Properties

Label 507.2.e.b
Level $507$
Weight $2$
Character orbit 507.e
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.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} -2 q^{5} -\zeta_{6} q^{6} -4 \zeta_{6} q^{7} + 3 q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} -2 q^{5} -\zeta_{6} q^{6} -4 \zeta_{6} q^{7} + 3 q^{8} -\zeta_{6} q^{9} + ( -2 + 2 \zeta_{6} ) q^{10} + ( 4 - 4 \zeta_{6} ) q^{11} + q^{12} -4 q^{14} + ( -2 + 2 \zeta_{6} ) q^{15} + ( 1 - \zeta_{6} ) q^{16} -2 \zeta_{6} q^{17} - q^{18} -2 \zeta_{6} q^{20} -4 q^{21} -4 \zeta_{6} q^{22} + ( 3 - 3 \zeta_{6} ) q^{24} - q^{25} - q^{27} + ( 4 - 4 \zeta_{6} ) q^{28} + ( 10 - 10 \zeta_{6} ) q^{29} + 2 \zeta_{6} q^{30} -4 q^{31} + 5 \zeta_{6} q^{32} -4 \zeta_{6} q^{33} -2 q^{34} + 8 \zeta_{6} q^{35} + ( 1 - \zeta_{6} ) q^{36} + ( -2 + 2 \zeta_{6} ) q^{37} -6 q^{40} + ( 6 - 6 \zeta_{6} ) q^{41} + ( -4 + 4 \zeta_{6} ) q^{42} + 12 \zeta_{6} q^{43} + 4 q^{44} + 2 \zeta_{6} q^{45} -\zeta_{6} q^{48} + ( -9 + 9 \zeta_{6} ) q^{49} + ( -1 + \zeta_{6} ) q^{50} -2 q^{51} + 6 q^{53} + ( -1 + \zeta_{6} ) q^{54} + ( -8 + 8 \zeta_{6} ) q^{55} -12 \zeta_{6} q^{56} -10 \zeta_{6} q^{58} + 12 \zeta_{6} q^{59} -2 q^{60} + 2 \zeta_{6} q^{61} + ( -4 + 4 \zeta_{6} ) q^{62} + ( -4 + 4 \zeta_{6} ) q^{63} + 7 q^{64} -4 q^{66} + ( -8 + 8 \zeta_{6} ) q^{67} + ( 2 - 2 \zeta_{6} ) q^{68} + 8 q^{70} -3 \zeta_{6} q^{72} -2 q^{73} + 2 \zeta_{6} q^{74} + ( -1 + \zeta_{6} ) q^{75} -16 q^{77} + 8 q^{79} + ( -2 + 2 \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -6 \zeta_{6} q^{82} -4 q^{83} -4 \zeta_{6} q^{84} + 4 \zeta_{6} q^{85} + 12 q^{86} -10 \zeta_{6} q^{87} + ( 12 - 12 \zeta_{6} ) q^{88} + ( -2 + 2 \zeta_{6} ) q^{89} + 2 q^{90} + ( -4 + 4 \zeta_{6} ) q^{93} + 5 q^{96} + 10 \zeta_{6} q^{97} + 9 \zeta_{6} q^{98} -4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{3} + q^{4} - 4q^{5} - q^{6} - 4q^{7} + 6q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} + q^{3} + q^{4} - 4q^{5} - q^{6} - 4q^{7} + 6q^{8} - q^{9} - 2q^{10} + 4q^{11} + 2q^{12} - 8q^{14} - 2q^{15} + q^{16} - 2q^{17} - 2q^{18} - 2q^{20} - 8q^{21} - 4q^{22} + 3q^{24} - 2q^{25} - 2q^{27} + 4q^{28} + 10q^{29} + 2q^{30} - 8q^{31} + 5q^{32} - 4q^{33} - 4q^{34} + 8q^{35} + q^{36} - 2q^{37} - 12q^{40} + 6q^{41} - 4q^{42} + 12q^{43} + 8q^{44} + 2q^{45} - q^{48} - 9q^{49} - q^{50} - 4q^{51} + 12q^{53} - q^{54} - 8q^{55} - 12q^{56} - 10q^{58} + 12q^{59} - 4q^{60} + 2q^{61} - 4q^{62} - 4q^{63} + 14q^{64} - 8q^{66} - 8q^{67} + 2q^{68} + 16q^{70} - 3q^{72} - 4q^{73} + 2q^{74} - q^{75} - 32q^{77} + 16q^{79} - 2q^{80} - q^{81} - 6q^{82} - 8q^{83} - 4q^{84} + 4q^{85} + 24q^{86} - 10q^{87} + 12q^{88} - 2q^{89} + 4q^{90} - 4q^{93} + 10q^{96} + 10q^{97} + 9q^{98} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i −2.00000 −0.500000 0.866025i −2.00000 3.46410i 3.00000 −0.500000 0.866025i −1.00000 + 1.73205i
