LMFDB - Embedded newform 8046.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$16$
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			1.1
Root			$-4.31154$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -4.31154 q^{5} -0.737290 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -4.31154 q^{5} -0.737290 q^{7} +1.00000 q^{8} -4.31154 q^{10} -0.384699 q^{11} -4.56661 q^{13} -0.737290 q^{14} +1.00000 q^{16} -2.02322 q^{17} -3.92509 q^{19} -4.31154 q^{20} -0.384699 q^{22} -0.886623 q^{23} +13.5894 q^{25} -4.56661 q^{26} -0.737290 q^{28} -2.08402 q^{29} -8.67502 q^{31} +1.00000 q^{32} -2.02322 q^{34} +3.17886 q^{35} +9.26515 q^{37} -3.92509 q^{38} -4.31154 q^{40} -11.0832 q^{41} -0.744652 q^{43} -0.384699 q^{44} -0.886623 q^{46} +8.10358 q^{47} -6.45640 q^{49} +13.5894 q^{50} -4.56661 q^{52} -6.86035 q^{53} +1.65865 q^{55} -0.737290 q^{56} -2.08402 q^{58} +1.42412 q^{59} +11.3117 q^{61} -8.67502 q^{62} +1.00000 q^{64} +19.6891 q^{65} +8.18125 q^{67} -2.02322 q^{68} +3.17886 q^{70} +1.67648 q^{71} -9.21460 q^{73} +9.26515 q^{74} -3.92509 q^{76} +0.283635 q^{77} +1.39377 q^{79} -4.31154 q^{80} -11.0832 q^{82} -14.7309 q^{83} +8.72319 q^{85} -0.744652 q^{86} -0.384699 q^{88} +6.64826 q^{89} +3.36692 q^{91} -0.886623 q^{92} +8.10358 q^{94} +16.9232 q^{95} +1.97758 q^{97} -6.45640 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.31154	−1.92818	−0.964091	−	0.265573i	$-0.914439\pi$
$5$	−4.31154	−1.92818	−0.964091	+	0.265573i	$0.914439\pi$
$6$	0	0
$7$	−0.737290	−0.278670	−0.139335	−	0.990245i	$-0.544496\pi$
$7$	−0.737290	−0.278670	−0.139335	+	0.990245i	$0.544496\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−4.31154	−1.36343
$11$	−0.384699	−0.115991	−0.0579955	−	0.998317i	$-0.518471\pi$
$11$	−0.384699	−0.115991	−0.0579955	+	0.998317i	$0.518471\pi$
$12$	0	0
$13$	−4.56661	−1.26655	−0.633275	−	0.773927i	$-0.718290\pi$
$13$	−4.56661	−1.26655	−0.633275	+	0.773927i	$0.718290\pi$
$14$	−0.737290	−0.197049
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.02322	−0.490702	−0.245351	−	0.969434i	$-0.578903\pi$
$17$	−2.02322	−0.490702	−0.245351	+	0.969434i	$0.578903\pi$
$18$	0	0
$19$	−3.92509	−0.900478	−0.450239	−	0.892908i	$-0.648661\pi$
$19$	−3.92509	−0.900478	−0.450239	+	0.892908i	$0.648661\pi$
$20$	−4.31154	−0.964091
$21$	0	0
$22$	−0.384699	−0.0820181
$23$	−0.886623	−0.184874	−0.0924369	−	0.995719i	$-0.529466\pi$
$23$	−0.886623	−0.184874	−0.0924369	+	0.995719i	$0.529466\pi$
$24$	0	0
$25$	13.5894	2.71788
$26$	−4.56661	−0.895586
$27$	0	0
$28$	−0.737290	−0.139335
$29$	−2.08402	−0.386993	−0.193497	−	0.981101i	$-0.561983\pi$
$29$	−2.08402	−0.386993	−0.193497	+	0.981101i	$0.561983\pi$
$30$	0	0
$31$	−8.67502	−1.55808	−0.779040	−	0.626974i	$-0.784293\pi$
$31$	−8.67502	−1.55808	−0.779040	+	0.626974i	$0.784293\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.02322	−0.346979
$35$	3.17886	0.537325
$36$	0	0
$37$	9.26515	1.52318	0.761590	−	0.648059i	$-0.224419\pi$
$37$	9.26515	1.52318	0.761590	+	0.648059i	$0.224419\pi$
$38$	−3.92509	−0.636734
$39$	0	0
$40$	−4.31154	−0.681715
$41$	−11.0832	−1.73090	−0.865451	−	0.500993i	$-0.832968\pi$
$41$	−11.0832	−1.73090	−0.865451	+	0.500993i	$0.832968\pi$
$42$	0	0
$43$	−0.744652	−0.113558	−0.0567792	−	0.998387i	$-0.518083\pi$
$43$	−0.744652	−0.113558	−0.0567792	+	0.998387i	$0.518083\pi$
$44$	−0.384699	−0.0579955
$45$	0	0
$46$	−0.886623	−0.130725
$47$	8.10358	1.18203	0.591014	−	0.806661i	$-0.298728\pi$
$47$	8.10358	1.18203	0.591014	+	0.806661i	$0.298728\pi$
$48$	0	0
$49$	−6.45640	−0.922343
$50$	13.5894	1.92183
$51$	0	0
$52$	−4.56661	−0.633275
$53$	−6.86035	−0.942342	−0.471171	−	0.882042i	$-0.656169\pi$
$53$	−6.86035	−0.942342	−0.471171	+	0.882042i	$0.656169\pi$
$54$	0	0
$55$	1.65865	0.223652
$56$	−0.737290	−0.0985246
$57$	0	0
$58$	−2.08402	−0.273646
$59$	1.42412	0.185404	0.0927022	−	0.995694i	$-0.470450\pi$
$59$	1.42412	0.185404	0.0927022	+	0.995694i	$0.470450\pi$
$60$	0	0
$61$	11.3117	1.44831	0.724155	−	0.689637i	$-0.242230\pi$
$61$	11.3117	1.44831	0.724155	+	0.689637i	$0.242230\pi$
$62$	−8.67502	−1.10173
$63$	0	0
$64$	1.00000	0.125000
$65$	19.6891	2.44214
$66$	0	0
$67$	8.18125	0.999499	0.499749	−	0.866170i	$-0.333425\pi$
$67$	8.18125	0.999499	0.499749	+	0.866170i	$0.333425\pi$
$68$	−2.02322	−0.245351
$69$	0	0
$70$	3.17886	0.379946
$71$	1.67648	0.198961	0.0994807	−	0.995039i	$-0.468282\pi$
$71$	1.67648	0.198961	0.0994807	+	0.995039i	$0.468282\pi$
$72$	0	0
$73$	−9.21460	−1.07849	−0.539243	−	0.842150i	$-0.681290\pi$
$73$	−9.21460	−1.07849	−0.539243	+	0.842150i	$0.681290\pi$
$74$	9.26515	1.07705
$75$	0	0
$76$	−3.92509	−0.450239
$77$	0.283635	0.0323232
$78$	0	0
$79$	1.39377	0.156812	0.0784059	−	0.996922i	$-0.475017\pi$
$79$	1.39377	0.156812	0.0784059	+	0.996922i	$0.475017\pi$
$80$	−4.31154	−0.482045
$81$	0	0
$82$	−11.0832	−1.22393
