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Properties

Label 3025.1.bl.c
Level $3025$
Weight $1$
Character orbit 3025.bl
Analytic conductor $1.510$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -55
Inner twists $16$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3025,1,Mod(493,3025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [8,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.50967166321\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content 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: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.275.1

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + (\zeta_{20}^{6} - \zeta_{20}) q^{2} - \zeta_{20}^{7} q^{4} + ( - \zeta_{20}^{7} - \zeta_{20}^{2}) q^{7} + \zeta_{20} q^{9} + (\zeta_{20}^{6} + \zeta_{20}) q^{13} + 2 \zeta_{20}^{3} q^{14} + \zeta_{20}^{4} q^{16} + \cdots + ( - \zeta_{20}^{5} + 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{7} + 2 q^{13} - 2 q^{16} + 2 q^{17} - 2 q^{18} - 4 q^{26} + 2 q^{28} - 8 q^{32} + 2 q^{36} - 8 q^{43} + 2 q^{52} + 2 q^{63} - 2 q^{68} + 4 q^{71} + 2 q^{73} + 2 q^{81} - 2 q^{83} - 4 q^{86}+ \cdots + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
0.642040 + 1.26007i 0 −0.587785 + 0.809017i 0 0 −1.39680 + 0.221232i 0 −0.951057 − 0.309017i 0
632.1 0.642040 − 1.26007i 0 −0.587785 − 0.809017i 0 0 −1.39680 − 0.221232i 0 −0.951057 + 0.309017i 0
807.1 0.221232 + 1.39680i 0 −0.951057 + 0.309017i 0 0 −0.642040 + 1.26007i 0 0.587785 − 0.809017i 0
1582.1 −1.26007 + 0.642040i 0 0.587785 − 0.809017i 0 0 −0.221232 − 1.39680i 0 0.951057 + 0.309017i 0
1818.1 0.221232 − 1.39680i 0 −0.951057 − 0.309017i 0 0 −0.642040 − 1.26007i 0 0.587785 + 0.809017i 0
2568.1 −1.26007 − 0.642040i 0 0.587785 + 0.809017i 0 0 −0.221232 + 1.39680i 0 0.951057 − 0.309017i 0
2743.1 1.39680 − 0.221232i 0 0.951057 − 0.309017i 0 0 1.26007 + 0.642040i 0 −0.587785 + 0.809017i 0
2907.1 1.39680 + 0.221232i 0 0.951057 + 0.309017i 0 0 1.26007 − 0.642040i 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
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.c odd 4 1 inner
11.c even 5 3 inner
55.e even 4 1 inner
55.h odd 10 3 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.c 8
5.b even 2 1 3025.1.bl.a 8
5.c odd 4 1 3025.1.bl.a 8
5.c odd 4 1 inner 3025.1.bl.c 8
11.b odd 2 1 3025.1.bl.a 8
11.c even 5 1 3025.1.f.a ✓ 2
11.c even 5 3 inner 3025.1.bl.c 8
11.d odd 10 1 3025.1.f.c yes 2
11.d odd 10 3 3025.1.bl.a 8
55.d odd 2 1 CM 3025.1.bl.c 8
55.e even 4 1 3025.1.bl.a 8
55.e even 4 1 inner 3025.1.bl.c 8
55.h odd 10 1 3025.1.f.a ✓ 2
55.h odd 10 3 inner 3025.1.bl.c 8
55.j even 10 1 3025.1.f.c yes 2
55.j even 10 3 3025.1.bl.a 8
55.k odd 20 1 3025.1.f.a ✓ 2
55.k odd 20 1 3025.1.f.c yes 2
55.k odd 20 3 3025.1.bl.a 8
55.k odd 20 3 inner 3025.1.bl.c 8
55.l even 20 1 3025.1.f.a ✓ 2
55.l even 20 1 3025.1.f.c yes 2
55.l even 20 3 3025.1.bl.a 8
55.l even 20 3 inner 3025.1.bl.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3025.1.f.a ✓ 2 11.c even 5 1
3025.1.f.a ✓ 2 55.h odd 10 1
3025.1.f.a ✓ 2 55.k odd 20 1
3025.1.f.a ✓ 2 55.l even 20 1
3025.1.f.c yes 2 11.d odd 10 1
3025.1.f.c yes 2 55.j even 10 1
3025.1.f.c yes 2 55.k odd 20 1
3025.1.f.c yes 2 55.l even 20 1
3025.1.bl.a 8 5.b even 2 1
3025.1.bl.a 8 5.c odd 4 1
3025.1.bl.a 8 11.b odd 2 1
3025.1.bl.a 8 11.d odd 10 3
3025.1.bl.a 8 55.e even 4 1
3025.1.bl.a 8 55.j even 10 3
3025.1.bl.a 8 55.k odd 20 3
3025.1.bl.a 8 55.l even 20 3
3025.1.bl.c 8 1.a even 1 1 trivial
3025.1.bl.c 8 5.c odd 4 1 inner
3025.1.bl.c 8 11.c even 5 3 inner
3025.1.bl.c 8 55.d odd 2 1 CM
3025.1.bl.c 8 55.e even 4 1 inner
3025.1.bl.c 8 55.h odd 10 3 inner
3025.1.bl.c 8 55.k odd 20 3 inner
3025.1.bl.c 8 55.l even 20 3 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2T_{2}^{7} + 2T_{2}^{6} - 4T_{2}^{4} + 8T_{2}^{2} - 16T_{2} + 16 \) acting on \(S_{1}^{\mathrm{new}}(3025, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - 4 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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