LMFDB - Embedded newform 8046.2.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.6
Root			$0.521020$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.521020 q^{5} -3.80450 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.521020 q^{5} -3.80450 q^{7} +1.00000 q^{8} +0.521020 q^{10} -2.59545 q^{11} -5.94316 q^{13} -3.80450 q^{14} +1.00000 q^{16} -4.15572 q^{17} +5.07861 q^{19} +0.521020 q^{20} -2.59545 q^{22} -3.81082 q^{23} -4.72854 q^{25} -5.94316 q^{26} -3.80450 q^{28} +3.56021 q^{29} -0.333783 q^{31} +1.00000 q^{32} -4.15572 q^{34} -1.98222 q^{35} +10.9514 q^{37} +5.07861 q^{38} +0.521020 q^{40} +8.72616 q^{41} -2.32902 q^{43} -2.59545 q^{44} -3.81082 q^{46} +4.62868 q^{47} +7.47422 q^{49} -4.72854 q^{50} -5.94316 q^{52} -6.24324 q^{53} -1.35228 q^{55} -3.80450 q^{56} +3.56021 q^{58} +14.2298 q^{59} +9.85892 q^{61} -0.333783 q^{62} +1.00000 q^{64} -3.09650 q^{65} +0.999107 q^{67} -4.15572 q^{68} -1.98222 q^{70} +2.76220 q^{71} -1.75101 q^{73} +10.9514 q^{74} +5.07861 q^{76} +9.87440 q^{77} +1.63063 q^{79} +0.521020 q^{80} +8.72616 q^{82} -15.4908 q^{83} -2.16521 q^{85} -2.32902 q^{86} -2.59545 q^{88} +4.25344 q^{89} +22.6107 q^{91} -3.81082 q^{92} +4.62868 q^{94} +2.64605 q^{95} +7.53987 q^{97} +7.47422 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.521020	0.233007	0.116504	−	0.993190i	$-0.462831\pi$
$5$	0.521020	0.233007	0.116504	+	0.993190i	$0.462831\pi$
$6$	0	0
$7$	−3.80450	−1.43797	−0.718983	−	0.695028i	$-0.755392\pi$
$7$	−3.80450	−1.43797	−0.718983	+	0.695028i	$0.755392\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.521020	0.164761
$11$	−2.59545	−0.782558	−0.391279	−	0.920272i	$-0.627967\pi$
$11$	−2.59545	−0.782558	−0.391279	+	0.920272i	$0.627967\pi$
$12$	0	0
$13$	−5.94316	−1.64834	−0.824168	−	0.566346i	$-0.808357\pi$
$13$	−5.94316	−1.64834	−0.824168	+	0.566346i	$0.808357\pi$
$14$	−3.80450	−1.01680
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.15572	−1.00791	−0.503955	−	0.863730i	$-0.668122\pi$
$17$	−4.15572	−1.00791	−0.503955	+	0.863730i	$0.668122\pi$
$18$	0	0
$19$	5.07861	1.16511	0.582556	−	0.812790i	$-0.302053\pi$
$19$	5.07861	1.16511	0.582556	+	0.812790i	$0.302053\pi$
$20$	0.521020	0.116504
$21$	0	0
$22$	−2.59545	−0.553352
$23$	−3.81082	−0.794611	−0.397305	−	0.917686i	$-0.630055\pi$
$23$	−3.81082	−0.794611	−0.397305	+	0.917686i	$0.630055\pi$
$24$	0	0
$25$	−4.72854	−0.945708
$26$	−5.94316	−1.16555
$27$	0	0
$28$	−3.80450	−0.718983
$29$	3.56021	0.661115	0.330557	−	0.943786i	$-0.392763\pi$
$29$	3.56021	0.661115	0.330557	+	0.943786i	$0.392763\pi$
$30$	0	0
$31$	−0.333783	−0.0599492	−0.0299746	−	0.999551i	$-0.509543\pi$
$31$	−0.333783	−0.0599492	−0.0299746	+	0.999551i	$0.509543\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.15572	−0.712700
$35$	−1.98222	−0.335056
$36$	0	0
$37$	10.9514	1.80039	0.900197	−	0.435482i	$-0.143422\pi$
$37$	10.9514	1.80039	0.900197	+	0.435482i	$0.143422\pi$
$38$	5.07861	0.823859
$39$	0	0
$40$	0.521020	0.0823805
$41$	8.72616	1.36280	0.681399	−	0.731912i	$-0.261372\pi$
$41$	8.72616	1.36280	0.681399	+	0.731912i	$0.261372\pi$
$42$	0	0
$43$	−2.32902	−0.355172	−0.177586	−	0.984105i	$-0.556829\pi$
$43$	−2.32902	−0.355172	−0.177586	+	0.984105i	$0.556829\pi$
$44$	−2.59545	−0.391279
$45$	0	0
$46$	−3.81082	−0.561875
$47$	4.62868	0.675163	0.337581	−	0.941296i	$-0.390391\pi$
$47$	4.62868	0.675163	0.337581	+	0.941296i	$0.390391\pi$
$48$	0	0
$49$	7.47422	1.06775
$50$	−4.72854	−0.668716
$51$	0	0
$52$	−5.94316	−0.824168
$53$	−6.24324	−0.857575	−0.428787	−	0.903405i	$-0.641059\pi$
$53$	−6.24324	−0.857575	−0.428787	+	0.903405i	$0.641059\pi$
$54$	0	0
$55$	−1.35228	−0.182342
$56$	−3.80450	−0.508398
$57$	0	0
$58$	3.56021	0.467479
$59$	14.2298	1.85256	0.926280	−	0.376836i	$-0.122988\pi$
$59$	14.2298	1.85256	0.926280	+	0.376836i	$0.122988\pi$
$60$	0	0
$61$	9.85892	1.26231	0.631153	−	0.775659i	$-0.282582\pi$
$61$	9.85892	1.26231	0.631153	+	0.775659i	$0.282582\pi$
$62$	−0.333783	−0.0423905
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.09650	−0.384074
$66$	0	0
$67$	0.999107	0.122060	0.0610301	−	0.998136i	$-0.480561\pi$
$67$	0.999107	0.122060	0.0610301	+	0.998136i	$0.480561\pi$
$68$	−4.15572	−0.503955
$69$	0	0
$70$	−1.98222	−0.236921
$71$	2.76220	0.327812	0.163906	−	0.986476i	$-0.447591\pi$
$71$	2.76220	0.327812	0.163906	+	0.986476i	$0.447591\pi$
$72$	0	0
$73$	−1.75101	−0.204941	−0.102470	−	0.994736i	$-0.532675\pi$
$73$	−1.75101	−0.204941	−0.102470	+	0.994736i	$0.532675\pi$
$74$	10.9514	1.27307
$75$	0	0
$76$	5.07861	0.582556
$77$	9.87440	1.12529
$78$	0	0
$79$	1.63063	0.183461	0.0917303	−	0.995784i	$-0.470760\pi$
$79$	1.63063	0.183461	0.0917303	+	0.995784i	$0.470760\pi$
$80$	0.521020	0.0582518
$81$	0	0
$82$	8.72616	0.963643
$83$	−15.4908	−1.70034	−0.850168	−	0.526511i	$-0.823500\pi$
$83$	−15.4908	−1.70034	−0.850168	+	0.526511i	$0.823500\pi$
$84$	0	0
$85$	−2.16521	−0.234850
$86$	−2.32902	−0.251145
$87$	0	0
$88$	−2.59545	−0.276676