$83$	−14.7309	−1.61693	−0.808463	−	0.588547i	$-0.799700\pi$
$83$	−14.7309	−1.61693	−0.808463	+	0.588547i	$0.799700\pi$
$84$	0	0
$85$	8.72319	0.946163
$86$	−0.744652	−0.0802979
$87$	0	0
$88$	−0.384699	−0.0410090
$89$	6.64826	0.704714	0.352357	−	0.935866i	$-0.385380\pi$
$89$	6.64826	0.704714	0.352357	+	0.935866i	$0.385380\pi$
$90$	0	0
$91$	3.36692	0.352949
$92$	−0.886623	−0.0924369
$93$	0	0
$94$	8.10358	0.835820
$95$	16.9232	1.73629
$96$	0	0
$97$	1.97758	0.200793	0.100397	−	0.994947i	$-0.467989\pi$
$97$	1.97758	0.200793	0.100397	+	0.994947i	$0.467989\pi$
$98$	−6.45640	−0.652195
$99$	0	0
$100$	13.5894	1.35894
$101$	−2.74572	−0.273209	−0.136605	−	0.990626i	$-0.543619\pi$
$101$	−2.74572	−0.273209	−0.136605	+	0.990626i	$0.543619\pi$
$102$	0	0
$103$	−2.40018	−0.236497	−0.118248	−	0.992984i	$-0.537728\pi$
$103$	−2.40018	−0.236497	−0.118248	+	0.992984i	$0.537728\pi$
$104$	−4.56661	−0.447793
$105$	0	0
$106$	−6.86035	−0.666336
$107$	10.8189	1.04590	0.522950	−	0.852363i	$-0.324831\pi$
$107$	10.8189	1.04590	0.522950	+	0.852363i	$0.324831\pi$
$108$	0	0
$109$	18.2862	1.75150	0.875750	−	0.482764i	$-0.160367\pi$
$109$	18.2862	1.75150	0.875750	+	0.482764i	$0.160367\pi$
$110$	1.65865	0.158146
$111$	0	0
$112$	−0.737290	−0.0696674
$113$	−15.0892	−1.41947	−0.709737	−	0.704467i	$-0.751186\pi$
$113$	−15.0892	−1.41947	−0.709737	+	0.704467i	$0.751186\pi$
$114$	0	0
$115$	3.82272	0.356470
$116$	−2.08402	−0.193497
$117$	0	0
$118$	1.42412	0.131101
$119$	1.49170	0.136744
$120$	0	0
$121$	−10.8520	−0.986546
$122$	11.3117	1.02411
$123$	0	0
$124$	−8.67502	−0.779040
$125$	−37.0337	−3.31239
$126$	0	0
$127$	22.2325	1.97282	0.986409	−	0.164309i	$-0.0525395\pi$
$127$	22.2325	1.97282	0.986409	+	0.164309i	$0.0525395\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	19.6891	1.72685
$131$	−13.3028	−1.16227	−0.581137	−	0.813805i	$-0.697392\pi$
$131$	−13.3028	−1.16227	−0.581137	+	0.813805i	$0.697392\pi$
$132$	0	0
$133$	2.89393	0.250936
$134$	8.18125	0.706752
$135$	0	0
$136$	−2.02322	−0.173489
$137$	17.2266	1.47177	0.735884	−	0.677107i	$-0.236767\pi$
$137$	17.2266	1.47177	0.735884	+	0.677107i	$0.236767\pi$
$138$	0	0
$139$	18.2223	1.54559	0.772796	−	0.634654i	$-0.218857\pi$
$139$	18.2223	1.54559	0.772796	+	0.634654i	$0.218857\pi$
$140$	3.17886	0.268663
$141$	0	0
$142$	1.67648	0.140687
$143$	1.75677	0.146908
$144$	0	0
$145$	8.98536	0.746194
$146$	−9.21460	−0.762605
$147$	0	0
$148$	9.26515	0.761590
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−9.11666	−0.741903	−0.370952	−	0.928652i	$-0.620968\pi$
$151$	−9.11666	−0.741903	−0.370952	+	0.928652i	$0.620968\pi$
$152$	−3.92509	−0.318367
$153$	0	0
$154$	0.283635	0.0228559
$155$	37.4027	3.00426
$156$	0	0
$157$	13.4052	1.06985	0.534925	−	0.844900i	$-0.320340\pi$
$157$	13.4052	1.06985	0.534925	+	0.844900i	$0.320340\pi$
$158$	1.39377	0.110883
$159$	0	0
$160$	−4.31154	−0.340858
$161$	0.653699	0.0515187
$162$	0	0
$163$	9.97283	0.781132	0.390566	−	0.920575i	$-0.372279\pi$
$163$	9.97283	0.781132	0.390566	+	0.920575i	$0.372279\pi$
$164$	−11.0832	−0.865451
$165$	0	0
$166$	−14.7309	−1.14334
$167$	10.2617	0.794070	0.397035	−	0.917803i	$-0.370039\pi$
$167$	10.2617	0.794070	0.397035	+	0.917803i	$0.370039\pi$
$168$	0	0
$169$	7.85394	0.604149
$170$	8.72319	0.669038
$171$	0	0
$172$	−0.744652	−0.0567792
$173$	−14.9430	−1.13610	−0.568048	−	0.822996i	$-0.692301\pi$
$173$	−14.9430	−1.13610	−0.568048	+	0.822996i	$0.692301\pi$
$174$	0	0
$175$	−10.0193	−0.757391
$176$	−0.384699	−0.0289978
$177$	0	0
$178$	6.64826	0.498308
$179$	−12.4760	−0.932499	−0.466249	−	0.884653i	$-0.654395\pi$
$179$	−12.4760	−0.932499	−0.466249	+	0.884653i	$0.654395\pi$
$180$	0	0
$181$	25.3030	1.88076	0.940378	−	0.340130i	$-0.110471\pi$
$181$	25.3030	1.88076	0.940378	+	0.340130i	$0.110471\pi$
$182$	3.36692	0.249573
$183$	0	0
$184$	−0.886623	−0.0653627
$185$	−39.9471	−2.93697
$186$	0	0
$187$	0.778329	0.0569171
$188$	8.10358	0.591014
$189$	0	0
$190$	16.9232	1.22774
$191$	19.6210	1.41973	0.709865	−	0.704338i	$-0.248756\pi$
$191$	19.6210	1.41973	0.709865	+	0.704338i	$0.248756\pi$
$192$	0	0
$193$	20.2653	1.45873	0.729366	−	0.684124i	$-0.239815\pi$
$193$	20.2653	1.45873	0.729366	+	0.684124i	$0.239815\pi$
$194$	1.97758	0.141982
$195$	0	0
$196$	−6.45640	−0.461172
$197$	−0.351995	−0.0250786	−0.0125393	−	0.999921i	$-0.503991\pi$
$197$	−0.351995	−0.0250786	−0.0125393	+	0.999921i	$0.503991\pi$
$198$	0	0
$199$	1.71637	0.121670	0.0608350	−	0.998148i	$-0.480624\pi$
$199$	1.71637	0.121670	0.0608350	+	0.998148i	$0.480624\pi$
$200$	13.5894	0.960917
$201$	0	0
$202$	−2.74572	−0.193188
$203$	1.53653	0.107843
$204$	0	0
$205$	47.7856	3.33749
$206$	−2.40018	−0.167228
$207$	0	0
$208$	−4.56661	−0.316638
$209$	1.50998	0.104447
$210$	0	0
$211$	19.2873	1.32779	0.663895	−	0.747826i	$-0.268902\pi$
$211$	19.2873	1.32779	0.663895	+	0.747826i	$0.268902\pi$
