LMFDB - Embedded newform 4004.2.a.j.1.1

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} - 2x^{9} - 18x^{8} + 32x^{7} + 93x^{6} - 119x^{5} - 174x^{4} + 107x^{3} + 51x^{2} - 17x - 2 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 93x^6 - 119x^5 - 174x^4 + 107x^3 + 51x^2 - 17*x - 2
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			1.1
Root			$3.21473$ 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)

$f(q)$	$=$	$q-3.21473 q^{3} -2.32236 q^{5} -1.00000 q^{7} +7.33449 q^{9} +O(q^{10})$	$q-3.21473 q^{3} -2.32236 q^{5} -1.00000 q^{7} +7.33449 q^{9} -1.00000 q^{11} -1.00000 q^{13} +7.46575 q^{15} +7.61912 q^{17} +7.80003 q^{19} +3.21473 q^{21} -9.32617 q^{23} +0.393340 q^{25} -13.9342 q^{27} +3.30504 q^{29} -9.29674 q^{31} +3.21473 q^{33} +2.32236 q^{35} +3.92918 q^{37} +3.21473 q^{39} -10.6878 q^{41} +7.32042 q^{43} -17.0333 q^{45} -10.8791 q^{47} +1.00000 q^{49} -24.4934 q^{51} -0.263461 q^{53} +2.32236 q^{55} -25.0750 q^{57} -7.99170 q^{59} -0.0316514 q^{61} -7.33449 q^{63} +2.32236 q^{65} +9.12418 q^{67} +29.9811 q^{69} +3.34749 q^{71} -14.3526 q^{73} -1.26448 q^{75} +1.00000 q^{77} +11.5685 q^{79} +22.7913 q^{81} -6.71902 q^{83} -17.6943 q^{85} -10.6248 q^{87} +10.8509 q^{89} +1.00000 q^{91} +29.8865 q^{93} -18.1145 q^{95} -7.56532 q^{97} -7.33449 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9	$ 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} + 9 q^{17} + 3 q^{19} + 2 q^{21} - 4 q^{23} + 22 q^{25} - 8 q^{27} + 13 q^{29} - 17 q^{31} + 2 q^{33} + 2 q^{35} + 11 q^{37} + 2 q^{39} + 8 q^{41} + 21 q^{43} - 15 q^{45} + q^{47} + 10 q^{49} + 19 q^{51} + 22 q^{53} + 2 q^{55} + 8 q^{57} - 24 q^{59} + 16 q^{61} - 10 q^{63} + 2 q^{65} + 19 q^{67} + 51 q^{69} - 25 q^{71} + 22 q^{73} + 28 q^{75} + 10 q^{77} + 41 q^{79} + 54 q^{81} + 8 q^{83} + 9 q^{85} - 5 q^{87} + 36 q^{89} + 10 q^{91} + 12 q^{93} + 13 q^{95} - 5 q^{97} - 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 + 9 * q^17 + 3 * q^19 + 2 * q^21 - 4 * q^23 + 22 * q^25 - 8 * q^27 + 13 * q^29 - 17 * q^31 + 2 * q^33 + 2 * q^35 + 11 * q^37 + 2 * q^39 + 8 * q^41 + 21 * q^43 - 15 * q^45 + q^47 + 10 * q^49 + 19 * q^51 + 22 * q^53 + 2 * q^55 + 8 * q^57 - 24 * q^59 + 16 * q^61 - 10 * q^63 + 2 * q^65 + 19 * q^67 + 51 * q^69 - 25 * q^71 + 22 * q^73 + 28 * q^75 + 10 * q^77 + 41 * q^79 + 54 * q^81 + 8 * q^83 + 9 * q^85 - 5 * q^87 + 36 * q^89 + 10 * q^91 + 12 * q^93 + 13 * q^95 - 5 * q^97 - 10 * 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
$3$	−3.21473	−1.85603	−0.928013	−	0.372549i	$-0.878484\pi$
$3$	−3.21473	−1.85603	−0.928013	+	0.372549i	$0.878484\pi$
$4$	0	0
$5$	−2.32236	−1.03859	−0.519295	−	0.854595i	$-0.673805\pi$
$5$	−2.32236	−1.03859	−0.519295	+	0.854595i	$0.673805\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.33449	2.44483
$10$	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
$15$	7.46575	1.92765
$16$	0	0
$17$	7.61912	1.84791	0.923954	−	0.382503i	$-0.124938\pi$
$17$	7.61912	1.84791	0.923954	+	0.382503i	$0.124938\pi$
$18$	0	0
$19$	7.80003	1.78945	0.894725	−	0.446617i	$-0.147371\pi$
$19$	7.80003	1.78945	0.894725	+	0.446617i	$0.147371\pi$
$20$	0	0
$21$	3.21473	0.701512
$22$	0	0
$23$	−9.32617	−1.94464	−0.972320	−	0.233652i	$-0.924933\pi$
$23$	−9.32617	−1.94464	−0.972320	+	0.233652i	$0.924933\pi$
$24$	0	0
$25$	0.393340	0.0786680
$26$	0	0
$27$	−13.9342	−2.68164
$28$	0	0
$29$	3.30504	0.613730	0.306865	−	0.951753i	$-0.400720\pi$
$29$	3.30504	0.613730	0.306865	+	0.951753i	$0.400720\pi$
$30$	0	0
$31$	−9.29674	−1.66974	−0.834872	−	0.550445i	$-0.814458\pi$
$31$	−9.29674	−1.66974	−0.834872	+	0.550445i	$0.814458\pi$
$32$	0	0
$33$	3.21473	0.559613
$34$	0	0
$35$	2.32236	0.392550
$36$	0	0
$37$	3.92918	0.645953	0.322976	−	0.946407i	$-0.395317\pi$
$37$	3.92918	0.645953	0.322976	+	0.946407i	$0.395317\pi$
$38$	0	0
$39$	3.21473	0.514769
$40$	0	0
$41$	−10.6878	−1.66915	−0.834577	−	0.550891i	$-0.814288\pi$
$41$	−10.6878	−1.66915	−0.834577	+	0.550891i	$0.814288\pi$
$42$	0	0
$43$	7.32042	1.11635	0.558177	−	0.829722i	$-0.311501\pi$
$43$	7.32042	1.11635	0.558177	+	0.829722i	$0.311501\pi$
$44$	0	0
$45$	−17.0333	−2.53917
$46$	0	0
$47$	−10.8791	−1.58687	−0.793437	−	0.608652i	$-0.791710\pi$
$47$	−10.8791	−1.58687	−0.793437	+	0.608652i	$0.791710\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−24.4934	−3.42976
$52$	0	0
$53$	−0.263461	−0.0361892	−0.0180946	−	0.999836i	$-0.505760\pi$
$53$	−0.263461	−0.0361892	−0.0180946	+	0.999836i	$0.505760\pi$
$54$	0	0
$55$	2.32236	0.313147
$56$	0	0
$57$	−25.0750	−3.32127
$58$	0	0
$59$	−7.99170	−1.04043	−0.520215	−	0.854035i	$-0.674148\pi$
$59$	−7.99170	−1.04043	−0.520215	+	0.854035i	$0.674148\pi$
$60$	0	0
