LMFDB - Embedded newform 6020.2.a.f.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.0699420168$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 12x^{6} + 26x^{5} + 55x^{4} - 52x^{3} - 82x^{2} + 22x + 27 $ x^8 - 3x^7 - 12x^6 + 26x^5 + 55x^4 - 52x^3 - 82x^2 + 22*x + 27
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			1.7
Root			$2.96620$ 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)

$f(q)$	$=$	$q+1.96620 q^{3} -1.00000 q^{5} +1.00000 q^{7} +0.865935 q^{9} +O(q^{10})$	$q+1.96620 q^{3} -1.00000 q^{5} +1.00000 q^{7} +0.865935 q^{9} +2.19143 q^{11} -5.77519 q^{13} -1.96620 q^{15} -1.81370 q^{17} -3.83213 q^{19} +1.96620 q^{21} +5.76131 q^{23} +1.00000 q^{25} -4.19600 q^{27} -0.573312 q^{29} +4.79741 q^{31} +4.30879 q^{33} -1.00000 q^{35} -4.95258 q^{37} -11.3552 q^{39} -7.21500 q^{41} -1.00000 q^{43} -0.865935 q^{45} -11.0576 q^{47} +1.00000 q^{49} -3.56608 q^{51} +5.76440 q^{53} -2.19143 q^{55} -7.53473 q^{57} -0.651924 q^{59} +0.841865 q^{61} +0.865935 q^{63} +5.77519 q^{65} +4.14510 q^{67} +11.3279 q^{69} -4.87383 q^{71} +11.0752 q^{73} +1.96620 q^{75} +2.19143 q^{77} -8.10003 q^{79} -10.8480 q^{81} -12.0981 q^{83} +1.81370 q^{85} -1.12725 q^{87} -16.6323 q^{89} -5.77519 q^{91} +9.43266 q^{93} +3.83213 q^{95} -8.76832 q^{97} +1.89764 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9	$ 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9} + 6 q^{11} - 13 q^{13} + 5 q^{15} - 2 q^{17} - 14 q^{19} - 5 q^{21} + 6 q^{23} + 8 q^{25} - 11 q^{27} - 7 q^{29} - 18 q^{31} - q^{33} - 8 q^{35} + 2 q^{37} + 9 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - q^{47} + 8 q^{49} - 19 q^{51} + 5 q^{53} - 6 q^{55} - 4 q^{57} - 12 q^{59} - 23 q^{61} + 11 q^{63} + 13 q^{65} + 8 q^{67} - 18 q^{69} + 20 q^{71} - 4 q^{73} - 5 q^{75} + 6 q^{77} + 24 q^{79} - 8 q^{81} - 14 q^{83} + 2 q^{85} + 10 q^{87} - 21 q^{89} - 13 q^{91} + q^{93} + 14 q^{95} + 7 q^{97} + 4 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9 + 6 * q^11 - 13 * q^13 + 5 * q^15 - 2 * q^17 - 14 * q^19 - 5 * q^21 + 6 * q^23 + 8 * q^25 - 11 * q^27 - 7 * q^29 - 18 * q^31 - q^33 - 8 * q^35 + 2 * q^37 + 9 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - q^47 + 8 * q^49 - 19 * q^51 + 5 * q^53 - 6 * q^55 - 4 * q^57 - 12 * q^59 - 23 * q^61 + 11 * q^63 + 13 * q^65 + 8 * q^67 - 18 * q^69 + 20 * q^71 - 4 * q^73 - 5 * q^75 + 6 * q^77 + 24 * q^79 - 8 * q^81 - 14 * q^83 + 2 * q^85 + 10 * q^87 - 21 * q^89 - 13 * q^91 + q^93 + 14 * q^95 + 7 * q^97 + 4 * 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
$3$	1.96620	1.13518	0.567592	−	0.823310i	$-0.307875\pi$
$3$	1.96620	1.13518	0.567592	+	0.823310i	$0.307875\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.865935	0.288645
$10$	0	0
$11$	2.19143	0.660742	0.330371	−	0.943851i	$-0.392826\pi$
$11$	2.19143	0.660742	0.330371	+	0.943851i	$0.392826\pi$
$12$	0	0
$13$	−5.77519	−1.60175	−0.800874	−	0.598833i	$-0.795631\pi$
$13$	−5.77519	−1.60175	−0.800874	+	0.598833i	$0.795631\pi$
$14$	0	0
$15$	−1.96620	−0.507670
$16$	0	0
$17$	−1.81370	−0.439886	−0.219943	−	0.975513i	$-0.570587\pi$
$17$	−1.81370	−0.439886	−0.219943	+	0.975513i	$0.570587\pi$
$18$	0	0
$19$	−3.83213	−0.879152	−0.439576	−	0.898206i	$-0.644871\pi$
$19$	−3.83213	−0.879152	−0.439576	+	0.898206i	$0.644871\pi$
$20$	0	0
$21$	1.96620	0.429060
$22$	0	0
$23$	5.76131	1.20132	0.600658	−	0.799506i	$-0.294905\pi$
$23$	5.76131	1.20132	0.600658	+	0.799506i	$0.294905\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.19600	−0.807520
$28$	0	0
$29$	−0.573312	−0.106461	−0.0532307	−	0.998582i	$-0.516952\pi$
$29$	−0.573312	−0.106461	−0.0532307	+	0.998582i	$0.516952\pi$
$30$	0	0
$31$	4.79741	0.861641	0.430820	−	0.902438i	$-0.358224\pi$
$31$	4.79741	0.861641	0.430820	+	0.902438i	$0.358224\pi$
$32$	0	0
$33$	4.30879	0.750065
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−4.95258	−0.814199	−0.407100	−	0.913384i	$-0.633460\pi$
$37$	−4.95258	−0.814199	−0.407100	+	0.913384i	$0.633460\pi$
$38$	0	0
$39$	−11.3552	−1.81828
$40$	0	0
$41$	−7.21500	−1.12679	−0.563397	−	0.826186i	$-0.690506\pi$
$41$	−7.21500	−1.12679	−0.563397	+	0.826186i	$0.690506\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	−0.865935	−0.129086
$46$	0	0
$47$	−11.0576	−1.61291	−0.806456	−	0.591294i	$-0.798617\pi$
$47$	−11.0576	−1.61291	−0.806456	+	0.591294i	$0.798617\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.56608	−0.499352
$52$	0	0
$53$	5.76440	0.791801	0.395900	−	0.918293i	$-0.370433\pi$
$53$	5.76440	0.791801	0.395900	+	0.918293i	$0.370433\pi$
$54$	0	0
$55$	−2.19143	−0.295493
$56$	0	0
$57$	−7.53473	−0.998000
$58$	0	0
$59$	−0.651924	−0.0848733	−0.0424367	−	0.999099i	$-0.513512\pi$
$59$	−0.651924	−0.0848733	−0.0424367	+	0.999099i	$0.513512\pi$
$60$	0	0
