LMFDB - Embedded newform 3840.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(1,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, 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("3840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3840 = 2^{8} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3840.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:	$30.6625543762$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	3840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -4.68585 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -4.68585 q^{7} +1.00000 q^{9} +2.29273 q^{11} +4.97858 q^{13} -1.00000 q^{15} -2.97858 q^{17} -2.68585 q^{19} -4.68585 q^{21} -2.68585 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -6.97858 q^{31} +2.29273 q^{33} +4.68585 q^{35} +4.39312 q^{37} +4.97858 q^{39} +11.3717 q^{41} +9.37169 q^{43} -1.00000 q^{45} +7.27131 q^{47} +14.9572 q^{49} -2.97858 q^{51} -2.00000 q^{53} -2.29273 q^{55} -2.68585 q^{57} +1.70727 q^{59} -4.58546 q^{61} -4.68585 q^{63} -4.97858 q^{65} -4.00000 q^{67} -2.68585 q^{69} -0.585462 q^{71} +6.00000 q^{73} +1.00000 q^{75} -10.7434 q^{77} +1.02142 q^{79} +1.00000 q^{81} +13.3717 q^{83} +2.97858 q^{85} -2.00000 q^{87} -3.37169 q^{89} -23.3288 q^{91} -6.97858 q^{93} +2.68585 q^{95} -3.95715 q^{97} +2.29273 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 4 q^{11} - 3 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 4 q^{23} + 3 q^{25} + 3 q^{27} - 6 q^{29} - 6 q^{31} + 4 q^{33} + 2 q^{35} + 4 q^{37} + 10 q^{41} + 4 q^{43} - 3 q^{45} + 4 q^{47} + 15 q^{49} + 6 q^{51} - 6 q^{53} - 4 q^{55} + 4 q^{57} + 8 q^{59} - 8 q^{61} - 2 q^{63} - 12 q^{67} + 4 q^{69} + 4 q^{71} + 18 q^{73} + 3 q^{75} + 16 q^{77} + 18 q^{79} + 3 q^{81} + 16 q^{83} - 6 q^{85} - 6 q^{87} + 14 q^{89} - 16 q^{91} - 6 q^{93} - 4 q^{95} + 18 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 + 4 * q^11 - 3 * q^15 + 6 * q^17 + 4 * q^19 - 2 * q^21 + 4 * q^23 + 3 * q^25 + 3 * q^27 - 6 * q^29 - 6 * q^31 + 4 * q^33 + 2 * q^35 + 4 * q^37 + 10 * q^41 + 4 * q^43 - 3 * q^45 + 4 * q^47 + 15 * q^49 + 6 * q^51 - 6 * q^53 - 4 * q^55 + 4 * q^57 + 8 * q^59 - 8 * q^61 - 2 * q^63 - 12 * q^67 + 4 * q^69 + 4 * q^71 + 18 * q^73 + 3 * q^75 + 16 * q^77 + 18 * q^79 + 3 * q^81 + 16 * q^83 - 6 * q^85 - 6 * q^87 + 14 * q^89 - 16 * q^91 - 6 * q^93 - 4 * q^95 + 18 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.68585	−1.77108	−0.885542	−	0.464560i	$-0.846213\pi$
$7$	−4.68585	−1.77108	−0.885542	+	0.464560i	$0.846213\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.29273	0.691284	0.345642	−	0.938366i	$-0.387661\pi$
$11$	2.29273	0.691284	0.345642	+	0.938366i	$0.387661\pi$
$12$	0	0
$13$	4.97858	1.38081	0.690404	−	0.723424i	$-0.257433\pi$
$13$	4.97858	1.38081	0.690404	+	0.723424i	$0.257433\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.97858	−0.722411	−0.361206	−	0.932486i	$-0.617635\pi$
$17$	−2.97858	−0.722411	−0.361206	+	0.932486i	$0.617635\pi$
$18$	0	0
$19$	−2.68585	−0.616175	−0.308088	−	0.951358i	$-0.599689\pi$
$19$	−2.68585	−0.616175	−0.308088	+	0.951358i	$0.599689\pi$
$20$	0	0
$21$	−4.68585	−1.02254
$22$	0	0
$23$	−2.68585	−0.560038	−0.280019	−	0.959995i	$-0.590341\pi$
$23$	−2.68585	−0.560038	−0.280019	+	0.959995i	$0.590341\pi$
$24$	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
$31$	−6.97858	−1.25339	−0.626695	−	0.779265i	$-0.715593\pi$
$31$	−6.97858	−1.25339	−0.626695	+	0.779265i	$0.715593\pi$
$32$	0	0
$33$	2.29273	0.399113
$34$	0	0
$35$	4.68585	0.792053
$36$	0	0
$37$	4.39312	0.722224	0.361112	−	0.932523i	$-0.382397\pi$
$37$	4.39312	0.722224	0.361112	+	0.932523i	$0.382397\pi$
$38$	0	0
$39$	4.97858	0.797210
$40$	0	0
$41$	11.3717	1.77596	0.887980	−	0.459882i	$-0.152108\pi$
$41$	11.3717	1.77596	0.887980	+	0.459882i	$0.152108\pi$
$42$	0	0
$43$	9.37169	1.42917	0.714585	−	0.699549i	$-0.246616\pi$
$43$	9.37169	1.42917	0.714585	+	0.699549i	$0.246616\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	7.27131	1.06063	0.530315	−	0.847801i	$-0.322074\pi$
$47$	7.27131	1.06063	0.530315	+	0.847801i	$0.322074\pi$
$48$	0	0
$49$	14.9572	2.13674
$50$	0	0
$51$	−2.97858	−0.417084
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−2.29273	−0.309152
$56$	0	0
$57$	−2.68585	−0.355749
$58$	0	0
$59$	1.70727	0.222267	0.111134	−	0.993805i	$-0.464552\pi$
$59$	1.70727	0.222267	0.111134	+	0.993805i	$0.464552\pi$
$60$	0	0
$61$	−4.58546	−0.587108	−0.293554	−	0.955942i	$-0.594838\pi$
$61$	−4.58546	−0.587108	−0.293554	+	0.955942i	$0.594838\pi$
$62$	0	0
$63$	−4.68585	−0.590361
$64$	0	0
$65$	−4.97858	−0.617516
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−2.68585	−0.323338
$70$	0	0
$71$	−0.585462	−0.0694816	−0.0347408	−	0.999396i	$-0.511061\pi$
$71$	−0.585462	−0.0694816	−0.0347408	+	0.999396i	$0.511061\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−10.7434	−1.22432