$61$	−0.0316514	−0.00405255	−0.00202627	−	0.999998i	$-0.500645\pi$
$61$	−0.0316514	−0.00405255	−0.00202627	+	0.999998i	$0.500645\pi$
$62$	0	0
$63$	−7.33449	−0.924059
$64$	0	0
$65$	2.32236	0.288053
$66$	0	0
$67$	9.12418	1.11470	0.557348	−	0.830279i	$-0.311819\pi$
$67$	9.12418	1.11470	0.557348	+	0.830279i	$0.311819\pi$
$68$	0	0
$69$	29.9811	3.60930
$70$	0	0
$71$	3.34749	0.397274	0.198637	−	0.980073i	$-0.436349\pi$
$71$	3.34749	0.397274	0.198637	+	0.980073i	$0.436349\pi$
$72$	0	0
$73$	−14.3526	−1.67985	−0.839923	−	0.542706i	$-0.817400\pi$
$73$	−14.3526	−1.67985	−0.839923	+	0.542706i	$0.817400\pi$
$74$	0	0
$75$	−1.26448	−0.146010
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	11.5685	1.30155	0.650777	−	0.759269i	$-0.274443\pi$
$79$	11.5685	1.30155	0.650777	+	0.759269i	$0.274443\pi$
$80$	0	0
$81$	22.7913	2.53236
$82$	0	0
$83$	−6.71902	−0.737508	−0.368754	−	0.929527i	$-0.620216\pi$
$83$	−6.71902	−0.737508	−0.368754	+	0.929527i	$0.620216\pi$
$84$	0	0
$85$	−17.6943	−1.91922
$86$	0	0
$87$	−10.6248	−1.13910
$88$	0	0
$89$	10.8509	1.15019	0.575095	−	0.818087i	$-0.304965\pi$
$89$	10.8509	1.15019	0.575095	+	0.818087i	$0.304965\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	29.8865	3.09909
$94$	0	0
$95$	−18.1145	−1.85850
$96$	0	0
$97$	−7.56532	−0.768141	−0.384071	−	0.923304i	$-0.625478\pi$
$97$	−7.56532	−0.768141	−0.384071	+	0.923304i	$0.625478\pi$
$98$	0	0
$99$	−7.33449	−0.737144
$100$	0	0
$101$	−7.50380	−0.746656	−0.373328	−	0.927699i	$-0.621783\pi$
$101$	−7.50380	−0.746656	−0.373328	+	0.927699i	$0.621783\pi$
$102$	0	0
$103$	0.752280	0.0741244	0.0370622	−	0.999313i	$-0.488200\pi$
$103$	0.752280	0.0741244	0.0370622	+	0.999313i	$0.488200\pi$
$104$	0	0
$105$	−7.46575	−0.728583
$106$	0	0
$107$	−11.2867	−1.09113	−0.545565	−	0.838069i	$-0.683685\pi$
$107$	−11.2867	−1.09113	−0.545565	+	0.838069i	$0.683685\pi$
$108$	0	0
$109$	1.22249	0.117093	0.0585466	−	0.998285i	$-0.481353\pi$
$109$	1.22249	0.117093	0.0585466	+	0.998285i	$0.481353\pi$
$110$	0	0
$111$	−12.6312	−1.19890
$112$	0	0
$113$	14.6311	1.37638	0.688189	−	0.725532i	$-0.258406\pi$
$113$	14.6311	1.37638	0.688189	+	0.725532i	$0.258406\pi$
$114$	0	0
$115$	21.6587	2.01968
$116$	0	0
$117$	−7.33449	−0.678074
$118$	0	0
$119$	−7.61912	−0.698444
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	34.3584	3.09799
$124$	0	0
$125$	10.6983	0.956886
$126$	0	0
$127$	−6.62675	−0.588029	−0.294014	−	0.955801i	$-0.594991\pi$
$127$	−6.62675	−0.588029	−0.294014	+	0.955801i	$0.594991\pi$
$128$	0	0
$129$	−23.5332	−2.07198
$130$	0	0
$131$	15.9794	1.39613	0.698065	−	0.716034i	$-0.254044\pi$
$131$	15.9794	1.39613	0.698065	+	0.716034i	$0.254044\pi$
$132$	0	0
$133$	−7.80003	−0.676349
$134$	0	0
$135$	32.3602	2.78512
$136$	0	0
$137$	−1.07940	−0.0922194	−0.0461097	−	0.998936i	$-0.514682\pi$
$137$	−1.07940	−0.0922194	−0.0461097	+	0.998936i	$0.514682\pi$
$138$	0	0
$139$	−10.6777	−0.905672	−0.452836	−	0.891594i	$-0.649588\pi$
$139$	−10.6777	−0.905672	−0.452836	+	0.891594i	$0.649588\pi$
$140$	0	0
$141$	34.9732	2.94528
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−7.67548	−0.637414
$146$	0	0
$147$	−3.21473	−0.265146
$148$	0	0
$149$	7.69415	0.630330	0.315165	−	0.949037i	$-0.397940\pi$
$149$	7.69415	0.630330	0.315165	+	0.949037i	$0.397940\pi$
$150$	0	0
$151$	−7.98828	−0.650077	−0.325038	−	0.945701i	$-0.605377\pi$
$151$	−7.98828	−0.650077	−0.325038	+	0.945701i	$0.605377\pi$
$152$	0	0
$153$	55.8824	4.51782
$154$	0	0
$155$	21.5903	1.73418
$156$	0	0
$157$	−10.2808	−0.820499	−0.410250	−	0.911973i	$-0.634558\pi$
$157$	−10.2808	−0.820499	−0.410250	+	0.911973i	$0.634558\pi$
$158$	0	0
$159$	0.846957	0.0671681
$160$	0	0
$161$	9.32617	0.735005
$162$	0	0
$163$	15.2380	1.19354	0.596768	−	0.802414i	$-0.296451\pi$
$163$	15.2380	1.19354	0.596768	+	0.802414i	$0.296451\pi$
$164$	0	0
$165$	−7.46575	−0.581208
$166$	0	0
$167$	1.92026	0.148595	0.0742973	−	0.997236i	$-0.476329\pi$
$167$	1.92026	0.148595	0.0742973	+	0.997236i	$0.476329\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	57.2093	4.37490
$172$	0	0
$173$	10.2962	0.782806	0.391403	−	0.920219i	$-0.371990\pi$
$173$	10.2962	0.782806	0.391403	+	0.920219i	$0.371990\pi$
$174$	0	0
$175$	−0.393340	−0.0297337
$176$	0	0
$177$	25.6912	1.93107
$178$	0	0
$179$	7.04939	0.526896	0.263448	−	0.964674i	$-0.415140\pi$
$179$	7.04939	0.526896	0.263448	+	0.964674i	$0.415140\pi$
$180$	0	0
$181$	5.29130	0.393299	0.196650	−	0.980474i	$-0.436994\pi$
$181$	5.29130	0.393299	0.196650	+	0.980474i	$0.436994\pi$
$182$	0	0
$183$	0.101751	0.00752163
$184$	0	0
$185$	−9.12495	−0.670880
$186$	0	0
$187$	−7.61912	−0.557165
$188$	0	0
$189$	13.9342	1.01356
$190$	0	0
$191$	0.652342	0.0472018	0.0236009	−	0.999721i	$-0.492487\pi$
