LMFDB - Embedded newform 6020.2.a.g.1.2

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 16x^{7} + 83x^{5} - 9x^{4} - 160x^{3} + 32x^{2} + 77x - 2 $ x^9 - 16x^7 + 83x^5 - 9x^4 - 160x^3 + 32x^2 + 77x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.42779$ 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)

$f(q)$	$=$	$q-2.42779 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.89414 q^{9} +O(q^{10})$	$q-2.42779 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.89414 q^{9} +5.60221 q^{11} +1.23392 q^{13} -2.42779 q^{15} -1.24812 q^{17} -4.75839 q^{19} +2.42779 q^{21} -2.51327 q^{23} +1.00000 q^{25} +0.257003 q^{27} -6.53192 q^{29} +6.58593 q^{31} -13.6010 q^{33} -1.00000 q^{35} -9.65052 q^{37} -2.99570 q^{39} +3.25849 q^{41} -1.00000 q^{43} +2.89414 q^{45} -1.17319 q^{47} +1.00000 q^{49} +3.03017 q^{51} -1.54141 q^{53} +5.60221 q^{55} +11.5524 q^{57} +0.324380 q^{59} -5.08535 q^{61} -2.89414 q^{63} +1.23392 q^{65} +1.53962 q^{67} +6.10168 q^{69} -5.31678 q^{71} -11.0687 q^{73} -2.42779 q^{75} -5.60221 q^{77} +1.64253 q^{79} -9.30637 q^{81} -1.00439 q^{83} -1.24812 q^{85} +15.8581 q^{87} +0.613215 q^{89} -1.23392 q^{91} -15.9892 q^{93} -4.75839 q^{95} +7.35290 q^{97} +16.2136 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9	$ 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9} + q^{11} - 14 q^{13} - 11 q^{17} - 2 q^{19} - 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 14 q^{33} - 9 q^{35} - 20 q^{37} - 6 q^{39} - 6 q^{41} - 9 q^{43} + 5 q^{45} + 9 q^{49} - 8 q^{51} - 31 q^{53} + q^{55} - 16 q^{57} + 2 q^{59} - 13 q^{61} - 5 q^{63} - 14 q^{65} - 10 q^{67} - 18 q^{69} + 12 q^{71} - 32 q^{73} - q^{77} + q^{79} - 27 q^{81} - 10 q^{83} - 11 q^{85} - 5 q^{87} - q^{89} + 14 q^{91} - 49 q^{93} - 2 q^{95} - 28 q^{97} - 14 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9 + q^11 - 14 * q^13 - 11 * q^17 - 2 * q^19 - 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 14 * q^33 - 9 * q^35 - 20 * q^37 - 6 * q^39 - 6 * q^41 - 9 * q^43 + 5 * q^45 + 9 * q^49 - 8 * q^51 - 31 * q^53 + q^55 - 16 * q^57 + 2 * q^59 - 13 * q^61 - 5 * q^63 - 14 * q^65 - 10 * q^67 - 18 * q^69 + 12 * q^71 - 32 * q^73 - q^77 + q^79 - 27 * q^81 - 10 * q^83 - 11 * q^85 - 5 * q^87 - q^89 + 14 * q^91 - 49 * q^93 - 2 * q^95 - 28 * q^97 - 14 * 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
$3$	−2.42779	−1.40168	−0.700841	−	0.713317i	$-0.747192\pi$
$3$	−2.42779	−1.40168	−0.700841	+	0.713317i	$0.747192\pi$
$4$	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
$9$	2.89414	0.964714
$10$	0	0
$11$	5.60221	1.68913	0.844565	−	0.535453i	$-0.179859\pi$
$11$	5.60221	1.68913	0.844565	+	0.535453i	$0.179859\pi$
$12$	0	0
$13$	1.23392	0.342229	0.171114	−	0.985251i	$-0.445263\pi$
$13$	1.23392	0.342229	0.171114	+	0.985251i	$0.445263\pi$
$14$	0	0
$15$	−2.42779	−0.626851
$16$	0	0
$17$	−1.24812	−0.302714	−0.151357	−	0.988479i	$-0.548364\pi$
$17$	−1.24812	−0.302714	−0.151357	+	0.988479i	$0.548364\pi$
$18$	0	0
$19$	−4.75839	−1.09165	−0.545825	−	0.837899i	$-0.683784\pi$
$19$	−4.75839	−1.09165	−0.545825	+	0.837899i	$0.683784\pi$
$20$	0	0
$21$	2.42779	0.529786
$22$	0	0
$23$	−2.51327	−0.524053	−0.262027	−	0.965061i	$-0.584391\pi$
$23$	−2.51327	−0.524053	−0.262027	+	0.965061i	$0.584391\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.257003	0.0494603
$28$	0	0
$29$	−6.53192	−1.21295	−0.606473	−	0.795104i	$-0.707416\pi$
$29$	−6.53192	−1.21295	−0.606473	+	0.795104i	$0.707416\pi$
$30$	0	0
$31$	6.58593	1.18287	0.591434	−	0.806353i	$-0.298562\pi$
$31$	6.58593	1.18287	0.591434	+	0.806353i	$0.298562\pi$
$32$	0	0
$33$	−13.6010	−2.36762
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−9.65052	−1.58653	−0.793267	−	0.608873i	$-0.791622\pi$
$37$	−9.65052	−1.58653	−0.793267	+	0.608873i	$0.791622\pi$
$38$	0	0
$39$	−2.99570	−0.479696
$40$	0	0
$41$	3.25849	0.508890	0.254445	−	0.967087i	$-0.418107\pi$
$41$	3.25849	0.508890	0.254445	+	0.967087i	$0.418107\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	2.89414	0.431433
$46$	0	0
$47$	−1.17319	−0.171127	−0.0855635	−	0.996333i	$-0.527269\pi$
$47$	−1.17319	−0.171127	−0.0855635	+	0.996333i	$0.527269\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.03017	0.424309
$52$	0	0
$53$	−1.54141	−0.211728	−0.105864	−	0.994381i	$-0.533761\pi$
$53$	−1.54141	−0.211728	−0.105864	+	0.994381i	$0.533761\pi$
$54$	0	0
$55$	5.60221	0.755402
$56$	0	0
$57$	11.5524	1.53015
$58$	0	0
$59$	0.324380	0.0422307	0.0211153	−	0.999777i	$-0.493278\pi$
$59$	0.324380	0.0422307	0.0211153	+	0.999777i	$0.493278\pi$
$60$	0	0
$61$	−5.08535	−0.651113	−0.325556	−	0.945523i	$-0.605552\pi$
$61$	−5.08535	−0.651113	−0.325556	+	0.945523i	$0.605552\pi$
$62$	0	0
$63$	−2.89414	−0.364627
$64$	0	0
