LMFDB - Embedded newform 4004.2.a.e.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.209057$ 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)

$f(q)$	$=$	$q+1.33826 q^{3} +1.82709 q^{5} -1.00000 q^{7} -1.20906 q^{9} +O(q^{10})$	$q+1.33826 q^{3} +1.82709 q^{5} -1.00000 q^{7} -1.20906 q^{9} +1.00000 q^{11} -1.00000 q^{13} +2.44512 q^{15} -5.73968 q^{17} -5.19236 q^{19} -1.33826 q^{21} +1.95630 q^{23} -1.66174 q^{25} -5.63282 q^{27} -0.627171 q^{29} +0.581886 q^{31} +1.33826 q^{33} -1.82709 q^{35} -6.27786 q^{37} -1.33826 q^{39} +8.73212 q^{41} -0.231398 q^{43} -2.20906 q^{45} -8.02859 q^{47} +1.00000 q^{49} -7.68119 q^{51} -8.69440 q^{53} +1.82709 q^{55} -6.94874 q^{57} -8.82051 q^{59} +0.601340 q^{61} +1.20906 q^{63} -1.82709 q^{65} -5.83623 q^{67} +2.61803 q^{69} -8.47214 q^{71} +13.6086 q^{73} -2.22384 q^{75} -1.00000 q^{77} +7.04082 q^{79} -3.91101 q^{81} +0.342932 q^{83} -10.4869 q^{85} -0.839318 q^{87} -9.91161 q^{89} +1.00000 q^{91} +0.778716 q^{93} -9.48692 q^{95} +12.0150 q^{97} -1.20906 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10}) $ 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9	$ 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} + q^{17} - 3 q^{19} - q^{21} - q^{23} - 11 q^{25} - 5 q^{27} + 3 q^{29} + 6 q^{31} + q^{33} - q^{35} + 4 q^{37} - q^{39} - 6 q^{41} - 3 q^{43} - 7 q^{45} - 7 q^{47} + 4 q^{49} - 11 q^{51} - 20 q^{53} + q^{55} - 2 q^{57} - 11 q^{59} - 18 q^{61} + 3 q^{63} - q^{65} - 16 q^{67} + 6 q^{69} - 16 q^{71} + 4 q^{73} + 6 q^{75} - 4 q^{77} + 9 q^{79} - 16 q^{81} - 14 q^{83} - 11 q^{85} - 3 q^{87} - 23 q^{89} + 4 q^{91} - q^{93} - 7 q^{95} + 14 q^{97} - 3 q^{99}+O(q^{100}) $ 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9 + 4 * q^11 - 4 * q^13 - q^15 + q^17 - 3 * q^19 - q^21 - q^23 - 11 * q^25 - 5 * q^27 + 3 * q^29 + 6 * q^31 + q^33 - q^35 + 4 * q^37 - q^39 - 6 * q^41 - 3 * q^43 - 7 * q^45 - 7 * q^47 + 4 * q^49 - 11 * q^51 - 20 * q^53 + q^55 - 2 * q^57 - 11 * q^59 - 18 * q^61 + 3 * q^63 - q^65 - 16 * q^67 + 6 * q^69 - 16 * q^71 + 4 * q^73 + 6 * q^75 - 4 * q^77 + 9 * q^79 - 16 * q^81 - 14 * q^83 - 11 * q^85 - 3 * q^87 - 23 * q^89 + 4 * q^91 - q^93 - 7 * q^95 + 14 * q^97 - 3 * 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
$3$	1.33826	0.772645	0.386323	−	0.922364i	$-0.373745\pi$
$3$	1.33826	0.772645	0.386323	+	0.922364i	$0.373745\pi$
$4$	0	0
$5$	1.82709	0.817100	0.408550	−	0.912736i	$-0.366035\pi$
$5$	1.82709	0.817100	0.408550	+	0.912736i	$0.366035\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.20906	−0.403019
$10$	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
$15$	2.44512	0.631329
$16$	0	0
$17$	−5.73968	−1.39208	−0.696039	−	0.718004i	$-0.745056\pi$
$17$	−5.73968	−1.39208	−0.696039	+	0.718004i	$0.745056\pi$
$18$	0	0
$19$	−5.19236	−1.19121	−0.595605	−	0.803278i	$-0.703088\pi$
$19$	−5.19236	−1.19121	−0.595605	+	0.803278i	$0.703088\pi$
$20$	0	0
$21$	−1.33826	−0.292033
$22$	0	0
$23$	1.95630	0.407916	0.203958	−	0.978980i	$-0.434619\pi$
$23$	1.95630	0.407916	0.203958	+	0.978980i	$0.434619\pi$
$24$	0	0
$25$	−1.66174	−0.332348
$26$	0	0
$27$	−5.63282	−1.08404
$28$	0	0
$29$	−0.627171	−0.116463	−0.0582313	−	0.998303i	$-0.518546\pi$
$29$	−0.627171	−0.116463	−0.0582313	+	0.998303i	$0.518546\pi$
$30$	0	0
$31$	0.581886	0.104510	0.0522549	−	0.998634i	$-0.483359\pi$
$31$	0.581886	0.104510	0.0522549	+	0.998634i	$0.483359\pi$
$32$	0	0
$33$	1.33826	0.232961
$34$	0	0
$35$	−1.82709	−0.308835
$36$	0	0
$37$	−6.27786	−1.03207	−0.516037	−	0.856566i	$-0.672593\pi$
$37$	−6.27786	−1.03207	−0.516037	+	0.856566i	$0.672593\pi$
$38$	0	0
$39$	−1.33826	−0.214293
$40$	0	0
$41$	8.73212	1.36373	0.681864	−	0.731479i	$-0.261169\pi$
$41$	8.73212	1.36373	0.681864	+	0.731479i	$0.261169\pi$
$42$	0	0
$43$	−0.231398	−0.0352878	−0.0176439	−	0.999844i	$-0.505617\pi$
$43$	−0.231398	−0.0352878	−0.0176439	+	0.999844i	$0.505617\pi$
$44$	0	0
$45$	−2.20906	−0.329307
$46$	0	0
$47$	−8.02859	−1.17109	−0.585545	−	0.810640i	$-0.699120\pi$
$47$	−8.02859	−1.17109	−0.585545	+	0.810640i	$0.699120\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.68119	−1.07558
$52$	0	0
$53$	−8.69440	−1.19427	−0.597134	−	0.802142i	$-0.703694\pi$
$53$	−8.69440	−1.19427	−0.597134	+	0.802142i	$0.703694\pi$
$54$	0	0
$55$	1.82709	0.246365
$56$	0	0
$57$	−6.94874	−0.920383
$58$	0	0
$59$	−8.82051	−1.14833	−0.574166	−	0.818739i	$-0.694674\pi$
$59$	−8.82051	−1.14833	−0.574166	+	0.818739i	$0.694674\pi$
$60$	0	0
$61$	0.601340	0.0769937	0.0384969	−	0.999259i	$-0.487743\pi$
$61$	0.601340	0.0769937	0.0384969	+	0.999259i	$0.487743\pi$
$62$	0	0
$63$	1.20906	0.152327
$64$	0	0
$65$	−1.82709	−0.226623
$66$	0	0
$67$	−5.83623	−0.713009	−0.356504	−	0.934294i	$-0.616032\pi$
