LMFDB - Embedded newform 8046.2.a.s.1.5

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
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			1.5
Root			$2.46314$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.46314 q^{5} +0.600028 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.46314 q^{5} +0.600028 q^{7} -1.00000 q^{8} +2.46314 q^{10} -0.522157 q^{11} +4.72035 q^{13} -0.600028 q^{14} +1.00000 q^{16} +6.66237 q^{17} +3.21884 q^{19} -2.46314 q^{20} +0.522157 q^{22} +7.53023 q^{23} +1.06705 q^{25} -4.72035 q^{26} +0.600028 q^{28} -9.92910 q^{29} +9.06917 q^{31} -1.00000 q^{32} -6.66237 q^{34} -1.47795 q^{35} +7.46104 q^{37} -3.21884 q^{38} +2.46314 q^{40} -1.73963 q^{41} -1.03105 q^{43} -0.522157 q^{44} -7.53023 q^{46} +10.6868 q^{47} -6.63997 q^{49} -1.06705 q^{50} +4.72035 q^{52} -3.23167 q^{53} +1.28615 q^{55} -0.600028 q^{56} +9.92910 q^{58} +5.24693 q^{59} -3.98285 q^{61} -9.06917 q^{62} +1.00000 q^{64} -11.6269 q^{65} +13.9646 q^{67} +6.66237 q^{68} +1.47795 q^{70} -9.65257 q^{71} -0.902486 q^{73} -7.46104 q^{74} +3.21884 q^{76} -0.313309 q^{77} +15.1054 q^{79} -2.46314 q^{80} +1.73963 q^{82} +2.97085 q^{83} -16.4103 q^{85} +1.03105 q^{86} +0.522157 q^{88} -7.74627 q^{89} +2.83234 q^{91} +7.53023 q^{92} -10.6868 q^{94} -7.92845 q^{95} -16.1901 q^{97} +6.63997 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8}+O(q^{10}) $ 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8	$ 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8} + 4 q^{10} - 6 q^{11} + 6 q^{13} - 6 q^{14} + 16 q^{16} - q^{17} + 10 q^{19} - 4 q^{20} + 6 q^{22} - 10 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 21 q^{31} - 16 q^{32} + q^{34} - 16 q^{35} + 17 q^{37} - 10 q^{38} + 4 q^{40} + 4 q^{41} + 16 q^{43} - 6 q^{44} + 10 q^{46} - 25 q^{47} + 36 q^{49} - 28 q^{50} + 6 q^{52} - 14 q^{53} + 19 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 23 q^{61} - 21 q^{62} + 16 q^{64} - 20 q^{65} + 22 q^{67} - q^{68} + 16 q^{70} - 10 q^{71} + 16 q^{73} - 17 q^{74} + 10 q^{76} + 2 q^{77} + 37 q^{79} - 4 q^{80} - 4 q^{82} - 33 q^{83} + 43 q^{85} - 16 q^{86} + 6 q^{88} + 3 q^{89} + 28 q^{91} - 10 q^{92} + 25 q^{94} - 14 q^{95} - 3 q^{97} - 36 q^{98}+O(q^{100}) $ 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8 + 4 * q^10 - 6 * q^11 + 6 * q^13 - 6 * q^14 + 16 * q^16 - q^17 + 10 * q^19 - 4 * q^20 + 6 * q^22 - 10 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 21 * q^31 - 16 * q^32 + q^34 - 16 * q^35 + 17 * q^37 - 10 * q^38 + 4 * q^40 + 4 * q^41 + 16 * q^43 - 6 * q^44 + 10 * q^46 - 25 * q^47 + 36 * q^49 - 28 * q^50 + 6 * q^52 - 14 * q^53 + 19 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 23 * q^61 - 21 * q^62 + 16 * q^64 - 20 * q^65 + 22 * q^67 - q^68 + 16 * q^70 - 10 * q^71 + 16 * q^73 - 17 * q^74 + 10 * q^76 + 2 * q^77 + 37 * q^79 - 4 * q^80 - 4 * q^82 - 33 * q^83 + 43 * q^85 - 16 * q^86 + 6 * q^88 + 3 * q^89 + 28 * q^91 - 10 * q^92 + 25 * q^94 - 14 * q^95 - 3 * q^97 - 36 * 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
$5$	−2.46314	−1.10155	−0.550775	−	0.834654i	$-0.685668\pi$
$5$	−2.46314	−1.10155	−0.550775	+	0.834654i	$0.685668\pi$
$6$	0	0
$7$	0.600028	0.226789	0.113395	−	0.993550i	$-0.463828\pi$
$7$	0.600028	0.226789	0.113395	+	0.993550i	$0.463828\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.46314	0.778913
$11$	−0.522157	−0.157436	−0.0787181	−	0.996897i	$-0.525083\pi$
$11$	−0.522157	−0.157436	−0.0787181	+	0.996897i	$0.525083\pi$
$12$	0	0
$13$	4.72035	1.30919	0.654594	−	0.755980i	$-0.272839\pi$
$13$	4.72035	1.30919	0.654594	+	0.755980i	$0.272839\pi$
$14$	−0.600028	−0.160364
$15$	0	0
$16$	1.00000	0.250000
$17$	6.66237	1.61586	0.807931	−	0.589277i	$-0.200587\pi$
$17$	6.66237	1.61586	0.807931	+	0.589277i	$0.200587\pi$
$18$	0	0
$19$	3.21884	0.738453	0.369226	−	0.929339i	$-0.379623\pi$
$19$	3.21884	0.738453	0.369226	+	0.929339i	$0.379623\pi$
$20$	−2.46314	−0.550775
$21$	0	0
$22$	0.522157	0.111324
$23$	7.53023	1.57016	0.785081	−	0.619393i	$-0.212621\pi$
$23$	7.53023	1.57016	0.785081	+	0.619393i	$0.212621\pi$
$24$	0	0
$25$	1.06705	0.213411
$26$	−4.72035	−0.925736
$27$	0	0
$28$	0.600028	0.113395
$29$	−9.92910	−1.84379	−0.921894	−	0.387442i	$-0.873359\pi$
$29$	−9.92910	−1.84379	−0.921894	+	0.387442i	$0.873359\pi$
$30$	0	0
$31$	9.06917	1.62887	0.814435	−	0.580254i	$-0.197047\pi$
$31$	9.06917	1.62887	0.814435	+	0.580254i	$0.197047\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.66237	−1.14259
$35$	−1.47795	−0.249820
$36$	0	0
$37$	7.46104	1.22659	0.613294	−	0.789855i	$-0.289844\pi$
$37$	7.46104	1.22659	0.613294	+	0.789855i	$0.289844\pi$
$38$	−3.21884	−0.522165
$39$	0	0
$40$	2.46314	0.389456
$41$	−1.73963	−0.271685	−0.135842	−	0.990730i	$-0.543374\pi$
$41$	−1.73963	−0.271685	−0.135842	+	0.990730i	$0.543374\pi$
$42$	0	0
$43$	−1.03105	−0.157234	−0.0786171	−	0.996905i	$-0.525050\pi$
$43$	−1.03105	−0.157234	−0.0786171	+	0.996905i	$0.525050\pi$
$44$	−0.522157	−0.0787181
$45$	0	0
$46$	−7.53023	−1.11027
$47$	10.6868	1.55883	0.779414	−	0.626509i	$-0.215517\pi$
$47$	10.6868	1.55883	0.779414	+	0.626509i	$0.215517\pi$
$48$	0	0
$49$	−6.63997	−0.948567
$50$	−1.06705	−0.150904
$51$	0	0
$52$	4.72035	0.654594