$65$	1.23392	0.153049
$66$	0	0
$67$	1.53962	0.188094	0.0940471	−	0.995568i	$-0.470020\pi$
$67$	1.53962	0.188094	0.0940471	+	0.995568i	$0.470020\pi$
$68$	0	0
$69$	6.10168	0.734556
$70$	0	0
$71$	−5.31678	−0.630986	−0.315493	−	0.948928i	$-0.602170\pi$
$71$	−5.31678	−0.630986	−0.315493	+	0.948928i	$0.602170\pi$
$72$	0	0
$73$	−11.0687	−1.29550	−0.647748	−	0.761855i	$-0.724289\pi$
$73$	−11.0687	−1.29550	−0.647748	+	0.761855i	$0.724289\pi$
$74$	0	0
$75$	−2.42779	−0.280336
$76$	0	0
$77$	−5.60221	−0.638431
$78$	0	0
$79$	1.64253	0.184800	0.0923998	−	0.995722i	$-0.470546\pi$
$79$	1.64253	0.184800	0.0923998	+	0.995722i	$0.470546\pi$
$80$	0	0
$81$	−9.30637	−1.03404
$82$	0	0
$83$	−1.00439	−0.110247	−0.0551233	−	0.998480i	$-0.517555\pi$
$83$	−1.00439	−0.110247	−0.0551233	+	0.998480i	$0.517555\pi$
$84$	0	0
$85$	−1.24812	−0.135378
$86$	0	0
$87$	15.8581	1.70017
$88$	0	0
$89$	0.613215	0.0650007	0.0325003	−	0.999472i	$-0.489653\pi$
$89$	0.613215	0.0650007	0.0325003	+	0.999472i	$0.489653\pi$
$90$	0	0
$91$	−1.23392	−0.129350
$92$	0	0
$93$	−15.9892	−1.65801
$94$	0	0
$95$	−4.75839	−0.488201
$96$	0	0
$97$	7.35290	0.746574	0.373287	−	0.927716i	$-0.378231\pi$
$97$	7.35290	0.746574	0.373287	+	0.927716i	$0.378231\pi$
$98$	0	0
$99$	16.2136	1.62953
$100$	0	0
$101$	19.6223	1.95249	0.976246	−	0.216665i	$-0.0695179\pi$
$101$	19.6223	1.95249	0.976246	+	0.216665i	$0.0695179\pi$
$102$	0	0
$103$	17.5055	1.72487	0.862435	−	0.506167i	$-0.168938\pi$
$103$	17.5055	1.72487	0.862435	+	0.506167i	$0.168938\pi$
$104$	0	0
$105$	2.42779	0.236928
$106$	0	0
$107$	11.0235	1.06568	0.532839	−	0.846217i	$-0.321125\pi$
$107$	11.0235	1.06568	0.532839	+	0.846217i	$0.321125\pi$
$108$	0	0
$109$	7.33529	0.702593	0.351296	−	0.936264i	$-0.385741\pi$
$109$	7.33529	0.702593	0.351296	+	0.936264i	$0.385741\pi$
$110$	0	0
$111$	23.4294	2.22382
$112$	0	0
$113$	−18.9702	−1.78457	−0.892283	−	0.451477i	$-0.850897\pi$
$113$	−18.9702	−1.78457	−0.892283	+	0.451477i	$0.850897\pi$
$114$	0	0
$115$	−2.51327	−0.234364
$116$	0	0
$117$	3.57115	0.330153
$118$	0	0
$119$	1.24812	0.114415
$120$	0	0
$121$	20.3848	1.85316
$122$	0	0
$123$	−7.91090	−0.713302
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.8019	−1.04725	−0.523626	−	0.851948i	$-0.675421\pi$
$127$	−11.8019	−1.04725	−0.523626	+	0.851948i	$0.675421\pi$
$128$	0	0
$129$	2.42779	0.213755
$130$	0	0
$131$	5.49820	0.480380	0.240190	−	0.970726i	$-0.422790\pi$
$131$	5.49820	0.480380	0.240190	+	0.970726i	$0.422790\pi$
$132$	0	0
$133$	4.75839	0.412605
$134$	0	0
$135$	0.257003	0.0221193
$136$	0	0
$137$	−12.9212	−1.10393	−0.551965	−	0.833867i	$-0.686122\pi$
$137$	−12.9212	−1.10393	−0.551965	+	0.833867i	$0.686122\pi$
$138$	0	0
$139$	7.44676	0.631626	0.315813	−	0.948821i	$-0.397723\pi$
$139$	7.44676	0.631626	0.315813	+	0.948821i	$0.397723\pi$
$140$	0	0
$141$	2.84825	0.239866
$142$	0	0
$143$	6.91270	0.578069
$144$	0	0
$145$	−6.53192	−0.542446
$146$	0	0
$147$	−2.42779	−0.200240
$148$	0	0
$149$	−14.3749	−1.17764	−0.588820	−	0.808264i	$-0.700407\pi$
$149$	−14.3749	−1.17764	−0.588820	+	0.808264i	$0.700407\pi$
$150$	0	0
$151$	−5.11005	−0.415850	−0.207925	−	0.978145i	$-0.566671\pi$
$151$	−5.11005	−0.415850	−0.207925	+	0.978145i	$0.566671\pi$
$152$	0	0
$153$	−3.61224	−0.292032
$154$	0	0
$155$	6.58593	0.528995
$156$	0	0
$157$	−7.12353	−0.568520	−0.284260	−	0.958747i	$-0.591748\pi$
$157$	−7.12353	−0.568520	−0.284260	+	0.958747i	$0.591748\pi$
$158$	0	0
$159$	3.74220	0.296776
$160$	0	0
$161$	2.51327	0.198073
$162$	0	0
$163$	2.26012	0.177026	0.0885131	−	0.996075i	$-0.471788\pi$
$163$	2.26012	0.177026	0.0885131	+	0.996075i	$0.471788\pi$
$164$	0	0
$165$	−13.6010	−1.05883
$166$	0	0
$167$	−16.9447	−1.31122	−0.655611	−	0.755098i	$-0.727589\pi$
$167$	−16.9447	−1.31122	−0.655611	+	0.755098i	$0.727589\pi$
$168$	0	0
$169$	−11.4774	−0.882879
$170$	0	0
$171$	−13.7715	−1.05313
$172$	0	0
$173$	−0.791377	−0.0601673	−0.0300836	−	0.999547i	$-0.509577\pi$
$173$	−0.791377	−0.0601673	−0.0300836	+	0.999547i	$0.509577\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−0.787525	−0.0591940
$178$	0	0
$179$	−4.25851	−0.318296	−0.159148	−	0.987255i	$-0.550875\pi$
$179$	−4.25851	−0.318296	−0.159148	+	0.987255i	$0.550875\pi$
$180$	0	0
$181$	−11.0913	−0.824409	−0.412204	−	0.911091i	$-0.635241\pi$
$181$	−11.0913	−0.824409	−0.412204	+	0.911091i	$0.635241\pi$
$182$	0	0
$183$	12.3461	0.912653
$184$	0	0
$185$	−9.65052	−0.709520
$186$	0	0
$187$	−6.99224	−0.511323
$188$	0	0
$189$	−0.257003	−0.0186942
$190$	0	0
$191$	17.2124	1.24545	0.622724	−	0.782441i	$-0.286026\pi$
$191$	17.2124	1.24545	0.622724	+	0.782441i	$0.286026\pi$
$192$	0	0
$193$	−8.78759	−0.632545	−0.316272	−	0.948668i	$-0.602431\pi$
$193$	−8.78759	−0.632545	−0.316272	+	0.948668i	$0.602431\pi$
