LMFDB - Embedded newform 8046.2.a.g.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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 $ x^9 - 3x^8 - 9x^7 + 25x^6 + 29x^5 - 58x^4 - 43x^3 + 34x^2 + 25x + 2
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			$-0.836154$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.63508 q^{5} +4.18187 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.63508 q^{5} +4.18187 q^{7} -1.00000 q^{8} -1.63508 q^{10} +0.197300 q^{11} -5.53420 q^{13} -4.18187 q^{14} +1.00000 q^{16} -6.24995 q^{17} -1.76358 q^{19} +1.63508 q^{20} -0.197300 q^{22} +1.25693 q^{23} -2.32651 q^{25} +5.53420 q^{26} +4.18187 q^{28} +6.46749 q^{29} -9.67005 q^{31} -1.00000 q^{32} +6.24995 q^{34} +6.83770 q^{35} -8.61274 q^{37} +1.76358 q^{38} -1.63508 q^{40} +7.00464 q^{41} +4.69520 q^{43} +0.197300 q^{44} -1.25693 q^{46} +6.92211 q^{47} +10.4880 q^{49} +2.32651 q^{50} -5.53420 q^{52} +5.96851 q^{53} +0.322602 q^{55} -4.18187 q^{56} -6.46749 q^{58} -4.41785 q^{59} +6.34679 q^{61} +9.67005 q^{62} +1.00000 q^{64} -9.04888 q^{65} -15.3102 q^{67} -6.24995 q^{68} -6.83770 q^{70} -8.52896 q^{71} -0.842454 q^{73} +8.61274 q^{74} -1.76358 q^{76} +0.825083 q^{77} -2.79367 q^{79} +1.63508 q^{80} -7.00464 q^{82} -1.68295 q^{83} -10.2192 q^{85} -4.69520 q^{86} -0.197300 q^{88} -0.518659 q^{89} -23.1433 q^{91} +1.25693 q^{92} -6.92211 q^{94} -2.88360 q^{95} +2.99290 q^{97} -10.4880 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	$ 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) $ 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.63508	0.731231	0.365615	−	0.930766i	$-0.380859\pi$
$5$	1.63508	0.731231	0.365615	+	0.930766i	$0.380859\pi$
$6$	0	0
$7$	4.18187	1.58060	0.790299	−	0.612721i	$-0.209925\pi$
$7$	4.18187	1.58060	0.790299	+	0.612721i	$0.209925\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.63508	−0.517058
$11$	0.197300	0.0594882	0.0297441	−	0.999558i	$-0.490531\pi$
$11$	0.197300	0.0594882	0.0297441	+	0.999558i	$0.490531\pi$
$12$	0	0
$13$	−5.53420	−1.53491	−0.767456	−	0.641101i	$-0.778478\pi$
$13$	−5.53420	−1.53491	−0.767456	+	0.641101i	$0.778478\pi$
$14$	−4.18187	−1.11765
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.24995	−1.51584	−0.757918	−	0.652350i	$-0.773783\pi$
$17$	−6.24995	−1.51584	−0.757918	+	0.652350i	$0.773783\pi$
$18$	0	0
$19$	−1.76358	−0.404593	−0.202297	−	0.979324i	$-0.564841\pi$
$19$	−1.76358	−0.404593	−0.202297	+	0.979324i	$0.564841\pi$
$20$	1.63508	0.365615
$21$	0	0
$22$	−0.197300	−0.0420645
$23$	1.25693	0.262088	0.131044	−	0.991377i	$-0.458167\pi$
$23$	1.25693	0.262088	0.131044	+	0.991377i	$0.458167\pi$
$24$	0	0
$25$	−2.32651	−0.465302
$26$	5.53420	1.08535
$27$	0	0
$28$	4.18187	0.790299
$29$	6.46749	1.20098	0.600492	−	0.799631i	$-0.294971\pi$
$29$	6.46749	1.20098	0.600492	+	0.799631i	$0.294971\pi$
$30$	0	0
$31$	−9.67005	−1.73679	−0.868396	−	0.495871i	$-0.834849\pi$
$31$	−9.67005	−1.73679	−0.868396	+	0.495871i	$0.834849\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.24995	1.07186
$35$	6.83770	1.15578
$36$	0	0
$37$	−8.61274	−1.41593	−0.707963	−	0.706250i	$-0.750386\pi$
$37$	−8.61274	−1.41593	−0.707963	+	0.706250i	$0.750386\pi$
$38$	1.76358	0.286091
$39$	0	0
$40$	−1.63508	−0.258529
$41$	7.00464	1.09394	0.546971	−	0.837152i	$-0.315781\pi$
$41$	7.00464	1.09394	0.546971	+	0.837152i	$0.315781\pi$
$42$	0	0
$43$	4.69520	0.716011	0.358006	−	0.933719i	$-0.383457\pi$
$43$	4.69520	0.716011	0.358006	+	0.933719i	$0.383457\pi$
$44$	0.197300	0.0297441
$45$	0	0
$46$	−1.25693	−0.185324
$47$	6.92211	1.00969	0.504847	−	0.863209i	$-0.331549\pi$
$47$	6.92211	1.00969	0.504847	+	0.863209i	$0.331549\pi$
$48$	0	0
$49$	10.4880	1.49829
$50$	2.32651	0.329018
$51$	0	0
$52$	−5.53420	−0.767456
$53$	5.96851	0.819838	0.409919	−	0.912122i	$-0.365557\pi$
$53$	5.96851	0.819838	0.409919	+	0.912122i	$0.365557\pi$
$54$	0	0
$55$	0.322602	0.0434996
$56$	−4.18187	−0.558826
$57$	0	0
$58$	−6.46749	−0.849224
$59$	−4.41785	−0.575155	−0.287577	−	0.957757i	$-0.592850\pi$
$59$	−4.41785	−0.575155	−0.287577	+	0.957757i	$0.592850\pi$
$60$	0	0
$61$	6.34679	0.812623	0.406311	−	0.913735i	$-0.366815\pi$
$61$	6.34679	0.812623	0.406311	+	0.913735i	$0.366815\pi$
$62$	9.67005	1.22810
$63$	0	0
$64$	1.00000	0.125000
$65$	−9.04888	−1.12237
$66$	0	0
$67$	−15.3102	−1.87044	−0.935222	−	0.354062i	$-0.884800\pi$
$67$	−15.3102	−1.87044	−0.935222	+	0.354062i	$0.884800\pi$
$68$	−6.24995	−0.757918
$69$	0	0
$70$	−6.83770	−0.817261
$71$	−8.52896	−1.01220	−0.506100	−	0.862475i	$-0.668913\pi$
$71$	−8.52896	−1.01220	−0.506100	+	0.862475i	$0.668913\pi$
$72$	0	0
$73$	−0.842454	−0.0986018	−0.0493009	−	0.998784i	$-0.515699\pi$
$73$	−0.842454	−0.0986018	−0.0493009	+	0.998784i	$0.515699\pi$
$74$	8.61274	1.00121
$75$	0	0
$76$	−1.76358	−0.202297
$77$	0.825083	0.0940269
$78$	0	0
$79$	−2.79367	−0.314313	−0.157156	−	0.987574i	$-0.550233\pi$
$79$	−2.79367	−0.314313	−0.157156	+	0.987574i	$0.550233\pi$
$80$	1.63508	0.182808
$81$	0	0
$82$	−7.00464	−0.773533
$83$	−1.68295	−0.184728	−0.0923641	−	0.995725i	$-0.529442\pi$
