LMFDB - Embedded newform 4004.2.a.g.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:	$6$
Coefficient field:	6.6.246302029.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 9x^{4} + 14x^{3} + 15x^{2} - 13x - 10 $ x^6 - 2x^5 - 9x^4 + 14x^3 + 15x^2 - 13*x - 10
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			$1.23083$ 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)

$f(q)$	$=$	$q-1.23083 q^{3} -3.39686 q^{5} +1.00000 q^{7} -1.48506 q^{9} +O(q^{10})$	$q-1.23083 q^{3} -3.39686 q^{5} +1.00000 q^{7} -1.48506 q^{9} -1.00000 q^{11} -1.00000 q^{13} +4.18095 q^{15} +3.16263 q^{17} -2.19508 q^{19} -1.23083 q^{21} +6.34698 q^{23} +6.53866 q^{25} +5.52034 q^{27} +1.18095 q^{29} -0.106521 q^{31} +1.23083 q^{33} -3.39686 q^{35} +8.25795 q^{37} +1.23083 q^{39} +2.65192 q^{41} -10.2788 q^{43} +5.04455 q^{45} -1.67533 q^{47} +1.00000 q^{49} -3.89265 q^{51} -1.94001 q^{53} +3.39686 q^{55} +2.70176 q^{57} +10.8964 q^{59} -3.90169 q^{61} -1.48506 q^{63} +3.39686 q^{65} +8.87600 q^{67} -7.81204 q^{69} -9.12203 q^{71} -10.7895 q^{73} -8.04796 q^{75} -1.00000 q^{77} +6.83428 q^{79} -2.33939 q^{81} -16.1030 q^{83} -10.7430 q^{85} -1.45354 q^{87} +18.4942 q^{89} -1.00000 q^{91} +0.131109 q^{93} +7.45637 q^{95} -8.62205 q^{97} +1.48506 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9	$ 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 20 q^{45} + 2 q^{47} + 6 q^{49} - 5 q^{51} - 3 q^{53} + 3 q^{55} - 22 q^{57} + 2 q^{59} - 26 q^{61} + 4 q^{63} + 3 q^{65} + 9 q^{67} - 11 q^{69} + 3 q^{71} - 7 q^{73} - 6 q^{75} - 6 q^{77} + 6 q^{81} - 15 q^{83} + q^{85} - 23 q^{87} - q^{89} - 6 q^{91} + 8 q^{93} + 12 q^{95} - 16 q^{97} - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * 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
$3$	−1.23083	−0.710618	−0.355309	−	0.934749i	$-0.615624\pi$
$3$	−1.23083	−0.710618	−0.355309	+	0.934749i	$0.615624\pi$
$4$	0	0
$5$	−3.39686	−1.51912	−0.759561	−	0.650436i	$-0.774586\pi$
$5$	−3.39686	−1.51912	−0.759561	+	0.650436i	$0.774586\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.48506	−0.495021
$10$	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
$15$	4.18095	1.07952
$16$	0	0
$17$	3.16263	0.767051	0.383525	−	0.923530i	$-0.374710\pi$
$17$	3.16263	0.767051	0.383525	+	0.923530i	$0.374710\pi$
$18$	0	0
$19$	−2.19508	−0.503585	−0.251793	−	0.967781i	$-0.581020\pi$
$19$	−2.19508	−0.503585	−0.251793	+	0.967781i	$0.581020\pi$
$20$	0	0
$21$	−1.23083	−0.268589
$22$	0	0
$23$	6.34698	1.32344	0.661718	−	0.749752i	$-0.269827\pi$
$23$	6.34698	1.32344	0.661718	+	0.749752i	$0.269827\pi$
$24$	0	0
$25$	6.53866	1.30773
$26$	0	0
$27$	5.52034	1.06239
$28$	0	0
$29$	1.18095	0.219296	0.109648	−	0.993970i	$-0.465028\pi$
$29$	1.18095	0.219296	0.109648	+	0.993970i	$0.465028\pi$
$30$	0	0
$31$	−0.106521	−0.0191318	−0.00956589	−	0.999954i	$-0.503045\pi$
$31$	−0.106521	−0.0191318	−0.00956589	+	0.999954i	$0.503045\pi$
$32$	0	0
$33$	1.23083	0.214260
$34$	0	0
$35$	−3.39686	−0.574174
$36$	0	0
$37$	8.25795	1.35760	0.678799	−	0.734324i	$-0.262501\pi$
$37$	8.25795	1.35760	0.678799	+	0.734324i	$0.262501\pi$
$38$	0	0
$39$	1.23083	0.197090
$40$	0	0
$41$	2.65192	0.414161	0.207080	−	0.978324i	$-0.433604\pi$
$41$	2.65192	0.414161	0.207080	+	0.978324i	$0.433604\pi$
$42$	0	0
$43$	−10.2788	−1.56750	−0.783750	−	0.621077i	$-0.786696\pi$
$43$	−10.2788	−1.56750	−0.783750	+	0.621077i	$0.786696\pi$
$44$	0	0
$45$	5.04455	0.751998
$46$	0	0
$47$	−1.67533	−0.244372	−0.122186	−	0.992507i	$-0.538991\pi$
$47$	−1.67533	−0.244372	−0.122186	+	0.992507i	$0.538991\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.89265	−0.545080
$52$	0	0
$53$	−1.94001	−0.266481	−0.133241	−	0.991084i	$-0.542538\pi$
$53$	−1.94001	−0.266481	−0.133241	+	0.991084i	$0.542538\pi$
$54$	0	0
$55$	3.39686	0.458032
$56$	0	0
$57$	2.70176	0.357857
$58$	0	0
$59$	10.8964	1.41859	0.709293	−	0.704914i	$-0.249014\pi$
$59$	10.8964	1.41859	0.709293	+	0.704914i	$0.249014\pi$
$60$	0	0
$61$	−3.90169	−0.499560	−0.249780	−	0.968303i	$-0.580358\pi$
$61$	−3.90169	−0.499560	−0.249780	+	0.968303i	$0.580358\pi$
$62$	0	0
$63$	−1.48506	−0.187101
$64$	0	0
$65$	3.39686	0.421329
$66$	0	0
$67$	8.87600	1.08438	0.542188	−	0.840257i	$-0.317596\pi$
$67$	8.87600	1.08438	0.542188	+	0.840257i	$0.317596\pi$
$68$	0	0
$69$	−7.81204	−0.940459
$70$	0	0
$71$	−9.12203	−1.08259	−0.541293	−	0.840834i	$-0.682065\pi$
$71$	−9.12203	−1.08259	−0.541293	+	0.840834i	$0.682065\pi$
$72$	0	0
$73$	−10.7895	−1.26281	−0.631407	−	0.775451i	$-0.717522\pi$
$73$	−10.7895	−1.26281	−0.631407	+	0.775451i	$0.717522\pi$
$74$	0	0
$75$	−8.04796	−0.929298