$194$	0	0
$195$	−2.99570	−0.214527
$196$	0	0
$197$	7.21834	0.514286	0.257143	−	0.966373i	$-0.417219\pi$
$197$	7.21834	0.514286	0.257143	+	0.966373i	$0.417219\pi$
$198$	0	0
$199$	19.8052	1.40395	0.701976	−	0.712200i	$-0.252301\pi$
$199$	19.8052	1.40395	0.701976	+	0.712200i	$0.252301\pi$
$200$	0	0
$201$	−3.73786	−0.263648
$202$	0	0
$203$	6.53192	0.458451
$204$	0	0
$205$	3.25849	0.227582
$206$	0	0
$207$	−7.27376	−0.505561
$208$	0	0
$209$	−26.6575	−1.84394
$210$	0	0
$211$	−11.0916	−0.763577	−0.381789	−	0.924250i	$-0.624692\pi$
$211$	−11.0916	−0.763577	−0.381789	+	0.924250i	$0.624692\pi$
$212$	0	0
$213$	12.9080	0.884442
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−6.58593	−0.447082
$218$	0	0
$219$	26.8725	1.81587
$220$	0	0
$221$	−1.54009	−0.103597
$222$	0	0
$223$	4.79436	0.321054	0.160527	−	0.987031i	$-0.448681\pi$
$223$	4.79436	0.321054	0.160527	+	0.987031i	$0.448681\pi$
$224$	0	0
$225$	2.89414	0.192943
$226$	0	0
$227$	−0.982827	−0.0652325	−0.0326163	−	0.999468i	$-0.510384\pi$
$227$	−0.982827	−0.0652325	−0.0326163	+	0.999468i	$0.510384\pi$
$228$	0	0
$229$	−6.28692	−0.415451	−0.207726	−	0.978187i	$-0.566606\pi$
$229$	−6.28692	−0.415451	−0.207726	+	0.978187i	$0.566606\pi$
$230$	0	0
$231$	13.6010	0.894878
$232$	0	0
$233$	11.8993	0.779547	0.389773	−	0.920911i	$-0.372553\pi$
$233$	11.8993	0.779547	0.389773	+	0.920911i	$0.372553\pi$
$234$	0	0
$235$	−1.17319	−0.0765304
$236$	0	0
$237$	−3.98772	−0.259030
$238$	0	0
$239$	−20.6417	−1.33520	−0.667601	−	0.744519i	$-0.732679\pi$
$239$	−20.6417	−1.33520	−0.667601	+	0.744519i	$0.732679\pi$
$240$	0	0
$241$	12.2542	0.789359	0.394680	−	0.918819i	$-0.370856\pi$
$241$	12.2542	0.789359	0.394680	+	0.918819i	$0.370856\pi$
$242$	0	0
$243$	21.8229	1.39994
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−5.87150	−0.373594
$248$	0	0
$249$	2.43845	0.154531
$250$	0	0
$251$	−16.6118	−1.04853	−0.524263	−	0.851556i	$-0.675659\pi$
$251$	−16.6118	−1.04853	−0.524263	+	0.851556i	$0.675659\pi$
$252$	0	0
$253$	−14.0799	−0.885194
$254$	0	0
$255$	3.03017	0.189757
$256$	0	0
$257$	−24.7804	−1.54576	−0.772880	−	0.634553i	$-0.781184\pi$
$257$	−24.7804	−1.54576	−0.772880	+	0.634553i	$0.781184\pi$
$258$	0	0
$259$	9.65052	0.599654
$260$	0	0
$261$	−18.9043	−1.17015
$262$	0	0
$263$	−9.38441	−0.578668	−0.289334	−	0.957228i	$-0.593434\pi$
$263$	−9.38441	−0.578668	−0.289334	+	0.957228i	$0.593434\pi$
$264$	0	0
$265$	−1.54141	−0.0946878
$266$	0	0
$267$	−1.48876	−0.0911103
$268$	0	0
$269$	0.417647	0.0254644	0.0127322	−	0.999919i	$-0.495947\pi$
$269$	0.417647	0.0254644	0.0127322	+	0.999919i	$0.495947\pi$
$270$	0	0
$271$	20.5145	1.24617	0.623085	−	0.782154i	$-0.285879\pi$
$271$	20.5145	1.24617	0.623085	+	0.782154i	$0.285879\pi$
$272$	0	0
$273$	2.99570	0.181308
$274$	0	0
$275$	5.60221	0.337826
$276$	0	0
$277$	−17.7293	−1.06525	−0.532625	−	0.846351i	$-0.678794\pi$
$277$	−17.7293	−1.06525	−0.532625	+	0.846351i	$0.678794\pi$
$278$	0	0
$279$	19.0606	1.14113
$280$	0	0
$281$	−19.0436	−1.13605	−0.568024	−	0.823012i	$-0.692292\pi$
$281$	−19.0436	−1.13605	−0.568024	+	0.823012i	$0.692292\pi$
$282$	0	0
$283$	−18.3384	−1.09011	−0.545054	−	0.838401i	$-0.683491\pi$
$283$	−18.3384	−1.09011	−0.545054	+	0.838401i	$0.683491\pi$
$284$	0	0
$285$	11.5524	0.684303
$286$	0	0
$287$	−3.25849	−0.192342
$288$	0	0
$289$	−15.4422	−0.908364
$290$	0	0
$291$	−17.8513	−1.04646
$292$	0	0
$293$	−3.08497	−0.180226	−0.0901131	−	0.995932i	$-0.528723\pi$
$293$	−3.08497	−0.180226	−0.0901131	+	0.995932i	$0.528723\pi$
$294$	0	0
$295$	0.324380	0.0188861
$296$	0	0
$297$	1.43979	0.0835449
$298$	0	0
$299$	−3.10118	−0.179346
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	−47.6387	−2.73677
$304$	0	0
$305$	−5.08535	−0.291186
$306$	0	0
$307$	−15.0514	−0.859027	−0.429513	−	0.903061i	$-0.641315\pi$
$307$	−15.0514	−0.859027	−0.429513	+	0.903061i	$0.641315\pi$
$308$	0	0
$309$	−42.4997	−2.41772
$310$	0	0
$311$	−12.9505	−0.734354	−0.367177	−	0.930151i	$-0.619676\pi$
$311$	−12.9505	−0.734354	−0.367177	+	0.930151i	$0.619676\pi$
$312$	0	0
$313$	18.6989	1.05692	0.528461	−	0.848957i	$-0.322769\pi$
$313$	18.6989	1.05692	0.528461	+	0.848957i	$0.322769\pi$
$314$	0	0
$315$	−2.89414	−0.163066
$316$	0	0
$317$	−12.2573	−0.688436	−0.344218	−	0.938890i	$-0.611856\pi$
$317$	−12.2573	−0.688436	−0.344218	+	0.938890i	$0.611856\pi$
$318$	0	0
$319$	−36.5932	−2.04883
$320$	0	0
$321$	−26.7626	−1.49374
$322$	0	0
$323$	5.93905	0.330458
$324$	0	0
$325$	1.23392	0.0684458
$326$	0	0
$327$	−17.8085	−0.984812
$328$	0	0
$329$	1.17319	0.0646800
$330$	0	0
$331$	6.27403	0.344852	0.172426	−	0.985022i	$-0.444839\pi$
$331$	6.27403	0.344852	0.172426	+	0.985022i	$0.444839\pi$
$332$	0	0
$333$	−27.9300	−1.53055
$334$	0	0
$335$	1.53962	0.0841183
