LMFDB - Embedded newform 8004.2.a.g.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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 $ x^12 - 3x^11 - 31x^10 + 80x^9 + 347x^8 - 697x^7 - 1714x^6 + 2146x^5 + 3304x^4 - 1156x^3 - 616x^2 + 136*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.413592$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.413592 q^{5} +2.75070 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.413592 q^{5} +2.75070 q^{7} +1.00000 q^{9} -5.40788 q^{11} +2.14893 q^{13} -0.413592 q^{15} +5.24934 q^{17} -3.94252 q^{19} -2.75070 q^{21} +1.00000 q^{23} -4.82894 q^{25} -1.00000 q^{27} -1.00000 q^{29} +4.35163 q^{31} +5.40788 q^{33} +1.13767 q^{35} -2.72398 q^{37} -2.14893 q^{39} -6.03343 q^{41} -3.64006 q^{43} +0.413592 q^{45} +2.47763 q^{47} +0.566323 q^{49} -5.24934 q^{51} -7.66902 q^{53} -2.23666 q^{55} +3.94252 q^{57} +7.67427 q^{59} -6.93585 q^{61} +2.75070 q^{63} +0.888782 q^{65} +10.2926 q^{67} -1.00000 q^{69} -0.981264 q^{71} -6.83799 q^{73} +4.82894 q^{75} -14.8754 q^{77} +3.02058 q^{79} +1.00000 q^{81} -0.833761 q^{83} +2.17109 q^{85} +1.00000 q^{87} -5.00216 q^{89} +5.91106 q^{91} -4.35163 q^{93} -1.63060 q^{95} -11.6816 q^{97} -5.40788 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * 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
$5$	0.413592	0.184964	0.0924820	−	0.995714i	$-0.470520\pi$
$5$	0.413592	0.184964	0.0924820	+	0.995714i	$0.470520\pi$
$6$	0	0
$7$	2.75070	1.03967	0.519833	−	0.854268i	$-0.325994\pi$
$7$	2.75070	1.03967	0.519833	+	0.854268i	$0.325994\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.40788	−1.63054	−0.815268	−	0.579083i	$-0.803411\pi$
$11$	−5.40788	−1.63054	−0.815268	+	0.579083i	$0.803411\pi$
$12$	0	0
$13$	2.14893	0.596007	0.298004	−	0.954565i	$-0.403679\pi$
$13$	2.14893	0.596007	0.298004	+	0.954565i	$0.403679\pi$
$14$	0	0
$15$	−0.413592	−0.106789
$16$	0	0
$17$	5.24934	1.27315	0.636576	−	0.771214i	$-0.280350\pi$
$17$	5.24934	1.27315	0.636576	+	0.771214i	$0.280350\pi$
$18$	0	0
$19$	−3.94252	−0.904477	−0.452238	−	0.891897i	$-0.649374\pi$
$19$	−3.94252	−0.904477	−0.452238	+	0.891897i	$0.649374\pi$
$20$	0	0
$21$	−2.75070	−0.600251
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.82894	−0.965788
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.35163	0.781576	0.390788	−	0.920481i	$-0.372202\pi$
$31$	4.35163	0.781576	0.390788	+	0.920481i	$0.372202\pi$
$32$	0	0
$33$	5.40788	0.941391
$34$	0	0
$35$	1.13767	0.192301
$36$	0	0
$37$	−2.72398	−0.447820	−0.223910	−	0.974610i	$-0.571882\pi$
$37$	−2.72398	−0.447820	−0.223910	+	0.974610i	$0.571882\pi$
$38$	0	0
$39$	−2.14893	−0.344105
$40$	0	0
$41$	−6.03343	−0.942264	−0.471132	−	0.882063i	$-0.656154\pi$
$41$	−6.03343	−0.942264	−0.471132	+	0.882063i	$0.656154\pi$
$42$	0	0
$43$	−3.64006	−0.555105	−0.277552	−	0.960711i	$-0.589523\pi$
$43$	−3.64006	−0.555105	−0.277552	+	0.960711i	$0.589523\pi$
$44$	0	0
$45$	0.413592	0.0616547
$46$	0	0
$47$	2.47763	0.361399	0.180700	−	0.983538i	$-0.442164\pi$
$47$	2.47763	0.361399	0.180700	+	0.983538i	$0.442164\pi$
$48$	0	0
$49$	0.566323	0.0809034
$50$	0	0
$51$	−5.24934	−0.735055
$52$	0	0
$53$	−7.66902	−1.05342	−0.526711	−	0.850045i	$-0.676575\pi$
$53$	−7.66902	−1.05342	−0.526711	+	0.850045i	$0.676575\pi$
$54$	0	0
$55$	−2.23666	−0.301591
$56$	0	0
$57$	3.94252	0.522200
$58$	0	0
$59$	7.67427	0.999105	0.499552	−	0.866284i	$-0.333498\pi$
$59$	7.67427	0.999105	0.499552	+	0.866284i	$0.333498\pi$
$60$	0	0
$61$	−6.93585	−0.888044	−0.444022	−	0.896016i	$-0.646449\pi$
$61$	−6.93585	−0.888044	−0.444022	+	0.896016i	$0.646449\pi$
$62$	0	0
$63$	2.75070	0.346555
$64$	0	0
$65$	0.888782	0.110240
$66$	0	0
$67$	10.2926	1.25744	0.628718	−	0.777634i	$-0.283580\pi$
$67$	10.2926	1.25744	0.628718	+	0.777634i	$0.283580\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−0.981264	−0.116455	−0.0582273	−	0.998303i	$-0.518545\pi$
$71$	−0.981264	−0.116455	−0.0582273	+	0.998303i	$0.518545\pi$
$72$	0	0
$73$	−6.83799	−0.800327	−0.400163	−	0.916444i	$-0.631047\pi$
$73$	−6.83799	−0.800327	−0.400163	+	0.916444i	$0.631047\pi$
$74$	0	0
$75$	4.82894	0.557598
$76$	0	0
$77$	−14.8754	−1.69521
$78$	0	0
$79$	3.02058	0.339842	0.169921	−	0.985458i	$-0.445649\pi$
$79$	3.02058	0.339842	0.169921	+	0.985458i	$0.445649\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.833761	−0.0915172	−0.0457586	−	0.998953i	$-0.514571\pi$
$83$	−0.833761	−0.0915172	−0.0457586	+	0.998953i	$0.514571\pi$
