Properties

Label 1638.2.bj.e
Level $1638$
Weight $2$
Character orbit 1638.bj
Analytic conductor $13.079$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
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.

\(f(q)\) \(=\) \( q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + \zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + ( \zeta_{12} - \zeta_{12}^{3} ) q^{2} + ( 1 - \zeta_{12}^{2} ) q^{4} + ( 2 - 4 \zeta_{12}^{2} ) q^{5} + \zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{10} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{11} + ( -3 \zeta_{12} - \zeta_{12}^{3} ) q^{13} + q^{14} -\zeta_{12}^{2} q^{16} + ( -3 + 2 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{17} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{19} + ( -2 - 2 \zeta_{12}^{2} ) q^{20} + ( 4 - 4 \zeta_{12}^{2} ) q^{22} + ( \zeta_{12} - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{23} -7 q^{25} + ( -3 - \zeta_{12}^{2} ) q^{26} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{28} + 8 \zeta_{12}^{2} q^{29} + ( -2 + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{31} -\zeta_{12} q^{32} + ( 2 - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{34} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{35} + ( -8 - 2 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{37} + ( -4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{38} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{40} + ( 8 - 4 \zeta_{12}^{2} ) q^{41} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{43} -4 \zeta_{12}^{3} q^{44} + ( 1 - 4 \zeta_{12} + \zeta_{12}^{2} ) q^{46} + ( 2 - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{47} + \zeta_{12}^{2} q^{49} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{50} + ( -4 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{52} + ( 10 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{53} + ( -8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{55} + ( 1 - \zeta_{12}^{2} ) q^{56} + 8 \zeta_{12} q^{58} + ( -1 - 8 \zeta_{12} - \zeta_{12}^{2} ) q^{59} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{61} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{62} - q^{64} + ( -10 \zeta_{12} + 14 \zeta_{12}^{3} ) q^{65} + ( 6 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{67} + ( -2 \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{68} + ( 2 - 4 \zeta_{12}^{2} ) q^{70} + ( -4 - 5 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{71} + ( 4 - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{73} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{74} + ( 4 - 4 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{76} + 4 q^{77} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{79} + ( -4 + 2 \zeta_{12}^{2} ) q^{80} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{82} + ( -1 + 2 \zeta_{12}^{2} ) q^{83} + ( 6 - 12 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{85} + ( -2 + 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{86} -4 \zeta_{12}^{2} q^{88} + ( 2 - 10 \zeta_{12} - \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{89} + ( 1 - 4 \zeta_{12}^{2} ) q^{91} + ( -4 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{92} + ( -2 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{94} + ( 12 - 8 \zeta_{12} - 12 \zeta_{12}^{2} + 16 \zeta_{12}^{3} ) q^{95} + ( 6 - 2 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{97} + \zeta_{12} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} + 4q^{14} - 2q^{16} - 6q^{17} + 12q^{19} - 12q^{20} + 8q^{22} - 8q^{23} - 28q^{25} - 14q^{26} + 16q^{29} - 24q^{37} - 16q^{38} + 24q^{41} + 2q^{43} + 6q^{46} + 2q^{49} + 40q^{53} + 2q^{56} - 6q^{59} - 6q^{62} - 4q^{64} + 18q^{67} + 6q^{68} - 24q^{71} - 4q^{74} + 12q^{76} + 16q^{77} - 24q^{79} - 12q^{80} + 36q^{85} - 8q^{88} + 6q^{89} - 4q^{91} - 16q^{92} - 8q^{94} + 24q^{95} + 36q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−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 3.46410i 0 −0.866025 0.500000i 1.00000i 0 1.73205 + 3.00000i
127.2 0.866025 0.500000i 0 0.500000 0.866025i 3.46410i 0 0.866025 + 0.500000i 1.00000i 0 −1.73205 3.00000i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 3.46410i 0 −0.866025 + 0.500000i 1.00000i 0 1.73205 3.00000i
1135.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 3.46410i 0 0.866025 0.500000i 1.00000i 0 −1.73205 + 3.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bj.e 4
3.b odd 2 1 546.2.s.c 4
13.e even 6 1 inner 1638.2.bj.e 4
39.h odd 6 1 546.2.s.c 4
39.k even 12 1 7098.2.a.bn 2
39.k even 12 1 7098.2.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.c 4 3.b odd 2 1
546.2.s.c 4 39.h odd 6 1
1638.2.bj.e 4 1.a even 1 1 trivial
1638.2.bj.e 4 13.e even 6 1 inner
7098.2.a.bn 2 39.k even 12 1
7098.2.a.bz 2 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{2} + 12 \)
\( T_{11}^{4} - 16 T_{11}^{2} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 12 + T^{2} )^{2} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( 256 - 16 T^{2} + T^{4} \)
$13$ \( 169 - T^{2} + T^{4} \)
$17$ \( 9 - 18 T + 39 T^{2} + 6 T^{3} + T^{4} \)
$19$ \( 16 + 48 T + 44 T^{2} - 12 T^{3} + T^{4} \)
$23$ \( 169 + 104 T + 51 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( ( 64 - 8 T + T^{2} )^{2} \)
$31$ \( 9 + 42 T^{2} + T^{4} \)
$37$ \( 1936 + 1056 T + 236 T^{2} + 24 T^{3} + T^{4} \)
$41$ \( ( 48 - 12 T + T^{2} )^{2} \)
$43$ \( 121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4} \)
$47$ \( 16 + 56 T^{2} + T^{4} \)
$53$ \( ( 97 - 20 T + T^{2} )^{2} \)
$59$ \( 3721 - 366 T - 49 T^{2} + 6 T^{3} + T^{4} \)
$61$ \( 729 + 27 T^{2} + T^{4} \)
$67$ \( 529 - 414 T + 131 T^{2} - 18 T^{3} + T^{4} \)
$71$ \( 529 + 552 T + 215 T^{2} + 24 T^{3} + T^{4} \)
$73$ \( 1024 + 128 T^{2} + T^{4} \)
$79$ \( ( 24 + 12 T + T^{2} )^{2} \)
$83$ \( ( 3 + T^{2} )^{2} \)
$89$ \( 9409 + 582 T - 85 T^{2} - 6 T^{3} + T^{4} \)
$97$ \( 10816 - 3744 T + 536 T^{2} - 36 T^{3} + T^{4} \)
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