LMFDB - Embedded newform 6040.2.a.p.1.12

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			1.12
Root			$-0.815820$ 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)

$f(q)$	$=$	$q+0.815820 q^{3} +1.00000 q^{5} -2.49382 q^{7} -2.33444 q^{9} +O(q^{10})$	$q+0.815820 q^{3} +1.00000 q^{5} -2.49382 q^{7} -2.33444 q^{9} +1.48661 q^{11} +5.26621 q^{13} +0.815820 q^{15} +1.04575 q^{17} -7.39355 q^{19} -2.03450 q^{21} +0.0259902 q^{23} +1.00000 q^{25} -4.35194 q^{27} +6.30898 q^{29} -8.70155 q^{31} +1.21281 q^{33} -2.49382 q^{35} +0.895610 q^{37} +4.29628 q^{39} +3.52055 q^{41} -8.45857 q^{43} -2.33444 q^{45} -4.55300 q^{47} -0.780881 q^{49} +0.853142 q^{51} -4.01326 q^{53} +1.48661 q^{55} -6.03180 q^{57} +4.73895 q^{59} -8.15857 q^{61} +5.82166 q^{63} +5.26621 q^{65} -8.54928 q^{67} +0.0212033 q^{69} -2.24512 q^{71} -6.46490 q^{73} +0.815820 q^{75} -3.70734 q^{77} +15.6805 q^{79} +3.45292 q^{81} +13.6215 q^{83} +1.04575 q^{85} +5.14699 q^{87} -1.61193 q^{89} -13.1330 q^{91} -7.09890 q^{93} -7.39355 q^{95} -4.71034 q^{97} -3.47041 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.815820	0.471014	0.235507	−	0.971873i	$-0.424325\pi$
$3$	0.815820	0.471014	0.235507	+	0.971873i	$0.424325\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.49382	−0.942574	−0.471287	−	0.881980i	$-0.656210\pi$
$7$	−2.49382	−0.942574	−0.471287	+	0.881980i	$0.656210\pi$
$8$	0	0
$9$	−2.33444	−0.778146
$10$	0	0
$11$	1.48661	0.448231	0.224115	−	0.974563i	$-0.428051\pi$
$11$	1.48661	0.448231	0.224115	+	0.974563i	$0.428051\pi$
$12$	0	0
$13$	5.26621	1.46058	0.730292	−	0.683135i	$-0.239384\pi$
$13$	5.26621	1.46058	0.730292	+	0.683135i	$0.239384\pi$
$14$	0	0
$15$	0.815820	0.210644
$16$	0	0
$17$	1.04575	0.253631	0.126816	−	0.991926i	$-0.459524\pi$
$17$	1.04575	0.253631	0.126816	+	0.991926i	$0.459524\pi$
$18$	0	0
$19$	−7.39355	−1.69620	−0.848098	−	0.529839i	$-0.822252\pi$
$19$	−7.39355	−1.69620	−0.848098	+	0.529839i	$0.822252\pi$
$20$	0	0
$21$	−2.03450	−0.443965
$22$	0	0
$23$	0.0259902	0.00541933	0.00270967	−	0.999996i	$-0.499137\pi$
$23$	0.0259902	0.00541933	0.00270967	+	0.999996i	$0.499137\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.35194	−0.837531
$28$	0	0
$29$	6.30898	1.17155	0.585774	−	0.810474i	$-0.300790\pi$
$29$	6.30898	1.17155	0.585774	+	0.810474i	$0.300790\pi$
$30$	0	0
$31$	−8.70155	−1.56284	−0.781422	−	0.624003i	$-0.785505\pi$
$31$	−8.70155	−1.56284	−0.781422	+	0.624003i	$0.785505\pi$
$32$	0	0
$33$	1.21281	0.211123
$34$	0	0
$35$	−2.49382	−0.421532
$36$	0	0
$37$	0.895610	0.147237	0.0736187	−	0.997286i	$-0.476545\pi$
$37$	0.895610	0.147237	0.0736187	+	0.997286i	$0.476545\pi$
$38$	0	0
$39$	4.29628	0.687955
$40$	0	0
$41$	3.52055	0.549817	0.274908	−	0.961470i	$-0.411352\pi$
$41$	3.52055	0.549817	0.274908	+	0.961470i	$0.411352\pi$
$42$	0	0
$43$	−8.45857	−1.28992	−0.644960	−	0.764216i	$-0.723126\pi$
$43$	−8.45857	−1.28992	−0.644960	+	0.764216i	$0.723126\pi$
$44$	0	0
$45$	−2.33444	−0.347998
$46$	0	0
$47$	−4.55300	−0.664123	−0.332062	−	0.943258i	$-0.607744\pi$
$47$	−4.55300	−0.664123	−0.332062	+	0.943258i	$0.607744\pi$
$48$	0	0
$49$	−0.780881	−0.111554
$50$	0	0
$51$	0.853142	0.119464
$52$	0	0
$53$	−4.01326	−0.551263	−0.275632	−	0.961263i	$-0.588887\pi$
$53$	−4.01326	−0.551263	−0.275632	+	0.961263i	$0.588887\pi$
$54$	0	0
$55$	1.48661	0.200455
$56$	0	0
$57$	−6.03180	−0.798932
$58$	0	0
$59$	4.73895	0.616958	0.308479	−	0.951231i	$-0.400180\pi$
$59$	4.73895	0.616958	0.308479	+	0.951231i	$0.400180\pi$
$60$	0	0
$61$	−8.15857	−1.04460	−0.522299	−	0.852763i	$-0.674925\pi$
$61$	−8.15857	−1.04460	−0.522299	+	0.852763i	$0.674925\pi$
$62$	0	0
$63$	5.82166	0.733460
$64$	0	0
$65$	5.26621	0.653193
$66$	0	0
$67$	−8.54928	−1.04446	−0.522231	−	0.852804i	$-0.674900\pi$
$67$	−8.54928	−1.04446	−0.522231	+	0.852804i	$0.674900\pi$
$68$	0	0
$69$	0.0212033	0.00255258
$70$	0	0
$71$	−2.24512	−0.266446	−0.133223	−	0.991086i	$-0.542533\pi$
$71$	−2.24512	−0.266446	−0.133223	+	0.991086i	$0.542533\pi$
$72$	0	0
$73$	−6.46490	−0.756659	−0.378329	−	0.925671i	$-0.623501\pi$
$73$	−6.46490	−0.756659	−0.378329	+	0.925671i	$0.623501\pi$
$74$	0	0
$75$	0.815820	0.0942027
$76$	0	0
$77$	−3.70734	−0.422490
$78$	0	0
$79$	15.6805	1.76420	0.882100	−	0.471063i	$-0.156130\pi$
$79$	15.6805	1.76420	0.882100	+	0.471063i	$0.156130\pi$
$80$	0	0
$81$	3.45292	0.383658
$82$	0	0
$83$	13.6215	1.49515	0.747577	−	0.664176i	$-0.231217\pi$
$83$	13.6215	1.49515	0.747577	+	0.664176i	$0.231217\pi$
