Properties

Label 507.2.b.d
Level $507$
Weight $2$
Character orbit 507.b
Analytic conductor $4.048$
Analytic rank $0$
Dimension $4$
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{3} - 3) q^{4} + (2 \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{3} - 3) q^{4} + (2 \beta_{2} + \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{2} + \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8} + q^{9} + ( - \beta_{3} - 3) q^{10} - 2 \beta_{2} q^{11} + (\beta_{3} - 3) q^{12} + (2 \beta_{3} - 6) q^{14} + (2 \beta_{2} + \beta_1) q^{15} + ( - 3 \beta_{3} + 3) q^{16} + (\beta_{3} - 1) q^{17} + \beta_1 q^{18} + ( - 4 \beta_{2} - 2 \beta_1) q^{19} - \beta_1 q^{20} + ( - \beta_{2} + \beta_1) q^{21} + (2 \beta_{3} - 2) q^{22} - 2 q^{23} + (4 \beta_{2} - \beta_1) q^{24} - 3 \beta_{3} q^{25} + q^{27} + (6 \beta_{2} - 4 \beta_1) q^{28} + (3 \beta_{3} - 1) q^{29} + ( - \beta_{3} - 3) q^{30} + ( - \beta_{2} - \beta_1) q^{31} + ( - 4 \beta_{2} + \beta_1) q^{32} - 2 \beta_{2} q^{33} + (4 \beta_{2} - \beta_1) q^{34} - 2 q^{35} + (\beta_{3} - 3) q^{36} + (6 \beta_{2} + \beta_1) q^{37} + (2 \beta_{3} + 6) q^{38} + ( - 3 \beta_{3} - 1) q^{40} + \beta_1 q^{41} + (2 \beta_{3} - 6) q^{42} + ( - \beta_{3} - 2) q^{43} + (4 \beta_{2} - 2 \beta_1) q^{44} + (2 \beta_{2} + \beta_1) q^{45} - 2 \beta_1 q^{46} + (2 \beta_{2} + 4 \beta_1) q^{47} + ( - 3 \beta_{3} + 3) q^{48} + (3 \beta_{3} - 1) q^{49} - 12 \beta_{2} q^{50} + (\beta_{3} - 1) q^{51} + ( - 3 \beta_{3} + 7) q^{53} + \beta_1 q^{54} + (2 \beta_{3} + 2) q^{55} + ( - 6 \beta_{3} + 14) q^{56} + ( - 4 \beta_{2} - 2 \beta_1) q^{57} + (12 \beta_{2} - \beta_1) q^{58} + ( - 6 \beta_{2} + 2 \beta_1) q^{59} - \beta_1 q^{60} + ( - 2 \beta_{3} + 9) q^{61} + 4 q^{62} + ( - \beta_{2} + \beta_1) q^{63} + ( - \beta_{3} - 3) q^{64} + (2 \beta_{3} - 2) q^{66} + ( - 3 \beta_{2} - \beta_1) q^{67} + ( - 3 \beta_{3} + 7) q^{68} - 2 q^{69} - 2 \beta_1 q^{70} - 14 \beta_{2} q^{71} + (4 \beta_{2} - \beta_1) q^{72} + ( - 7 \beta_{2} - 2 \beta_1) q^{73} + ( - 5 \beta_{3} + 1) q^{74} - 3 \beta_{3} q^{75} + 2 \beta_1 q^{76} + (2 \beta_{3} - 4) q^{77} + ( - \beta_{3} + 8) q^{79} + ( - 12 \beta_{2} - 3 \beta_1) q^{80} + q^{81} + (\beta_{3} - 5) q^{82} + (4 \beta_{2} - 2 \beta_1) q^{83} + (6 \beta_{2} - 4 \beta_1) q^{84} + (4 \beta_{2} + \beta_1) q^{85} + ( - 4 \beta_{2} - 2 \beta_1) q^{86} + (3 \beta_{3} - 1) q^{87} + ( - 2 \beta_{3} + 10) q^{88} + (8 \beta_{2} - 2 \beta_1) q^{89} + ( - \beta_{3} - 3) q^{90} + ( - 2 \beta_{3} + 6) q^{92} + ( - \beta_{2} - \beta_1) q^{93} + (2 \beta_{3} - 18) q^{94} + (6 \beta_{3} + 10) q^{95} + ( - 4 \beta_{2} + \beta_1) q^{96} + (7 \beta_{2} + \beta_1) q^{97} + (12 \beta_{2} - \beta_1) q^{98} - 2 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 10 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 10 q^{4} + 4 q^{9} - 14 q^{10} - 10 q^{12} - 20 q^{14} + 6 q^{16} - 2 q^{17} - 4 q^{22} - 8 q^{23} - 6 q^{25} + 4 q^{27} + 2 q^{29} - 14 q^{30} - 8 q^{35} - 10 q^{36} + 28 q^{38} - 10 q^{40} - 20 q^{42} - 10 q^{43} + 6 q^{48} + 2 q^{49} - 2 q^{51} + 22 q^{53} + 12 q^{55} + 44 q^{56} + 32 q^{61} + 16 q^{62} - 14 q^{64} - 4 q^{66} + 22 q^{68} - 8 q^{69} - 6 q^{74} - 6 q^{75} - 12 q^{77} + 30 q^{79} + 4 q^{81} - 18 q^{82} + 2 q^{87} + 36 q^{88} - 14 q^{90} + 20 q^{92} - 68 q^{94} + 52 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Character values

