LMFDB - Embedded newform 480.2.k.b.241.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,2,Mod(241,480)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("480.241"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	480.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.4
Root			$-0.671462 + 1.24464i$ of defining polynomial
Character	$\chi$	$=$	480.241
Dual form			480.2.k.b.241.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000i q^{5} -4.68585 q^{7} -1.00000 q^{9} -2.29273i q^{11} -4.97858i q^{13} +1.00000 q^{15} -2.97858 q^{17} -2.68585i q^{19} -4.68585i q^{21} -2.68585 q^{23} -1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +6.97858 q^{31} +2.29273 q^{33} +4.68585i q^{35} +4.39312i q^{37} +4.97858 q^{39} -11.3717 q^{41} -9.37169i q^{43} +1.00000i q^{45} -7.27131 q^{47} +14.9572 q^{49} -2.97858i q^{51} -2.00000i q^{53} -2.29273 q^{55} +2.68585 q^{57} -1.70727i q^{59} +4.58546i q^{61} +4.68585 q^{63} -4.97858 q^{65} -4.00000i q^{67} -2.68585i q^{69} -0.585462 q^{71} -6.00000 q^{73} -1.00000i q^{75} +10.7434i q^{77} -1.02142 q^{79} +1.00000 q^{81} +13.3717i q^{83} +2.97858i q^{85} -2.00000 q^{87} +3.37169 q^{89} +23.3288i q^{91} +6.97858i q^{93} -2.68585 q^{95} -3.95715 q^{97} +2.29273i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{7} - 6 q^{9} + 6 q^{15} + 12 q^{17} + 8 q^{23} - 6 q^{25} + 12 q^{31} + 8 q^{33} - 20 q^{41} - 8 q^{47} + 30 q^{49} - 8 q^{55} - 8 q^{57} + 4 q^{63} + 8 q^{71} - 36 q^{73} - 36 q^{79} + 6 q^{81}+ \cdots + 36 q^{97}+O(q^{100}) $ 6 * q - 4 * q^7 - 6 * q^9 + 6 * q^15 + 12 * q^17 + 8 * q^23 - 6 * q^25 + 12 * q^31 + 8 * q^33 - 20 * q^41 - 8 * q^47 + 30 * q^49 - 8 * q^55 - 8 * q^57 + 4 * q^63 + 8 * q^71 - 36 * q^73 - 36 * q^79 + 6 * q^81 - 12 * q^87 - 28 * q^89 + 8 * q^95 + 36 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	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
$11$	− 2.29273i	− 0.691284i	−0.938366	−	0.345642i	$-0.887661\pi$
$11$	− 2.29273i	− 0.691284i	0.938366	−	0.345642i	$-0.112339\pi$
$12$	0	0
$13$	− 4.97858i	− 1.38081i	−0.723424	−	0.690404i	$-0.757433\pi$
$13$	− 4.97858i	− 1.38081i	0.723424	−	0.690404i	$-0.242567\pi$
$14$	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
$19$	− 2.68585i	− 0.616175i	−0.951358	−	0.308088i	$-0.900311\pi$
$19$	− 2.68585i	− 0.616175i	0.951358	−	0.308088i	$-0.0996890\pi$
$20$	0	0
$21$	− 4.68585i	− 1.02254i
$22$	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
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	2.00000i	0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$	2.00000i	0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$	0	0
$31$	6.97858	1.25339	0.626695	−	0.779265i	$-0.284407\pi$
$31$	6.97858	1.25339	0.626695	+	0.779265i	$0.284407\pi$
$32$	0	0
$33$	2.29273	0.399113
$34$	0	0
$35$	4.68585i	0.792053i
$36$	0	0
$37$	4.39312i	0.722224i	0.932523	+	0.361112i	$0.117603\pi$
$37$	4.39312i	0.722224i	−0.932523	+	0.361112i	$0.882397\pi$
$38$	0	0
$39$	4.97858	0.797210
$40$	0	0
$41$	−11.3717	−1.77596	−0.887980	−	0.459882i	$-0.847892\pi$
$41$	−11.3717	−1.77596	−0.887980	+	0.459882i	$0.847892\pi$
$42$	0	0
$43$	− 9.37169i	− 1.42917i	−0.699549	−	0.714585i	$-0.746616\pi$
$43$	− 9.37169i	− 1.42917i	0.699549	−	0.714585i	$-0.253384\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	0	0
$47$	−7.27131	−1.06063	−0.530315	−	0.847801i	$-0.677926\pi$
$47$	−7.27131	−1.06063	−0.530315	+	0.847801i	$0.677926\pi$
$48$	0	0
$49$	14.9572	2.13674
$50$	0	0
$51$	− 2.97858i	− 0.417084i
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	−2.29273	−0.309152
$56$	0	0
$57$	2.68585	0.355749
$58$	0	0
$59$	− 1.70727i	− 0.222267i	−0.993805	−	0.111134i	$-0.964552\pi$
$59$	− 1.70727i	− 0.222267i	0.993805	−	0.111134i	$-0.0354482\pi$
$60$	0	0
$61$	4.58546i	0.587108i	0.955942	+	0.293554i	$0.0948381\pi$
$61$	4.58546i	0.587108i	−0.955942	+	0.293554i	$0.905162\pi$
$62$	0	0
$63$	4.68585	0.590361
$64$	0	0
$65$	−4.97858	−0.617516
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	− 2.68585i	− 0.323338i
$70$	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	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	10.7434i	1.22432i
$78$	0	0
$79$	−1.02142	−0.114919	−0.0574595	−	0.998348i	$-0.518300\pi$
