LMFDB - Embedded newform 8046.2.a.n.1.1

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} - 3 x^{11} - 31 x^{10} + 82 x^{9} + 334 x^{8} - 684 x^{7} - 1561 x^{6} + 1551 x^{5} + 3573 x^{4} + \cdots - 3 $ x^12 - 3x^11 - 31x^10 + 82x^9 + 334x^8 - 684x^7 - 1561x^6 + 1551x^5 + 3573x^4 + 345x^3 - 1607x^2 - 594*x - 3
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			1.1
Root			$4.06732$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -4.06732 q^{5} +1.57160 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -4.06732 q^{5} +1.57160 q^{7} +1.00000 q^{8} -4.06732 q^{10} -5.58596 q^{11} -4.93913 q^{13} +1.57160 q^{14} +1.00000 q^{16} +7.87299 q^{17} +7.88914 q^{19} -4.06732 q^{20} -5.58596 q^{22} -0.940894 q^{23} +11.5431 q^{25} -4.93913 q^{26} +1.57160 q^{28} -0.204779 q^{29} -0.902429 q^{31} +1.00000 q^{32} +7.87299 q^{34} -6.39221 q^{35} +0.171003 q^{37} +7.88914 q^{38} -4.06732 q^{40} -4.80594 q^{41} +7.68490 q^{43} -5.58596 q^{44} -0.940894 q^{46} +5.30779 q^{47} -4.53006 q^{49} +11.5431 q^{50} -4.93913 q^{52} +1.10798 q^{53} +22.7199 q^{55} +1.57160 q^{56} -0.204779 q^{58} +5.47835 q^{59} -14.9373 q^{61} -0.902429 q^{62} +1.00000 q^{64} +20.0890 q^{65} +0.418411 q^{67} +7.87299 q^{68} -6.39221 q^{70} -14.7301 q^{71} -0.739195 q^{73} +0.171003 q^{74} +7.88914 q^{76} -8.77892 q^{77} +4.33354 q^{79} -4.06732 q^{80} -4.80594 q^{82} -14.3148 q^{83} -32.0219 q^{85} +7.68490 q^{86} -5.58596 q^{88} +0.822766 q^{89} -7.76235 q^{91} -0.940894 q^{92} +5.30779 q^{94} -32.0876 q^{95} +4.95746 q^{97} -4.53006 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8} - 3 q^{10} - 14 q^{11} - 3 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} - 4 q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 11 q^{25} - 3 q^{26} - 6 q^{28} - 23 q^{29} - 14 q^{31} + 12 q^{32} - 8 q^{34} - 32 q^{35} - 19 q^{37} - 4 q^{38} - 3 q^{40} - 30 q^{41} - 15 q^{43} - 14 q^{44} - 13 q^{46} + q^{47} + 14 q^{49} + 11 q^{50} - 3 q^{52} - 16 q^{53} - 7 q^{55} - 6 q^{56} - 23 q^{58} - 26 q^{59} - 16 q^{61} - 14 q^{62} + 12 q^{64} - 8 q^{65} - 39 q^{67} - 8 q^{68} - 32 q^{70} - 15 q^{71} - 2 q^{73} - 19 q^{74} - 4 q^{76} - 34 q^{77} - 13 q^{79} - 3 q^{80} - 30 q^{82} - 6 q^{83} - 11 q^{85} - 15 q^{86} - 14 q^{88} - 18 q^{89} - 35 q^{91} - 13 q^{92} + q^{94} - 51 q^{95} + 19 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8 - 3 * q^10 - 14 * q^11 - 3 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 - 4 * q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 11 * q^25 - 3 * q^26 - 6 * q^28 - 23 * q^29 - 14 * q^31 + 12 * q^32 - 8 * q^34 - 32 * q^35 - 19 * q^37 - 4 * q^38 - 3 * q^40 - 30 * q^41 - 15 * q^43 - 14 * q^44 - 13 * q^46 + q^47 + 14 * q^49 + 11 * q^50 - 3 * q^52 - 16 * q^53 - 7 * q^55 - 6 * q^56 - 23 * q^58 - 26 * q^59 - 16 * q^61 - 14 * q^62 + 12 * q^64 - 8 * q^65 - 39 * q^67 - 8 * q^68 - 32 * q^70 - 15 * q^71 - 2 * q^73 - 19 * q^74 - 4 * q^76 - 34 * q^77 - 13 * q^79 - 3 * q^80 - 30 * q^82 - 6 * q^83 - 11 * q^85 - 15 * q^86 - 14 * q^88 - 18 * q^89 - 35 * q^91 - 13 * q^92 + q^94 - 51 * q^95 + 19 * 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
$5$	−4.06732	−1.81896	−0.909480	−	0.415748i	$-0.863520\pi$
$5$	−4.06732	−1.81896	−0.909480	+	0.415748i	$0.863520\pi$
$6$	0	0
$7$	1.57160	0.594010	0.297005	−	0.954876i	$-0.404012\pi$
$7$	1.57160	0.594010	0.297005	+	0.954876i	$0.404012\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−4.06732	−1.28620
$11$	−5.58596	−1.68423	−0.842116	−	0.539297i	$-0.818690\pi$
$11$	−5.58596	−1.68423	−0.842116	+	0.539297i	$0.818690\pi$
$12$	0	0
$13$	−4.93913	−1.36987	−0.684933	−	0.728606i	$-0.740169\pi$
$13$	−4.93913	−1.36987	−0.684933	+	0.728606i	$0.740169\pi$
$14$	1.57160	0.420029
$15$	0	0
$16$	1.00000	0.250000
$17$	7.87299	1.90948	0.954740	−	0.297441i	$-0.0961332\pi$
$17$	7.87299	1.90948	0.954740	+	0.297441i	$0.0961332\pi$
$18$	0	0
$19$	7.88914	1.80989	0.904946	−	0.425526i	$-0.139911\pi$
$19$	7.88914	1.80989	0.904946	+	0.425526i	$0.139911\pi$
$20$	−4.06732	−0.909480
$21$	0	0
$22$	−5.58596	−1.19093
$23$	−0.940894	−0.196190	−0.0980949	−	0.995177i	$-0.531275\pi$
$23$	−0.940894	−0.196190	−0.0980949	+	0.995177i	$0.531275\pi$
$24$	0	0
$25$	11.5431	2.30861
$26$	−4.93913	−0.968642
$27$	0	0
$28$	1.57160	0.297005
$29$	−0.204779	−0.0380265	−0.0190133	−	0.999819i	$-0.506052\pi$
$29$	−0.204779	−0.0380265	−0.0190133	+	0.999819i	$0.506052\pi$
$30$	0	0
$31$	−0.902429	−0.162081	−0.0810405	−	0.996711i	$-0.525824\pi$
$31$	−0.902429	−0.162081	−0.0810405	+	0.996711i	$0.525824\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.87299	1.35021
$35$	−6.39221	−1.08048
$36$	0	0
$37$	0.171003	0.0281128	0.0140564	−	0.999901i	$-0.495526\pi$
$37$	0.171003	0.0281128	0.0140564	+	0.999901i	$0.495526\pi$
$38$	7.88914	1.27979
$39$	0	0
$40$	−4.06732	−0.643099
$41$	−4.80594	−0.750561	−0.375281	−	0.926911i	$-0.622454\pi$
$41$	−4.80594	−0.750561	−0.375281	+	0.926911i	$0.622454\pi$
$42$	0	0
$43$	7.68490	1.17194	0.585968	−	0.810334i	$-0.300714\pi$
$43$	7.68490	1.17194	0.585968	+	0.810334i	$0.300714\pi$
$44$	−5.58596	−0.842116
$45$	0	0
$46$	−0.940894	−0.138727
$47$	5.30779	0.774221	0.387111	−	0.922033i	$-0.373473\pi$
$47$	5.30779	0.774221	0.387111	+	0.922033i	$0.373473\pi$