$83$	−1.68295	−0.184728	−0.0923641	+	0.995725i	$0.529442\pi$
$84$	0	0
$85$	−10.2192	−1.10843
$86$	−4.69520	−0.506296
$87$	0	0
$88$	−0.197300	−0.0210323
$89$	−0.518659	−0.0549777	−0.0274889	−	0.999622i	$-0.508751\pi$
$89$	−0.518659	−0.0549777	−0.0274889	+	0.999622i	$0.508751\pi$
$90$	0	0
$91$	−23.1433	−2.42608
$92$	1.25693	0.131044
$93$	0	0
$94$	−6.92211	−0.713961
$95$	−2.88360	−0.295851
$96$	0	0
$97$	2.99290	0.303883	0.151941	−	0.988390i	$-0.451447\pi$
$97$	2.99290	0.303883	0.151941	+	0.988390i	$0.451447\pi$
$98$	−10.4880	−1.05945
$99$	0	0
$100$	−2.32651	−0.232651
$101$	−4.86820	−0.484404	−0.242202	−	0.970226i	$-0.577870\pi$
$101$	−4.86820	−0.484404	−0.242202	+	0.970226i	$0.577870\pi$
$102$	0	0
$103$	7.84368	0.772860	0.386430	−	0.922319i	$-0.373708\pi$
$103$	7.84368	0.772860	0.386430	+	0.922319i	$0.373708\pi$
$104$	5.53420	0.542673
$105$	0	0
$106$	−5.96851	−0.579713
$107$	1.68404	0.162802	0.0814012	−	0.996681i	$-0.474060\pi$
$107$	1.68404	0.162802	0.0814012	+	0.996681i	$0.474060\pi$
$108$	0	0
$109$	−10.7457	−1.02925	−0.514627	−	0.857414i	$-0.672070\pi$
$109$	−10.7457	−1.02925	−0.514627	+	0.857414i	$0.672070\pi$
$110$	−0.322602	−0.0307589
$111$	0	0
$112$	4.18187	0.395149
$113$	−10.3413	−0.972826	−0.486413	−	0.873729i	$-0.661695\pi$
$113$	−10.3413	−0.972826	−0.486413	+	0.873729i	$0.661695\pi$
$114$	0	0
$115$	2.05519	0.191647
$116$	6.46749	0.600492
$117$	0	0
$118$	4.41785	0.406696
$119$	−26.1365	−2.39593
$120$	0	0
$121$	−10.9611	−0.996461
$122$	−6.34679	−0.574611
$123$	0	0
$124$	−9.67005	−0.868396
$125$	−11.9794	−1.07147
$126$	0	0
$127$	7.37977	0.654849	0.327424	−	0.944877i	$-0.393819\pi$
$127$	7.37977	0.654849	0.327424	+	0.944877i	$0.393819\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	9.04888	0.793639
$131$	9.07961	0.793289	0.396645	−	0.917972i	$-0.370175\pi$
$131$	9.07961	0.793289	0.396645	+	0.917972i	$0.370175\pi$
$132$	0	0
$133$	−7.37507	−0.639499
$134$	15.3102	1.32260
$135$	0	0
$136$	6.24995	0.535929
$137$	−12.7245	−1.08713	−0.543563	−	0.839368i	$-0.682925\pi$
$137$	−12.7245	−1.08713	−0.543563	+	0.839368i	$0.682925\pi$
$138$	0	0
$139$	1.66714	0.141405	0.0707026	−	0.997497i	$-0.477476\pi$
$139$	1.66714	0.141405	0.0707026	+	0.997497i	$0.477476\pi$
$140$	6.83770	0.577891
$141$	0	0
$142$	8.52896	0.715734
$143$	−1.09190	−0.0913092
$144$	0	0
$145$	10.5749	0.878196
$146$	0.842454	0.0697220
$147$	0	0
$148$	−8.61274	−0.707963
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−17.2820	−1.40639	−0.703193	−	0.710999i	$-0.748243\pi$
$151$	−17.2820	−1.40639	−0.703193	+	0.710999i	$0.748243\pi$
$152$	1.76358	0.143045
$153$	0	0
$154$	−0.825083	−0.0664871
$155$	−15.8113	−1.27000
$156$	0	0
$157$	−16.9018	−1.34891	−0.674457	−	0.738314i	$-0.735622\pi$
$157$	−16.9018	−1.34891	−0.674457	+	0.738314i	$0.735622\pi$
$158$	2.79367	0.222253
$159$	0	0
$160$	−1.63508	−0.129265
$161$	5.25632	0.414256
$162$	0	0
$163$	11.9163	0.933355	0.466678	−	0.884428i	$-0.345451\pi$
$163$	11.9163	0.933355	0.466678	+	0.884428i	$0.345451\pi$
$164$	7.00464	0.546971
$165$	0	0
$166$	1.68295	0.130623
$167$	−14.7560	−1.14186	−0.570929	−	0.821000i	$-0.693417\pi$
$167$	−14.7560	−1.14186	−0.570929	+	0.821000i	$0.693417\pi$
$168$	0	0
$169$	17.6274	1.35596
$170$	10.2192	0.783775
$171$	0	0
$172$	4.69520	0.358006
$173$	24.6303	1.87260	0.936302	−	0.351195i	$-0.114224\pi$
$173$	24.6303	1.87260	0.936302	+	0.351195i	$0.114224\pi$
$174$	0	0
$175$	−9.72915	−0.735455
$176$	0.197300	0.0148721
$177$	0	0
$178$	0.518659	0.0388751
$179$	−20.1909	−1.50914	−0.754568	−	0.656221i	$-0.772154\pi$
$179$	−20.1909	−1.50914	−0.754568	+	0.656221i	$0.772154\pi$
$180$	0	0
$181$	−18.5876	−1.38160	−0.690801	−	0.723045i	$-0.742742\pi$
$181$	−18.5876	−1.38160	−0.690801	+	0.723045i	$0.742742\pi$
$182$	23.1433	1.71550
$183$	0	0
$184$	−1.25693	−0.0926622
$185$	−14.0825	−1.03537
$186$	0	0
$187$	−1.23312	−0.0901743
$188$	6.92211	0.504847
$189$	0	0
$190$	2.88360	0.209198
$191$	−22.5264	−1.62995	−0.814976	−	0.579495i	$-0.803250\pi$
$191$	−22.5264	−1.62995	−0.814976	+	0.579495i	$0.803250\pi$
$192$	0	0
$193$	−3.61836	−0.260455	−0.130228	−	0.991484i	$-0.541571\pi$
$193$	−3.61836	−0.260455	−0.130228	+	0.991484i	$0.541571\pi$
$194$	−2.99290	−0.214878
$195$	0	0
$196$	10.4880	0.749145
$197$	22.2957	1.58851	0.794253	−	0.607587i	$-0.207862\pi$
$197$	22.2957	1.58851	0.794253	+	0.607587i	$0.207862\pi$
$198$	0	0
$199$	−11.0975	−0.786680	−0.393340	−	0.919393i	$-0.628680\pi$
$199$	−11.0975	−0.786680	−0.393340	+	0.919393i	$0.628680\pi$
$200$	2.32651	0.164509
$201$	0	0
$202$	4.86820	0.342525
$203$	27.0462	1.89827
$204$	0	0
$205$	11.4532	0.799923
$206$	−7.84368	−0.546495
$207$	0	0
$208$	−5.53420	−0.383728
$209$	−0.347955	−0.0240685
$210$	0	0
$211$	−12.6815	−0.873030	−0.436515	−	0.899697i	$-0.643787\pi$
$211$	−12.6815	−0.873030	−0.436515	+	0.899697i	$0.643787\pi$
$212$	5.96851	0.409919
$213$	0	0
$214$	−1.68404	−0.115119
$215$	7.67703	0.523569
$216$	0	0
$217$	−40.4389	−2.74517
$218$	10.7457	0.727793
$219$	0	0
$220$	0.322602	0.0217498
