Properties

Label 1008.4.a.be
Level $1008$
Weight $4$
Character orbit 1008.a
Self dual yes
Analytic conductor $59.474$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4739252858\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \(x^{2} - 19\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + 7 q^{7} +O(q^{10})\) \( q + \beta q^{5} + 7 q^{7} + 5 \beta q^{11} + 82 q^{13} -9 \beta q^{17} + 20 q^{19} + 15 \beta q^{23} -49 q^{25} + 28 \beta q^{29} -156 q^{31} + 7 \beta q^{35} + 186 q^{37} -19 \beta q^{41} -164 q^{43} -54 \beta q^{47} + 49 q^{49} -18 \beta q^{53} + 380 q^{55} -18 \beta q^{59} + 790 q^{61} + 82 \beta q^{65} + 44 q^{67} -51 \beta q^{71} + 126 q^{73} + 35 \beta q^{77} + 712 q^{79} + 168 \beta q^{83} -684 q^{85} -167 \beta q^{89} + 574 q^{91} + 20 \beta q^{95} + 798 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{7} + O(q^{10}) \) \( 2q + 14q^{7} + 164q^{13} + 40q^{19} - 98q^{25} - 312q^{31} + 372q^{37} - 328q^{43} + 98q^{49} + 760q^{55} + 1580q^{61} + 88q^{67} + 252q^{73} + 1424q^{79} - 1368q^{85} + 1148q^{91} + 1596q^{97} + O(q^{100}) \)

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
−4.35890
4.35890
0 0 0 −8.71780 0 7.00000 0 0 0
1.2 0 0 0 8.71780 0 7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.a.be 2
3.b odd 2 1 inner 1008.4.a.be 2
4.b odd 2 1 63.4.a.d 2
12.b even 2 1 63.4.a.d 2
20.d odd 2 1 1575.4.a.t 2
28.d even 2 1 441.4.a.q 2
28.f even 6 2 441.4.e.s 4
28.g odd 6 2 441.4.e.r 4
60.h even 2 1 1575.4.a.t 2
84.h odd 2 1 441.4.a.q 2
84.j odd 6 2 441.4.e.s 4
84.n even 6 2 441.4.e.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.a.d 2 4.b odd 2 1
63.4.a.d 2 12.b even 2 1
441.4.a.q 2 28.d even 2 1
441.4.a.q 2 84.h odd 2 1
441.4.e.r 4 28.g odd 6 2
441.4.e.r 4 84.n even 6 2
441.4.e.s 4 28.f even 6 2
441.4.e.s 4 84.j odd 6 2
1008.4.a.be 2 1.a even 1 1 trivial
1008.4.a.be 2 3.b odd 2 1 inner
1575.4.a.t 2 20.d odd 2 1
1575.4.a.t 2 60.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} - 76 \)
\( T_{11}^{2} - 1900 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -76 + T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -1900 + T^{2} \)
$13$ \( ( -82 + T )^{2} \)
$17$ \( -6156 + T^{2} \)
$19$ \( ( -20 + T )^{2} \)
$23$ \( -17100 + T^{2} \)
$29$ \( -59584 + T^{2} \)
$31$ \( ( 156 + T )^{2} \)
$37$ \( ( -186 + T )^{2} \)
$41$ \( -27436 + T^{2} \)
$43$ \( ( 164 + T )^{2} \)
$47$ \( -221616 + T^{2} \)
$53$ \( -24624 + T^{2} \)
$59$ \( -24624 + T^{2} \)
$61$ \( ( -790 + T )^{2} \)
$67$ \( ( -44 + T )^{2} \)
$71$ \( -197676 + T^{2} \)
$73$ \( ( -126 + T )^{2} \)
$79$ \( ( -712 + T )^{2} \)
$83$ \( -2145024 + T^{2} \)
$89$ \( -2119564 + T^{2} \)
$97$ \( ( -798 + T )^{2} \)
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