Properties

Label 4400.2.a.bx
Level $4400$
Weight $2$
Character orbit 4400.a
Self dual yes
Analytic conductor $35.134$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2200)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 6 x - 1\):

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

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
−2.36147
−0.167449
2.52892
0 −3.36147 0 0 0 0.576535 0 8.29947 0
1.2 0 −1.16745 0 0 0 −4.97196 0 −1.63706 0
1.3 0 1.52892 0 0 0 1.39543 0 −0.662410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(-1\)

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 4400.2.a.bx 3
4.b odd 2 1 2200.2.a.w yes 3
5.b even 2 1 4400.2.a.ca 3
5.c odd 4 2 4400.2.b.bc 6
20.d odd 2 1 2200.2.a.t 3
20.e even 4 2 2200.2.b.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.t 3 20.d odd 2 1
2200.2.a.w yes 3 4.b odd 2 1
2200.2.b.l 6 20.e even 4 2
4400.2.a.bx 3 1.a even 1 1 trivial
4400.2.a.ca 3 5.b even 2 1
4400.2.b.bc 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\):

\( T_{3}^{3} + 3 T_{3}^{2} - 3 T_{3} - 6 \)
\( T_{7}^{3} + 3 T_{7}^{2} - 9 T_{7} + 4 \)
\( T_{13}^{3} + 3 T_{13}^{2} - 21 T_{13} - 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -6 - 3 T + 3 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( 4 - 9 T + 3 T^{2} + T^{3} \)
$11$ \( ( -1 + T )^{3} \)
$13$ \( -15 - 21 T + 3 T^{2} + T^{3} \)
$17$ \( -144 - 45 T + 3 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( -5 + 6 T + 6 T^{2} + T^{3} \)
$29$ \( -31 + 36 T - 12 T^{2} + T^{3} \)
$31$ \( -73 - 45 T - 3 T^{2} + T^{3} \)
$37$ \( 4 + 27 T - 12 T^{2} + T^{3} \)
$41$ \( 20 - 3 T - 6 T^{2} + T^{3} \)
$43$ \( 12 - 12 T - 3 T^{2} + T^{3} \)
$47$ \( -8 - 9 T + 12 T^{2} + T^{3} \)
$53$ \( -542 - 69 T + 9 T^{2} + T^{3} \)
$59$ \( -100 + 135 T + 24 T^{2} + T^{3} \)
$61$ \( -346 - 111 T + 3 T^{2} + T^{3} \)
$67$ \( -128 - 48 T + 6 T^{2} + T^{3} \)
$71$ \( 1500 - 102 T - 15 T^{2} + T^{3} \)
$73$ \( 1184 + 351 T + 33 T^{2} + T^{3} \)
$79$ \( -172 + 21 T + 15 T^{2} + T^{3} \)
$83$ \( 13 - 102 T + 6 T^{2} + T^{3} \)
$89$ \( 453 - 102 T - 6 T^{2} + T^{3} \)
$97$ \( -2477 - 120 T + 18 T^{2} + T^{3} \)
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