861\pi$
$577$	17.4238	0.725364	0.362682	+	0.931913i	$0.381861\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.5359	−0.847607	−0.423804	−	0.905754i	$-0.639305\pi$
$587$	−20.5359	−0.847607	−0.423804	+	0.905754i	$0.639305\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	0	0
$593$	37.1244i	1.52451i	0.647274	+	0.762257i	$0.275909\pi$
$593$	37.1244i	1.52451i	−0.647274	+	0.762257i	$0.724091\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.0112	1.26708	0.633541	−	0.773709i	$-0.281601\pi$
$599$	31.0112	1.26708	0.633541	+	0.773709i	$0.281601\pi$
$600$	0	0
$601$	−6.31319	−0.257521	−0.128760	−	0.991676i	$-0.541100\pi$
$601$	−6.31319	−0.257521	−0.128760	+	0.991676i	$0.541100\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 19.1244i	− 0.777516i
$606$	0	0
$607$	16.9706i	0.688814i	0.938820	+	0.344407i	$0.111920\pi$
$607$	16.9706i	0.688814i	−0.938820	+	0.344407i	$0.888080\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	21.8695	0.884747
$612$	0	0
$613$	−19.6077	−0.791947	−0.395974	−	0.918262i	$-0.629593\pi$
$613$	−19.6077	−0.791947	−0.395974	+	0.918262i	$0.629593\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 31.4916i	− 1.26781i	−0.773413	−	0.633903i	$-0.781452\pi$
$617$	− 31.4916i	− 1.26781i	0.773413	−	0.633903i	$-0.218548\pi$
$618$	0	0
$619$	35.2538i	1.41697i	0.705726	+	0.708485i	$0.250621\pi$
$619$	35.2538i	1.41697i	−0.705726	+	0.708485i	$0.749379\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.53590i	0.101113i
$630$	0	0
$631$	− 7.85641i	− 0.312759i	−0.987697	−	0.156379i	$-0.950018\pi$
$631$	− 7.85641i	− 0.312759i	0.987697	−	0.156379i	$-0.0499822\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16.3923	−0.650509
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	43.9149i	1.73454i	0.497841	+	0.867268i	$0.334126\pi$
$641$	43.9149i	1.73454i	−0.497841	+	0.867268i	$0.665874\pi$
$642$	0	0
$643$	− 3.10583i	− 0.122482i	−0.998123	−	0.0612410i	$-0.980494\pi$
$643$	− 3.10583i	− 0.122482i	0.998123	−	0.0612410i	$-0.0195058\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.4641	0.607957	0.303978	−	0.952679i	$-0.401685\pi$
$647$	15.4641	0.607957	0.303978	+	0.952679i	$0.401685\pi$
$648$	0	0
$649$	44.0908	1.73072
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 22.0454i	− 0.862703i	−0.902184	−	0.431352i	$-0.858037\pi$
$653$	− 22.0454i	− 0.862703i	0.902184	−	0.431352i	$-0.141963\pi$
$654$	0	0
$655$	13.8564i	0.541415i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.7995	0.966052	0.483026	−	0.875606i	$-0.339538\pi$
$659$	24.7995	0.966052	0.483026	+	0.875606i	$0.339538\pi$
$660$	0	0
$661$	−11.6926	−0.454791	−0.227396	−	0.973802i	$-0.573021\pi$
$661$	−11.6926	−0.454791	−0.227396	+	0.973802i	$0.573021\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 22.3923i	− 0.867034i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−34.3923	−1.32770
$672$	0	0
$673$	−7.85641	−0.302842	−0.151421	−	0.988469i	$-0.548385\pi$
$673$	−7.85641	−0.302842	−0.151421	+	0.988469i	$0.548385\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.80385i	− 0.0693275i	−0.999399	−	0.0346637i	$-0.988964\pi$
$677$	− 1.80385i	− 0.0693275i	0.999399	−	0.0346637i	$-0.0110360\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−22.5259	−0.861929	−0.430964	−	0.902369i	$-0.641827\pi$
$683$	−22.5259	−0.861929	−0.430964	+	0.902369i	$0.641827\pi$
$684$	0	0
$685$	−16.4901	−0.630054
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 8.92820i	− 0.340137i
$690$	0	0
$691$	31.6675i	1.20469i	0.798236	+	0.602344i	$0.205767\pi$
