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Properties

Label 625.2.e.c
Level $625$
Weight $2$
Character orbit 625.e
Analytic conductor $4.991$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{5} + \zeta_{20}) q^{2} - \zeta_{20} q^{3} + (\zeta_{20}^{2} - 1) q^{4} + ( - \zeta_{20}^{6} - \zeta_{20}^{2}) q^{6} + (\zeta_{20}^{7} + \cdots + \zeta_{20}^{3}) q^{7} + \cdots - 2 \zeta_{20}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{5} + \zeta_{20}) q^{2} - \zeta_{20} q^{3} + (\zeta_{20}^{2} - 1) q^{4} + ( - \zeta_{20}^{6} - \zeta_{20}^{2}) q^{6} + (\zeta_{20}^{7} + \cdots + \zeta_{20}^{3}) q^{7} + \cdots + (4 \zeta_{20}^{6} - 4 \zeta_{20}^{4} - 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{4} - 4 q^{6} - 4 q^{9} + 16 q^{11} - 2 q^{14} + 18 q^{16} - 20 q^{19} + 6 q^{21} - 24 q^{26} + 6 q^{31} + 28 q^{34} + 8 q^{36} + 12 q^{39} - 4 q^{41} + 8 q^{44} - 24 q^{46} + 44 q^{49} - 24 q^{51} - 20 q^{54} - 10 q^{56} - 30 q^{59} - 34 q^{61} + 14 q^{64} - 28 q^{66} + 22 q^{69} + 6 q^{71} - 12 q^{74} + 40 q^{76} + 20 q^{79} - 2 q^{81} - 2 q^{84} - 24 q^{86} - 40 q^{89} + 6 q^{91} - 2 q^{94} - 14 q^{96} - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{20}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
−0.587785 + 0.809017i
0.587785 − 0.809017i
−0.951057 + 0.309017i
0.951057 − 0.309017i
−0.951057 − 0.309017i
0.951057 + 0.309017i
−0.587785 − 0.809017i
0.587785 + 0.809017i
−0.587785 − 0.190983i 0.587785 − 0.809017i −1.30902 − 0.951057i 0 −0.500000 + 0.363271i 1.61803i 1.31433 + 1.80902i 0.618034 + 1.90211i 0
124.2 0.587785 + 0.190983i −0.587785 + 0.809017i −1.30902 − 0.951057i 0 −0.500000 + 0.363271i − 1.61803i −1.31433 − 1.80902i 0.618034 + 1.90211i 0
249.1 −0.951057 + 1.30902i 0.951057 − 0.309017i −0.190983 − 0.587785i 0 −0.500000 + 1.53884i 0.618034i −2.12663 − 0.690983i −1.61803 + 1.17557i 0
249.2 0.951057 − 1.30902i −0.951057 + 0.309017i −0.190983 − 0.587785i 0 −0.500000 + 1.53884i − 0.618034i 2.12663 + 0.690983i −1.61803 + 1.17557i 0
374.1 −0.951057 − 1.30902i 0.951057 + 0.309017i −0.190983 + 0.587785i 0 −0.500000 − 1.53884i − 0.618034i −2.12663 + 0.690983i −1.61803 − 1.17557i 0
374.2 0.951057 + 1.30902i −0.951057 − 0.309017i −0.190983 + 0.587785i 0 −0.500000 − 1.53884i 0.618034i 2.12663 − 0.690983i −1.61803 − 1.17557i 0
499.1 −0.587785 + 0.190983i 0.587785 + 0.809017i −1.30902 + 0.951057i 0 −0.500000 − 0.363271i − 1.61803i 1.31433 − 1.80902i 0.618034 − 1.90211i 0
499.2 0.587785 − 0.190983i −0.587785 − 0.809017i −1.30902 + 0.951057i 0 −0.500000 − 0.363271i 1.61803i −1.31433 + 1.80902i 0.618034 − 1.90211i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.2
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.c 8
5.b even 2 1 inner 625.2.e.c 8
5.c odd 4 1 625.2.d.b 4
5.c odd 4 1 625.2.d.h 4
25.d even 5 2 125.2.e.a 8
25.d even 5 1 625.2.b.a 4
25.d even 5 1 inner 625.2.e.c 8
25.e even 10 2 125.2.e.a 8
25.e even 10 1 625.2.b.a 4
25.e even 10 1 inner 625.2.e.c 8
25.f odd 20 2 25.2.d.a ✓ 4
25.f odd 20 2 125.2.d.a 4
25.f odd 20 1 625.2.a.b 2
25.f odd 20 1 625.2.a.c 2
25.f odd 20 1 625.2.d.b 4
25.f odd 20 1 625.2.d.h 4
75.l even 20 2 225.2.h.b 4
75.l even 20 1 5625.2.a.d 2
75.l even 20 1 5625.2.a.f 2
100.l even 20 2 400.2.u.b 4
100.l even 20 1 10000.2.a.c 2
100.l even 20 1 10000.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a ✓ 4 25.f odd 20 2
125.2.d.a 4 25.f odd 20 2
125.2.e.a 8 25.d even 5 2
125.2.e.a 8 25.e even 10 2
225.2.h.b 4 75.l even 20 2
400.2.u.b 4 100.l even 20 2
625.2.a.b 2 25.f odd 20 1
625.2.a.c 2 25.f odd 20 1
625.2.b.a 4 25.d even 5 1
625.2.b.a 4 25.e even 10 1
625.2.d.b 4 5.c odd 4 1
625.2.d.b 4 25.f odd 20 1
625.2.d.h 4 5.c odd 4 1
625.2.d.h 4 25.f odd 20 1
625.2.e.c 8 1.a even 1 1 trivial
625.2.e.c 8 5.b even 2 1 inner
625.2.e.c 8 25.d even 5 1 inner
625.2.e.c 8 25.e even 10 1 inner
5625.2.a.d 2 75.l even 20 1
5625.2.a.f 2 75.l even 20 1
10000.2.a.c 2 100.l even 20 1
10000.2.a.l 2 100.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 \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{8} + T_{2}^{6} + 6T_{2}^{4} - 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} - T_{3}^{6} + T_{3}^{4} - T_{3}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 8 T^{3} + 24 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 36 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$17$ \( T^{8} - 44 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{4} + 10 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 101 T^{6} + \cdots + 923521 \) Copy content Toggle raw display
$29$ \( (T^{4} + 10 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 29 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} + 2 T^{3} + 24 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 27 T^{2} + 81)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{8} - T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$59$ \( (T^{4} + 15 T^{3} + \cdots + 2025)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 17 T^{3} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{6} + \cdots + 3748096 \) Copy content Toggle raw display
$71$ \( (T^{4} - 3 T^{3} + \cdots + 841)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 81 T^{6} + \cdots + 43046721 \) Copy content Toggle raw display
$79$ \( (T^{4} - 10 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 61 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$89$ \( (T^{4} + 20 T^{3} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 19 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
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