LMFDB - Embedded newform 6020.2.a.j.1.4

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} - x^{12} - 28 x^{11} + 26 x^{10} + 286 x^{9} - 235 x^{8} - 1298 x^{7} + 895 x^{6} + 2571 x^{5} + \cdots - 216 $ x^13 - x^12 - 28x^11 + 26x^10 + 286x^9 - 235x^8 - 1298x^7 + 895x^6 + 2571x^5 - 1548x^4 - 1833x^3 + 1119x^2 + 342*x - 216
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			1.4
Root			$1.72520$ 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)

$f(q)$	$=$	$q-1.72520 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.0236759 q^{9} +O(q^{10})$	$q-1.72520 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.0236759 q^{9} +1.64250 q^{11} +6.74628 q^{13} +1.72520 q^{15} +7.32820 q^{17} -1.33676 q^{19} +1.72520 q^{21} +3.73151 q^{23} +1.00000 q^{25} +5.21645 q^{27} +3.06534 q^{29} -6.07783 q^{31} -2.83365 q^{33} +1.00000 q^{35} +3.80756 q^{37} -11.6387 q^{39} -0.284168 q^{41} -1.00000 q^{43} +0.0236759 q^{45} -2.36027 q^{47} +1.00000 q^{49} -12.6426 q^{51} -10.2750 q^{53} -1.64250 q^{55} +2.30618 q^{57} -13.1870 q^{59} +12.1369 q^{61} +0.0236759 q^{63} -6.74628 q^{65} -11.6051 q^{67} -6.43762 q^{69} +15.0966 q^{71} -0.907421 q^{73} -1.72520 q^{75} -1.64250 q^{77} +0.835831 q^{79} -8.92841 q^{81} -8.20336 q^{83} -7.32820 q^{85} -5.28833 q^{87} -2.30093 q^{89} -6.74628 q^{91} +10.4855 q^{93} +1.33676 q^{95} -7.13319 q^{97} -0.0388877 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	$ 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) $ 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * 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
$3$	−1.72520	−0.996046	−0.498023	−	0.867164i	$-0.665941\pi$
$3$	−1.72520	−0.996046	−0.498023	+	0.867164i	$0.665941\pi$
$4$	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
$9$	−0.0236759	−0.00789195
$10$	0	0
$11$	1.64250	0.495233	0.247617	−	0.968858i	$-0.420353\pi$
$11$	1.64250	0.495233	0.247617	+	0.968858i	$0.420353\pi$
$12$	0	0
$13$	6.74628	1.87108	0.935540	−	0.353221i	$-0.114913\pi$
$13$	6.74628	1.87108	0.935540	+	0.353221i	$0.114913\pi$
$14$	0	0
$15$	1.72520	0.445445
$16$	0	0
$17$	7.32820	1.77735	0.888675	−	0.458538i	$-0.151627\pi$
$17$	7.32820	1.77735	0.888675	+	0.458538i	$0.151627\pi$
$18$	0	0
$19$	−1.33676	−0.306674	−0.153337	−	0.988174i	$-0.549002\pi$
$19$	−1.33676	−0.306674	−0.153337	+	0.988174i	$0.549002\pi$
$20$	0	0
$21$	1.72520	0.376470
$22$	0	0
$23$	3.73151	0.778075	0.389037	−	0.921222i	$-0.372808\pi$
$23$	3.73151	0.778075	0.389037	+	0.921222i	$0.372808\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.21645	1.00391
$28$	0	0
$29$	3.06534	0.569219	0.284609	−	0.958644i	$-0.408136\pi$
$29$	3.06534	0.569219	0.284609	+	0.958644i	$0.408136\pi$
$30$	0	0
$31$	−6.07783	−1.09161	−0.545805	−	0.837912i	$-0.683776\pi$
$31$	−6.07783	−1.09161	−0.545805	+	0.837912i	$0.683776\pi$
$32$	0	0
$33$	−2.83365	−0.493275
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	3.80756	0.625959	0.312979	−	0.949760i	$-0.398673\pi$
$37$	3.80756	0.625959	0.312979	+	0.949760i	$0.398673\pi$
$38$	0	0
$39$	−11.6387	−1.86368
$40$	0	0
$41$	−0.284168	−0.0443795	−0.0221898	−	0.999754i	$-0.507064\pi$
$41$	−0.284168	−0.0443795	−0.0221898	+	0.999754i	$0.507064\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	0.0236759	0.00352939
$46$	0	0
$47$	−2.36027	−0.344281	−0.172141	−	0.985072i	$-0.555068\pi$
$47$	−2.36027	−0.344281	−0.172141	+	0.985072i	$0.555068\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−12.6426	−1.77032
$52$	0	0
$53$	−10.2750	−1.41137	−0.705687	−	0.708524i	$-0.749361\pi$
$53$	−10.2750	−1.41137	−0.705687	+	0.708524i	$0.749361\pi$
$54$	0	0
$55$	−1.64250	−0.221475
$56$	0	0
$57$	2.30618	0.305461
$58$	0	0
$59$	−13.1870	−1.71679	−0.858397	−	0.512985i	$-0.828540\pi$
$59$	−13.1870	−1.71679	−0.858397	+	0.512985i	$0.828540\pi$
$60$	0	0
$61$	12.1369	1.55397	0.776984	−	0.629521i	$-0.216749\pi$
$61$	12.1369	1.55397	0.776984	+	0.629521i	$0.216749\pi$
$62$	0	0
$63$	0.0236759	0.00298288
$64$	0	0
$65$	−6.74628	−0.836772
$66$	0	0
$67$	−11.6051	−1.41778	−0.708892	−	0.705317i	$-0.750805\pi$
$67$	−11.6051	−1.41778	−0.708892	+	0.705317i	$0.750805\pi$
$68$	0	0
$69$	−6.43762	−0.774998
$70$	0	0
$71$	15.0966	1.79164	0.895820	−	0.444417i	$-0.146589\pi$
$71$	15.0966	1.79164	0.895820	+	0.444417i	$0.146589\pi$
$72$	0	0
$73$	−0.907421	−0.106206	−0.0531028	−	0.998589i	$-0.516911\pi$
$73$	−0.907421	−0.106206	−0.0531028	+	0.998589i	$0.516911\pi$
$74$	0	0
$75$	−1.72520	−0.199209
$76$	0	0
$77$	−1.64250	−0.187180
$78$	0	0
$79$	0.835831	0.0940383	0.0470191	−	0.998894i	$-0.485028\pi$
$79$	0.835831	0.0940383	0.0470191	+	0.998894i	$0.485028\pi$
$80$	0	0
$81$	−8.92841	−0.992046