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	6.83428	0.768916	0.384458	−	0.923142i	$-0.374388\pi$
$79$	6.83428	0.768916	0.384458	+	0.923142i	$0.374388\pi$
$80$	0	0
$81$	−2.33939	−0.259932
$82$	0	0
$83$	−16.1030	−1.76753	−0.883765	−	0.467932i	$-0.844999\pi$
$83$	−16.1030	−1.76753	−0.883765	+	0.467932i	$0.844999\pi$
$84$	0	0
$85$	−10.7430	−1.16524
$86$	0	0
$87$	−1.45354	−0.155836
$88$	0	0
$89$	18.4942	1.96038	0.980190	−	0.198062i	$-0.0634646\pi$
$89$	18.4942	1.96038	0.980190	+	0.198062i	$0.0634646\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0.131109	0.0135954
$94$	0	0
$95$	7.45637	0.765008
$96$	0	0
$97$	−8.62205	−0.875437	−0.437718	−	0.899112i	$-0.644213\pi$
$97$	−8.62205	−0.875437	−0.437718	+	0.899112i	$0.644213\pi$
$98$	0	0
$99$	1.48506	0.149255
$100$	0	0
$101$	−7.45344	−0.741645	−0.370823	−	0.928704i	$-0.620924\pi$
$101$	−7.45344	−0.741645	−0.370823	+	0.928704i	$0.620924\pi$
$102$	0	0
$103$	−14.5441	−1.43307	−0.716534	−	0.697552i	$-0.754273\pi$
$103$	−14.5441	−1.43307	−0.716534	+	0.697552i	$0.754273\pi$
$104$	0	0
$105$	4.18095	0.408019
$106$	0	0
$107$	−1.80526	−0.174521	−0.0872607	−	0.996186i	$-0.527811\pi$
$107$	−1.80526	−0.174521	−0.0872607	+	0.996186i	$0.527811\pi$
$108$	0	0
$109$	4.58771	0.439423	0.219711	−	0.975565i	$-0.429488\pi$
$109$	4.58771	0.439423	0.219711	+	0.975565i	$0.429488\pi$
$110$	0	0
$111$	−10.1641	−0.964735
$112$	0	0
$113$	1.78702	0.168108	0.0840541	−	0.996461i	$-0.473213\pi$
$113$	1.78702	0.168108	0.0840541	+	0.996461i	$0.473213\pi$
$114$	0	0
$115$	−21.5598	−2.01046
$116$	0	0
$117$	1.48506	0.137294
$118$	0	0
$119$	3.16263	0.289918
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.26406	−0.294310
$124$	0	0
$125$	−5.22660	−0.467481
$126$	0	0
$127$	−6.48925	−0.575828	−0.287914	−	0.957656i	$-0.592962\pi$
$127$	−6.48925	−0.575828	−0.287914	+	0.957656i	$0.592962\pi$
$128$	0	0
$129$	12.6514	1.11389
$130$	0	0
$131$	−4.39721	−0.384186	−0.192093	−	0.981377i	$-0.561528\pi$
$131$	−4.39721	−0.384186	−0.192093	+	0.981377i	$0.561528\pi$
$132$	0	0
$133$	−2.19508	−0.190337
$134$	0	0
$135$	−18.7518	−1.61390
$136$	0	0
$137$	15.8890	1.35749	0.678746	−	0.734373i	$-0.262524\pi$
$137$	15.8890	1.35749	0.678746	+	0.734373i	$0.262524\pi$
$138$	0	0
$139$	−5.11950	−0.434230	−0.217115	−	0.976146i	$-0.569665\pi$
$139$	−5.11950	−0.434230	−0.217115	+	0.976146i	$0.569665\pi$
$140$	0	0
$141$	2.06205	0.173656
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−4.01151	−0.333138
$146$	0	0
$147$	−1.23083	−0.101517
$148$	0	0
$149$	−11.4932	−0.941563	−0.470782	−	0.882250i	$-0.656028\pi$
$149$	−11.4932	−0.941563	−0.470782	+	0.882250i	$0.656028\pi$
$150$	0	0
$151$	18.4883	1.50456	0.752278	−	0.658845i	$-0.228955\pi$
$151$	18.4883	1.50456	0.752278	+	0.658845i	$0.228955\pi$
$152$	0	0
$153$	−4.69671	−0.379706
$154$	0	0
$155$	0.361838	0.0290635
$156$	0	0
$157$	−6.52315	−0.520604	−0.260302	−	0.965527i	$-0.583822\pi$
$157$	−6.52315	−0.520604	−0.260302	+	0.965527i	$0.583822\pi$
$158$	0	0
$159$	2.38782	0.189367
$160$	0	0
$161$	6.34698	0.500212
$162$	0	0
$163$	23.5579	1.84520	0.922598	−	0.385763i	$-0.126062\pi$
$163$	23.5579	1.84520	0.922598	+	0.385763i	$0.126062\pi$
$164$	0	0
$165$	−4.18095	−0.325486
$166$	0	0
$167$	−3.27032	−0.253065	−0.126532	−	0.991962i	$-0.540385\pi$
$167$	−3.27032	−0.253065	−0.126532	+	0.991962i	$0.540385\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.25983	0.249286
$172$	0	0
$173$	−10.2100	−0.776249	−0.388125	−	0.921607i	$-0.626877\pi$
$173$	−10.2100	−0.776249	−0.388125	+	0.921607i	$0.626877\pi$
$174$	0	0
$175$	6.53866	0.494276
$176$	0	0
$177$	−13.4115	−1.00807
$178$	0	0
$179$	−8.72388	−0.652053	−0.326027	−	0.945361i	$-0.605710\pi$
$179$	−8.72388	−0.652053	−0.326027	+	0.945361i	$0.605710\pi$
$180$	0	0
$181$	−9.72823	−0.723094	−0.361547	−	0.932354i	$-0.617751\pi$
$181$	−9.72823	−0.723094	−0.361547	+	0.932354i	$0.617751\pi$
$182$	0	0
$183$	4.80231	0.354997
$184$	0	0
$185$	−28.0511	−2.06236
$186$	0	0
$187$	−3.16263	−0.231274
$188$	0	0
$189$	5.52034	0.401546
$190$	0	0
$191$	10.2136	0.739029	0.369514	−	0.929225i	$-0.379524\pi$
$191$	10.2136	0.739029	0.369514	+	0.929225i	$0.379524\pi$
$192$	0	0
$193$	−11.3705	−0.818463	−0.409232	−	0.912430i	$-0.634203\pi$
$193$	−11.3705	−0.818463	−0.409232	+	0.912430i	$0.634203\pi$
$194$	0	0
$195$	−4.18095	−0.299404
$196$	0	0
$197$	−24.7295	−1.76191	−0.880954	−	0.473203i	$-0.843098\pi$
$197$	−24.7295	−1.76191	−0.880954	+	0.473203i	$0.843098\pi$
$198$	0	0
$199$	−20.4528	−1.44986	−0.724930	−	0.688822i	$-0.758128\pi$
$199$	−20.4528	−1.44986	−0.724930	+	0.688822i	$0.758128\pi$
$200$	0	0
$201$	−10.9248	−0.770578
$202$	0	0
$203$	1.18095	0.0828863
$204$	0	0
$205$	−9.00821	−0.629161
$206$	0	0
