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Properties

Label 80.11.p.c.33.2
Level $80$
Weight $11$
Character 80.33
Analytic conductor $50.829$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,11,Mod(17,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.17"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 80.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.8285802139\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 1148x^{3} + 68121x^{2} - 299628x + 658952 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 33.2
Root \(10.1043 + 10.1043i\) of defining polynomial
Character \(\chi\) \(=\) 80.33
Dual form 80.11.p.c.17.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.29207 - 4.29207i) q^{3} +(-2978.33 + 946.124i) q^{5} +(-21284.7 + 21284.7i) q^{7} -59012.2i q^{9} -155649. q^{11} +(-358614. - 358614. i) q^{13} +(16844.1 + 8722.40i) q^{15} +(-609397. + 609397. i) q^{17} +335226. i q^{19} +182711. q^{21} +(5.50132e6 + 5.50132e6i) q^{23} +(7.97532e6 - 5.63575e6i) q^{25} +(-506727. + 506727. i) q^{27} -1.33815e6i q^{29} -2.59306e7 q^{31} +(668055. + 668055. i) q^{33} +(4.32550e7 - 8.35308e7i) q^{35} +(-5.50890e7 + 5.50890e7i) q^{37} +3.07840e6i q^{39} +1.37064e8 q^{41} +(9.34290e7 + 9.34290e7i) q^{43} +(5.58328e7 + 1.75758e8i) q^{45} +(-1.14979e8 + 1.14979e8i) q^{47} -6.23600e8i q^{49} +5.23115e6 q^{51} +(-2.29978e7 - 2.29978e7i) q^{53} +(4.63574e8 - 1.47263e8i) q^{55} +(1.43882e6 - 1.43882e6i) q^{57} -8.73535e8i q^{59} +5.87905e8 q^{61} +(1.25605e9 + 1.25605e9i) q^{63} +(1.40737e9 + 7.28780e8i) q^{65} +(6.75197e8 - 6.75197e8i) q^{67} -4.72241e7i q^{69} +5.54597e8 q^{71} +(8.91747e8 + 8.91747e8i) q^{73} +(-5.84197e7 - 1.00416e7i) q^{75} +(3.31293e9 - 3.31293e9i) q^{77} -1.69149e9i q^{79} -3.48026e9 q^{81} +(-1.96163e9 - 1.96163e9i) q^{83} +(1.23842e9 - 2.39155e9i) q^{85} +(-5.74345e6 + 5.74345e6i) q^{87} +7.73241e9i q^{89} +1.52660e10 q^{91} +(1.11296e8 + 1.11296e8i) q^{93} +(-3.17166e8 - 9.98416e8i) q^{95} +(1.16280e10 - 1.16280e10i) q^{97} +9.18516e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 128 q^{3} + 5460 q^{5} - 13512 q^{7} - 647832 q^{11} - 742902 q^{13} - 1577720 q^{15} - 755118 q^{17} + 12277112 q^{21} + 15052992 q^{23} + 42644850 q^{25} + 47998120 q^{27} - 153847152 q^{31} - 173025784 q^{33}+ \cdots + 33281088582 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −4.29207 − 4.29207i −0.0176629 − 0.0176629i 0.698220 − 0.715883i \(-0.253976\pi\)