$221$	34.5885	2.32667
$222$	0	0
$223$	−4.65351	−0.311622	−0.155811	−	0.987787i	$-0.549799\pi$
$223$	−4.65351	−0.311622	−0.155811	+	0.987787i	$0.549799\pi$
$224$	−4.18187	−0.279413
$225$	0	0
$226$	10.3413	0.687892
$227$	−13.2197	−0.877425	−0.438713	−	0.898627i	$-0.644565\pi$
$227$	−13.2197	−0.877425	−0.438713	+	0.898627i	$0.644565\pi$
$228$	0	0
$229$	−4.02383	−0.265902	−0.132951	−	0.991123i	$-0.542445\pi$
$229$	−4.02383	−0.265902	−0.132951	+	0.991123i	$0.542445\pi$
$230$	−2.05519	−0.135515
$231$	0	0
$232$	−6.46749	−0.424612
$233$	7.44022	0.487425	0.243712	−	0.969848i	$-0.421635\pi$
$233$	7.44022	0.487425	0.243712	+	0.969848i	$0.421635\pi$
$234$	0	0
$235$	11.3182	0.738319
$236$	−4.41785	−0.287577
$237$	0	0
$238$	26.1365	1.69418
$239$	25.1146	1.62453	0.812263	−	0.583291i	$-0.198235\pi$
$239$	25.1146	1.62453	0.812263	+	0.583291i	$0.198235\pi$
$240$	0	0
$241$	−5.39031	−0.347220	−0.173610	−	0.984814i	$-0.555543\pi$
$241$	−5.39031	−0.347220	−0.173610	+	0.984814i	$0.555543\pi$
$242$	10.9611	0.704604
$243$	0	0
$244$	6.34679	0.406311
$245$	17.1488	1.09560
$246$	0	0
$247$	9.76002	0.621015
$248$	9.67005	0.614049
$249$	0	0
$250$	11.9794	0.757646
$251$	−27.5070	−1.73623	−0.868113	−	0.496366i	$-0.834667\pi$
$251$	−27.5070	−1.73623	−0.868113	+	0.496366i	$0.834667\pi$
$252$	0	0
$253$	0.247993	0.0155912
$254$	−7.37977	−0.463048
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.27868	0.516410	0.258205	−	0.966090i	$-0.416869\pi$
$257$	8.27868	0.516410	0.258205	+	0.966090i	$0.416869\pi$
$258$	0	0
$259$	−36.0173	−2.23801
$260$	−9.04888	−0.561187
$261$	0	0
$262$	−9.07961	−0.560940
$263$	−19.6577	−1.21214	−0.606072	−	0.795410i	$-0.707256\pi$
$263$	−19.6577	−1.21214	−0.606072	+	0.795410i	$0.707256\pi$
$264$	0	0
$265$	9.75900	0.599491
$266$	7.37507	0.452194
$267$	0	0
$268$	−15.3102	−0.935222
$269$	−3.61876	−0.220640	−0.110320	−	0.993896i	$-0.535188\pi$
$269$	−3.61876	−0.220640	−0.110320	+	0.993896i	$0.535188\pi$
$270$	0	0
$271$	16.2770	0.988756	0.494378	−	0.869247i	$-0.335396\pi$
$271$	16.2770	0.988756	0.494378	+	0.869247i	$0.335396\pi$
$272$	−6.24995	−0.378959
$273$	0	0
$274$	12.7245	0.768714
$275$	−0.459020	−0.0276800
$276$	0	0
$277$	9.87391	0.593266	0.296633	−	0.954992i	$-0.404136\pi$
$277$	9.87391	0.593266	0.296633	+	0.954992i	$0.404136\pi$
$278$	−1.66714	−0.0999886
$279$	0	0
$280$	−6.83770	−0.408630
$281$	5.71542	0.340953	0.170477	−	0.985362i	$-0.445469\pi$
$281$	5.71542	0.340953	0.170477	+	0.985362i	$0.445469\pi$
$282$	0	0
$283$	−11.6035	−0.689754	−0.344877	−	0.938648i	$-0.612079\pi$
$283$	−11.6035	−0.689754	−0.344877	+	0.938648i	$0.612079\pi$
$284$	−8.52896	−0.506100
$285$	0	0
$286$	1.09190	0.0645653
$287$	29.2925	1.72908
$288$	0	0
$289$	22.0619	1.29776
$290$	−10.5749	−0.620978
$291$	0	0
$292$	−0.842454	−0.0493009
$293$	−1.62266	−0.0947967	−0.0473983	−	0.998876i	$-0.515093\pi$
$293$	−1.62266	−0.0947967	−0.0473983	+	0.998876i	$0.515093\pi$
$294$	0	0
$295$	−7.22354	−0.420571
$296$	8.61274	0.500605
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−6.95612	−0.402283
$300$	0	0
$301$	19.6347	1.13173
$302$	17.2820	0.994465
$303$	0	0
$304$	−1.76358	−0.101148
$305$	10.3775	0.594215
$306$	0	0
$307$	−9.46804	−0.540369	−0.270185	−	0.962809i	$-0.587085\pi$
$307$	−9.46804	−0.540369	−0.270185	+	0.962809i	$0.587085\pi$
$308$	0.825083	0.0470135
$309$	0	0
$310$	15.8113	0.898023
$311$	18.2503	1.03488	0.517440	−	0.855720i	$-0.326885\pi$
$311$	18.2503	1.03488	0.517440	+	0.855720i	$0.326885\pi$
$312$	0	0
$313$	17.6571	0.998037	0.499018	−	0.866591i	$-0.333694\pi$
$313$	17.6571	0.998037	0.499018	+	0.866591i	$0.333694\pi$
$314$	16.9018	0.953826
$315$	0	0
$316$	−2.79367	−0.157156
$317$	15.3212	0.860526	0.430263	−	0.902704i	$-0.358421\pi$
$317$	15.3212	0.860526	0.430263	+	0.902704i	$0.358421\pi$
$318$	0	0
$319$	1.27604	0.0714444
$320$	1.63508	0.0914038
$321$	0	0
$322$	−5.25632	−0.292923
$323$	11.0223	0.613297
$324$	0	0
$325$	12.8754	0.714197
$326$	−11.9163	−0.659982
$327$	0	0
$328$	−7.00464	−0.386767
$329$	28.9474	1.59592
$330$	0	0
$331$	10.9330	0.600932	0.300466	−	0.953793i	$-0.402858\pi$
$331$	10.9330	0.600932	0.300466	+	0.953793i	$0.402858\pi$
$332$	−1.68295	−0.0923641
$333$	0	0
$334$	14.7560	0.807415
$335$	−25.0335	−1.36773
$336$	0	0
$337$	2.19251	0.119434	0.0597169	−	0.998215i	$-0.480980\pi$
$337$	2.19251	0.119434	0.0597169	+	0.998215i	$0.480980\pi$
$338$	−17.6274	−0.958806
$339$	0	0
$340$	−10.2192	−0.554213
$341$	−1.90790	−0.103319
$342$	0	0
$343$	14.5865	0.787595
$344$	−4.69520	−0.253148
$345$	0	0
$346$	−24.6303	−1.32413
$347$	22.7847	1.22315	0.611574	−	0.791187i	$-0.290537\pi$
$347$	22.7847	1.22315	0.611574	+	0.791187i	$0.290537\pi$
$348$	0	0
$349$	−19.0409	−1.01924	−0.509618	−	0.860401i	$-0.670213\pi$
$349$	−19.0409	−1.01924	−0.509618	+	0.860401i	$0.670213\pi$
$350$	9.72915	0.520045
$351$	0	0
$352$	−0.197300	−0.0105161
$353$	−15.2629	−0.812365	−0.406182	−	0.913792i	$-0.633140\pi$
$353$	−15.2629	−0.812365	−0.406182	+	0.913792i	$0.633140\pi$
$354$	0	0
$355$	−13.9455	−0.740152
$356$	−0.518659	−0.0274889