$207$	−9.42567	−0.655130
$208$	0	0
$209$	2.19508	0.151837
$210$	0	0
$211$	−11.7468	−0.808685	−0.404343	−	0.914608i	$-0.632500\pi$
$211$	−11.7468	−0.808685	−0.404343	+	0.914608i	$0.632500\pi$
$212$	0	0
$213$	11.2276	0.769306
$214$	0	0
$215$	34.9156	2.38122
$216$	0	0
$217$	−0.106521	−0.00723113
$218$	0	0
$219$	13.2800	0.897379
$220$	0	0
$221$	−3.16263	−0.212742
$222$	0	0
$223$	−7.01933	−0.470049	−0.235024	−	0.971989i	$-0.575517\pi$
$223$	−7.01933	−0.470049	−0.235024	+	0.971989i	$0.575517\pi$
$224$	0	0
$225$	−9.71032	−0.647355
$226$	0	0
$227$	−8.92569	−0.592419	−0.296210	−	0.955123i	$-0.595723\pi$
$227$	−8.92569	−0.592419	−0.296210	+	0.955123i	$0.595723\pi$
$228$	0	0
$229$	−26.3135	−1.73884	−0.869422	−	0.494070i	$-0.835509\pi$
$229$	−26.3135	−1.73884	−0.869422	+	0.494070i	$0.835509\pi$
$230$	0	0
$231$	1.23083	0.0809825
$232$	0	0
$233$	−12.2653	−0.803525	−0.401762	−	0.915744i	$-0.631602\pi$
$233$	−12.2653	−0.803525	−0.401762	+	0.915744i	$0.631602\pi$
$234$	0	0
$235$	5.69087	0.371231
$236$	0	0
$237$	−8.41182	−0.546406
$238$	0	0
$239$	14.8067	0.957767	0.478883	−	0.877879i	$-0.341042\pi$
$239$	14.8067	0.957767	0.478883	+	0.877879i	$0.341042\pi$
$240$	0	0
$241$	−0.291152	−0.0187547	−0.00937736	−	0.999956i	$-0.502985\pi$
$241$	−0.291152	−0.0187547	−0.00937736	+	0.999956i	$0.502985\pi$
$242$	0	0
$243$	−13.6816	−0.877677
$244$	0	0
$245$	−3.39686	−0.217017
$246$	0	0
$247$	2.19508	0.139669
$248$	0	0
$249$	19.8200	1.25604
$250$	0	0
$251$	−20.1644	−1.27276	−0.636382	−	0.771374i	$-0.719570\pi$
$251$	−20.1644	−1.27276	−0.636382	+	0.771374i	$0.719570\pi$
$252$	0	0
$253$	−6.34698	−0.399031
$254$	0	0
$255$	13.2228	0.828043
$256$	0	0
$257$	3.50338	0.218535	0.109267	−	0.994012i	$-0.465150\pi$
$257$	3.50338	0.218535	0.109267	+	0.994012i	$0.465150\pi$
$258$	0	0
$259$	8.25795	0.513124
$260$	0	0
$261$	−1.75378	−0.108556
$262$	0	0
$263$	11.0947	0.684131	0.342066	−	0.939676i	$-0.388873\pi$
$263$	11.0947	0.684131	0.342066	+	0.939676i	$0.388873\pi$
$264$	0	0
$265$	6.58996	0.404818
$266$	0	0
$267$	−22.7631	−1.39308
$268$	0	0
$269$	5.39862	0.329160	0.164580	−	0.986364i	$-0.447373\pi$
$269$	5.39862	0.329160	0.164580	+	0.986364i	$0.447373\pi$
$270$	0	0
$271$	10.0914	0.613011	0.306506	−	0.951869i	$-0.400840\pi$
$271$	10.0914	0.613011	0.306506	+	0.951869i	$0.400840\pi$
$272$	0	0
$273$	1.23083	0.0744931
$274$	0	0
$275$	−6.53866	−0.394296
$276$	0	0
$277$	−15.4817	−0.930207	−0.465104	−	0.885256i	$-0.653983\pi$
$277$	−15.4817	−0.930207	−0.465104	+	0.885256i	$0.653983\pi$
$278$	0	0
$279$	0.158191	0.00947064
$280$	0	0
$281$	7.35462	0.438740	0.219370	−	0.975642i	$-0.429600\pi$
$281$	7.35462	0.438740	0.219370	+	0.975642i	$0.429600\pi$
$282$	0	0
$283$	19.8687	1.18107	0.590536	−	0.807011i	$-0.298916\pi$
$283$	19.8687	1.18107	0.590536	+	0.807011i	$0.298916\pi$
$284$	0	0
$285$	−9.17750	−0.543628
$286$	0	0
$287$	2.65192	0.156538
$288$	0	0
$289$	−6.99777	−0.411633
$290$	0	0
$291$	10.6123	0.622101
$292$	0	0
$293$	31.6803	1.85078	0.925390	−	0.379016i	$-0.123737\pi$
$293$	31.6803	1.85078	0.925390	+	0.379016i	$0.123737\pi$
$294$	0	0
$295$	−37.0134	−2.15500
$296$	0	0
$297$	−5.52034	−0.320323
$298$	0	0
$299$	−6.34698	−0.367055
$300$	0	0
$301$	−10.2788	−0.592459
$302$	0	0
$303$	9.17390	0.527027
$304$	0	0
$305$	13.2535	0.758893
$306$	0	0
$307$	−27.8623	−1.59019	−0.795094	−	0.606487i	$-0.792578\pi$
$307$	−27.8623	−1.59019	−0.795094	+	0.606487i	$0.792578\pi$
$308$	0	0
$309$	17.9012	1.01837
$310$	0	0
$311$	23.0516	1.30714	0.653569	−	0.756867i	$-0.273271\pi$
$311$	23.0516	1.30714	0.653569	+	0.756867i	$0.273271\pi$
$312$	0	0
$313$	−19.0800	−1.07847	−0.539234	−	0.842156i	$-0.681286\pi$
$313$	−19.0800	−1.07847	−0.539234	+	0.842156i	$0.681286\pi$
$314$	0	0
$315$	5.04455	0.284228
$316$	0	0
$317$	−8.11045	−0.455528	−0.227764	−	0.973716i	$-0.573141\pi$
$317$	−8.11045	−0.455528	−0.227764	+	0.973716i	$0.573141\pi$
$318$	0	0
$319$	−1.18095	−0.0661204
$320$	0	0
$321$	2.22197	0.124018
$322$	0	0
$323$	−6.94222	−0.386275
$324$	0	0
$325$	−6.53866	−0.362699
$326$	0	0
$327$	−5.64668	−0.312262
$328$	0	0
$329$	−1.67533	−0.0923641
$330$	0	0
$331$	−13.0032	−0.714719	−0.357360	−	0.933967i	$-0.616323\pi$
$331$	−13.0032	−0.714719	−0.357360	+	0.933967i	$0.616323\pi$
$332$	0	0
$333$	−12.2636	−0.672040
$334$	0	0
$335$	−30.1505	−1.64730
$336$	0	0
$337$	−15.9549	−0.869118	−0.434559	−	0.900643i	$-0.643096\pi$
$337$	−15.9549	−0.869118	−0.434559	+	0.900643i	$0.643096\pi$
$338$	0	0
$339$	−2.19951	−0.119461
$340$	0	0
$341$	0.106521	0.00576845
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	26.5364	1.42867
$346$	0	0
$347$	−8.26941	−0.443925	−0.221962	−	0.975055i	$-0.571246\pi$
