LMFDB - Embedded newform 4004.2.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.6
Root			$0.418919$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.418919 q^{3} +0.0566209 q^{5} -1.00000 q^{7} -2.82451 q^{9} +O(q^{10})$	$q+0.418919 q^{3} +0.0566209 q^{5} -1.00000 q^{7} -2.82451 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.0237196 q^{15} +1.81578 q^{17} -6.32849 q^{19} -0.418919 q^{21} +9.33558 q^{23} -4.99679 q^{25} -2.43999 q^{27} +2.91553 q^{29} -5.95686 q^{31} +0.418919 q^{33} -0.0566209 q^{35} +10.2424 q^{37} +0.418919 q^{39} +4.73435 q^{41} -4.59450 q^{43} -0.159926 q^{45} +9.33558 q^{47} +1.00000 q^{49} +0.760664 q^{51} +3.56915 q^{53} +0.0566209 q^{55} -2.65112 q^{57} +5.04133 q^{59} -5.95989 q^{61} +2.82451 q^{63} +0.0566209 q^{65} +4.68606 q^{67} +3.91085 q^{69} -1.73501 q^{71} -3.12666 q^{73} -2.09325 q^{75} -1.00000 q^{77} -1.44270 q^{79} +7.45136 q^{81} +8.81091 q^{83} +0.102811 q^{85} +1.22137 q^{87} +3.69954 q^{89} -1.00000 q^{91} -2.49544 q^{93} -0.358325 q^{95} -12.6141 q^{97} -2.82451 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9}+O(q^{10}) $ 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9	$ 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9} + 10 q^{11} + 10 q^{13} + 3 q^{15} + 3 q^{17} + 6 q^{19} - q^{21} + 4 q^{23} + 22 q^{25} - 5 q^{27} + 10 q^{29} - q^{31} + q^{33} - 4 q^{35} + 20 q^{37} + q^{39} + 8 q^{45} + 4 q^{47} + 10 q^{49} + 11 q^{51} - 5 q^{53} + 4 q^{55} + 16 q^{57} + 11 q^{59} + 12 q^{61} - 17 q^{63} + 4 q^{65} - 2 q^{67} + 10 q^{69} + 28 q^{71} + 11 q^{73} - 6 q^{75} - 10 q^{77} - 10 q^{79} + 46 q^{81} + 7 q^{83} + 33 q^{85} - 47 q^{87} + 30 q^{89} - 10 q^{91} + 41 q^{93} - 2 q^{95} + 55 q^{97} + 17 q^{99}+O(q^{100}) $ 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9 + 10 * q^11 + 10 * q^13 + 3 * q^15 + 3 * q^17 + 6 * q^19 - q^21 + 4 * q^23 + 22 * q^25 - 5 * q^27 + 10 * q^29 - q^31 + q^33 - 4 * q^35 + 20 * q^37 + q^39 + 8 * q^45 + 4 * q^47 + 10 * q^49 + 11 * q^51 - 5 * q^53 + 4 * q^55 + 16 * q^57 + 11 * q^59 + 12 * q^61 - 17 * q^63 + 4 * q^65 - 2 * q^67 + 10 * q^69 + 28 * q^71 + 11 * q^73 - 6 * q^75 - 10 * q^77 - 10 * q^79 + 46 * q^81 + 7 * q^83 + 33 * q^85 - 47 * q^87 + 30 * q^89 - 10 * q^91 + 41 * q^93 - 2 * q^95 + 55 * q^97 + 17 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.418919	0.241863	0.120931	−	0.992661i	$-0.461412\pi$
$3$	0.418919	0.241863	0.120931	+	0.992661i	$0.461412\pi$
$4$	0	0
$5$	0.0566209	0.0253216	0.0126608	−	0.999920i	$-0.495970\pi$
$5$	0.0566209	0.0253216	0.0126608	+	0.999920i	$0.495970\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.82451	−0.941502
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.0237196	0.00612436
$16$	0	0
$17$	1.81578	0.440391	0.220196	−	0.975456i	$-0.429330\pi$
$17$	1.81578	0.440391	0.220196	+	0.975456i	$0.429330\pi$
$18$	0	0
$19$	−6.32849	−1.45185	−0.725927	−	0.687771i	$-0.758589\pi$
$19$	−6.32849	−1.45185	−0.725927	+	0.687771i	$0.758589\pi$
$20$	0	0
$21$	−0.418919	−0.0914156
$22$	0	0
$23$	9.33558	1.94660	0.973302	−	0.229529i	$-0.0737187\pi$
$23$	9.33558	1.94660	0.973302	+	0.229529i	$0.0737187\pi$
$24$	0	0
$25$	−4.99679	−0.999359
$26$	0	0
$27$	−2.43999	−0.469577
$28$	0	0
$29$	2.91553	0.541401	0.270701	−	0.962664i	$-0.412745\pi$
$29$	2.91553	0.541401	0.270701	+	0.962664i	$0.412745\pi$
$30$	0	0
$31$	−5.95686	−1.06988	−0.534942	−	0.844889i	$-0.679666\pi$
$31$	−5.95686	−1.06988	−0.534942	+	0.844889i	$0.679666\pi$
$32$	0	0
$33$	0.418919	0.0729244
$34$	0	0
$35$	−0.0566209	−0.00957068
$36$	0	0
$37$	10.2424	1.68383	0.841917	−	0.539607i	$-0.181427\pi$
$37$	10.2424	1.68383	0.841917	+	0.539607i	$0.181427\pi$
$38$	0	0
$39$	0.418919	0.0670807
$40$	0	0
$41$	4.73435	0.739380	0.369690	−	0.929155i	$-0.379464\pi$
$41$	4.73435	0.739380	0.369690	+	0.929155i	$0.379464\pi$
$42$	0	0
$43$	−4.59450	−0.700655	−0.350327	−	0.936627i	$-0.613930\pi$
$43$	−4.59450	−0.700655	−0.350327	+	0.936627i	$0.613930\pi$
$44$	0	0
$45$	−0.159926	−0.0238404
$46$	0	0
$47$	9.33558	1.36173	0.680867	−	0.732407i	$-0.261603\pi$
$47$	9.33558	1.36173	0.680867	+	0.732407i	$0.261603\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.760664	0.106514
$52$	0	0
$53$	3.56915	0.490261	0.245130	−	0.969490i	$-0.421169\pi$
$53$	3.56915	0.490261	0.245130	+	0.969490i	$0.421169\pi$
$54$	0	0
$55$	0.0566209	0.00763476
$56$	0	0
$57$	−2.65112	−0.351150
$58$	0	0
$59$	5.04133	0.656325	0.328162	−	0.944621i	$-0.393571\pi$
$59$	5.04133	0.656325	0.328162	+	0.944621i	$0.393571\pi$
$60$	0	0
$61$	−5.95989	−0.763086	−0.381543	−	0.924351i	$-0.624607\pi$
$61$	−5.95989	−0.763086	−0.381543	+	0.924351i	$0.624607\pi$
$62$	0	0
$63$	2.82451	0.355854
$64$	0	0
$65$	0.0566209	0.00702296
$66$	0	0
$67$	4.68606	0.572494	0.286247	−	0.958156i	$-0.407592\pi$
