LMFDB - Embedded newform 6040.2.a.p.1.1

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			1.1
Root			$3.33655$ 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)

$f(q)$	$=$	$q-3.33655 q^{3} +1.00000 q^{5} +2.35293 q^{7} +8.13258 q^{9} +O(q^{10})$	$q-3.33655 q^{3} +1.00000 q^{5} +2.35293 q^{7} +8.13258 q^{9} -5.41372 q^{11} +7.10664 q^{13} -3.33655 q^{15} -2.30666 q^{17} +3.33812 q^{19} -7.85068 q^{21} -2.59719 q^{23} +1.00000 q^{25} -17.1251 q^{27} -1.45979 q^{29} +2.14434 q^{31} +18.0631 q^{33} +2.35293 q^{35} -8.97284 q^{37} -23.7117 q^{39} +0.901406 q^{41} -1.97754 q^{43} +8.13258 q^{45} -11.7551 q^{47} -1.46372 q^{49} +7.69628 q^{51} +10.0416 q^{53} -5.41372 q^{55} -11.1378 q^{57} -4.21296 q^{59} -12.2881 q^{61} +19.1354 q^{63} +7.10664 q^{65} +0.385186 q^{67} +8.66565 q^{69} -12.4898 q^{71} -6.72268 q^{73} -3.33655 q^{75} -12.7381 q^{77} +12.8609 q^{79} +32.7411 q^{81} +2.69229 q^{83} -2.30666 q^{85} +4.87068 q^{87} +13.4454 q^{89} +16.7214 q^{91} -7.15471 q^{93} +3.33812 q^{95} +5.95427 q^{97} -44.0275 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.33655	−1.92636	−0.963180	−	0.268858i	$-0.913354\pi$
$3$	−3.33655	−1.92636	−0.963180	+	0.268858i	$0.913354\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.35293	0.889324	0.444662	−	0.895698i	$-0.353324\pi$
$7$	2.35293	0.889324	0.444662	+	0.895698i	$0.353324\pi$
$8$	0	0
$9$	8.13258	2.71086
$10$	0	0
$11$	−5.41372	−1.63230	−0.816148	−	0.577842i	$-0.803895\pi$
$11$	−5.41372	−1.63230	−0.816148	+	0.577842i	$0.803895\pi$
$12$	0	0
$13$	7.10664	1.97103	0.985513	−	0.169599i	$-0.0542472\pi$
$13$	7.10664	1.97103	0.985513	+	0.169599i	$0.0542472\pi$
$14$	0	0
$15$	−3.33655	−0.861494
$16$	0	0
$17$	−2.30666	−0.559446	−0.279723	−	0.960081i	$-0.590243\pi$
$17$	−2.30666	−0.559446	−0.279723	+	0.960081i	$0.590243\pi$
$18$	0	0
$19$	3.33812	0.765818	0.382909	−	0.923786i	$-0.374922\pi$
$19$	3.33812	0.765818	0.382909	+	0.923786i	$0.374922\pi$
$20$	0	0
$21$	−7.85068	−1.71316
$22$	0	0
$23$	−2.59719	−0.541551	−0.270775	−	0.962643i	$-0.587280\pi$
$23$	−2.59719	−0.541551	−0.270775	+	0.962643i	$0.587280\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−17.1251	−3.29573
$28$	0	0
$29$	−1.45979	−0.271077	−0.135538	−	0.990772i	$-0.543276\pi$
$29$	−1.45979	−0.271077	−0.135538	+	0.990772i	$0.543276\pi$
$30$	0	0
$31$	2.14434	0.385135	0.192568	−	0.981284i	$-0.438319\pi$
$31$	2.14434	0.385135	0.192568	+	0.981284i	$0.438319\pi$
$32$	0	0
$33$	18.0631	3.14439
$34$	0	0
$35$	2.35293	0.397718
$36$	0	0
$37$	−8.97284	−1.47513	−0.737563	−	0.675278i	$-0.764024\pi$
$37$	−8.97284	−1.47513	−0.737563	+	0.675278i	$0.764024\pi$
$38$	0	0
$39$	−23.7117	−3.79691
$40$	0	0
$41$	0.901406	0.140776	0.0703880	−	0.997520i	$-0.477576\pi$
$41$	0.901406	0.140776	0.0703880	+	0.997520i	$0.477576\pi$
$42$	0	0
$43$	−1.97754	−0.301572	−0.150786	−	0.988566i	$-0.548181\pi$
$43$	−1.97754	−0.301572	−0.150786	+	0.988566i	$0.548181\pi$
$44$	0	0
$45$	8.13258	1.21233
$46$	0	0
$47$	−11.7551	−1.71466	−0.857332	−	0.514764i	$-0.827880\pi$
$47$	−11.7551	−1.71466	−0.857332	+	0.514764i	$0.827880\pi$
$48$	0	0
$49$	−1.46372	−0.209103
$50$	0	0
$51$	7.69628	1.07769
$52$	0	0
$53$	10.0416	1.37932	0.689658	−	0.724135i	$-0.257761\pi$
$53$	10.0416	1.37932	0.689658	+	0.724135i	$0.257761\pi$
$54$	0	0
$55$	−5.41372	−0.729985
$56$	0	0
$57$	−11.1378	−1.47524
$58$	0	0
$59$	−4.21296	−0.548480	−0.274240	−	0.961661i	$-0.588426\pi$
$59$	−4.21296	−0.548480	−0.274240	+	0.961661i	$0.588426\pi$
$60$	0	0
$61$	−12.2881	−1.57333	−0.786664	−	0.617382i	$-0.788193\pi$
$61$	−12.2881	−1.57333	−0.786664	+	0.617382i	$0.788193\pi$
$62$	0	0
$63$	19.1354	2.41083
$64$	0	0
$65$	7.10664	0.881470
$66$	0	0
$67$	0.385186	0.0470579	0.0235290	−	0.999723i	$-0.492510\pi$
$67$	0.385186	0.0470579	0.0235290	+	0.999723i	$0.492510\pi$
$68$	0	0
$69$	8.66565	1.04322
$70$	0	0
$71$	−12.4898	−1.48226	−0.741131	−	0.671361i	$-0.765710\pi$
$71$	−12.4898	−1.48226	−0.741131	+	0.671361i	$0.765710\pi$
$72$	0	0
$73$	−6.72268	−0.786830	−0.393415	−	0.919361i	$-0.628707\pi$
$73$	−6.72268	−0.786830	−0.393415	+	0.919361i	$0.628707\pi$
$74$	0	0
$75$	−3.33655	−0.385272
$76$	0	0
$77$	−12.7381	−1.45164
$78$	0	0
$79$	12.8609	1.44697	0.723484	−	0.690341i	$-0.242540\pi$
$79$	12.8609	1.44697	0.723484	+	0.690341i	$0.242540\pi$
$80$	0	0
$81$	32.7411	3.63790
$82$	0	0
$83$	2.69229	0.295517	0.147759	−	0.989023i	$-0.452794\pi$
