LMFDB - Embedded newform 7056.2.h.l.4607.6

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			4607.6
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	7056.4607
Dual form			7056.2.h.l.4607.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051i q^{5} +O(q^{10})$	$q+0.732051i q^{5} +4.24264 q^{11} +3.48477 q^{13} +2.73205i q^{17} -1.79315i q^{19} +0.656339 q^{23} +4.46410 q^{25} -2.44949i q^{29} -6.69213i q^{31} +3.46410 q^{37} -6.19615i q^{41} +2.53590i q^{43} -3.46410 q^{47} +1.41421i q^{53} +3.10583i q^{55} +10.3923 q^{59} -3.20736 q^{61} +2.55103i q^{65} -0.928203i q^{67} +4.24264 q^{71} +3.20736 q^{73} -15.4641i q^{79} -9.46410 q^{83} -2.00000 q^{85} -7.66025i q^{89} +1.31268 q^{95} -10.1769 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.732051i	0.327383i	0.986512	+	0.163692i	$0.0523402\pi$
$5$	0.732051i	0.327383i	−0.986512	+	0.163692i	$0.947660\pi$
$6$	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	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	3.48477	0.966500	0.483250	−	0.875482i	$-0.339456\pi$
$13$	3.48477	0.966500	0.483250	+	0.875482i	$0.339456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.73205i	0.662620i	0.943522	+	0.331310i	$0.107491\pi$
$17$	2.73205i	0.662620i	−0.943522	+	0.331310i	$0.892509\pi$
$18$	0	0
$19$	− 1.79315i	− 0.411377i	−0.978618	−	0.205689i	$-0.934057\pi$
$19$	− 1.79315i	− 0.411377i	0.978618	−	0.205689i	$-0.0659434\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.656339	0.136856	0.0684280	−	0.997656i	$-0.478202\pi$
$23$	0.656339	0.136856	0.0684280	+	0.997656i	$0.478202\pi$
$24$	0	0
$25$	4.46410	0.892820
$26$	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
$31$	− 6.69213i	− 1.20194i	−0.799271	−	0.600971i	$-0.794781\pi$
$31$	− 6.69213i	− 1.20194i	0.799271	−	0.600971i	$-0.205219\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.46410	0.569495	0.284747	−	0.958603i	$-0.408090\pi$
$37$	3.46410	0.569495	0.284747	+	0.958603i	$0.408090\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.19615i	− 0.967676i	−0.875157	−	0.483838i	$-0.839242\pi$
$41$	− 6.19615i	− 0.967676i	0.875157	−	0.483838i	$-0.160758\pi$
$42$	0	0
$43$	2.53590i	0.386721i	0.981128	+	0.193360i	$0.0619387\pi$
$43$	2.53590i	0.386721i	−0.981128	+	0.193360i	$0.938061\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.41421i	0.194257i	0.995272	+	0.0971286i	$0.0309658\pi$
$53$	1.41421i	0.194257i	−0.995272	+	0.0971286i	$0.969034\pi$
$54$	0	0
$55$	3.10583i	0.418790i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	0	0
$61$	−3.20736	−0.410661	−0.205330	−	0.978693i	$-0.565827\pi$
$61$	−3.20736	−0.410661	−0.205330	+	0.978693i	$0.565827\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.55103i	0.316416i
$66$	0	0
$67$	− 0.928203i	− 0.113398i	−0.998391	−	0.0566990i	$-0.981942\pi$
$67$	− 0.928203i	− 0.113398i	0.998391	−	0.0566990i	$-0.0180575\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.24264	0.503509	0.251754	−	0.967791i	$-0.418992\pi$
$71$	4.24264	0.503509	0.251754	+	0.967791i	$0.418992\pi$
$72$	0	0
$73$	3.20736	0.375394	0.187697	−	0.982227i	$-0.439898\pi$
$73$	3.20736	0.375394	0.187697	+	0.982227i	$0.439898\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 15.4641i	− 1.73985i	−0.493186	−	0.869924i	$-0.664168\pi$
$79$	− 15.4641i	− 1.73985i	0.493186	−	0.869924i	$-0.335832\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.46410	−1.03882	−0.519410	−	0.854525i	$-0.673848\pi$
$83$	−9.46410	−1.03882	−0.519410	+	0.854525i	$0.673848\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 7.66025i	− 0.811985i	−0.913876	−	0.405993i	$-0.866926\pi$
$89$	− 7.66025i	− 0.811985i	0.913876	−	0.405993i	$-0.133074\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.31268	0.134678
$96$	0	0
$97$	−10.1769	−1.03331	−0.516654	−	0.856195i	$-0.672823\pi$
$97$	−10.1769	−1.03331	−0.516654	+	0.856195i	$0.672823\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.80385i	0.776512i	0.921552	+	0.388256i	$0.126922\pi$
