LMFDB - Embedded newform 7056.2.a.cp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(1,7056)]

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([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("7056.1");

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.a (trivial)

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:	yes
Analytic conductor:	$56.3424436662$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3528)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	7056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{5} +O(q^{10})$	$q-1.41421 q^{5} +1.41421 q^{13} +1.41421 q^{17} +4.00000 q^{23} -3.00000 q^{25} -6.00000 q^{29} -5.65685 q^{31} -4.00000 q^{37} +7.07107 q^{41} -4.00000 q^{43} +11.3137 q^{47} -4.00000 q^{53} -5.65685 q^{59} +4.24264 q^{61} -2.00000 q^{65} -4.00000 q^{67} +12.0000 q^{71} -7.07107 q^{73} -8.00000 q^{79} -5.65685 q^{83} -2.00000 q^{85} -12.7279 q^{89} +4.24264 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 8 q^{23} - 6 q^{25} - 12 q^{29} - 8 q^{37} - 8 q^{43} - 8 q^{53} - 4 q^{65} - 8 q^{67} + 24 q^{71} - 16 q^{79} - 4 q^{85}+O(q^{100}) $ 2 * q + 8 * q^23 - 6 * q^25 - 12 * q^29 - 8 * q^37 - 8 * q^43 - 8 * q^53 - 4 * q^65 - 8 * q^67 + 24 * q^71 - 16 * q^79 - 4 * q^85

Display 100 coefficients 10 coefficients

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
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.41421	0.342997	0.171499	−	0.985184i	$-0.445139\pi$
$17$	1.41421	0.342997	0.171499	+	0.985184i	$0.445139\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.07107	1.10432	0.552158	−	0.833740i	$-0.313805\pi$
$41$	7.07107	1.10432	0.552158	+	0.833740i	$0.313805\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.3137	1.65027	0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137	1.65027	0.825137	+	0.564933i	$0.191098\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.65685	−0.736460	−0.368230	−	0.929735i	$-0.620036\pi$
$59$	−5.65685	−0.736460	−0.368230	+	0.929735i	$0.620036\pi$
$60$	0	0
$61$	4.24264	0.543214	0.271607	−	0.962408i	$-0.412445\pi$
$61$	4.24264	0.543214	0.271607	+	0.962408i	$0.412445\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	−7.07107	−0.827606	−0.413803	−	0.910366i	$-0.635800\pi$
$73$	−7.07107	−0.827606	−0.413803	+	0.910366i	$0.635800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.65685	−0.620920	−0.310460	−	0.950586i	$-0.600483\pi$
$83$	−5.65685	−0.620920	−0.310460	+	0.950586i	$0.600483\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.7279	−1.34916	−0.674579	−	0.738203i	$-0.735675\pi$
$89$	−12.7279	−1.34916	−0.674579	+	0.738203i	$0.735675\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.24264	0.430775	0.215387	−	0.976529i	$-0.430899\pi$
$97$	4.24264	0.430775	0.215387	+	0.976529i	$0.430899\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.89949	0.985037	0.492518	−	0.870302i	$-0.336076\pi$
$101$	9.89949	0.985037	0.492518	+	0.870302i	$0.336076\pi$
$102$	0	0
$103$	11.3137	1.11477	0.557386	−	0.830253i	$-0.311804\pi$
$103$	11.3137	1.11477	0.557386	+	0.830253i	$0.311804\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.3137	1.01193
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	16.9706	1.43942	0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706	1.43942	0.719712	+	0.694273i	$0.244274\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.48528	0.704664
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−21.2132	−1.69300	−0.846499	−	0.532390i	$-0.821294\pi$
$157$	−21.2132	−1.69300	−0.846499	+	0.532390i	$0.821294\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−21.2132	−1.61281	−0.806405	−	0.591364i	$-0.798590\pi$
$173$	−21.2132	−1.61281	−0.806405	+	0.591364i	$0.798590\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.00000	0.597948	0.298974	−	0.954261i	$-0.403356\pi$
$179$	8.00000	0.597948	0.298974	+	0.954261i	$0.403356\pi$
$180$	0	0
$181$	4.24264	0.315353	0.157676	−	0.987491i	$-0.449600\pi$
$181$	4.24264	0.315353	0.157676	+	0.987491i	$0.449600\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.65685	0.415900
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−16.9706	−1.20301	−0.601506	−	0.798869i	$-0.705432\pi$
