LMFDB - Embedded newform 8004.2.a.h.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 27 x^{11} + 158 x^{10} + 180 x^{9} - 1652 x^{8} + 65 x^{7} + 7388 x^{6} + \cdots - 5832 $ x^13 - 5x^12 - 27x^11 + 158x^10 + 180x^9 - 1652x^8 + 65x^7 + 7388x^6 - 3259x^5 - 15445x^4 + 7832x^3 + 15102x^2 - 5184x - 5832
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.59293$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.59293 q^{5} +3.02509 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.59293 q^{5} +3.02509 q^{7} +1.00000 q^{9} +2.27286 q^{11} +6.72050 q^{13} -1.59293 q^{15} +0.393320 q^{17} -3.65181 q^{19} -3.02509 q^{21} -1.00000 q^{23} -2.46257 q^{25} -1.00000 q^{27} -1.00000 q^{29} +7.95831 q^{31} -2.27286 q^{33} +4.81877 q^{35} -4.01687 q^{37} -6.72050 q^{39} +5.18034 q^{41} +3.72762 q^{43} +1.59293 q^{45} +1.48865 q^{47} +2.15118 q^{49} -0.393320 q^{51} +6.75379 q^{53} +3.62051 q^{55} +3.65181 q^{57} +13.8164 q^{59} +8.85099 q^{61} +3.02509 q^{63} +10.7053 q^{65} -2.09309 q^{67} +1.00000 q^{69} -0.475402 q^{71} -4.12927 q^{73} +2.46257 q^{75} +6.87561 q^{77} -4.35662 q^{79} +1.00000 q^{81} -4.52898 q^{83} +0.626532 q^{85} +1.00000 q^{87} +9.75001 q^{89} +20.3301 q^{91} -7.95831 q^{93} -5.81708 q^{95} +5.15387 q^{97} +2.27286 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.59293	0.712381	0.356191	−	0.934413i	$-0.384075\pi$
$5$	1.59293	0.712381	0.356191	+	0.934413i	$0.384075\pi$
$6$	0	0
$7$	3.02509	1.14338	0.571689	−	0.820471i	$-0.306289\pi$
$7$	3.02509	1.14338	0.571689	+	0.820471i	$0.306289\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.27286	0.685293	0.342647	−	0.939464i	$-0.388677\pi$
$11$	2.27286	0.685293	0.342647	+	0.939464i	$0.388677\pi$
$12$	0	0
$13$	6.72050	1.86393	0.931966	−	0.362545i	$-0.118092\pi$
$13$	6.72050	1.86393	0.931966	+	0.362545i	$0.118092\pi$
$14$	0	0
$15$	−1.59293	−0.411293
$16$	0	0
$17$	0.393320	0.0953940	0.0476970	−	0.998862i	$-0.484812\pi$
$17$	0.393320	0.0953940	0.0476970	+	0.998862i	$0.484812\pi$
$18$	0	0
$19$	−3.65181	−0.837782	−0.418891	−	0.908037i	$-0.637581\pi$
$19$	−3.65181	−0.837782	−0.418891	+	0.908037i	$0.637581\pi$
$20$	0	0
$21$	−3.02509	−0.660129
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.46257	−0.492513
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	7.95831	1.42935	0.714677	−	0.699455i	$-0.246574\pi$
$31$	7.95831	1.42935	0.714677	+	0.699455i	$0.246574\pi$
$32$	0	0
$33$	−2.27286	−0.395654
$34$	0	0
$35$	4.81877	0.814520
$36$	0	0
$37$	−4.01687	−0.660370	−0.330185	−	0.943916i	$-0.607111\pi$
$37$	−4.01687	−0.660370	−0.330185	+	0.943916i	$0.607111\pi$
$38$	0	0
$39$	−6.72050	−1.07614
$40$	0	0
$41$	5.18034	0.809033	0.404516	−	0.914531i	$-0.367440\pi$
$41$	5.18034	0.809033	0.404516	+	0.914531i	$0.367440\pi$
$42$	0	0
$43$	3.72762	0.568457	0.284229	−	0.958757i	$-0.408263\pi$
$43$	3.72762	0.568457	0.284229	+	0.958757i	$0.408263\pi$
$44$	0	0
$45$	1.59293	0.237460
$46$	0	0
$47$	1.48865	0.217142	0.108571	−	0.994089i	$-0.465372\pi$
$47$	1.48865	0.217142	0.108571	+	0.994089i	$0.465372\pi$
$48$	0	0
$49$	2.15118	0.307312
$50$	0	0
$51$	−0.393320	−0.0550758
$52$	0	0
$53$	6.75379	0.927704	0.463852	−	0.885913i	$-0.346467\pi$
$53$	6.75379	0.927704	0.463852	+	0.885913i	$0.346467\pi$
$54$	0	0
$55$	3.62051	0.488190
$56$	0	0
$57$	3.65181	0.483694
$58$	0	0
$59$	13.8164	1.79874	0.899371	−	0.437186i	$-0.144025\pi$
$59$	13.8164	1.79874	0.899371	+	0.437186i	$0.144025\pi$
$60$	0	0
$61$	8.85099	1.13325	0.566627	−	0.823975i	$-0.308248\pi$
$61$	8.85099	1.13325	0.566627	+	0.823975i	$0.308248\pi$
$62$	0	0
$63$	3.02509	0.381126
$64$	0	0
$65$	10.7053	1.32783
$66$	0	0
$67$	−2.09309	−0.255712	−0.127856	−	0.991793i	$-0.540809\pi$
$67$	−2.09309	−0.255712	−0.127856	+	0.991793i	$0.540809\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−0.475402	−0.0564199	−0.0282099	−	0.999602i	$-0.508981\pi$
$71$	−0.475402	−0.0564199	−0.0282099	+	0.999602i	$0.508981\pi$
$72$	0	0
$73$	−4.12927	−0.483295	−0.241647	−	0.970364i	$-0.577688\pi$
$73$	−4.12927	−0.483295	−0.241647	+	0.970364i	$0.577688\pi$
$74$	0	0
$75$	2.46257	0.284353
$76$	0	0
$77$	6.87561	0.783549
$78$	0	0
$79$	−4.35662	−0.490159	−0.245079	−	0.969503i	$-0.578814\pi$
$79$	−4.35662	−0.490159	−0.245079	+	0.969503i	$0.578814\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.52898	−0.497120	−0.248560	−	0.968616i	$-0.579957\pi$
$83$	−4.52898	−0.497120	−0.248560	+	0.968616i	$0.579957\pi$
$84$	0	0
$85$	0.626532	0.0679569
