LMFDB - Embedded newform 4004.2.a.j.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:	$0$
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			1.3
Root			$2.16679$ 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)

$f(q)$	$=$	$q-2.16679 q^{3} -1.26627 q^{5} -1.00000 q^{7} +1.69497 q^{9} +O(q^{10})$	$q-2.16679 q^{3} -1.26627 q^{5} -1.00000 q^{7} +1.69497 q^{9} -1.00000 q^{11} -1.00000 q^{13} +2.74374 q^{15} -4.02352 q^{17} -5.92833 q^{19} +2.16679 q^{21} +4.27435 q^{23} -3.39656 q^{25} +2.82773 q^{27} -9.21632 q^{29} -0.386596 q^{31} +2.16679 q^{33} +1.26627 q^{35} +11.2499 q^{37} +2.16679 q^{39} -7.02502 q^{41} -3.45783 q^{43} -2.14628 q^{45} -9.82960 q^{47} +1.00000 q^{49} +8.71811 q^{51} +1.42216 q^{53} +1.26627 q^{55} +12.8454 q^{57} -11.6029 q^{59} +10.3432 q^{61} -1.69497 q^{63} +1.26627 q^{65} -10.6126 q^{67} -9.26160 q^{69} -0.110620 q^{71} -3.03553 q^{73} +7.35962 q^{75} +1.00000 q^{77} -0.962094 q^{79} -11.2120 q^{81} +4.40921 q^{83} +5.09487 q^{85} +19.9698 q^{87} -8.23230 q^{89} +1.00000 q^{91} +0.837670 q^{93} +7.50687 q^{95} +3.14917 q^{97} -1.69497 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.16679	−1.25099	−0.625497	−	0.780226i	$-0.715104\pi$
$3$	−2.16679	−1.25099	−0.625497	+	0.780226i	$0.715104\pi$
$4$	0	0
$5$	−1.26627	−0.566293	−0.283147	−	0.959077i	$-0.591378\pi$
$5$	−1.26627	−0.566293	−0.283147	+	0.959077i	$0.591378\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.69497	0.564988
$10$	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
$15$	2.74374	0.708430
$16$	0	0
$17$	−4.02352	−0.975847	−0.487924	−	0.872886i	$-0.662246\pi$
$17$	−4.02352	−0.975847	−0.487924	+	0.872886i	$0.662246\pi$
$18$	0	0
$19$	−5.92833	−1.36005	−0.680026	−	0.733188i	$-0.738032\pi$
$19$	−5.92833	−1.36005	−0.680026	+	0.733188i	$0.738032\pi$
$20$	0	0
$21$	2.16679	0.472832
$22$	0	0
$23$	4.27435	0.891263	0.445631	−	0.895217i	$-0.352979\pi$
$23$	4.27435	0.891263	0.445631	+	0.895217i	$0.352979\pi$
$24$	0	0
$25$	−3.39656	−0.679312
$26$	0	0
$27$	2.82773	0.544197
$28$	0	0
$29$	−9.21632	−1.71143	−0.855714	−	0.517449i	$-0.826882\pi$
$29$	−9.21632	−1.71143	−0.855714	+	0.517449i	$0.826882\pi$
$30$	0	0
$31$	−0.386596	−0.0694346	−0.0347173	−	0.999397i	$-0.511053\pi$
$31$	−0.386596	−0.0694346	−0.0347173	+	0.999397i	$0.511053\pi$
$32$	0	0
$33$	2.16679	0.377189
$34$	0	0
$35$	1.26627	0.214039
$36$	0	0
$37$	11.2499	1.84947	0.924737	−	0.380606i	$-0.124285\pi$
$37$	11.2499	1.84947	0.924737	+	0.380606i	$0.124285\pi$
$38$	0	0
$39$	2.16679	0.346964
$40$	0	0
$41$	−7.02502	−1.09712	−0.548562	−	0.836110i	$-0.684825\pi$
$41$	−7.02502	−1.09712	−0.548562	+	0.836110i	$0.684825\pi$
$42$	0	0
$43$	−3.45783	−0.527314	−0.263657	−	0.964616i	$-0.584929\pi$
$43$	−3.45783	−0.527314	−0.263657	+	0.964616i	$0.584929\pi$
$44$	0	0
$45$	−2.14628	−0.319949
$46$	0	0
$47$	−9.82960	−1.43379	−0.716897	−	0.697179i	$-0.754438\pi$
$47$	−9.82960	−1.43379	−0.716897	+	0.697179i	$0.754438\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.71811	1.22078
$52$	0	0
$53$	1.42216	0.195349	0.0976743	−	0.995218i	$-0.468860\pi$
$53$	1.42216	0.195349	0.0976743	+	0.995218i	$0.468860\pi$
$54$	0	0
$55$	1.26627	0.170744
$56$	0	0
$57$	12.8454	1.70142
$58$	0	0
$59$	−11.6029	−1.51057	−0.755286	−	0.655396i	$-0.772502\pi$
$59$	−11.6029	−1.51057	−0.755286	+	0.655396i	$0.772502\pi$
$60$	0	0
$61$	10.3432	1.32431	0.662153	−	0.749369i	$-0.269643\pi$
$61$	10.3432	1.32431	0.662153	+	0.749369i	$0.269643\pi$
$62$	0	0
$63$	−1.69497	−0.213546
$64$	0	0
$65$	1.26627	0.157062
$66$	0	0
$67$	−10.6126	−1.29654	−0.648270	−	0.761411i	$-0.724507\pi$
$67$	−10.6126	−1.29654	−0.648270	+	0.761411i	$0.724507\pi$
$68$	0	0
$69$	−9.26160	−1.11497
$70$	0	0
$71$	−0.110620	−0.0131282	−0.00656408	−	0.999978i	$-0.502089\pi$
$71$	−0.110620	−0.0131282	−0.00656408	+	0.999978i	$0.502089\pi$
$72$	0	0
$73$	−3.03553	−0.355282	−0.177641	−	0.984095i	$-0.556847\pi$
$73$	−3.03553	−0.355282	−0.177641	+	0.984095i	$0.556847\pi$
$74$	0	0
$75$	7.35962	0.849816
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−0.962094	−0.108244	−0.0541220	−	0.998534i	$-0.517236\pi$
$79$	−0.962094	−0.108244	−0.0541220	+	0.998534i	$0.517236\pi$
$80$	0	0
$81$	−11.2120	−1.24578
$82$	0	0
$83$	4.40921	0.483974	0.241987	−	0.970279i	$-0.422201\pi$
$83$	4.40921	0.483974	0.241987	+	0.970279i	$0.422201\pi$
$84$	0	0
$85$	5.09487	0.552616
$86$	0	0
$87$	19.9698	2.14099
$88$	0	0
$89$	−8.23230	−0.872622	−0.436311	−	0.899796i	$-0.643715\pi$
