LMFDB - Embedded newform 8496.2.a.bk.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1416)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14510$ 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)

$f(q)$	$=$	$q-2.74657 q^{5} +0.601466 q^{7} +O(q^{10})$	$q-2.74657 q^{5} +0.601466 q^{7} -3.00000 q^{11} -3.29021 q^{13} +2.14510 q^{17} +5.89167 q^{19} -1.60147 q^{23} +2.54364 q^{25} +5.14510 q^{29} +3.14510 q^{31} -1.65197 q^{35} +4.43531 q^{37} -11.1819 q^{41} +3.00000 q^{43} -1.65197 q^{47} -6.63824 q^{49} +6.74657 q^{53} +8.23970 q^{55} +1.00000 q^{59} +3.43531 q^{61} +9.03677 q^{65} -1.94950 q^{67} -4.70979 q^{71} +10.9284 q^{73} -1.80440 q^{77} -0.202931 q^{79} -3.39853 q^{83} -5.89167 q^{85} +12.6382 q^{89} -1.97895 q^{91} -16.1819 q^{95} -13.6172 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q+O(q^{10}) $ 3 * q	$ 3 q - 9 q^{11} + 3 q^{13} + 3 q^{19} - 3 q^{23} + 3 q^{25} + 9 q^{29} + 3 q^{31} - 15 q^{35} - 6 q^{37} - 6 q^{41} + 9 q^{43} - 15 q^{47} + 3 q^{49} + 12 q^{53} + 3 q^{59} - 9 q^{61} + 6 q^{65} + 6 q^{67} - 27 q^{71} - 3 q^{73} + 3 q^{79} - 12 q^{83} - 3 q^{85} + 15 q^{89} + 18 q^{91} - 21 q^{95} + 6 q^{97}+O(q^{100}) $ 3 * q - 9 * q^11 + 3 * q^13 + 3 * q^19 - 3 * q^23 + 3 * q^25 + 9 * q^29 + 3 * q^31 - 15 * q^35 - 6 * q^37 - 6 * q^41 + 9 * q^43 - 15 * q^47 + 3 * q^49 + 12 * q^53 + 3 * q^59 - 9 * q^61 + 6 * q^65 + 6 * q^67 - 27 * q^71 - 3 * q^73 + 3 * q^79 - 12 * q^83 - 3 * q^85 + 15 * q^89 + 18 * q^91 - 21 * q^95 + 6 * 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
$5$	−2.74657	−1.22830	−0.614151	−	0.789188i	$-0.710502\pi$
$5$	−2.74657	−1.22830	−0.614151	+	0.789188i	$0.710502\pi$
$6$	0	0
$7$	0.601466	0.227333	0.113666	−	0.993519i	$-0.463741\pi$
$7$	0.601466	0.227333	0.113666	+	0.993519i	$0.463741\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−3.29021	−0.912539	−0.456269	−	0.889842i	$-0.650815\pi$
$13$	−3.29021	−0.912539	−0.456269	+	0.889842i	$0.650815\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.14510	0.520264	0.260132	−	0.965573i	$-0.416234\pi$
$17$	2.14510	0.520264	0.260132	+	0.965573i	$0.416234\pi$
$18$	0	0
$19$	5.89167	1.35164	0.675821	−	0.737066i	$-0.263789\pi$
$19$	5.89167	1.35164	0.675821	+	0.737066i	$0.263789\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.60147	−0.333929	−0.166964	−	0.985963i	$-0.553396\pi$
$23$	−1.60147	−0.333929	−0.166964	+	0.985963i	$0.553396\pi$
$24$	0	0
$25$	2.54364	0.508727
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.14510	0.955422	0.477711	−	0.878517i	$-0.341467\pi$
$29$	5.14510	0.955422	0.477711	+	0.878517i	$0.341467\pi$
$30$	0	0
$31$	3.14510	0.564877	0.282439	−	0.959285i	$-0.408857\pi$
$31$	3.14510	0.564877	0.282439	+	0.959285i	$0.408857\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.65197	−0.279233
$36$	0	0
$37$	4.43531	0.729160	0.364580	−	0.931172i	$-0.381213\pi$
$37$	4.43531	0.729160	0.364580	+	0.931172i	$0.381213\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.1819	−1.74632	−0.873158	−	0.487438i	$-0.837932\pi$
$41$	−11.1819	−1.74632	−0.873158	+	0.487438i	$0.837932\pi$
$42$	0	0
$43$	3.00000	0.457496	0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000	0.457496	0.228748	+	0.973486i	$0.426537\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.65197	−0.240964	−0.120482	−	0.992716i	$-0.538444\pi$
$47$	−1.65197	−0.240964	−0.120482	+	0.992716i	$0.538444\pi$
$48$	0	0
$49$	−6.63824	−0.948320
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.74657	0.926712	0.463356	−	0.886172i	$-0.346645\pi$
$53$	6.74657	0.926712	0.463356	+	0.886172i	$0.346645\pi$
$54$	0	0
$55$	8.23970	1.11104
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	3.43531	0.439846	0.219923	−	0.975517i	$-0.429419\pi$
$61$	3.43531	0.439846	0.219923	+	0.975517i	$0.429419\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.03677	1.12087
$66$	0	0
$67$	−1.94950	−0.238169	−0.119085	−	0.992884i	$-0.537996\pi$
$67$	−1.94950	−0.238169	−0.119085	+	0.992884i	$0.537996\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.70979	−0.558950	−0.279475	−	0.960153i	$-0.590160\pi$
$71$	−4.70979	−0.558950	−0.279475	+	0.960153i	$0.590160\pi$
$72$	0	0
$73$	10.9284	1.27908	0.639539	−	0.768759i	$-0.279125\pi$
$73$	10.9284	1.27908	0.639539	+	0.768759i	$0.279125\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.80440	−0.205630
$78$	0	0
$79$	−0.202931	−0.0228315	−0.0114158	−	0.999935i	$-0.503634\pi$
$79$	−0.202931	−0.0228315	−0.0114158	+	0.999935i	$0.503634\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.39853	−0.373038	−0.186519	−	0.982451i	$-0.559721\pi$
