LMFDB - Embedded newform 6020.2.a.k.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.9
Root			$1.13823$ 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)

$f(q)$	$=$	$q+1.13823 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70443 q^{9} +O(q^{10})$	$q+1.13823 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70443 q^{9} +5.91678 q^{11} +6.81777 q^{13} +1.13823 q^{15} +1.83952 q^{17} +5.31499 q^{19} -1.13823 q^{21} -2.25542 q^{23} +1.00000 q^{25} -5.35473 q^{27} -4.83012 q^{29} -8.26236 q^{31} +6.73466 q^{33} -1.00000 q^{35} +4.41547 q^{37} +7.76019 q^{39} +9.47596 q^{41} +1.00000 q^{43} -1.70443 q^{45} -2.71613 q^{47} +1.00000 q^{49} +2.09380 q^{51} -0.586699 q^{53} +5.91678 q^{55} +6.04968 q^{57} +5.16899 q^{59} +11.0263 q^{61} +1.70443 q^{63} +6.81777 q^{65} +10.3817 q^{67} -2.56719 q^{69} -2.83554 q^{71} -6.55089 q^{73} +1.13823 q^{75} -5.91678 q^{77} -13.0439 q^{79} -0.981608 q^{81} +2.12724 q^{83} +1.83952 q^{85} -5.49778 q^{87} -9.79956 q^{89} -6.81777 q^{91} -9.40446 q^{93} +5.31499 q^{95} -0.563695 q^{97} -10.0848 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9}+O(q^{10}) $ 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9	$ 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9} + 11 q^{13} + 16 q^{17} + 3 q^{23} + 13 q^{25} + 6 q^{27} + 10 q^{29} - q^{31} + 14 q^{33} - 13 q^{35} + 16 q^{37} - 14 q^{39} + 23 q^{41} + 13 q^{43} + 15 q^{45} + 2 q^{47} + 13 q^{49} + 4 q^{51} + 20 q^{53} + 22 q^{57} + 2 q^{59} + 5 q^{61} - 15 q^{63} + 11 q^{65} + 19 q^{67} + 16 q^{69} + 4 q^{71} + 34 q^{73} - 15 q^{79} + 17 q^{81} + 27 q^{83} + 16 q^{85} - 5 q^{87} + 3 q^{89} - 11 q^{91} + 35 q^{93} + 45 q^{97} + q^{99}+O(q^{100}) $ 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9 + 11 * q^13 + 16 * q^17 + 3 * q^23 + 13 * q^25 + 6 * q^27 + 10 * q^29 - q^31 + 14 * q^33 - 13 * q^35 + 16 * q^37 - 14 * q^39 + 23 * q^41 + 13 * q^43 + 15 * q^45 + 2 * q^47 + 13 * q^49 + 4 * q^51 + 20 * q^53 + 22 * q^57 + 2 * q^59 + 5 * q^61 - 15 * q^63 + 11 * q^65 + 19 * q^67 + 16 * q^69 + 4 * q^71 + 34 * q^73 - 15 * q^79 + 17 * q^81 + 27 * q^83 + 16 * q^85 - 5 * q^87 + 3 * q^89 - 11 * q^91 + 35 * q^93 + 45 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.13823	0.657157	0.328579	−	0.944477i	$-0.393430\pi$
$3$	1.13823	0.657157	0.328579	+	0.944477i	$0.393430\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.70443	−0.568144
$10$	0	0
$11$	5.91678	1.78398	0.891989	−	0.452058i	$-0.149310\pi$
$11$	5.91678	1.78398	0.891989	+	0.452058i	$0.149310\pi$
$12$	0	0
$13$	6.81777	1.89091	0.945455	−	0.325754i	$-0.105618\pi$
$13$	6.81777	1.89091	0.945455	+	0.325754i	$0.105618\pi$
$14$	0	0
$15$	1.13823	0.293890
$16$	0	0
$17$	1.83952	0.446150	0.223075	−	0.974801i	$-0.428390\pi$
$17$	1.83952	0.446150	0.223075	+	0.974801i	$0.428390\pi$
$18$	0	0
$19$	5.31499	1.21934	0.609671	−	0.792655i	$-0.291302\pi$
$19$	5.31499	1.21934	0.609671	+	0.792655i	$0.291302\pi$
$20$	0	0
$21$	−1.13823	−0.248382
$22$	0	0
$23$	−2.25542	−0.470287	−0.235144	−	0.971961i	$-0.575556\pi$
$23$	−2.25542	−0.470287	−0.235144	+	0.971961i	$0.575556\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.35473	−1.03052
$28$	0	0
$29$	−4.83012	−0.896931	−0.448465	−	0.893800i	$-0.648029\pi$
$29$	−4.83012	−0.896931	−0.448465	+	0.893800i	$0.648029\pi$
$30$	0	0
$31$	−8.26236	−1.48396	−0.741981	−	0.670420i	$-0.766114\pi$
$31$	−8.26236	−1.48396	−0.741981	+	0.670420i	$0.766114\pi$
$32$	0	0
$33$	6.73466	1.17235
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	4.41547	0.725899	0.362950	−	0.931809i	$-0.381770\pi$
$37$	4.41547	0.725899	0.362950	+	0.931809i	$0.381770\pi$
$38$	0	0
$39$	7.76019	1.24262
$40$	0	0
$41$	9.47596	1.47990	0.739949	−	0.672663i	$-0.234850\pi$
$41$	9.47596	1.47990	0.739949	+	0.672663i	$0.234850\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	0	0
$45$	−1.70443	−0.254082
$46$	0	0
$47$	−2.71613	−0.396188	−0.198094	−	0.980183i	$-0.563475\pi$
$47$	−2.71613	−0.396188	−0.198094	+	0.980183i	$0.563475\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.09380	0.293191
$52$	0	0
$53$	−0.586699	−0.0805893	−0.0402946	−	0.999188i	$-0.512830\pi$
$53$	−0.586699	−0.0805893	−0.0402946	+	0.999188i	$0.512830\pi$
$54$	0	0
$55$	5.91678	0.797819
$56$	0	0
$57$	6.04968	0.801299
$58$	0	0
$59$	5.16899	0.672946	0.336473	−	0.941693i	$-0.390766\pi$
$59$	5.16899	0.672946	0.336473	+	0.941693i	$0.390766\pi$
$60$	0	0
$61$	11.0263	1.41177	0.705883	−	0.708328i	$-0.250550\pi$
$61$	11.0263	1.41177	0.705883	+	0.708328i	$0.250550\pi$
$62$	0	0
$63$	1.70443	0.214738
$64$	0	0
$65$	6.81777	0.845640
$66$	0	0
$67$	10.3817	1.26832	0.634161	−	0.773201i	$-0.281346\pi$
$67$	10.3817	1.26832	0.634161	+	0.773201i	$0.281346\pi$
