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Properties

Label 625.2.e.k
Level $625$
Weight $2$
Character orbit 625.e
Analytic conductor $4.991$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [625,2,Mod(124,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.124");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.e (of order \(10\), degree \(4\), not minimal)

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: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} - 6 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{4} - 6 q^{6} - 6 q^{9} + 4 q^{11} - 18 q^{14} - 28 q^{16} + 14 q^{21} - 20 q^{24} + 44 q^{26} + 20 q^{29} + 34 q^{31} + 2 q^{34} - 8 q^{36} + 18 q^{39} + 24 q^{41} - 98 q^{44} - 66 q^{46} + 16 q^{49} - 56 q^{51} + 60 q^{54} - 70 q^{56} - 40 q^{59} - 46 q^{61} + 56 q^{64} - 52 q^{66} - 12 q^{69} + 44 q^{71} + 72 q^{74} - 40 q^{76} - 150 q^{79} + 22 q^{81} + 62 q^{84} + 34 q^{86} - 10 q^{89} + 44 q^{91} + 102 q^{94} - 56 q^{96} + 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1 −2.35325 − 0.764617i 1.24419 − 1.71249i 3.33511 + 2.42310i 0 −4.23730 + 3.07858i 0.973070i −3.08683 − 4.24866i −0.457541 − 1.40817i 0
124.2 −2.21225 − 0.718805i −1.35327 + 1.86261i 2.75935 + 2.00479i 0 4.33262 − 3.14783i − 3.59425i −1.92884 − 2.65482i −0.710939 − 2.18805i 0
124.3 −1.60102 − 0.520202i −0.417528 + 0.574677i 0.674615 + 0.490137i 0 0.967418 − 0.702870i 4.59110i 1.15387 + 1.58816i 0.771126 + 2.37328i 0
124.4 −0.476469 − 0.154814i 1.81817 − 2.50250i −1.41498 − 1.02804i 0 −1.25372 + 0.910884i − 0.0237879i 1.10399 + 1.51951i −2.02970 − 6.24676i 0
124.5 0.476469 + 0.154814i −1.81817 + 2.50250i −1.41498 − 1.02804i 0 −1.25372 + 0.910884i 0.0237879i −1.10399 − 1.51951i −2.02970 − 6.24676i 0
124.6 1.60102 + 0.520202i 0.417528 − 0.574677i 0.674615 + 0.490137i 0 0.967418 − 0.702870i − 4.59110i −1.15387 − 1.58816i 0.771126 + 2.37328i 0
124.7 2.21225 + 0.718805i 1.35327 − 1.86261i 2.75935 + 2.00479i 0 4.33262 − 3.14783i 3.59425i 1.92884 + 2.65482i −0.710939 − 2.18805i 0
124.8 2.35325 + 0.764617i −1.24419 + 1.71249i 3.33511 + 2.42310i 0 −4.23730 + 3.07858i − 0.973070i 3.08683 + 4.24866i −0.457541 − 1.40817i 0
249.1 −1.56645 + 2.15604i −0.721930 + 0.234569i −1.57669 − 4.85257i 0 0.625130 − 1.92395i 2.04213i 7.86300 + 2.55484i −1.96089 + 1.42467i 0
249.2 −1.18361 + 1.62909i 2.87758 − 0.934982i −0.634989 − 1.95429i 0 −1.88274 + 5.79449i 0.369971i 0.105077 + 0.0341417i 4.97921 − 3.61761i 0
249.3 −0.621509 + 0.855434i 0.653760 − 0.212419i 0.272540 + 0.838792i 0 −0.224607 + 0.691269i − 1.01199i −2.89816 − 0.941671i −2.04477 + 1.48561i 0
249.4 −0.192059 + 0.264347i −1.63142 + 0.530081i 0.585041 + 1.80057i 0 0.173205 − 0.533069i 3.42409i −1.20986 − 0.393106i −0.0465016 + 0.0337854i 0