$691$	31.6675i	1.20469i	−0.798236	+	0.602344i	$0.794233\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	36.5665	1.38705
$696$	0	0
$697$	3.07180	0.116353
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29.2180i	1.10355i	0.833993	+	0.551775i	$0.186049\pi$
$701$	29.2180i	1.10355i	−0.833993	+	0.551775i	$0.813951\pi$
$702$	0	0
$703$	− 23.1822i	− 0.874334i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−11.0718	−0.415810	−0.207905	−	0.978149i	$-0.566664\pi$
$709$	−11.0718	−0.415810	−0.207905	+	0.978149i	$0.566664\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.3923i	0.613897i
$714$	0	0
$715$	73.1769i	2.73666i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.92820	0.258378	0.129189	−	0.991620i	$-0.458763\pi$
$719$	6.92820	0.258378	0.129189	+	0.991620i	$0.458763\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.03579i	0.224164i
$726$	0	0
$727$	44.2196i	1.64001i	0.572354	+	0.820006i	$0.306030\pi$
$727$	44.2196i	1.64001i	−0.572354	+	0.820006i	$0.693970\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.92820	0.256249
$732$	0	0
$733$	24.0416	0.887998	0.443999	−	0.896027i	$-0.353559\pi$
$733$	24.0416	0.887998	0.443999	+	0.896027i	$0.353559\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 54.8497i	− 2.02042i
$738$	0	0
$739$	− 33.7128i	− 1.24015i	−0.784544	−	0.620073i	$-0.787103\pi$
$739$	− 33.7128i	− 1.24015i	0.784544	−	0.620073i	$-0.212897\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.5259	−0.826394	−0.413197	−	0.910642i	$-0.635588\pi$
$743$	−22.5259	−0.826394	−0.413197	+	0.910642i	$0.635588\pi$
$744$	0	0
$745$	36.2891	1.32953
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 6.92820i	− 0.252814i	−0.991978	−	0.126407i	$-0.959656\pi$
$751$	− 6.92820i	− 0.252814i	0.991978	−	0.126407i	$-0.0403445\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	2.78461	0.101208	0.0506042	−	0.998719i	$-0.483885\pi$
$757$	2.78461	0.101208	0.0506042	+	0.998719i	$0.483885\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	50.0526i	1.81440i	0.420695	+	0.907202i	$0.361786\pi$
$761$	50.0526i	1.81440i	−0.420695	+	0.907202i	$0.638214\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−65.6086	−2.36899
$768$	0	0
$769$	29.4954	1.06363	0.531816	−	0.846860i	$-0.321510\pi$
$769$	29.4954	1.06363	0.531816	+	0.846860i	$0.321510\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15.2679i	0.549150i	0.961566	+	0.274575i	$0.0885372\pi$
$773$	15.2679i	0.549150i	−0.961566	+	0.274575i	$0.911463\pi$
$774$	0	0
$775$	− 4.41851i	− 0.158717i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−28.0812	−1.00611
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 18.0058i	− 0.642656i
$786$	0	0
$787$	24.4949i	0.873149i	0.899668	+	0.436574i	$0.143808\pi$
$787$	24.4949i	0.873149i	−0.899668	+	0.436574i	$0.856192\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	51.1769	1.81735
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.7321i	1.15943i	0.814820	+	0.579714i	$0.196836\pi$
$797$	32.7321i	1.15943i	−0.814820	+	0.579714i	$0.803164\pi$
$798$	0	0
$799$	− 2.53590i	− 0.0897136i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	34.3923	1.21368
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.62587i	0.268111i	0.990974	+	0.134056i	$0.0428001\pi$
$809$	7.62587i	0.268111i	−0.990974	+	0.134056i	$0.957200\pi$
$810$	0	0
$811$	34.2929i	1.20419i	0.798426	+	0.602093i	$0.205666\pi$
$811$	34.2929i	1.20419i	−0.798426	+	0.602093i	$0.794334\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	54.2487	1.90025
