LMFDB - Embedded newform 8004.2.a.j.1.9

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:	$16$
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} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.187793$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.187793 q^{5} +3.11113 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.187793 q^{5} +3.11113 q^{7} +1.00000 q^{9} +0.0152079 q^{11} +5.09199 q^{13} +0.187793 q^{15} +2.80593 q^{17} +4.44446 q^{19} +3.11113 q^{21} +1.00000 q^{23} -4.96473 q^{25} +1.00000 q^{27} -1.00000 q^{29} -3.85168 q^{31} +0.0152079 q^{33} +0.584250 q^{35} -2.44794 q^{37} +5.09199 q^{39} +4.07580 q^{41} -4.10517 q^{43} +0.187793 q^{45} +10.7652 q^{47} +2.67915 q^{49} +2.80593 q^{51} +8.69169 q^{53} +0.00285595 q^{55} +4.44446 q^{57} -8.68248 q^{59} +2.99638 q^{61} +3.11113 q^{63} +0.956242 q^{65} +11.2830 q^{67} +1.00000 q^{69} +4.10798 q^{71} +12.5759 q^{73} -4.96473 q^{75} +0.0473140 q^{77} +6.51135 q^{79} +1.00000 q^{81} -4.47591 q^{83} +0.526935 q^{85} -1.00000 q^{87} -9.25893 q^{89} +15.8419 q^{91} -3.85168 q^{93} +0.834639 q^{95} +7.78248 q^{97} +0.0152079 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * 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
$5$	0.187793	0.0839837	0.0419919	−	0.999118i	$-0.486630\pi$
$5$	0.187793	0.0839837	0.0419919	+	0.999118i	$0.486630\pi$
$6$	0	0
$7$	3.11113	1.17590	0.587949	−	0.808898i	$-0.299936\pi$
$7$	3.11113	1.17590	0.587949	+	0.808898i	$0.299936\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.0152079	0.00458537	0.00229268	−	0.999997i	$-0.499270\pi$
$11$	0.0152079	0.00458537	0.00229268	+	0.999997i	$0.499270\pi$
$12$	0	0
$13$	5.09199	1.41226	0.706132	−	0.708080i	$-0.250438\pi$
$13$	5.09199	1.41226	0.706132	+	0.708080i	$0.250438\pi$
$14$	0	0
$15$	0.187793	0.0484880
$16$	0	0
$17$	2.80593	0.680538	0.340269	−	0.940328i	$-0.389482\pi$
$17$	2.80593	0.680538	0.340269	+	0.940328i	$0.389482\pi$
$18$	0	0
$19$	4.44446	1.01963	0.509814	−	0.860285i	$-0.329714\pi$
$19$	4.44446	1.01963	0.509814	+	0.860285i	$0.329714\pi$
$20$	0	0
$21$	3.11113	0.678905
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.96473	−0.992947
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.85168	−0.691783	−0.345891	−	0.938275i	$-0.612423\pi$
$31$	−3.85168	−0.691783	−0.345891	+	0.938275i	$0.612423\pi$
$32$	0	0
$33$	0.0152079	0.00264736
$34$	0	0
$35$	0.584250	0.0987563
$36$	0	0
$37$	−2.44794	−0.402439	−0.201220	−	0.979546i	$-0.564491\pi$
$37$	−2.44794	−0.402439	−0.201220	+	0.979546i	$0.564491\pi$
$38$	0	0
$39$	5.09199	0.815371
$40$	0	0
$41$	4.07580	0.636533	0.318267	−	0.948001i	$-0.396899\pi$
$41$	4.07580	0.636533	0.318267	+	0.948001i	$0.396899\pi$
$42$	0	0
$43$	−4.10517	−0.626032	−0.313016	−	0.949748i	$-0.601339\pi$
$43$	−4.10517	−0.626032	−0.313016	+	0.949748i	$0.601339\pi$
$44$	0	0
$45$	0.187793	0.0279946
$46$	0	0
$47$	10.7652	1.57026	0.785130	−	0.619331i	$-0.212596\pi$
$47$	10.7652	1.57026	0.785130	+	0.619331i	$0.212596\pi$
$48$	0	0
$49$	2.67915	0.382736
$50$	0	0
$51$	2.80593	0.392909
$52$	0	0
$53$	8.69169	1.19390	0.596948	−	0.802280i	$-0.296380\pi$
$53$	8.69169	1.19390	0.596948	+	0.802280i	$0.296380\pi$
$54$	0	0
$55$	0.00285595	0.000385096 0
$56$	0	0
$57$	4.44446	0.588683
$58$	0	0
$59$	−8.68248	−1.13036	−0.565181	−	0.824967i	$-0.691194\pi$
$59$	−8.68248	−1.13036	−0.565181	+	0.824967i	$0.691194\pi$
$60$	0	0
$61$	2.99638	0.383647	0.191824	−	0.981429i	$-0.438560\pi$
$61$	2.99638	0.383647	0.191824	+	0.981429i	$0.438560\pi$
$62$	0	0
$63$	3.11113	0.391966
$64$	0	0
$65$	0.956242	0.118607
$66$	0	0
$67$	11.2830	1.37844	0.689220	−	0.724552i	$-0.257953\pi$
$67$	11.2830	1.37844	0.689220	+	0.724552i	$0.257953\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	4.10798	0.487528	0.243764	−	0.969835i	$-0.421618\pi$
$71$	4.10798	0.487528	0.243764	+	0.969835i	$0.421618\pi$
$72$	0	0
$73$	12.5759	1.47190	0.735949	−	0.677037i	$-0.236737\pi$
$73$	12.5759	1.47190	0.735949	+	0.677037i	$0.236737\pi$
$74$	0	0
$75$	−4.96473	−0.573278
$76$	0	0
$77$	0.0473140	0.00539193
$78$	0	0
$79$	6.51135	0.732584	0.366292	−	0.930500i	$-0.380627\pi$
$79$	6.51135	0.732584	0.366292	+	0.930500i	$0.380627\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.47591	−0.491295	−0.245647	−	0.969359i	$-0.579001\pi$
$83$	−4.47591	−0.491295	−0.245647	+	0.969359i	$0.579001\pi$
$84$	0	0
$85$	0.526935	0.0571541
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−9.25893	−0.981444	−0.490722	−	0.871316i	$-0.663267\pi$
