Properties

Label 507.2.j.c
Level $507$
Weight $2$
Character orbit 507.j
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.j (of order \(6\), 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + ( - \zeta_{6} + 2) q^{6} + ( - 2 \zeta_{6} + 4) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 1) q^{3} + \zeta_{6} q^{4} + ( - \zeta_{6} + 2) q^{6} + ( - 2 \zeta_{6} + 4) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} - 2) q^{11} + q^{12} + 6 q^{14} + ( - 5 \zeta_{6} + 5) q^{16} + 6 \zeta_{6} q^{17} + ( - 2 \zeta_{6} + 1) q^{18} + (2 \zeta_{6} - 4) q^{19} + ( - 4 \zeta_{6} + 2) q^{21} - 6 \zeta_{6} q^{22} + ( - \zeta_{6} - 1) q^{24} + 5 q^{25} - q^{27} + (2 \zeta_{6} + 2) q^{28} + (6 \zeta_{6} - 6) q^{29} + (4 \zeta_{6} - 2) q^{31} + ( - 3 \zeta_{6} + 6) q^{32} + (2 \zeta_{6} - 4) q^{33} + (12 \zeta_{6} - 6) q^{34} + ( - \zeta_{6} + 1) q^{36} + (4 \zeta_{6} + 4) q^{37} - 6 q^{38} + ( - 4 \zeta_{6} - 4) q^{41} + ( - 6 \zeta_{6} + 6) q^{42} + 4 \zeta_{6} q^{43} + ( - 4 \zeta_{6} + 2) q^{44} + ( - 4 \zeta_{6} + 2) q^{47} - 5 \zeta_{6} q^{48} + ( - 5 \zeta_{6} + 5) q^{49} + (5 \zeta_{6} + 5) q^{50} + 6 q^{51} + 6 q^{53} + ( - \zeta_{6} - 1) q^{54} - 6 \zeta_{6} q^{56} + (4 \zeta_{6} - 2) q^{57} + (6 \zeta_{6} - 12) q^{58} + (6 \zeta_{6} - 12) q^{59} + 2 \zeta_{6} q^{61} + (6 \zeta_{6} - 6) q^{62} + ( - 2 \zeta_{6} - 2) q^{63} - q^{64} - 6 q^{66} + ( - 6 \zeta_{6} - 6) q^{67} + (6 \zeta_{6} - 6) q^{68} + (2 \zeta_{6} - 4) q^{71} + (\zeta_{6} - 2) q^{72} + 12 \zeta_{6} q^{74} + ( - 5 \zeta_{6} + 5) q^{75} + ( - 2 \zeta_{6} - 2) q^{76} - 12 q^{77} - 8 q^{79} + (\zeta_{6} - 1) q^{81} - 12 \zeta_{6} q^{82} + (4 \zeta_{6} - 2) q^{83} + ( - 2 \zeta_{6} + 4) q^{84} + (8 \zeta_{6} - 4) q^{86} + 6 \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{88} + ( - 4 \zeta_{6} - 4) q^{89} + (2 \zeta_{6} + 2) q^{93} + ( - 6 \zeta_{6} + 6) q^{94} + ( - 6 \zeta_{6} + 3) q^{96} + ( - 8 \zeta_{6} + 16) q^{97} + ( - 5 \zeta_{6} + 10) q^{98} + (4 \zeta_{6} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + q^{3} + q^{4} + 3 q^{6} + 6 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + q^{3} + q^{4} + 3 q^{6} + 6 q^{7} - q^{9} - 6 q^{11} + 2 q^{12} + 12 q^{14} + 5 q^{16} + 6 q^{17} - 6 q^{19} - 6 q^{22} - 3 q^{24} + 10 q^{25} - 2 q^{27} + 6 q^{28} - 6 q^{29} + 9 q^{32} - 6 q^{33} + q^{36} + 12 q^{37} - 12 q^{38} - 12 q^{41} + 6 q^{42} + 4 q^{43} - 5 q^{48} + 5 q^{49} + 15 q^{50} + 12 q^{51} + 12 q^{53} - 3 q^{54} - 6 q^{56} - 18 q^{58} - 18 q^{59} + 2 q^{61} - 6 q^{62} - 6 q^{63} - 2 q^{64} - 12 q^{66} - 18 q^{67} - 6 q^{68} - 6 q^{71} - 3 q^{72} + 12 q^{74} + 5 q^{75} - 6 q^{76} - 24 q^{77} - 16 q^{79} - q^{81} - 12 q^{82} + 6 q^{84} + 6 q^{87} - 6 q^{88} - 12 q^{89} + 6 q^{93} + 6 q^{94} + 24 q^{97} + 15 q^{98}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