$84$	0	0
$85$	2.17109	0.235487
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−5.00216	−0.530228	−0.265114	−	0.964217i	$-0.585410\pi$
$89$	−5.00216	−0.530228	−0.265114	+	0.964217i	$0.585410\pi$
$90$	0	0
$91$	5.91106	0.619648
$92$	0	0
$93$	−4.35163	−0.451243
$94$	0	0
$95$	−1.63060	−0.167296
$96$	0	0
$97$	−11.6816	−1.18608	−0.593042	−	0.805171i	$-0.702073\pi$
$97$	−11.6816	−1.18608	−0.593042	+	0.805171i	$0.702073\pi$
$98$	0	0
$99$	−5.40788	−0.543512
$100$	0	0
$101$	−10.7048	−1.06517	−0.532586	−	0.846376i	$-0.678780\pi$
$101$	−10.7048	−1.06517	−0.532586	+	0.846376i	$0.678780\pi$
$102$	0	0
$103$	5.40129	0.532204	0.266102	−	0.963945i	$-0.414264\pi$
$103$	5.40129	0.532204	0.266102	+	0.963945i	$0.414264\pi$
$104$	0	0
$105$	−1.13767	−0.111025
$106$	0	0
$107$	6.56999	0.635144	0.317572	−	0.948234i	$-0.397132\pi$
$107$	6.56999	0.635144	0.317572	+	0.948234i	$0.397132\pi$
$108$	0	0
$109$	0.0782059	0.00749076	0.00374538	−	0.999993i	$-0.498808\pi$
$109$	0.0782059	0.00749076	0.00374538	+	0.999993i	$0.498808\pi$
$110$	0	0
$111$	2.72398	0.258549
$112$	0	0
$113$	−13.2337	−1.24492	−0.622460	−	0.782651i	$-0.713867\pi$
$113$	−13.2337	−1.24492	−0.622460	+	0.782651i	$0.713867\pi$
$114$	0	0
$115$	0.413592	0.0385677
$116$	0	0
$117$	2.14893	0.198669
$118$	0	0
$119$	14.4393	1.32365
$120$	0	0
$121$	18.2451	1.65865
$122$	0	0
$123$	6.03343	0.544016
$124$	0	0
$125$	−4.06517	−0.363600
$126$	0	0
$127$	4.49520	0.398884	0.199442	−	0.979910i	$-0.436087\pi$
$127$	4.49520	0.398884	0.199442	+	0.979910i	$0.436087\pi$
$128$	0	0
$129$	3.64006	0.320490
$130$	0	0
$131$	−16.6600	−1.45559	−0.727797	−	0.685793i	$-0.759456\pi$
$131$	−16.6600	−1.45559	−0.727797	+	0.685793i	$0.759456\pi$
$132$	0	0
$133$	−10.8447	−0.940353
$134$	0	0
$135$	−0.413592	−0.0355963
$136$	0	0
$137$	−2.27426	−0.194303	−0.0971514	−	0.995270i	$-0.530973\pi$
$137$	−2.27426	−0.194303	−0.0971514	+	0.995270i	$0.530973\pi$
$138$	0	0
$139$	−1.41693	−0.120183	−0.0600913	−	0.998193i	$-0.519139\pi$
$139$	−1.41693	−0.120183	−0.0600913	+	0.998193i	$0.519139\pi$
$140$	0	0
$141$	−2.47763	−0.208654
$142$	0	0
$143$	−11.6212	−0.971812
$144$	0	0
$145$	−0.413592	−0.0343470
$146$	0	0
$147$	−0.566323	−0.0467096
$148$	0	0
$149$	−7.50663	−0.614967	−0.307484	−	0.951553i	$-0.599487\pi$
$149$	−7.50663	−0.614967	−0.307484	+	0.951553i	$0.599487\pi$
$150$	0	0
$151$	19.0639	1.55140	0.775698	−	0.631104i	$-0.217398\pi$
$151$	19.0639	1.55140	0.775698	+	0.631104i	$0.217398\pi$
$152$	0	0
$153$	5.24934	0.424384
$154$	0	0
$155$	1.79980	0.144564
$156$	0	0
$157$	17.7593	1.41735	0.708673	−	0.705537i	$-0.249294\pi$
$157$	17.7593	1.41735	0.708673	+	0.705537i	$0.249294\pi$
$158$	0	0
$159$	7.66902	0.608193
$160$	0	0
$161$	2.75070	0.216785
$162$	0	0
$163$	−12.7907	−1.00184	−0.500921	−	0.865493i	$-0.667005\pi$
$163$	−12.7907	−1.00184	−0.500921	+	0.865493i	$0.667005\pi$
$164$	0	0
$165$	2.23666	0.174123
$166$	0	0
$167$	−6.05789	−0.468774	−0.234387	−	0.972143i	$-0.575308\pi$
$167$	−6.05789	−0.468774	−0.234387	+	0.972143i	$0.575308\pi$
$168$	0	0
$169$	−8.38208	−0.644775
$170$	0	0
$171$	−3.94252	−0.301492
$172$	0	0
$173$	−10.6465	−0.809442	−0.404721	−	0.914440i	$-0.632631\pi$
$173$	−10.6465	−0.809442	−0.404721	+	0.914440i	$0.632631\pi$
$174$	0	0
$175$	−13.2829	−1.00410
$176$	0	0
$177$	−7.67427	−0.576834
$178$	0	0
$179$	2.02536	0.151382	0.0756911	−	0.997131i	$-0.475884\pi$
$179$	2.02536	0.151382	0.0756911	+	0.997131i	$0.475884\pi$
$180$	0	0
$181$	8.61476	0.640330	0.320165	−	0.947362i	$-0.396262\pi$
$181$	8.61476	0.640330	0.320165	+	0.947362i	$0.396262\pi$
$182$	0	0
$183$	6.93585	0.512713
$184$	0	0
$185$	−1.12662	−0.0828306
$186$	0	0
$187$	−28.3878	−2.07592
$188$	0	0
$189$	−2.75070	−0.200084
$190$	0	0
$191$	−0.577000	−0.0417502	−0.0208751	−	0.999782i	$-0.506645\pi$
$191$	−0.577000	−0.0417502	−0.0208751	+	0.999782i	$0.506645\pi$
$192$	0	0
$193$	2.92378	0.210458	0.105229	−	0.994448i	$-0.466442\pi$
$193$	2.92378	0.210458	0.105229	+	0.994448i	$0.466442\pi$
$194$	0	0
$195$	−0.888782	−0.0636470
$196$	0	0
$197$	−20.4451	−1.45665	−0.728327	−	0.685230i	$-0.759702\pi$
$197$	−20.4451	−1.45665	−0.728327	+	0.685230i	$0.759702\pi$
$198$	0	0
$199$	−1.04208	−0.0738710	−0.0369355	−	0.999318i	$-0.511760\pi$
$199$	−1.04208	−0.0738710	−0.0369355	+	0.999318i	$0.511760\pi$
$200$	0	0
$201$	−10.2926	−0.725981
$202$	0	0
$203$	−2.75070	−0.193061
$204$	0	0
$205$	−2.49538	−0.174285
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	21.3207	1.47478
