Properties

Label 225.6.a.c
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 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{2} - 28 q^{4} + 132 q^{7} + 120 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 28 q^{4} + 132 q^{7} + 120 q^{8} - 472 q^{11} + 686 q^{13} - 264 q^{14} + 656 q^{16} - 1562 q^{17} - 2180 q^{19} + 944 q^{22} + 264 q^{23} - 1372 q^{26} - 3696 q^{28} - 170 q^{29} + 7272 q^{31} - 5152 q^{32} + 3124 q^{34} + 142 q^{37} + 4360 q^{38} + 16198 q^{41} + 10316 q^{43} + 13216 q^{44} - 528 q^{46} + 18568 q^{47} + 617 q^{49} - 19208 q^{52} + 21514 q^{53} + 15840 q^{56} + 340 q^{58} - 34600 q^{59} - 35738 q^{61} - 14544 q^{62} - 10688 q^{64} + 5772 q^{67} + 43736 q^{68} + 69088 q^{71} + 70526 q^{73} - 284 q^{74} + 61040 q^{76} - 62304 q^{77} + 47640 q^{79} - 32396 q^{82} + 74004 q^{83} - 20632 q^{86} - 56640 q^{88} + 90030 q^{89} + 90552 q^{91} - 7392 q^{92} - 37136 q^{94} + 33502 q^{97} - 1234 q^{98}+O(q^{100}) \) 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
−2.00000 0 −28.0000 0 0 132.000 120.000 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 225.6.a.c 1
3.b odd 2 1 75.6.a.c 1
5.b even 2 1 45.6.a.c 1
5.c odd 4 2 225.6.b.d 2
15.d odd 2 1 15.6.a.a 1
15.e even 4 2 75.6.b.d 2
20.d odd 2 1 720.6.a.w 1
60.h even 2 1 240.6.a.k 1
105.g even 2 1 735.6.a.a 1
120.i odd 2 1 960.6.a.v 1
120.m even 2 1 960.6.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.a.a 1 15.d odd 2 1
45.6.a.c 1 5.b even 2 1
75.6.a.c 1 3.b odd 2 1
75.6.b.d 2 15.e even 4 2
225.6.a.c 1 1.a even 1 1 trivial
225.6.b.d 2 5.c odd 4 2
240.6.a.k 1 60.h even 2 1
720.6.a.w 1 20.d odd 2 1
735.6.a.a 1 105.g even 2 1
960.6.a.m 1 120.m even 2 1
960.6.a.v 1 120.i odd 2 1

Hecke kernels

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

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{7} - 132 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 132 \) Copy content Toggle raw display
$11$ \( T + 472 \) Copy content Toggle raw display
$13$ \( T - 686 \) Copy content Toggle raw display
$17$ \( T + 1562 \) Copy content Toggle raw display
$19$ \( T + 2180 \) Copy content Toggle raw display
$23$ \( T - 264 \) Copy content Toggle raw display
$29$ \( T + 170 \) Copy content Toggle raw display
$31$ \( T - 7272 \) Copy content Toggle raw display
$37$ \( T - 142 \) Copy content Toggle raw display
$41$ \( T - 16198 \) Copy content Toggle raw display
$43$ \( T - 10316 \) Copy content Toggle raw display
$47$ \( T - 18568 \) Copy content Toggle raw display
$53$ \( T - 21514 \) Copy content Toggle raw display
$59$ \( T + 34600 \) Copy content Toggle raw display
$61$ \( T + 35738 \) Copy content Toggle raw display
$67$ \( T - 5772 \) Copy content Toggle raw display
$71$ \( T - 69088 \) Copy content Toggle raw display
$73$ \( T - 70526 \) Copy content Toggle raw display
$79$ \( T - 47640 \) Copy content Toggle raw display
$83$ \( T - 74004 \) Copy content Toggle raw display
$89$ \( T - 90030 \) Copy content Toggle raw display
$97$ \( T - 33502 \) Copy content Toggle raw display
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