$78$	0	0
$79$	1.02142	0.114919	0.0574595	−	0.998348i	$-0.481700\pi$
$79$	1.02142	0.114919	0.0574595	+	0.998348i	$0.481700\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.3717	1.46773	0.733867	−	0.679293i	$-0.237714\pi$
$83$	13.3717	1.46773	0.733867	+	0.679293i	$0.237714\pi$
$84$	0	0
$85$	2.97858	0.323072
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−3.37169	−0.357399	−0.178699	−	0.983904i	$-0.557189\pi$
$89$	−3.37169	−0.357399	−0.178699	+	0.983904i	$0.557189\pi$
$90$	0	0
$91$	−23.3288	−2.44553
$92$	0	0
$93$	−6.97858	−0.723645
$94$	0	0
$95$	2.68585	0.275562
$96$	0	0
$97$	−3.95715	−0.401788	−0.200894	−	0.979613i	$-0.564385\pi$
$97$	−3.95715	−0.401788	−0.200894	+	0.979613i	$0.564385\pi$
$98$	0	0
$99$	2.29273	0.230428
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	14.6430	1.44282	0.721409	−	0.692509i	$-0.243495\pi$
$103$	14.6430	1.44282	0.721409	+	0.692509i	$0.243495\pi$
$104$	0	0
$105$	4.68585	0.457292
$106$	0	0
$107$	−11.3288	−1.09520	−0.547600	−	0.836740i	$-0.684459\pi$
$107$	−11.3288	−1.09520	−0.547600	+	0.836740i	$0.684459\pi$
$108$	0	0
$109$	9.37169	0.897645	0.448823	−	0.893621i	$-0.351843\pi$
$109$	9.37169	0.897645	0.448823	+	0.893621i	$0.351843\pi$
$110$	0	0
$111$	4.39312	0.416976
$112$	0	0
$113$	19.7648	1.85932	0.929658	−	0.368423i	$-0.120102\pi$
$113$	19.7648	1.85932	0.929658	+	0.368423i	$0.120102\pi$
$114$	0	0
$115$	2.68585	0.250456
$116$	0	0
$117$	4.97858	0.460270
$118$	0	0
$119$	13.9572	1.27945
$120$	0	0
$121$	−5.74338	−0.522126
$122$	0	0
$123$	11.3717	1.02535
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.64300	0.589471	0.294735	−	0.955579i	$-0.404768\pi$
$127$	6.64300	0.589471	0.294735	+	0.955579i	$0.404768\pi$
$128$	0	0
$129$	9.37169	0.825132
$130$	0	0
$131$	−7.07896	−0.618492	−0.309246	−	0.950982i	$-0.600077\pi$
$131$	−7.07896	−0.618492	−0.309246	+	0.950982i	$0.600077\pi$
$132$	0	0
$133$	12.5855	1.09130
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	14.9786	1.27971	0.639853	−	0.768497i	$-0.278995\pi$
$137$	14.9786	1.27971	0.639853	+	0.768497i	$0.278995\pi$
$138$	0	0
$139$	4.64300	0.393814	0.196907	−	0.980422i	$-0.436910\pi$
$139$	4.64300	0.393814	0.196907	+	0.980422i	$0.436910\pi$
$140$	0	0
$141$	7.27131	0.612355
$142$	0	0
$143$	11.4145	0.954532
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	14.9572	1.23365
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	8.35027	0.679535	0.339768	−	0.940509i	$-0.389652\pi$
$151$	8.35027	0.679535	0.339768	+	0.940509i	$0.389652\pi$
$152$	0	0
$153$	−2.97858	−0.240804
$154$	0	0
$155$	6.97858	0.560533
$156$	0	0
$157$	22.3503	1.78375	0.891873	−	0.452286i	$-0.149391\pi$
$157$	22.3503	1.78375	0.891873	+	0.452286i	$0.149391\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	12.5855	0.991873
$162$	0	0
$163$	1.37169	0.107439	0.0537196	−	0.998556i	$-0.482892\pi$
$163$	1.37169	0.107439	0.0537196	+	0.998556i	$0.482892\pi$
$164$	0	0
$165$	−2.29273	−0.178489
$166$	0	0
$167$	−11.2713	−0.872200	−0.436100	−	0.899898i	$-0.643641\pi$
$167$	−11.2713	−0.872200	−0.436100	+	0.899898i	$0.643641\pi$
$168$	0	0
$169$	11.7862	0.906633
$170$	0	0
$171$	−2.68585	−0.205392
$172$	0	0
$173$	10.7862	0.820062	0.410031	−	0.912072i	$-0.365518\pi$
$173$	10.7862	0.820062	0.410031	+	0.912072i	$0.365518\pi$
$174$	0	0
$175$	−4.68585	−0.354217
$176$	0	0
$177$	1.70727	0.128326
$178$	0	0
$179$	−3.66442	−0.273892	−0.136946	−	0.990579i	$-0.543729\pi$
$179$	−3.66442	−0.273892	−0.136946	+	0.990579i	$0.543729\pi$
$180$	0	0
$181$	−6.62831	−0.492678	−0.246339	−	0.969184i	$-0.579228\pi$
$181$	−6.62831	−0.492678	−0.246339	+	0.969184i	$0.579228\pi$
$182$	0	0
$183$	−4.58546	−0.338967
$184$	0	0
$185$	−4.39312	−0.322988
$186$	0	0
$187$	−6.82908	−0.499392
$188$	0	0
$189$	−4.68585	−0.340845
$190$	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
$193$	1.21377	0.0873690	0.0436845	−	0.999045i	$-0.486090\pi$
$193$	1.21377	0.0873690	0.0436845	+	0.999045i	$0.486090\pi$
$194$	0	0
$195$	−4.97858	−0.356523
$196$	0	0
$197$	23.9572	1.70688	0.853438	−	0.521194i	$-0.174513\pi$
$197$	23.9572	1.70688	0.853438	+	0.521194i	$0.174513\pi$
$198$	0	0
$199$	−0.350269	−0.0248299	−0.0124150	−	0.999923i	$-0.503952\pi$
$199$	−0.350269	−0.0248299	−0.0124150	+	0.999923i	$0.503952\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	9.37169	0.657764
$204$	0	0
$205$	−11.3717	−0.794233
$206$	0	0
$207$	−2.68585	−0.186679
$208$	0	0
$209$	−6.15792	−0.425952
$210$	0	0
$211$	−14.1004	−0.970710	−0.485355	−	0.874317i	$-0.661310\pi$
$211$	−14.1004	−0.970710	−0.485355	+	0.874317i	$0.661310\pi$
$212$	0	0
$213$	−0.585462	−0.0401152
$214$	0	0
$215$	−9.37169	−0.639144
$216$	0	0
$217$	32.7005	2.21986
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−14.8291	−0.997512
$222$	0	0
$223$	−6.72869	−0.450587	−0.225293	−	0.974291i	$-0.572334\pi$
