LMFDB - Embedded newform 8496.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8496,2,Mod(1,8496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8496, 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("8496.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8496 = 2^{4} \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8496.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:	$67.8409015573$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	8496.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.68133 q^{5} -2.74590 q^{7} +O(q^{10})$	$q-1.68133 q^{5} -2.74590 q^{7} +2.18953 q^{11} +4.18953 q^{13} +6.29809 q^{17} -4.93543 q^{19} +3.44364 q^{23} -2.17313 q^{25} +9.12497 q^{29} +0.616763 q^{31} +4.61676 q^{35} +1.44364 q^{37} -4.10856 q^{41} -8.53579 q^{43} -6.48763 q^{47} +0.539958 q^{49} +0.664924 q^{53} -3.68133 q^{55} +1.00000 q^{59} -10.9958 q^{61} -7.04399 q^{65} +9.93126 q^{67} +6.82687 q^{71} +13.5040 q^{73} -6.01224 q^{77} +1.17313 q^{79} -15.3421 q^{83} -10.5892 q^{85} -1.97942 q^{89} -11.5040 q^{91} +8.29809 q^{95} +3.33508 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} - 9 q^{7}+O(q^{10}) $ 3 * q + 2 * q^5 - 9 * q^7	$ 3 q + 2 q^{5} - 9 q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} - 7 q^{19} + q^{23} - q^{25} + 11 q^{29} - 13 q^{31} - q^{35} - 5 q^{37} + q^{41} - 6 q^{43} + 11 q^{47} + 14 q^{49} - 2 q^{53} - 4 q^{55} + 3 q^{59} - q^{61} - 10 q^{67} + 26 q^{71} + 7 q^{73} + 17 q^{77} - 2 q^{79} - 3 q^{83} - 35 q^{85} + 23 q^{89} - q^{91} + 3 q^{95} + 14 q^{97}+O(q^{100}) $ 3 * q + 2 * q^5 - 9 * q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 - 7 * q^19 + q^23 - q^25 + 11 * q^29 - 13 * q^31 - q^35 - 5 * q^37 + q^41 - 6 * q^43 + 11 * q^47 + 14 * q^49 - 2 * q^53 - 4 * q^55 + 3 * q^59 - q^61 - 10 * q^67 + 26 * q^71 + 7 * q^73 + 17 * q^77 - 2 * q^79 - 3 * q^83 - 35 * q^85 + 23 * q^89 - q^91 + 3 * q^95 + 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.68133	−0.751914	−0.375957	−	0.926637i	$-0.622686\pi$
$5$	−1.68133	−0.751914	−0.375957	+	0.926637i	$0.622686\pi$
$6$	0	0
$7$	−2.74590	−1.03785	−0.518926	−	0.854819i	$-0.673668\pi$
$7$	−2.74590	−1.03785	−0.518926	+	0.854819i	$0.673668\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.18953	0.660169	0.330085	−	0.943951i	$-0.392923\pi$
$11$	2.18953	0.660169	0.330085	+	0.943951i	$0.392923\pi$
$12$	0	0
$13$	4.18953	1.16197	0.580984	−	0.813915i	$-0.302668\pi$
$13$	4.18953	1.16197	0.580984	+	0.813915i	$0.302668\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.29809	1.52751	0.763756	−	0.645505i	$-0.223353\pi$
$17$	6.29809	1.52751	0.763756	+	0.645505i	$0.223353\pi$
$18$	0	0
$19$	−4.93543	−1.13227	−0.566133	−	0.824314i	$-0.691561\pi$
$19$	−4.93543	−1.13227	−0.566133	+	0.824314i	$0.691561\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.44364	0.718048	0.359024	−	0.933328i	$-0.383110\pi$
$23$	3.44364	0.718048	0.359024	+	0.933328i	$0.383110\pi$
$24$	0	0
$25$	−2.17313	−0.434625
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.12497	1.69446	0.847232	−	0.531223i	$-0.178267\pi$
$29$	9.12497	1.69446	0.847232	+	0.531223i	$0.178267\pi$
$30$	0	0
$31$	0.616763	0.110774	0.0553870	−	0.998465i	$-0.482361\pi$
$31$	0.616763	0.110774	0.0553870	+	0.998465i	$0.482361\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.61676	0.780375
$36$	0	0
$37$	1.44364	0.237332	0.118666	−	0.992934i	$-0.462138\pi$
$37$	1.44364	0.237332	0.118666	+	0.992934i	$0.462138\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.10856	−0.641649	−0.320825	−	0.947139i	$-0.603960\pi$
$41$	−4.10856	−0.641649	−0.320825	+	0.947139i	$0.603960\pi$
$42$	0	0
$43$	−8.53579	−1.30170	−0.650848	−	0.759208i	$-0.725586\pi$
$43$	−8.53579	−1.30170	−0.650848	+	0.759208i	$0.725586\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.48763	−0.946318	−0.473159	−	0.880977i	$-0.656886\pi$
$47$	−6.48763	−0.946318	−0.473159	+	0.880977i	$0.656886\pi$
$48$	0	0
$49$	0.539958	0.0771368
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.664924	0.0913343	0.0456672	−	0.998957i	$-0.485459\pi$
$53$	0.664924	0.0913343	0.0456672	+	0.998957i	$0.485459\pi$
$54$	0	0
$55$	−3.68133	−0.496391
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−10.9958	−1.40787	−0.703936	−	0.710263i	$-0.748576\pi$
$61$	−10.9958	−1.40787	−0.703936	+	0.710263i	$0.748576\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.04399	−0.873700
$66$	0	0
$67$	9.93126	1.21330	0.606648	−	0.794970i	$-0.292514\pi$
$67$	9.93126	1.21330	0.606648	+	0.794970i	$0.292514\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.82687	0.810201	0.405100	−	0.914272i	$-0.367237\pi$
