LMFDB - Embedded newform 6020.2.a.j.1.7

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			1.7
Root			$0.612150$ 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)

$f(q)$	$=$	$q-0.612150 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.62527 q^{9} +O(q^{10})$	$q-0.612150 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.62527 q^{9} +5.91240 q^{11} -1.93529 q^{13} +0.612150 q^{15} -1.73933 q^{17} +4.81444 q^{19} +0.612150 q^{21} +2.86111 q^{23} +1.00000 q^{25} +3.44351 q^{27} -4.67656 q^{29} +3.67422 q^{31} -3.61927 q^{33} +1.00000 q^{35} -1.26771 q^{37} +1.18469 q^{39} -6.58150 q^{41} -1.00000 q^{43} +2.62527 q^{45} -4.00350 q^{47} +1.00000 q^{49} +1.06473 q^{51} -10.1944 q^{53} -5.91240 q^{55} -2.94716 q^{57} +5.53697 q^{59} -2.86135 q^{61} +2.62527 q^{63} +1.93529 q^{65} +5.99409 q^{67} -1.75143 q^{69} +0.451083 q^{71} -13.9914 q^{73} -0.612150 q^{75} -5.91240 q^{77} +1.98615 q^{79} +5.76788 q^{81} +1.16216 q^{83} +1.73933 q^{85} +2.86276 q^{87} +13.5463 q^{89} +1.93529 q^{91} -2.24917 q^{93} -4.81444 q^{95} +9.15426 q^{97} -15.5217 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.612150	−0.353425	−0.176712	−	0.984263i	$-0.556546\pi$
$3$	−0.612150	−0.353425	−0.176712	+	0.984263i	$0.556546\pi$
$4$	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
$9$	−2.62527	−0.875091
$10$	0	0
$11$	5.91240	1.78266	0.891328	−	0.453359i	$-0.149774\pi$
$11$	5.91240	1.78266	0.891328	+	0.453359i	$0.149774\pi$
$12$	0	0
$13$	−1.93529	−0.536753	−0.268377	−	0.963314i	$-0.586487\pi$
$13$	−1.93529	−0.536753	−0.268377	+	0.963314i	$0.586487\pi$
$14$	0	0
$15$	0.612150	0.158056
$16$	0	0
$17$	−1.73933	−0.421850	−0.210925	−	0.977502i	$-0.567648\pi$
$17$	−1.73933	−0.421850	−0.210925	+	0.977502i	$0.567648\pi$
$18$	0	0
$19$	4.81444	1.10451	0.552254	−	0.833676i	$-0.313768\pi$
$19$	4.81444	1.10451	0.552254	+	0.833676i	$0.313768\pi$
$20$	0	0
$21$	0.612150	0.133582
$22$	0	0
$23$	2.86111	0.596583	0.298291	−	0.954475i	$-0.403583\pi$
$23$	2.86111	0.596583	0.298291	+	0.954475i	$0.403583\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.44351	0.662703
$28$	0	0
$29$	−4.67656	−0.868416	−0.434208	−	0.900813i	$-0.642972\pi$
$29$	−4.67656	−0.868416	−0.434208	+	0.900813i	$0.642972\pi$
$30$	0	0
$31$	3.67422	0.659910	0.329955	−	0.943997i	$-0.392966\pi$
$31$	3.67422	0.659910	0.329955	+	0.943997i	$0.392966\pi$
$32$	0	0
$33$	−3.61927	−0.630035
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.26771	−0.208410	−0.104205	−	0.994556i	$-0.533230\pi$
$37$	−1.26771	−0.208410	−0.104205	+	0.994556i	$0.533230\pi$
$38$	0	0
$39$	1.18469	0.189702
$40$	0	0
$41$	−6.58150	−1.02786	−0.513929	−	0.857833i	$-0.671810\pi$
$41$	−6.58150	−1.02786	−0.513929	+	0.857833i	$0.671810\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	2.62527	0.391353
$46$	0	0
$47$	−4.00350	−0.583970	−0.291985	−	0.956423i	$-0.594316\pi$
$47$	−4.00350	−0.583970	−0.291985	+	0.956423i	$0.594316\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.06473	0.149092
$52$	0	0
$53$	−10.1944	−1.40031	−0.700157	−	0.713989i	$-0.746887\pi$
$53$	−10.1944	−1.40031	−0.700157	+	0.713989i	$0.746887\pi$
$54$	0	0
$55$	−5.91240	−0.797228
$56$	0	0
$57$	−2.94716	−0.390360
$58$	0	0
$59$	5.53697	0.720852	0.360426	−	0.932788i	$-0.382631\pi$
$59$	5.53697	0.720852	0.360426	+	0.932788i	$0.382631\pi$
$60$	0	0
$61$	−2.86135	−0.366359	−0.183179	−	0.983080i	$-0.558639\pi$
$61$	−2.86135	−0.366359	−0.183179	+	0.983080i	$0.558639\pi$
$62$	0	0
$63$	2.62527	0.330753
$64$	0	0
$65$	1.93529	0.240043
$66$	0	0
$67$	5.99409	0.732295	0.366148	−	0.930557i	$-0.380677\pi$
$67$	5.99409	0.732295	0.366148	+	0.930557i	$0.380677\pi$
$68$	0	0
$69$	−1.75143	−0.210847
$70$	0	0
$71$	0.451083	0.0535337	0.0267668	−	0.999642i	$-0.491479\pi$
$71$	0.451083	0.0535337	0.0267668	+	0.999642i	$0.491479\pi$
$72$	0	0
$73$	−13.9914	−1.63757	−0.818786	−	0.574099i	$-0.805352\pi$
$73$	−13.9914	−1.63757	−0.818786	+	0.574099i	$0.805352\pi$
$74$	0	0
$75$	−0.612150	−0.0706849
$76$	0	0
$77$	−5.91240	−0.673781
$78$	0	0
$79$	1.98615	0.223460	0.111730	−	0.993739i	$-0.464361\pi$
$79$	1.98615	0.223460	0.111730	+	0.993739i	$0.464361\pi$
$80$	0	0
$81$	5.76788	0.640875
$82$	0	0
$83$	1.16216	0.127564	0.0637819	−	0.997964i	$-0.479684\pi$
$83$	1.16216	0.127564	0.0637819	+	0.997964i	$0.479684\pi$
$84$	0	0
$85$	1.73933	0.188657
$86$	0	0
$87$	2.86276	0.306920
$88$	0	0
$89$	13.5463	1.43590	0.717951	−	0.696093i	$-0.245080\pi$
$89$	13.5463	1.43590	0.717951	+	0.696093i	$0.245080\pi$