$84$	0	0
$85$	1.04575	0.113427
$86$	0	0
$87$	5.14699	0.551816
$88$	0	0
$89$	−1.61193	−0.170864	−0.0854322	−	0.996344i	$-0.527227\pi$
$89$	−1.61193	−0.170864	−0.0854322	+	0.996344i	$0.527227\pi$
$90$	0	0
$91$	−13.1330	−1.37671
$92$	0	0
$93$	−7.09890	−0.736121
$94$	0	0
$95$	−7.39355	−0.758562
$96$	0	0
$97$	−4.71034	−0.478262	−0.239131	−	0.970987i	$-0.576863\pi$
$97$	−4.71034	−0.478262	−0.239131	+	0.970987i	$0.576863\pi$
$98$	0	0
$99$	−3.47041	−0.348789
$100$	0	0
$101$	2.90267	0.288826	0.144413	−	0.989517i	$-0.453871\pi$
$101$	2.90267	0.288826	0.144413	+	0.989517i	$0.453871\pi$
$102$	0	0
$103$	−16.0372	−1.58019	−0.790094	−	0.612985i	$-0.789969\pi$
$103$	−16.0372	−1.58019	−0.790094	+	0.612985i	$0.789969\pi$
$104$	0	0
$105$	−2.03450	−0.198547
$106$	0	0
$107$	15.6986	1.51764	0.758821	−	0.651299i	$-0.225776\pi$
$107$	15.6986	1.51764	0.758821	+	0.651299i	$0.225776\pi$
$108$	0	0
$109$	9.46068	0.906169	0.453085	−	0.891468i	$-0.350324\pi$
$109$	9.46068	0.906169	0.453085	+	0.891468i	$0.350324\pi$
$110$	0	0
$111$	0.730657	0.0693509
$112$	0	0
$113$	−10.8741	−1.02295	−0.511473	−	0.859300i	$-0.670900\pi$
$113$	−10.8741	−1.02295	−0.511473	+	0.859300i	$0.670900\pi$
$114$	0	0
$115$	0.0259902	0.00242360
$116$	0	0
$117$	−12.2936	−1.13655
$118$	0	0
$119$	−2.60790	−0.239066
$120$	0	0
$121$	−8.78998	−0.799089
$122$	0	0
$123$	2.87213	0.258971
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.83300	−0.606331	−0.303165	−	0.952938i	$-0.598043\pi$
$127$	−6.83300	−0.606331	−0.303165	+	0.952938i	$0.598043\pi$
$128$	0	0
$129$	−6.90067	−0.607570
$130$	0	0
$131$	−16.3567	−1.42909	−0.714544	−	0.699591i	$-0.753366\pi$
$131$	−16.3567	−1.42909	−0.714544	+	0.699591i	$0.753366\pi$
$132$	0	0
$133$	18.4382	1.59879
$134$	0	0
$135$	−4.35194	−0.374555
$136$	0	0
$137$	−14.8708	−1.27050	−0.635250	−	0.772307i	$-0.719103\pi$
$137$	−14.8708	−1.27050	−0.635250	+	0.772307i	$0.719103\pi$
$138$	0	0
$139$	17.4615	1.48106	0.740531	−	0.672022i	$-0.234574\pi$
$139$	17.4615	1.48106	0.740531	+	0.672022i	$0.234574\pi$
$140$	0	0
$141$	−3.71443	−0.312811
$142$	0	0
$143$	7.82882	0.654679
$144$	0	0
$145$	6.30898	0.523933
$146$	0	0
$147$	−0.637058	−0.0525437
$148$	0	0
$149$	17.5804	1.44024	0.720121	−	0.693849i	$-0.244086\pi$
$149$	17.5804	1.44024	0.720121	+	0.693849i	$0.244086\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−2.44123	−0.197362
$154$	0	0
$155$	−8.70155	−0.698925
$156$	0	0
$157$	−6.44378	−0.514270	−0.257135	−	0.966376i	$-0.582778\pi$
$157$	−6.44378	−0.514270	−0.257135	+	0.966376i	$0.582778\pi$
$158$	0	0
$159$	−3.27409	−0.259652
$160$	0	0
$161$	−0.0648148	−0.00510812
$162$	0	0
$163$	−24.4787	−1.91732	−0.958659	−	0.284558i	$-0.908153\pi$
$163$	−24.4787	−1.91732	−0.958659	+	0.284558i	$0.908153\pi$
$164$	0	0
$165$	1.21281	0.0944170
$166$	0	0
$167$	−4.35235	−0.336795	−0.168397	−	0.985719i	$-0.553859\pi$
$167$	−4.35235	−0.336795	−0.168397	+	0.985719i	$0.553859\pi$
$168$	0	0
$169$	14.7330	1.13331
$170$	0	0
$171$	17.2598	1.31989
$172$	0	0
$173$	15.3263	1.16524	0.582618	−	0.812746i	$-0.302028\pi$
$173$	15.3263	1.16524	0.582618	+	0.812746i	$0.302028\pi$
$174$	0	0
$175$	−2.49382	−0.188515
$176$	0	0
$177$	3.86613	0.290596
$178$	0	0
$179$	−5.48802	−0.410194	−0.205097	−	0.978742i	$-0.565751\pi$
$179$	−5.48802	−0.410194	−0.205097	+	0.978742i	$0.565751\pi$
$180$	0	0
$181$	−18.5589	−1.37947	−0.689734	−	0.724062i	$-0.742273\pi$
$181$	−18.5589	−1.37947	−0.689734	+	0.724062i	$0.742273\pi$
$182$	0	0
$183$	−6.65592	−0.492020
$184$	0	0
$185$	0.895610	0.0658466
$186$	0	0
$187$	1.55462	0.113685
$188$	0	0
$189$	10.8529	0.789435
$190$	0	0
$191$	−11.3552	−0.821633	−0.410816	−	0.911718i	$-0.634756\pi$
$191$	−11.3552	−0.821633	−0.410816	+	0.911718i	$0.634756\pi$
$192$	0	0
$193$	−6.49585	−0.467581	−0.233791	−	0.972287i	$-0.575113\pi$
$193$	−6.49585	−0.467581	−0.233791	+	0.972287i	$0.575113\pi$
$194$	0	0
$195$	4.29628	0.307663
$196$	0	0
$197$	−22.2838	−1.58765	−0.793827	−	0.608143i	$-0.791915\pi$
$197$	−22.2838	−1.58765	−0.793827	+	0.608143i	$0.791915\pi$
$198$	0	0
$199$	−22.6010	−1.60214	−0.801072	−	0.598568i	$-0.795736\pi$
$199$	−22.6010	−1.60214	−0.801072	+	0.598568i	$0.795736\pi$
$200$	0	0
$201$	−6.97467	−0.491955
$202$	0	0
$203$	−15.7334	−1.10427
$204$	0	0
$205$	3.52055	0.245886
$206$	0	0
$207$	−0.0606726	−0.00421703
$208$	0	0
$209$	−10.9913	−0.760287
$210$	0	0
$211$	−11.2261	−0.772834	−0.386417	−	0.922324i	$-0.626287\pi$
