LMFDB - Embedded newform 8046.2.a.q.1.12

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			1.12
Root			$-2.42130$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.42130 q^{5} -4.62365 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.42130 q^{5} -4.62365 q^{7} -1.00000 q^{8} -2.42130 q^{10} +5.75466 q^{11} -5.37930 q^{13} +4.62365 q^{14} +1.00000 q^{16} +2.77000 q^{17} +3.14018 q^{19} +2.42130 q^{20} -5.75466 q^{22} -1.27870 q^{23} +0.862695 q^{25} +5.37930 q^{26} -4.62365 q^{28} -0.713201 q^{29} -3.11809 q^{31} -1.00000 q^{32} -2.77000 q^{34} -11.1952 q^{35} -4.81173 q^{37} -3.14018 q^{38} -2.42130 q^{40} +9.85559 q^{41} +0.0559332 q^{43} +5.75466 q^{44} +1.27870 q^{46} +8.45189 q^{47} +14.3781 q^{49} -0.862695 q^{50} -5.37930 q^{52} -2.03001 q^{53} +13.9338 q^{55} +4.62365 q^{56} +0.713201 q^{58} +4.28447 q^{59} -5.69143 q^{61} +3.11809 q^{62} +1.00000 q^{64} -13.0249 q^{65} -4.54707 q^{67} +2.77000 q^{68} +11.1952 q^{70} -12.0967 q^{71} -4.39363 q^{73} +4.81173 q^{74} +3.14018 q^{76} -26.6075 q^{77} +8.53478 q^{79} +2.42130 q^{80} -9.85559 q^{82} +14.6651 q^{83} +6.70700 q^{85} -0.0559332 q^{86} -5.75466 q^{88} +5.81228 q^{89} +24.8720 q^{91} -1.27870 q^{92} -8.45189 q^{94} +7.60331 q^{95} +6.97235 q^{97} -14.3781 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.42130	1.08284	0.541419	−	0.840753i	$-0.317887\pi$
$5$	2.42130	1.08284	0.541419	+	0.840753i	$0.317887\pi$
$6$	0	0
$7$	−4.62365	−1.74757	−0.873787	−	0.486308i	$-0.838343\pi$
$7$	−4.62365	−1.74757	−0.873787	+	0.486308i	$0.838343\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.42130	−0.765682
$11$	5.75466	1.73510	0.867548	−	0.497353i	$-0.165695\pi$
$11$	5.75466	1.73510	0.867548	+	0.497353i	$0.165695\pi$
$12$	0	0
$13$	−5.37930	−1.49195	−0.745975	−	0.665974i	$-0.768016\pi$
$13$	−5.37930	−1.49195	−0.745975	+	0.665974i	$0.768016\pi$
$14$	4.62365	1.23572
$15$	0	0
$16$	1.00000	0.250000
$17$	2.77000	0.671823	0.335912	−	0.941894i	$-0.390956\pi$
$17$	2.77000	0.671823	0.335912	+	0.941894i	$0.390956\pi$
$18$	0	0
$19$	3.14018	0.720406	0.360203	−	0.932874i	$-0.382708\pi$
$19$	3.14018	0.720406	0.360203	+	0.932874i	$0.382708\pi$
$20$	2.42130	0.541419
$21$	0	0
$22$	−5.75466	−1.22690
$23$	−1.27870	−0.266627	−0.133314	−	0.991074i	$-0.542562\pi$
$23$	−1.27870	−0.266627	−0.133314	+	0.991074i	$0.542562\pi$
$24$	0	0
$25$	0.862695	0.172539
$26$	5.37930	1.05497
$27$	0	0
$28$	−4.62365	−0.873787
$29$	−0.713201	−0.132438	−0.0662191	−	0.997805i	$-0.521094\pi$
$29$	−0.713201	−0.132438	−0.0662191	+	0.997805i	$0.521094\pi$
$30$	0	0
$31$	−3.11809	−0.560026	−0.280013	−	0.959996i	$-0.590339\pi$
$31$	−3.11809	−0.560026	−0.280013	+	0.959996i	$0.590339\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.77000	−0.475051
$35$	−11.1952	−1.89234
$36$	0	0
$37$	−4.81173	−0.791044	−0.395522	−	0.918457i	$-0.629436\pi$
$37$	−4.81173	−0.791044	−0.395522	+	0.918457i	$0.629436\pi$
$38$	−3.14018	−0.509404
$39$	0	0
$40$	−2.42130	−0.382841
$41$	9.85559	1.53918	0.769592	−	0.638536i	$-0.220460\pi$
$41$	9.85559	1.53918	0.769592	+	0.638536i	$0.220460\pi$
$42$	0	0
$43$	0.0559332	0.00852973	0.00426487	−	0.999991i	$-0.498642\pi$
$43$	0.0559332	0.00852973	0.00426487	+	0.999991i	$0.498642\pi$
$44$	5.75466	0.867548
$45$	0	0
$46$	1.27870	0.188534
$47$	8.45189	1.23284	0.616418	−	0.787419i	$-0.288583\pi$
$47$	8.45189	1.23284	0.616418	+	0.787419i	$0.288583\pi$
$48$	0	0
$49$	14.3781	2.05402
$50$	−0.862695	−0.122003
$51$	0	0
$52$	−5.37930	−0.745975
$53$	−2.03001	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$53$	−2.03001	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$54$	0	0
$55$	13.9338	1.87883
$56$	4.62365	0.617861
$57$	0	0
$58$	0.713201	0.0936479
$59$	4.28447	0.557790	0.278895	−	0.960322i	$-0.410032\pi$
$59$	4.28447	0.557790	0.278895	+	0.960322i	$0.410032\pi$
$60$	0	0
$61$	−5.69143	−0.728713	−0.364356	−	0.931260i	$-0.618711\pi$
$61$	−5.69143	−0.728713	−0.364356	+	0.931260i	$0.618711\pi$
$62$	3.11809	0.395998
$63$	0	0
$64$	1.00000	0.125000
$65$	−13.0249	−1.61554
$66$	0	0
$67$	−4.54707	−0.555513	−0.277756	−	0.960652i	$-0.589591\pi$
$67$	−4.54707	−0.555513	−0.277756	+	0.960652i	$0.589591\pi$
$68$	2.77000	0.335912
$69$	0	0
$70$	11.1952	1.33809
$71$	−12.0967	−1.43562	−0.717808	−	0.696241i	$-0.754855\pi$
$71$	−12.0967	−1.43562	−0.717808	+	0.696241i	$0.754855\pi$
$72$	0	0
$73$	−4.39363	−0.514236	−0.257118	−	0.966380i	$-0.582773\pi$
$73$	−4.39363	−0.514236	−0.257118	+	0.966380i	$0.582773\pi$
$74$	4.81173	0.559353
$75$	0	0
$76$	3.14018	0.360203
$77$	−26.6075	−3.03221
$78$	0	0
$79$	8.53478	0.960237	0.480119	−	0.877204i	$-0.340594\pi$
$79$	8.53478	0.960237	0.480119	+	0.877204i	$0.340594\pi$
$80$	2.42130	0.270710
$81$	0	0
$82$	−9.85559	−1.08837
$83$	14.6651	1.60970	0.804851	−	0.593476i	$-0.202245\pi$