$90$	0	0
$91$	1.93529	0.202874
$92$	0	0
$93$	−2.24917	−0.233228
$94$	0	0
$95$	−4.81444	−0.493951
$96$	0	0
$97$	9.15426	0.929474	0.464737	−	0.885449i	$-0.346149\pi$
$97$	9.15426	0.929474	0.464737	+	0.885449i	$0.346149\pi$
$98$	0	0
$99$	−15.5217	−1.55999
$100$	0	0
$101$	−7.45022	−0.741325	−0.370662	−	0.928768i	$-0.620869\pi$
$101$	−7.45022	−0.741325	−0.370662	+	0.928768i	$0.620869\pi$
$102$	0	0
$103$	−6.01849	−0.593020	−0.296510	−	0.955030i	$-0.595823\pi$
$103$	−6.01849	−0.593020	−0.296510	+	0.955030i	$0.595823\pi$
$104$	0	0
$105$	−0.612150	−0.0597397
$106$	0	0
$107$	5.62049	0.543353	0.271677	−	0.962389i	$-0.412422\pi$
$107$	5.62049	0.543353	0.271677	+	0.962389i	$0.412422\pi$
$108$	0	0
$109$	−1.18483	−0.113486	−0.0567431	−	0.998389i	$-0.518072\pi$
$109$	−1.18483	−0.113486	−0.0567431	+	0.998389i	$0.518072\pi$
$110$	0	0
$111$	0.776029	0.0736574
$112$	0	0
$113$	0.192206	0.0180812	0.00904061	−	0.999959i	$-0.497122\pi$
$113$	0.192206	0.0180812	0.00904061	+	0.999959i	$0.497122\pi$
$114$	0	0
$115$	−2.86111	−0.266800
$116$	0	0
$117$	5.08067	0.469708
$118$	0	0
$119$	1.73933	0.159444
$120$	0	0
$121$	23.9565	2.17786
$122$	0	0
$123$	4.02886	0.363270
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.1524	1.16709	0.583543	−	0.812083i	$-0.301666\pi$
$127$	13.1524	1.16709	0.583543	+	0.812083i	$0.301666\pi$
$128$	0	0
$129$	0.612150	0.0538968
$130$	0	0
$131$	20.3017	1.77377	0.886884	−	0.461993i	$-0.152865\pi$
$131$	20.3017	1.77377	0.886884	+	0.461993i	$0.152865\pi$
$132$	0	0
$133$	−4.81444	−0.417465
$134$	0	0
$135$	−3.44351	−0.296370
$136$	0	0
$137$	16.3665	1.39828	0.699141	−	0.714984i	$-0.253566\pi$
$137$	16.3665	1.39828	0.699141	+	0.714984i	$0.253566\pi$
$138$	0	0
$139$	−12.6352	−1.07171	−0.535853	−	0.844312i	$-0.680010\pi$
$139$	−12.6352	−1.07171	−0.535853	+	0.844312i	$0.680010\pi$
$140$	0	0
$141$	2.45074	0.206390
$142$	0	0
$143$	−11.4422	−0.956846
$144$	0	0
$145$	4.67656	0.388368
$146$	0	0
$147$	−0.612150	−0.0504892
$148$	0	0
$149$	−0.321987	−0.0263782	−0.0131891	−	0.999913i	$-0.504198\pi$
$149$	−0.321987	−0.0263782	−0.0131891	+	0.999913i	$0.504198\pi$
$150$	0	0
$151$	23.2242	1.88996	0.944981	−	0.327125i	$-0.106080\pi$
$151$	23.2242	1.88996	0.944981	+	0.327125i	$0.106080\pi$
$152$	0	0
$153$	4.56622	0.369157
$154$	0	0
$155$	−3.67422	−0.295121
$156$	0	0
$157$	3.96266	0.316255	0.158127	−	0.987419i	$-0.449454\pi$
$157$	3.96266	0.316255	0.158127	+	0.987419i	$0.449454\pi$
$158$	0	0
$159$	6.24052	0.494906
$160$	0	0
$161$	−2.86111	−0.225487
$162$	0	0
$163$	−8.27993	−0.648534	−0.324267	−	0.945966i	$-0.605118\pi$
$163$	−8.27993	−0.648534	−0.324267	+	0.945966i	$0.605118\pi$
$164$	0	0
$165$	3.61927	0.281760
$166$	0	0
$167$	12.3967	0.959285	0.479643	−	0.877464i	$-0.340766\pi$
$167$	12.3967	0.959285	0.479643	+	0.877464i	$0.340766\pi$
$168$	0	0
$169$	−9.25465	−0.711896
$170$	0	0
$171$	−12.6392	−0.966545
$172$	0	0
$173$	−8.60332	−0.654099	−0.327049	−	0.945007i	$-0.606054\pi$
$173$	−8.60332	−0.654099	−0.327049	+	0.945007i	$0.606054\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−3.38945	−0.254767
$178$	0	0
$179$	13.1185	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$179$	13.1185	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$180$	0	0
$181$	5.27898	0.392384	0.196192	−	0.980566i	$-0.437142\pi$
$181$	5.27898	0.392384	0.196192	+	0.980566i	$0.437142\pi$
$182$	0	0
$183$	1.75158	0.129480
$184$	0	0
$185$	1.26771	0.0932040
$186$	0	0
$187$	−10.2836	−0.752013
$188$	0	0
$189$	−3.44351	−0.250478
$190$	0	0
$191$	3.26501	0.236248	0.118124	−	0.992999i	$-0.462312\pi$
$191$	3.26501	0.236248	0.118124	+	0.992999i	$0.462312\pi$
$192$	0	0
$193$	12.2624	0.882669	0.441334	−	0.897343i	$-0.354505\pi$
$193$	12.2624	0.882669	0.441334	+	0.897343i	$0.354505\pi$
$194$	0	0
$195$	−1.18469	−0.0848372
$196$	0	0
$197$	−1.50608	−0.107303	−0.0536517	−	0.998560i	$-0.517086\pi$
$197$	−1.50608	−0.107303	−0.0536517	+	0.998560i	$0.517086\pi$
$198$	0	0
$199$	8.32888	0.590419	0.295209	−	0.955433i	$-0.404611\pi$
$199$	8.32888	0.590419	0.295209	+	0.955433i	$0.404611\pi$
$200$	0	0
$201$	−3.66928	−0.258811
$202$	0	0
$203$	4.67656	0.328230
$204$	0	0
$205$	6.58150	0.459672
$206$	0	0
$207$	−7.51120	−0.522064
$208$	0	0
$209$	28.4649	1.96896
$210$	0	0
$211$	0.576601	0.0396949	0.0198474	−	0.999803i	$-0.493682\pi$
$211$	0.576601	0.0396949	0.0198474	+	0.999803i	$0.493682\pi$
$212$	0	0
$213$	−0.276130	−0.0189201
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−3.67422	−0.249422
