Properties

Label 507.2.e.c
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + q^{5} -\zeta_{6} q^{6} + 2 \zeta_{6} q^{7} + 3 q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} + \zeta_{6} q^{4} + q^{5} -\zeta_{6} q^{6} + 2 \zeta_{6} q^{7} + 3 q^{8} -\zeta_{6} q^{9} + ( 1 - \zeta_{6} ) q^{10} + ( -2 + 2 \zeta_{6} ) q^{11} + q^{12} + 2 q^{14} + ( 1 - \zeta_{6} ) q^{15} + ( 1 - \zeta_{6} ) q^{16} + 7 \zeta_{6} q^{17} - q^{18} -6 \zeta_{6} q^{19} + \zeta_{6} q^{20} + 2 q^{21} + 2 \zeta_{6} q^{22} + ( 6 - 6 \zeta_{6} ) q^{23} + ( 3 - 3 \zeta_{6} ) q^{24} -4 q^{25} - q^{27} + ( -2 + 2 \zeta_{6} ) q^{28} + ( 1 - \zeta_{6} ) q^{29} -\zeta_{6} q^{30} -4 q^{31} + 5 \zeta_{6} q^{32} + 2 \zeta_{6} q^{33} + 7 q^{34} + 2 \zeta_{6} q^{35} + ( 1 - \zeta_{6} ) q^{36} + ( 1 - \zeta_{6} ) q^{37} -6 q^{38} + 3 q^{40} + ( 9 - 9 \zeta_{6} ) q^{41} + ( 2 - 2 \zeta_{6} ) q^{42} -6 \zeta_{6} q^{43} -2 q^{44} -\zeta_{6} q^{45} -6 \zeta_{6} q^{46} -6 q^{47} -\zeta_{6} q^{48} + ( 3 - 3 \zeta_{6} ) q^{49} + ( -4 + 4 \zeta_{6} ) q^{50} + 7 q^{51} -9 q^{53} + ( -1 + \zeta_{6} ) q^{54} + ( -2 + 2 \zeta_{6} ) q^{55} + 6 \zeta_{6} q^{56} -6 q^{57} -\zeta_{6} q^{58} + q^{60} -\zeta_{6} q^{61} + ( -4 + 4 \zeta_{6} ) q^{62} + ( 2 - 2 \zeta_{6} ) q^{63} + 7 q^{64} + 2 q^{66} + ( -2 + 2 \zeta_{6} ) q^{67} + ( -7 + 7 \zeta_{6} ) q^{68} -6 \zeta_{6} q^{69} + 2 q^{70} + 6 \zeta_{6} q^{71} -3 \zeta_{6} q^{72} -11 q^{73} -\zeta_{6} q^{74} + ( -4 + 4 \zeta_{6} ) q^{75} + ( 6 - 6 \zeta_{6} ) q^{76} -4 q^{77} -4 q^{79} + ( 1 - \zeta_{6} ) q^{80} + ( -1 + \zeta_{6} ) q^{81} -9 \zeta_{6} q^{82} + 14 q^{83} + 2 \zeta_{6} q^{84} + 7 \zeta_{6} q^{85} -6 q^{86} -\zeta_{6} q^{87} + ( -6 + 6 \zeta_{6} ) q^{88} + ( -14 + 14 \zeta_{6} ) q^{89} - q^{90} + 6 q^{92} + ( -4 + 4 \zeta_{6} ) q^{93} + ( -6 + 6 \zeta_{6} ) q^{94} -6 \zeta_{6} q^{95} + 5 q^{96} -2 \zeta_{6} q^{97} -3 \zeta_{6} q^{98} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} + 2q^{7} + 6q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} + 2q^{7} + 6q^{8} - q^{9} + q^{10} - 2q^{11} + 2q^{12} + 4q^{14} + q^{15} + q^{16} + 7q^{17} - 2q^{18} - 6q^{19} + q^{20} + 4q^{21} + 2q^{22} + 6q^{23} + 3q^{24} - 8q^{25} - 2q^{27} - 2q^{28} + q^{29} - q^{30} - 8q^{31} + 5q^{32} + 2q^{33} + 14q^{34} + 2q^{35} + q^{36} + q^{37} - 12q^{38} + 6q^{40} + 9q^{41} + 2q^{42} - 6q^{43} - 4q^{44} - q^{45} - 6q^{46} - 12q^{47} - q^{48} + 3q^{49} - 4q^{50} + 14q^{51} - 18q^{53} - q^{54} - 2q^{55} + 6q^{56} - 12q^{57} - q^{58} + 2q^{60} - q^{61} - 4q^{62} + 2q^{63} + 14q^{64} + 4q^{66} - 2q^{67} - 7q^{68} - 6q^{69} + 4q^{70} + 6q^{71} - 3q^{72} - 22q^{73} - q^{74} - 4q^{75} + 6q^{76} - 8q^{77} - 8q^{79} + q^{80} - q^{81} - 9q^{82} + 28q^{83} + 2q^{84} + 7q^{85} - 12q^{86} - q^{87} - 6q^{88} - 14q^{89} - 2q^{90} + 12q^{92} - 4q^{93} - 6q^{94} - 6q^{95} + 10q^{96} - 2q^{97} - 3q^{98} + 4q^{99} + O(q^{100}) \)