$61$	0.841865	0.107790	0.0538949	−	0.998547i	$-0.482836\pi$
$61$	0.841865	0.107790	0.0538949	+	0.998547i	$0.482836\pi$
$62$	0	0
$63$	0.865935	0.109098
$64$	0	0
$65$	5.77519	0.716324
$66$	0	0
$67$	4.14510	0.506405	0.253202	−	0.967413i	$-0.418516\pi$
$67$	4.14510	0.506405	0.253202	+	0.967413i	$0.418516\pi$
$68$	0	0
$69$	11.3279	1.36372
$70$	0	0
$71$	−4.87383	−0.578417	−0.289209	−	0.957266i	$-0.593392\pi$
$71$	−4.87383	−0.578417	−0.289209	+	0.957266i	$0.593392\pi$
$72$	0	0
$73$	11.0752	1.29626	0.648129	−	0.761530i	$-0.275552\pi$
$73$	11.0752	1.29626	0.648129	+	0.761530i	$0.275552\pi$
$74$	0	0
$75$	1.96620	0.227037
$76$	0	0
$77$	2.19143	0.249737
$78$	0	0
$79$	−8.10003	−0.911325	−0.455663	−	0.890153i	$-0.650598\pi$
$79$	−8.10003	−0.911325	−0.455663	+	0.890153i	$0.650598\pi$
$80$	0	0
$81$	−10.8480	−1.20533
$82$	0	0
$83$	−12.0981	−1.32794	−0.663971	−	0.747758i	$-0.731130\pi$
$83$	−12.0981	−1.32794	−0.663971	+	0.747758i	$0.731130\pi$
$84$	0	0
$85$	1.81370	0.196723
$86$	0	0
$87$	−1.12725	−0.120853
$88$	0	0
$89$	−16.6323	−1.76302	−0.881511	−	0.472164i	$-0.843473\pi$
$89$	−16.6323	−1.76302	−0.881511	+	0.472164i	$0.843473\pi$
$90$	0	0
$91$	−5.77519	−0.605404
$92$	0	0
$93$	9.43266	0.978121
$94$	0	0
$95$	3.83213	0.393169
$96$	0	0
$97$	−8.76832	−0.890288	−0.445144	−	0.895459i	$-0.646847\pi$
$97$	−8.76832	−0.890288	−0.445144	+	0.895459i	$0.646847\pi$
$98$	0	0
$99$	1.89764	0.190720
$100$	0	0
$101$	−4.65380	−0.463071	−0.231535	−	0.972826i	$-0.574375\pi$
$101$	−4.65380	−0.463071	−0.231535	+	0.972826i	$0.574375\pi$
$102$	0	0
$103$	−10.6882	−1.05314	−0.526571	−	0.850131i	$-0.676523\pi$
$103$	−10.6882	−1.05314	−0.526571	+	0.850131i	$0.676523\pi$
$104$	0	0
$105$	−1.96620	−0.191881
$106$	0	0
$107$	6.66096	0.643939	0.321969	−	0.946750i	$-0.395655\pi$
$107$	6.66096	0.643939	0.321969	+	0.946750i	$0.395655\pi$
$108$	0	0
$109$	7.45077	0.713654	0.356827	−	0.934170i	$-0.383859\pi$
$109$	7.45077	0.713654	0.356827	+	0.934170i	$0.383859\pi$
$110$	0	0
$111$	−9.73776	−0.924267
$112$	0	0
$113$	4.39854	0.413780	0.206890	−	0.978364i	$-0.433666\pi$
$113$	4.39854	0.413780	0.206890	+	0.978364i	$0.433666\pi$
$114$	0	0
$115$	−5.76131	−0.537245
$116$	0	0
$117$	−5.00093	−0.462336
$118$	0	0
$119$	−1.81370	−0.166261
$120$	0	0
$121$	−6.19761	−0.563419
$122$	0	0
$123$	−14.1861	−1.27912
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.8845	−0.965847	−0.482923	−	0.875663i	$-0.660425\pi$
$127$	−10.8845	−0.965847	−0.482923	+	0.875663i	$0.660425\pi$
$128$	0	0
$129$	−1.96620	−0.173114
$130$	0	0
$131$	−12.9456	−1.13106	−0.565532	−	0.824727i	$-0.691329\pi$
$131$	−12.9456	−1.13106	−0.565532	+	0.824727i	$0.691329\pi$
$132$	0	0
$133$	−3.83213	−0.332288
$134$	0	0
$135$	4.19600	0.361134
$136$	0	0
$137$	7.64508	0.653163	0.326582	−	0.945169i	$-0.394103\pi$
$137$	7.64508	0.653163	0.326582	+	0.945169i	$0.394103\pi$
$138$	0	0
$139$	−2.30484	−0.195494	−0.0977471	−	0.995211i	$-0.531164\pi$
$139$	−2.30484	−0.195494	−0.0977471	+	0.995211i	$0.531164\pi$
$140$	0	0
$141$	−21.7414	−1.83095
$142$	0	0
$143$	−12.6559	−1.05834
$144$	0	0
$145$	0.573312	0.0476110
$146$	0	0
$147$	1.96620	0.162169
$148$	0	0
$149$	−9.93055	−0.813542	−0.406771	−	0.913530i	$-0.633345\pi$
$149$	−9.93055	−0.813542	−0.406771	+	0.913530i	$0.633345\pi$
$150$	0	0
$151$	12.6919	1.03285	0.516425	−	0.856332i	$-0.327262\pi$
$151$	12.6919	1.03285	0.516425	+	0.856332i	$0.327262\pi$
$152$	0	0
$153$	−1.57054	−0.126971
$154$	0	0
$155$	−4.79741	−0.385337
$156$	0	0
$157$	−0.419760	−0.0335005	−0.0167503	−	0.999860i	$-0.505332\pi$
$157$	−0.419760	−0.0335005	−0.0167503	+	0.999860i	$0.505332\pi$
$158$	0	0
$159$	11.3339	0.898840
$160$	0	0
$161$	5.76131	0.454055
$162$	0	0
$163$	−5.09016	−0.398692	−0.199346	−	0.979929i	$-0.563882\pi$
$163$	−5.09016	−0.398692	−0.199346	+	0.979929i	$0.563882\pi$
$164$	0	0
$165$	−4.30879	−0.335439
$166$	0	0
$167$	−8.02297	−0.620836	−0.310418	−	0.950600i	$-0.600469\pi$
$167$	−8.02297	−0.620836	−0.310418	+	0.950600i	$0.600469\pi$
$168$	0	0
$169$	20.3528	1.56560
$170$	0	0
$171$	−3.31838	−0.253763
$172$	0	0
$173$	5.00631	0.380623	0.190311	−	0.981724i	$-0.439050\pi$
$173$	5.00631	0.380623	0.190311	+	0.981724i	$0.439050\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−1.28181	−0.0963469
$178$	0	0
$179$	−14.3564	−1.07305	−0.536524	−	0.843885i	$-0.680263\pi$
$179$	−14.3564	−1.07305	−0.536524	+	0.843885i	$0.680263\pi$
$180$	0	0
$181$	1.28118	0.0952292	0.0476146	−	0.998866i	$-0.484838\pi$
$181$	1.28118	0.0952292	0.0476146	+	0.998866i	$0.484838\pi$
$182$	0	0
$183$	1.65527	0.122361
$184$	0	0
$185$	4.95258	0.364121
$186$	0	0
$187$	−3.97460	−0.290651
$188$	0	0
$189$	−4.19600	−0.305214
$190$	0	0
$191$	7.33829	0.530980	0.265490	−	0.964114i	$-0.414466\pi$
$191$	7.33829	0.530980	0.265490	+	0.964114i	$0.414466\pi$