$211$	−11.2261	−0.772834	−0.386417	+	0.922324i	$0.626287\pi$
$212$	0	0
$213$	−1.83161	−0.125500
$214$	0	0
$215$	−8.45857	−0.576870
$216$	0	0
$217$	21.7001	1.47310
$218$	0	0
$219$	−5.27419	−0.356397
$220$	0	0
$221$	5.50713	0.370450
$222$	0	0
$223$	1.96676	0.131704	0.0658519	−	0.997829i	$-0.479024\pi$
$223$	1.96676	0.131704	0.0658519	+	0.997829i	$0.479024\pi$
$224$	0	0
$225$	−2.33444	−0.155629
$226$	0	0
$227$	16.7316	1.11052	0.555259	−	0.831678i	$-0.312619\pi$
$227$	16.7316	1.11052	0.555259	+	0.831678i	$0.312619\pi$
$228$	0	0
$229$	9.09161	0.600790	0.300395	−	0.953815i	$-0.402881\pi$
$229$	9.09161	0.600790	0.300395	+	0.953815i	$0.402881\pi$
$230$	0	0
$231$	−3.02452	−0.198999
$232$	0	0
$233$	−26.3161	−1.72402	−0.862012	−	0.506888i	$-0.830796\pi$
$233$	−26.3161	−1.72402	−0.862012	+	0.506888i	$0.830796\pi$
$234$	0	0
$235$	−4.55300	−0.297005
$236$	0	0
$237$	12.7925	0.830962
$238$	0	0
$239$	−17.5753	−1.13685	−0.568425	−	0.822735i	$-0.692447\pi$
$239$	−17.5753	−1.13685	−0.568425	+	0.822735i	$0.692447\pi$
$240$	0	0
$241$	−20.8207	−1.34118	−0.670588	−	0.741830i	$-0.733958\pi$
$241$	−20.8207	−1.34118	−0.670588	+	0.741830i	$0.733958\pi$
$242$	0	0
$243$	15.8728	1.01824
$244$	0	0
$245$	−0.780881	−0.0498887
$246$	0	0
$247$	−38.9360	−2.47744
$248$	0	0
$249$	11.1127	0.704238
$250$	0	0
$251$	−14.5478	−0.918248	−0.459124	−	0.888372i	$-0.651837\pi$
$251$	−14.5478	−0.918248	−0.459124	+	0.888372i	$0.651837\pi$
$252$	0	0
$253$	0.0386374	0.00242911
$254$	0	0
$255$	0.853142	0.0534258
$256$	0	0
$257$	−0.407814	−0.0254388	−0.0127194	−	0.999919i	$-0.504049\pi$
$257$	−0.407814	−0.0254388	−0.0127194	+	0.999919i	$0.504049\pi$
$258$	0	0
$259$	−2.23349	−0.138782
$260$	0	0
$261$	−14.7279	−0.911636
$262$	0	0
$263$	22.5215	1.38873	0.694367	−	0.719621i	$-0.255685\pi$
$263$	22.5215	1.38873	0.694367	+	0.719621i	$0.255685\pi$
$264$	0	0
$265$	−4.01326	−0.246532
$266$	0	0
$267$	−1.31505	−0.0804795
$268$	0	0
$269$	−3.40350	−0.207515	−0.103757	−	0.994603i	$-0.533087\pi$
$269$	−3.40350	−0.207515	−0.103757	+	0.994603i	$0.533087\pi$
$270$	0	0
$271$	−16.5424	−1.00488	−0.502439	−	0.864613i	$-0.667564\pi$
$271$	−16.5424	−1.00488	−0.502439	+	0.864613i	$0.667564\pi$
$272$	0	0
$273$	−10.7141	−0.648448
$274$	0	0
$275$	1.48661	0.0896461
$276$	0	0
$277$	13.7340	0.825194	0.412597	−	0.910914i	$-0.364622\pi$
$277$	13.7340	0.825194	0.412597	+	0.910914i	$0.364622\pi$
$278$	0	0
$279$	20.3132	1.21612
$280$	0	0
$281$	−21.3283	−1.27234	−0.636170	−	0.771549i	$-0.719482\pi$
$281$	−21.3283	−1.27234	−0.636170	+	0.771549i	$0.719482\pi$
$282$	0	0
$283$	10.4259	0.619753	0.309876	−	0.950777i	$-0.399712\pi$
$283$	10.4259	0.619753	0.309876	+	0.950777i	$0.399712\pi$
$284$	0	0
$285$	−6.03180	−0.357293
$286$	0	0
$287$	−8.77959	−0.518243
$288$	0	0
$289$	−15.9064	−0.935671
$290$	0	0
$291$	−3.84278	−0.225268
$292$	0	0
$293$	27.0198	1.57851	0.789256	−	0.614064i	$-0.210466\pi$
$293$	27.0198	1.57851	0.789256	+	0.614064i	$0.210466\pi$
$294$	0	0
$295$	4.73895	0.275912
$296$	0	0
$297$	−6.46965	−0.375407
$298$	0	0
$299$	0.136870	0.00791539
$300$	0	0
$301$	21.0941	1.21584
$302$	0	0
$303$	2.36805	0.136041
$304$	0	0
$305$	−8.15857	−0.467158
$306$	0	0
$307$	−8.62153	−0.492057	−0.246028	−	0.969263i	$-0.579126\pi$
$307$	−8.62153	−0.492057	−0.246028	+	0.969263i	$0.579126\pi$
$308$	0	0
$309$	−13.0834	−0.744291
$310$	0	0
$311$	−26.2623	−1.48920	−0.744599	−	0.667512i	$-0.767359\pi$
$311$	−26.2623	−1.48920	−0.744599	+	0.667512i	$0.767359\pi$
$312$	0	0
$313$	20.6476	1.16707	0.583535	−	0.812088i	$-0.301669\pi$
$313$	20.6476	1.16707	0.583535	+	0.812088i	$0.301669\pi$
$314$	0	0
$315$	5.82166	0.328013
$316$	0	0
$317$	−5.14134	−0.288766	−0.144383	−	0.989522i	$-0.546120\pi$
$317$	−5.14134	−0.288766	−0.144383	+	0.989522i	$0.546120\pi$
$318$	0	0
$319$	9.37902	0.525124
$320$	0	0
$321$	12.8072	0.714830
$322$	0	0
$323$	−7.73179	−0.430208
$324$	0	0
$325$	5.26621	0.292117
$326$	0	0
$327$	7.71821	0.426818
$328$	0	0
$329$	11.3543	0.625985
$330$	0	0
$331$	22.6710	1.24611	0.623055	−	0.782178i	$-0.285891\pi$
$331$	22.6710	1.24611	0.623055	+	0.782178i	$0.285891\pi$
$332$	0	0
$333$	−2.09075	−0.114572
$334$	0	0
$335$	−8.54928	−0.467097
$336$	0	0
$337$	−8.13975	−0.443401	−0.221700	−	0.975115i	$-0.571161\pi$
$337$	−8.13975	−0.443401	−0.221700	+	0.975115i	$0.571161\pi$
$338$	0	0
$339$	−8.87127	−0.481821
$340$	0	0
$341$	−12.9358	−0.700515
$342$	0	0
$343$	19.4041	1.04772
$344$	0	0
$345$	0.0212033	0.00114155
$346$	0	0