$191$	0.652342	0.0472018	0.0236009	+	0.999721i	$0.492487\pi$
$192$	0	0
$193$	−4.64978	−0.334698	−0.167349	−	0.985898i	$-0.553521\pi$
$193$	−4.64978	−0.334698	−0.167349	+	0.985898i	$0.553521\pi$
$194$	0	0
$195$	−7.46575	−0.534633
$196$	0	0
$197$	−26.8614	−1.91380	−0.956898	−	0.290423i	$-0.906204\pi$
$197$	−26.8614	−1.91380	−0.956898	+	0.290423i	$0.906204\pi$
$198$	0	0
$199$	9.00651	0.638454	0.319227	−	0.947678i	$-0.396577\pi$
$199$	9.00651	0.638454	0.319227	+	0.947678i	$0.396577\pi$
$200$	0	0
$201$	−29.3318	−2.06890
$202$	0	0
$203$	−3.30504	−0.231968
$204$	0	0
$205$	24.8209	1.73357
$206$	0	0
$207$	−68.4027	−4.75432
$208$	0	0
$209$	−7.80003	−0.539540
$210$	0	0
$211$	−8.52687	−0.587014	−0.293507	−	0.955957i	$-0.594822\pi$
$211$	−8.52687	−0.587014	−0.293507	+	0.955957i	$0.594822\pi$
$212$	0	0
$213$	−10.7613	−0.737350
$214$	0	0
$215$	−17.0006	−1.15943
$216$	0	0
$217$	9.29674	0.631104
$218$	0	0
$219$	46.1398	3.11784
$220$	0	0
$221$	−7.61912	−0.512518
$222$	0	0
$223$	−17.8441	−1.19493	−0.597464	−	0.801896i	$-0.703825\pi$
$223$	−17.8441	−1.19493	−0.597464	+	0.801896i	$0.703825\pi$
$224$	0	0
$225$	2.88495	0.192330
$226$	0	0
$227$	1.13471	0.0753134	0.0376567	−	0.999291i	$-0.488011\pi$
$227$	1.13471	0.0753134	0.0376567	+	0.999291i	$0.488011\pi$
$228$	0	0
$229$	14.8718	0.982754	0.491377	−	0.870947i	$-0.336494\pi$
$229$	14.8718	0.982754	0.491377	+	0.870947i	$0.336494\pi$
$230$	0	0
$231$	−3.21473	−0.211514
$232$	0	0
$233$	4.47603	0.293234	0.146617	−	0.989193i	$-0.453161\pi$
$233$	4.47603	0.293234	0.146617	+	0.989193i	$0.453161\pi$
$234$	0	0
$235$	25.2651	1.64811
$236$	0	0
$237$	−37.1895	−2.41572
$238$	0	0
$239$	3.12851	0.202367	0.101183	−	0.994868i	$-0.467737\pi$
$239$	3.12851	0.202367	0.101183	+	0.994868i	$0.467737\pi$
$240$	0	0
$241$	20.0200	1.28960	0.644800	−	0.764351i	$-0.276941\pi$
$241$	20.0200	1.28960	0.644800	+	0.764351i	$0.276941\pi$
$242$	0	0
$243$	−31.4651	−2.01849
$244$	0	0
$245$	−2.32236	−0.148370
$246$	0	0
$247$	−7.80003	−0.496304
$248$	0	0
$249$	21.5998	1.36883
$250$	0	0
$251$	−7.06302	−0.445814	−0.222907	−	0.974840i	$-0.571555\pi$
$251$	−7.06302	−0.445814	−0.222907	+	0.974840i	$0.571555\pi$
$252$	0	0
$253$	9.32617	0.586331
$254$	0	0
$255$	56.8824	3.56212
$256$	0	0
$257$	9.60395	0.599078	0.299539	−	0.954084i	$-0.403167\pi$
$257$	9.60395	0.599078	0.299539	+	0.954084i	$0.403167\pi$
$258$	0	0
$259$	−3.92918	−0.244147
$260$	0	0
$261$	24.2408	1.50047
$262$	0	0
$263$	−25.7470	−1.58763	−0.793815	−	0.608159i	$-0.791908\pi$
$263$	−25.7470	−1.58763	−0.793815	+	0.608159i	$0.791908\pi$
$264$	0	0
$265$	0.611851	0.0375857
$266$	0	0
$267$	−34.8826	−2.13478
$268$	0	0
$269$	25.1199	1.53159	0.765793	−	0.643088i	$-0.222347\pi$
$269$	25.1199	1.53159	0.765793	+	0.643088i	$0.222347\pi$
$270$	0	0
$271$	7.75793	0.471260	0.235630	−	0.971843i	$-0.424285\pi$
$271$	7.75793	0.471260	0.235630	+	0.971843i	$0.424285\pi$
$272$	0	0
$273$	−3.21473	−0.194564
$274$	0	0
$275$	−0.393340	−0.0237193
$276$	0	0
$277$	5.19504	0.312140	0.156070	−	0.987746i	$-0.450117\pi$
$277$	5.19504	0.312140	0.156070	+	0.987746i	$0.450117\pi$
$278$	0	0
$279$	−68.1868	−4.08224
$280$	0	0
$281$	16.4855	0.983441	0.491721	−	0.870753i	$-0.336368\pi$
$281$	16.4855	0.983441	0.491721	+	0.870753i	$0.336368\pi$
$282$	0	0
$283$	−10.6838	−0.635084	−0.317542	−	0.948244i	$-0.602857\pi$
$283$	−10.6838	−0.635084	−0.317542	+	0.948244i	$0.602857\pi$
$284$	0	0
$285$	58.2331	3.44943
$286$	0	0
$287$	10.6878	0.630881
$288$	0	0
$289$	41.0510	2.41476
$290$	0	0
$291$	24.3204	1.42569
$292$	0	0
$293$	−6.03868	−0.352784	−0.176392	−	0.984320i	$-0.556443\pi$
$293$	−6.03868	−0.352784	−0.176392	+	0.984320i	$0.556443\pi$
$294$	0	0
$295$	18.5596	1.08058
$296$	0	0
$297$	13.9342	0.808545
$298$	0	0
$299$	9.32617	0.539346
$300$	0	0
$301$	−7.32042	−0.421942
$302$	0	0
$303$	24.1227	1.38581
$304$	0	0
$305$	0.0735058	0.00420893
$306$	0	0
$307$	4.38116	0.250046	0.125023	−	0.992154i	$-0.460100\pi$
$307$	4.38116	0.250046	0.125023	+	0.992154i	$0.460100\pi$
$308$	0	0
$309$	−2.41838	−0.137577
$310$	0	0
$311$	−24.4596	−1.38698	−0.693488	−	0.720468i	$-0.743927\pi$
$311$	−24.4596	−1.38698	−0.693488	+	0.720468i	$0.743927\pi$
$312$	0	0
$313$	6.79525	0.384090	0.192045	−	0.981386i	$-0.438488\pi$
$313$	6.79525	0.384090	0.192045	+	0.981386i	$0.438488\pi$
$314$	0	0
$315$	17.0333	0.959718
$316$	0	0
$317$	25.2154	1.41624	0.708119	−	0.706093i	$-0.249544\pi$
$317$	25.2154	1.41624	0.708119	+	0.706093i	$0.249544\pi$
$318$	0	0
$319$	−3.30504	−0.185047
$320$	0	0
$321$	36.2838	2.02516
$322$	0	0
$323$	59.4294	3.30674
$324$	0	0
$325$	−0.393340	−0.0218186
$326$	0	0
$327$	−3.92997	−0.217328
$328$	0	0
$329$	10.8791	0.599782
$330$	0	0
$331$	10.6879	0.587462	0.293731	−	0.955888i	$-0.405103\pi$
