Properties

Label 507.2.b.a
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: \(x^{2} + 1\)
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.

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

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 2.00000i 1.00000i 4.00000i 3.00000i 1.00000 −2.00000
337.2 1.00000i −1.00000 1.00000 2.00000i 1.00000i 4.00000i 3.00000i 1.00000 −2.00000
\(n\): 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.a 2
3.b odd 2 1 1521.2.b.b 2
13.b even 2 1 inner 507.2.b.a 2
13.c even 3 2 507.2.j.e 4
13.d odd 4 1 39.2.a.a 1
13.d odd 4 1 507.2.a.a 1
13.e even 6 2 507.2.j.e 4
13.f odd 12 2 507.2.e.a 2
13.f odd 12 2 507.2.e.b 2
39.d odd 2 1 1521.2.b.b 2
39.f even 4 1 117.2.a.a 1
39.f even 4 1 1521.2.a.e 1
52.f even 4 1 624.2.a.i 1
52.f even 4 1 8112.2.a.s 1
65.f even 4 1 975.2.c.f 2
65.g odd 4 1 975.2.a.f 1
65.k even 4 1 975.2.c.f 2
91.i even 4 1 1911.2.a.f 1
104.j odd 4 1 2496.2.a.q 1
104.m even 4 1 2496.2.a.e 1
117.y odd 12 2 1053.2.e.b 2
117.z even 12 2 1053.2.e.d 2
143.g even 4 1 4719.2.a.c 1
156.l odd 4 1 1872.2.a.h 1
195.j odd 4 1 2925.2.c.e 2
195.n even 4 1 2925.2.a.p 1
195.u odd 4 1 2925.2.c.e 2
273.o odd 4 1 5733.2.a.e 1
312.w odd 4 1 7488.2.a.by 1
312.y even 4 1 7488.2.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 13.d odd 4 1
117.2.a.a 1 39.f even 4 1
507.2.a.a 1 13.d odd 4 1
507.2.b.a 2 1.a even 1 1 trivial
507.2.b.a 2 13.b even 2 1 inner
507.2.e.a 2 13.f odd 12 2
507.2.e.b 2 13.f odd 12 2
507.2.j.e 4 13.c even 3 2
507.2.j.e 4 13.e even 6 2
624.2.a.i 1 52.f even 4 1
975.2.a.f 1 65.g odd 4 1
975.2.c.f 2 65.f even 4 1
975.2.c.f 2 65.k even 4 1
1053.2.e.b 2 117.y odd 12 2
1053.2.e.d 2 117.z even 12 2
1521.2.a.e 1 39.f even 4 1
1521.2.b.b 2 3.b odd 2 1
1521.2.b.b 2 39.d odd 2 1
1872.2.a.h 1 156.l odd 4 1
1911.2.a.f 1 91.i even 4 1
2496.2.a.e 1 104.m even 4 1
2496.2.a.q 1 104.j odd 4 1
2925.2.a.p 1 195.n even 4 1
2925.2.c.e 2 195.j odd 4 1
2925.2.c.e 2 195.u odd 4 1
4719.2.a.c 1 143.g even 4 1
5733.2.a.e 1 273.o odd 4 1
7488.2.a.bl 1 312.y even 4 1
7488.2.a.by 1 312.w odd 4 1
8112.2.a.s 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 \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{5}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 4 + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( 16 + T^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( 36 + T^{2} \)
$43$ \( ( -12 + T )^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -6 + T )^{2} \)
$59$ \( 144 + T^{2} \)
$61$ \( ( 2 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( 4 + T^{2} \)
$97$ \( 100 + T^{2} \)
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