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Properties

Label 378.2.k.a
Level $378$
Weight $2$
Character orbit 378.k
Analytic conductor $3.018$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [378,2,Mod(215,378)] 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("378.215"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(378, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,0,2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (3 \zeta_{12}^{2} - 1) q^{7} + \zeta_{12}^{3} q^{8} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{11} + (4 \zeta_{12}^{2} - 2) q^{13} + (3 \zeta_{12}^{3} - \zeta_{12}) q^{14}+ \cdots + (3 \zeta_{12}^{3} - 8 \zeta_{12}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{7} - 2 q^{16} + 12 q^{22} + 10 q^{25} - 8 q^{28} - 6 q^{31} + 16 q^{37} + 16 q^{43} - 26 q^{49} - 12 q^{52} + 18 q^{58} - 48 q^{61} - 4 q^{64} - 4 q^{67} + 18 q^{73} + 26 q^{79} - 36 q^{82}+ \cdots + 36 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i 0 0 0.500000 + 2.59808i − 1.00000i 0 0
215.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0.500000 + 2.59808i 1.00000i 0 0
269.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0 0 0.500000 − 2.59808i 1.00000i 0 0
269.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0 0 0.500000 − 2.59808i − 1.00000i 0 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.k.a ✓ 4
3.b odd 2 1 inner 378.2.k.a ✓ 4
7.c even 3 1 2646.2.d.c 4
7.d odd 6 1 inner 378.2.k.a ✓ 4
7.d odd 6 1 2646.2.d.c 4
9.c even 3 1 1134.2.l.d 4
9.c even 3 1 1134.2.t.a 4
9.d odd 6 1 1134.2.l.d 4
9.d odd 6 1 1134.2.t.a 4
21.g even 6 1 inner 378.2.k.a ✓ 4
21.g even 6 1 2646.2.d.c 4
21.h odd 6 1 2646.2.d.c 4
63.i even 6 1 1134.2.t.a 4
63.k odd 6 1 1134.2.l.d 4
63.s even 6 1 1134.2.l.d 4
63.t odd 6 1 1134.2.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.a ✓ 4 1.a even 1 1 trivial
378.2.k.a ✓ 4 3.b odd 2 1 inner
378.2.k.a ✓ 4 7.d odd 6 1 inner
378.2.k.a ✓ 4 21.g even 6 1 inner
1134.2.l.d 4 9.c even 3 1
1134.2.l.d 4 9.d odd 6 1
1134.2.l.d 4 63.k odd 6 1
1134.2.l.d 4 63.s even 6 1
1134.2.t.a 4 9.c even 3 1
1134.2.t.a 4 9.d odd 6 1
1134.2.t.a 4 63.i even 6 1
1134.2.t.a 4 63.t odd 6 1
2646.2.d.c 4 7.c even 3 1
2646.2.d.c 4 7.d odd 6 1
2646.2.d.c 4 21.g even 6 1
2646.2.d.c 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$61$ \( (T^{2} + 24 T + 192)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$97$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
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