$67$	−5.83623	−0.713009	−0.356504	+	0.934294i	$0.616032\pi$
$68$	0	0
$69$	2.61803	0.315174
$70$	0	0
$71$	−8.47214	−1.00546	−0.502729	−	0.864444i	$-0.667671\pi$
$71$	−8.47214	−1.00546	−0.502729	+	0.864444i	$0.667671\pi$
$72$	0	0
$73$	13.6086	1.59276	0.796381	−	0.604795i	$-0.206745\pi$
$73$	13.6086	1.59276	0.796381	+	0.604795i	$0.206745\pi$
$74$	0	0
$75$	−2.22384	−0.256787
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	7.04082	0.792154	0.396077	−	0.918217i	$-0.370371\pi$
$79$	7.04082	0.792154	0.396077	+	0.918217i	$0.370371\pi$
$80$	0	0
$81$	−3.91101	−0.434557
$82$	0	0
$83$	0.342932	0.0376416	0.0188208	−	0.999823i	$-0.494009\pi$
$83$	0.342932	0.0376416	0.0188208	+	0.999823i	$0.494009\pi$
$84$	0	0
$85$	−10.4869	−1.13747
$86$	0	0
$87$	−0.839318	−0.0899844
$88$	0	0
$89$	−9.91161	−1.05063	−0.525314	−	0.850908i	$-0.676052\pi$
$89$	−9.91161	−1.05063	−0.525314	+	0.850908i	$0.676052\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0.778716	0.0807491
$94$	0	0
$95$	−9.48692	−0.973337
$96$	0	0
$97$	12.0150	1.21994	0.609969	−	0.792426i	$-0.291182\pi$
$97$	12.0150	1.21994	0.609969	+	0.792426i	$0.291182\pi$
$98$	0	0
$99$	−1.20906	−0.121515
$100$	0	0
$101$	12.8874	1.28234	0.641170	−	0.767399i	$-0.278449\pi$
$101$	12.8874	1.28234	0.641170	+	0.767399i	$0.278449\pi$
$102$	0	0
$103$	−3.66332	−0.360958	−0.180479	−	0.983579i	$-0.557765\pi$
$103$	−3.66332	−0.360958	−0.180479	+	0.983579i	$0.557765\pi$
$104$	0	0
$105$	−2.44512	−0.238620
$106$	0	0
$107$	14.9459	1.44487	0.722435	−	0.691439i	$-0.243023\pi$
$107$	14.9459	1.44487	0.722435	+	0.691439i	$0.243023\pi$
$108$	0	0
$109$	−16.7805	−1.60728	−0.803640	−	0.595116i	$-0.797106\pi$
$109$	−16.7805	−1.60728	−0.803640	+	0.595116i	$0.797106\pi$
$110$	0	0
$111$	−8.40142	−0.797428
$112$	0	0
$113$	1.18014	0.111018	0.0555089	−	0.998458i	$-0.482322\pi$
$113$	1.18014	0.111018	0.0555089	+	0.998458i	$0.482322\pi$
$114$	0	0
$115$	3.57433	0.333308
$116$	0	0
$117$	1.20906	0.111777
$118$	0	0
$119$	5.73968	0.526156
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	11.6859	1.05368
$124$	0	0
$125$	−12.1716	−1.08866
$126$	0	0
$127$	6.48692	0.575621	0.287811	−	0.957687i	$-0.407073\pi$
$127$	6.48692	0.575621	0.287811	+	0.957687i	$0.407073\pi$
$128$	0	0
$129$	−0.309670	−0.0272650
$130$	0	0
$131$	13.6671	1.19410	0.597048	−	0.802205i	$-0.296340\pi$
$131$	13.6671	1.19410	0.597048	+	0.802205i	$0.296340\pi$
$132$	0	0
$133$	5.19236	0.450235
$134$	0	0
$135$	−10.2917	−0.885766
$136$	0	0
$137$	−6.20130	−0.529812	−0.264906	−	0.964274i	$-0.585341\pi$
$137$	−6.20130	−0.529812	−0.264906	+	0.964274i	$0.585341\pi$
$138$	0	0
$139$	−12.7947	−1.08523	−0.542615	−	0.839981i	$-0.682566\pi$
$139$	−12.7947	−1.08523	−0.542615	+	0.839981i	$0.682566\pi$
$140$	0	0
$141$	−10.7444	−0.904838
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−1.14590	−0.0951617
$146$	0	0
$147$	1.33826	0.110378
$148$	0	0
$149$	−21.7908	−1.78517	−0.892587	−	0.450876i	$-0.851112\pi$
$149$	−21.7908	−1.78517	−0.892587	+	0.450876i	$0.851112\pi$
$150$	0	0
$151$	11.6356	0.946890	0.473445	−	0.880823i	$-0.343010\pi$
$151$	11.6356	0.946890	0.473445	+	0.880823i	$0.343010\pi$
$152$	0	0
$153$	6.93960	0.561034
$154$	0	0
$155$	1.06316	0.0853950
$156$	0	0
$157$	−9.78806	−0.781172	−0.390586	−	0.920566i	$-0.627728\pi$
$157$	−9.78806	−0.781172	−0.390586	+	0.920566i	$0.627728\pi$
$158$	0	0
$159$	−11.6354	−0.922745
$160$	0	0
$161$	−1.95630	−0.154178
$162$	0	0
$163$	13.0179	1.01964	0.509822	−	0.860280i	$-0.329711\pi$
$163$	13.0179	1.01964	0.509822	+	0.860280i	$0.329711\pi$
$164$	0	0
$165$	2.44512	0.190353
$166$	0	0
$167$	−14.0314	−1.08578	−0.542889	−	0.839804i	$-0.682670\pi$
$167$	−14.0314	−1.08578	−0.542889	+	0.839804i	$0.682670\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.27786	0.480080
$172$	0	0
$173$	2.66994	0.202992	0.101496	−	0.994836i	$-0.467637\pi$
$173$	2.66994	0.202992	0.101496	+	0.994836i	$0.467637\pi$
$174$	0	0
$175$	1.66174	0.125616
$176$	0	0
$177$	−11.8041	−0.887254
$178$	0	0
$179$	−24.8056	−1.85406	−0.927029	−	0.374990i	$-0.877646\pi$
$179$	−24.8056	−1.85406	−0.927029	+	0.374990i	$0.877646\pi$
$180$	0	0
$181$	−7.61454	−0.565985	−0.282992	−	0.959122i	$-0.591327\pi$
$181$	−7.61454	−0.565985	−0.282992	+	0.959122i	$0.591327\pi$
$182$	0	0
$183$	0.804750	0.0594889
$184$	0	0
$185$	−11.4702	−0.843308
$186$	0	0
$187$	−5.73968	−0.419727
$188$	0	0
$189$	5.63282	0.409727
$190$	0	0
$191$	1.25146	0.0905521	0.0452761	−	0.998975i	$-0.485583\pi$
$191$	1.25146	0.0905521	0.0452761	+	0.998975i	$0.485583\pi$
$192$	0	0
$193$	22.8037	1.64145	0.820723	−	0.571327i	$-0.193571\pi$
$193$	22.8037	1.64145	0.820723	+	0.571327i	$0.193571\pi$
$194$	0	0
$195$	−2.44512	−0.175099
$196$	0	0