$357$	0	0
$358$	20.1909	1.06712
$359$	1.35972	0.0717635	0.0358817	−	0.999356i	$-0.488576\pi$
$359$	1.35972	0.0717635	0.0358817	+	0.999356i	$0.488576\pi$
$360$	0	0
$361$	−15.8898	−0.836304
$362$	18.5876	0.976941
$363$	0	0
$364$	−23.1433	−1.21304
$365$	−1.37748	−0.0721006
$366$	0	0
$367$	5.15305	0.268987	0.134494	−	0.990914i	$-0.457059\pi$
$367$	5.15305	0.268987	0.134494	+	0.990914i	$0.457059\pi$
$368$	1.25693	0.0655221
$369$	0	0
$370$	14.0825	0.732116
$371$	24.9595	1.29583
$372$	0	0
$373$	4.71232	0.243995	0.121997	−	0.992530i	$-0.461070\pi$
$373$	4.71232	0.243995	0.121997	+	0.992530i	$0.461070\pi$
$374$	1.23312	0.0637629
$375$	0	0
$376$	−6.92211	−0.356981
$377$	−35.7924	−1.84340
$378$	0	0
$379$	16.9403	0.870166	0.435083	−	0.900390i	$-0.356719\pi$
$379$	16.9403	0.870166	0.435083	+	0.900390i	$0.356719\pi$
$380$	−2.88360	−0.147926
$381$	0	0
$382$	22.5264	1.15255
$383$	6.39895	0.326971	0.163486	−	0.986546i	$-0.447726\pi$
$383$	6.39895	0.326971	0.163486	+	0.986546i	$0.447726\pi$
$384$	0	0
$385$	1.34908	0.0687554
$386$	3.61836	0.184170
$387$	0	0
$388$	2.99290	0.151941
$389$	−1.08628	−0.0550767	−0.0275384	−	0.999621i	$-0.508767\pi$
$389$	−1.08628	−0.0550767	−0.0275384	+	0.999621i	$0.508767\pi$
$390$	0	0
$391$	−7.85576	−0.397283
$392$	−10.4880	−0.529725
$393$	0	0
$394$	−22.2957	−1.12324
$395$	−4.56789	−0.229835
$396$	0	0
$397$	25.3461	1.27208	0.636042	−	0.771654i	$-0.280570\pi$
$397$	25.3461	1.27208	0.636042	+	0.771654i	$0.280570\pi$
$398$	11.0975	0.556267
$399$	0	0
$400$	−2.32651	−0.116325
$401$	−2.08150	−0.103945	−0.0519725	−	0.998649i	$-0.516551\pi$
$401$	−2.08150	−0.103945	−0.0519725	+	0.998649i	$0.516551\pi$
$402$	0	0
$403$	53.5161	2.66582
$404$	−4.86820	−0.242202
$405$	0	0
$406$	−27.0462	−1.34228
$407$	−1.69929	−0.0842309
$408$	0	0
$409$	1.78094	0.0880617	0.0440309	−	0.999030i	$-0.485980\pi$
$409$	1.78094	0.0880617	0.0440309	+	0.999030i	$0.485980\pi$
$410$	−11.4532	−0.565631
$411$	0	0
$412$	7.84368	0.386430
$413$	−18.4749	−0.909088
$414$	0	0
$415$	−2.75177	−0.135079
$416$	5.53420	0.271337
$417$	0	0
$418$	0.347955	0.0170190
$419$	−20.4936	−1.00118	−0.500589	−	0.865685i	$-0.666883\pi$
$419$	−20.4936	−1.00118	−0.500589	+	0.865685i	$0.666883\pi$
$420$	0	0
$421$	−13.6932	−0.667364	−0.333682	−	0.942686i	$-0.608291\pi$
$421$	−13.6932	−0.667364	−0.333682	+	0.942686i	$0.608291\pi$
$422$	12.6815	0.617326
$423$	0	0
$424$	−5.96851	−0.289857
$425$	14.5406	0.705321
$426$	0	0
$427$	26.5414	1.28443
$428$	1.68404	0.0814012
$429$	0	0
$430$	−7.67703	−0.370219
$431$	−22.2846	−1.07341	−0.536707	−	0.843769i	$-0.680332\pi$
$431$	−22.2846	−1.07341	−0.536707	+	0.843769i	$0.680332\pi$
$432$	0	0
$433$	−22.4806	−1.08035	−0.540175	−	0.841552i	$-0.681642\pi$
$433$	−22.4806	−1.08035	−0.540175	+	0.841552i	$0.681642\pi$
$434$	40.4389	1.94113
$435$	0	0
$436$	−10.7457	−0.514627
$437$	−2.21670	−0.106039
$438$	0	0
$439$	−24.1239	−1.15137	−0.575686	−	0.817671i	$-0.695265\pi$
$439$	−24.1239	−1.15137	−0.575686	+	0.817671i	$0.695265\pi$
$440$	−0.322602	−0.0153794
$441$	0	0
$442$	−34.5885	−1.64521
$443$	−29.6812	−1.41020	−0.705098	−	0.709110i	$-0.749097\pi$
$443$	−29.6812	−1.41020	−0.705098	+	0.709110i	$0.749097\pi$
$444$	0	0
$445$	−0.848050	−0.0402014
$446$	4.65351	0.220350
$447$	0	0
$448$	4.18187	0.197575
$449$	25.7344	1.21448	0.607240	−	0.794519i	$-0.292277\pi$
$449$	25.7344	1.21448	0.607240	+	0.794519i	$0.292277\pi$
$450$	0	0
$451$	1.38202	0.0650766
$452$	−10.3413	−0.486413
$453$	0	0
$454$	13.2197	0.620433
$455$	−37.8412	−1.77402
$456$	0	0
$457$	0.610859	0.0285748	0.0142874	−	0.999898i	$-0.495452\pi$
$457$	0.610859	0.0285748	0.0142874	+	0.999898i	$0.495452\pi$
$458$	4.02383	0.188021
$459$	0	0
$460$	2.05519	0.0958235
$461$	−11.9751	−0.557734	−0.278867	−	0.960330i	$-0.589959\pi$
$461$	−11.9751	−0.557734	−0.278867	+	0.960330i	$0.589959\pi$
$462$	0	0
$463$	−12.6330	−0.587103	−0.293552	−	0.955943i	$-0.594837\pi$
$463$	−12.6330	−0.587103	−0.293552	+	0.955943i	$0.594837\pi$
$464$	6.46749	0.300246
$465$	0	0
$466$	−7.44022	−0.344661
$467$	6.77417	0.313471	0.156736	−	0.987641i	$-0.449903\pi$
$467$	6.77417	0.313471	0.156736	+	0.987641i	$0.449903\pi$
$468$	0	0
$469$	−64.0254	−2.95642
$470$	−11.3182	−0.522070
$471$	0	0
$472$	4.41785	0.203348
$473$	0.926363	0.0425942
$474$	0	0
$475$	4.10299	0.188258
$476$	−26.1365	−1.19796
$477$	0	0
$478$	−25.1146	−1.14871
$479$	−12.1625	−0.555719	−0.277859	−	0.960622i	$-0.589625\pi$
$479$	−12.1625	−0.555719	−0.277859	+	0.960622i	$0.589625\pi$
$480$	0	0
$481$	47.6647	2.17332
$482$	5.39031	0.245522
$483$	0	0
$484$	−10.9611	−0.498231
$485$	4.89363	0.222208
$486$	0	0
$487$	−14.0747	−0.637783	−0.318892	−	0.947791i	$-0.603311\pi$
$487$	−14.0747	−0.637783	−0.318892	+	0.947791i	$0.603311\pi$
$488$	−6.34679	−0.287306
$489$	0	0
$490$	−17.1488	−0.774703
$491$	−19.2146	−0.867141	−0.433571	−	0.901120i	$-0.642747\pi$
$491$	−19.2146	−0.867141	−0.433571	+	0.901120i	$0.642747\pi$
$492$	0	0
$493$	−40.4215	−1.82049
$494$	−9.76002	−0.439124
$495$	0	0
$496$	−9.67005	−0.434198
$497$	−35.6670	−1.59988
$498$	0	0
