Properties

Label 600.3.c.a
Level $600$
Weight $3$
Character orbit 600.c
Analytic conductor $16.349$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(16.3488158616\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} -6 \zeta_{8}^{2} q^{7} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{3} -6 \zeta_{8}^{2} q^{7} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{9} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} -10 \zeta_{8}^{2} q^{13} + ( 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{17} -2 q^{19} + ( -6 + 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{21} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{23} + ( -10 \zeta_{8} - 23 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{27} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{29} -22 q^{31} + ( 4 \zeta_{8} - 16 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{33} -6 \zeta_{8}^{2} q^{37} + ( -10 + 20 \zeta_{8} + 20 \zeta_{8}^{3} ) q^{39} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{41} -82 \zeta_{8}^{2} q^{43} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{47} + 13 q^{49} + ( -64 - 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{51} + ( -44 \zeta_{8} + 44 \zeta_{8}^{3} ) q^{53} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{57} + ( -52 \zeta_{8} - 52 \zeta_{8}^{3} ) q^{59} -86 q^{61} + ( 24 \zeta_{8} - 42 \zeta_{8}^{2} - 24 \zeta_{8}^{3} ) q^{63} + 2 \zeta_{8}^{2} q^{67} + ( 32 + 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{69} + ( -88 \zeta_{8} - 88 \zeta_{8}^{3} ) q^{71} -82 \zeta_{8}^{2} q^{73} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{77} -10 q^{79} + ( 17 + 56 \zeta_{8} + 56 \zeta_{8}^{3} ) q^{81} + ( -52 \zeta_{8} + 52 \zeta_{8}^{3} ) q^{83} + ( -12 \zeta_{8} + 48 \zeta_{8}^{2} + 12 \zeta_{8}^{3} ) q^{87} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{89} -60 q^{91} + ( 44 \zeta_{8} + 22 \zeta_{8}^{2} - 44 \zeta_{8}^{3} ) q^{93} -94 \zeta_{8}^{2} q^{97} + ( -32 + 28 \zeta_{8} + 28 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 28q^{9} + O(q^{10}) \) \( 4q + 28q^{9} - 8q^{19} - 24q^{21} - 88q^{31} - 40q^{39} + 52q^{49} - 256q^{51} - 344q^{61} + 128q^{69} - 40q^{79} + 68q^{81} - 240q^{91} - 128q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0 −2.82843 1.00000i 0 0 0 6.00000i 0 7.00000 + 5.65685i 0
449.2 0 −2.82843 + 1.00000i 0 0 0 6.00000i 0 7.00000 5.65685i 0
449.3 0 2.82843 1.00000i 0 0 0 6.00000i 0 7.00000 5.65685i 0
449.4 0 2.82843 + 1.00000i 0 0 0 6.00000i 0 7.00000 + 5.65685i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.3.c.a 4
3.b odd 2 1 inner 600.3.c.a 4
4.b odd 2 1 1200.3.c.i 4
5.b even 2 1 inner 600.3.c.a 4
5.c odd 4 1 24.3.e.a 2
5.c odd 4 1 600.3.l.b 2
12.b even 2 1 1200.3.c.i 4
15.d odd 2 1 inner 600.3.c.a 4
15.e even 4 1 24.3.e.a 2
15.e even 4 1 600.3.l.b 2
20.d odd 2 1 1200.3.c.i 4
20.e even 4 1 48.3.e.b 2
20.e even 4 1 1200.3.l.n 2
35.f even 4 1 1176.3.d.a 2
40.i odd 4 1 192.3.e.c 2
40.k even 4 1 192.3.e.d 2
45.k odd 12 2 648.3.m.d 4
45.l even 12 2 648.3.m.d 4
60.h even 2 1 1200.3.c.i 4
60.l odd 4 1 48.3.e.b 2
60.l odd 4 1 1200.3.l.n 2
80.i odd 4 1 768.3.h.d 4
80.j even 4 1 768.3.h.c 4
80.s even 4 1 768.3.h.c 4
80.t odd 4 1 768.3.h.d 4
105.k odd 4 1 1176.3.d.a 2
120.q odd 4 1 192.3.e.d 2
120.w even 4 1 192.3.e.c 2
180.v odd 12 2 1296.3.q.e 4
180.x even 12 2 1296.3.q.e 4
240.z odd 4 1 768.3.h.c 4
240.bb even 4 1 768.3.h.d 4
240.bd odd 4 1 768.3.h.c 4
240.bf even 4 1 768.3.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 5.c odd 4 1
24.3.e.a 2 15.e even 4 1
48.3.e.b 2 20.e even 4 1
48.3.e.b 2 60.l odd 4 1
192.3.e.c 2 40.i odd 4 1
192.3.e.c 2 120.w even 4 1
192.3.e.d 2 40.k even 4 1
192.3.e.d 2 120.q odd 4 1
600.3.c.a 4 1.a even 1 1 trivial
600.3.c.a 4 3.b odd 2 1 inner
600.3.c.a 4 5.b even 2 1 inner
600.3.c.a 4 15.d odd 2 1 inner
600.3.l.b 2 5.c odd 4 1
600.3.l.b 2 15.e even 4 1
648.3.m.d 4 45.k odd 12 2
648.3.m.d 4 45.l even 12 2
768.3.h.c 4 80.j even 4 1
768.3.h.c 4 80.s even 4 1
768.3.h.c 4 240.z odd 4 1
768.3.h.c 4 240.bd odd 4 1
768.3.h.d 4 80.i odd 4 1
768.3.h.d 4 80.t odd 4 1
768.3.h.d 4 240.bb even 4 1
768.3.h.d 4 240.bf even 4 1
1176.3.d.a 2 35.f even 4 1
1176.3.d.a 2 105.k odd 4 1
1200.3.c.i 4 4.b odd 2 1
1200.3.c.i 4 12.b even 2 1
1200.3.c.i 4 20.d odd 2 1
1200.3.c.i 4 60.h even 2 1
1200.3.l.n 2 20.e even 4 1
1200.3.l.n 2 60.l odd 4 1
1296.3.q.e 4 180.v odd 12 2
1296.3.q.e 4 180.x even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 36 \) acting on \(S_{3}^{\mathrm{new}}(600, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 81 - 14 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 36 + T^{2} )^{2} \)
$11$ \( ( 32 + T^{2} )^{2} \)
$13$ \( ( 100 + T^{2} )^{2} \)
$17$ \( ( -512 + T^{2} )^{2} \)
$19$ \( ( 2 + T )^{4} \)
$23$ \( ( -128 + T^{2} )^{2} \)
$29$ \( ( 288 + T^{2} )^{2} \)
$31$ \( ( 22 + T )^{4} \)
$37$ \( ( 36 + T^{2} )^{2} \)
$41$ \( ( 1152 + T^{2} )^{2} \)
$43$ \( ( 6724 + T^{2} )^{2} \)
$47$ \( ( -4608 + T^{2} )^{2} \)
$53$ \( ( -3872 + T^{2} )^{2} \)
$59$ \( ( 5408 + T^{2} )^{2} \)
$61$ \( ( 86 + T )^{4} \)
$67$ \( ( 4 + T^{2} )^{2} \)
$71$ \( ( 15488 + T^{2} )^{2} \)
$73$ \( ( 6724 + T^{2} )^{2} \)
$79$ \( ( 10 + T )^{4} \)
$83$ \( ( -5408 + T^{2} )^{2} \)
$89$ \( ( 1152 + T^{2} )^{2} \)
$97$ \( ( 8836 + T^{2} )^{2} \)
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