LMFDB - Embedded newform 6040.2.a.o.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:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 $ x^15 - 5x^14 - 19x^13 + 119x^12 + 106x^11 - 1063x^10 - 48x^9 + 4510x^8 - 1130x^7 - 9512x^6 + 2839x^5 + 9383x^4 - 1781x^3 - 3692x^2 - 92x + 272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.77658$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77658 q^{3} +1.00000 q^{5} +2.60627 q^{7} +0.156227 q^{9} +O(q^{10})$	$q-1.77658 q^{3} +1.00000 q^{5} +2.60627 q^{7} +0.156227 q^{9} -2.28127 q^{11} -1.31177 q^{13} -1.77658 q^{15} +6.39339 q^{17} +7.78113 q^{19} -4.63023 q^{21} +1.97507 q^{23} +1.00000 q^{25} +5.05218 q^{27} +1.94551 q^{29} -3.61493 q^{31} +4.05285 q^{33} +2.60627 q^{35} -3.98185 q^{37} +2.33045 q^{39} -8.30160 q^{41} -3.07208 q^{43} +0.156227 q^{45} +1.41698 q^{47} -0.207373 q^{49} -11.3584 q^{51} +7.08653 q^{53} -2.28127 q^{55} -13.8238 q^{57} -0.598101 q^{59} +4.09476 q^{61} +0.407169 q^{63} -1.31177 q^{65} +6.99202 q^{67} -3.50886 q^{69} -1.07034 q^{71} +4.86142 q^{73} -1.77658 q^{75} -5.94559 q^{77} +5.97901 q^{79} -9.44427 q^{81} +7.40965 q^{83} +6.39339 q^{85} -3.45634 q^{87} +4.46666 q^{89} -3.41881 q^{91} +6.42220 q^{93} +7.78113 q^{95} -13.7303 q^{97} -0.356395 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.77658	−1.02571	−0.512854	−	0.858476i	$-0.671412\pi$
$3$	−1.77658	−1.02571	−0.512854	+	0.858476i	$0.671412\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.60627	0.985076	0.492538	−	0.870291i	$-0.336069\pi$
$7$	2.60627	0.985076	0.492538	+	0.870291i	$0.336069\pi$
$8$	0	0
$9$	0.156227	0.0520756
$10$	0	0
$11$	−2.28127	−0.687828	−0.343914	−	0.939001i	$-0.611753\pi$
$11$	−2.28127	−0.687828	−0.343914	+	0.939001i	$0.611753\pi$
$12$	0	0
$13$	−1.31177	−0.363818	−0.181909	−	0.983315i	$-0.558228\pi$
$13$	−1.31177	−0.363818	−0.181909	+	0.983315i	$0.558228\pi$
$14$	0	0
$15$	−1.77658	−0.458710
$16$	0	0
$17$	6.39339	1.55062	0.775312	−	0.631578i	$-0.217592\pi$
$17$	6.39339	1.55062	0.775312	+	0.631578i	$0.217592\pi$
$18$	0	0
$19$	7.78113	1.78511	0.892556	−	0.450936i	$-0.148910\pi$
$19$	7.78113	1.78511	0.892556	+	0.450936i	$0.148910\pi$
$20$	0	0
$21$	−4.63023	−1.01040
$22$	0	0
$23$	1.97507	0.411830	0.205915	−	0.978570i	$-0.433983\pi$
$23$	1.97507	0.411830	0.205915	+	0.978570i	$0.433983\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.05218	0.972293
$28$	0	0
$29$	1.94551	0.361271	0.180636	−	0.983550i	$-0.442185\pi$
$29$	1.94551	0.361271	0.180636	+	0.983550i	$0.442185\pi$
$30$	0	0
$31$	−3.61493	−0.649261	−0.324630	−	0.945841i	$-0.605240\pi$
$31$	−3.61493	−0.649261	−0.324630	+	0.945841i	$0.605240\pi$
$32$	0	0
$33$	4.05285	0.705510
$34$	0	0
$35$	2.60627	0.440540
$36$	0	0
$37$	−3.98185	−0.654613	−0.327306	−	0.944918i	$-0.606141\pi$
$37$	−3.98185	−0.654613	−0.327306	+	0.944918i	$0.606141\pi$
$38$	0	0
$39$	2.33045	0.373171
$40$	0	0
$41$	−8.30160	−1.29649	−0.648246	−	0.761431i	$-0.724497\pi$
$41$	−8.30160	−1.29649	−0.648246	+	0.761431i	$0.724497\pi$
$42$	0	0
$43$	−3.07208	−0.468488	−0.234244	−	0.972178i	$-0.575261\pi$
$43$	−3.07208	−0.468488	−0.234244	+	0.972178i	$0.575261\pi$
$44$	0	0
$45$	0.156227	0.0232889
$46$	0	0
$47$	1.41698	0.206688	0.103344	−	0.994646i	$-0.467046\pi$
$47$	1.41698	0.206688	0.103344	+	0.994646i	$0.467046\pi$
$48$	0	0
$49$	−0.207373	−0.0296247
$50$	0	0
$51$	−11.3584	−1.59049
$52$	0	0
$53$	7.08653	0.973410	0.486705	−	0.873567i	$-0.338199\pi$
$53$	7.08653	0.973410	0.486705	+	0.873567i	$0.338199\pi$
$54$	0	0
$55$	−2.28127	−0.307606
$56$	0	0
$57$	−13.8238	−1.83100
$58$	0	0
$59$	−0.598101	−0.0778661	−0.0389330	−	0.999242i	$-0.512396\pi$
$59$	−0.598101	−0.0778661	−0.0389330	+	0.999242i	$0.512396\pi$
$60$	0	0
$61$	4.09476	0.524281	0.262140	−	0.965030i	$-0.415572\pi$
$61$	4.09476	0.524281	0.262140	+	0.965030i	$0.415572\pi$
$62$	0	0
$63$	0.407169	0.0512985
$64$	0	0
$65$	−1.31177	−0.162704
$66$	0	0
$67$	6.99202	0.854211	0.427105	−	0.904202i	$-0.359533\pi$
$67$	6.99202	0.854211	0.427105	+	0.904202i	$0.359533\pi$
$68$	0	0
$69$	−3.50886	−0.422417
$70$	0	0
$71$	−1.07034	−0.127026	−0.0635129	−	0.997981i	$-0.520230\pi$
$71$	−1.07034	−0.127026	−0.0635129	+	0.997981i	$0.520230\pi$
$72$	0	0
$73$	4.86142	0.568986	0.284493	−	0.958678i	$-0.408175\pi$
$73$	4.86142	0.568986	0.284493	+	0.958678i	$0.408175\pi$
$74$	0	0
$75$	−1.77658	−0.205141
$76$	0	0
$77$	−5.94559	−0.677563
$78$	0	0