$53$	−3.23167	−0.443905	−0.221952	−	0.975058i	$-0.571243\pi$
$53$	−3.23167	−0.443905	−0.221952	+	0.975058i	$0.571243\pi$
$54$	0	0
$55$	1.28615	0.173424
$56$	−0.600028	−0.0801822
$57$	0	0
$58$	9.92910	1.30375
$59$	5.24693	0.683092	0.341546	−	0.939865i	$-0.389049\pi$
$59$	5.24693	0.683092	0.341546	+	0.939865i	$0.389049\pi$
$60$	0	0
$61$	−3.98285	−0.509951	−0.254976	−	0.966947i	$-0.582067\pi$
$61$	−3.98285	−0.509951	−0.254976	+	0.966947i	$0.582067\pi$
$62$	−9.06917	−1.15179
$63$	0	0
$64$	1.00000	0.125000
$65$	−11.6269	−1.44214
$66$	0	0
$67$	13.9646	1.70604	0.853021	−	0.521877i	$-0.174768\pi$
$67$	13.9646	1.70604	0.853021	+	0.521877i	$0.174768\pi$
$68$	6.66237	0.807931
$69$	0	0
$70$	1.47795	0.176649
$71$	−9.65257	−1.14555	−0.572775	−	0.819713i	$-0.694133\pi$
$71$	−9.65257	−1.14555	−0.572775	+	0.819713i	$0.694133\pi$
$72$	0	0
$73$	−0.902486	−0.105628	−0.0528140	−	0.998604i	$-0.516819\pi$
$73$	−0.902486	−0.105628	−0.0528140	+	0.998604i	$0.516819\pi$
$74$	−7.46104	−0.867328
$75$	0	0
$76$	3.21884	0.369226
$77$	−0.313309	−0.0357049
$78$	0	0
$79$	15.1054	1.69949	0.849743	−	0.527197i	$-0.176757\pi$
$79$	15.1054	1.69949	0.849743	+	0.527197i	$0.176757\pi$
$80$	−2.46314	−0.275387
$81$	0	0
$82$	1.73963	0.192110
$83$	2.97085	0.326093	0.163047	−	0.986618i	$-0.447868\pi$
$83$	2.97085	0.326093	0.163047	+	0.986618i	$0.447868\pi$
$84$	0	0
$85$	−16.4103	−1.77995
$86$	1.03105	0.111181
$87$	0	0
$88$	0.522157	0.0556621
$89$	−7.74627	−0.821103	−0.410551	−	0.911838i	$-0.634664\pi$
$89$	−7.74627	−0.821103	−0.410551	+	0.911838i	$0.634664\pi$
$90$	0	0
$91$	2.83234	0.296910
$92$	7.53023	0.785081
$93$	0	0
$94$	−10.6868	−1.10226
$95$	−7.92845	−0.813442
$96$	0	0
$97$	−16.1901	−1.64386	−0.821928	−	0.569591i	$-0.807102\pi$
$97$	−16.1901	−1.64386	−0.821928	+	0.569591i	$0.807102\pi$
$98$	6.63997	0.670738
$99$	0	0
$100$	1.06705	0.106705
$101$	−2.14499	−0.213434	−0.106717	−	0.994289i	$-0.534034\pi$
$101$	−2.14499	−0.213434	−0.106717	+	0.994289i	$0.534034\pi$
$102$	0	0
$103$	15.0984	1.48769	0.743845	−	0.668352i	$-0.233000\pi$
$103$	15.0984	1.48769	0.743845	+	0.668352i	$0.233000\pi$
$104$	−4.72035	−0.462868
$105$	0	0
$106$	3.23167	0.313888
$107$	−15.8743	−1.53462	−0.767311	−	0.641275i	$-0.778406\pi$
$107$	−15.8743	−1.53462	−0.767311	+	0.641275i	$0.778406\pi$
$108$	0	0
$109$	−6.36883	−0.610024	−0.305012	−	0.952349i	$-0.598660\pi$
$109$	−6.36883	−0.610024	−0.305012	+	0.952349i	$0.598660\pi$
$110$	−1.28615	−0.122629
$111$	0	0
$112$	0.600028	0.0566974
$113$	−0.712105	−0.0669892	−0.0334946	−	0.999439i	$-0.510664\pi$
$113$	−0.712105	−0.0669892	−0.0334946	+	0.999439i	$0.510664\pi$
$114$	0	0
$115$	−18.5480	−1.72961
$116$	−9.92910	−0.921894
$117$	0	0
$118$	−5.24693	−0.483019
$119$	3.99761	0.366460
$120$	0	0
$121$	−10.7274	−0.975214
$122$	3.98285	0.360590
$123$	0	0
$124$	9.06917	0.814435
$125$	9.68739	0.866467
$126$	0	0
$127$	−10.9793	−0.974255	−0.487128	−	0.873331i	$-0.661955\pi$
$127$	−10.9793	−0.974255	−0.487128	+	0.873331i	$0.661955\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	11.6269	1.01974
$131$	−18.6281	−1.62754	−0.813770	−	0.581187i	$-0.802589\pi$
$131$	−18.6281	−1.62754	−0.813770	+	0.581187i	$0.802589\pi$
$132$	0	0
$133$	1.93140	0.167473
$134$	−13.9646	−1.20635
$135$	0	0
$136$	−6.66237	−0.571293
$137$	7.71863	0.659447	0.329723	−	0.944078i	$-0.393045\pi$
$137$	7.71863	0.659447	0.329723	+	0.944078i	$0.393045\pi$
$138$	0	0
$139$	1.61317	0.136827	0.0684137	−	0.997657i	$-0.478206\pi$
$139$	1.61317	0.136827	0.0684137	+	0.997657i	$0.478206\pi$
$140$	−1.47795	−0.124910
$141$	0	0
$142$	9.65257	0.810026
$143$	−2.46476	−0.206114
$144$	0	0
$145$	24.4568	2.03102
$146$	0.902486	0.0746903
$147$	0	0
$148$	7.46104	0.613294
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	10.8003	0.878914	0.439457	−	0.898264i	$-0.355171\pi$
$151$	10.8003	0.878914	0.439457	+	0.898264i	$0.355171\pi$
$152$	−3.21884	−0.261083
$153$	0	0
$154$	0.313309	0.0252472
$155$	−22.3386	−1.79428
$156$	0	0
$157$	4.46308	0.356192	0.178096	−	0.984013i	$-0.443006\pi$
$157$	4.46308	0.356192	0.178096	+	0.984013i	$0.443006\pi$
$158$	−15.1054	−1.20172
$159$	0	0
$160$	2.46314	0.194728
$161$	4.51835	0.356096
$162$	0	0
$163$	−3.23968	−0.253751	−0.126876	−	0.991919i	$-0.540495\pi$
$163$	−3.23968	−0.253751	−0.126876	+	0.991919i	$0.540495\pi$
$164$	−1.73963	−0.135842
$165$	0	0
$166$	−2.97085	−0.230583
$167$	11.8022	0.913280	0.456640	−	0.889652i	$-0.349053\pi$
$167$	11.8022	0.913280	0.456640	+	0.889652i	$0.349053\pi$
$168$	0	0
$169$	9.28167	0.713975
$170$	16.4103	1.25862
$171$	0	0
$172$	−1.03105	−0.0786171
$173$	24.4344	1.85771	0.928855	−	0.370443i	$-0.120794\pi$
$173$	24.4344	1.85771	0.928855	+	0.370443i	$0.120794\pi$
$174$	0	0
$175$	0.640263	0.0483993
$176$	−0.522157	−0.0393591
$177$	0	0
$178$	7.74627	0.580607
$179$	−17.9810	−1.34396	−0.671981	−	0.740568i	$-0.734556\pi$
$179$	−17.9810	−1.34396	−0.671981	+	0.740568i	$0.734556\pi$
$180$	0	0
$181$	1.07757	0.0800950	0.0400475	−	0.999198i	$-0.487249\pi$
$181$	1.07757	0.0800950	0.0400475	+	0.999198i	$0.487249\pi$
$182$	−2.83234	−0.209947
$183$	0	0
$184$	−7.53023	−0.555136
$185$	−18.3776	−1.35115
$186$	0	0
$187$	−3.47880	−0.254395
$188$	10.6868	0.779414
$189$	0	0
$190$	7.92845	0.575190
