Properties

Label 378.2.d.a
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: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Character values

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

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

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 −2.50000 + 0.866025i 1.00000i 0 1.73205i
377.2 1.00000i 0 −1.00000 1.73205 0 −2.50000 0.866025i 1.00000i 0 1.73205i
377.3 1.00000i 0 −1.00000 −1.73205 0 −2.50000 0.866025i 1.00000i 0 1.73205i
377.4 1.00000i 0 −1.00000 1.73205 0 −2.50000 + 0.866025i 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.a 4
3.b odd 2 1 inner 378.2.d.a 4
4.b odd 2 1 3024.2.k.j 4
7.b odd 2 1 inner 378.2.d.a 4
9.c even 3 1 1134.2.m.e 4
9.c even 3 1 1134.2.m.f 4
9.d odd 6 1 1134.2.m.e 4
9.d odd 6 1 1134.2.m.f 4
12.b even 2 1 3024.2.k.j 4
21.c even 2 1 inner 378.2.d.a 4
28.d even 2 1 3024.2.k.j 4
63.l odd 6 1 1134.2.m.e 4
63.l odd 6 1 1134.2.m.f 4
63.o even 6 1 1134.2.m.e 4
63.o even 6 1 1134.2.m.f 4
84.h odd 2 1 3024.2.k.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.a 4 1.a even 1 1 trivial
378.2.d.a 4 3.b odd 2 1 inner
378.2.d.a 4 7.b odd 2 1 inner
378.2.d.a 4 21.c even 2 1 inner
1134.2.m.e 4 9.c even 3 1
1134.2.m.e 4 9.d odd 6 1
1134.2.m.e 4 63.l odd 6 1
1134.2.m.e 4 63.o even 6 1
1134.2.m.f 4 9.c even 3 1
1134.2.m.f 4 9.d odd 6 1
1134.2.m.f 4 63.l odd 6 1
1134.2.m.f 4 63.o even 6 1
3024.2.k.j 4 4.b odd 2 1
3024.2.k.j 4 12.b even 2 1
3024.2.k.j 4 28.d even 2 1
3024.2.k.j 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 \(S_{2}^{\mathrm{new}}(378, [\chi])\):

\( T_{5}^{2} - 3 \)
\( T_{13}^{2} + 48 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( \)
$5$ \( ( 1 + 7 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 5 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 13 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 14 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 2 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 82 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 97 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 106 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 2 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 2 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 163 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 70 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 47 T^{2} + 9409 T^{4} )^{2} \)
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