Properties

Label 378.2.g.h
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: \(\Q(\sqrt{-3}, \sqrt{7})\)
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{4} - 2q^{5} - 4q^{8} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{4} - 2q^{5} - 4q^{8} + 2q^{10} + 2q^{11} - 8q^{13} - 2q^{16} + 2q^{17} - 4q^{19} + 4q^{20} + 4q^{22} + 8q^{23} - 6q^{25} - 4q^{26} - 20q^{29} + 4q^{31} + 2q^{32} + 4q^{34} + 14q^{35} + 8q^{37} + 4q^{38} + 2q^{40} - 12q^{41} + 20q^{43} + 2q^{44} - 8q^{46} - 6q^{47} + 28q^{49} - 12q^{50} + 4q^{52} + 12q^{53} - 32q^{55} - 10q^{58} + 22q^{59} - 20q^{61} + 8q^{62} + 4q^{64} + 18q^{65} - 6q^{67} + 2q^{68} - 14q^{70} - 52q^{71} - 8q^{74} + 8q^{76} - 14q^{77} + 4q^{79} - 2q^{80} - 6q^{82} + 32q^{83} - 32q^{85} + 10q^{86} - 2q^{88} - 6q^{89} + 28q^{91} - 16q^{92} + 6q^{94} - 4q^{95} + 12q^{97} + 14q^{98} + O(q^{100}) \)

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 −1.82288 3.15731i 0 −2.64575 −1.00000 0 1.82288 3.15731i
109.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.822876 + 1.42526i 0 2.64575 −1.00000 0 −0.822876 + 1.42526i
163.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.82288 + 3.15731i 0 −2.64575 −1.00000 0 1.82288 + 3.15731i
163.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.822876 1.42526i 0 2.64575 −1.00000 0 −0.822876 1.42526i
\(n\): 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.h yes 4
3.b odd 2 1 378.2.g.g 4
7.c even 3 1 inner 378.2.g.h yes 4
7.c even 3 1 2646.2.a.bi 2
7.d odd 6 1 2646.2.a.bf 2
9.c even 3 1 1134.2.e.q 4
9.c even 3 1 1134.2.h.t 4
9.d odd 6 1 1134.2.e.t 4
9.d odd 6 1 1134.2.h.q 4
21.g even 6 1 2646.2.a.bo 2
21.h odd 6 1 378.2.g.g 4
21.h odd 6 1 2646.2.a.bl 2
63.g even 3 1 1134.2.e.q 4
63.h even 3 1 1134.2.h.t 4
63.j odd 6 1 1134.2.h.q 4
63.n odd 6 1 1134.2.e.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.g 4 3.b odd 2 1
378.2.g.g 4 21.h odd 6 1
378.2.g.h yes 4 1.a even 1 1 trivial
378.2.g.h yes 4 7.c even 3 1 inner
1134.2.e.q 4 9.c even 3 1
1134.2.e.q 4 63.g even 3 1
1134.2.e.t 4 9.d odd 6 1
1134.2.e.t 4 63.n odd 6 1
1134.2.h.q 4 9.d odd 6 1
1134.2.h.q 4 63.j odd 6 1
1134.2.h.t 4 9.c even 3 1
1134.2.h.t 4 63.h even 3 1
2646.2.a.bf 2 7.d odd 6 1
2646.2.a.bi 2 7.c even 3 1
2646.2.a.bl 2 21.h odd 6 1
2646.2.a.bo 2 21.g even 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

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( \)
$5$ \( 1 + 2 T - 12 T^{3} - 29 T^{4} - 60 T^{5} + 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 - 2 T - 12 T^{2} + 12 T^{3} + 91 T^{4} + 132 T^{5} - 1452 T^{6} - 2662 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 + 4 T + 23 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \)
$17$ \( 1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 204 T^{5} - 6936 T^{6} - 9826 T^{7} + 83521 T^{8} \)
$19$ \( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 8 T + 30 T^{2} + 96 T^{3} - 845 T^{4} + 2208 T^{5} + 15870 T^{6} - 97336 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 + 10 T + 76 T^{2} + 290 T^{3} + 841 T^{4} )^{2} \)
$31$ \( 1 - 4 T - 43 T^{2} + 12 T^{3} + 2024 T^{4} + 372 T^{5} - 41323 T^{6} - 119164 T^{7} + 923521 T^{8} \)
$37$ \( ( 1 - 8 T + 37 T^{2} )^{2}( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} ) \)
$41$ \( ( 1 + 6 T + 28 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{4} \)
$47$ \( 1 + 6 T - 4 T^{2} - 324 T^{3} - 2301 T^{4} - 15228 T^{5} - 8836 T^{6} + 622938 T^{7} + 4879681 T^{8} \)
$53$ \( ( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 22 T + 252 T^{2} - 2508 T^{3} + 21787 T^{4} - 147972 T^{5} + 877212 T^{6} - 4518338 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 20 T + 185 T^{2} + 1860 T^{3} + 18104 T^{4} + 113460 T^{5} + 688385 T^{6} + 4539620 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 6 T - 79 T^{2} - 114 T^{3} + 6324 T^{4} - 7638 T^{5} - 354631 T^{6} + 1804578 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 26 T + 304 T^{2} + 1846 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 34 T^{2} - 4173 T^{4} - 181186 T^{6} + 28398241 T^{8} \)
$79$ \( 1 - 4 T + 29 T^{2} + 684 T^{3} - 7336 T^{4} + 54036 T^{5} + 180989 T^{6} - 1972156 T^{7} + 38950081 T^{8} \)
$83$ \( ( 1 - 16 T + 202 T^{2} - 1328 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 + 6 T - 88 T^{2} - 324 T^{3} + 4251 T^{4} - 28836 T^{5} - 697048 T^{6} + 4229814 T^{7} + 62742241 T^{8} \)
$97$ \( ( 1 - 6 T + 91 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \)
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