LMFDB - Embedded newform 7056.2.a.ck.1.2

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:	$0$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 3528)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+2.82843 q^{5} +O(q^{10})$	$q+2.82843 q^{5} -2.00000 q^{11} +5.65685 q^{13} +2.82843 q^{17} -5.65685 q^{19} -6.00000 q^{23} +3.00000 q^{25} -4.00000 q^{29} +5.65685 q^{31} -2.00000 q^{37} -2.82843 q^{41} +4.00000 q^{43} +11.3137 q^{47} +12.0000 q^{53} -5.65685 q^{55} +11.3137 q^{59} +5.65685 q^{61} +16.0000 q^{65} +12.0000 q^{67} +6.00000 q^{71} -8.00000 q^{79} -11.3137 q^{83} +8.00000 q^{85} +8.48528 q^{89} -16.0000 q^{95} -11.3137 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 4 q^{11} - 12 q^{23} + 6 q^{25} - 8 q^{29} - 4 q^{37} + 8 q^{43} + 24 q^{53} + 32 q^{65} + 24 q^{67} + 12 q^{71} - 16 q^{79} + 16 q^{85} - 32 q^{95}+O(q^{100}) $ 2 * q - 4 * q^11 - 12 * q^23 + 6 * q^25 - 8 * q^29 - 4 * q^37 + 8 * q^43 + 24 * q^53 + 32 * q^65 + 24 * q^67 + 12 * q^71 - 16 * q^79 + 16 * q^85 - 32 * q^95

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
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.82843	−0.441726	−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843	−0.441726	−0.220863	+	0.975305i	$0.570887\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	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
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	5.65685	0.724286	0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685	0.724286	0.362143	+	0.932123i	$0.382045\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.0000	1.98456
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	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
$83$	−11.3137	−1.24184	−0.620920	−	0.783874i	$-0.713241\pi$
$83$	−11.3137	−1.24184	−0.620920	+	0.783874i	$0.713241\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.48528	0.899438	0.449719	−	0.893170i	$-0.351524\pi$
$89$	8.48528	0.899438	0.449719	+	0.893170i	$0.351524\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.0000	−1.64157
$96$	0	0
$97$	−11.3137	−1.14873	−0.574367	−	0.818598i	$-0.694752\pi$
$97$	−11.3137	−1.14873	−0.574367	+	0.818598i	$0.694752\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.48528	0.844317	0.422159	−	0.906522i	$-0.361273\pi$
$101$	8.48528	0.844317	0.422159	+	0.906522i	$0.361273\pi$
$102$	0	0
$103$	16.9706	1.67216	0.836080	−	0.548608i	$-0.184842\pi$
$103$	16.9706	1.67216	0.836080	+	0.548608i	$0.184842\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	−16.9706	−1.58251
$116$	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
$125$	−5.65685	−0.505964
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	11.3137	0.959616	0.479808	−	0.877373i	$-0.340706\pi$
$139$	11.3137	0.959616	0.479808	+	0.877373i	$0.340706\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−11.3137	−0.946100
$144$	0	0
$145$	−11.3137	−0.939552
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	16.0000	1.28515
$156$	0	0
$157$	−5.65685	−0.451466	−0.225733	−	0.974189i	$-0.572478\pi$
$157$	−5.65685	−0.451466	−0.225733	+	0.974189i	$0.572478\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.6274	1.75096	0.875481	−	0.483252i	$-0.160545\pi$
$167$	22.6274	1.75096	0.875481	+	0.483252i	$0.160545\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.82843	−0.215041	−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843	−0.215041	−0.107521	+	0.994203i	$0.534291\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.0000	−1.64436	−0.822179	−	0.569230i	$-0.807242\pi$
$179$	−22.0000	−1.64436	−0.822179	+	0.569230i	$0.807242\pi$
$180$	0	0
$181$	−16.9706	−1.26141	−0.630706	−	0.776022i	$-0.717235\pi$
$181$	−16.9706	−1.26141	−0.630706	+	0.776022i	$0.717235\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	−5.65685	−0.413670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−8.00000	−0.558744
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.3137	0.782586
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	11.3137	0.771589
