LMFDB - Embedded newform 8512.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8512,2,Mod(1,8512)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8512, 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("8512.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.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:	$67.9686622005$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.733.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 8 $ x^3 - x^2 - 7*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 532)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.17819$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17819 q^{3} -3.43366 q^{5} -1.00000 q^{7} -1.61186 q^{9} +O(q^{10})$	$q+1.17819 q^{3} -3.43366 q^{5} -1.00000 q^{7} -1.61186 q^{9} -2.61186 q^{11} -5.43366 q^{13} -4.04552 q^{15} -0.611859 q^{17} +1.00000 q^{19} -1.17819 q^{21} -1.43366 q^{23} +6.79005 q^{25} -5.43366 q^{27} -1.74453 q^{29} +0.255471 q^{31} -3.07728 q^{33} +3.43366 q^{35} -5.79005 q^{37} -6.40191 q^{39} +8.96825 q^{41} -7.58011 q^{43} +5.53458 q^{45} -10.6574 q^{47} +1.00000 q^{49} -0.720888 q^{51} -5.53458 q^{53} +8.96825 q^{55} +1.17819 q^{57} -4.30099 q^{59} +1.27911 q^{61} +1.61186 q^{63} +18.6574 q^{65} -0.611859 q^{67} -1.68914 q^{69} -1.07728 q^{71} -11.6891 q^{73} +8.00000 q^{75} +2.61186 q^{77} -12.4474 q^{79} -1.56634 q^{81} +10.4019 q^{83} +2.10092 q^{85} -2.05539 q^{87} -13.2237 q^{89} +5.43366 q^{91} +0.300994 q^{93} -3.43366 q^{95} -7.88110 q^{97} +4.20995 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) $ 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9	$ 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 8 q^{13} + 7 q^{15} + 9 q^{17} + 3 q^{19} - q^{21} + 4 q^{23} + 7 q^{25} - 8 q^{27} - 11 q^{29} - 5 q^{31} - 6 q^{33} + 2 q^{35} - 4 q^{37} + 5 q^{39} + 11 q^{41} + 4 q^{43} + 9 q^{45} - 2 q^{47} + 3 q^{49} - 4 q^{51} - 9 q^{53} + 11 q^{55} + q^{57} + 12 q^{59} + 2 q^{61} - 6 q^{63} + 26 q^{65} + 9 q^{67} + 9 q^{69} - 21 q^{73} + 24 q^{75} - 3 q^{77} + 6 q^{79} - 13 q^{81} + 7 q^{83} + 7 q^{85} - 26 q^{87} - 18 q^{89} + 8 q^{91} - 24 q^{93} - 2 q^{95} + 28 q^{97} + 26 q^{99}+O(q^{100}) $ 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 8 * q^13 + 7 * q^15 + 9 * q^17 + 3 * q^19 - q^21 + 4 * q^23 + 7 * q^25 - 8 * q^27 - 11 * q^29 - 5 * q^31 - 6 * q^33 + 2 * q^35 - 4 * q^37 + 5 * q^39 + 11 * q^41 + 4 * q^43 + 9 * q^45 - 2 * q^47 + 3 * q^49 - 4 * q^51 - 9 * q^53 + 11 * q^55 + q^57 + 12 * q^59 + 2 * q^61 - 6 * q^63 + 26 * q^65 + 9 * q^67 + 9 * q^69 - 21 * q^73 + 24 * q^75 - 3 * q^77 + 6 * q^79 - 13 * q^81 + 7 * q^83 + 7 * q^85 - 26 * q^87 - 18 * q^89 + 8 * q^91 - 24 * q^93 - 2 * q^95 + 28 * q^97 + 26 * q^99

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
$3$	1.17819	0.680231	0.340115	−	0.940384i	$-0.389534\pi$
$3$	1.17819	0.680231	0.340115	+	0.940384i	$0.389534\pi$
$4$	0	0
$5$	−3.43366	−1.53558	−0.767791	−	0.640701i	$-0.778644\pi$
$5$	−3.43366	−1.53558	−0.767791	+	0.640701i	$0.778644\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.61186	−0.537286
$10$	0	0
$11$	−2.61186	−0.787505	−0.393753	−	0.919216i	$-0.628823\pi$
$11$	−2.61186	−0.787505	−0.393753	+	0.919216i	$0.628823\pi$
$12$	0	0
$13$	−5.43366	−1.50703	−0.753514	−	0.657432i	$-0.771643\pi$
$13$	−5.43366	−1.50703	−0.753514	+	0.657432i	$0.771643\pi$
$14$	0	0
$15$	−4.04552	−1.04455
$16$	0	0
$17$	−0.611859	−0.148398	−0.0741988	−	0.997243i	$-0.523640\pi$
$17$	−0.611859	−0.148398	−0.0741988	+	0.997243i	$0.523640\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.17819	−0.257103
$22$	0	0
$23$	−1.43366	−0.298940	−0.149470	−	0.988766i	$-0.547757\pi$
$23$	−1.43366	−0.298940	−0.149470	+	0.988766i	$0.547757\pi$
$24$	0	0
$25$	6.79005	1.35801
$26$	0	0
$27$	−5.43366	−1.04571
$28$	0	0
$29$	−1.74453	−0.323951	−0.161975	−	0.986795i	$-0.551787\pi$
$29$	−1.74453	−0.323951	−0.161975	+	0.986795i	$0.551787\pi$
$30$	0	0
$31$	0.255471	0.0458839	0.0229419	−	0.999737i	$-0.492697\pi$
$31$	0.255471	0.0458839	0.0229419	+	0.999737i	$0.492697\pi$
$32$	0	0
$33$	−3.07728	−0.535685
$34$	0	0
$35$	3.43366	0.580395
$36$	0	0
$37$	−5.79005	−0.951879	−0.475939	−	0.879478i	$-0.657892\pi$
$37$	−5.79005	−0.951879	−0.475939	+	0.879478i	$0.657892\pi$
$38$	0	0
$39$	−6.40191	−1.02513
$40$	0	0
$41$	8.96825	1.40060	0.700302	−	0.713846i	$-0.253048\pi$
$41$	8.96825	1.40060	0.700302	+	0.713846i	$0.253048\pi$
$42$	0	0
$43$	−7.58011	−1.15596	−0.577978	−	0.816053i	$-0.696158\pi$
$43$	−7.58011	−1.15596	−0.577978	+	0.816053i	$0.696158\pi$
$44$	0	0
$45$	5.53458	0.825047
$46$	0	0
$47$	−10.6574	−1.55454	−0.777269	−	0.629168i	$-0.783396\pi$
$47$	−10.6574	−1.55454	−0.777269	+	0.629168i	$0.783396\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.720888	−0.100945
$52$	0	0
$53$	−5.53458	−0.760233	−0.380117	−	0.924939i	$-0.624116\pi$