$48$	0	0
$49$	−4.53006	−0.647152
$50$	11.5431	1.63244
$51$	0	0
$52$	−4.93913	−0.684933
$53$	1.10798	0.152192	0.0760962	−	0.997100i	$-0.475754\pi$
$53$	1.10798	0.152192	0.0760962	+	0.997100i	$0.475754\pi$
$54$	0	0
$55$	22.7199	3.06355
$56$	1.57160	0.210014
$57$	0	0
$58$	−0.204779	−0.0268888
$59$	5.47835	0.713220	0.356610	−	0.934253i	$-0.383932\pi$
$59$	5.47835	0.713220	0.356610	+	0.934253i	$0.383932\pi$
$60$	0	0
$61$	−14.9373	−1.91253	−0.956264	−	0.292503i	$-0.905512\pi$
$61$	−14.9373	−1.91253	−0.956264	+	0.292503i	$0.905512\pi$
$62$	−0.902429	−0.114609
$63$	0	0
$64$	1.00000	0.125000
$65$	20.0890	2.49173
$66$	0	0
$67$	0.418411	0.0511170	0.0255585	−	0.999673i	$-0.491864\pi$
$67$	0.418411	0.0511170	0.0255585	+	0.999673i	$0.491864\pi$
$68$	7.87299	0.954740
$69$	0	0
$70$	−6.39221	−0.764015
$71$	−14.7301	−1.74814	−0.874069	−	0.485803i	$-0.838527\pi$
$71$	−14.7301	−1.74814	−0.874069	+	0.485803i	$0.838527\pi$
$72$	0	0
$73$	−0.739195	−0.0865162	−0.0432581	−	0.999064i	$-0.513774\pi$
$73$	−0.739195	−0.0865162	−0.0432581	+	0.999064i	$0.513774\pi$
$74$	0.171003	0.0198787
$75$	0	0
$76$	7.88914	0.904946
$77$	−8.77892	−1.00045
$78$	0	0
$79$	4.33354	0.487562	0.243781	−	0.969830i	$-0.421612\pi$
$79$	4.33354	0.487562	0.243781	+	0.969830i	$0.421612\pi$
$80$	−4.06732	−0.454740
$81$	0	0
$82$	−4.80594	−0.530727
$83$	−14.3148	−1.57125	−0.785625	−	0.618703i	$-0.787659\pi$
$83$	−14.3148	−1.57125	−0.785625	+	0.618703i	$0.787659\pi$
$84$	0	0
$85$	−32.0219	−3.47327
$86$	7.68490	0.828684
$87$	0	0
$88$	−5.58596	−0.595466
$89$	0.822766	0.0872130	0.0436065	−	0.999049i	$-0.486115\pi$
$89$	0.822766	0.0872130	0.0436065	+	0.999049i	$0.486115\pi$
$90$	0	0
$91$	−7.76235	−0.813715
$92$	−0.940894	−0.0980949
$93$	0	0
$94$	5.30779	0.547457
$95$	−32.0876	−3.29212
$96$	0	0
$97$	4.95746	0.503354	0.251677	−	0.967811i	$-0.419018\pi$
$97$	4.95746	0.503354	0.251677	+	0.967811i	$0.419018\pi$
$98$	−4.53006	−0.457605
$99$	0	0
$100$	11.5431	1.15431
$101$	−0.290853	−0.0289409	−0.0144705	−	0.999895i	$-0.504606\pi$
$101$	−0.290853	−0.0289409	−0.0144705	+	0.999895i	$0.504606\pi$
$102$	0	0
$103$	5.56675	0.548509	0.274254	−	0.961657i	$-0.411569\pi$
$103$	5.56675	0.548509	0.274254	+	0.961657i	$0.411569\pi$
$104$	−4.93913	−0.484321
$105$	0	0
$106$	1.10798	0.107616
$107$	−18.6794	−1.80580	−0.902901	−	0.429849i	$-0.858567\pi$
$107$	−18.6794	−1.80580	−0.902901	+	0.429849i	$0.858567\pi$
$108$	0	0
$109$	6.52689	0.625163	0.312582	−	0.949891i	$-0.398806\pi$
$109$	6.52689	0.625163	0.312582	+	0.949891i	$0.398806\pi$
$110$	22.7199	2.16626
$111$	0	0
$112$	1.57160	0.148503
$113$	−9.46889	−0.890759	−0.445379	−	0.895342i	$-0.646931\pi$
$113$	−9.46889	−0.890759	−0.445379	+	0.895342i	$0.646931\pi$
$114$	0	0
$115$	3.82691	0.356861
$116$	−0.204779	−0.0190133
$117$	0	0
$118$	5.47835	0.504323
$119$	12.3732	1.13425
$120$	0	0
$121$	20.2030	1.83663
$122$	−14.9373	−1.35236
$123$	0	0
$124$	−0.902429	−0.0810405
$125$	−26.6127	−2.38032
$126$	0	0
$127$	−5.12874	−0.455102	−0.227551	−	0.973766i	$-0.573072\pi$
$127$	−5.12874	−0.455102	−0.227551	+	0.973766i	$0.573072\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	20.0890	1.76192
$131$	−11.8356	−1.03408	−0.517042	−	0.855960i	$-0.672967\pi$
$131$	−11.8356	−1.03408	−0.517042	+	0.855960i	$0.672967\pi$
$132$	0	0
$133$	12.3986	1.07509
$134$	0.418411	0.0361452
$135$	0	0
$136$	7.87299	0.675103
$137$	−2.13264	−0.182204	−0.0911020	−	0.995842i	$-0.529039\pi$
$137$	−2.13264	−0.182204	−0.0911020	+	0.995842i	$0.529039\pi$
$138$	0	0
$139$	−6.30669	−0.534926	−0.267463	−	0.963568i	$-0.586185\pi$
$139$	−6.30669	−0.534926	−0.267463	+	0.963568i	$0.586185\pi$
$140$	−6.39221	−0.540240
$141$	0	0
$142$	−14.7301	−1.23612
$143$	27.5898	2.30717
$144$	0	0
$145$	0.832902	0.0691687
$146$	−0.739195	−0.0611762
$147$	0	0
$148$	0.171003	0.0140564
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	6.79179	0.552708	0.276354	−	0.961056i	$-0.410874\pi$
$151$	6.79179	0.552708	0.276354	+	0.961056i	$0.410874\pi$
$152$	7.88914	0.639894
$153$	0	0
$154$	−8.77892	−0.707426
$155$	3.67046	0.294819
$156$	0	0
$157$	−5.44520	−0.434575	−0.217287	−	0.976108i	$-0.569721\pi$
$157$	−5.44520	−0.434575	−0.217287	+	0.976108i	$0.569721\pi$
$158$	4.33354	0.344758
$159$	0	0
$160$	−4.06732	−0.321550
$161$	−1.47871	−0.116539
$162$	0	0
$163$	−23.1832	−1.81585	−0.907924	−	0.419134i	$-0.862334\pi$
$163$	−23.1832	−1.81585	−0.907924	+	0.419134i	$0.862334\pi$
$164$	−4.80594	−0.375281
$165$	0	0
$166$	−14.3148	−1.11104
$167$	−2.70486	−0.209309	−0.104654	−	0.994509i	$-0.533374\pi$
$167$	−2.70486	−0.209309	−0.104654	+	0.994509i	$0.533374\pi$
$168$	0	0
$169$	11.3950	0.876535
$170$	−32.0219	−2.45597
$171$	0	0
$172$	7.68490	0.585968
$173$	−20.6227	−1.56791	−0.783956	−	0.620816i	$-0.786801\pi$
$173$	−20.6227	−1.56791	−0.783956	+	0.620816i	$0.786801\pi$
$174$	0	0
$175$	18.1411	1.37134
$176$	−5.58596	−0.421058
$177$	0	0
$178$	0.822766	0.0616689
$179$	−6.47021	−0.483607	−0.241803	−	0.970325i	$-0.577739\pi$
$179$	−6.47021	−0.483607	−0.241803	+	0.970325i	$0.577739\pi$
$180$	0	0
$181$	−17.9342	−1.33304	−0.666521	−	0.745486i	$-0.732217\pi$
$181$	−17.9342	−1.33304	−0.666521	+	0.745486i	$0.732217\pi$
$182$	−7.76235	−0.575384
$183$	0	0