$89$	−9.25893	−0.981444	−0.490722	+	0.871316i	$0.663267\pi$
$90$	0	0
$91$	15.8419	1.66068
$92$	0	0
$93$	−3.85168	−0.399401
$94$	0	0
$95$	0.834639	0.0856322
$96$	0	0
$97$	7.78248	0.790191	0.395095	−	0.918640i	$-0.370711\pi$
$97$	7.78248	0.790191	0.395095	+	0.918640i	$0.370711\pi$
$98$	0	0
$99$	0.0152079	0.00152846
$100$	0	0
$101$	−6.72116	−0.668780	−0.334390	−	0.942435i	$-0.608530\pi$
$101$	−6.72116	−0.668780	−0.334390	+	0.942435i	$0.608530\pi$
$102$	0	0
$103$	−14.4939	−1.42812	−0.714061	−	0.700083i	$-0.753146\pi$
$103$	−14.4939	−1.42812	−0.714061	+	0.700083i	$0.753146\pi$
$104$	0	0
$105$	0.584250	0.0570170
$106$	0	0
$107$	−9.80146	−0.947543	−0.473771	−	0.880648i	$-0.657108\pi$
$107$	−9.80146	−0.947543	−0.473771	+	0.880648i	$0.657108\pi$
$108$	0	0
$109$	−3.75215	−0.359391	−0.179695	−	0.983722i	$-0.557511\pi$
$109$	−3.75215	−0.359391	−0.179695	+	0.983722i	$0.557511\pi$
$110$	0	0
$111$	−2.44794	−0.232348
$112$	0	0
$113$	−12.1826	−1.14604	−0.573022	−	0.819540i	$-0.694229\pi$
$113$	−12.1826	−1.14604	−0.573022	+	0.819540i	$0.694229\pi$
$114$	0	0
$115$	0.187793	0.0175118
$116$	0	0
$117$	5.09199	0.470755
$118$	0	0
$119$	8.72962	0.800243
$120$	0	0
$121$	−10.9998	−0.999979
$122$	0	0
$123$	4.07580	0.367503
$124$	0	0
$125$	−1.87131	−0.167375
$126$	0	0
$127$	−14.7848	−1.31194	−0.655968	−	0.754788i	$-0.727740\pi$
$127$	−14.7848	−1.31194	−0.655968	+	0.754788i	$0.727740\pi$
$128$	0	0
$129$	−4.10517	−0.361440
$130$	0	0
$131$	−21.0448	−1.83869	−0.919346	−	0.393449i	$-0.871282\pi$
$131$	−21.0448	−1.83869	−0.919346	+	0.393449i	$0.871282\pi$
$132$	0	0
$133$	13.8273	1.19898
$134$	0	0
$135$	0.187793	0.0161627
$136$	0	0
$137$	−18.4369	−1.57517	−0.787583	−	0.616208i	$-0.788668\pi$
$137$	−18.4369	−1.57517	−0.787583	+	0.616208i	$0.788668\pi$
$138$	0	0
$139$	19.3555	1.64171	0.820856	−	0.571135i	$-0.193497\pi$
$139$	19.3555	1.64171	0.820856	+	0.571135i	$0.193497\pi$
$140$	0	0
$141$	10.7652	0.906590
$142$	0	0
$143$	0.0774388	0.00647575
$144$	0	0
$145$	−0.187793	−0.0155954
$146$	0	0
$147$	2.67915	0.220973
$148$	0	0
$149$	−5.36191	−0.439265	−0.219633	−	0.975583i	$-0.570486\pi$
$149$	−5.36191	−0.439265	−0.219633	+	0.975583i	$0.570486\pi$
$150$	0	0
$151$	4.92971	0.401174	0.200587	−	0.979676i	$-0.435715\pi$
$151$	4.92971	0.401174	0.200587	+	0.979676i	$0.435715\pi$
$152$	0	0
$153$	2.80593	0.226846
$154$	0	0
$155$	−0.723321	−0.0580985
$156$	0	0
$157$	14.5834	1.16388	0.581942	−	0.813230i	$-0.302293\pi$
$157$	14.5834	1.16388	0.581942	+	0.813230i	$0.302293\pi$
$158$	0	0
$159$	8.69169	0.689296
$160$	0	0
$161$	3.11113	0.245192
$162$	0	0
$163$	9.81554	0.768813	0.384406	−	0.923164i	$-0.374406\pi$
$163$	9.81554	0.768813	0.384406	+	0.923164i	$0.374406\pi$
$164$	0	0
$165$	0.00285595	0.000222336 0
$166$	0	0
$167$	15.8044	1.22298	0.611490	−	0.791252i	$-0.290570\pi$
$167$	15.8044	1.22298	0.611490	+	0.791252i	$0.290570\pi$
$168$	0	0
$169$	12.9284	0.994491
$170$	0	0
$171$	4.44446	0.339876
$172$	0	0
$173$	−3.48262	−0.264778	−0.132389	−	0.991198i	$-0.542265\pi$
$173$	−3.48262	−0.264778	−0.132389	+	0.991198i	$0.542265\pi$
$174$	0	0
$175$	−15.4459	−1.16760
$176$	0	0
$177$	−8.68248	−0.652615
$178$	0	0
$179$	−20.9975	−1.56942	−0.784712	−	0.619860i	$-0.787189\pi$
$179$	−20.9975	−1.56942	−0.784712	+	0.619860i	$0.787189\pi$
$180$	0	0
$181$	10.3852	0.771927	0.385964	−	0.922514i	$-0.373869\pi$
$181$	10.3852	0.771927	0.385964	+	0.922514i	$0.373869\pi$
$182$	0	0
$183$	2.99638	0.221499
$184$	0	0
$185$	−0.459707	−0.0337984
$186$	0	0
$187$	0.0426724	0.00312052
$188$	0	0
$189$	3.11113	0.226302
$190$	0	0
$191$	0.396221	0.0286695	0.0143348	−	0.999897i	$-0.495437\pi$
$191$	0.396221	0.0286695	0.0143348	+	0.999897i	$0.495437\pi$
$192$	0	0
$193$	3.34476	0.240761	0.120381	−	0.992728i	$-0.461589\pi$
$193$	3.34476	0.240761	0.120381	+	0.992728i	$0.461589\pi$
$194$	0	0
$195$	0.956242	0.0684779
$196$	0	0
$197$	−21.7540	−1.54991	−0.774955	−	0.632016i	$-0.782228\pi$
$197$	−21.7540	−1.54991	−0.774955	+	0.632016i	$0.782228\pi$
$198$	0	0
$199$	−27.1091	−1.92172	−0.960858	−	0.277041i	$-0.910646\pi$
$199$	−27.1091	−1.92172	−0.960858	+	0.277041i	$0.910646\pi$
$200$	0	0
$201$	11.2830	0.795843
$202$	0	0
$203$	−3.11113	−0.218359
$204$	0	0
$205$	0.765408	0.0534584
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0.0675911	0.00467537
$210$	0	0
$211$	12.1121	0.833832	0.416916	−	0.908945i	$-0.363111\pi$
$211$	12.1121	0.833832	0.416916	+	0.908945i	$0.363111\pi$