$347$	32.4079	1.73975	0.869874	−	0.493275i	$-0.164200\pi$
$347$	32.4079	1.73975	0.869874	+	0.493275i	$0.164200\pi$
$348$	0	0
$349$	−35.3292	−1.89113	−0.945563	−	0.325438i	$-0.894488\pi$
$349$	−35.3292	−1.89113	−0.945563	+	0.325438i	$0.894488\pi$
$350$	0	0
$351$	−22.9182	−1.22328
$352$	0	0
$353$	36.5765	1.94677	0.973386	−	0.229171i	$-0.0736015\pi$
$353$	36.5765	1.94677	0.973386	+	0.229171i	$0.0736015\pi$
$354$	0	0
$355$	−2.24512	−0.119158
$356$	0	0
$357$	−2.12758	−0.112603
$358$	0	0
$359$	14.0125	0.739554	0.369777	−	0.929121i	$-0.379434\pi$
$359$	14.0125	0.739554	0.369777	+	0.929121i	$0.379434\pi$
$360$	0	0
$361$	35.6646	1.87708
$362$	0	0
$363$	−7.17104	−0.376382
$364$	0	0
$365$	−6.46490	−0.338388
$366$	0	0
$367$	−24.4215	−1.27479	−0.637395	−	0.770537i	$-0.719988\pi$
$367$	−24.4215	−1.27479	−0.637395	+	0.770537i	$0.719988\pi$
$368$	0	0
$369$	−8.21850	−0.427838
$370$	0	0
$371$	10.0083	0.519606
$372$	0	0
$373$	−0.764317	−0.0395748	−0.0197874	−	0.999804i	$-0.506299\pi$
$373$	−0.764317	−0.0395748	−0.0197874	+	0.999804i	$0.506299\pi$
$374$	0	0
$375$	0.815820	0.0421287
$376$	0	0
$377$	33.2244	1.71115
$378$	0	0
$379$	−28.6707	−1.47271	−0.736357	−	0.676593i	$-0.763456\pi$
$379$	−28.6707	−1.47271	−0.736357	+	0.676593i	$0.763456\pi$
$380$	0	0
$381$	−5.57450	−0.285590
$382$	0	0
$383$	−11.3882	−0.581910	−0.290955	−	0.956737i	$-0.593973\pi$
$383$	−11.3882	−0.581910	−0.290955	+	0.956737i	$0.593973\pi$
$384$	0	0
$385$	−3.70734	−0.188943
$386$	0	0
$387$	19.7460	1.00375
$388$	0	0
$389$	−32.8184	−1.66396	−0.831980	−	0.554806i	$-0.812793\pi$
$389$	−32.8184	−1.66396	−0.831980	+	0.554806i	$0.812793\pi$
$390$	0	0
$391$	0.0271792	0.00137451
$392$	0	0
$393$	−13.3441	−0.673120
$394$	0	0
$395$	15.6805	0.788974
$396$	0	0
$397$	31.4177	1.57681	0.788405	−	0.615156i	$-0.210907\pi$
$397$	31.4177	1.57681	0.788405	+	0.615156i	$0.210907\pi$
$398$	0	0
$399$	15.0422	0.753052
$400$	0	0
$401$	13.9941	0.698834	0.349417	−	0.936967i	$-0.386380\pi$
$401$	13.9941	0.698834	0.349417	+	0.936967i	$0.386380\pi$
$402$	0	0
$403$	−45.8242	−2.28267
$404$	0	0
$405$	3.45292	0.171577
$406$	0	0
$407$	1.33143	0.0659963
$408$	0	0
$409$	28.3104	1.39986	0.699929	−	0.714213i	$-0.253215\pi$
$409$	28.3104	1.39986	0.699929	+	0.714213i	$0.253215\pi$
$410$	0	0
$411$	−12.1319	−0.598422
$412$	0	0
$413$	−11.8181	−0.581529
$414$	0	0
$415$	13.6215	0.668653
$416$	0	0
$417$	14.2454	0.697600
$418$	0	0
$419$	36.3234	1.77452	0.887258	−	0.461274i	$-0.152607\pi$
$419$	36.3234	1.77452	0.887258	+	0.461274i	$0.152607\pi$
$420$	0	0
$421$	26.2898	1.28128	0.640642	−	0.767840i	$-0.278668\pi$
$421$	26.2898	1.28128	0.640642	+	0.767840i	$0.278668\pi$
$422$	0	0
$423$	10.6287	0.516785
$424$	0	0
$425$	1.04575	0.0507262
$426$	0	0
$427$	20.3460	0.984610
$428$	0	0
$429$	6.38690	0.308363
$430$	0	0
$431$	−27.8740	−1.34264	−0.671321	−	0.741167i	$-0.734273\pi$
$431$	−27.8740	−1.34264	−0.671321	+	0.741167i	$0.734273\pi$
$432$	0	0
$433$	−5.44329	−0.261588	−0.130794	−	0.991410i	$-0.541753\pi$
$433$	−5.44329	−0.261588	−0.130794	+	0.991410i	$0.541753\pi$
$434$	0	0
$435$	5.14699	0.246779
$436$	0	0
$437$	−0.192160	−0.00919226
$438$	0	0
$439$	1.84567	0.0880890	0.0440445	−	0.999030i	$-0.485976\pi$
$439$	1.84567	0.0880890	0.0440445	+	0.999030i	$0.485976\pi$
$440$	0	0
$441$	1.82292	0.0868057
$442$	0	0
$443$	25.4715	1.21019	0.605094	−	0.796154i	$-0.293136\pi$
$443$	25.4715	1.21019	0.605094	+	0.796154i	$0.293136\pi$
$444$	0	0
$445$	−1.61193	−0.0764129
$446$	0	0
$447$	14.3424	0.678373
$448$	0	0
$449$	3.83174	0.180831	0.0904156	−	0.995904i	$-0.471180\pi$
$449$	3.83174	0.180831	0.0904156	+	0.995904i	$0.471180\pi$
$450$	0	0
$451$	5.23369	0.246445
$452$	0	0
$453$	0.815820	0.0383305
$454$	0	0
$455$	−13.1330	−0.615683
$456$	0	0
$457$	0.439109	0.0205407	0.0102703	−	0.999947i	$-0.496731\pi$
$457$	0.439109	0.0205407	0.0102703	+	0.999947i	$0.496731\pi$
$458$	0	0
$459$	−4.55103	−0.212424
$460$	0	0
$461$	−10.8733	−0.506422	−0.253211	−	0.967411i	$-0.581487\pi$
$461$	−10.8733	−0.506422	−0.253211	+	0.967411i	$0.581487\pi$
$462$	0	0
$463$	18.1320	0.842664	0.421332	−	0.906907i	$-0.361563\pi$
$463$	18.1320	0.842664	0.421332	+	0.906907i	$0.361563\pi$
$464$	0	0
$465$	−7.09890	−0.329203
$466$	0	0
$467$	−6.62096	−0.306381	−0.153191	−	0.988197i	$-0.548955\pi$
$467$	−6.62096	−0.306381	−0.153191	+	0.988197i	$0.548955\pi$
$468$	0	0
$469$	21.3203	0.984482
$470$	0	0
$471$	−5.25696	−0.242228
$472$	0	0
$473$	−12.5746	−0.578181
$474$	0	0
$475$	−7.39355	−0.339239