$68$	0	0
$69$	−2.56719	−0.309053
$70$	0	0
$71$	−2.83554	−0.336517	−0.168258	−	0.985743i	$-0.553814\pi$
$71$	−2.83554	−0.336517	−0.168258	+	0.985743i	$0.553814\pi$
$72$	0	0
$73$	−6.55089	−0.766724	−0.383362	−	0.923598i	$-0.625234\pi$
$73$	−6.55089	−0.766724	−0.383362	+	0.923598i	$0.625234\pi$
$74$	0	0
$75$	1.13823	0.131431
$76$	0	0
$77$	−5.91678	−0.674280
$78$	0	0
$79$	−13.0439	−1.46755	−0.733776	−	0.679391i	$-0.762244\pi$
$79$	−13.0439	−1.46755	−0.733776	+	0.679391i	$0.762244\pi$
$80$	0	0
$81$	−0.981608	−0.109068
$82$	0	0
$83$	2.12724	0.233495	0.116748	−	0.993162i	$-0.462753\pi$
$83$	2.12724	0.233495	0.116748	+	0.993162i	$0.462753\pi$
$84$	0	0
$85$	1.83952	0.199524
$86$	0	0
$87$	−5.49778	−0.589424
$88$	0	0
$89$	−9.79956	−1.03875	−0.519376	−	0.854546i	$-0.673835\pi$
$89$	−9.79956	−1.03875	−0.519376	+	0.854546i	$0.673835\pi$
$90$	0	0
$91$	−6.81777	−0.714697
$92$	0	0
$93$	−9.40446	−0.975197
$94$	0	0
$95$	5.31499	0.545306
$96$	0	0
$97$	−0.563695	−0.0572346	−0.0286173	−	0.999590i	$-0.509110\pi$
$97$	−0.563695	−0.0572346	−0.0286173	+	0.999590i	$0.509110\pi$
$98$	0	0
$99$	−10.0848	−1.01356
$100$	0	0
$101$	−5.26669	−0.524055	−0.262028	−	0.965060i	$-0.584391\pi$
$101$	−5.26669	−0.524055	−0.262028	+	0.965060i	$0.584391\pi$
$102$	0	0
$103$	13.6954	1.34945	0.674726	−	0.738068i	$-0.264262\pi$
$103$	13.6954	1.34945	0.674726	+	0.738068i	$0.264262\pi$
$104$	0	0
$105$	−1.13823	−0.111080
$106$	0	0
$107$	−14.4536	−1.39728	−0.698639	−	0.715474i	$-0.746211\pi$
$107$	−14.4536	−1.39728	−0.698639	+	0.715474i	$0.746211\pi$
$108$	0	0
$109$	−11.2456	−1.07713	−0.538565	−	0.842584i	$-0.681033\pi$
$109$	−11.2456	−1.07713	−0.538565	+	0.842584i	$0.681033\pi$
$110$	0	0
$111$	5.02582	0.477030
$112$	0	0
$113$	−5.05317	−0.475363	−0.237681	−	0.971343i	$-0.576387\pi$
$113$	−5.05317	−0.475363	−0.237681	+	0.971343i	$0.576387\pi$
$114$	0	0
$115$	−2.25542	−0.210319
$116$	0	0
$117$	−11.6204	−1.07431
$118$	0	0
$119$	−1.83952	−0.168629
$120$	0	0
$121$	24.0083	2.18257
$122$	0	0
$123$	10.7858	0.972525
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.74127	−0.686927	−0.343464	−	0.939166i	$-0.611600\pi$
$127$	−7.74127	−0.686927	−0.343464	+	0.939166i	$0.611600\pi$
$128$	0	0
$129$	1.13823	0.100216
$130$	0	0
$131$	15.4935	1.35367	0.676837	−	0.736133i	$-0.263350\pi$
$131$	15.4935	1.35367	0.676837	+	0.736133i	$0.263350\pi$
$132$	0	0
$133$	−5.31499	−0.460868
$134$	0	0
$135$	−5.35473	−0.460861
$136$	0	0
$137$	−11.0410	−0.943297	−0.471648	−	0.881787i	$-0.656341\pi$
$137$	−11.0410	−0.943297	−0.471648	+	0.881787i	$0.656341\pi$
$138$	0	0
$139$	−2.30409	−0.195430	−0.0977151	−	0.995214i	$-0.531153\pi$
$139$	−2.30409	−0.195430	−0.0977151	+	0.995214i	$0.531153\pi$
$140$	0	0
$141$	−3.09158	−0.260358
$142$	0	0
$143$	40.3393	3.37334
$144$	0	0
$145$	−4.83012	−0.401120
$146$	0	0
$147$	1.13823	0.0938796
$148$	0	0
$149$	13.4910	1.10523	0.552615	−	0.833437i	$-0.313630\pi$
$149$	13.4910	1.10523	0.552615	+	0.833437i	$0.313630\pi$
$150$	0	0
$151$	−4.39356	−0.357543	−0.178771	−	0.983891i	$-0.557212\pi$
$151$	−4.39356	−0.357543	−0.178771	+	0.983891i	$0.557212\pi$
$152$	0	0
$153$	−3.13535	−0.253478
$154$	0	0
$155$	−8.26236	−0.663648
$156$	0	0
$157$	−6.27198	−0.500559	−0.250279	−	0.968174i	$-0.580522\pi$
$157$	−6.27198	−0.500559	−0.250279	+	0.968174i	$0.580522\pi$
$158$	0	0
$159$	−0.667798	−0.0529598
$160$	0	0
$161$	2.25542	0.177752
$162$	0	0
$163$	23.1706	1.81486	0.907429	−	0.420206i	$-0.138042\pi$
$163$	23.1706	1.81486	0.907429	+	0.420206i	$0.138042\pi$
$164$	0	0
$165$	6.73466	0.524292
$166$	0	0
$167$	2.44189	0.188959	0.0944797	−	0.995527i	$-0.469881\pi$
$167$	2.44189	0.188959	0.0944797	+	0.995527i	$0.469881\pi$
$168$	0	0
$169$	33.4820	2.57554
$170$	0	0
$171$	−9.05904	−0.692762
$172$	0	0
$173$	10.3152	0.784254	0.392127	−	0.919911i	$-0.371739\pi$
$173$	10.3152	0.784254	0.392127	+	0.919911i	$0.371739\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	5.88350	0.442231
$178$	0	0
$179$	7.81492	0.584114	0.292057	−	0.956401i	$-0.405660\pi$
$179$	7.81492	0.584114	0.292057	+	0.956401i	$0.405660\pi$
$180$	0	0
$181$	−9.09582	−0.676087	−0.338044	−	0.941130i	$-0.609765\pi$
$181$	−9.09582	−0.676087	−0.338044	+	0.941130i	$0.609765\pi$
$182$	0	0
$183$	12.5504	0.927753
$184$	0	0
$185$	4.41547	0.324632
$186$	0	0
$187$	10.8841	0.795922
$188$	0	0
$189$	5.35473	0.389499
$190$	0	0
$191$	−10.5577	−0.763931	−0.381965	−	0.924177i	$-0.624753\pi$
$191$	−10.5577	−0.763931	−0.381965	+	0.924177i	$0.624753\pi$
$192$	0	0
$193$	−4.39564	−0.316405	−0.158203	−	0.987407i	$-0.550570\pi$
$193$	−4.39564	−0.316405	−0.158203	+	0.987407i	$0.550570\pi$
$194$	0	0
$195$	7.76019	0.555719
$196$	0	0
$197$	−11.2623	−0.802404	−0.401202	−	0.915990i	$-0.631407\pi$