$499$	35.5569	1.59175	0.795874	−	0.605463i	$-0.207012\pi$
$499$	35.5569	1.59175	0.795874	+	0.605463i	$0.207012\pi$
$500$	−11.9794	−0.535737
$501$	0	0
$502$	27.5070	1.22770
$503$	−7.48503	−0.333741	−0.166871	−	0.985979i	$-0.553366\pi$
$503$	−7.48503	−0.333741	−0.166871	+	0.985979i	$0.553366\pi$
$504$	0	0
$505$	−7.95990	−0.354211
$506$	−0.247993	−0.0110246
$507$	0	0
$508$	7.37977	0.327424
$509$	−6.17672	−0.273778	−0.136889	−	0.990586i	$-0.543710\pi$
$509$	−6.17672	−0.273778	−0.136889	+	0.990586i	$0.543710\pi$
$510$	0	0
$511$	−3.52303	−0.155850
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−8.27868	−0.365157
$515$	12.8251	0.565139
$516$	0	0
$517$	1.36573	0.0600649
$518$	36.0173	1.58251
$519$	0	0
$520$	9.04888	0.396819
$521$	−4.67763	−0.204930	−0.102465	−	0.994737i	$-0.532673\pi$
$521$	−4.67763	−0.204930	−0.102465	+	0.994737i	$0.532673\pi$
$522$	0	0
$523$	−6.80751	−0.297672	−0.148836	−	0.988862i	$-0.547553\pi$
$523$	−6.80751	−0.297672	−0.148836	+	0.988862i	$0.547553\pi$
$524$	9.07961	0.396645
$525$	0	0
$526$	19.6577	0.857115
$527$	60.4373	2.63269
$528$	0	0
$529$	−21.4201	−0.931310
$530$	−9.75900	−0.423904
$531$	0	0
$532$	−7.37507	−0.319750
$533$	−38.7651	−1.67910
$534$	0	0
$535$	2.75354	0.119046
$536$	15.3102	0.661302
$537$	0	0
$538$	3.61876	0.156016
$539$	2.06929	0.0891306
$540$	0	0
$541$	−37.6664	−1.61940	−0.809702	−	0.586841i	$-0.800371\pi$
$541$	−37.6664	−1.61940	−0.809702	+	0.586841i	$0.800371\pi$
$542$	−16.2770	−0.699156
$543$	0	0
$544$	6.24995	0.267964
$545$	−17.5702	−0.752623
$546$	0	0
$547$	−34.8485	−1.49001	−0.745007	−	0.667056i	$-0.767554\pi$
$547$	−34.8485	−1.49001	−0.745007	+	0.667056i	$0.767554\pi$
$548$	−12.7245	−0.543563
$549$	0	0
$550$	0.459020	0.0195727
$551$	−11.4060	−0.485910
$552$	0	0
$553$	−11.6828	−0.496802
$554$	−9.87391	−0.419502
$555$	0	0
$556$	1.66714	0.0707026
$557$	11.6360	0.493034	0.246517	−	0.969138i	$-0.420714\pi$
$557$	11.6360	0.493034	0.246517	+	0.969138i	$0.420714\pi$
$558$	0	0
$559$	−25.9842	−1.09901
$560$	6.83770	0.288945
$561$	0	0
$562$	−5.71542	−0.241090
$563$	24.7461	1.04292	0.521462	−	0.853274i	$-0.325387\pi$
$563$	24.7461	1.04292	0.521462	+	0.853274i	$0.325387\pi$
$564$	0	0
$565$	−16.9088	−0.711360
$566$	11.6035	0.487730
$567$	0	0
$568$	8.52896	0.357867
$569$	−8.14143	−0.341306	−0.170653	−	0.985331i	$-0.554588\pi$
$569$	−8.14143	−0.341306	−0.170653	+	0.985331i	$0.554588\pi$
$570$	0	0
$571$	1.63240	0.0683138	0.0341569	−	0.999416i	$-0.489125\pi$
$571$	1.63240	0.0683138	0.0341569	+	0.999416i	$0.489125\pi$
$572$	−1.09190	−0.0456546
$573$	0	0
$574$	−29.2925	−1.22264
$575$	−2.92426	−0.121950
$576$	0	0
$577$	9.01805	0.375426	0.187713	−	0.982224i	$-0.439892\pi$
$577$	9.01805	0.375426	0.187713	+	0.982224i	$0.439892\pi$
$578$	−22.0619	−0.917652
$579$	0	0
$580$	10.5749	0.439098
$581$	−7.03789	−0.291981
$582$	0	0
$583$	1.17759	0.0487707
$584$	0.842454	0.0348610
$585$	0	0
$586$	1.62266	0.0670314
$587$	42.4374	1.75158	0.875789	−	0.482694i	$-0.160342\pi$
$587$	42.4374	1.75158	0.875789	+	0.482694i	$0.160342\pi$
$588$	0	0
$589$	17.0539	0.702695
$590$	7.22354	0.297388
$591$	0	0
$592$	−8.61274	−0.353981
$593$	−40.5390	−1.66474	−0.832369	−	0.554222i	$-0.813016\pi$
$593$	−40.5390	−1.66474	−0.832369	+	0.554222i	$0.813016\pi$
$594$	0	0
$595$	−42.7352	−1.75197
$596$	1.00000	0.0409616
$597$	0	0
$598$	6.95612	0.284457
$599$	−11.4659	−0.468484	−0.234242	−	0.972178i	$-0.575261\pi$
$599$	−11.4659	−0.468484	−0.234242	+	0.972178i	$0.575261\pi$
$600$	0	0
$601$	36.0785	1.47167	0.735837	−	0.677159i	$-0.236789\pi$
$601$	36.0785	1.47167	0.735837	+	0.677159i	$0.236789\pi$
$602$	−19.6347	−0.800251
$603$	0	0
$604$	−17.2820	−0.703193
$605$	−17.9222	−0.728643
$606$	0	0
$607$	−37.2834	−1.51329	−0.756643	−	0.653828i	$-0.773162\pi$
$607$	−37.2834	−1.51329	−0.756643	+	0.653828i	$0.773162\pi$
$608$	1.76358	0.0715227
$609$	0	0
$610$	−10.3775	−0.420173
$611$	−38.3084	−1.54979
$612$	0	0
$613$	−0.666270	−0.0269104	−0.0134552	−	0.999909i	$-0.504283\pi$
$613$	−0.666270	−0.0269104	−0.0134552	+	0.999909i	$0.504283\pi$
$614$	9.46804	0.382099
$615$	0	0
$616$	−0.825083	−0.0332435
$617$	41.1430	1.65635	0.828177	−	0.560467i	$-0.189378\pi$
$617$	41.1430	1.65635	0.828177	+	0.560467i	$0.189378\pi$
$618$	0	0
$619$	−0.736346	−0.0295962	−0.0147981	−	0.999891i	$-0.504711\pi$
$619$	−0.736346	−0.0295962	−0.0147981	+	0.999891i	$0.504711\pi$
$620$	−15.8113	−0.634998
$621$	0	0
$622$	−18.2503	−0.731770
$623$	−2.16896	−0.0868977
$624$	0	0
$625$	−7.95481	−0.318193
$626$	−17.6571	−0.705719
$627$	0	0
$628$	−16.9018	−0.674457
$629$	53.8292	2.14631
$630$	0	0
$631$	46.8071	1.86336	0.931681	−	0.363277i	$-0.118342\pi$
$631$	46.8071	1.86336	0.931681	+	0.363277i	$0.118342\pi$
$632$	2.79367	0.111126
$633$	0	0
$634$	−15.3212	−0.608483
$635$	12.0665	0.478845
$636$	0	0
$637$	−58.0429	−2.29974
$638$	−1.27604	−0.0505188
$639$	0	0
$640$	−1.63508	−0.0646323
$641$	20.0532	0.792052	0.396026	−	0.918239i	$-0.370389\pi$
$641$	20.0532	0.792052	0.396026	+	0.918239i	$0.370389\pi$
$642$	0	0
$643$	48.7277	1.92163	0.960817	−	0.277182i	$-0.0894005\pi$
