LMFDB - Embedded newform 4020.2.a.i.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, 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("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 21x^{5} - 12x^{4} + 93x^{3} + 18x^{2} - 120x + 48 $ x^7 - 21x^5 - 12x^4 + 93x^3 + 18x^2 - 120*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-3.01567$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +4.86957 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +4.86957 q^{7} +1.00000 q^{9} +0.902547 q^{11} -0.581729 q^{13} +1.00000 q^{15} +4.01149 q^{17} -1.30402 q^{19} +4.86957 q^{21} +2.61420 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.25537 q^{29} +4.64605 q^{31} +0.902547 q^{33} +4.86957 q^{35} -6.64554 q^{37} -0.581729 q^{39} -10.9098 q^{41} -2.99531 q^{43} +1.00000 q^{45} -2.18621 q^{47} +16.7127 q^{49} +4.01149 q^{51} +2.97171 q^{53} +0.902547 q^{55} -1.30402 q^{57} +0.793753 q^{59} -14.7422 q^{61} +4.86957 q^{63} -0.581729 q^{65} -1.00000 q^{67} +2.61420 q^{69} +3.53140 q^{71} +6.74769 q^{73} +1.00000 q^{75} +4.39501 q^{77} +5.46330 q^{79} +1.00000 q^{81} -0.259232 q^{83} +4.01149 q^{85} +4.25537 q^{87} +12.3307 q^{89} -2.83277 q^{91} +4.64605 q^{93} -1.30402 q^{95} -9.07927 q^{97} +0.902547 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.86957	1.84052	0.920262	−	0.391302i	$-0.127975\pi$
$7$	4.86957	1.84052	0.920262	+	0.391302i	$0.127975\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.902547	0.272128	0.136064	−	0.990700i	$-0.456555\pi$
$11$	0.902547	0.272128	0.136064	+	0.990700i	$0.456555\pi$
$12$	0	0
$13$	−0.581729	−0.161342	−0.0806712	−	0.996741i	$-0.525706\pi$
$13$	−0.581729	−0.161342	−0.0806712	+	0.996741i	$0.525706\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.01149	0.972930	0.486465	−	0.873700i	$-0.338286\pi$
$17$	4.01149	0.972930	0.486465	+	0.873700i	$0.338286\pi$
$18$	0	0
$19$	−1.30402	−0.299164	−0.149582	−	0.988749i	$-0.547793\pi$
$19$	−1.30402	−0.299164	−0.149582	+	0.988749i	$0.547793\pi$
$20$	0	0
$21$	4.86957	1.06263
$22$	0	0
$23$	2.61420	0.545098	0.272549	−	0.962142i	$-0.412133\pi$
$23$	2.61420	0.545098	0.272549	+	0.962142i	$0.412133\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.25537	0.790203	0.395101	−	0.918637i	$-0.370709\pi$
$29$	4.25537	0.790203	0.395101	+	0.918637i	$0.370709\pi$
$30$	0	0
$31$	4.64605	0.834456	0.417228	−	0.908802i	$-0.363002\pi$
$31$	4.64605	0.834456	0.417228	+	0.908802i	$0.363002\pi$
$32$	0	0
$33$	0.902547	0.157113
$34$	0	0
$35$	4.86957	0.823108
$36$	0	0
$37$	−6.64554	−1.09252	−0.546260	−	0.837615i	$-0.683949\pi$
$37$	−6.64554	−1.09252	−0.546260	+	0.837615i	$0.683949\pi$
$38$	0	0
$39$	−0.581729	−0.0931511
$40$	0	0
$41$	−10.9098	−1.70382	−0.851909	−	0.523690i	$-0.824555\pi$
$41$	−10.9098	−1.70382	−0.851909	+	0.523690i	$0.824555\pi$
$42$	0	0
$43$	−2.99531	−0.456780	−0.228390	−	0.973570i	$-0.573346\pi$
$43$	−2.99531	−0.456780	−0.228390	+	0.973570i	$0.573346\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−2.18621	−0.318891	−0.159445	−	0.987207i	$-0.550971\pi$
$47$	−2.18621	−0.318891	−0.159445	+	0.987207i	$0.550971\pi$
$48$	0	0
$49$	16.7127	2.38753
$50$	0	0
$51$	4.01149	0.561721
$52$	0	0
$53$	2.97171	0.408196	0.204098	−	0.978950i	$-0.434574\pi$
$53$	2.97171	0.408196	0.204098	+	0.978950i	$0.434574\pi$
$54$	0	0
$55$	0.902547	0.121699
$56$	0	0
$57$	−1.30402	−0.172722
$58$	0	0
$59$	0.793753	0.103338	0.0516689	−	0.998664i	$-0.483546\pi$
$59$	0.793753	0.103338	0.0516689	+	0.998664i	$0.483546\pi$
$60$	0	0
$61$	−14.7422	−1.88755	−0.943773	−	0.330595i	$-0.892751\pi$
$61$	−14.7422	−1.88755	−0.943773	+	0.330595i	$0.892751\pi$
$62$	0	0
$63$	4.86957	0.613508
$64$	0	0
$65$	−0.581729	−0.0721546
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	2.61420	0.314712
$70$	0	0
$71$	3.53140	0.419100	0.209550	−	0.977798i	$-0.432800\pi$
$71$	3.53140	0.419100	0.209550	+	0.977798i	$0.432800\pi$
$72$	0	0
$73$	6.74769	0.789757	0.394879	−	0.918733i	$-0.370787\pi$
$73$	6.74769	0.789757	0.394879	+	0.918733i	$0.370787\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	4.39501	0.500858
$78$	0	0
$79$	5.46330	0.614669	0.307335	−	0.951602i	$-0.400563\pi$
$79$	5.46330	0.614669	0.307335	+	0.951602i	$0.400563\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.259232	−0.0284544	−0.0142272	−	0.999899i	$-0.504529\pi$
$83$	−0.259232	−0.0284544	−0.0142272	+	0.999899i	$0.504529\pi$
$84$	0	0
$85$	4.01149	0.435107
$86$	0	0
$87$	4.25537	0.456224