$197$	−11.2623	−0.802404	−0.401202	+	0.915990i	$0.631407\pi$
$198$	0	0
$199$	−2.45445	−0.173991	−0.0869956	−	0.996209i	$-0.527727\pi$
$199$	−2.45445	−0.173991	−0.0869956	+	0.996209i	$0.527727\pi$
$200$	0	0
$201$	11.8167	0.833486
$202$	0	0
$203$	4.83012	0.339008
$204$	0	0
$205$	9.47596	0.661830
$206$	0	0
$207$	3.84421	0.267191
$208$	0	0
$209$	31.4476	2.17528
$210$	0	0
$211$	−25.2678	−1.73950	−0.869752	−	0.493489i	$-0.835721\pi$
$211$	−25.2678	−1.73950	−0.869752	+	0.493489i	$0.835721\pi$
$212$	0	0
$213$	−3.22749	−0.221144
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	8.26236	0.560885
$218$	0	0
$219$	−7.45642	−0.503858
$220$	0	0
$221$	12.5415	0.843629
$222$	0	0
$223$	8.40191	0.562634	0.281317	−	0.959615i	$-0.409229\pi$
$223$	8.40191	0.562634	0.281317	+	0.959615i	$0.409229\pi$
$224$	0	0
$225$	−1.70443	−0.113629
$226$	0	0
$227$	7.60040	0.504456	0.252228	−	0.967668i	$-0.418837\pi$
$227$	7.60040	0.504456	0.252228	+	0.967668i	$0.418837\pi$
$228$	0	0
$229$	5.60474	0.370372	0.185186	−	0.982704i	$-0.440711\pi$
$229$	5.60474	0.370372	0.185186	+	0.982704i	$0.440711\pi$
$230$	0	0
$231$	−6.73466	−0.443108
$232$	0	0
$233$	2.72670	0.178632	0.0893160	−	0.996003i	$-0.471532\pi$
$233$	2.72670	0.178632	0.0893160	+	0.996003i	$0.471532\pi$
$234$	0	0
$235$	−2.71613	−0.177181
$236$	0	0
$237$	−14.8469	−0.964413
$238$	0	0
$239$	8.64606	0.559267	0.279634	−	0.960107i	$-0.409787\pi$
$239$	8.64606	0.559267	0.279634	+	0.960107i	$0.409787\pi$
$240$	0	0
$241$	16.3536	1.05343	0.526714	−	0.850042i	$-0.323424\pi$
$241$	16.3536	1.05343	0.526714	+	0.850042i	$0.323424\pi$
$242$	0	0
$243$	14.9469	0.958843
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	36.2364	2.30566
$248$	0	0
$249$	2.42129	0.153443
$250$	0	0
$251$	24.8946	1.57134	0.785668	−	0.618649i	$-0.212320\pi$
$251$	24.8946	1.57134	0.785668	+	0.618649i	$0.212320\pi$
$252$	0	0
$253$	−13.3448	−0.838982
$254$	0	0
$255$	2.09380	0.131119
$256$	0	0
$257$	9.97655	0.622320	0.311160	−	0.950358i	$-0.399282\pi$
$257$	9.97655	0.622320	0.311160	+	0.950358i	$0.399282\pi$
$258$	0	0
$259$	−4.41547	−0.274364
$260$	0	0
$261$	8.23261	0.509586
$262$	0	0
$263$	−16.5658	−1.02149	−0.510746	−	0.859732i	$-0.670631\pi$
$263$	−16.5658	−1.02149	−0.510746	+	0.859732i	$0.670631\pi$
$264$	0	0
$265$	−0.586699	−0.0360406
$266$	0	0
$267$	−11.1542	−0.682623
$268$	0	0
$269$	−12.0074	−0.732102	−0.366051	−	0.930595i	$-0.619290\pi$
$269$	−12.0074	−0.732102	−0.366051	+	0.930595i	$0.619290\pi$
$270$	0	0
$271$	27.7547	1.68598	0.842990	−	0.537929i	$-0.180793\pi$
$271$	27.7547	1.68598	0.842990	+	0.537929i	$0.180793\pi$
$272$	0	0
$273$	−7.76019	−0.469668
$274$	0	0
$275$	5.91678	0.356795
$276$	0	0
$277$	14.3182	0.860300	0.430150	−	0.902757i	$-0.358461\pi$
$277$	14.3182	0.860300	0.430150	+	0.902757i	$0.358461\pi$
$278$	0	0
$279$	14.0826	0.843105
$280$	0	0
$281$	−4.31468	−0.257392	−0.128696	−	0.991684i	$-0.541079\pi$
$281$	−4.31468	−0.257392	−0.128696	+	0.991684i	$0.541079\pi$
$282$	0	0
$283$	15.6495	0.930266	0.465133	−	0.885241i	$-0.346006\pi$
$283$	15.6495	0.930266	0.465133	+	0.885241i	$0.346006\pi$
$284$	0	0
$285$	6.04968	0.358352
$286$	0	0
$287$	−9.47596	−0.559349
$288$	0	0
$289$	−13.6162	−0.800950
$290$	0	0
$291$	−0.641614	−0.0376121
$292$	0	0
$293$	0.116097	0.00678247	0.00339123	−	0.999994i	$-0.498921\pi$
$293$	0.116097	0.00678247	0.00339123	+	0.999994i	$0.498921\pi$
$294$	0	0
$295$	5.16899	0.300950
$296$	0	0
$297$	−31.6827	−1.83842
$298$	0	0
$299$	−15.3769	−0.889271
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	−5.99470	−0.344387
$304$	0	0
$305$	11.0263	0.631361
$306$	0	0
$307$	−26.7693	−1.52781	−0.763903	−	0.645331i	$-0.776720\pi$
$307$	−26.7693	−1.52781	−0.763903	+	0.645331i	$0.776720\pi$
$308$	0	0
$309$	15.5886	0.886802
$310$	0	0
$311$	−25.3501	−1.43747	−0.718735	−	0.695284i	$-0.755279\pi$
$311$	−25.3501	−1.43747	−0.718735	+	0.695284i	$0.755279\pi$
$312$	0	0
$313$	30.0801	1.70023	0.850114	−	0.526599i	$-0.176533\pi$
$313$	30.0801	1.70023	0.850114	+	0.526599i	$0.176533\pi$
$314$	0	0
$315$	1.70443	0.0960339
$316$	0	0
$317$	26.6242	1.49536	0.747682	−	0.664057i	$-0.231167\pi$
$317$	26.6242	1.49536	0.747682	+	0.664057i	$0.231167\pi$
$318$	0	0
$319$	−28.5788	−1.60010
$320$	0	0
$321$	−16.4515	−0.918231
$322$	0	0
$323$	9.77705	0.544010
$324$	0	0
$325$	6.81777	0.378182
$326$	0	0
$327$	−12.8000	−0.707843
$328$	0	0
$329$	2.71613	0.149745
$330$	0	0
$331$	−11.9215	−0.655266	−0.327633	−	0.944805i	$-0.606251\pi$
$331$	−11.9215	−0.655266	−0.327633	+	0.944805i	$0.606251\pi$
$332$	0	0
$333$	−7.52588	−0.412416
$334$	0	0
$335$	10.3817	0.567210
$336$	0	0
$337$	21.2019	1.15494	0.577471	−	0.816411i	$-0.304040\pi$
$337$	21.2019	1.15494	0.577471	+	0.816411i	$0.304040\pi$
