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 $2$

Related objects

Downloads

Learn more

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: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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