$53$	−5.53458	−0.760233	−0.380117	+	0.924939i	$0.624116\pi$
$54$	0	0
$55$	8.96825	1.20928
$56$	0	0
$57$	1.17819	0.156056
$58$	0	0
$59$	−4.30099	−0.559942	−0.279971	−	0.960009i	$-0.590325\pi$
$59$	−4.30099	−0.559942	−0.279971	+	0.960009i	$0.590325\pi$
$60$	0	0
$61$	1.27911	0.163773	0.0818867	−	0.996642i	$-0.473905\pi$
$61$	1.27911	0.163773	0.0818867	+	0.996642i	$0.473905\pi$
$62$	0	0
$63$	1.61186	0.203075
$64$	0	0
$65$	18.6574	2.31416
$66$	0	0
$67$	−0.611859	−0.0747504	−0.0373752	−	0.999301i	$-0.511900\pi$
$67$	−0.611859	−0.0747504	−0.0373752	+	0.999301i	$0.511900\pi$
$68$	0	0
$69$	−1.68914	−0.203348
$70$	0	0
$71$	−1.07728	−0.127849	−0.0639246	−	0.997955i	$-0.520362\pi$
$71$	−1.07728	−0.127849	−0.0639246	+	0.997955i	$0.520362\pi$
$72$	0	0
$73$	−11.6891	−1.36811	−0.684055	−	0.729431i	$-0.739785\pi$
$73$	−11.6891	−1.36811	−0.684055	+	0.729431i	$0.739785\pi$
$74$	0	0
$75$	8.00000	0.923760
$76$	0	0
$77$	2.61186	0.297649
$78$	0	0
$79$	−12.4474	−1.40045	−0.700223	−	0.713924i	$-0.746916\pi$
$79$	−12.4474	−1.40045	−0.700223	+	0.713924i	$0.746916\pi$
$80$	0	0
$81$	−1.56634	−0.174037
$82$	0	0
$83$	10.4019	1.14176	0.570879	−	0.821034i	$-0.306603\pi$
$83$	10.4019	1.14176	0.570879	+	0.821034i	$0.306603\pi$
$84$	0	0
$85$	2.10092	0.227877
$86$	0	0
$87$	−2.05539	−0.220361
$88$	0	0
$89$	−13.2237	−1.40171	−0.700856	−	0.713303i	$-0.747198\pi$
$89$	−13.2237	−1.40171	−0.700856	+	0.713303i	$0.747198\pi$
$90$	0	0
$91$	5.43366	0.569603
$92$	0	0
$93$	0.300994	0.0312116
$94$	0	0
$95$	−3.43366	−0.352287
$96$	0	0
$97$	−7.88110	−0.800204	−0.400102	−	0.916471i	$-0.631025\pi$
$97$	−7.88110	−0.800204	−0.400102	+	0.916471i	$0.631025\pi$
$98$	0	0
$99$	4.20995	0.423116
$100$	0	0
$101$	16.2375	1.61569	0.807845	−	0.589395i	$-0.200634\pi$
$101$	16.2375	1.61569	0.807845	+	0.589395i	$0.200634\pi$
$102$	0	0
$103$	6.14644	0.605627	0.302813	−	0.953050i	$-0.402074\pi$
$103$	6.14644	0.605627	0.302813	+	0.953050i	$0.402074\pi$
$104$	0	0
$105$	4.04552	0.394803
$106$	0	0
$107$	−5.07728	−0.490839	−0.245419	−	0.969417i	$-0.578926\pi$
$107$	−5.07728	−0.490839	−0.245419	+	0.969417i	$0.578926\pi$
$108$	0	0
$109$	18.3010	1.75292	0.876459	−	0.481477i	$-0.159900\pi$
$109$	18.3010	1.75292	0.876459	+	0.481477i	$0.159900\pi$
$110$	0	0
$111$	−6.82181	−0.647497
$112$	0	0
$113$	3.17819	0.298979	0.149490	−	0.988763i	$-0.452237\pi$
$113$	3.17819	0.298979	0.149490	+	0.988763i	$0.452237\pi$
$114$	0	0
$115$	4.92272	0.459046
$116$	0	0
$117$	8.75830	0.809705
$118$	0	0
$119$	0.611859	0.0560890
$120$	0	0
$121$	−4.17819	−0.379836
$122$	0	0
$123$	10.5663	0.952734
$124$	0	0
$125$	−6.14644	−0.549754
$126$	0	0
$127$	10.0910	0.895436	0.447718	−	0.894175i	$-0.352237\pi$
$127$	10.0910	0.895436	0.447718	+	0.894175i	$0.352237\pi$
$128$	0	0
$129$	−8.93084	−0.786316
$130$	0	0
$131$	−10.1920	−0.890476	−0.445238	−	0.895412i	$-0.646881\pi$
$131$	−10.1920	−0.890476	−0.445238	+	0.895412i	$0.646881\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	18.6574	1.60577
$136$	0	0
$137$	11.7347	1.00256	0.501280	−	0.865285i	$-0.332863\pi$
$137$	11.7347	1.00256	0.501280	+	0.865285i	$0.332863\pi$
$138$	0	0
$139$	9.94461	0.843490	0.421745	−	0.906714i	$-0.361418\pi$
$139$	9.94461	0.843490	0.421745	+	0.906714i	$0.361418\pi$
$140$	0	0
$141$	−12.5565	−1.05745
$142$	0	0
$143$	14.1920	1.18679
$144$	0	0
$145$	5.99013	0.497453
$146$	0	0
$147$	1.17819	0.0971758
$148$	0	0
$149$	−13.0773	−1.07133	−0.535666	−	0.844430i	$-0.679939\pi$
$149$	−13.0773	−1.07133	−0.535666	+	0.844430i	$0.679939\pi$
$150$	0	0
$151$	−18.3465	−1.49302	−0.746509	−	0.665375i	$-0.768272\pi$
$151$	−18.3465	−1.49302	−0.746509	+	0.665375i	$0.768272\pi$
$152$	0	0
$153$	0.986230	0.0797320
$154$	0	0
$155$	−0.877200	−0.0704584
$156$	0	0
$157$	15.9901	1.27615	0.638076	−	0.769974i	$-0.279731\pi$
$157$	15.9901	1.27615	0.638076	+	0.769974i	$0.279731\pi$
$158$	0	0
$159$	−6.52081	−0.517134
$160$	0	0
$161$	1.43366	0.112989
$162$	0	0
$163$	3.33275	0.261041	0.130520	−	0.991446i	$-0.458335\pi$
$163$	3.33275	0.261041	0.130520	+	0.991446i	$0.458335\pi$
$164$	0	0
$165$	10.5663	0.822588
$166$	0	0
$167$	12.5028	0.967498	0.483749	−	0.875207i	$-0.339275\pi$
$167$	12.5028	0.967498	0.483749	+	0.875207i	$0.339275\pi$
$168$	0	0
$169$	16.5247	1.27113
$170$	0	0
$171$	−1.61186	−0.123262
$172$	0	0
$173$	−10.7763	−0.819306	−0.409653	−	0.912242i	$-0.634350\pi$
$173$	−10.7763	−0.819306	−0.409653	+	0.912242i	$0.634350\pi$
$174$	0	0
$175$	−6.79005	−0.513280
$176$	0	0
$177$	−5.06741	−0.380890
$178$	0	0
$179$	3.47919	0.260047	0.130023	−	0.991511i	$-0.458495\pi$
$179$	3.47919	0.260047	0.130023	+	0.991511i	$0.458495\pi$
$180$	0	0
$181$	7.83558	0.582414	0.291207	−	0.956660i	$-0.405943\pi$
$181$	7.83558	0.582414	0.291207	+	0.956660i	$0.405943\pi$
$182$	0	0
$183$	1.50704	0.111404
$184$	0	0
$185$	19.8811	1.46169
$186$	0	0
$187$	1.59809	0.116864
$188$	0	0
