LMFDB - Embedded newform 6040.2.a.o.1.10

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			1.10
Root			$1.18532$ 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)

$f(q)$	$=$	$q+1.18532 q^{3} +1.00000 q^{5} +1.34515 q^{7} -1.59502 q^{9} +O(q^{10})$	$q+1.18532 q^{3} +1.00000 q^{5} +1.34515 q^{7} -1.59502 q^{9} +4.11264 q^{11} +4.12181 q^{13} +1.18532 q^{15} -0.563717 q^{17} +3.58926 q^{19} +1.59443 q^{21} -7.93423 q^{23} +1.00000 q^{25} -5.44656 q^{27} -0.950537 q^{29} +5.09775 q^{31} +4.87478 q^{33} +1.34515 q^{35} -0.121735 q^{37} +4.88566 q^{39} +7.89513 q^{41} -1.45064 q^{43} -1.59502 q^{45} +7.26073 q^{47} -5.19057 q^{49} -0.668184 q^{51} +7.53966 q^{53} +4.11264 q^{55} +4.25441 q^{57} +8.56831 q^{59} -0.171816 q^{61} -2.14554 q^{63} +4.12181 q^{65} -0.197552 q^{67} -9.40460 q^{69} +1.03696 q^{71} +12.3068 q^{73} +1.18532 q^{75} +5.53212 q^{77} +4.90379 q^{79} -1.67085 q^{81} -15.3349 q^{83} -0.563717 q^{85} -1.12669 q^{87} +0.435248 q^{89} +5.54446 q^{91} +6.04246 q^{93} +3.58926 q^{95} -10.3461 q^{97} -6.55973 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.18532	0.684344	0.342172	−	0.939637i	$-0.388837\pi$
$3$	1.18532	0.684344	0.342172	+	0.939637i	$0.388837\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.34515	0.508419	0.254210	−	0.967149i	$-0.418185\pi$
$7$	1.34515	0.508419	0.254210	+	0.967149i	$0.418185\pi$
$8$	0	0
$9$	−1.59502	−0.531673
$10$	0	0
$11$	4.11264	1.24001	0.620003	−	0.784599i	$-0.287131\pi$
$11$	4.11264	1.24001	0.620003	+	0.784599i	$0.287131\pi$
$12$	0	0
$13$	4.12181	1.14319	0.571593	−	0.820537i	$-0.306326\pi$
$13$	4.12181	1.14319	0.571593	+	0.820537i	$0.306326\pi$
$14$	0	0
$15$	1.18532	0.306048
$16$	0	0
$17$	−0.563717	−0.136721	−0.0683607	−	0.997661i	$-0.521777\pi$
$17$	−0.563717	−0.136721	−0.0683607	+	0.997661i	$0.521777\pi$
$18$	0	0
$19$	3.58926	0.823432	0.411716	−	0.911312i	$-0.364930\pi$
$19$	3.58926	0.823432	0.411716	+	0.911312i	$0.364930\pi$
$20$	0	0
$21$	1.59443	0.347934
$22$	0	0
$23$	−7.93423	−1.65440	−0.827201	−	0.561906i	$-0.810068\pi$
$23$	−7.93423	−1.65440	−0.827201	+	0.561906i	$0.810068\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.44656	−1.04819
$28$	0	0
$29$	−0.950537	−0.176510	−0.0882551	−	0.996098i	$-0.528129\pi$
$29$	−0.950537	−0.176510	−0.0882551	+	0.996098i	$0.528129\pi$
$30$	0	0
$31$	5.09775	0.915583	0.457791	−	0.889060i	$-0.348641\pi$
$31$	5.09775	0.915583	0.457791	+	0.889060i	$0.348641\pi$
$32$	0	0
$33$	4.87478	0.848591
$34$	0	0
$35$	1.34515	0.227372
$36$	0	0
$37$	−0.121735	−0.0200131	−0.0100065	−	0.999950i	$-0.503185\pi$
$37$	−0.121735	−0.0200131	−0.0100065	+	0.999950i	$0.503185\pi$
$38$	0	0
$39$	4.88566	0.782332
$40$	0	0
$41$	7.89513	1.23301	0.616506	−	0.787350i	$-0.288548\pi$
$41$	7.89513	1.23301	0.616506	+	0.787350i	$0.288548\pi$
$42$	0	0
$43$	−1.45064	−0.221221	−0.110610	−	0.993864i	$-0.535281\pi$
$43$	−1.45064	−0.221221	−0.110610	+	0.993864i	$0.535281\pi$
$44$	0	0
$45$	−1.59502	−0.237771
$46$	0	0
$47$	7.26073	1.05909	0.529543	−	0.848283i	$-0.322363\pi$
$47$	7.26073	1.05909	0.529543	+	0.848283i	$0.322363\pi$
$48$	0	0
$49$	−5.19057	−0.741510
$50$	0	0
$51$	−0.668184	−0.0935645
$52$	0	0
$53$	7.53966	1.03565	0.517826	−	0.855486i	$-0.326741\pi$
$53$	7.53966	1.03565	0.517826	+	0.855486i	$0.326741\pi$
$54$	0	0
$55$	4.11264	0.554548
$56$	0	0
$57$	4.25441	0.563511
$58$	0	0
$59$	8.56831	1.11550	0.557750	−	0.830009i	$-0.311665\pi$
$59$	8.56831	1.11550	0.557750	+	0.830009i	$0.311665\pi$
$60$	0	0
$61$	−0.171816	−0.0219987	−0.0109994	−	0.999940i	$-0.503501\pi$
$61$	−0.171816	−0.0219987	−0.0109994	+	0.999940i	$0.503501\pi$
$62$	0	0
$63$	−2.14554	−0.270313
$64$	0	0
$65$	4.12181	0.511248
$66$	0	0
$67$	−0.197552	−0.0241348	−0.0120674	−	0.999927i	$-0.503841\pi$
$67$	−0.197552	−0.0241348	−0.0120674	+	0.999927i	$0.503841\pi$
$68$	0	0
$69$	−9.40460	−1.13218
$70$	0	0
$71$	1.03696	0.123064	0.0615322	−	0.998105i	$-0.480401\pi$
$71$	1.03696	0.123064	0.0615322	+	0.998105i	$0.480401\pi$
$72$	0	0
$73$	12.3068	1.44040	0.720200	−	0.693767i	$-0.244050\pi$
$73$	12.3068	1.44040	0.720200	+	0.693767i	$0.244050\pi$
$74$	0	0
$75$	1.18532	0.136869
$76$	0	0
$77$	5.53212	0.630443
$78$	0	0
$79$	4.90379	0.551719	0.275860	−	0.961198i	$-0.411038\pi$
$79$	4.90379	0.551719	0.275860	+	0.961198i	$0.411038\pi$
$80$	0	0
$81$	−1.67085	−0.185651
$82$	0	0
$83$	−15.3349	−1.68322	−0.841610	−	0.540085i	$-0.818392\pi$
$83$	−15.3349	−1.68322	−0.841610	+	0.540085i	$0.818392\pi$