$191$	−14.9599	−1.08246	−0.541230	−	0.840875i	$-0.682041\pi$
$191$	−14.9599	−1.08246	−0.541230	+	0.840875i	$0.682041\pi$
$192$	0	0
$193$	13.3320	0.959655	0.479828	−	0.877363i	$-0.340699\pi$
$193$	13.3320	0.959655	0.479828	+	0.877363i	$0.340699\pi$
$194$	16.1901	1.16238
$195$	0	0
$196$	−6.63997	−0.474283
$197$	12.4613	0.887828	0.443914	−	0.896069i	$-0.353590\pi$
$197$	12.4613	0.887828	0.443914	+	0.896069i	$0.353590\pi$
$198$	0	0
$199$	1.68669	0.119566	0.0597832	−	0.998211i	$-0.480959\pi$
$199$	1.68669	0.119566	0.0597832	+	0.998211i	$0.480959\pi$
$200$	−1.06705	−0.0754521
$201$	0	0
$202$	2.14499	0.150921
$203$	−5.95774	−0.418152
$204$	0	0
$205$	4.28495	0.299274
$206$	−15.0984	−1.05196
$207$	0	0
$208$	4.72035	0.327297
$209$	−1.68074	−0.116259
$210$	0	0
$211$	−11.0214	−0.758742	−0.379371	−	0.925245i	$-0.623860\pi$
$211$	−11.0214	−0.758742	−0.379371	+	0.925245i	$0.623860\pi$
$212$	−3.23167	−0.221952
$213$	0	0
$214$	15.8743	1.08514
$215$	2.53963	0.173201
$216$	0	0
$217$	5.44176	0.369411
$218$	6.36883	0.431352
$219$	0	0
$220$	1.28615	0.0867119
$221$	31.4487	2.11547
$222$	0	0
$223$	−18.3538	−1.22906	−0.614531	−	0.788893i	$-0.710655\pi$
$223$	−18.3538	−1.22906	−0.614531	+	0.788893i	$0.710655\pi$
$224$	−0.600028	−0.0400911
$225$	0	0
$226$	0.712105	0.0473685
$227$	23.7264	1.57477	0.787387	−	0.616459i	$-0.211433\pi$
$227$	23.7264	1.57477	0.787387	+	0.616459i	$0.211433\pi$
$228$	0	0
$229$	−11.6240	−0.768132	−0.384066	−	0.923306i	$-0.625477\pi$
$229$	−11.6240	−0.768132	−0.384066	+	0.923306i	$0.625477\pi$
$230$	18.5480	1.22302
$231$	0	0
$232$	9.92910	0.651877
$233$	−1.01374	−0.0664121	−0.0332061	−	0.999449i	$-0.510572\pi$
$233$	−1.01374	−0.0664121	−0.0332061	+	0.999449i	$0.510572\pi$
$234$	0	0
$235$	−26.3231	−1.71713
$236$	5.24693	0.341546
$237$	0	0
$238$	−3.99761	−0.259127
$239$	−12.6866	−0.820627	−0.410313	−	0.911945i	$-0.634581\pi$
$239$	−12.6866	−0.820627	−0.410313	+	0.911945i	$0.634581\pi$
$240$	0	0
$241$	−2.91650	−0.187868	−0.0939340	−	0.995578i	$-0.529944\pi$
$241$	−2.91650	−0.187868	−0.0939340	+	0.995578i	$0.529944\pi$
$242$	10.7274	0.689580
$243$	0	0
$244$	−3.98285	−0.254976
$245$	16.3552	1.04489
$246$	0	0
$247$	15.1940	0.966774
$248$	−9.06917	−0.575893
$249$	0	0
$250$	−9.68739	−0.612684
$251$	10.6258	0.670693	0.335347	−	0.942095i	$-0.391147\pi$
$251$	10.6258	0.670693	0.335347	+	0.942095i	$0.391147\pi$
$252$	0	0
$253$	−3.93196	−0.247200
$254$	10.9793	0.688902
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.46027	0.465359	0.232679	−	0.972553i	$-0.425251\pi$
$257$	7.46027	0.465359	0.232679	+	0.972553i	$0.425251\pi$
$258$	0	0
$259$	4.47684	0.278177
$260$	−11.6269	−0.721068
$261$	0	0
$262$	18.6281	1.15084
$263$	17.7053	1.09175	0.545877	−	0.837866i	$-0.316197\pi$
$263$	17.7053	1.09175	0.545877	+	0.837866i	$0.316197\pi$
$264$	0	0
$265$	7.96006	0.488983
$266$	−1.93140	−0.118422
$267$	0	0
$268$	13.9646	0.853021
$269$	9.47231	0.577537	0.288768	−	0.957399i	$-0.406754\pi$
$269$	9.47231	0.577537	0.288768	+	0.957399i	$0.406754\pi$
$270$	0	0
$271$	8.70903	0.529036	0.264518	−	0.964381i	$-0.414787\pi$
$271$	8.70903	0.529036	0.264518	+	0.964381i	$0.414787\pi$
$272$	6.66237	0.403965
$273$	0	0
$274$	−7.71863	−0.466299
$275$	−0.557170	−0.0335986
$276$	0	0
$277$	−28.1562	−1.69174	−0.845870	−	0.533389i	$-0.820918\pi$
$277$	−28.1562	−1.69174	−0.845870	+	0.533389i	$0.820918\pi$
$278$	−1.61317	−0.0967515
$279$	0	0
$280$	1.47795	0.0883246
$281$	21.9043	1.30670	0.653351	−	0.757055i	$-0.273363\pi$
$281$	21.9043	1.30670	0.653351	+	0.757055i	$0.273363\pi$
$282$	0	0
$283$	22.5153	1.33840	0.669199	−	0.743083i	$-0.266637\pi$
$283$	22.5153	1.33840	0.669199	+	0.743083i	$0.266637\pi$
$284$	−9.65257	−0.572775
$285$	0	0
$286$	2.46476	0.145744
$287$	−1.04383	−0.0616152
$288$	0	0
$289$	27.3872	1.61101
$290$	−24.4568	−1.43615
$291$	0	0
$292$	−0.902486	−0.0528140
$293$	−7.90118	−0.461592	−0.230796	−	0.973002i	$-0.574133\pi$
$293$	−7.90118	−0.461592	−0.230796	+	0.973002i	$0.574133\pi$
$294$	0	0
$295$	−12.9239	−0.752459
$296$	−7.46104	−0.433664
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	35.5453	2.05564
$300$	0	0
$301$	−0.618661	−0.0356590
$302$	−10.8003	−0.621486
$303$	0	0
$304$	3.21884	0.184613
$305$	9.81031	0.561737
$306$	0	0
$307$	−27.2200	−1.55353	−0.776763	−	0.629793i	$-0.783140\pi$
$307$	−27.2200	−1.55353	−0.776763	+	0.629793i	$0.783140\pi$
$308$	−0.313309	−0.0178524
$309$	0	0
$310$	22.3386	1.26875
$311$	−5.40526	−0.306504	−0.153252	−	0.988187i	$-0.548975\pi$
$311$	−5.40526	−0.306504	−0.153252	+	0.988187i	$0.548975\pi$
$312$	0	0
$313$	32.9443	1.86212	0.931061	−	0.364863i	$-0.118884\pi$
$313$	32.9443	1.86212	0.931061	+	0.364863i	$0.118884\pi$
$314$	−4.46308	−0.251866
$315$	0	0
$316$	15.1054	0.849743
$317$	−25.1368	−1.41182	−0.705910	−	0.708301i	$-0.749462\pi$
$317$	−25.1368	−1.41182	−0.705910	+	0.708301i	$0.749462\pi$
$318$	0	0
$319$	5.18455	0.290279
$320$	−2.46314	−0.137694
$321$	0	0
$322$	−4.51835	−0.251798
$323$	21.4451	1.19324
$324$	0	0
$325$	5.03687	0.279395
$326$	3.23968	0.179429
$327$	0	0
$328$	1.73963	0.0960551
$329$	6.41238	0.353526
$330$	0	0
$331$	4.09310	0.224977	0.112489	−	0.993653i	$-0.464118\pi$
$331$	4.09310	0.224977	0.112489	+	0.993653i	$0.464118\pi$
$332$	2.97085	0.163047
