LMFDB - Embedded newform 8046.2.a.r.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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 $ x^14 - 2x^13 - 42x^12 + 70x^11 + 680x^10 - 882x^9 - 5559x^8 + 5066x^7 + 24455x^6 - 12990x^5 - 55580x^4 + 9808x^3 + 53551x^2 + 6282*x - 7083
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			$-3.97637$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.97637 q^{5} +3.97870 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.97637 q^{5} +3.97870 q^{7} +1.00000 q^{8} -3.97637 q^{10} +4.91504 q^{11} +6.34583 q^{13} +3.97870 q^{14} +1.00000 q^{16} +3.91503 q^{17} +7.78524 q^{19} -3.97637 q^{20} +4.91504 q^{22} +0.326801 q^{23} +10.8115 q^{25} +6.34583 q^{26} +3.97870 q^{28} -2.16607 q^{29} +3.23819 q^{31} +1.00000 q^{32} +3.91503 q^{34} -15.8208 q^{35} -3.13933 q^{37} +7.78524 q^{38} -3.97637 q^{40} -2.94581 q^{41} -8.02158 q^{43} +4.91504 q^{44} +0.326801 q^{46} -0.363016 q^{47} +8.83003 q^{49} +10.8115 q^{50} +6.34583 q^{52} -8.28074 q^{53} -19.5440 q^{55} +3.97870 q^{56} -2.16607 q^{58} +1.35039 q^{59} +2.36771 q^{61} +3.23819 q^{62} +1.00000 q^{64} -25.2334 q^{65} +12.3873 q^{67} +3.91503 q^{68} -15.8208 q^{70} +16.4639 q^{71} -14.6270 q^{73} -3.13933 q^{74} +7.78524 q^{76} +19.5554 q^{77} -3.09750 q^{79} -3.97637 q^{80} -2.94581 q^{82} -12.8250 q^{83} -15.5676 q^{85} -8.02158 q^{86} +4.91504 q^{88} +7.48086 q^{89} +25.2482 q^{91} +0.326801 q^{92} -0.363016 q^{94} -30.9570 q^{95} -12.6702 q^{97} +8.83003 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8}+O(q^{10}) $ 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8	$ 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 4 q^{14} + 14 q^{16} + 9 q^{17} + 14 q^{19} + 2 q^{20} + 2 q^{22} + 30 q^{23} + 18 q^{25} + 4 q^{26} + 4 q^{28} + 6 q^{29} + 11 q^{31} + 14 q^{32} + 9 q^{34} - 18 q^{35} + 13 q^{37} + 14 q^{38} + 2 q^{40} - 2 q^{41} + 12 q^{43} + 2 q^{44} + 30 q^{46} + 21 q^{47} + 32 q^{49} + 18 q^{50} + 4 q^{52} + 22 q^{53} - 7 q^{55} + 4 q^{56} + 6 q^{58} + 14 q^{59} + 31 q^{61} + 11 q^{62} + 14 q^{64} + 24 q^{67} + 9 q^{68} - 18 q^{70} + 28 q^{71} + 24 q^{73} + 13 q^{74} + 14 q^{76} + 16 q^{77} + 65 q^{79} + 2 q^{80} - 2 q^{82} - 15 q^{83} - 19 q^{85} + 12 q^{86} + 2 q^{88} - 11 q^{89} + 68 q^{91} + 30 q^{92} + 21 q^{94} + 8 q^{95} + 23 q^{97} + 32 q^{98}+O(q^{100}) $ 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 4 * q^14 + 14 * q^16 + 9 * q^17 + 14 * q^19 + 2 * q^20 + 2 * q^22 + 30 * q^23 + 18 * q^25 + 4 * q^26 + 4 * q^28 + 6 * q^29 + 11 * q^31 + 14 * q^32 + 9 * q^34 - 18 * q^35 + 13 * q^37 + 14 * q^38 + 2 * q^40 - 2 * q^41 + 12 * q^43 + 2 * q^44 + 30 * q^46 + 21 * q^47 + 32 * q^49 + 18 * q^50 + 4 * q^52 + 22 * q^53 - 7 * q^55 + 4 * q^56 + 6 * q^58 + 14 * q^59 + 31 * q^61 + 11 * q^62 + 14 * q^64 + 24 * q^67 + 9 * q^68 - 18 * q^70 + 28 * q^71 + 24 * q^73 + 13 * q^74 + 14 * q^76 + 16 * q^77 + 65 * q^79 + 2 * q^80 - 2 * q^82 - 15 * q^83 - 19 * q^85 + 12 * q^86 + 2 * q^88 - 11 * q^89 + 68 * q^91 + 30 * q^92 + 21 * q^94 + 8 * q^95 + 23 * q^97 + 32 * 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
$5$	−3.97637	−1.77829	−0.889143	−	0.457629i	$-0.848699\pi$
$5$	−3.97637	−1.77829	−0.889143	+	0.457629i	$0.848699\pi$
$6$	0	0
$7$	3.97870	1.50381	0.751903	−	0.659274i	$-0.229136\pi$
$7$	3.97870	1.50381	0.751903	+	0.659274i	$0.229136\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.97637	−1.25744
$11$	4.91504	1.48194	0.740970	−	0.671538i	$-0.234366\pi$
$11$	4.91504	1.48194	0.740970	+	0.671538i	$0.234366\pi$
$12$	0	0
$13$	6.34583	1.76002	0.880009	−	0.474957i	$-0.157537\pi$
$13$	6.34583	1.76002	0.880009	+	0.474957i	$0.157537\pi$
$14$	3.97870	1.06335
$15$	0	0
$16$	1.00000	0.250000
$17$	3.91503	0.949534	0.474767	−	0.880112i	$-0.342532\pi$
$17$	3.91503	0.949534	0.474767	+	0.880112i	$0.342532\pi$
$18$	0	0
$19$	7.78524	1.78606	0.893028	−	0.450001i	$-0.148576\pi$
$19$	7.78524	1.78606	0.893028	+	0.450001i	$0.148576\pi$
$20$	−3.97637	−0.889143
$21$	0	0
$22$	4.91504	1.04789
$23$	0.326801	0.0681427	0.0340713	−	0.999419i	$-0.489153\pi$
$23$	0.326801	0.0681427	0.0340713	+	0.999419i	$0.489153\pi$
$24$	0	0
$25$	10.8115	2.16230
$26$	6.34583	1.24452
$27$	0	0
$28$	3.97870	0.751903
$29$	−2.16607	−0.402228	−0.201114	−	0.979568i	$-0.564456\pi$
$29$	−2.16607	−0.402228	−0.201114	+	0.979568i	$0.564456\pi$
$30$	0	0
$31$	3.23819	0.581596	0.290798	−	0.956784i	$-0.406079\pi$
$31$	3.23819	0.581596	0.290798	+	0.956784i	$0.406079\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.91503	0.671422
$35$	−15.8208	−2.67420
$36$	0	0
$37$	−3.13933	−0.516103	−0.258051	−	0.966131i	$-0.583080\pi$
$37$	−3.13933	−0.516103	−0.258051	+	0.966131i	$0.583080\pi$
$38$	7.78524	1.26293
$39$	0	0
$40$	−3.97637	−0.628719
$41$	−2.94581	−0.460058	−0.230029	−	0.973184i	$-0.573882\pi$
$41$	−2.94581	−0.460058	−0.230029	+	0.973184i	$0.573882\pi$
$42$	0	0
$43$	−8.02158	−1.22328	−0.611640	−	0.791136i	$-0.709490\pi$
$43$	−8.02158	−1.22328	−0.611640	+	0.791136i	$0.709490\pi$
$44$	4.91504	0.740970
$45$	0	0
$46$	0.326801	0.0481841
$47$	−0.363016	−0.0529513	−0.0264757	−	0.999649i	$-0.508428\pi$
$47$	−0.363016	−0.0529513	−0.0264757	+	0.999649i	$0.508428\pi$
$48$	0	0
