LMFDB - Embedded newform 8046.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.09224$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.09224 q^{5} -1.25253 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.09224 q^{5} -1.25253 q^{7} +1.00000 q^{8} -1.09224 q^{10} +0.173738 q^{11} +4.22835 q^{13} -1.25253 q^{14} +1.00000 q^{16} +4.61561 q^{17} +3.05127 q^{19} -1.09224 q^{20} +0.173738 q^{22} +0.515996 q^{23} -3.80700 q^{25} +4.22835 q^{26} -1.25253 q^{28} -5.77684 q^{29} +0.0135315 q^{31} +1.00000 q^{32} +4.61561 q^{34} +1.36807 q^{35} -7.73328 q^{37} +3.05127 q^{38} -1.09224 q^{40} +7.34603 q^{41} +5.19948 q^{43} +0.173738 q^{44} +0.515996 q^{46} +2.23044 q^{47} -5.43118 q^{49} -3.80700 q^{50} +4.22835 q^{52} +1.74794 q^{53} -0.189764 q^{55} -1.25253 q^{56} -5.77684 q^{58} +8.36601 q^{59} +6.66842 q^{61} +0.0135315 q^{62} +1.00000 q^{64} -4.61839 q^{65} -11.2370 q^{67} +4.61561 q^{68} +1.36807 q^{70} +5.70549 q^{71} +14.5008 q^{73} -7.73328 q^{74} +3.05127 q^{76} -0.217611 q^{77} -1.27461 q^{79} -1.09224 q^{80} +7.34603 q^{82} -4.98613 q^{83} -5.04138 q^{85} +5.19948 q^{86} +0.173738 q^{88} +3.60101 q^{89} -5.29613 q^{91} +0.515996 q^{92} +2.23044 q^{94} -3.33273 q^{95} -7.13298 q^{97} -5.43118 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.09224	−0.488467	−0.244233	−	0.969716i	$-0.578536\pi$
$5$	−1.09224	−0.488467	−0.244233	+	0.969716i	$0.578536\pi$
$6$	0	0
$7$	−1.25253	−0.473411	−0.236705	−	0.971581i	$-0.576068\pi$
$7$	−1.25253	−0.473411	−0.236705	+	0.971581i	$0.576068\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.09224	−0.345398
$11$	0.173738	0.0523839	0.0261920	−	0.999657i	$-0.491662\pi$
$11$	0.173738	0.0523839	0.0261920	+	0.999657i	$0.491662\pi$
$12$	0	0
$13$	4.22835	1.17273	0.586367	−	0.810046i	$-0.300558\pi$
$13$	4.22835	1.17273	0.586367	+	0.810046i	$0.300558\pi$
$14$	−1.25253	−0.334752
$15$	0	0
$16$	1.00000	0.250000
$17$	4.61561	1.11945	0.559725	−	0.828678i	$-0.310907\pi$
$17$	4.61561	1.11945	0.559725	+	0.828678i	$0.310907\pi$
$18$	0	0
$19$	3.05127	0.700009	0.350005	−	0.936748i	$-0.386180\pi$
$19$	3.05127	0.700009	0.350005	+	0.936748i	$0.386180\pi$
$20$	−1.09224	−0.244233
$21$	0	0
$22$	0.173738	0.0370410
$23$	0.515996	0.107593	0.0537963	−	0.998552i	$-0.482868\pi$
$23$	0.515996	0.107593	0.0537963	+	0.998552i	$0.482868\pi$
$24$	0	0
$25$	−3.80700	−0.761400
$26$	4.22835	0.829248
$27$	0	0
$28$	−1.25253	−0.236705
$29$	−5.77684	−1.07273	−0.536366	−	0.843985i	$-0.680203\pi$
$29$	−5.77684	−1.07273	−0.536366	+	0.843985i	$0.680203\pi$
$30$	0	0
$31$	0.0135315	0.00243032	0.00121516	−	0.999999i	$-0.499613\pi$
$31$	0.0135315	0.00243032	0.00121516	+	0.999999i	$0.499613\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.61561	0.791571
$35$	1.36807	0.231245
$36$	0	0
$37$	−7.73328	−1.27134	−0.635671	−	0.771960i	$-0.719277\pi$
$37$	−7.73328	−1.27134	−0.635671	+	0.771960i	$0.719277\pi$
$38$	3.05127	0.494981
$39$	0	0
$40$	−1.09224	−0.172699
$41$	7.34603	1.14726	0.573629	−	0.819115i	$-0.305535\pi$
$41$	7.34603	1.14726	0.573629	+	0.819115i	$0.305535\pi$
$42$	0	0
$43$	5.19948	0.792913	0.396457	−	0.918053i	$-0.370240\pi$
$43$	5.19948	0.792913	0.396457	+	0.918053i	$0.370240\pi$
$44$	0.173738	0.0261920
$45$	0	0
$46$	0.515996	0.0760794
$47$	2.23044	0.325343	0.162671	−	0.986680i	$-0.447989\pi$
$47$	2.23044	0.325343	0.162671	+	0.986680i	$0.447989\pi$
$48$	0	0
$49$	−5.43118	−0.775882
$50$	−3.80700	−0.538391
$51$	0	0
$52$	4.22835	0.586367
$53$	1.74794	0.240099	0.120049	−	0.992768i	$-0.461695\pi$
$53$	1.74794	0.240099	0.120049	+	0.992768i	$0.461695\pi$
$54$	0	0
$55$	−0.189764	−0.0255878
$56$	−1.25253	−0.167376
$57$	0	0
$58$	−5.77684	−0.758537
$59$	8.36601	1.08916	0.544581	−	0.838708i	$-0.316689\pi$
$59$	8.36601	1.08916	0.544581	+	0.838708i	$0.316689\pi$
$60$	0	0
$61$	6.66842	0.853804	0.426902	−	0.904298i	$-0.359605\pi$
$61$	6.66842	0.853804	0.426902	+	0.904298i	$0.359605\pi$
$62$	0.0135315	0.00171850
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.61839	−0.572841
$66$	0	0
$67$	−11.2370	−1.37281	−0.686406	−	0.727218i	$-0.740813\pi$
$67$	−11.2370	−1.37281	−0.686406	+	0.727218i	$0.740813\pi$
$68$	4.61561	0.559725
$69$	0	0
$70$	1.36807	0.163515
$71$	5.70549	0.677117	0.338558	−	0.940945i	$-0.390061\pi$
$71$	5.70549	0.677117	0.338558	+	0.940945i	$0.390061\pi$
$72$	0	0
$73$	14.5008	1.69719	0.848597	−	0.529040i	$-0.177448\pi$
$73$	14.5008	1.69719	0.848597	+	0.529040i	$0.177448\pi$
$74$	−7.73328	−0.898975
$75$	0	0
$76$	3.05127	0.350005
$77$	−0.217611	−0.0247991
$78$	0	0
$79$	−1.27461	−0.143405	−0.0717026	−	0.997426i	$-0.522843\pi$
$79$	−1.27461	−0.143405	−0.0717026	+	0.997426i	$0.522843\pi$
$80$	−1.09224	−0.122117
$81$	0	0
$82$	7.34603	0.811234
$83$	−4.98613	−0.547299	−0.273650	−	0.961829i	$-0.588231\pi$
$83$	−4.98613	−0.547299	−0.273650	+	0.961829i	$0.588231\pi$
