Properties

Label 441.2.a.h
Level $441$
Weight $2$
Character orbit 441.a
Self dual yes
Analytic conductor $3.521$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.52140272914\)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + 5 q^{4} + 3 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 5 q^{4} + 3 \beta q^{8} -2 \beta q^{11} + 11 q^{16} -14 q^{22} + 2 \beta q^{23} -5 q^{25} -4 \beta q^{29} + 5 \beta q^{32} + 6 q^{37} + 12 q^{43} -10 \beta q^{44} + 14 q^{46} -5 \beta q^{50} -4 \beta q^{53} -28 q^{58} + 13 q^{64} + 4 q^{67} -2 \beta q^{71} + 6 \beta q^{74} + 8 q^{79} + 12 \beta q^{86} -42 q^{88} + 10 \beta q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{4} + O(q^{10}) \) \( 2q + 10q^{4} + 22q^{16} - 28q^{22} - 10q^{25} + 12q^{37} + 24q^{43} + 28q^{46} - 56q^{58} + 26q^{64} + 8q^{67} + 16q^{79} - 84q^{88} + O(q^{100}) \)

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 5.00000 0 0 0 −7.93725 0 0
1.2 2.64575 0 5.00000 0 0 0 7.93725 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 441.2.a.h 2
3.b odd 2 1 inner 441.2.a.h 2
4.b odd 2 1 7056.2.a.co 2
7.b odd 2 1 CM 441.2.a.h 2
7.c even 3 2 441.2.e.h 4
7.d odd 6 2 441.2.e.h 4
12.b even 2 1 7056.2.a.co 2
21.c even 2 1 inner 441.2.a.h 2
21.g even 6 2 441.2.e.h 4
21.h odd 6 2 441.2.e.h 4
28.d even 2 1 7056.2.a.co 2
84.h odd 2 1 7056.2.a.co 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h 2 1.a even 1 1 trivial
441.2.a.h 2 3.b odd 2 1 inner
441.2.a.h 2 7.b odd 2 1 CM
441.2.a.h 2 21.c even 2 1 inner
441.2.e.h 4 7.c even 3 2
441.2.e.h 4 7.d odd 6 2
441.2.e.h 4 21.g even 6 2
441.2.e.h 4 21.h odd 6 2
7056.2.a.co 2 4.b odd 2 1
7056.2.a.co 2 12.b even 2 1
7056.2.a.co 2 28.d even 2 1
7056.2.a.co 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( ( 1 + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( 1 - 6 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 13 T^{2} )^{2} \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 + 18 T^{2} + 529 T^{4} \)
$29$ \( 1 - 54 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 6 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 12 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 - 6 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 114 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 97 T^{2} )^{2} \)
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