$79$	5.97901	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$79$	5.97901	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$80$	0	0
$81$	−9.44427	−1.04936
$82$	0	0
$83$	7.40965	0.813315	0.406657	−	0.913581i	$-0.366694\pi$
$83$	7.40965	0.813315	0.406657	+	0.913581i	$0.366694\pi$
$84$	0	0
$85$	6.39339	0.693461
$86$	0	0
$87$	−3.45634	−0.370559
$88$	0	0
$89$	4.46666	0.473465	0.236733	−	0.971575i	$-0.423923\pi$
$89$	4.46666	0.473465	0.236733	+	0.971575i	$0.423923\pi$
$90$	0	0
$91$	−3.41881	−0.358389
$92$	0	0
$93$	6.42220	0.665952
$94$	0	0
$95$	7.78113	0.798327
$96$	0	0
$97$	−13.7303	−1.39410	−0.697049	−	0.717024i	$-0.745504\pi$
$97$	−13.7303	−1.39410	−0.697049	+	0.717024i	$0.745504\pi$
$98$	0	0
$99$	−0.356395	−0.0358191
$100$	0	0
$101$	1.21546	0.120943	0.0604716	−	0.998170i	$-0.480740\pi$
$101$	1.21546	0.120943	0.0604716	+	0.998170i	$0.480740\pi$
$102$	0	0
$103$	8.59179	0.846574	0.423287	−	0.905996i	$-0.360876\pi$
$103$	8.59179	0.846574	0.423287	+	0.905996i	$0.360876\pi$
$104$	0	0
$105$	−4.63023	−0.451865
$106$	0	0
$107$	−1.33349	−0.128914	−0.0644568	−	0.997920i	$-0.520531\pi$
$107$	−1.33349	−0.128914	−0.0644568	+	0.997920i	$0.520531\pi$
$108$	0	0
$109$	4.25595	0.407646	0.203823	−	0.979008i	$-0.434663\pi$
$109$	4.25595	0.407646	0.203823	+	0.979008i	$0.434663\pi$
$110$	0	0
$111$	7.07407	0.671441
$112$	0	0
$113$	−19.3744	−1.82259	−0.911297	−	0.411750i	$-0.864918\pi$
$113$	−19.3744	−1.82259	−0.911297	+	0.411750i	$0.864918\pi$
$114$	0	0
$115$	1.97507	0.184176
$116$	0	0
$117$	−0.204933	−0.0189461
$118$	0	0
$119$	16.6629	1.52748
$120$	0	0
$121$	−5.79582	−0.526893
$122$	0	0
$123$	14.7484	1.32982
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.79746	−0.869384	−0.434692	−	0.900579i	$-0.643143\pi$
$127$	−9.79746	−0.869384	−0.434692	+	0.900579i	$0.643143\pi$
$128$	0	0
$129$	5.45779	0.480531
$130$	0	0
$131$	12.9794	1.13402	0.567009	−	0.823712i	$-0.308101\pi$
$131$	12.9794	1.13402	0.567009	+	0.823712i	$0.308101\pi$
$132$	0	0
$133$	20.2797	1.75847
$134$	0	0
$135$	5.05218	0.434823
$136$	0	0
$137$	3.74788	0.320203	0.160102	−	0.987101i	$-0.448818\pi$
$137$	3.74788	0.320203	0.160102	+	0.987101i	$0.448818\pi$
$138$	0	0
$139$	16.7483	1.42057	0.710285	−	0.703914i	$-0.248566\pi$
$139$	16.7483	1.42057	0.710285	+	0.703914i	$0.248566\pi$
$140$	0	0
$141$	−2.51738	−0.212002
$142$	0	0
$143$	2.99249	0.250244
$144$	0	0
$145$	1.94551	0.161565
$146$	0	0
$147$	0.368414	0.0303863
$148$	0	0
$149$	−9.53706	−0.781307	−0.390653	−	0.920538i	$-0.627751\pi$
$149$	−9.53706	−0.781307	−0.390653	+	0.920538i	$0.627751\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	0.998820	0.0807498
$154$	0	0
$155$	−3.61493	−0.290358
$156$	0	0
$157$	−2.54712	−0.203282	−0.101641	−	0.994821i	$-0.532409\pi$
$157$	−2.54712	−0.203282	−0.101641	+	0.994821i	$0.532409\pi$
$158$	0	0
$159$	−12.5898	−0.998433
$160$	0	0
$161$	5.14755	0.405684
$162$	0	0
$163$	−25.1170	−1.96732	−0.983658	−	0.180049i	$-0.942375\pi$
$163$	−25.1170	−1.96732	−0.983658	+	0.180049i	$0.942375\pi$
$164$	0	0
$165$	4.05285	0.315514
$166$	0	0
$167$	2.32205	0.179686	0.0898428	−	0.995956i	$-0.471364\pi$
$167$	2.32205	0.179686	0.0898428	+	0.995956i	$0.471364\pi$
$168$	0	0
$169$	−11.2793	−0.867636
$170$	0	0
$171$	1.21562	0.0929609
$172$	0	0
$173$	2.66069	0.202289	0.101144	−	0.994872i	$-0.467750\pi$
$173$	2.66069	0.202289	0.101144	+	0.994872i	$0.467750\pi$
$174$	0	0
$175$	2.60627	0.197015
$176$	0	0
$177$	1.06257	0.0798678
$178$	0	0
$179$	13.1030	0.979367	0.489684	−	0.871900i	$-0.337112\pi$
$179$	13.1030	0.979367	0.489684	+	0.871900i	$0.337112\pi$
$180$	0	0
$181$	4.71884	0.350749	0.175374	−	0.984502i	$-0.443886\pi$
$181$	4.71884	0.350749	0.175374	+	0.984502i	$0.443886\pi$
$182$	0	0
$183$	−7.27466	−0.537759
$184$	0	0
$185$	−3.98185	−0.292752
$186$	0	0
$187$	−14.5850	−1.06656
$188$	0	0
$189$	13.1673	0.957783
$190$	0	0
$191$	5.50796	0.398542	0.199271	−	0.979944i	$-0.436143\pi$
$191$	5.50796	0.398542	0.199271	+	0.979944i	$0.436143\pi$
$192$	0	0
$193$	4.50303	0.324135	0.162068	−	0.986780i	$-0.448184\pi$
$193$	4.50303	0.324135	0.162068	+	0.986780i	$0.448184\pi$
$194$	0	0
$195$	2.33045	0.166887
$196$	0	0
$197$	−7.49185	−0.533772	−0.266886	−	0.963728i	$-0.585995\pi$
$197$	−7.49185	−0.533772	−0.266886	+	0.963728i	$0.585995\pi$
$198$	0	0
$199$	−6.02087	−0.426808	−0.213404	−	0.976964i	$-0.568455\pi$
$199$	−6.02087	−0.426808	−0.213404	+	0.976964i	$0.568455\pi$
$200$	0	0
$201$	−12.4219	−0.876170
$202$	0	0
$203$	5.07051	0.355880
$204$	0	0
$205$	−8.30160	−0.579809
$206$	0	0
