LMFDB - Embedded newform 6020.2.a.j.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} - 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			1.3
Root			$2.78565$ 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)

$f(q)$	$=$	$q-2.78565 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.75984 q^{9} +O(q^{10})$	$q-2.78565 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.75984 q^{9} -4.82108 q^{11} +2.97436 q^{13} +2.78565 q^{15} +0.905847 q^{17} -2.18953 q^{19} +2.78565 q^{21} -8.65324 q^{23} +1.00000 q^{25} -4.90230 q^{27} +1.92768 q^{29} -1.24982 q^{31} +13.4298 q^{33} +1.00000 q^{35} -1.30038 q^{37} -8.28553 q^{39} -1.94655 q^{41} -1.00000 q^{43} -4.75984 q^{45} -10.4415 q^{47} +1.00000 q^{49} -2.52337 q^{51} +10.1678 q^{53} +4.82108 q^{55} +6.09926 q^{57} +5.71575 q^{59} -4.49867 q^{61} -4.75984 q^{63} -2.97436 q^{65} -8.65369 q^{67} +24.1049 q^{69} -4.78108 q^{71} -14.2866 q^{73} -2.78565 q^{75} +4.82108 q^{77} +15.3213 q^{79} -0.623426 q^{81} -8.82385 q^{83} -0.905847 q^{85} -5.36984 q^{87} -15.8266 q^{89} -2.97436 q^{91} +3.48156 q^{93} +2.18953 q^{95} -3.11321 q^{97} -22.9476 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.78565	−1.60830	−0.804148	−	0.594430i	$-0.797378\pi$
$3$	−2.78565	−1.60830	−0.804148	+	0.594430i	$0.797378\pi$
$4$	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
$9$	4.75984	1.58661
$10$	0	0
$11$	−4.82108	−1.45361	−0.726805	−	0.686844i	$-0.758996\pi$
$11$	−4.82108	−1.45361	−0.726805	+	0.686844i	$0.758996\pi$
$12$	0	0
$13$	2.97436	0.824939	0.412470	−	0.910971i	$-0.364666\pi$
$13$	2.97436	0.824939	0.412470	+	0.910971i	$0.364666\pi$
$14$	0	0
$15$	2.78565	0.719252
$16$	0	0
$17$	0.905847	0.219700	0.109850	−	0.993948i	$-0.464963\pi$
$17$	0.905847	0.219700	0.109850	+	0.993948i	$0.464963\pi$
$18$	0	0
$19$	−2.18953	−0.502313	−0.251156	−	0.967947i	$-0.580811\pi$
$19$	−2.18953	−0.502313	−0.251156	+	0.967947i	$0.580811\pi$
$20$	0	0
$21$	2.78565	0.607879
$22$	0	0
$23$	−8.65324	−1.80433	−0.902163	−	0.431396i	$-0.858021\pi$
$23$	−8.65324	−1.80433	−0.902163	+	0.431396i	$0.858021\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.90230	−0.943449
$28$	0	0
$29$	1.92768	0.357961	0.178981	−	0.983853i	$-0.442720\pi$
$29$	1.92768	0.357961	0.178981	+	0.983853i	$0.442720\pi$
$30$	0	0
$31$	−1.24982	−0.224475	−0.112237	−	0.993681i	$-0.535802\pi$
$31$	−1.24982	−0.224475	−0.112237	+	0.993681i	$0.535802\pi$
$32$	0	0
$33$	13.4298	2.33783
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.30038	−0.213780	−0.106890	−	0.994271i	$-0.534089\pi$
$37$	−1.30038	−0.213780	−0.106890	+	0.994271i	$0.534089\pi$
$38$	0	0
$39$	−8.28553	−1.32675
$40$	0	0
$41$	−1.94655	−0.304000	−0.152000	−	0.988381i	$-0.548571\pi$
$41$	−1.94655	−0.304000	−0.152000	+	0.988381i	$0.548571\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−4.75984	−0.709555
$46$	0	0
$47$	−10.4415	−1.52305	−0.761523	−	0.648138i	$-0.775548\pi$
$47$	−10.4415	−1.52305	−0.761523	+	0.648138i	$0.775548\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.52337	−0.353343
$52$	0	0
$53$	10.1678	1.39665	0.698327	−	0.715779i	$-0.253928\pi$
$53$	10.1678	1.39665	0.698327	+	0.715779i	$0.253928\pi$
$54$	0	0
$55$	4.82108	0.650074
$56$	0	0
$57$	6.09926	0.807867
$58$	0	0
$59$	5.71575	0.744127	0.372063	−	0.928207i	$-0.378650\pi$
$59$	5.71575	0.744127	0.372063	+	0.928207i	$0.378650\pi$
$60$	0	0
$61$	−4.49867	−0.575996	−0.287998	−	0.957631i	$-0.592990\pi$
$61$	−4.49867	−0.575996	−0.287998	+	0.957631i	$0.592990\pi$
$62$	0	0
$63$	−4.75984	−0.599684
$64$	0	0
$65$	−2.97436	−0.368924
$66$	0	0
$67$	−8.65369	−1.05722	−0.528608	−	0.848866i	$-0.677286\pi$
$67$	−8.65369	−1.05722	−0.528608	+	0.848866i	$0.677286\pi$
$68$	0	0
$69$	24.1049	2.90189
$70$	0	0
$71$	−4.78108	−0.567409	−0.283705	−	0.958912i	$-0.591564\pi$
$71$	−4.78108	−0.567409	−0.283705	+	0.958912i	$0.591564\pi$
$72$	0	0
$73$	−14.2866	−1.67212	−0.836058	−	0.548642i	$-0.815145\pi$
$73$	−14.2866	−1.67212	−0.836058	+	0.548642i	$0.815145\pi$
$74$	0	0
$75$	−2.78565	−0.321659
$76$	0	0
$77$	4.82108	0.549413
$78$	0	0
$79$	15.3213	1.72378	0.861890	−	0.507095i	$-0.169281\pi$
$79$	15.3213	1.72378	0.861890	+	0.507095i	$0.169281\pi$
$80$	0	0
$81$	−0.623426	−0.0692696
$82$	0	0
$83$	−8.82385	−0.968543	−0.484272	−	0.874918i	$-0.660915\pi$
$83$	−8.82385	−0.968543	−0.484272	+	0.874918i	$0.660915\pi$
$84$	0	0
$85$	−0.905847	−0.0982529
$86$	0	0
$87$	−5.36984	−0.575707
$88$	0	0