$89$	4.25344	0.450864	0.225432	−	0.974259i	$-0.427621\pi$
$89$	4.25344	0.450864	0.225432	+	0.974259i	$0.427621\pi$
$90$	0	0
$91$	22.6107	2.37025
$92$	−3.81082	−0.397305
$93$	0	0
$94$	4.62868	0.477412
$95$	2.64605	0.271480
$96$	0	0
$97$	7.53987	0.765558	0.382779	−	0.923840i	$-0.374967\pi$
$97$	7.53987	0.765558	0.382779	+	0.923840i	$0.374967\pi$
$98$	7.47422	0.755010
$99$	0	0
$100$	−4.72854	−0.472854
$101$	−4.29940	−0.427806	−0.213903	−	0.976855i	$-0.568618\pi$
$101$	−4.29940	−0.427806	−0.213903	+	0.976855i	$0.568618\pi$
$102$	0	0
$103$	18.6660	1.83922	0.919609	−	0.392836i	$-0.128506\pi$
$103$	18.6660	1.83922	0.919609	+	0.392836i	$0.128506\pi$
$104$	−5.94316	−0.582775
$105$	0	0
$106$	−6.24324	−0.606397
$107$	17.0297	1.64632	0.823161	−	0.567808i	$-0.192208\pi$
$107$	17.0297	1.64632	0.823161	+	0.567808i	$0.192208\pi$
$108$	0	0
$109$	7.62347	0.730196	0.365098	−	0.930969i	$-0.381035\pi$
$109$	7.62347	0.730196	0.365098	+	0.930969i	$0.381035\pi$
$110$	−1.35228	−0.128935
$111$	0	0
$112$	−3.80450	−0.359491
$113$	−0.881020	−0.0828794	−0.0414397	−	0.999141i	$-0.513194\pi$
$113$	−0.881020	−0.0828794	−0.0414397	+	0.999141i	$0.513194\pi$
$114$	0	0
$115$	−1.98551	−0.185150
$116$	3.56021	0.330557
$117$	0	0
$118$	14.2298	1.30996
$119$	15.8104	1.44934
$120$	0	0
$121$	−4.26363	−0.387603
$122$	9.85892	0.892585
$123$	0	0
$124$	−0.333783	−0.0299746
$125$	−5.06876	−0.453364
$126$	0	0
$127$	20.4932	1.81848	0.909239	−	0.416274i	$-0.136664\pi$
$127$	20.4932	1.81848	0.909239	+	0.416274i	$0.136664\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.09650	−0.271581
$131$	15.6511	1.36744	0.683720	−	0.729745i	$-0.260361\pi$
$131$	15.6511	1.36744	0.683720	+	0.729745i	$0.260361\pi$
$132$	0	0
$133$	−19.3216	−1.67539
$134$	0.999107	0.0863097
$135$	0	0
$136$	−4.15572	−0.356350
$137$	−16.6835	−1.42537	−0.712683	−	0.701486i	$-0.752520\pi$
$137$	−16.6835	−1.42537	−0.712683	+	0.701486i	$0.752520\pi$
$138$	0	0
$139$	−7.11331	−0.603343	−0.301671	−	0.953412i	$-0.597545\pi$
$139$	−7.11331	−0.603343	−0.301671	+	0.953412i	$0.597545\pi$
$140$	−1.98222	−0.167528
$141$	0	0
$142$	2.76220	0.231798
$143$	15.4252	1.28992
$144$	0	0
$145$	1.85494	0.154044
$146$	−1.75101	−0.144915
$147$	0	0
$148$	10.9514	0.900197
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−5.88124	−0.478609	−0.239304	−	0.970945i	$-0.576919\pi$
$151$	−5.88124	−0.478609	−0.239304	+	0.970945i	$0.576919\pi$
$152$	5.07861	0.411929
$153$	0	0
$154$	9.87440	0.795702
$155$	−0.173908	−0.0139686
$156$	0	0
$157$	16.5845	1.32359	0.661794	−	0.749686i	$-0.269795\pi$
$157$	16.5845	1.32359	0.661794	+	0.749686i	$0.269795\pi$
$158$	1.63063	0.129726
$159$	0	0
$160$	0.521020	0.0411902
$161$	14.4983	1.14262
$162$	0	0
$163$	−14.6879	−1.15044	−0.575221	−	0.817998i	$-0.695084\pi$
$163$	−14.6879	−1.15044	−0.575221	+	0.817998i	$0.695084\pi$
$164$	8.72616	0.681399
$165$	0	0
$166$	−15.4908	−1.20232
$167$	−11.3968	−0.881910	−0.440955	−	0.897529i	$-0.645360\pi$
$167$	−11.3968	−0.881910	−0.440955	+	0.897529i	$0.645360\pi$
$168$	0	0
$169$	22.3211	1.71701
$170$	−2.16521	−0.166064
$171$	0	0
$172$	−2.32902	−0.177586
$173$	1.64750	0.125257	0.0626285	−	0.998037i	$-0.480052\pi$
$173$	1.64750	0.125257	0.0626285	+	0.998037i	$0.480052\pi$
$174$	0	0
$175$	17.9897	1.35990
$176$	−2.59545	−0.195640
$177$	0	0
$178$	4.25344	0.318809
$179$	0.726639	0.0543116	0.0271558	−	0.999631i	$-0.491355\pi$
$179$	0.726639	0.0543116	0.0271558	+	0.999631i	$0.491355\pi$
$180$	0	0
$181$	−23.2314	−1.72677	−0.863387	−	0.504542i	$-0.831661\pi$
$181$	−23.2314	−1.72677	−0.863387	+	0.504542i	$0.831661\pi$
$182$	22.6107	1.67602
$183$	0	0
$184$	−3.81082	−0.280937
$185$	5.70588	0.419505
$186$	0	0
$187$	10.7860	0.788748
$188$	4.62868	0.337581
$189$	0	0
$190$	2.64605	0.191965
$191$	−1.37670	−0.0996145	−0.0498072	−	0.998759i	$-0.515861\pi$
$191$	−1.37670	−0.0996145	−0.0498072	+	0.998759i	$0.515861\pi$
$192$	0	0
$193$	−20.0379	−1.44236	−0.721180	−	0.692748i	$-0.756400\pi$
$193$	−20.0379	−1.44236	−0.721180	+	0.692748i	$0.756400\pi$
$194$	7.53987	0.541331
$195$	0	0
$196$	7.47422	0.533873
$197$	−14.1866	−1.01075	−0.505377	−	0.862899i	$-0.668646\pi$
$197$	−14.1866	−1.01075	−0.505377	+	0.862899i	$0.668646\pi$
$198$	0	0
$199$	−17.3796	−1.23200	−0.616002	−	0.787745i	$-0.711249\pi$
$199$	−17.3796	−1.23200	−0.616002	+	0.787745i	$0.711249\pi$
$200$	−4.72854	−0.334358
$201$	0	0
$202$	−4.29940	−0.302505
$203$	−13.5448	−0.950660
$204$	0	0
$205$	4.54650	0.317542
$206$	18.6660	1.30052
$207$	0	0
$208$	−5.94316	−0.412084
$209$	−13.1813	−0.911768
$210$	0	0
$211$	−19.6021	−1.34947	−0.674733	−	0.738062i	$-0.735741\pi$
$211$	−19.6021	−1.34947	−0.674733	+	0.738062i	$0.735741\pi$
$212$	−6.24324	−0.428787
$213$	0	0
$214$	17.0297	1.16413
$215$	−1.21347	−0.0827577
$216$	0	0
$217$	1.26988	0.0862049
$218$	7.62347	0.516327
$219$	0	0
$220$	−1.35228	−0.0911708
$221$	24.6981	1.66137
$222$	0	0
$223$	−16.1521	−1.08163	−0.540813	−	0.841143i	$-0.681883\pi$
$223$	−16.1521	−1.08163	−0.540813	+	0.841143i	$0.681883\pi$