$89$	−8.23230	−0.872622	−0.436311	+	0.899796i	$0.643715\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0.837670	0.0868624
$94$	0	0
$95$	7.50687	0.770189
$96$	0	0
$97$	3.14917	0.319750	0.159875	−	0.987137i	$-0.448891\pi$
$97$	3.14917	0.319750	0.159875	+	0.987137i	$0.448891\pi$
$98$	0	0
$99$	−1.69497	−0.170350
$100$	0	0
$101$	3.12028	0.310479	0.155240	−	0.987877i	$-0.450385\pi$
$101$	3.12028	0.310479	0.155240	+	0.987877i	$0.450385\pi$
$102$	0	0
$103$	−18.0568	−1.77919	−0.889595	−	0.456751i	$-0.849013\pi$
$103$	−18.0568	−1.77919	−0.889595	+	0.456751i	$0.849013\pi$
$104$	0	0
$105$	−2.74374	−0.267761
$106$	0	0
$107$	5.41959	0.523931	0.261966	−	0.965077i	$-0.415629\pi$
$107$	5.41959	0.523931	0.261966	+	0.965077i	$0.415629\pi$
$108$	0	0
$109$	15.3316	1.46850	0.734249	−	0.678880i	$-0.237535\pi$
$109$	15.3316	1.46850	0.734249	+	0.678880i	$0.237535\pi$
$110$	0	0
$111$	−24.3762	−2.31368
$112$	0	0
$113$	−6.18191	−0.581545	−0.290773	−	0.956792i	$-0.593912\pi$
$113$	−6.18191	−0.581545	−0.290773	+	0.956792i	$0.593912\pi$
$114$	0	0
$115$	−5.41248	−0.504716
$116$	0	0
$117$	−1.69497	−0.156700
$118$	0	0
$119$	4.02352	0.368836
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.2217	1.37250
$124$	0	0
$125$	10.6323	0.950983
$126$	0	0
$127$	−16.3527	−1.45107	−0.725535	−	0.688185i	$-0.758408\pi$
$127$	−16.3527	−1.45107	−0.725535	+	0.688185i	$0.758408\pi$
$128$	0	0
$129$	7.49239	0.659668
$130$	0	0
$131$	9.18222	0.802254	0.401127	−	0.916022i	$-0.368619\pi$
$131$	9.18222	0.802254	0.401127	+	0.916022i	$0.368619\pi$
$132$	0	0
$133$	5.92833	0.514052
$134$	0	0
$135$	−3.58067	−0.308175
$136$	0	0
$137$	9.38223	0.801578	0.400789	−	0.916170i	$-0.368736\pi$
$137$	9.38223	0.801578	0.400789	+	0.916170i	$0.368736\pi$
$138$	0	0
$139$	−2.32989	−0.197619	−0.0988094	−	0.995106i	$-0.531503\pi$
$139$	−2.32989	−0.197619	−0.0988094	+	0.995106i	$0.531503\pi$
$140$	0	0
$141$	21.2987	1.79367
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	11.6704	0.969170
$146$	0	0
$147$	−2.16679	−0.178714
$148$	0	0
$149$	11.4946	0.941676	0.470838	−	0.882220i	$-0.343952\pi$
$149$	11.4946	0.941676	0.470838	+	0.882220i	$0.343952\pi$
$150$	0	0
$151$	7.72516	0.628665	0.314332	−	0.949313i	$-0.398219\pi$
$151$	7.72516	0.628665	0.314332	+	0.949313i	$0.398219\pi$
$152$	0	0
$153$	−6.81973	−0.551342
$154$	0	0
$155$	0.489535	0.0393204
$156$	0	0
$157$	8.74921	0.698263	0.349131	−	0.937074i	$-0.386477\pi$
$157$	8.74921	0.698263	0.349131	+	0.937074i	$0.386477\pi$
$158$	0	0
$159$	−3.08151	−0.244380
$160$	0	0
$161$	−4.27435	−0.336866
$162$	0	0
$163$	−22.5300	−1.76468	−0.882342	−	0.470609i	$-0.844034\pi$
$163$	−22.5300	−1.76468	−0.882342	+	0.470609i	$0.844034\pi$
$164$	0	0
$165$	−2.74374	−0.213600
$166$	0	0
$167$	2.10426	0.162832	0.0814162	−	0.996680i	$-0.474056\pi$
$167$	2.10426	0.162832	0.0814162	+	0.996680i	$0.474056\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−10.0483	−0.768414
$172$	0	0
$173$	19.3806	1.47348	0.736741	−	0.676176i	$-0.236364\pi$
$173$	19.3806	1.47348	0.736741	+	0.676176i	$0.236364\pi$
$174$	0	0
$175$	3.39656	0.256756
$176$	0	0
$177$	25.1410	1.88972
$178$	0	0
$179$	2.59082	0.193647	0.0968233	−	0.995302i	$-0.469132\pi$
$179$	2.59082	0.193647	0.0968233	+	0.995302i	$0.469132\pi$
$180$	0	0
$181$	−5.89964	−0.438517	−0.219258	−	0.975667i	$-0.570364\pi$
$181$	−5.89964	−0.438517	−0.219258	+	0.975667i	$0.570364\pi$
$182$	0	0
$183$	−22.4114	−1.65670
$184$	0	0
$185$	−14.2454	−1.04735
$186$	0	0
$187$	4.02352	0.294229
$188$	0	0
$189$	−2.82773	−0.205687
$190$	0	0
$191$	8.43059	0.610016	0.305008	−	0.952350i	$-0.401341\pi$
$191$	8.43059	0.610016	0.305008	+	0.952350i	$0.401341\pi$
$192$	0	0
$193$	12.1056	0.871380	0.435690	−	0.900097i	$-0.356504\pi$
$193$	12.1056	0.871380	0.435690	+	0.900097i	$0.356504\pi$
$194$	0	0
$195$	−2.74374	−0.196483
$196$	0	0
$197$	17.0909	1.21768	0.608838	−	0.793295i	$-0.291636\pi$
$197$	17.0909	1.21768	0.608838	+	0.793295i	$0.291636\pi$
$198$	0	0
$199$	10.3031	0.730368	0.365184	−	0.930935i	$-0.381006\pi$
$199$	10.3031	0.730368	0.365184	+	0.930935i	$0.381006\pi$
$200$	0	0
$201$	22.9953	1.62196
$202$	0	0
$203$	9.21632	0.646859
$204$	0	0
$205$	8.89558	0.621294
$206$	0	0
$207$	7.24487	0.503553
$208$	0	0
$209$	5.92833	0.410071
$210$	0	0
$211$	20.4847	1.41023	0.705114	−	0.709094i	$-0.250896\pi$
$211$	20.4847	1.41023	0.705114	+	0.709094i	$0.250896\pi$
$212$	0	0
$213$	0.239690	0.0164233
$214$	0	0
$215$	4.37855	0.298615
$216$	0	0