$84$	0	0
$85$	−0.563717	−0.0611437
$86$	0	0
$87$	−1.12669	−0.120794
$88$	0	0
$89$	0.435248	0.0461362	0.0230681	−	0.999734i	$-0.492657\pi$
$89$	0.435248	0.0461362	0.0230681	+	0.999734i	$0.492657\pi$
$90$	0	0
$91$	5.54446	0.581218
$92$	0	0
$93$	6.04246	0.626574
$94$	0	0
$95$	3.58926	0.368250
$96$	0	0
$97$	−10.3461	−1.05049	−0.525246	−	0.850950i	$-0.676027\pi$
$97$	−10.3461	−1.05049	−0.525246	+	0.850950i	$0.676027\pi$
$98$	0	0
$99$	−6.55973	−0.659278
$100$	0	0
$101$	8.63328	0.859043	0.429522	−	0.903057i	$-0.358682\pi$
$101$	8.63328	0.859043	0.429522	+	0.903057i	$0.358682\pi$
$102$	0	0
$103$	−14.2628	−1.40536	−0.702678	−	0.711508i	$-0.748012\pi$
$103$	−14.2628	−1.40536	−0.702678	+	0.711508i	$0.748012\pi$
$104$	0	0
$105$	1.59443	0.155601
$106$	0	0
$107$	6.79607	0.657001	0.328500	−	0.944504i	$-0.393457\pi$
$107$	6.79607	0.657001	0.328500	+	0.944504i	$0.393457\pi$
$108$	0	0
$109$	4.47452	0.428582	0.214291	−	0.976770i	$-0.431256\pi$
$109$	4.47452	0.428582	0.214291	+	0.976770i	$0.431256\pi$
$110$	0	0
$111$	−0.144295	−0.0136958
$112$	0	0
$113$	−11.2776	−1.06090	−0.530452	−	0.847715i	$-0.677978\pi$
$113$	−11.2776	−1.06090	−0.530452	+	0.847715i	$0.677978\pi$
$114$	0	0
$115$	−7.93423	−0.739871
$116$	0	0
$117$	−6.57437	−0.607801
$118$	0	0
$119$	−0.758284	−0.0695118
$120$	0	0
$121$	5.91378	0.537616
$122$	0	0
$123$	9.35824	0.843804
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.9890	−1.24133	−0.620663	−	0.784077i	$-0.713137\pi$
$127$	−13.9890	−1.24133	−0.620663	+	0.784077i	$0.713137\pi$
$128$	0	0
$129$	−1.71947	−0.151391
$130$	0	0
$131$	18.3595	1.60408	0.802038	−	0.597273i	$-0.203749\pi$
$131$	18.3595	1.60408	0.802038	+	0.597273i	$0.203749\pi$
$132$	0	0
$133$	4.82809	0.418649
$134$	0	0
$135$	−5.44656	−0.468765
$136$	0	0
$137$	13.0376	1.11388	0.556940	−	0.830553i	$-0.311975\pi$
$137$	13.0376	1.11388	0.556940	+	0.830553i	$0.311975\pi$
$138$	0	0
$139$	−21.8540	−1.85363	−0.926816	−	0.375515i	$-0.877466\pi$
$139$	−21.8540	−1.85363	−0.926816	+	0.375515i	$0.877466\pi$
$140$	0	0
$141$	8.60628	0.724780
$142$	0	0
$143$	16.9515	1.41756
$144$	0	0
$145$	−0.950537	−0.0789378
$146$	0	0
$147$	−6.15248	−0.507448
$148$	0	0
$149$	−12.0523	−0.987364	−0.493682	−	0.869642i	$-0.664349\pi$
$149$	−12.0523	−0.987364	−0.493682	+	0.869642i	$0.664349\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	0.899139	0.0726911
$154$	0	0
$155$	5.09775	0.409461
$156$	0	0
$157$	−3.02343	−0.241296	−0.120648	−	0.992695i	$-0.538497\pi$
$157$	−3.02343	−0.241296	−0.120648	+	0.992695i	$0.538497\pi$
$158$	0	0
$159$	8.93691	0.708743
$160$	0	0
$161$	−10.6727	−0.841130
$162$	0	0
$163$	3.25576	0.255010	0.127505	−	0.991838i	$-0.459303\pi$
$163$	3.25576	0.255010	0.127505	+	0.991838i	$0.459303\pi$
$164$	0	0
$165$	4.87478	0.379501
$166$	0	0
$167$	16.6737	1.29025	0.645125	−	0.764077i	$-0.276805\pi$
$167$	16.6737	1.29025	0.645125	+	0.764077i	$0.276805\pi$
$168$	0	0
$169$	3.98935	0.306873
$170$	0	0
$171$	−5.72493	−0.437797
$172$	0	0
$173$	−15.3922	−1.17025	−0.585125	−	0.810943i	$-0.698954\pi$
$173$	−15.3922	−1.17025	−0.585125	+	0.810943i	$0.698954\pi$
$174$	0	0
$175$	1.34515	0.101684
$176$	0	0
$177$	10.1562	0.763386
$178$	0	0
$179$	−3.57998	−0.267580	−0.133790	−	0.991010i	$-0.542715\pi$
$179$	−3.57998	−0.267580	−0.133790	+	0.991010i	$0.542715\pi$
$180$	0	0
$181$	−8.40958	−0.625079	−0.312540	−	0.949905i	$-0.601180\pi$
$181$	−8.40958	−0.625079	−0.312540	+	0.949905i	$0.601180\pi$
$182$	0	0
$183$	−0.203656	−0.0150547
$184$	0	0
$185$	−0.121735	−0.00895012
$186$	0	0
$187$	−2.31836	−0.169535
$188$	0	0
$189$	−7.32645	−0.532921
$190$	0	0
$191$	20.0463	1.45050	0.725249	−	0.688487i	$-0.241725\pi$
$191$	20.0463	1.45050	0.725249	+	0.688487i	$0.241725\pi$
$192$	0	0
$193$	20.3315	1.46350	0.731748	−	0.681575i	$-0.238705\pi$
$193$	20.3315	1.46350	0.731748	+	0.681575i	$0.238705\pi$
$194$	0	0
$195$	4.88566	0.349870
$196$	0	0
$197$	−11.2001	−0.797971	−0.398986	−	0.916957i	$-0.630638\pi$
$197$	−11.2001	−0.797971	−0.398986	+	0.916957i	$0.630638\pi$
$198$	0	0
$199$	15.1371	1.07304	0.536520	−	0.843887i	$-0.319738\pi$
$199$	15.1371	1.07304	0.536520	+	0.843887i	$0.319738\pi$
$200$	0	0
$201$	−0.234162	−0.0165165
$202$	0	0
$203$	−1.27862	−0.0897412
$204$	0	0
$205$	7.89513	0.551420
$206$	0	0
$207$	12.6553	0.879601
$208$	0	0
$209$	14.7613	1.02106
$210$	0	0
$211$	−14.1103	−0.971391	−0.485696	−	0.874128i	$-0.661434\pi$
$211$	−14.1103	−0.971391	−0.485696	+	0.874128i	$0.661434\pi$
$212$	0	0