$218$	0	0
$219$	8.56484	0.578758
$220$	0	0
$221$	3.36611	0.226429
$222$	0	0
$223$	2.37617	0.159120	0.0795602	−	0.996830i	$-0.474648\pi$
$223$	2.37617	0.159120	0.0795602	+	0.996830i	$0.474648\pi$
$224$	0	0
$225$	−2.62527	−0.175018
$226$	0	0
$227$	−4.12569	−0.273832	−0.136916	−	0.990583i	$-0.543719\pi$
$227$	−4.12569	−0.273832	−0.136916	+	0.990583i	$0.543719\pi$
$228$	0	0
$229$	15.5208	1.02564	0.512822	−	0.858495i	$-0.328600\pi$
$229$	15.5208	1.02564	0.512822	+	0.858495i	$0.328600\pi$
$230$	0	0
$231$	3.61927	0.238131
$232$	0	0
$233$	13.3256	0.872992	0.436496	−	0.899706i	$-0.356219\pi$
$233$	13.3256	0.872992	0.436496	+	0.899706i	$0.356219\pi$
$234$	0	0
$235$	4.00350	0.261160
$236$	0	0
$237$	−1.21582	−0.0789762
$238$	0	0
$239$	25.2099	1.63069	0.815347	−	0.578972i	$-0.196546\pi$
$239$	25.2099	1.63069	0.815347	+	0.578972i	$0.196546\pi$
$240$	0	0
$241$	−18.1830	−1.17127	−0.585636	−	0.810574i	$-0.699155\pi$
$241$	−18.1830	−1.17127	−0.585636	+	0.810574i	$0.699155\pi$
$242$	0	0
$243$	−13.8613	−0.889205
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−9.31734	−0.592848
$248$	0	0
$249$	−0.711417	−0.0450842
$250$	0	0
$251$	11.3825	0.718460	0.359230	−	0.933249i	$-0.383039\pi$
$251$	11.3825	0.718460	0.359230	+	0.933249i	$0.383039\pi$
$252$	0	0
$253$	16.9160	1.06350
$254$	0	0
$255$	−1.06473	−0.0666760
$256$	0	0
$257$	−2.32866	−0.145258	−0.0726290	−	0.997359i	$-0.523139\pi$
$257$	−2.32866	−0.145258	−0.0726290	+	0.997359i	$0.523139\pi$
$258$	0	0
$259$	1.26771	0.0787717
$260$	0	0
$261$	12.2773	0.759943
$262$	0	0
$263$	−22.3664	−1.37917	−0.689585	−	0.724204i	$-0.742207\pi$
$263$	−22.3664	−1.37917	−0.689585	+	0.724204i	$0.742207\pi$
$264$	0	0
$265$	10.1944	0.626239
$266$	0	0
$267$	−8.29235	−0.507483
$268$	0	0
$269$	−22.2767	−1.35824	−0.679118	−	0.734029i	$-0.737638\pi$
$269$	−22.2767	−1.35824	−0.679118	+	0.734029i	$0.737638\pi$
$270$	0	0
$271$	16.8396	1.02293	0.511467	−	0.859303i	$-0.329102\pi$
$271$	16.8396	1.02293	0.511467	+	0.859303i	$0.329102\pi$
$272$	0	0
$273$	−1.18469	−0.0717006
$274$	0	0
$275$	5.91240	0.356531
$276$	0	0
$277$	23.2600	1.39756	0.698779	−	0.715337i	$-0.253727\pi$
$277$	23.2600	1.39756	0.698779	+	0.715337i	$0.253727\pi$
$278$	0	0
$279$	−9.64584	−0.577481
$280$	0	0
$281$	19.1400	1.14180	0.570899	−	0.821020i	$-0.306595\pi$
$281$	19.1400	1.14180	0.570899	+	0.821020i	$0.306595\pi$
$282$	0	0
$283$	−15.5803	−0.926152	−0.463076	−	0.886319i	$-0.653254\pi$
$283$	−15.5803	−0.926152	−0.463076	+	0.886319i	$0.653254\pi$
$284$	0	0
$285$	2.94716	0.174574
$286$	0	0
$287$	6.58150	0.388494
$288$	0	0
$289$	−13.9747	−0.822043
$290$	0	0
$291$	−5.60377	−0.328499
$292$	0	0
$293$	−7.73238	−0.451731	−0.225865	−	0.974159i	$-0.572521\pi$
$293$	−7.73238	−0.451731	−0.225865	+	0.974159i	$0.572521\pi$
$294$	0	0
$295$	−5.53697	−0.322375
$296$	0	0
$297$	20.3594	1.18137
$298$	0	0
$299$	−5.53708	−0.320218
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	4.56065	0.262002
$304$	0	0
$305$	2.86135	0.163841
$306$	0	0
$307$	5.88056	0.335621	0.167811	−	0.985819i	$-0.446330\pi$
$307$	5.88056	0.335621	0.167811	+	0.985819i	$0.446330\pi$
$308$	0	0
$309$	3.68422	0.209588
$310$	0	0
$311$	5.97674	0.338910	0.169455	−	0.985538i	$-0.445799\pi$
$311$	5.97674	0.338910	0.169455	+	0.985538i	$0.445799\pi$
$312$	0	0
$313$	16.3470	0.923985	0.461993	−	0.886884i	$-0.347135\pi$
$313$	16.3470	0.923985	0.461993	+	0.886884i	$0.347135\pi$
$314$	0	0
$315$	−2.62527	−0.147917
$316$	0	0
$317$	26.6650	1.49766	0.748829	−	0.662763i	$-0.230616\pi$
$317$	26.6650	1.49766	0.748829	+	0.662763i	$0.230616\pi$
$318$	0	0
$319$	−27.6497	−1.54809
$320$	0	0
$321$	−3.44058	−0.192035
$322$	0	0
$323$	−8.37390	−0.465936
$324$	0	0
$325$	−1.93529	−0.107351
$326$	0	0
$327$	0.725293	0.0401088
$328$	0	0
$329$	4.00350	0.220720
$330$	0	0
$331$	22.6620	1.24562	0.622809	−	0.782374i	$-0.285992\pi$
$331$	22.6620	1.24562	0.622809	+	0.782374i	$0.285992\pi$
$332$	0	0
$333$	3.32809	0.182378
$334$	0	0
$335$	−5.99409	−0.327492
$336$	0	0
$337$	−15.4653	−0.842450	−0.421225	−	0.906956i	$-0.638400\pi$
$337$	−15.4653	−0.842450	−0.421225	+	0.906956i	$0.638400\pi$
$338$	0	0
$339$	−0.117659	−0.00639035
$340$	0	0
$341$	21.7235	1.17639
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.75143	0.0942937
$346$	0	0
$347$	22.9582	1.23246	0.616231	−	0.787566i	$-0.288659\pi$
$347$	22.9582	1.23246	0.616231	+	0.787566i	$0.288659\pi$
$348$	0	0
$349$	0.607518	0.0325197	0.0162599	−	0.999868i	$-0.494824\pi$
