LMFDB - Embedded newform 6040.2.a.p.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + \cdots - 5628 $ x^19 - 5x^18 - 29x^17 + 165x^16 + 325x^15 - 2208x^14 - 1891x^13 + 15895x^12 + 6652x^11 - 67665x^10 - 17345x^9 + 174105x^8 + 41499x^7 - 262172x^6 - 80919x^5 + 206783x^4 + 91643x^3 - 59750x^2 - 40224x - 5628
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.40330$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.40330 q^{3} +1.00000 q^{5} +3.27194 q^{7} -1.03074 q^{9} +O(q^{10})$	$q-1.40330 q^{3} +1.00000 q^{5} +3.27194 q^{7} -1.03074 q^{9} +3.84502 q^{11} -5.20094 q^{13} -1.40330 q^{15} +0.810350 q^{17} -4.07576 q^{19} -4.59153 q^{21} +6.38337 q^{23} +1.00000 q^{25} +5.65635 q^{27} -7.98360 q^{29} -4.89556 q^{31} -5.39573 q^{33} +3.27194 q^{35} -7.95385 q^{37} +7.29849 q^{39} +10.5930 q^{41} -7.64500 q^{43} -1.03074 q^{45} -6.83414 q^{47} +3.70561 q^{49} -1.13717 q^{51} +1.56738 q^{53} +3.84502 q^{55} +5.71952 q^{57} +10.0522 q^{59} -7.68791 q^{61} -3.37252 q^{63} -5.20094 q^{65} -14.1120 q^{67} -8.95780 q^{69} -10.6571 q^{71} -8.04489 q^{73} -1.40330 q^{75} +12.5807 q^{77} -5.13669 q^{79} -4.84535 q^{81} +0.0490305 q^{83} +0.810350 q^{85} +11.2034 q^{87} +7.75451 q^{89} -17.0172 q^{91} +6.86996 q^{93} -4.07576 q^{95} +12.6883 q^{97} -3.96322 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.40330	−0.810197	−0.405099	−	0.914273i	$-0.632763\pi$
$3$	−1.40330	−0.810197	−0.405099	+	0.914273i	$0.632763\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.27194	1.23668	0.618339	−	0.785911i	$-0.287806\pi$
$7$	3.27194	1.23668	0.618339	+	0.785911i	$0.287806\pi$
$8$	0	0
$9$	−1.03074	−0.343580
$10$	0	0
$11$	3.84502	1.15932	0.579659	−	0.814859i	$-0.303186\pi$
$11$	3.84502	1.15932	0.579659	+	0.814859i	$0.303186\pi$
$12$	0	0
$13$	−5.20094	−1.44248	−0.721240	−	0.692685i	$-0.756428\pi$
$13$	−5.20094	−1.44248	−0.721240	+	0.692685i	$0.756428\pi$
$14$	0	0
$15$	−1.40330	−0.362331
$16$	0	0
$17$	0.810350	0.196539	0.0982694	−	0.995160i	$-0.468669\pi$
$17$	0.810350	0.196539	0.0982694	+	0.995160i	$0.468669\pi$
$18$	0	0
$19$	−4.07576	−0.935043	−0.467521	−	0.883982i	$-0.654853\pi$
$19$	−4.07576	−0.935043	−0.467521	+	0.883982i	$0.654853\pi$
$20$	0	0
$21$	−4.59153	−1.00195
$22$	0	0
$23$	6.38337	1.33102	0.665512	−	0.746387i	$-0.268213\pi$
$23$	6.38337	1.33102	0.665512	+	0.746387i	$0.268213\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65635	1.08857
$28$	0	0
$29$	−7.98360	−1.48252	−0.741258	−	0.671220i	$-0.765771\pi$
$29$	−7.98360	−1.48252	−0.741258	+	0.671220i	$0.765771\pi$
$30$	0	0
$31$	−4.89556	−0.879269	−0.439634	−	0.898177i	$-0.644892\pi$
$31$	−4.89556	−0.879269	−0.439634	+	0.898177i	$0.644892\pi$
$32$	0	0
$33$	−5.39573	−0.939276
$34$	0	0
$35$	3.27194	0.553059
$36$	0	0
$37$	−7.95385	−1.30760	−0.653802	−	0.756665i	$-0.726827\pi$
$37$	−7.95385	−1.30760	−0.653802	+	0.756665i	$0.726827\pi$
$38$	0	0
$39$	7.29849	1.16869
$40$	0	0
$41$	10.5930	1.65434	0.827172	−	0.561949i	$-0.189948\pi$
$41$	10.5930	1.65434	0.827172	+	0.561949i	$0.189948\pi$
$42$	0	0
$43$	−7.64500	−1.16585	−0.582926	−	0.812525i	$-0.698092\pi$
$43$	−7.64500	−1.16585	−0.582926	+	0.812525i	$0.698092\pi$
$44$	0	0
$45$	−1.03074	−0.153654
$46$	0	0
$47$	−6.83414	−0.996862	−0.498431	−	0.866929i	$-0.666090\pi$
$47$	−6.83414	−0.996862	−0.498431	+	0.866929i	$0.666090\pi$
$48$	0	0
$49$	3.70561	0.529373
$50$	0	0
$51$	−1.13717	−0.159235
$52$	0	0
$53$	1.56738	0.215296	0.107648	−	0.994189i	$-0.465668\pi$
$53$	1.56738	0.215296	0.107648	+	0.994189i	$0.465668\pi$
$54$	0	0
$55$	3.84502	0.518463
$56$	0	0
$57$	5.71952	0.757569
$58$	0	0
$59$	10.0522	1.30869	0.654345	−	0.756196i	$-0.272944\pi$
$59$	10.0522	1.30869	0.654345	+	0.756196i	$0.272944\pi$
$60$	0	0
$61$	−7.68791	−0.984336	−0.492168	−	0.870500i	$-0.663796\pi$
$61$	−7.68791	−0.984336	−0.492168	+	0.870500i	$0.663796\pi$
$62$	0	0
$63$	−3.37252	−0.424898
$64$	0	0
$65$	−5.20094	−0.645097
$66$	0	0
$67$	−14.1120	−1.72405	−0.862025	−	0.506866i	$-0.830804\pi$
$67$	−14.1120	−1.72405	−0.862025	+	0.506866i	$0.830804\pi$
$68$	0	0
$69$	−8.95780	−1.07839
$70$	0	0
$71$	−10.6571	−1.26476	−0.632381	−	0.774658i	$-0.717922\pi$
$71$	−10.6571	−1.26476	−0.632381	+	0.774658i	$0.717922\pi$
$72$	0	0
$73$	−8.04489	−0.941583	−0.470792	−	0.882244i	$-0.656032\pi$
$73$	−8.04489	−0.941583	−0.470792	+	0.882244i	$0.656032\pi$
$74$	0	0
$75$	−1.40330	−0.162039
$76$	0	0
$77$	12.5807	1.43370
$78$	0	0
$79$	−5.13669	−0.577922	−0.288961	−	0.957341i	$-0.593310\pi$