$79$	−1.02142	−0.114919	−0.0574595	+	0.998348i	$0.518300\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.3717i	1.46773i	0.679293	+	0.733867i	$0.262286\pi$
$83$	13.3717i	1.46773i	−0.679293	+	0.733867i	$0.737714\pi$
$84$	0	0
$85$	2.97858i	0.323072i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	3.37169	0.357399	0.178699	−	0.983904i	$-0.442811\pi$
$89$	3.37169	0.357399	0.178699	+	0.983904i	$0.442811\pi$
$90$	0	0
$91$	23.3288i	2.44553i
$92$	0	0
$93$	6.97858i	0.723645i
$94$	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
$99$	2.29273i	0.230428i
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	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
$107$	11.3288i	1.09520i	0.836740	+	0.547600i	$0.184459\pi$
$107$	11.3288i	1.09520i	−0.836740	+	0.547600i	$0.815541\pi$
$108$	0	0
$109$	− 9.37169i	− 0.897645i	−0.893621	−	0.448823i	$-0.851843\pi$
$109$	− 9.37169i	− 0.897645i	0.893621	−	0.448823i	$-0.148157\pi$
$110$	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
$115$	2.68585i	0.250456i
$116$	0	0
$117$	4.97858i	0.460270i
$118$	0	0
$119$	13.9572	1.27945
$120$	0	0
$121$	5.74338	0.522126
$122$	0	0
$123$	− 11.3717i	− 1.02535i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−6.64300	−0.589471	−0.294735	−	0.955579i	$-0.595232\pi$
$127$	−6.64300	−0.589471	−0.294735	+	0.955579i	$0.595232\pi$
$128$	0	0
$129$	9.37169	0.825132
$130$	0	0
$131$	− 7.07896i	− 0.618492i	−0.950982	−	0.309246i	$-0.899923\pi$
$131$	− 7.07896i	− 0.618492i	0.950982	−	0.309246i	$-0.100077\pi$
$132$	0	0
$133$	12.5855i	1.09130i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−14.9786	−1.27971	−0.639853	−	0.768497i	$-0.721005\pi$
$137$	−14.9786	−1.27971	−0.639853	+	0.768497i	$0.721005\pi$
$138$	0	0
$139$	− 4.64300i	− 0.393814i	−0.980422	−	0.196907i	$-0.936910\pi$
$139$	− 4.64300i	− 0.393814i	0.980422	−	0.196907i	$-0.0630897\pi$
$140$	0	0
$141$	− 7.27131i	− 0.612355i
$142$	0	0
$143$	−11.4145	−0.954532
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	14.9572i	1.23365i
$148$	0	0
$149$	2.00000i	0.163846i	0.996639	+	0.0819232i	$0.0261062\pi$
$149$	2.00000i	0.163846i	−0.996639	+	0.0819232i	$0.973894\pi$
$150$	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
$155$	− 6.97858i	− 0.560533i
$156$	0	0
$157$	− 22.3503i	− 1.78375i	−0.452286	−	0.891873i	$-0.649391\pi$
$157$	− 22.3503i	− 1.78375i	0.452286	−	0.891873i	$-0.350609\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	12.5855	0.991873
$162$	0	0
$163$	1.37169i	0.107439i	0.998556	+	0.0537196i	$0.0171077\pi$
$163$	1.37169i	0.107439i	−0.998556	+	0.0537196i	$0.982892\pi$
$164$	0	0
$165$	− 2.29273i	− 0.178489i
$166$	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
$171$	2.68585i	0.205392i
$172$	0	0
$173$	− 10.7862i	− 0.820062i	−0.912072	−	0.410031i	$-0.865518\pi$
$173$	− 10.7862i	− 0.820062i	0.912072	−	0.410031i	$-0.134482\pi$
$174$	0	0
$175$	4.68585	0.354217
$176$	0	0
$177$	1.70727	0.128326
$178$	0	0
$179$	− 3.66442i	− 0.273892i	−0.990579	−	0.136946i	$-0.956271\pi$
$179$	− 3.66442i	− 0.273892i	0.990579	−	0.136946i	$-0.0437287\pi$
$180$	0	0
$181$	− 6.62831i	− 0.492678i	−0.969184	−	0.246339i	$-0.920772\pi$
$181$	− 6.62831i	− 0.492678i	0.969184	−	0.246339i	$-0.0792277\pi$
$182$	0	0
$183$	−4.58546	−0.338967
$184$	0	0
$185$	4.39312	0.322988
$186$	0	0
$187$	6.82908i	0.499392i
$188$	0	0
$189$	4.68585i	0.340845i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	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
$195$	− 4.97858i	− 0.356523i
$196$	0	0
$197$	23.9572i	1.70688i	0.521194	+	0.853438i	$0.325487\pi$
$197$	23.9572i	1.70688i	−0.521194	+	0.853438i	$0.674513\pi$
$198$	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
$203$	− 9.37169i	− 0.657764i
$204$	0	0
$205$	11.3717i	0.794233i
$206$	0	0
$207$	2.68585	0.186679
$208$	0	0
$209$	−6.15792	−0.425952
$210$	0	0
$211$	− 14.1004i	− 0.970710i	−0.874317	−	0.485355i	$-0.838690\pi$
$211$	− 14.1004i	− 0.970710i	0.874317	−	0.485355i	$-0.161310\pi$
$212$	0	0
$213$	− 0.585462i	− 0.0401152i
$214$	0	0
$215$	−9.37169	−0.639144
$216$	0	0
$217$	−32.7005	−2.21986
$218$	0	0
$219$	− 6.00000i	− 0.405442i
$220$	0	0
$221$	14.8291i	0.997512i
$222$	0	0
$223$	6.72869	0.450587	0.225293	−	0.974291i	$-0.427666\pi$
$223$	6.72869	0.450587	0.225293	+	0.974291i	$0.427666\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 9.95715i	− 0.660880i	−0.943827	−	0.330440i	$-0.892803\pi$