$217$	0.386596	0.0262438
$218$	0	0
$219$	6.57735	0.444456
$220$	0	0
$221$	4.02352	0.270651
$222$	0	0
$223$	−6.52746	−0.437111	−0.218555	−	0.975825i	$-0.570134\pi$
$223$	−6.52746	−0.437111	−0.218555	+	0.975825i	$0.570134\pi$
$224$	0	0
$225$	−5.75705	−0.383803
$226$	0	0
$227$	1.19558	0.0793538	0.0396769	−	0.999213i	$-0.487367\pi$
$227$	1.19558	0.0793538	0.0396769	+	0.999213i	$0.487367\pi$
$228$	0	0
$229$	19.3904	1.28135	0.640676	−	0.767812i	$-0.278654\pi$
$229$	19.3904	1.28135	0.640676	+	0.767812i	$0.278654\pi$
$230$	0	0
$231$	−2.16679	−0.142564
$232$	0	0
$233$	24.1465	1.58189	0.790945	−	0.611888i	$-0.209589\pi$
$233$	24.1465	1.58189	0.790945	+	0.611888i	$0.209589\pi$
$234$	0	0
$235$	12.4469	0.811949
$236$	0	0
$237$	2.08465	0.135413
$238$	0	0
$239$	−17.5581	−1.13574	−0.567868	−	0.823119i	$-0.692232\pi$
$239$	−17.5581	−1.13574	−0.567868	+	0.823119i	$0.692232\pi$
$240$	0	0
$241$	−14.2266	−0.916418	−0.458209	−	0.888844i	$-0.651509\pi$
$241$	−14.2266	−0.916418	−0.458209	+	0.888844i	$0.651509\pi$
$242$	0	0
$243$	15.8108	1.01426
$244$	0	0
$245$	−1.26627	−0.0808991
$246$	0	0
$247$	5.92833	0.377211
$248$	0	0
$249$	−9.55383	−0.605449
$250$	0	0
$251$	−14.1949	−0.895977	−0.447989	−	0.894039i	$-0.647859\pi$
$251$	−14.1949	−0.895977	−0.447989	+	0.894039i	$0.647859\pi$
$252$	0	0
$253$	−4.27435	−0.268726
$254$	0	0
$255$	−11.0395	−0.691320
$256$	0	0
$257$	−4.71880	−0.294351	−0.147175	−	0.989110i	$-0.547018\pi$
$257$	−4.71880	−0.294351	−0.147175	+	0.989110i	$0.547018\pi$
$258$	0	0
$259$	−11.2499	−0.699036
$260$	0	0
$261$	−15.6213	−0.966937
$262$	0	0
$263$	13.7067	0.845189	0.422595	−	0.906319i	$-0.361119\pi$
$263$	13.7067	0.845189	0.422595	+	0.906319i	$0.361119\pi$
$264$	0	0
$265$	−1.80084	−0.110625
$266$	0	0
$267$	17.8376	1.09165
$268$	0	0
$269$	−8.63780	−0.526656	−0.263328	−	0.964706i	$-0.584820\pi$
$269$	−8.63780	−0.526656	−0.263328	+	0.964706i	$0.584820\pi$
$270$	0	0
$271$	30.2050	1.83482	0.917411	−	0.397941i	$-0.130275\pi$
$271$	30.2050	1.83482	0.917411	+	0.397941i	$0.130275\pi$
$272$	0	0
$273$	−2.16679	−0.131140
$274$	0	0
$275$	3.39656	0.204820
$276$	0	0
$277$	13.4586	0.808649	0.404325	−	0.914616i	$-0.367507\pi$
$277$	13.4586	0.808649	0.404325	+	0.914616i	$0.367507\pi$
$278$	0	0
$279$	−0.655266	−0.0392298
$280$	0	0
$281$	−2.55064	−0.152158	−0.0760791	−	0.997102i	$-0.524240\pi$
$281$	−2.55064	−0.152158	−0.0760791	+	0.997102i	$0.524240\pi$
$282$	0	0
$283$	−9.56393	−0.568517	−0.284258	−	0.958748i	$-0.591747\pi$
$283$	−9.56393	−0.568517	−0.284258	+	0.958748i	$0.591747\pi$
$284$	0	0
$285$	−16.2658	−0.963503
$286$	0	0
$287$	7.02502	0.414674
$288$	0	0
$289$	−0.811283	−0.0477225
$290$	0	0
$291$	−6.82358	−0.400005
$292$	0	0
$293$	−26.4263	−1.54384	−0.771920	−	0.635720i	$-0.780703\pi$
$293$	−26.4263	−1.54384	−0.771920	+	0.635720i	$0.780703\pi$
$294$	0	0
$295$	14.6924	0.855427
$296$	0	0
$297$	−2.82773	−0.164082
$298$	0	0
$299$	−4.27435	−0.247192
$300$	0	0
$301$	3.45783	0.199306
$302$	0	0
$303$	−6.76098	−0.388408
$304$	0	0
$305$	−13.0972	−0.749945
$306$	0	0
$307$	−15.8370	−0.903865	−0.451932	−	0.892052i	$-0.649265\pi$
$307$	−15.8370	−0.903865	−0.451932	+	0.892052i	$0.649265\pi$
$308$	0	0
$309$	39.1252	2.22576
$310$	0	0
$311$	23.3430	1.32366	0.661830	−	0.749654i	$-0.269780\pi$
$311$	23.3430	1.32366	0.661830	+	0.749654i	$0.269780\pi$
$312$	0	0
$313$	26.1084	1.47573	0.737866	−	0.674947i	$-0.235834\pi$
$313$	26.1084	1.47573	0.737866	+	0.674947i	$0.235834\pi$
$314$	0	0
$315$	2.14628	0.120929
$316$	0	0
$317$	−0.616309	−0.0346154	−0.0173077	−	0.999850i	$-0.505509\pi$
$317$	−0.616309	−0.0346154	−0.0173077	+	0.999850i	$0.505509\pi$
$318$	0	0
$319$	9.21632	0.516015
$320$	0	0
$321$	−11.7431	−0.655435
$322$	0	0
$323$	23.8528	1.32720
$324$	0	0
$325$	3.39656	0.188407
$326$	0	0
$327$	−33.2202	−1.83708
$328$	0	0
$329$	9.82960	0.541924
$330$	0	0
$331$	17.6186	0.968407	0.484203	−	0.874955i	$-0.339109\pi$
$331$	17.6186	0.968407	0.484203	+	0.874955i	$0.339109\pi$
$332$	0	0
$333$	19.0682	1.04493
$334$	0	0
$335$	13.4385	0.734222
$336$	0	0
$337$	−4.47690	−0.243872	−0.121936	−	0.992538i	$-0.538910\pi$
$337$	−4.47690	−0.243872	−0.121936	+	0.992538i	$0.538910\pi$
$338$	0	0
$339$	13.3949	0.727510
$340$	0	0
$341$	0.386596	0.0209353
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	11.7277	0.631398
$346$	0	0
$347$	−19.2072	−1.03110	−0.515549	−	0.856860i	$-0.672412\pi$
$347$	−19.2072	−1.03110	−0.515549	+	0.856860i	$0.672412\pi$
$348$	0	0
