LMFDB - Embedded newform 3120.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.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:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	3120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +5.56155 q^{11} +1.00000 q^{13} +1.00000 q^{15} -6.68466 q^{17} -3.12311 q^{19} -1.56155 q^{21} +5.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} +7.12311 q^{31} +5.56155 q^{33} -1.56155 q^{35} +9.80776 q^{37} +1.00000 q^{39} +2.68466 q^{41} +10.2462 q^{43} +1.00000 q^{45} -7.12311 q^{47} -4.56155 q^{49} -6.68466 q^{51} +3.56155 q^{53} +5.56155 q^{55} -3.12311 q^{57} +8.00000 q^{59} -10.6847 q^{61} -1.56155 q^{63} +1.00000 q^{65} +14.2462 q^{67} +5.56155 q^{69} -4.68466 q^{71} +16.2462 q^{73} +1.00000 q^{75} -8.68466 q^{77} -11.8078 q^{79} +1.00000 q^{81} +8.00000 q^{83} -6.68466 q^{85} -2.00000 q^{87} -14.6847 q^{89} -1.56155 q^{91} +7.12311 q^{93} -3.12311 q^{95} -17.8078 q^{97} +5.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 2 q^{13} + 2 q^{15} - q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 7 q^{33} + q^{35} - q^{37} + 2 q^{39} - 7 q^{41} + 4 q^{43} + 2 q^{45} - 6 q^{47} - 5 q^{49} - q^{51} + 3 q^{53} + 7 q^{55} + 2 q^{57} + 16 q^{59} - 9 q^{61} + q^{63} + 2 q^{65} + 12 q^{67} + 7 q^{69} + 3 q^{71} + 16 q^{73} + 2 q^{75} - 5 q^{77} - 3 q^{79} + 2 q^{81} + 16 q^{83} - q^{85} - 4 q^{87} - 17 q^{89} + q^{91} + 6 q^{93} + 2 q^{95} - 15 q^{97} + 7 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 2 * q^13 + 2 * q^15 - q^17 + 2 * q^19 + q^21 + 7 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 6 * q^31 + 7 * q^33 + q^35 - q^37 + 2 * q^39 - 7 * q^41 + 4 * q^43 + 2 * q^45 - 6 * q^47 - 5 * q^49 - q^51 + 3 * q^53 + 7 * q^55 + 2 * q^57 + 16 * q^59 - 9 * q^61 + q^63 + 2 * q^65 + 12 * q^67 + 7 * q^69 + 3 * q^71 + 16 * q^73 + 2 * q^75 - 5 * q^77 - 3 * q^79 + 2 * q^81 + 16 * q^83 - q^85 - 4 * q^87 - 17 * q^89 + q^91 + 6 * q^93 + 2 * q^95 - 15 * q^97 + 7 * 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
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.56155	1.67687	0.838436	−	0.545001i	$-0.183471\pi$
$11$	5.56155	1.67687	0.838436	+	0.545001i	$0.183471\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.68466	−1.62127	−0.810634	−	0.585553i	$-0.800877\pi$
$17$	−6.68466	−1.62127	−0.810634	+	0.585553i	$0.800877\pi$
$18$	0	0
$19$	−3.12311	−0.716490	−0.358245	−	0.933628i	$-0.616625\pi$
$19$	−3.12311	−0.716490	−0.358245	+	0.933628i	$0.616625\pi$
$20$	0	0
$21$	−1.56155	−0.340759
$22$	0	0
$23$	5.56155	1.15966	0.579832	−	0.814736i	$-0.303118\pi$
$23$	5.56155	1.15966	0.579832	+	0.814736i	$0.303118\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	7.12311	1.27935	0.639674	−	0.768647i	$-0.279069\pi$
$31$	7.12311	1.27935	0.639674	+	0.768647i	$0.279069\pi$
$32$	0	0
$33$	5.56155	0.968142
$34$	0	0
$35$	−1.56155	−0.263951
$36$	0	0
$37$	9.80776	1.61239	0.806193	−	0.591652i	$-0.201524\pi$
$37$	9.80776	1.61239	0.806193	+	0.591652i	$0.201524\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	2.68466	0.419273	0.209637	−	0.977779i	$-0.432772\pi$
$41$	2.68466	0.419273	0.209637	+	0.977779i	$0.432772\pi$
$42$	0	0
$43$	10.2462	1.56253	0.781266	−	0.624198i	$-0.214574\pi$
$43$	10.2462	1.56253	0.781266	+	0.624198i	$0.214574\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−7.12311	−1.03901	−0.519506	−	0.854467i	$-0.673884\pi$
$47$	−7.12311	−1.03901	−0.519506	+	0.854467i	$0.673884\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−6.68466	−0.936039
$52$	0	0
$53$	3.56155	0.489217	0.244608	−	0.969622i	$-0.421341\pi$
$53$	3.56155	0.489217	0.244608	+	0.969622i	$0.421341\pi$
$54$	0	0
$55$	5.56155	0.749920
$56$	0	0
$57$	−3.12311	−0.413665
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−10.6847	−1.36803	−0.684015	−	0.729468i	$-0.739768\pi$
$61$	−10.6847	−1.36803	−0.684015	+	0.729468i	$0.739768\pi$
$62$	0	0
$63$	−1.56155	−0.196737
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	14.2462	1.74045	0.870226	−	0.492653i	$-0.163973\pi$
$67$	14.2462	1.74045	0.870226	+	0.492653i	$0.163973\pi$
$68$	0	0
$69$	5.56155	0.669532
$70$	0	0
$71$	−4.68466	−0.555967	−0.277983	−	0.960586i	$-0.589666\pi$
$71$	−4.68466	−0.555967	−0.277983	+	0.960586i	$0.589666\pi$
$72$	0	0
$73$	16.2462	1.90148	0.950738	−	0.309997i	$-0.100328\pi$
$73$	16.2462	1.90148	0.950738	+	0.309997i	$0.100328\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−8.68466	−0.989709
$78$	0	0
