Properties

Label 600.2.m.a
Level 600
Weight 2
Character orbit 600.m
Analytic conductor 4.791
Analytic rank 0
Dimension 4
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 24)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + 2 q^{4} + ( 2 + \zeta_{8} + \zeta_{8}^{3} ) q^{6} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + 2 q^{4} + ( 2 + \zeta_{8} + \zeta_{8}^{3} ) q^{6} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{12} + 4 q^{16} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{17} + ( -\zeta_{8} - 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{18} -2 q^{19} + 4 \zeta_{8}^{2} q^{22} + ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{24} + ( \zeta_{8} - 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{27} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{32} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{33} -8 q^{34} + ( 2 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{36} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{38} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{41} -10 \zeta_{8}^{2} q^{43} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{44} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{48} -7 q^{49} + ( -8 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{51} + ( -2 + 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{54} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{57} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{59} + 8 q^{64} + ( 4 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{66} -14 \zeta_{8}^{2} q^{67} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{68} + ( -2 \zeta_{8} - 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{72} + 2 \zeta_{8}^{2} q^{73} -4 q^{76} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} + 16 \zeta_{8}^{2} q^{82} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{83} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{86} + 8 \zeta_{8}^{2} q^{88} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{89} + ( 8 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{96} + 10 \zeta_{8}^{2} q^{97} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{98} + ( 8 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{4} + 8q^{6} + 4q^{9} + O(q^{10}) \) \( 4q + 8q^{4} + 8q^{6} + 4q^{9} + 16q^{16} - 8q^{19} + 16q^{24} - 32q^{34} + 8q^{36} - 28q^{49} - 32q^{51} - 8q^{54} + 32q^{64} + 16q^{66} - 16q^{76} - 28q^{81} + 32q^{96} + 32q^{99} + O(q^{100}) \)

