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Properties

Label 3025.1.bf.a
Level $3025$
Weight $1$
Character orbit 3025.bf
Analytic conductor $1.510$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3025,1,Mod(202,3025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3025, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([1, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3025.202");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3025.bf (of order \(20\), degree \(8\), 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: \(1.50967166321\)
Analytic rank: \(0\)
Dimension: \(8\)
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: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{9} + 1) q^{3} + \zeta_{20}^{5} q^{4} + \zeta_{20}^{3} q^{5} + (\zeta_{20}^{9} - \zeta_{20}^{8} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{9} + 1) q^{3} + \zeta_{20}^{5} q^{4} + \zeta_{20}^{3} q^{5} + (\zeta_{20}^{9} - \zeta_{20}^{8} + 1) q^{9} + (\zeta_{20}^{5} - \zeta_{20}^{4}) q^{12} + (\zeta_{20}^{3} - \zeta_{20}^{2}) q^{15} - q^{16} + \zeta_{20}^{8} q^{20} + ( - \zeta_{20}^{9} + \zeta_{20}^{2}) q^{23} + \zeta_{20}^{6} q^{25} + (\zeta_{20}^{9} + \zeta_{20}^{8} + \cdots + 1) q^{27}+ \cdots + (\zeta_{20}^{6} + \zeta_{20}^{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 10 q^{9} + 2 q^{12} - 2 q^{15} - 8 q^{16} - 2 q^{20} + 2 q^{23} + 2 q^{25} + 10 q^{27} + 2 q^{36} + 2 q^{37} - 2 q^{45} - 2 q^{47} - 8 q^{48} - 8 q^{53} - 2 q^{60} - 2 q^{67} - 4 q^{71} + 2 q^{75} + 8 q^{81} - 2 q^{92} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3025\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(2301\)
\(\chi(n)\) \(-\zeta_{20}\) \(\zeta_{20}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
202.1
−0.951057 − 0.309017i
0.951057 + 0.309017i
0.951057 − 0.309017i
0.587785 − 0.809017i
−0.951057 + 0.309017i
−0.587785 − 0.809017i
−0.587785 + 0.809017i
0.587785 + 0.809017i
0 1.95106 − 0.309017i − 1.00000i −0.587785 − 0.809017i 0 0 0 2.76007 − 0.896802i 0
323.1 0 0.0489435 + 0.309017i 1.00000i 0.587785 + 0.809017i 0 0 0 0.857960 − 0.278768i 0
487.1 0 0.0489435 − 0.309017i − 1.00000i 0.587785 − 0.809017i 0 0 0 0.857960 + 0.278768i 0
753.1 0 0.412215 − 0.809017i 1.00000i −0.951057 − 0.309017i 0 0 0 0.103198 + 0.142040i 0
1213.1 0 1.95106 + 0.309017i 1.00000i −0.587785 + 0.809017i 0 0 0 2.76007 + 0.896802i 0
2308.1 0 1.58779 − 0.809017i 1.00000i 0.951057 − 0.309017i 0 0 0 1.27877 − 1.76007i 0
2447.1 0 1.58779 + 0.809017i − 1.00000i 0.951057 + 0.309017i 0 0 0 1.27877 + 1.76007i 0
2792.1 0 0.412215 + 0.809017i − 1.00000i −0.951057 + 0.309017i 0 0 0 0.103198 − 0.142040i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 202.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
275.be odd 20 1 inner
275.bf even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3025.1.bf.a 8
11.b odd 2 1 CM 3025.1.bf.a 8
11.c even 5 1 3025.1.bi.a 8
11.c even 5 1 3025.1.bj.a ✓ 8
11.c even 5 1 3025.1.bk.a 8
11.c even 5 1 3025.1.bq.a 8
11.d odd 10 1 3025.1.bi.a 8
11.d odd 10 1 3025.1.bj.a ✓ 8
11.d odd 10 1 3025.1.bk.a 8
11.d odd 10 1 3025.1.bq.a 8
25.f odd 20 1 3025.1.bi.a 8
275.be odd 20 1 inner 3025.1.bf.a 8
275.bf even 20 1 inner 3025.1.bf.a 8
275.bg even 20 1 3025.1.bk.a 8
275.bh odd 20 1 3025.1.bq.a 8
275.bj odd 20 1 3025.1.bj.a ✓ 8
275.bl even 20 1 3025.1.bq.a 8
275.bn even 20 1 3025.1.bj.a ✓ 8
275.bo even 20 1 3025.1.bi.a 8
275.bp odd 20 1 3025.1.bk.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3025.1.bf.a 8 1.a even 1 1 trivial
3025.1.bf.a 8 11.b odd 2 1 CM
3025.1.bf.a 8 275.be odd 20 1 inner
3025.1.bf.a 8 275.bf even 20 1 inner
3025.1.bi.a 8 11.c even 5 1
3025.1.bi.a 8 11.d odd 10 1
3025.1.bi.a 8 25.f odd 20 1
3025.1.bi.a 8 275.bo even 20 1
3025.1.bj.a ✓ 8 11.c even 5 1
3025.1.bj.a ✓ 8 11.d odd 10 1
3025.1.bj.a ✓ 8 275.bj odd 20 1
3025.1.bj.a ✓ 8 275.bn even 20 1
3025.1.bk.a 8 11.c even 5 1
3025.1.bk.a 8 11.d odd 10 1
3025.1.bk.a 8 275.bg even 20 1
3025.1.bk.a 8 275.bp odd 20 1
3025.1.bq.a 8 11.c even 5 1
3025.1.bq.a 8 11.d odd 10 1
3025.1.bq.a 8 275.bh odd 20 1
3025.1.bq.a 8 275.bl even 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3025, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$37$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( (T^{4} - 5 T + 5)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
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