Properties

Label 507.2.e.j
Level $507$
Weight $2$
Character orbit 507.e
Analytic conductor $4.048$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{3} + 1) q^{5} + ( - \beta_{2} + \beta_1) q^{6} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{3} - \beta_{2}) q^{8} - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{4} + ( - \beta_{3} + 1) q^{5} + ( - \beta_{2} + \beta_1) q^{6} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{3} - \beta_{2}) q^{8} - \beta_{5} q^{9} + ( - \beta_{5} - 2 \beta_1 + 1) q^{10} + (2 \beta_{5} + 4 \beta_{4} + 3 \beta_1 - 2) q^{11} + ( - \beta_{3} + \beta_{2} - 1) q^{12} + (4 \beta_{2} - 1) q^{14} + (\beta_{5} - \beta_{4} - 1) q^{15} + (3 \beta_{4} + 2 \beta_1) q^{16} + (3 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{17} + \beta_{2} q^{18} + ( - 3 \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{19} + ( - \beta_{2} + \beta_1) q^{20} + ( - \beta_{3} + 3) q^{21} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{22} + (3 \beta_{5} + 5 \beta_{4} + 2 \beta_1 - 3) q^{23} + ( - \beta_{4} + \beta_1) q^{24} + ( - 2 \beta_{3} - \beta_{2} - 2) q^{25} + q^{27} + ( - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_1 + 2) q^{28} + ( - 3 \beta_{5} - 2 \beta_{4} + \beta_1 + 3) q^{29} + (\beta_{5} - 2 \beta_{2} + 2 \beta_1) q^{30} + (2 \beta_{3} - 5 \beta_{2} + 5) q^{31} + ( - \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{32} + ( - 2 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{33} + ( - \beta_{3} - \beta_{2}) q^{34} + ( - 5 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1) q^{35} + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{36} + (4 \beta_{5} - \beta_{4} - \beta_1 - 4) q^{37} + ( - 3 \beta_{3} + 5 \beta_{2} + 4) q^{38} + ( - \beta_{3} - 3 \beta_{2} + 3) q^{40} - \beta_1 q^{41} + ( - \beta_{5} - 4 \beta_1 + 1) q^{42} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{43} + ( - 3 \beta_{3} - \beta_{2} - 6) q^{44} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{45} + ( - 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{46} + ( - 3 \beta_{3} + 9 \beta_{2} - 7) q^{47} + ( - 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{48} + (4 \beta_{5} - 6 \beta_{4} + \beta_1 - 4) q^{49} + ( - 3 \beta_{5} + \beta_{4} + \beta_1 + 3) q^{50} + ( - \beta_{3} + \beta_{2} - 3) q^{51} + (2 \beta_{3} + \beta_{2} - 4) q^{53} - \beta_1 q^{54} + ( - 3 \beta_{5} + 2 \beta_{4} + 2 \beta_1 + 3) q^{55} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{56} + (\beta_{3} + 3 \beta_{2} + 3) q^{57} + (3 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{58} + (8 \beta_{5} + 6 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{59} + \beta_{2} q^{60} + (7 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 5 \beta_1) q^{61} + ( - 3 \beta_{5} + 5 \beta_{4} + 2 \beta_1 + 3) q^{62} + (3 \beta_{5} - \beta_{4} - 3) q^{63} + ( - 3 \beta_{3} - 2 \beta_{2} + 1) q^{64} + (3 \beta_{3} - 5 \beta_{2} + 1) q^{66} + (4 \beta_{5} + 3 \beta_{4} + 4 \beta_1 - 4) q^{67} + (4 \beta_{5} + 3 \beta_{4} + 2 \beta_1 - 4) q^{68} + ( - 3 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{69} + (\beta_{3} + 8 \beta_{2} - 5) q^{70} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{71} + (\beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{72} + (\beta_{3} - 2 \beta_{2} + 