$83$	2.69229	0.295517	0.147759	+	0.989023i	$0.452794\pi$
$84$	0	0
$85$	−2.30666	−0.250192
$86$	0	0
$87$	4.87068	0.522191
$88$	0	0
$89$	13.4454	1.42521	0.712604	−	0.701567i	$-0.247516\pi$
$89$	13.4454	1.42521	0.712604	+	0.701567i	$0.247516\pi$
$90$	0	0
$91$	16.7214	1.75288
$92$	0	0
$93$	−7.15471	−0.741908
$94$	0	0
$95$	3.33812	0.342484
$96$	0	0
$97$	5.95427	0.604564	0.302282	−	0.953218i	$-0.402251\pi$
$97$	5.95427	0.604564	0.302282	+	0.953218i	$0.402251\pi$
$98$	0	0
$99$	−44.0275	−4.42493
$100$	0	0
$101$	−16.3540	−1.62728	−0.813642	−	0.581366i	$-0.802519\pi$
$101$	−16.3540	−1.62728	−0.813642	+	0.581366i	$0.802519\pi$
$102$	0	0
$103$	2.41825	0.238277	0.119139	−	0.992878i	$-0.461987\pi$
$103$	2.41825	0.238277	0.119139	+	0.992878i	$0.461987\pi$
$104$	0	0
$105$	−7.85068	−0.766148
$106$	0	0
$107$	7.76217	0.750397	0.375198	−	0.926944i	$-0.377575\pi$
$107$	7.76217	0.750397	0.375198	+	0.926944i	$0.377575\pi$
$108$	0	0
$109$	−16.4274	−1.57346	−0.786728	−	0.617300i	$-0.788226\pi$
$109$	−16.4274	−1.57346	−0.786728	+	0.617300i	$0.788226\pi$
$110$	0	0
$111$	29.9384	2.84162
$112$	0	0
$113$	−8.40511	−0.790687	−0.395343	−	0.918533i	$-0.629374\pi$
$113$	−8.40511	−0.790687	−0.395343	+	0.918533i	$0.629374\pi$
$114$	0	0
$115$	−2.59719	−0.242189
$116$	0	0
$117$	57.7953	5.34318
$118$	0	0
$119$	−5.42740	−0.497529
$120$	0	0
$121$	18.3083	1.66439
$122$	0	0
$123$	−3.00759	−0.271185
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.2710	1.71002	0.855010	−	0.518611i	$-0.173551\pi$
$127$	19.2710	1.71002	0.855010	+	0.518611i	$0.173551\pi$
$128$	0	0
$129$	6.59818	0.580937
$130$	0	0
$131$	−12.3384	−1.07801	−0.539006	−	0.842302i	$-0.681200\pi$
$131$	−12.3384	−1.07801	−0.539006	+	0.842302i	$0.681200\pi$
$132$	0	0
$133$	7.85437	0.681060
$134$	0	0
$135$	−17.1251	−1.47390
$136$	0	0
$137$	2.09541	0.179023	0.0895116	−	0.995986i	$-0.471469\pi$
$137$	2.09541	0.179023	0.0895116	+	0.995986i	$0.471469\pi$
$138$	0	0
$139$	−13.5877	−1.15249	−0.576246	−	0.817276i	$-0.695483\pi$
$139$	−13.5877	−1.15249	−0.576246	+	0.817276i	$0.695483\pi$
$140$	0	0
$141$	39.2217	3.30306
$142$	0	0
$143$	−38.4733	−3.21730
$144$	0	0
$145$	−1.45979	−0.121229
$146$	0	0
$147$	4.88377	0.402807
$148$	0	0
$149$	7.63807	0.625735	0.312867	−	0.949797i	$-0.398710\pi$
$149$	7.63807	0.625735	0.312867	+	0.949797i	$0.398710\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−18.7591	−1.51658
$154$	0	0
$155$	2.14434	0.172238
$156$	0	0
$157$	−3.64551	−0.290943	−0.145472	−	0.989362i	$-0.546470\pi$
$157$	−3.64551	−0.290943	−0.145472	+	0.989362i	$0.546470\pi$
$158$	0	0
$159$	−33.5043	−2.65706
$160$	0	0
$161$	−6.11100	−0.481614
$162$	0	0
$163$	16.2139	1.26997	0.634984	−	0.772525i	$-0.281007\pi$
$163$	16.2139	1.26997	0.634984	+	0.772525i	$0.281007\pi$
$164$	0	0
$165$	18.0631	1.40621
$166$	0	0
$167$	23.8658	1.84679	0.923396	−	0.383848i	$-0.125401\pi$
$167$	23.8658	1.84679	0.923396	+	0.383848i	$0.125401\pi$
$168$	0	0
$169$	37.5043	2.88494
$170$	0	0
$171$	27.1475	2.07602
$172$	0	0
$173$	−12.4529	−0.946773	−0.473387	−	0.880855i	$-0.656969\pi$
$173$	−12.4529	−0.946773	−0.473387	+	0.880855i	$0.656969\pi$
$174$	0	0
$175$	2.35293	0.177865
$176$	0	0
$177$	14.0567	1.05657
$178$	0	0
$179$	−11.8873	−0.888498	−0.444249	−	0.895903i	$-0.646529\pi$
$179$	−11.8873	−0.888498	−0.444249	+	0.895903i	$0.646529\pi$
$180$	0	0
$181$	−7.53084	−0.559763	−0.279881	−	0.960035i	$-0.590295\pi$
$181$	−7.53084	−0.559763	−0.279881	+	0.960035i	$0.590295\pi$
$182$	0	0
$183$	40.9998	3.03079
$184$	0	0
$185$	−8.97284	−0.659696
$186$	0	0
$187$	12.4876	0.913183
$188$	0	0
$189$	−40.2942	−2.93097
$190$	0	0
$191$	4.57380	0.330948	0.165474	−	0.986214i	$-0.447085\pi$
$191$	4.57380	0.330948	0.165474	+	0.986214i	$0.447085\pi$
$192$	0	0
$193$	5.68019	0.408869	0.204434	−	0.978880i	$-0.434464\pi$
$193$	5.68019	0.408869	0.204434	+	0.978880i	$0.434464\pi$
$194$	0	0
$195$	−23.7117	−1.69803
$196$	0	0
$197$	12.8905	0.918408	0.459204	−	0.888331i	$-0.348135\pi$
$197$	12.8905	0.918408	0.459204	+	0.888331i	$0.348135\pi$
$198$	0	0
$199$	−13.6389	−0.966834	−0.483417	−	0.875390i	$-0.660605\pi$
$199$	−13.6389	−0.966834	−0.483417	+	0.875390i	$0.660605\pi$
$200$	0	0
$201$	−1.28519	−0.0906505
$202$	0	0
$203$	−3.43479	−0.241075
$204$	0	0
$205$	0.901406	0.0629569
$206$	0	0
$207$	−21.1218	−1.46807
$208$	0	0
$209$	−18.0716	−1.25004
$210$	0	0
$211$	−25.3880	−1.74778	−0.873891	−	0.486122i	$-0.838411\pi$
