LMFDB - Embedded newform 6040.2.a.p.1.3

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			1.3
Root			$2.89238$ 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)

$f(q)$	$=$	$q-2.89238 q^{3} +1.00000 q^{5} +0.451178 q^{7} +5.36583 q^{9} +O(q^{10})$	$q-2.89238 q^{3} +1.00000 q^{5} +0.451178 q^{7} +5.36583 q^{9} -4.25329 q^{11} -4.52874 q^{13} -2.89238 q^{15} +2.97411 q^{17} -0.416600 q^{19} -1.30498 q^{21} +4.36636 q^{23} +1.00000 q^{25} -6.84288 q^{27} -1.71423 q^{29} +0.519238 q^{31} +12.3021 q^{33} +0.451178 q^{35} +5.90793 q^{37} +13.0988 q^{39} +7.30082 q^{41} +6.72872 q^{43} +5.36583 q^{45} -7.34363 q^{47} -6.79644 q^{49} -8.60224 q^{51} -8.53509 q^{53} -4.25329 q^{55} +1.20496 q^{57} -13.9094 q^{59} +12.3159 q^{61} +2.42095 q^{63} -4.52874 q^{65} -3.77950 q^{67} -12.6292 q^{69} +9.96010 q^{71} +13.1976 q^{73} -2.89238 q^{75} -1.91899 q^{77} -17.1764 q^{79} +3.69468 q^{81} -12.9481 q^{83} +2.97411 q^{85} +4.95821 q^{87} -2.27054 q^{89} -2.04327 q^{91} -1.50183 q^{93} -0.416600 q^{95} +14.6215 q^{97} -22.8225 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.89238	−1.66991	−0.834957	−	0.550315i	$-0.814507\pi$
$3$	−2.89238	−1.66991	−0.834957	+	0.550315i	$0.814507\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.451178	0.170529	0.0852646	−	0.996358i	$-0.472826\pi$
$7$	0.451178	0.170529	0.0852646	+	0.996358i	$0.472826\pi$
$8$	0	0
$9$	5.36583	1.78861
$10$	0	0
$11$	−4.25329	−1.28242	−0.641208	−	0.767367i	$-0.721566\pi$
$11$	−4.25329	−1.28242	−0.641208	+	0.767367i	$0.721566\pi$
$12$	0	0
$13$	−4.52874	−1.25605	−0.628023	−	0.778195i	$-0.716136\pi$
$13$	−4.52874	−1.25605	−0.628023	+	0.778195i	$0.716136\pi$
$14$	0	0
$15$	−2.89238	−0.746808
$16$	0	0
$17$	2.97411	0.721328	0.360664	−	0.932696i	$-0.382550\pi$
$17$	2.97411	0.721328	0.360664	+	0.932696i	$0.382550\pi$
$18$	0	0
$19$	−0.416600	−0.0955747	−0.0477873	−	0.998858i	$-0.515217\pi$
$19$	−0.416600	−0.0955747	−0.0477873	+	0.998858i	$0.515217\pi$
$20$	0	0
$21$	−1.30498	−0.284769
$22$	0	0
$23$	4.36636	0.910449	0.455225	−	0.890377i	$-0.349559\pi$
$23$	4.36636	0.910449	0.455225	+	0.890377i	$0.349559\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−6.84288	−1.31691
$28$	0	0
$29$	−1.71423	−0.318325	−0.159163	−	0.987252i	$-0.550879\pi$
$29$	−1.71423	−0.318325	−0.159163	+	0.987252i	$0.550879\pi$
$30$	0	0
$31$	0.519238	0.0932580	0.0466290	−	0.998912i	$-0.485152\pi$
$31$	0.519238	0.0932580	0.0466290	+	0.998912i	$0.485152\pi$
$32$	0	0
$33$	12.3021	2.14152
$34$	0	0
$35$	0.451178	0.0762630
$36$	0	0
$37$	5.90793	0.971257	0.485629	−	0.874165i	$-0.338591\pi$
$37$	5.90793	0.971257	0.485629	+	0.874165i	$0.338591\pi$
$38$	0	0
$39$	13.0988	2.09749
$40$	0	0
$41$	7.30082	1.14020	0.570098	−	0.821577i	$-0.306905\pi$
$41$	7.30082	1.14020	0.570098	+	0.821577i	$0.306905\pi$
$42$	0	0
$43$	6.72872	1.02612	0.513060	−	0.858353i	$-0.328512\pi$
$43$	6.72872	1.02612	0.513060	+	0.858353i	$0.328512\pi$
$44$	0	0
$45$	5.36583	0.799891
$46$	0	0
$47$	−7.34363	−1.07118	−0.535589	−	0.844479i	$-0.679910\pi$
$47$	−7.34363	−1.07118	−0.535589	+	0.844479i	$0.679910\pi$
$48$	0	0
$49$	−6.79644	−0.970920
$50$	0	0
$51$	−8.60224	−1.20455
$52$	0	0
$53$	−8.53509	−1.17238	−0.586192	−	0.810172i	$-0.699374\pi$
$53$	−8.53509	−1.17238	−0.586192	+	0.810172i	$0.699374\pi$
$54$	0	0
$55$	−4.25329	−0.573513
$56$	0	0
$57$	1.20496	0.159601
$58$	0	0
$59$	−13.9094	−1.81085	−0.905427	−	0.424503i	$-0.860449\pi$
$59$	−13.9094	−1.81085	−0.905427	+	0.424503i	$0.860449\pi$
$60$	0	0
$61$	12.3159	1.57688	0.788442	−	0.615109i	$-0.210888\pi$
$61$	12.3159	1.57688	0.788442	+	0.615109i	$0.210888\pi$
$62$	0	0
$63$	2.42095	0.305011
$64$	0	0
$65$	−4.52874	−0.561721
$66$	0	0
$67$	−3.77950	−0.461739	−0.230869	−	0.972985i	$-0.574157\pi$
$67$	−3.77950	−0.461739	−0.230869	+	0.972985i	$0.574157\pi$
$68$	0	0
$69$	−12.6292	−1.52037
$70$	0	0
$71$	9.96010	1.18205	0.591023	−	0.806654i	$-0.298724\pi$
$71$	9.96010	1.18205	0.591023	+	0.806654i	$0.298724\pi$
$72$	0	0
$73$	13.1976	1.54467	0.772333	−	0.635218i	$-0.219090\pi$
$73$	13.1976	1.54467	0.772333	+	0.635218i	$0.219090\pi$
$74$	0	0
$75$	−2.89238	−0.333983
$76$	0	0
$77$	−1.91899	−0.218689
$78$	0	0
$79$	−17.1764	−1.93250	−0.966248	−	0.257612i	$-0.917064\pi$
$79$	−17.1764	−1.93250	−0.966248	+	0.257612i	$0.917064\pi$
$80$	0	0
$81$	3.69468	0.410520
$82$	0	0
$83$	−12.9481	−1.42123	−0.710617	−	0.703579i	$-0.751584\pi$
$83$	−12.9481	−1.42123	−0.710617	+	0.703579i	$0.751584\pi$