$213$	1.22913	0.0842183
$214$	0	0
$215$	−1.45064	−0.0989329
$216$	0	0
$217$	6.85724	0.465500
$218$	0	0
$219$	14.5875	0.985729
$220$	0	0
$221$	−2.32354	−0.156298
$222$	0	0
$223$	20.8350	1.39522	0.697609	−	0.716479i	$-0.254247\pi$
$223$	20.8350	1.39522	0.697609	+	0.716479i	$0.254247\pi$
$224$	0	0
$225$	−1.59502	−0.106335
$226$	0	0
$227$	−6.28146	−0.416915	−0.208457	−	0.978031i	$-0.566844\pi$
$227$	−6.28146	−0.416915	−0.208457	+	0.978031i	$0.566844\pi$
$228$	0	0
$229$	−7.09454	−0.468820	−0.234410	−	0.972138i	$-0.575316\pi$
$229$	−7.09454	−0.468820	−0.234410	+	0.972138i	$0.575316\pi$
$230$	0	0
$231$	6.55732	0.431440
$232$	0	0
$233$	−0.388849	−0.0254743	−0.0127372	−	0.999919i	$-0.504054\pi$
$233$	−0.388849	−0.0254743	−0.0127372	+	0.999919i	$0.504054\pi$
$234$	0	0
$235$	7.26073	0.473638
$236$	0	0
$237$	5.81255	0.377566
$238$	0	0
$239$	4.19458	0.271325	0.135663	−	0.990755i	$-0.456684\pi$
$239$	4.19458	0.271325	0.135663	+	0.990755i	$0.456684\pi$
$240$	0	0
$241$	5.03199	0.324139	0.162070	−	0.986779i	$-0.448183\pi$
$241$	5.03199	0.324139	0.162070	+	0.986779i	$0.448183\pi$
$242$	0	0
$243$	14.3592	0.921143
$244$	0	0
$245$	−5.19057	−0.331613
$246$	0	0
$247$	14.7942	0.941335
$248$	0	0
$249$	−18.1767	−1.15190
$250$	0	0
$251$	2.81441	0.177644	0.0888221	−	0.996048i	$-0.471690\pi$
$251$	2.81441	0.177644	0.0888221	+	0.996048i	$0.471690\pi$
$252$	0	0
$253$	−32.6306	−2.05147
$254$	0	0
$255$	−0.668184	−0.0418433
$256$	0	0
$257$	30.9268	1.92916	0.964580	−	0.263789i	$-0.0849723\pi$
$257$	30.9268	1.92916	0.964580	+	0.263789i	$0.0849723\pi$
$258$	0	0
$259$	−0.163752	−0.0101750
$260$	0	0
$261$	1.51612	0.0938457
$262$	0	0
$263$	23.2812	1.43558	0.717791	−	0.696259i	$-0.245153\pi$
$263$	23.2812	1.43558	0.717791	+	0.696259i	$0.245153\pi$
$264$	0	0
$265$	7.53966	0.463158
$266$	0	0
$267$	0.515908	0.0315731
$268$	0	0
$269$	2.53839	0.154768	0.0773842	−	0.997001i	$-0.475343\pi$
$269$	2.53839	0.154768	0.0773842	+	0.997001i	$0.475343\pi$
$270$	0	0
$271$	−11.6995	−0.710694	−0.355347	−	0.934734i	$-0.615637\pi$
$271$	−11.6995	−0.710694	−0.355347	+	0.934734i	$0.615637\pi$
$272$	0	0
$273$	6.57195	0.397753
$274$	0	0
$275$	4.11264	0.248001
$276$	0	0
$277$	−1.51594	−0.0910841	−0.0455420	−	0.998962i	$-0.514501\pi$
$277$	−1.51594	−0.0910841	−0.0455420	+	0.998962i	$0.514501\pi$
$278$	0	0
$279$	−8.13101	−0.486791
$280$	0	0
$281$	−4.70596	−0.280734	−0.140367	−	0.990100i	$-0.544828\pi$
$281$	−4.70596	−0.280734	−0.140367	+	0.990100i	$0.544828\pi$
$282$	0	0
$283$	17.8545	1.06134	0.530671	−	0.847578i	$-0.321940\pi$
$283$	17.8545	1.06134	0.530671	+	0.847578i	$0.321940\pi$
$284$	0	0
$285$	4.25441	0.252010
$286$	0	0
$287$	10.6201	0.626887
$288$	0	0
$289$	−16.6822	−0.981307
$290$	0	0
$291$	−12.2635	−0.718898
$292$	0	0
$293$	5.39898	0.315412	0.157706	−	0.987486i	$-0.449590\pi$
$293$	5.39898	0.315412	0.157706	+	0.987486i	$0.449590\pi$
$294$	0	0
$295$	8.56831	0.498867
$296$	0	0
$297$	−22.3997	−1.29976
$298$	0	0
$299$	−32.7034	−1.89129
$300$	0	0
$301$	−1.95133	−0.112473
$302$	0	0
$303$	10.2332	0.587881
$304$	0	0
$305$	−0.171816	−0.00983814
$306$	0	0
$307$	32.6086	1.86107	0.930535	−	0.366202i	$-0.119342\pi$
$307$	32.6086	1.86107	0.930535	+	0.366202i	$0.119342\pi$
$308$	0	0
$309$	−16.9060	−0.961747
$310$	0	0
$311$	4.51834	0.256211	0.128106	−	0.991761i	$-0.459110\pi$
$311$	4.51834	0.256211	0.128106	+	0.991761i	$0.459110\pi$
$312$	0	0
$313$	7.44759	0.420962	0.210481	−	0.977598i	$-0.432497\pi$
$313$	7.44759	0.420962	0.210481	+	0.977598i	$0.432497\pi$
$314$	0	0
$315$	−2.14554	−0.120888
$316$	0	0
$317$	6.11300	0.343340	0.171670	−	0.985155i	$-0.445084\pi$
$317$	6.11300	0.343340	0.171670	+	0.985155i	$0.445084\pi$
$318$	0	0
$319$	−3.90921	−0.218874
$320$	0	0
$321$	8.05551	0.449615
$322$	0	0
$323$	−2.02332	−0.112581
$324$	0	0
$325$	4.12181	0.228637
$326$	0	0
$327$	5.30374	0.293297
$328$	0	0
$329$	9.76678	0.538460
$330$	0	0
$331$	24.1423	1.32698	0.663490	−	0.748186i	$-0.269075\pi$
$331$	24.1423	1.32698	0.663490	+	0.748186i	$0.269075\pi$
$332$	0	0
$333$	0.194169	0.0106404
$334$	0	0
$335$	−0.197552	−0.0107934
$336$	0	0
$337$	−28.9387	−1.57639	−0.788195	−	0.615426i	$-0.788984\pi$
$337$	−28.9387	−1.57639	−0.788195	+	0.615426i	$0.788984\pi$
$338$	0	0
$339$	−13.3675	−0.726024
$340$	0	0
$341$	20.9652	1.13533
$342$	0	0
$343$	−16.3982	−0.885417
$344$	0	0
$345$	−9.40460	−0.506326
$346$	0	0
$347$	−24.9397	−1.33883	−0.669417	−	0.742887i	$-0.733456\pi$
$347$	−24.9397	−1.33883	−0.669417	+	0.742887i	$0.733456\pi$