$82$	0	0
$83$	−8.20336	−0.900435	−0.450218	−	0.892919i	$-0.648654\pi$
$83$	−8.20336	−0.900435	−0.450218	+	0.892919i	$0.648654\pi$
$84$	0	0
$85$	−7.32820	−0.794855
$86$	0	0
$87$	−5.28833	−0.566968
$88$	0	0
$89$	−2.30093	−0.243898	−0.121949	−	0.992536i	$-0.538914\pi$
$89$	−2.30093	−0.243898	−0.121949	+	0.992536i	$0.538914\pi$
$90$	0	0
$91$	−6.74628	−0.707202
$92$	0	0
$93$	10.4855	1.08729
$94$	0	0
$95$	1.33676	0.137149
$96$	0	0
$97$	−7.13319	−0.724266	−0.362133	−	0.932126i	$-0.617951\pi$
$97$	−7.13319	−0.724266	−0.362133	+	0.932126i	$0.617951\pi$
$98$	0	0
$99$	−0.0388877	−0.00390836
$100$	0	0
$101$	−1.03103	−0.102591	−0.0512955	−	0.998684i	$-0.516335\pi$
$101$	−1.03103	−0.102591	−0.0512955	+	0.998684i	$0.516335\pi$
$102$	0	0
$103$	16.7964	1.65500	0.827498	−	0.561468i	$-0.189763\pi$
$103$	16.7964	1.65500	0.827498	+	0.561468i	$0.189763\pi$
$104$	0	0
$105$	−1.72520	−0.168363
$106$	0	0
$107$	10.8135	1.04538	0.522691	−	0.852522i	$-0.324928\pi$
$107$	10.8135	1.04538	0.522691	+	0.852522i	$0.324928\pi$
$108$	0	0
$109$	13.0927	1.25406	0.627028	−	0.778996i	$-0.284271\pi$
$109$	13.0927	1.25406	0.627028	+	0.778996i	$0.284271\pi$
$110$	0	0
$111$	−6.56881	−0.623484
$112$	0	0
$113$	0.197653	0.0185936	0.00929681	−	0.999957i	$-0.497041\pi$
$113$	0.197653	0.0185936	0.00929681	+	0.999957i	$0.497041\pi$
$114$	0	0
$115$	−3.73151	−0.347966
$116$	0	0
$117$	−0.159724	−0.0147665
$118$	0	0
$119$	−7.32820	−0.671775
$120$	0	0
$121$	−8.30219	−0.754744
$122$	0	0
$123$	0.490247	0.0442041
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.50612	−0.133647	−0.0668233	−	0.997765i	$-0.521286\pi$
$127$	−1.50612	−0.133647	−0.0668233	+	0.997765i	$0.521286\pi$
$128$	0	0
$129$	1.72520	0.151896
$130$	0	0
$131$	16.0859	1.40543	0.702716	−	0.711471i	$-0.251971\pi$
$131$	16.0859	1.40543	0.702716	+	0.711471i	$0.251971\pi$
$132$	0	0
$133$	1.33676	0.115912
$134$	0	0
$135$	−5.21645	−0.448961
$136$	0	0
$137$	11.5274	0.984851	0.492425	−	0.870355i	$-0.336110\pi$
$137$	11.5274	0.984851	0.492425	+	0.870355i	$0.336110\pi$
$138$	0	0
$139$	6.00986	0.509749	0.254875	−	0.966974i	$-0.417966\pi$
$139$	6.00986	0.509749	0.254875	+	0.966974i	$0.417966\pi$
$140$	0	0
$141$	4.07195	0.342920
$142$	0	0
$143$	11.0808	0.926621
$144$	0	0
$145$	−3.06534	−0.254562
$146$	0	0
$147$	−1.72520	−0.142292
$148$	0	0
$149$	5.87229	0.481077	0.240538	−	0.970640i	$-0.422676\pi$
$149$	5.87229	0.481077	0.240538	+	0.970640i	$0.422676\pi$
$150$	0	0
$151$	−13.4324	−1.09311	−0.546557	−	0.837422i	$-0.684062\pi$
$151$	−13.4324	−1.09311	−0.546557	+	0.837422i	$0.684062\pi$
$152$	0	0
$153$	−0.173502	−0.0140268
$154$	0	0
$155$	6.07783	0.488183
$156$	0	0
$157$	6.71075	0.535576	0.267788	−	0.963478i	$-0.413707\pi$
$157$	6.71075	0.535576	0.267788	+	0.963478i	$0.413707\pi$
$158$	0	0
$159$	17.7264	1.40579
$160$	0	0
$161$	−3.73151	−0.294085
$162$	0	0
$163$	−11.5421	−0.904049	−0.452024	−	0.892006i	$-0.649298\pi$
$163$	−11.5421	−0.904049	−0.452024	+	0.892006i	$0.649298\pi$
$164$	0	0
$165$	2.83365	0.220599
$166$	0	0
$167$	5.02275	0.388672	0.194336	−	0.980935i	$-0.437745\pi$
$167$	5.02275	0.388672	0.194336	+	0.980935i	$0.437745\pi$
$168$	0	0
$169$	32.5122	2.50094
$170$	0	0
$171$	0.0316489	0.00242026
$172$	0	0
$173$	−5.46273	−0.415324	−0.207662	−	0.978201i	$-0.566585\pi$
$173$	−5.46273	−0.415324	−0.207662	+	0.978201i	$0.566585\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	22.7502	1.71001
$178$	0	0
$179$	7.66160	0.572655	0.286327	−	0.958132i	$-0.407565\pi$
$179$	7.66160	0.572655	0.286327	+	0.958132i	$0.407565\pi$
$180$	0	0
$181$	−6.88280	−0.511594	−0.255797	−	0.966730i	$-0.582338\pi$
$181$	−6.88280	−0.511594	−0.255797	+	0.966730i	$0.582338\pi$
$182$	0	0
$183$	−20.9386	−1.54782
$184$	0	0
$185$	−3.80756	−0.279937
$186$	0	0
$187$	12.0366	0.880202
$188$	0	0
$189$	−5.21645	−0.379441
$190$	0	0
$191$	12.8856	0.932367	0.466183	−	0.884688i	$-0.345629\pi$
$191$	12.8856	0.932367	0.466183	+	0.884688i	$0.345629\pi$
$192$	0	0
$193$	−8.47755	−0.610227	−0.305114	−	0.952316i	$-0.598694\pi$
$193$	−8.47755	−0.610227	−0.305114	+	0.952316i	$0.598694\pi$
$194$	0	0
$195$	11.6387	0.833464
$196$	0	0
$197$	−4.62935	−0.329828	−0.164914	−	0.986308i	$-0.552735\pi$
$197$	−4.62935	−0.329828	−0.164914	+	0.986308i	$0.552735\pi$
$198$	0	0
$199$	5.31367	0.376676	0.188338	−	0.982104i	$-0.439690\pi$
$199$	5.31367	0.376676	0.188338	+	0.982104i	$0.439690\pi$
$200$	0	0
$201$	20.0211	1.41218
$202$	0	0
$203$	−3.06534	−0.215144
$204$	0	0
$205$	0.284168	0.0198471
$206$	0	0
$207$	−0.0883468	−0.00614053
$208$	0	0
$209$	−2.19563	−0.151875
$210$	0	0
$211$	14.5581	1.00222	0.501112	−	0.865383i	$-0.332925\pi$
$211$	14.5581	1.00222	0.501112	+	0.865383i	$0.332925\pi$