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	16.0000	1.07628
$222$	0	0
$223$	11.3137	0.757622	0.378811	−	0.925474i	$-0.376333\pi$
$223$	11.3137	0.757622	0.378811	+	0.925474i	$0.376333\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−16.9706	−1.12145	−0.560723	−	0.828003i	$-0.689477\pi$
$229$	−16.9706	−1.12145	−0.560723	+	0.828003i	$0.689477\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	32.0000	2.08745
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	11.3137	0.728780	0.364390	−	0.931246i	$-0.381278\pi$
$241$	11.3137	0.728780	0.364390	+	0.931246i	$0.381278\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−32.0000	−2.03611
$248$	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
$253$	12.0000	0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.4558	−1.58789	−0.793946	−	0.607988i	$-0.791977\pi$
$257$	−25.4558	−1.58789	−0.793946	+	0.607988i	$0.791977\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.0000	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$264$	0	0
$265$	33.9411	2.08499
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−31.1127	−1.89697	−0.948487	−	0.316815i	$-0.897387\pi$
$269$	−31.1127	−1.89697	−0.948487	+	0.316815i	$0.897387\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	28.2843	1.68133	0.840663	−	0.541559i	$-0.182166\pi$
$283$	28.2843	1.68133	0.840663	+	0.541559i	$0.182166\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.7990	1.15667	0.578335	−	0.815800i	$-0.303703\pi$
$293$	19.7990	1.15667	0.578335	+	0.815800i	$0.303703\pi$
$294$	0	0
$295$	32.0000	1.86311
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−33.9411	−1.96287
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	16.0000	0.916157
$306$	0	0
$307$	−16.9706	−0.968561	−0.484281	−	0.874913i	$-0.660919\pi$
$307$	−16.9706	−0.968561	−0.484281	+	0.874913i	$0.660919\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	33.9411	1.91847	0.959233	−	0.282617i	$-0.0912024\pi$
$313$	33.9411	1.91847	0.959233	+	0.282617i	$0.0912024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−16.0000	−0.890264
$324$	0	0
$325$	16.9706	0.941357
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	33.9411	1.85440
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.3137	−0.612672
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	0	0
$349$	16.9706	0.908413	0.454207	−	0.890896i	$-0.349923\pi$
$349$	16.9706	0.908413	0.454207	+	0.890896i	$0.349923\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.82843	0.150542	0.0752710	−	0.997163i	$-0.476018\pi$
$353$	2.82843	0.150542	0.0752710	+	0.997163i	$0.476018\pi$
$354$	0	0
$355$	16.9706	0.900704
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−22.6274	−1.16537
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.3137	−0.578103	−0.289052	−	0.957313i	$-0.593340\pi$
$383$	−11.3137	−0.578103	−0.289052	+	0.957313i	$0.593340\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−28.0000	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$390$	0	0
$391$	−16.9706	−0.858238
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−22.6274	−1.13851
$396$	0	0
$397$	16.9706	0.851728	0.425864	−	0.904787i	$-0.359970\pi$
$397$	16.9706	0.851728	0.425864	+	0.904787i	$0.359970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	−33.9411	−1.67828	−0.839140	−	0.543915i	$-0.816941\pi$
$409$	−33.9411	−1.67828	−0.839140	+	0.543915i	$0.816941\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−32.0000	−1.57082
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11.3137	−0.552711	−0.276355	−	0.961056i	$-0.589127\pi$
$419$	−11.3137	−0.552711	−0.276355	+	0.961056i	$0.589127\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.48528	0.411597
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	−11.3137	−0.543702	−0.271851	−	0.962339i	$-0.587636\pi$
$433$	−11.3137	−0.543702	−0.271851	+	0.962339i	$0.587636\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	33.9411	1.62362
$438$	0	0
