Properties

Label 225.12.a.f
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 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 78 q^{2} + 4036 q^{4} + 27760 q^{7} + 155064 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 78 q^{2} + 4036 q^{4} + 27760 q^{7} + 155064 q^{8} - 637836 q^{11} - 766214 q^{13} + 2165280 q^{14} + 3829264 q^{16} + 3084354 q^{17} - 19511404 q^{19} - 49751208 q^{22} + 15312360 q^{23} - 59764692 q^{26} + 112039360 q^{28} - 10751262 q^{29} - 50937400 q^{31} - 18888480 q^{32} + 240579612 q^{34} - 664740830 q^{37} - 1521889512 q^{38} - 898833450 q^{41} + 957947188 q^{43} - 2574306096 q^{44} + 1194364080 q^{46} - 1555741344 q^{47} - 1206709143 q^{49} - 3092439704 q^{52} + 3792417030 q^{53} + 4304576640 q^{56} - 838598436 q^{58} - 555306924 q^{59} + 4950420998 q^{61} - 3973117200 q^{62} - 9315634112 q^{64} - 5292399284 q^{67} + 12448452744 q^{68} + 14831086248 q^{71} - 13971005210 q^{73} - 51849784740 q^{74} - 78748026544 q^{76} - 17706327360 q^{77} + 3720542360 q^{79} - 70109009100 q^{82} + 8768454036 q^{83} + 74719880664 q^{86} - 98905401504 q^{88} + 25472769174 q^{89} - 21270100640 q^{91} + 61800684960 q^{92} - 121347824832 q^{94} + 39092494846 q^{97} - 94123313154 q^{98}+O(q^{100}) \) 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
78.0000 0 4036.00 0 0 27760.0 155064. 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.f 1
3.b odd 2 1 75.12.a.a 1
5.b even 2 1 9.12.a.a 1
5.c odd 4 2 225.12.b.a 2
15.d odd 2 1 3.12.a.a 1
15.e even 4 2 75.12.b.a 2
20.d odd 2 1 144.12.a.l 1
45.h odd 6 2 81.12.c.a 2
45.j even 6 2 81.12.c.e 2
60.h even 2 1 48.12.a.f 1
105.g even 2 1 147.12.a.c 1
120.i odd 2 1 192.12.a.q 1
120.m even 2 1 192.12.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 15.d odd 2 1
9.12.a.a 1 5.b even 2 1
48.12.a.f 1 60.h even 2 1
75.12.a.a 1 3.b odd 2 1
75.12.b.a 2 15.e even 4 2
81.12.c.a 2 45.h odd 6 2
81.12.c.e 2 45.j even 6 2
144.12.a.l 1 20.d odd 2 1
147.12.a.c 1 105.g even 2 1
192.12.a.g 1 120.m even 2 1
192.12.a.q 1 120.i odd 2 1
225.12.a.f 1 1.a even 1 1 trivial
225.12.b.a 2 5.c odd 4 2

Hecke kernels

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

\( T_{2} - 78 \) Copy content Toggle raw display
\( T_{7} - 27760 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 78 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 27760 \) Copy content Toggle raw display
$11$ \( T + 637836 \) Copy content Toggle raw display
$13$ \( T + 766214 \) Copy content Toggle raw display
$17$ \( T - 3084354 \) Copy content Toggle raw display
$19$ \( T + 19511404 \) Copy content Toggle raw display
$23$ \( T - 15312360 \) Copy content Toggle raw display
$29$ \( T + 10751262 \) Copy content Toggle raw display
$31$ \( T + 50937400 \) Copy content Toggle raw display
$37$ \( T + 664740830 \) Copy content Toggle raw display
$41$ \( T + 898833450 \) Copy content Toggle raw display
$43$ \( T - 957947188 \) Copy content Toggle raw display
$47$ \( T + 1555741344 \) Copy content Toggle raw display
$53$ \( T - 3792417030 \) Copy content Toggle raw display
$59$ \( T + 555306924 \) Copy content Toggle raw display
$61$ \( T - 4950420998 \) Copy content Toggle raw display
$67$ \( T + 5292399284 \) Copy content Toggle raw display
$71$ \( T - 14831086248 \) Copy content Toggle raw display
$73$ \( T + 13971005210 \) Copy content Toggle raw display
$79$ \( T - 3720542360 \) Copy content Toggle raw display
$83$ \( T - 8768454036 \) Copy content Toggle raw display
$89$ \( T - 25472769174 \) Copy content Toggle raw display
$97$ \( T - 39092494846 \) Copy content Toggle raw display
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