Properties

Label 600.2.r.e
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: \( x^{4} + 1 \) Copy content Toggle raw display
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.

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

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 0.292893 1.70711i 0 0 0 2.00000 + 2.00000i 0 −2.82843 1.00000i 0
257.2 0 1.70711 0.292893i 0 0 0 2.00000 + 2.00000i 0 2.82843 1.00000i 0
593.1 0 0.292893 + 1.70711i 0 0 0 2.00000 2.00000i 0 −2.82843 + 1.00000i 0
593.2 0 1.70711 + 0.292893i 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.e yes 4
3.b odd 2 1 inner 600.2.r.e yes 4
4.b odd 2 1 1200.2.v.a 4
5.b even 2 1 600.2.r.a 4
5.c odd 4 1 600.2.r.a 4
5.c odd 4 1 inner 600.2.r.e yes 4
12.b even 2 1 1200.2.v.a 4
15.d odd 2 1 600.2.r.a 4
15.e even 4 1 600.2.r.a 4
15.e even 4 1 inner 600.2.r.e yes 4
20.d odd 2 1 1200.2.v.k 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.k 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 5.b even 2 1
600.2.r.a 4 5.c odd 4 1
600.2.r.a 4 15.d odd 2 1
600.2.r.a 4 15.e even 4 1
600.2.r.e yes 4 1.a even 1 1 trivial
600.2.r.e yes 4 3.b odd 2 1 inner
600.2.r.e yes 4 5.c odd 4 1 inner
600.2.r.e yes 4 15.e even 4 1 inner
1200.2.v.a 4 4.b odd 2 1
1200.2.v.a 4 12.b even 2 1
1200.2.v.a 4 20.e even 4 1
1200.2.v.a 4 60.l odd 4 1
1200.2.v.k 4 20.d odd 2 1
1200.2.v.k 4 20.e even 4 1
1200.2.v.k 4 60.h even 2 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 \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{2} - 4T_{7} + 8 \) Copy content Toggle raw display
\( T_{17}^{4} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

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