LMFDB - Embedded newform 6040.2.a.p.1.5

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			1.5
Root			$2.34269$ 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)

$f(q)$	$=$	$q-2.34269 q^{3} +1.00000 q^{5} -1.53409 q^{7} +2.48818 q^{9} +O(q^{10})$	$q-2.34269 q^{3} +1.00000 q^{5} -1.53409 q^{7} +2.48818 q^{9} -2.13813 q^{11} -0.470591 q^{13} -2.34269 q^{15} +2.13955 q^{17} -4.98365 q^{19} +3.59390 q^{21} -2.33382 q^{23} +1.00000 q^{25} +1.19902 q^{27} +6.12425 q^{29} +6.34138 q^{31} +5.00898 q^{33} -1.53409 q^{35} +2.64294 q^{37} +1.10245 q^{39} +6.14093 q^{41} -7.94163 q^{43} +2.48818 q^{45} +10.8356 q^{47} -4.64656 q^{49} -5.01230 q^{51} +6.95012 q^{53} -2.13813 q^{55} +11.6751 q^{57} -9.95452 q^{59} -11.6416 q^{61} -3.81711 q^{63} -0.470591 q^{65} +1.12749 q^{67} +5.46741 q^{69} -8.62775 q^{71} -1.80753 q^{73} -2.34269 q^{75} +3.28010 q^{77} +5.78496 q^{79} -10.2735 q^{81} +10.7240 q^{83} +2.13955 q^{85} -14.3472 q^{87} -11.5253 q^{89} +0.721931 q^{91} -14.8559 q^{93} -4.98365 q^{95} -3.80804 q^{97} -5.32007 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.34269	−1.35255	−0.676276	−	0.736649i	$-0.736407\pi$
$3$	−2.34269	−1.35255	−0.676276	+	0.736649i	$0.736407\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.53409	−0.579833	−0.289916	−	0.957052i	$-0.593628\pi$
$7$	−1.53409	−0.579833	−0.289916	+	0.957052i	$0.593628\pi$
$8$	0	0
$9$	2.48818	0.829395
$10$	0	0
$11$	−2.13813	−0.644672	−0.322336	−	0.946625i	$-0.604468\pi$
$11$	−2.13813	−0.644672	−0.322336	+	0.946625i	$0.604468\pi$
$12$	0	0
$13$	−0.470591	−0.130518	−0.0652592	−	0.997868i	$-0.520787\pi$
$13$	−0.470591	−0.130518	−0.0652592	+	0.997868i	$0.520787\pi$
$14$	0	0
$15$	−2.34269	−0.604879
$16$	0	0
$17$	2.13955	0.518918	0.259459	−	0.965754i	$-0.416456\pi$
$17$	2.13955	0.518918	0.259459	+	0.965754i	$0.416456\pi$
$18$	0	0
$19$	−4.98365	−1.14333	−0.571663	−	0.820488i	$-0.693702\pi$
$19$	−4.98365	−1.14333	−0.571663	+	0.820488i	$0.693702\pi$
$20$	0	0
$21$	3.59390	0.784254
$22$	0	0
$23$	−2.33382	−0.486635	−0.243317	−	0.969947i	$-0.578236\pi$
$23$	−2.33382	−0.486635	−0.243317	+	0.969947i	$0.578236\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.19902	0.230752
$28$	0	0
$29$	6.12425	1.13725	0.568623	−	0.822599i	$-0.307477\pi$
$29$	6.12425	1.13725	0.568623	+	0.822599i	$0.307477\pi$
$30$	0	0
$31$	6.34138	1.13894	0.569472	−	0.822011i	$-0.307148\pi$
$31$	6.34138	1.13894	0.569472	+	0.822011i	$0.307148\pi$
$32$	0	0
$33$	5.00898	0.871952
$34$	0	0
$35$	−1.53409	−0.259309
$36$	0	0
$37$	2.64294	0.434497	0.217249	−	0.976116i	$-0.430292\pi$
$37$	2.64294	0.434497	0.217249	+	0.976116i	$0.430292\pi$
$38$	0	0
$39$	1.10245	0.176533
$40$	0	0
$41$	6.14093	0.959052	0.479526	−	0.877528i	$-0.340809\pi$
$41$	6.14093	0.959052	0.479526	+	0.877528i	$0.340809\pi$
$42$	0	0
$43$	−7.94163	−1.21109	−0.605544	−	0.795812i	$-0.707044\pi$
$43$	−7.94163	−1.21109	−0.605544	+	0.795812i	$0.707044\pi$
$44$	0	0
$45$	2.48818	0.370917
$46$	0	0
$47$	10.8356	1.58053	0.790267	−	0.612763i	$-0.209942\pi$
$47$	10.8356	1.58053	0.790267	+	0.612763i	$0.209942\pi$
$48$	0	0
$49$	−4.64656	−0.663794
$50$	0	0
$51$	−5.01230	−0.701863
$52$	0	0
$53$	6.95012	0.954673	0.477337	−	0.878721i	$-0.341602\pi$
$53$	6.95012	0.954673	0.477337	+	0.878721i	$0.341602\pi$
$54$	0	0
$55$	−2.13813	−0.288306
$56$	0	0
$57$	11.6751	1.54641
$58$	0	0
$59$	−9.95452	−1.29597	−0.647984	−	0.761654i	$-0.724388\pi$
$59$	−9.95452	−1.29597	−0.647984	+	0.761654i	$0.724388\pi$
$60$	0	0
$61$	−11.6416	−1.49056	−0.745280	−	0.666752i	$-0.767684\pi$
$61$	−11.6416	−1.49056	−0.745280	+	0.666752i	$0.767684\pi$
$62$	0	0
$63$	−3.81711	−0.480911
$64$	0	0
$65$	−0.470591	−0.0583696
$66$	0	0
$67$	1.12749	0.137745	0.0688723	−	0.997625i	$-0.478060\pi$
$67$	1.12749	0.137745	0.0688723	+	0.997625i	$0.478060\pi$
$68$	0	0
$69$	5.46741	0.658199
$70$	0	0
$71$	−8.62775	−1.02393	−0.511963	−	0.859008i	$-0.671081\pi$
$71$	−8.62775	−1.02393	−0.511963	+	0.859008i	$0.671081\pi$
$72$	0	0
$73$	−1.80753	−0.211555	−0.105778	−	0.994390i	$-0.533733\pi$
$73$	−1.80753	−0.211555	−0.105778	+	0.994390i	$0.533733\pi$
$74$	0	0
$75$	−2.34269	−0.270510
$76$	0	0
$77$	3.28010	0.373802
$78$	0	0
$79$	5.78496	0.650858	0.325429	−	0.945566i	$-0.394491\pi$
$79$	5.78496	0.650858	0.325429	+	0.945566i	$0.394491\pi$
$80$	0	0
$81$	−10.2735	−1.14150
$82$	0	0
$83$	10.7240	1.17711	0.588557	−	0.808456i	$-0.299696\pi$
$83$	10.7240	1.17711	0.588557	+	0.808456i	$0.299696\pi$
