Properties

Label 441.6.a.y
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{19}, \sqrt{69})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 71 x^{2} + 72 x - 15\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 44 q^{4} + \beta_{2} q^{5} + 12 \beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 44 q^{4} + \beta_{2} q^{5} + 12 \beta_{1} q^{8} -\beta_{3} q^{10} + 43 \beta_{1} q^{11} -\beta_{3} q^{13} -496 q^{16} -11 \beta_{2} q^{17} + \beta_{3} q^{19} + 44 \beta_{2} q^{20} + 3268 q^{22} + 341 \beta_{1} q^{23} + 6811 q^{25} + 76 \beta_{2} q^{26} + 518 \beta_{1} q^{29} + 11 \beta_{3} q^{31} -880 \beta_{1} q^{32} + 11 \beta_{3} q^{34} -5466 q^{37} -76 \beta_{2} q^{38} -12 \beta_{3} q^{40} + 99 \beta_{2} q^{41} + 12540 q^{43} + 1892 \beta_{1} q^{44} + 25916 q^{46} -100 \beta_{2} q^{47} + 6811 \beta_{1} q^{50} -44 \beta_{3} q^{52} -1738 \beta_{1} q^{53} -43 \beta_{3} q^{55} + 39368 q^{58} + 428 \beta_{2} q^{59} + 55 \beta_{3} q^{61} -836 \beta_{2} q^{62} -51008 q^{64} + 9936 \beta_{1} q^{65} -29996 q^{67} -484 \beta_{2} q^{68} + 7051 \beta_{1} q^{71} + 56 \beta_{3} q^{73} -5466 \beta_{1} q^{74} + 44 \beta_{3} q^{76} + 80168 q^{79} -496 \beta_{2} q^{80} -99 \beta_{3} q^{82} + 308 \beta_{2} q^{83} -109296 q^{85} + 12540 \beta_{1} q^{86} + 39216 q^{88} -209 \beta_{2} q^{89} + 15004 \beta_{1} q^{92} + 100 \beta_{3} q^{94} -9936 \beta_{1} q^{95} + 154 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 176 q^{4} + O(q^{10}) \) \( 4 q + 176 q^{4} - 1984 q^{16} + 13072 q^{22} + 27244 q^{25} - 21864 q^{37} + 50160 q^{43} + 103664 q^{46} + 157472 q^{58} - 204032 q^{64} - 119984 q^{67} + 320672 q^{79} - 437184 q^{85} + 156864 q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 71 x^{2} + 72 x - 15\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{3} - 6 \nu^{2} - 294 \nu + 148 \)\()/7\)
\(\beta_{2}\)\(=\)\((\)\( 48 \nu^{3} - 72 \nu^{2} - 3360 \nu + 1692 \)\()/7\)
\(\beta_{3}\)\(=\)\( 24 \nu^{2} - 24 \nu - 864 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 12 \beta_{1} + 12\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 12 \beta_{1} + 876\)\()/24\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{3} + 50 \beta_{2} - 572 \beta_{1} + 872\)\()/16\)

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
0.705587
9.01221
−8.01221
0.294413
−8.71780 0 44.0000 −99.6795 0 0 −104.614 0 868.986
1.2 −8.71780 0 44.0000 99.6795 0 0 −104.614 0 −868.986
1.3 8.71780 0 44.0000 −99.6795 0 0 104.614 0 −868.986
1.4 8.71780 0 44.0000 99.6795 0 0 104.614 0 868.986
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)

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 441.6.a.y 4
3.b odd 2 1 inner 441.6.a.y 4
7.b odd 2 1 inner 441.6.a.y 4
21.c even 2 1 inner 441.6.a.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.6.a.y 4 1.a even 1 1 trivial
441.6.a.y 4 3.b odd 2 1 inner
441.6.a.y 4 7.b odd 2 1 inner
441.6.a.y 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} - 76 \)
\( T_{5}^{2} - 9936 \)
\( T_{13}^{2} - 755136 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -76 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( ( -9936 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -140524 + T^{2} )^{2} \)
$13$ \( ( -755136 + T^{2} )^{2} \)
$17$ \( ( -1202256 + T^{2} )^{2} \)
$19$ \( ( -755136 + T^{2} )^{2} \)
$23$ \( ( -8837356 + T^{2} )^{2} \)
$29$ \( ( -20392624 + T^{2} )^{2} \)
$31$ \( ( -91371456 + T^{2} )^{2} \)
$37$ \( ( 5466 + T )^{4} \)
$41$ \( ( -97382736 + T^{2} )^{2} \)
$43$ \( ( -12540 + T )^{4} \)
$47$ \( ( -99360000 + T^{2} )^{2} \)
$53$ \( ( -229568944 + T^{2} )^{2} \)
$59$ \( ( -1820116224 + T^{2} )^{2} \)
$61$ \( ( -2284286400 + T^{2} )^{2} \)
$67$ \( ( 29996 + T )^{4} \)
$71$ \( ( -3778461676 + T^{2} )^{2} \)
$73$ \( ( -2368106496 + T^{2} )^{2} \)
$79$ \( ( -80168 + T )^{4} \)
$83$ \( ( -942568704 + T^{2} )^{2} \)
$89$ \( ( -434014416 + T^{2} )^{2} \)
$97$ \( ( -17908805376 + T^{2} )^{2} \)
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