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	9.75001	1.03350	0.516750	−	0.856137i	$-0.327142\pi$
$89$	9.75001	1.03350	0.516750	+	0.856137i	$0.327142\pi$
$90$	0	0
$91$	20.3301	2.13118
$92$	0	0
$93$	−7.95831	−0.825238
$94$	0	0
$95$	−5.81708	−0.596820
$96$	0	0
$97$	5.15387	0.523297	0.261648	−	0.965163i	$-0.415734\pi$
$97$	5.15387	0.523297	0.261648	+	0.965163i	$0.415734\pi$
$98$	0	0
$99$	2.27286	0.228431
$100$	0	0
$101$	4.91467	0.489028	0.244514	−	0.969646i	$-0.421372\pi$
$101$	4.91467	0.489028	0.244514	+	0.969646i	$0.421372\pi$
$102$	0	0
$103$	−15.4061	−1.51801	−0.759006	−	0.651083i	$-0.774315\pi$
$103$	−15.4061	−1.51801	−0.759006	+	0.651083i	$0.774315\pi$
$104$	0	0
$105$	−4.81877	−0.470264
$106$	0	0
$107$	−13.8283	−1.33683	−0.668415	−	0.743789i	$-0.733027\pi$
$107$	−13.8283	−1.33683	−0.668415	+	0.743789i	$0.733027\pi$
$108$	0	0
$109$	−14.8625	−1.42357	−0.711786	−	0.702396i	$-0.752113\pi$
$109$	−14.8625	−1.42357	−0.711786	+	0.702396i	$0.752113\pi$
$110$	0	0
$111$	4.01687	0.381265
$112$	0	0
$113$	10.6125	0.998344	0.499172	−	0.866503i	$-0.333638\pi$
$113$	10.6125	0.998344	0.499172	+	0.866503i	$0.333638\pi$
$114$	0	0
$115$	−1.59293	−0.148542
$116$	0	0
$117$	6.72050	0.621311
$118$	0	0
$119$	1.18983	0.109071
$120$	0	0
$121$	−5.83410	−0.530373
$122$	0	0
$123$	−5.18034	−0.467095
$124$	0	0
$125$	−11.8874	−1.06324
$126$	0	0
$127$	6.43569	0.571075	0.285537	−	0.958368i	$-0.407828\pi$
$127$	6.43569	0.571075	0.285537	+	0.958368i	$0.407828\pi$
$128$	0	0
$129$	−3.72762	−0.328199
$130$	0	0
$131$	−9.81384	−0.857439	−0.428720	−	0.903438i	$-0.641035\pi$
$131$	−9.81384	−0.857439	−0.428720	+	0.903438i	$0.641035\pi$
$132$	0	0
$133$	−11.0471	−0.957901
$134$	0	0
$135$	−1.59293	−0.137098
$136$	0	0
$137$	6.49231	0.554676	0.277338	−	0.960772i	$-0.410548\pi$
$137$	6.49231	0.554676	0.277338	+	0.960772i	$0.410548\pi$
$138$	0	0
$139$	−0.811714	−0.0688487	−0.0344243	−	0.999407i	$-0.510960\pi$
$139$	−0.811714	−0.0688487	−0.0344243	+	0.999407i	$0.510960\pi$
$140$	0	0
$141$	−1.48865	−0.125367
$142$	0	0
$143$	15.2748	1.27734
$144$	0	0
$145$	−1.59293	−0.132286
$146$	0	0
$147$	−2.15118	−0.177427
$148$	0	0
$149$	−14.1125	−1.15614	−0.578072	−	0.815986i	$-0.696195\pi$
$149$	−14.1125	−1.15614	−0.578072	+	0.815986i	$0.696195\pi$
$150$	0	0
$151$	−17.0350	−1.38629	−0.693146	−	0.720797i	$-0.743776\pi$
$151$	−17.0350	−1.38629	−0.693146	+	0.720797i	$0.743776\pi$
$152$	0	0
$153$	0.393320	0.0317980
$154$	0	0
$155$	12.6770	1.01824
$156$	0	0
$157$	11.2155	0.895097	0.447548	−	0.894260i	$-0.352297\pi$
$157$	11.2155	0.895097	0.447548	+	0.894260i	$0.352297\pi$
$158$	0	0
$159$	−6.75379	−0.535610
$160$	0	0
$161$	−3.02509	−0.238411
$162$	0	0
$163$	−15.0129	−1.17590	−0.587952	−	0.808896i	$-0.700066\pi$
$163$	−15.0129	−1.17590	−0.587952	+	0.808896i	$0.700066\pi$
$164$	0	0
$165$	−3.62051	−0.281857
$166$	0	0
$167$	6.33177	0.489967	0.244984	−	0.969527i	$-0.421217\pi$
$167$	6.33177	0.489967	0.244984	+	0.969527i	$0.421217\pi$
$168$	0	0
$169$	32.1652	2.47424
$170$	0	0
$171$	−3.65181	−0.279261
$172$	0	0
$173$	15.5100	1.17920	0.589601	−	0.807694i	$-0.299285\pi$
$173$	15.5100	1.17920	0.589601	+	0.807694i	$0.299285\pi$
$174$	0	0
$175$	−7.44949	−0.563128
$176$	0	0
$177$	−13.8164	−1.03850
$178$	0	0
$179$	−16.3219	−1.21996	−0.609980	−	0.792417i	$-0.708822\pi$
$179$	−16.3219	−1.21996	−0.609980	+	0.792417i	$0.708822\pi$
$180$	0	0
$181$	−10.4126	−0.773965	−0.386983	−	0.922087i	$-0.626483\pi$
$181$	−10.4126	−0.773965	−0.386983	+	0.922087i	$0.626483\pi$
$182$	0	0
$183$	−8.85099	−0.654284
$184$	0	0
$185$	−6.39861	−0.470435
$186$	0	0
$187$	0.893961	0.0653729
$188$	0	0
$189$	−3.02509	−0.220043
$190$	0	0
$191$	5.18679	0.375303	0.187652	−	0.982236i	$-0.439912\pi$
$191$	5.18679	0.375303	0.187652	+	0.982236i	$0.439912\pi$
$192$	0	0
$193$	−8.40729	−0.605170	−0.302585	−	0.953122i	$-0.597850\pi$
$193$	−8.40729	−0.605170	−0.302585	+	0.953122i	$0.597850\pi$
$194$	0	0
$195$	−10.7053	−0.766623
$196$	0	0
$197$	15.7786	1.12418	0.562088	−	0.827077i	$-0.309998\pi$
$197$	15.7786	1.12418	0.562088	+	0.827077i	$0.309998\pi$
$198$	0	0
$199$	9.77795	0.693141	0.346570	−	0.938024i	$-0.387346\pi$
$199$	9.77795	0.693141	0.346570	+	0.938024i	$0.387346\pi$
$200$	0	0
$201$	2.09309	0.147635
$202$	0	0
$203$	−3.02509	−0.212320
$204$	0	0
$205$	8.25193	0.576340
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−8.30005	−0.574127
$210$	0	0
$211$	−25.2712	−1.73974	−0.869871	−	0.493279i	$-0.835798\pi$