$207$	0.308559	0.0214463
$208$	0	0
$209$	−17.7508	−1.22785
$210$	0	0
$211$	25.5129	1.75638	0.878189	−	0.478314i	$-0.158752\pi$
$211$	25.5129	1.75638	0.878189	+	0.478314i	$0.158752\pi$
$212$	0	0
$213$	1.90154	0.130291
$214$	0	0
$215$	−3.07208	−0.209514
$216$	0	0
$217$	−9.42147	−0.639571
$218$	0	0
$219$	−8.63668	−0.583613
$220$	0	0
$221$	−8.38663	−0.564146
$222$	0	0
$223$	27.4597	1.83884	0.919420	−	0.393278i	$-0.128659\pi$
$223$	27.4597	1.83884	0.919420	+	0.393278i	$0.128659\pi$
$224$	0	0
$225$	0.156227	0.0104151
$226$	0	0
$227$	16.5059	1.09554	0.547768	−	0.836631i	$-0.315478\pi$
$227$	16.5059	1.09554	0.547768	+	0.836631i	$0.315478\pi$
$228$	0	0
$229$	9.46462	0.625440	0.312720	−	0.949845i	$-0.398760\pi$
$229$	9.46462	0.625440	0.312720	+	0.949845i	$0.398760\pi$
$230$	0	0
$231$	10.5628	0.694981
$232$	0	0
$233$	5.70473	0.373730	0.186865	−	0.982386i	$-0.440167\pi$
$233$	5.70473	0.373730	0.186865	+	0.982386i	$0.440167\pi$
$234$	0	0
$235$	1.41698	0.0924339
$236$	0	0
$237$	−10.6222	−0.689985
$238$	0	0
$239$	−11.3390	−0.733458	−0.366729	−	0.930328i	$-0.619522\pi$
$239$	−11.3390	−0.733458	−0.366729	+	0.930328i	$0.619522\pi$
$240$	0	0
$241$	16.4070	1.05687	0.528435	−	0.848974i	$-0.322779\pi$
$241$	16.4070	1.05687	0.528435	+	0.848974i	$0.322779\pi$
$242$	0	0
$243$	1.62193	0.104047
$244$	0	0
$245$	−0.207373	−0.0132486
$246$	0	0
$247$	−10.2070	−0.649456
$248$	0	0
$249$	−13.1638	−0.834223
$250$	0	0
$251$	16.6945	1.05375	0.526873	−	0.849944i	$-0.323364\pi$
$251$	16.6945	1.05375	0.526873	+	0.849944i	$0.323364\pi$
$252$	0	0
$253$	−4.50565	−0.283268
$254$	0	0
$255$	−11.3584	−0.711288
$256$	0	0
$257$	−1.58448	−0.0988374	−0.0494187	−	0.998778i	$-0.515737\pi$
$257$	−1.58448	−0.0988374	−0.0494187	+	0.998778i	$0.515737\pi$
$258$	0	0
$259$	−10.3778	−0.644843
$260$	0	0
$261$	0.303940	0.0188134
$262$	0	0
$263$	7.23665	0.446231	0.223115	−	0.974792i	$-0.428377\pi$
$263$	7.23665	0.446231	0.223115	+	0.974792i	$0.428377\pi$
$264$	0	0
$265$	7.08653	0.435322
$266$	0	0
$267$	−7.93537	−0.485637
$268$	0	0
$269$	22.8810	1.39508	0.697539	−	0.716546i	$-0.254278\pi$
$269$	22.8810	1.39508	0.697539	+	0.716546i	$0.254278\pi$
$270$	0	0
$271$	−6.28734	−0.381929	−0.190964	−	0.981597i	$-0.561161\pi$
$271$	−6.28734	−0.381929	−0.190964	+	0.981597i	$0.561161\pi$
$272$	0	0
$273$	6.07378	0.367602
$274$	0	0
$275$	−2.28127	−0.137566
$276$	0	0
$277$	−0.686440	−0.0412442	−0.0206221	−	0.999787i	$-0.506565\pi$
$277$	−0.686440	−0.0412442	−0.0206221	+	0.999787i	$0.506565\pi$
$278$	0	0
$279$	−0.564750	−0.0338107
$280$	0	0
$281$	−10.9812	−0.655084	−0.327542	−	0.944837i	$-0.606220\pi$
$281$	−10.9812	−0.655084	−0.327542	+	0.944837i	$0.606220\pi$
$282$	0	0
$283$	−23.3058	−1.38538	−0.692692	−	0.721234i	$-0.743575\pi$
$283$	−23.3058	−1.38538	−0.692692	+	0.721234i	$0.743575\pi$
$284$	0	0
$285$	−13.8238	−0.818850
$286$	0	0
$287$	−21.6362	−1.27714
$288$	0	0
$289$	23.8754	1.40444
$290$	0	0
$291$	24.3929	1.42994
$292$	0	0
$293$	11.0259	0.644137	0.322069	−	0.946716i	$-0.395622\pi$
$293$	11.0259	0.644137	0.322069	+	0.946716i	$0.395622\pi$
$294$	0	0
$295$	−0.598101	−0.0348228
$296$	0	0
$297$	−11.5254	−0.668770
$298$	0	0
$299$	−2.59082	−0.149831
$300$	0	0
$301$	−8.00666	−0.461496
$302$	0	0
$303$	−2.15937	−0.124052
$304$	0	0
$305$	4.09476	0.234465
$306$	0	0
$307$	20.3819	1.16326	0.581628	−	0.813455i	$-0.302416\pi$
$307$	20.3819	1.16326	0.581628	+	0.813455i	$0.302416\pi$
$308$	0	0
$309$	−15.2640	−0.868338
$310$	0	0
$311$	8.83013	0.500711	0.250356	−	0.968154i	$-0.419452\pi$
$311$	8.83013	0.500711	0.250356	+	0.968154i	$0.419452\pi$
$312$	0	0
$313$	−13.8996	−0.785651	−0.392825	−	0.919613i	$-0.628502\pi$
$313$	−13.8996	−0.785651	−0.392825	+	0.919613i	$0.628502\pi$
$314$	0	0
$315$	0.407169	0.0229414
$316$	0	0
$317$	5.11327	0.287190	0.143595	−	0.989637i	$-0.454134\pi$
$317$	5.11327	0.287190	0.143595	+	0.989637i	$0.454134\pi$
$318$	0	0
$319$	−4.43822	−0.248492
$320$	0	0
$321$	2.36905	0.132228
$322$	0	0
$323$	49.7478	2.76804
$324$	0	0
$325$	−1.31177	−0.0727636
$326$	0	0
$327$	−7.56103	−0.418126
$328$	0	0
$329$	3.69304	0.203604
$330$	0	0
$331$	−6.16245	−0.338719	−0.169360	−	0.985554i	$-0.554170\pi$
$331$	−6.16245	−0.338719	−0.169360	+	0.985554i	$0.554170\pi$
$332$	0	0
$333$	−0.622073	−0.0340894
$334$	0	0
$335$	6.99202	0.382015
$336$	0	0
$337$	7.60465	0.414251	0.207126	−	0.978314i	$-0.433589\pi$
$337$	7.60465	0.414251	0.207126	+	0.978314i	$0.433589\pi$
$338$	0	0
$339$	34.4202	1.86945
$340$	0	0
$341$	8.24662	0.446580