$212$	−6.86035	−0.471171
$213$	0	0
$214$	10.8189	0.739563
$215$	3.21060	0.218961
$216$	0	0
$217$	6.39601	0.434189
$218$	18.2862	1.23850
$219$	0	0
$220$	1.65865	0.111826
$221$	9.23925	0.621499
$222$	0	0
$223$	−25.6280	−1.71618	−0.858088	−	0.513502i	$-0.828348\pi$
$223$	−25.6280	−1.71618	−0.858088	+	0.513502i	$0.828348\pi$
$224$	−0.737290	−0.0492623
$225$	0	0
$226$	−15.0892	−1.00372
$227$	0.954268	0.0633370	0.0316685	−	0.999498i	$-0.489918\pi$
$227$	0.954268	0.0633370	0.0316685	+	0.999498i	$0.489918\pi$
$228$	0	0
$229$	0.0575982	0.00380620	0.00190310	−	0.999998i	$-0.499394\pi$
$229$	0.0575982	0.00380620	0.00190310	+	0.999998i	$0.499394\pi$
$230$	3.82272	0.252062
$231$	0	0
$232$	−2.08402	−0.136823
$233$	7.57013	0.495935	0.247968	−	0.968768i	$-0.420237\pi$
$233$	7.57013	0.495935	0.247968	+	0.968768i	$0.420237\pi$
$234$	0	0
$235$	−34.9389	−2.27917
$236$	1.42412	0.0927022
$237$	0	0
$238$	1.49170	0.0966924
$239$	−25.4935	−1.64904	−0.824518	−	0.565836i	$-0.808553\pi$
$239$	−25.4935	−1.64904	−0.824518	+	0.565836i	$0.808553\pi$
$240$	0	0
$241$	−4.83772	−0.311625	−0.155812	−	0.987787i	$-0.549800\pi$
$241$	−4.83772	−0.311625	−0.155812	+	0.987787i	$0.549800\pi$
$242$	−10.8520	−0.697593
$243$	0	0
$244$	11.3117	0.724155
$245$	27.8371	1.77845
$246$	0	0
$247$	17.9244	1.14050
$248$	−8.67502	−0.550864
$249$	0	0
$250$	−37.0337	−2.34221
$251$	−21.1499	−1.33497	−0.667486	−	0.744622i	$-0.732630\pi$
$251$	−21.1499	−1.33497	−0.667486	+	0.744622i	$0.732630\pi$
$252$	0	0
$253$	0.341083	0.0214437
$254$	22.2325	1.39499
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.4468	1.08830	0.544152	−	0.838987i	$-0.316851\pi$
$257$	17.4468	1.08830	0.544152	+	0.838987i	$0.316851\pi$
$258$	0	0
$259$	−6.83110	−0.424464
$260$	19.6891	1.22107
$261$	0	0
$262$	−13.3028	−0.821852
$263$	−20.3014	−1.25184	−0.625919	−	0.779888i	$-0.715276\pi$
$263$	−20.3014	−1.25184	−0.625919	+	0.779888i	$0.715276\pi$
$264$	0	0
$265$	29.5787	1.81701
$266$	2.89393	0.177438
$267$	0	0
$268$	8.18125	0.499749
$269$	7.90670	0.482080	0.241040	−	0.970515i	$-0.422512\pi$
$269$	7.90670	0.482080	0.241040	+	0.970515i	$0.422512\pi$
$270$	0	0
$271$	−6.19077	−0.376063	−0.188031	−	0.982163i	$-0.560211\pi$
$271$	−6.19077	−0.376063	−0.188031	+	0.982163i	$0.560211\pi$
$272$	−2.02322	−0.122676
$273$	0	0
$274$	17.2266	1.04070
$275$	−5.22783	−0.315250
$276$	0	0
$277$	−26.4793	−1.59098	−0.795492	−	0.605964i	$-0.792788\pi$
$277$	−26.4793	−1.59098	−0.795492	+	0.605964i	$0.792788\pi$
$278$	18.2223	1.09290
$279$	0	0
$280$	3.17886	0.189973
$281$	29.5435	1.76241	0.881207	−	0.472730i	$-0.156731\pi$
$281$	29.5435	1.76241	0.881207	+	0.472730i	$0.156731\pi$
$282$	0	0
$283$	19.7040	1.17128	0.585640	−	0.810571i	$-0.300843\pi$
$283$	19.7040	1.17128	0.585640	+	0.810571i	$0.300843\pi$
$284$	1.67648	0.0994807
$285$	0	0
$286$	1.75677	0.103880
$287$	8.17152	0.482350
$288$	0	0
$289$	−12.9066	−0.759211
$290$	8.98536	0.527639
$291$	0	0
$292$	−9.21460	−0.539243
$293$	−2.50747	−0.146488	−0.0732439	−	0.997314i	$-0.523335\pi$
$293$	−2.50747	−0.146488	−0.0732439	+	0.997314i	$0.523335\pi$
$294$	0	0
$295$	−6.14015	−0.357493
$296$	9.26515	0.538526
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	4.04886	0.234152
$300$	0	0
$301$	0.549024	0.0316452
$302$	−9.11666	−0.524605
$303$	0	0
$304$	−3.92509	−0.225120
$305$	−48.7708	−2.79261
$306$	0	0
$307$	−16.5485	−0.944473	−0.472236	−	0.881472i	$-0.656553\pi$
$307$	−16.5485	−0.944473	−0.472236	+	0.881472i	$0.656553\pi$
$308$	0.283635	0.0161616
$309$	0	0
$310$	37.4027	2.12433
$311$	12.7077	0.720588	0.360294	−	0.932839i	$-0.382676\pi$
$311$	12.7077	0.720588	0.360294	+	0.932839i	$0.382676\pi$
$312$	0	0
$313$	−3.47335	−0.196325	−0.0981627	−	0.995170i	$-0.531297\pi$
$313$	−3.47335	−0.196325	−0.0981627	+	0.995170i	$0.531297\pi$
$314$	13.4052	0.756498
$315$	0	0
$316$	1.39377	0.0784059
$317$	9.75919	0.548131	0.274065	−	0.961711i	$-0.411631\pi$
$317$	9.75919	0.548131	0.274065	+	0.961711i	$0.411631\pi$
$318$	0	0
$319$	0.801721	0.0448878
$320$	−4.31154	−0.241023
$321$	0	0
$322$	0.653699	0.0364292
$323$	7.94131	0.441867
$324$	0	0
$325$	−62.0576	−3.44234
$326$	9.97283	0.552344
$327$	0	0
$328$	−11.0832	−0.611966
$329$	−5.97469	−0.329395
$330$	0	0
$331$	25.1678	1.38335	0.691675	−	0.722209i	$-0.256873\pi$
$331$	25.1678	1.38335	0.691675	+	0.722209i	$0.256873\pi$
$332$	−14.7309	−0.808463
$333$	0	0
$334$	10.2617	0.561493
$335$	−35.2738	−1.92721
$336$	0	0
$337$	−13.2861	−0.723740	−0.361870	−	0.932229i	$-0.617862\pi$
$337$	−13.2861	−0.723740	−0.361870	+	0.932229i	$0.617862\pi$
$338$	7.85394	0.427198
$339$	0	0
$340$	8.72319	0.473081
$341$	3.33727	0.180723
$342$	0	0
$343$	9.92128	0.535699
$344$	−0.744652	−0.0401489
$345$	0	0
$346$	−14.9430	−0.803341
$347$	−0.872656	−0.0468466	−0.0234233	−	0.999726i	$-0.507457\pi$