$348$	0	0
$349$	2.37376	0.127064	0.0635322	−	0.997980i	$-0.479763\pi$
$349$	2.37376	0.127064	0.0635322	+	0.997980i	$0.479763\pi$
$350$	0	0
$351$	−22.4497	−1.19828
$352$	0	0
$353$	−0.137377	−0.00731183	−0.00365592	−	0.999993i	$-0.501164\pi$
$353$	−0.137377	−0.00731183	−0.00365592	+	0.999993i	$0.501164\pi$
$354$	0	0
$355$	1.03696	0.0550360
$356$	0	0
$357$	−0.898808	−0.0475700
$358$	0	0
$359$	18.8335	0.993995	0.496997	−	0.867752i	$-0.334436\pi$
$359$	18.8335	0.993995	0.496997	+	0.867752i	$0.334436\pi$
$360$	0	0
$361$	−6.11724	−0.321960
$362$	0	0
$363$	7.00971	0.367914
$364$	0	0
$365$	12.3068	0.644166
$366$	0	0
$367$	−11.4082	−0.595502	−0.297751	−	0.954644i	$-0.596237\pi$
$367$	−11.4082	−0.595502	−0.297751	+	0.954644i	$0.596237\pi$
$368$	0	0
$369$	−12.5929	−0.655559
$370$	0	0
$371$	10.1420	0.526546
$372$	0	0
$373$	−33.8746	−1.75396	−0.876980	−	0.480526i	$-0.840446\pi$
$373$	−33.8746	−1.75396	−0.876980	+	0.480526i	$0.840446\pi$
$374$	0	0
$375$	1.18532	0.0612096
$376$	0	0
$377$	−3.91793	−0.201784
$378$	0	0
$379$	−29.7177	−1.52650	−0.763248	−	0.646106i	$-0.776397\pi$
$379$	−29.7177	−1.52650	−0.763248	+	0.646106i	$0.776397\pi$
$380$	0	0
$381$	−16.5815	−0.849494
$382$	0	0
$383$	−32.1346	−1.64200	−0.821002	−	0.570926i	$-0.806584\pi$
$383$	−32.1346	−1.64200	−0.821002	+	0.570926i	$0.806584\pi$
$384$	0	0
$385$	5.53212	0.281943
$386$	0	0
$387$	2.31380	0.117617
$388$	0	0
$389$	−9.57317	−0.485379	−0.242689	−	0.970104i	$-0.578030\pi$
$389$	−9.57317	−0.485379	−0.242689	+	0.970104i	$0.578030\pi$
$390$	0	0
$391$	4.47266	0.226192
$392$	0	0
$393$	21.7618	1.09774
$394$	0	0
$395$	4.90379	0.246736
$396$	0	0
$397$	28.2759	1.41913	0.709564	−	0.704641i	$-0.248892\pi$
$397$	28.2759	1.41913	0.709564	+	0.704641i	$0.248892\pi$
$398$	0	0
$399$	5.72283	0.286500
$400$	0	0
$401$	−3.41539	−0.170557	−0.0852783	−	0.996357i	$-0.527178\pi$
$401$	−3.41539	−0.170557	−0.0852783	+	0.996357i	$0.527178\pi$
$402$	0	0
$403$	21.0120	1.04668
$404$	0	0
$405$	−1.67085	−0.0830254
$406$	0	0
$407$	−0.500651	−0.0248163
$408$	0	0
$409$	−27.3453	−1.35214	−0.676069	−	0.736838i	$-0.736318\pi$
$409$	−27.3453	−1.35214	−0.676069	+	0.736838i	$0.736318\pi$
$410$	0	0
$411$	15.4537	0.762277
$412$	0	0
$413$	11.5257	0.567142
$414$	0	0
$415$	−15.3349	−0.752759
$416$	0	0
$417$	−25.9040	−1.26852
$418$	0	0
$419$	11.1221	0.543352	0.271676	−	0.962389i	$-0.412422\pi$
$419$	11.1221	0.543352	0.271676	+	0.962389i	$0.412422\pi$
$420$	0	0
$421$	−3.53019	−0.172051	−0.0860253	−	0.996293i	$-0.527417\pi$
$421$	−3.53019	−0.172051	−0.0860253	+	0.996293i	$0.527417\pi$
$422$	0	0
$423$	−11.5810	−0.563088
$424$	0	0
$425$	−0.563717	−0.0273443
$426$	0	0
$427$	−0.231118	−0.0111846
$428$	0	0
$429$	20.0930	0.970097
$430$	0	0
$431$	−10.2268	−0.492605	−0.246303	−	0.969193i	$-0.579216\pi$
$431$	−10.2268	−0.492605	−0.246303	+	0.969193i	$0.579216\pi$
$432$	0	0
$433$	−14.0096	−0.673259	−0.336629	−	0.941637i	$-0.609287\pi$
$433$	−14.0096	−0.673259	−0.336629	+	0.941637i	$0.609287\pi$
$434$	0	0
$435$	−1.12669	−0.0540206
$436$	0	0
$437$	−28.4780	−1.36229
$438$	0	0
$439$	33.3698	1.59265	0.796327	−	0.604866i	$-0.206773\pi$
$439$	33.3698	1.59265	0.796327	+	0.604866i	$0.206773\pi$
$440$	0	0
$441$	8.27906	0.394241
$442$	0	0
$443$	−35.5126	−1.68725	−0.843627	−	0.536930i	$-0.819584\pi$
$443$	−35.5126	−1.68725	−0.843627	+	0.536930i	$0.819584\pi$
$444$	0	0
$445$	0.435248	0.0206328
$446$	0	0
$447$	−14.2858	−0.675697
$448$	0	0
$449$	13.4338	0.633982	0.316991	−	0.948428i	$-0.397327\pi$
$449$	13.4338	0.633982	0.316991	+	0.948428i	$0.397327\pi$
$450$	0	0
$451$	32.4698	1.52894
$452$	0	0
$453$	−1.18532	−0.0556911
$454$	0	0
$455$	5.54446	0.259928
$456$	0	0
$457$	−19.3560	−0.905437	−0.452718	−	0.891654i	$-0.649546\pi$
$457$	−19.3560	−0.905437	−0.452718	+	0.891654i	$0.649546\pi$
$458$	0	0
$459$	3.07032	0.143310
$460$	0	0
$461$	22.4899	1.04746	0.523729	−	0.851885i	$-0.324540\pi$
$461$	22.4899	1.04746	0.523729	+	0.851885i	$0.324540\pi$
$462$	0	0
$463$	−8.05006	−0.374118	−0.187059	−	0.982349i	$-0.559896\pi$
$463$	−8.05006	−0.374118	−0.187059	+	0.982349i	$0.559896\pi$
$464$	0	0
$465$	6.04246	0.280212
$466$	0	0
$467$	7.49589	0.346868	0.173434	−	0.984845i	$-0.444514\pi$
$467$	7.49589	0.346868	0.173434	+	0.984845i	$0.444514\pi$
$468$	0	0
$469$	−0.265737	−0.0122706
$470$	0	0
$471$	−3.58373	−0.165129
$472$	0	0
$473$	−5.96596	−0.274315
$474$	0	0
$475$	3.58926	0.164686
$476$	0	0
$477$	−12.0259	−0.550629