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.00000	0.0950229	0.0475114	−	0.998871i	$-0.484871\pi$
$443$	2.00000	0.0950229	0.0475114	+	0.998871i	$0.484871\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.0000	1.51017	0.755087	−	0.655625i	$-0.227595\pi$
$449$	32.0000	1.51017	0.755087	+	0.655625i	$0.227595\pi$
$450$	0	0
$451$	5.65685	0.266371
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.1421	−0.658665	−0.329332	−	0.944214i	$-0.606824\pi$
$461$	−14.1421	−0.658665	−0.329332	+	0.944214i	$0.606824\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−16.9706	−0.778663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	−11.3137	−0.515861
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−32.0000	−1.45305
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	0	0
$493$	−11.3137	−0.509544
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11.3137	−0.504453	−0.252227	−	0.967668i	$-0.581163\pi$
$503$	−11.3137	−0.504453	−0.252227	+	0.967668i	$0.581163\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.82843	0.125368	0.0626839	−	0.998033i	$-0.480034\pi$
$509$	2.82843	0.125368	0.0626839	+	0.998033i	$0.480034\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	48.0000	2.11513
$516$	0	0
$517$	−22.6274	−0.995153
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−31.1127	−1.36307	−0.681536	−	0.731785i	$-0.738688\pi$
$521$	−31.1127	−1.36307	−0.681536	+	0.731785i	$0.738688\pi$
$522$	0	0
$523$	22.6274	0.989428	0.494714	−	0.869056i	$-0.335273\pi$
$523$	22.6274	0.989428	0.494714	+	0.869056i	$0.335273\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.0000	0.696971
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−16.0000	−0.693037
$534$	0	0
$535$	−50.9117	−2.20110
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	39.5980	1.69619
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	22.6274	0.963960
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	22.6274	0.957038
$560$	0	0
$561$	0	0
$562$	0	0
$563$	33.9411	1.43045	0.715224	−	0.698895i	$-0.246325\pi$
$563$	33.9411	1.43045	0.715224	+	0.698895i	$0.246325\pi$
$564$	0	0
$565$	−22.6274	−0.951943
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−18.0000	−0.750652
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−24.0000	−0.993978
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.1421	0.580748	0.290374	−	0.956913i	$-0.406220\pi$
$593$	14.1421	0.580748	0.290374	+	0.956913i	$0.406220\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	22.0000	0.898896	0.449448	−	0.893307i	$-0.351621\pi$
$599$	22.0000	0.898896	0.449448	+	0.893307i	$0.351621\pi$
$600$	0	0
$601$	−22.6274	−0.922992	−0.461496	−	0.887142i	$-0.652687\pi$
$601$	−22.6274	−0.922992	−0.461496	+	0.887142i	$0.652687\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−19.7990	−0.804943
$606$	0	0
$607$	11.3137	0.459209	0.229605	−	0.973284i	$-0.426257\pi$
$607$	11.3137	0.459209	0.229605	+	0.973284i	$0.426257\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	64.0000	2.58916
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	48.0000	1.93241	0.966204	−	0.257780i	$-0.0829910\pi$
$617$	48.0000	1.93241	0.966204	+	0.257780i	$0.0829910\pi$
$618$	0	0
$619$	−11.3137	−0.454736	−0.227368	−	0.973809i	$-0.573012\pi$
$619$	−11.3137	−0.454736	−0.227368	+	0.973809i	$0.573012\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.65685	−0.225554
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	45.2548	1.79588
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	16.9706	0.669254	0.334627	−	0.942351i	$-0.391390\pi$
$643$	16.9706	0.669254	0.334627	+	0.942351i	$0.391390\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−22.6274	−0.888204
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.0000	−0.782660	−0.391330	−	0.920250i	$-0.627985\pi$
$653$	−20.0000	−0.782660	−0.391330	+	0.920250i	$0.627985\pi$
$654$	0	0
$655$	−32.0000	−1.25034
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	5.65685	0.220026	0.110013	−	0.993930i	$-0.464911\pi$
$661$	5.65685	0.220026	0.110013	+	0.993930i	$0.464911\pi$