−0.715883 + 0.698220i \(0.753976\pi\)
\(4\) 0 0
\(5\) −2978.33 + 946.124i −0.953067 + 0.302760i
\(6\) 0 0
\(7\) −21284.7 + 21284.7i −1.26642 + 1.26642i −0.318492 + 0.947926i \(0.603176\pi\)
−0.947926 + 0.318492i \(0.896824\pi\)
\(8\) 0 0
\(9\) − 59012.2i − 0.999376i
\(10\) 0 0
\(11\) −155649. −0.966456 −0.483228 − 0.875495i \(-0.660536\pi\)
−0.483228 + 0.875495i \(0.660536\pi\)
\(12\) 0 0
\(13\) −358614. − 358614.i −0.965853 − 0.965853i 0.0335834 − 0.999436i \(-0.489308\pi\)
−0.999436 + 0.0335834i \(0.989308\pi\)
\(14\) 0 0
\(15\) 16844.1 + 8722.40i 0.0221815 + 0.0114863i
\(16\) 0 0
\(17\) −609397. + 609397.i −0.429196 + 0.429196i −0.888354 − 0.459158i \(-0.848151\pi\)
0.459158 + 0.888354i \(0.348151\pi\)
\(18\) 0 0
\(19\) 335226.i 0.135385i 0.997706 + 0.0676924i \(0.0215637\pi\)
−0.997706 + 0.0676924i \(0.978436\pi\)
\(20\) 0 0
\(21\) 182711. 0.0447371
\(22\) 0 0
\(23\) 5.50132e6 + 5.50132e6i 0.854727 + 0.854727i 0.990711 − 0.135984i \(-0.0434196\pi\)
−0.135984 + 0.990711i \(0.543420\pi\)
\(24\) 0 0
\(25\) 7.97532e6 − 5.63575e6i 0.816673 − 0.577100i
\(26\) 0 0
\(27\) −506727. + 506727.i −0.0353147 + 0.0353147i
\(28\) 0 0
\(29\) − 1.33815e6i − 0.0652402i −0.999468 − 0.0326201i \(-0.989615\pi\)
0.999468 − 0.0326201i \(-0.0103851\pi\)
\(30\) 0 0
\(31\) −2.59306e7 −0.905743 −0.452871 − 0.891576i \(-0.649600\pi\)
−0.452871 + 0.891576i \(0.649600\pi\)
\(32\) 0 0
\(33\) 668055. + 668055.i 0.0170704 + 0.0170704i
\(34\) 0 0
\(35\) 4.32550e7 − 8.35308e7i 0.823561 − 1.59040i
\(36\) 0 0
\(37\) −5.50890e7 + 5.50890e7i −0.794431 + 0.794431i −0.982211 − 0.187780i \(-0.939871\pi\)
0.187780 + 0.982211i \(0.439871\pi\)
\(38\) 0 0
\(39\) 3.07840e6i 0.0341194i
\(40\) 0 0
\(41\) 1.37064e8 1.18305 0.591525 − 0.806287i \(-0.298526\pi\)
0.591525 + 0.806287i \(0.298526\pi\)
\(42\) 0 0
\(43\) 9.34290e7 + 9.34290e7i 0.635535 + 0.635535i 0.949451 − 0.313916i \(-0.101641\pi\)
−0.313916 + 0.949451i \(0.601641\pi\)
\(44\) 0 0
\(45\) 5.58328e7 + 1.75758e8i 0.302571 + 0.952472i
\(46\) 0 0
\(47\) −1.14979e8 + 1.14979e8i −0.501334 + 0.501334i −0.911852 − 0.410518i \(-0.865348\pi\)
0.410518 + 0.911852i \(0.365348\pi\)
\(48\) 0 0
\(49\) − 6.23600e8i − 2.20763i
\(50\) 0 0
\(51\) 5.23115e6 0.0151616
\(52\) 0 0
\(53\) −2.29978e7 − 2.29978e7i −0.0549930 − 0.0549930i 0.679075 − 0.734068i \(-0.262381\pi\)
−0.734068 + 0.679075i \(0.762381\pi\)
\(54\) 0 0
\(55\) 4.63574e8 − 1.47263e8i 0.921097 − 0.292604i
\(56\) 0 0