$84$	0	0
$85$	−5.04138	−0.546814
$86$	5.19948	0.560674
$87$	0	0
$88$	0.173738	0.0185205
$89$	3.60101	0.381706	0.190853	−	0.981619i	$-0.438875\pi$
$89$	3.60101	0.381706	0.190853	+	0.981619i	$0.438875\pi$
$90$	0	0
$91$	−5.29613	−0.555185
$92$	0.515996	0.0537963
$93$	0	0
$94$	2.23044	0.230052
$95$	−3.33273	−0.341931
$96$	0	0
$97$	−7.13298	−0.724244	−0.362122	−	0.932131i	$-0.617948\pi$
$97$	−7.13298	−0.724244	−0.362122	+	0.932131i	$0.617948\pi$
$98$	−5.43118	−0.548632
$99$	0	0
$100$	−3.80700	−0.380700
$101$	12.5976	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$101$	12.5976	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$102$	0	0
$103$	0.697965	0.0687725	0.0343862	−	0.999409i	$-0.489052\pi$
$103$	0.697965	0.0687725	0.0343862	+	0.999409i	$0.489052\pi$
$104$	4.22835	0.414624
$105$	0	0
$106$	1.74794	0.169775
$107$	−8.15974	−0.788832	−0.394416	−	0.918932i	$-0.629053\pi$
$107$	−8.15974	−0.788832	−0.394416	+	0.918932i	$0.629053\pi$
$108$	0	0
$109$	13.8923	1.33064	0.665319	−	0.746559i	$-0.268295\pi$
$109$	13.8923	1.33064	0.665319	+	0.746559i	$0.268295\pi$
$110$	−0.189764	−0.0180933
$111$	0	0
$112$	−1.25253	−0.118353
$113$	4.70720	0.442816	0.221408	−	0.975181i	$-0.428935\pi$
$113$	4.70720	0.442816	0.221408	+	0.975181i	$0.428935\pi$
$114$	0	0
$115$	−0.563594	−0.0525554
$116$	−5.77684	−0.536366
$117$	0	0
$118$	8.36601	0.770154
$119$	−5.78118	−0.529960
$120$	0	0
$121$	−10.9698	−0.997256
$122$	6.66842	0.603730
$123$	0	0
$124$	0.0135315	0.00121516
$125$	9.61940	0.860385
$126$	0	0
$127$	8.14529	0.722777	0.361389	−	0.932415i	$-0.382303\pi$
$127$	8.14529	0.722777	0.361389	+	0.932415i	$0.382303\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.61839	−0.405060
$131$	1.84870	0.161521	0.0807607	−	0.996734i	$-0.474265\pi$
$131$	1.84870	0.161521	0.0807607	+	0.996734i	$0.474265\pi$
$132$	0	0
$133$	−3.82180	−0.331392
$134$	−11.2370	−0.970725
$135$	0	0
$136$	4.61561	0.395785
$137$	−12.4658	−1.06502	−0.532511	−	0.846423i	$-0.678751\pi$
$137$	−12.4658	−1.06502	−0.532511	+	0.846423i	$0.678751\pi$
$138$	0	0
$139$	9.51802	0.807308	0.403654	−	0.914912i	$-0.367740\pi$
$139$	9.51802	0.807308	0.403654	+	0.914912i	$0.367740\pi$
$140$	1.36807	0.115623
$141$	0	0
$142$	5.70549	0.478794
$143$	0.734624	0.0614324
$144$	0	0
$145$	6.30973	0.523994
$146$	14.5008	1.20010
$147$	0	0
$148$	−7.73328	−0.635671
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−10.4423	−0.849782	−0.424891	−	0.905244i	$-0.639688\pi$
$151$	−10.4423	−0.849782	−0.424891	+	0.905244i	$0.639688\pi$
$152$	3.05127	0.247491
$153$	0	0
$154$	−0.217611	−0.0175356
$155$	−0.0147797	−0.00118713
$156$	0	0
$157$	−8.12291	−0.648278	−0.324139	−	0.946009i	$-0.605075\pi$
$157$	−8.12291	−0.648278	−0.324139	+	0.946009i	$0.605075\pi$
$158$	−1.27461	−0.101403
$159$	0	0
$160$	−1.09224	−0.0863495
$161$	−0.646299	−0.0509355
$162$	0	0
$163$	−0.0916217	−0.00717637	−0.00358818	−	0.999994i	$-0.501142\pi$
$163$	−0.0916217	−0.00717637	−0.00358818	+	0.999994i	$0.501142\pi$
$164$	7.34603	0.573629
$165$	0	0
$166$	−4.98613	−0.386999
$167$	13.1457	1.01725	0.508624	−	0.860989i	$-0.330154\pi$
$167$	13.1457	1.01725	0.508624	+	0.860989i	$0.330154\pi$
$168$	0	0
$169$	4.87896	0.375305
$170$	−5.04138	−0.386656
$171$	0	0
$172$	5.19948	0.396457
$173$	6.71040	0.510182	0.255091	−	0.966917i	$-0.417895\pi$
$173$	6.71040	0.510182	0.255091	+	0.966917i	$0.417895\pi$
$174$	0	0
$175$	4.76837	0.360455
$176$	0.173738	0.0130960
$177$	0	0
$178$	3.60101	0.269907
$179$	18.5877	1.38931	0.694655	−	0.719343i	$-0.255557\pi$
$179$	18.5877	1.38931	0.694655	+	0.719343i	$0.255557\pi$
$180$	0	0
$181$	−6.34620	−0.471709	−0.235855	−	0.971788i	$-0.575789\pi$
$181$	−6.34620	−0.471709	−0.235855	+	0.971788i	$0.575789\pi$
$182$	−5.29613	−0.392575
$183$	0	0
$184$	0.515996	0.0380397
$185$	8.44663	0.621009
$186$	0	0
$187$	0.801906	0.0586412
$188$	2.23044	0.162671
$189$	0	0
$190$	−3.33273	−0.241782
$191$	0.581014	0.0420407	0.0210203	−	0.999779i	$-0.493309\pi$
$191$	0.581014	0.0420407	0.0210203	+	0.999779i	$0.493309\pi$
$192$	0	0
$193$	20.8034	1.49746	0.748729	−	0.662876i	$-0.230664\pi$
$193$	20.8034	1.49746	0.748729	+	0.662876i	$0.230664\pi$
$194$	−7.13298	−0.512118
$195$	0	0
$196$	−5.43118	−0.387941
$197$	10.3409	0.736756	0.368378	−	0.929676i	$-0.379913\pi$
$197$	10.3409	0.736756	0.368378	+	0.929676i	$0.379913\pi$
$198$	0	0
$199$	−9.52240	−0.675025	−0.337513	−	0.941321i	$-0.609586\pi$
$199$	−9.52240	−0.675025	−0.337513	+	0.941321i	$0.609586\pi$
$200$	−3.80700	−0.269196
$201$	0	0
$202$	12.5976	0.886361
$203$	7.23565	0.507843
$204$	0	0
$205$	−8.02367	−0.560397
$206$	0.697965	0.0486295
$207$	0	0
$208$	4.22835	0.293183
$209$	0.530121	0.0366692
$210$	0	0
$211$	−5.92251	−0.407722	−0.203861	−	0.979000i	$-0.565349\pi$
$211$	−5.92251	−0.407722	−0.203861	+	0.979000i	$0.565349\pi$
$212$	1.74794	0.120049
