Properties

Label 1225.4.a.h
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.2773397570\)
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

\(f(q)\) \(=\) \( q - q^{2} + 7 q^{3} - 7 q^{4} - 7 q^{6} + 15 q^{8} + 22 q^{9} + O(q^{10}) \) \( q - q^{2} + 7 q^{3} - 7 q^{4} - 7 q^{6} + 15 q^{8} + 22 q^{9} - 43 q^{11} - 49 q^{12} - 28 q^{13} + 41 q^{16} + 91 q^{17} - 22 q^{18} + 35 q^{19} + 43 q^{22} - 162 q^{23} + 105 q^{24} + 28 q^{26} - 35 q^{27} + 160 q^{29} - 42 q^{31} - 161 q^{32} - 301 q^{33} - 91 q^{34} - 154 q^{36} + 314 q^{37} - 35 q^{38} - 196 q^{39} + 203 q^{41} - 92 q^{43} + 301 q^{44} + 162 q^{46} + 196 q^{47} + 287 q^{48} + 637 q^{51} + 196 q^{52} - 82 q^{53} + 35 q^{54} + 245 q^{57} - 160 q^{58} + 280 q^{59} + 518 q^{61} + 42 q^{62} - 167 q^{64} + 301 q^{66} - 141 q^{67} - 637 q^{68} - 1134 q^{69} + 412 q^{71} + 330 q^{72} - 763 q^{73} - 314 q^{74} - 245 q^{76} + 196 q^{78} + 510 q^{79} - 839 q^{81} - 203 q^{82} + 777 q^{83} + 92 q^{86} + 1120 q^{87} - 645 q^{88} + 945 q^{89} + 1134 q^{92} - 294 q^{93} - 196 q^{94} - 1127 q^{96} + 1246 q^{97} - 946 q^{99} + O(q^{100}) \)

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 7.00000 −7.00000 0 −7.00000 0 15.0000 22.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)

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 1225.4.a.h 1
5.b even 2 1 1225.4.a.i 1
7.b odd 2 1 25.4.a.a 1
21.c even 2 1 225.4.a.e 1
28.d even 2 1 400.4.a.s 1
35.c odd 2 1 25.4.a.b yes 1
35.f even 4 2 25.4.b.b 2
56.e even 2 1 1600.4.a.h 1
56.h odd 2 1 1600.4.a.bt 1
105.g even 2 1 225.4.a.c 1
105.k odd 4 2 225.4.b.f 2
140.c even 2 1 400.4.a.c 1
140.j odd 4 2 400.4.c.e 2
280.c odd 2 1 1600.4.a.i 1
280.n 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 7.b odd 2 1
25.4.a.b yes 1 35.c odd 2 1
25.4.b.b 2 35.f even 4 2
225.4.a.c 1 105.g even 2 1
225.4.a.e 1 21.c even 2 1
225.4.b.f 2 105.k odd 4 2
400.4.a.c 1 140.c even 2 1
400.4.a.s 1 28.d even 2 1
400.4.c.e 2 140.j odd 4 2
1225.4.a.h 1 1.a even 1 1 trivial
1225.4.a.i 1 5.b even 2 1
1600.4.a.h 1 56.e even 2 1
1600.4.a.i 1 280.c odd 2 1
1600.4.a.bs 1 280.n even 2 1
1600.4.a.bt 1 56.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2} + 1 \)
\( T_{3} - 7 \)
\( T_{19} - 35 \)

Hecke characteristic polynomials

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