$83$	−3.39853	−0.373038	−0.186519	+	0.982451i	$0.559721\pi$
$84$	0	0
$85$	−5.89167	−0.639041
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.6382	1.33965	0.669825	−	0.742519i	$-0.266369\pi$
$89$	12.6382	1.33965	0.669825	+	0.742519i	$0.266369\pi$
$90$	0	0
$91$	−1.97895	−0.207450
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−16.1819	−1.66023
$96$	0	0
$97$	−13.6172	−1.38262	−0.691308	−	0.722560i	$-0.742965\pi$
$97$	−13.6172	−1.38262	−0.691308	+	0.722560i	$0.742965\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.8201	1.37515	0.687576	−	0.726112i	$-0.258675\pi$
$101$	13.8201	1.37515	0.687576	+	0.726112i	$0.258675\pi$
$102$	0	0
$103$	3.88434	0.382736	0.191368	−	0.981518i	$-0.438708\pi$
$103$	3.88434	0.382736	0.191368	+	0.981518i	$0.438708\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.81812	−0.175765	−0.0878823	−	0.996131i	$-0.528010\pi$
$107$	−1.81812	−0.175765	−0.0878823	+	0.996131i	$0.528010\pi$
$108$	0	0
$109$	9.38481	0.898902	0.449451	−	0.893305i	$-0.351620\pi$
$109$	9.38481	0.898902	0.449451	+	0.893305i	$0.351620\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.1240	−0.952390	−0.476195	−	0.879340i	$-0.657984\pi$
$113$	−10.1240	−0.952390	−0.476195	+	0.879340i	$0.657984\pi$
$114$	0	0
$115$	4.39853	0.410165
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.29021	0.118273
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.74657	0.603431
$126$	0	0
$127$	−1.25343	−0.111224	−0.0556120	−	0.998452i	$-0.517711\pi$
$127$	−1.25343	−0.111224	−0.0556120	+	0.998452i	$0.517711\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−0.216658	−0.0189295	−0.00946475	−	0.999955i	$-0.503013\pi$
$131$	−0.216658	−0.0189295	−0.00946475	+	0.999955i	$0.503013\pi$
$132$	0	0
$133$	3.54364	0.307272
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.29021	−0.793716	−0.396858	−	0.917880i	$-0.629899\pi$
$137$	−9.29021	−0.793716	−0.396858	+	0.917880i	$0.629899\pi$
$138$	0	0
$139$	−10.5299	−0.893135	−0.446568	−	0.894750i	$-0.647354\pi$
$139$	−10.5299	−0.893135	−0.446568	+	0.894750i	$0.647354\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.87062	0.825422
$144$	0	0
$145$	−14.1314	−1.17355
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.0441	1.31438	0.657192	−	0.753723i	$-0.271744\pi$
$149$	16.0441	1.31438	0.657192	+	0.753723i	$0.271744\pi$
$150$	0	0
$151$	−16.2324	−1.32097	−0.660486	−	0.750838i	$-0.729650\pi$
$151$	−16.2324	−1.32097	−0.660486	+	0.750838i	$0.729650\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.63824	−0.693840
$156$	0	0
$157$	3.94950	0.315204	0.157602	−	0.987503i	$-0.449624\pi$
$157$	3.94950	0.315204	0.157602	+	0.987503i	$0.449624\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.963226	−0.0759129
$162$	0	0
$163$	0.348034	0.0272601	0.0136301	−	0.999907i	$-0.495661\pi$
$163$	0.348034	0.0272601	0.0136301	+	0.999907i	$0.495661\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.5226	−1.27856	−0.639278	−	0.768976i	$-0.720767\pi$
$167$	−16.5226	−1.27856	−0.639278	+	0.768976i	$0.720767\pi$
$168$	0	0
$169$	−2.17455	−0.167273
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.77601	−0.515171	−0.257585	−	0.966256i	$-0.582927\pi$
$173$	−6.77601	−0.515171	−0.257585	+	0.966256i	$0.582927\pi$
$174$	0	0
$175$	1.52991	0.115650
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−22.2765	−1.66502	−0.832511	−	0.554008i	$-0.813098\pi$
$179$	−22.2765	−1.66502	−0.832511	+	0.554008i	$0.813098\pi$
$180$	0	0
$181$	−8.92844	−0.663646	−0.331823	−	0.943342i	$-0.607664\pi$
$181$	−8.92844	−0.663646	−0.331823	+	0.943342i	$0.607664\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.1819	−0.895629
$186$	0	0
$187$	−6.43531	−0.470596
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.2260	−1.39114	−0.695571	−	0.718457i	$-0.744849\pi$
$191$	−19.2260	−1.39114	−0.695571	+	0.718457i	$0.744849\pi$
$192$	0	0
$193$	5.45636	0.392758	0.196379	−	0.980528i	$-0.437082\pi$
$193$	5.45636	0.392758	0.196379	+	0.980528i	$0.437082\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.1892	0.725951	0.362975	−	0.931799i	$-0.381761\pi$
$197$	10.1892	0.725951	0.362975	+	0.931799i	$0.381761\pi$
$198$	0	0
$199$	−25.2186	−1.78770	−0.893851	−	0.448363i	$-0.852007\pi$
$199$	−25.2186	−1.78770	−0.893851	+	0.448363i	$0.852007\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.09460	0.217198
$204$	0	0
$205$	30.7118	2.14500
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−17.6750	−1.22261
$210$	0	0