$338$	0	0
$339$	−5.75167	−0.312388
$340$	0	0
$341$	−48.8866	−2.64736
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.56719	−0.138213
$346$	0	0
$347$	−19.7804	−1.06187	−0.530934	−	0.847413i	$-0.678159\pi$
$347$	−19.7804	−1.06187	−0.530934	+	0.847413i	$0.678159\pi$
$348$	0	0
$349$	−10.7285	−0.574283	−0.287141	−	0.957888i	$-0.592705\pi$
$349$	−10.7285	−0.574283	−0.287141	+	0.957888i	$0.592705\pi$
$350$	0	0
$351$	−36.5073	−1.94862
$352$	0	0
$353$	24.6214	1.31047	0.655233	−	0.755427i	$-0.272571\pi$
$353$	24.6214	1.31047	0.655233	+	0.755427i	$0.272571\pi$
$354$	0	0
$355$	−2.83554	−0.150495
$356$	0	0
$357$	−2.09380	−0.110816
$358$	0	0
$359$	9.94648	0.524955	0.262478	−	0.964938i	$-0.415460\pi$
$359$	9.94648	0.524955	0.262478	+	0.964938i	$0.415460\pi$
$360$	0	0
$361$	9.24910	0.486795
$362$	0	0
$363$	27.3270	1.43429
$364$	0	0
$365$	−6.55089	−0.342889
$366$	0	0
$367$	−20.9406	−1.09309	−0.546546	−	0.837429i	$-0.684058\pi$
$367$	−20.9406	−1.09309	−0.546546	+	0.837429i	$0.684058\pi$
$368$	0	0
$369$	−16.1511	−0.840795
$370$	0	0
$371$	0.586699	0.0304599
$372$	0	0
$373$	13.4588	0.696871	0.348436	−	0.937333i	$-0.386713\pi$
$373$	13.4588	0.696871	0.348436	+	0.937333i	$0.386713\pi$
$374$	0	0
$375$	1.13823	0.0587779
$376$	0	0
$377$	−32.9306	−1.69601
$378$	0	0
$379$	−27.4825	−1.41168	−0.705841	−	0.708370i	$-0.749431\pi$
$379$	−27.4825	−1.41168	−0.705841	+	0.708370i	$0.749431\pi$
$380$	0	0
$381$	−8.81135	−0.451419
$382$	0	0
$383$	−2.54945	−0.130271	−0.0651355	−	0.997876i	$-0.520748\pi$
$383$	−2.54945	−0.130271	−0.0651355	+	0.997876i	$0.520748\pi$
$384$	0	0
$385$	−5.91678	−0.301547
$386$	0	0
$387$	−1.70443	−0.0866412
$388$	0	0
$389$	−2.01472	−0.102151	−0.0510753	−	0.998695i	$-0.516265\pi$
$389$	−2.01472	−0.102151	−0.0510753	+	0.998695i	$0.516265\pi$
$390$	0	0
$391$	−4.14890	−0.209819
$392$	0	0
$393$	17.6352	0.889577
$394$	0	0
$395$	−13.0439	−0.656309
$396$	0	0
$397$	−7.07336	−0.355002	−0.177501	−	0.984121i	$-0.556801\pi$
$397$	−7.07336	−0.355002	−0.177501	+	0.984121i	$0.556801\pi$
$398$	0	0
$399$	−6.04968	−0.302863
$400$	0	0
$401$	29.4582	1.47107	0.735536	−	0.677486i	$-0.236931\pi$
$401$	29.4582	1.47107	0.735536	+	0.677486i	$0.236931\pi$
$402$	0	0
$403$	−56.3308	−2.80604
$404$	0	0
$405$	−0.981608	−0.0487765
$406$	0	0
$407$	26.1254	1.29499
$408$	0	0
$409$	8.96723	0.443401	0.221701	−	0.975115i	$-0.428839\pi$
$409$	8.96723	0.443401	0.221701	+	0.975115i	$0.428839\pi$
$410$	0	0
$411$	−12.5672	−0.619894
$412$	0	0
$413$	−5.16899	−0.254350
$414$	0	0
$415$	2.12724	0.104422
$416$	0	0
$417$	−2.62258	−0.128428
$418$	0	0
$419$	−9.16201	−0.447593	−0.223797	−	0.974636i	$-0.571845\pi$
$419$	−9.16201	−0.447593	−0.223797	+	0.974636i	$0.571845\pi$
$420$	0	0
$421$	9.21048	0.448891	0.224446	−	0.974487i	$-0.427943\pi$
$421$	9.21048	0.448891	0.224446	+	0.974487i	$0.427943\pi$
$422$	0	0
$423$	4.62946	0.225092
$424$	0	0
$425$	1.83952	0.0892300
$426$	0	0
$427$	−11.0263	−0.533598
$428$	0	0
$429$	45.9154	2.21681
$430$	0	0
$431$	32.9284	1.58610	0.793052	−	0.609154i	$-0.208491\pi$
$431$	32.9284	1.58610	0.793052	+	0.609154i	$0.208491\pi$
$432$	0	0
$433$	−12.6558	−0.608197	−0.304099	−	0.952641i	$-0.598355\pi$
$433$	−12.6558	−0.608197	−0.304099	+	0.952641i	$0.598355\pi$
$434$	0	0
$435$	−5.49778	−0.263599
$436$	0	0
$437$	−11.9875	−0.573441
$438$	0	0
$439$	−26.1030	−1.24583	−0.622914	−	0.782290i	$-0.714051\pi$
$439$	−26.1030	−1.24583	−0.622914	+	0.782290i	$0.714051\pi$
$440$	0	0
$441$	−1.70443	−0.0811635
$442$	0	0
$443$	−29.8711	−1.41922	−0.709609	−	0.704595i	$-0.751129\pi$
$443$	−29.8711	−1.41922	−0.709609	+	0.704595i	$0.751129\pi$
$444$	0	0
$445$	−9.79956	−0.464544
$446$	0	0
$447$	15.3559	0.726310
$448$	0	0
$449$	−0.496917	−0.0234510	−0.0117255	−	0.999931i	$-0.503732\pi$
$449$	−0.496917	−0.0234510	−0.0117255	+	0.999931i	$0.503732\pi$
$450$	0	0
$451$	56.0672	2.64010
$452$	0	0
$453$	−5.00088	−0.234962
$454$	0	0
$455$	−6.81777	−0.319622
$456$	0	0
$457$	2.52820	0.118264	0.0591321	−	0.998250i	$-0.481167\pi$
$457$	2.52820	0.118264	0.0591321	+	0.998250i	$0.481167\pi$
$458$	0	0
$459$	−9.85015	−0.459765
$460$	0	0
$461$	3.88572	0.180976	0.0904879	−	0.995898i	$-0.471157\pi$
$461$	3.88572	0.180976	0.0904879	+	0.995898i	$0.471157\pi$
$462$	0	0
$463$	32.6521	1.51747	0.758736	−	0.651398i	$-0.225817\pi$
$463$	32.6521	1.51747	0.758736	+	0.651398i	$0.225817\pi$
$464$	0	0
$465$	−9.40446	−0.436121
$466$	0	0
$467$	20.6405	0.955130	0.477565	−	0.878597i	$-0.341520\pi$
$467$	20.6405	0.955130	0.477565	+	0.878597i	$0.341520\pi$
$468$	0	0
$469$	−10.3817	−0.479380
$470$	0	0
$471$	−7.13896	−0.328946
$472$	0	0
$473$	5.91678	0.272054
$474$	0	0
$475$	5.31499	0.243868
$476$	0	0
$477$	0.999989	0.0457863