$331$	10.6879	0.587462	0.293731	+	0.955888i	$0.405103\pi$
$332$	0	0
$333$	28.8185	1.57924
$334$	0	0
$335$	−21.1896	−1.15771
$336$	0	0
$337$	3.31287	0.180464	0.0902318	−	0.995921i	$-0.471239\pi$
$337$	3.31287	0.180464	0.0902318	+	0.995921i	$0.471239\pi$
$338$	0	0
$339$	−47.0350	−2.55459
$340$	0	0
$341$	9.29674	0.503447
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−69.6268	−3.74858
$346$	0	0
$347$	31.5814	1.69538	0.847690	−	0.530491i	$-0.177993\pi$
$347$	31.5814	1.69538	0.847690	+	0.530491i	$0.177993\pi$
$348$	0	0
$349$	4.44159	0.237753	0.118876	−	0.992909i	$-0.462071\pi$
$349$	4.44159	0.237753	0.118876	+	0.992909i	$0.462071\pi$
$350$	0	0
$351$	13.9342	0.743753
$352$	0	0
$353$	−17.2499	−0.918118	−0.459059	−	0.888406i	$-0.651813\pi$
$353$	−17.2499	−0.918118	−0.459059	+	0.888406i	$0.651813\pi$
$354$	0	0
$355$	−7.77406	−0.412604
$356$	0	0
$357$	24.4934	1.29633
$358$	0	0
$359$	2.58716	0.136545	0.0682726	−	0.997667i	$-0.478251\pi$
$359$	2.58716	0.136545	0.0682726	+	0.997667i	$0.478251\pi$
$360$	0	0
$361$	41.8405	2.20213
$362$	0	0
$363$	−3.21473	−0.168730
$364$	0	0
$365$	33.3319	1.74467
$366$	0	0
$367$	23.8694	1.24597	0.622985	−	0.782233i	$-0.285920\pi$
$367$	23.8694	1.24597	0.622985	+	0.782233i	$0.285920\pi$
$368$	0	0
$369$	−78.3896	−4.08080
$370$	0	0
$371$	0.263461	0.0136782
$372$	0	0
$373$	8.71369	0.451178	0.225589	−	0.974223i	$-0.427569\pi$
$373$	8.71369	0.451178	0.225589	+	0.974223i	$0.427569\pi$
$374$	0	0
$375$	−34.3922	−1.77600
$376$	0	0
$377$	−3.30504	−0.170218
$378$	0	0
$379$	−5.74904	−0.295308	−0.147654	−	0.989039i	$-0.547172\pi$
$379$	−5.74904	−0.295308	−0.147654	+	0.989039i	$0.547172\pi$
$380$	0	0
$381$	21.3032	1.09140
$382$	0	0
$383$	4.83845	0.247233	0.123617	−	0.992330i	$-0.460551\pi$
$383$	4.83845	0.247233	0.123617	+	0.992330i	$0.460551\pi$
$384$	0	0
$385$	−2.32236	−0.118358
$386$	0	0
$387$	53.6916	2.72930
$388$	0	0
$389$	−0.769207	−0.0390004	−0.0195002	−	0.999810i	$-0.506207\pi$
$389$	−0.769207	−0.0390004	−0.0195002	+	0.999810i	$0.506207\pi$
$390$	0	0
$391$	−71.0572	−3.59352
$392$	0	0
$393$	−51.3696	−2.59125
$394$	0	0
$395$	−26.8661	−1.35178
$396$	0	0
$397$	−2.96203	−0.148660	−0.0743300	−	0.997234i	$-0.523682\pi$
$397$	−2.96203	−0.148660	−0.0743300	+	0.997234i	$0.523682\pi$
$398$	0	0
$399$	25.0750	1.25532
$400$	0	0
$401$	−38.5674	−1.92597	−0.962983	−	0.269563i	$-0.913121\pi$
$401$	−38.5674	−1.92597	−0.962983	+	0.269563i	$0.913121\pi$
$402$	0	0
$403$	9.29674	0.463103
$404$	0	0
$405$	−52.9294	−2.63009
$406$	0	0
$407$	−3.92918	−0.194762
$408$	0	0
$409$	23.8884	1.18120	0.590602	−	0.806963i	$-0.298890\pi$
$409$	23.8884	1.18120	0.590602	+	0.806963i	$0.298890\pi$
$410$	0	0
$411$	3.46998	0.171162
$412$	0	0
$413$	7.99170	0.393246
$414$	0	0
$415$	15.6040	0.765969
$416$	0	0
$417$	34.3260	1.68095
$418$	0	0
$419$	13.5141	0.660207	0.330104	−	0.943945i	$-0.392916\pi$
$419$	13.5141	0.660207	0.330104	+	0.943945i	$0.392916\pi$
$420$	0	0
$421$	14.1621	0.690220	0.345110	−	0.938562i	$-0.387842\pi$
$421$	14.1621	0.690220	0.345110	+	0.938562i	$0.387842\pi$
$422$	0	0
$423$	−79.7923	−3.87964
$424$	0	0
$425$	2.99691	0.145371
$426$	0	0
$427$	0.0316514	0.00153172
$428$	0	0
$429$	−3.21473	−0.155209
$430$	0	0
$431$	−14.9317	−0.719235	−0.359617	−	0.933100i	$-0.617093\pi$
$431$	−14.9317	−0.719235	−0.359617	+	0.933100i	$0.617093\pi$
$432$	0	0
$433$	27.6671	1.32960	0.664799	−	0.747022i	$-0.268517\pi$
$433$	27.6671	1.32960	0.664799	+	0.747022i	$0.268517\pi$
$434$	0	0
$435$	24.6746	1.18306
$436$	0	0
$437$	−72.7444	−3.47984
$438$	0	0
$439$	12.8124	0.611503	0.305752	−	0.952111i	$-0.401092\pi$
$439$	12.8124	0.611503	0.305752	+	0.952111i	$0.401092\pi$
$440$	0	0
$441$	7.33449	0.349261
$442$	0	0
$443$	−19.9609	−0.948369	−0.474185	−	0.880425i	$-0.657257\pi$
$443$	−19.9609	−0.948369	−0.474185	+	0.880425i	$0.657257\pi$
$444$	0	0
$445$	−25.1996	−1.19458
$446$	0	0
$447$	−24.7346	−1.16991
$448$	0	0
$449$	−3.34012	−0.157630	−0.0788150	−	0.996889i	$-0.525114\pi$
$449$	−3.34012	−0.157630	−0.0788150	+	0.996889i	$0.525114\pi$
$450$	0	0
$451$	10.6878	0.503269
$452$	0	0
$453$	25.6802	1.20656
$454$	0	0
$455$	−2.32236	−0.108874
$456$	0	0
$457$	2.15297	0.100712	0.0503558	−	0.998731i	$-0.483964\pi$
$457$	2.15297	0.100712	0.0503558	+	0.998731i	$0.483964\pi$
$458$	0	0
$459$	−106.166	−4.95543
$460$	0	0
$461$	7.81564	0.364011	0.182005	−	0.983298i	$-0.441741\pi$
$461$	7.81564	0.364011	0.182005	+	0.983298i	$0.441741\pi$
$462$	0	0
$463$	−10.5469	−0.490156	−0.245078	−	0.969503i	$-0.578814\pi$
$463$	−10.5469	−0.490156	−0.245078	+	0.969503i	$0.578814\pi$
$464$	0	0
$465$	−69.4071	−3.21868
$466$	0	0
$467$	22.2528	1.02974	0.514868	−	0.857269i	$-0.327841\pi$
$467$	22.2528	1.02974	0.514868	+	0.857269i	$0.327841\pi$