$643$	48.7277	1.92163	0.960817	+	0.277182i	$0.0894005\pi$
$644$	5.25632	0.207128
$645$	0	0
$646$	−11.0223	−0.433666
$647$	21.8927	0.860692	0.430346	−	0.902664i	$-0.358392\pi$
$647$	21.8927	0.860692	0.430346	+	0.902664i	$0.358392\pi$
$648$	0	0
$649$	−0.871641	−0.0342149
$650$	−12.8754	−0.505014
$651$	0	0
$652$	11.9163	0.466678
$653$	−29.9921	−1.17368	−0.586840	−	0.809703i	$-0.699628\pi$
$653$	−29.9921	−1.17368	−0.586840	+	0.809703i	$0.699628\pi$
$654$	0	0
$655$	14.8459	0.580077
$656$	7.00464	0.273485
$657$	0	0
$658$	−28.9474	−1.12849
$659$	42.8025	1.66735	0.833675	−	0.552255i	$-0.186233\pi$
$659$	42.8025	1.66735	0.833675	+	0.552255i	$0.186233\pi$
$660$	0	0
$661$	37.7037	1.46651	0.733253	−	0.679956i	$-0.238001\pi$
$661$	37.7037	1.46651	0.733253	+	0.679956i	$0.238001\pi$
$662$	−10.9330	−0.424923
$663$	0	0
$664$	1.68295	0.0653113
$665$	−12.0588	−0.467621
$666$	0	0
$667$	8.12920	0.314764
$668$	−14.7560	−0.570929
$669$	0	0
$670$	25.0335	0.967128
$671$	1.25222	0.0483415
$672$	0	0
$673$	−7.65651	−0.295137	−0.147568	−	0.989052i	$-0.547145\pi$
$673$	−7.65651	−0.295137	−0.147568	+	0.989052i	$0.547145\pi$
$674$	−2.19251	−0.0844525
$675$	0	0
$676$	17.6274	0.677978
$677$	51.3670	1.97420	0.987098	−	0.160118i	$-0.0511875\pi$
$677$	51.3670	1.97420	0.987098	+	0.160118i	$0.0511875\pi$
$678$	0	0
$679$	12.5159	0.480317
$680$	10.2192	0.391887
$681$	0	0
$682$	1.90790	0.0730573
$683$	−6.92554	−0.264998	−0.132499	−	0.991183i	$-0.542300\pi$
$683$	−6.92554	−0.264998	−0.132499	+	0.991183i	$0.542300\pi$
$684$	0	0
$685$	−20.8056	−0.794940
$686$	−14.5865	−0.556914
$687$	0	0
$688$	4.69520	0.179003
$689$	−33.0310	−1.25838
$690$	0	0
$691$	9.16011	0.348467	0.174233	−	0.984704i	$-0.444255\pi$
$691$	9.16011	0.348467	0.174233	+	0.984704i	$0.444255\pi$
$692$	24.6303	0.936302
$693$	0	0
$694$	−22.7847	−0.864896
$695$	2.72591	0.103400
$696$	0	0
$697$	−43.7787	−1.65823
$698$	19.0409	0.720709
$699$	0	0
$700$	−9.72915	−0.367727
$701$	−14.6149	−0.551995	−0.275998	−	0.961158i	$-0.589008\pi$
$701$	−14.6149	−0.551995	−0.275998	+	0.961158i	$0.589008\pi$
$702$	0	0
$703$	15.1893	0.572874
$704$	0.197300	0.00743603
$705$	0	0
$706$	15.2629	0.574429
$707$	−20.3582	−0.765648
$708$	0	0
$709$	−42.3835	−1.59175	−0.795873	−	0.605464i	$-0.792988\pi$
$709$	−42.3835	−1.59175	−0.795873	+	0.605464i	$0.792988\pi$
$710$	13.9455	0.523367
$711$	0	0
$712$	0.518659	0.0194376
$713$	−12.1546	−0.455193
$714$	0	0
$715$	−1.78534	−0.0667681
$716$	−20.1909	−0.754568
$717$	0	0
$718$	−1.35972	−0.0507444
$719$	20.0259	0.746841	0.373420	−	0.927662i	$-0.378185\pi$
$719$	20.0259	0.746841	0.373420	+	0.927662i	$0.378185\pi$
$720$	0	0
$721$	32.8012	1.22158
$722$	15.8898	0.591356
$723$	0	0
$724$	−18.5876	−0.690801
$725$	−15.0467	−0.558820
$726$	0	0
$727$	10.9223	0.405088	0.202544	−	0.979273i	$-0.435079\pi$
$727$	10.9223	0.405088	0.202544	+	0.979273i	$0.435079\pi$
$728$	23.1433	0.857748
$729$	0	0
$730$	1.37748	0.0509828
$731$	−29.3448	−1.08535
$732$	0	0
$733$	33.8316	1.24960	0.624799	−	0.780785i	$-0.285181\pi$
$733$	33.8316	1.24960	0.624799	+	0.780785i	$0.285181\pi$
$734$	−5.15305	−0.190203
$735$	0	0
$736$	−1.25693	−0.0463311
$737$	−3.02071	−0.111269
$738$	0	0
$739$	−18.9760	−0.698045	−0.349023	−	0.937114i	$-0.613486\pi$
$739$	−18.9760	−0.698045	−0.349023	+	0.937114i	$0.613486\pi$
$740$	−14.0825	−0.517684
$741$	0	0
$742$	−24.9595	−0.916293
$743$	15.2042	0.557786	0.278893	−	0.960322i	$-0.410032\pi$
$743$	15.2042	0.557786	0.278893	+	0.960322i	$0.410032\pi$
$744$	0	0
$745$	1.63508	0.0599047
$746$	−4.71232	−0.172530
$747$	0	0
$748$	−1.23312	−0.0450872
$749$	7.04244	0.257325
$750$	0	0
$751$	−3.34506	−0.122063	−0.0610316	−	0.998136i	$-0.519439\pi$
$751$	−3.34506	−0.122063	−0.0610316	+	0.998136i	$0.519439\pi$
$752$	6.92211	0.252423
$753$	0	0
$754$	35.7924	1.30348
$755$	−28.2574	−1.02839
$756$	0	0
$757$	−42.2280	−1.53480	−0.767401	−	0.641167i	$-0.778450\pi$
$757$	−42.2280	−1.53480	−0.767401	+	0.641167i	$0.778450\pi$
$758$	−16.9403	−0.615301
$759$	0	0
$760$	2.88360	0.104599
$761$	27.8189	1.00844	0.504218	−	0.863577i	$-0.331781\pi$
$761$	27.8189	1.00844	0.504218	+	0.863577i	$0.331781\pi$
$762$	0	0
$763$	−44.9373	−1.62684
$764$	−22.5264	−0.814976
$765$	0	0
$766$	−6.39895	−0.231203
$767$	24.4493	0.882812
$768$	0	0
$769$	35.7072	1.28764	0.643818	−	0.765179i	$-0.277349\pi$
$769$	35.7072	1.28764	0.643818	+	0.765179i	$0.277349\pi$
$770$	−1.34908	−0.0486174
$771$	0	0
$772$	−3.61836	−0.130228
$773$	52.1349	1.87516	0.937580	−	0.347769i	$-0.113061\pi$
$773$	52.1349	1.87516	0.937580	+	0.347769i	$0.113061\pi$
$774$	0	0
$775$	22.4975	0.808133
$776$	−2.99290	−0.107439
$777$	0	0
$778$	1.08628	0.0389451
$779$	−12.3533	−0.442601
$780$	0	0
$781$	−1.68276	−0.0602140
$782$	7.85576	0.280921
$783$	0	0
$784$	10.4880	0.374572
$785$	−27.6359	−0.986367
$786$	0	0
$787$	−29.2235	−1.04170	−0.520852	−	0.853647i	$-0.674386\pi$
$787$	−29.2235	−1.04170	−0.520852	+	0.853647i	$0.674386\pi$
$788$	22.2957	0.794253
$789$	0	0
$790$	4.56789	0.162518
$791$	−43.2459	−1.53765
$792$	0	0