$224$	−3.80450	−0.254199
$225$	0	0
$226$	−0.881020	−0.0586046
$227$	13.8099	0.916597	0.458298	−	0.888798i	$-0.348459\pi$
$227$	13.8099	0.916597	0.458298	+	0.888798i	$0.348459\pi$
$228$	0	0
$229$	1.21160	0.0800645	0.0400322	−	0.999198i	$-0.487254\pi$
$229$	1.21160	0.0800645	0.0400322	+	0.999198i	$0.487254\pi$
$230$	−1.98551	−0.130921
$231$	0	0
$232$	3.56021	0.233739
$233$	16.2688	1.06581	0.532904	−	0.846176i	$-0.321101\pi$
$233$	16.2688	1.06581	0.532904	+	0.846176i	$0.321101\pi$
$234$	0	0
$235$	2.41164	0.157318
$236$	14.2298	0.926280
$237$	0	0
$238$	15.8104	1.02484
$239$	19.9292	1.28912	0.644558	−	0.764556i	$-0.277042\pi$
$239$	19.9292	1.28912	0.644558	+	0.764556i	$0.277042\pi$
$240$	0	0
$241$	18.3050	1.17913	0.589564	−	0.807722i	$-0.299300\pi$
$241$	18.3050	1.17913	0.589564	+	0.807722i	$0.299300\pi$
$242$	−4.26363	−0.274076
$243$	0	0
$244$	9.85892	0.631153
$245$	3.89422	0.248792
$246$	0	0
$247$	−30.1830	−1.92050
$248$	−0.333783	−0.0211952
$249$	0	0
$250$	−5.06876	−0.320577
$251$	8.58652	0.541976	0.270988	−	0.962583i	$-0.412650\pi$
$251$	8.58652	0.541976	0.270988	+	0.962583i	$0.412650\pi$
$252$	0	0
$253$	9.89080	0.621829
$254$	20.4932	1.28586
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.6142	1.41064	0.705318	−	0.708891i	$-0.250804\pi$
$257$	22.6142	1.41064	0.705318	+	0.708891i	$0.250804\pi$
$258$	0	0
$259$	−41.6645	−2.58891
$260$	−3.09650	−0.192037
$261$	0	0
$262$	15.6511	0.966926
$263$	−4.52340	−0.278925	−0.139462	−	0.990227i	$-0.544537\pi$
$263$	−4.52340	−0.278925	−0.139462	+	0.990227i	$0.544537\pi$
$264$	0	0
$265$	−3.25285	−0.199821
$266$	−19.3216	−1.18468
$267$	0	0
$268$	0.999107	0.0610301
$269$	14.5857	0.889308	0.444654	−	0.895702i	$-0.353327\pi$
$269$	14.5857	0.889308	0.444654	+	0.895702i	$0.353327\pi$
$270$	0	0
$271$	25.4451	1.54568	0.772839	−	0.634602i	$-0.218836\pi$
$271$	25.4451	1.54568	0.772839	+	0.634602i	$0.218836\pi$
$272$	−4.15572	−0.251977
$273$	0	0
$274$	−16.6835	−1.00789
$275$	12.2727	0.740071
$276$	0	0
$277$	−29.2019	−1.75457	−0.877285	−	0.479970i	$-0.840647\pi$
$277$	−29.2019	−1.75457	−0.877285	+	0.479970i	$0.840647\pi$
$278$	−7.11331	−0.426628
$279$	0	0
$280$	−1.98222	−0.118460
$281$	12.2619	0.731486	0.365743	−	0.930716i	$-0.380815\pi$
$281$	12.2619	0.731486	0.365743	+	0.930716i	$0.380815\pi$
$282$	0	0
$283$	−31.8598	−1.89387	−0.946935	−	0.321426i	$-0.895838\pi$
$283$	−31.8598	−1.89387	−0.946935	+	0.321426i	$0.895838\pi$
$284$	2.76220	0.163906
$285$	0	0
$286$	15.4252	0.912110
$287$	−33.1987	−1.95966
$288$	0	0
$289$	0.269980	0.0158812
$290$	1.85494	0.108926
$291$	0	0
$292$	−1.75101	−0.102470
$293$	−6.41534	−0.374788	−0.187394	−	0.982285i	$-0.560004\pi$
$293$	−6.41534	−0.374788	−0.187394	+	0.982285i	$0.560004\pi$
$294$	0	0
$295$	7.41400	0.431660
$296$	10.9514	0.636536
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	22.6483	1.30979
$300$	0	0
$301$	8.86076	0.510725
$302$	−5.88124	−0.338427
$303$	0	0
$304$	5.07861	0.291278
$305$	5.13669	0.294126
$306$	0	0
$307$	22.0849	1.26045	0.630225	−	0.776412i	$-0.282962\pi$
$307$	22.0849	1.26045	0.630225	+	0.776412i	$0.282962\pi$
$308$	9.87440	0.562646
$309$	0	0
$310$	−0.173908	−0.00987729
$311$	−20.3773	−1.15549	−0.577744	−	0.816218i	$-0.696067\pi$
$311$	−20.3773	−1.15549	−0.577744	+	0.816218i	$0.696067\pi$
$312$	0	0
$313$	2.20781	0.124793	0.0623965	−	0.998051i	$-0.480126\pi$
$313$	2.20781	0.124793	0.0623965	+	0.998051i	$0.480126\pi$
$314$	16.5845	0.935918
$315$	0	0
$316$	1.63063	0.0917303
$317$	10.7150	0.601813	0.300906	−	0.953654i	$-0.402711\pi$
$317$	10.7150	0.601813	0.300906	+	0.953654i	$0.402711\pi$
$318$	0	0
$319$	−9.24036	−0.517361
$320$	0.521020	0.0291259
$321$	0	0
$322$	14.4983	0.807957
$323$	−21.1052	−1.17433
$324$	0	0
$325$	28.1024	1.55884
$326$	−14.6879	−0.813485
$327$	0	0
$328$	8.72616	0.481822
$329$	−17.6098	−0.970861
$330$	0	0
$331$	−1.73930	−0.0956004	−0.0478002	−	0.998857i	$-0.515221\pi$
$331$	−1.73930	−0.0956004	−0.0478002	+	0.998857i	$0.515221\pi$
$332$	−15.4908	−0.850168
$333$	0	0
$334$	−11.3968	−0.623605
$335$	0.520554	0.0284409
$336$	0	0
$337$	−4.99432	−0.272058	−0.136029	−	0.990705i	$-0.543434\pi$
$337$	−4.99432	−0.272058	−0.136029	+	0.990705i	$0.543434\pi$
$338$	22.3211	1.21411
$339$	0	0
$340$	−2.16521	−0.117425
$341$	0.866318	0.0469137
$342$	0	0
$343$	−1.80417	−0.0974160
$344$	−2.32902	−0.125572
$345$	0	0
$346$	1.64750	0.0885701
$347$	25.6139	1.37502	0.687512	−	0.726173i	$-0.258703\pi$
$347$	25.6139	1.37502	0.687512	+	0.726173i	$0.258703\pi$
$348$	0	0
$349$	0.0786096	0.00420787	0.00210394	−	0.999998i	$-0.499330\pi$
$349$	0.0786096	0.00420787	0.00210394	+	0.999998i	$0.499330\pi$
$350$	17.9897	0.961591
$351$	0	0
$352$	−2.59545	−0.138338
$353$	−10.3044	−0.548446	−0.274223	−	0.961666i	$-0.588421\pi$
$353$	−10.3044	−0.548446	−0.274223	+	0.961666i	$0.588421\pi$
$354$	0	0
$355$	1.43916	0.0763826
$356$	4.25344	0.225432
$357$	0	0
$358$	0.726639	0.0384041
$359$	27.2468	1.43803	0.719016	−	0.694993i	$-0.244593\pi$