$212$	0	0
$213$	−26.0447	−1.78456
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	6.07783	0.412590
$218$	0	0
$219$	1.56549	0.105786
$220$	0	0
$221$	49.4381	3.32556
$222$	0	0
$223$	−20.3596	−1.36338	−0.681689	−	0.731642i	$-0.738754\pi$
$223$	−20.3596	−1.36338	−0.681689	+	0.731642i	$0.738754\pi$
$224$	0	0
$225$	−0.0236759	−0.00157839
$226$	0	0
$227$	−2.33319	−0.154859	−0.0774297	−	0.996998i	$-0.524671\pi$
$227$	−2.33319	−0.154859	−0.0774297	+	0.996998i	$0.524671\pi$
$228$	0	0
$229$	−6.04063	−0.399176	−0.199588	−	0.979880i	$-0.563960\pi$
$229$	−6.04063	−0.399176	−0.199588	+	0.979880i	$0.563960\pi$
$230$	0	0
$231$	2.83365	0.186440
$232$	0	0
$233$	14.4627	0.947484	0.473742	−	0.880664i	$-0.342903\pi$
$233$	14.4627	0.947484	0.473742	+	0.880664i	$0.342903\pi$
$234$	0	0
$235$	2.36027	0.153967
$236$	0	0
$237$	−1.44198	−0.0936665
$238$	0	0
$239$	−9.20173	−0.595210	−0.297605	−	0.954689i	$-0.596188\pi$
$239$	−9.20173	−0.595210	−0.297605	+	0.954689i	$0.596188\pi$
$240$	0	0
$241$	24.2924	1.56481	0.782404	−	0.622771i	$-0.213993\pi$
$241$	24.2924	1.56481	0.782404	+	0.622771i	$0.213993\pi$
$242$	0	0
$243$	−0.246041	−0.0157835
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−9.01815	−0.573811
$248$	0	0
$249$	14.1525	0.896875
$250$	0	0
$251$	8.74334	0.551875	0.275937	−	0.961176i	$-0.411012\pi$
$251$	8.74334	0.551875	0.275937	+	0.961176i	$0.411012\pi$
$252$	0	0
$253$	6.12902	0.385328
$254$	0	0
$255$	12.6426	0.791712
$256$	0	0
$257$	−24.1755	−1.50803	−0.754014	−	0.656858i	$-0.771885\pi$
$257$	−24.1755	−1.50803	−0.754014	+	0.656858i	$0.771885\pi$
$258$	0	0
$259$	−3.80756	−0.236590
$260$	0	0
$261$	−0.0725745	−0.00449225
$262$	0	0
$263$	3.90229	0.240626	0.120313	−	0.992736i	$-0.461610\pi$
$263$	3.90229	0.240626	0.120313	+	0.992736i	$0.461610\pi$
$264$	0	0
$265$	10.2750	0.631185
$266$	0	0
$267$	3.96956	0.242933
$268$	0	0
$269$	−1.59871	−0.0974753	−0.0487377	−	0.998812i	$-0.515520\pi$
$269$	−1.59871	−0.0974753	−0.0487377	+	0.998812i	$0.515520\pi$
$270$	0	0
$271$	−1.97390	−0.119906	−0.0599529	−	0.998201i	$-0.519095\pi$
$271$	−1.97390	−0.119906	−0.0599529	+	0.998201i	$0.519095\pi$
$272$	0	0
$273$	11.6387	0.704406
$274$	0	0
$275$	1.64250	0.0990466
$276$	0	0
$277$	5.34341	0.321054	0.160527	−	0.987031i	$-0.448681\pi$
$277$	5.34341	0.321054	0.160527	+	0.987031i	$0.448681\pi$
$278$	0	0
$279$	0.143898	0.00861493
$280$	0	0
$281$	25.6699	1.53134	0.765668	−	0.643236i	$-0.222409\pi$
$281$	25.6699	1.53134	0.765668	+	0.643236i	$0.222409\pi$
$282$	0	0
$283$	−22.7236	−1.35078	−0.675388	−	0.737462i	$-0.736024\pi$
$283$	−22.7236	−1.35078	−0.675388	+	0.737462i	$0.736024\pi$
$284$	0	0
$285$	−2.30618	−0.136606
$286$	0	0
$287$	0.284168	0.0167739
$288$	0	0
$289$	36.7025	2.15897
$290$	0	0
$291$	12.3062	0.721402
$292$	0	0
$293$	26.7179	1.56088	0.780438	−	0.625233i	$-0.214996\pi$
$293$	26.7179	1.56088	0.780438	+	0.625233i	$0.214996\pi$
$294$	0	0
$295$	13.1870	0.767774
$296$	0	0
$297$	8.56804	0.497168
$298$	0	0
$299$	25.1738	1.45584
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	1.77873	0.102185
$304$	0	0
$305$	−12.1369	−0.694955
$306$	0	0
$307$	14.5072	0.827969	0.413984	−	0.910284i	$-0.364137\pi$
$307$	14.5072	0.827969	0.413984	+	0.910284i	$0.364137\pi$
$308$	0	0
$309$	−28.9772	−1.64845
$310$	0	0
$311$	−30.1976	−1.71235	−0.856174	−	0.516687i	$-0.827165\pi$
$311$	−30.1976	−1.71235	−0.856174	+	0.516687i	$0.827165\pi$
$312$	0	0
$313$	11.8985	0.672544	0.336272	−	0.941765i	$-0.390834\pi$
$313$	11.8985	0.672544	0.336272	+	0.941765i	$0.390834\pi$
$314$	0	0
$315$	−0.0236759	−0.00133398
$316$	0	0
$317$	−20.1955	−1.13429	−0.567145	−	0.823618i	$-0.691952\pi$
$317$	−20.1955	−1.13429	−0.567145	+	0.823618i	$0.691952\pi$
$318$	0	0
$319$	5.03482	0.281896
$320$	0	0
$321$	−18.6555	−1.04125
$322$	0	0
$323$	−9.79605	−0.545067
$324$	0	0
$325$	6.74628	0.374216
$326$	0	0
$327$	−22.5876	−1.24910
$328$	0	0
$329$	2.36027	0.130126
$330$	0	0
$331$	22.3872	1.23051	0.615256	−	0.788328i	$-0.289053\pi$
$331$	22.3872	1.23051	0.615256	+	0.788328i	$0.289053\pi$
$332$	0	0
$333$	−0.0901472	−0.00494004
$334$	0	0
$335$	11.6051	0.634052
$336$	0	0
$337$	4.56246	0.248533	0.124266	−	0.992249i	$-0.460342\pi$
$337$	4.56246	0.248533	0.124266	+	0.992249i	$0.460342\pi$
$338$	0	0
$339$	−0.340991	−0.0185201
$340$	0	0
$341$	−9.98284	−0.540601
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	6.43762	0.346590
$346$	0	0
$347$	−20.7346	−1.11309	−0.556547	−	0.830816i	$-0.687874\pi$
$347$	−20.7346	−1.11309	−0.556547	+	0.830816i	$0.687874\pi$
$348$	0	0
$349$	13.0617	0.699176	0.349588	−	0.936903i	$-0.386322\pi$
$349$	13.0617	0.699176	0.349588	+	0.936903i	$0.386322\pi$
