Properties

Label 456.2.bf.b
Level $456$
Weight $2$
Character orbit 456.bf
Analytic conductor $3.641$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( -2 - \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{7} + ( -2 - \beta_{1} + \beta_{3} ) q^{9} + ( 2 - 4 \beta_{2} ) q^{11} + ( -\beta_{2} - 2 \beta_{3} ) q^{13} + ( -2 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{15} + ( 1 - \beta_{1} ) q^{17} + ( -3 - 2 \beta_{2} ) q^{19} + ( -2 - \beta_{1} + 5 \beta_{2} ) q^{21} + ( -1 + 3 \beta_{2} + 5 \beta_{3} ) q^{23} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{25} + ( -5 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} + ( -1 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{29} + ( 3 + \beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{31} + ( 2 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{33} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -5 - \beta_{1} + 11 \beta_{2} + \beta_{3} ) q^{37} + ( 5 \beta_{2} - \beta_{3} ) q^{39} + ( 3 + \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -5 - 2 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{43} + ( -5 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{45} + ( 3 - \beta_{2} + \beta_{3} ) q^{47} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{49} + ( -2 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{51} + ( 1 - 2 \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{53} + ( 2 + 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{55} + ( -3 + 3 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{57} + ( 5 - \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{59} + \beta_{2} q^{61} + ( -5 + 2 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} ) q^{63} + ( 6 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{65} + ( 6 - \beta_{2} + 4 \beta_{3} ) q^{67} + ( -1 + \beta_{1} - 12 \beta_{2} + 2 \beta_{3} ) q^{69} + ( 9 + \beta_{1} - 10 \beta_{2} - 2 \beta_{3} ) q^{71} + ( 1 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{73} + ( -5 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{75} + ( -2 + 6 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{77} + ( -5 + 8 \beta_{1} + 3 \beta_{2} ) q^{79} + ( 1 + 5 \beta_{1} - 5 \beta_{3} ) q^{81} + ( -2 + 4 \beta_{2} ) q^{83} + ( 1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{85} + ( 5 + \beta_{1} + 2 \beta_{3} ) q^{87} + ( 1 - 2 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{89} + ( 11 - 6 \beta_{2} - \beta_{3} ) q^{91} + ( 6 - 3 \beta_{1} - 7 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -3 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{95} + ( 3 + 3 \beta_{1} + 6 \beta_{2} ) q^{97} + ( -4 - 2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 3 q^{5} + 2 q^{7} - 10 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{3} + 3 q^{5} + 2 q^{7} - 10 q^{9} - 4 q^{15} + 3 q^{17} - 16 q^{19} + q^{21} - 3 q^{23} - 3 q^{25} - 16 q^{27} + 5 q^{29} + 18 q^{35} + 11 q^{39} + 7 q^{41} - 12 q^{43} - 13 q^{45} + 9 q^{47} + 6 q^{49} - 4 q^{51} - 17 q^{53} + 6 q^{55} - 8 q^{57} + 9 q^{59} + 2 q^{61} - 5 q^{63} + 22 q^{65} + 18 q^{67} - 29 q^{69} + 19 q^{71} - 18 q^{75} - 6 q^{79} + 14 q^{81} + 7 q^{85} + 19 q^{87} - 9 q^{89} + 33 q^{91} + 11 q^{93} - 9 q^{95} + 27 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(343\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
1.68614 + 0.396143i
−1.18614 1.26217i
−1.18614 + 1.26217i
1.68614 0.396143i
0 0.500000 1.65831i 0 −0.686141 0.396143i 0 −2.37228 0 −2.50000 1.65831i 0
65.2 0 0.500000 + 1.65831i 0 2.18614 + 1.26217i 0 3.37228 0 −2.50000 + 1.65831i 0
449.1 0 0.500000 1.65831i 0 2.18614 1.26217i 0 3.37228 0 −2.50000 1.65831i 0
449.2 0 0.500000 + 1.65831i 0 −0.686141 + 0.396143i 0 −2.37228 0 −2.50000 + 1.65831i 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
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 456.2.bf.b yes 4
3.b odd 2 1 456.2.bf.a 4
4.b odd 2 1 912.2.bn.i 4
12.b even 2 1 912.2.bn.j 4
19.d odd 6 1 456.2.bf.a 4
57.f even 6 1 inner 456.2.bf.b yes 4
76.f even 6 1 912.2.bn.j 4
228.n odd 6 1 912.2.bn.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bf.a 4 3.b odd 2 1
456.2.bf.a 4 19.d odd 6 1
456.2.bf.b yes 4 1.a even 1 1 trivial
456.2.bf.b yes 4 57.f even 6 1 inner
912.2.bn.i 4 4.b odd 2 1
912.2.bn.i 4 228.n odd 6 1
912.2.bn.j 4 12.b even 2 1
912.2.bn.j 4 76.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 3 T_{5}^{3} + T_{5}^{2} + 6 T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(456, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 - T + T^{2} )^{2} \)
$5$ \( 4 + 6 T + T^{2} - 3 T^{3} + T^{4} \)
$7$ \( ( -8 - T + T^{2} )^{2} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( 121 - 11 T^{2} + T^{4} \)
$17$ \( 4 + 6 T + T^{2} - 3 T^{3} + T^{4} \)
$19$ \( ( 19 + 8 T + T^{2} )^{2} \)
$23$ \( 4624 - 204 T - 65 T^{2} + 3 T^{3} + T^{4} \)
$29$ \( 4 + 10 T + 27 T^{2} - 5 T^{3} + T^{4} \)
$31$ \( 1156 + 79 T^{2} + T^{4} \)
$37$ \( 7744 + 187 T^{2} + T^{4} \)
$41$ \( 16 - 28 T + 45 T^{2} - 7 T^{3} + T^{4} \)
$43$ \( 9 + 36 T + 141 T^{2} + 12 T^{3} + T^{4} \)
$47$ \( 16 - 36 T + 31 T^{2} - 9 T^{3} + T^{4} \)
$53$ \( 4096 + 1088 T + 225 T^{2} + 17 T^{3} + T^{4} \)
$59$ \( 144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( 289 + 306 T + 91 T^{2} - 18 T^{3} + T^{4} \)
$71$ \( 6724 - 1558 T + 279 T^{2} - 19 T^{3} + T^{4} \)
$73$ \( 1089 + 33 T^{2} + T^{4} \)
$79$ \( 29929 - 1038 T - 161 T^{2} + 6 T^{3} + T^{4} \)
$83$ \( ( 12 + T^{2} )^{2} \)
$89$ \( 144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4} \)
$97$ \( 1296 - 972 T + 279 T^{2} - 27 T^{3} + T^{4} \)
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