$227$	− 9.95715i	− 0.660880i	0.943827	−	0.330440i	$-0.107197\pi$
$228$	0	0
$229$	− 11.3288i	− 0.748631i	−0.927301	−	0.374316i	$-0.877878\pi$
$229$	− 11.3288i	− 0.748631i	0.927301	−	0.374316i	$-0.122122\pi$
$230$	0	0
$231$	−10.7434	−0.706863
$232$	0	0
$233$	−18.9786	−1.24333	−0.621664	−	0.783284i	$-0.713543\pi$
$233$	−18.9786	−1.24333	−0.621664	+	0.783284i	$0.713543\pi$
$234$	0	0
$235$	7.27131i	0.474328i
$236$	0	0
$237$	− 1.02142i	− 0.0663485i
$238$	0	0
$239$	−2.62831	−0.170011	−0.0850055	−	0.996380i	$-0.527091\pi$
$239$	−2.62831	−0.170011	−0.0850055	+	0.996380i	$0.527091\pi$
$240$	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
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 14.9572i	− 0.955578i
$246$	0	0
$247$	−13.3717	−0.850820
$248$	0	0
$249$	−13.3717	−0.847397
$250$	0	0
$251$	− 30.9933i	− 1.95628i	−0.207952	−	0.978139i	$-0.566680\pi$
$251$	− 30.9933i	− 1.95628i	0.207952	−	0.978139i	$-0.433320\pi$
$252$	0	0
$253$	6.15792i	0.387145i
$254$	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
$259$	− 20.5855i	− 1.27912i
$260$	0	0
$261$	− 2.00000i	− 0.123797i
$262$	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
$267$	3.37169i	0.206344i
$268$	0	0
$269$	− 24.7434i	− 1.50863i	−0.656512	−	0.754315i	$-0.727969\pi$
$269$	− 24.7434i	− 1.50863i	0.656512	−	0.754315i	$-0.272031\pi$
$270$	0	0
$271$	−27.5640	−1.67440	−0.837198	−	0.546900i	$-0.815808\pi$
$271$	−27.5640	−1.67440	−0.837198	+	0.546900i	$0.815808\pi$
$272$	0	0
$273$	−23.3288	−1.41193
$274$	0	0
$275$	2.29273i	0.138257i
$276$	0	0
$277$	− 20.3074i	− 1.22015i	−0.792342	−	0.610077i	$-0.791138\pi$
$277$	− 20.3074i	− 1.22015i	0.792342	−	0.610077i	$-0.208862\pi$
$278$	0	0
$279$	−6.97858	−0.417796
$280$	0	0
$281$	−10.7862	−0.643453	−0.321726	−	0.946833i	$-0.604263\pi$
$281$	−10.7862	−0.643453	−0.321726	+	0.946833i	$0.604263\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	− 2.68585i	− 0.159096i
$286$	0	0
$287$	53.2860	3.14537
$288$	0	0
$289$	−8.12808	−0.478122
$290$	0	0
$291$	− 3.95715i	− 0.231972i
$292$	0	0
$293$	− 21.9143i	− 1.28025i	−0.768272	−	0.640124i	$-0.778883\pi$
$293$	− 21.9143i	− 1.28025i	0.768272	−	0.640124i	$-0.221117\pi$
$294$	0	0
$295$	−1.70727	−0.0994010
$296$	0	0
$297$	−2.29273	−0.133038
$298$	0	0
$299$	13.3717i	0.773305i
$300$	0	0
$301$	43.9143i	2.53118i
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	4.58546	0.262563
$306$	0	0
$307$	− 26.5426i	− 1.51487i	−0.652912	−	0.757434i	$-0.726453\pi$
$307$	− 26.5426i	− 1.51487i	0.652912	−	0.757434i	$-0.273547\pi$
$308$	0	0
$309$	14.6430i	0.833011i
$310$	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	$-0.351076\pi$
$313$	15.9572	0.901952	0.450976	+	0.892536i	$0.351076\pi$
$314$	0	0
$315$	− 4.68585i	− 0.264018i
$316$	0	0
$317$	− 33.5296i	− 1.88321i	−0.336718	−	0.941605i	$-0.609317\pi$
$317$	− 33.5296i	− 1.88321i	0.336718	−	0.941605i	$-0.390683\pi$
$318$	0	0
$319$	4.58546	0.256737
$320$	0	0
$321$	−11.3288	−0.632315
$322$	0	0
$323$	8.00000i	0.445132i
$324$	0	0
$325$	4.97858i	0.276162i
$326$	0	0
$327$	9.37169	0.518256
$328$	0	0
$329$	34.0722	1.87846
$330$	0	0
$331$	19.8568i	1.09143i	0.837972	+	0.545713i	$0.183741\pi$
$331$	19.8568i	1.09143i	−0.837972	+	0.545713i	$0.816259\pi$
$332$	0	0
$333$	− 4.39312i	− 0.240741i
$334$	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
$339$	19.7648i	1.07348i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	−37.2860	−2.01325
$344$	0	0
$345$	−2.68585	−0.144601
$346$	0	0
$347$	− 0.786230i	− 0.0422071i	−0.999777	−	0.0211035i	$-0.993282\pi$
$347$	− 0.786230i	− 0.0422071i	0.999777	−	0.0211035i	$-0.00671796\pi$
$348$	0	0
$349$	6.15792i	0.329626i	0.986325	+	0.164813i	$0.0527021\pi$
$349$	6.15792i	0.329626i	−0.986325	+	0.164813i	$0.947298\pi$
$350$	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
$355$	0.585462i	0.0310731i
$356$	0	0
$357$	13.9572i	0.738691i
$358$	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
$363$	5.74338i	0.301450i
$364$	0	0
$365$	6.00000i	0.314054i
$366$	0	0
$367$	0.485078	0.0253209	0.0126604	−	0.999920i	$-0.495970\pi$
$367$	0.485078	0.0253209	0.0126604	+	0.999920i	$0.495970\pi$
$368$	0	0
$369$	11.3717	0.591987
$370$	0	0
$371$	9.37169i	0.486554i
$372$	0	0
$373$	− 12.3931i	− 0.641691i	−0.947132	−	0.320846i	$-0.896033\pi$