$212$	0	0
$213$	4.10798	0.281474
$214$	0	0
$215$	−0.770923	−0.0525765
$216$	0	0
$217$	−11.9831	−0.813466
$218$	0	0
$219$	12.5759	0.849800
$220$	0	0
$221$	14.2878	0.961099
$222$	0	0
$223$	14.3444	0.960572	0.480286	−	0.877112i	$-0.340533\pi$
$223$	14.3444	0.960572	0.480286	+	0.877112i	$0.340533\pi$
$224$	0	0
$225$	−4.96473	−0.330982
$226$	0	0
$227$	23.3773	1.55161	0.775804	−	0.630974i	$-0.217344\pi$
$227$	23.3773	1.55161	0.775804	+	0.630974i	$0.217344\pi$
$228$	0	0
$229$	−1.31074	−0.0866164	−0.0433082	−	0.999062i	$-0.513790\pi$
$229$	−1.31074	−0.0866164	−0.0433082	+	0.999062i	$0.513790\pi$
$230$	0	0
$231$	0.0473140	0.00311303
$232$	0	0
$233$	13.3088	0.871891	0.435945	−	0.899973i	$-0.356414\pi$
$233$	13.3088	0.871891	0.435945	+	0.899973i	$0.356414\pi$
$234$	0	0
$235$	2.02163	0.131876
$236$	0	0
$237$	6.51135	0.422958
$238$	0	0
$239$	−6.21912	−0.402281	−0.201141	−	0.979562i	$-0.564465\pi$
$239$	−6.21912	−0.402281	−0.201141	+	0.979562i	$0.564465\pi$
$240$	0	0
$241$	13.0556	0.840984	0.420492	−	0.907296i	$-0.361858\pi$
$241$	13.0556	0.840984	0.420492	+	0.907296i	$0.361858\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.503126	0.0321436
$246$	0	0
$247$	22.6311	1.43998
$248$	0	0
$249$	−4.47591	−0.283649
$250$	0	0
$251$	−6.35262	−0.400974	−0.200487	−	0.979696i	$-0.564252\pi$
$251$	−6.35262	−0.400974	−0.200487	+	0.979696i	$0.564252\pi$
$252$	0	0
$253$	0.0152079	0.000956116 0
$254$	0	0
$255$	0.526935	0.0329979
$256$	0	0
$257$	−4.20856	−0.262523	−0.131261	−	0.991348i	$-0.541903\pi$
$257$	−4.20856	−0.262523	−0.131261	+	0.991348i	$0.541903\pi$
$258$	0	0
$259$	−7.61588	−0.473228
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	9.56002	0.589496	0.294748	−	0.955575i	$-0.404764\pi$
$263$	9.56002	0.589496	0.294748	+	0.955575i	$0.404764\pi$
$264$	0	0
$265$	1.63224	0.100268
$266$	0	0
$267$	−9.25893	−0.566637
$268$	0	0
$269$	14.6114	0.890872	0.445436	−	0.895314i	$-0.353049\pi$
$269$	14.6114	0.890872	0.445436	+	0.895314i	$0.353049\pi$
$270$	0	0
$271$	−5.99455	−0.364143	−0.182071	−	0.983285i	$-0.558280\pi$
$271$	−5.99455	−0.364143	−0.182071	+	0.983285i	$0.558280\pi$
$272$	0	0
$273$	15.8419	0.958793
$274$	0	0
$275$	−0.0755034	−0.00455303
$276$	0	0
$277$	13.5161	0.812102	0.406051	−	0.913850i	$-0.366906\pi$
$277$	13.5161	0.812102	0.406051	+	0.913850i	$0.366906\pi$
$278$	0	0
$279$	−3.85168	−0.230594
$280$	0	0
$281$	21.8604	1.30408	0.652041	−	0.758184i	$-0.273913\pi$
$281$	21.8604	1.30408	0.652041	+	0.758184i	$0.273913\pi$
$282$	0	0
$283$	15.8875	0.944415	0.472208	−	0.881487i	$-0.343457\pi$
$283$	15.8875	0.944415	0.472208	+	0.881487i	$0.343457\pi$
$284$	0	0
$285$	0.834639	0.0494398
$286$	0	0
$287$	12.6804	0.748498
$288$	0	0
$289$	−9.12676	−0.536868
$290$	0	0
$291$	7.78248	0.456217
$292$	0	0
$293$	21.9988	1.28518	0.642592	−	0.766209i	$-0.277859\pi$
$293$	21.9988	1.28518	0.642592	+	0.766209i	$0.277859\pi$
$294$	0	0
$295$	−1.63051	−0.0949320
$296$	0	0
$297$	0.0152079	0.000882455 0
$298$	0	0
$299$	5.09199	0.294478
$300$	0	0
$301$	−12.7717	−0.736150
$302$	0	0
$303$	−6.72116	−0.386120
$304$	0	0
$305$	0.562700	0.0322201
$306$	0	0
$307$	6.94886	0.396592	0.198296	−	0.980142i	$-0.436459\pi$
$307$	6.94886	0.396592	0.198296	+	0.980142i	$0.436459\pi$
$308$	0	0
$309$	−14.4939	−0.824527
$310$	0	0
$311$	−24.1375	−1.36871	−0.684357	−	0.729147i	$-0.739917\pi$
$311$	−24.1375	−1.36871	−0.684357	+	0.729147i	$0.739917\pi$
$312$	0	0
$313$	12.5436	0.709004	0.354502	−	0.935055i	$-0.384650\pi$
$313$	12.5436	0.709004	0.354502	+	0.935055i	$0.384650\pi$
$314$	0	0
$315$	0.584250	0.0329188
$316$	0	0
$317$	−10.6497	−0.598149	−0.299074	−	0.954230i	$-0.596678\pi$
$317$	−10.6497	−0.598149	−0.299074	+	0.954230i	$0.596678\pi$
$318$	0	0
$319$	−0.0152079	−0.000851482 0
$320$	0	0
$321$	−9.80146	−0.547064
$322$	0	0
$323$	12.4708	0.693895
$324$	0	0
$325$	−25.2804	−1.40230
$326$	0	0
$327$	−3.75215	−0.207494
$328$	0	0
$329$	33.4918	1.84647
$330$	0	0
$331$	−32.5359	−1.78834	−0.894169	−	0.447730i	$-0.852232\pi$
$331$	−32.5359	−1.78834	−0.894169	+	0.447730i	$0.852232\pi$
$332$	0	0
$333$	−2.44794	−0.134146
$334$	0	0
$335$	2.11887	0.115767
$336$	0	0
$337$	10.9431	0.596110	0.298055	−	0.954549i	$-0.403662\pi$
$337$	10.9431	0.596110	0.298055	+	0.954549i	$0.403662\pi$
$338$	0	0
$339$	−12.1826	−0.661669
$340$	0	0
$341$	−0.0585762	−0.00317208
$342$	0	0
$343$	−13.4427	−0.725840
$344$	0	0
$345$	0.187793	0.0101105
$346$	0	0
$347$	−25.5705	−1.37270	−0.686350	−	0.727272i	$-0.740788\pi$
