Properties

Label 225.4.a.f
Level $225$
Weight $4$
Character orbit 225.a
Self dual yes
Analytic conductor $13.275$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.2754297513\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 7q^{4} + 24q^{7} - 15q^{8} + O(q^{10}) \) \( q + q^{2} - 7q^{4} + 24q^{7} - 15q^{8} - 52q^{11} - 22q^{13} + 24q^{14} + 41q^{16} - 14q^{17} - 20q^{19} - 52q^{22} - 168q^{23} - 22q^{26} - 168q^{28} - 230q^{29} - 288q^{31} + 161q^{32} - 14q^{34} + 34q^{37} - 20q^{38} - 122q^{41} + 188q^{43} + 364q^{44} - 168q^{46} + 256q^{47} + 233q^{49} + 154q^{52} - 338q^{53} - 360q^{56} - 230q^{58} - 100q^{59} + 742q^{61} - 288q^{62} - 167q^{64} + 84q^{67} + 98q^{68} + 328q^{71} + 38q^{73} + 34q^{74} + 140q^{76} - 1248q^{77} - 240q^{79} - 122q^{82} + 1212q^{83} + 188q^{86} + 780q^{88} - 330q^{89} - 528q^{91} + 1176q^{92} + 256q^{94} - 866q^{97} + 233q^{98} + O(q^{100}) \)

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 0 0 24.0000 −15.0000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

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 225.4.a.f 1
3.b odd 2 1 75.4.a.b 1
5.b even 2 1 45.4.a.c 1
5.c odd 4 2 225.4.b.e 2
12.b even 2 1 1200.4.a.t 1
15.d odd 2 1 15.4.a.a 1
15.e even 4 2 75.4.b.b 2
20.d odd 2 1 720.4.a.n 1
35.c odd 2 1 2205.4.a.l 1
45.h odd 6 2 405.4.e.g 2
45.j even 6 2 405.4.e.i 2
60.h even 2 1 240.4.a.e 1
60.l odd 4 2 1200.4.f.b 2
105.g even 2 1 735.4.a.e 1
120.i odd 2 1 960.4.a.b 1
120.m even 2 1 960.4.a.ba 1
165.d even 2 1 1815.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 15.d odd 2 1
45.4.a.c 1 5.b even 2 1
75.4.a.b 1 3.b odd 2 1
75.4.b.b 2 15.e even 4 2
225.4.a.f 1 1.a even 1 1 trivial
225.4.b.e 2 5.c odd 4 2
240.4.a.e 1 60.h even 2 1
405.4.e.g 2 45.h odd 6 2
405.4.e.i 2 45.j even 6 2
720.4.a.n 1 20.d odd 2 1
735.4.a.e 1 105.g even 2 1
960.4.a.b 1 120.i odd 2 1
960.4.a.ba 1 120.m even 2 1
1200.4.a.t 1 12.b even 2 1
1200.4.f.b 2 60.l odd 4 2
1815.4.a.e 1 165.d even 2 1
2205.4.a.l 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} - 1 \)
\( T_{7} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -24 + T \)
$11$ \( 52 + T \)
$13$ \( 22 + T \)
$17$ \( 14 + T \)
$19$ \( 20 + T \)
$23$ \( 168 + T \)
$29$ \( 230 + T \)
$31$ \( 288 + T \)
$37$ \( -34 + T \)
$41$ \( 122 + T \)
$43$ \( -188 + T \)
$47$ \( -256 + T \)
$53$ \( 338 + T \)
$59$ \( 100 + T \)
$61$ \( -742 + T \)
$67$ \( -84 + T \)
$71$ \( -328 + T \)
$73$ \( -38 + T \)
$79$ \( 240 + T \)
$83$ \( -1212 + T \)
$89$ \( 330 + T \)
$97$ \( 866 + T \)
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