LMFDB - Embedded newform 8004.2.a.g.1.11

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			1.11
Root			$-2.64977$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.64977 q^{5} -1.93393 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.64977 q^{5} -1.93393 q^{7} +1.00000 q^{9} -1.24093 q^{11} -1.48851 q^{13} -2.64977 q^{15} -4.24112 q^{17} +2.85565 q^{19} +1.93393 q^{21} +1.00000 q^{23} +2.02127 q^{25} -1.00000 q^{27} -1.00000 q^{29} +8.48867 q^{31} +1.24093 q^{33} -5.12446 q^{35} +0.788034 q^{37} +1.48851 q^{39} -12.2376 q^{41} +11.0116 q^{43} +2.64977 q^{45} -4.51365 q^{47} -3.25992 q^{49} +4.24112 q^{51} +10.2067 q^{53} -3.28817 q^{55} -2.85565 q^{57} -1.07637 q^{59} -1.98032 q^{61} -1.93393 q^{63} -3.94421 q^{65} +0.801106 q^{67} -1.00000 q^{69} +1.24243 q^{71} +1.47031 q^{73} -2.02127 q^{75} +2.39986 q^{77} -12.7792 q^{79} +1.00000 q^{81} +3.32257 q^{83} -11.2380 q^{85} +1.00000 q^{87} +2.02980 q^{89} +2.87868 q^{91} -8.48867 q^{93} +7.56682 q^{95} +9.39099 q^{97} -1.24093 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.64977	1.18501	0.592506	−	0.805566i	$-0.298139\pi$
$5$	2.64977	1.18501	0.592506	+	0.805566i	$0.298139\pi$
$6$	0	0
$7$	−1.93393	−0.730956	−0.365478	−	0.930820i	$-0.619094\pi$
$7$	−1.93393	−0.730956	−0.365478	+	0.930820i	$0.619094\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.24093	−0.374154	−0.187077	−	0.982345i	$-0.559901\pi$
$11$	−1.24093	−0.374154	−0.187077	+	0.982345i	$0.559901\pi$
$12$	0	0
$13$	−1.48851	−0.412839	−0.206420	−	0.978464i	$-0.566181\pi$
$13$	−1.48851	−0.412839	−0.206420	+	0.978464i	$0.566181\pi$
$14$	0	0
$15$	−2.64977	−0.684167
$16$	0	0
$17$	−4.24112	−1.02862	−0.514311	−	0.857604i	$-0.671952\pi$
$17$	−4.24112	−1.02862	−0.514311	+	0.857604i	$0.671952\pi$
$18$	0	0
$19$	2.85565	0.655132	0.327566	−	0.944828i	$-0.393772\pi$
$19$	2.85565	0.655132	0.327566	+	0.944828i	$0.393772\pi$
$20$	0	0
$21$	1.93393	0.422018
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.02127	0.404254
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	8.48867	1.52461	0.762305	−	0.647217i	$-0.224067\pi$
$31$	8.48867	1.52461	0.762305	+	0.647217i	$0.224067\pi$
$32$	0	0
$33$	1.24093	0.216018
$34$	0	0
$35$	−5.12446	−0.866192
$36$	0	0
$37$	0.788034	0.129552	0.0647760	−	0.997900i	$-0.479367\pi$
$37$	0.788034	0.129552	0.0647760	+	0.997900i	$0.479367\pi$
$38$	0	0
$39$	1.48851	0.238353
$40$	0	0
$41$	−12.2376	−1.91119	−0.955597	−	0.294676i	$-0.904788\pi$
$41$	−12.2376	−1.91119	−0.955597	+	0.294676i	$0.904788\pi$
$42$	0	0
$43$	11.0116	1.67925	0.839625	−	0.543166i	$-0.182775\pi$
$43$	11.0116	1.67925	0.839625	+	0.543166i	$0.182775\pi$
$44$	0	0
$45$	2.64977	0.395004
$46$	0	0
$47$	−4.51365	−0.658383	−0.329192	−	0.944263i	$-0.606776\pi$
$47$	−4.51365	−0.658383	−0.329192	+	0.944263i	$0.606776\pi$
$48$	0	0
$49$	−3.25992	−0.465703
$50$	0	0
$51$	4.24112	0.593876
$52$	0	0
$53$	10.2067	1.40199	0.700997	−	0.713164i	$-0.252739\pi$
$53$	10.2067	1.40199	0.700997	+	0.713164i	$0.252739\pi$
$54$	0	0
$55$	−3.28817	−0.443377
$56$	0	0
$57$	−2.85565	−0.378241
$58$	0	0
$59$	−1.07637	−0.140131	−0.0700656	−	0.997542i	$-0.522321\pi$
$59$	−1.07637	−0.140131	−0.0700656	+	0.997542i	$0.522321\pi$
$60$	0	0
$61$	−1.98032	−0.253554	−0.126777	−	0.991931i	$-0.540463\pi$
$61$	−1.98032	−0.253554	−0.126777	+	0.991931i	$0.540463\pi$
$62$	0	0
$63$	−1.93393	−0.243652
$64$	0	0
$65$	−3.94421	−0.489219
$66$	0	0
$67$	0.801106	0.0978707	0.0489353	−	0.998802i	$-0.484417\pi$
$67$	0.801106	0.0978707	0.0489353	+	0.998802i	$0.484417\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	1.24243	0.147449	0.0737246	−	0.997279i	$-0.476511\pi$
$71$	1.24243	0.147449	0.0737246	+	0.997279i	$0.476511\pi$
$72$	0	0
$73$	1.47031	0.172087	0.0860436	−	0.996291i	$-0.472578\pi$
$73$	1.47031	0.172087	0.0860436	+	0.996291i	$0.472578\pi$
$74$	0	0
$75$	−2.02127	−0.233396
$76$	0	0
$77$	2.39986	0.273490
$78$	0	0
$79$	−12.7792	−1.43777	−0.718885	−	0.695129i	$-0.755347\pi$
$79$	−12.7792	−1.43777	−0.718885	+	0.695129i	$0.755347\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.32257	0.364699	0.182350	−	0.983234i	$-0.441630\pi$
$83$	3.32257	0.364699	0.182350	+	0.983234i	$0.441630\pi$
$84$	0	0
$85$	−11.2380	−1.21893
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	2.02980	0.215158	0.107579	−	0.994197i	$-0.465690\pi$
$89$	2.02980	0.215158	0.107579	+	0.994197i	$0.465690\pi$
$90$	0	0
$91$	2.87868	0.301767
$92$	0	0