$349$	0.607518	0.0325197	0.0162599	+	0.999868i	$0.494824\pi$
$350$	0	0
$351$	−6.66419	−0.355708
$352$	0	0
$353$	10.8989	0.580088	0.290044	−	0.957013i	$-0.406330\pi$
$353$	10.8989	0.580088	0.290044	+	0.957013i	$0.406330\pi$
$354$	0	0
$355$	−0.451083	−0.0239410
$356$	0	0
$357$	−1.06473	−0.0563515
$358$	0	0
$359$	−5.61860	−0.296539	−0.148269	−	0.988947i	$-0.547370\pi$
$359$	−5.61860	−0.296539	−0.148269	+	0.988947i	$0.547370\pi$
$360$	0	0
$361$	4.17880	0.219937
$362$	0	0
$363$	−14.6650	−0.769711
$364$	0	0
$365$	13.9914	0.732344
$366$	0	0
$367$	−24.4392	−1.27571	−0.637857	−	0.770155i	$-0.720179\pi$
$367$	−24.4392	−1.27571	−0.637857	+	0.770155i	$0.720179\pi$
$368$	0	0
$369$	17.2782	0.899469
$370$	0	0
$371$	10.1944	0.529269
$372$	0	0
$373$	34.0383	1.76244	0.881218	−	0.472710i	$-0.156724\pi$
$373$	34.0383	1.76244	0.881218	+	0.472710i	$0.156724\pi$
$374$	0	0
$375$	0.612150	0.0316113
$376$	0	0
$377$	9.05051	0.466125
$378$	0	0
$379$	26.1141	1.34139	0.670695	−	0.741733i	$-0.265996\pi$
$379$	26.1141	1.34139	0.670695	+	0.741733i	$0.265996\pi$
$380$	0	0
$381$	−8.05123	−0.412477
$382$	0	0
$383$	−14.4909	−0.740451	−0.370226	−	0.928942i	$-0.620720\pi$
$383$	−14.4909	−0.740451	−0.370226	+	0.928942i	$0.620720\pi$
$384$	0	0
$385$	5.91240	0.301324
$386$	0	0
$387$	2.62527	0.133450
$388$	0	0
$389$	1.97200	0.0999843	0.0499922	−	0.998750i	$-0.484080\pi$
$389$	1.97200	0.0999843	0.0499922	+	0.998750i	$0.484080\pi$
$390$	0	0
$391$	−4.97642	−0.251668
$392$	0	0
$393$	−12.4277	−0.626893
$394$	0	0
$395$	−1.98615	−0.0999343
$396$	0	0
$397$	−30.1482	−1.51309	−0.756547	−	0.653940i	$-0.773115\pi$
$397$	−30.1482	−1.51309	−0.756547	+	0.653940i	$0.773115\pi$
$398$	0	0
$399$	2.94716	0.147542
$400$	0	0
$401$	11.3590	0.567242	0.283621	−	0.958937i	$-0.408464\pi$
$401$	11.3590	0.567242	0.283621	+	0.958937i	$0.408464\pi$
$402$	0	0
$403$	−7.11069	−0.354209
$404$	0	0
$405$	−5.76788	−0.286608
$406$	0	0
$407$	−7.49522	−0.371524
$408$	0	0
$409$	17.6786	0.874151	0.437076	−	0.899425i	$-0.356014\pi$
$409$	17.6786	0.874151	0.437076	+	0.899425i	$0.356014\pi$
$410$	0	0
$411$	−10.0187	−0.494187
$412$	0	0
$413$	−5.53697	−0.272456
$414$	0	0
$415$	−1.16216	−0.0570483
$416$	0	0
$417$	7.73464	0.378767
$418$	0	0
$419$	−0.731781	−0.0357498	−0.0178749	−	0.999840i	$-0.505690\pi$
$419$	−0.731781	−0.0357498	−0.0178749	+	0.999840i	$0.505690\pi$
$420$	0	0
$421$	−22.8001	−1.11121	−0.555605	−	0.831446i	$-0.687513\pi$
$421$	−22.8001	−1.11121	−0.555605	+	0.831446i	$0.687513\pi$
$422$	0	0
$423$	10.5103	0.511027
$424$	0	0
$425$	−1.73933	−0.0843700
$426$	0	0
$427$	2.86135	0.138471
$428$	0	0
$429$	7.00435	0.338173
$430$	0	0
$431$	−2.48131	−0.119521	−0.0597603	−	0.998213i	$-0.519034\pi$
$431$	−2.48131	−0.119521	−0.0597603	+	0.998213i	$0.519034\pi$
$432$	0	0
$433$	14.5626	0.699835	0.349917	−	0.936781i	$-0.386210\pi$
$433$	14.5626	0.699835	0.349917	+	0.936781i	$0.386210\pi$
$434$	0	0
$435$	−2.86276	−0.137259
$436$	0	0
$437$	13.7746	0.658930
$438$	0	0
$439$	−3.60942	−0.172268	−0.0861341	−	0.996284i	$-0.527451\pi$
$439$	−3.60942	−0.172268	−0.0861341	+	0.996284i	$0.527451\pi$
$440$	0	0
$441$	−2.62527	−0.125013
$442$	0	0
$443$	12.3587	0.587179	0.293590	−	0.955932i	$-0.405150\pi$
$443$	12.3587	0.587179	0.293590	+	0.955932i	$0.405150\pi$
$444$	0	0
$445$	−13.5463	−0.642155
$446$	0	0
$447$	0.197104	0.00932270
$448$	0	0
$449$	−19.8351	−0.936075	−0.468038	−	0.883709i	$-0.655039\pi$
$449$	−19.8351	−0.936075	−0.468038	+	0.883709i	$0.655039\pi$
$450$	0	0
$451$	−38.9125	−1.83232
$452$	0	0
$453$	−14.2167	−0.667959
$454$	0	0
$455$	−1.93529	−0.0907279
$456$	0	0
$457$	33.5718	1.57042	0.785210	−	0.619229i	$-0.212555\pi$
$457$	33.5718	1.57042	0.785210	+	0.619229i	$0.212555\pi$
$458$	0	0
$459$	−5.98940	−0.279561
$460$	0	0
$461$	4.94717	0.230413	0.115206	−	0.993342i	$-0.463247\pi$
$461$	4.94717	0.230413	0.115206	+	0.993342i	$0.463247\pi$
$462$	0	0
$463$	5.48570	0.254942	0.127471	−	0.991842i	$-0.459314\pi$
$463$	5.48570	0.254942	0.127471	+	0.991842i	$0.459314\pi$
$464$	0	0
$465$	2.24917	0.104303
$466$	0	0
$467$	1.04621	0.0484128	0.0242064	−	0.999707i	$-0.492294\pi$
$467$	1.04621	0.0484128	0.0242064	+	0.999707i	$0.492294\pi$
$468$	0	0
$469$	−5.99409	−0.276782
$470$	0	0
$471$	−2.42574	−0.111772
$472$	0	0
$473$	−5.91240	−0.271853
$474$	0	0
$475$	4.81444	0.220901
$476$	0	0
$477$	26.7632	1.22540
$478$	0	0
$479$	−7.13192	−0.325866	−0.162933	−	0.986637i	$-0.552095\pi$
