LMFDB - Embedded newform 7488.2.a.ct.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7488,2,Mod(1,7488)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7488, 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("7488.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7488 = 2^{6} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7488.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:	$59.7919810335$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7488.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607 q^{5} +3.23607 q^{7} +O(q^{10})$	$q+3.23607 q^{5} +3.23607 q^{7} +2.00000 q^{11} +1.00000 q^{13} +4.47214 q^{17} -0.763932 q^{19} -6.47214 q^{23} +5.47214 q^{25} -4.47214 q^{29} -5.70820 q^{31} +10.4721 q^{35} +8.47214 q^{37} +3.23607 q^{41} +2.47214 q^{43} +10.9443 q^{47} +3.47214 q^{49} -0.472136 q^{53} +6.47214 q^{55} -0.472136 q^{59} +3.52786 q^{61} +3.23607 q^{65} -11.2361 q^{67} +4.47214 q^{71} -8.47214 q^{73} +6.47214 q^{77} +8.94427 q^{79} +16.4721 q^{83} +14.4721 q^{85} -12.1803 q^{89} +3.23607 q^{91} -2.47214 q^{95} +4.47214 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 2 * q^7	$ 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} + 2 q^{13} - 6 q^{19} - 4 q^{23} + 2 q^{25} + 2 q^{31} + 12 q^{35} + 8 q^{37} + 2 q^{41} - 4 q^{43} + 4 q^{47} - 2 q^{49} + 8 q^{53} + 4 q^{55} + 8 q^{59} + 16 q^{61} + 2 q^{65} - 18 q^{67} - 8 q^{73} + 4 q^{77} + 24 q^{83} + 20 q^{85} - 2 q^{89} + 2 q^{91} + 4 q^{95}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 + 2 * q^13 - 6 * q^19 - 4 * q^23 + 2 * q^25 + 2 * q^31 + 12 * q^35 + 8 * q^37 + 2 * q^41 - 4 * q^43 + 4 * q^47 - 2 * q^49 + 8 * q^53 + 4 * q^55 + 8 * q^59 + 16 * q^61 + 2 * q^65 - 18 * q^67 - 8 * q^73 + 4 * q^77 + 24 * q^83 + 20 * q^85 - 2 * q^89 + 2 * q^91 + 4 * 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
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	−0.763932	−0.175258	−0.0876290	−	0.996153i	$-0.527929\pi$
$19$	−0.763932	−0.175258	−0.0876290	+	0.996153i	$0.527929\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.47214	−1.34953	−0.674767	−	0.738031i	$-0.735756\pi$
$23$	−6.47214	−1.34953	−0.674767	+	0.738031i	$0.735756\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	−5.70820	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$31$	−5.70820	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.4721	1.77011
$36$	0	0
$37$	8.47214	1.39281	0.696405	−	0.717649i	$-0.254782\pi$
$37$	8.47214	1.39281	0.696405	+	0.717649i	$0.254782\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.23607	0.505389	0.252694	−	0.967546i	$-0.418683\pi$
$41$	3.23607	0.505389	0.252694	+	0.967546i	$0.418683\pi$
$42$	0	0
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.9443	1.59639	0.798193	−	0.602402i	$-0.205789\pi$
$47$	10.9443	1.59639	0.798193	+	0.602402i	$0.205789\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	6.47214	0.872703
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.472136	−0.0614669	−0.0307334	−	0.999528i	$-0.509784\pi$
$59$	−0.472136	−0.0614669	−0.0307334	+	0.999528i	$0.509784\pi$
$60$	0	0
$61$	3.52786	0.451697	0.225848	−	0.974162i	$-0.427485\pi$
$61$	3.52786	0.451697	0.225848	+	0.974162i	$0.427485\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.23607	0.401385
$66$	0	0
$67$	−11.2361	−1.37270	−0.686352	−	0.727269i	$-0.740789\pi$
$67$	−11.2361	−1.37270	−0.686352	+	0.727269i	$0.740789\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.47214	0.530745	0.265372	−	0.964146i	$-0.414505\pi$
$71$	4.47214	0.530745	0.265372	+	0.964146i	$0.414505\pi$
$72$	0	0
$73$	−8.47214	−0.991589	−0.495794	−	0.868440i	$-0.665123\pi$
$73$	−8.47214	−0.991589	−0.495794	+	0.868440i	$0.665123\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.47214	0.737568
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.4721	1.80805	0.904026	−	0.427478i	$-0.140598\pi$
$83$	16.4721	1.80805	0.904026	+	0.427478i	$0.140598\pi$
$84$	0	0
$85$	14.4721	1.56972
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−12.1803	−1.29111	−0.645557	−	0.763712i	$-0.723375\pi$
$89$	−12.1803	−1.29111	−0.645557	+	0.763712i	$0.723375\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.47214	−0.253636