$184$	−0.940894	−0.0693636
$185$	−0.695525	−0.0511360
$186$	0	0
$187$	−43.9782	−3.21601
$188$	5.30779	0.387111
$189$	0	0
$190$	−32.0876	−2.32788
$191$	−15.3668	−1.11190	−0.555950	−	0.831216i	$-0.687645\pi$
$191$	−15.3668	−1.11190	−0.555950	+	0.831216i	$0.687645\pi$
$192$	0	0
$193$	−5.51075	−0.396672	−0.198336	−	0.980134i	$-0.563554\pi$
$193$	−5.51075	−0.396672	−0.198336	+	0.980134i	$0.563554\pi$
$194$	4.95746	0.355925
$195$	0	0
$196$	−4.53006	−0.323576
$197$	−13.4647	−0.959318	−0.479659	−	0.877455i	$-0.659240\pi$
$197$	−13.4647	−0.959318	−0.479659	+	0.877455i	$0.659240\pi$
$198$	0	0
$199$	9.35376	0.663070	0.331535	−	0.943443i	$-0.392433\pi$
$199$	9.35376	0.663070	0.331535	+	0.943443i	$0.392433\pi$
$200$	11.5431	0.816218
$201$	0	0
$202$	−0.290853	−0.0204643
$203$	−0.321832	−0.0225881
$204$	0	0
$205$	19.5473	1.36524
$206$	5.56675	0.387854
$207$	0	0
$208$	−4.93913	−0.342467
$209$	−44.0684	−3.04828
$210$	0	0
$211$	23.2407	1.59996	0.799978	−	0.600030i	$-0.204845\pi$
$211$	23.2407	1.59996	0.799978	+	0.600030i	$0.204845\pi$
$212$	1.10798	0.0760962
$213$	0	0
$214$	−18.6794	−1.27689
$215$	−31.2569	−2.13171
$216$	0	0
$217$	−1.41826	−0.0962778
$218$	6.52689	0.442057
$219$	0	0
$220$	22.7199	1.53177
$221$	−38.8857	−2.61573
$222$	0	0
$223$	−2.84518	−0.190527	−0.0952636	−	0.995452i	$-0.530369\pi$
$223$	−2.84518	−0.190527	−0.0952636	+	0.995452i	$0.530369\pi$
$224$	1.57160	0.105007
$225$	0	0
$226$	−9.46889	−0.629862
$227$	3.85562	0.255906	0.127953	−	0.991780i	$-0.459159\pi$
$227$	3.85562	0.255906	0.127953	+	0.991780i	$0.459159\pi$
$228$	0	0
$229$	−23.3427	−1.54253	−0.771264	−	0.636516i	$-0.780375\pi$
$229$	−23.3427	−1.54253	−0.771264	+	0.636516i	$0.780375\pi$
$230$	3.82691	0.252339
$231$	0	0
$232$	−0.204779	−0.0134444
$233$	23.1635	1.51749	0.758744	−	0.651389i	$-0.225813\pi$
$233$	23.1635	1.51749	0.758744	+	0.651389i	$0.225813\pi$
$234$	0	0
$235$	−21.5885	−1.40828
$236$	5.47835	0.356610
$237$	0	0
$238$	12.3732	0.802037
$239$	15.2898	0.989012	0.494506	−	0.869174i	$-0.335349\pi$
$239$	15.2898	0.989012	0.494506	+	0.869174i	$0.335349\pi$
$240$	0	0
$241$	0.945306	0.0608925	0.0304462	−	0.999536i	$-0.490307\pi$
$241$	0.945306	0.0608925	0.0304462	+	0.999536i	$0.490307\pi$
$242$	20.2030	1.29870
$243$	0	0
$244$	−14.9373	−0.956264
$245$	18.4252	1.17714
$246$	0	0
$247$	−38.9654	−2.47931
$248$	−0.902429	−0.0573043
$249$	0	0
$250$	−26.6127	−1.68314
$251$	15.1651	0.957211	0.478605	−	0.878030i	$-0.341142\pi$
$251$	15.1651	0.957211	0.478605	+	0.878030i	$0.341142\pi$
$252$	0	0
$253$	5.25580	0.330429
$254$	−5.12874	−0.321806
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.11895	−0.568824	−0.284412	−	0.958702i	$-0.591798\pi$
$257$	−9.11895	−0.568824	−0.284412	+	0.958702i	$0.591798\pi$
$258$	0	0
$259$	0.268750	0.0166993
$260$	20.0890	1.24587
$261$	0	0
$262$	−11.8356	−0.731207
$263$	16.2778	1.00373	0.501865	−	0.864946i	$-0.332648\pi$
$263$	16.2778	1.00373	0.501865	+	0.864946i	$0.332648\pi$
$264$	0	0
$265$	−4.50650	−0.276832
$266$	12.3986	0.760207
$267$	0	0
$268$	0.418411	0.0255585
$269$	27.0890	1.65165	0.825823	−	0.563930i	$-0.190711\pi$
$269$	27.0890	1.65165	0.825823	+	0.563930i	$0.190711\pi$
$270$	0	0
$271$	−23.4258	−1.42302	−0.711509	−	0.702677i	$-0.751988\pi$
$271$	−23.4258	−1.42302	−0.711509	+	0.702677i	$0.751988\pi$
$272$	7.87299	0.477370
$273$	0	0
$274$	−2.13264	−0.128838
$275$	−64.4792	−3.88824
$276$	0	0
$277$	0.534771	0.0321313	0.0160656	−	0.999871i	$-0.494886\pi$
$277$	0.534771	0.0321313	0.0160656	+	0.999871i	$0.494886\pi$
$278$	−6.30669	−0.378250
$279$	0	0
$280$	−6.39221	−0.382008
$281$	−12.2586	−0.731288	−0.365644	−	0.930755i	$-0.619151\pi$
$281$	−12.2586	−0.731288	−0.365644	+	0.930755i	$0.619151\pi$
$282$	0	0
$283$	14.2319	0.845996	0.422998	−	0.906131i	$-0.360978\pi$
$283$	14.2319	0.845996	0.422998	+	0.906131i	$0.360978\pi$
$284$	−14.7301	−0.874069
$285$	0	0
$286$	27.5898	1.63142
$287$	−7.55303	−0.445841
$288$	0	0
$289$	44.9840	2.64612
$290$	0.832902	0.0489097
$291$	0	0
$292$	−0.739195	−0.0432581
$293$	−24.3487	−1.42247	−0.711234	−	0.702955i	$-0.751863\pi$
$293$	−24.3487	−1.42247	−0.711234	+	0.702955i	$0.751863\pi$
$294$	0	0
$295$	−22.2822	−1.29732
$296$	0.171003	0.00993937
$297$	0	0
$298$	1.00000	0.0579284
$299$	4.64719	0.268754
$300$	0	0
$301$	12.0776	0.696143
$302$	6.79179	0.390824
$303$	0	0
$304$	7.88914	0.452473
$305$	60.7549	3.47881
$306$	0	0
$307$	32.5283	1.85649	0.928245	−	0.371970i	$-0.121318\pi$
$307$	32.5283	1.85649	0.928245	+	0.371970i	$0.121318\pi$
$308$	−8.77892	−0.500225
$309$	0	0
$310$	3.67046	0.208468
$311$	3.03297	0.171984	0.0859920	−	0.996296i	$-0.472594\pi$
$311$	3.03297	0.171984	0.0859920	+	0.996296i	$0.472594\pi$
$312$	0	0
$313$	−12.0503	−0.681126	−0.340563	−	0.940222i	$-0.610618\pi$
$313$	−12.0503	−0.681126	−0.340563	+	0.940222i	$0.610618\pi$
$314$	−5.44520	−0.307291
$315$	0	0
$316$	4.33354	0.243781
$317$	22.9636	1.28976	0.644882	−	0.764282i	$-0.276906\pi$
$317$	22.9636	1.28976	0.644882	+	0.764282i	$0.276906\pi$
$318$	0	0
$319$	1.14389	0.0640454
$320$	−4.06732	−0.227370
$321$	0	0
$322$	−1.47871	−0.0824054
$323$	62.1111	3.45595
$324$	0	0
$325$	−57.0127	−3.16249
$326$	−23.1832	−1.28400
$327$	0	0
$328$	−4.80594	−0.265363
$329$	8.34175	0.459896