$210$	0	0
$211$	16.4609	1.13322	0.566608	−	0.823988i	$-0.308255\pi$
$211$	16.4609	1.13322	0.566608	+	0.823988i	$0.308255\pi$
$212$	0	0
$213$	0.981264	0.0672351
$214$	0	0
$215$	−1.50550	−0.102674
$216$	0	0
$217$	11.9700	0.812578
$218$	0	0
$219$	6.83799	0.462069
$220$	0	0
$221$	11.2805	0.758808
$222$	0	0
$223$	18.7123	1.25307	0.626535	−	0.779393i	$-0.284472\pi$
$223$	18.7123	1.25307	0.626535	+	0.779393i	$0.284472\pi$
$224$	0	0
$225$	−4.82894	−0.321929
$226$	0	0
$227$	2.90845	0.193040	0.0965202	−	0.995331i	$-0.469229\pi$
$227$	2.90845	0.193040	0.0965202	+	0.995331i	$0.469229\pi$
$228$	0	0
$229$	7.60770	0.502731	0.251365	−	0.967892i	$-0.419120\pi$
$229$	7.60770	0.502731	0.251365	+	0.967892i	$0.419120\pi$
$230$	0	0
$231$	14.8754	0.978731
$232$	0	0
$233$	−28.7075	−1.88069	−0.940344	−	0.340225i	$-0.889497\pi$
$233$	−28.7075	−1.88069	−0.940344	+	0.340225i	$0.889497\pi$
$234$	0	0
$235$	1.02473	0.0668459
$236$	0	0
$237$	−3.02058	−0.196208
$238$	0	0
$239$	6.02058	0.389439	0.194719	−	0.980859i	$-0.437620\pi$
$239$	6.02058	0.389439	0.194719	+	0.980859i	$0.437620\pi$
$240$	0	0
$241$	18.4403	1.18785	0.593923	−	0.804522i	$-0.297578\pi$
$241$	18.4403	1.18785	0.593923	+	0.804522i	$0.297578\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.234227	0.0149642
$246$	0	0
$247$	−8.47222	−0.539075
$248$	0	0
$249$	0.833761	0.0528375
$250$	0	0
$251$	−9.57148	−0.604146	−0.302073	−	0.953285i	$-0.597679\pi$
$251$	−9.57148	−0.604146	−0.302073	+	0.953285i	$0.597679\pi$
$252$	0	0
$253$	−5.40788	−0.339990
$254$	0	0
$255$	−2.17109	−0.135959
$256$	0	0
$257$	15.1906	0.947563	0.473781	−	0.880642i	$-0.342889\pi$
$257$	15.1906	0.947563	0.473781	+	0.880642i	$0.342889\pi$
$258$	0	0
$259$	−7.49285	−0.465583
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−27.9622	−1.72422	−0.862110	−	0.506721i	$-0.830857\pi$
$263$	−27.9622	−1.72422	−0.862110	+	0.506721i	$0.830857\pi$
$264$	0	0
$265$	−3.17185	−0.194845
$266$	0	0
$267$	5.00216	0.306127
$268$	0	0
$269$	−22.3628	−1.36348	−0.681742	−	0.731592i	$-0.738777\pi$
$269$	−22.3628	−1.36348	−0.681742	+	0.731592i	$0.738777\pi$
$270$	0	0
$271$	−2.45290	−0.149003	−0.0745017	−	0.997221i	$-0.523737\pi$
$271$	−2.45290	−0.149003	−0.0745017	+	0.997221i	$0.523737\pi$
$272$	0	0
$273$	−5.91106	−0.357754
$274$	0	0
$275$	26.1143	1.57475
$276$	0	0
$277$	1.25987	0.0756982	0.0378491	−	0.999283i	$-0.487949\pi$
$277$	1.25987	0.0756982	0.0378491	+	0.999283i	$0.487949\pi$
$278$	0	0
$279$	4.35163	0.260525
$280$	0	0
$281$	16.9691	1.01229	0.506145	−	0.862449i	$-0.331070\pi$
$281$	16.9691	1.01229	0.506145	+	0.862449i	$0.331070\pi$
$282$	0	0
$283$	−13.9411	−0.828715	−0.414357	−	0.910114i	$-0.635994\pi$
$283$	−13.9411	−0.828715	−0.414357	+	0.910114i	$0.635994\pi$
$284$	0	0
$285$	1.63060	0.0965882
$286$	0	0
$287$	−16.5961	−0.979639
$288$	0	0
$289$	10.5556	0.620918
$290$	0	0
$291$	11.6816	0.684786
$292$	0	0
$293$	−22.0522	−1.28831	−0.644153	−	0.764897i	$-0.722790\pi$
$293$	−22.0522	−1.28831	−0.644153	+	0.764897i	$0.722790\pi$
$294$	0	0
$295$	3.17402	0.184798
$296$	0	0
$297$	5.40788	0.313797
$298$	0	0
$299$	2.14893	0.124276
$300$	0	0
$301$	−10.0127	−0.577123
$302$	0	0
$303$	10.7048	0.614977
$304$	0	0
$305$	−2.86861	−0.164256
$306$	0	0
$307$	32.1760	1.83638	0.918190	−	0.396139i	$-0.129650\pi$
$307$	32.1760	1.83638	0.918190	+	0.396139i	$0.129650\pi$
$308$	0	0
$309$	−5.40129	−0.307268
$310$	0	0
$311$	3.57851	0.202919	0.101459	−	0.994840i	$-0.467649\pi$
$311$	3.57851	0.202919	0.101459	+	0.994840i	$0.467649\pi$
$312$	0	0
$313$	−0.885822	−0.0500696	−0.0250348	−	0.999687i	$-0.507970\pi$
$313$	−0.885822	−0.0500696	−0.0250348	+	0.999687i	$0.507970\pi$
$314$	0	0
$315$	1.13767	0.0641002
$316$	0	0
$317$	2.62770	0.147586	0.0737932	−	0.997274i	$-0.476490\pi$
$317$	2.62770	0.147586	0.0737932	+	0.997274i	$0.476490\pi$
$318$	0	0
$319$	5.40788	0.302783
$320$	0	0
$321$	−6.56999	−0.366701
$322$	0	0
$323$	−20.6957	−1.15154
$324$	0	0
$325$	−10.3771	−0.575617
$326$	0	0
$327$	−0.0782059	−0.00432479
$328$	0	0
$329$	6.81520	0.375734
$330$	0	0
$331$	−9.42233	−0.517898	−0.258949	−	0.965891i	$-0.583376\pi$
$331$	−9.42233	−0.517898	−0.258949	+	0.965891i	$0.583376\pi$
$332$	0	0
$333$	−2.72398	−0.149273
$334$	0	0
$335$	4.25692	0.232580
$336$	0	0
$337$	8.82736	0.480857	0.240428	−	0.970667i	$-0.422712\pi$
$337$	8.82736	0.480857	0.240428	+	0.970667i	$0.422712\pi$
$338$	0	0
$339$	13.2337	0.718755
$340$	0	0
$341$	−23.5331	−1.27439
$342$	0	0
$343$	−17.6971	−0.955553