$79$	−11.8078	−1.32848	−0.664239	−	0.747521i	$-0.731244\pi$
$79$	−11.8078	−1.32848	−0.664239	+	0.747521i	$0.731244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	−6.68466	−0.725053
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−14.6847	−1.55657	−0.778285	−	0.627911i	$-0.783910\pi$
$89$	−14.6847	−1.55657	−0.778285	+	0.627911i	$0.783910\pi$
$90$	0	0
$91$	−1.56155	−0.163695
$92$	0	0
$93$	7.12311	0.738632
$94$	0	0
$95$	−3.12311	−0.320424
$96$	0	0
$97$	−17.8078	−1.80810	−0.904052	−	0.427422i	$-0.859422\pi$
$97$	−17.8078	−1.80810	−0.904052	+	0.427422i	$0.859422\pi$
$98$	0	0
$99$	5.56155	0.558957
$100$	0	0
$101$	1.12311	0.111753	0.0558766	−	0.998438i	$-0.482205\pi$
$101$	1.12311	0.111753	0.0558766	+	0.998438i	$0.482205\pi$
$102$	0	0
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	0	0
$105$	−1.56155	−0.152392
$106$	0	0
$107$	15.8078	1.52819	0.764097	−	0.645101i	$-0.223185\pi$
$107$	15.8078	1.52819	0.764097	+	0.645101i	$0.223185\pi$
$108$	0	0
$109$	−13.1231	−1.25697	−0.628483	−	0.777824i	$-0.716324\pi$
$109$	−13.1231	−1.25697	−0.628483	+	0.777824i	$0.716324\pi$
$110$	0	0
$111$	9.80776	0.930912
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	5.56155	0.518617
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	10.4384	0.956891
$120$	0	0
$121$	19.9309	1.81190
$122$	0	0
$123$	2.68466	0.242067
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.3693	1.54128	0.770639	−	0.637272i	$-0.219937\pi$
$127$	17.3693	1.54128	0.770639	+	0.637272i	$0.219937\pi$
$128$	0	0
$129$	10.2462	0.902129
$130$	0	0
$131$	15.1231	1.32131	0.660656	−	0.750689i	$-0.270278\pi$
$131$	15.1231	1.32131	0.660656	+	0.750689i	$0.270278\pi$
$132$	0	0
$133$	4.87689	0.422880
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	16.2462	1.38801	0.694004	−	0.719971i	$-0.255845\pi$
$137$	16.2462	1.38801	0.694004	+	0.719971i	$0.255845\pi$
$138$	0	0
$139$	9.56155	0.811000	0.405500	−	0.914095i	$-0.367097\pi$
$139$	9.56155	0.811000	0.405500	+	0.914095i	$0.367097\pi$
$140$	0	0
$141$	−7.12311	−0.599874
$142$	0	0
$143$	5.56155	0.465080
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	−4.56155	−0.376231
$148$	0	0
$149$	−12.4384	−1.01900	−0.509499	−	0.860471i	$-0.670169\pi$
$149$	−12.4384	−1.01900	−0.509499	+	0.860471i	$0.670169\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	−6.68466	−0.540423
$154$	0	0
$155$	7.12311	0.572142
$156$	0	0
$157$	10.4924	0.837386	0.418693	−	0.908128i	$-0.362488\pi$
$157$	10.4924	0.837386	0.418693	+	0.908128i	$0.362488\pi$
$158$	0	0
$159$	3.56155	0.282450
$160$	0	0
$161$	−8.68466	−0.684447
$162$	0	0
$163$	−13.5616	−1.06222	−0.531111	−	0.847302i	$-0.678225\pi$
$163$	−13.5616	−1.06222	−0.531111	+	0.847302i	$0.678225\pi$
$164$	0	0
$165$	5.56155	0.432966
$166$	0	0
$167$	5.75379	0.445242	0.222621	−	0.974905i	$-0.428539\pi$
$167$	5.75379	0.445242	0.222621	+	0.974905i	$0.428539\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−3.12311	−0.238830
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	23.1231	1.72830	0.864151	−	0.503233i	$-0.167856\pi$
$179$	23.1231	1.72830	0.864151	+	0.503233i	$0.167856\pi$
$180$	0	0
$181$	−8.93087	−0.663826	−0.331913	−	0.943310i	$-0.607694\pi$
$181$	−8.93087	−0.663826	−0.331913	+	0.943310i	$0.607694\pi$
$182$	0	0
$183$	−10.6847	−0.789833
$184$	0	0
$185$	9.80776	0.721081
$186$	0	0
$187$	−37.1771	−2.71866
$188$	0	0
$189$	−1.56155	−0.113586
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	9.31534	0.670533	0.335266	−	0.942123i	$-0.391174\pi$
$193$	9.31534	0.670533	0.335266	+	0.942123i	$0.391174\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	7.36932	0.525042	0.262521	−	0.964926i	$-0.415446\pi$
$197$	7.36932	0.525042	0.262521	+	0.964926i	$0.415446\pi$
$198$	0	0
$199$	−1.75379	−0.124323	−0.0621614	−	0.998066i	$-0.519799\pi$
$199$	−1.75379	−0.124323	−0.0621614	+	0.998066i	$0.519799\pi$
$200$	0	0
$201$	14.2462	1.00485
$202$	0	0
$203$	3.12311	0.219199
$204$	0	0
$205$	2.68466	0.187505
$206$	0	0
$207$	5.56155	0.386555
$208$	0	0
$209$	−17.3693	−1.20146
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	−4.68466	−0.320988
$214$	0	0
$215$	10.2462	0.698786
$216$	0	0
$217$	−11.1231	−0.755086
$218$	0	0
$219$	16.2462	1.09782
$220$	0	0
$221$	−6.68466	−0.449659
$222$	0	0
$223$	2.24621	0.150417	0.0752087	−	0.997168i	$-0.476038\pi$
$223$	2.24621	0.150417	0.0752087	+	0.997168i	$0.476038\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−25.3693	−1.68382	−0.841910	−	0.539617i	$-0.818569\pi$
