Properties

Label 507.2.a.e
Level $507$
Weight $2$
Character orbit 507.a
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - q^{3} -2 q^{4} + 2 \beta q^{5} + \beta q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} -2 q^{4} + 2 \beta q^{5} + \beta q^{7} + q^{9} -2 \beta q^{11} + 2 q^{12} -2 \beta q^{15} + 4 q^{16} + 2 \beta q^{19} -4 \beta q^{20} -\beta q^{21} + 6 q^{23} + 7 q^{25} - q^{27} -2 \beta q^{28} + 6 q^{29} + \beta q^{31} + 2 \beta q^{33} + 6 q^{35} -2 q^{36} + 4 \beta q^{41} + q^{43} + 4 \beta q^{44} + 2 \beta q^{45} -2 \beta q^{47} -4 q^{48} -4 q^{49} + 12 q^{53} -12 q^{55} -2 \beta q^{57} + 2 \beta q^{59} + 4 \beta q^{60} + q^{61} + \beta q^{63} -8 q^{64} -5 \beta q^{67} -6 q^{69} -6 \beta q^{71} -\beta q^{73} -7 q^{75} -4 \beta q^{76} -6 q^{77} -11 q^{79} + 8 \beta q^{80} + q^{81} -8 \beta q^{83} + 2 \beta q^{84} -6 q^{87} + 4 \beta q^{89} -12 q^{92} -\beta q^{93} + 12 q^{95} -3 \beta q^{97} -2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 4q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 4q^{4} + 2q^{9} + 4q^{12} + 8q^{16} + 12q^{23} + 14q^{25} - 2q^{27} + 12q^{29} + 12q^{35} - 4q^{36} + 2q^{43} - 8q^{48} - 8q^{49} + 24q^{53} - 24q^{55} + 2q^{61} - 16q^{64} - 12q^{69} - 14q^{75} - 12q^{77} - 22q^{79} + 2q^{81} - 12q^{87} - 24q^{92} + 24q^{95} + O(q^{100}) \)

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
0 −1.00000 −2.00000 −3.46410 0 −1.73205 0 1.00000 0
1.2 0 −1.00000 −2.00000 3.46410 0 1.73205 0 1.00000 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 507.2.a.e 2
3.b odd 2 1 1521.2.a.h 2
4.b odd 2 1 8112.2.a.bu 2
13.b even 2 1 inner 507.2.a.e 2
13.c even 3 2 507.2.e.f 4
13.d odd 4 2 507.2.b.c 2
13.e even 6 2 507.2.e.f 4
13.f odd 12 2 39.2.j.a 2
13.f odd 12 2 507.2.j.b 2
39.d odd 2 1 1521.2.a.h 2
39.f even 4 2 1521.2.b.f 2
39.k even 12 2 117.2.q.a 2
52.b odd 2 1 8112.2.a.bu 2
52.l even 12 2 624.2.bv.b 2
65.o even 12 2 975.2.w.d 4
65.s odd 12 2 975.2.bc.c 2
65.t even 12 2 975.2.w.d 4
156.v odd 12 2 1872.2.by.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 13.f odd 12 2
117.2.q.a 2 39.k even 12 2
507.2.a.e 2 1.a even 1 1 trivial
507.2.a.e 2 13.b even 2 1 inner
507.2.b.c 2 13.d odd 4 2
507.2.e.f 4 13.c even 3 2
507.2.e.f 4 13.e even 6 2
507.2.j.b 2 13.f odd 12 2
624.2.bv.b 2 52.l even 12 2
975.2.w.d 4 65.o even 12 2
975.2.w.d 4 65.t even 12 2
975.2.bc.c 2 65.s odd 12 2
1521.2.a.h 2 3.b odd 2 1
1521.2.a.h 2 39.d odd 2 1
1521.2.b.f 2 39.f even 4 2
1872.2.by.f 2 156.v odd 12 2
8112.2.a.bu 2 4.b odd 2 1
8112.2.a.bu 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(507))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( -12 + T^{2} \)
$7$ \( -3 + T^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( -12 + T^{2} \)
$23$ \( ( -6 + T )^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( -3 + T^{2} \)
$37$ \( T^{2} \)
$41$ \( -48 + T^{2} \)
$43$ \( ( -1 + T )^{2} \)
$47$ \( -12 + T^{2} \)
$53$ \( ( -12 + T )^{2} \)
$59$ \( -12 + T^{2} \)
$61$ \( ( -1 + T )^{2} \)
$67$ \( -75 + T^{2} \)
$71$ \( -108 + T^{2} \)
$73$ \( -3 + T^{2} \)
$79$ \( ( 11 + T )^{2} \)
$83$ \( -192 + T^{2} \)
$89$ \( -48 + T^{2} \)
$97$ \( -27 + T^{2} \)
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