$88$	0	0
$89$	12.3307	1.30705	0.653525	−	0.756905i	$-0.273290\pi$
$89$	12.3307	1.30705	0.653525	+	0.756905i	$0.273290\pi$
$90$	0	0
$91$	−2.83277	−0.296955
$92$	0	0
$93$	4.64605	0.481773
$94$	0	0
$95$	−1.30402	−0.133790
$96$	0	0
$97$	−9.07927	−0.921860	−0.460930	−	0.887436i	$-0.652484\pi$
$97$	−9.07927	−0.921860	−0.460930	+	0.887436i	$0.652484\pi$
$98$	0	0
$99$	0.902547	0.0907094
$100$	0	0
$101$	6.39381	0.636208	0.318104	−	0.948056i	$-0.396954\pi$
$101$	6.39381	0.636208	0.318104	+	0.948056i	$0.396954\pi$
$102$	0	0
$103$	−16.8065	−1.65600	−0.827998	−	0.560731i	$-0.810520\pi$
$103$	−16.8065	−1.65600	−0.827998	+	0.560731i	$0.810520\pi$
$104$	0	0
$105$	4.86957	0.475221
$106$	0	0
$107$	−1.43097	−0.138337	−0.0691685	−	0.997605i	$-0.522035\pi$
$107$	−1.43097	−0.138337	−0.0691685	+	0.997605i	$0.522035\pi$
$108$	0	0
$109$	−9.53526	−0.913312	−0.456656	−	0.889643i	$-0.650953\pi$
$109$	−9.53526	−0.913312	−0.456656	+	0.889643i	$0.650953\pi$
$110$	0	0
$111$	−6.64554	−0.630767
$112$	0	0
$113$	10.1136	0.951408	0.475704	−	0.879606i	$-0.342193\pi$
$113$	10.1136	0.951408	0.475704	+	0.879606i	$0.342193\pi$
$114$	0	0
$115$	2.61420	0.243775
$116$	0	0
$117$	−0.581729	−0.0537808
$118$	0	0
$119$	19.5342	1.79070
$120$	0	0
$121$	−10.1854	−0.925946
$122$	0	0
$123$	−10.9098	−0.983699
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.94581	−0.261399	−0.130699	−	0.991422i	$-0.541722\pi$
$127$	−2.94581	−0.261399	−0.130699	+	0.991422i	$0.541722\pi$
$128$	0	0
$129$	−2.99531	−0.263722
$130$	0	0
$131$	3.61154	0.315541	0.157771	−	0.987476i	$-0.449569\pi$
$131$	3.61154	0.315541	0.157771	+	0.987476i	$0.449569\pi$
$132$	0	0
$133$	−6.35004	−0.550618
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−12.7757	−1.09150	−0.545749	−	0.837949i	$-0.683755\pi$
$137$	−12.7757	−1.09150	−0.545749	+	0.837949i	$0.683755\pi$
$138$	0	0
$139$	14.3344	1.21583	0.607914	−	0.794003i	$-0.292006\pi$
$139$	14.3344	1.21583	0.607914	+	0.794003i	$0.292006\pi$
$140$	0	0
$141$	−2.18621	−0.184112
$142$	0	0
$143$	−0.525037	−0.0439058
$144$	0	0
$145$	4.25537	0.353390
$146$	0	0
$147$	16.7127	1.37844
$148$	0	0
$149$	−13.3584	−1.09436	−0.547180	−	0.837015i	$-0.684299\pi$
$149$	−13.3584	−1.09436	−0.547180	+	0.837015i	$0.684299\pi$
$150$	0	0
$151$	7.93687	0.645894	0.322947	−	0.946417i	$-0.395327\pi$
$151$	7.93687	0.645894	0.322947	+	0.946417i	$0.395327\pi$
$152$	0	0
$153$	4.01149	0.324310
$154$	0	0
$155$	4.64605	0.373180
$156$	0	0
$157$	−12.9708	−1.03518	−0.517590	−	0.855628i	$-0.673171\pi$
$157$	−12.9708	−1.03518	−0.517590	+	0.855628i	$0.673171\pi$
$158$	0	0
$159$	2.97171	0.235672
$160$	0	0
$161$	12.7300	1.00327
$162$	0	0
$163$	−23.5564	−1.84508	−0.922539	−	0.385905i	$-0.873889\pi$
$163$	−23.5564	−1.84508	−0.922539	+	0.385905i	$0.873889\pi$
$164$	0	0
$165$	0.902547	0.0702632
$166$	0	0
$167$	−11.6408	−0.900789	−0.450394	−	0.892830i	$-0.648717\pi$
$167$	−11.6408	−0.900789	−0.450394	+	0.892830i	$0.648717\pi$
$168$	0	0
$169$	−12.6616	−0.973969
$170$	0	0
$171$	−1.30402	−0.0997212
$172$	0	0
$173$	7.16397	0.544666	0.272333	−	0.962203i	$-0.412205\pi$
$173$	7.16397	0.544666	0.272333	+	0.962203i	$0.412205\pi$
$174$	0	0
$175$	4.86957	0.368105
$176$	0	0
$177$	0.793753	0.0596621
$178$	0	0
$179$	−8.30426	−0.620689	−0.310345	−	0.950624i	$-0.600444\pi$
$179$	−8.30426	−0.620689	−0.310345	+	0.950624i	$0.600444\pi$
$180$	0	0
$181$	−11.9945	−0.891545	−0.445773	−	0.895146i	$-0.647071\pi$
$181$	−11.9945	−0.891545	−0.445773	+	0.895146i	$0.647071\pi$
$182$	0	0
$183$	−14.7422	−1.08977
$184$	0	0
$185$	−6.64554	−0.488590
$186$	0	0
$187$	3.62056	0.264761
$188$	0	0
$189$	4.86957	0.354209
$190$	0	0
$191$	−0.260316	−0.0188358	−0.00941789	−	0.999956i	$-0.502998\pi$
$191$	−0.260316	−0.0188358	−0.00941789	+	0.999956i	$0.502998\pi$
$192$	0	0
$193$	2.09344	0.150689	0.0753447	−	0.997158i	$-0.475994\pi$
$193$	2.09344	0.150689	0.0753447	+	0.997158i	$0.475994\pi$
$194$	0	0
$195$	−0.581729	−0.0416585
$196$	0	0
$197$	14.9652	1.06623	0.533115	−	0.846043i	$-0.321021\pi$
$197$	14.9652	1.06623	0.533115	+	0.846043i	$0.321021\pi$
$198$	0	0
$199$	14.3883	1.01996	0.509981	−	0.860186i	$-0.329653\pi$
$199$	14.3883	1.01996	0.509981	+	0.860186i	$0.329653\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	20.7218	1.45439
$204$	0	0
$205$	−10.9098	−0.761970
$206$	0	0
$207$	2.61420	0.181699
$208$	0	0
$209$	−1.17694	−0.0814108
$210$	0	0
$211$	19.7552	1.36000	0.680002	−	0.733211i	$-0.261979\pi$