$350$	0	0
$351$	35.1916	1.87839
$352$	0	0
$353$	−31.4617	−1.67454	−0.837269	−	0.546791i	$-0.815849\pi$
$353$	−31.4617	−1.67454	−0.837269	+	0.546791i	$0.815849\pi$
$354$	0	0
$355$	−15.0966	−0.801246
$356$	0	0
$357$	12.6426	0.669119
$358$	0	0
$359$	−31.1283	−1.64289	−0.821446	−	0.570287i	$-0.806832\pi$
$359$	−31.1283	−1.64289	−0.821446	+	0.570287i	$0.806832\pi$
$360$	0	0
$361$	−17.2131	−0.905951
$362$	0	0
$363$	14.3230	0.751760
$364$	0	0
$365$	0.907421	0.0474966
$366$	0	0
$367$	20.2843	1.05883	0.529416	−	0.848362i	$-0.322411\pi$
$367$	20.2843	1.05883	0.529416	+	0.848362i	$0.322411\pi$
$368$	0	0
$369$	0.00672792	0.000350241 0
$370$	0	0
$371$	10.2750	0.533449
$372$	0	0
$373$	1.82053	0.0942635	0.0471318	−	0.998889i	$-0.484992\pi$
$373$	1.82053	0.0942635	0.0471318	+	0.998889i	$0.484992\pi$
$374$	0	0
$375$	1.72520	0.0890891
$376$	0	0
$377$	20.6796	1.06505
$378$	0	0
$379$	−7.26579	−0.373218	−0.186609	−	0.982434i	$-0.559750\pi$
$379$	−7.26579	−0.373218	−0.186609	+	0.982434i	$0.559750\pi$
$380$	0	0
$381$	2.59836	0.133118
$382$	0	0
$383$	1.37373	0.0701942	0.0350971	−	0.999384i	$-0.488826\pi$
$383$	1.37373	0.0701942	0.0350971	+	0.999384i	$0.488826\pi$
$384$	0	0
$385$	1.64250	0.0837097
$386$	0	0
$387$	0.0236759	0.00120351
$388$	0	0
$389$	13.5056	0.684762	0.342381	−	0.939561i	$-0.388767\pi$
$389$	13.5056	0.684762	0.342381	+	0.939561i	$0.388767\pi$
$390$	0	0
$391$	27.3453	1.38291
$392$	0	0
$393$	−27.7514	−1.39987
$394$	0	0
$395$	−0.835831	−0.0420552
$396$	0	0
$397$	−17.7660	−0.891649	−0.445825	−	0.895120i	$-0.647089\pi$
$397$	−17.7660	−0.891649	−0.445825	+	0.895120i	$0.647089\pi$
$398$	0	0
$399$	−2.30618	−0.115454
$400$	0	0
$401$	25.8687	1.29182	0.645910	−	0.763414i	$-0.276478\pi$
$401$	25.8687	1.29182	0.645910	+	0.763414i	$0.276478\pi$
$402$	0	0
$403$	−41.0027	−2.04249
$404$	0	0
$405$	8.92841	0.443656
$406$	0	0
$407$	6.25392	0.309995
$408$	0	0
$409$	−33.4082	−1.65193	−0.825966	−	0.563720i	$-0.809370\pi$
$409$	−33.4082	−1.65193	−0.825966	+	0.563720i	$0.809370\pi$
$410$	0	0
$411$	−19.8871	−0.980957
$412$	0	0
$413$	13.1870	0.648888
$414$	0	0
$415$	8.20336	0.402687
$416$	0	0
$417$	−10.3682	−0.507734
$418$	0	0
$419$	23.2688	1.13676	0.568379	−	0.822767i	$-0.307571\pi$
$419$	23.2688	1.13676	0.568379	+	0.822767i	$0.307571\pi$
$420$	0	0
$421$	31.7827	1.54899	0.774497	−	0.632578i	$-0.218003\pi$
$421$	31.7827	1.54899	0.774497	+	0.632578i	$0.218003\pi$
$422$	0	0
$423$	0.0558815	0.00271705
$424$	0	0
$425$	7.32820	0.355470
$426$	0	0
$427$	−12.1369	−0.587345
$428$	0	0
$429$	−19.1166	−0.922957
$430$	0	0
$431$	20.1181	0.969053	0.484527	−	0.874776i	$-0.338992\pi$
$431$	20.1181	0.969053	0.484527	+	0.874776i	$0.338992\pi$
$432$	0	0
$433$	32.1061	1.54292	0.771460	−	0.636277i	$-0.219527\pi$
$433$	32.1061	1.54292	0.771460	+	0.636277i	$0.219527\pi$
$434$	0	0
$435$	5.28833	0.253556
$436$	0	0
$437$	−4.98814	−0.238615
$438$	0	0
$439$	15.9171	0.759680	0.379840	−	0.925052i	$-0.375979\pi$
$439$	15.9171	0.759680	0.379840	+	0.925052i	$0.375979\pi$
$440$	0	0
$441$	−0.0236759	−0.00112742
$442$	0	0
$443$	9.36380	0.444888	0.222444	−	0.974946i	$-0.428597\pi$
$443$	9.36380	0.444888	0.222444	+	0.974946i	$0.428597\pi$
$444$	0	0
$445$	2.30093	0.109074
$446$	0	0
$447$	−10.1309	−0.479175
$448$	0	0
$449$	−33.4891	−1.58045	−0.790224	−	0.612818i	$-0.790036\pi$
$449$	−33.4891	−1.58045	−0.790224	+	0.612818i	$0.790036\pi$
$450$	0	0
$451$	−0.466746	−0.0219782
$452$	0	0
$453$	23.1736	1.08879
$454$	0	0
$455$	6.74628	0.316270
$456$	0	0
$457$	23.5936	1.10366	0.551831	−	0.833956i	$-0.313929\pi$
$457$	23.5936	1.10366	0.551831	+	0.833956i	$0.313929\pi$
$458$	0	0
$459$	38.2272	1.78429
$460$	0	0
$461$	−37.9159	−1.76592	−0.882960	−	0.469448i	$-0.844453\pi$
$461$	−37.9159	−1.76592	−0.882960	+	0.469448i	$0.844453\pi$
$462$	0	0
$463$	10.5488	0.490247	0.245123	−	0.969492i	$-0.421172\pi$
$463$	10.5488	0.490247	0.245123	+	0.969492i	$0.421172\pi$
$464$	0	0
$465$	−10.4855	−0.486253
$466$	0	0
$467$	−25.6179	−1.18545	−0.592727	−	0.805403i	$-0.701949\pi$
$467$	−25.6179	−1.18545	−0.592727	+	0.805403i	$0.701949\pi$
$468$	0	0
$469$	11.6051	0.535872
$470$	0	0
$471$	−11.5774	−0.533459
$472$	0	0
$473$	−1.64250	−0.0755223
$474$	0	0
$475$	−1.33676	−0.0613348
$476$	0	0
$477$	0.243268	0.0111385
$478$	0	0
$479$	−15.0718	−0.688647	−0.344324	−	0.938851i	$-0.611892\pi$
$479$	−15.0718	−0.688647	−0.344324	+	0.938851i	$0.611892\pi$
$480$	0	0
$481$	25.6868	1.17122
$482$	0	0
$483$	6.43762	0.292922
$484$	0	0
$485$	7.13319	0.323902
$486$	0	0
$487$	0.0361065	0.00163614	0.000818071	−	1.00000i	$-0.499740\pi$
$487$	0.0361065	0.00163614	0.000818071		1.00000i	$0.499740\pi$
$488$	0	0