$84$	0	0
$85$	2.97411	0.322588
$86$	0	0
$87$	4.95821	0.531575
$88$	0	0
$89$	−2.27054	−0.240676	−0.120338	−	0.992733i	$-0.538398\pi$
$89$	−2.27054	−0.240676	−0.120338	+	0.992733i	$0.538398\pi$
$90$	0	0
$91$	−2.04327	−0.214193
$92$	0	0
$93$	−1.50183	−0.155733
$94$	0	0
$95$	−0.416600	−0.0427423
$96$	0	0
$97$	14.6215	1.48459	0.742295	−	0.670073i	$-0.233737\pi$
$97$	14.6215	1.48459	0.742295	+	0.670073i	$0.233737\pi$
$98$	0	0
$99$	−22.8225	−2.29374
$100$	0	0
$101$	8.49145	0.844931	0.422465	−	0.906379i	$-0.361165\pi$
$101$	8.49145	0.844931	0.422465	+	0.906379i	$0.361165\pi$
$102$	0	0
$103$	−1.32253	−0.130313	−0.0651565	−	0.997875i	$-0.520755\pi$
$103$	−1.32253	−0.130313	−0.0651565	+	0.997875i	$0.520755\pi$
$104$	0	0
$105$	−1.30498	−0.127353
$106$	0	0
$107$	8.30393	0.802771	0.401386	−	0.915909i	$-0.368529\pi$
$107$	8.30393	0.802771	0.401386	+	0.915909i	$0.368529\pi$
$108$	0	0
$109$	−8.11894	−0.777654	−0.388827	−	0.921311i	$-0.627120\pi$
$109$	−8.11894	−0.777654	−0.388827	+	0.921311i	$0.627120\pi$
$110$	0	0
$111$	−17.0879	−1.62192
$112$	0	0
$113$	5.99555	0.564014	0.282007	−	0.959412i	$-0.409000\pi$
$113$	5.99555	0.564014	0.282007	+	0.959412i	$0.409000\pi$
$114$	0	0
$115$	4.36636	0.407165
$116$	0	0
$117$	−24.3005	−2.24658
$118$	0	0
$119$	1.34185	0.123007
$120$	0	0
$121$	7.09047	0.644588
$122$	0	0
$123$	−21.1167	−1.90403
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	3.64749	0.323662	0.161831	−	0.986818i	$-0.448260\pi$
$127$	3.64749	0.323662	0.161831	+	0.986818i	$0.448260\pi$
$128$	0	0
$129$	−19.4620	−1.71353
$130$	0	0
$131$	7.37463	0.644324	0.322162	−	0.946684i	$-0.395590\pi$
$131$	7.37463	0.644324	0.322162	+	0.946684i	$0.395590\pi$
$132$	0	0
$133$	−0.187961	−0.0162983
$134$	0	0
$135$	−6.84288	−0.588942
$136$	0	0
$137$	22.7441	1.94316	0.971579	−	0.236714i	$-0.0760706\pi$
$137$	22.7441	1.94316	0.971579	+	0.236714i	$0.0760706\pi$
$138$	0	0
$139$	14.9027	1.26403	0.632014	−	0.774957i	$-0.282229\pi$
$139$	14.9027	1.26403	0.632014	+	0.774957i	$0.282229\pi$
$140$	0	0
$141$	21.2405	1.78878
$142$	0	0
$143$	19.2620	1.61077
$144$	0	0
$145$	−1.71423	−0.142359
$146$	0	0
$147$	19.6579	1.62135
$148$	0	0
$149$	1.10295	0.0903569	0.0451785	−	0.998979i	$-0.485614\pi$
$149$	1.10295	0.0903569	0.0451785	+	0.998979i	$0.485614\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	15.9586	1.29018
$154$	0	0
$155$	0.519238	0.0417062
$156$	0	0
$157$	−9.71155	−0.775066	−0.387533	−	0.921856i	$-0.626673\pi$
$157$	−9.71155	−0.775066	−0.387533	+	0.921856i	$0.626673\pi$
$158$	0	0
$159$	24.6867	1.95778
$160$	0	0
$161$	1.97001	0.155258
$162$	0	0
$163$	−9.78801	−0.766656	−0.383328	−	0.923612i	$-0.625222\pi$
$163$	−9.78801	−0.766656	−0.383328	+	0.923612i	$0.625222\pi$
$164$	0	0
$165$	12.3021	0.957718
$166$	0	0
$167$	4.81946	0.372941	0.186471	−	0.982461i	$-0.440295\pi$
$167$	4.81946	0.372941	0.186471	+	0.982461i	$0.440295\pi$
$168$	0	0
$169$	7.50945	0.577650
$170$	0	0
$171$	−2.23541	−0.170946
$172$	0	0
$173$	19.0278	1.44666	0.723329	−	0.690503i	$-0.242611\pi$
$173$	19.0278	1.44666	0.723329	+	0.690503i	$0.242611\pi$
$174$	0	0
$175$	0.451178	0.0341059
$176$	0	0
$177$	40.2313	3.02397
$178$	0	0
$179$	−13.0012	−0.971756	−0.485878	−	0.874027i	$-0.661500\pi$
$179$	−13.0012	−0.971756	−0.485878	+	0.874027i	$0.661500\pi$
$180$	0	0
$181$	−20.5584	−1.52809	−0.764045	−	0.645163i	$-0.776790\pi$
$181$	−20.5584	−1.52809	−0.764045	+	0.645163i	$0.776790\pi$
$182$	0	0
$183$	−35.6221	−2.63326
$184$	0	0
$185$	5.90793	0.434359
$186$	0	0
$187$	−12.6498	−0.925041
$188$	0	0
$189$	−3.08736	−0.224572
$190$	0	0
$191$	−18.9544	−1.37149	−0.685747	−	0.727840i	$-0.740524\pi$
$191$	−18.9544	−1.37149	−0.685747	+	0.727840i	$0.740524\pi$
$192$	0	0
$193$	−27.7140	−1.99490	−0.997449	−	0.0713838i	$-0.977258\pi$
$193$	−27.7140	−1.99490	−0.997449	+	0.0713838i	$0.977258\pi$
$194$	0	0
$195$	13.0988	0.938025
$196$	0	0
$197$	7.09673	0.505621	0.252810	−	0.967516i	$-0.418645\pi$
$197$	7.09673	0.505621	0.252810	+	0.967516i	$0.418645\pi$
$198$	0	0
$199$	−22.7907	−1.61559	−0.807794	−	0.589465i	$-0.799338\pi$
$199$	−22.7907	−1.61559	−0.807794	+	0.589465i	$0.799338\pi$
$200$	0	0
$201$	10.9317	0.771064
$202$	0	0
$203$	−0.773424	−0.0542838
$204$	0	0
$205$	7.30082	0.509911
$206$	0	0
$207$	23.4292	1.62844
$208$	0	0
$209$	1.77192	0.122566
$210$	0	0
$211$	17.0733	1.17537	0.587687	−	0.809089i	$-0.300039\pi$
$211$	17.0733	1.17537	0.587687	+	0.809089i	$0.300039\pi$