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
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
−1.41421 −1.41421 1.00000i 2.00000 0 2.00000 + 1.41421i 0 −2.82843 1.00000 + 2.82843i 0
299.2 −1.41421 −1.41421 + 1.00000i 2.00000 0 2.00000 1.41421i 0 −2.82843 1.00000 2.82843i 0
299.3 1.41421 1.41421 1.00000i 2.00000 0 2.00000 1.41421i 0 2.82843 1.00000 2.82843i 0
299.4 1.41421 1.41421 + 1.00000i 2.00000 0 2.00000 + 1.41421i 0 2.82843 1.00000 + 2.82843i 0
\(n\): e.g. 2-40 or 990-1000
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.a 4
3.b odd 2 1 inner 600.2.m.a 4
4.b odd 2 1 2400.2.m.a 4
5.b even 2 1 inner 600.2.m.a 4
5.c odd 4 1 24.2.f.a 2
5.c odd 4 1 600.2.b.a 2
8.b even 2 1 2400.2.m.a 4
8.d odd 2 1 CM 600.2.m.a 4
12.b even 2 1 2400.2.m.a 4
15.d odd 2 1 inner 600.2.m.a 4
15.e even 4 1 24.2.f.a 2
15.e even 4 1 600.2.b.a 2
20.d odd 2 1 2400.2.m.a 4
20.e even 4 1 96.2.f.a 2
20.e even 4 1 2400.2.b.a 2
24.f even 2 1 inner 600.2.m.a 4
24.h odd 2 1 2400.2.m.a 4
40.e odd 2 1 inner 600.2.m.a 4
40.f even 2 1 2400.2.m.a 4
40.i odd 4 1 96.2.f.a 2
40.i odd 4 1 2400.2.b.a 2
40.k even 4 1 24.2.f.a 2
40.k even 4 1 600.2.b.a 2
45.k odd 12 2 648.2.l.b 4
45.l even 12 2 648.2.l.b 4
60.h even 2 1 2400.2.m.a 4
60.l odd 4 1 96.2.f.a 2
60.l odd 4 1 2400.2.b.a 2
80.i odd 4 1 768.2.c.h 4
80.j even 4 1 768.2.c.h 4
80.s even 4 1 768.2.c.h 4
80.t odd 4 1 768.2.c.h 4
120.i odd 2 1 2400.2.m.a 4
120.m even 2 1 inner 600.2.m.a 4
120.q odd 4 1 24.2.f.a 2
120.q odd 4 1 600.2.b.a 2
120.w even 4 1 96.2.f.a 2
120.w even 4 1 2400.2.b.a 2
180.v odd 12 2 2592.2.p.b 4
180.x even 12 2 2592.2.p.b 4
240.z odd 4 1 768.2.c.h 4
240.bb even 4 1 768.2.c.h 4
240.bd odd 4 1 768.2.c.h 4
240.bf even 4 1 768.2.c.h 4
360.bo even 12 2 648.2.l.b 4
360.br even 12 2 2592.2.p.b 4
360.bt odd 12 2 648.2.l.b 4
360.bu odd 12 2 2592.2.p.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 5.c odd 4 1
24.2.f.a 2 15.e even 4 1
24.2.f.a 2 40.k even 4 1
24.2.f.a 2 120.q odd 4 1
96.2.f.a 2 20.e even 4 1
96.2.f.a 2 40.i odd 4 1
96.2.f.a 2 60.l odd 4 1
96.2.f.a 2 120.w even 4 1
600.2.b.a 2 5.c odd 4 1
600.2.b.a 2 15.e even 4 1
600.2.b.a 2 40.k even 4 1
600.2.b.a 2 120.q odd 4 1
600.2.m.a 4 1.a even 1 1 trivial
600.2.m.a 4 3.b odd 2 1 inner
600.2.m.a 4 5.b even 2 1 inner
600.2.m.a 4 8.d odd 2 1 CM
600.2.m.a 4 15.d odd 2 1 inner
600.2.m.a 4 24.f even 2 1 inner
600.2.m.a 4 40.e odd 2 1 inner
600.2.m.a 4 120.m even 2 1 inner
648.2.l.b 4 45.k odd 12 2
648.2.l.b 4 45.l even 12 2
648.2.l.b 4 360.bo even 12 2
648.2.l.b 4 360.bt odd 12 2
768.2.c.h 4 80.i odd 4 1
768.2.c.h 4 80.j even 4 1
768.2.c.h 4 80.s even 4 1
768.2.c.h 4 80.t odd 4 1
768.2.c.h 4 240.z odd 4 1
768.2.c.h 4 240.bb even 4 1
768.2.c.h 4 240.bd odd 4 1
768.2.c.h 4 240.bf even 4 1
2400.2.b.a 2 20.e even 4 1
2400.2.b.a 2 40.i odd 4 1
2400.2.b.a 2 60.l odd 4 1
2400.2.b.a 2 120.w even 4 1
2400.2.m.a 4 4.b odd 2 1
2400.2.m.a 4 8.b even 2 1
2400.2.m.a 4 12.b even 2 1
2400.2.m.a 4 20.d odd 2 1
2400.2.m.a 4 24.h odd 2 1
2400.2.m.a 4 40.f even 2 1
2400.2.m.a 4 60.h even 2 1
2400.2.m.a 4 120.i odd 2 1
2592.2.p.b 4 180.v odd 12 2
2592.2.p.b 4 180.x even 12 2
2592.2.p.b 4 360.br even 12 2
2592.2.p.b 4 360.bu odd 12 2

Hecke kernels

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

\( T_{7} \)
\( T_{11}^{2} + 8 \)
\( T_{29} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} )^{2} \)
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ 1
$7$ \( ( 1 + 7 T^{2} )^{4} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 + 2 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 2 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 23 T^{2} )^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2}( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 47 T^{2} )^{4} \)
$53$ \( ( 1 - 53 T^{2} )^{4} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )^{2}( 1 + 6 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 + 62 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 142 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 18 T + 89 T^{2} )^{2}( 1 + 18 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 94 T^{2} + 9409 T^{4} )^{2} \)
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