$373$	− 12.3931i	− 0.641691i	0.947132	−	0.320846i	$-0.103967\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	9.95715	0.512820
$378$	0	0
$379$	26.0147i	1.33629i	0.744033	+	0.668143i	$0.232910\pi$
$379$	26.0147i	1.33629i	−0.744033	+	0.668143i	$0.767090\pi$
$380$	0	0
$381$	− 6.64300i	− 0.340331i
$382$	0	0
$383$	−6.68585	−0.341631	−0.170815	−	0.985303i	$-0.554640\pi$
$383$	−6.68585	−0.341631	−0.170815	+	0.985303i	$0.554640\pi$
$384$	0	0
$385$	10.7434	0.547534
$386$	0	0
$387$	9.37169i	0.476390i
$388$	0	0
$389$	29.9143i	1.51672i	0.651838	+	0.758358i	$0.273998\pi$
$389$	29.9143i	1.51672i	−0.651838	+	0.758358i	$0.726002\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	7.07896	0.357086
$394$	0	0
$395$	1.02142i	0.0513934i
$396$	0	0
$397$	9.76481i	0.490082i	0.969513	+	0.245041i	$0.0788014\pi$
$397$	9.76481i	0.490082i	−0.969513	+	0.245041i	$0.921199\pi$
$398$	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
$403$	− 34.7434i	− 1.73069i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	10.0722	0.499262
$408$	0	0
$409$	−25.9143	−1.28138	−0.640690	−	0.767800i	$-0.721352\pi$
$409$	−25.9143	−1.28138	−0.640690	+	0.767800i	$0.721352\pi$
$410$	0	0
$411$	− 14.9786i	− 0.738839i
$412$	0	0
$413$	8.00000i	0.393654i
$414$	0	0
$415$	13.3717	0.656391
$416$	0	0
$417$	4.64300	0.227369
$418$	0	0
$419$	12.2499i	0.598446i	0.954183	+	0.299223i	$0.0967275\pi$
$419$	12.2499i	0.598446i	−0.954183	+	0.299223i	$0.903273\pi$
$420$	0	0
$421$	− 4.67115i	− 0.227658i	−0.993500	−	0.113829i	$-0.963688\pi$
$421$	− 4.67115i	− 0.227658i	0.993500	−	0.113829i	$-0.0363116\pi$
$422$	0	0
$423$	7.27131	0.353543
$424$	0	0
$425$	2.97858	0.144482
$426$	0	0
$427$	− 21.4868i	− 1.03982i
$428$	0	0
$429$	− 11.4145i	− 0.551099i
$430$	0	0
$431$	−0.585462	−0.0282007	−0.0141004	−	0.999901i	$-0.504488\pi$
$431$	−0.585462	−0.0282007	−0.0141004	+	0.999901i	$0.504488\pi$
$432$	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
$435$	2.00000i	0.0958927i
$436$	0	0
$437$	7.21377i	0.345081i
$438$	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
$443$	20.7005i	0.983512i	0.870733	+	0.491756i	$0.163645\pi$
$443$	20.7005i	0.983512i	−0.870733	+	0.491756i	$0.836355\pi$
$444$	0	0
$445$	− 3.37169i	− 0.159834i
$446$	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
$451$	26.0722i	1.22769i
$452$	0	0
$453$	8.35027i	0.392330i
$454$	0	0
$455$	23.3288	1.09367
$456$	0	0
$457$	38.7005	1.81033	0.905167	−	0.425055i	$-0.139745\pi$
$457$	38.7005	1.81033	0.905167	+	0.425055i	$0.139745\pi$
$458$	0	0
$459$	2.97858i	0.139028i
$460$	0	0
$461$	− 4.74338i	− 0.220921i	−0.993880	−	0.110461i	$-0.964767\pi$
$461$	− 4.74338i	− 0.220921i	0.993880	−	0.110461i	$-0.0352326\pi$
$462$	0	0
$463$	−15.3142	−0.711709	−0.355855	−	0.934541i	$-0.615810\pi$
$463$	−15.3142	−0.711709	−0.355855	+	0.934541i	$0.615810\pi$
$464$	0	0
$465$	6.97858	0.323624
$466$	0	0
$467$	30.5426i	1.41334i	0.707541	+	0.706672i	$0.249804\pi$
$467$	30.5426i	1.41334i	−0.707541	+	0.706672i	$0.750196\pi$
$468$	0	0
$469$	18.7434i	0.865489i
$470$	0	0
$471$	22.3503	1.02985
$472$	0	0
$473$	−21.4868	−0.987963
$474$	0	0
$475$	2.68585i	0.123235i
$476$	0	0
$477$	2.00000i	0.0915737i
$478$	0	0
$479$	−3.32885	−0.152099	−0.0760494	−	0.997104i	$-0.524231\pi$
$479$	−3.32885	−0.152099	−0.0760494	+	0.997104i	$0.524231\pi$
$480$	0	0
$481$	21.8715	0.997253
$482$	0	0
$483$	12.5855i	0.572658i
$484$	0	0
$485$	3.95715i	0.179685i
$486$	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
$491$	− 14.2927i	− 0.645022i	−0.946566	−	0.322511i	$-0.895473\pi$
$491$	− 14.2927i	− 0.645022i	0.946566	−	0.322511i	$-0.104527\pi$
$492$	0	0
$493$	− 5.95715i	− 0.268297i
$494$	0	0
$495$	2.29273	0.103051
$496$	0	0
$497$	2.74338	0.123058
$498$	0	0
$499$	9.22846i	0.413123i	0.978434	+	0.206561i	$0.0662273\pi$
$499$	9.22846i	0.413123i	−0.978434	+	0.206561i	$0.933773\pi$
$500$	0	0
$501$	− 11.2713i	− 0.503565i
$502$	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
$507$	− 11.7862i	− 0.523445i
$508$	0	0
$509$	43.4011i	1.92372i	0.273544	+	0.961859i	$0.411804\pi$
$509$	43.4011i	1.92372i	−0.273544	+	0.961859i	$0.588196\pi$
$510$	0	0
$511$	28.1151	1.24374
$512$	0	0
$513$	−2.68585	−0.118583
$514$	0	0
$515$	− 14.6430i	− 0.645248i
$516$	0	0
$517$	16.6712i	0.733196i
$518$	0	0
$519$	10.7862	0.473463
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	13.5725i	0.593482i	0.954958	+	0.296741i	$0.0958998\pi$
