Properties

Label 25.4.a.a
Level 25
Weight 4
Character orbit 25.a
Self dual yes
Analytic conductor 1.475
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.47504775014\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 7q^{3} - 7q^{4} + 7q^{6} - 6q^{7} + 15q^{8} + 22q^{9} + O(q^{10}) \) \( q - q^{2} - 7q^{3} - 7q^{4} + 7q^{6} - 6q^{7} + 15q^{8} + 22q^{9} - 43q^{11} + 49q^{12} + 28q^{13} + 6q^{14} + 41q^{16} - 91q^{17} - 22q^{18} - 35q^{19} + 42q^{21} + 43q^{22} - 162q^{23} - 105q^{24} - 28q^{26} + 35q^{27} + 42q^{28} + 160q^{29} + 42q^{31} - 161q^{32} + 301q^{33} + 91q^{34} - 154q^{36} + 314q^{37} + 35q^{38} - 196q^{39} - 203q^{41} - 42q^{42} - 92q^{43} + 301q^{44} + 162q^{46} - 196q^{47} - 287q^{48} - 307q^{49} + 637q^{51} - 196q^{52} - 82q^{53} - 35q^{54} - 90q^{56} + 245q^{57} - 160q^{58} - 280q^{59} - 518q^{61} - 42q^{62} - 132q^{63} - 167q^{64} - 301q^{66} - 141q^{67} + 637q^{68} + 1134q^{69} + 412q^{71} + 330q^{72} + 763q^{73} - 314q^{74} + 245q^{76} + 258q^{77} + 196q^{78} + 510q^{79} - 839q^{81} + 203q^{82} - 777q^{83} - 294q^{84} + 92q^{86} - 1120q^{87} - 645q^{88} - 945q^{89} - 168q^{91} + 1134q^{92} - 294q^{93} + 196q^{94} + 1127q^{96} - 1246q^{97} + 307q^{98} - 946q^{99} + O(q^{100}) \)

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 −6.00000 15.0000 22.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 25.4.a.a 1
3.b odd 2 1 225.4.a.e 1
4.b odd 2 1 400.4.a.s 1
5.b even 2 1 25.4.a.b yes 1
5.c odd 4 2 25.4.b.b 2
7.b odd 2 1 1225.4.a.h 1
8.b even 2 1 1600.4.a.bt 1
8.d odd 2 1 1600.4.a.h 1
15.d odd 2 1 225.4.a.c 1
15.e even 4 2 225.4.b.f 2
20.d odd 2 1 400.4.a.c 1
20.e even 4 2 400.4.c.e 2
35.c odd 2 1 1225.4.a.i 1
40.e odd 2 1 1600.4.a.bs 1
40.f even 2 1 1600.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 1.a even 1 1 trivial
25.4.a.b yes 1 5.b even 2 1
25.4.b.b 2 5.c odd 4 2
225.4.a.c 1 15.d odd 2 1
225.4.a.e 1 3.b odd 2 1
225.4.b.f 2 15.e even 4 2
400.4.a.c 1 20.d odd 2 1
400.4.a.s 1 4.b odd 2 1
400.4.c.e 2 20.e even 4 2
1225.4.a.h 1 7.b odd 2 1
1225.4.a.i 1 35.c odd 2 1
1600.4.a.h 1 8.d odd 2 1
1600.4.a.i 1 40.f even 2 1
1600.4.a.bs 1 40.e odd 2 1
1600.4.a.bt 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 8 T^{2} \)
$3$ \( 1 + 7 T + 27 T^{2} \)
$5$ 1
$7$ \( 1 + 6 T + 343 T^{2} \)
$11$ \( 1 + 43 T + 1331 T^{2} \)
$13$ \( 1 - 28 T + 2197 T^{2} \)
$17$ \( 1 + 91 T + 4913 T^{2} \)
$19$ \( 1 + 35 T + 6859 T^{2} \)
$23$ \( 1 + 162 T + 12167 T^{2} \)
$29$ \( 1 - 160 T + 24389 T^{2} \)
$31$ \( 1 - 42 T + 29791 T^{2} \)
$37$ \( 1 - 314 T + 50653 T^{2} \)
$41$ \( 1 + 203 T + 68921 T^{2} \)
$43$ \( 1 + 92 T + 79507 T^{2} \)
$47$ \( 1 + 196 T + 103823 T^{2} \)
$53$ \( 1 + 82 T + 148877 T^{2} \)
$59$ \( 1 + 280 T + 205379 T^{2} \)
$61$ \( 1 + 518 T + 226981 T^{2} \)
$67$ \( 1 + 141 T + 300763 T^{2} \)
$71$ \( 1 - 412 T + 357911 T^{2} \)
$73$ \( 1 - 763 T + 389017 T^{2} \)
$79$ \( 1 - 510 T + 493039 T^{2} \)
$83$ \( 1 + 777 T + 571787 T^{2} \)
$89$ \( 1 + 945 T + 704969 T^{2} \)
$97$ \( 1 + 1246 T + 912673 T^{2} \)
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