$211$	−25.3880	−1.74778	−0.873891	+	0.486122i	$0.838411\pi$
$212$	0	0
$213$	41.6727	2.85537
$214$	0	0
$215$	−1.97754	−0.134867
$216$	0	0
$217$	5.04549	0.342510
$218$	0	0
$219$	22.4306	1.51572
$220$	0	0
$221$	−16.3926	−1.10268
$222$	0	0
$223$	22.6645	1.51773	0.758864	−	0.651249i	$-0.225755\pi$
$223$	22.6645	1.51773	0.758864	+	0.651249i	$0.225755\pi$
$224$	0	0
$225$	8.13258	0.542172
$226$	0	0
$227$	−20.5011	−1.36071	−0.680353	−	0.732885i	$-0.738173\pi$
$227$	−20.5011	−1.36071	−0.680353	+	0.732885i	$0.738173\pi$
$228$	0	0
$229$	−7.31552	−0.483423	−0.241712	−	0.970348i	$-0.577709\pi$
$229$	−7.31552	−0.483423	−0.241712	+	0.970348i	$0.577709\pi$
$230$	0	0
$231$	42.5013	2.79638
$232$	0	0
$233$	−9.36300	−0.613390	−0.306695	−	0.951808i	$-0.599223\pi$
$233$	−9.36300	−0.613390	−0.306695	+	0.951808i	$0.599223\pi$
$234$	0	0
$235$	−11.7551	−0.766821
$236$	0	0
$237$	−42.9112	−2.78738
$238$	0	0
$239$	−14.1741	−0.916847	−0.458424	−	0.888734i	$-0.651586\pi$
$239$	−14.1741	−0.916847	−0.458424	+	0.888734i	$0.651586\pi$
$240$	0	0
$241$	−7.16536	−0.461561	−0.230781	−	0.973006i	$-0.574128\pi$
$241$	−7.16536	−0.461561	−0.230781	+	0.973006i	$0.574128\pi$
$242$	0	0
$243$	−57.8671	−3.71218
$244$	0	0
$245$	−1.46372	−0.0935135
$246$	0	0
$247$	23.7228	1.50945
$248$	0	0
$249$	−8.98296	−0.569272
$250$	0	0
$251$	−8.50114	−0.536587	−0.268294	−	0.963337i	$-0.586460\pi$
$251$	−8.50114	−0.536587	−0.268294	+	0.963337i	$0.586460\pi$
$252$	0	0
$253$	14.0604	0.883972
$254$	0	0
$255$	7.69628	0.481960
$256$	0	0
$257$	8.65320	0.539772	0.269886	−	0.962892i	$-0.413014\pi$
$257$	8.65320	0.539772	0.269886	+	0.962892i	$0.413014\pi$
$258$	0	0
$259$	−21.1125	−1.31187
$260$	0	0
$261$	−11.8719	−0.734851
$262$	0	0
$263$	−14.4377	−0.890268	−0.445134	−	0.895464i	$-0.646844\pi$
$263$	−14.4377	−0.890268	−0.445134	+	0.895464i	$0.646844\pi$
$264$	0	0
$265$	10.0416	0.616849
$266$	0	0
$267$	−44.8612	−2.74546
$268$	0	0
$269$	−17.3952	−1.06061	−0.530303	−	0.847808i	$-0.677922\pi$
$269$	−17.3952	−1.06061	−0.530303	+	0.847808i	$0.677922\pi$
$270$	0	0
$271$	30.2168	1.83554	0.917770	−	0.397112i	$-0.129987\pi$
$271$	30.2168	1.83554	0.917770	+	0.397112i	$0.129987\pi$
$272$	0	0
$273$	−55.7919	−3.37668
$274$	0	0
$275$	−5.41372	−0.326459
$276$	0	0
$277$	1.14217	0.0686261	0.0343131	−	0.999411i	$-0.489076\pi$
$277$	1.14217	0.0686261	0.0343131	+	0.999411i	$0.489076\pi$
$278$	0	0
$279$	17.4390	1.04405
$280$	0	0
$281$	−16.4049	−0.978634	−0.489317	−	0.872106i	$-0.662754\pi$
$281$	−16.4049	−0.978634	−0.489317	+	0.872106i	$0.662754\pi$
$282$	0	0
$283$	−10.6711	−0.634328	−0.317164	−	0.948371i	$-0.602731\pi$
$283$	−10.6711	−0.634328	−0.317164	+	0.948371i	$0.602731\pi$
$284$	0	0
$285$	−11.1378	−0.659747
$286$	0	0
$287$	2.12095	0.125195
$288$	0	0
$289$	−11.6793	−0.687020
$290$	0	0
$291$	−19.8667	−1.16461
$292$	0	0
$293$	6.92639	0.404644	0.202322	−	0.979319i	$-0.435151\pi$
$293$	6.92639	0.404644	0.202322	+	0.979319i	$0.435151\pi$
$294$	0	0
$295$	−4.21296	−0.245288
$296$	0	0
$297$	92.7106	5.37961
$298$	0	0
$299$	−18.4573	−1.06741
$300$	0	0
$301$	−4.65302	−0.268196
$302$	0	0
$303$	54.5660	3.13474
$304$	0	0
$305$	−12.2881	−0.703613
$306$	0	0
$307$	5.20357	0.296984	0.148492	−	0.988914i	$-0.452558\pi$
$307$	5.20357	0.296984	0.148492	+	0.988914i	$0.452558\pi$
$308$	0	0
$309$	−8.06862	−0.459008
$310$	0	0
$311$	6.77414	0.384127	0.192063	−	0.981383i	$-0.438482\pi$
$311$	6.77414	0.384127	0.192063	+	0.981383i	$0.438482\pi$
$312$	0	0
$313$	2.80023	0.158278	0.0791391	−	0.996864i	$-0.474783\pi$
$313$	2.80023	0.158278	0.0791391	+	0.996864i	$0.474783\pi$
$314$	0	0
$315$	19.1354	1.07816
$316$	0	0
$317$	−19.7137	−1.10723	−0.553616	−	0.832772i	$-0.686752\pi$
$317$	−19.7137	−1.10723	−0.553616	+	0.832772i	$0.686752\pi$
$318$	0	0
$319$	7.90290	0.442478
$320$	0	0
$321$	−25.8989	−1.44553
$322$	0	0
$323$	−7.69990	−0.428434
$324$	0	0
$325$	7.10664	0.394205
$326$	0	0
$327$	54.8107	3.03104
$328$	0	0
$329$	−27.6590	−1.52489
$330$	0	0
$331$	2.82468	0.155258	0.0776291	−	0.996982i	$-0.475265\pi$
$331$	2.82468	0.155258	0.0776291	+	0.996982i	$0.475265\pi$
$332$	0	0
$333$	−72.9724	−3.99886
$334$	0	0
$335$	0.385186	0.0210449
$336$	0	0
$337$	−26.5951	−1.44873	−0.724363	−	0.689419i	$-0.757866\pi$
$337$	−26.5951	−1.44873	−0.724363	+	0.689419i	$0.757866\pi$
$338$	0	0
$339$	28.0441	1.52315
$340$	0	0
$341$	−11.6089	−0.628655
$342$	0	0
$343$	−19.9145	−1.07528
$344$	0	0
$345$	8.66565	0.466543
$346$	0	0