\(57\) 1.43882e6 − 1.43882e6i 0.00239128 − 0.00239128i
\(58\) 0 0
\(59\) − 8.73535e8i − 1.22186i −0.791686 − 0.610929i \(-0.790796\pi\)
0.791686 − 0.610929i \(-0.209204\pi\)
\(60\) 0 0
\(61\) 5.87905e8 0.696079 0.348039 − 0.937480i \(-0.386848\pi\)
0.348039 + 0.937480i \(0.386848\pi\)
\(62\) 0 0
\(63\) 1.25605e9 + 1.25605e9i 1.26563 + 1.26563i
\(64\) 0 0
\(65\) 1.40737e9 + 7.28780e8i 1.21294 + 0.628101i
\(66\) 0 0
\(67\) 6.75197e8 − 6.75197e8i 0.500100 − 0.500100i −0.411369 − 0.911469i \(-0.634949\pi\)
0.911469 + 0.411369i \(0.134949\pi\)
\(68\) 0 0
\(69\) − 4.72241e7i − 0.0301938i
\(70\) 0 0
\(71\) 5.54597e8 0.307387 0.153694 − 0.988119i \(-0.450883\pi\)
0.153694 + 0.988119i \(0.450883\pi\)
\(72\) 0 0
\(73\) 8.91747e8 + 8.91747e8i 0.430158 + 0.430158i 0.888682 − 0.458524i \(-0.151622\pi\)
−0.458524 + 0.888682i \(0.651622\pi\)
\(74\) 0 0
\(75\) −5.84197e7 − 1.00416e7i −0.0246180 − 0.00423154i
\(76\) 0 0
\(77\) 3.31293e9 − 3.31293e9i 1.22394 − 1.22394i
\(78\) 0 0
\(79\) − 1.69149e9i − 0.549711i −0.961486 − 0.274856i \(-0.911370\pi\)
0.961486 − 0.274856i \(-0.0886300\pi\)
\(80\) 0 0
\(81\) −3.48026e9 −0.998129
\(82\) 0 0
\(83\) −1.96163e9 − 1.96163e9i −0.497996 − 0.497996i 0.412818 − 0.910814i \(-0.364545\pi\)
−0.910814 + 0.412818i \(0.864545\pi\)
\(84\) 0 0
\(85\) 1.23842e9 − 2.39155e9i 0.279109 − 0.538996i
\(86\) 0 0
\(87\) −5.74345e6 + 5.74345e6i −0.00115233 + 0.00115233i
\(88\) 0 0
\(89\) 7.73241e9i 1.38473i 0.721548 + 0.692364i \(0.243431\pi\)
−0.721548 + 0.692364i \(0.756569\pi\)
\(90\) 0 0
\(91\) 1.52660e10 2.44635
\(92\) 0 0
\(93\) 1.11296e8 + 1.11296e8i 0.0159980 + 0.0159980i
\(94\) 0 0
\(95\) −3.17166e8 − 9.98416e8i −0.0409891 − 0.129031i
\(96\) 0 0
\(97\) 1.16280e10 − 1.16280e10i 1.35408 − 1.35408i 0.473039 − 0.881041i \(-0.343157\pi\)
0.881041 − 0.473039i \(-0.156843\pi\)
\(98\) 0 0
\(99\) 9.18516e9i 0.965853i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 80.11.p.c.33.2 6
4.3 odd 2 10.11.c.c.3.2 ✓ 6
5.2 odd 4 inner 80.11.p.c.17.2 6
12.11 even 2 90.11.g.c.73.3 6
20.3 even 4 50.11.c.e.7.2 6
20.7 even 4 10.11.c.c.7.2 yes 6
20.19 odd 2 50.11.c.e.43.2 6
60.47 odd 4 90.11.g.c.37.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.11.c.c.3.2 ✓ 6 4.3 odd 2
10.11.c.c.7.2 yes 6 20.7 even 4
50.11.c.e.7.2 6 20.3 even 4
50.11.c.e.43.2 6 20.19 odd 2
80.11.p.c.17.2 6 5.2 odd 4 inner
80.11.p.c.33.2 6 1.1 even 1 trivial
90.11.g.c.37.3 6 60.47 odd 4
90.11.g.c.73.3 6 12.11 even 2