$344$	0	0
$345$	−0.413592	−0.0222670
$346$	0	0
$347$	−9.49207	−0.509561	−0.254781	−	0.966999i	$-0.582003\pi$
$347$	−9.49207	−0.509561	−0.254781	+	0.966999i	$0.582003\pi$
$348$	0	0
$349$	−9.91108	−0.530528	−0.265264	−	0.964176i	$-0.585459\pi$
$349$	−9.91108	−0.530528	−0.265264	+	0.964176i	$0.585459\pi$
$350$	0	0
$351$	−2.14893	−0.114702
$352$	0	0
$353$	−9.79466	−0.521317	−0.260658	−	0.965431i	$-0.583940\pi$
$353$	−9.79466	−0.521317	−0.260658	+	0.965431i	$0.583940\pi$
$354$	0	0
$355$	−0.405843	−0.0215399
$356$	0	0
$357$	−14.4393	−0.764211
$358$	0	0
$359$	18.3697	0.969513	0.484757	−	0.874649i	$-0.338908\pi$
$359$	18.3697	0.969513	0.484757	+	0.874649i	$0.338908\pi$
$360$	0	0
$361$	−3.45651	−0.181922
$362$	0	0
$363$	−18.2451	−0.957622
$364$	0	0
$365$	−2.82814	−0.148032
$366$	0	0
$367$	−30.7192	−1.60353	−0.801766	−	0.597639i	$-0.796106\pi$
$367$	−30.7192	−1.60353	−0.801766	+	0.597639i	$0.796106\pi$
$368$	0	0
$369$	−6.03343	−0.314088
$370$	0	0
$371$	−21.0951	−1.09521
$372$	0	0
$373$	9.55347	0.494660	0.247330	−	0.968931i	$-0.420447\pi$
$373$	9.55347	0.494660	0.247330	+	0.968931i	$0.420447\pi$
$374$	0	0
$375$	4.06517	0.209925
$376$	0	0
$377$	−2.14893	−0.110676
$378$	0	0
$379$	−18.9145	−0.971571	−0.485786	−	0.874078i	$-0.661466\pi$
$379$	−18.9145	−0.971571	−0.485786	+	0.874078i	$0.661466\pi$
$380$	0	0
$381$	−4.49520	−0.230296
$382$	0	0
$383$	−5.97880	−0.305502	−0.152751	−	0.988265i	$-0.548813\pi$
$383$	−5.97880	−0.305502	−0.152751	+	0.988265i	$0.548813\pi$
$384$	0	0
$385$	−6.15236	−0.313553
$386$	0	0
$387$	−3.64006	−0.185035
$388$	0	0
$389$	−30.1804	−1.53021	−0.765103	−	0.643908i	$-0.777312\pi$
$389$	−30.1804	−1.53021	−0.765103	+	0.643908i	$0.777312\pi$
$390$	0	0
$391$	5.24934	0.265471
$392$	0	0
$393$	16.6600	0.840388
$394$	0	0
$395$	1.24929	0.0628585
$396$	0	0
$397$	−24.2430	−1.21672	−0.608361	−	0.793660i	$-0.708173\pi$
$397$	−24.2430	−1.21672	−0.608361	+	0.793660i	$0.708173\pi$
$398$	0	0
$399$	10.8447	0.542913
$400$	0	0
$401$	12.0298	0.600739	0.300370	−	0.953823i	$-0.402890\pi$
$401$	12.0298	0.600739	0.300370	+	0.953823i	$0.402890\pi$
$402$	0	0
$403$	9.35138	0.465825
$404$	0	0
$405$	0.413592	0.0205516
$406$	0	0
$407$	14.7310	0.730187
$408$	0	0
$409$	11.8879	0.587821	0.293910	−	0.955833i	$-0.405043\pi$
$409$	11.8879	0.587821	0.293910	+	0.955833i	$0.405043\pi$
$410$	0	0
$411$	2.27426	0.112181
$412$	0	0
$413$	21.1096	1.03873
$414$	0	0
$415$	−0.344837	−0.0169274
$416$	0	0
$417$	1.41693	0.0693874
$418$	0	0
$419$	15.8820	0.775888	0.387944	−	0.921683i	$-0.373185\pi$
$419$	15.8820	0.775888	0.387944	+	0.921683i	$0.373185\pi$
$420$	0	0
$421$	−15.7258	−0.766430	−0.383215	−	0.923659i	$-0.625183\pi$
$421$	−15.7258	−0.766430	−0.383215	+	0.923659i	$0.625183\pi$
$422$	0	0
$423$	2.47763	0.120466
$424$	0	0
$425$	−25.3488	−1.22960
$426$	0	0
$427$	−19.0784	−0.923269
$428$	0	0
$429$	11.6212	0.561076
$430$	0	0
$431$	−2.48721	−0.119805	−0.0599024	−	0.998204i	$-0.519079\pi$
$431$	−2.48721	−0.119805	−0.0599024	+	0.998204i	$0.519079\pi$
$432$	0	0
$433$	22.8419	1.09771	0.548857	−	0.835917i	$-0.315063\pi$
$433$	22.8419	1.09771	0.548857	+	0.835917i	$0.315063\pi$
$434$	0	0
$435$	0.413592	0.0198302
$436$	0	0
$437$	−3.94252	−0.188596
$438$	0	0
$439$	3.32717	0.158797	0.0793987	−	0.996843i	$-0.474700\pi$
$439$	3.32717	0.158797	0.0793987	+	0.996843i	$0.474700\pi$
$440$	0	0
$441$	0.566323	0.0269678
$442$	0	0
$443$	−4.05630	−0.192721	−0.0963604	−	0.995347i	$-0.530720\pi$
$443$	−4.05630	−0.192721	−0.0963604	+	0.995347i	$0.530720\pi$
$444$	0	0
$445$	−2.06885	−0.0980731
$446$	0	0
$447$	7.50663	0.355051
$448$	0	0
$449$	36.8958	1.74122	0.870610	−	0.491974i	$-0.163724\pi$
$449$	36.8958	1.74122	0.870610	+	0.491974i	$0.163724\pi$
$450$	0	0
$451$	32.6281	1.53640
$452$	0	0
$453$	−19.0639	−0.895699
$454$	0	0
$455$	2.44477	0.114613
$456$	0	0
$457$	22.2602	1.04129	0.520643	−	0.853774i	$-0.325692\pi$
$457$	22.2602	1.04129	0.520643	+	0.853774i	$0.325692\pi$
$458$	0	0
$459$	−5.24934	−0.245018
$460$	0	0
$461$	−10.4495	−0.486684	−0.243342	−	0.969941i	$-0.578244\pi$
$461$	−10.4495	−0.486684	−0.243342	+	0.969941i	$0.578244\pi$
$462$	0	0
$463$	−40.0311	−1.86041	−0.930203	−	0.367047i	$-0.880369\pi$
$463$	−40.0311	−1.86041	−0.930203	+	0.367047i	$0.880369\pi$
$464$	0	0
$465$	−1.79980	−0.0834638
$466$	0	0
$467$	−15.3735	−0.711399	−0.355700	−	0.934600i	$-0.615757\pi$
$467$	−15.3735	−0.711399	−0.355700	+	0.934600i	$0.615757\pi$
$468$	0	0
$469$	28.3117	1.30731
