Properties

Label 1350.4.a.ba
Level $1350$
Weight $4$
Character orbit 1350.a
Self dual yes
Analytic conductor $79.653$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.6525785077\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 23 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 23 q^{7} + 8 q^{8} - 30 q^{11} - 34 q^{13} + 46 q^{14} + 16 q^{16} + 42 q^{17} - 139 q^{19} - 60 q^{22} - 192 q^{23} - 68 q^{26} + 92 q^{28} - 234 q^{29} - 55 q^{31} + 32 q^{32} + 84 q^{34} + 191 q^{37} - 278 q^{38} - 138 q^{41} + 53 q^{43} - 120 q^{44} - 384 q^{46} - 366 q^{47} + 186 q^{49} - 136 q^{52} + 330 q^{53} + 184 q^{56} - 468 q^{58} + 396 q^{59} + 23 q^{61} - 110 q^{62} + 64 q^{64} + 452 q^{67} + 168 q^{68} - 204 q^{71} - 691 q^{73} + 382 q^{74} - 556 q^{76} - 690 q^{77} - 709 q^{79} - 276 q^{82} - 1098 q^{83} + 106 q^{86} - 240 q^{88} + 816 q^{89} - 782 q^{91} - 768 q^{92} - 732 q^{94} + 905 q^{97} + 372 q^{98}+O(q^{100}) \) 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 4.00000 0 0 23.0000 8.00000 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 1350.4.a.ba yes 1
3.b odd 2 1 1350.4.a.m yes 1
5.b even 2 1 1350.4.a.b 1
5.c odd 4 2 1350.4.c.e 2
15.d odd 2 1 1350.4.a.p yes 1
15.e even 4 2 1350.4.c.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1350.4.a.b 1 5.b even 2 1
1350.4.a.m yes 1 3.b odd 2 1
1350.4.a.p yes 1 15.d odd 2 1
1350.4.a.ba yes 1 1.a even 1 1 trivial
1350.4.c.e 2 5.c odd 4 2
1350.4.c.p 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1350))\):

\( T_{7} - 23 \) Copy content Toggle raw display
\( T_{11} + 30 \) Copy content Toggle raw display
\( T_{17} - 42 \) 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
$7$ \( T - 23 \) Copy content Toggle raw display
$11$ \( T + 30 \) Copy content Toggle raw display
$13$ \( T + 34 \) Copy content Toggle raw display
$17$ \( T - 42 \) Copy content Toggle raw display
$19$ \( T + 139 \) Copy content Toggle raw display
$23$ \( T + 192 \) Copy content Toggle raw display
$29$ \( T + 234 \) Copy content Toggle raw display
$31$ \( T + 55 \) Copy content Toggle raw display
$37$ \( T - 191 \) Copy content Toggle raw display
$41$ \( T + 138 \) Copy content Toggle raw display
$43$ \( T - 53 \) Copy content Toggle raw display
$47$ \( T + 366 \) Copy content Toggle raw display
$53$ \( T - 330 \) Copy content Toggle raw display
$59$ \( T - 396 \) Copy content Toggle raw display
$61$ \( T - 23 \) Copy content Toggle raw display
$67$ \( T - 452 \) Copy content Toggle raw display
$71$ \( T + 204 \) Copy content Toggle raw display
$73$ \( T + 691 \) Copy content Toggle raw display
$79$ \( T + 709 \) Copy content Toggle raw display
$83$ \( T + 1098 \) Copy content Toggle raw display
$89$ \( T - 816 \) Copy content Toggle raw display
$97$ \( T - 905 \) Copy content Toggle raw display
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