$468$	0	0
$469$	−9.12418	−0.421316
$470$	0	0
$471$	33.0501	1.52287
$472$	0	0
$473$	−7.32042	−0.336593
$474$	0	0
$475$	3.06807	0.140773
$476$	0	0
$477$	−1.93235	−0.0884764
$478$	0	0
$479$	14.7498	0.673933	0.336967	−	0.941517i	$-0.390599\pi$
$479$	14.7498	0.673933	0.336967	+	0.941517i	$0.390599\pi$
$480$	0	0
$481$	−3.92918	−0.179155
$482$	0	0
$483$	−29.9811	−1.36419
$484$	0	0
$485$	17.5694	0.797784
$486$	0	0
$487$	11.2792	0.511111	0.255556	−	0.966794i	$-0.417742\pi$
$487$	11.2792	0.511111	0.255556	+	0.966794i	$0.417742\pi$
$488$	0	0
$489$	−48.9862	−2.21523
$490$	0	0
$491$	10.3197	0.465721	0.232861	−	0.972510i	$-0.425191\pi$
$491$	10.3197	0.465721	0.232861	+	0.972510i	$0.425191\pi$
$492$	0	0
$493$	25.1815	1.13412
$494$	0	0
$495$	17.0333	0.765590
$496$	0	0
$497$	−3.34749	−0.150155
$498$	0	0
$499$	13.9587	0.624879	0.312440	−	0.949938i	$-0.398854\pi$
$499$	13.9587	0.624879	0.312440	+	0.949938i	$0.398854\pi$
$500$	0	0
$501$	−6.17313	−0.275795
$502$	0	0
$503$	11.2367	0.501018	0.250509	−	0.968114i	$-0.419402\pi$
$503$	11.2367	0.501018	0.250509	+	0.968114i	$0.419402\pi$
$504$	0	0
$505$	17.4265	0.775469
$506$	0	0
$507$	−3.21473	−0.142771
$508$	0	0
$509$	−17.7545	−0.786953	−0.393477	−	0.919335i	$-0.628728\pi$
$509$	−17.7545	−0.786953	−0.393477	+	0.919335i	$0.628728\pi$
$510$	0	0
$511$	14.3526	0.634922
$512$	0	0
$513$	−108.687	−4.79866
$514$	0	0
$515$	−1.74706	−0.0769848
$516$	0	0
$517$	10.8791	0.478461
$518$	0	0
$519$	−33.0995	−1.45291
$520$	0	0
$521$	−27.2360	−1.19323	−0.596616	−	0.802527i	$-0.703489\pi$
$521$	−27.2360	−1.19323	−0.596616	+	0.802527i	$0.703489\pi$
$522$	0	0
$523$	−21.6701	−0.947568	−0.473784	−	0.880641i	$-0.657112\pi$
$523$	−21.6701	−0.947568	−0.473784	+	0.880641i	$0.657112\pi$
$524$	0	0
$525$	1.26448	0.0551865
$526$	0	0
$527$	−70.8330	−3.08553
$528$	0	0
$529$	63.9774	2.78163
$530$	0	0
$531$	−58.6150	−2.54368
$532$	0	0
$533$	10.6878	0.462940
$534$	0	0
$535$	26.2118	1.13324
$536$	0	0
$537$	−22.6619	−0.977932
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	15.3907	0.661699	0.330850	−	0.943683i	$-0.392665\pi$
$541$	15.3907	0.661699	0.330850	+	0.943683i	$0.392665\pi$
$542$	0	0
$543$	−17.0101	−0.729973
$544$	0	0
$545$	−2.83906	−0.121612
$546$	0	0
$547$	3.24887	0.138912	0.0694559	−	0.997585i	$-0.477874\pi$
$547$	3.24887	0.138912	0.0694559	+	0.997585i	$0.477874\pi$
$548$	0	0
$549$	−0.232147	−0.00990779
$550$	0	0
$551$	25.7794	1.09824
$552$	0	0
$553$	−11.5685	−0.491941
$554$	0	0
$555$	29.3342	1.24517
$556$	0	0
$557$	−31.2255	−1.32307	−0.661533	−	0.749916i	$-0.730094\pi$
$557$	−31.2255	−1.32307	−0.661533	+	0.749916i	$0.730094\pi$
$558$	0	0
$559$	−7.32042	−0.309621
$560$	0	0
$561$	24.4934	1.03411
$562$	0	0
$563$	−17.0900	−0.720259	−0.360130	−	0.932902i	$-0.617268\pi$
$563$	−17.0900	−0.720259	−0.360130	+	0.932902i	$0.617268\pi$
$564$	0	0
$565$	−33.9786	−1.42949
$566$	0	0
$567$	−22.7913	−0.957143
$568$	0	0
$569$	39.1469	1.64112	0.820561	−	0.571559i	$-0.193661\pi$
$569$	39.1469	1.64112	0.820561	+	0.571559i	$0.193661\pi$
$570$	0	0
$571$	28.0171	1.17248	0.586239	−	0.810138i	$-0.300608\pi$
$571$	28.0171	1.17248	0.586239	+	0.810138i	$0.300608\pi$
$572$	0	0
$573$	−2.09710	−0.0876077
$574$	0	0
$575$	−3.66836	−0.152981
$576$	0	0
$577$	12.4714	0.519191	0.259595	−	0.965717i	$-0.416411\pi$
$577$	12.4714	0.519191	0.259595	+	0.965717i	$0.416411\pi$
$578$	0	0
$579$	14.9478	0.621209
$580$	0	0
$581$	6.71902	0.278752
$582$	0	0
$583$	0.263461	0.0109115
$584$	0	0
$585$	17.0333	0.704240
$586$	0	0
$587$	−4.30523	−0.177696	−0.0888479	−	0.996045i	$-0.528318\pi$
$587$	−4.30523	−0.177696	−0.0888479	+	0.996045i	$0.528318\pi$
$588$	0	0
$589$	−72.5149	−2.98792
$590$	0	0
$591$	86.3522	3.55206
$592$	0	0
$593$	26.3981	1.08404	0.542021	−	0.840365i	$-0.317659\pi$
$593$	26.3981	1.08404	0.542021	+	0.840365i	$0.317659\pi$
$594$	0	0
$595$	17.6943	0.725396
$596$	0	0
$597$	−28.9535	−1.18499
$598$	0	0
$599$	43.6137	1.78201	0.891004	−	0.453995i	$-0.150002\pi$
$599$	43.6137	1.78201	0.891004	+	0.453995i	$0.150002\pi$
$600$	0	0
$601$	−35.8016	−1.46038	−0.730188	−	0.683246i	$-0.760568\pi$
$601$	−35.8016	−1.46038	−0.730188	+	0.683246i	$0.760568\pi$
$602$	0	0
$603$	66.9212	2.72524
$604$	0	0
$605$	−2.32236	−0.0944172
$606$	0	0
$607$	−11.5258	−0.467817	−0.233909	−	0.972259i	$-0.575152\pi$
$607$	−11.5258	−0.467817	−0.233909	+	0.972259i	$0.575152\pi$
$608$	0	0
$609$	10.6248	0.430539
$610$	0	0
$611$	10.8791	0.440120
$612$	0	0
$613$	−12.3169	−0.497473	−0.248737	−	0.968571i	$-0.580015\pi$
$613$	−12.3169	−0.497473	−0.248737	+	0.968571i	$0.580015\pi$
$614$	0	0
$615$	−79.7925	−3.21754
$616$	0	0
$617$	−37.4137	−1.50622	−0.753110	−	0.657895i	$-0.771447\pi$
