Properties

Label 1521.4.a.d
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q - 3 q^{2} + q^{4} + 9 q^{5} + 15 q^{7} + 21 q^{8} + O(q^{10}) \) \( q - 3 q^{2} + q^{4} + 9 q^{5} + 15 q^{7} + 21 q^{8} - 27 q^{10} + 48 q^{11} - 45 q^{14} - 71 q^{16} - 45 q^{17} + 6 q^{19} + 9 q^{20} - 144 q^{22} + 162 q^{23} - 44 q^{25} + 15 q^{28} + 144 q^{29} + 264 q^{31} + 45 q^{32} + 135 q^{34} + 135 q^{35} + 303 q^{37} - 18 q^{38} + 189 q^{40} + 192 q^{41} + 97 q^{43} + 48 q^{44} - 486 q^{46} - 111 q^{47} - 118 q^{49} + 132 q^{50} + 414 q^{53} + 432 q^{55} + 315 q^{56} - 432 q^{58} - 522 q^{59} + 376 q^{61} - 792 q^{62} + 433 q^{64} - 36 q^{67} - 45 q^{68} - 405 q^{70} - 357 q^{71} - 1098 q^{73} - 909 q^{74} + 6 q^{76} + 720 q^{77} - 830 q^{79} - 639 q^{80} - 576 q^{82} + 438 q^{83} - 405 q^{85} - 291 q^{86} + 1008 q^{88} + 438 q^{89} + 162 q^{92} + 333 q^{94} + 54 q^{95} - 852 q^{97} + 354 q^{98} + O(q^{100}) \)

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
−3.00000 0 1.00000 9.00000 0 15.0000 21.0000 0 −27.0000
\(n\): 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.d 1
3.b odd 2 1 169.4.a.c 1
13.b even 2 1 1521.4.a.i 1
13.d odd 4 2 117.4.b.a 2
39.d odd 2 1 169.4.a.b 1
39.f even 4 2 13.4.b.a 2
39.h odd 6 2 169.4.c.c 2
39.i odd 6 2 169.4.c.b 2
39.k even 12 4 169.4.e.d 4
156.l odd 4 2 208.4.f.b 2
195.j odd 4 2 325.4.d.b 2
195.n even 4 2 325.4.c.b 2
195.u odd 4 2 325.4.d.a 2
312.w odd 4 2 832.4.f.c 2
312.y even 4 2 832.4.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 39.f even 4 2
117.4.b.a 2 13.d odd 4 2
169.4.a.b 1 39.d odd 2 1
169.4.a.c 1 3.b odd 2 1
169.4.c.b 2 39.i odd 6 2
169.4.c.c 2 39.h odd 6 2
169.4.e.d 4 39.k even 12 4
208.4.f.b 2 156.l odd 4 2
325.4.c.b 2 195.n even 4 2
325.4.d.a 2 195.u odd 4 2
325.4.d.b 2 195.j odd 4 2
832.4.f.c 2 312.w odd 4 2
832.4.f.e 2 312.y even 4 2
1521.4.a.d 1 1.a even 1 1 trivial
1521.4.a.i 1 13.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2} + 3 \)
\( T_{5} - 9 \)
\( T_{7} - 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + T \)
$3$ \( T \)
$5$ \( -9 + T \)
$7$ \( -15 + T \)
$11$ \( -48 + T \)
$13$ \( T \)
$17$ \( 45 + T \)
$19$ \( -6 + T \)
$23$ \( -162 + T \)
$29$ \( -144 + T \)
$31$ \( -264 + T \)
$37$ \( -303 + T \)
$41$ \( -192 + T \)
$43$ \( -97 + T \)
$47$ \( 111 + T \)
$53$ \( -414 + T \)
$59$ \( 522 + T \)
$61$ \( -376 + T \)
$67$ \( 36 + T \)
$71$ \( 357 + T \)
$73$ \( 1098 + T \)
$79$ \( 830 + T \)
$83$ \( -438 + T \)
$89$ \( -438 + T \)
$97$ \( 852 + T \)
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