$67$	4.68606	0.572494	0.286247	+	0.958156i	$0.407592\pi$
$68$	0	0
$69$	3.91085	0.470811
$70$	0	0
$71$	−1.73501	−0.205908	−0.102954	−	0.994686i	$-0.532829\pi$
$71$	−1.73501	−0.205908	−0.102954	+	0.994686i	$0.532829\pi$
$72$	0	0
$73$	−3.12666	−0.365948	−0.182974	−	0.983118i	$-0.558572\pi$
$73$	−3.12666	−0.365948	−0.182974	+	0.983118i	$0.558572\pi$
$74$	0	0
$75$	−2.09325	−0.241708
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−1.44270	−0.162316	−0.0811580	−	0.996701i	$-0.525862\pi$
$79$	−1.44270	−0.162316	−0.0811580	+	0.996701i	$0.525862\pi$
$80$	0	0
$81$	7.45136	0.827929
$82$	0	0
$83$	8.81091	0.967123	0.483562	−	0.875310i	$-0.339343\pi$
$83$	8.81091	0.967123	0.483562	+	0.875310i	$0.339343\pi$
$84$	0	0
$85$	0.102811	0.0111514
$86$	0	0
$87$	1.22137	0.130945
$88$	0	0
$89$	3.69954	0.392150	0.196075	−	0.980589i	$-0.437180\pi$
$89$	3.69954	0.392150	0.196075	+	0.980589i	$0.437180\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−2.49544	−0.258765
$94$	0	0
$95$	−0.358325	−0.0367633
$96$	0	0
$97$	−12.6141	−1.28077	−0.640383	−	0.768056i	$-0.721224\pi$
$97$	−12.6141	−1.28077	−0.640383	+	0.768056i	$0.721224\pi$
$98$	0	0
$99$	−2.82451	−0.283874
$100$	0	0
$101$	5.29238	0.526611	0.263306	−	0.964712i	$-0.415187\pi$
$101$	5.29238	0.526611	0.263306	+	0.964712i	$0.415187\pi$
$102$	0	0
$103$	9.28393	0.914773	0.457386	−	0.889268i	$-0.348786\pi$
$103$	9.28393	0.914773	0.457386	+	0.889268i	$0.348786\pi$
$104$	0	0
$105$	−0.0237196	−0.00231479
$106$	0	0
$107$	7.69542	0.743944	0.371972	−	0.928244i	$-0.378682\pi$
$107$	7.69542	0.743944	0.371972	+	0.928244i	$0.378682\pi$
$108$	0	0
$109$	17.6849	1.69391	0.846953	−	0.531667i	$-0.178434\pi$
$109$	17.6849	1.69391	0.846953	+	0.531667i	$0.178434\pi$
$110$	0	0
$111$	4.29072	0.407257
$112$	0	0
$113$	−4.02454	−0.378597	−0.189299	−	0.981920i	$-0.560621\pi$
$113$	−4.02454	−0.378597	−0.189299	+	0.981920i	$0.560621\pi$
$114$	0	0
$115$	0.528589	0.0492912
$116$	0	0
$117$	−2.82451	−0.261126
$118$	0	0
$119$	−1.81578	−0.166452
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.98331	0.178829
$124$	0	0
$125$	−0.566028	−0.0506270
$126$	0	0
$127$	15.4738	1.37308	0.686538	−	0.727094i	$-0.259130\pi$
$127$	15.4738	1.37308	0.686538	+	0.727094i	$0.259130\pi$
$128$	0	0
$129$	−1.92472	−0.169462
$130$	0	0
$131$	−0.261624	−0.0228582	−0.0114291	−	0.999935i	$-0.503638\pi$
$131$	−0.261624	−0.0228582	−0.0114291	+	0.999935i	$0.503638\pi$
$132$	0	0
$133$	6.32849	0.548750
$134$	0	0
$135$	−0.138155	−0.0118905
$136$	0	0
$137$	19.2449	1.64420	0.822101	−	0.569342i	$-0.192802\pi$
$137$	19.2449	1.64420	0.822101	+	0.569342i	$0.192802\pi$
$138$	0	0
$139$	−5.68670	−0.482339	−0.241170	−	0.970483i	$-0.577531\pi$
$139$	−5.68670	−0.482339	−0.241170	+	0.970483i	$0.577531\pi$
$140$	0	0
$141$	3.91085	0.329353
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.165080	0.0137092
$146$	0	0
$147$	0.418919	0.0345518
$148$	0	0
$149$	−10.6120	−0.869369	−0.434684	−	0.900583i	$-0.643140\pi$
$149$	−10.6120	−0.869369	−0.434684	+	0.900583i	$0.643140\pi$
$150$	0	0
$151$	4.92908	0.401123	0.200561	−	0.979681i	$-0.435723\pi$
$151$	4.92908	0.401123	0.200561	+	0.979681i	$0.435723\pi$
$152$	0	0
$153$	−5.12868	−0.414629
$154$	0	0
$155$	−0.337283	−0.0270912
$156$	0	0
$157$	20.6411	1.64734	0.823669	−	0.567071i	$-0.191924\pi$
$157$	20.6411	1.64734	0.823669	+	0.567071i	$0.191924\pi$
$158$	0	0
$159$	1.49518	0.118576
$160$	0	0
$161$	−9.33558	−0.735747
$162$	0	0
$163$	−13.5361	−1.06023	−0.530116	−	0.847925i	$-0.677852\pi$
$163$	−13.5361	−1.06023	−0.530116	+	0.847925i	$0.677852\pi$
$164$	0	0
$165$	0.0237196	0.00184656
$166$	0	0
$167$	11.5312	0.892312	0.446156	−	0.894955i	$-0.352793\pi$
$167$	11.5312	0.892312	0.446156	+	0.894955i	$0.352793\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	17.8749	1.36692
$172$	0	0
$173$	11.7525	0.893523	0.446762	−	0.894653i	$-0.352577\pi$
$173$	11.7525	0.893523	0.446762	+	0.894653i	$0.352577\pi$
$174$	0	0
$175$	4.99679	0.377722
$176$	0	0
$177$	2.11191	0.158741
$178$	0	0
$179$	−1.28804	−0.0962726	−0.0481363	−	0.998841i	$-0.515328\pi$
$179$	−1.28804	−0.0962726	−0.0481363	+	0.998841i	$0.515328\pi$
$180$	0	0
$181$	−8.30314	−0.617168	−0.308584	−	0.951197i	$-0.599855\pi$
$181$	−8.30314	−0.617168	−0.308584	+	0.951197i	$0.599855\pi$
$182$	0	0
$183$	−2.49671	−0.184562
$184$	0	0
$185$	0.579932	0.0426374
$186$	0	0
$187$	1.81578	0.132783
$188$	0	0
$189$	2.43999	0.177484
$190$	0	0
$191$	−12.8885	−0.932581	−0.466291	−	0.884632i	$-0.654410\pi$
$191$	−12.8885	−0.932581	−0.466291	+	0.884632i	$0.654410\pi$
$192$	0	0
$193$	18.8981	1.36032	0.680159	−	0.733065i	$-0.261911\pi$
$193$	18.8981	1.36032	0.680159	+	0.733065i	$0.261911\pi$
$194$	0	0
$195$	0.0237196	0.00169859
$196$	0	0