$189$	5.43366	0.395241
$190$	0	0
$191$	23.4238	1.69489	0.847443	−	0.530886i	$-0.178141\pi$
$191$	23.4238	1.69489	0.847443	+	0.530886i	$0.178141\pi$
$192$	0	0
$193$	21.4157	1.54153	0.770767	−	0.637117i	$-0.219873\pi$
$193$	21.4157	1.54153	0.770767	+	0.637117i	$0.219873\pi$
$194$	0	0
$195$	21.9820	1.57416
$196$	0	0
$197$	21.8275	1.55514	0.777571	−	0.628795i	$-0.216451\pi$
$197$	21.8275	1.55514	0.777571	+	0.628795i	$0.216451\pi$
$198$	0	0
$199$	−24.3920	−1.72911	−0.864553	−	0.502542i	$-0.832398\pi$
$199$	−24.3920	−1.72911	−0.864553	+	0.502542i	$0.832398\pi$
$200$	0	0
$201$	−0.720888	−0.0508475
$202$	0	0
$203$	1.74453	0.122442
$204$	0	0
$205$	−30.7940	−2.15074
$206$	0	0
$207$	2.31086	0.160616
$208$	0	0
$209$	−2.61186	−0.180666
$210$	0	0
$211$	5.62563	0.387284	0.193642	−	0.981072i	$-0.437970\pi$
$211$	5.62563	0.387284	0.193642	+	0.981072i	$0.437970\pi$
$212$	0	0
$213$	−1.26924	−0.0869669
$214$	0	0
$215$	26.0275	1.77506
$216$	0	0
$217$	−0.255471	−0.0173425
$218$	0	0
$219$	−13.7721	−0.930630
$220$	0	0
$221$	3.32464	0.223639
$222$	0	0
$223$	19.3702	1.29712	0.648561	−	0.761163i	$-0.275371\pi$
$223$	19.3702	1.29712	0.648561	+	0.761163i	$0.275371\pi$
$224$	0	0
$225$	−10.9446	−0.729640
$226$	0	0
$227$	15.2138	1.00978	0.504889	−	0.863184i	$-0.331533\pi$
$227$	15.2138	1.00978	0.504889	+	0.863184i	$0.331533\pi$
$228$	0	0
$229$	0.265341	0.0175343	0.00876713	−	0.999962i	$-0.497209\pi$
$229$	0.265341	0.0175343	0.00876713	+	0.999962i	$0.497209\pi$
$230$	0	0
$231$	3.07728	0.202470
$232$	0	0
$233$	20.6119	1.35033	0.675164	−	0.737668i	$-0.264073\pi$
$233$	20.6119	1.35033	0.675164	+	0.737668i	$0.264073\pi$
$234$	0	0
$235$	36.5939	2.38712
$236$	0	0
$237$	−14.6655	−0.952626
$238$	0	0
$239$	0.00811199	0.000524721 0	0.000262360	−	1.00000i	$-0.499916\pi$
$239$	0.00811199	0.000524721 0	0.000262360		1.00000i	$0.499916\pi$
$240$	0	0
$241$	−20.0910	−1.29418	−0.647089	−	0.762414i	$-0.724014\pi$
$241$	−20.0910	−1.29418	−0.647089	+	0.762414i	$0.724014\pi$
$242$	0	0
$243$	14.4555	0.927324
$244$	0	0
$245$	−3.43366	−0.219369
$246$	0	0
$247$	−5.43366	−0.345736
$248$	0	0
$249$	12.2555	0.776659
$250$	0	0
$251$	24.3465	1.53674	0.768369	−	0.640007i	$-0.221068\pi$
$251$	24.3465	1.53674	0.768369	+	0.640007i	$0.221068\pi$
$252$	0	0
$253$	3.74453	0.235417
$254$	0	0
$255$	2.47529	0.155009
$256$	0	0
$257$	−23.2057	−1.44753	−0.723767	−	0.690044i	$-0.757591\pi$
$257$	−23.2057	−1.44753	−0.723767	+	0.690044i	$0.757591\pi$
$258$	0	0
$259$	5.79005	0.359776
$260$	0	0
$261$	2.81193	0.174054
$262$	0	0
$263$	23.3327	1.43876	0.719379	−	0.694617i	$-0.244426\pi$
$263$	23.3327	1.43876	0.719379	+	0.694617i	$0.244426\pi$
$264$	0	0
$265$	19.0039	1.16740
$266$	0	0
$267$	−15.5801	−0.953487
$268$	0	0
$269$	−25.9820	−1.58415	−0.792076	−	0.610423i	$-0.791000\pi$
$269$	−25.9820	−1.58415	−0.792076	+	0.610423i	$0.791000\pi$
$270$	0	0
$271$	−25.1503	−1.52777	−0.763887	−	0.645350i	$-0.776712\pi$
$271$	−25.1503	−1.52777	−0.763887	+	0.645350i	$0.776712\pi$
$272$	0	0
$273$	6.40191	0.387461
$274$	0	0
$275$	−17.7347	−1.06944
$276$	0	0
$277$	19.2237	1.15504	0.577521	−	0.816376i	$-0.304020\pi$
$277$	19.2237	1.15504	0.577521	+	0.816376i	$0.304020\pi$
$278$	0	0
$279$	−0.411782	−0.0246528
$280$	0	0
$281$	−1.27911	−0.0763054	−0.0381527	−	0.999272i	$-0.512147\pi$
$281$	−1.27911	−0.0763054	−0.0381527	+	0.999272i	$0.512147\pi$
$282$	0	0
$283$	−23.1147	−1.37403	−0.687013	−	0.726645i	$-0.741078\pi$
$283$	−23.1147	−1.37403	−0.687013	+	0.726645i	$0.741078\pi$
$284$	0	0
$285$	−4.04552	−0.239636
$286$	0	0
$287$	−8.96825	−0.529379
$288$	0	0
$289$	−16.6256	−0.977978
$290$	0	0
$291$	−9.28546	−0.544324
$292$	0	0
$293$	−23.7901	−1.38983	−0.694915	−	0.719092i	$-0.744558\pi$
$293$	−23.7901	−1.38983	−0.694915	+	0.719092i	$0.744558\pi$
$294$	0	0
$295$	14.7682	0.859836
$296$	0	0
$297$	14.1920	0.823501
$298$	0	0
$299$	7.79005	0.450510
$300$	0	0
$301$	7.58011	0.436910
$302$	0	0
$303$	19.1309	1.09904
$304$	0	0
$305$	−4.39204	−0.251488
$306$	0	0
$307$	−18.5011	−1.05591	−0.527956	−	0.849272i	$-0.677041\pi$
$307$	−18.5011	−1.05591	−0.527956	+	0.849272i	$0.677041\pi$
$308$	0	0
$309$	7.24170	0.411966
$310$	0	0
$311$	12.3465	0.700107	0.350053	−	0.936730i	$-0.386163\pi$
$311$	12.3465	0.700107	0.350053	+	0.936730i	$0.386163\pi$
$312$	0	0
$313$	−5.67115	−0.320552	−0.160276	−	0.987072i	$-0.551239\pi$
$313$	−5.67115	−0.320552	−0.160276	+	0.987072i	$0.551239\pi$
$314$	0	0
$315$	−5.53458	−0.311838
$316$	0	0
$317$	15.0219	0.843713	0.421856	−	0.906663i	$-0.361379\pi$
$317$	15.0219	0.843713	0.421856	+	0.906663i	$0.361379\pi$
$318$	0	0
$319$	4.55646	0.255113
$320$	0	0
$321$	−5.98202	−0.333884
$322$	0	0
$323$	−0.611859	−0.0340447
$324$	0	0
$325$	−36.8949	−2.04656
$326$	0	0
$327$	21.5621	1.19239
$328$	0	0
$329$	10.6574	0.587560
$330$	0	0
$331$	−8.73076	−0.479886	−0.239943	−	0.970787i	$-0.577129\pi$
