LMFDB - Embedded newform 4004.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 23x^{8} + 23x^{7} + 170x^{6} - 165x^{5} - 411x^{4} + 360x^{3} + 111x^{2} - 48x - 12 $ x^10 - x^9 - 23x^8 + 23x^7 + 170x^6 - 165x^5 - 411x^4 + 360x^3 + 111x^2 - 48x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.290239$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.290239 q^{3} +0.725041 q^{5} -1.00000 q^{7} -2.91576 q^{9} +O(q^{10})$	$q-0.290239 q^{3} +0.725041 q^{5} -1.00000 q^{7} -2.91576 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.210435 q^{15} +5.87760 q^{17} +6.34044 q^{19} +0.290239 q^{21} -1.90152 q^{23} -4.47432 q^{25} +1.71699 q^{27} -1.28218 q^{29} +6.23051 q^{31} -0.290239 q^{33} -0.725041 q^{35} -11.8703 q^{37} -0.290239 q^{39} +1.13457 q^{41} -3.11520 q^{43} -2.11405 q^{45} -1.90152 q^{47} +1.00000 q^{49} -1.70591 q^{51} +2.01493 q^{53} +0.725041 q^{55} -1.84024 q^{57} -2.94832 q^{59} +9.69135 q^{61} +2.91576 q^{63} +0.725041 q^{65} +3.12485 q^{67} +0.551895 q^{69} -10.9036 q^{71} -2.27504 q^{73} +1.29862 q^{75} -1.00000 q^{77} +3.11965 q^{79} +8.24895 q^{81} +7.10427 q^{83} +4.26150 q^{85} +0.372140 q^{87} +14.8253 q^{89} -1.00000 q^{91} -1.80834 q^{93} +4.59708 q^{95} +10.6590 q^{97} -2.91576 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9}+O(q^{10}) $ 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9	$ 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9} + 10 q^{11} + 10 q^{13} + 3 q^{15} + 3 q^{17} + 6 q^{19} - q^{21} + 4 q^{23} + 22 q^{25} - 5 q^{27} + 10 q^{29} - q^{31} + q^{33} - 4 q^{35} + 20 q^{37} + q^{39} + 8 q^{45} + 4 q^{47} + 10 q^{49} + 11 q^{51} - 5 q^{53} + 4 q^{55} + 16 q^{57} + 11 q^{59} + 12 q^{61} - 17 q^{63} + 4 q^{65} - 2 q^{67} + 10 q^{69} + 28 q^{71} + 11 q^{73} - 6 q^{75} - 10 q^{77} - 10 q^{79} + 46 q^{81} + 7 q^{83} + 33 q^{85} - 47 q^{87} + 30 q^{89} - 10 q^{91} + 41 q^{93} - 2 q^{95} + 55 q^{97} + 17 q^{99}+O(q^{100}) $ 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9 + 10 * q^11 + 10 * q^13 + 3 * q^15 + 3 * q^17 + 6 * q^19 - q^21 + 4 * q^23 + 22 * q^25 - 5 * q^27 + 10 * q^29 - q^31 + q^33 - 4 * q^35 + 20 * q^37 + q^39 + 8 * q^45 + 4 * q^47 + 10 * q^49 + 11 * q^51 - 5 * q^53 + 4 * q^55 + 16 * q^57 + 11 * q^59 + 12 * q^61 - 17 * q^63 + 4 * q^65 - 2 * q^67 + 10 * q^69 + 28 * q^71 + 11 * q^73 - 6 * q^75 - 10 * q^77 - 10 * q^79 + 46 * q^81 + 7 * q^83 + 33 * q^85 - 47 * q^87 + 30 * q^89 - 10 * q^91 + 41 * q^93 - 2 * q^95 + 55 * q^97 + 17 * 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
$3$	−0.290239	−0.167570	−0.0837848	−	0.996484i	$-0.526701\pi$
$3$	−0.290239	−0.167570	−0.0837848	+	0.996484i	$0.526701\pi$
$4$	0	0
$5$	0.725041	0.324248	0.162124	−	0.986770i	$-0.448166\pi$
$5$	0.725041	0.324248	0.162124	+	0.986770i	$0.448166\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.91576	−0.971920
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.210435	−0.0543342
$16$	0	0
$17$	5.87760	1.42553	0.712763	−	0.701405i	$-0.247443\pi$
$17$	5.87760	1.42553	0.712763	+	0.701405i	$0.247443\pi$
$18$	0	0
$19$	6.34044	1.45460	0.727298	−	0.686322i	$-0.240776\pi$
$19$	6.34044	1.45460	0.727298	+	0.686322i	$0.240776\pi$
$20$	0	0
$21$	0.290239	0.0633354
$22$	0	0
$23$	−1.90152	−0.396494	−0.198247	−	0.980152i	$-0.563525\pi$
$23$	−1.90152	−0.396494	−0.198247	+	0.980152i	$0.563525\pi$
$24$	0	0
$25$	−4.47432	−0.894863
$26$	0	0
$27$	1.71699	0.330434
$28$	0	0
$29$	−1.28218	−0.238096	−0.119048	−	0.992889i	$-0.537984\pi$
$29$	−1.28218	−0.238096	−0.119048	+	0.992889i	$0.537984\pi$
$30$	0	0
$31$	6.23051	1.11903	0.559516	−	0.828820i	$-0.310987\pi$
$31$	6.23051	1.11903	0.559516	+	0.828820i	$0.310987\pi$
$32$	0	0
$33$	−0.290239	−0.0505242
$34$	0	0
$35$	−0.725041	−0.122554
$36$	0	0
$37$	−11.8703	−1.95146	−0.975729	−	0.218981i	$-0.929727\pi$
$37$	−11.8703	−1.95146	−0.975729	+	0.218981i	$0.929727\pi$
$38$	0	0
$39$	−0.290239	−0.0464755
$40$	0	0
$41$	1.13457	0.177190	0.0885950	−	0.996068i	$-0.471762\pi$
$41$	1.13457	0.177190	0.0885950	+	0.996068i	$0.471762\pi$
$42$	0	0
$43$	−3.11520	−0.475064	−0.237532	−	0.971380i	$-0.576338\pi$
$43$	−3.11520	−0.475064	−0.237532	+	0.971380i	$0.576338\pi$
$44$	0	0
$45$	−2.11405	−0.315143
$46$	0	0
$47$	−1.90152	−0.277365	−0.138683	−	0.990337i	$-0.544287\pi$
$47$	−1.90152	−0.277365	−0.138683	+	0.990337i	$0.544287\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.70591	−0.238875
$52$	0	0
$53$	2.01493	0.276772	0.138386	−	0.990378i	$-0.455808\pi$
$53$	2.01493	0.276772	0.138386	+	0.990378i	$0.455808\pi$
$54$	0	0
$55$	0.725041	0.0977645
$56$	0	0
$57$	−1.84024	−0.243746
$58$	0	0
$59$	−2.94832	−0.383839	−0.191919	−	0.981411i	$-0.561471\pi$
$59$	−2.94832	−0.383839	−0.191919	+	0.981411i	$0.561471\pi$
$60$	0	0
$61$	9.69135	1.24085	0.620425	−	0.784266i	$-0.286960\pi$
