Properties

Label 1521.4.a.i
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

\(f(q)\) \(=\) \( q + 3 q^{2} + q^{4} - 9 q^{5} - 15 q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + q^{4} - 9 q^{5} - 15 q^{7} - 21 q^{8} - 27 q^{10} - 48 q^{11} - 45 q^{14} - 71 q^{16} - 45 q^{17} - 6 q^{19} - 9 q^{20} - 144 q^{22} + 162 q^{23} - 44 q^{25} - 15 q^{28} + 144 q^{29} - 264 q^{31} - 45 q^{32} - 135 q^{34} + 135 q^{35} - 303 q^{37} - 18 q^{38} + 189 q^{40} - 192 q^{41} + 97 q^{43} - 48 q^{44} + 486 q^{46} + 111 q^{47} - 118 q^{49} - 132 q^{50} + 414 q^{53} + 432 q^{55} + 315 q^{56} + 432 q^{58} + 522 q^{59} + 376 q^{61} - 792 q^{62} + 433 q^{64} + 36 q^{67} - 45 q^{68} + 405 q^{70} + 357 q^{71} + 1098 q^{73} - 909 q^{74} - 6 q^{76} + 720 q^{77} - 830 q^{79} + 639 q^{80} - 576 q^{82} - 438 q^{83} + 405 q^{85} + 291 q^{86} + 1008 q^{88} - 438 q^{89} + 162 q^{92} + 333 q^{94} + 54 q^{95} + 852 q^{97} - 354 q^{98}+O(q^{100}) \) 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
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.i 1
3.b odd 2 1 169.4.a.b 1
13.b even 2 1 1521.4.a.d 1
13.d odd 4 2 117.4.b.a 2
39.d odd 2 1 169.4.a.c 1
39.f even 4 2 13.4.b.a 2
39.h odd 6 2 169.4.c.b 2
39.i odd 6 2 169.4.c.c 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.a 2
195.n even 4 2 325.4.c.b 2
195.u odd 4 2 325.4.d.b 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 3.b odd 2 1
169.4.a.c 1 39.d odd 2 1
169.4.c.b 2 39.h odd 6 2
169.4.c.c 2 39.i 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.j odd 4 2
325.4.d.b 2 195.u 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 13.b even 2 1
1521.4.a.i 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{5} + 9 \) Copy content Toggle raw display
\( T_{7} + 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

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