$192$	0	0
$193$	12.7656	0.918885	0.459443	−	0.888207i	$-0.348049\pi$
$193$	12.7656	0.918885	0.459443	+	0.888207i	$0.348049\pi$
$194$	0	0
$195$	11.3552	0.813160
$196$	0	0
$197$	14.1097	1.00527	0.502636	−	0.864498i	$-0.332363\pi$
$197$	14.1097	1.00527	0.502636	+	0.864498i	$0.332363\pi$
$198$	0	0
$199$	22.0719	1.56464	0.782318	−	0.622880i	$-0.214037\pi$
$199$	22.0719	1.56464	0.782318	+	0.622880i	$0.214037\pi$
$200$	0	0
$201$	8.15009	0.574863
$202$	0	0
$203$	−0.573312	−0.0402386
$204$	0	0
$205$	7.21500	0.503918
$206$	0	0
$207$	4.98892	0.346754
$208$	0	0
$209$	−8.39787	−0.580893
$210$	0	0
$211$	−6.16163	−0.424184	−0.212092	−	0.977250i	$-0.568028\pi$
$211$	−6.16163	−0.424184	−0.212092	+	0.977250i	$0.568028\pi$
$212$	0	0
$213$	−9.58291	−0.656610
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	4.79741	0.325670
$218$	0	0
$219$	21.7761	1.47149
$220$	0	0
$221$	10.4744	0.704586
$222$	0	0
$223$	−6.10730	−0.408975	−0.204488	−	0.978869i	$-0.565553\pi$
$223$	−6.10730	−0.408975	−0.204488	+	0.978869i	$0.565553\pi$
$224$	0	0
$225$	0.865935	0.0577290
$226$	0	0
$227$	9.09527	0.603674	0.301837	−	0.953359i	$-0.402400\pi$
$227$	9.09527	0.603674	0.301837	+	0.953359i	$0.402400\pi$
$228$	0	0
$229$	−15.7449	−1.04045	−0.520225	−	0.854029i	$-0.674152\pi$
$229$	−15.7449	−1.04045	−0.520225	+	0.854029i	$0.674152\pi$
$230$	0	0
$231$	4.30879	0.283498
$232$	0	0
$233$	10.1874	0.667396	0.333698	−	0.942680i	$-0.391704\pi$
$233$	10.1874	0.667396	0.333698	+	0.942680i	$0.391704\pi$
$234$	0	0
$235$	11.0576	0.721316
$236$	0	0
$237$	−15.9263	−1.03452
$238$	0	0
$239$	10.6509	0.688950	0.344475	−	0.938795i	$-0.388057\pi$
$239$	10.6509	0.688950	0.344475	+	0.938795i	$0.388057\pi$
$240$	0	0
$241$	−10.5208	−0.677702	−0.338851	−	0.940840i	$-0.610038\pi$
$241$	−10.5208	−0.677702	−0.338851	+	0.940840i	$0.610038\pi$
$242$	0	0
$243$	−8.74125	−0.560752
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	22.1313	1.40818
$248$	0	0
$249$	−23.7873	−1.50746
$250$	0	0
$251$	−20.3519	−1.28460	−0.642300	−	0.766454i	$-0.722020\pi$
$251$	−20.3519	−1.28460	−0.642300	+	0.766454i	$0.722020\pi$
$252$	0	0
$253$	12.6255	0.793760
$254$	0	0
$255$	3.56608	0.223317
$256$	0	0
$257$	−3.11152	−0.194091	−0.0970456	−	0.995280i	$-0.530939\pi$
$257$	−3.11152	−0.194091	−0.0970456	+	0.995280i	$0.530939\pi$
$258$	0	0
$259$	−4.95258	−0.307738
$260$	0	0
$261$	−0.496451	−0.0307295
$262$	0	0
$263$	−10.3070	−0.635554	−0.317777	−	0.948166i	$-0.602936\pi$
$263$	−10.3070	−0.635554	−0.317777	+	0.948166i	$0.602936\pi$
$264$	0	0
$265$	−5.76440	−0.354104
$266$	0	0
$267$	−32.7024	−2.00136
$268$	0	0
$269$	−9.89403	−0.603250	−0.301625	−	0.953427i	$-0.597529\pi$
$269$	−9.89403	−0.603250	−0.301625	+	0.953427i	$0.597529\pi$
$270$	0	0
$271$	3.31412	0.201318	0.100659	−	0.994921i	$-0.467905\pi$
$271$	3.31412	0.201318	0.100659	+	0.994921i	$0.467905\pi$
$272$	0	0
$273$	−11.3552	−0.687245
$274$	0	0
$275$	2.19143	0.132148
$276$	0	0
$277$	4.54693	0.273199	0.136599	−	0.990626i	$-0.456383\pi$
$277$	4.54693	0.273199	0.136599	+	0.990626i	$0.456383\pi$
$278$	0	0
$279$	4.15425	0.248708
$280$	0	0
$281$	16.9506	1.01119	0.505594	−	0.862772i	$-0.331274\pi$
$281$	16.9506	1.01119	0.505594	+	0.862772i	$0.331274\pi$
$282$	0	0
$283$	−6.12462	−0.364071	−0.182035	−	0.983292i	$-0.558269\pi$
$283$	−6.12462	−0.364071	−0.182035	+	0.983292i	$0.558269\pi$
$284$	0	0
$285$	7.53473	0.446319
$286$	0	0
$287$	−7.21500	−0.425888
$288$	0	0
$289$	−13.7105	−0.806500
$290$	0	0
$291$	−17.2402	−1.01064
$292$	0	0
$293$	−11.7907	−0.688820	−0.344410	−	0.938819i	$-0.611921\pi$
$293$	−11.7907	−0.688820	−0.344410	+	0.938819i	$0.611921\pi$
$294$	0	0
$295$	0.651924	0.0379565
$296$	0	0
$297$	−9.19525	−0.533562
$298$	0	0
$299$	−33.2726	−1.92421
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	−9.15030	−0.525671
$304$	0	0
$305$	−0.841865	−0.0482050
$306$	0	0
$307$	−13.8279	−0.789198	−0.394599	−	0.918853i	$-0.629116\pi$
$307$	−13.8279	−0.789198	−0.394599	+	0.918853i	$0.629116\pi$
$308$	0	0
$309$	−21.0152	−1.19551
$310$	0	0
$311$	3.85645	0.218679	0.109340	−	0.994004i	$-0.465126\pi$
$311$	3.85645	0.218679	0.109340	+	0.994004i	$0.465126\pi$
$312$	0	0
$313$	27.9379	1.57915	0.789573	−	0.613657i	$-0.210302\pi$
$313$	27.9379	1.57915	0.789573	+	0.613657i	$0.210302\pi$
$314$	0	0
$315$	−0.865935	−0.0487899
$316$	0	0
$317$	5.40031	0.303312	0.151656	−	0.988433i	$-0.451539\pi$
$317$	5.40031	0.303312	0.151656	+	0.988433i	$0.451539\pi$
$318$	0	0
$319$	−1.25638	−0.0703436
$320$	0	0
$321$	13.0968	0.730990
$322$	0	0
$323$	6.95032	0.386726
$324$	0	0
$325$	−5.77519	−0.320350
$326$	0	0
$327$	14.6497	0.810130
$328$	0	0
$329$	−11.0576	−0.609623
$330$	0	0
$331$	−16.5291	−0.908520	−0.454260	−	0.890869i	$-0.650096\pi$
$331$	−16.5291	−0.908520	−0.454260	+	0.890869i	$0.650096\pi$