$349$	24.7928	1.32713	0.663564	−	0.748119i	$-0.269043\pi$
$349$	24.7928	1.32713	0.663564	+	0.748119i	$0.269043\pi$
$350$	0	0
$351$	−2.82773	−0.150933
$352$	0	0
$353$	−27.9445	−1.48733	−0.743667	−	0.668550i	$-0.766915\pi$
$353$	−27.9445	−1.48733	−0.743667	+	0.668550i	$0.766915\pi$
$354$	0	0
$355$	0.140075	0.00743439
$356$	0	0
$357$	−8.71811	−0.461411
$358$	0	0
$359$	−1.25709	−0.0663468	−0.0331734	−	0.999450i	$-0.510561\pi$
$359$	−1.25709	−0.0663468	−0.0331734	+	0.999450i	$0.510561\pi$
$360$	0	0
$361$	16.1451	0.849744
$362$	0	0
$363$	−2.16679	−0.113727
$364$	0	0
$365$	3.84380	0.201194
$366$	0	0
$367$	0.580457	0.0302996	0.0151498	−	0.999885i	$-0.495177\pi$
$367$	0.580457	0.0302996	0.0151498	+	0.999885i	$0.495177\pi$
$368$	0	0
$369$	−11.9072	−0.619862
$370$	0	0
$371$	−1.42216	−0.0738348
$372$	0	0
$373$	11.3570	0.588045	0.294023	−	0.955798i	$-0.405006\pi$
$373$	11.3570	0.588045	0.294023	+	0.955798i	$0.405006\pi$
$374$	0	0
$375$	−23.0380	−1.18968
$376$	0	0
$377$	9.21632	0.474665
$378$	0	0
$379$	28.8713	1.48302	0.741509	−	0.670943i	$-0.234111\pi$
$379$	28.8713	1.48302	0.741509	+	0.670943i	$0.234111\pi$
$380$	0	0
$381$	35.4329	1.81528
$382$	0	0
$383$	−36.9921	−1.89021	−0.945103	−	0.326772i	$-0.894039\pi$
$383$	−36.9921	−1.89021	−0.945103	+	0.326772i	$0.894039\pi$
$384$	0	0
$385$	−1.26627	−0.0645351
$386$	0	0
$387$	−5.86091	−0.297927
$388$	0	0
$389$	−17.3019	−0.877241	−0.438620	−	0.898672i	$-0.644533\pi$
$389$	−17.3019	−0.877241	−0.438620	+	0.898672i	$0.644533\pi$
$390$	0	0
$391$	−17.1979	−0.869736
$392$	0	0
$393$	−19.8959	−1.00362
$394$	0	0
$395$	1.21827	0.0612979
$396$	0	0
$397$	−37.6443	−1.88932	−0.944658	−	0.328057i	$-0.893606\pi$
$397$	−37.6443	−1.88932	−0.944658	+	0.328057i	$0.893606\pi$
$398$	0	0
$399$	−12.8454	−0.643076
$400$	0	0
$401$	14.3907	0.718637	0.359319	−	0.933215i	$-0.383009\pi$
$401$	14.3907	0.718637	0.359319	+	0.933215i	$0.383009\pi$
$402$	0	0
$403$	0.386596	0.0192577
$404$	0	0
$405$	14.1974	0.705475
$406$	0	0
$407$	−11.2499	−0.557637
$408$	0	0
$409$	15.4941	0.766134	0.383067	−	0.923720i	$-0.374868\pi$
$409$	15.4941	0.766134	0.383067	+	0.923720i	$0.374868\pi$
$410$	0	0
$411$	−20.3293	−1.00277
$412$	0	0
$413$	11.6029	0.570942
$414$	0	0
$415$	−5.58326	−0.274071
$416$	0	0
$417$	5.04838	0.247220
$418$	0	0
$419$	−5.97766	−0.292028	−0.146014	−	0.989283i	$-0.546644\pi$
$419$	−5.97766	−0.292028	−0.146014	+	0.989283i	$0.546644\pi$
$420$	0	0
$421$	8.40791	0.409776	0.204888	−	0.978785i	$-0.434317\pi$
$421$	8.40791	0.409776	0.204888	+	0.978785i	$0.434317\pi$
$422$	0	0
$423$	−16.6608	−0.810077
$424$	0	0
$425$	13.6661	0.662904
$426$	0	0
$427$	−10.3432	−0.500540
$428$	0	0
$429$	−2.16679	−0.104613
$430$	0	0
$431$	−0.683402	−0.0329183	−0.0164592	−	0.999865i	$-0.505239\pi$
$431$	−0.683402	−0.0329183	−0.0164592	+	0.999865i	$0.505239\pi$
$432$	0	0
$433$	−25.1528	−1.20877	−0.604384	−	0.796693i	$-0.706581\pi$
$433$	−25.1528	−1.20877	−0.604384	+	0.796693i	$0.706581\pi$
$434$	0	0
$435$	−25.2872	−1.21243
$436$	0	0
$437$	−25.3398	−1.21216
$438$	0	0
$439$	29.8307	1.42374	0.711872	−	0.702309i	$-0.247848\pi$
$439$	29.8307	1.42374	0.711872	+	0.702309i	$0.247848\pi$
$440$	0	0
$441$	1.69497	0.0807126
$442$	0	0
$443$	−19.1914	−0.911811	−0.455905	−	0.890028i	$-0.650684\pi$
$443$	−19.1914	−0.911811	−0.455905	+	0.890028i	$0.650684\pi$
$444$	0	0
$445$	10.4243	0.494160
$446$	0	0
$447$	−24.9064	−1.17803
$448$	0	0
$449$	−13.8690	−0.654518	−0.327259	−	0.944935i	$-0.606125\pi$
$449$	−13.8690	−0.654518	−0.327259	+	0.944935i	$0.606125\pi$
$450$	0	0
$451$	7.02502	0.330795
$452$	0	0
$453$	−16.7388	−0.786457
$454$	0	0
$455$	−1.26627	−0.0593637
$456$	0	0
$457$	8.93340	0.417887	0.208943	−	0.977928i	$-0.432998\pi$
$457$	8.93340	0.417887	0.208943	+	0.977928i	$0.432998\pi$
$458$	0	0
$459$	−11.3774	−0.531053
$460$	0	0
$461$	−5.08117	−0.236653	−0.118327	−	0.992975i	$-0.537753\pi$
$461$	−5.08117	−0.236653	−0.118327	+	0.992975i	$0.537753\pi$
$462$	0	0
$463$	5.00344	0.232530	0.116265	−	0.993218i	$-0.462908\pi$
$463$	5.00344	0.232530	0.116265	+	0.993218i	$0.462908\pi$
$464$	0	0
$465$	−1.06072	−0.0491896
$466$	0	0
$467$	14.0959	0.652282	0.326141	−	0.945321i	$-0.394252\pi$
$467$	14.0959	0.652282	0.326141	+	0.945321i	$0.394252\pi$
$468$	0	0
$469$	10.6126	0.490046
$470$	0	0
$471$	−18.9577	−0.873523
$472$	0	0
$473$	3.45783	0.158991
$474$	0	0
$475$	20.1359	0.923900
$476$	0	0
$477$	2.41051	0.110370
$478$	0	0
$479$	−19.5024	−0.891087	−0.445543	−	0.895260i	$-0.646989\pi$