$101$	7.80385i	0.776512i	−0.921552	+	0.388256i	$0.873078\pi$
$102$	0	0
$103$	11.5911i	1.14211i	0.820913	+	0.571053i	$0.193465\pi$
$103$	11.5911i	1.14211i	−0.820913	+	0.571053i	$0.806535\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.9396	1.83096	0.915479	−	0.402365i	$-0.131812\pi$
$107$	18.9396	1.83096	0.915479	+	0.402365i	$0.131812\pi$
$108$	0	0
$109$	−7.85641	−0.752507	−0.376254	−	0.926517i	$-0.622788\pi$
$109$	−7.85641	−0.752507	−0.376254	+	0.926517i	$0.622788\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.1112i	1.51561i	0.652481	+	0.757805i	$0.273728\pi$
$113$	16.1112i	1.51561i	−0.652481	+	0.757805i	$0.726272\pi$
$114$	0	0
$115$	0.480473i	0.0448044i
$116$	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
$131$	−18.9282	−1.65376	−0.826882	−	0.562375i	$-0.809888\pi$
$131$	−18.9282	−1.65376	−0.826882	+	0.562375i	$0.809888\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9348i	0.934221i	0.884199	+	0.467110i	$0.154705\pi$
$137$	10.9348i	0.934221i	−0.884199	+	0.467110i	$0.845295\pi$
$138$	0	0
$139$	− 3.58630i	− 0.304186i	−0.988366	−	0.152093i	$-0.951399\pi$
$139$	− 3.58630i	− 0.304186i	0.988366	−	0.152093i	$-0.0486014\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	14.7846	1.23635
$144$	0	0
$145$	1.79315	0.148913
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.1112i	1.31988i	0.751320	+	0.659939i	$0.229418\pi$
$149$	16.1112i	1.31988i	−0.751320	+	0.659939i	$0.770582\pi$
$150$	0	0
$151$	− 16.3923i	− 1.33399i	−0.745064	−	0.666993i	$-0.767581\pi$
$151$	− 16.3923i	− 1.33399i	0.745064	−	0.666993i	$-0.232419\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.89898	0.393496
$156$	0	0
$157$	17.9043	1.42892	0.714459	−	0.699677i	$-0.246673\pi$
$157$	17.9043	1.42892	0.714459	+	0.699677i	$0.246673\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 7.85641i	− 0.615361i	−0.951490	−	0.307681i	$-0.900447\pi$
$163$	− 7.85641i	− 0.615361i	0.951490	−	0.307681i	$-0.0995528\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	0	0
$169$	−0.856406	−0.0658774
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 17.2679i	− 1.31286i	−0.754388	−	0.656429i	$-0.772066\pi$
$173$	− 17.2679i	− 1.31286i	0.754388	−	0.656429i	$-0.227934\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.4543	0.781391	0.390695	−	0.920520i	$-0.372235\pi$
$179$	10.4543	0.781391	0.390695	+	0.920520i	$0.372235\pi$
$180$	0	0
$181$	−7.07107	−0.525588	−0.262794	−	0.964852i	$-0.584644\pi$
$181$	−7.07107	−0.525588	−0.262794	+	0.964852i	$0.584644\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.53590i	0.186443i
$186$	0	0
$187$	11.5911i	0.847626i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.3533	1.11092	0.555462	−	0.831542i	$-0.312541\pi$
$191$	15.3533	1.11092	0.555462	+	0.831542i	$0.312541\pi$
$192$	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
$197$	− 16.1112i	− 1.14787i	−0.818900	−	0.573936i	$-0.805416\pi$
$197$	− 16.1112i	− 1.14787i	0.818900	−	0.573936i	$-0.194584\pi$
$198$	0	0
$199$	− 28.0812i	− 1.99062i	−0.0967191	−	0.995312i	$-0.530835\pi$
$199$	− 28.0812i	− 1.99062i	0.0967191	−	0.995312i	$-0.469165\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.53590	0.316801
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 7.60770i	− 0.526235i
$210$	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
$215$	−1.85641	−0.126606
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.52056i	0.640422i
$222$	0	0
$223$	12.0716i	0.808373i	0.914677	+	0.404187i	$0.132445\pi$
$223$	12.0716i	0.808373i	−0.914677	+	0.404187i	$0.867555\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	29.3205	1.94607	0.973035	−	0.230657i	$-0.0740874\pi$
$227$	29.3205	1.94607	0.973035	+	0.230657i	$0.0740874\pi$
$228$	0	0
$229$	−10.6574	−0.704259	−0.352129	−	0.935951i	$-0.614542\pi$
$229$	−10.6574	−0.704259	−0.352129	+	0.935951i	$0.614542\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.5211i	0.951307i	0.879633	+	0.475654i	$0.157788\pi$