$71$	6.82687	0.810201	0.405100	+	0.914272i	$0.367237\pi$
$72$	0	0
$73$	13.5040	1.58053	0.790264	−	0.612767i	$-0.209943\pi$
$73$	13.5040	1.58053	0.790264	+	0.612767i	$0.209943\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.01224	−0.685158
$78$	0	0
$79$	1.17313	0.131987	0.0659936	−	0.997820i	$-0.478978\pi$
$79$	1.17313	0.131987	0.0659936	+	0.997820i	$0.478978\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−15.3421	−1.68401	−0.842006	−	0.539468i	$-0.818626\pi$
$83$	−15.3421	−1.68401	−0.842006	+	0.539468i	$0.818626\pi$
$84$	0	0
$85$	−10.5892	−1.14856
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.97942	−0.209819	−0.104909	−	0.994482i	$-0.533455\pi$
$89$	−1.97942	−0.209819	−0.104909	+	0.994482i	$0.533455\pi$
$90$	0	0
$91$	−11.5040	−1.20595
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.29809	0.851366
$96$	0	0
$97$	3.33508	0.338626	0.169313	−	0.985562i	$-0.445845\pi$
$97$	3.33508	0.338626	0.169313	+	0.985562i	$0.445845\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.49180	0.347447	0.173723	−	0.984794i	$-0.444420\pi$
$101$	3.49180	0.347447	0.173723	+	0.984794i	$0.444420\pi$
$102$	0	0
$103$	−8.21712	−0.809657	−0.404828	−	0.914393i	$-0.632669\pi$
$103$	−8.21712	−0.809657	−0.404828	+	0.914393i	$0.632669\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.7899	1.71981	0.859907	−	0.510451i	$-0.170522\pi$
$107$	17.7899	1.71981	0.859907	+	0.510451i	$0.170522\pi$
$108$	0	0
$109$	−18.2981	−1.75264	−0.876320	−	0.481730i	$-0.840009\pi$
$109$	−18.2981	−1.75264	−0.876320	+	0.481730i	$0.840009\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.71414	0.725686	0.362843	−	0.931850i	$-0.381806\pi$
$113$	7.71414	0.725686	0.362843	+	0.931850i	$0.381806\pi$
$114$	0	0
$115$	−5.78989	−0.539910
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−17.2939	−1.58533
$120$	0	0
$121$	−6.20594	−0.564176
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0604	1.07871
$126$	0	0
$127$	−6.18953	−0.549232	−0.274616	−	0.961554i	$-0.588551\pi$
$127$	−6.18953	−0.549232	−0.274616	+	0.961554i	$0.588551\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.5798	−1.36121	−0.680606	−	0.732650i	$-0.738283\pi$
$131$	−15.5798	−1.36121	−0.680606	+	0.732650i	$0.738283\pi$
$132$	0	0
$133$	13.5522	1.17512
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.664924	−0.0568083	−0.0284041	−	0.999597i	$-0.509043\pi$
$137$	−0.664924	−0.0568083	−0.0284041	+	0.999597i	$0.509043\pi$
$138$	0	0
$139$	18.7857	1.59338	0.796692	−	0.604385i	$-0.206581\pi$
$139$	18.7857	1.59338	0.796692	+	0.604385i	$0.206581\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.17313	0.767095
$144$	0	0
$145$	−15.3421	−1.27409
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.04816	0.659331	0.329666	−	0.944098i	$-0.393064\pi$
$149$	8.04816	0.659331	0.329666	+	0.944098i	$0.393064\pi$
$150$	0	0
$151$	17.8503	1.45264	0.726318	−	0.687359i	$-0.241230\pi$
$151$	17.8503	1.45264	0.726318	+	0.687359i	$0.241230\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.03698	−0.0832924
$156$	0	0
$157$	10.4395	0.833160	0.416580	−	0.909099i	$-0.363229\pi$
$157$	10.4395	0.833160	0.416580	+	0.909099i	$0.363229\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.45587	−0.745227
$162$	0	0
$163$	3.76231	0.294686	0.147343	−	0.989085i	$-0.452928\pi$
$163$	3.76231	0.294686	0.147343	+	0.989085i	$0.452928\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.1086	1.40128	0.700641	−	0.713514i	$-0.252897\pi$
$167$	18.1086	1.40128	0.700641	+	0.713514i	$0.252897\pi$
$168$	0	0
$169$	4.55220	0.350169
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.66598	0.734891	0.367446	−	0.930045i	$-0.380232\pi$
$173$	9.66598	0.734891	0.367446	+	0.930045i	$0.380232\pi$
$174$	0	0
$175$	5.96719	0.451077
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.95601	−0.669403	−0.334702	−	0.942324i	$-0.608636\pi$
$179$	−8.95601	−0.669403	−0.334702	+	0.942324i	$0.608636\pi$
$180$	0	0
$181$	−0.487628	−0.0362451	−0.0181225	−	0.999836i	$-0.505769\pi$
$181$	−0.487628	−0.0362451	−0.0181225	+	0.999836i	$0.505769\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.42723	−0.178453
$186$	0	0
$187$	13.7899	1.00842
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.07681	0.367345	0.183672	−	0.982988i	$-0.441201\pi$
$191$	5.07681	0.367345	0.183672	+	0.982988i	$0.441201\pi$
$192$	0	0
$193$	9.36266	0.673939	0.336970	−	0.941516i	$-0.390598\pi$
$193$	9.36266	0.673939	0.336970	+	0.941516i	$0.390598\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.2499	1.15776	0.578880	−	0.815413i	$-0.303490\pi$