$213$	0	0
$214$	−8.15974	−0.557789
$215$	−5.67911	−0.387312
$216$	0	0
$217$	−0.0169485	−0.00115054
$218$	13.8923	0.940904
$219$	0	0
$220$	−0.189764	−0.0127939
$221$	19.5164	1.31282
$222$	0	0
$223$	21.7359	1.45554	0.727771	−	0.685820i	$-0.240556\pi$
$223$	21.7359	1.45554	0.727771	+	0.685820i	$0.240556\pi$
$224$	−1.25253	−0.0836880
$225$	0	0
$226$	4.70720	0.313118
$227$	−0.0539413	−0.00358021	−0.00179010	−	0.999998i	$-0.500570\pi$
$227$	−0.0539413	−0.00358021	−0.00179010	+	0.999998i	$0.500570\pi$
$228$	0	0
$229$	15.4074	1.01815	0.509073	−	0.860723i	$-0.329988\pi$
$229$	15.4074	1.01815	0.509073	+	0.860723i	$0.329988\pi$
$230$	−0.563594	−0.0371623
$231$	0	0
$232$	−5.77684	−0.379268
$233$	−15.3630	−1.00646	−0.503232	−	0.864151i	$-0.667856\pi$
$233$	−15.3630	−1.00646	−0.503232	+	0.864151i	$0.667856\pi$
$234$	0	0
$235$	−2.43618	−0.158919
$236$	8.36601	0.544581
$237$	0	0
$238$	−5.78118	−0.374738
$239$	25.0515	1.62045	0.810225	−	0.586119i	$-0.199345\pi$
$239$	25.0515	1.62045	0.810225	+	0.586119i	$0.199345\pi$
$240$	0	0
$241$	1.66193	0.107054	0.0535270	−	0.998566i	$-0.482954\pi$
$241$	1.66193	0.107054	0.0535270	+	0.998566i	$0.482954\pi$
$242$	−10.9698	−0.705166
$243$	0	0
$244$	6.66842	0.426902
$245$	5.93217	0.378993
$246$	0	0
$247$	12.9018	0.820924
$248$	0.0135315	0.000859248 0
$249$	0	0
$250$	9.61940	0.608384
$251$	14.0249	0.885244	0.442622	−	0.896708i	$-0.354048\pi$
$251$	14.0249	0.885244	0.442622	+	0.896708i	$0.354048\pi$
$252$	0	0
$253$	0.0896480	0.00563612
$254$	8.14529	0.511081
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.25186	−0.577115	−0.288557	−	0.957463i	$-0.593176\pi$
$257$	−9.25186	−0.577115	−0.288557	+	0.957463i	$0.593176\pi$
$258$	0	0
$259$	9.68614	0.601868
$260$	−4.61839	−0.286421
$261$	0	0
$262$	1.84870	0.114213
$263$	3.86172	0.238124	0.119062	−	0.992887i	$-0.462011\pi$
$263$	3.86172	0.238124	0.119062	+	0.992887i	$0.462011\pi$
$264$	0	0
$265$	−1.90918	−0.117280
$266$	−3.82180	−0.234329
$267$	0	0
$268$	−11.2370	−0.686406
$269$	5.13433	0.313046	0.156523	−	0.987674i	$-0.449972\pi$
$269$	5.13433	0.313046	0.156523	+	0.987674i	$0.449972\pi$
$270$	0	0
$271$	10.0580	0.610981	0.305490	−	0.952195i	$-0.401180\pi$
$271$	10.0580	0.610981	0.305490	+	0.952195i	$0.401180\pi$
$272$	4.61561	0.279863
$273$	0	0
$274$	−12.4658	−0.753084
$275$	−0.661420	−0.0398851
$276$	0	0
$277$	−19.1060	−1.14797	−0.573986	−	0.818865i	$-0.694603\pi$
$277$	−19.1060	−1.14797	−0.573986	+	0.818865i	$0.694603\pi$
$278$	9.51802	0.570853
$279$	0	0
$280$	1.36807	0.0817576
$281$	14.8839	0.887900	0.443950	−	0.896052i	$-0.353577\pi$
$281$	14.8839	0.887900	0.443950	+	0.896052i	$0.353577\pi$
$282$	0	0
$283$	25.8463	1.53640	0.768201	−	0.640208i	$-0.221152\pi$
$283$	25.8463	1.53640	0.768201	+	0.640208i	$0.221152\pi$
$284$	5.70549	0.338558
$285$	0	0
$286$	0.734624	0.0434393
$287$	−9.20111	−0.543124
$288$	0	0
$289$	4.30388	0.253169
$290$	6.30973	0.370520
$291$	0	0
$292$	14.5008	0.848597
$293$	21.3302	1.24612	0.623061	−	0.782173i	$-0.285889\pi$
$293$	21.3302	1.24612	0.623061	+	0.782173i	$0.285889\pi$
$294$	0	0
$295$	−9.13773	−0.532019
$296$	−7.73328	−0.449488
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	2.18181	0.126177
$300$	0	0
$301$	−6.51249	−0.375374
$302$	−10.4423	−0.600887
$303$	0	0
$304$	3.05127	0.175002
$305$	−7.28355	−0.417055
$306$	0	0
$307$	−16.9755	−0.968845	−0.484423	−	0.874834i	$-0.660970\pi$
$307$	−16.9755	−0.968845	−0.484423	+	0.874834i	$0.660970\pi$
$308$	−0.217611	−0.0123996
$309$	0	0
$310$	−0.0147797	−0.000839428 0
$311$	33.1623	1.88046	0.940230	−	0.340540i	$-0.110610\pi$
$311$	33.1623	1.88046	0.940230	+	0.340540i	$0.110610\pi$
$312$	0	0
$313$	10.8522	0.613405	0.306702	−	0.951805i	$-0.400774\pi$
$313$	10.8522	0.613405	0.306702	+	0.951805i	$0.400774\pi$
$314$	−8.12291	−0.458402
$315$	0	0
$316$	−1.27461	−0.0717026
$317$	27.2821	1.53231	0.766157	−	0.642654i	$-0.222167\pi$
$317$	27.2821	1.53231	0.766157	+	0.642654i	$0.222167\pi$
$318$	0	0
$319$	−1.00366	−0.0561939
$320$	−1.09224	−0.0610583
$321$	0	0
$322$	−0.646299	−0.0360168
$323$	14.0835	0.783625
$324$	0	0
$325$	−16.0973	−0.892920
$326$	−0.0916217	−0.00507446
$327$	0	0
$328$	7.34603	0.405617
$329$	−2.79369	−0.154021
$330$	0	0
$331$	15.9144	0.874733	0.437366	−	0.899283i	$-0.355911\pi$
$331$	15.9144	0.874733	0.437366	+	0.899283i	$0.355911\pi$
$332$	−4.98613	−0.273650
$333$	0	0
$334$	13.1457	0.719303
$335$	12.2735	0.670573
$336$	0	0
$337$	0.182766	0.00995592	0.00497796	−	0.999988i	$-0.498415\pi$
$337$	0.182766	0.00995592	0.00497796	+	0.999988i	$0.498415\pi$
$338$	4.87896	0.265380
$339$	0	0
$340$	−5.04138	−0.273407
$341$	0.00235092	0.000127310 0
$342$	0	0
$343$	15.5704	0.840722
$344$	5.19948	0.280337
$345$	0	0
$346$	6.71040	0.360753
$347$	7.90087	0.424141	0.212071	−	0.977254i	$-0.431979\pi$
$347$	7.90087	0.424141	0.212071	+	0.977254i	$0.431979\pi$