$342$	0	0
$343$	−18.7843	−1.01426
$344$	0	0
$345$	−3.50886	−0.188911
$346$	0	0
$347$	25.5261	1.37031	0.685157	−	0.728395i	$-0.259733\pi$
$347$	25.5261	1.37031	0.685157	+	0.728395i	$0.259733\pi$
$348$	0	0
$349$	−18.4024	−0.985057	−0.492529	−	0.870296i	$-0.663927\pi$
$349$	−18.4024	−0.985057	−0.492529	+	0.870296i	$0.663927\pi$
$350$	0	0
$351$	−6.62728	−0.353738
$352$	0	0
$353$	0.897815	0.0477859	0.0238929	−	0.999715i	$-0.492394\pi$
$353$	0.897815	0.0477859	0.0238929	+	0.999715i	$0.492394\pi$
$354$	0	0
$355$	−1.07034	−0.0568077
$356$	0	0
$357$	−29.6029	−1.56675
$358$	0	0
$359$	27.9689	1.47614	0.738070	−	0.674724i	$-0.235737\pi$
$359$	27.9689	1.47614	0.738070	+	0.674724i	$0.235737\pi$
$360$	0	0
$361$	41.5459	2.18663
$362$	0	0
$363$	10.2967	0.540438
$364$	0	0
$365$	4.86142	0.254458
$366$	0	0
$367$	−3.91692	−0.204462	−0.102231	−	0.994761i	$-0.532598\pi$
$367$	−3.91692	−0.204462	−0.102231	+	0.994761i	$0.532598\pi$
$368$	0	0
$369$	−1.29693	−0.0675157
$370$	0	0
$371$	18.4694	0.958883
$372$	0	0
$373$	−14.3592	−0.743493	−0.371747	−	0.928334i	$-0.621241\pi$
$373$	−14.3592	−0.743493	−0.371747	+	0.928334i	$0.621241\pi$
$374$	0	0
$375$	−1.77658	−0.0917421
$376$	0	0
$377$	−2.55205	−0.131437
$378$	0	0
$379$	−5.89768	−0.302943	−0.151472	−	0.988462i	$-0.548401\pi$
$379$	−5.89768	−0.302943	−0.151472	+	0.988462i	$0.548401\pi$
$380$	0	0
$381$	17.4059	0.891734
$382$	0	0
$383$	13.1426	0.671558	0.335779	−	0.941941i	$-0.391001\pi$
$383$	13.1426	0.671558	0.335779	+	0.941941i	$0.391001\pi$
$384$	0	0
$385$	−5.94559	−0.303015
$386$	0	0
$387$	−0.479941	−0.0243968
$388$	0	0
$389$	−6.99546	−0.354684	−0.177342	−	0.984149i	$-0.556750\pi$
$389$	−6.99546	−0.354684	−0.177342	+	0.984149i	$0.556750\pi$
$390$	0	0
$391$	12.6274	0.638594
$392$	0	0
$393$	−23.0590	−1.16317
$394$	0	0
$395$	5.97901	0.300837
$396$	0	0
$397$	14.7249	0.739022	0.369511	−	0.929226i	$-0.379525\pi$
$397$	14.7249	0.739022	0.369511	+	0.929226i	$0.379525\pi$
$398$	0	0
$399$	−36.0284	−1.80368
$400$	0	0
$401$	17.5737	0.877587	0.438794	−	0.898588i	$-0.355406\pi$
$401$	17.5737	0.877587	0.438794	+	0.898588i	$0.355406\pi$
$402$	0	0
$403$	4.74194	0.236213
$404$	0	0
$405$	−9.44427	−0.469290
$406$	0	0
$407$	9.08367	0.450261
$408$	0	0
$409$	−1.98121	−0.0979646	−0.0489823	−	0.998800i	$-0.515598\pi$
$409$	−1.98121	−0.0979646	−0.0489823	+	0.998800i	$0.515598\pi$
$410$	0	0
$411$	−6.65840	−0.328435
$412$	0	0
$413$	−1.55881	−0.0767040
$414$	0	0
$415$	7.40965	0.363725
$416$	0	0
$417$	−29.7546	−1.45709
$418$	0	0
$419$	−8.45705	−0.413154	−0.206577	−	0.978430i	$-0.566232\pi$
$419$	−8.45705	−0.413154	−0.206577	+	0.978430i	$0.566232\pi$
$420$	0	0
$421$	18.0771	0.881024	0.440512	−	0.897747i	$-0.354797\pi$
$421$	18.0771	0.881024	0.440512	+	0.897747i	$0.354797\pi$
$422$	0	0
$423$	0.221371	0.0107634
$424$	0	0
$425$	6.39339	0.310125
$426$	0	0
$427$	10.6720	0.516457
$428$	0	0
$429$	−5.31638	−0.256677
$430$	0	0
$431$	−8.84546	−0.426071	−0.213035	−	0.977045i	$-0.568335\pi$
$431$	−8.84546	−0.426071	−0.213035	+	0.977045i	$0.568335\pi$
$432$	0	0
$433$	17.5709	0.844402	0.422201	−	0.906502i	$-0.361258\pi$
$433$	17.5709	0.844402	0.422201	+	0.906502i	$0.361258\pi$
$434$	0	0
$435$	−3.45634	−0.165719
$436$	0	0
$437$	15.3682	0.735163
$438$	0	0
$439$	39.1439	1.86824	0.934119	−	0.356961i	$-0.116187\pi$
$439$	39.1439	1.86824	0.934119	+	0.356961i	$0.116187\pi$
$440$	0	0
$441$	−0.0323972	−0.00154273
$442$	0	0
$443$	33.1911	1.57696	0.788478	−	0.615063i	$-0.210869\pi$
$443$	33.1911	1.57696	0.788478	+	0.615063i	$0.210869\pi$
$444$	0	0
$445$	4.46666	0.211740
$446$	0	0
$447$	16.9433	0.801392
$448$	0	0
$449$	25.3519	1.19643	0.598214	−	0.801336i	$-0.295877\pi$
$449$	25.3519	1.19643	0.598214	+	0.801336i	$0.295877\pi$
$450$	0	0
$451$	18.9382	0.891763
$452$	0	0
$453$	1.77658	0.0834709
$454$	0	0
$455$	−3.41881	−0.160276
$456$	0	0
$457$	−34.8617	−1.63076	−0.815380	−	0.578926i	$-0.803472\pi$
$457$	−34.8617	−1.63076	−0.815380	+	0.578926i	$0.803472\pi$
$458$	0	0
$459$	32.3006	1.50766
$460$	0	0
$461$	11.4912	0.535198	0.267599	−	0.963530i	$-0.413770\pi$
$461$	11.4912	0.535198	0.267599	+	0.963530i	$0.413770\pi$
$462$	0	0
$463$	−21.2178	−0.986074	−0.493037	−	0.870008i	$-0.664113\pi$
$463$	−21.2178	−0.986074	−0.493037	+	0.870008i	$0.664113\pi$
$464$	0	0
$465$	6.42220	0.297823
$466$	0	0
$467$	23.4549	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$467$	23.4549	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$468$	0	0