$197$	18.8233	1.34110	0.670551	−	0.741863i	$-0.266058\pi$
$197$	18.8233	1.34110	0.670551	+	0.741863i	$0.266058\pi$
$198$	0	0
$199$	7.31303	0.518407	0.259204	−	0.965823i	$-0.416540\pi$
$199$	7.31303	0.518407	0.259204	+	0.965823i	$0.416540\pi$
$200$	0	0
$201$	−7.81040	−0.550903
$202$	0	0
$203$	0.627171	0.0440188
$204$	0	0
$205$	15.9544	1.11430
$206$	0	0
$207$	−2.36527	−0.164398
$208$	0	0
$209$	−5.19236	−0.359163
$210$	0	0
$211$	−7.89334	−0.543400	−0.271700	−	0.962382i	$-0.587586\pi$
$211$	−7.89334	−0.543400	−0.271700	+	0.962382i	$0.587586\pi$
$212$	0	0
$213$	−11.3379	−0.776862
$214$	0	0
$215$	−0.422784	−0.0288337
$216$	0	0
$217$	−0.581886	−0.0395010
$218$	0	0
$219$	18.2118	1.23064
$220$	0	0
$221$	5.73968	0.386093
$222$	0	0
$223$	5.28326	0.353793	0.176897	−	0.984229i	$-0.443394\pi$
$223$	5.28326	0.353793	0.176897	+	0.984229i	$0.443394\pi$
$224$	0	0
$225$	2.00914	0.133942
$226$	0	0
$227$	27.5227	1.82675	0.913374	−	0.407121i	$-0.133467\pi$
$227$	27.5227	1.82675	0.913374	+	0.407121i	$0.133467\pi$
$228$	0	0
$229$	−15.8766	−1.04916	−0.524579	−	0.851362i	$-0.675777\pi$
$229$	−15.8766	−1.04916	−0.524579	+	0.851362i	$0.675777\pi$
$230$	0	0
$231$	−1.33826	−0.0880511
$232$	0	0
$233$	−9.43508	−0.618113	−0.309056	−	0.951044i	$-0.600013\pi$
$233$	−9.43508	−0.618113	−0.309056	+	0.951044i	$0.600013\pi$
$234$	0	0
$235$	−14.6690	−0.956898
$236$	0	0
$237$	9.42245	0.612054
$238$	0	0
$239$	22.4763	1.45387	0.726935	−	0.686707i	$-0.240944\pi$
$239$	22.4763	1.45387	0.726935	+	0.686707i	$0.240944\pi$
$240$	0	0
$241$	−29.4175	−1.89494	−0.947472	−	0.319838i	$-0.896372\pi$
$241$	−29.4175	−1.89494	−0.947472	+	0.319838i	$0.896372\pi$
$242$	0	0
$243$	11.6645	0.748278
$244$	0	0
$245$	1.82709	0.116729
$246$	0	0
$247$	5.19236	0.330382
$248$	0	0
$249$	0.458932	0.0290836
$250$	0	0
$251$	15.6025	0.984822	0.492411	−	0.870363i	$-0.336116\pi$
$251$	15.6025	0.984822	0.492411	+	0.870363i	$0.336116\pi$
$252$	0	0
$253$	1.95630	0.122991
$254$	0	0
$255$	−14.0342	−0.878858
$256$	0	0
$257$	−4.37941	−0.273180	−0.136590	−	0.990628i	$-0.543614\pi$
$257$	−4.37941	−0.273180	−0.136590	+	0.990628i	$0.543614\pi$
$258$	0	0
$259$	6.27786	0.390087
$260$	0	0
$261$	0.758285	0.0469367
$262$	0	0
$263$	−7.62973	−0.470469	−0.235235	−	0.971939i	$-0.575586\pi$
$263$	−7.62973	−0.470469	−0.235235	+	0.971939i	$0.575586\pi$
$264$	0	0
$265$	−15.8855	−0.975836
$266$	0	0
$267$	−13.2643	−0.811764
$268$	0	0
$269$	−6.56966	−0.400559	−0.200280	−	0.979739i	$-0.564185\pi$
$269$	−6.56966	−0.400559	−0.200280	+	0.979739i	$0.564185\pi$
$270$	0	0
$271$	12.0535	0.732197	0.366099	−	0.930576i	$-0.380693\pi$
$271$	12.0535	0.732197	0.366099	+	0.930576i	$0.380693\pi$
$272$	0	0
$273$	1.33826	0.0809953
$274$	0	0
$275$	−1.66174	−0.100207
$276$	0	0
$277$	−2.02254	−0.121523	−0.0607615	−	0.998152i	$-0.519353\pi$
$277$	−2.02254	−0.121523	−0.0607615	+	0.998152i	$0.519353\pi$
$278$	0	0
$279$	−0.703533	−0.0421194
$280$	0	0
$281$	−2.02327	−0.120698	−0.0603492	−	0.998177i	$-0.519221\pi$
$281$	−2.02327	−0.120698	−0.0603492	+	0.998177i	$0.519221\pi$
$282$	0	0
$283$	9.90076	0.588539	0.294270	−	0.955722i	$-0.404924\pi$
$283$	9.90076	0.588539	0.294270	+	0.955722i	$0.404924\pi$
$284$	0	0
$285$	−12.6960	−0.752045
$286$	0	0
$287$	−8.73212	−0.515441
$288$	0	0
$289$	15.9439	0.937879
$290$	0	0
$291$	16.0792	0.942579
$292$	0	0
$293$	16.5096	0.964501	0.482250	−	0.876033i	$-0.339820\pi$
$293$	16.5096	0.964501	0.482250	+	0.876033i	$0.339820\pi$
$294$	0	0
$295$	−16.1159	−0.938302
$296$	0	0
$297$	−5.63282	−0.326849
$298$	0	0
$299$	−1.95630	−0.113135
$300$	0	0
$301$	0.231398	0.0133375
$302$	0	0
$303$	17.2467	0.990795
$304$	0	0
$305$	1.09870	0.0629116
$306$	0	0
$307$	−27.4577	−1.56709	−0.783546	−	0.621334i	$-0.786591\pi$
$307$	−27.4577	−1.56709	−0.783546	+	0.621334i	$0.786591\pi$
$308$	0	0
$309$	−4.90248	−0.278892
$310$	0	0
$311$	11.6695	0.661719	0.330860	−	0.943680i	$-0.392661\pi$
$311$	11.6695	0.661719	0.330860	+	0.943680i	$0.392661\pi$
$312$	0	0
$313$	−33.1499	−1.87374	−0.936872	−	0.349673i	$-0.886293\pi$
$313$	−33.1499	−1.87374	−0.936872	+	0.349673i	$0.886293\pi$
$314$	0	0
$315$	2.20906	0.124466
$316$	0	0
$317$	26.7524	1.50257	0.751283	−	0.659980i	$-0.229435\pi$
$317$	26.7524	1.50257	0.751283	+	0.659980i	$0.229435\pi$
$318$	0	0
$319$	−0.627171	−0.0351148
$320$	0	0
$321$	20.0015	1.11637
$322$	0	0
$323$	29.8025	1.65826
$324$	0	0
$325$	1.66174	0.0921767
$326$	0	0
$327$	−22.4567	−1.24186
$328$	0	0
$329$	8.02859	0.442631
$330$	0	0
$331$	22.7553	1.25075	0.625373	−	0.780326i	$-0.284947\pi$
$331$	22.7553	1.25075	0.625373	+	0.780326i	$0.284947\pi$
$332$	0	0
$333$	7.59029	0.415946
$334$	0	0
$335$	−10.6633	−0.582599
$336$	0	0
$337$	−15.5281	−0.845868	−0.422934	−	0.906161i	$-0.639000\pi$