$348$	0	0
$349$	−9.57512	−0.512544	−0.256272	−	0.966605i	$-0.582494\pi$
$349$	−9.57512	−0.512544	−0.256272	+	0.966605i	$0.582494\pi$
$350$	4.76837	0.254880
$351$	0	0
$352$	0.173738	0.00926025
$353$	6.58919	0.350707	0.175354	−	0.984506i	$-0.443893\pi$
$353$	6.58919	0.350707	0.175354	+	0.984506i	$0.443893\pi$
$354$	0	0
$355$	−6.23179	−0.330749
$356$	3.60101	0.190853
$357$	0	0
$358$	18.5877	0.982390
$359$	−11.2505	−0.593777	−0.296889	−	0.954912i	$-0.595949\pi$
$359$	−11.2505	−0.593777	−0.296889	+	0.954912i	$0.595949\pi$
$360$	0	0
$361$	−9.68976	−0.509987
$362$	−6.34620	−0.333549
$363$	0	0
$364$	−5.29613	−0.277592
$365$	−15.8384	−0.829022
$366$	0	0
$367$	9.21281	0.480905	0.240452	−	0.970661i	$-0.422704\pi$
$367$	9.21281	0.480905	0.240452	+	0.970661i	$0.422704\pi$
$368$	0.515996	0.0268981
$369$	0	0
$370$	8.44663	0.439119
$371$	−2.18935	−0.113665
$372$	0	0
$373$	−34.9485	−1.80956	−0.904782	−	0.425874i	$-0.859967\pi$
$373$	−34.9485	−1.80956	−0.904782	+	0.425874i	$0.859967\pi$
$374$	0.801906	0.0414656
$375$	0	0
$376$	2.23044	0.115026
$377$	−24.4265	−1.25803
$378$	0	0
$379$	2.91395	0.149680	0.0748398	−	0.997196i	$-0.476155\pi$
$379$	2.91395	0.149680	0.0748398	+	0.997196i	$0.476155\pi$
$380$	−3.33273	−0.170966
$381$	0	0
$382$	0.581014	0.0297272
$383$	16.4250	0.839280	0.419640	−	0.907691i	$-0.362156\pi$
$383$	16.4250	0.839280	0.419640	+	0.907691i	$0.362156\pi$
$384$	0	0
$385$	0.237685	0.0121135
$386$	20.8034	1.05886
$387$	0	0
$388$	−7.13298	−0.362122
$389$	−5.41838	−0.274723	−0.137361	−	0.990521i	$-0.543862\pi$
$389$	−5.41838	−0.274723	−0.137361	+	0.990521i	$0.543862\pi$
$390$	0	0
$391$	2.38164	0.120445
$392$	−5.43118	−0.274316
$393$	0	0
$394$	10.3409	0.520965
$395$	1.39219	0.0700486
$396$	0	0
$397$	23.7221	1.19058	0.595288	−	0.803513i	$-0.297038\pi$
$397$	23.7221	1.19058	0.595288	+	0.803513i	$0.297038\pi$
$398$	−9.52240	−0.477315
$399$	0	0
$400$	−3.80700	−0.190350
$401$	−13.7355	−0.685919	−0.342959	−	0.939350i	$-0.611429\pi$
$401$	−13.7355	−0.685919	−0.342959	+	0.939350i	$0.611429\pi$
$402$	0	0
$403$	0.0572158	0.00285012
$404$	12.5976	0.626752
$405$	0	0
$406$	7.23565	0.359099
$407$	−1.34356	−0.0665979
$408$	0	0
$409$	−17.3492	−0.857863	−0.428932	−	0.903337i	$-0.641110\pi$
$409$	−17.3492	−0.857863	−0.428932	+	0.903337i	$0.641110\pi$
$410$	−8.02367	−0.396261
$411$	0	0
$412$	0.697965	0.0343862
$413$	−10.4787	−0.515621
$414$	0	0
$415$	5.44608	0.267337
$416$	4.22835	0.207312
$417$	0	0
$418$	0.530121	0.0259290
$419$	0.488645	0.0238719	0.0119359	−	0.999929i	$-0.496201\pi$
$419$	0.488645	0.0238719	0.0119359	+	0.999929i	$0.496201\pi$
$420$	0	0
$421$	−30.8098	−1.50158	−0.750789	−	0.660542i	$-0.770326\pi$
$421$	−30.8098	−1.50158	−0.750789	+	0.660542i	$0.770326\pi$
$422$	−5.92251	−0.288303
$423$	0	0
$424$	1.74794	0.0848877
$425$	−17.5716	−0.852350
$426$	0	0
$427$	−8.35238	−0.404200
$428$	−8.15974	−0.394416
$429$	0	0
$430$	−5.67911	−0.273871
$431$	−23.2831	−1.12151	−0.560755	−	0.827982i	$-0.689489\pi$
$431$	−23.2831	−1.12151	−0.560755	+	0.827982i	$0.689489\pi$
$432$	0	0
$433$	−34.3793	−1.65216	−0.826082	−	0.563551i	$-0.809435\pi$
$433$	−34.3793	−1.65216	−0.826082	+	0.563551i	$0.809435\pi$
$434$	−0.0169485	−0.000813555 0
$435$	0	0
$436$	13.8923	0.665319
$437$	1.57444	0.0753158
$438$	0	0
$439$	−4.22947	−0.201862	−0.100931	−	0.994893i	$-0.532182\pi$
$439$	−4.22947	−0.201862	−0.100931	+	0.994893i	$0.532182\pi$
$440$	−0.189764	−0.00904665
$441$	0	0
$442$	19.5164	0.928302
$443$	−24.0179	−1.14113	−0.570563	−	0.821254i	$-0.693275\pi$
$443$	−24.0179	−1.14113	−0.570563	+	0.821254i	$0.693275\pi$
$444$	0	0
$445$	−3.93318	−0.186451
$446$	21.7359	1.02922
$447$	0	0
$448$	−1.25253	−0.0591764
$449$	13.7355	0.648218	0.324109	−	0.946020i	$-0.394936\pi$
$449$	13.7355	0.648218	0.324109	+	0.946020i	$0.394936\pi$
$450$	0	0
$451$	1.27628	0.0600978
$452$	4.70720	0.221408
$453$	0	0
$454$	−0.0539413	−0.00253159
$455$	5.78467	0.271189
$456$	0	0
$457$	−21.2382	−0.993483	−0.496741	−	0.867899i	$-0.665470\pi$
$457$	−21.2382	−0.993483	−0.496741	+	0.867899i	$0.665470\pi$
$458$	15.4074	0.719939
$459$	0	0
$460$	−0.563594	−0.0262777
$461$	25.1134	1.16965	0.584825	−	0.811160i	$-0.301163\pi$
$461$	25.1134	1.16965	0.584825	+	0.811160i	$0.301163\pi$
$462$	0	0
$463$	36.1754	1.68122	0.840608	−	0.541644i	$-0.182198\pi$
$463$	36.1754	1.68122	0.840608	+	0.541644i	$0.182198\pi$
$464$	−5.77684	−0.268183
$465$	0	0
$466$	−15.3630	−0.711677
$467$	6.71738	0.310843	0.155422	−	0.987848i	$-0.450326\pi$
$467$	6.71738	0.310843	0.155422	+	0.987848i	$0.450326\pi$
$468$	0	0
$469$	14.0746	0.649904
$470$	−2.43618	−0.112373
$471$	0	0
$472$	8.36601	0.385077
$473$	0.903346	0.0415359
$474$	0	0
$475$	−11.6162	−0.532987
$476$	−5.78118	−0.264980
$477$	0	0
$478$	25.0515	1.14583
$479$	−34.5320	−1.57781	−0.788903	−	0.614518i	$-0.789351\pi$