$93$	−8.48867	−0.880234
$94$	0	0
$95$	7.56682	0.776339
$96$	0	0
$97$	9.39099	0.953511	0.476755	−	0.879036i	$-0.341813\pi$
$97$	9.39099	0.953511	0.476755	+	0.879036i	$0.341813\pi$
$98$	0	0
$99$	−1.24093	−0.124718
$100$	0	0
$101$	4.57021	0.454753	0.227376	−	0.973807i	$-0.426985\pi$
$101$	4.57021	0.454753	0.227376	+	0.973807i	$0.426985\pi$
$102$	0	0
$103$	−17.8586	−1.75966	−0.879828	−	0.475292i	$-0.842342\pi$
$103$	−17.8586	−1.75966	−0.879828	+	0.475292i	$0.842342\pi$
$104$	0	0
$105$	5.12446	0.500096
$106$	0	0
$107$	0.391113	0.0378104	0.0189052	−	0.999821i	$-0.493982\pi$
$107$	0.391113	0.0378104	0.0189052	+	0.999821i	$0.493982\pi$
$108$	0	0
$109$	4.81471	0.461166	0.230583	−	0.973053i	$-0.425937\pi$
$109$	4.81471	0.461166	0.230583	+	0.973053i	$0.425937\pi$
$110$	0	0
$111$	−0.788034	−0.0747968
$112$	0	0
$113$	−15.6910	−1.47609	−0.738044	−	0.674752i	$-0.764251\pi$
$113$	−15.6910	−1.47609	−0.738044	+	0.674752i	$0.764251\pi$
$114$	0	0
$115$	2.64977	0.247092
$116$	0	0
$117$	−1.48851	−0.137613
$118$	0	0
$119$	8.20202	0.751878
$120$	0	0
$121$	−9.46010	−0.860009
$122$	0	0
$123$	12.2376	1.10343
$124$	0	0
$125$	−7.89294	−0.705966
$126$	0	0
$127$	−17.7894	−1.57856	−0.789279	−	0.614035i	$-0.789546\pi$
$127$	−17.7894	−1.57856	−0.789279	+	0.614035i	$0.789546\pi$
$128$	0	0
$129$	−11.0116	−0.969516
$130$	0	0
$131$	−5.98647	−0.523040	−0.261520	−	0.965198i	$-0.584224\pi$
$131$	−5.98647	−0.523040	−0.261520	+	0.965198i	$0.584224\pi$
$132$	0	0
$133$	−5.52263	−0.478873
$134$	0	0
$135$	−2.64977	−0.228056
$136$	0	0
$137$	−15.5240	−1.32630	−0.663151	−	0.748486i	$-0.730781\pi$
$137$	−15.5240	−1.32630	−0.663151	+	0.748486i	$0.730781\pi$
$138$	0	0
$139$	−15.7224	−1.33355	−0.666777	−	0.745257i	$-0.732327\pi$
$139$	−15.7224	−1.33355	−0.666777	+	0.745257i	$0.732327\pi$
$140$	0	0
$141$	4.51365	0.380118
$142$	0	0
$143$	1.84714	0.154465
$144$	0	0
$145$	−2.64977	−0.220051
$146$	0	0
$147$	3.25992	0.268874
$148$	0	0
$149$	−8.87792	−0.727307	−0.363654	−	0.931534i	$-0.618471\pi$
$149$	−8.87792	−0.727307	−0.363654	+	0.931534i	$0.618471\pi$
$150$	0	0
$151$	4.12277	0.335506	0.167753	−	0.985829i	$-0.446349\pi$
$151$	4.12277	0.335506	0.167753	+	0.985829i	$0.446349\pi$
$152$	0	0
$153$	−4.24112	−0.342874
$154$	0	0
$155$	22.4930	1.80668
$156$	0	0
$157$	−12.9881	−1.03656	−0.518281	−	0.855211i	$-0.673428\pi$
$157$	−12.9881	−1.03656	−0.518281	+	0.855211i	$0.673428\pi$
$158$	0	0
$159$	−10.2067	−0.809441
$160$	0	0
$161$	−1.93393	−0.152415
$162$	0	0
$163$	6.56405	0.514136	0.257068	−	0.966393i	$-0.417244\pi$
$163$	6.56405	0.514136	0.257068	+	0.966393i	$0.417244\pi$
$164$	0	0
$165$	3.28817	0.255984
$166$	0	0
$167$	19.7215	1.52610	0.763050	−	0.646340i	$-0.223701\pi$
$167$	19.7215	1.52610	0.763050	+	0.646340i	$0.223701\pi$
$168$	0	0
$169$	−10.7843	−0.829564
$170$	0	0
$171$	2.85565	0.218377
$172$	0	0
$173$	18.6036	1.41440	0.707202	−	0.707012i	$-0.249957\pi$
$173$	18.6036	1.41440	0.707202	+	0.707012i	$0.249957\pi$
$174$	0	0
$175$	−3.90899	−0.295492
$176$	0	0
$177$	1.07637	0.0809047
$178$	0	0
$179$	12.7500	0.952976	0.476488	−	0.879181i	$-0.341909\pi$
$179$	12.7500	0.952976	0.476488	+	0.879181i	$0.341909\pi$
$180$	0	0
$181$	−20.1955	−1.50112	−0.750561	−	0.660801i	$-0.770217\pi$
$181$	−20.1955	−1.50112	−0.750561	+	0.660801i	$0.770217\pi$
$182$	0	0
$183$	1.98032	0.146389
$184$	0	0
$185$	2.08811	0.153521
$186$	0	0
$187$	5.26292	0.384863
$188$	0	0
$189$	1.93393	0.140673
$190$	0	0
$191$	−2.95262	−0.213644	−0.106822	−	0.994278i	$-0.534068\pi$
$191$	−2.95262	−0.213644	−0.106822	+	0.994278i	$0.534068\pi$
$192$	0	0
$193$	10.9562	0.788648	0.394324	−	0.918972i	$-0.370979\pi$
$193$	10.9562	0.788648	0.394324	+	0.918972i	$0.370979\pi$
$194$	0	0
$195$	3.94421	0.282451
$196$	0	0
$197$	−27.2203	−1.93936	−0.969682	−	0.244372i	$-0.921418\pi$
$197$	−27.2203	−1.93936	−0.969682	+	0.244372i	$0.921418\pi$
$198$	0	0
$199$	−1.73732	−0.123156	−0.0615778	−	0.998102i	$-0.519613\pi$
$199$	−1.73732	−0.123156	−0.0615778	+	0.998102i	$0.519613\pi$
$200$	0	0
$201$	−0.801106	−0.0565057
$202$	0	0
$203$	1.93393	0.135735
$204$	0	0
$205$	−32.4268	−2.26479
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−3.54366	−0.245120
$210$	0	0
$211$	15.0518	1.03621	0.518105	−	0.855317i	$-0.326638\pi$
$211$	15.0518	1.03621	0.518105	+	0.855317i	$0.326638\pi$
$212$	0	0
$213$	−1.24243	−0.0851298
$214$	0	0
$215$	29.1781	1.98993
$216$	0	0
$217$	−16.4165	−1.11442
$218$	0	0