$337$	−15.5281	−0.845868	−0.422934	+	0.906161i	$0.639000\pi$
$338$	0	0
$339$	1.57933	0.0857774
$340$	0	0
$341$	0.581886	0.0315109
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.78339	0.257529
$346$	0	0
$347$	−21.1118	−1.13334	−0.566670	−	0.823945i	$-0.691769\pi$
$347$	−21.1118	−1.13334	−0.566670	+	0.823945i	$0.691769\pi$
$348$	0	0
$349$	12.3995	0.663731	0.331865	−	0.943327i	$-0.392322\pi$
$349$	12.3995	0.663731	0.331865	+	0.943327i	$0.392322\pi$
$350$	0	0
$351$	5.63282	0.300658
$352$	0	0
$353$	−12.0300	−0.640291	−0.320145	−	0.947368i	$-0.603732\pi$
$353$	−12.0300	−0.640291	−0.320145	+	0.947368i	$0.603732\pi$
$354$	0	0
$355$	−15.4794	−0.821559
$356$	0	0
$357$	7.68119	0.406532
$358$	0	0
$359$	1.66739	0.0880012	0.0440006	−	0.999032i	$-0.485990\pi$
$359$	1.66739	0.0880012	0.0440006	+	0.999032i	$0.485990\pi$
$360$	0	0
$361$	7.96064	0.418981
$362$	0	0
$363$	1.33826	0.0702405
$364$	0	0
$365$	24.8641	1.30145
$366$	0	0
$367$	2.73303	0.142663	0.0713315	−	0.997453i	$-0.477275\pi$
$367$	2.73303	0.142663	0.0713315	+	0.997453i	$0.477275\pi$
$368$	0	0
$369$	−10.5576	−0.549609
$370$	0	0
$371$	8.69440	0.451391
$372$	0	0
$373$	−22.7125	−1.17601	−0.588005	−	0.808857i	$-0.700087\pi$
$373$	−22.7125	−1.17601	−0.588005	+	0.808857i	$0.700087\pi$
$374$	0	0
$375$	−16.2888	−0.841149
$376$	0	0
$377$	0.627171	0.0323009
$378$	0	0
$379$	29.4386	1.51216	0.756080	−	0.654479i	$-0.227112\pi$
$379$	29.4386	1.51216	0.756080	+	0.654479i	$0.227112\pi$
$380$	0	0
$381$	8.68119	0.444751
$382$	0	0
$383$	19.9742	1.02063	0.510316	−	0.859987i	$-0.329528\pi$
$383$	19.9742	1.02063	0.510316	+	0.859987i	$0.329528\pi$
$384$	0	0
$385$	−1.82709	−0.0931172
$386$	0	0
$387$	0.279773	0.0142216
$388$	0	0
$389$	−11.1113	−0.563367	−0.281683	−	0.959507i	$-0.590893\pi$
$389$	−11.1113	−0.563367	−0.281683	+	0.959507i	$0.590893\pi$
$390$	0	0
$391$	−11.2285	−0.567850
$392$	0	0
$393$	18.2901	0.922613
$394$	0	0
$395$	12.8642	0.647269
$396$	0	0
$397$	4.65503	0.233629	0.116815	−	0.993154i	$-0.462732\pi$
$397$	4.65503	0.233629	0.116815	+	0.993154i	$0.462732\pi$
$398$	0	0
$399$	6.94874	0.347872
$400$	0	0
$401$	−29.0419	−1.45028	−0.725141	−	0.688600i	$-0.758225\pi$
$401$	−29.0419	−1.45028	−0.725141	+	0.688600i	$0.758225\pi$
$402$	0	0
$403$	−0.581886	−0.0289858
$404$	0	0
$405$	−7.14577	−0.355076
$406$	0	0
$407$	−6.27786	−0.311182
$408$	0	0
$409$	−6.73559	−0.333053	−0.166527	−	0.986037i	$-0.553255\pi$
$409$	−6.73559	−0.333053	−0.166527	+	0.986037i	$0.553255\pi$
$410$	0	0
$411$	−8.29895	−0.409357
$412$	0	0
$413$	8.82051	0.434029
$414$	0	0
$415$	0.626567	0.0307570
$416$	0	0
$417$	−17.1226	−0.838498
$418$	0	0
$419$	−35.5777	−1.73809	−0.869043	−	0.494736i	$-0.835265\pi$
$419$	−35.5777	−1.73809	−0.869043	+	0.494736i	$0.835265\pi$
$420$	0	0
$421$	−4.46454	−0.217588	−0.108794	−	0.994064i	$-0.534699\pi$
$421$	−4.46454	−0.217588	−0.108794	+	0.994064i	$0.534699\pi$
$422$	0	0
$423$	9.70702	0.471972
$424$	0	0
$425$	9.53785	0.462654
$426$	0	0
$427$	−0.601340	−0.0291009
$428$	0	0
$429$	−1.33826	−0.0646119
$430$	0	0
$431$	−31.0917	−1.49763	−0.748816	−	0.662777i	$-0.769378\pi$
$431$	−31.0917	−1.49763	−0.748816	+	0.662777i	$0.769378\pi$
$432$	0	0
$433$	6.65479	0.319809	0.159904	−	0.987133i	$-0.448881\pi$
$433$	6.65479	0.319809	0.159904	+	0.987133i	$0.448881\pi$
$434$	0	0
$435$	−1.53351	−0.0735262
$436$	0	0
$437$	−10.1578	−0.485913
$438$	0	0
$439$	32.7058	1.56096	0.780480	−	0.625180i	$-0.214975\pi$
$439$	32.7058	1.56096	0.780480	+	0.625180i	$0.214975\pi$
$440$	0	0
$441$	−1.20906	−0.0575741
$442$	0	0
$443$	4.02332	0.191154	0.0955768	−	0.995422i	$-0.469530\pi$
$443$	4.02332	0.191154	0.0955768	+	0.995422i	$0.469530\pi$
$444$	0	0
$445$	−18.1094	−0.858469
$446$	0	0
$447$	−29.1618	−1.37931
$448$	0	0
$449$	−18.9958	−0.896470	−0.448235	−	0.893916i	$-0.647947\pi$
$449$	−18.9958	−0.896470	−0.448235	+	0.893916i	$0.647947\pi$
$450$	0	0
$451$	8.73212	0.411180
$452$	0	0
$453$	15.5714	0.731610
$454$	0	0
$455$	1.82709	0.0856553
$456$	0	0
$457$	−29.2589	−1.36867	−0.684336	−	0.729166i	$-0.739908\pi$
$457$	−29.2589	−1.36867	−0.684336	+	0.729166i	$0.739908\pi$
$458$	0	0
$459$	32.3306	1.50906
$460$	0	0
$461$	−38.1545	−1.77703	−0.888515	−	0.458848i	$-0.848262\pi$
$461$	−38.1545	−1.77703	−0.888515	+	0.458848i	$0.848262\pi$
$462$	0	0
$463$	−23.2935	−1.08254	−0.541270	−	0.840849i	$-0.682056\pi$
$463$	−23.2935	−1.08254	−0.541270	+	0.840849i	$0.682056\pi$
$464$	0	0
$465$	1.42278	0.0659800
$466$	0	0
$467$	−24.6805	−1.14208	−0.571038	−	0.820924i	$-0.693459\pi$
$467$	−24.6805	−1.14208	−0.571038	+	0.820924i	$0.693459\pi$
$468$	0	0
$469$	5.83623	0.269492
$470$	0	0
$471$	−13.0990	−0.603569
$472$	0	0
$473$	−0.231398	−0.0106397
$474$	0	0
$475$	8.62835	0.395896
$476$	0	0