$469$	18.2231	0.841463
$470$	0	0
$471$	4.52515	0.208508
$472$	0	0
$473$	7.00823	0.322239
$474$	0	0
$475$	7.78113	0.357023
$476$	0	0
$477$	1.10711	0.0506909
$478$	0	0
$479$	−7.49971	−0.342670	−0.171335	−	0.985213i	$-0.554808\pi$
$479$	−7.49971	−0.342670	−0.171335	+	0.985213i	$0.554808\pi$
$480$	0	0
$481$	5.22326	0.238160
$482$	0	0
$483$	−9.14502	−0.416113
$484$	0	0
$485$	−13.7303	−0.623459
$486$	0	0
$487$	17.7308	0.803459	0.401730	−	0.915758i	$-0.368409\pi$
$487$	17.7308	0.803459	0.401730	+	0.915758i	$0.368409\pi$
$488$	0	0
$489$	44.6223	2.01789
$490$	0	0
$491$	−1.77034	−0.0798942	−0.0399471	−	0.999202i	$-0.512719\pi$
$491$	−1.77034	−0.0798942	−0.0399471	+	0.999202i	$0.512719\pi$
$492$	0	0
$493$	12.4384	0.560196
$494$	0	0
$495$	−0.356395	−0.0160188
$496$	0	0
$497$	−2.78959	−0.125130
$498$	0	0
$499$	28.6543	1.28274	0.641372	−	0.767230i	$-0.278366\pi$
$499$	28.6543	1.28274	0.641372	+	0.767230i	$0.278366\pi$
$500$	0	0
$501$	−4.12530	−0.184305
$502$	0	0
$503$	−1.46154	−0.0651669	−0.0325834	−	0.999469i	$-0.510373\pi$
$503$	−1.46154	−0.0651669	−0.0325834	+	0.999469i	$0.510373\pi$
$504$	0	0
$505$	1.21546	0.0540874
$506$	0	0
$507$	20.0385	0.889941
$508$	0	0
$509$	−38.1045	−1.68895	−0.844475	−	0.535595i	$-0.820088\pi$
$509$	−38.1045	−1.68895	−0.844475	+	0.535595i	$0.820088\pi$
$510$	0	0
$511$	12.6701	0.560494
$512$	0	0
$513$	39.3117	1.73565
$514$	0	0
$515$	8.59179	0.378600
$516$	0	0
$517$	−3.23252	−0.142166
$518$	0	0
$519$	−4.72693	−0.207489
$520$	0	0
$521$	1.72108	0.0754019	0.0377010	−	0.999289i	$-0.487997\pi$
$521$	1.72108	0.0754019	0.0377010	+	0.999289i	$0.487997\pi$
$522$	0	0
$523$	−34.4644	−1.50702	−0.753511	−	0.657436i	$-0.771641\pi$
$523$	−34.4644	−1.50702	−0.753511	+	0.657436i	$0.771641\pi$
$524$	0	0
$525$	−4.63023	−0.202080
$526$	0	0
$527$	−23.1117	−1.00676
$528$	0	0
$529$	−19.0991	−0.830396
$530$	0	0
$531$	−0.0934395	−0.00405493
$532$	0	0
$533$	10.8898	0.471687
$534$	0	0
$535$	−1.33349	−0.0576519
$536$	0	0
$537$	−23.2786	−1.00454
$538$	0	0
$539$	0.473073	0.0203767
$540$	0	0
$541$	24.8794	1.06965	0.534824	−	0.844963i	$-0.320378\pi$
$541$	24.8794	1.06965	0.534824	+	0.844963i	$0.320378\pi$
$542$	0	0
$543$	−8.38339	−0.359766
$544$	0	0
$545$	4.25595	0.182305
$546$	0	0
$547$	30.0464	1.28469	0.642346	−	0.766415i	$-0.277961\pi$
$547$	30.0464	1.28469	0.642346	+	0.766415i	$0.277961\pi$
$548$	0	0
$549$	0.639712	0.0273023
$550$	0	0
$551$	15.1382	0.644910
$552$	0	0
$553$	15.5829	0.662653
$554$	0	0
$555$	7.07407	0.300278
$556$	0	0
$557$	−24.7375	−1.04816	−0.524081	−	0.851669i	$-0.675591\pi$
$557$	−24.7375	−1.04816	−0.524081	+	0.851669i	$0.675591\pi$
$558$	0	0
$559$	4.02985	0.170444
$560$	0	0
$561$	25.9114	1.09398
$562$	0	0
$563$	27.2108	1.14680	0.573399	−	0.819277i	$-0.305625\pi$
$563$	27.2108	1.14680	0.573399	+	0.819277i	$0.305625\pi$
$564$	0	0
$565$	−19.3744	−0.815089
$566$	0	0
$567$	−24.6143	−1.03370
$568$	0	0
$569$	4.41551	0.185108	0.0925539	−	0.995708i	$-0.470497\pi$
$569$	4.41551	0.185108	0.0925539	+	0.995708i	$0.470497\pi$
$570$	0	0
$571$	−35.0818	−1.46813	−0.734063	−	0.679081i	$-0.762378\pi$
$571$	−35.0818	−1.46813	−0.734063	+	0.679081i	$0.762378\pi$
$572$	0	0
$573$	−9.78532	−0.408788
$574$	0	0
$575$	1.97507	0.0823660
$576$	0	0
$577$	−28.9006	−1.20315	−0.601575	−	0.798817i	$-0.705460\pi$
$577$	−28.9006	−1.20315	−0.601575	+	0.798817i	$0.705460\pi$
$578$	0	0
$579$	−7.99998	−0.332468
$580$	0	0
$581$	19.3115	0.801177
$582$	0	0
$583$	−16.1663	−0.669538
$584$	0	0
$585$	−0.204933	−0.00847294
$586$	0	0
$587$	30.1795	1.24564	0.622821	−	0.782364i	$-0.285986\pi$
$587$	30.1795	1.24564	0.622821	+	0.782364i	$0.285986\pi$
$588$	0	0
$589$	−28.1282	−1.15900
$590$	0	0
$591$	13.3098	0.547494
$592$	0	0
$593$	30.4806	1.25169	0.625844	−	0.779949i	$-0.284755\pi$
$593$	30.4806	1.25169	0.625844	+	0.779949i	$0.284755\pi$
$594$	0	0
$595$	16.6629	0.683112
$596$	0	0
$597$	10.6965	0.437780
$598$	0	0
$599$	35.8167	1.46343	0.731716	−	0.681610i	$-0.238720\pi$
$599$	35.8167	1.46343	0.731716	+	0.681610i	$0.238720\pi$
$600$	0	0
$601$	35.7172	1.45693	0.728467	−	0.685081i	$-0.240233\pi$
$601$	35.7172	1.45693	0.728467	+	0.685081i	$0.240233\pi$
$602$	0	0
$603$	1.09234	0.0444836
$604$	0	0
$605$	−5.79582	−0.235634
$606$	0	0
$607$	−32.8792	−1.33452	−0.667262	−	0.744823i	$-0.732534\pi$
$607$	−32.8792	−1.33452	−0.667262	+	0.744823i	$0.732534\pi$
$608$	0	0
$609$	−9.00815	−0.365028
$610$	0	0
$611$	−1.85875	−0.0751970
$612$	0	0
$613$	30.2128	1.22028	0.610142	−	0.792292i	$-0.291113\pi$