$49$	8.83003	1.26143
$50$	10.8115	1.52898
$51$	0	0
$52$	6.34583	0.880009
$53$	−8.28074	−1.13745	−0.568723	−	0.822529i	$-0.692562\pi$
$53$	−8.28074	−1.13745	−0.568723	+	0.822529i	$0.692562\pi$
$54$	0	0
$55$	−19.5440	−2.63531
$56$	3.97870	0.531676
$57$	0	0
$58$	−2.16607	−0.284418
$59$	1.35039	0.175806	0.0879030	−	0.996129i	$-0.471983\pi$
$59$	1.35039	0.175806	0.0879030	+	0.996129i	$0.471983\pi$
$60$	0	0
$61$	2.36771	0.303154	0.151577	−	0.988445i	$-0.451565\pi$
$61$	2.36771	0.303154	0.151577	+	0.988445i	$0.451565\pi$
$62$	3.23819	0.411250
$63$	0	0
$64$	1.00000	0.125000
$65$	−25.2334	−3.12981
$66$	0	0
$67$	12.3873	1.51335	0.756673	−	0.653794i	$-0.226824\pi$
$67$	12.3873	1.51335	0.756673	+	0.653794i	$0.226824\pi$
$68$	3.91503	0.474767
$69$	0	0
$70$	−15.8208	−1.89094
$71$	16.4639	1.95391	0.976955	−	0.213447i	$-0.0684690\pi$
$71$	16.4639	1.95391	0.976955	+	0.213447i	$0.0684690\pi$
$72$	0	0
$73$	−14.6270	−1.71196	−0.855978	−	0.517012i	$-0.827044\pi$
$73$	−14.6270	−1.71196	−0.855978	+	0.517012i	$0.827044\pi$
$74$	−3.13933	−0.364940
$75$	0	0
$76$	7.78524	0.893028
$77$	19.5554	2.22855
$78$	0	0
$79$	−3.09750	−0.348496	−0.174248	−	0.984702i	$-0.555749\pi$
$79$	−3.09750	−0.348496	−0.174248	+	0.984702i	$0.555749\pi$
$80$	−3.97637	−0.444572
$81$	0	0
$82$	−2.94581	−0.325310
$83$	−12.8250	−1.40773	−0.703866	−	0.710333i	$-0.748544\pi$
$83$	−12.8250	−1.40773	−0.703866	+	0.710333i	$0.748544\pi$
$84$	0	0
$85$	−15.5676	−1.68854
$86$	−8.02158	−0.864989
$87$	0	0
$88$	4.91504	0.523945
$89$	7.48086	0.792969	0.396485	−	0.918041i	$-0.370230\pi$
$89$	7.48086	0.792969	0.396485	+	0.918041i	$0.370230\pi$
$90$	0	0
$91$	25.2482	2.64673
$92$	0.326801	0.0340713
$93$	0	0
$94$	−0.363016	−0.0374422
$95$	−30.9570	−3.17612
$96$	0	0
$97$	−12.6702	−1.28646	−0.643230	−	0.765673i	$-0.722406\pi$
$97$	−12.6702	−1.28646	−0.643230	+	0.765673i	$0.722406\pi$
$98$	8.83003	0.891968
$99$	0	0
$100$	10.8115	1.08115
$101$	−5.33340	−0.530693	−0.265346	−	0.964153i	$-0.585486\pi$
$101$	−5.33340	−0.530693	−0.265346	+	0.964153i	$0.585486\pi$
$102$	0	0
$103$	−10.3245	−1.01730	−0.508652	−	0.860972i	$-0.669856\pi$
$103$	−10.3245	−1.01730	−0.508652	+	0.860972i	$0.669856\pi$
$104$	6.34583	0.622260
$105$	0	0
$106$	−8.28074	−0.804296
$107$	−5.23190	−0.505786	−0.252893	−	0.967494i	$-0.581382\pi$
$107$	−5.23190	−0.505786	−0.252893	+	0.967494i	$0.581382\pi$
$108$	0	0
$109$	7.81093	0.748151	0.374076	−	0.927398i	$-0.377960\pi$
$109$	7.81093	0.748151	0.374076	+	0.927398i	$0.377960\pi$
$110$	−19.5440	−1.86345
$111$	0	0
$112$	3.97870	0.375952
$113$	−4.70760	−0.442854	−0.221427	−	0.975177i	$-0.571071\pi$
$113$	−4.70760	−0.442854	−0.221427	+	0.975177i	$0.571071\pi$
$114$	0	0
$115$	−1.29948	−0.121177
$116$	−2.16607	−0.201114
$117$	0	0
$118$	1.35039	0.124314
$119$	15.5767	1.42792
$120$	0	0
$121$	13.1576	1.19614
$122$	2.36771	0.214362
$123$	0	0
$124$	3.23819	0.290798
$125$	−23.1087	−2.06691
$126$	0	0
$127$	−13.0016	−1.15371	−0.576854	−	0.816847i	$-0.695720\pi$
$127$	−13.0016	−1.15371	−0.576854	+	0.816847i	$0.695720\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−25.2334	−2.21311
$131$	−2.29609	−0.200610	−0.100305	−	0.994957i	$-0.531982\pi$
$131$	−2.29609	−0.200610	−0.100305	+	0.994957i	$0.531982\pi$
$132$	0	0
$133$	30.9751	2.68588
$134$	12.3873	1.07010
$135$	0	0
$136$	3.91503	0.335711
$137$	−16.7768	−1.43334	−0.716671	−	0.697411i	$-0.754335\pi$
$137$	−16.7768	−1.43334	−0.716671	+	0.697411i	$0.754335\pi$
$138$	0	0
$139$	19.4538	1.65005	0.825026	−	0.565095i	$-0.191160\pi$
$139$	19.4538	1.65005	0.825026	+	0.565095i	$0.191160\pi$
$140$	−15.8208	−1.33710
$141$	0	0
$142$	16.4639	1.38162
$143$	31.1900	2.60824
$144$	0	0
$145$	8.61308	0.715277
$146$	−14.6270	−1.21054
$147$	0	0
$148$	−3.13933	−0.258051
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−17.1986	−1.39960	−0.699800	−	0.714339i	$-0.746727\pi$
$151$	−17.1986	−1.39960	−0.699800	+	0.714339i	$0.746727\pi$
$152$	7.78524	0.631466
$153$	0	0
$154$	19.5554	1.57582
$155$	−12.8762	−1.03424
$156$	0	0
$157$	2.78965	0.222638	0.111319	−	0.993785i	$-0.464492\pi$
$157$	2.78965	0.222638	0.111319	+	0.993785i	$0.464492\pi$
$158$	−3.09750	−0.246424
$159$	0	0
$160$	−3.97637	−0.314360
$161$	1.30024	0.102473
$162$	0	0
$163$	−19.5308	−1.52977	−0.764887	−	0.644165i	$-0.777205\pi$
$163$	−19.5308	−1.52977	−0.764887	+	0.644165i	$0.777205\pi$
$164$	−2.94581	−0.230029
$165$	0	0
$166$	−12.8250	−0.995416
$167$	19.1414	1.48121	0.740603	−	0.671942i	$-0.234540\pi$
$167$	19.1414	1.48121	0.740603	+	0.671942i	$0.234540\pi$
$168$	0	0
$169$	27.2696	2.09766
$170$	−15.5676	−1.19398
$171$	0	0
$172$	−8.02158	−0.611640
$173$	−5.18504	−0.394211	−0.197106	−	0.980382i	$-0.563154\pi$
$173$	−5.18504	−0.394211	−0.197106	+	0.980382i	$0.563154\pi$
$174$	0	0
$175$	43.0157	3.25168
$176$	4.91504	0.370485
$177$	0	0
$178$	7.48086	0.560714
$179$	−7.36999	−0.550859	−0.275430	−	0.961321i	$-0.588820\pi$
$179$	−7.36999	−0.550859	−0.275430	+	0.961321i	$0.588820\pi$
$180$	0	0
$181$	15.0483	1.11853	0.559266	−	0.828988i	$-0.311083\pi$
$181$	15.0483	1.11853	0.559266	+	0.828988i	$0.311083\pi$
$182$	25.2482	1.87152
$183$	0	0
$184$	0.326801	0.0240921
$185$	12.4831	0.917778
$186$	0	0
$187$	19.2425	1.40715
