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Properties

Label 5120.2.a.t.1.5
Level $5120$
Weight $2$
Character 5120.1
Self dual yes
Analytic conductor $40.883$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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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] = [5120,2,Mod(1,5120)] 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("5120.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5120 = 2^{10} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5120.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-4,0,8,0,-4,0,8,0,-8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.8834058349\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 80)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.731397\) of defining polynomial
Character \(\chi\) \(=\) 5120.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.169718 q^{3} +1.00000 q^{5} +2.66881 q^{7} -2.97120 q^{9} +4.94571 q^{11} -4.15881 q^{13} +0.169718 q^{15} -1.85116 q^{17} -4.87701 q^{19} +0.452946 q^{21} +0.707288 q^{23} +1.00000 q^{25} -1.01342 q^{27} -4.94847 q^{29} -6.84272 q^{31} +0.839377 q^{33} +2.66881 q^{35} +0.137894 q^{37} -0.705826 q^{39} -10.2052 q^{41} -6.27690 q^{43} -2.97120 q^{45} -1.89428 q^{47} +0.122561 q^{49} -0.314175 q^{51} -10.5203 q^{53} +4.94571 q^{55} -0.827717 q^{57} +1.35704 q^{59} +9.18991 q^{61} -7.92956 q^{63} -4.15881 q^{65} -4.94538 q^{67} +0.120040 q^{69} +7.86777 q^{71} +15.6564 q^{73} +0.169718 q^{75} +13.1992 q^{77} -6.70212 q^{79} +8.74159 q^{81} -5.47763 q^{83} -1.85116 q^{85} -0.839845 q^{87} -10.5055 q^{89} -11.0991 q^{91} -1.16133 q^{93} -4.87701 q^{95} +4.79937 q^{97} -14.6947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{15} - 16 q^{19} - 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} + 8 q^{45} - 20 q^{47} + 8 q^{49} - 24 q^{51} - 8 q^{55} - 16 q^{59}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.169718 0.0979869 0.0489934 − 0.998799i \(-0.484399\pi\)
0.0489934 + 0.998799i \(0.484399\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.66881 1.00872 0.504358 − 0.863495i \(-0.331729\pi\)
0.504358 + 0.863495i \(0.331729\pi\)
\(8\) 0 0
\(9\) −2.97120 −0.990399
\(10\) 0 0
\(11\) 4.94571 1.49119 0.745594 − 0.666401i \(-0.232166\pi\)
0.745594 + 0.666401i \(0.232166\pi\)
\(12\) 0 0
\(13\) −4.15881 −1.15345 −0.576723 − 0.816939i \(-0.695669\pi\)
−0.576723 + 0.816939i \(0.695669\pi\)
\(14\) 0 0
\(15\) 0.169718 0.0438211
\(16\) 0 0
\(17\) −1.85116 −0.448971 −0.224486 − 0.974477i \(-0.572070\pi\)
−0.224486 + 0.974477i \(0.572070\pi\)
\(18\) 0 0
\(19\) −4.87701 −1.11886 −0.559431 − 0.828877i \(-0.688980\pi\)
−0.559431 + 0.828877i \(0.688980\pi\)
\(20\) 0 0
\(21\) 0.452946 0.0988409
\(22\) 0 0
\(23\) 0.707288 0.147480 0.0737399 − 0.997278i \(-0.476507\pi\)
0.0737399 + 0.997278i \(0.476507\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.01342 −0.195033
\(28\) 0 0
\(29\) −4.94847 −0.918907 −0.459454 − 0.888202i \(-0.651955\pi\)
−0.459454 + 0.888202i \(0.651955\pi\)
\(30\) 0 0
\(31\) −6.84272 −1.22899 −0.614494 − 0.788921i \(-0.710640\pi\)
−0.614494 + 0.788921i \(0.710640\pi\)
\(32\) 0 0
\(33\) 0.839377 0.146117
\(34\) 0 0
\(35\) 2.66881 0.451112
\(36\) 0 0
\(37\) 0.137894 0.0226697 0.0113348 − 0.999936i \(-0.496392\pi\)
0.0113348 + 0.999936i \(0.496392\pi\)
\(38\) 0 0
\(39\) −0.705826 −0.113023
\(40\) 0 0
\(41\) −10.2052 −1.59379 −0.796896 − 0.604117i \(-0.793526\pi\)
−0.796896 + 0.604117i \(0.793526\pi\)
\(42\) 0 0
\(43\) −6.27690 −0.957218 −0.478609 − 0.878028i \(-0.658859\pi\)
−0.478609 + 0.878028i \(0.658859\pi\)
\(44\) 0 0
\(45\) −2.97120 −0.442920
\(46\) 0 0
