Properties

Label 378.2.k.b
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: \( x^{4} - x^{2} + 1 \) 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

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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 3) q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 3) q^{7} + \zeta_{12}^{3} q^{8} + (2 \zeta_{12}^{2} - 4) q^{10} + (6 \zeta_{12}^{2} - 3) q^{13} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{17} + ( - 2 \zeta_{12}^{2} - 2) q^{19} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{20} - 6 \zeta_{12} q^{23} - 7 \zeta_{12}^{2} q^{25} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{26} + (\zeta_{12}^{2} + 2) q^{28} + ( - 5 \zeta_{12}^{2} + 10) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + (8 \zeta_{12}^{2} - 4) q^{34} + (8 \zeta_{12}^{3} + 2 \zeta_{12}) q^{35} + ( - 5 \zeta_{12}^{2} + 5) q^{37} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{38} + ( - 2 \zeta_{12}^{2} - 2) q^{40} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{41} + q^{43} - 6 \zeta_{12}^{2} q^{46} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{2} + 5) q^{49} - 7 \zeta_{12}^{3} q^{50} + (3 \zeta_{12}^{2} - 6) q^{52} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{53} + (\zeta_{12}^{3} + 2 \zeta_{12}) q^{56} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{59} + ( - \zeta_{12}^{2} - 1) q^{61} + ( - 5 \zeta_{12}^{3} + 10 \zeta_{12}) q^{62} - q^{64} - 18 \zeta_{12} q^{65} + 13 \zeta_{12}^{2} q^{67} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{68} + (10 \zeta_{12}^{2} - 8) q^{70} - 6 \zeta_{12}^{3} q^{71} + (4 \zeta_{12}^{2} - 8) q^{73} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{74} + ( - 4 \zeta_{12}^{2} + 2) q^{76} + ( - 7 \zeta_{12}^{2} + 7) q^{79} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{80} + (4 \zeta_{12}^{2} + 4) q^{82} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{83} - 24 q^{85} + \zeta_{12} q^{86} + (12 \zeta_{12}^{3} - 6 \zeta_{12}) q^{89} + (12 \zeta_{12}^{2} + 3) q^{91} - 6 \zeta_{12}^{3} q^{92} + ( - 4 \zeta_{12}^{2} + 8) q^{94} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{95} + ( - 2 \zeta_{12}^{2} + 1) q^{97} + ( - 8 \zeta_{12}^{3} + 5 \zeta_{12}) q^{98} +O(q^{100}) \) 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 \( 4 q + 2 q^{4} + 8 q^{7} - 12 q^{10} - 2 q^{16} - 12 q^{19} - 14 q^{25} + 10 q^{28} + 30 q^{31} + 10 q^{37} - 12 q^{40} + 4 q^{43} - 12 q^{46} + 4 q^{49} - 18 q^{52} - 6 q^{61} - 4 q^{64} + 26 q^{67} - 12 q^{70} - 24 q^{73} + 14 q^{79} + 24 q^{82} - 96 q^{85} + 36 q^{91} + 24 q^{94}+O(q^{100}) \) 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.

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 1.73205 3.00000i 0 2.00000 1.73205i 1.00000i 0 −3.00000 + 1.73205i
215.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.73205 + 3.00000i 0 2.00000 1.73205i 1.00000i 0 −3.00000 + 1.73205i
269.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.73205 + 3.00000i 0 2.00000 + 1.73205i 1.00000i 0 −3.00000 1.73205i
269.2 0.866025 0.500000i 0 0.500000 0.866025i −1.73205 3.00000i 0 2.00000 + 1.73205i 1.00000i 0 −3.00000 1.73205i
\(n\): 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.b 4
3.b odd 2 1 inner 378.2.k.b 4
7.c even 3 1 2646.2.d.b 4
7.d odd 6 1 inner 378.2.k.b 4
7.d odd 6 1 2646.2.d.b 4
9.c even 3 1 1134.2.l.a 4
9.c even 3 1 1134.2.t.b 4
9.d odd 6 1 1134.2.l.a 4
9.d odd 6 1 1134.2.t.b 4
21.g even 6 1 inner 378.2.k.b 4
21.g even 6 1 2646.2.d.b 4
21.h odd 6 1 2646.2.d.b 4
63.i even 6 1 1134.2.t.b 4
63.k odd 6 1 1134.2.l.a 4
63.s even 6 1 1134.2.l.a 4
63.t odd 6 1 1134.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.b 4 1.a even 1 1 trivial
378.2.k.b 4 3.b odd 2 1 inner
378.2.k.b 4 7.d odd 6 1 inner
378.2.k.b 4 21.g even 6 1 inner
1134.2.l.a 4 9.c even 3 1
1134.2.l.a 4 9.d odd 6 1
1134.2.l.a 4 63.k odd 6 1
1134.2.l.a 4 63.s even 6 1
1134.2.t.b 4 9.c even 3 1
1134.2.t.b 4 9.d odd 6 1
1134.2.t.b 4 63.i even 6 1
1134.2.t.b 4 63.t odd 6 1
2646.2.d.b 4 7.c even 3 1
2646.2.d.b 4 7.d odd 6 1
2646.2.d.b 4 21.g even 6 1
2646.2.d.b 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 12T_{5}^{2} + 144 \) 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
$5$ \( T^{4} + 12T^{2} + 144 \) 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
$13$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$97$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
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