$219$	−1.47031	−0.0993546
$220$	0	0
$221$	6.31296	0.424656
$222$	0	0
$223$	−9.96082	−0.667026	−0.333513	−	0.942746i	$-0.608234\pi$
$223$	−9.96082	−0.667026	−0.333513	+	0.942746i	$0.608234\pi$
$224$	0	0
$225$	2.02127	0.134751
$226$	0	0
$227$	−6.11865	−0.406109	−0.203054	−	0.979167i	$-0.565087\pi$
$227$	−6.11865	−0.406109	−0.203054	+	0.979167i	$0.565087\pi$
$228$	0	0
$229$	−14.5804	−0.963500	−0.481750	−	0.876309i	$-0.659999\pi$
$229$	−14.5804	−0.963500	−0.481750	+	0.876309i	$0.659999\pi$
$230$	0	0
$231$	−2.39986	−0.157899
$232$	0	0
$233$	−15.2715	−1.00047	−0.500234	−	0.865890i	$-0.666753\pi$
$233$	−15.2715	−1.00047	−0.500234	+	0.865890i	$0.666753\pi$
$234$	0	0
$235$	−11.9601	−0.780192
$236$	0	0
$237$	12.7792	0.830097
$238$	0	0
$239$	−5.61982	−0.363516	−0.181758	−	0.983343i	$-0.558179\pi$
$239$	−5.61982	−0.363516	−0.181758	+	0.983343i	$0.558179\pi$
$240$	0	0
$241$	15.6192	1.00612	0.503059	−	0.864252i	$-0.332208\pi$
$241$	15.6192	1.00612	0.503059	+	0.864252i	$0.332208\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−8.63804	−0.551864
$246$	0	0
$247$	−4.25068	−0.270464
$248$	0	0
$249$	−3.32257	−0.210559
$250$	0	0
$251$	−10.7037	−0.675613	−0.337807	−	0.941216i	$-0.609685\pi$
$251$	−10.7037	−0.675613	−0.337807	+	0.941216i	$0.609685\pi$
$252$	0	0
$253$	−1.24093	−0.0780164
$254$	0	0
$255$	11.2380	0.703750
$256$	0	0
$257$	−3.36904	−0.210155	−0.105077	−	0.994464i	$-0.533509\pi$
$257$	−3.36904	−0.210155	−0.105077	+	0.994464i	$0.533509\pi$
$258$	0	0
$259$	−1.52400	−0.0946967
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−16.2821	−1.00399	−0.501997	−	0.864869i	$-0.667401\pi$
$263$	−16.2821	−1.00399	−0.501997	+	0.864869i	$0.667401\pi$
$264$	0	0
$265$	27.0453	1.66138
$266$	0	0
$267$	−2.02980	−0.124222
$268$	0	0
$269$	−1.07002	−0.0652406	−0.0326203	−	0.999468i	$-0.510385\pi$
$269$	−1.07002	−0.0652406	−0.0326203	+	0.999468i	$0.510385\pi$
$270$	0	0
$271$	27.4839	1.66953	0.834765	−	0.550606i	$-0.185603\pi$
$271$	27.4839	1.66953	0.834765	+	0.550606i	$0.185603\pi$
$272$	0	0
$273$	−2.87868	−0.174225
$274$	0	0
$275$	−2.50825	−0.151253
$276$	0	0
$277$	15.6321	0.939245	0.469622	−	0.882867i	$-0.344390\pi$
$277$	15.6321	0.939245	0.469622	+	0.882867i	$0.344390\pi$
$278$	0	0
$279$	8.48867	0.508204
$280$	0	0
$281$	−2.26311	−0.135006	−0.0675028	−	0.997719i	$-0.521503\pi$
$281$	−2.26311	−0.135006	−0.0675028	+	0.997719i	$0.521503\pi$
$282$	0	0
$283$	31.0378	1.84501	0.922503	−	0.385991i	$-0.126140\pi$
$283$	31.0378	1.84501	0.922503	+	0.385991i	$0.126140\pi$
$284$	0	0
$285$	−7.56682	−0.448220
$286$	0	0
$287$	23.6667	1.39700
$288$	0	0
$289$	0.987100	0.0580647
$290$	0	0
$291$	−9.39099	−0.550510
$292$	0	0
$293$	−7.01724	−0.409952	−0.204976	−	0.978767i	$-0.565712\pi$
$293$	−7.01724	−0.409952	−0.204976	+	0.978767i	$0.565712\pi$
$294$	0	0
$295$	−2.85212	−0.166057
$296$	0	0
$297$	1.24093	0.0720059
$298$	0	0
$299$	−1.48851	−0.0860829
$300$	0	0
$301$	−21.2956	−1.22746
$302$	0	0
$303$	−4.57021	−0.262552
$304$	0	0
$305$	−5.24739	−0.300464
$306$	0	0
$307$	7.12070	0.406400	0.203200	−	0.979137i	$-0.434866\pi$
$307$	7.12070	0.406400	0.203200	+	0.979137i	$0.434866\pi$
$308$	0	0
$309$	17.8586	1.01594
$310$	0	0
$311$	3.14032	0.178071	0.0890357	−	0.996028i	$-0.471621\pi$
$311$	3.14032	0.178071	0.0890357	+	0.996028i	$0.471621\pi$
$312$	0	0
$313$	21.0426	1.18940	0.594700	−	0.803947i	$-0.297271\pi$
$313$	21.0426	1.18940	0.594700	+	0.803947i	$0.297271\pi$
$314$	0	0
$315$	−5.12446	−0.288731
$316$	0	0
$317$	7.50308	0.421415	0.210708	−	0.977549i	$-0.432423\pi$
$317$	7.50308	0.421415	0.210708	+	0.977549i	$0.432423\pi$
$318$	0	0
$319$	1.24093	0.0694786
$320$	0	0
$321$	−0.391113	−0.0218298
$322$	0	0
$323$	−12.1112	−0.673884
$324$	0	0
$325$	−3.00869	−0.166892
$326$	0	0
$327$	−4.81471	−0.266254
$328$	0	0
$329$	8.72907	0.481249
$330$	0	0
$331$	10.2367	0.562661	0.281331	−	0.959611i	$-0.409224\pi$
$331$	10.2367	0.562661	0.281331	+	0.959611i	$0.409224\pi$
$332$	0	0
$333$	0.788034	0.0431840
$334$	0	0
$335$	2.12274	0.115978
$336$	0	0
$337$	−23.9208	−1.30305	−0.651525	−	0.758628i	$-0.725870\pi$
$337$	−23.9208	−1.30305	−0.651525	+	0.758628i	$0.725870\pi$
$338$	0	0
$339$	15.6910	0.852220
$340$	0	0
$341$	−10.5338	−0.570439
$342$	0	0
$343$	19.8420	1.07136
$344$	0	0
$345$	−2.64977	−0.142659
$346$	0	0
$347$	31.6145	1.69716	0.848578	−	0.529071i	$-0.177459\pi$
$347$	31.6145	1.69716	0.848578	+	0.529071i	$0.177459\pi$
$348$	0	0
