Properties

Label 1521.4.a.a
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

\(f(q)\) \(=\) \( q - 5 q^{2} + 17 q^{4} - 7 q^{5} + 13 q^{7} - 45 q^{8} + O(q^{10}) \) \( q - 5 q^{2} + 17 q^{4} - 7 q^{5} + 13 q^{7} - 45 q^{8} + 35 q^{10} - 26 q^{11} - 65 q^{14} + 89 q^{16} - 77 q^{17} + 126 q^{19} - 119 q^{20} + 130 q^{22} + 96 q^{23} - 76 q^{25} + 221 q^{28} + 82 q^{29} - 196 q^{31} - 85 q^{32} + 385 q^{34} - 91 q^{35} + 131 q^{37} - 630 q^{38} + 315 q^{40} + 336 q^{41} - 201 q^{43} - 442 q^{44} - 480 q^{46} - 105 q^{47} - 174 q^{49} + 380 q^{50} + 432 q^{53} + 182 q^{55} - 585 q^{56} - 410 q^{58} - 294 q^{59} - 56 q^{61} + 980 q^{62} - 287 q^{64} - 478 q^{67} - 1309 q^{68} + 455 q^{70} + 9 q^{71} - 98 q^{73} - 655 q^{74} + 2142 q^{76} - 338 q^{77} + 1304 q^{79} - 623 q^{80} - 1680 q^{82} - 308 q^{83} + 539 q^{85} + 1005 q^{86} + 1170 q^{88} - 1190 q^{89} + 1632 q^{92} + 525 q^{94} - 882 q^{95} - 70 q^{97} + 870 q^{98} + O(q^{100}) \)

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
−5.00000 0 17.0000 −7.00000 0 13.0000 −45.0000 0 35.0000
\(n\): 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.a 1
3.b odd 2 1 169.4.a.e 1
13.b even 2 1 117.4.a.b 1
39.d odd 2 1 13.4.a.a 1
39.f even 4 2 169.4.b.a 2
39.h odd 6 2 169.4.c.e 2
39.i odd 6 2 169.4.c.a 2
39.k even 12 4 169.4.e.e 4
52.b odd 2 1 1872.4.a.k 1
156.h even 2 1 208.4.a.g 1
195.e odd 2 1 325.4.a.d 1
195.s even 4 2 325.4.b.b 2
273.g even 2 1 637.4.a.a 1
312.b odd 2 1 832.4.a.r 1
312.h even 2 1 832.4.a.a 1
429.e even 2 1 1573.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 39.d odd 2 1
117.4.a.b 1 13.b even 2 1
169.4.a.e 1 3.b odd 2 1
169.4.b.a 2 39.f even 4 2
169.4.c.a 2 39.i odd 6 2
169.4.c.e 2 39.h odd 6 2
169.4.e.e 4 39.k even 12 4
208.4.a.g 1 156.h even 2 1
325.4.a.d 1 195.e odd 2 1
325.4.b.b 2 195.s even 4 2
637.4.a.a 1 273.g even 2 1
832.4.a.a 1 312.h even 2 1
832.4.a.r 1 312.b odd 2 1
1521.4.a.a 1 1.a even 1 1 trivial
1573.4.a.a 1 429.e even 2 1
1872.4.a.k 1 52.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2} + 5 \)
\( T_{5} + 7 \)
\( T_{7} - 13 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5 + T \)
$3$ \( T \)
$5$ \( 7 + T \)
$7$ \( -13 + T \)
$11$ \( 26 + T \)
$13$ \( T \)
$17$ \( 77 + T \)
$19$ \( -126 + T \)
$23$ \( -96 + T \)
$29$ \( -82 + T \)
$31$ \( 196 + T \)
$37$ \( -131 + T \)
$41$ \( -336 + T \)
$43$ \( 201 + T \)
$47$ \( 105 + T \)
$53$ \( -432 + T \)
$59$ \( 294 + T \)
$61$ \( 56 + T \)
$67$ \( 478 + T \)
$71$ \( -9 + T \)
$73$ \( 98 + T \)
$79$ \( -1304 + T \)
$83$ \( 308 + T \)
$89$ \( 1190 + T \)
$97$ \( 70 + T \)
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