$223$	−6.72869	−0.450587	−0.225293	+	0.974291i	$0.572334\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−9.95715	−0.660880	−0.330440	−	0.943827i	$-0.607197\pi$
$227$	−9.95715	−0.660880	−0.330440	+	0.943827i	$0.607197\pi$
$228$	0	0
$229$	−11.3288	−0.748631	−0.374316	−	0.927301i	$-0.622122\pi$
$229$	−11.3288	−0.748631	−0.374316	+	0.927301i	$0.622122\pi$
$230$	0	0
$231$	−10.7434	−0.706863
$232$	0	0
$233$	18.9786	1.24333	0.621664	−	0.783284i	$-0.286457\pi$
$233$	18.9786	1.24333	0.621664	+	0.783284i	$0.286457\pi$
$234$	0	0
$235$	−7.27131	−0.474328
$236$	0	0
$237$	1.02142	0.0663485
$238$	0	0
$239$	2.62831	0.170011	0.0850055	−	0.996380i	$-0.472909\pi$
$239$	2.62831	0.170011	0.0850055	+	0.996380i	$0.472909\pi$
$240$	0	0
$241$	10.7862	0.694802	0.347401	−	0.937717i	$-0.387064\pi$
$241$	10.7862	0.694802	0.347401	+	0.937717i	$0.387064\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.9572	−0.955578
$246$	0	0
$247$	−13.3717	−0.850820
$248$	0	0
$249$	13.3717	0.847397
$250$	0	0
$251$	30.9933	1.95628	0.978139	−	0.207952i	$-0.0666799\pi$
$251$	30.9933	1.95628	0.978139	+	0.207952i	$0.0666799\pi$
$252$	0	0
$253$	−6.15792	−0.387145
$254$	0	0
$255$	2.97858	0.186526
$256$	0	0
$257$	20.9357	1.30594	0.652968	−	0.757386i	$-0.273524\pi$
$257$	20.9357	1.30594	0.652968	+	0.757386i	$0.273524\pi$
$258$	0	0
$259$	−20.5855	−1.27912
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	19.2713	1.18832	0.594160	−	0.804347i	$-0.297485\pi$
$263$	19.2713	1.18832	0.594160	+	0.804347i	$0.297485\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	−3.37169	−0.206344
$268$	0	0
$269$	24.7434	1.50863	0.754315	−	0.656512i	$-0.227969\pi$
$269$	24.7434	1.50863	0.754315	+	0.656512i	$0.227969\pi$
$270$	0	0
$271$	27.5640	1.67440	0.837198	−	0.546900i	$-0.184192\pi$
$271$	27.5640	1.67440	0.837198	+	0.546900i	$0.184192\pi$
$272$	0	0
$273$	−23.3288	−1.41193
$274$	0	0
$275$	2.29273	0.138257
$276$	0	0
$277$	−20.3074	−1.22015	−0.610077	−	0.792342i	$-0.708862\pi$
$277$	−20.3074	−1.22015	−0.610077	+	0.792342i	$0.708862\pi$
$278$	0	0
$279$	−6.97858	−0.417796
$280$	0	0
$281$	10.7862	0.643453	0.321726	−	0.946833i	$-0.395737\pi$
$281$	10.7862	0.643453	0.321726	+	0.946833i	$0.395737\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	2.68585	0.159096
$286$	0	0
$287$	−53.2860	−3.14537
$288$	0	0
$289$	−8.12808	−0.478122
$290$	0	0
$291$	−3.95715	−0.231972
$292$	0	0
$293$	−21.9143	−1.28025	−0.640124	−	0.768272i	$-0.721117\pi$
$293$	−21.9143	−1.28025	−0.640124	+	0.768272i	$0.721117\pi$
$294$	0	0
$295$	−1.70727	−0.0994010
$296$	0	0
$297$	2.29273	0.133038
$298$	0	0
$299$	−13.3717	−0.773305
$300$	0	0
$301$	−43.9143	−2.53118
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	4.58546	0.262563
$306$	0	0
$307$	−26.5426	−1.51487	−0.757434	−	0.652912i	$-0.773547\pi$
$307$	−26.5426	−1.51487	−0.757434	+	0.652912i	$0.773547\pi$
$308$	0	0
$309$	14.6430	0.833011
$310$	0	0
$311$	−12.2008	−0.691842	−0.345921	−	0.938264i	$-0.612434\pi$
$311$	−12.2008	−0.691842	−0.345921	+	0.938264i	$0.612434\pi$
$312$	0	0
$313$	−15.9572	−0.901952	−0.450976	−	0.892536i	$-0.648924\pi$
$313$	−15.9572	−0.901952	−0.450976	+	0.892536i	$0.648924\pi$
$314$	0	0
$315$	4.68585	0.264018
$316$	0	0
$317$	33.5296	1.88321	0.941605	−	0.336718i	$-0.109317\pi$
$317$	33.5296	1.88321	0.941605	+	0.336718i	$0.109317\pi$
$318$	0	0
$319$	−4.58546	−0.256737
$320$	0	0
$321$	−11.3288	−0.632315
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	4.97858	0.276162
$326$	0	0
$327$	9.37169	0.518256
$328$	0	0
$329$	−34.0722	−1.87846
$330$	0	0
$331$	−19.8568	−1.09143	−0.545713	−	0.837972i	$-0.683741\pi$
$331$	−19.8568	−1.09143	−0.545713	+	0.837972i	$0.683741\pi$
$332$	0	0
$333$	4.39312	0.240741
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	7.17092	0.390625	0.195313	−	0.980741i	$-0.437428\pi$
$337$	7.17092	0.390625	0.195313	+	0.980741i	$0.437428\pi$
$338$	0	0
$339$	19.7648	1.07348
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	−37.2860	−2.01325
$344$	0	0
$345$	2.68585	0.144601
$346$	0	0
$347$	0.786230	0.0422071	0.0211035	−	0.999777i	$-0.493282\pi$
$347$	0.786230	0.0422071	0.0211035	+	0.999777i	$0.493282\pi$
$348$	0	0
$349$	−6.15792	−0.329626	−0.164813	−	0.986325i	$-0.552702\pi$
$349$	−6.15792	−0.329626	−0.164813	+	0.986325i	$0.552702\pi$
$350$	0	0
$351$	4.97858	0.265737
$352$	0	0
$353$	21.7220	1.15614	0.578072	−	0.815986i	$-0.303805\pi$
$353$	21.7220	1.15614	0.578072	+	0.815986i	$0.303805\pi$
$354$	0	0
$355$	0.585462	0.0310731
$356$	0	0
$357$	13.9572	0.738691
$358$	0	0
$359$	−0.585462	−0.0308995	−0.0154498	−	0.999881i	$-0.504918\pi$
$359$	−0.585462	−0.0308995	−0.0154498	+	0.999881i	$0.504918\pi$
$360$	0	0
$361$	−11.7862	−0.620328
$362$	0	0
$363$	−5.74338	−0.301450
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−0.485078	−0.0253209	−0.0126604	−	0.999920i	$-0.504030\pi$
$367$	−0.485078	−0.0253209	−0.0126604	+	0.999920i	$0.504030\pi$
$368$	0	0
$369$	11.3717	0.591987
