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: \(\Q(\sqrt{-3}) \)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -3 \zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + ( -3 + 3 \zeta_{6} ) q^{10} -4 q^{13} + ( 2 + \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + 3 q^{20} -6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} -3 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} + 6 q^{34} + ( 6 + 3 \zeta_{6} ) q^{35} -8 \zeta_{6} q^{37} + ( 4 - 4 \zeta_{6} ) q^{38} -3 \zeta_{6} q^{40} -6 q^{41} + 8 q^{43} + ( -6 + 6 \zeta_{6} ) q^{46} -6 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + 4 q^{50} + ( 4 - 4 \zeta_{6} ) q^{52} + ( -9 + 9 \zeta_{6} ) q^{53} + ( -3 + 2 \zeta_{6} ) q^{56} + 3 \zeta_{6} q^{58} + ( 3 - 3 \zeta_{6} ) q^{59} + 10 \zeta_{6} q^{61} + 8 q^{62} + q^{64} + 12 \zeta_{6} q^{65} + ( 10 - 10 \zeta_{6} ) q^{67} -6 \zeta_{6} q^{68} + ( 3 - 9 \zeta_{6} ) q^{70} + 6 q^{71} + ( 7 - 7 \zeta_{6} ) q^{73} + ( -8 + 8 \zeta_{6} ) q^{74} -4 q^{76} -17 \zeta_{6} q^{79} + ( -3 + 3 \zeta_{6} ) q^{80} + 6 \zeta_{6} q^{82} -12 q^{83} + 18 q^{85} -8 \zeta_{6} q^{86} -6 \zeta_{6} q^{89} + ( 12 - 8 \zeta_{6} ) q^{91} + 6 q^{92} + ( -6 + 6 \zeta_{6} ) q^{94} + ( 12 - 12 \zeta_{6} ) q^{95} -10 q^{97} + ( -8 + 3 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 3q^{5} - 4q^{7} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 3q^{5} - 4q^{7} + 2q^{8} - 3q^{10} - 8q^{13} + 5q^{14} - q^{16} - 6q^{17} + 4q^{19} + 6q^{20} - 6q^{23} - 4q^{25} + 4q^{26} - q^{28} - 6q^{29} - 8q^{31} - q^{32} + 12q^{34} + 15q^{35} - 8q^{37} + 4q^{38} - 3q^{40} - 12q^{41} + 16q^{43} - 6q^{46} - 6q^{47} + 2q^{49} + 8q^{50} + 4q^{52} - 9q^{53} - 4q^{56} + 3q^{58} + 3q^{59} + 10q^{61} + 16q^{62} + 2q^{64} + 12q^{65} + 10q^{67} - 6q^{68} - 3q^{70} + 12q^{71} + 7q^{73} - 8q^{74} - 8q^{76} - 17q^{79} - 3q^{80} + 6q^{82} - 24q^{83} + 36q^{85} - 8q^{86} - 6q^{89} + 16q^{91} + 12q^{92} - 6q^{94} + 12q^{95} - 20q^{97} - 13q^{98} + O(q^{100}) \)

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 −1.50000 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 \(S_{2}^{\mathrm{new}}(378, [\chi])\):

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( \)
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \)
$23$ \( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 3 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 6 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \)
$79$ \( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \)
$83$ \( ( 1 + 12 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 10 T + 97 T^{2} )^{2} \)
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