$61$	9.69135	1.24085	0.620425	+	0.784266i	$0.286960\pi$
$62$	0	0
$63$	2.91576	0.367351
$64$	0	0
$65$	0.725041	0.0899303
$66$	0	0
$67$	3.12485	0.381761	0.190880	−	0.981613i	$-0.438866\pi$
$67$	3.12485	0.381761	0.190880	+	0.981613i	$0.438866\pi$
$68$	0	0
$69$	0.551895	0.0664404
$70$	0	0
$71$	−10.9036	−1.29402	−0.647012	−	0.762480i	$-0.723981\pi$
$71$	−10.9036	−1.29402	−0.647012	+	0.762480i	$0.723981\pi$
$72$	0	0
$73$	−2.27504	−0.266274	−0.133137	−	0.991098i	$-0.542505\pi$
$73$	−2.27504	−0.266274	−0.133137	+	0.991098i	$0.542505\pi$
$74$	0	0
$75$	1.29862	0.149952
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	3.11965	0.350988	0.175494	−	0.984481i	$-0.443848\pi$
$79$	3.11965	0.350988	0.175494	+	0.984481i	$0.443848\pi$
$80$	0	0
$81$	8.24895	0.916550
$82$	0	0
$83$	7.10427	0.779795	0.389898	−	0.920858i	$-0.372510\pi$
$83$	7.10427	0.779795	0.389898	+	0.920858i	$0.372510\pi$
$84$	0	0
$85$	4.26150	0.462224
$86$	0	0
$87$	0.372140	0.0398976
$88$	0	0
$89$	14.8253	1.57148	0.785739	−	0.618559i	$-0.212283\pi$
$89$	14.8253	1.57148	0.785739	+	0.618559i	$0.212283\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−1.80834	−0.187516
$94$	0	0
$95$	4.59708	0.471650
$96$	0	0
$97$	10.6590	1.08226	0.541128	−	0.840940i	$-0.317997\pi$
$97$	10.6590	1.08226	0.541128	+	0.840940i	$0.317997\pi$
$98$	0	0
$99$	−2.91576	−0.293045
$100$	0	0
$101$	9.69285	0.964475	0.482237	−	0.876041i	$-0.339824\pi$
$101$	9.69285	0.964475	0.482237	+	0.876041i	$0.339824\pi$
$102$	0	0
$103$	−6.56619	−0.646986	−0.323493	−	0.946231i	$-0.604857\pi$
$103$	−6.56619	−0.646986	−0.323493	+	0.946231i	$0.604857\pi$
$104$	0	0
$105$	0.210435	0.0205364
$106$	0	0
$107$	−1.52571	−0.147496	−0.0737479	−	0.997277i	$-0.523496\pi$
$107$	−1.52571	−0.147496	−0.0737479	+	0.997277i	$0.523496\pi$
$108$	0	0
$109$	14.1696	1.35721	0.678603	−	0.734505i	$-0.262586\pi$
$109$	14.1696	1.35721	0.678603	+	0.734505i	$0.262586\pi$
$110$	0	0
$111$	3.44521	0.327005
$112$	0	0
$113$	13.4051	1.26104	0.630521	−	0.776172i	$-0.282841\pi$
$113$	13.4051	1.26104	0.630521	+	0.776172i	$0.282841\pi$
$114$	0	0
$115$	−1.37868	−0.128563
$116$	0	0
$117$	−2.91576	−0.269562
$118$	0	0
$119$	−5.87760	−0.538798
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.329297	−0.0296917
$124$	0	0
$125$	−6.86927	−0.614406
$126$	0	0
$127$	8.61242	0.764229	0.382114	−	0.924115i	$-0.375196\pi$
$127$	8.61242	0.764229	0.382114	+	0.924115i	$0.375196\pi$
$128$	0	0
$129$	0.904154	0.0796063
$130$	0	0
$131$	20.5354	1.79419	0.897094	−	0.441840i	$-0.145674\pi$
$131$	20.5354	1.79419	0.897094	+	0.441840i	$0.145674\pi$
$132$	0	0
$133$	−6.34044	−0.549786
$134$	0	0
$135$	1.24489	0.107143
$136$	0	0
$137$	14.9497	1.27724	0.638620	−	0.769523i	$-0.279506\pi$
$137$	14.9497	1.27724	0.638620	+	0.769523i	$0.279506\pi$
$138$	0	0
$139$	5.27533	0.447448	0.223724	−	0.974653i	$-0.428179\pi$
$139$	5.27533	0.447448	0.223724	+	0.974653i	$0.428179\pi$
$140$	0	0
$141$	0.551895	0.0464780
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−0.929636	−0.0772021
$146$	0	0
$147$	−0.290239	−0.0239385
$148$	0	0
$149$	17.6511	1.44603	0.723017	−	0.690830i	$-0.242755\pi$
$149$	17.6511	1.44603	0.723017	+	0.690830i	$0.242755\pi$
$150$	0	0
$151$	−11.9198	−0.970022	−0.485011	−	0.874508i	$-0.661184\pi$
$151$	−11.9198	−0.970022	−0.485011	+	0.874508i	$0.661184\pi$
$152$	0	0
$153$	−17.1377	−1.38550
$154$	0	0
$155$	4.51737	0.362844
$156$	0	0
$157$	17.3825	1.38728	0.693639	−	0.720323i	$-0.256007\pi$
$157$	17.3825	1.38728	0.693639	+	0.720323i	$0.256007\pi$
$158$	0	0
$159$	−0.584812	−0.0463786
$160$	0	0
$161$	1.90152	0.149861
$162$	0	0
$163$	−8.68282	−0.680091	−0.340046	−	0.940409i	$-0.610443\pi$
$163$	−8.68282	−0.680091	−0.340046	+	0.940409i	$0.610443\pi$
$164$	0	0
$165$	−0.210435	−0.0163824
$166$	0	0
$167$	−1.15812	−0.0896179	−0.0448090	−	0.998996i	$-0.514268\pi$
$167$	−1.15812	−0.0896179	−0.0448090	+	0.998996i	$0.514268\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−18.4872	−1.41375
$172$	0	0
$173$	1.45911	0.110934	0.0554672	−	0.998461i	$-0.482335\pi$
$173$	1.45911	0.110934	0.0554672	+	0.998461i	$0.482335\pi$
$174$	0	0
$175$	4.47432	0.338226
$176$	0	0
$177$	0.855718	0.0643197
$178$	0	0
$179$	−17.2077	−1.28617	−0.643083	−	0.765797i	$-0.722345\pi$
$179$	−17.2077	−1.28617	−0.643083	+	0.765797i	$0.722345\pi$
$180$	0	0
$181$	−12.1911	−0.906155	−0.453078	−	0.891471i	$-0.649674\pi$
$181$	−12.1911	−0.906155	−0.453078	+	0.891471i	$0.649674\pi$
$182$	0	0
$183$	−2.81281	−0.207929
$184$	0	0
$185$	−8.60642	−0.632757
$186$	0	0
$187$	5.87760	0.429812
$188$	0	0
$189$	−1.71699	−0.124892
$190$	0	0
$191$	9.18097	0.664311	0.332156	−	0.943225i	$-0.392224\pi$
$191$	9.18097	0.664311	0.332156	+	0.943225i	$0.392224\pi$
$192$	0	0
$193$	7.03573	0.506443	0.253222	−	0.967408i	$-0.418510\pi$