$330$	0	0
$331$	−2.62133	−0.144081	−0.0720406	−	0.997402i	$-0.522951\pi$
$331$	−2.62133	−0.144081	−0.0720406	+	0.997402i	$0.522951\pi$
$332$	−14.3148	−0.785625
$333$	0	0
$334$	−2.70486	−0.148004
$335$	−1.70181	−0.0929798
$336$	0	0
$337$	−20.5905	−1.12164	−0.560818	−	0.827939i	$-0.689513\pi$
$337$	−20.5905	−1.12164	−0.560818	+	0.827939i	$0.689513\pi$
$338$	11.3950	0.619804
$339$	0	0
$340$	−32.0219	−1.73663
$341$	5.04093	0.272982
$342$	0	0
$343$	−18.1207	−0.978425
$344$	7.68490	0.414342
$345$	0	0
$346$	−20.6227	−1.10868
$347$	−24.2425	−1.30141	−0.650704	−	0.759332i	$-0.725526\pi$
$347$	−24.2425	−1.30141	−0.650704	+	0.759332i	$0.725526\pi$
$348$	0	0
$349$	16.5003	0.883240	0.441620	−	0.897202i	$-0.354404\pi$
$349$	16.5003	0.883240	0.441620	+	0.897202i	$0.354404\pi$
$350$	18.1411	0.969684
$351$	0	0
$352$	−5.58596	−0.297733
$353$	13.1514	0.699977	0.349989	−	0.936754i	$-0.386185\pi$
$353$	13.1514	0.699977	0.349989	+	0.936754i	$0.386185\pi$
$354$	0	0
$355$	59.9118	3.17979
$356$	0.822766	0.0436065
$357$	0	0
$358$	−6.47021	−0.341961
$359$	−2.41049	−0.127221	−0.0636103	−	0.997975i	$-0.520261\pi$
$359$	−2.41049	−0.127221	−0.0636103	+	0.997975i	$0.520261\pi$
$360$	0	0
$361$	43.2385	2.27571
$362$	−17.9342	−0.942603
$363$	0	0
$364$	−7.76235	−0.406858
$365$	3.00654	0.157369
$366$	0	0
$367$	−30.2400	−1.57852	−0.789258	−	0.614062i	$-0.789534\pi$
$367$	−30.2400	−1.57852	−0.789258	+	0.614062i	$0.789534\pi$
$368$	−0.940894	−0.0490475
$369$	0	0
$370$	−0.695525	−0.0361586
$371$	1.74130	0.0904039
$372$	0	0
$373$	−31.0658	−1.60853	−0.804263	−	0.594273i	$-0.797440\pi$
$373$	−31.0658	−1.60853	−0.804263	+	0.594273i	$0.797440\pi$
$374$	−43.9782	−2.27406
$375$	0	0
$376$	5.30779	0.273729
$377$	1.01143	0.0520913
$378$	0	0
$379$	23.3362	1.19870	0.599351	−	0.800486i	$-0.295425\pi$
$379$	23.3362	1.19870	0.599351	+	0.800486i	$0.295425\pi$
$380$	−32.0876	−1.64606
$381$	0	0
$382$	−15.3668	−0.786231
$383$	22.4995	1.14967	0.574835	−	0.818269i	$-0.305066\pi$
$383$	22.4995	1.14967	0.574835	+	0.818269i	$0.305066\pi$
$384$	0	0
$385$	35.7067	1.81978
$386$	−5.51075	−0.280490
$387$	0	0
$388$	4.95746	0.251677
$389$	−24.5906	−1.24679	−0.623396	−	0.781906i	$-0.714248\pi$
$389$	−24.5906	−1.24679	−0.623396	+	0.781906i	$0.714248\pi$
$390$	0	0
$391$	−7.40765	−0.374621
$392$	−4.53006	−0.228803
$393$	0	0
$394$	−13.4647	−0.678340
$395$	−17.6259	−0.886855
$396$	0	0
$397$	−30.5635	−1.53394	−0.766970	−	0.641683i	$-0.778236\pi$
$397$	−30.5635	−1.53394	−0.766970	+	0.641683i	$0.778236\pi$
$398$	9.35376	0.468861
$399$	0	0
$400$	11.5431	0.577154
$401$	0.864928	0.0431925	0.0215962	−	0.999767i	$-0.493125\pi$
$401$	0.864928	0.0431925	0.0215962	+	0.999767i	$0.493125\pi$
$402$	0	0
$403$	4.45721	0.222029
$404$	−0.290853	−0.0144705
$405$	0	0
$406$	−0.321832	−0.0159722
$407$	−0.955219	−0.0473484
$408$	0	0
$409$	11.7599	0.581488	0.290744	−	0.956801i	$-0.406097\pi$
$409$	11.7599	0.581488	0.290744	+	0.956801i	$0.406097\pi$
$410$	19.5473	0.965371
$411$	0	0
$412$	5.56675	0.274254
$413$	8.60979	0.423660
$414$	0	0
$415$	58.2227	2.85804
$416$	−4.93913	−0.242161
$417$	0	0
$418$	−44.0684	−2.15546
$419$	−16.0270	−0.782972	−0.391486	−	0.920184i	$-0.628039\pi$
$419$	−16.0270	−0.782972	−0.391486	+	0.920184i	$0.628039\pi$
$420$	0	0
$421$	34.1099	1.66241	0.831207	−	0.555964i	$-0.187651\pi$
$421$	34.1099	1.66241	0.831207	+	0.555964i	$0.187651\pi$
$422$	23.2407	1.13134
$423$	0	0
$424$	1.10798	0.0538081
$425$	90.8785	4.40825
$426$	0	0
$427$	−23.4756	−1.13606
$428$	−18.6794	−0.902901
$429$	0	0
$430$	−31.2569	−1.50734
$431$	−13.0066	−0.626507	−0.313253	−	0.949670i	$-0.601419\pi$
$431$	−13.0066	−0.626507	−0.313253	+	0.949670i	$0.601419\pi$
$432$	0	0
$433$	−19.9670	−0.959551	−0.479775	−	0.877391i	$-0.659282\pi$
$433$	−19.9670	−0.959551	−0.479775	+	0.877391i	$0.659282\pi$
$434$	−1.41826	−0.0680787
$435$	0	0
$436$	6.52689	0.312582
$437$	−7.42284	−0.355083
$438$	0	0
$439$	−18.9555	−0.904695	−0.452348	−	0.891842i	$-0.649413\pi$
$439$	−18.9555	−0.904695	−0.452348	+	0.891842i	$0.649413\pi$
$440$	22.7199	1.08313
$441$	0	0
$442$	−38.8857	−1.84960
$443$	1.34509	0.0639072	0.0319536	−	0.999489i	$-0.489827\pi$
$443$	1.34509	0.0639072	0.0319536	+	0.999489i	$0.489827\pi$
$444$	0	0
$445$	−3.34645	−0.158637
$446$	−2.84518	−0.134723
$447$	0	0
$448$	1.57160	0.0742513
$449$	32.1823	1.51878	0.759389	−	0.650637i	$-0.225498\pi$
$449$	32.1823	1.51878	0.759389	+	0.650637i	$0.225498\pi$
$450$	0	0
$451$	26.8458	1.26412
$452$	−9.46889	−0.445379
$453$	0	0
$454$	3.85562	0.180953
$455$	31.5719	1.48012
$456$	0	0
$457$	−21.4473	−1.00326	−0.501632	−	0.865081i	$-0.667267\pi$
$457$	−21.4473	−1.00326	−0.501632	+	0.865081i	$0.667267\pi$
$458$	−23.3427	−1.09073
$459$	0	0
$460$	3.82691	0.178431
$461$	−30.5421	−1.42249	−0.711243	−	0.702947i	$-0.751867\pi$
$461$	−30.5421	−1.42249	−0.711243	+	0.702947i	$0.751867\pi$
$462$	0	0
$463$	7.10281	0.330095	0.165048	−	0.986286i	$-0.447222\pi$
$463$	7.10281	0.330095	0.165048	+	0.986286i	$0.447222\pi$
$464$	−0.204779	−0.00950663
$465$	0	0
$466$	23.1635	1.07303
$467$	6.65218	0.307826	0.153913	−	0.988084i	$-0.450812\pi$
$467$	6.65218	0.307826	0.153913	+	0.988084i	$0.450812\pi$
$468$	0	0
$469$	0.657576	0.0303641
$470$	−21.5885	−0.995803