$347$	−0.872656	−0.0468466	−0.0234233	+	0.999726i	$0.507457\pi$
$348$	0	0
$349$	29.6249	1.58579	0.792893	−	0.609361i	$-0.208574\pi$
$349$	29.6249	1.58579	0.792893	+	0.609361i	$0.208574\pi$
$350$	−10.0193	−0.535557
$351$	0	0
$352$	−0.384699	−0.0205045
$353$	11.7865	0.627334	0.313667	−	0.949533i	$-0.398442\pi$
$353$	11.7865	0.627334	0.313667	+	0.949533i	$0.398442\pi$
$354$	0	0
$355$	−7.22821	−0.383634
$356$	6.64826	0.352357
$357$	0	0
$358$	−12.4760	−0.659376
$359$	−15.8002	−0.833903	−0.416952	−	0.908929i	$-0.636902\pi$
$359$	−15.8002	−0.833903	−0.416952	+	0.908929i	$0.636902\pi$
$360$	0	0
$361$	−3.59364	−0.189139
$362$	25.3030	1.32990
$363$	0	0
$364$	3.36692	0.176474
$365$	39.7291	2.07952
$366$	0	0
$367$	−11.5574	−0.603294	−0.301647	−	0.953420i	$-0.597536\pi$
$367$	−11.5574	−0.603294	−0.301647	+	0.953420i	$0.597536\pi$
$368$	−0.886623	−0.0462184
$369$	0	0
$370$	−39.9471	−2.07675
$371$	5.05807	0.262602
$372$	0	0
$373$	1.24655	0.0645442	0.0322721	−	0.999479i	$-0.489726\pi$
$373$	1.24655	0.0645442	0.0322721	+	0.999479i	$0.489726\pi$
$374$	0.778329	0.0402464
$375$	0	0
$376$	8.10358	0.417910
$377$	9.51693	0.490147
$378$	0	0
$379$	−36.7886	−1.88971	−0.944853	−	0.327496i	$-0.893795\pi$
$379$	−36.7886	−1.88971	−0.944853	+	0.327496i	$0.893795\pi$
$380$	16.9232	0.868143
$381$	0	0
$382$	19.6210	1.00390
$383$	16.7444	0.855599	0.427800	−	0.903874i	$-0.359289\pi$
$383$	16.7444	0.855599	0.427800	+	0.903874i	$0.359289\pi$
$384$	0	0
$385$	−1.22290	−0.0623249
$386$	20.2653	1.03148
$387$	0	0
$388$	1.97758	0.100397
$389$	24.3679	1.23550	0.617751	−	0.786374i	$-0.288044\pi$
$389$	24.3679	1.23550	0.617751	+	0.786374i	$0.288044\pi$
$390$	0	0
$391$	1.79383	0.0907180
$392$	−6.45640	−0.326098
$393$	0	0
$394$	−0.351995	−0.0177332
$395$	−6.00932	−0.302362
$396$	0	0
$397$	6.13938	0.308126	0.154063	−	0.988061i	$-0.450764\pi$
$397$	6.13938	0.308126	0.154063	+	0.988061i	$0.450764\pi$
$398$	1.71637	0.0860336
$399$	0	0
$400$	13.5894	0.679471
$401$	1.78173	0.0889753	0.0444876	−	0.999010i	$-0.485834\pi$
$401$	1.78173	0.0889753	0.0444876	+	0.999010i	$0.485834\pi$
$402$	0	0
$403$	39.6155	1.97339
$404$	−2.74572	−0.136605
$405$	0	0
$406$	1.53653	0.0762567
$407$	−3.56429	−0.176675
$408$	0	0
$409$	37.5612	1.85728	0.928641	−	0.370979i	$-0.120978\pi$
$409$	37.5612	1.85728	0.928641	+	0.370979i	$0.120978\pi$
$410$	47.7856	2.35996
$411$	0	0
$412$	−2.40018	−0.118248
$413$	−1.04999	−0.0516665
$414$	0	0
$415$	63.5129	3.11773
$416$	−4.56661	−0.223897
$417$	0	0
$418$	1.50998	0.0738555
$419$	15.5966	0.761942	0.380971	−	0.924587i	$-0.375590\pi$
$419$	15.5966	0.761942	0.380971	+	0.924587i	$0.375590\pi$
$420$	0	0
$421$	−26.3847	−1.28591	−0.642955	−	0.765904i	$-0.722292\pi$
$421$	−26.3847	−1.28591	−0.642955	+	0.765904i	$0.722292\pi$
$422$	19.2873	0.938889
$423$	0	0
$424$	−6.86035	−0.333168
$425$	−27.4943	−1.33367
$426$	0	0
$427$	−8.33998	−0.403600
$428$	10.8189	0.522950
$429$	0	0
$430$	3.21060	0.154829
$431$	−9.31506	−0.448691	−0.224345	−	0.974510i	$-0.572024\pi$
$431$	−9.31506	−0.448691	−0.224345	+	0.974510i	$0.572024\pi$
$432$	0	0
$433$	6.04329	0.290422	0.145211	−	0.989401i	$-0.453614\pi$
$433$	6.04329	0.290422	0.145211	+	0.989401i	$0.453614\pi$
$434$	6.39601	0.307018
$435$	0	0
$436$	18.2862	0.875750
$437$	3.48008	0.166475
$438$	0	0
$439$	20.9123	0.998091	0.499045	−	0.866576i	$-0.333684\pi$
$439$	20.9123	0.998091	0.499045	+	0.866576i	$0.333684\pi$
$440$	1.65865	0.0790728
$441$	0	0
$442$	9.23925	0.439466
$443$	16.6125	0.789284	0.394642	−	0.918835i	$-0.370869\pi$
$443$	16.6125	0.789284	0.394642	+	0.918835i	$0.370869\pi$
$444$	0	0
$445$	−28.6643	−1.35882
$446$	−25.6280	−1.21352
$447$	0	0
$448$	−0.737290	−0.0348337
$449$	−22.4072	−1.05746	−0.528731	−	0.848789i	$-0.677332\pi$
$449$	−22.4072	−1.05746	−0.528731	+	0.848789i	$0.677332\pi$
$450$	0	0
$451$	4.26369	0.200769
$452$	−15.0892	−0.709737
$453$	0	0
$454$	0.954268	0.0447860
$455$	−14.5166	−0.680550
$456$	0	0
$457$	1.01282	0.0473776	0.0236888	−	0.999719i	$-0.492459\pi$
$457$	1.01282	0.0473776	0.0236888	+	0.999719i	$0.492459\pi$
$458$	0.0575982	0.00269139
$459$	0	0
$460$	3.82272	0.178235
$461$	30.7155	1.43056	0.715282	−	0.698836i	$-0.246298\pi$
$461$	30.7155	1.43056	0.715282	+	0.698836i	$0.246298\pi$
$462$	0	0
$463$	−15.9290	−0.740283	−0.370142	−	0.928975i	$-0.620691\pi$
$463$	−15.9290	−0.740283	−0.370142	+	0.928975i	$0.620691\pi$
$464$	−2.08402	−0.0967484
$465$	0	0
$466$	7.57013	0.350679
$467$	−0.107431	−0.00497133	−0.00248567	−	0.999997i	$-0.500791\pi$
$467$	−0.107431	−0.00497133	−0.00248567	+	0.999997i	$0.500791\pi$
$468$	0	0
$469$	−6.03196	−0.278530
$470$	−34.9389	−1.61161
$471$	0	0
$472$	1.42412	0.0655503
$473$	0.286467	0.0131717
$474$	0	0
$475$	−53.3397	−2.44739
$476$	1.49170	0.0683719
$477$	0	0
$478$	−25.4935	−1.16604
$479$	−9.90961	−0.452782	−0.226391	−	0.974037i	$-0.572693\pi$