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

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

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
2.56155i
1.56155i
1.56155i
2.56155i
2.56155i 1.00000 −4.56155 0.561553i 2.56155i 3.56155i 6.56155i 1.00000 −1.43845
337.2 1.56155i 1.00000 −0.438447 3.56155i 1.56155i 0.561553i 2.43845i 1.00000 −5.56155
337.3 1.56155i 1.00000 −0.438447 3.56155i 1.56155i 0.561553i 2.43845i 1.00000 −5.56155
337.4 2.56155i 1.00000 −4.56155 0.561553i 2.56155i 3.56155i 6.56155i 1.00000 −1.43845
\(n\): 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 507.2.b.d 4
3.b odd 2 1 1521.2.b.h 4
13.b even 2 1 inner 507.2.b.d 4
13.c even 3 2 507.2.j.g 8
13.d odd 4 1 507.2.a.d 2
13.d odd 4 1 507.2.a.g 2
13.e even 6 2 507.2.j.g 8
13.f odd 12 2 39.2.e.b 4
13.f odd 12 2 507.2.e.g 4
39.d odd 2 1 1521.2.b.h 4
39.f even 4 1 1521.2.a.g 2
39.f even 4 1 1521.2.a.m 2
39.k even 12 2 117.2.g.c 4
52.f even 4 1 8112.2.a.bk 2
52.f even 4 1 8112.2.a.bo 2
52.l even 12 2 624.2.q.h 4
65.o even 12 2 975.2.bb.i 8
65.s odd 12 2 975.2.i.k 4
65.t even 12 2 975.2.bb.i 8
156.v odd 12 2 1872.2.t.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 13.f odd 12 2
117.2.g.c 4 39.k even 12 2
507.2.a.d 2 13.d odd 4 1
507.2.a.g 2 13.d odd 4 1
507.2.b.d 4 1.a even 1 1 trivial
507.2.b.d 4 13.b even 2 1 inner
507.2.e.g 4 13.f odd 12 2
507.2.j.g 8 13.c even 3 2
507.2.j.g 8 13.e even 6 2
624.2.q.h 4 52.l even 12 2
975.2.i.k 4 65.s odd 12 2
975.2.bb.i 8 65.o even 12 2
975.2.bb.i 8 65.t even 12 2
1521.2.a.g 2 39.f even 4 1
1521.2.a.m 2 39.f even 4 1
1521.2.b.h 4 3.b odd 2 1
1521.2.b.h 4 39.d odd 2 1
1872.2.t.r 4 156.v odd 12 2
8112.2.a.bk 2 52.f even 4 1
8112.2.a.bo 2 52.f even 4 1

Hecke kernels

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

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} + 13T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 52T^{2} + 64 \) Copy content Toggle raw display
$23$ \( (T + 2)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 38)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 69T^{2} + 676 \) Copy content Toggle raw display
$41$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T + 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 68)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 11 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 132T^{2} + 1024 \) Copy content Toggle raw display
$61$ \( (T^{2} - 16 T + 47)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 106T^{2} + 361 \) Copy content Toggle raw display
$79$ \( (T^{2} - 15 T + 52)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 84T^{2} + 64 \) Copy content Toggle raw display
$89$ \( T^{4} + 196T^{2} + 4096 \) Copy content Toggle raw display
$97$ \( T^{4} + 93T^{2} + 1444 \) Copy content Toggle raw display
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