$197$	16.2499	1.15776	0.578880	+	0.815413i	$0.303490\pi$
$198$	0	0
$199$	−6.64958	−0.471376	−0.235688	−	0.971829i	$-0.575734\pi$
$199$	−6.64958	−0.471376	−0.235688	+	0.971829i	$0.575734\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−25.0562	−1.75860
$204$	0	0
$205$	6.90785	0.482465
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.8063	−0.747487
$210$	0	0
$211$	−13.1526	−0.905459	−0.452729	−	0.891648i	$-0.649550\pi$
$211$	−13.1526	−0.905459	−0.452729	+	0.891648i	$0.649550\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	14.3515	0.978763
$216$	0	0
$217$	−1.69357	−0.114967
$218$	0	0
$219$	0	0
$220$	0	0
$221$	26.3861	1.77492
$222$	0	0
$223$	9.70892	0.650157	0.325079	−	0.945687i	$-0.394609\pi$
$223$	9.70892	0.650157	0.325079	+	0.945687i	$0.394609\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.16896	0.409448	0.204724	−	0.978820i	$-0.434370\pi$
$227$	6.16896	0.409448	0.204724	+	0.978820i	$0.434370\pi$
$228$	0	0
$229$	25.8175	1.70607	0.853033	−	0.521856i	$-0.174760\pi$
$229$	25.8175	1.70607	0.853033	+	0.521856i	$0.174760\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.2499	−0.802520	−0.401260	−	0.915964i	$-0.631428\pi$
$233$	−12.2499	−0.802520	−0.401260	+	0.915964i	$0.631428\pi$
$234$	0	0
$235$	10.9078	0.711549
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.9477	−1.48436	−0.742181	−	0.670200i	$-0.766208\pi$
$239$	−22.9477	−1.48436	−0.742181	+	0.670200i	$0.766208\pi$
$240$	0	0
$241$	4.67015	0.300831	0.150415	−	0.988623i	$-0.451939\pi$
$241$	4.67015	0.300831	0.150415	+	0.988623i	$0.451939\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.907847	−0.0580002
$246$	0	0
$247$	−20.6772	−1.31566
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.4067	1.41430	0.707148	−	0.707066i	$-0.249982\pi$
$251$	22.4067	1.41430	0.707148	+	0.707066i	$0.249982\pi$
$252$	0	0
$253$	7.53996	0.474033
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−31.1372	−1.94229	−0.971143	−	0.238499i	$-0.923345\pi$
$257$	−31.1372	−1.94229	−0.971143	+	0.238499i	$0.923345\pi$
$258$	0	0
$259$	−3.96408	−0.246316
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.9547	1.53877	0.769386	−	0.638784i	$-0.220562\pi$
$263$	24.9547	1.53877	0.769386	+	0.638784i	$0.220562\pi$
$264$	0	0
$265$	−1.11796	−0.0686755
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.99477	−0.365508	−0.182754	−	0.983159i	$-0.558501\pi$
$269$	−5.99477	−0.365508	−0.182754	+	0.983159i	$0.558501\pi$
$270$	0	0
$271$	1.97241	0.119816	0.0599078	−	0.998204i	$-0.480919\pi$
$271$	1.97241	0.119816	0.0599078	+	0.998204i	$0.480919\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.75814	−0.286926
$276$	0	0
$277$	−2.53579	−0.152361	−0.0761804	−	0.997094i	$-0.524272\pi$
$277$	−2.53579	−0.152361	−0.0761804	+	0.997094i	$0.524272\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.8545	1.12476	0.562381	−	0.826878i	$-0.309885\pi$
$281$	18.8545	1.12476	0.562381	+	0.826878i	$0.309885\pi$
$282$	0	0
$283$	5.78989	0.344173	0.172087	−	0.985082i	$-0.444949\pi$
$283$	5.78989	0.344173	0.172087	+	0.985082i	$0.444949\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.2817	0.665937
$288$	0	0
$289$	22.6660	1.33329
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.73472	−0.451867	−0.225934	−	0.974143i	$-0.572543\pi$
$293$	−7.73472	−0.451867	−0.225934	+	0.974143i	$0.572543\pi$
$294$	0	0
$295$	−1.68133	−0.0978909
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.4272	0.834348
$300$	0	0
$301$	23.4384	1.35097
$302$	0	0
$303$	0	0
$304$	0	0
$305$	18.4876	1.05860
$306$	0	0
$307$	−16.4999	−0.941697	−0.470849	−	0.882214i	$-0.656052\pi$
$307$	−16.4999	−0.941697	−0.470849	+	0.882214i	$0.656052\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.9354	1.41396	0.706979	−	0.707234i	$-0.250057\pi$
$311$	24.9354	1.41396	0.706979	+	0.707234i	$0.250057\pi$
$312$	0	0
$313$	−24.1690	−1.36611	−0.683055	−	0.730367i	$-0.739349\pi$
$313$	−24.1690	−1.36611	−0.683055	+	0.730367i	$0.739349\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.1812	1.02116	0.510579	−	0.859831i	$-0.329431\pi$
$317$	18.1812	1.02116	0.510579	+	0.859831i	$0.329431\pi$
$318$	0	0
$319$	19.9794	1.11863
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−31.0838	−1.72955
$324$	0	0
$325$	−9.10439	−0.505021
$326$	0	0
$327$	0	0
$328$	0	0
$329$	17.8144	0.982138
$330$	0	0
$331$	14.6290	0.804083	0.402041	−	0.915622i	$-0.368301\pi$
$331$	14.6290	0.804083	0.402041	+	0.915622i	$0.368301\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−16.6977	−0.912295
$336$	0	0
$337$	−19.1526	−1.04331	−0.521653	−	0.853158i	$-0.674684\pi$
