Properties

Label 2268.2.i.i
Level $2268$
Weight $2$
Character orbit 2268.i
Analytic conductor $18.110$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{10})\)
Defining polynomial: \(x^{4} + 10 x^{2} + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{11} -\beta_{1} q^{17} -7 \beta_{2} q^{19} -\beta_{1} q^{23} + 5 \beta_{2} q^{25} + \beta_{1} q^{29} + 3 q^{31} + ( -2 \beta_{1} + \beta_{3} ) q^{35} -4 \beta_{2} q^{37} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{41} + ( -5 - 5 \beta_{2} ) q^{43} + 3 \beta_{3} q^{47} + ( 3 - 5 \beta_{2} ) q^{49} + 3 \beta_{1} q^{53} + 20 q^{55} + 4 \beta_{3} q^{59} -3 q^{61} + 10 q^{67} -4 \beta_{3} q^{71} + ( 5 + 5 \beta_{2} ) q^{73} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{77} + 12 q^{79} + 2 \beta_{1} q^{83} -10 \beta_{2} q^{85} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{89} -7 \beta_{3} q^{95} + ( 5 + 5 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{7} + O(q^{10}) \) \( 4 q - 10 q^{7} + 14 q^{19} - 10 q^{25} + 12 q^{31} + 8 q^{37} - 10 q^{43} + 22 q^{49} + 80 q^{55} - 12 q^{61} + 40 q^{67} + 10 q^{73} + 48 q^{79} + 20 q^{85} + 10 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 10 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/10\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(10 \beta_{2}\)
\(\nu^{3}\)\(=\)\(10 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
−1.58114 + 2.73861i
1.58114 2.73861i
−1.58114 2.73861i
1.58114 + 2.73861i
0 0 0 −1.58114 + 2.73861i 0 −2.50000 0.866025i 0 0 0
865.2 0 0 0 1.58114 2.73861i 0 −2.50000 0.866025i 0 0 0
2053.1 0 0 0 −1.58114 2.73861i 0 −2.50000 + 0.866025i 0 0 0
2053.2 0 0 0 1.58114 + 2.73861i 0 −2.50000 + 0.866025i 0 0 0
\(n\): 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
63.h even 3 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.i.i 4
3.b odd 2 1 inner 2268.2.i.i 4
7.c even 3 1 2268.2.l.i 4
9.c even 3 1 756.2.k.d 4
9.c even 3 1 2268.2.l.i 4
9.d odd 6 1 756.2.k.d 4
9.d odd 6 1 2268.2.l.i 4
21.h odd 6 1 2268.2.l.i 4
63.g even 3 1 756.2.k.d 4
63.h even 3 1 inner 2268.2.i.i 4
63.h even 3 1 5292.2.a.q 2
63.i even 6 1 5292.2.a.r 2
63.j odd 6 1 inner 2268.2.i.i 4
63.j odd 6 1 5292.2.a.q 2
63.n odd 6 1 756.2.k.d 4
63.t odd 6 1 5292.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.d 4 9.c even 3 1
756.2.k.d 4 9.d odd 6 1
756.2.k.d 4 63.g even 3 1
756.2.k.d 4 63.n odd 6 1
2268.2.i.i 4 1.a even 1 1 trivial
2268.2.i.i 4 3.b odd 2 1 inner
2268.2.i.i 4 63.h even 3 1 inner
2268.2.i.i 4 63.j odd 6 1 inner
2268.2.l.i 4 7.c even 3 1
2268.2.l.i 4 9.c even 3 1
2268.2.l.i 4 9.d odd 6 1
2268.2.l.i 4 21.h odd 6 1
5292.2.a.q 2 63.h even 3 1
5292.2.a.q 2 63.j odd 6 1
5292.2.a.r 2 63.i even 6 1
5292.2.a.r 2 63.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{4} + 10 T_{5}^{2} + 100 \)
\( T_{13} \)
\( T_{19}^{2} - 7 T_{19} + 49 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 100 + 10 T^{2} + T^{4} \)
$7$ \( ( 7 + 5 T + T^{2} )^{2} \)
$11$ \( 1600 + 40 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 100 + 10 T^{2} + T^{4} \)
$19$ \( ( 49 - 7 T + T^{2} )^{2} \)
$23$ \( 100 + 10 T^{2} + T^{4} \)
$29$ \( 100 + 10 T^{2} + T^{4} \)
$31$ \( ( -3 + T )^{4} \)
$37$ \( ( 16 - 4 T + T^{2} )^{2} \)
$41$ \( 8100 + 90 T^{2} + T^{4} \)
$43$ \( ( 25 + 5 T + T^{2} )^{2} \)
$47$ \( ( -90 + T^{2} )^{2} \)
$53$ \( 8100 + 90 T^{2} + T^{4} \)
$59$ \( ( -160 + T^{2} )^{2} \)
$61$ \( ( 3 + T )^{4} \)
$67$ \( ( -10 + T )^{4} \)
$71$ \( ( -160 + T^{2} )^{2} \)
$73$ \( ( 25 - 5 T + T^{2} )^{2} \)
$79$ \( ( -12 + T )^{4} \)
$83$ \( 1600 + 40 T^{2} + T^{4} \)
$89$ \( 8100 + 90 T^{2} + T^{4} \)
$97$ \( ( 25 - 5 T + T^{2} )^{2} \)
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