$233$	14.5211i	0.951307i	−0.879633	+	0.475654i	$0.842212\pi$
$234$	0	0
$235$	− 2.53590i	− 0.165424i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.82894	−0.506412	−0.253206	−	0.967412i	$-0.581485\pi$
$239$	−7.82894	−0.506412	−0.253206	+	0.967412i	$0.581485\pi$
$240$	0	0
$241$	−0.933740	−0.0601475	−0.0300737	−	0.999548i	$-0.509574\pi$
$241$	−0.933740	−0.0601475	−0.0300737	+	0.999548i	$0.509574\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.24871i	− 0.397596i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.85641	−0.495892	−0.247946	−	0.968774i	$-0.579756\pi$
$251$	−7.85641	−0.495892	−0.247946	+	0.968774i	$0.579756\pi$
$252$	0	0
$253$	2.78461	0.175067
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.0526i	1.50036i	0.661235	+	0.750179i	$0.270033\pi$
$257$	24.0526i	1.50036i	−0.661235	+	0.750179i	$0.729967\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.1122	1.61015	0.805073	−	0.593176i	$-0.202126\pi$
$263$	26.1122	1.61015	0.805073	+	0.593176i	$0.202126\pi$
$264$	0	0
$265$	−1.03528	−0.0635965
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.6603i	1.68648i	0.537541	+	0.843238i	$0.319353\pi$
$269$	27.6603i	1.68648i	−0.537541	+	0.843238i	$0.680647\pi$
$270$	0	0
$271$	− 1.79315i	− 0.108926i	−0.998516	−	0.0544631i	$-0.982655\pi$
$271$	− 1.79315i	− 0.108926i	0.998516	−	0.0544631i	$-0.0173447\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.9396	1.14210
$276$	0	0
$277$	22.7846	1.36899	0.684497	−	0.729015i	$-0.260022\pi$
$277$	22.7846	1.36899	0.684497	+	0.729015i	$0.260022\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.72311i	0.281757i	0.990027	+	0.140879i	$0.0449927\pi$
$281$	4.72311i	0.281757i	−0.990027	+	0.140879i	$0.955007\pi$
$282$	0	0
$283$	− 24.9754i	− 1.48463i	−0.670050	−	0.742316i	$-0.733727\pi$
$283$	− 24.9754i	− 1.48463i	0.670050	−	0.742316i	$-0.266273\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	9.53590	0.560935
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 18.0526i	− 1.05464i	−0.849666	−	0.527321i	$-0.823197\pi$
$293$	− 18.0526i	− 1.05464i	0.849666	−	0.527321i	$-0.176803\pi$
$294$	0	0
$295$	7.60770i	0.442937i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.28719	0.132271
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.34795i	− 0.134443i
$306$	0	0
$307$	29.8744i	1.70502i	0.522712	+	0.852510i	$0.324920\pi$
$307$	29.8744i	1.70502i	−0.522712	+	0.852510i	$0.675080\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.2487	1.37502	0.687509	−	0.726176i	$-0.258704\pi$
$311$	24.2487	1.37502	0.687509	+	0.726176i	$0.258704\pi$
$312$	0	0
$313$	−3.48477	−0.196971	−0.0984853	−	0.995139i	$-0.531400\pi$
$313$	−3.48477	−0.196971	−0.0984853	+	0.995139i	$0.531400\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 7.62587i	− 0.428312i	−0.976800	−	0.214156i	$-0.931300\pi$
$317$	− 7.62587i	− 0.428312i	0.976800	−	0.214156i	$-0.0687000\pi$
$318$	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
$331$	− 8.78461i	− 0.482846i	−0.970420	−	0.241423i	$-0.922386\pi$
$331$	− 8.78461i	− 0.482846i	0.970420	−	0.241423i	$-0.0776141\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.679492	0.0371246
$336$	0	0
$337$	22.3923	1.21979	0.609893	−	0.792484i	$-0.291212\pi$
$337$	22.3923	1.21979	0.609893	+	0.792484i	$0.291212\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 28.3923i	− 1.53753i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.3533	0.824207	0.412104	−	0.911137i	$-0.364794\pi$
$347$	15.3533	0.824207	0.412104	+	0.911137i	$0.364794\pi$
$348$	0	0
$349$	21.4906	1.15037	0.575183	−	0.818025i	$-0.304931\pi$
$349$	21.4906	1.15037	0.575183	+	0.818025i	$0.304931\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.9808i	1.32959i	0.747025	+	0.664796i	$0.231481\pi$
$353$	24.9808i	1.32959i	−0.747025	+	0.664796i	$0.768519\pi$
$354$	0	0
$355$	3.10583i	0.164840i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	31.0112	1.63671	0.818353	−	0.574716i	$-0.194887\pi$
