Properties

Label 507.2.a.g
Level $507$
Weight $2$
Character orbit 507.a
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.56155
2.56155
−1.56155 1.00000 0.438447 −3.56155 −1.56155 0.561553 2.43845 1.00000 5.56155
1.2 2.56155 1.00000 4.56155 0.561553 2.56155 −3.56155 6.56155 1.00000 1.43845
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.a.g 2
3.b odd 2 1 1521.2.a.g 2
4.b odd 2 1 8112.2.a.bk 2
13.b even 2 1 507.2.a.d 2
13.c even 3 2 39.2.e.b 4
13.d odd 4 2 507.2.b.d 4
13.e even 6 2 507.2.e.g 4
13.f odd 12 4 507.2.j.g 8
39.d odd 2 1 1521.2.a.m 2
39.f even 4 2 1521.2.b.h 4
39.i odd 6 2 117.2.g.c 4
52.b odd 2 1 8112.2.a.bo 2
52.j odd 6 2 624.2.q.h 4
65.n even 6 2 975.2.i.k 4
65.q odd 12 4 975.2.bb.i 8
156.p even 6 2 1872.2.t.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 13.c even 3 2
117.2.g.c 4 39.i odd 6 2
507.2.a.d 2 13.b even 2 1
507.2.a.g 2 1.a even 1 1 trivial
507.2.b.d 4 13.d odd 4 2
507.2.e.g 4 13.e even 6 2
507.2.j.g 8 13.f odd 12 4
624.2.q.h 4 52.j odd 6 2
975.2.i.k 4 65.n even 6 2
975.2.bb.i 8 65.q odd 12 4
1521.2.a.g 2 3.b odd 2 1
1521.2.a.m 2 39.d odd 2 1
1521.2.b.h 4 39.f even 4 2
1872.2.t.r 4 156.p even 6 2
8112.2.a.bk 2 4.b odd 2 1
8112.2.a.bo 2 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(507))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 2 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$41$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$47$ \( T^{2} - 68 \) Copy content Toggle raw display
$53$ \( T^{2} - 11T - 8 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
$61$ \( T^{2} - 16T + 47 \) Copy content Toggle raw display
$67$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$71$ \( (T - 14)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 19 \) Copy content Toggle raw display
$79$ \( T^{2} - 15T + 52 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$97$ \( T^{2} + 13T + 38 \) Copy content Toggle raw display
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