$211$	19.7552	1.36000	0.680002	+	0.733211i	$0.261979\pi$
$212$	0	0
$213$	3.53140	0.241967
$214$	0	0
$215$	−2.99531	−0.204278
$216$	0	0
$217$	22.6243	1.53584
$218$	0	0
$219$	6.74769	0.455967
$220$	0	0
$221$	−2.33360	−0.156975
$222$	0	0
$223$	19.2085	1.28629	0.643147	−	0.765742i	$-0.277628\pi$
$223$	19.2085	1.28629	0.643147	+	0.765742i	$0.277628\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−7.26953	−0.482495	−0.241248	−	0.970464i	$-0.577557\pi$
$227$	−7.26953	−0.482495	−0.241248	+	0.970464i	$0.577557\pi$
$228$	0	0
$229$	−9.20742	−0.608443	−0.304222	−	0.952601i	$-0.598396\pi$
$229$	−9.20742	−0.608443	−0.304222	+	0.952601i	$0.598396\pi$
$230$	0	0
$231$	4.39501	0.289171
$232$	0	0
$233$	−10.3916	−0.680778	−0.340389	−	0.940285i	$-0.610559\pi$
$233$	−10.3916	−0.680778	−0.340389	+	0.940285i	$0.610559\pi$
$234$	0	0
$235$	−2.18621	−0.142612
$236$	0	0
$237$	5.46330	0.354879
$238$	0	0
$239$	−6.21202	−0.401822	−0.200911	−	0.979610i	$-0.564390\pi$
$239$	−6.21202	−0.401822	−0.200911	+	0.979610i	$0.564390\pi$
$240$	0	0
$241$	13.8765	0.893866	0.446933	−	0.894568i	$-0.352516\pi$
$241$	13.8765	0.893866	0.446933	+	0.894568i	$0.352516\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	16.7127	1.06774
$246$	0	0
$247$	0.758588	0.0482678
$248$	0	0
$249$	−0.259232	−0.0164281
$250$	0	0
$251$	3.39049	0.214006	0.107003	−	0.994259i	$-0.465875\pi$
$251$	3.39049	0.214006	0.107003	+	0.994259i	$0.465875\pi$
$252$	0	0
$253$	2.35943	0.148336
$254$	0	0
$255$	4.01149	0.251209
$256$	0	0
$257$	14.4877	0.903720	0.451860	−	0.892089i	$-0.350761\pi$
$257$	14.4877	0.903720	0.451860	+	0.892089i	$0.350761\pi$
$258$	0	0
$259$	−32.3609	−2.01081
$260$	0	0
$261$	4.25537	0.263401
$262$	0	0
$263$	25.1276	1.54944	0.774719	−	0.632306i	$-0.217892\pi$
$263$	25.1276	1.54944	0.774719	+	0.632306i	$0.217892\pi$
$264$	0	0
$265$	2.97171	0.182551
$266$	0	0
$267$	12.3307	0.754625
$268$	0	0
$269$	32.3638	1.97326	0.986629	−	0.162984i	$-0.0521119\pi$
$269$	32.3638	1.97326	0.986629	+	0.162984i	$0.0521119\pi$
$270$	0	0
$271$	−11.8524	−0.719981	−0.359991	−	0.932956i	$-0.617220\pi$
$271$	−11.8524	−0.719981	−0.359991	+	0.932956i	$0.617220\pi$
$272$	0	0
$273$	−2.83277	−0.171447
$274$	0	0
$275$	0.902547	0.0544256
$276$	0	0
$277$	18.9472	1.13843	0.569213	−	0.822190i	$-0.307248\pi$
$277$	18.9472	1.13843	0.569213	+	0.822190i	$0.307248\pi$
$278$	0	0
$279$	4.64605	0.278152
$280$	0	0
$281$	18.0110	1.07445	0.537224	−	0.843440i	$-0.319473\pi$
$281$	18.0110	1.07445	0.537224	+	0.843440i	$0.319473\pi$
$282$	0	0
$283$	1.47795	0.0878553	0.0439276	−	0.999035i	$-0.486013\pi$
$283$	1.47795	0.0878553	0.0439276	+	0.999035i	$0.486013\pi$
$284$	0	0
$285$	−1.30402	−0.0772437
$286$	0	0
$287$	−53.1258	−3.13592
$288$	0	0
$289$	−0.907933	−0.0534078
$290$	0	0
$291$	−9.07927	−0.532236
$292$	0	0
$293$	−18.1602	−1.06093	−0.530466	−	0.847706i	$-0.677983\pi$
$293$	−18.1602	−1.06093	−0.530466	+	0.847706i	$0.677983\pi$
$294$	0	0
$295$	0.793753	0.0462141
$296$	0	0
$297$	0.902547	0.0523711
$298$	0	0
$299$	−1.52075	−0.0879474
$300$	0	0
$301$	−14.5859	−0.840715
$302$	0	0
$303$	6.39381	0.367315
$304$	0	0
$305$	−14.7422	−0.844136
$306$	0	0
$307$	−3.89241	−0.222152	−0.111076	−	0.993812i	$-0.535430\pi$
$307$	−3.89241	−0.222152	−0.111076	+	0.993812i	$0.535430\pi$
$308$	0	0
$309$	−16.8065	−0.956090
$310$	0	0
$311$	1.10230	0.0625055	0.0312527	−	0.999512i	$-0.490050\pi$
$311$	1.10230	0.0625055	0.0312527	+	0.999512i	$0.490050\pi$
$312$	0	0
$313$	12.6573	0.715434	0.357717	−	0.933830i	$-0.383555\pi$
$313$	12.6573	0.715434	0.357717	+	0.933830i	$0.383555\pi$
$314$	0	0
$315$	4.86957	0.274369
$316$	0	0
$317$	−14.6581	−0.823281	−0.411641	−	0.911346i	$-0.635044\pi$
$317$	−14.6581	−0.823281	−0.411641	+	0.911346i	$0.635044\pi$
$318$	0	0
$319$	3.84067	0.215036
$320$	0	0
$321$	−1.43097	−0.0798689
$322$	0	0
$323$	−5.23108	−0.291065
$324$	0	0
$325$	−0.581729	−0.0322685
$326$	0	0
$327$	−9.53526	−0.527301
$328$	0	0
$329$	−10.6459	−0.586926
$330$	0	0
$331$	2.04174	0.112224	0.0561122	−	0.998424i	$-0.482130\pi$
$331$	2.04174	0.112224	0.0561122	+	0.998424i	$0.482130\pi$
$332$	0	0
$333$	−6.64554	−0.364174
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	29.2003	1.59064	0.795320	−	0.606190i	$-0.207303\pi$
$337$	29.2003	1.59064	0.795320	+	0.606190i	$0.207303\pi$
$338$	0	0
$339$	10.1136	0.549295
$340$	0	0
$341$	4.19328	0.227079
$342$	0	0
$343$	47.2967	2.55378
$344$	0	0
$345$	2.61420	0.140744
$346$	0	0
