Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} -\beta_{3} q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 - \beta_{2} ) q^{2} + \beta_{2} q^{4} + ( 1 - \beta_{1} + \beta_{2} ) q^{5} -\beta_{3} q^{7} + q^{8} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( -2 - \beta_{3} ) q^{13} -\beta_{1} q^{14} + ( -1 - \beta_{2} ) q^{16} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( -2 - 2 \beta_{2} ) q^{19} + ( -1 - \beta_{3} ) q^{20} + ( 1 + \beta_{3} ) q^{22} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{23} + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{25} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{26} + ( \beta_{1} + \beta_{3} ) q^{28} + ( 5 - \beta_{3} ) q^{29} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{31} + \beta_{2} q^{32} + ( 1 + \beta_{3} ) q^{34} + ( -7 + \beta_{1} - 7 \beta_{2} ) q^{35} + ( 4 - 3 \beta_{1} + 4 \beta_{2} ) q^{37} + 2 \beta_{2} q^{38} + ( 1 - \beta_{1} + \beta_{2} ) q^{40} + ( 3 + 3 \beta_{3} ) q^{41} + 5 q^{43} + ( -1 + \beta_{1} - \beta_{2} ) q^{44} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{46} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{47} + 7 q^{49} + ( 3 + 2 \beta_{3} ) q^{50} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{52} + 6 \beta_{2} q^{53} + ( -8 - 2 \beta_{3} ) q^{55} -\beta_{3} q^{56} + ( -5 - \beta_{1} - 5 \beta_{2} ) q^{58} + ( \beta_{1} + 11 \beta_{2} + \beta_{3} ) q^{59} + ( -10 - \beta_{1} - 10 \beta_{2} ) q^{61} + ( -2 - \beta_{3} ) q^{62} + q^{64} + ( -9 + 3 \beta_{1} - 9 \beta_{2} ) q^{65} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -1 + \beta_{1} - \beta_{2} ) q^{68} + ( -\beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{70} + ( 13 + \beta_{3} ) q^{71} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{73} + ( 3 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{74} + 2 q^{76} + ( \beta_{1} - 7 \beta_{2} + \beta_{3} ) q^{77} + ( 2 + 5 \beta_{1} + 2 \beta_{2} ) q^{79} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{80} + ( -3 + 3 \beta_{1} - 3 \beta_{2} ) q^{82} + ( -8 - 2 \beta_{3} ) q^{83} + ( -8 - 2 \beta_{3} ) q^{85} + ( -5 - 5 \beta_{2} ) q^{86} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{88} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 7 + 2 \beta_{3} ) q^{91} + ( 4 - 2 \beta_{3} ) q^{92} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{94} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{95} + ( 3 + 4 \beta_{3} ) q^{97} + ( -7 - 7 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 4 q^{8} + O(q^{10}) \) \( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 4 q^{8} + 2 q^{10} - 2 q^{11} - 8 q^{13} - 2 q^{16} - 2 q^{17} - 4 q^{19} - 4 q^{20} + 4 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{26} + 20 q^{29} + 4 q^{31} - 2 q^{32} + 4 q^{34} - 14 q^{35} + 8 q^{37} - 4 q^{38} + 2 q^{40} + 12 q^{41} + 20 q^{43} - 2 q^{44} - 8 q^{46} + 6 q^{47} + 28 q^{49} + 12 q^{50} + 4 q^{52} - 12 q^{53} - 32 q^{55} - 10 q^{58} - 22 q^{59} - 20 q^{61} - 8 q^{62} + 4 q^{64} - 18 q^{65} - 6 q^{67} - 2 q^{68} - 14 q^{70} + 52 q^{71} + 8 q^{74} + 8 q^{76} + 14 q^{77} + 4 q^{79} + 2 q^{80} - 6 q^{82} - 32 q^{83} - 32 q^{85} - 10 q^{86} - 2 q^{88} + 6 q^{89} + 28 q^{91} + 16 q^{92} + 6 q^{94} + 4 q^{95} + 12 q^{97} - 14 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/7\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(7 \beta_{2}\)
