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: \( x^{4} - 2x^{3} - 71x^{2} + 72x - 15 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 44 q^{4} + \beta_{2} q^{5} + 12 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 44 q^{4} + \beta_{2} q^{5} + 12 \beta_1 q^{8} - \beta_{3} q^{10} + 43 \beta_1 q^{11} - \beta_{3} q^{13} - 496 q^{16} - 11 \beta_{2} q^{17} + \beta_{3} q^{19} + 44 \beta_{2} q^{20} + 3268 q^{22} + 341 \beta_1 q^{23} + 6811 q^{25} + 76 \beta_{2} q^{26} + 518 \beta_1 q^{29} + 11 \beta_{3} q^{31} - 880 \beta_1 q^{32} + 11 \beta_{3} q^{34} - 5466 q^{37} - 76 \beta_{2} q^{38} - 12 \beta_{3} q^{40} + 99 \beta_{2} q^{41} + 12540 q^{43} + 1892 \beta_1 q^{44} + 25916 q^{46} - 100 \beta_{2} q^{47} + 6811 \beta_1 q^{50} - 44 \beta_{3} q^{52} - 1738 \beta_1 q^{53} - 43 \beta_{3} q^{55} + 39368 q^{58} + 428 \beta_{2} q^{59} + 55 \beta_{3} q^{61} - 836 \beta_{2} q^{62} - 51008 q^{64} + 9936 \beta_1 q^{65} - 29996 q^{67} - 484 \beta_{2} q^{68} + 7051 \beta_1 q^{71} + 56 \beta_{3} q^{73} - 5466 \beta_1 q^{74} + 44 \beta_{3} q^{76} + 80168 q^{79} - 496 \beta_{2} q^{80} - 99 \beta_{3} q^{82} + 308 \beta_{2} q^{83} - 109296 q^{85} + 12540 \beta_1 q^{86} + 39216 q^{88} - 209 \beta_{2} q^{89} + 15004 \beta_1 q^{92} + 100 \beta_{3} q^{94} - 9936 \beta_1 q^{95} + 154 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 176 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 176 q^{4} - 1984 q^{16} + 13072 q^{22} + 27244 q^{25} - 21864 q^{37} + 50160 q^{43} + 103664 q^{46} + 157472 q^{58} - 204032 q^{64} - 119984 q^{67} + 320672 q^{79} - 437184 q^{85} + 156864 q^{88}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 4\nu^{3} - 6\nu^{2} - 294\nu + 148 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48\nu^{3} - 72\nu^{2} - 3360\nu + 1692 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 24\nu^{2} - 24\nu - 864 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 12\beta _1 + 12 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} - 12\beta _1 + 876 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 50\beta_{2} - 572\beta _1 + 872 ) / 16 \) Copy content Toggle raw display

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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} - 76 \) Copy content Toggle raw display
\( T_{5}^{2} - 9936 \) Copy content Toggle raw display
\( T_{13}^{2} - 755136 \) Copy content Toggle raw display

Hecke characteristic polynomials

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