Properties

Label 600.2.m.b
Level 600
Weight 2
Character orbit 600.m
Analytic conductor 4.791
Analytic rank 0
Dimension 8
CM discriminant -8
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 600.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{2} + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{3} + 2 q^{4} + ( \zeta_{24} - 2 \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{6} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( -1 - 2 \zeta_{24}^{3} + \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{2} + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{3} + 2 q^{4} + ( \zeta_{24} - 2 \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{6} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( -1 - 2 \zeta_{24}^{3} + \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{9} + ( 3 - \zeta_{24} + \zeta_{24}^{3} - 6 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{11} + ( 2 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{12} + 4 q^{16} + ( -2 \zeta_{24} - 6 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{17} + ( 4 \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{18} + ( 1 - 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{19} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{22} + ( 2 \zeta_{24} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{24} + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 5 \zeta_{24}^{6} ) q^{27} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{32} + ( 2 \zeta_{24} + \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 5 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{33} + ( 4 + 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{34} + ( -2 - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{36} + ( -\zeta_{24} + 12 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{38} + ( -3 - 4 \zeta_{24} + 4 \zeta_{24}^{3} + 6 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{41} -10 \zeta_{24}^{6} q^{43} + ( 6 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 12 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{44} + ( 4 \zeta_{24} - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{48} -7 q^{49} + ( 6 + 5 \zeta_{24} - 6 \zeta_{24}^{3} - 7 \zeta_{24}^{4} - 6 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{51} + ( -2 - 5 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} ) q^{54} + ( 4 \zeta_{24} - 7 \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 5 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{57} + ( 10 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} ) q^{59} + 8 q^{64} + ( -12 + 5 \zeta_{24} - 6 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 6 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{66} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 7 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{67} + ( -4 \zeta_{24} - 12 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{68} + ( 8 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{72} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - \zeta_{24}^{6} - 12 \zeta_{24}^{7} ) q^{73} + ( 2 - 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{76} + ( 4 \zeta_{24} + 7 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{81} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 8 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{82} + ( -\zeta_{24} - 18 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + 9 \zeta_{24}^{6} ) q^{83} + ( -10 \zeta_{24} + 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} ) q^{86} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{88} + ( -9 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 18 \zeta_{24}^{4} + 2 \zeta_{24}^{5} ) q^{89} + ( 4 \zeta_{24} - 8 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{96} + 10 \zeta_{24}^{6} q^{97} + ( 7 \zeta_{24} + 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} ) q^{98} + ( 7 - 12 \zeta_{24} + 5 \zeta_{24}^{3} - \zeta_{24}^{4} + 5 \zeta_{24}^{5} + 7 \zeta_{24}^{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{4} - 8q^{6} - 4q^{9} + O(q^{10}) \) \( 8q + 16q^{4} - 8q^{6} - 4q^{9} + 32q^{16} + 8q^{19} - 16q^{24} + 32q^{34} - 8q^{36} - 56q^{49} + 20q^{51} - 16q^{54} + 64q^{64} - 64q^{66} + 16q^{76} + 28q^{81} - 32q^{96} + 52q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