$332$	0	0
$333$	−4.28861	−0.235014
$334$	0	0
$335$	−4.14510	−0.226471
$336$	0	0
$337$	−28.9154	−1.57512	−0.787562	−	0.616235i	$-0.788657\pi$
$337$	−28.9154	−1.57512	−0.787562	+	0.616235i	$0.788657\pi$
$338$	0	0
$339$	8.64840	0.469717
$340$	0	0
$341$	10.5132	0.569322
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−11.3279	−0.609872
$346$	0	0
$347$	−17.2908	−0.928220	−0.464110	−	0.885778i	$-0.653626\pi$
$347$	−17.2908	−0.928220	−0.464110	+	0.885778i	$0.653626\pi$
$348$	0	0
$349$	35.0124	1.87417	0.937085	−	0.349100i	$-0.113513\pi$
$349$	35.0124	1.87417	0.937085	+	0.349100i	$0.113513\pi$
$350$	0	0
$351$	24.2326	1.29344
$352$	0	0
$353$	23.7890	1.26616	0.633079	−	0.774087i	$-0.281791\pi$
$353$	23.7890	1.26616	0.633079	+	0.774087i	$0.281791\pi$
$354$	0	0
$355$	4.87383	0.258676
$356$	0	0
$357$	−3.56608	−0.188737
$358$	0	0
$359$	30.3959	1.60423	0.802117	−	0.597167i	$-0.203707\pi$
$359$	30.3959	1.60423	0.802117	+	0.597167i	$0.203707\pi$
$360$	0	0
$361$	−4.31476	−0.227093
$362$	0	0
$363$	−12.1857	−0.639585
$364$	0	0
$365$	−11.0752	−0.579704
$366$	0	0
$367$	7.28574	0.380312	0.190156	−	0.981754i	$-0.439101\pi$
$367$	7.28574	0.380312	0.190156	+	0.981754i	$0.439101\pi$
$368$	0	0
$369$	−6.24772	−0.325243
$370$	0	0
$371$	5.76440	0.299273
$372$	0	0
$373$	−14.3069	−0.740781	−0.370391	−	0.928876i	$-0.620776\pi$
$373$	−14.3069	−0.740781	−0.370391	+	0.928876i	$0.620776\pi$
$374$	0	0
$375$	−1.96620	−0.101534
$376$	0	0
$377$	3.31098	0.170524
$378$	0	0
$379$	−13.6497	−0.701137	−0.350569	−	0.936537i	$-0.614012\pi$
$379$	−13.6497	−0.701137	−0.350569	+	0.936537i	$0.614012\pi$
$380$	0	0
$381$	−21.4012	−1.09641
$382$	0	0
$383$	3.26563	0.166866	0.0834330	−	0.996513i	$-0.473412\pi$
$383$	3.26563	0.166866	0.0834330	+	0.996513i	$0.473412\pi$
$384$	0	0
$385$	−2.19143	−0.111686
$386$	0	0
$387$	−0.865935	−0.0440179
$388$	0	0
$389$	−25.8699	−1.31165	−0.655827	−	0.754911i	$-0.727680\pi$
$389$	−25.8699	−1.31165	−0.655827	+	0.754911i	$0.727680\pi$
$390$	0	0
$391$	−10.4493	−0.528442
$392$	0	0
$393$	−25.4536	−1.28397
$394$	0	0
$395$	8.10003	0.407557
$396$	0	0
$397$	16.1397	0.810027	0.405014	−	0.914311i	$-0.367267\pi$
$397$	16.1397	0.810027	0.405014	+	0.914311i	$0.367267\pi$
$398$	0	0
$399$	−7.53473	−0.377208
$400$	0	0
$401$	30.4209	1.51915	0.759573	−	0.650422i	$-0.225408\pi$
$401$	30.4209	1.51915	0.759573	+	0.650422i	$0.225408\pi$
$402$	0	0
$403$	−27.7059	−1.38013
$404$	0	0
$405$	10.8480	0.539040
$406$	0	0
$407$	−10.8533	−0.537976
$408$	0	0
$409$	−4.39347	−0.217243	−0.108622	−	0.994083i	$-0.534644\pi$
$409$	−4.39347	−0.217243	−0.108622	+	0.994083i	$0.534644\pi$
$410$	0	0
$411$	15.0317	0.741461
$412$	0	0
$413$	−0.651924	−0.0320791
$414$	0	0
$415$	12.0981	0.593874
$416$	0	0
$417$	−4.53178	−0.221922
$418$	0	0
$419$	10.2055	0.498572	0.249286	−	0.968430i	$-0.419804\pi$
$419$	10.2055	0.498572	0.249286	+	0.968430i	$0.419804\pi$
$420$	0	0
$421$	−34.7509	−1.69366	−0.846828	−	0.531868i	$-0.821490\pi$
$421$	−34.7509	−1.69366	−0.846828	+	0.531868i	$0.821490\pi$
$422$	0	0
$423$	−9.57513	−0.465559
$424$	0	0
$425$	−1.81370	−0.0879772
$426$	0	0
$427$	0.841865	0.0407407
$428$	0	0
$429$	−24.8841	−1.20141
$430$	0	0
$431$	19.4464	0.936700	0.468350	−	0.883543i	$-0.344849\pi$
$431$	19.4464	0.936700	0.468350	+	0.883543i	$0.344849\pi$
$432$	0	0
$433$	26.6731	1.28183	0.640913	−	0.767613i	$-0.278556\pi$
$433$	26.6731	1.28183	0.640913	+	0.767613i	$0.278556\pi$
$434$	0	0
$435$	1.12725	0.0540473
$436$	0	0
$437$	−22.0781	−1.05614
$438$	0	0
$439$	−13.9452	−0.665570	−0.332785	−	0.943003i	$-0.607988\pi$
$439$	−13.9452	−0.665570	−0.332785	+	0.943003i	$0.607988\pi$
$440$	0	0
$441$	0.865935	0.0412350
$442$	0	0
$443$	19.6521	0.933701	0.466851	−	0.884336i	$-0.345389\pi$
$443$	19.6521	0.933701	0.466851	+	0.884336i	$0.345389\pi$
$444$	0	0
$445$	16.6323	0.788447
$446$	0	0
$447$	−19.5254	−0.923521
$448$	0	0
$449$	−10.9642	−0.517431	−0.258715	−	0.965954i	$-0.583299\pi$
$449$	−10.9642	−0.517431	−0.258715	+	0.965954i	$0.583299\pi$
$450$	0	0
$451$	−15.8112	−0.744521
$452$	0	0
$453$	24.9547	1.17248
$454$	0	0
$455$	5.77519	0.270745
$456$	0	0
$457$	−32.4974	−1.52017	−0.760083	−	0.649826i	$-0.774842\pi$
$457$	−32.4974	−1.52017	−0.760083	+	0.649826i	$0.774842\pi$
$458$	0	0
$459$	7.61026	0.355216
$460$	0	0
$461$	35.5772	1.65700	0.828499	−	0.559991i	$-0.189195\pi$
$461$	35.5772	1.65700	0.828499	+	0.559991i	$0.189195\pi$
$462$	0	0
$463$	26.4623	1.22981	0.614904	−	0.788602i	$-0.289195\pi$
$463$	26.4623	1.22981	0.614904	+	0.788602i	$0.289195\pi$
$464$	0	0
$465$	−9.43266	−0.437429
$466$	0	0
$467$	−15.7765	−0.730047	−0.365024	−	0.930998i	$-0.618939\pi$
$467$	−15.7765	−0.730047	−0.365024	+	0.930998i	$0.618939\pi$
$468$	0	0
$469$	4.14510	0.191403
$470$	0	0
