Properties

Label 600.2.r.a
Level $600$
Weight $2$
Character orbit 600.r
Analytic conductor $4.791$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 + 2 \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} + ( -2 + 2 \zeta_{8}^{2} ) q^{7} + ( -2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{9} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} -4 \zeta_{8} q^{17} + 4 \zeta_{8}^{2} q^{19} + ( 4 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21} + 6 \zeta_{8}^{3} q^{23} + ( -1 + 5 \zeta_{8} + \zeta_{8}^{2} ) q^{27} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{29} + 8 q^{31} + ( 4 + 4 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{33} + ( 8 - 8 \zeta_{8}^{2} ) q^{37} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + ( -2 - 2 \zeta_{8}^{2} ) q^{43} + 2 \zeta_{8} q^{47} -\zeta_{8}^{2} q^{49} + ( -4 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{51} + 8 \zeta_{8}^{3} q^{53} + ( 4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{57} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{59} -6 q^{61} + ( -2 - 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{63} + ( -6 + 6 \zeta_{8}^{2} ) q^{67} + ( 6 \zeta_{8} + 6 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{69} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{71} + ( 8 + 8 \zeta_{8}^{2} ) q^{73} -16 \zeta_{8} q^{77} + ( 7 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{81} + 14 \zeta_{8}^{3} q^{83} + ( -4 + 8 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{87} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{89} + ( -8 - 8 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{93} + ( -8 + 8 \zeta_{8}^{2} ) q^{97} + ( -4 \zeta_{8} - 16 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{7} + O(q^{10}) \) \( 4 q - 4 q^{3} - 8 q^{7} + 16 q^{21} - 4 q^{27} + 32 q^{31} + 16 q^{33} + 32 q^{37} - 8 q^{43} - 16 q^{51} + 16 q^{57} - 24 q^{61} - 8 q^{63} - 24 q^{67} + 32 q^{73} + 28 q^{81} - 16 q^{87} - 32 q^{93} - 32 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\zeta_{8}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 −1.70711 + 0.292893i 0 0 0 −2.00000 2.00000i 0 2.82843 1.00000i 0
257.2 0 −0.292893 + 1.70711i 0 0 0 −2.00000 2.00000i 0 −2.82843 1.00000i 0
593.1 0 −1.70711 0.292893i 0 0 0 −2.00000 + 2.00000i 0 2.82843 + 1.00000i 0
593.2 0 −0.292893 1.70711i 0 0 0 −2.00000 + 2.00000i 0 −2.82843 + 1.00000i 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.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.r.a 4
3.b odd 2 1 inner 600.2.r.a 4
4.b odd 2 1 1200.2.v.k 4
5.b even 2 1 600.2.r.e yes 4
5.c odd 4 1 inner 600.2.r.a 4
5.c odd 4 1 600.2.r.e yes 4
12.b even 2 1 1200.2.v.k 4
15.d odd 2 1 600.2.r.e yes 4
15.e even 4 1 inner 600.2.r.a 4
15.e even 4 1 600.2.r.e yes 4
20.d odd 2 1 1200.2.v.a 4
20.e even 4 1 1200.2.v.a 4
20.e even 4 1 1200.2.v.k 4
60.h even 2 1 1200.2.v.a 4
60.l odd 4 1 1200.2.v.a 4
60.l odd 4 1 1200.2.v.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.r.a 4 1.a even 1 1 trivial
600.2.r.a 4 3.b odd 2 1 inner
600.2.r.a 4 5.c odd 4 1 inner
600.2.r.a 4 15.e even 4 1 inner
600.2.r.e yes 4 5.b even 2 1
600.2.r.e yes 4 5.c odd 4 1
600.2.r.e yes 4 15.d odd 2 1
600.2.r.e yes 4 15.e even 4 1
1200.2.v.a 4 20.d odd 2 1
1200.2.v.a 4 20.e even 4 1
1200.2.v.a 4 60.h even 2 1
1200.2.v.a 4 60.l odd 4 1
1200.2.v.k 4 4.b odd 2 1
1200.2.v.k 4 12.b even 2 1
1200.2.v.k 4 20.e even 4 1
1200.2.v.k 4 60.l odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 4 T_{7} + 8 \)
\( T_{17}^{4} + 256 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 12 T + 8 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 8 + 4 T + T^{2} )^{2} \)
$11$ \( ( 32 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( 256 + T^{4} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( 1296 + T^{4} \)
$29$ \( ( -32 + T^{2} )^{2} \)
$31$ \( ( -8 + T )^{4} \)
$37$ \( ( 128 - 16 T + T^{2} )^{2} \)
$41$ \( ( 32 + T^{2} )^{2} \)
$43$ \( ( 8 + 4 T + T^{2} )^{2} \)
$47$ \( 16 + T^{4} \)
$53$ \( 4096 + T^{4} \)
$59$ \( ( -32 + T^{2} )^{2} \)
$61$ \( ( 6 + T )^{4} \)
$67$ \( ( 72 + 12 T + T^{2} )^{2} \)
$71$ \( ( 128 + T^{2} )^{2} \)
$73$ \( ( 128 - 16 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( 38416 + T^{4} \)
$89$ \( ( -128 + T^{2} )^{2} \)
$97$ \( ( 128 + 16 T + T^{2} )^{2} \)
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