$479$	−34.5320	−1.57781	−0.788903	+	0.614518i	$0.789351\pi$
$480$	0	0
$481$	−32.6990	−1.49095
$482$	1.66193	0.0756987
$483$	0	0
$484$	−10.9698	−0.498628
$485$	7.79096	0.353769
$486$	0	0
$487$	14.7774	0.669628	0.334814	−	0.942284i	$-0.391327\pi$
$487$	14.7774	0.669628	0.334814	+	0.942284i	$0.391327\pi$
$488$	6.66842	0.301865
$489$	0	0
$490$	5.93217	0.267988
$491$	−25.0893	−1.13227	−0.566133	−	0.824314i	$-0.691561\pi$
$491$	−25.0893	−1.13227	−0.566133	+	0.824314i	$0.691561\pi$
$492$	0	0
$493$	−26.6637	−1.20087
$494$	12.9018	0.580481
$495$	0	0
$496$	0.0135315	0.000607580 0
$497$	−7.14628	−0.320554
$498$	0	0
$499$	11.2941	0.505592	0.252796	−	0.967520i	$-0.418650\pi$
$499$	11.2941	0.505592	0.252796	+	0.967520i	$0.418650\pi$
$500$	9.61940	0.430193
$501$	0	0
$502$	14.0249	0.625962
$503$	−26.2467	−1.17028	−0.585141	−	0.810932i	$-0.698961\pi$
$503$	−26.2467	−1.17028	−0.585141	+	0.810932i	$0.698961\pi$
$504$	0	0
$505$	−13.7596	−0.612295
$506$	0.0896480	0.00398534
$507$	0	0
$508$	8.14529	0.361389
$509$	−2.41347	−0.106975	−0.0534875	−	0.998569i	$-0.517034\pi$
$509$	−2.41347	−0.106975	−0.0534875	+	0.998569i	$0.517034\pi$
$510$	0	0
$511$	−18.1627	−0.803470
$512$	1.00000	0.0441942
$513$	0	0
$514$	−9.25186	−0.408082
$515$	−0.762348	−0.0335931
$516$	0	0
$517$	0.387511	0.0170427
$518$	9.68614	0.425585
$519$	0	0
$520$	−4.61839	−0.202530
$521$	14.3007	0.626527	0.313263	−	0.949666i	$-0.398578\pi$
$521$	14.3007	0.626527	0.313263	+	0.949666i	$0.398578\pi$
$522$	0	0
$523$	25.7551	1.12619	0.563096	−	0.826392i	$-0.309610\pi$
$523$	25.7551	1.12619	0.563096	+	0.826392i	$0.309610\pi$
$524$	1.84870	0.0807607
$525$	0	0
$526$	3.86172	0.168379
$527$	0.0624560	0.00272062
$528$	0	0
$529$	−22.7337	−0.988424
$530$	−1.90918	−0.0829296
$531$	0	0
$532$	−3.82180	−0.165696
$533$	31.0616	1.34543
$534$	0	0
$535$	8.91244	0.385318
$536$	−11.2370	−0.485363
$537$	0	0
$538$	5.13433	0.221357
$539$	−0.943600	−0.0406437
$540$	0	0
$541$	7.64575	0.328716	0.164358	−	0.986401i	$-0.447445\pi$
$541$	7.64575	0.328716	0.164358	+	0.986401i	$0.447445\pi$
$542$	10.0580	0.432029
$543$	0	0
$544$	4.61561	0.197893
$545$	−15.1738	−0.649973
$546$	0	0
$547$	−15.0339	−0.642803	−0.321402	−	0.946943i	$-0.604154\pi$
$547$	−15.0339	−0.642803	−0.321402	+	0.946943i	$0.604154\pi$
$548$	−12.4658	−0.532511
$549$	0	0
$550$	−0.661420	−0.0282030
$551$	−17.6267	−0.750923
$552$	0	0
$553$	1.59649	0.0678895
$554$	−19.1060	−0.811738
$555$	0	0
$556$	9.51802	0.403654
$557$	21.8515	0.925878	0.462939	−	0.886390i	$-0.346795\pi$
$557$	21.8515	0.925878	0.462939	+	0.886390i	$0.346795\pi$
$558$	0	0
$559$	21.9852	0.929876
$560$	1.36807	0.0578114
$561$	0	0
$562$	14.8839	0.627840
$563$	−15.5931	−0.657173	−0.328586	−	0.944474i	$-0.606572\pi$
$563$	−15.5931	−0.657173	−0.328586	+	0.944474i	$0.606572\pi$
$564$	0	0
$565$	−5.14142	−0.216301
$566$	25.8463	1.08640
$567$	0	0
$568$	5.70549	0.239397
$569$	−0.929346	−0.0389602	−0.0194801	−	0.999810i	$-0.506201\pi$
$569$	−0.929346	−0.0389602	−0.0194801	+	0.999810i	$0.506201\pi$
$570$	0	0
$571$	−24.7144	−1.03426	−0.517132	−	0.855906i	$-0.673000\pi$
$571$	−24.7144	−1.03426	−0.517132	+	0.855906i	$0.673000\pi$
$572$	0.734624	0.0307162
$573$	0	0
$574$	−9.20111	−0.384047
$575$	−1.96440	−0.0819210
$576$	0	0
$577$	−5.56144	−0.231526	−0.115763	−	0.993277i	$-0.536931\pi$
$577$	−5.56144	−0.231526	−0.115763	+	0.993277i	$0.536931\pi$
$578$	4.30388	0.179018
$579$	0	0
$580$	6.30973	0.261997
$581$	6.24527	0.259097
$582$	0	0
$583$	0.303684	0.0125773
$584$	14.5008	0.600048
$585$	0	0
$586$	21.3302	0.881141
$587$	−1.85575	−0.0765952	−0.0382976	−	0.999266i	$-0.512193\pi$
$587$	−1.85575	−0.0765952	−0.0382976	+	0.999266i	$0.512193\pi$
$588$	0	0
$589$	0.0412881	0.00170125
$590$	−9.13773	−0.376194
$591$	0	0
$592$	−7.73328	−0.317836
$593$	−24.9148	−1.02313	−0.511564	−	0.859245i	$-0.670934\pi$
$593$	−24.9148	−1.02313	−0.511564	+	0.859245i	$0.670934\pi$
$594$	0	0
$595$	6.31446	0.258868
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	2.18181	0.0892209
$599$	18.6921	0.763740	0.381870	−	0.924216i	$-0.375280\pi$
$599$	18.6921	0.763740	0.381870	+	0.924216i	$0.375280\pi$
$600$	0	0
$601$	−30.2749	−1.23494	−0.617470	−	0.786595i	$-0.711842\pi$
$601$	−30.2749	−1.23494	−0.617470	+	0.786595i	$0.711842\pi$
$602$	−6.51249	−0.265429
$603$	0	0
$604$	−10.4423	−0.424891
$605$	11.9817	0.487126
$606$	0	0
$607$	3.04930	0.123767	0.0618837	−	0.998083i	$-0.480289\pi$
$607$	3.04930	0.123767	0.0618837	+	0.998083i	$0.480289\pi$
$608$	3.05127	0.123745
$609$	0	0
$610$	−7.28355	−0.294902
$611$	9.43108	0.381541
$612$	0	0
$613$	22.9276	0.926037	0.463019	−	0.886349i	$-0.346766\pi$
$613$	22.9276	0.926037	0.463019	+	0.886349i	$0.346766\pi$
$614$	−16.9755	−0.685077
$615$	0	0
$616$	−0.217611	−0.00876781
$617$	17.9432	0.722366	0.361183	−	0.932495i	$-0.382373\pi$