$523$	13.5725i	0.593482i	−0.954958	+	0.296741i	$0.904100\pi$
$524$	0	0
$525$	4.68585i	0.204507i
$526$	0	0
$527$	−20.7862	−0.905462
$528$	0	0
$529$	−15.7862	−0.686358
$530$	0	0
$531$	1.70727i	0.0740892i
$532$	0	0
$533$	56.6148i	2.45226i
$534$	0	0
$535$	11.3288	0.489789
$536$	0	0
$537$	3.66442	0.158132
$538$	0	0
$539$	− 34.2927i	− 1.47709i
$540$	0	0
$541$	37.2860i	1.60305i	0.597961	+	0.801525i	$0.295978\pi$
$541$	37.2860i	1.60305i	−0.597961	+	0.801525i	$0.704022\pi$
$542$	0	0
$543$	6.62831	0.284448
$544$	0	0
$545$	−9.37169	−0.401439
$546$	0	0
$547$	− 0.200768i	− 0.00858424i	−0.999991	−	0.00429212i	$-0.998634\pi$
$547$	− 0.200768i	− 0.00858424i	0.999991	−	0.00429212i	$-0.00136623\pi$
$548$	0	0
$549$	− 4.58546i	− 0.195703i
$550$	0	0
$551$	5.37169	0.228842
$552$	0	0
$553$	4.78623	0.203531
$554$	0	0
$555$	4.39312i	0.186477i
$556$	0	0
$557$	− 9.21377i	− 0.390400i	−0.980763	−	0.195200i	$-0.937464\pi$
$557$	− 9.21377i	− 0.390400i	0.980763	−	0.195200i	$-0.0625356\pi$
$558$	0	0
$559$	−46.6577	−1.97341
$560$	0	0
$561$	−6.82908	−0.288324
$562$	0	0
$563$	− 36.7005i	− 1.54674i	−0.633953	−	0.773372i	$-0.718569\pi$
$563$	− 36.7005i	− 1.54674i	0.633953	−	0.773372i	$-0.281431\pi$
$564$	0	0
$565$	− 19.7648i	− 0.831512i
$566$	0	0
$567$	−4.68585	−0.196787
$568$	0	0
$569$	13.4145	0.562367	0.281183	−	0.959654i	$-0.409273\pi$
$569$	13.4145	0.562367	0.281183	+	0.959654i	$0.409273\pi$
$570$	0	0
$571$	18.6858i	0.781978i	0.920395	+	0.390989i	$0.127867\pi$
$571$	18.6858i	0.781978i	−0.920395	+	0.390989i	$0.872133\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	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
$579$	1.21377i	0.0504425i
$580$	0	0
$581$	− 62.6577i	− 2.59948i
$582$	0	0
$583$	−4.58546	−0.189910
$584$	0	0
$585$	4.97858	0.205839
$586$	0	0
$587$	− 27.3288i	− 1.12798i	−0.825781	−	0.563991i	$-0.809265\pi$
$587$	− 27.3288i	− 1.12798i	0.825781	−	0.563991i	$-0.190735\pi$
$588$	0	0
$589$	− 18.7434i	− 0.772308i
$590$	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
$595$	− 13.9572i	− 0.572188i
$596$	0	0
$597$	− 0.350269i	− 0.0143356i
$598$	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	$-0.602879\pi$
$601$	−15.5725	−0.635214	−0.317607	+	0.948222i	$0.602879\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	− 5.74338i	− 0.233502i
$606$	0	0
$607$	31.2285	1.26752	0.633762	−	0.773528i	$-0.281510\pi$
$607$	31.2285	1.26752	0.633762	+	0.773528i	$0.281510\pi$
$608$	0	0
$609$	9.37169	0.379760
$610$	0	0
$611$	36.2008i	1.46453i
$612$	0	0
$613$	0.978577i	0.0395244i	0.999805	+	0.0197622i	$0.00629091\pi$
$613$	0.978577i	0.0395244i	−0.999805	+	0.0197622i	$0.993709\pi$
$614$	0	0
$615$	−11.3717	−0.458551
$616$	0	0
$617$	32.9357	1.32594	0.662971	−	0.748645i	$-0.269295\pi$
$617$	32.9357	1.32594	0.662971	+	0.748645i	$0.269295\pi$
$618$	0	0
$619$	3.35700i	0.134929i	0.997722	+	0.0674646i	$0.0214910\pi$
$619$	3.35700i	0.134929i	−0.997722	+	0.0674646i	$0.978509\pi$
$620$	0	0
$621$	2.68585i	0.107779i
$622$	0	0
$623$	−15.7992	−0.632983
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 6.15792i	− 0.245924i
$628$	0	0
$629$	− 13.0852i	− 0.521742i
$630$	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
$635$	6.64300i	0.263619i
$636$	0	0
$637$	− 74.4653i	− 2.95042i
$638$	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
$643$	− 29.2860i	− 1.15493i	−0.816416	−	0.577464i	$-0.804043\pi$
$643$	− 29.2860i	− 1.15493i	0.816416	−	0.577464i	$-0.195957\pi$
$644$	0	0
$645$	− 9.37169i	− 0.369010i
$646$	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
$651$	− 32.7005i	− 1.28164i
$652$	0	0
$653$	17.5296i	0.685987i	0.939338	+	0.342993i	$0.111441\pi$
$653$	17.5296i	0.685987i	−0.939338	+	0.342993i	$0.888559\pi$
$654$	0	0
$655$	−7.07896	−0.276598
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	23.8652i	0.929656i	0.885401	+	0.464828i	$0.153884\pi$
$659$	23.8652i	0.929656i	−0.885401	+	0.464828i	$0.846116\pi$
$660$	0	0
$661$	− 30.1579i	− 1.17301i	−0.809947	−	0.586504i	$-0.800504\pi$
$661$	− 30.1579i	− 1.17301i	0.809947	−	0.586504i	$-0.199496\pi$
$662$	0	0
$663$	−14.8291	−0.575914
$664$	0	0
$665$	12.5855	0.488043
$666$	0	0
$667$	− 5.37169i	− 0.207993i
$668$	0	0
$669$	6.72869i	0.260146i
$670$	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