$212$	0	0
$213$	−28.8084	−1.97392
$214$	0	0
$215$	6.72872	0.458895
$216$	0	0
$217$	0.234269	0.0159032
$218$	0	0
$219$	−38.1725	−2.57946
$220$	0	0
$221$	−13.4690	−0.906020
$222$	0	0
$223$	−1.97058	−0.131960	−0.0659800	−	0.997821i	$-0.521017\pi$
$223$	−1.97058	−0.131960	−0.0659800	+	0.997821i	$0.521017\pi$
$224$	0	0
$225$	5.36583	0.357722
$226$	0	0
$227$	−20.8242	−1.38215	−0.691074	−	0.722784i	$-0.742862\pi$
$227$	−20.8242	−1.38215	−0.691074	+	0.722784i	$0.742862\pi$
$228$	0	0
$229$	−11.3583	−0.750580	−0.375290	−	0.926908i	$-0.622457\pi$
$229$	−11.3583	−0.750580	−0.375290	+	0.926908i	$0.622457\pi$
$230$	0	0
$231$	5.55044	0.365192
$232$	0	0
$233$	−5.39183	−0.353231	−0.176615	−	0.984280i	$-0.556515\pi$
$233$	−5.39183	−0.353231	−0.176615	+	0.984280i	$0.556515\pi$
$234$	0	0
$235$	−7.34363	−0.479046
$236$	0	0
$237$	49.6806	3.22710
$238$	0	0
$239$	−2.49168	−0.161173	−0.0805865	−	0.996748i	$-0.525679\pi$
$239$	−2.49168	−0.161173	−0.0805865	+	0.996748i	$0.525679\pi$
$240$	0	0
$241$	−26.3996	−1.70055	−0.850275	−	0.526339i	$-0.823564\pi$
$241$	−26.3996	−1.70055	−0.850275	+	0.526339i	$0.823564\pi$
$242$	0	0
$243$	9.84225	0.631380
$244$	0	0
$245$	−6.79644	−0.434209
$246$	0	0
$247$	1.88667	0.120046
$248$	0	0
$249$	37.4506	2.37334
$250$	0	0
$251$	−4.13027	−0.260700	−0.130350	−	0.991468i	$-0.541610\pi$
$251$	−4.13027	−0.260700	−0.130350	+	0.991468i	$0.541610\pi$
$252$	0	0
$253$	−18.5714	−1.16757
$254$	0	0
$255$	−8.60224	−0.538693
$256$	0	0
$257$	−9.57443	−0.597237	−0.298618	−	0.954373i	$-0.596526\pi$
$257$	−9.57443	−0.597237	−0.298618	+	0.954373i	$0.596526\pi$
$258$	0	0
$259$	2.66553	0.165628
$260$	0	0
$261$	−9.19829	−0.569360
$262$	0	0
$263$	1.21730	0.0750620	0.0375310	−	0.999295i	$-0.488051\pi$
$263$	1.21730	0.0750620	0.0375310	+	0.999295i	$0.488051\pi$
$264$	0	0
$265$	−8.53509	−0.524307
$266$	0	0
$267$	6.56724	0.401909
$268$	0	0
$269$	−29.3682	−1.79061	−0.895304	−	0.445456i	$-0.853042\pi$
$269$	−29.3682	−1.79061	−0.895304	+	0.445456i	$0.853042\pi$
$270$	0	0
$271$	20.6624	1.25515	0.627577	−	0.778554i	$-0.284047\pi$
$271$	20.6624	1.25515	0.627577	+	0.778554i	$0.284047\pi$
$272$	0	0
$273$	5.90989	0.357683
$274$	0	0
$275$	−4.25329	−0.256483
$276$	0	0
$277$	−23.5077	−1.41244	−0.706221	−	0.707992i	$-0.749601\pi$
$277$	−23.5077	−1.41244	−0.706221	+	0.707992i	$0.749601\pi$
$278$	0	0
$279$	2.78615	0.166802
$280$	0	0
$281$	−26.8257	−1.60028	−0.800142	−	0.599810i	$-0.795243\pi$
$281$	−26.8257	−1.60028	−0.800142	+	0.599810i	$0.795243\pi$
$282$	0	0
$283$	25.2537	1.50118	0.750589	−	0.660769i	$-0.229770\pi$
$283$	25.2537	1.50118	0.750589	+	0.660769i	$0.229770\pi$
$284$	0	0
$285$	1.20496	0.0713759
$286$	0	0
$287$	3.29397	0.194437
$288$	0	0
$289$	−8.15467	−0.479686
$290$	0	0
$291$	−42.2909	−2.47914
$292$	0	0
$293$	3.08761	0.180380	0.0901900	−	0.995925i	$-0.471253\pi$
$293$	3.08761	0.180380	0.0901900	+	0.995925i	$0.471253\pi$
$294$	0	0
$295$	−13.9094	−0.809838
$296$	0	0
$297$	29.1048	1.68883
$298$	0	0
$299$	−19.7741	−1.14357
$300$	0	0
$301$	3.03585	0.174984
$302$	0	0
$303$	−24.5605	−1.41096
$304$	0	0
$305$	12.3159	0.705204
$306$	0	0
$307$	−20.5478	−1.17273	−0.586363	−	0.810048i	$-0.699441\pi$
$307$	−20.5478	−1.17273	−0.586363	+	0.810048i	$0.699441\pi$
$308$	0	0
$309$	3.82526	0.217611
$310$	0	0
$311$	15.8529	0.898933	0.449467	−	0.893297i	$-0.351614\pi$
$311$	15.8529	0.898933	0.449467	+	0.893297i	$0.351614\pi$
$312$	0	0
$313$	27.8519	1.57428	0.787140	−	0.616775i	$-0.211561\pi$
$313$	27.8519	1.57428	0.787140	+	0.616775i	$0.211561\pi$
$314$	0	0
$315$	2.42095	0.136405
$316$	0	0
$317$	3.01896	0.169562	0.0847808	−	0.996400i	$-0.472981\pi$
$317$	3.01896	0.169562	0.0847808	+	0.996400i	$0.472981\pi$
$318$	0	0
$319$	7.29113	0.408225
$320$	0	0
$321$	−24.0181	−1.34056
$322$	0	0
$323$	−1.23902	−0.0689406
$324$	0	0
$325$	−4.52874	−0.251209
$326$	0	0
$327$	23.4830	1.29861
$328$	0	0
$329$	−3.31329	−0.182667
$330$	0	0
$331$	25.2754	1.38926	0.694631	−	0.719366i	$-0.255568\pi$
$331$	25.2754	1.38926	0.694631	+	0.719366i	$0.255568\pi$
$332$	0	0
$333$	31.7010	1.73720
$334$	0	0
$335$	−3.77950	−0.206496
$336$	0	0
$337$	−11.0237	−0.600499	−0.300249	−	0.953861i	$-0.597070\pi$
$337$	−11.0237	−0.600499	−0.300249	+	0.953861i	$0.597070\pi$
$338$	0	0
$339$	−17.3414	−0.941855
$340$	0	0
$341$	−2.20847	−0.119595
$342$	0	0
$343$	−6.22465	−0.336100
$344$	0	0
$345$	−12.6292	−0.679931
$346$	0	0
$347$	−9.11433	−0.489283	−0.244641	−	0.969614i	$-0.578670\pi$