$476$	0	0
$477$	9.36870	0.428963
$478$	0	0
$479$	−8.44168	−0.385710	−0.192855	−	0.981227i	$-0.561775\pi$
$479$	−8.44168	−0.385710	−0.192855	+	0.981227i	$0.561775\pi$
$480$	0	0
$481$	4.71647	0.215053
$482$	0	0
$483$	−0.0528772	−0.00240600
$484$	0	0
$485$	−4.71034	−0.213885
$486$	0	0
$487$	19.2073	0.870366	0.435183	−	0.900342i	$-0.356684\pi$
$487$	19.2073	0.870366	0.435183	+	0.900342i	$0.356684\pi$
$488$	0	0
$489$	−19.9702	−0.903083
$490$	0	0
$491$	24.0758	1.08653	0.543263	−	0.839563i	$-0.317189\pi$
$491$	24.0758	1.08653	0.543263	+	0.839563i	$0.317189\pi$
$492$	0	0
$493$	6.59761	0.297141
$494$	0	0
$495$	−3.47041	−0.155983
$496$	0	0
$497$	5.59891	0.251145
$498$	0	0
$499$	−42.2199	−1.89002	−0.945011	−	0.327037i	$-0.893950\pi$
$499$	−42.2199	−1.89002	−0.945011	+	0.327037i	$0.893950\pi$
$500$	0	0
$501$	−3.55073	−0.158635
$502$	0	0
$503$	10.0451	0.447888	0.223944	−	0.974602i	$-0.428107\pi$
$503$	10.0451	0.447888	0.223944	+	0.974602i	$0.428107\pi$
$504$	0	0
$505$	2.90267	0.129167
$506$	0	0
$507$	12.0195	0.533803
$508$	0	0
$509$	−9.94121	−0.440636	−0.220318	−	0.975428i	$-0.570710\pi$
$509$	−9.94121	−0.440636	−0.220318	+	0.975428i	$0.570710\pi$
$510$	0	0
$511$	16.1223	0.713207
$512$	0	0
$513$	32.1763	1.42062
$514$	0	0
$515$	−16.0372	−0.706682
$516$	0	0
$517$	−6.76855	−0.297680
$518$	0	0
$519$	12.5035	0.548842
$520$	0	0
$521$	33.7158	1.47711	0.738557	−	0.674191i	$-0.235508\pi$
$521$	33.7158	1.47711	0.738557	+	0.674191i	$0.235508\pi$
$522$	0	0
$523$	1.08746	0.0475513	0.0237757	−	0.999717i	$-0.492431\pi$
$523$	1.08746	0.0475513	0.0237757	+	0.999717i	$0.492431\pi$
$524$	0	0
$525$	−2.03450	−0.0887930
$526$	0	0
$527$	−9.09963	−0.396386
$528$	0	0
$529$	−22.9993	−0.999971
$530$	0	0
$531$	−11.0628	−0.480084
$532$	0	0
$533$	18.5399	0.803054
$534$	0	0
$535$	15.6986	0.678710
$536$	0	0
$537$	−4.47723	−0.193207
$538$	0	0
$539$	−1.16087	−0.0500021
$540$	0	0
$541$	34.8695	1.49916	0.749579	−	0.661915i	$-0.230256\pi$
$541$	34.8695	1.49916	0.749579	+	0.661915i	$0.230256\pi$
$542$	0	0
$543$	−15.1407	−0.649749
$544$	0	0
$545$	9.46068	0.405251
$546$	0	0
$547$	−18.3571	−0.784895	−0.392447	−	0.919774i	$-0.628372\pi$
$547$	−18.3571	−0.784895	−0.392447	+	0.919774i	$0.628372\pi$
$548$	0	0
$549$	19.0457	0.812849
$550$	0	0
$551$	−46.6458	−1.98718
$552$	0	0
$553$	−39.1044	−1.66289
$554$	0	0
$555$	0.730657	0.0310146
$556$	0	0
$557$	−0.866545	−0.0367167	−0.0183583	−	0.999831i	$-0.505844\pi$
$557$	−0.866545	−0.0367167	−0.0183583	+	0.999831i	$0.505844\pi$
$558$	0	0
$559$	−44.5446	−1.88404
$560$	0	0
$561$	1.26829	0.0535473
$562$	0	0
$563$	34.6019	1.45830	0.729148	−	0.684356i	$-0.239917\pi$
$563$	34.6019	1.45830	0.729148	+	0.684356i	$0.239917\pi$
$564$	0	0
$565$	−10.8741	−0.457475
$566$	0	0
$567$	−8.61094	−0.361626
$568$	0	0
$569$	−28.0389	−1.17545	−0.587725	−	0.809061i	$-0.699976\pi$
$569$	−28.0389	−1.17545	−0.587725	+	0.809061i	$0.699976\pi$
$570$	0	0
$571$	19.7871	0.828064	0.414032	−	0.910262i	$-0.364120\pi$
$571$	19.7871	0.828064	0.414032	+	0.910262i	$0.364120\pi$
$572$	0	0
$573$	−9.26378	−0.387000
$574$	0	0
$575$	0.0259902	0.00108387
$576$	0	0
$577$	30.1597	1.25557	0.627783	−	0.778388i	$-0.283963\pi$
$577$	30.1597	1.25557	0.627783	+	0.778388i	$0.283963\pi$
$578$	0	0
$579$	−5.29944	−0.220237
$580$	0	0
$581$	−33.9695	−1.40929
$582$	0	0
$583$	−5.96616	−0.247093
$584$	0	0
$585$	−12.2936	−0.508280
$586$	0	0
$587$	−9.49212	−0.391782	−0.195891	−	0.980626i	$-0.562760\pi$
$587$	−9.49212	−0.391782	−0.195891	+	0.980626i	$0.562760\pi$
$588$	0	0
$589$	64.3353	2.65089
$590$	0	0
$591$	−18.1796	−0.747807
$592$	0	0
$593$	5.21125	0.214000	0.107000	−	0.994259i	$-0.465875\pi$
$593$	5.21125	0.214000	0.107000	+	0.994259i	$0.465875\pi$
$594$	0	0
$595$	−2.60790	−0.106914
$596$	0	0
$597$	−18.4384	−0.754631
$598$	0	0
$599$	−38.0809	−1.55594	−0.777971	−	0.628300i	$-0.783751\pi$
$599$	−38.0809	−1.55594	−0.777971	+	0.628300i	$0.783751\pi$
$600$	0	0
$601$	−16.8393	−0.686890	−0.343445	−	0.939173i	$-0.611594\pi$
$601$	−16.8393	−0.686890	−0.343445	+	0.939173i	$0.611594\pi$
$602$	0	0
$603$	19.9578	0.812743
$604$	0	0
$605$	−8.78998	−0.357364
$606$	0	0
$607$	−32.8649	−1.33395	−0.666974	−	0.745081i	$-0.732411\pi$
$607$	−32.8649	−1.33395	−0.666974	+	0.745081i	$0.732411\pi$
$608$	0	0
$609$	−12.8357	−0.520127
$610$	0	0
$611$	−23.9771	−0.970008
$612$	0	0
$613$	−6.55646	−0.264813	−0.132406	−	0.991196i	$-0.542270\pi$
$613$	−6.55646	−0.264813	−0.132406	+	0.991196i	$0.542270\pi$