$479$	−9.90961	−0.452782	−0.226391	+	0.974037i	$0.572693\pi$
$480$	0	0
$481$	−42.3103	−1.92918
$482$	−4.83772	−0.220352
$483$	0	0
$484$	−10.8520	−0.493273
$485$	−8.52644	−0.387166
$486$	0	0
$487$	12.1588	0.550968	0.275484	−	0.961306i	$-0.411162\pi$
$487$	12.1588	0.550968	0.275484	+	0.961306i	$0.411162\pi$
$488$	11.3117	0.512055
$489$	0	0
$490$	27.8371	1.25755
$491$	−35.2613	−1.59132	−0.795661	−	0.605743i	$-0.792876\pi$
$491$	−35.2613	−1.59132	−0.795661	+	0.605743i	$0.792876\pi$
$492$	0	0
$493$	4.21643	0.189899
$494$	17.9244	0.806456
$495$	0	0
$496$	−8.67502	−0.389520
$497$	−1.23605	−0.0554445
$498$	0	0
$499$	−10.1476	−0.454269	−0.227134	−	0.973863i	$-0.572936\pi$
$499$	−10.1476	−0.454269	−0.227134	+	0.973863i	$0.572936\pi$
$500$	−37.0337	−1.65620
$501$	0	0
$502$	−21.1499	−0.943968
$503$	33.9494	1.51373	0.756865	−	0.653571i	$-0.226730\pi$
$503$	33.9494	1.51373	0.756865	+	0.653571i	$0.226730\pi$
$504$	0	0
$505$	11.8383	0.526797
$506$	0.341083	0.0151630
$507$	0	0
$508$	22.2325	0.986409
$509$	14.7198	0.652442	0.326221	−	0.945294i	$-0.394225\pi$
$509$	14.7198	0.652442	0.326221	+	0.945294i	$0.394225\pi$
$510$	0	0
$511$	6.79383	0.300541
$512$	1.00000	0.0441942
$513$	0	0
$514$	17.4468	0.769547
$515$	10.3485	0.456009
$516$	0	0
$517$	−3.11744	−0.137105
$518$	−6.83110	−0.300141
$519$	0	0
$520$	19.6891	0.863426
$521$	−7.87978	−0.345219	−0.172610	−	0.984990i	$-0.555220\pi$
$521$	−7.87978	−0.345219	−0.172610	+	0.984990i	$0.555220\pi$
$522$	0	0
$523$	13.0081	0.568803	0.284402	−	0.958705i	$-0.408205\pi$
$523$	13.0081	0.568803	0.284402	+	0.958705i	$0.408205\pi$
$524$	−13.3028	−0.581137
$525$	0	0
$526$	−20.3014	−0.885183
$527$	17.5515	0.764553
$528$	0	0
$529$	−22.2139	−0.965822
$530$	29.5787	1.28482
$531$	0	0
$532$	2.89393	0.125468
$533$	50.6126	2.19227
$534$	0	0
$535$	−46.6461	−2.01668
$536$	8.18125	0.353376
$537$	0	0
$538$	7.90670	0.340882
$539$	2.48377	0.106984
$540$	0	0
$541$	40.2454	1.73028	0.865142	−	0.501527i	$-0.167228\pi$
$541$	40.2454	1.73028	0.865142	+	0.501527i	$0.167228\pi$
$542$	−6.19077	−0.265916
$543$	0	0
$544$	−2.02322	−0.0867447
$545$	−78.8418	−3.37721
$546$	0	0
$547$	25.0996	1.07318	0.536591	−	0.843843i	$-0.319712\pi$
$547$	25.0996	1.07318	0.536591	+	0.843843i	$0.319712\pi$
$548$	17.2266	0.735884
$549$	0	0
$550$	−5.22783	−0.222916
$551$	8.17999	0.348479
$552$	0	0
$553$	−1.02762	−0.0436987
$554$	−26.4793	−1.12500
$555$	0	0
$556$	18.2223	0.772796
$557$	−0.595764	−0.0252433	−0.0126217	−	0.999920i	$-0.504018\pi$
$557$	−0.595764	−0.0252433	−0.0126217	+	0.999920i	$0.504018\pi$
$558$	0	0
$559$	3.40053	0.143827
$560$	3.17886	0.134331
$561$	0	0
$562$	29.5435	1.24622
$563$	12.3115	0.518868	0.259434	−	0.965761i	$-0.416464\pi$
$563$	12.3115	0.518868	0.259434	+	0.965761i	$0.416464\pi$
$564$	0	0
$565$	65.0578	2.73700
$566$	19.7040	0.828220
$567$	0	0
$568$	1.67648	0.0703435
$569$	−34.5032	−1.44645	−0.723225	−	0.690613i	$-0.757341\pi$
$569$	−34.5032	−1.44645	−0.723225	+	0.690613i	$0.757341\pi$
$570$	0	0
$571$	−28.7066	−1.20133	−0.600667	−	0.799499i	$-0.705098\pi$
$571$	−28.7066	−1.20133	−0.600667	+	0.799499i	$0.705098\pi$
$572$	1.75677	0.0734542
$573$	0	0
$574$	8.17152	0.341073
$575$	−12.0487	−0.502465
$576$	0	0
$577$	−8.32222	−0.346458	−0.173229	−	0.984882i	$-0.555420\pi$
$577$	−8.32222	−0.346458	−0.173229	+	0.984882i	$0.555420\pi$
$578$	−12.9066	−0.536844
$579$	0	0
$580$	8.98536	0.373097
$581$	10.8609	0.450588
$582$	0	0
$583$	2.63917	0.109303
$584$	−9.21460	−0.381303
$585$	0	0
$586$	−2.50747	−0.103582
$587$	−16.8560	−0.695722	−0.347861	−	0.937546i	$-0.613092\pi$
$587$	−16.8560	−0.695722	−0.347861	+	0.937546i	$0.613092\pi$
$588$	0	0
$589$	34.0503	1.40302
$590$	−6.14015	−0.252786
$591$	0	0
$592$	9.26515	0.380795
$593$	33.3330	1.36882	0.684411	−	0.729096i	$-0.260059\pi$
$593$	33.3330	1.36882	0.684411	+	0.729096i	$0.260059\pi$
$594$	0	0
$595$	−6.43152	−0.263667
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	4.04886	0.165570
$599$	−13.2495	−0.541358	−0.270679	−	0.962670i	$-0.587248\pi$
$599$	−13.2495	−0.541358	−0.270679	+	0.962670i	$0.587248\pi$
$600$	0	0
$601$	14.0509	0.573148	0.286574	−	0.958058i	$-0.407484\pi$
$601$	14.0509	0.573148	0.286574	+	0.958058i	$0.407484\pi$
$602$	0.549024	0.0223766
$603$	0	0
$604$	−9.11666	−0.370952
$605$	46.7889	1.90224
$606$	0	0
$607$	−33.8753	−1.37496	−0.687478	−	0.726205i	$-0.741282\pi$
$607$	−33.8753	−1.37496	−0.687478	+	0.726205i	$0.741282\pi$
$608$	−3.92509	−0.159184
$609$	0	0
$610$	−48.7708	−1.97467
$611$	−37.0059	−1.49710
$612$	0	0
$613$	−10.5426	−0.425810	−0.212905	−	0.977073i	$-0.568293\pi$
$613$	−10.5426	−0.425810	−0.212905	+	0.977073i	$0.568293\pi$
$614$	−16.5485	−0.667843
$615$	0	0
$616$	0.283635	0.0114280
$617$	6.43807	0.259187	0.129594	−	0.991567i	$-0.458633\pi$
$617$	6.43807	0.259187	0.129594	+	0.991567i	$0.458633\pi$