$83$	14.6651	1.60970	0.804851	+	0.593476i	$0.202245\pi$
$84$	0	0
$85$	6.70700	0.727476
$86$	−0.0559332	−0.00603143
$87$	0	0
$88$	−5.75466	−0.613449
$89$	5.81228	0.616101	0.308050	−	0.951370i	$-0.400323\pi$
$89$	5.81228	0.616101	0.308050	+	0.951370i	$0.400323\pi$
$90$	0	0
$91$	24.8720	2.60729
$92$	−1.27870	−0.133314
$93$	0	0
$94$	−8.45189	−0.871746
$95$	7.60331	0.780083
$96$	0	0
$97$	6.97235	0.707935	0.353968	−	0.935258i	$-0.384832\pi$
$97$	6.97235	0.707935	0.353968	+	0.935258i	$0.384832\pi$
$98$	−14.3781	−1.45241
$99$	0	0
$100$	0.862695	0.0862695
$101$	−1.35743	−0.135070	−0.0675349	−	0.997717i	$-0.521513\pi$
$101$	−1.35743	−0.135070	−0.0675349	+	0.997717i	$0.521513\pi$
$102$	0	0
$103$	−18.8772	−1.86003	−0.930014	−	0.367523i	$-0.880206\pi$
$103$	−18.8772	−1.86003	−0.930014	+	0.367523i	$0.880206\pi$
$104$	5.37930	0.527484
$105$	0	0
$106$	2.03001	0.197172
$107$	1.66501	0.160963	0.0804813	−	0.996756i	$-0.474354\pi$
$107$	1.66501	0.160963	0.0804813	+	0.996756i	$0.474354\pi$
$108$	0	0
$109$	16.3709	1.56805	0.784023	−	0.620732i	$-0.213164\pi$
$109$	16.3709	1.56805	0.784023	+	0.620732i	$0.213164\pi$
$110$	−13.9338	−1.32853
$111$	0	0
$112$	−4.62365	−0.436894
$113$	5.29780	0.498375	0.249188	−	0.968455i	$-0.419836\pi$
$113$	5.29780	0.498375	0.249188	+	0.968455i	$0.419836\pi$
$114$	0	0
$115$	−3.09611	−0.288714
$116$	−0.713201	−0.0662191
$117$	0	0
$118$	−4.28447	−0.394417
$119$	−12.8075	−1.17406
$120$	0	0
$121$	22.1161	2.01056
$122$	5.69143	0.515278
$123$	0	0
$124$	−3.11809	−0.280013
$125$	−10.0177	−0.896007
$126$	0	0
$127$	−3.69254	−0.327660	−0.163830	−	0.986489i	$-0.552385\pi$
$127$	−3.69254	−0.327660	−0.163830	+	0.986489i	$0.552385\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	13.0249	1.14236
$131$	−18.6771	−1.63183	−0.815915	−	0.578172i	$-0.803766\pi$
$131$	−18.6771	−1.63183	−0.815915	+	0.578172i	$0.803766\pi$
$132$	0	0
$133$	−14.5191	−1.25896
$134$	4.54707	0.392807
$135$	0	0
$136$	−2.77000	−0.237525
$137$	11.5769	0.989078	0.494539	−	0.869155i	$-0.335337\pi$
$137$	11.5769	0.989078	0.494539	+	0.869155i	$0.335337\pi$
$138$	0	0
$139$	−3.51803	−0.298395	−0.149198	−	0.988807i	$-0.547669\pi$
$139$	−3.51803	−0.298395	−0.149198	+	0.988807i	$0.547669\pi$
$140$	−11.1952	−0.946171
$141$	0	0
$142$	12.0967	1.01513
$143$	−30.9561	−2.58868
$144$	0	0
$145$	−1.72687	−0.143409
$146$	4.39363	0.363620
$147$	0	0
$148$	−4.81173	−0.395522
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	22.1146	1.79966	0.899831	−	0.436239i	$-0.143690\pi$
$151$	22.1146	1.79966	0.899831	+	0.436239i	$0.143690\pi$
$152$	−3.14018	−0.254702
$153$	0	0
$154$	26.6075	2.14410
$155$	−7.54984	−0.606418
$156$	0	0
$157$	−9.16866	−0.731739	−0.365869	−	0.930666i	$-0.619228\pi$
$157$	−9.16866	−0.731739	−0.365869	+	0.930666i	$0.619228\pi$
$158$	−8.53478	−0.678990
$159$	0	0
$160$	−2.42130	−0.191421
$161$	5.91225	0.465951
$162$	0	0
$163$	−2.38667	−0.186938	−0.0934692	−	0.995622i	$-0.529796\pi$
$163$	−2.38667	−0.186938	−0.0934692	+	0.995622i	$0.529796\pi$
$164$	9.85559	0.769592
$165$	0	0
$166$	−14.6651	−1.13823
$167$	8.36126	0.647014	0.323507	−	0.946226i	$-0.395138\pi$
$167$	8.36126	0.647014	0.323507	+	0.946226i	$0.395138\pi$
$168$	0	0
$169$	15.9369	1.22592
$170$	−6.70700	−0.514403
$171$	0	0
$172$	0.0559332	0.00426487
$173$	−13.1646	−1.00088	−0.500441	−	0.865770i	$-0.666829\pi$
$173$	−13.1646	−1.00088	−0.500441	+	0.865770i	$0.666829\pi$
$174$	0	0
$175$	−3.98880	−0.301525
$176$	5.75466	0.433774
$177$	0	0
$178$	−5.81228	−0.435649
$179$	−5.86503	−0.438373	−0.219187	−	0.975683i	$-0.570340\pi$
$179$	−5.86503	−0.438373	−0.219187	+	0.975683i	$0.570340\pi$
$180$	0	0
$181$	21.8824	1.62650	0.813252	−	0.581911i	$-0.197695\pi$
$181$	21.8824	1.62650	0.813252	+	0.581911i	$0.197695\pi$
$182$	−24.8720	−1.84364
$183$	0	0
$184$	1.27870	0.0942669
$185$	−11.6506	−0.856573
$186$	0	0
$187$	15.9404	1.16568
$188$	8.45189	0.616418
$189$	0	0
$190$	−7.60331	−0.551602
$191$	0.591279	0.0427835	0.0213917	−	0.999771i	$-0.493190\pi$
$191$	0.591279	0.0427835	0.0213917	+	0.999771i	$0.493190\pi$
$192$	0	0
$193$	26.1244	1.88048	0.940239	−	0.340516i	$-0.110602\pi$
$193$	26.1244	1.88048	0.940239	+	0.340516i	$0.110602\pi$
$194$	−6.97235	−0.500586
$195$	0	0
$196$	14.3781	1.02701
$197$	−11.8377	−0.843399	−0.421700	−	0.906736i	$-0.638566\pi$
$197$	−11.8377	−0.843399	−0.421700	+	0.906736i	$0.638566\pi$
$198$	0	0
$199$	−10.4355	−0.739750	−0.369875	−	0.929082i	$-0.620599\pi$
$199$	−10.4355	−0.739750	−0.369875	+	0.929082i	$0.620599\pi$
$200$	−0.862695	−0.0610017
$201$	0	0
$202$	1.35743	0.0955087
$203$	3.29759	0.231446
$204$	0	0
$205$	23.8633	1.66669
$206$	18.8772	1.31524
$207$	0	0
$208$	−5.37930	−0.372988
$209$	18.0706	1.24997
$210$	0	0
$211$	−2.16223	−0.148854	−0.0744270	−	0.997226i	$-0.523713\pi$
$211$	−2.16223	−0.148854	−0.0744270	+	0.997226i	$0.523713\pi$
