Properties

Label 1350.2.a.r
Level $1350$
Weight $2$
Character orbit 1350.a
Self dual yes
Analytic conductor $10.780$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{7} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} + q^{7} + q^{8} - 3q^{11} + 4q^{13} + q^{14} + q^{16} + 2q^{19} - 3q^{22} + 6q^{23} + 4q^{26} + q^{28} + 6q^{29} + 5q^{31} + q^{32} - 2q^{37} + 2q^{38} - 6q^{41} + 10q^{43} - 3q^{44} + 6q^{46} - 6q^{47} - 6q^{49} + 4q^{52} - 9q^{53} + q^{56} + 6q^{58} + 12q^{59} + 8q^{61} + 5q^{62} + q^{64} - 14q^{67} + 7q^{73} - 2q^{74} + 2q^{76} - 3q^{77} + 8q^{79} - 6q^{82} + 3q^{83} + 10q^{86} - 3q^{88} - 18q^{89} + 4q^{91} + 6q^{92} - 6q^{94} + q^{97} - 6q^{98} + O(q^{100}) \)

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
1.00000 0 1.00000 0 0 1.00000 1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

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 1350.2.a.r 1
3.b odd 2 1 1350.2.a.h 1
5.b even 2 1 54.2.a.a 1
5.c odd 4 2 1350.2.c.b 2
15.d odd 2 1 54.2.a.b yes 1
15.e even 4 2 1350.2.c.k 2
20.d odd 2 1 432.2.a.g 1
35.c odd 2 1 2646.2.a.a 1
40.e odd 2 1 1728.2.a.d 1
40.f even 2 1 1728.2.a.c 1
45.h odd 6 2 162.2.c.b 2
45.j even 6 2 162.2.c.c 2
55.d odd 2 1 6534.2.a.bc 1
60.h even 2 1 432.2.a.b 1
65.d even 2 1 9126.2.a.u 1
105.g even 2 1 2646.2.a.bd 1
120.i odd 2 1 1728.2.a.y 1
120.m even 2 1 1728.2.a.z 1
165.d even 2 1 6534.2.a.b 1
180.n even 6 2 1296.2.i.o 2
180.p odd 6 2 1296.2.i.c 2
195.e odd 2 1 9126.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.2.a.a 1 5.b even 2 1
54.2.a.b yes 1 15.d odd 2 1
162.2.c.b 2 45.h odd 6 2
162.2.c.c 2 45.j even 6 2
432.2.a.b 1 60.h even 2 1
432.2.a.g 1 20.d odd 2 1
1296.2.i.c 2 180.p odd 6 2
1296.2.i.o 2 180.n even 6 2
1350.2.a.h 1 3.b odd 2 1
1350.2.a.r 1 1.a even 1 1 trivial
1350.2.c.b 2 5.c odd 4 2
1350.2.c.k 2 15.e even 4 2
1728.2.a.c 1 40.f even 2 1
1728.2.a.d 1 40.e odd 2 1
1728.2.a.y 1 120.i odd 2 1
1728.2.a.z 1 120.m even 2 1
2646.2.a.a 1 35.c odd 2 1
2646.2.a.bd 1 105.g even 2 1
6534.2.a.b 1 165.d even 2 1
6534.2.a.bc 1 55.d odd 2 1
9126.2.a.r 1 195.e odd 2 1
9126.2.a.u 1 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\):

\( T_{7} - 1 \)
\( T_{11} + 3 \)
\( T_{13} - 4 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( -1 + T \)
$11$ \( 3 + T \)
$13$ \( -4 + T \)
$17$ \( T \)
$19$ \( -2 + T \)
$23$ \( -6 + T \)
$29$ \( -6 + T \)
$31$ \( -5 + T \)
$37$ \( 2 + T \)
$41$ \( 6 + T \)
$43$ \( -10 + T \)
$47$ \( 6 + T \)
$53$ \( 9 + T \)
$59$ \( -12 + T \)
$61$ \( -8 + T \)
$67$ \( 14 + T \)
$71$ \( T \)
$73$ \( -7 + T \)
$79$ \( -8 + T \)
$83$ \( -3 + T \)
$89$ \( 18 + T \)
$97$ \( -1 + T \)
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