LMFDB - Embedded newform 8046.2.a.m.1.9

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:	$1$
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} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 $ x^12 - 5x^11 - 21x^10 + 116x^9 + 106x^8 - 774x^7 - 63x^6 + 2013x^5 - 417x^4 - 2249x^3 + 761x^2 + 910*x - 375
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.17050$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.17050 q^{5} +0.506529 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.17050 q^{5} +0.506529 q^{7} +1.00000 q^{8} +1.17050 q^{10} +3.88070 q^{11} -3.61369 q^{13} +0.506529 q^{14} +1.00000 q^{16} -6.95227 q^{17} -2.33695 q^{19} +1.17050 q^{20} +3.88070 q^{22} +1.72649 q^{23} -3.62993 q^{25} -3.61369 q^{26} +0.506529 q^{28} -4.21919 q^{29} -3.29279 q^{31} +1.00000 q^{32} -6.95227 q^{34} +0.592892 q^{35} -7.04259 q^{37} -2.33695 q^{38} +1.17050 q^{40} -10.5485 q^{41} -6.84084 q^{43} +3.88070 q^{44} +1.72649 q^{46} +4.17683 q^{47} -6.74343 q^{49} -3.62993 q^{50} -3.61369 q^{52} -12.8841 q^{53} +4.54236 q^{55} +0.506529 q^{56} -4.21919 q^{58} +7.05273 q^{59} +11.2400 q^{61} -3.29279 q^{62} +1.00000 q^{64} -4.22983 q^{65} -5.00991 q^{67} -6.95227 q^{68} +0.592892 q^{70} +0.923816 q^{71} +2.46929 q^{73} -7.04259 q^{74} -2.33695 q^{76} +1.96569 q^{77} -7.80783 q^{79} +1.17050 q^{80} -10.5485 q^{82} +8.00835 q^{83} -8.13763 q^{85} -6.84084 q^{86} +3.88070 q^{88} -15.6581 q^{89} -1.83044 q^{91} +1.72649 q^{92} +4.17683 q^{94} -2.73540 q^{95} +1.02168 q^{97} -6.74343 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 10 q^{11} - q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 10 q^{19} - 5 q^{20} - 10 q^{22} - 15 q^{23} + 7 q^{25} - q^{26} - 6 q^{28} - 33 q^{29} - 6 q^{31} + 12 q^{32} - 6 q^{34} - 16 q^{35} - 13 q^{37} - 10 q^{38} - 5 q^{40} - 20 q^{41} - 11 q^{43} - 10 q^{44} - 15 q^{46} - 15 q^{47} + 2 q^{49} + 7 q^{50} - q^{52} - 4 q^{53} - 17 q^{55} - 6 q^{56} - 33 q^{58} - 10 q^{59} - 12 q^{61} - 6 q^{62} + 12 q^{64} - 40 q^{65} - 19 q^{67} - 6 q^{68} - 16 q^{70} - 47 q^{71} - 2 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{77} - 15 q^{79} - 5 q^{80} - 20 q^{82} - 18 q^{83} - 25 q^{85} - 11 q^{86} - 10 q^{88} - 24 q^{89} - 3 q^{91} - 15 q^{92} - 15 q^{94} + 3 q^{95} - 25 q^{97} + 2 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 - 10 * q^11 - q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 - 10 * q^19 - 5 * q^20 - 10 * q^22 - 15 * q^23 + 7 * q^25 - q^26 - 6 * q^28 - 33 * q^29 - 6 * q^31 + 12 * q^32 - 6 * q^34 - 16 * q^35 - 13 * q^37 - 10 * q^38 - 5 * q^40 - 20 * q^41 - 11 * q^43 - 10 * q^44 - 15 * q^46 - 15 * q^47 + 2 * q^49 + 7 * q^50 - q^52 - 4 * q^53 - 17 * q^55 - 6 * q^56 - 33 * q^58 - 10 * q^59 - 12 * q^61 - 6 * q^62 + 12 * q^64 - 40 * q^65 - 19 * q^67 - 6 * q^68 - 16 * q^70 - 47 * q^71 - 2 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^77 - 15 * q^79 - 5 * q^80 - 20 * q^82 - 18 * q^83 - 25 * q^85 - 11 * q^86 - 10 * q^88 - 24 * q^89 - 3 * q^91 - 15 * q^92 - 15 * q^94 + 3 * q^95 - 25 * q^97 + 2 * 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
$5$	1.17050	0.523463	0.261732	−	0.965141i	$-0.415706\pi$
$5$	1.17050	0.523463	0.261732	+	0.965141i	$0.415706\pi$
$6$	0	0
$7$	0.506529	0.191450	0.0957249	−	0.995408i	$-0.469483\pi$
$7$	0.506529	0.191450	0.0957249	+	0.995408i	$0.469483\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.17050	0.370144
$11$	3.88070	1.17008	0.585038	−	0.811006i	$-0.301080\pi$
$11$	3.88070	1.17008	0.585038	+	0.811006i	$0.301080\pi$
$12$	0	0
$13$	−3.61369	−1.00226	−0.501129	−	0.865372i	$-0.667082\pi$
$13$	−3.61369	−1.00226	−0.501129	+	0.865372i	$0.667082\pi$
$14$	0.506529	0.135375
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.95227	−1.68617	−0.843087	−	0.537777i	$-0.819264\pi$
$17$	−6.95227	−1.68617	−0.843087	+	0.537777i	$0.819264\pi$
$18$	0	0
$19$	−2.33695	−0.536133	−0.268066	−	0.963400i	$-0.586385\pi$
$19$	−2.33695	−0.536133	−0.268066	+	0.963400i	$0.586385\pi$
$20$	1.17050	0.261732
$21$	0	0
$22$	3.88070	0.827369
$23$	1.72649	0.359997	0.179999	−	0.983667i	$-0.442391\pi$
$23$	1.72649	0.359997	0.179999	+	0.983667i	$0.442391\pi$
$24$	0	0
$25$	−3.62993	−0.725986
$26$	−3.61369	−0.708704
$27$	0	0
$28$	0.506529	0.0957249
$29$	−4.21919	−0.783483	−0.391742	−	0.920075i	$-0.628127\pi$
$29$	−4.21919	−0.783483	−0.391742	+	0.920075i	$0.628127\pi$
$30$	0	0
$31$	−3.29279	−0.591403	−0.295702	−	0.955280i	$-0.595553\pi$
$31$	−3.29279	−0.591403	−0.295702	+	0.955280i	$0.595553\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.95227	−1.19231
$35$	0.592892	0.100217
$36$	0	0
$37$	−7.04259	−1.15779	−0.578897	−	0.815400i	$-0.696517\pi$
$37$	−7.04259	−1.15779	−0.578897	+	0.815400i	$0.696517\pi$
$38$	−2.33695	−0.379103
$39$	0	0
$40$	1.17050	0.185072
$41$	−10.5485	−1.64740	−0.823700	−	0.567026i	$-0.808094\pi$
$41$	−10.5485	−1.64740	−0.823700	+	0.567026i	$0.808094\pi$
$42$	0	0
$43$	−6.84084	−1.04322	−0.521609	−	0.853184i	$-0.674668\pi$
$43$	−6.84084	−1.04322	−0.521609	+	0.853184i	$0.674668\pi$
$44$	3.88070	0.585038
$45$	0	0
$46$	1.72649	0.254557
$47$	4.17683	0.609253	0.304627	−	0.952472i	$-0.401468\pi$
$47$	4.17683	0.609253	0.304627	+	0.952472i	$0.401468\pi$