$359$	31.0112	1.63671	0.818353	+	0.574716i	$0.194887\pi$
$360$	0	0
$361$	15.7846	0.830769
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.34795i	0.122898i
$366$	0	0
$367$	7.17260i	0.374407i	0.982321	+	0.187203i	$0.0599424\pi$
$367$	7.17260i	0.374407i	−0.982321	+	0.187203i	$0.940058\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.7846	−0.558406	−0.279203	−	0.960232i	$-0.590070\pi$
$373$	−10.7846	−0.558406	−0.279203	+	0.960232i	$0.590070\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.53590i	− 0.439621i
$378$	0	0
$379$	13.8564i	0.711756i	0.934532	+	0.355878i	$0.115818\pi$
$379$	13.8564i	0.711756i	−0.934532	+	0.355878i	$0.884182\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−30.9282	−1.58036	−0.790179	−	0.612877i	$-0.790012\pi$
$383$	−30.9282	−1.58036	−0.790179	+	0.612877i	$0.790012\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.13681i	− 0.0576387i	−0.999585	−	0.0288193i	$-0.990825\pi$
$389$	− 1.13681i	− 0.0576387i	0.999585	−	0.0288193i	$-0.00917475\pi$
$390$	0	0
$391$	1.79315i	0.0906835i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.3205	0.569597
$396$	0	0
$397$	−22.8033	−1.14446	−0.572232	−	0.820092i	$-0.693922\pi$
$397$	−22.8033	−1.14446	−0.572232	+	0.820092i	$0.693922\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	29.2180i	1.45908i	0.683939	+	0.729539i	$0.260265\pi$
$401$	29.2180i	1.45908i	−0.683939	+	0.729539i	$0.739735\pi$
$402$	0	0
$403$	− 23.3205i	− 1.16168i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.6969	0.728500
$408$	0	0
$409$	−15.0759	−0.745454	−0.372727	−	0.927941i	$-0.621577\pi$
$409$	−15.0759	−0.745454	−0.372727	+	0.927941i	$0.621577\pi$
$410$	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
$419$	11.0718	0.540893	0.270446	−	0.962735i	$-0.412829\pi$
$419$	11.0718	0.540893	0.270446	+	0.962735i	$0.412829\pi$
$420$	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
$425$	12.1962i	0.591600i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.96902	0.0948442	0.0474221	−	0.998875i	$-0.484899\pi$
$431$	1.96902	0.0948442	0.0474221	+	0.998875i	$0.484899\pi$
$432$	0	0
$433$	−21.9711	−1.05586	−0.527931	−	0.849287i	$-0.677032\pi$
$433$	−21.9711	−1.05586	−0.527931	+	0.849287i	$0.677032\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.17691i	− 0.0562995i
$438$	0	0
$439$	23.1822i	1.10643i	0.833040	+	0.553213i	$0.186599\pi$
$439$	23.1822i	1.10643i	−0.833040	+	0.553213i	$0.813401\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.86800	0.326308	0.163154	−	0.986601i	$-0.447833\pi$
$443$	6.86800	0.326308	0.163154	+	0.986601i	$0.447833\pi$
$444$	0	0
$445$	5.60770	0.265830
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.89949i	0.467186i	0.972334	+	0.233593i	$0.0750483\pi$
$449$	9.89949i	0.467186i	−0.972334	+	0.233593i	$0.924952\pi$
$450$	0	0
$451$	− 26.2880i	− 1.23786i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	36.7846	1.72071	0.860356	−	0.509694i	$-0.170241\pi$
$457$	36.7846	1.72071	0.860356	+	0.509694i	$0.170241\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	11.8038i	0.549760i	0.961479	+	0.274880i	$0.0886381\pi$
$461$	11.8038i	0.549760i	−0.961479	+	0.274880i	$0.911362\pi$
$462$	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
$467$	−11.0718	−0.512342	−0.256171	−	0.966632i	$-0.582461\pi$
$467$	−11.0718	−0.512342	−0.256171	+	0.966632i	$0.582461\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.7589i	0.494695i
$474$	0	0
$475$	− 8.00481i	− 0.367286i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19.8564	0.907262	0.453631	−	0.891190i	$-0.350128\pi$
$479$	19.8564	0.907262	0.453631	+	0.891190i	$0.350128\pi$
$480$	0	0
$481$	12.0716	0.550417
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 7.45001i	− 0.338287i
$486$	0	0
$487$	28.3923i	1.28658i	0.765623	+	0.643289i	$0.222431\pi$
$487$	28.3923i	1.28658i	−0.765623	+	0.643289i	$0.777569\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.82894	0.353315	0.176658	−	0.984272i	$-0.443471\pi$