\(\nu^{3}\)\(=\)\(7 \beta_{3}\)

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.32288 + 2.29129i
−1.32288 2.29129i
1.32288 2.29129i
−1.32288 + 2.29129i
−0.500000 0.866025i 0 −0.500000 + 0.866025i −0.822876 1.42526i 0 2.64575 1.00000 0 −0.822876 + 1.42526i
109.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.82288 + 3.15731i 0 −2.64575 1.00000 0 1.82288 3.15731i
163.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −0.822876 + 1.42526i 0 2.64575 1.00000 0 −0.822876 1.42526i
163.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.82288 3.15731i 0 −2.64575 1.00000 0 1.82288 + 3.15731i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.g.g 4
3.b odd 2 1 378.2.g.h yes 4
7.c even 3 1 inner 378.2.g.g 4
7.c even 3 1 2646.2.a.bl 2
7.d odd 6 1 2646.2.a.bo 2
9.c even 3 1 1134.2.e.t 4
9.c even 3 1 1134.2.h.q 4
9.d odd 6 1 1134.2.e.q 4
9.d odd 6 1 1134.2.h.t 4
21.g even 6 1 2646.2.a.bf 2
21.h odd 6 1 378.2.g.h yes 4
21.h odd 6 1 2646.2.a.bi 2
63.g even 3 1 1134.2.e.t 4
63.h even 3 1 1134.2.h.q 4
63.j odd 6 1 1134.2.h.t 4
63.n odd 6 1 1134.2.e.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.g 4 1.a even 1 1 trivial
378.2.g.g 4 7.c even 3 1 inner
378.2.g.h yes 4 3.b odd 2 1
378.2.g.h yes 4 21.h odd 6 1
1134.2.e.q 4 9.d odd 6 1
1134.2.e.q 4 63.n odd 6 1
1134.2.e.t 4 9.c even 3 1
1134.2.e.t 4 63.g even 3 1
1134.2.h.q 4 9.c even 3 1
1134.2.h.q 4 63.h even 3 1
1134.2.h.t 4 9.d odd 6 1
1134.2.h.t 4 63.j odd 6 1
2646.2.a.bf 2 21.g even 6 1
2646.2.a.bi 2 21.h odd 6 1
2646.2.a.bl 2 7.c even 3 1
2646.2.a.bo 2 7.d odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 2 T_{5}^{3} + 10 T_{5}^{2} + 12 T_{5} + 36 \)
\( T_{11}^{4} + 2 T_{11}^{3} + 10 T_{11}^{2} - 12 T_{11} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( ( -7 + T^{2} )^{2} \)
$11$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$13$ \( ( -3 + 4 T + T^{2} )^{2} \)
$17$ \( 36 - 12 T + 10 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( ( 4 + 2 T + T^{2} )^{2} \)
$23$ \( 144 - 96 T + 76 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( ( 18 - 10 T + T^{2} )^{2} \)
$31$ \( 9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4} \)
$37$ \( 2209 + 376 T + 111 T^{2} - 8 T^{3} + T^{4} \)
$41$ \( ( -54 - 6 T + T^{2} )^{2} \)
$43$ \( ( -5 + T )^{4} \)
$47$ \( 2916 + 324 T + 90 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( ( 36 + 6 T + T^{2} )^{2} \)
$59$ \( 12996 + 2508 T + 370 T^{2} + 22 T^{3} + T^{4} \)
$61$ \( 8649 + 1860 T + 307 T^{2} + 20 T^{3} + T^{4} \)
$67$ \( 361 - 114 T + 55 T^{2} + 6 T^{3} + T^{4} \)
$71$ \( ( 162 - 26 T + T^{2} )^{2} \)
$73$ \( 12544 + 112 T^{2} + T^{4} \)
$79$ \( 29241 + 684 T + 187 T^{2} - 4 T^{3} + T^{4} \)
$83$ \( ( 36 + 16 T + T^{2} )^{2} \)
$89$ \( 2916 + 324 T + 90 T^{2} - 6 T^{3} + T^{4} \)
$97$ \( ( -103 - 6 T + T^{2} )^{2} \)
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