$227$	−25.3693	−1.68382	−0.841910	+	0.539617i	$0.818569\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	−8.68466	−0.571409
$232$	0	0
$233$	18.6847	1.22407	0.612036	−	0.790830i	$-0.290351\pi$
$233$	18.6847	1.22407	0.612036	+	0.790830i	$0.290351\pi$
$234$	0	0
$235$	−7.12311	−0.464660
$236$	0	0
$237$	−11.8078	−0.766997
$238$	0	0
$239$	3.31534	0.214452	0.107226	−	0.994235i	$-0.465803\pi$
$239$	3.31534	0.214452	0.107226	+	0.994235i	$0.465803\pi$
$240$	0	0
$241$	11.7538	0.757128	0.378564	−	0.925575i	$-0.376418\pi$
$241$	11.7538	0.757128	0.378564	+	0.925575i	$0.376418\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.56155	−0.291427
$246$	0	0
$247$	−3.12311	−0.198718
$248$	0	0
$249$	8.00000	0.506979
$250$	0	0
$251$	−8.49242	−0.536037	−0.268018	−	0.963414i	$-0.586369\pi$
$251$	−8.49242	−0.536037	−0.268018	+	0.963414i	$0.586369\pi$
$252$	0	0
$253$	30.9309	1.94461
$254$	0	0
$255$	−6.68466	−0.418610
$256$	0	0
$257$	14.4924	0.904012	0.452006	−	0.892015i	$-0.350708\pi$
$257$	14.4924	0.904012	0.452006	+	0.892015i	$0.350708\pi$
$258$	0	0
$259$	−15.3153	−0.951649
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−28.4924	−1.75692	−0.878459	−	0.477818i	$-0.841428\pi$
$263$	−28.4924	−1.75692	−0.878459	+	0.477818i	$0.841428\pi$
$264$	0	0
$265$	3.56155	0.218784
$266$	0	0
$267$	−14.6847	−0.898687
$268$	0	0
$269$	−13.1231	−0.800130	−0.400065	−	0.916487i	$-0.631012\pi$
$269$	−13.1231	−0.800130	−0.400065	+	0.916487i	$0.631012\pi$
$270$	0	0
$271$	5.36932	0.326163	0.163081	−	0.986613i	$-0.447857\pi$
$271$	5.36932	0.326163	0.163081	+	0.986613i	$0.447857\pi$
$272$	0	0
$273$	−1.56155	−0.0945095
$274$	0	0
$275$	5.56155	0.335374
$276$	0	0
$277$	−22.4924	−1.35144	−0.675719	−	0.737159i	$-0.736167\pi$
$277$	−22.4924	−1.35144	−0.675719	+	0.737159i	$0.736167\pi$
$278$	0	0
$279$	7.12311	0.426449
$280$	0	0
$281$	3.75379	0.223932	0.111966	−	0.993712i	$-0.464285\pi$
$281$	3.75379	0.223932	0.111966	+	0.993712i	$0.464285\pi$
$282$	0	0
$283$	−30.7386	−1.82722	−0.913611	−	0.406589i	$-0.866718\pi$
$283$	−30.7386	−1.82722	−0.913611	+	0.406589i	$0.866718\pi$
$284$	0	0
$285$	−3.12311	−0.184997
$286$	0	0
$287$	−4.19224	−0.247460
$288$	0	0
$289$	27.6847	1.62851
$290$	0	0
$291$	−17.8078	−1.04391
$292$	0	0
$293$	−17.6155	−1.02911	−0.514555	−	0.857457i	$-0.672043\pi$
$293$	−17.6155	−1.02911	−0.514555	+	0.857457i	$0.672043\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	5.56155	0.322714
$298$	0	0
$299$	5.56155	0.321633
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	1.12311	0.0645207
$304$	0	0
$305$	−10.6847	−0.611802
$306$	0	0
$307$	−13.5616	−0.773999	−0.386999	−	0.922080i	$-0.626488\pi$
$307$	−13.5616	−0.773999	−0.386999	+	0.922080i	$0.626488\pi$
$308$	0	0
$309$	−14.2462	−0.810439
$310$	0	0
$311$	26.7386	1.51621	0.758104	−	0.652133i	$-0.226126\pi$
$311$	26.7386	1.51621	0.758104	+	0.652133i	$0.226126\pi$
$312$	0	0
$313$	−24.7386	−1.39831	−0.699155	−	0.714970i	$-0.746440\pi$
$313$	−24.7386	−1.39831	−0.699155	+	0.714970i	$0.746440\pi$
$314$	0	0
$315$	−1.56155	−0.0879835
$316$	0	0
$317$	2.87689	0.161582	0.0807912	−	0.996731i	$-0.474255\pi$
$317$	2.87689	0.161582	0.0807912	+	0.996731i	$0.474255\pi$
$318$	0	0
$319$	−11.1231	−0.622774
$320$	0	0
$321$	15.8078	0.882303
$322$	0	0
$323$	20.8769	1.16162
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−13.1231	−0.725709
$328$	0	0
$329$	11.1231	0.613237
$330$	0	0
$331$	−9.75379	−0.536117	−0.268058	−	0.963403i	$-0.586382\pi$
$331$	−9.75379	−0.536117	−0.268058	+	0.963403i	$0.586382\pi$
$332$	0	0
$333$	9.80776	0.537462
$334$	0	0
$335$	14.2462	0.778354
$336$	0	0
$337$	−2.87689	−0.156714	−0.0783572	−	0.996925i	$-0.524967\pi$
$337$	−2.87689	−0.156714	−0.0783572	+	0.996925i	$0.524967\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	39.6155	2.14530
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	5.56155	0.299424
$346$	0	0
$347$	6.05398	0.324994	0.162497	−	0.986709i	$-0.448045\pi$
$347$	6.05398	0.324994	0.162497	+	0.986709i	$0.448045\pi$
$348$	0	0
$349$	−0.630683	−0.0337597	−0.0168798	−	0.999858i	$-0.505373\pi$
$349$	−0.630683	−0.0337597	−0.0168798	+	0.999858i	$0.505373\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−4.24621	−0.226003	−0.113002	−	0.993595i	$-0.536046\pi$
$353$	−4.24621	−0.226003	−0.113002	+	0.993595i	$0.536046\pi$
$354$	0	0
$355$	−4.68466	−0.248636
$356$	0	0
$357$	10.4384	0.552461
$358$	0	0
$359$	2.24621	0.118550	0.0592752	−	0.998242i	$-0.481121\pi$