$479$	−19.5024	−0.891087	−0.445543	+	0.895260i	$0.646989\pi$
$480$	0	0
$481$	−11.2499	−0.512952
$482$	0	0
$483$	9.26160	0.421417
$484$	0	0
$485$	−3.98770	−0.181072
$486$	0	0
$487$	−22.4696	−1.01819	−0.509096	−	0.860710i	$-0.670020\pi$
$487$	−22.4696	−1.01819	−0.509096	+	0.860710i	$0.670020\pi$
$488$	0	0
$489$	48.8176	2.20761
$490$	0	0
$491$	3.66861	0.165562	0.0827810	−	0.996568i	$-0.473620\pi$
$491$	3.66861	0.165562	0.0827810	+	0.996568i	$0.473620\pi$
$492$	0	0
$493$	37.0821	1.67009
$494$	0	0
$495$	2.14628	0.0964683
$496$	0	0
$497$	0.110620	0.00496198
$498$	0	0
$499$	18.5223	0.829173	0.414586	−	0.910010i	$-0.363926\pi$
$499$	18.5223	0.829173	0.414586	+	0.910010i	$0.363926\pi$
$500$	0	0
$501$	−4.55948	−0.203703
$502$	0	0
$503$	−7.36591	−0.328430	−0.164215	−	0.986425i	$-0.552509\pi$
$503$	−7.36591	−0.328430	−0.164215	+	0.986425i	$0.552509\pi$
$504$	0	0
$505$	−3.95112	−0.175822
$506$	0	0
$507$	−2.16679	−0.0962304
$508$	0	0
$509$	25.7009	1.13917	0.569587	−	0.821931i	$-0.307103\pi$
$509$	25.7009	1.13917	0.569587	+	0.821931i	$0.307103\pi$
$510$	0	0
$511$	3.03553	0.134284
$512$	0	0
$513$	−16.7637	−0.740137
$514$	0	0
$515$	22.8648	1.00754
$516$	0	0
$517$	9.82960	0.432305
$518$	0	0
$519$	−41.9937	−1.84332
$520$	0	0
$521$	−20.4838	−0.897410	−0.448705	−	0.893680i	$-0.648115\pi$
$521$	−20.4838	−0.897410	−0.448705	+	0.893680i	$0.648115\pi$
$522$	0	0
$523$	24.4451	1.06891	0.534455	−	0.845197i	$-0.320517\pi$
$523$	24.4451	1.06891	0.534455	+	0.845197i	$0.320517\pi$
$524$	0	0
$525$	−7.35962	−0.321200
$526$	0	0
$527$	1.55548	0.0677576
$528$	0	0
$529$	−4.72996	−0.205650
$530$	0	0
$531$	−19.6665	−0.853455
$532$	0	0
$533$	7.02502	0.304287
$534$	0	0
$535$	−6.86266	−0.296699
$536$	0	0
$537$	−5.61375	−0.242251
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	10.6128	0.456280	0.228140	−	0.973628i	$-0.426736\pi$
$541$	10.6128	0.456280	0.228140	+	0.973628i	$0.426736\pi$
$542$	0	0
$543$	12.7833	0.548582
$544$	0	0
$545$	−19.4139	−0.831601
$546$	0	0
$547$	10.8945	0.465815	0.232907	−	0.972499i	$-0.425176\pi$
$547$	10.8945	0.465815	0.232907	+	0.972499i	$0.425176\pi$
$548$	0	0
$549$	17.5313	0.748217
$550$	0	0
$551$	54.6374	2.32763
$552$	0	0
$553$	0.962094	0.0409124
$554$	0	0
$555$	30.8668	1.31022
$556$	0	0
$557$	−8.34509	−0.353593	−0.176796	−	0.984247i	$-0.556573\pi$
$557$	−8.34509	−0.353593	−0.176796	+	0.984247i	$0.556573\pi$
$558$	0	0
$559$	3.45783	0.146251
$560$	0	0
$561$	−8.71811	−0.368079
$562$	0	0
$563$	29.1620	1.22903	0.614517	−	0.788904i	$-0.289351\pi$
$563$	29.1620	1.22903	0.614517	+	0.788904i	$0.289351\pi$
$564$	0	0
$565$	7.82797	0.329325
$566$	0	0
$567$	11.2120	0.470859
$568$	0	0
$569$	−14.2396	−0.596957	−0.298478	−	0.954416i	$-0.596479\pi$
$569$	−14.2396	−0.596957	−0.298478	+	0.954416i	$0.596479\pi$
$570$	0	0
$571$	−24.7058	−1.03390	−0.516952	−	0.856014i	$-0.672934\pi$
$571$	−24.7058	−1.03390	−0.516952	+	0.856014i	$0.672934\pi$
$572$	0	0
$573$	−18.2673	−0.763127
$574$	0	0
$575$	−14.5181	−0.605445
$576$	0	0
$577$	19.7337	0.821523	0.410761	−	0.911743i	$-0.365263\pi$
$577$	19.7337	0.821523	0.410761	+	0.911743i	$0.365263\pi$
$578$	0	0
$579$	−26.2303	−1.09009
$580$	0	0
$581$	−4.40921	−0.182925
$582$	0	0
$583$	−1.42216	−0.0588998
$584$	0	0
$585$	2.14628	0.0887379
$586$	0	0
$587$	−2.03951	−0.0841796	−0.0420898	−	0.999114i	$-0.513402\pi$
$587$	−2.03951	−0.0841796	−0.0420898	+	0.999114i	$0.513402\pi$
$588$	0	0
$589$	2.29187	0.0944348
$590$	0	0
$591$	−37.0323	−1.52331
$592$	0	0
$593$	16.0760	0.660160	0.330080	−	0.943953i	$-0.392924\pi$
$593$	16.0760	0.660160	0.330080	+	0.943953i	$0.392924\pi$
$594$	0	0
$595$	−5.09487	−0.208869
$596$	0	0
$597$	−22.3246	−0.913686
$598$	0	0
$599$	−30.1955	−1.23376	−0.616878	−	0.787059i	$-0.711603\pi$
$599$	−30.1955	−1.23376	−0.616878	+	0.787059i	$0.711603\pi$
$600$	0	0
$601$	−1.64091	−0.0669343	−0.0334671	−	0.999440i	$-0.510655\pi$
$601$	−1.64091	−0.0669343	−0.0334671	+	0.999440i	$0.510655\pi$
$602$	0	0
$603$	−17.9880	−0.732530
$604$	0	0
$605$	−1.26627	−0.0514812
$606$	0	0
$607$	29.7180	1.20622	0.603109	−	0.797659i	$-0.293928\pi$
$607$	29.7180	1.20622	0.603109	+	0.797659i	$0.293928\pi$
$608$	0	0
$609$	−19.9698	−0.809217
$610$	0	0
$611$	9.82960	0.397663
$612$	0	0
$613$	1.65761	0.0669504	0.0334752	−	0.999440i	$-0.489343\pi$
$613$	1.65761	0.0669504	0.0334752	+	0.999440i	$0.489343\pi$
$614$	0	0
$615$	−19.2748	−0.777236
$616$	0	0
$617$	−44.4303	−1.78869	−0.894347	−	0.447373i	$-0.852360\pi$