$96$	0	0
$97$	4.47214	0.454077	0.227038	−	0.973886i	$-0.427096\pi$
$97$	4.47214	0.454077	0.227038	+	0.973886i	$0.427096\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.9443	1.48701	0.743505	−	0.668730i	$-0.233162\pi$
$101$	14.9443	1.48701	0.743505	+	0.668730i	$0.233162\pi$
$102$	0	0
$103$	−10.4721	−1.03185	−0.515925	−	0.856634i	$-0.672552\pi$
$103$	−10.4721	−1.03185	−0.515925	+	0.856634i	$0.672552\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.52786	0.534399	0.267199	−	0.963641i	$-0.413902\pi$
$107$	5.52786	0.534399	0.267199	+	0.963641i	$0.413902\pi$
$108$	0	0
$109$	18.9443	1.81453	0.907266	−	0.420557i	$-0.138165\pi$
$109$	18.9443	1.81453	0.907266	+	0.420557i	$0.138165\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−20.9443	−1.95306
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14.4721	1.32666
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.52786	0.136656
$126$	0	0
$127$	−2.47214	−0.219367	−0.109683	−	0.993967i	$-0.534984\pi$
$127$	−2.47214	−0.219367	−0.109683	+	0.993967i	$0.534984\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.4164	1.34694	0.673469	−	0.739216i	$-0.264804\pi$
$131$	15.4164	1.34694	0.673469	+	0.739216i	$0.264804\pi$
$132$	0	0
$133$	−2.47214	−0.214361
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.2361	0.959962	0.479981	−	0.877279i	$-0.340644\pi$
$137$	11.2361	0.959962	0.479981	+	0.877279i	$0.340644\pi$
$138$	0	0
$139$	−21.8885	−1.85656	−0.928281	−	0.371879i	$-0.878713\pi$
$139$	−21.8885	−1.85656	−0.928281	+	0.371879i	$0.878713\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	−14.4721	−1.20185
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.7082	1.77841	0.889203	−	0.457513i	$-0.151260\pi$
$149$	21.7082	1.77841	0.889203	+	0.457513i	$0.151260\pi$
$150$	0	0
$151$	−23.2361	−1.89092	−0.945462	−	0.325732i	$-0.894389\pi$
$151$	−23.2361	−1.89092	−0.945462	+	0.325732i	$0.894389\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−18.4721	−1.48372
$156$	0	0
$157$	−13.4164	−1.07075	−0.535373	−	0.844616i	$-0.679829\pi$
$157$	−13.4164	−1.07075	−0.535373	+	0.844616i	$0.679829\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−20.9443	−1.65064
$162$	0	0
$163$	−7.23607	−0.566773	−0.283386	−	0.959006i	$-0.591458\pi$
$163$	−7.23607	−0.566773	−0.283386	+	0.959006i	$0.591458\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−19.8885	−1.53902	−0.769511	−	0.638634i	$-0.779500\pi$
$167$	−19.8885	−1.53902	−0.769511	+	0.638634i	$0.779500\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	17.7082	1.33861
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.4164	−1.45125	−0.725625	−	0.688090i	$-0.758449\pi$
$179$	−19.4164	−1.45125	−0.725625	+	0.688090i	$0.758449\pi$
$180$	0	0
$181$	17.4164	1.29455	0.647276	−	0.762256i	$-0.275908\pi$
$181$	17.4164	1.29455	0.647276	+	0.762256i	$0.275908\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	27.4164	2.01569
$186$	0	0
$187$	8.94427	0.654070
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.4721	0.757737	0.378869	−	0.925450i	$-0.376313\pi$
$191$	10.4721	0.757737	0.378869	+	0.925450i	$0.376313\pi$
$192$	0	0
$193$	−15.8885	−1.14368	−0.571841	−	0.820364i	$-0.693771\pi$
$193$	−15.8885	−1.14368	−0.571841	+	0.820364i	$0.693771\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27.2361	1.94049	0.970245	−	0.242126i	$-0.0778448\pi$
$197$	27.2361	1.94049	0.970245	+	0.242126i	$0.0778448\pi$
$198$	0	0
$199$	−18.4721	−1.30945	−0.654727	−	0.755865i	$-0.727217\pi$
$199$	−18.4721	−1.30945	−0.654727	+	0.755865i	$0.727217\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−14.4721	−1.01574
$204$	0	0
$205$	10.4721	0.731406
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.52786	−0.105685
$210$	0	0
$211$	0.944272	0.0650064	0.0325032	−	0.999472i	$-0.489652\pi$
$211$	0.944272	0.0650064	0.0325032	+	0.999472i	$0.489652\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.00000	0.545595
$216$	0	0
$217$	−18.4721	−1.25397
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.47214	0.300828
$222$	0	0
$223$	−26.6525	−1.78478	−0.892391	−	0.451263i	$-0.850974\pi$