\(47\) −1.89428 −0.276310 −0.138155 − 0.990411i \(-0.544117\pi\)
−0.138155 + 0.990411i \(0.544117\pi\)
\(48\) 0 0
\(49\) 0.122561 0.0175087
\(50\) 0 0
\(51\) −0.314175 −0.0439933
\(52\) 0 0
\(53\) −10.5203 −1.44507 −0.722537 − 0.691332i \(-0.757024\pi\)
−0.722537 + 0.691332i \(0.757024\pi\)
\(54\) 0 0
\(55\) 4.94571 0.666879
\(56\) 0 0
\(57\) −0.827717 −0.109634
\(58\) 0 0
\(59\) 1.35704 0.176672 0.0883359 − 0.996091i \(-0.471845\pi\)
0.0883359 + 0.996091i \(0.471845\pi\)
\(60\) 0 0
\(61\) 9.18991 1.17665 0.588324 − 0.808625i \(-0.299788\pi\)
0.588324 + 0.808625i \(0.299788\pi\)
\(62\) 0 0
\(63\) −7.92956 −0.999031
\(64\) 0 0
\(65\) −4.15881 −0.515837
\(66\) 0 0
\(67\) −4.94538 −0.604174 −0.302087 − 0.953280i \(-0.597683\pi\)
−0.302087 + 0.953280i \(0.597683\pi\)
\(68\) 0 0
\(69\) 0.120040 0.0144511
\(70\) 0 0
\(71\) 7.86777 0.933733 0.466866 − 0.884328i \(-0.345383\pi\)
0.466866 + 0.884328i \(0.345383\pi\)
\(72\) 0 0
\(73\) 15.6564 1.83244 0.916220 − 0.400675i \(-0.131224\pi\)
0.916220 + 0.400675i \(0.131224\pi\)
\(74\) 0 0
\(75\) 0.169718 0.0195974
\(76\) 0 0
\(77\) 13.1992 1.50418
\(78\) 0 0
\(79\) −6.70212 −0.754047 −0.377024 − 0.926204i \(-0.623052\pi\)
−0.377024 + 0.926204i \(0.623052\pi\)
\(80\) 0 0
\(81\) 8.74159 0.971288
\(82\) 0 0
\(83\) −5.47763 −0.601248 −0.300624 − 0.953743i \(-0.597195\pi\)
−0.300624 + 0.953743i \(0.597195\pi\)
\(84\) 0 0
\(85\) −1.85116 −0.200786
\(86\) 0 0
\(87\) −0.839845 −0.0900408
\(88\) 0 0
\(89\) −10.5055 −1.11358 −0.556790 − 0.830653i \(-0.687967\pi\)
−0.556790 + 0.830653i \(0.687967\pi\)
\(90\) 0 0
\(91\) −11.0991 −1.16350
\(92\) 0 0
\(93\) −1.16133 −0.120425
\(94\) 0 0
\(95\) −4.87701 −0.500370
\(96\) 0 0
\(97\) 4.79937 0.487303 0.243651 − 0.969863i \(-0.421655\pi\)
0.243651 + 0.969863i \(0.421655\pi\)
\(98\) 0 0
\(99\) −14.6947 −1.47687
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 5120.2.a.t.1.5 8
4.3 odd 2 5120.2.a.v.1.4 8
8.3 odd 2 5120.2.a.s.1.5 8
8.5 even 2 5120.2.a.u.1.4 8
32.3 odd 8 80.2.l.a.61.7 yes 16
32.5 even 8 640.2.l.a.481.4 16
32.11 odd 8 80.2.l.a.21.7 ✓ 16
32.13 even 8 640.2.l.a.161.4 16
32.19 odd 8 640.2.l.b.161.5 16
32.21 even 8 320.2.l.a.241.5 16
32.27 odd 8 640.2.l.b.481.5 16
32.29 even 8 320.2.l.a.81.5 16
96.11 even 8 720.2.t.c.181.2 16
96.29 odd 8 2880.2.t.c.721.6 16
96.35 even 8 720.2.t.c.541.2 16
96.53 odd 8 2880.2.t.c.2161.7 16
160.3 even 8 400.2.q.g.349.3 16
160.29 even 8 1600.2.l.i.401.4 16
160.43 even 8 400.2.q.h.149.6 16
160.53 odd 8 1600.2.q.g.49.5 16
160.67 even 8 400.2.q.h.349.6 16
160.93 odd 8 1600.2.q.h.849.4 16
160.99 odd 8 400.2.l.h.301.2 16
160.107 even 8 400.2.q.g.149.3 16
160.117 odd 8 1600.2.q.h.49.4 16
160.139 odd 8 400.2.l.h.101.2 16
160.149 even 8 1600.2.l.i.1201.4 16
160.157 odd 8 1600.2.q.g.849.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.7 ✓ 16 32.11 odd 8
80.2.l.a.61.7 yes 16 32.3 odd 8
320.2.l.a.81.5 16 32.29 even 8
320.2.l.a.241.5 16 32.21 even 8
400.2.l.h.101.2 16 160.139 odd 8
400.2.l.h.301.2 16 160.99 odd 8
400.2.q.g.149.3 16 160.107 even 8
400.2.q.g.349.3 16 160.3 even 8
400.2.q.h.149.6 16 160.43 even 8
400.2.q.h.349.6 16 160.67 even 8
640.2.l.a.161.4 16 32.13 even 8
640.2.l.a.481.4 16 32.5 even 8
640.2.l.b.161.5 16 32.19 odd 8
640.2.l.b.481.5 16 32.27 odd 8
720.2.t.c.181.2 16 96.11 even 8
720.2.t.c.541.2 16 96.35 even 8
1600.2.l.i.401.4 16 160.29 even 8
1600.2.l.i.1201.4 16 160.149 even 8
1600.2.q.g.49.5 16 160.53 odd 8
1600.2.q.g.849.5 16 160.157 odd 8
1600.2.q.h.49.4 16 160.117 odd 8
1600.2.q.h.849.4 16 160.93 odd 8
2880.2.t.c.721.6 16 96.29 odd 8
2880.2.t.c.2161.7 16 96.53 odd 8
5120.2.a.s.1.5 8 8.3 odd 2
5120.2.a.t.1.5 8 1.1 even 1 trivial
5120.2.a.u.1.4 8 8.5 even 2
5120.2.a.v.1.4 8 4.3 odd 2