$199$	−16.9706	−1.20301	−0.601506	+	0.798869i	$0.705432\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−10.0000	−0.698430
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	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
$221$	2.00000	0.134535
$222$	0	0
$223$	−5.65685	−0.378811	−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685	−0.378811	−0.189405	+	0.981899i	$0.560656\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.6274	−1.50183	−0.750917	−	0.660396i	$-0.770388\pi$
$227$	−22.6274	−1.50183	−0.750917	+	0.660396i	$0.770388\pi$
$228$	0	0
$229$	4.24264	0.280362	0.140181	−	0.990126i	$-0.455232\pi$
$229$	4.24264	0.280362	0.140181	+	0.990126i	$0.455232\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	−16.0000	−1.04372
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−15.5563	−1.00207	−0.501036	−	0.865426i	$-0.667048\pi$
$241$	−15.5563	−1.00207	−0.501036	+	0.865426i	$0.667048\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−22.6274	−1.42823	−0.714115	−	0.700028i	$-0.753171\pi$
$251$	−22.6274	−1.42823	−0.714115	+	0.700028i	$0.753171\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.24264	−0.264649	−0.132324	−	0.991206i	$-0.542244\pi$
$257$	−4.24264	−0.264649	−0.132324	+	0.991206i	$0.542244\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.00000	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$263$	−4.00000	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$264$	0	0
$265$	5.65685	0.347498
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.89949	−0.603583	−0.301791	−	0.953374i	$-0.597585\pi$
$269$	−9.89949	−0.603583	−0.301791	+	0.953374i	$0.597585\pi$
$270$	0	0
$271$	22.6274	1.37452	0.687259	−	0.726413i	$-0.258814\pi$
$271$	22.6274	1.37452	0.687259	+	0.726413i	$0.258814\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	−16.9706	−1.00880	−0.504398	−	0.863472i	$-0.668285\pi$
$283$	−16.9706	−1.00880	−0.504398	+	0.863472i	$0.668285\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.0000	−0.882353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.41421	−0.0826192	−0.0413096	−	0.999146i	$-0.513153\pi$
$293$	−1.41421	−0.0826192	−0.0413096	+	0.999146i	$0.513153\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	11.3137	0.645707	0.322854	−	0.946449i	$-0.395358\pi$
$307$	11.3137	0.645707	0.322854	+	0.946449i	$0.395358\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.3137	0.641542	0.320771	−	0.947157i	$-0.396058\pi$
$311$	11.3137	0.641542	0.320771	+	0.947157i	$0.396058\pi$
$312$	0	0
$313$	12.7279	0.719425	0.359712	−	0.933063i	$-0.382875\pi$
$313$	12.7279	0.719425	0.359712	+	0.933063i	$0.382875\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.00000	0.224662	0.112331	−	0.993671i	$-0.464168\pi$
$317$	4.00000	0.224662	0.112331	+	0.993671i	$0.464168\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.24264	−0.235339
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.65685	0.309067
$336$	0	0
$337$	16.0000	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$337$	16.0000	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	12.7279	0.681310	0.340655	−	0.940188i	$-0.389351\pi$
$349$	12.7279	0.681310	0.340655	+	0.940188i	$0.389351\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.5563	0.827981	0.413990	−	0.910281i	$-0.364135\pi$
$353$	15.5563	0.827981	0.413990	+	0.910281i	$0.364135\pi$
$354$	0	0
$355$	−16.9706	−0.900704
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	−28.2843	−1.47643	−0.738213	−	0.674567i	$-0.764330\pi$
$367$	−28.2843	−1.47643	−0.738213	+	0.674567i	$0.764330\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.48528	−0.437014
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.65685	−0.289052	−0.144526	−	0.989501i	$-0.546166\pi$
$383$	−5.65685	−0.289052	−0.144526	+	0.989501i	$0.546166\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.3137	0.569254
$396$	0	0
$397$	9.89949	0.496841	0.248421	−	0.968652i	$-0.420088\pi$
$397$	9.89949	0.496841	0.248421	+	0.968652i	$0.420088\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−12.7279	−0.629355	−0.314678	−	0.949199i	$-0.601896\pi$