$489$	19.9125	0.900474
$490$	0	0
$491$	−7.56912	−0.341590	−0.170795	−	0.985307i	$-0.554634\pi$
$491$	−7.56912	−0.341590	−0.170795	+	0.985307i	$0.554634\pi$
$492$	0	0
$493$	22.4634	1.01170
$494$	0	0
$495$	0.0388877	0.00174787
$496$	0	0
$497$	−15.0966	−0.677176
$498$	0	0
$499$	10.5611	0.472779	0.236390	−	0.971658i	$-0.424036\pi$
$499$	10.5611	0.472779	0.236390	+	0.971658i	$0.424036\pi$
$500$	0	0
$501$	−8.66526	−0.387135
$502$	0	0
$503$	−19.2074	−0.856414	−0.428207	−	0.903681i	$-0.640855\pi$
$503$	−19.2074	−0.856414	−0.428207	+	0.903681i	$0.640855\pi$
$504$	0	0
$505$	1.03103	0.0458801
$506$	0	0
$507$	−56.0902	−2.49105
$508$	0	0
$509$	−38.7365	−1.71696	−0.858482	−	0.512843i	$-0.828592\pi$
$509$	−38.7365	−1.71696	−0.858482	+	0.512843i	$0.828592\pi$
$510$	0	0
$511$	0.907421	0.0401419
$512$	0	0
$513$	−6.97315	−0.307872
$514$	0	0
$515$	−16.7964	−0.740137
$516$	0	0
$517$	−3.87675	−0.170499
$518$	0	0
$519$	9.42432	0.413682
$520$	0	0
$521$	2.01464	0.0882630	0.0441315	−	0.999026i	$-0.485948\pi$
$521$	2.01464	0.0882630	0.0441315	+	0.999026i	$0.485948\pi$
$522$	0	0
$523$	9.03292	0.394982	0.197491	−	0.980305i	$-0.436721\pi$
$523$	9.03292	0.394982	0.197491	+	0.980305i	$0.436721\pi$
$524$	0	0
$525$	1.72520	0.0752940
$526$	0	0
$527$	−44.5395	−1.94017
$528$	0	0
$529$	−9.07580	−0.394600
$530$	0	0
$531$	0.312212	0.0135489
$532$	0	0
$533$	−1.91707	−0.0830377
$534$	0	0
$535$	−10.8135	−0.467509
$536$	0	0
$537$	−13.2178	−0.570391
$538$	0	0
$539$	1.64250	0.0707476
$540$	0	0
$541$	44.7844	1.92543	0.962716	−	0.270513i	$-0.0871933\pi$
$541$	44.7844	1.92543	0.962716	+	0.270513i	$0.0871933\pi$
$542$	0	0
$543$	11.8742	0.509572
$544$	0	0
$545$	−13.0927	−0.560831
$546$	0	0
$547$	22.2971	0.953354	0.476677	−	0.879078i	$-0.341841\pi$
$547$	22.2971	0.953354	0.476677	+	0.879078i	$0.341841\pi$
$548$	0	0
$549$	−0.287351	−0.0122638
$550$	0	0
$551$	−4.09762	−0.174564
$552$	0	0
$553$	−0.835831	−0.0355431
$554$	0	0
$555$	6.56881	0.278830
$556$	0	0
$557$	15.5993	0.660966	0.330483	−	0.943812i	$-0.392788\pi$
$557$	15.5993	0.660966	0.330483	+	0.943812i	$0.392788\pi$
$558$	0	0
$559$	−6.74628	−0.285337
$560$	0	0
$561$	−20.7656	−0.876722
$562$	0	0
$563$	−7.01870	−0.295803	−0.147901	−	0.989002i	$-0.547252\pi$
$563$	−7.01870	−0.295803	−0.147901	+	0.989002i	$0.547252\pi$
$564$	0	0
$565$	−0.197653	−0.00831532
$566$	0	0
$567$	8.92841	0.374958
$568$	0	0
$569$	−9.95397	−0.417292	−0.208646	−	0.977991i	$-0.566906\pi$
$569$	−9.95397	−0.417292	−0.208646	+	0.977991i	$0.566906\pi$
$570$	0	0
$571$	38.1796	1.59777	0.798883	−	0.601487i	$-0.205425\pi$
$571$	38.1796	1.59777	0.798883	+	0.601487i	$0.205425\pi$
$572$	0	0
$573$	−22.2302	−0.928681
$574$	0	0
$575$	3.73151	0.155615
$576$	0	0
$577$	5.46885	0.227671	0.113836	−	0.993500i	$-0.463686\pi$
$577$	5.46885	0.227671	0.113836	+	0.993500i	$0.463686\pi$
$578$	0	0
$579$	14.6255	0.607814
$580$	0	0
$581$	8.20336	0.340333
$582$	0	0
$583$	−16.8766	−0.698959
$584$	0	0
$585$	0.159724	0.00660377
$586$	0	0
$587$	30.5451	1.26073	0.630366	−	0.776298i	$-0.282905\pi$
$587$	30.5451	1.26073	0.630366	+	0.776298i	$0.282905\pi$
$588$	0	0
$589$	8.12459	0.334768
$590$	0	0
$591$	7.98657	0.328524
$592$	0	0
$593$	−4.28238	−0.175856	−0.0879281	−	0.996127i	$-0.528025\pi$
$593$	−4.28238	−0.175856	−0.0879281	+	0.996127i	$0.528025\pi$
$594$	0	0
$595$	7.32820	0.300427
$596$	0	0
$597$	−9.16715	−0.375187
$598$	0	0
$599$	10.2832	0.420160	0.210080	−	0.977684i	$-0.432627\pi$
$599$	10.2832	0.420160	0.210080	+	0.977684i	$0.432627\pi$
$600$	0	0
$601$	5.36035	0.218653	0.109327	−	0.994006i	$-0.465131\pi$
$601$	5.36035	0.218653	0.109327	+	0.994006i	$0.465131\pi$
$602$	0	0
$603$	0.274760	0.0111891
$604$	0	0
$605$	8.30219	0.337532
$606$	0	0
$607$	−32.5444	−1.32094	−0.660468	−	0.750854i	$-0.729642\pi$
$607$	−32.5444	−1.32094	−0.660468	+	0.750854i	$0.729642\pi$
$608$	0	0
$609$	5.28833	0.214294
$610$	0	0
$611$	−15.9231	−0.644178
$612$	0	0
$613$	16.0166	0.646906	0.323453	−	0.946244i	$-0.395156\pi$
$613$	16.0166	0.646906	0.323453	+	0.946244i	$0.395156\pi$
$614$	0	0
$615$	−0.490247	−0.0197687
$616$	0	0
$617$	36.5571	1.47173	0.735866	−	0.677127i	$-0.236775\pi$
$617$	36.5571	1.47173	0.735866	+	0.677127i	$0.236775\pi$
$618$	0	0
$619$	−9.86347	−0.396446	−0.198223	−	0.980157i	$-0.563517\pi$
$619$	−9.86347	−0.396446	−0.198223	+	0.980157i	$0.563517\pi$
$620$	0	0
$621$	19.4653	0.781114
$622$	0	0
$623$	2.30093	0.0921847
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.78791	0.151275
$628$	0	0
$629$	27.9025	1.11255
$630$	0	0
$631$	44.6840	1.77884	0.889421	−	0.457090i	$-0.151108\pi$
$631$	44.6840	1.77884	0.889421	+	0.457090i	$0.151108\pi$
$632$	0	0
$633$	−25.1157	−0.998260
