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Properties

Label 3025.1.bl.b
Level $3025$
Weight $1$
Character orbit 3025.bl
Analytic conductor $1.510$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -11
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3025,1,Mod(493,3025)]
 
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([15, 12]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3025.493");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3025.bl (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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 605)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.1375.1
Artin image: $D_4:C_{20}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{80} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{20}^{8} - \zeta_{20}^{3}) q^{3} + \zeta_{20}^{7} q^{4} - \zeta_{20} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{20}^{8} - \zeta_{20}^{3}) q^{3} + \zeta_{20}^{7} q^{4} - \zeta_{20} q^{9} + (\zeta_{20}^{5} + 1) q^{12} - \zeta_{20}^{4} q^{16} + ( - \zeta_{20}^{5} + 1) q^{23} - \zeta_{20}^{8} q^{36} + ( - \zeta_{20}^{7} - \zeta_{20}^{2}) q^{37} + ( - \zeta_{20}^{8} + \zeta_{20}^{3}) q^{47} + (\zeta_{20}^{7} - \zeta_{20}^{2}) q^{48} - \zeta_{20}^{9} q^{49} + ( - \zeta_{20}^{6} - \zeta_{20}) q^{53} + \zeta_{20} q^{64} + ( - \zeta_{20}^{5} - 1) q^{67} - 2 \zeta_{20}^{3} q^{69} + 2 \zeta_{20}^{4} q^{71} - \zeta_{20}^{2} q^{81} - 2 \zeta_{20}^{5} q^{89} + (\zeta_{20}^{7} + \zeta_{20}^{2}) q^{92} + ( - \zeta_{20}^{6} + \zeta_{20}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 8 q^{12} + 2 q^{16} + 8 q^{23} + 2 q^{36} - 2 q^{37} + 2 q^{47} - 2 q^{48} - 2 q^{53} - 8 q^{67} - 4 q^{71} - 2 q^{81} + 2 q^{92} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(727\) \(2301\)
\(\chi(n)\) \(\zeta_{20}^{5}\) \(-\zeta_{20}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
493.1
−0.951057 − 0.309017i
−0.951057 + 0.309017i
0.587785 − 0.809017i
0.951057 + 0.309017i
0.587785 + 0.809017i
0.951057 − 0.309017i
−0.587785 + 0.809017i
−0.587785 − 0.809017i
0 1.39680 + 0.221232i 0.587785 − 0.809017i 0 0 0 0 0.951057 + 0.309017i 0
632.1 0 1.39680 − 0.221232i 0.587785 + 0.809017i 0 0 0 0 0.951057 − 0.309017i 0
807.1 0 0.642040 + 1.26007i 0.951057 − 0.309017i 0 0 0 0 −0.587785 + 0.809017i 0
1582.1 0 0.221232 − 1.39680i −0.587785 + 0.809017i 0 0 0 0 −0.951057 − 0.309017i 0
1818.1 0 0.642040 − 1.26007i 0.951057 + 0.309017i 0 0 0 0 −0.587785 − 0.809017i 0
2568.1 0 0.221232 + 1.39680i −0.587785 − 0.809017i 0 0 0 0 −0.951057 + 0.309017i 0
2743.1 0 −1.26007 + 0.642040i −0.951057 + 0.309017i 0 0 0 0 0.587785 − 0.809017i 0
2907.1 0 −1.26007 − 0.642040i −0.951057 − 0.309017i 0 0 0 0 0.587785 + 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 493.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.c odd 4 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.e even 4 1 inner
55.k odd 20 3 inner
55.l even 20 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3025.1.bl.b 8
5.b even 2 1 605.1.l.a 8
5.c odd 4 1 605.1.l.a 8
5.c odd 4 1 inner 3025.1.bl.b 8
11.b odd 2 1 CM 3025.1.bl.b 8
11.c even 5 1 3025.1.f.b 2
11.c even 5 3 inner 3025.1.bl.b 8
11.d odd 10 1 3025.1.f.b 2
11.d odd 10 3 inner 3025.1.bl.b 8
55.d odd 2 1 605.1.l.a 8
55.e even 4 1 605.1.l.a 8
55.e even 4 1 inner 3025.1.bl.b 8
55.h odd 10 1 605.1.f.a ✓ 2
55.h odd 10 3 605.1.l.a 8
55.j even 10 1 605.1.f.a ✓ 2
55.j even 10 3 605.1.l.a 8
55.k odd 20 1 605.1.f.a ✓ 2
55.k odd 20 3 605.1.l.a 8
55.k odd 20 1 3025.1.f.b 2
55.k odd 20 3 inner 3025.1.bl.b 8
55.l even 20 1 605.1.f.a ✓ 2
55.l even 20 3 605.1.l.a 8
55.l even 20 1 3025.1.f.b 2
55.l even 20 3 inner 3025.1.bl.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.1.f.a ✓ 2 55.h odd 10 1
605.1.f.a ✓ 2 55.j even 10 1
605.1.f.a ✓ 2 55.k odd 20 1
605.1.f.a ✓ 2 55.l even 20 1
605.1.l.a 8 5.b even 2 1
605.1.l.a 8 5.c odd 4 1
605.1.l.a 8 55.d odd 2 1
605.1.l.a 8 55.e even 4 1
605.1.l.a 8 55.h odd 10 3
605.1.l.a 8 55.j even 10 3
605.1.l.a 8 55.k odd 20 3
605.1.l.a 8 55.l even 20 3
3025.1.f.b 2 11.c even 5 1
3025.1.f.b 2 11.d odd 10 1
3025.1.f.b 2 55.k odd 20 1
3025.1.f.b 2 55.l even 20 1
3025.1.bl.b 8 1.a even 1 1 trivial
3025.1.bl.b 8 5.c odd 4 1 inner
3025.1.bl.b 8 11.b odd 2 1 CM
3025.1.bl.b 8 11.c even 5 3 inner
3025.1.bl.b 8 11.d odd 10 3 inner
3025.1.bl.b 8 55.e even 4 1 inner
3025.1.bl.b 8 55.k odd 20 3 inner
3025.1.bl.b 8 55.l even 20 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(3025, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) 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
$23$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) 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
$53$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) 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
$89$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
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