249.5 0.192059 − 0.264347i 1.63142 − 0.530081i 0.585041 + 1.80057i 0 0.173205 − 0.533069i − 3.42409i 1.20986 + 0.393106i −0.0465016 + 0.0337854i 0
249.6 0.621509 − 0.855434i −0.653760 + 0.212419i 0.272540 + 0.838792i 0 −0.224607 + 0.691269i 1.01199i 2.89816 + 0.941671i −2.04477 + 1.48561i 0
249.7 1.18361 − 1.62909i −2.87758 + 0.934982i −0.634989 − 1.95429i 0 −1.88274 + 5.79449i − 0.369971i −0.105077 − 0.0341417i 4.97921 − 3.61761i 0
249.8 1.56645 − 2.15604i 0.721930 − 0.234569i −1.57669 − 4.85257i 0 0.625130 − 1.92395i − 2.04213i −7.86300 − 2.55484i −1.96089 + 1.42467i 0
374.1 −1.56645 − 2.15604i −0.721930 − 0.234569i −1.57669 + 4.85257i 0 0.625130 + 1.92395i − 2.04213i 7.86300 − 2.55484i −1.96089 − 1.42467i 0
374.2 −1.18361 − 1.62909i 2.87758 + 0.934982i −0.634989 + 1.95429i 0 −1.88274 − 5.79449i − 0.369971i 0.105077 − 0.0341417i 4.97921 + 3.61761i 0
374.3 −0.621509 − 0.855434i 0.653760 + 0.212419i 0.272540 − 0.838792i 0 −0.224607 − 0.691269i 1.01199i −2.89816 + 0.941671i −2.04477 − 1.48561i 0
374.4 −0.192059 − 0.264347i −1.63142 − 0.530081i 0.585041 − 1.80057i 0 0.173205 + 0.533069i − 3.42409i −1.20986 + 0.393106i −0.0465016 − 0.0337854i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.e.k 32
5.b even 2 1 inner 625.2.e.k 32
5.c odd 4 1 625.2.d.n 16
5.c odd 4 1 625.2.d.p 16
25.d even 5 1 625.2.b.d 16
25.d even 5 2 625.2.e.j 32
25.d even 5 1 inner 625.2.e.k 32
25.e even 10 1 625.2.b.d 16
25.e even 10 2 625.2.e.j 32
25.e even 10 1 inner 625.2.e.k 32
25.f odd 20 1 625.2.a.e ✓ 8
25.f odd 20 1 625.2.a.g yes 8
25.f odd 20 2 625.2.d.m 16
25.f odd 20 1 625.2.d.n 16
25.f odd 20 1 625.2.d.p 16
25.f odd 20 2 625.2.d.q 16
75.l even 20 1 5625.2.a.s 8
75.l even 20 1 5625.2.a.be 8
100.l even 20 1 10000.2.a.be 8
100.l even 20 1 10000.2.a.bn 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e ✓ 8 25.f odd 20 1
625.2.a.g yes 8 25.f odd 20 1
625.2.b.d 16 25.d even 5 1
625.2.b.d 16 25.e even 10 1
625.2.d.m 16 25.f odd 20 2
625.2.d.n 16 5.c odd 4 1
625.2.d.n 16 25.f odd 20 1
625.2.d.p 16 5.c odd 4 1
625.2.d.p 16 25.f odd 20 1
625.2.d.q 16 25.f odd 20 2
625.2.e.j 32 25.d even 5 2
625.2.e.j 32 25.e even 10 2
625.2.e.k 32 1.a even 1 1 trivial
625.2.e.k 32 5.b even 2 1 inner
625.2.e.k 32 25.d even 5 1 inner
625.2.e.k 32 25.e even 10 1 inner
5625.2.a.s 8 75.l even 20 1
5625.2.a.be 8 75.l even 20 1
10000.2.a.be 8 100.l even 20 1
10000.2.a.bn 8 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{32} - 16 T_{2}^{30} + 160 T_{2}^{28} - 1340 T_{2}^{26} + 10170 T_{2}^{24} - 57088 T_{2}^{22} + \cdots + 6561 \) Copy content Toggle raw display
\( T_{3}^{32} - 9 T_{3}^{30} + 125 T_{3}^{28} - 1320 T_{3}^{26} + 12600 T_{3}^{24} - 49887 T_{3}^{22} + \cdots + 707281 \) Copy content Toggle raw display