Properties

Label 441.6.a.o
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $2$
CM discriminant -7
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: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} -25 q^{4} -57 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} -25 q^{4} -57 \beta q^{8} -302 \beta q^{11} + 401 q^{16} -2114 q^{22} -418 \beta q^{23} -3125 q^{25} + 2036 \beta q^{29} + 2225 \beta q^{32} + 8886 q^{37} -11748 q^{43} + 7550 \beta q^{44} -2926 q^{46} -3125 \beta q^{50} -12364 \beta q^{53} + 14252 q^{58} + 2743 q^{64} + 69364 q^{67} + 32098 \beta q^{71} + 8886 \beta q^{74} + 80168 q^{79} -11748 \beta q^{86} + 120498 q^{88} + 10450 \beta q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 50 q^{4} + O(q^{10}) \) \( 2 q - 50 q^{4} + 802 q^{16} - 4228 q^{22} - 6250 q^{25} + 17772 q^{37} - 23496 q^{43} - 5852 q^{46} + 28504 q^{58} + 5486 q^{64} + 138728 q^{67} + 160336 q^{79} + 240996 q^{88} + O(q^{100}) \)

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
−2.64575
2.64575
−2.64575 0 −25.0000 0 0 0 150.808 0 0
1.2 2.64575 0 −25.0000 0 0 0 −150.808 0 0
\(n\): 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.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.o 2
3.b odd 2 1 inner 441.6.a.o 2
7.b odd 2 1 CM 441.6.a.o 2
21.c even 2 1 inner 441.6.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.6.a.o 2 1.a even 1 1 trivial
441.6.a.o 2 3.b odd 2 1 inner
441.6.a.o 2 7.b odd 2 1 CM
441.6.a.o 2 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} - 7 \)
\( T_{5} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -7 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -638428 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( -1223068 + T^{2} \)
$29$ \( -29017072 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -8886 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( 11748 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( -1070079472 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( -69364 + T )^{2} \)
$71$ \( -7211971228 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( -80168 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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