$84$	0	0
$85$	2.13955	0.232067
$86$	0	0
$87$	−14.3472	−1.53818
$88$	0	0
$89$	−11.5253	−1.22168	−0.610841	−	0.791753i	$-0.709169\pi$
$89$	−11.5253	−1.22168	−0.610841	+	0.791753i	$0.709169\pi$
$90$	0	0
$91$	0.721931	0.0756789
$92$	0	0
$93$	−14.8559	−1.54048
$94$	0	0
$95$	−4.98365	−0.511311
$96$	0	0
$97$	−3.80804	−0.386648	−0.193324	−	0.981135i	$-0.561927\pi$
$97$	−3.80804	−0.386648	−0.193324	+	0.981135i	$0.561927\pi$
$98$	0	0
$99$	−5.32007	−0.534687
$100$	0	0
$101$	16.4634	1.63817	0.819086	−	0.573671i	$-0.194481\pi$
$101$	16.4634	1.63817	0.819086	+	0.573671i	$0.194481\pi$
$102$	0	0
$103$	8.93869	0.880755	0.440377	−	0.897813i	$-0.354845\pi$
$103$	8.93869	0.880755	0.440377	+	0.897813i	$0.354845\pi$
$104$	0	0
$105$	3.59390	0.350729
$106$	0	0
$107$	−9.36125	−0.904986	−0.452493	−	0.891768i	$-0.649465\pi$
$107$	−9.36125	−0.904986	−0.452493	+	0.891768i	$0.649465\pi$
$108$	0	0
$109$	6.95632	0.666294	0.333147	−	0.942875i	$-0.391889\pi$
$109$	6.95632	0.666294	0.333147	+	0.942875i	$0.391889\pi$
$110$	0	0
$111$	−6.19159	−0.587680
$112$	0	0
$113$	7.75915	0.729919	0.364960	−	0.931023i	$-0.381083\pi$
$113$	7.75915	0.729919	0.364960	+	0.931023i	$0.381083\pi$
$114$	0	0
$115$	−2.33382	−0.217630
$116$	0	0
$117$	−1.17092	−0.108251
$118$	0	0
$119$	−3.28227	−0.300886
$120$	0	0
$121$	−6.42838	−0.584398
$122$	0	0
$123$	−14.3863	−1.29717
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.2593	0.910367	0.455184	−	0.890398i	$-0.349574\pi$
$127$	10.2593	0.910367	0.455184	+	0.890398i	$0.349574\pi$
$128$	0	0
$129$	18.6048	1.63806
$130$	0	0
$131$	4.40919	0.385232	0.192616	−	0.981274i	$-0.438303\pi$
$131$	4.40919	0.385232	0.192616	+	0.981274i	$0.438303\pi$
$132$	0	0
$133$	7.64538	0.662938
$134$	0	0
$135$	1.19902	0.103195
$136$	0	0
$137$	17.3442	1.48181	0.740906	−	0.671609i	$-0.234396\pi$
$137$	17.3442	1.48181	0.740906	+	0.671609i	$0.234396\pi$
$138$	0	0
$139$	−18.7375	−1.58929	−0.794646	−	0.607073i	$-0.792344\pi$
$139$	−18.7375	−1.58929	−0.794646	+	0.607073i	$0.792344\pi$
$140$	0	0
$141$	−25.3844	−2.13775
$142$	0	0
$143$	1.00619	0.0841415
$144$	0	0
$145$	6.12425	0.508591
$146$	0	0
$147$	10.8854	0.897815
$148$	0	0
$149$	−13.3543	−1.09402	−0.547012	−	0.837125i	$-0.684235\pi$
$149$	−13.3543	−1.09402	−0.547012	+	0.837125i	$0.684235\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	5.32360	0.430388
$154$	0	0
$155$	6.34138	0.509352
$156$	0	0
$157$	−2.26259	−0.180574	−0.0902871	−	0.995916i	$-0.528778\pi$
$157$	−2.26259	−0.180574	−0.0902871	+	0.995916i	$0.528778\pi$
$158$	0	0
$159$	−16.2820	−1.29124
$160$	0	0
$161$	3.58030	0.282167
$162$	0	0
$163$	1.08367	0.0848794	0.0424397	−	0.999099i	$-0.486487\pi$
$163$	1.08367	0.0848794	0.0424397	+	0.999099i	$0.486487\pi$
$164$	0	0
$165$	5.00898	0.389949
$166$	0	0
$167$	−6.51511	−0.504155	−0.252077	−	0.967707i	$-0.581114\pi$
$167$	−6.51511	−0.504155	−0.252077	+	0.967707i	$0.581114\pi$
$168$	0	0
$169$	−12.7785	−0.982965
$170$	0	0
$171$	−12.4002	−0.948269
$172$	0	0
$173$	−3.76879	−0.286536	−0.143268	−	0.989684i	$-0.545761\pi$
$173$	−3.76879	−0.286536	−0.143268	+	0.989684i	$0.545761\pi$
$174$	0	0
$175$	−1.53409	−0.115967
$176$	0	0
$177$	23.3203	1.75286
$178$	0	0
$179$	−2.53535	−0.189501	−0.0947506	−	0.995501i	$-0.530205\pi$
$179$	−2.53535	−0.189501	−0.0947506	+	0.995501i	$0.530205\pi$
$180$	0	0
$181$	5.12147	0.380676	0.190338	−	0.981719i	$-0.439042\pi$
$181$	5.12147	0.380676	0.190338	+	0.981719i	$0.439042\pi$
$182$	0	0
$183$	27.2727	2.01606
$184$	0	0
$185$	2.64294	0.194313
$186$	0	0
$187$	−4.57465	−0.334532
$188$	0	0
$189$	−1.83941	−0.133798
$190$	0	0
$191$	23.9806	1.73517	0.867587	−	0.497286i	$-0.165670\pi$
$191$	23.9806	1.73517	0.867587	+	0.497286i	$0.165670\pi$
$192$	0	0
$193$	18.3581	1.32144	0.660722	−	0.750631i	$-0.270250\pi$
$193$	18.3581	1.32144	0.660722	+	0.750631i	$0.270250\pi$
$194$	0	0
$195$	1.10245	0.0789479
$196$	0	0
$197$	−25.3522	−1.80627	−0.903133	−	0.429360i	$-0.858739\pi$
$197$	−25.3522	−1.80627	−0.903133	+	0.429360i	$0.858739\pi$
$198$	0	0
$199$	1.52208	0.107897	0.0539486	−	0.998544i	$-0.482819\pi$
$199$	1.52208	0.107897	0.0539486	+	0.998544i	$0.482819\pi$
$200$	0	0
$201$	−2.64135	−0.186307
$202$	0	0
$203$	−9.39518	−0.659412
$204$	0	0
$205$	6.14093	0.428901
$206$	0	0
$207$	−5.80697	−0.403612
$208$	0	0
$209$	10.6557	0.737070
$210$	0	0
$211$	5.84374	0.402300	0.201150	−	0.979560i	$-0.435532\pi$