$347$	−9.11433	−0.489283	−0.244641	+	0.969614i	$0.578670\pi$
$348$	0	0
$349$	−30.4330	−1.62904	−0.814522	−	0.580133i	$-0.803001\pi$
$349$	−30.4330	−1.62904	−0.814522	+	0.580133i	$0.803001\pi$
$350$	0	0
$351$	30.9896	1.65410
$352$	0	0
$353$	20.9558	1.11536	0.557682	−	0.830054i	$-0.311691\pi$
$353$	20.9558	1.11536	0.557682	+	0.830054i	$0.311691\pi$
$354$	0	0
$355$	9.96010	0.528627
$356$	0	0
$357$	−3.88114	−0.205412
$358$	0	0
$359$	−20.8189	−1.09878	−0.549390	−	0.835566i	$-0.685140\pi$
$359$	−20.8189	−1.09878	−0.549390	+	0.835566i	$0.685140\pi$
$360$	0	0
$361$	−18.8264	−0.990865
$362$	0	0
$363$	−20.5083	−1.07641
$364$	0	0
$365$	13.1976	0.690796
$366$	0	0
$367$	−10.6761	−0.557289	−0.278645	−	0.960394i	$-0.589885\pi$
$367$	−10.6761	−0.557289	−0.278645	+	0.960394i	$0.589885\pi$
$368$	0	0
$369$	39.1750	2.03937
$370$	0	0
$371$	−3.85085	−0.199926
$372$	0	0
$373$	23.0816	1.19512	0.597559	−	0.801825i	$-0.296137\pi$
$373$	23.0816	1.19512	0.597559	+	0.801825i	$0.296137\pi$
$374$	0	0
$375$	−2.89238	−0.149362
$376$	0	0
$377$	7.76331	0.399831
$378$	0	0
$379$	10.8675	0.558224	0.279112	−	0.960259i	$-0.409960\pi$
$379$	10.8675	0.558224	0.279112	+	0.960259i	$0.409960\pi$
$380$	0	0
$381$	−10.5499	−0.540488
$382$	0	0
$383$	−17.6815	−0.903482	−0.451741	−	0.892149i	$-0.649197\pi$
$383$	−17.6815	−0.903482	−0.451741	+	0.892149i	$0.649197\pi$
$384$	0	0
$385$	−1.91899	−0.0978008
$386$	0	0
$387$	36.1052	1.83533
$388$	0	0
$389$	−19.8691	−1.00740	−0.503701	−	0.863878i	$-0.668029\pi$
$389$	−19.8691	−1.00740	−0.503701	+	0.863878i	$0.668029\pi$
$390$	0	0
$391$	12.9860	0.656732
$392$	0	0
$393$	−21.3302	−1.07597
$394$	0	0
$395$	−17.1764	−0.864239
$396$	0	0
$397$	31.7296	1.59246	0.796231	−	0.604993i	$-0.206824\pi$
$397$	31.7296	1.59246	0.796231	+	0.604993i	$0.206824\pi$
$398$	0	0
$399$	0.543653	0.0272167
$400$	0	0
$401$	−35.3388	−1.76474	−0.882368	−	0.470560i	$-0.844052\pi$
$401$	−35.3388	−1.76474	−0.882368	+	0.470560i	$0.844052\pi$
$402$	0	0
$403$	−2.35149	−0.117136
$404$	0	0
$405$	3.69468	0.183590
$406$	0	0
$407$	−25.1281	−1.24555
$408$	0	0
$409$	−19.4214	−0.960326	−0.480163	−	0.877179i	$-0.659422\pi$
$409$	−19.4214	−0.960326	−0.480163	+	0.877179i	$0.659422\pi$
$410$	0	0
$411$	−65.7844	−3.24491
$412$	0	0
$413$	−6.27563	−0.308803
$414$	0	0
$415$	−12.9481	−0.635595
$416$	0	0
$417$	−43.1041	−2.11082
$418$	0	0
$419$	−4.14059	−0.202281	−0.101141	−	0.994872i	$-0.532249\pi$
$419$	−4.14059	−0.202281	−0.101141	+	0.994872i	$0.532249\pi$
$420$	0	0
$421$	3.31113	0.161374	0.0806872	−	0.996739i	$-0.474289\pi$
$421$	3.31113	0.161374	0.0806872	+	0.996739i	$0.474289\pi$
$422$	0	0
$423$	−39.4047	−1.91592
$424$	0	0
$425$	2.97411	0.144266
$426$	0	0
$427$	5.55664	0.268905
$428$	0	0
$429$	−55.7130	−2.68985
$430$	0	0
$431$	−18.9816	−0.914312	−0.457156	−	0.889387i	$-0.651132\pi$
$431$	−18.9816	−0.914312	−0.457156	+	0.889387i	$0.651132\pi$
$432$	0	0
$433$	−18.9959	−0.912884	−0.456442	−	0.889753i	$-0.650876\pi$
$433$	−18.9959	−0.912884	−0.456442	+	0.889753i	$0.650876\pi$
$434$	0	0
$435$	4.95821	0.237728
$436$	0	0
$437$	−1.81903	−0.0870159
$438$	0	0
$439$	11.5187	0.549759	0.274879	−	0.961479i	$-0.411362\pi$
$439$	11.5187	0.549759	0.274879	+	0.961479i	$0.411362\pi$
$440$	0	0
$441$	−36.4686	−1.73660
$442$	0	0
$443$	−36.6022	−1.73902	−0.869512	−	0.493911i	$-0.835567\pi$
$443$	−36.6022	−1.73902	−0.869512	+	0.493911i	$0.835567\pi$
$444$	0	0
$445$	−2.27054	−0.107634
$446$	0	0
$447$	−3.19014	−0.150888
$448$	0	0
$449$	20.0117	0.944412	0.472206	−	0.881488i	$-0.343458\pi$
$449$	20.0117	0.944412	0.472206	+	0.881488i	$0.343458\pi$
$450$	0	0
$451$	−31.0525	−1.46220
$452$	0	0
$453$	−2.89238	−0.135896
$454$	0	0
$455$	−2.04327	−0.0957898
$456$	0	0
$457$	1.42946	0.0668672	0.0334336	−	0.999441i	$-0.489356\pi$
$457$	1.42946	0.0668672	0.0334336	+	0.999441i	$0.489356\pi$
$458$	0	0
$459$	−20.3515	−0.949926
$460$	0	0
$461$	−22.2114	−1.03449	−0.517244	−	0.855838i	$-0.673042\pi$
$461$	−22.2114	−1.03449	−0.517244	+	0.855838i	$0.673042\pi$
$462$	0	0
$463$	15.8050	0.734522	0.367261	−	0.930118i	$-0.380296\pi$
$463$	15.8050	0.734522	0.367261	+	0.930118i	$0.380296\pi$
$464$	0	0
$465$	−1.50183	−0.0696458
$466$	0	0
$467$	−10.1175	−0.468182	−0.234091	−	0.972215i	$-0.575211\pi$
$467$	−10.1175	−0.468182	−0.234091	+	0.972215i	$0.575211\pi$
$468$	0	0
$469$	−1.70523	−0.0787400
$470$	0	0
$471$	28.0895	1.29429
$472$	0	0
$473$	−28.6192	−1.31591
$474$	0	0
