Properties

Label 507.4.b.a
Level $507$
Weight $4$
Character orbit 507.b
Analytic conductor $29.914$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + 8 q^{4} + 3 \beta q^{5} - 6 \beta q^{7} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + 8 q^{4} + 3 \beta q^{5} - 6 \beta q^{7} + 9 q^{9} + 30 \beta q^{11} - 24 q^{12} - 9 \beta q^{15} + 64 q^{16} - 117 q^{17} + 14 \beta q^{19} + 24 \beta q^{20} + 18 \beta q^{21} + 18 q^{23} + 98 q^{25} - 27 q^{27} - 48 \beta q^{28} - 99 q^{29} + 112 \beta q^{31} - 90 \beta q^{33} + 54 q^{35} + 72 q^{36} + 65 \beta q^{37} - 21 \beta q^{41} - 82 q^{43} + 240 \beta q^{44} + 27 \beta q^{45} - 42 \beta q^{47} - 192 q^{48} + 235 q^{49} + 351 q^{51} - 261 q^{53} - 270 q^{55} - 42 \beta q^{57} + 456 \beta q^{59} - 72 \beta q^{60} - 719 q^{61} - 54 \beta q^{63} + 512 q^{64} - 406 \beta q^{67} - 936 q^{68} - 54 q^{69} + 270 \beta q^{71} + 395 \beta q^{73} - 294 q^{75} + 112 \beta q^{76} + 540 q^{77} - 440 q^{79} + 192 \beta q^{80} + 81 q^{81} + 690 \beta q^{83} + 144 \beta q^{84} - 351 \beta q^{85} + 297 q^{87} + 876 \beta q^{89} + 144 q^{92} - 336 \beta q^{93} - 126 q^{95} + 668 \beta q^{97} + 270 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9} - 48 q^{12} + 128 q^{16} - 234 q^{17} + 36 q^{23} + 196 q^{25} - 54 q^{27} - 198 q^{29} + 108 q^{35} + 144 q^{36} - 164 q^{43} - 384 q^{48} + 470 q^{49} + 702 q^{51} - 522 q^{53} - 540 q^{55} - 1438 q^{61} + 1024 q^{64} - 1872 q^{68} - 108 q^{69} - 588 q^{75} + 1080 q^{77} - 880 q^{79} + 162 q^{81} + 594 q^{87} + 288 q^{92} - 252 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −3.00000 8.00000 5.19615i 0 10.3923i 0 9.00000 0
337.2 0 −3.00000 8.00000 5.19615i 0 10.3923i 0 9.00000 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.4.b.a 2
13.b even 2 1 inner 507.4.b.a 2
13.c even 3 1 39.4.j.a 2
13.d odd 4 2 507.4.a.g 2
13.e even 6 1 39.4.j.a 2
39.f even 4 2 1521.4.a.m 2
39.h odd 6 1 117.4.q.b 2
39.i odd 6 1 117.4.q.b 2
52.i odd 6 1 624.4.bv.a 2
52.j odd 6 1 624.4.bv.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.a 2 13.c even 3 1
39.4.j.a 2 13.e even 6 1
117.4.q.b 2 39.h odd 6 1
117.4.q.b 2 39.i odd 6 1
507.4.a.g 2 13.d odd 4 2
507.4.b.a 2 1.a even 1 1 trivial
507.4.b.a 2 13.b even 2 1 inner
624.4.bv.a 2 52.i odd 6 1
624.4.bv.a 2 52.j odd 6 1
1521.4.a.m 2 39.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(507, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{2} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 27 \) Copy content Toggle raw display
$7$ \( T^{2} + 108 \) Copy content Toggle raw display
$11$ \( T^{2} + 2700 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 117)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 588 \) Copy content Toggle raw display
$23$ \( (T - 18)^{2} \) Copy content Toggle raw display
$29$ \( (T + 99)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 37632 \) Copy content Toggle raw display
$37$ \( T^{2} + 12675 \) Copy content Toggle raw display
$41$ \( T^{2} + 1323 \) Copy content Toggle raw display
$43$ \( (T + 82)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 5292 \) Copy content Toggle raw display
$53$ \( (T + 261)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 623808 \) Copy content Toggle raw display
$61$ \( (T + 719)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 494508 \) Copy content Toggle raw display
$71$ \( T^{2} + 218700 \) Copy content Toggle raw display
$73$ \( T^{2} + 468075 \) Copy content Toggle raw display
$79$ \( (T + 440)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1428300 \) Copy content Toggle raw display
$89$ \( T^{2} + 2302128 \) Copy content Toggle raw display
$97$ \( T^{2} + 1338672 \) Copy content Toggle raw display
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