$347$	−25.5705	−1.37270	−0.686350	+	0.727272i	$0.740788\pi$
$348$	0	0
$349$	−24.9768	−1.33698	−0.668490	−	0.743721i	$-0.733059\pi$
$349$	−24.9768	−1.33698	−0.668490	+	0.743721i	$0.733059\pi$
$350$	0	0
$351$	5.09199	0.271790
$352$	0	0
$353$	−3.60990	−0.192136	−0.0960678	−	0.995375i	$-0.530627\pi$
$353$	−3.60990	−0.192136	−0.0960678	+	0.995375i	$0.530627\pi$
$354$	0	0
$355$	0.771452	0.0409444
$356$	0	0
$357$	8.72962	0.462020
$358$	0	0
$359$	35.1320	1.85420	0.927098	−	0.374818i	$-0.122295\pi$
$359$	35.1320	1.85420	0.927098	+	0.374818i	$0.122295\pi$
$360$	0	0
$361$	0.753189	0.0396415
$362$	0	0
$363$	−10.9998	−0.577338
$364$	0	0
$365$	2.36167	0.123615
$366$	0	0
$367$	30.0290	1.56750	0.783750	−	0.621077i	$-0.213305\pi$
$367$	30.0290	1.56750	0.783750	+	0.621077i	$0.213305\pi$
$368$	0	0
$369$	4.07580	0.212178
$370$	0	0
$371$	27.0410	1.40390
$372$	0	0
$373$	30.9812	1.60415	0.802073	−	0.597225i	$-0.203730\pi$
$373$	30.9812	1.60415	0.802073	+	0.597225i	$0.203730\pi$
$374$	0	0
$375$	−1.87131	−0.0966341
$376$	0	0
$377$	−5.09199	−0.262251
$378$	0	0
$379$	13.2264	0.679394	0.339697	−	0.940535i	$-0.389676\pi$
$379$	13.2264	0.679394	0.339697	+	0.940535i	$0.389676\pi$
$380$	0	0
$381$	−14.7848	−0.757447
$382$	0	0
$383$	10.9631	0.560189	0.280094	−	0.959972i	$-0.409634\pi$
$383$	10.9631	0.560189	0.280094	+	0.959972i	$0.409634\pi$
$384$	0	0
$385$	0.00888525	0.000452834 0
$386$	0	0
$387$	−4.10517	−0.208677
$388$	0	0
$389$	−7.37047	−0.373698	−0.186849	−	0.982389i	$-0.559827\pi$
$389$	−7.37047	−0.373698	−0.186849	+	0.982389i	$0.559827\pi$
$390$	0	0
$391$	2.80593	0.141902
$392$	0	0
$393$	−21.0448	−1.06157
$394$	0	0
$395$	1.22279	0.0615251
$396$	0	0
$397$	11.9862	0.601569	0.300785	−	0.953692i	$-0.402751\pi$
$397$	11.9862	0.601569	0.300785	+	0.953692i	$0.402751\pi$
$398$	0	0
$399$	13.8273	0.692231
$400$	0	0
$401$	3.73312	0.186423	0.0932115	−	0.995646i	$-0.470287\pi$
$401$	3.73312	0.186423	0.0932115	+	0.995646i	$0.470287\pi$
$402$	0	0
$403$	−19.6127	−0.976981
$404$	0	0
$405$	0.187793	0.00933153
$406$	0	0
$407$	−0.0372282	−0.00184533
$408$	0	0
$409$	−6.21178	−0.307153	−0.153576	−	0.988137i	$-0.549079\pi$
$409$	−6.21178	−0.307153	−0.153576	+	0.988137i	$0.549079\pi$
$410$	0	0
$411$	−18.4369	−0.909423
$412$	0	0
$413$	−27.0123	−1.32919
$414$	0	0
$415$	−0.840546	−0.0412608
$416$	0	0
$417$	19.3555	0.947843
$418$	0	0
$419$	−13.5022	−0.659625	−0.329812	−	0.944047i	$-0.606985\pi$
$419$	−13.5022	−0.659625	−0.329812	+	0.944047i	$0.606985\pi$
$420$	0	0
$421$	−34.2427	−1.66888	−0.834442	−	0.551095i	$-0.814210\pi$
$421$	−34.2427	−1.66888	−0.834442	+	0.551095i	$0.814210\pi$
$422$	0	0
$423$	10.7652	0.523420
$424$	0	0
$425$	−13.9307	−0.675738
$426$	0	0
$427$	9.32214	0.451130
$428$	0	0
$429$	0.0774388	0.00373878
$430$	0	0
$431$	−8.29898	−0.399748	−0.199874	−	0.979822i	$-0.564053\pi$
$431$	−8.29898	−0.399748	−0.199874	+	0.979822i	$0.564053\pi$
$432$	0	0
$433$	−7.21981	−0.346962	−0.173481	−	0.984837i	$-0.555502\pi$
$433$	−7.21981	−0.346962	−0.173481	+	0.984837i	$0.555502\pi$
$434$	0	0
$435$	−0.187793	−0.00900400
$436$	0	0
$437$	4.44446	0.212607
$438$	0	0
$439$	14.3686	0.685774	0.342887	−	0.939377i	$-0.388595\pi$
$439$	14.3686	0.685774	0.342887	+	0.939377i	$0.388595\pi$
$440$	0	0
$441$	2.67915	0.127579
$442$	0	0
$443$	−27.6642	−1.31437	−0.657183	−	0.753731i	$-0.728252\pi$
$443$	−27.6642	−1.31437	−0.657183	+	0.753731i	$0.728252\pi$
$444$	0	0
$445$	−1.73876	−0.0824253
$446$	0	0
$447$	−5.36191	−0.253610
$448$	0	0
$449$	−35.9054	−1.69448	−0.847240	−	0.531211i	$-0.821737\pi$
$449$	−35.9054	−1.69448	−0.847240	+	0.531211i	$0.821737\pi$
$450$	0	0
$451$	0.0619846	0.00291874
$452$	0	0
$453$	4.92971	0.231618
$454$	0	0
$455$	2.97500	0.139470
$456$	0	0
$457$	3.92186	0.183457	0.0917285	−	0.995784i	$-0.470761\pi$
$457$	3.92186	0.183457	0.0917285	+	0.995784i	$0.470761\pi$
$458$	0	0
$459$	2.80593	0.130970
$460$	0	0
$461$	9.40397	0.437986	0.218993	−	0.975726i	$-0.429723\pi$
$461$	9.40397	0.437986	0.218993	+	0.975726i	$0.429723\pi$
$462$	0	0
$463$	10.3463	0.480835	0.240418	−	0.970670i	$-0.422716\pi$
$463$	10.3463	0.480835	0.240418	+	0.970670i	$0.422716\pi$
$464$	0	0
$465$	−0.723321	−0.0335432
$466$	0	0
$467$	−7.31033	−0.338282	−0.169141	−	0.985592i	$-0.554099\pi$
$467$	−7.31033	−0.338282	−0.169141	+	0.985592i	$0.554099\pi$
$468$	0	0
$469$	35.1030	1.62090
$470$	0	0
$471$	14.5834	0.671969
$472$	0	0
$473$	−0.0624312	−0.00287059
$474$	0	0
$475$	−22.0655	−1.01244
