Properties

Label 225.4.a.e
Level $225$
Weight $4$
Character orbit 225.a
Self dual yes
Analytic conductor $13.275$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 7 q^{4} - 6 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - 7 q^{4} - 6 q^{7} - 15 q^{8} + 43 q^{11} + 28 q^{13} - 6 q^{14} + 41 q^{16} + 91 q^{17} - 35 q^{19} + 43 q^{22} + 162 q^{23} + 28 q^{26} + 42 q^{28} - 160 q^{29} + 42 q^{31} + 161 q^{32} + 91 q^{34} + 314 q^{37} - 35 q^{38} + 203 q^{41} - 92 q^{43} - 301 q^{44} + 162 q^{46} + 196 q^{47} - 307 q^{49} - 196 q^{52} + 82 q^{53} + 90 q^{56} - 160 q^{58} + 280 q^{59} - 518 q^{61} + 42 q^{62} - 167 q^{64} - 141 q^{67} - 637 q^{68} - 412 q^{71} + 763 q^{73} + 314 q^{74} + 245 q^{76} - 258 q^{77} + 510 q^{79} + 203 q^{82} + 777 q^{83} - 92 q^{86} - 645 q^{88} + 945 q^{89} - 168 q^{91} - 1134 q^{92} + 196 q^{94} - 1246 q^{97} - 307 q^{98}+O(q^{100}) \) 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
0
1.00000 0 −7.00000 0 0 −6.00000 −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.e 1
3.b odd 2 1 25.4.a.a 1
5.b even 2 1 225.4.a.c 1
5.c odd 4 2 225.4.b.f 2
12.b even 2 1 400.4.a.s 1
15.d odd 2 1 25.4.a.b yes 1
15.e even 4 2 25.4.b.b 2
21.c even 2 1 1225.4.a.h 1
24.f even 2 1 1600.4.a.h 1
24.h odd 2 1 1600.4.a.bt 1
60.h even 2 1 400.4.a.c 1
60.l odd 4 2 400.4.c.e 2
105.g even 2 1 1225.4.a.i 1
120.i odd 2 1 1600.4.a.i 1
120.m even 2 1 1600.4.a.bs 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 3.b odd 2 1
25.4.a.b yes 1 15.d odd 2 1
25.4.b.b 2 15.e even 4 2
225.4.a.c 1 5.b even 2 1
225.4.a.e 1 1.a even 1 1 trivial
225.4.b.f 2 5.c odd 4 2
400.4.a.c 1 60.h even 2 1
400.4.a.s 1 12.b even 2 1
400.4.c.e 2 60.l odd 4 2
1225.4.a.h 1 21.c even 2 1
1225.4.a.i 1 105.g even 2 1
1600.4.a.h 1 24.f even 2 1
1600.4.a.i 1 120.i odd 2 1
1600.4.a.bs 1 120.m even 2 1
1600.4.a.bt 1 24.h odd 2 1

Hecke kernels

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

\( T_{2} - 1 \) Copy content Toggle raw display
\( T_{7} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 6 \) Copy content Toggle raw display
$11$ \( T - 43 \) Copy content Toggle raw display
$13$ \( T - 28 \) Copy content Toggle raw display
$17$ \( T - 91 \) Copy content Toggle raw display
$19$ \( T + 35 \) Copy content Toggle raw display
$23$ \( T - 162 \) Copy content Toggle raw display
$29$ \( T + 160 \) Copy content Toggle raw display
$31$ \( T - 42 \) Copy content Toggle raw display
$37$ \( T - 314 \) Copy content Toggle raw display
$41$ \( T - 203 \) Copy content Toggle raw display
$43$ \( T + 92 \) Copy content Toggle raw display
$47$ \( T - 196 \) Copy content Toggle raw display
$53$ \( T - 82 \) Copy content Toggle raw display
$59$ \( T - 280 \) Copy content Toggle raw display
$61$ \( T + 518 \) Copy content Toggle raw display
$67$ \( T + 141 \) Copy content Toggle raw display
$71$ \( T + 412 \) Copy content Toggle raw display
$73$ \( T - 763 \) Copy content Toggle raw display
$79$ \( T - 510 \) Copy content Toggle raw display
$83$ \( T - 777 \) Copy content Toggle raw display
$89$ \( T - 945 \) Copy content Toggle raw display
$97$ \( T + 1246 \) Copy content Toggle raw display
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