$223$	−26.6525	−1.78478	−0.892391	+	0.451263i	$0.850974\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.94427	−0.460908	−0.230454	−	0.973083i	$-0.574021\pi$
$227$	−6.94427	−0.460908	−0.230454	+	0.973083i	$0.574021\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.41641	−0.616889	−0.308445	−	0.951242i	$-0.599808\pi$
$233$	−9.41641	−0.616889	−0.308445	+	0.951242i	$0.599808\pi$
$234$	0	0
$235$	35.4164	2.31031
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.52786	−0.228199	−0.114099	−	0.993469i	$-0.536398\pi$
$239$	−3.52786	−0.228199	−0.114099	+	0.993469i	$0.536398\pi$
$240$	0	0
$241$	22.3607	1.44038	0.720189	−	0.693778i	$-0.244055\pi$
$241$	22.3607	1.44038	0.720189	+	0.693778i	$0.244055\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	11.2361	0.717846
$246$	0	0
$247$	−0.763932	−0.0486078
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.0557	−0.697831	−0.348916	−	0.937154i	$-0.613450\pi$
$251$	−11.0557	−0.697831	−0.348916	+	0.937154i	$0.613450\pi$
$252$	0	0
$253$	−12.9443	−0.813799
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.52786	0.220062	0.110031	−	0.993928i	$-0.464905\pi$
$257$	3.52786	0.220062	0.110031	+	0.993928i	$0.464905\pi$
$258$	0	0
$259$	27.4164	1.70357
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.94427	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$264$	0	0
$265$	−1.52786	−0.0938559
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.5279	0.702866	0.351433	−	0.936213i	$-0.385694\pi$
$269$	11.5279	0.702866	0.351433	+	0.936213i	$0.385694\pi$
$270$	0	0
$271$	21.1246	1.28323	0.641614	−	0.767027i	$-0.278265\pi$
$271$	21.1246	1.28323	0.641614	+	0.767027i	$0.278265\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	10.9443	0.659964
$276$	0	0
$277$	−2.94427	−0.176904	−0.0884521	−	0.996080i	$-0.528192\pi$
$277$	−2.94427	−0.176904	−0.0884521	+	0.996080i	$0.528192\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.70820	−0.579143	−0.289571	−	0.957156i	$-0.593513\pi$
$281$	−9.70820	−0.579143	−0.289571	+	0.957156i	$0.593513\pi$
$282$	0	0
$283$	13.5279	0.804148	0.402074	−	0.915607i	$-0.368289\pi$
$283$	13.5279	0.804148	0.402074	+	0.915607i	$0.368289\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.4721	0.618151
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−12.1803	−0.711583	−0.355792	−	0.934565i	$-0.615789\pi$
$293$	−12.1803	−0.711583	−0.355792	+	0.934565i	$0.615789\pi$
$294$	0	0
$295$	−1.52786	−0.0889557
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.47214	−0.374293
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.4164	0.653702
$306$	0	0
$307$	11.2361	0.641276	0.320638	−	0.947202i	$-0.396103\pi$
$307$	11.2361	0.641276	0.320638	+	0.947202i	$0.396103\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.4164	1.55464	0.777321	−	0.629104i	$-0.216578\pi$
$311$	27.4164	1.55464	0.777321	+	0.629104i	$0.216578\pi$
$312$	0	0
$313$	−3.88854	−0.219793	−0.109897	−	0.993943i	$-0.535052\pi$
$313$	−3.88854	−0.219793	−0.109897	+	0.993943i	$0.535052\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.76393	−0.267569	−0.133785	−	0.991010i	$-0.542713\pi$
$317$	−4.76393	−0.267569	−0.133785	+	0.991010i	$0.542713\pi$
$318$	0	0
$319$	−8.94427	−0.500783
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.41641	−0.190094
$324$	0	0
$325$	5.47214	0.303539
$326$	0	0
$327$	0	0
$328$	0	0
$329$	35.4164	1.95257
$330$	0	0
$331$	−8.18034	−0.449632	−0.224816	−	0.974401i	$-0.572178\pi$
$331$	−8.18034	−0.449632	−0.224816	+	0.974401i	$0.572178\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−36.3607	−1.98660
$336$	0	0
$337$	10.9443	0.596172	0.298086	−	0.954539i	$-0.403652\pi$
$337$	10.9443	0.596172	0.298086	+	0.954539i	$0.403652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.4164	−0.618233
$342$	0	0
$343$	−11.4164	−0.616428
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	−30.3607	−1.62517	−0.812585	−	0.582843i	$-0.801940\pi$
$349$	−30.3607	−1.62517	−0.812585	+	0.582843i	$0.801940\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−27.5967	−1.46883	−0.734413	−	0.678702i	$-0.762543\pi$
$353$	−27.5967	−1.46883	−0.734413	+	0.678702i	$0.762543\pi$
$354$	0	0
$355$	14.4721	0.768101
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.8328	−0.888402	−0.444201	−	0.895927i	$-0.646512\pi$