$193$	7.03573	0.506443	0.253222	+	0.967408i	$0.418510\pi$
$194$	0	0
$195$	−0.210435	−0.0150696
$196$	0	0
$197$	2.45044	0.174587	0.0872933	−	0.996183i	$-0.472178\pi$
$197$	2.45044	0.174587	0.0872933	+	0.996183i	$0.472178\pi$
$198$	0	0
$199$	21.5873	1.53028	0.765141	−	0.643863i	$-0.222669\pi$
$199$	21.5873	1.53028	0.765141	+	0.643863i	$0.222669\pi$
$200$	0	0
$201$	−0.906953	−0.0639715
$202$	0	0
$203$	1.28218	0.0899917
$204$	0	0
$205$	0.822610	0.0574536
$206$	0	0
$207$	5.54438	0.385361
$208$	0	0
$209$	6.34044	0.438577
$210$	0	0
$211$	16.9458	1.16660	0.583298	−	0.812258i	$-0.301762\pi$
$211$	16.9458	1.16660	0.583298	+	0.812258i	$0.301762\pi$
$212$	0	0
$213$	3.16466	0.216839
$214$	0	0
$215$	−2.25865	−0.154039
$216$	0	0
$217$	−6.23051	−0.422954
$218$	0	0
$219$	0.660307	0.0446194
$220$	0	0
$221$	5.87760	0.395370
$222$	0	0
$223$	−7.33382	−0.491109	−0.245555	−	0.969383i	$-0.578970\pi$
$223$	−7.33382	−0.491109	−0.245555	+	0.969383i	$0.578970\pi$
$224$	0	0
$225$	13.0460	0.869736
$226$	0	0
$227$	−13.9554	−0.926249	−0.463125	−	0.886293i	$-0.653272\pi$
$227$	−13.9554	−0.926249	−0.463125	+	0.886293i	$0.653272\pi$
$228$	0	0
$229$	−23.4922	−1.55241	−0.776204	−	0.630482i	$-0.782857\pi$
$229$	−23.4922	−1.55241	−0.776204	+	0.630482i	$0.782857\pi$
$230$	0	0
$231$	0.290239	0.0190963
$232$	0	0
$233$	−4.31252	−0.282523	−0.141261	−	0.989972i	$-0.545116\pi$
$233$	−4.31252	−0.282523	−0.141261	+	0.989972i	$0.545116\pi$
$234$	0	0
$235$	−1.37868	−0.0899351
$236$	0	0
$237$	−0.905443	−0.0588149
$238$	0	0
$239$	−14.1648	−0.916245	−0.458122	−	0.888889i	$-0.651478\pi$
$239$	−14.1648	−0.916245	−0.458122	+	0.888889i	$0.651478\pi$
$240$	0	0
$241$	−2.70893	−0.174498	−0.0872488	−	0.996187i	$-0.527808\pi$
$241$	−2.70893	−0.174498	−0.0872488	+	0.996187i	$0.527808\pi$
$242$	0	0
$243$	−7.54512	−0.484020
$244$	0	0
$245$	0.725041	0.0463212
$246$	0	0
$247$	6.34044	0.403432
$248$	0	0
$249$	−2.06194	−0.130670
$250$	0	0
$251$	−15.1169	−0.954168	−0.477084	−	0.878858i	$-0.658306\pi$
$251$	−15.1169	−0.954168	−0.477084	+	0.878858i	$0.658306\pi$
$252$	0	0
$253$	−1.90152	−0.119547
$254$	0	0
$255$	−1.23685	−0.0774548
$256$	0	0
$257$	14.4371	0.900562	0.450281	−	0.892887i	$-0.351324\pi$
$257$	14.4371	0.900562	0.450281	+	0.892887i	$0.351324\pi$
$258$	0	0
$259$	11.8703	0.737582
$260$	0	0
$261$	3.73854	0.231410
$262$	0	0
$263$	−3.06104	−0.188752	−0.0943759	−	0.995537i	$-0.530086\pi$
$263$	−3.06104	−0.188752	−0.0943759	+	0.995537i	$0.530086\pi$
$264$	0	0
$265$	1.46091	0.0897429
$266$	0	0
$267$	−4.30288	−0.263332
$268$	0	0
$269$	−27.9895	−1.70655	−0.853275	−	0.521461i	$-0.825387\pi$
$269$	−27.9895	−1.70655	−0.853275	+	0.521461i	$0.825387\pi$
$270$	0	0
$271$	26.2080	1.59202	0.796012	−	0.605281i	$-0.206939\pi$
$271$	26.2080	1.59202	0.796012	+	0.605281i	$0.206939\pi$
$272$	0	0
$273$	0.290239	0.0175661
$274$	0	0
$275$	−4.47432	−0.269811
$276$	0	0
$277$	−4.37582	−0.262918	−0.131459	−	0.991322i	$-0.541966\pi$
$277$	−4.37582	−0.262918	−0.131459	+	0.991322i	$0.541966\pi$
$278$	0	0
$279$	−18.1667	−1.08761
$280$	0	0
$281$	11.2865	0.673297	0.336649	−	0.941630i	$-0.390707\pi$
$281$	11.2865	0.673297	0.336649	+	0.941630i	$0.390707\pi$
$282$	0	0
$283$	1.36041	0.0808681	0.0404340	−	0.999182i	$-0.487126\pi$
$283$	1.36041	0.0808681	0.0404340	+	0.999182i	$0.487126\pi$
$284$	0	0
$285$	−1.33425	−0.0790343
$286$	0	0
$287$	−1.13457	−0.0669715
$288$	0	0
$289$	17.5461	1.03212
$290$	0	0
$291$	−3.09366	−0.181353
$292$	0	0
$293$	7.32355	0.427846	0.213923	−	0.976851i	$-0.431376\pi$
$293$	7.32355	0.427846	0.213923	+	0.976851i	$0.431376\pi$
$294$	0	0
$295$	−2.13765	−0.124459
$296$	0	0
$297$	1.71699	0.0996296
$298$	0	0
$299$	−1.90152	−0.109968
$300$	0	0
$301$	3.11520	0.179557
$302$	0	0
$303$	−2.81325	−0.161617
$304$	0	0
$305$	7.02662	0.402343
$306$	0	0
$307$	−18.9566	−1.08191	−0.540955	−	0.841051i	$-0.681937\pi$
$307$	−18.9566	−1.08191	−0.540955	+	0.841051i	$0.681937\pi$
$308$	0	0
$309$	1.90577	0.108415
$310$	0	0
$311$	−11.0696	−0.627700	−0.313850	−	0.949473i	$-0.601619\pi$
$311$	−11.0696	−0.627700	−0.313850	+	0.949473i	$0.601619\pi$
$312$	0	0
$313$	13.3831	0.756458	0.378229	−	0.925712i	$-0.376533\pi$
$313$	13.3831	0.756458	0.378229	+	0.925712i	$0.376533\pi$
$314$	0	0
$315$	2.11405	0.119113
$316$	0	0
$317$	−19.6417	−1.10319	−0.551593	−	0.834113i	$-0.685980\pi$
$317$	−19.6417	−1.10319	−0.551593	+	0.834113i	$0.685980\pi$
$318$	0	0
$319$	−1.28218	−0.0717885
$320$	0	0
$321$	0.442820	0.0247158
$322$	0	0
$323$	37.2665	2.07356
$324$	0	0
$325$	−4.47432	−0.248190
$326$	0	0
$327$	−4.11259	−0.227427
$328$	0	0
$329$	1.90152	0.104834
$330$	0	0
$331$	34.8540	1.91575	0.957876	−	0.287183i	$-0.0927186\pi$
$331$	34.8540	1.91575	0.957876	+	0.287183i	$0.0927186\pi$
$332$	0	0
$333$	34.6108	1.89666