$477$	10.5120	0.481312
$478$	0	0
$479$	−32.0042	−1.46231	−0.731155	−	0.682211i	$-0.761018\pi$
$479$	−32.0042	−1.46231	−0.731155	+	0.682211i	$0.761018\pi$
$480$	0	0
$481$	6.27786	0.286246
$482$	0	0
$483$	−2.61803	−0.119125
$484$	0	0
$485$	21.9525	0.996811
$486$	0	0
$487$	27.9776	1.26779	0.633893	−	0.773421i	$-0.281456\pi$
$487$	27.9776	1.26779	0.633893	+	0.773421i	$0.281456\pi$
$488$	0	0
$489$	17.4214	0.787823
$490$	0	0
$491$	7.85094	0.354308	0.177154	−	0.984183i	$-0.443311\pi$
$491$	7.85094	0.354308	0.177154	+	0.984183i	$0.443311\pi$
$492$	0	0
$493$	3.59976	0.162125
$494$	0	0
$495$	−2.20906	−0.0992897
$496$	0	0
$497$	8.47214	0.380027
$498$	0	0
$499$	16.0213	0.717211	0.358606	−	0.933489i	$-0.383252\pi$
$499$	16.0213	0.717211	0.358606	+	0.933489i	$0.383252\pi$
$500$	0	0
$501$	−18.7776	−0.838922
$502$	0	0
$503$	−9.77997	−0.436067	−0.218034	−	0.975941i	$-0.569964\pi$
$503$	−9.77997	−0.436067	−0.218034	+	0.975941i	$0.569964\pi$
$504$	0	0
$505$	23.5464	1.04780
$506$	0	0
$507$	1.33826	0.0594343
$508$	0	0
$509$	5.19525	0.230275	0.115138	−	0.993350i	$-0.463269\pi$
$509$	5.19525	0.230275	0.115138	+	0.993350i	$0.463269\pi$
$510$	0	0
$511$	−13.6086	−0.602008
$512$	0	0
$513$	29.2476	1.29131
$514$	0	0
$515$	−6.69322	−0.294938
$516$	0	0
$517$	−8.02859	−0.353097
$518$	0	0
$519$	3.57308	0.156841
$520$	0	0
$521$	18.0314	0.789968	0.394984	−	0.918688i	$-0.370750\pi$
$521$	18.0314	0.789968	0.394984	+	0.918688i	$0.370750\pi$
$522$	0	0
$523$	−35.1067	−1.53511	−0.767555	−	0.640983i	$-0.778527\pi$
$523$	−35.1067	−1.53511	−0.767555	+	0.640983i	$0.778527\pi$
$524$	0	0
$525$	2.22384	0.0970564
$526$	0	0
$527$	−3.33984	−0.145486
$528$	0	0
$529$	−19.1729	−0.833605
$530$	0	0
$531$	10.6645	0.462800
$532$	0	0
$533$	−8.73212	−0.378230
$534$	0	0
$535$	27.3074	1.18060
$536$	0	0
$537$	−33.1964	−1.43253
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	12.8285	0.551542	0.275771	−	0.961223i	$-0.411067\pi$
$541$	12.8285	0.551542	0.275771	+	0.961223i	$0.411067\pi$
$542$	0	0
$543$	−10.1902	−0.437305
$544$	0	0
$545$	−30.6595	−1.31331
$546$	0	0
$547$	4.64642	0.198667	0.0993333	−	0.995054i	$-0.468329\pi$
$547$	4.64642	0.198667	0.0993333	+	0.995054i	$0.468329\pi$
$548$	0	0
$549$	−0.727055	−0.0310299
$550$	0	0
$551$	3.25650	0.138731
$552$	0	0
$553$	−7.04082	−0.299406
$554$	0	0
$555$	−15.3502	−0.651578
$556$	0	0
$557$	11.1491	0.472401	0.236200	−	0.971704i	$-0.424098\pi$
$557$	11.1491	0.472401	0.236200	+	0.971704i	$0.424098\pi$
$558$	0	0
$559$	0.231398	0.00978707
$560$	0	0
$561$	−7.68119	−0.324300
$562$	0	0
$563$	−40.9548	−1.72604	−0.863019	−	0.505171i	$-0.831429\pi$
$563$	−40.9548	−1.72604	−0.863019	+	0.505171i	$0.831429\pi$
$564$	0	0
$565$	2.15622	0.0907127
$566$	0	0
$567$	3.91101	0.164247
$568$	0	0
$569$	−17.7472	−0.744003	−0.372001	−	0.928232i	$-0.621328\pi$
$569$	−17.7472	−0.744003	−0.372001	+	0.928232i	$0.621328\pi$
$570$	0	0
$571$	−43.8168	−1.83367	−0.916837	−	0.399261i	$-0.869267\pi$
$571$	−43.8168	−1.83367	−0.916837	+	0.399261i	$0.869267\pi$
$572$	0	0
$573$	1.67477	0.0699647
$574$	0	0
$575$	−3.25085	−0.135570
$576$	0	0
$577$	34.3512	1.43006	0.715030	−	0.699094i	$-0.246413\pi$
$577$	34.3512	1.43006	0.715030	+	0.699094i	$0.246413\pi$
$578$	0	0
$579$	30.5173	1.26826
$580$	0	0
$581$	−0.342932	−0.0142272
$582$	0	0
$583$	−8.69440	−0.360085
$584$	0	0
$585$	2.20906	0.0913333
$586$	0	0
$587$	25.9232	1.06996	0.534982	−	0.844863i	$-0.320318\pi$
$587$	25.9232	1.06996	0.534982	+	0.844863i	$0.320318\pi$
$588$	0	0
$589$	−3.02136	−0.124493
$590$	0	0
$591$	25.1905	1.03620
$592$	0	0
$593$	17.3859	0.713954	0.356977	−	0.934113i	$-0.383808\pi$
$593$	17.3859	0.713954	0.356977	+	0.934113i	$0.383808\pi$
$594$	0	0
$595$	10.4869	0.429922
$596$	0	0
$597$	9.78675	0.400545
$598$	0	0
$599$	22.0590	0.901305	0.450652	−	0.892699i	$-0.351191\pi$
$599$	22.0590	0.901305	0.450652	+	0.892699i	$0.351191\pi$
$600$	0	0
$601$	8.99463	0.366898	0.183449	−	0.983029i	$-0.441274\pi$
$601$	8.99463	0.366898	0.183449	+	0.983029i	$0.441274\pi$
$602$	0	0
$603$	7.05633	0.287356
$604$	0	0
$605$	1.82709	0.0742818
$606$	0	0
$607$	22.3318	0.906418	0.453209	−	0.891404i	$-0.350279\pi$
$607$	22.3318	0.906418	0.453209	+	0.891404i	$0.350279\pi$
$608$	0	0
$609$	0.839318	0.0340109
$610$	0	0
$611$	8.02859	0.324802
$612$	0	0
$613$	−1.90634	−0.0769964	−0.0384982	−	0.999259i	$-0.512257\pi$
$613$	−1.90634	−0.0769964	−0.0384982	+	0.999259i	$0.512257\pi$
$614$	0	0
$615$	21.3511	0.860961
$616$	0	0
$617$	13.2489	0.533381	0.266690	−	0.963782i	$-0.414070\pi$
$617$	13.2489	0.533381	0.266690	+	0.963782i	$0.414070\pi$
$618$	0	0
$619$	−8.40837	−0.337961	−0.168981	−	0.985619i	$-0.554048\pi$
$619$	−8.40837	−0.337961	−0.168981	+	0.985619i	$0.554048\pi$
$620$	0	0
$621$	−11.0195	−0.442195