$79$	−5.13669	−0.577922	−0.288961	+	0.957341i	$0.593310\pi$
$80$	0	0
$81$	−4.84535	−0.538373
$82$	0	0
$83$	0.0490305	0.00538180	0.00269090	−	0.999996i	$-0.499143\pi$
$83$	0.0490305	0.00538180	0.00269090	+	0.999996i	$0.499143\pi$
$84$	0	0
$85$	0.810350	0.0878948
$86$	0	0
$87$	11.2034	1.20113
$88$	0	0
$89$	7.75451	0.821976	0.410988	−	0.911641i	$-0.365184\pi$
$89$	7.75451	0.821976	0.410988	+	0.911641i	$0.365184\pi$
$90$	0	0
$91$	−17.0172	−1.78388
$92$	0	0
$93$	6.86996	0.712381
$94$	0	0
$95$	−4.07576	−0.418164
$96$	0	0
$97$	12.6883	1.28830	0.644152	−	0.764898i	$-0.277211\pi$
$97$	12.6883	1.28830	0.644152	+	0.764898i	$0.277211\pi$
$98$	0	0
$99$	−3.96322	−0.398319
$100$	0	0
$101$	−11.4164	−1.13598	−0.567988	−	0.823037i	$-0.692278\pi$
$101$	−11.4164	−1.13598	−0.567988	+	0.823037i	$0.692278\pi$
$102$	0	0
$103$	−13.2005	−1.30069	−0.650344	−	0.759640i	$-0.725375\pi$
$103$	−13.2005	−1.30069	−0.650344	+	0.759640i	$0.725375\pi$
$104$	0	0
$105$	−4.59153	−0.448087
$106$	0	0
$107$	4.57558	0.442338	0.221169	−	0.975236i	$-0.429013\pi$
$107$	4.57558	0.442338	0.221169	+	0.975236i	$0.429013\pi$
$108$	0	0
$109$	13.2630	1.27037	0.635184	−	0.772361i	$-0.280924\pi$
$109$	13.2630	1.27037	0.635184	+	0.772361i	$0.280924\pi$
$110$	0	0
$111$	11.1617	1.05942
$112$	0	0
$113$	−0.00584044	−0.000549422 0	−0.000274711	−	1.00000i	$-0.500087\pi$
$113$	−0.00584044	−0.000549422 0	−0.000274711		1.00000i	$0.500087\pi$
$114$	0	0
$115$	6.38337	0.595252
$116$	0	0
$117$	5.36082	0.495608
$118$	0	0
$119$	2.65142	0.243055
$120$	0	0
$121$	3.78419	0.344018
$122$	0	0
$123$	−14.8651	−1.34035
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.40279	0.568156	0.284078	−	0.958801i	$-0.408313\pi$
$127$	6.40279	0.568156	0.284078	+	0.958801i	$0.408313\pi$
$128$	0	0
$129$	10.7283	0.944570
$130$	0	0
$131$	14.6034	1.27591	0.637953	−	0.770075i	$-0.279781\pi$
$131$	14.6034	1.27591	0.637953	+	0.770075i	$0.279781\pi$
$132$	0	0
$133$	−13.3356	−1.15635
$134$	0	0
$135$	5.65635	0.486821
$136$	0	0
$137$	21.5765	1.84341	0.921704	−	0.387893i	$-0.126797\pi$
$137$	21.5765	1.84341	0.921704	+	0.387893i	$0.126797\pi$
$138$	0	0
$139$	−7.42104	−0.629444	−0.314722	−	0.949184i	$-0.601911\pi$
$139$	−7.42104	−0.629444	−0.314722	+	0.949184i	$0.601911\pi$
$140$	0	0
$141$	9.59037	0.807655
$142$	0	0
$143$	−19.9977	−1.67229
$144$	0	0
$145$	−7.98360	−0.663002
$146$	0	0
$147$	−5.20009	−0.428897
$148$	0	0
$149$	−23.4817	−1.92369	−0.961847	−	0.273588i	$-0.911790\pi$
$149$	−23.4817	−1.92369	−0.961847	+	0.273588i	$0.911790\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−0.835260	−0.0675268
$154$	0	0
$155$	−4.89556	−0.393221
$156$	0	0
$157$	15.4023	1.22924	0.614619	−	0.788824i	$-0.289310\pi$
$157$	15.4023	1.22924	0.614619	+	0.788824i	$0.289310\pi$
$158$	0	0
$159$	−2.19951	−0.174432
$160$	0	0
$161$	20.8860	1.64605
$162$	0	0
$163$	−5.19282	−0.406733	−0.203367	−	0.979103i	$-0.565188\pi$
$163$	−5.19282	−0.406733	−0.203367	+	0.979103i	$0.565188\pi$
$164$	0	0
$165$	−5.39573	−0.420057
$166$	0	0
$167$	−16.4222	−1.27078	−0.635392	−	0.772190i	$-0.719162\pi$
$167$	−16.4222	−1.27078	−0.635392	+	0.772190i	$0.719162\pi$
$168$	0	0
$169$	14.0498	1.08075
$170$	0	0
$171$	4.20105	0.321262
$172$	0	0
$173$	−14.6971	−1.11740	−0.558701	−	0.829369i	$-0.688700\pi$
$173$	−14.6971	−1.11740	−0.558701	+	0.829369i	$0.688700\pi$
$174$	0	0
$175$	3.27194	0.247336
$176$	0	0
$177$	−14.1063	−1.06030
$178$	0	0
$179$	−5.69508	−0.425670	−0.212835	−	0.977088i	$-0.568270\pi$
$179$	−5.69508	−0.425670	−0.212835	+	0.977088i	$0.568270\pi$
$180$	0	0
$181$	−21.1956	−1.57546	−0.787729	−	0.616022i	$-0.788743\pi$
$181$	−21.1956	−1.57546	−0.787729	+	0.616022i	$0.788743\pi$
$182$	0	0
$183$	10.7885	0.797507
$184$	0	0
$185$	−7.95385	−0.584779
$186$	0	0
$187$	3.11581	0.227851
$188$	0	0
$189$	18.5073	1.34620
$190$	0	0
$191$	1.74064	0.125948	0.0629742	−	0.998015i	$-0.479941\pi$
$191$	1.74064	0.125948	0.0629742	+	0.998015i	$0.479941\pi$
$192$	0	0
$193$	20.8045	1.49754	0.748772	−	0.662828i	$-0.230644\pi$
$193$	20.8045	1.49754	0.748772	+	0.662828i	$0.230644\pi$
$194$	0	0
$195$	7.29849	0.522656
$196$	0	0
$197$	13.1987	0.940372	0.470186	−	0.882567i	$-0.344187\pi$
$197$	13.1987	0.940372	0.470186	+	0.882567i	$0.344187\pi$
$198$	0	0
$199$	−6.29581	−0.446298	−0.223149	−	0.974784i	$-0.571634\pi$
$199$	−6.29581	−0.446298	−0.223149	+	0.974784i	$0.571634\pi$
$200$	0	0
$201$	19.8034	1.39682
$202$	0	0
$203$	−26.1219	−1.83340
$204$	0	0