$48$	0	0
$49$	−6.74343	−0.963347
$50$	−3.62993	−0.513350
$51$	0	0
$52$	−3.61369	−0.501129
$53$	−12.8841	−1.76976	−0.884880	−	0.465819i	$-0.845760\pi$
$53$	−12.8841	−1.76976	−0.884880	+	0.465819i	$0.845760\pi$
$54$	0	0
$55$	4.54236	0.612492
$56$	0.506529	0.0676877
$57$	0	0
$58$	−4.21919	−0.554006
$59$	7.05273	0.918188	0.459094	−	0.888388i	$-0.348174\pi$
$59$	7.05273	0.918188	0.459094	+	0.888388i	$0.348174\pi$
$60$	0	0
$61$	11.2400	1.43913	0.719566	−	0.694424i	$-0.244341\pi$
$61$	11.2400	1.43913	0.719566	+	0.694424i	$0.244341\pi$
$62$	−3.29279	−0.418185
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.22983	−0.524646
$66$	0	0
$67$	−5.00991	−0.612058	−0.306029	−	0.952022i	$-0.599000\pi$
$67$	−5.00991	−0.612058	−0.306029	+	0.952022i	$0.599000\pi$
$68$	−6.95227	−0.843087
$69$	0	0
$70$	0.592892	0.0708641
$71$	0.923816	0.109637	0.0548184	−	0.998496i	$-0.482542\pi$
$71$	0.923816	0.109637	0.0548184	+	0.998496i	$0.482542\pi$
$72$	0	0
$73$	2.46929	0.289009	0.144505	−	0.989504i	$-0.453841\pi$
$73$	2.46929	0.289009	0.144505	+	0.989504i	$0.453841\pi$
$74$	−7.04259	−0.818685
$75$	0	0
$76$	−2.33695	−0.268066
$77$	1.96569	0.224011
$78$	0	0
$79$	−7.80783	−0.878450	−0.439225	−	0.898377i	$-0.644747\pi$
$79$	−7.80783	−0.878450	−0.439225	+	0.898377i	$0.644747\pi$
$80$	1.17050	0.130866
$81$	0	0
$82$	−10.5485	−1.16489
$83$	8.00835	0.879030	0.439515	−	0.898235i	$-0.355150\pi$
$83$	8.00835	0.879030	0.439515	+	0.898235i	$0.355150\pi$
$84$	0	0
$85$	−8.13763	−0.882650
$86$	−6.84084	−0.737667
$87$	0	0
$88$	3.88070	0.413684
$89$	−15.6581	−1.65975	−0.829877	−	0.557946i	$-0.811590\pi$
$89$	−15.6581	−1.65975	−0.829877	+	0.557946i	$0.811590\pi$
$90$	0	0
$91$	−1.83044	−0.191882
$92$	1.72649	0.179999
$93$	0	0
$94$	4.17683	0.430807
$95$	−2.73540	−0.280646
$96$	0	0
$97$	1.02168	0.103736	0.0518681	−	0.998654i	$-0.483482\pi$
$97$	1.02168	0.103736	0.0518681	+	0.998654i	$0.483482\pi$
$98$	−6.74343	−0.681189
$99$	0	0
$100$	−3.62993	−0.362993
$101$	4.32214	0.430069	0.215035	−	0.976606i	$-0.431014\pi$
$101$	4.32214	0.430069	0.215035	+	0.976606i	$0.431014\pi$
$102$	0	0
$103$	6.94103	0.683920	0.341960	−	0.939715i	$-0.388909\pi$
$103$	6.94103	0.683920	0.341960	+	0.939715i	$0.388909\pi$
$104$	−3.61369	−0.354352
$105$	0	0
$106$	−12.8841	−1.25141
$107$	14.3087	1.38327	0.691636	−	0.722246i	$-0.256890\pi$
$107$	14.3087	1.38327	0.691636	+	0.722246i	$0.256890\pi$
$108$	0	0
$109$	−7.75669	−0.742957	−0.371478	−	0.928442i	$-0.621149\pi$
$109$	−7.75669	−0.742957	−0.371478	+	0.928442i	$0.621149\pi$
$110$	4.54236	0.433097
$111$	0	0
$112$	0.506529	0.0478625
$113$	1.71654	0.161479	0.0807393	−	0.996735i	$-0.474272\pi$
$113$	1.71654	0.161479	0.0807393	+	0.996735i	$0.474272\pi$
$114$	0	0
$115$	2.02085	0.188445
$116$	−4.21919	−0.391742
$117$	0	0
$118$	7.05273	0.649257
$119$	−3.52153	−0.322818
$120$	0	0
$121$	4.05985	0.369077
$122$	11.2400	1.01762
$123$	0	0
$124$	−3.29279	−0.295702
$125$	−10.1013	−0.903490
$126$	0	0
$127$	3.86129	0.342634	0.171317	−	0.985216i	$-0.445198\pi$
$127$	3.86129	0.342634	0.171317	+	0.985216i	$0.445198\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.22983	−0.370980
$131$	7.45036	0.650941	0.325470	−	0.945552i	$-0.394477\pi$
$131$	7.45036	0.650941	0.325470	+	0.945552i	$0.394477\pi$
$132$	0	0
$133$	−1.18373	−0.102642
$134$	−5.00991	−0.432790
$135$	0	0
$136$	−6.95227	−0.596153
$137$	11.7309	1.00223	0.501117	−	0.865379i	$-0.332923\pi$
$137$	11.7309	1.00223	0.501117	+	0.865379i	$0.332923\pi$
$138$	0	0
$139$	−0.614537	−0.0521244	−0.0260622	−	0.999660i	$-0.508297\pi$
$139$	−0.614537	−0.0521244	−0.0260622	+	0.999660i	$0.508297\pi$
$140$	0.592892	0.0501085
$141$	0	0
$142$	0.923816	0.0775249
$143$	−14.0237	−1.17272
$144$	0	0
$145$	−4.93855	−0.410125
$146$	2.46929	0.204360
$147$	0	0
$148$	−7.04259	−0.578897
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	17.5323	1.42676	0.713378	−	0.700779i	$-0.247164\pi$
$151$	17.5323	1.42676	0.713378	+	0.700779i	$0.247164\pi$
$152$	−2.33695	−0.189551
$153$	0	0
$154$	1.96569	0.158400
$155$	−3.85422	−0.309578
$156$	0	0
$157$	22.2770	1.77790	0.888950	−	0.458005i	$-0.151436\pi$
$157$	22.2770	1.77790	0.888950	+	0.458005i	$0.151436\pi$
$158$	−7.80783	−0.621158
$159$	0	0
$160$	1.17050	0.0925361
$161$	0.874515	0.0689214
$162$	0	0
$163$	13.4831	1.05608	0.528038	−	0.849221i	$-0.322928\pi$
$163$	13.4831	1.05608	0.528038	+	0.849221i	$0.322928\pi$
$164$	−10.5485	−0.823700
$165$	0	0
$166$	8.00835	0.621568
$167$	−12.6208	−0.976625	−0.488313	−	0.872669i	$-0.662387\pi$
$167$	−12.6208	−0.976625	−0.488313	+	0.872669i	$0.662387\pi$
$168$	0	0
$169$	0.0587896	0.00452227
$170$	−8.13763	−0.624128
$171$	0	0
$172$	−6.84084	−0.521609
$173$	−13.5303	−1.02869	−0.514346	−	0.857583i	$-0.671965\pi$
$173$	−13.5303	−1.02869	−0.514346	+	0.857583i	$0.671965\pi$
$174$	0	0
$175$	−1.83866	−0.138990
$176$	3.88070	0.292519
$177$	0	0
$178$	−15.6581	−1.17362
$179$	7.23353	0.540659	0.270330	−	0.962768i	$-0.412867\pi$
$179$	7.23353	0.540659	0.270330	+	0.962768i	$0.412867\pi$
$180$	0	0
$181$	3.30723	0.245824	0.122912	−	0.992418i	$-0.460777\pi$
$181$	3.30723	0.245824	0.122912	+	0.992418i	$0.460777\pi$
$182$	−1.83044	−0.135681
$183$	0	0
$184$	1.72649	0.127278
$185$	−8.24335	−0.606063