$212$	−2.03001	−0.139422
$213$	0	0
$214$	−1.66501	−0.113818
$215$	0.135431	0.00923632
$216$	0	0
$217$	14.4170	0.978688
$218$	−16.3709	−1.10878
$219$	0	0
$220$	13.9338	0.939414
$221$	−14.9007	−1.00233
$222$	0	0
$223$	20.0738	1.34424	0.672121	−	0.740441i	$-0.265383\pi$
$223$	20.0738	1.34424	0.672121	+	0.740441i	$0.265383\pi$
$224$	4.62365	0.308931
$225$	0	0
$226$	−5.29780	−0.352405
$227$	27.4930	1.82478	0.912389	−	0.409325i	$-0.134236\pi$
$227$	27.4930	1.82478	0.912389	+	0.409325i	$0.134236\pi$
$228$	0	0
$229$	−6.33695	−0.418758	−0.209379	−	0.977835i	$-0.567144\pi$
$229$	−6.33695	−0.418758	−0.209379	+	0.977835i	$0.567144\pi$
$230$	3.09611	0.204152
$231$	0	0
$232$	0.713201	0.0468239
$233$	22.0167	1.44236	0.721182	−	0.692746i	$-0.243599\pi$
$233$	22.0167	1.44236	0.721182	+	0.692746i	$0.243599\pi$
$234$	0	0
$235$	20.4646	1.33496
$236$	4.28447	0.278895
$237$	0	0
$238$	12.8075	0.830187
$239$	19.3365	1.25078	0.625388	−	0.780314i	$-0.284941\pi$
$239$	19.3365	1.25078	0.625388	+	0.780314i	$0.284941\pi$
$240$	0	0
$241$	15.0621	0.970237	0.485119	−	0.874448i	$-0.338776\pi$
$241$	15.0621	0.970237	0.485119	+	0.874448i	$0.338776\pi$
$242$	−22.1161	−1.42168
$243$	0	0
$244$	−5.69143	−0.364356
$245$	34.8138	2.22417
$246$	0	0
$247$	−16.8920	−1.07481
$248$	3.11809	0.197999
$249$	0	0
$250$	10.0177	0.633572
$251$	−12.2993	−0.776322	−0.388161	−	0.921592i	$-0.626890\pi$
$251$	−12.2993	−0.776322	−0.388161	+	0.921592i	$0.626890\pi$
$252$	0	0
$253$	−7.35848	−0.462623
$254$	3.69254	0.231690
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.58586	0.597949	0.298975	−	0.954261i	$-0.403355\pi$
$257$	9.58586	0.597949	0.298975	+	0.954261i	$0.403355\pi$
$258$	0	0
$259$	22.2478	1.38241
$260$	−13.0249	−0.807771
$261$	0	0
$262$	18.6771	1.15388
$263$	10.6930	0.659357	0.329679	−	0.944093i	$-0.393060\pi$
$263$	10.6930	0.659357	0.329679	+	0.944093i	$0.393060\pi$
$264$	0	0
$265$	−4.91528	−0.301943
$266$	14.5191	0.890221
$267$	0	0
$268$	−4.54707	−0.277756
$269$	4.75865	0.290140	0.145070	−	0.989421i	$-0.453659\pi$
$269$	4.75865	0.290140	0.145070	+	0.989421i	$0.453659\pi$
$270$	0	0
$271$	20.5355	1.24744	0.623721	−	0.781647i	$-0.285620\pi$
$271$	20.5355	1.24744	0.623721	+	0.781647i	$0.285620\pi$
$272$	2.77000	0.167956
$273$	0	0
$274$	−11.5769	−0.699384
$275$	4.96452	0.299372
$276$	0	0
$277$	6.22873	0.374248	0.187124	−	0.982336i	$-0.440083\pi$
$277$	6.22873	0.374248	0.187124	+	0.982336i	$0.440083\pi$
$278$	3.51803	0.210997
$279$	0	0
$280$	11.1952	0.669044
$281$	16.3809	0.977202	0.488601	−	0.872507i	$-0.337507\pi$
$281$	16.3809	0.977202	0.488601	+	0.872507i	$0.337507\pi$
$282$	0	0
$283$	−22.9033	−1.36146	−0.680730	−	0.732535i	$-0.738337\pi$
$283$	−22.9033	−1.36146	−0.680730	+	0.732535i	$0.738337\pi$
$284$	−12.0967	−0.717808
$285$	0	0
$286$	30.9561	1.83047
$287$	−45.5688	−2.68984
$288$	0	0
$289$	−9.32711	−0.548653
$290$	1.72687	0.101406
$291$	0	0
$292$	−4.39363	−0.257118
$293$	20.6725	1.20770	0.603850	−	0.797098i	$-0.293633\pi$
$293$	20.6725	1.20770	0.603850	+	0.797098i	$0.293633\pi$
$294$	0	0
$295$	10.3740	0.603996
$296$	4.81173	0.279676
$297$	0	0
$298$	1.00000	0.0579284
$299$	6.87851	0.397794
$300$	0	0
$301$	−0.258615	−0.0149063
$302$	−22.1146	−1.27255
$303$	0	0
$304$	3.14018	0.180101
$305$	−13.7807	−0.789078
$306$	0	0
$307$	−22.4597	−1.28185	−0.640923	−	0.767606i	$-0.721448\pi$
$307$	−22.4597	−1.28185	−0.640923	+	0.767606i	$0.721448\pi$
$308$	−26.6075	−1.51611
$309$	0	0
$310$	7.54984	0.428802
$311$	−10.3800	−0.588597	−0.294299	−	0.955714i	$-0.595086\pi$
$311$	−10.3800	−0.588597	−0.294299	+	0.955714i	$0.595086\pi$
$312$	0	0
$313$	21.4476	1.21229	0.606144	−	0.795355i	$-0.292716\pi$
$313$	21.4476	1.21229	0.606144	+	0.795355i	$0.292716\pi$
$314$	9.16866	0.517418
$315$	0	0
$316$	8.53478	0.480119
$317$	20.4942	1.15107	0.575536	−	0.817777i	$-0.304794\pi$
$317$	20.4942	1.15107	0.575536	+	0.817777i	$0.304794\pi$
$318$	0	0
$319$	−4.10423	−0.229793
$320$	2.42130	0.135355
$321$	0	0
$322$	−5.91225	−0.329477
$323$	8.69828	0.483985
$324$	0	0
$325$	−4.64070	−0.257420
$326$	2.38667	0.132185
$327$	0	0
$328$	−9.85559	−0.544184
$329$	−39.0786	−2.15447
$330$	0	0
$331$	−14.5565	−0.800097	−0.400048	−	0.916494i	$-0.631007\pi$
$331$	−14.5565	−0.800097	−0.400048	+	0.916494i	$0.631007\pi$
$332$	14.6651	0.804851
$333$	0	0
$334$	−8.36126	−0.457508
$335$	−11.0098	−0.601531
$336$	0	0
$337$	−15.6907	−0.854729	−0.427365	−	0.904079i	$-0.640558\pi$
$337$	−15.6907	−0.854729	−0.427365	+	0.904079i	$0.640558\pi$
$338$	−15.9369	−0.866853
$339$	0	0
$340$	6.70700	0.363738
$341$	−17.9436	−0.971699
$342$	0	0
$343$	−34.1139	−1.84198
$344$	−0.0559332	−0.00301572
$345$	0	0
$346$	13.1646	0.707731
$347$	−36.6514	−1.96755	−0.983775	−	0.179409i	$-0.942582\pi$
$347$	−36.6514	−1.96755	−0.983775	+	0.179409i	$0.942582\pi$
