Properties

Label 507.2.b.b
Level $507$
Weight $2$
Character orbit 507.b
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{3} + q^{4} - i q^{5} - i q^{6} - 2 i q^{7} + 3 i q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - q^{3} + q^{4} - i q^{5} - i q^{6} - 2 i q^{7} + 3 i q^{8} + q^{9} + q^{10} + 2 i q^{11} - q^{12} + 2 q^{14} + i q^{15} - q^{16} + 7 q^{17} + i q^{18} - 6 i q^{19} - i q^{20} + 2 i q^{21} - 2 q^{22} + 6 q^{23} - 3 i q^{24} + 4 q^{25} - q^{27} - 2 i q^{28} - q^{29} - q^{30} + 4 i q^{31} + 5 i q^{32} - 2 i q^{33} + 7 i q^{34} - 2 q^{35} + q^{36} - i q^{37} + 6 q^{38} + 3 q^{40} + 9 i q^{41} - 2 q^{42} - 6 q^{43} + 2 i q^{44} - i q^{45} + 6 i q^{46} - 6 i q^{47} + q^{48} + 3 q^{49} + 4 i q^{50} - 7 q^{51} - 9 q^{53} - i q^{54} + 2 q^{55} + 6 q^{56} + 6 i q^{57} - i q^{58} + i q^{60} + q^{61} - 4 q^{62} - 2 i q^{63} - 7 q^{64} + 2 q^{66} - 2 i q^{67} + 7 q^{68} - 6 q^{69} - 2 i q^{70} + 6 i q^{71} + 3 i q^{72} - 11 i q^{73} + q^{74} - 4 q^{75} - 6 i q^{76} + 4 q^{77} - 4 q^{79} + i q^{80} + q^{81} - 9 q^{82} - 14 i q^{83} + 2 i q^{84} - 7 i q^{85} - 6 i q^{86} + q^{87} - 6 q^{88} + 14 i q^{89} + q^{90} + 6 q^{92} - 4 i q^{93} + 6 q^{94} - 6 q^{95} - 5 i q^{96} - 2 i q^{97} + 3 i q^{98} + 2 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} + 4 q^{14} - 2 q^{16} + 14 q^{17} - 4 q^{22} + 12 q^{23} + 8 q^{25} - 2 q^{27} - 2 q^{29} - 2 q^{30} - 4 q^{35} + 2 q^{36} + 12 q^{38} + 6 q^{40} - 4 q^{42} - 12 q^{43} + 2 q^{48} + 6 q^{49} - 14 q^{51} - 18 q^{53} + 4 q^{55} + 12 q^{56} + 2 q^{61} - 8 q^{62} - 14 q^{64} + 4 q^{66} + 14 q^{68} - 12 q^{69} + 2 q^{74} - 8 q^{75} + 8 q^{77} - 8 q^{79} + 2 q^{81} - 18 q^{82} + 2 q^{87} - 12 q^{88} + 2 q^{90} + 12 q^{92} + 12 q^{94} - 12 q^{95}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i −1.00000 1.00000 1.00000i 1.00000i 2.00000i 3.00000i 1.00000 1.00000
337.2 1.00000i −1.00000 1.00000 1.00000i 1.00000i 2.00000i 3.00000i 1.00000 1.00000
\(n\): 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.b 2
3.b odd 2 1 1521.2.b.c 2
13.b even 2 1 inner 507.2.b.b 2
13.c even 3 2 507.2.j.d 4
13.d odd 4 1 507.2.a.b 1
13.d odd 4 1 507.2.a.c 1
13.e even 6 2 507.2.j.d 4
13.f odd 12 2 39.2.e.a 2
13.f odd 12 2 507.2.e.c 2
39.d odd 2 1 1521.2.b.c 2
39.f even 4 1 1521.2.a.a 1
39.f even 4 1 1521.2.a.d 1
39.k even 12 2 117.2.g.b 2
52.f even 4 1 8112.2.a.w 1
52.f even 4 1 8112.2.a.bc 1
52.l even 12 2 624.2.q.c 2
65.o even 12 2 975.2.bb.d 4
65.s odd 12 2 975.2.i.f 2
65.t even 12 2 975.2.bb.d 4
156.v odd 12 2 1872.2.t.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 13.f odd 12 2
117.2.g.b 2 39.k even 12 2
507.2.a.b 1 13.d odd 4 1
507.2.a.c 1 13.d odd 4 1
507.2.b.b 2 1.a even 1 1 trivial
507.2.b.b 2 13.b even 2 1 inner
507.2.e.c 2 13.f odd 12 2
507.2.j.d 4 13.c even 3 2
507.2.j.d 4 13.e even 6 2
624.2.q.c 2 52.l even 12 2
975.2.i.f 2 65.s odd 12 2
975.2.bb.d 4 65.o even 12 2
975.2.bb.d 4 65.t even 12 2
1521.2.a.a 1 39.f even 4 1
1521.2.a.d 1 39.f even 4 1
1521.2.b.c 2 3.b odd 2 1
1521.2.b.c 2 39.d odd 2 1
1872.2.t.j 2 156.v odd 12 2
8112.2.a.w 1 52.f even 4 1
8112.2.a.bc 1 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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 81 \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 196 \) Copy content Toggle raw display
$89$ \( T^{2} + 196 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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