$347$	−9.82612	−0.527494	−0.263747	−	0.964592i	$-0.584958\pi$
$347$	−9.82612	−0.527494	−0.263747	+	0.964592i	$0.584958\pi$
$348$	0	0
$349$	−4.55042	−0.243579	−0.121789	−	0.992556i	$-0.538863\pi$
$349$	−4.55042	−0.243579	−0.121789	+	0.992556i	$0.538863\pi$
$350$	0	0
$351$	−0.581729	−0.0310504
$352$	0	0
$353$	36.9807	1.96828	0.984142	−	0.177383i	$-0.0567630\pi$
$353$	36.9807	1.96828	0.984142	+	0.177383i	$0.0567630\pi$
$354$	0	0
$355$	3.53140	0.187427
$356$	0	0
$357$	19.5342	1.03386
$358$	0	0
$359$	−16.1583	−0.852803	−0.426402	−	0.904534i	$-0.640219\pi$
$359$	−16.1583	−0.852803	−0.426402	+	0.904534i	$0.640219\pi$
$360$	0	0
$361$	−17.2995	−0.910501
$362$	0	0
$363$	−10.1854	−0.534595
$364$	0	0
$365$	6.74769	0.353190
$366$	0	0
$367$	−7.45628	−0.389214	−0.194607	−	0.980881i	$-0.562343\pi$
$367$	−7.45628	−0.389214	−0.194607	+	0.980881i	$0.562343\pi$
$368$	0	0
$369$	−10.9098	−0.567939
$370$	0	0
$371$	14.4710	0.751295
$372$	0	0
$373$	8.02874	0.415712	0.207856	−	0.978159i	$-0.433351\pi$
$373$	8.02874	0.415712	0.207856	+	0.978159i	$0.433351\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−2.47547	−0.127493
$378$	0	0
$379$	10.5688	0.542885	0.271442	−	0.962455i	$-0.412499\pi$
$379$	10.5688	0.542885	0.271442	+	0.962455i	$0.412499\pi$
$380$	0	0
$381$	−2.94581	−0.150919
$382$	0	0
$383$	34.5393	1.76488	0.882438	−	0.470429i	$-0.155901\pi$
$383$	34.5393	1.76488	0.882438	+	0.470429i	$0.155901\pi$
$384$	0	0
$385$	4.39501	0.223991
$386$	0	0
$387$	−2.99531	−0.152260
$388$	0	0
$389$	33.1635	1.68146	0.840728	−	0.541458i	$-0.182127\pi$
$389$	33.1635	1.68146	0.840728	+	0.541458i	$0.182127\pi$
$390$	0	0
$391$	10.4868	0.530342
$392$	0	0
$393$	3.61154	0.182178
$394$	0	0
$395$	5.46330	0.274888
$396$	0	0
$397$	−24.8059	−1.24497	−0.622486	−	0.782631i	$-0.713877\pi$
$397$	−24.8059	−1.24497	−0.622486	+	0.782631i	$0.713877\pi$
$398$	0	0
$399$	−6.35004	−0.317899
$400$	0	0
$401$	15.1747	0.757791	0.378895	−	0.925440i	$-0.376304\pi$
$401$	15.1747	0.757791	0.378895	+	0.925440i	$0.376304\pi$
$402$	0	0
$403$	−2.70274	−0.134633
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−5.99791	−0.297306
$408$	0	0
$409$	−4.03041	−0.199291	−0.0996454	−	0.995023i	$-0.531771\pi$
$409$	−4.03041	−0.199291	−0.0996454	+	0.995023i	$0.531771\pi$
$410$	0	0
$411$	−12.7757	−0.630176
$412$	0	0
$413$	3.86524	0.190196
$414$	0	0
$415$	−0.259232	−0.0127252
$416$	0	0
$417$	14.3344	0.701959
$418$	0	0
$419$	35.7146	1.74477	0.872385	−	0.488819i	$-0.162572\pi$
$419$	35.7146	1.74477	0.872385	+	0.488819i	$0.162572\pi$
$420$	0	0
$421$	−5.07047	−0.247120	−0.123560	−	0.992337i	$-0.539431\pi$
$421$	−5.07047	−0.247120	−0.123560	+	0.992337i	$0.539431\pi$
$422$	0	0
$423$	−2.18621	−0.106297
$424$	0	0
$425$	4.01149	0.194586
$426$	0	0
$427$	−71.7882	−3.47407
$428$	0	0
$429$	−0.525037	−0.0253490
$430$	0	0
$431$	−27.8932	−1.34357	−0.671784	−	0.740747i	$-0.734472\pi$
$431$	−27.8932	−1.34357	−0.671784	+	0.740747i	$0.734472\pi$
$432$	0	0
$433$	21.1856	1.01811	0.509057	−	0.860733i	$-0.329994\pi$
$433$	21.1856	1.01811	0.509057	+	0.860733i	$0.329994\pi$
$434$	0	0
$435$	4.25537	0.204030
$436$	0	0
$437$	−3.40898	−0.163073
$438$	0	0
$439$	−29.1116	−1.38942	−0.694712	−	0.719288i	$-0.744468\pi$
$439$	−29.1116	−1.38942	−0.694712	+	0.719288i	$0.744468\pi$
$440$	0	0
$441$	16.7127	0.795843
$442$	0	0
$443$	−10.1251	−0.481056	−0.240528	−	0.970642i	$-0.577321\pi$
$443$	−10.1251	−0.481056	−0.240528	+	0.970642i	$0.577321\pi$
$444$	0	0
$445$	12.3307	0.584530
$446$	0	0
$447$	−13.3584	−0.631829
$448$	0	0
$449$	−8.23413	−0.388593	−0.194296	−	0.980943i	$-0.562242\pi$
$449$	−8.23413	−0.388593	−0.194296	+	0.980943i	$0.562242\pi$
$450$	0	0
$451$	−9.84656	−0.463657
$452$	0	0
$453$	7.93687	0.372907
$454$	0	0
$455$	−2.83277	−0.132802
$456$	0	0
$457$	11.0852	0.518542	0.259271	−	0.965805i	$-0.416518\pi$
$457$	11.0852	0.518542	0.259271	+	0.965805i	$0.416518\pi$
$458$	0	0
$459$	4.01149	0.187240
$460$	0	0
$461$	24.8305	1.15647	0.578236	−	0.815869i	$-0.303741\pi$
$461$	24.8305	1.15647	0.578236	+	0.815869i	$0.303741\pi$
$462$	0	0
$463$	−32.4984	−1.51033	−0.755164	−	0.655536i	$-0.772443\pi$
$463$	−32.4984	−1.51033	−0.755164	+	0.655536i	$0.772443\pi$
$464$	0	0
$465$	4.64605	0.215456
$466$	0	0
$467$	−30.5590	−1.41410	−0.707052	−	0.707162i	$-0.749975\pi$
$467$	−30.5590	−1.41410	−0.707052	+	0.707162i	$0.749975\pi$
$468$	0	0
$469$	−4.86957	−0.224856
$470$	0	0
$471$	−12.9708	−0.597662
$472$	0	0
$473$	−2.70341	−0.124303