$614$	0	0
$615$	2.87213	0.115815
$616$	0	0
$617$	−30.2120	−1.21629	−0.608144	−	0.793827i	$-0.708086\pi$
$617$	−30.2120	−1.21629	−0.608144	+	0.793827i	$0.708086\pi$
$618$	0	0
$619$	32.5398	1.30789	0.653943	−	0.756544i	$-0.273114\pi$
$619$	32.5398	1.30789	0.653943	+	0.756544i	$0.273114\pi$
$620$	0	0
$621$	−0.113108	−0.00453886
$622$	0	0
$623$	4.01986	0.161052
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.96695	−0.358106
$628$	0	0
$629$	0.936583	0.0373440
$630$	0	0
$631$	13.9013	0.553402	0.276701	−	0.960956i	$-0.410759\pi$
$631$	13.9013	0.553402	0.276701	+	0.960956i	$0.410759\pi$
$632$	0	0
$633$	−9.15844	−0.364015
$634$	0	0
$635$	−6.83300	−0.271159
$636$	0	0
$637$	−4.11229	−0.162935
$638$	0	0
$639$	5.24108	0.207334
$640$	0	0
$641$	34.0600	1.34529	0.672645	−	0.739965i	$-0.265158\pi$
$641$	34.0600	1.34529	0.672645	+	0.739965i	$0.265158\pi$
$642$	0	0
$643$	12.1253	0.478176	0.239088	−	0.970998i	$-0.423152\pi$
$643$	12.1253	0.478176	0.239088	+	0.970998i	$0.423152\pi$
$644$	0	0
$645$	−6.90067	−0.271713
$646$	0	0
$647$	−8.82321	−0.346876	−0.173438	−	0.984845i	$-0.555488\pi$
$647$	−8.82321	−0.346876	−0.173438	+	0.984845i	$0.555488\pi$
$648$	0	0
$649$	7.04498	0.276540
$650$	0	0
$651$	17.7033	0.693849
$652$	0	0
$653$	−30.1111	−1.17834	−0.589169	−	0.808010i	$-0.700545\pi$
$653$	−30.1111	−1.17834	−0.589169	+	0.808010i	$0.700545\pi$
$654$	0	0
$655$	−16.3567	−0.639107
$656$	0	0
$657$	15.0919	0.588791
$658$	0	0
$659$	50.7203	1.97578	0.987891	−	0.155151i	$-0.0495865\pi$
$659$	50.7203	1.97578	0.987891	+	0.155151i	$0.0495865\pi$
$660$	0	0
$661$	10.0303	0.390135	0.195068	−	0.980790i	$-0.437507\pi$
$661$	10.0303	0.390135	0.195068	+	0.980790i	$0.437507\pi$
$662$	0	0
$663$	4.49282	0.174487
$664$	0	0
$665$	18.4382	0.715001
$666$	0	0
$667$	0.163972	0.00634902
$668$	0	0
$669$	1.60452	0.0620343
$670$	0	0
$671$	−12.1286	−0.468221
$672$	0	0
$673$	36.7156	1.41528	0.707641	−	0.706572i	$-0.249759\pi$
$673$	36.7156	1.41528	0.707641	+	0.706572i	$0.249759\pi$
$674$	0	0
$675$	−4.35194	−0.167506
$676$	0	0
$677$	12.0158	0.461806	0.230903	−	0.972977i	$-0.425832\pi$
$677$	12.0158	0.461806	0.230903	+	0.972977i	$0.425832\pi$
$678$	0	0
$679$	11.7467	0.450797
$680$	0	0
$681$	13.6500	0.523069
$682$	0	0
$683$	1.66763	0.0638100	0.0319050	−	0.999491i	$-0.489843\pi$
$683$	1.66763	0.0638100	0.0319050	+	0.999491i	$0.489843\pi$
$684$	0	0
$685$	−14.8708	−0.568185
$686$	0	0
$687$	7.41711	0.282980
$688$	0	0
$689$	−21.1346	−0.805166
$690$	0	0
$691$	−29.8533	−1.13567	−0.567837	−	0.823141i	$-0.692220\pi$
$691$	−29.8533	−1.13567	−0.567837	+	0.823141i	$0.692220\pi$
$692$	0	0
$693$	8.65455	0.328759
$694$	0	0
$695$	17.4615	0.662351
$696$	0	0
$697$	3.68160	0.139451
$698$	0	0
$699$	−21.4692	−0.812039
$700$	0	0
$701$	−28.6608	−1.08250	−0.541251	−	0.840861i	$-0.682049\pi$
$701$	−28.6608	−1.08250	−0.541251	+	0.840861i	$0.682049\pi$
$702$	0	0
$703$	−6.62174	−0.249744
$704$	0	0
$705$	−3.71443	−0.139893
$706$	0	0
$707$	−7.23872	−0.272240
$708$	0	0
$709$	−41.6945	−1.56587	−0.782935	−	0.622103i	$-0.786278\pi$
$709$	−41.6945	−1.56587	−0.782935	+	0.622103i	$0.786278\pi$
$710$	0	0
$711$	−36.6053	−1.37280
$712$	0	0
$713$	−0.226155	−0.00846958
$714$	0	0
$715$	7.82882	0.292781
$716$	0	0
$717$	−14.3383	−0.535472
$718$	0	0
$719$	40.3489	1.50476	0.752380	−	0.658730i	$-0.228906\pi$
$719$	40.3489	1.50476	0.752380	+	0.658730i	$0.228906\pi$
$720$	0	0
$721$	39.9937	1.48944
$722$	0	0
$723$	−16.9859	−0.631712
$724$	0	0
$725$	6.30898	0.234310
$726$	0	0
$727$	−44.7788	−1.66076	−0.830378	−	0.557201i	$-0.811875\pi$
$727$	−44.7788	−1.66076	−0.830378	+	0.557201i	$0.811875\pi$
$728$	0	0
$729$	2.59057	0.0959470
$730$	0	0
$731$	−8.84553	−0.327164
$732$	0	0
$733$	−28.6441	−1.05800	−0.528998	−	0.848623i	$-0.677432\pi$
$733$	−28.6441	−1.05800	−0.528998	+	0.848623i	$0.677432\pi$
$734$	0	0
$735$	−0.637058	−0.0234983
$736$	0	0
$737$	−12.7095	−0.468159
$738$	0	0
$739$	16.7894	0.617608	0.308804	−	0.951126i	$-0.400071\pi$
$739$	16.7894	0.617608	0.308804	+	0.951126i	$0.400071\pi$
$740$	0	0
$741$	−31.7647	−1.16691
$742$	0	0
$743$	−26.6824	−0.978882	−0.489441	−	0.872036i	$-0.662799\pi$
$743$	−26.6824	−0.978882	−0.489441	+	0.872036i	$0.662799\pi$
$744$	0	0
$745$	17.5804	0.644096
$746$	0	0
$747$	−31.7985	−1.16345
$748$	0	0
$749$	−39.1495	−1.43049
$750$	0	0
$751$	−31.8648	−1.16276	−0.581382	−	0.813631i	$-0.697488\pi$
$751$	−31.8648	−1.16276	−0.581382	+	0.813631i	$0.697488\pi$
$752$	0	0
$753$	−11.8684	−0.432508