$471$	−0.825332	−0.0380293
$472$	0	0
$473$	−2.19143	−0.100762
$474$	0	0
$475$	−3.83213	−0.175830
$476$	0	0
$477$	4.99159	0.228549
$478$	0	0
$479$	10.0699	0.460104	0.230052	−	0.973178i	$-0.426110\pi$
$479$	10.0699	0.460104	0.230052	+	0.973178i	$0.426110\pi$
$480$	0	0
$481$	28.6021	1.30414
$482$	0	0
$483$	11.3279	0.515436
$484$	0	0
$485$	8.76832	0.398149
$486$	0	0
$487$	−30.8774	−1.39919	−0.699594	−	0.714540i	$-0.746636\pi$
$487$	−30.8774	−1.39919	−0.699594	+	0.714540i	$0.746636\pi$
$488$	0	0
$489$	−10.0083	−0.452590
$490$	0	0
$491$	20.6209	0.930609	0.465305	−	0.885151i	$-0.345945\pi$
$491$	20.6209	0.930609	0.465305	+	0.885151i	$0.345945\pi$
$492$	0	0
$493$	1.03981	0.0468309
$494$	0	0
$495$	−1.89764	−0.0852925
$496$	0	0
$497$	−4.87383	−0.218621
$498$	0	0
$499$	17.7962	0.796665	0.398333	−	0.917241i	$-0.369589\pi$
$499$	17.7962	0.796665	0.398333	+	0.917241i	$0.369589\pi$
$500$	0	0
$501$	−15.7747	−0.704764
$502$	0	0
$503$	0.341244	0.0152153	0.00760766	−	0.999971i	$-0.497578\pi$
$503$	0.341244	0.0152153	0.00760766	+	0.999971i	$0.497578\pi$
$504$	0	0
$505$	4.65380	0.207092
$506$	0	0
$507$	40.0176	1.77724
$508$	0	0
$509$	8.90859	0.394866	0.197433	−	0.980316i	$-0.436739\pi$
$509$	8.90859	0.394866	0.197433	+	0.980316i	$0.436739\pi$
$510$	0	0
$511$	11.0752	0.489940
$512$	0	0
$513$	16.0796	0.709932
$514$	0	0
$515$	10.6882	0.470979
$516$	0	0
$517$	−24.2319	−1.06572
$518$	0	0
$519$	9.84339	0.432077
$520$	0	0
$521$	30.8074	1.34970	0.674849	−	0.737956i	$-0.264209\pi$
$521$	30.8074	1.34970	0.674849	+	0.737956i	$0.264209\pi$
$522$	0	0
$523$	−20.6500	−0.902960	−0.451480	−	0.892281i	$-0.649104\pi$
$523$	−20.6500	−0.902960	−0.451480	+	0.892281i	$0.649104\pi$
$524$	0	0
$525$	1.96620	0.0858119
$526$	0	0
$527$	−8.70104	−0.379023
$528$	0	0
$529$	10.1927	0.443160
$530$	0	0
$531$	−0.564524	−0.0244982
$532$	0	0
$533$	41.6680	1.80484
$534$	0	0
$535$	−6.66096	−0.287978
$536$	0	0
$537$	−28.2275	−1.21811
$538$	0	0
$539$	2.19143	0.0943918
$540$	0	0
$541$	23.3455	1.00370	0.501850	−	0.864955i	$-0.332653\pi$
$541$	23.3455	1.00370	0.501850	+	0.864955i	$0.332653\pi$
$542$	0	0
$543$	2.51905	0.108103
$544$	0	0
$545$	−7.45077	−0.319156
$546$	0	0
$547$	−15.3839	−0.657770	−0.328885	−	0.944370i	$-0.606673\pi$
$547$	−15.3839	−0.657770	−0.328885	+	0.944370i	$0.606673\pi$
$548$	0	0
$549$	0.729000	0.0311130
$550$	0	0
$551$	2.19701	0.0935957
$552$	0	0
$553$	−8.10003	−0.344449
$554$	0	0
$555$	9.73776	0.413345
$556$	0	0
$557$	9.31065	0.394505	0.197253	−	0.980353i	$-0.436798\pi$
$557$	9.31065	0.394505	0.197253	+	0.980353i	$0.436798\pi$
$558$	0	0
$559$	5.77519	0.244264
$560$	0	0
$561$	−7.81484	−0.329943
$562$	0	0
$563$	16.7667	0.706632	0.353316	−	0.935504i	$-0.385054\pi$
$563$	16.7667	0.706632	0.353316	+	0.935504i	$0.385054\pi$
$564$	0	0
$565$	−4.39854	−0.185048
$566$	0	0
$567$	−10.8480	−0.455572
$568$	0	0
$569$	−1.77924	−0.0745896	−0.0372948	−	0.999304i	$-0.511874\pi$
$569$	−1.77924	−0.0745896	−0.0372948	+	0.999304i	$0.511874\pi$
$570$	0	0
$571$	−40.5822	−1.69831	−0.849157	−	0.528141i	$-0.822889\pi$
$571$	−40.5822	−1.69831	−0.849157	+	0.528141i	$0.822889\pi$
$572$	0	0
$573$	14.4285	0.602760
$574$	0	0
$575$	5.76131	0.240263
$576$	0	0
$577$	−13.9641	−0.581333	−0.290667	−	0.956824i	$-0.593877\pi$
$577$	−13.9641	−0.581333	−0.290667	+	0.956824i	$0.593877\pi$
$578$	0	0
$579$	25.0996	1.04310
$580$	0	0
$581$	−12.0981	−0.501915
$582$	0	0
$583$	12.6323	0.523176
$584$	0	0
$585$	5.00093	0.206763
$586$	0	0
$587$	25.1679	1.03879	0.519395	−	0.854534i	$-0.326157\pi$
$587$	25.1679	1.03879	0.519395	+	0.854534i	$0.326157\pi$
$588$	0	0
$589$	−18.3843	−0.757513
$590$	0	0
$591$	27.7424	1.14117
$592$	0	0
$593$	−15.7803	−0.648021	−0.324010	−	0.946054i	$-0.605031\pi$
$593$	−15.7803	−0.648021	−0.324010	+	0.946054i	$0.605031\pi$
$594$	0	0
$595$	1.81370	0.0743543
$596$	0	0
$597$	43.3977	1.77615
$598$	0	0
$599$	−6.61352	−0.270221	−0.135111	−	0.990831i	$-0.543139\pi$
$599$	−6.61352	−0.270221	−0.135111	+	0.990831i	$0.543139\pi$
$600$	0	0
$601$	−32.1973	−1.31335	−0.656677	−	0.754172i	$-0.728039\pi$
$601$	−32.1973	−1.31335	−0.656677	+	0.754172i	$0.728039\pi$
$602$	0	0
$603$	3.58939	0.146171
$604$	0	0
$605$	6.19761	0.251969
$606$	0	0
$607$	−13.8793	−0.563342	−0.281671	−	0.959511i	$-0.590889\pi$
$607$	−13.8793	−0.563342	−0.281671	+	0.959511i	$0.590889\pi$
$608$	0	0
$609$	−1.12725	−0.0456783
$610$	0	0
$611$	63.8595	2.58348
$612$	0	0
$613$	−39.4165	−1.59202	−0.796008	−	0.605286i	$-0.793059\pi$
$613$	−39.4165	−1.59202	−0.796008	+	0.605286i	$0.793059\pi$
$614$	0	0
$615$	14.1861	0.572040
$616$	0	0
$617$	−1.67001	−0.0672319	−0.0336159	−	0.999435i	$-0.510702\pi$
$617$	−1.67001	−0.0672319	−0.0336159	+	0.999435i	$0.510702\pi$
$618$	0	0
$619$	8.37283	0.336533	0.168266	−	0.985742i	$-0.446183\pi$