$186$	0	0
$187$	−26.9797	−1.97295
$188$	4.17683	0.304627
$189$	0	0
$190$	−2.73540	−0.198446
$191$	−12.7158	−0.920083	−0.460042	−	0.887897i	$-0.652165\pi$
$191$	−12.7158	−0.920083	−0.460042	+	0.887897i	$0.652165\pi$
$192$	0	0
$193$	−3.36901	−0.242507	−0.121253	−	0.992622i	$-0.538691\pi$
$193$	−3.36901	−0.242507	−0.121253	+	0.992622i	$0.538691\pi$
$194$	1.02168	0.0733526
$195$	0	0
$196$	−6.74343	−0.481673
$197$	0.815685	0.0581151	0.0290576	−	0.999578i	$-0.490749\pi$
$197$	0.815685	0.0581151	0.0290576	+	0.999578i	$0.490749\pi$
$198$	0	0
$199$	5.35685	0.379737	0.189868	−	0.981810i	$-0.439194\pi$
$199$	5.35685	0.379737	0.189868	+	0.981810i	$0.439194\pi$
$200$	−3.62993	−0.256675
$201$	0	0
$202$	4.32214	0.304105
$203$	−2.13714	−0.149998
$204$	0	0
$205$	−12.3470	−0.862353
$206$	6.94103	0.483604
$207$	0	0
$208$	−3.61369	−0.250565
$209$	−9.06900	−0.627316
$210$	0	0
$211$	13.3550	0.919398	0.459699	−	0.888075i	$-0.347957\pi$
$211$	13.3550	0.919398	0.459699	+	0.888075i	$0.347957\pi$
$212$	−12.8841	−0.884880
$213$	0	0
$214$	14.3087	0.978122
$215$	−8.00720	−0.546087
$216$	0	0
$217$	−1.66790	−0.113224
$218$	−7.75669	−0.525350
$219$	0	0
$220$	4.54236	0.306246
$221$	25.1234	1.68998
$222$	0	0
$223$	18.7279	1.25411	0.627057	−	0.778974i	$-0.284259\pi$
$223$	18.7279	1.25411	0.627057	+	0.778974i	$0.284259\pi$
$224$	0.506529	0.0338439
$225$	0	0
$226$	1.71654	0.114183
$227$	−8.54460	−0.567125	−0.283562	−	0.958954i	$-0.591516\pi$
$227$	−8.54460	−0.567125	−0.283562	+	0.958954i	$0.591516\pi$
$228$	0	0
$229$	−5.34208	−0.353015	−0.176507	−	0.984299i	$-0.556480\pi$
$229$	−5.34208	−0.353015	−0.176507	+	0.984299i	$0.556480\pi$
$230$	2.02085	0.133251
$231$	0	0
$232$	−4.21919	−0.277003
$233$	−14.8123	−0.970387	−0.485194	−	0.874407i	$-0.661251\pi$
$233$	−14.8123	−0.970387	−0.485194	+	0.874407i	$0.661251\pi$
$234$	0	0
$235$	4.88898	0.318922
$236$	7.05273	0.459094
$237$	0	0
$238$	−3.52153	−0.228267
$239$	−6.91374	−0.447212	−0.223606	−	0.974680i	$-0.571783\pi$
$239$	−6.91374	−0.447212	−0.223606	+	0.974680i	$0.571783\pi$
$240$	0	0
$241$	8.56369	0.551636	0.275818	−	0.961210i	$-0.411051\pi$
$241$	8.56369	0.551636	0.275818	+	0.961210i	$0.411051\pi$
$242$	4.05985	0.260977
$243$	0	0
$244$	11.2400	0.719566
$245$	−7.89318	−0.504277
$246$	0	0
$247$	8.44501	0.537343
$248$	−3.29279	−0.209093
$249$	0	0
$250$	−10.1013	−0.638864
$251$	−16.1199	−1.01748	−0.508741	−	0.860920i	$-0.669889\pi$
$251$	−16.1199	−1.01748	−0.508741	+	0.860920i	$0.669889\pi$
$252$	0	0
$253$	6.69998	0.421224
$254$	3.86129	0.242279
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.39190	−0.149203	−0.0746013	−	0.997213i	$-0.523768\pi$
$257$	−2.39190	−0.149203	−0.0746013	+	0.997213i	$0.523768\pi$
$258$	0	0
$259$	−3.56727	−0.221660
$260$	−4.22983	−0.262323
$261$	0	0
$262$	7.45036	0.460285
$263$	28.7218	1.77106	0.885532	−	0.464579i	$-0.153794\pi$
$263$	28.7218	1.77106	0.885532	+	0.464579i	$0.153794\pi$
$264$	0	0
$265$	−15.0808	−0.926405
$266$	−1.18373	−0.0725792
$267$	0	0
$268$	−5.00991	−0.306029
$269$	10.6420	0.648853	0.324427	−	0.945911i	$-0.394829\pi$
$269$	10.6420	0.648853	0.324427	+	0.945911i	$0.394829\pi$
$270$	0	0
$271$	−16.2311	−0.985969	−0.492984	−	0.870038i	$-0.664094\pi$
$271$	−16.2311	−0.985969	−0.492984	+	0.870038i	$0.664094\pi$
$272$	−6.95227	−0.421544
$273$	0	0
$274$	11.7309	0.708687
$275$	−14.0867	−0.849459
$276$	0	0
$277$	−17.4271	−1.04709	−0.523545	−	0.851998i	$-0.675391\pi$
$277$	−17.4271	−1.04709	−0.523545	+	0.851998i	$0.675391\pi$
$278$	−0.614537	−0.0368575
$279$	0	0
$280$	0.592892	0.0354321
$281$	−10.0480	−0.599414	−0.299707	−	0.954031i	$-0.596889\pi$
$281$	−10.0480	−0.599414	−0.299707	+	0.954031i	$0.596889\pi$
$282$	0	0
$283$	−2.11210	−0.125551	−0.0627756	−	0.998028i	$-0.519995\pi$
$283$	−2.11210	−0.125551	−0.0627756	+	0.998028i	$0.519995\pi$
$284$	0.923816	0.0548184
$285$	0	0
$286$	−14.0237	−0.829237
$287$	−5.34312	−0.315394
$288$	0	0
$289$	31.3341	1.84318
$290$	−4.93855	−0.290002
$291$	0	0
$292$	2.46929	0.144505
$293$	17.4877	1.02164	0.510820	−	0.859687i	$-0.329342\pi$
$293$	17.4877	1.02164	0.510820	+	0.859687i	$0.329342\pi$
$294$	0	0
$295$	8.25522	0.480638
$296$	−7.04259	−0.409342
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−6.23900	−0.360810
$300$	0	0
$301$	−3.46508	−0.199724
$302$	17.5323	1.00887
$303$	0	0
$304$	−2.33695	−0.134033
$305$	13.1564	0.753333
$306$	0	0
$307$	14.4985	0.827474	0.413737	−	0.910396i	$-0.364223\pi$
$307$	14.4985	0.827474	0.413737	+	0.910396i	$0.364223\pi$
$308$	1.96569	0.112005
$309$	0	0
$310$	−3.85422	−0.218905
$311$	27.0815	1.53565	0.767826	−	0.640659i	$-0.221339\pi$
$311$	27.0815	1.53565	0.767826	+	0.640659i	$0.221339\pi$
$312$	0	0
$313$	20.4400	1.15534	0.577668	−	0.816272i	$-0.303963\pi$
$313$	20.4400	1.15534	0.577668	+	0.816272i	$0.303963\pi$
$314$	22.2770	1.25716
$315$	0	0
$316$	−7.80783	−0.439225
$317$	10.5261	0.591206	0.295603	−	0.955311i	$-0.404479\pi$
$317$	10.5261	0.591206	0.295603	+	0.955311i	$0.404479\pi$
$318$	0	0
$319$	−16.3734	−0.916735
$320$	1.17050	0.0654329
$321$	0	0
$322$	0.874515	0.0487348
$323$	16.2471	0.904013
$324$	0	0
$325$	13.1175	0.727626
$326$	13.4831	0.746758
$327$	0	0
$328$	−10.5485	−0.582444
$329$	2.11568	0.116641
$330$	0	0