$188$	−0.363016	−0.0264757
$189$	0	0
$190$	−30.9570	−2.24585
$191$	−18.3141	−1.32516	−0.662581	−	0.748991i	$-0.730539\pi$
$191$	−18.3141	−1.32516	−0.662581	+	0.748991i	$0.730539\pi$
$192$	0	0
$193$	3.28497	0.236457	0.118229	−	0.992986i	$-0.462278\pi$
$193$	3.28497	0.236457	0.118229	+	0.992986i	$0.462278\pi$
$194$	−12.6702	−0.909665
$195$	0	0
$196$	8.83003	0.630717
$197$	9.35608	0.666593	0.333297	−	0.942822i	$-0.391839\pi$
$197$	9.35608	0.666593	0.333297	+	0.942822i	$0.391839\pi$
$198$	0	0
$199$	−6.64643	−0.471153	−0.235576	−	0.971856i	$-0.575698\pi$
$199$	−6.64643	−0.471153	−0.235576	+	0.971856i	$0.575698\pi$
$200$	10.8115	0.764489
$201$	0	0
$202$	−5.33340	−0.375257
$203$	−8.61812	−0.604874
$204$	0	0
$205$	11.7136	0.818114
$206$	−10.3245	−0.719342
$207$	0	0
$208$	6.34583	0.440004
$209$	38.2647	2.64683
$210$	0	0
$211$	22.1497	1.52485	0.762424	−	0.647078i	$-0.224009\pi$
$211$	22.1497	1.52485	0.762424	+	0.647078i	$0.224009\pi$
$212$	−8.28074	−0.568723
$213$	0	0
$214$	−5.23190	−0.357645
$215$	31.8968	2.17534
$216$	0	0
$217$	12.8838	0.874608
$218$	7.81093	0.529023
$219$	0	0
$220$	−19.5440	−1.31766
$221$	24.8441	1.67120
$222$	0	0
$223$	−3.49546	−0.234074	−0.117037	−	0.993128i	$-0.537340\pi$
$223$	−3.49546	−0.234074	−0.117037	+	0.993128i	$0.537340\pi$
$224$	3.97870	0.265838
$225$	0	0
$226$	−4.70760	−0.313145
$227$	−27.8165	−1.84625	−0.923123	−	0.384504i	$-0.874372\pi$
$227$	−27.8165	−1.84625	−0.923123	+	0.384504i	$0.874372\pi$
$228$	0	0
$229$	3.86757	0.255576	0.127788	−	0.991801i	$-0.459212\pi$
$229$	3.86757	0.255576	0.127788	+	0.991801i	$0.459212\pi$
$230$	−1.29948	−0.0856852
$231$	0	0
$232$	−2.16607	−0.142209
$233$	−18.7741	−1.22993	−0.614966	−	0.788554i	$-0.710830\pi$
$233$	−18.7741	−1.22993	−0.614966	+	0.788554i	$0.710830\pi$
$234$	0	0
$235$	1.44349	0.0941626
$236$	1.35039	0.0879030
$237$	0	0
$238$	15.5767	1.00969
$239$	28.6493	1.85317	0.926584	−	0.376088i	$-0.122731\pi$
$239$	28.6493	1.85317	0.926584	+	0.376088i	$0.122731\pi$
$240$	0	0
$241$	−19.7815	−1.27424	−0.637120	−	0.770765i	$-0.719874\pi$
$241$	−19.7815	−1.27424	−0.637120	+	0.770765i	$0.719874\pi$
$242$	13.1576	0.845802
$243$	0	0
$244$	2.36771	0.151577
$245$	−35.1115	−2.24319
$246$	0	0
$247$	49.4038	3.14349
$248$	3.23819	0.205625
$249$	0	0
$250$	−23.1087	−1.46152
$251$	3.15876	0.199379	0.0996896	−	0.995019i	$-0.468215\pi$
$251$	3.15876	0.199379	0.0996896	+	0.995019i	$0.468215\pi$
$252$	0	0
$253$	1.60624	0.100983
$254$	−13.0016	−0.815794
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.9377	−1.05655	−0.528273	−	0.849075i	$-0.677160\pi$
$257$	−16.9377	−1.05655	−0.528273	+	0.849075i	$0.677160\pi$
$258$	0	0
$259$	−12.4904	−0.776119
$260$	−25.2334	−1.56491
$261$	0	0
$262$	−2.29609	−0.141853
$263$	5.37727	0.331576	0.165788	−	0.986161i	$-0.446983\pi$
$263$	5.37727	0.331576	0.165788	+	0.986161i	$0.446983\pi$
$264$	0	0
$265$	32.9273	2.02271
$266$	30.9751	1.89921
$267$	0	0
$268$	12.3873	0.756673
$269$	17.5221	1.06834	0.534171	−	0.845377i	$-0.320624\pi$
$269$	17.5221	1.06834	0.534171	+	0.845377i	$0.320624\pi$
$270$	0	0
$271$	−12.6339	−0.767452	−0.383726	−	0.923447i	$-0.625359\pi$
$271$	−12.6339	−0.767452	−0.383726	+	0.923447i	$0.625359\pi$
$272$	3.91503	0.237384
$273$	0	0
$274$	−16.7768	−1.01353
$275$	53.1390	3.20440
$276$	0	0
$277$	−4.04656	−0.243134	−0.121567	−	0.992583i	$-0.538792\pi$
$277$	−4.04656	−0.243134	−0.121567	+	0.992583i	$0.538792\pi$
$278$	19.4538	1.16676
$279$	0	0
$280$	−15.8208	−0.945472
$281$	19.0295	1.13520	0.567602	−	0.823303i	$-0.307871\pi$
$281$	19.0295	1.13520	0.567602	+	0.823303i	$0.307871\pi$
$282$	0	0
$283$	−28.8527	−1.71511	−0.857557	−	0.514389i	$-0.828019\pi$
$283$	−28.8527	−1.71511	−0.857557	+	0.514389i	$0.828019\pi$
$284$	16.4639	0.976955
$285$	0	0
$286$	31.1900	1.84430
$287$	−11.7205	−0.691838
$288$	0	0
$289$	−1.67255	−0.0983850
$290$	8.61308	0.505777
$291$	0	0
$292$	−14.6270	−0.855978
$293$	22.5050	1.31476	0.657379	−	0.753560i	$-0.271665\pi$
$293$	22.5050	1.31476	0.657379	+	0.753560i	$0.271665\pi$
$294$	0	0
$295$	−5.36966	−0.312633
$296$	−3.13933	−0.182470
$297$	0	0
$298$	1.00000	0.0579284
$299$	2.07382	0.119932
$300$	0	0
$301$	−31.9154	−1.83958
$302$	−17.1986	−0.989666
$303$	0	0
$304$	7.78524	0.446514
$305$	−9.41489	−0.539095
$306$	0	0
$307$	−18.9722	−1.08280	−0.541402	−	0.840764i	$-0.682106\pi$
$307$	−18.9722	−1.08280	−0.541402	+	0.840764i	$0.682106\pi$
$308$	19.5554	1.11428
$309$	0	0
$310$	−12.8762	−0.731321
$311$	20.0019	1.13420	0.567101	−	0.823649i	$-0.308065\pi$
$311$	20.0019	1.13420	0.567101	+	0.823649i	$0.308065\pi$
$312$	0	0
$313$	0.815058	0.0460698	0.0230349	−	0.999735i	$-0.492667\pi$
$313$	0.815058	0.0460698	0.0230349	+	0.999735i	$0.492667\pi$
$314$	2.78965	0.157429
$315$	0	0
$316$	−3.09750	−0.174248
$317$	5.53542	0.310900	0.155450	−	0.987844i	$-0.450317\pi$
$317$	5.53542	0.310900	0.155450	+	0.987844i	$0.450317\pi$
$318$	0	0
$319$	−10.6463	−0.596078
$320$	−3.97637	−0.222286
$321$	0	0
$322$	1.30024	0.0724596
$323$	30.4794	1.69592
$324$	0	0
$325$	68.6080	3.80569
$326$	−19.5308	−1.08171
$327$	0	0
$328$	−2.94581	−0.162655
$329$	−1.44433	−0.0796285
$330$	0	0
$331$	−12.5667	−0.690727	−0.345363	−	0.938469i	$-0.612244\pi$