$359$	27.2468	1.43803	0.719016	+	0.694993i	$0.244593\pi$
$360$	0	0
$361$	6.79224	0.357487
$362$	−23.2314	−1.22101
$363$	0	0
$364$	22.6107	1.18512
$365$	−0.912313	−0.0477526
$366$	0	0
$367$	0.529145	0.0276211	0.0138106	−	0.999905i	$-0.495604\pi$
$367$	0.529145	0.0276211	0.0138106	+	0.999905i	$0.495604\pi$
$368$	−3.81082	−0.198653
$369$	0	0
$370$	5.70588	0.296635
$371$	23.7524	1.23316
$372$	0	0
$373$	−5.98357	−0.309818	−0.154909	−	0.987929i	$-0.549508\pi$
$373$	−5.98357	−0.309818	−0.154909	+	0.987929i	$0.549508\pi$
$374$	10.7860	0.557729
$375$	0	0
$376$	4.62868	0.238706
$377$	−21.1589	−1.08974
$378$	0	0
$379$	6.57408	0.337688	0.168844	−	0.985643i	$-0.445997\pi$
$379$	6.57408	0.337688	0.168844	+	0.985643i	$0.445997\pi$
$380$	2.64605	0.135740
$381$	0	0
$382$	−1.37670	−0.0704381
$383$	19.3332	0.987880	0.493940	−	0.869496i	$-0.335556\pi$
$383$	19.3332	0.987880	0.493940	+	0.869496i	$0.335556\pi$
$384$	0	0
$385$	5.14476	0.262201
$386$	−20.0379	−1.01990
$387$	0	0
$388$	7.53987	0.382779
$389$	21.1093	1.07029	0.535143	−	0.844761i	$-0.320258\pi$
$389$	21.1093	1.07029	0.535143	+	0.844761i	$0.320258\pi$
$390$	0	0
$391$	15.8367	0.800896
$392$	7.47422	0.377505
$393$	0	0
$394$	−14.1866	−0.714711
$395$	0.849593	0.0427476
$396$	0	0
$397$	−24.9547	−1.25244	−0.626219	−	0.779647i	$-0.715399\pi$
$397$	−24.9547	−1.25244	−0.626219	+	0.779647i	$0.715399\pi$
$398$	−17.3796	−0.871159
$399$	0	0
$400$	−4.72854	−0.236427
$401$	12.8772	0.643054	0.321527	−	0.946900i	$-0.395804\pi$
$401$	12.8772	0.643054	0.321527	+	0.946900i	$0.395804\pi$
$402$	0	0
$403$	1.98373	0.0988164
$404$	−4.29940	−0.213903
$405$	0	0
$406$	−13.5448	−0.672218
$407$	−28.4238	−1.40891
$408$	0	0
$409$	−32.1682	−1.59061	−0.795307	−	0.606207i	$-0.792690\pi$
$409$	−32.1682	−1.59061	−0.795307	+	0.606207i	$0.792690\pi$
$410$	4.54650	0.224536
$411$	0	0
$412$	18.6660	0.919609
$413$	−54.1372	−2.66392
$414$	0	0
$415$	−8.07102	−0.396191
$416$	−5.94316	−0.291387
$417$	0	0
$418$	−13.1813	−0.644717
$419$	−34.0954	−1.66567	−0.832835	−	0.553521i	$-0.813284\pi$
$419$	−34.0954	−1.66567	−0.832835	+	0.553521i	$0.813284\pi$
$420$	0	0
$421$	−34.5529	−1.68400	−0.842002	−	0.539475i	$-0.818623\pi$
$421$	−34.5529	−1.68400	−0.842002	+	0.539475i	$0.818623\pi$
$422$	−19.6021	−0.954216
$423$	0	0
$424$	−6.24324	−0.303198
$425$	19.6505	0.953188
$426$	0	0
$427$	−37.5083	−1.81515
$428$	17.0297	0.823161
$429$	0	0
$430$	−1.21347	−0.0585185
$431$	38.6876	1.86352	0.931758	−	0.363081i	$-0.118275\pi$
$431$	38.6876	1.86352	0.931758	+	0.363081i	$0.118275\pi$
$432$	0	0
$433$	12.0904	0.581026	0.290513	−	0.956871i	$-0.406174\pi$
$433$	12.0904	0.581026	0.290513	+	0.956871i	$0.406174\pi$
$434$	1.26988	0.0609561
$435$	0	0
$436$	7.62347	0.365098
$437$	−19.3537	−0.925811
$438$	0	0
$439$	−9.69013	−0.462485	−0.231242	−	0.972896i	$-0.574279\pi$
$439$	−9.69013	−0.462485	−0.231242	+	0.972896i	$0.574279\pi$
$440$	−1.35228	−0.0644675
$441$	0	0
$442$	24.6981	1.17477
$443$	−9.80151	−0.465684	−0.232842	−	0.972515i	$-0.574802\pi$
$443$	−9.80151	−0.465684	−0.232842	+	0.972515i	$0.574802\pi$
$444$	0	0
$445$	2.21613	0.105054
$446$	−16.1521	−0.764826
$447$	0	0
$448$	−3.80450	−0.179746
$449$	8.14566	0.384418	0.192209	−	0.981354i	$-0.438435\pi$
$449$	8.14566	0.384418	0.192209	+	0.981354i	$0.438435\pi$
$450$	0	0
$451$	−22.6483	−1.06647
$452$	−0.881020	−0.0414397
$453$	0	0
$454$	13.8099	0.648132
$455$	11.7806	0.552285
$456$	0	0
$457$	−2.08233	−0.0974074	−0.0487037	−	0.998813i	$-0.515509\pi$
$457$	−2.08233	−0.0974074	−0.0487037	+	0.998813i	$0.515509\pi$
$458$	1.21160	0.0566141
$459$	0	0
$460$	−1.98551	−0.0925750
$461$	−4.50128	−0.209646	−0.104823	−	0.994491i	$-0.533428\pi$
$461$	−4.50128	−0.209646	−0.104823	+	0.994491i	$0.533428\pi$
$462$	0	0
$463$	38.3084	1.78034	0.890170	−	0.455628i	$-0.150585\pi$
$463$	38.3084	1.78034	0.890170	+	0.455628i	$0.150585\pi$
$464$	3.56021	0.165279
$465$	0	0
$466$	16.2688	0.753640
$467$	−39.1511	−1.81170	−0.905848	−	0.423603i	$-0.860765\pi$
$467$	−39.1511	−1.81170	−0.905848	+	0.423603i	$0.860765\pi$
$468$	0	0
$469$	−3.80110	−0.175519
$470$	2.41164	0.111240
$471$	0	0
$472$	14.2298	0.654979
$473$	6.04486	0.277943
$474$	0	0
$475$	−24.0144	−1.10186
$476$	15.8104	0.724670
$477$	0	0
$478$	19.9292	0.911542
$479$	6.94569	0.317357	0.158678	−	0.987330i	$-0.449277\pi$
$479$	6.94569	0.317357	0.158678	+	0.987330i	$0.449277\pi$
$480$	0	0
$481$	−65.0857	−2.96765
$482$	18.3050	0.833769
$483$	0	0
$484$	−4.26363	−0.193801
$485$	3.92842	0.178380
$486$	0	0
$487$	25.5902	1.15960	0.579802	−	0.814758i	$-0.303130\pi$
$487$	25.5902	1.15960	0.579802	+	0.814758i	$0.303130\pi$
$488$	9.85892	0.446292
$489$	0	0
$490$	3.89422	0.175923
$491$	−12.5096	−0.564548	−0.282274	−	0.959334i	$-0.591089\pi$
$491$	−12.5096	−0.564548	−0.282274	+	0.959334i	$0.591089\pi$
$492$	0	0
$493$	−14.7952	−0.666344
$494$	−30.1830	−1.35800
$495$	0	0
$496$	−0.333783	−0.0149873
$497$	−10.5088	−0.471383
$498$	0	0
$499$	16.7637	0.750448	0.375224	−	0.926934i	$-0.377566\pi$
