Properties

Label 1350.4.a.r
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: no (minimal twist has level 270)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} - 14 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} - 14 q^{7} + 8 q^{8} + 3 q^{11} - 47 q^{13} - 28 q^{14} + 16 q^{16} + 39 q^{17} + 32 q^{19} + 6 q^{22} + 99 q^{23} - 94 q^{26} - 56 q^{28} + 51 q^{29} + 83 q^{31} + 32 q^{32} + 78 q^{34} - 314 q^{37} + 64 q^{38} - 108 q^{41} - 299 q^{43} + 12 q^{44} + 198 q^{46} - 531 q^{47} - 147 q^{49} - 188 q^{52} - 564 q^{53} - 112 q^{56} + 102 q^{58} + 12 q^{59} + 230 q^{61} + 166 q^{62} + 64 q^{64} + 268 q^{67} + 156 q^{68} + 120 q^{71} - 1106 q^{73} - 628 q^{74} + 128 q^{76} - 42 q^{77} - 739 q^{79} - 216 q^{82} - 1086 q^{83} - 598 q^{86} + 24 q^{88} - 120 q^{89} + 658 q^{91} + 396 q^{92} - 1062 q^{94} + 1642 q^{97} - 294 q^{98}+O(q^{100}) \) 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 −14.0000 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.r 1
3.b odd 2 1 1350.4.a.e 1
5.b even 2 1 270.4.a.f 1
5.c odd 4 2 1350.4.c.k 2
15.d odd 2 1 270.4.a.j yes 1
15.e even 4 2 1350.4.c.j 2
20.d odd 2 1 2160.4.a.l 1
45.h odd 6 2 810.4.e.f 2
45.j even 6 2 810.4.e.n 2
60.h even 2 1 2160.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.f 1 5.b even 2 1
270.4.a.j yes 1 15.d odd 2 1
810.4.e.f 2 45.h odd 6 2
810.4.e.n 2 45.j even 6 2
1350.4.a.e 1 3.b odd 2 1
1350.4.a.r 1 1.a even 1 1 trivial
1350.4.c.j 2 15.e even 4 2
1350.4.c.k 2 5.c odd 4 2
2160.4.a.b 1 60.h even 2 1
2160.4.a.l 1 20.d odd 2 1

Hecke kernels

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

\( T_{7} + 14 \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{17} - 39 \) 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
$7$ \( T + 14 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T + 47 \) Copy content Toggle raw display
$17$ \( T - 39 \) Copy content Toggle raw display
$19$ \( T - 32 \) Copy content Toggle raw display
$23$ \( T - 99 \) Copy content Toggle raw display
$29$ \( T - 51 \) Copy content Toggle raw display
$31$ \( T - 83 \) Copy content Toggle raw display
$37$ \( T + 314 \) Copy content Toggle raw display
$41$ \( T + 108 \) Copy content Toggle raw display
$43$ \( T + 299 \) Copy content Toggle raw display
$47$ \( T + 531 \) Copy content Toggle raw display
$53$ \( T + 564 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 230 \) Copy content Toggle raw display
$67$ \( T - 268 \) Copy content Toggle raw display
$71$ \( T - 120 \) Copy content Toggle raw display
$73$ \( T + 1106 \) Copy content Toggle raw display
$79$ \( T + 739 \) Copy content Toggle raw display
$83$ \( T + 1086 \) Copy content Toggle raw display
$89$ \( T + 120 \) Copy content Toggle raw display
$97$ \( T - 1642 \) Copy content Toggle raw display
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