$617$	−37.4137	−1.50622	−0.753110	+	0.657895i	$0.771447\pi$
$618$	0	0
$619$	11.0903	0.445755	0.222877	−	0.974846i	$-0.428455\pi$
$619$	11.0903	0.445755	0.222877	+	0.974846i	$0.428455\pi$
$620$	0	0
$621$	129.953	5.21483
$622$	0	0
$623$	−10.8509	−0.434731
$624$	0	0
$625$	−26.8120	−1.07248
$626$	0	0
$627$	25.0750	1.00140
$628$	0	0
$629$	29.9369	1.19366
$630$	0	0
$631$	29.0142	1.15504	0.577519	−	0.816377i	$-0.304021\pi$
$631$	29.0142	1.15504	0.577519	+	0.816377i	$0.304021\pi$
$632$	0	0
$633$	27.4116	1.08951
$634$	0	0
$635$	15.3897	0.610720
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	24.5521	0.971267
$640$	0	0
$641$	−47.7689	−1.88676	−0.943379	−	0.331717i	$-0.892372\pi$
$641$	−47.7689	−1.88676	−0.943379	+	0.331717i	$0.892372\pi$
$642$	0	0
$643$	10.2059	0.402480	0.201240	−	0.979542i	$-0.435503\pi$
$643$	10.2059	0.402480	0.201240	+	0.979542i	$0.435503\pi$
$644$	0	0
$645$	54.6524	2.15194
$646$	0	0
$647$	−21.8958	−0.860813	−0.430407	−	0.902635i	$-0.641630\pi$
$647$	−21.8958	−0.860813	−0.430407	+	0.902635i	$0.641630\pi$
$648$	0	0
$649$	7.99170	0.313702
$650$	0	0
$651$	−29.8865	−1.17134
$652$	0	0
$653$	−36.9237	−1.44493	−0.722467	−	0.691405i	$-0.756992\pi$
$653$	−36.9237	−1.44493	−0.722467	+	0.691405i	$0.756992\pi$
$654$	0	0
$655$	−37.1100	−1.45001
$656$	0	0
$657$	−105.269	−4.10694
$658$	0	0
$659$	−20.7114	−0.806802	−0.403401	−	0.915023i	$-0.632172\pi$
$659$	−20.7114	−0.806802	−0.403401	+	0.915023i	$0.632172\pi$
$660$	0	0
$661$	−2.89506	−0.112605	−0.0563024	−	0.998414i	$-0.517931\pi$
$661$	−2.89506	−0.112605	−0.0563024	+	0.998414i	$0.517931\pi$
$662$	0	0
$663$	24.4934	0.951245
$664$	0	0
$665$	18.1145	0.702449
$666$	0	0
$667$	−30.8233	−1.19348
$668$	0	0
$669$	57.3639	2.21782
$670$	0	0
$671$	0.0316514	0.00122189
$672$	0	0
$673$	−22.7168	−0.875668	−0.437834	−	0.899056i	$-0.644254\pi$
$673$	−22.7168	−0.875668	−0.437834	+	0.899056i	$0.644254\pi$
$674$	0	0
$675$	−5.48089	−0.210959
$676$	0	0
$677$	22.7768	0.875385	0.437692	−	0.899125i	$-0.355796\pi$
$677$	22.7768	0.875385	0.437692	+	0.899125i	$0.355796\pi$
$678$	0	0
$679$	7.56532	0.290330
$680$	0	0
$681$	−3.64779	−0.139784
$682$	0	0
$683$	−19.4559	−0.744458	−0.372229	−	0.928141i	$-0.621406\pi$
$683$	−19.4559	−0.744458	−0.372229	+	0.928141i	$0.621406\pi$
$684$	0	0
$685$	2.50675	0.0957781
$686$	0	0
$687$	−47.8087	−1.82402
$688$	0	0
$689$	0.263461	0.0100371
$690$	0	0
$691$	37.1334	1.41262	0.706311	−	0.707902i	$-0.250358\pi$
$691$	37.1334	1.41262	0.706311	+	0.707902i	$0.250358\pi$
$692$	0	0
$693$	7.33449	0.278614
$694$	0	0
$695$	24.7975	0.940622
$696$	0	0
$697$	−81.4317	−3.08444
$698$	0	0
$699$	−14.3892	−0.544250
$700$	0	0
$701$	−32.9134	−1.24312	−0.621561	−	0.783366i	$-0.713501\pi$
$701$	−32.9134	−1.24312	−0.621561	+	0.783366i	$0.713501\pi$
$702$	0	0
$703$	30.6477	1.15590
$704$	0	0
$705$	−81.2203	−3.05893
$706$	0	0
$707$	7.50380	0.282210
$708$	0	0
$709$	−3.08800	−0.115972	−0.0579861	−	0.998317i	$-0.518468\pi$
$709$	−3.08800	−0.115972	−0.0579861	+	0.998317i	$0.518468\pi$
$710$	0	0
$711$	84.8488	3.18208
$712$	0	0
$713$	86.7029	3.24705
$714$	0	0
$715$	−2.32236	−0.0868512
$716$	0	0
$717$	−10.0573	−0.375598
$718$	0	0
$719$	−8.32604	−0.310509	−0.155254	−	0.987875i	$-0.549620\pi$
$719$	−8.32604	−0.310509	−0.155254	+	0.987875i	$0.549620\pi$
$720$	0	0
$721$	−0.752280	−0.0280164
$722$	0	0
$723$	−64.3588	−2.39353
$724$	0	0
$725$	1.30000	0.0482809
$726$	0	0
$727$	−21.4453	−0.795363	−0.397681	−	0.917524i	$-0.630185\pi$
$727$	−21.4453	−0.795363	−0.397681	+	0.917524i	$0.630185\pi$
$728$	0	0
$729$	32.7781	1.21400
$730$	0	0
$731$	55.7752	2.06292
$732$	0	0
$733$	17.4800	0.645639	0.322820	−	0.946461i	$-0.395369\pi$
$733$	17.4800	0.645639	0.322820	+	0.946461i	$0.395369\pi$
$734$	0	0
$735$	7.46575	0.275378
$736$	0	0
$737$	−9.12418	−0.336094
$738$	0	0
$739$	27.8932	1.02607	0.513034	−	0.858368i	$-0.328521\pi$
$739$	27.8932	1.02607	0.513034	+	0.858368i	$0.328521\pi$
$740$	0	0
$741$	25.0750	0.921153
$742$	0	0
$743$	−31.8612	−1.16887	−0.584437	−	0.811439i	$-0.698684\pi$
$743$	−31.8612	−1.16887	−0.584437	+	0.811439i	$0.698684\pi$
$744$	0	0
$745$	−17.8686	−0.654654
$746$	0	0
$747$	−49.2806	−1.80308
$748$	0	0
$749$	11.2867	0.412408
$750$	0	0
$751$	−30.7619	−1.12252	−0.561260	−	0.827640i	$-0.689683\pi$
$751$	−30.7619	−1.12252	−0.561260	+	0.827640i	$0.689683\pi$
$752$	0	0
$753$	22.7057	0.827441
$754$	0	0
$755$	18.5516	0.675163
$756$	0	0
$757$	42.9708	1.56180	0.780901	−	0.624655i	$-0.214761\pi$
$757$	42.9708	1.56180	0.780901	+	0.624655i	$0.214761\pi$
$758$	0	0
$759$	−29.9811	−1.08825
$760$	0	0
$761$	24.0241	0.870874	0.435437	−	0.900219i	$-0.356594\pi$
$761$	24.0241	0.870874	0.435437	+	0.900219i	$0.356594\pi$
$762$	0	0
$763$	−1.22249	−0.0442571
$764$	0	0