$337$	−19.1526	−1.04331	−0.521653	+	0.853158i	$0.674684\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.35042	0.0731295
$342$	0	0
$343$	17.7386	0.957795
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.4301	1.57989	0.789944	−	0.613178i	$-0.210109\pi$
$347$	29.4301	1.57989	0.789944	+	0.613178i	$0.210109\pi$
$348$	0	0
$349$	28.9065	1.54733	0.773665	−	0.633595i	$-0.218421\pi$
$349$	28.9065	1.54733	0.773665	+	0.633595i	$0.218421\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.0481609	−0.00256335	−0.00128167	−	0.999999i	$-0.500408\pi$
$353$	−0.0481609	−0.00256335	−0.00128167	+	0.999999i	$0.500408\pi$
$354$	0	0
$355$	−11.4782	−0.609201
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.7417	0.619705	0.309852	−	0.950785i	$-0.399720\pi$
$359$	11.7417	0.619705	0.309852	+	0.950785i	$0.399720\pi$
$360$	0	0
$361$	5.35849	0.282026
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−22.7047	−1.18842
$366$	0	0
$367$	7.04399	0.367693	0.183847	−	0.982955i	$-0.441145\pi$
$367$	7.04399	0.367693	0.183847	+	0.982955i	$0.441145\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.82581	−0.0947915
$372$	0	0
$373$	31.5316	1.63265	0.816323	−	0.577596i	$-0.196009\pi$
$373$	31.5316	1.63265	0.816323	+	0.577596i	$0.196009\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	38.2294	1.96891
$378$	0	0
$379$	10.5030	0.539502	0.269751	−	0.962930i	$-0.413059\pi$
$379$	10.5030	0.539502	0.269751	+	0.962930i	$0.413059\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.3023	1.19069	0.595345	−	0.803470i	$-0.297015\pi$
$383$	23.3023	1.19069	0.595345	+	0.803470i	$0.297015\pi$
$384$	0	0
$385$	10.1086	0.515180
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.4835	1.69768	0.848839	−	0.528651i	$-0.177302\pi$
$389$	33.4835	1.69768	0.848839	+	0.528651i	$0.177302\pi$
$390$	0	0
$391$	21.6883	1.09683
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.97241	−0.0992430
$396$	0	0
$397$	−10.6977	−0.536904	−0.268452	−	0.963293i	$-0.586512\pi$
$397$	−10.6977	−0.536904	−0.268452	+	0.963293i	$0.586512\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.3145	1.16427	0.582135	−	0.813092i	$-0.302217\pi$
$401$	23.3145	1.16427	0.582135	+	0.813092i	$0.302217\pi$
$402$	0	0
$403$	2.58395	0.128716
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.16089	0.156679
$408$	0	0
$409$	−8.65970	−0.428194	−0.214097	−	0.976812i	$-0.568681\pi$
$409$	−8.65970	−0.428194	−0.214097	+	0.976812i	$0.568681\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.74590	−0.135117
$414$	0	0
$415$	25.7951	1.26623
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.63734	0.226549	0.113274	−	0.993564i	$-0.463866\pi$
$419$	4.63734	0.226549	0.113274	+	0.993564i	$0.463866\pi$
$420$	0	0
$421$	1.68133	0.0819430	0.0409715	−	0.999160i	$-0.486955\pi$
$421$	1.68133	0.0819430	0.0409715	+	0.999160i	$0.486955\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13.6866	−0.663896
$426$	0	0
$427$	30.1934	1.46116
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.5040	0.554130	0.277065	−	0.960851i	$-0.410638\pi$
$431$	11.5040	0.554130	0.277065	+	0.960851i	$0.410638\pi$
$432$	0	0
$433$	24.5069	1.17773	0.588863	−	0.808233i	$-0.299576\pi$
$433$	24.5069	1.17773	0.588863	+	0.808233i	$0.299576\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.9958	−0.813021
$438$	0	0
$439$	5.63840	0.269106	0.134553	−	0.990906i	$-0.457040\pi$
$439$	5.63840	0.269106	0.134553	+	0.990906i	$0.457040\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−38.0070	−1.80577	−0.902884	−	0.429885i	$-0.858554\pi$
$443$	−38.0070	−1.80577	−0.902884	+	0.429885i	$0.858554\pi$
$444$	0	0
$445$	3.32807	0.157765
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.0357	0.520805	0.260402	−	0.965500i	$-0.416145\pi$
$449$	11.0357	0.520805	0.260402	+	0.965500i	$0.416145\pi$
$450$	0	0
$451$	−8.99583	−0.423597
$452$	0	0
$453$	0	0
$454$	0	0
$455$	19.3421	0.906771
$456$	0	0
$457$	−32.3913	−1.51520	−0.757601	−	0.652718i	$-0.773629\pi$
$457$	−32.3913	−1.51520	−0.757601	+	0.652718i	$0.773629\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−34.9599	−1.62825	−0.814123	−	0.580693i	$-0.802782\pi$
$461$	−34.9599	−1.62825	−0.814123	+	0.580693i	$0.802782\pi$
$462$	0	0
$463$	22.8063	1.05990	0.529949	−	0.848029i	$-0.322211\pi$
$463$	22.8063	1.05990	0.529949	+	0.848029i	$0.322211\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.2416	−1.67706	−0.838530	−	0.544855i	$-0.816585\pi$
$467$	−36.2416	−1.67706	−0.838530	+	0.544855i	$0.816585\pi$
$468$	0	0
$469$	−27.2702	−1.25922
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−18.6894	−0.859340
$474$	0	0
$475$	10.7253	0.492112