$479$	−7.13192	−0.325866	−0.162933	+	0.986637i	$0.552095\pi$
$480$	0	0
$481$	2.45339	0.111865
$482$	0	0
$483$	1.75143	0.0796927
$484$	0	0
$485$	−9.15426	−0.415673
$486$	0	0
$487$	−10.8039	−0.489573	−0.244786	−	0.969577i	$-0.578718\pi$
$487$	−10.8039	−0.489573	−0.244786	+	0.969577i	$0.578718\pi$
$488$	0	0
$489$	5.06856	0.229208
$490$	0	0
$491$	31.4092	1.41748	0.708739	−	0.705471i	$-0.249264\pi$
$491$	31.4092	1.41748	0.708739	+	0.705471i	$0.249264\pi$
$492$	0	0
$493$	8.13410	0.366341
$494$	0	0
$495$	15.5217	0.697647
$496$	0	0
$497$	−0.451083	−0.0202338
$498$	0	0
$499$	−11.0186	−0.493260	−0.246630	−	0.969110i	$-0.579323\pi$
$499$	−11.0186	−0.493260	−0.246630	+	0.969110i	$0.579323\pi$
$500$	0	0
$501$	−7.58863	−0.339035
$502$	0	0
$503$	7.39982	0.329942	0.164971	−	0.986298i	$-0.447247\pi$
$503$	7.39982	0.329942	0.164971	+	0.986298i	$0.447247\pi$
$504$	0	0
$505$	7.45022	0.331530
$506$	0	0
$507$	5.66523	0.251602
$508$	0	0
$509$	−35.3909	−1.56868	−0.784338	−	0.620334i	$-0.786997\pi$
$509$	−35.3909	−1.56868	−0.784338	+	0.620334i	$0.786997\pi$
$510$	0	0
$511$	13.9914	0.618944
$512$	0	0
$513$	16.5786	0.731961
$514$	0	0
$515$	6.01849	0.265206
$516$	0	0
$517$	−23.6703	−1.04102
$518$	0	0
$519$	5.26652	0.231175
$520$	0	0
$521$	4.88820	0.214156	0.107078	−	0.994251i	$-0.465851\pi$
$521$	4.88820	0.214156	0.107078	+	0.994251i	$0.465851\pi$
$522$	0	0
$523$	16.7470	0.732293	0.366147	−	0.930557i	$-0.380677\pi$
$523$	16.7470	0.732293	0.366147	+	0.930557i	$0.380677\pi$
$524$	0	0
$525$	0.612150	0.0267164
$526$	0	0
$527$	−6.39069	−0.278383
$528$	0	0
$529$	−14.8140	−0.644089
$530$	0	0
$531$	−14.5360	−0.630811
$532$	0	0
$533$	12.7371	0.551706
$534$	0	0
$535$	−5.62049	−0.242995
$536$	0	0
$537$	−8.03048	−0.346541
$538$	0	0
$539$	5.91240	0.254665
$540$	0	0
$541$	45.6698	1.96350	0.981748	−	0.190185i	$-0.0609087\pi$
$541$	45.6698	1.96350	0.981748	+	0.190185i	$0.0609087\pi$
$542$	0	0
$543$	−3.23153	−0.138678
$544$	0	0
$545$	1.18483	0.0507525
$546$	0	0
$547$	−10.7977	−0.461674	−0.230837	−	0.972992i	$-0.574146\pi$
$547$	−10.7977	−0.461674	−0.230837	+	0.972992i	$0.574146\pi$
$548$	0	0
$549$	7.51183	0.320597
$550$	0	0
$551$	−22.5150	−0.959172
$552$	0	0
$553$	−1.98615	−0.0844599
$554$	0	0
$555$	−0.776029	−0.0329406
$556$	0	0
$557$	−23.4994	−0.995700	−0.497850	−	0.867263i	$-0.665877\pi$
$557$	−23.4994	−0.995700	−0.497850	+	0.867263i	$0.665877\pi$
$558$	0	0
$559$	1.93529	0.0818541
$560$	0	0
$561$	6.29512	0.265780
$562$	0	0
$563$	−0.597972	−0.0252015	−0.0126007	−	0.999921i	$-0.504011\pi$
$563$	−0.597972	−0.0252015	−0.0126007	+	0.999921i	$0.504011\pi$
$564$	0	0
$565$	−0.192206	−0.00808617
$566$	0	0
$567$	−5.76788	−0.242228
$568$	0	0
$569$	−0.952842	−0.0399452	−0.0199726	−	0.999801i	$-0.506358\pi$
$569$	−0.952842	−0.0399452	−0.0199726	+	0.999801i	$0.506358\pi$
$570$	0	0
$571$	−27.5126	−1.15137	−0.575683	−	0.817673i	$-0.695264\pi$
$571$	−27.5126	−1.15137	−0.575683	+	0.817673i	$0.695264\pi$
$572$	0	0
$573$	−1.99867	−0.0834957
$574$	0	0
$575$	2.86111	0.119317
$576$	0	0
$577$	14.9925	0.624145	0.312072	−	0.950058i	$-0.398977\pi$
$577$	14.9925	0.624145	0.312072	+	0.950058i	$0.398977\pi$
$578$	0	0
$579$	−7.50644	−0.311957
$580$	0	0
$581$	−1.16216	−0.0482146
$582$	0	0
$583$	−60.2736	−2.49628
$584$	0	0
$585$	−5.08067	−0.210060
$586$	0	0
$587$	−2.05582	−0.0848527	−0.0424263	−	0.999100i	$-0.513509\pi$
$587$	−2.05582	−0.0848527	−0.0424263	+	0.999100i	$0.513509\pi$
$588$	0	0
$589$	17.6893	0.728875
$590$	0	0
$591$	0.921943	0.0379237
$592$	0	0
$593$	−34.6660	−1.42356	−0.711780	−	0.702402i	$-0.752111\pi$
$593$	−34.6660	−1.42356	−0.711780	+	0.702402i	$0.752111\pi$
$594$	0	0
$595$	−1.73933	−0.0713056
$596$	0	0
$597$	−5.09852	−0.208669
$598$	0	0
$599$	−25.4355	−1.03927	−0.519633	−	0.854390i	$-0.673931\pi$
$599$	−25.4355	−1.03927	−0.519633	+	0.854390i	$0.673931\pi$
$600$	0	0
$601$	−29.7024	−1.21159	−0.605794	−	0.795622i	$-0.707144\pi$
$601$	−29.7024	−1.21159	−0.605794	+	0.795622i	$0.707144\pi$
$602$	0	0
$603$	−15.7361	−0.640825
$604$	0	0
$605$	−23.9565	−0.973970
$606$	0	0
$607$	26.0432	1.05706	0.528530	−	0.848915i	$-0.322744\pi$
$607$	26.0432	1.05706	0.528530	+	0.848915i	$0.322744\pi$
$608$	0	0
$609$	−2.86276	−0.116005
$610$	0	0
$611$	7.74794	0.313448
$612$	0	0
$613$	6.23846	0.251969	0.125984	−	0.992032i	$-0.459791\pi$
$613$	6.23846	0.251969	0.125984	+	0.992032i	$0.459791\pi$
$614$	0	0
$615$	−4.02886	−0.162459
$616$	0	0
$617$	−13.6555	−0.549750	−0.274875	−	0.961480i	$-0.588637\pi$