$619$	8.37283	0.336533	0.168266	+	0.985742i	$0.446183\pi$
$620$	0	0
$621$	−24.1744	−0.970086
$622$	0	0
$623$	−16.6323	−0.666359
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−16.5119	−0.659421
$628$	0	0
$629$	8.98247	0.358155
$630$	0	0
$631$	33.0576	1.31600	0.658000	−	0.753018i	$-0.271403\pi$
$631$	33.0576	1.31600	0.658000	+	0.753018i	$0.271403\pi$
$632$	0	0
$633$	−12.1150	−0.481528
$634$	0	0
$635$	10.8845	0.431940
$636$	0	0
$637$	−5.77519	−0.228821
$638$	0	0
$639$	−4.22042	−0.166957
$640$	0	0
$641$	−40.7193	−1.60832	−0.804158	−	0.594416i	$-0.797383\pi$
$641$	−40.7193	−1.60832	−0.804158	+	0.594416i	$0.797383\pi$
$642$	0	0
$643$	34.2373	1.35019	0.675094	−	0.737732i	$-0.264103\pi$
$643$	34.2373	1.35019	0.675094	+	0.737732i	$0.264103\pi$
$644$	0	0
$645$	1.96620	0.0774190
$646$	0	0
$647$	−21.7533	−0.855211	−0.427605	−	0.903965i	$-0.640643\pi$
$647$	−21.7533	−0.855211	−0.427605	+	0.903965i	$0.640643\pi$
$648$	0	0
$649$	−1.42865	−0.0560794
$650$	0	0
$651$	9.43266	0.369695
$652$	0	0
$653$	23.2305	0.909079	0.454539	−	0.890727i	$-0.349804\pi$
$653$	23.2305	0.909079	0.454539	+	0.890727i	$0.349804\pi$
$654$	0	0
$655$	12.9456	0.505827
$656$	0	0
$657$	9.59043	0.374158
$658$	0	0
$659$	−22.5863	−0.879836	−0.439918	−	0.898038i	$-0.644993\pi$
$659$	−22.5863	−0.879836	−0.439918	+	0.898038i	$0.644993\pi$
$660$	0	0
$661$	34.3146	1.33468	0.667341	−	0.744752i	$-0.267432\pi$
$661$	34.3146	1.33468	0.667341	+	0.744752i	$0.267432\pi$
$662$	0	0
$663$	20.5948	0.799836
$664$	0	0
$665$	3.83213	0.148604
$666$	0	0
$667$	−3.30303	−0.127894
$668$	0	0
$669$	−12.0082	−0.464262
$670$	0	0
$671$	1.84489	0.0712213
$672$	0	0
$673$	−20.3195	−0.783261	−0.391630	−	0.920123i	$-0.628089\pi$
$673$	−20.3195	−0.783261	−0.391630	+	0.920123i	$0.628089\pi$
$674$	0	0
$675$	−4.19600	−0.161504
$676$	0	0
$677$	−0.490312	−0.0188442	−0.00942212	−	0.999956i	$-0.502999\pi$
$677$	−0.490312	−0.0188442	−0.00942212	+	0.999956i	$0.502999\pi$
$678$	0	0
$679$	−8.76832	−0.336497
$680$	0	0
$681$	17.8831	0.685282
$682$	0	0
$683$	12.2532	0.468854	0.234427	−	0.972134i	$-0.424679\pi$
$683$	12.2532	0.468854	0.234427	+	0.972134i	$0.424679\pi$
$684$	0	0
$685$	−7.64508	−0.292104
$686$	0	0
$687$	−30.9575	−1.18110
$688$	0	0
$689$	−33.2905	−1.26827
$690$	0	0
$691$	−20.5774	−0.782800	−0.391400	−	0.920221i	$-0.628009\pi$
$691$	−20.5774	−0.782800	−0.391400	+	0.920221i	$0.628009\pi$
$692$	0	0
$693$	1.89764	0.0720853
$694$	0	0
$695$	2.30484	0.0874277
$696$	0	0
$697$	13.0858	0.495661
$698$	0	0
$699$	20.0303	0.757617
$700$	0	0
$701$	9.22651	0.348481	0.174240	−	0.984703i	$-0.444253\pi$
$701$	9.22651	0.348481	0.174240	+	0.984703i	$0.444253\pi$
$702$	0	0
$703$	18.9789	0.715805
$704$	0	0
$705$	21.7414	0.818827
$706$	0	0
$707$	−4.65380	−0.175024
$708$	0	0
$709$	12.0565	0.452791	0.226395	−	0.974036i	$-0.427306\pi$
$709$	12.0565	0.452791	0.226395	+	0.974036i	$0.427306\pi$
$710$	0	0
$711$	−7.01410	−0.263049
$712$	0	0
$713$	27.6394	1.03510
$714$	0	0
$715$	12.6559	0.473305
$716$	0	0
$717$	20.9418	0.782086
$718$	0	0
$719$	−7.39452	−0.275769	−0.137885	−	0.990448i	$-0.544030\pi$
$719$	−7.39452	−0.275769	−0.137885	+	0.990448i	$0.544030\pi$
$720$	0	0
$721$	−10.6882	−0.398050
$722$	0	0
$723$	−20.6859	−0.769318
$724$	0	0
$725$	−0.573312	−0.0212923
$726$	0	0
$727$	27.8317	1.03222	0.516110	−	0.856522i	$-0.327380\pi$
$727$	27.8317	1.03222	0.516110	+	0.856522i	$0.327380\pi$
$728$	0	0
$729$	15.3568	0.568772
$730$	0	0
$731$	1.81370	0.0670820
$732$	0	0
$733$	0.918365	0.0339206	0.0169603	−	0.999856i	$-0.494601\pi$
$733$	0.918365	0.0339206	0.0169603	+	0.999856i	$0.494601\pi$
$734$	0	0
$735$	−1.96620	−0.0725243
$736$	0	0
$737$	9.08372	0.334603
$738$	0	0
$739$	−1.24893	−0.0459425	−0.0229713	−	0.999736i	$-0.507313\pi$
$739$	−1.24893	−0.0459425	−0.0229713	+	0.999736i	$0.507313\pi$
$740$	0	0
$741$	43.5145	1.59854
$742$	0	0
$743$	28.0885	1.03047	0.515233	−	0.857050i	$-0.327705\pi$
$743$	28.0885	1.03047	0.515233	+	0.857050i	$0.327705\pi$
$744$	0	0
$745$	9.93055	0.363827
$746$	0	0
$747$	−10.4762	−0.383304
$748$	0	0
$749$	6.66096	0.243386
$750$	0	0
$751$	4.85755	0.177254	0.0886272	−	0.996065i	$-0.471752\pi$
$751$	4.85755	0.177254	0.0886272	+	0.996065i	$0.471752\pi$
$752$	0	0
$753$	−40.0158	−1.45826
$754$	0	0
$755$	−12.6919	−0.461905
$756$	0	0
$757$	4.91023	0.178465	0.0892326	−	0.996011i	$-0.471559\pi$
$757$	4.91023	0.178465	0.0892326	+	0.996011i	$0.471559\pi$
$758$	0	0
$759$	24.8243	0.901065
$760$	0	0
$761$	−2.05197	−0.0743839	−0.0371919	−	0.999308i	$-0.511841\pi$
$761$	−2.05197	−0.0743839	−0.0371919	+	0.999308i	$0.511841\pi$
$762$	0	0
$763$	7.45077	0.269736
$764$	0	0
$765$	1.57054	0.0567831
$766$	0	0
$767$	3.76498	0.135946
$768$	0	0
$769$	−11.6648	−0.420642	−0.210321	−	0.977632i	$-0.567451\pi$