$331$	−8.73076	−0.479886	−0.239943	+	0.970787i	$0.577129\pi$
$332$	0	0
$333$	9.33275	0.511431
$334$	0	0
$335$	2.10092	0.114785
$336$	0	0
$337$	−22.7502	−1.23928	−0.619641	−	0.784885i	$-0.712722\pi$
$337$	−22.7502	−1.23928	−0.619641	+	0.784885i	$0.712722\pi$
$338$	0	0
$339$	3.74453	0.203375
$340$	0	0
$341$	−0.667253	−0.0361338
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.79992	0.312257
$346$	0	0
$347$	−18.0731	−0.970213	−0.485106	−	0.874455i	$-0.661219\pi$
$347$	−18.0731	−0.970213	−0.485106	+	0.874455i	$0.661219\pi$
$348$	0	0
$349$	−10.3384	−0.553402	−0.276701	−	0.960956i	$-0.589241\pi$
$349$	−10.3384	−0.553402	−0.276701	+	0.960956i	$0.589241\pi$
$350$	0	0
$351$	29.5247	1.57591
$352$	0	0
$353$	−17.0318	−0.906509	−0.453254	−	0.891381i	$-0.649737\pi$
$353$	−17.0318	−0.906509	−0.453254	+	0.891381i	$0.649737\pi$
$354$	0	0
$355$	3.69901	0.196323
$356$	0	0
$357$	0.720888	0.0381535
$358$	0	0
$359$	−16.8575	−0.889703	−0.444851	−	0.895604i	$-0.646743\pi$
$359$	−16.8575	−0.889703	−0.444851	+	0.895604i	$0.646743\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.92272	−0.258376
$364$	0	0
$365$	40.1366	2.10084
$366$	0	0
$367$	34.7484	1.81385	0.906927	−	0.421289i	$-0.138422\pi$
$367$	34.7484	1.81385	0.906927	+	0.421289i	$0.138422\pi$
$368$	0	0
$369$	−14.4555	−0.752526
$370$	0	0
$371$	5.53458	0.287341
$372$	0	0
$373$	29.0039	1.50176	0.750882	−	0.660436i	$-0.229628\pi$
$373$	29.0039	1.50176	0.750882	+	0.660436i	$0.229628\pi$
$374$	0	0
$375$	−7.24170	−0.373960
$376$	0	0
$377$	9.47919	0.488203
$378$	0	0
$379$	−9.88110	−0.507558	−0.253779	−	0.967262i	$-0.581674\pi$
$379$	−9.88110	−0.507558	−0.253779	+	0.967262i	$0.581674\pi$
$380$	0	0
$381$	11.8892	0.609103
$382$	0	0
$383$	19.5882	1.00091	0.500456	−	0.865762i	$-0.333166\pi$
$383$	19.5882	1.00091	0.500456	+	0.865762i	$0.333166\pi$
$384$	0	0
$385$	−8.96825	−0.457064
$386$	0	0
$387$	12.2181	0.621079
$388$	0	0
$389$	−16.6119	−0.842255	−0.421127	−	0.907002i	$-0.638365\pi$
$389$	−16.6119	−0.842255	−0.421127	+	0.907002i	$0.638365\pi$
$390$	0	0
$391$	0.877200	0.0443619
$392$	0	0
$393$	−12.0081	−0.605729
$394$	0	0
$395$	42.7403	2.15050
$396$	0	0
$397$	−19.1129	−0.959250	−0.479625	−	0.877473i	$-0.659227\pi$
$397$	−19.1129	−0.959250	−0.479625	+	0.877473i	$0.659227\pi$
$398$	0	0
$399$	−1.17819	−0.0589835
$400$	0	0
$401$	6.96825	0.347978	0.173989	−	0.984748i	$-0.444334\pi$
$401$	6.96825	0.347978	0.173989	+	0.984748i	$0.444334\pi$
$402$	0	0
$403$	−1.38814	−0.0691482
$404$	0	0
$405$	5.37827	0.267248
$406$	0	0
$407$	15.1228	0.749609
$408$	0	0
$409$	−25.6891	−1.27025	−0.635123	−	0.772411i	$-0.719051\pi$
$409$	−25.6891	−1.27025	−0.635123	+	0.772411i	$0.719051\pi$
$410$	0	0
$411$	13.8257	0.681972
$412$	0	0
$413$	4.30099	0.211638
$414$	0	0
$415$	−35.7167	−1.75326
$416$	0	0
$417$	11.7167	0.573768
$418$	0	0
$419$	−0.519053	−0.0253574	−0.0126787	−	0.999920i	$-0.504036\pi$
$419$	−0.519053	−0.0253574	−0.0126787	+	0.999920i	$0.504036\pi$
$420$	0	0
$421$	−36.2375	−1.76611	−0.883054	−	0.469272i	$-0.844516\pi$
$421$	−36.2375	−1.76611	−0.883054	+	0.469272i	$0.844516\pi$
$422$	0	0
$423$	17.1782	0.835232
$424$	0	0
$425$	−4.15455	−0.201525
$426$	0	0
$427$	−1.27911	−0.0619006
$428$	0	0
$429$	16.7209	0.807292
$430$	0	0
$431$	−3.73466	−0.179892	−0.0899461	−	0.995947i	$-0.528669\pi$
$431$	−3.73466	−0.179892	−0.0899461	+	0.995947i	$0.528669\pi$
$432$	0	0
$433$	−3.79005	−0.182138	−0.0910692	−	0.995845i	$-0.529028\pi$
$433$	−3.79005	−0.182138	−0.0910692	+	0.995845i	$0.529028\pi$
$434$	0	0
$435$	7.05753	0.338383
$436$	0	0
$437$	−1.43366	−0.0685815
$438$	0	0
$439$	6.57445	0.313781	0.156891	−	0.987616i	$-0.449853\pi$
$439$	6.57445	0.313781	0.156891	+	0.987616i	$0.449853\pi$
$440$	0	0
$441$	−1.61186	−0.0767552
$442$	0	0
$443$	23.4238	1.11290	0.556449	−	0.830882i	$-0.312163\pi$
$443$	23.4238	1.11290	0.556449	+	0.830882i	$0.312163\pi$
$444$	0	0
$445$	45.4058	2.15244
$446$	0	0
$447$	−15.4076	−0.728753
$448$	0	0
$449$	−32.2001	−1.51962	−0.759808	−	0.650148i	$-0.774707\pi$
$449$	−32.2001	−1.51962	−0.759808	+	0.650148i	$0.774707\pi$
$450$	0	0
$451$	−23.4238	−1.10298
$452$	0	0
$453$	−21.6158	−1.01560
$454$	0	0
$455$	−18.6574	−0.874672
$456$	0	0
$457$	19.0674	0.891936	0.445968	−	0.895049i	$-0.352860\pi$
$457$	19.0674	0.891936	0.445968	+	0.895049i	$0.352860\pi$
$458$	0	0
$459$	3.32464	0.155181
$460$	0	0
$461$	−19.8712	−0.925495	−0.462748	−	0.886490i	$-0.653136\pi$
$461$	−19.8712	−0.925495	−0.462748	+	0.886490i	$0.653136\pi$
$462$	0	0
$463$	−24.5028	−1.13874	−0.569372	−	0.822080i	$-0.692813\pi$
$463$	−24.5028	−1.13874	−0.569372	+	0.822080i	$0.692813\pi$
$464$	0	0
$465$	−1.03351	−0.0479280
$466$	0	0
$467$	29.7802	1.37806	0.689031	−	0.724732i	$-0.258036\pi$
$467$	29.7802	1.37806	0.689031	+	0.724732i	$0.258036\pi$
$468$	0	0
$469$	0.611859	0.0282530
$470$	0	0
$471$	18.8395	0.868077