$478$	0	0
$479$	33.2239	1.51804	0.759020	−	0.651067i	$-0.225678\pi$
$479$	33.2239	1.51804	0.759020	+	0.651067i	$0.225678\pi$
$480$	0	0
$481$	30.1037	1.37261
$482$	0	0
$483$	2.56719	0.116811
$484$	0	0
$485$	−0.563695	−0.0255961
$486$	0	0
$487$	32.0884	1.45406	0.727031	−	0.686604i	$-0.240899\pi$
$487$	32.0884	1.45406	0.727031	+	0.686604i	$0.240899\pi$
$488$	0	0
$489$	26.3734	1.19265
$490$	0	0
$491$	−31.5787	−1.42513	−0.712563	−	0.701608i	$-0.752466\pi$
$491$	−31.5787	−1.42513	−0.712563	+	0.701608i	$0.752466\pi$
$492$	0	0
$493$	−8.88512	−0.400166
$494$	0	0
$495$	−10.0848	−0.453276
$496$	0	0
$497$	2.83554	0.127191
$498$	0	0
$499$	−4.32049	−0.193412	−0.0967058	−	0.995313i	$-0.530831\pi$
$499$	−4.32049	−0.193412	−0.0967058	+	0.995313i	$0.530831\pi$
$500$	0	0
$501$	2.77944	0.124176
$502$	0	0
$503$	−13.4434	−0.599412	−0.299706	−	0.954032i	$-0.596889\pi$
$503$	−13.4434	−0.599412	−0.299706	+	0.954032i	$0.596889\pi$
$504$	0	0
$505$	−5.26669	−0.234365
$506$	0	0
$507$	38.1102	1.69253
$508$	0	0
$509$	7.79874	0.345673	0.172836	−	0.984951i	$-0.444707\pi$
$509$	7.79874	0.345673	0.172836	+	0.984951i	$0.444707\pi$
$510$	0	0
$511$	6.55089	0.289794
$512$	0	0
$513$	−28.4603	−1.25655
$514$	0	0
$515$	13.6954	0.603493
$516$	0	0
$517$	−16.0707	−0.706791
$518$	0	0
$519$	11.7411	0.515378
$520$	0	0
$521$	25.0780	1.09869	0.549343	−	0.835597i	$-0.314878\pi$
$521$	25.0780	1.09869	0.549343	+	0.835597i	$0.314878\pi$
$522$	0	0
$523$	−23.4994	−1.02756	−0.513779	−	0.857923i	$-0.671755\pi$
$523$	−23.4994	−1.02756	−0.513779	+	0.857923i	$0.671755\pi$
$524$	0	0
$525$	−1.13823	−0.0496764
$526$	0	0
$527$	−15.1988	−0.662070
$528$	0	0
$529$	−17.9131	−0.778830
$530$	0	0
$531$	−8.81020	−0.382330
$532$	0	0
$533$	64.6050	2.79835
$534$	0	0
$535$	−14.4536	−0.624882
$536$	0	0
$537$	8.89517	0.383855
$538$	0	0
$539$	5.91678	0.254854
$540$	0	0
$541$	−8.54398	−0.367335	−0.183667	−	0.982988i	$-0.558797\pi$
$541$	−8.54398	−0.367335	−0.183667	+	0.982988i	$0.558797\pi$
$542$	0	0
$543$	−10.3531	−0.444296
$544$	0	0
$545$	−11.2456	−0.481707
$546$	0	0
$547$	−5.91183	−0.252772	−0.126386	−	0.991981i	$-0.540338\pi$
$547$	−5.91183	−0.252772	−0.126386	+	0.991981i	$0.540338\pi$
$548$	0	0
$549$	−18.7935	−0.802087
$550$	0	0
$551$	−25.6720	−1.09366
$552$	0	0
$553$	13.0439	0.554683
$554$	0	0
$555$	5.02582	0.213334
$556$	0	0
$557$	−43.1002	−1.82621	−0.913106	−	0.407722i	$-0.866323\pi$
$557$	−43.1002	−1.82621	−0.913106	+	0.407722i	$0.866323\pi$
$558$	0	0
$559$	6.81777	0.288361
$560$	0	0
$561$	12.3886	0.523046
$562$	0	0
$563$	−19.0801	−0.804129	−0.402064	−	0.915611i	$-0.631707\pi$
$563$	−19.0801	−0.804129	−0.402064	+	0.915611i	$0.631707\pi$
$564$	0	0
$565$	−5.05317	−0.212589
$566$	0	0
$567$	0.981608	0.0412237
$568$	0	0
$569$	28.1367	1.17955	0.589775	−	0.807568i	$-0.299216\pi$
$569$	28.1367	1.17955	0.589775	+	0.807568i	$0.299216\pi$
$570$	0	0
$571$	−12.0569	−0.504566	−0.252283	−	0.967653i	$-0.581181\pi$
$571$	−12.0569	−0.504566	−0.252283	+	0.967653i	$0.581181\pi$
$572$	0	0
$573$	−12.0171	−0.502023
$574$	0	0
$575$	−2.25542	−0.0940575
$576$	0	0
$577$	13.4955	0.561824	0.280912	−	0.959733i	$-0.409363\pi$
$577$	13.4955	0.561824	0.280912	+	0.959733i	$0.409363\pi$
$578$	0	0
$579$	−5.00325	−0.207928
$580$	0	0
$581$	−2.12724	−0.0882529
$582$	0	0
$583$	−3.47137	−0.143769
$584$	0	0
$585$	−11.6204	−0.480446
$586$	0	0
$587$	−26.1767	−1.08043	−0.540214	−	0.841528i	$-0.681657\pi$
$587$	−26.1767	−1.08043	−0.540214	+	0.841528i	$0.681657\pi$
$588$	0	0
$589$	−43.9143	−1.80946
$590$	0	0
$591$	−12.8191	−0.527306
$592$	0	0
$593$	−30.8269	−1.26591	−0.632954	−	0.774189i	$-0.718158\pi$
$593$	−30.8269	−1.26591	−0.632954	+	0.774189i	$0.718158\pi$
$594$	0	0
$595$	−1.83952	−0.0754131
$596$	0	0
$597$	−2.79373	−0.114340
$598$	0	0
$599$	−3.56090	−0.145495	−0.0727473	−	0.997350i	$-0.523177\pi$
$599$	−3.56090	−0.145495	−0.0727473	+	0.997350i	$0.523177\pi$
$600$	0	0
$601$	24.7176	1.00825	0.504127	−	0.863630i	$-0.331814\pi$
$601$	24.7176	1.00825	0.504127	+	0.863630i	$0.331814\pi$
$602$	0	0
$603$	−17.6948	−0.720590
$604$	0	0
$605$	24.0083	0.976077
$606$	0	0
$607$	22.0074	0.893254	0.446627	−	0.894720i	$-0.352625\pi$
$607$	22.0074	0.893254	0.446627	+	0.894720i	$0.352625\pi$
$608$	0	0
$609$	5.49778	0.222781
$610$	0	0
$611$	−18.5179	−0.749156
$612$	0	0
$613$	11.5454	0.466315	0.233158	−	0.972439i	$-0.425094\pi$
$613$	11.5454	0.466315	0.233158	+	0.972439i	$0.425094\pi$
$614$	0	0
$615$	10.7858	0.434926
$616$	0	0
$617$	−1.50071	−0.0604162	−0.0302081	−	0.999544i	$-0.509617\pi$
$617$	−1.50071	−0.0604162	−0.0302081	+	0.999544i	$0.509617\pi$
$618$	0	0
$619$	−43.5034	−1.74855	−0.874275	−	0.485432i	$-0.838662\pi$
$619$	−43.5034	−1.74855	−0.874275	+	0.485432i	$0.838662\pi$