$333$	0	0
$334$	−11.8022	−0.645786
$335$	−34.3966	−1.87929
$336$	0	0
$337$	26.0313	1.41802	0.709008	−	0.705200i	$-0.249143\pi$
$337$	26.0313	1.41802	0.709008	+	0.705200i	$0.249143\pi$
$338$	−9.28167	−0.504857
$339$	0	0
$340$	−16.4103	−0.889976
$341$	−4.73553	−0.256443
$342$	0	0
$343$	−8.18437	−0.441914
$344$	1.03105	0.0555907
$345$	0	0
$346$	−24.4344	−1.31360
$347$	6.53114	0.350610	0.175305	−	0.984514i	$-0.443909\pi$
$347$	6.53114	0.350610	0.175305	+	0.984514i	$0.443909\pi$
$348$	0	0
$349$	23.5618	1.26124	0.630618	−	0.776094i	$-0.282802\pi$
$349$	23.5618	1.26124	0.630618	+	0.776094i	$0.282802\pi$
$350$	−0.640263	−0.0342235
$351$	0	0
$352$	0.522157	0.0278311
$353$	5.05539	0.269072	0.134536	−	0.990909i	$-0.457046\pi$
$353$	5.05539	0.269072	0.134536	+	0.990909i	$0.457046\pi$
$354$	0	0
$355$	23.7756	1.26188
$356$	−7.74627	−0.410551
$357$	0	0
$358$	17.9810	0.950325
$359$	−3.77664	−0.199323	−0.0996616	−	0.995021i	$-0.531776\pi$
$359$	−3.77664	−0.199323	−0.0996616	+	0.995021i	$0.531776\pi$
$360$	0	0
$361$	−8.63906	−0.454687
$362$	−1.07757	−0.0566357
$363$	0	0
$364$	2.83234	0.148455
$365$	2.22295	0.116354
$366$	0	0
$367$	3.87803	0.202431	0.101216	−	0.994865i	$-0.467727\pi$
$367$	3.87803	0.202431	0.101216	+	0.994865i	$0.467727\pi$
$368$	7.53023	0.392541
$369$	0	0
$370$	18.3776	0.955405
$371$	−1.93910	−0.100673
$372$	0	0
$373$	−11.4620	−0.593479	−0.296739	−	0.954958i	$-0.595899\pi$
$373$	−11.4620	−0.593479	−0.296739	+	0.954958i	$0.595899\pi$
$374$	3.47880	0.179885
$375$	0	0
$376$	−10.6868	−0.551129
$377$	−46.8688	−2.41387
$378$	0	0
$379$	1.68630	0.0866194	0.0433097	−	0.999062i	$-0.486210\pi$
$379$	1.68630	0.0866194	0.0433097	+	0.999062i	$0.486210\pi$
$380$	−7.92845	−0.406721
$381$	0	0
$382$	14.9599	0.765415
$383$	30.6373	1.56549	0.782747	−	0.622341i	$-0.213818\pi$
$383$	30.6373	1.56549	0.782747	+	0.622341i	$0.213818\pi$
$384$	0	0
$385$	0.771724	0.0393307
$386$	−13.3320	−0.678579
$387$	0	0
$388$	−16.1901	−0.821928
$389$	−12.1259	−0.614807	−0.307403	−	0.951579i	$-0.599460\pi$
$389$	−12.1259	−0.614807	−0.307403	+	0.951579i	$0.599460\pi$
$390$	0	0
$391$	50.1692	2.53717
$392$	6.63997	0.335369
$393$	0	0
$394$	−12.4613	−0.627789
$395$	−37.2066	−1.87207
$396$	0	0
$397$	7.03616	0.353135	0.176567	−	0.984289i	$-0.443501\pi$
$397$	7.03616	0.353135	0.176567	+	0.984289i	$0.443501\pi$
$398$	−1.68669	−0.0845462
$399$	0	0
$400$	1.06705	0.0533527
$401$	−21.3385	−1.06559	−0.532797	−	0.846243i	$-0.678859\pi$
$401$	−21.3385	−1.06559	−0.532797	+	0.846243i	$0.678859\pi$
$402$	0	0
$403$	42.8096	2.13250
$404$	−2.14499	−0.106717
$405$	0	0
$406$	5.95774	0.295678
$407$	−3.89583	−0.193109
$408$	0	0
$409$	12.7311	0.629513	0.314756	−	0.949172i	$-0.398077\pi$
$409$	12.7311	0.629513	0.314756	+	0.949172i	$0.398077\pi$
$410$	−4.28495	−0.211619
$411$	0	0
$412$	15.0984	0.743845
$413$	3.14831	0.154918
$414$	0	0
$415$	−7.31762	−0.359208
$416$	−4.72035	−0.231434
$417$	0	0
$418$	1.68074	0.0822077
$419$	14.6289	0.714667	0.357333	−	0.933977i	$-0.383686\pi$
$419$	14.6289	0.714667	0.357333	+	0.933977i	$0.383686\pi$
$420$	0	0
$421$	−28.6508	−1.39635	−0.698176	−	0.715926i	$-0.746005\pi$
$421$	−28.6508	−1.39635	−0.698176	+	0.715926i	$0.746005\pi$
$422$	11.0214	0.536512
$423$	0	0
$424$	3.23167	0.156944
$425$	7.10911	0.344842
$426$	0	0
$427$	−2.38982	−0.115652
$428$	−15.8743	−0.767311
$429$	0	0
$430$	−2.53963	−0.122472
$431$	16.0721	0.774164	0.387082	−	0.922045i	$-0.373483\pi$
$431$	16.0721	0.774164	0.387082	+	0.922045i	$0.373483\pi$
$432$	0	0
$433$	−7.06619	−0.339580	−0.169790	−	0.985480i	$-0.554309\pi$
$433$	−7.06619	−0.339580	−0.169790	+	0.985480i	$0.554309\pi$
$434$	−5.44176	−0.261213
$435$	0	0
$436$	−6.36883	−0.305012
$437$	24.2386	1.15949
$438$	0	0
$439$	13.0150	0.621173	0.310587	−	0.950545i	$-0.399475\pi$
$439$	13.0150	0.621173	0.310587	+	0.950545i	$0.399475\pi$
$440$	−1.28615	−0.0613146
$441$	0	0
$442$	−31.4487	−1.49586
$443$	−24.4272	−1.16057	−0.580286	−	0.814412i	$-0.697059\pi$
$443$	−24.4272	−1.16057	−0.580286	+	0.814412i	$0.697059\pi$
$444$	0	0
$445$	19.0801	0.904485
$446$	18.3538	0.869078
$447$	0	0
$448$	0.600028	0.0283487
$449$	−6.27709	−0.296234	−0.148117	−	0.988970i	$-0.547321\pi$
$449$	−6.27709	−0.296234	−0.148117	+	0.988970i	$0.547321\pi$
$450$	0	0
$451$	0.908360	0.0427730
$452$	−0.712105	−0.0334946
$453$	0	0
$454$	−23.7264	−1.11353
$455$	−6.97645	−0.327061
$456$	0	0
$457$	26.2213	1.22658	0.613291	−	0.789857i	$-0.289845\pi$
$457$	26.2213	1.22658	0.613291	+	0.789857i	$0.289845\pi$
$458$	11.6240	0.543152
$459$	0	0
$460$	−18.5480	−0.864806
$461$	−37.3461	−1.73938	−0.869691	−	0.493596i	$-0.835682\pi$
$461$	−37.3461	−1.73938	−0.869691	+	0.493596i	$0.835682\pi$
$462$	0	0
$463$	−16.5565	−0.769446	−0.384723	−	0.923032i	$-0.625703\pi$
$463$	−16.5565	−0.769446	−0.384723	+	0.923032i	$0.625703\pi$
$464$	−9.92910	−0.460947
$465$	0	0
$466$	1.01374	0.0469605
$467$	35.3201	1.63442	0.817210	−	0.576340i	$-0.195520\pi$
$467$	35.3201	1.63442	0.817210	+	0.576340i	$0.195520\pi$
$468$	0	0
$469$	8.37913	0.386912
$470$	26.3231	1.21419
$471$	0	0
$472$	−5.24693	−0.241509
$473$	0.538371	0.0247543
$474$	0	0
$475$	3.43468	0.157594
$476$	3.99761	0.183230
$477$	0	0
$478$	12.6866	0.580271
