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Properties

Label 315.1.bg.b
Level $315$
Weight $1$
Character orbit 315.bg
Analytic conductor $0.157$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -35
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,1,Mod(34,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3, 3])) 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("315.34"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 315.bg (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.157205478979\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2835.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.3472875.1

$q$-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.

\(f(q)\) \(=\) \( q + q^{3} + \zeta_{6}^{2} q^{4} + \zeta_{6}^{2} q^{5} - \zeta_{6} q^{7} + q^{9} + \zeta_{6} q^{11} + \zeta_{6}^{2} q^{12} - \zeta_{6}^{2} q^{13} + \zeta_{6}^{2} q^{15} - \zeta_{6} q^{16} - q^{17} - \zeta_{6} q^{20} + \cdots + \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{4} - q^{5} - q^{7} + 2 q^{9} + q^{11} - q^{12} + q^{13} - q^{15} - q^{16} - 2 q^{17} - q^{20} - q^{21} - q^{25} + 2 q^{27} + 2 q^{28} - 2 q^{29} + q^{33} + 2 q^{35} - q^{36}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 1.00000 −0.500000 − 0.866025i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 1.00000 0
139.1 0 1.00000 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
9.c even 3 1 inner
315.bg odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.1.bg.b yes 2
3.b odd 2 1 945.1.bg.b 2
5.b even 2 1 315.1.bg.a ✓ 2
5.c odd 4 2 1575.1.y.a 4
7.b odd 2 1 315.1.bg.a ✓ 2
7.c even 3 1 2205.1.q.a 2
7.c even 3 1 2205.1.bn.a 2
7.d odd 6 1 2205.1.q.b 2
7.d odd 6 1 2205.1.bn.b 2
9.c even 3 1 inner 315.1.bg.b yes 2
9.c even 3 1 2835.1.e.d 1
9.d odd 6 1 945.1.bg.b 2
9.d odd 6 1 2835.1.e.b 1
15.d odd 2 1 945.1.bg.a 2
21.c even 2 1 945.1.bg.a 2
35.c odd 2 1 CM 315.1.bg.b yes 2
35.f even 4 2 1575.1.y.a 4
35.i odd 6 1 2205.1.q.a 2
35.i odd 6 1 2205.1.bn.a 2
35.j even 6 1 2205.1.q.b 2
35.j even 6 1 2205.1.bn.b 2
45.h odd 6 1 945.1.bg.a 2
45.h odd 6 1 2835.1.e.c 1
45.j even 6 1 315.1.bg.a ✓ 2
45.j even 6 1 2835.1.e.a 1
45.k odd 12 2 1575.1.y.a 4
63.g even 3 1 2205.1.q.a 2
63.h even 3 1 2205.1.bn.a 2
63.k odd 6 1 2205.1.q.b 2
63.l odd 6 1 315.1.bg.a ✓ 2
63.l odd 6 1 2835.1.e.a 1
63.o even 6 1 945.1.bg.a 2
63.o even 6 1 2835.1.e.c 1
63.t odd 6 1 2205.1.bn.b 2
105.g even 2 1 945.1.bg.b 2
315.q odd 6 1 2205.1.bn.a 2
315.r even 6 1 2205.1.bn.b 2
315.z even 6 1 945.1.bg.b 2
315.z even 6 1 2835.1.e.b 1
315.bg odd 6 1 inner 315.1.bg.b yes 2
315.bg odd 6 1 2835.1.e.d 1
315.bn odd 6 1 2205.1.q.a 2
315.bo even 6 1 2205.1.q.b 2
315.cb even 12 2 1575.1.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.1.bg.a ✓ 2 5.b even 2 1
315.1.bg.a ✓ 2 7.b odd 2 1
315.1.bg.a ✓ 2 45.j even 6 1
315.1.bg.a ✓ 2 63.l odd 6 1
315.1.bg.b yes 2 1.a even 1 1 trivial
315.1.bg.b yes 2 9.c even 3 1 inner
315.1.bg.b yes 2 35.c odd 2 1 CM
315.1.bg.b yes 2 315.bg odd 6 1 inner
945.1.bg.a 2 15.d odd 2 1
945.1.bg.a 2 21.c even 2 1
945.1.bg.a 2 45.h odd 6 1
945.1.bg.a 2 63.o even 6 1
945.1.bg.b 2 3.b odd 2 1
945.1.bg.b 2 9.d odd 6 1
945.1.bg.b 2 105.g even 2 1
945.1.bg.b 2 315.z even 6 1
1575.1.y.a 4 5.c odd 4 2
1575.1.y.a 4 35.f even 4 2
1575.1.y.a 4 45.k odd 12 2
1575.1.y.a 4 315.cb even 12 2
2205.1.q.a 2 7.c even 3 1
2205.1.q.a 2 35.i odd 6 1
2205.1.q.a 2 63.g even 3 1
2205.1.q.a 2 315.bn odd 6 1
2205.1.q.b 2 7.d odd 6 1
2205.1.q.b 2 35.j even 6 1
2205.1.q.b 2 63.k odd 6 1
2205.1.q.b 2 315.bo even 6 1
2205.1.bn.a 2 7.c even 3 1
2205.1.bn.a 2 35.i odd 6 1
2205.1.bn.a 2 63.h even 3 1
2205.1.bn.a 2 315.q odd 6 1
2205.1.bn.b 2 7.d odd 6 1
2205.1.bn.b 2 35.j even 6 1
2205.1.bn.b 2 63.t odd 6 1
2205.1.bn.b 2 315.r even 6 1
2835.1.e.a 1 45.j even 6 1
2835.1.e.a 1 63.l odd 6 1
2835.1.e.b 1 9.d odd 6 1
2835.1.e.b 1 315.z even 6 1
2835.1.e.c 1 45.h odd 6 1
2835.1.e.c 1 63.o even 6 1
2835.1.e.d 1 9.c even 3 1
2835.1.e.d 1 315.bg odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} - T_{13} + 1 \) acting on \(S_{1}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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