$370$	0	0
$371$	9.37169	0.486554
$372$	0	0
$373$	−12.3931	−0.641691	−0.320846	−	0.947132i	$-0.603967\pi$
$373$	−12.3931	−0.641691	−0.320846	+	0.947132i	$0.603967\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−9.95715	−0.512820
$378$	0	0
$379$	−26.0147	−1.33629	−0.668143	−	0.744033i	$-0.732910\pi$
$379$	−26.0147	−1.33629	−0.668143	+	0.744033i	$0.732910\pi$
$380$	0	0
$381$	6.64300	0.340331
$382$	0	0
$383$	6.68585	0.341631	0.170815	−	0.985303i	$-0.445360\pi$
$383$	6.68585	0.341631	0.170815	+	0.985303i	$0.445360\pi$
$384$	0	0
$385$	10.7434	0.547534
$386$	0	0
$387$	9.37169	0.476390
$388$	0	0
$389$	29.9143	1.51672	0.758358	−	0.651838i	$-0.226002\pi$
$389$	29.9143	1.51672	0.758358	+	0.651838i	$0.226002\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	−7.07896	−0.357086
$394$	0	0
$395$	−1.02142	−0.0513934
$396$	0	0
$397$	−9.76481	−0.490082	−0.245041	−	0.969513i	$-0.578801\pi$
$397$	−9.76481	−0.490082	−0.245041	+	0.969513i	$0.578801\pi$
$398$	0	0
$399$	12.5855	0.630061
$400$	0	0
$401$	−6.58546	−0.328862	−0.164431	−	0.986389i	$-0.552579\pi$
$401$	−6.58546	−0.328862	−0.164431	+	0.986389i	$0.552579\pi$
$402$	0	0
$403$	−34.7434	−1.73069
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	10.0722	0.499262
$408$	0	0
$409$	25.9143	1.28138	0.640690	−	0.767800i	$-0.278648\pi$
$409$	25.9143	1.28138	0.640690	+	0.767800i	$0.278648\pi$
$410$	0	0
$411$	14.9786	0.738839
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	−13.3717	−0.656391
$416$	0	0
$417$	4.64300	0.227369
$418$	0	0
$419$	12.2499	0.598446	0.299223	−	0.954183i	$-0.403273\pi$
$419$	12.2499	0.598446	0.299223	+	0.954183i	$0.403273\pi$
$420$	0	0
$421$	−4.67115	−0.227658	−0.113829	−	0.993500i	$-0.536312\pi$
$421$	−4.67115	−0.227658	−0.113829	+	0.993500i	$0.536312\pi$
$422$	0	0
$423$	7.27131	0.353543
$424$	0	0
$425$	−2.97858	−0.144482
$426$	0	0
$427$	21.4868	1.03982
$428$	0	0
$429$	11.4145	0.551099
$430$	0	0
$431$	0.585462	0.0282007	0.0141004	−	0.999901i	$-0.495512\pi$
$431$	0.585462	0.0282007	0.0141004	+	0.999901i	$0.495512\pi$
$432$	0	0
$433$	−21.9143	−1.05313	−0.526567	−	0.850133i	$-0.676521\pi$
$433$	−21.9143	−1.05313	−0.526567	+	0.850133i	$0.676521\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	7.21377	0.345081
$438$	0	0
$439$	2.39312	0.114217	0.0571086	−	0.998368i	$-0.481812\pi$
$439$	2.39312	0.114217	0.0571086	+	0.998368i	$0.481812\pi$
$440$	0	0
$441$	14.9572	0.712245
$442$	0	0
$443$	−20.7005	−0.983512	−0.491756	−	0.870733i	$-0.663645\pi$
$443$	−20.7005	−0.983512	−0.491756	+	0.870733i	$0.663645\pi$
$444$	0	0
$445$	3.37169	0.159834
$446$	0	0
$447$	2.00000	0.0945968
$448$	0	0
$449$	−37.9143	−1.78929	−0.894643	−	0.446781i	$-0.852570\pi$
$449$	−37.9143	−1.78929	−0.894643	+	0.446781i	$0.852570\pi$
$450$	0	0
$451$	26.0722	1.22769
$452$	0	0
$453$	8.35027	0.392330
$454$	0	0
$455$	23.3288	1.09367
$456$	0	0
$457$	−38.7005	−1.81033	−0.905167	−	0.425055i	$-0.860255\pi$
$457$	−38.7005	−1.81033	−0.905167	+	0.425055i	$0.860255\pi$
$458$	0	0
$459$	−2.97858	−0.139028
$460$	0	0
$461$	4.74338	0.220921	0.110461	−	0.993880i	$-0.464767\pi$
$461$	4.74338	0.220921	0.110461	+	0.993880i	$0.464767\pi$
$462$	0	0
$463$	15.3142	0.711709	0.355855	−	0.934541i	$-0.384190\pi$
$463$	15.3142	0.711709	0.355855	+	0.934541i	$0.384190\pi$
$464$	0	0
$465$	6.97858	0.323624
$466$	0	0
$467$	30.5426	1.41334	0.706672	−	0.707541i	$-0.250196\pi$
$467$	30.5426	1.41334	0.706672	+	0.707541i	$0.250196\pi$
$468$	0	0
$469$	18.7434	0.865489
$470$	0	0
$471$	22.3503	1.02985
$472$	0	0
$473$	21.4868	0.987963
$474$	0	0
$475$	−2.68585	−0.123235
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	3.32885	0.152099	0.0760494	−	0.997104i	$-0.475769\pi$
$479$	3.32885	0.152099	0.0760494	+	0.997104i	$0.475769\pi$
$480$	0	0
$481$	21.8715	0.997253
$482$	0	0
$483$	12.5855	0.572658
$484$	0	0
$485$	3.95715	0.179685
$486$	0	0
$487$	12.1004	0.548321	0.274160	−	0.961684i	$-0.411600\pi$
$487$	12.1004	0.548321	0.274160	+	0.961684i	$0.411600\pi$
$488$	0	0
$489$	1.37169	0.0620301
$490$	0	0
$491$	14.2927	0.645022	0.322511	−	0.946566i	$-0.395473\pi$
$491$	14.2927	0.645022	0.322511	+	0.946566i	$0.395473\pi$
$492$	0	0
$493$	5.95715	0.268297
$494$	0	0
$495$	−2.29273	−0.103051
$496$	0	0
$497$	2.74338	0.123058
$498$	0	0
$499$	9.22846	0.413123	0.206561	−	0.978434i	$-0.433773\pi$
$499$	9.22846	0.413123	0.206561	+	0.978434i	$0.433773\pi$
$500$	0	0
$501$	−11.2713	−0.503565
$502$	0	0
$503$	14.1004	0.628705	0.314353	−	0.949306i	$-0.398213\pi$
$503$	14.1004	0.628705	0.314353	+	0.949306i	$0.398213\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	11.7862	0.523445
$508$	0	0
$509$	−43.4011	−1.92372	−0.961859	−	0.273544i	$-0.911804\pi$
$509$	−43.4011	−1.92372	−0.961859	+	0.273544i	$0.911804\pi$
$510$	0	0
$511$	−28.1151	−1.24374
$512$	0	0
$513$	−2.68585	−0.118583
$514$	0	0
$515$	−14.6430	−0.645248
$516$	0	0
$517$	16.6712	0.733196