$348$	0	0
$349$	−26.6932	−1.42885	−0.714426	−	0.699711i	$-0.753312\pi$
$349$	−26.6932	−1.42885	−0.714426	+	0.699711i	$0.753312\pi$
$350$	3.98880	0.213210
$351$	0	0
$352$	−5.75466	−0.306725
$353$	14.9235	0.794299	0.397149	−	0.917754i	$-0.369999\pi$
$353$	14.9235	0.794299	0.397149	+	0.917754i	$0.369999\pi$
$354$	0	0
$355$	−29.2898	−1.55454
$356$	5.81228	0.308050
$357$	0	0
$358$	5.86503	0.309977
$359$	13.7520	0.725803	0.362901	−	0.931828i	$-0.381786\pi$
$359$	13.7520	0.725803	0.362901	+	0.931828i	$0.381786\pi$
$360$	0	0
$361$	−9.13930	−0.481016
$362$	−21.8824	−1.15011
$363$	0	0
$364$	24.8720	1.30365
$365$	−10.6383	−0.556834
$366$	0	0
$367$	30.7516	1.60522	0.802610	−	0.596504i	$-0.203444\pi$
$367$	30.7516	1.60522	0.802610	+	0.596504i	$0.203444\pi$
$368$	−1.27870	−0.0666568
$369$	0	0
$370$	11.6506	0.605688
$371$	9.38608	0.487301
$372$	0	0
$373$	13.1146	0.679046	0.339523	−	0.940598i	$-0.389734\pi$
$373$	13.1146	0.679046	0.339523	+	0.940598i	$0.389734\pi$
$374$	−15.9404	−0.824259
$375$	0	0
$376$	−8.45189	−0.435873
$377$	3.83653	0.197591
$378$	0	0
$379$	1.99676	0.102567	0.0512833	−	0.998684i	$-0.483669\pi$
$379$	1.99676	0.102567	0.0512833	+	0.998684i	$0.483669\pi$
$380$	7.60331	0.390041
$381$	0	0
$382$	−0.591279	−0.0302525
$383$	11.8887	0.607486	0.303743	−	0.952754i	$-0.401764\pi$
$383$	11.8887	0.607486	0.303743	+	0.952754i	$0.401764\pi$
$384$	0	0
$385$	−64.4248	−3.28339
$386$	−26.1244	−1.32970
$387$	0	0
$388$	6.97235	0.353968
$389$	32.7574	1.66086	0.830432	−	0.557120i	$-0.188094\pi$
$389$	32.7574	1.66086	0.830432	+	0.557120i	$0.188094\pi$
$390$	0	0
$391$	−3.54199	−0.179126
$392$	−14.3781	−0.726205
$393$	0	0
$394$	11.8377	0.596373
$395$	20.6653	1.03978
$396$	0	0
$397$	−0.448753	−0.0225223	−0.0112611	−	0.999937i	$-0.503585\pi$
$397$	−0.448753	−0.0225223	−0.0112611	+	0.999937i	$0.503585\pi$
$398$	10.4355	0.523082
$399$	0	0
$400$	0.862695	0.0431347
$401$	12.9635	0.647366	0.323683	−	0.946166i	$-0.395079\pi$
$401$	12.9635	0.647366	0.323683	+	0.946166i	$0.395079\pi$
$402$	0	0
$403$	16.7732	0.835531
$404$	−1.35743	−0.0675349
$405$	0	0
$406$	−3.29759	−0.163657
$407$	−27.6899	−1.37254
$408$	0	0
$409$	−26.0203	−1.28662	−0.643311	−	0.765605i	$-0.722440\pi$
$409$	−26.0203	−1.28662	−0.643311	+	0.765605i	$0.722440\pi$
$410$	−23.8633	−1.17853
$411$	0	0
$412$	−18.8772	−0.930014
$413$	−19.8099	−0.974780
$414$	0	0
$415$	35.5086	1.74305
$416$	5.37930	0.263742
$417$	0	0
$418$	−18.0706	−0.883864
$419$	8.84720	0.432214	0.216107	−	0.976370i	$-0.430664\pi$
$419$	8.84720	0.432214	0.216107	+	0.976370i	$0.430664\pi$
$420$	0	0
$421$	−22.1463	−1.07934	−0.539672	−	0.841875i	$-0.681452\pi$
$421$	−22.1463	−1.07934	−0.539672	+	0.841875i	$0.681452\pi$
$422$	2.16223	0.105256
$423$	0	0
$424$	2.03001	0.0985862
$425$	2.38966	0.115916
$426$	0	0
$427$	26.3152	1.27348
$428$	1.66501	0.0804813
$429$	0	0
$430$	−0.135431	−0.00653107
$431$	16.3398	0.787062	0.393531	−	0.919311i	$-0.371253\pi$
$431$	16.3398	0.787062	0.393531	+	0.919311i	$0.371253\pi$
$432$	0	0
$433$	−15.5368	−0.746651	−0.373326	−	0.927700i	$-0.621783\pi$
$433$	−15.5368	−0.746651	−0.373326	+	0.927700i	$0.621783\pi$
$434$	−14.4170	−0.692037
$435$	0	0
$436$	16.3709	0.784023
$437$	−4.01534	−0.192080
$438$	0	0
$439$	−16.4230	−0.783829	−0.391914	−	0.920002i	$-0.628187\pi$
$439$	−16.4230	−0.783829	−0.391914	+	0.920002i	$0.628187\pi$
$440$	−13.9338	−0.664266
$441$	0	0
$442$	14.9007	0.708752
$443$	−9.94687	−0.472590	−0.236295	−	0.971681i	$-0.575933\pi$
$443$	−9.94687	−0.472590	−0.236295	+	0.971681i	$0.575933\pi$
$444$	0	0
$445$	14.0733	0.667137
$446$	−20.0738	−0.950523
$447$	0	0
$448$	−4.62365	−0.218447
$449$	29.6325	1.39844	0.699221	−	0.714906i	$-0.253530\pi$
$449$	29.6325	1.39844	0.699221	+	0.714906i	$0.253530\pi$
$450$	0	0
$451$	56.7156	2.67063
$452$	5.29780	0.249188
$453$	0	0
$454$	−27.4930	−1.29031
$455$	60.2226	2.82328
$456$	0	0
$457$	−32.0007	−1.49693	−0.748466	−	0.663173i	$-0.769209\pi$
$457$	−32.0007	−1.49693	−0.748466	+	0.663173i	$0.769209\pi$
$458$	6.33695	0.296106
$459$	0	0
$460$	−3.09611	−0.144357
$461$	10.0526	0.468194	0.234097	−	0.972213i	$-0.424787\pi$
$461$	10.0526	0.468194	0.234097	+	0.972213i	$0.424787\pi$
$462$	0	0
$463$	34.7035	1.61281	0.806404	−	0.591365i	$-0.201411\pi$
$463$	34.7035	1.61281	0.806404	+	0.591365i	$0.201411\pi$
$464$	−0.713201	−0.0331095
$465$	0	0
$466$	−22.0167	−1.01990
$467$	11.1085	0.514039	0.257019	−	0.966406i	$-0.417260\pi$
$467$	11.1085	0.514039	0.257019	+	0.966406i	$0.417260\pi$
$468$	0	0
$469$	21.0240	0.970800
$470$	−20.4646	−0.943960
$471$	0	0
$472$	−4.28447	−0.197209
$473$	0.321877	0.0147999
$474$	0	0
$475$	2.70901	0.124298
$476$	−12.8075	−0.587031
$477$	0	0
$478$	−19.3365	−0.884432
$479$	−5.66215	−0.258710	−0.129355	−	0.991598i	$-0.541291\pi$