$476$	0	0
$477$	8.69169	0.397965
$478$	0	0
$479$	−12.1642	−0.555799	−0.277899	−	0.960610i	$-0.589638\pi$
$479$	−12.1642	−0.555799	−0.277899	+	0.960610i	$0.589638\pi$
$480$	0	0
$481$	−12.4649	−0.568351
$482$	0	0
$483$	3.11113	0.141561
$484$	0	0
$485$	1.46150	0.0663632
$486$	0	0
$487$	15.1124	0.684806	0.342403	−	0.939553i	$-0.388759\pi$
$487$	15.1124	0.684806	0.342403	+	0.939553i	$0.388759\pi$
$488$	0	0
$489$	9.81554	0.443874
$490$	0	0
$491$	24.3352	1.09823	0.549116	−	0.835746i	$-0.314964\pi$
$491$	24.3352	1.09823	0.549116	+	0.835746i	$0.314964\pi$
$492$	0	0
$493$	−2.80593	−0.126373
$494$	0	0
$495$	0.00285595	0.000128365 0
$496$	0	0
$497$	12.7805	0.573283
$498$	0	0
$499$	37.8616	1.69492	0.847458	−	0.530862i	$-0.178132\pi$
$499$	37.8616	1.69492	0.847458	+	0.530862i	$0.178132\pi$
$500$	0	0
$501$	15.8044	0.706088
$502$	0	0
$503$	−3.15047	−0.140473	−0.0702364	−	0.997530i	$-0.522375\pi$
$503$	−3.15047	−0.140473	−0.0702364	+	0.997530i	$0.522375\pi$
$504$	0	0
$505$	−1.26219	−0.0561667
$506$	0	0
$507$	12.9284	0.574170
$508$	0	0
$509$	−7.42367	−0.329048	−0.164524	−	0.986373i	$-0.552609\pi$
$509$	−7.42367	−0.329048	−0.164524	+	0.986373i	$0.552609\pi$
$510$	0	0
$511$	39.1253	1.73080
$512$	0	0
$513$	4.44446	0.196228
$514$	0	0
$515$	−2.72185	−0.119939
$516$	0	0
$517$	0.163716	0.00720022
$518$	0	0
$519$	−3.48262	−0.152870
$520$	0	0
$521$	−2.65840	−0.116467	−0.0582333	−	0.998303i	$-0.518547\pi$
$521$	−2.65840	−0.116467	−0.0582333	+	0.998303i	$0.518547\pi$
$522$	0	0
$523$	−15.0408	−0.657686	−0.328843	−	0.944385i	$-0.606659\pi$
$523$	−15.0408	−0.657686	−0.328843	+	0.944385i	$0.606659\pi$
$524$	0	0
$525$	−15.4459	−0.674116
$526$	0	0
$527$	−10.8076	−0.470784
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−8.68248	−0.376787
$532$	0	0
$533$	20.7540	0.898954
$534$	0	0
$535$	−1.84065	−0.0795782
$536$	0	0
$537$	−20.9975	−0.906108
$538$	0	0
$539$	0.0407444	0.00175498
$540$	0	0
$541$	24.5939	1.05737	0.528686	−	0.848817i	$-0.322685\pi$
$541$	24.5939	1.05737	0.528686	+	0.848817i	$0.322685\pi$
$542$	0	0
$543$	10.3852	0.445672
$544$	0	0
$545$	−0.704629	−0.0301830
$546$	0	0
$547$	6.45639	0.276055	0.138028	−	0.990428i	$-0.455924\pi$
$547$	6.45639	0.276055	0.138028	+	0.990428i	$0.455924\pi$
$548$	0	0
$549$	2.99638	0.127882
$550$	0	0
$551$	−4.44446	−0.189340
$552$	0	0
$553$	20.2577	0.861444
$554$	0	0
$555$	−0.459707	−0.0195135
$556$	0	0
$557$	−15.5826	−0.660255	−0.330127	−	0.943936i	$-0.607092\pi$
$557$	−15.5826	−0.660255	−0.330127	+	0.943936i	$0.607092\pi$
$558$	0	0
$559$	−20.9035	−0.884123
$560$	0	0
$561$	0.0426724	0.00180163
$562$	0	0
$563$	28.0751	1.18322	0.591612	−	0.806223i	$-0.298492\pi$
$563$	28.0751	1.18322	0.591612	+	0.806223i	$0.298492\pi$
$564$	0	0
$565$	−2.28781	−0.0962490
$566$	0	0
$567$	3.11113	0.130655
$568$	0	0
$569$	−5.35305	−0.224412	−0.112206	−	0.993685i	$-0.535792\pi$
$569$	−5.35305	−0.224412	−0.112206	+	0.993685i	$0.535792\pi$
$570$	0	0
$571$	−37.6630	−1.57615	−0.788073	−	0.615582i	$-0.788921\pi$
$571$	−37.6630	−1.57615	−0.788073	+	0.615582i	$0.788921\pi$
$572$	0	0
$573$	0.396221	0.0165524
$574$	0	0
$575$	−4.96473	−0.207044
$576$	0	0
$577$	26.2753	1.09385	0.546927	−	0.837180i	$-0.315797\pi$
$577$	26.2753	1.09385	0.546927	+	0.837180i	$0.315797\pi$
$578$	0	0
$579$	3.34476	0.139003
$580$	0	0
$581$	−13.9252	−0.577713
$582$	0	0
$583$	0.132183	0.00547445
$584$	0	0
$585$	0.956242	0.0395358
$586$	0	0
$587$	32.8088	1.35417	0.677083	−	0.735907i	$-0.263244\pi$
$587$	32.8088	1.35417	0.677083	+	0.735907i	$0.263244\pi$
$588$	0	0
$589$	−17.1186	−0.705361
$590$	0	0
$591$	−21.7540	−0.894842
$592$	0	0
$593$	33.2183	1.36411	0.682056	−	0.731300i	$-0.261086\pi$
$593$	33.2183	1.36411	0.682056	+	0.731300i	$0.261086\pi$
$594$	0	0
$595$	1.63936	0.0672074
$596$	0	0
$597$	−27.1091	−1.10950
$598$	0	0
$599$	15.8213	0.646441	0.323221	−	0.946324i	$-0.395234\pi$
$599$	15.8213	0.646441	0.323221	+	0.946324i	$0.395234\pi$
$600$	0	0
$601$	−5.10952	−0.208422	−0.104211	−	0.994555i	$-0.533232\pi$
$601$	−5.10952	−0.208422	−0.104211	+	0.994555i	$0.533232\pi$
$602$	0	0
$603$	11.2830	0.459480
$604$	0	0
$605$	−2.06568	−0.0839820
$606$	0	0
$607$	−8.83452	−0.358582	−0.179291	−	0.983796i	$-0.557380\pi$
$607$	−8.83452	−0.358582	−0.179291	+	0.983796i	$0.557380\pi$
$608$	0	0
$609$	−3.11113	−0.126069
$610$	0	0
$611$	54.8161	2.21762
$612$	0	0
$613$	13.7895	0.556954	0.278477	−	0.960443i	$-0.410170\pi$
$613$	13.7895	0.556954	0.278477	+	0.960443i	$0.410170\pi$