$331$	−10.1909	−0.560140	−0.280070	−	0.959980i	$-0.590358\pi$
$331$	−10.1909	−0.560140	−0.280070	+	0.959980i	$0.590358\pi$
$332$	8.00835	0.439515
$333$	0	0
$334$	−12.6208	−0.690578
$335$	−5.86410	−0.320390
$336$	0	0
$337$	−14.6134	−0.796042	−0.398021	−	0.917376i	$-0.630303\pi$
$337$	−14.6134	−0.796042	−0.398021	+	0.917376i	$0.630303\pi$
$338$	0.0587896	0.00319773
$339$	0	0
$340$	−8.13763	−0.441325
$341$	−12.7784	−0.691987
$342$	0	0
$343$	−6.96144	−0.375882
$344$	−6.84084	−0.368833
$345$	0	0
$346$	−13.5303	−0.727395
$347$	−18.1962	−0.976822	−0.488411	−	0.872614i	$-0.662423\pi$
$347$	−18.1962	−0.976822	−0.488411	+	0.872614i	$0.662423\pi$
$348$	0	0
$349$	18.9456	1.01413	0.507067	−	0.861907i	$-0.330730\pi$
$349$	18.9456	1.01413	0.507067	+	0.861907i	$0.330730\pi$
$350$	−1.83866	−0.0982807
$351$	0	0
$352$	3.88070	0.206842
$353$	−36.1335	−1.92319	−0.961596	−	0.274468i	$-0.911498\pi$
$353$	−36.1335	−1.92319	−0.961596	+	0.274468i	$0.911498\pi$
$354$	0	0
$355$	1.08133	0.0573908
$356$	−15.6581	−0.829877
$357$	0	0
$358$	7.23353	0.382304
$359$	−13.2222	−0.697842	−0.348921	−	0.937152i	$-0.613452\pi$
$359$	−13.2222	−0.697842	−0.348921	+	0.937152i	$0.613452\pi$
$360$	0	0
$361$	−13.5387	−0.712562
$362$	3.30723	0.173824
$363$	0	0
$364$	−1.83044	−0.0959411
$365$	2.89031	0.151286
$366$	0	0
$367$	−2.09001	−0.109098	−0.0545488	−	0.998511i	$-0.517372\pi$
$367$	−2.09001	−0.109098	−0.0545488	+	0.998511i	$0.517372\pi$
$368$	1.72649	0.0899993
$369$	0	0
$370$	−8.24335	−0.428551
$371$	−6.52614	−0.338820
$372$	0	0
$373$	−15.6737	−0.811552	−0.405776	−	0.913973i	$-0.632999\pi$
$373$	−15.6737	−0.811552	−0.405776	+	0.913973i	$0.632999\pi$
$374$	−26.9797	−1.39509
$375$	0	0
$376$	4.17683	0.215404
$377$	15.2468	0.785253
$378$	0	0
$379$	29.1521	1.49744	0.748722	−	0.662884i	$-0.230668\pi$
$379$	29.1521	1.49744	0.748722	+	0.662884i	$0.230668\pi$
$380$	−2.73540	−0.140323
$381$	0	0
$382$	−12.7158	−0.650597
$383$	13.1248	0.670648	0.335324	−	0.942103i	$-0.391154\pi$
$383$	13.1248	0.670648	0.335324	+	0.942103i	$0.391154\pi$
$384$	0	0
$385$	2.30084	0.117261
$386$	−3.36901	−0.171478
$387$	0	0
$388$	1.02168	0.0518681
$389$	−32.1642	−1.63079	−0.815394	−	0.578907i	$-0.803479\pi$
$389$	−32.1642	−1.63079	−0.815394	+	0.578907i	$0.803479\pi$
$390$	0	0
$391$	−12.0030	−0.607018
$392$	−6.74343	−0.340595
$393$	0	0
$394$	0.815685	0.0410936
$395$	−9.13906	−0.459836
$396$	0	0
$397$	33.8252	1.69764	0.848818	−	0.528685i	$-0.177315\pi$
$397$	33.8252	1.69764	0.848818	+	0.528685i	$0.177315\pi$
$398$	5.35685	0.268514
$399$	0	0
$400$	−3.62993	−0.181497
$401$	9.59093	0.478948	0.239474	−	0.970903i	$-0.423025\pi$
$401$	9.59093	0.478948	0.239474	+	0.970903i	$0.423025\pi$
$402$	0	0
$403$	11.8992	0.592739
$404$	4.32214	0.215035
$405$	0	0
$406$	−2.13714	−0.106064
$407$	−27.3302	−1.35471
$408$	0	0
$409$	−6.89619	−0.340995	−0.170497	−	0.985358i	$-0.554537\pi$
$409$	−6.89619	−0.340995	−0.170497	+	0.985358i	$0.554537\pi$
$410$	−12.3470	−0.609776
$411$	0	0
$412$	6.94103	0.341960
$413$	3.57241	0.175787
$414$	0	0
$415$	9.37377	0.460140
$416$	−3.61369	−0.177176
$417$	0	0
$418$	−9.06900	−0.443579
$419$	−3.01240	−0.147165	−0.0735827	−	0.997289i	$-0.523443\pi$
$419$	−3.01240	−0.147165	−0.0735827	+	0.997289i	$0.523443\pi$
$420$	0	0
$421$	−38.9520	−1.89840	−0.949201	−	0.314670i	$-0.898106\pi$
$421$	−38.9520	−1.89840	−0.949201	+	0.314670i	$0.898106\pi$
$422$	13.3550	0.650113
$423$	0	0
$424$	−12.8841	−0.625705
$425$	25.2363	1.22414
$426$	0	0
$427$	5.69337	0.275522
$428$	14.3087	0.691636
$429$	0	0
$430$	−8.00720	−0.386142
$431$	5.42792	0.261454	0.130727	−	0.991418i	$-0.458269\pi$
$431$	5.42792	0.261454	0.130727	+	0.991418i	$0.458269\pi$
$432$	0	0
$433$	22.8723	1.09917	0.549587	−	0.835436i	$-0.314785\pi$
$433$	22.8723	1.09917	0.549587	+	0.835436i	$0.314785\pi$
$434$	−1.66790	−0.0800615
$435$	0	0
$436$	−7.75669	−0.371478
$437$	−4.03471	−0.193006
$438$	0	0
$439$	−7.90666	−0.377364	−0.188682	−	0.982038i	$-0.560422\pi$
$439$	−7.90666	−0.377364	−0.188682	+	0.982038i	$0.560422\pi$
$440$	4.54236	0.216549
$441$	0	0
$442$	25.1234	1.19500
$443$	10.0631	0.478110	0.239055	−	0.971006i	$-0.423162\pi$
$443$	10.0631	0.478110	0.239055	+	0.971006i	$0.423162\pi$
$444$	0	0
$445$	−18.3278	−0.868821
$446$	18.7279	0.886792
$447$	0	0
$448$	0.506529	0.0239312
$449$	−13.1764	−0.621831	−0.310915	−	0.950438i	$-0.600636\pi$
$449$	−13.1764	−0.621831	−0.310915	+	0.950438i	$0.600636\pi$
$450$	0	0
$451$	−40.9356	−1.92758
$452$	1.71654	0.0807393
$453$	0	0
$454$	−8.54460	−0.401018
$455$	−2.14253	−0.100443
$456$	0	0
$457$	15.0444	0.703749	0.351874	−	0.936047i	$-0.385544\pi$
$457$	15.0444	0.703749	0.351874	+	0.936047i	$0.385544\pi$
$458$	−5.34208	−0.249619
$459$	0	0
$460$	2.02085	0.0942227
$461$	0.361142	0.0168201	0.00841003	−	0.999965i	$-0.497323\pi$
$461$	0.361142	0.0168201	0.00841003	+	0.999965i	$0.497323\pi$
$462$	0	0
$463$	3.07957	0.143120	0.0715599	−	0.997436i	$-0.477202\pi$
$463$	3.07957	0.143120	0.0715599	+	0.997436i	$0.477202\pi$
$464$	−4.21919	−0.195871
$465$	0	0
$466$	−14.8123	−0.686167
$467$	−25.3161	−1.17149	−0.585746	−	0.810495i	$-0.699198\pi$
$467$	−25.3161	−1.17149	−0.585746	+	0.810495i	$0.699198\pi$
$468$	0	0
$469$	−2.53766	−0.117178
$470$	4.88898	0.225512