7) q^{73} + (\beta_{4} + \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{74} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_1 + 2) q^{75} + ( - 4 \beta_{5} - 7 \beta_{4} - 6 \beta_1 + 4) q^{76} + (10 \beta_{3} - 8 \beta_{2} + 1) q^{77} + (5 \beta_{3} - 4 \beta_{2}) q^{79} + ( - 4 \beta_{5} + 3 \beta_{4} + \beta_1 + 4) q^{80} + (\beta_{5} - 1) q^{81} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{82} + ( - 3 \beta_{3} - \beta_{2} - 6) q^{83} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{84} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1) q^{85} + (4 \beta_{3} - 5 \beta_{2} - 2) q^{86} + (3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1) q^{87} + ( - 6 \beta_{5} - 5 \beta_{4} - 6 \beta_1 + 6) q^{88} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_1 + 2) q^{89} + (2 \beta_{2} - 1) q^{90} + ( - 6 \beta_{3} + \beta_{2} - 8) q^{92} + (5 \beta_{5} + 2 \beta_{4} + 5 \beta_1 - 5) q^{93} + (6 \beta_{5} - 9 \beta_{4} - 5 \beta_1 - 6) q^{94} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} + 7 \beta_1) q^{95} + (4 \beta_{3} - 3 \beta_{2} + 1) q^{96} + (3 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{97} + (7 \beta_{5} - \beta_{4} - \beta_{3} - 9 \beta_{2} + 9 \beta_1) q^{98} + (4 \beta_{3} - 3 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 3 q^{3} + q^{4} + 8 q^{5} - q^{6} - 10 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} - 3 q^{3} + q^{4} + 8 q^{5} - q^{6} - 10 q^{7} - 3 q^{9} + q^{10} + q^{11} - 2 q^{12} + 2 q^{14} - 4 q^{15} + 5 q^{16} + 7 q^{17} + 2 q^{18} - 11 q^{19} - q^{20} + 20 q^{21} + 5 q^{22} - 2 q^{23} - 10 q^{25} + 6 q^{27} + q^{28} + 8 q^{29} + q^{30} + 16 q^{31} + 4 q^{32} + q^{33} - 18 q^{35} + q^{36} - 14 q^{37} + 40 q^{38} + 14 q^{40} - q^{41} - q^{42} + 3 q^{43} - 32 q^{44} - 4 q^{45} - 3 q^{46} - 18 q^{47} + 5 q^{48} - 17 q^{49} + 11 q^{50} - 14 q^{51} - 26 q^{53} - q^{54} + 13 q^{55} - 7 q^{56} + 22 q^{57} + 12 q^{58} + 14 q^{59} + 2 q^{60} + 13 q^{61} + 16 q^{62} - 10 q^{63} + 8 q^{64} - 10 q^{66} - 5 q^{67} - 7 q^{68} - 2 q^{69} - 16 q^{70} + 6 q^{71} + 36 q^{73} - 7 q^{74} + 5 q^{75} - q^{76} - 30 q^{77} - 18 q^{79} + 16 q^{80} - 3 q^{81} - 5 q^{82} - 32 q^{83} + q^{84} + 7 q^{85} - 30 q^{86} + 8 q^{87} + 7 q^{88} + 5 q^{89} - 2 q^{90} - 34 q^{92} - 8 q^{93} - 32 q^{94} - 3 q^{95} - 8 q^{96} - 5 q^{97} + 13 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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
0.900969 1.56052i
0.222521 0.385418i
−0.623490 + 1.07992i
0.900969 + 1.56052i
0.222521 + 0.385418i
−0.623490 1.07992i
−0.900969 + 1.56052i −0.500000 + 0.866025i −0.623490 1.07992i 1.44504 −0.900969 1.56052i −1.72252 2.98349i −1.35690 −0.500000 0.866025i −1.30194 + 2.25502i
22.2 −0.222521 + 0.385418i −0.500000 + 0.866025i 0.900969 + 1.56052i −0.246980 −0.222521 0.385418i −0.876510 1.51816i −1.69202 −0.500000 0.866025i 0.0549581 0.0951903i
22.3 0.623490 1.07992i −0.500000 + 0.866025i 0.222521 + 0.385418i 2.80194 0.623490 + 1.07992i −2.40097 4.15860i 3.04892 −0.500000 0.866025i 1.74698 3.02586i
484.1 −0.900969 1.56052i −0.500000 0.866025i −0.623490 + 1.07992i 1.44504 −0.900969 + 1.56052i −1.72252 + 2.98349i −1.35690 −0.500000 + 0.866025i −1.30194 2.25502i