$347$	−8.26941	−0.443925	−0.221962	+	0.975055i	$0.571246\pi$
$348$	0	0
$349$	−23.7238	−1.26990	−0.634952	−	0.772551i	$-0.718980\pi$
$349$	−23.7238	−1.26990	−0.634952	+	0.772551i	$0.718980\pi$
$350$	0	0
$351$	−5.52034	−0.294654
$352$	0	0
$353$	5.53541	0.294620	0.147310	−	0.989090i	$-0.452938\pi$
$353$	5.53541	0.294620	0.147310	+	0.989090i	$0.452938\pi$
$354$	0	0
$355$	30.9863	1.64458
$356$	0	0
$357$	−3.89265	−0.206021
$358$	0	0
$359$	21.4074	1.12984	0.564920	−	0.825146i	$-0.308907\pi$
$359$	21.4074	1.12984	0.564920	+	0.825146i	$0.308907\pi$
$360$	0	0
$361$	−14.1816	−0.746402
$362$	0	0
$363$	−1.23083	−0.0646017
$364$	0	0
$365$	36.6504	1.91837
$366$	0	0
$367$	10.1018	0.527310	0.263655	−	0.964617i	$-0.415072\pi$
$367$	10.1018	0.527310	0.263655	+	0.964617i	$0.415072\pi$
$368$	0	0
$369$	−3.93828	−0.205018
$370$	0	0
$371$	−1.94001	−0.100721
$372$	0	0
$373$	−32.7466	−1.69556	−0.847778	−	0.530351i	$-0.822060\pi$
$373$	−32.7466	−1.69556	−0.847778	+	0.530351i	$0.822060\pi$
$374$	0	0
$375$	6.43304	0.332201
$376$	0	0
$377$	−1.18095	−0.0608219
$378$	0	0
$379$	12.0694	0.619963	0.309981	−	0.950743i	$-0.399677\pi$
$379$	12.0694	0.619963	0.309981	+	0.950743i	$0.399677\pi$
$380$	0	0
$381$	7.98715	0.409194
$382$	0	0
$383$	25.7858	1.31759	0.658797	−	0.752321i	$-0.271066\pi$
$383$	25.7858	1.31759	0.658797	+	0.752321i	$0.271066\pi$
$384$	0	0
$385$	3.39686	0.173120
$386$	0	0
$387$	15.2647	0.775946
$388$	0	0
$389$	24.3502	1.23460	0.617301	−	0.786727i	$-0.288226\pi$
$389$	24.3502	1.23460	0.617301	+	0.786727i	$0.288226\pi$
$390$	0	0
$391$	20.0732	1.01514
$392$	0	0
$393$	5.41221	0.273010
$394$	0	0
$395$	−23.2151	−1.16808
$396$	0	0
$397$	−2.07699	−0.104241	−0.0521206	−	0.998641i	$-0.516598\pi$
$397$	−2.07699	−0.104241	−0.0521206	+	0.998641i	$0.516598\pi$
$398$	0	0
$399$	2.70176	0.135257
$400$	0	0
$401$	2.13586	0.106660	0.0533300	−	0.998577i	$-0.483016\pi$
$401$	2.13586	0.106660	0.0533300	+	0.998577i	$0.483016\pi$
$402$	0	0
$403$	0.106521	0.00530620
$404$	0	0
$405$	7.94659	0.394869
$406$	0	0
$407$	−8.25795	−0.409331
$408$	0	0
$409$	−29.5348	−1.46040	−0.730202	−	0.683231i	$-0.760574\pi$
$409$	−29.5348	−1.46040	−0.730202	+	0.683231i	$0.760574\pi$
$410$	0	0
$411$	−19.5567	−0.964659
$412$	0	0
$413$	10.8964	0.536175
$414$	0	0
$415$	54.6995	2.68509
$416$	0	0
$417$	6.30122	0.308572
$418$	0	0
$419$	−29.7109	−1.45147	−0.725737	−	0.687972i	$-0.758501\pi$
$419$	−29.7109	−1.45147	−0.725737	+	0.687972i	$0.758501\pi$
$420$	0	0
$421$	−1.90761	−0.0929712	−0.0464856	−	0.998919i	$-0.514802\pi$
$421$	−1.90761	−0.0929712	−0.0464856	+	0.998919i	$0.514802\pi$
$422$	0	0
$423$	2.48798	0.120970
$424$	0	0
$425$	20.6794	1.00310
$426$	0	0
$427$	−3.90169	−0.188816
$428$	0	0
$429$	−1.23083	−0.0594249
$430$	0	0
$431$	−22.2763	−1.07301	−0.536506	−	0.843896i	$-0.680256\pi$
$431$	−22.2763	−1.07301	−0.536506	+	0.843896i	$0.680256\pi$
$432$	0	0
$433$	−3.57208	−0.171663	−0.0858315	−	0.996310i	$-0.527355\pi$
$433$	−3.57208	−0.171663	−0.0858315	+	0.996310i	$0.527355\pi$
$434$	0	0
$435$	4.93748	0.236734
$436$	0	0
$437$	−13.9321	−0.666463
$438$	0	0
$439$	−2.06151	−0.0983907	−0.0491953	−	0.998789i	$-0.515666\pi$
$439$	−2.06151	−0.0983907	−0.0491953	+	0.998789i	$0.515666\pi$
$440$	0	0
$441$	−1.48506	−0.0707173
$442$	0	0
$443$	0.867037	0.0411942	0.0205971	−	0.999788i	$-0.493443\pi$
$443$	0.867037	0.0411942	0.0205971	+	0.999788i	$0.493443\pi$
$444$	0	0
$445$	−62.8221	−2.97805
$446$	0	0
$447$	14.1462	0.669092
$448$	0	0
$449$	8.70183	0.410665	0.205332	−	0.978692i	$-0.434172\pi$
$449$	8.70183	0.410665	0.205332	+	0.978692i	$0.434172\pi$
$450$	0	0
$451$	−2.65192	−0.124874
$452$	0	0
$453$	−22.7559	−1.06917
$454$	0	0
$455$	3.39686	0.159247
$456$	0	0
$457$	28.6837	1.34177	0.670884	−	0.741562i	$-0.265915\pi$
$457$	28.6837	1.34177	0.670884	+	0.741562i	$0.265915\pi$
$458$	0	0
$459$	17.4588	0.814907
$460$	0	0
$461$	−3.27569	−0.152564	−0.0762821	−	0.997086i	$-0.524305\pi$
$461$	−3.27569	−0.152564	−0.0762821	+	0.997086i	$0.524305\pi$
$462$	0	0
$463$	7.36592	0.342323	0.171162	−	0.985243i	$-0.445248\pi$
$463$	7.36592	0.342323	0.171162	+	0.985243i	$0.445248\pi$
$464$	0	0
$465$	−0.445360	−0.0206531
$466$	0	0
$467$	27.9263	1.29228	0.646138	−	0.763220i	$-0.276383\pi$
$467$	27.9263	1.29228	0.646138	+	0.763220i	$0.276383\pi$
$468$	0	0
$469$	8.87600	0.409856
$470$	0	0
$471$	8.02887	0.369951
$472$	0	0
$473$	10.2788	0.472619
$474$	0	0
$475$	−14.3529	−0.658554
$476$	0	0
$477$	2.88105	0.131914
$478$	0	0
$479$	15.1212	0.690905	0.345452	−	0.938436i	$-0.387725\pi$
$479$	15.1212	0.690905	0.345452	+	0.938436i	$0.387725\pi$
$480$	0	0
$481$	−8.25795	−0.376530
$482$	0	0
$483$	−7.81204	−0.355460
$484$	0	0