$474$	0	0
$475$	−1.30402	−0.0598327
$476$	0	0
$477$	2.97171	0.136065
$478$	0	0
$479$	−29.0256	−1.32621	−0.663106	−	0.748525i	$-0.730762\pi$
$479$	−29.0256	−1.32621	−0.663106	+	0.748525i	$0.730762\pi$
$480$	0	0
$481$	3.86590	0.176270
$482$	0	0
$483$	12.7300	0.579236
$484$	0	0
$485$	−9.07927	−0.412268
$486$	0	0
$487$	1.96861	0.0892062	0.0446031	−	0.999005i	$-0.485798\pi$
$487$	1.96861	0.0892062	0.0446031	+	0.999005i	$0.485798\pi$
$488$	0	0
$489$	−23.5564	−1.06526
$490$	0	0
$491$	−38.2195	−1.72482	−0.862412	−	0.506208i	$-0.831047\pi$
$491$	−38.2195	−1.72482	−0.862412	+	0.506208i	$0.831047\pi$
$492$	0	0
$493$	17.0704	0.768812
$494$	0	0
$495$	0.902547	0.0405665
$496$	0	0
$497$	17.1964	0.771364
$498$	0	0
$499$	−38.3841	−1.71831	−0.859153	−	0.511718i	$-0.829009\pi$
$499$	−38.3841	−1.71831	−0.859153	+	0.511718i	$0.829009\pi$
$500$	0	0
$501$	−11.6408	−0.520071
$502$	0	0
$503$	−11.2753	−0.502739	−0.251370	−	0.967891i	$-0.580881\pi$
$503$	−11.2753	−0.502739	−0.251370	+	0.967891i	$0.580881\pi$
$504$	0	0
$505$	6.39381	0.284521
$506$	0	0
$507$	−12.6616	−0.562321
$508$	0	0
$509$	−9.20491	−0.408000	−0.204000	−	0.978971i	$-0.565394\pi$
$509$	−9.20491	−0.408000	−0.204000	+	0.978971i	$0.565394\pi$
$510$	0	0
$511$	32.8583	1.45357
$512$	0	0
$513$	−1.30402	−0.0575741
$514$	0	0
$515$	−16.8065	−0.740584
$516$	0	0
$517$	−1.97315	−0.0867792
$518$	0	0
$519$	7.16397	0.314463
$520$	0	0
$521$	−37.2813	−1.63332	−0.816662	−	0.577116i	$-0.804178\pi$
$521$	−37.2813	−1.63332	−0.816662	+	0.577116i	$0.804178\pi$
$522$	0	0
$523$	−34.5487	−1.51071	−0.755355	−	0.655316i	$-0.772535\pi$
$523$	−34.5487	−1.51071	−0.755355	+	0.655316i	$0.772535\pi$
$524$	0	0
$525$	4.86957	0.212525
$526$	0	0
$527$	18.6376	0.811867
$528$	0	0
$529$	−16.1660	−0.702869
$530$	0	0
$531$	0.793753	0.0344459
$532$	0	0
$533$	6.34652	0.274898
$534$	0	0
$535$	−1.43097	−0.0618662
$536$	0	0
$537$	−8.30426	−0.358355
$538$	0	0
$539$	15.0840	0.649714
$540$	0	0
$541$	3.42255	0.147147	0.0735735	−	0.997290i	$-0.476560\pi$
$541$	3.42255	0.147147	0.0735735	+	0.997290i	$0.476560\pi$
$542$	0	0
$543$	−11.9945	−0.514734
$544$	0	0
$545$	−9.53526	−0.408446
$546$	0	0
$547$	−9.76905	−0.417694	−0.208847	−	0.977948i	$-0.566971\pi$
$547$	−9.76905	−0.417694	−0.208847	+	0.977948i	$0.566971\pi$
$548$	0	0
$549$	−14.7422	−0.629182
$550$	0	0
$551$	−5.54911	−0.236400
$552$	0	0
$553$	26.6039	1.13131
$554$	0	0
$555$	−6.64554	−0.282088
$556$	0	0
$557$	−36.3002	−1.53809	−0.769044	−	0.639196i	$-0.779267\pi$
$557$	−36.3002	−1.53809	−0.769044	+	0.639196i	$0.779267\pi$
$558$	0	0
$559$	1.74246	0.0736981
$560$	0	0
$561$	3.62056	0.152860
$562$	0	0
$563$	27.7886	1.17115	0.585574	−	0.810619i	$-0.300869\pi$
$563$	27.7886	1.17115	0.585574	+	0.810619i	$0.300869\pi$
$564$	0	0
$565$	10.1136	0.425482
$566$	0	0
$567$	4.86957	0.204503
$568$	0	0
$569$	22.7992	0.955791	0.477895	−	0.878417i	$-0.341400\pi$
$569$	22.7992	0.955791	0.477895	+	0.878417i	$0.341400\pi$
$570$	0	0
$571$	−9.72074	−0.406800	−0.203400	−	0.979096i	$-0.565199\pi$
$571$	−9.72074	−0.406800	−0.203400	+	0.979096i	$0.565199\pi$
$572$	0	0
$573$	−0.260316	−0.0108748
$574$	0	0
$575$	2.61420	0.109020
$576$	0	0
$577$	−11.6653	−0.485632	−0.242816	−	0.970072i	$-0.578071\pi$
$577$	−11.6653	−0.485632	−0.242816	+	0.970072i	$0.578071\pi$
$578$	0	0
$579$	2.09344	0.0870005
$580$	0	0
$581$	−1.26235	−0.0523709
$582$	0	0
$583$	2.68211	0.111082
$584$	0	0
$585$	−0.581729	−0.0240515
$586$	0	0
$587$	−6.59025	−0.272009	−0.136004	−	0.990708i	$-0.543426\pi$
$587$	−6.59025	−0.272009	−0.136004	+	0.990708i	$0.543426\pi$
$588$	0	0
$589$	−6.05857	−0.249639
$590$	0	0
$591$	14.9652	0.615588
$592$	0	0
$593$	2.35991	0.0969098	0.0484549	−	0.998825i	$-0.484570\pi$
$593$	2.35991	0.0969098	0.0484549	+	0.998825i	$0.484570\pi$
$594$	0	0
$595$	19.5342	0.800826
$596$	0	0
$597$	14.3883	0.588875
$598$	0	0
$599$	37.8903	1.54815	0.774077	−	0.633091i	$-0.218214\pi$
$599$	37.8903	1.54815	0.774077	+	0.633091i	$0.218214\pi$
$600$	0	0
$601$	−10.2228	−0.416997	−0.208499	−	0.978023i	$-0.566858\pi$
$601$	−10.2228	−0.416997	−0.208499	+	0.978023i	$0.566858\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−10.1854	−0.414096
$606$	0	0
$607$	15.5888	0.632731	0.316365	−	0.948637i	$-0.397537\pi$
$607$	15.5888	0.632731	0.316365	+	0.948637i	$0.397537\pi$
$608$	0	0
$609$	20.7218	0.839691
$610$	0	0
$611$	1.27178	0.0514506
$612$	0	0
$613$	39.1619	1.58173	0.790867	−	0.611989i	$-0.209630\pi$