$197$	−8.42610	−0.600335	−0.300167	−	0.953887i	$-0.597043\pi$
$197$	−8.42610	−0.600335	−0.300167	+	0.953887i	$0.597043\pi$
$198$	0	0
$199$	−10.1415	−0.718911	−0.359455	−	0.933162i	$-0.617038\pi$
$199$	−10.1415	−0.718911	−0.359455	+	0.933162i	$0.617038\pi$
$200$	0	0
$201$	1.96308	0.138465
$202$	0	0
$203$	−2.91553	−0.204630
$204$	0	0
$205$	0.268063	0.0187223
$206$	0	0
$207$	−26.3684	−1.83273
$208$	0	0
$209$	−6.32849	−0.437751
$210$	0	0
$211$	−18.9089	−1.30174	−0.650870	−	0.759189i	$-0.725596\pi$
$211$	−18.9089	−1.30174	−0.650870	+	0.759189i	$0.725596\pi$
$212$	0	0
$213$	−0.726829	−0.0498015
$214$	0	0
$215$	−0.260145	−0.0177417
$216$	0	0
$217$	5.95686	0.404378
$218$	0	0
$219$	−1.30982	−0.0885092
$220$	0	0
$221$	1.81578	0.122143
$222$	0	0
$223$	15.6017	1.04477	0.522383	−	0.852711i	$-0.325043\pi$
$223$	15.6017	1.04477	0.522383	+	0.852711i	$0.325043\pi$
$224$	0	0
$225$	14.1135	0.940899
$226$	0	0
$227$	23.2506	1.54319	0.771597	−	0.636111i	$-0.219458\pi$
$227$	23.2506	1.54319	0.771597	+	0.636111i	$0.219458\pi$
$228$	0	0
$229$	12.5033	0.826244	0.413122	−	0.910676i	$-0.364438\pi$
$229$	12.5033	0.826244	0.413122	+	0.910676i	$0.364438\pi$
$230$	0	0
$231$	−0.418919	−0.0275628
$232$	0	0
$233$	−18.7263	−1.22680	−0.613401	−	0.789771i	$-0.710199\pi$
$233$	−18.7263	−1.22680	−0.613401	+	0.789771i	$0.710199\pi$
$234$	0	0
$235$	0.528589	0.0344814
$236$	0	0
$237$	−0.604372	−0.0392582
$238$	0	0
$239$	−6.85126	−0.443171	−0.221586	−	0.975141i	$-0.571123\pi$
$239$	−6.85126	−0.443171	−0.221586	+	0.975141i	$0.571123\pi$
$240$	0	0
$241$	6.18660	0.398514	0.199257	−	0.979947i	$-0.436147\pi$
$241$	6.18660	0.398514	0.199257	+	0.979947i	$0.436147\pi$
$242$	0	0
$243$	10.4415	0.669823
$244$	0	0
$245$	0.0566209	0.00361738
$246$	0	0
$247$	−6.32849	−0.402672
$248$	0	0
$249$	3.69106	0.233911
$250$	0	0
$251$	−21.9707	−1.38678	−0.693389	−	0.720563i	$-0.743883\pi$
$251$	−21.9707	−1.38678	−0.693389	+	0.720563i	$0.743883\pi$
$252$	0	0
$253$	9.33558	0.586923
$254$	0	0
$255$	0.0430695	0.00269712
$256$	0	0
$257$	−2.24105	−0.139793	−0.0698963	−	0.997554i	$-0.522267\pi$
$257$	−2.24105	−0.139793	−0.0698963	+	0.997554i	$0.522267\pi$
$258$	0	0
$259$	−10.2424	−0.636429
$260$	0	0
$261$	−8.23495	−0.509730
$262$	0	0
$263$	9.30926	0.574034	0.287017	−	0.957926i	$-0.407336\pi$
$263$	9.30926	0.574034	0.287017	+	0.957926i	$0.407336\pi$
$264$	0	0
$265$	0.202089	0.0124142
$266$	0	0
$267$	1.54981	0.0948466
$268$	0	0
$269$	17.0006	1.03654	0.518272	−	0.855216i	$-0.326575\pi$
$269$	17.0006	1.03654	0.518272	+	0.855216i	$0.326575\pi$
$270$	0	0
$271$	−6.88321	−0.418125	−0.209063	−	0.977902i	$-0.567041\pi$
$271$	−6.88321	−0.418125	−0.209063	+	0.977902i	$0.567041\pi$
$272$	0	0
$273$	−0.418919	−0.0253541
$274$	0	0
$275$	−4.99679	−0.301318
$276$	0	0
$277$	23.1933	1.39355	0.696777	−	0.717288i	$-0.254617\pi$
$277$	23.1933	1.39355	0.696777	+	0.717288i	$0.254617\pi$
$278$	0	0
$279$	16.8252	1.00730
$280$	0	0
$281$	2.12565	0.126806	0.0634029	−	0.997988i	$-0.479805\pi$
$281$	2.12565	0.126806	0.0634029	+	0.997988i	$0.479805\pi$
$282$	0	0
$283$	−16.8370	−1.00086	−0.500429	−	0.865778i	$-0.666824\pi$
$283$	−16.8370	−1.00086	−0.500429	+	0.865778i	$0.666824\pi$
$284$	0	0
$285$	−0.150109	−0.00889169
$286$	0	0
$287$	−4.73435	−0.279460
$288$	0	0
$289$	−13.7029	−0.806056
$290$	0	0
$291$	−5.28427	−0.309770
$292$	0	0
$293$	−28.3695	−1.65736	−0.828682	−	0.559720i	$-0.810909\pi$
$293$	−28.3695	−1.65736	−0.828682	+	0.559720i	$0.810909\pi$
$294$	0	0
$295$	0.285444	0.0166192
$296$	0	0
$297$	−2.43999	−0.141583
$298$	0	0
$299$	9.33558	0.539891
$300$	0	0
$301$	4.59450	0.264823
$302$	0	0
$303$	2.21708	0.127368
$304$	0	0
$305$	−0.337454	−0.0193226
$306$	0	0
$307$	4.98998	0.284793	0.142397	−	0.989810i	$-0.454519\pi$
$307$	4.98998	0.284793	0.142397	+	0.989810i	$0.454519\pi$
$308$	0	0
$309$	3.88921	0.221249
$310$	0	0
$311$	−12.5219	−0.710052	−0.355026	−	0.934856i	$-0.615528\pi$
$311$	−12.5219	−0.710052	−0.355026	+	0.934856i	$0.615528\pi$
$312$	0	0
$313$	20.6401	1.16665	0.583323	−	0.812240i	$-0.301752\pi$
$313$	20.6401	1.16665	0.583323	+	0.812240i	$0.301752\pi$
$314$	0	0
$315$	0.159926	0.00901082
$316$	0	0
$317$	25.4402	1.42887	0.714433	−	0.699704i	$-0.246685\pi$
$317$	25.4402	1.42887	0.714433	+	0.699704i	$0.246685\pi$
$318$	0	0
$319$	2.91553	0.163239
$320$	0	0
$321$	3.22376	0.179932
$322$	0	0
$323$	−11.4911	−0.639384
$324$	0	0
$325$	−4.99679	−0.277172
$326$	0	0
$327$	7.40854	0.409693
$328$	0	0
$329$	−9.33558	−0.514687
$330$	0	0
$331$	6.14959	0.338012	0.169006	−	0.985615i	$-0.445944\pi$
$331$	6.14959	0.338012	0.169006	+	0.985615i	$0.445944\pi$
$332$	0	0
$333$	−28.9296	−1.58533
$334$	0	0
$335$	0.265329	0.0144965
$336$	0	0
$337$	28.4778	1.55129	0.775643	−	0.631172i	$-0.217426\pi$
$337$	28.4778	1.55129	0.775643	+	0.631172i	$0.217426\pi$