$634$	0	0
$635$	1.50612	0.0597686
$636$	0	0
$637$	6.74628	0.267297
$638$	0	0
$639$	−0.357426	−0.0141395
$640$	0	0
$641$	36.6320	1.44688	0.723439	−	0.690388i	$-0.242560\pi$
$641$	36.6320	1.44688	0.723439	+	0.690388i	$0.242560\pi$
$642$	0	0
$643$	−43.1609	−1.70210	−0.851049	−	0.525086i	$-0.824033\pi$
$643$	−43.1609	−1.70210	−0.851049	+	0.525086i	$0.824033\pi$
$644$	0	0
$645$	−1.72520	−0.0679298
$646$	0	0
$647$	6.79545	0.267157	0.133578	−	0.991038i	$-0.457353\pi$
$647$	6.79545	0.267157	0.133578	+	0.991038i	$0.457353\pi$
$648$	0	0
$649$	−21.6596	−0.850214
$650$	0	0
$651$	−10.4855	−0.410958
$652$	0	0
$653$	35.2433	1.37918	0.689589	−	0.724201i	$-0.257791\pi$
$653$	35.2433	1.37918	0.689589	+	0.724201i	$0.257791\pi$
$654$	0	0
$655$	−16.0859	−0.628528
$656$	0	0
$657$	0.0214840	0.000838170 0
$658$	0	0
$659$	21.7561	0.847496	0.423748	−	0.905780i	$-0.360714\pi$
$659$	21.7561	0.847496	0.423748	+	0.905780i	$0.360714\pi$
$660$	0	0
$661$	44.2640	1.72167	0.860835	−	0.508885i	$-0.169942\pi$
$661$	44.2640	1.72167	0.860835	+	0.508885i	$0.169942\pi$
$662$	0	0
$663$	−85.2907	−3.31242
$664$	0	0
$665$	−1.33676	−0.0518373
$666$	0	0
$667$	11.4383	0.442895
$668$	0	0
$669$	35.1244	1.35799
$670$	0	0
$671$	19.9348	0.769576
$672$	0	0
$673$	19.0490	0.734286	0.367143	−	0.930165i	$-0.380336\pi$
$673$	19.0490	0.734286	0.367143	+	0.930165i	$0.380336\pi$
$674$	0	0
$675$	5.21645	0.200781
$676$	0	0
$677$	16.5448	0.635868	0.317934	−	0.948113i	$-0.397011\pi$
$677$	16.5448	0.635868	0.317934	+	0.948113i	$0.397011\pi$
$678$	0	0
$679$	7.13319	0.273747
$680$	0	0
$681$	4.02523	0.154247
$682$	0	0
$683$	24.8350	0.950285	0.475143	−	0.879909i	$-0.342396\pi$
$683$	24.8350	0.950285	0.475143	+	0.879909i	$0.342396\pi$
$684$	0	0
$685$	−11.5274	−0.440439
$686$	0	0
$687$	10.4213	0.397598
$688$	0	0
$689$	−69.3177	−2.64079
$690$	0	0
$691$	−47.2083	−1.79589	−0.897944	−	0.440110i	$-0.854940\pi$
$691$	−47.2083	−1.79589	−0.897944	+	0.440110i	$0.854940\pi$
$692$	0	0
$693$	0.0388877	0.00147722
$694$	0	0
$695$	−6.00986	−0.227967
$696$	0	0
$697$	−2.08244	−0.0788780
$698$	0	0
$699$	−24.9511	−0.943738
$700$	0	0
$701$	42.7808	1.61581	0.807903	−	0.589315i	$-0.200602\pi$
$701$	42.7808	1.61581	0.807903	+	0.589315i	$0.200602\pi$
$702$	0	0
$703$	−5.08979	−0.191965
$704$	0	0
$705$	−4.07195	−0.153359
$706$	0	0
$707$	1.03103	0.0387758
$708$	0	0
$709$	−11.1990	−0.420588	−0.210294	−	0.977638i	$-0.567442\pi$
$709$	−11.1990	−0.420588	−0.210294	+	0.977638i	$0.567442\pi$
$710$	0	0
$711$	−0.0197890	−0.000742146 0
$712$	0	0
$713$	−22.6795	−0.849354
$714$	0	0
$715$	−11.0808	−0.414397
$716$	0	0
$717$	15.8748	0.592857
$718$	0	0
$719$	0.713969	0.0266266	0.0133133	−	0.999911i	$-0.495762\pi$
$719$	0.713969	0.0266266	0.0133133	+	0.999911i	$0.495762\pi$
$720$	0	0
$721$	−16.7964	−0.625530
$722$	0	0
$723$	−41.9092	−1.55862
$724$	0	0
$725$	3.06534	0.113844
$726$	0	0
$727$	38.7015	1.43536	0.717680	−	0.696373i	$-0.245204\pi$
$727$	38.7015	1.43536	0.717680	+	0.696373i	$0.245204\pi$
$728$	0	0
$729$	27.2097	1.00777
$730$	0	0
$731$	−7.32820	−0.271043
$732$	0	0
$733$	26.8865	0.993077	0.496539	−	0.868015i	$-0.334604\pi$
$733$	26.8865	0.993077	0.496539	+	0.868015i	$0.334604\pi$
$734$	0	0
$735$	1.72520	0.0636351
$736$	0	0
$737$	−19.0613	−0.702134
$738$	0	0
$739$	−5.09246	−0.187329	−0.0936646	−	0.995604i	$-0.529858\pi$
$739$	−5.09246	−0.187329	−0.0936646	+	0.995604i	$0.529858\pi$
$740$	0	0
$741$	15.5581	0.571543
$742$	0	0
$743$	−48.8127	−1.79077	−0.895383	−	0.445298i	$-0.853098\pi$
$743$	−48.8127	−1.79077	−0.895383	+	0.445298i	$0.853098\pi$
$744$	0	0
$745$	−5.87229	−0.215144
$746$	0	0
$747$	0.194222	0.00710619
$748$	0	0
$749$	−10.8135	−0.395117
$750$	0	0
$751$	27.4445	1.00146	0.500732	−	0.865602i	$-0.333064\pi$
$751$	27.4445	1.00146	0.500732	+	0.865602i	$0.333064\pi$
$752$	0	0
$753$	−15.0840	−0.549693
$754$	0	0
$755$	13.4324	0.488856
$756$	0	0
$757$	31.5322	1.14606	0.573029	−	0.819535i	$-0.305768\pi$
$757$	31.5322	1.14606	0.573029	+	0.819535i	$0.305768\pi$
$758$	0	0
$759$	−10.5738	−0.383805
$760$	0	0
$761$	−24.1946	−0.877055	−0.438527	−	0.898718i	$-0.644500\pi$
$761$	−24.1946	−0.877055	−0.438527	+	0.898718i	$0.644500\pi$
$762$	0	0
$763$	−13.0927	−0.473989
$764$	0	0
$765$	0.173502	0.00627296
$766$	0	0
$767$	−88.9628	−3.21226
$768$	0	0
$769$	−18.9987	−0.685111	−0.342556	−	0.939498i	$-0.611293\pi$
$769$	−18.9987	−0.685111	−0.342556	+	0.939498i	$0.611293\pi$
$770$	0	0
$771$	41.7077	1.50207
$772$	0	0
$773$	13.5737	0.488211	0.244105	−	0.969749i	$-0.421506\pi$
$773$	13.5737	0.488211	0.244105	+	0.969749i	$0.421506\pi$
$774$	0	0
$775$	−6.07783	−0.218322
$776$	0	0
$777$	6.56881	0.235655
$778$	0	0