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 1.00000 −0.500000 0.866025i 1.00000 + 1.73205i 3.00000 −0.500000 0.866025i 0.500000 0.866025i
484.1 0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 1.00000 1.73205i 3.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
\(n\): 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.c 2
13.b even 2 1 39.2.e.a 2
13.c even 3 1 507.2.a.b 1
13.c even 3 1 inner 507.2.e.c 2
13.d odd 4 2 507.2.j.d 4
13.e even 6 1 39.2.e.a 2
13.e even 6 1 507.2.a.c 1
13.f odd 12 2 507.2.b.b 2
13.f odd 12 2 507.2.j.d 4
39.d odd 2 1 117.2.g.b 2
39.h odd 6 1 117.2.g.b 2
39.h odd 6 1 1521.2.a.a 1
39.i odd 6 1 1521.2.a.d 1
39.k even 12 2 1521.2.b.c 2
52.b odd 2 1 624.2.q.c 2
52.i odd 6 1 624.2.q.c 2
52.i odd 6 1 8112.2.a.w 1
52.j odd 6 1 8112.2.a.bc 1
65.d even 2 1 975.2.i.f 2
65.h odd 4 2 975.2.bb.d 4
65.l even 6 1 975.2.i.f 2
65.r odd 12 2 975.2.bb.d 4
156.h even 2 1 1872.2.t.j 2
156.r even 6 1 1872.2.t.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 13.b even 2 1
39.2.e.a 2 13.e even 6 1
117.2.g.b 2 39.d odd 2 1
117.2.g.b 2 39.h odd 6 1
507.2.a.b 1 13.c even 3 1
507.2.a.c 1 13.e even 6 1
507.2.b.b 2 13.f odd 12 2
507.2.e.c 2 1.a even 1 1 trivial
507.2.e.c 2 13.c even 3 1 inner
507.2.j.d 4 13.d odd 4 2
507.2.j.d 4 13.f odd 12 2
624.2.q.c 2 52.b odd 2 1
624.2.q.c 2 52.i odd 6 1
975.2.i.f 2 65.d even 2 1
975.2.i.f 2 65.l even 6 1
975.2.bb.d 4 65.h odd 4 2
975.2.bb.d 4 65.r odd 12 2
1521.2.a.a 1 39.h odd 6 1
1521.2.a.d 1 39.i odd 6 1
1521.2.b.c 2 39.k even 12 2
1872.2.t.j 2 156.h even 2 1
1872.2.t.j 2 156.r even 6 1
8112.2.a.w 1 52.i odd 6 1
8112.2.a.bc 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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( 4 - 2 T + T^{2} \)
$11$ \( 4 + 2 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( 49 - 7 T + T^{2} \)
$19$ \( 36 + 6 T + T^{2} \)
$23$ \( 36 - 6 T + T^{2} \)
$29$ \( 1 - T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 1 - T + T^{2} \)
$41$ \( 81 - 9 T + T^{2} \)
$43$ \( 36 + 6 T + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( ( 9 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 4 + 2 T + T^{2} \)
$71$ \( 36 - 6 T + T^{2} \)
$73$ \( ( 11 + T )^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( ( -14 + T )^{2} \)
$89$ \( 196 + 14 T + T^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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