$617$	−13.6555	−0.549750	−0.274875	+	0.961480i	$0.588637\pi$
$618$	0	0
$619$	16.2947	0.654938	0.327469	−	0.944862i	$-0.393804\pi$
$619$	16.2947	0.654938	0.327469	+	0.944862i	$0.393804\pi$
$620$	0	0
$621$	9.85226	0.395357
$622$	0	0
$623$	−13.5463	−0.542720
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.4248	−0.695878
$628$	0	0
$629$	2.20497	0.0879179
$630$	0	0
$631$	28.8150	1.14711	0.573553	−	0.819168i	$-0.305565\pi$
$631$	28.8150	1.14711	0.573553	+	0.819168i	$0.305565\pi$
$632$	0	0
$633$	−0.352966	−0.0140291
$634$	0	0
$635$	−13.1524	−0.521936
$636$	0	0
$637$	−1.93529	−0.0766790
$638$	0	0
$639$	−1.18422	−0.0468468
$640$	0	0
$641$	−14.0801	−0.556131	−0.278066	−	0.960562i	$-0.589693\pi$
$641$	−14.0801	−0.556131	−0.278066	+	0.960562i	$0.589693\pi$
$642$	0	0
$643$	27.4713	1.08336	0.541682	−	0.840584i	$-0.317788\pi$
$643$	27.4713	1.08336	0.541682	+	0.840584i	$0.317788\pi$
$644$	0	0
$645$	−0.612150	−0.0241034
$646$	0	0
$647$	−19.6329	−0.771851	−0.385925	−	0.922530i	$-0.626118\pi$
$647$	−19.6329	−0.771851	−0.385925	+	0.922530i	$0.626118\pi$
$648$	0	0
$649$	32.7368	1.28503
$650$	0	0
$651$	2.24917	0.0881521
$652$	0	0
$653$	−14.0202	−0.548651	−0.274326	−	0.961637i	$-0.588455\pi$
$653$	−14.0202	−0.548651	−0.274326	+	0.961637i	$0.588455\pi$
$654$	0	0
$655$	−20.3017	−0.793253
$656$	0	0
$657$	36.7313	1.43302
$658$	0	0
$659$	3.21766	0.125342	0.0626712	−	0.998034i	$-0.480038\pi$
$659$	3.21766	0.125342	0.0626712	+	0.998034i	$0.480038\pi$
$660$	0	0
$661$	41.2206	1.60330	0.801649	−	0.597796i	$-0.203957\pi$
$661$	41.2206	1.60330	0.801649	+	0.597796i	$0.203957\pi$
$662$	0	0
$663$	−2.06056	−0.0800257
$664$	0	0
$665$	4.81444	0.186696
$666$	0	0
$667$	−13.3802	−0.518082
$668$	0	0
$669$	−1.45457	−0.0562371
$670$	0	0
$671$	−16.9175	−0.653092
$672$	0	0
$673$	19.2769	0.743070	0.371535	−	0.928419i	$-0.378832\pi$
$673$	19.2769	0.743070	0.371535	+	0.928419i	$0.378832\pi$
$674$	0	0
$675$	3.44351	0.132541
$676$	0	0
$677$	3.60391	0.138510	0.0692548	−	0.997599i	$-0.477938\pi$
$677$	3.60391	0.138510	0.0692548	+	0.997599i	$0.477938\pi$
$678$	0	0
$679$	−9.15426	−0.351308
$680$	0	0
$681$	2.52554	0.0967789
$682$	0	0
$683$	12.0674	0.461744	0.230872	−	0.972984i	$-0.425842\pi$
$683$	12.0674	0.461744	0.230872	+	0.972984i	$0.425842\pi$
$684$	0	0
$685$	−16.3665	−0.625331
$686$	0	0
$687$	−9.50106	−0.362488
$688$	0	0
$689$	19.7292	0.751623
$690$	0	0
$691$	29.5161	1.12284	0.561422	−	0.827529i	$-0.310254\pi$
$691$	29.5161	1.12284	0.561422	+	0.827529i	$0.310254\pi$
$692$	0	0
$693$	15.5217	0.589619
$694$	0	0
$695$	12.6352	0.479281
$696$	0	0
$697$	11.4474	0.433601
$698$	0	0
$699$	−8.15729	−0.308537
$700$	0	0
$701$	21.4849	0.811473	0.405737	−	0.913990i	$-0.367015\pi$
$701$	21.4849	0.811473	0.405737	+	0.913990i	$0.367015\pi$
$702$	0	0
$703$	−6.10331	−0.230191
$704$	0	0
$705$	−2.45074	−0.0923002
$706$	0	0
$707$	7.45022	0.280194
$708$	0	0
$709$	−8.95431	−0.336286	−0.168143	−	0.985763i	$-0.553777\pi$
$709$	−8.95431	−0.336286	−0.168143	+	0.985763i	$0.553777\pi$
$710$	0	0
$711$	−5.21420	−0.195548
$712$	0	0
$713$	10.5124	0.393691
$714$	0	0
$715$	11.4422	0.427915
$716$	0	0
$717$	−15.4322	−0.576328
$718$	0	0
$719$	31.5690	1.17732	0.588662	−	0.808379i	$-0.299655\pi$
$719$	31.5690	1.17732	0.588662	+	0.808379i	$0.299655\pi$
$720$	0	0
$721$	6.01849	0.224140
$722$	0	0
$723$	11.1307	0.413956
$724$	0	0
$725$	−4.67656	−0.173683
$726$	0	0
$727$	14.5380	0.539185	0.269592	−	0.962975i	$-0.413111\pi$
$727$	14.5380	0.539185	0.269592	+	0.962975i	$0.413111\pi$
$728$	0	0
$729$	−8.81843	−0.326608
$730$	0	0
$731$	1.73933	0.0643315
$732$	0	0
$733$	40.7106	1.50368	0.751839	−	0.659346i	$-0.229167\pi$
$733$	40.7106	1.50368	0.751839	+	0.659346i	$0.229167\pi$
$734$	0	0
$735$	0.612150	0.0225795
$736$	0	0
$737$	35.4395	1.30543
$738$	0	0
$739$	−9.31559	−0.342679	−0.171340	−	0.985212i	$-0.554810\pi$
$739$	−9.31559	−0.342679	−0.171340	+	0.985212i	$0.554810\pi$
$740$	0	0
$741$	5.70360	0.209527
$742$	0	0
$743$	18.5606	0.680924	0.340462	−	0.940258i	$-0.389417\pi$
$743$	18.5606	0.680924	0.340462	+	0.940258i	$0.389417\pi$
$744$	0	0
$745$	0.321987	0.0117967
$746$	0	0
$747$	−3.05099	−0.111630
$748$	0	0
$749$	−5.62049	−0.205368
$750$	0	0
$751$	−34.5884	−1.26215	−0.631074	−	0.775723i	$-0.717385\pi$
$751$	−34.5884	−1.26215	−0.631074	+	0.775723i	$0.717385\pi$
$752$	0	0
$753$	−6.96782	−0.253921
$754$	0	0
$755$	−23.2242	−0.845217
$756$	0	0
$757$	−10.0280	−0.364472	−0.182236	−	0.983255i	$-0.558334\pi$