$620$	0	0
$621$	12.0772	0.484639
$622$	0	0
$623$	9.79956	0.392611
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	35.7946	1.42950
$628$	0	0
$629$	8.12237	0.323860
$630$	0	0
$631$	−20.1841	−0.803515	−0.401757	−	0.915746i	$-0.631601\pi$
$631$	−20.1841	−0.803515	−0.401757	+	0.915746i	$0.631601\pi$
$632$	0	0
$633$	−28.7605	−1.14313
$634$	0	0
$635$	−7.74127	−0.307203
$636$	0	0
$637$	6.81777	0.270130
$638$	0	0
$639$	4.83299	0.191190
$640$	0	0
$641$	−32.6360	−1.28904	−0.644521	−	0.764586i	$-0.722943\pi$
$641$	−32.6360	−1.28904	−0.644521	+	0.764586i	$0.722943\pi$
$642$	0	0
$643$	−15.5466	−0.613099	−0.306550	−	0.951855i	$-0.599175\pi$
$643$	−15.5466	−0.613099	−0.306550	+	0.951855i	$0.599175\pi$
$644$	0	0
$645$	1.13823	0.0448178
$646$	0	0
$647$	−36.2426	−1.42484	−0.712422	−	0.701751i	$-0.752402\pi$
$647$	−36.2426	−1.42484	−0.712422	+	0.701751i	$0.752402\pi$
$648$	0	0
$649$	30.5838	1.20052
$650$	0	0
$651$	9.40446	0.368590
$652$	0	0
$653$	24.2358	0.948419	0.474210	−	0.880412i	$-0.342734\pi$
$653$	24.2358	0.948419	0.474210	+	0.880412i	$0.342734\pi$
$654$	0	0
$655$	15.4935	0.605382
$656$	0	0
$657$	11.1656	0.435610
$658$	0	0
$659$	39.0556	1.52139	0.760696	−	0.649108i	$-0.224858\pi$
$659$	39.0556	1.52139	0.760696	+	0.649108i	$0.224858\pi$
$660$	0	0
$661$	6.09864	0.237210	0.118605	−	0.992942i	$-0.462158\pi$
$661$	6.09864	0.237210	0.118605	+	0.992942i	$0.462158\pi$
$662$	0	0
$663$	14.2751	0.554397
$664$	0	0
$665$	−5.31499	−0.206106
$666$	0	0
$667$	10.8939	0.421815
$668$	0	0
$669$	9.56330	0.369739
$670$	0	0
$671$	65.2399	2.51856
$672$	0	0
$673$	−45.5821	−1.75706	−0.878530	−	0.477687i	$-0.841475\pi$
$673$	−45.5821	−1.75706	−0.878530	+	0.477687i	$0.841475\pi$
$674$	0	0
$675$	−5.35473	−0.206103
$676$	0	0
$677$	−33.3359	−1.28120	−0.640602	−	0.767873i	$-0.721315\pi$
$677$	−33.3359	−1.28120	−0.640602	+	0.767873i	$0.721315\pi$
$678$	0	0
$679$	0.563695	0.0216326
$680$	0	0
$681$	8.65100	0.331507
$682$	0	0
$683$	32.8541	1.25713	0.628564	−	0.777758i	$-0.283643\pi$
$683$	32.8541	1.25713	0.628564	+	0.777758i	$0.283643\pi$
$684$	0	0
$685$	−11.0410	−0.421855
$686$	0	0
$687$	6.37948	0.243393
$688$	0	0
$689$	−3.99998	−0.152387
$690$	0	0
$691$	−31.9869	−1.21684	−0.608419	−	0.793616i	$-0.708196\pi$
$691$	−31.9869	−1.21684	−0.608419	+	0.793616i	$0.708196\pi$
$692$	0	0
$693$	10.0848	0.383088
$694$	0	0
$695$	−2.30409	−0.0873990
$696$	0	0
$697$	17.4313	0.660256
$698$	0	0
$699$	3.10361	0.117389
$700$	0	0
$701$	−37.7126	−1.42439	−0.712193	−	0.701984i	$-0.752298\pi$
$701$	−37.7126	−1.42439	−0.712193	+	0.701984i	$0.752298\pi$
$702$	0	0
$703$	23.4682	0.885119
$704$	0	0
$705$	−3.09158	−0.116436
$706$	0	0
$707$	5.26669	0.198074
$708$	0	0
$709$	−10.3476	−0.388614	−0.194307	−	0.980941i	$-0.562246\pi$
$709$	−10.3476	−0.388614	−0.194307	+	0.980941i	$0.562246\pi$
$710$	0	0
$711$	22.2324	0.833782
$712$	0	0
$713$	18.6351	0.697889
$714$	0	0
$715$	40.3393	1.50860
$716$	0	0
$717$	9.84120	0.367526
$718$	0	0
$719$	−25.6279	−0.955761	−0.477880	−	0.878425i	$-0.658595\pi$
$719$	−25.6279	−0.955761	−0.477880	+	0.878425i	$0.658595\pi$
$720$	0	0
$721$	−13.6954	−0.510045
$722$	0	0
$723$	18.6142	0.692268
$724$	0	0
$725$	−4.83012	−0.179386
$726$	0	0
$727$	11.4193	0.423517	0.211758	−	0.977322i	$-0.432081\pi$
$727$	11.4193	0.423517	0.211758	+	0.977322i	$0.432081\pi$
$728$	0	0
$729$	19.9578	0.739178
$730$	0	0
$731$	1.83952	0.0680373
$732$	0	0
$733$	−23.9065	−0.883008	−0.441504	−	0.897259i	$-0.645555\pi$
$733$	−23.9065	−0.883008	−0.441504	+	0.897259i	$0.645555\pi$
$734$	0	0
$735$	1.13823	0.0419842
$736$	0	0
$737$	61.4260	2.26266
$738$	0	0
$739$	−28.1324	−1.03487	−0.517434	−	0.855723i	$-0.673113\pi$
$739$	−28.1324	−1.03487	−0.517434	+	0.855723i	$0.673113\pi$
$740$	0	0
$741$	41.2453	1.51518
$742$	0	0
$743$	−3.66824	−0.134575	−0.0672873	−	0.997734i	$-0.521434\pi$
$743$	−3.66824	−0.134575	−0.0672873	+	0.997734i	$0.521434\pi$
$744$	0	0
$745$	13.4910	0.494274
$746$	0	0
$747$	−3.62574	−0.132659
$748$	0	0
$749$	14.4536	0.528121
$750$	0	0
$751$	−28.7518	−1.04917	−0.524585	−	0.851358i	$-0.675779\pi$
$751$	−28.7518	−1.04917	−0.524585	+	0.851358i	$0.675779\pi$
$752$	0	0
$753$	28.3358	1.03261
$754$	0	0
$755$	−4.39356	−0.159898
$756$	0	0
$757$	32.2975	1.17387	0.586937	−	0.809633i	$-0.300334\pi$
$757$	32.2975	1.17387	0.586937	+	0.809633i	$0.300334\pi$
$758$	0	0
$759$	−15.1895	−0.551343
$760$	0	0
$761$	−47.6310	−1.72662	−0.863311	−	0.504673i	$-0.831613\pi$
$761$	−47.6310	−1.72662	−0.863311	+	0.504673i	$0.831613\pi$
$762$	0	0
$763$	11.2456	0.407117
$764$	0	0
$765$	−3.13535	−0.113359
$766$	0	0
$767$	35.2410	1.27248
$768$	0	0
$769$	31.6031	1.13964	0.569818	−	0.821771i	$-0.307014\pi$
$769$	31.6031	1.13964	0.569818	+	0.821771i	$0.307014\pi$