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−49.0643	−1.78327	−0.891636	−	0.452753i	$-0.850442\pi$
$757$	−49.0643	−1.78327	−0.891636	+	0.452753i	$0.850442\pi$
$758$	0	0
$759$	0.0315211	0.00114414
$760$	0	0
$761$	−33.1581	−1.20198	−0.600991	−	0.799256i	$-0.705227\pi$
$761$	−33.1581	−1.20198	−0.600991	+	0.799256i	$0.705227\pi$
$762$	0	0
$763$	−23.5932	−0.854131
$764$	0	0
$765$	−2.44123	−0.0882630
$766$	0	0
$767$	24.9563	0.901120
$768$	0	0
$769$	−30.3930	−1.09600	−0.548000	−	0.836478i	$-0.684611\pi$
$769$	−30.3930	−1.09600	−0.548000	+	0.836478i	$0.684611\pi$
$770$	0	0
$771$	−0.332703	−0.0119820
$772$	0	0
$773$	19.7199	0.709275	0.354637	−	0.935004i	$-0.384604\pi$
$773$	19.7199	0.709275	0.354637	+	0.935004i	$0.384604\pi$
$774$	0	0
$775$	−8.70155	−0.312569
$776$	0	0
$777$	−1.82212	−0.0653683
$778$	0	0
$779$	−26.0293	−0.932597
$780$	0	0
$781$	−3.33762	−0.119429
$782$	0	0
$783$	−27.4563	−0.981209
$784$	0	0
$785$	−6.44378	−0.229988
$786$	0	0
$787$	2.88388	0.102799	0.0513996	−	0.998678i	$-0.483632\pi$
$787$	2.88388	0.102799	0.0513996	+	0.998678i	$0.483632\pi$
$788$	0	0
$789$	18.3735	0.654113
$790$	0	0
$791$	27.1179	0.964201
$792$	0	0
$793$	−42.9647	−1.52572
$794$	0	0
$795$	−3.27409	−0.116120
$796$	0	0
$797$	45.4239	1.60900	0.804498	−	0.593955i	$-0.202434\pi$
$797$	45.4239	1.60900	0.804498	+	0.593955i	$0.202434\pi$
$798$	0	0
$799$	−4.76129	−0.168442
$800$	0	0
$801$	3.76295	0.132957
$802$	0	0
$803$	−9.61080	−0.339158
$804$	0	0
$805$	−0.0648148	−0.00228442
$806$	0	0
$807$	−2.77664	−0.0977423
$808$	0	0
$809$	−3.54331	−0.124576	−0.0622880	−	0.998058i	$-0.519840\pi$
$809$	−3.54331	−0.124576	−0.0622880	+	0.998058i	$0.519840\pi$
$810$	0	0
$811$	14.4017	0.505712	0.252856	−	0.967504i	$-0.418630\pi$
$811$	14.4017	0.505712	0.252856	+	0.967504i	$0.418630\pi$
$812$	0	0
$813$	−13.4956	−0.473311
$814$	0	0
$815$	−24.4787	−0.857450
$816$	0	0
$817$	62.5388	2.18796
$818$	0	0
$819$	30.6581	1.07128
$820$	0	0
$821$	−45.0725	−1.57304	−0.786520	−	0.617565i	$-0.788119\pi$
$821$	−45.0725	−1.57304	−0.786520	+	0.617565i	$0.788119\pi$
$822$	0	0
$823$	14.9721	0.521894	0.260947	−	0.965353i	$-0.415965\pi$
$823$	14.9721	0.521894	0.260947	+	0.965353i	$0.415965\pi$
$824$	0	0
$825$	1.21281	0.0422245
$826$	0	0
$827$	17.7163	0.616058	0.308029	−	0.951377i	$-0.400331\pi$
$827$	17.7163	0.616058	0.308029	+	0.951377i	$0.400331\pi$
$828$	0	0
$829$	49.3964	1.71561	0.857803	−	0.513978i	$-0.171829\pi$
$829$	49.3964	1.71561	0.857803	+	0.513978i	$0.171829\pi$
$830$	0	0
$831$	11.2044	0.388678
$832$	0	0
$833$	−0.816605	−0.0282937
$834$	0	0
$835$	−4.35235	−0.150619
$836$	0	0
$837$	37.8686	1.30893
$838$	0	0
$839$	34.1615	1.17939	0.589693	−	0.807627i	$-0.299249\pi$
$839$	34.1615	1.17939	0.589693	+	0.807627i	$0.299249\pi$
$840$	0	0
$841$	10.8033	0.372527
$842$	0	0
$843$	−17.4000	−0.599289
$844$	0	0
$845$	14.7330	0.506830
$846$	0	0
$847$	21.9206	0.753201
$848$	0	0
$849$	8.50562	0.291912
$850$	0	0
$851$	0.0232771	0.000797929 0
$852$	0	0
$853$	−27.5569	−0.943531	−0.471765	−	0.881724i	$-0.656383\pi$
$853$	−27.5569	−0.943531	−0.471765	+	0.881724i	$0.656383\pi$
$854$	0	0
$855$	17.2598	0.590272
$856$	0	0
$857$	53.2308	1.81833	0.909165	−	0.416437i	$-0.136721\pi$
$857$	53.2308	1.81833	0.909165	+	0.416437i	$0.136721\pi$
$858$	0	0
$859$	20.7430	0.707741	0.353870	−	0.935294i	$-0.384865\pi$
$859$	20.7430	0.707741	0.353870	+	0.935294i	$0.384865\pi$
$860$	0	0
$861$	−7.16256	−0.244100
$862$	0	0
$863$	−14.9469	−0.508800	−0.254400	−	0.967099i	$-0.581878\pi$
$863$	−14.9469	−0.508800	−0.254400	+	0.967099i	$0.581878\pi$
$864$	0	0
$865$	15.3263	0.521110
$866$	0	0
$867$	−12.9768	−0.440714
$868$	0	0
$869$	23.3109	0.790768
$870$	0	0
$871$	−45.0223	−1.52552
$872$	0	0
$873$	10.9960	0.372158
$874$	0	0
$875$	−2.49382	−0.0843064
$876$	0	0
$877$	29.0485	0.980897	0.490449	−	0.871470i	$-0.336833\pi$
$877$	29.0485	0.980897	0.490449	+	0.871470i	$0.336833\pi$
$878$	0	0
$879$	22.0433	0.743501
$880$	0	0
$881$	9.34637	0.314887	0.157444	−	0.987528i	$-0.449675\pi$
$881$	9.34637	0.314887	0.157444	+	0.987528i	$0.449675\pi$
$882$	0	0
$883$	43.2527	1.45557	0.727784	−	0.685806i	$-0.240550\pi$
$883$	43.2527	1.45557	0.727784	+	0.685806i	$0.240550\pi$
$884$	0	0
$885$	3.86613	0.129958
$886$	0	0
$887$	−46.4026	−1.55805	−0.779023	−	0.626996i	$-0.784284\pi$
$887$	−46.4026	−1.55805	−0.779023	+	0.626996i	$0.784284\pi$
$888$	0	0
$889$	17.0402	0.571512
$890$	0	0
$891$	5.13315	0.171967
$892$	0	0
