Properties

Label 600.1.bj.b
Level 600
Weight 1
Character orbit 600.bj
Analytic conductor 0.299
Analytic rank 0
Dimension 4
Projective image \(D_{5}\)
CM discriminant -24
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 600.bj (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.299439007580\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.225000000.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{10} q^{2} -\zeta_{10}^{2} q^{3} + \zeta_{10}^{2} q^{4} + \zeta_{10} q^{5} -\zeta_{10}^{3} q^{6} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7} + \zeta_{10}^{3} q^{8} + \zeta_{10}^{4} q^{9} +O(q^{10})\) \( q + \zeta_{10} q^{2} -\zeta_{10}^{2} q^{3} + \zeta_{10}^{2} q^{4} + \zeta_{10} q^{5} -\zeta_{10}^{3} q^{6} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{7} + \zeta_{10}^{3} q^{8} + \zeta_{10}^{4} q^{9} + \zeta_{10}^{2} q^{10} + ( -1 - \zeta_{10}^{2} ) q^{11} -\zeta_{10}^{4} q^{12} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{14} -\zeta_{10}^{3} q^{15} + \zeta_{10}^{4} q^{16} - q^{18} + \zeta_{10}^{3} q^{20} + ( -1 - \zeta_{10}^{4} ) q^{21} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{22} + q^{24} + \zeta_{10}^{2} q^{25} + \zeta_{10} q^{27} + ( 1 + \zeta_{10}^{4} ) q^{28} -2 \zeta_{10}^{2} q^{29} -\zeta_{10}^{4} q^{30} + ( 1 - \zeta_{10} ) q^{31} - q^{32} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{33} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{35} -\zeta_{10} q^{36} + \zeta_{10}^{4} q^{40} + ( 1 - \zeta_{10} ) q^{42} + ( -\zeta_{10}^{2} - \zeta_{10}^{4} ) q^{44} - q^{45} + \zeta_{10} q^{48} + ( 1 - \zeta_{10} + \zeta_{10}^{4} ) q^{49} + \zeta_{10}^{3} q^{50} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{53} + \zeta_{10}^{2} q^{54} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{55} + ( -1 + \zeta_{10} ) q^{56} -2 \zeta_{10}^{3} q^{58} + ( -1 + \zeta_{10}^{3} ) q^{59} + q^{60} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{62} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{63} -\zeta_{10} q^{64} + ( -1 + \zeta_{10}^{3} ) q^{66} + ( 1 + \zeta_{10}^{4} ) q^{70} -\zeta_{10}^{2} q^{72} -2 \zeta_{10} q^{73} -\zeta_{10}^{4} q^{75} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{77} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{79} - q^{80} -\zeta_{10}^{3} q^{81} + ( -1 + \zeta_{10} ) q^{83} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{84} + 2 \zeta_{10}^{4} q^{87} + ( 1 - \zeta_{10}^{3} ) q^{88} -\zeta_{10} q^{90} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{93} + \zeta_{10}^{2} q^{96} + ( 1 + \zeta_{10}^{4} ) q^{97} + ( -1 + \zeta_{10} - \zeta_{10}^{2} ) q^{98} + ( \zeta_{10} - \zeta_{10}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 2q^{7} + q^{8} - q^{9} + O(q^{10}) \) \( 4q + q^{2} + q^{3} - q^{4} + q^{5} - q^{6} - 2q^{7} + q^{8} - q^{9} - q^{10} - 3q^{11} + q^{12} + 2q^{14} - q^{15} - q^{16} - 4q^{18} + q^{20} - 3q^{21} - 2q^{22} + 4q^{24} - q^{25} + q^{27} + 3q^{28} + 2q^{29} + q^{30} + 3q^{31} - 4q^{32} - 2q^{33} + 2q^{35} - q^{36} - q^{40} + 3q^{42} + 2q^{44} - 4q^{45} + q^{48} + 2q^{49} + q^{50} + 2q^{53} - q^{54} - 2q^{55} - 3q^{56} - 2q^{58} - 3q^{59} + 4q^{60} + 2q^{62} - 2q^{63} - q^{64} - 3q^{66} + 3q^{70} + q^{72} - 2q^{73} + q^{75} - q^{77} - 2q^{79} - 4q^{80} - q^{81} - 3q^{83} + 2q^{84} - 2q^{87} + 3q^{88} - q^{90} + 2q^{93} - q^{96} + 3q^{97} - 2q^{98} + 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{10}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i 0.809017 + 0.587785i −0.809017 0.587785i −0.309017 + 0.951057i −0.809017 + 0.587785i −1.61803 0.809017 0.587785i 0.309017 + 0.951057i −0.809017 0.587785i