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	9.61531i	0.369546i	0.982781	+	0.184773i	$0.0591550\pi$
$677$	9.61531i	0.369546i	−0.982781	+	0.184773i	$0.940845\pi$
$678$	0	0
$679$	18.5426	0.711600
$680$	0	0
$681$	9.95715	0.381559
$682$	0	0
$683$	− 18.6283i	− 0.712792i	−0.934335	−	0.356396i	$-0.884005\pi$
$683$	− 18.6283i	− 0.712792i	0.934335	−	0.356396i	$-0.115995\pi$
$684$	0	0
$685$	14.9786i	0.572302i
$686$	0	0
$687$	11.3288	0.432222
$688$	0	0
$689$	−9.95715	−0.379337
$690$	0	0
$691$	13.4292i	0.510872i	0.966826	+	0.255436i	$0.0822190\pi$
$691$	13.4292i	0.510872i	−0.966826	+	0.255436i	$0.917781\pi$
$692$	0	0
$693$	− 10.7434i	− 0.408107i
$694$	0	0
$695$	−4.64300	−0.176119
$696$	0	0
$697$	33.8715	1.28297
$698$	0	0
$699$	− 18.9786i	− 0.717836i
$700$	0	0
$701$	19.1709i	0.724076i	0.932163	+	0.362038i	$0.117919\pi$
$701$	19.1709i	0.724076i	−0.932163	+	0.362038i	$0.882081\pi$
$702$	0	0
$703$	11.7992	0.445016
$704$	0	0
$705$	−7.27131	−0.273853
$706$	0	0
$707$	9.37169i	0.352459i
$708$	0	0
$709$	15.4145i	0.578905i	0.957192	+	0.289453i	$0.0934732\pi$
$709$	15.4145i	0.578905i	−0.957192	+	0.289453i	$0.906527\pi$
$710$	0	0
$711$	1.02142	0.0383063
$712$	0	0
$713$	−18.7434	−0.701945
$714$	0	0
$715$	11.4145i	0.426880i
$716$	0	0
$717$	− 2.62831i	− 0.0981559i
$718$	0	0
$719$	−20.7862	−0.775196	−0.387598	−	0.921829i	$-0.626695\pi$
$719$	−20.7862	−0.775196	−0.387598	+	0.921829i	$0.626695\pi$
$720$	0	0
$721$	−68.6148	−2.55535
$722$	0	0
$723$	10.7862i	0.401144i
$724$	0	0
$725$	− 2.00000i	− 0.0742781i
$726$	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
$731$	27.9143i	1.03245i
$732$	0	0
$733$	− 35.9227i	− 1.32684i	−0.748249	−	0.663418i	$-0.769105\pi$
$733$	− 35.9227i	− 1.32684i	0.748249	−	0.663418i	$-0.230895\pi$
$734$	0	0
$735$	14.9572	0.551703
$736$	0	0
$737$	−9.17092	−0.337815
$738$	0	0
$739$	− 29.0277i	− 1.06780i	−0.845547	−	0.533900i	$-0.820726\pi$
$739$	− 29.0277i	− 1.06780i	0.845547	−	0.533900i	$-0.179274\pi$
$740$	0	0
$741$	− 13.3717i	− 0.491221i
$742$	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
$747$	− 13.3717i	− 0.489245i
$748$	0	0
$749$	− 53.0852i	− 1.93969i
$750$	0	0
$751$	10.8929	0.397487	0.198744	−	0.980052i	$-0.436314\pi$
$751$	10.8929	0.397487	0.198744	+	0.980052i	$0.436314\pi$
$752$	0	0
$753$	30.9933	1.12946
$754$	0	0
$755$	− 8.35027i	− 0.303897i
$756$	0	0
$757$	34.3503i	1.24848i	0.781232	+	0.624241i	$0.214592\pi$
$757$	34.3503i	1.24848i	−0.781232	+	0.624241i	$0.785408\pi$
$758$	0	0
$759$	−6.15792	−0.223518
$760$	0	0
$761$	−19.0852	−0.691839	−0.345920	−	0.938264i	$-0.612433\pi$
$761$	−19.0852	−0.691839	−0.345920	+	0.938264i	$0.612433\pi$
$762$	0	0
$763$	43.9143i	1.58980i
$764$	0	0
$765$	− 2.97858i	− 0.107691i
$766$	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
$771$	20.9357i	0.753982i
$772$	0	0
$773$	− 11.9572i	− 0.430069i	−0.976606	−	0.215034i	$-0.931014\pi$
$773$	− 11.9572i	− 0.430069i	0.976606	−	0.215034i	$-0.0689864\pi$
$774$	0	0
$775$	−6.97858	−0.250678
$776$	0	0
$777$	20.5855	0.738499
$778$	0	0
$779$	30.5426i	1.09430i
$780$	0	0
$781$	1.34231i	0.0480315i
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−22.3503	−0.797715
$786$	0	0
$787$	− 33.0852i	− 1.17936i	−0.807637	−	0.589681i	$-0.799254\pi$
$787$	− 33.0852i	− 1.17936i	0.807637	−	0.589681i	$-0.200746\pi$
$788$	0	0
$789$	19.2713i	0.686077i
$790$	0	0
$791$	−92.6148	−3.29300
$792$	0	0
$793$	22.8291	0.810684
$794$	0	0
$795$	− 2.00000i	− 0.0709327i
$796$	0	0
$797$	− 10.0000i	− 0.354218i	−0.984191	−	0.177109i	$-0.943325\pi$
$797$	− 10.0000i	− 0.354218i	0.984191	−	0.177109i	$-0.0566745\pi$
$798$	0	0
$799$	21.6582	0.766210
$800$	0	0
$801$	−3.37169	−0.119133
$802$	0	0
$803$	13.7564i	0.485452i
$804$	0	0
$805$	− 12.5855i	− 0.443579i
$806$	0	0
$807$	24.7434	0.871008
$808$	0	0
$809$	−30.6148	−1.07636	−0.538180	−	0.842830i	$-0.680888\pi$
$809$	−30.6148	−1.07636	−0.538180	+	0.842830i	$0.680888\pi$
$810$	0	0
$811$	− 53.9290i	− 1.89370i	−0.321670	−	0.946852i	$-0.604244\pi$
$811$	− 53.9290i	− 1.89370i	0.321670	−	0.946852i	$-0.395756\pi$
$812$	0	0
$813$	− 27.5640i	− 0.966713i
$814$	0	0
$815$	1.37169	0.0480483
$816$	0	0
$817$	−25.1709	−0.880619
$818$	0	0
$819$	− 23.3288i	− 0.815176i
$820$	0	0
$821$	12.6577i	0.441757i	0.975301	+	0.220878i	$0.0708924\pi$
$821$	12.6577i	0.441757i	−0.975301	+	0.220878i	$0.929108\pi$