$769$	−11.6648	−0.420642	−0.210321	+	0.977632i	$0.567451\pi$
$770$	0	0
$771$	−6.11786	−0.220329
$772$	0	0
$773$	32.9163	1.18392	0.591958	−	0.805969i	$-0.298355\pi$
$773$	32.9163	1.18392	0.591958	+	0.805969i	$0.298355\pi$
$774$	0	0
$775$	4.79741	0.172328
$776$	0	0
$777$	−9.73776	−0.349340
$778$	0	0
$779$	27.6488	0.990623
$780$	0	0
$781$	−10.6807	−0.382185
$782$	0	0
$783$	2.40562	0.0859697
$784$	0	0
$785$	0.419760	0.0149819
$786$	0	0
$787$	−38.4250	−1.36970	−0.684851	−	0.728683i	$-0.740133\pi$
$787$	−38.4250	−1.36970	−0.684851	+	0.728683i	$0.740133\pi$
$788$	0	0
$789$	−20.2655	−0.721471
$790$	0	0
$791$	4.39854	0.156394
$792$	0	0
$793$	−4.86193	−0.172652
$794$	0	0
$795$	−11.3339	−0.401974
$796$	0	0
$797$	40.0924	1.42015	0.710073	−	0.704128i	$-0.248662\pi$
$797$	40.0924	1.42015	0.710073	+	0.704128i	$0.248662\pi$
$798$	0	0
$799$	20.0551	0.709497
$800$	0	0
$801$	−14.4025	−0.508887
$802$	0	0
$803$	24.2707	0.856493
$804$	0	0
$805$	−5.76131	−0.203059
$806$	0	0
$807$	−19.4536	−0.684800
$808$	0	0
$809$	−29.7239	−1.04504	−0.522518	−	0.852628i	$-0.675007\pi$
$809$	−29.7239	−1.04504	−0.522518	+	0.852628i	$0.675007\pi$
$810$	0	0
$811$	−22.9705	−0.806602	−0.403301	−	0.915067i	$-0.632137\pi$
$811$	−22.9705	−0.806602	−0.403301	+	0.915067i	$0.632137\pi$
$812$	0	0
$813$	6.51622	0.228534
$814$	0	0
$815$	5.09016	0.178301
$816$	0	0
$817$	3.83213	0.134069
$818$	0	0
$819$	−5.00093	−0.174747
$820$	0	0
$821$	19.1535	0.668460	0.334230	−	0.942491i	$-0.391524\pi$
$821$	19.1535	0.668460	0.334230	+	0.942491i	$0.391524\pi$
$822$	0	0
$823$	32.9821	1.14968	0.574841	−	0.818265i	$-0.305064\pi$
$823$	32.9821	1.14968	0.574841	+	0.818265i	$0.305064\pi$
$824$	0	0
$825$	4.30879	0.150013
$826$	0	0
$827$	−30.5155	−1.06113	−0.530563	−	0.847645i	$-0.678019\pi$
$827$	−30.5155	−1.06113	−0.530563	+	0.847645i	$0.678019\pi$
$828$	0	0
$829$	−16.3230	−0.566921	−0.283460	−	0.958984i	$-0.591482\pi$
$829$	−16.3230	−0.566921	−0.283460	+	0.958984i	$0.591482\pi$
$830$	0	0
$831$	8.94017	0.310131
$832$	0	0
$833$	−1.81370	−0.0628408
$834$	0	0
$835$	8.02297	0.277646
$836$	0	0
$837$	−20.1299	−0.695792
$838$	0	0
$839$	−53.2886	−1.83973	−0.919864	−	0.392237i	$-0.871701\pi$
$839$	−53.2886	−1.83973	−0.919864	+	0.392237i	$0.871701\pi$
$840$	0	0
$841$	−28.6713	−0.988666
$842$	0	0
$843$	33.3282	1.14788
$844$	0	0
$845$	−20.3528	−0.700156
$846$	0	0
$847$	−6.19761	−0.212953
$848$	0	0
$849$	−12.0422	−0.413287
$850$	0	0
$851$	−28.5333	−0.978111
$852$	0	0
$853$	−6.62411	−0.226805	−0.113403	−	0.993549i	$-0.536175\pi$
$853$	−6.62411	−0.226805	−0.113403	+	0.993549i	$0.536175\pi$
$854$	0	0
$855$	3.31838	0.113486
$856$	0	0
$857$	49.8890	1.70418	0.852088	−	0.523398i	$-0.175336\pi$
$857$	49.8890	1.70418	0.852088	+	0.523398i	$0.175336\pi$
$858$	0	0
$859$	−46.1228	−1.57369	−0.786845	−	0.617151i	$-0.788287\pi$
$859$	−46.1228	−1.57369	−0.786845	+	0.617151i	$0.788287\pi$
$860$	0	0
$861$	−14.1861	−0.483462
$862$	0	0
$863$	−38.4907	−1.31024	−0.655120	−	0.755525i	$-0.727382\pi$
$863$	−38.4907	−1.31024	−0.655120	+	0.755525i	$0.727382\pi$
$864$	0	0
$865$	−5.00631	−0.170220
$866$	0	0
$867$	−26.9576	−0.915527
$868$	0	0
$869$	−17.7507	−0.602151
$870$	0	0
$871$	−23.9387	−0.811133
$872$	0	0
$873$	−7.59279	−0.256977
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−11.8194	−0.399112	−0.199556	−	0.979886i	$-0.563950\pi$
$877$	−11.8194	−0.399112	−0.199556	+	0.979886i	$0.563950\pi$
$878$	0	0
$879$	−23.1829	−0.781939
$880$	0	0
$881$	−10.4320	−0.351462	−0.175731	−	0.984438i	$-0.556229\pi$
$881$	−10.4320	−0.351462	−0.175731	+	0.984438i	$0.556229\pi$
$882$	0	0
$883$	−19.3155	−0.650019	−0.325009	−	0.945711i	$-0.605367\pi$
$883$	−19.3155	−0.650019	−0.325009	+	0.945711i	$0.605367\pi$
$884$	0	0
$885$	1.28181	0.0430876
$886$	0	0
$887$	50.5114	1.69601	0.848004	−	0.529990i	$-0.177804\pi$
$887$	50.5114	1.69601	0.848004	+	0.529990i	$0.177804\pi$
$888$	0	0
$889$	−10.8845	−0.365056
$890$	0	0
$891$	−23.7726	−0.796412
$892$	0	0
$893$	42.3741	1.41799
$894$	0	0
$895$	14.3564	0.479881
$896$	0	0
$897$	−65.4206	−2.18433
$898$	0	0
$899$	−2.75042	−0.0917315
$900$	0	0
$901$	−10.4549	−0.348302
$902$	0	0
$903$	−1.96620	−0.0654310
$904$	0	0
$905$	−1.28118	−0.0425878
$906$	0	0
$907$	37.4059	1.24204	0.621020	−	0.783794i	$-0.286718\pi$
$907$	37.4059	1.24204	0.621020	+	0.783794i	$0.286718\pi$
$908$	0	0
$909$	−4.02989	−0.133663
$910$	0	0
$911$	57.9960	1.92149	0.960746	−	0.277429i	$-0.0894824\pi$
$911$	57.9960	1.92149	0.960746	+	0.277429i	$0.0894824\pi$
$912$	0	0
$913$	−26.5122	−0.877427
$914$	0	0
$915$	−1.65527	−0.0547216
$916$	0	0
$917$	−12.9456	−0.427502
$918$	0	0
$919$	6.29207	0.207556	0.103778	−	0.994600i	$-0.466907\pi$
$919$	6.29207	0.207556	0.103778	+	0.994600i	$0.466907\pi$
$920$	0	0
$921$	−27.1883	−0.895886
$922$	0	0
$923$	28.1473	0.926479