$347$	1.89543	0.101752	0.0508760	−	0.998705i	$-0.483799\pi$
$347$	1.89543	0.101752	0.0508760	+	0.998705i	$0.483799\pi$
$348$	0	0
$349$	11.5730	0.619487	0.309744	−	0.950820i	$-0.399757\pi$
$349$	11.5730	0.619487	0.309744	+	0.950820i	$0.399757\pi$
$350$	0	0
$351$	−121.702	−6.49598
$352$	0	0
$353$	−30.5809	−1.62766	−0.813829	−	0.581105i	$-0.802621\pi$
$353$	−30.5809	−1.62766	−0.813829	+	0.581105i	$0.802621\pi$
$354$	0	0
$355$	−12.4898	−0.662888
$356$	0	0
$357$	18.1088	0.958420
$358$	0	0
$359$	−3.93418	−0.207638	−0.103819	−	0.994596i	$-0.533106\pi$
$359$	−3.93418	−0.207638	−0.103819	+	0.994596i	$0.533106\pi$
$360$	0	0
$361$	−7.85694	−0.413523
$362$	0	0
$363$	−61.0867	−3.20622
$364$	0	0
$365$	−6.72268	−0.351881
$366$	0	0
$367$	−22.7904	−1.18965	−0.594824	−	0.803856i	$-0.702778\pi$
$367$	−22.7904	−1.18965	−0.594824	+	0.803856i	$0.702778\pi$
$368$	0	0
$369$	7.33076	0.381624
$370$	0	0
$371$	23.6271	1.22666
$372$	0	0
$373$	−30.2646	−1.56704	−0.783520	−	0.621367i	$-0.786578\pi$
$373$	−30.2646	−1.56704	−0.783520	+	0.621367i	$0.786578\pi$
$374$	0	0
$375$	−3.33655	−0.172299
$376$	0	0
$377$	−10.3742	−0.534299
$378$	0	0
$379$	13.9979	0.719025	0.359513	−	0.933140i	$-0.382943\pi$
$379$	13.9979	0.719025	0.359513	+	0.933140i	$0.382943\pi$
$380$	0	0
$381$	−64.2985	−3.29411
$382$	0	0
$383$	20.6813	1.05677	0.528384	−	0.849006i	$-0.322798\pi$
$383$	20.6813	1.05677	0.528384	+	0.849006i	$0.322798\pi$
$384$	0	0
$385$	−12.7381	−0.649194
$386$	0	0
$387$	−16.0825	−0.817521
$388$	0	0
$389$	−8.90209	−0.451354	−0.225677	−	0.974202i	$-0.572459\pi$
$389$	−8.90209	−0.451354	−0.225677	+	0.974202i	$0.572459\pi$
$390$	0	0
$391$	5.99082	0.302969
$392$	0	0
$393$	41.1678	2.07664
$394$	0	0
$395$	12.8609	0.647104
$396$	0	0
$397$	32.1091	1.61151	0.805756	−	0.592248i	$-0.201759\pi$
$397$	32.1091	1.61151	0.805756	+	0.592248i	$0.201759\pi$
$398$	0	0
$399$	−26.2065	−1.31197
$400$	0	0
$401$	18.7681	0.937234	0.468617	−	0.883401i	$-0.344752\pi$
$401$	18.7681	0.937234	0.468617	+	0.883401i	$0.344752\pi$
$402$	0	0
$403$	15.2391	0.759111
$404$	0	0
$405$	32.7411	1.62692
$406$	0	0
$407$	48.5764	2.40784
$408$	0	0
$409$	−19.6949	−0.973849	−0.486924	−	0.873444i	$-0.661881\pi$
$409$	−19.6949	−0.973849	−0.486924	+	0.873444i	$0.661881\pi$
$410$	0	0
$411$	−6.99146	−0.344863
$412$	0	0
$413$	−9.91279	−0.487777
$414$	0	0
$415$	2.69229	0.132159
$416$	0	0
$417$	45.3360	2.22011
$418$	0	0
$419$	2.27818	0.111296	0.0556482	−	0.998450i	$-0.482277\pi$
$419$	2.27818	0.111296	0.0556482	+	0.998450i	$0.482277\pi$
$420$	0	0
$421$	−10.4056	−0.507136	−0.253568	−	0.967318i	$-0.581604\pi$
$421$	−10.4056	−0.507136	−0.253568	+	0.967318i	$0.581604\pi$
$422$	0	0
$423$	−95.5997	−4.64822
$424$	0	0
$425$	−2.30666	−0.111889
$426$	0	0
$427$	−28.9130	−1.39920
$428$	0	0
$429$	128.368	6.19768
$430$	0	0
$431$	−7.99886	−0.385291	−0.192646	−	0.981268i	$-0.561707\pi$
$431$	−7.99886	−0.385291	−0.192646	+	0.981268i	$0.561707\pi$
$432$	0	0
$433$	29.5222	1.41875	0.709374	−	0.704832i	$-0.248978\pi$
$433$	29.5222	1.41875	0.709374	+	0.704832i	$0.248978\pi$
$434$	0	0
$435$	4.87068	0.233531
$436$	0	0
$437$	−8.66972	−0.414729
$438$	0	0
$439$	−10.7474	−0.512946	−0.256473	−	0.966551i	$-0.582560\pi$
$439$	−10.7474	−0.512946	−0.256473	+	0.966551i	$0.582560\pi$
$440$	0	0
$441$	−11.9038	−0.566848
$442$	0	0
$443$	−4.61008	−0.219031	−0.109516	−	0.993985i	$-0.534930\pi$
$443$	−4.61008	−0.219031	−0.109516	+	0.993985i	$0.534930\pi$
$444$	0	0
$445$	13.4454	0.637372
$446$	0	0
$447$	−25.4848	−1.20539
$448$	0	0
$449$	41.1548	1.94222	0.971109	−	0.238638i	$-0.0767008\pi$
$449$	41.1548	1.94222	0.971109	+	0.238638i	$0.0767008\pi$
$450$	0	0
$451$	−4.87996	−0.229788
$452$	0	0
$453$	−3.33655	−0.156765
$454$	0	0
$455$	16.7214	0.783912
$456$	0	0
$457$	16.5181	0.772686	0.386343	−	0.922355i	$-0.373738\pi$
$457$	16.5181	0.772686	0.386343	+	0.922355i	$0.373738\pi$
$458$	0	0
$459$	39.5018	1.84379
$460$	0	0
$461$	0.908549	0.0423154	0.0211577	−	0.999776i	$-0.493265\pi$
$461$	0.908549	0.0423154	0.0211577	+	0.999776i	$0.493265\pi$
$462$	0	0
$463$	−39.7878	−1.84910	−0.924548	−	0.381066i	$-0.875557\pi$
$463$	−39.7878	−1.84910	−0.924548	+	0.381066i	$0.875557\pi$
$464$	0	0
$465$	−7.15471	−0.331792
$466$	0	0
$467$	−14.1345	−0.654065	−0.327032	−	0.945013i	$-0.606049\pi$
$467$	−14.1345	−0.654065	−0.327032	+	0.945013i	$0.606049\pi$
$468$	0	0
$469$	0.906315	0.0418498
$470$	0	0
$471$	12.1634	0.560461
$472$	0	0
