LMFDB - Embedded newform 8624.2.a.bq.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8624,2,Mod(1,8624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,-2,0,0,0,-4,0,0,0,0,0,0,0,12,0,-6,0,0,0, -12,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.8629867032$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4312)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{15} -5.65685 q^{17} +6.00000 q^{23} -3.00000 q^{25} +5.65685 q^{27} -6.00000 q^{29} +7.07107 q^{31} +1.41421 q^{33} +6.00000 q^{37} +8.00000 q^{43} -1.41421 q^{45} +1.41421 q^{47} +8.00000 q^{51} -1.41421 q^{55} +9.89949 q^{59} -8.48528 q^{61} -14.0000 q^{67} -8.48528 q^{69} -2.00000 q^{71} +2.82843 q^{73} +4.24264 q^{75} -5.00000 q^{81} -5.65685 q^{83} -8.00000 q^{85} +8.48528 q^{87} +18.3848 q^{89} -10.0000 q^{93} +9.89949 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 2 q^{11} - 4 q^{15} + 12 q^{23} - 6 q^{25} - 12 q^{29} + 12 q^{37} + 16 q^{43} + 16 q^{51} - 28 q^{67} - 4 q^{71} - 10 q^{81} - 16 q^{85} - 20 q^{93} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 2 * q^11 - 4 * q^15 + 12 * q^23 - 6 * q^25 - 12 * q^29 + 12 * q^37 + 16 * q^43 + 16 * q^51 - 28 * q^67 - 4 * q^71 - 10 * q^81 - 16 * q^85 - 20 * q^93 + 2 * q^99

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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	0	0
$17$	−5.65685	−1.37199	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	−5.65685	−1.37199	−0.685994	+	0.727607i	$0.740633\pi$
$18$	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
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	5.65685	1.08866
$28$	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
$31$	7.07107	1.27000	0.635001	−	0.772512i	$-0.281000\pi$
$31$	7.07107	1.27000	0.635001	+	0.772512i	$0.281000\pi$
$32$	0	0
$33$	1.41421	0.246183
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−1.41421	−0.210819
$46$	0	0
$47$	1.41421	0.206284	0.103142	−	0.994667i	$-0.467110\pi$
$47$	1.41421	0.206284	0.103142	+	0.994667i	$0.467110\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.00000	1.12022
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.89949	1.28880	0.644402	−	0.764687i	$-0.277106\pi$
$59$	9.89949	1.28880	0.644402	+	0.764687i	$0.277106\pi$
$60$	0	0
$61$	−8.48528	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$61$	−8.48528	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−14.0000	−1.71037	−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000	−1.71037	−0.855186	+	0.518321i	$0.826557\pi$
$68$	0	0
$69$	−8.48528	−1.02151
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	2.82843	0.331042	0.165521	−	0.986206i	$-0.447069\pi$
$73$	2.82843	0.331042	0.165521	+	0.986206i	$0.447069\pi$
$74$	0	0
$75$	4.24264	0.489898
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	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
$85$	−8.00000	−0.867722
$86$	0	0
$87$	8.48528	0.909718
$88$	0	0
$89$	18.3848	1.94878	0.974391	−	0.224860i	$-0.0721923\pi$
$89$	18.3848	1.94878	0.974391	+	0.224860i	$0.0721923\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.89949	1.00514	0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949	1.00514	0.502571	+	0.864536i	$0.332388\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−5.65685	−0.562878	−0.281439	−	0.959579i	$-0.590812\pi$
$101$	−5.65685	−0.562878	−0.281439	+	0.959579i	$0.590812\pi$
$102$	0	0
$103$	−12.7279	−1.25412	−0.627060	−	0.778971i	$-0.715742\pi$
$103$	−12.7279	−1.25412	−0.627060	+	0.778971i	$0.715742\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−8.48528	−0.805387
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	8.48528	0.791257
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	−11.3137	−0.996116
$130$	0	0
$131$	−2.82843	−0.247121	−0.123560	−	0.992337i	$-0.539431\pi$
$131$	−2.82843	−0.247121	−0.123560	+	0.992337i	$0.539431\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.00000	0.688530