$471$	0	0
$472$	5.47835	0.252161
$473$	−42.9276	−1.97381
$474$	0	0
$475$	91.0649	4.17834
$476$	12.3732	0.567126
$477$	0	0
$478$	15.2898	0.699337
$479$	35.2090	1.60874	0.804371	−	0.594128i	$-0.202503\pi$
$479$	35.2090	1.60874	0.804371	+	0.594128i	$0.202503\pi$
$480$	0	0
$481$	−0.844608	−0.0385108
$482$	0.945306	0.0430575
$483$	0	0
$484$	20.2030	0.918317
$485$	−20.1636	−0.915580
$486$	0	0
$487$	−1.65086	−0.0748077	−0.0374038	−	0.999300i	$-0.511909\pi$
$487$	−1.65086	−0.0748077	−0.0374038	+	0.999300i	$0.511909\pi$
$488$	−14.9373	−0.676181
$489$	0	0
$490$	18.4252	0.832366
$491$	27.4459	1.23861	0.619307	−	0.785149i	$-0.287414\pi$
$491$	27.4459	1.23861	0.619307	+	0.785149i	$0.287414\pi$
$492$	0	0
$493$	−1.61222	−0.0726109
$494$	−38.9654	−1.75314
$495$	0	0
$496$	−0.902429	−0.0405203
$497$	−23.1498	−1.03841
$498$	0	0
$499$	0.983371	0.0440217	0.0220109	−	0.999758i	$-0.492993\pi$
$499$	0.983371	0.0440217	0.0220109	+	0.999758i	$0.492993\pi$
$500$	−26.6127	−1.19016
$501$	0	0
$502$	15.1651	0.676850
$503$	−5.25232	−0.234189	−0.117095	−	0.993121i	$-0.537358\pi$
$503$	−5.25232	−0.234189	−0.117095	+	0.993121i	$0.537358\pi$
$504$	0	0
$505$	1.18299	0.0526423
$506$	5.25580	0.233649
$507$	0	0
$508$	−5.12874	−0.227551
$509$	27.1329	1.20264	0.601322	−	0.799007i	$-0.294641\pi$
$509$	27.1329	1.20264	0.601322	+	0.799007i	$0.294641\pi$
$510$	0	0
$511$	−1.16172	−0.0513915
$512$	1.00000	0.0441942
$513$	0	0
$514$	−9.11895	−0.402220
$515$	−22.6418	−0.997715
$516$	0	0
$517$	−29.6491	−1.30397
$518$	0.268750	0.0118082
$519$	0	0
$520$	20.0890	0.880961
$521$	−29.5472	−1.29449	−0.647244	−	0.762283i	$-0.724078\pi$
$521$	−29.5472	−1.29449	−0.647244	+	0.762283i	$0.724078\pi$
$522$	0	0
$523$	20.5749	0.899676	0.449838	−	0.893110i	$-0.351482\pi$
$523$	20.5749	0.899676	0.449838	+	0.893110i	$0.351482\pi$
$524$	−11.8356	−0.517042
$525$	0	0
$526$	16.2778	0.709744
$527$	−7.10481	−0.309491
$528$	0	0
$529$	−22.1147	−0.961510
$530$	−4.50650	−0.195750
$531$	0	0
$532$	12.3986	0.537547
$533$	23.7371	1.02817
$534$	0	0
$535$	75.9749	3.28468
$536$	0.418411	0.0180726
$537$	0	0
$538$	27.0890	1.16789
$539$	25.3048	1.08995
$540$	0	0
$541$	−23.6273	−1.01582	−0.507909	−	0.861411i	$-0.669581\pi$
$541$	−23.6273	−1.01582	−0.507909	+	0.861411i	$0.669581\pi$
$542$	−23.4258	−1.00623
$543$	0	0
$544$	7.87299	0.337552
$545$	−26.5469	−1.13715
$546$	0	0
$547$	−39.7753	−1.70067	−0.850335	−	0.526242i	$-0.823601\pi$
$547$	−39.7753	−1.70067	−0.850335	+	0.526242i	$0.823601\pi$
$548$	−2.13264	−0.0911020
$549$	0	0
$550$	−64.4792	−2.74940
$551$	−1.61553	−0.0688239
$552$	0	0
$553$	6.81061	0.289617
$554$	0.534771	0.0227202
$555$	0	0
$556$	−6.30669	−0.267463
$557$	26.1257	1.10698	0.553491	−	0.832855i	$-0.313295\pi$
$557$	26.1257	1.10698	0.553491	+	0.832855i	$0.313295\pi$
$558$	0	0
$559$	−37.9567	−1.60540
$560$	−6.39221	−0.270120
$561$	0	0
$562$	−12.2586	−0.517099
$563$	−13.8429	−0.583408	−0.291704	−	0.956509i	$-0.594222\pi$
$563$	−13.8429	−0.583408	−0.291704	+	0.956509i	$0.594222\pi$
$564$	0	0
$565$	38.5130	1.62025
$566$	14.2319	0.598210
$567$	0	0
$568$	−14.7301	−0.618060
$569$	−34.2689	−1.43663	−0.718314	−	0.695719i	$-0.755086\pi$
$569$	−34.2689	−1.43663	−0.718314	+	0.695719i	$0.755086\pi$
$570$	0	0
$571$	−11.5344	−0.482701	−0.241351	−	0.970438i	$-0.577590\pi$
$571$	−11.5344	−0.482701	−0.241351	+	0.970438i	$0.577590\pi$
$572$	27.5898	1.15359
$573$	0	0
$574$	−7.55303	−0.315257
$575$	−10.8608	−0.452927
$576$	0	0
$577$	−2.54351	−0.105888	−0.0529438	−	0.998597i	$-0.516860\pi$
$577$	−2.54351	−0.105888	−0.0529438	+	0.998597i	$0.516860\pi$
$578$	44.9840	1.87109
$579$	0	0
$580$	0.832902	0.0345844
$581$	−22.4972	−0.933339
$582$	0	0
$583$	−6.18912	−0.256327
$584$	−0.739195	−0.0305881
$585$	0	0
$586$	−24.3487	−1.00584
$587$	14.0884	0.581492	0.290746	−	0.956800i	$-0.406097\pi$
$587$	14.0884	0.581492	0.290746	+	0.956800i	$0.406097\pi$
$588$	0	0
$589$	−7.11939	−0.293349
$590$	−22.2822	−0.917343
$591$	0	0
$592$	0.171003	0.00702820
$593$	4.79677	0.196980	0.0984898	−	0.995138i	$-0.468599\pi$
$593$	4.79677	0.196980	0.0984898	+	0.995138i	$0.468599\pi$
$594$	0	0
$595$	−50.3258	−2.06316
$596$	1.00000	0.0409616
$597$	0	0
$598$	4.64719	0.190038
$599$	−21.2415	−0.867902	−0.433951	−	0.900936i	$-0.642881\pi$
$599$	−21.2415	−0.867902	−0.433951	+	0.900936i	$0.642881\pi$
$600$	0	0
$601$	−19.1482	−0.781071	−0.390535	−	0.920588i	$-0.627710\pi$
$601$	−19.1482	−0.781071	−0.390535	+	0.920588i	$0.627710\pi$
$602$	12.0776	0.492247
$603$	0	0
$604$	6.79179	0.276354
$605$	−82.1719	−3.34076
$606$	0	0
$607$	−6.70131	−0.271998	−0.135999	−	0.990709i	$-0.543424\pi$
$607$	−6.70131	−0.271998	−0.135999	+	0.990709i	$0.543424\pi$
$608$	7.88914	0.319947
$609$	0	0
$610$	60.7549	2.45989
$611$	−26.2159	−1.06058
$612$	0	0
$613$	−17.8887	−0.722519	−0.361259	−	0.932465i	$-0.617653\pi$
$613$	−17.8887	−0.722519	−0.361259	+	0.932465i	$0.617653\pi$
$614$	32.5283	1.31274
$615$	0	0
$616$	−8.77892	−0.353713
$617$	−11.7352	−0.472442	−0.236221	−	0.971699i	$-0.575909\pi$
$617$	−11.7352	−0.472442	−0.236221	+	0.971699i	$0.575909\pi$
$618$	0	0
$619$	44.2609	1.77899	0.889497	−	0.456940i	$-0.151055\pi$
$619$	44.2609	1.77899	0.889497	+	0.456940i	$0.151055\pi$
$620$	3.67046	0.147409