$334$	0	0
$335$	2.26564	0.123785
$336$	0	0
$337$	−26.1781	−1.42601	−0.713006	−	0.701158i	$-0.752667\pi$
$337$	−26.1781	−1.42601	−0.713006	+	0.701158i	$0.752667\pi$
$338$	0	0
$339$	−3.89068	−0.211313
$340$	0	0
$341$	6.23051	0.337401
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.400147	0.0215432
$346$	0	0
$347$	24.6324	1.32234	0.661170	−	0.750236i	$-0.270060\pi$
$347$	24.6324	1.32234	0.661170	+	0.750236i	$0.270060\pi$
$348$	0	0
$349$	−4.53465	−0.242734	−0.121367	−	0.992608i	$-0.538728\pi$
$349$	−4.53465	−0.242734	−0.121367	+	0.992608i	$0.538728\pi$
$350$	0	0
$351$	1.71699	0.0916459
$352$	0	0
$353$	31.3092	1.66642	0.833211	−	0.552955i	$-0.186500\pi$
$353$	31.3092	1.66642	0.833211	+	0.552955i	$0.186500\pi$
$354$	0	0
$355$	−7.90558	−0.419585
$356$	0	0
$357$	1.70591	0.0902862
$358$	0	0
$359$	11.9658	0.631528	0.315764	−	0.948838i	$-0.397739\pi$
$359$	11.9658	0.631528	0.315764	+	0.948838i	$0.397739\pi$
$360$	0	0
$361$	21.2011	1.11585
$362$	0	0
$363$	−0.290239	−0.0152336
$364$	0	0
$365$	−1.64950	−0.0863388
$366$	0	0
$367$	30.0194	1.56700	0.783500	−	0.621392i	$-0.213432\pi$
$367$	30.0194	1.56700	0.783500	+	0.621392i	$0.213432\pi$
$368$	0	0
$369$	−3.30813	−0.172215
$370$	0	0
$371$	−2.01493	−0.104610
$372$	0	0
$373$	−20.3855	−1.05552	−0.527760	−	0.849393i	$-0.676968\pi$
$373$	−20.3855	−1.05552	−0.527760	+	0.849393i	$0.676968\pi$
$374$	0	0
$375$	1.99373	0.102956
$376$	0	0
$377$	−1.28218	−0.0660358
$378$	0	0
$379$	−20.5271	−1.05441	−0.527203	−	0.849739i	$-0.676759\pi$
$379$	−20.5271	−1.05441	−0.527203	+	0.849739i	$0.676759\pi$
$380$	0	0
$381$	−2.49966	−0.128062
$382$	0	0
$383$	−14.3287	−0.732163	−0.366081	−	0.930583i	$-0.619301\pi$
$383$	−14.3287	−0.732163	−0.366081	+	0.930583i	$0.619301\pi$
$384$	0	0
$385$	−0.725041	−0.0369515
$386$	0	0
$387$	9.08319	0.461724
$388$	0	0
$389$	18.1012	0.917767	0.458884	−	0.888496i	$-0.348250\pi$
$389$	18.1012	0.917767	0.458884	+	0.888496i	$0.348250\pi$
$390$	0	0
$391$	−11.1764	−0.565213
$392$	0	0
$393$	−5.96018	−0.300651
$394$	0	0
$395$	2.26187	0.113807
$396$	0	0
$397$	29.0563	1.45829	0.729147	−	0.684357i	$-0.239917\pi$
$397$	29.0563	1.45829	0.729147	+	0.684357i	$0.239917\pi$
$398$	0	0
$399$	1.84024	0.0921274
$400$	0	0
$401$	16.2459	0.811281	0.405640	−	0.914033i	$-0.367049\pi$
$401$	16.2459	0.811281	0.405640	+	0.914033i	$0.367049\pi$
$402$	0	0
$403$	6.23051	0.310364
$404$	0	0
$405$	5.98083	0.297190
$406$	0	0
$407$	−11.8703	−0.588387
$408$	0	0
$409$	1.98813	0.0983068	0.0491534	−	0.998791i	$-0.484348\pi$
$409$	1.98813	0.0983068	0.0491534	+	0.998791i	$0.484348\pi$
$410$	0	0
$411$	−4.33899	−0.214027
$412$	0	0
$413$	2.94832	0.145077
$414$	0	0
$415$	5.15089	0.252847
$416$	0	0
$417$	−1.53111	−0.0749787
$418$	0	0
$419$	−33.3657	−1.63002	−0.815009	−	0.579448i	$-0.803268\pi$
$419$	−33.3657	−1.63002	−0.815009	+	0.579448i	$0.803268\pi$
$420$	0	0
$421$	10.9632	0.534314	0.267157	−	0.963653i	$-0.413916\pi$
$421$	10.9632	0.534314	0.267157	+	0.963653i	$0.413916\pi$
$422$	0	0
$423$	5.54438	0.269577
$424$	0	0
$425$	−26.2982	−1.27565
$426$	0	0
$427$	−9.69135	−0.468997
$428$	0	0
$429$	−0.290239	−0.0140129
$430$	0	0
$431$	−11.5953	−0.558526	−0.279263	−	0.960215i	$-0.590090\pi$
$431$	−11.5953	−0.558526	−0.279263	+	0.960215i	$0.590090\pi$
$432$	0	0
$433$	5.92558	0.284765	0.142383	−	0.989812i	$-0.454524\pi$
$433$	5.92558	0.284765	0.142383	+	0.989812i	$0.454524\pi$
$434$	0	0
$435$	0.269817	0.0129367
$436$	0	0
$437$	−12.0565	−0.576739
$438$	0	0
$439$	29.0133	1.38473	0.692366	−	0.721547i	$-0.256569\pi$
$439$	29.0133	1.38473	0.692366	+	0.721547i	$0.256569\pi$
$440$	0	0
$441$	−2.91576	−0.138846
$442$	0	0
$443$	−22.9326	−1.08956	−0.544780	−	0.838579i	$-0.683387\pi$
$443$	−22.9326	−1.08956	−0.544780	+	0.838579i	$0.683387\pi$
$444$	0	0
$445$	10.7489	0.509549
$446$	0	0
$447$	−5.12304	−0.242311
$448$	0	0
$449$	22.4326	1.05866	0.529330	−	0.848416i	$-0.322443\pi$
$449$	22.4326	1.05866	0.529330	+	0.848416i	$0.322443\pi$
$450$	0	0
$451$	1.13457	0.0534248
$452$	0	0
$453$	3.45960	0.162546
$454$	0	0
$455$	−0.725041	−0.0339904
$456$	0	0
$457$	18.1431	0.848697	0.424349	−	0.905499i	$-0.360503\pi$
$457$	18.1431	0.848697	0.424349	+	0.905499i	$0.360503\pi$
$458$	0	0
$459$	10.0917	0.471042
$460$	0	0
$461$	28.8828	1.34520	0.672602	−	0.740004i	$-0.265177\pi$
$461$	28.8828	1.34520	0.672602	+	0.740004i	$0.265177\pi$
$462$	0	0
$463$	−40.4237	−1.87865	−0.939324	−	0.343030i	$-0.888547\pi$
$463$	−40.4237	−1.87865	−0.939324	+	0.343030i	$0.888547\pi$
$464$	0	0
$465$	−1.31112	−0.0608017
$466$	0	0
$467$	16.7859	0.776761	0.388380	−	0.921499i	$-0.373035\pi$
$467$	16.7859	0.776761	0.388380	+	0.921499i	$0.373035\pi$
$468$	0	0
$469$	−3.12485	−0.144292
$470$	0	0
$471$	−5.04509	−0.232466
$472$	0	0
$473$	−3.11520	−0.143237
$474$	0	0
$475$	−28.3691	−1.30166