$478$	0	0
$479$	11.5324	0.526927	0.263464	−	0.964669i	$-0.415135\pi$
$479$	11.5324	0.526927	0.263464	+	0.964669i	$0.415135\pi$
$480$	0	0
$481$	−0.501768	−0.0228787
$482$	0	0
$483$	−12.6506	−0.575622
$484$	0	0
$485$	−10.3461	−0.469794
$486$	0	0
$487$	11.9096	0.539676	0.269838	−	0.962906i	$-0.413030\pi$
$487$	11.9096	0.539676	0.269838	+	0.962906i	$0.413030\pi$
$488$	0	0
$489$	3.85911	0.174515
$490$	0	0
$491$	8.93517	0.403239	0.201619	−	0.979464i	$-0.435380\pi$
$491$	8.93517	0.403239	0.201619	+	0.979464i	$0.435380\pi$
$492$	0	0
$493$	0.535833	0.0241327
$494$	0	0
$495$	−6.55973	−0.294838
$496$	0	0
$497$	1.39487	0.0625683
$498$	0	0
$499$	8.49487	0.380283	0.190141	−	0.981757i	$-0.439105\pi$
$499$	8.49487	0.380283	0.190141	+	0.981757i	$0.439105\pi$
$500$	0	0
$501$	19.7637	0.882975
$502$	0	0
$503$	19.8284	0.884106	0.442053	−	0.896989i	$-0.354250\pi$
$503$	19.8284	0.884106	0.442053	+	0.896989i	$0.354250\pi$
$504$	0	0
$505$	8.63328	0.384176
$506$	0	0
$507$	4.72865	0.210007
$508$	0	0
$509$	5.13769	0.227724	0.113862	−	0.993497i	$-0.463678\pi$
$509$	5.13769	0.227724	0.113862	+	0.993497i	$0.463678\pi$
$510$	0	0
$511$	16.5545	0.732327
$512$	0	0
$513$	−19.5491	−0.863114
$514$	0	0
$515$	−14.2628	−0.628494
$516$	0	0
$517$	29.8607	1.31327
$518$	0	0
$519$	−18.2447	−0.800853
$520$	0	0
$521$	−2.59407	−0.113648	−0.0568241	−	0.998384i	$-0.518097\pi$
$521$	−2.59407	−0.113648	−0.0568241	+	0.998384i	$0.518097\pi$
$522$	0	0
$523$	33.4376	1.46212	0.731062	−	0.682311i	$-0.239025\pi$
$523$	33.4376	1.46212	0.731062	+	0.682311i	$0.239025\pi$
$524$	0	0
$525$	1.59443	0.0695868
$526$	0	0
$527$	−2.87369	−0.125180
$528$	0	0
$529$	39.9521	1.73705
$530$	0	0
$531$	−13.6666	−0.593081
$532$	0	0
$533$	32.5422	1.40956
$534$	0	0
$535$	6.79607	0.293820
$536$	0	0
$537$	−4.24342	−0.183117
$538$	0	0
$539$	−21.3469	−0.919477
$540$	0	0
$541$	−16.0473	−0.689926	−0.344963	−	0.938616i	$-0.612108\pi$
$541$	−16.0473	−0.689926	−0.344963	+	0.938616i	$0.612108\pi$
$542$	0	0
$543$	−9.96804	−0.427769
$544$	0	0
$545$	4.47452	0.191668
$546$	0	0
$547$	−2.23868	−0.0957192	−0.0478596	−	0.998854i	$-0.515240\pi$
$547$	−2.23868	−0.0957192	−0.0478596	+	0.998854i	$0.515240\pi$
$548$	0	0
$549$	0.274049	0.0116961
$550$	0	0
$551$	−3.41172	−0.145344
$552$	0	0
$553$	6.59633	0.280505
$554$	0	0
$555$	−0.144295	−0.00612496
$556$	0	0
$557$	−26.2546	−1.11244	−0.556222	−	0.831034i	$-0.687750\pi$
$557$	−26.2546	−1.11244	−0.556222	+	0.831034i	$0.687750\pi$
$558$	0	0
$559$	−5.97927	−0.252896
$560$	0	0
$561$	−2.74800	−0.116021
$562$	0	0
$563$	−11.4216	−0.481364	−0.240682	−	0.970604i	$-0.577371\pi$
$563$	−11.4216	−0.481364	−0.240682	+	0.970604i	$0.577371\pi$
$564$	0	0
$565$	−11.2776	−0.474451
$566$	0	0
$567$	−2.24755	−0.0943883
$568$	0	0
$569$	−17.4430	−0.731247	−0.365623	−	0.930763i	$-0.619144\pi$
$569$	−17.4430	−0.731247	−0.365623	+	0.930763i	$0.619144\pi$
$570$	0	0
$571$	−4.60319	−0.192637	−0.0963187	−	0.995351i	$-0.530707\pi$
$571$	−4.60319	−0.192637	−0.0963187	+	0.995351i	$0.530707\pi$
$572$	0	0
$573$	23.7612	0.992640
$574$	0	0
$575$	−7.93423	−0.330880
$576$	0	0
$577$	19.8627	0.826895	0.413448	−	0.910528i	$-0.364324\pi$
$577$	19.8627	0.826895	0.413448	+	0.910528i	$0.364324\pi$
$578$	0	0
$579$	24.0994	1.00154
$580$	0	0
$581$	−20.6277	−0.855782
$582$	0	0
$583$	31.0079	1.28422
$584$	0	0
$585$	−6.57437	−0.271817
$586$	0	0
$587$	21.8334	0.901160	0.450580	−	0.892736i	$-0.351217\pi$
$587$	21.8334	0.901160	0.450580	+	0.892736i	$0.351217\pi$
$588$	0	0
$589$	18.2971	0.753920
$590$	0	0
$591$	−13.2756	−0.546087
$592$	0	0
$593$	−31.5169	−1.29425	−0.647123	−	0.762386i	$-0.724028\pi$
$593$	−31.5169	−1.29425	−0.647123	+	0.762386i	$0.724028\pi$
$594$	0	0
$595$	−0.758284	−0.0310866
$596$	0	0
$597$	17.9423	0.734329
$598$	0	0
$599$	5.75634	0.235198	0.117599	−	0.993061i	$-0.462480\pi$
$599$	5.75634	0.235198	0.117599	+	0.993061i	$0.462480\pi$
$600$	0	0
$601$	5.18438	0.211475	0.105738	−	0.994394i	$-0.466280\pi$
$601$	5.18438	0.211475	0.105738	+	0.994394i	$0.466280\pi$
$602$	0	0
$603$	0.315099	0.0128318
$604$	0	0
$605$	5.91378	0.240429
$606$	0	0
$607$	14.8460	0.602581	0.301291	−	0.953532i	$-0.402583\pi$
$607$	14.8460	0.602581	0.301291	+	0.953532i	$0.402583\pi$
$608$	0	0
$609$	−1.51557	−0.0614139
$610$	0	0
$611$	29.9274	1.21073
$612$	0	0
$613$	−23.4499	−0.947132	−0.473566	−	0.880758i	$-0.657033\pi$
$613$	−23.4499	−0.947132	−0.473566	+	0.880758i	$0.657033\pi$
$614$	0	0
$615$	9.35824	0.377361
$616$	0	0
