Properties

Label 2646.2.d.a
Level $2646$
Weight $2$
Character orbit 2646.d
Analytic conductor $21.128$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{3} q^{2} - q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} -\zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12}^{3} q^{2} - q^{4} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{5} -\zeta_{12}^{3} q^{8} + ( -1 + 2 \zeta_{12}^{2} ) q^{10} + q^{16} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{17} + ( 4 - 8 \zeta_{12}^{2} ) q^{19} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{20} -6 \zeta_{12}^{3} q^{23} -2 q^{25} -9 \zeta_{12}^{3} q^{29} + ( -2 + 4 \zeta_{12}^{2} ) q^{31} + \zeta_{12}^{3} q^{32} + ( 2 - 4 \zeta_{12}^{2} ) q^{34} + 4 q^{37} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{38} + ( 1 - 2 \zeta_{12}^{2} ) q^{40} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} -8 q^{43} + 6 q^{46} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} -2 \zeta_{12}^{3} q^{50} + 3 \zeta_{12}^{3} q^{53} + 9 q^{58} + ( -14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{59} + ( 2 - 4 \zeta_{12}^{2} ) q^{61} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{62} - q^{64} + 14 q^{67} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + 6 \zeta_{12}^{3} q^{71} + ( 7 - 14 \zeta_{12}^{2} ) q^{73} + 4 \zeta_{12}^{3} q^{74} + ( -4 + 8 \zeta_{12}^{2} ) q^{76} + 11 q^{79} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{80} + ( -2 + 4 \zeta_{12}^{2} ) q^{82} + ( -20 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{83} -6 q^{85} -8 \zeta_{12}^{3} q^{86} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{89} + 6 \zeta_{12}^{3} q^{92} + ( 2 - 4 \zeta_{12}^{2} ) q^{94} -12 \zeta_{12}^{3} q^{95} + ( 4 - 8 \zeta_{12}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{16} - 8q^{25} + 16q^{37} - 32q^{43} + 24q^{46} + 36q^{58} - 4q^{64} + 56q^{67} + 44q^{79} - 24q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2645.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
1.00000i 0 −1.00000 −1.73205 0 0 1.00000i 0 1.73205i
2645.2 1.00000i 0 −1.00000 1.73205 0 0 1.00000i 0 1.73205i
2645.3 1.00000i 0 −1.00000 −1.73205 0 0 1.00000i 0 1.73205i
2645.4 1.00000i 0 −1.00000 1.73205 0 0 1.00000i 0 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( -3 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( ( 48 + T^{2} )^{2} \)
$23$ \( ( 36 + T^{2} )^{2} \)
$29$ \( ( 81 + T^{2} )^{2} \)
$31$ \( ( 12 + T^{2} )^{2} \)
$37$ \( ( -4 + T )^{4} \)
$41$ \( ( -12 + T^{2} )^{2} \)
$43$ \( ( 8 + T )^{4} \)
$47$ \( ( -12 + T^{2} )^{2} \)
$53$ \( ( 9 + T^{2} )^{2} \)
$59$ \( ( -147 + T^{2} )^{2} \)
$61$ \( ( 12 + T^{2} )^{2} \)
$67$ \( ( -14 + T )^{4} \)
$71$ \( ( 36 + T^{2} )^{2} \)
$73$ \( ( 147 + T^{2} )^{2} \)
$79$ \( ( -11 + T )^{4} \)
$83$ \( ( -300 + T^{2} )^{2} \)
$89$ \( ( -108 + T^{2} )^{2} \)
$97$ \( ( 48 + T^{2} )^{2} \)
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