$472$	0	0
$473$	19.7982	0.910321
$474$	0	0
$475$	6.79005	0.311549
$476$	0	0
$477$	8.92096	0.408463
$478$	0	0
$479$	−17.4792	−0.798644	−0.399322	−	0.916811i	$-0.630755\pi$
$479$	−17.4792	−0.798644	−0.399322	+	0.916811i	$0.630755\pi$
$480$	0	0
$481$	31.4612	1.43451
$482$	0	0
$483$	1.68914	0.0768583
$484$	0	0
$485$	27.0611	1.22878
$486$	0	0
$487$	34.0829	1.54445	0.772223	−	0.635352i	$-0.219145\pi$
$487$	34.0829	1.54445	0.772223	+	0.635352i	$0.219145\pi$
$488$	0	0
$489$	3.92662	0.177568
$490$	0	0
$491$	−32.8949	−1.48452	−0.742262	−	0.670109i	$-0.766247\pi$
$491$	−32.8949	−1.48452	−0.742262	+	0.670109i	$0.766247\pi$
$492$	0	0
$493$	1.06741	0.0480735
$494$	0	0
$495$	−14.4555	−0.649729
$496$	0	0
$497$	1.07728	0.0483225
$498$	0	0
$499$	35.5346	1.59075	0.795373	−	0.606120i	$-0.207275\pi$
$499$	35.5346	1.59075	0.795373	+	0.606120i	$0.207275\pi$
$500$	0	0
$501$	14.7308	0.658122
$502$	0	0
$503$	32.5858	1.45293	0.726464	−	0.687205i	$-0.241163\pi$
$503$	32.5858	1.45293	0.726464	+	0.687205i	$0.241163\pi$
$504$	0	0
$505$	−55.7541	−2.48102
$506$	0	0
$507$	19.4693	0.864663
$508$	0	0
$509$	19.7622	0.875944	0.437972	−	0.898989i	$-0.355697\pi$
$509$	19.7622	0.875944	0.437972	+	0.898989i	$0.355697\pi$
$510$	0	0
$511$	11.6891	0.517097
$512$	0	0
$513$	−5.43366	−0.239902
$514$	0	0
$515$	−21.1048	−0.929989
$516$	0	0
$517$	27.8356	1.22421
$518$	0	0
$519$	−12.6966	−0.557317
$520$	0	0
$521$	−26.9503	−1.18071	−0.590356	−	0.807143i	$-0.701013\pi$
$521$	−26.9503	−1.18071	−0.590356	+	0.807143i	$0.701013\pi$
$522$	0	0
$523$	−15.6793	−0.685606	−0.342803	−	0.939407i	$-0.611376\pi$
$523$	−15.6793	−0.685606	−0.342803	+	0.939407i	$0.611376\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	0	0
$527$	−0.156312	−0.00680905
$528$	0	0
$529$	−20.9446	−0.910635
$530$	0	0
$531$	6.93259	0.300849
$532$	0	0
$533$	−48.7304	−2.11075
$534$	0	0
$535$	17.4337	0.753723
$536$	0	0
$537$	4.09916	0.176892
$538$	0	0
$539$	−2.61186	−0.112501
$540$	0	0
$541$	−24.6295	−1.05891	−0.529453	−	0.848339i	$-0.677603\pi$
$541$	−24.6295	−1.05891	−0.529453	+	0.848339i	$0.677603\pi$
$542$	0	0
$543$	9.23183	0.396176
$544$	0	0
$545$	−62.8395	−2.69175
$546$	0	0
$547$	0.603747	0.0258143	0.0129072	−	0.999917i	$-0.495891\pi$
$547$	0.603747	0.0258143	0.0129072	+	0.999917i	$0.495891\pi$
$548$	0	0
$549$	−2.06175	−0.0879932
$550$	0	0
$551$	−1.74453	−0.0743195
$552$	0	0
$553$	12.4474	0.529319
$554$	0	0
$555$	23.4238	0.994285
$556$	0	0
$557$	−27.6810	−1.17288	−0.586441	−	0.809992i	$-0.699472\pi$
$557$	−27.6810	−1.17288	−0.586441	+	0.809992i	$0.699472\pi$
$558$	0	0
$559$	41.1878	1.74206
$560$	0	0
$561$	1.88286	0.0794943
$562$	0	0
$563$	0.831677	0.0350510	0.0175255	−	0.999846i	$-0.494421\pi$
$563$	0.831677	0.0350510	0.0175255	+	0.999846i	$0.494421\pi$
$564$	0	0
$565$	−10.9129	−0.459107
$566$	0	0
$567$	1.56634	0.0657799
$568$	0	0
$569$	39.4139	1.65232	0.826159	−	0.563437i	$-0.190521\pi$
$569$	39.4139	1.65232	0.826159	+	0.563437i	$0.190521\pi$
$570$	0	0
$571$	−9.63550	−0.403233	−0.201617	−	0.979465i	$-0.564619\pi$
$571$	−9.63550	−0.403233	−0.201617	+	0.979465i	$0.564619\pi$
$572$	0	0
$573$	27.5978	1.15291
$574$	0	0
$575$	−9.73466	−0.405963
$576$	0	0
$577$	7.48730	0.311700	0.155850	−	0.987781i	$-0.450188\pi$
$577$	7.48730	0.311700	0.155850	+	0.987781i	$0.450188\pi$
$578$	0	0
$579$	25.2318	1.04860
$580$	0	0
$581$	−10.4019	−0.431544
$582$	0	0
$583$	14.4555	0.598688
$584$	0	0
$585$	−30.0731	−1.24337
$586$	0	0
$587$	28.6020	1.18053	0.590265	−	0.807209i	$-0.299023\pi$
$587$	28.6020	1.18053	0.590265	+	0.807209i	$0.299023\pi$
$588$	0	0
$589$	0.255471	0.0105265
$590$	0	0
$591$	25.7170	1.05786
$592$	0	0
$593$	−18.0829	−0.742577	−0.371289	−	0.928518i	$-0.621084\pi$
$593$	−18.0829	−0.742577	−0.371289	+	0.928518i	$0.621084\pi$
$594$	0	0
$595$	−2.10092	−0.0861292
$596$	0	0
$597$	−28.7386	−1.17619
$598$	0	0
$599$	22.9601	0.938126	0.469063	−	0.883165i	$-0.344592\pi$
$599$	22.9601	0.938126	0.469063	+	0.883165i	$0.344592\pi$
$600$	0	0
$601$	−11.8437	−0.483114	−0.241557	−	0.970387i	$-0.577658\pi$
$601$	−11.8437	−0.483114	−0.241557	+	0.970387i	$0.577658\pi$
$602$	0	0
$603$	0.986230	0.0401624
$604$	0	0
$605$	14.3465	0.583269
$606$	0	0
$607$	−40.5939	−1.64765	−0.823827	−	0.566841i	$-0.808165\pi$
$607$	−40.5939	−1.64765	−0.823827	+	0.566841i	$0.808165\pi$
$608$	0	0
$609$	2.05539	0.0832888
$610$	0	0
$611$	57.9086	2.34273
$612$	0	0
$613$	36.8021	1.48642	0.743211	−	0.669057i	$-0.233302\pi$
$613$	36.8021	1.48642	0.743211	+	0.669057i	$0.233302\pi$
$614$	0	0
$615$	−36.2813	−1.46300
$616$	0	0
$617$	−5.41568	−0.218027	−0.109014	−	0.994040i	$-0.534769\pi$
$617$	−5.41568	−0.218027	−0.109014	+	0.994040i	$0.534769\pi$
$618$	0	0
$619$	36.4295	1.46422	0.732112	−	0.681185i	$-0.238535\pi$
$619$	36.4295	1.46422	0.732112	+	0.681185i	$0.238535\pi$
$620$	0	0
$621$	7.79005	0.312604
$622$	0	0