$211$	5.84374	0.402300	0.201150	+	0.979560i	$0.435532\pi$
$212$	0	0
$213$	20.2121	1.38491
$214$	0	0
$215$	−7.94163	−0.541615
$216$	0	0
$217$	−9.72826	−0.660398
$218$	0	0
$219$	4.23447	0.286139
$220$	0	0
$221$	−1.00685	−0.0677283
$222$	0	0
$223$	−26.8074	−1.79515	−0.897577	−	0.440858i	$-0.854674\pi$
$223$	−26.8074	−1.79515	−0.897577	+	0.440858i	$0.854674\pi$
$224$	0	0
$225$	2.48818	0.165879
$226$	0	0
$227$	−7.51585	−0.498844	−0.249422	−	0.968395i	$-0.580241\pi$
$227$	−7.51585	−0.498844	−0.249422	+	0.968395i	$0.580241\pi$
$228$	0	0
$229$	−14.7307	−0.973435	−0.486717	−	0.873559i	$-0.661806\pi$
$229$	−14.7307	−0.973435	−0.486717	+	0.873559i	$0.661806\pi$
$230$	0	0
$231$	−7.68425	−0.505586
$232$	0	0
$233$	−21.5943	−1.41469	−0.707344	−	0.706870i	$-0.750107\pi$
$233$	−21.5943	−1.41469	−0.707344	+	0.706870i	$0.750107\pi$
$234$	0	0
$235$	10.8356	0.706836
$236$	0	0
$237$	−13.5523	−0.880319
$238$	0	0
$239$	−3.63938	−0.235412	−0.117706	−	0.993048i	$-0.537554\pi$
$239$	−3.63938	−0.235412	−0.117706	+	0.993048i	$0.537554\pi$
$240$	0	0
$241$	−13.8607	−0.892844	−0.446422	−	0.894822i	$-0.647302\pi$
$241$	−13.8607	−0.892844	−0.446422	+	0.894822i	$0.647302\pi$
$242$	0	0
$243$	20.4705	1.31318
$244$	0	0
$245$	−4.64656	−0.296858
$246$	0	0
$247$	2.34526	0.149225
$248$	0	0
$249$	−25.1230	−1.59211
$250$	0	0
$251$	−17.7326	−1.11927	−0.559637	−	0.828738i	$-0.689059\pi$
$251$	−17.7326	−1.11927	−0.559637	+	0.828738i	$0.689059\pi$
$252$	0	0
$253$	4.99002	0.313720
$254$	0	0
$255$	−5.01230	−0.313883
$256$	0	0
$257$	−12.6072	−0.786414	−0.393207	−	0.919450i	$-0.628634\pi$
$257$	−12.6072	−0.786414	−0.393207	+	0.919450i	$0.628634\pi$
$258$	0	0
$259$	−4.05452	−0.251936
$260$	0	0
$261$	15.2383	0.943225
$262$	0	0
$263$	−29.0394	−1.79065	−0.895324	−	0.445416i	$-0.853056\pi$
$263$	−29.0394	−1.79065	−0.895324	+	0.445416i	$0.853056\pi$
$264$	0	0
$265$	6.95012	0.426943
$266$	0	0
$267$	27.0002	1.65239
$268$	0	0
$269$	26.8894	1.63947	0.819737	−	0.572740i	$-0.194119\pi$
$269$	26.8894	1.63947	0.819737	+	0.572740i	$0.194119\pi$
$270$	0	0
$271$	−12.7656	−0.775455	−0.387727	−	0.921774i	$-0.626740\pi$
$271$	−12.7656	−0.775455	−0.387727	+	0.921774i	$0.626740\pi$
$272$	0	0
$273$	−1.69126	−0.102360
$274$	0	0
$275$	−2.13813	−0.128934
$276$	0	0
$277$	29.7716	1.78880	0.894400	−	0.447269i	$-0.147603\pi$
$277$	29.7716	1.78880	0.894400	+	0.447269i	$0.147603\pi$
$278$	0	0
$279$	15.7785	0.944635
$280$	0	0
$281$	11.7299	0.699750	0.349875	−	0.936796i	$-0.386224\pi$
$281$	11.7299	0.699750	0.349875	+	0.936796i	$0.386224\pi$
$282$	0	0
$283$	−24.8547	−1.47746	−0.738729	−	0.674002i	$-0.764574\pi$
$283$	−24.8547	−1.47746	−0.738729	+	0.674002i	$0.764574\pi$
$284$	0	0
$285$	11.6751	0.691575
$286$	0	0
$287$	−9.42076	−0.556090
$288$	0	0
$289$	−12.4223	−0.730724
$290$	0	0
$291$	8.92105	0.522961
$292$	0	0
$293$	−22.7840	−1.33105	−0.665526	−	0.746374i	$-0.731793\pi$
$293$	−22.7840	−1.33105	−0.665526	+	0.746374i	$0.731793\pi$
$294$	0	0
$295$	−9.95452	−0.579575
$296$	0	0
$297$	−2.56367	−0.148759
$298$	0	0
$299$	1.09827	0.0635148
$300$	0	0
$301$	12.1832	0.702229
$302$	0	0
$303$	−38.5687	−2.21571
$304$	0	0
$305$	−11.6416	−0.666599
$306$	0	0
$307$	−17.8743	−1.02014	−0.510070	−	0.860133i	$-0.670381\pi$
$307$	−17.8743	−1.02014	−0.510070	+	0.860133i	$0.670381\pi$
$308$	0	0
$309$	−20.9406	−1.19127
$310$	0	0
$311$	−14.3909	−0.816035	−0.408017	−	0.912974i	$-0.633780\pi$
$311$	−14.3909	−0.816035	−0.408017	+	0.912974i	$0.633780\pi$
$312$	0	0
$313$	−5.94964	−0.336294	−0.168147	−	0.985762i	$-0.553778\pi$
$313$	−5.94964	−0.336294	−0.168147	+	0.985762i	$0.553778\pi$
$314$	0	0
$315$	−3.81711	−0.215070
$316$	0	0
$317$	21.2783	1.19511	0.597555	−	0.801828i	$-0.296139\pi$
$317$	21.2783	1.19511	0.597555	+	0.801828i	$0.296139\pi$
$318$	0	0
$319$	−13.0945	−0.733150
$320$	0	0
$321$	21.9305	1.22404
$322$	0	0
$323$	−10.6628	−0.593293
$324$	0	0
$325$	−0.470591	−0.0261037
$326$	0	0
$327$	−16.2965	−0.901197
$328$	0	0
$329$	−16.6228	−0.916446
$330$	0	0
$331$	0.843628	0.0463700	0.0231850	−	0.999731i	$-0.492619\pi$
$331$	0.843628	0.0463700	0.0231850	+	0.999731i	$0.492619\pi$
$332$	0	0
$333$	6.57613	0.360370
$334$	0	0
$335$	1.12749	0.0616012
$336$	0	0
$337$	29.7351	1.61977	0.809886	−	0.586587i	$-0.199529\pi$
$337$	29.7351	1.61977	0.809886	+	0.586587i	$0.199529\pi$
$338$	0	0
$339$	−18.1773	−0.987253
$340$	0	0
$341$	−13.5587	−0.734245
$342$	0	0
$343$	17.8669	0.964722
$344$	0	0
$345$	5.46741	0.294355