$349$	−10.5683	−0.565710	−0.282855	−	0.959163i	$-0.591282\pi$
$349$	−10.5683	−0.565710	−0.282855	+	0.959163i	$0.591282\pi$
$350$	0	0
$351$	1.48851	0.0794509
$352$	0	0
$353$	−18.2584	−0.971794	−0.485897	−	0.874016i	$-0.661507\pi$
$353$	−18.2584	−0.971794	−0.485897	+	0.874016i	$0.661507\pi$
$354$	0	0
$355$	3.29215	0.174729
$356$	0	0
$357$	−8.20202	−0.434097
$358$	0	0
$359$	−24.9980	−1.31935	−0.659673	−	0.751553i	$-0.729305\pi$
$359$	−24.9980	−1.31935	−0.659673	+	0.751553i	$0.729305\pi$
$360$	0	0
$361$	−10.8452	−0.570802
$362$	0	0
$363$	9.46010	0.496526
$364$	0	0
$365$	3.89599	0.203925
$366$	0	0
$367$	−11.2238	−0.585879	−0.292939	−	0.956131i	$-0.594633\pi$
$367$	−11.2238	−0.585879	−0.292939	+	0.956131i	$0.594633\pi$
$368$	0	0
$369$	−12.2376	−0.637065
$370$	0	0
$371$	−19.7390	−1.02480
$372$	0	0
$373$	17.8707	0.925307	0.462654	−	0.886539i	$-0.346897\pi$
$373$	17.8707	0.925307	0.462654	+	0.886539i	$0.346897\pi$
$374$	0	0
$375$	7.89294	0.407590
$376$	0	0
$377$	1.48851	0.0766623
$378$	0	0
$379$	−11.7746	−0.604820	−0.302410	−	0.953178i	$-0.597791\pi$
$379$	−11.7746	−0.604820	−0.302410	+	0.953178i	$0.597791\pi$
$380$	0	0
$381$	17.7894	0.911381
$382$	0	0
$383$	−11.8807	−0.607076	−0.303538	−	0.952819i	$-0.598168\pi$
$383$	−11.8807	−0.607076	−0.303538	+	0.952819i	$0.598168\pi$
$384$	0	0
$385$	6.35908	0.324089
$386$	0	0
$387$	11.0116	0.559750
$388$	0	0
$389$	−34.3701	−1.74264	−0.871318	−	0.490719i	$-0.836734\pi$
$389$	−34.3701	−1.74264	−0.871318	+	0.490719i	$0.836734\pi$
$390$	0	0
$391$	−4.24112	−0.214483
$392$	0	0
$393$	5.98647	0.301978
$394$	0	0
$395$	−33.8619	−1.70378
$396$	0	0
$397$	−26.4584	−1.32791	−0.663955	−	0.747772i	$-0.731123\pi$
$397$	−26.4584	−1.32791	−0.663955	+	0.747772i	$0.731123\pi$
$398$	0	0
$399$	5.52263	0.276477
$400$	0	0
$401$	27.3683	1.36671	0.683354	−	0.730087i	$-0.260520\pi$
$401$	27.3683	1.36671	0.683354	+	0.730087i	$0.260520\pi$
$402$	0	0
$403$	−12.6355	−0.629419
$404$	0	0
$405$	2.64977	0.131668
$406$	0	0
$407$	−0.977892	−0.0484723
$408$	0	0
$409$	2.46181	0.121729	0.0608644	−	0.998146i	$-0.480614\pi$
$409$	2.46181	0.121729	0.0608644	+	0.998146i	$0.480614\pi$
$410$	0	0
$411$	15.5240	0.765741
$412$	0	0
$413$	2.08162	0.102430
$414$	0	0
$415$	8.80403	0.432173
$416$	0	0
$417$	15.7224	0.769928
$418$	0	0
$419$	6.50269	0.317677	0.158839	−	0.987305i	$-0.449225\pi$
$419$	6.50269	0.317677	0.158839	+	0.987305i	$0.449225\pi$
$420$	0	0
$421$	−18.3109	−0.892419	−0.446210	−	0.894928i	$-0.647226\pi$
$421$	−18.3109	−0.892419	−0.446210	+	0.894928i	$0.647226\pi$
$422$	0	0
$423$	−4.51365	−0.219461
$424$	0	0
$425$	−8.57245	−0.415825
$426$	0	0
$427$	3.82979	0.185337
$428$	0	0
$429$	−1.84714	−0.0891806
$430$	0	0
$431$	−16.9720	−0.817513	−0.408757	−	0.912643i	$-0.634038\pi$
$431$	−16.9720	−0.817513	−0.408757	+	0.912643i	$0.634038\pi$
$432$	0	0
$433$	−24.3836	−1.17180	−0.585900	−	0.810383i	$-0.699259\pi$
$433$	−24.3836	−1.17180	−0.585900	+	0.810383i	$0.699259\pi$
$434$	0	0
$435$	2.64977	0.127047
$436$	0	0
$437$	2.85565	0.136604
$438$	0	0
$439$	−4.70699	−0.224653	−0.112326	−	0.993671i	$-0.535830\pi$
$439$	−4.70699	−0.224653	−0.112326	+	0.993671i	$0.535830\pi$
$440$	0	0
$441$	−3.25992	−0.155234
$442$	0	0
$443$	−8.69144	−0.412943	−0.206471	−	0.978453i	$-0.566198\pi$
$443$	−8.69144	−0.412943	−0.206471	+	0.978453i	$0.566198\pi$
$444$	0	0
$445$	5.37850	0.254965
$446$	0	0
$447$	8.87792	0.419911
$448$	0	0
$449$	−34.5410	−1.63009	−0.815045	−	0.579398i	$-0.803288\pi$
$449$	−34.5410	−1.63009	−0.815045	+	0.579398i	$0.803288\pi$
$450$	0	0
$451$	15.1860	0.715080
$452$	0	0
$453$	−4.12277	−0.193705
$454$	0	0
$455$	7.62782	0.357598
$456$	0	0
$457$	−26.5728	−1.24302	−0.621511	−	0.783405i	$-0.713481\pi$
$457$	−26.5728	−1.24302	−0.621511	+	0.783405i	$0.713481\pi$
$458$	0	0
$459$	4.24112	0.197959
$460$	0	0
$461$	13.7477	0.640295	0.320147	−	0.947368i	$-0.396268\pi$
$461$	13.7477	0.640295	0.320147	+	0.947368i	$0.396268\pi$
$462$	0	0
$463$	25.9111	1.20419	0.602095	−	0.798424i	$-0.294333\pi$
$463$	25.9111	1.20419	0.602095	+	0.798424i	$0.294333\pi$
$464$	0	0
$465$	−22.4930	−1.04309
$466$	0	0
$467$	21.1330	0.977920	0.488960	−	0.872306i	$-0.337376\pi$
$467$	21.1330	0.977920	0.488960	+	0.872306i	$0.337376\pi$
$468$	0	0
$469$	−1.54928	−0.0715391
$470$	0	0
$471$	12.9881	0.598459
$472$	0	0
$473$	−13.6646	−0.628298
$474$	0	0
$475$	5.77205	0.264840
$476$	0	0
$477$	10.2067	0.467331
$478$	0	0
$479$	−20.4547	−0.934597	−0.467299	−	0.884100i	$-0.654773\pi$