$770$	0	0
$771$	11.3556	0.408962
$772$	0	0
$773$	24.6754	0.887512	0.443756	−	0.896148i	$-0.353646\pi$
$773$	24.6754	0.887512	0.443756	+	0.896148i	$0.353646\pi$
$774$	0	0
$775$	−8.26236	−0.296793
$776$	0	0
$777$	−5.02582	−0.180300
$778$	0	0
$779$	50.3646	1.80450
$780$	0	0
$781$	−16.7773	−0.600338
$782$	0	0
$783$	25.8640	0.924303
$784$	0	0
$785$	−6.27198	−0.223857
$786$	0	0
$787$	−12.7454	−0.454323	−0.227161	−	0.973857i	$-0.572944\pi$
$787$	−12.7454	−0.454323	−0.227161	+	0.973857i	$0.572944\pi$
$788$	0	0
$789$	−18.8557	−0.671281
$790$	0	0
$791$	5.05317	0.179670
$792$	0	0
$793$	75.1745	2.66952
$794$	0	0
$795$	−0.667798	−0.0236843
$796$	0	0
$797$	−55.4073	−1.96263	−0.981313	−	0.192416i	$-0.938368\pi$
$797$	−55.4073	−1.96263	−0.981313	+	0.192416i	$0.938368\pi$
$798$	0	0
$799$	−4.99639	−0.176759
$800$	0	0
$801$	16.7027	0.590161
$802$	0	0
$803$	−38.7602	−1.36782
$804$	0	0
$805$	2.25542	0.0794931
$806$	0	0
$807$	−13.6671	−0.481106
$808$	0	0
$809$	−29.3170	−1.03073	−0.515365	−	0.856971i	$-0.672344\pi$
$809$	−29.3170	−1.03073	−0.515365	+	0.856971i	$0.672344\pi$
$810$	0	0
$811$	−17.6371	−0.619322	−0.309661	−	0.950847i	$-0.600215\pi$
$811$	−17.6371	−0.619322	−0.309661	+	0.950847i	$0.600215\pi$
$812$	0	0
$813$	31.5913	1.10795
$814$	0	0
$815$	23.1706	0.811629
$816$	0	0
$817$	5.31499	0.185948
$818$	0	0
$819$	11.6204	0.406051
$820$	0	0
$821$	−42.8466	−1.49536	−0.747678	−	0.664061i	$-0.768831\pi$
$821$	−42.8466	−1.49536	−0.747678	+	0.664061i	$0.768831\pi$
$822$	0	0
$823$	−27.0082	−0.941448	−0.470724	−	0.882280i	$-0.656007\pi$
$823$	−27.0082	−0.941448	−0.470724	+	0.882280i	$0.656007\pi$
$824$	0	0
$825$	6.73466	0.234471
$826$	0	0
$827$	36.7195	1.27686	0.638430	−	0.769680i	$-0.279584\pi$
$827$	36.7195	1.27686	0.638430	+	0.769680i	$0.279584\pi$
$828$	0	0
$829$	10.5121	0.365102	0.182551	−	0.983196i	$-0.441565\pi$
$829$	10.5121	0.365102	0.182551	+	0.983196i	$0.441565\pi$
$830$	0	0
$831$	16.2975	0.565352
$832$	0	0
$833$	1.83952	0.0637357
$834$	0	0
$835$	2.44189	0.0845053
$836$	0	0
$837$	44.2426	1.52925
$838$	0	0
$839$	−0.705867	−0.0243692	−0.0121846	−	0.999926i	$-0.503879\pi$
$839$	−0.705867	−0.0243692	−0.0121846	+	0.999926i	$0.503879\pi$
$840$	0	0
$841$	−5.66995	−0.195516
$842$	0	0
$843$	−4.91109	−0.169147
$844$	0	0
$845$	33.4820	1.15182
$846$	0	0
$847$	−24.0083	−0.824936
$848$	0	0
$849$	17.8127	0.611331
$850$	0	0
$851$	−9.95874	−0.341381
$852$	0	0
$853$	6.98490	0.239158	0.119579	−	0.992825i	$-0.461845\pi$
$853$	6.98490	0.239158	0.119579	+	0.992825i	$0.461845\pi$
$854$	0	0
$855$	−9.05904	−0.309813
$856$	0	0
$857$	32.9101	1.12419	0.562094	−	0.827073i	$-0.309996\pi$
$857$	32.9101	1.12419	0.562094	+	0.827073i	$0.309996\pi$
$858$	0	0
$859$	−33.2859	−1.13570	−0.567850	−	0.823132i	$-0.692225\pi$
$859$	−33.2859	−1.13570	−0.567850	+	0.823132i	$0.692225\pi$
$860$	0	0
$861$	−10.7858	−0.367580
$862$	0	0
$863$	−15.0643	−0.512794	−0.256397	−	0.966572i	$-0.582535\pi$
$863$	−15.0643	−0.512794	−0.256397	+	0.966572i	$0.582535\pi$
$864$	0	0
$865$	10.3152	0.350729
$866$	0	0
$867$	−15.4983	−0.526350
$868$	0	0
$869$	−77.1779	−2.61808
$870$	0	0
$871$	70.7797	2.39828
$872$	0	0
$873$	0.960780	0.0325175
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	6.86636	0.231860	0.115930	−	0.993257i	$-0.463015\pi$
$877$	6.86636	0.231860	0.115930	+	0.993257i	$0.463015\pi$
$878$	0	0
$879$	0.132145	0.00445715
$880$	0	0
$881$	−29.4524	−0.992276	−0.496138	−	0.868244i	$-0.665249\pi$
$881$	−29.4524	−0.992276	−0.496138	+	0.868244i	$0.665249\pi$
$882$	0	0
$883$	−48.7052	−1.63906	−0.819531	−	0.573035i	$-0.805766\pi$
$883$	−48.7052	−1.63906	−0.819531	+	0.573035i	$0.805766\pi$
$884$	0	0
$885$	5.88350	0.197772
$886$	0	0
$887$	−44.2822	−1.48685	−0.743425	−	0.668819i	$-0.766800\pi$
$887$	−44.2822	−1.48685	−0.743425	+	0.668819i	$0.766800\pi$
$888$	0	0
$889$	7.74127	0.259634
$890$	0	0
$891$	−5.80796	−0.194574
$892$	0	0
$893$	−14.4362	−0.483089
$894$	0	0
$895$	7.81492	0.261224
$896$	0	0
$897$	−17.5025	−0.584391
$898$	0	0
$899$	39.9082	1.33101
$900$	0	0
$901$	−1.07925	−0.0359549
$902$	0	0
$903$	−1.13823	−0.0378779
$904$	0	0
$905$	−9.09582	−0.302355
$906$	0	0
$907$	−6.23222	−0.206937	−0.103469	−	0.994633i	$-0.532994\pi$
$907$	−6.23222	−0.206937	−0.103469	+	0.994633i	$0.532994\pi$
$908$	0	0
$909$	8.97672	0.297739
$910$	0	0
$911$	−47.2863	−1.56666	−0.783332	−	0.621603i	$-0.786482\pi$
$911$	−47.2863	−1.56666	−0.783332	+	0.621603i	$0.786482\pi$
$912$	0	0
$913$	12.5864	0.416550
$914$	0	0
$915$	12.5504	0.414904
$916$	0	0
$917$	−15.4935	−0.511641
$918$	0	0
$919$	5.95895	0.196568	0.0982839	−	0.995158i	$-0.468665\pi$
$919$	5.95895	0.196568	0.0982839	+	0.995158i	$0.468665\pi$
$920$	0	0
$921$	−30.4696	−1.00401