$211$	−24.2186	−1.66728	−0.833640	−	0.552308i	$-0.813747\pi$
$211$	−24.2186	−1.66728	−0.833640	+	0.552308i	$0.813747\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.23970	−0.561943
$216$	0	0
$217$	1.89167	0.128415
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.05783	−0.474761
$222$	0	0
$223$	−13.9632	−0.935047	−0.467523	−	0.883981i	$-0.654854\pi$
$223$	−13.9632	−0.935047	−0.467523	+	0.883981i	$0.654854\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.6382	−1.03795	−0.518973	−	0.854791i	$-0.673686\pi$
$227$	−15.6382	−1.03795	−0.518973	+	0.854791i	$0.673686\pi$
$228$	0	0
$229$	5.29753	0.350071	0.175035	−	0.984562i	$-0.443996\pi$
$229$	5.29753	0.350071	0.175035	+	0.984562i	$0.443996\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.7696	−1.22964	−0.614819	−	0.788668i	$-0.710771\pi$
$233$	−18.7696	−1.22964	−0.614819	+	0.788668i	$0.710771\pi$
$234$	0	0
$235$	4.53724	0.295977
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.2397	0.791721	0.395860	−	0.918311i	$-0.370446\pi$
$239$	12.2397	0.791721	0.395860	+	0.918311i	$0.370446\pi$
$240$	0	0
$241$	3.12405	0.201238	0.100619	−	0.994925i	$-0.467918\pi$
$241$	3.12405	0.201238	0.100619	+	0.994925i	$0.467918\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	18.2324	1.16482
$246$	0	0
$247$	−19.3848	−1.23343
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.44264	0.469775	0.234888	−	0.972023i	$-0.424528\pi$
$251$	7.44264	0.469775	0.234888	+	0.972023i	$0.424528\pi$
$252$	0	0
$253$	4.80440	0.302050
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.9358	−1.24356	−0.621780	−	0.783192i	$-0.713590\pi$
$257$	−19.9358	−1.24356	−0.621780	+	0.783192i	$0.713590\pi$
$258$	0	0
$259$	2.66769	0.165762
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.2829	0.695732	0.347866	−	0.937544i	$-0.386906\pi$
$263$	11.2829	0.695732	0.347866	+	0.937544i	$0.386906\pi$
$264$	0	0
$265$	−18.5299	−1.13828
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.68141	0.468344	0.234172	−	0.972195i	$-0.424762\pi$
$269$	7.68141	0.468344	0.234172	+	0.972195i	$0.424762\pi$
$270$	0	0
$271$	3.68141	0.223630	0.111815	−	0.993729i	$-0.464334\pi$
$271$	3.68141	0.223630	0.111815	+	0.993729i	$0.464334\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.63091	−0.460161
$276$	0	0
$277$	31.6172	1.89969	0.949846	−	0.312717i	$-0.101239\pi$
$277$	31.6172	1.89969	0.949846	+	0.312717i	$0.101239\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	9.41425	0.561607	0.280804	−	0.959765i	$-0.409399\pi$
$281$	9.41425	0.561607	0.280804	+	0.959765i	$0.409399\pi$
$282$	0	0
$283$	6.72551	0.399790	0.199895	−	0.979817i	$-0.435940\pi$
$283$	6.72551	0.399790	0.199895	+	0.979817i	$0.435940\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.72551	−0.396995
$288$	0	0
$289$	−12.3985	−0.729326
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.181876	0.0106253	0.00531266	−	0.999986i	$-0.498309\pi$
$293$	0.181876	0.0106253	0.00531266	+	0.999986i	$0.498309\pi$
$294$	0	0
$295$	−2.74657	−0.159911
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.26915	0.304723
$300$	0	0
$301$	1.80440	0.104004
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.43531	−0.540264
$306$	0	0
$307$	−5.08727	−0.290346	−0.145173	−	0.989406i	$-0.546374\pi$
$307$	−5.08727	−0.290346	−0.145173	+	0.989406i	$0.546374\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.8412	1.06838	0.534192	−	0.845363i	$-0.320616\pi$
$311$	18.8412	1.06838	0.534192	+	0.845363i	$0.320616\pi$
$312$	0	0
$313$	4.00733	0.226508	0.113254	−	0.993566i	$-0.463873\pi$
$313$	4.00733	0.226508	0.113254	+	0.993566i	$0.463873\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.15243	0.0647269	0.0323635	−	0.999476i	$-0.489697\pi$
$317$	1.15243	0.0647269	0.0323635	+	0.999476i	$0.489697\pi$
$318$	0	0
$319$	−15.4353	−0.864211
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.6382	0.703210
$324$	0	0
$325$	−8.36909	−0.464234
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.993601	−0.0547790
$330$	0	0
$331$	4.52152	0.248525	0.124263	−	0.992249i	$-0.460343\pi$
$331$	4.52152	0.248525	0.124263	+	0.992249i	$0.460343\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.35443	0.292544
$336$	0	0
$337$	11.7760	0.641480	0.320740	−	0.947167i	$-0.396068\pi$
$337$	11.7760	0.641480	0.320740	+	0.947167i	$0.396068\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.43531	−0.510951
$342$	0	0
$343$	−8.20293	−0.442917
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.173485	0.00931318	0.00465659	−	0.999989i	$-0.498518\pi$