$924$	0	0
$925$	−4.95258	−0.162840
$926$	0	0
$927$	−9.25530	−0.303984
$928$	0	0
$929$	37.2417	1.22186	0.610930	−	0.791685i	$-0.290796\pi$
$929$	37.2417	1.22186	0.610930	+	0.791685i	$0.290796\pi$
$930$	0	0
$931$	−3.83213	−0.125593
$932$	0	0
$933$	7.58254	0.248241
$934$	0	0
$935$	3.97460	0.129983
$936$	0	0
$937$	−1.44747	−0.0472868	−0.0236434	−	0.999720i	$-0.507527\pi$
$937$	−1.44747	−0.0472868	−0.0236434	+	0.999720i	$0.507527\pi$
$938$	0	0
$939$	54.9315	1.79262
$940$	0	0
$941$	−17.7983	−0.580209	−0.290104	−	0.956995i	$-0.593690\pi$
$941$	−17.7983	−0.580209	−0.290104	+	0.956995i	$0.593690\pi$
$942$	0	0
$943$	−41.5679	−1.35364
$944$	0	0
$945$	4.19600	0.136496
$946$	0	0
$947$	7.44688	0.241991	0.120995	−	0.992653i	$-0.461391\pi$
$947$	7.44688	0.241991	0.120995	+	0.992653i	$0.461391\pi$
$948$	0	0
$949$	−63.9616	−2.07628
$950$	0	0
$951$	10.6181	0.344315
$952$	0	0
$953$	17.3143	0.560865	0.280433	−	0.959874i	$-0.409522\pi$
$953$	17.3143	0.560865	0.280433	+	0.959874i	$0.409522\pi$
$954$	0	0
$955$	−7.33829	−0.237461
$956$	0	0
$957$	−2.47028	−0.0798530
$958$	0	0
$959$	7.64508	0.246873
$960$	0	0
$961$	−7.98484	−0.257575
$962$	0	0
$963$	5.76795	0.185870
$964$	0	0
$965$	−12.7656	−0.410938
$966$	0	0
$967$	−24.7965	−0.797402	−0.398701	−	0.917081i	$-0.630539\pi$
$967$	−24.7965	−0.797402	−0.398701	+	0.917081i	$0.630539\pi$
$968$	0	0
$969$	13.6657	0.439006
$970$	0	0
$971$	20.0464	0.643319	0.321659	−	0.946855i	$-0.395759\pi$
$971$	20.0464	0.643319	0.321659	+	0.946855i	$0.395759\pi$
$972$	0	0
$973$	−2.30484	−0.0738899
$974$	0	0
$975$	−11.3552	−0.363656
$976$	0	0
$977$	−28.9610	−0.926544	−0.463272	−	0.886216i	$-0.653325\pi$
$977$	−28.9610	−0.926544	−0.463272	+	0.886216i	$0.653325\pi$
$978$	0	0
$979$	−36.4486	−1.16490
$980$	0	0
$981$	6.45188	0.205993
$982$	0	0
$983$	35.8256	1.14266	0.571330	−	0.820720i	$-0.306428\pi$
$983$	35.8256	1.14266	0.571330	+	0.820720i	$0.306428\pi$
$984$	0	0
$985$	−14.1097	−0.449572
$986$	0	0
$987$	−21.7414	−0.692035
$988$	0	0
$989$	−5.76131	−0.183199
$990$	0	0
$991$	46.0117	1.46161	0.730805	−	0.682586i	$-0.239145\pi$
$991$	46.0117	1.46161	0.730805	+	0.682586i	$0.239145\pi$
$992$	0	0
$993$	−32.4994	−1.03134
$994$	0	0
$995$	−22.0719	−0.699726
$996$	0	0
$997$	−55.9371	−1.77154	−0.885772	−	0.464120i	$-0.846371\pi$
$997$	−55.9371	−1.77154	−0.885772	+	0.464120i	$0.846371\pi$
$998$	0	0
$999$	20.7810	0.657482

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6020.2.a.f.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6020.2.a.f.1.7	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.96620 q^{3} -1.00000 q^{5} +1.00000 q^{7} +0.865935 q^{9} +O(q^{10})\)	\(q+1.96620 q^{3} -1.00000 q^{5} +1.00000 q^{7} +0.865935 q^{9} +2.19143 q^{11} -5.77519 q^{13} -1.96620 q^{15} -1.81370 q^{17} -3.83213 q^{19} +1.96620 q^{21} +5.76131 q^{23} +1.00000 q^{25} -4.19600 q^{27} -0.573312 q^{29} +4.79741 q^{31} +4.30879 q^{33} -1.00000 q^{35} -4.95258 q^{37} -11.3552 q^{39} -7.21500 q^{41} -1.00000 q^{43} -0.865935 q^{45} -11.0576 q^{47} +1.00000 q^{49} -3.56608 q^{51} +5.76440 q^{53} -2.19143 q^{55} -7.53473 q^{57} -0.651924 q^{59} +0.841865 q^{61} +0.865935 q^{63} +5.77519 q^{65} +4.14510 q^{67} +11.3279 q^{69} -4.87383 q^{71} +11.0752 q^{73} +1.96620 q^{75} +2.19143 q^{77} -8.10003 q^{79} -10.8480 q^{81} -12.0981 q^{83} +1.81370 q^{85} -1.12725 q^{87} -16.6323 q^{89} -5.77519 q^{91} +9.43266 q^{93} +3.83213 q^{95} -8.76832 q^{97} +1.89764 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9	\( 8 q - 5 q^{3} - 8 q^{5} + 8 q^{7} + 11 q^{9} + 6 q^{11} - 13 q^{13} + 5 q^{15} - 2 q^{17} - 14 q^{19} - 5 q^{21} + 6 q^{23} + 8 q^{25} - 11 q^{27} - 7 q^{29} - 18 q^{31} - q^{33} - 8 q^{35} + 2 q^{37} + 9 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - q^{47} + 8 q^{49} - 19 q^{51} + 5 q^{53} - 6 q^{55} - 4 q^{57} - 12 q^{59} - 23 q^{61} + 11 q^{63} + 13 q^{65} + 8 q^{67} - 18 q^{69} + 20 q^{71} - 4 q^{73} - 5 q^{75} + 6 q^{77} + 24 q^{79} - 8 q^{81} - 14 q^{83} + 2 q^{85} + 10 q^{87} - 21 q^{89} - 13 q^{91} + q^{93} + 14 q^{95} + 7 q^{97} + 4 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 8 * q^5 + 8 * q^7 + 11 * q^9 + 6 * q^11 - 13 * q^13 + 5 * q^15 - 2 * q^17 - 14 * q^19 - 5 * q^21 + 6 * q^23 + 8 * q^25 - 11 * q^27 - 7 * q^29 - 18 * q^31 - q^33 - 8 * q^35 + 2 * q^37 + 9 * q^39 - 18 * q^41 - 8 * q^43 - 11 * q^45 - q^47 + 8 * q^49 - 19 * q^51 + 5 * q^53 - 6 * q^55 - 4 * q^57 - 12 * q^59 - 23 * q^61 + 11 * q^63 + 13 * q^65 + 8 * q^67 - 18 * q^69 + 20 * q^71 - 4 * q^73 - 5 * q^75 + 6 * q^77 + 24 * q^79 - 8 * q^81 - 14 * q^83 + 2 * q^85 + 10 * q^87 - 21 * q^89 - 13 * q^91 + q^93 + 14 * q^95 + 7 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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