$470$	0	0
$471$	−17.7593	−0.818305
$472$	0	0
$473$	19.6850	0.905119
$474$	0	0
$475$	19.0382	0.873533
$476$	0	0
$477$	−7.66902	−0.351140
$478$	0	0
$479$	−11.5928	−0.529688	−0.264844	−	0.964291i	$-0.585320\pi$
$479$	−11.5928	−0.529688	−0.264844	+	0.964291i	$0.585320\pi$
$480$	0	0
$481$	−5.85366	−0.266904
$482$	0	0
$483$	−2.75070	−0.125161
$484$	0	0
$485$	−4.83141	−0.219383
$486$	0	0
$487$	−29.4897	−1.33631	−0.668154	−	0.744023i	$-0.732915\pi$
$487$	−29.4897	−1.33631	−0.668154	+	0.744023i	$0.732915\pi$
$488$	0	0
$489$	12.7907	0.578414
$490$	0	0
$491$	11.7766	0.531472	0.265736	−	0.964046i	$-0.414385\pi$
$491$	11.7766	0.531472	0.265736	+	0.964046i	$0.414385\pi$
$492$	0	0
$493$	−5.24934	−0.236419
$494$	0	0
$495$	−2.23666	−0.100530
$496$	0	0
$497$	−2.69916	−0.121074
$498$	0	0
$499$	−29.4491	−1.31832	−0.659160	−	0.752002i	$-0.729088\pi$
$499$	−29.4491	−1.31832	−0.659160	+	0.752002i	$0.729088\pi$
$500$	0	0
$501$	6.05789	0.270647
$502$	0	0
$503$	4.57702	0.204079	0.102040	−	0.994780i	$-0.467463\pi$
$503$	4.57702	0.204079	0.102040	+	0.994780i	$0.467463\pi$
$504$	0	0
$505$	−4.42744	−0.197018
$506$	0	0
$507$	8.38208	0.372261
$508$	0	0
$509$	−27.0861	−1.20057	−0.600285	−	0.799786i	$-0.704946\pi$
$509$	−27.0861	−1.20057	−0.600285	+	0.799786i	$0.704946\pi$
$510$	0	0
$511$	−18.8092	−0.832072
$512$	0	0
$513$	3.94252	0.174067
$514$	0	0
$515$	2.23393	0.0984387
$516$	0	0
$517$	−13.3987	−0.589275
$518$	0	0
$519$	10.6465	0.467331
$520$	0	0
$521$	−39.6318	−1.73630	−0.868150	−	0.496303i	$-0.834691\pi$
$521$	−39.6318	−1.73630	−0.868150	+	0.496303i	$0.834691\pi$
$522$	0	0
$523$	9.48188	0.414614	0.207307	−	0.978276i	$-0.433530\pi$
$523$	9.48188	0.414614	0.207307	+	0.978276i	$0.433530\pi$
$524$	0	0
$525$	13.2829	0.579715
$526$	0	0
$527$	22.8432	0.995066
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	7.67427	0.333035
$532$	0	0
$533$	−12.9655	−0.561596
$534$	0	0
$535$	2.71729	0.117479
$536$	0	0
$537$	−2.02536	−0.0874005
$538$	0	0
$539$	−3.06261	−0.131916
$540$	0	0
$541$	−11.0625	−0.475613	−0.237806	−	0.971313i	$-0.576428\pi$
$541$	−11.0625	−0.475613	−0.237806	+	0.971313i	$0.576428\pi$
$542$	0	0
$543$	−8.61476	−0.369695
$544$	0	0
$545$	0.0323453	0.00138552
$546$	0	0
$547$	−30.6278	−1.30955	−0.654775	−	0.755824i	$-0.727237\pi$
$547$	−30.6278	−1.30955	−0.654775	+	0.755824i	$0.727237\pi$
$548$	0	0
$549$	−6.93585	−0.296015
$550$	0	0
$551$	3.94252	0.167957
$552$	0	0
$553$	8.30869	0.353322
$554$	0	0
$555$	1.12662	0.0478223
$556$	0	0
$557$	3.17601	0.134572	0.0672860	−	0.997734i	$-0.478566\pi$
$557$	3.17601	0.134572	0.0672860	+	0.997734i	$0.478566\pi$
$558$	0	0
$559$	−7.82226	−0.330846
$560$	0	0
$561$	28.3878	1.19853
$562$	0	0
$563$	−12.8333	−0.540857	−0.270429	−	0.962740i	$-0.587165\pi$
$563$	−12.8333	−0.540857	−0.270429	+	0.962740i	$0.587165\pi$
$564$	0	0
$565$	−5.47335	−0.230266
$566$	0	0
$567$	2.75070	0.115518
$568$	0	0
$569$	26.4461	1.10868	0.554338	−	0.832292i	$-0.312971\pi$
$569$	26.4461	1.10868	0.554338	+	0.832292i	$0.312971\pi$
$570$	0	0
$571$	6.77280	0.283433	0.141716	−	0.989907i	$-0.454738\pi$
$571$	6.77280	0.283433	0.141716	+	0.989907i	$0.454738\pi$
$572$	0	0
$573$	0.577000	0.0241045
$574$	0	0
$575$	−4.82894	−0.201381
$576$	0	0
$577$	1.17680	0.0489907	0.0244954	−	0.999700i	$-0.492202\pi$
$577$	1.17680	0.0489907	0.0244954	+	0.999700i	$0.492202\pi$
$578$	0	0
$579$	−2.92378	−0.121508
$580$	0	0
$581$	−2.29342	−0.0951472
$582$	0	0
$583$	41.4732	1.71764
$584$	0	0
$585$	0.888782	0.0367466
$586$	0	0
$587$	−11.6449	−0.480635	−0.240318	−	0.970694i	$-0.577252\pi$
$587$	−11.6449	−0.480635	−0.240318	+	0.970694i	$0.577252\pi$
$588$	0	0
$589$	−17.1564	−0.706918
$590$	0	0
$591$	20.4451	0.840999
$592$	0	0
$593$	−4.15677	−0.170698	−0.0853490	−	0.996351i	$-0.527201\pi$
$593$	−4.15677	−0.170698	−0.0853490	+	0.996351i	$0.527201\pi$
$594$	0	0
$595$	5.97200	0.244828
$596$	0	0
$597$	1.04208	0.0426495
$598$	0	0
$599$	6.35672	0.259728	0.129864	−	0.991532i	$-0.458546\pi$
$599$	6.35672	0.259728	0.129864	+	0.991532i	$0.458546\pi$
$600$	0	0
$601$	−13.7488	−0.560826	−0.280413	−	0.959880i	$-0.590471\pi$
$601$	−13.7488	−0.560826	−0.280413	+	0.959880i	$0.590471\pi$
$602$	0	0
$603$	10.2926	0.419145
$604$	0	0
$605$	7.54605	0.306791
$606$	0	0
$607$	−6.97089	−0.282940	−0.141470	−	0.989943i	$-0.545183\pi$
$607$	−6.97089	−0.282940	−0.141470	+	0.989943i	$0.545183\pi$
$608$	0	0
$609$	2.75070	0.111464
$610$	0	0
$611$	5.32426	0.215397
$612$	0	0