$89$	−15.8266	−1.67762	−0.838809	−	0.544426i	$-0.816748\pi$
$89$	−15.8266	−1.67762	−0.838809	+	0.544426i	$0.816748\pi$
$90$	0	0
$91$	−2.97436	−0.311798
$92$	0	0
$93$	3.48156	0.361021
$94$	0	0
$95$	2.18953	0.224641
$96$	0	0
$97$	−3.11321	−0.316098	−0.158049	−	0.987431i	$-0.550520\pi$
$97$	−3.11321	−0.316098	−0.158049	+	0.987431i	$0.550520\pi$
$98$	0	0
$99$	−22.9476	−2.30632
$100$	0	0
$101$	2.67591	0.266263	0.133131	−	0.991098i	$-0.457497\pi$
$101$	2.67591	0.266263	0.133131	+	0.991098i	$0.457497\pi$
$102$	0	0
$103$	−8.91361	−0.878284	−0.439142	−	0.898418i	$-0.644718\pi$
$103$	−8.91361	−0.878284	−0.439142	+	0.898418i	$0.644718\pi$
$104$	0	0
$105$	−2.78565	−0.271852
$106$	0	0
$107$	−11.8559	−1.14615	−0.573077	−	0.819501i	$-0.694251\pi$
$107$	−11.8559	−1.14615	−0.573077	+	0.819501i	$0.694251\pi$
$108$	0	0
$109$	−19.6687	−1.88392	−0.941960	−	0.335725i	$-0.891019\pi$
$109$	−19.6687	−1.88392	−0.941960	+	0.335725i	$0.891019\pi$
$110$	0	0
$111$	3.62239	0.343822
$112$	0	0
$113$	7.21444	0.678678	0.339339	−	0.940664i	$-0.389797\pi$
$113$	7.21444	0.678678	0.339339	+	0.940664i	$0.389797\pi$
$114$	0	0
$115$	8.65324	0.806919
$116$	0	0
$117$	14.1575	1.30886
$118$	0	0
$119$	−0.905847	−0.0830389
$120$	0	0
$121$	12.2428	1.11298
$122$	0	0
$123$	5.42240	0.488921
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.94305	0.882303	0.441151	−	0.897433i	$-0.354570\pi$
$127$	9.94305	0.882303	0.441151	+	0.897433i	$0.354570\pi$
$128$	0	0
$129$	2.78565	0.245263
$130$	0	0
$131$	−0.347352	−0.0303483	−0.0151741	−	0.999885i	$-0.504830\pi$
$131$	−0.347352	−0.0303483	−0.0151741	+	0.999885i	$0.504830\pi$
$132$	0	0
$133$	2.18953	0.189856
$134$	0	0
$135$	4.90230	0.421923
$136$	0	0
$137$	−2.58571	−0.220912	−0.110456	−	0.993881i	$-0.535231\pi$
$137$	−2.58571	−0.220912	−0.110456	+	0.993881i	$0.535231\pi$
$138$	0	0
$139$	16.0300	1.35964	0.679822	−	0.733378i	$-0.262057\pi$
$139$	16.0300	1.35964	0.679822	+	0.733378i	$0.262057\pi$
$140$	0	0
$141$	29.0863	2.44951
$142$	0	0
$143$	−14.3396	−1.19914
$144$	0	0
$145$	−1.92768	−0.160085
$146$	0	0
$147$	−2.78565	−0.229756
$148$	0	0
$149$	−14.1057	−1.15558	−0.577790	−	0.816185i	$-0.696085\pi$
$149$	−14.1057	−1.15558	−0.577790	+	0.816185i	$0.696085\pi$
$150$	0	0
$151$	16.7762	1.36523	0.682615	−	0.730779i	$-0.260843\pi$
$151$	16.7762	1.36523	0.682615	+	0.730779i	$0.260843\pi$
$152$	0	0
$153$	4.31169	0.348579
$154$	0	0
$155$	1.24982	0.100388
$156$	0	0
$157$	5.29162	0.422317	0.211159	−	0.977452i	$-0.432276\pi$
$157$	5.29162	0.422317	0.211159	+	0.977452i	$0.432276\pi$
$158$	0	0
$159$	−28.3239	−2.24623
$160$	0	0
$161$	8.65324	0.681971
$162$	0	0
$163$	15.6473	1.22559	0.612794	−	0.790243i	$-0.290046\pi$
$163$	15.6473	1.22559	0.612794	+	0.790243i	$0.290046\pi$
$164$	0	0
$165$	−13.4298	−1.04551
$166$	0	0
$167$	−20.0701	−1.55307	−0.776534	−	0.630076i	$-0.783024\pi$
$167$	−20.0701	−1.55307	−0.776534	+	0.630076i	$0.783024\pi$
$168$	0	0
$169$	−4.15318	−0.319475
$170$	0	0
$171$	−10.4218	−0.796976
$172$	0	0
$173$	−6.33571	−0.481695	−0.240847	−	0.970563i	$-0.577425\pi$
$173$	−6.33571	−0.481695	−0.240847	+	0.970563i	$0.577425\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−15.9221	−1.19678
$178$	0	0
$179$	24.4288	1.82590	0.912948	−	0.408076i	$-0.133800\pi$
$179$	24.4288	1.82590	0.912948	+	0.408076i	$0.133800\pi$
$180$	0	0
$181$	−1.70134	−0.126459	−0.0632297	−	0.997999i	$-0.520140\pi$
$181$	−1.70134	−0.126459	−0.0632297	+	0.997999i	$0.520140\pi$
$182$	0	0
$183$	12.5317	0.926371
$184$	0	0
$185$	1.30038	0.0956055
$186$	0	0
$187$	−4.36716	−0.319358
$188$	0	0
$189$	4.90230	0.356590
$190$	0	0
$191$	−18.3038	−1.32442	−0.662209	−	0.749319i	$-0.730381\pi$
$191$	−18.3038	−1.32442	−0.662209	+	0.749319i	$0.730381\pi$
$192$	0	0
$193$	−6.66620	−0.479844	−0.239922	−	0.970792i	$-0.577122\pi$
$193$	−6.66620	−0.479844	−0.239922	+	0.970792i	$0.577122\pi$
$194$	0	0
$195$	8.28553	0.593339
$196$	0	0
$197$	−0.258111	−0.0183897	−0.00919484	−	0.999958i	$-0.502927\pi$
$197$	−0.258111	−0.0183897	−0.00919484	+	0.999958i	$0.502927\pi$
$198$	0	0
$199$	3.55224	0.251812	0.125906	−	0.992042i	$-0.459816\pi$
$199$	3.55224	0.251812	0.125906	+	0.992042i	$0.459816\pi$
$200$	0	0
$201$	24.1061	1.70032
$202$	0	0
$203$	−1.92768	−0.135297
$204$	0	0
$205$	1.94655	0.135953
$206$	0	0
$207$	−41.1881	−2.86277
$208$	0	0
$209$	10.5559	0.730166
$210$	0	0
$211$	3.41946	0.235406	0.117703	−	0.993049i	$-0.462447\pi$
$211$	3.41946	0.235406	0.117703	+	0.993049i	$0.462447\pi$
