LMFDB - Embedded newform 6020.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6020,2,Mod(1,6020)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, 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("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6020.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:	$48.0699420168$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 27 x^{11} - 2 x^{10} + 268 x^{9} + 37 x^{8} - 1201 x^{7} - 189 x^{6} + 2384 x^{5} + 231 x^{4} + \cdots - 4 $ x^13 - 27x^11 - 2x^10 + 268x^9 + 37x^8 - 1201x^7 - 189x^6 + 2384x^5 + 231x^4 - 1729x^3 + 20x^2 + 105*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.12558$ of defining polynomial
Character	$\chi$	$=$	6020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12558 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.51810 q^{9} +O(q^{10})$	$q-2.12558 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.51810 q^{9} -3.61637 q^{11} +6.86597 q^{13} -2.12558 q^{15} +0.203512 q^{17} +1.68823 q^{19} +2.12558 q^{21} -2.05405 q^{23} +1.00000 q^{25} +3.14991 q^{27} -0.977923 q^{29} +9.91993 q^{31} +7.68690 q^{33} -1.00000 q^{35} -7.37711 q^{37} -14.5942 q^{39} -1.14788 q^{41} +1.00000 q^{43} +1.51810 q^{45} -12.3853 q^{47} +1.00000 q^{49} -0.432581 q^{51} -4.07093 q^{53} -3.61637 q^{55} -3.58847 q^{57} +1.41508 q^{59} +9.13417 q^{61} -1.51810 q^{63} +6.86597 q^{65} -4.02784 q^{67} +4.36604 q^{69} -2.28033 q^{71} +16.2886 q^{73} -2.12558 q^{75} +3.61637 q^{77} -15.6772 q^{79} -11.2497 q^{81} -3.75006 q^{83} +0.203512 q^{85} +2.07865 q^{87} +0.248283 q^{89} -6.86597 q^{91} -21.0856 q^{93} +1.68823 q^{95} +12.1966 q^{97} -5.49001 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9}+O(q^{10}) $ 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9	$ 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9} + 11 q^{13} + 16 q^{17} + 3 q^{23} + 13 q^{25} + 6 q^{27} + 10 q^{29} - q^{31} + 14 q^{33} - 13 q^{35} + 16 q^{37} - 14 q^{39} + 23 q^{41} + 13 q^{43} + 15 q^{45} + 2 q^{47} + 13 q^{49} + 4 q^{51} + 20 q^{53} + 22 q^{57} + 2 q^{59} + 5 q^{61} - 15 q^{63} + 11 q^{65} + 19 q^{67} + 16 q^{69} + 4 q^{71} + 34 q^{73} - 15 q^{79} + 17 q^{81} + 27 q^{83} + 16 q^{85} - 5 q^{87} + 3 q^{89} - 11 q^{91} + 35 q^{93} + 45 q^{97} + q^{99}+O(q^{100}) $ 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9 + 11 * q^13 + 16 * q^17 + 3 * q^23 + 13 * q^25 + 6 * q^27 + 10 * q^29 - q^31 + 14 * q^33 - 13 * q^35 + 16 * q^37 - 14 * q^39 + 23 * q^41 + 13 * q^43 + 15 * q^45 + 2 * q^47 + 13 * q^49 + 4 * q^51 + 20 * q^53 + 22 * q^57 + 2 * q^59 + 5 * q^61 - 15 * q^63 + 11 * q^65 + 19 * q^67 + 16 * q^69 + 4 * q^71 + 34 * q^73 - 15 * q^79 + 17 * q^81 + 27 * q^83 + 16 * q^85 - 5 * q^87 + 3 * q^89 - 11 * q^91 + 35 * q^93 + 45 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.12558	−1.22721	−0.613603	−	0.789615i	$-0.710280\pi$
$3$	−2.12558	−1.22721	−0.613603	+	0.789615i	$0.710280\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.51810	0.506032
$10$	0	0
$11$	−3.61637	−1.09038	−0.545189	−	0.838313i	$-0.683542\pi$
$11$	−3.61637	−1.09038	−0.545189	+	0.838313i	$0.683542\pi$
$12$	0	0
$13$	6.86597	1.90428	0.952139	−	0.305665i	$-0.0988789\pi$
$13$	6.86597	1.90428	0.952139	+	0.305665i	$0.0988789\pi$
$14$	0	0
$15$	−2.12558	−0.548823
$16$	0	0
$17$	0.203512	0.0493589	0.0246794	−	0.999695i	$-0.492143\pi$
$17$	0.203512	0.0493589	0.0246794	+	0.999695i	$0.492143\pi$
$18$	0	0
$19$	1.68823	0.387306	0.193653	−	0.981070i	$-0.437966\pi$
$19$	1.68823	0.387306	0.193653	+	0.981070i	$0.437966\pi$
$20$	0	0
$21$	2.12558	0.463840
$22$	0	0
$23$	−2.05405	−0.428298	−0.214149	−	0.976801i	$-0.568698\pi$
$23$	−2.05405	−0.428298	−0.214149	+	0.976801i	$0.568698\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.14991	0.606200
$28$	0	0
$29$	−0.977923	−0.181596	−0.0907979	−	0.995869i	$-0.528942\pi$
$29$	−0.977923	−0.181596	−0.0907979	+	0.995869i	$0.528942\pi$
$30$	0	0
$31$	9.91993	1.78167	0.890836	−	0.454325i	$-0.150119\pi$
$31$	9.91993	1.78167	0.890836	+	0.454325i	$0.150119\pi$
$32$	0	0
$33$	7.68690	1.33812
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−7.37711	−1.21279	−0.606395	−	0.795164i	$-0.707385\pi$
$37$	−7.37711	−1.21279	−0.606395	+	0.795164i	$0.707385\pi$
$38$	0	0
$39$	−14.5942	−2.33694
$40$	0	0
$41$	−1.14788	−0.179269	−0.0896346	−	0.995975i	$-0.528570\pi$
$41$	−1.14788	−0.179269	−0.0896346	+	0.995975i	$0.528570\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	0	0
$45$	1.51810	0.226305
$46$	0	0
$47$	−12.3853	−1.80658	−0.903288	−	0.429034i	$-0.858854\pi$
$47$	−12.3853	−1.80658	−0.903288	+	0.429034i	$0.858854\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.432581	−0.0605735
$52$	0	0
$53$	−4.07093	−0.559185	−0.279592	−	0.960119i	$-0.590199\pi$