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−11.3137	−0.959616	−0.479808	−	0.877373i	$-0.659294\pi$
$139$	−11.3137	−0.959616	−0.479808	+	0.877373i	$0.659294\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−8.48528	−0.704664
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	5.65685	0.457330
$154$	0	0
$155$	10.0000	0.803219
$156$	0	0
$157$	−9.89949	−0.790066	−0.395033	−	0.918667i	$-0.629267\pi$
$157$	−9.89949	−0.790066	−0.395033	+	0.918667i	$0.629267\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	2.00000	0.155700
$166$	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
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.3137	0.860165	0.430083	−	0.902790i	$-0.358484\pi$
$173$	11.3137	0.860165	0.430083	+	0.902790i	$0.358484\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.0000	−1.05230
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	12.7279	0.946059	0.473029	−	0.881047i	$-0.343160\pi$
$181$	12.7279	0.946059	0.473029	+	0.881047i	$0.343160\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	8.48528	0.623850
$186$	0	0
$187$	5.65685	0.413670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−4.24264	−0.300753	−0.150376	−	0.988629i	$-0.548049\pi$
$199$	−4.24264	−0.300753	−0.150376	+	0.988629i	$0.548049\pi$
$200$	0	0
$201$	19.7990	1.39651
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	2.82843	0.193801
$214$	0	0
$215$	11.3137	0.771589
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−7.07107	−0.473514	−0.236757	−	0.971569i	$-0.576084\pi$
$223$	−7.07107	−0.473514	−0.236757	+	0.971569i	$0.576084\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	14.1421	0.938647	0.469323	−	0.883026i	$-0.344498\pi$
$227$	14.1421	0.938647	0.469323	+	0.883026i	$0.344498\pi$
$228$	0	0
$229$	18.3848	1.21490	0.607450	−	0.794358i	$-0.292192\pi$
$229$	18.3848	1.21490	0.607450	+	0.794358i	$0.292192\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−28.0000	−1.81117	−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000	−1.81117	−0.905585	+	0.424165i	$0.860568\pi$
$240$	0	0
$241$	−11.3137	−0.728780	−0.364390	−	0.931246i	$-0.618722\pi$
$241$	−11.3137	−0.728780	−0.364390	+	0.931246i	$0.618722\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	4.24264	0.267793	0.133897	−	0.990995i	$-0.457251\pi$
$251$	4.24264	0.267793	0.133897	+	0.990995i	$0.457251\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	0	0
$255$	11.3137	0.708492
$256$	0	0
$257$	9.89949	0.617514	0.308757	−	0.951141i	$-0.400087\pi$
$257$	9.89949	0.617514	0.308757	+	0.951141i	$0.400087\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−26.0000	−1.59117
$268$	0	0
$269$	21.2132	1.29339	0.646696	−	0.762748i	$-0.276150\pi$
$269$	21.2132	1.29339	0.646696	+	0.762748i	$0.276150\pi$
$270$	0	0
$271$	−2.82843	−0.171815	−0.0859074	−	0.996303i	$-0.527379\pi$
$271$	−2.82843	−0.171815	−0.0859074	+	0.996303i	$0.527379\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.00000	0.180907
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	0	0
$279$	−7.07107	−0.423334
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	19.7990	1.17693	0.588464	−	0.808523i	$-0.299733\pi$
$283$	19.7990	1.17693	0.588464	+	0.808523i	$0.299733\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	−14.0000	−0.820695
$292$	0	0
$293$	−2.82843	−0.165238	−0.0826192	−	0.996581i	$-0.526329\pi$
$293$	−2.82843	−0.165238	−0.0826192	+	0.996581i	$0.526329\pi$
$294$	0	0
$295$	14.0000	0.815112
$296$	0	0
$297$	−5.65685	−0.328244
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	8.00000	0.459588
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	2.82843	0.161427	0.0807134	−	0.996737i	$-0.474280\pi$
$307$	2.82843	0.161427	0.0807134	+	0.996737i	$0.474280\pi$
$308$	0	0
$309$	18.0000	1.02398
$310$	0	0
$311$	−1.41421	−0.0801927	−0.0400963	−	0.999196i	$-0.512766\pi$