$491$	7.82894	0.353315	0.176658	+	0.984272i	$0.443471\pi$
$492$	0	0
$493$	6.69213	0.301398
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 33.4641i	− 1.49806i	−0.662537	−	0.749029i	$-0.730520\pi$
$499$	− 33.4641i	− 1.49806i	0.662537	−	0.749029i	$-0.269480\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−26.5359	−1.18318	−0.591589	−	0.806240i	$-0.701499\pi$
$503$	−26.5359	−1.18318	−0.591589	+	0.806240i	$0.701499\pi$
$504$	0	0
$505$	−5.71281	−0.254217
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 34.1962i	− 1.51572i	−0.652418	−	0.757859i	$-0.726246\pi$
$509$	− 34.1962i	− 1.51572i	0.652418	−	0.757859i	$-0.273754\pi$
$510$	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
$521$	− 16.5885i	− 0.726754i	−0.931642	−	0.363377i	$-0.881624\pi$
$521$	− 16.5885i	− 0.726754i	0.931642	−	0.363377i	$-0.118376\pi$
$522$	0	0
$523$	20.5569i	0.898889i	0.893308	+	0.449444i	$0.148378\pi$
$523$	20.5569i	0.898889i	−0.893308	+	0.449444i	$0.851622\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18.2832	0.796430
$528$	0	0
$529$	−22.5692	−0.981270
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 21.5921i	− 0.935259i
$534$	0	0
$535$	13.8647i	0.599425i
$536$	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
$545$	− 5.75129i	− 0.246358i
$546$	0	0
$547$	− 29.3205i	− 1.25365i	−0.779158	−	0.626827i	$-0.784353\pi$
$547$	− 29.3205i	− 1.25365i	0.779158	−	0.626827i	$-0.215647\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.39230	−0.187118
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.07107i	0.299611i	0.988716	+	0.149805i	$0.0478647\pi$
$557$	7.07107i	0.299611i	−0.988716	+	0.149805i	$0.952135\pi$
$558$	0	0
$559$	8.83701i	0.373766i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.67949	−0.281507	−0.140754	−	0.990045i	$-0.544953\pi$
$563$	−6.67949	−0.281507	−0.140754	+	0.990045i	$0.544953\pi$
$564$	0	0
$565$	−11.7942	−0.496185
$566$	0	0
$567$	0	0
$568$	0	0
$569$	35.4297i	1.48529i	0.669686	+	0.742644i	$0.266429\pi$
$569$	35.4297i	1.48529i	−0.669686	+	0.742644i	$0.733571\pi$
$570$	0	0
$571$	− 17.3205i	− 0.724841i	−0.932015	−	0.362420i	$-0.881950\pi$
$571$	− 17.3205i	− 0.724841i	0.932015	−	0.362420i	$-0.118050\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.92996	0.122188
$576$	0	0
$577$	31.5660	1.31411	0.657054	−	0.753843i	$-0.271802\pi$
$577$	31.5660	1.31411	0.657054	+	0.753843i	$0.271802\pi$
$578$	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
$587$	−27.4641	−1.13356	−0.566782	−	0.823868i	$-0.691812\pi$
$587$	−27.4641	−1.13356	−0.566782	+	0.823868i	$0.691812\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.8756i	0.528739i	0.964421	+	0.264370i	$0.0851639\pi$
$593$	12.8756i	0.528739i	−0.964421	+	0.264370i	$0.914836\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−11.4152	−0.466414	−0.233207	−	0.972427i	$-0.574922\pi$
$599$	−11.4152	−0.466414	−0.233207	+	0.972427i	$0.574922\pi$
$600$	0	0
$601$	−3.48477	−0.142147	−0.0710733	−	0.997471i	$-0.522642\pi$
$601$	−3.48477	−0.142147	−0.0710733	+	0.997471i	$0.522642\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.12436i	0.208335i
$606$	0	0
$607$	− 16.9706i	− 0.688814i	−0.938820	−	0.344407i	$-0.888080\pi$
$607$	− 16.9706i	− 0.688814i	0.938820	−	0.344407i	$-0.111920\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.0716	−0.488364
$612$	0	0
$613$	−40.3923	−1.63143	−0.815715	−	0.578454i	$-0.803656\pi$
$613$	−40.3923	−1.63143	−0.815715	+	0.578454i	$0.803656\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.3906i	1.46503i	0.680750	+	0.732516i	$0.261654\pi$
$617$	36.3906i	1.46503i	−0.680750	+	0.732516i	$0.738346\pi$
$618$	0	0
$619$	− 15.6579i	− 0.629344i	−0.949201	−	0.314672i	$-0.898106\pi$
$619$	− 15.6579i	− 0.629344i	0.949201	−	0.314672i	$-0.101894\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.46410i	0.377358i
$630$	0	0
$631$	19.8564i	0.790471i	0.918580	+	0.395236i	$0.129337\pi$
$631$	19.8564i	0.790471i	−0.918580	+	0.395236i	$0.870663\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.39230	0.174303