$479$	−20.4547	−0.934597	−0.467299	+	0.884100i	$0.654773\pi$
$480$	0	0
$481$	−1.17300	−0.0534841
$482$	0	0
$483$	1.93393	0.0879968
$484$	0	0
$485$	24.8840	1.12992
$486$	0	0
$487$	−8.57065	−0.388373	−0.194187	−	0.980965i	$-0.562207\pi$
$487$	−8.57065	−0.388373	−0.194187	+	0.980965i	$0.562207\pi$
$488$	0	0
$489$	−6.56405	−0.296837
$490$	0	0
$491$	−40.2177	−1.81500	−0.907500	−	0.420051i	$-0.862012\pi$
$491$	−40.2177	−1.81500	−0.907500	+	0.420051i	$0.862012\pi$
$492$	0	0
$493$	4.24112	0.191010
$494$	0	0
$495$	−3.28817	−0.147792
$496$	0	0
$497$	−2.40277	−0.107779
$498$	0	0
$499$	−26.5415	−1.18816	−0.594080	−	0.804406i	$-0.702484\pi$
$499$	−26.5415	−1.18816	−0.594080	+	0.804406i	$0.702484\pi$
$500$	0	0
$501$	−19.7215	−0.881094
$502$	0	0
$503$	1.45303	0.0647872	0.0323936	−	0.999475i	$-0.489687\pi$
$503$	1.45303	0.0647872	0.0323936	+	0.999475i	$0.489687\pi$
$504$	0	0
$505$	12.1100	0.538887
$506$	0	0
$507$	10.7843	0.478949
$508$	0	0
$509$	−36.2225	−1.60553	−0.802767	−	0.596293i	$-0.796640\pi$
$509$	−36.2225	−1.60553	−0.802767	+	0.596293i	$0.796640\pi$
$510$	0	0
$511$	−2.84348	−0.125788
$512$	0	0
$513$	−2.85565	−0.126080
$514$	0	0
$515$	−47.3210	−2.08521
$516$	0	0
$517$	5.60111	0.246337
$518$	0	0
$519$	−18.6036	−0.816606
$520$	0	0
$521$	31.9431	1.39945	0.699727	−	0.714410i	$-0.253305\pi$
$521$	31.9431	1.39945	0.699727	+	0.714410i	$0.253305\pi$
$522$	0	0
$523$	−9.46920	−0.414059	−0.207030	−	0.978335i	$-0.566380\pi$
$523$	−9.46920	−0.414059	−0.207030	+	0.978335i	$0.566380\pi$
$524$	0	0
$525$	3.90899	0.170602
$526$	0	0
$527$	−36.0015	−1.56825
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.07637	−0.0467104
$532$	0	0
$533$	18.2158	0.789016
$534$	0	0
$535$	1.03636	0.0448057
$536$	0	0
$537$	−12.7500	−0.550201
$538$	0	0
$539$	4.04533	0.174245
$540$	0	0
$541$	−21.0013	−0.902917	−0.451458	−	0.892292i	$-0.649096\pi$
$541$	−21.0013	−0.902917	−0.451458	+	0.892292i	$0.649096\pi$
$542$	0	0
$543$	20.1955	0.866673
$544$	0	0
$545$	12.7579	0.546487
$546$	0	0
$547$	10.7155	0.458161	0.229080	−	0.973408i	$-0.426428\pi$
$547$	10.7155	0.458161	0.229080	+	0.973408i	$0.426428\pi$
$548$	0	0
$549$	−1.98032	−0.0845180
$550$	0	0
$551$	−2.85565	−0.121655
$552$	0	0
$553$	24.7140	1.05095
$554$	0	0
$555$	−2.08811	−0.0886352
$556$	0	0
$557$	−3.64952	−0.154635	−0.0773175	−	0.997007i	$-0.524636\pi$
$557$	−3.64952	−0.154635	−0.0773175	+	0.997007i	$0.524636\pi$
$558$	0	0
$559$	−16.3909	−0.693260
$560$	0	0
$561$	−5.26292	−0.222201
$562$	0	0
$563$	31.4639	1.32605	0.663023	−	0.748599i	$-0.269273\pi$
$563$	31.4639	1.32605	0.663023	+	0.748599i	$0.269273\pi$
$564$	0	0
$565$	−41.5776	−1.74918
$566$	0	0
$567$	−1.93393	−0.0812173
$568$	0	0
$569$	7.53105	0.315718	0.157859	−	0.987462i	$-0.449541\pi$
$569$	7.53105	0.315718	0.157859	+	0.987462i	$0.449541\pi$
$570$	0	0
$571$	−15.5206	−0.649519	−0.324759	−	0.945797i	$-0.605283\pi$
$571$	−15.5206	−0.649519	−0.324759	+	0.945797i	$0.605283\pi$
$572$	0	0
$573$	2.95262	0.123348
$574$	0	0
$575$	2.02127	0.0842928
$576$	0	0
$577$	−28.8794	−1.20226	−0.601132	−	0.799150i	$-0.705283\pi$
$577$	−28.8794	−1.20226	−0.601132	+	0.799150i	$0.705283\pi$
$578$	0	0
$579$	−10.9562	−0.455326
$580$	0	0
$581$	−6.42560	−0.266579
$582$	0	0
$583$	−12.6657	−0.524561
$584$	0	0
$585$	−3.94421	−0.163073
$586$	0	0
$587$	8.03989	0.331842	0.165921	−	0.986139i	$-0.446940\pi$
$587$	8.03989	0.331842	0.165921	+	0.986139i	$0.446940\pi$
$588$	0	0
$589$	24.2407	0.998821
$590$	0	0
$591$	27.2203	1.11969
$592$	0	0
$593$	−38.0061	−1.56073	−0.780363	−	0.625327i	$-0.784966\pi$
$593$	−38.0061	−1.56073	−0.780363	+	0.625327i	$0.784966\pi$
$594$	0	0
$595$	21.7334	0.890985
$596$	0	0
$597$	1.73732	0.0711039
$598$	0	0
$599$	23.9171	0.977225	0.488613	−	0.872501i	$-0.337503\pi$
$599$	23.9171	0.977225	0.488613	+	0.872501i	$0.337503\pi$
$600$	0	0
$601$	29.4240	1.20023	0.600116	−	0.799913i	$-0.295121\pi$
$601$	29.4240	1.20023	0.600116	+	0.799913i	$0.295121\pi$
$602$	0	0
$603$	0.801106	0.0326236
$604$	0	0
$605$	−25.0671	−1.01912
$606$	0	0
$607$	42.3530	1.71905	0.859527	−	0.511090i	$-0.170758\pi$
$607$	42.3530	1.71905	0.859527	+	0.511090i	$0.170758\pi$
$608$	0	0
$609$	−1.93393	−0.0783667
$610$	0	0
$611$	6.71862	0.271806
$612$	0	0
$613$	31.1855	1.25957	0.629784	−	0.776770i	$-0.283143\pi$
$613$	31.1855	1.25957	0.629784	+	0.776770i	$0.283143\pi$
$614$	0	0
$615$	32.4268	1.30758
$616$	0	0