$479$	0.176656	0.00807160	0.00403580	−	0.999992i	$-0.498715\pi$
$479$	0.176656	0.00807160	0.00403580	+	0.999992i	$0.498715\pi$
$480$	0	0
$481$	35.2187	1.60583
$482$	2.91650	0.132843
$483$	0	0
$484$	−10.7274	−0.487607
$485$	39.8785	1.81079
$486$	0	0
$487$	−34.0268	−1.54190	−0.770951	−	0.636895i	$-0.780219\pi$
$487$	−34.0268	−1.54190	−0.770951	+	0.636895i	$0.780219\pi$
$488$	3.98285	0.180295
$489$	0	0
$490$	−16.3552	−0.738851
$491$	12.0506	0.543837	0.271919	−	0.962320i	$-0.412342\pi$
$491$	12.0506	0.543837	0.271919	+	0.962320i	$0.412342\pi$
$492$	0	0
$493$	−66.1513	−2.97931
$494$	−15.1940	−0.683612
$495$	0	0
$496$	9.06917	0.407218
$497$	−5.79182	−0.259799
$498$	0	0
$499$	−28.6773	−1.28377	−0.641886	−	0.766800i	$-0.721848\pi$
$499$	−28.6773	−1.28377	−0.641886	+	0.766800i	$0.721848\pi$
$500$	9.68739	0.433233
$501$	0	0
$502$	−10.6258	−0.474252
$503$	26.3648	1.17555	0.587774	−	0.809025i	$-0.300005\pi$
$503$	26.3648	1.17555	0.587774	+	0.809025i	$0.300005\pi$
$504$	0	0
$505$	5.28340	0.235108
$506$	3.93196	0.174797
$507$	0	0
$508$	−10.9793	−0.487128
$509$	37.6836	1.67030	0.835148	−	0.550025i	$-0.185382\pi$
$509$	37.6836	1.67030	0.835148	+	0.550025i	$0.185382\pi$
$510$	0	0
$511$	−0.541517	−0.0239553
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−7.46027	−0.329058
$515$	−37.1895	−1.63876
$516$	0	0
$517$	−5.58018	−0.245416
$518$	−4.47684	−0.196701
$519$	0	0
$520$	11.6269	0.509872
$521$	−6.59184	−0.288794	−0.144397	−	0.989520i	$-0.546124\pi$
$521$	−6.59184	−0.288794	−0.144397	+	0.989520i	$0.546124\pi$
$522$	0	0
$523$	44.0936	1.92808	0.964039	−	0.265763i	$-0.0856237\pi$
$523$	44.0936	1.92808	0.964039	+	0.265763i	$0.0856237\pi$
$524$	−18.6281	−0.813770
$525$	0	0
$526$	−17.7053	−0.771986
$527$	60.4221	2.63203
$528$	0	0
$529$	33.7044	1.46541
$530$	−7.96006	−0.345763
$531$	0	0
$532$	1.93140	0.0837367
$533$	−8.21166	−0.355687
$534$	0	0
$535$	39.1005	1.69046
$536$	−13.9646	−0.603177
$537$	0	0
$538$	−9.47231	−0.408380
$539$	3.46710	0.149339
$540$	0	0
$541$	−34.1904	−1.46996	−0.734980	−	0.678089i	$-0.762808\pi$
$541$	−34.1904	−1.46996	−0.734980	+	0.678089i	$0.762808\pi$
$542$	−8.70903	−0.374085
$543$	0	0
$544$	−6.66237	−0.285647
$545$	15.6873	0.671971
$546$	0	0
$547$	32.3970	1.38520	0.692599	−	0.721323i	$-0.256466\pi$
$547$	32.3970	1.38520	0.692599	+	0.721323i	$0.256466\pi$
$548$	7.71863	0.329723
$549$	0	0
$550$	0.557170	0.0237578
$551$	−31.9602	−1.36155
$552$	0	0
$553$	9.06365	0.385425
$554$	28.1562	1.19624
$555$	0	0
$556$	1.61317	0.0684137
$557$	−21.8150	−0.924330	−0.462165	−	0.886794i	$-0.652927\pi$
$557$	−21.8150	−0.924330	−0.462165	+	0.886794i	$0.652927\pi$
$558$	0	0
$559$	−4.86693	−0.205849
$560$	−1.47795	−0.0624549
$561$	0	0
$562$	−21.9043	−0.923978
$563$	−28.6884	−1.20907	−0.604537	−	0.796577i	$-0.706642\pi$
$563$	−28.6884	−1.20907	−0.604537	+	0.796577i	$0.706642\pi$
$564$	0	0
$565$	1.75401	0.0737919
$566$	−22.5153	−0.946390
$567$	0	0
$568$	9.65257	0.405013
$569$	28.5487	1.19683	0.598413	−	0.801188i	$-0.295798\pi$
$569$	28.5487	1.19683	0.598413	+	0.801188i	$0.295798\pi$
$570$	0	0
$571$	21.1167	0.883705	0.441852	−	0.897088i	$-0.354321\pi$
$571$	21.1167	0.883705	0.441852	+	0.897088i	$0.354321\pi$
$572$	−2.46476	−0.103057
$573$	0	0
$574$	1.04383	0.0435685
$575$	8.03517	0.335090
$576$	0	0
$577$	26.2408	1.09242	0.546210	−	0.837648i	$-0.316070\pi$
$577$	26.2408	1.09242	0.546210	+	0.837648i	$0.316070\pi$
$578$	−27.3872	−1.13916
$579$	0	0
$580$	24.4568	1.01551
$581$	1.78259	0.0739545
$582$	0	0
$583$	1.68744	0.0698867
$584$	0.902486	0.0373451
$585$	0	0
$586$	7.90118	0.326395
$587$	−27.7026	−1.14341	−0.571705	−	0.820459i	$-0.693718\pi$
$587$	−27.7026	−1.14341	−0.571705	+	0.820459i	$0.693718\pi$
$588$	0	0
$589$	29.1922	1.20284
$590$	12.9239	0.532069
$591$	0	0
$592$	7.46104	0.306647
$593$	13.0519	0.535976	0.267988	−	0.963422i	$-0.413641\pi$
$593$	13.0519	0.535976	0.267988	+	0.963422i	$0.413641\pi$
$594$	0	0
$595$	−9.84667	−0.403674
$596$	1.00000	0.0409616
$597$	0	0
$598$	−35.5453	−1.45356
$599$	−36.2545	−1.48132	−0.740659	−	0.671881i	$-0.765487\pi$
$599$	−36.2545	−1.48132	−0.740659	+	0.671881i	$0.765487\pi$
$600$	0	0
$601$	−31.8745	−1.30019	−0.650094	−	0.759854i	$-0.725271\pi$
$601$	−31.8745	−1.30019	−0.650094	+	0.759854i	$0.725271\pi$
$602$	0.618661	0.0252147
$603$	0	0
$604$	10.8003	0.439457
$605$	26.4230	1.07425
$606$	0	0
$607$	−6.08572	−0.247012	−0.123506	−	0.992344i	$-0.539414\pi$
$607$	−6.08572	−0.247012	−0.123506	+	0.992344i	$0.539414\pi$
$608$	−3.21884	−0.130541
$609$	0	0
$610$	−9.81031	−0.397208
$611$	50.4454	2.04080
$612$	0	0
$613$	−3.79463	−0.153264	−0.0766319	−	0.997059i	$-0.524417\pi$
$613$	−3.79463	−0.153264	−0.0766319	+	0.997059i	$0.524417\pi$
$614$	27.2200	1.09851
$615$	0	0
$616$	0.313309	0.0126236
$617$	38.8946	1.56584	0.782919	−	0.622123i	$-0.213730\pi$
$617$	38.8946	1.56584	0.782919	+	0.622123i	$0.213730\pi$
$618$	0	0
$619$	−19.6002	−0.787797	−0.393899	−	0.919154i	$-0.628874\pi$
$619$	−19.6002	−0.787797	−0.393899	+	0.919154i	$0.628874\pi$
$620$	−22.3386	−0.897141
$621$	0	0
$622$	5.40526	0.216731
$623$	−4.64798	−0.186217
$624$	0	0
$625$	−29.1967	−1.16787
$626$	−32.9443	−1.31672
$627$	0	0
$628$	4.46308	0.178096
$629$	49.7082	1.98200
$630$	0	0
