Properties

Label 1521.2.a.g
Level $1521$
Weight $2$
Character orbit 1521.a
Self dual yes
Analytic conductor $12.145$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + ( 2 + \beta ) q^{4} + ( 2 - \beta ) q^{5} + ( -1 - \beta ) q^{7} + ( -4 - \beta ) q^{8} +O(q^{10})\) \( q -\beta q^{2} + ( 2 + \beta ) q^{4} + ( 2 - \beta ) q^{5} + ( -1 - \beta ) q^{7} + ( -4 - \beta ) q^{8} + ( 4 - \beta ) q^{10} + 2 q^{11} + ( 4 + 2 \beta ) q^{14} + 3 \beta q^{16} -\beta q^{17} + ( 4 - 2 \beta ) q^{19} -\beta q^{20} -2 \beta q^{22} -2 q^{23} + ( 3 - 3 \beta ) q^{25} + ( -6 - 4 \beta ) q^{28} + ( -2 + 3 \beta ) q^{29} + ( 1 - \beta ) q^{31} + ( -4 - \beta ) q^{32} + ( 4 + \beta ) q^{34} + 2 q^{35} + ( 6 - \beta ) q^{37} + ( 8 - 2 \beta ) q^{38} + ( -4 + 3 \beta ) q^{40} -\beta q^{41} + ( 3 - \beta ) q^{43} + ( 4 + 2 \beta ) q^{44} + 2 \beta q^{46} + ( -2 + 4 \beta ) q^{47} + ( -2 + 3 \beta ) q^{49} + 12 q^{50} + ( -4 - 3 \beta ) q^{53} + ( 4 - 2 \beta ) q^{55} + ( 8 + 6 \beta ) q^{56} + ( -12 - \beta ) q^{58} + ( 6 + 2 \beta ) q^{59} + ( 7 + 2 \beta ) q^{61} + 4 q^{62} + ( 4 - \beta ) q^{64} + ( 3 - \beta ) q^{67} + ( -4 - 3 \beta ) q^{68} -2 \beta q^{70} -14 q^{71} + ( -7 + 2 \beta ) q^{73} + ( 4 - 5 \beta ) q^{74} -2 \beta q^{76} + ( -2 - 2 \beta ) q^{77} + ( 7 + \beta ) q^{79} + ( -12 + 3 \beta ) q^{80} + ( 4 + \beta ) q^{82} + ( 4 + 2 \beta ) q^{83} + ( 4 - \beta ) q^{85} + ( 4 - 2 \beta ) q^{86} + ( -8 - 2 \beta ) q^{88} + ( -8 - 2 \beta ) q^{89} + ( -4 - 2 \beta ) q^{92} + ( -16 - 2 \beta ) q^{94} + ( 16 - 6 \beta ) q^{95} + ( -7 + \beta ) q^{97} + ( -12 - \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 5q^{4} + 3q^{5} - 3q^{7} - 9q^{8} + O(q^{10}) \) \( 2q - q^{2} + 5q^{4} + 3q^{5} - 3q^{7} - 9q^{8} + 7q^{10} + 4q^{11} + 10q^{14} + 3q^{16} - q^{17} + 6q^{19} - q^{20} - 2q^{22} - 4q^{23} + 3q^{25} - 16q^{28} - q^{29} + q^{31} - 9q^{32} + 9q^{34} + 4q^{35} + 11q^{37} + 14q^{38} - 5q^{40} - q^{41} + 5q^{43} + 10q^{44} + 2q^{46} - q^{49} + 24q^{50} - 11q^{53} + 6q^{55} + 22q^{56} - 25q^{58} + 14q^{59} + 16q^{61} + 8q^{62} + 7q^{64} + 5q^{67} - 11q^{68} - 2q^{70} - 28q^{71} - 12q^{73} + 3q^{74} - 2q^{76} - 6q^{77} + 15q^{79} - 21q^{80} + 9q^{82} + 10q^{83} + 7q^{85} + 6q^{86} - 18q^{88} - 18q^{89} - 10q^{92} - 34q^{94} + 26q^{95} - 13q^{97} - 25q^{98} + O(q^{100}) \)

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
2.56155
−1.56155
−2.56155 0 4.56155 −0.561553 0 −3.56155 −6.56155 0 1.43845
1.2 1.56155 0 0.438447 3.56155 0 0.561553 −2.43845 0 5.56155
\(n\): 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 1521.2.a.g 2
3.b odd 2 1 507.2.a.g 2
12.b even 2 1 8112.2.a.bk 2
13.b even 2 1 1521.2.a.m 2
13.c even 3 2 117.2.g.c 4
13.d odd 4 2 1521.2.b.h 4
39.d odd 2 1 507.2.a.d 2
39.f even 4 2 507.2.b.d 4
39.h odd 6 2 507.2.e.g 4
39.i odd 6 2 39.2.e.b 4
39.k even 12 4 507.2.j.g 8
52.j odd 6 2 1872.2.t.r 4
156.h even 2 1 8112.2.a.bo 2
156.p even 6 2 624.2.q.h 4
195.x odd 6 2 975.2.i.k 4
195.bl even 12 4 975.2.bb.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 39.i odd 6 2
117.2.g.c 4 13.c even 3 2
507.2.a.d 2 39.d odd 2 1
507.2.a.g 2 3.b odd 2 1
507.2.b.d 4 39.f even 4 2
507.2.e.g 4 39.h odd 6 2
507.2.j.g 8 39.k even 12 4
624.2.q.h 4 156.p even 6 2
975.2.i.k 4 195.x odd 6 2
975.2.bb.i 8 195.bl even 12 4
1521.2.a.g 2 1.a even 1 1 trivial
1521.2.a.m 2 13.b even 2 1
1521.2.b.h 4 13.d odd 4 2
1872.2.t.r 4 52.j odd 6 2
8112.2.a.bk 2 12.b even 2 1
8112.2.a.bo 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\):

\( T_{2}^{2} + T_{2} - 4 \)
\( T_{5}^{2} - 3 T_{5} - 2 \)
\( T_{7}^{2} + 3 T_{7} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -2 - 3 T + T^{2} \)
$7$ \( -2 + 3 T + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( -4 + T + T^{2} \)
$19$ \( -8 - 6 T + T^{2} \)
$23$ \( ( 2 + T )^{2} \)
$29$ \( -38 + T + T^{2} \)
$31$ \( -4 - T + T^{2} \)
$37$ \( 26 - 11 T + T^{2} \)
$41$ \( -4 + T + T^{2} \)
$43$ \( 2 - 5 T + T^{2} \)
$47$ \( -68 + T^{2} \)
$53$ \( -8 + 11 T + T^{2} \)
$59$ \( 32 - 14 T + T^{2} \)
$61$ \( 47 - 16 T + T^{2} \)
$67$ \( 2 - 5 T + T^{2} \)
$71$ \( ( 14 + T )^{2} \)
$73$ \( 19 + 12 T + T^{2} \)
$79$ \( 52 - 15 T + T^{2} \)
$83$ \( 8 - 10 T + T^{2} \)
$89$ \( 64 + 18 T + T^{2} \)
$97$ \( 38 + 13 T + T^{2} \)
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