$479$	−5.66215	−0.258710	−0.129355	+	0.991598i	$0.541291\pi$
$480$	0	0
$481$	25.8838	1.18020
$482$	−15.0621	−0.686061
$483$	0	0
$484$	22.1161	1.00528
$485$	16.8822	0.766579
$486$	0	0
$487$	−35.5368	−1.61032	−0.805162	−	0.593055i	$-0.797922\pi$
$487$	−35.5368	−1.61032	−0.805162	+	0.593055i	$0.797922\pi$
$488$	5.69143	0.257639
$489$	0	0
$490$	−34.8138	−1.57273
$491$	1.32518	0.0598043	0.0299022	−	0.999553i	$-0.490480\pi$
$491$	1.32518	0.0598043	0.0299022	+	0.999553i	$0.490480\pi$
$492$	0	0
$493$	−1.97557	−0.0889750
$494$	16.8920	0.760005
$495$	0	0
$496$	−3.11809	−0.140007
$497$	55.9310	2.50885
$498$	0	0
$499$	16.8764	0.755489	0.377745	−	0.925910i	$-0.376700\pi$
$499$	16.8764	0.755489	0.377745	+	0.925910i	$0.376700\pi$
$500$	−10.0177	−0.448003
$501$	0	0
$502$	12.2993	0.548942
$503$	−10.6307	−0.473999	−0.237000	−	0.971510i	$-0.576164\pi$
$503$	−10.6307	−0.473999	−0.237000	+	0.971510i	$0.576164\pi$
$504$	0	0
$505$	−3.28676	−0.146259
$506$	7.35848	0.327124
$507$	0	0
$508$	−3.69254	−0.163830
$509$	18.0273	0.799045	0.399522	−	0.916723i	$-0.369176\pi$
$509$	18.0273	0.799045	0.399522	+	0.916723i	$0.369176\pi$
$510$	0	0
$511$	20.3146	0.898666
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−9.58586	−0.422814
$515$	−45.7074	−2.01411
$516$	0	0
$517$	48.6378	2.13909
$518$	−22.2478	−0.977510
$519$	0	0
$520$	13.0249	0.571180
$521$	0.237005	0.0103834	0.00519169	−	0.999987i	$-0.498347\pi$
$521$	0.237005	0.0103834	0.00519169	+	0.999987i	$0.498347\pi$
$522$	0	0
$523$	42.3389	1.85135	0.925674	−	0.378321i	$-0.123499\pi$
$523$	42.3389	1.85135	0.925674	+	0.378321i	$0.123499\pi$
$524$	−18.6771	−0.815915
$525$	0	0
$526$	−10.6930	−0.466236
$527$	−8.63711	−0.376239
$528$	0	0
$529$	−21.3649	−0.928910
$530$	4.91528	0.213506
$531$	0	0
$532$	−14.5191	−0.629481
$533$	−53.0162	−2.29639
$534$	0	0
$535$	4.03149	0.174296
$536$	4.54707	0.196403
$537$	0	0
$538$	−4.75865	−0.205160
$539$	82.7413	3.56392
$540$	0	0
$541$	−27.1339	−1.16658	−0.583288	−	0.812265i	$-0.698234\pi$
$541$	−27.1339	−1.16658	−0.583288	+	0.812265i	$0.698234\pi$
$542$	−20.5355	−0.882075
$543$	0	0
$544$	−2.77000	−0.118763
$545$	39.6388	1.69794
$546$	0	0
$547$	−16.0471	−0.686123	−0.343062	−	0.939313i	$-0.611464\pi$
$547$	−16.0471	−0.686123	−0.343062	+	0.939313i	$0.611464\pi$
$548$	11.5769	0.494539
$549$	0	0
$550$	−4.96452	−0.211688
$551$	−2.23958	−0.0954092
$552$	0	0
$553$	−39.4618	−1.67809
$554$	−6.22873	−0.264633
$555$	0	0
$556$	−3.51803	−0.149198
$557$	−24.9616	−1.05766	−0.528828	−	0.848729i	$-0.677368\pi$
$557$	−24.9616	−1.05766	−0.528828	+	0.848729i	$0.677368\pi$
$558$	0	0
$559$	−0.300882	−0.0127259
$560$	−11.1952	−0.473085
$561$	0	0
$562$	−16.3809	−0.690986
$563$	−7.49303	−0.315794	−0.157897	−	0.987456i	$-0.550471\pi$
$563$	−7.49303	−0.315794	−0.157897	+	0.987456i	$0.550471\pi$
$564$	0	0
$565$	12.8276	0.539660
$566$	22.9033	0.962697
$567$	0	0
$568$	12.0967	0.507567
$569$	33.7700	1.41571	0.707856	−	0.706357i	$-0.249663\pi$
$569$	33.7700	1.41571	0.707856	+	0.706357i	$0.249663\pi$
$570$	0	0
$571$	−0.236568	−0.00990006	−0.00495003	−	0.999988i	$-0.501576\pi$
$571$	−0.236568	−0.00990006	−0.00495003	+	0.999988i	$0.501576\pi$
$572$	−30.9561	−1.29434
$573$	0	0
$574$	45.5688	1.90200
$575$	−1.10313	−0.0460036
$576$	0	0
$577$	30.5626	1.27234	0.636169	−	0.771549i	$-0.280518\pi$
$577$	30.5626	1.27234	0.636169	+	0.771549i	$0.280518\pi$
$578$	9.32711	0.387957
$579$	0	0
$580$	−1.72687	−0.0717045
$581$	−67.8062	−2.81308
$582$	0	0
$583$	−11.6821	−0.483821
$584$	4.39363	0.181810
$585$	0	0
$586$	−20.6725	−0.853973
$587$	2.09669	0.0865397	0.0432698	−	0.999063i	$-0.486222\pi$
$587$	2.09669	0.0865397	0.0432698	+	0.999063i	$0.486222\pi$
$588$	0	0
$589$	−9.79136	−0.403446
$590$	−10.3740	−0.427090
$591$	0	0
$592$	−4.81173	−0.197761
$593$	23.7017	0.973312	0.486656	−	0.873594i	$-0.338216\pi$
$593$	23.7017	0.973312	0.486656	+	0.873594i	$0.338216\pi$
$594$	0	0
$595$	−31.0108	−1.27132
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−6.87851	−0.281283
$599$	−46.2157	−1.88832	−0.944161	−	0.329484i	$-0.893125\pi$
$599$	−46.2157	−1.88832	−0.944161	+	0.329484i	$0.893125\pi$
$600$	0	0
$601$	20.7365	0.845861	0.422930	−	0.906162i	$-0.361001\pi$
$601$	20.7365	0.845861	0.422930	+	0.906162i	$0.361001\pi$
$602$	0.258615	0.0105404
$603$	0	0
$604$	22.1146	0.899831
$605$	53.5498	2.17711
$606$	0	0
$607$	31.8192	1.29150	0.645751	−	0.763548i	$-0.276545\pi$
$607$	31.8192	1.29150	0.645751	+	0.763548i	$0.276545\pi$
$608$	−3.14018	−0.127351
$609$	0	0
$610$	13.7807	0.557962
$611$	−45.4653	−1.83933
$612$	0	0
$613$	31.1593	1.25851	0.629256	−	0.777198i	$-0.283360\pi$
$613$	31.1593	1.25851	0.629256	+	0.777198i	$0.283360\pi$
$614$	22.4597	0.906401
$615$	0	0
$616$	26.6075	1.07205
$617$	−11.0620	−0.445339	−0.222670	−	0.974894i	$-0.571477\pi$