$347$	0.173485	0.00931318	0.00465659	+	0.999989i	$0.498518\pi$
$348$	0	0
$349$	−31.5530	−1.68899	−0.844496	−	0.535563i	$-0.820100\pi$
$349$	−31.5530	−1.68899	−0.844496	+	0.535563i	$0.820100\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.55936	0.508793	0.254397	−	0.967100i	$-0.418123\pi$
$353$	9.55936	0.508793	0.254397	+	0.967100i	$0.418123\pi$
$354$	0	0
$355$	12.9358	0.686560
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.6172	−0.982577	−0.491289	−	0.870997i	$-0.663474\pi$
$359$	−18.6172	−0.982577	−0.491289	+	0.870997i	$0.663474\pi$
$360$	0	0
$361$	15.7118	0.826936
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−30.0157	−1.57109
$366$	0	0
$367$	−15.5583	−0.812136	−0.406068	−	0.913843i	$-0.633100\pi$
$367$	−15.5583	−0.812136	−0.406068	+	0.913843i	$0.633100\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.05783	0.210672
$372$	0	0
$373$	−0.297533	−0.0154057	−0.00770284	−	0.999970i	$-0.502452\pi$
$373$	−0.297533	−0.0154057	−0.00770284	+	0.999970i	$0.502452\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.9284	−0.871859
$378$	0	0
$379$	24.6456	1.26596	0.632979	−	0.774169i	$-0.281832\pi$
$379$	24.6456	1.26596	0.632979	+	0.774169i	$0.281832\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.66769	0.136312	0.0681562	−	0.997675i	$-0.478288\pi$
$383$	2.66769	0.136312	0.0681562	+	0.997675i	$0.478288\pi$
$384$	0	0
$385$	4.95590	0.252576
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.56668	−0.282242	−0.141121	−	0.989992i	$-0.545071\pi$
$389$	−5.56668	−0.282242	−0.141121	+	0.989992i	$0.545071\pi$
$390$	0	0
$391$	−3.43531	−0.173731
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.557364	0.0280440
$396$	0	0
$397$	−10.9358	−0.548851	−0.274425	−	0.961608i	$-0.588488\pi$
$397$	−10.9358	−0.548851	−0.274425	+	0.961608i	$0.588488\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.47208	0.323200	0.161600	−	0.986856i	$-0.448335\pi$
$401$	6.47208	0.323200	0.161600	+	0.986856i	$0.448335\pi$
$402$	0	0
$403$	−10.3480	−0.515472
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.3059	−0.659550
$408$	0	0
$409$	−22.5804	−1.11653	−0.558265	−	0.829663i	$-0.688533\pi$
$409$	−22.5804	−1.11653	−0.558265	+	0.829663i	$0.688533\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.601466	0.0295962
$414$	0	0
$415$	9.33431	0.458203
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−17.3554	−0.847865	−0.423932	−	0.905694i	$-0.639351\pi$
$419$	−17.3554	−0.847865	−0.423932	+	0.905694i	$0.639351\pi$
$420$	0	0
$421$	−29.7275	−1.44883	−0.724415	−	0.689364i	$-0.757890\pi$
$421$	−29.7275	−1.44883	−0.724415	+	0.689364i	$0.757890\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.45636	0.264672
$426$	0	0
$427$	2.06622	0.0999914
$428$	0	0
$429$	0	0
$430$	0	0
$431$	27.6981	1.33417	0.667084	−	0.744982i	$-0.267542\pi$
$431$	27.6981	1.33417	0.667084	+	0.744982i	$0.267542\pi$
$432$	0	0
$433$	−3.97055	−0.190813	−0.0954063	−	0.995438i	$-0.530415\pi$
$433$	−3.97055	−0.190813	−0.0954063	+	0.995438i	$0.530415\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.43531	−0.451352
$438$	0	0
$439$	−7.44904	−0.355523	−0.177762	−	0.984074i	$-0.556886\pi$
$439$	−7.44904	−0.355523	−0.177762	+	0.984074i	$0.556886\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.1314	−1.47910	−0.739548	−	0.673104i	$-0.764961\pi$
$443$	−31.1314	−1.47910	−0.739548	+	0.673104i	$0.764961\pi$
$444$	0	0
$445$	−34.7118	−1.64550
$446$	0	0
$447$	0	0
$448$	0	0
$449$	26.4657	1.24899	0.624496	−	0.781028i	$-0.285304\pi$
$449$	26.4657	1.24899	0.624496	+	0.781028i	$0.285304\pi$
$450$	0	0
$451$	33.5456	1.57960
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.43531	0.254811
$456$	0	0
$457$	30.5961	1.43123	0.715613	−	0.698497i	$-0.246147\pi$
$457$	30.5961	1.43123	0.715613	+	0.698497i	$0.246147\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11.1535	−0.519470	−0.259735	−	0.965680i	$-0.583635\pi$
$461$	−11.1535	−0.519470	−0.259735	+	0.965680i	$0.583635\pi$
$462$	0	0
$463$	−26.9515	−1.25254	−0.626271	−	0.779605i	$-0.715420\pi$
$463$	−26.9515	−1.25254	−0.626271	+	0.779605i	$0.715420\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−30.6172	−1.41679	−0.708397	−	0.705814i	$-0.750581\pi$
$467$	−30.6172	−1.41679	−0.708397	+	0.705814i	$0.750581\pi$
$468$	0	0
$469$	−1.17256	−0.0541436
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−9.00000	−0.413820
$474$	0	0
$475$	14.9863	0.687617
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.3711	0.656631	0.328316	−	0.944568i	$-0.393519\pi$