$485$	29.2879	1.32990
$486$	0	0
$487$	−42.4809	−1.92499	−0.962496	−	0.271294i	$-0.912548\pi$
$487$	−42.4809	−1.92499	−0.962496	+	0.271294i	$0.912548\pi$
$488$	0	0
$489$	−28.9957	−1.31123
$490$	0	0
$491$	−1.38616	−0.0625565	−0.0312782	−	0.999511i	$-0.509958\pi$
$491$	−1.38616	−0.0625565	−0.0312782	+	0.999511i	$0.509958\pi$
$492$	0	0
$493$	3.73490	0.168212
$494$	0	0
$495$	−5.04455	−0.226736
$496$	0	0
$497$	−9.12203	−0.409179
$498$	0	0
$499$	−4.29544	−0.192290	−0.0961451	−	0.995367i	$-0.530651\pi$
$499$	−4.29544	−0.192290	−0.0961451	+	0.995367i	$0.530651\pi$
$500$	0	0
$501$	4.02520	0.179832
$502$	0	0
$503$	15.2735	0.681014	0.340507	−	0.940242i	$-0.389401\pi$
$503$	15.2735	0.681014	0.340507	+	0.940242i	$0.389401\pi$
$504$	0	0
$505$	25.3183	1.12665
$506$	0	0
$507$	−1.23083	−0.0546630
$508$	0	0
$509$	−27.0065	−1.19704	−0.598521	−	0.801107i	$-0.704245\pi$
$509$	−27.0065	−1.19704	−0.598521	+	0.801107i	$0.704245\pi$
$510$	0	0
$511$	−10.7895	−0.477299
$512$	0	0
$513$	−12.1176	−0.535004
$514$	0	0
$515$	49.4041	2.17701
$516$	0	0
$517$	1.67533	0.0736810
$518$	0	0
$519$	12.5667	0.551617
$520$	0	0
$521$	−23.1359	−1.01360	−0.506801	−	0.862063i	$-0.669172\pi$
$521$	−23.1359	−1.01360	−0.506801	+	0.862063i	$0.669172\pi$
$522$	0	0
$523$	25.3734	1.10950	0.554751	−	0.832016i	$-0.312813\pi$
$523$	25.3734	1.10950	0.554751	+	0.832016i	$0.312813\pi$
$524$	0	0
$525$	−8.04796	−0.351242
$526$	0	0
$527$	−0.336887	−0.0146750
$528$	0	0
$529$	17.2842	0.751485
$530$	0	0
$531$	−16.1818	−0.702230
$532$	0	0
$533$	−2.65192	−0.114868
$534$	0	0
$535$	6.13223	0.265119
$536$	0	0
$537$	10.7376	0.463361
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	27.5079	1.18266	0.591328	−	0.806431i	$-0.298604\pi$
$541$	27.5079	1.18266	0.591328	+	0.806431i	$0.298604\pi$
$542$	0	0
$543$	11.9738	0.513844
$544$	0	0
$545$	−15.5838	−0.667537
$546$	0	0
$547$	36.5131	1.56119	0.780593	−	0.625040i	$-0.214917\pi$
$547$	36.5131	1.56119	0.780593	+	0.625040i	$0.214917\pi$
$548$	0	0
$549$	5.79426	0.247293
$550$	0	0
$551$	−2.59227	−0.110434
$552$	0	0
$553$	6.83428	0.290623
$554$	0	0
$555$	34.5261	1.46555
$556$	0	0
$557$	25.7885	1.09269	0.546346	−	0.837559i	$-0.316018\pi$
$557$	25.7885	1.09269	0.546346	+	0.837559i	$0.316018\pi$
$558$	0	0
$559$	10.2788	0.434746
$560$	0	0
$561$	3.89265	0.164348
$562$	0	0
$563$	21.3642	0.900393	0.450197	−	0.892929i	$-0.351354\pi$
$563$	21.3642	0.900393	0.450197	+	0.892929i	$0.351354\pi$
$564$	0	0
$565$	−6.07024	−0.255377
$566$	0	0
$567$	−2.33939	−0.0982452
$568$	0	0
$569$	−42.3797	−1.77665	−0.888325	−	0.459216i	$-0.848131\pi$
$569$	−42.3797	−1.77665	−0.888325	+	0.459216i	$0.848131\pi$
$570$	0	0
$571$	25.4943	1.06690	0.533451	−	0.845831i	$-0.320895\pi$
$571$	25.4943	1.06690	0.533451	+	0.845831i	$0.320895\pi$
$572$	0	0
$573$	−12.5712	−0.525167
$574$	0	0
$575$	41.5007	1.73070
$576$	0	0
$577$	−12.5199	−0.521211	−0.260606	−	0.965445i	$-0.583922\pi$
$577$	−12.5199	−0.521211	−0.260606	+	0.965445i	$0.583922\pi$
$578$	0	0
$579$	13.9951	0.581615
$580$	0	0
$581$	−16.1030	−0.668063
$582$	0	0
$583$	1.94001	0.0803472
$584$	0	0
$585$	−5.04455	−0.208567
$586$	0	0
$587$	40.3933	1.66721	0.833605	−	0.552361i	$-0.186273\pi$
$587$	40.3933	1.66721	0.833605	+	0.552361i	$0.186273\pi$
$588$	0	0
$589$	0.233822	0.00963448
$590$	0	0
$591$	30.4378	1.25204
$592$	0	0
$593$	−30.0975	−1.23596	−0.617979	−	0.786195i	$-0.712048\pi$
$593$	−30.0975	−1.23596	−0.617979	+	0.786195i	$0.712048\pi$
$594$	0	0
$595$	−10.7430	−0.440421
$596$	0	0
$597$	25.1739	1.03030
$598$	0	0
$599$	17.8893	0.730935	0.365468	−	0.930824i	$-0.380909\pi$
$599$	17.8893	0.730935	0.365468	+	0.930824i	$0.380909\pi$
$600$	0	0
$601$	−21.2120	−0.865256	−0.432628	−	0.901573i	$-0.642414\pi$
$601$	−21.2120	−0.865256	−0.432628	+	0.901573i	$0.642414\pi$
$602$	0	0
$603$	−13.1814	−0.536790
$604$	0	0
$605$	−3.39686	−0.138102
$606$	0	0
$607$	3.63526	0.147551	0.0737753	−	0.997275i	$-0.476495\pi$
$607$	3.63526	0.147551	0.0737753	+	0.997275i	$0.476495\pi$
$608$	0	0
$609$	−1.45354	−0.0589005
$610$	0	0
$611$	1.67533	0.0677767
$612$	0	0
$613$	17.6492	0.712845	0.356423	−	0.934325i	$-0.383996\pi$
$613$	17.6492	0.712845	0.356423	+	0.934325i	$0.383996\pi$
$614$	0	0
$615$	11.0876	0.447093
$616$	0	0
$617$	24.6499	0.992366	0.496183	−	0.868218i	$-0.334735\pi$
$617$	24.6499	0.992366	0.496183	+	0.868218i	$0.334735\pi$
$618$	0	0
$619$	−31.5610	−1.26854	−0.634271	−	0.773111i	$-0.718700\pi$
$619$	−31.5610	−1.26854	−0.634271	+	0.773111i	$0.718700\pi$
$620$	0	0
$621$	35.0375	1.40601
$622$	0	0
$623$	18.4942	0.740954
$624$	0	0
$625$	−14.9393	−0.597570
$626$	0	0
$627$	−2.70176	−0.107898
$628$	0	0
$629$	26.1168	1.04135
$630$	0	0