$471$	0	0
$472$	7.05273	0.324628
$473$	−26.5473	−1.22064
$474$	0	0
$475$	8.48296	0.389225
$476$	−3.52153	−0.161409
$477$	0	0
$478$	−6.91374	−0.316227
$479$	−11.9963	−0.548123	−0.274062	−	0.961712i	$-0.588367\pi$
$479$	−11.9963	−0.548123	−0.274062	+	0.961712i	$0.588367\pi$
$480$	0	0
$481$	25.4498	1.16041
$482$	8.56369	0.390065
$483$	0	0
$484$	4.05985	0.184539
$485$	1.19588	0.0543021
$486$	0	0
$487$	−15.3061	−0.693585	−0.346792	−	0.937942i	$-0.612729\pi$
$487$	−15.3061	−0.693585	−0.346792	+	0.937942i	$0.612729\pi$
$488$	11.2400	0.508810
$489$	0	0
$490$	−7.89318	−0.356578
$491$	−18.7839	−0.847706	−0.423853	−	0.905731i	$-0.639323\pi$
$491$	−18.7839	−0.847706	−0.423853	+	0.905731i	$0.639323\pi$
$492$	0	0
$493$	29.3329	1.32109
$494$	8.44501	0.379959
$495$	0	0
$496$	−3.29279	−0.147851
$497$	0.467939	0.0209899
$498$	0	0
$499$	−2.22731	−0.0997079	−0.0498540	−	0.998757i	$-0.515876\pi$
$499$	−2.22731	−0.0997079	−0.0498540	+	0.998757i	$0.515876\pi$
$500$	−10.1013	−0.451745
$501$	0	0
$502$	−16.1199	−0.719468
$503$	−10.4225	−0.464716	−0.232358	−	0.972630i	$-0.574644\pi$
$503$	−10.4225	−0.464716	−0.232358	+	0.972630i	$0.574644\pi$
$504$	0	0
$505$	5.05907	0.225126
$506$	6.69998	0.297850
$507$	0	0
$508$	3.86129	0.171317
$509$	−25.1510	−1.11480	−0.557400	−	0.830244i	$-0.688201\pi$
$509$	−25.1510	−1.11480	−0.557400	+	0.830244i	$0.688201\pi$
$510$	0	0
$511$	1.25077	0.0553307
$512$	1.00000	0.0441942
$513$	0	0
$514$	−2.39190	−0.105502
$515$	8.12447	0.358007
$516$	0	0
$517$	16.2090	0.712873
$518$	−3.56727	−0.156737
$519$	0	0
$520$	−4.22983	−0.185490
$521$	−30.0875	−1.31816	−0.659078	−	0.752074i	$-0.729053\pi$
$521$	−30.0875	−1.31816	−0.659078	+	0.752074i	$0.729053\pi$
$522$	0	0
$523$	−23.8359	−1.04227	−0.521137	−	0.853473i	$-0.674492\pi$
$523$	−23.8359	−1.04227	−0.521137	+	0.853473i	$0.674492\pi$
$524$	7.45036	0.325470
$525$	0	0
$526$	28.7218	1.25233
$527$	22.8924	0.997209
$528$	0	0
$529$	−20.0192	−0.870402
$530$	−15.0808	−0.655067
$531$	0	0
$532$	−1.18373	−0.0513212
$533$	38.1191	1.65112
$534$	0	0
$535$	16.7483	0.724093
$536$	−5.00991	−0.216395
$537$	0	0
$538$	10.6420	0.458808
$539$	−26.1692	−1.12719
$540$	0	0
$541$	−16.0897	−0.691750	−0.345875	−	0.938281i	$-0.612418\pi$
$541$	−16.0897	−0.691750	−0.345875	+	0.938281i	$0.612418\pi$
$542$	−16.2311	−0.697185
$543$	0	0
$544$	−6.95227	−0.298076
$545$	−9.07921	−0.388910
$546$	0	0
$547$	−1.17274	−0.0501429	−0.0250715	−	0.999686i	$-0.507981\pi$
$547$	−1.17274	−0.0501429	−0.0250715	+	0.999686i	$0.507981\pi$
$548$	11.7309	0.501117
$549$	0	0
$550$	−14.0867	−0.600658
$551$	9.86001	0.420051
$552$	0	0
$553$	−3.95489	−0.168179
$554$	−17.4271	−0.740404
$555$	0	0
$556$	−0.614537	−0.0260622
$557$	32.1956	1.36417	0.682086	−	0.731272i	$-0.261073\pi$
$557$	32.1956	1.36417	0.682086	+	0.731272i	$0.261073\pi$
$558$	0	0
$559$	24.7207	1.04557
$560$	0.592892	0.0250542
$561$	0	0
$562$	−10.0480	−0.423850
$563$	−39.9085	−1.68194	−0.840970	−	0.541081i	$-0.818015\pi$
$563$	−39.9085	−1.68194	−0.840970	+	0.541081i	$0.818015\pi$
$564$	0	0
$565$	2.00921	0.0845282
$566$	−2.11210	−0.0887780
$567$	0	0
$568$	0.923816	0.0387624
$569$	−45.1626	−1.89331	−0.946657	−	0.322244i	$-0.895563\pi$
$569$	−45.1626	−1.89331	−0.946657	+	0.322244i	$0.895563\pi$
$570$	0	0
$571$	8.79540	0.368076	0.184038	−	0.982919i	$-0.441083\pi$
$571$	8.79540	0.368076	0.184038	+	0.982919i	$0.441083\pi$
$572$	−14.0237	−0.586359
$573$	0	0
$574$	−5.34312	−0.223018
$575$	−6.26703	−0.261353
$576$	0	0
$577$	20.4382	0.850855	0.425427	−	0.904993i	$-0.360124\pi$
$577$	20.4382	0.850855	0.425427	+	0.904993i	$0.360124\pi$
$578$	31.3341	1.30333
$579$	0	0
$580$	−4.93855	−0.205062
$581$	4.05646	0.168290
$582$	0	0
$583$	−49.9992	−2.07075
$584$	2.46929	0.102180
$585$	0	0
$586$	17.4877	0.722409
$587$	40.4128	1.66801	0.834007	−	0.551753i	$-0.186041\pi$
$587$	40.4128	1.66801	0.834007	+	0.551753i	$0.186041\pi$
$588$	0	0
$589$	7.69509	0.317071
$590$	8.25522	0.339862
$591$	0	0
$592$	−7.04259	−0.289449
$593$	−17.1642	−0.704850	−0.352425	−	0.935840i	$-0.614643\pi$
$593$	−17.1642	−0.704850	−0.352425	+	0.935840i	$0.614643\pi$
$594$	0	0
$595$	−4.12195	−0.168983
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−6.23900	−0.255131
$599$	−11.0793	−0.452689	−0.226345	−	0.974047i	$-0.572678\pi$
$599$	−11.0793	−0.452689	−0.226345	+	0.974047i	$0.572678\pi$
$600$	0	0
$601$	−11.3103	−0.461358	−0.230679	−	0.973030i	$-0.574095\pi$
$601$	−11.3103	−0.461358	−0.230679	+	0.973030i	$0.574095\pi$
$602$	−3.46508	−0.141226
$603$	0	0
$604$	17.5323	0.713378
$605$	4.75206	0.193199
$606$	0	0
$607$	−10.9521	−0.444533	−0.222267	−	0.974986i	$-0.571346\pi$
$607$	−10.9521	−0.444533	−0.222267	+	0.974986i	$0.571346\pi$
$608$	−2.33695	−0.0947757
$609$	0	0
$610$	13.1564	0.532687
$611$	−15.0938	−0.610629
$612$	0	0
$613$	6.68464	0.269990	0.134995	−	0.990846i	$-0.456898\pi$
$613$	6.68464	0.269990	0.134995	+	0.990846i	$0.456898\pi$
$614$	14.4985	0.585112
$615$	0	0
$616$	1.96569	0.0791998
$617$	−21.3178	−0.858222	−0.429111	−	0.903252i	$-0.641173\pi$
$617$	−21.3178	−0.858222	−0.429111	+	0.903252i	$0.641173\pi$
$618$	0	0
$619$	−27.6883	−1.11289	−0.556443	−	0.830885i	$-0.687834\pi$
$619$	−27.6883	−1.11289	−0.556443	+	0.830885i	$0.687834\pi$