$631$	4.29603	0.171022	0.0855110	−	0.996337i	$-0.472748\pi$
$631$	4.29603	0.171022	0.0855110	+	0.996337i	$0.472748\pi$
$632$	−15.1054	−0.600859
$633$	0	0
$634$	25.1368	0.998308
$635$	27.0435	1.07319
$636$	0	0
$637$	−31.3429	−1.24185
$638$	−5.18455	−0.205258
$639$	0	0
$640$	2.46314	0.0973641
$641$	−8.61934	−0.340444	−0.170222	−	0.985406i	$-0.554448\pi$
$641$	−8.61934	−0.340444	−0.170222	+	0.985406i	$0.554448\pi$
$642$	0	0
$643$	14.0450	0.553880	0.276940	−	0.960887i	$-0.410680\pi$
$643$	14.0450	0.553880	0.276940	+	0.960887i	$0.410680\pi$
$644$	4.51835	0.178048
$645$	0	0
$646$	−21.4451	−0.843746
$647$	−17.6982	−0.695789	−0.347895	−	0.937534i	$-0.613103\pi$
$647$	−17.6982	−0.695789	−0.347895	+	0.937534i	$0.613103\pi$
$648$	0	0
$649$	−2.73972	−0.107543
$650$	−5.03687	−0.197562
$651$	0	0
$652$	−3.23968	−0.126876
$653$	39.3708	1.54070	0.770350	−	0.637621i	$-0.220082\pi$
$653$	39.3708	1.54070	0.770350	+	0.637621i	$0.220082\pi$
$654$	0	0
$655$	45.8835	1.79282
$656$	−1.73963	−0.0679212
$657$	0	0
$658$	−6.41238	−0.249981
$659$	−10.6738	−0.415794	−0.207897	−	0.978151i	$-0.566662\pi$
$659$	−10.6738	−0.415794	−0.207897	+	0.978151i	$0.566662\pi$
$660$	0	0
$661$	36.1676	1.40676	0.703378	−	0.710816i	$-0.251674\pi$
$661$	36.1676	1.40676	0.703378	+	0.710816i	$0.251674\pi$
$662$	−4.09310	−0.159083
$663$	0	0
$664$	−2.97085	−0.115291
$665$	−4.75730	−0.184480
$666$	0	0
$667$	−74.7685	−2.89505
$668$	11.8022	0.456640
$669$	0	0
$670$	34.3966	1.32886
$671$	2.07967	0.0802848
$672$	0	0
$673$	−6.02184	−0.232125	−0.116062	−	0.993242i	$-0.537027\pi$
$673$	−6.02184	−0.232125	−0.116062	+	0.993242i	$0.537027\pi$
$674$	−26.0313	−1.00269
$675$	0	0
$676$	9.28167	0.356987
$677$	18.4988	0.710966	0.355483	−	0.934683i	$-0.384316\pi$
$677$	18.4988	0.710966	0.355483	+	0.934683i	$0.384316\pi$
$678$	0	0
$679$	−9.71453	−0.372809
$680$	16.4103	0.629308
$681$	0	0
$682$	4.73553	0.181333
$683$	28.9413	1.10741	0.553703	−	0.832714i	$-0.313214\pi$
$683$	28.9413	1.10741	0.553703	+	0.832714i	$0.313214\pi$
$684$	0	0
$685$	−19.0121	−0.726413
$686$	8.18437	0.312481
$687$	0	0
$688$	−1.03105	−0.0393085
$689$	−15.2546	−0.581155
$690$	0	0
$691$	45.8506	1.74424	0.872120	−	0.489292i	$-0.162745\pi$
$691$	45.8506	1.74424	0.872120	+	0.489292i	$0.162745\pi$
$692$	24.4344	0.928855
$693$	0	0
$694$	−6.53114	−0.247919
$695$	−3.97346	−0.150722
$696$	0	0
$697$	−11.5901	−0.439005
$698$	−23.5618	−0.891828
$699$	0	0
$700$	0.640263	0.0241997
$701$	−16.7174	−0.631409	−0.315704	−	0.948858i	$-0.602241\pi$
$701$	−16.7174	−0.631409	−0.315704	+	0.948858i	$0.602241\pi$
$702$	0	0
$703$	24.0159	0.905777
$704$	−0.522157	−0.0196795
$705$	0	0
$706$	−5.05539	−0.190262
$707$	−1.28705	−0.0484046
$708$	0	0
$709$	−18.3919	−0.690721	−0.345360	−	0.938470i	$-0.612243\pi$
$709$	−18.3919	−0.690721	−0.345360	+	0.938470i	$0.612243\pi$
$710$	−23.7756	−0.892283
$711$	0	0
$712$	7.74627	0.290304
$713$	68.2930	2.55759
$714$	0	0
$715$	6.07105	0.227044
$716$	−17.9810	−0.671981
$717$	0	0
$718$	3.77664	0.140943
$719$	−39.9438	−1.48965	−0.744826	−	0.667259i	$-0.767467\pi$
$719$	−39.9438	−1.48965	−0.744826	+	0.667259i	$0.767467\pi$
$720$	0	0
$721$	9.05948	0.337392
$722$	8.63906	0.321513
$723$	0	0
$724$	1.07757	0.0400475
$725$	−10.5949	−0.393484
$726$	0	0
$727$	4.26721	0.158262	0.0791310	−	0.996864i	$-0.474785\pi$
$727$	4.26721	0.158262	0.0791310	+	0.996864i	$0.474785\pi$
$728$	−2.83234	−0.104974
$729$	0	0
$730$	−2.22295	−0.0822750
$731$	−6.86925	−0.254069
$732$	0	0
$733$	−10.3156	−0.381015	−0.190507	−	0.981686i	$-0.561013\pi$
$733$	−10.3156	−0.381015	−0.190507	+	0.981686i	$0.561013\pi$
$734$	−3.87803	−0.143141
$735$	0	0
$736$	−7.53023	−0.277568
$737$	−7.29169	−0.268593
$738$	0	0
$739$	−50.9389	−1.87382	−0.936909	−	0.349573i	$-0.886327\pi$
$739$	−50.9389	−1.87382	−0.936909	+	0.349573i	$0.886327\pi$
$740$	−18.3776	−0.675573
$741$	0	0
$742$	1.93910	0.0711865
$743$	19.1062	0.700939	0.350469	−	0.936574i	$-0.386022\pi$
$743$	19.1062	0.700939	0.350469	+	0.936574i	$0.386022\pi$
$744$	0	0
$745$	−2.46314	−0.0902424
$746$	11.4620	0.419653
$747$	0	0
$748$	−3.47880	−0.127198
$749$	−9.52500	−0.348036
$750$	0	0
$751$	12.1583	0.443662	0.221831	−	0.975085i	$-0.428797\pi$
$751$	12.1583	0.443662	0.221831	+	0.975085i	$0.428797\pi$
$752$	10.6868	0.389707
$753$	0	0
$754$	46.8688	1.70686
$755$	−26.6026	−0.968167
$756$	0	0
$757$	37.4410	1.36082	0.680408	−	0.732833i	$-0.261802\pi$
$757$	37.4410	1.36082	0.680408	+	0.732833i	$0.261802\pi$
$758$	−1.68630	−0.0612491
$759$	0	0
$760$	7.92845	0.287595
$761$	18.9245	0.686012	0.343006	−	0.939333i	$-0.388555\pi$
$761$	18.9245	0.686012	0.343006	+	0.939333i	$0.388555\pi$
$762$	0	0
$763$	−3.82148	−0.138347
$764$	−14.9599	−0.541230
$765$	0	0
$766$	−30.6373	−1.10697
$767$	24.7673	0.894296
$768$	0	0
$769$	−14.4264	−0.520229	−0.260115	−	0.965578i	$-0.583760\pi$
$769$	−14.4264	−0.520229	−0.260115	+	0.965578i	$0.583760\pi$
$770$	−0.771724	−0.0278110
$771$	0	0
$772$	13.3320	0.479828
$773$	−20.7385	−0.745911	−0.372956	−	0.927849i	$-0.621656\pi$
$773$	−20.7385	−0.745911	−0.372956	+	0.927849i	$0.621656\pi$
$774$	0	0
$775$	9.67729	0.347619
$776$	16.1901	0.581191
$777$	0	0
$778$	12.1259	0.434734
$779$	−5.59960	−0.200626
$780$	0	0
$781$	5.04016	0.180351