$617$	−11.0620	−0.445339	−0.222670	+	0.974894i	$0.571477\pi$
$618$	0	0
$619$	−22.5979	−0.908285	−0.454142	−	0.890929i	$-0.650054\pi$
$619$	−22.5979	−0.908285	−0.454142	+	0.890929i	$0.650054\pi$
$620$	−7.54984	−0.303209
$621$	0	0
$622$	10.3800	0.416201
$623$	−26.8740	−1.07668
$624$	0	0
$625$	−28.5692	−1.14277
$626$	−21.4476	−0.857217
$627$	0	0
$628$	−9.16866	−0.365869
$629$	−13.3285	−0.531442
$630$	0	0
$631$	−0.468730	−0.0186598	−0.00932992	−	0.999956i	$-0.502970\pi$
$631$	−0.468730	−0.0186598	−0.00932992	+	0.999956i	$0.502970\pi$
$632$	−8.53478	−0.339495
$633$	0	0
$634$	−20.4942	−0.813930
$635$	−8.94074	−0.354803
$636$	0	0
$637$	−77.3443	−3.06449
$638$	4.10423	0.162488
$639$	0	0
$640$	−2.42130	−0.0957103
$641$	30.9786	1.22358	0.611792	−	0.791019i	$-0.290449\pi$
$641$	30.9786	1.22358	0.611792	+	0.791019i	$0.290449\pi$
$642$	0	0
$643$	−36.4591	−1.43781	−0.718903	−	0.695111i	$-0.755355\pi$
$643$	−36.4591	−1.43781	−0.718903	+	0.695111i	$0.755355\pi$
$644$	5.91225	0.232975
$645$	0	0
$646$	−8.69828	−0.342229
$647$	20.9279	0.822760	0.411380	−	0.911464i	$-0.365047\pi$
$647$	20.9279	0.822760	0.411380	+	0.911464i	$0.365047\pi$
$648$	0	0
$649$	24.6557	0.967819
$650$	4.64070	0.182023
$651$	0	0
$652$	−2.38667	−0.0934692
$653$	23.6389	0.925062	0.462531	−	0.886603i	$-0.346941\pi$
$653$	23.6389	0.925062	0.462531	+	0.886603i	$0.346941\pi$
$654$	0	0
$655$	−45.2230	−1.76701
$656$	9.85559	0.384796
$657$	0	0
$658$	39.0786	1.52344
$659$	19.5364	0.761030	0.380515	−	0.924775i	$-0.375747\pi$
$659$	19.5364	0.761030	0.380515	+	0.924775i	$0.375747\pi$
$660$	0	0
$661$	−7.81474	−0.303958	−0.151979	−	0.988384i	$-0.548565\pi$
$661$	−7.81474	−0.303958	−0.151979	+	0.988384i	$0.548565\pi$
$662$	14.5565	0.565754
$663$	0	0
$664$	−14.6651	−0.569116
$665$	−35.1550	−1.36325
$666$	0	0
$667$	0.911969	0.0353116
$668$	8.36126	0.323507
$669$	0	0
$670$	11.0098	0.425346
$671$	−32.7522	−1.26439
$672$	0	0
$673$	21.2026	0.817300	0.408650	−	0.912691i	$-0.366000\pi$
$673$	21.2026	0.817300	0.408650	+	0.912691i	$0.366000\pi$
$674$	15.6907	0.604385
$675$	0	0
$676$	15.9369	0.612958
$677$	3.83407	0.147355	0.0736776	−	0.997282i	$-0.476526\pi$
$677$	3.83407	0.147355	0.0736776	+	0.997282i	$0.476526\pi$
$678$	0	0
$679$	−32.2377	−1.23717
$680$	−6.70700	−0.257202
$681$	0	0
$682$	17.9436	0.687095
$683$	−18.3118	−0.700681	−0.350341	−	0.936622i	$-0.613934\pi$
$683$	−18.3118	−0.700681	−0.350341	+	0.936622i	$0.613934\pi$
$684$	0	0
$685$	28.0311	1.07101
$686$	34.1139	1.30247
$687$	0	0
$688$	0.0559332	0.00213243
$689$	10.9201	0.416021
$690$	0	0
$691$	19.0354	0.724142	0.362071	−	0.932151i	$-0.382070\pi$
$691$	19.0354	0.724142	0.362071	+	0.932151i	$0.382070\pi$
$692$	−13.1646	−0.500441
$693$	0	0
$694$	36.6514	1.39127
$695$	−8.51820	−0.323114
$696$	0	0
$697$	27.3000	1.03406
$698$	26.6932	1.01035
$699$	0	0
$700$	−3.98880	−0.150762
$701$	18.7586	0.708504	0.354252	−	0.935150i	$-0.384735\pi$
$701$	18.7586	0.708504	0.354252	+	0.935150i	$0.384735\pi$
$702$	0	0
$703$	−15.1097	−0.569872
$704$	5.75466	0.216887
$705$	0	0
$706$	−14.9235	−0.561654
$707$	6.27630	0.236044
$708$	0	0
$709$	48.2259	1.81116	0.905581	−	0.424173i	$-0.139435\pi$
$709$	48.2259	1.81116	0.905581	+	0.424173i	$0.139435\pi$
$710$	29.2898	1.09923
$711$	0	0
$712$	−5.81228	−0.217824
$713$	3.98710	0.149318
$714$	0	0
$715$	−74.9540	−2.80312
$716$	−5.86503	−0.219187
$717$	0	0
$718$	−13.7520	−0.513220
$719$	16.7886	0.626108	0.313054	−	0.949735i	$-0.398648\pi$
$719$	16.7886	0.626108	0.313054	+	0.949735i	$0.398648\pi$
$720$	0	0
$721$	87.2817	3.25054
$722$	9.13930	0.340130
$723$	0	0
$724$	21.8824	0.813252
$725$	−0.615275	−0.0228507
$726$	0	0
$727$	−21.2002	−0.786273	−0.393137	−	0.919480i	$-0.628610\pi$
$727$	−21.2002	−0.786273	−0.393137	+	0.919480i	$0.628610\pi$
$728$	−24.8720	−0.921818
$729$	0	0
$730$	10.6383	0.393741
$731$	0.154935	0.00573047
$732$	0	0
$733$	25.6128	0.946032	0.473016	−	0.881054i	$-0.343165\pi$
$733$	25.6128	0.946032	0.473016	+	0.881054i	$0.343165\pi$
$734$	−30.7516	−1.13506
$735$	0	0
$736$	1.27870	0.0471334
$737$	−26.1668	−0.963868
$738$	0	0
$739$	16.9591	0.623850	0.311925	−	0.950107i	$-0.399026\pi$
$739$	16.9591	0.623850	0.311925	+	0.950107i	$0.399026\pi$
$740$	−11.6506	−0.428286
$741$	0	0
$742$	−9.38608	−0.344574
$743$	−40.1918	−1.47450	−0.737248	−	0.675622i	$-0.763875\pi$
$743$	−40.1918	−1.47450	−0.737248	+	0.675622i	$0.763875\pi$
$744$	0	0
$745$	−2.42130	−0.0887096
$746$	−13.1146	−0.480158
$747$	0	0
$748$	15.9404	0.582839
$749$	−7.69842	−0.281294
$750$	0	0
$751$	35.4352	1.29305	0.646524	−	0.762893i	$-0.276222\pi$
$751$	35.4352	1.29305	0.646524	+	0.762893i	$0.276222\pi$
$752$	8.45189	0.308209
$753$	0	0
$754$	−3.83653	−0.139718
$755$	53.5461	1.94874
$756$	0	0
$757$	14.3642	0.522075	0.261037	−	0.965329i	$-0.415935\pi$