$617$	−24.3940	−0.982065	−0.491032	−	0.871141i	$-0.663380\pi$
$617$	−24.3940	−0.982065	−0.491032	+	0.871141i	$0.663380\pi$
$618$	0	0
$619$	−0.273497	−0.0109928	−0.00549639	−	0.999985i	$-0.501750\pi$
$619$	−0.273497	−0.0109928	−0.00549639	+	0.999985i	$0.501750\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−3.92549	−0.157271
$624$	0	0
$625$	−31.0208	−1.24083
$626$	0	0
$627$	3.54366	0.141520
$628$	0	0
$629$	−3.34214	−0.133260
$630$	0	0
$631$	−5.91729	−0.235564	−0.117782	−	0.993039i	$-0.537578\pi$
$631$	−5.91729	−0.235564	−0.117782	+	0.993039i	$0.537578\pi$
$632$	0	0
$633$	−15.0518	−0.598256
$634$	0	0
$635$	−47.1379	−1.87061
$636$	0	0
$637$	4.85244	0.192261
$638$	0	0
$639$	1.24243	0.0491497
$640$	0	0
$641$	−29.9256	−1.18199	−0.590994	−	0.806676i	$-0.701264\pi$
$641$	−29.9256	−1.18199	−0.590994	+	0.806676i	$0.701264\pi$
$642$	0	0
$643$	20.9811	0.827413	0.413706	−	0.910410i	$-0.364234\pi$
$643$	20.9811	0.827413	0.413706	+	0.910410i	$0.364234\pi$
$644$	0	0
$645$	−29.1781	−1.14889
$646$	0	0
$647$	−15.5729	−0.612236	−0.306118	−	0.951994i	$-0.599030\pi$
$647$	−15.5729	−0.612236	−0.306118	+	0.951994i	$0.599030\pi$
$648$	0	0
$649$	1.33569	0.0524306
$650$	0	0
$651$	16.4165	0.643413
$652$	0	0
$653$	33.4932	1.31069	0.655345	−	0.755330i	$-0.272523\pi$
$653$	33.4932	1.31069	0.655345	+	0.755330i	$0.272523\pi$
$654$	0	0
$655$	−15.8628	−0.619809
$656$	0	0
$657$	1.47031	0.0573624
$658$	0	0
$659$	36.6739	1.42861	0.714305	−	0.699834i	$-0.246743\pi$
$659$	36.6739	1.42861	0.714305	+	0.699834i	$0.246743\pi$
$660$	0	0
$661$	−19.4626	−0.757007	−0.378504	−	0.925600i	$-0.623561\pi$
$661$	−19.4626	−0.757007	−0.378504	+	0.925600i	$0.623561\pi$
$662$	0	0
$663$	−6.31296	−0.245175
$664$	0	0
$665$	−14.6337	−0.567470
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	9.96082	0.385108
$670$	0	0
$671$	2.45743	0.0948681
$672$	0	0
$673$	7.20582	0.277764	0.138882	−	0.990309i	$-0.455649\pi$
$673$	7.20582	0.277764	0.138882	+	0.990309i	$0.455649\pi$
$674$	0	0
$675$	−2.02127	−0.0777987
$676$	0	0
$677$	−19.7475	−0.758959	−0.379479	−	0.925200i	$-0.623897\pi$
$677$	−19.7475	−0.758959	−0.379479	+	0.925200i	$0.623897\pi$
$678$	0	0
$679$	−18.1615	−0.696974
$680$	0	0
$681$	6.11865	0.234467
$682$	0	0
$683$	−17.0182	−0.651183	−0.325592	−	0.945510i	$-0.605563\pi$
$683$	−17.0182	−0.651183	−0.325592	+	0.945510i	$0.605563\pi$
$684$	0	0
$685$	−41.1349	−1.57168
$686$	0	0
$687$	14.5804	0.556277
$688$	0	0
$689$	−15.1927	−0.578798
$690$	0	0
$691$	−6.06240	−0.230625	−0.115312	−	0.993329i	$-0.536787\pi$
$691$	−6.06240	−0.230625	−0.115312	+	0.993329i	$0.536787\pi$
$692$	0	0
$693$	2.39986	0.0911633
$694$	0	0
$695$	−41.6606	−1.58028
$696$	0	0
$697$	51.9012	1.96590
$698$	0	0
$699$	15.2715	0.577621
$700$	0	0
$701$	10.3633	0.391415	0.195708	−	0.980662i	$-0.437300\pi$
$701$	10.3633	0.391415	0.195708	+	0.980662i	$0.437300\pi$
$702$	0	0
$703$	2.25035	0.0848736
$704$	0	0
$705$	11.9601	0.450444
$706$	0	0
$707$	−8.83845	−0.332404
$708$	0	0
$709$	21.2000	0.796184	0.398092	−	0.917345i	$-0.369672\pi$
$709$	21.2000	0.796184	0.398092	+	0.917345i	$0.369672\pi$
$710$	0	0
$711$	−12.7792	−0.479257
$712$	0	0
$713$	8.48867	0.317903
$714$	0	0
$715$	4.89448	0.183043
$716$	0	0
$717$	5.61982	0.209876
$718$	0	0
$719$	24.3265	0.907225	0.453612	−	0.891199i	$-0.350135\pi$
$719$	24.3265	0.907225	0.453612	+	0.891199i	$0.350135\pi$
$720$	0	0
$721$	34.5372	1.28623
$722$	0	0
$723$	−15.6192	−0.580883
$724$	0	0
$725$	−2.02127	−0.0750681
$726$	0	0
$727$	47.5490	1.76350	0.881748	−	0.471720i	$-0.156367\pi$
$727$	47.5490	1.76350	0.881748	+	0.471720i	$0.156367\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−46.7014	−1.72732
$732$	0	0
$733$	43.8561	1.61986	0.809931	−	0.586526i	$-0.199505\pi$
$733$	43.8561	1.61986	0.809931	+	0.586526i	$0.199505\pi$
$734$	0	0
$735$	8.63804	0.318619
$736$	0	0
$737$	−0.994114	−0.0366187
$738$	0	0
$739$	−34.7321	−1.27764	−0.638821	−	0.769356i	$-0.720577\pi$
$739$	−34.7321	−1.27764	−0.638821	+	0.769356i	$0.720577\pi$
$740$	0	0
$741$	4.25068	0.156152
$742$	0	0
$743$	22.0528	0.809040	0.404520	−	0.914529i	$-0.367439\pi$
$743$	22.0528	0.809040	0.404520	+	0.914529i	$0.367439\pi$
$744$	0	0
$745$	−23.5244	−0.861868
$746$	0	0
$747$	3.32257	0.121566
$748$	0	0
$749$	−0.756385	−0.0276377
$750$	0	0
$751$	−39.6205	−1.44577	−0.722887	−	0.690966i	$-0.757185\pi$
$751$	−39.6205	−1.44577	−0.722887	+	0.690966i	$0.757185\pi$
$752$	0	0
$753$	10.7037	0.390065
$754$	0	0