$613$	30.2128	1.22028	0.610142	+	0.792292i	$0.291113\pi$
$614$	0	0
$615$	14.7484	0.594714
$616$	0	0
$617$	2.83344	0.114070	0.0570349	−	0.998372i	$-0.481835\pi$
$617$	2.83344	0.114070	0.0570349	+	0.998372i	$0.481835\pi$
$618$	0	0
$619$	26.1383	1.05059	0.525293	−	0.850921i	$-0.323956\pi$
$619$	26.1383	1.05059	0.525293	+	0.850921i	$0.323956\pi$
$620$	0	0
$621$	9.97840	0.400419
$622$	0	0
$623$	11.6413	0.466399
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	31.5357	1.25941
$628$	0	0
$629$	−25.4575	−1.01506
$630$	0	0
$631$	−15.3007	−0.609113	−0.304557	−	0.952494i	$-0.598508\pi$
$631$	−15.3007	−0.609113	−0.304557	+	0.952494i	$0.598508\pi$
$632$	0	0
$633$	−45.3256	−1.80153
$634$	0	0
$635$	−9.79746	−0.388800
$636$	0	0
$637$	0.272025	0.0107780
$638$	0	0
$639$	−0.167216	−0.00661495
$640$	0	0
$641$	−9.66862	−0.381887	−0.190944	−	0.981601i	$-0.561155\pi$
$641$	−9.66862	−0.381887	−0.190944	+	0.981601i	$0.561155\pi$
$642$	0	0
$643$	−7.64711	−0.301573	−0.150786	−	0.988566i	$-0.548181\pi$
$643$	−7.64711	−0.301573	−0.150786	+	0.988566i	$0.548181\pi$
$644$	0	0
$645$	5.45779	0.214900
$646$	0	0
$647$	−34.2097	−1.34492	−0.672461	−	0.740133i	$-0.734763\pi$
$647$	−34.2097	−1.34492	−0.672461	+	0.740133i	$0.734763\pi$
$648$	0	0
$649$	1.36443	0.0535585
$650$	0	0
$651$	16.7380	0.656013
$652$	0	0
$653$	−41.8239	−1.63670	−0.818348	−	0.574723i	$-0.805110\pi$
$653$	−41.8239	−1.63670	−0.818348	+	0.574723i	$0.805110\pi$
$654$	0	0
$655$	12.9794	0.507148
$656$	0	0
$657$	0.759484	0.0296303
$658$	0	0
$659$	−4.80142	−0.187037	−0.0935185	−	0.995618i	$-0.529811\pi$
$659$	−4.80142	−0.187037	−0.0935185	+	0.995618i	$0.529811\pi$
$660$	0	0
$661$	−4.53489	−0.176387	−0.0881933	−	0.996103i	$-0.528109\pi$
$661$	−4.53489	−0.176387	−0.0881933	+	0.996103i	$0.528109\pi$
$662$	0	0
$663$	14.8995	0.578648
$664$	0	0
$665$	20.2797	0.786413
$666$	0	0
$667$	3.84250	0.148782
$668$	0	0
$669$	−48.7843	−1.88611
$670$	0	0
$671$	−9.34125	−0.360615
$672$	0	0
$673$	19.3398	0.745494	0.372747	−	0.927933i	$-0.378416\pi$
$673$	19.3398	0.745494	0.372747	+	0.927933i	$0.378416\pi$
$674$	0	0
$675$	5.05218	0.194459
$676$	0	0
$677$	9.39476	0.361070	0.180535	−	0.983569i	$-0.442217\pi$
$677$	9.39476	0.361070	0.180535	+	0.983569i	$0.442217\pi$
$678$	0	0
$679$	−35.7847	−1.37329
$680$	0	0
$681$	−29.3240	−1.12370
$682$	0	0
$683$	13.0122	0.497897	0.248948	−	0.968517i	$-0.419915\pi$
$683$	13.0122	0.497897	0.248948	+	0.968517i	$0.419915\pi$
$684$	0	0
$685$	3.74788	0.143199
$686$	0	0
$687$	−16.8146	−0.641518
$688$	0	0
$689$	−9.29586	−0.354144
$690$	0	0
$691$	−17.5166	−0.666361	−0.333181	−	0.942863i	$-0.608122\pi$
$691$	−17.5166	−0.666361	−0.333181	+	0.942863i	$0.608122\pi$
$692$	0	0
$693$	−0.928861	−0.0352845
$694$	0	0
$695$	16.7483	0.635299
$696$	0	0
$697$	−53.0754	−2.01037
$698$	0	0
$699$	−10.1349	−0.383337
$700$	0	0
$701$	−23.6114	−0.891790	−0.445895	−	0.895085i	$-0.647115\pi$
$701$	−23.6114	−0.891790	−0.445895	+	0.895085i	$0.647115\pi$
$702$	0	0
$703$	−30.9833	−1.16856
$704$	0	0
$705$	−2.51738	−0.0948101
$706$	0	0
$707$	3.16782	0.119138
$708$	0	0
$709$	45.7833	1.71943	0.859714	−	0.510776i	$-0.170642\pi$
$709$	45.7833	1.71943	0.859714	+	0.510776i	$0.170642\pi$
$710$	0	0
$711$	0.934083	0.0350309
$712$	0	0
$713$	−7.13973	−0.267385
$714$	0	0
$715$	2.99249	0.111913
$716$	0	0
$717$	20.1446	0.752313
$718$	0	0
$719$	27.0994	1.01064	0.505318	−	0.862933i	$-0.331375\pi$
$719$	27.0994	1.01064	0.505318	+	0.862933i	$0.331375\pi$
$720$	0	0
$721$	22.3925	0.833940
$722$	0	0
$723$	−29.1484	−1.08404
$724$	0	0
$725$	1.94551	0.0722542
$726$	0	0
$727$	−17.6759	−0.655564	−0.327782	−	0.944753i	$-0.606301\pi$
$727$	−17.6759	−0.655564	−0.327782	+	0.944753i	$0.606301\pi$
$728$	0	0
$729$	25.4513	0.942642
$730$	0	0
$731$	−19.6410	−0.726449
$732$	0	0
$733$	−14.1675	−0.523289	−0.261644	−	0.965164i	$-0.584265\pi$
$733$	−14.1675	−0.523289	−0.261644	+	0.965164i	$0.584265\pi$
$734$	0	0
$735$	0.368414	0.0135892
$736$	0	0
$737$	−15.9507	−0.587550
$738$	0	0
$739$	−32.6833	−1.20227	−0.601137	−	0.799146i	$-0.705286\pi$
$739$	−32.6833	−1.20227	−0.601137	+	0.799146i	$0.705286\pi$
$740$	0	0
$741$	18.1335	0.666152
$742$	0	0
$743$	−32.1033	−1.17776	−0.588878	−	0.808222i	$-0.700430\pi$
$743$	−32.1033	−1.17776	−0.588878	+	0.808222i	$0.700430\pi$
$744$	0	0
$745$	−9.53706	−0.349411
$746$	0	0
$747$	1.15759	0.0423539
$748$	0	0
$749$	−3.47544	−0.126990
$750$	0	0
$751$	11.9149	0.434780	0.217390	−	0.976085i	$-0.430246\pi$