$212$	0	0
$213$	13.3184	0.912562
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	1.24982	0.0848434
$218$	0	0
$219$	39.7973	2.68926
$220$	0	0
$221$	2.69432	0.181239
$222$	0	0
$223$	14.4443	0.967259	0.483629	−	0.875273i	$-0.339318\pi$
$223$	14.4443	0.967259	0.483629	+	0.875273i	$0.339318\pi$
$224$	0	0
$225$	4.75984	0.317323
$226$	0	0
$227$	−10.6439	−0.706462	−0.353231	−	0.935536i	$-0.614917\pi$
$227$	−10.6439	−0.706462	−0.353231	+	0.935536i	$0.614917\pi$
$228$	0	0
$229$	14.1972	0.938179	0.469090	−	0.883151i	$-0.344582\pi$
$229$	14.1972	0.938179	0.469090	+	0.883151i	$0.344582\pi$
$230$	0	0
$231$	−13.4298	−0.883618
$232$	0	0
$233$	2.67275	0.175098	0.0875488	−	0.996160i	$-0.472097\pi$
$233$	2.67275	0.175098	0.0875488	+	0.996160i	$0.472097\pi$
$234$	0	0
$235$	10.4415	0.681127
$236$	0	0
$237$	−42.6798	−2.77235
$238$	0	0
$239$	27.1542	1.75646	0.878228	−	0.478241i	$-0.158726\pi$
$239$	27.1542	1.75646	0.878228	+	0.478241i	$0.158726\pi$
$240$	0	0
$241$	19.1301	1.23228	0.616138	−	0.787638i	$-0.288696\pi$
$241$	19.1301	1.23228	0.616138	+	0.787638i	$0.288696\pi$
$242$	0	0
$243$	16.4436	1.05485
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−6.51245	−0.414377
$248$	0	0
$249$	24.5801	1.55770
$250$	0	0
$251$	14.9334	0.942585	0.471293	−	0.881977i	$-0.343788\pi$
$251$	14.9334	0.942585	0.471293	+	0.881977i	$0.343788\pi$
$252$	0	0
$253$	41.7180	2.62279
$254$	0	0
$255$	2.52337	0.158020
$256$	0	0
$257$	−10.7292	−0.669270	−0.334635	−	0.942348i	$-0.608613\pi$
$257$	−10.7292	−0.669270	−0.334635	+	0.942348i	$0.608613\pi$
$258$	0	0
$259$	1.30038	0.0808014
$260$	0	0
$261$	9.17545	0.567946
$262$	0	0
$263$	−14.8577	−0.916167	−0.458083	−	0.888909i	$-0.651464\pi$
$263$	−14.8577	−0.916167	−0.458083	+	0.888909i	$0.651464\pi$
$264$	0	0
$265$	−10.1678	−0.624603
$266$	0	0
$267$	44.0874	2.69811
$268$	0	0
$269$	−14.3463	−0.874710	−0.437355	−	0.899289i	$-0.644085\pi$
$269$	−14.3463	−0.874710	−0.437355	+	0.899289i	$0.644085\pi$
$270$	0	0
$271$	−8.36218	−0.507966	−0.253983	−	0.967209i	$-0.581741\pi$
$271$	−8.36218	−0.507966	−0.253983	+	0.967209i	$0.581741\pi$
$272$	0	0
$273$	8.28553	0.501463
$274$	0	0
$275$	−4.82108	−0.290722
$276$	0	0
$277$	20.3661	1.22368	0.611840	−	0.790982i	$-0.290430\pi$
$277$	20.3661	1.22368	0.611840	+	0.790982i	$0.290430\pi$
$278$	0	0
$279$	−5.94895	−0.356154
$280$	0	0
$281$	25.8714	1.54336	0.771679	−	0.636013i	$-0.219417\pi$
$281$	25.8714	1.54336	0.771679	+	0.636013i	$0.219417\pi$
$282$	0	0
$283$	−16.4671	−0.978869	−0.489434	−	0.872040i	$-0.662797\pi$
$283$	−16.4671	−0.978869	−0.489434	+	0.872040i	$0.662797\pi$
$284$	0	0
$285$	−6.09926	−0.361289
$286$	0	0
$287$	1.94655	0.114901
$288$	0	0
$289$	−16.1794	−0.951732
$290$	0	0
$291$	8.67230	0.508379
$292$	0	0
$293$	3.71304	0.216918	0.108459	−	0.994101i	$-0.465408\pi$
$293$	3.71304	0.216918	0.108459	+	0.994101i	$0.465408\pi$
$294$	0	0
$295$	−5.71575	−0.332784
$296$	0	0
$297$	23.6344	1.37141
$298$	0	0
$299$	−25.7379	−1.48846
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	−7.45414	−0.428229
$304$	0	0
$305$	4.49867	0.257593
$306$	0	0
$307$	−3.33595	−0.190393	−0.0951965	−	0.995459i	$-0.530348\pi$
$307$	−3.33595	−0.190393	−0.0951965	+	0.995459i	$0.530348\pi$
$308$	0	0
$309$	24.8302	1.41254
$310$	0	0
$311$	26.5120	1.50336	0.751679	−	0.659530i	$-0.229244\pi$
$311$	26.5120	1.50336	0.751679	+	0.659530i	$0.229244\pi$
$312$	0	0
$313$	−3.62990	−0.205174	−0.102587	−	0.994724i	$-0.532712\pi$
$313$	−3.62990	−0.205174	−0.102587	+	0.994724i	$0.532712\pi$
$314$	0	0
$315$	4.75984	0.268187
$316$	0	0
$317$	−24.9774	−1.40287	−0.701435	−	0.712733i	$-0.747457\pi$
$317$	−24.9774	−1.40287	−0.701435	+	0.712733i	$0.747457\pi$
$318$	0	0
$319$	−9.29350	−0.520336
$320$	0	0
$321$	33.0264	1.84336
$322$	0	0
$323$	−1.98338	−0.110358
$324$	0	0
$325$	2.97436	0.164988
$326$	0	0
$327$	54.7901	3.02990
$328$	0	0
$329$	10.4415	0.575658
$330$	0	0
$331$	24.2382	1.33225	0.666125	−	0.745840i	$-0.267952\pi$
$331$	24.2382	1.33225	0.666125	+	0.745840i	$0.267952\pi$
$332$	0	0
$333$	−6.18958	−0.339187
$334$	0	0
$335$	8.65369	0.472802
$336$	0	0
$337$	23.4639	1.27816	0.639080	−	0.769140i	$-0.279315\pi$
$337$	23.4639	1.27816	0.639080	+	0.769140i	$0.279315\pi$
$338$	0	0
$339$	−20.0969	−1.09151
$340$	0	0
$341$	6.02549	0.326298
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−24.1049	−1.29776
$346$	0	0
$347$	21.6197	1.16060	0.580302	−	0.814401i	$-0.302934\pi$