$336$	0	0
$337$	5.51835	0.300604	0.150302	−	0.988640i	$-0.451975\pi$
$337$	5.51835	0.300604	0.150302	+	0.988640i	$0.451975\pi$
$338$	0	0
$339$	46.0556	2.50139
$340$	0	0
$341$	36.8958	1.99802
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	6.10168	0.328503
$346$	0	0
$347$	1.05235	0.0564929	0.0282465	−	0.999601i	$-0.491008\pi$
$347$	1.05235	0.0564929	0.0282465	+	0.999601i	$0.491008\pi$
$348$	0	0
$349$	−6.70919	−0.359135	−0.179567	−	0.983746i	$-0.557470\pi$
$349$	−6.70919	−0.359135	−0.179567	+	0.983746i	$0.557470\pi$
$350$	0	0
$351$	0.317123	0.0169268
$352$	0	0
$353$	−17.2381	−0.917489	−0.458745	−	0.888568i	$-0.651701\pi$
$353$	−17.2381	−0.917489	−0.458745	+	0.888568i	$0.651701\pi$
$354$	0	0
$355$	−5.31678	−0.282185
$356$	0	0
$357$	−3.03017	−0.160374
$358$	0	0
$359$	13.4138	0.707954	0.353977	−	0.935254i	$-0.384829\pi$
$359$	13.4138	0.707954	0.353977	+	0.935254i	$0.384829\pi$
$360$	0	0
$361$	3.64232	0.191701
$362$	0	0
$363$	−49.4899	−2.59754
$364$	0	0
$365$	−11.0687	−0.579363
$366$	0	0
$367$	0.798978	0.0417063	0.0208532	−	0.999783i	$-0.493362\pi$
$367$	0.798978	0.0417063	0.0208532	+	0.999783i	$0.493362\pi$
$368$	0	0
$369$	9.43052	0.490933
$370$	0	0
$371$	1.54141	0.0800258
$372$	0	0
$373$	−30.9722	−1.60368	−0.801839	−	0.597541i	$-0.796145\pi$
$373$	−30.9722	−1.60368	−0.801839	+	0.597541i	$0.796145\pi$
$374$	0	0
$375$	−2.42779	−0.125370
$376$	0	0
$377$	−8.05989	−0.415105
$378$	0	0
$379$	−31.7156	−1.62912	−0.814561	−	0.580078i	$-0.803022\pi$
$379$	−31.7156	−1.62912	−0.814561	+	0.580078i	$0.803022\pi$
$380$	0	0
$381$	28.6525	1.46791
$382$	0	0
$383$	−6.40771	−0.327419	−0.163709	−	0.986509i	$-0.552346\pi$
$383$	−6.40771	−0.327419	−0.163709	+	0.986509i	$0.552346\pi$
$384$	0	0
$385$	−5.60221	−0.285515
$386$	0	0
$387$	−2.89414	−0.147117
$388$	0	0
$389$	23.0143	1.16687	0.583435	−	0.812160i	$-0.301708\pi$
$389$	23.0143	1.16687	0.583435	+	0.812160i	$0.301708\pi$
$390$	0	0
$391$	3.13687	0.158638
$392$	0	0
$393$	−13.3485	−0.673341
$394$	0	0
$395$	1.64253	0.0826449
$396$	0	0
$397$	3.25116	0.163171	0.0815856	−	0.996666i	$-0.474002\pi$
$397$	3.25116	0.163171	0.0815856	+	0.996666i	$0.474002\pi$
$398$	0	0
$399$	−11.5524	−0.578341
$400$	0	0
$401$	1.76108	0.0879439	0.0439719	−	0.999033i	$-0.485999\pi$
$401$	1.76108	0.0879439	0.0439719	+	0.999033i	$0.485999\pi$
$402$	0	0
$403$	8.12654	0.404812
$404$	0	0
$405$	−9.30637	−0.462437
$406$	0	0
$407$	−54.0642	−2.67986
$408$	0	0
$409$	23.4505	1.15955	0.579775	−	0.814776i	$-0.303140\pi$
$409$	23.4505	1.15955	0.579775	+	0.814776i	$0.303140\pi$
$410$	0	0
$411$	31.3698	1.54736
$412$	0	0
$413$	−0.324380	−0.0159617
$414$	0	0
$415$	−1.00439	−0.0493038
$416$	0	0
$417$	−18.0791	−0.885339
$418$	0	0
$419$	−39.8389	−1.94626	−0.973129	−	0.230263i	$-0.926041\pi$
$419$	−39.8389	−1.94626	−0.973129	+	0.230263i	$0.926041\pi$
$420$	0	0
$421$	23.4035	1.14062	0.570308	−	0.821431i	$-0.306824\pi$
$421$	23.4035	1.14062	0.570308	+	0.821431i	$0.306824\pi$
$422$	0	0
$423$	−3.39537	−0.165089
$424$	0	0
$425$	−1.24812	−0.0605428
$426$	0	0
$427$	5.08535	0.246097
$428$	0	0
$429$	−16.7826	−0.810269
$430$	0	0
$431$	10.9087	0.525452	0.262726	−	0.964870i	$-0.415378\pi$
$431$	10.9087	0.525452	0.262726	+	0.964870i	$0.415378\pi$
$432$	0	0
$433$	−22.2495	−1.06924	−0.534622	−	0.845091i	$-0.679546\pi$
$433$	−22.2495	−1.06924	−0.534622	+	0.845091i	$0.679546\pi$
$434$	0	0
$435$	15.8581	0.760337
$436$	0	0
$437$	11.9591	0.572083
$438$	0	0
$439$	−2.54253	−0.121349	−0.0606743	−	0.998158i	$-0.519325\pi$
$439$	−2.54253	−0.121349	−0.0606743	+	0.998158i	$0.519325\pi$
$440$	0	0
$441$	2.89414	0.137816
$442$	0	0
$443$	−12.4985	−0.593824	−0.296912	−	0.954905i	$-0.595957\pi$
$443$	−12.4985	−0.593824	−0.296912	+	0.954905i	$0.595957\pi$
$444$	0	0
$445$	0.613215	0.0290692
$446$	0	0
$447$	34.8993	1.65068
$448$	0	0
$449$	−24.6170	−1.16175	−0.580873	−	0.813994i	$-0.697289\pi$
$449$	−24.6170	−1.16175	−0.580873	+	0.813994i	$0.697289\pi$
$450$	0	0
$451$	18.2547	0.859581
$452$	0	0
$453$	12.4061	0.582890
$454$	0	0
$455$	−1.23392	−0.0578472
$456$	0	0
$457$	21.7327	1.01661	0.508307	−	0.861176i	$-0.330272\pi$
$457$	21.7327	1.01661	0.508307	+	0.861176i	$0.330272\pi$
$458$	0	0
$459$	−0.320771	−0.0149723
$460$	0	0
$461$	−19.6572	−0.915525	−0.457763	−	0.889075i	$-0.651349\pi$
$461$	−19.6572	−0.915525	−0.457763	+	0.889075i	$0.651349\pi$
$462$	0	0
$463$	25.5061	1.18537	0.592684	−	0.805435i	$-0.298068\pi$
$463$	25.5061	1.18537	0.592684	+	0.805435i	$0.298068\pi$
$464$	0	0
$465$	−15.9892	−0.741483
$466$	0	0
$467$	−38.0276	−1.75971	−0.879853	−	0.475246i	$-0.842359\pi$
$467$	−38.0276	−1.75971	−0.879853	+	0.475246i	$0.842359\pi$
$468$	0	0
$469$	−1.53962	−0.0710929
$470$	0	0
$471$	17.2944	0.796884
$472$	0	0
$473$	−5.60221	−0.257590