$346$	0	0
$347$	5.71094	0.306579	0.153290	−	0.988181i	$-0.451013\pi$
$347$	5.71094	0.306579	0.153290	+	0.988181i	$0.451013\pi$
$348$	0	0
$349$	−22.9550	−1.22875	−0.614377	−	0.789013i	$-0.710593\pi$
$349$	−22.9550	−1.22875	−0.614377	+	0.789013i	$0.710593\pi$
$350$	0	0
$351$	−0.564249	−0.0301174
$352$	0	0
$353$	−28.1580	−1.49870	−0.749349	−	0.662176i	$-0.769633\pi$
$353$	−28.1580	−1.49870	−0.749349	+	0.662176i	$0.769633\pi$
$354$	0	0
$355$	−8.62775	−0.457913
$356$	0	0
$357$	7.68934	0.406963
$358$	0	0
$359$	8.39658	0.443154	0.221577	−	0.975143i	$-0.428880\pi$
$359$	8.39658	0.443154	0.221577	+	0.975143i	$0.428880\pi$
$360$	0	0
$361$	5.83672	0.307196
$362$	0	0
$363$	15.0597	0.790429
$364$	0	0
$365$	−1.80753	−0.0946104
$366$	0	0
$367$	17.4444	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$367$	17.4444	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$368$	0	0
$369$	15.2798	0.795433
$370$	0	0
$371$	−10.6621	−0.553551
$372$	0	0
$373$	−8.12399	−0.420644	−0.210322	−	0.977632i	$-0.567451\pi$
$373$	−8.12399	−0.420644	−0.210322	+	0.977632i	$0.567451\pi$
$374$	0	0
$375$	−2.34269	−0.120976
$376$	0	0
$377$	−2.88202	−0.148431
$378$	0	0
$379$	−17.4033	−0.893950	−0.446975	−	0.894547i	$-0.647499\pi$
$379$	−17.4033	−0.893950	−0.446975	+	0.894547i	$0.647499\pi$
$380$	0	0
$381$	−24.0344	−1.23132
$382$	0	0
$383$	15.6984	0.802153	0.401076	−	0.916045i	$-0.368636\pi$
$383$	15.6984	0.802153	0.401076	+	0.916045i	$0.368636\pi$
$384$	0	0
$385$	3.28010	0.167169
$386$	0	0
$387$	−19.7603	−1.00447
$388$	0	0
$389$	−16.2270	−0.822741	−0.411370	−	0.911468i	$-0.634950\pi$
$389$	−16.2270	−0.822741	−0.411370	+	0.911468i	$0.634950\pi$
$390$	0	0
$391$	−4.99333	−0.252524
$392$	0	0
$393$	−10.3293	−0.521047
$394$	0	0
$395$	5.78496	0.291073
$396$	0	0
$397$	−31.5789	−1.58490	−0.792450	−	0.609937i	$-0.791195\pi$
$397$	−31.5789	−1.58490	−0.792450	+	0.609937i	$0.791195\pi$
$398$	0	0
$399$	−17.9107	−0.896658
$400$	0	0
$401$	−2.88267	−0.143954	−0.0719768	−	0.997406i	$-0.522931\pi$
$401$	−2.88267	−0.143954	−0.0719768	+	0.997406i	$0.522931\pi$
$402$	0	0
$403$	−2.98419	−0.148653
$404$	0	0
$405$	−10.2735	−0.510494
$406$	0	0
$407$	−5.65097	−0.280108
$408$	0	0
$409$	17.9093	0.885557	0.442778	−	0.896631i	$-0.353993\pi$
$409$	17.9093	0.885557	0.442778	+	0.896631i	$0.353993\pi$
$410$	0	0
$411$	−40.6320	−2.00423
$412$	0	0
$413$	15.2712	0.751445
$414$	0	0
$415$	10.7240	0.526421
$416$	0	0
$417$	43.8961	2.14960
$418$	0	0
$419$	6.13161	0.299549	0.149774	−	0.988720i	$-0.452145\pi$
$419$	6.13161	0.299549	0.149774	+	0.988720i	$0.452145\pi$
$420$	0	0
$421$	−35.7932	−1.74445	−0.872227	−	0.489100i	$-0.837325\pi$
$421$	−35.7932	−1.74445	−0.872227	+	0.489100i	$0.837325\pi$
$422$	0	0
$423$	26.9610	1.31089
$424$	0	0
$425$	2.13955	0.103784
$426$	0	0
$427$	17.8594	0.864276
$428$	0	0
$429$	−2.35718	−0.113806
$430$	0	0
$431$	−32.5146	−1.56617	−0.783087	−	0.621912i	$-0.786356\pi$
$431$	−32.5146	−1.56617	−0.783087	+	0.621912i	$0.786356\pi$
$432$	0	0
$433$	27.5934	1.32606	0.663028	−	0.748595i	$-0.269271\pi$
$433$	27.5934	1.32606	0.663028	+	0.748595i	$0.269271\pi$
$434$	0	0
$435$	−14.3472	−0.687896
$436$	0	0
$437$	11.6309	0.556383
$438$	0	0
$439$	11.1400	0.531685	0.265843	−	0.964016i	$-0.414350\pi$
$439$	11.1400	0.531685	0.265843	+	0.964016i	$0.414350\pi$
$440$	0	0
$441$	−11.5615	−0.550547
$442$	0	0
$443$	5.43580	0.258262	0.129131	−	0.991628i	$-0.458781\pi$
$443$	5.43580	0.258262	0.129131	+	0.991628i	$0.458781\pi$
$444$	0	0
$445$	−11.5253	−0.546353
$446$	0	0
$447$	31.2849	1.47972
$448$	0	0
$449$	−18.3551	−0.866230	−0.433115	−	0.901339i	$-0.642586\pi$
$449$	−18.3551	−0.866230	−0.433115	+	0.901339i	$0.642586\pi$
$450$	0	0
$451$	−13.1301	−0.618274
$452$	0	0
$453$	−2.34269	−0.110069
$454$	0	0
$455$	0.721931	0.0338446
$456$	0	0
$457$	−9.54149	−0.446332	−0.223166	−	0.974780i	$-0.571639\pi$
$457$	−9.54149	−0.446332	−0.223166	+	0.974780i	$0.571639\pi$
$458$	0	0
$459$	2.56537	0.119741
$460$	0	0
$461$	−32.4864	−1.51304	−0.756521	−	0.653970i	$-0.773102\pi$
$461$	−32.4864	−1.51304	−0.756521	+	0.653970i	$0.773102\pi$
$462$	0	0
$463$	−7.99549	−0.371582	−0.185791	−	0.982589i	$-0.559485\pi$
$463$	−7.99549	−0.371582	−0.185791	+	0.982589i	$0.559485\pi$
$464$	0	0
$465$	−14.8559	−0.688924
$466$	0	0
$467$	20.3813	0.943136	0.471568	−	0.881830i	$-0.343688\pi$
$467$	20.3813	0.943136	0.471568	+	0.881830i	$0.343688\pi$
$468$	0	0
$469$	−1.72967	−0.0798688
$470$	0	0