$409$	−12.7279	−0.629355	−0.314678	+	0.949199i	$0.601896\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.65685	0.276355	0.138178	−	0.990407i	$-0.455875\pi$
$419$	5.65685	0.276355	0.138178	+	0.990407i	$0.455875\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.24264	−0.205798
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	15.5563	0.747590	0.373795	−	0.927511i	$-0.378056\pi$
$433$	15.5563	0.747590	0.373795	+	0.927511i	$0.378056\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	33.9411	1.61992	0.809961	−	0.586484i	$-0.199488\pi$
$439$	33.9411	1.61992	0.809961	+	0.586484i	$0.199488\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9.89949	−0.461065	−0.230533	−	0.973065i	$-0.574047\pi$
$461$	−9.89949	−0.461065	−0.230533	+	0.973065i	$0.574047\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−33.9411	−1.57061	−0.785304	−	0.619110i	$-0.787493\pi$
$467$	−33.9411	−1.57061	−0.785304	+	0.619110i	$0.787493\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11.3137	−0.516937	−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137	−0.516937	−0.258468	+	0.966020i	$0.583218\pi$
$480$	0	0
$481$	−5.65685	−0.257930
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	0	0
$493$	−8.48528	−0.382158
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.0416	1.06563	0.532813	−	0.846233i	$-0.321135\pi$
$509$	24.0416	1.06563	0.532813	+	0.846233i	$0.321135\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−41.0122	−1.79678	−0.898388	−	0.439202i	$-0.855261\pi$
$521$	−41.0122	−1.79678	−0.898388	+	0.439202i	$0.855261\pi$
$522$	0	0
$523$	−22.6274	−0.989428	−0.494714	−	0.869056i	$-0.664727\pi$
$523$	−22.6274	−0.989428	−0.494714	+	0.869056i	$0.664727\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.0000	0.433148
$534$	0	0
$535$	−22.6274	−0.978269
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.65685	0.242313
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.0000	−1.52537	−0.762684	−	0.646771i	$-0.776119\pi$
$557$	−36.0000	−1.52537	−0.762684	+	0.646771i	$0.776119\pi$
$558$	0	0
$559$	−5.65685	−0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.9706	0.715224	0.357612	−	0.933870i	$-0.383591\pi$
$563$	16.9706	0.715224	0.357612	+	0.933870i	$0.383591\pi$
$564$	0	0
$565$	22.6274	0.951943
$566$	0	0
$567$	0	0
$568$	0	0
$569$	46.0000	1.92842	0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000	1.92842	0.964210	+	0.265139i	$0.0854179\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	−12.7279	−0.529870	−0.264935	−	0.964266i	$-0.585351\pi$
$577$	−12.7279	−0.529870	−0.264935	+	0.964266i	$0.585351\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.9411	1.40090	0.700450	−	0.713701i	$-0.252983\pi$
$587$	33.9411	1.40090	0.700450	+	0.713701i	$0.252983\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	41.0122	1.68417	0.842084	−	0.539346i	$-0.181328\pi$
$593$	41.0122	1.68417	0.842084	+	0.539346i	$0.181328\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−26.8701	−1.09605	−0.548026	−	0.836461i	$-0.684621\pi$
$601$	−26.8701	−1.09605	−0.548026	+	0.836461i	$0.684621\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.5563	0.632456
$606$	0	0
$607$	39.5980	1.60723	0.803616	−	0.595148i	$-0.202907\pi$
$607$	39.5980	1.60723	0.803616	+	0.595148i	$0.202907\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	−12.0000	−0.484675	−0.242338	−	0.970192i	$-0.577914\pi$
$613$	−12.0000	−0.484675	−0.242338	+	0.970192i	$0.577914\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	33.9411	1.36421	0.682105	−	0.731255i	$-0.261065\pi$
$619$	33.9411	1.36421	0.682105	+	0.731255i	$0.261065\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.65685	−0.225554
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.3137	−0.448971
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	39.5980	1.56159	0.780796	−	0.624786i	$-0.214814\pi$
$643$	39.5980	1.56159	0.780796	+	0.624786i	$0.214814\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.9706	0.667182	0.333591	−	0.942718i	$-0.391740\pi$
$647$	16.9706	0.667182	0.333591	+	0.942718i	$0.391740\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−9.89949	−0.385046	−0.192523	−	0.981292i	$-0.561667\pi$