$53$	−4.07093	−0.559185	−0.279592	+	0.960119i	$0.590199\pi$
$54$	0	0
$55$	−3.61637	−0.487632
$56$	0	0
$57$	−3.58847	−0.475304
$58$	0	0
$59$	1.41508	0.184228	0.0921138	−	0.995748i	$-0.470638\pi$
$59$	1.41508	0.184228	0.0921138	+	0.995748i	$0.470638\pi$
$60$	0	0
$61$	9.13417	1.16951	0.584755	−	0.811210i	$-0.301191\pi$
$61$	9.13417	1.16951	0.584755	+	0.811210i	$0.301191\pi$
$62$	0	0
$63$	−1.51810	−0.191262
$64$	0	0
$65$	6.86597	0.851619
$66$	0	0
$67$	−4.02784	−0.492079	−0.246040	−	0.969260i	$-0.579129\pi$
$67$	−4.02784	−0.492079	−0.246040	+	0.969260i	$0.579129\pi$
$68$	0	0
$69$	4.36604	0.525610
$70$	0	0
$71$	−2.28033	−0.270625	−0.135313	−	0.990803i	$-0.543204\pi$
$71$	−2.28033	−0.270625	−0.135313	+	0.990803i	$0.543204\pi$
$72$	0	0
$73$	16.2886	1.90643	0.953216	−	0.302290i	$-0.0977510\pi$
$73$	16.2886	1.90643	0.953216	+	0.302290i	$0.0977510\pi$
$74$	0	0
$75$	−2.12558	−0.245441
$76$	0	0
$77$	3.61637	0.412124
$78$	0	0
$79$	−15.6772	−1.76382	−0.881909	−	0.471419i	$-0.843742\pi$
$79$	−15.6772	−1.76382	−0.881909	+	0.471419i	$0.843742\pi$
$80$	0	0
$81$	−11.2497	−1.24996
$82$	0	0
$83$	−3.75006	−0.411622	−0.205811	−	0.978592i	$-0.565983\pi$
$83$	−3.75006	−0.411622	−0.205811	+	0.978592i	$0.565983\pi$
$84$	0	0
$85$	0.203512	0.0220740
$86$	0	0
$87$	2.07865	0.222855
$88$	0	0
$89$	0.248283	0.0263179	0.0131590	−	0.999913i	$-0.495811\pi$
$89$	0.248283	0.0263179	0.0131590	+	0.999913i	$0.495811\pi$
$90$	0	0
$91$	−6.86597	−0.719749
$92$	0	0
$93$	−21.0856	−2.18648
$94$	0	0
$95$	1.68823	0.173209
$96$	0	0
$97$	12.1966	1.23838	0.619189	−	0.785242i	$-0.287462\pi$
$97$	12.1966	1.23838	0.619189	+	0.785242i	$0.287462\pi$
$98$	0	0
$99$	−5.49001	−0.551767
$100$	0	0
$101$	−18.4880	−1.83963	−0.919813	−	0.392358i	$-0.871659\pi$
$101$	−18.4880	−1.83963	−0.919813	+	0.392358i	$0.871659\pi$
$102$	0	0
$103$	6.73437	0.663557	0.331778	−	0.943357i	$-0.392351\pi$
$103$	6.73437	0.663557	0.331778	+	0.943357i	$0.392351\pi$
$104$	0	0
$105$	2.12558	0.207436
$106$	0	0
$107$	14.5158	1.40330	0.701648	−	0.712524i	$-0.252448\pi$
$107$	14.5158	1.40330	0.701648	+	0.712524i	$0.252448\pi$
$108$	0	0
$109$	10.9226	1.04620	0.523098	−	0.852273i	$-0.324776\pi$
$109$	10.9226	1.04620	0.523098	+	0.852273i	$0.324776\pi$
$110$	0	0
$111$	15.6806	1.48834
$112$	0	0
$113$	6.74515	0.634531	0.317265	−	0.948337i	$-0.397235\pi$
$113$	6.74515	0.634531	0.317265	+	0.948337i	$0.397235\pi$
$114$	0	0
$115$	−2.05405	−0.191541
$116$	0	0
$117$	10.4232	0.963626
$118$	0	0
$119$	−0.203512	−0.0186559
$120$	0	0
$121$	2.07817	0.188924
$122$	0	0
$123$	2.43992	0.220000
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.46506	0.484945	0.242473	−	0.970158i	$-0.422042\pi$
$127$	5.46506	0.484945	0.242473	+	0.970158i	$0.422042\pi$
$128$	0	0
$129$	−2.12558	−0.187147
$130$	0	0
$131$	−14.3730	−1.25578	−0.627888	−	0.778303i	$-0.716081\pi$
$131$	−14.3730	−1.25578	−0.627888	+	0.778303i	$0.716081\pi$
$132$	0	0
$133$	−1.68823	−0.146388
$134$	0	0
$135$	3.14991	0.271101
$136$	0	0
$137$	10.6087	0.906360	0.453180	−	0.891419i	$-0.350289\pi$
$137$	10.6087	0.906360	0.453180	+	0.891419i	$0.350289\pi$
$138$	0	0
$139$	4.88193	0.414080	0.207040	−	0.978332i	$-0.433617\pi$
$139$	4.88193	0.414080	0.207040	+	0.978332i	$0.433617\pi$
$140$	0	0
$141$	26.3259	2.21704
$142$	0	0
$143$	−24.8299	−2.07638
$144$	0	0
$145$	−0.977923	−0.0812121
$146$	0	0
$147$	−2.12558	−0.175315
$148$	0	0
$149$	15.2119	1.24621	0.623106	−	0.782138i	$-0.285871\pi$
$149$	15.2119	1.24621	0.623106	+	0.782138i	$0.285871\pi$
$150$	0	0
$151$	6.13405	0.499182	0.249591	−	0.968351i	$-0.419704\pi$
$151$	6.13405	0.499182	0.249591	+	0.968351i	$0.419704\pi$
$152$	0	0
$153$	0.308951	0.0249772
$154$	0	0
$155$	9.91993	0.796788
$156$	0	0
$157$	−5.23127	−0.417501	−0.208750	−	0.977969i	$-0.566940\pi$
$157$	−5.23127	−0.417501	−0.208750	+	0.977969i	$0.566940\pi$
$158$	0	0
$159$	8.65309	0.686234
$160$	0	0
$161$	2.05405	0.161881
$162$	0	0
$163$	−7.17696	−0.562143	−0.281071	−	0.959687i	$-0.590690\pi$
$163$	−7.17696	−0.562143	−0.281071	+	0.959687i	$0.590690\pi$
$164$	0	0
$165$	7.68690	0.598424
$166$	0	0
$167$	−5.79717	−0.448598	−0.224299	−	0.974520i	$-0.572009\pi$
$167$	−5.79717	−0.448598	−0.224299	+	0.974520i	$0.572009\pi$
$168$	0	0
$169$	34.1416	2.62627
$170$	0	0
$171$	2.56289	0.195989
$172$	0	0
$173$	−19.3171	−1.46865	−0.734326	−	0.678797i	$-0.762502\pi$
$173$	−19.3171	−1.46865	−0.734326	+	0.678797i	$0.762502\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−3.00786	−0.226085
$178$	0	0
$179$	−11.0231	−0.823909	−0.411954	−	0.911204i	$-0.635154\pi$
$179$	−11.0231	−0.823909	−0.411954	+	0.911204i	$0.635154\pi$
$180$	0	0
$181$	−0.341921	−0.0254148	−0.0127074	−	0.999919i	$-0.504045\pi$
$181$	−0.341921	−0.0254148	−0.0127074	+	0.999919i	$0.504045\pi$
$182$	0	0
$183$	−19.4154	−1.43523
$184$	0	0