$473$	10.7059	0.492256
$474$	0	0
$475$	3.33812	0.153164
$476$	0	0
$477$	81.6640	3.73914
$478$	0	0
$479$	−37.5386	−1.71518	−0.857592	−	0.514331i	$-0.828040\pi$
$479$	−37.5386	−1.71518	−0.857592	+	0.514331i	$0.828040\pi$
$480$	0	0
$481$	−63.7667	−2.90751
$482$	0	0
$483$	20.3897	0.927762
$484$	0	0
$485$	5.95427	0.270369
$486$	0	0
$487$	15.3571	0.695898	0.347949	−	0.937513i	$-0.386878\pi$
$487$	15.3571	0.695898	0.347949	+	0.937513i	$0.386878\pi$
$488$	0	0
$489$	−54.0984	−2.44641
$490$	0	0
$491$	−10.8559	−0.489918	−0.244959	−	0.969533i	$-0.578775\pi$
$491$	−10.8559	−0.489918	−0.244959	+	0.969533i	$0.578775\pi$
$492$	0	0
$493$	3.36724	0.151653
$494$	0	0
$495$	−44.0275	−1.97889
$496$	0	0
$497$	−29.3875	−1.31821
$498$	0	0
$499$	−6.90608	−0.309159	−0.154579	−	0.987980i	$-0.549402\pi$
$499$	−6.90608	−0.309159	−0.154579	+	0.987980i	$0.549402\pi$
$500$	0	0
$501$	−79.6295	−3.55759
$502$	0	0
$503$	−36.2667	−1.61705	−0.808526	−	0.588461i	$-0.799734\pi$
$503$	−36.2667	−1.61705	−0.808526	+	0.588461i	$0.799734\pi$
$504$	0	0
$505$	−16.3540	−0.727744
$506$	0	0
$507$	−125.135	−5.55744
$508$	0	0
$509$	−6.83221	−0.302833	−0.151416	−	0.988470i	$-0.548383\pi$
$509$	−6.83221	−0.302833	−0.151416	+	0.988470i	$0.548383\pi$
$510$	0	0
$511$	−15.8180	−0.699747
$512$	0	0
$513$	−57.1658	−2.52393
$514$	0	0
$515$	2.41825	0.106561
$516$	0	0
$517$	63.6390	2.79884
$518$	0	0
$519$	41.5496	1.82383
$520$	0	0
$521$	19.5686	0.857314	0.428657	−	0.903467i	$-0.358987\pi$
$521$	19.5686	0.857314	0.428657	+	0.903467i	$0.358987\pi$
$522$	0	0
$523$	0.236246	0.0103303	0.00516515	−	0.999987i	$-0.498356\pi$
$523$	0.236246	0.0103303	0.00516515	+	0.999987i	$0.498356\pi$
$524$	0	0
$525$	−7.85068	−0.342632
$526$	0	0
$527$	−4.94626	−0.215462
$528$	0	0
$529$	−16.2546	−0.706723
$530$	0	0
$531$	−34.2622	−1.48685
$532$	0	0
$533$	6.40597	0.277473
$534$	0	0
$535$	7.76217	0.335588
$536$	0	0
$537$	39.6626	1.71157
$538$	0	0
$539$	7.92415	0.341317
$540$	0	0
$541$	−10.1429	−0.436078	−0.218039	−	0.975940i	$-0.569966\pi$
$541$	−10.1429	−0.436078	−0.218039	+	0.975940i	$0.569966\pi$
$542$	0	0
$543$	25.1270	1.07830
$544$	0	0
$545$	−16.4274	−0.703671
$546$	0	0
$547$	−27.6631	−1.18279	−0.591394	−	0.806383i	$-0.701422\pi$
$547$	−27.6631	−1.18279	−0.591394	+	0.806383i	$0.701422\pi$
$548$	0	0
$549$	−99.9338	−4.26507
$550$	0	0
$551$	−4.87297	−0.207595
$552$	0	0
$553$	30.2609	1.28682
$554$	0	0
$555$	29.9384	1.27081
$556$	0	0
$557$	39.7970	1.68625	0.843127	−	0.537715i	$-0.180712\pi$
$557$	39.7970	1.68625	0.843127	+	0.537715i	$0.180712\pi$
$558$	0	0
$559$	−14.0537	−0.594407
$560$	0	0
$561$	−41.6655	−1.75912
$562$	0	0
$563$	8.13579	0.342883	0.171441	−	0.985194i	$-0.445158\pi$
$563$	8.13579	0.342883	0.171441	+	0.985194i	$0.445158\pi$
$564$	0	0
$565$	−8.40511	−0.353606
$566$	0	0
$567$	77.0376	3.23528
$568$	0	0
$569$	−8.78964	−0.368481	−0.184241	−	0.982881i	$-0.558983\pi$
$569$	−8.78964	−0.368481	−0.184241	+	0.982881i	$0.558983\pi$
$570$	0	0
$571$	10.4366	0.436760	0.218380	−	0.975864i	$-0.429923\pi$
$571$	10.4366	0.436760	0.218380	+	0.975864i	$0.429923\pi$
$572$	0	0
$573$	−15.2607	−0.637525
$574$	0	0
$575$	−2.59719	−0.108310
$576$	0	0
$577$	−16.4780	−0.685988	−0.342994	−	0.939338i	$-0.611441\pi$
$577$	−16.4780	−0.685988	−0.342994	+	0.939338i	$0.611441\pi$
$578$	0	0
$579$	−18.9523	−0.787629
$580$	0	0
$581$	6.33477	0.262810
$582$	0	0
$583$	−54.3623	−2.25145
$584$	0	0
$585$	57.7953	2.38954
$586$	0	0
$587$	5.32677	0.219859	0.109930	−	0.993939i	$-0.464937\pi$
$587$	5.32677	0.219859	0.109930	+	0.993939i	$0.464937\pi$
$588$	0	0
$589$	7.15807	0.294943
$590$	0	0
$591$	−43.0097	−1.76918
$592$	0	0
$593$	28.2745	1.16109	0.580547	−	0.814227i	$-0.302839\pi$
$593$	28.2745	1.16109	0.580547	+	0.814227i	$0.302839\pi$
$594$	0	0
$595$	−5.42740	−0.222502
$596$	0	0
$597$	45.5068	1.86247
$598$	0	0
$599$	−20.6103	−0.842115	−0.421057	−	0.907034i	$-0.638341\pi$
$599$	−20.6103	−0.842115	−0.421057	+	0.907034i	$0.638341\pi$
$600$	0	0
$601$	−33.8576	−1.38108	−0.690541	−	0.723293i	$-0.742627\pi$
$601$	−33.8576	−1.38108	−0.690541	+	0.723293i	$0.742627\pi$
$602$	0	0
$603$	3.13255	0.127567
$604$	0	0
$605$	18.3083	0.744339
$606$	0	0
$607$	18.0293	0.731787	0.365893	−	0.930657i	$-0.380764\pi$
$607$	18.0293	0.731787	0.365893	+	0.930657i	$0.380764\pi$
$608$	0	0
$609$	11.4604	0.464397
$610$	0	0
$611$	−83.5395	−3.37965
$612$	0	0
$613$	39.1057	1.57946	0.789731	−	0.613453i	$-0.210220\pi$