$779$	0.379864	0.0136100
$780$	0	0
$781$	24.7962	0.887279
$782$	0	0
$783$	15.9902	0.571443
$784$	0	0
$785$	−6.71075	−0.239517
$786$	0	0
$787$	4.15501	0.148110	0.0740550	−	0.997254i	$-0.476406\pi$
$787$	4.15501	0.148110	0.0740550	+	0.997254i	$0.476406\pi$
$788$	0	0
$789$	−6.73225	−0.239674
$790$	0	0
$791$	−0.197653	−0.00702773
$792$	0	0
$793$	81.8787	2.90760
$794$	0	0
$795$	−17.7264	−0.628690
$796$	0	0
$797$	0.280590	0.00993900	0.00496950	−	0.999988i	$-0.498418\pi$
$797$	0.280590	0.00993900	0.00496950	+	0.999988i	$0.498418\pi$
$798$	0	0
$799$	−17.2966	−0.611908
$800$	0	0
$801$	0.0544764	0.00192483
$802$	0	0
$803$	−1.49044	−0.0525965
$804$	0	0
$805$	3.73151	0.131519
$806$	0	0
$807$	2.75811	0.0970900
$808$	0	0
$809$	−20.6982	−0.727711	−0.363856	−	0.931455i	$-0.618540\pi$
$809$	−20.6982	−0.727711	−0.363856	+	0.931455i	$0.618540\pi$
$810$	0	0
$811$	13.3643	0.469285	0.234642	−	0.972082i	$-0.424608\pi$
$811$	13.3643	0.469285	0.234642	+	0.972082i	$0.424608\pi$
$812$	0	0
$813$	3.40538	0.119432
$814$	0	0
$815$	11.5421	0.404303
$816$	0	0
$817$	1.33676	0.0467673
$818$	0	0
$819$	0.159724	0.00558120
$820$	0	0
$821$	28.6459	0.999748	0.499874	−	0.866098i	$-0.333380\pi$
$821$	28.6459	0.999748	0.499874	+	0.866098i	$0.333380\pi$
$822$	0	0
$823$	0.234251	0.00816548	0.00408274	−	0.999992i	$-0.498700\pi$
$823$	0.234251	0.00816548	0.00408274	+	0.999992i	$0.498700\pi$
$824$	0	0
$825$	−2.83365	−0.0986550
$826$	0	0
$827$	40.0794	1.39370	0.696849	−	0.717218i	$-0.254585\pi$
$827$	40.0794	1.39370	0.696849	+	0.717218i	$0.254585\pi$
$828$	0	0
$829$	−18.1991	−0.632080	−0.316040	−	0.948746i	$-0.602353\pi$
$829$	−18.1991	−0.632080	−0.316040	+	0.948746i	$0.602353\pi$
$830$	0	0
$831$	−9.21846	−0.319785
$832$	0	0
$833$	7.32820	0.253907
$834$	0	0
$835$	−5.02275	−0.173819
$836$	0	0
$837$	−31.7047	−1.09587
$838$	0	0
$839$	−30.9805	−1.06957	−0.534784	−	0.844989i	$-0.679607\pi$
$839$	−30.9805	−1.06957	−0.534784	+	0.844989i	$0.679607\pi$
$840$	0	0
$841$	−19.6037	−0.675990
$842$	0	0
$843$	−44.2857	−1.52528
$844$	0	0
$845$	−32.5122	−1.11845
$846$	0	0
$847$	8.30219	0.285267
$848$	0	0
$849$	39.2028	1.34544
$850$	0	0
$851$	14.2080	0.487042
$852$	0	0
$853$	1.14847	0.0393227	0.0196614	−	0.999807i	$-0.493741\pi$
$853$	1.14847	0.0393227	0.0196614	+	0.999807i	$0.493741\pi$
$854$	0	0
$855$	−0.0316489	−0.00108237
$856$	0	0
$857$	30.5526	1.04366	0.521829	−	0.853050i	$-0.325250\pi$
$857$	30.5526	1.04366	0.521829	+	0.853050i	$0.325250\pi$
$858$	0	0
$859$	36.7529	1.25399	0.626996	−	0.779022i	$-0.284284\pi$
$859$	36.7529	1.25399	0.626996	+	0.779022i	$0.284284\pi$
$860$	0	0
$861$	−0.490247	−0.0167076
$862$	0	0
$863$	−33.5593	−1.14237	−0.571186	−	0.820821i	$-0.693516\pi$
$863$	−33.5593	−1.14237	−0.571186	+	0.820821i	$0.693516\pi$
$864$	0	0
$865$	5.46273	0.185738
$866$	0	0
$867$	−63.3193	−2.15044
$868$	0	0
$869$	1.37285	0.0465709
$870$	0	0
$871$	−78.2909	−2.65279
$872$	0	0
$873$	0.168885	0.00571587
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	3.20061	0.108077	0.0540385	−	0.998539i	$-0.482791\pi$
$877$	3.20061	0.108077	0.0540385	+	0.998539i	$0.482791\pi$
$878$	0	0
$879$	−46.0938	−1.55470
$880$	0	0
$881$	−20.5857	−0.693550	−0.346775	−	0.937948i	$-0.612723\pi$
$881$	−20.5857	−0.693550	−0.346775	+	0.937948i	$0.612723\pi$
$882$	0	0
$883$	−36.4131	−1.22540	−0.612699	−	0.790316i	$-0.709916\pi$
$883$	−36.4131	−1.22540	−0.612699	+	0.790316i	$0.709916\pi$
$884$	0	0
$885$	−22.7502	−0.764738
$886$	0	0
$887$	−25.0704	−0.841782	−0.420891	−	0.907111i	$-0.638283\pi$
$887$	−25.0704	−0.841782	−0.420891	+	0.907111i	$0.638283\pi$
$888$	0	0
$889$	1.50612	0.0505137
$890$	0	0
$891$	−14.6649	−0.491294
$892$	0	0
$893$	3.15512	0.105582
$894$	0	0
$895$	−7.66160	−0.256099
$896$	0	0
$897$	−43.4299	−1.45008
$898$	0	0
$899$	−18.6306	−0.621365
$900$	0	0
$901$	−75.2969	−2.50850
$902$	0	0
$903$	−1.72520	−0.0574111
$904$	0	0
$905$	6.88280	0.228792
$906$	0	0
$907$	37.1645	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$907$	37.1645	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$908$	0	0
$909$	0.0244105	0.000809644 0
$910$	0	0
$911$	−22.3343	−0.739969	−0.369985	−	0.929038i	$-0.620637\pi$
$911$	−22.3343	−0.739969	−0.369985	+	0.929038i	$0.620637\pi$
$912$	0	0
$913$	−13.4740	−0.445925
$914$	0	0
$915$	20.9386	0.692208
$916$	0	0
$917$	−16.0859	−0.531203
$918$	0	0
$919$	4.50072	0.148465	0.0742326	−	0.997241i	$-0.476349\pi$
$919$	4.50072	0.148465	0.0742326	+	0.997241i	$0.476349\pi$
$920$	0	0
$921$	−25.0278	−0.824695
$922$	0	0
$923$	101.846	3.35230
$924$	0	0
$925$	3.80756	0.125192
$926$	0	0
$927$	−0.397669	−0.0130612
$928$	0	0
$929$	−19.6631	−0.645127	−0.322563	−	0.946548i	$-0.604545\pi$
