LMFDB - Embedded newform 8046.2.a.l.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} - 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			1.4
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	$-0.584294\pi$
$5$	−1.17050	−0.523463	−0.261732	+	0.965141i	$0.584294\pi$
$6$	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	$-0.698920\pi$
$11$	−3.88070	−1.17008	−0.585038	+	0.811006i	$0.698920\pi$
$12$	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	$-0.180736\pi$
$17$	6.95227	1.68617	0.843087	+	0.537777i	$0.180736\pi$
$18$	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	$-0.557609\pi$
$23$	−1.72649	−0.359997	−0.179999	+	0.983667i	$0.557609\pi$
$24$	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	$-0.371873\pi$
$29$	4.21919	0.783483	0.391742	+	0.920075i	$0.371873\pi$
$30$	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	$-0.191906\pi$
$41$	10.5485	1.64740	0.823700	+	0.567026i	$0.191906\pi$
$42$	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	$-0.598532\pi$
$47$	−4.17683	−0.609253	−0.304627	+	0.952472i	$0.598532\pi$
$48$	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	$-0.154240\pi$
$53$	12.8841	1.76976	0.884880	+	0.465819i	$0.154240\pi$
$54$	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	$-0.651826\pi$
$59$	−7.05273	−0.918188	−0.459094	+	0.888388i	$0.651826\pi$
$60$	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	$-0.517458\pi$
$71$	−0.923816	−0.109637	−0.0548184	+	0.998496i	$0.517458\pi$
$72$	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	$-0.644850\pi$
$83$	−8.00835	−0.879030	−0.439515	+	0.898235i	$0.644850\pi$
$84$	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	$-0.188410\pi$
$89$	15.6581	1.65975	0.829877	+	0.557946i	$0.188410\pi$
$90$	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	$-0.568986\pi$
$101$	−4.32214	−0.430069	−0.215035	+	0.976606i	$0.568986\pi$
$102$	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	$-0.743110\pi$
$107$	−14.3087	−1.38327	−0.691636	+	0.722246i	$0.743110\pi$
$108$	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	$-0.525728\pi$
$113$	−1.71654	−0.161479	−0.0807393	+	0.996735i	$0.525728\pi$
$114$	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	$-0.605523\pi$
$131$	−7.45036	−0.650941	−0.325470	+	0.945552i	$0.605523\pi$
$132$	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	$-0.667077\pi$
$137$	−11.7309	−1.00223	−0.501117	+	0.865379i	$0.667077\pi$
$138$	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	$-0.337613\pi$
$167$	12.6208	0.976625	0.488313	+	0.872669i	$0.337613\pi$
$168$	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	$-0.328035\pi$
$173$	13.5303	1.02869	0.514346	+	0.857583i	$0.328035\pi$
$174$	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	$-0.587133\pi$
$179$	−7.23353	−0.540659	−0.270330	+	0.962768i	$0.587133\pi$
$180$	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	$-0.347835\pi$
$191$	12.7158	0.920083	0.460042	+	0.887897i	$0.347835\pi$
$192$	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	$-0.509251\pi$
$197$	−0.815685	−0.0581151	−0.0290576	+	0.999578i	$0.509251\pi$
$198$	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	$-0.408484\pi$
$227$	8.54460	0.567125	0.283562	+	0.958954i	$0.408484\pi$
$228$	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	$-0.338749\pi$
$233$	14.8123	0.970387	0.485194	+	0.874407i	$0.338749\pi$
$234$	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	$-0.428217\pi$
$239$	6.91374	0.447212	0.223606	+	0.974680i	$0.428217\pi$
$240$	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	$-0.330111\pi$
$251$	16.1199	1.01748	0.508741	+	0.860920i	$0.330111\pi$
$252$	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	$-0.476232\pi$
$257$	2.39190	0.149203	0.0746013	+	0.997213i	$0.476232\pi$
$258$	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	$-0.846206\pi$
$263$	−28.7218	−1.77106	−0.885532	+	0.464579i	$0.846206\pi$
$264$	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	$-0.605171\pi$