$662$	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
$671$	−11.3137	−0.436761
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.48528	0.326116	0.163058	−	0.986616i	$-0.447864\pi$
$677$	8.48528	0.326116	0.163058	+	0.986616i	$0.447864\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14.0000	−0.535695	−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000	−0.535695	−0.267848	+	0.963461i	$0.586312\pi$
$684$	0	0
$685$	22.6274	0.864549
$686$	0	0
$687$	0	0
$688$	0	0
$689$	67.8823	2.58611
$690$	0	0
$691$	22.6274	0.860788	0.430394	−	0.902641i	$-0.358375\pi$
$691$	22.6274	0.860788	0.430394	+	0.902641i	$0.358375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	32.0000	1.21383
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.00000	0.151078	0.0755390	−	0.997143i	$-0.475932\pi$
$701$	4.00000	0.151078	0.0755390	+	0.997143i	$0.475932\pi$
$702$	0	0
$703$	11.3137	0.426705
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−33.9411	−1.27111
$714$	0	0
$715$	−32.0000	−1.19673
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.3137	0.421930	0.210965	−	0.977494i	$-0.432339\pi$
$719$	11.3137	0.421930	0.210965	+	0.977494i	$0.432339\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.0000	−0.445669
$726$	0	0
$727$	39.5980	1.46861	0.734304	−	0.678821i	$-0.237509\pi$
$727$	39.5980	1.46861	0.734304	+	0.678821i	$0.237509\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.3137	0.418453
$732$	0	0
$733$	16.9706	0.626822	0.313411	−	0.949618i	$-0.398528\pi$
$733$	16.9706	0.626822	0.313411	+	0.949618i	$0.398528\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.0000	−0.953847	−0.476924	−	0.878945i	$-0.658248\pi$
$743$	−26.0000	−0.953847	−0.476924	+	0.878945i	$0.658248\pi$
$744$	0	0
$745$	56.5685	2.07251
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−48.0000	−1.75154	−0.875772	−	0.482724i	$-0.839647\pi$
$751$	−48.0000	−1.75154	−0.875772	+	0.482724i	$0.839647\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.6274	0.823496
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31.1127	1.12783	0.563917	−	0.825831i	$-0.309294\pi$
$761$	31.1127	1.12783	0.563917	+	0.825831i	$0.309294\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	64.0000	2.31091
$768$	0	0
$769$	−33.9411	−1.22395	−0.611974	−	0.790878i	$-0.709624\pi$
$769$	−33.9411	−1.22395	−0.611974	+	0.790878i	$0.709624\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.1127	1.11905	0.559523	−	0.828815i	$-0.310984\pi$
$773$	31.1127	1.11905	0.559523	+	0.828815i	$0.310984\pi$
$774$	0	0
$775$	16.9706	0.609601
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.0000	−0.571064
$786$	0	0
$787$	33.9411	1.20987	0.604935	−	0.796275i	$-0.293199\pi$
$787$	33.9411	1.20987	0.604935	+	0.796275i	$0.293199\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	32.0000	1.13635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.4558	−0.901692	−0.450846	−	0.892602i	$-0.648878\pi$
$797$	−25.4558	−0.901692	−0.450846	+	0.892602i	$0.648878\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	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
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	22.6274	0.794556	0.397278	−	0.917698i	$-0.369955\pi$
$811$	22.6274	0.794556	0.397278	+	0.917698i	$0.369955\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.3137	0.396302
$816$	0	0
$817$	−22.6274	−0.791633
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.0000	−1.73867	−0.869335	−	0.494223i	$-0.835453\pi$
$827$	−50.0000	−1.73867	−0.869335	+	0.494223i	$0.835453\pi$
$828$	0	0
$829$	−5.65685	−0.196471	−0.0982353	−	0.995163i	$-0.531320\pi$
$829$	−5.65685	−0.196471	−0.0982353	+	0.995163i	$0.531320\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	64.0000	2.21481
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−45.2548	−1.56237	−0.781185	−	0.624299i	$-0.785385\pi$
$839$	−45.2548	−1.56237	−0.781185	+	0.624299i	$0.785385\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	53.7401	1.84872