$661$	−9.89949	−0.385046	−0.192523	+	0.981292i	$0.561667\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−8.00000	−0.308377	−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000	−0.308377	−0.154189	+	0.988041i	$0.549276\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	29.6985	1.14141	0.570703	−	0.821157i	$-0.306671\pi$
$677$	29.6985	1.14141	0.570703	+	0.821157i	$0.306671\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	2.82843	0.108069
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.65685	−0.215509
$690$	0	0
$691$	−28.2843	−1.07598	−0.537992	−	0.842950i	$-0.680817\pi$
$691$	−28.2843	−1.07598	−0.537992	+	0.842950i	$0.680817\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.0000	−0.910372
$696$	0	0
$697$	10.0000	0.378777
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−22.6274	−0.847403
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−39.5980	−1.47676	−0.738378	−	0.674387i	$-0.764408\pi$
$719$	−39.5980	−1.47676	−0.738378	+	0.674387i	$0.764408\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	18.0000	0.668503
$726$	0	0
$727$	−33.9411	−1.25881	−0.629403	−	0.777079i	$-0.716701\pi$
$727$	−33.9411	−1.25881	−0.629403	+	0.777079i	$0.716701\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.65685	−0.209226
$732$	0	0
$733$	−29.6985	−1.09694	−0.548469	−	0.836171i	$-0.684789\pi$
$733$	−29.6985	−1.09694	−0.548469	+	0.836171i	$0.684789\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	16.9706	0.621753
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.3137	0.411748
$756$	0	0
$757$	−4.00000	−0.145382	−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000	−0.145382	−0.0726912	+	0.997354i	$0.523159\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	15.5563	0.563917	0.281959	−	0.959427i	$-0.409016\pi$
$761$	15.5563	0.563917	0.281959	+	0.959427i	$0.409016\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	38.1838	1.37694	0.688471	−	0.725264i	$-0.258282\pi$
$769$	38.1838	1.37694	0.688471	+	0.725264i	$0.258282\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−32.5269	−1.16991	−0.584956	−	0.811065i	$-0.698888\pi$
$773$	−32.5269	−1.16991	−0.584956	+	0.811065i	$0.698888\pi$
$774$	0	0
$775$	16.9706	0.609601
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	30.0000	1.07075
$786$	0	0
$787$	5.65685	0.201645	0.100823	−	0.994904i	$-0.467853\pi$
$787$	5.65685	0.201645	0.100823	+	0.994904i	$0.467853\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.0416	−0.851598	−0.425799	−	0.904818i	$-0.640007\pi$
$797$	−24.0416	−0.851598	−0.425799	+	0.904818i	$0.640007\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−40.0000	−1.40633	−0.703163	−	0.711029i	$-0.748229\pi$
$809$	−40.0000	−1.40633	−0.703163	+	0.711029i	$0.748229\pi$
$810$	0	0
$811$	11.3137	0.397278	0.198639	−	0.980073i	$-0.436348\pi$
$811$	11.3137	0.397278	0.198639	+	0.980073i	$0.436348\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	28.2843	0.990755
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	36.0000	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$821$	36.0000	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	0	0
$829$	−4.24264	−0.147353	−0.0736765	−	0.997282i	$-0.523473\pi$
$829$	−4.24264	−0.147353	−0.0736765	+	0.997282i	$0.523473\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	39.5980	1.36707	0.683537	−	0.729916i	$-0.260441\pi$
$839$	39.5980	1.36707	0.683537	+	0.729916i	$0.260441\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15.5563	0.535155
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	−38.1838	−1.30739	−0.653694	−	0.756759i	$-0.726781\pi$
$853$	−38.1838	−1.30739	−0.653694	+	0.756759i	$0.726781\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.8701	0.917864	0.458932	−	0.888471i	$-0.348232\pi$
$857$	26.8701	0.917864	0.458932	+	0.888471i	$0.348232\pi$
$858$	0	0
$859$	−56.5685	−1.93009	−0.965047	−	0.262077i	$-0.915592\pi$
$859$	−56.5685	−1.93009	−0.965047	+	0.262077i	$0.915592\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	0	0
$865$	30.0000	1.02003
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−5.65685	−0.191675