$338$	0	0
$339$	−1.68596	−0.0915686
$340$	0	0
$341$	−5.95686	−0.322582
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.221436	0.0119217
$346$	0	0
$347$	10.8967	0.584968	0.292484	−	0.956270i	$-0.405518\pi$
$347$	10.8967	0.584968	0.292484	+	0.956270i	$0.405518\pi$
$348$	0	0
$349$	−11.8351	−0.633519	−0.316759	−	0.948506i	$-0.602595\pi$
$349$	−11.8351	−0.633519	−0.316759	+	0.948506i	$0.602595\pi$
$350$	0	0
$351$	−2.43999	−0.130237
$352$	0	0
$353$	−24.6948	−1.31437	−0.657186	−	0.753729i	$-0.728253\pi$
$353$	−24.6948	−1.31437	−0.657186	+	0.753729i	$0.728253\pi$
$354$	0	0
$355$	−0.0982380	−0.00521393
$356$	0	0
$357$	−0.760664	−0.0402586
$358$	0	0
$359$	−23.5972	−1.24541	−0.622705	−	0.782457i	$-0.713966\pi$
$359$	−23.5972	−1.24541	−0.622705	+	0.782457i	$0.713966\pi$
$360$	0	0
$361$	21.0498	1.10788
$362$	0	0
$363$	0.418919	0.0219875
$364$	0	0
$365$	−0.177034	−0.00926640
$366$	0	0
$367$	−3.70635	−0.193470	−0.0967349	−	0.995310i	$-0.530840\pi$
$367$	−3.70635	−0.193470	−0.0967349	+	0.995310i	$0.530840\pi$
$368$	0	0
$369$	−13.3722	−0.696128
$370$	0	0
$371$	−3.56915	−0.185301
$372$	0	0
$373$	−8.91203	−0.461448	−0.230724	−	0.973019i	$-0.574109\pi$
$373$	−8.91203	−0.461448	−0.230724	+	0.973019i	$0.574109\pi$
$374$	0	0
$375$	−0.237120	−0.0122448
$376$	0	0
$377$	2.91553	0.150158
$378$	0	0
$379$	−8.99922	−0.462259	−0.231129	−	0.972923i	$-0.574242\pi$
$379$	−8.99922	−0.462259	−0.231129	+	0.972923i	$0.574242\pi$
$380$	0	0
$381$	6.48226	0.332096
$382$	0	0
$383$	16.5128	0.843764	0.421882	−	0.906651i	$-0.361370\pi$
$383$	16.5128	0.843764	0.421882	+	0.906651i	$0.361370\pi$
$384$	0	0
$385$	−0.0566209	−0.00288567
$386$	0	0
$387$	12.9772	0.659668
$388$	0	0
$389$	−17.4141	−0.882930	−0.441465	−	0.897279i	$-0.645541\pi$
$389$	−17.4141	−0.882930	−0.441465	+	0.897279i	$0.645541\pi$
$390$	0	0
$391$	16.9514	0.857267
$392$	0	0
$393$	−0.109599	−0.00552855
$394$	0	0
$395$	−0.0816868	−0.00411011
$396$	0	0
$397$	28.5614	1.43346	0.716729	−	0.697352i	$-0.245639\pi$
$397$	28.5614	1.43346	0.716729	+	0.697352i	$0.245639\pi$
$398$	0	0
$399$	2.65112	0.132722
$400$	0	0
$401$	34.9501	1.74532	0.872662	−	0.488325i	$-0.162392\pi$
$401$	34.9501	1.74532	0.872662	+	0.488325i	$0.162392\pi$
$402$	0	0
$403$	−5.95686	−0.296732
$404$	0	0
$405$	0.421903	0.0209645
$406$	0	0
$407$	10.2424	0.507695
$408$	0	0
$409$	7.19996	0.356015	0.178007	−	0.984029i	$-0.443035\pi$
$409$	7.19996	0.356015	0.178007	+	0.984029i	$0.443035\pi$
$410$	0	0
$411$	8.06204	0.397671
$412$	0	0
$413$	−5.04133	−0.248067
$414$	0	0
$415$	0.498882	0.0244891
$416$	0	0
$417$	−2.38226	−0.116660
$418$	0	0
$419$	−13.7425	−0.671363	−0.335681	−	0.941976i	$-0.608967\pi$
$419$	−13.7425	−0.671363	−0.335681	+	0.941976i	$0.608967\pi$
$420$	0	0
$421$	−4.38174	−0.213553	−0.106777	−	0.994283i	$-0.534053\pi$
$421$	−4.38174	−0.213553	−0.106777	+	0.994283i	$0.534053\pi$
$422$	0	0
$423$	−26.3684	−1.28208
$424$	0	0
$425$	−9.07308	−0.440109
$426$	0	0
$427$	5.95989	0.288419
$428$	0	0
$429$	0.418919	0.0202256
$430$	0	0
$431$	−12.2767	−0.591348	−0.295674	−	0.955289i	$-0.595544\pi$
$431$	−12.2767	−0.591348	−0.295674	+	0.955289i	$0.595544\pi$
$432$	0	0
$433$	7.81593	0.375610	0.187805	−	0.982206i	$-0.439863\pi$
$433$	7.81593	0.375610	0.187805	+	0.982206i	$0.439863\pi$
$434$	0	0
$435$	0.0691552	0.00331574
$436$	0	0
$437$	−59.0801	−2.82619
$438$	0	0
$439$	−13.9010	−0.663458	−0.331729	−	0.943375i	$-0.607632\pi$
$439$	−13.9010	−0.663458	−0.331729	+	0.943375i	$0.607632\pi$
$440$	0	0
$441$	−2.82451	−0.134500
$442$	0	0
$443$	17.4844	0.830707	0.415353	−	0.909660i	$-0.363658\pi$
$443$	17.4844	0.830707	0.415353	+	0.909660i	$0.363658\pi$
$444$	0	0
$445$	0.209471	0.00992989
$446$	0	0
$447$	−4.44556	−0.210268
$448$	0	0
$449$	27.9550	1.31928	0.659639	−	0.751582i	$-0.270709\pi$
$449$	27.9550	1.31928	0.659639	+	0.751582i	$0.270709\pi$
$450$	0	0
$451$	4.73435	0.222932
$452$	0	0
$453$	2.06488	0.0970167
$454$	0	0
$455$	−0.0566209	−0.00265443
$456$	0	0
$457$	22.7994	1.06651	0.533257	−	0.845954i	$-0.320968\pi$
$457$	22.7994	1.06651	0.533257	+	0.845954i	$0.320968\pi$
$458$	0	0
$459$	−4.43049	−0.206798
$460$	0	0
$461$	−6.13027	−0.285515	−0.142758	−	0.989758i	$-0.545597\pi$
$461$	−6.13027	−0.285515	−0.142758	+	0.989758i	$0.545597\pi$
$462$	0	0
$463$	−20.2969	−0.943276	−0.471638	−	0.881792i	$-0.656337\pi$
$463$	−20.2969	−0.943276	−0.471638	+	0.881792i	$0.656337\pi$
$464$	0	0
$465$	−0.141294	−0.00655236
$466$	0	0
$467$	30.9285	1.43120	0.715600	−	0.698511i	$-0.246153\pi$
$467$	30.9285	1.43120	0.715600	+	0.698511i	$0.246153\pi$
$468$	0	0
$469$	−4.68606	−0.216382
$470$	0	0
$471$	8.64693	0.398430
$472$	0	0
$473$	−4.59450	−0.211255
$474$	0	0
$475$	31.6222	1.45092
$476$	0	0
$477$	−10.0811	−0.461582
$478$	0	0