$751$	11.9149	0.434780	0.217390	+	0.976085i	$0.430246\pi$
$752$	0	0
$753$	−29.6590	−1.08083
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−30.1553	−1.09601	−0.548006	−	0.836474i	$-0.684613\pi$
$757$	−30.1553	−1.09601	−0.548006	+	0.836474i	$0.684613\pi$
$758$	0	0
$759$	8.00464	0.290550
$760$	0	0
$761$	34.7183	1.25854	0.629269	−	0.777187i	$-0.283354\pi$
$761$	34.7183	1.25854	0.629269	+	0.777187i	$0.283354\pi$
$762$	0	0
$763$	11.0921	0.401563
$764$	0	0
$765$	0.998820	0.0361124
$766$	0	0
$767$	0.784568	0.0283291
$768$	0	0
$769$	−4.83288	−0.174278	−0.0871391	−	0.996196i	$-0.527772\pi$
$769$	−4.83288	−0.174278	−0.0871391	+	0.996196i	$0.527772\pi$
$770$	0	0
$771$	2.81496	0.101378
$772$	0	0
$773$	−17.2308	−0.619749	−0.309874	−	0.950778i	$-0.600287\pi$
$773$	−17.2308	−0.619749	−0.309874	+	0.950778i	$0.600287\pi$
$774$	0	0
$775$	−3.61493	−0.129852
$776$	0	0
$777$	18.4369	0.661421
$778$	0	0
$779$	−64.5958	−2.31438
$780$	0	0
$781$	2.44173	0.0873719
$782$	0	0
$783$	9.82905	0.351262
$784$	0	0
$785$	−2.54712	−0.0909105
$786$	0	0
$787$	−31.1552	−1.11056	−0.555281	−	0.831663i	$-0.687389\pi$
$787$	−31.1552	−1.11056	−0.555281	+	0.831663i	$0.687389\pi$
$788$	0	0
$789$	−12.8565	−0.457702
$790$	0	0
$791$	−50.4949	−1.79539
$792$	0	0
$793$	−5.37137	−0.190743
$794$	0	0
$795$	−12.5898	−0.446513
$796$	0	0
$797$	−36.2245	−1.28314	−0.641569	−	0.767065i	$-0.721716\pi$
$797$	−36.2245	−1.28314	−0.641569	+	0.767065i	$0.721716\pi$
$798$	0	0
$799$	9.05933	0.320496
$800$	0	0
$801$	0.697813	0.0246560
$802$	0	0
$803$	−11.0902	−0.391364
$804$	0	0
$805$	5.14755	0.181427
$806$	0	0
$807$	−40.6499	−1.43094
$808$	0	0
$809$	−11.6819	−0.410713	−0.205356	−	0.978687i	$-0.565835\pi$
$809$	−11.6819	−0.410713	−0.205356	+	0.978687i	$0.565835\pi$
$810$	0	0
$811$	−22.7055	−0.797298	−0.398649	−	0.917104i	$-0.630521\pi$
$811$	−22.7055	−0.797298	−0.398649	+	0.917104i	$0.630521\pi$
$812$	0	0
$813$	11.1699	0.391747
$814$	0	0
$815$	−25.1170	−0.879810
$816$	0	0
$817$	−23.9042	−0.836303
$818$	0	0
$819$	−0.534110	−0.0186633
$820$	0	0
$821$	38.9704	1.36008	0.680038	−	0.733177i	$-0.261963\pi$
$821$	38.9704	1.36008	0.680038	+	0.733177i	$0.261963\pi$
$822$	0	0
$823$	−29.1324	−1.01549	−0.507747	−	0.861506i	$-0.669521\pi$
$823$	−29.1324	−1.01549	−0.507747	+	0.861506i	$0.669521\pi$
$824$	0	0
$825$	4.05285	0.141102
$826$	0	0
$827$	16.7134	0.581182	0.290591	−	0.956847i	$-0.406148\pi$
$827$	16.7134	0.581182	0.290591	+	0.956847i	$0.406148\pi$
$828$	0	0
$829$	47.9687	1.66602	0.833012	−	0.553255i	$-0.186615\pi$
$829$	47.9687	1.66602	0.833012	+	0.553255i	$0.186615\pi$
$830$	0	0
$831$	1.21951	0.0423045
$832$	0	0
$833$	−1.32582	−0.0459368
$834$	0	0
$835$	2.32205	0.0803578
$836$	0	0
$837$	−18.2633	−0.631272
$838$	0	0
$839$	26.3673	0.910301	0.455151	−	0.890414i	$-0.349585\pi$
$839$	26.3673	0.910301	0.455151	+	0.890414i	$0.349585\pi$
$840$	0	0
$841$	−25.2150	−0.869483
$842$	0	0
$843$	19.5090	0.671925
$844$	0	0
$845$	−11.2793	−0.388019
$846$	0	0
$847$	−15.1055	−0.519030
$848$	0	0
$849$	41.4045	1.42100
$850$	0	0
$851$	−7.86442	−0.269589
$852$	0	0
$853$	12.4866	0.427534	0.213767	−	0.976885i	$-0.431427\pi$
$853$	12.4866	0.427534	0.213767	+	0.976885i	$0.431427\pi$
$854$	0	0
$855$	1.21562	0.0415734
$856$	0	0
$857$	−11.6568	−0.398189	−0.199094	−	0.979980i	$-0.563800\pi$
$857$	−11.6568	−0.398189	−0.199094	+	0.979980i	$0.563800\pi$
$858$	0	0
$859$	−22.9909	−0.784439	−0.392219	−	0.919872i	$-0.628293\pi$
$859$	−22.9909	−0.784439	−0.392219	+	0.919872i	$0.628293\pi$
$860$	0	0
$861$	38.4384	1.30998
$862$	0	0
$863$	−54.9248	−1.86966	−0.934831	−	0.355092i	$-0.884449\pi$
$863$	−54.9248	−1.86966	−0.934831	+	0.355092i	$0.884449\pi$
$864$	0	0
$865$	2.66069	0.0904663
$866$	0	0
$867$	−42.4166	−1.44054
$868$	0	0
$869$	−13.6397	−0.462696
$870$	0	0
$871$	−9.17188	−0.310777
$872$	0	0
$873$	−2.14504	−0.0725985
$874$	0	0
$875$	2.60627	0.0881079
$876$	0	0
$877$	−14.8662	−0.501997	−0.250999	−	0.967987i	$-0.580759\pi$
$877$	−14.8662	−0.501997	−0.250999	+	0.967987i	$0.580759\pi$
$878$	0	0
$879$	−19.5883	−0.660696
$880$	0	0
$881$	−17.2001	−0.579487	−0.289744	−	0.957104i	$-0.593570\pi$
$881$	−17.2001	−0.579487	−0.289744	+	0.957104i	$0.593570\pi$
$882$	0	0
$883$	−20.2709	−0.682171	−0.341085	−	0.940032i	$-0.610795\pi$
$883$	−20.2709	−0.682171	−0.341085	+	0.940032i	$0.610795\pi$
$884$	0	0
$885$	1.06257	0.0357180
$886$	0	0
$887$	−23.7899	−0.798786	−0.399393	−	0.916780i	$-0.630779\pi$
$887$	−23.7899	−0.798786	−0.399393	+	0.916780i	$0.630779\pi$