$623$	13.2237	0.529797
$624$	0	0
$625$	−12.8454	−0.513818
$626$	0	0
$627$	−3.07728	−0.122895
$628$	0	0
$629$	3.54269	0.141256
$630$	0	0
$631$	40.7957	1.62405	0.812026	−	0.583622i	$-0.198365\pi$
$631$	40.7957	1.62405	0.812026	+	0.583622i	$0.198365\pi$
$632$	0	0
$633$	6.62808	0.263443
$634$	0	0
$635$	−34.6493	−1.37501
$636$	0	0
$637$	−5.43366	−0.215290
$638$	0	0
$639$	1.73642	0.0686916
$640$	0	0
$641$	−4.39380	−0.173545	−0.0867723	−	0.996228i	$-0.527655\pi$
$641$	−4.39380	−0.173545	−0.0867723	+	0.996228i	$0.527655\pi$
$642$	0	0
$643$	−20.4831	−0.807774	−0.403887	−	0.914809i	$-0.632341\pi$
$643$	−20.4831	−0.807774	−0.403887	+	0.914809i	$0.632341\pi$
$644$	0	0
$645$	30.6655	1.20745
$646$	0	0
$647$	−11.6793	−0.459159	−0.229580	−	0.973290i	$-0.573735\pi$
$647$	−11.6793	−0.459159	−0.229580	+	0.973290i	$0.573735\pi$
$648$	0	0
$649$	11.2336	0.440957
$650$	0	0
$651$	−0.300994	−0.0117969
$652$	0	0
$653$	14.1902	0.555306	0.277653	−	0.960681i	$-0.410444\pi$
$653$	14.1902	0.555306	0.277653	+	0.960681i	$0.410444\pi$
$654$	0	0
$655$	34.9958	1.36740
$656$	0	0
$657$	18.8412	0.735067
$658$	0	0
$659$	−14.2392	−0.554682	−0.277341	−	0.960772i	$-0.589453\pi$
$659$	−14.2392	−0.554682	−0.277341	+	0.960772i	$0.589453\pi$
$660$	0	0
$661$	28.2929	1.10047	0.550233	−	0.835011i	$-0.314539\pi$
$661$	28.2929	1.10047	0.550233	+	0.835011i	$0.314539\pi$
$662$	0	0
$663$	3.91707	0.152126
$664$	0	0
$665$	3.43366	0.133152
$666$	0	0
$667$	2.50107	0.0968418
$668$	0	0
$669$	22.8218	0.882342
$670$	0	0
$671$	−3.34086	−0.128972
$672$	0	0
$673$	−38.1203	−1.46943	−0.734716	−	0.678375i	$-0.762684\pi$
$673$	−38.1203	−1.46943	−0.734716	+	0.678375i	$0.762684\pi$
$674$	0	0
$675$	−36.8949	−1.42008
$676$	0	0
$677$	−30.2911	−1.16418	−0.582091	−	0.813123i	$-0.697765\pi$
$677$	−30.2911	−1.16418	−0.582091	+	0.813123i	$0.697765\pi$
$678$	0	0
$679$	7.88110	0.302449
$680$	0	0
$681$	17.9249	0.686882
$682$	0	0
$683$	13.9721	0.534629	0.267315	−	0.963609i	$-0.413864\pi$
$683$	13.9721	0.534629	0.267315	+	0.963609i	$0.413864\pi$
$684$	0	0
$685$	−40.2929	−1.53951
$686$	0	0
$687$	0.312624	0.0119273
$688$	0	0
$689$	30.0731	1.14569
$690$	0	0
$691$	−46.9422	−1.78576	−0.892882	−	0.450291i	$-0.851320\pi$
$691$	−46.9422	−1.78576	−0.892882	+	0.450291i	$0.851320\pi$
$692$	0	0
$693$	−4.20995	−0.159923
$694$	0	0
$695$	−34.1464	−1.29525
$696$	0	0
$697$	−5.48730	−0.207846
$698$	0	0
$699$	24.2848	0.918534
$700$	0	0
$701$	−19.3148	−0.729509	−0.364754	−	0.931104i	$-0.618847\pi$
$701$	−19.3148	−0.729509	−0.364754	+	0.931104i	$0.618847\pi$
$702$	0	0
$703$	−5.79005	−0.218376
$704$	0	0
$705$	43.1147	1.62379
$706$	0	0
$707$	−16.2375	−0.610674
$708$	0	0
$709$	18.1902	0.683148	0.341574	−	0.939855i	$-0.389040\pi$
$709$	18.1902	0.683148	0.341574	+	0.939855i	$0.389040\pi$
$710$	0	0
$711$	20.0635	0.752440
$712$	0	0
$713$	−0.366259	−0.0137165
$714$	0	0
$715$	−48.7304	−1.82242
$716$	0	0
$717$	0.00955750	0.000356931 0
$718$	0	0
$719$	−30.1383	−1.12397	−0.561985	−	0.827147i	$-0.689962\pi$
$719$	−30.1383	−1.12397	−0.561985	+	0.827147i	$0.689962\pi$
$720$	0	0
$721$	−6.14644	−0.228905
$722$	0	0
$723$	−23.6712	−0.880340
$724$	0	0
$725$	−11.8454	−0.439929
$726$	0	0
$727$	−20.1902	−0.748813	−0.374407	−	0.927265i	$-0.622154\pi$
$727$	−20.1902	−0.748813	−0.374407	+	0.927265i	$0.622154\pi$
$728$	0	0
$729$	21.7304	0.804831
$730$	0	0
$731$	4.63795	0.171541
$732$	0	0
$733$	−39.1878	−1.44743	−0.723716	−	0.690098i	$-0.757568\pi$
$733$	−39.1878	−1.44743	−0.723716	+	0.690098i	$0.757568\pi$
$734$	0	0
$735$	−4.04552	−0.149221
$736$	0	0
$737$	1.59809	0.0588663
$738$	0	0
$739$	−21.4337	−0.788450	−0.394225	−	0.919014i	$-0.628987\pi$
$739$	−21.4337	−0.788450	−0.394225	+	0.919014i	$0.628987\pi$
$740$	0	0
$741$	−6.40191	−0.235180
$742$	0	0
$743$	11.5409	0.423396	0.211698	−	0.977335i	$-0.432101\pi$
$743$	11.5409	0.423396	0.211698	+	0.977335i	$0.432101\pi$
$744$	0	0
$745$	44.9030	1.64512
$746$	0	0
$747$	−16.7664	−0.613451
$748$	0	0
$749$	5.07728	0.185520
$750$	0	0
$751$	−50.5205	−1.84352	−0.921760	−	0.387762i	$-0.873248\pi$
$751$	−50.5205	−1.84352	−0.921760	+	0.387762i	$0.873248\pi$
$752$	0	0
$753$	28.6849	1.04534
$754$	0	0
$755$	62.9958	2.29265
$756$	0	0
$757$	−4.97812	−0.180933	−0.0904664	−	0.995900i	$-0.528836\pi$
$757$	−4.97812	−0.180933	−0.0904664	+	0.995900i	$0.528836\pi$
$758$	0	0
$759$	4.41178	0.160138
$760$	0	0
$761$	28.8592	1.04615	0.523073	−	0.852288i	$-0.324786\pi$
$761$	28.8592	1.04615	0.523073	+	0.852288i	$0.324786\pi$
$762$	0	0
$763$	−18.3010	−0.662540
$764$	0	0
$765$	−3.38638	−0.122435
$766$	0	0
$767$	23.3702	0.843848
$768$	0	0
$769$	−15.6239	−0.563411	−0.281706	−	0.959501i	$-0.590900\pi$
$769$	−15.6239	−0.563411	−0.281706	+	0.959501i	$0.590900\pi$
$770$	0	0
$771$	−27.3409	−0.984657
$772$	0	0
$773$	−27.9640	−1.00580	−0.502898	−	0.864346i	$-0.667733\pi$
$773$	−27.9640	−1.00580	−0.502898	+	0.864346i	$0.667733\pi$
