Properties

Label 378.2.k.b
Level 378
Weight 2
Character orbit 378.k
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.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} + ( 3 - 2 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{5} + ( 3 - 2 \zeta_{12}^{2} ) q^{7} + \zeta_{12}^{3} q^{8} + ( -4 + 2 \zeta_{12}^{2} ) q^{10} + ( -3 + 6 \zeta_{12}^{2} ) q^{13} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{17} + ( -2 - 2 \zeta_{12}^{2} ) q^{19} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{20} -6 \zeta_{12} q^{23} -7 \zeta_{12}^{2} q^{25} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{26} + ( 2 + \zeta_{12}^{2} ) q^{28} + ( 10 - 5 \zeta_{12}^{2} ) q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + ( -4 + 8 \zeta_{12}^{2} ) q^{34} + ( 2 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{35} + ( 5 - 5 \zeta_{12}^{2} ) q^{37} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{38} + ( -2 - 2 \zeta_{12}^{2} ) q^{40} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + q^{43} -6 \zeta_{12}^{2} q^{46} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{47} + ( 5 - 8 \zeta_{12}^{2} ) q^{49} -7 \zeta_{12}^{3} q^{50} + ( -6 + 3 \zeta_{12}^{2} ) q^{52} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{56} + ( -4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{59} + ( -1 - \zeta_{12}^{2} ) q^{61} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{62} - q^{64} -18 \zeta_{12} q^{65} + 13 \zeta_{12}^{2} q^{67} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{68} + ( -8 + 10 \zeta_{12}^{2} ) q^{70} -6 \zeta_{12}^{3} q^{71} + ( -8 + 4 \zeta_{12}^{2} ) q^{73} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{74} + ( 2 - 4 \zeta_{12}^{2} ) q^{76} + ( 7 - 7 \zeta_{12}^{2} ) q^{79} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{80} + ( 4 + 4 \zeta_{12}^{2} ) q^{82} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{83} -24 q^{85} + \zeta_{12} q^{86} + ( -6 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{89} + ( 3 + 12 \zeta_{12}^{2} ) q^{91} -6 \zeta_{12}^{3} q^{92} + ( 8 - 4 \zeta_{12}^{2} ) q^{94} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{95} + ( 1 - 2 \zeta_{12}^{2} ) q^{97} + ( 5 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 8q^{7} + O(q^{10}) \) \( 4q + 2q^{4} + 8q^{7} - 12q^{10} - 2q^{16} - 12q^{19} - 14q^{25} + 10q^{28} + 30q^{31} + 10q^{37} - 12q^{40} + 4q^{43} - 12q^{46} + 4q^{49} - 18q^{52} - 6q^{61} - 4q^{64} + 26q^{67} - 12q^{70} - 24q^{73} + 14q^{79} + 24q^{82} - 96q^{85} + 36q^{91} + 24q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1 - \zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 1.73205 3.00000i 0 2.00000 1.73205i 1.00000i 0 −3.00000 + 1.73205i
215.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.73205 + 3.00000i 0 2.00000 1.73205i 1.00000i 0 −3.00000 + 1.73205i
269.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.73205 + 3.00000i 0 2.00000 + 1.73205i 1.00000i 0 −3.00000 1.73205i
269.2 0.866025 0.500000i 0 0.500000 0.866025i −1.73205 3.00000i 0 2.00000 + 1.73205i 1.00000i 0 −3.00000 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.k.b 4
3.b odd 2 1 inner 378.2.k.b 4
7.c even 3 1 2646.2.d.b 4
7.d odd 6 1 inner 378.2.k.b 4
7.d odd 6 1 2646.2.d.b 4
9.c even 3 1 1134.2.l.a 4
9.c even 3 1 1134.2.t.b 4
9.d odd 6 1 1134.2.l.a 4
9.d odd 6 1 1134.2.t.b 4
21.g even 6 1 inner 378.2.k.b 4
21.g even 6 1 2646.2.d.b 4
21.h odd 6 1 2646.2.d.b 4
63.i even 6 1 1134.2.t.b 4
63.k odd 6 1 1134.2.l.a 4
63.s even 6 1 1134.2.l.a 4
63.t odd 6 1 1134.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.b 4 1.a even 1 1 trivial
378.2.k.b 4 3.b odd 2 1 inner
378.2.k.b 4 7.d odd 6 1 inner
378.2.k.b 4 21.g even 6 1 inner
1134.2.l.a 4 9.c even 3 1
1134.2.l.a 4 9.d odd 6 1
1134.2.l.a 4 63.k odd 6 1
1134.2.l.a 4 63.s even 6 1
1134.2.t.b 4 9.c even 3 1
1134.2.t.b 4 9.d odd 6 1
1134.2.t.b 4 63.i even 6 1
1134.2.t.b 4 63.t odd 6 1
2646.2.d.b 4 7.c even 3 1
2646.2.d.b 4 7.d odd 6 1
2646.2.d.b 4 21.g even 6 1
2646.2.d.b 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 12 T_{5}^{2} + 144 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( \)
$5$ \( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - 4 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 + 11 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 14 T^{2} - 93 T^{4} + 4046 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 29 T^{2} )^{4} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )^{2}( 1 - 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 34 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - T + 43 T^{2} )^{4} \)
$47$ \( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 70 T^{2} + 1419 T^{4} - 243670 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 3 T + 64 T^{2} + 183 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 106 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 7 T - 30 T^{2} - 553 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 154 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 191 T^{2} + 9409 T^{4} )^{2} \)
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