$888$	0	0
$889$	−25.5348	−0.856410
$890$	0	0
$891$	21.5449	0.721782
$892$	0	0
$893$	11.0257	0.368962
$894$	0	0
$895$	13.1030	0.437986
$896$	0	0
$897$	4.60280	0.153683
$898$	0	0
$899$	−7.03287	−0.234559
$900$	0	0
$901$	45.3069	1.50939
$902$	0	0
$903$	14.2244	0.473360
$904$	0	0
$905$	4.71884	0.156860
$906$	0	0
$907$	−0.962436	−0.0319572	−0.0159786	−	0.999872i	$-0.505086\pi$
$907$	−0.962436	−0.0319572	−0.0159786	+	0.999872i	$0.505086\pi$
$908$	0	0
$909$	0.189888	0.00629820
$910$	0	0
$911$	−11.4227	−0.378450	−0.189225	−	0.981934i	$-0.560598\pi$
$911$	−11.4227	−0.378450	−0.189225	+	0.981934i	$0.560598\pi$
$912$	0	0
$913$	−16.9034	−0.559421
$914$	0	0
$915$	−7.27466	−0.240493
$916$	0	0
$917$	33.8278	1.11709
$918$	0	0
$919$	50.6472	1.67070	0.835348	−	0.549721i	$-0.185266\pi$
$919$	50.6472	1.67070	0.835348	+	0.549721i	$0.185266\pi$
$920$	0	0
$921$	−36.2100	−1.19316
$922$	0	0
$923$	1.40403	0.0462143
$924$	0	0
$925$	−3.98185	−0.130923
$926$	0	0
$927$	1.34227	0.0440859
$928$	0	0
$929$	44.2876	1.45303	0.726515	−	0.687151i	$-0.241139\pi$
$929$	44.2876	1.45303	0.726515	+	0.687151i	$0.241139\pi$
$930$	0	0
$931$	−1.61359	−0.0528834
$932$	0	0
$933$	−15.6874	−0.513583
$934$	0	0
$935$	−14.5850	−0.476981
$936$	0	0
$937$	2.81875	0.0920847	0.0460423	−	0.998939i	$-0.485339\pi$
$937$	2.81875	0.0920847	0.0460423	+	0.998939i	$0.485339\pi$
$938$	0	0
$939$	24.6937	0.805848
$940$	0	0
$941$	31.6578	1.03201	0.516007	−	0.856584i	$-0.327418\pi$
$941$	31.6578	1.03201	0.516007	+	0.856584i	$0.327418\pi$
$942$	0	0
$943$	−16.3962	−0.533934
$944$	0	0
$945$	13.1673	0.428333
$946$	0	0
$947$	−44.3073	−1.43979	−0.719896	−	0.694081i	$-0.755811\pi$
$947$	−44.3073	−1.43979	−0.719896	+	0.694081i	$0.755811\pi$
$948$	0	0
$949$	−6.37704	−0.207007
$950$	0	0
$951$	−9.08412	−0.294573
$952$	0	0
$953$	−9.15817	−0.296662	−0.148331	−	0.988938i	$-0.547390\pi$
$953$	−9.15817	−0.296662	−0.148331	+	0.988938i	$0.547390\pi$
$954$	0	0
$955$	5.50796	0.178233
$956$	0	0
$957$	7.88483	0.254881
$958$	0	0
$959$	9.76798	0.315424
$960$	0	0
$961$	−17.9323	−0.578460
$962$	0	0
$963$	−0.208327	−0.00671326
$964$	0	0
$965$	4.50303	0.144958
$966$	0	0
$967$	13.9811	0.449601	0.224801	−	0.974405i	$-0.427827\pi$
$967$	13.9811	0.449601	0.224801	+	0.974405i	$0.427827\pi$
$968$	0	0
$969$	−88.3808	−2.83920
$970$	0	0
$971$	−17.9393	−0.575699	−0.287850	−	0.957676i	$-0.592940\pi$
$971$	−17.9393	−0.575699	−0.287850	+	0.957676i	$0.592940\pi$
$972$	0	0
$973$	43.6505	1.39937
$974$	0	0
$975$	2.33045	0.0746342
$976$	0	0
$977$	3.22398	0.103144	0.0515722	−	0.998669i	$-0.483577\pi$
$977$	3.22398	0.103144	0.0515722	+	0.998669i	$0.483577\pi$
$978$	0	0
$979$	−10.1896	−0.325663
$980$	0	0
$981$	0.664894	0.0212284
$982$	0	0
$983$	41.8529	1.33490	0.667450	−	0.744655i	$-0.267386\pi$
$983$	41.8529	1.33490	0.667450	+	0.744655i	$0.267386\pi$
$984$	0	0
$985$	−7.49185	−0.238710
$986$	0	0
$987$	−6.56097	−0.208838
$988$	0	0
$989$	−6.06756	−0.192937
$990$	0	0
$991$	27.4828	0.873021	0.436510	−	0.899699i	$-0.356214\pi$
$991$	27.4828	0.873021	0.436510	+	0.899699i	$0.356214\pi$
$992$	0	0
$993$	10.9481	0.347427
$994$	0	0
$995$	−6.02087	−0.190874
$996$	0	0
$997$	27.2188	0.862029	0.431014	−	0.902345i	$-0.358156\pi$
$997$	27.2188	0.862029	0.431014	+	0.902345i	$0.358156\pi$
$998$	0	0
$999$	−20.1170	−0.636475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.3	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.3	✓	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.77658 q^{3} +1.00000 q^{5} +2.60627 q^{7} +0.156227 q^{9} +O(q^{10})\)	\(q-1.77658 q^{3} +1.00000 q^{5} +2.60627 q^{7} +0.156227 q^{9} -2.28127 q^{11} -1.31177 q^{13} -1.77658 q^{15} +6.39339 q^{17} +7.78113 q^{19} -4.63023 q^{21} +1.97507 q^{23} +1.00000 q^{25} +5.05218 q^{27} +1.94551 q^{29} -3.61493 q^{31} +4.05285 q^{33} +2.60627 q^{35} -3.98185 q^{37} +2.33045 q^{39} -8.30160 q^{41} -3.07208 q^{43} +0.156227 q^{45} +1.41698 q^{47} -0.207373 q^{49} -11.3584 q^{51} +7.08653 q^{53} -2.28127 q^{55} -13.8238 q^{57} -0.598101 q^{59} +4.09476 q^{61} +0.407169 q^{63} -1.31177 q^{65} +6.99202 q^{67} -3.50886 q^{69} -1.07034 q^{71} +4.86142 q^{73} -1.77658 q^{75} -5.94559 q^{77} +5.97901 q^{79} -9.44427 q^{81} +7.40965 q^{83} +6.39339 q^{85} -3.45634 q^{87} +4.46666 q^{89} -3.41881 q^{91} +6.42220 q^{93} +7.78113 q^{95} -13.7303 q^{97} -0.356395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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