$311$	−1.41421	−0.0801927	−0.0400963	+	0.999196i	$0.512766\pi$
$312$	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
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.1421	0.782062
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	−19.7990	−1.08173
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	2.82843	0.153619
$340$	0	0
$341$	−7.07107	−0.382920
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−12.0000	−0.646058
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	−2.82843	−0.151402	−0.0757011	−	0.997131i	$-0.524119\pi$
$349$	−2.82843	−0.151402	−0.0757011	+	0.997131i	$0.524119\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.2132	1.12906	0.564532	−	0.825411i	$-0.309057\pi$
$353$	21.2132	1.12906	0.564532	+	0.825411i	$0.309057\pi$
$354$	0	0
$355$	−2.82843	−0.150117
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−1.41421	−0.0742270
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	4.24264	0.221464	0.110732	−	0.993850i	$-0.464680\pi$
$367$	4.24264	0.221464	0.110732	+	0.993850i	$0.464680\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	0	0
$375$	16.0000	0.826236
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−22.0000	−1.13006	−0.565032	−	0.825069i	$-0.691136\pi$
$379$	−22.0000	−1.13006	−0.565032	+	0.825069i	$0.691136\pi$
$380$	0	0
$381$	22.6274	1.15924
$382$	0	0
$383$	1.41421	0.0722629	0.0361315	−	0.999347i	$-0.488496\pi$
$383$	1.41421	0.0722629	0.0361315	+	0.999347i	$0.488496\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	−33.9411	−1.71648
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−18.3848	−0.922705	−0.461353	−	0.887217i	$-0.652636\pi$
$397$	−18.3848	−0.922705	−0.461353	+	0.887217i	$0.652636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−7.07107	−0.351364
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−8.48528	−0.419570	−0.209785	−	0.977748i	$-0.567276\pi$
$409$	−8.48528	−0.419570	−0.209785	+	0.977748i	$0.567276\pi$
$410$	0	0
$411$	25.4558	1.25564
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	−24.0416	−1.17451	−0.587255	−	0.809402i	$-0.699792\pi$
$419$	−24.0416	−1.17451	−0.587255	+	0.809402i	$0.699792\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	0	0
$423$	−1.41421	−0.0687614
$424$	0	0
$425$	16.9706	0.823193
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	0	0
$433$	−21.2132	−1.01944	−0.509721	−	0.860340i	$-0.670251\pi$
$433$	−21.2132	−1.01944	−0.509721	+	0.860340i	$0.670251\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−8.48528	−0.404980	−0.202490	−	0.979284i	$-0.564903\pi$
$439$	−8.48528	−0.404980	−0.202490	+	0.979284i	$0.564903\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	26.0000	1.23252
$446$	0	0
$447$	2.82843	0.133780
$448$	0	0
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−5.65685	−0.265782
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	−32.0000	−1.49363
$460$	0	0
$461$	−36.7696	−1.71253	−0.856264	−	0.516538i	$-0.827221\pi$
$461$	−36.7696	−1.71253	−0.856264	+	0.516538i	$0.827221\pi$
$462$	0	0
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	0	0
$465$	−14.1421	−0.655826
$466$	0	0
$467$	−29.6985	−1.37428	−0.687141	−	0.726524i	$-0.741135\pi$
$467$	−29.6985	−1.37428	−0.687141	+	0.726524i	$0.741135\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	−8.00000	−0.367840
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−36.7696	−1.68004	−0.840022	−	0.542553i	$-0.817458\pi$
$479$	−36.7696	−1.68004	−0.840022	+	0.542553i	$0.817458\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.0000	0.635707
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−8.48528	−0.383718
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	33.9411	1.52863
$494$	0	0
$495$	1.41421	0.0635642
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	2.82843	0.126113	0.0630567	−	0.998010i	$-0.479915\pi$
$503$	2.82843	0.126113	0.0630567	+	0.998010i	$0.479915\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	18.3848	0.816497