$479$	−11.8307	−0.540560	−0.270280	−	0.962782i	$-0.587116\pi$
$479$	−11.8307	−0.540560	−0.270280	+	0.962782i	$0.587116\pi$
$480$	0	0
$481$	10.2424	0.467011
$482$	0	0
$483$	−3.91085	−0.177950
$484$	0	0
$485$	−0.714221	−0.0324311
$486$	0	0
$487$	23.4721	1.06362	0.531811	−	0.846863i	$-0.321512\pi$
$487$	23.4721	1.06362	0.531811	+	0.846863i	$0.321512\pi$
$488$	0	0
$489$	−5.67054	−0.256431
$490$	0	0
$491$	2.71097	0.122345	0.0611723	−	0.998127i	$-0.480516\pi$
$491$	2.71097	0.122345	0.0611723	+	0.998127i	$0.480516\pi$
$492$	0	0
$493$	5.29397	0.238428
$494$	0	0
$495$	−0.159926	−0.00718815
$496$	0	0
$497$	1.73501	0.0778260
$498$	0	0
$499$	27.4537	1.22900	0.614498	−	0.788919i	$-0.289359\pi$
$499$	27.4537	1.22900	0.614498	+	0.788919i	$0.289359\pi$
$500$	0	0
$501$	4.83064	0.215817
$502$	0	0
$503$	−7.01204	−0.312652	−0.156326	−	0.987706i	$-0.549965\pi$
$503$	−7.01204	−0.312652	−0.156326	+	0.987706i	$0.549965\pi$
$504$	0	0
$505$	0.299659	0.0133347
$506$	0	0
$507$	0.418919	0.0186048
$508$	0	0
$509$	8.54801	0.378884	0.189442	−	0.981892i	$-0.439332\pi$
$509$	8.54801	0.378884	0.189442	+	0.981892i	$0.439332\pi$
$510$	0	0
$511$	3.12666	0.138315
$512$	0	0
$513$	15.4415	0.681758
$514$	0	0
$515$	0.525664	0.0231635
$516$	0	0
$517$	9.33558	0.410578
$518$	0	0
$519$	4.92333	0.216110
$520$	0	0
$521$	6.82344	0.298940	0.149470	−	0.988766i	$-0.452243\pi$
$521$	6.82344	0.298940	0.149470	+	0.988766i	$0.452243\pi$
$522$	0	0
$523$	−26.8264	−1.17303	−0.586517	−	0.809937i	$-0.699501\pi$
$523$	−26.8264	−1.17303	−0.586517	+	0.809937i	$0.699501\pi$
$524$	0	0
$525$	2.09325	0.0913569
$526$	0	0
$527$	−10.8163	−0.471167
$528$	0	0
$529$	64.1531	2.78926
$530$	0	0
$531$	−14.2393	−0.617931
$532$	0	0
$533$	4.73435	0.205067
$534$	0	0
$535$	0.435722	0.0188379
$536$	0	0
$537$	−0.539584	−0.0232848
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−35.5375	−1.52788	−0.763939	−	0.645288i	$-0.776737\pi$
$541$	−35.5375	−1.52788	−0.763939	+	0.645288i	$0.776737\pi$
$542$	0	0
$543$	−3.47834	−0.149270
$544$	0	0
$545$	1.00134	0.0428925
$546$	0	0
$547$	−41.1832	−1.76087	−0.880434	−	0.474168i	$-0.842749\pi$
$547$	−41.1832	−1.76087	−0.880434	+	0.474168i	$0.842749\pi$
$548$	0	0
$549$	16.8338	0.718447
$550$	0	0
$551$	−18.4509	−0.786036
$552$	0	0
$553$	1.44270	0.0613497
$554$	0	0
$555$	0.242944	0.0103124
$556$	0	0
$557$	17.8182	0.754983	0.377491	−	0.926013i	$-0.376787\pi$
$557$	17.8182	0.754983	0.377491	+	0.926013i	$0.376787\pi$
$558$	0	0
$559$	−4.59450	−0.194327
$560$	0	0
$561$	0.760664	0.0321153
$562$	0	0
$563$	−21.7191	−0.915353	−0.457676	−	0.889119i	$-0.651318\pi$
$563$	−21.7191	−0.915353	−0.457676	+	0.889119i	$0.651318\pi$
$564$	0	0
$565$	−0.227873	−0.00958670
$566$	0	0
$567$	−7.45136	−0.312928
$568$	0	0
$569$	27.5843	1.15640	0.578198	−	0.815897i	$-0.303756\pi$
$569$	27.5843	1.15640	0.578198	+	0.815897i	$0.303756\pi$
$570$	0	0
$571$	31.1378	1.30308	0.651539	−	0.758615i	$-0.274124\pi$
$571$	31.1378	1.30308	0.651539	+	0.758615i	$0.274124\pi$
$572$	0	0
$573$	−5.39925	−0.225557
$574$	0	0
$575$	−46.6480	−1.94536
$576$	0	0
$577$	−17.0710	−0.710675	−0.355338	−	0.934738i	$-0.615634\pi$
$577$	−17.0710	−0.710675	−0.355338	+	0.934738i	$0.615634\pi$
$578$	0	0
$579$	7.91678	0.329010
$580$	0	0
$581$	−8.81091	−0.365538
$582$	0	0
$583$	3.56915	0.147819
$584$	0	0
$585$	−0.159926	−0.00661213
$586$	0	0
$587$	1.33484	0.0550947	0.0275473	−	0.999621i	$-0.491230\pi$
$587$	1.33484	0.0550947	0.0275473	+	0.999621i	$0.491230\pi$
$588$	0	0
$589$	37.6979	1.55332
$590$	0	0
$591$	−3.52985	−0.145199
$592$	0	0
$593$	1.46179	0.0600287	0.0300143	−	0.999549i	$-0.490445\pi$
$593$	1.46179	0.0600287	0.0300143	+	0.999549i	$0.490445\pi$
$594$	0	0
$595$	−0.102811	−0.00421484
$596$	0	0
$597$	−4.24846	−0.173878
$598$	0	0
$599$	0.884487	0.0361392	0.0180696	−	0.999837i	$-0.494248\pi$
$599$	0.884487	0.0361392	0.0180696	+	0.999837i	$0.494248\pi$
$600$	0	0
$601$	−31.2464	−1.27457	−0.637284	−	0.770629i	$-0.719942\pi$
$601$	−31.2464	−1.27457	−0.637284	+	0.770629i	$0.719942\pi$
$602$	0	0
$603$	−13.2358	−0.539004
$604$	0	0
$605$	0.0566209	0.00230197
$606$	0	0
$607$	−34.4337	−1.39762	−0.698810	−	0.715307i	$-0.746287\pi$
$607$	−34.4337	−1.39762	−0.698810	+	0.715307i	$0.746287\pi$
$608$	0	0
$609$	−1.22137	−0.0494925
$610$	0	0
$611$	9.33558	0.377677
$612$	0	0
$613$	−23.0226	−0.929876	−0.464938	−	0.885343i	$-0.653923\pi$
$613$	−23.0226	−0.929876	−0.464938	+	0.885343i	$0.653923\pi$
$614$	0	0
$615$	0.112297	0.00452823
$616$	0	0
$617$	−17.8069	−0.716878	−0.358439	−	0.933553i	$-0.616691\pi$
$617$	−17.8069	−0.716878	−0.358439	+	0.933553i	$0.616691\pi$
$618$	0	0
$619$	−44.4849	−1.78800	−0.894000	−	0.448068i	$-0.852112\pi$
$619$	−44.4849	−1.78800	−0.894000	+	0.448068i	$0.852112\pi$
$620$	0	0
$621$	−22.7788	−0.914081
$622$	0	0
