Properties

Label 441.4.a.e
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{2} - 4q^{4} + 7q^{5} + 24q^{8} + O(q^{10}) \) \( q - 2q^{2} - 4q^{4} + 7q^{5} + 24q^{8} - 14q^{10} + 5q^{11} + 14q^{13} - 16q^{16} - 21q^{17} - 49q^{19} - 28q^{20} - 10q^{22} + 159q^{23} - 76q^{25} - 28q^{26} - 58q^{29} - 147q^{31} - 160q^{32} + 42q^{34} + 219q^{37} + 98q^{38} + 168q^{40} + 350q^{41} - 124q^{43} - 20q^{44} - 318q^{46} + 525q^{47} + 152q^{50} - 56q^{52} - 303q^{53} + 35q^{55} + 116q^{58} - 105q^{59} + 413q^{61} + 294q^{62} + 448q^{64} + 98q^{65} + 415q^{67} + 84q^{68} + 432q^{71} + 1113q^{73} - 438q^{74} + 196q^{76} - 103q^{79} - 112q^{80} - 700q^{82} + 1092q^{83} - 147q^{85} + 248q^{86} + 120q^{88} - 329q^{89} - 636q^{92} - 1050q^{94} - 343q^{95} + 882q^{97} + O(q^{100}) \)

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
0
−2.00000 0 −4.00000 7.00000 0 0 24.0000 0 −14.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.a.e 1
3.b odd 2 1 49.4.a.c 1
7.b odd 2 1 441.4.a.d 1
7.c even 3 2 441.4.e.k 2
7.d odd 6 2 63.4.e.b 2
12.b even 2 1 784.4.a.r 1
15.d odd 2 1 1225.4.a.d 1
21.c even 2 1 49.4.a.d 1
21.g even 6 2 7.4.c.a 2
21.h odd 6 2 49.4.c.a 2
84.h odd 2 1 784.4.a.b 1
84.j odd 6 2 112.4.i.c 2
105.g even 2 1 1225.4.a.c 1
105.p even 6 2 175.4.e.a 2
105.w odd 12 4 175.4.k.a 4
168.ba even 6 2 448.4.i.f 2
168.be odd 6 2 448.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 21.g even 6 2
49.4.a.c 1 3.b odd 2 1
49.4.a.d 1 21.c even 2 1
49.4.c.a 2 21.h odd 6 2
63.4.e.b 2 7.d odd 6 2
112.4.i.c 2 84.j odd 6 2
175.4.e.a 2 105.p even 6 2
175.4.k.a 4 105.w odd 12 4
441.4.a.d 1 7.b odd 2 1
441.4.a.e 1 1.a even 1 1 trivial
441.4.e.k 2 7.c even 3 2
448.4.i.a 2 168.be odd 6 2
448.4.i.f 2 168.ba even 6 2
784.4.a.b 1 84.h odd 2 1
784.4.a.r 1 12.b even 2 1
1225.4.a.c 1 105.g even 2 1
1225.4.a.d 1 15.d odd 2 1

Hecke kernels

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

\( T_{2} + 2 \)
\( T_{5} - 7 \)
\( T_{13} - 14 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T \)
$3$ \( T \)
$5$ \( -7 + T \)
$7$ \( T \)
$11$ \( -5 + T \)
$13$ \( -14 + T \)
$17$ \( 21 + T \)
$19$ \( 49 + T \)
$23$ \( -159 + T \)
$29$ \( 58 + T \)
$31$ \( 147 + T \)
$37$ \( -219 + T \)
$41$ \( -350 + T \)
$43$ \( 124 + T \)
$47$ \( -525 + T \)
$53$ \( 303 + T \)
$59$ \( 105 + T \)
$61$ \( -413 + T \)
$67$ \( -415 + T \)
$71$ \( -432 + T \)
$73$ \( -1113 + T \)
$79$ \( 103 + T \)
$83$ \( -1092 + T \)
$89$ \( 329 + T \)
$97$ \( -882 + T \)
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