Properties

Label 507.2.f.c
Level $507$
Weight $2$
Character orbit 507.f
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{4} + ( 2 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{4} + ( 2 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( -2 + 4 \zeta_{12}^{2} ) q^{12} -4 q^{16} + ( -5 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{19} + ( 5 + \zeta_{12} - \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{21} + 5 \zeta_{12}^{3} q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -2 + 6 \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{28} + ( -6 + 5 \zeta_{12} + 5 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{31} + 6 \zeta_{12}^{3} q^{36} + ( -7 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{37} + ( 1 - 2 \zeta_{12}^{2} ) q^{43} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{48} + ( 3 - 6 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{49} + ( 7 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{57} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{61} + ( 6 + 9 \zeta_{12} - 9 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{63} -8 \zeta_{12}^{3} q^{64} + ( -2 - 7 \zeta_{12} - 7 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{67} + ( -8 + \zeta_{12} - \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{73} + ( -5 + 10 \zeta_{12}^{2} ) q^{75} + ( 6 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{76} + ( -14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{79} + 9 q^{81} + ( 8 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{84} + ( 11 - 7 \zeta_{12} - 7 \zeta_{12}^{2} + 11 \zeta_{12}^{3} ) q^{93} + ( 8 - 11 \zeta_{12} - 11 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{7} + 12q^{9} + O(q^{10}) \) \( 4q + 2q^{7} + 12q^{9} - 16q^{16} - 16q^{19} + 18q^{21} + 4q^{28} - 14q^{31} - 20q^{37} + 12q^{57} + 6q^{63} - 22q^{67} - 34q^{73} + 32q^{76} + 36q^{81} + 36q^{84} + 30q^{93} + 10q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 −1.73205 2.00000i 0 0 −2.09808 2.09808i 0 3.00000 0
239.2 0 1.73205 2.00000i 0 0 3.09808 + 3.09808i 0 3.00000 0
437.1 0 −1.73205 2.00000i 0 0 −2.09808 + 2.09808i 0 3.00000 0
437.2 0 1.73205 2.00000i 0 0 3.09808 3.09808i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.f.c 4
3.b odd 2 1 CM 507.2.f.c 4
13.b even 2 1 507.2.f.b 4
13.c even 3 1 39.2.k.a 4
13.c even 3 1 507.2.k.b 4
13.d odd 4 1 507.2.f.b 4
13.d odd 4 1 inner 507.2.f.c 4
13.e even 6 1 507.2.k.a 4
13.e even 6 1 507.2.k.c 4
13.f odd 12 1 39.2.k.a 4
13.f odd 12 1 507.2.k.a 4
13.f odd 12 1 507.2.k.b 4
13.f odd 12 1 507.2.k.c 4
39.d odd 2 1 507.2.f.b 4
39.f even 4 1 507.2.f.b 4
39.f even 4 1 inner 507.2.f.c 4
39.h odd 6 1 507.2.k.a 4
39.h odd 6 1 507.2.k.c 4
39.i odd 6 1 39.2.k.a 4
39.i odd 6 1 507.2.k.b 4
39.k even 12 1 39.2.k.a 4
39.k even 12 1 507.2.k.a 4
39.k even 12 1 507.2.k.b 4
39.k even 12 1 507.2.k.c 4
52.j odd 6 1 624.2.cn.b 4
52.l even 12 1 624.2.cn.b 4
65.n even 6 1 975.2.bo.c 4
65.o even 12 1 975.2.bp.a 4
65.q odd 12 1 975.2.bp.a 4
65.q odd 12 1 975.2.bp.d 4
65.s odd 12 1 975.2.bo.c 4
65.t even 12 1 975.2.bp.d 4
156.p even 6 1 624.2.cn.b 4
156.v odd 12 1 624.2.cn.b 4
195.x odd 6 1 975.2.bo.c 4
195.bc odd 12 1 975.2.bp.d 4
195.bh even 12 1 975.2.bo.c 4
195.bl even 12 1 975.2.bp.a 4
195.bl even 12 1 975.2.bp.d 4
195.bn odd 12 1 975.2.bp.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.a 4 13.c even 3 1
39.2.k.a 4 13.f odd 12 1
39.2.k.a 4 39.i odd 6 1
39.2.k.a 4 39.k even 12 1
507.2.f.b 4 13.b even 2 1
507.2.f.b 4 13.d odd 4 1
507.2.f.b 4 39.d odd 2 1
507.2.f.b 4 39.f even 4 1
507.2.f.c 4 1.a even 1 1 trivial
507.2.f.c 4 3.b odd 2 1 CM
507.2.f.c 4 13.d odd 4 1 inner
507.2.f.c 4 39.f even 4 1 inner
507.2.k.a 4 13.e even 6 1
507.2.k.a 4 13.f odd 12 1
507.2.k.a 4 39.h odd 6 1
507.2.k.a 4 39.k even 12 1
507.2.k.b 4 13.c even 3 1
507.2.k.b 4 13.f odd 12 1
507.2.k.b 4 39.i odd 6 1
507.2.k.b 4 39.k even 12 1
507.2.k.c 4 13.e even 6 1
507.2.k.c 4 13.f odd 12 1
507.2.k.c 4 39.h odd 6 1
507.2.k.c 4 39.k even 12 1
624.2.cn.b 4 52.j odd 6 1
624.2.cn.b 4 52.l even 12 1
624.2.cn.b 4 156.p even 6 1
624.2.cn.b 4 156.v odd 12 1
975.2.bo.c 4 65.n even 6 1
975.2.bo.c 4 65.s odd 12 1
975.2.bo.c 4 195.x odd 6 1
975.2.bo.c 4 195.bh even 12 1
975.2.bp.a 4 65.o even 12 1
975.2.bp.a 4 65.q odd 12 1
975.2.bp.a 4 195.bl even 12 1
975.2.bp.a 4 195.bn odd 12 1
975.2.bp.d 4 65.q odd 12 1
975.2.bp.d 4 65.t even 12 1
975.2.bp.d 4 195.bc odd 12 1
975.2.bp.d 4 195.bl even 12 1

Hecke kernels

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

\( T_{2} \)
\( T_{5} \)
\( T_{7}^{4} - 2 T_{7}^{3} + 2 T_{7}^{2} + 26 T_{7} + 169 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( 169 + 26 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( 676 + 416 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( 169 - 182 T + 98 T^{2} + 14 T^{3} + T^{4} \)
$37$ \( 676 + 520 T + 200 T^{2} + 20 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 3 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -75 + T^{2} )^{2} \)
$67$ \( 169 - 286 T + 242 T^{2} + 22 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( 20449 + 4862 T + 578 T^{2} + 34 T^{3} + T^{4} \)
$79$ \( ( -147 + T^{2} )^{2} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( 28561 + 1690 T + 50 T^{2} - 10 T^{3} + T^{4} \)
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