$757$	14.3642	0.522075	0.261037	+	0.965329i	$0.415935\pi$
$758$	−1.99676	−0.0725256
$759$	0	0
$760$	−7.60331	−0.275801
$761$	14.8591	0.538642	0.269321	−	0.963050i	$-0.413201\pi$
$761$	14.8591	0.538642	0.269321	+	0.963050i	$0.413201\pi$
$762$	0	0
$763$	−75.6932	−2.74028
$764$	0.591279	0.0213917
$765$	0	0
$766$	−11.8887	−0.429558
$767$	−23.0474	−0.832195
$768$	0	0
$769$	22.1003	0.796957	0.398478	−	0.917178i	$-0.369538\pi$
$769$	22.1003	0.796957	0.398478	+	0.917178i	$0.369538\pi$
$770$	64.4248	2.32171
$771$	0	0
$772$	26.1244	0.940239
$773$	−8.03972	−0.289169	−0.144584	−	0.989492i	$-0.546185\pi$
$773$	−8.03972	−0.289169	−0.144584	+	0.989492i	$0.546185\pi$
$774$	0	0
$775$	−2.68996	−0.0966264
$776$	−6.97235	−0.250293
$777$	0	0
$778$	−32.7574	−1.17441
$779$	30.9483	1.10884
$780$	0	0
$781$	−69.6125	−2.49093
$782$	3.54199	0.126661
$783$	0	0
$784$	14.3781	0.513505
$785$	−22.2001	−0.792355
$786$	0	0
$787$	33.6592	1.19982	0.599910	−	0.800067i	$-0.295203\pi$
$787$	33.6592	1.19982	0.599910	+	0.800067i	$0.295203\pi$
$788$	−11.8377	−0.421700
$789$	0	0
$790$	−20.6653	−0.735237
$791$	−24.4952	−0.870948
$792$	0	0
$793$	30.6159	1.08720
$794$	0.448753	0.0159256
$795$	0	0
$796$	−10.4355	−0.369875
$797$	−8.64942	−0.306378	−0.153189	−	0.988197i	$-0.548954\pi$
$797$	−8.64942	−0.306378	−0.153189	+	0.988197i	$0.548954\pi$
$798$	0	0
$799$	23.4117	0.828247
$800$	−0.862695	−0.0305009
$801$	0	0
$802$	−12.9635	−0.457757
$803$	−25.2839	−0.892249
$804$	0	0
$805$	14.3153	0.504549
$806$	−16.7732	−0.590810
$807$	0	0
$808$	1.35743	0.0477544
$809$	16.2679	0.571950	0.285975	−	0.958237i	$-0.407683\pi$
$809$	16.2679	0.571950	0.285975	+	0.958237i	$0.407683\pi$
$810$	0	0
$811$	27.3764	0.961315	0.480657	−	0.876908i	$-0.340398\pi$
$811$	27.3764	0.961315	0.480657	+	0.876908i	$0.340398\pi$
$812$	3.29759	0.115723
$813$	0	0
$814$	27.6899	0.970530
$815$	−5.77885	−0.202424
$816$	0	0
$817$	0.175640	0.00614487
$818$	26.0203	0.909779
$819$	0	0
$820$	23.8633	0.833344
$821$	30.8637	1.07715	0.538576	−	0.842577i	$-0.318963\pi$
$821$	30.8637	1.07715	0.538576	+	0.842577i	$0.318963\pi$
$822$	0	0
$823$	−14.9591	−0.521442	−0.260721	−	0.965414i	$-0.583960\pi$
$823$	−14.9591	−0.521442	−0.260721	+	0.965414i	$0.583960\pi$
$824$	18.8772	0.657619
$825$	0	0
$826$	19.8099	0.689273
$827$	29.6460	1.03089	0.515446	−	0.856922i	$-0.327626\pi$
$827$	29.6460	1.03089	0.515446	+	0.856922i	$0.327626\pi$
$828$	0	0
$829$	32.0478	1.11307	0.556533	−	0.830826i	$-0.312131\pi$
$829$	32.0478	1.11307	0.556533	+	0.830826i	$0.312131\pi$
$830$	−35.5086	−1.23252
$831$	0	0
$832$	−5.37930	−0.186494
$833$	39.8274	1.37994
$834$	0	0
$835$	20.2451	0.700611
$836$	18.0706	0.624986
$837$	0	0
$838$	−8.84720	−0.305621
$839$	−15.1014	−0.521358	−0.260679	−	0.965426i	$-0.583946\pi$
$839$	−15.1014	−0.521358	−0.260679	+	0.965426i	$0.583946\pi$
$840$	0	0
$841$	−28.4913	−0.982460
$842$	22.1463	0.763211
$843$	0	0
$844$	−2.16223	−0.0744270
$845$	38.5880	1.32747
$846$	0	0
$847$	−102.257	−3.51360
$848$	−2.03001	−0.0697110
$849$	0	0
$850$	−2.38966	−0.0819648
$851$	6.15275	0.210914
$852$	0	0
$853$	8.69277	0.297635	0.148817	−	0.988865i	$-0.452453\pi$
$853$	8.69277	0.297635	0.148817	+	0.988865i	$0.452453\pi$
$854$	−26.3152	−0.900486
$855$	0	0
$856$	−1.66501	−0.0569089
$857$	32.6649	1.11581	0.557905	−	0.829905i	$-0.311605\pi$
$857$	32.6649	1.11581	0.557905	+	0.829905i	$0.311605\pi$
$858$	0	0
$859$	20.9796	0.715815	0.357908	−	0.933757i	$-0.383490\pi$
$859$	20.9796	0.715815	0.357908	+	0.933757i	$0.383490\pi$
$860$	0.135431	0.00461816
$861$	0	0
$862$	−16.3398	−0.556537
$863$	−20.6305	−0.702270	−0.351135	−	0.936325i	$-0.614204\pi$
$863$	−20.6305	−0.702270	−0.351135	+	0.936325i	$0.614204\pi$
$864$	0	0
$865$	−31.8754	−1.08379
$866$	15.5368	0.527962
$867$	0	0
$868$	14.4170	0.489344
$869$	49.1148	1.66610
$870$	0	0
$871$	24.4601	0.828797
$872$	−16.3709	−0.554388
$873$	0	0
$874$	4.01534	0.135821
$875$	46.3181	1.56584
$876$	0	0
$877$	−8.51539	−0.287544	−0.143772	−	0.989611i	$-0.545923\pi$
$877$	−8.51539	−0.287544	−0.143772	+	0.989611i	$0.545923\pi$
$878$	16.4230	0.554251
$879$	0	0
$880$	13.9338	0.469707
$881$	2.56517	0.0864229	0.0432114	−	0.999066i	$-0.486241\pi$
$881$	2.56517	0.0864229	0.0432114	+	0.999066i	$0.486241\pi$
$882$	0	0
$883$	−42.9484	−1.44533	−0.722664	−	0.691200i	$-0.757083\pi$
$883$	−42.9484	−1.44533	−0.722664	+	0.691200i	$0.757083\pi$
$884$	−14.9007	−0.501163
$885$	0	0
$886$	9.94687	0.334171
$887$	11.2289	0.377031	0.188515	−	0.982070i	$-0.439633\pi$
$887$	11.2289	0.377031	0.188515	+	0.982070i	$0.439633\pi$
$888$	0	0
$889$	17.0730	0.572610
$890$	−14.0733	−0.471737
$891$	0	0
$892$	20.0738	0.672121
$893$	26.5404	0.888141
$894$	0	0
$895$	−14.2010	−0.474687
$896$	4.62365	0.154465
$897$	0	0
$898$	−29.6325	−0.988848
$899$	2.22383	0.0741688
$900$	0	0
$901$	−5.62314	−0.187334