$617$	17.9432	0.722366	0.361183	+	0.932495i	$0.382373\pi$
$618$	0	0
$619$	30.5288	1.22706	0.613528	−	0.789673i	$-0.289750\pi$
$619$	30.5288	1.22706	0.613528	+	0.789673i	$0.289750\pi$
$620$	−0.0147797	−0.000593565 0
$621$	0	0
$622$	33.1623	1.32969
$623$	−4.51036	−0.180704
$624$	0	0
$625$	8.52827	0.341131
$626$	10.8522	0.433743
$627$	0	0
$628$	−8.12291	−0.324139
$629$	−35.6938	−1.42321
$630$	0	0
$631$	−27.6736	−1.10167	−0.550834	−	0.834615i	$-0.685690\pi$
$631$	−27.6736	−1.10167	−0.550834	+	0.834615i	$0.685690\pi$
$632$	−1.27461	−0.0507014
$633$	0	0
$634$	27.2821	1.08351
$635$	−8.89664	−0.353053
$636$	0	0
$637$	−22.9649	−0.909903
$638$	−1.00366	−0.0397351
$639$	0	0
$640$	−1.09224	−0.0431748
$641$	−18.5127	−0.731208	−0.365604	−	0.930771i	$-0.619137\pi$
$641$	−18.5127	−0.731208	−0.365604	+	0.930771i	$0.619137\pi$
$642$	0	0
$643$	−30.8776	−1.21769	−0.608847	−	0.793287i	$-0.708368\pi$
$643$	−30.8776	−1.21769	−0.608847	+	0.793287i	$0.708368\pi$
$644$	−0.646299	−0.0254677
$645$	0	0
$646$	14.0835	0.554107
$647$	16.5474	0.650544	0.325272	−	0.945621i	$-0.394544\pi$
$647$	16.5474	0.650544	0.325272	+	0.945621i	$0.394544\pi$
$648$	0	0
$649$	1.45349	0.0570546
$650$	−16.0973	−0.631390
$651$	0	0
$652$	−0.0916217	−0.00358818
$653$	9.00280	0.352307	0.176153	−	0.984363i	$-0.443635\pi$
$653$	9.00280	0.352307	0.176153	+	0.984363i	$0.443635\pi$
$654$	0	0
$655$	−2.01923	−0.0788978
$656$	7.34603	0.286814
$657$	0	0
$658$	−2.79369	−0.108909
$659$	−10.7181	−0.417516	−0.208758	−	0.977967i	$-0.566942\pi$
$659$	−10.7181	−0.417516	−0.208758	+	0.977967i	$0.566942\pi$
$660$	0	0
$661$	−4.29076	−0.166891	−0.0834456	−	0.996512i	$-0.526592\pi$
$661$	−4.29076	−0.166891	−0.0834456	+	0.996512i	$0.526592\pi$
$662$	15.9144	0.618529
$663$	0	0
$664$	−4.98613	−0.193500
$665$	4.17434	0.161874
$666$	0	0
$667$	−2.98083	−0.115418
$668$	13.1457	0.508624
$669$	0	0
$670$	12.2735	0.474167
$671$	1.15856	0.0447256
$672$	0	0
$673$	8.41744	0.324469	0.162234	−	0.986752i	$-0.448130\pi$
$673$	8.41744	0.324469	0.162234	+	0.986752i	$0.448130\pi$
$674$	0.182766	0.00703990
$675$	0	0
$676$	4.87896	0.187652
$677$	17.4120	0.669197	0.334599	−	0.942361i	$-0.391399\pi$
$677$	17.4120	0.669197	0.334599	+	0.942361i	$0.391399\pi$
$678$	0	0
$679$	8.93425	0.342865
$680$	−5.04138	−0.193328
$681$	0	0
$682$	0.00235092	9.00216e−5 0
$683$	−22.7209	−0.869392	−0.434696	−	0.900577i	$-0.643144\pi$
$683$	−22.7209	−0.869392	−0.434696	+	0.900577i	$0.643144\pi$
$684$	0	0
$685$	13.6157	0.520228
$686$	15.5704	0.594480
$687$	0	0
$688$	5.19948	0.198228
$689$	7.39092	0.281572
$690$	0	0
$691$	−21.1774	−0.805625	−0.402812	−	0.915283i	$-0.631967\pi$
$691$	−21.1774	−0.805625	−0.402812	+	0.915283i	$0.631967\pi$
$692$	6.71040	0.255091
$693$	0	0
$694$	7.90087	0.299913
$695$	−10.3960	−0.394343
$696$	0	0
$697$	33.9064	1.28430
$698$	−9.57512	−0.362424
$699$	0	0
$700$	4.76837	0.180228
$701$	−29.7208	−1.12254	−0.561269	−	0.827633i	$-0.689687\pi$
$701$	−29.7208	−1.12254	−0.561269	+	0.827633i	$0.689687\pi$
$702$	0	0
$703$	−23.5963	−0.889952
$704$	0.173738	0.00654799
$705$	0	0
$706$	6.58919	0.247987
$707$	−15.7788	−0.593422
$708$	0	0
$709$	−42.9661	−1.61363	−0.806814	−	0.590806i	$-0.798810\pi$
$709$	−42.9661	−1.61363	−0.806814	+	0.590806i	$0.798810\pi$
$710$	−6.23179	−0.233875
$711$	0	0
$712$	3.60101	0.134953
$713$	0.00698217	0.000261484 0
$714$	0	0
$715$	−0.802390	−0.0300077
$716$	18.5877	0.694655
$717$	0	0
$718$	−11.2505	−0.419864
$719$	−5.77459	−0.215356	−0.107678	−	0.994186i	$-0.534342\pi$
$719$	−5.77459	−0.215356	−0.107678	+	0.994186i	$0.534342\pi$
$720$	0	0
$721$	−0.874220	−0.0325576
$722$	−9.68976	−0.360616
$723$	0	0
$724$	−6.34620	−0.235855
$725$	21.9924	0.816779
$726$	0	0
$727$	−1.49085	−0.0552924	−0.0276462	−	0.999618i	$-0.508801\pi$
$727$	−1.49085	−0.0552924	−0.0276462	+	0.999618i	$0.508801\pi$
$728$	−5.29613	−0.196288
$729$	0	0
$730$	−15.8384	−0.586207
$731$	23.9988	0.887627
$732$	0	0
$733$	−39.7439	−1.46797	−0.733987	−	0.679164i	$-0.762343\pi$
$733$	−39.7439	−1.46797	−0.733987	+	0.679164i	$0.762343\pi$
$734$	9.21281	0.340051
$735$	0	0
$736$	0.515996	0.0190199
$737$	−1.95228	−0.0719133
$738$	0	0
$739$	−43.2872	−1.59235	−0.796173	−	0.605069i	$-0.793145\pi$
$739$	−43.2872	−1.59235	−0.796173	+	0.605069i	$0.793145\pi$
$740$	8.44663	0.310504
$741$	0	0
$742$	−2.18935	−0.0803735
$743$	10.2674	0.376676	0.188338	−	0.982104i	$-0.439690\pi$
$743$	10.2674	0.376676	0.188338	+	0.982104i	$0.439690\pi$
$744$	0	0
$745$	1.09224	0.0400167
$746$	−34.9485	−1.27956
$747$	0	0
$748$	0.801906	0.0293206
$749$	10.2203	0.373442
$750$	0	0
$751$	−12.1914	−0.444870	−0.222435	−	0.974948i	$-0.571401\pi$
$751$	−12.1914	−0.444870	−0.222435	+	0.974948i	$0.571401\pi$
$752$	2.23044	0.0813357
$753$	0	0
$754$	−24.4265	−0.889562
$755$	11.4055	0.415090
$756$	0	0
$757$	−3.51744	−0.127843	−0.0639217	−	0.997955i	$-0.520361\pi$