$331$	−12.5667	−0.690727	−0.345363	+	0.938469i	$0.612244\pi$
$332$	−12.8250	−0.703866
$333$	0	0
$334$	19.1414	1.04737
$335$	−49.2563	−2.69116
$336$	0	0
$337$	16.9323	0.922362	0.461181	−	0.887306i	$-0.347426\pi$
$337$	16.9323	0.922362	0.461181	+	0.887306i	$0.347426\pi$
$338$	27.2696	1.48327
$339$	0	0
$340$	−15.5676	−0.844272
$341$	15.9158	0.861890
$342$	0	0
$343$	7.28115	0.393145
$344$	−8.02158	−0.432495
$345$	0	0
$346$	−5.18504	−0.278749
$347$	−24.8180	−1.33230	−0.666149	−	0.745818i	$-0.732059\pi$
$347$	−24.8180	−1.33230	−0.666149	+	0.745818i	$0.732059\pi$
$348$	0	0
$349$	−30.7917	−1.64824	−0.824122	−	0.566412i	$-0.808331\pi$
$349$	−30.7917	−1.64824	−0.824122	+	0.566412i	$0.808331\pi$
$350$	43.0157	2.29929
$351$	0	0
$352$	4.91504	0.261972
$353$	2.21283	0.117777	0.0588885	−	0.998265i	$-0.481244\pi$
$353$	2.21283	0.117777	0.0588885	+	0.998265i	$0.481244\pi$
$354$	0	0
$355$	−65.4667	−3.47461
$356$	7.48086	0.396485
$357$	0	0
$358$	−7.36999	−0.389516
$359$	−19.0335	−1.00455	−0.502275	−	0.864708i	$-0.667503\pi$
$359$	−19.0335	−1.00455	−0.502275	+	0.864708i	$0.667503\pi$
$360$	0	0
$361$	41.6099	2.19000
$362$	15.0483	0.790922
$363$	0	0
$364$	25.2482	1.32336
$365$	58.1622	3.04435
$366$	0	0
$367$	−9.32625	−0.486826	−0.243413	−	0.969923i	$-0.578267\pi$
$367$	−9.32625	−0.486826	−0.243413	+	0.969923i	$0.578267\pi$
$368$	0.326801	0.0170357
$369$	0	0
$370$	12.4831	0.648967
$371$	−32.9465	−1.71050
$372$	0	0
$373$	−1.00989	−0.0522902	−0.0261451	−	0.999658i	$-0.508323\pi$
$373$	−1.00989	−0.0522902	−0.0261451	+	0.999658i	$0.508323\pi$
$374$	19.2425	0.995007
$375$	0	0
$376$	−0.363016	−0.0187211
$377$	−13.7455	−0.707929
$378$	0	0
$379$	−15.4253	−0.792346	−0.396173	−	0.918176i	$-0.629662\pi$
$379$	−15.4253	−0.792346	−0.396173	+	0.918176i	$0.629662\pi$
$380$	−30.9570	−1.58806
$381$	0	0
$382$	−18.3141	−0.937030
$383$	1.71294	0.0875272	0.0437636	−	0.999042i	$-0.486065\pi$
$383$	1.71294	0.0875272	0.0437636	+	0.999042i	$0.486065\pi$
$384$	0	0
$385$	−77.7597	−3.96300
$386$	3.28497	0.167201
$387$	0	0
$388$	−12.6702	−0.643230
$389$	−7.81808	−0.396392	−0.198196	−	0.980162i	$-0.563508\pi$
$389$	−7.81808	−0.396392	−0.198196	+	0.980162i	$0.563508\pi$
$390$	0	0
$391$	1.27943	0.0647038
$392$	8.83003	0.445984
$393$	0	0
$394$	9.35608	0.471353
$395$	12.3168	0.619725
$396$	0	0
$397$	34.9062	1.75189	0.875946	−	0.482409i	$-0.160238\pi$
$397$	34.9062	1.75189	0.875946	+	0.482409i	$0.160238\pi$
$398$	−6.64643	−0.333155
$399$	0	0
$400$	10.8115	0.540575
$401$	23.1660	1.15685	0.578427	−	0.815734i	$-0.303667\pi$
$401$	23.1660	1.15685	0.578427	+	0.815734i	$0.303667\pi$
$402$	0	0
$403$	20.5490	1.02362
$404$	−5.33340	−0.265346
$405$	0	0
$406$	−8.61812	−0.427710
$407$	−15.4299	−0.764833
$408$	0	0
$409$	11.4998	0.568627	0.284313	−	0.958731i	$-0.408234\pi$
$409$	11.4998	0.568627	0.284313	+	0.958731i	$0.408234\pi$
$410$	11.7136	0.578494
$411$	0	0
$412$	−10.3245	−0.508652
$413$	5.37280	0.264378
$414$	0	0
$415$	50.9971	2.50335
$416$	6.34583	0.311130
$417$	0	0
$418$	38.2647	1.87159
$419$	−22.9792	−1.12261	−0.561304	−	0.827610i	$-0.689700\pi$
$419$	−22.9792	−1.12261	−0.561304	+	0.827610i	$0.689700\pi$
$420$	0	0
$421$	13.0764	0.637304	0.318652	−	0.947872i	$-0.396770\pi$
$421$	13.0764	0.637304	0.318652	+	0.947872i	$0.396770\pi$
$422$	22.1497	1.07823
$423$	0	0
$424$	−8.28074	−0.402148
$425$	42.3274	2.05318
$426$	0	0
$427$	9.42040	0.455885
$428$	−5.23190	−0.252893
$429$	0	0
$430$	31.8968	1.53820
$431$	−10.5093	−0.506215	−0.253107	−	0.967438i	$-0.581453\pi$
$431$	−10.5093	−0.506215	−0.253107	+	0.967438i	$0.581453\pi$
$432$	0	0
$433$	−18.5679	−0.892318	−0.446159	−	0.894954i	$-0.647208\pi$
$433$	−18.5679	−0.892318	−0.446159	+	0.894954i	$0.647208\pi$
$434$	12.8838	0.618441
$435$	0	0
$436$	7.81093	0.374076
$437$	2.54422	0.121707
$438$	0	0
$439$	−7.42108	−0.354189	−0.177094	−	0.984194i	$-0.556670\pi$
$439$	−7.42108	−0.354189	−0.177094	+	0.984194i	$0.556670\pi$
$440$	−19.5440	−0.931724
$441$	0	0
$442$	24.8441	1.18171
$443$	35.6040	1.69160	0.845798	−	0.533504i	$-0.179125\pi$
$443$	35.6040	1.69160	0.845798	+	0.533504i	$0.179125\pi$
$444$	0	0
$445$	−29.7466	−1.41013
$446$	−3.49546	−0.165515
$447$	0	0
$448$	3.97870	0.187976
$449$	10.5148	0.496223	0.248111	−	0.968731i	$-0.420190\pi$
$449$	10.5148	0.496223	0.248111	+	0.968731i	$0.420190\pi$
$450$	0	0
$451$	−14.4788	−0.681778
$452$	−4.70760	−0.221427
$453$	0	0
$454$	−27.8165	−1.30549
$455$	−100.396	−4.70664
$456$	0	0
$457$	21.8345	1.02137	0.510687	−	0.859767i	$-0.329391\pi$
$457$	21.8345	1.02137	0.510687	+	0.859767i	$0.329391\pi$
$458$	3.86757	0.180720
$459$	0	0
$460$	−1.29948	−0.0605886
$461$	−3.20249	−0.149155	−0.0745775	−	0.997215i	$-0.523761\pi$
$461$	−3.20249	−0.149155	−0.0745775	+	0.997215i	$0.523761\pi$
$462$	0	0
$463$	8.35376	0.388232	0.194116	−	0.980979i	$-0.437816\pi$
$463$	8.35376	0.388232	0.194116	+	0.980979i	$0.437816\pi$
$464$	−2.16607	−0.100557
$465$	0	0
$466$	−18.7741	−0.869693
$467$	−23.7621	−1.09958	−0.549789	−	0.835304i	$-0.685292\pi$
$467$	−23.7621	−1.09958	−0.549789	+	0.835304i	$0.685292\pi$
$468$	0	0
$469$	49.2852	2.27578
$470$	1.44349	0.0665830
$471$	0	0
$472$	1.35039	0.0621568
$473$	−39.4264	−1.81283
$474$	0	0
$475$	84.1702	3.86199
$476$	15.5767	0.713958