$614$	0	0
$615$	0.765408	0.0308642
$616$	0	0
$617$	−6.49692	−0.261556	−0.130778	−	0.991412i	$-0.541748\pi$
$617$	−6.49692	−0.261556	−0.130778	+	0.991412i	$0.541748\pi$
$618$	0	0
$619$	27.9394	1.12298	0.561490	−	0.827484i	$-0.310228\pi$
$619$	27.9394	1.12298	0.561490	+	0.827484i	$0.310228\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−28.8058	−1.15408
$624$	0	0
$625$	24.4722	0.978890
$626$	0	0
$627$	0.0675911	0.00269933
$628$	0	0
$629$	−6.86876	−0.273875
$630$	0	0
$631$	11.8703	0.472548	0.236274	−	0.971686i	$-0.424074\pi$
$631$	11.8703	0.472548	0.236274	+	0.971686i	$0.424074\pi$
$632$	0	0
$633$	12.1121	0.481413
$634$	0	0
$635$	−2.77648	−0.110181
$636$	0	0
$637$	13.6422	0.540524
$638$	0	0
$639$	4.10798	0.162509
$640$	0	0
$641$	−14.8700	−0.587328	−0.293664	−	0.955909i	$-0.594875\pi$
$641$	−14.8700	−0.587328	−0.293664	+	0.955909i	$0.594875\pi$
$642$	0	0
$643$	−36.4858	−1.43886	−0.719430	−	0.694565i	$-0.755597\pi$
$643$	−36.4858	−1.43886	−0.719430	+	0.694565i	$0.755597\pi$
$644$	0	0
$645$	−0.770923	−0.0303551
$646$	0	0
$647$	22.8891	0.899863	0.449932	−	0.893063i	$-0.351448\pi$
$647$	22.8891	0.899863	0.449932	+	0.893063i	$0.351448\pi$
$648$	0	0
$649$	−0.132043	−0.00518313
$650$	0	0
$651$	−11.9831	−0.469655
$652$	0	0
$653$	−36.1865	−1.41609	−0.708044	−	0.706168i	$-0.750422\pi$
$653$	−36.1865	−1.41609	−0.708044	+	0.706168i	$0.750422\pi$
$654$	0	0
$655$	−3.95207	−0.154420
$656$	0	0
$657$	12.5759	0.490632
$658$	0	0
$659$	−9.45417	−0.368282	−0.184141	−	0.982900i	$-0.558950\pi$
$659$	−9.45417	−0.368282	−0.184141	+	0.982900i	$0.558950\pi$
$660$	0	0
$661$	−6.62737	−0.257775	−0.128887	−	0.991659i	$-0.541141\pi$
$661$	−6.62737	−0.257775	−0.128887	+	0.991659i	$0.541141\pi$
$662$	0	0
$663$	14.2878	0.554891
$664$	0	0
$665$	2.59667	0.100695
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	14.3444	0.554587
$670$	0	0
$671$	0.0455688	0.00175916
$672$	0	0
$673$	−38.3436	−1.47804	−0.739018	−	0.673685i	$-0.764710\pi$
$673$	−38.3436	−1.47804	−0.739018	+	0.673685i	$0.764710\pi$
$674$	0	0
$675$	−4.96473	−0.191093
$676$	0	0
$677$	−5.45076	−0.209490	−0.104745	−	0.994499i	$-0.533403\pi$
$677$	−5.45076	−0.209490	−0.104745	+	0.994499i	$0.533403\pi$
$678$	0	0
$679$	24.2123	0.929184
$680$	0	0
$681$	23.3773	0.895822
$682$	0	0
$683$	−18.1931	−0.696140	−0.348070	−	0.937469i	$-0.613163\pi$
$683$	−18.1931	−0.696140	−0.348070	+	0.937469i	$0.613163\pi$
$684$	0	0
$685$	−3.46232	−0.132288
$686$	0	0
$687$	−1.31074	−0.0500080
$688$	0	0
$689$	44.2580	1.68610
$690$	0	0
$691$	14.7569	0.561378	0.280689	−	0.959799i	$-0.409437\pi$
$691$	14.7569	0.561378	0.280689	+	0.959799i	$0.409437\pi$
$692$	0	0
$693$	0.0473140	0.00179731
$694$	0	0
$695$	3.63483	0.137877
$696$	0	0
$697$	11.4364	0.433185
$698$	0	0
$699$	13.3088	0.503386
$700$	0	0
$701$	4.42193	0.167014	0.0835069	−	0.996507i	$-0.473388\pi$
$701$	4.42193	0.167014	0.0835069	+	0.996507i	$0.473388\pi$
$702$	0	0
$703$	−10.8798	−0.410339
$704$	0	0
$705$	2.02163	0.0761388
$706$	0	0
$707$	−20.9104	−0.786417
$708$	0	0
$709$	−21.3150	−0.800503	−0.400251	−	0.916405i	$-0.631077\pi$
$709$	−21.3150	−0.800503	−0.400251	+	0.916405i	$0.631077\pi$
$710$	0	0
$711$	6.51135	0.244195
$712$	0	0
$713$	−3.85168	−0.144247
$714$	0	0
$715$	0.0145425	0.000543858 0
$716$	0	0
$717$	−6.21912	−0.232257
$718$	0	0
$719$	−12.5714	−0.468835	−0.234417	−	0.972136i	$-0.575318\pi$
$719$	−12.5714	−0.468835	−0.234417	+	0.972136i	$0.575318\pi$
$720$	0	0
$721$	−45.0923	−1.67933
$722$	0	0
$723$	13.0556	0.485542
$724$	0	0
$725$	4.96473	0.184386
$726$	0	0
$727$	−5.10913	−0.189487	−0.0947436	−	0.995502i	$-0.530203\pi$
$727$	−5.10913	−0.189487	−0.0947436	+	0.995502i	$0.530203\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.5188	−0.426038
$732$	0	0
$733$	32.5291	1.20149	0.600745	−	0.799441i	$-0.294871\pi$
$733$	32.5291	1.20149	0.600745	+	0.799441i	$0.294871\pi$
$734$	0	0
$735$	0.503126	0.0185581
$736$	0	0
$737$	0.171592	0.00632065
$738$	0	0
$739$	32.3091	1.18851	0.594254	−	0.804277i	$-0.297447\pi$
$739$	32.3091	1.18851	0.594254	+	0.804277i	$0.297447\pi$
$740$	0	0
$741$	22.6311	0.831376
$742$	0	0
$743$	−15.8188	−0.580337	−0.290168	−	0.956976i	$-0.593711\pi$
$743$	−15.8188	−0.580337	−0.290168	+	0.956976i	$0.593711\pi$
$744$	0	0
$745$	−1.00693	−0.0368911
$746$	0	0
$747$	−4.47591	−0.163765
$748$	0	0
$749$	−30.4936	−1.11421
$750$	0	0
$751$	−37.3597	−1.36328	−0.681638	−	0.731690i	$-0.738732\pi$
$751$	−37.3597	−1.36328	−0.681638	+	0.731690i	$0.738732\pi$
