Properties

Label 1368.2.s.j
Level $1368$
Weight $2$
Character orbit 1368.s
Analytic conductor $10.924$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.9235349965\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 456)
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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{5} + ( 1 - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{7} +O(q^{10})\) \( q + ( -2 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{5} + ( 1 - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{7} + ( 2 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{11} + ( -1 + \zeta_{18}^{3} ) q^{13} + ( -4 \zeta_{18}^{2} - 4 \zeta_{18}^{4} ) q^{17} + ( 1 + 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{19} + ( 2 \zeta_{18} - 2 \zeta_{18}^{2} ) q^{23} + ( -3 + 4 \zeta_{18} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{25} + ( 4 + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{29} + ( 5 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{31} + ( -2 \zeta_{18} + 6 \zeta_{18}^{2} - 8 \zeta_{18}^{3} + 6 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{35} + ( -1 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{5} ) q^{37} + ( -4 \zeta_{18} + 4 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{41} + ( 2 \zeta_{18} - 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{43} + ( -6 + 6 \zeta_{18}^{3} ) q^{47} + ( 2 - 4 \zeta_{18} - 4 \zeta_{18}^{2} - 4 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{49} + ( -2 \zeta_{18} + 6 \zeta_{18}^{2} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{53} + ( -12 \zeta_{18} + 8 \zeta_{18}^{2} + 8 \zeta_{18}^{4} - 12 \zeta_{18}^{5} ) q^{55} + ( 6 \zeta_{18} - 6 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 6 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{59} + ( 1 - 4 \zeta_{18} + 4 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{61} + ( -2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{65} + ( 1 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{67} + ( -4 \zeta_{18} + 8 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 8 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{71} + ( -4 \zeta_{18} + 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{73} + ( 2 - 10 \zeta_{18}^{4} + 10 \zeta_{18}^{5} ) q^{77} + ( -6 \zeta_{18} + 2 \zeta_{18}^{2} - 9 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{79} + ( 4 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 8 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{83} + ( 8 - 8 \zeta_{18} + 8 \zeta_{18}^{2} - 8 \zeta_{18}^{3} ) q^{85} + ( 6 - 2 \zeta_{18} - 2 \zeta_{18}^{2} - 6 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{89} + ( -1 + 2 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{91} + ( -8 + 10 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{95} + ( 4 \zeta_{18} - 8 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 8 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{7} + O(q^{10}) \) \( 6 q + 6 q^{7} + 12 q^{11} - 3 q^{13} - 9 q^{25} + 12 q^{29} + 30 q^{31} - 24 q^{35} - 6 q^{37} + 12 q^{41} - 9 q^{43} - 18 q^{47} + 12 q^{49} - 6 q^{59} + 3 q^{61} + 3 q^{67} + 6 q^{71} + 9 q^{73} + 12 q^{77} - 27 q^{79} + 24 q^{83} + 24 q^{85} + 18 q^{89} - 3 q^{91} - 48 q^{95} - 6 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \zeta_{18}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
−0.766044 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
0 0 0 −1.87939 + 3.25519i 0 4.75877 0 0 0
505.2 0 0 0 0.347296 0.601535i 0 0.305407 0 0 0
505.3 0 0 0 1.53209 2.65366i 0 −2.06418 0 0 0
577.1 0 0 0 −1.87939 3.25519i 0 4.75877 0 0 0
577.2 0 0 0 0.347296 + 0.601535i 0 0.305407 0 0 0
577.3 0 0 0 1.53209 + 2.65366i 0 −2.06418 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.s.j 6
3.b odd 2 1 456.2.q.f 6
4.b odd 2 1 2736.2.s.x 6
12.b even 2 1 912.2.q.k 6
19.c even 3 1 inner 1368.2.s.j 6
57.f even 6 1 8664.2.a.z 3
57.h odd 6 1 456.2.q.f 6
57.h odd 6 1 8664.2.a.x 3
76.g odd 6 1 2736.2.s.x 6
228.m even 6 1 912.2.q.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.f 6 3.b odd 2 1
456.2.q.f 6 57.h odd 6 1
912.2.q.k 6 12.b even 2 1
912.2.q.k 6 228.m even 6 1
1368.2.s.j 6 1.a even 1 1 trivial
1368.2.s.j 6 19.c even 3 1 inner
2736.2.s.x 6 4.b odd 2 1
2736.2.s.x 6 76.g odd 6 1
8664.2.a.x 3 57.h odd 6 1
8664.2.a.z 3 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1368, [\chi])\):

\( T_{5}^{6} + 12 T_{5}^{4} - 16 T_{5}^{3} + 144 T_{5}^{2} - 96 T_{5} + 64 \)
\( T_{7}^{3} - 3 T_{7}^{2} - 9 T_{7} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 64 - 96 T + 144 T^{2} - 16 T^{3} + 12 T^{4} + T^{6} \)
$7$ \( ( 3 - 9 T - 3 T^{2} + T^{3} )^{2} \)
$11$ \( ( 136 - 24 T - 6 T^{2} + T^{3} )^{2} \)
$13$ \( ( 1 + T + T^{2} )^{3} \)
$17$ \( 4096 - 3072 T + 2304 T^{2} - 128 T^{3} + 48 T^{4} + T^{6} \)
$19$ \( 6859 + 171 T^{2} + 64 T^{3} + 9 T^{4} + T^{6} \)
$23$ \( 64 - 96 T + 144 T^{2} - 16 T^{3} + 12 T^{4} + T^{6} \)
$29$ \( 36864 + 2304 T^{2} - 384 T^{3} + 144 T^{4} - 12 T^{5} + T^{6} \)
$31$ \( ( 127 + 39 T - 15 T^{2} + T^{3} )^{2} \)
$37$ \( ( -111 - 45 T + 3 T^{2} + T^{3} )^{2} \)
$41$ \( 36864 + 2304 T^{2} - 384 T^{3} + 144 T^{4} - 12 T^{5} + T^{6} \)
$43$ \( 289 - 255 T + 378 T^{2} + 169 T^{3} + 66 T^{4} + 9 T^{5} + T^{6} \)
$47$ \( ( 36 + 6 T + T^{2} )^{3} \)
$53$ \( 87616 - 24864 T + 7056 T^{2} - 592 T^{3} + 84 T^{4} + T^{6} \)
$59$ \( 179776 + 40704 T + 11760 T^{2} + 272 T^{3} + 132 T^{4} + 6 T^{5} + T^{6} \)
$61$ \( 289 + 765 T + 1974 T^{2} + 169 T^{3} + 54 T^{4} - 3 T^{5} + T^{6} \)
$67$ \( 1369 + 1221 T + 978 T^{2} + 173 T^{3} + 42 T^{4} - 3 T^{5} + T^{6} \)
$71$ \( 732736 - 112992 T + 22560 T^{2} - 920 T^{3} + 168 T^{4} - 6 T^{5} + T^{6} \)
$73$ \( 32761 - 3801 T + 2070 T^{2} - 173 T^{3} + 102 T^{4} - 9 T^{5} + T^{6} \)
$79$ \( 104329 - 51357 T + 34002 T^{2} + 4939 T^{3} + 570 T^{4} + 27 T^{5} + T^{6} \)
$83$ \( ( -64 - 96 T - 12 T^{2} + T^{3} )^{2} \)
$89$ \( 5184 + 5184 T + 6480 T^{2} - 1440 T^{3} + 252 T^{4} - 18 T^{5} + T^{6} \)
$97$ \( 732736 + 112992 T + 22560 T^{2} + 920 T^{3} + 168 T^{4} + 6 T^{5} + T^{6} \)
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