$613$	39.1619	1.58173	0.790867	+	0.611989i	$0.209630\pi$
$614$	0	0
$615$	−10.9098	−0.439924
$616$	0	0
$617$	20.3113	0.817701	0.408850	−	0.912601i	$-0.365930\pi$
$617$	20.3113	0.817701	0.408850	+	0.912601i	$0.365930\pi$
$618$	0	0
$619$	−24.3352	−0.978113	−0.489057	−	0.872252i	$-0.662659\pi$
$619$	−24.3352	−0.978113	−0.489057	+	0.872252i	$0.662659\pi$
$620$	0	0
$621$	2.61420	0.104904
$622$	0	0
$623$	60.0451	2.40566
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.17694	−0.0470026
$628$	0	0
$629$	−26.6585	−1.06295
$630$	0	0
$631$	−29.9751	−1.19329	−0.596644	−	0.802506i	$-0.703500\pi$
$631$	−29.9751	−1.19329	−0.596644	+	0.802506i	$0.703500\pi$
$632$	0	0
$633$	19.7552	0.785198
$634$	0	0
$635$	−2.94581	−0.116901
$636$	0	0
$637$	−9.72226	−0.385210
$638$	0	0
$639$	3.53140	0.139700
$640$	0	0
$641$	−13.0274	−0.514552	−0.257276	−	0.966338i	$-0.582825\pi$
$641$	−13.0274	−0.514552	−0.257276	+	0.966338i	$0.582825\pi$
$642$	0	0
$643$	−26.8495	−1.05884	−0.529421	−	0.848359i	$-0.677591\pi$
$643$	−26.8495	−1.05884	−0.529421	+	0.848359i	$0.677591\pi$
$644$	0	0
$645$	−2.99531	−0.117940
$646$	0	0
$647$	−23.7266	−0.932787	−0.466394	−	0.884577i	$-0.654447\pi$
$647$	−23.7266	−0.932787	−0.466394	+	0.884577i	$0.654447\pi$
$648$	0	0
$649$	0.716399	0.0281211
$650$	0	0
$651$	22.6243	0.886716
$652$	0	0
$653$	23.3155	0.912405	0.456203	−	0.889876i	$-0.349209\pi$
$653$	23.3155	0.912405	0.456203	+	0.889876i	$0.349209\pi$
$654$	0	0
$655$	3.61154	0.141114
$656$	0	0
$657$	6.74769	0.263252
$658$	0	0
$659$	−28.3244	−1.10336	−0.551681	−	0.834055i	$-0.686013\pi$
$659$	−28.3244	−1.10336	−0.551681	+	0.834055i	$0.686013\pi$
$660$	0	0
$661$	−6.61366	−0.257242	−0.128621	−	0.991694i	$-0.541055\pi$
$661$	−6.61366	−0.257242	−0.128621	+	0.991694i	$0.541055\pi$
$662$	0	0
$663$	−2.33360	−0.0906295
$664$	0	0
$665$	−6.35004	−0.246244
$666$	0	0
$667$	11.1244	0.430738
$668$	0	0
$669$	19.2085	0.742643
$670$	0	0
$671$	−13.3055	−0.513654
$672$	0	0
$673$	−32.9553	−1.27033	−0.635166	−	0.772375i	$-0.719068\pi$
$673$	−32.9553	−1.27033	−0.635166	+	0.772375i	$0.719068\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	13.4700	0.517695	0.258848	−	0.965918i	$-0.416657\pi$
$677$	13.4700	0.517695	0.258848	+	0.965918i	$0.416657\pi$
$678$	0	0
$679$	−44.2121	−1.69671
$680$	0	0
$681$	−7.26953	−0.278569
$682$	0	0
$683$	6.85721	0.262384	0.131192	−	0.991357i	$-0.458120\pi$
$683$	6.85721	0.262384	0.131192	+	0.991357i	$0.458120\pi$
$684$	0	0
$685$	−12.7757	−0.488133
$686$	0	0
$687$	−9.20742	−0.351285
$688$	0	0
$689$	−1.72873	−0.0658594
$690$	0	0
$691$	−17.9655	−0.683441	−0.341721	−	0.939802i	$-0.611010\pi$
$691$	−17.9655	−0.683441	−0.341721	+	0.939802i	$0.611010\pi$
$692$	0	0
$693$	4.39501	0.166953
$694$	0	0
$695$	14.3344	0.543735
$696$	0	0
$697$	−43.7644	−1.65769
$698$	0	0
$699$	−10.3916	−0.393047
$700$	0	0
$701$	5.25231	0.198377	0.0991885	−	0.995069i	$-0.468375\pi$
$701$	5.25231	0.198377	0.0991885	+	0.995069i	$0.468375\pi$
$702$	0	0
$703$	8.66595	0.326843
$704$	0	0
$705$	−2.18621	−0.0823373
$706$	0	0
$707$	31.1351	1.17096
$708$	0	0
$709$	−21.9679	−0.825023	−0.412511	−	0.910952i	$-0.635348\pi$
$709$	−21.9679	−0.825023	−0.412511	+	0.910952i	$0.635348\pi$
$710$	0	0
$711$	5.46330	0.204890
$712$	0	0
$713$	12.1457	0.454860
$714$	0	0
$715$	−0.525037	−0.0196353
$716$	0	0
$717$	−6.21202	−0.231992
$718$	0	0
$719$	−41.3454	−1.54192	−0.770962	−	0.636881i	$-0.780224\pi$
$719$	−41.3454	−1.54192	−0.770962	+	0.636881i	$0.780224\pi$
$720$	0	0
$721$	−81.8406	−3.04790
$722$	0	0
$723$	13.8765	0.516074
$724$	0	0
$725$	4.25537	0.158041
$726$	0	0
$727$	11.2451	0.417058	0.208529	−	0.978016i	$-0.433132\pi$
$727$	11.2451	0.417058	0.208529	+	0.978016i	$0.433132\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.0157	−0.444415
$732$	0	0
$733$	−44.4296	−1.64105	−0.820523	−	0.571614i	$-0.806317\pi$
$733$	−44.4296	−1.64105	−0.820523	+	0.571614i	$0.806317\pi$
$734$	0	0
$735$	16.7127	0.616458
$736$	0	0
$737$	−0.902547	−0.0332457
$738$	0	0
$739$	15.4451	0.568159	0.284079	−	0.958801i	$-0.408312\pi$
$739$	15.4451	0.568159	0.284079	+	0.958801i	$0.408312\pi$
$740$	0	0
$741$	0.758588	0.0278674
$742$	0	0
$743$	12.0567	0.442317	0.221159	−	0.975238i	$-0.429016\pi$
$743$	12.0567	0.442317	0.221159	+	0.975238i	$0.429016\pi$
$744$	0	0
$745$	−13.3584	−0.489412
$746$	0	0
$747$	−0.259232	−0.00948479
$748$	0	0
$749$	−6.96820	−0.254613
$750$	0	0
$751$	−2.11592	−0.0772112	−0.0386056	−	0.999255i	$-0.512292\pi$