$471$	5.30054	0.244236
$472$	0	0
$473$	16.9803	0.780754
$474$	0	0
$475$	−4.98365	−0.228665
$476$	0	0
$477$	17.2932	0.791801
$478$	0	0
$479$	11.9464	0.545845	0.272923	−	0.962036i	$-0.412010\pi$
$479$	11.9464	0.545845	0.272923	+	0.962036i	$0.412010\pi$
$480$	0	0
$481$	−1.24375	−0.0567099
$482$	0	0
$483$	−8.38752	−0.381645
$484$	0	0
$485$	−3.80804	−0.172914
$486$	0	0
$487$	−18.3090	−0.829661	−0.414830	−	0.909899i	$-0.636159\pi$
$487$	−18.3090	−0.829661	−0.414830	+	0.909899i	$0.636159\pi$
$488$	0	0
$489$	−2.53870	−0.114804
$490$	0	0
$491$	−38.3267	−1.72966	−0.864831	−	0.502063i	$-0.832574\pi$
$491$	−38.3267	−1.72966	−0.864831	+	0.502063i	$0.832574\pi$
$492$	0	0
$493$	13.1032	0.590137
$494$	0	0
$495$	−5.32007	−0.239120
$496$	0	0
$497$	13.2358	0.593706
$498$	0	0
$499$	−26.2404	−1.17468	−0.587340	−	0.809340i	$-0.699825\pi$
$499$	−26.2404	−1.17468	−0.587340	+	0.809340i	$0.699825\pi$
$500$	0	0
$501$	15.2629	0.681895
$502$	0	0
$503$	14.4877	0.645973	0.322986	−	0.946404i	$-0.395313\pi$
$503$	14.4877	0.645973	0.322986	+	0.946404i	$0.395313\pi$
$504$	0	0
$505$	16.4634	0.732613
$506$	0	0
$507$	29.9361	1.32951
$508$	0	0
$509$	35.2685	1.56325	0.781624	−	0.623750i	$-0.214392\pi$
$509$	35.2685	1.56325	0.781624	+	0.623750i	$0.214392\pi$
$510$	0	0
$511$	2.77292	0.122667
$512$	0	0
$513$	−5.97550	−0.263825
$514$	0	0
$515$	8.93869	0.393886
$516$	0	0
$517$	−23.1680	−1.01893
$518$	0	0
$519$	8.82911	0.387555
$520$	0	0
$521$	−32.6900	−1.43217	−0.716087	−	0.698011i	$-0.754069\pi$
$521$	−32.6900	−1.43217	−0.716087	+	0.698011i	$0.754069\pi$
$522$	0	0
$523$	15.0223	0.656880	0.328440	−	0.944525i	$-0.393477\pi$
$523$	15.0223	0.656880	0.328440	+	0.944525i	$0.393477\pi$
$524$	0	0
$525$	3.59390	0.156851
$526$	0	0
$527$	13.5677	0.591019
$528$	0	0
$529$	−17.5533	−0.763187
$530$	0	0
$531$	−24.7687	−1.07487
$532$	0	0
$533$	−2.88987	−0.125174
$534$	0	0
$535$	−9.36125	−0.404722
$536$	0	0
$537$	5.93954	0.256310
$538$	0	0
$539$	9.93496	0.427929
$540$	0	0
$541$	23.3271	1.00291	0.501455	−	0.865184i	$-0.332798\pi$
$541$	23.3271	1.00291	0.501455	+	0.865184i	$0.332798\pi$
$542$	0	0
$543$	−11.9980	−0.514883
$544$	0	0
$545$	6.95632	0.297976
$546$	0	0
$547$	−16.2255	−0.693754	−0.346877	−	0.937911i	$-0.612758\pi$
$547$	−16.2255	−0.693754	−0.346877	+	0.937911i	$0.612758\pi$
$548$	0	0
$549$	−28.9666	−1.23626
$550$	0	0
$551$	−30.5211	−1.30024
$552$	0	0
$553$	−8.87467	−0.377389
$554$	0	0
$555$	−6.19159	−0.262819
$556$	0	0
$557$	25.3055	1.07223	0.536115	−	0.844145i	$-0.319891\pi$
$557$	25.3055	1.07223	0.536115	+	0.844145i	$0.319891\pi$
$558$	0	0
$559$	3.73726	0.158069
$560$	0	0
$561$	10.7170	0.452471
$562$	0	0
$563$	12.4154	0.523248	0.261624	−	0.965170i	$-0.415742\pi$
$563$	12.4154	0.523248	0.261624	+	0.965170i	$0.415742\pi$
$564$	0	0
$565$	7.75915	0.326430
$566$	0	0
$567$	15.7605	0.661879
$568$	0	0
$569$	46.9428	1.96794	0.983972	−	0.178322i	$-0.0570669\pi$
$569$	46.9428	1.96794	0.983972	+	0.178322i	$0.0570669\pi$
$570$	0	0
$571$	−26.6173	−1.11390	−0.556950	−	0.830546i	$-0.688028\pi$
$571$	−26.6173	−1.11390	−0.556950	+	0.830546i	$0.688028\pi$
$572$	0	0
$573$	−56.1790	−2.34691
$574$	0	0
$575$	−2.33382	−0.0973270
$576$	0	0
$577$	2.77259	0.115424	0.0577122	−	0.998333i	$-0.481619\pi$
$577$	2.77259	0.115424	0.0577122	+	0.998333i	$0.481619\pi$
$578$	0	0
$579$	−43.0072	−1.78732
$580$	0	0
$581$	−16.4516	−0.682529
$582$	0	0
$583$	−14.8603	−0.615451
$584$	0	0
$585$	−1.17092	−0.0484115
$586$	0	0
$587$	−42.9921	−1.77447	−0.887237	−	0.461313i	$-0.847378\pi$
$587$	−42.9921	−1.77447	−0.887237	+	0.461313i	$0.847378\pi$
$588$	0	0
$589$	−31.6032	−1.30219
$590$	0	0
$591$	59.3922	2.44307
$592$	0	0
$593$	−11.2655	−0.462617	−0.231308	−	0.972880i	$-0.574301\pi$
$593$	−11.2655	−0.462617	−0.231308	+	0.972880i	$0.574301\pi$
$594$	0	0
$595$	−3.28227	−0.134560
$596$	0	0
$597$	−3.56575	−0.145937
$598$	0	0
$599$	−23.8803	−0.975724	−0.487862	−	0.872921i	$-0.662223\pi$
$599$	−23.8803	−0.975724	−0.487862	+	0.872921i	$0.662223\pi$
$600$	0	0
$601$	−5.39691	−0.220145	−0.110072	−	0.993924i	$-0.535108\pi$
$601$	−5.39691	−0.220145	−0.110072	+	0.993924i	$0.535108\pi$
$602$	0	0
$603$	2.80540	0.114245
$604$	0	0
$605$	−6.42838	−0.261351
$606$	0	0
$607$	7.16153	0.290678	0.145339	−	0.989382i	$-0.453573\pi$
$607$	7.16153	0.290678	0.145339	+	0.989382i	$0.453573\pi$
$608$	0	0
$609$	22.0100	0.891889
$610$	0	0
$611$	−5.09913	−0.206289
$612$	0	0