$347$	21.6197	1.16060	0.580302	+	0.814401i	$0.302934\pi$
$348$	0	0
$349$	10.8252	0.579459	0.289729	−	0.957109i	$-0.406435\pi$
$349$	10.8252	0.579459	0.289729	+	0.957109i	$0.406435\pi$
$350$	0	0
$351$	−14.5812	−0.778288
$352$	0	0
$353$	−2.28126	−0.121419	−0.0607097	−	0.998155i	$-0.519336\pi$
$353$	−2.28126	−0.121419	−0.0607097	+	0.998155i	$0.519336\pi$
$354$	0	0
$355$	4.78108	0.253753
$356$	0	0
$357$	2.52337	0.133551
$358$	0	0
$359$	9.55725	0.504413	0.252206	−	0.967673i	$-0.418844\pi$
$359$	9.55725	0.504413	0.252206	+	0.967673i	$0.418844\pi$
$360$	0	0
$361$	−14.2060	−0.747682
$362$	0	0
$363$	−34.1041	−1.79000
$364$	0	0
$365$	14.2866	0.747793
$366$	0	0
$367$	−30.9039	−1.61317	−0.806585	−	0.591118i	$-0.798687\pi$
$367$	−30.9039	−1.61317	−0.806585	+	0.591118i	$0.798687\pi$
$368$	0	0
$369$	−9.26526	−0.482330
$370$	0	0
$371$	−10.1678	−0.527886
$372$	0	0
$373$	24.1622	1.25107	0.625535	−	0.780196i	$-0.284881\pi$
$373$	24.1622	1.25107	0.625535	+	0.780196i	$0.284881\pi$
$374$	0	0
$375$	2.78565	0.143850
$376$	0	0
$377$	5.73361	0.295296
$378$	0	0
$379$	25.0415	1.28630	0.643149	−	0.765741i	$-0.277628\pi$
$379$	25.0415	1.28630	0.643149	+	0.765741i	$0.277628\pi$
$380$	0	0
$381$	−27.6978	−1.41900
$382$	0	0
$383$	11.4582	0.585486	0.292743	−	0.956191i	$-0.405432\pi$
$383$	11.4582	0.585486	0.292743	+	0.956191i	$0.405432\pi$
$384$	0	0
$385$	−4.82108	−0.245705
$386$	0	0
$387$	−4.75984	−0.241956
$388$	0	0
$389$	38.3038	1.94208	0.971041	−	0.238914i	$-0.0767913\pi$
$389$	38.3038	1.94208	0.971041	+	0.238914i	$0.0767913\pi$
$390$	0	0
$391$	−7.83851	−0.396411
$392$	0	0
$393$	0.967600	0.0488090
$394$	0	0
$395$	−15.3213	−0.770898
$396$	0	0
$397$	22.9860	1.15363	0.576816	−	0.816874i	$-0.304295\pi$
$397$	22.9860	1.15363	0.576816	+	0.816874i	$0.304295\pi$
$398$	0	0
$399$	−6.09926	−0.305345
$400$	0	0
$401$	−18.1870	−0.908218	−0.454109	−	0.890946i	$-0.650042\pi$
$401$	−18.1870	−0.908218	−0.454109	+	0.890946i	$0.650042\pi$
$402$	0	0
$403$	−3.71742	−0.185178
$404$	0	0
$405$	0.623426	0.0309783
$406$	0	0
$407$	6.26921	0.310753
$408$	0	0
$409$	−12.5556	−0.620837	−0.310418	−	0.950600i	$-0.600469\pi$
$409$	−12.5556	−0.620837	−0.310418	+	0.950600i	$0.600469\pi$
$410$	0	0
$411$	7.20288	0.355292
$412$	0	0
$413$	−5.71575	−0.281254
$414$	0	0
$415$	8.82385	0.433146
$416$	0	0
$417$	−44.6538	−2.18671
$418$	0	0
$419$	−26.2217	−1.28102	−0.640508	−	0.767952i	$-0.721276\pi$
$419$	−26.2217	−1.28102	−0.640508	+	0.767952i	$0.721276\pi$
$420$	0	0
$421$	30.5078	1.48686	0.743429	−	0.668815i	$-0.233198\pi$
$421$	30.5078	1.48686	0.743429	+	0.668815i	$0.233198\pi$
$422$	0	0
$423$	−49.6998	−2.41649
$424$	0	0
$425$	0.905847	0.0439400
$426$	0	0
$427$	4.49867	0.217706
$428$	0	0
$429$	39.9452	1.92857
$430$	0	0
$431$	−7.88770	−0.379937	−0.189969	−	0.981790i	$-0.560839\pi$
$431$	−7.88770	−0.379937	−0.189969	+	0.981790i	$0.560839\pi$
$432$	0	0
$433$	−31.2935	−1.50387	−0.751934	−	0.659238i	$-0.770879\pi$
$433$	−31.2935	−1.50387	−0.751934	+	0.659238i	$0.770879\pi$
$434$	0	0
$435$	5.36984	0.257464
$436$	0	0
$437$	18.9465	0.906335
$438$	0	0
$439$	−8.65101	−0.412890	−0.206445	−	0.978458i	$-0.566189\pi$
$439$	−8.65101	−0.412890	−0.206445	+	0.978458i	$0.566189\pi$
$440$	0	0
$441$	4.75984	0.226659
$442$	0	0
$443$	40.9466	1.94543	0.972716	−	0.231998i	$-0.0745263\pi$
$443$	40.9466	1.94543	0.972716	+	0.231998i	$0.0745263\pi$
$444$	0	0
$445$	15.8266	0.750254
$446$	0	0
$447$	39.2934	1.85852
$448$	0	0
$449$	−37.6790	−1.77818	−0.889092	−	0.457728i	$-0.848663\pi$
$449$	−37.6790	−1.77818	−0.889092	+	0.457728i	$0.848663\pi$
$450$	0	0
$451$	9.38446	0.441897
$452$	0	0
$453$	−46.7327	−2.19569
$454$	0	0
$455$	2.97436	0.139440
$456$	0	0
$457$	−1.81777	−0.0850317	−0.0425159	−	0.999096i	$-0.513537\pi$
$457$	−1.81777	−0.0850317	−0.0425159	+	0.999096i	$0.513537\pi$
$458$	0	0
$459$	−4.44074	−0.207276
$460$	0	0
$461$	40.4352	1.88326	0.941628	−	0.336655i	$-0.109296\pi$
$461$	40.4352	1.88326	0.941628	+	0.336655i	$0.109296\pi$
$462$	0	0
$463$	−31.5420	−1.46588	−0.732941	−	0.680292i	$-0.761853\pi$
$463$	−31.5420	−1.46588	−0.732941	+	0.680292i	$0.761853\pi$
$464$	0	0
$465$	−3.48156	−0.161454
$466$	0	0
$467$	4.05364	0.187580	0.0937899	−	0.995592i	$-0.470102\pi$
$467$	4.05364	0.187580	0.0937899	+	0.995592i	$0.470102\pi$
$468$	0	0
$469$	8.65369	0.399590
$470$	0	0
$471$	−14.7406	−0.679211
$472$	0	0
$473$	4.82108	0.221673
$474$	0	0
$475$	−2.18953	−0.100463
$476$	0	0
$477$	48.3971	2.21595
$478$	0	0