$752$	0	0
$753$	−6.35262	−0.231502
$754$	0	0
$755$	0.925768	0.0336921
$756$	0	0
$757$	−22.0675	−0.802056	−0.401028	−	0.916066i	$-0.631347\pi$
$757$	−22.0675	−0.802056	−0.401028	+	0.916066i	$0.631347\pi$
$758$	0	0
$759$	0.0152079	0.000552014 0
$760$	0	0
$761$	38.4308	1.39311	0.696557	−	0.717501i	$-0.254714\pi$
$761$	38.4308	1.39311	0.696557	+	0.717501i	$0.254714\pi$
$762$	0	0
$763$	−11.6734	−0.422607
$764$	0	0
$765$	0.526935	0.0190514
$766$	0	0
$767$	−44.2111	−1.59637
$768$	0	0
$769$	28.1201	1.01404	0.507019	−	0.861935i	$-0.330747\pi$
$769$	28.1201	1.01404	0.507019	+	0.861935i	$0.330747\pi$
$770$	0	0
$771$	−4.20856	−0.151568
$772$	0	0
$773$	46.7516	1.68154	0.840769	−	0.541394i	$-0.182103\pi$
$773$	46.7516	1.68154	0.840769	+	0.541394i	$0.182103\pi$
$774$	0	0
$775$	19.1226	0.686904
$776$	0	0
$777$	−7.61588	−0.273218
$778$	0	0
$779$	18.1147	0.649027
$780$	0	0
$781$	0.0624740	0.00223550
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	2.73867	0.0977474
$786$	0	0
$787$	−22.7176	−0.809794	−0.404897	−	0.914362i	$-0.632693\pi$
$787$	−22.7176	−0.809794	−0.404897	+	0.914362i	$0.632693\pi$
$788$	0	0
$789$	9.56002	0.340346
$790$	0	0
$791$	−37.9017	−1.34763
$792$	0	0
$793$	15.2575	0.541811
$794$	0	0
$795$	1.63224	0.0578897
$796$	0	0
$797$	21.2517	0.752773	0.376387	−	0.926463i	$-0.377166\pi$
$797$	21.2517	0.752773	0.376387	+	0.926463i	$0.377166\pi$
$798$	0	0
$799$	30.2063	1.06862
$800$	0	0
$801$	−9.25893	−0.327148
$802$	0	0
$803$	0.191254	0.00674919
$804$	0	0
$805$	0.584250	0.0205921
$806$	0	0
$807$	14.6114	0.514345
$808$	0	0
$809$	−32.3017	−1.13567	−0.567833	−	0.823144i	$-0.692218\pi$
$809$	−32.3017	−1.13567	−0.567833	+	0.823144i	$0.692218\pi$
$810$	0	0
$811$	−10.5968	−0.372104	−0.186052	−	0.982540i	$-0.559569\pi$
$811$	−10.5968	−0.372104	−0.186052	+	0.982540i	$0.559569\pi$
$812$	0	0
$813$	−5.99455	−0.210238
$814$	0	0
$815$	1.84329	0.0645678
$816$	0	0
$817$	−18.2452	−0.638320
$818$	0	0
$819$	15.8419	0.553560
$820$	0	0
$821$	−21.3525	−0.745208	−0.372604	−	0.927990i	$-0.621535\pi$
$821$	−21.3525	−0.745208	−0.372604	+	0.927990i	$0.621535\pi$
$822$	0	0
$823$	10.9692	0.382361	0.191181	−	0.981555i	$-0.438768\pi$
$823$	10.9692	0.382361	0.191181	+	0.981555i	$0.438768\pi$
$824$	0	0
$825$	−0.0755034	−0.00262869
$826$	0	0
$827$	−50.2907	−1.74878	−0.874390	−	0.485224i	$-0.838738\pi$
$827$	−50.2907	−1.74878	−0.874390	+	0.485224i	$0.838738\pi$
$828$	0	0
$829$	25.4590	0.884229	0.442114	−	0.896959i	$-0.354229\pi$
$829$	25.4590	0.884229	0.442114	+	0.896959i	$0.354229\pi$
$830$	0	0
$831$	13.5161	0.468867
$832$	0	0
$833$	7.51750	0.260466
$834$	0	0
$835$	2.96796	0.102710
$836$	0	0
$837$	−3.85168	−0.133134
$838$	0	0
$839$	−32.8811	−1.13518	−0.567590	−	0.823311i	$-0.692124\pi$
$839$	−32.8811	−1.13518	−0.567590	+	0.823311i	$0.692124\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	21.8604	0.752912
$844$	0	0
$845$	2.42787	0.0835211
$846$	0	0
$847$	−34.2217	−1.17587
$848$	0	0
$849$	15.8875	0.545259
$850$	0	0
$851$	−2.44794	−0.0839144
$852$	0	0
$853$	47.4857	1.62588	0.812939	−	0.582349i	$-0.197866\pi$
$853$	47.4857	1.62588	0.812939	+	0.582349i	$0.197866\pi$
$854$	0	0
$855$	0.834639	0.0285441
$856$	0	0
$857$	2.90915	0.0993745	0.0496873	−	0.998765i	$-0.484178\pi$
$857$	2.90915	0.0993745	0.0496873	+	0.998765i	$0.484178\pi$
$858$	0	0
$859$	−33.1860	−1.13229	−0.566146	−	0.824305i	$-0.691566\pi$
$859$	−33.1860	−1.13229	−0.566146	+	0.824305i	$0.691566\pi$
$860$	0	0
$861$	12.6804	0.432146
$862$	0	0
$863$	−32.7369	−1.11438	−0.557189	−	0.830386i	$-0.688120\pi$
$863$	−32.7369	−1.11438	−0.557189	+	0.830386i	$0.688120\pi$
$864$	0	0
$865$	−0.654012	−0.0222371
$866$	0	0
$867$	−9.12676	−0.309961
$868$	0	0
$869$	0.0990243	0.00335917
$870$	0	0
$871$	57.4530	1.94672
$872$	0	0
$873$	7.78248	0.263397
$874$	0	0
$875$	−5.82190	−0.196816
$876$	0	0
$877$	−25.6359	−0.865664	−0.432832	−	0.901474i	$-0.642486\pi$
$877$	−25.6359	−0.865664	−0.432832	+	0.901474i	$0.642486\pi$
$878$	0	0
$879$	21.9988	0.742001
$880$	0	0
$881$	4.12625	0.139017	0.0695084	−	0.997581i	$-0.477857\pi$
$881$	4.12625	0.139017	0.0695084	+	0.997581i	$0.477857\pi$
$882$	0	0
$883$	5.82512	0.196031	0.0980155	−	0.995185i	$-0.468751\pi$
$883$	5.82512	0.196031	0.0980155	+	0.995185i	$0.468751\pi$
$884$	0	0
$885$	−1.63051	−0.0548090
$886$	0	0
$887$	−9.53824	−0.320263	−0.160131	−	0.987096i	$-0.551192\pi$
$887$	−9.53824	−0.320263	−0.160131	+	0.987096i	$0.551192\pi$
$888$	0	0