$613$	−14.5629	−0.588191	−0.294096	−	0.955776i	$-0.595018\pi$
$613$	−14.5629	−0.588191	−0.294096	+	0.955776i	$0.595018\pi$
$614$	0	0
$615$	2.49538	0.100623
$616$	0	0
$617$	10.4608	0.421138	0.210569	−	0.977579i	$-0.432468\pi$
$617$	10.4608	0.421138	0.210569	+	0.977579i	$0.432468\pi$
$618$	0	0
$619$	−35.8262	−1.43997	−0.719987	−	0.693987i	$-0.755852\pi$
$619$	−35.8262	−1.43997	−0.719987	+	0.693987i	$0.755852\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−13.7594	−0.551259
$624$	0	0
$625$	22.4634	0.898535
$626$	0	0
$627$	−21.3207	−0.851466
$628$	0	0
$629$	−14.2991	−0.570144
$630$	0	0
$631$	−22.4116	−0.892193	−0.446097	−	0.894985i	$-0.647186\pi$
$631$	−22.4116	−0.892193	−0.446097	+	0.894985i	$0.647186\pi$
$632$	0	0
$633$	−16.4609	−0.654262
$634$	0	0
$635$	1.85918	0.0737793
$636$	0	0
$637$	1.21699	0.0482190
$638$	0	0
$639$	−0.981264	−0.0388182
$640$	0	0
$641$	42.2654	1.66938	0.834692	−	0.550717i	$-0.185646\pi$
$641$	42.2654	1.66938	0.834692	+	0.550717i	$0.185646\pi$
$642$	0	0
$643$	−25.9281	−1.02250	−0.511251	−	0.859431i	$-0.670818\pi$
$643$	−25.9281	−1.02250	−0.511251	+	0.859431i	$0.670818\pi$
$644$	0	0
$645$	1.50550	0.0592791
$646$	0	0
$647$	−30.8953	−1.21462	−0.607310	−	0.794465i	$-0.707751\pi$
$647$	−30.8953	−1.21462	−0.607310	+	0.794465i	$0.707751\pi$
$648$	0	0
$649$	−41.5015	−1.62908
$650$	0	0
$651$	−11.9700	−0.469142
$652$	0	0
$653$	18.1136	0.708839	0.354419	−	0.935087i	$-0.384679\pi$
$653$	18.1136	0.708839	0.354419	+	0.935087i	$0.384679\pi$
$654$	0	0
$655$	−6.89046	−0.269233
$656$	0	0
$657$	−6.83799	−0.266776
$658$	0	0
$659$	22.9131	0.892567	0.446283	−	0.894892i	$-0.352747\pi$
$659$	22.9131	0.892567	0.446283	+	0.894892i	$0.352747\pi$
$660$	0	0
$661$	−24.2173	−0.941942	−0.470971	−	0.882149i	$-0.656096\pi$
$661$	−24.2173	−0.941942	−0.470971	+	0.882149i	$0.656096\pi$
$662$	0	0
$663$	−11.2805	−0.438098
$664$	0	0
$665$	−4.48527	−0.173931
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−18.7123	−0.723460
$670$	0	0
$671$	37.5082	1.44799
$672$	0	0
$673$	−47.7524	−1.84072	−0.920359	−	0.391073i	$-0.872104\pi$
$673$	−47.7524	−1.84072	−0.920359	+	0.391073i	$0.872104\pi$
$674$	0	0
$675$	4.82894	0.185866
$676$	0	0
$677$	−2.08884	−0.0802808	−0.0401404	−	0.999194i	$-0.512781\pi$
$677$	−2.08884	−0.0802808	−0.0401404	+	0.999194i	$0.512781\pi$
$678$	0	0
$679$	−32.1325	−1.23313
$680$	0	0
$681$	−2.90845	−0.111452
$682$	0	0
$683$	−16.0856	−0.615498	−0.307749	−	0.951468i	$-0.599576\pi$
$683$	−16.0856	−0.615498	−0.307749	+	0.951468i	$0.599576\pi$
$684$	0	0
$685$	−0.940614	−0.0359390
$686$	0	0
$687$	−7.60770	−0.290252
$688$	0	0
$689$	−16.4802	−0.627847
$690$	0	0
$691$	−14.3136	−0.544517	−0.272258	−	0.962224i	$-0.587771\pi$
$691$	−14.3136	−0.544517	−0.272258	+	0.962224i	$0.587771\pi$
$692$	0	0
$693$	−14.8754	−0.565071
$694$	0	0
$695$	−0.586031	−0.0222294
$696$	0	0
$697$	−31.6716	−1.19965
$698$	0	0
$699$	28.7075	1.08582
$700$	0	0
$701$	−34.8008	−1.31441	−0.657205	−	0.753712i	$-0.728261\pi$
$701$	−34.8008	−1.31441	−0.657205	+	0.753712i	$0.728261\pi$
$702$	0	0
$703$	10.7394	0.405043
$704$	0	0
$705$	−1.02473	−0.0385935
$706$	0	0
$707$	−29.4457	−1.10742
$708$	0	0
$709$	−10.2207	−0.383846	−0.191923	−	0.981410i	$-0.561472\pi$
$709$	−10.2207	−0.383846	−0.191923	+	0.981410i	$0.561472\pi$
$710$	0	0
$711$	3.02058	0.113281
$712$	0	0
$713$	4.35163	0.162970
$714$	0	0
$715$	−4.80643	−0.179750
$716$	0	0
$717$	−6.02058	−0.224843
$718$	0	0
$719$	−12.2122	−0.455439	−0.227719	−	0.973727i	$-0.573127\pi$
$719$	−12.2122	−0.455439	−0.227719	+	0.973727i	$0.573127\pi$
$720$	0	0
$721$	14.8573	0.553314
$722$	0	0
$723$	−18.4403	−0.685804
$724$	0	0
$725$	4.82894	0.179342
$726$	0	0
$727$	27.0485	1.00317	0.501586	−	0.865108i	$-0.332750\pi$
$727$	27.0485	1.00317	0.501586	+	0.865108i	$0.332750\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.1079	−0.706733
$732$	0	0
$733$	11.0713	0.408928	0.204464	−	0.978874i	$-0.434455\pi$
$733$	11.0713	0.408928	0.204464	+	0.978874i	$0.434455\pi$
$734$	0	0
$735$	−0.234227	−0.00863959
$736$	0	0
$737$	−55.6609	−2.05029
$738$	0	0
$739$	−26.0664	−0.958869	−0.479434	−	0.877578i	$-0.659158\pi$
$739$	−26.0664	−0.958869	−0.479434	+	0.877578i	$0.659158\pi$
$740$	0	0
$741$	8.47222	0.311235
$742$	0	0
$743$	−40.1217	−1.47192	−0.735962	−	0.677023i	$-0.763270\pi$
$743$	−40.1217	−1.47192	−0.735962	+	0.677023i	$0.763270\pi$
$744$	0	0
$745$	−3.10468	−0.113747
$746$	0	0
$747$	−0.833761	−0.0305057
$748$	0	0
$749$	18.0720	0.660337