$893$	33.6628	1.12648
$894$	0	0
$895$	−5.48802	−0.183444
$896$	0	0
$897$	0.111661	0.00372826
$898$	0	0
$899$	−54.8979	−1.83095
$900$	0	0
$901$	−4.19685	−0.139817
$902$	0	0
$903$	17.2090	0.572679
$904$	0	0
$905$	−18.5589	−0.616917
$906$	0	0
$907$	2.55941	0.0849838	0.0424919	−	0.999097i	$-0.486470\pi$
$907$	2.55941	0.0849838	0.0424919	+	0.999097i	$0.486470\pi$
$908$	0	0
$909$	−6.77610	−0.224749
$910$	0	0
$911$	57.4927	1.90482	0.952410	−	0.304820i	$-0.0985964\pi$
$911$	57.4927	1.90482	0.952410	+	0.304820i	$0.0985964\pi$
$912$	0	0
$913$	20.2499	0.670173
$914$	0	0
$915$	−6.65592	−0.220038
$916$	0	0
$917$	40.7905	1.34702
$918$	0	0
$919$	35.4036	1.16786	0.583929	−	0.811805i	$-0.301515\pi$
$919$	35.4036	1.16786	0.583929	+	0.811805i	$0.301515\pi$
$920$	0	0
$921$	−7.03361	−0.231765
$922$	0	0
$923$	−11.8233	−0.389167
$924$	0	0
$925$	0.895610	0.0294475
$926$	0	0
$927$	37.4378	1.22962
$928$	0	0
$929$	49.6502	1.62897	0.814486	−	0.580184i	$-0.197019\pi$
$929$	49.6502	1.62897	0.814486	+	0.580184i	$0.197019\pi$
$930$	0	0
$931$	5.77348	0.189218
$932$	0	0
$933$	−21.4253	−0.701432
$934$	0	0
$935$	1.55462	0.0508416
$936$	0	0
$937$	25.4318	0.830821	0.415410	−	0.909634i	$-0.363638\pi$
$937$	25.4318	0.830821	0.415410	+	0.909634i	$0.363638\pi$
$938$	0	0
$939$	16.8447	0.549706
$940$	0	0
$941$	−8.10387	−0.264179	−0.132089	−	0.991238i	$-0.542169\pi$
$941$	−8.10387	−0.264179	−0.132089	+	0.991238i	$0.542169\pi$
$942$	0	0
$943$	0.0914997	0.00297964
$944$	0	0
$945$	10.8529	0.353046
$946$	0	0
$947$	−18.2598	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$947$	−18.2598	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$948$	0	0
$949$	−34.0455	−1.10516
$950$	0	0
$951$	−4.19440	−0.136013
$952$	0	0
$953$	17.2527	0.558871	0.279435	−	0.960164i	$-0.409853\pi$
$953$	17.2527	0.558871	0.279435	+	0.960164i	$0.409853\pi$
$954$	0	0
$955$	−11.3552	−0.367445
$956$	0	0
$957$	7.65158	0.247341
$958$	0	0
$959$	37.0851	1.19754
$960$	0	0
$961$	44.7170	1.44248
$962$	0	0
$963$	−36.6475	−1.18095
$964$	0	0
$965$	−6.49585	−0.209109
$966$	0	0
$967$	−14.1667	−0.455572	−0.227786	−	0.973711i	$-0.573149\pi$
$967$	−14.1667	−0.455572	−0.227786	+	0.973711i	$0.573149\pi$
$968$	0	0
$969$	−6.30774	−0.202634
$970$	0	0
$971$	−22.0791	−0.708553	−0.354276	−	0.935141i	$-0.615273\pi$
$971$	−22.0791	−0.708553	−0.354276	+	0.935141i	$0.615273\pi$
$972$	0	0
$973$	−43.5457	−1.39601
$974$	0	0
$975$	4.29628	0.137591
$976$	0	0
$977$	47.7424	1.52742	0.763708	−	0.645562i	$-0.223377\pi$
$977$	47.7424	1.52742	0.763708	+	0.645562i	$0.223377\pi$
$978$	0	0
$979$	−2.39632	−0.0765867
$980$	0	0
$981$	−22.0854	−0.705132
$982$	0	0
$983$	−5.77646	−0.184240	−0.0921202	−	0.995748i	$-0.529364\pi$
$983$	−5.77646	−0.184240	−0.0921202	+	0.995748i	$0.529364\pi$
$984$	0	0
$985$	−22.2838	−0.710021
$986$	0	0
$987$	9.26309	0.294848
$988$	0	0
$989$	−0.219840	−0.00699051
$990$	0	0
$991$	49.6709	1.57785	0.788924	−	0.614491i	$-0.210639\pi$
$991$	49.6709	1.57785	0.788924	+	0.614491i	$0.210639\pi$
$992$	0	0
$993$	18.4954	0.586935
$994$	0	0
$995$	−22.6010	−0.716500
$996$	0	0
$997$	38.2765	1.21223	0.606114	−	0.795378i	$-0.292727\pi$
$997$	38.2765	1.21223	0.606114	+	0.795378i	$0.292727\pi$
$998$	0	0
$999$	−3.89764	−0.123316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.12	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.12	✓	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)

\(f(q)\)	\(=\)	\(q+0.815820 q^{3} +1.00000 q^{5} -2.49382 q^{7} -2.33444 q^{9} +O(q^{10})\)	\(q+0.815820 q^{3} +1.00000 q^{5} -2.49382 q^{7} -2.33444 q^{9} +1.48661 q^{11} +5.26621 q^{13} +0.815820 q^{15} +1.04575 q^{17} -7.39355 q^{19} -2.03450 q^{21} +0.0259902 q^{23} +1.00000 q^{25} -4.35194 q^{27} +6.30898 q^{29} -8.70155 q^{31} +1.21281 q^{33} -2.49382 q^{35} +0.895610 q^{37} +4.29628 q^{39} +3.52055 q^{41} -8.45857 q^{43} -2.33444 q^{45} -4.55300 q^{47} -0.780881 q^{49} +0.853142 q^{51} -4.01326 q^{53} +1.48661 q^{55} -6.03180 q^{57} +4.73895 q^{59} -8.15857 q^{61} +5.82166 q^{63} +5.26621 q^{65} -8.54928 q^{67} +0.0212033 q^{69} -2.24512 q^{71} -6.46490 q^{73} +0.815820 q^{75} -3.70734 q^{77} +15.6805 q^{79} +3.45292 q^{81} +13.6215 q^{83} +1.04575 q^{85} +5.14699 q^{87} -1.61193 q^{89} -13.1330 q^{91} -7.09890 q^{93} -7.39355 q^{95} -4.71034 q^{97} -3.47041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 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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