$822$	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
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	0	0
$829$	− 41.3717i	− 1.43690i	−0.695580	−	0.718449i	$-0.744852\pi$
$829$	− 41.3717i	− 1.43690i	0.695580	−	0.718449i	$-0.255148\pi$
$830$	0	0
$831$	20.3074	0.704457
$832$	0	0
$833$	−44.5510	−1.54360
$834$	0	0
$835$	11.2713i	0.390060i
$836$	0	0
$837$	− 6.97858i	− 0.241215i
$838$	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
$843$	− 10.7862i	− 0.371498i
$844$	0	0
$845$	11.7862i	0.405459i
$846$	0	0
$847$	−26.9126	−0.924728
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	− 11.7992i	− 0.404472i
$852$	0	0
$853$	6.63673i	0.227237i	0.993524	+	0.113619i	$0.0362442\pi$
$853$	6.63673i	0.227237i	−0.993524	+	0.113619i	$0.963756\pi$
$854$	0	0
$855$	2.68585	0.0918540
$856$	0	0
$857$	9.80765	0.335023	0.167512	−	0.985870i	$-0.446427\pi$
$857$	9.80765	0.335023	0.167512	+	0.985870i	$0.446427\pi$
$858$	0	0
$859$	− 39.3864i	− 1.34385i	−0.740621	−	0.671923i	$-0.765469\pi$
$859$	− 39.3864i	− 1.34385i	0.740621	−	0.671923i	$-0.234531\pi$
$860$	0	0
$861$	53.2860i	1.81598i
$862$	0	0
$863$	−7.07054	−0.240684	−0.120342	−	0.992732i	$-0.538399\pi$
$863$	−7.07054	−0.240684	−0.120342	+	0.992732i	$0.538399\pi$
$864$	0	0
$865$	−10.7862	−0.366743
$866$	0	0
$867$	− 8.12808i	− 0.276044i
$868$	0	0
$869$	2.34185i	0.0794417i
$870$	0	0
$871$	−19.9143	−0.674771
$872$	0	0
$873$	3.95715	0.133929
$874$	0	0
$875$	− 4.68585i	− 0.158411i
$876$	0	0
$877$	23.1365i	0.781264i	0.920547	+	0.390632i	$0.127744\pi$
$877$	23.1365i	0.781264i	−0.920547	+	0.390632i	$0.872256\pi$
$878$	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
$883$	41.2003i	1.38650i	0.720697	+	0.693250i	$0.243822\pi$
$883$	41.2003i	1.38650i	−0.720697	+	0.693250i	$0.756178\pi$
$884$	0	0
$885$	− 1.70727i	− 0.0573892i
$886$	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
$891$	− 2.29273i	− 0.0768094i
$892$	0	0
$893$	19.5296i	0.653534i
$894$	0	0
$895$	−3.66442	−0.122488
$896$	0	0
$897$	−13.3717	−0.446468
$898$	0	0
$899$	13.9572i	0.465497i
$900$	0	0
$901$	5.95715i	0.198462i
$902$	0	0
$903$	−43.9143	−1.46138
$904$	0	0
$905$	−6.62831	−0.220332
$906$	0	0
$907$	50.6577i	1.68206i	0.540988	+	0.841031i	$0.318051\pi$
$907$	50.6577i	1.68206i	−0.540988	+	0.841031i	$0.681949\pi$
$908$	0	0
$909$	2.00000i	0.0663358i
$910$	0	0
$911$	26.4569	0.876557	0.438279	−	0.898839i	$-0.355588\pi$
$911$	26.4569	0.876557	0.438279	+	0.898839i	$0.355588\pi$
$912$	0	0
$913$	30.6577	1.01462
$914$	0	0
$915$	4.58546i	0.151591i
$916$	0	0
$917$	33.1709i	1.09540i
$918$	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
$923$	2.91477i	0.0959407i
$924$	0	0
$925$	− 4.39312i	− 0.144445i
$926$	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
$931$	− 40.1726i	− 1.31660i
$932$	0	0
$933$	− 12.2008i	− 0.399435i
$934$	0	0
$935$	6.82908	0.223335
$936$	0	0
$937$	−42.2302	−1.37960	−0.689799	−	0.724000i	$-0.742301\pi$
$937$	−42.2302	−1.37960	−0.689799	+	0.724000i	$0.742301\pi$
$938$	0	0
$939$	15.9572i	0.520742i
$940$	0	0
$941$	32.7434i	1.06740i	0.845673	+	0.533702i	$0.179200\pi$
$941$	32.7434i	1.06740i	−0.845673	+	0.533702i	$0.820800\pi$
$942$	0	0
$943$	30.5426	0.994604
$944$	0	0
$945$	4.68585	0.152431
$946$	0	0
$947$	15.9143i	0.517146i	0.965992	+	0.258573i	$0.0832522\pi$
$947$	15.9143i	0.517146i	−0.965992	+	0.258573i	$0.916748\pi$
$948$	0	0
$949$	29.8715i	0.969669i
$950$	0	0
$951$	33.5296	1.08727
$952$	0	0
$953$	55.6791	1.80362	0.901812	−	0.432129i	$-0.142238\pi$
$953$	55.6791	1.80362	0.901812	+	0.432129i	$0.142238\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	4.58546i	0.148227i
$958$	0	0
$959$	70.1873	2.26647
$960$	0	0
$961$	17.7005	0.570985
$962$	0	0
$963$	− 11.3288i	− 0.365067i
$964$	0	0
$965$	− 1.21377i	− 0.0390726i
$966$	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
$971$	30.3221i	0.973083i	0.873657	+	0.486542i	$0.161742\pi$
$971$	30.3221i	0.973083i	−0.873657	+	0.486542i	$0.838258\pi$
$972$	0	0
$973$	21.7564i	0.697478i
$974$	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
$979$	− 7.73038i	− 0.247064i
$980$	0	0
$981$	9.37169i	0.299215i
$982$	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
$987$	34.0722i	1.08453i
$988$	0	0
$989$	25.1709i	0.800389i
$990$	0	0
$991$	−19.0937	−0.606530	−0.303265	−	0.952906i	$-0.598077\pi$