$499$	16.7637	0.750448	0.375224	+	0.926934i	$0.377566\pi$
$500$	−5.06876	−0.226682
$501$	0	0
$502$	8.58652	0.383235
$503$	3.36683	0.150119	0.0750597	−	0.997179i	$-0.476085\pi$
$503$	3.36683	0.150119	0.0750597	+	0.997179i	$0.476085\pi$
$504$	0	0
$505$	−2.24007	−0.0996820
$506$	9.89080	0.439700
$507$	0	0
$508$	20.4932	0.909239
$509$	16.3124	0.723035	0.361518	−	0.932365i	$-0.382259\pi$
$509$	16.3124	0.723035	0.361518	+	0.932365i	$0.382259\pi$
$510$	0	0
$511$	6.66173	0.294698
$512$	1.00000	0.0441942
$513$	0	0
$514$	22.6142	0.997470
$515$	9.72537	0.428551
$516$	0	0
$517$	−12.0135	−0.528354
$518$	−41.6645	−1.83063
$519$	0	0
$520$	−3.09650	−0.135791
$521$	6.52575	0.285898	0.142949	−	0.989730i	$-0.454341\pi$
$521$	6.52575	0.285898	0.142949	+	0.989730i	$0.454341\pi$
$522$	0	0
$523$	−7.96252	−0.348177	−0.174088	−	0.984730i	$-0.555698\pi$
$523$	−7.96252	−0.348177	−0.174088	+	0.984730i	$0.555698\pi$
$524$	15.6511	0.683720
$525$	0	0
$526$	−4.52340	−0.197230
$527$	1.38711	0.0604234
$528$	0	0
$529$	−8.47765	−0.368594
$530$	−3.25285	−0.141295
$531$	0	0
$532$	−19.3216	−0.837696
$533$	−51.8610	−2.24635
$534$	0	0
$535$	8.87280	0.383605
$536$	0.999107	0.0431548
$537$	0	0
$538$	14.5857	0.628836
$539$	−19.3990	−0.835573
$540$	0	0
$541$	−4.51098	−0.193942	−0.0969711	−	0.995287i	$-0.530915\pi$
$541$	−4.51098	−0.193942	−0.0969711	+	0.995287i	$0.530915\pi$
$542$	25.4451	1.09296
$543$	0	0
$544$	−4.15572	−0.178175
$545$	3.97198	0.170141
$546$	0	0
$547$	1.79031	0.0765482	0.0382741	−	0.999267i	$-0.487814\pi$
$547$	1.79031	0.0765482	0.0382741	+	0.999267i	$0.487814\pi$
$548$	−16.6835	−0.712683
$549$	0	0
$550$	12.2727	0.523309
$551$	18.0809	0.770273
$552$	0	0
$553$	−6.20375	−0.263810
$554$	−29.2019	−1.24067
$555$	0	0
$556$	−7.11331	−0.301671
$557$	35.2862	1.49513	0.747563	−	0.664191i	$-0.231224\pi$
$557$	35.2862	1.49513	0.747563	+	0.664191i	$0.231224\pi$
$558$	0	0
$559$	13.8417	0.585443
$560$	−1.98222	−0.0837641
$561$	0	0
$562$	12.2619	0.517239
$563$	−23.4692	−0.989111	−0.494555	−	0.869146i	$-0.664669\pi$
$563$	−23.4692	−0.989111	−0.494555	+	0.869146i	$0.664669\pi$
$564$	0	0
$565$	−0.459029	−0.0193115
$566$	−31.8598	−1.33917
$567$	0	0
$568$	2.76220	0.115899
$569$	9.61156	0.402937	0.201469	−	0.979495i	$-0.435429\pi$
$569$	9.61156	0.402937	0.201469	+	0.979495i	$0.435429\pi$
$570$	0	0
$571$	29.4788	1.23365	0.616826	−	0.787100i	$-0.288418\pi$
$571$	29.4788	1.23365	0.616826	+	0.787100i	$0.288418\pi$
$572$	15.4252	0.644959
$573$	0	0
$574$	−33.1987	−1.38569
$575$	18.0196	0.751470
$576$	0	0
$577$	−35.0951	−1.46103	−0.730514	−	0.682898i	$-0.760719\pi$
$577$	−35.0951	−1.46103	−0.730514	+	0.682898i	$0.760719\pi$
$578$	0.269980	0.0112297
$579$	0	0
$580$	1.85494	0.0770222
$581$	58.9348	2.44503
$582$	0	0
$583$	16.2040	0.671102
$584$	−1.75101	−0.0724575
$585$	0	0
$586$	−6.41534	−0.265015
$587$	36.7742	1.51784	0.758918	−	0.651187i	$-0.225728\pi$
$587$	36.7742	1.51784	0.758918	+	0.651187i	$0.225728\pi$
$588$	0	0
$589$	−1.69515	−0.0698476
$590$	7.41400	0.305230
$591$	0	0
$592$	10.9514	0.450099
$593$	7.34045	0.301436	0.150718	−	0.988577i	$-0.451841\pi$
$593$	7.34045	0.301436	0.150718	+	0.988577i	$0.451841\pi$
$594$	0	0
$595$	8.23754	0.337706
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	22.6483	0.926158
$599$	31.9550	1.30565	0.652824	−	0.757510i	$-0.273584\pi$
$599$	31.9550	1.30565	0.652824	+	0.757510i	$0.273584\pi$
$600$	0	0
$601$	29.0846	1.18639	0.593193	−	0.805060i	$-0.297867\pi$
$601$	29.0846	1.18639	0.593193	+	0.805060i	$0.297867\pi$
$602$	8.86076	0.361137
$603$	0	0
$604$	−5.88124	−0.239304
$605$	−2.22143	−0.0903142
$606$	0	0
$607$	40.5189	1.64461	0.822306	−	0.569045i	$-0.192687\pi$
$607$	40.5189	1.64461	0.822306	+	0.569045i	$0.192687\pi$
$608$	5.07861	0.205965
$609$	0	0
$610$	5.13669	0.207979
$611$	−27.5090	−1.11289
$612$	0	0
$613$	9.46247	0.382186	0.191093	−	0.981572i	$-0.438797\pi$
$613$	9.46247	0.382186	0.191093	+	0.981572i	$0.438797\pi$
$614$	22.0849	0.891273
$615$	0	0
$616$	9.87440	0.397851
$617$	30.0965	1.21164	0.605820	−	0.795601i	$-0.292845\pi$
$617$	30.0965	1.21164	0.605820	+	0.795601i	$0.292845\pi$
$618$	0	0
$619$	−12.5706	−0.505255	−0.252628	−	0.967564i	$-0.581295\pi$
$619$	−12.5706	−0.505255	−0.252628	+	0.967564i	$0.581295\pi$
$620$	−0.173908	−0.00698430
$621$	0	0
$622$	−20.3773	−0.817054
$623$	−16.1822	−0.648327
$624$	0	0
$625$	21.0018	0.840071
$626$	2.20781	0.0882420
$627$	0	0
$628$	16.5845	0.661794
$629$	−45.5108	−1.81463
$630$	0	0
$631$	18.3630	0.731019	0.365510	−	0.930808i	$-0.380895\pi$
$631$	18.3630	0.731019	0.365510	+	0.930808i	$0.380895\pi$
$632$	1.63063	0.0648631
$633$	0	0
$634$	10.7150	0.425546
$635$	10.6774	0.423719
$636$	0	0
$637$	−44.4205	−1.76000
$638$	−9.24036	−0.365829
$639$	0	0
$640$	0.521020	0.0205951
$641$	−31.4572	−1.24249	−0.621243	−	0.783618i	$-0.713372\pi$
$641$	−31.4572	−1.24249	−0.621243	+	0.783618i	$0.713372\pi$
$642$	0	0
$643$	20.0596	0.791072	0.395536	−	0.918451i	$-0.370559\pi$
$643$	20.0596	0.791072	0.395536	+	0.918451i	$0.370559\pi$
