Properties

Label 225.12.a.b
Level $225$
Weight $12$
Character orbit 225.a
Self dual yes
Analytic conductor $172.877$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.877215626\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 24 q^{2} - 1472 q^{4} + 16744 q^{7} + 84480 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 24 q^{2} - 1472 q^{4} + 16744 q^{7} + 84480 q^{8} - 534612 q^{11} + 577738 q^{13} - 401856 q^{14} + 987136 q^{16} - 6905934 q^{17} + 10661420 q^{19} + 12830688 q^{22} + 18643272 q^{23} - 13865712 q^{26} - 24647168 q^{28} - 128406630 q^{29} - 52843168 q^{31} - 196706304 q^{32} + 165742416 q^{34} + 182213314 q^{37} - 255874080 q^{38} - 308120442 q^{41} + 17125708 q^{43} + 786948864 q^{44} - 447438528 q^{46} + 2687348496 q^{47} - 1696965207 q^{49} - 850430336 q^{52} - 1596055698 q^{53} + 1414533120 q^{56} + 3081759120 q^{58} + 5189203740 q^{59} + 6956478662 q^{61} + 1268236032 q^{62} + 2699296768 q^{64} + 15481826884 q^{67} + 10165534848 q^{68} - 9791485272 q^{71} - 1463791322 q^{73} - 4373119536 q^{74} - 15693610240 q^{76} - 8951543328 q^{77} + 38116845680 q^{79} + 7394890608 q^{82} - 29335099668 q^{83} - 411016992 q^{86} - 45164021760 q^{88} + 24992917110 q^{89} + 9673645072 q^{91} - 27442896384 q^{92} - 64496363904 q^{94} - 75013568546 q^{97} + 40727164968 q^{98}+O(q^{100}) \) 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
−24.0000 0 −1472.00 0 0 16744.0 84480.0 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 225.12.a.b 1
3.b odd 2 1 25.12.a.b 1
5.b even 2 1 9.12.a.b 1
5.c odd 4 2 225.12.b.d 2
15.d odd 2 1 1.12.a.a 1
15.e even 4 2 25.12.b.b 2
20.d odd 2 1 144.12.a.d 1
45.h odd 6 2 81.12.c.d 2
45.j even 6 2 81.12.c.b 2
60.h even 2 1 16.12.a.a 1
105.g even 2 1 49.12.a.a 1
105.o odd 6 2 49.12.c.b 2
105.p even 6 2 49.12.c.c 2
120.i odd 2 1 64.12.a.b 1
120.m even 2 1 64.12.a.f 1
165.d even 2 1 121.12.a.b 1
195.e odd 2 1 169.12.a.a 1
240.t even 4 2 256.12.b.c 2
240.bm odd 4 2 256.12.b.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 15.d odd 2 1
9.12.a.b 1 5.b even 2 1
16.12.a.a 1 60.h even 2 1
25.12.a.b 1 3.b odd 2 1
25.12.b.b 2 15.e even 4 2
49.12.a.a 1 105.g even 2 1
49.12.c.b 2 105.o odd 6 2
49.12.c.c 2 105.p even 6 2
64.12.a.b 1 120.i odd 2 1
64.12.a.f 1 120.m even 2 1
81.12.c.b 2 45.j even 6 2
81.12.c.d 2 45.h odd 6 2
121.12.a.b 1 165.d even 2 1
144.12.a.d 1 20.d odd 2 1
169.12.a.a 1 195.e odd 2 1
225.12.a.b 1 1.a even 1 1 trivial
225.12.b.d 2 5.c odd 4 2
256.12.b.c 2 240.t even 4 2
256.12.b.e 2 240.bm odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 24 \) Copy content Toggle raw display
\( T_{7} - 16744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 24 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 16744 \) Copy content Toggle raw display
$11$ \( T + 534612 \) Copy content Toggle raw display
$13$ \( T - 577738 \) Copy content Toggle raw display
$17$ \( T + 6905934 \) Copy content Toggle raw display
$19$ \( T - 10661420 \) Copy content Toggle raw display
$23$ \( T - 18643272 \) Copy content Toggle raw display
$29$ \( T + 128406630 \) Copy content Toggle raw display
$31$ \( T + 52843168 \) Copy content Toggle raw display
$37$ \( T - 182213314 \) Copy content Toggle raw display
$41$ \( T + 308120442 \) Copy content Toggle raw display
$43$ \( T - 17125708 \) Copy content Toggle raw display
$47$ \( T - 2687348496 \) Copy content Toggle raw display
$53$ \( T + 1596055698 \) Copy content Toggle raw display
$59$ \( T - 5189203740 \) Copy content Toggle raw display
$61$ \( T - 6956478662 \) Copy content Toggle raw display
$67$ \( T - 15481826884 \) Copy content Toggle raw display
$71$ \( T + 9791485272 \) Copy content Toggle raw display
$73$ \( T + 1463791322 \) Copy content Toggle raw display
$79$ \( T - 38116845680 \) Copy content Toggle raw display
$83$ \( T + 29335099668 \) Copy content Toggle raw display
$89$ \( T - 24992917110 \) Copy content Toggle raw display
$97$ \( T + 75013568546 \) Copy content Toggle raw display
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