$359$	−16.8328	−0.888402	−0.444201	+	0.895927i	$0.646512\pi$
$360$	0	0
$361$	−18.4164	−0.969285
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−27.4164	−1.43504
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.52786	−0.0793227
$372$	0	0
$373$	34.9443	1.80935	0.904673	−	0.426107i	$-0.140115\pi$
$373$	34.9443	1.80935	0.904673	+	0.426107i	$0.140115\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.47214	−0.230327
$378$	0	0
$379$	−23.5967	−1.21208	−0.606042	−	0.795433i	$-0.707244\pi$
$379$	−23.5967	−1.21208	−0.606042	+	0.795433i	$0.707244\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−30.0000	−1.53293	−0.766464	−	0.642287i	$-0.777986\pi$
$383$	−30.0000	−1.53293	−0.766464	+	0.642287i	$0.777986\pi$
$384$	0	0
$385$	20.9443	1.06742
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.8885	−0.602773	−0.301387	−	0.953502i	$-0.597449\pi$
$389$	−11.8885	−0.602773	−0.301387	+	0.953502i	$0.597449\pi$
$390$	0	0
$391$	−28.9443	−1.46377
$392$	0	0
$393$	0	0
$394$	0	0
$395$	28.9443	1.45634
$396$	0	0
$397$	16.4721	0.826713	0.413356	−	0.910569i	$-0.364356\pi$
$397$	16.4721	0.826713	0.413356	+	0.910569i	$0.364356\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.18034	−0.208756	−0.104378	−	0.994538i	$-0.533285\pi$
$401$	−4.18034	−0.208756	−0.104378	+	0.994538i	$0.533285\pi$
$402$	0	0
$403$	−5.70820	−0.284346
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.9443	0.839896
$408$	0	0
$409$	−10.3607	−0.512303	−0.256151	−	0.966637i	$-0.582455\pi$
$409$	−10.3607	−0.512303	−0.256151	+	0.966637i	$0.582455\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.52786	−0.0751813
$414$	0	0
$415$	53.3050	2.61664
$416$	0	0
$417$	0	0
$418$	0	0
$419$	29.5279	1.44253	0.721265	−	0.692659i	$-0.243561\pi$
$419$	29.5279	1.44253	0.721265	+	0.692659i	$0.243561\pi$
$420$	0	0
$421$	9.05573	0.441349	0.220675	−	0.975347i	$-0.429174\pi$
$421$	9.05573	0.441349	0.220675	+	0.975347i	$0.429174\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	24.4721	1.18707
$426$	0	0
$427$	11.4164	0.552479
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	0	0
$433$	0.472136	0.0226894	0.0113447	−	0.999936i	$-0.496389\pi$
$433$	0.472136	0.0226894	0.0113447	+	0.999936i	$0.496389\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.94427	0.236517
$438$	0	0
$439$	13.5279	0.645650	0.322825	−	0.946459i	$-0.395368\pi$
$439$	13.5279	0.645650	0.322825	+	0.946459i	$0.395368\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−21.5279	−1.02282	−0.511410	−	0.859337i	$-0.670877\pi$
$443$	−21.5279	−1.02282	−0.511410	+	0.859337i	$0.670877\pi$
$444$	0	0
$445$	−39.4164	−1.86852
$446$	0	0
$447$	0	0
$448$	0	0
$449$	36.5410	1.72448	0.862239	−	0.506502i	$-0.169062\pi$
$449$	36.5410	1.72448	0.862239	+	0.506502i	$0.169062\pi$
$450$	0	0
$451$	6.47214	0.304761
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.4721	0.490941
$456$	0	0
$457$	−34.3607	−1.60732	−0.803662	−	0.595085i	$-0.797118\pi$
$457$	−34.3607	−1.60732	−0.803662	+	0.595085i	$0.797118\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.6525	−1.05503	−0.527515	−	0.849545i	$-0.676876\pi$
$461$	−22.6525	−1.05503	−0.527515	+	0.849545i	$0.676876\pi$
$462$	0	0
$463$	−17.7082	−0.822970	−0.411485	−	0.911417i	$-0.634990\pi$
$463$	−17.7082	−0.822970	−0.411485	+	0.911417i	$0.634990\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.88854	0.0873914	0.0436957	−	0.999045i	$-0.486087\pi$
$467$	1.88854	0.0873914	0.0436957	+	0.999045i	$0.486087\pi$
$468$	0	0
$469$	−36.3607	−1.67898
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.94427	0.227338
$474$	0	0
$475$	−4.18034	−0.191807
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.3050	1.06483	0.532415	−	0.846483i	$-0.321285\pi$
$479$	23.3050	1.06483	0.532415	+	0.846483i	$0.321285\pi$
$480$	0	0
$481$	8.47214	0.386296
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.4721	0.657146
$486$	0	0
$487$	26.2918	1.19140	0.595698	−	0.803209i	$-0.296876\pi$
$487$	26.2918	1.19140	0.595698	+	0.803209i	$0.296876\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−11.0557	−0.498938	−0.249469	−	0.968383i	$-0.580256\pi$
