Properties

Label 600.1.z.a
Level 600
Weight 1
Character orbit 600.z
Analytic conductor 0.299
Analytic rank 0
Dimension 8
Projective image \(D_{10}\)
CM discriminant -24
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.299439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{10}\)
Projective field Galois closure of 10.0.759375000000000000.15

$q$-expansion

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

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

Character values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{20}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.951057 + 0.309017i
−0.951057 0.309017i
0.951057 0.309017i
−0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 0.809017i
0.587785 0.809017i
−0.587785 + 0.809017i
−0.951057 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0.951057 + 0.309017i 0.809017 0.587785i 1.17557i −0.587785 0.809017i −0.309017 0.951057i −0.809017 0.587785i
29.2 0.951057 + 0.309017i 0.587785 0.809017i 0.809017 + 0.587785i −0.951057 0.309017i 0.809017 0.587785i 1.17557i 0.587785 + 0.809017i −0.309017 0.951057i −0.809017 0.587785i
269.1 −0.951057 + 0.309017i −0.587785 0.809017i 0.809017 0.587785i 0.951057 0.309017i 0.809017 + 0.587785i 1.17557i −0.587785 + 0.809017i −0.309017 + 0.951057i −0.809017 + 0.587785i
269.2 0.951057 0.309017i 0.587785 + 0.809017i 0.809017 0.587785i −0.951057 + 0.309017i 0.809017 + 0.587785i 1.17557i 0.587785 0.809017i −0.309017 + 0.951057i −0.809017 + 0.587785i
389.1 −0.587785 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0.587785 + 0.809017i −0.309017 0.951057i 1.90211i 0.951057 0.309017i 0.809017 + 0.587785i 0.309017 0.951057i
389.2 0.587785 + 0.809017i −0.951057 0.309017i −0.309017 + 0.951057i −0.587785 0.809017i −0.309017 0.951057i 1.90211i −0.951057 + 0.309017i 0.809017 + 0.587785i 0.309017 0.951057i
509.1 −0.587785 + 0.809017i 0.951057 0.309017i −0.309017 0.951057i 0.587785 0.809017i −0.309017 + 0.951057i 1.90211i 0.951057 + 0.309017i 0.809017 0.587785i 0.309017 + 0.951057i
509.2 0.587785 0.809017i −0.951057 + 0.309017i −0.309017 0.951057i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.90211i −0.951057 0.309017i 0.809017 0.587785i 0.309017 + 0.951057i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 509.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner
200.o even 10 1 inner
600.z 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.z.a 8
3.b odd 2 1 inner 600.1.z.a 8
4.b odd 2 1 2400.1.cf.a 8
5.b even 2 1 3000.1.z.c 8
5.c odd 4 1 3000.1.bj.c 8
5.c odd 4 1 3000.1.bj.d 8
8.b even 2 1 inner 600.1.z.a 8
8.d odd 2 1 2400.1.cf.a 8
12.b even 2 1 2400.1.cf.a 8
15.d odd 2 1 3000.1.z.c 8
15.e even 4 1 3000.1.bj.c 8
15.e even 4 1 3000.1.bj.d 8
24.f even 2 1 2400.1.cf.a 8
24.h odd 2 1 CM 600.1.z.a 8
25.d even 5 1 3000.1.z.c 8
25.e even 10 1 inner 600.1.z.a 8
25.f odd 20 1 3000.1.bj.c 8
25.f odd 20 1 3000.1.bj.d 8
40.f even 2 1 3000.1.z.c 8
40.i odd 4 1 3000.1.bj.c 8
40.i odd 4 1 3000.1.bj.d 8
75.h odd 10 1 inner 600.1.z.a 8
75.j odd 10 1 3000.1.z.c 8
75.l even 20 1 3000.1.bj.c 8
75.l even 20 1 3000.1.bj.d 8
100.h odd 10 1 2400.1.cf.a 8
120.i odd 2 1 3000.1.z.c 8
120.w even 4 1 3000.1.bj.c 8
120.w even 4 1 3000.1.bj.d 8
200.o even 10 1 inner 600.1.z.a 8
200.s odd 10 1 2400.1.cf.a 8
200.t even 10 1 3000.1.z.c 8
200.x odd 20 1 3000.1.bj.c 8
200.x odd 20 1 3000.1.bj.d 8
300.r even 10 1 2400.1.cf.a 8
600.z odd 10 1 inner 600.1.z.a 8
600.bj odd 10 1 3000.1.z.c 8
600.bk even 10 1 2400.1.cf.a 8
600.bp even 20 1 3000.1.bj.c 8
600.bp even 20 1 3000.1.bj.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.1.z.a 8 1.a even 1 1 trivial
600.1.z.a 8 3.b odd 2 1 inner
600.1.z.a 8 8.b even 2 1 inner
600.1.z.a 8 24.h odd 2 1 CM
600.1.z.a 8 25.e even 10 1 inner
600.1.z.a 8 75.h odd 10 1 inner
600.1.z.a 8 200.o even 10 1 inner
600.1.z.a 8 600.z odd 10 1 inner
2400.1.cf.a 8 4.b odd 2 1
2400.1.cf.a 8 8.d odd 2 1
2400.1.cf.a 8 12.b even 2 1
2400.1.cf.a 8 24.f even 2 1
2400.1.cf.a 8 100.h odd 10 1
2400.1.cf.a 8 200.s odd 10 1
2400.1.cf.a 8 300.r even 10 1
2400.1.cf.a 8 600.bk even 10 1
3000.1.z.c 8 5.b even 2 1
3000.1.z.c 8 15.d odd 2 1
3000.1.z.c 8 25.d even 5 1
3000.1.z.c 8 40.f even 2 1
3000.1.z.c 8 75.j odd 10 1
3000.1.z.c 8 120.i odd 2 1
3000.1.z.c 8 200.t even 10 1
3000.1.z.c 8 600.bj odd 10 1
3000.1.bj.c 8 5.c odd 4 1
3000.1.bj.c 8 15.e even 4 1
3000.1.bj.c 8 25.f odd 20 1
3000.1.bj.c 8 40.i odd 4 1
3000.1.bj.c 8 75.l even 20 1
3000.1.bj.c 8 120.w even 4 1
3000.1.bj.c 8 200.x odd 20 1
3000.1.bj.c 8 600.bp even 20 1
3000.1.bj.d 8 5.c odd 4 1
3000.1.bj.d 8 15.e even 4 1
3000.1.bj.d 8 25.f odd 20 1
3000.1.bj.d 8 40.i odd 4 1
3000.1.bj.d 8 75.l even 20 1
3000.1.bj.d 8 120.w even 4 1
3000.1.bj.d 8 200.x odd 20 1
3000.1.bj.d 8 600.bp even 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(600, [\chi])\).

Hecke Characteristic Polynomials

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