$620$	−3.85422	−0.154789
$621$	0	0
$622$	27.0815	1.08587
$623$	−7.93127	−0.317760
$624$	0	0
$625$	6.32605	0.253042
$626$	20.4400	0.816946
$627$	0	0
$628$	22.2770	0.888950
$629$	48.9620	1.95224
$630$	0	0
$631$	37.1206	1.47775	0.738874	−	0.673843i	$-0.235358\pi$
$631$	37.1206	1.47775	0.738874	+	0.673843i	$0.235358\pi$
$632$	−7.80783	−0.310579
$633$	0	0
$634$	10.5261	0.418046
$635$	4.51964	0.179356
$636$	0	0
$637$	24.3687	0.965523
$638$	−16.3734	−0.648229
$639$	0	0
$640$	1.17050	0.0462681
$641$	8.02175	0.316840	0.158420	−	0.987372i	$-0.449360\pi$
$641$	8.02175	0.316840	0.158420	+	0.987372i	$0.449360\pi$
$642$	0	0
$643$	1.00334	0.0395677	0.0197839	−	0.999804i	$-0.493702\pi$
$643$	1.00334	0.0395677	0.0197839	+	0.999804i	$0.493702\pi$
$644$	0.874515	0.0344607
$645$	0	0
$646$	16.2471	0.639234
$647$	−27.4169	−1.07787	−0.538935	−	0.842347i	$-0.681173\pi$
$647$	−27.4169	−1.07787	−0.538935	+	0.842347i	$0.681173\pi$
$648$	0	0
$649$	27.3696	1.07435
$650$	13.1175	0.514509
$651$	0	0
$652$	13.4831	0.528038
$653$	−39.3088	−1.53827	−0.769137	−	0.639084i	$-0.779314\pi$
$653$	−39.3088	−1.53827	−0.769137	+	0.639084i	$0.779314\pi$
$654$	0	0
$655$	8.72064	0.340744
$656$	−10.5485	−0.411850
$657$	0	0
$658$	2.11568	0.0824780
$659$	−36.5849	−1.42515	−0.712573	−	0.701598i	$-0.752470\pi$
$659$	−36.5849	−1.42515	−0.712573	+	0.701598i	$0.752470\pi$
$660$	0	0
$661$	−13.5700	−0.527810	−0.263905	−	0.964549i	$-0.585011\pi$
$661$	−13.5700	−0.527810	−0.263905	+	0.964549i	$0.585011\pi$
$662$	−10.1909	−0.396079
$663$	0	0
$664$	8.00835	0.310784
$665$	−1.38556	−0.0537296
$666$	0	0
$667$	−7.28437	−0.282052
$668$	−12.6208	−0.488313
$669$	0	0
$670$	−5.86410	−0.226550
$671$	43.6190	1.68389
$672$	0	0
$673$	−1.54169	−0.0594279	−0.0297139	−	0.999558i	$-0.509460\pi$
$673$	−1.54169	−0.0594279	−0.0297139	+	0.999558i	$0.509460\pi$
$674$	−14.6134	−0.562886
$675$	0	0
$676$	0.0587896	0.00226114
$677$	2.97532	0.114351	0.0571755	−	0.998364i	$-0.481791\pi$
$677$	2.97532	0.114351	0.0571755	+	0.998364i	$0.481791\pi$
$678$	0	0
$679$	0.517512	0.0198603
$680$	−8.13763	−0.312064
$681$	0	0
$682$	−12.7784	−0.489309
$683$	−6.60805	−0.252850	−0.126425	−	0.991976i	$-0.540350\pi$
$683$	−6.60805	−0.252850	−0.126425	+	0.991976i	$0.540350\pi$
$684$	0	0
$685$	13.7310	0.524633
$686$	−6.96144	−0.265789
$687$	0	0
$688$	−6.84084	−0.260805
$689$	46.5590	1.77376
$690$	0	0
$691$	−16.9869	−0.646212	−0.323106	−	0.946363i	$-0.604727\pi$
$691$	−16.9869	−0.646212	−0.323106	+	0.946363i	$0.604727\pi$
$692$	−13.5303	−0.514346
$693$	0	0
$694$	−18.1962	−0.690717
$695$	−0.719316	−0.0272852
$696$	0	0
$697$	73.3361	2.77780
$698$	18.9456	0.717101
$699$	0	0
$700$	−1.83866	−0.0694950
$701$	−16.5146	−0.623747	−0.311873	−	0.950124i	$-0.600956\pi$
$701$	−16.5146	−0.623747	−0.311873	+	0.950124i	$0.600956\pi$
$702$	0	0
$703$	16.4582	0.620731
$704$	3.88070	0.146259
$705$	0	0
$706$	−36.1335	−1.35990
$707$	2.18929	0.0823367
$708$	0	0
$709$	−1.25383	−0.0470885	−0.0235442	−	0.999723i	$-0.507495\pi$
$709$	−1.25383	−0.0470885	−0.0235442	+	0.999723i	$0.507495\pi$
$710$	1.08133	0.0405814
$711$	0	0
$712$	−15.6581	−0.586812
$713$	−5.68497	−0.212904
$714$	0	0
$715$	−16.4147	−0.613875
$716$	7.23353	0.270330
$717$	0	0
$718$	−13.2222	−0.493449
$719$	−50.6632	−1.88942	−0.944709	−	0.327909i	$-0.893656\pi$
$719$	−50.6632	−1.88942	−0.944709	+	0.327909i	$0.893656\pi$
$720$	0	0
$721$	3.51583	0.130936
$722$	−13.5387	−0.503857
$723$	0	0
$724$	3.30723	0.122912
$725$	15.3154	0.568798
$726$	0	0
$727$	20.3402	0.754378	0.377189	−	0.926136i	$-0.376891\pi$
$727$	20.3402	0.754378	0.377189	+	0.926136i	$0.376891\pi$
$728$	−1.83044	−0.0678406
$729$	0	0
$730$	2.89031	0.106975
$731$	47.5594	1.75905
$732$	0	0
$733$	17.1935	0.635058	0.317529	−	0.948249i	$-0.397147\pi$
$733$	17.1935	0.635058	0.317529	+	0.948249i	$0.397147\pi$
$734$	−2.09001	−0.0771437
$735$	0	0
$736$	1.72649	0.0636391
$737$	−19.4420	−0.716154
$738$	0	0
$739$	40.2873	1.48199	0.740996	−	0.671509i	$-0.234354\pi$
$739$	40.2873	1.48199	0.740996	+	0.671509i	$0.234354\pi$
$740$	−8.24335	−0.303032
$741$	0	0
$742$	−6.52614	−0.239582
$743$	13.2514	0.486145	0.243073	−	0.970008i	$-0.421845\pi$
$743$	13.2514	0.486145	0.243073	+	0.970008i	$0.421845\pi$
$744$	0	0
$745$	−1.17050	−0.0428838
$746$	−15.6737	−0.573854
$747$	0	0
$748$	−26.9797	−0.986476
$749$	7.24776	0.264827
$750$	0	0
$751$	38.9147	1.42002	0.710010	−	0.704192i	$-0.248691\pi$
$751$	38.9147	1.42002	0.710010	+	0.704192i	$0.248691\pi$
$752$	4.17683	0.152313
$753$	0	0
$754$	15.2468	0.555257
$755$	20.5215	0.746855
$756$	0	0
$757$	5.51490	0.200442	0.100221	−	0.994965i	$-0.468045\pi$
$757$	5.51490	0.200442	0.100221	+	0.994965i	$0.468045\pi$
$758$	29.1521	1.05885
$759$	0	0
$760$	−2.73540	−0.0992232
$761$	40.3355	1.46216	0.731081	−	0.682291i	$-0.239016\pi$
$761$	40.3355	1.46216	0.731081	+	0.682291i	$0.239016\pi$
$762$	0	0
$763$	−3.92899	−0.142239
$764$	−12.7158	−0.460042
$765$	0	0
$766$	13.1248	0.474219
$767$	−25.4864	−0.920262
$768$	0	0
$769$	15.9355	0.574648	0.287324	−	0.957833i	$-0.407234\pi$
$769$	15.9355	0.574648	0.287324	+	0.957833i	$0.407234\pi$
$770$	2.30084	0.0829164
$771$	0	0
$772$	−3.36901	−0.121253
$773$	26.9572	0.969584	0.484792	−	0.874629i	$-0.338895\pi$