341.1 0.809017 + 0.587785i −0.309017 0.951057i 0.309017 + 0.951057i 0.809017 + 0.587785i 0.309017 0.951057i 0.618034 −0.309017 + 0.951057i −0.809017 + 0.587785i 0.309017 + 0.951057i
461.1 0.809017 0.587785i −0.309017 + 0.951057i 0.309017 0.951057i 0.809017 0.587785i 0.309017 + 0.951057i 0.618034 −0.309017 0.951057i −0.809017 0.587785i 0.309017 0.951057i
581.1 −0.309017 0.951057i 0.809017 0.587785i −0.809017 + 0.587785i −0.309017 0.951057i −0.809017 0.587785i −1.61803 0.809017 + 0.587785i 0.309017 0.951057i −0.809017 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
25.d even 5 1 inner
600.bj odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.1.bj.b yes 4
3.b odd 2 1 600.1.bj.a 4
4.b odd 2 1 2400.1.cp.a 4
5.b even 2 1 3000.1.bj.a 4
5.c odd 4 2 3000.1.z.a 8
8.b even 2 1 600.1.bj.a 4
8.d odd 2 1 2400.1.cp.b 4
12.b even 2 1 2400.1.cp.b 4
15.d odd 2 1 3000.1.bj.b 4
15.e even 4 2 3000.1.z.b 8
24.f even 2 1 2400.1.cp.a 4
24.h odd 2 1 CM 600.1.bj.b yes 4
25.d even 5 1 inner 600.1.bj.b yes 4
25.e even 10 1 3000.1.bj.a 4
25.f odd 20 2 3000.1.z.a 8
40.f even 2 1 3000.1.bj.b 4
40.i odd 4 2 3000.1.z.b 8
75.h odd 10 1 3000.1.bj.b 4
75.j odd 10 1 600.1.bj.a 4
75.l even 20 2 3000.1.z.b 8
100.j odd 10 1 2400.1.cp.a 4
120.i odd 2 1 3000.1.bj.a 4
120.w even 4 2 3000.1.z.a 8
200.n odd 10 1 2400.1.cp.b 4
200.o even 10 1 3000.1.bj.b 4
200.t even 10 1 600.1.bj.a 4
200.x odd 20 2 3000.1.z.b 8
300.n even 10 1 2400.1.cp.b 4
600.z odd 10 1 3000.1.bj.a 4
600.bg even 10 1 2400.1.cp.a 4
600.bj odd 10 1 inner 600.1.bj.b yes 4
600.bp even 20 2 3000.1.z.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.1.bj.a 4 3.b odd 2 1
600.1.bj.a 4 8.b even 2 1
600.1.bj.a 4 75.j odd 10 1
600.1.bj.a 4 200.t even 10 1
600.1.bj.b yes 4 1.a even 1 1 trivial
600.1.bj.b yes 4 24.h odd 2 1 CM
600.1.bj.b yes 4 25.d even 5 1 inner
600.1.bj.b yes 4 600.bj odd 10 1 inner
2400.1.cp.a 4 4.b odd 2 1
2400.1.cp.a 4 24.f even 2 1
2400.1.cp.a 4 100.j odd 10 1
2400.1.cp.a 4 600.bg even 10 1
2400.1.cp.b 4 8.d odd 2 1
2400.1.cp.b 4 12.b even 2 1
2400.1.cp.b 4 200.n odd 10 1
2400.1.cp.b 4 300.n even 10 1
3000.1.z.a 8 5.c odd 4 2
3000.1.z.a 8 25.f odd 20 2
3000.1.z.a 8 120.w even 4 2
3000.1.z.a 8 600.bp even 20 2
3000.1.z.b 8 15.e even 4 2
3000.1.z.b 8 40.i odd 4 2
3000.1.z.b 8 75.l even 20 2
3000.1.z.b 8 200.x odd 20 2
3000.1.bj.a 4 5.b even 2 1
3000.1.bj.a 4 25.e even 10 1
3000.1.bj.a 4 120.i odd 2 1
3000.1.bj.a 4 600.z odd 10 1
3000.1.bj.b 4 15.d odd 2 1
3000.1.bj.b 4 40.f even 2 1
3000.1.bj.b 4 75.h odd 10 1
3000.1.bj.b 4 200.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 3 T_{11}^{3} + 4 T_{11}^{2} + 2 T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(600, [\chi])\).

Hecke characteristic polynomials

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