$479$	−35.0713	−1.60245	−0.801225	−	0.598363i	$-0.795818\pi$
$479$	−35.0713	−1.60245	−0.801225	+	0.598363i	$0.795818\pi$
$480$	0	0
$481$	−3.86779	−0.176356
$482$	0	0
$483$	−24.1049	−1.09681
$484$	0	0
$485$	3.11321	0.141363
$486$	0	0
$487$	−2.42365	−0.109826	−0.0549129	−	0.998491i	$-0.517488\pi$
$487$	−2.42365	−0.109826	−0.0549129	+	0.998491i	$0.517488\pi$
$488$	0	0
$489$	−43.5878	−1.97111
$490$	0	0
$491$	25.3798	1.14537	0.572687	−	0.819774i	$-0.305901\pi$
$491$	25.3798	1.14537	0.572687	+	0.819774i	$0.305901\pi$
$492$	0	0
$493$	1.74618	0.0786441
$494$	0	0
$495$	22.9476	1.03142
$496$	0	0
$497$	4.78108	0.214461
$498$	0	0
$499$	22.2000	0.993809	0.496904	−	0.867805i	$-0.334470\pi$
$499$	22.2000	0.993809	0.496904	+	0.867805i	$0.334470\pi$
$500$	0	0
$501$	55.9081	2.49779
$502$	0	0
$503$	35.3338	1.57546	0.787728	−	0.616023i	$-0.211257\pi$
$503$	35.3338	1.57546	0.787728	+	0.616023i	$0.211257\pi$
$504$	0	0
$505$	−2.67591	−0.119076
$506$	0	0
$507$	11.5693	0.513811
$508$	0	0
$509$	−33.2917	−1.47563	−0.737813	−	0.675005i	$-0.764142\pi$
$509$	−33.2917	−1.47563	−0.737813	+	0.675005i	$0.764142\pi$
$510$	0	0
$511$	14.2866	0.632000
$512$	0	0
$513$	10.7337	0.473906
$514$	0	0
$515$	8.91361	0.392781
$516$	0	0
$517$	50.3392	2.21392
$518$	0	0
$519$	17.6491	0.774708
$520$	0	0
$521$	41.4187	1.81459	0.907293	−	0.420499i	$-0.138145\pi$
$521$	41.4187	1.81459	0.907293	+	0.420499i	$0.138145\pi$
$522$	0	0
$523$	1.51076	0.0660611	0.0330306	−	0.999454i	$-0.489484\pi$
$523$	1.51076	0.0660611	0.0330306	+	0.999454i	$0.489484\pi$
$524$	0	0
$525$	2.78565	0.121576
$526$	0	0
$527$	−1.13215	−0.0493171
$528$	0	0
$529$	51.8786	2.25559
$530$	0	0
$531$	27.2061	1.18064
$532$	0	0
$533$	−5.78974	−0.250781
$534$	0	0
$535$	11.8559	0.512576
$536$	0	0
$537$	−68.0501	−2.93658
$538$	0	0
$539$	−4.82108	−0.207659
$540$	0	0
$541$	12.3490	0.530925	0.265462	−	0.964121i	$-0.414475\pi$
$541$	12.3490	0.530925	0.265462	+	0.964121i	$0.414475\pi$
$542$	0	0
$543$	4.73933	0.203384
$544$	0	0
$545$	19.6687	0.842515
$546$	0	0
$547$	−2.64423	−0.113059	−0.0565295	−	0.998401i	$-0.518003\pi$
$547$	−2.64423	−0.113059	−0.0565295	+	0.998401i	$0.518003\pi$
$548$	0	0
$549$	−21.4130	−0.913883
$550$	0	0
$551$	−4.22071	−0.179808
$552$	0	0
$553$	−15.3213	−0.651528
$554$	0	0
$555$	−3.62239	−0.153762
$556$	0	0
$557$	−3.39571	−0.143881	−0.0719403	−	0.997409i	$-0.522919\pi$
$557$	−3.39571	−0.143881	−0.0719403	+	0.997409i	$0.522919\pi$
$558$	0	0
$559$	−2.97436	−0.125802
$560$	0	0
$561$	12.1654	0.513622
$562$	0	0
$563$	−0.0807578	−0.00340354	−0.00170177	−	0.999999i	$-0.500542\pi$
$563$	−0.0807578	−0.00340354	−0.00170177	+	0.999999i	$0.500542\pi$
$564$	0	0
$565$	−7.21444	−0.303514
$566$	0	0
$567$	0.623426	0.0261814
$568$	0	0
$569$	21.8030	0.914031	0.457015	−	0.889459i	$-0.348918\pi$
$569$	21.8030	0.914031	0.457015	+	0.889459i	$0.348918\pi$
$570$	0	0
$571$	20.8391	0.872089	0.436045	−	0.899925i	$-0.356379\pi$
$571$	20.8391	0.872089	0.436045	+	0.899925i	$0.356379\pi$
$572$	0	0
$573$	50.9880	2.13005
$574$	0	0
$575$	−8.65324	−0.360865
$576$	0	0
$577$	27.1026	1.12830	0.564149	−	0.825673i	$-0.309204\pi$
$577$	27.1026	1.12830	0.564149	+	0.825673i	$0.309204\pi$
$578$	0	0
$579$	18.5697	0.771730
$580$	0	0
$581$	8.82385	0.366075
$582$	0	0
$583$	−49.0197	−2.03019
$584$	0	0
$585$	−14.1575	−0.585340
$586$	0	0
$587$	−12.6062	−0.520314	−0.260157	−	0.965566i	$-0.583774\pi$
$587$	−12.6062	−0.520314	−0.260157	+	0.965566i	$0.583774\pi$
$588$	0	0
$589$	2.73652	0.112756
$590$	0	0
$591$	0.719008	0.0295760
$592$	0	0
$593$	−1.17297	−0.0481682	−0.0240841	−	0.999710i	$-0.507667\pi$
$593$	−1.17297	−0.0481682	−0.0240841	+	0.999710i	$0.507667\pi$
$594$	0	0
$595$	0.905847	0.0371361
$596$	0	0
$597$	−9.89531	−0.404988
$598$	0	0
$599$	42.0175	1.71679	0.858394	−	0.512991i	$-0.171463\pi$
$599$	42.0175	1.71679	0.858394	+	0.512991i	$0.171463\pi$
$600$	0	0
$601$	34.9752	1.42667	0.713334	−	0.700824i	$-0.247184\pi$
$601$	34.9752	1.42667	0.713334	+	0.700824i	$0.247184\pi$
$602$	0	0
$603$	−41.1902	−1.67739
$604$	0	0
$605$	−12.2428	−0.497740
$606$	0	0
$607$	8.55487	0.347232	0.173616	−	0.984813i	$-0.444455\pi$
$607$	8.55487	0.347232	0.173616	+	0.984813i	$0.444455\pi$
$608$	0	0
$609$	5.36984	0.217597
$610$	0	0
$611$	−31.0567	−1.25642
$612$	0	0
$613$	1.24094	0.0501213	0.0250606	−	0.999686i	$-0.492022\pi$
$613$	1.24094	0.0501213	0.0250606	+	0.999686i	$0.492022\pi$
$614$	0	0
$615$	−5.42240	−0.218652
$616$	0	0
$617$	−34.0611	−1.37125	−0.685625	−	0.727955i	$-0.740471\pi$