$205$	10.5930	0.739845
$206$	0	0
$207$	−6.57960	−0.457314
$208$	0	0
$209$	−15.6714	−1.08401
$210$	0	0
$211$	−9.48657	−0.653082	−0.326541	−	0.945183i	$-0.605883\pi$
$211$	−9.48657	−0.653082	−0.326541	+	0.945183i	$0.605883\pi$
$212$	0	0
$213$	14.9551	1.02471
$214$	0	0
$215$	−7.64500	−0.521385
$216$	0	0
$217$	−16.0180	−1.08737
$218$	0	0
$219$	11.2894	0.762868
$220$	0	0
$221$	−4.21458	−0.283503
$222$	0	0
$223$	14.5509	0.974401	0.487200	−	0.873290i	$-0.338018\pi$
$223$	14.5509	0.974401	0.487200	+	0.873290i	$0.338018\pi$
$224$	0	0
$225$	−1.03074	−0.0687160
$226$	0	0
$227$	9.07610	0.602402	0.301201	−	0.953561i	$-0.402613\pi$
$227$	9.07610	0.602402	0.301201	+	0.953561i	$0.402613\pi$
$228$	0	0
$229$	−12.5140	−0.826945	−0.413473	−	0.910517i	$-0.635684\pi$
$229$	−12.5140	−0.826945	−0.413473	+	0.910517i	$0.635684\pi$
$230$	0	0
$231$	−17.6545	−1.16158
$232$	0	0
$233$	4.53632	0.297184	0.148592	−	0.988899i	$-0.452526\pi$
$233$	4.53632	0.297184	0.148592	+	0.988899i	$0.452526\pi$
$234$	0	0
$235$	−6.83414	−0.445810
$236$	0	0
$237$	7.20833	0.468231
$238$	0	0
$239$	1.41068	0.0912491	0.0456245	−	0.998959i	$-0.485472\pi$
$239$	1.41068	0.0912491	0.0456245	+	0.998959i	$0.485472\pi$
$240$	0	0
$241$	−15.6998	−1.01131	−0.505657	−	0.862735i	$-0.668750\pi$
$241$	−15.6998	−1.01131	−0.505657	+	0.862735i	$0.668750\pi$
$242$	0	0
$243$	−10.1696	−0.652377
$244$	0	0
$245$	3.70561	0.236743
$246$	0	0
$247$	21.1978	1.34878
$248$	0	0
$249$	−0.0688047	−0.00436032
$250$	0	0
$251$	25.7100	1.62280	0.811399	−	0.584493i	$-0.198706\pi$
$251$	25.7100	1.62280	0.811399	+	0.584493i	$0.198706\pi$
$252$	0	0
$253$	24.5442	1.54308
$254$	0	0
$255$	−1.13717	−0.0712121
$256$	0	0
$257$	−10.3265	−0.644150	−0.322075	−	0.946714i	$-0.604380\pi$
$257$	−10.3265	−0.644150	−0.322075	+	0.946714i	$0.604380\pi$
$258$	0	0
$259$	−26.0245	−1.61709
$260$	0	0
$261$	8.22902	0.509363
$262$	0	0
$263$	−6.04732	−0.372894	−0.186447	−	0.982465i	$-0.559697\pi$
$263$	−6.04732	−0.372894	−0.186447	+	0.982465i	$0.559697\pi$
$264$	0	0
$265$	1.56738	0.0962832
$266$	0	0
$267$	−10.8819	−0.665963
$268$	0	0
$269$	22.8039	1.39038	0.695189	−	0.718827i	$-0.255321\pi$
$269$	22.8039	1.39038	0.695189	+	0.718827i	$0.255321\pi$
$270$	0	0
$271$	7.21320	0.438170	0.219085	−	0.975706i	$-0.429693\pi$
$271$	7.21320	0.438170	0.219085	+	0.975706i	$0.429693\pi$
$272$	0	0
$273$	23.8803	1.44530
$274$	0	0
$275$	3.84502	0.231864
$276$	0	0
$277$	5.93640	0.356684	0.178342	−	0.983969i	$-0.442927\pi$
$277$	5.93640	0.356684	0.178342	+	0.983969i	$0.442927\pi$
$278$	0	0
$279$	5.04605	0.302099
$280$	0	0
$281$	−25.9424	−1.54759	−0.773796	−	0.633435i	$-0.781644\pi$
$281$	−25.9424	−1.54759	−0.773796	+	0.633435i	$0.781644\pi$
$282$	0	0
$283$	−27.1237	−1.61234	−0.806168	−	0.591687i	$-0.798462\pi$
$283$	−27.1237	−1.61234	−0.806168	+	0.591687i	$0.798462\pi$
$284$	0	0
$285$	5.71952	0.338795
$286$	0	0
$287$	34.6596	2.04589
$288$	0	0
$289$	−16.3433	−0.961373
$290$	0	0
$291$	−17.8056	−1.04378
$292$	0	0
$293$	−10.1144	−0.590891	−0.295446	−	0.955360i	$-0.595468\pi$
$293$	−10.1144	−0.590891	−0.295446	+	0.955360i	$0.595468\pi$
$294$	0	0
$295$	10.0522	0.585264
$296$	0	0
$297$	21.7488	1.26199
$298$	0	0
$299$	−33.1995	−1.91998
$300$	0	0
$301$	−25.0140	−1.44178
$302$	0	0
$303$	16.0207	0.920365
$304$	0	0
$305$	−7.68791	−0.440209
$306$	0	0
$307$	25.2546	1.44135	0.720677	−	0.693271i	$-0.243831\pi$
$307$	25.2546	1.44135	0.720677	+	0.693271i	$0.243831\pi$
$308$	0	0
$309$	18.5244	1.05381
$310$	0	0
$311$	−21.4464	−1.21611	−0.608056	−	0.793894i	$-0.708050\pi$
$311$	−21.4464	−1.21611	−0.608056	+	0.793894i	$0.708050\pi$
$312$	0	0
$313$	−5.28376	−0.298656	−0.149328	−	0.988788i	$-0.547711\pi$
$313$	−5.28376	−0.298656	−0.149328	+	0.988788i	$0.547711\pi$
$314$	0	0
$315$	−3.37252	−0.190020
$316$	0	0
$317$	23.3760	1.31292	0.656462	−	0.754359i	$-0.272052\pi$
$317$	23.3760	1.31292	0.656462	+	0.754359i	$0.272052\pi$
$318$	0	0
$319$	−30.6971	−1.71871
$320$	0	0
$321$	−6.42092	−0.358381
$322$	0	0
$323$	−3.30279	−0.183772
$324$	0	0
$325$	−5.20094	−0.288496
$326$	0	0
$327$	−18.6121	−1.02925
$328$	0	0
$329$	−22.3609	−1.23280
$330$	0	0
$331$	−24.7162	−1.35853	−0.679263	−	0.733895i	$-0.737700\pi$
$331$	−24.7162	−1.35853	−0.679263	+	0.733895i	$0.737700\pi$
$332$	0	0
$333$	8.19835	0.449267
$334$	0	0
$335$	−14.1120	−0.771019
$336$	0	0
$337$	−11.6223	−0.633106	−0.316553	−	0.948575i	$-0.602525\pi$
$337$	−11.6223	−0.633106	−0.316553	+	0.948575i	$0.602525\pi$
$338$	0	0