$211$	−25.2712	−1.73974	−0.869871	+	0.493279i	$0.835798\pi$
$212$	0	0
$213$	0.475402	0.0325740
$214$	0	0
$215$	5.93785	0.404958
$216$	0	0
$217$	24.0746	1.63429
$218$	0	0
$219$	4.12927	0.279030
$220$	0	0
$221$	2.64331	0.177808
$222$	0	0
$223$	7.56939	0.506884	0.253442	−	0.967351i	$-0.418437\pi$
$223$	7.56939	0.506884	0.253442	+	0.967351i	$0.418437\pi$
$224$	0	0
$225$	−2.46257	−0.164171
$226$	0	0
$227$	−4.15942	−0.276070	−0.138035	−	0.990427i	$-0.544079\pi$
$227$	−4.15942	−0.276070	−0.138035	+	0.990427i	$0.544079\pi$
$228$	0	0
$229$	13.3711	0.883588	0.441794	−	0.897117i	$-0.354342\pi$
$229$	13.3711	0.883588	0.441794	+	0.897117i	$0.354342\pi$
$230$	0	0
$231$	−6.87561	−0.452382
$232$	0	0
$233$	−27.4032	−1.79524	−0.897622	−	0.440767i	$-0.854707\pi$
$233$	−27.4032	−1.79524	−0.897622	+	0.440767i	$0.854707\pi$
$234$	0	0
$235$	2.37132	0.154688
$236$	0	0
$237$	4.35662	0.282993
$238$	0	0
$239$	−13.1708	−0.851947	−0.425974	−	0.904736i	$-0.640068\pi$
$239$	−13.1708	−0.851947	−0.425974	+	0.904736i	$0.640068\pi$
$240$	0	0
$241$	15.2578	0.982843	0.491421	−	0.870922i	$-0.336477\pi$
$241$	15.2578	0.982843	0.491421	+	0.870922i	$0.336477\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.42669	0.218923
$246$	0	0
$247$	−24.5420	−1.56157
$248$	0	0
$249$	4.52898	0.287012
$250$	0	0
$251$	21.1598	1.33559	0.667797	−	0.744343i	$-0.267237\pi$
$251$	21.1598	1.33559	0.667797	+	0.744343i	$0.267237\pi$
$252$	0	0
$253$	−2.27286	−0.142894
$254$	0	0
$255$	−0.626532	−0.0392349
$256$	0	0
$257$	2.85039	0.177802	0.0889011	−	0.996040i	$-0.471664\pi$
$257$	2.85039	0.177802	0.0889011	+	0.996040i	$0.471664\pi$
$258$	0	0
$259$	−12.1514	−0.755052
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	12.3594	0.762111	0.381055	−	0.924552i	$-0.375561\pi$
$263$	12.3594	0.762111	0.381055	+	0.924552i	$0.375561\pi$
$264$	0	0
$265$	10.7583	0.660879
$266$	0	0
$267$	−9.75001	−0.596691
$268$	0	0
$269$	29.1296	1.77606	0.888030	−	0.459785i	$-0.152074\pi$
$269$	29.1296	1.77606	0.888030	+	0.459785i	$0.152074\pi$
$270$	0	0
$271$	14.3718	0.873023	0.436511	−	0.899699i	$-0.356214\pi$
$271$	14.3718	0.873023	0.436511	+	0.899699i	$0.356214\pi$
$272$	0	0
$273$	−20.3301	−1.23044
$274$	0	0
$275$	−5.59707	−0.337516
$276$	0	0
$277$	22.6868	1.36312	0.681559	−	0.731763i	$-0.261302\pi$
$277$	22.6868	1.36312	0.681559	+	0.731763i	$0.261302\pi$
$278$	0	0
$279$	7.95831	0.476451
$280$	0	0
$281$	−19.0446	−1.13611	−0.568054	−	0.822992i	$-0.692303\pi$
$281$	−19.0446	−1.13611	−0.568054	+	0.822992i	$0.692303\pi$
$282$	0	0
$283$	−11.9840	−0.712374	−0.356187	−	0.934415i	$-0.615923\pi$
$283$	−11.9840	−0.712374	−0.356187	+	0.934415i	$0.615923\pi$
$284$	0	0
$285$	5.81708	0.344574
$286$	0	0
$287$	15.6710	0.925030
$288$	0	0
$289$	−16.8453	−0.990900
$290$	0	0
$291$	−5.15387	−0.302125
$292$	0	0
$293$	−13.8173	−0.807216	−0.403608	−	0.914932i	$-0.632244\pi$
$293$	−13.8173	−0.807216	−0.403608	+	0.914932i	$0.632244\pi$
$294$	0	0
$295$	22.0086	1.28139
$296$	0	0
$297$	−2.27286	−0.131885
$298$	0	0
$299$	−6.72050	−0.388657
$300$	0	0
$301$	11.2764	0.649961
$302$	0	0
$303$	−4.91467	−0.282340
$304$	0	0
$305$	14.0990	0.807308
$306$	0	0
$307$	−16.6941	−0.952784	−0.476392	−	0.879233i	$-0.658056\pi$
$307$	−16.6941	−0.952784	−0.476392	+	0.879233i	$0.658056\pi$
$308$	0	0
$309$	15.4061	0.876425
$310$	0	0
$311$	7.13885	0.404807	0.202404	−	0.979302i	$-0.435125\pi$
$311$	7.13885	0.404807	0.202404	+	0.979302i	$0.435125\pi$
$312$	0	0
$313$	−13.2545	−0.749186	−0.374593	−	0.927189i	$-0.622218\pi$
$313$	−13.2545	−0.749186	−0.374593	+	0.927189i	$0.622218\pi$
$314$	0	0
$315$	4.81877	0.271507
$316$	0	0
$317$	8.67751	0.487378	0.243689	−	0.969853i	$-0.421642\pi$
$317$	8.67751	0.487378	0.243689	+	0.969853i	$0.421642\pi$
$318$	0	0
$319$	−2.27286	−0.127256
$320$	0	0
$321$	13.8283	0.771819
$322$	0	0
$323$	−1.43633	−0.0799194
$324$	0	0
$325$	−16.5497	−0.918011
$326$	0	0
$327$	14.8625	0.821900
$328$	0	0
$329$	4.50331	0.248276
$330$	0	0
$331$	−1.92292	−0.105693	−0.0528467	−	0.998603i	$-0.516829\pi$
$331$	−1.92292	−0.105693	−0.0528467	+	0.998603i	$0.516829\pi$
$332$	0	0
$333$	−4.01687	−0.220123
$334$	0	0
$335$	−3.33415	−0.182164
$336$	0	0
$337$	6.45926	0.351858	0.175929	−	0.984403i	$-0.443707\pi$
$337$	6.45926	0.351858	0.175929	+	0.984403i	$0.443707\pi$
$338$	0	0
$339$	−10.6125	−0.576394
$340$	0	0
$341$	18.0881	0.979527
$342$	0	0
$343$	−14.6681	−0.792004
$344$	0	0
$345$	1.59293	0.0857606
$346$	0	0