$508$	0	0
$509$	−18.3848	−0.814891	−0.407445	−	0.913230i	$-0.633580\pi$
$509$	−18.3848	−0.814891	−0.407445	+	0.913230i	$0.633580\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−18.0000	−0.793175
$516$	0	0
$517$	−1.41421	−0.0621970
$518$	0	0
$519$	−16.0000	−0.702322
$520$	0	0
$521$	−35.3553	−1.54895	−0.774473	−	0.632607i	$-0.781985\pi$
$521$	−35.3553	−1.54895	−0.774473	+	0.632607i	$0.781985\pi$
$522$	0	0
$523$	14.1421	0.618392	0.309196	−	0.950998i	$-0.399940\pi$
$523$	14.1421	0.618392	0.309196	+	0.950998i	$0.399940\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−40.0000	−1.74243
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−9.89949	−0.429601
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−16.9706	−0.732334
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	−14.1421	−0.605783
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	8.48528	0.362143
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−12.0000	−0.509372
$556$	0	0
$557$	−38.0000	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	22.6274	0.953632	0.476816	−	0.879003i	$-0.341791\pi$
$563$	22.6274	0.953632	0.476816	+	0.879003i	$0.341791\pi$
$564$	0	0
$565$	−2.82843	−0.118993
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	0	0
$573$	−11.3137	−0.472637
$574$	0	0
$575$	−18.0000	−0.750652
$576$	0	0
$577$	−15.5563	−0.647619	−0.323810	−	0.946122i	$-0.604964\pi$
$577$	−15.5563	−0.647619	−0.323810	+	0.946122i	$0.604964\pi$
$578$	0	0
$579$	31.1127	1.29300
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.07107	−0.291854	−0.145927	−	0.989295i	$-0.546616\pi$
$587$	−7.07107	−0.291854	−0.145927	+	0.989295i	$0.546616\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	2.82843	0.116346
$592$	0	0
$593$	36.7696	1.50994	0.754972	−	0.655757i	$-0.227650\pi$
$593$	36.7696	1.50994	0.754972	+	0.655757i	$0.227650\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.00000	0.245564
$598$	0	0
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−16.9706	−0.692244	−0.346122	−	0.938190i	$-0.612502\pi$
$601$	−16.9706	−0.692244	−0.346122	+	0.938190i	$0.612502\pi$
$602$	0	0
$603$	14.0000	0.570124
$604$	0	0
$605$	1.41421	0.0574960
$606$	0	0
$607$	28.2843	1.14802	0.574012	−	0.818847i	$-0.305386\pi$
$607$	28.2843	1.14802	0.574012	+	0.818847i	$0.305386\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	46.6690	1.87579	0.937894	−	0.346923i	$-0.112773\pi$
$619$	46.6690	1.87579	0.937894	+	0.346923i	$0.112773\pi$
$620$	0	0
$621$	33.9411	1.36201
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−33.9411	−1.35332
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	11.3137	0.449680
$634$	0	0
$635$	−22.6274	−0.897942
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.00000	0.0791188
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	15.5563	0.613483	0.306741	−	0.951793i	$-0.400761\pi$
$643$	15.5563	0.613483	0.306741	+	0.951793i	$0.400761\pi$
$644$	0	0
$645$	−16.0000	−0.629999
$646$	0	0
$647$	−35.3553	−1.38996	−0.694981	−	0.719028i	$-0.744587\pi$
$647$	−35.3553	−1.38996	−0.694981	+	0.719028i	$0.744587\pi$
$648$	0	0
$649$	−9.89949	−0.388589
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	−2.82843	−0.110347
$658$	0	0
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	38.1838	1.48518	0.742588	−	0.669748i	$-0.233598\pi$
$661$	38.1838	1.48518	0.742588	+	0.669748i	$0.233598\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−36.0000	−1.39393
$668$	0	0
$669$	10.0000	0.386622
$670$	0	0
$671$	8.48528	0.327571
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	−16.9706	−0.653197
$676$	0	0
$677$	−28.2843	−1.08705	−0.543526	−	0.839392i	$-0.682911\pi$
$677$	−28.2843	−1.08705	−0.543526	+	0.839392i	$0.682911\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−25.4558	−0.972618
$686$	0	0
$687$	−26.0000	−0.991962
$688$	0	0
$689$	0	0
$690$	0	0