$782$	−50.1692	−1.79405
$783$	0	0
$784$	−6.63997	−0.237142
$785$	−10.9932	−0.392363
$786$	0	0
$787$	−41.0218	−1.46227	−0.731135	−	0.682232i	$-0.761009\pi$
$787$	−41.0218	−1.46227	−0.731135	+	0.682232i	$0.761009\pi$
$788$	12.4613	0.443914
$789$	0	0
$790$	37.2066	1.32375
$791$	−0.427283	−0.0151924
$792$	0	0
$793$	−18.8004	−0.667623
$794$	−7.03616	−0.249704
$795$	0	0
$796$	1.68669	0.0597832
$797$	16.3198	0.578078	0.289039	−	0.957317i	$-0.406664\pi$
$797$	16.3198	0.578078	0.289039	+	0.957317i	$0.406664\pi$
$798$	0	0
$799$	71.1993	2.51885
$800$	−1.06705	−0.0377261
$801$	0	0
$802$	21.3385	0.753489
$803$	0.471239	0.0166297
$804$	0	0
$805$	−11.1293	−0.392258
$806$	−42.8096	−1.50790
$807$	0	0
$808$	2.14499	0.0754603
$809$	54.9188	1.93084	0.965421	−	0.260697i	$-0.0839525\pi$
$809$	54.9188	1.93084	0.965421	+	0.260697i	$0.0839525\pi$
$810$	0	0
$811$	0.686568	0.0241087	0.0120543	−	0.999927i	$-0.496163\pi$
$811$	0.686568	0.0241087	0.0120543	+	0.999927i	$0.496163\pi$
$812$	−5.95774	−0.209076
$813$	0	0
$814$	3.89583	0.136549
$815$	7.97978	0.279519
$816$	0	0
$817$	−3.31880	−0.116110
$818$	−12.7311	−0.445133
$819$	0	0
$820$	4.28495	0.149637
$821$	−2.49104	−0.0869379	−0.0434690	−	0.999055i	$-0.513841\pi$
$821$	−2.49104	−0.0869379	−0.0434690	+	0.999055i	$0.513841\pi$
$822$	0	0
$823$	−3.99573	−0.139282	−0.0696412	−	0.997572i	$-0.522185\pi$
$823$	−3.99573	−0.139282	−0.0696412	+	0.997572i	$0.522185\pi$
$824$	−15.0984	−0.525978
$825$	0	0
$826$	−3.14831	−0.109544
$827$	42.0246	1.46134	0.730669	−	0.682732i	$-0.239209\pi$
$827$	42.0246	1.46134	0.730669	+	0.682732i	$0.239209\pi$
$828$	0	0
$829$	−29.4324	−1.02223	−0.511114	−	0.859513i	$-0.670767\pi$
$829$	−29.4324	−1.02223	−0.511114	+	0.859513i	$0.670767\pi$
$830$	7.31762	0.253998
$831$	0	0
$832$	4.72035	0.163649
$833$	−44.2379	−1.53275
$834$	0	0
$835$	−29.0704	−1.00602
$836$	−1.68074	−0.0581296
$837$	0	0
$838$	−14.6289	−0.505346
$839$	6.24113	0.215468	0.107734	−	0.994180i	$-0.465641\pi$
$839$	6.24113	0.215468	0.107734	+	0.994180i	$0.465641\pi$
$840$	0	0
$841$	69.5871	2.39955
$842$	28.6508	0.987371
$843$	0	0
$844$	−11.0214	−0.379371
$845$	−22.8621	−0.786479
$846$	0	0
$847$	−6.43672	−0.221168
$848$	−3.23167	−0.110976
$849$	0	0
$850$	−7.10911	−0.243840
$851$	56.1834	1.92594
$852$	0	0
$853$	−17.4115	−0.596157	−0.298079	−	0.954541i	$-0.596346\pi$
$853$	−17.4115	−0.596157	−0.298079	+	0.954541i	$0.596346\pi$
$854$	2.38982	0.0817780
$855$	0	0
$856$	15.8743	0.542571
$857$	−32.4849	−1.10966	−0.554832	−	0.831963i	$-0.687217\pi$
$857$	−32.4849	−1.10966	−0.554832	+	0.831963i	$0.687217\pi$
$858$	0	0
$859$	1.94308	0.0662969	0.0331484	−	0.999450i	$-0.489447\pi$
$859$	1.94308	0.0662969	0.0331484	+	0.999450i	$0.489447\pi$
$860$	2.53963	0.0866006
$861$	0	0
$862$	−16.0721	−0.547416
$863$	−16.9605	−0.577343	−0.288671	−	0.957428i	$-0.593214\pi$
$863$	−16.9605	−0.577343	−0.288671	+	0.957428i	$0.593214\pi$
$864$	0	0
$865$	−60.1852	−2.04636
$866$	7.06619	0.240119
$867$	0	0
$868$	5.44176	0.184705
$869$	−7.88737	−0.267561
$870$	0	0
$871$	65.9175	2.23353
$872$	6.36883	0.215676
$873$	0	0
$874$	−24.2386	−0.819884
$875$	5.81271	0.196506
$876$	0	0
$877$	−34.7323	−1.17282	−0.586412	−	0.810013i	$-0.699460\pi$
$877$	−34.7323	−1.17282	−0.586412	+	0.810013i	$0.699460\pi$
$878$	−13.0150	−0.439236
$879$	0	0
$880$	1.28615	0.0433559
$881$	38.5324	1.29819	0.649095	−	0.760708i	$-0.275148\pi$
$881$	38.5324	1.29819	0.649095	+	0.760708i	$0.275148\pi$
$882$	0	0
$883$	−21.6947	−0.730085	−0.365042	−	0.930991i	$-0.618945\pi$
$883$	−21.6947	−0.730085	−0.365042	+	0.930991i	$0.618945\pi$
$884$	31.4487	1.05773
$885$	0	0
$886$	24.4272	0.820649
$887$	−5.08356	−0.170689	−0.0853446	−	0.996351i	$-0.527199\pi$
$887$	−5.08356	−0.170689	−0.0853446	+	0.996351i	$0.527199\pi$
$888$	0	0
$889$	−6.58789	−0.220951
$890$	−19.0801	−0.639567
$891$	0	0
$892$	−18.3538	−0.614531
$893$	34.3991	1.15112
$894$	0	0
$895$	44.2897	1.48044
$896$	−0.600028	−0.0200455
$897$	0	0
$898$	6.27709	0.209469
$899$	−90.0487	−3.00329
$900$	0	0
$901$	−21.5306	−0.717288
$902$	−0.908360	−0.0302451
$903$	0	0
$904$	0.712105	0.0236843
$905$	−2.65420	−0.0882286
$906$	0	0
$907$	−37.8592	−1.25709	−0.628547	−	0.777771i	$-0.716350\pi$
$907$	−37.8592	−1.25709	−0.628547	+	0.777771i	$0.716350\pi$
$908$	23.7264	0.787387
$909$	0	0
$910$	6.97645	0.231267
$911$	−17.8921	−0.592792	−0.296396	−	0.955065i	$-0.595785\pi$
$911$	−17.8921	−0.592792	−0.296396	+	0.955065i	$0.595785\pi$
$912$	0	0
$913$	−1.55125	−0.0513389
$914$	−26.2213	−0.867325
$915$	0	0
$916$	−11.6240	−0.384066
$917$	−11.1774	−0.369109
$918$	0	0
$919$	−25.7468	−0.849307	−0.424654	−	0.905356i	$-0.639604\pi$
$919$	−25.7468	−0.849307	−0.424654	+	0.905356i	$0.639604\pi$
$920$	18.5480	0.611510
$921$	0	0
$922$	37.3461	1.22993
$923$	−45.5635	−1.49974
$924$	0	0
$925$	7.96133	0.261767
$926$	16.5565	0.544080
$927$	0	0
$928$	9.92910	0.325939
$929$	8.26304	0.271101	0.135551	−	0.990770i	$-0.456720\pi$
$929$	8.26304	0.271101	0.135551	+	0.990770i	$0.456720\pi$
$930$	0	0
$931$	−21.3730	−0.700472
$932$	−1.01374	−0.0332061
$933$	0	0
$934$	−35.3201	−1.15571
$935$	8.56877	0.280229
$936$	0	0
$937$	−0.507727	−0.0165867	−0.00829336	−	0.999966i	$-0.502640\pi$
$937$	−0.507727	−0.0165867	−0.00829336	+	0.999966i	$0.502640\pi$
