Properties

Label 600.2.f.c
Level 600
Weight 2
Character orbit 600.f
Analytic conductor 4.791
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} -4 i q^{7} - q^{9} +O(q^{10})\) \( q + i q^{3} -4 i q^{7} - q^{9} -6 i q^{13} + 2 i q^{17} -4 q^{19} + 4 q^{21} -8 i q^{23} -i q^{27} + 6 q^{29} + 6 i q^{37} + 6 q^{39} + 10 q^{41} -4 i q^{43} -8 i q^{47} -9 q^{49} -2 q^{51} + 10 i q^{53} -4 i q^{57} + 6 q^{61} + 4 i q^{63} + 4 i q^{67} + 8 q^{69} -14 i q^{73} -16 q^{79} + q^{81} + 12 i q^{83} + 6 i q^{87} -2 q^{89} -24 q^{91} -2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 8q^{19} + 8q^{21} + 12q^{29} + 12q^{39} + 20q^{41} - 18q^{49} - 4q^{51} + 12q^{61} + 16q^{69} - 32q^{79} + 2q^{81} - 4q^{89} - 48q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 1.00000i 0 0 0 4.00000i 0 −1.00000 0
49.2 0 1.00000i 0 0 0 4.00000i 0 −1.00000 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
5.b 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.f.c 2
3.b odd 2 1 1800.2.f.g 2
4.b odd 2 1 1200.2.f.f 2
5.b even 2 1 inner 600.2.f.c 2
5.c odd 4 1 120.2.a.a 1
5.c odd 4 1 600.2.a.a 1
8.b even 2 1 4800.2.f.u 2
8.d odd 2 1 4800.2.f.n 2
12.b even 2 1 3600.2.f.l 2
15.d odd 2 1 1800.2.f.g 2
15.e even 4 1 360.2.a.e 1
15.e even 4 1 1800.2.a.c 1
20.d odd 2 1 1200.2.f.f 2
20.e even 4 1 240.2.a.a 1
20.e even 4 1 1200.2.a.r 1
35.f even 4 1 5880.2.a.p 1
40.e odd 2 1 4800.2.f.n 2
40.f even 2 1 4800.2.f.u 2
40.i odd 4 1 960.2.a.g 1
40.i odd 4 1 4800.2.a.bl 1
40.k even 4 1 960.2.a.n 1
40.k even 4 1 4800.2.a.bh 1
45.k odd 12 2 3240.2.q.m 2
45.l even 12 2 3240.2.q.a 2
60.h even 2 1 3600.2.f.l 2
60.l odd 4 1 720.2.a.f 1
60.l odd 4 1 3600.2.a.bo 1
80.i odd 4 1 3840.2.k.a 2
80.j even 4 1 3840.2.k.z 2
80.s even 4 1 3840.2.k.z 2
80.t odd 4 1 3840.2.k.a 2
120.q odd 4 1 2880.2.a.b 1
120.w even 4 1 2880.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 5.c odd 4 1
240.2.a.a 1 20.e even 4 1
360.2.a.e 1 15.e even 4 1
600.2.a.a 1 5.c odd 4 1
600.2.f.c 2 1.a even 1 1 trivial
600.2.f.c 2 5.b even 2 1 inner
720.2.a.f 1 60.l odd 4 1
960.2.a.g 1 40.i odd 4 1
960.2.a.n 1 40.k even 4 1
1200.2.a.r 1 20.e even 4 1
1200.2.f.f 2 4.b odd 2 1
1200.2.f.f 2 20.d odd 2 1
1800.2.a.c 1 15.e even 4 1
1800.2.f.g 2 3.b odd 2 1
1800.2.f.g 2 15.d odd 2 1
2880.2.a.b 1 120.q odd 4 1
2880.2.a.r 1 120.w even 4 1
3240.2.q.a 2 45.l even 12 2
3240.2.q.m 2 45.k odd 12 2
3600.2.a.bo 1 60.l odd 4 1
3600.2.f.l 2 12.b even 2 1
3600.2.f.l 2 60.h even 2 1
3840.2.k.a 2 80.i odd 4 1
3840.2.k.a 2 80.t odd 4 1
3840.2.k.z 2 80.j even 4 1
3840.2.k.z 2 80.s even 4 1
4800.2.a.bh 1 40.k even 4 1
4800.2.a.bl 1 40.i odd 4 1
4800.2.f.n 2 8.d odd 2 1
4800.2.f.n 2 40.e odd 2 1
4800.2.f.u 2 8.b even 2 1
4800.2.f.u 2 40.f even 2 1
5880.2.a.p 1 35.f even 4 1

Hecke kernels

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

\( T_{7}^{2} + 16 \)
\( T_{11} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 18 T^{2} + 529 T^{4} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 10 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 6 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 6 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 + 50 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 16 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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