Properties

Label 1521.4.a.k
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4 q^{2} + 8 q^{4} + 17 q^{5} - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 8 q^{4} + 17 q^{5} - 20 q^{7} + 68 q^{10} - 32 q^{11} - 80 q^{14} - 64 q^{16} + 13 q^{17} - 30 q^{19} + 136 q^{20} - 128 q^{22} - 78 q^{23} + 164 q^{25} - 160 q^{28} - 197 q^{29} + 74 q^{31} - 256 q^{32} + 52 q^{34} - 340 q^{35} + 227 q^{37} - 120 q^{38} - 165 q^{41} - 156 q^{43} - 256 q^{44} - 312 q^{46} - 162 q^{47} + 57 q^{49} + 656 q^{50} - 93 q^{53} - 544 q^{55} - 788 q^{58} - 864 q^{59} + 145 q^{61} + 296 q^{62} - 512 q^{64} - 862 q^{67} + 104 q^{68} - 1360 q^{70} + 654 q^{71} - 215 q^{73} + 908 q^{74} - 240 q^{76} + 640 q^{77} - 76 q^{79} - 1088 q^{80} - 660 q^{82} + 628 q^{83} + 221 q^{85} - 624 q^{86} - 266 q^{89} - 624 q^{92} - 648 q^{94} - 510 q^{95} - 238 q^{97} + 228 q^{98}+O(q^{100}) \) 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
4.00000 0 8.00000 17.0000 0 −20.0000 0 0 68.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)

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 1521.4.a.k 1
3.b odd 2 1 169.4.a.a 1
13.b even 2 1 1521.4.a.b 1
13.e even 6 2 117.4.g.c 2
39.d odd 2 1 169.4.a.d 1
39.f even 4 2 169.4.b.c 2
39.h odd 6 2 13.4.c.a 2
39.i odd 6 2 169.4.c.d 2
39.k even 12 4 169.4.e.c 4
156.r even 6 2 208.4.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a 2 39.h odd 6 2
117.4.g.c 2 13.e even 6 2
169.4.a.a 1 3.b odd 2 1
169.4.a.d 1 39.d odd 2 1
169.4.b.c 2 39.f even 4 2
169.4.c.d 2 39.i odd 6 2
169.4.e.c 4 39.k even 12 4
208.4.i.b 2 156.r even 6 2
1521.4.a.b 1 13.b even 2 1
1521.4.a.k 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2} - 4 \) Copy content Toggle raw display
\( T_{5} - 17 \) Copy content Toggle raw display
\( T_{7} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 17 \) Copy content Toggle raw display
$7$ \( T + 20 \) Copy content Toggle raw display
$11$ \( T + 32 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 13 \) Copy content Toggle raw display
$19$ \( T + 30 \) Copy content Toggle raw display
$23$ \( T + 78 \) Copy content Toggle raw display
$29$ \( T + 197 \) Copy content Toggle raw display
$31$ \( T - 74 \) Copy content Toggle raw display
$37$ \( T - 227 \) Copy content Toggle raw display
$41$ \( T + 165 \) Copy content Toggle raw display
$43$ \( T + 156 \) Copy content Toggle raw display
$47$ \( T + 162 \) Copy content Toggle raw display
$53$ \( T + 93 \) Copy content Toggle raw display
$59$ \( T + 864 \) Copy content Toggle raw display
$61$ \( T - 145 \) Copy content Toggle raw display
$67$ \( T + 862 \) Copy content Toggle raw display
$71$ \( T - 654 \) Copy content Toggle raw display
$73$ \( T + 215 \) Copy content Toggle raw display
$79$ \( T + 76 \) Copy content Toggle raw display
$83$ \( T - 628 \) Copy content Toggle raw display
$89$ \( T + 266 \) Copy content Toggle raw display
$97$ \( T + 238 \) Copy content Toggle raw display
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