$929$	−19.6631	−0.645127	−0.322563	+	0.946548i	$0.604545\pi$
$930$	0	0
$931$	−1.33676	−0.0438105
$932$	0	0
$933$	52.0970	1.70558
$934$	0	0
$935$	−12.0366	−0.393639
$936$	0	0
$937$	−24.8588	−0.812103	−0.406051	−	0.913850i	$-0.633095\pi$
$937$	−24.8588	−0.812103	−0.406051	+	0.913850i	$0.633095\pi$
$938$	0	0
$939$	−20.5273	−0.669884
$940$	0	0
$941$	2.28099	0.0743580	0.0371790	−	0.999309i	$-0.488163\pi$
$941$	2.28099	0.0743580	0.0371790	+	0.999309i	$0.488163\pi$
$942$	0	0
$943$	−1.06038	−0.0345306
$944$	0	0
$945$	5.21645	0.169691
$946$	0	0
$947$	20.6956	0.672515	0.336258	−	0.941770i	$-0.390839\pi$
$947$	20.6956	0.672515	0.336258	+	0.941770i	$0.390839\pi$
$948$	0	0
$949$	−6.12171	−0.198719
$950$	0	0
$951$	34.8413	1.12981
$952$	0	0
$953$	−15.4607	−0.500821	−0.250410	−	0.968140i	$-0.580566\pi$
$953$	−15.4607	−0.500821	−0.250410	+	0.968140i	$0.580566\pi$
$954$	0	0
$955$	−12.8856	−0.416967
$956$	0	0
$957$	−8.68609	−0.280781
$958$	0	0
$959$	−11.5274	−0.372239
$960$	0	0
$961$	5.93996	0.191612
$962$	0	0
$963$	−0.256019	−0.00825011
$964$	0	0
$965$	8.47755	0.272902
$966$	0	0
$967$	−18.1149	−0.582537	−0.291268	−	0.956641i	$-0.594077\pi$
$967$	−18.1149	−0.582537	−0.291268	+	0.956641i	$0.594077\pi$
$968$	0	0
$969$	16.9002	0.542912
$970$	0	0
$971$	−22.2878	−0.715248	−0.357624	−	0.933866i	$-0.616413\pi$
$971$	−22.2878	−0.715248	−0.357624	+	0.933866i	$0.616413\pi$
$972$	0	0
$973$	−6.00986	−0.192667
$974$	0	0
$975$	−11.6387	−0.372736
$976$	0	0
$977$	34.0009	1.08779	0.543893	−	0.839154i	$-0.316950\pi$
$977$	34.0009	1.08779	0.543893	+	0.839154i	$0.316950\pi$
$978$	0	0
$979$	−3.77928	−0.120786
$980$	0	0
$981$	−0.309982	−0.00989696
$982$	0	0
$983$	−10.8422	−0.345811	−0.172905	−	0.984938i	$-0.555316\pi$
$983$	−10.8422	−0.345811	−0.172905	+	0.984938i	$0.555316\pi$
$984$	0	0
$985$	4.62935	0.147503
$986$	0	0
$987$	−4.07195	−0.129612
$988$	0	0
$989$	−3.73151	−0.118655
$990$	0	0
$991$	28.0418	0.890777	0.445388	−	0.895338i	$-0.353066\pi$
$991$	28.0418	0.890777	0.445388	+	0.895338i	$0.353066\pi$
$992$	0	0
$993$	−38.6224	−1.22565
$994$	0	0
$995$	−5.31367	−0.168455
$996$	0	0
$997$	−2.78623	−0.0882408	−0.0441204	−	0.999026i	$-0.514049\pi$
$997$	−2.78623	−0.0882408	−0.0441204	+	0.999026i	$0.514049\pi$
$998$	0	0
$999$	19.8619	0.628404

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.j.1.4	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.j.1.4	✓	13	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:	\( N \)	\(=\)	\( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.72520 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.0236759 q^{9} +O(q^{10})\)	\(q-1.72520 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.0236759 q^{9} +1.64250 q^{11} +6.74628 q^{13} +1.72520 q^{15} +7.32820 q^{17} -1.33676 q^{19} +1.72520 q^{21} +3.73151 q^{23} +1.00000 q^{25} +5.21645 q^{27} +3.06534 q^{29} -6.07783 q^{31} -2.83365 q^{33} +1.00000 q^{35} +3.80756 q^{37} -11.6387 q^{39} -0.284168 q^{41} -1.00000 q^{43} +0.0236759 q^{45} -2.36027 q^{47} +1.00000 q^{49} -12.6426 q^{51} -10.2750 q^{53} -1.64250 q^{55} +2.30618 q^{57} -13.1870 q^{59} +12.1369 q^{61} +0.0236759 q^{63} -6.74628 q^{65} -11.6051 q^{67} -6.43762 q^{69} +15.0966 q^{71} -0.907421 q^{73} -1.72520 q^{75} -1.64250 q^{77} +0.835831 q^{79} -8.92841 q^{81} -8.20336 q^{83} -7.32820 q^{85} -5.28833 q^{87} -2.30093 q^{89} -6.74628 q^{91} +10.4855 q^{93} +1.33676 q^{95} -7.13319 q^{97} -0.0388877 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9} + 6 q^{11} + q^{13} + q^{15} - 6 q^{17} + 2 q^{19} + q^{21} - 6 q^{23} + 13 q^{25} - q^{27} + 19 q^{29} - 24 q^{31} + 17 q^{33} + 13 q^{35} + 15 q^{39} + 4 q^{41} - 13 q^{43} - 18 q^{45} - 3 q^{47} + 13 q^{49} + 7 q^{51} - 5 q^{53} - 6 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 18 q^{63} - q^{65} - 2 q^{67} + 20 q^{69} + 18 q^{71} + 14 q^{73} - q^{75} - 6 q^{77} + 12 q^{79} + 37 q^{81} + 2 q^{83} + 6 q^{85} - 2 q^{87} + 17 q^{89} - q^{91} + 15 q^{93} - 2 q^{95} + 17 q^{97} + 80 q^{99}+O(q^{100}) \) 13 * q - q^3 - 13 * q^5 - 13 * q^7 + 18 * q^9 + 6 * q^11 + q^13 + q^15 - 6 * q^17 + 2 * q^19 + q^21 - 6 * q^23 + 13 * q^25 - q^27 + 19 * q^29 - 24 * q^31 + 17 * q^33 + 13 * q^35 + 15 * q^39 + 4 * q^41 - 13 * q^43 - 18 * q^45 - 3 * q^47 + 13 * q^49 + 7 * q^51 - 5 * q^53 - 6 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 18 * q^63 - q^65 - 2 * q^67 + 20 * q^69 + 18 * q^71 + 14 * q^73 - q^75 - 6 * q^77 + 12 * q^79 + 37 * q^81 + 2 * q^83 + 6 * q^85 - 2 * q^87 + 17 * q^89 - q^91 + 15 * q^93 - 2 * q^95 + 17 * q^97 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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