$491$	−11.0557	−0.498938	−0.249469	+	0.968383i	$0.580256\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.4721	0.649164
$498$	0	0
$499$	−40.1803	−1.79872	−0.899360	−	0.437210i	$-0.855967\pi$
$499$	−40.1803	−1.79872	−0.899360	+	0.437210i	$0.855967\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	39.7771	1.77357	0.886786	−	0.462180i	$-0.152932\pi$
$503$	39.7771	1.77357	0.886786	+	0.462180i	$0.152932\pi$
$504$	0	0
$505$	48.3607	2.15202
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.1803	0.717181	0.358590	−	0.933495i	$-0.383257\pi$
$509$	16.1803	0.717181	0.358590	+	0.933495i	$0.383257\pi$
$510$	0	0
$511$	−27.4164	−1.21283
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−33.8885	−1.49331
$516$	0	0
$517$	21.8885	0.962657
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.9443	−0.829964	−0.414982	−	0.909830i	$-0.636212\pi$
$521$	−18.9443	−0.829964	−0.414982	+	0.909830i	$0.636212\pi$
$522$	0	0
$523$	−23.0557	−1.00816	−0.504078	−	0.863658i	$-0.668168\pi$
$523$	−23.0557	−1.00816	−0.504078	+	0.863658i	$0.668168\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.5279	−1.11201
$528$	0	0
$529$	18.8885	0.821241
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.23607	0.140170
$534$	0	0
$535$	17.8885	0.773389
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.94427	0.299111
$540$	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
$545$	61.3050	2.62602
$546$	0	0
$547$	−0.944272	−0.0403742	−0.0201871	−	0.999796i	$-0.506426\pi$
$547$	−0.944272	−0.0403742	−0.0201871	+	0.999796i	$0.506426\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.41641	0.145544
$552$	0	0
$553$	28.9443	1.23084
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.7082	0.580835	0.290418	−	0.956900i	$-0.406206\pi$
$557$	13.7082	0.580835	0.290418	+	0.956900i	$0.406206\pi$
$558$	0	0
$559$	2.47214	0.104560
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.52786	0.0643918	0.0321959	−	0.999482i	$-0.489750\pi$
$563$	1.52786	0.0643918	0.0321959	+	0.999482i	$0.489750\pi$
$564$	0	0
$565$	6.47214	0.272285
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−30.3607	−1.27279	−0.636393	−	0.771365i	$-0.719574\pi$
$569$	−30.3607	−1.27279	−0.636393	+	0.771365i	$0.719574\pi$
$570$	0	0
$571$	34.4721	1.44261	0.721307	−	0.692615i	$-0.243542\pi$
$571$	34.4721	1.44261	0.721307	+	0.692615i	$0.243542\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−35.4164	−1.47697
$576$	0	0
$577$	−18.9443	−0.788660	−0.394330	−	0.918969i	$-0.629023\pi$
$577$	−18.9443	−0.788660	−0.394330	+	0.918969i	$0.629023\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	53.3050	2.21146
$582$	0	0
$583$	−0.944272	−0.0391077
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.58359	0.271734	0.135867	−	0.990727i	$-0.456618\pi$
$587$	6.58359	0.271734	0.135867	+	0.990727i	$0.456618\pi$
$588$	0	0
$589$	4.36068	0.179679
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15.2361	−0.625670	−0.312835	−	0.949807i	$-0.601279\pi$
$593$	−15.2361	−0.625670	−0.312835	+	0.949807i	$0.601279\pi$
$594$	0	0
$595$	46.8328	1.91996
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.9443	1.01920	0.509598	−	0.860413i	$-0.329794\pi$
$599$	24.9443	1.01920	0.509598	+	0.860413i	$0.329794\pi$
$600$	0	0
$601$	47.8885	1.95341	0.976707	−	0.214576i	$-0.0688371\pi$
$601$	47.8885	1.95341	0.976707	+	0.214576i	$0.0688371\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−22.6525	−0.920954
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10.9443	0.442758
$612$	0	0
$613$	−30.3607	−1.22626	−0.613128	−	0.789983i	$-0.710089\pi$
$613$	−30.3607	−1.22626	−0.613128	+	0.789983i	$0.710089\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.59675	−0.144800	−0.0723998	−	0.997376i	$-0.523066\pi$
$617$	−3.59675	−0.144800	−0.0723998	+	0.997376i	$0.523066\pi$
$618$	0	0
$619$	−38.6525	−1.55357	−0.776787	−	0.629763i	$-0.783152\pi$
$619$	−38.6525	−1.55357	−0.776787	+	0.629763i	$0.783152\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−39.4164	−1.57919
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	37.8885	1.51072
$630$	0	0
$631$	18.6525	0.742543	0.371272	−	0.928524i	$-0.378922\pi$
$631$	18.6525	0.742543	0.371272	+	0.928524i	$0.378922\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	3.47214	0.137571
