Properties

Label 378.2.g.d
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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

\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 5T_{11} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

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