$622$	0	0
$623$	9.91161	0.397100
$624$	0	0
$625$	−13.9299	−0.557197
$626$	0	0
$627$	−6.94874	−0.277506
$628$	0	0
$629$	36.0329	1.43673
$630$	0	0
$631$	−21.2983	−0.847872	−0.423936	−	0.905692i	$-0.639352\pi$
$631$	−21.2983	−0.847872	−0.423936	+	0.905692i	$0.639352\pi$
$632$	0	0
$633$	−10.5634	−0.419856
$634$	0	0
$635$	11.8522	0.470340
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	10.2433	0.405218
$640$	0	0
$641$	14.5790	0.575836	0.287918	−	0.957655i	$-0.407037\pi$
$641$	14.5790	0.575836	0.287918	+	0.957655i	$0.407037\pi$
$642$	0	0
$643$	−21.8684	−0.862407	−0.431203	−	0.902255i	$-0.641911\pi$
$643$	−21.8684	−0.862407	−0.431203	+	0.902255i	$0.641911\pi$
$644$	0	0
$645$	−0.565796	−0.0222782
$646$	0	0
$647$	34.9589	1.37438	0.687188	−	0.726479i	$-0.258845\pi$
$647$	34.9589	1.37438	0.687188	+	0.726479i	$0.258845\pi$
$648$	0	0
$649$	−8.82051	−0.346235
$650$	0	0
$651$	−0.778716	−0.0305203
$652$	0	0
$653$	5.06003	0.198014	0.0990071	−	0.995087i	$-0.468433\pi$
$653$	5.06003	0.198014	0.0990071	+	0.995087i	$0.468433\pi$
$654$	0	0
$655$	24.9710	0.975696
$656$	0	0
$657$	−16.4535	−0.641913
$658$	0	0
$659$	26.9805	1.05101	0.525505	−	0.850790i	$-0.323876\pi$
$659$	26.9805	1.05101	0.525505	+	0.850790i	$0.323876\pi$
$660$	0	0
$661$	47.0793	1.83117	0.915587	−	0.402121i	$-0.131727\pi$
$661$	47.0793	1.83117	0.915587	+	0.402121i	$0.131727\pi$
$662$	0	0
$663$	7.68119	0.298313
$664$	0	0
$665$	9.48692	0.367887
$666$	0	0
$667$	−1.22693	−0.0475070
$668$	0	0
$669$	7.07039	0.273357
$670$	0	0
$671$	0.601340	0.0232145
$672$	0	0
$673$	−38.0212	−1.46561	−0.732804	−	0.680440i	$-0.761789\pi$
$673$	−38.0212	−1.46561	−0.732804	+	0.680440i	$0.761789\pi$
$674$	0	0
$675$	9.36027	0.360277
$676$	0	0
$677$	9.03793	0.347356	0.173678	−	0.984802i	$-0.444435\pi$
$677$	9.03793	0.347356	0.173678	+	0.984802i	$0.444435\pi$
$678$	0	0
$679$	−12.0150	−0.461093
$680$	0	0
$681$	36.8326	1.41143
$682$	0	0
$683$	−36.0775	−1.38047	−0.690233	−	0.723587i	$-0.742492\pi$
$683$	−36.0775	−1.38047	−0.690233	+	0.723587i	$0.742492\pi$
$684$	0	0
$685$	−11.3303	−0.432910
$686$	0	0
$687$	−21.2471	−0.810627
$688$	0	0
$689$	8.69440	0.331230
$690$	0	0
$691$	24.7687	0.942246	0.471123	−	0.882067i	$-0.343849\pi$
$691$	24.7687	0.942246	0.471123	+	0.882067i	$0.343849\pi$
$692$	0	0
$693$	1.20906	0.0459283
$694$	0	0
$695$	−23.3770	−0.886742
$696$	0	0
$697$	−50.1196	−1.89842
$698$	0	0
$699$	−12.6266	−0.477582
$700$	0	0
$701$	30.1960	1.14049	0.570243	−	0.821476i	$-0.306849\pi$
$701$	30.1960	1.14049	0.570243	+	0.821476i	$0.306849\pi$
$702$	0	0
$703$	32.5969	1.22942
$704$	0	0
$705$	−19.6309	−0.739343
$706$	0	0
$707$	−12.8874	−0.484679
$708$	0	0
$709$	−21.5645	−0.809873	−0.404936	−	0.914345i	$-0.632706\pi$
$709$	−21.5645	−0.809873	−0.404936	+	0.914345i	$0.632706\pi$
$710$	0	0
$711$	−8.51275	−0.319253
$712$	0	0
$713$	1.13834	0.0426312
$714$	0	0
$715$	−1.82709	−0.0683293
$716$	0	0
$717$	30.0791	1.12333
$718$	0	0
$719$	24.2548	0.904551	0.452276	−	0.891878i	$-0.350612\pi$
$719$	24.2548	0.904551	0.452276	+	0.891878i	$0.350612\pi$
$720$	0	0
$721$	3.66332	0.136429
$722$	0	0
$723$	−39.3682	−1.46412
$724$	0	0
$725$	1.04219	0.0387061
$726$	0	0
$727$	32.6233	1.20993	0.604965	−	0.796252i	$-0.293187\pi$
$727$	32.6233	1.20993	0.604965	+	0.796252i	$0.293187\pi$
$728$	0	0
$729$	27.3432	1.01271
$730$	0	0
$731$	1.32815	0.0491233
$732$	0	0
$733$	−4.27568	−0.157926	−0.0789629	−	0.996878i	$-0.525161\pi$
$733$	−4.27568	−0.157926	−0.0789629	+	0.996878i	$0.525161\pi$
$734$	0	0
$735$	2.44512	0.0901898
$736$	0	0
$737$	−5.83623	−0.214980
$738$	0	0
$739$	−7.47063	−0.274811	−0.137406	−	0.990515i	$-0.543876\pi$
$739$	−7.47063	−0.274811	−0.137406	+	0.990515i	$0.543876\pi$
$740$	0	0
$741$	6.94874	0.255268
$742$	0	0
$743$	24.9579	0.915616	0.457808	−	0.889051i	$-0.348635\pi$
$743$	24.9579	0.915616	0.457808	+	0.889051i	$0.348635\pi$
$744$	0	0
$745$	−39.8138	−1.45866
$746$	0	0
$747$	−0.414624	−0.0151703
$748$	0	0
$749$	−14.9459	−0.546110
$750$	0	0
$751$	16.2803	0.594078	0.297039	−	0.954865i	$-0.404001\pi$
$751$	16.2803	0.594078	0.297039	+	0.954865i	$0.404001\pi$
$752$	0	0
$753$	20.8802	0.760919
$754$	0	0
$755$	21.2593	0.773704
$756$	0	0
$757$	19.8897	0.722903	0.361451	−	0.932391i	$-0.382281\pi$
$757$	19.8897	0.722903	0.361451	+	0.932391i	$0.382281\pi$
$758$	0	0
$759$	2.61803	0.0950286
$760$	0	0
$761$	−14.0194	−0.508202	−0.254101	−	0.967178i	$-0.581780\pi$
$761$	−14.0194	−0.508202	−0.254101	+	0.967178i	$0.581780\pi$
$762$	0	0
$763$	16.7805	0.607495
$764$	0	0
$765$	12.6793	0.458420
$766$	0	0
$767$	8.82051	0.318490
$768$	0	0
$769$	−0.908397	−0.0327576	−0.0163788	−	0.999866i	$-0.505214\pi$
$769$	−0.908397	−0.0327576	−0.0163788	+	0.999866i	$0.505214\pi$
$770$	0	0