$773$	26.9572	0.969584	0.484792	+	0.874629i	$0.338895\pi$
$774$	0	0
$775$	11.9526	0.429351
$776$	1.02168	0.0366763
$777$	0	0
$778$	−32.1642	−1.15314
$779$	24.6513	0.883225
$780$	0	0
$781$	3.58505	0.128283
$782$	−12.0030	−0.429227
$783$	0	0
$784$	−6.74343	−0.240837
$785$	26.0752	0.930665
$786$	0	0
$787$	51.2529	1.82697	0.913484	−	0.406875i	$-0.133381\pi$
$787$	51.2529	1.82697	0.913484	+	0.406875i	$0.133381\pi$
$788$	0.815685	0.0290576
$789$	0	0
$790$	−9.13906	−0.325153
$791$	0.869478	0.0309151
$792$	0	0
$793$	−40.6179	−1.44238
$794$	33.8252	1.20041
$795$	0	0
$796$	5.35685	0.189868
$797$	10.0808	0.357082	0.178541	−	0.983933i	$-0.442862\pi$
$797$	10.0808	0.357082	0.178541	+	0.983933i	$0.442862\pi$
$798$	0	0
$799$	−29.0385	−1.02731
$800$	−3.62993	−0.128337
$801$	0	0
$802$	9.59093	0.338667
$803$	9.58260	0.338163
$804$	0	0
$805$	1.02362	0.0360778
$806$	11.8992	0.419130
$807$	0	0
$808$	4.32214	0.152052
$809$	8.55995	0.300952	0.150476	−	0.988614i	$-0.451919\pi$
$809$	8.55995	0.300952	0.150476	+	0.988614i	$0.451919\pi$
$810$	0	0
$811$	−7.01809	−0.246439	−0.123219	−	0.992379i	$-0.539322\pi$
$811$	−7.01809	−0.246439	−0.123219	+	0.992379i	$0.539322\pi$
$812$	−2.13714	−0.0749989
$813$	0	0
$814$	−27.3302	−0.957923
$815$	15.7819	0.552817
$816$	0	0
$817$	15.9867	0.559303
$818$	−6.89619	−0.241120
$819$	0	0
$820$	−12.3470	−0.431177
$821$	16.3421	0.570344	0.285172	−	0.958476i	$-0.407949\pi$
$821$	16.3421	0.570344	0.285172	+	0.958476i	$0.407949\pi$
$822$	0	0
$823$	30.5614	1.06530	0.532652	−	0.846335i	$-0.321196\pi$
$823$	30.5614	1.06530	0.532652	+	0.846335i	$0.321196\pi$
$824$	6.94103	0.241802
$825$	0	0
$826$	3.57241	0.124300
$827$	3.44532	0.119805	0.0599027	−	0.998204i	$-0.480921\pi$
$827$	3.44532	0.119805	0.0599027	+	0.998204i	$0.480921\pi$
$828$	0	0
$829$	−12.4705	−0.433120	−0.216560	−	0.976269i	$-0.569484\pi$
$829$	−12.4705	−0.433120	−0.216560	+	0.976269i	$0.569484\pi$
$830$	9.37377	0.325368
$831$	0	0
$832$	−3.61369	−0.125282
$833$	46.8822	1.62437
$834$	0	0
$835$	−14.7726	−0.511228
$836$	−9.06900	−0.313658
$837$	0	0
$838$	−3.01240	−0.104062
$839$	−30.3546	−1.04796	−0.523979	−	0.851731i	$-0.675553\pi$
$839$	−30.3546	−1.04796	−0.523979	+	0.851731i	$0.675553\pi$
$840$	0	0
$841$	−11.1985	−0.386154
$842$	−38.9520	−1.34237
$843$	0	0
$844$	13.3550	0.459699
$845$	0.0688132	0.00236724
$846$	0	0
$847$	2.05643	0.0706598
$848$	−12.8841	−0.442440
$849$	0	0
$850$	25.2363	0.865597
$851$	−12.1589	−0.416803
$852$	0	0
$853$	−41.6101	−1.42470	−0.712351	−	0.701824i	$-0.752369\pi$
$853$	−41.6101	−1.42470	−0.712351	+	0.701824i	$0.752369\pi$
$854$	5.69337	0.194823
$855$	0	0
$856$	14.3087	0.489061
$857$	−39.6564	−1.35464	−0.677318	−	0.735690i	$-0.736858\pi$
$857$	−39.6564	−1.35464	−0.677318	+	0.735690i	$0.736858\pi$
$858$	0	0
$859$	46.6299	1.59099	0.795497	−	0.605958i	$-0.207210\pi$
$859$	46.6299	1.59099	0.795497	+	0.605958i	$0.207210\pi$
$860$	−8.00720	−0.273043
$861$	0	0
$862$	5.42792	0.184876
$863$	−32.8049	−1.11669	−0.558346	−	0.829608i	$-0.688564\pi$
$863$	−32.8049	−1.11669	−0.558346	+	0.829608i	$0.688564\pi$
$864$	0	0
$865$	−15.8372	−0.538482
$866$	22.8723	0.777234
$867$	0	0
$868$	−1.66790	−0.0566121
$869$	−30.2999	−1.02785
$870$	0	0
$871$	18.1043	0.613440
$872$	−7.75669	−0.262675
$873$	0	0
$874$	−4.03471	−0.136476
$875$	−5.11661	−0.172973
$876$	0	0
$877$	−39.7704	−1.34295	−0.671475	−	0.741027i	$-0.734339\pi$
$877$	−39.7704	−1.34295	−0.671475	+	0.741027i	$0.734339\pi$
$878$	−7.90666	−0.266837
$879$	0	0
$880$	4.54236	0.153123
$881$	−27.5405	−0.927861	−0.463931	−	0.885871i	$-0.653561\pi$
$881$	−27.5405	−0.927861	−0.463931	+	0.885871i	$0.653561\pi$
$882$	0	0
$883$	47.8260	1.60947	0.804737	−	0.593632i	$-0.202306\pi$
$883$	47.8260	1.60947	0.804737	+	0.593632i	$0.202306\pi$
$884$	25.1234	0.844991
$885$	0	0
$886$	10.0631	0.338075
$887$	45.4546	1.52622	0.763108	−	0.646271i	$-0.223673\pi$
$887$	45.4546	1.52622	0.763108	+	0.646271i	$0.223673\pi$
$888$	0	0
$889$	1.95585	0.0655972
$890$	−18.3278	−0.614349
$891$	0	0
$892$	18.7279	0.627057
$893$	−9.76103	−0.326640
$894$	0	0
$895$	8.46684	0.283015
$896$	0.506529	0.0169219
$897$	0	0
$898$	−13.1764	−0.439701
$899$	13.8929	0.463355
$900$	0	0
$901$	89.5735	2.98412
$902$	−40.9356	−1.36301
$903$	0	0
$904$	1.71654	0.0570913
$905$	3.87111	0.128680
$906$	0	0
$907$	−12.7377	−0.422947	−0.211473	−	0.977384i	$-0.567826\pi$
$907$	−12.7377	−0.422947	−0.211473	+	0.977384i	$0.567826\pi$
$908$	−8.54460	−0.283562
$909$	0	0
$910$	−2.14253	−0.0710242
$911$	−39.3617	−1.30411	−0.652055	−	0.758171i	$-0.726093\pi$
$911$	−39.3617	−1.30411	−0.652055	+	0.758171i	$0.726093\pi$
$912$	0	0
$913$	31.0780	1.02853
$914$	15.0444	0.497626
$915$	0	0
$916$	−5.34208	−0.176507
$917$	3.77382	0.124623
$918$	0	0
$919$	−21.2378	−0.700570	−0.350285	−	0.936643i	$-0.613915\pi$
$919$	−21.2378	−0.700570	−0.350285	+	0.936643i	$0.613915\pi$
$920$	2.02085	0.0666255
$921$	0	0
$922$	0.361142	0.0118936
$923$	−3.33839	−0.109884
$924$	0	0
$925$	25.5641	0.840543
$926$	3.07957	0.101201
$927$	0	0
$928$	−4.21919	−0.138502
$929$	−28.3154	−0.928997	−0.464498	−	0.885574i	$-0.653765\pi$
$929$	−28.3154	−0.928997	−0.464498	+	0.885574i	$0.653765\pi$
$930$	0	0
$931$	15.7590	0.516482
$932$	−14.8123	−0.485194