$691$	29.6985	1.12978	0.564892	−	0.825165i	$-0.308918\pi$
$691$	29.6985	1.12978	0.564892	+	0.825165i	$0.308918\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.0000	−0.606915
$696$	0	0
$697$	0	0
$698$	0	0
$699$	2.82843	0.106981
$700$	0	0
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−2.82843	−0.106525
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	42.4264	1.58888
$714$	0	0
$715$	0	0
$716$	0	0
$717$	39.5980	1.47881
$718$	0	0
$719$	−46.6690	−1.74046	−0.870231	−	0.492644i	$-0.836030\pi$
$719$	−46.6690	−1.74046	−0.870231	+	0.492644i	$0.836030\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	16.0000	0.595046
$724$	0	0
$725$	18.0000	0.668503
$726$	0	0
$727$	−7.07107	−0.262251	−0.131126	−	0.991366i	$-0.541859\pi$
$727$	−7.07107	−0.262251	−0.131126	+	0.991366i	$0.541859\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−45.2548	−1.67381
$732$	0	0
$733$	−48.0833	−1.77600	−0.887998	−	0.459848i	$-0.847904\pi$
$733$	−48.0833	−1.77600	−0.887998	+	0.459848i	$0.847904\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.0000	0.515697
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	−2.82843	−0.103626
$746$	0	0
$747$	5.65685	0.206973
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	5.65685	0.205874
$756$	0	0
$757$	−32.0000	−1.16306	−0.581530	−	0.813525i	$-0.697546\pi$
$757$	−32.0000	−1.16306	−0.581530	+	0.813525i	$0.697546\pi$
$758$	0	0
$759$	8.48528	0.307996
$760$	0	0
$761$	−36.7696	−1.33290	−0.666448	−	0.745552i	$-0.732186\pi$
$761$	−36.7696	−1.33290	−0.666448	+	0.745552i	$0.732186\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.00000	0.289241
$766$	0	0
$767$	0	0
$768$	0	0
$769$	36.7696	1.32594	0.662972	−	0.748644i	$-0.269295\pi$
$769$	36.7696	1.32594	0.662972	+	0.748644i	$0.269295\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	26.8701	0.966449	0.483224	−	0.875497i	$-0.339466\pi$
$773$	26.8701	0.966449	0.483224	+	0.875497i	$0.339466\pi$
$774$	0	0
$775$	−21.2132	−0.762001
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	2.00000	0.0715656
$782$	0	0
$783$	−33.9411	−1.21296
$784$	0	0
$785$	−14.0000	−0.499681
$786$	0	0
$787$	−39.5980	−1.41152	−0.705758	−	0.708453i	$-0.749393\pi$
$787$	−39.5980	−1.41152	−0.705758	+	0.708453i	$0.749393\pi$
$788$	0	0
$789$	33.9411	1.20834
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	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
$799$	−8.00000	−0.283020
$800$	0	0
$801$	−18.3848	−0.649594
$802$	0	0
$803$	−2.82843	−0.0998130
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	8.48528	0.297959	0.148979	−	0.988840i	$-0.452401\pi$
$811$	8.48528	0.297959	0.148979	+	0.988840i	$0.452401\pi$
$812$	0	0
$813$	4.00000	0.140286
$814$	0	0
$815$	8.48528	0.297226
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	0	0
$825$	−4.24264	−0.147710
$826$	0	0
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	0	0
$829$	−35.3553	−1.22794	−0.613971	−	0.789329i	$-0.710429\pi$
$829$	−35.3553	−1.22794	−0.613971	+	0.789329i	$0.710429\pi$
$830$	0	0
$831$	36.7696	1.27552
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	40.0000	1.38260
$838$	0	0
$839$	49.4975	1.70884	0.854421	−	0.519581i	$-0.173912\pi$
$839$	49.4975	1.70884	0.854421	+	0.519581i	$0.173912\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	42.4264	1.46124
$844$	0	0
$845$	−18.3848	−0.632456
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	36.0000	1.23406
$852$	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	$-0.422346\pi$
$857$	14.1421	0.483086	0.241543	+	0.970390i	$0.422346\pi$
$858$	0	0
$859$	−46.6690	−1.59233	−0.796164	−	0.605081i	$-0.793141\pi$
$859$	−46.6690	−1.59233	−0.796164	+	0.605081i	$0.793141\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	16.0000	0.544016
$866$	0	0
$867$	−21.2132	−0.720438
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−9.89949	−0.335047
$874$	0	0
$875$	0	0
$876$	0	0
$877$	58.0000	1.95852	0.979260	−	0.202606i	$-0.0649409\pi$