$185$	−7.37711	−0.542376
$186$	0	0
$187$	−0.735975	−0.0538198
$188$	0	0
$189$	−3.14991	−0.229122
$190$	0	0
$191$	−7.15274	−0.517554	−0.258777	−	0.965937i	$-0.583319\pi$
$191$	−7.15274	−0.517554	−0.258777	+	0.965937i	$0.583319\pi$
$192$	0	0
$193$	12.8627	0.925878	0.462939	−	0.886390i	$-0.346795\pi$
$193$	12.8627	0.925878	0.462939	+	0.886390i	$0.346795\pi$
$194$	0	0
$195$	−14.5942	−1.04511
$196$	0	0
$197$	12.2981	0.876204	0.438102	−	0.898925i	$-0.355651\pi$
$197$	12.2981	0.876204	0.438102	+	0.898925i	$0.355651\pi$
$198$	0	0
$199$	−15.5229	−1.10039	−0.550194	−	0.835037i	$-0.685446\pi$
$199$	−15.5229	−1.10039	−0.550194	+	0.835037i	$0.685446\pi$
$200$	0	0
$201$	8.56151	0.603882
$202$	0	0
$203$	0.977923	0.0686367
$204$	0	0
$205$	−1.14788	−0.0801716
$206$	0	0
$207$	−3.11824	−0.216733
$208$	0	0
$209$	−6.10526	−0.422310
$210$	0	0
$211$	−10.9476	−0.753662	−0.376831	−	0.926282i	$-0.622986\pi$
$211$	−10.9476	−0.753662	−0.376831	+	0.926282i	$0.622986\pi$
$212$	0	0
$213$	4.84702	0.332112
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−9.91993	−0.673409
$218$	0	0
$219$	−34.6227	−2.33958
$220$	0	0
$221$	1.39731	0.0939930
$222$	0	0
$223$	18.3398	1.22813	0.614063	−	0.789257i	$-0.289534\pi$
$223$	18.3398	1.22813	0.614063	+	0.789257i	$0.289534\pi$
$224$	0	0
$225$	1.51810	0.101206
$226$	0	0
$227$	−1.82427	−0.121081	−0.0605405	−	0.998166i	$-0.519282\pi$
$227$	−1.82427	−0.121081	−0.0605405	+	0.998166i	$0.519282\pi$
$228$	0	0
$229$	14.2104	0.939051	0.469526	−	0.882919i	$-0.344425\pi$
$229$	14.2104	0.939051	0.469526	+	0.882919i	$0.344425\pi$
$230$	0	0
$231$	−7.68690	−0.505761
$232$	0	0
$233$	−17.3160	−1.13441	−0.567206	−	0.823576i	$-0.691976\pi$
$233$	−17.3160	−1.13441	−0.567206	+	0.823576i	$0.691976\pi$
$234$	0	0
$235$	−12.3853	−0.807925
$236$	0	0
$237$	33.3231	2.16457
$238$	0	0
$239$	6.84055	0.442479	0.221239	−	0.975220i	$-0.428990\pi$
$239$	6.84055	0.442479	0.221239	+	0.975220i	$0.428990\pi$
$240$	0	0
$241$	−7.61536	−0.490548	−0.245274	−	0.969454i	$-0.578878\pi$
$241$	−7.61536	−0.490548	−0.245274	+	0.969454i	$0.578878\pi$
$242$	0	0
$243$	14.4624	0.927762
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	11.5913	0.737538
$248$	0	0
$249$	7.97105	0.505145
$250$	0	0
$251$	11.5603	0.729679	0.364839	−	0.931071i	$-0.381124\pi$
$251$	11.5603	0.729679	0.364839	+	0.931071i	$0.381124\pi$
$252$	0	0
$253$	7.42820	0.467007
$254$	0	0
$255$	−0.432581	−0.0270893
$256$	0	0
$257$	27.6186	1.72280	0.861399	−	0.507929i	$-0.169589\pi$
$257$	27.6186	1.72280	0.861399	+	0.507929i	$0.169589\pi$
$258$	0	0
$259$	7.37711	0.458391
$260$	0	0
$261$	−1.48458	−0.0918933
$262$	0	0
$263$	18.8017	1.15936	0.579682	−	0.814843i	$-0.303177\pi$
$263$	18.8017	1.15936	0.579682	+	0.814843i	$0.303177\pi$
$264$	0	0
$265$	−4.07093	−0.250075
$266$	0	0
$267$	−0.527746	−0.0322975
$268$	0	0
$269$	−4.85832	−0.296217	−0.148109	−	0.988971i	$-0.547318\pi$
$269$	−4.85832	−0.296217	−0.148109	+	0.988971i	$0.547318\pi$
$270$	0	0
$271$	2.47478	0.150332	0.0751662	−	0.997171i	$-0.476051\pi$
$271$	2.47478	0.150332	0.0751662	+	0.997171i	$0.476051\pi$
$272$	0	0
$273$	14.5942	0.883280
$274$	0	0
$275$	−3.61637	−0.218076
$276$	0	0
$277$	27.3870	1.64552	0.822762	−	0.568386i	$-0.192432\pi$
$277$	27.3870	1.64552	0.822762	+	0.568386i	$0.192432\pi$
$278$	0	0
$279$	15.0594	0.901584
$280$	0	0
$281$	24.3145	1.45048	0.725240	−	0.688496i	$-0.241729\pi$
$281$	24.3145	1.45048	0.725240	+	0.688496i	$0.241729\pi$
$282$	0	0
$283$	31.0053	1.84308	0.921538	−	0.388288i	$-0.126933\pi$
$283$	31.0053	1.84308	0.921538	+	0.388288i	$0.126933\pi$
$284$	0	0
$285$	−3.58847	−0.212562
$286$	0	0
$287$	1.14788	0.0677574
$288$	0	0
$289$	−16.9586	−0.997564
$290$	0	0
$291$	−25.9249	−1.51974
$292$	0	0
$293$	24.1171	1.40894	0.704468	−	0.709736i	$-0.251186\pi$
$293$	24.1171	1.40894	0.704468	+	0.709736i	$0.251186\pi$
$294$	0	0
$295$	1.41508	0.0823891
$296$	0	0
$297$	−11.3912	−0.660987
$298$	0	0
$299$	−14.1030	−0.815599
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	39.2978	2.25760
$304$	0	0
$305$	9.13417	0.523021
$306$	0	0
$307$	17.1224	0.977225	0.488613	−	0.872501i	$-0.337503\pi$
$307$	17.1224	0.977225	0.488613	+	0.872501i	$0.337503\pi$
$308$	0	0
$309$	−14.3144	−0.814320
$310$	0	0
$311$	6.22624	0.353057	0.176529	−	0.984295i	$-0.443513\pi$
$311$	6.22624	0.353057	0.176529	+	0.984295i	$0.443513\pi$
$312$	0	0
$313$	3.86565	0.218499	0.109250	−	0.994014i	$-0.465155\pi$
$313$	3.86565	0.218499	0.109250	+	0.994014i	$0.465155\pi$
$314$	0	0
$315$	−1.51810	−0.0855351
$316$	0	0
$317$	30.3126	1.70252	0.851262	−	0.524741i	$-0.175838\pi$
$317$	30.3126	1.70252	0.851262	+	0.524741i	$0.175838\pi$
$318$	0	0
$319$	3.53654	0.198008
$320$	0	0
$321$	−30.8545	−1.72213
$322$	0	0
$323$	0.343574	0.0191170
$324$	0	0
$325$	6.86597	0.380856
$326$	0	0