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28.0000	0.945493	0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000	0.945493	0.472746	+	0.881199i	$0.343263\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29.6985	1.00057	0.500284	−	0.865862i	$-0.333229\pi$
$881$	29.6985	1.00057	0.500284	+	0.865862i	$0.333229\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.2843	−0.949693	−0.474846	−	0.880069i	$-0.657496\pi$
$887$	−28.2843	−0.949693	−0.474846	+	0.880069i	$0.657496\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.9411	1.13200
$900$	0	0
$901$	−5.65685	−0.188457
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−60.0000	−1.98789	−0.993944	−	0.109885i	$-0.964952\pi$
$911$	−60.0000	−1.98789	−0.993944	+	0.109885i	$0.964952\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	16.9706	0.558593
$924$	0	0
$925$	12.0000	0.394558
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.89949	0.324792	0.162396	−	0.986726i	$-0.448078\pi$
$929$	9.89949	0.324792	0.162396	+	0.986726i	$0.448078\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.1838	−1.24741	−0.623705	−	0.781660i	$-0.714373\pi$
$937$	−38.1838	−1.24741	−0.623705	+	0.781660i	$0.714373\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.8701	−0.875939	−0.437969	−	0.898990i	$-0.644302\pi$
$941$	−26.8701	−0.875939	−0.437969	+	0.898990i	$0.644302\pi$
$942$	0	0
$943$	28.2843	0.921063
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	0	0
$955$	5.65685	0.183052
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.82843	0.0910503
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.6274	−0.726148	−0.363074	−	0.931760i	$-0.618273\pi$
$971$	−22.6274	−0.726148	−0.363074	+	0.931760i	$0.618273\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−45.2548	−1.44341	−0.721703	−	0.692203i	$-0.756640\pi$
$983$	−45.2548	−1.44341	−0.721703	+	0.692203i	$0.756640\pi$
$984$	0	0
$985$	16.9706	0.540727
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	24.0000	0.760851
$996$	0	0
$997$	46.6690	1.47802	0.739012	−	0.673693i	$-0.235293\pi$
$997$	46.6690	1.47802	0.739012	+	0.673693i	$0.235293\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7056.2.a.cp.1.1		2
3.2	odd	2		7056.2.a.cn.1.2		2
4.3	odd	2		3528.2.a.bg.1.1	✓	2
7.6	odd	2	inner	7056.2.a.cp.1.2		2
12.11	even	2		3528.2.a.bh.1.2	yes	2
21.20	even	2		7056.2.a.cn.1.1		2
28.3	even	6		3528.2.s.bh.3313.1		4
28.11	odd	6		3528.2.s.bh.3313.2		4
28.19	even	6		3528.2.s.bh.361.1		4
28.23	odd	6		3528.2.s.bh.361.2		4
28.27	even	2		3528.2.a.bg.1.2	yes	2
84.11	even	6		3528.2.s.bg.3313.1		4
84.23	even	6		3528.2.s.bg.361.1		4
84.47	odd	6		3528.2.s.bg.361.2		4
84.59	odd	6		3528.2.s.bg.3313.2		4
84.83	odd	2		3528.2.a.bh.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3528.2.a.bg.1.1	✓	2	4.3	odd	2
3528.2.a.bg.1.2	yes	2	28.27	even	2
3528.2.a.bh.1.1	yes	2	84.83	odd	2
3528.2.a.bh.1.2	yes	2	12.11	even	2
3528.2.s.bg.361.1		4	84.23	even	6
3528.2.s.bg.361.2		4	84.47	odd	6
3528.2.s.bg.3313.1		4	84.11	even	6
3528.2.s.bg.3313.2		4	84.59	odd	6
3528.2.s.bh.361.1		4	28.19	even	6
3528.2.s.bh.361.2		4	28.23	odd	6
3528.2.s.bh.3313.1		4	28.3	even	6
3528.2.s.bh.3313.2		4	28.11	odd	6
7056.2.a.cn.1.1		2	21.20	even	2
7056.2.a.cn.1.2		2	3.2	odd	2
7056.2.a.cp.1.1		2	1.1	even	1	trivial
7056.2.a.cp.1.2		2	7.6	odd	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.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{5} +O(q^{10})\)	\(q-1.41421 q^{5} +1.41421 q^{13} +1.41421 q^{17} +4.00000 q^{23} -3.00000 q^{25} -6.00000 q^{29} -5.65685 q^{31} -4.00000 q^{37} +7.07107 q^{41} -4.00000 q^{43} +11.3137 q^{47} -4.00000 q^{53} -5.65685 q^{59} +4.24264 q^{61} -2.00000 q^{65} -4.00000 q^{67} +12.0000 q^{71} -7.07107 q^{73} -8.00000 q^{79} -5.65685 q^{83} -2.00000 q^{85} -12.7279 q^{89} +4.24264 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 8 q^{23} - 6 q^{25} - 12 q^{29} - 8 q^{37} - 8 q^{43} - 8 q^{53} - 4 q^{65} - 8 q^{67} + 24 q^{71} - 16 q^{79} - 4 q^{85}+O(q^{100}) \) 2 * q + 8 * q^23 - 6 * q^25 - 12 * q^29 - 8 * q^37 - 8 * q^43 - 8 * q^53 - 4 * q^65 - 8 * q^67 + 24 * q^71 - 16 * q^79 - 4 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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