Properties

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

Related objects

Downloads

Learn more

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

\(f(q)\) \(=\) \( q - 2 q^{2} + 4 q^{4} - 5 q^{5} + 8 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 4 q^{4} - 5 q^{5} + 8 q^{7} - 8 q^{8} + 10 q^{10} - 18 q^{11} + 8 q^{13} - 16 q^{14} + 16 q^{16} - 15 q^{17} + 23 q^{19} - 20 q^{20} + 36 q^{22} - 63 q^{23} + 25 q^{25} - 16 q^{26} + 32 q^{28} - 156 q^{29} - 85 q^{31} - 32 q^{32} + 30 q^{34} - 40 q^{35} + 74 q^{37} - 46 q^{38} + 40 q^{40} - 246 q^{41} - 190 q^{43} - 72 q^{44} + 126 q^{46} - 288 q^{47} - 279 q^{49} - 50 q^{50} + 32 q^{52} + 177 q^{53} + 90 q^{55} - 64 q^{56} + 312 q^{58} - 792 q^{59} - 907 q^{61} + 170 q^{62} + 64 q^{64} - 40 q^{65} - 322 q^{67} - 60 q^{68} + 80 q^{70} + 270 q^{71} + 254 q^{73} - 148 q^{74} + 92 q^{76} - 144 q^{77} - 1123 q^{79} - 80 q^{80} + 492 q^{82} + 771 q^{83} + 75 q^{85} + 380 q^{86} + 144 q^{88} + 198 q^{89} + 64 q^{91} - 252 q^{92} + 576 q^{94} - 115 q^{95} - 1192 q^{97} + 558 q^{98}+O(q^{100}) \) 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 8.00000 −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.b 1
3.b odd 2 1 270.4.a.l yes 1
4.b odd 2 1 2160.4.a.c 1
5.b even 2 1 1350.4.a.t 1
5.c odd 4 2 1350.4.c.g 2
9.c even 3 2 810.4.e.v 2
9.d odd 6 2 810.4.e.b 2
12.b even 2 1 2160.4.a.m 1
15.d odd 2 1 1350.4.a.f 1
15.e even 4 2 1350.4.c.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.b 1 1.a even 1 1 trivial
270.4.a.l yes 1 3.b odd 2 1
810.4.e.b 2 9.d odd 6 2
810.4.e.v 2 9.c even 3 2
1350.4.a.f 1 15.d odd 2 1
1350.4.a.t 1 5.b even 2 1
1350.4.c.g 2 5.c odd 4 2
1350.4.c.n 2 15.e even 4 2
2160.4.a.c 1 4.b odd 2 1
2160.4.a.m 1 12.b even 2 1

Hecke kernels

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

\( T_{7} - 8 \) Copy content Toggle raw display
\( T_{11} + 18 \) 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
$7$ \( T - 8 \) Copy content Toggle raw display
$11$ \( T + 18 \) Copy content Toggle raw display
$13$ \( T - 8 \) Copy content Toggle raw display
$17$ \( T + 15 \) Copy content Toggle raw display
$19$ \( T - 23 \) Copy content Toggle raw display
$23$ \( T + 63 \) Copy content Toggle raw display
$29$ \( T + 156 \) Copy content Toggle raw display
$31$ \( T + 85 \) Copy content Toggle raw display
$37$ \( T - 74 \) Copy content Toggle raw display
$41$ \( T + 246 \) Copy content Toggle raw display
$43$ \( T + 190 \) Copy content Toggle raw display
$47$ \( T + 288 \) Copy content Toggle raw display
$53$ \( T - 177 \) Copy content Toggle raw display
$59$ \( T + 792 \) Copy content Toggle raw display
$61$ \( T + 907 \) Copy content Toggle raw display
$67$ \( T + 322 \) Copy content Toggle raw display
$71$ \( T - 270 \) Copy content Toggle raw display
$73$ \( T - 254 \) Copy content Toggle raw display
$79$ \( T + 1123 \) Copy content Toggle raw display
$83$ \( T - 771 \) Copy content Toggle raw display
$89$ \( T - 198 \) Copy content Toggle raw display
$97$ \( T + 1192 \) Copy content Toggle raw display
show more
show less