$518$	0	0
$519$	10.7862	0.473463
$520$	0	0
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−13.5725	−0.593482	−0.296741	−	0.954958i	$-0.595900\pi$
$523$	−13.5725	−0.593482	−0.296741	+	0.954958i	$0.595900\pi$
$524$	0	0
$525$	−4.68585	−0.204507
$526$	0	0
$527$	20.7862	0.905462
$528$	0	0
$529$	−15.7862	−0.686358
$530$	0	0
$531$	1.70727	0.0740892
$532$	0	0
$533$	56.6148	2.45226
$534$	0	0
$535$	11.3288	0.489789
$536$	0	0
$537$	−3.66442	−0.158132
$538$	0	0
$539$	34.2927	1.47709
$540$	0	0
$541$	−37.2860	−1.60305	−0.801525	−	0.597961i	$-0.795978\pi$
$541$	−37.2860	−1.60305	−0.801525	+	0.597961i	$0.795978\pi$
$542$	0	0
$543$	−6.62831	−0.284448
$544$	0	0
$545$	−9.37169	−0.401439
$546$	0	0
$547$	−0.200768	−0.00858424	−0.00429212	−	0.999991i	$-0.501366\pi$
$547$	−0.200768	−0.00858424	−0.00429212	+	0.999991i	$0.501366\pi$
$548$	0	0
$549$	−4.58546	−0.195703
$550$	0	0
$551$	5.37169	0.228842
$552$	0	0
$553$	−4.78623	−0.203531
$554$	0	0
$555$	−4.39312	−0.186477
$556$	0	0
$557$	9.21377	0.390400	0.195200	−	0.980763i	$-0.437464\pi$
$557$	9.21377	0.390400	0.195200	+	0.980763i	$0.437464\pi$
$558$	0	0
$559$	46.6577	1.97341
$560$	0	0
$561$	−6.82908	−0.288324
$562$	0	0
$563$	−36.7005	−1.54674	−0.773372	−	0.633953i	$-0.781431\pi$
$563$	−36.7005	−1.54674	−0.773372	+	0.633953i	$0.781431\pi$
$564$	0	0
$565$	−19.7648	−0.831512
$566$	0	0
$567$	−4.68585	−0.196787
$568$	0	0
$569$	−13.4145	−0.562367	−0.281183	−	0.959654i	$-0.590727\pi$
$569$	−13.4145	−0.562367	−0.281183	+	0.959654i	$0.590727\pi$
$570$	0	0
$571$	−18.6858	−0.781978	−0.390989	−	0.920395i	$-0.627867\pi$
$571$	−18.6858	−0.781978	−0.390989	+	0.920395i	$0.627867\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	−2.68585	−0.112008
$576$	0	0
$577$	−2.78623	−0.115992	−0.0579961	−	0.998317i	$-0.518471\pi$
$577$	−2.78623	−0.115992	−0.0579961	+	0.998317i	$0.518471\pi$
$578$	0	0
$579$	1.21377	0.0504425
$580$	0	0
$581$	−62.6577	−2.59948
$582$	0	0
$583$	−4.58546	−0.189910
$584$	0	0
$585$	−4.97858	−0.205839
$586$	0	0
$587$	27.3288	1.12798	0.563991	−	0.825781i	$-0.309265\pi$
$587$	27.3288	1.12798	0.563991	+	0.825781i	$0.309265\pi$
$588$	0	0
$589$	18.7434	0.772308
$590$	0	0
$591$	23.9572	0.985466
$592$	0	0
$593$	−6.97858	−0.286576	−0.143288	−	0.989681i	$-0.545767\pi$
$593$	−6.97858	−0.286576	−0.143288	+	0.989681i	$0.545767\pi$
$594$	0	0
$595$	−13.9572	−0.572188
$596$	0	0
$597$	−0.350269	−0.0143356
$598$	0	0
$599$	−36.4998	−1.49134	−0.745670	−	0.666315i	$-0.767870\pi$
$599$	−36.4998	−1.49134	−0.745670	+	0.666315i	$0.767870\pi$
$600$	0	0
$601$	15.5725	0.635214	0.317607	−	0.948222i	$-0.397121\pi$
$601$	15.5725	0.635214	0.317607	+	0.948222i	$0.397121\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	5.74338	0.233502
$606$	0	0
$607$	−31.2285	−1.26752	−0.633762	−	0.773528i	$-0.718490\pi$
$607$	−31.2285	−1.26752	−0.633762	+	0.773528i	$0.718490\pi$
$608$	0	0
$609$	9.37169	0.379760
$610$	0	0
$611$	36.2008	1.46453
$612$	0	0
$613$	0.978577	0.0395244	0.0197622	−	0.999805i	$-0.493709\pi$
$613$	0.978577	0.0395244	0.0197622	+	0.999805i	$0.493709\pi$
$614$	0	0
$615$	−11.3717	−0.458551
$616$	0	0
$617$	−32.9357	−1.32594	−0.662971	−	0.748645i	$-0.730705\pi$
$617$	−32.9357	−1.32594	−0.662971	+	0.748645i	$0.730705\pi$
$618$	0	0
$619$	−3.35700	−0.134929	−0.0674646	−	0.997722i	$-0.521491\pi$
$619$	−3.35700	−0.134929	−0.0674646	+	0.997722i	$0.521491\pi$
$620$	0	0
$621$	−2.68585	−0.107779
$622$	0	0
$623$	15.7992	0.632983
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.15792	−0.245924
$628$	0	0
$629$	−13.0852	−0.521742
$630$	0	0
$631$	27.7648	1.10530	0.552650	−	0.833414i	$-0.313617\pi$
$631$	27.7648	1.10530	0.552650	+	0.833414i	$0.313617\pi$
$632$	0	0
$633$	−14.1004	−0.560440
$634$	0	0
$635$	−6.64300	−0.263619
$636$	0	0
$637$	74.4653	2.95042
$638$	0	0
$639$	−0.585462	−0.0231605
$640$	0	0
$641$	−21.1281	−0.834509	−0.417254	−	0.908790i	$-0.637008\pi$
$641$	−21.1281	−0.834509	−0.417254	+	0.908790i	$0.637008\pi$
$642$	0	0
$643$	−29.2860	−1.15493	−0.577464	−	0.816416i	$-0.695957\pi$
$643$	−29.2860	−1.15493	−0.577464	+	0.816416i	$0.695957\pi$
$644$	0	0
$645$	−9.37169	−0.369010
$646$	0	0
$647$	−15.6728	−0.616163	−0.308082	−	0.951360i	$-0.599687\pi$
$647$	−15.6728	−0.616163	−0.308082	+	0.951360i	$0.599687\pi$
$648$	0	0
$649$	3.91431	0.153650
$650$	0	0
$651$	32.7005	1.28164
$652$	0	0
$653$	−17.5296	−0.685987	−0.342993	−	0.939338i	$-0.611441\pi$
$653$	−17.5296	−0.685987	−0.342993	+	0.939338i	$0.611441\pi$
$654$	0	0
$655$	7.07896	0.276598
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	23.8652	0.929656	0.464828	−	0.885401i	$-0.346116\pi$
$659$	23.8652	0.929656	0.464828	+	0.885401i	$0.346116\pi$
$660$	0	0
$661$	−30.1579	−1.17301	−0.586504	−	0.809947i	$-0.699496\pi$
$661$	−30.1579	−1.17301	−0.586504	+	0.809947i	$0.699496\pi$
$662$	0	0
$663$	−14.8291	−0.575914
$664$	0	0
$665$	−12.5855	−0.488043
$666$	0	0