$618$	0	0
$619$	−2.70749	−0.108823	−0.0544116	−	0.998519i	$-0.517328\pi$
$619$	−2.70749	−0.108823	−0.0544116	+	0.998519i	$0.517328\pi$
$620$	37.4027	1.50213
$621$	0	0
$622$	12.7077	0.509532
$623$	−4.90170	−0.196382
$624$	0	0
$625$	91.7252	3.66901
$626$	−3.47335	−0.138823
$627$	0	0
$628$	13.4052	0.534925
$629$	−18.7454	−0.747428
$630$	0	0
$631$	−20.5707	−0.818906	−0.409453	−	0.912331i	$-0.634280\pi$
$631$	−20.5707	−0.818906	−0.409453	+	0.912331i	$0.634280\pi$
$632$	1.39377	0.0554414
$633$	0	0
$634$	9.75919	0.387587
$635$	−95.8565	−3.80395
$636$	0	0
$637$	29.4839	1.16819
$638$	0.801721	0.0317405
$639$	0	0
$640$	−4.31154	−0.170429
$641$	−21.6781	−0.856232	−0.428116	−	0.903724i	$-0.640823\pi$
$641$	−21.6781	−0.856232	−0.428116	+	0.903724i	$0.640823\pi$
$642$	0	0
$643$	16.5623	0.653152	0.326576	−	0.945171i	$-0.394105\pi$
$643$	16.5623	0.653152	0.326576	+	0.945171i	$0.394105\pi$
$644$	0.653699	0.0257593
$645$	0	0
$646$	7.94131	0.312447
$647$	31.1219	1.22353	0.611764	−	0.791040i	$-0.290460\pi$
$647$	31.1219	1.22353	0.611764	+	0.791040i	$0.290460\pi$
$648$	0	0
$649$	−0.547856	−0.0215052
$650$	−62.0576	−2.43410
$651$	0	0
$652$	9.97283	0.390566
$653$	−29.7608	−1.16463	−0.582314	−	0.812964i	$-0.697853\pi$
$653$	−29.7608	−1.16463	−0.582314	+	0.812964i	$0.697853\pi$
$654$	0	0
$655$	57.3558	2.24108
$656$	−11.0832	−0.432726
$657$	0	0
$658$	−5.97469	−0.232918
$659$	−4.89599	−0.190721	−0.0953604	−	0.995443i	$-0.530400\pi$
$659$	−4.89599	−0.190721	−0.0953604	+	0.995443i	$0.530400\pi$
$660$	0	0
$661$	11.4854	0.446730	0.223365	−	0.974735i	$-0.428296\pi$
$661$	11.4854	0.446730	0.223365	+	0.974735i	$0.428296\pi$
$662$	25.1678	0.978176
$663$	0	0
$664$	−14.7309	−0.571670
$665$	−12.4773	−0.483850
$666$	0	0
$667$	1.84774	0.0715449
$668$	10.2617	0.397035
$669$	0	0
$670$	−35.2738	−1.36275
$671$	−4.35158	−0.167991
$672$	0	0
$673$	18.2687	0.704208	0.352104	−	0.935961i	$-0.385466\pi$
$673$	18.2687	0.704208	0.352104	+	0.935961i	$0.385466\pi$
$674$	−13.2861	−0.511761
$675$	0	0
$676$	7.85394	0.302075
$677$	20.1443	0.774207	0.387103	−	0.922036i	$-0.373476\pi$
$677$	20.1443	0.774207	0.387103	+	0.922036i	$0.373476\pi$
$678$	0	0
$679$	−1.45805	−0.0559550
$680$	8.72319	0.334519
$681$	0	0
$682$	3.33727	0.127791
$683$	−14.9293	−0.571256	−0.285628	−	0.958341i	$-0.592202\pi$
$683$	−14.9293	−0.571256	−0.285628	+	0.958341i	$0.592202\pi$
$684$	0	0
$685$	−74.2733	−2.83784
$686$	9.92128	0.378796
$687$	0	0
$688$	−0.744652	−0.0283896
$689$	31.3286	1.19352
$690$	0	0
$691$	−24.5907	−0.935476	−0.467738	−	0.883867i	$-0.654931\pi$
$691$	−24.5907	−0.935476	−0.467738	+	0.883867i	$0.654931\pi$
$692$	−14.9430	−0.568048
$693$	0	0
$694$	−0.872656	−0.0331256
$695$	−78.5661	−2.98018
$696$	0	0
$697$	22.4237	0.849358
$698$	29.6249	1.12132
$699$	0	0
$700$	−10.0193	−0.378696
$701$	17.2015	0.649692	0.324846	−	0.945767i	$-0.394688\pi$
$701$	17.2015	0.649692	0.324846	+	0.945767i	$0.394688\pi$
$702$	0	0
$703$	−36.3666	−1.37159
$704$	−0.384699	−0.0144989
$705$	0	0
$706$	11.7865	0.443592
$707$	2.02439	0.0761350
$708$	0	0
$709$	−32.9687	−1.23816	−0.619082	−	0.785326i	$-0.712495\pi$
$709$	−32.9687	−1.23816	−0.619082	+	0.785326i	$0.712495\pi$
$710$	−7.22821	−0.271270
$711$	0	0
$712$	6.64826	0.249154
$713$	7.69148	0.288048
$714$	0	0
$715$	−7.57439	−0.283266
$716$	−12.4760	−0.466249
$717$	0	0
$718$	−15.8002	−0.589658
$719$	−3.98039	−0.148444	−0.0742218	−	0.997242i	$-0.523647\pi$
$719$	−3.98039	−0.148444	−0.0742218	+	0.997242i	$0.523647\pi$
$720$	0	0
$721$	1.76963	0.0659044
$722$	−3.59364	−0.133742
$723$	0	0
$724$	25.3030	0.940378
$725$	−28.3207	−1.05180
$726$	0	0
$727$	−1.86140	−0.0690354	−0.0345177	−	0.999404i	$-0.510990\pi$
$727$	−1.86140	−0.0690354	−0.0345177	+	0.999404i	$0.510990\pi$
$728$	3.36692	0.124786
$729$	0	0
$730$	39.7291	1.47044
$731$	1.50659	0.0557233
$732$	0	0
$733$	−38.2753	−1.41373	−0.706865	−	0.707349i	$-0.749891\pi$
$733$	−38.2753	−1.41373	−0.706865	+	0.707349i	$0.749891\pi$
$734$	−11.5574	−0.426593
$735$	0	0
$736$	−0.886623	−0.0326814
$737$	−3.14732	−0.115933
$738$	0	0
$739$	−15.5181	−0.570841	−0.285421	−	0.958402i	$-0.592133\pi$
$739$	−15.5181	−0.570841	−0.285421	+	0.958402i	$0.592133\pi$
$740$	−39.9471	−1.46848
$741$	0	0
$742$	5.05807	0.185688
$743$	−10.8576	−0.398328	−0.199164	−	0.979966i	$-0.563823\pi$
$743$	−10.8576	−0.398328	−0.199164	+	0.979966i	$0.563823\pi$
$744$	0	0
$745$	4.31154	0.157963
$746$	1.24655	0.0456396
$747$	0	0
$748$	0.778329	0.0284585
$749$	−7.97665	−0.291460
$750$	0	0
$751$	11.5130	0.420115	0.210058	−	0.977689i	$-0.432635\pi$
$751$	11.5130	0.420115	0.210058	+	0.977689i	$0.432635\pi$
$752$	8.10358	0.295507
$753$	0	0
$754$	9.51693	0.346586
$755$	39.3069	1.43052
$756$	0	0
$757$	−14.3921	−0.523091	−0.261545	−	0.965191i	$-0.584232\pi$
$757$	−14.3921	−0.523091	−0.261545	+	0.965191i	$0.584232\pi$