$477$	0	0
$478$	28.6493	1.31039
$479$	31.2679	1.42867	0.714333	−	0.699806i	$-0.246730\pi$
$479$	31.2679	1.42867	0.714333	+	0.699806i	$0.246730\pi$
$480$	0	0
$481$	−19.9217	−0.908350
$482$	−19.7815	−0.901023
$483$	0	0
$484$	13.1576	0.598072
$485$	50.3812	2.28769
$486$	0	0
$487$	−19.6414	−0.890038	−0.445019	−	0.895521i	$-0.646803\pi$
$487$	−19.6414	−0.890038	−0.445019	+	0.895521i	$0.646803\pi$
$488$	2.36771	0.107181
$489$	0	0
$490$	−35.1115	−1.58617
$491$	38.1504	1.72170	0.860851	−	0.508857i	$-0.169932\pi$
$491$	38.1504	1.72170	0.860851	+	0.508857i	$0.169932\pi$
$492$	0	0
$493$	−8.48021	−0.381930
$494$	49.4038	2.22278
$495$	0	0
$496$	3.23819	0.145399
$497$	65.5050	2.93830
$498$	0	0
$499$	−21.7587	−0.974054	−0.487027	−	0.873387i	$-0.661919\pi$
$499$	−21.7587	−0.974054	−0.487027	+	0.873387i	$0.661919\pi$
$500$	−23.1087	−1.03345
$501$	0	0
$502$	3.15876	0.140982
$503$	−21.4893	−0.958161	−0.479081	−	0.877771i	$-0.659030\pi$
$503$	−21.4893	−0.958161	−0.479081	+	0.877771i	$0.659030\pi$
$504$	0	0
$505$	21.2076	0.943724
$506$	1.60624	0.0714060
$507$	0	0
$508$	−13.0016	−0.576854
$509$	−1.25423	−0.0555930	−0.0277965	−	0.999614i	$-0.508849\pi$
$509$	−1.25423	−0.0555930	−0.0277965	+	0.999614i	$0.508849\pi$
$510$	0	0
$511$	−58.1963	−2.57445
$512$	1.00000	0.0441942
$513$	0	0
$514$	−16.9377	−0.747091
$515$	41.0540	1.80906
$516$	0	0
$517$	−1.78424	−0.0784706
$518$	−12.4904	−0.548799
$519$	0	0
$520$	−25.2334	−1.10656
$521$	−21.1219	−0.925368	−0.462684	−	0.886523i	$-0.653113\pi$
$521$	−21.1219	−0.925368	−0.462684	+	0.886523i	$0.653113\pi$
$522$	0	0
$523$	−27.9767	−1.22333	−0.611667	−	0.791115i	$-0.709501\pi$
$523$	−27.9767	−1.22333	−0.611667	+	0.791115i	$0.709501\pi$
$524$	−2.29609	−0.100305
$525$	0	0
$526$	5.37727	0.234460
$527$	12.6776	0.552245
$528$	0	0
$529$	−22.8932	−0.995357
$530$	32.9273	1.43027
$531$	0	0
$532$	30.9751	1.34294
$533$	−18.6936	−0.809710
$534$	0	0
$535$	20.8039	0.899433
$536$	12.3873	0.535048
$537$	0	0
$538$	17.5221	0.755432
$539$	43.3999	1.86937
$540$	0	0
$541$	19.3166	0.830484	0.415242	−	0.909711i	$-0.363697\pi$
$541$	19.3166	0.830484	0.415242	+	0.909711i	$0.363697\pi$
$542$	−12.6339	−0.542671
$543$	0	0
$544$	3.91503	0.167855
$545$	−31.0591	−1.33043
$546$	0	0
$547$	39.0015	1.66758	0.833792	−	0.552079i	$-0.186165\pi$
$547$	39.0015	1.66758	0.833792	+	0.552079i	$0.186165\pi$
$548$	−16.7768	−0.716671
$549$	0	0
$550$	53.1390	2.26585
$551$	−16.8633	−0.718403
$552$	0	0
$553$	−12.3240	−0.524070
$554$	−4.04656	−0.171922
$555$	0	0
$556$	19.4538	0.825026
$557$	10.7342	0.454821	0.227410	−	0.973799i	$-0.426974\pi$
$557$	10.7342	0.454821	0.227410	+	0.973799i	$0.426974\pi$
$558$	0	0
$559$	−50.9036	−2.15299
$560$	−15.8208	−0.668549
$561$	0	0
$562$	19.0295	0.802710
$563$	−20.7387	−0.874033	−0.437016	−	0.899453i	$-0.643965\pi$
$563$	−20.7387	−0.874033	−0.437016	+	0.899453i	$0.643965\pi$
$564$	0	0
$565$	18.7192	0.787521
$566$	−28.8527	−1.21277
$567$	0	0
$568$	16.4639	0.690811
$569$	−33.5489	−1.40644	−0.703221	−	0.710971i	$-0.748256\pi$
$569$	−33.5489	−1.40644	−0.703221	+	0.710971i	$0.748256\pi$
$570$	0	0
$571$	−11.9612	−0.500561	−0.250281	−	0.968173i	$-0.580523\pi$
$571$	−11.9612	−0.500561	−0.250281	+	0.968173i	$0.580523\pi$
$572$	31.1900	1.30412
$573$	0	0
$574$	−11.7205	−0.489203
$575$	3.53321	0.147345
$576$	0	0
$577$	14.0032	0.582962	0.291481	−	0.956577i	$-0.405852\pi$
$577$	14.0032	0.582962	0.291481	+	0.956577i	$0.405852\pi$
$578$	−1.67255	−0.0695687
$579$	0	0
$580$	8.61308	0.357639
$581$	−51.0270	−2.11696
$582$	0	0
$583$	−40.7001	−1.68563
$584$	−14.6270	−0.605268
$585$	0	0
$586$	22.5050	0.929674
$587$	16.1231	0.665470	0.332735	−	0.943020i	$-0.392029\pi$
$587$	16.1231	0.665470	0.332735	+	0.943020i	$0.392029\pi$
$588$	0	0
$589$	25.2101	1.03876
$590$	−5.36966	−0.221065
$591$	0	0
$592$	−3.13933	−0.129026
$593$	−15.9631	−0.655525	−0.327763	−	0.944760i	$-0.606295\pi$
$593$	−15.9631	−0.655525	−0.327763	+	0.944760i	$0.606295\pi$
$594$	0	0
$595$	−61.9388	−2.53924
$596$	1.00000	0.0409616
$597$	0	0
$598$	2.07382	0.0848049
$599$	−20.0682	−0.819966	−0.409983	−	0.912093i	$-0.634465\pi$
$599$	−20.0682	−0.819966	−0.409983	+	0.912093i	$0.634465\pi$
$600$	0	0
$601$	−5.58689	−0.227894	−0.113947	−	0.993487i	$-0.536349\pi$
$601$	−5.58689	−0.227894	−0.113947	+	0.993487i	$0.536349\pi$
$602$	−31.9154	−1.30078
$603$	0	0
$604$	−17.1986	−0.699800
$605$	−52.3194	−2.12709
$606$	0	0
$607$	−29.8348	−1.21096	−0.605478	−	0.795862i	$-0.707018\pi$
$607$	−29.8348	−1.21096	−0.605478	+	0.795862i	$0.707018\pi$
$608$	7.78524	0.315733
$609$	0	0
$610$	−9.41489	−0.381198
$611$	−2.30364	−0.0931952
$612$	0	0
$613$	−17.4395	−0.704377	−0.352188	−	0.935929i	$-0.614562\pi$
$613$	−17.4395	−0.704377	−0.352188	+	0.935929i	$0.614562\pi$
$614$	−18.9722	−0.765657
$615$	0	0
$616$	19.5554	0.787911
$617$	−24.8736	−1.00137	−0.500687	−	0.865629i	$-0.666919\pi$
$617$	−24.8736	−1.00137	−0.500687	+	0.865629i	$0.666919\pi$
$618$	0	0
$619$	−14.4597	−0.581183	−0.290591	−	0.956847i	$-0.593852\pi$
$619$	−14.4597	−0.581183	−0.290591	+	0.956847i	$0.593852\pi$
$620$	−12.8762	−0.517122
$621$	0	0
$622$	20.0019	0.802001
$623$	29.7641	1.19247
$624$	0	0
$625$	37.8312	1.51325
$626$	0.815058	0.0325763
$627$	0	0