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.97382i	0.393942i	0.980409	+	0.196971i	$0.0631105\pi$
$641$	9.97382i	0.393942i	−0.980409	+	0.196971i	$0.936889\pi$
$642$	0	0
$643$	− 11.5911i	− 0.457109i	−0.973531	−	0.228554i	$-0.926600\pi$
$643$	− 11.5911i	− 0.457109i	0.973531	−	0.228554i	$-0.0733999\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.53590	0.335581	0.167790	−	0.985823i	$-0.446337\pi$
$647$	8.53590	0.335581	0.167790	+	0.985823i	$0.446337\pi$
$648$	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
$659$	−34.5975	−1.34773	−0.673863	−	0.738856i	$-0.735366\pi$
$659$	−34.5975	−1.34773	−0.673863	+	0.738856i	$0.735366\pi$
$660$	0	0
$661$	16.5916	0.645339	0.322670	−	0.946512i	$-0.395420\pi$
$661$	16.5916	0.645339	0.322670	+	0.946512i	$0.395420\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.60770i	− 0.0622502i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−13.6077	−0.525319
$672$	0	0
$673$	19.8564	0.765408	0.382704	−	0.923871i	$-0.374993\pi$
$673$	19.8564	0.765408	0.382704	+	0.923871i	$0.374993\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 12.1962i	− 0.468736i	−0.972148	−	0.234368i	$-0.924698\pi$
$677$	− 12.1962i	− 0.468736i	0.972148	−	0.234368i	$-0.0753021\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.92996	0.112112	0.0560560	−	0.998428i	$-0.482147\pi$
$683$	2.92996	0.112112	0.0560560	+	0.998428i	$0.482147\pi$
$684$	0	0
$685$	−8.00481	−0.305848
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.92820i	0.187750i
$690$	0	0
$691$	− 2.27362i	− 0.0864927i	−0.999064	−	0.0432464i	$-0.986230\pi$
$691$	− 2.27362i	− 0.0864927i	0.999064	−	0.0432464i	$-0.0137700\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.62536	0.0995854
$696$	0	0
$697$	16.9282	0.641201
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4.72311i	− 0.178390i	−0.996014	−	0.0891948i	$-0.971571\pi$
$701$	− 4.72311i	− 0.178390i	0.996014	−	0.0891948i	$-0.0284294\pi$
$702$	0	0
$703$	− 6.21166i	− 0.234277i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.9282	−0.936198	−0.468099	−	0.883676i	$-0.655061\pi$
$709$	−24.9282	−0.936198	−0.468099	+	0.883676i	$0.655061\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 4.39230i	− 0.164493i
$714$	0	0
$715$	10.8231i	0.404760i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.92820	−0.258378	−0.129189	−	0.991620i	$-0.541237\pi$
$719$	−6.92820	−0.258378	−0.129189	+	0.991620i	$0.541237\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 10.9348i	− 0.406107i
$726$	0	0
$727$	− 49.1185i	− 1.82171i	−0.412731	−	0.910853i	$-0.635425\pi$
$727$	− 49.1185i	− 1.82171i	0.412731	−	0.910853i	$-0.364575\pi$
$728$	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	$-0.646441\pi$
$733$	−24.0416	−0.887998	−0.443999	+	0.896027i	$0.646441\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.93803i	− 0.145059i
$738$	0	0
$739$	21.7128i	0.798719i	0.916794	+	0.399359i	$0.130767\pi$
$739$	21.7128i	0.798719i	−0.916794	+	0.399359i	$0.869233\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.92996	0.107490	0.0537450	−	0.998555i	$-0.482884\pi$
$743$	2.92996	0.107490	0.0537450	+	0.998555i	$0.482884\pi$
$744$	0	0
$745$	−11.7942	−0.432105
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	6.92820i	0.252814i	0.991978	+	0.126407i	$0.0403445\pi$
$751$	6.92820i	0.252814i	−0.991978	+	0.126407i	$0.959656\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	−38.7846	−1.40965	−0.704825	−	0.709381i	$-0.748975\pi$
$757$	−38.7846	−1.40965	−0.704825	+	0.709381i	$0.748975\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.9474i	0.433094i	0.976272	+	0.216547i	$0.0694795\pi$
$761$	11.9474i	0.433094i	−0.976272	+	0.216547i	$0.930520\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	36.2147	1.30764
$768$	0	0
$769$	9.69642	0.349662	0.174831	−	0.984598i	$-0.444062\pi$
$769$	9.69642	0.349662	0.174831	+	0.984598i	$0.444062\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.7321i	0.673745i	0.941550	+	0.336873i	$0.109369\pi$