$757$	−10.0280	−0.364472	−0.182236	+	0.983255i	$0.558334\pi$
$758$	0	0
$759$	−10.3551	−0.375868
$760$	0	0
$761$	0.398102	0.0144312	0.00721559	−	0.999974i	$-0.497703\pi$
$761$	0.398102	0.0144312	0.00721559	+	0.999974i	$0.497703\pi$
$762$	0	0
$763$	1.18483	0.0428937
$764$	0	0
$765$	−4.56622	−0.165092
$766$	0	0
$767$	−10.7156	−0.386919
$768$	0	0
$769$	50.5288	1.82211	0.911057	−	0.412280i	$-0.135268\pi$
$769$	50.5288	1.82211	0.911057	+	0.412280i	$0.135268\pi$
$770$	0	0
$771$	1.42549	0.0513377
$772$	0	0
$773$	−37.1675	−1.33682	−0.668410	−	0.743793i	$-0.733025\pi$
$773$	−37.1675	−1.33682	−0.668410	+	0.743793i	$0.733025\pi$
$774$	0	0
$775$	3.67422	0.131982
$776$	0	0
$777$	−0.776029	−0.0278399
$778$	0	0
$779$	−31.6862	−1.13528
$780$	0	0
$781$	2.66698	0.0954321
$782$	0	0
$783$	−16.1038	−0.575502
$784$	0	0
$785$	−3.96266	−0.141433
$786$	0	0
$787$	−14.2634	−0.508435	−0.254218	−	0.967147i	$-0.581818\pi$
$787$	−14.2634	−0.508435	−0.254218	+	0.967147i	$0.581818\pi$
$788$	0	0
$789$	13.6916	0.487433
$790$	0	0
$791$	−0.192206	−0.00683406
$792$	0	0
$793$	5.53755	0.196644
$794$	0	0
$795$	−6.24052	−0.221328
$796$	0	0
$797$	−22.3028	−0.790007	−0.395004	−	0.918680i	$-0.629257\pi$
$797$	−22.3028	−0.790007	−0.395004	+	0.918680i	$0.629257\pi$
$798$	0	0
$799$	6.96341	0.246348
$800$	0	0
$801$	−35.5627	−1.25655
$802$	0	0
$803$	−82.7229	−2.91923
$804$	0	0
$805$	2.86111	0.100841
$806$	0	0
$807$	13.6367	0.480034
$808$	0	0
$809$	23.4188	0.823360	0.411680	−	0.911329i	$-0.364942\pi$
$809$	23.4188	0.823360	0.411680	+	0.911329i	$0.364942\pi$
$810$	0	0
$811$	19.9649	0.701061	0.350530	−	0.936551i	$-0.386001\pi$
$811$	19.9649	0.701061	0.350530	+	0.936551i	$0.386001\pi$
$812$	0	0
$813$	−10.3084	−0.361530
$814$	0	0
$815$	8.27993	0.290033
$816$	0	0
$817$	−4.81444	−0.168436
$818$	0	0
$819$	−5.08067	−0.177533
$820$	0	0
$821$	28.2744	0.986784	0.493392	−	0.869807i	$-0.335757\pi$
$821$	28.2744	0.986784	0.493392	+	0.869807i	$0.335757\pi$
$822$	0	0
$823$	8.94801	0.311908	0.155954	−	0.987764i	$-0.450155\pi$
$823$	8.94801	0.311908	0.155954	+	0.987764i	$0.450155\pi$
$824$	0	0
$825$	−3.61927	−0.126007
$826$	0	0
$827$	0.380717	0.0132388	0.00661942	−	0.999978i	$-0.497893\pi$
$827$	0.380717	0.0132388	0.00661942	+	0.999978i	$0.497893\pi$
$828$	0	0
$829$	−15.1248	−0.525304	−0.262652	−	0.964891i	$-0.584597\pi$
$829$	−15.1248	−0.525304	−0.262652	+	0.964891i	$0.584597\pi$
$830$	0	0
$831$	−14.2386	−0.493932
$832$	0	0
$833$	−1.73933	−0.0602643
$834$	0	0
$835$	−12.3967	−0.429005
$836$	0	0
$837$	12.6522	0.437325
$838$	0	0
$839$	17.2650	0.596054	0.298027	−	0.954557i	$-0.403671\pi$
$839$	17.2650	0.596054	0.298027	+	0.954557i	$0.403671\pi$
$840$	0	0
$841$	−7.12975	−0.245853
$842$	0	0
$843$	−11.7166	−0.403540
$844$	0	0
$845$	9.25465	0.318370
$846$	0	0
$847$	−23.9565	−0.823155
$848$	0	0
$849$	9.53747	0.327325
$850$	0	0
$851$	−3.62706	−0.124334
$852$	0	0
$853$	49.8966	1.70843	0.854214	−	0.519922i	$-0.174039\pi$
$853$	49.8966	1.70843	0.854214	+	0.519922i	$0.174039\pi$
$854$	0	0
$855$	12.6392	0.432252
$856$	0	0
$857$	−12.9721	−0.443119	−0.221560	−	0.975147i	$-0.571115\pi$
$857$	−12.9721	−0.443119	−0.221560	+	0.975147i	$0.571115\pi$
$858$	0	0
$859$	−6.30942	−0.215275	−0.107637	−	0.994190i	$-0.534329\pi$
$859$	−6.30942	−0.215275	−0.107637	+	0.994190i	$0.534329\pi$
$860$	0	0
$861$	−4.02886	−0.137303
$862$	0	0
$863$	22.9208	0.780233	0.390117	−	0.920765i	$-0.372435\pi$
$863$	22.9208	0.780233	0.390117	+	0.920765i	$0.372435\pi$
$864$	0	0
$865$	8.60332	0.292522
$866$	0	0
$867$	8.55462	0.290530
$868$	0	0
$869$	11.7429	0.398352
$870$	0	0
$871$	−11.6003	−0.393062
$872$	0	0
$873$	−24.0324	−0.813374
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−36.1273	−1.21993	−0.609967	−	0.792427i	$-0.708817\pi$
$877$	−36.1273	−1.21993	−0.609967	+	0.792427i	$0.708817\pi$
$878$	0	0
$879$	4.73337	0.159653
$880$	0	0
$881$	−2.04541	−0.0689116	−0.0344558	−	0.999406i	$-0.510970\pi$
$881$	−2.04541	−0.0689116	−0.0344558	+	0.999406i	$0.510970\pi$
$882$	0	0
$883$	38.6963	1.30224	0.651118	−	0.758977i	$-0.274300\pi$
$883$	38.6963	1.30224	0.651118	+	0.758977i	$0.274300\pi$
$884$	0	0
$885$	3.38945	0.113935
$886$	0	0
$887$	−41.9214	−1.40758	−0.703791	−	0.710407i	$-0.748511\pi$
$887$	−41.9214	−1.40758	−0.703791	+	0.710407i	$0.748511\pi$
$888$	0	0
$889$	−13.1524	−0.441117
$890$	0	0
$891$	34.1020	1.14246
$892$	0	0
$893$	−19.2746	−0.645000
$894$	0	0
$895$	−13.1185	−0.438503
$896$	0	0
$897$	3.38952	0.113173
$898$	0	0
$899$	−17.1827	−0.573076