$628$	2.78965	0.111319
$629$	−12.2906	−0.490057
$630$	0	0
$631$	39.7281	1.58155	0.790775	−	0.612107i	$-0.209678\pi$
$631$	39.7281	1.58155	0.790775	+	0.612107i	$0.209678\pi$
$632$	−3.09750	−0.123212
$633$	0	0
$634$	5.53542	0.219840
$635$	51.6992	2.05162
$636$	0	0
$637$	56.0339	2.22014
$638$	−10.6463	−0.421491
$639$	0	0
$640$	−3.97637	−0.157180
$641$	−27.5770	−1.08923	−0.544613	−	0.838688i	$-0.683323\pi$
$641$	−27.5770	−1.08923	−0.544613	+	0.838688i	$0.683323\pi$
$642$	0	0
$643$	42.0236	1.65725	0.828625	−	0.559804i	$-0.189123\pi$
$643$	42.0236	1.65725	0.828625	+	0.559804i	$0.189123\pi$
$644$	1.30024	0.0512367
$645$	0	0
$646$	30.4794	1.19920
$647$	−40.8551	−1.60618	−0.803089	−	0.595859i	$-0.796812\pi$
$647$	−40.8551	−1.60618	−0.803089	+	0.595859i	$0.796812\pi$
$648$	0	0
$649$	6.63723	0.260534
$650$	68.6080	2.69103
$651$	0	0
$652$	−19.5308	−0.764887
$653$	11.4322	0.447375	0.223687	−	0.974661i	$-0.428191\pi$
$653$	11.4322	0.447375	0.223687	+	0.974661i	$0.428191\pi$
$654$	0	0
$655$	9.13008	0.356742
$656$	−2.94581	−0.115014
$657$	0	0
$658$	−1.44433	−0.0563059
$659$	−6.50796	−0.253514	−0.126757	−	0.991934i	$-0.540457\pi$
$659$	−6.50796	−0.253514	−0.126757	+	0.991934i	$0.540457\pi$
$660$	0	0
$661$	−8.58449	−0.333898	−0.166949	−	0.985966i	$-0.553392\pi$
$661$	−8.58449	−0.333898	−0.166949	+	0.985966i	$0.553392\pi$
$662$	−12.5667	−0.488418
$663$	0	0
$664$	−12.8250	−0.497708
$665$	−123.168	−4.77627
$666$	0	0
$667$	−0.707872	−0.0274089
$668$	19.1414	0.740603
$669$	0	0
$670$	−49.2563	−1.90294
$671$	11.6374	0.449256
$672$	0	0
$673$	29.2440	1.12727	0.563636	−	0.826023i	$-0.309402\pi$
$673$	29.2440	1.12727	0.563636	+	0.826023i	$0.309402\pi$
$674$	16.9323	0.652208
$675$	0	0
$676$	27.2696	1.04883
$677$	−13.4525	−0.517022	−0.258511	−	0.966008i	$-0.583232\pi$
$677$	−13.4525	−0.517022	−0.258511	+	0.966008i	$0.583232\pi$
$678$	0	0
$679$	−50.4107	−1.93459
$680$	−15.5676	−0.596990
$681$	0	0
$682$	15.9158	0.609448
$683$	−22.3612	−0.855626	−0.427813	−	0.903867i	$-0.640716\pi$
$683$	−22.3612	−0.855626	−0.427813	+	0.903867i	$0.640716\pi$
$684$	0	0
$685$	66.7109	2.54889
$686$	7.28115	0.277995
$687$	0	0
$688$	−8.02158	−0.305820
$689$	−52.5482	−2.00193
$690$	0	0
$691$	−31.5684	−1.20092	−0.600458	−	0.799656i	$-0.705015\pi$
$691$	−31.5684	−1.20092	−0.600458	+	0.799656i	$0.705015\pi$
$692$	−5.18504	−0.197106
$693$	0	0
$694$	−24.8180	−0.942077
$695$	−77.3556	−2.93426
$696$	0	0
$697$	−11.5329	−0.436841
$698$	−30.7917	−1.16548
$699$	0	0
$700$	43.0157	1.62584
$701$	−4.20560	−0.158843	−0.0794216	−	0.996841i	$-0.525307\pi$
$701$	−4.20560	−0.158843	−0.0794216	+	0.996841i	$0.525307\pi$
$702$	0	0
$703$	−24.4404	−0.921788
$704$	4.91504	0.185242
$705$	0	0
$706$	2.21283	0.0832809
$707$	−21.2200	−0.798059
$708$	0	0
$709$	28.9493	1.08721	0.543607	−	0.839340i	$-0.317058\pi$
$709$	28.9493	1.08721	0.543607	+	0.839340i	$0.317058\pi$
$710$	−65.4667	−2.45692
$711$	0	0
$712$	7.48086	0.280357
$713$	1.05824	0.0396315
$714$	0	0
$715$	−124.023	−4.63820
$716$	−7.36999	−0.275430
$717$	0	0
$718$	−19.0335	−0.710324
$719$	30.2377	1.12768	0.563838	−	0.825886i	$-0.309324\pi$
$719$	30.2377	1.12768	0.563838	+	0.825886i	$0.309324\pi$
$720$	0	0
$721$	−41.0781	−1.52983
$722$	41.6099	1.54856
$723$	0	0
$724$	15.0483	0.559266
$725$	−23.4185	−0.869739
$726$	0	0
$727$	−4.40245	−0.163278	−0.0816389	−	0.996662i	$-0.526015\pi$
$727$	−4.40245	−0.163278	−0.0816389	+	0.996662i	$0.526015\pi$
$728$	25.2482	0.935759
$729$	0	0
$730$	58.1622	2.15268
$731$	−31.4047	−1.16155
$732$	0	0
$733$	10.8311	0.400055	0.200028	−	0.979790i	$-0.435897\pi$
$733$	10.8311	0.400055	0.200028	+	0.979790i	$0.435897\pi$
$734$	−9.32625	−0.344238
$735$	0	0
$736$	0.326801	0.0120460
$737$	60.8839	2.24269
$738$	0	0
$739$	24.2015	0.890265	0.445132	−	0.895465i	$-0.353157\pi$
$739$	24.2015	0.890265	0.445132	+	0.895465i	$0.353157\pi$
$740$	12.4831	0.458889
$741$	0	0
$742$	−32.9465	−1.20951
$743$	36.7444	1.34802	0.674011	−	0.738721i	$-0.264570\pi$
$743$	36.7444	1.34802	0.674011	+	0.738721i	$0.264570\pi$
$744$	0	0
$745$	−3.97637	−0.145683
$746$	−1.00989	−0.0369748
$747$	0	0
$748$	19.2425	0.703576
$749$	−20.8161	−0.760605
$750$	0	0
$751$	35.2073	1.28473	0.642366	−	0.766398i	$-0.277953\pi$
$751$	35.2073	1.28473	0.642366	+	0.766398i	$0.277953\pi$
$752$	−0.363016	−0.0132378
$753$	0	0
$754$	−13.7455	−0.500582
$755$	68.3879	2.48889
$756$	0	0
$757$	35.3709	1.28558	0.642789	−	0.766043i	$-0.277777\pi$
$757$	35.3709	1.28558	0.642789	+	0.766043i	$0.277777\pi$
$758$	−15.4253	−0.560273
$759$	0	0
$760$	−30.9570	−1.12293
$761$	24.3913	0.884184	0.442092	−	0.896970i	$-0.354236\pi$
$761$	24.3913	0.884184	0.442092	+	0.896970i	$0.354236\pi$
$762$	0	0
$763$	31.0773	1.12507
$764$	−18.3141	−0.662581
$765$	0	0
$766$	1.71294	0.0618911
$767$	8.56936	0.309422
$768$	0	0
$769$	−5.97514	−0.215469	−0.107735	−	0.994180i	$-0.534360\pi$
$769$	−5.97514	−0.215469	−0.107735	+	0.994180i	$0.534360\pi$
$770$	−77.7597	−2.80226
$771$	0	0
$772$	3.28497	0.118229
$773$	−30.0234	−1.07987	−0.539933	−	0.841708i	$-0.681551\pi$
$773$	−30.0234	−1.07987	−0.539933	+	0.841708i	$0.681551\pi$
$774$	0	0
$775$	35.0097	1.25759
$776$	−12.6702	−0.454832
$777$	0	0
$778$	−7.81808	−0.280292