0.500000 + 0.866025i
0.500000 0.866025i
1.50000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i 0 1.50000 0.866025i 3.00000 1.73205i 1.73205i −0.500000 0.866025i 0
361.1 1.50000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 0 1.50000 + 0.866025i 3.00000 + 1.73205i 1.73205i −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
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.c 2
13.b even 2 1 507.2.j.a 2
13.c even 3 1 39.2.b.a 2
13.c even 3 1 507.2.j.a 2
13.d odd 4 2 507.2.e.e 4
13.e even 6 1 39.2.b.a 2
13.e even 6 1 inner 507.2.j.c 2
13.f odd 12 2 507.2.a.f 2
13.f odd 12 2 507.2.e.e 4
39.h odd 6 1 117.2.b.a 2
39.i odd 6 1 117.2.b.a 2
39.k even 12 2 1521.2.a.l 2
52.i odd 6 1 624.2.c.e 2
52.j odd 6 1 624.2.c.e 2
52.l even 12 2 8112.2.a.bv 2
65.l even 6 1 975.2.b.d 2
65.n even 6 1 975.2.b.d 2
65.q odd 12 2 975.2.h.f 4
65.r odd 12 2 975.2.h.f 4
91.n odd 6 1 1911.2.c.d 2
91.t odd 6 1 1911.2.c.d 2
104.n odd 6 1 2496.2.c.d 2
104.p odd 6 1 2496.2.c.d 2
104.r even 6 1 2496.2.c.k 2
104.s even 6 1 2496.2.c.k 2
156.p even 6 1 1872.2.c.e 2
156.r even 6 1 1872.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 13.c even 3 1
39.2.b.a 2 13.e even 6 1
117.2.b.a 2 39.h odd 6 1
117.2.b.a 2 39.i odd 6 1
507.2.a.f 2 13.f odd 12 2
507.2.e.e 4 13.d odd 4 2
507.2.e.e 4 13.f odd 12 2
507.2.j.a 2 13.b even 2 1
507.2.j.a 2 13.c even 3 1
507.2.j.c 2 1.a even 1 1 trivial
507.2.j.c 2 13.e even 6 1 inner
624.2.c.e 2 52.i odd 6 1
624.2.c.e 2 52.j odd 6 1
975.2.b.d 2 65.l even 6 1
975.2.b.d 2 65.n even 6 1
975.2.h.f 4 65.q odd 12 2
975.2.h.f 4 65.r odd 12 2
1521.2.a.l 2 39.k even 12 2
1872.2.c.e 2 156.p even 6 1
1872.2.c.e 2 156.r even 6 1
1911.2.c.d 2 91.n odd 6 1
1911.2.c.d 2 91.t odd 6 1
2496.2.c.d 2 104.n odd 6 1
2496.2.c.d 2 104.p odd 6 1
2496.2.c.k 2 104.r even 6 1
2496.2.c.k 2 104.s even 6 1
8112.2.a.bv 2 52.l even 12 2

Hecke kernels

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

\( T_{2}^{2} - 3T_{2} + 3 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 12 \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 18T + 108 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 24T + 192 \) Copy content Toggle raw display
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