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	16.9706	0.581061	0.290531	−	0.956866i	$-0.406168\pi$
$853$	16.9706	0.581061	0.290531	+	0.956866i	$0.406168\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.1421	−0.483086	−0.241543	−	0.970390i	$-0.577654\pi$
$857$	−14.1421	−0.483086	−0.241543	+	0.970390i	$0.577654\pi$
$858$	0	0
$859$	−50.9117	−1.73708	−0.868542	−	0.495615i	$-0.834943\pi$
$859$	−50.9117	−1.73708	−0.868542	+	0.495615i	$0.834943\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	−8.00000	−0.272008
$866$	0	0
$867$	0	0
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	67.8823	2.30010
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−8.48528	−0.285876	−0.142938	−	0.989732i	$-0.545655\pi$
$881$	−8.48528	−0.285876	−0.142938	+	0.989732i	$0.545655\pi$
$882$	0	0
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−45.2548	−1.51951	−0.759754	−	0.650210i	$-0.774681\pi$
$887$	−45.2548	−1.51951	−0.759754	+	0.650210i	$0.774681\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−64.0000	−2.14168
$894$	0	0
$895$	−62.2254	−2.07997
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−22.6274	−0.754667
$900$	0	0
$901$	33.9411	1.13074
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−48.0000	−1.59557
$906$	0	0
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46.0000	1.52405	0.762024	−	0.647549i	$-0.224206\pi$
$911$	46.0000	1.52405	0.762024	+	0.647549i	$0.224206\pi$
$912$	0	0
$913$	22.6274	0.748858
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	33.9411	1.11719
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−36.7696	−1.20637	−0.603185	−	0.797601i	$-0.706102\pi$
$929$	−36.7696	−1.20637	−0.603185	+	0.797601i	$0.706102\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−16.0000	−0.523256
$936$	0	0
$937$	45.2548	1.47841	0.739205	−	0.673480i	$-0.235201\pi$
$937$	45.2548	1.47841	0.739205	+	0.673480i	$0.235201\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−19.7990	−0.645429	−0.322714	−	0.946496i	$-0.604595\pi$
$941$	−19.7990	−0.645429	−0.322714	+	0.946496i	$0.604595\pi$
$942$	0	0
$943$	16.9706	0.552638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.0000	−1.36482	−0.682408	−	0.730971i	$-0.739067\pi$
$947$	−42.0000	−1.36482	−0.682408	+	0.730971i	$0.739067\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	−50.9117	−1.64746
$956$	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
$965$	62.2254	2.00311
$966$	0	0
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.6274	0.726148	0.363074	−	0.931760i	$-0.381727\pi$
$971$	22.6274	0.726148	0.363074	+	0.931760i	$0.381727\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	−16.9706	−0.542382
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.6274	0.721703	0.360851	−	0.932623i	$-0.382486\pi$
$983$	22.6274	0.721703	0.360851	+	0.932623i	$0.382486\pi$
$984$	0	0
$985$	11.3137	0.360485
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−39.5980	−1.25408	−0.627040	−	0.778987i	$-0.715734\pi$
$997$	−39.5980	−1.25408	−0.627040	+	0.778987i	$0.715734\pi$
$998$	0	0
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82843 q^{5} +O(q^{10})\)	\(q+2.82843 q^{5} -2.00000 q^{11} +5.65685 q^{13} +2.82843 q^{17} -5.65685 q^{19} -6.00000 q^{23} +3.00000 q^{25} -4.00000 q^{29} +5.65685 q^{31} -2.00000 q^{37} -2.82843 q^{41} +4.00000 q^{43} +11.3137 q^{47} +12.0000 q^{53} -5.65685 q^{55} +11.3137 q^{59} +5.65685 q^{61} +16.0000 q^{65} +12.0000 q^{67} +6.00000 q^{71} -8.00000 q^{79} -11.3137 q^{83} +8.00000 q^{85} +8.48528 q^{89} -16.0000 q^{95} -11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 4 q^{11} - 12 q^{23} + 6 q^{25} - 8 q^{29} - 4 q^{37} + 8 q^{43} + 24 q^{53} + 32 q^{65} + 24 q^{67} + 12 q^{71} - 16 q^{79} + 16 q^{85} - 32 q^{95}+O(q^{100}) \) 2 * q - 4 * q^11 - 12 * q^23 + 6 * q^25 - 8 * q^29 - 4 * q^37 + 8 * q^43 + 24 * q^53 + 32 * q^65 + 24 * q^67 + 12 * q^71 - 16 * q^79 + 16 * q^85 - 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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