$774$	0	0
$775$	1.73466	0.0623108
$776$	0	0
$777$	6.82181	0.244731
$778$	0	0
$779$	8.96825	0.321321
$780$	0	0
$781$	2.81369	0.100682
$782$	0	0
$783$	9.47919	0.338759
$784$	0	0
$785$	−54.9047	−1.95963
$786$	0	0
$787$	−18.5744	−0.662108	−0.331054	−	0.943612i	$-0.607404\pi$
$787$	−18.5744	−0.662108	−0.331054	+	0.943612i	$0.607404\pi$
$788$	0	0
$789$	27.4905	0.978688
$790$	0	0
$791$	−3.17819	−0.113004
$792$	0	0
$793$	−6.95026	−0.246811
$794$	0	0
$795$	22.3903	0.794101
$796$	0	0
$797$	21.8631	0.774431	0.387216	−	0.921989i	$-0.373437\pi$
$797$	21.8631	0.774431	0.387216	+	0.921989i	$0.373437\pi$
$798$	0	0
$799$	6.52081	0.230690
$800$	0	0
$801$	21.3148	0.753120
$802$	0	0
$803$	30.5304	1.07739
$804$	0	0
$805$	−4.92272	−0.173503
$806$	0	0
$807$	−30.6119	−1.07759
$808$	0	0
$809$	35.0138	1.23102	0.615509	−	0.788130i	$-0.288950\pi$
$809$	35.0138	1.23102	0.615509	+	0.788130i	$0.288950\pi$
$810$	0	0
$811$	−36.6020	−1.28527	−0.642635	−	0.766173i	$-0.722159\pi$
$811$	−36.6020	−1.28527	−0.642635	+	0.766173i	$0.722159\pi$
$812$	0	0
$813$	−29.6320	−1.03924
$814$	0	0
$815$	−11.4435	−0.400850
$816$	0	0
$817$	−7.58011	−0.265194
$818$	0	0
$819$	−8.75830	−0.306040
$820$	0	0
$821$	18.4199	0.642859	0.321429	−	0.946934i	$-0.395837\pi$
$821$	18.4199	0.642859	0.321429	+	0.946934i	$0.395837\pi$
$822$	0	0
$823$	−33.8257	−1.17909	−0.589545	−	0.807736i	$-0.700693\pi$
$823$	−33.8257	−1.17909	−0.589545	+	0.807736i	$0.700693\pi$
$824$	0	0
$825$	−20.8949	−0.727466
$826$	0	0
$827$	−32.5304	−1.13119	−0.565596	−	0.824683i	$-0.691354\pi$
$827$	−32.5304	−1.13119	−0.565596	+	0.824683i	$0.691354\pi$
$828$	0	0
$829$	13.9168	0.483349	0.241674	−	0.970357i	$-0.422303\pi$
$829$	13.9168	0.483349	0.241674	+	0.970357i	$0.422303\pi$
$830$	0	0
$831$	22.6493	0.785694
$832$	0	0
$833$	−0.611859	−0.0211996
$834$	0	0
$835$	−42.9305	−1.48567
$836$	0	0
$837$	−1.38814	−0.0479812
$838$	0	0
$839$	−27.2872	−0.942060	−0.471030	−	0.882117i	$-0.656118\pi$
$839$	−27.2872	−0.942060	−0.471030	+	0.882117i	$0.656118\pi$
$840$	0	0
$841$	−25.9566	−0.895056
$842$	0	0
$843$	−1.50704	−0.0519053
$844$	0	0
$845$	−56.7403	−1.95193
$846$	0	0
$847$	4.17819	0.143564
$848$	0	0
$849$	−27.2336	−0.934654
$850$	0	0
$851$	8.30099	0.284554
$852$	0	0
$853$	−9.95448	−0.340835	−0.170417	−	0.985372i	$-0.554512\pi$
$853$	−9.95448	−0.340835	−0.170417	+	0.985372i	$0.554512\pi$
$854$	0	0
$855$	5.53458	0.189279
$856$	0	0
$857$	20.1644	0.688804	0.344402	−	0.938822i	$-0.388082\pi$
$857$	20.1644	0.688804	0.344402	+	0.938822i	$0.388082\pi$
$858$	0	0
$859$	−9.78829	−0.333972	−0.166986	−	0.985959i	$-0.553404\pi$
$859$	−9.78829	−0.333972	−0.166986	+	0.985959i	$0.553404\pi$
$860$	0	0
$861$	−10.5663	−0.360100
$862$	0	0
$863$	41.0039	1.39579	0.697894	−	0.716201i	$-0.254120\pi$
$863$	41.0039	1.39579	0.697894	+	0.716201i	$0.254120\pi$
$864$	0	0
$865$	37.0021	1.25811
$866$	0	0
$867$	−19.5882	−0.665251
$868$	0	0
$869$	32.5109	1.10286
$870$	0	0
$871$	3.32464	0.112651
$872$	0	0
$873$	12.7032	0.429939
$874$	0	0
$875$	6.14644	0.207788
$876$	0	0
$877$	−21.2513	−0.717604	−0.358802	−	0.933414i	$-0.616815\pi$
$877$	−21.2513	−0.717604	−0.358802	+	0.933414i	$0.616815\pi$
$878$	0	0
$879$	−28.0293	−0.945405
$880$	0	0
$881$	−9.66159	−0.325507	−0.162754	−	0.986667i	$-0.552038\pi$
$881$	−9.66159	−0.325507	−0.162754	+	0.986667i	$0.552038\pi$
$882$	0	0
$883$	27.7703	0.934545	0.467273	−	0.884113i	$-0.345237\pi$
$883$	27.7703	0.934545	0.467273	+	0.884113i	$0.345237\pi$
$884$	0	0
$885$	17.3998	0.584887
$886$	0	0
$887$	−10.3483	−0.347461	−0.173731	−	0.984793i	$-0.555582\pi$
$887$	−10.3483	−0.347461	−0.173731	+	0.984793i	$0.555582\pi$
$888$	0	0
$889$	−10.0910	−0.338443
$890$	0	0
$891$	4.09105	0.137055
$892$	0	0
$893$	−10.6574	−0.356636
$894$	0	0
$895$	−11.9464	−0.399323
$896$	0	0
$897$	9.17819	0.306451
$898$	0	0
$899$	−0.445676	−0.0148641
$900$	0	0
$901$	3.38638	0.112817
$902$	0	0
$903$	8.93084	0.297200
$904$	0	0
$905$	−26.9047	−0.894344
$906$	0	0
$907$	13.2791	0.440926	0.220463	−	0.975395i	$-0.429243\pi$
$907$	13.2791	0.440926	0.220463	+	0.975395i	$0.429243\pi$
$908$	0	0
$909$	−26.1725	−0.868088
$910$	0	0
$911$	10.1821	0.337348	0.168674	−	0.985672i	$-0.446052\pi$
$911$	10.1821	0.337348	0.168674	+	0.985672i	$0.446052\pi$
$912$	0	0
$913$	−27.1683	−0.899140
$914$	0	0
$915$	−5.17468	−0.171070
$916$	0	0
$917$	10.1920	0.336568
$918$	0	0
$919$	−11.0773	−0.365406	−0.182703	−	0.983168i	$-0.558485\pi$
$919$	−11.0773	−0.365406	−0.182703	+	0.983168i	$0.558485\pi$
$920$	0	0
$921$	−21.7979	−0.718264
$922$	0	0
$923$	5.85356	0.192672
$924$	0	0
$925$	−39.3148	−1.29266
$926$	0	0
$927$	−9.90719	−0.325395
$928$	0	0
$929$	23.0512	0.756285	0.378142	−	0.925747i	$-0.376563\pi$
$929$	23.0512	0.756285	0.378142	+	0.925747i	$0.376563\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	14.5466	0.476234
