Properties

Label 225.4.a.g
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

\(f(q)\) \(=\) \( q + 3 q^{2} + q^{4} - 20 q^{7} - 21 q^{8} + O(q^{10}) \) \( q + 3 q^{2} + q^{4} - 20 q^{7} - 21 q^{8} + 24 q^{11} - 74 q^{13} - 60 q^{14} - 71 q^{16} + 54 q^{17} - 124 q^{19} + 72 q^{22} - 120 q^{23} - 222 q^{26} - 20 q^{28} + 78 q^{29} + 200 q^{31} - 45 q^{32} + 162 q^{34} + 70 q^{37} - 372 q^{38} - 330 q^{41} - 92 q^{43} + 24 q^{44} - 360 q^{46} - 24 q^{47} + 57 q^{49} - 74 q^{52} + 450 q^{53} + 420 q^{56} + 234 q^{58} - 24 q^{59} - 322 q^{61} + 600 q^{62} + 433 q^{64} + 196 q^{67} + 54 q^{68} + 288 q^{71} + 430 q^{73} + 210 q^{74} - 124 q^{76} - 480 q^{77} - 520 q^{79} - 990 q^{82} + 156 q^{83} - 276 q^{86} - 504 q^{88} - 1026 q^{89} + 1480 q^{91} - 120 q^{92} - 72 q^{94} + 286 q^{97} + 171 q^{98} + O(q^{100}) \)

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
3.00000 0 1.00000 0 0 −20.0000 −21.0000 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.g 1
3.b odd 2 1 75.4.a.a 1
5.b even 2 1 45.4.a.b 1
5.c odd 4 2 225.4.b.d 2
12.b even 2 1 1200.4.a.o 1
15.d odd 2 1 15.4.a.b 1
15.e even 4 2 75.4.b.a 2
20.d odd 2 1 720.4.a.r 1
35.c odd 2 1 2205.4.a.c 1
45.h odd 6 2 405.4.e.d 2
45.j even 6 2 405.4.e.k 2
60.h even 2 1 240.4.a.f 1
60.l odd 4 2 1200.4.f.m 2
105.g even 2 1 735.4.a.i 1
120.i odd 2 1 960.4.a.bi 1
120.m even 2 1 960.4.a.l 1
165.d even 2 1 1815.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 15.d odd 2 1
45.4.a.b 1 5.b even 2 1
75.4.a.a 1 3.b odd 2 1
75.4.b.a 2 15.e even 4 2
225.4.a.g 1 1.a even 1 1 trivial
225.4.b.d 2 5.c odd 4 2
240.4.a.f 1 60.h even 2 1
405.4.e.d 2 45.h odd 6 2
405.4.e.k 2 45.j even 6 2
720.4.a.r 1 20.d odd 2 1
735.4.a.i 1 105.g even 2 1
960.4.a.l 1 120.m even 2 1
960.4.a.bi 1 120.i odd 2 1
1200.4.a.o 1 12.b even 2 1
1200.4.f.m 2 60.l odd 4 2
1815.4.a.a 1 165.d even 2 1
2205.4.a.c 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} - 3 \)
\( T_{7} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 20 + T \)
$11$ \( -24 + T \)
$13$ \( 74 + T \)
$17$ \( -54 + T \)
$19$ \( 124 + T \)
$23$ \( 120 + T \)
$29$ \( -78 + T \)
$31$ \( -200 + T \)
$37$ \( -70 + T \)
$41$ \( 330 + T \)
$43$ \( 92 + T \)
$47$ \( 24 + T \)
$53$ \( -450 + T \)
$59$ \( 24 + T \)
$61$ \( 322 + T \)
$67$ \( -196 + T \)
$71$ \( -288 + T \)
$73$ \( -430 + T \)
$79$ \( 520 + T \)
$83$ \( -156 + T \)
$89$ \( 1026 + T \)
$97$ \( -286 + T \)
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