$474$	0	0
$475$	−4.75839	−0.218330
$476$	0	0
$477$	−4.46105	−0.204257
$478$	0	0
$479$	−5.11684	−0.233794	−0.116897	−	0.993144i	$-0.537295\pi$
$479$	−5.11684	−0.233794	−0.116897	+	0.993144i	$0.537295\pi$
$480$	0	0
$481$	−11.9080	−0.542958
$482$	0	0
$483$	−6.10168	−0.277636
$484$	0	0
$485$	7.35290	0.333878
$486$	0	0
$487$	−18.0579	−0.818281	−0.409141	−	0.912471i	$-0.634171\pi$
$487$	−18.0579	−0.818281	−0.409141	+	0.912471i	$0.634171\pi$
$488$	0	0
$489$	−5.48708	−0.248134
$490$	0	0
$491$	−10.8296	−0.488733	−0.244367	−	0.969683i	$-0.578580\pi$
$491$	−10.8296	−0.488733	−0.244367	+	0.969683i	$0.578580\pi$
$492$	0	0
$493$	8.15263	0.367176
$494$	0	0
$495$	16.2136	0.728747
$496$	0	0
$497$	5.31678	0.238490
$498$	0	0
$499$	34.2121	1.53154	0.765772	−	0.643112i	$-0.222357\pi$
$499$	34.2121	1.53154	0.765772	+	0.643112i	$0.222357\pi$
$500$	0	0
$501$	41.1382	1.83792
$502$	0	0
$503$	29.8207	1.32964	0.664819	−	0.747005i	$-0.268509\pi$
$503$	29.8207	1.32964	0.664819	+	0.747005i	$0.268509\pi$
$504$	0	0
$505$	19.6223	0.873181
$506$	0	0
$507$	27.8647	1.23752
$508$	0	0
$509$	−14.9121	−0.660968	−0.330484	−	0.943812i	$-0.607212\pi$
$509$	−14.9121	−0.660968	−0.330484	+	0.943812i	$0.607212\pi$
$510$	0	0
$511$	11.0687	0.489651
$512$	0	0
$513$	−1.22292	−0.0539934
$514$	0	0
$515$	17.5055	0.771386
$516$	0	0
$517$	−6.57245	−0.289056
$518$	0	0
$519$	1.92129	0.0843354
$520$	0	0
$521$	−4.70230	−0.206011	−0.103006	−	0.994681i	$-0.532846\pi$
$521$	−4.70230	−0.206011	−0.103006	+	0.994681i	$0.532846\pi$
$522$	0	0
$523$	−9.12953	−0.399207	−0.199603	−	0.979877i	$-0.563965\pi$
$523$	−9.12953	−0.399207	−0.199603	+	0.979877i	$0.563965\pi$
$524$	0	0
$525$	2.42779	0.105957
$526$	0	0
$527$	−8.22004	−0.358071
$528$	0	0
$529$	−16.6835	−0.725368
$530$	0	0
$531$	0.938801	0.0407405
$532$	0	0
$533$	4.02072	0.174157
$534$	0	0
$535$	11.0235	0.476586
$536$	0	0
$537$	10.3387	0.446150
$538$	0	0
$539$	5.60221	0.241304
$540$	0	0
$541$	8.82256	0.379312	0.189656	−	0.981851i	$-0.439263\pi$
$541$	8.82256	0.379312	0.189656	+	0.981851i	$0.439263\pi$
$542$	0	0
$543$	26.9273	1.15556
$544$	0	0
$545$	7.33529	0.314209
$546$	0	0
$547$	12.0655	0.515882	0.257941	−	0.966161i	$-0.416956\pi$
$547$	12.0655	0.515882	0.257941	+	0.966161i	$0.416956\pi$
$548$	0	0
$549$	−14.7177	−0.628137
$550$	0	0
$551$	31.0814	1.32411
$552$	0	0
$553$	−1.64253	−0.0698477
$554$	0	0
$555$	23.4294	0.994522
$556$	0	0
$557$	−10.9168	−0.462561	−0.231281	−	0.972887i	$-0.574292\pi$
$557$	−10.9168	−0.462561	−0.231281	+	0.972887i	$0.574292\pi$
$558$	0	0
$559$	−1.23392	−0.0521894
$560$	0	0
$561$	16.9757	0.716713
$562$	0	0
$563$	−21.4973	−0.906002	−0.453001	−	0.891510i	$-0.649647\pi$
$563$	−21.4973	−0.906002	−0.453001	+	0.891510i	$0.649647\pi$
$564$	0	0
$565$	−18.9702	−0.798082
$566$	0	0
$567$	9.30637	0.390831
$568$	0	0
$569$	−22.5986	−0.947382	−0.473691	−	0.880691i	$-0.657079\pi$
$569$	−22.5986	−0.947382	−0.473691	+	0.880691i	$0.657079\pi$
$570$	0	0
$571$	−44.7125	−1.87116	−0.935580	−	0.353116i	$-0.885122\pi$
$571$	−44.7125	−1.87116	−0.935580	+	0.353116i	$0.885122\pi$
$572$	0	0
$573$	−41.7881	−1.74572
$574$	0	0
$575$	−2.51327	−0.104811
$576$	0	0
$577$	−33.2495	−1.38419	−0.692097	−	0.721804i	$-0.743313\pi$
$577$	−33.2495	−1.38419	−0.692097	+	0.721804i	$0.743313\pi$
$578$	0	0
$579$	21.3344	0.886627
$580$	0	0
$581$	1.00439	0.0416693
$582$	0	0
$583$	−8.63528	−0.357637
$584$	0	0
$585$	3.57115	0.147649
$586$	0	0
$587$	44.5725	1.83970	0.919851	−	0.392267i	$-0.128309\pi$
$587$	44.5725	1.83970	0.919851	+	0.392267i	$0.128309\pi$
$588$	0	0
$589$	−31.3385	−1.29128
$590$	0	0
$591$	−17.5246	−0.720865
$592$	0	0
$593$	41.5242	1.70520	0.852598	−	0.522567i	$-0.175026\pi$
$593$	41.5242	1.70520	0.852598	+	0.522567i	$0.175026\pi$
$594$	0	0
$595$	1.24812	0.0511680
$596$	0	0
$597$	−48.0828	−1.96790
$598$	0	0
$599$	24.3345	0.994281	0.497141	−	0.867670i	$-0.334383\pi$
$599$	24.3345	0.994281	0.497141	+	0.867670i	$0.334383\pi$
$600$	0	0
$601$	−15.1913	−0.619668	−0.309834	−	0.950791i	$-0.600273\pi$
$601$	−15.1913	−0.619668	−0.309834	+	0.950791i	$0.600273\pi$
$602$	0	0
$603$	4.45587	0.181457
$604$	0	0
$605$	20.3848	0.828759
$606$	0	0
$607$	−24.9852	−1.01412	−0.507058	−	0.861912i	$-0.669267\pi$
$607$	−24.9852	−1.01412	−0.507058	+	0.861912i	$0.669267\pi$
$608$	0	0
$609$	−15.8581	−0.642602
$610$	0	0
$611$	−1.44762	−0.0585646
$612$	0	0
$613$	−7.57181	−0.305822	−0.152911	−	0.988240i	$-0.548865\pi$
$613$	−7.57181	−0.305822	−0.152911	+	0.988240i	$0.548865\pi$
$614$	0	0
$615$	−7.91090	−0.318998
$616$	0	0
$617$	−33.3908	−1.34426	−0.672131	−	0.740432i	$-0.734621\pi$
$617$	−33.3908	−1.34426	−0.672131	+	0.740432i	$0.734621\pi$
$618$	0	0
$619$	−4.47465	−0.179851	−0.0899256	−	0.995948i	$-0.528663\pi$