$475$	−0.416600	−0.0191149
$476$	0	0
$477$	−45.7979	−2.09694
$478$	0	0
$479$	−4.25747	−0.194529	−0.0972644	−	0.995259i	$-0.531009\pi$
$479$	−4.25747	−0.194529	−0.0972644	+	0.995259i	$0.531009\pi$
$480$	0	0
$481$	−26.7554	−1.21994
$482$	0	0
$483$	−5.69800	−0.259268
$484$	0	0
$485$	14.6215	0.663929
$486$	0	0
$487$	−22.5398	−1.02138	−0.510689	−	0.859766i	$-0.670610\pi$
$487$	−22.5398	−1.02138	−0.510689	+	0.859766i	$0.670610\pi$
$488$	0	0
$489$	28.3106	1.28025
$490$	0	0
$491$	34.1919	1.54306	0.771529	−	0.636194i	$-0.219492\pi$
$491$	34.1919	1.54306	0.771529	+	0.636194i	$0.219492\pi$
$492$	0	0
$493$	−5.09832	−0.229617
$494$	0	0
$495$	−22.8225	−1.02579
$496$	0	0
$497$	4.49378	0.201574
$498$	0	0
$499$	−20.9147	−0.936273	−0.468136	−	0.883656i	$-0.655074\pi$
$499$	−20.9147	−0.936273	−0.468136	+	0.883656i	$0.655074\pi$
$500$	0	0
$501$	−13.9397	−0.622780
$502$	0	0
$503$	8.21327	0.366212	0.183106	−	0.983093i	$-0.441385\pi$
$503$	8.21327	0.366212	0.183106	+	0.983093i	$0.441385\pi$
$504$	0	0
$505$	8.49145	0.377865
$506$	0	0
$507$	−21.7201	−0.964626
$508$	0	0
$509$	6.57506	0.291434	0.145717	−	0.989326i	$-0.453451\pi$
$509$	6.57506	0.291434	0.145717	+	0.989326i	$0.453451\pi$
$510$	0	0
$511$	5.95448	0.263411
$512$	0	0
$513$	2.85075	0.125864
$514$	0	0
$515$	−1.32253	−0.0582777
$516$	0	0
$517$	31.2346	1.37370
$518$	0	0
$519$	−55.0356	−2.41580
$520$	0	0
$521$	22.9988	1.00760	0.503798	−	0.863822i	$-0.331936\pi$
$521$	22.9988	1.00760	0.503798	+	0.863822i	$0.331936\pi$
$522$	0	0
$523$	−18.6672	−0.816260	−0.408130	−	0.912924i	$-0.633819\pi$
$523$	−18.6672	−0.816260	−0.408130	+	0.912924i	$0.633819\pi$
$524$	0	0
$525$	−1.30498	−0.0569538
$526$	0	0
$527$	1.54427	0.0672696
$528$	0	0
$529$	−3.93490	−0.171082
$530$	0	0
$531$	−74.6357	−3.23891
$532$	0	0
$533$	−33.0635	−1.43214
$534$	0	0
$535$	8.30393	0.359010
$536$	0	0
$537$	37.6044	1.62275
$538$	0	0
$539$	28.9072	1.24512
$540$	0	0
$541$	0.192302	0.00826770	0.00413385	−	0.999991i	$-0.498684\pi$
$541$	0.192302	0.00826770	0.00413385	+	0.999991i	$0.498684\pi$
$542$	0	0
$543$	59.4625	2.55178
$544$	0	0
$545$	−8.11894	−0.347777
$546$	0	0
$547$	34.2014	1.46235	0.731174	−	0.682191i	$-0.238973\pi$
$547$	34.2014	1.46235	0.731174	+	0.682191i	$0.238973\pi$
$548$	0	0
$549$	66.0848	2.82043
$550$	0	0
$551$	0.714150	0.0304238
$552$	0	0
$553$	−7.74962	−0.329547
$554$	0	0
$555$	−17.0879	−0.725343
$556$	0	0
$557$	−23.5418	−0.997497	−0.498749	−	0.866747i	$-0.666207\pi$
$557$	−23.5418	−0.997497	−0.498749	+	0.866747i	$0.666207\pi$
$558$	0	0
$559$	−30.4726	−1.28885
$560$	0	0
$561$	36.5878	1.54474
$562$	0	0
$563$	1.68157	0.0708695	0.0354348	−	0.999372i	$-0.488718\pi$
$563$	1.68157	0.0708695	0.0354348	+	0.999372i	$0.488718\pi$
$564$	0	0
$565$	5.99555	0.252235
$566$	0	0
$567$	1.66696	0.0700057
$568$	0	0
$569$	−5.29042	−0.221786	−0.110893	−	0.993832i	$-0.535371\pi$
$569$	−5.29042	−0.221786	−0.110893	+	0.993832i	$0.535371\pi$
$570$	0	0
$571$	21.0892	0.882556	0.441278	−	0.897371i	$-0.354525\pi$
$571$	21.0892	0.882556	0.441278	+	0.897371i	$0.354525\pi$
$572$	0	0
$573$	54.8233	2.29028
$574$	0	0
$575$	4.36636	0.182090
$576$	0	0
$577$	−10.4200	−0.433791	−0.216895	−	0.976195i	$-0.569593\pi$
$577$	−10.4200	−0.433791	−0.216895	+	0.976195i	$0.569593\pi$
$578$	0	0
$579$	80.1593	3.33131
$580$	0	0
$581$	−5.84188	−0.242362
$582$	0	0
$583$	36.3022	1.50348
$584$	0	0
$585$	−24.3005	−1.00470
$586$	0	0
$587$	−43.7873	−1.80730	−0.903648	−	0.428275i	$-0.859121\pi$
$587$	−43.7873	−1.80730	−0.903648	+	0.428275i	$0.859121\pi$
$588$	0	0
$589$	−0.216315	−0.00891310
$590$	0	0
$591$	−20.5264	−0.844343
$592$	0	0
$593$	−14.2309	−0.584393	−0.292197	−	0.956358i	$-0.594386\pi$
$593$	−14.2309	−0.584393	−0.292197	+	0.956358i	$0.594386\pi$
$594$	0	0
$595$	1.34185	0.0550106
$596$	0	0
$597$	65.9192	2.69789
$598$	0	0
$599$	19.0831	0.779716	0.389858	−	0.920875i	$-0.372524\pi$
$599$	19.0831	0.779716	0.389858	+	0.920875i	$0.372524\pi$
$600$	0	0
$601$	−8.90747	−0.363343	−0.181672	−	0.983359i	$-0.558151\pi$
$601$	−8.90747	−0.363343	−0.181672	+	0.983359i	$0.558151\pi$
$602$	0	0
$603$	−20.2801	−0.825871
$604$	0	0
$605$	7.09047	0.288269
$606$	0	0
$607$	−5.79536	−0.235227	−0.117613	−	0.993059i	$-0.537524\pi$
$607$	−5.79536	−0.235227	−0.117613	+	0.993059i	$0.537524\pi$
$608$	0	0
$609$	2.23703	0.0906492
$610$	0	0
$611$	33.2574	1.34545
$612$	0	0
$613$	32.9712	1.33169	0.665847	−	0.746089i	$-0.268071\pi$
$613$	32.9712	1.33169	0.665847	+	0.746089i	$0.268071\pi$
