Properties

Label 1521.2.a.l
Level $1521$
Weight $2$
Character orbit 1521.a
Self dual yes
Analytic conductor $12.145$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} -2 \beta q^{7} -\beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + q^{4} -2 \beta q^{7} -\beta q^{8} + 2 \beta q^{11} -6 q^{14} -5 q^{16} -6 q^{17} -2 \beta q^{19} + 6 q^{22} -5 q^{25} -2 \beta q^{28} -6 q^{29} + 2 \beta q^{31} -3 \beta q^{32} -6 \beta q^{34} + 4 \beta q^{37} -6 q^{38} -4 \beta q^{41} + 4 q^{43} + 2 \beta q^{44} -2 \beta q^{47} + 5 q^{49} -5 \beta q^{50} -6 q^{53} + 6 q^{56} -6 \beta q^{58} -6 \beta q^{59} -2 q^{61} + 6 q^{62} + q^{64} + 6 \beta q^{67} -6 q^{68} + 2 \beta q^{71} + 12 q^{74} -2 \beta q^{76} -12 q^{77} -8 q^{79} -12 q^{82} -2 \beta q^{83} + 4 \beta q^{86} -6 q^{88} + 4 \beta q^{89} -6 q^{94} + 8 \beta q^{97} + 5 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + O(q^{10}) \) \( 2q + 2q^{4} - 12q^{14} - 10q^{16} - 12q^{17} + 12q^{22} - 10q^{25} - 12q^{29} - 12q^{38} + 8q^{43} + 10q^{49} - 12q^{53} + 12q^{56} - 4q^{61} + 12q^{62} + 2q^{64} - 12q^{68} + 24q^{74} - 24q^{77} - 16q^{79} - 24q^{82} - 12q^{88} - 12q^{94} + O(q^{100}) \)

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
−1.73205
1.73205
−1.73205 0 1.00000 0 0 3.46410 1.73205 0 0
1.2 1.73205 0 1.00000 0 0 −3.46410 −1.73205 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 1521.2.a.l 2
3.b odd 2 1 507.2.a.f 2
12.b even 2 1 8112.2.a.bv 2
13.b even 2 1 inner 1521.2.a.l 2
13.d odd 4 2 117.2.b.a 2
39.d odd 2 1 507.2.a.f 2
39.f even 4 2 39.2.b.a 2
39.h odd 6 2 507.2.e.e 4
39.i odd 6 2 507.2.e.e 4
39.k even 12 2 507.2.j.a 2
39.k even 12 2 507.2.j.c 2
52.f even 4 2 1872.2.c.e 2
156.h even 2 1 8112.2.a.bv 2
156.l odd 4 2 624.2.c.e 2
195.j odd 4 2 975.2.h.f 4
195.n even 4 2 975.2.b.d 2
195.u odd 4 2 975.2.h.f 4
273.o odd 4 2 1911.2.c.d 2
312.w odd 4 2 2496.2.c.d 2
312.y even 4 2 2496.2.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 39.f even 4 2
117.2.b.a 2 13.d odd 4 2
507.2.a.f 2 3.b odd 2 1
507.2.a.f 2 39.d odd 2 1
507.2.e.e 4 39.h odd 6 2
507.2.e.e 4 39.i odd 6 2
507.2.j.a 2 39.k even 12 2
507.2.j.c 2 39.k even 12 2
624.2.c.e 2 156.l odd 4 2
975.2.b.d 2 195.n even 4 2
975.2.h.f 4 195.j odd 4 2
975.2.h.f 4 195.u odd 4 2
1521.2.a.l 2 1.a even 1 1 trivial
1521.2.a.l 2 13.b even 2 1 inner
1872.2.c.e 2 52.f even 4 2
1911.2.c.d 2 273.o odd 4 2
2496.2.c.d 2 312.w odd 4 2
2496.2.c.k 2 312.y even 4 2
8112.2.a.bv 2 12.b even 2 1
8112.2.a.bv 2 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{2} - 3 \)
\( T_{5} \)
\( T_{7}^{2} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -12 + T^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 6 + T )^{2} \)
$19$ \( -12 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( -12 + T^{2} \)
$37$ \( -48 + T^{2} \)
$41$ \( -48 + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( -12 + T^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( -108 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( -108 + T^{2} \)
$71$ \( -12 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( -12 + T^{2} \)
$89$ \( -48 + T^{2} \)
$97$ \( -192 + T^{2} \)
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