$339$	0.00819590	0.000445140 0
$340$	0	0
$341$	−18.8235	−1.01935
$342$	0	0
$343$	−10.7791	−0.582014
$344$	0	0
$345$	−8.95780	−0.482272
$346$	0	0
$347$	−7.21844	−0.387506	−0.193753	−	0.981050i	$-0.562066\pi$
$347$	−7.21844	−0.387506	−0.193753	+	0.981050i	$0.562066\pi$
$348$	0	0
$349$	23.6872	1.26795	0.633975	−	0.773354i	$-0.281422\pi$
$349$	23.6872	1.26795	0.633975	+	0.773354i	$0.281422\pi$
$350$	0	0
$351$	−29.4183	−1.57023
$352$	0	0
$353$	−7.95333	−0.423313	−0.211657	−	0.977344i	$-0.567886\pi$
$353$	−7.95333	−0.423313	−0.211657	+	0.977344i	$0.567886\pi$
$354$	0	0
$355$	−10.6571	−0.565619
$356$	0	0
$357$	−3.72074	−0.196923
$358$	0	0
$359$	−23.7509	−1.25352	−0.626762	−	0.779211i	$-0.715620\pi$
$359$	−23.7509	−1.25352	−0.626762	+	0.779211i	$0.715620\pi$
$360$	0	0
$361$	−2.38820	−0.125695
$362$	0	0
$363$	−5.31037	−0.278722
$364$	0	0
$365$	−8.04489	−0.421089
$366$	0	0
$367$	−6.27946	−0.327785	−0.163893	−	0.986478i	$-0.552405\pi$
$367$	−6.27946	−0.327785	−0.163893	+	0.986478i	$0.552405\pi$
$368$	0	0
$369$	−10.9186	−0.568400
$370$	0	0
$371$	5.12837	0.266252
$372$	0	0
$373$	7.67701	0.397501	0.198750	−	0.980050i	$-0.436312\pi$
$373$	7.67701	0.397501	0.198750	+	0.980050i	$0.436312\pi$
$374$	0	0
$375$	−1.40330	−0.0724663
$376$	0	0
$377$	41.5222	2.13850
$378$	0	0
$379$	−30.8889	−1.58666	−0.793328	−	0.608795i	$-0.791653\pi$
$379$	−30.8889	−1.58666	−0.793328	+	0.608795i	$0.791653\pi$
$380$	0	0
$381$	−8.98506	−0.460319
$382$	0	0
$383$	−30.3203	−1.54930	−0.774648	−	0.632392i	$-0.782073\pi$
$383$	−30.3203	−1.54930	−0.774648	+	0.632392i	$0.782073\pi$
$384$	0	0
$385$	12.5807	0.641171
$386$	0	0
$387$	7.88001	0.400563
$388$	0	0
$389$	−19.4655	−0.986939	−0.493469	−	0.869763i	$-0.664272\pi$
$389$	−19.4655	−0.986939	−0.493469	+	0.869763i	$0.664272\pi$
$390$	0	0
$391$	5.17276	0.261598
$392$	0	0
$393$	−20.4930	−1.03374
$394$	0	0
$395$	−5.13669	−0.258455
$396$	0	0
$397$	−4.70594	−0.236185	−0.118092	−	0.993003i	$-0.537678\pi$
$397$	−4.70594	−0.236185	−0.118092	+	0.993003i	$0.537678\pi$
$398$	0	0
$399$	18.7140	0.936870
$400$	0	0
$401$	15.1758	0.757842	0.378921	−	0.925429i	$-0.376295\pi$
$401$	15.1758	0.757842	0.378921	+	0.925429i	$0.376295\pi$
$402$	0	0
$403$	25.4615	1.26833
$404$	0	0
$405$	−4.84535	−0.240768
$406$	0	0
$407$	−30.5827	−1.51593
$408$	0	0
$409$	14.4829	0.716132	0.358066	−	0.933696i	$-0.383436\pi$
$409$	14.4829	0.716132	0.358066	+	0.933696i	$0.383436\pi$
$410$	0	0
$411$	−30.2784	−1.49353
$412$	0	0
$413$	32.8904	1.61843
$414$	0	0
$415$	0.0490305	0.00240681
$416$	0	0
$417$	10.4140	0.509974
$418$	0	0
$419$	−31.2731	−1.52779	−0.763897	−	0.645339i	$-0.776716\pi$
$419$	−31.2731	−1.52779	−0.763897	+	0.645339i	$0.776716\pi$
$420$	0	0
$421$	−20.6704	−1.00741	−0.503706	−	0.863875i	$-0.668031\pi$
$421$	−20.6704	−1.00741	−0.503706	+	0.863875i	$0.668031\pi$
$422$	0	0
$423$	7.04422	0.342502
$424$	0	0
$425$	0.810350	0.0393077
$426$	0	0
$427$	−25.1544	−1.21731
$428$	0	0
$429$	28.0629	1.35489
$430$	0	0
$431$	−20.6994	−0.997056	−0.498528	−	0.866874i	$-0.666126\pi$
$431$	−20.6994	−0.997056	−0.498528	+	0.866874i	$0.666126\pi$
$432$	0	0
$433$	−0.0645117	−0.00310023	−0.00155012	−	0.999999i	$-0.500493\pi$
$433$	−0.0645117	−0.00310023	−0.00155012	+	0.999999i	$0.500493\pi$
$434$	0	0
$435$	11.2034	0.537162
$436$	0	0
$437$	−26.0171	−1.24457
$438$	0	0
$439$	−24.3477	−1.16205	−0.581026	−	0.813885i	$-0.697349\pi$
$439$	−24.3477	−1.16205	−0.581026	+	0.813885i	$0.697349\pi$
$440$	0	0
$441$	−3.81952	−0.181882
$442$	0	0
$443$	−3.21879	−0.152929	−0.0764647	−	0.997072i	$-0.524363\pi$
$443$	−3.21879	−0.152929	−0.0764647	+	0.997072i	$0.524363\pi$
$444$	0	0
$445$	7.75451	0.367599
$446$	0	0
$447$	32.9519	1.55857
$448$	0	0
$449$	3.15584	0.148933	0.0744667	−	0.997224i	$-0.476275\pi$
$449$	3.15584	0.148933	0.0744667	+	0.997224i	$0.476275\pi$
$450$	0	0
$451$	40.7302	1.91791
$452$	0	0
$453$	−1.40330	−0.0659329
$454$	0	0
$455$	−17.0172	−0.797777
$456$	0	0
$457$	12.1301	0.567424	0.283712	−	0.958910i	$-0.408434\pi$
$457$	12.1301	0.567424	0.283712	+	0.958910i	$0.408434\pi$
$458$	0	0
$459$	4.58362	0.213945
$460$	0	0
$461$	22.5679	1.05109	0.525546	−	0.850765i	$-0.323861\pi$
$461$	22.5679	1.05109	0.525546	+	0.850765i	$0.323861\pi$
$462$	0	0
$463$	18.6841	0.868322	0.434161	−	0.900835i	$-0.357045\pi$
$463$	18.6841	0.868322	0.434161	+	0.900835i	$0.357045\pi$
$464$	0	0
$465$	6.86996	0.318587
$466$	0	0
$467$	−14.2947	−0.661481	−0.330740	−	0.943722i	$-0.607298\pi$
