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: \( x^{2} - 7 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{4} - 9 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{4} - 9 \beta q^{8} + 10 \beta q^{11} - 55 q^{16} + 70 q^{22} - 82 \beta q^{23} - 125 q^{25} - 100 \beta q^{29} + 17 \beta q^{32} - 450 q^{37} + 180 q^{43} - 10 \beta q^{44} - 574 q^{46} - 125 \beta q^{50} + 188 \beta q^{53} - 700 q^{58} + 559 q^{64} - 740 q^{67} + 370 \beta q^{71} - 450 \beta q^{74} - 1384 q^{79} + 180 \beta q^{86} - 630 q^{88} + 82 \beta q^{92} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 110 q^{16} + 140 q^{22} - 250 q^{25} - 900 q^{37} + 360 q^{43} - 1148 q^{46} - 1400 q^{58} + 1118 q^{64} - 1480 q^{67} - 2768 q^{79} - 1260 q^{88}+O(q^{100}) \) 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
−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 \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} - 7 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

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