$359$	2.24621	0.118550	0.0592752	+	0.998242i	$0.481121\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	19.9309	1.04610
$364$	0	0
$365$	16.2462	0.850366
$366$	0	0
$367$	−14.2462	−0.743646	−0.371823	−	0.928304i	$-0.621267\pi$
$367$	−14.2462	−0.743646	−0.371823	+	0.928304i	$0.621267\pi$
$368$	0	0
$369$	2.68466	0.139758
$370$	0	0
$371$	−5.56155	−0.288741
$372$	0	0
$373$	−14.8769	−0.770296	−0.385148	−	0.922855i	$-0.625850\pi$
$373$	−14.8769	−0.770296	−0.385148	+	0.922855i	$0.625850\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−11.1231	−0.571356	−0.285678	−	0.958326i	$-0.592219\pi$
$379$	−11.1231	−0.571356	−0.285678	+	0.958326i	$0.592219\pi$
$380$	0	0
$381$	17.3693	0.889857
$382$	0	0
$383$	−2.63068	−0.134422	−0.0672108	−	0.997739i	$-0.521410\pi$
$383$	−2.63068	−0.134422	−0.0672108	+	0.997739i	$0.521410\pi$
$384$	0	0
$385$	−8.68466	−0.442611
$386$	0	0
$387$	10.2462	0.520844
$388$	0	0
$389$	35.8617	1.81826	0.909131	−	0.416510i	$-0.136747\pi$
$389$	35.8617	1.81826	0.909131	+	0.416510i	$0.136747\pi$
$390$	0	0
$391$	−37.1771	−1.88013
$392$	0	0
$393$	15.1231	0.762860
$394$	0	0
$395$	−11.8078	−0.594113
$396$	0	0
$397$	33.8078	1.69676	0.848382	−	0.529385i	$-0.177577\pi$
$397$	33.8078	1.69676	0.848382	+	0.529385i	$0.177577\pi$
$398$	0	0
$399$	4.87689	0.244150
$400$	0	0
$401$	−4.24621	−0.212046	−0.106023	−	0.994364i	$-0.533812\pi$
$401$	−4.24621	−0.212046	−0.106023	+	0.994364i	$0.533812\pi$
$402$	0	0
$403$	7.12311	0.354827
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	54.5464	2.70376
$408$	0	0
$409$	−18.4924	−0.914391	−0.457196	−	0.889366i	$-0.651146\pi$
$409$	−18.4924	−0.914391	−0.457196	+	0.889366i	$0.651146\pi$
$410$	0	0
$411$	16.2462	0.801367
$412$	0	0
$413$	−12.4924	−0.614712
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	9.56155	0.468231
$418$	0	0
$419$	−0.876894	−0.0428391	−0.0214195	−	0.999771i	$-0.506819\pi$
$419$	−0.876894	−0.0428391	−0.0214195	+	0.999771i	$0.506819\pi$
$420$	0	0
$421$	19.8617	0.968002	0.484001	−	0.875067i	$-0.339183\pi$
$421$	19.8617	0.968002	0.484001	+	0.875067i	$0.339183\pi$
$422$	0	0
$423$	−7.12311	−0.346337
$424$	0	0
$425$	−6.68466	−0.324254
$426$	0	0
$427$	16.6847	0.807427
$428$	0	0
$429$	5.56155	0.268514
$430$	0	0
$431$	−13.7538	−0.662497	−0.331248	−	0.943544i	$-0.607470\pi$
$431$	−13.7538	−0.662497	−0.331248	+	0.943544i	$0.607470\pi$
$432$	0	0
$433$	35.3693	1.69974	0.849870	−	0.526992i	$-0.176680\pi$
$433$	35.3693	1.69974	0.849870	+	0.526992i	$0.176680\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	−17.3693	−0.830887
$438$	0	0
$439$	−6.93087	−0.330792	−0.165396	−	0.986227i	$-0.552890\pi$
$439$	−6.93087	−0.330792	−0.165396	+	0.986227i	$0.552890\pi$
$440$	0	0
$441$	−4.56155	−0.217217
$442$	0	0
$443$	3.31534	0.157517	0.0787583	−	0.996894i	$-0.474904\pi$
$443$	3.31534	0.157517	0.0787583	+	0.996894i	$0.474904\pi$
$444$	0	0
$445$	−14.6847	−0.696120
$446$	0	0
$447$	−12.4384	−0.588318
$448$	0	0
$449$	10.6847	0.504240	0.252120	−	0.967696i	$-0.418872\pi$
$449$	10.6847	0.504240	0.252120	+	0.967696i	$0.418872\pi$
$450$	0	0
$451$	14.9309	0.703067
$452$	0	0
$453$	−4.00000	−0.187936
$454$	0	0
$455$	−1.56155	−0.0732067
$456$	0	0
$457$	−35.5616	−1.66350	−0.831750	−	0.555151i	$-0.812660\pi$
$457$	−35.5616	−1.66350	−0.831750	+	0.555151i	$0.812660\pi$
$458$	0	0
$459$	−6.68466	−0.312013
$460$	0	0
$461$	−15.5616	−0.724774	−0.362387	−	0.932028i	$-0.618038\pi$
$461$	−15.5616	−0.724774	−0.362387	+	0.932028i	$0.618038\pi$
$462$	0	0
$463$	6.43845	0.299220	0.149610	−	0.988745i	$-0.452198\pi$
$463$	6.43845	0.299220	0.149610	+	0.988745i	$0.452198\pi$
$464$	0	0
$465$	7.12311	0.330326
$466$	0	0
$467$	−25.1771	−1.16506	−0.582528	−	0.812811i	$-0.697936\pi$
$467$	−25.1771	−1.16506	−0.582528	+	0.812811i	$0.697936\pi$
$468$	0	0
$469$	−22.2462	−1.02723
$470$	0	0
$471$	10.4924	0.483465
$472$	0	0
$473$	56.9848	2.62017
$474$	0	0
$475$	−3.12311	−0.143298
$476$	0	0
$477$	3.56155	0.163072
$478$	0	0
$479$	−34.5464	−1.57847	−0.789233	−	0.614094i	$-0.789521\pi$
$479$	−34.5464	−1.57847	−0.789233	+	0.614094i	$0.789521\pi$
$480$	0	0
$481$	9.80776	0.447196
$482$	0	0
$483$	−8.68466	−0.395166
$484$	0	0
$485$	−17.8078	−0.808609
$486$	0	0
$487$	−4.68466	−0.212282	−0.106141	−	0.994351i	$-0.533850\pi$
$487$	−4.68466	−0.212282	−0.106141	+	0.994351i	$0.533850\pi$
$488$	0	0
$489$	−13.5616	−0.613275
$490$	0	0
$491$	7.50758	0.338812	0.169406	−	0.985546i	$-0.445815\pi$