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
0.258819 + 0.965926i
−1.41421 −0.158919 1.72474i 2.00000 0 0.224745 + 2.43916i 0 −2.82843 −2.94949 + 0.548188i 0
299.2 −1.41421 −0.158919 + 1.72474i 2.00000 0 0.224745 2.43916i 0 −2.82843 −2.94949 0.548188i 0
299.3 −1.41421 1.57313 0.724745i 2.00000 0 −2.22474 + 1.02494i 0 −2.82843 1.94949 2.28024i 0
299.4 −1.41421 1.57313 + 0.724745i 2.00000 0 −2.22474 1.02494i 0 −2.82843 1.94949 + 2.28024i 0
299.5 1.41421 −1.57313 0.724745i 2.00000 0 −2.22474 1.02494i 0 2.82843 1.94949 + 2.28024i 0
299.6 1.41421 −1.57313 + 0.724745i 2.00000 0 −2.22474 + 1.02494i 0 2.82843 1.94949 2.28024i 0
299.7 1.41421 0.158919 1.72474i 2.00000 0 0.224745 2.43916i 0 2.82843 −2.94949 0.548188i 0
299.8 1.41421 0.158919 + 1.72474i 2.00000 0 0.224745 + 2.43916i 0 2.82843 −2.94949 + 0.548188i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.8
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
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.m.b 8
3.b odd 2 1 inner 600.2.m.b 8
4.b odd 2 1 2400.2.m.b 8
5.b even 2 1 inner 600.2.m.b 8
5.c odd 4 1 600.2.b.b 4
5.c odd 4 1 600.2.b.d yes 4
8.b even 2 1 2400.2.m.b 8
8.d odd 2 1 CM 600.2.m.b 8
12.b even 2 1 2400.2.m.b 8
15.d odd 2 1 inner 600.2.m.b 8
15.e even 4 1 600.2.b.b 4
15.e even 4 1 600.2.b.d yes 4
20.d odd 2 1 2400.2.m.b 8
20.e even 4 1 2400.2.b.b 4
20.e even 4 1 2400.2.b.d 4
24.f even 2 1 inner 600.2.m.b 8
24.h odd 2 1 2400.2.m.b 8
40.e odd 2 1 inner 600.2.m.b 8
40.f even 2 1 2400.2.m.b 8
40.i odd 4 1 2400.2.b.b 4
40.i odd 4 1 2400.2.b.d 4
40.k even 4 1 600.2.b.b 4
40.k even 4 1 600.2.b.d yes 4
60.h even 2 1 2400.2.m.b 8
60.l odd 4 1 2400.2.b.b 4
60.l odd 4 1 2400.2.b.d 4
120.i odd 2 1 2400.2.m.b 8
120.m even 2 1 inner 600.2.m.b 8
120.q odd 4 1 600.2.b.b 4
120.q odd 4 1 600.2.b.d yes 4
120.w even 4 1 2400.2.b.b 4
120.w even 4 1 2400.2.b.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.b.b 4 5.c odd 4 1
600.2.b.b 4 15.e even 4 1
600.2.b.b 4 40.k even 4 1
600.2.b.b 4 120.q odd 4 1
600.2.b.d yes 4 5.c odd 4 1
600.2.b.d yes 4 15.e even 4 1
600.2.b.d yes 4 40.k even 4 1
600.2.b.d yes 4 120.q odd 4 1
600.2.m.b 8 1.a even 1 1 trivial
600.2.m.b 8 3.b odd 2 1 inner
600.2.m.b 8 5.b even 2 1 inner
600.2.m.b 8 8.d odd 2 1 CM
600.2.m.b 8 15.d odd 2 1 inner
600.2.m.b 8 24.f even 2 1 inner
600.2.m.b 8 40.e odd 2 1 inner
600.2.m.b 8 120.m even 2 1 inner
2400.2.b.b 4 20.e even 4 1
2400.2.b.b 4 40.i odd 4 1
2400.2.b.b 4 60.l odd 4 1
2400.2.b.b 4 120.w even 4 1
2400.2.b.d 4 20.e even 4 1
2400.2.b.d 4 40.i odd 4 1
2400.2.b.d 4 60.l odd 4 1
2400.2.b.d 4 120.w even 4 1
2400.2.m.b 8 4.b odd 2 1
2400.2.m.b 8 8.b even 2 1
2400.2.m.b 8 12.b even 2 1
2400.2.m.b 8 20.d odd 2 1
2400.2.m.b 8 24.h odd 2 1
2400.2.m.b 8 40.f even 2 1
2400.2.m.b 8 60.h even 2 1
2400.2.m.b 8 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7} \)
\( T_{11}^{4} + 58 T_{11}^{2} + 625 \)
\( T_{29} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{4} \)
$3$ \( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \)
$5$ 1
$7$ \( ( 1 + 7 T^{2} )^{8} \)
$11$ \( ( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} )^{2}( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{8} \)
$17$ \( ( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 23 T^{2} )^{8} \)
$29$ \( ( 1 + 29 T^{2} )^{8} \)
$31$ \( ( 1 - 31 T^{2} )^{8} \)
$37$ \( ( 1 + 37 T^{2} )^{8} \)
$41$ \( ( 1 - 6 T - 5 T^{2} - 246 T^{3} + 1681 T^{4} )^{2}( 1 + 6 T - 5 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 14 T^{2} + 1849 T^{4} )^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{8} \)
$53$ \( ( 1 - 53 T^{2} )^{8} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )^{4}( 1 + 6 T + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{8} \)
$67$ \( ( 1 - 62 T^{2} - 645 T^{4} - 278318 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{8} \)
$73$ \( ( 1 + 142 T^{2} + 14835 T^{4} + 756718 T^{6} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{8} \)
$83$ \( ( 1 - 158 T^{2} + 18075 T^{4} - 1088462 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4} )^{2}( 1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 94 T^{2} + 9409 T^{4} )^{4} \)
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