$619$	−4.47465	−0.179851	−0.0899256	+	0.995948i	$0.528663\pi$
$620$	0	0
$621$	−0.645919	−0.0259198
$622$	0	0
$623$	−0.613215	−0.0245680
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	64.7188	2.58462
$628$	0	0
$629$	12.0450	0.480266
$630$	0	0
$631$	38.6293	1.53781	0.768905	−	0.639363i	$-0.220802\pi$
$631$	38.6293	1.53781	0.768905	+	0.639363i	$0.220802\pi$
$632$	0	0
$633$	26.9280	1.07029
$634$	0	0
$635$	−11.8019	−0.468345
$636$	0	0
$637$	1.23392	0.0488898
$638$	0	0
$639$	−15.3875	−0.608721
$640$	0	0
$641$	41.3066	1.63151	0.815757	−	0.578395i	$-0.196321\pi$
$641$	41.3066	1.63151	0.815757	+	0.578395i	$0.196321\pi$
$642$	0	0
$643$	23.6706	0.933478	0.466739	−	0.884395i	$-0.345429\pi$
$643$	23.6706	0.933478	0.466739	+	0.884395i	$0.345429\pi$
$644$	0	0
$645$	2.42779	0.0955939
$646$	0	0
$647$	−2.86005	−0.112440	−0.0562201	−	0.998418i	$-0.517905\pi$
$647$	−2.86005	−0.112440	−0.0562201	+	0.998418i	$0.517905\pi$
$648$	0	0
$649$	1.81724	0.0713331
$650$	0	0
$651$	15.9892	0.626667
$652$	0	0
$653$	−33.0761	−1.29437	−0.647183	−	0.762334i	$-0.724053\pi$
$653$	−33.0761	−1.29437	−0.647183	+	0.762334i	$0.724053\pi$
$654$	0	0
$655$	5.49820	0.214833
$656$	0	0
$657$	−32.0344	−1.24978
$658$	0	0
$659$	18.1146	0.705646	0.352823	−	0.935690i	$-0.385222\pi$
$659$	18.1146	0.705646	0.352823	+	0.935690i	$0.385222\pi$
$660$	0	0
$661$	−36.9558	−1.43741	−0.718706	−	0.695314i	$-0.755265\pi$
$661$	−36.9558	−1.43741	−0.718706	+	0.695314i	$0.755265\pi$
$662$	0	0
$663$	3.73900	0.145211
$664$	0	0
$665$	4.75839	0.184523
$666$	0	0
$667$	16.4165	0.635648
$668$	0	0
$669$	−11.6397	−0.450016
$670$	0	0
$671$	−28.4892	−1.09981
$672$	0	0
$673$	−30.4626	−1.17425	−0.587124	−	0.809497i	$-0.699740\pi$
$673$	−30.4626	−1.17425	−0.587124	+	0.809497i	$0.699740\pi$
$674$	0	0
$675$	0.257003	0.00989207
$676$	0	0
$677$	32.8358	1.26198	0.630991	−	0.775790i	$-0.282648\pi$
$677$	32.8358	1.26198	0.630991	+	0.775790i	$0.282648\pi$
$678$	0	0
$679$	−7.35290	−0.282178
$680$	0	0
$681$	2.38609	0.0914353
$682$	0	0
$683$	−28.9844	−1.10906	−0.554529	−	0.832164i	$-0.687102\pi$
$683$	−28.9844	−1.10906	−0.554529	+	0.832164i	$0.687102\pi$
$684$	0	0
$685$	−12.9212	−0.493693
$686$	0	0
$687$	15.2633	0.582331
$688$	0	0
$689$	−1.90198	−0.0724596
$690$	0	0
$691$	−4.88809	−0.185952	−0.0929759	−	0.995668i	$-0.529638\pi$
$691$	−4.88809	−0.185952	−0.0929759	+	0.995668i	$0.529638\pi$
$692$	0	0
$693$	−16.2136	−0.615903
$694$	0	0
$695$	7.44676	0.282472
$696$	0	0
$697$	−4.06698	−0.154048
$698$	0	0
$699$	−28.8889	−1.09268
$700$	0	0
$701$	1.02374	0.0386660	0.0193330	−	0.999813i	$-0.493846\pi$
$701$	1.02374	0.0386660	0.0193330	+	0.999813i	$0.493846\pi$
$702$	0	0
$703$	45.9210	1.73194
$704$	0	0
$705$	2.84825	0.107271
$706$	0	0
$707$	−19.6223	−0.737973
$708$	0	0
$709$	−26.6961	−1.00259	−0.501297	−	0.865275i	$-0.667144\pi$
$709$	−26.6961	−1.00259	−0.501297	+	0.865275i	$0.667144\pi$
$710$	0	0
$711$	4.75373	0.178279
$712$	0	0
$713$	−16.5522	−0.619886
$714$	0	0
$715$	6.91270	0.258520
$716$	0	0
$717$	50.1137	1.87153
$718$	0	0
$719$	−33.2279	−1.23919	−0.619596	−	0.784921i	$-0.712704\pi$
$719$	−33.2279	−1.23919	−0.619596	+	0.784921i	$0.712704\pi$
$720$	0	0
$721$	−17.5055	−0.651940
$722$	0	0
$723$	−29.7505	−1.10643
$724$	0	0
$725$	−6.53192	−0.242589
$726$	0	0
$727$	−21.0368	−0.780213	−0.390106	−	0.920770i	$-0.627562\pi$
$727$	−21.0368	−0.780213	−0.390106	+	0.920770i	$0.627562\pi$
$728$	0	0
$729$	−25.0621	−0.928226
$730$	0	0
$731$	1.24812	0.0461634
$732$	0	0
$733$	−8.96438	−0.331107	−0.165553	−	0.986201i	$-0.552941\pi$
$733$	−8.96438	−0.331107	−0.165553	+	0.986201i	$0.552941\pi$
$734$	0	0
$735$	−2.42779	−0.0895502
$736$	0	0
$737$	8.62526	0.317716
$738$	0	0
$739$	37.7550	1.38884	0.694420	−	0.719570i	$-0.255661\pi$
$739$	37.7550	1.38884	0.694420	+	0.719570i	$0.255661\pi$
$740$	0	0
$741$	14.2547	0.523661
$742$	0	0
$743$	10.7746	0.395282	0.197641	−	0.980274i	$-0.436672\pi$
$743$	10.7746	0.395282	0.197641	+	0.980274i	$0.436672\pi$
$744$	0	0
$745$	−14.3749	−0.526657
$746$	0	0
$747$	−2.90686	−0.106356
$748$	0	0
$749$	−11.0235	−0.402788
$750$	0	0
$751$	−4.45726	−0.162648	−0.0813238	−	0.996688i	$-0.525915\pi$
$751$	−4.45726	−0.162648	−0.0813238	+	0.996688i	$0.525915\pi$
$752$	0	0
$753$	40.3299	1.46970
$754$	0	0
$755$	−5.11005	−0.185974
$756$	0	0
$757$	47.6797	1.73295	0.866475	−	0.499221i	$-0.166380\pi$
$757$	47.6797	1.73295	0.866475	+	0.499221i	$0.166380\pi$
$758$	0	0
$759$	34.1829	1.24076
$760$	0	0
$761$	18.0901	0.655766	0.327883	−	0.944718i	$-0.393665\pi$
$761$	18.0901	0.655766	0.327883	+	0.944718i	$0.393665\pi$
$762$	0	0
$763$	−7.33529	−0.265555
$764$	0	0
$765$	−3.61224	−0.130601
$766$	0	0
$767$	0.400260	0.0144526
$768$	0	0
$769$	−50.1208	−1.80740	−0.903701	−	0.428164i	$-0.859160\pi$