$491$	7.50758	0.338812	0.169406	+	0.985546i	$0.445815\pi$
$492$	0	0
$493$	13.3693	0.602124
$494$	0	0
$495$	5.56155	0.249973
$496$	0	0
$497$	7.31534	0.328138
$498$	0	0
$499$	37.8617	1.69492	0.847462	−	0.530856i	$-0.178129\pi$
$499$	37.8617	1.69492	0.847462	+	0.530856i	$0.178129\pi$
$500$	0	0
$501$	5.75379	0.257060
$502$	0	0
$503$	−18.7386	−0.835514	−0.417757	−	0.908559i	$-0.637184\pi$
$503$	−18.7386	−0.835514	−0.417757	+	0.908559i	$0.637184\pi$
$504$	0	0
$505$	1.12311	0.0499775
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−42.6847	−1.89196	−0.945982	−	0.324219i	$-0.894899\pi$
$509$	−42.6847	−1.89196	−0.945982	+	0.324219i	$0.894899\pi$
$510$	0	0
$511$	−25.3693	−1.12227
$512$	0	0
$513$	−3.12311	−0.137888
$514$	0	0
$515$	−14.2462	−0.627763
$516$	0	0
$517$	−39.6155	−1.74229
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−13.7538	−0.601411	−0.300706	−	0.953717i	$-0.597222\pi$
$523$	−13.7538	−0.601411	−0.300706	+	0.953717i	$0.597222\pi$
$524$	0	0
$525$	−1.56155	−0.0681518
$526$	0	0
$527$	−47.6155	−2.07416
$528$	0	0
$529$	7.93087	0.344820
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	2.68466	0.116285
$534$	0	0
$535$	15.8078	0.683429
$536$	0	0
$537$	23.1231	0.997836
$538$	0	0
$539$	−25.3693	−1.09273
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	−8.93087	−0.383260
$544$	0	0
$545$	−13.1231	−0.562132
$546$	0	0
$547$	6.73863	0.288123	0.144062	−	0.989569i	$-0.453984\pi$
$547$	6.73863	0.288123	0.144062	+	0.989569i	$0.453984\pi$
$548$	0	0
$549$	−10.6847	−0.456010
$550$	0	0
$551$	6.24621	0.266098
$552$	0	0
$553$	18.4384	0.784083
$554$	0	0
$555$	9.80776	0.416316
$556$	0	0
$557$	−34.0000	−1.44063	−0.720313	−	0.693649i	$-0.756002\pi$
$557$	−34.0000	−1.44063	−0.720313	+	0.693649i	$0.756002\pi$
$558$	0	0
$559$	10.2462	0.433369
$560$	0	0
$561$	−37.1771	−1.56962
$562$	0	0
$563$	25.5616	1.07729	0.538646	−	0.842533i	$-0.318936\pi$
$563$	25.5616	1.07729	0.538646	+	0.842533i	$0.318936\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	−1.56155	−0.0655791
$568$	0	0
$569$	−1.12311	−0.0470830	−0.0235415	−	0.999723i	$-0.507494\pi$
$569$	−1.12311	−0.0470830	−0.0235415	+	0.999723i	$0.507494\pi$
$570$	0	0
$571$	−25.5616	−1.06972	−0.534859	−	0.844941i	$-0.679635\pi$
$571$	−25.5616	−1.06972	−0.534859	+	0.844941i	$0.679635\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	5.56155	0.231933
$576$	0	0
$577$	−36.5464	−1.52145	−0.760723	−	0.649076i	$-0.775156\pi$
$577$	−36.5464	−1.52145	−0.760723	+	0.649076i	$0.775156\pi$
$578$	0	0
$579$	9.31534	0.387132
$580$	0	0
$581$	−12.4924	−0.518273
$582$	0	0
$583$	19.8078	0.820354
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	17.3693	0.716908	0.358454	−	0.933547i	$-0.383304\pi$
$587$	17.3693	0.716908	0.358454	+	0.933547i	$0.383304\pi$
$588$	0	0
$589$	−22.2462	−0.916639
$590$	0	0
$591$	7.36932	0.303133
$592$	0	0
$593$	−5.61553	−0.230602	−0.115301	−	0.993331i	$-0.536783\pi$
$593$	−5.61553	−0.230602	−0.115301	+	0.993331i	$0.536783\pi$
$594$	0	0
$595$	10.4384	0.427935
$596$	0	0
$597$	−1.75379	−0.0717778
$598$	0	0
$599$	18.7386	0.765640	0.382820	−	0.923823i	$-0.374953\pi$
$599$	18.7386	0.765640	0.382820	+	0.923823i	$0.374953\pi$
$600$	0	0
$601$	−3.56155	−0.145279	−0.0726394	−	0.997358i	$-0.523142\pi$
$601$	−3.56155	−0.145279	−0.0726394	+	0.997358i	$0.523142\pi$
$602$	0	0
$603$	14.2462	0.580151
$604$	0	0
$605$	19.9309	0.810305
$606$	0	0
$607$	−26.7386	−1.08529	−0.542644	−	0.839963i	$-0.682577\pi$
$607$	−26.7386	−1.08529	−0.542644	+	0.839963i	$0.682577\pi$
$608$	0	0
$609$	3.12311	0.126555
$610$	0	0
$611$	−7.12311	−0.288170
$612$	0	0
$613$	4.93087	0.199156	0.0995780	−	0.995030i	$-0.468251\pi$
$613$	4.93087	0.199156	0.0995780	+	0.995030i	$0.468251\pi$
$614$	0	0
$615$	2.68466	0.108256
$616$	0	0
$617$	0.246211	0.00991209	0.00495605	−	0.999988i	$-0.498422\pi$
$617$	0.246211	0.00991209	0.00495605	+	0.999988i	$0.498422\pi$
$618$	0	0
$619$	−6.63068	−0.266510	−0.133255	−	0.991082i	$-0.542543\pi$
$619$	−6.63068	−0.266510	−0.133255	+	0.991082i	$0.542543\pi$
$620$	0	0
$621$	5.56155	0.223177
$622$	0	0
$623$	22.9309	0.918706
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.3693	−0.693664
$628$	0	0
$629$	−65.5616	−2.61411
$630$	0	0
$631$	−26.2462	−1.04485	−0.522423	−	0.852687i	$-0.674972\pi$
$631$	−26.2462	−1.04485	−0.522423	+	0.852687i	$0.674972\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	17.3693	0.689280
$636$	0	0
$637$	−4.56155	−0.180735
$638$	0	0
$639$	−4.68466	−0.185322
$640$	0	0