$614$	0	0
$615$	−21.1167	−0.851508
$616$	0	0
$617$	−10.2643	−0.413225	−0.206613	−	0.978423i	$-0.566244\pi$
$617$	−10.2643	−0.413225	−0.206613	+	0.978423i	$0.566244\pi$
$618$	0	0
$619$	2.77427	0.111507	0.0557536	−	0.998445i	$-0.482244\pi$
$619$	2.77427	0.111507	0.0557536	+	0.998445i	$0.482244\pi$
$620$	0	0
$621$	−29.8785	−1.19898
$622$	0	0
$623$	−1.02442	−0.0410424
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.12506	−0.204675
$628$	0	0
$629$	17.5708	0.700595
$630$	0	0
$631$	−28.6155	−1.13917	−0.569583	−	0.821934i	$-0.692895\pi$
$631$	−28.6155	−1.13917	−0.569583	+	0.821934i	$0.692895\pi$
$632$	0	0
$633$	−49.3823	−1.96277
$634$	0	0
$635$	3.64749	0.144746
$636$	0	0
$637$	30.7793	1.21952
$638$	0	0
$639$	53.4443	2.11422
$640$	0	0
$641$	−43.5514	−1.72018	−0.860088	−	0.510145i	$-0.829592\pi$
$641$	−43.5514	−1.72018	−0.860088	+	0.510145i	$0.829592\pi$
$642$	0	0
$643$	47.9037	1.88914	0.944569	−	0.328314i	$-0.106480\pi$
$643$	47.9037	1.88914	0.944569	+	0.328314i	$0.106480\pi$
$644$	0	0
$645$	−19.4620	−0.766315
$646$	0	0
$647$	14.2047	0.558443	0.279221	−	0.960227i	$-0.409924\pi$
$647$	14.2047	0.558443	0.279221	+	0.960227i	$0.409924\pi$
$648$	0	0
$649$	59.1608	2.32227
$650$	0	0
$651$	−0.677594	−0.0265570
$652$	0	0
$653$	−38.9930	−1.52591	−0.762956	−	0.646450i	$-0.776253\pi$
$653$	−38.9930	−1.52591	−0.762956	+	0.646450i	$0.776253\pi$
$654$	0	0
$655$	7.37463	0.288151
$656$	0	0
$657$	70.8163	2.76281
$658$	0	0
$659$	−35.1488	−1.36920	−0.684602	−	0.728917i	$-0.740024\pi$
$659$	−35.1488	−1.36920	−0.684602	+	0.728917i	$0.740024\pi$
$660$	0	0
$661$	−14.5166	−0.564629	−0.282314	−	0.959322i	$-0.591102\pi$
$661$	−14.5166	−0.564629	−0.282314	+	0.959322i	$0.591102\pi$
$662$	0	0
$663$	38.9573	1.51298
$664$	0	0
$665$	−0.187961	−0.00728881
$666$	0	0
$667$	−7.48496	−0.289819
$668$	0	0
$669$	5.69967	0.220362
$670$	0	0
$671$	−52.3829	−2.02222
$672$	0	0
$673$	−9.14709	−0.352594	−0.176297	−	0.984337i	$-0.556412\pi$
$673$	−9.14709	−0.352594	−0.176297	+	0.984337i	$0.556412\pi$
$674$	0	0
$675$	−6.84288	−0.263383
$676$	0	0
$677$	35.8250	1.37687	0.688433	−	0.725300i	$-0.258299\pi$
$677$	35.8250	1.37687	0.688433	+	0.725300i	$0.258299\pi$
$678$	0	0
$679$	6.59691	0.253166
$680$	0	0
$681$	60.2313	2.30807
$682$	0	0
$683$	−40.9629	−1.56740	−0.783701	−	0.621138i	$-0.786671\pi$
$683$	−40.9629	−1.56740	−0.783701	+	0.621138i	$0.786671\pi$
$684$	0	0
$685$	22.7441	0.869007
$686$	0	0
$687$	32.8526	1.25340
$688$	0	0
$689$	38.6532	1.47257
$690$	0	0
$691$	−46.1245	−1.75466	−0.877330	−	0.479888i	$-0.840677\pi$
$691$	−46.1245	−1.75466	−0.877330	+	0.479888i	$0.840677\pi$
$692$	0	0
$693$	−10.2970	−0.391150
$694$	0	0
$695$	14.9027	0.565291
$696$	0	0
$697$	21.7134	0.822455
$698$	0	0
$699$	15.5952	0.589865
$700$	0	0
$701$	−8.19095	−0.309368	−0.154684	−	0.987964i	$-0.549436\pi$
$701$	−8.19095	−0.309368	−0.154684	+	0.987964i	$0.549436\pi$
$702$	0	0
$703$	−2.46124	−0.0928276
$704$	0	0
$705$	21.2405	0.799965
$706$	0	0
$707$	3.83116	0.144085
$708$	0	0
$709$	7.12455	0.267568	0.133784	−	0.991010i	$-0.457287\pi$
$709$	7.12455	0.267568	0.133784	+	0.991010i	$0.457287\pi$
$710$	0	0
$711$	−92.1658	−3.45649
$712$	0	0
$713$	2.26718	0.0849067
$714$	0	0
$715$	19.2620	0.720359
$716$	0	0
$717$	7.20686	0.269145
$718$	0	0
$719$	33.0778	1.23359	0.616797	−	0.787122i	$-0.288430\pi$
$719$	33.0778	1.23359	0.616797	+	0.787122i	$0.288430\pi$
$720$	0	0
$721$	−0.596697	−0.0222222
$722$	0	0
$723$	76.3577	2.83977
$724$	0	0
$725$	−1.71423	−0.0636650
$726$	0	0
$727$	−28.5515	−1.05892	−0.529458	−	0.848336i	$-0.677605\pi$
$727$	−28.5515	−1.05892	−0.529458	+	0.848336i	$0.677605\pi$
$728$	0	0
$729$	−39.5515	−1.46487
$730$	0	0
$731$	20.0120	0.740169
$732$	0	0
$733$	3.73152	0.137827	0.0689134	−	0.997623i	$-0.478047\pi$
$733$	3.73152	0.137827	0.0689134	+	0.997623i	$0.478047\pi$
$734$	0	0
$735$	19.6579	0.725091
$736$	0	0
$737$	16.0753	0.592141
$738$	0	0
$739$	−15.5110	−0.570582	−0.285291	−	0.958441i	$-0.592090\pi$
$739$	−15.5110	−0.570582	−0.285291	+	0.958441i	$0.592090\pi$
$740$	0	0
$741$	−5.45697	−0.200467
$742$	0	0
$743$	−43.0936	−1.58095	−0.790475	−	0.612494i	$-0.790166\pi$
$743$	−43.0936	−1.58095	−0.790475	+	0.612494i	$0.790166\pi$
$744$	0	0
$745$	1.10295	0.0404089
$746$	0	0
$747$	−69.4771	−2.54204
$748$	0	0
$749$	3.74655	0.136896
$750$	0	0
$751$	28.8861	1.05407	0.527034	−	0.849844i	$-0.323304\pi$
$751$	28.8861	1.05407	0.527034	+	0.849844i	$0.323304\pi$