$476$	0	0
$477$	−5.87506	−0.269001
$478$	0	0
$479$	−32.9297	−1.50460	−0.752299	−	0.658822i	$-0.771055\pi$
$479$	−32.9297	−1.50460	−0.752299	+	0.658822i	$0.771055\pi$
$480$	0	0
$481$	−11.8703	−0.541237
$482$	0	0
$483$	−0.551895	−0.0251121
$484$	0	0
$485$	7.72820	0.350920
$486$	0	0
$487$	−11.5744	−0.524488	−0.262244	−	0.965002i	$-0.584463\pi$
$487$	−11.5744	−0.524488	−0.262244	+	0.965002i	$0.584463\pi$
$488$	0	0
$489$	2.52010	0.113963
$490$	0	0
$491$	−32.2917	−1.45730	−0.728652	−	0.684884i	$-0.759853\pi$
$491$	−32.2917	−1.45730	−0.728652	+	0.684884i	$0.759853\pi$
$492$	0	0
$493$	−7.53616	−0.339411
$494$	0	0
$495$	−2.11405	−0.0950193
$496$	0	0
$497$	10.9036	0.489095
$498$	0	0
$499$	−29.6005	−1.32510	−0.662550	−	0.749017i	$-0.730526\pi$
$499$	−29.6005	−1.32510	−0.662550	+	0.749017i	$0.730526\pi$
$500$	0	0
$501$	0.336131	0.0150172
$502$	0	0
$503$	12.5340	0.558862	0.279431	−	0.960166i	$-0.409854\pi$
$503$	12.5340	0.558862	0.279431	+	0.960166i	$0.409854\pi$
$504$	0	0
$505$	7.02772	0.312729
$506$	0	0
$507$	−0.290239	−0.0128900
$508$	0	0
$509$	34.2093	1.51630	0.758151	−	0.652079i	$-0.226103\pi$
$509$	34.2093	1.51630	0.758151	+	0.652079i	$0.226103\pi$
$510$	0	0
$511$	2.27504	0.100642
$512$	0	0
$513$	10.8864	0.480648
$514$	0	0
$515$	−4.76076	−0.209784
$516$	0	0
$517$	−1.90152	−0.0836287
$518$	0	0
$519$	−0.423492	−0.0185892
$520$	0	0
$521$	−13.0953	−0.573717	−0.286858	−	0.957973i	$-0.592611\pi$
$521$	−13.0953	−0.573717	−0.286858	+	0.957973i	$0.592611\pi$
$522$	0	0
$523$	−14.4882	−0.633526	−0.316763	−	0.948505i	$-0.602596\pi$
$523$	−14.4882	−0.633526	−0.316763	+	0.948505i	$0.602596\pi$
$524$	0	0
$525$	−1.29862	−0.0566765
$526$	0	0
$527$	36.6204	1.59521
$528$	0	0
$529$	−19.3842	−0.842792
$530$	0	0
$531$	8.59660	0.373061
$532$	0	0
$533$	1.13457	0.0491437
$534$	0	0
$535$	−1.10620	−0.0478253
$536$	0	0
$537$	4.99435	0.215522
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−5.84431	−0.251266	−0.125633	−	0.992077i	$-0.540096\pi$
$541$	−5.84431	−0.251266	−0.125633	+	0.992077i	$0.540096\pi$
$542$	0	0
$543$	3.53833	0.151844
$544$	0	0
$545$	10.2736	0.440072
$546$	0	0
$547$	−18.5565	−0.793419	−0.396710	−	0.917944i	$-0.629848\pi$
$547$	−18.5565	−0.793419	−0.396710	+	0.917944i	$0.629848\pi$
$548$	0	0
$549$	−28.2577	−1.20601
$550$	0	0
$551$	−8.12960	−0.346333
$552$	0	0
$553$	−3.11965	−0.132661
$554$	0	0
$555$	2.49792	0.106031
$556$	0	0
$557$	−21.8331	−0.925099	−0.462549	−	0.886594i	$-0.653065\pi$
$557$	−21.8331	−0.925099	−0.462549	+	0.886594i	$0.653065\pi$
$558$	0	0
$559$	−3.11520	−0.131759
$560$	0	0
$561$	−1.70591	−0.0720235
$562$	0	0
$563$	−38.1972	−1.60982	−0.804910	−	0.593398i	$-0.797786\pi$
$563$	−38.1972	−1.60982	−0.804910	+	0.593398i	$0.797786\pi$
$564$	0	0
$565$	9.71923	0.408891
$566$	0	0
$567$	−8.24895	−0.346423
$568$	0	0
$569$	−28.2285	−1.18340	−0.591699	−	0.806159i	$-0.701543\pi$
$569$	−28.2285	−1.18340	−0.591699	+	0.806159i	$0.701543\pi$
$570$	0	0
$571$	−28.2085	−1.18049	−0.590245	−	0.807224i	$-0.700969\pi$
$571$	−28.2085	−1.18049	−0.590245	+	0.807224i	$0.700969\pi$
$572$	0	0
$573$	−2.66468	−0.111318
$574$	0	0
$575$	8.50800	0.354808
$576$	0	0
$577$	6.54293	0.272386	0.136193	−	0.990682i	$-0.456513\pi$
$577$	6.54293	0.272386	0.136193	+	0.990682i	$0.456513\pi$
$578$	0	0
$579$	−2.04205	−0.0848645
$580$	0	0
$581$	−7.10427	−0.294735
$582$	0	0
$583$	2.01493	0.0834500
$584$	0	0
$585$	−2.11405	−0.0874051
$586$	0	0
$587$	36.4797	1.50568	0.752839	−	0.658205i	$-0.228684\pi$
$587$	36.4797	1.50568	0.752839	+	0.658205i	$0.228684\pi$
$588$	0	0
$589$	39.5041	1.62774
$590$	0	0
$591$	−0.711213	−0.0292554
$592$	0	0
$593$	−25.4782	−1.04627	−0.523133	−	0.852251i	$-0.675237\pi$
$593$	−25.4782	−1.04627	−0.523133	+	0.852251i	$0.675237\pi$
$594$	0	0
$595$	−4.26150	−0.174704
$596$	0	0
$597$	−6.26548	−0.256429
$598$	0	0
$599$	3.43053	0.140168	0.0700838	−	0.997541i	$-0.477673\pi$
$599$	3.43053	0.140168	0.0700838	+	0.997541i	$0.477673\pi$
$600$	0	0
$601$	−5.90108	−0.240710	−0.120355	−	0.992731i	$-0.538403\pi$
$601$	−5.90108	−0.240710	−0.120355	+	0.992731i	$0.538403\pi$
$602$	0	0
$603$	−9.11131	−0.371041
$604$	0	0
$605$	0.725041	0.0294771
$606$	0	0
$607$	−0.972862	−0.0394872	−0.0197436	−	0.999805i	$-0.506285\pi$
$607$	−0.972862	−0.0394872	−0.0197436	+	0.999805i	$0.506285\pi$
$608$	0	0
$609$	−0.372140	−0.0150799
$610$	0	0
$611$	−1.90152	−0.0769272
$612$	0	0
$613$	39.4709	1.59422	0.797108	−	0.603837i	$-0.206362\pi$
$613$	39.4709	1.59422	0.797108	+	0.603837i	$0.206362\pi$
$614$	0	0
$615$	−0.238754	−0.00962747
$616$	0	0
$617$	−45.1264	−1.81672	−0.908361	−	0.418187i	$-0.862666\pi$
$617$	−45.1264	−1.81672	−0.908361	+	0.418187i	$0.862666\pi$
$618$	0	0
$619$	24.0419	0.966325	0.483162	−	0.875531i	$-0.339488\pi$
$619$	24.0419	0.966325	0.483162	+	0.875531i	$0.339488\pi$