$938$	−8.37913	−0.273588
$939$	0	0
$940$	−26.3231	−0.858563
$941$	47.7315	1.55600	0.778002	−	0.628262i	$-0.216234\pi$
$941$	47.7315	1.55600	0.778002	+	0.628262i	$0.216234\pi$
$942$	0	0
$943$	−13.0998	−0.426589
$944$	5.24693	0.170773
$945$	0	0
$946$	−0.538371	−0.0175040
$947$	−32.1295	−1.04407	−0.522035	−	0.852924i	$-0.674827\pi$
$947$	−32.1295	−1.04407	−0.522035	+	0.852924i	$0.674827\pi$
$948$	0	0
$949$	−4.26005	−0.138287
$950$	−3.43468	−0.111436
$951$	0	0
$952$	−3.99761	−0.129563
$953$	−32.7953	−1.06234	−0.531172	−	0.847264i	$-0.678248\pi$
$953$	−32.7953	−1.06234	−0.531172	+	0.847264i	$0.678248\pi$
$954$	0	0
$955$	36.8483	1.19238
$956$	−12.6866	−0.410313
$957$	0	0
$958$	−0.176656	−0.00570748
$959$	4.63140	0.149556
$960$	0	0
$961$	51.2498	1.65322
$962$	−35.2187	−1.13550
$963$	0	0
$964$	−2.91650	−0.0939340
$965$	−32.8385	−1.05711
$966$	0	0
$967$	−18.4519	−0.593373	−0.296686	−	0.954975i	$-0.595882\pi$
$967$	−18.4519	−0.593373	−0.296686	+	0.954975i	$0.595882\pi$
$968$	10.7274	0.344790
$969$	0	0
$970$	−39.8785	−1.28042
$971$	28.4774	0.913883	0.456942	−	0.889497i	$-0.348945\pi$
$971$	28.4774	0.913883	0.456942	+	0.889497i	$0.348945\pi$
$972$	0	0
$973$	0.967948	0.0310310
$974$	34.0268	1.09029
$975$	0	0
$976$	−3.98285	−0.127488
$977$	−8.04117	−0.257260	−0.128630	−	0.991693i	$-0.541058\pi$
$977$	−8.04117	−0.257260	−0.128630	+	0.991693i	$0.541058\pi$
$978$	0	0
$979$	4.04477	0.129271
$980$	16.3552	0.522446
$981$	0	0
$982$	−12.0506	−0.384551
$983$	25.8162	0.823410	0.411705	−	0.911317i	$-0.364933\pi$
$983$	25.8162	0.823410	0.411705	+	0.911317i	$0.364933\pi$
$984$	0	0
$985$	−30.6938	−0.977987
$986$	66.1513	2.10669
$987$	0	0
$988$	15.1940	0.483387
$989$	−7.76407	−0.246883
$990$	0	0
$991$	46.8393	1.48790	0.743949	−	0.668236i	$-0.232950\pi$
$991$	46.8393	1.48790	0.743949	+	0.668236i	$0.232950\pi$
$992$	−9.06917	−0.287946
$993$	0	0
$994$	5.79182	0.183705
$995$	−4.15455	−0.131708
$996$	0	0
$997$	−17.1360	−0.542702	−0.271351	−	0.962480i	$-0.587470\pi$
$997$	−17.1360	−0.542702	−0.271351	+	0.962480i	$0.587470\pi$
$998$	28.6773	0.907764
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.s.1.5	✓	16
3.2	odd	2		8046.2.a.t.1.12	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.5	✓	16	1.1	even	1	trivial
8046.2.a.t.1.12	yes	16	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.46314 q^{5} +0.600028 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.46314 q^{5} +0.600028 q^{7} -1.00000 q^{8} +2.46314 q^{10} -0.522157 q^{11} +4.72035 q^{13} -0.600028 q^{14} +1.00000 q^{16} +6.66237 q^{17} +3.21884 q^{19} -2.46314 q^{20} +0.522157 q^{22} +7.53023 q^{23} +1.06705 q^{25} -4.72035 q^{26} +0.600028 q^{28} -9.92910 q^{29} +9.06917 q^{31} -1.00000 q^{32} -6.66237 q^{34} -1.47795 q^{35} +7.46104 q^{37} -3.21884 q^{38} +2.46314 q^{40} -1.73963 q^{41} -1.03105 q^{43} -0.522157 q^{44} -7.53023 q^{46} +10.6868 q^{47} -6.63997 q^{49} -1.06705 q^{50} +4.72035 q^{52} -3.23167 q^{53} +1.28615 q^{55} -0.600028 q^{56} +9.92910 q^{58} +5.24693 q^{59} -3.98285 q^{61} -9.06917 q^{62} +1.00000 q^{64} -11.6269 q^{65} +13.9646 q^{67} +6.66237 q^{68} +1.47795 q^{70} -9.65257 q^{71} -0.902486 q^{73} -7.46104 q^{74} +3.21884 q^{76} -0.313309 q^{77} +15.1054 q^{79} -2.46314 q^{80} +1.73963 q^{82} +2.97085 q^{83} -16.4103 q^{85} +1.03105 q^{86} +0.522157 q^{88} -7.74627 q^{89} +2.83234 q^{91} +7.53023 q^{92} -10.6868 q^{94} -7.92845 q^{95} -16.1901 q^{97} +6.63997 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8}+O(q^{10}) \) 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8	\( 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8} + 4 q^{10} - 6 q^{11} + 6 q^{13} - 6 q^{14} + 16 q^{16} - q^{17} + 10 q^{19} - 4 q^{20} + 6 q^{22} - 10 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 21 q^{31} - 16 q^{32} + q^{34} - 16 q^{35} + 17 q^{37} - 10 q^{38} + 4 q^{40} + 4 q^{41} + 16 q^{43} - 6 q^{44} + 10 q^{46} - 25 q^{47} + 36 q^{49} - 28 q^{50} + 6 q^{52} - 14 q^{53} + 19 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 23 q^{61} - 21 q^{62} + 16 q^{64} - 20 q^{65} + 22 q^{67} - q^{68} + 16 q^{70} - 10 q^{71} + 16 q^{73} - 17 q^{74} + 10 q^{76} + 2 q^{77} + 37 q^{79} - 4 q^{80} - 4 q^{82} - 33 q^{83} + 43 q^{85} - 16 q^{86} + 6 q^{88} + 3 q^{89} + 28 q^{91} - 10 q^{92} + 25 q^{94} - 14 q^{95} - 3 q^{97} - 36 q^{98}+O(q^{100}) \) 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8 + 4 * q^10 - 6 * q^11 + 6 * q^13 - 6 * q^14 + 16 * q^16 - q^17 + 10 * q^19 - 4 * q^20 + 6 * q^22 - 10 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 21 * q^31 - 16 * q^32 + q^34 - 16 * q^35 + 17 * q^37 - 10 * q^38 + 4 * q^40 + 4 * q^41 + 16 * q^43 - 6 * q^44 + 10 * q^46 - 25 * q^47 + 36 * q^49 - 28 * q^50 + 6 * q^52 - 14 * q^53 + 19 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 23 * q^61 - 21 * q^62 + 16 * q^64 - 20 * q^65 + 22 * q^67 - q^68 + 16 * q^70 - 10 * q^71 + 16 * q^73 - 17 * q^74 + 10 * q^76 + 2 * q^77 + 37 * q^79 - 4 * q^80 - 4 * q^82 - 33 * q^83 + 43 * q^85 - 16 * q^86 + 6 * q^88 + 3 * q^89 + 28 * q^91 - 10 * q^92 + 25 * q^94 - 14 * q^95 - 3 * q^97 - 36 * q^98

Introduction

L-functions

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