$644$	14.4983	0.571312
$645$	0	0
$646$	−21.1052	−0.830375
$647$	12.6291	0.496499	0.248250	−	0.968696i	$-0.420145\pi$
$647$	12.6291	0.496499	0.248250	+	0.968696i	$0.420145\pi$
$648$	0	0
$649$	−36.9327	−1.44974
$650$	28.1024	1.10227
$651$	0	0
$652$	−14.6879	−0.575221
$653$	15.6465	0.612295	0.306148	−	0.951984i	$-0.400960\pi$
$653$	15.6465	0.612295	0.306148	+	0.951984i	$0.400960\pi$
$654$	0	0
$655$	8.15452	0.318623
$656$	8.72616	0.340699
$657$	0	0
$658$	−17.6098	−0.686502
$659$	22.3652	0.871223	0.435612	−	0.900135i	$-0.356532\pi$
$659$	22.3652	0.871223	0.435612	+	0.900135i	$0.356532\pi$
$660$	0	0
$661$	3.01110	0.117118	0.0585591	−	0.998284i	$-0.481349\pi$
$661$	3.01110	0.117118	0.0585591	+	0.998284i	$0.481349\pi$
$662$	−1.73930	−0.0675997
$663$	0	0
$664$	−15.4908	−0.601160
$665$	−10.0669	−0.390378
$666$	0	0
$667$	−13.5673	−0.525329
$668$	−11.3968	−0.440955
$669$	0	0
$670$	0.520554	0.0201108
$671$	−25.5884	−0.987828
$672$	0	0
$673$	−27.4388	−1.05769	−0.528843	−	0.848720i	$-0.677374\pi$
$673$	−27.4388	−1.05769	−0.528843	+	0.848720i	$0.677374\pi$
$674$	−4.99432	−0.192374
$675$	0	0
$676$	22.3211	0.858505
$677$	47.8154	1.83770	0.918848	−	0.394611i	$-0.129121\pi$
$677$	47.8154	1.83770	0.918848	+	0.394611i	$0.129121\pi$
$678$	0	0
$679$	−28.6854	−1.10085
$680$	−2.16521	−0.0830320
$681$	0	0
$682$	0.866318	0.0331730
$683$	29.8137	1.14079	0.570395	−	0.821370i	$-0.306790\pi$
$683$	29.8137	1.14079	0.570395	+	0.821370i	$0.306790\pi$
$684$	0	0
$685$	−8.69243	−0.332121
$686$	−1.80417	−0.0688835
$687$	0	0
$688$	−2.32902	−0.0887930
$689$	37.1045	1.41357
$690$	0	0
$691$	40.1758	1.52836	0.764180	−	0.645003i	$-0.223144\pi$
$691$	40.1758	1.52836	0.764180	+	0.645003i	$0.223144\pi$
$692$	1.64750	0.0626285
$693$	0	0
$694$	25.6139	0.972289
$695$	−3.70618	−0.140583
$696$	0	0
$697$	−36.2635	−1.37358
$698$	0.0786096	0.00297542
$699$	0	0
$700$	17.9897	0.679948
$701$	44.4409	1.67851	0.839255	−	0.543739i	$-0.182992\pi$
$701$	44.4409	1.67851	0.839255	+	0.543739i	$0.182992\pi$
$702$	0	0
$703$	55.6177	2.09766
$704$	−2.59545	−0.0978198
$705$	0	0
$706$	−10.3044	−0.387810
$707$	16.3571	0.615171
$708$	0	0
$709$	−2.83559	−0.106493	−0.0532465	−	0.998581i	$-0.516957\pi$
$709$	−2.83559	−0.106493	−0.0532465	+	0.998581i	$0.516957\pi$
$710$	1.43916	0.0540107
$711$	0	0
$712$	4.25344	0.159404
$713$	1.27199	0.0476363
$714$	0	0
$715$	8.03683	0.300560
$716$	0.726639	0.0271558
$717$	0	0
$718$	27.2468	1.01684
$719$	−25.4873	−0.950514	−0.475257	−	0.879847i	$-0.657645\pi$
$719$	−25.4873	−0.950514	−0.475257	+	0.879847i	$0.657645\pi$
$720$	0	0
$721$	−71.0149	−2.64473
$722$	6.79224	0.252781
$723$	0	0
$724$	−23.2314	−0.863387
$725$	−16.8346	−0.625221
$726$	0	0
$727$	−16.2163	−0.601430	−0.300715	−	0.953714i	$-0.597225\pi$
$727$	−16.2163	−0.601430	−0.300715	+	0.953714i	$0.597225\pi$
$728$	22.6107	0.838010
$729$	0	0
$730$	−0.912313	−0.0337662
$731$	9.67875	0.357981
$732$	0	0
$733$	15.3690	0.567666	0.283833	−	0.958874i	$-0.408394\pi$
$733$	15.3690	0.567666	0.283833	+	0.958874i	$0.408394\pi$
$734$	0.529145	0.0195311
$735$	0	0
$736$	−3.81082	−0.140469
$737$	−2.59313	−0.0955193
$738$	0	0
$739$	18.6468	0.685932	0.342966	−	0.939348i	$-0.388568\pi$
$739$	18.6468	0.685932	0.342966	+	0.939348i	$0.388568\pi$
$740$	5.70588	0.209752
$741$	0	0
$742$	23.7524	0.871978
$743$	−43.4329	−1.59340	−0.796699	−	0.604377i	$-0.793422\pi$
$743$	−43.4329	−1.59340	−0.796699	+	0.604377i	$0.793422\pi$
$744$	0	0
$745$	−0.521020	−0.0190887
$746$	−5.98357	−0.219074
$747$	0	0
$748$	10.7860	0.394374
$749$	−64.7894	−2.36735
$750$	0	0
$751$	−15.1200	−0.551737	−0.275868	−	0.961195i	$-0.588965\pi$
$751$	−15.1200	−0.551737	−0.275868	+	0.961195i	$0.588965\pi$
$752$	4.62868	0.168791
$753$	0	0
$754$	−21.1589	−0.770562
$755$	−3.06424	−0.111519
$756$	0	0
$757$	−42.9383	−1.56062	−0.780309	−	0.625394i	$-0.784938\pi$
$757$	−42.9383	−1.56062	−0.780309	+	0.625394i	$0.784938\pi$
$758$	6.57408	0.238781
$759$	0	0
$760$	2.64605	0.0959825
$761$	39.4095	1.42859	0.714297	−	0.699843i	$-0.246747\pi$
$761$	39.4095	1.42859	0.714297	+	0.699843i	$0.246747\pi$
$762$	0	0
$763$	−29.0035	−1.05000
$764$	−1.37670	−0.0498072
$765$	0	0
$766$	19.3332	0.698536
$767$	−84.5698	−3.05364
$768$	0	0
$769$	11.4272	0.412074	0.206037	−	0.978544i	$-0.433943\pi$
$769$	11.4272	0.412074	0.206037	+	0.978544i	$0.433943\pi$
$770$	5.14476	0.185404
$771$	0	0
$772$	−20.0379	−0.721180
$773$	−38.2800	−1.37684	−0.688418	−	0.725314i	$-0.741694\pi$
$773$	−38.2800	−1.37684	−0.688418	+	0.725314i	$0.741694\pi$
$774$	0	0
$775$	1.57831	0.0566944
$776$	7.53987	0.270666
$777$	0	0
$778$	21.1093	0.756807
$779$	44.3167	1.58781
$780$	0	0
$781$	−7.16915	−0.256532
$782$	15.8367	0.566319
$783$	0	0
$784$	7.47422	0.266936
$785$	8.64086	0.308405
$786$	0	0
$787$	40.6786	1.45004	0.725018	−	0.688730i	$-0.241831\pi$
$787$	40.6786	1.45004	0.725018	+	0.688730i	$0.241831\pi$
$788$	−14.1866	−0.505377
$789$	0	0
$790$	0.849593	0.0302272
$791$	3.35184	0.119178
$792$	0	0