$758$	−36.7886	−1.33622
$759$	0	0
$760$	16.9232	0.613870
$761$	−22.8443	−0.828106	−0.414053	−	0.910253i	$-0.635887\pi$
$761$	−22.8443	−0.828106	−0.414053	+	0.910253i	$0.635887\pi$
$762$	0	0
$763$	−13.4822	−0.488090
$764$	19.6210	0.709865
$765$	0	0
$766$	16.7444	0.605000
$767$	−6.50339	−0.234824
$768$	0	0
$769$	−16.1575	−0.582655	−0.291328	−	0.956623i	$-0.594097\pi$
$769$	−16.1575	−0.582655	−0.291328	+	0.956623i	$0.594097\pi$
$770$	−1.22290	−0.0440704
$771$	0	0
$772$	20.2653	0.729366
$773$	−33.6610	−1.21070	−0.605351	−	0.795958i	$-0.706967\pi$
$773$	−33.6610	−1.21070	−0.605351	+	0.795958i	$0.706967\pi$
$774$	0	0
$775$	−117.888	−4.23468
$776$	1.97758	0.0709911
$777$	0	0
$778$	24.3679	0.873632
$779$	43.5025	1.55864
$780$	0	0
$781$	−0.644939	−0.0230777
$782$	1.79383	0.0641473
$783$	0	0
$784$	−6.45640	−0.230586
$785$	−57.7970	−2.06286
$786$	0	0
$787$	−31.9509	−1.13893	−0.569463	−	0.822017i	$-0.692849\pi$
$787$	−31.9509	−1.13893	−0.569463	+	0.822017i	$0.692849\pi$
$788$	−0.351995	−0.0125393
$789$	0	0
$790$	−6.00932	−0.213802
$791$	11.1251	0.395564
$792$	0	0
$793$	−51.6560	−1.83436
$794$	6.13938	0.217878
$795$	0	0
$796$	1.71637	0.0608350
$797$	21.4741	0.760650	0.380325	−	0.924853i	$-0.375812\pi$
$797$	21.4741	0.760650	0.380325	+	0.924853i	$0.375812\pi$
$798$	0	0
$799$	−16.3953	−0.580024
$800$	13.5894	0.480458
$801$	0	0
$802$	1.78173	0.0629150
$803$	3.54484	0.125095
$804$	0	0
$805$	−2.81845	−0.0993374
$806$	39.6155	1.39539
$807$	0	0
$808$	−2.74572	−0.0965940
$809$	−6.34665	−0.223136	−0.111568	−	0.993757i	$-0.535587\pi$
$809$	−6.34665	−0.223136	−0.111568	+	0.993757i	$0.535587\pi$
$810$	0	0
$811$	30.1167	1.05754	0.528769	−	0.848765i	$-0.322654\pi$
$811$	30.1167	1.05754	0.528769	+	0.848765i	$0.322654\pi$
$812$	1.53653	0.0539216
$813$	0	0
$814$	−3.56429	−0.124928
$815$	−42.9983	−1.50616
$816$	0	0
$817$	2.92283	0.102257
$818$	37.5612	1.31330
$819$	0	0
$820$	47.7856	1.66875
$821$	−40.6192	−1.41762	−0.708810	−	0.705400i	$-0.750767\pi$
$821$	−40.6192	−1.41762	−0.708810	+	0.705400i	$0.750767\pi$
$822$	0	0
$823$	47.9339	1.67087	0.835435	−	0.549590i	$-0.185216\pi$
$823$	47.9339	1.67087	0.835435	+	0.549590i	$0.185216\pi$
$824$	−2.40018	−0.0836142
$825$	0	0
$826$	−1.04999	−0.0365338
$827$	−22.5280	−0.783374	−0.391687	−	0.920099i	$-0.628108\pi$
$827$	−22.5280	−0.783374	−0.391687	+	0.920099i	$0.628108\pi$
$828$	0	0
$829$	33.5222	1.16427	0.582137	−	0.813091i	$-0.302217\pi$
$829$	33.5222	1.16427	0.582137	+	0.813091i	$0.302217\pi$
$830$	63.5129	2.20457
$831$	0	0
$832$	−4.56661	−0.158319
$833$	13.0627	0.452596
$834$	0	0
$835$	−44.2436	−1.53111
$836$	1.50998	0.0522237
$837$	0	0
$838$	15.5966	0.538775
$839$	31.5650	1.08975	0.544873	−	0.838518i	$-0.316578\pi$
$839$	31.5650	1.08975	0.544873	+	0.838518i	$0.316578\pi$
$840$	0	0
$841$	−24.6568	−0.850236
$842$	−26.3847	−0.909276
$843$	0	0
$844$	19.2873	0.663895
$845$	−33.8626	−1.16491
$846$	0	0
$847$	8.00108	0.274920
$848$	−6.86035	−0.235586
$849$	0	0
$850$	−27.4943	−0.943048
$851$	−8.21470	−0.281596
$852$	0	0
$853$	10.3635	0.354838	0.177419	−	0.984135i	$-0.443225\pi$
$853$	10.3635	0.354838	0.177419	+	0.984135i	$0.443225\pi$
$854$	−8.33998	−0.285388
$855$	0	0
$856$	10.8189	0.369781
$857$	−40.7818	−1.39308	−0.696540	−	0.717518i	$-0.745278\pi$
$857$	−40.7818	−1.39308	−0.696540	+	0.717518i	$0.745278\pi$
$858$	0	0
$859$	30.9884	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$859$	30.9884	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$860$	3.21060	0.109481
$861$	0	0
$862$	−9.31506	−0.317272
$863$	−19.0267	−0.647677	−0.323839	−	0.946112i	$-0.604973\pi$
$863$	−19.0267	−0.647677	−0.323839	+	0.946112i	$0.604973\pi$
$864$	0	0
$865$	64.4274	2.19060
$866$	6.04329	0.205359
$867$	0	0
$868$	6.39601	0.217095
$869$	−0.536183	−0.0181888
$870$	0	0
$871$	−37.3606	−1.26592
$872$	18.2862	0.619249
$873$	0	0
$874$	3.48008	0.117715
$875$	27.3046	0.923063
$876$	0	0
$877$	27.3644	0.924031	0.462016	−	0.886872i	$-0.347126\pi$
$877$	27.3644	0.924031	0.462016	+	0.886872i	$0.347126\pi$
$878$	20.9123	0.705757
$879$	0	0
$880$	1.65865	0.0559129
$881$	−22.6708	−0.763798	−0.381899	−	0.924204i	$-0.624730\pi$
$881$	−22.6708	−0.763798	−0.381899	+	0.924204i	$0.624730\pi$
$882$	0	0
$883$	−32.8171	−1.10438	−0.552191	−	0.833717i	$-0.686208\pi$
$883$	−32.8171	−1.10438	−0.552191	+	0.833717i	$0.686208\pi$
$884$	9.23925	0.310749
$885$	0	0
$886$	16.6125	0.558108
$887$	56.1562	1.88554	0.942771	−	0.333441i	$-0.108210\pi$
$887$	56.1562	1.88554	0.942771	+	0.333441i	$0.108210\pi$
$888$	0	0
$889$	−16.3918	−0.549764
$890$	−28.6643	−0.960829
$891$	0	0
$892$	−25.6280	−0.858088
$893$	−31.8073	−1.06439
$894$	0	0
$895$	53.7908	1.79803
$896$	−0.737290	−0.0246311
$897$	0	0
$898$	−22.4072	−0.747739
$899$	18.0789	0.602967
$900$	0	0
$901$	13.8800	0.462409
$902$	4.26369	0.141965
$903$	0	0
