Properties

Label 504.2.r.a
Level $504$
Weight $2$
Character orbit 504.r
Analytic conductor $4.024$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} -\zeta_{6} q^{7} -3 q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{5} -\zeta_{6} q^{7} -3 q^{9} + 6 \zeta_{6} q^{11} + ( -6 + 6 \zeta_{6} ) q^{13} + ( 1 + \zeta_{6} ) q^{15} -2 q^{17} + 7 q^{19} + ( 2 - \zeta_{6} ) q^{21} + ( 1 - \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} -2 \zeta_{6} q^{29} + ( -10 + 10 \zeta_{6} ) q^{31} + ( -12 + 6 \zeta_{6} ) q^{33} - q^{35} -6 q^{37} + ( -6 - 6 \zeta_{6} ) q^{39} + ( 8 - 8 \zeta_{6} ) q^{41} + 10 \zeta_{6} q^{43} + ( -3 + 3 \zeta_{6} ) q^{45} -8 \zeta_{6} q^{47} + ( -1 + \zeta_{6} ) q^{49} + ( 2 - 4 \zeta_{6} ) q^{51} + 2 q^{53} + 6 q^{55} + ( -7 + 14 \zeta_{6} ) q^{57} -7 \zeta_{6} q^{61} + 3 \zeta_{6} q^{63} + 6 \zeta_{6} q^{65} + ( 12 - 12 \zeta_{6} ) q^{67} + ( 1 + \zeta_{6} ) q^{69} + 15 q^{71} -2 q^{73} + ( -8 + 4 \zeta_{6} ) q^{75} + ( 6 - 6 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + 9 q^{81} -12 \zeta_{6} q^{83} + ( -2 + 2 \zeta_{6} ) q^{85} + ( 4 - 2 \zeta_{6} ) q^{87} + 4 q^{89} + 6 q^{91} + ( -10 - 10 \zeta_{6} ) q^{93} + ( 7 - 7 \zeta_{6} ) q^{95} + 2 \zeta_{6} q^{97} -18 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} - q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + q^{5} - q^{7} - 6q^{9} + 6q^{11} - 6q^{13} + 3q^{15} - 4q^{17} + 14q^{19} + 3q^{21} + q^{23} + 4q^{25} - 2q^{29} - 10q^{31} - 18q^{33} - 2q^{35} - 12q^{37} - 18q^{39} + 8q^{41} + 10q^{43} - 3q^{45} - 8q^{47} - q^{49} + 4q^{53} + 12q^{55} - 7q^{61} + 3q^{63} + 6q^{65} + 12q^{67} + 3q^{69} + 30q^{71} - 4q^{73} - 12q^{75} + 6q^{77} - q^{79} + 18q^{81} - 12q^{83} - 2q^{85} + 6q^{87} + 8q^{89} + 12q^{91} - 30q^{93} + 7q^{95} + 2q^{97} - 18q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
0.500000 0.866025i
0.500000 + 0.866025i
0 1.73205i 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −3.00000 0
337.1 0 1.73205i 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.r.a 2
3.b odd 2 1 1512.2.r.a 2
4.b odd 2 1 1008.2.r.c 2
9.c even 3 1 inner 504.2.r.a 2
9.c even 3 1 4536.2.a.d 1
9.d odd 6 1 1512.2.r.a 2
9.d odd 6 1 4536.2.a.g 1
12.b even 2 1 3024.2.r.b 2
36.f odd 6 1 1008.2.r.c 2
36.f odd 6 1 9072.2.a.i 1
36.h even 6 1 3024.2.r.b 2
36.h even 6 1 9072.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.a 2 1.a even 1 1 trivial
504.2.r.a 2 9.c even 3 1 inner
1008.2.r.c 2 4.b odd 2 1
1008.2.r.c 2 36.f odd 6 1
1512.2.r.a 2 3.b odd 2 1
1512.2.r.a 2 9.d odd 6 1
3024.2.r.b 2 12.b even 2 1
3024.2.r.b 2 36.h even 6 1
4536.2.a.d 1 9.c even 3 1
4536.2.a.g 1 9.d odd 6 1
9072.2.a.i 1 36.f odd 6 1
9072.2.a.n 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( 1 + T + T^{2} \)
$11$ \( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \)
$13$ \( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} \)
$29$ \( 1 + 2 T - 25 T^{2} + 58 T^{3} + 841 T^{4} \)
$31$ \( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 8 T + 23 T^{2} - 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 10 T + 57 T^{2} - 430 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 2 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 7 T - 12 T^{2} + 427 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 15 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 2 T + 73 T^{2} )^{2} \)
$79$ \( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 4 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 2 T - 93 T^{2} - 194 T^{3} + 9409 T^{4} \)
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