$347$	6.19328	0.332472	0.166236	−	0.986086i	$-0.446839\pi$
$347$	6.19328	0.332472	0.166236	+	0.986086i	$0.446839\pi$
$348$	0	0
$349$	19.0179	1.01800	0.509001	−	0.860766i	$-0.330015\pi$
$349$	19.0179	1.01800	0.509001	+	0.860766i	$0.330015\pi$
$350$	0	0
$351$	−6.72050	−0.358714
$352$	0	0
$353$	−4.89584	−0.260579	−0.130290	−	0.991476i	$-0.541591\pi$
$353$	−4.89584	−0.260579	−0.130290	+	0.991476i	$0.541591\pi$
$354$	0	0
$355$	−0.757284	−0.0401924
$356$	0	0
$357$	−1.18983	−0.0629724
$358$	0	0
$359$	20.9974	1.10820	0.554100	−	0.832450i	$-0.313062\pi$
$359$	20.9974	1.10820	0.554100	+	0.832450i	$0.313062\pi$
$360$	0	0
$361$	−5.66430	−0.298121
$362$	0	0
$363$	5.83410	0.306211
$364$	0	0
$365$	−6.57766	−0.344290
$366$	0	0
$367$	20.8226	1.08693	0.543465	−	0.839432i	$-0.317112\pi$
$367$	20.8226	1.08693	0.543465	+	0.839432i	$0.317112\pi$
$368$	0	0
$369$	5.18034	0.269678
$370$	0	0
$371$	20.4308	1.06072
$372$	0	0
$373$	9.99821	0.517688	0.258844	−	0.965919i	$-0.416659\pi$
$373$	9.99821	0.517688	0.258844	+	0.965919i	$0.416659\pi$
$374$	0	0
$375$	11.8874	0.613861
$376$	0	0
$377$	−6.72050	−0.346124
$378$	0	0
$379$	25.0145	1.28491	0.642453	−	0.766325i	$-0.277917\pi$
$379$	25.0145	1.28491	0.642453	+	0.766325i	$0.277917\pi$
$380$	0	0
$381$	−6.43569	−0.329710
$382$	0	0
$383$	7.08191	0.361869	0.180934	−	0.983495i	$-0.442088\pi$
$383$	7.08191	0.361869	0.180934	+	0.983495i	$0.442088\pi$
$384$	0	0
$385$	10.9524	0.558186
$386$	0	0
$387$	3.72762	0.189486
$388$	0	0
$389$	6.91327	0.350517	0.175258	−	0.984522i	$-0.443924\pi$
$389$	6.91327	0.350517	0.175258	+	0.984522i	$0.443924\pi$
$390$	0	0
$391$	−0.393320	−0.0198910
$392$	0	0
$393$	9.81384	0.495043
$394$	0	0
$395$	−6.93981	−0.349180
$396$	0	0
$397$	−17.9404	−0.900402	−0.450201	−	0.892927i	$-0.648648\pi$
$397$	−17.9404	−0.900402	−0.450201	+	0.892927i	$0.648648\pi$
$398$	0	0
$399$	11.0471	0.553044
$400$	0	0
$401$	36.9089	1.84314	0.921571	−	0.388210i	$-0.126906\pi$
$401$	36.9089	1.84314	0.921571	+	0.388210i	$0.126906\pi$
$402$	0	0
$403$	53.4838	2.66422
$404$	0	0
$405$	1.59293	0.0791535
$406$	0	0
$407$	−9.12980	−0.452547
$408$	0	0
$409$	−21.9253	−1.08413	−0.542067	−	0.840335i	$-0.682358\pi$
$409$	−21.9253	−1.08413	−0.542067	+	0.840335i	$0.682358\pi$
$410$	0	0
$411$	−6.49231	−0.320242
$412$	0	0
$413$	41.7959	2.05664
$414$	0	0
$415$	−7.21436	−0.354139
$416$	0	0
$417$	0.811714	0.0397498
$418$	0	0
$419$	0.589779	0.0288126	0.0144063	−	0.999896i	$-0.495414\pi$
$419$	0.589779	0.0288126	0.0144063	+	0.999896i	$0.495414\pi$
$420$	0	0
$421$	−8.11577	−0.395539	−0.197769	−	0.980249i	$-0.563370\pi$
$421$	−8.11577	−0.395539	−0.197769	+	0.980249i	$0.563370\pi$
$422$	0	0
$423$	1.48865	0.0723808
$424$	0	0
$425$	−0.968576	−0.0469828
$426$	0	0
$427$	26.7751	1.29574
$428$	0	0
$429$	−15.2748	−0.737473
$430$	0	0
$431$	−6.67268	−0.321412	−0.160706	−	0.987002i	$-0.551377\pi$
$431$	−6.67268	−0.321412	−0.160706	+	0.987002i	$0.551377\pi$
$432$	0	0
$433$	17.4815	0.840106	0.420053	−	0.907500i	$-0.362012\pi$
$433$	17.4815	0.840106	0.420053	+	0.907500i	$0.362012\pi$
$434$	0	0
$435$	1.59293	0.0763753
$436$	0	0
$437$	3.65181	0.174690
$438$	0	0
$439$	−19.7055	−0.940492	−0.470246	−	0.882535i	$-0.655835\pi$
$439$	−19.7055	−0.940492	−0.470246	+	0.882535i	$0.655835\pi$
$440$	0	0
$441$	2.15118	0.102437
$442$	0	0
$443$	0.251744	0.0119607	0.00598036	−	0.999982i	$-0.498096\pi$
$443$	0.251744	0.0119607	0.00598036	+	0.999982i	$0.498096\pi$
$444$	0	0
$445$	15.5311	0.736245
$446$	0	0
$447$	14.1125	0.667500
$448$	0	0
$449$	−27.6358	−1.30421	−0.652107	−	0.758127i	$-0.726115\pi$
$449$	−27.6358	−1.30421	−0.652107	+	0.758127i	$0.726115\pi$
$450$	0	0
$451$	11.7742	0.554425
$452$	0	0
$453$	17.0350	0.800376
$454$	0	0
$455$	32.3846	1.51821
$456$	0	0
$457$	22.6077	1.05754	0.528771	−	0.848765i	$-0.322653\pi$
$457$	22.6077	1.05754	0.528771	+	0.848765i	$0.322653\pi$
$458$	0	0
$459$	−0.393320	−0.0183586
$460$	0	0
$461$	5.49742	0.256040	0.128020	−	0.991772i	$-0.459138\pi$
$461$	5.49742	0.256040	0.128020	+	0.991772i	$0.459138\pi$
$462$	0	0
$463$	−42.2292	−1.96256	−0.981279	−	0.192591i	$-0.938311\pi$
$463$	−42.2292	−1.96256	−0.981279	+	0.192591i	$0.938311\pi$
$464$	0	0
$465$	−12.6770	−0.587884
$466$	0	0
$467$	0.358764	0.0166016	0.00830080	−	0.999966i	$-0.497358\pi$
$467$	0.358764	0.0166016	0.00830080	+	0.999966i	$0.497358\pi$
$468$	0	0
$469$	−6.33179	−0.292375
$470$	0	0
$471$	−11.2155	−0.516784
$472$	0	0
$473$	8.47237	0.389560
$474$	0	0