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−21.4164	−0.845897	−0.422949	−	0.906154i	$-0.639005\pi$
$641$	−21.4164	−0.845897	−0.422949	+	0.906154i	$0.639005\pi$
$642$	0	0
$643$	−29.1246	−1.14856	−0.574281	−	0.818658i	$-0.694718\pi$
$643$	−29.1246	−1.14856	−0.574281	+	0.818658i	$0.694718\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−0.944272	−0.0370659
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.3607	0.405445	0.202722	−	0.979236i	$-0.435021\pi$
$653$	10.3607	0.405445	0.202722	+	0.979236i	$0.435021\pi$
$654$	0	0
$655$	49.8885	1.94931
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.8328	1.35689	0.678447	−	0.734649i	$-0.262653\pi$
$659$	34.8328	1.35689	0.678447	+	0.734649i	$0.262653\pi$
$660$	0	0
$661$	36.2492	1.40993	0.704966	−	0.709241i	$-0.250962\pi$
$661$	36.2492	1.40993	0.704966	+	0.709241i	$0.250962\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	28.9443	1.12073
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.05573	0.272383
$672$	0	0
$673$	18.5836	0.716345	0.358172	−	0.933655i	$-0.383400\pi$
$673$	18.5836	0.716345	0.358172	+	0.933655i	$0.383400\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.3607	−1.32059	−0.660294	−	0.751007i	$-0.729568\pi$
$677$	−34.3607	−1.32059	−0.660294	+	0.751007i	$0.729568\pi$
$678$	0	0
$679$	14.4721	0.555390
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10.0000	0.382639	0.191320	−	0.981528i	$-0.438723\pi$
$683$	10.0000	0.382639	0.191320	+	0.981528i	$0.438723\pi$
$684$	0	0
$685$	36.3607	1.38927
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−0.472136	−0.0179869
$690$	0	0
$691$	27.5967	1.04983	0.524915	−	0.851155i	$-0.324097\pi$
$691$	27.5967	1.04983	0.524915	+	0.851155i	$0.324097\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−70.8328	−2.68684
$696$	0	0
$697$	14.4721	0.548171
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	−6.47214	−0.244101
$704$	0	0
$705$	0	0
$706$	0	0
$707$	48.3607	1.81879
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	36.9443	1.38357
$714$	0	0
$715$	6.47214	0.242044
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6.47214	−0.241370	−0.120685	−	0.992691i	$-0.538509\pi$
$719$	−6.47214	−0.241370	−0.120685	+	0.992691i	$0.538509\pi$
$720$	0	0
$721$	−33.8885	−1.26208
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−24.4721	−0.908872
$726$	0	0
$727$	10.4721	0.388390	0.194195	−	0.980963i	$-0.437791\pi$
$727$	10.4721	0.388390	0.194195	+	0.980963i	$0.437791\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.0557	0.408911
$732$	0	0
$733$	7.88854	0.291370	0.145685	−	0.989331i	$-0.453461\pi$
$733$	7.88854	0.291370	0.145685	+	0.989331i	$0.453461\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22.4721	−0.827772
$738$	0	0
$739$	−23.8197	−0.876220	−0.438110	−	0.898921i	$-0.644352\pi$
$739$	−23.8197	−0.876220	−0.438110	+	0.898921i	$0.644352\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.52786	0.276171	0.138085	−	0.990420i	$-0.455905\pi$
$743$	7.52786	0.276171	0.138085	+	0.990420i	$0.455905\pi$
$744$	0	0
$745$	70.2492	2.57373
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17.8885	0.653633
$750$	0	0
$751$	−12.3607	−0.451048	−0.225524	−	0.974238i	$-0.572409\pi$
$751$	−12.3607	−0.451048	−0.225524	+	0.974238i	$0.572409\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−75.1935	−2.73657
$756$	0	0
$757$	−10.5836	−0.384667	−0.192334	−	0.981330i	$-0.561606\pi$
$757$	−10.5836	−0.384667	−0.192334	+	0.981330i	$0.561606\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.1803	1.16654	0.583268	−	0.812280i	$-0.301774\pi$
$761$	32.1803	1.16654	0.583268	+	0.812280i	$0.301774\pi$
$762$	0	0
$763$	61.3050	2.21939
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.472136	−0.0170478
$768$	0	0
$769$	0.832816	0.0300321	0.0150161	−	0.999887i	$-0.495220\pi$
$769$	0.832816	0.0300321	0.0150161	+	0.999887i	$0.495220\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	21.7082	0.780790	0.390395	−	0.920647i	$-0.372338\pi$
$773$	21.7082	0.780790	0.390395	+	0.920647i	$0.372338\pi$
$774$	0	0
$775$	−31.2361	−1.12203
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.47214	−0.0885735
$780$	0	0
$781$	8.94427	0.320051