$327$	−23.2169	−1.28390
$328$	0	0
$329$	12.3853	0.682822
$330$	0	0
$331$	12.4759	0.685739	0.342869	−	0.939383i	$-0.388601\pi$
$331$	12.4759	0.685739	0.342869	+	0.939383i	$0.388601\pi$
$332$	0	0
$333$	−11.1992	−0.613711
$334$	0	0
$335$	−4.02784	−0.220065
$336$	0	0
$337$	−35.0062	−1.90691	−0.953453	−	0.301540i	$-0.902499\pi$
$337$	−35.0062	−1.90691	−0.953453	+	0.301540i	$0.902499\pi$
$338$	0	0
$339$	−14.3374	−0.778699
$340$	0	0
$341$	−35.8742	−1.94270
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.36604	0.235060
$346$	0	0
$347$	14.2228	0.763522	0.381761	−	0.924261i	$-0.375318\pi$
$347$	14.2228	0.763522	0.381761	+	0.924261i	$0.375318\pi$
$348$	0	0
$349$	2.14192	0.114655	0.0573273	−	0.998355i	$-0.481742\pi$
$349$	2.14192	0.114655	0.0573273	+	0.998355i	$0.481742\pi$
$350$	0	0
$351$	21.6272	1.15437
$352$	0	0
$353$	24.3442	1.29571	0.647854	−	0.761764i	$-0.275667\pi$
$353$	24.3442	1.29571	0.647854	+	0.761764i	$0.275667\pi$
$354$	0	0
$355$	−2.28033	−0.121027
$356$	0	0
$357$	0.432581	0.0228946
$358$	0	0
$359$	26.4991	1.39857	0.699284	−	0.714844i	$-0.253502\pi$
$359$	26.4991	1.39857	0.699284	+	0.714844i	$0.253502\pi$
$360$	0	0
$361$	−16.1499	−0.849994
$362$	0	0
$363$	−4.41731	−0.231849
$364$	0	0
$365$	16.2886	0.852582
$366$	0	0
$367$	−11.8277	−0.617402	−0.308701	−	0.951159i	$-0.599894\pi$
$367$	−11.8277	−0.617402	−0.308701	+	0.951159i	$0.599894\pi$
$368$	0	0
$369$	−1.74260	−0.0907160
$370$	0	0
$371$	4.07093	0.211352
$372$	0	0
$373$	−28.7320	−1.48769	−0.743844	−	0.668353i	$-0.766999\pi$
$373$	−28.7320	−1.48769	−0.743844	+	0.668353i	$0.766999\pi$
$374$	0	0
$375$	−2.12558	−0.109765
$376$	0	0
$377$	−6.71439	−0.345809
$378$	0	0
$379$	26.1656	1.34403	0.672017	−	0.740536i	$-0.265428\pi$
$379$	26.1656	1.34403	0.672017	+	0.740536i	$0.265428\pi$
$380$	0	0
$381$	−11.6164	−0.595127
$382$	0	0
$383$	−18.3510	−0.937690	−0.468845	−	0.883280i	$-0.655330\pi$
$383$	−18.3510	−0.937690	−0.468845	+	0.883280i	$0.655330\pi$
$384$	0	0
$385$	3.61637	0.184308
$386$	0	0
$387$	1.51810	0.0771692
$388$	0	0
$389$	−15.1945	−0.770392	−0.385196	−	0.922835i	$-0.625866\pi$
$389$	−15.1945	−0.770392	−0.385196	+	0.922835i	$0.625866\pi$
$390$	0	0
$391$	−0.418023	−0.0211403
$392$	0	0
$393$	30.5510	1.54110
$394$	0	0
$395$	−15.6772	−0.788804
$396$	0	0
$397$	18.0222	0.904511	0.452255	−	0.891889i	$-0.350620\pi$
$397$	18.0222	0.904511	0.452255	+	0.891889i	$0.350620\pi$
$398$	0	0
$399$	3.58847	0.179648
$400$	0	0
$401$	−21.5994	−1.07863	−0.539313	−	0.842106i	$-0.681316\pi$
$401$	−21.5994	−1.07863	−0.539313	+	0.842106i	$0.681316\pi$
$402$	0	0
$403$	68.1100	3.39280
$404$	0	0
$405$	−11.2497	−0.559001
$406$	0	0
$407$	26.6784	1.32240
$408$	0	0
$409$	31.6355	1.56427	0.782137	−	0.623106i	$-0.214130\pi$
$409$	31.6355	1.56427	0.782137	+	0.623106i	$0.214130\pi$
$410$	0	0
$411$	−22.5496	−1.11229
$412$	0	0
$413$	−1.41508	−0.0696315
$414$	0	0
$415$	−3.75006	−0.184083
$416$	0	0
$417$	−10.3769	−0.508161
$418$	0	0
$419$	10.4322	0.509648	0.254824	−	0.966987i	$-0.417982\pi$
$419$	10.4322	0.509648	0.254824	+	0.966987i	$0.417982\pi$
$420$	0	0
$421$	22.8508	1.11368	0.556839	−	0.830621i	$-0.312014\pi$
$421$	22.8508	1.11368	0.556839	+	0.830621i	$0.312014\pi$
$422$	0	0
$423$	−18.8020	−0.914186
$424$	0	0
$425$	0.203512	0.00987178
$426$	0	0
$427$	−9.13417	−0.442033
$428$	0	0
$429$	52.7780	2.54815
$430$	0	0
$431$	−18.3644	−0.884582	−0.442291	−	0.896872i	$-0.645834\pi$
$431$	−18.3644	−0.884582	−0.442291	+	0.896872i	$0.645834\pi$
$432$	0	0
$433$	−5.50016	−0.264321	−0.132160	−	0.991228i	$-0.542191\pi$
$433$	−5.50016	−0.264321	−0.132160	+	0.991228i	$0.542191\pi$
$434$	0	0
$435$	2.07865	0.0996639
$436$	0	0
$437$	−3.46770	−0.165882
$438$	0	0
$439$	−30.9685	−1.47805	−0.739023	−	0.673680i	$-0.764712\pi$
$439$	−30.9685	−1.47805	−0.739023	+	0.673680i	$0.764712\pi$
$440$	0	0
$441$	1.51810	0.0722903
$442$	0	0
$443$	24.2609	1.15267	0.576334	−	0.817214i	$-0.304483\pi$
$443$	24.2609	1.15267	0.576334	+	0.817214i	$0.304483\pi$
$444$	0	0
$445$	0.248283	0.0117697
$446$	0	0
$447$	−32.3342	−1.52936
$448$	0	0
$449$	−2.41757	−0.114092	−0.0570462	−	0.998372i	$-0.518168\pi$
$449$	−2.41757	−0.114092	−0.0570462	+	0.998372i	$0.518168\pi$
$450$	0	0
$451$	4.15117	0.195471
$452$	0	0
$453$	−13.0384	−0.612598
$454$	0	0
$455$	−6.86597	−0.321882
$456$	0	0
$457$	−11.2929	−0.528261	−0.264131	−	0.964487i	$-0.585085\pi$
$457$	−11.2929	−0.528261	−0.264131	+	0.964487i	$0.585085\pi$
$458$	0	0
$459$	0.641043	0.0299213
$460$	0	0
$461$	6.84977	0.319026	0.159513	−	0.987196i	$-0.449008\pi$
$461$	6.84977	0.319026	0.159513	+	0.987196i	$0.449008\pi$
$462$	0	0
$463$	22.8080	1.05998	0.529989	−	0.848004i	$-0.322196\pi$
$463$	22.8080	1.05998	0.529989	+	0.848004i	$0.322196\pi$
$464$	0	0
$465$	−21.0856	−0.977823
$466$	0	0