$793$	−58.5931	−2.08070
$794$	−24.9547	−0.885608
$795$	0	0
$796$	−17.3796	−0.616002
$797$	−13.6102	−0.482100	−0.241050	−	0.970513i	$-0.577492\pi$
$797$	−13.6102	−0.482100	−0.241050	+	0.970513i	$0.577492\pi$
$798$	0	0
$799$	−19.2355	−0.680503
$800$	−4.72854	−0.167179
$801$	0	0
$802$	12.8772	0.454708
$803$	4.54467	0.160378
$804$	0	0
$805$	7.55388	0.266239
$806$	1.98373	0.0698737
$807$	0	0
$808$	−4.29940	−0.151252
$809$	19.9821	0.702533	0.351267	−	0.936275i	$-0.385751\pi$
$809$	19.9821	0.702533	0.351267	+	0.936275i	$0.385751\pi$
$810$	0	0
$811$	21.0760	0.740077	0.370039	−	0.929016i	$-0.379344\pi$
$811$	21.0760	0.740077	0.370039	+	0.929016i	$0.379344\pi$
$812$	−13.5448	−0.475330
$813$	0	0
$814$	−28.4238	−0.996252
$815$	−7.65266	−0.268061
$816$	0	0
$817$	−11.8282	−0.413815
$818$	−32.1682	−1.12473
$819$	0	0
$820$	4.54650	0.158771
$821$	16.2202	0.566090	0.283045	−	0.959107i	$-0.408655\pi$
$821$	16.2202	0.566090	0.283045	+	0.959107i	$0.408655\pi$
$822$	0	0
$823$	31.0918	1.08379	0.541896	−	0.840446i	$-0.317707\pi$
$823$	31.0918	1.08379	0.541896	+	0.840446i	$0.317707\pi$
$824$	18.6660	0.650262
$825$	0	0
$826$	−54.1372	−1.88367
$827$	−3.20445	−0.111430	−0.0557149	−	0.998447i	$-0.517744\pi$
$827$	−3.20445	−0.111430	−0.0557149	+	0.998447i	$0.517744\pi$
$828$	0	0
$829$	23.2585	0.807801	0.403901	−	0.914803i	$-0.367654\pi$
$829$	23.2585	0.807801	0.403901	+	0.914803i	$0.367654\pi$
$830$	−8.07102	−0.280149
$831$	0	0
$832$	−5.94316	−0.206042
$833$	−31.0607	−1.07619
$834$	0	0
$835$	−5.93795	−0.205491
$836$	−13.1813	−0.455884
$837$	0	0
$838$	−34.0954	−1.17781
$839$	−38.6945	−1.33588	−0.667941	−	0.744214i	$-0.732824\pi$
$839$	−38.6945	−1.33588	−0.667941	+	0.744214i	$0.732824\pi$
$840$	0	0
$841$	−16.3249	−0.562927
$842$	−34.5529	−1.19077
$843$	0	0
$844$	−19.6021	−0.674733
$845$	11.6297	0.400075
$846$	0	0
$847$	16.2210	0.557359
$848$	−6.24324	−0.214394
$849$	0	0
$850$	19.6505	0.674005
$851$	−41.7337	−1.43061
$852$	0	0
$853$	−27.2112	−0.931695	−0.465848	−	0.884865i	$-0.654251\pi$
$853$	−27.2112	−0.931695	−0.465848	+	0.884865i	$0.654251\pi$
$854$	−37.5083	−1.28351
$855$	0	0
$856$	17.0297	0.582063
$857$	45.0972	1.54049	0.770245	−	0.637748i	$-0.220134\pi$
$857$	45.0972	1.54049	0.770245	+	0.637748i	$0.220134\pi$
$858$	0	0
$859$	−6.16692	−0.210413	−0.105206	−	0.994450i	$-0.533550\pi$
$859$	−6.16692	−0.210413	−0.105206	+	0.994450i	$0.533550\pi$
$860$	−1.21347	−0.0413788
$861$	0	0
$862$	38.6876	1.31770
$863$	20.3374	0.692292	0.346146	−	0.938181i	$-0.387490\pi$
$863$	20.3374	0.692292	0.346146	+	0.938181i	$0.387490\pi$
$864$	0	0
$865$	0.858379	0.0291858
$866$	12.0904	0.410848
$867$	0	0
$868$	1.26988	0.0431025
$869$	−4.23223	−0.143569
$870$	0	0
$871$	−5.93785	−0.201196
$872$	7.62347	0.258163
$873$	0	0
$874$	−19.3537	−0.654647
$875$	19.2841	0.651922
$876$	0	0
$877$	19.6879	0.664812	0.332406	−	0.943136i	$-0.392140\pi$
$877$	19.6879	0.664812	0.332406	+	0.943136i	$0.392140\pi$
$878$	−9.69013	−0.327026
$879$	0	0
$880$	−1.35228	−0.0455854
$881$	−17.0297	−0.573746	−0.286873	−	0.957969i	$-0.592616\pi$
$881$	−17.0297	−0.573746	−0.286873	+	0.957969i	$0.592616\pi$
$882$	0	0
$883$	−23.9168	−0.804866	−0.402433	−	0.915449i	$-0.631835\pi$
$883$	−23.9168	−0.804866	−0.402433	+	0.915449i	$0.631835\pi$
$884$	24.6981	0.830686
$885$	0	0
$886$	−9.80151	−0.329288
$887$	−10.4639	−0.351343	−0.175671	−	0.984449i	$-0.556210\pi$
$887$	−10.4639	−0.351343	−0.175671	+	0.984449i	$0.556210\pi$
$888$	0	0
$889$	−77.9664	−2.61491
$890$	2.21613	0.0742847
$891$	0	0
$892$	−16.1521	−0.540813
$893$	23.5073	0.786640
$894$	0	0
$895$	0.378594	0.0126550
$896$	−3.80450	−0.127099
$897$	0	0
$898$	8.14566	0.271824
$899$	−1.18834	−0.0396333
$900$	0	0
$901$	25.9451	0.864357
$902$	−22.6483	−0.754107
$903$	0	0
$904$	−0.881020	−0.0293023
$905$	−12.1040	−0.402351
$906$	0	0
$907$	−13.9978	−0.464789	−0.232395	−	0.972622i	$-0.574656\pi$
$907$	−13.9978	−0.464789	−0.232395	+	0.972622i	$0.574656\pi$
$908$	13.8099	0.458298
$909$	0	0
$910$	11.7806	0.390525
$911$	−13.4593	−0.445927	−0.222963	−	0.974827i	$-0.571573\pi$
$911$	−13.4593	−0.445927	−0.222963	+	0.974827i	$0.571573\pi$
$912$	0	0
$913$	40.2056	1.33061
$914$	−2.08233	−0.0688774
$915$	0	0
$916$	1.21160	0.0400322
$917$	−59.5445	−1.96633
$918$	0	0
$919$	16.8509	0.555860	0.277930	−	0.960601i	$-0.410352\pi$
$919$	16.8509	0.555860	0.277930	+	0.960601i	$0.410352\pi$
$920$	−1.98551	−0.0654604
$921$	0	0
$922$	−4.50128	−0.148242
$923$	−16.4162	−0.540345
$924$	0	0
$925$	−51.7840	−1.70265
$926$	38.3084	1.25889
$927$	0	0
$928$	3.56021	0.116870
$929$	−32.1070	−1.05340	−0.526699	−	0.850052i	$-0.676570\pi$
$929$	−32.1070	−1.05340	−0.526699	+	0.850052i	$0.676570\pi$
$930$	0	0
$931$	37.9586	1.24404
$932$	16.2688	0.532904
$933$	0	0
$934$	−39.1511	−1.28106
$935$	5.61970	0.183784
$936$	0	0
$937$	2.23263	0.0729367	0.0364683	−	0.999335i	$-0.488389\pi$
$937$	2.23263	0.0729367	0.0364683	+	0.999335i	$0.488389\pi$
$938$	−3.80110	−0.124110
$939$	0	0
$940$	2.41164	0.0786589