$617$	−34.0611	−1.37125	−0.685625	+	0.727955i	$0.740471\pi$
$618$	0	0
$619$	10.8041	0.434252	0.217126	−	0.976144i	$-0.430332\pi$
$619$	10.8041	0.434252	0.217126	+	0.976144i	$0.430332\pi$
$620$	0	0
$621$	42.4208	1.70229
$622$	0	0
$623$	15.8266	0.634080
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−29.4050	−1.17432
$628$	0	0
$629$	−1.17794	−0.0469676
$630$	0	0
$631$	33.2723	1.32455	0.662274	−	0.749261i	$-0.269591\pi$
$631$	33.2723	1.32455	0.662274	+	0.749261i	$0.269591\pi$
$632$	0	0
$633$	−9.52543	−0.378602
$634$	0	0
$635$	−9.94305	−0.394578
$636$	0	0
$637$	2.97436	0.117848
$638$	0	0
$639$	−22.7572	−0.900260
$640$	0	0
$641$	5.62832	0.222305	0.111153	−	0.993803i	$-0.464546\pi$
$641$	5.62832	0.222305	0.111153	+	0.993803i	$0.464546\pi$
$642$	0	0
$643$	−37.8054	−1.49090	−0.745450	−	0.666561i	$-0.767766\pi$
$643$	−37.8054	−1.49090	−0.745450	+	0.666561i	$0.767766\pi$
$644$	0	0
$645$	−2.78565	−0.109685
$646$	0	0
$647$	1.52980	0.0601426	0.0300713	−	0.999548i	$-0.490427\pi$
$647$	1.52980	0.0601426	0.0300713	+	0.999548i	$0.490427\pi$
$648$	0	0
$649$	−27.5561	−1.08167
$650$	0	0
$651$	−3.48156	−0.136453
$652$	0	0
$653$	−5.75004	−0.225016	−0.112508	−	0.993651i	$-0.535888\pi$
$653$	−5.75004	−0.225016	−0.112508	+	0.993651i	$0.535888\pi$
$654$	0	0
$655$	0.347352	0.0135722
$656$	0	0
$657$	−68.0018	−2.65300
$658$	0	0
$659$	−39.4749	−1.53772	−0.768862	−	0.639415i	$-0.779177\pi$
$659$	−39.4749	−1.53772	−0.768862	+	0.639415i	$0.779177\pi$
$660$	0	0
$661$	−17.2686	−0.671673	−0.335836	−	0.941920i	$-0.609019\pi$
$661$	−17.2686	−0.671673	−0.335836	+	0.941920i	$0.609019\pi$
$662$	0	0
$663$	−7.50542	−0.291486
$664$	0	0
$665$	−2.18953	−0.0849063
$666$	0	0
$667$	−16.6807	−0.645878
$668$	0	0
$669$	−40.2366	−1.55564
$670$	0	0
$671$	21.6884	0.837273
$672$	0	0
$673$	−30.3016	−1.16804	−0.584020	−	0.811739i	$-0.698521\pi$
$673$	−30.3016	−1.16804	−0.584020	+	0.811739i	$0.698521\pi$
$674$	0	0
$675$	−4.90230	−0.188690
$676$	0	0
$677$	−18.8600	−0.724848	−0.362424	−	0.932013i	$-0.618051\pi$
$677$	−18.8600	−0.724848	−0.362424	+	0.932013i	$0.618051\pi$
$678$	0	0
$679$	3.11321	0.119474
$680$	0	0
$681$	29.6502	1.13620
$682$	0	0
$683$	−5.78496	−0.221356	−0.110678	−	0.993856i	$-0.535302\pi$
$683$	−5.78496	−0.221356	−0.110678	+	0.993856i	$0.535302\pi$
$684$	0	0
$685$	2.58571	0.0987949
$686$	0	0
$687$	−39.5485	−1.50887
$688$	0	0
$689$	30.2427	1.15215
$690$	0	0
$691$	2.33639	0.0888806	0.0444403	−	0.999012i	$-0.485850\pi$
$691$	2.33639	0.0888806	0.0444403	+	0.999012i	$0.485850\pi$
$692$	0	0
$693$	22.9476	0.871706
$694$	0	0
$695$	−16.0300	−0.608051
$696$	0	0
$697$	−1.76327	−0.0667888
$698$	0	0
$699$	−7.44534	−0.281609
$700$	0	0
$701$	−1.02160	−0.0385854	−0.0192927	−	0.999814i	$-0.506141\pi$
$701$	−1.02160	−0.0385854	−0.0192927	+	0.999814i	$0.506141\pi$
$702$	0	0
$703$	2.84721	0.107385
$704$	0	0
$705$	−29.0863	−1.09545
$706$	0	0
$707$	−2.67591	−0.100638
$708$	0	0
$709$	27.1962	1.02137	0.510687	−	0.859766i	$-0.329391\pi$
$709$	27.1962	1.02137	0.510687	+	0.859766i	$0.329391\pi$
$710$	0	0
$711$	72.9270	2.73497
$712$	0	0
$713$	10.8150	0.405025
$714$	0	0
$715$	14.3396	0.536272
$716$	0	0
$717$	−75.6420	−2.82490
$718$	0	0
$719$	−2.85170	−0.106351	−0.0531753	−	0.998585i	$-0.516934\pi$
$719$	−2.85170	−0.106351	−0.0531753	+	0.998585i	$0.516934\pi$
$720$	0	0
$721$	8.91361	0.331960
$722$	0	0
$723$	−53.2897	−1.98186
$724$	0	0
$725$	1.92768	0.0715922
$726$	0	0
$727$	22.6672	0.840679	0.420339	−	0.907367i	$-0.361911\pi$
$727$	22.6672	0.840679	0.420339	+	0.907367i	$0.361911\pi$
$728$	0	0
$729$	−43.9357	−1.62725
$730$	0	0
$731$	−0.905847	−0.0335040
$732$	0	0
$733$	−36.4041	−1.34462	−0.672308	−	0.740272i	$-0.734697\pi$
$733$	−36.4041	−1.34462	−0.672308	+	0.740272i	$0.734697\pi$
$734$	0	0
$735$	2.78565	0.102750
$736$	0	0
$737$	41.7201	1.53678
$738$	0	0
$739$	19.2573	0.708392	0.354196	−	0.935171i	$-0.384755\pi$
$739$	19.2573	0.708392	0.354196	+	0.935171i	$0.384755\pi$
$740$	0	0
$741$	18.1414	0.666441
$742$	0	0
$743$	−12.0302	−0.441347	−0.220673	−	0.975348i	$-0.570826\pi$
$743$	−12.0302	−0.441347	−0.220673	+	0.975348i	$0.570826\pi$
$744$	0	0
$745$	14.1057	0.516791
$746$	0	0
$747$	−42.0001	−1.53670
$748$	0	0
$749$	11.8559	0.433206
$750$	0	0
$751$	−1.85998	−0.0678717	−0.0339358	−	0.999424i	$-0.510804\pi$
$751$	−1.85998	−0.0678717	−0.0339358	+	0.999424i	$0.510804\pi$
$752$	0	0
$753$	−41.5991	−1.51596
$754$	0	0
$755$	−16.7762	−0.610549
