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

\(f(q)\) \(=\) \( q - 24q^{2} - 1472q^{4} + 16744q^{7} + 84480q^{8} + O(q^{10}) \) \( q - 24q^{2} - 1472q^{4} + 16744q^{7} + 84480q^{8} - 534612q^{11} + 577738q^{13} - 401856q^{14} + 987136q^{16} - 6905934q^{17} + 10661420q^{19} + 12830688q^{22} + 18643272q^{23} - 13865712q^{26} - 24647168q^{28} - 128406630q^{29} - 52843168q^{31} - 196706304q^{32} + 165742416q^{34} + 182213314q^{37} - 255874080q^{38} - 308120442q^{41} + 17125708q^{43} + 786948864q^{44} - 447438528q^{46} + 2687348496q^{47} - 1696965207q^{49} - 850430336q^{52} - 1596055698q^{53} + 1414533120q^{56} + 3081759120q^{58} + 5189203740q^{59} + 6956478662q^{61} + 1268236032q^{62} + 2699296768q^{64} + 15481826884q^{67} + 10165534848q^{68} - 9791485272q^{71} - 1463791322q^{73} - 4373119536q^{74} - 15693610240q^{76} - 8951543328q^{77} + 38116845680q^{79} + 7394890608q^{82} - 29335099668q^{83} - 411016992q^{86} - 45164021760q^{88} + 24992917110q^{89} + 9673645072q^{91} - 27442896384q^{92} - 64496363904q^{94} - 75013568546q^{97} + 40727164968q^{98} + O(q^{100}) \)

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
−24.0000 0 −1472.00 0 0 16744.0 84480.0 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 225.12.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

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 \(S_{12}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2} + 24 \)
\( T_{7} - 16744 \)