$623$	−3.69954	−0.148219
$624$	0	0
$625$	24.9519	0.998077
$626$	0	0
$627$	−2.65112	−0.105876
$628$	0	0
$629$	18.5979	0.741546
$630$	0	0
$631$	−22.6405	−0.901305	−0.450653	−	0.892699i	$-0.648809\pi$
$631$	−22.6405	−0.901305	−0.450653	+	0.892699i	$0.648809\pi$
$632$	0	0
$633$	−7.92127	−0.314842
$634$	0	0
$635$	0.876139	0.0347685
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	4.90056	0.193863
$640$	0	0
$641$	2.84624	0.112420	0.0562099	−	0.998419i	$-0.482098\pi$
$641$	2.84624	0.112420	0.0562099	+	0.998419i	$0.482098\pi$
$642$	0	0
$643$	−38.0141	−1.49913	−0.749565	−	0.661930i	$-0.769738\pi$
$643$	−38.0141	−1.49913	−0.749565	+	0.661930i	$0.769738\pi$
$644$	0	0
$645$	−0.108980	−0.00429106
$646$	0	0
$647$	−29.6321	−1.16496	−0.582479	−	0.812846i	$-0.697917\pi$
$647$	−29.6321	−1.16496	−0.582479	+	0.812846i	$0.697917\pi$
$648$	0	0
$649$	5.04133	0.197889
$650$	0	0
$651$	2.49544	0.0978040
$652$	0	0
$653$	31.8057	1.24465	0.622327	−	0.782757i	$-0.286187\pi$
$653$	31.8057	1.24465	0.622327	+	0.782757i	$0.286187\pi$
$654$	0	0
$655$	−0.0148134	−0.000578807 0
$656$	0	0
$657$	8.83128	0.344541
$658$	0	0
$659$	−23.7123	−0.923700	−0.461850	−	0.886958i	$-0.652814\pi$
$659$	−23.7123	−0.923700	−0.461850	+	0.886958i	$0.652814\pi$
$660$	0	0
$661$	7.65516	0.297751	0.148876	−	0.988856i	$-0.452435\pi$
$661$	7.65516	0.297751	0.148876	+	0.988856i	$0.452435\pi$
$662$	0	0
$663$	0.760664	0.0295417
$664$	0	0
$665$	0.358325	0.0138952
$666$	0	0
$667$	27.2182	1.05389
$668$	0	0
$669$	6.53584	0.252690
$670$	0	0
$671$	−5.95989	−0.230079
$672$	0	0
$673$	−8.06827	−0.311009	−0.155504	−	0.987835i	$-0.549700\pi$
$673$	−8.06827	−0.311009	−0.155504	+	0.987835i	$0.549700\pi$
$674$	0	0
$675$	12.1922	0.469276
$676$	0	0
$677$	−0.595456	−0.0228852	−0.0114426	−	0.999935i	$-0.503642\pi$
$677$	−0.595456	−0.0228852	−0.0114426	+	0.999935i	$0.503642\pi$
$678$	0	0
$679$	12.6141	0.484084
$680$	0	0
$681$	9.74010	0.373241
$682$	0	0
$683$	4.06747	0.155637	0.0778187	−	0.996968i	$-0.475204\pi$
$683$	4.06747	0.155637	0.0778187	+	0.996968i	$0.475204\pi$
$684$	0	0
$685$	1.08966	0.0416339
$686$	0	0
$687$	5.23788	0.199838
$688$	0	0
$689$	3.56915	0.135974
$690$	0	0
$691$	−16.8785	−0.642089	−0.321045	−	0.947064i	$-0.604034\pi$
$691$	−16.8785	−0.642089	−0.321045	+	0.947064i	$0.604034\pi$
$692$	0	0
$693$	2.82451	0.107294
$694$	0	0
$695$	−0.321986	−0.0122136
$696$	0	0
$697$	8.59653	0.325617
$698$	0	0
$699$	−7.84481	−0.296718
$700$	0	0
$701$	−9.07847	−0.342889	−0.171445	−	0.985194i	$-0.554843\pi$
$701$	−9.07847	−0.342889	−0.171445	+	0.985194i	$0.554843\pi$
$702$	0	0
$703$	−64.8187	−2.44468
$704$	0	0
$705$	0.221436	0.00833976
$706$	0	0
$707$	−5.29238	−0.199040
$708$	0	0
$709$	−48.9329	−1.83771	−0.918857	−	0.394591i	$-0.870886\pi$
$709$	−48.9329	−1.83771	−0.918857	+	0.394591i	$0.870886\pi$
$710$	0	0
$711$	4.07491	0.152821
$712$	0	0
$713$	−55.6108	−2.08264
$714$	0	0
$715$	0.0566209	0.00211750
$716$	0	0
$717$	−2.87012	−0.107187
$718$	0	0
$719$	−33.3055	−1.24209	−0.621043	−	0.783776i	$-0.713291\pi$
$719$	−33.3055	−1.24209	−0.621043	+	0.783776i	$0.713291\pi$
$720$	0	0
$721$	−9.28393	−0.345752
$722$	0	0
$723$	2.59168	0.0963858
$724$	0	0
$725$	−14.5683	−0.541054
$726$	0	0
$727$	33.9940	1.26077	0.630385	−	0.776283i	$-0.282897\pi$
$727$	33.9940	1.26077	0.630385	+	0.776283i	$0.282897\pi$
$728$	0	0
$729$	−17.9799	−0.665924
$730$	0	0
$731$	−8.34260	−0.308562
$732$	0	0
$733$	−42.2264	−1.55967	−0.779834	−	0.625987i	$-0.784696\pi$
$733$	−42.2264	−1.55967	−0.779834	+	0.625987i	$0.784696\pi$
$734$	0	0
$735$	0.0237196	0.000874909 0
$736$	0	0
$737$	4.68606	0.172613
$738$	0	0
$739$	10.9293	0.402042	0.201021	−	0.979587i	$-0.435574\pi$
$739$	10.9293	0.402042	0.201021	+	0.979587i	$0.435574\pi$
$740$	0	0
$741$	−2.65112	−0.0973914
$742$	0	0
$743$	23.0522	0.845703	0.422852	−	0.906199i	$-0.361029\pi$
$743$	23.0522	0.845703	0.422852	+	0.906199i	$0.361029\pi$
$744$	0	0
$745$	−0.600861	−0.0220138
$746$	0	0
$747$	−24.8865	−0.910549
$748$	0	0
$749$	−7.69542	−0.281185
$750$	0	0
$751$	−9.83961	−0.359052	−0.179526	−	0.983753i	$-0.557456\pi$
$751$	−9.83961	−0.359052	−0.179526	+	0.983753i	$0.557456\pi$
$752$	0	0
$753$	−9.20394	−0.335410
$754$	0	0
$755$	0.279089	0.0101571
$756$	0	0
$757$	−9.66410	−0.351248	−0.175624	−	0.984457i	$-0.556194\pi$
$757$	−9.66410	−0.351248	−0.175624	+	0.984457i	$0.556194\pi$
$758$	0	0
$759$	3.91085	0.141955
$760$	0	0
$761$	−13.2223	−0.479310	−0.239655	−	0.970858i	$-0.577034\pi$
$761$	−13.2223	−0.479310	−0.239655	+	0.970858i	$0.577034\pi$
$762$	0	0
$763$	−17.6849	−0.640237
$764$	0	0
$765$	−0.290391	−0.0104991
$766$	0	0
$767$	5.04133	0.182032
$768$	0	0
$769$	28.0815	1.01265	0.506323	−	0.862344i	$-0.331005\pi$
$769$	28.0815	1.01265	0.506323	+	0.862344i	$0.331005\pi$
$770$	0	0