$934$	0	0
$935$	−5.48730	−0.179454
$936$	0	0
$937$	6.36626	0.207977	0.103988	−	0.994579i	$-0.466840\pi$
$937$	6.36626	0.207977	0.103988	+	0.994579i	$0.466840\pi$
$938$	0	0
$939$	−6.68172	−0.218050
$940$	0	0
$941$	2.15455	0.0702364	0.0351182	−	0.999383i	$-0.488819\pi$
$941$	2.15455	0.0702364	0.0351182	+	0.999383i	$0.488819\pi$
$942$	0	0
$943$	−12.8575	−0.418696
$944$	0	0
$945$	−18.6574	−0.606925
$946$	0	0
$947$	24.7745	0.805064	0.402532	−	0.915406i	$-0.368130\pi$
$947$	24.7745	0.805064	0.402532	+	0.915406i	$0.368130\pi$
$948$	0	0
$949$	63.5148	2.06178
$950$	0	0
$951$	17.6987	0.573919
$952$	0	0
$953$	−4.11714	−0.133367	−0.0666836	−	0.997774i	$-0.521242\pi$
$953$	−4.11714	−0.133367	−0.0666836	+	0.997774i	$0.521242\pi$
$954$	0	0
$955$	−80.4295	−2.60264
$956$	0	0
$957$	5.36840	0.173536
$958$	0	0
$959$	−11.7347	−0.378932
$960$	0	0
$961$	−30.9347	−0.997895
$962$	0	0
$963$	8.18385	0.263721
$964$	0	0
$965$	−73.5343	−2.36715
$966$	0	0
$967$	−18.9047	−0.607935	−0.303968	−	0.952682i	$-0.598312\pi$
$967$	−18.9047	−0.607935	−0.303968	+	0.952682i	$0.598312\pi$
$968$	0	0
$969$	−0.720888	−0.0231583
$970$	0	0
$971$	−7.57199	−0.242997	−0.121498	−	0.992592i	$-0.538770\pi$
$971$	−7.57199	−0.242997	−0.121498	+	0.992592i	$0.538770\pi$
$972$	0	0
$973$	−9.94461	−0.318809
$974$	0	0
$975$	−43.4693	−1.39213
$976$	0	0
$977$	44.4277	1.42137	0.710684	−	0.703511i	$-0.248386\pi$
$977$	44.4277	1.42137	0.710684	+	0.703511i	$0.248386\pi$
$978$	0	0
$979$	34.5385	1.10385
$980$	0	0
$981$	−29.4986	−0.941818
$982$	0	0
$983$	−49.3780	−1.57491	−0.787456	−	0.616371i	$-0.788602\pi$
$983$	−49.3780	−1.57491	−0.787456	+	0.616371i	$0.788602\pi$
$984$	0	0
$985$	−74.9482	−2.38805
$986$	0	0
$987$	12.5565	0.399677
$988$	0	0
$989$	10.8673	0.345561
$990$	0	0
$991$	53.1405	1.68806	0.844031	−	0.536294i	$-0.180176\pi$
$991$	53.1405	1.68806	0.844031	+	0.536294i	$0.180176\pi$
$992$	0	0
$993$	−10.2865	−0.326433
$994$	0	0
$995$	83.7541	2.65518
$996$	0	0
$997$	22.3546	0.707978	0.353989	−	0.935250i	$-0.384825\pi$
$997$	22.3546	0.707978	0.353989	+	0.935250i	$0.384825\pi$
$998$	0	0
$999$	31.4612	0.995388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.bn.1.2		3
4.3	odd	2		8512.2.a.bl.1.2		3
8.3	odd	2		2128.2.a.r.1.2		3
8.5	even	2		532.2.a.e.1.2	✓	3
24.5	odd	2		4788.2.a.o.1.1		3
56.13	odd	2		3724.2.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.a.e.1.2	✓	3	8.5	even	2
2128.2.a.r.1.2		3	8.3	odd	2
3724.2.a.i.1.2		3	56.13	odd	2
4788.2.a.o.1.1		3	24.5	odd	2
8512.2.a.bl.1.2		3	4.3	odd	2
8512.2.a.bn.1.2		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:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17819 q^{3} -3.43366 q^{5} -1.00000 q^{7} -1.61186 q^{9} +O(q^{10})\)	\(q+1.17819 q^{3} -3.43366 q^{5} -1.00000 q^{7} -1.61186 q^{9} -2.61186 q^{11} -5.43366 q^{13} -4.04552 q^{15} -0.611859 q^{17} +1.00000 q^{19} -1.17819 q^{21} -1.43366 q^{23} +6.79005 q^{25} -5.43366 q^{27} -1.74453 q^{29} +0.255471 q^{31} -3.07728 q^{33} +3.43366 q^{35} -5.79005 q^{37} -6.40191 q^{39} +8.96825 q^{41} -7.58011 q^{43} +5.53458 q^{45} -10.6574 q^{47} +1.00000 q^{49} -0.720888 q^{51} -5.53458 q^{53} +8.96825 q^{55} +1.17819 q^{57} -4.30099 q^{59} +1.27911 q^{61} +1.61186 q^{63} +18.6574 q^{65} -0.611859 q^{67} -1.68914 q^{69} -1.07728 q^{71} -11.6891 q^{73} +8.00000 q^{75} +2.61186 q^{77} -12.4474 q^{79} -1.56634 q^{81} +10.4019 q^{83} +2.10092 q^{85} -2.05539 q^{87} -13.2237 q^{89} +5.43366 q^{91} +0.300994 q^{93} -3.43366 q^{95} -7.88110 q^{97} +4.20995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9}+O(q^{10}) \) 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9	\( 3 q + q^{3} - 2 q^{5} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 8 q^{13} + 7 q^{15} + 9 q^{17} + 3 q^{19} - q^{21} + 4 q^{23} + 7 q^{25} - 8 q^{27} - 11 q^{29} - 5 q^{31} - 6 q^{33} + 2 q^{35} - 4 q^{37} + 5 q^{39} + 11 q^{41} + 4 q^{43} + 9 q^{45} - 2 q^{47} + 3 q^{49} - 4 q^{51} - 9 q^{53} + 11 q^{55} + q^{57} + 12 q^{59} + 2 q^{61} - 6 q^{63} + 26 q^{65} + 9 q^{67} + 9 q^{69} - 21 q^{73} + 24 q^{75} - 3 q^{77} + 6 q^{79} - 13 q^{81} + 7 q^{83} + 7 q^{85} - 26 q^{87} - 18 q^{89} + 8 q^{91} - 24 q^{93} - 2 q^{95} + 28 q^{97} + 26 q^{99}+O(q^{100}) \) 3 * q + q^3 - 2 * q^5 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 8 * q^13 + 7 * q^15 + 9 * q^17 + 3 * q^19 - q^21 + 4 * q^23 + 7 * q^25 - 8 * q^27 - 11 * q^29 - 5 * q^31 - 6 * q^33 + 2 * q^35 - 4 * q^37 + 5 * q^39 + 11 * q^41 + 4 * q^43 + 9 * q^45 - 2 * q^47 + 3 * q^49 - 4 * q^51 - 9 * q^53 + 11 * q^55 + q^57 + 12 * q^59 + 2 * q^61 - 6 * q^63 + 26 * q^65 + 9 * q^67 + 9 * q^69 - 21 * q^73 + 24 * q^75 - 3 * q^77 + 6 * q^79 - 13 * q^81 + 7 * q^83 + 7 * q^85 - 26 * q^87 - 18 * q^89 + 8 * q^91 - 24 * q^93 - 2 * q^95 + 28 * q^97 + 26 * q^99

Introduction

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