$922$	0	0
$923$	−19.3321	−0.636322
$924$	0	0
$925$	4.41547	0.145180
$926$	0	0
$927$	−23.3430	−0.766684
$928$	0	0
$929$	−0.163535	−0.00536541	−0.00268270	−	0.999996i	$-0.500854\pi$
$929$	−0.163535	−0.00536541	−0.00268270	+	0.999996i	$0.500854\pi$
$930$	0	0
$931$	5.31499	0.174192
$932$	0	0
$933$	−28.8542	−0.944644
$934$	0	0
$935$	10.8841	0.355947
$936$	0	0
$937$	26.1374	0.853873	0.426937	−	0.904282i	$-0.359593\pi$
$937$	26.1374	0.853873	0.426937	+	0.904282i	$0.359593\pi$
$938$	0	0
$939$	34.2381	1.11732
$940$	0	0
$941$	−41.8188	−1.36325	−0.681627	−	0.731700i	$-0.738727\pi$
$941$	−41.8188	−1.36325	−0.681627	+	0.731700i	$0.738727\pi$
$942$	0	0
$943$	−21.3723	−0.695977
$944$	0	0
$945$	5.35473	0.174189
$946$	0	0
$947$	−3.56033	−0.115695	−0.0578476	−	0.998325i	$-0.518424\pi$
$947$	−3.56033	−0.115695	−0.0578476	+	0.998325i	$0.518424\pi$
$948$	0	0
$949$	−44.6625	−1.44981
$950$	0	0
$951$	30.3045	0.982689
$952$	0	0
$953$	−3.20226	−0.103732	−0.0518658	−	0.998654i	$-0.516517\pi$
$953$	−3.20226	−0.103732	−0.0518658	+	0.998654i	$0.516517\pi$
$954$	0	0
$955$	−10.5577	−0.341640
$956$	0	0
$957$	−32.5292	−1.05152
$958$	0	0
$959$	11.0410	0.356533
$960$	0	0
$961$	37.2665	1.20215
$962$	0	0
$963$	24.6351	0.793856
$964$	0	0
$965$	−4.39564	−0.141501
$966$	0	0
$967$	36.1921	1.16386	0.581929	−	0.813239i	$-0.302298\pi$
$967$	36.1921	1.16386	0.581929	+	0.813239i	$0.302298\pi$
$968$	0	0
$969$	11.1285	0.357500
$970$	0	0
$971$	7.19761	0.230982	0.115491	−	0.993309i	$-0.463156\pi$
$971$	7.19761	0.230982	0.115491	+	0.993309i	$0.463156\pi$
$972$	0	0
$973$	2.30409	0.0738657
$974$	0	0
$975$	7.76019	0.248525
$976$	0	0
$977$	−5.40458	−0.172908	−0.0864539	−	0.996256i	$-0.527554\pi$
$977$	−5.40458	−0.172908	−0.0864539	+	0.996256i	$0.527554\pi$
$978$	0	0
$979$	−57.9819	−1.85311
$980$	0	0
$981$	19.1673	0.611965
$982$	0	0
$983$	39.0229	1.24464	0.622318	−	0.782765i	$-0.286191\pi$
$983$	39.0229	1.24464	0.622318	+	0.782765i	$0.286191\pi$
$984$	0	0
$985$	−11.2623	−0.358846
$986$	0	0
$987$	3.09158	0.0984061
$988$	0	0
$989$	−2.25542	−0.0717182
$990$	0	0
$991$	−31.6862	−1.00655	−0.503273	−	0.864127i	$-0.667871\pi$
$991$	−31.6862	−1.00655	−0.503273	+	0.864127i	$0.667871\pi$
$992$	0	0
$993$	−13.5694	−0.430613
$994$	0	0
$995$	−2.45445	−0.0778112
$996$	0	0
$997$	−41.2650	−1.30687	−0.653437	−	0.756981i	$-0.726674\pi$
$997$	−41.2650	−1.30687	−0.653437	+	0.756981i	$0.726674\pi$
$998$	0	0
$999$	−23.6436	−0.748052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.k.1.9	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.k.1.9	✓	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)

\(f(q)\)	\(=\)	\(q+1.13823 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70443 q^{9} +O(q^{10})\)	\(q+1.13823 q^{3} +1.00000 q^{5} -1.00000 q^{7} -1.70443 q^{9} +5.91678 q^{11} +6.81777 q^{13} +1.13823 q^{15} +1.83952 q^{17} +5.31499 q^{19} -1.13823 q^{21} -2.25542 q^{23} +1.00000 q^{25} -5.35473 q^{27} -4.83012 q^{29} -8.26236 q^{31} +6.73466 q^{33} -1.00000 q^{35} +4.41547 q^{37} +7.76019 q^{39} +9.47596 q^{41} +1.00000 q^{43} -1.70443 q^{45} -2.71613 q^{47} +1.00000 q^{49} +2.09380 q^{51} -0.586699 q^{53} +5.91678 q^{55} +6.04968 q^{57} +5.16899 q^{59} +11.0263 q^{61} +1.70443 q^{63} +6.81777 q^{65} +10.3817 q^{67} -2.56719 q^{69} -2.83554 q^{71} -6.55089 q^{73} +1.13823 q^{75} -5.91678 q^{77} -13.0439 q^{79} -0.981608 q^{81} +2.12724 q^{83} +1.83952 q^{85} -5.49778 q^{87} -9.79956 q^{89} -6.81777 q^{91} -9.40446 q^{93} +5.31499 q^{95} -0.563695 q^{97} -10.0848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9}+O(q^{10}) \) 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9	\( 13 q + 13 q^{5} - 13 q^{7} + 15 q^{9} + 11 q^{13} + 16 q^{17} + 3 q^{23} + 13 q^{25} + 6 q^{27} + 10 q^{29} - q^{31} + 14 q^{33} - 13 q^{35} + 16 q^{37} - 14 q^{39} + 23 q^{41} + 13 q^{43} + 15 q^{45} + 2 q^{47} + 13 q^{49} + 4 q^{51} + 20 q^{53} + 22 q^{57} + 2 q^{59} + 5 q^{61} - 15 q^{63} + 11 q^{65} + 19 q^{67} + 16 q^{69} + 4 q^{71} + 34 q^{73} - 15 q^{79} + 17 q^{81} + 27 q^{83} + 16 q^{85} - 5 q^{87} + 3 q^{89} - 11 q^{91} + 35 q^{93} + 45 q^{97} + q^{99}+O(q^{100}) \) 13 * q + 13 * q^5 - 13 * q^7 + 15 * q^9 + 11 * q^13 + 16 * q^17 + 3 * q^23 + 13 * q^25 + 6 * q^27 + 10 * q^29 - q^31 + 14 * q^33 - 13 * q^35 + 16 * q^37 - 14 * q^39 + 23 * q^41 + 13 * q^43 + 15 * q^45 + 2 * q^47 + 13 * q^49 + 4 * q^51 + 20 * q^53 + 22 * q^57 + 2 * q^59 + 5 * q^61 - 15 * q^63 + 11 * q^65 + 19 * q^67 + 16 * q^69 + 4 * q^71 + 34 * q^73 - 15 * q^79 + 17 * q^81 + 27 * q^83 + 16 * q^85 - 5 * q^87 + 3 * q^89 - 11 * q^91 + 35 * q^93 + 45 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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