$750$	0	0
$751$	21.5597	0.786726	0.393363	−	0.919383i	$-0.371312\pi$
$751$	21.5597	0.786726	0.393363	+	0.919383i	$0.371312\pi$
$752$	0	0
$753$	9.57148	0.348804
$754$	0	0
$755$	7.88467	0.286952
$756$	0	0
$757$	35.2200	1.28009	0.640047	−	0.768336i	$-0.278915\pi$
$757$	35.2200	1.28009	0.640047	+	0.768336i	$0.278915\pi$
$758$	0	0
$759$	5.40788	0.196294
$760$	0	0
$761$	−35.3427	−1.28117	−0.640585	−	0.767887i	$-0.721308\pi$
$761$	−35.3427	−1.28117	−0.640585	+	0.767887i	$0.721308\pi$
$762$	0	0
$763$	0.215120	0.00778788
$764$	0	0
$765$	2.17109	0.0784958
$766$	0	0
$767$	16.4915	0.595474
$768$	0	0
$769$	17.3197	0.624564	0.312282	−	0.949989i	$-0.398907\pi$
$769$	17.3197	0.624564	0.312282	+	0.949989i	$0.398907\pi$
$770$	0	0
$771$	−15.1906	−0.547076
$772$	0	0
$773$	−43.0548	−1.54857	−0.774287	−	0.632834i	$-0.781892\pi$
$773$	−43.0548	−1.54857	−0.774287	+	0.632834i	$0.781892\pi$
$774$	0	0
$775$	−21.0138	−0.754837
$776$	0	0
$777$	7.49285	0.268804
$778$	0	0
$779$	23.7869	0.852256
$780$	0	0
$781$	5.30656	0.189883
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	7.34511	0.262158
$786$	0	0
$787$	−15.6413	−0.557552	−0.278776	−	0.960356i	$-0.589929\pi$
$787$	−15.6413	−0.557552	−0.278776	+	0.960356i	$0.589929\pi$
$788$	0	0
$789$	27.9622	0.995479
$790$	0	0
$791$	−36.4018	−1.29430
$792$	0	0
$793$	−14.9047	−0.529281
$794$	0	0
$795$	3.17185	0.112494
$796$	0	0
$797$	0.738555	0.0261610	0.0130805	−	0.999914i	$-0.495836\pi$
$797$	0.738555	0.0261610	0.0130805	+	0.999914i	$0.495836\pi$
$798$	0	0
$799$	13.0059	0.460116
$800$	0	0
$801$	−5.00216	−0.176743
$802$	0	0
$803$	36.9790	1.30496
$804$	0	0
$805$	1.13767	0.0400974
$806$	0	0
$807$	22.3628	0.787208
$808$	0	0
$809$	0.366427	0.0128829	0.00644145	−	0.999979i	$-0.497950\pi$
$809$	0.366427	0.0128829	0.00644145	+	0.999979i	$0.497950\pi$
$810$	0	0
$811$	4.82115	0.169294	0.0846468	−	0.996411i	$-0.473024\pi$
$811$	4.82115	0.169294	0.0846468	+	0.996411i	$0.473024\pi$
$812$	0	0
$813$	2.45290	0.0860271
$814$	0	0
$815$	−5.29012	−0.185305
$816$	0	0
$817$	14.3510	0.502079
$818$	0	0
$819$	5.91106	0.206549
$820$	0	0
$821$	47.3002	1.65079	0.825394	−	0.564558i	$-0.190953\pi$
$821$	47.3002	1.65079	0.825394	+	0.564558i	$0.190953\pi$
$822$	0	0
$823$	−14.4791	−0.504709	−0.252354	−	0.967635i	$-0.581205\pi$
$823$	−14.4791	−0.504709	−0.252354	+	0.967635i	$0.581205\pi$
$824$	0	0
$825$	−26.1143	−0.909184
$826$	0	0
$827$	34.3250	1.19360	0.596798	−	0.802392i	$-0.296439\pi$
$827$	34.3250	1.19360	0.596798	+	0.802392i	$0.296439\pi$
$828$	0	0
$829$	8.30962	0.288605	0.144302	−	0.989534i	$-0.453906\pi$
$829$	8.30962	0.288605	0.144302	+	0.989534i	$0.453906\pi$
$830$	0	0
$831$	−1.25987	−0.0437044
$832$	0	0
$833$	2.97283	0.103002
$834$	0	0
$835$	−2.50550	−0.0867063
$836$	0	0
$837$	−4.35163	−0.150414
$838$	0	0
$839$	−48.9839	−1.69111	−0.845556	−	0.533887i	$-0.820731\pi$
$839$	−48.9839	−1.69111	−0.845556	+	0.533887i	$0.820731\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−16.9691	−0.584445
$844$	0	0
$845$	−3.46676	−0.119260
$846$	0	0
$847$	50.1868	1.72444
$848$	0	0
$849$	13.9411	0.478459
$850$	0	0
$851$	−2.72398	−0.0933770
$852$	0	0
$853$	−21.7630	−0.745152	−0.372576	−	0.928002i	$-0.621525\pi$
$853$	−21.7630	−0.745152	−0.372576	+	0.928002i	$0.621525\pi$
$854$	0	0
$855$	−1.63060	−0.0557652
$856$	0	0
$857$	−9.99374	−0.341380	−0.170690	−	0.985325i	$-0.554600\pi$
$857$	−9.99374	−0.341380	−0.170690	+	0.985325i	$0.554600\pi$
$858$	0	0
$859$	51.9696	1.77318	0.886590	−	0.462557i	$-0.153068\pi$
$859$	51.9696	1.77318	0.886590	+	0.462557i	$0.153068\pi$
$860$	0	0
$861$	16.5961	0.565595
$862$	0	0
$863$	26.2719	0.894305	0.447152	−	0.894458i	$-0.352438\pi$
$863$	26.2719	0.894305	0.447152	+	0.894458i	$0.352438\pi$
$864$	0	0
$865$	−4.40333	−0.149718
$866$	0	0
$867$	−10.5556	−0.358487
$868$	0	0
$869$	−16.3349	−0.554124
$870$	0	0
$871$	22.1180	0.749441
$872$	0	0
$873$	−11.6816	−0.395362
$874$	0	0
$875$	−11.1820	−0.378022
$876$	0	0
$877$	20.5178	0.692837	0.346418	−	0.938080i	$-0.387398\pi$
$877$	20.5178	0.692837	0.346418	+	0.938080i	$0.387398\pi$
$878$	0	0
$879$	22.0522	0.743804
$880$	0	0
$881$	−30.1070	−1.01433	−0.507165	−	0.861849i	$-0.669306\pi$
$881$	−30.1070	−1.01433	−0.507165	+	0.861849i	$0.669306\pi$
$882$	0	0
$883$	39.3130	1.32299	0.661493	−	0.749951i	$-0.269923\pi$
$883$	39.3130	1.32299	0.661493	+	0.749951i	$0.269923\pi$
$884$	0	0
$885$	−3.17402	−0.106693
$886$	0	0
$887$	−56.9041	−1.91065	−0.955326	−	0.295555i	$-0.904495\pi$