$475$	8.99282	0.412619
$476$	0	0
$477$	6.75379	0.309235
$478$	0	0
$479$	−5.88797	−0.269028	−0.134514	−	0.990912i	$-0.542947\pi$
$479$	−5.88797	−0.269028	−0.134514	+	0.990912i	$0.542947\pi$
$480$	0	0
$481$	−26.9954	−1.23089
$482$	0	0
$483$	3.02509	0.137646
$484$	0	0
$485$	8.20977	0.372787
$486$	0	0
$487$	17.5314	0.794421	0.397211	−	0.917727i	$-0.369978\pi$
$487$	17.5314	0.794421	0.397211	+	0.917727i	$0.369978\pi$
$488$	0	0
$489$	15.0129	0.678908
$490$	0	0
$491$	12.0120	0.542093	0.271046	−	0.962566i	$-0.412630\pi$
$491$	12.0120	0.542093	0.271046	+	0.962566i	$0.412630\pi$
$492$	0	0
$493$	−0.393320	−0.0177142
$494$	0	0
$495$	3.62051	0.162730
$496$	0	0
$497$	−1.43814	−0.0645092
$498$	0	0
$499$	24.4220	1.09328	0.546640	−	0.837367i	$-0.315906\pi$
$499$	24.4220	1.09328	0.546640	+	0.837367i	$0.315906\pi$
$500$	0	0
$501$	−6.33177	−0.282883
$502$	0	0
$503$	−2.00288	−0.0893038	−0.0446519	−	0.999003i	$-0.514218\pi$
$503$	−2.00288	−0.0893038	−0.0446519	+	0.999003i	$0.514218\pi$
$504$	0	0
$505$	7.82874	0.348374
$506$	0	0
$507$	−32.1652	−1.42851
$508$	0	0
$509$	−26.5246	−1.17568	−0.587842	−	0.808976i	$-0.700022\pi$
$509$	−26.5246	−1.17568	−0.587842	+	0.808976i	$0.700022\pi$
$510$	0	0
$511$	−12.4914	−0.552589
$512$	0	0
$513$	3.65181	0.161231
$514$	0	0
$515$	−24.5410	−1.08140
$516$	0	0
$517$	3.38350	0.148806
$518$	0	0
$519$	−15.5100	−0.680813
$520$	0	0
$521$	−13.2734	−0.581520	−0.290760	−	0.956796i	$-0.593908\pi$
$521$	−13.2734	−0.581520	−0.290760	+	0.956796i	$0.593908\pi$
$522$	0	0
$523$	−16.3181	−0.713539	−0.356769	−	0.934192i	$-0.616122\pi$
$523$	−16.3181	−0.713539	−0.356769	+	0.934192i	$0.616122\pi$
$524$	0	0
$525$	7.44949	0.325122
$526$	0	0
$527$	3.13016	0.136352
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	13.8164	0.599581
$532$	0	0
$533$	34.8145	1.50798
$534$	0	0
$535$	−22.0275	−0.952332
$536$	0	0
$537$	16.3219	0.704344
$538$	0	0
$539$	4.88934	0.210599
$540$	0	0
$541$	−5.52434	−0.237510	−0.118755	−	0.992924i	$-0.537890\pi$
$541$	−5.52434	−0.237510	−0.118755	+	0.992924i	$0.537890\pi$
$542$	0	0
$543$	10.4126	0.446849
$544$	0	0
$545$	−23.6750	−1.01413
$546$	0	0
$547$	−18.2169	−0.778899	−0.389450	−	0.921048i	$-0.627335\pi$
$547$	−18.2169	−0.778899	−0.389450	+	0.921048i	$0.627335\pi$
$548$	0	0
$549$	8.85099	0.377751
$550$	0	0
$551$	3.65181	0.155572
$552$	0	0
$553$	−13.1792	−0.560436
$554$	0	0
$555$	6.39861	0.271606
$556$	0	0
$557$	5.08038	0.215263	0.107631	−	0.994191i	$-0.465673\pi$
$557$	5.08038	0.215263	0.107631	+	0.994191i	$0.465673\pi$
$558$	0	0
$559$	25.0515	1.05957
$560$	0	0
$561$	−0.893961	−0.0377431
$562$	0	0
$563$	−35.8223	−1.50973	−0.754865	−	0.655880i	$-0.772298\pi$
$563$	−35.8223	−1.50973	−0.754865	+	0.655880i	$0.772298\pi$
$564$	0	0
$565$	16.9051	0.711202
$566$	0	0
$567$	3.02509	0.127042
$568$	0	0
$569$	−15.7356	−0.659669	−0.329834	−	0.944039i	$-0.606993\pi$
$569$	−15.7356	−0.659669	−0.329834	+	0.944039i	$0.606993\pi$
$570$	0	0
$571$	−5.94280	−0.248699	−0.124349	−	0.992239i	$-0.539684\pi$
$571$	−5.94280	−0.248699	−0.124349	+	0.992239i	$0.539684\pi$
$572$	0	0
$573$	−5.18679	−0.216681
$574$	0	0
$575$	2.46257	0.102696
$576$	0	0
$577$	−10.9888	−0.457469	−0.228735	−	0.973489i	$-0.573459\pi$
$577$	−10.9888	−0.457469	−0.228735	+	0.973489i	$0.573459\pi$
$578$	0	0
$579$	8.40729	0.349395
$580$	0	0
$581$	−13.7006	−0.568396
$582$	0	0
$583$	15.3504	0.635750
$584$	0	0
$585$	10.7053	0.442610
$586$	0	0
$587$	−2.98785	−0.123322	−0.0616609	−	0.998097i	$-0.519640\pi$
$587$	−2.98785	−0.123322	−0.0616609	+	0.998097i	$0.519640\pi$
$588$	0	0
$589$	−29.0622	−1.19749
$590$	0	0
$591$	−15.7786	−0.649043
$592$	0	0
$593$	−29.1607	−1.19749	−0.598743	−	0.800941i	$-0.704333\pi$
$593$	−29.1607	−1.19749	−0.598743	+	0.800941i	$0.704333\pi$
$594$	0	0
$595$	1.89532	0.0777004
$596$	0	0
$597$	−9.77795	−0.400185
$598$	0	0
$599$	12.9779	0.530263	0.265131	−	0.964212i	$-0.414585\pi$
$599$	12.9779	0.530263	0.265131	+	0.964212i	$0.414585\pi$
$600$	0	0
$601$	−10.4006	−0.424250	−0.212125	−	0.977242i	$-0.568038\pi$
$601$	−10.4006	−0.424250	−0.212125	+	0.977242i	$0.568038\pi$
$602$	0	0
$603$	−2.09309	−0.0852372
$604$	0	0
$605$	−9.29333	−0.377828
$606$	0	0
$607$	−2.79225	−0.113334	−0.0566669	−	0.998393i	$-0.518047\pi$
$607$	−2.79225	−0.113334	−0.0566669	+	0.998393i	$0.518047\pi$
$608$	0	0
$609$	3.02509	0.122583
$610$	0	0
$611$	10.0045	0.404739
$612$	0	0
$613$	−2.81573	−0.113726	−0.0568632	−	0.998382i	$-0.518110\pi$