$631$	13.2757	0.528498	0.264249	−	0.964455i	$-0.414876\pi$
$631$	13.2757	0.528498	0.264249	+	0.964455i	$0.414876\pi$
$632$	0	0
$633$	14.4583	0.574667
$634$	0	0
$635$	22.0431	0.874753
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	13.5468	0.535903
$640$	0	0
$641$	7.44289	0.293977	0.146988	−	0.989138i	$-0.453042\pi$
$641$	7.44289	0.293977	0.146988	+	0.989138i	$0.453042\pi$
$642$	0	0
$643$	−29.8155	−1.17581	−0.587904	−	0.808931i	$-0.700047\pi$
$643$	−29.8155	−1.17581	−0.587904	+	0.808931i	$0.700047\pi$
$644$	0	0
$645$	−42.9751	−1.69214
$646$	0	0
$647$	4.91853	0.193367	0.0966836	−	0.995315i	$-0.469177\pi$
$647$	4.91853	0.193367	0.0966836	+	0.995315i	$0.469177\pi$
$648$	0	0
$649$	−10.8964	−0.427720
$650$	0	0
$651$	0.131109	0.00513858
$652$	0	0
$653$	22.6141	0.884957	0.442479	−	0.896779i	$-0.354099\pi$
$653$	22.6141	0.884957	0.442479	+	0.896779i	$0.354099\pi$
$654$	0	0
$655$	14.9367	0.583626
$656$	0	0
$657$	16.0231	0.625120
$658$	0	0
$659$	−8.34513	−0.325080	−0.162540	−	0.986702i	$-0.551969\pi$
$659$	−8.34513	−0.325080	−0.162540	+	0.986702i	$0.551969\pi$
$660$	0	0
$661$	−11.1078	−0.432043	−0.216021	−	0.976389i	$-0.569308\pi$
$661$	−11.1078	−0.432043	−0.216021	+	0.976389i	$0.569308\pi$
$662$	0	0
$663$	3.89265	0.151178
$664$	0	0
$665$	7.45637	0.289146
$666$	0	0
$667$	7.49545	0.290225
$668$	0	0
$669$	8.63958	0.334025
$670$	0	0
$671$	3.90169	0.150623
$672$	0	0
$673$	−42.8896	−1.65327	−0.826636	−	0.562738i	$-0.809748\pi$
$673$	−42.8896	−1.65327	−0.826636	+	0.562738i	$0.809748\pi$
$674$	0	0
$675$	36.0956	1.38932
$676$	0	0
$677$	11.1029	0.426720	0.213360	−	0.976974i	$-0.431559\pi$
$677$	11.1029	0.426720	0.213360	+	0.976974i	$0.431559\pi$
$678$	0	0
$679$	−8.62205	−0.330884
$680$	0	0
$681$	10.9860	0.420984
$682$	0	0
$683$	22.0974	0.845535	0.422768	−	0.906238i	$-0.361059\pi$
$683$	22.0974	0.845535	0.422768	+	0.906238i	$0.361059\pi$
$684$	0	0
$685$	−53.9728	−2.06220
$686$	0	0
$687$	32.3874	1.23566
$688$	0	0
$689$	1.94001	0.0739087
$690$	0	0
$691$	−31.5899	−1.20174	−0.600868	−	0.799348i	$-0.705178\pi$
$691$	−31.5899	−1.20174	−0.600868	+	0.799348i	$0.705178\pi$
$692$	0	0
$693$	1.48506	0.0564129
$694$	0	0
$695$	17.3902	0.659648
$696$	0	0
$697$	8.38705	0.317682
$698$	0	0
$699$	15.0964	0.571000
$700$	0	0
$701$	−21.5646	−0.814484	−0.407242	−	0.913320i	$-0.633509\pi$
$701$	−21.5646	−0.814484	−0.407242	+	0.913320i	$0.633509\pi$
$702$	0	0
$703$	−18.1268	−0.683667
$704$	0	0
$705$	−7.00448	−0.263804
$706$	0	0
$707$	−7.45344	−0.280316
$708$	0	0
$709$	−5.12248	−0.192379	−0.0961894	−	0.995363i	$-0.530665\pi$
$709$	−5.12248	−0.192379	−0.0961894	+	0.995363i	$0.530665\pi$
$710$	0	0
$711$	−10.1493	−0.380630
$712$	0	0
$713$	−0.676088	−0.0253197
$714$	0	0
$715$	−3.39686	−0.127035
$716$	0	0
$717$	−18.2245	−0.680607
$718$	0	0
$719$	22.5114	0.839534	0.419767	−	0.907632i	$-0.362112\pi$
$719$	22.5114	0.839534	0.419767	+	0.907632i	$0.362112\pi$
$720$	0	0
$721$	−14.5441	−0.541649
$722$	0	0
$723$	0.358357	0.0133275
$724$	0	0
$725$	7.72181	0.286781
$726$	0	0
$727$	−15.9162	−0.590299	−0.295150	−	0.955451i	$-0.595369\pi$
$727$	−15.9162	−0.590299	−0.295150	+	0.955451i	$0.595369\pi$
$728$	0	0
$729$	23.8579	0.883626
$730$	0	0
$731$	−32.5080	−1.20235
$732$	0	0
$733$	9.51957	0.351613	0.175807	−	0.984425i	$-0.443747\pi$
$733$	9.51957	0.351613	0.175807	+	0.984425i	$0.443747\pi$
$734$	0	0
$735$	4.18095	0.154217
$736$	0	0
$737$	−8.87600	−0.326952
$738$	0	0
$739$	−32.9373	−1.21162	−0.605808	−	0.795611i	$-0.707150\pi$
$739$	−32.9373	−1.21162	−0.605808	+	0.795611i	$0.707150\pi$
$740$	0	0
$741$	−2.70176	−0.0992517
$742$	0	0
$743$	−47.1311	−1.72907	−0.864536	−	0.502571i	$-0.832387\pi$
$743$	−47.1311	−1.72907	−0.864536	+	0.502571i	$0.832387\pi$
$744$	0	0
$745$	39.0409	1.43035
$746$	0	0
$747$	23.9139	0.874965
$748$	0	0
$749$	−1.80526	−0.0659629
$750$	0	0
$751$	−17.7852	−0.648991	−0.324496	−	0.945887i	$-0.605195\pi$
$751$	−17.7852	−0.648991	−0.324496	+	0.945887i	$0.605195\pi$
$752$	0	0
$753$	24.8188	0.904449
$754$	0	0
$755$	−62.8022	−2.28560
$756$	0	0
$757$	−10.8860	−0.395657	−0.197829	−	0.980237i	$-0.563389\pi$
$757$	−10.8860	−0.395657	−0.197829	+	0.980237i	$0.563389\pi$
$758$	0	0
$759$	7.81204	0.283559
$760$	0	0
$761$	−10.0129	−0.362968	−0.181484	−	0.983394i	$-0.558090\pi$
$761$	−10.0129	−0.362968	−0.181484	+	0.983394i	$0.558090\pi$
$762$	0	0
$763$	4.58771	0.166086
$764$	0	0
$765$	15.9541	0.576820
$766$	0	0
$767$	−10.8964	−0.393445
$768$	0	0
$769$	9.07971	0.327423	0.163711	−	0.986508i	$-0.447653\pi$
$769$	9.07971	0.327423	0.163711	+	0.986508i	$0.447653\pi$
$770$	0	0
$771$	−4.31206	−0.155295
$772$	0	0
$773$	−5.55431	−0.199775	−0.0998873	−	0.994999i	$-0.531848\pi$
