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Properties

Label 2420.1.q.a
Level $2420$
Weight $1$
Character orbit 2420.q
Analytic conductor $1.208$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -11
Inner twists $16$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2420,1,Mod(1129,2420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2420.1129"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2420, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 5, 7])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2420.q (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.20773733057\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{15})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 220)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.242000.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{30}^{7} + \zeta_{30}^{2}) q^{3} + \zeta_{30}^{11} q^{5} + (\zeta_{30}^{14} + \cdots + \zeta_{30}^{4}) q^{9} + (\zeta_{30}^{13} - \zeta_{30}^{3}) q^{15} + (\zeta_{30}^{10} + \zeta_{30}^{5}) q^{23}+ \cdots + ( - \zeta_{30}^{14} + \zeta_{30}^{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{5} + 4 q^{9} - 3 q^{15} + q^{25} - 2 q^{31} - 8 q^{45} + 2 q^{49} + 2 q^{59} - 6 q^{69} + 2 q^{71} - 3 q^{75} - 2 q^{81} + 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2420\mathbb{Z}\right)^\times\).

\(n\) \(1211\) \(1937\) \(2301\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{30}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1129.1
−0.104528 − 0.994522i
0.913545 + 0.406737i
−0.104528 + 0.994522i
0.913545 − 0.406737i
0.669131 + 0.743145i
−0.978148 + 0.207912i
0.669131 − 0.743145i
−0.978148 − 0.207912i
0 −1.64728 − 0.535233i 0 −0.913545 − 0.406737i 0 0 0 1.61803 + 1.17557i 0
1129.2 0 1.64728 + 0.535233i 0 0.104528 + 0.994522i 0 0 0 1.61803 + 1.17557i 0
1449.1 0 −1.64728 + 0.535233i 0 −0.913545 + 0.406737i 0 0 0 1.61803 − 1.17557i 0
1449.2 0 1.64728 − 0.535233i 0 0.104528 − 0.994522i 0 0 0 1.61803 − 1.17557i 0
1909.1 0 −1.01807 + 1.40126i 0 0.978148 − 0.207912i 0 0 0 −0.618034 − 1.90211i 0
1909.2 0 1.01807 − 1.40126i 0 −0.669131 − 0.743145i 0 0 0 −0.618034 − 1.90211i 0
2169.1 0 −1.01807 − 1.40126i 0 0.978148 + 0.207912i 0 0 0 −0.618034 + 1.90211i 0
2169.2 0 1.01807 + 1.40126i 0 −0.669131 + 0.743145i 0 0 0 −0.618034 + 1.90211i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1129.2
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.b even 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.d odd 2 1 inner
55.h odd 10 3 inner
55.j even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2420.1.q.a 8
5.b even 2 1 inner 2420.1.q.a 8
11.b odd 2 1 CM 2420.1.q.a 8
11.c even 5 1 220.1.e.a ✓ 2
11.c even 5 3 inner 2420.1.q.a 8
11.d odd 10 1 220.1.e.a ✓ 2
11.d odd 10 3 inner 2420.1.q.a 8
33.f even 10 1 1980.1.p.a 2
33.h odd 10 1 1980.1.p.a 2
44.g even 10 1 880.1.i.b 2
44.h odd 10 1 880.1.i.b 2
55.d odd 2 1 inner 2420.1.q.a 8
55.h odd 10 1 220.1.e.a ✓ 2
55.h odd 10 3 inner 2420.1.q.a 8
55.j even 10 1 220.1.e.a ✓ 2
55.j even 10 3 inner 2420.1.q.a 8
55.k odd 20 2 1100.1.f.b 2
55.l even 20 2 1100.1.f.b 2
88.k even 10 1 3520.1.i.c 2
88.l odd 10 1 3520.1.i.c 2
88.o even 10 1 3520.1.i.d 2
88.p odd 10 1 3520.1.i.d 2
165.o odd 10 1 1980.1.p.a 2
165.r even 10 1 1980.1.p.a 2
220.n odd 10 1 880.1.i.b 2
220.o even 10 1 880.1.i.b 2
440.ba odd 10 1 3520.1.i.d 2
440.bd even 10 1 3520.1.i.d 2
440.bh odd 10 1 3520.1.i.c 2
440.bm even 10 1 3520.1.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.1.e.a ✓ 2 11.c even 5 1
220.1.e.a ✓ 2 11.d odd 10 1
220.1.e.a ✓ 2 55.h odd 10 1
220.1.e.a ✓ 2 55.j even 10 1
880.1.i.b 2 44.g even 10 1
880.1.i.b 2 44.h odd 10 1
880.1.i.b 2 220.n odd 10 1
880.1.i.b 2 220.o even 10 1
1100.1.f.b 2 55.k odd 20 2
1100.1.f.b 2 55.l even 20 2
1980.1.p.a 2 33.f even 10 1
1980.1.p.a 2 33.h odd 10 1
1980.1.p.a 2 165.o odd 10 1
1980.1.p.a 2 165.r even 10 1
2420.1.q.a 8 1.a even 1 1 trivial
2420.1.q.a 8 5.b even 2 1 inner
2420.1.q.a 8 11.b odd 2 1 CM
2420.1.q.a 8 11.c even 5 3 inner
2420.1.q.a 8 11.d odd 10 3 inner
2420.1.q.a 8 55.d odd 2 1 inner
2420.1.q.a 8 55.h odd 10 3 inner
2420.1.q.a 8 55.j even 10 3 inner
3520.1.i.c 2 88.k even 10 1
3520.1.i.c 2 88.l odd 10 1
3520.1.i.c 2 440.bh odd 10 1
3520.1.i.c 2 440.bm even 10 1
3520.1.i.d 2 88.o even 10 1
3520.1.i.d 2 88.p odd 10 1
3520.1.i.d 2 440.ba odd 10 1
3520.1.i.d 2 440.bd even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2420, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) 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
$23$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) 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
$89$ \( (T - 1)^{8} \) Copy content Toggle raw display
$97$ \( T^{8} - 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
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