$641$	0.630683	0.0249105	0.0124552	−	0.999922i	$-0.496035\pi$
$641$	0.630683	0.0249105	0.0124552	+	0.999922i	$0.496035\pi$
$642$	0	0
$643$	−11.8078	−0.465653	−0.232826	−	0.972518i	$-0.574797\pi$
$643$	−11.8078	−0.465653	−0.232826	+	0.972518i	$0.574797\pi$
$644$	0	0
$645$	10.2462	0.403444
$646$	0	0
$647$	0.684658	0.0269167	0.0134584	−	0.999909i	$-0.495716\pi$
$647$	0.684658	0.0269167	0.0134584	+	0.999909i	$0.495716\pi$
$648$	0	0
$649$	44.4924	1.74648
$650$	0	0
$651$	−11.1231	−0.435949
$652$	0	0
$653$	23.7538	0.929558	0.464779	−	0.885427i	$-0.346134\pi$
$653$	23.7538	0.929558	0.464779	+	0.885427i	$0.346134\pi$
$654$	0	0
$655$	15.1231	0.590909
$656$	0	0
$657$	16.2462	0.633825
$658$	0	0
$659$	−8.49242	−0.330818	−0.165409	−	0.986225i	$-0.552894\pi$
$659$	−8.49242	−0.330818	−0.165409	+	0.986225i	$0.552894\pi$
$660$	0	0
$661$	−30.8769	−1.20097	−0.600486	−	0.799635i	$-0.705026\pi$
$661$	−30.8769	−1.20097	−0.600486	+	0.799635i	$0.705026\pi$
$662$	0	0
$663$	−6.68466	−0.259611
$664$	0	0
$665$	4.87689	0.189118
$666$	0	0
$667$	−11.1231	−0.430688
$668$	0	0
$669$	2.24621	0.0868435
$670$	0	0
$671$	−59.4233	−2.29401
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	7.06913	0.271689	0.135844	−	0.990730i	$-0.456625\pi$
$677$	7.06913	0.271689	0.135844	+	0.990730i	$0.456625\pi$
$678$	0	0
$679$	27.8078	1.06716
$680$	0	0
$681$	−25.3693	−0.972154
$682$	0	0
$683$	−4.49242	−0.171898	−0.0859489	−	0.996300i	$-0.527392\pi$
$683$	−4.49242	−0.171898	−0.0859489	+	0.996300i	$0.527392\pi$
$684$	0	0
$685$	16.2462	0.620736
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	3.56155	0.135684
$690$	0	0
$691$	−28.8769	−1.09853	−0.549264	−	0.835649i	$-0.685092\pi$
$691$	−28.8769	−1.09853	−0.549264	+	0.835649i	$0.685092\pi$
$692$	0	0
$693$	−8.68466	−0.329903
$694$	0	0
$695$	9.56155	0.362690
$696$	0	0
$697$	−17.9460	−0.679754
$698$	0	0
$699$	18.6847	0.706719
$700$	0	0
$701$	−19.3693	−0.731569	−0.365785	−	0.930700i	$-0.619199\pi$
$701$	−19.3693	−0.731569	−0.365785	+	0.930700i	$0.619199\pi$
$702$	0	0
$703$	−30.6307	−1.15526
$704$	0	0
$705$	−7.12311	−0.268272
$706$	0	0
$707$	−1.75379	−0.0659580
$708$	0	0
$709$	17.1231	0.643072	0.321536	−	0.946897i	$-0.395801\pi$
$709$	17.1231	0.643072	0.321536	+	0.946897i	$0.395801\pi$
$710$	0	0
$711$	−11.8078	−0.442826
$712$	0	0
$713$	39.6155	1.48361
$714$	0	0
$715$	5.56155	0.207990
$716$	0	0
$717$	3.31534	0.123814
$718$	0	0
$719$	−9.75379	−0.363755	−0.181877	−	0.983321i	$-0.558217\pi$
$719$	−9.75379	−0.363755	−0.181877	+	0.983321i	$0.558217\pi$
$720$	0	0
$721$	22.2462	0.828492
$722$	0	0
$723$	11.7538	0.437128
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	22.6307	0.839326	0.419663	−	0.907680i	$-0.362148\pi$
$727$	22.6307	0.839326	0.419663	+	0.907680i	$0.362148\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−68.4924	−2.53328
$732$	0	0
$733$	−29.4233	−1.08677	−0.543387	−	0.839482i	$-0.682858\pi$
$733$	−29.4233	−1.08677	−0.543387	+	0.839482i	$0.682858\pi$
$734$	0	0
$735$	−4.56155	−0.168255
$736$	0	0
$737$	79.2311	2.91851
$738$	0	0
$739$	−29.8617	−1.09848	−0.549241	−	0.835664i	$-0.685083\pi$
$739$	−29.8617	−1.09848	−0.549241	+	0.835664i	$0.685083\pi$
$740$	0	0
$741$	−3.12311	−0.114730
$742$	0	0
$743$	8.87689	0.325662	0.162831	−	0.986654i	$-0.447938\pi$
$743$	8.87689	0.325662	0.162831	+	0.986654i	$0.447938\pi$
$744$	0	0
$745$	−12.4384	−0.455709
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	−24.6847	−0.901958
$750$	0	0
$751$	7.31534	0.266941	0.133470	−	0.991053i	$-0.457388\pi$
$751$	7.31534	0.266941	0.133470	+	0.991053i	$0.457388\pi$
$752$	0	0
$753$	−8.49242	−0.309481
$754$	0	0
$755$	−4.00000	−0.145575
$756$	0	0
$757$	−14.8769	−0.540710	−0.270355	−	0.962761i	$-0.587141\pi$
$757$	−14.8769	−0.540710	−0.270355	+	0.962761i	$0.587141\pi$
$758$	0	0
$759$	30.9309	1.12272
$760$	0	0
$761$	22.4924	0.815350	0.407675	−	0.913127i	$-0.366340\pi$
$761$	22.4924	0.815350	0.407675	+	0.913127i	$0.366340\pi$
$762$	0	0
$763$	20.4924	0.741876
$764$	0	0
$765$	−6.68466	−0.241684
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	25.6155	0.923720	0.461860	−	0.886953i	$-0.347182\pi$
$769$	25.6155	0.923720	0.461860	+	0.886953i	$0.347182\pi$
$770$	0	0
$771$	14.4924	0.521932
$772$	0	0
$773$	−21.1231	−0.759745	−0.379873	−	0.925039i	$-0.624032\pi$
$773$	−21.1231	−0.759745	−0.379873	+	0.925039i	$0.624032\pi$
$774$	0	0
$775$	7.12311	0.255870
$776$	0	0
$777$	−15.3153	−0.549435
$778$	0	0
$779$	−8.38447	−0.300405
$780$	0	0