$269$	−10.6420	−0.648853	−0.324427	+	0.945911i	$0.605171\pi$
$270$	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	$-0.403111\pi$
$281$	10.0480	0.599414	0.299707	+	0.954031i	$0.403111\pi$
$282$	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	$-0.670658\pi$
$293$	−17.4877	−1.02164	−0.510820	+	0.859687i	$0.670658\pi$
$294$	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	$-0.778661\pi$
$311$	−27.0815	−1.53565	−0.767826	+	0.640659i	$0.778661\pi$
$312$	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	$-0.595521\pi$
$317$	−10.5261	−0.591206	−0.295603	+	0.955311i	$0.595521\pi$
$318$	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	$-0.337577\pi$
$347$	18.1962	0.976822	0.488411	+	0.872614i	$0.337577\pi$
$348$	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	$-0.0885018\pi$
$353$	36.1335	1.92319	0.961596	+	0.274468i	$0.0885018\pi$
$354$	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	$-0.386548\pi$
$359$	13.2222	0.697842	0.348921	+	0.937152i	$0.386548\pi$
$360$	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	$-0.608846\pi$
$383$	−13.1248	−0.670648	−0.335324	+	0.942103i	$0.608846\pi$
$384$	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	$-0.196521\pi$
$389$	32.1642	1.63079	0.815394	+	0.578907i	$0.196521\pi$
$390$	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	$-0.576975\pi$
$401$	−9.59093	−0.478948	−0.239474	+	0.970903i	$0.576975\pi$
$402$	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	$-0.476557\pi$
$419$	3.01240	0.147165	0.0735827	+	0.997289i	$0.476557\pi$
$420$	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	$-0.541731\pi$
$431$	−5.42792	−0.261454	−0.130727	+	0.991418i	$0.541731\pi$
$432$	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	$-0.576838\pi$
$443$	−10.0631	−0.478110	−0.239055	+	0.971006i	$0.576838\pi$
$444$	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	$-0.399364\pi$
$449$	13.1764	0.621831	0.310915	+	0.950438i	$0.399364\pi$
$450$	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	$-0.502677\pi$
$461$	−0.361142	−0.0168201	−0.00841003	+	0.999965i	$0.502677\pi$
$462$	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	$-0.300802\pi$
$467$	25.3161	1.17149	0.585746	+	0.810495i	$0.300802\pi$
$468$	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	$-0.411633\pi$
$479$	11.9963	0.548123	0.274062	+	0.961712i	$0.411633\pi$
$480$	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	$-0.360677\pi$
$491$	18.7839	0.847706	0.423853	+	0.905731i	$0.360677\pi$
$492$	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	$-0.425356\pi$
$503$	10.4225	0.464716	0.232358	+	0.972630i	$0.425356\pi$
$504$	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	$-0.311799\pi$
$509$	25.1510	1.11480	0.557400	+	0.830244i	$0.311799\pi$
$510$	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	$-0.270947\pi$
$521$	30.0875	1.31816	0.659078	+	0.752074i	$0.270947\pi$
$522$	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	$-0.738927\pi$
$557$	−32.1956	−1.36417	−0.682086	+	0.731272i	$0.738927\pi$
$558$	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	$-0.181985\pi$
$563$	39.9085	1.68194	0.840970	+	0.541081i	$0.181985\pi$
$564$	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	$-0.104437\pi$
$569$	45.1626	1.89331	0.946657	+	0.322244i	$0.104437\pi$
$570$	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	$-0.813959\pi$
$587$	−40.4128	−1.66801	−0.834007	+	0.551753i	$0.813959\pi$
$588$	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	$-0.385357\pi$
$593$	17.1642	0.704850	0.352425	+	0.935840i	$0.385357\pi$
$594$	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	$-0.427322\pi$
$599$	11.0793	0.452689	0.226345	+	0.974047i	$0.427322\pi$
$600$	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	$-0.358827\pi$
$617$	21.3178	0.858222	0.429111	+	0.903252i	$0.358827\pi$
$618$	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	$-0.550640\pi$
$641$	−8.02175	−0.316840	−0.158420	+	0.987372i	$0.550640\pi$