$620$	0	0
$621$	−3.26488	−0.131015
$622$	0	0
$623$	−14.8253	−0.593963
$624$	0	0
$625$	17.3911	0.695643
$626$	0	0
$627$	−1.84024	−0.0734922
$628$	0	0
$629$	−69.7686	−2.78186
$630$	0	0
$631$	36.6951	1.46081	0.730404	−	0.683015i	$-0.239332\pi$
$631$	36.6951	1.46081	0.730404	+	0.683015i	$0.239332\pi$
$632$	0	0
$633$	−4.91833	−0.195486
$634$	0	0
$635$	6.24436	0.247800
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	31.7924	1.25769
$640$	0	0
$641$	36.3299	1.43494	0.717472	−	0.696587i	$-0.245299\pi$
$641$	36.3299	1.43494	0.717472	+	0.696587i	$0.245299\pi$
$642$	0	0
$643$	34.6648	1.36705	0.683523	−	0.729929i	$-0.260447\pi$
$643$	34.6648	1.36705	0.683523	+	0.729929i	$0.260447\pi$
$644$	0	0
$645$	0.655549	0.0258122
$646$	0	0
$647$	−13.6418	−0.536316	−0.268158	−	0.963375i	$-0.586415\pi$
$647$	−13.6418	−0.536316	−0.268158	+	0.963375i	$0.586415\pi$
$648$	0	0
$649$	−2.94832	−0.115732
$650$	0	0
$651$	1.80834	0.0708743
$652$	0	0
$653$	18.1253	0.709299	0.354649	−	0.934999i	$-0.384600\pi$
$653$	18.1253	0.709299	0.354649	+	0.934999i	$0.384600\pi$
$654$	0	0
$655$	14.8890	0.581762
$656$	0	0
$657$	6.63349	0.258797
$658$	0	0
$659$	15.0955	0.588037	0.294018	−	0.955800i	$-0.405007\pi$
$659$	15.0955	0.588037	0.294018	+	0.955800i	$0.405007\pi$
$660$	0	0
$661$	−23.8396	−0.927253	−0.463627	−	0.886031i	$-0.653452\pi$
$661$	−23.8396	−0.927253	−0.463627	+	0.886031i	$0.653452\pi$
$662$	0	0
$663$	−1.70591	−0.0662520
$664$	0	0
$665$	−4.59708	−0.178267
$666$	0	0
$667$	2.43810	0.0944035
$668$	0	0
$669$	2.12856	0.0822950
$670$	0	0
$671$	9.69135	0.374130
$672$	0	0
$673$	−8.22144	−0.316913	−0.158457	−	0.987366i	$-0.550652\pi$
$673$	−8.22144	−0.316913	−0.158457	+	0.987366i	$0.550652\pi$
$674$	0	0
$675$	−7.68234	−0.295693
$676$	0	0
$677$	38.4803	1.47892	0.739458	−	0.673202i	$-0.235082\pi$
$677$	38.4803	1.47892	0.739458	+	0.673202i	$0.235082\pi$
$678$	0	0
$679$	−10.6590	−0.409054
$680$	0	0
$681$	4.05039	0.155211
$682$	0	0
$683$	−4.77857	−0.182847	−0.0914234	−	0.995812i	$-0.529142\pi$
$683$	−4.77857	−0.182847	−0.0914234	+	0.995812i	$0.529142\pi$
$684$	0	0
$685$	10.8391	0.414142
$686$	0	0
$687$	6.81836	0.260136
$688$	0	0
$689$	2.01493	0.0767628
$690$	0	0
$691$	26.7100	1.01610	0.508048	−	0.861329i	$-0.330367\pi$
$691$	26.7100	1.01610	0.508048	+	0.861329i	$0.330367\pi$
$692$	0	0
$693$	2.91576	0.110761
$694$	0	0
$695$	3.82483	0.145084
$696$	0	0
$697$	6.66854	0.252589
$698$	0	0
$699$	1.25166	0.0473422
$700$	0	0
$701$	−30.4697	−1.15083	−0.575413	−	0.817863i	$-0.695159\pi$
$701$	−30.4697	−1.15083	−0.575413	+	0.817863i	$0.695159\pi$
$702$	0	0
$703$	−75.2626	−2.83858
$704$	0	0
$705$	0.400147	0.0150704
$706$	0	0
$707$	−9.69285	−0.364537
$708$	0	0
$709$	−24.7243	−0.928539	−0.464269	−	0.885694i	$-0.653683\pi$
$709$	−24.7243	−0.928539	−0.464269	+	0.885694i	$0.653683\pi$
$710$	0	0
$711$	−9.09614	−0.341132
$712$	0	0
$713$	−11.8474	−0.443690
$714$	0	0
$715$	0.725041	0.0271150
$716$	0	0
$717$	4.11118	0.153535
$718$	0	0
$719$	−0.393816	−0.0146869	−0.00734344	−	0.999973i	$-0.502338\pi$
$719$	−0.393816	−0.0146869	−0.00734344	+	0.999973i	$0.502338\pi$
$720$	0	0
$721$	6.56619	0.244538
$722$	0	0
$723$	0.786238	0.0292405
$724$	0	0
$725$	5.73689	0.213063
$726$	0	0
$727$	−30.5156	−1.13176	−0.565880	−	0.824487i	$-0.691464\pi$
$727$	−30.5156	−1.13176	−0.565880	+	0.824487i	$0.691464\pi$
$728$	0	0
$729$	−22.5570	−0.835443
$730$	0	0
$731$	−18.3099	−0.677216
$732$	0	0
$733$	29.6878	1.09655	0.548273	−	0.836300i	$-0.315286\pi$
$733$	29.6878	1.09655	0.548273	+	0.836300i	$0.315286\pi$
$734$	0	0
$735$	−0.210435	−0.00776202
$736$	0	0
$737$	3.12485	0.115105
$738$	0	0
$739$	−51.3150	−1.88765	−0.943826	−	0.330442i	$-0.892802\pi$
$739$	−51.3150	−1.88765	−0.943826	+	0.330442i	$0.892802\pi$
$740$	0	0
$741$	−1.84024	−0.0676030
$742$	0	0
$743$	−26.0106	−0.954237	−0.477118	−	0.878839i	$-0.658319\pi$
$743$	−26.0106	−0.954237	−0.477118	+	0.878839i	$0.658319\pi$
$744$	0	0
$745$	12.7978	0.468874
$746$	0	0
$747$	−20.7144	−0.757899
$748$	0	0
$749$	1.52571	0.0557482
$750$	0	0
$751$	−21.0159	−0.766880	−0.383440	−	0.923566i	$-0.625261\pi$
$751$	−21.0159	−0.766880	−0.383440	+	0.923566i	$0.625261\pi$
$752$	0	0
$753$	4.38751	0.159890
$754$	0	0
$755$	−8.64237	−0.314528
$756$	0	0
$757$	18.7407	0.681141	0.340570	−	0.940219i	$-0.389380\pi$
$757$	18.7407	0.681141	0.340570	+	0.940219i	$0.389380\pi$
$758$	0	0
$759$	0.551895	0.0200325
$760$	0	0
$761$	−36.6787	−1.32960	−0.664801	−	0.747021i	$-0.731484\pi$
$761$	−36.6787	−1.32960	−0.664801	+	0.747021i	$0.731484\pi$
$762$	0	0
$763$	−14.1696	−0.512976
$764$	0	0
$765$	−12.4255	−0.449245
$766$	0	0
$767$	−2.94832	−0.106458
$768$	0	0
$769$	−17.5403	−0.632520	−0.316260	−	0.948673i	$-0.602427\pi$
$769$	−17.5403	−0.632520	−0.316260	+	0.948673i	$0.602427\pi$