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17.0562	0.779319	0.389660	−	0.920959i	$-0.372593\pi$
$479$	17.0562	0.779319	0.389660	+	0.920959i	$0.372593\pi$
$480$	0	0
$481$	6.04816	0.275772
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.60737	−0.254617
$486$	0	0
$487$	−15.3473	−0.695453	−0.347727	−	0.937596i	$-0.613046\pi$
$487$	−15.3473	−0.695453	−0.347727	+	0.937596i	$0.613046\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.0122	0.812881	0.406440	−	0.913677i	$-0.366770\pi$
$491$	18.0122	0.812881	0.406440	+	0.913677i	$0.366770\pi$
$492$	0	0
$493$	57.4699	2.58831
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18.7459	−0.840868
$498$	0	0
$499$	21.6074	0.967279	0.483639	−	0.875267i	$-0.339315\pi$
$499$	21.6074	0.967279	0.483639	+	0.875267i	$0.339315\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.76231	0.346104	0.173052	−	0.984913i	$-0.444637\pi$
$503$	7.76231	0.346104	0.173052	+	0.984913i	$0.444637\pi$
$504$	0	0
$505$	−5.87086	−0.261250
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−40.7704	−1.80712	−0.903558	−	0.428467i	$-0.859054\pi$
$509$	−40.7704	−1.80712	−0.903558	+	0.428467i	$0.859054\pi$
$510$	0	0
$511$	−37.0807	−1.64035
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.8157	0.608792
$516$	0	0
$517$	−14.2049	−0.624730
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−41.0427	−1.79811	−0.899056	−	0.437834i	$-0.855746\pi$
$521$	−41.0427	−1.79811	−0.899056	+	0.437834i	$0.855746\pi$
$522$	0	0
$523$	22.5686	0.986856	0.493428	−	0.869787i	$-0.335744\pi$
$523$	22.5686	0.986856	0.493428	+	0.869787i	$0.335744\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.88443	0.169208
$528$	0	0
$529$	−11.1414	−0.484408
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−17.2130	−0.745576
$534$	0	0
$535$	−29.9107	−1.29315
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.18226	0.0509233
$540$	0	0
$541$	35.0357	1.50630	0.753150	−	0.657849i	$-0.228533\pi$
$541$	35.0357	1.50630	0.753150	+	0.657849i	$0.228533\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.7651	1.31783
$546$	0	0
$547$	−3.70191	−0.158282	−0.0791410	−	0.996863i	$-0.525218\pi$
$547$	−3.70191	−0.158282	−0.0791410	+	0.996863i	$0.525218\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−45.0357	−1.91858
$552$	0	0
$553$	−3.22129	−0.136983
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.7253	0.708675	0.354337	−	0.935118i	$-0.384706\pi$
$557$	16.7253	0.708675	0.354337	+	0.935118i	$0.384706\pi$
$558$	0	0
$559$	−35.7610	−1.51253
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.9149	1.05004	0.525018	−	0.851091i	$-0.324059\pi$
$563$	24.9149	1.05004	0.525018	+	0.851091i	$0.324059\pi$
$564$	0	0
$565$	−12.9700	−0.545653
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.4548	−1.77980	−0.889899	−	0.456157i	$-0.849225\pi$
$569$	−42.4548	−1.77980	−0.889899	+	0.456157i	$0.849225\pi$
$570$	0	0
$571$	32.6977	1.36836	0.684179	−	0.729314i	$-0.260161\pi$
$571$	32.6977	1.36836	0.684179	+	0.729314i	$0.260161\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.48346	−0.312082
$576$	0	0
$577$	−13.2716	−0.552503	−0.276251	−	0.961085i	$-0.589092\pi$
$577$	−13.2716	−0.552503	−0.276251	+	0.961085i	$0.589092\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	42.1278	1.74776
$582$	0	0
$583$	1.45587	0.0602961
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.4067	−0.594626	−0.297313	−	0.954780i	$-0.596091\pi$
$587$	−14.4067	−0.594626	−0.297313	+	0.954780i	$0.596091\pi$
$588$	0	0
$589$	−3.04399	−0.125426
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.85863	0.0763247	0.0381623	−	0.999272i	$-0.487850\pi$
$593$	1.85863	0.0763247	0.0381623	+	0.999272i	$0.487850\pi$
$594$	0	0
$595$	29.0768	1.19203
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.2171	−0.826049	−0.413025	−	0.910720i	$-0.635528\pi$
$599$	−20.2171	−0.826049	−0.413025	+	0.910720i	$0.635528\pi$
$600$	0	0
$601$	−29.6608	−1.20989	−0.604944	−	0.796268i	$-0.706804\pi$
$601$	−29.6608	−1.20989	−0.604944	+	0.796268i	$0.706804\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10.4342	0.424212
$606$	0	0
$607$	−8.06040	−0.327161	−0.163581	−	0.986530i	$-0.552304\pi$
$607$	−8.06040	−0.327161	−0.163581	+	0.986530i	$0.552304\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−27.1801	−1.09959
$612$	0	0
$613$	−35.7886	−1.44549	−0.722743	−	0.691117i	$-0.757119\pi$
$613$	−35.7886	−1.44549	−0.722743	+	0.691117i	$0.757119\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.7159	−0.874250	−0.437125	−	0.899401i	$-0.644003\pi$
$617$	−21.7159	−0.874250	−0.437125	+	0.899401i	$0.644003\pi$
$618$	0	0