$621$	0	0
$622$	3.03297	0.121611
$623$	1.29306	0.0518054
$624$	0	0
$625$	50.5271	2.02108
$626$	−12.0503	−0.481629
$627$	0	0
$628$	−5.44520	−0.217287
$629$	1.34631	0.0536808
$630$	0	0
$631$	−44.3503	−1.76556	−0.882778	−	0.469790i	$-0.844330\pi$
$631$	−44.3503	−1.76556	−0.882778	+	0.469790i	$0.844330\pi$
$632$	4.33354	0.172379
$633$	0	0
$634$	22.9636	0.912001
$635$	20.8602	0.827813
$636$	0	0
$637$	22.3745	0.886512
$638$	1.14389	0.0452870
$639$	0	0
$640$	−4.06732	−0.160775
$641$	−4.93583	−0.194954	−0.0974768	−	0.995238i	$-0.531077\pi$
$641$	−4.93583	−0.194954	−0.0974768	+	0.995238i	$0.531077\pi$
$642$	0	0
$643$	39.1187	1.54269	0.771345	−	0.636418i	$-0.219584\pi$
$643$	39.1187	1.54269	0.771345	+	0.636418i	$0.219584\pi$
$644$	−1.47871	−0.0582694
$645$	0	0
$646$	62.1111	2.44373
$647$	32.7294	1.28673	0.643363	−	0.765561i	$-0.277539\pi$
$647$	32.7294	1.28673	0.643363	+	0.765561i	$0.277539\pi$
$648$	0	0
$649$	−30.6018	−1.20123
$650$	−57.0127	−2.23622
$651$	0	0
$652$	−23.1832	−0.907924
$653$	−28.8564	−1.12924	−0.564619	−	0.825352i	$-0.690977\pi$
$653$	−28.8564	−1.12924	−0.564619	+	0.825352i	$0.690977\pi$
$654$	0	0
$655$	48.1392	1.88096
$656$	−4.80594	−0.187640
$657$	0	0
$658$	8.34175	0.325195
$659$	−37.7166	−1.46923	−0.734616	−	0.678483i	$-0.762638\pi$
$659$	−37.7166	−1.46923	−0.734616	+	0.678483i	$0.762638\pi$
$660$	0	0
$661$	−17.7652	−0.690985	−0.345493	−	0.938421i	$-0.612288\pi$
$661$	−17.7652	−0.690985	−0.345493	+	0.938421i	$0.612288\pi$
$662$	−2.62133	−0.101881
$663$	0	0
$664$	−14.3148	−0.555521
$665$	−50.4290	−1.95555
$666$	0	0
$667$	0.192675	0.00746042
$668$	−2.70486	−0.104654
$669$	0	0
$670$	−1.70181	−0.0657467
$671$	83.4394	3.22114
$672$	0	0
$673$	−25.4007	−0.979124	−0.489562	−	0.871969i	$-0.662843\pi$
$673$	−25.4007	−0.979124	−0.489562	+	0.871969i	$0.662843\pi$
$674$	−20.5905	−0.793117
$675$	0	0
$676$	11.3950	0.438268
$677$	−0.307918	−0.0118343	−0.00591713	−	0.999982i	$-0.501883\pi$
$677$	−0.307918	−0.0118343	−0.00591713	+	0.999982i	$0.501883\pi$
$678$	0	0
$679$	7.79116	0.298997
$680$	−32.0219	−1.22799
$681$	0	0
$682$	5.04093	0.193027
$683$	38.4608	1.47166	0.735830	−	0.677166i	$-0.236792\pi$
$683$	38.4608	1.47166	0.735830	+	0.677166i	$0.236792\pi$
$684$	0	0
$685$	8.67413	0.331422
$686$	−18.1207	−0.691851
$687$	0	0
$688$	7.68490	0.292984
$689$	−5.47244	−0.208483
$690$	0	0
$691$	10.6513	0.405195	0.202598	−	0.979262i	$-0.435062\pi$
$691$	10.6513	0.405195	0.202598	+	0.979262i	$0.435062\pi$
$692$	−20.6227	−0.783956
$693$	0	0
$694$	−24.2425	−0.920234
$695$	25.6513	0.973009
$696$	0	0
$697$	−37.8371	−1.43318
$698$	16.5003	0.624545
$699$	0	0
$700$	18.1411	0.685670
$701$	−37.6353	−1.42147	−0.710733	−	0.703461i	$-0.751637\pi$
$701$	−37.6353	−1.42147	−0.710733	+	0.703461i	$0.751637\pi$
$702$	0	0
$703$	1.34907	0.0508811
$704$	−5.58596	−0.210529
$705$	0	0
$706$	13.1514	0.494959
$707$	−0.457105	−0.0171912
$708$	0	0
$709$	34.4146	1.29247	0.646233	−	0.763140i	$-0.276343\pi$
$709$	34.4146	1.29247	0.646233	+	0.763140i	$0.276343\pi$
$710$	59.9118	2.24845
$711$	0	0
$712$	0.822766	0.0308345
$713$	0.849090	0.0317987
$714$	0	0
$715$	−112.216	−4.19665
$716$	−6.47021	−0.241803
$717$	0	0
$718$	−2.41049	−0.0899585
$719$	−39.8338	−1.48555	−0.742776	−	0.669541i	$-0.766491\pi$
$719$	−39.8338	−1.48555	−0.742776	+	0.669541i	$0.766491\pi$
$720$	0	0
$721$	8.74873	0.325820
$722$	43.2385	1.60917
$723$	0	0
$724$	−17.9342	−0.666521
$725$	−2.36378	−0.0877886
$726$	0	0
$727$	−47.2444	−1.75220	−0.876099	−	0.482132i	$-0.839863\pi$
$727$	−47.2444	−1.75220	−0.876099	+	0.482132i	$0.839863\pi$
$728$	−7.76235	−0.287692
$729$	0	0
$730$	3.00654	0.111277
$731$	60.5032	2.23779
$732$	0	0
$733$	10.1050	0.373237	0.186618	−	0.982432i	$-0.440247\pi$
$733$	10.1050	0.373237	0.186618	+	0.982432i	$0.440247\pi$
$734$	−30.2400	−1.11618
$735$	0	0
$736$	−0.940894	−0.0346818
$737$	−2.33723	−0.0860929
$738$	0	0
$739$	−39.4342	−1.45061	−0.725306	−	0.688427i	$-0.758302\pi$
$739$	−39.4342	−1.45061	−0.725306	+	0.688427i	$0.758302\pi$
$740$	−0.695525	−0.0255680
$741$	0	0
$742$	1.74130	0.0639252
$743$	16.2517	0.596219	0.298109	−	0.954532i	$-0.403644\pi$
$743$	16.2517	0.596219	0.298109	+	0.954532i	$0.403644\pi$
$744$	0	0
$745$	−4.06732	−0.149015
$746$	−31.0658	−1.13740
$747$	0	0
$748$	−43.9782	−1.60800
$749$	−29.3565	−1.07266
$750$	0	0
$751$	−15.5666	−0.568032	−0.284016	−	0.958820i	$-0.591667\pi$
$751$	−15.5666	−0.568032	−0.284016	+	0.958820i	$0.591667\pi$
$752$	5.30779	0.193555
$753$	0	0
$754$	1.01143	0.0368341
$755$	−27.6244	−1.00535
$756$	0	0
$757$	−32.8617	−1.19438	−0.597189	−	0.802100i	$-0.703716\pi$
$757$	−32.8617	−1.19438	−0.597189	+	0.802100i	$0.703716\pi$
$758$	23.3362	0.847611
$759$	0	0
$760$	−32.0876	−1.16394
$761$	−14.3275	−0.519373	−0.259686	−	0.965693i	$-0.583619\pi$
$761$	−14.3275	−0.519373	−0.259686	+	0.965693i	$0.583619\pi$
$762$	0	0
$763$	10.2577	0.371353
$764$	−15.3668	−0.555950
$765$	0	0
$766$	22.4995	0.812940
$767$	−27.0582	−0.977017
$768$	0	0
$769$	29.1063	1.04960	0.524801	−	0.851225i	$-0.324140\pi$
$769$	29.1063	1.04960	0.524801	+	0.851225i	$0.324140\pi$
$770$	35.7067	1.28678
$771$	0	0
$772$	−5.51075	−0.198336
$773$	−9.82416	−0.353351	−0.176675	−	0.984269i	$-0.556534\pi$