$991$	−19.0937	−0.606530	−0.303265	+	0.952906i	$0.598077\pi$
$992$	0	0
$993$	−19.8568	−0.630136
$994$	0	0
$995$	0.350269i	0.0111043i
$996$	0	0
$997$	38.8500i	1.23039i	0.788374	+	0.615197i	$0.210923\pi$
$997$	38.8500i	1.23039i	−0.788374	+	0.615197i	$0.789077\pi$
$998$	0	0
$999$	4.39312	0.138992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.k.b.241.4		6
3.2	odd	2		1440.2.k.f.721.4		6
4.3	odd	2		120.2.k.b.61.2	yes	6
5.2	odd	4		2400.2.d.f.49.1		6
5.3	odd	4		2400.2.d.e.49.6		6
5.4	even	2		2400.2.k.c.1201.3		6
8.3	odd	2		120.2.k.b.61.1	✓	6
8.5	even	2	inner	480.2.k.b.241.1		6
12.11	even	2		360.2.k.f.181.5		6
15.2	even	4		7200.2.d.q.2449.1		6
15.8	even	4		7200.2.d.r.2449.6		6
15.14	odd	2		7200.2.k.p.3601.6		6
16.3	odd	4		3840.2.a.bq.1.1		3
16.5	even	4		3840.2.a.br.1.3		3
16.11	odd	4		3840.2.a.bp.1.1		3
16.13	even	4		3840.2.a.bo.1.3		3
20.3	even	4		600.2.d.f.349.6		6
20.7	even	4		600.2.d.e.349.1		6
20.19	odd	2		600.2.k.c.301.5		6
24.5	odd	2		1440.2.k.f.721.1		6
24.11	even	2		360.2.k.f.181.6		6
40.3	even	4		600.2.d.e.349.2		6
40.13	odd	4		2400.2.d.f.49.6		6
40.19	odd	2		600.2.k.c.301.6		6
40.27	even	4		600.2.d.f.349.5		6
40.29	even	2		2400.2.k.c.1201.6		6
40.37	odd	4		2400.2.d.e.49.1		6
60.23	odd	4		1800.2.d.r.1549.1		6
60.47	odd	4		1800.2.d.q.1549.6		6
60.59	even	2		1800.2.k.p.901.2		6
120.29	odd	2		7200.2.k.p.3601.5		6
120.53	even	4		7200.2.d.q.2449.6		6
120.59	even	2		1800.2.k.p.901.1		6
120.77	even	4		7200.2.d.r.2449.1		6
120.83	odd	4		1800.2.d.q.1549.5		6
120.107	odd	4		1800.2.d.r.1549.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.1	✓	6	8.3	odd	2
120.2.k.b.61.2	yes	6	4.3	odd	2
360.2.k.f.181.5		6	12.11	even	2
360.2.k.f.181.6		6	24.11	even	2
480.2.k.b.241.1		6	8.5	even	2	inner
480.2.k.b.241.4		6	1.1	even	1	trivial
600.2.d.e.349.1		6	20.7	even	4
600.2.d.e.349.2		6	40.3	even	4
600.2.d.f.349.5		6	40.27	even	4
600.2.d.f.349.6		6	20.3	even	4
600.2.k.c.301.5		6	20.19	odd	2
600.2.k.c.301.6		6	40.19	odd	2
1440.2.k.f.721.1		6	24.5	odd	2
1440.2.k.f.721.4		6	3.2	odd	2
1800.2.d.q.1549.5		6	120.83	odd	4
1800.2.d.q.1549.6		6	60.47	odd	4
1800.2.d.r.1549.1		6	60.23	odd	4
1800.2.d.r.1549.2		6	120.107	odd	4
1800.2.k.p.901.1		6	120.59	even	2
1800.2.k.p.901.2		6	60.59	even	2
2400.2.d.e.49.1		6	40.37	odd	4
2400.2.d.e.49.6		6	5.3	odd	4
2400.2.d.f.49.1		6	5.2	odd	4
2400.2.d.f.49.6		6	40.13	odd	4
2400.2.k.c.1201.3		6	5.4	even	2
2400.2.k.c.1201.6		6	40.29	even	2
3840.2.a.bo.1.3		3	16.13	even	4
3840.2.a.bp.1.1		3	16.11	odd	4
3840.2.a.bq.1.1		3	16.3	odd	4
3840.2.a.br.1.3		3	16.5	even	4
7200.2.d.q.2449.1		6	15.2	even	4
7200.2.d.q.2449.6		6	120.53	even	4
7200.2.d.r.2449.1		6	120.77	even	4
7200.2.d.r.2449.6		6	15.8	even	4
7200.2.k.p.3601.5		6	120.29	odd	2
7200.2.k.p.3601.6		6	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	480.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000i q^{5} -4.68585 q^{7} -1.00000 q^{9} -2.29273i q^{11} -4.97858i q^{13} +1.00000 q^{15} -2.97858 q^{17} -2.68585i q^{19} -4.68585i q^{21} -2.68585 q^{23} -1.00000 q^{25} -1.00000i q^{27} +2.00000i q^{29} +6.97858 q^{31} +2.29273 q^{33} +4.68585i q^{35} +4.39312i q^{37} +4.97858 q^{39} -11.3717 q^{41} -9.37169i q^{43} +1.00000i q^{45} -7.27131 q^{47} +14.9572 q^{49} -2.97858i q^{51} -2.00000i q^{53} -2.29273 q^{55} +2.68585 q^{57} -1.70727i q^{59} +4.58546i q^{61} +4.68585 q^{63} -4.97858 q^{65} -4.00000i q^{67} -2.68585i q^{69} -0.585462 q^{71} -6.00000 q^{73} -1.00000i q^{75} +10.7434i q^{77} -1.02142 q^{79} +1.00000 q^{81} +13.3717i q^{83} +2.97858i q^{85} -2.00000 q^{87} +3.37169 q^{89} +23.3288i q^{91} +6.97858i q^{93} -2.68585 q^{95} -3.95715 q^{97} +2.29273i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{7} - 6 q^{9} + 6 q^{15} + 12 q^{17} + 8 q^{23} - 6 q^{25} + 12 q^{31} + 8 q^{33} - 20 q^{41} - 8 q^{47} + 30 q^{49} - 8 q^{55} - 8 q^{57} + 4 q^{63} + 8 q^{71} - 36 q^{73} - 36 q^{79} + 6 q^{81}+ \cdots + 36 q^{97}+O(q^{100}) \) 6 * q - 4 * q^7 - 6 * q^9 + 6 * q^15 + 12 * q^17 + 8 * q^23 - 6 * q^25 + 12 * q^31 + 8 * q^33 - 20 * q^41 - 8 * q^47 + 30 * q^49 - 8 * q^55 - 8 * q^57 + 4 * q^63 + 8 * q^71 - 36 * q^73 - 36 * q^79 + 6 * q^81 - 12 * q^87 - 28 * q^89 + 8 * q^95 + 36 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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