$619$	15.3679	0.617688	0.308844	−	0.951113i	$-0.400058\pi$
$619$	15.3679	0.617688	0.308844	+	0.951113i	$0.400058\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.43530	0.217761
$624$	0	0
$625$	−9.41188	−0.376475
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.09215	0.362528
$630$	0	0
$631$	24.5358	0.976754	0.488377	−	0.872633i	$-0.337589\pi$
$631$	24.5358	0.976754	0.488377	+	0.872633i	$0.337589\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.4067	0.412975
$636$	0	0
$637$	2.26217	0.0896305
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.0521	0.634018	0.317009	−	0.948422i	$-0.397321\pi$
$641$	16.0521	0.634018	0.317009	+	0.948422i	$0.397321\pi$
$642$	0	0
$643$	9.47645	0.373715	0.186857	−	0.982387i	$-0.440170\pi$
$643$	9.47645	0.373715	0.186857	+	0.982387i	$0.440170\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.1924	1.61944	0.809720	−	0.586817i	$-0.199619\pi$
$647$	41.1924	1.61944	0.809720	+	0.586817i	$0.199619\pi$
$648$	0	0
$649$	2.18953	0.0859467
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14.0757	−0.550827	−0.275413	−	0.961326i	$-0.588815\pi$
$653$	−14.0757	−0.550827	−0.275413	+	0.961326i	$0.588815\pi$
$654$	0	0
$655$	26.1948	1.02351
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.6220	−0.803319	−0.401659	−	0.915789i	$-0.631566\pi$
$659$	−20.6220	−0.803319	−0.401659	+	0.915789i	$0.631566\pi$
$660$	0	0
$661$	−30.1414	−1.17236	−0.586182	−	0.810180i	$-0.699370\pi$
$661$	−30.1414	−1.17236	−0.586182	+	0.810180i	$0.699370\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−22.7857	−0.883592
$666$	0	0
$667$	31.4231	1.21671
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.0757	−0.929434
$672$	0	0
$673$	3.27468	0.126230	0.0631148	−	0.998006i	$-0.479897\pi$
$673$	3.27468	0.126230	0.0631148	+	0.998006i	$0.479897\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.6126	−0.753773	−0.376887	−	0.926259i	$-0.623005\pi$
$677$	−19.6126	−0.753773	−0.376887	+	0.926259i	$0.623005\pi$
$678$	0	0
$679$	−9.15778	−0.351443
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.1812	0.466101	0.233050	−	0.972465i	$-0.425129\pi$
$683$	12.1812	0.466101	0.233050	+	0.972465i	$0.425129\pi$
$684$	0	0
$685$	1.11796	0.0427149
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.78572	0.106128
$690$	0	0
$691$	13.8052	0.525176	0.262588	−	0.964908i	$-0.415424\pi$
$691$	13.8052	0.525176	0.262588	+	0.964908i	$0.415424\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−31.5850	−1.19809
$696$	0	0
$697$	−25.8761	−0.980127
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.06980	0.342561	0.171281	−	0.985222i	$-0.445209\pi$
$701$	9.06980	0.342561	0.171281	+	0.985222i	$0.445209\pi$
$702$	0	0
$703$	−7.12497	−0.268723
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.58812	−0.360598
$708$	0	0
$709$	26.2171	0.984605	0.492302	−	0.870424i	$-0.336155\pi$
$709$	26.2171	0.984605	0.492302	+	0.870424i	$0.336155\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.12391	0.0795409
$714$	0	0
$715$	−15.4231	−0.576790
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−37.5972	−1.40214	−0.701070	−	0.713092i	$-0.747294\pi$
$719$	−37.5972	−1.40214	−0.701070	+	0.713092i	$0.747294\pi$
$720$	0	0
$721$	22.5634	0.840304
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−19.8297	−0.736457
$726$	0	0
$727$	−42.3463	−1.57054	−0.785268	−	0.619156i	$-0.787475\pi$
$727$	−42.3463	−1.57054	−0.785268	+	0.619156i	$0.787475\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−53.7592	−1.98836
$732$	0	0
$733$	−5.96408	−0.220288	−0.110144	−	0.993916i	$-0.535131\pi$
$733$	−5.96408	−0.220288	−0.110144	+	0.993916i	$0.535131\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.7448	0.800981
$738$	0	0
$739$	−39.9435	−1.46935	−0.734673	−	0.678422i	$-0.762664\pi$
$739$	−39.9435	−1.46935	−0.734673	+	0.678422i	$0.762664\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22.7407	0.834274	0.417137	−	0.908844i	$-0.363033\pi$
$743$	22.7407	0.834274	0.417137	+	0.908844i	$0.363033\pi$
$744$	0	0
$745$	−13.5316	−0.495760
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−48.8492	−1.78491
$750$	0	0
$751$	−17.8052	−0.649722	−0.324861	−	0.945762i	$-0.605318\pi$
$751$	−17.8052	−0.649722	−0.324861	+	0.945762i	$0.605318\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−30.0122	−1.09226
$756$	0	0
$757$	27.4353	0.997153	0.498576	−	0.866846i	$-0.333856\pi$
$757$	27.4353	0.997153	0.498576	+	0.866846i	$0.333856\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10.1812	−0.369068	−0.184534	−	0.982826i	$-0.559078\pi$
$761$	−10.1812	−0.369068	−0.184534	+	0.982826i	$0.559078\pi$
$762$	0	0