$773$	−9.82416	−0.353351	−0.176675	+	0.984269i	$0.556534\pi$
$774$	0	0
$775$	−10.4168	−0.374183
$776$	4.95746	0.177962
$777$	0	0
$778$	−24.5906	−0.881615
$779$	−37.9147	−1.35844
$780$	0	0
$781$	82.2816	2.94427
$782$	−7.40765	−0.264897
$783$	0	0
$784$	−4.53006	−0.161788
$785$	22.1474	0.790474
$786$	0	0
$787$	43.0361	1.53407	0.767035	−	0.641605i	$-0.221731\pi$
$787$	43.0361	1.53407	0.767035	+	0.641605i	$0.221731\pi$
$788$	−13.4647	−0.479659
$789$	0	0
$790$	−17.6259	−0.627101
$791$	−14.8814	−0.529120
$792$	0	0
$793$	73.7773	2.61991
$794$	−30.5635	−1.08466
$795$	0	0
$796$	9.35376	0.331535
$797$	8.59255	0.304364	0.152182	−	0.988353i	$-0.451370\pi$
$797$	8.59255	0.304364	0.152182	+	0.988353i	$0.451370\pi$
$798$	0	0
$799$	41.7882	1.47836
$800$	11.5431	0.408109
$801$	0	0
$802$	0.864928	0.0305417
$803$	4.12911	0.145713
$804$	0	0
$805$	6.01439	0.211979
$806$	4.45721	0.156999
$807$	0	0
$808$	−0.290853	−0.0102322
$809$	−38.6972	−1.36052	−0.680261	−	0.732970i	$-0.738134\pi$
$809$	−38.6972	−1.36052	−0.680261	+	0.732970i	$0.738134\pi$
$810$	0	0
$811$	−37.4120	−1.31371	−0.656857	−	0.754015i	$-0.728115\pi$
$811$	−37.4120	−1.31371	−0.656857	+	0.754015i	$0.728115\pi$
$812$	−0.321832	−0.0112941
$813$	0	0
$814$	−0.955219	−0.0334804
$815$	94.2934	3.30295
$816$	0	0
$817$	60.6273	2.12108
$818$	11.7599	0.411174
$819$	0	0
$820$	19.5473	0.682620
$821$	−50.2676	−1.75435	−0.877175	−	0.480170i	$-0.840575\pi$
$821$	−50.2676	−1.75435	−0.877175	+	0.480170i	$0.840575\pi$
$822$	0	0
$823$	−23.7737	−0.828700	−0.414350	−	0.910118i	$-0.635991\pi$
$823$	−23.7737	−0.828700	−0.414350	+	0.910118i	$0.635991\pi$
$824$	5.56675	0.193927
$825$	0	0
$826$	8.60979	0.299573
$827$	5.62101	0.195462	0.0977308	−	0.995213i	$-0.468842\pi$
$827$	5.62101	0.195462	0.0977308	+	0.995213i	$0.468842\pi$
$828$	0	0
$829$	55.0028	1.91033	0.955163	−	0.296081i	$-0.0956798\pi$
$829$	55.0028	1.91033	0.955163	+	0.296081i	$0.0956798\pi$
$830$	58.2227	2.02094
$831$	0	0
$832$	−4.93913	−0.171233
$833$	−35.6651	−1.23572
$834$	0	0
$835$	11.0015	0.380724
$836$	−44.0684	−1.52414
$837$	0	0
$838$	−16.0270	−0.553645
$839$	35.7616	1.23463	0.617314	−	0.786717i	$-0.288221\pi$
$839$	35.7616	1.23463	0.617314	+	0.786717i	$0.288221\pi$
$840$	0	0
$841$	−28.9581	−0.998554
$842$	34.1099	1.17550
$843$	0	0
$844$	23.2407	0.799978
$845$	−46.3469	−1.59438
$846$	0	0
$847$	31.7511	1.09098
$848$	1.10798	0.0380481
$849$	0	0
$850$	90.8785	3.11711
$851$	−0.160896	−0.00551545
$852$	0	0
$853$	−36.4867	−1.24928	−0.624640	−	0.780913i	$-0.714754\pi$
$853$	−36.4867	−1.24928	−0.624640	+	0.780913i	$0.714754\pi$
$854$	−23.4756	−0.803317
$855$	0	0
$856$	−18.6794	−0.638447
$857$	−16.5977	−0.566967	−0.283483	−	0.958977i	$-0.591490\pi$
$857$	−16.5977	−0.566967	−0.283483	+	0.958977i	$0.591490\pi$
$858$	0	0
$859$	−41.2851	−1.40863	−0.704315	−	0.709888i	$-0.748746\pi$
$859$	−41.2851	−1.40863	−0.704315	+	0.709888i	$0.748746\pi$
$860$	−31.2569	−1.06585
$861$	0	0
$862$	−13.0066	−0.443007
$863$	−31.1869	−1.06162	−0.530808	−	0.847492i	$-0.678111\pi$
$863$	−31.1869	−1.06162	−0.530808	+	0.847492i	$0.678111\pi$
$864$	0	0
$865$	83.8790	2.85197
$866$	−19.9670	−0.678505
$867$	0	0
$868$	−1.41826	−0.0481389
$869$	−24.2070	−0.821166
$870$	0	0
$871$	−2.06658	−0.0700235
$872$	6.52689	0.221029
$873$	0	0
$874$	−7.42284	−0.251081
$875$	−41.8247	−1.41393
$876$	0	0
$877$	−5.63482	−0.190274	−0.0951372	−	0.995464i	$-0.530329\pi$
$877$	−5.63482	−0.190274	−0.0951372	+	0.995464i	$0.530329\pi$
$878$	−18.9555	−0.639716
$879$	0	0
$880$	22.7199	0.765887
$881$	32.7369	1.10293	0.551467	−	0.834197i	$-0.314068\pi$
$881$	32.7369	1.10293	0.551467	+	0.834197i	$0.314068\pi$
$882$	0	0
$883$	−24.7743	−0.833720	−0.416860	−	0.908971i	$-0.636869\pi$
$883$	−24.7743	−0.833720	−0.416860	+	0.908971i	$0.636869\pi$
$884$	−38.8857	−1.30787
$885$	0	0
$886$	1.34509	0.0451892
$887$	4.93459	0.165687	0.0828437	−	0.996563i	$-0.473600\pi$
$887$	4.93459	0.165687	0.0828437	+	0.996563i	$0.473600\pi$
$888$	0	0
$889$	−8.06035	−0.270336
$890$	−3.34645	−0.112173
$891$	0	0
$892$	−2.84518	−0.0952636
$893$	41.8739	1.40126
$894$	0	0
$895$	26.3164	0.879661
$896$	1.57160	0.0525036
$897$	0	0
$898$	32.1823	1.07394
$899$	0.184799	0.00616338
$900$	0	0
$901$	8.72309	0.290608
$902$	26.8458	0.893867
$903$	0	0
$904$	−9.46889	−0.314931
$905$	72.9443	2.42475
$906$	0	0
$907$	23.9279	0.794514	0.397257	−	0.917707i	$-0.369962\pi$
$907$	23.9279	0.794514	0.397257	+	0.917707i	$0.369962\pi$
$908$	3.85562	0.127953
$909$	0	0
$910$	31.5719	1.04660
$911$	3.24441	0.107492	0.0537460	−	0.998555i	$-0.482884\pi$
$911$	3.24441	0.107492	0.0537460	+	0.998555i	$0.482884\pi$
$912$	0	0
$913$	79.9618	2.64635
$914$	−21.4473	−0.709415
$915$	0	0
$916$	−23.3427	−0.771264
$917$	−18.6009	−0.614256
$918$	0	0
$919$	−8.38134	−0.276475	−0.138237	−	0.990399i	$-0.544144\pi$
$919$	−8.38134	−0.276475	−0.138237	+	0.990399i	$0.544144\pi$
$920$	3.82691	0.126170
$921$	0	0
$922$	−30.5421	−1.00585
$923$	72.7536	2.39472
$924$	0	0
$925$	1.97391	0.0649016
$926$	7.10281	0.233413
$927$	0	0
$928$	−0.204779	−0.00672220
$929$	33.0020	1.08276	0.541381	−	0.840777i	$-0.317902\pi$
$929$	33.0020	1.08276	0.541381	+	0.840777i	$0.317902\pi$
$930$	0	0
$931$	−35.7383	−1.17127
$932$	23.1635	0.758744
$933$	0	0