$752$	0	0
$753$	11.9463	0.435347
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−12.5168	−0.454931	−0.227466	−	0.973786i	$-0.573044\pi$
$757$	−12.5168	−0.454931	−0.227466	+	0.973786i	$0.573044\pi$
$758$	0	0
$759$	53.7154	1.94975
$760$	0	0
$761$	11.1642	0.404701	0.202351	−	0.979313i	$-0.435142\pi$
$761$	11.1642	0.404701	0.202351	+	0.979313i	$0.435142\pi$
$762$	0	0
$763$	−3.66309	−0.132613
$764$	0	0
$765$	15.9586	0.576984
$766$	0	0
$767$	62.9921	2.27451
$768$	0	0
$769$	26.9100	0.970398	0.485199	−	0.874404i	$-0.338747\pi$
$769$	26.9100	0.970398	0.485199	+	0.874404i	$0.338747\pi$
$770$	0	0
$771$	27.6928	0.997333
$772$	0	0
$773$	−53.4862	−1.92377	−0.961883	−	0.273463i	$-0.911831\pi$
$773$	−53.4862	−1.92377	−0.961883	+	0.273463i	$0.911831\pi$
$774$	0	0
$775$	0.519238	0.0186516
$776$	0	0
$777$	−7.70970	−0.276584
$778$	0	0
$779$	−3.04152	−0.108974
$780$	0	0
$781$	−42.3632	−1.51587
$782$	0	0
$783$	11.7303	0.419207
$784$	0	0
$785$	−9.71155	−0.346620
$786$	0	0
$787$	−9.94048	−0.354340	−0.177170	−	0.984180i	$-0.556694\pi$
$787$	−9.94048	−0.354340	−0.177170	+	0.984180i	$0.556694\pi$
$788$	0	0
$789$	−3.52089	−0.125347
$790$	0	0
$791$	2.70506	0.0961809
$792$	0	0
$793$	−55.7753	−1.98064
$794$	0	0
$795$	24.6867	0.875547
$796$	0	0
$797$	10.3057	0.365045	0.182523	−	0.983202i	$-0.441574\pi$
$797$	10.3057	0.365045	0.182523	+	0.983202i	$0.441574\pi$
$798$	0	0
$799$	−21.8408	−0.772671
$800$	0	0
$801$	−12.1833	−0.430476
$802$	0	0
$803$	−56.1333	−1.98090
$804$	0	0
$805$	1.97001	0.0694336
$806$	0	0
$807$	84.9437	2.99016
$808$	0	0
$809$	−10.6334	−0.373850	−0.186925	−	0.982374i	$-0.559852\pi$
$809$	−10.6334	−0.373850	−0.186925	+	0.982374i	$0.559852\pi$
$810$	0	0
$811$	−2.58363	−0.0907235	−0.0453617	−	0.998971i	$-0.514444\pi$
$811$	−2.58363	−0.0907235	−0.0453617	+	0.998971i	$0.514444\pi$
$812$	0	0
$813$	−59.7636	−2.09600
$814$	0	0
$815$	−9.78801	−0.342859
$816$	0	0
$817$	−2.80319	−0.0980711
$818$	0	0
$819$	−10.9638	−0.383107
$820$	0	0
$821$	5.20798	0.181760	0.0908799	−	0.995862i	$-0.471032\pi$
$821$	5.20798	0.181760	0.0908799	+	0.995862i	$0.471032\pi$
$822$	0	0
$823$	−5.58046	−0.194523	−0.0972614	−	0.995259i	$-0.531008\pi$
$823$	−5.58046	−0.194523	−0.0972614	+	0.995259i	$0.531008\pi$
$824$	0	0
$825$	12.3021	0.428305
$826$	0	0
$827$	31.7547	1.10422	0.552109	−	0.833772i	$-0.313823\pi$
$827$	31.7547	1.10422	0.552109	+	0.833772i	$0.313823\pi$
$828$	0	0
$829$	6.92092	0.240373	0.120187	−	0.992751i	$-0.461651\pi$
$829$	6.92092	0.240373	0.120187	+	0.992751i	$0.461651\pi$
$830$	0	0
$831$	67.9931	2.35866
$832$	0	0
$833$	−20.2134	−0.700351
$834$	0	0
$835$	4.81946	0.166784
$836$	0	0
$837$	−3.55309	−0.122813
$838$	0	0
$839$	−24.7877	−0.855767	−0.427883	−	0.903834i	$-0.640741\pi$
$839$	−24.7877	−0.855767	−0.427883	+	0.903834i	$0.640741\pi$
$840$	0	0
$841$	−26.0614	−0.898669
$842$	0	0
$843$	77.5899	2.67234
$844$	0	0
$845$	7.50945	0.258333
$846$	0	0
$847$	3.19907	0.109921
$848$	0	0
$849$	−73.0433	−2.50684
$850$	0	0
$851$	25.7961	0.884280
$852$	0	0
$853$	−0.0282306	−0.000966597 0	−0.000483298	−	1.00000i	$-0.500154\pi$
$853$	−0.0282306	−0.000966597 0	−0.000483298		1.00000i	$0.500154\pi$
$854$	0	0
$855$	−2.23541	−0.0764494
$856$	0	0
$857$	−29.5176	−1.00830	−0.504151	−	0.863615i	$-0.668195\pi$
$857$	−29.5176	−1.00830	−0.504151	+	0.863615i	$0.668195\pi$
$858$	0	0
$859$	3.60108	0.122867	0.0614336	−	0.998111i	$-0.480433\pi$
$859$	3.60108	0.122867	0.0614336	+	0.998111i	$0.480433\pi$
$860$	0	0
$861$	−9.52739	−0.324693
$862$	0	0
$863$	44.2763	1.50718	0.753592	−	0.657342i	$-0.228319\pi$
$863$	44.2763	1.50718	0.753592	+	0.657342i	$0.228319\pi$
$864$	0	0
$865$	19.0278	0.646966
$866$	0	0
$867$	23.5864	0.801035
$868$	0	0
$869$	73.0562	2.47826
$870$	0	0
$871$	17.1163	0.579965
$872$	0	0
$873$	78.4566	2.65536
$874$	0	0
$875$	0.451178	0.0152526
$876$	0	0
$877$	24.5938	0.830474	0.415237	−	0.909713i	$-0.363699\pi$
$877$	24.5938	0.830474	0.415237	+	0.909713i	$0.363699\pi$
$878$	0	0
$879$	−8.93052	−0.301219
$880$	0	0
$881$	18.4874	0.622857	0.311428	−	0.950270i	$-0.399193\pi$
$881$	18.4874	0.622857	0.311428	+	0.950270i	$0.399193\pi$
$882$	0	0
$883$	3.37235	0.113489	0.0567443	−	0.998389i	$-0.481928\pi$
$883$	3.37235	0.113489	0.0567443	+	0.998389i	$0.481928\pi$
$884$	0	0
$885$	40.2313	1.35236
$886$	0	0
$887$	−3.21471	−0.107939	−0.0539696	−	0.998543i	$-0.517187\pi$
$887$	−3.21471	−0.107939	−0.0539696	+	0.998543i	$0.517187\pi$