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−43.4164	−1.54960
$786$	0	0
$787$	−17.7082	−0.631229	−0.315615	−	0.948887i	$-0.602211\pi$
$787$	−17.7082	−0.631229	−0.315615	+	0.948887i	$0.602211\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.47214	0.230123
$792$	0	0
$793$	3.52786	0.125278
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	48.9443	1.73152
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16.9443	−0.597950
$804$	0	0
$805$	−67.7771	−2.38883
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13.0557	0.459015	0.229507	−	0.973307i	$-0.426288\pi$
$809$	13.0557	0.459015	0.229507	+	0.973307i	$0.426288\pi$
$810$	0	0
$811$	−3.81966	−0.134126	−0.0670632	−	0.997749i	$-0.521363\pi$
$811$	−3.81966	−0.134126	−0.0670632	+	0.997749i	$0.521363\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23.4164	−0.820241
$816$	0	0
$817$	−1.88854	−0.0660718
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.12461	−0.318451	−0.159226	−	0.987242i	$-0.550900\pi$
$821$	−9.12461	−0.318451	−0.159226	+	0.987242i	$0.550900\pi$
$822$	0	0
$823$	17.3050	0.603213	0.301606	−	0.953433i	$-0.402477\pi$
$823$	17.3050	0.603213	0.301606	+	0.953433i	$0.402477\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.7771	1.31364	0.656819	−	0.754048i	$-0.271902\pi$
$827$	37.7771	1.31364	0.656819	+	0.754048i	$0.271902\pi$
$828$	0	0
$829$	−39.5279	−1.37286	−0.686430	−	0.727196i	$-0.740823\pi$
$829$	−39.5279	−1.37286	−0.686430	+	0.727196i	$0.740823\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15.5279	0.538009
$834$	0	0
$835$	−64.3607	−2.22729
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.3607	−0.495786	−0.247893	−	0.968787i	$-0.579738\pi$
$839$	−14.3607	−0.495786	−0.247893	+	0.968787i	$0.579738\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.23607	0.111324
$846$	0	0
$847$	−22.6525	−0.778348
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−54.8328	−1.87964
$852$	0	0
$853$	11.5279	0.394707	0.197353	−	0.980332i	$-0.436765\pi$
$853$	11.5279	0.394707	0.197353	+	0.980332i	$0.436765\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.4164	1.27812	0.639060	−	0.769157i	$-0.279324\pi$
$857$	37.4164	1.27812	0.639060	+	0.769157i	$0.279324\pi$
$858$	0	0
$859$	32.9443	1.12404	0.562022	−	0.827122i	$-0.310024\pi$
$859$	32.9443	1.12404	0.562022	+	0.827122i	$0.310024\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.8885	−1.63014	−0.815072	−	0.579359i	$-0.803303\pi$
$863$	−47.8885	−1.63014	−0.815072	+	0.579359i	$0.803303\pi$
$864$	0	0
$865$	6.47214	0.220059
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17.8885	0.606827
$870$	0	0
$871$	−11.2361	−0.380720
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.94427	0.167147
$876$	0	0
$877$	−27.3050	−0.922023	−0.461011	−	0.887394i	$-0.652513\pi$
$877$	−27.3050	−0.922023	−0.461011	+	0.887394i	$0.652513\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.63932	−0.189994	−0.0949968	−	0.995478i	$-0.530284\pi$
$881$	−5.63932	−0.189994	−0.0949968	+	0.995478i	$0.530284\pi$
$882$	0	0
$883$	3.63932	0.122473	0.0612364	−	0.998123i	$-0.480496\pi$
$883$	3.63932	0.122473	0.0612364	+	0.998123i	$0.480496\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.3050	0.581043	0.290522	−	0.956868i	$-0.406171\pi$
$887$	17.3050	0.581043	0.290522	+	0.956868i	$0.406171\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.36068	−0.279779
$894$	0	0
$895$	−62.8328	−2.10027
$896$	0	0
$897$	0	0
$898$	0	0
$899$	25.5279	0.851402
$900$	0	0
$901$	−2.11146	−0.0703428
$902$	0	0
$903$	0	0
$904$	0	0
$905$	56.3607	1.87349
$906$	0	0
$907$	−20.3607	−0.676065	−0.338033	−	0.941134i	$-0.609761\pi$
$907$	−20.3607	−0.676065	−0.338033	+	0.941134i	$0.609761\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.5279	1.50841	0.754203	−	0.656642i	$-0.228024\pi$
$911$	45.5279	1.50841	0.754203	+	0.656642i	$0.228024\pi$
$912$	0	0
$913$	32.9443	1.09030
$914$	0	0
$915$	0	0
$916$	0	0
$917$	49.8885	1.64746
$918$	0	0
$919$	−13.8885	−0.458141	−0.229070	−	0.973410i	$-0.573569\pi$
$919$	−13.8885	−0.458141	−0.229070	+	0.973410i	$0.573569\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.47214	0.147202
$924$	0	0
$925$	46.3607	1.52433
