Properties

Label 507.1.i.a
Level $507$
Weight $1$
Character orbit 507.i
Analytic conductor $0.253$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -39, 13
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 507.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.253025961405\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\)
Artin image: $C_3\times D_4$
Artin field: Galois closure of 12.6.48388330638999.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{9} - q^{12} + \zeta_{6}^{2} q^{16} + q^{25} - q^{27} + \zeta_{6}^{2} q^{36} + 2 \zeta_{6} q^{43} + \zeta_{6} q^{48} -\zeta_{6}^{2} q^{49} + 2 \zeta_{6} q^{61} + q^{64} -\zeta_{6}^{2} q^{75} -2 q^{79} + \zeta_{6}^{2} q^{81} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} - q^{4} - q^{9} + O(q^{10}) \) \( 2q + q^{3} - q^{4} - q^{9} - 2q^{12} - q^{16} + 2q^{25} - 2q^{27} - q^{36} + 2q^{43} + q^{48} + q^{49} + 2q^{61} + 2q^{64} + q^{75} - 4q^{79} - q^{81} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 0 0 −0.500000 + 0.866025i 0
191.1 0 0.500000 0.866025i −0.500000 0.866025i 0 0 0 0 −0.500000 0.866025i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.b even 2 1 RM by \(\Q(\sqrt{13}) \)
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
13.c even 3 1 inner
13.e even 6 1 inner
39.h odd 6 1 inner
39.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.1.i.a 2
3.b odd 2 1 CM 507.1.i.a 2
13.b even 2 1 RM 507.1.i.a 2
13.c even 3 1 507.1.c.a 1
13.c even 3 1 inner 507.1.i.a 2
13.d odd 4 2 507.1.h.a 2
13.e even 6 1 507.1.c.a 1
13.e even 6 1 inner 507.1.i.a 2
13.f odd 12 2 39.1.d.a 1
13.f odd 12 2 507.1.h.a 2
39.d odd 2 1 CM 507.1.i.a 2
39.f even 4 2 507.1.h.a 2
39.h odd 6 1 507.1.c.a 1
39.h odd 6 1 inner 507.1.i.a 2
39.i odd 6 1 507.1.c.a 1
39.i odd 6 1 inner 507.1.i.a 2
39.k even 12 2 39.1.d.a 1
39.k even 12 2 507.1.h.a 2
52.l even 12 2 624.1.l.a 1
65.o even 12 2 975.1.e.a 2
65.s odd 12 2 975.1.g.a 1
65.t even 12 2 975.1.e.a 2
91.w even 12 2 1911.1.w.a 2
91.x odd 12 2 1911.1.w.b 2
91.ba even 12 2 1911.1.w.a 2
91.bc even 12 2 1911.1.h.a 1
91.bd odd 12 2 1911.1.w.b 2
104.u even 12 2 2496.1.l.a 1
104.x odd 12 2 2496.1.l.b 1
117.w odd 12 2 1053.1.n.b 2
117.x even 12 2 1053.1.n.b 2
117.bb odd 12 2 1053.1.n.b 2
117.bc even 12 2 1053.1.n.b 2
156.v odd 12 2 624.1.l.a 1
195.bc odd 12 2 975.1.e.a 2
195.bh even 12 2 975.1.g.a 1
195.bn odd 12 2 975.1.e.a 2
273.bs odd 12 2 1911.1.w.a 2
273.bv even 12 2 1911.1.w.b 2
273.bw even 12 2 1911.1.w.b 2
273.ca odd 12 2 1911.1.h.a 1
273.ch odd 12 2 1911.1.w.a 2
312.bo even 12 2 2496.1.l.b 1
312.bq odd 12 2 2496.1.l.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 13.f odd 12 2
39.1.d.a 1 39.k even 12 2
507.1.c.a 1 13.c even 3 1
507.1.c.a 1 13.e even 6 1
507.1.c.a 1 39.h odd 6 1
507.1.c.a 1 39.i odd 6 1
507.1.h.a 2 13.d odd 4 2
507.1.h.a 2 13.f odd 12 2
507.1.h.a 2 39.f even 4 2
507.1.h.a 2 39.k even 12 2
507.1.i.a 2 1.a even 1 1 trivial
507.1.i.a 2 3.b odd 2 1 CM
507.1.i.a 2 13.b even 2 1 RM
507.1.i.a 2 13.c even 3 1 inner
507.1.i.a 2 13.e even 6 1 inner
507.1.i.a 2 39.d odd 2 1 CM
507.1.i.a 2 39.h odd 6 1 inner
507.1.i.a 2 39.i odd 6 1 inner
624.1.l.a 1 52.l even 12 2
624.1.l.a 1 156.v odd 12 2
975.1.e.a 2 65.o even 12 2
975.1.e.a 2 65.t even 12 2
975.1.e.a 2 195.bc odd 12 2
975.1.e.a 2 195.bn odd 12 2
975.1.g.a 1 65.s odd 12 2
975.1.g.a 1 195.bh even 12 2
1053.1.n.b 2 117.w odd 12 2
1053.1.n.b 2 117.x even 12 2
1053.1.n.b 2 117.bb odd 12 2
1053.1.n.b 2 117.bc even 12 2
1911.1.h.a 1 91.bc even 12 2
1911.1.h.a 1 273.ca odd 12 2
1911.1.w.a 2 91.w even 12 2
1911.1.w.a 2 91.ba even 12 2
1911.1.w.a 2 273.bs odd 12 2
1911.1.w.a 2 273.ch odd 12 2
1911.1.w.b 2 91.x odd 12 2
1911.1.w.b 2 91.bd odd 12 2
1911.1.w.b 2 273.bv even 12 2
1911.1.w.b 2 273.bw even 12 2
2496.1.l.a 1 104.u even 12 2
2496.1.l.a 1 312.bq odd 12 2
2496.1.l.b 1 104.x odd 12 2
2496.1.l.b 1 312.bo even 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(507, [\chi])\).

Hecke characteristic polynomials

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