$667$	5.37169	0.207993
$668$	0	0
$669$	−6.72869	−0.260146
$670$	0	0
$671$	−10.5132	−0.405859
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	9.61531	0.369546	0.184773	−	0.982781i	$-0.440845\pi$
$677$	9.61531	0.369546	0.184773	+	0.982781i	$0.440845\pi$
$678$	0	0
$679$	18.5426	0.711600
$680$	0	0
$681$	−9.95715	−0.381559
$682$	0	0
$683$	18.6283	0.712792	0.356396	−	0.934335i	$-0.384005\pi$
$683$	18.6283	0.712792	0.356396	+	0.934335i	$0.384005\pi$
$684$	0	0
$685$	−14.9786	−0.572302
$686$	0	0
$687$	−11.3288	−0.432222
$688$	0	0
$689$	−9.95715	−0.379337
$690$	0	0
$691$	13.4292	0.510872	0.255436	−	0.966826i	$-0.417781\pi$
$691$	13.4292	0.510872	0.255436	+	0.966826i	$0.417781\pi$
$692$	0	0
$693$	−10.7434	−0.408107
$694$	0	0
$695$	−4.64300	−0.176119
$696$	0	0
$697$	−33.8715	−1.28297
$698$	0	0
$699$	18.9786	0.717836
$700$	0	0
$701$	−19.1709	−0.724076	−0.362038	−	0.932163i	$-0.617919\pi$
$701$	−19.1709	−0.724076	−0.362038	+	0.932163i	$0.617919\pi$
$702$	0	0
$703$	−11.7992	−0.445016
$704$	0	0
$705$	−7.27131	−0.273853
$706$	0	0
$707$	9.37169	0.352459
$708$	0	0
$709$	15.4145	0.578905	0.289453	−	0.957192i	$-0.406527\pi$
$709$	15.4145	0.578905	0.289453	+	0.957192i	$0.406527\pi$
$710$	0	0
$711$	1.02142	0.0383063
$712$	0	0
$713$	18.7434	0.701945
$714$	0	0
$715$	−11.4145	−0.426880
$716$	0	0
$717$	2.62831	0.0981559
$718$	0	0
$719$	20.7862	0.775196	0.387598	−	0.921829i	$-0.373305\pi$
$719$	20.7862	0.775196	0.387598	+	0.921829i	$0.373305\pi$
$720$	0	0
$721$	−68.6148	−2.55535
$722$	0	0
$723$	10.7862	0.401144
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−12.3012	−0.456224	−0.228112	−	0.973635i	$-0.573255\pi$
$727$	−12.3012	−0.456224	−0.228112	+	0.973635i	$0.573255\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−27.9143	−1.03245
$732$	0	0
$733$	35.9227	1.32684	0.663418	−	0.748249i	$-0.269105\pi$
$733$	35.9227	1.32684	0.663418	+	0.748249i	$0.269105\pi$
$734$	0	0
$735$	−14.9572	−0.551703
$736$	0	0
$737$	−9.17092	−0.337815
$738$	0	0
$739$	−29.0277	−1.06780	−0.533900	−	0.845547i	$-0.679274\pi$
$739$	−29.0277	−1.06780	−0.533900	+	0.845547i	$0.679274\pi$
$740$	0	0
$741$	−13.3717	−0.491221
$742$	0	0
$743$	−2.60015	−0.0953904	−0.0476952	−	0.998862i	$-0.515188\pi$
$743$	−2.60015	−0.0953904	−0.0476952	+	0.998862i	$0.515188\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	0	0
$747$	13.3717	0.489245
$748$	0	0
$749$	53.0852	1.93969
$750$	0	0
$751$	−10.8929	−0.397487	−0.198744	−	0.980052i	$-0.563686\pi$
$751$	−10.8929	−0.397487	−0.198744	+	0.980052i	$0.563686\pi$
$752$	0	0
$753$	30.9933	1.12946
$754$	0	0
$755$	−8.35027	−0.303897
$756$	0	0
$757$	34.3503	1.24848	0.624241	−	0.781232i	$-0.285408\pi$
$757$	34.3503	1.24848	0.624241	+	0.781232i	$0.285408\pi$
$758$	0	0
$759$	−6.15792	−0.223518
$760$	0	0
$761$	19.0852	0.691839	0.345920	−	0.938264i	$-0.387567\pi$
$761$	19.0852	0.691839	0.345920	+	0.938264i	$0.387567\pi$
$762$	0	0
$763$	−43.9143	−1.58980
$764$	0	0
$765$	2.97858	0.107691
$766$	0	0
$767$	8.49977	0.306909
$768$	0	0
$769$	31.8715	1.14931	0.574657	−	0.818394i	$-0.305135\pi$
$769$	31.8715	1.14931	0.574657	+	0.818394i	$0.305135\pi$
$770$	0	0
$771$	20.9357	0.753982
$772$	0	0
$773$	−11.9572	−0.430069	−0.215034	−	0.976606i	$-0.568986\pi$
$773$	−11.9572	−0.430069	−0.215034	+	0.976606i	$0.568986\pi$
$774$	0	0
$775$	−6.97858	−0.250678
$776$	0	0
$777$	−20.5855	−0.738499
$778$	0	0
$779$	−30.5426	−1.09430
$780$	0	0
$781$	−1.34231	−0.0480315
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−22.3503	−0.797715
$786$	0	0
$787$	−33.0852	−1.17936	−0.589681	−	0.807637i	$-0.700746\pi$
$787$	−33.0852	−1.17936	−0.589681	+	0.807637i	$0.700746\pi$
$788$	0	0
$789$	19.2713	0.686077
$790$	0	0
$791$	−92.6148	−3.29300
$792$	0	0
$793$	−22.8291	−0.810684
$794$	0	0
$795$	2.00000	0.0709327
$796$	0	0
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	0	0
$799$	−21.6582	−0.766210
$800$	0	0
$801$	−3.37169	−0.119133
$802$	0	0
$803$	13.7564	0.485452
$804$	0	0
$805$	−12.5855	−0.443579
$806$	0	0
$807$	24.7434	0.871008
$808$	0	0
$809$	30.6148	1.07636	0.538180	−	0.842830i	$-0.319112\pi$
$809$	30.6148	1.07636	0.538180	+	0.842830i	$0.319112\pi$
$810$	0	0
$811$	53.9290	1.89370	0.946852	−	0.321670i	$-0.104244\pi$
$811$	53.9290	1.89370	0.946852	+	0.321670i	$0.104244\pi$
$812$	0	0
$813$	27.5640	0.966713
$814$	0	0
$815$	−1.37169	−0.0480483
$816$	0	0
$817$	−25.1709	−0.880619
$818$	0	0
$819$	−23.3288	−0.815176
$820$	0	0
$821$	12.6577	0.441757	0.220878	−	0.975301i	$-0.429108\pi$
$821$	12.6577	0.441757	0.220878	+	0.975301i	$0.429108\pi$
$822$	0	0
$823$	19.8139	0.690670	0.345335	−	0.938479i	$-0.387765\pi$
$823$	19.8139	0.690670	0.345335	+	0.938479i	$0.387765\pi$
$824$	0	0
$825$	2.29273	0.0798226
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	41.3717	1.43690	0.718449	−	0.695580i	$-0.244852\pi$
$829$	41.3717	1.43690	0.718449	+	0.695580i	$0.244852\pi$
