Properties

Label 4416.2.a.bs
Level $4416$
Weight $2$
Character orbit 4416.a
Self dual yes
Analytic conductor $35.262$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4416 = 2^{6} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4416.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2619375326\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 552)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{3} -\beta_{2} q^{5} + ( -1 - \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta_{2} q^{5} + ( -1 - \beta_{1} ) q^{7} + q^{9} + ( 1 - \beta_{1} - \beta_{2} ) q^{11} -2 q^{13} -\beta_{2} q^{15} + ( 1 - \beta_{1} ) q^{17} + ( 2 - \beta_{2} ) q^{19} + ( -1 - \beta_{1} ) q^{21} + q^{23} + ( 5 - 2 \beta_{1} - 2 \beta_{2} ) q^{25} + q^{27} -2 \beta_{1} q^{29} + ( 2 + 2 \beta_{1} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} ) q^{33} + ( -2 + 2 \beta_{1} ) q^{35} + ( -5 - \beta_{1} + \beta_{2} ) q^{37} -2 q^{39} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 2 \beta_{1} + \beta_{2} ) q^{43} -\beta_{2} q^{45} + ( 6 + 2 \beta_{1} ) q^{47} + ( 3 + 2 \beta_{1} + 2 \beta_{2} ) q^{49} + ( 1 - \beta_{1} ) q^{51} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{53} + ( 8 - 4 \beta_{2} ) q^{55} + ( 2 - \beta_{2} ) q^{57} + ( -4 + 2 \beta_{2} ) q^{59} + ( -7 + \beta_{1} - \beta_{2} ) q^{61} + ( -1 - \beta_{1} ) q^{63} + 2 \beta_{2} q^{65} + ( -2 \beta_{1} - \beta_{2} ) q^{67} + q^{69} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{71} + 2 q^{73} + ( 5 - 2 \beta_{1} - 2 \beta_{2} ) q^{75} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 9 + \beta_{1} ) q^{79} + q^{81} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{85} -2 \beta_{1} q^{87} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{89} + ( 2 + 2 \beta_{1} ) q^{91} + ( 2 + 2 \beta_{1} ) q^{93} + ( 10 - 2 \beta_{1} - 4 \beta_{2} ) q^{95} + ( 6 + 2 \beta_{2} ) q^{97} + ( 1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - 2q^{7} + 3q^{9} + 4q^{11} - 6q^{13} + 4q^{17} + 6q^{19} - 2q^{21} + 3q^{23} + 17q^{25} + 3q^{27} + 2q^{29} + 4q^{31} + 4q^{33} - 8q^{35} - 14q^{37} - 6q^{39} - 2q^{41} - 2q^{43} + 16q^{47} + 7q^{49} + 4q^{51} + 4q^{53} + 24q^{55} + 6q^{57} - 12q^{59} - 22q^{61} - 2q^{63} + 2q^{67} + 3q^{69} + 8q^{71} + 6q^{73} + 17q^{75} + 16q^{77} + 26q^{79} + 3q^{81} - 4q^{83} - 8q^{85} + 2q^{87} - 8q^{89} + 4q^{91} + 4q^{93} + 32q^{95} + 18q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.48119
2.17009
0.311108
0 1.00000 0 −3.35026 0 2.96239 0 1.00000 0
1.2 0 1.00000 0 −1.07838 0 −4.34017 0 1.00000 0
1.3 0 1.00000 0 4.42864 0 −0.622216 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4416.2.a.bs 3
4.b odd 2 1 4416.2.a.bp 3
8.b even 2 1 1104.2.a.o 3
8.d odd 2 1 552.2.a.g 3
24.f even 2 1 1656.2.a.n 3
24.h odd 2 1 3312.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
552.2.a.g 3 8.d odd 2 1
1104.2.a.o 3 8.b even 2 1
1656.2.a.n 3 24.f even 2 1
3312.2.a.bf 3 24.h odd 2 1
4416.2.a.bp 3 4.b odd 2 1
4416.2.a.bs 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4416))\):

\( T_{5}^{3} - 16 T_{5} - 16 \)
\( T_{7}^{3} + 2 T_{7}^{2} - 12 T_{7} - 8 \)
\( T_{11}^{3} - 4 T_{11}^{2} - 16 T_{11} + 32 \)
\( T_{17}^{3} - 4 T_{17}^{2} - 8 T_{17} + 16 \)
\( T_{19}^{3} - 6 T_{19}^{2} - 4 T_{19} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( -16 - 16 T + T^{3} \)
$7$ \( -8 - 12 T + 2 T^{2} + T^{3} \)
$11$ \( 32 - 16 T - 4 T^{2} + T^{3} \)
$13$ \( ( 2 + T )^{3} \)
$17$ \( 16 - 8 T - 4 T^{2} + T^{3} \)
$19$ \( 8 - 4 T - 6 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( 40 - 52 T - 2 T^{2} + T^{3} \)
$31$ \( 64 - 48 T - 4 T^{2} + T^{3} \)
$37$ \( -152 + 28 T + 14 T^{2} + T^{3} \)
$41$ \( -104 - 84 T + 2 T^{2} + T^{3} \)
$43$ \( -184 - 52 T + 2 T^{2} + T^{3} \)
$47$ \( 128 + 32 T - 16 T^{2} + T^{3} \)
$53$ \( 592 - 144 T - 4 T^{2} + T^{3} \)
$59$ \( -64 - 16 T + 12 T^{2} + T^{3} \)
$61$ \( 200 + 124 T + 22 T^{2} + T^{3} \)
$67$ \( 184 - 52 T - 2 T^{2} + T^{3} \)
$71$ \( -256 - 128 T - 8 T^{2} + T^{3} \)
$73$ \( ( -2 + T )^{3} \)
$79$ \( -536 + 212 T - 26 T^{2} + T^{3} \)
$83$ \( -160 - 176 T + 4 T^{2} + T^{3} \)
$89$ \( -304 - 40 T + 8 T^{2} + T^{3} \)
$97$ \( 296 + 44 T - 18 T^{2} + T^{3} \)
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