$770$	0	0
$771$	−4.19021	−0.150907
$772$	0	0
$773$	−26.3987	−0.949497	−0.474748	−	0.880122i	$-0.657461\pi$
$773$	−26.3987	−0.949497	−0.474748	+	0.880122i	$0.657461\pi$
$774$	0	0
$775$	−27.8772	−1.00138
$776$	0	0
$777$	−3.44521	−0.123596
$778$	0	0
$779$	7.19367	0.257740
$780$	0	0
$781$	−10.9036	−0.390163
$782$	0	0
$783$	−2.20149	−0.0786749
$784$	0	0
$785$	12.6030	0.449822
$786$	0	0
$787$	−2.54466	−0.0907074	−0.0453537	−	0.998971i	$-0.514441\pi$
$787$	−2.54466	−0.0907074	−0.0453537	+	0.998971i	$0.514441\pi$
$788$	0	0
$789$	0.888434	0.0316291
$790$	0	0
$791$	−13.4051	−0.476629
$792$	0	0
$793$	9.69135	0.344150
$794$	0	0
$795$	−0.424013	−0.0150382
$796$	0	0
$797$	−21.4638	−0.760285	−0.380143	−	0.924928i	$-0.624125\pi$
$797$	−21.4638	−0.760285	−0.380143	+	0.924928i	$0.624125\pi$
$798$	0	0
$799$	−11.1764	−0.395391
$800$	0	0
$801$	−43.2270	−1.52735
$802$	0	0
$803$	−2.27504	−0.0802846
$804$	0	0
$805$	1.37868	0.0485921
$806$	0	0
$807$	8.12365	0.285966
$808$	0	0
$809$	−40.2173	−1.41397	−0.706983	−	0.707230i	$-0.749944\pi$
$809$	−40.2173	−1.41397	−0.706983	+	0.707230i	$0.749944\pi$
$810$	0	0
$811$	35.7217	1.25436	0.627180	−	0.778875i	$-0.284209\pi$
$811$	35.7217	1.25436	0.627180	+	0.778875i	$0.284209\pi$
$812$	0	0
$813$	−7.60659	−0.266775
$814$	0	0
$815$	−6.29540	−0.220518
$816$	0	0
$817$	−19.7517	−0.691026
$818$	0	0
$819$	2.91576	0.101885
$820$	0	0
$821$	−19.8954	−0.694355	−0.347177	−	0.937799i	$-0.612860\pi$
$821$	−19.8954	−0.694355	−0.347177	+	0.937799i	$0.612860\pi$
$822$	0	0
$823$	35.5930	1.24069	0.620347	−	0.784327i	$-0.286992\pi$
$823$	35.5930	1.24069	0.620347	+	0.784327i	$0.286992\pi$
$824$	0	0
$825$	1.29862	0.0452122
$826$	0	0
$827$	−16.3126	−0.567244	−0.283622	−	0.958936i	$-0.591536\pi$
$827$	−16.3126	−0.567244	−0.283622	+	0.958936i	$0.591536\pi$
$828$	0	0
$829$	32.0673	1.11374	0.556872	−	0.830598i	$-0.312001\pi$
$829$	32.0673	1.11374	0.556872	+	0.830598i	$0.312001\pi$
$830$	0	0
$831$	1.27004	0.0440570
$832$	0	0
$833$	5.87760	0.203647
$834$	0	0
$835$	−0.839684	−0.0290585
$836$	0	0
$837$	10.6977	0.369766
$838$	0	0
$839$	7.18675	0.248114	0.124057	−	0.992275i	$-0.460409\pi$
$839$	7.18675	0.248114	0.124057	+	0.992275i	$0.460409\pi$
$840$	0	0
$841$	−27.3560	−0.943311
$842$	0	0
$843$	−3.27579	−0.112824
$844$	0	0
$845$	0.725041	0.0249422
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−0.394845	−0.0135510
$850$	0	0
$851$	22.5715	0.773742
$852$	0	0
$853$	−8.04185	−0.275348	−0.137674	−	0.990478i	$-0.543963\pi$
$853$	−8.04185	−0.275348	−0.137674	+	0.990478i	$0.543963\pi$
$854$	0	0
$855$	−13.4040	−0.458406
$856$	0	0
$857$	−18.4261	−0.629422	−0.314711	−	0.949187i	$-0.601908\pi$
$857$	−18.4261	−0.629422	−0.314711	+	0.949187i	$0.601908\pi$
$858$	0	0
$859$	30.7615	1.04957	0.524785	−	0.851235i	$-0.324146\pi$
$859$	30.7615	1.04957	0.524785	+	0.851235i	$0.324146\pi$
$860$	0	0
$861$	0.329297	0.0112224
$862$	0	0
$863$	−54.0999	−1.84158	−0.920790	−	0.390058i	$-0.872455\pi$
$863$	−54.0999	−1.84158	−0.920790	+	0.390058i	$0.872455\pi$
$864$	0	0
$865$	1.05792	0.0359703
$866$	0	0
$867$	−5.09257	−0.172953
$868$	0	0
$869$	3.11965	0.105827
$870$	0	0
$871$	3.12485	0.105881
$872$	0	0
$873$	−31.0791	−1.05187
$874$	0	0
$875$	6.86927	0.232224
$876$	0	0
$877$	17.4058	0.587752	0.293876	−	0.955844i	$-0.405055\pi$
$877$	17.4058	0.587752	0.293876	+	0.955844i	$0.405055\pi$
$878$	0	0
$879$	−2.12558	−0.0716940
$880$	0	0
$881$	−52.2399	−1.76001	−0.880004	−	0.474966i	$-0.842460\pi$
$881$	−52.2399	−1.76001	−0.880004	+	0.474966i	$0.842460\pi$
$882$	0	0
$883$	−9.29545	−0.312817	−0.156408	−	0.987692i	$-0.549992\pi$
$883$	−9.29545	−0.312817	−0.156408	+	0.987692i	$0.549992\pi$
$884$	0	0
$885$	0.620431	0.0208556
$886$	0	0
$887$	−25.2407	−0.847501	−0.423750	−	0.905779i	$-0.639287\pi$
$887$	−25.2407	−0.847501	−0.423750	+	0.905779i	$0.639287\pi$
$888$	0	0
$889$	−8.61242	−0.288851
$890$	0	0
$891$	8.24895	0.276350
$892$	0	0
$893$	−12.0565	−0.403454
$894$	0	0
$895$	−12.4763	−0.417037
$896$	0	0
$897$	0.551895	0.0184272
$898$	0	0
$899$	−7.98865	−0.266436
$900$	0	0
$901$	11.8430	0.394546
$902$	0	0
$903$	−0.904154	−0.0300883
$904$	0	0
$905$	−8.83903	−0.293819
$906$	0	0
$907$	47.9797	1.59314	0.796571	−	0.604545i	$-0.206645\pi$
$907$	47.9797	1.59314	0.796571	+	0.604545i	$0.206645\pi$
$908$	0	0
$909$	−28.2620	−0.937393
$910$	0	0
$911$	−42.4401	−1.40610	−0.703052	−	0.711138i	$-0.748180\pi$
$911$	−42.4401	−1.40610	−0.703052	+	0.711138i	$0.748180\pi$
$912$	0	0
$913$	7.10427	0.235117
$914$	0	0
$915$	−2.03940	−0.0674205
$916$	0	0
$917$	−20.5354	−0.678139
$918$	0	0
$919$	30.8930	1.01907	0.509533	−	0.860451i	$-0.329818\pi$
$919$	30.8930	1.01907	0.509533	+	0.860451i	$0.329818\pi$
$920$	0	0
$921$	5.50195	0.181295
$922$	0	0
$923$	−10.9036	−0.358897
$924$	0	0