$642$	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	$-0.318827\pi$
$647$	27.4169	1.07787	0.538935	+	0.842347i	$0.318827\pi$
$648$	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	$-0.220686\pi$
$653$	39.3088	1.53827	0.769137	+	0.639084i	$0.220686\pi$
$654$	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	$-0.247530\pi$
$659$	36.5849	1.42515	0.712573	+	0.701598i	$0.247530\pi$
$660$	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	$-0.518209\pi$
$677$	−2.97532	−0.114351	−0.0571755	+	0.998364i	$0.518209\pi$
$678$	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	$-0.459650\pi$
$683$	6.60805	0.252850	0.126425	+	0.991976i	$0.459650\pi$
$684$	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	$-0.399044\pi$
$701$	16.5146	0.623747	0.311873	+	0.950124i	$0.399044\pi$
$702$	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	$-0.106344\pi$
$719$	50.6632	1.88942	0.944709	+	0.327909i	$0.106344\pi$
$720$	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	$-0.578155\pi$
$743$	−13.2514	−0.486145	−0.243073	+	0.970008i	$0.578155\pi$
$744$	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	$-0.760984\pi$
$761$	−40.3355	−1.46216	−0.731081	+	0.682291i	$0.760984\pi$
$762$	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	$-0.661105\pi$
$773$	−26.9572	−0.969584	−0.484792	+	0.874629i	$0.661105\pi$
$774$	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	$-0.557138\pi$
$797$	−10.0808	−0.357082	−0.178541	+	0.983933i	$0.557138\pi$
$798$	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	$-0.548081\pi$
$809$	−8.55995	−0.300952	−0.150476	+	0.988614i	$0.548081\pi$
$810$	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	$-0.592051\pi$
$821$	−16.3421	−0.570344	−0.285172	+	0.958476i	$0.592051\pi$
$822$	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	$-0.519079\pi$
$827$	−3.44532	−0.119805	−0.0599027	+	0.998204i	$0.519079\pi$
$828$	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	$-0.324447\pi$
$839$	30.3546	1.04796	0.523979	+	0.851731i	$0.324447\pi$
$840$	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	$-0.263142\pi$
$857$	39.6564	1.35464	0.677318	+	0.735690i	$0.263142\pi$
$858$	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	$-0.311436\pi$
$863$	32.8049	1.11669	0.558346	+	0.829608i	$0.311436\pi$
$864$	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	$-0.346439\pi$
$881$	27.5405	0.927861	0.463931	+	0.885871i	$0.346439\pi$
$882$	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	$-0.776327\pi$
$887$	−45.4546	−1.52622	−0.763108	+	0.646271i	$0.776327\pi$
$888$	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	$-0.273907\pi$
$911$	39.3617	1.30411	0.652055	+	0.758171i	$0.273907\pi$
$912$	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	$-0.346235\pi$
$929$	28.3154	0.928997	0.464498	+	0.885574i	$0.346235\pi$
$930$	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	$-0.586420\pi$
$941$	−16.4529	−0.536349	−0.268174	+	0.963370i	$0.586420\pi$
$942$	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	$-0.591109\pi$
$947$	−17.3768	−0.564671	−0.282335	+	0.959316i	$0.591109\pi$
$948$	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	$-0.777453\pi$
$953$	−47.2562	−1.53078	−0.765389	+	0.643568i	$0.777453\pi$
$954$	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	$-0.463889\pi$
$971$	7.05507	0.226408	0.113204	+	0.993572i	$0.463889\pi$
$972$	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	$-0.530020\pi$
$977$	−5.88708	−0.188344	−0.0941721	+	0.995556i	$0.530020\pi$
$978$	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	$-0.610189\pi$
$983$	−21.2758	−0.678593	−0.339297	+	0.940679i	$0.610189\pi$
$984$	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.l.1.4	✓	12
3.2	odd	2		8046.2.a.m.1.9	yes	12

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
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

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Database

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