$887$	−56.9041	−1.91065	−0.955326	+	0.295555i	$0.904495\pi$
$888$	0	0
$889$	12.3649	0.414706
$890$	0	0
$891$	−5.40788	−0.181171
$892$	0	0
$893$	−9.76811	−0.326877
$894$	0	0
$895$	0.837671	0.0280003
$896$	0	0
$897$	−2.14893	−0.0717508
$898$	0	0
$899$	−4.35163	−0.145135
$900$	0	0
$901$	−40.2573	−1.34117
$902$	0	0
$903$	10.0127	0.333202
$904$	0	0
$905$	3.56300	0.118438
$906$	0	0
$907$	24.9777	0.829371	0.414685	−	0.909965i	$-0.363892\pi$
$907$	24.9777	0.829371	0.414685	+	0.909965i	$0.363892\pi$
$908$	0	0
$909$	−10.7048	−0.355057
$910$	0	0
$911$	30.1327	0.998340	0.499170	−	0.866504i	$-0.333638\pi$
$911$	30.1327	0.998340	0.499170	+	0.866504i	$0.333638\pi$
$912$	0	0
$913$	4.50888	0.149222
$914$	0	0
$915$	2.86861	0.0948334
$916$	0	0
$917$	−45.8267	−1.51333
$918$	0	0
$919$	29.8994	0.986291	0.493146	−	0.869947i	$-0.335847\pi$
$919$	29.8994	0.986291	0.493146	+	0.869947i	$0.335847\pi$
$920$	0	0
$921$	−32.1760	−1.06024
$922$	0	0
$923$	−2.10867	−0.0694078
$924$	0	0
$925$	13.1540	0.432500
$926$	0	0
$927$	5.40129	0.177401
$928$	0	0
$929$	33.0203	1.08336	0.541681	−	0.840584i	$-0.317788\pi$
$929$	33.0203	1.08336	0.541681	+	0.840584i	$0.317788\pi$
$930$	0	0
$931$	−2.23274	−0.0731752
$932$	0	0
$933$	−3.57851	−0.117155
$934$	0	0
$935$	−11.7410	−0.383971
$936$	0	0
$937$	−27.4957	−0.898245	−0.449122	−	0.893470i	$-0.648263\pi$
$937$	−27.4957	−0.898245	−0.449122	+	0.893470i	$0.648263\pi$
$938$	0	0
$939$	0.885822	0.0289077
$940$	0	0
$941$	−6.16565	−0.200994	−0.100497	−	0.994937i	$-0.532043\pi$
$941$	−6.16565	−0.200994	−0.100497	+	0.994937i	$0.532043\pi$
$942$	0	0
$943$	−6.03343	−0.196476
$944$	0	0
$945$	−1.13767	−0.0370083
$946$	0	0
$947$	30.2342	0.982479	0.491239	−	0.871025i	$-0.336544\pi$
$947$	30.2342	0.982479	0.491239	+	0.871025i	$0.336544\pi$
$948$	0	0
$949$	−14.6944	−0.477000
$950$	0	0
$951$	−2.62770	−0.0852090
$952$	0	0
$953$	−11.3177	−0.366616	−0.183308	−	0.983056i	$-0.558681\pi$
$953$	−11.3177	−0.366616	−0.183308	+	0.983056i	$0.558681\pi$
$954$	0	0
$955$	−0.238643	−0.00772229
$956$	0	0
$957$	−5.40788	−0.174812
$958$	0	0
$959$	−6.25578	−0.202010
$960$	0	0
$961$	−12.0633	−0.389138
$962$	0	0
$963$	6.56999	0.211715
$964$	0	0
$965$	1.20925	0.0389272
$966$	0	0
$967$	34.5548	1.11121	0.555603	−	0.831448i	$-0.312487\pi$
$967$	34.5548	1.11121	0.555603	+	0.831448i	$0.312487\pi$
$968$	0	0
$969$	20.6957	0.664840
$970$	0	0
$971$	17.7659	0.570135	0.285068	−	0.958507i	$-0.407984\pi$
$971$	17.7659	0.570135	0.285068	+	0.958507i	$0.407984\pi$
$972$	0	0
$973$	−3.89755	−0.124950
$974$	0	0
$975$	10.3771	0.332333
$976$	0	0
$977$	39.9876	1.27932	0.639659	−	0.768659i	$-0.279076\pi$
$977$	39.9876	1.27932	0.639659	+	0.768659i	$0.279076\pi$
$978$	0	0
$979$	27.0511	0.864556
$980$	0	0
$981$	0.0782059	0.00249692
$982$	0	0
$983$	40.6221	1.29564	0.647822	−	0.761792i	$-0.275680\pi$
$983$	40.6221	1.29564	0.647822	+	0.761792i	$0.275680\pi$
$984$	0	0
$985$	−8.45593	−0.269428
$986$	0	0
$987$	−6.81520	−0.216930
$988$	0	0
$989$	−3.64006	−0.115747
$990$	0	0
$991$	61.4622	1.95241	0.976205	−	0.216850i	$-0.0695782\pi$
$991$	61.4622	1.95241	0.976205	+	0.216850i	$0.0695782\pi$
$992$	0	0
$993$	9.42233	0.299009
$994$	0	0
$995$	−0.430996	−0.0136635
$996$	0	0
$997$	−16.5556	−0.524322	−0.262161	−	0.965024i	$-0.584435\pi$
$997$	−16.5556	−0.524322	−0.262161	+	0.965024i	$0.584435\pi$
$998$	0	0
$999$	2.72398	0.0861830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.8	✓	12	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.413592 q^{5} +2.75070 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.413592 q^{5} +2.75070 q^{7} +1.00000 q^{9} -5.40788 q^{11} +2.14893 q^{13} -0.413592 q^{15} +5.24934 q^{17} -3.94252 q^{19} -2.75070 q^{21} +1.00000 q^{23} -4.82894 q^{25} -1.00000 q^{27} -1.00000 q^{29} +4.35163 q^{31} +5.40788 q^{33} +1.13767 q^{35} -2.72398 q^{37} -2.14893 q^{39} -6.03343 q^{41} -3.64006 q^{43} +0.413592 q^{45} +2.47763 q^{47} +0.566323 q^{49} -5.24934 q^{51} -7.66902 q^{53} -2.23666 q^{55} +3.94252 q^{57} +7.67427 q^{59} -6.93585 q^{61} +2.75070 q^{63} +0.888782 q^{65} +10.2926 q^{67} -1.00000 q^{69} -0.981264 q^{71} -6.83799 q^{73} +4.82894 q^{75} -14.8754 q^{77} +3.02058 q^{79} +1.00000 q^{81} -0.833761 q^{83} +2.17109 q^{85} +1.00000 q^{87} -5.00216 q^{89} +5.91106 q^{91} -4.35163 q^{93} -1.63060 q^{95} -11.6816 q^{97} -5.40788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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