$902$	−56.7156	−1.88842
$903$	0	0
$904$	−5.29780	−0.176202
$905$	52.9838	1.76124
$906$	0	0
$907$	−10.1389	−0.336656	−0.168328	−	0.985731i	$-0.553837\pi$
$907$	−10.1389	−0.336656	−0.168328	+	0.985731i	$0.553837\pi$
$908$	27.4930	0.912389
$909$	0	0
$910$	−60.2226	−1.99636
$911$	−42.4151	−1.40528	−0.702638	−	0.711548i	$-0.747995\pi$
$911$	−42.4151	−1.40528	−0.702638	+	0.711548i	$0.747995\pi$
$912$	0	0
$913$	84.3927	2.79299
$914$	32.0007	1.05849
$915$	0	0
$916$	−6.33695	−0.209379
$917$	86.3566	2.85175
$918$	0	0
$919$	52.0686	1.71758	0.858792	−	0.512324i	$-0.171215\pi$
$919$	52.0686	1.71758	0.858792	+	0.512324i	$0.171215\pi$
$920$	3.09611	0.102076
$921$	0	0
$922$	−10.0526	−0.331063
$923$	65.0719	2.14187
$924$	0	0
$925$	−4.15106	−0.136486
$926$	−34.7035	−1.14043
$927$	0	0
$928$	0.713201	0.0234120
$929$	35.7499	1.17291	0.586457	−	0.809980i	$-0.300522\pi$
$929$	35.7499	1.17291	0.586457	+	0.809980i	$0.300522\pi$
$930$	0	0
$931$	45.1498	1.47973
$932$	22.0167	0.721182
$933$	0	0
$934$	−11.1085	−0.363480
$935$	38.5965	1.26224
$936$	0	0
$937$	−34.5548	−1.12886	−0.564428	−	0.825483i	$-0.690903\pi$
$937$	−34.5548	−1.12886	−0.564428	+	0.825483i	$0.690903\pi$
$938$	−21.0240	−0.686459
$939$	0	0
$940$	20.4646	0.667481
$941$	31.9704	1.04221	0.521103	−	0.853494i	$-0.325521\pi$
$941$	31.9704	1.04221	0.521103	+	0.853494i	$0.325521\pi$
$942$	0	0
$943$	−12.6023	−0.410388
$944$	4.28447	0.139448
$945$	0	0
$946$	−0.321877	−0.0104651
$947$	−46.5930	−1.51407	−0.757034	−	0.653376i	$-0.773352\pi$
$947$	−46.5930	−1.51407	−0.757034	+	0.653376i	$0.773352\pi$
$948$	0	0
$949$	23.6347	0.767214
$950$	−2.70901	−0.0878920
$951$	0	0
$952$	12.8075	0.415093
$953$	49.7322	1.61098	0.805492	−	0.592607i	$-0.201901\pi$
$953$	49.7322	1.61098	0.805492	+	0.592607i	$0.201901\pi$
$954$	0	0
$955$	1.43166	0.0463276
$956$	19.3365	0.625388
$957$	0	0
$958$	5.66215	0.182936
$959$	−53.5273	−1.72849
$960$	0	0
$961$	−21.2775	−0.686371
$962$	−25.8838	−0.834526
$963$	0	0
$964$	15.0621	0.485119
$965$	63.2551	2.03625
$966$	0	0
$967$	−6.87967	−0.221235	−0.110618	−	0.993863i	$-0.535283\pi$
$967$	−6.87967	−0.221235	−0.110618	+	0.993863i	$0.535283\pi$
$968$	−22.1161	−0.710840
$969$	0	0
$970$	−16.8822	−0.542053
$971$	−20.1481	−0.646584	−0.323292	−	0.946299i	$-0.604790\pi$
$971$	−20.1481	−0.646584	−0.323292	+	0.946299i	$0.604790\pi$
$972$	0	0
$973$	16.2661	0.521468
$974$	35.5368	1.13867
$975$	0	0
$976$	−5.69143	−0.182178
$977$	−15.0212	−0.480572	−0.240286	−	0.970702i	$-0.577241\pi$
$977$	−15.0212	−0.480572	−0.240286	+	0.970702i	$0.577241\pi$
$978$	0	0
$979$	33.4477	1.06899
$980$	34.8138	1.11208
$981$	0	0
$982$	−1.32518	−0.0422881
$983$	27.3895	0.873589	0.436795	−	0.899561i	$-0.356114\pi$
$983$	27.3895	0.873589	0.436795	+	0.899561i	$0.356114\pi$
$984$	0	0
$985$	−28.6626	−0.913265
$986$	1.97557	0.0629148
$987$	0	0
$988$	−16.8920	−0.537405
$989$	−0.0715217	−0.00227426
$990$	0	0
$991$	−14.3919	−0.457174	−0.228587	−	0.973523i	$-0.573411\pi$
$991$	−14.3919	−0.457174	−0.228587	+	0.973523i	$0.573411\pi$
$992$	3.11809	0.0989996
$993$	0	0
$994$	−55.9310	−1.77402
$995$	−25.2674	−0.801029
$996$	0	0
$997$	49.4860	1.56724	0.783619	−	0.621241i	$-0.213371\pi$
$997$	49.4860	1.56724	0.783619	+	0.621241i	$0.213371\pi$
$998$	−16.8764	−0.534212
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.q.1.12	✓	14
3.2	odd	2		8046.2.a.r.1.3	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.q.1.12	✓	14	1.1	even	1	trivial
8046.2.a.r.1.3	yes	14	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.42130 q^{5} -4.62365 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.42130 q^{5} -4.62365 q^{7} -1.00000 q^{8} -2.42130 q^{10} +5.75466 q^{11} -5.37930 q^{13} +4.62365 q^{14} +1.00000 q^{16} +2.77000 q^{17} +3.14018 q^{19} +2.42130 q^{20} -5.75466 q^{22} -1.27870 q^{23} +0.862695 q^{25} +5.37930 q^{26} -4.62365 q^{28} -0.713201 q^{29} -3.11809 q^{31} -1.00000 q^{32} -2.77000 q^{34} -11.1952 q^{35} -4.81173 q^{37} -3.14018 q^{38} -2.42130 q^{40} +9.85559 q^{41} +0.0559332 q^{43} +5.75466 q^{44} +1.27870 q^{46} +8.45189 q^{47} +14.3781 q^{49} -0.862695 q^{50} -5.37930 q^{52} -2.03001 q^{53} +13.9338 q^{55} +4.62365 q^{56} +0.713201 q^{58} +4.28447 q^{59} -5.69143 q^{61} +3.11809 q^{62} +1.00000 q^{64} -13.0249 q^{65} -4.54707 q^{67} +2.77000 q^{68} +11.1952 q^{70} -12.0967 q^{71} -4.39363 q^{73} +4.81173 q^{74} +3.14018 q^{76} -26.6075 q^{77} +8.53478 q^{79} +2.42130 q^{80} -9.85559 q^{82} +14.6651 q^{83} +6.70700 q^{85} -0.0559332 q^{86} -5.75466 q^{88} +5.81228 q^{89} +24.8720 q^{91} -1.27870 q^{92} -8.45189 q^{94} +7.60331 q^{95} +6.97235 q^{97} -14.3781 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) \) 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

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