$888$	0	0
$889$	1.64567	0.0551939
$890$	0	0
$891$	−15.7145	−0.526457
$892$	0	0
$893$	3.05936	0.102378
$894$	0	0
$895$	−13.0012	−0.434582
$896$	0	0
$897$	57.1941	1.90966
$898$	0	0
$899$	−0.890096	−0.0296864
$900$	0	0
$901$	−25.3843	−0.845674
$902$	0	0
$903$	−8.78082	−0.292208
$904$	0	0
$905$	−20.5584	−0.683383
$906$	0	0
$907$	−8.28771	−0.275189	−0.137594	−	0.990489i	$-0.543937\pi$
$907$	−8.28771	−0.275189	−0.137594	+	0.990489i	$0.543937\pi$
$908$	0	0
$909$	45.5637	1.51125
$910$	0	0
$911$	−38.1356	−1.26349	−0.631744	−	0.775177i	$-0.717661\pi$
$911$	−38.1356	−1.26349	−0.631744	+	0.775177i	$0.717661\pi$
$912$	0	0
$913$	55.0718	1.82261
$914$	0	0
$915$	−35.6221	−1.17763
$916$	0	0
$917$	3.32727	0.109876
$918$	0	0
$919$	29.4109	0.970177	0.485088	−	0.874465i	$-0.338787\pi$
$919$	29.4109	0.970177	0.485088	+	0.874465i	$0.338787\pi$
$920$	0	0
$921$	59.4320	1.95835
$922$	0	0
$923$	−45.1067	−1.48470
$924$	0	0
$925$	5.90793	0.194251
$926$	0	0
$927$	−7.09649	−0.233079
$928$	0	0
$929$	26.1640	0.858412	0.429206	−	0.903207i	$-0.358793\pi$
$929$	26.1640	0.858412	0.429206	+	0.903207i	$0.358793\pi$
$930$	0	0
$931$	2.83140	0.0927953
$932$	0	0
$933$	−45.8524	−1.50114
$934$	0	0
$935$	−12.6498	−0.413691
$936$	0	0
$937$	−20.8460	−0.681010	−0.340505	−	0.940243i	$-0.610598\pi$
$937$	−20.8460	−0.681010	−0.340505	+	0.940243i	$0.610598\pi$
$938$	0	0
$939$	−80.5580	−2.62891
$940$	0	0
$941$	23.2085	0.756575	0.378288	−	0.925688i	$-0.376513\pi$
$941$	23.2085	0.756575	0.378288	+	0.925688i	$0.376513\pi$
$942$	0	0
$943$	31.8780	1.03809
$944$	0	0
$945$	−3.08736	−0.100432
$946$	0	0
$947$	2.78439	0.0904805	0.0452402	−	0.998976i	$-0.485595\pi$
$947$	2.78439	0.0904805	0.0452402	+	0.998976i	$0.485595\pi$
$948$	0	0
$949$	−59.7686	−1.94017
$950$	0	0
$951$	−8.73196	−0.283153
$952$	0	0
$953$	−46.9890	−1.52212	−0.761061	−	0.648680i	$-0.775321\pi$
$953$	−46.9890	−1.52212	−0.761061	+	0.648680i	$0.775321\pi$
$954$	0	0
$955$	−18.9544	−0.613351
$956$	0	0
$957$	−21.0887	−0.681700
$958$	0	0
$959$	10.2616	0.331365
$960$	0	0
$961$	−30.7304	−0.991303
$962$	0	0
$963$	44.5575	1.43585
$964$	0	0
$965$	−27.7140	−0.892145
$966$	0	0
$967$	45.3169	1.45729	0.728647	−	0.684890i	$-0.240150\pi$
$967$	45.3169	1.45729	0.728647	+	0.684890i	$0.240150\pi$
$968$	0	0
$969$	3.58370	0.115125
$970$	0	0
$971$	60.7065	1.94816	0.974082	−	0.226194i	$-0.0726284\pi$
$971$	60.7065	1.94816	0.974082	+	0.226194i	$0.0726284\pi$
$972$	0	0
$973$	6.72376	0.215554
$974$	0	0
$975$	13.0988	0.419497
$976$	0	0
$977$	−41.0911	−1.31462	−0.657311	−	0.753619i	$-0.728306\pi$
$977$	−41.0911	−1.31462	−0.657311	+	0.753619i	$0.728306\pi$
$978$	0	0
$979$	9.65725	0.308647
$980$	0	0
$981$	−43.5649	−1.39092
$982$	0	0
$983$	19.7730	0.630661	0.315330	−	0.948982i	$-0.397885\pi$
$983$	19.7730	0.630661	0.315330	+	0.948982i	$0.397885\pi$
$984$	0	0
$985$	7.09673	0.226120
$986$	0	0
$987$	9.58326	0.305039
$988$	0	0
$989$	29.3800	0.934231
$990$	0	0
$991$	7.08562	0.225082	0.112541	−	0.993647i	$-0.464101\pi$
$991$	7.08562	0.225082	0.112541	+	0.993647i	$0.464101\pi$
$992$	0	0
$993$	−73.1060	−2.31995
$994$	0	0
$995$	−22.7907	−0.722513
$996$	0	0
$997$	−3.07362	−0.0973424	−0.0486712	−	0.998815i	$-0.515499\pi$
$997$	−3.07362	−0.0973424	−0.0486712	+	0.998815i	$0.515499\pi$
$998$	0	0
$999$	−40.4273	−1.27906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.3	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-2.89238 q^{3} +1.00000 q^{5} +0.451178 q^{7} +5.36583 q^{9} +O(q^{10})\)	\(q-2.89238 q^{3} +1.00000 q^{5} +0.451178 q^{7} +5.36583 q^{9} -4.25329 q^{11} -4.52874 q^{13} -2.89238 q^{15} +2.97411 q^{17} -0.416600 q^{19} -1.30498 q^{21} +4.36636 q^{23} +1.00000 q^{25} -6.84288 q^{27} -1.71423 q^{29} +0.519238 q^{31} +12.3021 q^{33} +0.451178 q^{35} +5.90793 q^{37} +13.0988 q^{39} +7.30082 q^{41} +6.72872 q^{43} +5.36583 q^{45} -7.34363 q^{47} -6.79644 q^{49} -8.60224 q^{51} -8.53509 q^{53} -4.25329 q^{55} +1.20496 q^{57} -13.9094 q^{59} +12.3159 q^{61} +2.42095 q^{63} -4.52874 q^{65} -3.77950 q^{67} -12.6292 q^{69} +9.96010 q^{71} +13.1976 q^{73} -2.89238 q^{75} -1.91899 q^{77} -17.1764 q^{79} +3.69468 q^{81} -12.9481 q^{83} +2.97411 q^{85} +4.95821 q^{87} -2.27054 q^{89} -2.04327 q^{91} -1.50183 q^{93} -0.416600 q^{95} +14.6215 q^{97} -22.8225 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 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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