$889$	−45.9974	−1.54270
$890$	0	0
$891$	0.0152079	0.000509485 0
$892$	0	0
$893$	47.8453	1.60108
$894$	0	0
$895$	−3.94319	−0.131806
$896$	0	0
$897$	5.09199	0.170017
$898$	0	0
$899$	3.85168	0.128461
$900$	0	0
$901$	24.3883	0.812491
$902$	0	0
$903$	−12.7717	−0.425016
$904$	0	0
$905$	1.95027	0.0648293
$906$	0	0
$907$	−48.0271	−1.59471	−0.797357	−	0.603507i	$-0.793769\pi$
$907$	−48.0271	−1.59471	−0.797357	+	0.603507i	$0.793769\pi$
$908$	0	0
$909$	−6.72116	−0.222927
$910$	0	0
$911$	−8.95354	−0.296644	−0.148322	−	0.988939i	$-0.547387\pi$
$911$	−8.95354	−0.296644	−0.148322	+	0.988939i	$0.547387\pi$
$912$	0	0
$913$	−0.0680694	−0.00225277
$914$	0	0
$915$	0.562700	0.0186023
$916$	0	0
$917$	−65.4732	−2.16211
$918$	0	0
$919$	−43.9536	−1.44990	−0.724948	−	0.688804i	$-0.758136\pi$
$919$	−43.9536	−1.44990	−0.724948	+	0.688804i	$0.758136\pi$
$920$	0	0
$921$	6.94886	0.228973
$922$	0	0
$923$	20.9178	0.688519
$924$	0	0
$925$	12.1534	0.399601
$926$	0	0
$927$	−14.4939	−0.476041
$928$	0	0
$929$	37.2230	1.22125	0.610623	−	0.791921i	$-0.290919\pi$
$929$	37.2230	1.22125	0.610623	+	0.791921i	$0.290919\pi$
$930$	0	0
$931$	11.9074	0.390248
$932$	0	0
$933$	−24.1375	−0.790228
$934$	0	0
$935$	0.00801360	0.000262073 0
$936$	0	0
$937$	31.1645	1.01810	0.509049	−	0.860737i	$-0.329997\pi$
$937$	31.1645	1.01810	0.509049	+	0.860737i	$0.329997\pi$
$938$	0	0
$939$	12.5436	0.409344
$940$	0	0
$941$	−42.5579	−1.38735	−0.693674	−	0.720289i	$-0.744009\pi$
$941$	−42.5579	−1.38735	−0.693674	+	0.720289i	$0.744009\pi$
$942$	0	0
$943$	4.07580	0.132726
$944$	0	0
$945$	0.584250	0.0190057
$946$	0	0
$947$	56.5593	1.83793	0.918966	−	0.394338i	$-0.129026\pi$
$947$	56.5593	1.83793	0.918966	+	0.394338i	$0.129026\pi$
$948$	0	0
$949$	64.0364	2.07871
$950$	0	0
$951$	−10.6497	−0.345341
$952$	0	0
$953$	−18.3682	−0.595003	−0.297501	−	0.954721i	$-0.596153\pi$
$953$	−18.3682	−0.595003	−0.297501	+	0.954721i	$0.596153\pi$
$954$	0	0
$955$	0.0744076	0.00240777
$956$	0	0
$957$	−0.0152079	−0.000491603 0
$958$	0	0
$959$	−57.3595	−1.85224
$960$	0	0
$961$	−16.1645	−0.521436
$962$	0	0
$963$	−9.80146	−0.315848
$964$	0	0
$965$	0.628123	0.0202200
$966$	0	0
$967$	10.7246	0.344880	0.172440	−	0.985020i	$-0.444835\pi$
$967$	10.7246	0.344880	0.172440	+	0.985020i	$0.444835\pi$
$968$	0	0
$969$	12.4708	0.400621
$970$	0	0
$971$	−52.8094	−1.69474	−0.847368	−	0.531007i	$-0.821814\pi$
$971$	−52.8094	−1.69474	−0.847368	+	0.531007i	$0.821814\pi$
$972$	0	0
$973$	60.2175	1.93049
$974$	0	0
$975$	−25.2804	−0.809620
$976$	0	0
$977$	7.04878	0.225510	0.112755	−	0.993623i	$-0.464032\pi$
$977$	7.04878	0.225510	0.112755	+	0.993623i	$0.464032\pi$
$978$	0	0
$979$	−0.140809	−0.00450028
$980$	0	0
$981$	−3.75215	−0.119797
$982$	0	0
$983$	6.66905	0.212710	0.106355	−	0.994328i	$-0.466082\pi$
$983$	6.66905	0.212710	0.106355	+	0.994328i	$0.466082\pi$
$984$	0	0
$985$	−4.08526	−0.130167
$986$	0	0
$987$	33.4918	1.06606
$988$	0	0
$989$	−4.10517	−0.130537
$990$	0	0
$991$	−43.2315	−1.37330	−0.686648	−	0.726990i	$-0.740918\pi$
$991$	−43.2315	−1.37330	−0.686648	+	0.726990i	$0.740918\pi$
$992$	0	0
$993$	−32.5359	−1.03250
$994$	0	0
$995$	−5.09092	−0.161393
$996$	0	0
$997$	19.0251	0.602531	0.301266	−	0.953540i	$-0.402591\pi$
$997$	19.0251	0.602531	0.301266	+	0.953540i	$0.402591\pi$
$998$	0	0
$999$	−2.44794	−0.0774495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.9	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.9	✓	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.187793 q^{5} +3.11113 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.187793 q^{5} +3.11113 q^{7} +1.00000 q^{9} +0.0152079 q^{11} +5.09199 q^{13} +0.187793 q^{15} +2.80593 q^{17} +4.44446 q^{19} +3.11113 q^{21} +1.00000 q^{23} -4.96473 q^{25} +1.00000 q^{27} -1.00000 q^{29} -3.85168 q^{31} +0.0152079 q^{33} +0.584250 q^{35} -2.44794 q^{37} +5.09199 q^{39} +4.07580 q^{41} -4.10517 q^{43} +0.187793 q^{45} +10.7652 q^{47} +2.67915 q^{49} +2.80593 q^{51} +8.69169 q^{53} +0.00285595 q^{55} +4.44446 q^{57} -8.68248 q^{59} +2.99638 q^{61} +3.11113 q^{63} +0.956242 q^{65} +11.2830 q^{67} +1.00000 q^{69} +4.10798 q^{71} +12.5759 q^{73} -4.96473 q^{75} +0.0473140 q^{77} +6.51135 q^{79} +1.00000 q^{81} -4.47591 q^{83} +0.526935 q^{85} -1.00000 q^{87} -9.25893 q^{89} +15.8419 q^{91} -3.85168 q^{93} +0.834639 q^{95} +7.78248 q^{97} +0.0152079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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