$479$	14.3711	0.656631	0.328316	+	0.944568i	$0.393519\pi$
$480$	0	0
$481$	−14.5931	−0.665387
$482$	0	0
$483$	0	0
$484$	0	0
$485$	37.4005	1.69827
$486$	0	0
$487$	−18.6520	−0.845201	−0.422601	−	0.906316i	$-0.638883\pi$
$487$	−18.6520	−0.845201	−0.422601	+	0.906316i	$0.638883\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.46369	−0.201444	−0.100722	−	0.994915i	$-0.532115\pi$
$491$	−4.46369	−0.201444	−0.100722	+	0.994915i	$0.532115\pi$
$492$	0	0
$493$	11.0368	0.497071
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.83278	−0.127068
$498$	0	0
$499$	−7.55829	−0.338356	−0.169178	−	0.985586i	$-0.554111\pi$
$499$	−7.55829	−0.338356	−0.169178	+	0.985586i	$0.554111\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.1451	0.942814	0.471407	−	0.881916i	$-0.343746\pi$
$503$	21.1451	0.942814	0.471407	+	0.881916i	$0.343746\pi$
$504$	0	0
$505$	−37.9579	−1.68910
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.5687	−1.17764	−0.588818	−	0.808266i	$-0.700406\pi$
$509$	−26.5687	−1.17764	−0.588818	+	0.808266i	$0.700406\pi$
$510$	0	0
$511$	6.57308	0.290776
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.6686	−0.470115
$516$	0	0
$517$	4.95590	0.217960
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.84117	0.431149	0.215575	−	0.976487i	$-0.430838\pi$
$521$	9.84117	0.431149	0.215575	+	0.976487i	$0.430838\pi$
$522$	0	0
$523$	31.3270	1.36983	0.684917	−	0.728621i	$-0.259839\pi$
$523$	31.3270	1.36983	0.684917	+	0.728621i	$0.259839\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.74657	0.293885
$528$	0	0
$529$	−20.4353	−0.888492
$530$	0	0
$531$	0	0
$532$	0	0
$533$	36.7907	1.59358
$534$	0	0
$535$	4.99360	0.215892
$536$	0	0
$537$	0	0
$538$	0	0
$539$	19.9147	0.857788
$540$	0	0
$541$	−31.5383	−1.35594	−0.677969	−	0.735091i	$-0.737140\pi$
$541$	−31.5383	−1.35594	−0.677969	+	0.735091i	$0.737140\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−25.7760	−1.10412
$546$	0	0
$547$	−41.8642	−1.78998	−0.894992	−	0.446082i	$-0.852819\pi$
$547$	−41.8642	−1.78998	−0.894992	+	0.446082i	$0.852819\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	30.3133	1.29139
$552$	0	0
$553$	−0.122056	−0.00519035
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.4931	−1.08018	−0.540089	−	0.841608i	$-0.681610\pi$
$557$	−25.4931	−1.08018	−0.540089	+	0.841608i	$0.681610\pi$
$558$	0	0
$559$	−9.87062	−0.417483
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.47009	−0.230537	−0.115268	−	0.993334i	$-0.536773\pi$
$563$	−5.47009	−0.230537	−0.115268	+	0.993334i	$0.536773\pi$
$564$	0	0
$565$	27.8064	1.16982
$566$	0	0
$567$	0	0
$568$	0	0
$569$	42.4216	1.77841	0.889203	−	0.457514i	$-0.151260\pi$
$569$	42.4216	1.77841	0.889203	+	0.457514i	$0.151260\pi$
$570$	0	0
$571$	−3.42798	−0.143457	−0.0717283	−	0.997424i	$-0.522851\pi$
$571$	−3.42798	−0.143457	−0.0717283	+	0.997424i	$0.522851\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.07355	−0.169879
$576$	0	0
$577$	−43.6265	−1.81620	−0.908098	−	0.418759i	$-0.862465\pi$
$577$	−43.6265	−1.81620	−0.908098	+	0.418759i	$0.862465\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.04410	−0.0848036
$582$	0	0
$583$	−20.2397	−0.838243
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.29021	−0.135801	−0.0679007	−	0.997692i	$-0.521630\pi$
$587$	−3.29021	−0.135801	−0.0679007	+	0.997692i	$0.521630\pi$
$588$	0	0
$589$	18.5299	0.763512
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−13.7255	−0.563639	−0.281820	−	0.959467i	$-0.590938\pi$
$593$	−13.7255	−0.563639	−0.281820	+	0.959467i	$0.590938\pi$
$594$	0	0
$595$	−3.54364	−0.145275
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.6318	1.00643	0.503215	−	0.864161i	$-0.332150\pi$
$599$	24.6318	1.00643	0.503215	+	0.864161i	$0.332150\pi$
$600$	0	0
$601$	−3.60147	−0.146907	−0.0734534	−	0.997299i	$-0.523402\pi$
$601$	−3.60147	−0.146907	−0.0734534	+	0.997299i	$0.523402\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.49314	0.223328
$606$	0	0
$607$	−47.4520	−1.92602	−0.963008	−	0.269474i	$-0.913150\pi$
$607$	−47.4520	−1.92602	−0.963008	+	0.269474i	$0.913150\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.43531	0.219889
$612$	0	0
$613$	−10.9348	−0.441654	−0.220827	−	0.975313i	$-0.570876\pi$
$613$	−10.9348	−0.441654	−0.220827	+	0.975313i	$0.570876\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.5961	1.27201	0.636006	−	0.771684i	$-0.280585\pi$
$617$	31.5961	1.27201	0.636006	+	0.771684i	$0.280585\pi$
$618$	0	0
$619$	−33.0829	−1.32971	−0.664856	−	0.746971i	$-0.731507\pi$