$781$	−26.0540	−0.932285
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	10.4924	0.374491
$786$	0	0
$787$	18.7386	0.667960	0.333980	−	0.942580i	$-0.391608\pi$
$787$	18.7386	0.667960	0.333980	+	0.942580i	$0.391608\pi$
$788$	0	0
$789$	−28.4924	−1.01436
$790$	0	0
$791$	−3.12311	−0.111045
$792$	0	0
$793$	−10.6847	−0.379423
$794$	0	0
$795$	3.56155	0.126315
$796$	0	0
$797$	41.4233	1.46729	0.733644	−	0.679534i	$-0.237818\pi$
$797$	41.4233	1.46729	0.733644	+	0.679534i	$0.237818\pi$
$798$	0	0
$799$	47.6155	1.68452
$800$	0	0
$801$	−14.6847	−0.518857
$802$	0	0
$803$	90.3542	3.18853
$804$	0	0
$805$	−8.68466	−0.306094
$806$	0	0
$807$	−13.1231	−0.461955
$808$	0	0
$809$	−18.4924	−0.650159	−0.325079	−	0.945687i	$-0.605391\pi$
$809$	−18.4924	−0.650159	−0.325079	+	0.945687i	$0.605391\pi$
$810$	0	0
$811$	−40.9848	−1.43917	−0.719586	−	0.694403i	$-0.755669\pi$
$811$	−40.9848	−1.43917	−0.719586	+	0.694403i	$0.755669\pi$
$812$	0	0
$813$	5.36932	0.188310
$814$	0	0
$815$	−13.5616	−0.475040
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	−1.56155	−0.0545651
$820$	0	0
$821$	−33.3153	−1.16271	−0.581357	−	0.813649i	$-0.697478\pi$
$821$	−33.3153	−1.16271	−0.581357	+	0.813649i	$0.697478\pi$
$822$	0	0
$823$	11.5076	0.401129	0.200564	−	0.979681i	$-0.435722\pi$
$823$	11.5076	0.401129	0.200564	+	0.979681i	$0.435722\pi$
$824$	0	0
$825$	5.56155	0.193628
$826$	0	0
$827$	−22.6307	−0.786946	−0.393473	−	0.919336i	$-0.628727\pi$
$827$	−22.6307	−0.786946	−0.393473	+	0.919336i	$0.628727\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	0	0
$831$	−22.4924	−0.780253
$832$	0	0
$833$	30.4924	1.05650
$834$	0	0
$835$	5.75379	0.199118
$836$	0	0
$837$	7.12311	0.246211
$838$	0	0
$839$	31.8078	1.09813	0.549063	−	0.835781i	$-0.314985\pi$
$839$	31.8078	1.09813	0.549063	+	0.835781i	$0.314985\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	3.75379	0.129287
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−31.1231	−1.06940
$848$	0	0
$849$	−30.7386	−1.05495
$850$	0	0
$851$	54.5464	1.86983
$852$	0	0
$853$	−39.5616	−1.35456	−0.677281	−	0.735725i	$-0.736842\pi$
$853$	−39.5616	−1.35456	−0.677281	+	0.735725i	$0.736842\pi$
$854$	0	0
$855$	−3.12311	−0.106808
$856$	0	0
$857$	−34.7926	−1.18849	−0.594246	−	0.804283i	$-0.702550\pi$
$857$	−34.7926	−1.18849	−0.594246	+	0.804283i	$0.702550\pi$
$858$	0	0
$859$	−32.1922	−1.09838	−0.549192	−	0.835696i	$-0.685065\pi$
$859$	−32.1922	−1.09838	−0.549192	+	0.835696i	$0.685065\pi$
$860$	0	0
$861$	−4.19224	−0.142871
$862$	0	0
$863$	−1.86174	−0.0633743	−0.0316872	−	0.999498i	$-0.510088\pi$
$863$	−1.86174	−0.0633743	−0.0316872	+	0.999498i	$0.510088\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	27.6847	0.940220
$868$	0	0
$869$	−65.6695	−2.22769
$870$	0	0
$871$	14.2462	0.482714
$872$	0	0
$873$	−17.8078	−0.602701
$874$	0	0
$875$	−1.56155	−0.0527901
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	0	0
$879$	−17.6155	−0.594157
$880$	0	0
$881$	8.24621	0.277822	0.138911	−	0.990305i	$-0.455640\pi$
$881$	8.24621	0.277822	0.138911	+	0.990305i	$0.455640\pi$
$882$	0	0
$883$	−32.4924	−1.09346	−0.546729	−	0.837310i	$-0.684127\pi$
$883$	−32.4924	−1.09346	−0.546729	+	0.837310i	$0.684127\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	−26.0540	−0.874807	−0.437403	−	0.899265i	$-0.644102\pi$
$887$	−26.0540	−0.874807	−0.437403	+	0.899265i	$0.644102\pi$
$888$	0	0
$889$	−27.1231	−0.909680
$890$	0	0
$891$	5.56155	0.186319
$892$	0	0
$893$	22.2462	0.744441
$894$	0	0
$895$	23.1231	0.772920
$896$	0	0
$897$	5.56155	0.185695
$898$	0	0
$899$	−14.2462	−0.475138
$900$	0	0
$901$	−23.8078	−0.793152
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	−8.93087	−0.296872
$906$	0	0
$907$	47.1231	1.56470	0.782349	−	0.622841i	$-0.214022\pi$
$907$	47.1231	1.56470	0.782349	+	0.622841i	$0.214022\pi$
$908$	0	0
$909$	1.12311	0.0372511
$910$	0	0
$911$	−17.7538	−0.588209	−0.294105	−	0.955773i	$-0.595021\pi$
$911$	−17.7538	−0.588209	−0.294105	+	0.955773i	$0.595021\pi$
$912$	0	0
$913$	44.4924	1.47248
$914$	0	0
$915$	−10.6847	−0.353224
$916$	0	0
$917$	−23.6155	−0.779853
$918$	0	0
$919$	12.1922	0.402185	0.201092	−	0.979572i	$-0.435551\pi$
$919$	12.1922	0.402185	0.201092	+	0.979572i	$0.435551\pi$
$920$	0	0
$921$	−13.5616	−0.446868
$922$	0	0
$923$	−4.68466	−0.154197
$924$	0	0
$925$	9.80776	0.322477
$926$	0	0
$927$	−14.2462	−0.467907
$928$	0	0
$929$	18.3002	0.600410	0.300205	−	0.953875i	$-0.402945\pi$
$929$	18.3002	0.600410	0.300205	+	0.953875i	$0.402945\pi$
