Properties

Label 1638.2.bj.d
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: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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 2.73205i 0 −0.866025 0.500000i 1.00000i 0 −1.36603 2.36603i
127.2 0.866025 0.500000i 0 0.500000 0.866025i 0.732051i 0 0.866025 + 0.500000i 1.00000i 0 0.366025 + 0.633975i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.73205i 0 −0.866025 + 0.500000i 1.00000i 0 −1.36603 + 2.36603i
1135.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.732051i 0 0.866025 0.500000i 1.00000i 0 0.366025 0.633975i
\(n\): 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.d 4
3.b odd 2 1 546.2.s.d 4
13.e even 6 1 inner 1638.2.bj.d 4
39.h odd 6 1 546.2.s.d 4
39.k even 12 1 7098.2.a.bj 2
39.k even 12 1 7098.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.s.d 4 3.b odd 2 1
546.2.s.d 4 39.h odd 6 1
1638.2.bj.d 4 1.a even 1 1 trivial
1638.2.bj.d 4 13.e even 6 1 inner
7098.2.a.bj 2 39.k even 12 1
7098.2.a.bs 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 \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{4} + 8T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 6T_{11}^{3} + 11T_{11}^{2} + 6T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + 15 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} + 59 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 152T^{2} + 5476 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$41$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$43$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$47$ \( T^{4} + 26T^{2} + 121 \) Copy content Toggle raw display
$53$ \( (T^{2} - 14 T + 37)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 24 T^{3} + 204 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$61$ \( T^{4} - 20 T^{3} + 303 T^{2} + \cdots + 9409 \) Copy content Toggle raw display
$67$ \( T^{4} + 24 T^{3} + 204 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$71$ \( T^{4} + 18 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$73$ \( T^{4} + 224T^{2} + 7744 \) Copy content Toggle raw display
$79$ \( (T^{2} + 6 T - 183)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 168T^{2} + 4356 \) Copy content Toggle raw display
$89$ \( T^{4} + 12 T^{3} + 51 T^{2} + 36 T + 9 \) Copy content Toggle raw display
$97$ \( T^{4} - 6 T^{3} + 14 T^{2} - 12 T + 4 \) Copy content Toggle raw display
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