$613$	−2.81573	−0.113726	−0.0568632	+	0.998382i	$0.518110\pi$
$614$	0	0
$615$	−8.25193	−0.332750
$616$	0	0
$617$	9.31011	0.374811	0.187405	−	0.982283i	$-0.439992\pi$
$617$	9.31011	0.374811	0.187405	+	0.982283i	$0.439992\pi$
$618$	0	0
$619$	16.9065	0.679528	0.339764	−	0.940511i	$-0.389653\pi$
$619$	16.9065	0.679528	0.339764	+	0.940511i	$0.389653\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	29.4947	1.18168
$624$	0	0
$625$	−6.62294	−0.264918
$626$	0	0
$627$	8.30005	0.331472
$628$	0	0
$629$	−1.57992	−0.0629954
$630$	0	0
$631$	−42.2196	−1.68073	−0.840367	−	0.542018i	$-0.817661\pi$
$631$	−42.2196	−1.68073	−0.840367	+	0.542018i	$0.817661\pi$
$632$	0	0
$633$	25.2712	1.00444
$634$	0	0
$635$	10.2516	0.406823
$636$	0	0
$637$	14.4570	0.572808
$638$	0	0
$639$	−0.475402	−0.0188066
$640$	0	0
$641$	−7.18655	−0.283852	−0.141926	−	0.989877i	$-0.545329\pi$
$641$	−7.18655	−0.283852	−0.141926	+	0.989877i	$0.545329\pi$
$642$	0	0
$643$	35.3534	1.39420	0.697101	−	0.716973i	$-0.254473\pi$
$643$	35.3534	1.39420	0.697101	+	0.716973i	$0.254473\pi$
$644$	0	0
$645$	−5.93785	−0.233803
$646$	0	0
$647$	5.08926	0.200079	0.100040	−	0.994983i	$-0.468103\pi$
$647$	5.08926	0.200079	0.100040	+	0.994983i	$0.468103\pi$
$648$	0	0
$649$	31.4028	1.23267
$650$	0	0
$651$	−24.0746	−0.943558
$652$	0	0
$653$	50.2303	1.96566	0.982832	−	0.184502i	$-0.0590673\pi$
$653$	50.2303	1.96566	0.982832	+	0.184502i	$0.0590673\pi$
$654$	0	0
$655$	−15.6328	−0.610824
$656$	0	0
$657$	−4.12927	−0.161098
$658$	0	0
$659$	−31.9896	−1.24614	−0.623069	−	0.782167i	$-0.714114\pi$
$659$	−31.9896	−1.24614	−0.623069	+	0.782167i	$0.714114\pi$
$660$	0	0
$661$	20.3109	0.790001	0.395000	−	0.918681i	$-0.370744\pi$
$661$	20.3109	0.790001	0.395000	+	0.918681i	$0.370744\pi$
$662$	0	0
$663$	−2.64331	−0.102658
$664$	0	0
$665$	−17.5972	−0.682391
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−7.56939	−0.292649
$670$	0	0
$671$	20.1171	0.776611
$672$	0	0
$673$	31.6281	1.21917	0.609586	−	0.792720i	$-0.291336\pi$
$673$	31.6281	1.21917	0.609586	+	0.792720i	$0.291336\pi$
$674$	0	0
$675$	2.46257	0.0947842
$676$	0	0
$677$	47.9089	1.84129	0.920644	−	0.390403i	$-0.127664\pi$
$677$	47.9089	1.84129	0.920644	+	0.390403i	$0.127664\pi$
$678$	0	0
$679$	15.5909	0.598325
$680$	0	0
$681$	4.15942	0.159389
$682$	0	0
$683$	−3.50159	−0.133985	−0.0669923	−	0.997753i	$-0.521340\pi$
$683$	−3.50159	−0.133985	−0.0669923	+	0.997753i	$0.521340\pi$
$684$	0	0
$685$	10.3418	0.395141
$686$	0	0
$687$	−13.3711	−0.510140
$688$	0	0
$689$	45.3889	1.72918
$690$	0	0
$691$	6.80249	0.258779	0.129390	−	0.991594i	$-0.458698\pi$
$691$	6.80249	0.258779	0.129390	+	0.991594i	$0.458698\pi$
$692$	0	0
$693$	6.87561	0.261183
$694$	0	0
$695$	−1.29301	−0.0490465
$696$	0	0
$697$	2.03753	0.0771769
$698$	0	0
$699$	27.4032	1.03648
$700$	0	0
$701$	7.57172	0.285980	0.142990	−	0.989724i	$-0.454328\pi$
$701$	7.57172	0.285980	0.142990	+	0.989724i	$0.454328\pi$
$702$	0	0
$703$	14.6689	0.553246
$704$	0	0
$705$	−2.37132	−0.0893092
$706$	0	0
$707$	14.8673	0.559143
$708$	0	0
$709$	−43.5554	−1.63576	−0.817880	−	0.575389i	$-0.804851\pi$
$709$	−43.5554	−1.63576	−0.817880	+	0.575389i	$0.804851\pi$
$710$	0	0
$711$	−4.35662	−0.163386
$712$	0	0
$713$	−7.95831	−0.298041
$714$	0	0
$715$	24.3317	0.909953
$716$	0	0
$717$	13.1708	0.491872
$718$	0	0
$719$	−22.6035	−0.842968	−0.421484	−	0.906836i	$-0.638491\pi$
$719$	−22.6035	−0.842968	−0.421484	+	0.906836i	$0.638491\pi$
$720$	0	0
$721$	−46.6050	−1.73566
$722$	0	0
$723$	−15.2578	−0.567445
$724$	0	0
$725$	2.46257	0.0914574
$726$	0	0
$727$	48.2501	1.78950	0.894748	−	0.446571i	$-0.147355\pi$
$727$	48.2501	1.78950	0.894748	+	0.446571i	$0.147355\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.46615	0.0542274
$732$	0	0
$733$	25.1376	0.928478	0.464239	−	0.885710i	$-0.346328\pi$
$733$	25.1376	0.928478	0.464239	+	0.885710i	$0.346328\pi$
$734$	0	0
$735$	−3.42669	−0.126395
$736$	0	0
$737$	−4.75730	−0.175237
$738$	0	0
$739$	−24.3545	−0.895896	−0.447948	−	0.894060i	$-0.647845\pi$
$739$	−24.3545	−0.895896	−0.447948	+	0.894060i	$0.647845\pi$
$740$	0	0
$741$	24.5420	0.901573
$742$	0	0
$743$	−2.17696	−0.0798648	−0.0399324	−	0.999202i	$-0.512714\pi$
$743$	−2.17696	−0.0798648	−0.0399324	+	0.999202i	$0.512714\pi$
$744$	0	0
$745$	−22.4803	−0.823615
$746$	0	0
$747$	−4.52898	−0.165707
$748$	0	0
$749$	−41.8318	−1.52850
$750$	0	0
$751$	−28.0053	−1.02193	−0.510964	−	0.859602i	$-0.670711\pi$
$751$	−28.0053	−1.02193	−0.510964	+	0.859602i	$0.670711\pi$