$757$	−3.51744	−0.127843	−0.0639217	+	0.997955i	$0.520361\pi$
$758$	2.91395	0.105839
$759$	0	0
$760$	−3.33273	−0.120891
$761$	44.1493	1.60041	0.800206	−	0.599726i	$-0.204724\pi$
$761$	44.1493	1.60041	0.800206	+	0.599726i	$0.204724\pi$
$762$	0	0
$763$	−17.4005	−0.629939
$764$	0.581014	0.0210203
$765$	0	0
$766$	16.4250	0.593461
$767$	35.3744	1.27730
$768$	0	0
$769$	15.0500	0.542716	0.271358	−	0.962478i	$-0.412527\pi$
$769$	15.0500	0.542716	0.271358	+	0.962478i	$0.412527\pi$
$770$	0.237685	0.00856556
$771$	0	0
$772$	20.8034	0.748729
$773$	14.8464	0.533989	0.266995	−	0.963698i	$-0.413969\pi$
$773$	14.8464	0.533989	0.266995	+	0.963698i	$0.413969\pi$
$774$	0	0
$775$	−0.0515143	−0.00185045
$776$	−7.13298	−0.256059
$777$	0	0
$778$	−5.41838	−0.194258
$779$	22.4147	0.803091
$780$	0	0
$781$	0.991259	0.0354700
$782$	2.38164	0.0851671
$783$	0	0
$784$	−5.43118	−0.193971
$785$	8.87220	0.316662
$786$	0	0
$787$	−50.1456	−1.78750	−0.893749	−	0.448566i	$-0.851935\pi$
$787$	−50.1456	−1.78750	−0.893749	+	0.448566i	$0.851935\pi$
$788$	10.3409	0.368378
$789$	0	0
$790$	1.39219	0.0495318
$791$	−5.89590	−0.209634
$792$	0	0
$793$	28.1964	1.00128
$794$	23.7221	0.841864
$795$	0	0
$796$	−9.52240	−0.337513
$797$	41.8347	1.48186	0.740931	−	0.671582i	$-0.234385\pi$
$797$	41.8347	1.48186	0.740931	+	0.671582i	$0.234385\pi$
$798$	0	0
$799$	10.2948	0.364205
$800$	−3.80700	−0.134598
$801$	0	0
$802$	−13.7355	−0.485018
$803$	2.51934	0.0889056
$804$	0	0
$805$	0.705916	0.0248803
$806$	0.0572158	0.00201534
$807$	0	0
$808$	12.5976	0.443181
$809$	12.5882	0.442578	0.221289	−	0.975208i	$-0.428974\pi$
$809$	12.5882	0.442578	0.221289	+	0.975208i	$0.428974\pi$
$810$	0	0
$811$	−30.3817	−1.06684	−0.533422	−	0.845849i	$-0.679094\pi$
$811$	−30.3817	−1.06684	−0.533422	+	0.845849i	$0.679094\pi$
$812$	7.23565	0.253922
$813$	0	0
$814$	−1.34356	−0.0470918
$815$	0.100073	0.00350542
$816$	0	0
$817$	15.8650	0.555047
$818$	−17.3492	−0.606601
$819$	0	0
$820$	−8.02367	−0.280199
$821$	−13.9419	−0.486577	−0.243289	−	0.969954i	$-0.578226\pi$
$821$	−13.9419	−0.486577	−0.243289	+	0.969954i	$0.578226\pi$
$822$	0	0
$823$	18.9495	0.660538	0.330269	−	0.943887i	$-0.392860\pi$
$823$	18.9495	0.660538	0.330269	+	0.943887i	$0.392860\pi$
$824$	0.697965	0.0243148
$825$	0	0
$826$	−10.4787	−0.364599
$827$	−9.25287	−0.321754	−0.160877	−	0.986974i	$-0.551432\pi$
$827$	−9.25287	−0.321754	−0.160877	+	0.986974i	$0.551432\pi$
$828$	0	0
$829$	−48.9325	−1.69950	−0.849748	−	0.527190i	$-0.823246\pi$
$829$	−48.9325	−1.69950	−0.849748	+	0.527190i	$0.823246\pi$
$830$	5.44608	0.189036
$831$	0	0
$832$	4.22835	0.146592
$833$	−25.0682	−0.868562
$834$	0	0
$835$	−14.3584	−0.496892
$836$	0.530121	0.0183346
$837$	0	0
$838$	0.488645	0.0168800
$839$	53.3647	1.84236	0.921178	−	0.389142i	$-0.127229\pi$
$839$	53.3647	1.84236	0.921178	+	0.389142i	$0.127229\pi$
$840$	0	0
$841$	4.37191	0.150756
$842$	−30.8098	−1.06178
$843$	0	0
$844$	−5.92251	−0.203861
$845$	−5.32902	−0.183324
$846$	0	0
$847$	13.7400	0.472112
$848$	1.74794	0.0600247
$849$	0	0
$850$	−17.5716	−0.602702
$851$	−3.99034	−0.136787
$852$	0	0
$853$	−9.04264	−0.309614	−0.154807	−	0.987945i	$-0.549476\pi$
$853$	−9.04264	−0.309614	−0.154807	+	0.987945i	$0.549476\pi$
$854$	−8.35238	−0.285813
$855$	0	0
$856$	−8.15974	−0.278894
$857$	−31.8134	−1.08673	−0.543363	−	0.839498i	$-0.682849\pi$
$857$	−31.8134	−1.08673	−0.543363	+	0.839498i	$0.682849\pi$
$858$	0	0
$859$	5.01136	0.170986	0.0854928	−	0.996339i	$-0.472754\pi$
$859$	5.01136	0.170986	0.0854928	+	0.996339i	$0.472754\pi$
$860$	−5.67911	−0.193656
$861$	0	0
$862$	−23.2831	−0.793027
$863$	−29.2328	−0.995096	−0.497548	−	0.867437i	$-0.665766\pi$
$863$	−29.2328	−0.995096	−0.497548	+	0.867437i	$0.665766\pi$
$864$	0	0
$865$	−7.32939	−0.249207
$866$	−34.3793	−1.16826
$867$	0	0
$868$	−0.0169485	−0.000575270 0
$869$	−0.221448	−0.00751212
$870$	0	0
$871$	−47.5138	−1.60994
$872$	13.8923	0.470452
$873$	0	0
$874$	1.57444	0.0532563
$875$	−12.0486	−0.407316
$876$	0	0
$877$	−15.2618	−0.515354	−0.257677	−	0.966231i	$-0.582957\pi$
$877$	−15.2618	−0.515354	−0.257677	+	0.966231i	$0.582957\pi$
$878$	−4.22947	−0.142738
$879$	0	0
$880$	−0.189764	−0.00639695
$881$	58.0930	1.95720	0.978601	−	0.205766i	$-0.0659685\pi$
$881$	58.0930	1.95720	0.978601	+	0.205766i	$0.0659685\pi$
$882$	0	0
$883$	36.3203	1.22227	0.611137	−	0.791525i	$-0.290712\pi$
$883$	36.3203	1.22227	0.611137	+	0.791525i	$0.290712\pi$
$884$	19.5164	0.656409
$885$	0	0
$886$	−24.0179	−0.806898
$887$	9.97904	0.335063	0.167532	−	0.985867i	$-0.446420\pi$
$887$	9.97904	0.335063	0.167532	+	0.985867i	$0.446420\pi$
$888$	0	0
$889$	−10.2022	−0.342171
$890$	−3.93318	−0.131841
$891$	0	0
$892$	21.7359	0.727771
$893$	6.80567	0.227743
$894$	0	0
$895$	−20.3023	−0.678631
$896$	−1.25253	−0.0418440
$897$	0	0
$898$	13.7355	0.458359
