Properties

Label 1350.4.a.u
Level $1350$
Weight $4$
Character orbit 1350.a
Self dual yes
Analytic conductor $79.653$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.6525785077\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 4 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 4 q^{7} + 8 q^{8} - 42 q^{11} - 20 q^{13} + 8 q^{14} + 16 q^{16} + 93 q^{17} + 59 q^{19} - 84 q^{22} + 9 q^{23} - 40 q^{26} + 16 q^{28} - 120 q^{29} + 47 q^{31} + 32 q^{32} + 186 q^{34} + 262 q^{37} + 118 q^{38} - 126 q^{41} + 178 q^{43} - 168 q^{44} + 18 q^{46} + 144 q^{47} - 327 q^{49} - 80 q^{52} + 741 q^{53} + 32 q^{56} - 240 q^{58} + 444 q^{59} + 221 q^{61} + 94 q^{62} + 64 q^{64} + 538 q^{67} + 372 q^{68} - 690 q^{71} + 1126 q^{73} + 524 q^{74} + 236 q^{76} - 168 q^{77} + 665 q^{79} - 252 q^{82} + 75 q^{83} + 356 q^{86} - 336 q^{88} + 1086 q^{89} - 80 q^{91} + 36 q^{92} + 288 q^{94} - 1544 q^{97} - 654 q^{98}+O(q^{100}) \) 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 4.00000 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.u 1
3.b odd 2 1 1350.4.a.g 1
5.b even 2 1 270.4.a.e 1
5.c odd 4 2 1350.4.c.d 2
15.d odd 2 1 270.4.a.i yes 1
15.e even 4 2 1350.4.c.q 2
20.d odd 2 1 2160.4.a.o 1
45.h odd 6 2 810.4.e.h 2
45.j even 6 2 810.4.e.q 2
60.h even 2 1 2160.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.e 1 5.b even 2 1
270.4.a.i yes 1 15.d odd 2 1
810.4.e.h 2 45.h odd 6 2
810.4.e.q 2 45.j even 6 2
1350.4.a.g 1 3.b odd 2 1
1350.4.a.u 1 1.a even 1 1 trivial
1350.4.c.d 2 5.c odd 4 2
1350.4.c.q 2 15.e even 4 2
2160.4.a.e 1 60.h even 2 1
2160.4.a.o 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1350))\):

\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} + 42 \) Copy content Toggle raw display
\( T_{17} - 93 \) 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
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T + 42 \) Copy content Toggle raw display
$13$ \( T + 20 \) Copy content Toggle raw display
$17$ \( T - 93 \) Copy content Toggle raw display
$19$ \( T - 59 \) Copy content Toggle raw display
$23$ \( T - 9 \) Copy content Toggle raw display
$29$ \( T + 120 \) Copy content Toggle raw display
$31$ \( T - 47 \) Copy content Toggle raw display
$37$ \( T - 262 \) Copy content Toggle raw display
$41$ \( T + 126 \) Copy content Toggle raw display
$43$ \( T - 178 \) Copy content Toggle raw display
$47$ \( T - 144 \) Copy content Toggle raw display
$53$ \( T - 741 \) Copy content Toggle raw display
$59$ \( T - 444 \) Copy content Toggle raw display
$61$ \( T - 221 \) Copy content Toggle raw display
$67$ \( T - 538 \) Copy content Toggle raw display
$71$ \( T + 690 \) Copy content Toggle raw display
$73$ \( T - 1126 \) Copy content Toggle raw display
$79$ \( T - 665 \) Copy content Toggle raw display
$83$ \( T - 75 \) Copy content Toggle raw display
$89$ \( T - 1086 \) Copy content Toggle raw display
$97$ \( T + 1544 \) Copy content Toggle raw display
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