$773$	−5.55431	−0.199775	−0.0998873	+	0.994999i	$0.531848\pi$
$774$	0	0
$775$	−0.696506	−0.0250192
$776$	0	0
$777$	−10.1641	−0.364635
$778$	0	0
$779$	−5.82118	−0.208565
$780$	0	0
$781$	9.12203	0.326412
$782$	0	0
$783$	6.51923	0.232978
$784$	0	0
$785$	22.1582	0.790861
$786$	0	0
$787$	29.1397	1.03872	0.519359	−	0.854556i	$-0.326171\pi$
$787$	29.1397	1.03872	0.519359	+	0.854556i	$0.326171\pi$
$788$	0	0
$789$	−13.6557	−0.486156
$790$	0	0
$791$	1.78702	0.0635389
$792$	0	0
$793$	3.90169	0.138553
$794$	0	0
$795$	−8.11110	−0.287671
$796$	0	0
$797$	3.50193	0.124045	0.0620224	−	0.998075i	$-0.480245\pi$
$797$	3.50193	0.124045	0.0620224	+	0.998075i	$0.480245\pi$
$798$	0	0
$799$	−5.29846	−0.187446
$800$	0	0
$801$	−27.4650	−0.970430
$802$	0	0
$803$	10.7895	0.380753
$804$	0	0
$805$	−21.5598	−0.759883
$806$	0	0
$807$	−6.64477	−0.233907
$808$	0	0
$809$	26.4886	0.931290	0.465645	−	0.884972i	$-0.345822\pi$
$809$	26.4886	0.931290	0.465645	+	0.884972i	$0.345822\pi$
$810$	0	0
$811$	−38.8727	−1.36500	−0.682502	−	0.730884i	$-0.739108\pi$
$811$	−38.8727	−1.36500	−0.682502	+	0.730884i	$0.739108\pi$
$812$	0	0
$813$	−12.4208	−0.435617
$814$	0	0
$815$	−80.0228	−2.80308
$816$	0	0
$817$	22.5627	0.789370
$818$	0	0
$819$	1.48506	0.0518923
$820$	0	0
$821$	−22.2718	−0.777293	−0.388646	−	0.921387i	$-0.627057\pi$
$821$	−22.2718	−0.777293	−0.388646	+	0.921387i	$0.627057\pi$
$822$	0	0
$823$	42.5999	1.48494	0.742470	−	0.669879i	$-0.233654\pi$
$823$	42.5999	1.48494	0.742470	+	0.669879i	$0.233654\pi$
$824$	0	0
$825$	8.04796	0.280194
$826$	0	0
$827$	−47.6752	−1.65783	−0.828915	−	0.559374i	$-0.811042\pi$
$827$	−47.6752	−1.65783	−0.828915	+	0.559374i	$0.811042\pi$
$828$	0	0
$829$	−39.3203	−1.36565	−0.682824	−	0.730582i	$-0.739249\pi$
$829$	−39.3203	−1.36565	−0.682824	+	0.730582i	$0.739249\pi$
$830$	0	0
$831$	19.0553	0.661022
$832$	0	0
$833$	3.16263	0.109579
$834$	0	0
$835$	11.1088	0.384436
$836$	0	0
$837$	−0.588033	−0.0203254
$838$	0	0
$839$	42.7861	1.47714	0.738570	−	0.674177i	$-0.235502\pi$
$839$	42.7861	1.47714	0.738570	+	0.674177i	$0.235502\pi$
$840$	0	0
$841$	−27.6054	−0.951909
$842$	0	0
$843$	−9.05226	−0.311777
$844$	0	0
$845$	−3.39686	−0.116856
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−24.4549	−0.839291
$850$	0	0
$851$	52.4130	1.79670
$852$	0	0
$853$	38.2924	1.31111	0.655554	−	0.755148i	$-0.272435\pi$
$853$	38.2924	1.31111	0.655554	+	0.755148i	$0.272435\pi$
$854$	0	0
$855$	−11.0732	−0.378695
$856$	0	0
$857$	−37.4605	−1.27962	−0.639812	−	0.768531i	$-0.720988\pi$
$857$	−37.4605	−1.27962	−0.639812	+	0.768531i	$0.720988\pi$
$858$	0	0
$859$	25.2995	0.863209	0.431605	−	0.902063i	$-0.357948\pi$
$859$	25.2995	0.863209	0.431605	+	0.902063i	$0.357948\pi$
$860$	0	0
$861$	−3.26406	−0.111239
$862$	0	0
$863$	−6.55106	−0.223001	−0.111500	−	0.993764i	$-0.535566\pi$
$863$	−6.55106	−0.223001	−0.111500	+	0.993764i	$0.535566\pi$
$864$	0	0
$865$	34.6818	1.17922
$866$	0	0
$867$	8.61304	0.292514
$868$	0	0
$869$	−6.83428	−0.231837
$870$	0	0
$871$	−8.87600	−0.300752
$872$	0	0
$873$	12.8043	0.433360
$874$	0	0
$875$	−5.22660	−0.176691
$876$	0	0
$877$	−5.01202	−0.169244	−0.0846219	−	0.996413i	$-0.526968\pi$
$877$	−5.01202	−0.169244	−0.0846219	+	0.996413i	$0.526968\pi$
$878$	0	0
$879$	−38.9929	−1.31520
$880$	0	0
$881$	−36.9417	−1.24460	−0.622298	−	0.782780i	$-0.713801\pi$
$881$	−36.9417	−1.24460	−0.622298	+	0.782780i	$0.713801\pi$
$882$	0	0
$883$	46.3187	1.55875	0.779374	−	0.626559i	$-0.215537\pi$
$883$	46.3187	1.55875	0.779374	+	0.626559i	$0.215537\pi$
$884$	0	0
$885$	45.5571	1.53139
$886$	0	0
$887$	2.27654	0.0764386	0.0382193	−	0.999269i	$-0.487831\pi$
$887$	2.27654	0.0764386	0.0382193	+	0.999269i	$0.487831\pi$
$888$	0	0
$889$	−6.48925	−0.217642
$890$	0	0
$891$	2.33939	0.0783726
$892$	0	0
$893$	3.67749	0.123062
$894$	0	0
$895$	29.6338	0.990549
$896$	0	0
$897$	7.81204	0.260836
$898$	0	0
$899$	−0.125796	−0.00419553
$900$	0	0
$901$	−6.13555	−0.204405
$902$	0	0
$903$	12.6514	0.421012
$904$	0	0
$905$	33.0454	1.09847
$906$	0	0
$907$	−15.8866	−0.527506	−0.263753	−	0.964590i	$-0.584960\pi$
$907$	−15.8866	−0.527506	−0.263753	+	0.964590i	$0.584960\pi$
$908$	0	0
$909$	11.0688	0.367130
$910$	0	0
$911$	2.97392	0.0985304	0.0492652	−	0.998786i	$-0.484312\pi$
$911$	2.97392	0.0985304	0.0492652	+	0.998786i	$0.484312\pi$
$912$	0	0
$913$	16.1030	0.532930
$914$	0	0
$915$	−16.3128	−0.539283
$916$	0	0
$917$	−4.39721	−0.145209
$918$	0	0
$919$	−9.78177	−0.322671	−0.161335	−	0.986900i	$-0.551580\pi$
$919$	−9.78177	−0.322671	−0.161335	+	0.986900i	$0.551580\pi$
$920$	0	0
$921$	34.2937	1.13002
$922$	0	0
$923$	9.12203	0.300255
$924$	0	0
$925$	53.9959	1.77537
$926$	0	0
