Properties

Label 378.2.k.a
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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Newspace parameters

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

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -1 + 3 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} + ( -2 + 4 \zeta_{12}^{2} ) q^{13} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 3 q^{22} + 5 \zeta_{12}^{2} q^{25} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{26} + ( -3 + 2 \zeta_{12}^{2} ) q^{28} -9 \zeta_{12}^{3} q^{29} + ( -2 + \zeta_{12}^{2} ) q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( 8 - 8 \zeta_{12}^{2} ) q^{37} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{41} + 4 q^{43} + 3 \zeta_{12} q^{44} + ( 6 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{47} + ( -8 + 3 \zeta_{12}^{2} ) q^{49} + 5 \zeta_{12}^{3} q^{50} + ( -4 + 2 \zeta_{12}^{2} ) q^{52} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} + ( -3 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{56} + ( 9 - 9 \zeta_{12}^{2} ) q^{58} + ( 3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{59} + ( -8 - 8 \zeta_{12}^{2} ) q^{61} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{62} - q^{64} -2 \zeta_{12}^{2} q^{67} -12 \zeta_{12}^{3} q^{71} + ( 6 - 3 \zeta_{12}^{2} ) q^{73} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{74} + ( 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{77} + ( 13 - 13 \zeta_{12}^{2} ) q^{79} + ( -6 - 6 \zeta_{12}^{2} ) q^{82} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{83} + 4 \zeta_{12} q^{86} + 3 \zeta_{12}^{2} q^{88} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{89} + ( -10 + 2 \zeta_{12}^{2} ) q^{91} + ( 12 - 6 \zeta_{12}^{2} ) q^{94} + ( 5 - 10 \zeta_{12}^{2} ) q^{97} + ( -8 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 2q^{7} + O(q^{10}) \) \( 4q + 2q^{4} + 2q^{7} - 2q^{16} + 12q^{22} + 10q^{25} - 8q^{28} - 6q^{31} + 16q^{37} + 16q^{43} - 26q^{49} - 12q^{52} + 18q^{58} - 48q^{61} - 4q^{64} - 4q^{67} + 18q^{73} + 26q^{79} - 36q^{82} + 6q^{88} - 36q^{91} + 36q^{94} + O(q^{100}) \)

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.

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 \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke characteristic polynomials

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