$763$	50.2447	1.81898
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.18953	0.151275
$768$	0	0
$769$	38.0398	1.37175	0.685876	−	0.727719i	$-0.259419\pi$
$769$	38.0398	1.37175	0.685876	+	0.727719i	$0.259419\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.1676	1.58860	0.794300	−	0.607526i	$-0.207838\pi$
$773$	44.1676	1.58860	0.794300	+	0.607526i	$0.207838\pi$
$774$	0	0
$775$	−1.34030	−0.0481452
$776$	0	0
$777$	0	0
$778$	0	0
$779$	20.2775	0.726517
$780$	0	0
$781$	14.9477	0.534870
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17.5522	−0.626465
$786$	0	0
$787$	16.3379	0.582384	0.291192	−	0.956665i	$-0.405948\pi$
$787$	16.3379	0.582384	0.291192	+	0.956665i	$0.405948\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−21.1823	−0.753154
$792$	0	0
$793$	−46.0674	−1.63590
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.9424	1.16688	0.583441	−	0.812156i	$-0.301706\pi$
$797$	32.9424	1.16688	0.583441	+	0.812156i	$0.301706\pi$
$798$	0	0
$799$	−40.8597	−1.44551
$800$	0	0
$801$	0	0
$802$	0	0
$803$	29.5675	1.04342
$804$	0	0
$805$	15.8984	0.560347
$806$	0	0
$807$	0	0
$808$	0	0
$809$	51.9659	1.82702	0.913511	−	0.406814i	$-0.133360\pi$
$809$	51.9659	1.82702	0.913511	+	0.406814i	$0.133360\pi$
$810$	0	0
$811$	−26.9424	−0.946077	−0.473039	−	0.881042i	$-0.656843\pi$
$811$	−26.9424	−0.946077	−0.473039	+	0.881042i	$0.656843\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.32568	−0.221579
$816$	0	0
$817$	42.1278	1.47387
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27.8914	0.973418	0.486709	−	0.873564i	$-0.338197\pi$
$821$	27.8914	0.973418	0.486709	+	0.873564i	$0.338197\pi$
$822$	0	0
$823$	−18.3103	−0.638258	−0.319129	−	0.947711i	$-0.603390\pi$
$823$	−18.3103	−0.638258	−0.319129	+	0.947711i	$0.603390\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.37384	−0.152093	−0.0760467	−	0.997104i	$-0.524230\pi$
$827$	−4.37384	−0.152093	−0.0760467	+	0.997104i	$0.524230\pi$
$828$	0	0
$829$	30.5204	1.06002	0.530009	−	0.847992i	$-0.322188\pi$
$829$	30.5204	1.06002	0.530009	+	0.847992i	$0.322188\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.40070	0.117827
$834$	0	0
$835$	−30.4465	−1.05364
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−0.859686	−0.0296797	−0.0148398	−	0.999890i	$-0.504724\pi$
$839$	−0.859686	−0.0296797	−0.0148398	+	0.999890i	$0.504724\pi$
$840$	0	0
$841$	54.2650	1.87121
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.65375	−0.263297
$846$	0	0
$847$	17.0409	0.585532
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.97136	0.170416
$852$	0	0
$853$	50.4741	1.72820	0.864099	−	0.503321i	$-0.167889\pi$
$853$	50.4741	1.72820	0.864099	+	0.503321i	$0.167889\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38.4946	−1.31495	−0.657476	−	0.753476i	$-0.728376\pi$
$857$	−38.4946	−1.31495	−0.657476	+	0.753476i	$0.728376\pi$
$858$	0	0
$859$	27.1044	0.924790	0.462395	−	0.886674i	$-0.346990\pi$
$859$	27.1044	0.924790	0.462395	+	0.886674i	$0.346990\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.6443	0.362338	0.181169	−	0.983452i	$-0.442012\pi$
$863$	10.6443	0.362338	0.181169	+	0.983452i	$0.442012\pi$
$864$	0	0
$865$	−16.2517	−0.552575
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.56860	0.0871339
$870$	0	0
$871$	41.6074	1.40981
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−33.1166	−1.11955
$876$	0	0
$877$	40.0328	1.35181	0.675906	−	0.736988i	$-0.263753\pi$
$877$	40.0328	1.35181	0.675906	+	0.736988i	$0.263753\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−29.9588	−1.00934	−0.504670	−	0.863313i	$-0.668386\pi$
$881$	−29.9588	−1.00934	−0.504670	+	0.863313i	$0.668386\pi$
$882$	0	0
$883$	12.3239	0.414732	0.207366	−	0.978263i	$-0.433511\pi$
$883$	12.3239	0.414732	0.207366	+	0.978263i	$0.433511\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.4559	0.854725	0.427362	−	0.904080i	$-0.359443\pi$
$887$	25.4559	0.854725	0.427362	+	0.904080i	$0.359443\pi$
$888$	0	0
$889$	16.9958	0.570022
$890$	0	0
$891$	0	0
$892$	0	0
$893$	32.0192	1.07148
$894$	0	0
$895$	15.0580	0.503334
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.62794	0.187702
$900$	0	0
$901$	4.18775	0.139514
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.819863	0.0272532
$906$	0	0
$907$	−38.4863	−1.27792	−0.638958	−	0.769241i	$-0.720634\pi$
$907$	−38.4863	−1.27792	−0.638958	+	0.769241i	$0.720634\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	21.2234	0.703163	0.351581	−	0.936157i	$-0.385644\pi$
$911$	21.2234	0.703163	0.351581	+	0.936157i	$0.385644\pi$
$912$	0	0
$913$	−33.5920	−1.11173
$914$	0	0
$915$	0	0
$916$	0	0