$773$	18.7321i	0.673745i	−0.941550	+	0.336873i	$0.890631\pi$
$774$	0	0
$775$	− 29.8744i	− 1.07312i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.1106	−0.398080
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	0	0
$784$	0	0
$785$	13.1069i	0.467804i
$786$	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
$793$	−11.1769	−0.396904
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29.2679i	1.03672i	0.855161	+	0.518362i	$0.173458\pi$
$797$	29.2679i	1.03672i	−0.855161	+	0.518362i	$0.826542\pi$
$798$	0	0
$799$	− 9.46410i	− 0.334816i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13.6077	0.480205
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.7680i	0.765322i	0.923889	+	0.382661i	$0.124992\pi$
$809$	21.7680i	0.765322i	−0.923889	+	0.382661i	$0.875008\pi$
$810$	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
$815$	5.75129	0.201459
$816$	0	0
$817$	4.54725	0.159088
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 33.0817i	− 1.15456i	−0.816546	−	0.577280i	$-0.804114\pi$
$821$	− 33.0817i	− 1.15456i	0.816546	−	0.577280i	$-0.195886\pi$
$822$	0	0
$823$	− 15.4641i	− 0.539045i	−0.962994	−	0.269522i	$-0.913134\pi$
$823$	− 15.4641i	− 0.539045i	0.962994	−	0.269522i	$-0.0868658\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.8386	0.828948	0.414474	−	0.910061i	$-0.363966\pi$
$827$	23.8386	0.828948	0.414474	+	0.910061i	$0.363966\pi$
$828$	0	0
$829$	−22.3228	−0.775303	−0.387652	−	0.921806i	$-0.626714\pi$
$829$	−22.3228	−0.775303	−0.387652	+	0.921806i	$0.626714\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 7.60770i	− 0.263275i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	48.9282	1.68919	0.844595	−	0.535406i	$-0.179842\pi$
$839$	48.9282	1.68919	0.844595	+	0.535406i	$0.179842\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 0.626933i	− 0.0215672i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.27362	0.0779388
$852$	0	0
$853$	36.1875	1.23904	0.619519	−	0.784982i	$-0.287328\pi$
$853$	36.1875	1.23904	0.619519	+	0.784982i	$0.287328\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 48.0526i	− 1.64144i	−0.571328	−	0.820722i	$-0.693571\pi$
$857$	− 48.0526i	− 1.64144i	0.571328	−	0.820722i	$-0.306429\pi$
$858$	0	0
$859$	9.31749i	0.317909i	0.987286	+	0.158954i	$0.0508122\pi$
$859$	9.31749i	0.317909i	−0.987286	+	0.158954i	$0.949188\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−49.2944	−1.67800	−0.839000	−	0.544131i	$-0.816860\pi$
$863$	−49.2944	−1.67800	−0.839000	+	0.544131i	$0.816860\pi$
$864$	0	0
$865$	12.6410	0.429807
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 65.6086i	− 2.22562i
$870$	0	0
$871$	− 3.23457i	− 0.109599i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.1051	1.08411	0.542056	−	0.840342i	$-0.317646\pi$
$877$	32.1051	1.08411	0.542056	+	0.840342i	$0.317646\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 57.7654i	− 1.94616i	−0.230457	−	0.973082i	$-0.574022\pi$
$881$	− 57.7654i	− 1.94616i	0.230457	−	0.973082i	$-0.425978\pi$
$882$	0	0
$883$	37.1769i	1.25110i	0.780183	+	0.625551i	$0.215126\pi$
$883$	37.1769i	1.25110i	−0.780183	+	0.625551i	$0.784874\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	43.1769	1.44974	0.724869	−	0.688886i	$-0.241900\pi$
$887$	43.1769	1.44974	0.724869	+	0.688886i	$0.241900\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.21166i	0.207865i
$894$	0	0
$895$	7.65308i	0.255814i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.3923	−0.546714
$900$	0	0
$901$	−3.86370	−0.128719
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 5.17638i	− 0.172069i
$906$	0	0
$907$	0.679492i	0.0225622i	0.999936	+	0.0112811i	$0.00359096\pi$
$907$	0.679492i	0.0225622i	−0.999936	+	0.0112811i	$0.996409\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0.656339	0.0217455	0.0108727	−	0.999941i	$-0.496539\pi$
$911$	0.656339	0.0217455	0.0108727	+	0.999941i	$0.496539\pi$