$613$	39.1057	1.57946	0.789731	+	0.613453i	$0.210220\pi$
$614$	0	0
$615$	−3.00759	−0.121278
$616$	0	0
$617$	−27.0518	−1.08907	−0.544533	−	0.838740i	$-0.683293\pi$
$617$	−27.0518	−1.08907	−0.544533	+	0.838740i	$0.683293\pi$
$618$	0	0
$619$	−15.3853	−0.618389	−0.309194	−	0.950999i	$-0.600059\pi$
$619$	−15.3853	−0.618389	−0.309194	+	0.950999i	$0.600059\pi$
$620$	0	0
$621$	44.4771	1.78481
$622$	0	0
$623$	31.6360	1.26747
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	60.2970	2.40803
$628$	0	0
$629$	20.6973	0.825254
$630$	0	0
$631$	31.9743	1.27288	0.636439	−	0.771327i	$-0.280407\pi$
$631$	31.9743	1.27288	0.636439	+	0.771327i	$0.280407\pi$
$632$	0	0
$633$	84.7084	3.36686
$634$	0	0
$635$	19.2710	0.764744
$636$	0	0
$637$	−10.4021	−0.412147
$638$	0	0
$639$	−101.574	−4.01820
$640$	0	0
$641$	−13.2599	−0.523736	−0.261868	−	0.965104i	$-0.584339\pi$
$641$	−13.2599	−0.523736	−0.261868	+	0.965104i	$0.584339\pi$
$642$	0	0
$643$	6.66670	0.262909	0.131454	−	0.991322i	$-0.458035\pi$
$643$	6.66670	0.262909	0.131454	+	0.991322i	$0.458035\pi$
$644$	0	0
$645$	6.59818	0.259803
$646$	0	0
$647$	7.25281	0.285138	0.142569	−	0.989785i	$-0.454464\pi$
$647$	7.25281	0.285138	0.142569	+	0.989785i	$0.454464\pi$
$648$	0	0
$649$	22.8077	0.895282
$650$	0	0
$651$	−16.8345	−0.659797
$652$	0	0
$653$	36.3764	1.42352	0.711760	−	0.702423i	$-0.247898\pi$
$653$	36.3764	1.42352	0.711760	+	0.702423i	$0.247898\pi$
$654$	0	0
$655$	−12.3384	−0.482102
$656$	0	0
$657$	−54.6728	−2.13299
$658$	0	0
$659$	−31.2710	−1.21815	−0.609073	−	0.793114i	$-0.708458\pi$
$659$	−31.2710	−1.21815	−0.609073	+	0.793114i	$0.708458\pi$
$660$	0	0
$661$	−23.3459	−0.908050	−0.454025	−	0.890989i	$-0.650012\pi$
$661$	−23.3459	−0.908050	−0.454025	+	0.890989i	$0.650012\pi$
$662$	0	0
$663$	54.6947	2.12416
$664$	0	0
$665$	7.85437	0.304579
$666$	0	0
$667$	3.79135	0.146802
$668$	0	0
$669$	−75.6214	−2.92369
$670$	0	0
$671$	66.5242	2.56814
$672$	0	0
$673$	−1.23928	−0.0477706	−0.0238853	−	0.999715i	$-0.507604\pi$
$673$	−1.23928	−0.0477706	−0.0238853	+	0.999715i	$0.507604\pi$
$674$	0	0
$675$	−17.1251	−0.659146
$676$	0	0
$677$	28.5718	1.09810	0.549051	−	0.835789i	$-0.314989\pi$
$677$	28.5718	1.09810	0.549051	+	0.835789i	$0.314989\pi$
$678$	0	0
$679$	14.0100	0.537654
$680$	0	0
$681$	68.4030	2.62121
$682$	0	0
$683$	18.5979	0.711629	0.355814	−	0.934557i	$-0.384204\pi$
$683$	18.5979	0.711629	0.355814	+	0.934557i	$0.384204\pi$
$684$	0	0
$685$	2.09541	0.0800616
$686$	0	0
$687$	24.4086	0.931247
$688$	0	0
$689$	71.3619	2.71867
$690$	0	0
$691$	24.8610	0.945759	0.472879	−	0.881127i	$-0.343215\pi$
$691$	24.8610	0.945759	0.472879	+	0.881127i	$0.343215\pi$
$692$	0	0
$693$	−103.594	−3.93520
$694$	0	0
$695$	−13.5877	−0.515410
$696$	0	0
$697$	−2.07923	−0.0787566
$698$	0	0
$699$	31.2401	1.18161
$700$	0	0
$701$	−38.5861	−1.45738	−0.728689	−	0.684845i	$-0.759870\pi$
$701$	−38.5861	−1.45738	−0.728689	+	0.684845i	$0.759870\pi$
$702$	0	0
$703$	−29.9524	−1.12968
$704$	0	0
$705$	39.2217	1.47717
$706$	0	0
$707$	−38.4799	−1.44718
$708$	0	0
$709$	−9.89928	−0.371775	−0.185888	−	0.982571i	$-0.559516\pi$
$709$	−9.89928	−0.371775	−0.185888	+	0.982571i	$0.559516\pi$
$710$	0	0
$711$	104.593	3.92253
$712$	0	0
$713$	−5.56925	−0.208570
$714$	0	0
$715$	−38.4733	−1.43882
$716$	0	0
$717$	47.2927	1.76618
$718$	0	0
$719$	9.55542	0.356357	0.178179	−	0.983998i	$-0.442980\pi$
$719$	9.55542	0.356357	0.178179	+	0.983998i	$0.442980\pi$
$720$	0	0
$721$	5.68998	0.211906
$722$	0	0
$723$	23.9076	0.889133
$724$	0	0
$725$	−1.45979	−0.0542153
$726$	0	0
$727$	2.10017	0.0778911	0.0389455	−	0.999241i	$-0.487600\pi$
$727$	2.10017	0.0778911	0.0389455	+	0.999241i	$0.487600\pi$
$728$	0	0
$729$	94.8533	3.51309
$730$	0	0
$731$	4.56151	0.168714
$732$	0	0
$733$	−3.35961	−0.124090	−0.0620451	−	0.998073i	$-0.519762\pi$
$733$	−3.35961	−0.124090	−0.0620451	+	0.998073i	$0.519762\pi$
$734$	0	0
$735$	4.88377	0.180141
$736$	0	0
$737$	−2.08529	−0.0768125
$738$	0	0
$739$	−12.1420	−0.446652	−0.223326	−	0.974744i	$-0.571691\pi$
$739$	−12.1420	−0.446652	−0.223326	+	0.974744i	$0.571691\pi$
$740$	0	0
$741$	−79.1524	−2.90774
$742$	0	0
$743$	18.3929	0.674772	0.337386	−	0.941366i	$-0.390457\pi$
$743$	18.3929	0.674772	0.337386	+	0.941366i	$0.390457\pi$
$744$	0	0
$745$	7.63807	0.279837
$746$	0	0
$747$	21.8953	0.801106
$748$	0	0
$749$	18.2638	0.667346
$750$	0	0
$751$	30.9814	1.13053	0.565264	−	0.824910i	$-0.308774\pi$