484.2 −0.222521 0.385418i −0.500000 0.866025i 0.900969 1.56052i −0.246980 −0.222521 + 0.385418i −0.876510 + 1.51816i −1.69202 −0.500000 + 0.866025i 0.0549581 + 0.0951903i
484.3 0.623490 + 1.07992i −0.500000 0.866025i 0.222521 0.385418i 2.80194 0.623490 1.07992i −2.40097 + 4.15860i 3.04892 −0.500000 + 0.866025i 1.74698 + 3.02586i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 484.3
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.j 6
13.b even 2 1 507.2.e.k 6
13.c even 3 1 507.2.a.k yes 3
13.c even 3 1 inner 507.2.e.j 6
13.d odd 4 2 507.2.j.h 12
13.e even 6 1 507.2.a.j 3
13.e even 6 1 507.2.e.k 6
13.f odd 12 2 507.2.b.g 6
13.f odd 12 2 507.2.j.h 12
39.h odd 6 1 1521.2.a.q 3
39.i odd 6 1 1521.2.a.p 3
39.k even 12 2 1521.2.b.m 6
52.i odd 6 1 8112.2.a.by 3
52.j odd 6 1 8112.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 13.e even 6 1
507.2.a.k yes 3 13.c even 3 1
507.2.b.g 6 13.f odd 12 2
507.2.e.j 6 1.a even 1 1 trivial
507.2.e.j 6 13.c even 3 1 inner
507.2.e.k 6 13.b even 2 1
507.2.e.k 6 13.e even 6 1
507.2.j.h 12 13.d odd 4 2
507.2.j.h 12 13.f odd 12 2
1521.2.a.p 3 39.i odd 6 1
1521.2.a.q 3 39.h odd 6 1
1521.2.b.m 6 39.k even 12 2
8112.2.a.by 3 52.i odd 6 1
8112.2.a.cf 3 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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{6} + T_{2}^{5} + 3T_{2}^{4} + 5T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 4T_{5}^{2} + 3T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + 3 T^{4} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{3} - 4 T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 10 T^{5} + 69 T^{4} + \cdots + 841 \) Copy content Toggle raw display
$11$ \( T^{6} - T^{5} + 31 T^{4} - 56 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 7 T^{5} + 35 T^{4} - 84 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$19$ \( T^{6} + 11 T^{5} + 111 T^{4} + \cdots + 12769 \) Copy content Toggle raw display
$23$ \( T^{6} + 2 T^{5} + 47 T^{4} + \cdots + 6889 \) Copy content Toggle raw display
$29$ \( T^{6} - 8 T^{5} + 59 T^{4} + \cdots + 1849 \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} - 23 T + 197)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 14 T^{5} + 133 T^{4} + \cdots + 8281 \) Copy content Toggle raw display
$41$ \( T^{6} + T^{5} + 3 T^{4} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + 34 T^{4} + 133 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$47$ \( (T^{3} + 9 T^{2} - 120 T - 911)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 13 T^{2} + 40 T + 29)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 14 T^{5} + 196 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$61$ \( T^{6} - 13 T^{5} + 157 T^{4} + \cdots + 49729 \) Copy content Toggle raw display
$67$ \( T^{6} + 5 T^{5} + 47 T^{4} + \cdots + 9409 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} + 115 T^{4} + \cdots + 212521 \) Copy content Toggle raw display
$73$ \( (T^{3} - 18 T^{2} + 101 T - 167)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 9 T^{2} - 22 T - 169)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 16 T^{2} + 55 T - 43)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 5 T^{5} + 33 T^{4} + 42 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{6} + 5 T^{5} + 194 T^{4} + \cdots + 1413721 \) Copy content Toggle raw display
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