$617$	−33.0825	−1.33185	−0.665925	−	0.746019i	$-0.731963\pi$
$617$	−33.0825	−1.33185	−0.665925	+	0.746019i	$0.731963\pi$
$618$	0	0
$619$	−47.7904	−1.92086	−0.960428	−	0.278528i	$-0.910154\pi$
$619$	−47.7904	−1.92086	−0.960428	+	0.278528i	$0.910154\pi$
$620$	0	0
$621$	43.2143	1.73413
$622$	0	0
$623$	0.585475	0.0234566
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	17.4968	0.698757
$628$	0	0
$629$	0.0686239	0.00273622
$630$	0	0
$631$	18.3846	0.731879	0.365939	−	0.930639i	$-0.380748\pi$
$631$	18.3846	0.731879	0.365939	+	0.930639i	$0.380748\pi$
$632$	0	0
$633$	−16.7252	−0.664766
$634$	0	0
$635$	−13.9890	−0.555138
$636$	0	0
$637$	−21.3946	−0.847683
$638$	0	0
$639$	−1.65397	−0.0654300
$640$	0	0
$641$	−12.7960	−0.505412	−0.252706	−	0.967543i	$-0.581321\pi$
$641$	−12.7960	−0.505412	−0.252706	+	0.967543i	$0.581321\pi$
$642$	0	0
$643$	19.6963	0.776747	0.388373	−	0.921502i	$-0.373037\pi$
$643$	19.6963	0.776747	0.388373	+	0.921502i	$0.373037\pi$
$644$	0	0
$645$	−1.71947	−0.0677041
$646$	0	0
$647$	2.38253	0.0936667	0.0468334	−	0.998903i	$-0.485087\pi$
$647$	2.38253	0.0936667	0.0468334	+	0.998903i	$0.485087\pi$
$648$	0	0
$649$	35.2384	1.38323
$650$	0	0
$651$	8.12802	0.318562
$652$	0	0
$653$	−42.9186	−1.67953	−0.839767	−	0.542947i	$-0.817308\pi$
$653$	−42.9186	−1.67953	−0.839767	+	0.542947i	$0.817308\pi$
$654$	0	0
$655$	18.3595	0.717365
$656$	0	0
$657$	−19.6295	−0.765822
$658$	0	0
$659$	−26.6338	−1.03751	−0.518753	−	0.854924i	$-0.673604\pi$
$659$	−26.6338	−1.03751	−0.518753	+	0.854924i	$0.673604\pi$
$660$	0	0
$661$	8.38878	0.326286	0.163143	−	0.986602i	$-0.447837\pi$
$661$	8.38878	0.326286	0.163143	+	0.986602i	$0.447837\pi$
$662$	0	0
$663$	−2.75413	−0.106962
$664$	0	0
$665$	4.82809	0.187225
$666$	0	0
$667$	7.54178	0.292019
$668$	0	0
$669$	24.6962	0.954809
$670$	0	0
$671$	−0.706615	−0.0272786
$672$	0	0
$673$	−7.13146	−0.274898	−0.137449	−	0.990509i	$-0.543890\pi$
$673$	−7.13146	−0.274898	−0.137449	+	0.990509i	$0.543890\pi$
$674$	0	0
$675$	−5.44656	−0.209638
$676$	0	0
$677$	−30.9174	−1.18825	−0.594127	−	0.804371i	$-0.702502\pi$
$677$	−30.9174	−1.18825	−0.594127	+	0.804371i	$0.702502\pi$
$678$	0	0
$679$	−13.9171	−0.534090
$680$	0	0
$681$	−7.44553	−0.285313
$682$	0	0
$683$	−7.01793	−0.268534	−0.134267	−	0.990945i	$-0.542868\pi$
$683$	−7.01793	−0.268534	−0.134267	+	0.990945i	$0.542868\pi$
$684$	0	0
$685$	13.0376	0.498142
$686$	0	0
$687$	−8.40929	−0.320834
$688$	0	0
$689$	31.0771	1.18394
$690$	0	0
$691$	22.0731	0.839700	0.419850	−	0.907594i	$-0.362083\pi$
$691$	22.0731	0.839700	0.419850	+	0.907594i	$0.362083\pi$
$692$	0	0
$693$	−8.82383	−0.335190
$694$	0	0
$695$	−21.8540	−0.828970
$696$	0	0
$697$	−4.45062	−0.168579
$698$	0	0
$699$	−0.460910	−0.0174332
$700$	0	0
$701$	5.78291	0.218417	0.109209	−	0.994019i	$-0.465168\pi$
$701$	5.78291	0.218417	0.109209	+	0.994019i	$0.465168\pi$
$702$	0	0
$703$	−0.436937	−0.0164794
$704$	0	0
$705$	8.60628	0.324131
$706$	0	0
$707$	11.6131	0.436754
$708$	0	0
$709$	−15.0167	−0.563964	−0.281982	−	0.959420i	$-0.590992\pi$
$709$	−15.0167	−0.563964	−0.281982	+	0.959420i	$0.590992\pi$
$710$	0	0
$711$	−7.82163	−0.293334
$712$	0	0
$713$	−40.4467	−1.51474
$714$	0	0
$715$	16.9515	0.633951
$716$	0	0
$717$	4.97192	0.185680
$718$	0	0
$719$	14.0927	0.525569	0.262784	−	0.964855i	$-0.415359\pi$
$719$	14.0927	0.525569	0.262784	+	0.964855i	$0.415359\pi$
$720$	0	0
$721$	−19.1856	−0.714510
$722$	0	0
$723$	5.96451	0.221823
$724$	0	0
$725$	−0.950537	−0.0353020
$726$	0	0
$727$	−19.0185	−0.705357	−0.352679	−	0.935744i	$-0.614729\pi$
$727$	−19.0185	−0.705357	−0.352679	+	0.935744i	$0.614729\pi$
$728$	0	0
$729$	22.0328	0.816029
$730$	0	0
$731$	0.817750	0.0302456
$732$	0	0
$733$	13.1259	0.484817	0.242409	−	0.970174i	$-0.422063\pi$
$733$	13.1259	0.484817	0.242409	+	0.970174i	$0.422063\pi$
$734$	0	0
$735$	−6.15248	−0.226938
$736$	0	0
$737$	−0.812459	−0.0299273
$738$	0	0
$739$	−43.3121	−1.59326	−0.796632	−	0.604465i	$-0.793387\pi$
$739$	−43.3121	−1.59326	−0.796632	+	0.604465i	$0.793387\pi$
$740$	0	0
$741$	17.5359	0.644197
$742$	0	0
$743$	−25.8022	−0.946590	−0.473295	−	0.880904i	$-0.656936\pi$
$743$	−25.8022	−0.946590	−0.473295	+	0.880904i	$0.656936\pi$
$744$	0	0
$745$	−12.0523	−0.441563
$746$	0	0
$747$	24.4594	0.894923
$748$	0	0
$749$	9.14174	0.334032
$750$	0	0
$751$	17.9113	0.653591	0.326796	−	0.945095i	$-0.394031\pi$
$751$	17.9113	0.653591	0.326796	+	0.945095i	$0.394031\pi$
$752$	0	0
$753$	3.33598	0.121570
