Properties

Label 378.2.g.a
Level $378$
Weight $2$
Character orbit 378.g
Analytic conductor $3.018$
Analytic rank $1$
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: \(1\)
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.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 3) q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{4} - 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 3) q^{7} + q^{8} + (3 \zeta_{6} - 3) q^{10} - 4 q^{13} + (\zeta_{6} + 2) q^{14} - \zeta_{6} q^{16} + (6 \zeta_{6} - 6) q^{17} + 4 \zeta_{6} q^{19} + 3 q^{20} - 6 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + 4 \zeta_{6} q^{26} + ( - 3 \zeta_{6} + 1) q^{28} - 3 q^{29} + (8 \zeta_{6} - 8) q^{31} + (\zeta_{6} - 1) q^{32} + 6 q^{34} + (3 \zeta_{6} + 6) q^{35} - 8 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} - 3 \zeta_{6} q^{40} - 6 q^{41} + 8 q^{43} + (6 \zeta_{6} - 6) q^{46} - 6 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + 4 q^{50} + ( - 4 \zeta_{6} + 4) q^{52} + (9 \zeta_{6} - 9) q^{53} + (2 \zeta_{6} - 3) q^{56} + 3 \zeta_{6} q^{58} + ( - 3 \zeta_{6} + 3) q^{59} + 10 \zeta_{6} q^{61} + 8 q^{62} + q^{64} + 12 \zeta_{6} q^{65} + ( - 10 \zeta_{6} + 10) q^{67} - 6 \zeta_{6} q^{68} + ( - 9 \zeta_{6} + 3) q^{70} + 6 q^{71} + ( - 7 \zeta_{6} + 7) q^{73} + (8 \zeta_{6} - 8) q^{74} - 4 q^{76} - 17 \zeta_{6} q^{79} + (3 \zeta_{6} - 3) q^{80} + 6 \zeta_{6} q^{82} - 12 q^{83} + 18 q^{85} - 8 \zeta_{6} q^{86} - 6 \zeta_{6} q^{89} + ( - 8 \zeta_{6} + 12) q^{91} + 6 q^{92} + (6 \zeta_{6} - 6) q^{94} + ( - 12 \zeta_{6} + 12) q^{95} - 10 q^{97} + (3 \zeta_{6} - 8) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 3 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - 3 q^{5} - 4 q^{7} + 2 q^{8} - 3 q^{10} - 8 q^{13} + 5 q^{14} - q^{16} - 6 q^{17} + 4 q^{19} + 6 q^{20} - 6 q^{23} - 4 q^{25} + 4 q^{26} - q^{28} - 6 q^{29} - 8 q^{31} - q^{32} + 12 q^{34} + 15 q^{35} - 8 q^{37} + 4 q^{38} - 3 q^{40} - 12 q^{41} + 16 q^{43} - 6 q^{46} - 6 q^{47} + 2 q^{49} + 8 q^{50} + 4 q^{52} - 9 q^{53} - 4 q^{56} + 3 q^{58} + 3 q^{59} + 10 q^{61} + 16 q^{62} + 2 q^{64} + 12 q^{65} + 10 q^{67} - 6 q^{68} - 3 q^{70} + 12 q^{71} + 7 q^{73} - 8 q^{74} - 8 q^{76} - 17 q^{79} - 3 q^{80} + 6 q^{82} - 24 q^{83} + 36 q^{85} - 8 q^{86} - 6 q^{89} + 16 q^{91} + 12 q^{92} - 6 q^{94} + 12 q^{95} - 20 q^{97} - 13 q^{98}+O(q^{100}) \) 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 −1.50000 2.59808i 0 −2.00000 + 1.73205i 1.00000 0 −1.50000 + 2.59808i
163.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i −1.50000 + 2.59808i 0 −2.00000 1.73205i 1.00000 0 −1.50000 2.59808i
\(n\): 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.a 2
3.b odd 2 1 378.2.g.f yes 2
7.c even 3 1 inner 378.2.g.a 2
7.c even 3 1 2646.2.a.bc 1
7.d odd 6 1 2646.2.a.r 1
9.c even 3 1 1134.2.e.j 2
9.c even 3 1 1134.2.h.g 2
9.d odd 6 1 1134.2.e.f 2
9.d odd 6 1 1134.2.h.k 2
21.g even 6 1 2646.2.a.m 1
21.h odd 6 1 378.2.g.f yes 2
21.h odd 6 1 2646.2.a.b 1
63.g even 3 1 1134.2.e.j 2
63.h even 3 1 1134.2.h.g 2
63.j odd 6 1 1134.2.h.k 2
63.n odd 6 1 1134.2.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.a 2 1.a even 1 1 trivial
378.2.g.a 2 7.c even 3 1 inner
378.2.g.f yes 2 3.b odd 2 1
378.2.g.f yes 2 21.h odd 6 1
1134.2.e.f 2 9.d odd 6 1
1134.2.e.f 2 63.n odd 6 1
1134.2.e.j 2 9.c even 3 1
1134.2.e.j 2 63.g even 3 1
1134.2.h.g 2 9.c even 3 1
1134.2.h.g 2 63.h even 3 1
1134.2.h.k 2 9.d odd 6 1
1134.2.h.k 2 63.j odd 6 1
2646.2.a.b 1 21.h odd 6 1
2646.2.a.m 1 21.g even 6 1
2646.2.a.r 1 7.d odd 6 1
2646.2.a.bc 1 7.c even 3 1

Hecke kernels

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

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11} \) 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
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) 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
$53$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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