$467$	8.78758	0.406641	0.203320	−	0.979112i	$-0.434827\pi$
$467$	8.78758	0.406641	0.203320	+	0.979112i	$0.434827\pi$
$468$	0	0
$469$	4.02784	0.185989
$470$	0	0
$471$	11.1195	0.512359
$472$	0	0
$473$	−3.61637	−0.166281
$474$	0	0
$475$	1.68823	0.0774612
$476$	0	0
$477$	−6.18006	−0.282966
$478$	0	0
$479$	3.86387	0.176545	0.0882723	−	0.996096i	$-0.471865\pi$
$479$	3.86387	0.176545	0.0882723	+	0.996096i	$0.471865\pi$
$480$	0	0
$481$	−50.6510	−2.30949
$482$	0	0
$483$	−4.36604	−0.198662
$484$	0	0
$485$	12.1966	0.553819
$486$	0	0
$487$	27.7107	1.25569	0.627846	−	0.778337i	$-0.283937\pi$
$487$	27.7107	1.25569	0.627846	+	0.778337i	$0.283937\pi$
$488$	0	0
$489$	15.2552	0.689864
$490$	0	0
$491$	−20.8622	−0.941499	−0.470749	−	0.882267i	$-0.656016\pi$
$491$	−20.8622	−0.941499	−0.470749	+	0.882267i	$0.656016\pi$
$492$	0	0
$493$	−0.199019	−0.00896336
$494$	0	0
$495$	−5.49001	−0.246758
$496$	0	0
$497$	2.28033	0.102287
$498$	0	0
$499$	−21.8346	−0.977453	−0.488727	−	0.872437i	$-0.662538\pi$
$499$	−21.8346	−0.977453	−0.488727	+	0.872437i	$0.662538\pi$
$500$	0	0
$501$	12.3223	0.550522
$502$	0	0
$503$	40.0214	1.78447	0.892233	−	0.451575i	$-0.149138\pi$
$503$	40.0214	1.78447	0.892233	+	0.451575i	$0.149138\pi$
$504$	0	0
$505$	−18.4880	−0.822705
$506$	0	0
$507$	−72.5707	−3.22298
$508$	0	0
$509$	35.9440	1.59319	0.796595	−	0.604513i	$-0.206632\pi$
$509$	35.9440	1.59319	0.796595	+	0.604513i	$0.206632\pi$
$510$	0	0
$511$	−16.2886	−0.720564
$512$	0	0
$513$	5.31776	0.234785
$514$	0	0
$515$	6.73437	0.296752
$516$	0	0
$517$	44.7898	1.96985
$518$	0	0
$519$	41.0601	1.80234
$520$	0	0
$521$	−19.1027	−0.836904	−0.418452	−	0.908239i	$-0.637427\pi$
$521$	−19.1027	−0.836904	−0.418452	+	0.908239i	$0.637427\pi$
$522$	0	0
$523$	−2.58018	−0.112823	−0.0564116	−	0.998408i	$-0.517966\pi$
$523$	−2.58018	−0.112823	−0.0564116	+	0.998408i	$0.517966\pi$
$524$	0	0
$525$	2.12558	0.0927680
$526$	0	0
$527$	2.01882	0.0879414
$528$	0	0
$529$	−18.7809	−0.816561
$530$	0	0
$531$	2.14823	0.0932251
$532$	0	0
$533$	−7.88133	−0.341378
$534$	0	0
$535$	14.5158	0.627573
$536$	0	0
$537$	23.4306	1.01110
$538$	0	0
$539$	−3.61637	−0.155768
$540$	0	0
$541$	7.50617	0.322716	0.161358	−	0.986896i	$-0.448413\pi$
$541$	7.50617	0.322716	0.161358	+	0.986896i	$0.448413\pi$
$542$	0	0
$543$	0.726781	0.0311892
$544$	0	0
$545$	10.9226	0.467873
$546$	0	0
$547$	−4.71626	−0.201653	−0.100826	−	0.994904i	$-0.532149\pi$
$547$	−4.71626	−0.201653	−0.100826	+	0.994904i	$0.532149\pi$
$548$	0	0
$549$	13.8666	0.591810
$550$	0	0
$551$	−1.65096	−0.0703331
$552$	0	0
$553$	15.6772	0.666661
$554$	0	0
$555$	15.6806	0.665606
$556$	0	0
$557$	34.1054	1.44509	0.722547	−	0.691322i	$-0.242971\pi$
$557$	34.1054	1.44509	0.722547	+	0.691322i	$0.242971\pi$
$558$	0	0
$559$	6.86597	0.290400
$560$	0	0
$561$	1.56438	0.0660480
$562$	0	0
$563$	20.6417	0.869943	0.434972	−	0.900444i	$-0.356758\pi$
$563$	20.6417	0.869943	0.434972	+	0.900444i	$0.356758\pi$
$564$	0	0
$565$	6.74515	0.283771
$566$	0	0
$567$	11.2497	0.472442
$568$	0	0
$569$	18.0816	0.758018	0.379009	−	0.925393i	$-0.376265\pi$
$569$	18.0816	0.758018	0.379009	+	0.925393i	$0.376265\pi$
$570$	0	0
$571$	−15.4261	−0.645561	−0.322781	−	0.946474i	$-0.604618\pi$
$571$	−15.4261	−0.645561	−0.322781	+	0.946474i	$0.604618\pi$
$572$	0	0
$573$	15.2037	0.635145
$574$	0	0
$575$	−2.05405	−0.0856596
$576$	0	0
$577$	−17.0376	−0.709283	−0.354641	−	0.935002i	$-0.615397\pi$
$577$	−17.0376	−0.709283	−0.354641	+	0.935002i	$0.615397\pi$
$578$	0	0
$579$	−27.3407	−1.13624
$580$	0	0
$581$	3.75006	0.155579
$582$	0	0
$583$	14.7220	0.609723
$584$	0	0
$585$	10.4232	0.430947
$586$	0	0
$587$	32.4346	1.33872	0.669359	−	0.742939i	$-0.266569\pi$
$587$	32.4346	1.33872	0.669359	+	0.742939i	$0.266569\pi$
$588$	0	0
$589$	16.7471	0.690053
$590$	0	0
$591$	−26.1406	−1.07528
$592$	0	0
$593$	−5.47020	−0.224634	−0.112317	−	0.993672i	$-0.535827\pi$
$593$	−5.47020	−0.224634	−0.112317	+	0.993672i	$0.535827\pi$
$594$	0	0
$595$	−0.203512	−0.00834317
$596$	0	0
$597$	32.9951	1.35040
$598$	0	0
$599$	−2.00986	−0.0821209	−0.0410604	−	0.999157i	$-0.513074\pi$
$599$	−2.00986	−0.0821209	−0.0410604	+	0.999157i	$0.513074\pi$
$600$	0	0
$601$	−34.9542	−1.42581	−0.712906	−	0.701260i	$-0.752621\pi$
$601$	−34.9542	−1.42581	−0.712906	+	0.701260i	$0.752621\pi$
$602$	0	0
$603$	−6.11466	−0.249008
$604$	0	0
$605$	2.07817	0.0844895
$606$	0	0
$607$	39.9094	1.61987	0.809936	−	0.586518i	$-0.199502\pi$
$607$	39.9094	1.61987	0.809936	+	0.586518i	$0.199502\pi$
$608$	0	0
$609$	−2.07865	−0.0842313
$610$	0	0
$611$	−85.0369	−3.44022
$612$	0	0
$613$	7.92330	0.320019	0.160009	−	0.987115i	$-0.448848\pi$
$613$	7.92330	0.320019	0.160009	+	0.987115i	$0.448848\pi$
$614$	0	0
$615$	2.43992	0.0983870
$616$	0	0
$617$	−46.0919	−1.85559	−0.927795	−	0.373090i	$-0.878298\pi$