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−9.91917	−0.360519	−0.180259	−	0.983619i	$-0.557694\pi$
$757$	−9.91917	−0.360519	−0.180259	+	0.983619i	$0.557694\pi$
$758$	0	0
$759$	−38.6777	−1.40391
$760$	0	0
$761$	−53.0736	−1.92392	−0.961958	−	0.273196i	$-0.911919\pi$
$761$	−53.0736	−1.92392	−0.961958	+	0.273196i	$0.911919\pi$
$762$	0	0
$763$	6.01891	0.217899
$764$	0	0
$765$	0.899139	0.0325084
$766$	0	0
$767$	35.3170	1.27522
$768$	0	0
$769$	−27.5177	−0.992312	−0.496156	−	0.868233i	$-0.665256\pi$
$769$	−27.5177	−0.992312	−0.496156	+	0.868233i	$0.665256\pi$
$770$	0	0
$771$	36.6581	1.32021
$772$	0	0
$773$	−14.9909	−0.539186	−0.269593	−	0.962974i	$-0.586889\pi$
$773$	−14.9909	−0.539186	−0.269593	+	0.962974i	$0.586889\pi$
$774$	0	0
$775$	5.09775	0.183117
$776$	0	0
$777$	−0.194098	−0.00696323
$778$	0	0
$779$	28.3376	1.01530
$780$	0	0
$781$	4.26463	0.152601
$782$	0	0
$783$	5.17716	0.185017
$784$	0	0
$785$	−3.02343	−0.107911
$786$	0	0
$787$	3.01506	0.107475	0.0537377	−	0.998555i	$-0.482887\pi$
$787$	3.01506	0.107475	0.0537377	+	0.998555i	$0.482887\pi$
$788$	0	0
$789$	27.5957	0.982432
$790$	0	0
$791$	−15.1700	−0.539384
$792$	0	0
$793$	−0.708192	−0.0251486
$794$	0	0
$795$	8.93691	0.316959
$796$	0	0
$797$	37.8437	1.34049	0.670247	−	0.742138i	$-0.266188\pi$
$797$	37.8437	1.34049	0.670247	+	0.742138i	$0.266188\pi$
$798$	0	0
$799$	−4.09300	−0.144800
$800$	0	0
$801$	−0.694230	−0.0245294
$802$	0	0
$803$	50.6133	1.78610
$804$	0	0
$805$	−10.6727	−0.376165
$806$	0	0
$807$	3.00880	0.105915
$808$	0	0
$809$	36.2423	1.27421	0.637105	−	0.770777i	$-0.280132\pi$
$809$	36.2423	1.27421	0.637105	+	0.770777i	$0.280132\pi$
$810$	0	0
$811$	−18.9981	−0.667114	−0.333557	−	0.942730i	$-0.608249\pi$
$811$	−18.9981	−0.667114	−0.333557	+	0.942730i	$0.608249\pi$
$812$	0	0
$813$	−13.8676	−0.486359
$814$	0	0
$815$	3.25576	0.114044
$816$	0	0
$817$	−5.20672	−0.182160
$818$	0	0
$819$	−8.84352	−0.309018
$820$	0	0
$821$	23.1629	0.808391	0.404196	−	0.914673i	$-0.367551\pi$
$821$	23.1629	0.808391	0.404196	+	0.914673i	$0.367551\pi$
$822$	0	0
$823$	−26.8891	−0.937295	−0.468648	−	0.883385i	$-0.655259\pi$
$823$	−26.8891	−0.937295	−0.468648	+	0.883385i	$0.655259\pi$
$824$	0	0
$825$	4.87478	0.169718
$826$	0	0
$827$	−31.3964	−1.09176	−0.545879	−	0.837864i	$-0.683804\pi$
$827$	−31.3964	−1.09176	−0.545879	+	0.837864i	$0.683804\pi$
$828$	0	0
$829$	−8.30386	−0.288405	−0.144202	−	0.989548i	$-0.546062\pi$
$829$	−8.30386	−0.288405	−0.144202	+	0.989548i	$0.546062\pi$
$830$	0	0
$831$	−1.79687	−0.0623328
$832$	0	0
$833$	2.92601	0.101380
$834$	0	0
$835$	16.6737	0.577017
$836$	0	0
$837$	−27.7652	−0.959706
$838$	0	0
$839$	32.4113	1.11896	0.559481	−	0.828843i	$-0.311001\pi$
$839$	32.4113	1.11896	0.559481	+	0.828843i	$0.311001\pi$
$840$	0	0
$841$	−28.0965	−0.968844
$842$	0	0
$843$	−5.57806	−0.192119
$844$	0	0
$845$	3.98935	0.137238
$846$	0	0
$847$	7.95492	0.273334
$848$	0	0
$849$	21.1633	0.726323
$850$	0	0
$851$	0.965872	0.0331097
$852$	0	0
$853$	18.9399	0.648490	0.324245	−	0.945973i	$-0.394890\pi$
$853$	18.9399	0.648490	0.324245	+	0.945973i	$0.394890\pi$
$854$	0	0
$855$	−5.72493	−0.195789
$856$	0	0
$857$	−25.8599	−0.883357	−0.441678	−	0.897174i	$-0.645617\pi$
$857$	−25.8599	−0.883357	−0.441678	+	0.897174i	$0.645617\pi$
$858$	0	0
$859$	44.4320	1.51600	0.757999	−	0.652255i	$-0.226177\pi$
$859$	44.4320	1.51600	0.757999	+	0.652255i	$0.226177\pi$
$860$	0	0
$861$	12.5883	0.429006
$862$	0	0
$863$	28.0579	0.955102	0.477551	−	0.878604i	$-0.341525\pi$
$863$	28.0579	0.955102	0.477551	+	0.878604i	$0.341525\pi$
$864$	0	0
$865$	−15.3922	−0.523352
$866$	0	0
$867$	−19.7738	−0.671552
$868$	0	0
$869$	20.1675	0.684135
$870$	0	0
$871$	−0.814272	−0.0275906
$872$	0	0
$873$	16.5023	0.558518
$874$	0	0
$875$	1.34515	0.0454744
$876$	0	0
$877$	−41.6128	−1.40516	−0.702582	−	0.711603i	$-0.747970\pi$
$877$	−41.6128	−1.40516	−0.702582	+	0.711603i	$0.747970\pi$
$878$	0	0
$879$	6.39951	0.215850
$880$	0	0
$881$	44.6270	1.50352	0.751760	−	0.659436i	$-0.229205\pi$
$881$	44.6270	1.50352	0.751760	+	0.659436i	$0.229205\pi$
$882$	0	0
$883$	38.1385	1.28346	0.641731	−	0.766930i	$-0.278216\pi$
$883$	38.1385	1.28346	0.641731	+	0.766930i	$0.278216\pi$
$884$	0	0
$885$	10.1562	0.341396
$886$	0	0
$887$	31.6355	1.06222	0.531109	−	0.847304i	$-0.321776\pi$
$887$	31.6355	1.06222	0.531109	+	0.847304i	$0.321776\pi$
$888$	0	0
$889$	−18.8174	−0.631114
$890$	0	0
$891$	−6.87162	−0.230208
$892$	0	0
$893$	26.0606	0.872085
$894$	0	0