$467$	−14.2947	−0.661481	−0.330740	+	0.943722i	$0.607298\pi$
$468$	0	0
$469$	−46.1735	−2.13210
$470$	0	0
$471$	−21.6141	−0.995926
$472$	0	0
$473$	−29.3952	−1.35159
$474$	0	0
$475$	−4.07576	−0.187009
$476$	0	0
$477$	−1.61556	−0.0739714
$478$	0	0
$479$	31.2260	1.42675	0.713377	−	0.700780i	$-0.247165\pi$
$479$	31.2260	1.42675	0.713377	+	0.700780i	$0.247165\pi$
$480$	0	0
$481$	41.3675	1.88620
$482$	0	0
$483$	−29.3094	−1.33362
$484$	0	0
$485$	12.6883	0.576147
$486$	0	0
$487$	2.68911	0.121855	0.0609276	−	0.998142i	$-0.480594\pi$
$487$	2.68911	0.121855	0.0609276	+	0.998142i	$0.480594\pi$
$488$	0	0
$489$	7.28710	0.329534
$490$	0	0
$491$	14.2689	0.643948	0.321974	−	0.946749i	$-0.395654\pi$
$491$	14.2689	0.643948	0.321974	+	0.946749i	$0.395654\pi$
$492$	0	0
$493$	−6.46951	−0.291372
$494$	0	0
$495$	−3.96322	−0.178133
$496$	0	0
$497$	−34.8693	−1.56410
$498$	0	0
$499$	8.96012	0.401110	0.200555	−	0.979682i	$-0.435725\pi$
$499$	8.96012	0.401110	0.200555	+	0.979682i	$0.435725\pi$
$500$	0	0
$501$	23.0453	1.02959
$502$	0	0
$503$	30.5544	1.36236	0.681178	−	0.732118i	$-0.261468\pi$
$503$	30.5544	1.36236	0.681178	+	0.732118i	$0.261468\pi$
$504$	0	0
$505$	−11.4164	−0.508024
$506$	0	0
$507$	−19.7161	−0.875622
$508$	0	0
$509$	−29.3372	−1.30035	−0.650174	−	0.759785i	$-0.725304\pi$
$509$	−29.3372	−1.30035	−0.650174	+	0.759785i	$0.725304\pi$
$510$	0	0
$511$	−26.3224	−1.16444
$512$	0	0
$513$	−23.0539	−1.01786
$514$	0	0
$515$	−13.2005	−0.581685
$516$	0	0
$517$	−26.2774	−1.15568
$518$	0	0
$519$	20.6245	0.905317
$520$	0	0
$521$	−39.1930	−1.71707	−0.858537	−	0.512751i	$-0.828626\pi$
$521$	−39.1930	−1.71707	−0.858537	+	0.512751i	$0.828626\pi$
$522$	0	0
$523$	−6.88164	−0.300913	−0.150456	−	0.988617i	$-0.548074\pi$
$523$	−6.88164	−0.300913	−0.150456	+	0.988617i	$0.548074\pi$
$524$	0	0
$525$	−4.59153	−0.200391
$526$	0	0
$527$	−3.96712	−0.172810
$528$	0	0
$529$	17.7474	0.771627
$530$	0	0
$531$	−10.3613	−0.449640
$532$	0	0
$533$	−55.0934	−2.38636
$534$	0	0
$535$	4.57558	0.197820
$536$	0	0
$537$	7.99192	0.344877
$538$	0	0
$539$	14.2482	0.613711
$540$	0	0
$541$	−0.545100	−0.0234357	−0.0117178	−	0.999931i	$-0.503730\pi$
$541$	−0.545100	−0.0234357	−0.0117178	+	0.999931i	$0.503730\pi$
$542$	0	0
$543$	29.7439	1.27643
$544$	0	0
$545$	13.2630	0.568126
$546$	0	0
$547$	−9.61966	−0.411307	−0.205653	−	0.978625i	$-0.565932\pi$
$547$	−9.61966	−0.411307	−0.205653	+	0.978625i	$0.565932\pi$
$548$	0	0
$549$	7.92424	0.338198
$550$	0	0
$551$	32.5392	1.38622
$552$	0	0
$553$	−16.8069	−0.714704
$554$	0	0
$555$	11.1617	0.473786
$556$	0	0
$557$	−22.9413	−0.972054	−0.486027	−	0.873944i	$-0.661554\pi$
$557$	−22.9413	−0.972054	−0.486027	+	0.873944i	$0.661554\pi$
$558$	0	0
$559$	39.7612	1.68172
$560$	0	0
$561$	−4.37243	−0.184604
$562$	0	0
$563$	−15.8451	−0.667790	−0.333895	−	0.942610i	$-0.608363\pi$
$563$	−15.8451	−0.667790	−0.333895	+	0.942610i	$0.608363\pi$
$564$	0	0
$565$	−0.00584044	−0.000245709 0
$566$	0	0
$567$	−15.8537	−0.665794
$568$	0	0
$569$	−19.0538	−0.798776	−0.399388	−	0.916782i	$-0.630777\pi$
$569$	−19.0538	−0.798776	−0.399388	+	0.916782i	$0.630777\pi$
$570$	0	0
$571$	20.5077	0.858219	0.429110	−	0.903252i	$-0.358827\pi$
$571$	20.5077	0.858219	0.429110	+	0.903252i	$0.358827\pi$
$572$	0	0
$573$	−2.44265	−0.102043
$574$	0	0
$575$	6.38337	0.266205
$576$	0	0
$577$	6.35981	0.264762	0.132381	−	0.991199i	$-0.457738\pi$
$577$	6.35981	0.264762	0.132381	+	0.991199i	$0.457738\pi$
$578$	0	0
$579$	−29.1951	−1.21331
$580$	0	0
$581$	0.160425	0.00665556
$582$	0	0
$583$	6.02660	0.249596
$584$	0	0
$585$	5.36082	0.221643
$586$	0	0
$587$	32.1029	1.32503	0.662515	−	0.749049i	$-0.269489\pi$
$587$	32.1029	1.32503	0.662515	+	0.749049i	$0.269489\pi$
$588$	0	0
$589$	19.9531	0.822154
$590$	0	0
$591$	−18.5218	−0.761887
$592$	0	0
$593$	18.5175	0.760424	0.380212	−	0.924899i	$-0.375851\pi$
$593$	18.5175	0.760424	0.380212	+	0.924899i	$0.375851\pi$
$594$	0	0
$595$	2.65142	0.108698
$596$	0	0
$597$	8.83493	0.361589
$598$	0	0
$599$	−5.55831	−0.227107	−0.113553	−	0.993532i	$-0.536223\pi$
$599$	−5.55831	−0.227107	−0.113553	+	0.993532i	$0.536223\pi$
$600$	0	0
$601$	−23.2149	−0.946954	−0.473477	−	0.880806i	$-0.657001\pi$
$601$	−23.2149	−0.946954	−0.473477	+	0.880806i	$0.657001\pi$
$602$	0	0
$603$	14.5458	0.592349
$604$	0	0
$605$	3.78419	0.153849
$606$	0	0
$607$	3.25348	0.132055	0.0660273	−	0.997818i	$-0.478968\pi$
$607$	3.25348	0.132055	0.0660273	+	0.997818i	$0.478968\pi$
$608$	0	0
$609$	36.6569	1.48541