$751$	30.9814	1.13053	0.565264	+	0.824910i	$0.308774\pi$
$752$	0	0
$753$	28.3645	1.03366
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−45.2346	−1.64408	−0.822040	−	0.569429i	$-0.807164\pi$
$757$	−45.2346	−1.64408	−0.822040	+	0.569429i	$0.807164\pi$
$758$	0	0
$759$	−46.9134	−1.70285
$760$	0	0
$761$	−21.2717	−0.771100	−0.385550	−	0.922687i	$-0.625988\pi$
$761$	−21.2717	−0.771100	−0.385550	+	0.922687i	$0.625988\pi$
$762$	0	0
$763$	−38.6524	−1.39931
$764$	0	0
$765$	−18.7591	−0.678236
$766$	0	0
$767$	−29.9399	−1.08107
$768$	0	0
$769$	10.1576	0.366292	0.183146	−	0.983086i	$-0.441372\pi$
$769$	10.1576	0.366292	0.183146	+	0.983086i	$0.441372\pi$
$770$	0	0
$771$	−28.8719	−1.03980
$772$	0	0
$773$	20.4688	0.736210	0.368105	−	0.929784i	$-0.380007\pi$
$773$	20.4688	0.736210	0.368105	+	0.929784i	$0.380007\pi$
$774$	0	0
$775$	2.14434	0.0770270
$776$	0	0
$777$	70.4429	2.52712
$778$	0	0
$779$	3.00900	0.107809
$780$	0	0
$781$	67.6160	2.41949
$782$	0	0
$783$	24.9991	0.893396
$784$	0	0
$785$	−3.64551	−0.130114
$786$	0	0
$787$	−4.89209	−0.174384	−0.0871921	−	0.996192i	$-0.527789\pi$
$787$	−4.89209	−0.174384	−0.0871921	+	0.996192i	$0.527789\pi$
$788$	0	0
$789$	48.1722	1.71498
$790$	0	0
$791$	−19.7766	−0.703177
$792$	0	0
$793$	−87.3269	−3.10107
$794$	0	0
$795$	−33.5043	−1.18827
$796$	0	0
$797$	−19.0782	−0.675784	−0.337892	−	0.941185i	$-0.609714\pi$
$797$	−19.0782	−0.675784	−0.337892	+	0.941185i	$0.609714\pi$
$798$	0	0
$799$	27.1151	0.959263
$800$	0	0
$801$	109.346	3.86354
$802$	0	0
$803$	36.3947	1.28434
$804$	0	0
$805$	−6.11100	−0.215384
$806$	0	0
$807$	58.0401	2.04311
$808$	0	0
$809$	33.6655	1.18362	0.591809	−	0.806078i	$-0.298414\pi$
$809$	33.6655	1.18362	0.591809	+	0.806078i	$0.298414\pi$
$810$	0	0
$811$	22.8424	0.802104	0.401052	−	0.916055i	$-0.368645\pi$
$811$	22.8424	0.802104	0.401052	+	0.916055i	$0.368645\pi$
$812$	0	0
$813$	−100.820	−3.53591
$814$	0	0
$815$	16.2139	0.567947
$816$	0	0
$817$	−6.60128	−0.230950
$818$	0	0
$819$	135.988	4.75182
$820$	0	0
$821$	0.517548	0.0180625	0.00903127	−	0.999959i	$-0.497125\pi$
$821$	0.517548	0.0180625	0.00903127	+	0.999959i	$0.497125\pi$
$822$	0	0
$823$	9.91645	0.345666	0.172833	−	0.984951i	$-0.444708\pi$
$823$	9.91645	0.345666	0.172833	+	0.984951i	$0.444708\pi$
$824$	0	0
$825$	18.0631	0.628878
$826$	0	0
$827$	52.8207	1.83676	0.918378	−	0.395704i	$-0.129499\pi$
$827$	52.8207	1.83676	0.918378	+	0.395704i	$0.129499\pi$
$828$	0	0
$829$	29.5853	1.02754	0.513770	−	0.857928i	$-0.328248\pi$
$829$	29.5853	1.02754	0.513770	+	0.857928i	$0.328248\pi$
$830$	0	0
$831$	−3.81090	−0.132199
$832$	0	0
$833$	3.37629	0.116982
$834$	0	0
$835$	23.8658	0.825911
$836$	0	0
$837$	−36.7221	−1.26930
$838$	0	0
$839$	9.83903	0.339681	0.169840	−	0.985472i	$-0.445675\pi$
$839$	9.83903	0.339681	0.169840	+	0.985472i	$0.445675\pi$
$840$	0	0
$841$	−26.8690	−0.926517
$842$	0	0
$843$	54.7358	1.88520
$844$	0	0
$845$	37.5043	1.29019
$846$	0	0
$847$	43.0782	1.48019
$848$	0	0
$849$	35.6045	1.22194
$850$	0	0
$851$	23.3041	0.798856
$852$	0	0
$853$	−0.0584971	−0.00200290	−0.00100145	−	0.999999i	$-0.500319\pi$
$853$	−0.0584971	−0.00200290	−0.00100145	+	0.999999i	$0.500319\pi$
$854$	0	0
$855$	27.1475	0.928427
$856$	0	0
$857$	−53.8637	−1.83995	−0.919974	−	0.391980i	$-0.871790\pi$
$857$	−53.8637	−1.83995	−0.919974	+	0.391980i	$0.871790\pi$
$858$	0	0
$859$	−24.7577	−0.844721	−0.422361	−	0.906428i	$-0.638798\pi$
$859$	−24.7577	−0.844721	−0.422361	+	0.906428i	$0.638798\pi$
$860$	0	0
$861$	−7.07665	−0.241172
$862$	0	0
$863$	6.23688	0.212306	0.106153	−	0.994350i	$-0.466147\pi$
$863$	6.23688	0.212306	0.106153	+	0.994350i	$0.466147\pi$
$864$	0	0
$865$	−12.4529	−0.423410
$866$	0	0
$867$	38.9687	1.32345
$868$	0	0
$869$	−69.6254	−2.36188
$870$	0	0
$871$	2.73738	0.0927524
$872$	0	0
$873$	48.4236	1.63889
$874$	0	0
$875$	2.35293	0.0795436
$876$	0	0
$877$	−30.8283	−1.04100	−0.520499	−	0.853862i	$-0.674254\pi$
$877$	−30.8283	−1.04100	−0.520499	+	0.853862i	$0.674254\pi$
$878$	0	0
$879$	−23.1103	−0.779490
$880$	0	0
$881$	7.43062	0.250344	0.125172	−	0.992135i	$-0.460052\pi$
$881$	7.43062	0.250344	0.125172	+	0.992135i	$0.460052\pi$
$882$	0	0
$883$	−29.2605	−0.984695	−0.492347	−	0.870399i	$-0.663861\pi$
$883$	−29.2605	−0.984695	−0.492347	+	0.870399i	$0.663861\pi$
$884$	0	0
$885$	14.0567	0.472512
$886$	0	0
$887$	38.9733	1.30860	0.654298	−	0.756237i	$-0.272964\pi$