$617$	−46.0919	−1.85559	−0.927795	+	0.373090i	$0.878298\pi$
$618$	0	0
$619$	32.7439	1.31609	0.658044	−	0.752979i	$-0.271384\pi$
$619$	32.7439	1.31609	0.658044	+	0.752979i	$0.271384\pi$
$620$	0	0
$621$	−6.47005	−0.259634
$622$	0	0
$623$	−0.248283	−0.00994725
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	12.9772	0.518261
$628$	0	0
$629$	−1.50133	−0.0598619
$630$	0	0
$631$	12.7513	0.507623	0.253811	−	0.967254i	$-0.418316\pi$
$631$	12.7513	0.507623	0.253811	+	0.967254i	$0.418316\pi$
$632$	0	0
$633$	23.2700	0.924898
$634$	0	0
$635$	5.46506	0.216874
$636$	0	0
$637$	6.86597	0.272040
$638$	0	0
$639$	−3.46176	−0.136945
$640$	0	0
$641$	39.8189	1.57275	0.786375	−	0.617749i	$-0.211955\pi$
$641$	39.8189	1.57275	0.786375	+	0.617749i	$0.211955\pi$
$642$	0	0
$643$	15.0219	0.592406	0.296203	−	0.955125i	$-0.404280\pi$
$643$	15.0219	0.592406	0.296203	+	0.955125i	$0.404280\pi$
$644$	0	0
$645$	−2.12558	−0.0836947
$646$	0	0
$647$	−12.8881	−0.506683	−0.253341	−	0.967377i	$-0.581530\pi$
$647$	−12.8881	−0.506683	−0.253341	+	0.967377i	$0.581530\pi$
$648$	0	0
$649$	−5.11745	−0.200878
$650$	0	0
$651$	21.0856	0.826411
$652$	0	0
$653$	34.1888	1.33791	0.668956	−	0.743302i	$-0.266741\pi$
$653$	34.1888	1.33791	0.668956	+	0.743302i	$0.266741\pi$
$654$	0	0
$655$	−14.3730	−0.561600
$656$	0	0
$657$	24.7276	0.964716
$658$	0	0
$659$	30.5051	1.18831	0.594156	−	0.804350i	$-0.297486\pi$
$659$	30.5051	1.18831	0.594156	+	0.804350i	$0.297486\pi$
$660$	0	0
$661$	−7.52970	−0.292871	−0.146436	−	0.989220i	$-0.546780\pi$
$661$	−7.52970	−0.292871	−0.146436	+	0.989220i	$0.546780\pi$
$662$	0	0
$663$	−2.97009	−0.115349
$664$	0	0
$665$	−1.68823	−0.0654667
$666$	0	0
$667$	2.00870	0.0777771
$668$	0	0
$669$	−38.9828	−1.50716
$670$	0	0
$671$	−33.0326	−1.27521
$672$	0	0
$673$	−0.477892	−0.0184214	−0.00921069	−	0.999958i	$-0.502932\pi$
$673$	−0.477892	−0.0184214	−0.00921069	+	0.999958i	$0.502932\pi$
$674$	0	0
$675$	3.14991	0.121240
$676$	0	0
$677$	−39.9785	−1.53650	−0.768249	−	0.640151i	$-0.778872\pi$
$677$	−39.9785	−1.53650	−0.768249	+	0.640151i	$0.778872\pi$
$678$	0	0
$679$	−12.1966	−0.468063
$680$	0	0
$681$	3.87763	0.148591
$682$	0	0
$683$	37.3581	1.42947	0.714734	−	0.699396i	$-0.246548\pi$
$683$	37.3581	1.42947	0.714734	+	0.699396i	$0.246548\pi$
$684$	0	0
$685$	10.6087	0.405337
$686$	0	0
$687$	−30.2054	−1.15241
$688$	0	0
$689$	−27.9509	−1.06484
$690$	0	0
$691$	−0.576982	−0.0219494	−0.0109747	−	0.999940i	$-0.503493\pi$
$691$	−0.576982	−0.0219494	−0.0109747	+	0.999940i	$0.503493\pi$
$692$	0	0
$693$	5.49001	0.208548
$694$	0	0
$695$	4.88193	0.185182
$696$	0	0
$697$	−0.233608	−0.00884852
$698$	0	0
$699$	36.8067	1.39216
$700$	0	0
$701$	−45.2920	−1.71065	−0.855327	−	0.518089i	$-0.826644\pi$
$701$	−45.2920	−1.71065	−0.855327	+	0.518089i	$0.826644\pi$
$702$	0	0
$703$	−12.4542	−0.469721
$704$	0	0
$705$	26.3259	0.991490
$706$	0	0
$707$	18.4880	0.695313
$708$	0	0
$709$	25.9603	0.974960	0.487480	−	0.873134i	$-0.337916\pi$
$709$	25.9603	0.974960	0.487480	+	0.873134i	$0.337916\pi$
$710$	0	0
$711$	−23.7994	−0.892549
$712$	0	0
$713$	−20.3760	−0.763087
$714$	0	0
$715$	−24.8299	−0.928587
$716$	0	0
$717$	−14.5402	−0.543012
$718$	0	0
$719$	5.79814	0.216234	0.108117	−	0.994138i	$-0.465518\pi$
$719$	5.79814	0.216234	0.108117	+	0.994138i	$0.465518\pi$
$720$	0	0
$721$	−6.73437	−0.250801
$722$	0	0
$723$	16.1871	0.602004
$724$	0	0
$725$	−0.977923	−0.0363191
$726$	0	0
$727$	−27.1023	−1.00517	−0.502585	−	0.864528i	$-0.667618\pi$
$727$	−27.1023	−1.00517	−0.502585	+	0.864528i	$0.667618\pi$
$728$	0	0
$729$	3.00805	0.111409
$730$	0	0
$731$	0.203512	0.00752716
$732$	0	0
$733$	−48.6633	−1.79742	−0.898710	−	0.438542i	$-0.855495\pi$
$733$	−48.6633	−1.79742	−0.898710	+	0.438542i	$0.855495\pi$
$734$	0	0
$735$	−2.12558	−0.0784033
$736$	0	0
$737$	14.5662	0.536553
$738$	0	0
$739$	−16.0323	−0.589756	−0.294878	−	0.955535i	$-0.595279\pi$
$739$	−16.0323	−0.589756	−0.294878	+	0.955535i	$0.595279\pi$
$740$	0	0
$741$	−24.6383	−0.905111
$742$	0	0
$743$	34.3944	1.26181	0.630904	−	0.775861i	$-0.282684\pi$
$743$	34.3944	1.26181	0.630904	+	0.775861i	$0.282684\pi$
$744$	0	0
$745$	15.2119	0.557323
$746$	0	0
$747$	−5.69295	−0.208294
$748$	0	0
$749$	−14.5158	−0.530396
$750$	0	0
$751$	19.7932	0.722263	0.361131	−	0.932515i	$-0.382391\pi$
$751$	19.7932	0.722263	0.361131	+	0.932515i	$0.382391\pi$
$752$	0	0
$753$	−24.5723	−0.895465
$754$	0	0
$755$	6.13405	0.223241
$756$	0	0
$757$	−32.9775	−1.19859	−0.599294	−	0.800529i	$-0.704552\pi$
$757$	−32.9775	−1.19859	−0.599294	+	0.800529i	$0.704552\pi$
$758$	0	0
$759$	−15.7892	−0.573113
$760$	0	0
$761$	32.1442	1.16523	0.582614	−	0.812749i	$-0.302030\pi$
$761$	32.1442	1.16523	0.582614	+	0.812749i	$0.302030\pi$
$762$	0	0
$763$	−10.9226	−0.395425
$764$	0	0
$765$	0.308951	0.0111701
$766$	0	0
$767$	9.71589	0.350820