$755$	10.9244	0.397579
$756$	0	0
$757$	−16.6428	−0.604893	−0.302447	−	0.953166i	$-0.597803\pi$
$757$	−16.6428	−0.604893	−0.302447	+	0.953166i	$0.597803\pi$
$758$	0	0
$759$	1.24093	0.0450428
$760$	0	0
$761$	10.9063	0.395354	0.197677	−	0.980267i	$-0.436660\pi$
$761$	10.9063	0.395354	0.197677	+	0.980267i	$0.436660\pi$
$762$	0	0
$763$	−9.31130	−0.337092
$764$	0	0
$765$	−11.2380	−0.406310
$766$	0	0
$767$	1.60219	0.0578516
$768$	0	0
$769$	−43.4487	−1.56680	−0.783400	−	0.621518i	$-0.786516\pi$
$769$	−43.4487	−1.56680	−0.783400	+	0.621518i	$0.786516\pi$
$770$	0	0
$771$	3.36904	0.121333
$772$	0	0
$773$	−36.5429	−1.31436	−0.657178	−	0.753736i	$-0.728250\pi$
$773$	−36.5429	−1.31436	−0.657178	+	0.753736i	$0.728250\pi$
$774$	0	0
$775$	17.1579	0.616330
$776$	0	0
$777$	1.52400	0.0546732
$778$	0	0
$779$	−34.9464	−1.25208
$780$	0	0
$781$	−1.54176	−0.0551687
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−34.4154	−1.22834
$786$	0	0
$787$	−33.1757	−1.18259	−0.591293	−	0.806457i	$-0.701382\pi$
$787$	−33.1757	−1.18259	−0.591293	+	0.806457i	$0.701382\pi$
$788$	0	0
$789$	16.2821	0.579657
$790$	0	0
$791$	30.3453	1.07896
$792$	0	0
$793$	2.94773	0.104677
$794$	0	0
$795$	−27.0453	−0.959198
$796$	0	0
$797$	28.4906	1.00919	0.504595	−	0.863356i	$-0.331642\pi$
$797$	28.4906	1.00919	0.504595	+	0.863356i	$0.331642\pi$
$798$	0	0
$799$	19.1429	0.677228
$800$	0	0
$801$	2.02980	0.0717195
$802$	0	0
$803$	−1.82455	−0.0643871
$804$	0	0
$805$	−5.12446	−0.180613
$806$	0	0
$807$	1.07002	0.0376667
$808$	0	0
$809$	28.0357	0.985684	0.492842	−	0.870119i	$-0.335958\pi$
$809$	28.0357	0.985684	0.492842	+	0.870119i	$0.335958\pi$
$810$	0	0
$811$	4.54499	0.159596	0.0797981	−	0.996811i	$-0.474572\pi$
$811$	4.54499	0.159596	0.0797981	+	0.996811i	$0.474572\pi$
$812$	0	0
$813$	−27.4839	−0.963904
$814$	0	0
$815$	17.3932	0.609258
$816$	0	0
$817$	31.4453	1.10013
$818$	0	0
$819$	2.87868	0.100589
$820$	0	0
$821$	−16.8343	−0.587522	−0.293761	−	0.955879i	$-0.594907\pi$
$821$	−16.8343	−0.587522	−0.293761	+	0.955879i	$0.594907\pi$
$822$	0	0
$823$	−11.0980	−0.386850	−0.193425	−	0.981115i	$-0.561960\pi$
$823$	−11.0980	−0.386850	−0.193425	+	0.981115i	$0.561960\pi$
$824$	0	0
$825$	2.50825	0.0873261
$826$	0	0
$827$	22.3267	0.776375	0.388188	−	0.921580i	$-0.373101\pi$
$827$	22.3267	0.776375	0.388188	+	0.921580i	$0.373101\pi$
$828$	0	0
$829$	−20.6403	−0.716867	−0.358433	−	0.933555i	$-0.616689\pi$
$829$	−20.6403	−0.716867	−0.358433	+	0.933555i	$0.616689\pi$
$830$	0	0
$831$	−15.6321	−0.542273
$832$	0	0
$833$	13.8257	0.479033
$834$	0	0
$835$	52.2575	1.80845
$836$	0	0
$837$	−8.48867	−0.293411
$838$	0	0
$839$	−39.9378	−1.37881	−0.689403	−	0.724378i	$-0.742127\pi$
$839$	−39.9378	−1.37881	−0.689403	+	0.724378i	$0.742127\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	2.26311	0.0779455
$844$	0	0
$845$	−28.5760	−0.983043
$846$	0	0
$847$	18.2951	0.628629
$848$	0	0
$849$	−31.0378	−1.06521
$850$	0	0
$851$	0.788034	0.0270134
$852$	0	0
$853$	11.6903	0.400270	0.200135	−	0.979768i	$-0.435862\pi$
$853$	11.6903	0.400270	0.200135	+	0.979768i	$0.435862\pi$
$854$	0	0
$855$	7.56682	0.258780
$856$	0	0
$857$	−53.8039	−1.83791	−0.918953	−	0.394367i	$-0.870964\pi$
$857$	−53.8039	−1.83791	−0.918953	+	0.394367i	$0.870964\pi$
$858$	0	0
$859$	−9.42914	−0.321718	−0.160859	−	0.986977i	$-0.551426\pi$
$859$	−9.42914	−0.321718	−0.160859	+	0.986977i	$0.551426\pi$
$860$	0	0
$861$	−23.6667	−0.806558
$862$	0	0
$863$	6.06532	0.206466	0.103233	−	0.994657i	$-0.467081\pi$
$863$	6.06532	0.206466	0.103233	+	0.994657i	$0.467081\pi$
$864$	0	0
$865$	49.2951	1.67609
$866$	0	0
$867$	−0.987100	−0.0335237
$868$	0	0
$869$	15.8580	0.537947
$870$	0	0
$871$	−1.19246	−0.0404048
$872$	0	0
$873$	9.39099	0.317837
$874$	0	0
$875$	15.2644	0.516030
$876$	0	0
$877$	55.5669	1.87636	0.938180	−	0.346148i	$-0.112510\pi$
$877$	55.5669	1.87636	0.938180	+	0.346148i	$0.112510\pi$
$878$	0	0
$879$	7.01724	0.236686
$880$	0	0
$881$	−52.5638	−1.77092	−0.885460	−	0.464715i	$-0.846157\pi$
$881$	−52.5638	−1.77092	−0.885460	+	0.464715i	$0.846157\pi$
$882$	0	0
$883$	1.12724	0.0379346	0.0189673	−	0.999820i	$-0.493962\pi$
$883$	1.12724	0.0379346	0.0189673	+	0.999820i	$0.493962\pi$
$884$	0	0
$885$	2.85212	0.0958731
$886$	0	0
$887$	−20.6619	−0.693759	−0.346879	−	0.937910i	$-0.612759\pi$
$887$	−20.6619	−0.693759	−0.346879	+	0.937910i	$0.612759\pi$
$888$	0	0
$889$	34.4035	1.15386
$890$	0	0
$891$	−1.24093	−0.0415726