$933$	0	0
$934$	−25.3161	−0.828369
$935$	−31.5797	−1.03277
$936$	0	0
$937$	40.8706	1.33519	0.667593	−	0.744527i	$-0.267325\pi$
$937$	40.8706	1.33519	0.667593	+	0.744527i	$0.267325\pi$
$938$	−2.53766	−0.0828576
$939$	0	0
$940$	4.88898	0.159461
$941$	16.4529	0.536349	0.268174	−	0.963370i	$-0.413580\pi$
$941$	16.4529	0.536349	0.268174	+	0.963370i	$0.413580\pi$
$942$	0	0
$943$	−18.2119	−0.593060
$944$	7.05273	0.229547
$945$	0	0
$946$	−26.5473	−0.863126
$947$	17.3768	0.564671	0.282335	−	0.959316i	$-0.408891\pi$
$947$	17.3768	0.564671	0.282335	+	0.959316i	$0.408891\pi$
$948$	0	0
$949$	−8.92328	−0.289662
$950$	8.48296	0.275223
$951$	0	0
$952$	−3.52153	−0.114133
$953$	47.2562	1.53078	0.765389	−	0.643568i	$-0.222547\pi$
$953$	47.2562	1.53078	0.765389	+	0.643568i	$0.222547\pi$
$954$	0	0
$955$	−14.8838	−0.481630
$956$	−6.91374	−0.223606
$957$	0	0
$958$	−11.9963	−0.387582
$959$	5.94202	0.191878
$960$	0	0
$961$	−20.1575	−0.650242
$962$	25.4498	0.820534
$963$	0	0
$964$	8.56369	0.275818
$965$	−3.94342	−0.126943
$966$	0	0
$967$	−45.1307	−1.45131	−0.725653	−	0.688061i	$-0.758462\pi$
$967$	−45.1307	−1.45131	−0.725653	+	0.688061i	$0.758462\pi$
$968$	4.05985	0.130489
$969$	0	0
$970$	1.19588	0.0383974
$971$	−7.05507	−0.226408	−0.113204	−	0.993572i	$-0.536111\pi$
$971$	−7.05507	−0.226408	−0.113204	+	0.993572i	$0.536111\pi$
$972$	0	0
$973$	−0.311281	−0.00997921
$974$	−15.3061	−0.490438
$975$	0	0
$976$	11.2400	0.359783
$977$	5.88708	0.188344	0.0941721	−	0.995556i	$-0.469980\pi$
$977$	5.88708	0.188344	0.0941721	+	0.995556i	$0.469980\pi$
$978$	0	0
$979$	−60.7644	−1.94204
$980$	−7.89318	−0.252138
$981$	0	0
$982$	−18.7839	−0.599419
$983$	21.2758	0.678593	0.339297	−	0.940679i	$-0.389811\pi$
$983$	21.2758	0.678593	0.339297	+	0.940679i	$0.389811\pi$
$984$	0	0
$985$	0.954758	0.0304211
$986$	29.3329	0.934151
$987$	0	0
$988$	8.44501	0.268672
$989$	−11.8106	−0.375556
$990$	0	0
$991$	−33.3295	−1.05875	−0.529373	−	0.848389i	$-0.677573\pi$
$991$	−33.3295	−1.05875	−0.529373	+	0.848389i	$0.677573\pi$
$992$	−3.29279	−0.104546
$993$	0	0
$994$	0.467939	0.0148421
$995$	6.27019	0.198778
$996$	0	0
$997$	−2.18494	−0.0691979	−0.0345989	−	0.999401i	$-0.511015\pi$
$997$	−2.18494	−0.0691979	−0.0345989	+	0.999401i	$0.511015\pi$
$998$	−2.22731	−0.0705042
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.l.1.4	✓	12	3.2	odd	2
8046.2.a.m.1.9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.17050 q^{5} +0.506529 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.17050 q^{5} +0.506529 q^{7} +1.00000 q^{8} +1.17050 q^{10} +3.88070 q^{11} -3.61369 q^{13} +0.506529 q^{14} +1.00000 q^{16} -6.95227 q^{17} -2.33695 q^{19} +1.17050 q^{20} +3.88070 q^{22} +1.72649 q^{23} -3.62993 q^{25} -3.61369 q^{26} +0.506529 q^{28} -4.21919 q^{29} -3.29279 q^{31} +1.00000 q^{32} -6.95227 q^{34} +0.592892 q^{35} -7.04259 q^{37} -2.33695 q^{38} +1.17050 q^{40} -10.5485 q^{41} -6.84084 q^{43} +3.88070 q^{44} +1.72649 q^{46} +4.17683 q^{47} -6.74343 q^{49} -3.62993 q^{50} -3.61369 q^{52} -12.8841 q^{53} +4.54236 q^{55} +0.506529 q^{56} -4.21919 q^{58} +7.05273 q^{59} +11.2400 q^{61} -3.29279 q^{62} +1.00000 q^{64} -4.22983 q^{65} -5.00991 q^{67} -6.95227 q^{68} +0.592892 q^{70} +0.923816 q^{71} +2.46929 q^{73} -7.04259 q^{74} -2.33695 q^{76} +1.96569 q^{77} -7.80783 q^{79} +1.17050 q^{80} -10.5485 q^{82} +8.00835 q^{83} -8.13763 q^{85} -6.84084 q^{86} +3.88070 q^{88} -15.6581 q^{89} -1.83044 q^{91} +1.72649 q^{92} +4.17683 q^{94} -2.73540 q^{95} +1.02168 q^{97} -6.74343 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} - 5 q^{5} - 6 q^{7} + 12 q^{8} - 5 q^{10} - 10 q^{11} - q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} - 10 q^{19} - 5 q^{20} - 10 q^{22} - 15 q^{23} + 7 q^{25} - q^{26} - 6 q^{28} - 33 q^{29} - 6 q^{31} + 12 q^{32} - 6 q^{34} - 16 q^{35} - 13 q^{37} - 10 q^{38} - 5 q^{40} - 20 q^{41} - 11 q^{43} - 10 q^{44} - 15 q^{46} - 15 q^{47} + 2 q^{49} + 7 q^{50} - q^{52} - 4 q^{53} - 17 q^{55} - 6 q^{56} - 33 q^{58} - 10 q^{59} - 12 q^{61} - 6 q^{62} + 12 q^{64} - 40 q^{65} - 19 q^{67} - 6 q^{68} - 16 q^{70} - 47 q^{71} - 2 q^{73} - 13 q^{74} - 10 q^{76} + 6 q^{77} - 15 q^{79} - 5 q^{80} - 20 q^{82} - 18 q^{83} - 25 q^{85} - 11 q^{86} - 10 q^{88} - 24 q^{89} - 3 q^{91} - 15 q^{92} - 15 q^{94} + 3 q^{95} - 25 q^{97} + 2 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 - 5 * q^5 - 6 * q^7 + 12 * q^8 - 5 * q^10 - 10 * q^11 - q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 - 10 * q^19 - 5 * q^20 - 10 * q^22 - 15 * q^23 + 7 * q^25 - q^26 - 6 * q^28 - 33 * q^29 - 6 * q^31 + 12 * q^32 - 6 * q^34 - 16 * q^35 - 13 * q^37 - 10 * q^38 - 5 * q^40 - 20 * q^41 - 11 * q^43 - 10 * q^44 - 15 * q^46 - 15 * q^47 + 2 * q^49 + 7 * q^50 - q^52 - 4 * q^53 - 17 * q^55 - 6 * q^56 - 33 * q^58 - 10 * q^59 - 12 * q^61 - 6 * q^62 + 12 * q^64 - 40 * q^65 - 19 * q^67 - 6 * q^68 - 16 * q^70 - 47 * q^71 - 2 * q^73 - 13 * q^74 - 10 * q^76 + 6 * q^77 - 15 * q^79 - 5 * q^80 - 20 * q^82 - 18 * q^83 - 25 * q^85 - 11 * q^86 - 10 * q^88 - 24 * q^89 - 3 * q^91 - 15 * q^92 - 15 * q^94 + 3 * q^95 - 25 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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