$617$	−44.4303	−1.78869	−0.894347	+	0.447373i	$0.852360\pi$
$618$	0	0
$619$	−9.75763	−0.392192	−0.196096	−	0.980585i	$-0.562826\pi$
$619$	−9.75763	−0.392192	−0.196096	+	0.980585i	$0.562826\pi$
$620$	0	0
$621$	12.0867	0.485023
$622$	0	0
$623$	8.23230	0.329820
$624$	0	0
$625$	3.51941	0.140776
$626$	0	0
$627$	−12.8454	−0.512997
$628$	0	0
$629$	−45.2643	−1.80480
$630$	0	0
$631$	6.50867	0.259106	0.129553	−	0.991572i	$-0.458646\pi$
$631$	6.50867	0.259106	0.129553	+	0.991572i	$0.458646\pi$
$632$	0	0
$633$	−44.3861	−1.76419
$634$	0	0
$635$	20.7070	0.821732
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−0.187497	−0.00741726
$640$	0	0
$641$	14.3554	0.567005	0.283503	−	0.958971i	$-0.408504\pi$
$641$	14.3554	0.567005	0.283503	+	0.958971i	$0.408504\pi$
$642$	0	0
$643$	−34.7551	−1.37061	−0.685304	−	0.728257i	$-0.740331\pi$
$643$	−34.7551	−1.37061	−0.685304	+	0.728257i	$0.740331\pi$
$644$	0	0
$645$	−9.48739	−0.373566
$646$	0	0
$647$	−19.7842	−0.777797	−0.388899	−	0.921281i	$-0.627144\pi$
$647$	−19.7842	−0.777797	−0.388899	+	0.921281i	$0.627144\pi$
$648$	0	0
$649$	11.6029	0.455454
$650$	0	0
$651$	−0.837670	−0.0328309
$652$	0	0
$653$	3.07113	0.120183	0.0600913	−	0.998193i	$-0.480861\pi$
$653$	3.07113	0.120183	0.0600913	+	0.998193i	$0.480861\pi$
$654$	0	0
$655$	−11.6272	−0.454311
$656$	0	0
$657$	−5.14512	−0.200730
$658$	0	0
$659$	44.4879	1.73300	0.866501	−	0.499176i	$-0.166364\pi$
$659$	44.4879	1.73300	0.866501	+	0.499176i	$0.166364\pi$
$660$	0	0
$661$	−27.1088	−1.05441	−0.527206	−	0.849738i	$-0.676760\pi$
$661$	−27.1088	−1.05441	−0.527206	+	0.849738i	$0.676760\pi$
$662$	0	0
$663$	−8.71811	−0.338583
$664$	0	0
$665$	−7.50687	−0.291104
$666$	0	0
$667$	−39.3938	−1.52533
$668$	0	0
$669$	14.1436	0.546824
$670$	0	0
$671$	−10.3432	−0.399293
$672$	0	0
$673$	12.1972	0.470168	0.235084	−	0.971975i	$-0.424463\pi$
$673$	12.1972	0.470168	0.235084	+	0.971975i	$0.424463\pi$
$674$	0	0
$675$	−9.60456	−0.369680
$676$	0	0
$677$	−33.3082	−1.28014	−0.640070	−	0.768317i	$-0.721095\pi$
$677$	−33.3082	−1.28014	−0.640070	+	0.768317i	$0.721095\pi$
$678$	0	0
$679$	−3.14917	−0.120854
$680$	0	0
$681$	−2.59058	−0.0992712
$682$	0	0
$683$	−30.6220	−1.17172	−0.585859	−	0.810413i	$-0.699243\pi$
$683$	−30.6220	−1.17172	−0.585859	+	0.810413i	$0.699243\pi$
$684$	0	0
$685$	−11.8804	−0.453928
$686$	0	0
$687$	−42.0148	−1.60296
$688$	0	0
$689$	−1.42216	−0.0541799
$690$	0	0
$691$	18.7175	0.712048	0.356024	−	0.934477i	$-0.384132\pi$
$691$	18.7175	0.712048	0.356024	+	0.934477i	$0.384132\pi$
$692$	0	0
$693$	1.69497	0.0643864
$694$	0	0
$695$	2.95027	0.111910
$696$	0	0
$697$	28.2653	1.07063
$698$	0	0
$699$	−52.3203	−1.97894
$700$	0	0
$701$	8.28526	0.312930	0.156465	−	0.987684i	$-0.449990\pi$
$701$	8.28526	0.312930	0.156465	+	0.987684i	$0.449990\pi$
$702$	0	0
$703$	−66.6932	−2.51538
$704$	0	0
$705$	−26.9699	−1.01574
$706$	0	0
$707$	−3.12028	−0.117350
$708$	0	0
$709$	−34.1126	−1.28112	−0.640562	−	0.767906i	$-0.721299\pi$
$709$	−34.1126	−1.28112	−0.640562	+	0.767906i	$0.721299\pi$
$710$	0	0
$711$	−1.63072	−0.0611566
$712$	0	0
$713$	−1.65244	−0.0618845
$714$	0	0
$715$	−1.26627	−0.0473558
$716$	0	0
$717$	38.0446	1.42080
$718$	0	0
$719$	−43.8645	−1.63587	−0.817934	−	0.575312i	$-0.804881\pi$
$719$	−43.8645	−1.63587	−0.817934	+	0.575312i	$0.804881\pi$
$720$	0	0
$721$	18.0568	0.672470
$722$	0	0
$723$	30.8261	1.14643
$724$	0	0
$725$	31.3038	1.16259
$726$	0	0
$727$	−49.5628	−1.83818	−0.919091	−	0.394046i	$-0.871075\pi$
$727$	−49.5628	−1.83818	−0.919091	+	0.394046i	$0.871075\pi$
$728$	0	0
$729$	−0.622653	−0.0230612
$730$	0	0
$731$	13.9127	0.514578
$732$	0	0
$733$	6.94857	0.256651	0.128326	−	0.991732i	$-0.459040\pi$
$733$	6.94857	0.256651	0.128326	+	0.991732i	$0.459040\pi$
$734$	0	0
$735$	2.74374	0.101204
$736$	0	0
$737$	10.6126	0.390921
$738$	0	0
$739$	21.1168	0.776794	0.388397	−	0.921492i	$-0.373029\pi$
$739$	21.1168	0.776794	0.388397	+	0.921492i	$0.373029\pi$
$740$	0	0
$741$	−12.8454	−0.471889
$742$	0	0
$743$	−16.2453	−0.595984	−0.297992	−	0.954568i	$-0.596317\pi$
$743$	−16.2453	−0.595984	−0.297992	+	0.954568i	$0.596317\pi$
$744$	0	0
$745$	−14.5553	−0.533265
$746$	0	0
$747$	7.47347	0.273440
$748$	0	0
$749$	−5.41959	−0.198027
$750$	0	0
$751$	29.3475	1.07091	0.535453	−	0.844565i	$-0.320141\pi$
$751$	29.3475	1.07091	0.535453	+	0.844565i	$0.320141\pi$
$752$	0	0
$753$	30.7574	1.12086
$754$	0	0
$755$	−9.78215	−0.356009
$756$	0	0
$757$	35.9554	1.30682	0.653410	−	0.757004i	$-0.273338\pi$