$610$	0	0
$611$	35.5439	1.43795
$612$	0	0
$613$	−15.4078	−0.622315	−0.311158	−	0.950358i	$-0.600717\pi$
$613$	−15.4078	−0.622315	−0.311158	+	0.950358i	$0.600717\pi$
$614$	0	0
$615$	−14.8651	−0.599421
$616$	0	0
$617$	38.1339	1.53521	0.767606	−	0.640922i	$-0.221448\pi$
$617$	38.1339	1.53521	0.767606	+	0.640922i	$0.221448\pi$
$618$	0	0
$619$	42.1189	1.69290	0.846451	−	0.532466i	$-0.178735\pi$
$619$	42.1189	1.69290	0.846451	+	0.532466i	$0.178735\pi$
$620$	0	0
$621$	36.1066	1.44891
$622$	0	0
$623$	25.3723	1.01652
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	21.9917	0.878264
$628$	0	0
$629$	−6.44540	−0.256995
$630$	0	0
$631$	−29.8991	−1.19027	−0.595133	−	0.803627i	$-0.702901\pi$
$631$	−29.8991	−1.19027	−0.595133	+	0.803627i	$0.702901\pi$
$632$	0	0
$633$	13.3125	0.529126
$634$	0	0
$635$	6.40279	0.254087
$636$	0	0
$637$	−19.2727	−0.763610
$638$	0	0
$639$	10.9847	0.434547
$640$	0	0
$641$	−31.8436	−1.25775	−0.628874	−	0.777507i	$-0.716484\pi$
$641$	−31.8436	−1.25775	−0.628874	+	0.777507i	$0.716484\pi$
$642$	0	0
$643$	−16.3952	−0.646564	−0.323282	−	0.946303i	$-0.604786\pi$
$643$	−16.3952	−0.646564	−0.323282	+	0.946303i	$0.604786\pi$
$644$	0	0
$645$	10.7283	0.422425
$646$	0	0
$647$	−8.62054	−0.338908	−0.169454	−	0.985538i	$-0.554200\pi$
$647$	−8.62054	−0.338908	−0.169454	+	0.985538i	$0.554200\pi$
$648$	0	0
$649$	38.6511	1.51719
$650$	0	0
$651$	22.4781	0.880986
$652$	0	0
$653$	−27.1705	−1.06326	−0.531632	−	0.846975i	$-0.678421\pi$
$653$	−27.1705	−1.06326	−0.531632	+	0.846975i	$0.678421\pi$
$654$	0	0
$655$	14.6034	0.570603
$656$	0	0
$657$	8.29219	0.323509
$658$	0	0
$659$	−20.8605	−0.812609	−0.406304	−	0.913738i	$-0.633183\pi$
$659$	−20.8605	−0.812609	−0.406304	+	0.913738i	$0.633183\pi$
$660$	0	0
$661$	19.2152	0.747383	0.373692	−	0.927553i	$-0.378092\pi$
$661$	19.2152	0.747383	0.373692	+	0.927553i	$0.378092\pi$
$662$	0	0
$663$	5.91433	0.229694
$664$	0	0
$665$	−13.3356	−0.517134
$666$	0	0
$667$	−50.9623	−1.97327
$668$	0	0
$669$	−20.4193	−0.789457
$670$	0	0
$671$	−29.5602	−1.14116
$672$	0	0
$673$	−29.6437	−1.14268	−0.571341	−	0.820713i	$-0.693576\pi$
$673$	−29.6437	−1.14268	−0.571341	+	0.820713i	$0.693576\pi$
$674$	0	0
$675$	5.65635	0.217713
$676$	0	0
$677$	10.0487	0.386204	0.193102	−	0.981179i	$-0.438145\pi$
$677$	10.0487	0.386204	0.193102	+	0.981179i	$0.438145\pi$
$678$	0	0
$679$	41.5154	1.59322
$680$	0	0
$681$	−12.7365	−0.488064
$682$	0	0
$683$	−1.27859	−0.0489238	−0.0244619	−	0.999701i	$-0.507787\pi$
$683$	−1.27859	−0.0489238	−0.0244619	+	0.999701i	$0.507787\pi$
$684$	0	0
$685$	21.5765	0.824397
$686$	0	0
$687$	17.5609	0.669989
$688$	0	0
$689$	−8.15184	−0.310560
$690$	0	0
$691$	38.6689	1.47103	0.735517	−	0.677506i	$-0.236939\pi$
$691$	38.6689	1.47103	0.735517	+	0.677506i	$0.236939\pi$
$692$	0	0
$693$	−12.9674	−0.492592
$694$	0	0
$695$	−7.42104	−0.281496
$696$	0	0
$697$	8.58401	0.325143
$698$	0	0
$699$	−6.36583	−0.240778
$700$	0	0
$701$	20.4895	0.773877	0.386938	−	0.922106i	$-0.373533\pi$
$701$	20.4895	0.773877	0.386938	+	0.922106i	$0.373533\pi$
$702$	0	0
$703$	32.4180	1.22267
$704$	0	0
$705$	9.59037	0.361194
$706$	0	0
$707$	−37.3539	−1.40484
$708$	0	0
$709$	6.92461	0.260059	0.130030	−	0.991510i	$-0.458493\pi$
$709$	6.92461	0.260059	0.130030	+	0.991510i	$0.458493\pi$
$710$	0	0
$711$	5.29459	0.198563
$712$	0	0
$713$	−31.2502	−1.17033
$714$	0	0
$715$	−19.9977	−0.747872
$716$	0	0
$717$	−1.97961	−0.0739297
$718$	0	0
$719$	7.56860	0.282261	0.141131	−	0.989991i	$-0.454926\pi$
$719$	7.56860	0.282261	0.141131	+	0.989991i	$0.454926\pi$
$720$	0	0
$721$	−43.1914	−1.60853
$722$	0	0
$723$	22.0316	0.819364
$724$	0	0
$725$	−7.98360	−0.296503
$726$	0	0
$727$	21.7850	0.807960	0.403980	−	0.914768i	$-0.367627\pi$
$727$	21.7850	0.807960	0.403980	+	0.914768i	$0.367627\pi$
$728$	0	0
$729$	28.8070	1.06693
$730$	0	0
$731$	−6.19513	−0.229135
$732$	0	0
$733$	−37.2890	−1.37730	−0.688651	−	0.725093i	$-0.741797\pi$
$733$	−37.2890	−1.37730	−0.688651	+	0.725093i	$0.741797\pi$
$734$	0	0
$735$	−5.20009	−0.191808
$736$	0	0
$737$	−54.2608	−1.99872
$738$	0	0
$739$	−52.9735	−1.94866	−0.974330	−	0.225123i	$-0.927722\pi$
$739$	−52.9735	−1.94866	−0.974330	+	0.225123i	$0.927722\pi$
$740$	0	0
$741$	−29.7469	−1.09278
$742$	0	0
$743$	39.6643	1.45514	0.727571	−	0.686032i	$-0.240649\pi$
$743$	39.6643	1.45514	0.727571	+	0.686032i	$0.240649\pi$
$744$	0	0
$745$	−23.4817	−0.860302
$746$	0	0
$747$	−0.0505378	−0.00184908
$748$	0	0
$749$	14.9710	0.547030
$750$	0	0