$613$	−21.7198	−0.877256	−0.438628	−	0.898669i	$-0.644535\pi$
$613$	−21.7198	−0.877256	−0.438628	+	0.898669i	$0.644535\pi$
$614$	0	0
$615$	−14.3863	−0.580111
$616$	0	0
$617$	−17.8005	−0.716622	−0.358311	−	0.933602i	$-0.616647\pi$
$617$	−17.8005	−0.716622	−0.358311	+	0.933602i	$0.616647\pi$
$618$	0	0
$619$	0.0443852	0.00178399	0.000891996	−	1.00000i	$-0.499716\pi$
$619$	0.0443852	0.00178399	0.000891996		1.00000i	$0.499716\pi$
$620$	0	0
$621$	−2.79830	−0.112292
$622$	0	0
$623$	17.6809	0.708372
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−24.9630	−0.996925
$628$	0	0
$629$	5.65472	0.225468
$630$	0	0
$631$	14.1559	0.563537	0.281769	−	0.959482i	$-0.409079\pi$
$631$	14.1559	0.563537	0.281769	+	0.959482i	$0.409079\pi$
$632$	0	0
$633$	−13.6901	−0.544131
$634$	0	0
$635$	10.2593	0.407129
$636$	0	0
$637$	2.18663	0.0866373
$638$	0	0
$639$	−21.4674	−0.849239
$640$	0	0
$641$	−27.6008	−1.09017	−0.545083	−	0.838382i	$-0.683502\pi$
$641$	−27.6008	−1.09017	−0.545083	+	0.838382i	$0.683502\pi$
$642$	0	0
$643$	9.08990	0.358471	0.179235	−	0.983806i	$-0.442638\pi$
$643$	9.08990	0.358471	0.179235	+	0.983806i	$0.442638\pi$
$644$	0	0
$645$	18.6048	0.732562
$646$	0	0
$647$	45.8102	1.80099	0.900493	−	0.434870i	$-0.143206\pi$
$647$	45.8102	1.80099	0.900493	+	0.434870i	$0.143206\pi$
$648$	0	0
$649$	21.2841	0.835474
$650$	0	0
$651$	22.7903	0.893222
$652$	0	0
$653$	−25.8298	−1.01080	−0.505398	−	0.862886i	$-0.668654\pi$
$653$	−25.8298	−1.01080	−0.505398	+	0.862886i	$0.668654\pi$
$654$	0	0
$655$	4.40919	0.172281
$656$	0	0
$657$	−4.49747	−0.175463
$658$	0	0
$659$	−33.3393	−1.29871	−0.649357	−	0.760484i	$-0.724962\pi$
$659$	−33.3393	−1.29871	−0.649357	+	0.760484i	$0.724962\pi$
$660$	0	0
$661$	−28.2987	−1.10069	−0.550346	−	0.834937i	$-0.685504\pi$
$661$	−28.2987	−1.10069	−0.550346	+	0.834937i	$0.685504\pi$
$662$	0	0
$663$	2.35874	0.0916061
$664$	0	0
$665$	7.64538	0.296475
$666$	0	0
$667$	−14.2929	−0.553423
$668$	0	0
$669$	62.8013	2.42804
$670$	0	0
$671$	24.8914	0.960922
$672$	0	0
$673$	−11.3107	−0.435996	−0.217998	−	0.975949i	$-0.569953\pi$
$673$	−11.3107	−0.435996	−0.217998	+	0.975949i	$0.569953\pi$
$674$	0	0
$675$	1.19902	0.0461504
$676$	0	0
$677$	47.1099	1.81058	0.905290	−	0.424795i	$-0.139654\pi$
$677$	47.1099	1.81058	0.905290	+	0.424795i	$0.139654\pi$
$678$	0	0
$679$	5.84189	0.224191
$680$	0	0
$681$	17.6073	0.674713
$682$	0	0
$683$	11.2475	0.430372	0.215186	−	0.976573i	$-0.430964\pi$
$683$	11.2475	0.430372	0.215186	+	0.976573i	$0.430964\pi$
$684$	0	0
$685$	17.3442	0.662687
$686$	0	0
$687$	34.5095	1.31662
$688$	0	0
$689$	−3.27067	−0.124602
$690$	0	0
$691$	26.7880	1.01906	0.509531	−	0.860452i	$-0.329819\pi$
$691$	26.7880	1.01906	0.509531	+	0.860452i	$0.329819\pi$
$692$	0	0
$693$	8.16149	0.310029
$694$	0	0
$695$	−18.7375	−0.710753
$696$	0	0
$697$	13.1388	0.497669
$698$	0	0
$699$	50.5886	1.91344
$700$	0	0
$701$	−12.2160	−0.461391	−0.230696	−	0.973026i	$-0.574100\pi$
$701$	−12.2160	−0.461391	−0.230696	+	0.973026i	$0.574100\pi$
$702$	0	0
$703$	−13.1715	−0.496772
$704$	0	0
$705$	−25.3844	−0.956032
$706$	0	0
$707$	−25.2564	−0.949866
$708$	0	0
$709$	−10.0659	−0.378034	−0.189017	−	0.981974i	$-0.560530\pi$
$709$	−10.0659	−0.378034	−0.189017	+	0.981974i	$0.560530\pi$
$710$	0	0
$711$	14.3940	0.539819
$712$	0	0
$713$	−14.7996	−0.554250
$714$	0	0
$715$	1.00619	0.0376292
$716$	0	0
$717$	8.52593	0.318407
$718$	0	0
$719$	24.2850	0.905679	0.452840	−	0.891592i	$-0.350411\pi$
$719$	24.2850	0.905679	0.452840	+	0.891592i	$0.350411\pi$
$720$	0	0
$721$	−13.7128	−0.510691
$722$	0	0
$723$	32.4712	1.20762
$724$	0	0
$725$	6.12425	0.227449
$726$	0	0
$727$	31.7157	1.17627	0.588135	−	0.808762i	$-0.299862\pi$
$727$	31.7157	1.17627	0.588135	+	0.808762i	$0.299862\pi$
$728$	0	0
$729$	−17.1355	−0.634649
$730$	0	0
$731$	−16.9915	−0.628455
$732$	0	0
$733$	−10.1203	−0.373801	−0.186900	−	0.982379i	$-0.559844\pi$
$733$	−10.1203	−0.373801	−0.186900	+	0.982379i	$0.559844\pi$
$734$	0	0
$735$	10.8854	0.401515
$736$	0	0
$737$	−2.41072	−0.0888000
$738$	0	0
$739$	−45.9129	−1.68893	−0.844467	−	0.535608i	$-0.820082\pi$
$739$	−45.9129	−1.68893	−0.844467	+	0.535608i	$0.820082\pi$
$740$	0	0
$741$	−5.49421	−0.201835
$742$	0	0
$743$	−14.0301	−0.514714	−0.257357	−	0.966316i	$-0.582852\pi$
$743$	−14.0301	−0.514714	−0.257357	+	0.966316i	$0.582852\pi$
$744$	0	0
$745$	−13.3543	−0.489263
$746$	0	0
$747$	26.6833	0.976292
$748$	0	0
$749$	14.3610	0.524741
$750$	0	0