$757$	35.9554	1.30682	0.653410	+	0.757004i	$0.273338\pi$
$758$	0	0
$759$	9.26160	0.336175
$760$	0	0
$761$	40.5514	1.46999	0.734994	−	0.678074i	$-0.237185\pi$
$761$	40.5514	1.46999	0.734994	+	0.678074i	$0.237185\pi$
$762$	0	0
$763$	−15.3316	−0.555040
$764$	0	0
$765$	8.63562	0.312222
$766$	0	0
$767$	11.6029	0.418957
$768$	0	0
$769$	29.6679	1.06985	0.534926	−	0.844899i	$-0.320340\pi$
$769$	29.6679	1.06985	0.534926	+	0.844899i	$0.320340\pi$
$770$	0	0
$771$	10.2246	0.368232
$772$	0	0
$773$	19.3481	0.695904	0.347952	−	0.937512i	$-0.386877\pi$
$773$	19.3481	0.695904	0.347952	+	0.937512i	$0.386877\pi$
$774$	0	0
$775$	1.31309	0.0471678
$776$	0	0
$777$	24.3762	0.874490
$778$	0	0
$779$	41.6467	1.49215
$780$	0	0
$781$	0.110620	0.00395829
$782$	0	0
$783$	−26.0613	−0.931354
$784$	0	0
$785$	−11.0789	−0.395422
$786$	0	0
$787$	−37.8709	−1.34995	−0.674975	−	0.737840i	$-0.735846\pi$
$787$	−37.8709	−1.34995	−0.674975	+	0.737840i	$0.735846\pi$
$788$	0	0
$789$	−29.6994	−1.05733
$790$	0	0
$791$	6.18191	0.219803
$792$	0	0
$793$	−10.3432	−0.367296
$794$	0	0
$795$	3.90203	0.138391
$796$	0	0
$797$	−52.2540	−1.85093	−0.925465	−	0.378833i	$-0.876325\pi$
$797$	−52.2540	−1.85093	−0.925465	+	0.378833i	$0.876325\pi$
$798$	0	0
$799$	39.5496	1.39916
$800$	0	0
$801$	−13.9535	−0.493021
$802$	0	0
$803$	3.03553	0.107122
$804$	0	0
$805$	5.41248	0.190765
$806$	0	0
$807$	18.7163	0.658844
$808$	0	0
$809$	−19.4286	−0.683073	−0.341537	−	0.939868i	$-0.610947\pi$
$809$	−19.4286	−0.683073	−0.341537	+	0.939868i	$0.610947\pi$
$810$	0	0
$811$	4.28637	0.150515	0.0752573	−	0.997164i	$-0.476022\pi$
$811$	4.28637	0.150515	0.0752573	+	0.997164i	$0.476022\pi$
$812$	0	0
$813$	−65.4478	−2.29535
$814$	0	0
$815$	28.5290	0.999329
$816$	0	0
$817$	20.4992	0.717176
$818$	0	0
$819$	1.69497	0.0592269
$820$	0	0
$821$	23.8955	0.833960	0.416980	−	0.908916i	$-0.363088\pi$
$821$	23.8955	0.833960	0.416980	+	0.908916i	$0.363088\pi$
$822$	0	0
$823$	3.73641	0.130243	0.0651215	−	0.997877i	$-0.479256\pi$
$823$	3.73641	0.130243	0.0651215	+	0.997877i	$0.479256\pi$
$824$	0	0
$825$	−7.35962	−0.256229
$826$	0	0
$827$	−37.5279	−1.30497	−0.652487	−	0.757800i	$-0.726274\pi$
$827$	−37.5279	−1.30497	−0.652487	+	0.757800i	$0.726274\pi$
$828$	0	0
$829$	6.15568	0.213796	0.106898	−	0.994270i	$-0.465908\pi$
$829$	6.15568	0.213796	0.106898	+	0.994270i	$0.465908\pi$
$830$	0	0
$831$	−29.1619	−1.01162
$832$	0	0
$833$	−4.02352	−0.139407
$834$	0	0
$835$	−2.66456	−0.0922110
$836$	0	0
$837$	−1.09319	−0.0377861
$838$	0	0
$839$	−36.8832	−1.27335	−0.636675	−	0.771132i	$-0.719691\pi$
$839$	−36.8832	−1.27335	−0.636675	+	0.771132i	$0.719691\pi$
$840$	0	0
$841$	55.9406	1.92899
$842$	0	0
$843$	5.52669	0.190349
$844$	0	0
$845$	−1.26627	−0.0435610
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	20.7230	0.711212
$850$	0	0
$851$	48.0860	1.64837
$852$	0	0
$853$	−31.9892	−1.09529	−0.547645	−	0.836711i	$-0.684476\pi$
$853$	−31.9892	−1.09529	−0.547645	+	0.836711i	$0.684476\pi$
$854$	0	0
$855$	12.7239	0.435148
$856$	0	0
$857$	14.3815	0.491263	0.245631	−	0.969363i	$-0.421005\pi$
$857$	14.3815	0.491263	0.245631	+	0.969363i	$0.421005\pi$
$858$	0	0
$859$	−40.2207	−1.37231	−0.686156	−	0.727455i	$-0.740703\pi$
$859$	−40.2207	−1.37231	−0.686156	+	0.727455i	$0.740703\pi$
$860$	0	0
$861$	−15.2217	−0.518755
$862$	0	0
$863$	−16.6596	−0.567101	−0.283550	−	0.958957i	$-0.591512\pi$
$863$	−16.6596	−0.567101	−0.283550	+	0.958957i	$0.591512\pi$
$864$	0	0
$865$	−24.5411	−0.834423
$866$	0	0
$867$	1.75788	0.0597006
$868$	0	0
$869$	0.962094	0.0326368
$870$	0	0
$871$	10.6126	0.359595
$872$	0	0
$873$	5.33773	0.180655
$874$	0	0
$875$	−10.6323	−0.359438
$876$	0	0
$877$	−33.1133	−1.11816	−0.559079	−	0.829115i	$-0.688845\pi$
$877$	−33.1133	−1.11816	−0.559079	+	0.829115i	$0.688845\pi$
$878$	0	0
$879$	57.2601	1.93134
$880$	0	0
$881$	52.5883	1.77175	0.885873	−	0.463928i	$-0.153560\pi$
$881$	52.5883	1.77175	0.885873	+	0.463928i	$0.153560\pi$
$882$	0	0
$883$	−8.94639	−0.301070	−0.150535	−	0.988605i	$-0.548100\pi$
$883$	−8.94639	−0.301070	−0.150535	+	0.988605i	$0.548100\pi$
$884$	0	0
$885$	−31.8354	−1.07013
$886$	0	0
$887$	−2.42416	−0.0813954	−0.0406977	−	0.999172i	$-0.512958\pi$
$887$	−2.42416	−0.0813954	−0.0406977	+	0.999172i	$0.512958\pi$
$888$	0	0
$889$	16.3527	0.548453
$890$	0	0
$891$	11.2120	0.375616
$892$	0	0
$893$	58.2732	1.95004
$894$	0	0
$895$	−3.28067	−0.109661
$896$	0	0
$897$	9.26160	0.309236
$898$	0	0
$899$	3.56299	0.118832
$900$	0	0