$917$	42.7805	1.41274
$918$	0	0
$919$	−46.3051	−1.52746	−0.763732	−	0.645533i	$-0.776635\pi$
$919$	−46.3051	−1.52746	−0.763732	+	0.645533i	$0.776635\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.6014	0.941427
$924$	0	0
$925$	−3.13720	−0.103151
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−31.9313	−1.04763	−0.523815	−	0.851832i	$-0.675492\pi$
$929$	−31.9313	−1.04763	−0.523815	+	0.851832i	$0.675492\pi$
$930$	0	0
$931$	−2.66492	−0.0873394
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−23.1854	−0.758243
$936$	0	0
$937$	−33.6813	−1.10032	−0.550161	−	0.835059i	$-0.685433\pi$
$937$	−33.6813	−1.10032	−0.550161	+	0.835059i	$0.685433\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.6977	0.739925	0.369963	−	0.929047i	$-0.379371\pi$
$941$	22.6977	0.739925	0.369963	+	0.929047i	$0.379371\pi$
$942$	0	0
$943$	−14.1484	−0.460735
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.5533	0.732882	0.366441	−	0.930441i	$-0.380576\pi$
$947$	22.5533	0.732882	0.366441	+	0.930441i	$0.380576\pi$
$948$	0	0
$949$	56.5756	1.83652
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.3791	0.336211	0.168105	−	0.985769i	$-0.446235\pi$
$953$	10.3791	0.336211	0.168105	+	0.985769i	$0.446235\pi$
$954$	0	0
$955$	−8.53579	−0.276212
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.82581	0.0589586
$960$	0	0
$961$	−30.6196	−0.987729
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−15.7417	−0.506744
$966$	0	0
$967$	−32.1913	−1.03520	−0.517601	−	0.855622i	$-0.673175\pi$
$967$	−32.1913	−1.03520	−0.517601	+	0.855622i	$0.673175\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−33.1114	−1.06260	−0.531298	−	0.847185i	$-0.678295\pi$
$971$	−33.1114	−1.06260	−0.531298	+	0.847185i	$0.678295\pi$
$972$	0	0
$973$	−51.5837	−1.65370
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−12.2307	−0.391294	−0.195647	−	0.980674i	$-0.562681\pi$
$977$	−12.2307	−0.391294	−0.195647	+	0.980674i	$0.562681\pi$
$978$	0	0
$979$	−4.33402	−0.138516
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−7.26111	−0.231593	−0.115797	−	0.993273i	$-0.536942\pi$
$983$	−7.26111	−0.231593	−0.115797	+	0.993273i	$0.536942\pi$
$984$	0	0
$985$	−27.3215	−0.870536
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−29.3941	−0.934679
$990$	0	0
$991$	20.1054	0.638671	0.319335	−	0.947642i	$-0.396540\pi$
$991$	20.1054	0.638671	0.319335	+	0.947642i	$0.396540\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.1801	0.354434
$996$	0	0
$997$	−42.3173	−1.34020	−0.670102	−	0.742269i	$-0.733750\pi$
$997$	−42.3173	−1.34020	−0.670102	+	0.742269i	$0.733750\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8496.2.a.bl.1.1		3
3.2	odd	2		2832.2.a.t.1.3		3
4.3	odd	2		531.2.a.d.1.2		3
12.11	even	2		177.2.a.d.1.2	✓	3
60.59	even	2		4425.2.a.w.1.2		3
84.83	odd	2		8673.2.a.s.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.d.1.2	✓	3	12.11	even	2
531.2.a.d.1.2		3	4.3	odd	2
2832.2.a.t.1.3		3	3.2	odd	2
4425.2.a.w.1.2		3	60.59	even	2
8496.2.a.bl.1.1		3	1.1	even	1	trivial
8673.2.a.s.1.2		3	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8496 = 2^{4} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8496.a (trivial)

\(f(q)\)	\(=\)	\(q-1.68133 q^{5} -2.74590 q^{7} +O(q^{10})\)	\(q-1.68133 q^{5} -2.74590 q^{7} +2.18953 q^{11} +4.18953 q^{13} +6.29809 q^{17} -4.93543 q^{19} +3.44364 q^{23} -2.17313 q^{25} +9.12497 q^{29} +0.616763 q^{31} +4.61676 q^{35} +1.44364 q^{37} -4.10856 q^{41} -8.53579 q^{43} -6.48763 q^{47} +0.539958 q^{49} +0.664924 q^{53} -3.68133 q^{55} +1.00000 q^{59} -10.9958 q^{61} -7.04399 q^{65} +9.93126 q^{67} +6.82687 q^{71} +13.5040 q^{73} -6.01224 q^{77} +1.17313 q^{79} -15.3421 q^{83} -10.5892 q^{85} -1.97942 q^{89} -11.5040 q^{91} +8.29809 q^{95} +3.33508 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} - 9 q^{7}+O(q^{10}) \) 3 * q + 2 * q^5 - 9 * q^7	\( 3 q + 2 q^{5} - 9 q^{7} - 2 q^{11} + 4 q^{13} - 3 q^{17} - 7 q^{19} + q^{23} - q^{25} + 11 q^{29} - 13 q^{31} - q^{35} - 5 q^{37} + q^{41} - 6 q^{43} + 11 q^{47} + 14 q^{49} - 2 q^{53} - 4 q^{55} + 3 q^{59} - q^{61} - 10 q^{67} + 26 q^{71} + 7 q^{73} + 17 q^{77} - 2 q^{79} - 3 q^{83} - 35 q^{85} + 23 q^{89} - q^{91} + 3 q^{95} + 14 q^{97}+O(q^{100}) \) 3 * q + 2 * q^5 - 9 * q^7 - 2 * q^11 + 4 * q^13 - 3 * q^17 - 7 * q^19 + q^23 - q^25 + 11 * q^29 - 13 * q^31 - q^35 - 5 * q^37 + q^41 - 6 * q^43 + 11 * q^47 + 14 * q^49 - 2 * q^53 - 4 * q^55 + 3 * q^59 - q^61 - 10 * q^67 + 26 * q^71 + 7 * q^73 + 17 * q^77 - 2 * q^79 - 3 * q^83 - 35 * q^85 + 23 * q^89 - q^91 + 3 * q^95 + 14 * q^97

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