$887$	38.9733	1.30860	0.654298	+	0.756237i	$0.272964\pi$
$888$	0	0
$889$	45.3432	1.52076
$890$	0	0
$891$	−177.251	−5.93814
$892$	0	0
$893$	−39.2401	−1.31312
$894$	0	0
$895$	−11.8873	−0.397348
$896$	0	0
$897$	61.5836	2.05622
$898$	0	0
$899$	−3.13029	−0.104401
$900$	0	0
$901$	−23.1625	−0.771654
$902$	0	0
$903$	15.5250	0.516641
$904$	0	0
$905$	−7.53084	−0.250334
$906$	0	0
$907$	13.5692	0.450557	0.225279	−	0.974294i	$-0.427671\pi$
$907$	13.5692	0.450557	0.225279	+	0.974294i	$0.427671\pi$
$908$	0	0
$909$	−133.000	−4.41134
$910$	0	0
$911$	−22.2093	−0.735827	−0.367913	−	0.929860i	$-0.619928\pi$
$911$	−22.2093	−0.735827	−0.367913	+	0.929860i	$0.619928\pi$
$912$	0	0
$913$	−14.5753	−0.482372
$914$	0	0
$915$	40.9998	1.35541
$916$	0	0
$917$	−29.0314	−0.958703
$918$	0	0
$919$	6.83245	0.225382	0.112691	−	0.993630i	$-0.464053\pi$
$919$	6.83245	0.225382	0.112691	+	0.993630i	$0.464053\pi$
$920$	0	0
$921$	−17.3620	−0.572097
$922$	0	0
$923$	−88.7602	−2.92158
$924$	0	0
$925$	−8.97284	−0.295025
$926$	0	0
$927$	19.6666	0.645937
$928$	0	0
$929$	−28.6125	−0.938745	−0.469372	−	0.883000i	$-0.655520\pi$
$929$	−28.6125	−0.938745	−0.469372	+	0.883000i	$0.655520\pi$
$930$	0	0
$931$	−4.88607	−0.160134
$932$	0	0
$933$	−22.6023	−0.739966
$934$	0	0
$935$	12.4876	0.408388
$936$	0	0
$937$	−4.48793	−0.146614	−0.0733072	−	0.997309i	$-0.523355\pi$
$937$	−4.48793	−0.146614	−0.0733072	+	0.997309i	$0.523355\pi$
$938$	0	0
$939$	−9.34311	−0.304901
$940$	0	0
$941$	54.4634	1.77546	0.887728	−	0.460369i	$-0.152283\pi$
$941$	54.4634	1.77546	0.887728	+	0.460369i	$0.152283\pi$
$942$	0	0
$943$	−2.34112	−0.0762374
$944$	0	0
$945$	−40.2942	−1.31077
$946$	0	0
$947$	−35.4643	−1.15244	−0.576218	−	0.817296i	$-0.695472\pi$
$947$	−35.4643	−1.15244	−0.576218	+	0.817296i	$0.695472\pi$
$948$	0	0
$949$	−47.7757	−1.55086
$950$	0	0
$951$	65.7758	2.13293
$952$	0	0
$953$	−2.68115	−0.0868511	−0.0434256	−	0.999057i	$-0.513827\pi$
$953$	−2.68115	−0.0868511	−0.0434256	+	0.999057i	$0.513827\pi$
$954$	0	0
$955$	4.57380	0.148005
$956$	0	0
$957$	−26.3685	−0.852371
$958$	0	0
$959$	4.93036	0.159210
$960$	0	0
$961$	−26.4018	−0.851671
$962$	0	0
$963$	63.1265	2.03422
$964$	0	0
$965$	5.68019	0.182852
$966$	0	0
$967$	−45.1482	−1.45187	−0.725933	−	0.687765i	$-0.758592\pi$
$967$	−45.1482	−1.45187	−0.725933	+	0.687765i	$0.758592\pi$
$968$	0	0
$969$	25.6911	0.825318
$970$	0	0
$971$	−25.8203	−0.828614	−0.414307	−	0.910137i	$-0.635976\pi$
$971$	−25.8203	−0.828614	−0.414307	+	0.910137i	$0.635976\pi$
$972$	0	0
$973$	−31.9709	−1.02494
$974$	0	0
$975$	−23.7117	−0.759381
$976$	0	0
$977$	14.5484	0.465445	0.232722	−	0.972543i	$-0.425237\pi$
$977$	14.5484	0.465445	0.232722	+	0.972543i	$0.425237\pi$
$978$	0	0
$979$	−72.7895	−2.32636
$980$	0	0
$981$	−133.597	−4.26542
$982$	0	0
$983$	33.8003	1.07806	0.539032	−	0.842285i	$-0.318790\pi$
$983$	33.8003	1.07806	0.539032	+	0.842285i	$0.318790\pi$
$984$	0	0
$985$	12.8905	0.410725
$986$	0	0
$987$	92.2858	2.93749
$988$	0	0
$989$	5.13605	0.163317
$990$	0	0
$991$	11.5476	0.366822	0.183411	−	0.983036i	$-0.441286\pi$
$991$	11.5476	0.366822	0.183411	+	0.983036i	$0.441286\pi$
$992$	0	0
$993$	−9.42468	−0.299083
$994$	0	0
$995$	−13.6389	−0.432381
$996$	0	0
$997$	49.0557	1.55361	0.776804	−	0.629742i	$-0.216839\pi$
$997$	49.0557	1.55361	0.776804	+	0.629742i	$0.216839\pi$
$998$	0	0
$999$	153.661	4.86162

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-3.33655 q^{3} +1.00000 q^{5} +2.35293 q^{7} +8.13258 q^{9} +O(q^{10})\)	\(q-3.33655 q^{3} +1.00000 q^{5} +2.35293 q^{7} +8.13258 q^{9} -5.41372 q^{11} +7.10664 q^{13} -3.33655 q^{15} -2.30666 q^{17} +3.33812 q^{19} -7.85068 q^{21} -2.59719 q^{23} +1.00000 q^{25} -17.1251 q^{27} -1.45979 q^{29} +2.14434 q^{31} +18.0631 q^{33} +2.35293 q^{35} -8.97284 q^{37} -23.7117 q^{39} +0.901406 q^{41} -1.97754 q^{43} +8.13258 q^{45} -11.7551 q^{47} -1.46372 q^{49} +7.69628 q^{51} +10.0416 q^{53} -5.41372 q^{55} -11.1378 q^{57} -4.21296 q^{59} -12.2881 q^{61} +19.1354 q^{63} +7.10664 q^{65} +0.385186 q^{67} +8.66565 q^{69} -12.4898 q^{71} -6.72268 q^{73} -3.33655 q^{75} -12.7381 q^{77} +12.8609 q^{79} +32.7411 q^{81} +2.69229 q^{83} -2.30666 q^{85} +4.87068 q^{87} +13.4454 q^{89} +16.7214 q^{91} -7.15471 q^{93} +3.33812 q^{95} +5.95427 q^{97} -44.0275 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 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

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