Properties

Label 507.2.e.h
Level $507$
Weight $2$
Character orbit 507.e
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.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 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.

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

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

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

Character values

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

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−0.207107 + 0.358719i −0.500000 + 0.866025i 0.914214 + 1.58346i −2.82843 −0.207107 0.358719i −1.41421 2.44949i −1.58579 −0.500000 0.866025i 0.585786 1.01461i
22.2 1.20711 2.09077i −0.500000 + 0.866025i −1.91421 3.31552i 2.82843 1.20711 + 2.09077i 1.41421 + 2.44949i −4.41421 −0.500000 0.866025i 3.41421 5.91359i
484.1 −0.207107 0.358719i −0.500000 0.866025i 0.914214 1.58346i −2.82843 −0.207107 + 0.358719i −1.41421 + 2.44949i −1.58579 −0.500000 + 0.866025i 0.585786 + 1.01461i
484.2 1.20711 + 2.09077i −0.500000 0.866025i −1.91421 + 3.31552i 2.82843 1.20711 2.09077i 1.41421 2.44949i −4.41421 −0.500000 + 0.866025i 3.41421 + 5.91359i
\(n\): 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.h 4
13.b even 2 1 507.2.e.d 4
13.c even 3 1 39.2.a.b 2
13.c even 3 1 inner 507.2.e.h 4
13.d odd 4 2 507.2.j.f 8
13.e even 6 1 507.2.a.h 2
13.e even 6 1 507.2.e.d 4
13.f odd 12 2 507.2.b.e 4
13.f odd 12 2 507.2.j.f 8
39.h odd 6 1 1521.2.a.f 2
39.i odd 6 1 117.2.a.c 2
39.k even 12 2 1521.2.b.j 4
52.i odd 6 1 8112.2.a.bm 2
52.j odd 6 1 624.2.a.k 2
65.n even 6 1 975.2.a.l 2
65.q odd 12 2 975.2.c.h 4
91.n odd 6 1 1911.2.a.h 2
104.n odd 6 1 2496.2.a.bi 2
104.r even 6 1 2496.2.a.bf 2
117.f even 3 1 1053.2.e.m 4
117.h even 3 1 1053.2.e.m 4
117.k odd 6 1 1053.2.e.e 4
117.u odd 6 1 1053.2.e.e 4
143.k odd 6 1 4719.2.a.p 2
156.p even 6 1 1872.2.a.w 2
195.x odd 6 1 2925.2.a.v 2
195.bl even 12 2 2925.2.c.u 4
273.bn even 6 1 5733.2.a.u 2
312.bh odd 6 1 7488.2.a.cl 2
312.bn even 6 1 7488.2.a.co 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 13.c even 3 1
117.2.a.c 2 39.i odd 6 1
507.2.a.h 2 13.e even 6 1
507.2.b.e 4 13.f odd 12 2
507.2.e.d 4 13.b even 2 1
507.2.e.d 4 13.e even 6 1
507.2.e.h 4 1.a even 1 1 trivial
507.2.e.h 4 13.c even 3 1 inner
507.2.j.f 8 13.d odd 4 2
507.2.j.f 8 13.f odd 12 2
624.2.a.k 2 52.j odd 6 1
975.2.a.l 2 65.n even 6 1
975.2.c.h 4 65.q odd 12 2
1053.2.e.e 4 117.k odd 6 1
1053.2.e.e 4 117.u odd 6 1
1053.2.e.m 4 117.f even 3 1
1053.2.e.m 4 117.h even 3 1
1521.2.a.f 2 39.h odd 6 1
1521.2.b.j 4 39.k even 12 2
1872.2.a.w 2 156.p even 6 1
1911.2.a.h 2 91.n odd 6 1
2496.2.a.bf 2 104.r even 6 1
2496.2.a.bi 2 104.n odd 6 1
2925.2.a.v 2 195.x odd 6 1
2925.2.c.u 4 195.bl even 12 2
4719.2.a.p 2 143.k odd 6 1
5733.2.a.u 2 273.bn even 6 1
7488.2.a.cl 2 312.bh odd 6 1
7488.2.a.co 2 312.bn even 6 1
8112.2.a.bm 2 52.i odd 6 1

Hecke kernels

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

\( T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + 44 T^{2} - 112 T + 784 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784 \) Copy content Toggle raw display
$41$ \( T^{4} + 16 T^{3} + 200 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$43$ \( T^{4} + 8 T^{3} + 80 T^{2} - 128 T + 256 \) Copy content Toggle raw display
$47$ \( (T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T + 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 4 T^{3} + 44 T^{2} - 112 T + 784 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + 140 T^{2} + \cdots + 15376 \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64 \) Copy content Toggle raw display
$71$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 24 T^{3} + 440 T^{2} + \cdots + 18496 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + 44 T^{2} + 112 T + 784 \) Copy content Toggle raw display
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