$925$	53.1113	1.74629
$926$	0	0
$927$	19.1454	0.628819
$928$	0	0
$929$	39.6612	1.30124	0.650621	−	0.759403i	$-0.274509\pi$
$929$	39.6612	1.30124	0.650621	+	0.759403i	$0.274509\pi$
$930$	0	0
$931$	6.34044	0.207799
$932$	0	0
$933$	3.21283	0.105183
$934$	0	0
$935$	4.26150	0.139366
$936$	0	0
$937$	20.6337	0.674074	0.337037	−	0.941491i	$-0.390575\pi$
$937$	20.6337	0.674074	0.337037	+	0.941491i	$0.390575\pi$
$938$	0	0
$939$	−3.88430	−0.126759
$940$	0	0
$941$	−11.1034	−0.361959	−0.180979	−	0.983487i	$-0.557927\pi$
$941$	−11.1034	−0.361959	−0.180979	+	0.983487i	$0.557927\pi$
$942$	0	0
$943$	−2.15741	−0.0702548
$944$	0	0
$945$	−1.24489	−0.0404961
$946$	0	0
$947$	36.7277	1.19349	0.596745	−	0.802431i	$-0.296460\pi$
$947$	36.7277	1.19349	0.596745	+	0.802431i	$0.296460\pi$
$948$	0	0
$949$	−2.27504	−0.0738511
$950$	0	0
$951$	5.70078	0.184860
$952$	0	0
$953$	41.5838	1.34703	0.673515	−	0.739174i	$-0.264784\pi$
$953$	41.5838	1.34703	0.673515	+	0.739174i	$0.264784\pi$
$954$	0	0
$955$	6.65658	0.215402
$956$	0	0
$957$	0.372140	0.0120296
$958$	0	0
$959$	−14.9497	−0.482751
$960$	0	0
$961$	7.81920	0.252232
$962$	0	0
$963$	4.44860	0.143354
$964$	0	0
$965$	5.10120	0.164213
$966$	0	0
$967$	−42.4475	−1.36502	−0.682510	−	0.730876i	$-0.739112\pi$
$967$	−42.4475	−1.36502	−0.682510	+	0.730876i	$0.739112\pi$
$968$	0	0
$969$	−10.8162	−0.347467
$970$	0	0
$971$	−9.68898	−0.310934	−0.155467	−	0.987841i	$-0.549688\pi$
$971$	−9.68898	−0.310934	−0.155467	+	0.987841i	$0.549688\pi$
$972$	0	0
$973$	−5.27533	−0.169119
$974$	0	0
$975$	1.29862	0.0415892
$976$	0	0
$977$	5.63840	0.180388	0.0901941	−	0.995924i	$-0.471251\pi$
$977$	5.63840	0.180388	0.0901941	+	0.995924i	$0.471251\pi$
$978$	0	0
$979$	14.8253	0.473818
$980$	0	0
$981$	−41.3153	−1.31910
$982$	0	0
$983$	−20.3721	−0.649768	−0.324884	−	0.945754i	$-0.605325\pi$
$983$	−20.3721	−0.649768	−0.324884	+	0.945754i	$0.605325\pi$
$984$	0	0
$985$	1.77667	0.0566094
$986$	0	0
$987$	−0.551895	−0.0175670
$988$	0	0
$989$	5.92362	0.188360
$990$	0	0
$991$	38.4581	1.22166	0.610830	−	0.791762i	$-0.290836\pi$
$991$	38.4581	1.22166	0.610830	+	0.791762i	$0.290836\pi$
$992$	0	0
$993$	−10.1160	−0.321022
$994$	0	0
$995$	15.6517	0.496191
$996$	0	0
$997$	16.9652	0.537292	0.268646	−	0.963239i	$-0.413424\pi$
$997$	16.9652	0.537292	0.268646	+	0.963239i	$0.413424\pi$
$998$	0	0
$999$	−20.3811	−0.644828

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.k.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.k.1.4	✓	10	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q-0.290239 q^{3} +0.725041 q^{5} -1.00000 q^{7} -2.91576 q^{9} +O(q^{10})\)	\(q-0.290239 q^{3} +0.725041 q^{5} -1.00000 q^{7} -2.91576 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.210435 q^{15} +5.87760 q^{17} +6.34044 q^{19} +0.290239 q^{21} -1.90152 q^{23} -4.47432 q^{25} +1.71699 q^{27} -1.28218 q^{29} +6.23051 q^{31} -0.290239 q^{33} -0.725041 q^{35} -11.8703 q^{37} -0.290239 q^{39} +1.13457 q^{41} -3.11520 q^{43} -2.11405 q^{45} -1.90152 q^{47} +1.00000 q^{49} -1.70591 q^{51} +2.01493 q^{53} +0.725041 q^{55} -1.84024 q^{57} -2.94832 q^{59} +9.69135 q^{61} +2.91576 q^{63} +0.725041 q^{65} +3.12485 q^{67} +0.551895 q^{69} -10.9036 q^{71} -2.27504 q^{73} +1.29862 q^{75} -1.00000 q^{77} +3.11965 q^{79} +8.24895 q^{81} +7.10427 q^{83} +4.26150 q^{85} +0.372140 q^{87} +14.8253 q^{89} -1.00000 q^{91} -1.80834 q^{93} +4.59708 q^{95} +10.6590 q^{97} -2.91576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9}+O(q^{10}) \) 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9	\( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9} + 10 q^{11} + 10 q^{13} + 3 q^{15} + 3 q^{17} + 6 q^{19} - q^{21} + 4 q^{23} + 22 q^{25} - 5 q^{27} + 10 q^{29} - q^{31} + q^{33} - 4 q^{35} + 20 q^{37} + q^{39} + 8 q^{45} + 4 q^{47} + 10 q^{49} + 11 q^{51} - 5 q^{53} + 4 q^{55} + 16 q^{57} + 11 q^{59} + 12 q^{61} - 17 q^{63} + 4 q^{65} - 2 q^{67} + 10 q^{69} + 28 q^{71} + 11 q^{73} - 6 q^{75} - 10 q^{77} - 10 q^{79} + 46 q^{81} + 7 q^{83} + 33 q^{85} - 47 q^{87} + 30 q^{89} - 10 q^{91} + 41 q^{93} - 2 q^{95} + 55 q^{97} + 17 q^{99}+O(q^{100}) \) 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9 + 10 * q^11 + 10 * q^13 + 3 * q^15 + 3 * q^17 + 6 * q^19 - q^21 + 4 * q^23 + 22 * q^25 - 5 * q^27 + 10 * q^29 - q^31 + q^33 - 4 * q^35 + 20 * q^37 + q^39 + 8 * q^45 + 4 * q^47 + 10 * q^49 + 11 * q^51 - 5 * q^53 + 4 * q^55 + 16 * q^57 + 11 * q^59 + 12 * q^61 - 17 * q^63 + 4 * q^65 - 2 * q^67 + 10 * q^69 + 28 * q^71 + 11 * q^73 - 6 * q^75 - 10 * q^77 - 10 * q^79 + 46 * q^81 + 7 * q^83 + 33 * q^85 - 47 * q^87 + 30 * q^89 - 10 * q^91 + 41 * q^93 - 2 * q^95 + 55 * q^97 + 17 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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