$771$	−5.86079	−0.211071
$772$	0	0
$773$	35.3320	1.27080	0.635401	−	0.772182i	$-0.280835\pi$
$773$	35.3320	1.27080	0.635401	+	0.772182i	$0.280835\pi$
$774$	0	0
$775$	−0.966943	−0.0347336
$776$	0	0
$777$	8.40142	0.301399
$778$	0	0
$779$	−45.3404	−1.62449
$780$	0	0
$781$	−8.47214	−0.303157
$782$	0	0
$783$	3.53274	0.126250
$784$	0	0
$785$	−17.8837	−0.638296
$786$	0	0
$787$	20.0502	0.714711	0.357355	−	0.933968i	$-0.383678\pi$
$787$	20.0502	0.714711	0.357355	+	0.933968i	$0.383678\pi$
$788$	0	0
$789$	−10.2106	−0.363506
$790$	0	0
$791$	−1.18014	−0.0419608
$792$	0	0
$793$	−0.601340	−0.0213542
$794$	0	0
$795$	−21.2589	−0.753975
$796$	0	0
$797$	47.7951	1.69299	0.846495	−	0.532397i	$-0.178709\pi$
$797$	47.7951	1.69299	0.846495	+	0.532397i	$0.178709\pi$
$798$	0	0
$799$	46.0816	1.63025
$800$	0	0
$801$	11.9837	0.423423
$802$	0	0
$803$	13.6086	0.480236
$804$	0	0
$805$	−3.57433	−0.125979
$806$	0	0
$807$	−8.79192	−0.309490
$808$	0	0
$809$	−2.12429	−0.0746860	−0.0373430	−	0.999303i	$-0.511889\pi$
$809$	−2.12429	−0.0746860	−0.0373430	+	0.999303i	$0.511889\pi$
$810$	0	0
$811$	46.1032	1.61890	0.809452	−	0.587186i	$-0.199764\pi$
$811$	46.1032	1.61890	0.809452	+	0.587186i	$0.199764\pi$
$812$	0	0
$813$	16.1307	0.565729
$814$	0	0
$815$	23.7850	0.833151
$816$	0	0
$817$	1.20150	0.0420352
$818$	0	0
$819$	−1.20906	−0.0422479
$820$	0	0
$821$	−26.2755	−0.917021	−0.458511	−	0.888689i	$-0.651617\pi$
$821$	−26.2755	−0.917021	−0.458511	+	0.888689i	$0.651617\pi$
$822$	0	0
$823$	−13.5086	−0.470879	−0.235439	−	0.971889i	$-0.575653\pi$
$823$	−13.5086	−0.470879	−0.235439	+	0.971889i	$0.575653\pi$
$824$	0	0
$825$	−2.22384	−0.0774242
$826$	0	0
$827$	47.0202	1.63505	0.817526	−	0.575892i	$-0.195345\pi$
$827$	47.0202	1.63505	0.817526	+	0.575892i	$0.195345\pi$
$828$	0	0
$829$	−46.1976	−1.60451	−0.802254	−	0.596983i	$-0.796366\pi$
$829$	−46.1976	−1.60451	−0.802254	+	0.596983i	$0.796366\pi$
$830$	0	0
$831$	−2.70669	−0.0938941
$832$	0	0
$833$	−5.73968	−0.198868
$834$	0	0
$835$	−25.6366	−0.887190
$836$	0	0
$837$	−3.27766	−0.113292
$838$	0	0
$839$	27.2528	0.940872	0.470436	−	0.882434i	$-0.344097\pi$
$839$	27.2528	0.940872	0.470436	+	0.882434i	$0.344097\pi$
$840$	0	0
$841$	−28.6067	−0.986436
$842$	0	0
$843$	−2.70767	−0.0932571
$844$	0	0
$845$	1.82709	0.0628538
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	13.2498	0.454732
$850$	0	0
$851$	−12.2814	−0.420999
$852$	0	0
$853$	44.4026	1.52032	0.760158	−	0.649738i	$-0.225121\pi$
$853$	44.4026	1.52032	0.760158	+	0.649738i	$0.225121\pi$
$854$	0	0
$855$	11.4702	0.392273
$856$	0	0
$857$	−29.0812	−0.993394	−0.496697	−	0.867924i	$-0.665454\pi$
$857$	−29.0812	−0.993394	−0.496697	+	0.867924i	$0.665454\pi$
$858$	0	0
$859$	1.01511	0.0346352	0.0173176	−	0.999850i	$-0.494487\pi$
$859$	1.01511	0.0346352	0.0173176	+	0.999850i	$0.494487\pi$
$860$	0	0
$861$	−11.6859	−0.398253
$862$	0	0
$863$	−9.65479	−0.328653	−0.164326	−	0.986406i	$-0.552545\pi$
$863$	−9.65479	−0.328653	−0.164326	+	0.986406i	$0.552545\pi$
$864$	0	0
$865$	4.87823	0.165865
$866$	0	0
$867$	21.3372	0.724648
$868$	0	0
$869$	7.04082	0.238843
$870$	0	0
$871$	5.83623	0.197753
$872$	0	0
$873$	−14.5268	−0.491658
$874$	0	0
$875$	12.1716	0.411475
$876$	0	0
$877$	−24.6211	−0.831396	−0.415698	−	0.909503i	$-0.636463\pi$
$877$	−24.6211	−0.831396	−0.415698	+	0.909503i	$0.636463\pi$
$878$	0	0
$879$	22.0941	0.745217
$880$	0	0
$881$	39.4554	1.32929	0.664643	−	0.747161i	$-0.268584\pi$
$881$	39.4554	1.32929	0.664643	+	0.747161i	$0.268584\pi$
$882$	0	0
$883$	18.6241	0.626753	0.313376	−	0.949629i	$-0.398540\pi$
$883$	18.6241	0.626753	0.313376	+	0.949629i	$0.398540\pi$
$884$	0	0
$885$	−21.5672	−0.724975
$886$	0	0
$887$	−25.1262	−0.843655	−0.421827	−	0.906676i	$-0.638611\pi$
$887$	−25.1262	−0.843655	−0.421827	+	0.906676i	$0.638611\pi$
$888$	0	0
$889$	−6.48692	−0.217564
$890$	0	0
$891$	−3.91101	−0.131024
$892$	0	0
$893$	41.6874	1.39501
$894$	0	0
$895$	−45.3221	−1.51495
$896$	0	0
$897$	−2.61803	−0.0874136
$898$	0	0
$899$	−0.364942	−0.0121715
$900$	0	0
$901$	49.9031	1.66251
$902$	0	0
$903$	0.309670	0.0103052
$904$	0	0
$905$	−13.9125	−0.462466
$906$	0	0
$907$	−53.7812	−1.78578	−0.892888	−	0.450279i	$-0.851325\pi$
$907$	−53.7812	−1.78578	−0.892888	+	0.450279i	$0.851325\pi$
$908$	0	0
$909$	−15.5816	−0.516808
$910$	0	0
$911$	43.9248	1.45529	0.727647	−	0.685952i	$-0.240614\pi$
$911$	43.9248	1.45529	0.727647	+	0.685952i	$0.240614\pi$
$912$	0	0
$913$	0.342932	0.0113494
$914$	0	0
$915$	1.47035	0.0486083
$916$	0	0
$917$	−13.6671	−0.451326
$918$	0	0
$919$	−29.4543	−0.971606	−0.485803	−	0.874068i	$-0.661473\pi$
$919$	−29.4543	−0.971606	−0.485803	+	0.874068i	$0.661473\pi$
$920$	0	0
$921$	−36.7455	−1.21081
$922$	0	0
$923$	8.47214	0.278864