$901$	−5.72209	−0.190630
$902$	0	0
$903$	−7.49239	−0.249331
$904$	0	0
$905$	7.47054	0.248329
$906$	0	0
$907$	19.0627	0.632967	0.316484	−	0.948598i	$-0.397498\pi$
$907$	19.0627	0.632967	0.316484	+	0.948598i	$0.397498\pi$
$908$	0	0
$909$	5.28876	0.175417
$910$	0	0
$911$	−23.0989	−0.765302	−0.382651	−	0.923893i	$-0.624989\pi$
$911$	−23.0989	−0.765302	−0.382651	+	0.923893i	$0.624989\pi$
$912$	0	0
$913$	−4.40921	−0.145924
$914$	0	0
$915$	28.3789	0.938178
$916$	0	0
$917$	−9.18222	−0.303224
$918$	0	0
$919$	36.8042	1.21406	0.607030	−	0.794679i	$-0.292361\pi$
$919$	36.8042	1.21406	0.607030	+	0.794679i	$0.292361\pi$
$920$	0	0
$921$	34.3154	1.13073
$922$	0	0
$923$	0.110620	0.00364110
$924$	0	0
$925$	−38.2110	−1.25637
$926$	0	0
$927$	−30.6056	−1.00522
$928$	0	0
$929$	22.0856	0.724604	0.362302	−	0.932061i	$-0.381991\pi$
$929$	22.0856	0.724604	0.362302	+	0.932061i	$0.381991\pi$
$930$	0	0
$931$	−5.92833	−0.194293
$932$	0	0
$933$	−50.5793	−1.65589
$934$	0	0
$935$	−5.09487	−0.166620
$936$	0	0
$937$	−43.3523	−1.41626	−0.708129	−	0.706083i	$-0.750461\pi$
$937$	−43.3523	−1.41626	−0.708129	+	0.706083i	$0.750461\pi$
$938$	0	0
$939$	−56.5713	−1.84613
$940$	0	0
$941$	−51.0017	−1.66261	−0.831304	−	0.555818i	$-0.812405\pi$
$941$	−51.0017	−1.66261	−0.831304	+	0.555818i	$0.812405\pi$
$942$	0	0
$943$	−30.0274	−0.977826
$944$	0	0
$945$	3.58067	0.116479
$946$	0	0
$947$	−8.04943	−0.261571	−0.130786	−	0.991411i	$-0.541750\pi$
$947$	−8.04943	−0.261571	−0.130786	+	0.991411i	$0.541750\pi$
$948$	0	0
$949$	3.03553	0.0985375
$950$	0	0
$951$	1.33541	0.0433036
$952$	0	0
$953$	−39.4760	−1.27875	−0.639377	−	0.768894i	$-0.720807\pi$
$953$	−39.4760	−1.27875	−0.639377	+	0.768894i	$0.720807\pi$
$954$	0	0
$955$	−10.6754	−0.345448
$956$	0	0
$957$	−19.9698	−0.645532
$958$	0	0
$959$	−9.38223	−0.302968
$960$	0	0
$961$	−30.8505	−0.995179
$962$	0	0
$963$	9.18601	0.296015
$964$	0	0
$965$	−15.3290	−0.493457
$966$	0	0
$967$	26.8622	0.863829	0.431914	−	0.901915i	$-0.357838\pi$
$967$	26.8622	0.863829	0.431914	+	0.901915i	$0.357838\pi$
$968$	0	0
$969$	−51.6839	−1.66032
$970$	0	0
$971$	−45.4602	−1.45889	−0.729443	−	0.684041i	$-0.760221\pi$
$971$	−45.4602	−1.45889	−0.729443	+	0.684041i	$0.760221\pi$
$972$	0	0
$973$	2.32989	0.0746929
$974$	0	0
$975$	−7.35962	−0.235696
$976$	0	0
$977$	11.8298	0.378468	0.189234	−	0.981932i	$-0.439400\pi$
$977$	11.8298	0.378468	0.189234	+	0.981932i	$0.439400\pi$
$978$	0	0
$979$	8.23230	0.263105
$980$	0	0
$981$	25.9865	0.829684
$982$	0	0
$983$	11.8826	0.378995	0.189497	−	0.981881i	$-0.439314\pi$
$983$	11.8826	0.378995	0.189497	+	0.981881i	$0.439314\pi$
$984$	0	0
$985$	−21.6417	−0.689562
$986$	0	0
$987$	−21.2987	−0.677944
$988$	0	0
$989$	−14.7800	−0.469976
$990$	0	0
$991$	40.0821	1.27325	0.636625	−	0.771173i	$-0.280330\pi$
$991$	40.0821	1.27325	0.636625	+	0.771173i	$0.280330\pi$
$992$	0	0
$993$	−38.1758	−1.21147
$994$	0	0
$995$	−13.0465	−0.413602
$996$	0	0
$997$	14.1962	0.449599	0.224800	−	0.974405i	$-0.427827\pi$
$997$	14.1962	0.449599	0.224800	+	0.974405i	$0.427827\pi$
$998$	0	0
$999$	31.8117	1.00648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.j.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.j.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-2.16679 q^{3} -1.26627 q^{5} -1.00000 q^{7} +1.69497 q^{9} +O(q^{10})\)	\(q-2.16679 q^{3} -1.26627 q^{5} -1.00000 q^{7} +1.69497 q^{9} -1.00000 q^{11} -1.00000 q^{13} +2.74374 q^{15} -4.02352 q^{17} -5.92833 q^{19} +2.16679 q^{21} +4.27435 q^{23} -3.39656 q^{25} +2.82773 q^{27} -9.21632 q^{29} -0.386596 q^{31} +2.16679 q^{33} +1.26627 q^{35} +11.2499 q^{37} +2.16679 q^{39} -7.02502 q^{41} -3.45783 q^{43} -2.14628 q^{45} -9.82960 q^{47} +1.00000 q^{49} +8.71811 q^{51} +1.42216 q^{53} +1.26627 q^{55} +12.8454 q^{57} -11.6029 q^{59} +10.3432 q^{61} -1.69497 q^{63} +1.26627 q^{65} -10.6126 q^{67} -9.26160 q^{69} -0.110620 q^{71} -3.03553 q^{73} +7.35962 q^{75} +1.00000 q^{77} -0.962094 q^{79} -11.2120 q^{81} +4.40921 q^{83} +5.09487 q^{85} +19.9698 q^{87} -8.23230 q^{89} +1.00000 q^{91} +0.837670 q^{93} +7.50687 q^{95} +3.14917 q^{97} -1.69497 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 2 q^{5} - 10 q^{7} + 10 q^{9}+O(q^{10}) \) 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

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Representations

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Database

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