$934$	6.65218	0.217666
$935$	178.873	5.84978
$936$	0	0
$937$	−15.2388	−0.497828	−0.248914	−	0.968526i	$-0.580074\pi$
$937$	−15.2388	−0.497828	−0.248914	+	0.968526i	$0.580074\pi$
$938$	0.657576	0.0214706
$939$	0	0
$940$	−21.5885	−0.704139
$941$	−30.1722	−0.983585	−0.491792	−	0.870713i	$-0.663658\pi$
$941$	−30.1722	−0.983585	−0.491792	+	0.870713i	$0.663658\pi$
$942$	0	0
$943$	4.52188	0.147253
$944$	5.47835	0.178305
$945$	0	0
$946$	−42.9276	−1.39570
$947$	−22.2033	−0.721509	−0.360754	−	0.932661i	$-0.617481\pi$
$947$	−22.2033	−0.721509	−0.360754	+	0.932661i	$0.617481\pi$
$948$	0	0
$949$	3.65097	0.118516
$950$	91.0649	2.95453
$951$	0	0
$952$	12.3732	0.401018
$953$	11.5820	0.375179	0.187589	−	0.982248i	$-0.439933\pi$
$953$	11.5820	0.375179	0.187589	+	0.982248i	$0.439933\pi$
$954$	0	0
$955$	62.5015	2.02250
$956$	15.2898	0.494506
$957$	0	0
$958$	35.2090	1.13755
$959$	−3.35167	−0.108231
$960$	0	0
$961$	−30.1856	−0.973730
$962$	−0.844608	−0.0272312
$963$	0	0
$964$	0.945306	0.0304462
$965$	22.4140	0.721531
$966$	0	0
$967$	44.1692	1.42038	0.710192	−	0.704008i	$-0.248608\pi$
$967$	44.1692	1.42038	0.710192	+	0.704008i	$0.248608\pi$
$968$	20.2030	0.649348
$969$	0	0
$970$	−20.1636	−0.647413
$971$	43.0939	1.38295	0.691475	−	0.722401i	$-0.256961\pi$
$971$	43.0939	1.38295	0.691475	+	0.722401i	$0.256961\pi$
$972$	0	0
$973$	−9.91162	−0.317752
$974$	−1.65086	−0.0528970
$975$	0	0
$976$	−14.9373	−0.478132
$977$	−7.07297	−0.226284	−0.113142	−	0.993579i	$-0.536092\pi$
$977$	−7.07297	−0.226284	−0.113142	+	0.993579i	$0.536092\pi$
$978$	0	0
$979$	−4.59594	−0.146887
$980$	18.4252	0.588571
$981$	0	0
$982$	27.4459	0.875833
$983$	−4.09871	−0.130728	−0.0653642	−	0.997861i	$-0.520821\pi$
$983$	−4.09871	−0.130728	−0.0653642	+	0.997861i	$0.520821\pi$
$984$	0	0
$985$	54.7651	1.74496
$986$	−1.61222	−0.0513437
$987$	0	0
$988$	−38.9654	−1.23966
$989$	−7.23068	−0.229922
$990$	0	0
$991$	21.0888	0.669908	0.334954	−	0.942234i	$-0.391279\pi$
$991$	21.0888	0.669908	0.334954	+	0.942234i	$0.391279\pi$
$992$	−0.902429	−0.0286521
$993$	0	0
$994$	−23.1498	−0.734268
$995$	−38.0447	−1.20610
$996$	0	0
$997$	−3.58311	−0.113478	−0.0567391	−	0.998389i	$-0.518070\pi$
$997$	−3.58311	−0.113478	−0.0567391	+	0.998389i	$0.518070\pi$
$998$	0.983371	0.0311281
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.n.1.1	yes	12
3.2	odd	2		8046.2.a.k.1.12	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.k.1.12	✓	12	3.2	odd	2
8046.2.a.n.1.1	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.06732 q^{5} +1.57160 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.06732 q^{5} +1.57160 q^{7} +1.00000 q^{8} -4.06732 q^{10} -5.58596 q^{11} -4.93913 q^{13} +1.57160 q^{14} +1.00000 q^{16} +7.87299 q^{17} +7.88914 q^{19} -4.06732 q^{20} -5.58596 q^{22} -0.940894 q^{23} +11.5431 q^{25} -4.93913 q^{26} +1.57160 q^{28} -0.204779 q^{29} -0.902429 q^{31} +1.00000 q^{32} +7.87299 q^{34} -6.39221 q^{35} +0.171003 q^{37} +7.88914 q^{38} -4.06732 q^{40} -4.80594 q^{41} +7.68490 q^{43} -5.58596 q^{44} -0.940894 q^{46} +5.30779 q^{47} -4.53006 q^{49} +11.5431 q^{50} -4.93913 q^{52} +1.10798 q^{53} +22.7199 q^{55} +1.57160 q^{56} -0.204779 q^{58} +5.47835 q^{59} -14.9373 q^{61} -0.902429 q^{62} +1.00000 q^{64} +20.0890 q^{65} +0.418411 q^{67} +7.87299 q^{68} -6.39221 q^{70} -14.7301 q^{71} -0.739195 q^{73} +0.171003 q^{74} +7.88914 q^{76} -8.77892 q^{77} +4.33354 q^{79} -4.06732 q^{80} -4.80594 q^{82} -14.3148 q^{83} -32.0219 q^{85} +7.68490 q^{86} -5.58596 q^{88} +0.822766 q^{89} -7.76235 q^{91} -0.940894 q^{92} +5.30779 q^{94} -32.0876 q^{95} +4.95746 q^{97} -4.53006 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} - 3 q^{5} - 6 q^{7} + 12 q^{8} - 3 q^{10} - 14 q^{11} - 3 q^{13} - 6 q^{14} + 12 q^{16} - 8 q^{17} - 4 q^{19} - 3 q^{20} - 14 q^{22} - 13 q^{23} + 11 q^{25} - 3 q^{26} - 6 q^{28} - 23 q^{29} - 14 q^{31} + 12 q^{32} - 8 q^{34} - 32 q^{35} - 19 q^{37} - 4 q^{38} - 3 q^{40} - 30 q^{41} - 15 q^{43} - 14 q^{44} - 13 q^{46} + q^{47} + 14 q^{49} + 11 q^{50} - 3 q^{52} - 16 q^{53} - 7 q^{55} - 6 q^{56} - 23 q^{58} - 26 q^{59} - 16 q^{61} - 14 q^{62} + 12 q^{64} - 8 q^{65} - 39 q^{67} - 8 q^{68} - 32 q^{70} - 15 q^{71} - 2 q^{73} - 19 q^{74} - 4 q^{76} - 34 q^{77} - 13 q^{79} - 3 q^{80} - 30 q^{82} - 6 q^{83} - 11 q^{85} - 15 q^{86} - 14 q^{88} - 18 q^{89} - 35 q^{91} - 13 q^{92} + q^{94} - 51 q^{95} + 19 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 - 3 * q^5 - 6 * q^7 + 12 * q^8 - 3 * q^10 - 14 * q^11 - 3 * q^13 - 6 * q^14 + 12 * q^16 - 8 * q^17 - 4 * q^19 - 3 * q^20 - 14 * q^22 - 13 * q^23 + 11 * q^25 - 3 * q^26 - 6 * q^28 - 23 * q^29 - 14 * q^31 + 12 * q^32 - 8 * q^34 - 32 * q^35 - 19 * q^37 - 4 * q^38 - 3 * q^40 - 30 * q^41 - 15 * q^43 - 14 * q^44 - 13 * q^46 + q^47 + 14 * q^49 + 11 * q^50 - 3 * q^52 - 16 * q^53 - 7 * q^55 - 6 * q^56 - 23 * q^58 - 26 * q^59 - 16 * q^61 - 14 * q^62 + 12 * q^64 - 8 * q^65 - 39 * q^67 - 8 * q^68 - 32 * q^70 - 15 * q^71 - 2 * q^73 - 19 * q^74 - 4 * q^76 - 34 * q^77 - 13 * q^79 - 3 * q^80 - 30 * q^82 - 6 * q^83 - 11 * q^85 - 15 * q^86 - 14 * q^88 - 18 * q^89 - 35 * q^91 - 13 * q^92 + q^94 - 51 * q^95 + 19 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more