$751$	−2.11592	−0.0772112	−0.0386056	+	0.999255i	$0.512292\pi$
$752$	0	0
$753$	3.39049	0.123556
$754$	0	0
$755$	7.93687	0.288852
$756$	0	0
$757$	39.7664	1.44533	0.722667	−	0.691196i	$-0.242916\pi$
$757$	39.7664	1.44533	0.722667	+	0.691196i	$0.242916\pi$
$758$	0	0
$759$	2.35943	0.0856420
$760$	0	0
$761$	42.8266	1.55246	0.776231	−	0.630448i	$-0.217129\pi$
$761$	42.8266	1.55246	0.776231	+	0.630448i	$0.217129\pi$
$762$	0	0
$763$	−46.4326	−1.68097
$764$	0	0
$765$	4.01149	0.145036
$766$	0	0
$767$	−0.461749	−0.0166728
$768$	0	0
$769$	20.5902	0.742501	0.371251	−	0.928533i	$-0.378929\pi$
$769$	20.5902	0.742501	0.371251	+	0.928533i	$0.378929\pi$
$770$	0	0
$771$	14.4877	0.521763
$772$	0	0
$773$	10.5761	0.380397	0.190198	−	0.981746i	$-0.439087\pi$
$773$	10.5761	0.380397	0.190198	+	0.981746i	$0.439087\pi$
$774$	0	0
$775$	4.64605	0.166891
$776$	0	0
$777$	−32.3609	−1.16094
$778$	0	0
$779$	14.2266	0.509720
$780$	0	0
$781$	3.18725	0.114049
$782$	0	0
$783$	4.25537	0.152075
$784$	0	0
$785$	−12.9708	−0.462947
$786$	0	0
$787$	−29.7386	−1.06007	−0.530033	−	0.847977i	$-0.677821\pi$
$787$	−29.7386	−1.06007	−0.530033	+	0.847977i	$0.677821\pi$
$788$	0	0
$789$	25.1276	0.894568
$790$	0	0
$791$	49.2489	1.75109
$792$	0	0
$793$	8.57596	0.304541
$794$	0	0
$795$	2.97171	0.105396
$796$	0	0
$797$	7.80357	0.276417	0.138208	−	0.990403i	$-0.455866\pi$
$797$	7.80357	0.276417	0.138208	+	0.990403i	$0.455866\pi$
$798$	0	0
$799$	−8.76995	−0.310258
$800$	0	0
$801$	12.3307	0.435683
$802$	0	0
$803$	6.09010	0.214915
$804$	0	0
$805$	12.7300	0.448674
$806$	0	0
$807$	32.3638	1.13926
$808$	0	0
$809$	−36.8248	−1.29469	−0.647345	−	0.762197i	$-0.724121\pi$
$809$	−36.8248	−1.29469	−0.647345	+	0.762197i	$0.724121\pi$
$810$	0	0
$811$	−13.1710	−0.462497	−0.231248	−	0.972895i	$-0.574281\pi$
$811$	−13.1710	−0.462497	−0.231248	+	0.972895i	$0.574281\pi$
$812$	0	0
$813$	−11.8524	−0.415681
$814$	0	0
$815$	−23.5564	−0.825144
$816$	0	0
$817$	3.90596	0.136652
$818$	0	0
$819$	−2.83277	−0.0989849
$820$	0	0
$821$	−38.9809	−1.36044	−0.680222	−	0.733007i	$-0.738116\pi$
$821$	−38.9809	−1.36044	−0.680222	+	0.733007i	$0.738116\pi$
$822$	0	0
$823$	−15.9445	−0.555790	−0.277895	−	0.960612i	$-0.589637\pi$
$823$	−15.9445	−0.555790	−0.277895	+	0.960612i	$0.589637\pi$
$824$	0	0
$825$	0.902547	0.0314226
$826$	0	0
$827$	9.48231	0.329732	0.164866	−	0.986316i	$-0.447281\pi$
$827$	9.48231	0.329732	0.164866	+	0.986316i	$0.447281\pi$
$828$	0	0
$829$	20.7504	0.720692	0.360346	−	0.932819i	$-0.382659\pi$
$829$	20.7504	0.720692	0.360346	+	0.932819i	$0.382659\pi$
$830$	0	0
$831$	18.9472	0.657270
$832$	0	0
$833$	67.0429	2.32290
$834$	0	0
$835$	−11.6408	−0.402845
$836$	0	0
$837$	4.64605	0.160591
$838$	0	0
$839$	−14.2259	−0.491131	−0.245566	−	0.969380i	$-0.578974\pi$
$839$	−14.2259	−0.491131	−0.245566	+	0.969380i	$0.578974\pi$
$840$	0	0
$841$	−10.8918	−0.375579
$842$	0	0
$843$	18.0110	0.620333
$844$	0	0
$845$	−12.6616	−0.435572
$846$	0	0
$847$	−49.5986	−1.70423
$848$	0	0
$849$	1.47795	0.0507233
$850$	0	0
$851$	−17.3728	−0.595531
$852$	0	0
$853$	−1.78178	−0.0610071	−0.0305036	−	0.999535i	$-0.509711\pi$
$853$	−1.78178	−0.0610071	−0.0305036	+	0.999535i	$0.509711\pi$
$854$	0	0
$855$	−1.30402	−0.0445967
$856$	0	0
$857$	9.97852	0.340860	0.170430	−	0.985370i	$-0.445484\pi$
$857$	9.97852	0.340860	0.170430	+	0.985370i	$0.445484\pi$
$858$	0	0
$859$	34.5035	1.17724	0.588622	−	0.808409i	$-0.299671\pi$
$859$	34.5035	1.17724	0.588622	+	0.808409i	$0.299671\pi$
$860$	0	0
$861$	−53.1258	−1.81052
$862$	0	0
$863$	37.6265	1.28082	0.640411	−	0.768033i	$-0.278764\pi$
$863$	37.6265	1.28082	0.640411	+	0.768033i	$0.278764\pi$
$864$	0	0
$865$	7.16397	0.243582
$866$	0	0
$867$	−0.907933	−0.0308350
$868$	0	0
$869$	4.93088	0.167269
$870$	0	0
$871$	0.581729	0.0197111
$872$	0	0
$873$	−9.07927	−0.307287
$874$	0	0
$875$	4.86957	0.164622
$876$	0	0
$877$	31.3370	1.05818	0.529088	−	0.848567i	$-0.322534\pi$
$877$	31.3370	1.05818	0.529088	+	0.848567i	$0.322534\pi$
$878$	0	0
$879$	−18.1602	−0.612530
$880$	0	0
$881$	27.3218	0.920496	0.460248	−	0.887790i	$-0.347761\pi$
$881$	27.3218	0.920496	0.460248	+	0.887790i	$0.347761\pi$
$882$	0	0
$883$	23.7217	0.798300	0.399150	−	0.916886i	$-0.369305\pi$
$883$	23.7217	0.798300	0.399150	+	0.916886i	$0.369305\pi$
$884$	0	0
$885$	0.793753	0.0266817
$886$	0	0
$887$	−18.7257	−0.628746	−0.314373	−	0.949299i	$-0.601794\pi$
$887$	−18.7257	−0.628746	−0.314373	+	0.949299i	$0.601794\pi$