$619$	−33.0829	−1.32971	−0.664856	+	0.746971i	$0.731507\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.60147	0.304546
$624$	0	0
$625$	−31.2481	−1.24992
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.51419	0.379356
$630$	0	0
$631$	−17.1157	−0.681364	−0.340682	−	0.940179i	$-0.610658\pi$
$631$	−17.1157	−0.681364	−0.340682	+	0.940179i	$0.610658\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.44264	0.136617
$636$	0	0
$637$	21.8412	0.865379
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−48.9083	−1.93176	−0.965881	−	0.258985i	$-0.916612\pi$
$641$	−48.9083	−1.93176	−0.965881	+	0.258985i	$0.916612\pi$
$642$	0	0
$643$	49.1260	1.93734	0.968671	−	0.248348i	$-0.0798876\pi$
$643$	49.1260	1.93734	0.968671	+	0.248348i	$0.0798876\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	41.1608	1.61820	0.809099	−	0.587672i	$-0.199955\pi$
$647$	41.1608	1.61820	0.809099	+	0.587672i	$0.199955\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.0618	1.48947	0.744737	−	0.667358i	$-0.232575\pi$
$653$	38.0618	1.48947	0.744737	+	0.667358i	$0.232575\pi$
$654$	0	0
$655$	0.595066	0.0232512
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−9.90633	−0.385896	−0.192948	−	0.981209i	$-0.561805\pi$
$659$	−9.90633	−0.385896	−0.192948	+	0.981209i	$0.561805\pi$
$660$	0	0
$661$	1.14510	0.0445393	0.0222697	−	0.999752i	$-0.492911\pi$
$661$	1.14510	0.0445393	0.0222697	+	0.999752i	$0.492911\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.73284	−0.377423
$666$	0	0
$667$	−8.23970	−0.319043
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.3059	−0.397856
$672$	0	0
$673$	8.40586	0.324022	0.162011	−	0.986789i	$-0.448202\pi$
$673$	8.40586	0.324022	0.162011	+	0.986789i	$0.448202\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.69607	−0.334217	−0.167109	−	0.985939i	$-0.553443\pi$
$677$	−8.69607	−0.334217	−0.167109	+	0.985939i	$0.553443\pi$
$678$	0	0
$679$	−8.19027	−0.314314
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−42.5246	−1.62716	−0.813579	−	0.581455i	$-0.802484\pi$
$683$	−42.5246	−1.62716	−0.813579	+	0.581455i	$0.802484\pi$
$684$	0	0
$685$	25.5162	0.974923
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−22.1976	−0.845661
$690$	0	0
$691$	11.4143	0.434219	0.217109	−	0.976147i	$-0.430337\pi$
$691$	11.4143	0.434219	0.217109	+	0.976147i	$0.430337\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28.9211	1.09704
$696$	0	0
$697$	−23.9863	−0.908545
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−21.3250	−0.805434	−0.402717	−	0.915325i	$-0.631934\pi$
$701$	−21.3250	−0.805434	−0.402717	+	0.915325i	$0.631934\pi$
$702$	0	0
$703$	26.1314	0.985563
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.31232	0.312617
$708$	0	0
$709$	−30.4941	−1.14523	−0.572614	−	0.819825i	$-0.694071\pi$
$709$	−30.4941	−1.14523	−0.572614	+	0.819825i	$0.694071\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.03677	−0.188629
$714$	0	0
$715$	−27.1103	−1.01387
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−44.3436	−1.65374	−0.826869	−	0.562394i	$-0.809880\pi$
$719$	−44.3436	−1.65374	−0.826869	+	0.562394i	$0.809880\pi$
$720$	0	0
$721$	2.33630	0.0870083
$722$	0	0
$723$	0	0
$724$	0	0
$725$	13.0873	0.486049
$726$	0	0
$727$	−12.0368	−0.446419	−0.223210	−	0.974770i	$-0.571653\pi$
$727$	−12.0368	−0.446419	−0.223210	+	0.974770i	$0.571653\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.43531	0.238018
$732$	0	0
$733$	13.6907	0.505679	0.252839	−	0.967508i	$-0.418636\pi$
$733$	13.6907	0.505679	0.252839	+	0.967508i	$0.418636\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.84850	0.215432
$738$	0	0
$739$	−26.0441	−0.958048	−0.479024	−	0.877802i	$-0.659009\pi$
$739$	−26.0441	−0.958048	−0.479024	+	0.877802i	$0.659009\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.09460	0.333649	0.166824	−	0.985987i	$-0.446649\pi$
$743$	9.09460	0.333649	0.166824	+	0.985987i	$0.446649\pi$
$744$	0	0
$745$	−44.0662	−1.61446
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.09354	−0.0399570
$750$	0	0
$751$	23.7098	0.865183	0.432591	−	0.901590i	$-0.357599\pi$
$751$	23.7098	0.865183	0.432591	+	0.901590i	$0.357599\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	44.5833	1.62255
$756$	0	0
$757$	−31.0671	−1.12915	−0.564577	−	0.825380i	$-0.690961\pi$
$757$	−31.0671	−1.12915	−0.564577	+	0.825380i	$0.690961\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.42798	−0.0517642	−0.0258821	−	0.999665i	$-0.508239\pi$
$761$	−1.42798	−0.0517642	−0.0258821	+	0.999665i	$0.508239\pi$
$762$	0	0