$900$	0	0
$901$	17.7315	0.590722
$902$	0	0
$903$	−0.612150	−0.0203711
$904$	0	0
$905$	−5.27898	−0.175479
$906$	0	0
$907$	15.3374	0.509269	0.254635	−	0.967037i	$-0.418045\pi$
$907$	15.3374	0.509269	0.254635	+	0.967037i	$0.418045\pi$
$908$	0	0
$909$	19.5589	0.648726
$910$	0	0
$911$	18.5317	0.613982	0.306991	−	0.951712i	$-0.400678\pi$
$911$	18.5317	0.613982	0.306991	+	0.951712i	$0.400678\pi$
$912$	0	0
$913$	6.87117	0.227402
$914$	0	0
$915$	−1.75158	−0.0579053
$916$	0	0
$917$	−20.3017	−0.670421
$918$	0	0
$919$	−6.49503	−0.214251	−0.107126	−	0.994245i	$-0.534165\pi$
$919$	−6.49503	−0.214251	−0.107126	+	0.994245i	$0.534165\pi$
$920$	0	0
$921$	−3.59978	−0.118617
$922$	0	0
$923$	−0.872977	−0.0287344
$924$	0	0
$925$	−1.26771	−0.0416821
$926$	0	0
$927$	15.8002	0.518946
$928$	0	0
$929$	−17.9152	−0.587780	−0.293890	−	0.955839i	$-0.594950\pi$
$929$	−17.9152	−0.587780	−0.293890	+	0.955839i	$0.594950\pi$
$930$	0	0
$931$	4.81444	0.157787
$932$	0	0
$933$	−3.65866	−0.119779
$934$	0	0
$935$	10.2836	0.336311
$936$	0	0
$937$	39.4893	1.29006	0.645030	−	0.764158i	$-0.276845\pi$
$937$	39.4893	1.29006	0.645030	+	0.764158i	$0.276845\pi$
$938$	0	0
$939$	−10.0068	−0.326559
$940$	0	0
$941$	3.56203	0.116119	0.0580594	−	0.998313i	$-0.481509\pi$
$941$	3.56203	0.116119	0.0580594	+	0.998313i	$0.481509\pi$
$942$	0	0
$943$	−18.8304	−0.613202
$944$	0	0
$945$	3.44351	0.112017
$946$	0	0
$947$	−11.1936	−0.363744	−0.181872	−	0.983322i	$-0.558216\pi$
$947$	−11.1936	−0.363744	−0.181872	+	0.983322i	$0.558216\pi$
$948$	0	0
$949$	27.0775	0.878972
$950$	0	0
$951$	−16.3230	−0.529309
$952$	0	0
$953$	6.53703	0.211755	0.105878	−	0.994379i	$-0.466235\pi$
$953$	6.53703	0.211755	0.105878	+	0.994379i	$0.466235\pi$
$954$	0	0
$955$	−3.26501	−0.105653
$956$	0	0
$957$	16.9258	0.547132
$958$	0	0
$959$	−16.3665	−0.528501
$960$	0	0
$961$	−17.5001	−0.564519
$962$	0	0
$963$	−14.7553	−0.475484
$964$	0	0
$965$	−12.2624	−0.394741
$966$	0	0
$967$	25.6015	0.823290	0.411645	−	0.911344i	$-0.364954\pi$
$967$	25.6015	0.823290	0.411645	+	0.911344i	$0.364954\pi$
$968$	0	0
$969$	5.12608	0.164673
$970$	0	0
$971$	−46.3007	−1.48586	−0.742931	−	0.669368i	$-0.766565\pi$
$971$	−46.3007	−1.48586	−0.742931	+	0.669368i	$0.766565\pi$
$972$	0	0
$973$	12.6352	0.405067
$974$	0	0
$975$	1.18469	0.0379404
$976$	0	0
$977$	−56.4192	−1.80501	−0.902505	−	0.430680i	$-0.858274\pi$
$977$	−56.4192	−1.80501	−0.902505	+	0.430680i	$0.858274\pi$
$978$	0	0
$979$	80.0910	2.55972
$980$	0	0
$981$	3.11050	0.0993107
$982$	0	0
$983$	51.7887	1.65180	0.825902	−	0.563814i	$-0.190667\pi$
$983$	51.7887	1.65180	0.825902	+	0.563814i	$0.190667\pi$
$984$	0	0
$985$	1.50608	0.0479875
$986$	0	0
$987$	−2.45074	−0.0780079
$988$	0	0
$989$	−2.86111	−0.0909780
$990$	0	0
$991$	4.84924	0.154041	0.0770207	−	0.997029i	$-0.475459\pi$
$991$	4.84924	0.154041	0.0770207	+	0.997029i	$0.475459\pi$
$992$	0	0
$993$	−13.8725	−0.440232
$994$	0	0
$995$	−8.32888	−0.264043
$996$	0	0
$997$	−43.9197	−1.39095	−0.695476	−	0.718550i	$-0.744806\pi$
$997$	−43.9197	−1.39095	−0.695476	+	0.718550i	$0.744806\pi$
$998$	0	0
$999$	−4.36537	−0.138114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.j.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.j.1.7	✓	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)

\(f(q)\)	\(=\)	\(q-0.612150 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.62527 q^{9} +O(q^{10})\)	\(q-0.612150 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.62527 q^{9} +5.91240 q^{11} -1.93529 q^{13} +0.612150 q^{15} -1.73933 q^{17} +4.81444 q^{19} +0.612150 q^{21} +2.86111 q^{23} +1.00000 q^{25} +3.44351 q^{27} -4.67656 q^{29} +3.67422 q^{31} -3.61927 q^{33} +1.00000 q^{35} -1.26771 q^{37} +1.18469 q^{39} -6.58150 q^{41} -1.00000 q^{43} +2.62527 q^{45} -4.00350 q^{47} +1.00000 q^{49} +1.06473 q^{51} -10.1944 q^{53} -5.91240 q^{55} -2.94716 q^{57} +5.53697 q^{59} -2.86135 q^{61} +2.62527 q^{63} +1.93529 q^{65} +5.99409 q^{67} -1.75143 q^{69} +0.451083 q^{71} -13.9914 q^{73} -0.612150 q^{75} -5.91240 q^{77} +1.98615 q^{79} +5.76788 q^{81} +1.16216 q^{83} +1.73933 q^{85} +2.86276 q^{87} +13.5463 q^{89} +1.93529 q^{91} -2.24917 q^{93} -4.81444 q^{95} +9.15426 q^{97} -15.5217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) \) 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

Fields

Representations

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Database

Properties

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