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8.54102	−0.280222	−0.140111	−	0.990136i	$-0.544746\pi$
$929$	−8.54102	−0.280222	−0.140111	+	0.990136i	$0.544746\pi$
$930$	0	0
$931$	−2.65248	−0.0869314
$932$	0	0
$933$	0	0
$934$	0	0
$935$	28.9443	0.946579
$936$	0	0
$937$	31.3050	1.02269	0.511344	−	0.859376i	$-0.329148\pi$
$937$	31.3050	1.02269	0.511344	+	0.859376i	$0.329148\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31.2361	−1.01827	−0.509133	−	0.860688i	$-0.670034\pi$
$941$	−31.2361	−1.01827	−0.509133	+	0.860688i	$0.670034\pi$
$942$	0	0
$943$	−20.9443	−0.682039
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.4164	1.34585	0.672926	−	0.739710i	$-0.265037\pi$
$947$	41.4164	1.34585	0.672926	+	0.739710i	$0.265037\pi$
$948$	0	0
$949$	−8.47214	−0.275017
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−31.5279	−1.02129	−0.510644	−	0.859792i	$-0.670593\pi$
$953$	−31.5279	−1.02129	−0.510644	+	0.859792i	$0.670593\pi$
$954$	0	0
$955$	33.8885	1.09661
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.3607	1.17415
$960$	0	0
$961$	1.58359	0.0510836
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−51.4164	−1.65515
$966$	0	0
$967$	15.2361	0.489959	0.244979	−	0.969528i	$-0.421219\pi$
$967$	15.2361	0.489959	0.244979	+	0.969528i	$0.421219\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−50.8328	−1.63130	−0.815651	−	0.578544i	$-0.803621\pi$
$971$	−50.8328	−1.63130	−0.815651	+	0.578544i	$0.803621\pi$
$972$	0	0
$973$	−70.8328	−2.27080
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.7082	0.438564	0.219282	−	0.975661i	$-0.429628\pi$
$977$	13.7082	0.438564	0.219282	+	0.975661i	$0.429628\pi$
$978$	0	0
$979$	−24.3607	−0.778571
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.5836	0.720305	0.360152	−	0.932893i	$-0.382725\pi$
$983$	22.5836	0.720305	0.360152	+	0.932893i	$0.382725\pi$
$984$	0	0
$985$	88.1378	2.80830
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−40.9443	−1.30064	−0.650319	−	0.759661i	$-0.725365\pi$
$991$	−40.9443	−1.30064	−0.650319	+	0.759661i	$0.725365\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−59.7771	−1.89506
$996$	0	0
$997$	59.8885	1.89669	0.948345	−	0.317242i	$-0.102757\pi$
$997$	59.8885	1.89669	0.948345	+	0.317242i	$0.102757\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7488.2.a.ct.1.2		2
3.2	odd	2		2496.2.a.be.1.1		2
4.3	odd	2		7488.2.a.cs.1.2		2
8.3	odd	2		3744.2.a.r.1.1		2
8.5	even	2		3744.2.a.s.1.1		2
12.11	even	2		2496.2.a.bh.1.1		2
24.5	odd	2		1248.2.a.n.1.2	yes	2
24.11	even	2		1248.2.a.l.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.a.l.1.2	✓	2	24.11	even	2
1248.2.a.n.1.2	yes	2	24.5	odd	2
2496.2.a.be.1.1		2	3.2	odd	2
2496.2.a.bh.1.1		2	12.11	even	2
3744.2.a.r.1.1		2	8.3	odd	2
3744.2.a.s.1.1		2	8.5	even	2
7488.2.a.cs.1.2		2	4.3	odd	2
7488.2.a.ct.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7488.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23607 q^{5} +3.23607 q^{7} +O(q^{10})\)	\(q+3.23607 q^{5} +3.23607 q^{7} +2.00000 q^{11} +1.00000 q^{13} +4.47214 q^{17} -0.763932 q^{19} -6.47214 q^{23} +5.47214 q^{25} -4.47214 q^{29} -5.70820 q^{31} +10.4721 q^{35} +8.47214 q^{37} +3.23607 q^{41} +2.47214 q^{43} +10.9443 q^{47} +3.47214 q^{49} -0.472136 q^{53} +6.47214 q^{55} -0.472136 q^{59} +3.52786 q^{61} +3.23607 q^{65} -11.2361 q^{67} +4.47214 q^{71} -8.47214 q^{73} +6.47214 q^{77} +8.94427 q^{79} +16.4721 q^{83} +14.4721 q^{85} -12.1803 q^{89} +3.23607 q^{91} -2.47214 q^{95} +4.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} + 2 q^{13} - 6 q^{19} - 4 q^{23} + 2 q^{25} + 2 q^{31} + 12 q^{35} + 8 q^{37} + 2 q^{41} - 4 q^{43} + 4 q^{47} - 2 q^{49} + 8 q^{53} + 4 q^{55} + 8 q^{59} + 16 q^{61} + 2 q^{65} - 18 q^{67} - 8 q^{73} + 4 q^{77} + 24 q^{83} + 20 q^{85} - 2 q^{89} + 2 q^{91} + 4 q^{95}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 + 2 * q^13 - 6 * q^19 - 4 * q^23 + 2 * q^25 + 2 * q^31 + 12 * q^35 + 8 * q^37 + 2 * q^41 - 4 * q^43 + 4 * q^47 - 2 * q^49 + 8 * q^53 + 4 * q^55 + 8 * q^59 + 16 * q^61 + 2 * q^65 - 18 * q^67 - 8 * q^73 + 4 * q^77 + 24 * q^83 + 20 * q^85 - 2 * q^89 + 2 * q^91 + 4 * q^95

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