$763$	5.64464	0.204350
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.29021	−0.118802
$768$	0	0
$769$	−42.1965	−1.52165	−0.760823	−	0.648960i	$-0.775204\pi$
$769$	−42.1965	−1.52165	−0.760823	+	0.648960i	$0.775204\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	31.2765	1.12494	0.562468	−	0.826819i	$-0.309852\pi$
$773$	31.2765	1.12494	0.562468	+	0.826819i	$0.309852\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−65.8799	−2.36039
$780$	0	0
$781$	14.1294	0.505589
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.8476	−0.387166
$786$	0	0
$787$	−23.4236	−0.834960	−0.417480	−	0.908686i	$-0.637087\pi$
$787$	−23.4236	−0.834960	−0.417480	+	0.908686i	$0.637087\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.08927	−0.216509
$792$	0	0
$793$	−11.3029	−0.401377
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.8064	0.737000	0.368500	−	0.929628i	$-0.379871\pi$
$797$	20.8064	0.737000	0.368500	+	0.929628i	$0.379871\pi$
$798$	0	0
$799$	−3.54364	−0.125365
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−32.7853	−1.15697
$804$	0	0
$805$	2.64557	0.0932440
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.61519	0.162262	0.0811308	−	0.996703i	$-0.474147\pi$
$809$	4.61519	0.162262	0.0811308	+	0.996703i	$0.474147\pi$
$810$	0	0
$811$	50.2260	1.76367	0.881836	−	0.471556i	$-0.156307\pi$
$811$	50.2260	1.76367	0.881836	+	0.471556i	$0.156307\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.955899	−0.0334837
$816$	0	0
$817$	17.6750	0.618370
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.3427	−0.360963	−0.180481	−	0.983578i	$-0.557766\pi$
$821$	−10.3427	−0.360963	−0.180481	+	0.983578i	$0.557766\pi$
$822$	0	0
$823$	6.19760	0.216035	0.108017	−	0.994149i	$-0.465550\pi$
$823$	6.19760	0.216035	0.108017	+	0.994149i	$0.465550\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.0093	1.28694	0.643470	−	0.765471i	$-0.277494\pi$
$827$	37.0093	1.28694	0.643470	+	0.765471i	$0.277494\pi$
$828$	0	0
$829$	21.5824	0.749588	0.374794	−	0.927108i	$-0.377713\pi$
$829$	21.5824	0.749588	0.374794	+	0.927108i	$0.377713\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14.2397	−0.493377
$834$	0	0
$835$	45.3804	1.57045
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.19760	0.283012	0.141506	−	0.989937i	$-0.454805\pi$
$839$	8.19760	0.283012	0.141506	+	0.989937i	$0.454805\pi$
$840$	0	0
$841$	−2.52792	−0.0871696
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.97255	0.205462
$846$	0	0
$847$	−1.20293	−0.0413332
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−7.10299	−0.243487
$852$	0	0
$853$	−11.1681	−0.382390	−0.191195	−	0.981552i	$-0.561236\pi$
$853$	−11.1681	−0.382390	−0.191195	+	0.981552i	$0.561236\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.84850	0.131462	0.0657311	−	0.997837i	$-0.479062\pi$
$857$	3.84850	0.131462	0.0657311	+	0.997837i	$0.479062\pi$
$858$	0	0
$859$	−14.0735	−0.480183	−0.240092	−	0.970750i	$-0.577177\pi$
$859$	−14.0735	−0.480183	−0.240092	+	0.970750i	$0.577177\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−20.5456	−0.699381	−0.349691	−	0.936865i	$-0.613713\pi$
$863$	−20.5456	−0.699381	−0.349691	+	0.936865i	$0.613713\pi$
$864$	0	0
$865$	18.6108	0.632786
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.608793	0.0206519
$870$	0	0
$871$	6.41425	0.217339
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.05783	0.137180
$876$	0	0
$877$	26.8853	0.907851	0.453926	−	0.891040i	$-0.350023\pi$
$877$	26.8853	0.907851	0.453926	+	0.891040i	$0.350023\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.6686	0.426816	0.213408	−	0.976963i	$-0.431544\pi$
$881$	12.6686	0.426816	0.213408	+	0.976963i	$0.431544\pi$
$882$	0	0
$883$	−50.9021	−1.71299	−0.856495	−	0.516155i	$-0.827363\pi$
$883$	−50.9021	−1.71299	−0.856495	+	0.516155i	$0.827363\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.9809	−0.939508	−0.469754	−	0.882797i	$-0.655657\pi$
$887$	−27.9809	−0.939508	−0.469754	+	0.882797i	$0.655657\pi$
$888$	0	0
$889$	−0.753896	−0.0252849
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.73284	−0.325697
$894$	0	0
$895$	61.1839	2.04515
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16.1819	0.539696
$900$	0	0
$901$	14.4721	0.482135
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.5226	0.815158
$906$	0	0
$907$	22.7927	0.756818	0.378409	−	0.925639i	$-0.376471\pi$
$907$	22.7927	0.756818	0.378409	+	0.925639i	$0.376471\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.9338	0.693567	0.346784	−	0.937945i	$-0.387274\pi$