$765$	−129.779	−4.69216
$766$	0	0
$767$	7.99170	0.288564
$768$	0	0
$769$	49.1252	1.77150	0.885750	−	0.464163i	$-0.153645\pi$
$769$	49.1252	1.77150	0.885750	+	0.464163i	$0.153645\pi$
$770$	0	0
$771$	−30.8741	−1.11190
$772$	0	0
$773$	20.7109	0.744921	0.372460	−	0.928048i	$-0.378514\pi$
$773$	20.7109	0.744921	0.372460	+	0.928048i	$0.378514\pi$
$774$	0	0
$775$	−3.65678	−0.131355
$776$	0	0
$777$	12.6312	0.453143
$778$	0	0
$779$	−83.3652	−2.98687
$780$	0	0
$781$	−3.34749	−0.119783
$782$	0	0
$783$	−46.0531	−1.64580
$784$	0	0
$785$	23.8757	0.852162
$786$	0	0
$787$	26.4745	0.943713	0.471857	−	0.881675i	$-0.343584\pi$
$787$	26.4745	0.943713	0.471857	+	0.881675i	$0.343584\pi$
$788$	0	0
$789$	82.7698	2.94668
$790$	0	0
$791$	−14.6311	−0.520222
$792$	0	0
$793$	0.0316514	0.00112397
$794$	0	0
$795$	−1.96694	−0.0697600
$796$	0	0
$797$	33.0401	1.17034	0.585169	−	0.810911i	$-0.301028\pi$
$797$	33.0401	1.17034	0.585169	+	0.810911i	$0.301028\pi$
$798$	0	0
$799$	−82.8889	−2.93240
$800$	0	0
$801$	79.5856	2.81202
$802$	0	0
$803$	14.3526	0.506492
$804$	0	0
$805$	−21.6587	−0.763369
$806$	0	0
$807$	−80.7536	−2.84266
$808$	0	0
$809$	18.7333	0.658628	0.329314	−	0.944220i	$-0.393183\pi$
$809$	18.7333	0.658628	0.329314	+	0.944220i	$0.393183\pi$
$810$	0	0
$811$	26.6881	0.937146	0.468573	−	0.883425i	$-0.344768\pi$
$811$	26.6881	0.937146	0.468573	+	0.883425i	$0.344768\pi$
$812$	0	0
$813$	−24.9396	−0.874671
$814$	0	0
$815$	−35.3882	−1.23959
$816$	0	0
$817$	57.0996	1.99766
$818$	0	0
$819$	7.33449	0.256288
$820$	0	0
$821$	4.31411	0.150563	0.0752817	−	0.997162i	$-0.476014\pi$
$821$	4.31411	0.150563	0.0752817	+	0.997162i	$0.476014\pi$
$822$	0	0
$823$	26.9016	0.937731	0.468865	−	0.883270i	$-0.344663\pi$
$823$	26.9016	0.937731	0.468865	+	0.883270i	$0.344663\pi$
$824$	0	0
$825$	1.26448	0.0440236
$826$	0	0
$827$	−17.2063	−0.598322	−0.299161	−	0.954203i	$-0.596707\pi$
$827$	−17.2063	−0.598322	−0.299161	+	0.954203i	$0.596707\pi$
$828$	0	0
$829$	−22.0565	−0.766055	−0.383028	−	0.923737i	$-0.625119\pi$
$829$	−22.0565	−0.766055	−0.383028	+	0.923737i	$0.625119\pi$
$830$	0	0
$831$	−16.7007	−0.579339
$832$	0	0
$833$	7.61912	0.263987
$834$	0	0
$835$	−4.45954	−0.154329
$836$	0	0
$837$	129.543	4.47765
$838$	0	0
$839$	16.2143	0.559781	0.279890	−	0.960032i	$-0.409702\pi$
$839$	16.2143	0.559781	0.279890	+	0.960032i	$0.409702\pi$
$840$	0	0
$841$	−18.0767	−0.623335
$842$	0	0
$843$	−52.9964	−1.82529
$844$	0	0
$845$	−2.32236	−0.0798915
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	34.3454	1.17873
$850$	0	0
$851$	−36.6442	−1.25615
$852$	0	0
$853$	−10.7332	−0.367496	−0.183748	−	0.982973i	$-0.558823\pi$
$853$	−10.7332	−0.367496	−0.183748	+	0.982973i	$0.558823\pi$
$854$	0	0
$855$	−132.860	−4.54373
$856$	0	0
$857$	33.3678	1.13982	0.569912	−	0.821706i	$-0.306977\pi$
$857$	33.3678	1.13982	0.569912	+	0.821706i	$0.306977\pi$
$858$	0	0
$859$	36.0562	1.23022	0.615111	−	0.788440i	$-0.289111\pi$
$859$	36.0562	1.23022	0.615111	+	0.788440i	$0.289111\pi$
$860$	0	0
$861$	−34.3584	−1.17093
$862$	0	0
$863$	28.6809	0.976310	0.488155	−	0.872757i	$-0.337670\pi$
$863$	28.6809	0.976310	0.488155	+	0.872757i	$0.337670\pi$
$864$	0	0
$865$	−23.9115	−0.813014
$866$	0	0
$867$	−131.968	−4.48186
$868$	0	0
$869$	−11.5685	−0.392434
$870$	0	0
$871$	−9.12418	−0.309161
$872$	0	0
$873$	−55.4877	−1.87797
$874$	0	0
$875$	−10.6983	−0.361669
$876$	0	0
$877$	37.7520	1.27479	0.637397	−	0.770536i	$-0.280011\pi$
$877$	37.7520	1.27479	0.637397	+	0.770536i	$0.280011\pi$
$878$	0	0
$879$	19.4127	0.654775
$880$	0	0
$881$	−8.05877	−0.271507	−0.135753	−	0.990743i	$-0.543346\pi$
$881$	−8.05877	−0.271507	−0.135753	+	0.990743i	$0.543346\pi$
$882$	0	0
$883$	45.8441	1.54278	0.771389	−	0.636364i	$-0.219563\pi$
$883$	45.8441	1.54278	0.771389	+	0.636364i	$0.219563\pi$
$884$	0	0
$885$	−59.6640	−2.00558
$886$	0	0
$887$	−7.60900	−0.255485	−0.127743	−	0.991807i	$-0.540773\pi$
$887$	−7.60900	−0.255485	−0.127743	+	0.991807i	$0.540773\pi$
$888$	0	0
$889$	6.62675	0.222254
$890$	0	0
$891$	−22.7913	−0.763536
$892$	0	0
$893$	−84.8570	−2.83963
$894$	0	0
$895$	−16.3712	−0.547228
$896$	0	0
$897$	−29.9811	−1.00104
$898$	0	0
$899$	−30.7261	−1.02477
$900$	0	0
$901$	−2.00734	−0.0668743
$902$	0	0
$903$	23.5332	0.783135
$904$	0	0
$905$	−12.2883	−0.408476
$906$	0	0
$907$	17.0073	0.564718	0.282359	−	0.959309i	$-0.408883\pi$
$907$	17.0073	0.564718	0.282359	+	0.959309i	$0.408883\pi$
$908$	0	0
$909$	−55.0366	−1.82545
$910$	0	0
$911$	41.9745	1.39068	0.695338	−	0.718683i	$-0.255255\pi$
$911$	41.9745	1.39068	0.695338	+	0.718683i	$0.255255\pi$
$912$	0	0
$913$	6.71902	0.222367
$914$	0	0
$915$	−0.236301	−0.00781188
$916$	0	0
$917$	−15.9794	−0.527688
$918$	0	0
$919$	1.82068	0.0600586	0.0300293	−	0.999549i	$-0.490440\pi$