$768$	0	0
$769$	13.2409	0.477480	0.238740	−	0.971084i	$-0.423266\pi$
$769$	13.2409	0.477480	0.238740	+	0.971084i	$0.423266\pi$
$770$	0	0
$771$	−58.7055	−2.11423
$772$	0	0
$773$	12.2934	0.442163	0.221082	−	0.975255i	$-0.429041\pi$
$773$	12.2934	0.442163	0.221082	+	0.975255i	$0.429041\pi$
$774$	0	0
$775$	9.91993	0.356335
$776$	0	0
$777$	−15.6806	−0.562540
$778$	0	0
$779$	−1.93789	−0.0694320
$780$	0	0
$781$	8.24652	0.295084
$782$	0	0
$783$	−3.08036	−0.110083
$784$	0	0
$785$	−5.23127	−0.186712
$786$	0	0
$787$	−41.8768	−1.49275	−0.746374	−	0.665527i	$-0.768207\pi$
$787$	−41.8768	−1.49275	−0.746374	+	0.665527i	$0.768207\pi$
$788$	0	0
$789$	−39.9646	−1.42278
$790$	0	0
$791$	−6.74515	−0.239830
$792$	0	0
$793$	62.7149	2.22707
$794$	0	0
$795$	8.65309	0.306893
$796$	0	0
$797$	37.8438	1.34049	0.670247	−	0.742138i	$-0.266188\pi$
$797$	37.8438	1.34049	0.670247	+	0.742138i	$0.266188\pi$
$798$	0	0
$799$	−2.52055	−0.0891706
$800$	0	0
$801$	0.376918	0.0133177
$802$	0	0
$803$	−58.9056	−2.07873
$804$	0	0
$805$	2.05405	0.0723956
$806$	0	0
$807$	10.3268	0.363519
$808$	0	0
$809$	25.4553	0.894962	0.447481	−	0.894293i	$-0.352321\pi$
$809$	25.4553	0.894962	0.447481	+	0.894293i	$0.352321\pi$
$810$	0	0
$811$	−0.878965	−0.0308646	−0.0154323	−	0.999881i	$-0.504912\pi$
$811$	−0.878965	−0.0308646	−0.0154323	+	0.999881i	$0.504912\pi$
$812$	0	0
$813$	−5.26035	−0.184489
$814$	0	0
$815$	−7.17696	−0.251398
$816$	0	0
$817$	1.68823	0.0590636
$818$	0	0
$819$	−10.4232	−0.364216
$820$	0	0
$821$	−20.7666	−0.724760	−0.362380	−	0.932030i	$-0.618036\pi$
$821$	−20.7666	−0.724760	−0.362380	+	0.932030i	$0.618036\pi$
$822$	0	0
$823$	−22.8948	−0.798062	−0.399031	−	0.916937i	$-0.630653\pi$
$823$	−22.8948	−0.798062	−0.399031	+	0.916937i	$0.630653\pi$
$824$	0	0
$825$	7.68690	0.267623
$826$	0	0
$827$	−52.6428	−1.83057	−0.915284	−	0.402808i	$-0.868034\pi$
$827$	−52.6428	−1.83057	−0.915284	+	0.402808i	$0.868034\pi$
$828$	0	0
$829$	−35.4647	−1.23174	−0.615871	−	0.787847i	$-0.711196\pi$
$829$	−35.4647	−1.23174	−0.615871	+	0.787847i	$0.711196\pi$
$830$	0	0
$831$	−58.2133	−2.01940
$832$	0	0
$833$	0.203512	0.00705127
$834$	0	0
$835$	−5.79717	−0.200619
$836$	0	0
$837$	31.2469	1.08005
$838$	0	0
$839$	25.4716	0.879377	0.439688	−	0.898150i	$-0.355089\pi$
$839$	25.4716	0.879377	0.439688	+	0.898150i	$0.355089\pi$
$840$	0	0
$841$	−28.0437	−0.967023
$842$	0	0
$843$	−51.6824	−1.78004
$844$	0	0
$845$	34.1416	1.17451
$846$	0	0
$847$	−2.07817	−0.0714067
$848$	0	0
$849$	−65.9044	−2.26183
$850$	0	0
$851$	15.1529	0.519435
$852$	0	0
$853$	46.9553	1.60772	0.803859	−	0.594820i	$-0.202777\pi$
$853$	46.9553	1.60772	0.803859	+	0.594820i	$0.202777\pi$
$854$	0	0
$855$	2.56289	0.0876491
$856$	0	0
$857$	2.91674	0.0996339	0.0498170	−	0.998758i	$-0.484136\pi$
$857$	2.91674	0.0996339	0.0498170	+	0.998758i	$0.484136\pi$
$858$	0	0
$859$	4.92713	0.168112	0.0840558	−	0.996461i	$-0.473213\pi$
$859$	4.92713	0.168112	0.0840558	+	0.996461i	$0.473213\pi$
$860$	0	0
$861$	−2.43992	−0.0831522
$862$	0	0
$863$	−15.5366	−0.528872	−0.264436	−	0.964403i	$-0.585186\pi$
$863$	−15.5366	−0.528872	−0.264436	+	0.964403i	$0.585186\pi$
$864$	0	0
$865$	−19.3171	−0.656801
$866$	0	0
$867$	36.0469	1.22422
$868$	0	0
$869$	56.6945	1.92323
$870$	0	0
$871$	−27.6551	−0.937056
$872$	0	0
$873$	18.5156	0.626659
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	39.7847	1.34343	0.671717	−	0.740808i	$-0.265557\pi$
$877$	39.7847	1.34343	0.671717	+	0.740808i	$0.265557\pi$
$878$	0	0
$879$	−51.2629	−1.72905
$880$	0	0
$881$	−36.5325	−1.23081	−0.615406	−	0.788211i	$-0.711008\pi$
$881$	−36.5325	−1.23081	−0.615406	+	0.788211i	$0.711008\pi$
$882$	0	0
$883$	4.30919	0.145016	0.0725079	−	0.997368i	$-0.476900\pi$
$883$	4.30919	0.145016	0.0725079	+	0.997368i	$0.476900\pi$
$884$	0	0
$885$	−3.00786	−0.101108
$886$	0	0
$887$	27.8062	0.933640	0.466820	−	0.884352i	$-0.345400\pi$
$887$	27.8062	0.933640	0.466820	+	0.884352i	$0.345400\pi$
$888$	0	0
$889$	−5.46506	−0.183292
$890$	0	0
$891$	40.6830	1.36293
$892$	0	0
$893$	−20.9091	−0.699698
$894$	0	0
$895$	−11.0231	−0.368463
$896$	0	0
$897$	29.9771	1.00091
$898$	0	0
$899$	−9.70093	−0.323544
$900$	0	0
$901$	−0.828482	−0.0276007
$902$	0	0
$903$	2.12558	0.0707349
$904$	0	0
$905$	−0.341921	−0.0113658
$906$	0	0
$907$	15.6464	0.519529	0.259765	−	0.965672i	$-0.416355\pi$
$907$	15.6464	0.519529	0.259765	+	0.965672i	$0.416355\pi$
$908$	0	0
$909$	−28.0666	−0.930910
$910$	0	0
$911$	−37.6468	−1.24729	−0.623647	−	0.781706i	$-0.714350\pi$
$911$	−37.6468	−1.24729	−0.623647	+	0.781706i	$0.714350\pi$
$912$	0	0
$913$	13.5616	0.448824
$914$	0	0
$915$	−19.4154	−0.641854
$916$	0	0
$917$	14.3730	0.474639
$918$	0	0
$919$	−12.8261	−0.423096	−0.211548	−	0.977368i	$-0.567850\pi$
$919$	−12.8261	−0.423096	−0.211548	+	0.977368i	$0.567850\pi$