$752$	0	0
$753$	−21.1598	−0.771106
$754$	0	0
$755$	−27.1357	−0.987568
$756$	0	0
$757$	23.2265	0.844180	0.422090	−	0.906554i	$-0.361297\pi$
$757$	23.2265	0.844180	0.422090	+	0.906554i	$0.361297\pi$
$758$	0	0
$759$	2.27286	0.0824996
$760$	0	0
$761$	47.0553	1.70575	0.852877	−	0.522111i	$-0.174855\pi$
$761$	47.0553	1.70575	0.852877	+	0.522111i	$0.174855\pi$
$762$	0	0
$763$	−44.9605	−1.62768
$764$	0	0
$765$	0.626532	0.0226523
$766$	0	0
$767$	92.8532	3.35273
$768$	0	0
$769$	−32.9177	−1.18704	−0.593521	−	0.804819i	$-0.702262\pi$
$769$	−32.9177	−1.18704	−0.593521	+	0.804819i	$0.702262\pi$
$770$	0	0
$771$	−2.85039	−0.102654
$772$	0	0
$773$	−19.5165	−0.701960	−0.350980	−	0.936383i	$-0.614151\pi$
$773$	−19.5165	−0.701960	−0.350980	+	0.936383i	$0.614151\pi$
$774$	0	0
$775$	−19.5979	−0.703976
$776$	0	0
$777$	12.1514	0.435930
$778$	0	0
$779$	−18.9176	−0.677793
$780$	0	0
$781$	−1.08052	−0.0386642
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	17.8656	0.637650
$786$	0	0
$787$	11.7164	0.417644	0.208822	−	0.977954i	$-0.433037\pi$
$787$	11.7164	0.417644	0.208822	+	0.977954i	$0.433037\pi$
$788$	0	0
$789$	−12.3594	−0.440005
$790$	0	0
$791$	32.1039	1.14148
$792$	0	0
$793$	59.4831	2.11231
$794$	0	0
$795$	−10.7583	−0.381559
$796$	0	0
$797$	1.65264	0.0585395	0.0292698	−	0.999572i	$-0.490682\pi$
$797$	1.65264	0.0585395	0.0292698	+	0.999572i	$0.490682\pi$
$798$	0	0
$799$	0.585517	0.0207141
$800$	0	0
$801$	9.75001	0.344500
$802$	0	0
$803$	−9.38527	−0.331199
$804$	0	0
$805$	−4.81877	−0.169839
$806$	0	0
$807$	−29.1296	−1.02541
$808$	0	0
$809$	−28.3685	−0.997382	−0.498691	−	0.866780i	$-0.666186\pi$
$809$	−28.3685	−0.997382	−0.498691	+	0.866780i	$0.666186\pi$
$810$	0	0
$811$	−14.4077	−0.505923	−0.252961	−	0.967476i	$-0.581405\pi$
$811$	−14.4077	−0.505923	−0.252961	+	0.967476i	$0.581405\pi$
$812$	0	0
$813$	−14.3718	−0.504040
$814$	0	0
$815$	−23.9146	−0.837692
$816$	0	0
$817$	−13.6126	−0.476243
$818$	0	0
$819$	20.3301	0.710393
$820$	0	0
$821$	25.2639	0.881716	0.440858	−	0.897577i	$-0.354674\pi$
$821$	25.2639	0.881716	0.440858	+	0.897577i	$0.354674\pi$
$822$	0	0
$823$	−34.7557	−1.21151	−0.605754	−	0.795652i	$-0.707128\pi$
$823$	−34.7557	−1.21151	−0.605754	+	0.795652i	$0.707128\pi$
$824$	0	0
$825$	5.59707	0.194865
$826$	0	0
$827$	−26.5233	−0.922305	−0.461152	−	0.887321i	$-0.652564\pi$
$827$	−26.5233	−0.922305	−0.461152	+	0.887321i	$0.652564\pi$
$828$	0	0
$829$	4.29688	0.149237	0.0746185	−	0.997212i	$-0.476226\pi$
$829$	4.29688	0.149237	0.0746185	+	0.997212i	$0.476226\pi$
$830$	0	0
$831$	−22.6868	−0.786997
$832$	0	0
$833$	0.846102	0.0293157
$834$	0	0
$835$	10.0861	0.349043
$836$	0	0
$837$	−7.95831	−0.275079
$838$	0	0
$839$	47.4571	1.63840	0.819201	−	0.573507i	$-0.194417\pi$
$839$	47.4571	1.63840	0.819201	+	0.573507i	$0.194417\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	19.0446	0.655932
$844$	0	0
$845$	51.2370	1.76261
$846$	0	0
$847$	−17.6487	−0.606416
$848$	0	0
$849$	11.9840	0.411290
$850$	0	0
$851$	4.01687	0.137697
$852$	0	0
$853$	50.8015	1.73941	0.869705	−	0.493571i	$-0.164309\pi$
$853$	50.8015	1.73941	0.869705	+	0.493571i	$0.164309\pi$
$854$	0	0
$855$	−5.81708	−0.198940
$856$	0	0
$857$	24.0135	0.820285	0.410142	−	0.912022i	$-0.365479\pi$
$857$	24.0135	0.820285	0.410142	+	0.912022i	$0.365479\pi$
$858$	0	0
$859$	−39.5855	−1.35064	−0.675321	−	0.737524i	$-0.735995\pi$
$859$	−39.5855	−1.35064	−0.675321	+	0.737524i	$0.735995\pi$
$860$	0	0
$861$	−15.6710	−0.534066
$862$	0	0
$863$	44.2743	1.50711	0.753557	−	0.657382i	$-0.228336\pi$
$863$	44.2743	1.50711	0.753557	+	0.657382i	$0.228336\pi$
$864$	0	0
$865$	24.7064	0.840042
$866$	0	0
$867$	16.8453	0.572096
$868$	0	0
$869$	−9.90200	−0.335902
$870$	0	0
$871$	−14.0666	−0.476629
$872$	0	0
$873$	5.15387	0.174432
$874$	0	0
$875$	−35.9604	−1.21568
$876$	0	0
$877$	4.39672	0.148467	0.0742334	−	0.997241i	$-0.476349\pi$
$877$	4.39672	0.148467	0.0742334	+	0.997241i	$0.476349\pi$
$878$	0	0
$879$	13.8173	0.466046
$880$	0	0
$881$	33.1604	1.11720	0.558601	−	0.829436i	$-0.311338\pi$
$881$	33.1604	1.11720	0.558601	+	0.829436i	$0.311338\pi$
$882$	0	0
$883$	14.3455	0.482765	0.241383	−	0.970430i	$-0.422399\pi$
$883$	14.3455	0.482765	0.241383	+	0.970430i	$0.422399\pi$
$884$	0	0
$885$	−22.0086	−0.739811
$886$	0	0
$887$	−1.66238	−0.0558172	−0.0279086	−	0.999610i	$-0.508885\pi$
$887$	−1.66238	−0.0558172	−0.0279086	+	0.999610i	$0.508885\pi$
$888$	0	0
$889$	19.4685	0.652954