$899$	−0.0781691	−0.00260708
$900$	0	0
$901$	8.06783	0.268778
$902$	1.27628	0.0424956
$903$	0	0
$904$	4.70720	0.156559
$905$	6.93160	0.230414
$906$	0	0
$907$	−14.0426	−0.466276	−0.233138	−	0.972444i	$-0.574899\pi$
$907$	−14.0426	−0.466276	−0.233138	+	0.972444i	$0.574899\pi$
$908$	−0.0539413	−0.00179010
$909$	0	0
$910$	5.78467	0.191760
$911$	−26.0282	−0.862354	−0.431177	−	0.902267i	$-0.641902\pi$
$911$	−26.0282	−0.862354	−0.431177	+	0.902267i	$0.641902\pi$
$912$	0	0
$913$	−0.866280	−0.0286697
$914$	−21.2382	−0.702498
$915$	0	0
$916$	15.4074	0.509073
$917$	−2.31554	−0.0764660
$918$	0	0
$919$	23.3909	0.771593	0.385797	−	0.922584i	$-0.373927\pi$
$919$	23.3909	0.771593	0.385797	+	0.922584i	$0.373927\pi$
$920$	−0.563594	−0.0185811
$921$	0	0
$922$	25.1134	0.827067
$923$	24.1248	0.794078
$924$	0	0
$925$	29.4406	0.968001
$926$	36.1754	1.18880
$927$	0	0
$928$	−5.77684	−0.189634
$929$	−55.6486	−1.82577	−0.912886	−	0.408215i	$-0.866152\pi$
$929$	−55.6486	−1.82577	−0.912886	+	0.408215i	$0.866152\pi$
$930$	0	0
$931$	−16.5720	−0.543125
$932$	−15.3630	−0.503232
$933$	0	0
$934$	6.71738	0.219799
$935$	−0.875878	−0.0286443
$936$	0	0
$937$	−4.22699	−0.138090	−0.0690449	−	0.997614i	$-0.521995\pi$
$937$	−4.22699	−0.138090	−0.0690449	+	0.997614i	$0.521995\pi$
$938$	14.0746	0.459552
$939$	0	0
$940$	−2.43618	−0.0794596
$941$	−19.0791	−0.621960	−0.310980	−	0.950416i	$-0.600657\pi$
$941$	−19.0791	−0.621960	−0.310980	+	0.950416i	$0.600657\pi$
$942$	0	0
$943$	3.79052	0.123436
$944$	8.36601	0.272291
$945$	0	0
$946$	0.903346	0.0293703
$947$	−19.0693	−0.619669	−0.309835	−	0.950790i	$-0.600274\pi$
$947$	−19.0693	−0.619669	−0.309835	+	0.950790i	$0.600274\pi$
$948$	0	0
$949$	61.3146	1.99036
$950$	−11.6162	−0.376879
$951$	0	0
$952$	−5.78118	−0.187369
$953$	−38.7971	−1.25676	−0.628380	−	0.777906i	$-0.716282\pi$
$953$	−38.7971	−1.25676	−0.628380	+	0.777906i	$0.716282\pi$
$954$	0	0
$955$	−0.634609	−0.0205355
$956$	25.0515	0.810225
$957$	0	0
$958$	−34.5320	−1.11568
$959$	15.6137	0.504193
$960$	0	0
$961$	−30.9998	−0.999994
$962$	−32.6990	−1.05426
$963$	0	0
$964$	1.66193	0.0535270
$965$	−22.7224	−0.731459
$966$	0	0
$967$	−33.0907	−1.06413	−0.532063	−	0.846705i	$-0.678583\pi$
$967$	−33.0907	−1.06413	−0.532063	+	0.846705i	$0.678583\pi$
$968$	−10.9698	−0.352583
$969$	0	0
$970$	7.79096	0.250153
$971$	36.7261	1.17860	0.589299	−	0.807915i	$-0.299404\pi$
$971$	36.7261	1.17860	0.589299	+	0.807915i	$0.299404\pi$
$972$	0	0
$973$	−11.9216	−0.382188
$974$	14.7774	0.473498
$975$	0	0
$976$	6.66842	0.213451
$977$	8.47740	0.271216	0.135608	−	0.990763i	$-0.456701\pi$
$977$	8.47740	0.271216	0.135608	+	0.990763i	$0.456701\pi$
$978$	0	0
$979$	0.625631	0.0199953
$980$	5.93217	0.189496
$981$	0	0
$982$	−25.0893	−0.800633
$983$	12.1207	0.386590	0.193295	−	0.981141i	$-0.438083\pi$
$983$	12.1207	0.386590	0.193295	+	0.981141i	$0.438083\pi$
$984$	0	0
$985$	−11.2948	−0.359881
$986$	−26.6637	−0.849144
$987$	0	0
$988$	12.9018	0.410462
$989$	2.68291	0.0853116
$990$	0	0
$991$	−7.69712	−0.244507	−0.122254	−	0.992499i	$-0.539012\pi$
$991$	−7.69712	−0.244507	−0.122254	+	0.992499i	$0.539012\pi$
$992$	0.0135315	0.000429624 0
$993$	0	0
$994$	−7.14628	−0.226666
$995$	10.4008	0.329727
$996$	0	0
$997$	−23.3270	−0.738774	−0.369387	−	0.929276i	$-0.620432\pi$
$997$	−23.3270	−0.738774	−0.369387	+	0.929276i	$0.620432\pi$
$998$	11.2941	0.357507
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.p.1.4	yes	12
3.2	odd	2		8046.2.a.i.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.9	✓	12	3.2	odd	2
8046.2.a.p.1.4	yes	12	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.09224 q^{5} -1.25253 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.09224 q^{5} -1.25253 q^{7} +1.00000 q^{8} -1.09224 q^{10} +0.173738 q^{11} +4.22835 q^{13} -1.25253 q^{14} +1.00000 q^{16} +4.61561 q^{17} +3.05127 q^{19} -1.09224 q^{20} +0.173738 q^{22} +0.515996 q^{23} -3.80700 q^{25} +4.22835 q^{26} -1.25253 q^{28} -5.77684 q^{29} +0.0135315 q^{31} +1.00000 q^{32} +4.61561 q^{34} +1.36807 q^{35} -7.73328 q^{37} +3.05127 q^{38} -1.09224 q^{40} +7.34603 q^{41} +5.19948 q^{43} +0.173738 q^{44} +0.515996 q^{46} +2.23044 q^{47} -5.43118 q^{49} -3.80700 q^{50} +4.22835 q^{52} +1.74794 q^{53} -0.189764 q^{55} -1.25253 q^{56} -5.77684 q^{58} +8.36601 q^{59} +6.66842 q^{61} +0.0135315 q^{62} +1.00000 q^{64} -4.61839 q^{65} -11.2370 q^{67} +4.61561 q^{68} +1.36807 q^{70} +5.70549 q^{71} +14.5008 q^{73} -7.73328 q^{74} +3.05127 q^{76} -0.217611 q^{77} -1.27461 q^{79} -1.09224 q^{80} +7.34603 q^{82} -4.98613 q^{83} -5.04138 q^{85} +5.19948 q^{86} +0.173738 q^{88} +3.60101 q^{89} -5.29613 q^{91} +0.515996 q^{92} +2.23044 q^{94} -3.33273 q^{95} -7.13298 q^{97} -5.43118 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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