Properties

Label 270.4.a.e
Level $270$
Weight $4$
Character orbit 270.a
Self dual yes
Analytic conductor $15.931$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 270.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9305157015\)
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

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 4 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} + 5 q^{5} - 4 q^{7} - 8 q^{8} - 10 q^{10} - 42 q^{11} + 20 q^{13} + 8 q^{14} + 16 q^{16} - 93 q^{17} + 59 q^{19} + 20 q^{20} + 84 q^{22} - 9 q^{23} + 25 q^{25} - 40 q^{26} - 16 q^{28} - 120 q^{29} + 47 q^{31} - 32 q^{32} + 186 q^{34} - 20 q^{35} - 262 q^{37} - 118 q^{38} - 40 q^{40} - 126 q^{41} - 178 q^{43} - 168 q^{44} + 18 q^{46} - 144 q^{47} - 327 q^{49} - 50 q^{50} + 80 q^{52} - 741 q^{53} - 210 q^{55} + 32 q^{56} + 240 q^{58} + 444 q^{59} + 221 q^{61} - 94 q^{62} + 64 q^{64} + 100 q^{65} - 538 q^{67} - 372 q^{68} + 40 q^{70} - 690 q^{71} - 1126 q^{73} + 524 q^{74} + 236 q^{76} + 168 q^{77} + 665 q^{79} + 80 q^{80} + 252 q^{82} - 75 q^{83} - 465 q^{85} + 356 q^{86} + 336 q^{88} + 1086 q^{89} - 80 q^{91} - 36 q^{92} + 288 q^{94} + 295 q^{95} + 1544 q^{97} + 654 q^{98}+O(q^{100}) \) 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 5.00000 0 −4.00000 −8.00000 0 −10.0000
\(n\): 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 270.4.a.e 1
3.b odd 2 1 270.4.a.i yes 1
4.b odd 2 1 2160.4.a.o 1
5.b even 2 1 1350.4.a.u 1
5.c odd 4 2 1350.4.c.d 2
9.c even 3 2 810.4.e.q 2
9.d odd 6 2 810.4.e.h 2
12.b even 2 1 2160.4.a.e 1
15.d odd 2 1 1350.4.a.g 1
15.e even 4 2 1350.4.c.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.e 1 1.a even 1 1 trivial
270.4.a.i yes 1 3.b odd 2 1
810.4.e.h 2 9.d odd 6 2
810.4.e.q 2 9.c even 3 2
1350.4.a.g 1 15.d odd 2 1
1350.4.a.u 1 5.b even 2 1
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 12.b even 2 1
2160.4.a.o 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(270))\):

\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{11} + 42 \) 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
$5$ \( T - 5 \) 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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