$920$	0	0
$921$	−36.3950	−1.19926
$922$	0	0
$923$	−15.6567	−0.515345
$924$	0	0
$925$	−7.37711	−0.242558
$926$	0	0
$927$	10.2234	0.335781
$928$	0	0
$929$	−5.99315	−0.196629	−0.0983144	−	0.995155i	$-0.531345\pi$
$929$	−5.99315	−0.196629	−0.0983144	+	0.995155i	$0.531345\pi$
$930$	0	0
$931$	1.68823	0.0553294
$932$	0	0
$933$	−13.2344	−0.433274
$934$	0	0
$935$	−0.735975	−0.0240690
$936$	0	0
$937$	−13.4848	−0.440530	−0.220265	−	0.975440i	$-0.570692\pi$
$937$	−13.4848	−0.440530	−0.220265	+	0.975440i	$0.570692\pi$
$938$	0	0
$939$	−8.21675	−0.268144
$940$	0	0
$941$	4.56502	0.148815	0.0744077	−	0.997228i	$-0.476293\pi$
$941$	4.56502	0.148815	0.0744077	+	0.997228i	$0.476293\pi$
$942$	0	0
$943$	2.35780	0.0767806
$944$	0	0
$945$	−3.14991	−0.102466
$946$	0	0
$947$	−38.9354	−1.26523	−0.632615	−	0.774466i	$-0.718019\pi$
$947$	−38.9354	−1.26523	−0.632615	+	0.774466i	$0.718019\pi$
$948$	0	0
$949$	111.837	3.63038
$950$	0	0
$951$	−64.4319	−2.08935
$952$	0	0
$953$	2.28226	0.0739297	0.0369648	−	0.999317i	$-0.488231\pi$
$953$	2.28226	0.0739297	0.0369648	+	0.999317i	$0.488231\pi$
$954$	0	0
$955$	−7.15274	−0.231457
$956$	0	0
$957$	−7.51720	−0.242996
$958$	0	0
$959$	−10.6087	−0.342572
$960$	0	0
$961$	67.4051	2.17436
$962$	0	0
$963$	22.0364	0.710113
$964$	0	0
$965$	12.8627	0.414065
$966$	0	0
$967$	22.6559	0.728566	0.364283	−	0.931288i	$-0.381314\pi$
$967$	22.6559	0.728566	0.364283	+	0.931288i	$0.381314\pi$
$968$	0	0
$969$	−0.730296	−0.0234605
$970$	0	0
$971$	24.5994	0.789432	0.394716	−	0.918803i	$-0.370843\pi$
$971$	24.5994	0.789432	0.394716	+	0.918803i	$0.370843\pi$
$972$	0	0
$973$	−4.88193	−0.156508
$974$	0	0
$975$	−14.5942	−0.467388
$976$	0	0
$977$	32.2717	1.03246	0.516232	−	0.856449i	$-0.327334\pi$
$977$	32.2717	1.03246	0.516232	+	0.856449i	$0.327334\pi$
$978$	0	0
$979$	−0.897884	−0.0286965
$980$	0	0
$981$	16.5816	0.529409
$982$	0	0
$983$	14.0159	0.447039	0.223520	−	0.974699i	$-0.428245\pi$
$983$	14.0159	0.447039	0.223520	+	0.974699i	$0.428245\pi$
$984$	0	0
$985$	12.2981	0.391850
$986$	0	0
$987$	−26.3259	−0.837962
$988$	0	0
$989$	−2.05405	−0.0653148
$990$	0	0
$991$	10.0795	0.320185	0.160093	−	0.987102i	$-0.448821\pi$
$991$	10.0795	0.320185	0.160093	+	0.987102i	$0.448821\pi$
$992$	0	0
$993$	−26.5186	−0.841542
$994$	0	0
$995$	−15.5229	−0.492108
$996$	0	0
$997$	−4.82168	−0.152704	−0.0763520	−	0.997081i	$-0.524327\pi$
$997$	−4.82168	−0.152704	−0.0763520	+	0.997081i	$0.524327\pi$
$998$	0	0
$999$	−23.2372	−0.735192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.k.1.3	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.k.1.3	✓	13	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12558 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.51810 q^{9} +O(q^{10})\)	\(q-2.12558 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.51810 q^{9} -3.61637 q^{11} +6.86597 q^{13} -2.12558 q^{15} +0.203512 q^{17} +1.68823 q^{19} +2.12558 q^{21} -2.05405 q^{23} +1.00000 q^{25} +3.14991 q^{27} -0.977923 q^{29} +9.91993 q^{31} +7.68690 q^{33} -1.00000 q^{35} -7.37711 q^{37} -14.5942 q^{39} -1.14788 q^{41} +1.00000 q^{43} +1.51810 q^{45} -12.3853 q^{47} +1.00000 q^{49} -0.432581 q^{51} -4.07093 q^{53} -3.61637 q^{55} -3.58847 q^{57} +1.41508 q^{59} +9.13417 q^{61} -1.51810 q^{63} +6.86597 q^{65} -4.02784 q^{67} +4.36604 q^{69} -2.28033 q^{71} +16.2886 q^{73} -2.12558 q^{75} +3.61637 q^{77} -15.6772 q^{79} -11.2497 q^{81} -3.75006 q^{83} +0.203512 q^{85} +2.07865 q^{87} +0.248283 q^{89} -6.86597 q^{91} -21.0856 q^{93} +1.68823 q^{95} +12.1966 q^{97} -5.49001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9}+O(q^{10}) \) 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9	\( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9} + 11 q^{13} + 16 q^{17} + 3 q^{23} + 13 q^{25} + 6 q^{27} + 10 q^{29} - q^{31} + 14 q^{33} - 13 q^{35} + 16 q^{37} - 14 q^{39} + 23 q^{41} + 13 q^{43} + 15 q^{45} + 2 q^{47} + 13 q^{49} + 4 q^{51} + 20 q^{53} + 22 q^{57} + 2 q^{59} + 5 q^{61} - 15 q^{63} + 11 q^{65} + 19 q^{67} + 16 q^{69} + 4 q^{71} + 34 q^{73} - 15 q^{79} + 17 q^{81} + 27 q^{83} + 16 q^{85} - 5 q^{87} + 3 q^{89} - 11 q^{91} + 35 q^{93} + 45 q^{97} + q^{99}+O(q^{100}) \) 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9 + 11 * q^13 + 16 * q^17 + 3 * q^23 + 13 * q^25 + 6 * q^27 + 10 * q^29 - q^31 + 14 * q^33 - 13 * q^35 + 16 * q^37 - 14 * q^39 + 23 * q^41 + 13 * q^43 + 15 * q^45 + 2 * q^47 + 13 * q^49 + 4 * q^51 + 20 * q^53 + 22 * q^57 + 2 * q^59 + 5 * q^61 - 15 * q^63 + 11 * q^65 + 19 * q^67 + 16 * q^69 + 4 * q^71 + 34 * q^73 - 15 * q^79 + 17 * q^81 + 27 * q^83 + 16 * q^85 - 5 * q^87 + 3 * q^89 - 11 * q^91 + 35 * q^93 + 45 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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