Properties

Label 5.8.a.a
Level $5$
Weight $8$
Character orbit 5.a
Self dual yes
Analytic conductor $1.562$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5,8,Mod(1,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(1.56192512742\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 14 q^{2} - 48 q^{3} + 68 q^{4} + 125 q^{5} + 672 q^{6} - 1644 q^{7} + 840 q^{8} + 117 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 14 q^{2} - 48 q^{3} + 68 q^{4} + 125 q^{5} + 672 q^{6} - 1644 q^{7} + 840 q^{8} + 117 q^{9} - 1750 q^{10} + 172 q^{11} - 3264 q^{12} + 3862 q^{13} + 23016 q^{14} - 6000 q^{15} - 20464 q^{16} - 12254 q^{17} - 1638 q^{18} - 25940 q^{19} + 8500 q^{20} + 78912 q^{21} - 2408 q^{22} + 12972 q^{23} - 40320 q^{24} + 15625 q^{25} - 54068 q^{26} + 99360 q^{27} - 111792 q^{28} - 81610 q^{29} + 84000 q^{30} - 156888 q^{31} + 178976 q^{32} - 8256 q^{33} + 171556 q^{34} - 205500 q^{35} + 7956 q^{36} + 110126 q^{37} + 363160 q^{38} - 185376 q^{39} + 105000 q^{40} + 467882 q^{41} - 1104768 q^{42} - 499208 q^{43} + 11696 q^{44} + 14625 q^{45} - 181608 q^{46} - 396884 q^{47} + 982272 q^{48} + 1879193 q^{49} - 218750 q^{50} + 588192 q^{51} + 262616 q^{52} - 1280498 q^{53} - 1391040 q^{54} + 21500 q^{55} - 1380960 q^{56} + 1245120 q^{57} + 1142540 q^{58} - 1337420 q^{59} - 408000 q^{60} - 923978 q^{61} + 2196432 q^{62} - 192348 q^{63} + 113728 q^{64} + 482750 q^{65} + 115584 q^{66} - 797304 q^{67} - 833272 q^{68} - 622656 q^{69} + 2877000 q^{70} + 5103392 q^{71} + 98280 q^{72} - 4267478 q^{73} - 1541764 q^{74} - 750000 q^{75} - 1763920 q^{76} - 282768 q^{77} + 2595264 q^{78} - 960 q^{79} - 2558000 q^{80} - 5025159 q^{81} - 6550348 q^{82} + 6140832 q^{83} + 5366016 q^{84} - 1531750 q^{85} + 6988912 q^{86} + 3917280 q^{87} + 144480 q^{88} + 2010570 q^{89} - 204750 q^{90} - 6349128 q^{91} + 882096 q^{92} + 7530624 q^{93} + 5556376 q^{94} - 3242500 q^{95} - 8590848 q^{96} - 4881934 q^{97} - 26308702 q^{98} + 20124 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−14.0000 −48.0000 68.0000 125.000 672.000 −1644.00 840.000 117.000 −1750.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.8.a.a 1
3.b odd 2 1 45.8.a.f 1
4.b odd 2 1 80.8.a.d 1
5.b even 2 1 25.8.a.a 1
5.c odd 4 2 25.8.b.a 2
7.b odd 2 1 245.8.a.a 1
8.b even 2 1 320.8.a.h 1
8.d odd 2 1 320.8.a.a 1
11.b odd 2 1 605.8.a.c 1
15.d odd 2 1 225.8.a.b 1
15.e even 4 2 225.8.b.b 2
20.d odd 2 1 400.8.a.e 1
20.e even 4 2 400.8.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.a.a 1 1.a even 1 1 trivial
25.8.a.a 1 5.b even 2 1
25.8.b.a 2 5.c odd 4 2
45.8.a.f 1 3.b odd 2 1
80.8.a.d 1 4.b odd 2 1
225.8.a.b 1 15.d odd 2 1
225.8.b.b 2 15.e even 4 2
245.8.a.a 1 7.b odd 2 1
320.8.a.a 1 8.d odd 2 1
320.8.a.h 1 8.b even 2 1
400.8.a.e 1 20.d odd 2 1
400.8.c.e 2 20.e even 4 2
605.8.a.c 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 14 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 14 \) Copy content Toggle raw display
$3$ \( T + 48 \) Copy content Toggle raw display
$5$ \( T - 125 \) Copy content Toggle raw display
$7$ \( T + 1644 \) Copy content Toggle raw display
$11$ \( T - 172 \) Copy content Toggle raw display
$13$ \( T - 3862 \) Copy content Toggle raw display
$17$ \( T + 12254 \) Copy content Toggle raw display
$19$ \( T + 25940 \) Copy content Toggle raw display
$23$ \( T - 12972 \) Copy content Toggle raw display
$29$ \( T + 81610 \) Copy content Toggle raw display
$31$ \( T + 156888 \) Copy content Toggle raw display
$37$ \( T - 110126 \) Copy content Toggle raw display
$41$ \( T - 467882 \) Copy content Toggle raw display
$43$ \( T + 499208 \) Copy content Toggle raw display
$47$ \( T + 396884 \) Copy content Toggle raw display
$53$ \( T + 1280498 \) Copy content Toggle raw display
$59$ \( T + 1337420 \) Copy content Toggle raw display
$61$ \( T + 923978 \) Copy content Toggle raw display
$67$ \( T + 797304 \) Copy content Toggle raw display
$71$ \( T - 5103392 \) Copy content Toggle raw display
$73$ \( T + 4267478 \) Copy content Toggle raw display
$79$ \( T + 960 \) Copy content Toggle raw display
$83$ \( T - 6140832 \) Copy content Toggle raw display
$89$ \( T - 2010570 \) Copy content Toggle raw display
$97$ \( T + 4881934 \) Copy content Toggle raw display
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