$919$	1.82068	0.0600586	0.0300293	+	0.999549i	$0.490440\pi$
$920$	0	0
$921$	−14.0843	−0.464092
$922$	0	0
$923$	−3.34749	−0.110184
$924$	0	0
$925$	1.54550	0.0508158
$926$	0	0
$927$	5.51759	0.181222
$928$	0	0
$929$	−39.6077	−1.29949	−0.649743	−	0.760154i	$-0.725124\pi$
$929$	−39.6077	−1.29949	−0.649743	+	0.760154i	$0.725124\pi$
$930$	0	0
$931$	7.80003	0.255636
$932$	0	0
$933$	78.6310	2.57426
$934$	0	0
$935$	17.6943	0.578666
$936$	0	0
$937$	52.0877	1.70163	0.850815	−	0.525465i	$-0.176109\pi$
$937$	52.0877	1.70163	0.850815	+	0.525465i	$0.176109\pi$
$938$	0	0
$939$	−21.8449	−0.712882
$940$	0	0
$941$	−3.21629	−0.104848	−0.0524240	−	0.998625i	$-0.516695\pi$
$941$	−3.21629	−0.104848	−0.0524240	+	0.998625i	$0.516695\pi$
$942$	0	0
$943$	99.6763	3.24591
$944$	0	0
$945$	−32.3602	−1.05268
$946$	0	0
$947$	−17.0436	−0.553842	−0.276921	−	0.960893i	$-0.589314\pi$
$947$	−17.0436	−0.553842	−0.276921	+	0.960893i	$0.589314\pi$
$948$	0	0
$949$	14.3526	0.465905
$950$	0	0
$951$	−81.0607	−2.62857
$952$	0	0
$953$	5.45131	0.176585	0.0882926	−	0.996095i	$-0.471859\pi$
$953$	5.45131	0.176585	0.0882926	+	0.996095i	$0.471859\pi$
$954$	0	0
$955$	−1.51497	−0.0490233
$956$	0	0
$957$	10.6248	0.343451
$958$	0	0
$959$	1.07940	0.0348557
$960$	0	0
$961$	55.4293	1.78804
$962$	0	0
$963$	−82.7824	−2.66763
$964$	0	0
$965$	10.7984	0.347614
$966$	0	0
$967$	9.50097	0.305531	0.152765	−	0.988262i	$-0.451182\pi$
$967$	9.50097	0.305531	0.152765	+	0.988262i	$0.451182\pi$
$968$	0	0
$969$	−191.049	−6.13739
$970$	0	0
$971$	21.1050	0.677291	0.338645	−	0.940914i	$-0.390031\pi$
$971$	21.1050	0.677291	0.338645	+	0.940914i	$0.390031\pi$
$972$	0	0
$973$	10.6777	0.342312
$974$	0	0
$975$	1.26448	0.0404958
$976$	0	0
$977$	18.1636	0.581104	0.290552	−	0.956859i	$-0.406161\pi$
$977$	18.1636	0.581104	0.290552	+	0.956859i	$0.406161\pi$
$978$	0	0
$979$	−10.8509	−0.346795
$980$	0	0
$981$	8.96634	0.286273
$982$	0	0
$983$	−53.0122	−1.69083	−0.845413	−	0.534114i	$-0.820645\pi$
$983$	−53.0122	−1.69083	−0.845413	+	0.534114i	$0.820645\pi$
$984$	0	0
$985$	62.3818	1.98765
$986$	0	0
$987$	−34.9732	−1.11321
$988$	0	0
$989$	−68.2715	−2.17091
$990$	0	0
$991$	−12.1416	−0.385690	−0.192845	−	0.981229i	$-0.561772\pi$
$991$	−12.1416	−0.385690	−0.192845	+	0.981229i	$0.561772\pi$
$992$	0	0
$993$	−34.3588	−1.09034
$994$	0	0
$995$	−20.9163	−0.663092
$996$	0	0
$997$	−55.7433	−1.76541	−0.882704	−	0.469929i	$-0.844279\pi$
$997$	−55.7433	−1.76541	−0.882704	+	0.469929i	$0.844279\pi$
$998$	0	0
$999$	−54.7500	−1.73221

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.21473 q^{3} -2.32236 q^{5} -1.00000 q^{7} +7.33449 q^{9} +O(q^{10})\)	\(q-3.21473 q^{3} -2.32236 q^{5} -1.00000 q^{7} +7.33449 q^{9} -1.00000 q^{11} -1.00000 q^{13} +7.46575 q^{15} +7.61912 q^{17} +7.80003 q^{19} +3.21473 q^{21} -9.32617 q^{23} +0.393340 q^{25} -13.9342 q^{27} +3.30504 q^{29} -9.29674 q^{31} +3.21473 q^{33} +2.32236 q^{35} +3.92918 q^{37} +3.21473 q^{39} -10.6878 q^{41} +7.32042 q^{43} -17.0333 q^{45} -10.8791 q^{47} +1.00000 q^{49} -24.4934 q^{51} -0.263461 q^{53} +2.32236 q^{55} -25.0750 q^{57} -7.99170 q^{59} -0.0316514 q^{61} -7.33449 q^{63} +2.32236 q^{65} +9.12418 q^{67} +29.9811 q^{69} +3.34749 q^{71} -14.3526 q^{73} -1.26448 q^{75} +1.00000 q^{77} +11.5685 q^{79} +22.7913 q^{81} -6.71902 q^{83} -17.6943 q^{85} -10.6248 q^{87} +10.8509 q^{89} +1.00000 q^{91} +29.8865 q^{93} -18.1145 q^{95} -7.56532 q^{97} -7.33449 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} + 9 q^{17} + 3 q^{19} + 2 q^{21} - 4 q^{23} + 22 q^{25} - 8 q^{27} + 13 q^{29} - 17 q^{31} + 2 q^{33} + 2 q^{35} + 11 q^{37} + 2 q^{39} + 8 q^{41} + 21 q^{43} - 15 q^{45} + q^{47} + 10 q^{49} + 19 q^{51} + 22 q^{53} + 2 q^{55} + 8 q^{57} - 24 q^{59} + 16 q^{61} - 10 q^{63} + 2 q^{65} + 19 q^{67} + 51 q^{69} - 25 q^{71} + 22 q^{73} + 28 q^{75} + 10 q^{77} + 41 q^{79} + 54 q^{81} + 8 q^{83} + 9 q^{85} - 5 q^{87} + 36 q^{89} + 10 q^{91} + 12 q^{93} + 13 q^{95} - 5 q^{97} - 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - 2 * q^5 - 10 * q^7 + 10 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 + 9 * q^17 + 3 * q^19 + 2 * q^21 - 4 * q^23 + 22 * q^25 - 8 * q^27 + 13 * q^29 - 17 * q^31 + 2 * q^33 + 2 * q^35 + 11 * q^37 + 2 * q^39 + 8 * q^41 + 21 * q^43 - 15 * q^45 + q^47 + 10 * q^49 + 19 * q^51 + 22 * q^53 + 2 * q^55 + 8 * q^57 - 24 * q^59 + 16 * q^61 - 10 * q^63 + 2 * q^65 + 19 * q^67 + 51 * q^69 - 25 * q^71 + 22 * q^73 + 28 * q^75 + 10 * q^77 + 41 * q^79 + 54 * q^81 + 8 * q^83 + 9 * q^85 - 5 * q^87 + 36 * q^89 + 10 * q^91 + 12 * q^93 + 13 * q^95 - 5 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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