Properties

Label 441.4.a.i
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

\(f(q)\) \(=\) \( q + q^{2} - 7q^{4} + 16q^{5} - 15q^{8} + O(q^{10}) \) \( q + q^{2} - 7q^{4} + 16q^{5} - 15q^{8} + 16q^{10} + 8q^{11} - 28q^{13} + 41q^{16} + 54q^{17} + 110q^{19} - 112q^{20} + 8q^{22} - 48q^{23} + 131q^{25} - 28q^{26} + 110q^{29} - 12q^{31} + 161q^{32} + 54q^{34} - 246q^{37} + 110q^{38} - 240q^{40} + 182q^{41} + 128q^{43} - 56q^{44} - 48q^{46} + 324q^{47} + 131q^{50} + 196q^{52} + 162q^{53} + 128q^{55} + 110q^{58} + 810q^{59} + 488q^{61} - 12q^{62} - 167q^{64} - 448q^{65} + 244q^{67} - 378q^{68} + 768q^{71} + 702q^{73} - 246q^{74} - 770q^{76} + 440q^{79} + 656q^{80} + 182q^{82} - 1302q^{83} + 864q^{85} + 128q^{86} - 120q^{88} + 730q^{89} + 336q^{92} + 324q^{94} + 1760q^{95} - 294q^{97} + O(q^{100}) \)

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
1.00000 0 −7.00000 16.0000 0 0 −15.0000 0 16.0000
\(n\): 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.i 1
3.b odd 2 1 49.4.a.b 1
7.b odd 2 1 63.4.a.b 1
7.c even 3 2 441.4.e.e 2
7.d odd 6 2 441.4.e.h 2
12.b even 2 1 784.4.a.g 1
15.d odd 2 1 1225.4.a.j 1
21.c even 2 1 7.4.a.a 1
21.g even 6 2 49.4.c.c 2
21.h odd 6 2 49.4.c.b 2
28.d even 2 1 1008.4.a.c 1
35.c odd 2 1 1575.4.a.e 1
84.h odd 2 1 112.4.a.f 1
105.g even 2 1 175.4.a.b 1
105.k odd 4 2 175.4.b.b 2
168.e odd 2 1 448.4.a.e 1
168.i even 2 1 448.4.a.i 1
231.h odd 2 1 847.4.a.b 1
273.g even 2 1 1183.4.a.b 1
357.c even 2 1 2023.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 21.c even 2 1
49.4.a.b 1 3.b odd 2 1
49.4.c.b 2 21.h odd 6 2
49.4.c.c 2 21.g even 6 2
63.4.a.b 1 7.b odd 2 1
112.4.a.f 1 84.h odd 2 1
175.4.a.b 1 105.g even 2 1
175.4.b.b 2 105.k odd 4 2
441.4.a.i 1 1.a even 1 1 trivial
441.4.e.e 2 7.c even 3 2
441.4.e.h 2 7.d odd 6 2
448.4.a.e 1 168.e odd 2 1
448.4.a.i 1 168.i even 2 1
784.4.a.g 1 12.b even 2 1
847.4.a.b 1 231.h odd 2 1
1008.4.a.c 1 28.d even 2 1
1183.4.a.b 1 273.g even 2 1
1225.4.a.j 1 15.d odd 2 1
1575.4.a.e 1 35.c odd 2 1
2023.4.a.a 1 357.c even 2 1

Hecke kernels

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

\( T_{2} - 1 \)
\( T_{5} - 16 \)
\( T_{13} + 28 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( -16 + T \)
$7$ \( T \)
$11$ \( -8 + T \)
$13$ \( 28 + T \)
$17$ \( -54 + T \)
$19$ \( -110 + T \)
$23$ \( 48 + T \)
$29$ \( -110 + T \)
$31$ \( 12 + T \)
$37$ \( 246 + T \)
$41$ \( -182 + T \)
$43$ \( -128 + T \)
$47$ \( -324 + T \)
$53$ \( -162 + T \)
$59$ \( -810 + T \)
$61$ \( -488 + T \)
$67$ \( -244 + T \)
$71$ \( -768 + T \)
$73$ \( -702 + T \)
$79$ \( -440 + T \)
$83$ \( 1302 + T \)
$89$ \( -730 + T \)
$97$ \( 294 + T \)
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