$924$	0	0
$925$	10.4322	0.343008
$926$	0	0
$927$	4.42916	0.145473
$928$	0	0
$929$	−10.6037	−0.347896	−0.173948	−	0.984755i	$-0.555652\pi$
$929$	−10.6037	−0.347896	−0.173948	+	0.984755i	$0.555652\pi$
$930$	0	0
$931$	−5.19236	−0.170173
$932$	0	0
$933$	15.6169	0.511274
$934$	0	0
$935$	−10.4869	−0.342959
$936$	0	0
$937$	−57.7592	−1.88691	−0.943455	−	0.331500i	$-0.892445\pi$
$937$	−57.7592	−1.88691	−0.943455	+	0.331500i	$0.892445\pi$
$938$	0	0
$939$	−44.3632	−1.44774
$940$	0	0
$941$	−59.4944	−1.93946	−0.969731	−	0.244174i	$-0.921483\pi$
$941$	−59.4944	−1.93946	−0.969731	+	0.244174i	$0.921483\pi$
$942$	0	0
$943$	17.0826	0.556286
$944$	0	0
$945$	10.2917	0.334788
$946$	0	0
$947$	−12.4309	−0.403951	−0.201976	−	0.979391i	$-0.564736\pi$
$947$	−12.4309	−0.403951	−0.201976	+	0.979391i	$0.564736\pi$
$948$	0	0
$949$	−13.6086	−0.441753
$950$	0	0
$951$	35.8017	1.16095
$952$	0	0
$953$	29.9466	0.970065	0.485032	−	0.874496i	$-0.338808\pi$
$953$	29.9466	0.970065	0.485032	+	0.874496i	$0.338808\pi$
$954$	0	0
$955$	2.28652	0.0739901
$956$	0	0
$957$	−0.839318	−0.0271313
$958$	0	0
$959$	6.20130	0.200250
$960$	0	0
$961$	−30.6614	−0.989078
$962$	0	0
$963$	−18.0704	−0.582310
$964$	0	0
$965$	41.6644	1.34122
$966$	0	0
$967$	−5.16600	−0.166127	−0.0830637	−	0.996544i	$-0.526470\pi$
$967$	−5.16600	−0.166127	−0.0830637	+	0.996544i	$0.526470\pi$
$968$	0	0
$969$	39.8835	1.28124
$970$	0	0
$971$	21.5721	0.692281	0.346141	−	0.938183i	$-0.387492\pi$
$971$	21.5721	0.692281	0.346141	+	0.938183i	$0.387492\pi$
$972$	0	0
$973$	12.7947	0.410179
$974$	0	0
$975$	2.22384	0.0712199
$976$	0	0
$977$	10.2790	0.328854	0.164427	−	0.986389i	$-0.447423\pi$
$977$	10.2790	0.328854	0.164427	+	0.986389i	$0.447423\pi$
$978$	0	0
$979$	−9.91161	−0.316777
$980$	0	0
$981$	20.2886	0.647764
$982$	0	0
$983$	−23.9249	−0.763085	−0.381543	−	0.924351i	$-0.624607\pi$
$983$	−23.9249	−0.763085	−0.381543	+	0.924351i	$0.624607\pi$
$984$	0	0
$985$	34.3918	1.09581
$986$	0	0
$987$	10.7444	0.341996
$988$	0	0
$989$	−0.452682	−0.0143944
$990$	0	0
$991$	−53.3220	−1.69383	−0.846915	−	0.531729i	$-0.821543\pi$
$991$	−53.3220	−1.69383	−0.846915	+	0.531729i	$0.821543\pi$
$992$	0	0
$993$	30.4526	0.966383
$994$	0	0
$995$	13.3616	0.423591
$996$	0	0
$997$	28.6889	0.908586	0.454293	−	0.890852i	$-0.349892\pi$
$997$	28.6889	0.908586	0.454293	+	0.890852i	$0.349892\pi$
$998$	0	0
$999$	35.3621	1.11881

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.e.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.e.1.3	✓	4	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q+1.33826 q^{3} +1.82709 q^{5} -1.00000 q^{7} -1.20906 q^{9} +O(q^{10})\)	\(q+1.33826 q^{3} +1.82709 q^{5} -1.00000 q^{7} -1.20906 q^{9} +1.00000 q^{11} -1.00000 q^{13} +2.44512 q^{15} -5.73968 q^{17} -5.19236 q^{19} -1.33826 q^{21} +1.95630 q^{23} -1.66174 q^{25} -5.63282 q^{27} -0.627171 q^{29} +0.581886 q^{31} +1.33826 q^{33} -1.82709 q^{35} -6.27786 q^{37} -1.33826 q^{39} +8.73212 q^{41} -0.231398 q^{43} -2.20906 q^{45} -8.02859 q^{47} +1.00000 q^{49} -7.68119 q^{51} -8.69440 q^{53} +1.82709 q^{55} -6.94874 q^{57} -8.82051 q^{59} +0.601340 q^{61} +1.20906 q^{63} -1.82709 q^{65} -5.83623 q^{67} +2.61803 q^{69} -8.47214 q^{71} +13.6086 q^{73} -2.22384 q^{75} -1.00000 q^{77} +7.04082 q^{79} -3.91101 q^{81} +0.342932 q^{83} -10.4869 q^{85} -0.839318 q^{87} -9.91161 q^{89} +1.00000 q^{91} +0.778716 q^{93} -9.48692 q^{95} +12.0150 q^{97} -1.20906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9}+O(q^{10}) \) 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9	\( 4 q + q^{3} + q^{5} - 4 q^{7} - 3 q^{9} + 4 q^{11} - 4 q^{13} - q^{15} + q^{17} - 3 q^{19} - q^{21} - q^{23} - 11 q^{25} - 5 q^{27} + 3 q^{29} + 6 q^{31} + q^{33} - q^{35} + 4 q^{37} - q^{39} - 6 q^{41} - 3 q^{43} - 7 q^{45} - 7 q^{47} + 4 q^{49} - 11 q^{51} - 20 q^{53} + q^{55} - 2 q^{57} - 11 q^{59} - 18 q^{61} + 3 q^{63} - q^{65} - 16 q^{67} + 6 q^{69} - 16 q^{71} + 4 q^{73} + 6 q^{75} - 4 q^{77} + 9 q^{79} - 16 q^{81} - 14 q^{83} - 11 q^{85} - 3 q^{87} - 23 q^{89} + 4 q^{91} - q^{93} - 7 q^{95} + 14 q^{97} - 3 q^{99}+O(q^{100}) \) 4 * q + q^3 + q^5 - 4 * q^7 - 3 * q^9 + 4 * q^11 - 4 * q^13 - q^15 + q^17 - 3 * q^19 - q^21 - q^23 - 11 * q^25 - 5 * q^27 + 3 * q^29 + 6 * q^31 + q^33 - q^35 + 4 * q^37 - q^39 - 6 * q^41 - 3 * q^43 - 7 * q^45 - 7 * q^47 + 4 * q^49 - 11 * q^51 - 20 * q^53 + q^55 - 2 * q^57 - 11 * q^59 - 18 * q^61 + 3 * q^63 - q^65 - 16 * q^67 + 6 * q^69 - 16 * q^71 + 4 * q^73 + 6 * q^75 - 4 * q^77 + 9 * q^79 - 16 * q^81 - 14 * q^83 - 11 * q^85 - 3 * q^87 - 23 * q^89 + 4 * q^91 - q^93 - 7 * q^95 + 14 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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