$892$	0	0
$893$	−12.8894	−0.431328
$894$	0	0
$895$	33.7844	1.12929
$896$	0	0
$897$	1.48851	0.0497000
$898$	0	0
$899$	−8.48867	−0.283113
$900$	0	0
$901$	−43.2877	−1.44212
$902$	0	0
$903$	21.2956	0.708673
$904$	0	0
$905$	−53.5135	−1.77885
$906$	0	0
$907$	24.7191	0.820784	0.410392	−	0.911909i	$-0.365392\pi$
$907$	24.7191	0.820784	0.410392	+	0.911909i	$0.365392\pi$
$908$	0	0
$909$	4.57021	0.151584
$910$	0	0
$911$	−39.4313	−1.30642	−0.653208	−	0.757178i	$-0.726577\pi$
$911$	−39.4313	−1.30642	−0.653208	+	0.757178i	$0.726577\pi$
$912$	0	0
$913$	−4.12307	−0.136454
$914$	0	0
$915$	5.24739	0.173473
$916$	0	0
$917$	11.5774	0.382319
$918$	0	0
$919$	43.6803	1.44088	0.720440	−	0.693518i	$-0.243940\pi$
$919$	43.6803	1.44088	0.720440	+	0.693518i	$0.243940\pi$
$920$	0	0
$921$	−7.12070	−0.234635
$922$	0	0
$923$	−1.84937	−0.0608728
$924$	0	0
$925$	1.59283	0.0523719
$926$	0	0
$927$	−17.8586	−0.586552
$928$	0	0
$929$	0.457221	0.0150009	0.00750047	−	0.999972i	$-0.497613\pi$
$929$	0.457221	0.0150009	0.00750047	+	0.999972i	$0.497613\pi$
$930$	0	0
$931$	−9.30921	−0.305097
$932$	0	0
$933$	−3.14032	−0.102810
$934$	0	0
$935$	13.9455	0.456067
$936$	0	0
$937$	45.8896	1.49915	0.749574	−	0.661920i	$-0.230258\pi$
$937$	45.8896	1.49915	0.749574	+	0.661920i	$0.230258\pi$
$938$	0	0
$939$	−21.0426	−0.686701
$940$	0	0
$941$	21.0845	0.687336	0.343668	−	0.939091i	$-0.388331\pi$
$941$	21.0845	0.687336	0.343668	+	0.939091i	$0.388331\pi$
$942$	0	0
$943$	−12.2376	−0.398512
$944$	0	0
$945$	5.12446	0.166699
$946$	0	0
$947$	−30.3315	−0.985640	−0.492820	−	0.870131i	$-0.664034\pi$
$947$	−30.3315	−0.985640	−0.492820	+	0.870131i	$0.664034\pi$
$948$	0	0
$949$	−2.18858	−0.0710443
$950$	0	0
$951$	−7.50308	−0.243304
$952$	0	0
$953$	−28.0954	−0.910100	−0.455050	−	0.890466i	$-0.650379\pi$
$953$	−28.0954	−0.910100	−0.455050	+	0.890466i	$0.650379\pi$
$954$	0	0
$955$	−7.82376	−0.253171
$956$	0	0
$957$	−1.24093	−0.0401135
$958$	0	0
$959$	30.0222	0.969468
$960$	0	0
$961$	41.0576	1.32444
$962$	0	0
$963$	0.391113	0.0126035
$964$	0	0
$965$	29.0315	0.934557
$966$	0	0
$967$	27.4178	0.881697	0.440848	−	0.897582i	$-0.354678\pi$
$967$	27.4178	0.881697	0.440848	+	0.897582i	$0.354678\pi$
$968$	0	0
$969$	12.1112	0.389067
$970$	0	0
$971$	−8.24362	−0.264550	−0.132275	−	0.991213i	$-0.542228\pi$
$971$	−8.24362	−0.264550	−0.132275	+	0.991213i	$0.542228\pi$
$972$	0	0
$973$	30.4059	0.974769
$974$	0	0
$975$	3.00869	0.0963551
$976$	0	0
$977$	28.5065	0.912004	0.456002	−	0.889979i	$-0.349281\pi$
$977$	28.5065	0.912004	0.456002	+	0.889979i	$0.349281\pi$
$978$	0	0
$979$	−2.51884	−0.0805023
$980$	0	0
$981$	4.81471	0.153722
$982$	0	0
$983$	−51.7100	−1.64929	−0.824646	−	0.565649i	$-0.808626\pi$
$983$	−51.7100	−1.64929	−0.824646	+	0.565649i	$0.808626\pi$
$984$	0	0
$985$	−72.1274	−2.29817
$986$	0	0
$987$	−8.72907	−0.277849
$988$	0	0
$989$	11.0116	0.350148
$990$	0	0
$991$	−39.0995	−1.24204	−0.621019	−	0.783796i	$-0.713281\pi$
$991$	−39.0995	−1.24204	−0.621019	+	0.783796i	$0.713281\pi$
$992$	0	0
$993$	−10.2367	−0.324853
$994$	0	0
$995$	−4.60351	−0.145941
$996$	0	0
$997$	60.2082	1.90681	0.953407	−	0.301688i	$-0.0975500\pi$
$997$	60.2082	1.90681	0.953407	+	0.301688i	$0.0975500\pi$
$998$	0	0
$999$	−0.788034	−0.0249323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.11	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.64977 q^{5} -1.93393 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.64977 q^{5} -1.93393 q^{7} +1.00000 q^{9} -1.24093 q^{11} -1.48851 q^{13} -2.64977 q^{15} -4.24112 q^{17} +2.85565 q^{19} +1.93393 q^{21} +1.00000 q^{23} +2.02127 q^{25} -1.00000 q^{27} -1.00000 q^{29} +8.48867 q^{31} +1.24093 q^{33} -5.12446 q^{35} +0.788034 q^{37} +1.48851 q^{39} -12.2376 q^{41} +11.0116 q^{43} +2.64977 q^{45} -4.51365 q^{47} -3.25992 q^{49} +4.24112 q^{51} +10.2067 q^{53} -3.28817 q^{55} -2.85565 q^{57} -1.07637 q^{59} -1.98032 q^{61} -1.93393 q^{63} -3.94421 q^{65} +0.801106 q^{67} -1.00000 q^{69} +1.24243 q^{71} +1.47031 q^{73} -2.02127 q^{75} +2.39986 q^{77} -12.7792 q^{79} +1.00000 q^{81} +3.32257 q^{83} -11.2380 q^{85} +1.00000 q^{87} +2.02980 q^{89} +2.87868 q^{91} -8.48867 q^{93} +7.56682 q^{95} +9.39099 q^{97} -1.24093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 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

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Database

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