$793$	−35.1244	−1.24730
$794$	−25.3461	−0.899500
$795$	0	0
$796$	−11.0975	−0.393340
$797$	−16.9257	−0.599538	−0.299769	−	0.954012i	$-0.596910\pi$
$797$	−16.9257	−0.599538	−0.299769	+	0.954012i	$0.596910\pi$
$798$	0	0
$799$	−43.2628	−1.53053
$800$	2.32651	0.0822545
$801$	0	0
$802$	2.08150	0.0735002
$803$	−0.166216	−0.00586564
$804$	0	0
$805$	8.59452	0.302917
$806$	−53.5161	−1.88502
$807$	0	0
$808$	4.86820	0.171263
$809$	33.6153	1.18185	0.590926	−	0.806726i	$-0.298763\pi$
$809$	33.6153	1.18185	0.590926	+	0.806726i	$0.298763\pi$
$810$	0	0
$811$	50.8551	1.78577	0.892883	−	0.450289i	$-0.148679\pi$
$811$	50.8551	1.78577	0.892883	+	0.450289i	$0.148679\pi$
$812$	27.0462	0.949136
$813$	0	0
$814$	1.69929	0.0595602
$815$	19.4841	0.682498
$816$	0	0
$817$	−8.28037	−0.289693
$818$	−1.78094	−0.0622691
$819$	0	0
$820$	11.4532	0.399962
$821$	−10.0302	−0.350056	−0.175028	−	0.984563i	$-0.556002\pi$
$821$	−10.0302	−0.350056	−0.175028	+	0.984563i	$0.556002\pi$
$822$	0	0
$823$	−17.0828	−0.595470	−0.297735	−	0.954649i	$-0.596231\pi$
$823$	−17.0828	−0.595470	−0.297735	+	0.954649i	$0.596231\pi$
$824$	−7.84368	−0.273247
$825$	0	0
$826$	18.4749	0.642823
$827$	−33.6331	−1.16954	−0.584768	−	0.811201i	$-0.698814\pi$
$827$	−33.6331	−1.16954	−0.584768	+	0.811201i	$0.698814\pi$
$828$	0	0
$829$	−29.4770	−1.02378	−0.511889	−	0.859052i	$-0.671054\pi$
$829$	−29.4770	−1.02378	−0.511889	+	0.859052i	$0.671054\pi$
$830$	2.75177	0.0955152
$831$	0	0
$832$	−5.53420	−0.191864
$833$	−65.5496	−2.27116
$834$	0	0
$835$	−24.1273	−0.834961
$836$	−0.347955	−0.0120343
$837$	0	0
$838$	20.4936	0.707940
$839$	−22.2593	−0.768476	−0.384238	−	0.923234i	$-0.625536\pi$
$839$	−22.2593	−0.768476	−0.384238	+	0.923234i	$0.625536\pi$
$840$	0	0
$841$	12.8285	0.442362
$842$	13.6932	0.471897
$843$	0	0
$844$	−12.6815	−0.436515
$845$	28.8223	0.991516
$846$	0	0
$847$	−45.8378	−1.57500
$848$	5.96851	0.204960
$849$	0	0
$850$	−14.5406	−0.498737
$851$	−10.8256	−0.371098
$852$	0	0
$853$	−6.65495	−0.227861	−0.113931	−	0.993489i	$-0.536344\pi$
$853$	−6.65495	−0.227861	−0.113931	+	0.993489i	$0.536344\pi$
$854$	−26.5414	−0.908229
$855$	0	0
$856$	−1.68404	−0.0575593
$857$	−47.2482	−1.61397	−0.806983	−	0.590574i	$-0.798901\pi$
$857$	−47.2482	−1.61397	−0.806983	+	0.590574i	$0.798901\pi$
$858$	0	0
$859$	33.1021	1.12943	0.564714	−	0.825287i	$-0.308987\pi$
$859$	33.1021	1.12943	0.564714	+	0.825287i	$0.308987\pi$
$860$	7.67703	0.261785
$861$	0	0
$862$	22.2846	0.759018
$863$	−5.50067	−0.187245	−0.0936225	−	0.995608i	$-0.529845\pi$
$863$	−5.50067	−0.187245	−0.0936225	+	0.995608i	$0.529845\pi$
$864$	0	0
$865$	40.2725	1.36931
$866$	22.4806	0.763923
$867$	0	0
$868$	−40.4389	−1.37259
$869$	−0.551192	−0.0186979
$870$	0	0
$871$	84.7300	2.87097
$872$	10.7457	0.363897
$873$	0	0
$874$	2.21670	0.0749810
$875$	−50.0964	−1.69357
$876$	0	0
$877$	22.7687	0.768844	0.384422	−	0.923157i	$-0.374401\pi$
$877$	22.7687	0.768844	0.384422	+	0.923157i	$0.374401\pi$
$878$	24.1239	0.814143
$879$	0	0
$880$	0.322602	0.0108749
$881$	35.0854	1.18206	0.591029	−	0.806650i	$-0.298722\pi$
$881$	35.0854	1.18206	0.591029	+	0.806650i	$0.298722\pi$
$882$	0	0
$883$	−13.7453	−0.462565	−0.231283	−	0.972887i	$-0.574292\pi$
$883$	−13.7453	−0.462565	−0.231283	+	0.972887i	$0.574292\pi$
$884$	34.5885	1.16334
$885$	0	0
$886$	29.6812	0.997159
$887$	27.4555	0.921865	0.460932	−	0.887435i	$-0.347515\pi$
$887$	27.4555	0.921865	0.460932	+	0.887435i	$0.347515\pi$
$888$	0	0
$889$	30.8612	1.03505
$890$	0.848050	0.0284267
$891$	0	0
$892$	−4.65351	−0.155811
$893$	−12.2077	−0.408515
$894$	0	0
$895$	−33.0137	−1.10353
$896$	−4.18187	−0.139706
$897$	0	0
$898$	−25.7344	−0.858767
$899$	−62.5410	−2.08586
$900$	0	0
$901$	−37.3029	−1.24274
$902$	−1.38202	−0.0460161
$903$	0	0
$904$	10.3413	0.343946
$905$	−30.3922	−1.01027
$906$	0	0
$907$	2.38802	0.0792927	0.0396464	−	0.999214i	$-0.487377\pi$
$907$	2.38802	0.0792927	0.0396464	+	0.999214i	$0.487377\pi$
$908$	−13.2197	−0.438713
$909$	0	0
$910$	37.8412	1.25442
$911$	27.0356	0.895729	0.447864	−	0.894101i	$-0.352185\pi$
$911$	27.0356	0.895729	0.447864	+	0.894101i	$0.352185\pi$
$912$	0	0
$913$	−0.332047	−0.0109891
$914$	−0.610859	−0.0202054
$915$	0	0
$916$	−4.02383	−0.132951
$917$	37.9697	1.25387
$918$	0	0
$919$	3.17380	0.104694	0.0523470	−	0.998629i	$-0.483330\pi$
$919$	3.17380	0.104694	0.0523470	+	0.998629i	$0.483330\pi$
$920$	−2.05519	−0.0677575
$921$	0	0
$922$	11.9751	0.394377
$923$	47.2010	1.55364
$924$	0	0
$925$	20.0376	0.658833
$926$	12.6330	0.415145
$927$	0	0
$928$	−6.46749	−0.212306
$929$	−56.2318	−1.84490	−0.922452	−	0.386112i	$-0.873818\pi$
$929$	−56.2318	−1.84490	−0.922452	+	0.386112i	$0.873818\pi$
$930$	0	0
$931$	−18.4965	−0.606198
$932$	7.44022	0.243712
$933$	0	0
$934$	−6.77417	−0.221658
$935$	−2.01624	−0.0659382
$936$	0	0
$937$	−27.0832	−0.884770	−0.442385	−	0.896825i	$-0.645868\pi$
$937$	−27.0832	−0.884770	−0.442385	+	0.896825i	$0.645868\pi$
$938$	64.0254	2.09050
$939$	0	0
$940$	11.3182	0.369159
$941$	−37.6898	−1.22865	−0.614326	−	0.789052i	$-0.710572\pi$