$890$	0	0
$891$	2.27286	0.0761437
$892$	0	0
$893$	−5.43627	−0.181918
$894$	0	0
$895$	−25.9998	−0.869076
$896$	0	0
$897$	6.72050	0.224391
$898$	0	0
$899$	−7.95831	−0.265424
$900$	0	0
$901$	2.65640	0.0884975
$902$	0	0
$903$	−11.2764	−0.375255
$904$	0	0
$905$	−16.5866	−0.551358
$906$	0	0
$907$	17.3161	0.574973	0.287486	−	0.957785i	$-0.407180\pi$
$907$	17.3161	0.574973	0.287486	+	0.957785i	$0.407180\pi$
$908$	0	0
$909$	4.91467	0.163009
$910$	0	0
$911$	22.8472	0.756963	0.378481	−	0.925609i	$-0.376446\pi$
$911$	22.8472	0.756963	0.378481	+	0.925609i	$0.376446\pi$
$912$	0	0
$913$	−10.2937	−0.340673
$914$	0	0
$915$	−14.0990	−0.466100
$916$	0	0
$917$	−29.6878	−0.980377
$918$	0	0
$919$	−25.6199	−0.845122	−0.422561	−	0.906335i	$-0.638869\pi$
$919$	−25.6199	−0.845122	−0.422561	+	0.906335i	$0.638869\pi$
$920$	0	0
$921$	16.6941	0.550090
$922$	0	0
$923$	−3.19494	−0.105163
$924$	0	0
$925$	9.89182	0.325241
$926$	0	0
$927$	−15.4061	−0.506004
$928$	0	0
$929$	48.7579	1.59970	0.799848	−	0.600203i	$-0.204914\pi$
$929$	48.7579	1.59970	0.799848	+	0.600203i	$0.204914\pi$
$930$	0	0
$931$	−7.85570	−0.257460
$932$	0	0
$933$	−7.13885	−0.233715
$934$	0	0
$935$	1.42402	0.0465704
$936$	0	0
$937$	15.1713	0.495624	0.247812	−	0.968808i	$-0.420288\pi$
$937$	15.1713	0.495624	0.247812	+	0.968808i	$0.420288\pi$
$938$	0	0
$939$	13.2545	0.432543
$940$	0	0
$941$	47.6056	1.55190	0.775948	−	0.630796i	$-0.217272\pi$
$941$	47.6056	1.55190	0.775948	+	0.630796i	$0.217272\pi$
$942$	0	0
$943$	−5.18034	−0.168695
$944$	0	0
$945$	−4.81877	−0.156755
$946$	0	0
$947$	−54.2595	−1.76320	−0.881598	−	0.472001i	$-0.843532\pi$
$947$	−54.2595	−1.76320	−0.881598	+	0.472001i	$0.843532\pi$
$948$	0	0
$949$	−27.7508	−0.900829
$950$	0	0
$951$	−8.67751	−0.281388
$952$	0	0
$953$	38.1626	1.23621	0.618104	−	0.786096i	$-0.287901\pi$
$953$	38.1626	1.23621	0.618104	+	0.786096i	$0.287901\pi$
$954$	0	0
$955$	8.26221	0.267359
$956$	0	0
$957$	2.27286	0.0734712
$958$	0	0
$959$	19.6398	0.634204
$960$	0	0
$961$	32.3347	1.04305
$962$	0	0
$963$	−13.8283	−0.445610
$964$	0	0
$965$	−13.3923	−0.431112
$966$	0	0
$967$	−31.7841	−1.02211	−0.511054	−	0.859549i	$-0.670745\pi$
$967$	−31.7841	−1.02211	−0.511054	+	0.859549i	$0.670745\pi$
$968$	0	0
$969$	1.43633	0.0461415
$970$	0	0
$971$	0.0667786	0.00214303	0.00107151	−	0.999999i	$-0.499659\pi$
$971$	0.0667786	0.00214303	0.00107151	+	0.999999i	$0.499659\pi$
$972$	0	0
$973$	−2.45551	−0.0787200
$974$	0	0
$975$	16.5497	0.530014
$976$	0	0
$977$	−24.3839	−0.780109	−0.390054	−	0.920792i	$-0.627544\pi$
$977$	−24.3839	−0.780109	−0.390054	+	0.920792i	$0.627544\pi$
$978$	0	0
$979$	22.1604	0.708250
$980$	0	0
$981$	−14.8625	−0.474524
$982$	0	0
$983$	18.8181	0.600203	0.300101	−	0.953907i	$-0.402979\pi$
$983$	18.8181	0.600203	0.300101	+	0.953907i	$0.402979\pi$
$984$	0	0
$985$	25.1342	0.800842
$986$	0	0
$987$	−4.50331	−0.143342
$988$	0	0
$989$	−3.72762	−0.118531
$990$	0	0
$991$	−0.338307	−0.0107467	−0.00537334	−	0.999986i	$-0.501710\pi$
$991$	−0.338307	−0.0107467	−0.00537334	+	0.999986i	$0.501710\pi$
$992$	0	0
$993$	1.92292	0.0610222
$994$	0	0
$995$	15.5756	0.493780
$996$	0	0
$997$	30.3875	0.962381	0.481190	−	0.876616i	$-0.340205\pi$
$997$	30.3875	0.962381	0.481190	+	0.876616i	$0.340205\pi$
$998$	0	0
$999$	4.01687	0.127088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.h.1.8	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.h.1.8	✓	13	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.59293 q^{5} +3.02509 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.59293 q^{5} +3.02509 q^{7} +1.00000 q^{9} +2.27286 q^{11} +6.72050 q^{13} -1.59293 q^{15} +0.393320 q^{17} -3.65181 q^{19} -3.02509 q^{21} -1.00000 q^{23} -2.46257 q^{25} -1.00000 q^{27} -1.00000 q^{29} +7.95831 q^{31} -2.27286 q^{33} +4.81877 q^{35} -4.01687 q^{37} -6.72050 q^{39} +5.18034 q^{41} +3.72762 q^{43} +1.59293 q^{45} +1.48865 q^{47} +2.15118 q^{49} -0.393320 q^{51} +6.75379 q^{53} +3.62051 q^{55} +3.65181 q^{57} +13.8164 q^{59} +8.85099 q^{61} +3.02509 q^{63} +10.7053 q^{65} -2.09309 q^{67} +1.00000 q^{69} -0.475402 q^{71} -4.12927 q^{73} +2.46257 q^{75} +6.87561 q^{77} -4.35662 q^{79} +1.00000 q^{81} -4.52898 q^{83} +0.626532 q^{85} +1.00000 q^{87} +9.75001 q^{89} +20.3301 q^{91} -7.95831 q^{93} -5.81708 q^{95} +5.15387 q^{97} +2.27286 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

Introduction

L-functions

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