$751$	7.04609	0.257115	0.128558	−	0.991702i	$-0.458965\pi$
$751$	7.04609	0.257115	0.128558	+	0.991702i	$0.458965\pi$
$752$	0	0
$753$	41.5420	1.51388
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	15.9434	0.579473	0.289737	−	0.957106i	$-0.406432\pi$
$757$	15.9434	0.579473	0.289737	+	0.957106i	$0.406432\pi$
$758$	0	0
$759$	−11.6901	−0.424322
$760$	0	0
$761$	8.02167	0.290785	0.145393	−	0.989374i	$-0.453555\pi$
$761$	8.02167	0.290785	0.145393	+	0.989374i	$0.453555\pi$
$762$	0	0
$763$	−10.6716	−0.386339
$764$	0	0
$765$	5.32360	0.192475
$766$	0	0
$767$	4.68451	0.169148
$768$	0	0
$769$	9.10965	0.328502	0.164251	−	0.986419i	$-0.447479\pi$
$769$	9.10965	0.328502	0.164251	+	0.986419i	$0.447479\pi$
$770$	0	0
$771$	29.5347	1.06367
$772$	0	0
$773$	−43.3037	−1.55753	−0.778763	−	0.627318i	$-0.784152\pi$
$773$	−43.3037	−1.55753	−0.778763	+	0.627318i	$0.784152\pi$
$774$	0	0
$775$	6.34138	0.227789
$776$	0	0
$777$	9.49848	0.340756
$778$	0	0
$779$	−30.6042	−1.09651
$780$	0	0
$781$	18.4473	0.660096
$782$	0	0
$783$	7.34312	0.262422
$784$	0	0
$785$	−2.26259	−0.0807552
$786$	0	0
$787$	−2.73232	−0.0973967	−0.0486984	−	0.998814i	$-0.515507\pi$
$787$	−2.73232	−0.0973967	−0.0486984	+	0.998814i	$0.515507\pi$
$788$	0	0
$789$	68.0303	2.42194
$790$	0	0
$791$	−11.9033	−0.423231
$792$	0	0
$793$	5.47845	0.194546
$794$	0	0
$795$	−16.2820	−0.577462
$796$	0	0
$797$	51.7344	1.83253	0.916263	−	0.400577i	$-0.131190\pi$
$797$	51.7344	1.83253	0.916263	+	0.400577i	$0.131190\pi$
$798$	0	0
$799$	23.1833	0.820167
$800$	0	0
$801$	−28.6771	−1.01326
$802$	0	0
$803$	3.86474	0.136384
$804$	0	0
$805$	3.58030	0.126189
$806$	0	0
$807$	−62.9934	−2.21747
$808$	0	0
$809$	−15.1644	−0.533154	−0.266577	−	0.963814i	$-0.585893\pi$
$809$	−15.1644	−0.533154	−0.266577	+	0.963814i	$0.585893\pi$
$810$	0	0
$811$	29.6580	1.04143	0.520716	−	0.853730i	$-0.325665\pi$
$811$	29.6580	1.04143	0.520716	+	0.853730i	$0.325665\pi$
$812$	0	0
$813$	29.9058	1.04884
$814$	0	0
$815$	1.08367	0.0379592
$816$	0	0
$817$	39.5783	1.38467
$818$	0	0
$819$	1.79630	0.0627677
$820$	0	0
$821$	−39.9429	−1.39402	−0.697008	−	0.717063i	$-0.745486\pi$
$821$	−39.9429	−1.39402	−0.697008	+	0.717063i	$0.745486\pi$
$822$	0	0
$823$	−10.8775	−0.379164	−0.189582	−	0.981865i	$-0.560713\pi$
$823$	−10.8775	−0.379164	−0.189582	+	0.981865i	$0.560713\pi$
$824$	0	0
$825$	5.00898	0.174390
$826$	0	0
$827$	−2.37489	−0.0825831	−0.0412915	−	0.999147i	$-0.513147\pi$
$827$	−2.37489	−0.0825831	−0.0412915	+	0.999147i	$0.513147\pi$
$828$	0	0
$829$	−21.6713	−0.752677	−0.376338	−	0.926482i	$-0.622817\pi$
$829$	−21.6713	−0.752677	−0.376338	+	0.926482i	$0.622817\pi$
$830$	0	0
$831$	−69.7454	−2.41944
$832$	0	0
$833$	−9.94155	−0.344454
$834$	0	0
$835$	−6.51511	−0.225465
$836$	0	0
$837$	7.60345	0.262814
$838$	0	0
$839$	27.6715	0.955328	0.477664	−	0.878543i	$-0.341484\pi$
$839$	27.6715	0.955328	0.477664	+	0.878543i	$0.341484\pi$
$840$	0	0
$841$	8.50646	0.293326
$842$	0	0
$843$	−27.4796	−0.946447
$844$	0	0
$845$	−12.7785	−0.439595
$846$	0	0
$847$	9.86174	0.338853
$848$	0	0
$849$	58.2268	1.99834
$850$	0	0
$851$	−6.16815	−0.211442
$852$	0	0
$853$	−48.6219	−1.66478	−0.832390	−	0.554190i	$-0.813028\pi$
$853$	−48.6219	−1.66478	−0.832390	+	0.554190i	$0.813028\pi$
$854$	0	0
$855$	−12.4002	−0.424079
$856$	0	0
$857$	−31.0486	−1.06060	−0.530299	−	0.847810i	$-0.677920\pi$
$857$	−31.0486	−1.06060	−0.530299	+	0.847810i	$0.677920\pi$
$858$	0	0
$859$	−17.5514	−0.598847	−0.299424	−	0.954120i	$-0.596794\pi$
$859$	−17.5514	−0.598847	−0.299424	+	0.954120i	$0.596794\pi$
$860$	0	0
$861$	22.0699	0.752140
$862$	0	0
$863$	15.0347	0.511787	0.255894	−	0.966705i	$-0.417630\pi$
$863$	15.0347	0.511787	0.255894	+	0.966705i	$0.417630\pi$
$864$	0	0
$865$	−3.76879	−0.128143
$866$	0	0
$867$	29.1016	0.988342
$868$	0	0
$869$	−12.3690	−0.419590
$870$	0	0
$871$	−0.530585	−0.0179782
$872$	0	0
$873$	−9.47511	−0.320684
$874$	0	0
$875$	−1.53409	−0.0518618
$876$	0	0
$877$	13.0582	0.440943	0.220472	−	0.975393i	$-0.429240\pi$
$877$	13.0582	0.440943	0.220472	+	0.975393i	$0.429240\pi$
$878$	0	0
$879$	53.3757	1.80032
$880$	0	0
$881$	−24.2230	−0.816093	−0.408047	−	0.912961i	$-0.633790\pi$
$881$	−24.2230	−0.816093	−0.408047	+	0.912961i	$0.633790\pi$
$882$	0	0
$883$	36.6556	1.23356	0.616779	−	0.787136i	$-0.288437\pi$
$883$	36.6556	1.23356	0.616779	+	0.787136i	$0.288437\pi$
$884$	0	0
$885$	23.3203	0.783904
$886$	0	0
$887$	45.0955	1.51416	0.757079	−	0.653324i	$-0.226626\pi$