$888$	0	0
$889$	−14.3448	−0.481110
$890$	0	0
$891$	0.902547	0.0302365
$892$	0	0
$893$	2.85086	0.0954006
$894$	0	0
$895$	−8.30426	−0.277581
$896$	0	0
$897$	−1.52075	−0.0507765
$898$	0	0
$899$	19.7707	0.659389
$900$	0	0
$901$	11.9210	0.397146
$902$	0	0
$903$	−14.5859	−0.485387
$904$	0	0
$905$	−11.9945	−0.398711
$906$	0	0
$907$	1.06139	0.0352429	0.0176214	−	0.999845i	$-0.494391\pi$
$907$	1.06139	0.0352429	0.0176214	+	0.999845i	$0.494391\pi$
$908$	0	0
$909$	6.39381	0.212069
$910$	0	0
$911$	30.6850	1.01664	0.508319	−	0.861169i	$-0.330267\pi$
$911$	30.6850	1.01664	0.508319	+	0.861169i	$0.330267\pi$
$912$	0	0
$913$	−0.233969	−0.00774323
$914$	0	0
$915$	−14.7422	−0.487362
$916$	0	0
$917$	17.5866	0.580761
$918$	0	0
$919$	10.0658	0.332039	0.166020	−	0.986122i	$-0.446909\pi$
$919$	10.0658	0.332039	0.166020	+	0.986122i	$0.446909\pi$
$920$	0	0
$921$	−3.89241	−0.128259
$922$	0	0
$923$	−2.05432	−0.0676186
$924$	0	0
$925$	−6.64554	−0.218504
$926$	0	0
$927$	−16.8065	−0.551999
$928$	0	0
$929$	−17.4991	−0.574127	−0.287064	−	0.957912i	$-0.592679\pi$
$929$	−17.4991	−0.574127	−0.287064	+	0.957912i	$0.592679\pi$
$930$	0	0
$931$	−21.7938	−0.714262
$932$	0	0
$933$	1.10230	0.0360875
$934$	0	0
$935$	3.62056	0.118405
$936$	0	0
$937$	−56.6357	−1.85021	−0.925105	−	0.379712i	$-0.876023\pi$
$937$	−56.6357	−1.85021	−0.925105	+	0.379712i	$0.876023\pi$
$938$	0	0
$939$	12.6573	0.413056
$940$	0	0
$941$	1.08772	0.0354585	0.0177293	−	0.999843i	$-0.494356\pi$
$941$	1.08772	0.0354585	0.0177293	+	0.999843i	$0.494356\pi$
$942$	0	0
$943$	−28.5202	−0.928747
$944$	0	0
$945$	4.86957	0.158407
$946$	0	0
$947$	−30.9974	−1.00728	−0.503641	−	0.863913i	$-0.668006\pi$
$947$	−30.9974	−1.00728	−0.503641	+	0.863913i	$0.668006\pi$
$948$	0	0
$949$	−3.92532	−0.127421
$950$	0	0
$951$	−14.6581	−0.475322
$952$	0	0
$953$	54.9897	1.78129	0.890646	−	0.454697i	$-0.150253\pi$
$953$	54.9897	1.78129	0.890646	+	0.454697i	$0.150253\pi$
$954$	0	0
$955$	−0.260316	−0.00842362
$956$	0	0
$957$	3.84067	0.124151
$958$	0	0
$959$	−62.2119	−2.00893
$960$	0	0
$961$	−9.41419	−0.303683
$962$	0	0
$963$	−1.43097	−0.0461123
$964$	0	0
$965$	2.09344	0.0673903
$966$	0	0
$967$	2.80907	0.0903335	0.0451667	−	0.998979i	$-0.485618\pi$
$967$	2.80907	0.0903335	0.0451667	+	0.998979i	$0.485618\pi$
$968$	0	0
$969$	−5.23108	−0.168047
$970$	0	0
$971$	−35.3367	−1.13401	−0.567005	−	0.823714i	$-0.691898\pi$
$971$	−35.3367	−1.13401	−0.567005	+	0.823714i	$0.691898\pi$
$972$	0	0
$973$	69.8024	2.23776
$974$	0	0
$975$	−0.581729	−0.0186302
$976$	0	0
$977$	5.44113	0.174077	0.0870386	−	0.996205i	$-0.472260\pi$
$977$	5.44113	0.174077	0.0870386	+	0.996205i	$0.472260\pi$
$978$	0	0
$979$	11.1290	0.355685
$980$	0	0
$981$	−9.53526	−0.304437
$982$	0	0
$983$	14.3637	0.458131	0.229066	−	0.973411i	$-0.426433\pi$
$983$	14.3637	0.458131	0.229066	+	0.973411i	$0.426433\pi$
$984$	0	0
$985$	14.9652	0.476832
$986$	0	0
$987$	−10.6459	−0.338862
$988$	0	0
$989$	−7.83033	−0.248990
$990$	0	0
$991$	−51.7443	−1.64371	−0.821855	−	0.569696i	$-0.807061\pi$
$991$	−51.7443	−1.64371	−0.821855	+	0.569696i	$0.807061\pi$
$992$	0	0
$993$	2.04174	0.0647928
$994$	0	0
$995$	14.3883	0.456140
$996$	0	0
$997$	41.7517	1.32229	0.661145	−	0.750258i	$-0.270071\pi$
$997$	41.7517	1.32229	0.661145	+	0.750258i	$0.270071\pi$
$998$	0	0
$999$	−6.64554	−0.210256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.i.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.i.1.7	✓	7	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +4.86957 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +4.86957 q^{7} +1.00000 q^{9} +0.902547 q^{11} -0.581729 q^{13} +1.00000 q^{15} +4.01149 q^{17} -1.30402 q^{19} +4.86957 q^{21} +2.61420 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.25537 q^{29} +4.64605 q^{31} +0.902547 q^{33} +4.86957 q^{35} -6.64554 q^{37} -0.581729 q^{39} -10.9098 q^{41} -2.99531 q^{43} +1.00000 q^{45} -2.18621 q^{47} +16.7127 q^{49} +4.01149 q^{51} +2.97171 q^{53} +0.902547 q^{55} -1.30402 q^{57} +0.793753 q^{59} -14.7422 q^{61} +4.86957 q^{63} -0.581729 q^{65} -1.00000 q^{67} +2.61420 q^{69} +3.53140 q^{71} +6.74769 q^{73} +1.00000 q^{75} +4.39501 q^{77} +5.46330 q^{79} +1.00000 q^{81} -0.259232 q^{83} +4.01149 q^{85} +4.25537 q^{87} +12.3307 q^{89} -2.83277 q^{91} +4.64605 q^{93} -1.30402 q^{95} -9.07927 q^{97} +0.902547 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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