$771$	−0.938816	−0.0338106
$772$	0	0
$773$	27.8811	1.00281	0.501406	−	0.865212i	$-0.332816\pi$
$773$	27.8811	1.00281	0.501406	+	0.865212i	$0.332816\pi$
$774$	0	0
$775$	29.7652	1.06920
$776$	0	0
$777$	−4.29072	−0.153929
$778$	0	0
$779$	−29.9612	−1.07347
$780$	0	0
$781$	−1.73501	−0.0620837
$782$	0	0
$783$	−7.11389	−0.254230
$784$	0	0
$785$	1.16872	0.0417133
$786$	0	0
$787$	13.6918	0.488061	0.244030	−	0.969768i	$-0.421530\pi$
$787$	13.6918	0.488061	0.244030	+	0.969768i	$0.421530\pi$
$788$	0	0
$789$	3.89982	0.138837
$790$	0	0
$791$	4.02454	0.143096
$792$	0	0
$793$	−5.95989	−0.211642
$794$	0	0
$795$	0.0846587	0.00300253
$796$	0	0
$797$	37.5959	1.33172	0.665858	−	0.746078i	$-0.268066\pi$
$797$	37.5959	1.33172	0.665858	+	0.746078i	$0.268066\pi$
$798$	0	0
$799$	16.9514	0.599696
$800$	0	0
$801$	−10.4494	−0.369211
$802$	0	0
$803$	−3.12666	−0.110337
$804$	0	0
$805$	−0.528589	−0.0186303
$806$	0	0
$807$	7.12186	0.250701
$808$	0	0
$809$	33.9041	1.19201	0.596003	−	0.802982i	$-0.296755\pi$
$809$	33.9041	1.19201	0.596003	+	0.802982i	$0.296755\pi$
$810$	0	0
$811$	23.8202	0.836440	0.418220	−	0.908346i	$-0.362654\pi$
$811$	23.8202	0.836440	0.418220	+	0.908346i	$0.362654\pi$
$812$	0	0
$813$	−2.88351	−0.101129
$814$	0	0
$815$	−0.766429	−0.0268468
$816$	0	0
$817$	29.0762	1.01725
$818$	0	0
$819$	2.82451	0.0986963
$820$	0	0
$821$	−2.82211	−0.0984922	−0.0492461	−	0.998787i	$-0.515682\pi$
$821$	−2.82211	−0.0984922	−0.0492461	+	0.998787i	$0.515682\pi$
$822$	0	0
$823$	−7.22434	−0.251825	−0.125912	−	0.992041i	$-0.540186\pi$
$823$	−7.22434	−0.251825	−0.125912	+	0.992041i	$0.540186\pi$
$824$	0	0
$825$	−2.09325	−0.0728776
$826$	0	0
$827$	−37.4824	−1.30339	−0.651695	−	0.758481i	$-0.725942\pi$
$827$	−37.4824	−1.30339	−0.651695	+	0.758481i	$0.725942\pi$
$828$	0	0
$829$	36.1820	1.25665	0.628326	−	0.777950i	$-0.283740\pi$
$829$	36.1820	1.25665	0.628326	+	0.777950i	$0.283740\pi$
$830$	0	0
$831$	9.71613	0.337049
$832$	0	0
$833$	1.81578	0.0629130
$834$	0	0
$835$	0.652907	0.0225948
$836$	0	0
$837$	14.5347	0.502393
$838$	0	0
$839$	−15.5455	−0.536689	−0.268344	−	0.963323i	$-0.586477\pi$
$839$	−15.5455	−0.536689	−0.268344	+	0.963323i	$0.586477\pi$
$840$	0	0
$841$	−20.4997	−0.706885
$842$	0	0
$843$	0.890476	0.0306696
$844$	0	0
$845$	0.0566209	0.00194782
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−7.05334	−0.242070
$850$	0	0
$851$	95.6184	3.27776
$852$	0	0
$853$	6.53270	0.223675	0.111838	−	0.993726i	$-0.464326\pi$
$853$	6.53270	0.223675	0.111838	+	0.993726i	$0.464326\pi$
$854$	0	0
$855$	1.01209	0.0346128
$856$	0	0
$857$	−32.7932	−1.12020	−0.560098	−	0.828427i	$-0.689236\pi$
$857$	−32.7932	−1.12020	−0.560098	+	0.828427i	$0.689236\pi$
$858$	0	0
$859$	5.50252	0.187743	0.0938717	−	0.995584i	$-0.470076\pi$
$859$	5.50252	0.187743	0.0938717	+	0.995584i	$0.470076\pi$
$860$	0	0
$861$	−1.98331	−0.0675909
$862$	0	0
$863$	13.4254	0.457006	0.228503	−	0.973543i	$-0.426617\pi$
$863$	13.4254	0.457006	0.228503	+	0.973543i	$0.426617\pi$
$864$	0	0
$865$	0.665435	0.0226255
$866$	0	0
$867$	−5.74042	−0.194955
$868$	0	0
$869$	−1.44270	−0.0489401
$870$	0	0
$871$	4.68606	0.158781
$872$	0	0
$873$	35.6286	1.20584
$874$	0	0
$875$	0.566028	0.0191352
$876$	0	0
$877$	−20.4989	−0.692199	−0.346099	−	0.938198i	$-0.612494\pi$
$877$	−20.4989	−0.692199	−0.346099	+	0.938198i	$0.612494\pi$
$878$	0	0
$879$	−11.8845	−0.400855
$880$	0	0
$881$	35.4258	1.19353	0.596763	−	0.802418i	$-0.296453\pi$
$881$	35.4258	1.19353	0.596763	+	0.802418i	$0.296453\pi$
$882$	0	0
$883$	15.0965	0.508037	0.254019	−	0.967199i	$-0.418248\pi$
$883$	15.0965	0.508037	0.254019	+	0.967199i	$0.418248\pi$
$884$	0	0
$885$	0.119578	0.00401957
$886$	0	0
$887$	−40.8672	−1.37219	−0.686093	−	0.727514i	$-0.740676\pi$
$887$	−40.8672	−1.37219	−0.686093	+	0.727514i	$0.740676\pi$
$888$	0	0
$889$	−15.4738	−0.518974
$890$	0	0
$891$	7.45136	0.249630
$892$	0	0
$893$	−59.0801	−1.97704
$894$	0	0
$895$	−0.0729300	−0.00243778
$896$	0	0
$897$	3.91085	0.130579
$898$	0	0
$899$	−17.3674	−0.579236
$900$	0	0
$901$	6.48079	0.215907
$902$	0	0
$903$	1.92472	0.0640508
$904$	0	0
$905$	−0.470131	−0.0156277
$906$	0	0
$907$	−4.52627	−0.150292	−0.0751462	−	0.997173i	$-0.523942\pi$
$907$	−4.52627	−0.150292	−0.0751462	+	0.997173i	$0.523942\pi$
$908$	0	0
$909$	−14.9484	−0.495806
$910$	0	0
$911$	42.4030	1.40487	0.702437	−	0.711746i	$-0.252095\pi$
$911$	42.4030	1.40487	0.702437	+	0.711746i	$0.252095\pi$
$912$	0	0
$913$	8.81091	0.291599
$914$	0	0
$915$	−0.141366	−0.00467341
$916$	0	0
$917$	0.261624	0.00863959
$918$	0	0
$919$	−36.0044	−1.18768	−0.593838	−	0.804585i	$-0.702388\pi$
$919$	−36.0044	−1.18768	−0.593838	+	0.804585i	$0.702388\pi$
$920$	0	0
$921$	2.09040	0.0688809
$922$	0	0
$923$	−1.73501	−0.0571087
$924$	0	0
$925$	−51.1790	−1.68275
$926$	0	0