$751$	34.0092	1.24101	0.620507	−	0.784201i	$-0.286927\pi$
$751$	34.0092	1.24101	0.620507	+	0.784201i	$0.286927\pi$
$752$	0	0
$753$	−36.0789	−1.31479
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−9.21701	−0.334998	−0.167499	−	0.985872i	$-0.553569\pi$
$757$	−9.21701	−0.334998	−0.167499	+	0.985872i	$0.553569\pi$
$758$	0	0
$759$	−34.4429	−1.25020
$760$	0	0
$761$	31.3311	1.13575	0.567875	−	0.823115i	$-0.307766\pi$
$761$	31.3311	1.13575	0.567875	+	0.823115i	$0.307766\pi$
$762$	0	0
$763$	43.3959	1.57104
$764$	0	0
$765$	−0.835260	−0.0301989
$766$	0	0
$767$	−52.2811	−1.88776
$768$	0	0
$769$	7.21347	0.260124	0.130062	−	0.991506i	$-0.458482\pi$
$769$	7.21347	0.260124	0.130062	+	0.991506i	$0.458482\pi$
$770$	0	0
$771$	14.4912	0.521888
$772$	0	0
$773$	55.0219	1.97900	0.989499	−	0.144536i	$-0.0461691\pi$
$773$	55.0219	1.97900	0.989499	+	0.144536i	$0.0461691\pi$
$774$	0	0
$775$	−4.89556	−0.175854
$776$	0	0
$777$	36.5203	1.31016
$778$	0	0
$779$	−43.1744	−1.54688
$780$	0	0
$781$	−40.9767	−1.46626
$782$	0	0
$783$	−45.1580	−1.61382
$784$	0	0
$785$	15.4023	0.549732
$786$	0	0
$787$	−1.86384	−0.0664387	−0.0332193	−	0.999448i	$-0.510576\pi$
$787$	−1.86384	−0.0664387	−0.0332193	+	0.999448i	$0.510576\pi$
$788$	0	0
$789$	8.48623	0.302118
$790$	0	0
$791$	−0.0191096	−0.000679458 0
$792$	0	0
$793$	39.9844	1.41989
$794$	0	0
$795$	−2.19951	−0.0780084
$796$	0	0
$797$	48.3216	1.71164	0.855820	−	0.517274i	$-0.173053\pi$
$797$	48.3216	1.71164	0.855820	+	0.517274i	$0.173053\pi$
$798$	0	0
$799$	−5.53805	−0.195922
$800$	0	0
$801$	−7.99288	−0.282415
$802$	0	0
$803$	−30.9328	−1.09159
$804$	0	0
$805$	20.8860	0.736136
$806$	0	0
$807$	−32.0008	−1.12648
$808$	0	0
$809$	−27.8708	−0.979884	−0.489942	−	0.871755i	$-0.662982\pi$
$809$	−27.8708	−0.979884	−0.489942	+	0.871755i	$0.662982\pi$
$810$	0	0
$811$	−16.7951	−0.589756	−0.294878	−	0.955535i	$-0.595279\pi$
$811$	−16.7951	−0.589756	−0.294878	+	0.955535i	$0.595279\pi$
$812$	0	0
$813$	−10.1223	−0.355005
$814$	0	0
$815$	−5.19282	−0.181897
$816$	0	0
$817$	31.1592	1.09012
$818$	0	0
$819$	17.5403	0.612907
$820$	0	0
$821$	−41.9196	−1.46300	−0.731502	−	0.681839i	$-0.761180\pi$
$821$	−41.9196	−1.46300	−0.731502	+	0.681839i	$0.761180\pi$
$822$	0	0
$823$	42.9448	1.49696	0.748480	−	0.663157i	$-0.230784\pi$
$823$	42.9448	1.49696	0.748480	+	0.663157i	$0.230784\pi$
$824$	0	0
$825$	−5.39573	−0.187855
$826$	0	0
$827$	−13.8783	−0.482596	−0.241298	−	0.970451i	$-0.577573\pi$
$827$	−13.8783	−0.482596	−0.241298	+	0.970451i	$0.577573\pi$
$828$	0	0
$829$	−9.64561	−0.335006	−0.167503	−	0.985872i	$-0.553570\pi$
$829$	−9.64561	−0.335006	−0.167503	+	0.985872i	$0.553570\pi$
$830$	0	0
$831$	−8.33057	−0.288984
$832$	0	0
$833$	3.00284	0.104042
$834$	0	0
$835$	−16.4222	−0.568312
$836$	0	0
$837$	−27.6910	−0.957141
$838$	0	0
$839$	−42.6996	−1.47415	−0.737077	−	0.675808i	$-0.763795\pi$
$839$	−42.6996	−1.47415	−0.737077	+	0.675808i	$0.763795\pi$
$840$	0	0
$841$	34.7378	1.19786
$842$	0	0
$843$	36.4050	1.25386
$844$	0	0
$845$	14.0498	0.483327
$846$	0	0
$847$	12.3817	0.425439
$848$	0	0
$849$	38.0628	1.30631
$850$	0	0
$851$	−50.7724	−1.74045
$852$	0	0
$853$	47.9867	1.64303	0.821517	−	0.570184i	$-0.193128\pi$
$853$	47.9867	1.64303	0.821517	+	0.570184i	$0.193128\pi$
$854$	0	0
$855$	4.20105	0.143673
$856$	0	0
$857$	35.4416	1.21066	0.605331	−	0.795974i	$-0.293041\pi$
$857$	35.4416	1.21066	0.605331	+	0.795974i	$0.293041\pi$
$858$	0	0
$859$	22.5526	0.769486	0.384743	−	0.923024i	$-0.374290\pi$
$859$	22.5526	0.769486	0.384743	+	0.923024i	$0.374290\pi$
$860$	0	0
$861$	−48.6379	−1.65758
$862$	0	0
$863$	12.5448	0.427028	0.213514	−	0.976940i	$-0.431509\pi$
$863$	12.5448	0.427028	0.213514	+	0.976940i	$0.431509\pi$
$864$	0	0
$865$	−14.6971	−0.499718
$866$	0	0
$867$	22.9347	0.778902
$868$	0	0
$869$	−19.7507	−0.669996
$870$	0	0
$871$	73.3954	2.48691
$872$	0	0
$873$	−13.0784	−0.442635
$874$	0	0
$875$	3.27194	0.110612
$876$	0	0
$877$	30.5541	1.03174	0.515869	−	0.856668i	$-0.327469\pi$
$877$	30.5541	1.03174	0.515869	+	0.856668i	$0.327469\pi$
$878$	0	0
$879$	14.1936	0.478738
$880$	0	0
$881$	48.4175	1.63123	0.815614	−	0.578597i	$-0.196399\pi$
$881$	48.4175	1.63123	0.815614	+	0.578597i	$0.196399\pi$
$882$	0	0
$883$	3.78906	0.127512	0.0637560	−	0.997966i	$-0.479692\pi$
$883$	3.78906	0.127512	0.0637560	+	0.997966i	$0.479692\pi$
$884$	0	0
$885$	−14.1063	−0.474180
$886$	0	0
$887$	−12.4598	−0.418361	−0.209180	−	0.977877i	$-0.567080\pi$
