Properties

Label 1521.4.a.m
Level $1521$
Weight $4$
Character orbit 1521.a
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{4} - 3 \beta q^{5} + 6 \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{4} - 3 \beta q^{5} + 6 \beta q^{7} + 30 \beta q^{11} + 64 q^{16} - 117 q^{17} + 14 \beta q^{19} + 24 \beta q^{20} + 18 q^{23} - 98 q^{25} - 48 \beta q^{28} + 99 q^{29} + 112 \beta q^{31} - 54 q^{35} - 65 \beta q^{37} + 21 \beta q^{41} + 82 q^{43} - 240 \beta q^{44} - 42 \beta q^{47} - 235 q^{49} + 261 q^{53} - 270 q^{55} + 456 \beta q^{59} - 719 q^{61} - 512 q^{64} - 406 \beta q^{67} + 936 q^{68} - 270 \beta q^{71} - 395 \beta q^{73} - 112 \beta q^{76} + 540 q^{77} - 440 q^{79} - 192 \beta q^{80} - 690 \beta q^{83} + 351 \beta q^{85} + 876 \beta q^{89} - 144 q^{92} - 126 q^{95} + 668 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{4} + 128 q^{16} - 234 q^{17} + 36 q^{23} - 196 q^{25} + 198 q^{29} - 108 q^{35} + 164 q^{43} - 470 q^{49} + 522 q^{53} - 540 q^{55} - 1438 q^{61} - 1024 q^{64} + 1872 q^{68} + 1080 q^{77} - 880 q^{79} - 288 q^{92} - 252 q^{95}+O(q^{100}) \) 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
1.73205
−1.73205
0 0 −8.00000 −5.19615 0 10.3923 0 0 0
1.2 0 0 −8.00000 5.19615 0 −10.3923 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.4.a.m 2
3.b odd 2 1 507.4.a.g 2
13.b even 2 1 inner 1521.4.a.m 2
13.f odd 12 2 117.4.q.b 2
39.d odd 2 1 507.4.a.g 2
39.f even 4 2 507.4.b.a 2
39.k even 12 2 39.4.j.a 2
156.v odd 12 2 624.4.bv.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.a 2 39.k even 12 2
117.4.q.b 2 13.f odd 12 2
507.4.a.g 2 3.b odd 2 1
507.4.a.g 2 39.d odd 2 1
507.4.b.a 2 39.f even 4 2
624.4.bv.a 2 156.v odd 12 2
1521.4.a.m 2 1.a even 1 1 trivial
1521.4.a.m 2 13.b even 2 1 inner

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{2} - 27 \) Copy content Toggle raw display
\( T_{7}^{2} - 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 27 \) Copy content Toggle raw display
$7$ \( T^{2} - 108 \) Copy content Toggle raw display
$11$ \( T^{2} - 2700 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 117)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 588 \) Copy content Toggle raw display
$23$ \( (T - 18)^{2} \) Copy content Toggle raw display
$29$ \( (T - 99)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 37632 \) Copy content Toggle raw display
$37$ \( T^{2} - 12675 \) Copy content Toggle raw display
$41$ \( T^{2} - 1323 \) Copy content Toggle raw display
$43$ \( (T - 82)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 5292 \) Copy content Toggle raw display
$53$ \( (T - 261)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 623808 \) Copy content Toggle raw display
$61$ \( (T + 719)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 494508 \) Copy content Toggle raw display
$71$ \( T^{2} - 218700 \) Copy content Toggle raw display
$73$ \( T^{2} - 468075 \) Copy content Toggle raw display
$79$ \( (T + 440)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 1428300 \) Copy content Toggle raw display
$89$ \( T^{2} - 2302128 \) Copy content Toggle raw display
$97$ \( T^{2} - 1338672 \) Copy content Toggle raw display
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