Properties

Label 378.2.d.b
Level $378$
Weight $2$
Character orbit 378.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} - \beta_{3} q^{5} + ( - \beta_{2} + 2) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} - \beta_{3} q^{5} + ( - \beta_{2} + 2) q^{7} + \beta_1 q^{8} + \beta_{2} q^{10} - 3 \beta_1 q^{11} - 2 \beta_{2} q^{13} + ( - \beta_{3} - 2 \beta_1) q^{14} + q^{16} - 4 \beta_{3} q^{17} - \beta_{2} q^{19} + \beta_{3} q^{20} - 3 q^{22} - 3 \beta_1 q^{23} - 2 q^{25} - 2 \beta_{3} q^{26} + (\beta_{2} - 2) q^{28} - 6 \beta_1 q^{29} + 3 \beta_{2} q^{31} - \beta_1 q^{32} + 4 \beta_{2} q^{34} + ( - 2 \beta_{3} + 3 \beta_1) q^{35} + 7 q^{37} - \beta_{3} q^{38} - \beta_{2} q^{40} + 7 \beta_{3} q^{41} - 2 q^{43} + 3 \beta_1 q^{44} - 3 q^{46} + 2 \beta_{3} q^{47} + ( - 4 \beta_{2} + 1) q^{49} + 2 \beta_1 q^{50} + 2 \beta_{2} q^{52} + 12 \beta_1 q^{53} + 3 \beta_{2} q^{55} + (\beta_{3} + 2 \beta_1) q^{56} - 6 q^{58} - 2 \beta_{3} q^{59} - 4 \beta_{2} q^{61} + 3 \beta_{3} q^{62} - q^{64} + 6 \beta_1 q^{65} + 2 q^{67} + 4 \beta_{3} q^{68} + (2 \beta_{2} + 3) q^{70} + 3 \beta_1 q^{71} + 2 \beta_{2} q^{73} - 7 \beta_1 q^{74} + \beta_{2} q^{76} + ( - 3 \beta_{3} - 6 \beta_1) q^{77} - 10 q^{79} - \beta_{3} q^{80} - 7 \beta_{2} q^{82} + 10 \beta_{3} q^{83} + 12 q^{85} + 2 \beta_1 q^{86} + 3 q^{88} + 3 \beta_{3} q^{89} + ( - 4 \beta_{2} - 6) q^{91} + 3 \beta_1 q^{92} - 2 \beta_{2} q^{94} + 3 \beta_1 q^{95} - 8 \beta_{2} q^{97} + ( - 4 \beta_{3} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{7} + 4 q^{16} - 12 q^{22} - 8 q^{25} - 8 q^{28} + 28 q^{37} - 8 q^{43} - 12 q^{46} + 4 q^{49} - 24 q^{58} - 4 q^{64} + 8 q^{67} + 12 q^{70} - 40 q^{79} + 48 q^{85} + 12 q^{88} - 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
1.00000i 0 −1.00000 −1.73205 0 2.00000 1.73205i 1.00000i 0 1.73205i
377.2 1.00000i 0 −1.00000 1.73205 0 2.00000 + 1.73205i 1.00000i 0 1.73205i
377.3 1.00000i 0 −1.00000 −1.73205 0 2.00000 + 1.73205i 1.00000i 0 1.73205i
377.4 1.00000i 0 −1.00000 1.73205 0 2.00000 1.73205i 1.00000i 0 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.d.b 4
3.b odd 2 1 inner 378.2.d.b 4
4.b odd 2 1 3024.2.k.e 4
7.b odd 2 1 inner 378.2.d.b 4
9.c even 3 1 1134.2.m.b 4
9.c even 3 1 1134.2.m.c 4
9.d odd 6 1 1134.2.m.b 4
9.d odd 6 1 1134.2.m.c 4
12.b even 2 1 3024.2.k.e 4
21.c even 2 1 inner 378.2.d.b 4
28.d even 2 1 3024.2.k.e 4
63.l odd 6 1 1134.2.m.b 4
63.l odd 6 1 1134.2.m.c 4
63.o even 6 1 1134.2.m.b 4
63.o even 6 1 1134.2.m.c 4
84.h odd 2 1 3024.2.k.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.b 4 1.a even 1 1 trivial
378.2.d.b 4 3.b odd 2 1 inner
378.2.d.b 4 7.b odd 2 1 inner
378.2.d.b 4 21.c even 2 1 inner
1134.2.m.b 4 9.c even 3 1
1134.2.m.b 4 9.d odd 6 1
1134.2.m.b 4 63.l odd 6 1
1134.2.m.b 4 63.o even 6 1
1134.2.m.c 4 9.c even 3 1
1134.2.m.c 4 9.d odd 6 1
1134.2.m.c 4 63.l odd 6 1
1134.2.m.c 4 63.o even 6 1
3024.2.k.e 4 4.b odd 2 1
3024.2.k.e 4 12.b even 2 1
3024.2.k.e 4 28.d even 2 1
3024.2.k.e 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\):

\( T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$37$ \( (T - 7)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T - 2)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$79$ \( (T + 10)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
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