484.1 0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −2.00000 −0.500000 + 0.866025i −2.00000 + 3.46410i 3.00000 −0.500000 + 0.866025i −1.00000 1.73205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.e.b 2
13.b even 2 1 507.2.e.a 2
13.c even 3 1 507.2.a.a 1
13.c even 3 1 inner 507.2.e.b 2
13.d odd 4 2 507.2.j.e 4
13.e even 6 1 39.2.a.a 1
13.e even 6 1 507.2.e.a 2
13.f odd 12 2 507.2.b.a 2
13.f odd 12 2 507.2.j.e 4
39.h odd 6 1 117.2.a.a 1
39.i odd 6 1 1521.2.a.e 1
39.k even 12 2 1521.2.b.b 2
52.i odd 6 1 624.2.a.i 1
52.j odd 6 1 8112.2.a.s 1
65.l even 6 1 975.2.a.f 1
65.r odd 12 2 975.2.c.f 2
91.t odd 6 1 1911.2.a.f 1
104.p odd 6 1 2496.2.a.e 1
104.s even 6 1 2496.2.a.q 1
117.l even 6 1 1053.2.e.b 2
117.m odd 6 1 1053.2.e.d 2
117.r even 6 1 1053.2.e.b 2
117.v odd 6 1 1053.2.e.d 2
143.i odd 6 1 4719.2.a.c 1
156.r even 6 1 1872.2.a.h 1
195.y odd 6 1 2925.2.a.p 1
195.bf even 12 2 2925.2.c.e 2
273.u even 6 1 5733.2.a.e 1
312.ba even 6 1 7488.2.a.by 1
312.bg odd 6 1 7488.2.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 13.e even 6 1
117.2.a.a 1 39.h odd 6 1
507.2.a.a 1 13.c even 3 1
507.2.b.a 2 13.f odd 12 2
507.2.e.a 2 13.b even 2 1
507.2.e.a 2 13.e even 6 1
507.2.e.b 2 1.a even 1 1 trivial
507.2.e.b 2 13.c even 3 1 inner
507.2.j.e 4 13.d odd 4 2
507.2.j.e 4 13.f odd 12 2
624.2.a.i 1 52.i odd 6 1
975.2.a.f 1 65.l even 6 1
975.2.c.f 2 65.r odd 12 2
1053.2.e.b 2 117.l even 6 1
1053.2.e.b 2 117.r even 6 1
1053.2.e.d 2 117.m odd 6 1
1053.2.e.d 2 117.v odd 6 1
1521.2.a.e 1 39.i odd 6 1
1521.2.b.b 2 39.k even 12 2
1872.2.a.h 1 156.r even 6 1
1911.2.a.f 1 91.t odd 6 1
2496.2.a.e 1 104.p odd 6 1
2496.2.a.q 1 104.s even 6 1
2925.2.a.p 1 195.y odd 6 1
2925.2.c.e 2 195.bf even 12 2
4719.2.a.c 1 143.i odd 6 1
5733.2.a.e 1 273.u even 6 1
7488.2.a.bl 1 312.bg odd 6 1
7488.2.a.by 1 312.ba even 6 1
8112.2.a.s 1 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \)
\( T_{5} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( ( 2 + T )^{2} \)
$7$ \( 16 + 4 T + T^{2} \)
$11$ \( 16 - 4 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 4 + 2 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 100 - 10 T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 4 + 2 T + T^{2} \)
$41$ \( 36 - 6 T + T^{2} \)
$43$ \( 144 - 12 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 144 - 12 T + T^{2} \)
$61$ \( 4 - 2 T + T^{2} \)
$67$ \( 64 + 8 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( 4 + 2 T + T^{2} \)
$97$ \( 100 - 10 T + T^{2} \)
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