$816$	0	0
$817$	−63.3350	−2.21581
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.68784i	0.128706i	0.997927	+	0.0643532i	$0.0204984\pi$
$821$	3.68784i	0.128706i	−0.997927	+	0.0643532i	$0.979502\pi$
$822$	0	0
$823$	− 8.53590i	− 0.297543i	−0.988872	−	0.148771i	$-0.952468\pi$
$823$	− 8.53590i	− 0.297543i	0.988872	−	0.148771i	$-0.0475318\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.3533	0.533886	0.266943	−	0.963712i	$-0.413986\pi$
$827$	15.3533	0.533886	0.266943	+	0.963712i	$0.413986\pi$
$828$	0	0
$829$	−36.4649	−1.26648	−0.633240	−	0.773956i	$-0.718275\pi$
$829$	−36.4649	−1.26648	−0.633240	+	0.773956i	$0.718275\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 28.3923i	− 0.982556i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	35.0718	1.21081	0.605406	−	0.795916i	$-0.293011\pi$
$839$	35.0718	1.21081	0.605406	+	0.795916i	$0.293011\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 73.3731i	− 2.52411i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−31.6675	−1.08555
$852$	0	0
$853$	7.90327	0.270603	0.135301	−	0.990804i	$-0.456800\pi$
$853$	7.90327	0.270603	0.135301	+	0.990804i	$0.456800\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.94744i	− 0.339798i	−0.985461	−	0.169899i	$-0.945656\pi$
$857$	− 9.94744i	− 0.339798i	0.985461	−	0.169899i	$-0.0543442\pi$
$858$	0	0
$859$	34.7733i	1.18645i	0.805036	+	0.593225i	$0.202146\pi$
$859$	34.7733i	1.18645i	−0.805036	+	0.593225i	$0.797854\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.1026	0.343895	0.171948	−	0.985106i	$-0.444994\pi$
$863$	10.1026	0.343895	0.171948	+	0.985106i	$0.444994\pi$
$864$	0	0
$865$	−56.6410	−1.92585
$866$	0	0
$867$	0	0
$868$	0	0
$869$	36.2147i	1.22850i
$870$	0	0
$871$	81.6182i	2.76553i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−44.1051	−1.48932	−0.744662	−	0.667442i	$-0.767389\pi$
$877$	−44.1051	−1.48932	−0.744662	+	0.667442i	$0.767389\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35.7654i	1.20497i	0.798132	+	0.602483i	$0.205822\pi$
$881$	35.7654i	1.20497i	−0.798132	+	0.602483i	$0.794178\pi$
$882$	0	0
$883$	− 25.1769i	− 0.847271i	−0.905833	−	0.423635i	$-0.860754\pi$
$883$	− 25.1769i	− 0.847271i	0.905833	−	0.423635i	$-0.139246\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.1769	−0.643898	−0.321949	−	0.946757i	$-0.604338\pi$
$887$	−19.1769	−0.643898	−0.321949	+	0.946757i	$0.604338\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	23.1822i	0.775763i
$894$	0	0
$895$	− 51.7439i	− 1.72961i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.39230	0.146492
$900$	0	0
$901$	−1.03528	−0.0344901
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 19.3185i	− 0.642169i
$906$	0	0
$907$	35.3205i	1.17280i	0.810022	+	0.586399i	$0.199455\pi$
$907$	35.3205i	1.17280i	−0.810022	+	0.586399i	$0.800545\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.14162	0.302875	0.151438	−	0.988467i	$-0.451610\pi$
$911$	9.14162	0.302875	0.151438	+	0.988467i	$0.451610\pi$
$912$	0	0
$913$	10.7589	0.356068
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 49.1769i	− 1.62220i	−0.584910	−	0.811098i	$-0.698870\pi$
$919$	− 49.1769i	− 1.62220i	0.584910	−	0.811098i	$-0.301130\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−26.7846	−0.881626
$924$	0	0
$925$	8.53590	0.280659
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 10.4449i	− 0.342685i	−0.985212	−	0.171342i	$-0.945190\pi$
$929$	− 10.4449i	− 0.342685i	0.985212	−	0.171342i	$-0.0548104\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.48528	0.277498