$779$	−22.9338	−0.821689
$780$	0	0
$781$	80.9208	2.89558
$782$	1.27943	0.0457525
$783$	0	0
$784$	8.83003	0.315358
$785$	−11.0927	−0.395914
$786$	0	0
$787$	−24.5778	−0.876104	−0.438052	−	0.898950i	$-0.644331\pi$
$787$	−24.5778	−0.876104	−0.438052	+	0.898950i	$0.644331\pi$
$788$	9.35608	0.333297
$789$	0	0
$790$	12.3168	0.438212
$791$	−18.7301	−0.665966
$792$	0	0
$793$	15.0251	0.533557
$794$	34.9062	1.23878
$795$	0	0
$796$	−6.64643	−0.235576
$797$	18.5470	0.656970	0.328485	−	0.944509i	$-0.393462\pi$
$797$	18.5470	0.656970	0.328485	+	0.944509i	$0.393462\pi$
$798$	0	0
$799$	−1.42122	−0.0502791
$800$	10.8115	0.382245
$801$	0	0
$802$	23.1660	0.818019
$803$	−71.8921	−2.53702
$804$	0	0
$805$	−5.17024	−0.182227
$806$	20.5490	0.723808
$807$	0	0
$808$	−5.33340	−0.187628
$809$	−18.2710	−0.642375	−0.321187	−	0.947016i	$-0.604082\pi$
$809$	−18.2710	−0.642375	−0.321187	+	0.947016i	$0.604082\pi$
$810$	0	0
$811$	24.2156	0.850323	0.425162	−	0.905117i	$-0.360217\pi$
$811$	24.2156	0.850323	0.425162	+	0.905117i	$0.360217\pi$
$812$	−8.61812	−0.302437
$813$	0	0
$814$	−15.4299	−0.540819
$815$	77.6618	2.72038
$816$	0	0
$817$	−62.4499	−2.18485
$818$	11.4998	0.402080
$819$	0	0
$820$	11.7136	0.409057
$821$	37.7645	1.31799	0.658995	−	0.752147i	$-0.270982\pi$
$821$	37.7645	1.31799	0.658995	+	0.752147i	$0.270982\pi$
$822$	0	0
$823$	−43.6502	−1.52155	−0.760776	−	0.649015i	$-0.775181\pi$
$823$	−43.6502	−1.52155	−0.760776	+	0.649015i	$0.775181\pi$
$824$	−10.3245	−0.359671
$825$	0	0
$826$	5.37280	0.186944
$827$	7.99599	0.278048	0.139024	−	0.990289i	$-0.455604\pi$
$827$	7.99599	0.278048	0.139024	+	0.990289i	$0.455604\pi$
$828$	0	0
$829$	35.8206	1.24410	0.622051	−	0.782977i	$-0.286300\pi$
$829$	35.8206	1.24410	0.622051	+	0.782977i	$0.286300\pi$
$830$	50.9971	1.77014
$831$	0	0
$832$	6.34583	0.220002
$833$	34.5698	1.19777
$834$	0	0
$835$	−76.1133	−2.63401
$836$	38.2647	1.32341
$837$	0	0
$838$	−22.9792	−0.793804
$839$	18.9876	0.655524	0.327762	−	0.944760i	$-0.393706\pi$
$839$	18.9876	0.655524	0.327762	+	0.944760i	$0.393706\pi$
$840$	0	0
$841$	−24.3082	−0.838212
$842$	13.0764	0.450642
$843$	0	0
$844$	22.1497	0.762424
$845$	−108.434	−3.73024
$846$	0	0
$847$	52.3501	1.79877
$848$	−8.28074	−0.284362
$849$	0	0
$850$	42.3274	1.45182
$851$	−1.02594	−0.0351686
$852$	0	0
$853$	31.2321	1.06937	0.534683	−	0.845053i	$-0.320431\pi$
$853$	31.2321	1.06937	0.534683	+	0.845053i	$0.320431\pi$
$854$	9.42040	0.322360
$855$	0	0
$856$	−5.23190	−0.178823
$857$	26.7287	0.913034	0.456517	−	0.889715i	$-0.349097\pi$
$857$	26.7287	0.913034	0.456517	+	0.889715i	$0.349097\pi$
$858$	0	0
$859$	37.3634	1.27482	0.637412	−	0.770523i	$-0.280005\pi$
$859$	37.3634	1.27482	0.637412	+	0.770523i	$0.280005\pi$
$860$	31.8968	1.08767
$861$	0	0
$862$	−10.5093	−0.357948
$863$	46.7170	1.59027	0.795133	−	0.606435i	$-0.207401\pi$
$863$	46.7170	1.59027	0.795133	+	0.606435i	$0.207401\pi$
$864$	0	0
$865$	20.6176	0.701020
$866$	−18.5679	−0.630964
$867$	0	0
$868$	12.8838	0.437304
$869$	−15.2243	−0.516450
$870$	0	0
$871$	78.6075	2.66351
$872$	7.81093	0.264511
$873$	0	0
$874$	2.54422	0.0860596
$875$	−91.9425	−3.10823
$876$	0	0
$877$	5.21103	0.175964	0.0879819	−	0.996122i	$-0.471958\pi$
$877$	5.21103	0.175964	0.0879819	+	0.996122i	$0.471958\pi$
$878$	−7.42108	−0.250449
$879$	0	0
$880$	−19.5440	−0.658828
$881$	−4.11533	−0.138649	−0.0693246	−	0.997594i	$-0.522084\pi$
$881$	−4.11533	−0.138649	−0.0693246	+	0.997594i	$0.522084\pi$
$882$	0	0
$883$	8.48855	0.285663	0.142831	−	0.989747i	$-0.454379\pi$
$883$	8.48855	0.285663	0.142831	+	0.989747i	$0.454379\pi$
$884$	24.8441	0.835598
$885$	0	0
$886$	35.6040	1.19614
$887$	−28.6805	−0.962996	−0.481498	−	0.876447i	$-0.659907\pi$
$887$	−28.6805	−0.962996	−0.481498	+	0.876447i	$0.659907\pi$
$888$	0	0
$889$	−51.7295	−1.73495
$890$	−29.7466	−0.997110
$891$	0	0
$892$	−3.49546	−0.117037
$893$	−2.82616	−0.0945740
$894$	0	0
$895$	29.3058	0.979585
$896$	3.97870	0.132919
$897$	0	0
$898$	10.5148	0.350883
$899$	−7.01413	−0.233934
$900$	0	0
$901$	−32.4193	−1.08004
$902$	−14.4788	−0.482090
$903$	0	0
$904$	−4.70760	−0.156572
$905$	−59.8377	−1.98907
$906$	0	0
$907$	29.3899	0.975875	0.487937	−	0.872879i	$-0.337749\pi$
$907$	29.3899	0.975875	0.487937	+	0.872879i	$0.337749\pi$
$908$	−27.8165	−0.923123
$909$	0	0
$910$	−100.396	−3.32809
$911$	−16.2438	−0.538180	−0.269090	−	0.963115i	$-0.586723\pi$
$911$	−16.2438	−0.538180	−0.269090	+	0.963115i	$0.586723\pi$
$912$	0	0
$913$	−63.0356	−2.08617
$914$	21.8345	0.722221
$915$	0	0
$916$	3.86757	0.127788
$917$	−9.13543	−0.301678
$918$	0	0
$919$	−4.69765	−0.154961	−0.0774806	−	0.996994i	$-0.524688\pi$
$919$	−4.69765	−0.154961	−0.0774806	+	0.996994i	$0.524688\pi$
$920$	−1.29948	−0.0428426
$921$	0	0
$922$	−3.20249	−0.105468
$923$	104.477	3.43891
$924$	0	0
$925$	−33.9409	−1.11597
$926$	8.35376	0.274522
$927$	0	0
$928$	−2.16607	−0.0711046
$929$	−41.7030	−1.36823	−0.684116	−	0.729373i	$-0.739812\pi$
$929$	−41.7030	−1.36823	−0.684116	+	0.729373i	$0.739812\pi$
$930$	0	0
$931$	68.7439	2.25299
$932$	−18.7741	−0.614966
$933$	0	0
$934$	−23.7621	−0.777519
$935$	−76.5153	−2.50232
$936$	0	0
$937$	−42.4747	−1.38759	−0.693794	−	0.720173i	$-0.744062\pi$
$937$	−42.4747	−1.38759	−0.693794	+	0.720173i	$0.744062\pi$