$830$	0	0
$831$	−20.3074	−0.704457
$832$	0	0
$833$	−44.5510	−1.54360
$834$	0	0
$835$	11.2713	0.390060
$836$	0	0
$837$	−6.97858	−0.241215
$838$	0	0
$839$	−20.9013	−0.721593	−0.360797	−	0.932645i	$-0.617495\pi$
$839$	−20.9013	−0.721593	−0.360797	+	0.932645i	$0.617495\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	10.7862	0.371498
$844$	0	0
$845$	−11.7862	−0.405459
$846$	0	0
$847$	26.9126	0.924728
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−11.7992	−0.404472
$852$	0	0
$853$	6.63673	0.227237	0.113619	−	0.993524i	$-0.463756\pi$
$853$	6.63673	0.227237	0.113619	+	0.993524i	$0.463756\pi$
$854$	0	0
$855$	2.68585	0.0918540
$856$	0	0
$857$	−9.80765	−0.335023	−0.167512	−	0.985870i	$-0.553573\pi$
$857$	−9.80765	−0.335023	−0.167512	+	0.985870i	$0.553573\pi$
$858$	0	0
$859$	39.3864	1.34385	0.671923	−	0.740621i	$-0.265469\pi$
$859$	39.3864	1.34385	0.671923	+	0.740621i	$0.265469\pi$
$860$	0	0
$861$	−53.2860	−1.81598
$862$	0	0
$863$	7.07054	0.240684	0.120342	−	0.992732i	$-0.461601\pi$
$863$	7.07054	0.240684	0.120342	+	0.992732i	$0.461601\pi$
$864$	0	0
$865$	−10.7862	−0.366743
$866$	0	0
$867$	−8.12808	−0.276044
$868$	0	0
$869$	2.34185	0.0794417
$870$	0	0
$871$	−19.9143	−0.674771
$872$	0	0
$873$	−3.95715	−0.133929
$874$	0	0
$875$	4.68585	0.158411
$876$	0	0
$877$	−23.1365	−0.781264	−0.390632	−	0.920547i	$-0.627744\pi$
$877$	−23.1365	−0.781264	−0.390632	+	0.920547i	$0.627744\pi$
$878$	0	0
$879$	−21.9143	−0.739151
$880$	0	0
$881$	28.4569	0.958738	0.479369	−	0.877613i	$-0.340866\pi$
$881$	28.4569	0.958738	0.479369	+	0.877613i	$0.340866\pi$
$882$	0	0
$883$	41.2003	1.38650	0.693250	−	0.720697i	$-0.256178\pi$
$883$	41.2003	1.38650	0.693250	+	0.720697i	$0.256178\pi$
$884$	0	0
$885$	−1.70727	−0.0573892
$886$	0	0
$887$	3.55777	0.119458	0.0597291	−	0.998215i	$-0.480976\pi$
$887$	3.55777	0.119458	0.0597291	+	0.998215i	$0.480976\pi$
$888$	0	0
$889$	−31.1281	−1.04400
$890$	0	0
$891$	2.29273	0.0768094
$892$	0	0
$893$	−19.5296	−0.653534
$894$	0	0
$895$	3.66442	0.122488
$896$	0	0
$897$	−13.3717	−0.446468
$898$	0	0
$899$	13.9572	0.465497
$900$	0	0
$901$	5.95715	0.198462
$902$	0	0
$903$	−43.9143	−1.46138
$904$	0	0
$905$	6.62831	0.220332
$906$	0	0
$907$	−50.6577	−1.68206	−0.841031	−	0.540988i	$-0.818051\pi$
$907$	−50.6577	−1.68206	−0.841031	+	0.540988i	$0.818051\pi$
$908$	0	0
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	−26.4569	−0.876557	−0.438279	−	0.898839i	$-0.644412\pi$
$911$	−26.4569	−0.876557	−0.438279	+	0.898839i	$0.644412\pi$
$912$	0	0
$913$	30.6577	1.01462
$914$	0	0
$915$	4.58546	0.151591
$916$	0	0
$917$	33.1709	1.09540
$918$	0	0
$919$	−29.8077	−0.983264	−0.491632	−	0.870803i	$-0.663599\pi$
$919$	−29.8077	−0.983264	−0.491632	+	0.870803i	$0.663599\pi$
$920$	0	0
$921$	−26.5426	−0.874609
$922$	0	0
$923$	−2.91477	−0.0959407
$924$	0	0
$925$	4.39312	0.144445
$926$	0	0
$927$	14.6430	0.480939
$928$	0	0
$929$	−8.82908	−0.289673	−0.144836	−	0.989456i	$-0.546266\pi$
$929$	−8.82908	−0.289673	−0.144836	+	0.989456i	$0.546266\pi$
$930$	0	0
$931$	−40.1726	−1.31660
$932$	0	0
$933$	−12.2008	−0.399435
$934$	0	0
$935$	6.82908	0.223335
$936$	0	0
$937$	42.2302	1.37960	0.689799	−	0.724000i	$-0.257699\pi$
$937$	42.2302	1.37960	0.689799	+	0.724000i	$0.257699\pi$
$938$	0	0
$939$	−15.9572	−0.520742
$940$	0	0
$941$	−32.7434	−1.06740	−0.533702	−	0.845673i	$-0.679200\pi$
$941$	−32.7434	−1.06740	−0.533702	+	0.845673i	$0.679200\pi$
$942$	0	0
$943$	−30.5426	−0.994604
$944$	0	0
$945$	4.68585	0.152431
$946$	0	0
$947$	15.9143	0.517146	0.258573	−	0.965992i	$-0.416748\pi$
$947$	15.9143	0.517146	0.258573	+	0.965992i	$0.416748\pi$
$948$	0	0
$949$	29.8715	0.969669
$950$	0	0
$951$	33.5296	1.08727
$952$	0	0
$953$	−55.6791	−1.80362	−0.901812	−	0.432129i	$-0.857762\pi$
$953$	−55.6791	−1.80362	−0.901812	+	0.432129i	$0.857762\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	−4.58546	−0.148227
$958$	0	0
$959$	−70.1873	−2.26647
$960$	0	0
$961$	17.7005	0.570985
$962$	0	0
$963$	−11.3288	−0.365067
$964$	0	0
$965$	−1.21377	−0.0390726
$966$	0	0
$967$	54.7581	1.76090	0.880451	−	0.474138i	$-0.157240\pi$
$967$	54.7581	1.76090	0.880451	+	0.474138i	$0.157240\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	−30.3221	−0.973083	−0.486542	−	0.873657i	$-0.661742\pi$
$971$	−30.3221	−0.973083	−0.486542	+	0.873657i	$0.661742\pi$
$972$	0	0
$973$	−21.7564	−0.697478
$974$	0	0
$975$	4.97858	0.159442
$976$	0	0
$977$	−29.7220	−0.950890	−0.475445	−	0.879746i	$-0.657713\pi$
$977$	−29.7220	−0.950890	−0.475445	+	0.879746i	$0.657713\pi$
$978$	0	0
$979$	−7.73038	−0.247064
$980$	0	0
$981$	9.37169	0.299215
$982$	0	0
$983$	49.5443	1.58022	0.790109	−	0.612966i	$-0.210024\pi$
$983$	49.5443	1.58022	0.790109	+	0.612966i	$0.210024\pi$
$984$	0	0
$985$	−23.9572	−0.763338
$986$	0	0
$987$	−34.0722	−1.08453
$988$	0	0
$989$	−25.1709	−0.800389
$990$	0	0
$991$	19.0937	0.606530	0.303265	−	0.952906i	$-0.401923\pi$