$912$	0	0
$913$	−40.1528	−1.32886
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	13.1769i	0.434666i	0.976097	+	0.217333i	$0.0697358\pi$
$919$	13.1769i	0.434666i	−0.976097	+	0.217333i	$0.930264\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	14.7846	0.486641
$924$	0	0
$925$	15.4641	0.508457
$926$	0	0
$927$	0	0
$928$	0	0
$929$	48.4449i	1.58942i	0.606986	+	0.794712i	$0.292378\pi$
$929$	48.4449i	1.58942i	−0.606986	+	0.794712i	$0.707622\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.48528	−0.277498
$936$	0	0
$937$	−12.9310	−0.422437	−0.211219	−	0.977439i	$-0.567743\pi$
$937$	−12.9310	−0.422437	−0.211219	+	0.977439i	$0.567743\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21.9090i	0.714212i	0.934064	+	0.357106i	$0.116236\pi$
$941$	21.9090i	0.714212i	−0.934064	+	0.357106i	$0.883764\pi$
$942$	0	0
$943$	− 4.06678i	− 0.132432i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.24264	−0.137867	−0.0689336	−	0.997621i	$-0.521960\pi$
$947$	−4.24264	−0.137867	−0.0689336	+	0.997621i	$0.521960\pi$
$948$	0	0
$949$	11.1769	0.362818
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.7386i	1.25487i	0.778671	+	0.627433i	$0.215894\pi$
$953$	38.7386i	1.25487i	−0.778671	+	0.627433i	$0.784106\pi$
$954$	0	0
$955$	11.2394i	0.363698i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−13.7846	−0.444665
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.92820i	0.0942622i
$966$	0	0
$967$	11.0718i	0.356045i	0.984026	+	0.178022i	$0.0569700\pi$
$967$	11.0718i	0.356045i	−0.984026	+	0.178022i	$0.943030\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.6795	−0.406904	−0.203452	−	0.979085i	$-0.565216\pi$
$971$	−12.6795	−0.406904	−0.203452	+	0.979085i	$0.565216\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 47.5013i	− 1.51970i	−0.650099	−	0.759850i	$-0.725272\pi$
$977$	− 47.5013i	− 1.51970i	0.650099	−	0.759850i	$-0.274728\pi$
$978$	0	0
$979$	− 32.4997i	− 1.03870i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−35.3205	−1.12655	−0.563275	−	0.826270i	$-0.690459\pi$
$983$	−35.3205	−1.12655	−0.563275	+	0.826270i	$0.690459\pi$
$984$	0	0
$985$	11.7942	0.375794
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.66441i	0.0529251i
$990$	0	0
$991$	44.1051i	1.40105i	0.713630	+	0.700523i	$0.247050\pi$
$991$	44.1051i	1.40105i	−0.713630	+	0.700523i	$0.752950\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	20.5569	0.651696
$996$	0	0
$997$	0.933740	0.0295718	0.0147859	−	0.999891i	$-0.495293\pi$
$997$	0.933740	0.0295718	0.0147859	+	0.999891i	$0.495293\pi$
$998$	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.4607.6	yes	8
3.2	odd	2		7056.2.h.m.4607.3	yes	8
4.3	odd	2		7056.2.h.m.4607.5	yes	8
7.6	odd	2		7056.2.h.m.4607.4	yes	8
12.11	even	2	inner	7056.2.h.l.4607.4	yes	8
21.20	even	2	inner	7056.2.h.l.4607.5	yes	8
28.27	even	2	inner	7056.2.h.l.4607.3	✓	8
84.83	odd	2		7056.2.h.m.4607.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7056.2.h.l.4607.3	✓	8	28.27	even	2	inner
7056.2.h.l.4607.4	yes	8	12.11	even	2	inner
7056.2.h.l.4607.5	yes	8	21.20	even	2	inner
7056.2.h.l.4607.6	yes	8	1.1	even	1	trivial
7056.2.h.m.4607.3	yes	8	3.2	odd	2
7056.2.h.m.4607.4	yes	8	7.6	odd	2
7056.2.h.m.4607.5	yes	8	4.3	odd	2
7056.2.h.m.4607.6	yes	8	84.83	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)

\(f(q)\)	\(=\)	\(q+0.732051i q^{5} +O(q^{10})\)	\(q+0.732051i q^{5} +4.24264 q^{11} +3.48477 q^{13} +2.73205i q^{17} -1.79315i q^{19} +0.656339 q^{23} +4.46410 q^{25} -2.44949i q^{29} -6.69213i q^{31} +3.46410 q^{37} -6.19615i q^{41} +2.53590i q^{43} -3.46410 q^{47} +1.41421i q^{53} +3.10583i q^{55} +10.3923 q^{59} -3.20736 q^{61} +2.55103i q^{65} -0.928203i q^{67} +4.24264 q^{71} +3.20736 q^{73} -15.4641i q^{79} -9.46410 q^{83} -2.00000 q^{85} -7.66025i q^{89} +1.31268 q^{95} -10.1769 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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

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