$769$	−50.1208	−1.80740	−0.903701	+	0.428164i	$0.859160\pi$
$770$	0	0
$771$	60.1615	2.16666
$772$	0	0
$773$	28.9133	1.03994	0.519970	−	0.854184i	$-0.325943\pi$
$773$	28.9133	1.03994	0.519970	+	0.854184i	$0.325943\pi$
$774$	0	0
$775$	6.58593	0.236574
$776$	0	0
$777$	−23.4294	−0.840524
$778$	0	0
$779$	−15.5052	−0.555530
$780$	0	0
$781$	−29.7857	−1.06582
$782$	0	0
$783$	−1.67873	−0.0599927
$784$	0	0
$785$	−7.12353	−0.254250
$786$	0	0
$787$	−36.4445	−1.29911	−0.649553	−	0.760316i	$-0.725044\pi$
$787$	−36.4445	−1.29911	−0.649553	+	0.760316i	$0.725044\pi$
$788$	0	0
$789$	22.7833	0.811108
$790$	0	0
$791$	18.9702	0.674502
$792$	0	0
$793$	−6.27494	−0.222830
$794$	0	0
$795$	3.74220	0.132722
$796$	0	0
$797$	19.6900	0.697455	0.348727	−	0.937224i	$-0.386614\pi$
$797$	19.6900	0.697455	0.348727	+	0.937224i	$0.386614\pi$
$798$	0	0
$799$	1.46428	0.0518025
$800$	0	0
$801$	1.77473	0.0627071
$802$	0	0
$803$	−62.0093	−2.18826
$804$	0	0
$805$	2.51327	0.0885811
$806$	0	0
$807$	−1.01396	−0.0356930
$808$	0	0
$809$	−21.7256	−0.763830	−0.381915	−	0.924197i	$-0.624735\pi$
$809$	−21.7256	−0.763830	−0.381915	+	0.924197i	$0.624735\pi$
$810$	0	0
$811$	−4.79162	−0.168257	−0.0841284	−	0.996455i	$-0.526811\pi$
$811$	−4.79162	−0.168257	−0.0841284	+	0.996455i	$0.526811\pi$
$812$	0	0
$813$	−49.8049	−1.74673
$814$	0	0
$815$	2.26012	0.0791685
$816$	0	0
$817$	4.75839	0.166475
$818$	0	0
$819$	−3.57115	−0.124786
$820$	0	0
$821$	−7.07438	−0.246897	−0.123449	−	0.992351i	$-0.539395\pi$
$821$	−7.07438	−0.246897	−0.123449	+	0.992351i	$0.539395\pi$
$822$	0	0
$823$	−10.1070	−0.352306	−0.176153	−	0.984363i	$-0.556365\pi$
$823$	−10.1070	−0.352306	−0.176153	+	0.984363i	$0.556365\pi$
$824$	0	0
$825$	−13.6010	−0.473525
$826$	0	0
$827$	−24.3634	−0.847200	−0.423600	−	0.905849i	$-0.639234\pi$
$827$	−24.3634	−0.847200	−0.423600	+	0.905849i	$0.639234\pi$
$828$	0	0
$829$	−33.4518	−1.16183	−0.580914	−	0.813965i	$-0.697305\pi$
$829$	−33.4518	−1.16183	−0.580914	+	0.813965i	$0.697305\pi$
$830$	0	0
$831$	43.0429	1.49314
$832$	0	0
$833$	−1.24812	−0.0432448
$834$	0	0
$835$	−16.9447	−0.586397
$836$	0	0
$837$	1.69261	0.0585050
$838$	0	0
$839$	20.6677	0.713528	0.356764	−	0.934195i	$-0.383880\pi$
$839$	20.6677	0.713528	0.356764	+	0.934195i	$0.383880\pi$
$840$	0	0
$841$	13.6660	0.471240
$842$	0	0
$843$	46.2339	1.59238
$844$	0	0
$845$	−11.4774	−0.394836
$846$	0	0
$847$	−20.3848	−0.700429
$848$	0	0
$849$	44.5218	1.52798
$850$	0	0
$851$	24.2544	0.831428
$852$	0	0
$853$	−12.6134	−0.431873	−0.215937	−	0.976407i	$-0.569280\pi$
$853$	−12.6134	−0.431873	−0.215937	+	0.976407i	$0.569280\pi$
$854$	0	0
$855$	−13.7715	−0.470974
$856$	0	0
$857$	55.8359	1.90732	0.953658	−	0.300892i	$-0.0972843\pi$
$857$	55.8359	1.90732	0.953658	+	0.300892i	$0.0972843\pi$
$858$	0	0
$859$	−6.40051	−0.218382	−0.109191	−	0.994021i	$-0.534826\pi$
$859$	−6.40051	−0.218382	−0.109191	+	0.994021i	$0.534826\pi$
$860$	0	0
$861$	7.91090	0.269603
$862$	0	0
$863$	40.9220	1.39300	0.696500	−	0.717557i	$-0.254740\pi$
$863$	40.9220	1.39300	0.696500	+	0.717557i	$0.254740\pi$
$864$	0	0
$865$	−0.791377	−0.0269076
$866$	0	0
$867$	37.4903	1.27324
$868$	0	0
$869$	9.20183	0.312151
$870$	0	0
$871$	1.89977	0.0643713
$872$	0	0
$873$	21.2803	0.720230
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−32.5889	−1.10045	−0.550224	−	0.835017i	$-0.685458\pi$
$877$	−32.5889	−1.10045	−0.550224	+	0.835017i	$0.685458\pi$
$878$	0	0
$879$	7.48966	0.252620
$880$	0	0
$881$	10.4170	0.350957	0.175478	−	0.984483i	$-0.443853\pi$
$881$	10.4170	0.350957	0.175478	+	0.984483i	$0.443853\pi$
$882$	0	0
$883$	12.9794	0.436793	0.218397	−	0.975860i	$-0.429917\pi$
$883$	12.9794	0.436793	0.218397	+	0.975860i	$0.429917\pi$
$884$	0	0
$885$	−0.787525	−0.0264724
$886$	0	0
$887$	37.8263	1.27008	0.635042	−	0.772477i	$-0.280983\pi$
$887$	37.8263	1.27008	0.635042	+	0.772477i	$0.280983\pi$
$888$	0	0
$889$	11.8019	0.395824
$890$	0	0
$891$	−52.1363	−1.74663
$892$	0	0
$893$	5.58249	0.186811
$894$	0	0
$895$	−4.25851	−0.142346
$896$	0	0
$897$	7.52901	0.251386
$898$	0	0
$899$	−43.0188	−1.43476
$900$	0	0
$901$	1.92386	0.0640931
$902$	0	0
$903$	−2.42779	−0.0807916
$904$	0	0
$905$	−11.0913	−0.368687
$906$	0	0
$907$	1.07136	0.0355738	0.0177869	−	0.999842i	$-0.494338\pi$
$907$	1.07136	0.0355738	0.0177869	+	0.999842i	$0.494338\pi$
$908$	0	0
$909$	56.7897	1.88360
$910$	0	0
$911$	−34.4095	−1.14004	−0.570019	−	0.821632i	$-0.693064\pi$
$911$	−34.4095	−1.14004	−0.570019	+	0.821632i	$0.693064\pi$
$912$	0	0
$913$	−5.62683	−0.186221
$914$	0	0
$915$	12.3461	0.408151
$916$	0	0
$917$	−5.49820	−0.181567
$918$	0	0
$919$	42.3801	1.39799	0.698995	−	0.715126i	$-0.253631\pi$
$919$	42.3801	1.39799	0.698995	+	0.715126i	$0.253631\pi$
$920$	0	0
$921$	36.5415	1.20408
