Properties

Label 441.4.a.p
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{4} -9 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} - q^{4} -9 \beta q^{8} + 10 \beta q^{11} -55 q^{16} + 70 q^{22} -82 \beta q^{23} -125 q^{25} -100 \beta q^{29} + 17 \beta q^{32} -450 q^{37} + 180 q^{43} -10 \beta q^{44} -574 q^{46} -125 \beta q^{50} + 188 \beta q^{53} -700 q^{58} + 559 q^{64} -740 q^{67} + 370 \beta q^{71} -450 \beta q^{74} -1384 q^{79} + 180 \beta q^{86} -630 q^{88} + 82 \beta q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} - 110q^{16} + 140q^{22} - 250q^{25} - 900q^{37} + 360q^{43} - 1148q^{46} - 1400q^{58} + 1118q^{64} - 1480q^{67} - 2768q^{79} - 1260q^{88} + O(q^{100}) \)

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 −1.00000 0 0 0 23.8118 0 0
1.2 2.64575 0 −1.00000 0 0 0 −23.8118 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 441.4.a.p 2
3.b odd 2 1 inner 441.4.a.p 2
7.b odd 2 1 CM 441.4.a.p 2
7.c even 3 2 441.4.e.t 4
7.d odd 6 2 441.4.e.t 4
21.c even 2 1 inner 441.4.a.p 2
21.g even 6 2 441.4.e.t 4
21.h odd 6 2 441.4.e.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.a.p 2 1.a even 1 1 trivial
441.4.a.p 2 3.b odd 2 1 inner
441.4.a.p 2 7.b odd 2 1 CM
441.4.a.p 2 21.c even 2 1 inner
441.4.e.t 4 7.c even 3 2
441.4.e.t 4 7.d odd 6 2
441.4.e.t 4 21.g even 6 2
441.4.e.t 4 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -7 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -700 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( -47068 + T^{2} \)
$29$ \( -70000 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 450 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( ( -180 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( -247408 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( ( 740 + T )^{2} \)
$71$ \( -958300 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 1384 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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