$941$	52.3267	1.70580	0.852901	−	0.522072i	$-0.174841\pi$
$941$	52.3267	1.70580	0.852901	+	0.522072i	$0.174841\pi$
$942$	0	0
$943$	−33.2538	−1.08289
$944$	14.2298	0.463140
$945$	0	0
$946$	6.04486	0.196535
$947$	4.09470	0.133060	0.0665299	−	0.997784i	$-0.478807\pi$
$947$	4.09470	0.133060	0.0665299	+	0.997784i	$0.478807\pi$
$948$	0	0
$949$	10.4066	0.337811
$950$	−24.0144	−0.779130
$951$	0	0
$952$	15.8104	0.512419
$953$	24.6103	0.797206	0.398603	−	0.917124i	$-0.369495\pi$
$953$	24.6103	0.797206	0.398603	+	0.917124i	$0.369495\pi$
$954$	0	0
$955$	−0.717288	−0.0232109
$956$	19.9292	0.644558
$957$	0	0
$958$	6.94569	0.224405
$959$	63.4723	2.04963
$960$	0	0
$961$	−30.8886	−0.996406
$962$	−65.0857	−2.09845
$963$	0	0
$964$	18.3050	0.589564
$965$	−10.4401	−0.336080
$966$	0	0
$967$	46.6103	1.49889	0.749443	−	0.662069i	$-0.230322\pi$
$967$	46.6103	1.49889	0.749443	+	0.662069i	$0.230322\pi$
$968$	−4.26363	−0.137038
$969$	0	0
$970$	3.92842	0.126134
$971$	23.5060	0.754344	0.377172	−	0.926143i	$-0.376896\pi$
$971$	23.5060	0.754344	0.377172	+	0.926143i	$0.376896\pi$
$972$	0	0
$973$	27.0626	0.867587
$974$	25.5902	0.819963
$975$	0	0
$976$	9.85892	0.315576
$977$	−46.2457	−1.47953	−0.739766	−	0.672864i	$-0.765064\pi$
$977$	−46.2457	−1.47953	−0.739766	+	0.672864i	$0.765064\pi$
$978$	0	0
$979$	−11.0396	−0.352827
$980$	3.89422	0.124396
$981$	0	0
$982$	−12.5096	−0.399196
$983$	41.5238	1.32440	0.662202	−	0.749325i	$-0.269622\pi$
$983$	41.5238	1.32440	0.662202	+	0.749325i	$0.269622\pi$
$984$	0	0
$985$	−7.39150	−0.235513
$986$	−14.7952	−0.471176
$987$	0	0
$988$	−30.1830	−0.960248
$989$	8.87547	0.282224
$990$	0	0
$991$	−56.8466	−1.80579	−0.902896	−	0.429859i	$-0.858563\pi$
$991$	−56.8466	−1.80579	−0.902896	+	0.429859i	$0.858563\pi$
$992$	−0.333783	−0.0105976
$993$	0	0
$994$	−10.5088	−0.333318
$995$	−9.05510	−0.287066
$996$	0	0
$997$	31.1355	0.986071	0.493035	−	0.870009i	$-0.335887\pi$
$997$	31.1355	0.986071	0.493035	+	0.870009i	$0.335887\pi$
$998$	16.7637	0.530647
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.p.1.6	yes	12
3.2	odd	2		8046.2.a.i.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.7	✓	12	3.2	odd	2
8046.2.a.p.1.6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.521020 q^{5} -3.80450 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.521020 q^{5} -3.80450 q^{7} +1.00000 q^{8} +0.521020 q^{10} -2.59545 q^{11} -5.94316 q^{13} -3.80450 q^{14} +1.00000 q^{16} -4.15572 q^{17} +5.07861 q^{19} +0.521020 q^{20} -2.59545 q^{22} -3.81082 q^{23} -4.72854 q^{25} -5.94316 q^{26} -3.80450 q^{28} +3.56021 q^{29} -0.333783 q^{31} +1.00000 q^{32} -4.15572 q^{34} -1.98222 q^{35} +10.9514 q^{37} +5.07861 q^{38} +0.521020 q^{40} +8.72616 q^{41} -2.32902 q^{43} -2.59545 q^{44} -3.81082 q^{46} +4.62868 q^{47} +7.47422 q^{49} -4.72854 q^{50} -5.94316 q^{52} -6.24324 q^{53} -1.35228 q^{55} -3.80450 q^{56} +3.56021 q^{58} +14.2298 q^{59} +9.85892 q^{61} -0.333783 q^{62} +1.00000 q^{64} -3.09650 q^{65} +0.999107 q^{67} -4.15572 q^{68} -1.98222 q^{70} +2.76220 q^{71} -1.75101 q^{73} +10.9514 q^{74} +5.07861 q^{76} +9.87440 q^{77} +1.63063 q^{79} +0.521020 q^{80} +8.72616 q^{82} -15.4908 q^{83} -2.16521 q^{85} -2.32902 q^{86} -2.59545 q^{88} +4.25344 q^{89} +22.6107 q^{91} -3.81082 q^{92} +4.62868 q^{94} +2.64605 q^{95} +7.53987 q^{97} +7.47422 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8} + 5 q^{10} + 6 q^{11} + 3 q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 8 q^{19} + 5 q^{20} + 6 q^{22} + 11 q^{23} + 11 q^{25} + 3 q^{26} + 6 q^{28} + 29 q^{29} + 2 q^{31} + 12 q^{32} + 6 q^{34} + 4 q^{35} + 5 q^{37} + 8 q^{38} + 5 q^{40} + 22 q^{41} + 9 q^{43} + 6 q^{44} + 11 q^{46} + 15 q^{47} + 14 q^{49} + 11 q^{50} + 3 q^{52} + 12 q^{53} + 13 q^{55} + 6 q^{56} + 29 q^{58} + 34 q^{59} - 4 q^{61} + 2 q^{62} + 12 q^{64} + 12 q^{65} + q^{67} + 6 q^{68} + 4 q^{70} + 21 q^{71} - 2 q^{73} + 5 q^{74} + 8 q^{76} + 34 q^{77} + 9 q^{79} + 5 q^{80} + 22 q^{82} + 10 q^{83} + 5 q^{85} + 9 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 11 q^{92} + 15 q^{94} + 69 q^{95} - 13 q^{97} + 14 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 5 * q^5 + 6 * q^7 + 12 * q^8 + 5 * q^10 + 6 * q^11 + 3 * q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 8 * q^19 + 5 * q^20 + 6 * q^22 + 11 * q^23 + 11 * q^25 + 3 * q^26 + 6 * q^28 + 29 * q^29 + 2 * q^31 + 12 * q^32 + 6 * q^34 + 4 * q^35 + 5 * q^37 + 8 * q^38 + 5 * q^40 + 22 * q^41 + 9 * q^43 + 6 * q^44 + 11 * q^46 + 15 * q^47 + 14 * q^49 + 11 * q^50 + 3 * q^52 + 12 * q^53 + 13 * q^55 + 6 * q^56 + 29 * q^58 + 34 * q^59 - 4 * q^61 + 2 * q^62 + 12 * q^64 + 12 * q^65 + q^67 + 6 * q^68 + 4 * q^70 + 21 * q^71 - 2 * q^73 + 5 * q^74 + 8 * q^76 + 34 * q^77 + 9 * q^79 + 5 * q^80 + 22 * q^82 + 10 * q^83 + 5 * q^85 + 9 * q^86 + 6 * q^88 - 2 * q^89 + 17 * q^91 + 11 * q^92 + 15 * q^94 + 69 * q^95 - 13 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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