Properties

Label 378.2.g.c
Level $378$
Weight $2$
Character orbit 378.g
Analytic conductor $3.018$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 1.00000 + 1.73205i 0 0.500000 2.59808i 1.00000 0 1.00000 1.73205i
163.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.00000 1.73205i 0 0.500000 + 2.59808i 1.00000 0 1.00000 + 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
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.c 2
3.b odd 2 1 378.2.g.d yes 2
7.c even 3 1 inner 378.2.g.c 2
7.c even 3 1 2646.2.a.t 1
7.d odd 6 1 2646.2.a.bb 1
9.c even 3 1 1134.2.e.o 2
9.c even 3 1 1134.2.h.b 2
9.d odd 6 1 1134.2.e.b 2
9.d odd 6 1 1134.2.h.o 2
21.g even 6 1 2646.2.a.c 1
21.h odd 6 1 378.2.g.d yes 2
21.h odd 6 1 2646.2.a.k 1
63.g even 3 1 1134.2.e.o 2
63.h even 3 1 1134.2.h.b 2
63.j odd 6 1 1134.2.h.o 2
63.n odd 6 1 1134.2.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.c 2 1.a even 1 1 trivial
378.2.g.c 2 7.c even 3 1 inner
378.2.g.d yes 2 3.b odd 2 1
378.2.g.d yes 2 21.h odd 6 1
1134.2.e.b 2 9.d odd 6 1
1134.2.e.b 2 63.n odd 6 1
1134.2.e.o 2 9.c even 3 1
1134.2.e.o 2 63.g even 3 1
1134.2.h.b 2 9.c even 3 1
1134.2.h.b 2 63.h even 3 1
1134.2.h.o 2 9.d odd 6 1
1134.2.h.o 2 63.j odd 6 1
2646.2.a.c 1 21.g even 6 1
2646.2.a.k 1 21.h odd 6 1
2646.2.a.t 1 7.c even 3 1
2646.2.a.bb 1 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 \(S_{2}^{\mathrm{new}}(378, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} + 5 T_{11} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( 7 - T + T^{2} \)
$11$ \( 25 + 5 T + T^{2} \)
$13$ \( ( -6 + T )^{2} \)
$17$ \( 16 - 4 T + T^{2} \)
$19$ \( 16 - 4 T + T^{2} \)
$23$ \( 16 - 4 T + T^{2} \)
$29$ \( ( -7 + T )^{2} \)
$31$ \( 9 + 3 T + T^{2} \)
$37$ \( 64 + 8 T + T^{2} \)
$41$ \( ( 6 + T )^{2} \)
$43$ \( ( -8 + T )^{2} \)
$47$ \( 36 + 6 T + T^{2} \)
$53$ \( 36 - 6 T + T^{2} \)
$59$ \( 49 + 7 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 100 + 10 T + T^{2} \)
$71$ \( ( 4 + T )^{2} \)
$73$ \( 169 + 13 T + T^{2} \)
$79$ \( 9 - 3 T + T^{2} \)
$83$ \( ( 7 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( ( 5 + T )^{2} \)
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