$887$	−12.4598	−0.418361	−0.209180	+	0.977877i	$0.567080\pi$
$888$	0	0
$889$	20.9496	0.702626
$890$	0	0
$891$	−18.6305	−0.624145
$892$	0	0
$893$	27.8543	0.932109
$894$	0	0
$895$	−5.69508	−0.190365
$896$	0	0
$897$	46.5890	1.55556
$898$	0	0
$899$	39.0842	1.30353
$900$	0	0
$901$	1.27012	0.0423140
$902$	0	0
$903$	35.1022	1.16813
$904$	0	0
$905$	−21.1956	−0.704566
$906$	0	0
$907$	−26.7569	−0.888447	−0.444223	−	0.895916i	$-0.646520\pi$
$907$	−26.7569	−0.888447	−0.444223	+	0.895916i	$0.646520\pi$
$908$	0	0
$909$	11.7674	0.390299
$910$	0	0
$911$	48.1522	1.59535	0.797677	−	0.603085i	$-0.206062\pi$
$911$	48.1522	1.59535	0.797677	+	0.603085i	$0.206062\pi$
$912$	0	0
$913$	0.188524	0.00623922
$914$	0	0
$915$	10.7885	0.356656
$916$	0	0
$917$	47.7815	1.57789
$918$	0	0
$919$	33.8920	1.11799	0.558996	−	0.829170i	$-0.311187\pi$
$919$	33.8920	1.11799	0.558996	+	0.829170i	$0.311187\pi$
$920$	0	0
$921$	−35.4398	−1.16778
$922$	0	0
$923$	55.4268	1.82439
$924$	0	0
$925$	−7.95385	−0.261521
$926$	0	0
$927$	13.6063	0.446890
$928$	0	0
$929$	−40.4302	−1.32647	−0.663236	−	0.748410i	$-0.730817\pi$
$929$	−40.4302	−1.32647	−0.663236	+	0.748410i	$0.730817\pi$
$930$	0	0
$931$	−15.1032	−0.494986
$932$	0	0
$933$	30.0957	0.985290
$934$	0	0
$935$	3.11581	0.101898
$936$	0	0
$937$	22.6483	0.739888	0.369944	−	0.929054i	$-0.379377\pi$
$937$	22.6483	0.739888	0.369944	+	0.929054i	$0.379377\pi$
$938$	0	0
$939$	7.41472	0.241970
$940$	0	0
$941$	25.0363	0.816160	0.408080	−	0.912946i	$-0.366198\pi$
$941$	25.0363	0.816160	0.408080	+	0.912946i	$0.366198\pi$
$942$	0	0
$943$	67.6188	2.20197
$944$	0	0
$945$	18.5073	0.602041
$946$	0	0
$947$	−22.6428	−0.735792	−0.367896	−	0.929867i	$-0.619922\pi$
$947$	−22.6428	−0.735792	−0.367896	+	0.929867i	$0.619922\pi$
$948$	0	0
$949$	41.8410	1.35822
$950$	0	0
$951$	−32.8036	−1.06373
$952$	0	0
$953$	1.93946	0.0628253	0.0314126	−	0.999507i	$-0.489999\pi$
$953$	1.93946	0.0628253	0.0314126	+	0.999507i	$0.489999\pi$
$954$	0	0
$955$	1.74064	0.0563258
$956$	0	0
$957$	43.0773	1.39249
$958$	0	0
$959$	70.5972	2.27970
$960$	0	0
$961$	−7.03349	−0.226887
$962$	0	0
$963$	−4.71623	−0.151979
$964$	0	0
$965$	20.8045	0.669722
$966$	0	0
$967$	−11.8997	−0.382669	−0.191335	−	0.981525i	$-0.561282\pi$
$967$	−11.8997	−0.382669	−0.191335	+	0.981525i	$0.561282\pi$
$968$	0	0
$969$	4.63482	0.148892
$970$	0	0
$971$	19.5792	0.628327	0.314163	−	0.949369i	$-0.398276\pi$
$971$	19.5792	0.628327	0.314163	+	0.949369i	$0.398276\pi$
$972$	0	0
$973$	−24.2812	−0.778420
$974$	0	0
$975$	7.29849	0.233739
$976$	0	0
$977$	60.3027	1.92925	0.964626	−	0.263621i	$-0.0849168\pi$
$977$	60.3027	1.92925	0.964626	+	0.263621i	$0.0849168\pi$
$978$	0	0
$979$	29.8162	0.952931
$980$	0	0
$981$	−13.6707	−0.436473
$982$	0	0
$983$	−49.9657	−1.59366	−0.796829	−	0.604205i	$-0.793491\pi$
$983$	−49.9657	−1.59366	−0.796829	+	0.604205i	$0.793491\pi$
$984$	0	0
$985$	13.1987	0.420547
$986$	0	0
$987$	31.3791	0.998809
$988$	0	0
$989$	−48.8009	−1.55178
$990$	0	0
$991$	37.4437	1.18944	0.594720	−	0.803933i	$-0.297263\pi$
$991$	37.4437	1.18944	0.594720	+	0.803933i	$0.297263\pi$
$992$	0	0
$993$	34.6843	1.10067
$994$	0	0
$995$	−6.29581	−0.199591
$996$	0	0
$997$	−47.5262	−1.50517	−0.752585	−	0.658495i	$-0.771193\pi$
$997$	−47.5262	−1.50517	−0.752585	+	0.658495i	$0.771193\pi$
$998$	0	0
$999$	−44.9898	−1.42341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.8	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.8	✓	19	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.40330 q^{3} +1.00000 q^{5} +3.27194 q^{7} -1.03074 q^{9} +O(q^{10})\)	\(q-1.40330 q^{3} +1.00000 q^{5} +3.27194 q^{7} -1.03074 q^{9} +3.84502 q^{11} -5.20094 q^{13} -1.40330 q^{15} +0.810350 q^{17} -4.07576 q^{19} -4.59153 q^{21} +6.38337 q^{23} +1.00000 q^{25} +5.65635 q^{27} -7.98360 q^{29} -4.89556 q^{31} -5.39573 q^{33} +3.27194 q^{35} -7.95385 q^{37} +7.29849 q^{39} +10.5930 q^{41} -7.64500 q^{43} -1.03074 q^{45} -6.83414 q^{47} +3.70561 q^{49} -1.13717 q^{51} +1.56738 q^{53} +3.84502 q^{55} +5.71952 q^{57} +10.0522 q^{59} -7.68791 q^{61} -3.37252 q^{63} -5.20094 q^{65} -14.1120 q^{67} -8.95780 q^{69} -10.6571 q^{71} -8.04489 q^{73} -1.40330 q^{75} +12.5807 q^{77} -5.13669 q^{79} -4.84535 q^{81} +0.0490305 q^{83} +0.810350 q^{85} +11.2034 q^{87} +7.75451 q^{89} -17.0172 q^{91} +6.86996 q^{93} -4.07576 q^{95} +12.6883 q^{97} -3.96322 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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