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Properties

Label 378.2.k.c
Level $378$
Weight $2$
Character orbit 378.k
Analytic conductor $3.018$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,2,Mod(215,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

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: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 3) q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 3) q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 2) q^{10} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{17} + (4 \zeta_{12}^{2} + 4) q^{19} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{20} - 6 \zeta_{12} q^{23} + 2 \zeta_{12}^{2} q^{25} + (\zeta_{12}^{2} + 2) q^{28} + 9 \zeta_{12}^{3} q^{29} + ( - 2 \zeta_{12}^{2} + 4) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + ( - 4 \zeta_{12}^{2} + 2) q^{34} + ( - 4 \zeta_{12}^{3} - \zeta_{12}) q^{35} + (4 \zeta_{12}^{2} - 4) q^{37} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{38} + (\zeta_{12}^{2} + 1) q^{40} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{41} - 8 q^{43} - 6 \zeta_{12}^{2} q^{46} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{2} + 5) q^{49} + 2 \zeta_{12}^{3} q^{50} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{53} + (\zeta_{12}^{3} + 2 \zeta_{12}) q^{56} + (9 \zeta_{12}^{2} - 9) q^{58} + ( - 7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{59} + (2 \zeta_{12}^{2} + 2) q^{61} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{62} - q^{64} - 14 \zeta_{12}^{2} q^{67} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{68} + ( - 5 \zeta_{12}^{2} + 4) q^{70} - 6 \zeta_{12}^{3} q^{71} + (7 \zeta_{12}^{2} - 14) q^{73} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{74} + (8 \zeta_{12}^{2} - 4) q^{76} + (11 \zeta_{12}^{2} - 11) q^{79} + (\zeta_{12}^{3} + \zeta_{12}) q^{80} + ( - 2 \zeta_{12}^{2} - 2) q^{82} + ( - 10 \zeta_{12}^{3} + 20 \zeta_{12}) q^{83} - 6 q^{85} - 8 \zeta_{12} q^{86} + (12 \zeta_{12}^{3} - 6 \zeta_{12}) q^{89} - 6 \zeta_{12}^{3} q^{92} + (2 \zeta_{12}^{2} - 4) q^{94} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{95} + ( - 8 \zeta_{12}^{2} + 4) q^{97} + ( - 8 \zeta_{12}^{3} + 5 \zeta_{12}) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 8 q^{7} + 6 q^{10} - 2 q^{16} + 24 q^{19} + 4 q^{25} + 10 q^{28} + 12 q^{31} - 8 q^{37} + 6 q^{40} - 32 q^{43} - 12 q^{46} + 4 q^{49} - 18 q^{58} + 12 q^{61} - 4 q^{64} - 28 q^{67} + 6 q^{70} - 42 q^{73} - 22 q^{79} - 12 q^{82} - 24 q^{85} - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 −0.866025 + 1.50000i 0 2.00000 − 1.73205i − 1.00000i 0 1.50000 − 0.866025i
215.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.866025 − 1.50000i 0 2.00000 − 1.73205i 1.00000i 0 1.50000 − 0.866025i
269.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.866025 − 1.50000i 0 2.00000 + 1.73205i 1.00000i 0 1.50000 + 0.866025i
269.2 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.866025 + 1.50000i 0 2.00000 + 1.73205i − 1.00000i 0 1.50000 + 0.866025i
\(n\): 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.c ✓ 4
3.b odd 2 1 inner 378.2.k.c ✓ 4
7.c even 3 1 2646.2.d.a 4
7.d odd 6 1 inner 378.2.k.c ✓ 4
7.d odd 6 1 2646.2.d.a 4
9.c even 3 1 1134.2.l.b 4
9.c even 3 1 1134.2.t.c 4
9.d odd 6 1 1134.2.l.b 4
9.d odd 6 1 1134.2.t.c 4
21.g even 6 1 inner 378.2.k.c ✓ 4
21.g even 6 1 2646.2.d.a 4
21.h odd 6 1 2646.2.d.a 4
63.i even 6 1 1134.2.t.c 4
63.k odd 6 1 1134.2.l.b 4
63.s even 6 1 1134.2.l.b 4
63.t odd 6 1 1134.2.t.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.c ✓ 4 1.a even 1 1 trivial
378.2.k.c ✓ 4 3.b odd 2 1 inner
378.2.k.c ✓ 4 7.d odd 6 1 inner
378.2.k.c ✓ 4 21.g even 6 1 inner
1134.2.l.b 4 9.c even 3 1
1134.2.l.b 4 9.d odd 6 1
1134.2.l.b 4 63.k odd 6 1
1134.2.l.b 4 63.s even 6 1
1134.2.t.c 4 9.c even 3 1
1134.2.t.c 4 9.d odd 6 1
1134.2.t.c 4 63.i even 6 1
1134.2.t.c 4 63.t odd 6 1
2646.2.d.a 4 7.c even 3 1
2646.2.d.a 4 7.d odd 6 1
2646.2.d.a 4 21.g even 6 1
2646.2.d.a 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 3T_{5}^{2} + 9 \) 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
$5$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$19$ \( (T^{2} - 12 T + 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$59$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 21 T + 147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$97$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
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