$911$	20.9338	0.693567	0.346784	+	0.937945i	$0.387274\pi$
$912$	0	0
$913$	10.1956	0.337425
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.130312	−0.00430329
$918$	0	0
$919$	46.8432	1.54521	0.772607	−	0.634885i	$-0.218953\pi$
$919$	46.8432	1.54521	0.772607	+	0.634885i	$0.218953\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15.4962	0.510063
$924$	0	0
$925$	11.2818	0.370944
$926$	0	0
$927$	0	0
$928$	0	0
$929$	53.8799	1.76774	0.883872	−	0.467729i	$-0.154928\pi$
$929$	53.8799	1.76774	0.883872	+	0.467729i	$0.154928\pi$
$930$	0	0
$931$	−39.1103	−1.28179
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17.6750	0.578035
$936$	0	0
$937$	34.6035	1.13045	0.565223	−	0.824938i	$-0.308790\pi$
$937$	34.6035	1.13045	0.565223	+	0.824938i	$0.308790\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.9358	−1.00848	−0.504239	−	0.863564i	$-0.668227\pi$
$941$	−30.9358	−1.00848	−0.504239	+	0.863564i	$0.668227\pi$
$942$	0	0
$943$	17.9074	0.583145
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.26076	−0.235943	−0.117971	−	0.993017i	$-0.537639\pi$
$947$	−7.26076	−0.235943	−0.117971	+	0.993017i	$0.537639\pi$
$948$	0	0
$949$	−35.9568	−1.16721
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−15.7780	−0.511100	−0.255550	−	0.966796i	$-0.582257\pi$
$953$	−15.7780	−0.511100	−0.255550	+	0.966796i	$0.582257\pi$
$954$	0	0
$955$	52.8055	1.70874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.58774	−0.180437
$960$	0	0
$961$	−21.1083	−0.680914
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−14.9863	−0.482425
$966$	0	0
$967$	31.0986	1.00006	0.500032	−	0.866007i	$-0.333322\pi$
$967$	31.0986	1.00006	0.500032	+	0.866007i	$0.333322\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.16815	0.294220	0.147110	−	0.989120i	$-0.453003\pi$
$971$	9.16815	0.294220	0.147110	+	0.989120i	$0.453003\pi$
$972$	0	0
$973$	−6.33338	−0.203039
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.3123	−0.873799	−0.436899	−	0.899510i	$-0.643923\pi$
$977$	−27.3123	−0.873799	−0.436899	+	0.899510i	$0.643923\pi$
$978$	0	0
$979$	−37.9147	−1.21176
$980$	0	0
$981$	0	0
$982$	0	0
$983$	57.5897	1.83683	0.918414	−	0.395622i	$-0.129471\pi$
$983$	57.5897	1.83683	0.918414	+	0.395622i	$0.129471\pi$
$984$	0	0
$985$	−27.9853	−0.891687
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.80440	−0.152771
$990$	0	0
$991$	−3.84956	−0.122285	−0.0611427	−	0.998129i	$-0.519474\pi$
$991$	−3.84956	−0.122285	−0.0611427	+	0.998129i	$0.519474\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	69.2647	2.19584
$996$	0	0
$997$	8.63824	0.273576	0.136788	−	0.990600i	$-0.456322\pi$
$997$	8.63824	0.273576	0.136788	+	0.990600i	$0.456322\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8496.2.a.bk.1.1		3
3.2	odd	2		2832.2.a.u.1.3		3
4.3	odd	2		4248.2.a.m.1.1		3
12.11	even	2		1416.2.a.d.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1416.2.a.d.1.3	✓	3	12.11	even	2
2832.2.a.u.1.3		3	3.2	odd	2
4248.2.a.m.1.1		3	4.3	odd	2
8496.2.a.bk.1.1		3	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:	\( N \)	\(=\)	\( 8496 = 2^{4} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8496.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74657 q^{5} +0.601466 q^{7} +O(q^{10})\)	\(q-2.74657 q^{5} +0.601466 q^{7} -3.00000 q^{11} -3.29021 q^{13} +2.14510 q^{17} +5.89167 q^{19} -1.60147 q^{23} +2.54364 q^{25} +5.14510 q^{29} +3.14510 q^{31} -1.65197 q^{35} +4.43531 q^{37} -11.1819 q^{41} +3.00000 q^{43} -1.65197 q^{47} -6.63824 q^{49} +6.74657 q^{53} +8.23970 q^{55} +1.00000 q^{59} +3.43531 q^{61} +9.03677 q^{65} -1.94950 q^{67} -4.70979 q^{71} +10.9284 q^{73} -1.80440 q^{77} -0.202931 q^{79} -3.39853 q^{83} -5.89167 q^{85} +12.6382 q^{89} -1.97895 q^{91} -16.1819 q^{95} -13.6172 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q+O(q^{10}) \) 3 * q	\( 3 q - 9 q^{11} + 3 q^{13} + 3 q^{19} - 3 q^{23} + 3 q^{25} + 9 q^{29} + 3 q^{31} - 15 q^{35} - 6 q^{37} - 6 q^{41} + 9 q^{43} - 15 q^{47} + 3 q^{49} + 12 q^{53} + 3 q^{59} - 9 q^{61} + 6 q^{65} + 6 q^{67} - 27 q^{71} - 3 q^{73} + 3 q^{79} - 12 q^{83} - 3 q^{85} + 15 q^{89} + 18 q^{91} - 21 q^{95} + 6 q^{97}+O(q^{100}) \) 3 * q - 9 * q^11 + 3 * q^13 + 3 * q^19 - 3 * q^23 + 3 * q^25 + 9 * q^29 + 3 * q^31 - 15 * q^35 - 6 * q^37 - 6 * q^41 + 9 * q^43 - 15 * q^47 + 3 * q^49 + 12 * q^53 + 3 * q^59 - 9 * q^61 + 6 * q^65 + 6 * q^67 - 27 * q^71 - 3 * q^73 + 3 * q^79 - 12 * q^83 - 3 * q^85 + 15 * q^89 + 18 * q^91 - 21 * q^95 + 6 * q^97

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