$756$	0	0
$757$	10.8234	0.393382	0.196691	−	0.980465i	$-0.436980\pi$
$757$	10.8234	0.393382	0.196691	+	0.980465i	$0.436980\pi$
$758$	0	0
$759$	−116.212	−4.21821
$760$	0	0
$761$	−21.0173	−0.761876	−0.380938	−	0.924601i	$-0.624399\pi$
$761$	−21.0173	−0.761876	−0.380938	+	0.924601i	$0.624399\pi$
$762$	0	0
$763$	19.6687	0.712055
$764$	0	0
$765$	−4.31169	−0.155889
$766$	0	0
$767$	17.0007	0.613859
$768$	0	0
$769$	−33.2226	−1.19804	−0.599019	−	0.800735i	$-0.704442\pi$
$769$	−33.2226	−1.19804	−0.599019	+	0.800735i	$0.704442\pi$
$770$	0	0
$771$	29.8878	1.07638
$772$	0	0
$773$	−29.6669	−1.06704	−0.533521	−	0.845787i	$-0.679132\pi$
$773$	−29.6669	−1.06704	−0.533521	+	0.845787i	$0.679132\pi$
$774$	0	0
$775$	−1.24982	−0.0448949
$776$	0	0
$777$	−3.62239	−0.129953
$778$	0	0
$779$	4.26202	0.152703
$780$	0	0
$781$	23.0499	0.824792
$782$	0	0
$783$	−9.45007	−0.337718
$784$	0	0
$785$	−5.29162	−0.188866
$786$	0	0
$787$	4.03633	0.143880	0.0719399	−	0.997409i	$-0.477081\pi$
$787$	4.03633	0.143880	0.0719399	+	0.997409i	$0.477081\pi$
$788$	0	0
$789$	41.3884	1.47347
$790$	0	0
$791$	−7.21444	−0.256516
$792$	0	0
$793$	−13.3807	−0.475161
$794$	0	0
$795$	28.3239	1.00455
$796$	0	0
$797$	0.735898	0.0260668	0.0130334	−	0.999915i	$-0.495851\pi$
$797$	0.735898	0.0260668	0.0130334	+	0.999915i	$0.495851\pi$
$798$	0	0
$799$	−9.45838	−0.334614
$800$	0	0
$801$	−75.3322	−2.66173
$802$	0	0
$803$	68.8766	2.43060
$804$	0	0
$805$	−8.65324	−0.304987
$806$	0	0
$807$	39.9638	1.40679
$808$	0	0
$809$	−13.3928	−0.470864	−0.235432	−	0.971891i	$-0.575651\pi$
$809$	−13.3928	−0.470864	−0.235432	+	0.971891i	$0.575651\pi$
$810$	0	0
$811$	−13.2103	−0.463876	−0.231938	−	0.972731i	$-0.574507\pi$
$811$	−13.2103	−0.463876	−0.231938	+	0.972731i	$0.574507\pi$
$812$	0	0
$813$	23.2941	0.816960
$814$	0	0
$815$	−15.6473	−0.548100
$816$	0	0
$817$	2.18953	0.0766019
$818$	0	0
$819$	−14.1575	−0.494703
$820$	0	0
$821$	23.3292	0.814195	0.407098	−	0.913385i	$-0.366541\pi$
$821$	23.3292	0.814195	0.407098	+	0.913385i	$0.366541\pi$
$822$	0	0
$823$	2.15760	0.0752092	0.0376046	−	0.999293i	$-0.488027\pi$
$823$	2.15760	0.0752092	0.0376046	+	0.999293i	$0.488027\pi$
$824$	0	0
$825$	13.4298	0.467567
$826$	0	0
$827$	20.9062	0.726979	0.363489	−	0.931598i	$-0.381585\pi$
$827$	20.9062	0.726979	0.363489	+	0.931598i	$0.381585\pi$
$828$	0	0
$829$	−5.74562	−0.199554	−0.0997769	−	0.995010i	$-0.531813\pi$
$829$	−5.74562	−0.199554	−0.0997769	+	0.995010i	$0.531813\pi$
$830$	0	0
$831$	−56.7328	−1.96804
$832$	0	0
$833$	0.905847	0.0313857
$834$	0	0
$835$	20.0701	0.694553
$836$	0	0
$837$	6.12700	0.211780
$838$	0	0
$839$	−14.1818	−0.489612	−0.244806	−	0.969572i	$-0.578724\pi$
$839$	−14.1818	−0.489612	−0.244806	+	0.969572i	$0.578724\pi$
$840$	0	0
$841$	−25.2841	−0.871864
$842$	0	0
$843$	−72.0686	−2.48217
$844$	0	0
$845$	4.15318	0.142874
$846$	0	0
$847$	−12.2428	−0.420667
$848$	0	0
$849$	45.8716	1.57431
$850$	0	0
$851$	11.2525	0.385730
$852$	0	0
$853$	−28.1939	−0.965340	−0.482670	−	0.875802i	$-0.660333\pi$
$853$	−28.1939	−0.965340	−0.482670	+	0.875802i	$0.660333\pi$
$854$	0	0
$855$	10.4218	0.356419
$856$	0	0
$857$	10.0741	0.344126	0.172063	−	0.985086i	$-0.444957\pi$
$857$	10.0741	0.344126	0.172063	+	0.985086i	$0.444957\pi$
$858$	0	0
$859$	−40.7464	−1.39025	−0.695125	−	0.718889i	$-0.744651\pi$
$859$	−40.7464	−1.39025	−0.695125	+	0.718889i	$0.744651\pi$
$860$	0	0
$861$	−5.42240	−0.184795
$862$	0	0
$863$	11.6004	0.394883	0.197441	−	0.980315i	$-0.436737\pi$
$863$	11.6004	0.394883	0.197441	+	0.980315i	$0.436737\pi$
$864$	0	0
$865$	6.33571	0.215421
$866$	0	0
$867$	45.0703	1.53067
$868$	0	0
$869$	−73.8652	−2.50570
$870$	0	0
$871$	−25.7392	−0.872139
$872$	0	0
$873$	−14.8184	−0.501526
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	28.3606	0.957670	0.478835	−	0.877905i	$-0.341059\pi$
$877$	28.3606	0.957670	0.478835	+	0.877905i	$0.341059\pi$
$878$	0	0
$879$	−10.3432	−0.348868
$880$	0	0
$881$	55.5643	1.87201	0.936005	−	0.351987i	$-0.114494\pi$
$881$	55.5643	1.87201	0.936005	+	0.351987i	$0.114494\pi$
$882$	0	0
$883$	−11.6990	−0.393703	−0.196851	−	0.980433i	$-0.563072\pi$
$883$	−11.6990	−0.393703	−0.196851	+	0.980433i	$0.563072\pi$
$884$	0	0
$885$	15.9221	0.535214
$886$	0	0
$887$	2.65872	0.0892709	0.0446355	−	0.999003i	$-0.485787\pi$
$887$	2.65872	0.0892709	0.0446355	+	0.999003i	$0.485787\pi$
$888$	0	0
$889$	−9.94305	−0.333479
$890$	0	0
$891$	3.00559	0.100691
$892$	0	0
$893$	22.8619	0.765045