$941$	−37.6898	−1.22865	−0.614326	+	0.789052i	$0.710572\pi$
$942$	0	0
$943$	8.80436	0.286709
$944$	−4.41785	−0.143789
$945$	0	0
$946$	−0.926363	−0.0301187
$947$	42.1198	1.36871	0.684354	−	0.729150i	$-0.260084\pi$
$947$	42.1198	1.36871	0.684354	+	0.729150i	$0.260084\pi$
$948$	0	0
$949$	4.66231	0.151345
$950$	−4.10299	−0.133118
$951$	0	0
$952$	26.1365	0.847088
$953$	6.25412	0.202591	0.101295	−	0.994856i	$-0.467701\pi$
$953$	6.25412	0.202591	0.101295	+	0.994856i	$0.467701\pi$
$954$	0	0
$955$	−36.8325	−1.19187
$956$	25.1146	0.812263
$957$	0	0
$958$	12.1625	0.392952
$959$	−53.2121	−1.71831
$960$	0	0
$961$	62.5099	2.01645
$962$	−47.6647	−1.53677
$963$	0	0
$964$	−5.39031	−0.173610
$965$	−5.91631	−0.190453
$966$	0	0
$967$	31.9859	1.02860	0.514298	−	0.857611i	$-0.328052\pi$
$967$	31.9859	1.02860	0.514298	+	0.857611i	$0.328052\pi$
$968$	10.9611	0.352302
$969$	0	0
$970$	−4.89363	−0.157125
$971$	−33.2640	−1.06749	−0.533747	−	0.845644i	$-0.679217\pi$
$971$	−33.2640	−1.06749	−0.533747	+	0.845644i	$0.679217\pi$
$972$	0	0
$973$	6.97177	0.223505
$974$	14.0747	0.450981
$975$	0	0
$976$	6.34679	0.203156
$977$	−15.1642	−0.485145	−0.242573	−	0.970133i	$-0.577991\pi$
$977$	−15.1642	−0.485145	−0.242573	+	0.970133i	$0.577991\pi$
$978$	0	0
$979$	−0.102331	−0.00327053
$980$	17.1488	0.547798
$981$	0	0
$982$	19.2146	0.613162
$983$	−20.6179	−0.657609	−0.328804	−	0.944398i	$-0.606646\pi$
$983$	−20.6179	−0.657609	−0.328804	+	0.944398i	$0.606646\pi$
$984$	0	0
$985$	36.4554	1.16156
$986$	40.4215	1.28728
$987$	0	0
$988$	9.76002	0.310508
$989$	5.90155	0.187658
$990$	0	0
$991$	36.9640	1.17420	0.587101	−	0.809514i	$-0.300269\pi$
$991$	36.9640	1.17420	0.587101	+	0.809514i	$0.300269\pi$
$992$	9.67005	0.307024
$993$	0	0
$994$	35.6670	1.13129
$995$	−18.1453	−0.575244
$996$	0	0
$997$	36.7817	1.16489	0.582444	−	0.812871i	$-0.302096\pi$
$997$	36.7817	1.16489	0.582444	+	0.812871i	$0.302096\pi$
$998$	−35.5569	−1.12554
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.g.1.6	✓	9
3.2	odd	2		8046.2.a.h.1.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.g.1.6	✓	9	1.1	even	1	trivial
8046.2.a.h.1.4	yes	9	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.63508 q^{5} +4.18187 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.63508 q^{5} +4.18187 q^{7} -1.00000 q^{8} -1.63508 q^{10} +0.197300 q^{11} -5.53420 q^{13} -4.18187 q^{14} +1.00000 q^{16} -6.24995 q^{17} -1.76358 q^{19} +1.63508 q^{20} -0.197300 q^{22} +1.25693 q^{23} -2.32651 q^{25} +5.53420 q^{26} +4.18187 q^{28} +6.46749 q^{29} -9.67005 q^{31} -1.00000 q^{32} +6.24995 q^{34} +6.83770 q^{35} -8.61274 q^{37} +1.76358 q^{38} -1.63508 q^{40} +7.00464 q^{41} +4.69520 q^{43} +0.197300 q^{44} -1.25693 q^{46} +6.92211 q^{47} +10.4880 q^{49} +2.32651 q^{50} -5.53420 q^{52} +5.96851 q^{53} +0.322602 q^{55} -4.18187 q^{56} -6.46749 q^{58} -4.41785 q^{59} +6.34679 q^{61} +9.67005 q^{62} +1.00000 q^{64} -9.04888 q^{65} -15.3102 q^{67} -6.24995 q^{68} -6.83770 q^{70} -8.52896 q^{71} -0.842454 q^{73} +8.61274 q^{74} -1.76358 q^{76} +0.825083 q^{77} -2.79367 q^{79} +1.63508 q^{80} -7.00464 q^{82} -1.68295 q^{83} -10.2192 q^{85} -4.69520 q^{86} -0.197300 q^{88} -0.518659 q^{89} -23.1433 q^{91} +1.25693 q^{92} -6.92211 q^{94} -2.88360 q^{95} +2.99290 q^{97} -10.4880 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8	\( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{5} - 4 q^{7} - 9 q^{8} - 4 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} + 9 q^{16} + q^{17} - 10 q^{19} + 4 q^{20} - 4 q^{22} + 8 q^{23} - 3 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} - 17 q^{31} - 9 q^{32} - q^{34} + 10 q^{35} - 11 q^{37} + 10 q^{38} - 4 q^{40} - 16 q^{43} + 4 q^{44} - 8 q^{46} + 7 q^{47} - 5 q^{49} + 3 q^{50} - 8 q^{52} + 12 q^{53} - 23 q^{55} + 4 q^{56} - 4 q^{58} + 6 q^{59} - 13 q^{61} + 17 q^{62} + 9 q^{64} - 24 q^{65} - 14 q^{67} + q^{68} - 10 q^{70} + 30 q^{71} - 12 q^{73} + 11 q^{74} - 10 q^{76} + 12 q^{77} - 35 q^{79} + 4 q^{80} + 5 q^{83} - 27 q^{85} + 16 q^{86} - 4 q^{88} + 23 q^{89} - 28 q^{91} + 8 q^{92} - 7 q^{94} + 32 q^{95} - 21 q^{97} + 5 q^{98}+O(q^{100}) \) 9 * q - 9 * q^2 + 9 * q^4 + 4 * q^5 - 4 * q^7 - 9 * q^8 - 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 9 * q^16 + q^17 - 10 * q^19 + 4 * q^20 - 4 * q^22 + 8 * q^23 - 3 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 - 17 * q^31 - 9 * q^32 - q^34 + 10 * q^35 - 11 * q^37 + 10 * q^38 - 4 * q^40 - 16 * q^43 + 4 * q^44 - 8 * q^46 + 7 * q^47 - 5 * q^49 + 3 * q^50 - 8 * q^52 + 12 * q^53 - 23 * q^55 + 4 * q^56 - 4 * q^58 + 6 * q^59 - 13 * q^61 + 17 * q^62 + 9 * q^64 - 24 * q^65 - 14 * q^67 + q^68 - 10 * q^70 + 30 * q^71 - 12 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 - 35 * q^79 + 4 * q^80 + 5 * q^83 - 27 * q^85 + 16 * q^86 - 4 * q^88 + 23 * q^89 - 28 * q^91 + 8 * q^92 - 7 * q^94 + 32 * q^95 - 21 * q^97 + 5 * q^98

Introduction

L-functions

Modular forms

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Database

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