Properties

Label 600.1.q.b
Level 600
Weight 1
Character orbit 600.q
Analytic conductor 0.299
Analytic rank 0
Dimension 8
Projective image \(D_{6}\)
CM discriminant -8
Inner twists 16

Related objects

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.299439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.5400000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{24}^{3} q^{2} -\zeta_{24} q^{3} + \zeta_{24}^{6} q^{4} -\zeta_{24}^{4} q^{6} + \zeta_{24}^{9} q^{8} + \zeta_{24}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{24}^{3} q^{2} -\zeta_{24} q^{3} + \zeta_{24}^{6} q^{4} -\zeta_{24}^{4} q^{6} + \zeta_{24}^{9} q^{8} + \zeta_{24}^{2} q^{9} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{11} -\zeta_{24}^{7} q^{12} - q^{16} + \zeta_{24}^{3} q^{17} + \zeta_{24}^{5} q^{18} -\zeta_{24}^{6} q^{19} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{22} -\zeta_{24}^{10} q^{24} -\zeta_{24}^{3} q^{27} -\zeta_{24}^{3} q^{32} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{33} + \zeta_{24}^{6} q^{34} + \zeta_{24}^{8} q^{36} -\zeta_{24}^{9} q^{38} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{41} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{44} + \zeta_{24} q^{48} -\zeta_{24}^{6} q^{49} -\zeta_{24}^{4} q^{51} -\zeta_{24}^{6} q^{54} + \zeta_{24}^{7} q^{57} -\zeta_{24}^{6} q^{64} + ( 1 - \zeta_{24}^{8} ) q^{66} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{67} + \zeta_{24}^{9} q^{68} + \zeta_{24}^{11} q^{72} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{73} + q^{76} + \zeta_{24}^{4} q^{81} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{82} -\zeta_{24}^{9} q^{83} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{88} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{89} + \zeta_{24}^{4} q^{96} -\zeta_{24}^{9} q^{98} + ( \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{6} + O(q^{10}) \) \( 8q - 4q^{6} - 8q^{16} - 4q^{36} - 4q^{51} + 12q^{66} + 8q^{76} + 4q^{81} + 4q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 0.965926i
−0.707107 + 0.707107i −0.258819 + 0.965926i 1.00000i 0 −0.500000 0.866025i 0 0.707107 + 0.707107i −0.866025 0.500000i 0
107.2 −0.707107 + 0.707107i 0.965926 0.258819i 1.00000i 0 −0.500000 + 0.866025i 0 0.707107 + 0.707107i 0.866025 0.500000i 0
107.3 0.707107 0.707107i −0.965926 + 0.258819i 1.00000i 0 −0.500000 + 0.866025i 0 −0.707107 0.707107i 0.866025 0.500000i 0
107.4 0.707107 0.707107i 0.258819 0.965926i 1.00000i 0 −0.500000 0.866025i 0 −0.707107 0.707107i −0.866025 0.500000i 0
443.1 −0.707107 0.707107i −0.258819 0.965926i 1.00000i 0 −0.500000 + 0.866025i 0 0.707107 0.707107i −0.866025 + 0.500000i 0
443.2 −0.707107 0.707107i 0.965926 + 0.258819i 1.00000i 0 −0.500000 0.866025i 0 0.707107 0.707107i 0.866025 + 0.500000i 0
443.3 0.707107 + 0.707107i −0.965926 0.258819i 1.00000i 0 −0.500000 0.866025i 0 −0.707107 + 0.707107i 0.866025 + 0.500000i 0
443.4 0.707107 + 0.707107i 0.258819 + 0.965926i 1.00000i 0 −0.500000 + 0.866025i 0 −0.707107 + 0.707107i −0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 443.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner
24.f even 2 1 inner
40.e odd 2 1 inner
40.k even 4 2 inner
120.m even 2 1 inner
120.q odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.1.q.b 8
3.b odd 2 1 inner 600.1.q.b 8
4.b odd 2 1 2400.1.u.b 8
5.b even 2 1 inner 600.1.q.b 8
5.c odd 4 2 inner 600.1.q.b 8
8.b even 2 1 2400.1.u.b 8
8.d odd 2 1 CM 600.1.q.b 8
12.b even 2 1 2400.1.u.b 8
15.d odd 2 1 inner 600.1.q.b 8
15.e even 4 2 inner 600.1.q.b 8
20.d odd 2 1 2400.1.u.b 8
20.e even 4 2 2400.1.u.b 8
24.f even 2 1 inner 600.1.q.b 8
24.h odd 2 1 2400.1.u.b 8
40.e odd 2 1 inner 600.1.q.b 8
40.f even 2 1 2400.1.u.b 8
40.i odd 4 2 2400.1.u.b 8
40.k even 4 2 inner 600.1.q.b 8
60.h even 2 1 2400.1.u.b 8
60.l odd 4 2 2400.1.u.b 8
120.i odd 2 1 2400.1.u.b 8
120.m even 2 1 inner 600.1.q.b 8
120.q odd 4 2 inner 600.1.q.b 8
120.w even 4 2 2400.1.u.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.1.q.b 8 1.a even 1 1 trivial
600.1.q.b 8 3.b odd 2 1 inner
600.1.q.b 8 5.b even 2 1 inner
600.1.q.b 8 5.c odd 4 2 inner
600.1.q.b 8 8.d odd 2 1 CM
600.1.q.b 8 15.d odd 2 1 inner
600.1.q.b 8 15.e even 4 2 inner
600.1.q.b 8 24.f even 2 1 inner
600.1.q.b 8 40.e odd 2 1 inner
600.1.q.b 8 40.k even 4 2 inner
600.1.q.b 8 120.m even 2 1 inner
600.1.q.b 8 120.q odd 4 2 inner
2400.1.u.b 8 4.b odd 2 1
2400.1.u.b 8 8.b even 2 1
2400.1.u.b 8 12.b even 2 1
2400.1.u.b 8 20.d odd 2 1
2400.1.u.b 8 20.e even 4 2
2400.1.u.b 8 24.h odd 2 1
2400.1.u.b 8 40.f even 2 1
2400.1.u.b 8 40.i odd 4 2
2400.1.u.b 8 60.h even 2 1
2400.1.u.b 8 60.l odd 4 2
2400.1.u.b 8 120.i odd 2 1
2400.1.u.b 8 120.w even 4 2

Hecke kernels

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

Hecke Characteristic Polynomials

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