$887$	45.0955	1.51416	0.757079	+	0.653324i	$0.226626\pi$
$888$	0	0
$889$	−15.7388	−0.527861
$890$	0	0
$891$	21.9661	0.735892
$892$	0	0
$893$	−54.0008	−1.80707
$894$	0	0
$895$	−2.53535	−0.0847475
$896$	0	0
$897$	−2.57291	−0.0859070
$898$	0	0
$899$	38.8362	1.29526
$900$	0	0
$901$	14.8702	0.495397
$902$	0	0
$903$	−28.5415	−0.949800
$904$	0	0
$905$	5.12147	0.170243
$906$	0	0
$907$	−15.3556	−0.509874	−0.254937	−	0.966958i	$-0.582055\pi$
$907$	−15.3556	−0.509874	−0.254937	+	0.966958i	$0.582055\pi$
$908$	0	0
$909$	40.9640	1.35869
$910$	0	0
$911$	−51.6696	−1.71189	−0.855944	−	0.517068i	$-0.827023\pi$
$911$	−51.6696	−1.71189	−0.855944	+	0.517068i	$0.827023\pi$
$912$	0	0
$913$	−22.9294	−0.758852
$914$	0	0
$915$	27.2727	0.901609
$916$	0	0
$917$	−6.76411	−0.223370
$918$	0	0
$919$	33.9576	1.12016	0.560080	−	0.828439i	$-0.310771\pi$
$919$	33.9576	1.12016	0.560080	+	0.828439i	$0.310771\pi$
$920$	0	0
$921$	41.8739	1.37979
$922$	0	0
$923$	4.06014	0.133641
$924$	0	0
$925$	2.64294	0.0868995
$926$	0	0
$927$	22.2411	0.730494
$928$	0	0
$929$	59.3695	1.94785	0.973925	−	0.226872i	$-0.0728497\pi$
$929$	59.3695	1.94785	0.973925	+	0.226872i	$0.0728497\pi$
$930$	0	0
$931$	23.1568	0.758933
$932$	0	0
$933$	33.7134	1.10373
$934$	0	0
$935$	−4.57465	−0.149607
$936$	0	0
$937$	53.0524	1.73315	0.866573	−	0.499050i	$-0.166317\pi$
$937$	53.0524	1.73315	0.866573	+	0.499050i	$0.166317\pi$
$938$	0	0
$939$	13.9381	0.454854
$940$	0	0
$941$	−21.4469	−0.699150	−0.349575	−	0.936908i	$-0.613674\pi$
$941$	−21.4469	−0.699150	−0.349575	+	0.936908i	$0.613674\pi$
$942$	0	0
$943$	−14.3318	−0.466708
$944$	0	0
$945$	−1.83941	−0.0598361
$946$	0	0
$947$	−34.0827	−1.10754	−0.553770	−	0.832670i	$-0.686811\pi$
$947$	−34.0827	−1.10754	−0.553770	+	0.832670i	$0.686811\pi$
$948$	0	0
$949$	0.850607	0.0276119
$950$	0	0
$951$	−49.8485	−1.61645
$952$	0	0
$953$	−58.1867	−1.88485	−0.942426	−	0.334416i	$-0.891461\pi$
$953$	−58.1867	−1.88485	−0.942426	+	0.334416i	$0.891461\pi$
$954$	0	0
$955$	23.9806	0.775993
$956$	0	0
$957$	30.6763	0.991623
$958$	0	0
$959$	−26.6076	−0.859204
$960$	0	0
$961$	9.21304	0.297195
$962$	0	0
$963$	−23.2925	−0.750591
$964$	0	0
$965$	18.3581	0.590967
$966$	0	0
$967$	−26.3432	−0.847139	−0.423570	−	0.905864i	$-0.639223\pi$
$967$	−26.3432	−0.847139	−0.423570	+	0.905864i	$0.639223\pi$
$968$	0	0
$969$	24.9795	0.802459
$970$	0	0
$971$	25.5703	0.820588	0.410294	−	0.911953i	$-0.365426\pi$
$971$	25.5703	0.820588	0.410294	+	0.911953i	$0.365426\pi$
$972$	0	0
$973$	28.7450	0.921524
$974$	0	0
$975$	1.10245	0.0353066
$976$	0	0
$977$	−0.109403	−0.00350011	−0.00175006	−	0.999998i	$-0.500557\pi$
$977$	−0.109403	−0.00350011	−0.00175006	+	0.999998i	$0.500557\pi$
$978$	0	0
$979$	24.6427	0.787584
$980$	0	0
$981$	17.3086	0.552621
$982$	0	0
$983$	16.9658	0.541124	0.270562	−	0.962703i	$-0.412790\pi$
$983$	16.9658	0.541124	0.270562	+	0.962703i	$0.412790\pi$
$984$	0	0
$985$	−25.3522	−0.807787
$986$	0	0
$987$	38.9421	1.23954
$988$	0	0
$989$	18.5343	0.589357
$990$	0	0
$991$	−22.0384	−0.700073	−0.350036	−	0.936736i	$-0.613831\pi$
$991$	−22.0384	−0.700073	−0.350036	+	0.936736i	$0.613831\pi$
$992$	0	0
$993$	−1.97636	−0.0627178
$994$	0	0
$995$	1.52208	0.0482531
$996$	0	0
$997$	24.7975	0.785346	0.392673	−	0.919678i	$-0.371550\pi$
$997$	24.7975	0.785346	0.392673	+	0.919678i	$0.371550\pi$
$998$	0	0
$999$	3.16895	0.100261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.5	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-2.34269 q^{3} +1.00000 q^{5} -1.53409 q^{7} +2.48818 q^{9} +O(q^{10})\)	\(q-2.34269 q^{3} +1.00000 q^{5} -1.53409 q^{7} +2.48818 q^{9} -2.13813 q^{11} -0.470591 q^{13} -2.34269 q^{15} +2.13955 q^{17} -4.98365 q^{19} +3.59390 q^{21} -2.33382 q^{23} +1.00000 q^{25} +1.19902 q^{27} +6.12425 q^{29} +6.34138 q^{31} +5.00898 q^{33} -1.53409 q^{35} +2.64294 q^{37} +1.10245 q^{39} +6.14093 q^{41} -7.94163 q^{43} +2.48818 q^{45} +10.8356 q^{47} -4.64656 q^{49} -5.01230 q^{51} +6.95012 q^{53} -2.13813 q^{55} +11.6751 q^{57} -9.95452 q^{59} -11.6416 q^{61} -3.81711 q^{63} -0.470591 q^{65} +1.12749 q^{67} +5.46741 q^{69} -8.62775 q^{71} -1.80753 q^{73} -2.34269 q^{75} +3.28010 q^{77} +5.78496 q^{79} -10.2735 q^{81} +10.7240 q^{83} +2.13955 q^{85} -14.3472 q^{87} -11.5253 q^{89} +0.721931 q^{91} -14.8559 q^{93} -4.98365 q^{95} -3.80804 q^{97} -5.32007 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more