$927$	−26.2225	−0.861261
$928$	0	0
$929$	−15.5035	−0.508655	−0.254327	−	0.967118i	$-0.581854\pi$
$929$	−15.5035	−0.508655	−0.254327	+	0.967118i	$0.581854\pi$
$930$	0	0
$931$	−6.32849	−0.207408
$932$	0	0
$933$	−5.24566	−0.171735
$934$	0	0
$935$	0.102811	0.00336228
$936$	0	0
$937$	−48.6811	−1.59034	−0.795171	−	0.606385i	$-0.792619\pi$
$937$	−48.6811	−1.59034	−0.795171	+	0.606385i	$0.792619\pi$
$938$	0	0
$939$	8.64651	0.282168
$940$	0	0
$941$	18.4000	0.599824	0.299912	−	0.953967i	$-0.403043\pi$
$941$	18.4000	0.599824	0.299912	+	0.953967i	$0.403043\pi$
$942$	0	0
$943$	44.1979	1.43928
$944$	0	0
$945$	0.138155	0.00449417
$946$	0	0
$947$	−2.14903	−0.0698340	−0.0349170	−	0.999390i	$-0.511117\pi$
$947$	−2.14903	−0.0698340	−0.0349170	+	0.999390i	$0.511117\pi$
$948$	0	0
$949$	−3.12666	−0.101496
$950$	0	0
$951$	10.6574	0.345589
$952$	0	0
$953$	−52.0388	−1.68570	−0.842851	−	0.538147i	$-0.819125\pi$
$953$	−52.0388	−1.68570	−0.842851	+	0.538147i	$0.819125\pi$
$954$	0	0
$955$	−0.729760	−0.0236145
$956$	0	0
$957$	1.22137	0.0394813
$958$	0	0
$959$	−19.2449	−0.621450
$960$	0	0
$961$	4.48419	0.144651
$962$	0	0
$963$	−21.7358	−0.700425
$964$	0	0
$965$	1.07003	0.0344455
$966$	0	0
$967$	20.8887	0.671736	0.335868	−	0.941909i	$-0.390970\pi$
$967$	20.8887	0.671736	0.335868	+	0.941909i	$0.390970\pi$
$968$	0	0
$969$	−4.81385	−0.154643
$970$	0	0
$971$	−8.37099	−0.268638	−0.134319	−	0.990938i	$-0.542885\pi$
$971$	−8.37099	−0.268638	−0.134319	+	0.990938i	$0.542885\pi$
$972$	0	0
$973$	5.68670	0.182307
$974$	0	0
$975$	−2.09325	−0.0670377
$976$	0	0
$977$	21.1749	0.677444	0.338722	−	0.940886i	$-0.390005\pi$
$977$	21.1749	0.677444	0.338722	+	0.940886i	$0.390005\pi$
$978$	0	0
$979$	3.69954	0.118238
$980$	0	0
$981$	−49.9511	−1.59482
$982$	0	0
$983$	20.4621	0.652640	0.326320	−	0.945259i	$-0.394191\pi$
$983$	20.4621	0.652640	0.326320	+	0.945259i	$0.394191\pi$
$984$	0	0
$985$	−0.477093	−0.0152015
$986$	0	0
$987$	−3.91085	−0.124484
$988$	0	0
$989$	−42.8923	−1.36390
$990$	0	0
$991$	−53.4550	−1.69805	−0.849026	−	0.528351i	$-0.822811\pi$
$991$	−53.4550	−1.69805	−0.849026	+	0.528351i	$0.822811\pi$
$992$	0	0
$993$	2.57618	0.0817526
$994$	0	0
$995$	−0.574220	−0.0182040
$996$	0	0
$997$	−20.1708	−0.638816	−0.319408	−	0.947617i	$-0.603484\pi$
$997$	−20.1708	−0.638816	−0.319408	+	0.947617i	$0.603484\pi$
$998$	0	0
$999$	−24.9913	−0.790690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.k.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.k.1.6	✓	10	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.418919 q^{3} +0.0566209 q^{5} -1.00000 q^{7} -2.82451 q^{9} +O(q^{10})\)	\(q+0.418919 q^{3} +0.0566209 q^{5} -1.00000 q^{7} -2.82451 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.0237196 q^{15} +1.81578 q^{17} -6.32849 q^{19} -0.418919 q^{21} +9.33558 q^{23} -4.99679 q^{25} -2.43999 q^{27} +2.91553 q^{29} -5.95686 q^{31} +0.418919 q^{33} -0.0566209 q^{35} +10.2424 q^{37} +0.418919 q^{39} +4.73435 q^{41} -4.59450 q^{43} -0.159926 q^{45} +9.33558 q^{47} +1.00000 q^{49} +0.760664 q^{51} +3.56915 q^{53} +0.0566209 q^{55} -2.65112 q^{57} +5.04133 q^{59} -5.95989 q^{61} +2.82451 q^{63} +0.0566209 q^{65} +4.68606 q^{67} +3.91085 q^{69} -1.73501 q^{71} -3.12666 q^{73} -2.09325 q^{75} -1.00000 q^{77} -1.44270 q^{79} +7.45136 q^{81} +8.81091 q^{83} +0.102811 q^{85} +1.22137 q^{87} +3.69954 q^{89} -1.00000 q^{91} -2.49544 q^{93} -0.358325 q^{95} -12.6141 q^{97} -2.82451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9}+O(q^{10}) \) 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9	\( 10 q + q^{3} + 4 q^{5} - 10 q^{7} + 17 q^{9} + 10 q^{11} + 10 q^{13} + 3 q^{15} + 3 q^{17} + 6 q^{19} - q^{21} + 4 q^{23} + 22 q^{25} - 5 q^{27} + 10 q^{29} - q^{31} + q^{33} - 4 q^{35} + 20 q^{37} + q^{39} + 8 q^{45} + 4 q^{47} + 10 q^{49} + 11 q^{51} - 5 q^{53} + 4 q^{55} + 16 q^{57} + 11 q^{59} + 12 q^{61} - 17 q^{63} + 4 q^{65} - 2 q^{67} + 10 q^{69} + 28 q^{71} + 11 q^{73} - 6 q^{75} - 10 q^{77} - 10 q^{79} + 46 q^{81} + 7 q^{83} + 33 q^{85} - 47 q^{87} + 30 q^{89} - 10 q^{91} + 41 q^{93} - 2 q^{95} + 55 q^{97} + 17 q^{99}+O(q^{100}) \) 10 * q + q^3 + 4 * q^5 - 10 * q^7 + 17 * q^9 + 10 * q^11 + 10 * q^13 + 3 * q^15 + 3 * q^17 + 6 * q^19 - q^21 + 4 * q^23 + 22 * q^25 - 5 * q^27 + 10 * q^29 - q^31 + q^33 - 4 * q^35 + 20 * q^37 + q^39 + 8 * q^45 + 4 * q^47 + 10 * q^49 + 11 * q^51 - 5 * q^53 + 4 * q^55 + 16 * q^57 + 11 * q^59 + 12 * q^61 - 17 * q^63 + 4 * q^65 - 2 * q^67 + 10 * q^69 + 28 * q^71 + 11 * q^73 - 6 * q^75 - 10 * q^77 - 10 * q^79 + 46 * q^81 + 7 * q^83 + 33 * q^85 - 47 * q^87 + 30 * q^89 - 10 * q^91 + 41 * q^93 - 2 * q^95 + 55 * q^97 + 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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