$938$	49.2852	1.60922
$939$	0	0
$940$	1.44349	0.0470813
$941$	−19.7033	−0.642311	−0.321155	−	0.947027i	$-0.604071\pi$
$941$	−19.7033	−0.642311	−0.321155	+	0.947027i	$0.604071\pi$
$942$	0	0
$943$	−0.962692	−0.0313496
$944$	1.35039	0.0439515
$945$	0	0
$946$	−39.4264	−1.28186
$947$	15.1530	0.492406	0.246203	−	0.969218i	$-0.420817\pi$
$947$	15.1530	0.492406	0.246203	+	0.969218i	$0.420817\pi$
$948$	0	0
$949$	−92.8203	−3.01307
$950$	84.1702	2.73084
$951$	0	0
$952$	15.5767	0.504844
$953$	−8.88250	−0.287732	−0.143866	−	0.989597i	$-0.545953\pi$
$953$	−8.88250	−0.287732	−0.143866	+	0.989597i	$0.545953\pi$
$954$	0	0
$955$	72.8236	2.35652
$956$	28.6493	0.926584
$957$	0	0
$958$	31.2679	1.01022
$959$	−66.7500	−2.15547
$960$	0	0
$961$	−20.5141	−0.661746
$962$	−19.9217	−0.642300
$963$	0	0
$964$	−19.7815	−0.637120
$965$	−13.0623	−0.420489
$966$	0	0
$967$	−45.3412	−1.45807	−0.729037	−	0.684474i	$-0.760032\pi$
$967$	−45.3412	−1.45807	−0.729037	+	0.684474i	$0.760032\pi$
$968$	13.1576	0.422901
$969$	0	0
$970$	50.3812	1.61764
$971$	−11.3316	−0.363649	−0.181824	−	0.983331i	$-0.558200\pi$
$971$	−11.3316	−0.363649	−0.181824	+	0.983331i	$0.558200\pi$
$972$	0	0
$973$	77.4009	2.48136
$974$	−19.6414	−0.629352
$975$	0	0
$976$	2.36771	0.0757886
$977$	47.1107	1.50720	0.753602	−	0.657331i	$-0.228315\pi$
$977$	47.1107	1.50720	0.753602	+	0.657331i	$0.228315\pi$
$978$	0	0
$979$	36.7687	1.17513
$980$	−35.1115	−1.12159
$981$	0	0
$982$	38.1504	1.21743
$983$	−7.85591	−0.250565	−0.125282	−	0.992121i	$-0.539984\pi$
$983$	−7.85591	−0.250565	−0.125282	+	0.992121i	$0.539984\pi$
$984$	0	0
$985$	−37.2032	−1.18539
$986$	−8.48021	−0.270065
$987$	0	0
$988$	49.4038	1.57174
$989$	−2.62146	−0.0833576
$990$	0	0
$991$	−36.2906	−1.15281	−0.576404	−	0.817165i	$-0.695545\pi$
$991$	−36.2906	−1.15281	−0.576404	+	0.817165i	$0.695545\pi$
$992$	3.23819	0.102813
$993$	0	0
$994$	65.5050	2.07769
$995$	26.4286	0.837844
$996$	0	0
$997$	−6.21216	−0.196741	−0.0983705	−	0.995150i	$-0.531363\pi$
$997$	−6.21216	−0.196741	−0.0983705	+	0.995150i	$0.531363\pi$
$998$	−21.7587	−0.688760
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.r.1.1	yes	14
3.2	odd	2		8046.2.a.q.1.14	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.q.1.14	✓	14	3.2	odd	2
8046.2.a.r.1.1	yes	14	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.97637 q^{5} +3.97870 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.97637 q^{5} +3.97870 q^{7} +1.00000 q^{8} -3.97637 q^{10} +4.91504 q^{11} +6.34583 q^{13} +3.97870 q^{14} +1.00000 q^{16} +3.91503 q^{17} +7.78524 q^{19} -3.97637 q^{20} +4.91504 q^{22} +0.326801 q^{23} +10.8115 q^{25} +6.34583 q^{26} +3.97870 q^{28} -2.16607 q^{29} +3.23819 q^{31} +1.00000 q^{32} +3.91503 q^{34} -15.8208 q^{35} -3.13933 q^{37} +7.78524 q^{38} -3.97637 q^{40} -2.94581 q^{41} -8.02158 q^{43} +4.91504 q^{44} +0.326801 q^{46} -0.363016 q^{47} +8.83003 q^{49} +10.8115 q^{50} +6.34583 q^{52} -8.28074 q^{53} -19.5440 q^{55} +3.97870 q^{56} -2.16607 q^{58} +1.35039 q^{59} +2.36771 q^{61} +3.23819 q^{62} +1.00000 q^{64} -25.2334 q^{65} +12.3873 q^{67} +3.91503 q^{68} -15.8208 q^{70} +16.4639 q^{71} -14.6270 q^{73} -3.13933 q^{74} +7.78524 q^{76} +19.5554 q^{77} -3.09750 q^{79} -3.97637 q^{80} -2.94581 q^{82} -12.8250 q^{83} -15.5676 q^{85} -8.02158 q^{86} +4.91504 q^{88} +7.48086 q^{89} +25.2482 q^{91} +0.326801 q^{92} -0.363016 q^{94} -30.9570 q^{95} -12.6702 q^{97} +8.83003 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8}+O(q^{10}) \) 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8	\( 14 q + 14 q^{2} + 14 q^{4} + 2 q^{5} + 4 q^{7} + 14 q^{8} + 2 q^{10} + 2 q^{11} + 4 q^{13} + 4 q^{14} + 14 q^{16} + 9 q^{17} + 14 q^{19} + 2 q^{20} + 2 q^{22} + 30 q^{23} + 18 q^{25} + 4 q^{26} + 4 q^{28} + 6 q^{29} + 11 q^{31} + 14 q^{32} + 9 q^{34} - 18 q^{35} + 13 q^{37} + 14 q^{38} + 2 q^{40} - 2 q^{41} + 12 q^{43} + 2 q^{44} + 30 q^{46} + 21 q^{47} + 32 q^{49} + 18 q^{50} + 4 q^{52} + 22 q^{53} - 7 q^{55} + 4 q^{56} + 6 q^{58} + 14 q^{59} + 31 q^{61} + 11 q^{62} + 14 q^{64} + 24 q^{67} + 9 q^{68} - 18 q^{70} + 28 q^{71} + 24 q^{73} + 13 q^{74} + 14 q^{76} + 16 q^{77} + 65 q^{79} + 2 q^{80} - 2 q^{82} - 15 q^{83} - 19 q^{85} + 12 q^{86} + 2 q^{88} - 11 q^{89} + 68 q^{91} + 30 q^{92} + 21 q^{94} + 8 q^{95} + 23 q^{97} + 32 q^{98}+O(q^{100}) \) 14 * q + 14 * q^2 + 14 * q^4 + 2 * q^5 + 4 * q^7 + 14 * q^8 + 2 * q^10 + 2 * q^11 + 4 * q^13 + 4 * q^14 + 14 * q^16 + 9 * q^17 + 14 * q^19 + 2 * q^20 + 2 * q^22 + 30 * q^23 + 18 * q^25 + 4 * q^26 + 4 * q^28 + 6 * q^29 + 11 * q^31 + 14 * q^32 + 9 * q^34 - 18 * q^35 + 13 * q^37 + 14 * q^38 + 2 * q^40 - 2 * q^41 + 12 * q^43 + 2 * q^44 + 30 * q^46 + 21 * q^47 + 32 * q^49 + 18 * q^50 + 4 * q^52 + 22 * q^53 - 7 * q^55 + 4 * q^56 + 6 * q^58 + 14 * q^59 + 31 * q^61 + 11 * q^62 + 14 * q^64 + 24 * q^67 + 9 * q^68 - 18 * q^70 + 28 * q^71 + 24 * q^73 + 13 * q^74 + 14 * q^76 + 16 * q^77 + 65 * q^79 + 2 * q^80 - 2 * q^82 - 15 * q^83 - 19 * q^85 + 12 * q^86 + 2 * q^88 - 11 * q^89 + 68 * q^91 + 30 * q^92 + 21 * q^94 + 8 * q^95 + 23 * q^97 + 32 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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