$991$	19.0937	0.606530	0.303265	+	0.952906i	$0.401923\pi$
$992$	0	0
$993$	−19.8568	−0.630136
$994$	0	0
$995$	0.350269	0.0111043
$996$	0	0
$997$	38.8500	1.23039	0.615197	−	0.788374i	$-0.289077\pi$
$997$	38.8500	1.23039	0.615197	+	0.788374i	$0.289077\pi$
$998$	0	0
$999$	4.39312	0.138992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3840.2.a.bq.1.1		3
4.3	odd	2		3840.2.a.bo.1.3		3
8.3	odd	2		3840.2.a.br.1.3		3
8.5	even	2		3840.2.a.bp.1.1		3
16.3	odd	4		480.2.k.b.241.1		6
16.5	even	4		120.2.k.b.61.2	yes	6
16.11	odd	4		480.2.k.b.241.4		6
16.13	even	4		120.2.k.b.61.1	✓	6
48.5	odd	4		360.2.k.f.181.5		6
48.11	even	4		1440.2.k.f.721.4		6
48.29	odd	4		360.2.k.f.181.6		6
48.35	even	4		1440.2.k.f.721.1		6
80.3	even	4		2400.2.d.f.49.6		6
80.13	odd	4		600.2.d.e.349.2		6
80.19	odd	4		2400.2.k.c.1201.6		6
80.27	even	4		2400.2.d.f.49.1		6
80.29	even	4		600.2.k.c.301.6		6
80.37	odd	4		600.2.d.e.349.1		6
80.43	even	4		2400.2.d.e.49.6		6
80.53	odd	4		600.2.d.f.349.6		6
80.59	odd	4		2400.2.k.c.1201.3		6
80.67	even	4		2400.2.d.e.49.1		6
80.69	even	4		600.2.k.c.301.5		6
80.77	odd	4		600.2.d.f.349.5		6
240.29	odd	4		1800.2.k.p.901.1		6
240.53	even	4		1800.2.d.r.1549.1		6
240.59	even	4		7200.2.k.p.3601.6		6
240.77	even	4		1800.2.d.r.1549.2		6
240.83	odd	4		7200.2.d.q.2449.6		6
240.107	odd	4		7200.2.d.q.2449.1		6
240.149	odd	4		1800.2.k.p.901.2		6
240.173	even	4		1800.2.d.q.1549.5		6
240.179	even	4		7200.2.k.p.3601.5		6
240.197	even	4		1800.2.d.q.1549.6		6
240.203	odd	4		7200.2.d.r.2449.6		6
240.227	odd	4		7200.2.d.r.2449.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.1	✓	6	16.13	even	4
120.2.k.b.61.2	yes	6	16.5	even	4
360.2.k.f.181.5		6	48.5	odd	4
360.2.k.f.181.6		6	48.29	odd	4
480.2.k.b.241.1		6	16.3	odd	4
480.2.k.b.241.4		6	16.11	odd	4
600.2.d.e.349.1		6	80.37	odd	4
600.2.d.e.349.2		6	80.13	odd	4
600.2.d.f.349.5		6	80.77	odd	4
600.2.d.f.349.6		6	80.53	odd	4
600.2.k.c.301.5		6	80.69	even	4
600.2.k.c.301.6		6	80.29	even	4
1440.2.k.f.721.1		6	48.35	even	4
1440.2.k.f.721.4		6	48.11	even	4
1800.2.d.q.1549.5		6	240.173	even	4
1800.2.d.q.1549.6		6	240.197	even	4
1800.2.d.r.1549.1		6	240.53	even	4
1800.2.d.r.1549.2		6	240.77	even	4
1800.2.k.p.901.1		6	240.29	odd	4
1800.2.k.p.901.2		6	240.149	odd	4
2400.2.d.e.49.1		6	80.67	even	4
2400.2.d.e.49.6		6	80.43	even	4
2400.2.d.f.49.1		6	80.27	even	4
2400.2.d.f.49.6		6	80.3	even	4
2400.2.k.c.1201.3		6	80.59	odd	4
2400.2.k.c.1201.6		6	80.19	odd	4
3840.2.a.bo.1.3		3	4.3	odd	2
3840.2.a.bp.1.1		3	8.5	even	2
3840.2.a.bq.1.1		3	1.1	even	1	trivial
3840.2.a.br.1.3		3	8.3	odd	2
7200.2.d.q.2449.1		6	240.107	odd	4
7200.2.d.q.2449.6		6	240.83	odd	4
7200.2.d.r.2449.1		6	240.227	odd	4
7200.2.d.r.2449.6		6	240.203	odd	4
7200.2.k.p.3601.5		6	240.179	even	4
7200.2.k.p.3601.6		6	240.59	even	4

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:	\( N \)	\(=\)	\( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -4.68585 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -4.68585 q^{7} +1.00000 q^{9} +2.29273 q^{11} +4.97858 q^{13} -1.00000 q^{15} -2.97858 q^{17} -2.68585 q^{19} -4.68585 q^{21} -2.68585 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -6.97858 q^{31} +2.29273 q^{33} +4.68585 q^{35} +4.39312 q^{37} +4.97858 q^{39} +11.3717 q^{41} +9.37169 q^{43} -1.00000 q^{45} +7.27131 q^{47} +14.9572 q^{49} -2.97858 q^{51} -2.00000 q^{53} -2.29273 q^{55} -2.68585 q^{57} +1.70727 q^{59} -4.58546 q^{61} -4.68585 q^{63} -4.97858 q^{65} -4.00000 q^{67} -2.68585 q^{69} -0.585462 q^{71} +6.00000 q^{73} +1.00000 q^{75} -10.7434 q^{77} +1.02142 q^{79} +1.00000 q^{81} +13.3717 q^{83} +2.97858 q^{85} -2.00000 q^{87} -3.37169 q^{89} -23.3288 q^{91} -6.97858 q^{93} +2.68585 q^{95} -3.95715 q^{97} +2.29273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 4 q^{11} - 3 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 4 q^{23} + 3 q^{25} + 3 q^{27} - 6 q^{29} - 6 q^{31} + 4 q^{33} + 2 q^{35} + 4 q^{37} + 10 q^{41} + 4 q^{43} - 3 q^{45} + 4 q^{47} + 15 q^{49} + 6 q^{51} - 6 q^{53} - 4 q^{55} + 4 q^{57} + 8 q^{59} - 8 q^{61} - 2 q^{63} - 12 q^{67} + 4 q^{69} + 4 q^{71} + 18 q^{73} + 3 q^{75} + 16 q^{77} + 18 q^{79} + 3 q^{81} + 16 q^{83} - 6 q^{85} - 6 q^{87} + 14 q^{89} - 16 q^{91} - 6 q^{93} - 4 q^{95} + 18 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 + 4 * q^11 - 3 * q^15 + 6 * q^17 + 4 * q^19 - 2 * q^21 + 4 * q^23 + 3 * q^25 + 3 * q^27 - 6 * q^29 - 6 * q^31 + 4 * q^33 + 2 * q^35 + 4 * q^37 + 10 * q^41 + 4 * q^43 - 3 * q^45 + 4 * q^47 + 15 * q^49 + 6 * q^51 - 6 * q^53 - 4 * q^55 + 4 * q^57 + 8 * q^59 - 8 * q^61 - 2 * q^63 - 12 * q^67 + 4 * q^69 + 4 * q^71 + 18 * q^73 + 3 * q^75 + 16 * q^77 + 18 * q^79 + 3 * q^81 + 16 * q^83 - 6 * q^85 - 6 * q^87 + 14 * q^89 - 16 * q^91 - 6 * q^93 - 4 * q^95 + 18 * q^97 + 4 * q^99

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