Properties

Label 225.6.a.f
Level 225
Weight 6
Character orbit 225.a
Self dual yes
Analytic conductor 36.086
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} - 28q^{4} - 192q^{7} - 120q^{8} + O(q^{10}) \) \( q + 2q^{2} - 28q^{4} - 192q^{7} - 120q^{8} + 148q^{11} - 286q^{13} - 384q^{14} + 656q^{16} - 1678q^{17} + 1060q^{19} + 296q^{22} + 2976q^{23} - 572q^{26} + 5376q^{28} + 3410q^{29} - 2448q^{31} + 5152q^{32} - 3356q^{34} - 182q^{37} + 2120q^{38} + 9398q^{41} + 1244q^{43} - 4144q^{44} + 5952q^{46} - 12088q^{47} + 20057q^{49} + 8008q^{52} + 23846q^{53} + 23040q^{56} + 6820q^{58} + 20020q^{59} + 32302q^{61} - 4896q^{62} - 10688q^{64} - 60972q^{67} + 46984q^{68} + 32648q^{71} + 38774q^{73} - 364q^{74} - 29680q^{76} - 28416q^{77} - 33360q^{79} + 18796q^{82} + 16716q^{83} + 2488q^{86} - 17760q^{88} - 101370q^{89} + 54912q^{91} - 83328q^{92} - 24176q^{94} + 119038q^{97} + 40114q^{98} + O(q^{100}) \)

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
2.00000 0 −28.0000 0 0 −192.000 −120.000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 225.6.a.f 1
3.b odd 2 1 25.6.a.a 1
5.b even 2 1 45.6.a.b 1
5.c odd 4 2 225.6.b.e 2
12.b even 2 1 400.6.a.g 1
15.d odd 2 1 5.6.a.a 1
15.e even 4 2 25.6.b.a 2
20.d odd 2 1 720.6.a.a 1
60.h even 2 1 80.6.a.e 1
60.l odd 4 2 400.6.c.j 2
105.g even 2 1 245.6.a.b 1
120.i odd 2 1 320.6.a.j 1
120.m even 2 1 320.6.a.g 1
165.d even 2 1 605.6.a.a 1
195.e odd 2 1 845.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 15.d odd 2 1
25.6.a.a 1 3.b odd 2 1
25.6.b.a 2 15.e even 4 2
45.6.a.b 1 5.b even 2 1
80.6.a.e 1 60.h even 2 1
225.6.a.f 1 1.a even 1 1 trivial
225.6.b.e 2 5.c odd 4 2
245.6.a.b 1 105.g even 2 1
320.6.a.g 1 120.m even 2 1
320.6.a.j 1 120.i odd 2 1
400.6.a.g 1 12.b even 2 1
400.6.c.j 2 60.l odd 4 2
605.6.a.a 1 165.d even 2 1
720.6.a.a 1 20.d odd 2 1
845.6.a.b 1 195.e odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} - 2 \)
\( T_{7} + 192 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 32 T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 + 192 T + 16807 T^{2} \)
$11$ \( 1 - 148 T + 161051 T^{2} \)
$13$ \( 1 + 286 T + 371293 T^{2} \)
$17$ \( 1 + 1678 T + 1419857 T^{2} \)
$19$ \( 1 - 1060 T + 2476099 T^{2} \)
$23$ \( 1 - 2976 T + 6436343 T^{2} \)
$29$ \( 1 - 3410 T + 20511149 T^{2} \)
$31$ \( 1 + 2448 T + 28629151 T^{2} \)
$37$ \( 1 + 182 T + 69343957 T^{2} \)
$41$ \( 1 - 9398 T + 115856201 T^{2} \)
$43$ \( 1 - 1244 T + 147008443 T^{2} \)
$47$ \( 1 + 12088 T + 229345007 T^{2} \)
$53$ \( 1 - 23846 T + 418195493 T^{2} \)
$59$ \( 1 - 20020 T + 714924299 T^{2} \)
$61$ \( 1 - 32302 T + 844596301 T^{2} \)
$67$ \( 1 + 60972 T + 1350125107 T^{2} \)
$71$ \( 1 - 32648 T + 1804229351 T^{2} \)
$73$ \( 1 - 38774 T + 2073071593 T^{2} \)
$79$ \( 1 + 33360 T + 3077056399 T^{2} \)
$83$ \( 1 - 16716 T + 3939040643 T^{2} \)
$89$ \( 1 + 101370 T + 5584059449 T^{2} \)
$97$ \( 1 - 119038 T + 8587340257 T^{2} \)
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