$936$	0	0
$937$	52.1228	1.70278	0.851389	−	0.524534i	$-0.175761\pi$
$937$	52.1228	1.70278	0.851389	+	0.524534i	$0.175761\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 43.9090i	− 1.43139i	−0.698412	−	0.715696i	$-0.746110\pi$
$941$	− 43.9090i	− 1.43139i	0.698412	−	0.715696i	$-0.253890\pi$
$942$	0	0
$943$	38.3596i	1.24916i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.24264	0.137867	0.0689336	−	0.997621i	$-0.478040\pi$
$947$	4.24264	0.137867	0.0689336	+	0.997621i	$0.478040\pi$
$948$	0	0
$949$	−51.1769	−1.66127
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 9.34469i	− 0.302704i	−0.988480	−	0.151352i	$-0.951637\pi$
$953$	− 9.34469i	− 0.302704i	0.988480	−	0.151352i	$-0.0483627\pi$
$954$	0	0
$955$	− 65.1282i	− 2.10750i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	27.7846	0.896278
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 10.9282i	− 0.351791i
$966$	0	0
$967$	24.9282i	0.801637i	0.916157	+	0.400818i	$0.131274\pi$
$967$	24.9282i	0.801637i	−0.916157	+	0.400818i	$0.868726\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−47.3205	−1.51859	−0.759294	−	0.650748i	$-0.774455\pi$
$971$	−47.3205	−1.51859	−0.759294	+	0.650748i	$0.774455\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.41044i	0.109110i	0.998511	+	0.0545548i	$0.0173739\pi$
$977$	3.41044i	0.109110i	−0.998511	+	0.0545548i	$0.982626\pi$
$978$	0	0
$979$	− 40.9850i	− 1.30989i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−0.679492	−0.0216724	−0.0108362	−	0.999941i	$-0.503449\pi$
$983$	−0.679492	−0.0216724	−0.0108362	+	0.999941i	$0.503449\pi$
$984$	0	0
$985$	−36.2891	−1.15627
$986$	0	0
$987$	0	0
$988$	0	0
$989$	86.5172i	2.75109i
$990$	0	0
$991$	− 32.1051i	− 1.01985i	−0.860218	−	0.509926i	$-0.829673\pi$
$991$	− 32.1051i	− 1.01985i	0.860218	−	0.509926i	$-0.170327\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−30.3548	−0.962313
$996$	0	0
$997$	23.5612	0.746189	0.373095	−	0.927793i	$-0.378297\pi$
$997$	23.5612	0.746189	0.373095	+	0.927793i	$0.378297\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.h.l.4607.1	✓	8
3.2	odd	2		7056.2.h.m.4607.8	yes	8
4.3	odd	2		7056.2.h.m.4607.2	yes	8
7.6	odd	2		7056.2.h.m.4607.7	yes	8
12.11	even	2	inner	7056.2.h.l.4607.7	yes	8
21.20	even	2	inner	7056.2.h.l.4607.2	yes	8
28.27	even	2	inner	7056.2.h.l.4607.8	yes	8
84.83	odd	2		7056.2.h.m.4607.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7056.2.h.l.4607.1	✓	8	1.1	even	1	trivial
7056.2.h.l.4607.2	yes	8	21.20	even	2	inner
7056.2.h.l.4607.7	yes	8	12.11	even	2	inner
7056.2.h.l.4607.8	yes	8	28.27	even	2	inner
7056.2.h.m.4607.1	yes	8	84.83	odd	2
7056.2.h.m.4607.2	yes	8	4.3	odd	2
7056.2.h.m.4607.7	yes	8	7.6	odd	2
7056.2.h.m.4607.8	yes	8	3.2	odd	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:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.73205i q^{5} +O(q^{10})\)	\(q-2.73205i q^{5} -4.24264 q^{11} +6.31319 q^{13} -0.732051i q^{17} +6.69213i q^{19} +9.14162 q^{23} -2.46410 q^{25} -2.44949i q^{29} +1.79315i q^{31} -3.46410 q^{37} +4.19615i q^{41} +9.46410i q^{43} +3.46410 q^{47} -1.41421i q^{53} +11.5911i q^{55} -10.3923 q^{59} +8.10634 q^{61} -17.2480i q^{65} +12.9282i q^{67} -4.24264 q^{71} -8.10634 q^{73} -8.53590i q^{79} -2.53590 q^{83} -2.00000 q^{85} +9.66025i q^{89} +18.2832 q^{95} -4.52004 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{25} - 48 q^{83} - 16 q^{85}+O(q^{100}) \) 8 * q + 8 * q^25 - 48 * q^83 - 16 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more