$927$	21.5989	0.709400
$928$	0	0
$929$	30.5744	1.00311	0.501557	−	0.865124i	$-0.332761\pi$
$929$	30.5744	1.00311	0.501557	+	0.865124i	$0.332761\pi$
$930$	0	0
$931$	−2.19508	−0.0719408
$932$	0	0
$933$	−28.3726	−0.928877
$934$	0	0
$935$	10.7430	0.351334
$936$	0	0
$937$	−34.9339	−1.14124	−0.570620	−	0.821214i	$-0.693297\pi$
$937$	−34.9339	−1.14124	−0.570620	+	0.821214i	$0.693297\pi$
$938$	0	0
$939$	23.4842	0.766379
$940$	0	0
$941$	3.81311	0.124304	0.0621519	−	0.998067i	$-0.480204\pi$
$941$	3.81311	0.124304	0.0621519	+	0.998067i	$0.480204\pi$
$942$	0	0
$943$	16.8317	0.548116
$944$	0	0
$945$	−18.7518	−0.609997
$946$	0	0
$947$	31.4864	1.02317	0.511585	−	0.859233i	$-0.329059\pi$
$947$	31.4864	1.02317	0.511585	+	0.859233i	$0.329059\pi$
$948$	0	0
$949$	10.7895	0.350242
$950$	0	0
$951$	9.98256	0.323707
$952$	0	0
$953$	−27.6842	−0.896780	−0.448390	−	0.893838i	$-0.648002\pi$
$953$	−27.6842	−0.896780	−0.448390	+	0.893838i	$0.648002\pi$
$954$	0	0
$955$	−34.6941	−1.12267
$956$	0	0
$957$	1.45354	0.0469864
$958$	0	0
$959$	15.8890	0.513084
$960$	0	0
$961$	−30.9887	−0.999634
$962$	0	0
$963$	2.68093	0.0863918
$964$	0	0
$965$	38.6239	1.24335
$966$	0	0
$967$	−3.76838	−0.121183	−0.0605915	−	0.998163i	$-0.519299\pi$
$967$	−3.76838	−0.121183	−0.0605915	+	0.998163i	$0.519299\pi$
$968$	0	0
$969$	8.54467	0.274494
$970$	0	0
$971$	−32.2161	−1.03386	−0.516932	−	0.856027i	$-0.672926\pi$
$971$	−32.2161	−1.03386	−0.516932	+	0.856027i	$0.672926\pi$
$972$	0	0
$973$	−5.11950	−0.164124
$974$	0	0
$975$	8.04796	0.257741
$976$	0	0
$977$	−47.9400	−1.53374	−0.766869	−	0.641804i	$-0.778186\pi$
$977$	−47.9400	−1.53374	−0.766869	+	0.641804i	$0.778186\pi$
$978$	0	0
$979$	−18.4942	−0.591077
$980$	0	0
$981$	−6.81304	−0.217524
$982$	0	0
$983$	−38.9194	−1.24134	−0.620668	−	0.784074i	$-0.713138\pi$
$983$	−38.9194	−1.24134	−0.620668	+	0.784074i	$0.713138\pi$
$984$	0	0
$985$	84.0028	2.67655
$986$	0	0
$987$	2.06205	0.0656356
$988$	0	0
$989$	−65.2392	−2.07449
$990$	0	0
$991$	−25.1033	−0.797432	−0.398716	−	0.917074i	$-0.630544\pi$
$991$	−25.1033	−0.797432	−0.398716	+	0.917074i	$0.630544\pi$
$992$	0	0
$993$	16.0047	0.507893
$994$	0	0
$995$	69.4753	2.20252
$996$	0	0
$997$	22.5013	0.712622	0.356311	−	0.934367i	$-0.384034\pi$
$997$	22.5013	0.712622	0.356311	+	0.934367i	$0.384034\pi$
$998$	0	0
$999$	45.5867	1.44230

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.g.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.g.1.3	✓	6	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q-1.23083 q^{3} -3.39686 q^{5} +1.00000 q^{7} -1.48506 q^{9} +O(q^{10})\)	\(q-1.23083 q^{3} -3.39686 q^{5} +1.00000 q^{7} -1.48506 q^{9} -1.00000 q^{11} -1.00000 q^{13} +4.18095 q^{15} +3.16263 q^{17} -2.19508 q^{19} -1.23083 q^{21} +6.34698 q^{23} +6.53866 q^{25} +5.52034 q^{27} +1.18095 q^{29} -0.106521 q^{31} +1.23083 q^{33} -3.39686 q^{35} +8.25795 q^{37} +1.23083 q^{39} +2.65192 q^{41} -10.2788 q^{43} +5.04455 q^{45} -1.67533 q^{47} +1.00000 q^{49} -3.89265 q^{51} -1.94001 q^{53} +3.39686 q^{55} +2.70176 q^{57} +10.8964 q^{59} -3.90169 q^{61} -1.48506 q^{63} +3.39686 q^{65} +8.87600 q^{67} -7.81204 q^{69} -9.12203 q^{71} -10.7895 q^{73} -8.04796 q^{75} -1.00000 q^{77} +6.83428 q^{79} -2.33939 q^{81} -16.1030 q^{83} -10.7430 q^{85} -1.45354 q^{87} +18.4942 q^{89} -1.00000 q^{91} +0.131109 q^{93} +7.45637 q^{95} -8.62205 q^{97} +1.48506 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9	\( 6 q - 2 q^{3} - 3 q^{5} + 6 q^{7} + 4 q^{9} - 6 q^{11} - 6 q^{13} + 4 q^{15} - q^{17} - 12 q^{19} - 2 q^{21} + 5 q^{23} - 3 q^{25} - 8 q^{27} - 14 q^{29} - 4 q^{31} + 2 q^{33} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 6 q^{41} - 14 q^{43} - 20 q^{45} + 2 q^{47} + 6 q^{49} - 5 q^{51} - 3 q^{53} + 3 q^{55} - 22 q^{57} + 2 q^{59} - 26 q^{61} + 4 q^{63} + 3 q^{65} + 9 q^{67} - 11 q^{69} + 3 q^{71} - 7 q^{73} - 6 q^{75} - 6 q^{77} + 6 q^{81} - 15 q^{83} + q^{85} - 23 q^{87} - q^{89} - 6 q^{91} + 8 q^{93} + 12 q^{95} - 16 q^{97} - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 3 * q^5 + 6 * q^7 + 4 * q^9 - 6 * q^11 - 6 * q^13 + 4 * q^15 - q^17 - 12 * q^19 - 2 * q^21 + 5 * q^23 - 3 * q^25 - 8 * q^27 - 14 * q^29 - 4 * q^31 + 2 * q^33 - 3 * q^35 - 3 * q^37 + 2 * q^39 + 6 * q^41 - 14 * q^43 - 20 * q^45 + 2 * q^47 + 6 * q^49 - 5 * q^51 - 3 * q^53 + 3 * q^55 - 22 * q^57 + 2 * q^59 - 26 * q^61 + 4 * q^63 + 3 * q^65 + 9 * q^67 - 11 * q^69 + 3 * q^71 - 7 * q^73 - 6 * q^75 - 6 * q^77 + 6 * q^81 - 15 * q^83 + q^85 - 23 * q^87 - q^89 - 6 * q^91 + 8 * q^93 + 12 * q^95 - 16 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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