$877$	58.0000	1.95852	0.979260	+	0.202606i	$0.0649409\pi$
$878$	0	0
$879$	4.00000	0.134917
$880$	0	0
$881$	−38.1838	−1.28644	−0.643222	−	0.765680i	$-0.722403\pi$
$881$	−38.1838	−1.28644	−0.643222	+	0.765680i	$0.722403\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	−19.7990	−0.665536
$886$	0	0
$887$	−2.82843	−0.0949693	−0.0474846	−	0.998872i	$-0.515121\pi$
$887$	−2.82843	−0.0949693	−0.0474846	+	0.998872i	$0.515121\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	0	0
$894$	0	0
$895$	16.9706	0.567263
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−42.4264	−1.41500
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	0	0
$909$	5.65685	0.187626
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	5.65685	0.187215
$914$	0	0
$915$	16.9706	0.561029
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−18.0000	−0.591836
$926$	0	0
$927$	12.7279	0.418040
$928$	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
$933$	2.00000	0.0654771
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	31.1127	1.01641	0.508204	−	0.861237i	$-0.330310\pi$
$937$	31.1127	1.01641	0.508204	+	0.861237i	$0.330310\pi$
$938$	0	0
$939$	−18.0000	−0.587408
$940$	0	0
$941$	−42.4264	−1.38306	−0.691531	−	0.722347i	$-0.743063\pi$
$941$	−42.4264	−1.38306	−0.691531	+	0.722347i	$0.743063\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	25.4558	0.825462
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	0	0
$955$	11.3137	0.366103
$956$	0	0
$957$	−8.48528	−0.274290
$958$	0	0
$959$	0	0
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−31.1127	−1.00155
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−7.07107	−0.226921	−0.113461	−	0.993542i	$-0.536194\pi$
$971$	−7.07107	−0.226921	−0.113461	+	0.993542i	$0.536194\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.0000	−0.895799	−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000	−0.895799	−0.447900	+	0.894084i	$0.647828\pi$
$978$	0	0
$979$	−18.3848	−0.587580
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−29.6985	−0.947235	−0.473617	−	0.880731i	$-0.657052\pi$
$983$	−29.6985	−0.947235	−0.473617	+	0.880731i	$0.657052\pi$
$984$	0	0
$985$	−2.82843	−0.0901212
$986$	0	0
$987$	0	0
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	18.0000	0.571789	0.285894	−	0.958261i	$-0.407709\pi$
$991$	18.0000	0.571789	0.285894	+	0.958261i	$0.407709\pi$
$992$	0	0
$993$	−5.65685	−0.179515
$994$	0	0
$995$	−6.00000	−0.190213
$996$	0	0
$997$	50.9117	1.61239	0.806195	−	0.591650i	$-0.201523\pi$
$997$	50.9117	1.61239	0.806195	+	0.591650i	$0.201523\pi$
$998$	0	0
$999$	33.9411	1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.bq.1.1		2
4.3	odd	2		4312.2.a.s.1.2	yes	2
7.6	odd	2	inner	8624.2.a.bq.1.2		2
28.27	even	2		4312.2.a.s.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4312.2.a.s.1.1	✓	2	28.27	even	2
4312.2.a.s.1.2	yes	2	4.3	odd	2
8624.2.a.bq.1.1		2	1.1	even	1	trivial
8624.2.a.bq.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{15} -5.65685 q^{17} +6.00000 q^{23} -3.00000 q^{25} +5.65685 q^{27} -6.00000 q^{29} +7.07107 q^{31} +1.41421 q^{33} +6.00000 q^{37} +8.00000 q^{43} -1.41421 q^{45} +1.41421 q^{47} +8.00000 q^{51} -1.41421 q^{55} +9.89949 q^{59} -8.48528 q^{61} -14.0000 q^{67} -8.48528 q^{69} -2.00000 q^{71} +2.82843 q^{73} +4.24264 q^{75} -5.00000 q^{81} -5.65685 q^{83} -8.00000 q^{85} +8.48528 q^{87} +18.3848 q^{89} -10.0000 q^{93} +9.89949 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 2 q^{11} - 4 q^{15} + 12 q^{23} - 6 q^{25} - 12 q^{29} + 12 q^{37} + 16 q^{43} + 16 q^{51} - 28 q^{67} - 4 q^{71} - 10 q^{81} - 16 q^{85} - 20 q^{93} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 2 * q^11 - 4 * q^15 + 12 * q^23 - 6 * q^25 - 12 * q^29 + 12 * q^37 + 16 * q^43 + 16 * q^51 - 28 * q^67 - 4 * q^71 - 10 * q^81 - 16 * q^85 - 20 * q^93 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more