$922$	0	0
$923$	−6.56050	−0.215942
$924$	0	0
$925$	−9.65052	−0.317307
$926$	0	0
$927$	50.6635	1.66401
$928$	0	0
$929$	−2.11566	−0.0694126	−0.0347063	−	0.999398i	$-0.511050\pi$
$929$	−2.11566	−0.0694126	−0.0347063	+	0.999398i	$0.511050\pi$
$930$	0	0
$931$	−4.75839	−0.155950
$932$	0	0
$933$	31.4409	1.02933
$934$	0	0
$935$	−6.99224	−0.228671
$936$	0	0
$937$	−11.1551	−0.364421	−0.182210	−	0.983260i	$-0.558325\pi$
$937$	−11.1551	−0.364421	−0.182210	+	0.983260i	$0.558325\pi$
$938$	0	0
$939$	−45.3968	−1.48147
$940$	0	0
$941$	38.4631	1.25386	0.626930	−	0.779075i	$-0.284311\pi$
$941$	38.4631	1.25386	0.626930	+	0.779075i	$0.284311\pi$
$942$	0	0
$943$	−8.18945	−0.266685
$944$	0	0
$945$	−0.257003	−0.00836032
$946$	0	0
$947$	1.68741	0.0548333	0.0274167	−	0.999624i	$-0.491272\pi$
$947$	1.68741	0.0548333	0.0274167	+	0.999624i	$0.491272\pi$
$948$	0	0
$949$	−13.6580	−0.443356
$950$	0	0
$951$	29.7580	0.964968
$952$	0	0
$953$	−22.6435	−0.733495	−0.366748	−	0.930321i	$-0.619529\pi$
$953$	−22.6435	−0.733495	−0.366748	+	0.930321i	$0.619529\pi$
$954$	0	0
$955$	17.2124	0.556982
$956$	0	0
$957$	88.8404	2.87180
$958$	0	0
$959$	12.9212	0.417247
$960$	0	0
$961$	12.3745	0.399177
$962$	0	0
$963$	31.9034	1.02807
$964$	0	0
$965$	−8.78759	−0.282883
$966$	0	0
$967$	0.937152	0.0301368	0.0150684	−	0.999886i	$-0.495203\pi$
$967$	0.937152	0.0301368	0.0150684	+	0.999886i	$0.495203\pi$
$968$	0	0
$969$	−14.4187	−0.463197
$970$	0	0
$971$	−40.0053	−1.28383	−0.641916	−	0.766775i	$-0.721860\pi$
$971$	−40.0053	−1.28383	−0.641916	+	0.766775i	$0.721860\pi$
$972$	0	0
$973$	−7.44676	−0.238732
$974$	0	0
$975$	−2.99570	−0.0959392
$976$	0	0
$977$	54.2414	1.73534	0.867668	−	0.497144i	$-0.165618\pi$
$977$	54.2414	1.73534	0.867668	+	0.497144i	$0.165618\pi$
$978$	0	0
$979$	3.43536	0.109795
$980$	0	0
$981$	21.2293	0.677801
$982$	0	0
$983$	−0.0852251	−0.00271826	−0.00135913	−	0.999999i	$-0.500433\pi$
$983$	−0.0852251	−0.00271826	−0.00135913	+	0.999999i	$0.500433\pi$
$984$	0	0
$985$	7.21834	0.229996
$986$	0	0
$987$	−2.84825	−0.0906608
$988$	0	0
$989$	2.51327	0.0799173
$990$	0	0
$991$	−37.6441	−1.19580	−0.597902	−	0.801569i	$-0.703999\pi$
$991$	−37.6441	−1.19580	−0.597902	+	0.801569i	$0.703999\pi$
$992$	0	0
$993$	−15.2320	−0.483373
$994$	0	0
$995$	19.8052	0.627867
$996$	0	0
$997$	27.7871	0.880028	0.440014	−	0.897991i	$-0.354974\pi$
$997$	27.7871	0.880028	0.440014	+	0.897991i	$0.354974\pi$
$998$	0	0
$999$	−2.48022	−0.0784705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.g.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.g.1.2	✓	9	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)

\(f(q)\)	\(=\)	\(q-2.42779 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.89414 q^{9} +O(q^{10})\)	\(q-2.42779 q^{3} +1.00000 q^{5} -1.00000 q^{7} +2.89414 q^{9} +5.60221 q^{11} +1.23392 q^{13} -2.42779 q^{15} -1.24812 q^{17} -4.75839 q^{19} +2.42779 q^{21} -2.51327 q^{23} +1.00000 q^{25} +0.257003 q^{27} -6.53192 q^{29} +6.58593 q^{31} -13.6010 q^{33} -1.00000 q^{35} -9.65052 q^{37} -2.99570 q^{39} +3.25849 q^{41} -1.00000 q^{43} +2.89414 q^{45} -1.17319 q^{47} +1.00000 q^{49} +3.03017 q^{51} -1.54141 q^{53} +5.60221 q^{55} +11.5524 q^{57} +0.324380 q^{59} -5.08535 q^{61} -2.89414 q^{63} +1.23392 q^{65} +1.53962 q^{67} +6.10168 q^{69} -5.31678 q^{71} -11.0687 q^{73} -2.42779 q^{75} -5.60221 q^{77} +1.64253 q^{79} -9.30637 q^{81} -1.00439 q^{83} -1.24812 q^{85} +15.8581 q^{87} +0.613215 q^{89} -1.23392 q^{91} -15.9892 q^{93} -4.75839 q^{95} +7.35290 q^{97} +16.2136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) \) 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9	\( 9 q + 9 q^{5} - 9 q^{7} + 5 q^{9} + q^{11} - 14 q^{13} - 11 q^{17} - 2 q^{19} - 6 q^{23} + 9 q^{25} - 6 q^{29} + 6 q^{31} - 14 q^{33} - 9 q^{35} - 20 q^{37} - 6 q^{39} - 6 q^{41} - 9 q^{43} + 5 q^{45} + 9 q^{49} - 8 q^{51} - 31 q^{53} + q^{55} - 16 q^{57} + 2 q^{59} - 13 q^{61} - 5 q^{63} - 14 q^{65} - 10 q^{67} - 18 q^{69} + 12 q^{71} - 32 q^{73} - q^{77} + q^{79} - 27 q^{81} - 10 q^{83} - 11 q^{85} - 5 q^{87} - q^{89} + 14 q^{91} - 49 q^{93} - 2 q^{95} - 28 q^{97} - 14 q^{99}+O(q^{100}) \) 9 * q + 9 * q^5 - 9 * q^7 + 5 * q^9 + q^11 - 14 * q^13 - 11 * q^17 - 2 * q^19 - 6 * q^23 + 9 * q^25 - 6 * q^29 + 6 * q^31 - 14 * q^33 - 9 * q^35 - 20 * q^37 - 6 * q^39 - 6 * q^41 - 9 * q^43 + 5 * q^45 + 9 * q^49 - 8 * q^51 - 31 * q^53 + q^55 - 16 * q^57 + 2 * q^59 - 13 * q^61 - 5 * q^63 - 14 * q^65 - 10 * q^67 - 18 * q^69 + 12 * q^71 - 32 * q^73 - q^77 + q^79 - 27 * q^81 - 10 * q^83 - 11 * q^85 - 5 * q^87 - q^89 + 14 * q^91 - 49 * q^93 - 2 * q^95 - 28 * q^97 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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