$894$	0	0
$895$	−24.4288	−0.816566
$896$	0	0
$897$	71.6967	2.39388
$898$	0	0
$899$	−2.40926	−0.0803532
$900$	0	0
$901$	9.21047	0.306845
$902$	0	0
$903$	−2.78565	−0.0927006
$904$	0	0
$905$	1.70134	0.0565544
$906$	0	0
$907$	39.1029	1.29839	0.649195	−	0.760622i	$-0.275106\pi$
$907$	39.1029	1.29839	0.649195	+	0.760622i	$0.275106\pi$
$908$	0	0
$909$	12.7369	0.422456
$910$	0	0
$911$	19.0335	0.630608	0.315304	−	0.948991i	$-0.397894\pi$
$911$	19.0335	0.630608	0.315304	+	0.948991i	$0.397894\pi$
$912$	0	0
$913$	42.5405	1.40788
$914$	0	0
$915$	−12.5317	−0.414286
$916$	0	0
$917$	0.347352	0.0114706
$918$	0	0
$919$	−48.6299	−1.60415	−0.802076	−	0.597222i	$-0.796271\pi$
$919$	−48.6299	−1.60415	−0.802076	+	0.597222i	$0.796271\pi$
$920$	0	0
$921$	9.29280	0.306208
$922$	0	0
$923$	−14.2206	−0.468078
$924$	0	0
$925$	−1.30038	−0.0427561
$926$	0	0
$927$	−42.4274	−1.39350
$928$	0	0
$929$	−2.31471	−0.0759433	−0.0379716	−	0.999279i	$-0.512090\pi$
$929$	−2.31471	−0.0759433	−0.0379716	+	0.999279i	$0.512090\pi$
$930$	0	0
$931$	−2.18953	−0.0717589
$932$	0	0
$933$	−73.8531	−2.41784
$934$	0	0
$935$	4.36716	0.142821
$936$	0	0
$937$	39.3180	1.28446	0.642232	−	0.766510i	$-0.278008\pi$
$937$	39.3180	1.28446	0.642232	+	0.766510i	$0.278008\pi$
$938$	0	0
$939$	10.1116	0.329981
$940$	0	0
$941$	−15.6126	−0.508958	−0.254479	−	0.967078i	$-0.581904\pi$
$941$	−15.6126	−0.508958	−0.254479	+	0.967078i	$0.581904\pi$
$942$	0	0
$943$	16.8439	0.548514
$944$	0	0
$945$	−4.90230	−0.159472
$946$	0	0
$947$	3.26133	0.105979	0.0529895	−	0.998595i	$-0.483125\pi$
$947$	3.26133	0.105979	0.0529895	+	0.998595i	$0.483125\pi$
$948$	0	0
$949$	−42.4934	−1.37939
$950$	0	0
$951$	69.5783	2.25623
$952$	0	0
$953$	−39.0902	−1.26626	−0.633128	−	0.774047i	$-0.718229\pi$
$953$	−39.0902	−1.26626	−0.633128	+	0.774047i	$0.718229\pi$
$954$	0	0
$955$	18.3038	0.592298
$956$	0	0
$957$	25.8884	0.836854
$958$	0	0
$959$	2.58571	0.0834969
$960$	0	0
$961$	−29.4379	−0.949611
$962$	0	0
$963$	−56.4323	−1.81851
$964$	0	0
$965$	6.66620	0.214593
$966$	0	0
$967$	3.98691	0.128210	0.0641051	−	0.997943i	$-0.479581\pi$
$967$	3.98691	0.128210	0.0641051	+	0.997943i	$0.479581\pi$
$968$	0	0
$969$	5.52500	0.177488
$970$	0	0
$971$	−3.02098	−0.0969479	−0.0484739	−	0.998824i	$-0.515436\pi$
$971$	−3.02098	−0.0969479	−0.0484739	+	0.998824i	$0.515436\pi$
$972$	0	0
$973$	−16.0300	−0.513897
$974$	0	0
$975$	−8.28553	−0.265349
$976$	0	0
$977$	−45.1274	−1.44375	−0.721877	−	0.692021i	$-0.756720\pi$
$977$	−45.1274	−1.44375	−0.721877	+	0.692021i	$0.756720\pi$
$978$	0	0
$979$	76.3014	2.43860
$980$	0	0
$981$	−93.6199	−2.98905
$982$	0	0
$983$	−58.6285	−1.86996	−0.934979	−	0.354703i	$-0.884582\pi$
$983$	−58.6285	−1.86996	−0.934979	+	0.354703i	$0.884582\pi$
$984$	0	0
$985$	0.258111	0.00822412
$986$	0	0
$987$	−29.0863	−0.925827
$988$	0	0
$989$	8.65324	0.275157
$990$	0	0
$991$	−3.90191	−0.123948	−0.0619740	−	0.998078i	$-0.519740\pi$
$991$	−3.90191	−0.123948	−0.0619740	+	0.998078i	$0.519740\pi$
$992$	0	0
$993$	−67.5190	−2.14265
$994$	0	0
$995$	−3.55224	−0.112614
$996$	0	0
$997$	15.3535	0.486249	0.243125	−	0.969995i	$-0.421828\pi$
$997$	15.3535	0.486249	0.243125	+	0.969995i	$0.421828\pi$
$998$	0	0
$999$	6.37484	0.201691

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.j.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-2.78565 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.75984 q^{9} +O(q^{10})\)	\(q-2.78565 q^{3} -1.00000 q^{5} -1.00000 q^{7} +4.75984 q^{9} -4.82108 q^{11} +2.97436 q^{13} +2.78565 q^{15} +0.905847 q^{17} -2.18953 q^{19} +2.78565 q^{21} -8.65324 q^{23} +1.00000 q^{25} -4.90230 q^{27} +1.92768 q^{29} -1.24982 q^{31} +13.4298 q^{33} +1.00000 q^{35} -1.30038 q^{37} -8.28553 q^{39} -1.94655 q^{41} -1.00000 q^{43} -4.75984 q^{45} -10.4415 q^{47} +1.00000 q^{49} -2.52337 q^{51} +10.1678 q^{53} +4.82108 q^{55} +6.09926 q^{57} +5.71575 q^{59} -4.49867 q^{61} -4.75984 q^{63} -2.97436 q^{65} -8.65369 q^{67} +24.1049 q^{69} -4.78108 q^{71} -14.2866 q^{73} -2.78565 q^{75} +4.82108 q^{77} +15.3213 q^{79} -0.623426 q^{81} -8.82385 q^{83} -0.905847 q^{85} -5.36984 q^{87} -15.8266 q^{89} -2.97436 q^{91} +3.48156 q^{93} +2.18953 q^{95} -3.11321 q^{97} -22.9476 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{3} - 13 q^{5} - 13 q^{7} + 18 q^{9}+O(q^{10}) \) 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

Groups

Database

Properties

Related objects

Downloads

Learn more