$904$	−15.0892	−0.501860
$905$	−109.095	−3.62644
$906$	0	0
$907$	−17.8626	−0.593119	−0.296560	−	0.955014i	$-0.595839\pi$
$907$	−17.8626	−0.593119	−0.296560	+	0.955014i	$0.595839\pi$
$908$	0.954268	0.0316685
$909$	0	0
$910$	−14.5166	−0.481221
$911$	−9.56757	−0.316988	−0.158494	−	0.987360i	$-0.550664\pi$
$911$	−9.56757	−0.316988	−0.158494	+	0.987360i	$0.550664\pi$
$912$	0	0
$913$	5.66696	0.187549
$914$	1.01282	0.0335010
$915$	0	0
$916$	0.0575982	0.00190310
$917$	9.80806	0.323891
$918$	0	0
$919$	29.2427	0.964627	0.482314	−	0.875999i	$-0.339797\pi$
$919$	29.2427	0.964627	0.482314	+	0.875999i	$0.339797\pi$
$920$	3.82272	0.126031
$921$	0	0
$922$	30.7155	1.01156
$923$	−7.65582	−0.251994
$924$	0	0
$925$	125.908	4.13983
$926$	−15.9290	−0.523459
$927$	0	0
$928$	−2.08402	−0.0684114
$929$	5.82338	0.191059	0.0955295	−	0.995427i	$-0.469546\pi$
$929$	5.82338	0.191059	0.0955295	+	0.995427i	$0.469546\pi$
$930$	0	0
$931$	25.3420	0.830550
$932$	7.57013	0.247968
$933$	0	0
$934$	−0.107431	−0.00351526
$935$	−3.35580	−0.109746
$936$	0	0
$937$	−14.8174	−0.484064	−0.242032	−	0.970268i	$-0.577814\pi$
$937$	−14.8174	−0.484064	−0.242032	+	0.970268i	$0.577814\pi$
$938$	−6.03196	−0.196950
$939$	0	0
$940$	−34.9389	−1.13958
$941$	30.4980	0.994207	0.497103	−	0.867691i	$-0.334397\pi$
$941$	30.4980	0.994207	0.497103	+	0.867691i	$0.334397\pi$
$942$	0	0
$943$	9.82661	0.319998
$944$	1.42412	0.0463511
$945$	0	0
$946$	0.286467	0.00931383
$947$	35.7716	1.16242	0.581210	−	0.813754i	$-0.302579\pi$
$947$	35.7716	1.16242	0.581210	+	0.813754i	$0.302579\pi$
$948$	0	0
$949$	42.0795	1.36596
$950$	−53.3397	−1.73057
$951$	0	0
$952$	1.49170	0.0483462
$953$	19.9355	0.645774	0.322887	−	0.946438i	$-0.395347\pi$
$953$	19.9355	0.645774	0.322887	+	0.946438i	$0.395347\pi$
$954$	0	0
$955$	−84.5970	−2.73750
$956$	−25.4935	−0.824518
$957$	0	0
$958$	−9.90961	−0.320165
$959$	−12.7010	−0.410137
$960$	0	0
$961$	44.2560	1.42761
$962$	−42.3103	−1.36414
$963$	0	0
$964$	−4.83772	−0.155812
$965$	−87.3749	−2.81270
$966$	0	0
$967$	−11.9849	−0.385408	−0.192704	−	0.981257i	$-0.561726\pi$
$967$	−11.9849	−0.385408	−0.192704	+	0.981257i	$0.561726\pi$
$968$	−10.8520	−0.348797
$969$	0	0
$970$	−8.52644	−0.273768
$971$	−56.5448	−1.81461	−0.907304	−	0.420475i	$-0.861863\pi$
$971$	−56.5448	−1.81461	−0.907304	+	0.420475i	$0.861863\pi$
$972$	0	0
$973$	−13.4351	−0.430710
$974$	12.1588	0.389593
$975$	0	0
$976$	11.3117	0.362078
$977$	−48.0582	−1.53752	−0.768759	−	0.639539i	$-0.779125\pi$
$977$	−48.0582	−1.53752	−0.768759	+	0.639539i	$0.779125\pi$
$978$	0	0
$979$	−2.55758	−0.0817405
$980$	27.8371	0.889223
$981$	0	0
$982$	−35.2613	−1.12523
$983$	13.7738	0.439316	0.219658	−	0.975577i	$-0.429506\pi$
$983$	13.7738	0.439316	0.219658	+	0.975577i	$0.429506\pi$
$984$	0	0
$985$	1.51764	0.0483561
$986$	4.21643	0.134279
$987$	0	0
$988$	17.9244	0.570250
$989$	0.660226	0.0209940
$990$	0	0
$991$	16.5277	0.525020	0.262510	−	0.964929i	$-0.415450\pi$
$991$	16.5277	0.525020	0.262510	+	0.964929i	$0.415450\pi$
$992$	−8.67502	−0.275432
$993$	0	0
$994$	−1.23605	−0.0392052
$995$	−7.40019	−0.234602
$996$	0	0
$997$	48.5367	1.53717	0.768586	−	0.639747i	$-0.220961\pi$
$997$	48.5367	1.53717	0.768586	+	0.639747i	$0.220961\pi$
$998$	−10.1476	−0.321216
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.t.1.1	yes	16
3.2	odd	2		8046.2.a.s.1.16	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.16	✓	16	3.2	odd	2
8046.2.a.t.1.1	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.31154 q^{5} -0.737290 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -4.31154 q^{5} -0.737290 q^{7} +1.00000 q^{8} -4.31154 q^{10} -0.384699 q^{11} -4.56661 q^{13} -0.737290 q^{14} +1.00000 q^{16} -2.02322 q^{17} -3.92509 q^{19} -4.31154 q^{20} -0.384699 q^{22} -0.886623 q^{23} +13.5894 q^{25} -4.56661 q^{26} -0.737290 q^{28} -2.08402 q^{29} -8.67502 q^{31} +1.00000 q^{32} -2.02322 q^{34} +3.17886 q^{35} +9.26515 q^{37} -3.92509 q^{38} -4.31154 q^{40} -11.0832 q^{41} -0.744652 q^{43} -0.384699 q^{44} -0.886623 q^{46} +8.10358 q^{47} -6.45640 q^{49} +13.5894 q^{50} -4.56661 q^{52} -6.86035 q^{53} +1.65865 q^{55} -0.737290 q^{56} -2.08402 q^{58} +1.42412 q^{59} +11.3117 q^{61} -8.67502 q^{62} +1.00000 q^{64} +19.6891 q^{65} +8.18125 q^{67} -2.02322 q^{68} +3.17886 q^{70} +1.67648 q^{71} -9.21460 q^{73} +9.26515 q^{74} -3.92509 q^{76} +0.283635 q^{77} +1.39377 q^{79} -4.31154 q^{80} -11.0832 q^{82} -14.7309 q^{83} +8.72319 q^{85} -0.744652 q^{86} -0.384699 q^{88} +6.64826 q^{89} +3.36692 q^{91} -0.886623 q^{92} +8.10358 q^{94} +16.9232 q^{95} +1.97758 q^{97} -6.45640 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) 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

Modular forms

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Database

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