$895$	−3.57998	−0.119666
$896$	0	0
$897$	−38.7640	−1.29429
$898$	0	0
$899$	−4.84560	−0.161610
$900$	0	0
$901$	−4.25023	−0.141596
$902$	0	0
$903$	−2.31295	−0.0769701
$904$	0	0
$905$	−8.40958	−0.279544
$906$	0	0
$907$	−52.7368	−1.75110	−0.875548	−	0.483131i	$-0.839500\pi$
$907$	−52.7368	−1.75110	−0.875548	+	0.483131i	$0.839500\pi$
$908$	0	0
$909$	−13.7702	−0.456730
$910$	0	0
$911$	−33.4869	−1.10947	−0.554735	−	0.832027i	$-0.687180\pi$
$911$	−33.4869	−1.10947	−0.554735	+	0.832027i	$0.687180\pi$
$912$	0	0
$913$	−63.0668	−2.08720
$914$	0	0
$915$	−0.203656	−0.00673267
$916$	0	0
$917$	24.6963	0.815543
$918$	0	0
$919$	−40.7379	−1.34382	−0.671910	−	0.740633i	$-0.734526\pi$
$919$	−40.7379	−1.34382	−0.671910	+	0.740633i	$0.734526\pi$
$920$	0	0
$921$	38.6516	1.27361
$922$	0	0
$923$	4.27415	0.140685
$924$	0	0
$925$	−0.121735	−0.00400262
$926$	0	0
$927$	22.7494	0.747190
$928$	0	0
$929$	−38.9791	−1.27886	−0.639431	−	0.768848i	$-0.720830\pi$
$929$	−38.9791	−1.27886	−0.639431	+	0.768848i	$0.720830\pi$
$930$	0	0
$931$	−18.6303	−0.610583
$932$	0	0
$933$	5.35567	0.175337
$934$	0	0
$935$	−2.31836	−0.0758185
$936$	0	0
$937$	−45.1328	−1.47443	−0.737213	−	0.675661i	$-0.763859\pi$
$937$	−45.1328	−1.47443	−0.737213	+	0.675661i	$0.763859\pi$
$938$	0	0
$939$	8.82776	0.288083
$940$	0	0
$941$	−3.40497	−0.110999	−0.0554994	−	0.998459i	$-0.517675\pi$
$941$	−3.40497	−0.110999	−0.0554994	+	0.998459i	$0.517675\pi$
$942$	0	0
$943$	−62.6418	−2.03990
$944$	0	0
$945$	−7.32645	−0.238329
$946$	0	0
$947$	−29.4468	−0.956893	−0.478447	−	0.878117i	$-0.658800\pi$
$947$	−29.4468	−0.956893	−0.478447	+	0.878117i	$0.658800\pi$
$948$	0	0
$949$	50.7262	1.64664
$950$	0	0
$951$	7.24585	0.234963
$952$	0	0
$953$	−50.4385	−1.63386	−0.816931	−	0.576735i	$-0.804326\pi$
$953$	−50.4385	−1.63386	−0.816931	+	0.576735i	$0.804326\pi$
$954$	0	0
$955$	20.0463	0.648682
$956$	0	0
$957$	−4.63366	−0.149785
$958$	0	0
$959$	17.5376	0.566318
$960$	0	0
$961$	−5.01296	−0.161708
$962$	0	0
$963$	−10.8399	−0.349310
$964$	0	0
$965$	20.3315	0.654496
$966$	0	0
$967$	45.4418	1.46131	0.730655	−	0.682747i	$-0.239215\pi$
$967$	45.4418	1.46131	0.730655	+	0.682747i	$0.239215\pi$
$968$	0	0
$969$	−2.39828	−0.0770440
$970$	0	0
$971$	2.44943	0.0786061	0.0393031	−	0.999227i	$-0.487486\pi$
$971$	2.44943	0.0786061	0.0393031	+	0.999227i	$0.487486\pi$
$972$	0	0
$973$	−29.3969	−0.942423
$974$	0	0
$975$	4.88566	0.156466
$976$	0	0
$977$	1.37697	0.0440532	0.0220266	−	0.999757i	$-0.492988\pi$
$977$	1.37697	0.0440532	0.0220266	+	0.999757i	$0.492988\pi$
$978$	0	0
$979$	1.79002	0.0572092
$980$	0	0
$981$	−7.13695	−0.227865
$982$	0	0
$983$	−1.43294	−0.0457037	−0.0228519	−	0.999739i	$-0.507275\pi$
$983$	−1.43294	−0.0457037	−0.0228519	+	0.999739i	$0.507275\pi$
$984$	0	0
$985$	−11.2001	−0.356864
$986$	0	0
$987$	11.5767	0.368492
$988$	0	0
$989$	11.5097	0.365988
$990$	0	0
$991$	−29.5826	−0.939722	−0.469861	−	0.882740i	$-0.655696\pi$
$991$	−29.5826	−0.939722	−0.469861	+	0.882740i	$0.655696\pi$
$992$	0	0
$993$	28.6163	0.908110
$994$	0	0
$995$	15.1371	0.479878
$996$	0	0
$997$	−45.2398	−1.43276	−0.716379	−	0.697711i	$-0.754202\pi$
$997$	−45.2398	−1.43276	−0.716379	+	0.697711i	$0.754202\pi$
$998$	0	0
$999$	0.663036	0.0209775

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.10	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.10	✓	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)

\(f(q)\)	\(=\)	\(q+1.18532 q^{3} +1.00000 q^{5} +1.34515 q^{7} -1.59502 q^{9} +O(q^{10})\)	\(q+1.18532 q^{3} +1.00000 q^{5} +1.34515 q^{7} -1.59502 q^{9} +4.11264 q^{11} +4.12181 q^{13} +1.18532 q^{15} -0.563717 q^{17} +3.58926 q^{19} +1.59443 q^{21} -7.93423 q^{23} +1.00000 q^{25} -5.44656 q^{27} -0.950537 q^{29} +5.09775 q^{31} +4.87478 q^{33} +1.34515 q^{35} -0.121735 q^{37} +4.88566 q^{39} +7.89513 q^{41} -1.45064 q^{43} -1.59502 q^{45} +7.26073 q^{47} -5.19057 q^{49} -0.668184 q^{51} +7.53966 q^{53} +4.11264 q^{55} +4.25441 q^{57} +8.56831 q^{59} -0.171816 q^{61} -2.14554 q^{63} +4.12181 q^{65} -0.197552 q^{67} -9.40460 q^{69} +1.03696 q^{71} +12.3068 q^{73} +1.18532 q^{75} +5.53212 q^{77} +4.90379 q^{79} -1.67085 q^{81} -15.3349 q^{83} -0.563717 q^{85} -1.12669 q^{87} +0.435248 q^{89} +5.54446 q^{91} +6.04246 q^{93} +3.58926 q^{95} -10.3461 q^{97} -6.55973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 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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