$930$	0	0
$931$	14.2462	0.466901
$932$	0	0
$933$	26.7386	0.875384
$934$	0	0
$935$	−37.1771	−1.21582
$936$	0	0
$937$	−31.7538	−1.03735	−0.518676	−	0.854971i	$-0.673575\pi$
$937$	−31.7538	−1.03735	−0.518676	+	0.854971i	$0.673575\pi$
$938$	0	0
$939$	−24.7386	−0.807315
$940$	0	0
$941$	20.5464	0.669793	0.334897	−	0.942255i	$-0.391299\pi$
$941$	20.5464	0.669793	0.334897	+	0.942255i	$0.391299\pi$
$942$	0	0
$943$	14.9309	0.486216
$944$	0	0
$945$	−1.56155	−0.0507973
$946$	0	0
$947$	50.7386	1.64878	0.824392	−	0.566019i	$-0.191517\pi$
$947$	50.7386	1.64878	0.824392	+	0.566019i	$0.191517\pi$
$948$	0	0
$949$	16.2462	0.527374
$950$	0	0
$951$	2.87689	0.0932897
$952$	0	0
$953$	29.8078	0.965568	0.482784	−	0.875739i	$-0.339626\pi$
$953$	29.8078	0.965568	0.482784	+	0.875739i	$0.339626\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	−11.1231	−0.359559
$958$	0	0
$959$	−25.3693	−0.819218
$960$	0	0
$961$	19.7386	0.636730
$962$	0	0
$963$	15.8078	0.509398
$964$	0	0
$965$	9.31534	0.299871
$966$	0	0
$967$	42.2462	1.35855	0.679273	−	0.733885i	$-0.262295\pi$
$967$	42.2462	1.35855	0.679273	+	0.733885i	$0.262295\pi$
$968$	0	0
$969$	20.8769	0.670662
$970$	0	0
$971$	8.49242	0.272535	0.136267	−	0.990672i	$-0.456489\pi$
$971$	8.49242	0.272535	0.136267	+	0.990672i	$0.456489\pi$
$972$	0	0
$973$	−14.9309	−0.478662
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−43.8617	−1.40326	−0.701631	−	0.712541i	$-0.747544\pi$
$977$	−43.8617	−1.40326	−0.701631	+	0.712541i	$0.747544\pi$
$978$	0	0
$979$	−81.6695	−2.61017
$980$	0	0
$981$	−13.1231	−0.418989
$982$	0	0
$983$	52.9848	1.68995	0.844977	−	0.534803i	$-0.179614\pi$
$983$	52.9848	1.68995	0.844977	+	0.534803i	$0.179614\pi$
$984$	0	0
$985$	7.36932	0.234806
$986$	0	0
$987$	11.1231	0.354052
$988$	0	0
$989$	56.9848	1.81201
$990$	0	0
$991$	34.0540	1.08176	0.540880	−	0.841100i	$-0.318091\pi$
$991$	34.0540	1.08176	0.540880	+	0.841100i	$0.318091\pi$
$992$	0	0
$993$	−9.75379	−0.309527
$994$	0	0
$995$	−1.75379	−0.0555988
$996$	0	0
$997$	10.8769	0.344475	0.172237	−	0.985055i	$-0.444900\pi$
$997$	10.8769	0.344475	0.172237	+	0.985055i	$0.444900\pi$
$998$	0	0
$999$	9.80776	0.310304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.bg.1.1		2
3.2	odd	2		9360.2.a.ci.1.1		2
4.3	odd	2		1560.2.a.o.1.2	✓	2
12.11	even	2		4680.2.a.y.1.2		2
20.19	odd	2		7800.2.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.a.o.1.2	✓	2	4.3	odd	2
3120.2.a.bg.1.1		2	1.1	even	1	trivial
4680.2.a.y.1.2		2	12.11	even	2
7800.2.a.bd.1.1		2	20.19	odd	2
9360.2.a.ci.1.1		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +5.56155 q^{11} +1.00000 q^{13} +1.00000 q^{15} -6.68466 q^{17} -3.12311 q^{19} -1.56155 q^{21} +5.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} +7.12311 q^{31} +5.56155 q^{33} -1.56155 q^{35} +9.80776 q^{37} +1.00000 q^{39} +2.68466 q^{41} +10.2462 q^{43} +1.00000 q^{45} -7.12311 q^{47} -4.56155 q^{49} -6.68466 q^{51} +3.56155 q^{53} +5.56155 q^{55} -3.12311 q^{57} +8.00000 q^{59} -10.6847 q^{61} -1.56155 q^{63} +1.00000 q^{65} +14.2462 q^{67} +5.56155 q^{69} -4.68466 q^{71} +16.2462 q^{73} +1.00000 q^{75} -8.68466 q^{77} -11.8078 q^{79} +1.00000 q^{81} +8.00000 q^{83} -6.68466 q^{85} -2.00000 q^{87} -14.6847 q^{89} -1.56155 q^{91} +7.12311 q^{93} -3.12311 q^{95} -17.8078 q^{97} +5.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 2 q^{13} + 2 q^{15} - q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 6 q^{31} + 7 q^{33} + q^{35} - q^{37} + 2 q^{39} - 7 q^{41} + 4 q^{43} + 2 q^{45} - 6 q^{47} - 5 q^{49} - q^{51} + 3 q^{53} + 7 q^{55} + 2 q^{57} + 16 q^{59} - 9 q^{61} + q^{63} + 2 q^{65} + 12 q^{67} + 7 q^{69} + 3 q^{71} + 16 q^{73} + 2 q^{75} - 5 q^{77} - 3 q^{79} + 2 q^{81} + 16 q^{83} - q^{85} - 4 q^{87} - 17 q^{89} + q^{91} + 6 q^{93} + 2 q^{95} - 15 q^{97} + 7 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + q^7 + 2 * q^9 + 7 * q^11 + 2 * q^13 + 2 * q^15 - q^17 + 2 * q^19 + q^21 + 7 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 6 * q^31 + 7 * q^33 + q^35 - q^37 + 2 * q^39 - 7 * q^41 + 4 * q^43 + 2 * q^45 - 6 * q^47 - 5 * q^49 - q^51 + 3 * q^53 + 7 * q^55 + 2 * q^57 + 16 * q^59 - 9 * q^61 + q^63 + 2 * q^65 + 12 * q^67 + 7 * q^69 + 3 * q^71 + 16 * q^73 + 2 * q^75 - 5 * q^77 - 3 * q^79 + 2 * q^81 + 16 * q^83 - q^85 - 4 * q^87 - 17 * q^89 + q^91 + 6 * q^93 + 2 * q^95 - 15 * q^97 + 7 * q^99

Introduction

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