Properties

Label 600.2.r.c
Level $600$
Weight $2$
Character orbit 600.r
Analytic conductor $4.791$
Analytic rank $0$
Dimension $4$
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.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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{7} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{7} + ( - 2 \beta_{2} + 1) q^{9} + (2 \beta_{2} + 2 \beta_1) q^{11} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{17} + (2 \beta_{2} + 2 \beta_1) q^{19} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{21} + (\beta_{3} + \beta_{2} - 3 \beta_1 - 3) q^{23} + (\beta_{3} + 5 \beta_1) q^{27} + ( - 4 \beta_{3} - 2) q^{29} - 4 \beta_{3} q^{31} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{33} + ( - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 3) q^{37} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{39} + 4 \beta_{2} q^{41} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 - 5) q^{47} + ( - 4 \beta_{2} - \beta_1) q^{49} + ( - \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 3) q^{51} + (3 \beta_1 + 3) q^{53} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{57} - 4 q^{59} + (4 \beta_{3} + 6) q^{61} + (\beta_{3} - 3 \beta_{2} + 5 \beta_1 - 3) q^{63} + (5 \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 3) q^{67} + (2 \beta_{3} + 4 \beta_{2} - 5 \beta_1 + 1) q^{69} + (6 \beta_{2} - 2 \beta_1) q^{71} + ( - 4 \beta_{3} + 4 \beta_{2} + \beta_1 - 1) q^{73} + ( - 2 \beta_1 + 2) q^{77} + ( - 2 \beta_{2} - 2 \beta_1) q^{79} + ( - 4 \beta_{2} - 7) q^{81} + (3 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{83} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 8) q^{87} + ( - 4 \beta_{3} + 10) q^{89} + (6 \beta_{3} - 10) q^{91} + ( - 4 \beta_{2} + 8) q^{93} + ( - \beta_1 - 1) q^{97} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} + 4 q^{9} - 4 q^{13} - 4 q^{17} + 4 q^{21} - 12 q^{23} - 8 q^{29} - 8 q^{33} + 12 q^{37} - 20 q^{39} - 4 q^{43} - 20 q^{47} + 12 q^{51} + 12 q^{53} - 8 q^{57} - 16 q^{59} + 24 q^{61} - 12 q^{63} - 12 q^{67} + 4 q^{69} - 4 q^{73} + 8 q^{77} - 28 q^{81} + 4 q^{83} + 32 q^{87} + 40 q^{89} - 40 q^{91} + 32 q^{93} - 4 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 −1.41421 + 1.00000i 0 0 0 −0.414214 0.414214i 0 1.00000 2.82843i 0
257.2 0 1.41421 + 1.00000i 0 0 0 2.41421 + 2.41421i 0 1.00000 + 2.82843i 0
593.1 0 −1.41421 1.00000i 0 0 0 −0.414214 + 0.414214i 0 1.00000 + 2.82843i 0
593.2 0 1.41421 1.00000i 0 0 0 2.41421 2.41421i 0 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
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.c 4
3.b odd 2 1 600.2.r.b 4
4.b odd 2 1 1200.2.v.d 4
5.b even 2 1 120.2.r.b 4
5.c odd 4 1 120.2.r.c yes 4
5.c odd 4 1 600.2.r.b 4
12.b even 2 1 1200.2.v.j 4
15.d odd 2 1 120.2.r.c yes 4
15.e even 4 1 120.2.r.b 4
15.e even 4 1 inner 600.2.r.c 4
20.d odd 2 1 240.2.v.c 4
20.e even 4 1 240.2.v.a 4
20.e even 4 1 1200.2.v.j 4
40.e odd 2 1 960.2.v.g 4
40.f even 2 1 960.2.v.f 4
40.i odd 4 1 960.2.v.a 4
40.k even 4 1 960.2.v.i 4
60.h even 2 1 240.2.v.a 4
60.l odd 4 1 240.2.v.c 4
60.l odd 4 1 1200.2.v.d 4
120.i odd 2 1 960.2.v.a 4
120.m even 2 1 960.2.v.i 4
120.q odd 4 1 960.2.v.g 4
120.w even 4 1 960.2.v.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.r.b 4 5.b even 2 1
120.2.r.b 4 15.e even 4 1
120.2.r.c yes 4 5.c odd 4 1
120.2.r.c yes 4 15.d odd 2 1
240.2.v.a 4 20.e even 4 1
240.2.v.a 4 60.h even 2 1
240.2.v.c 4 20.d odd 2 1
240.2.v.c 4 60.l odd 4 1
600.2.r.b 4 3.b odd 2 1
600.2.r.b 4 5.c odd 4 1
600.2.r.c 4 1.a even 1 1 trivial
600.2.r.c 4 15.e even 4 1 inner
960.2.v.a 4 40.i odd 4 1
960.2.v.a 4 120.i odd 2 1
960.2.v.f 4 40.f even 2 1
960.2.v.f 4 120.w even 4 1
960.2.v.g 4 40.e odd 2 1
960.2.v.g 4 120.q odd 4 1
960.2.v.i 4 40.k even 4 1
960.2.v.i 4 120.m even 2 1
1200.2.v.d 4 4.b odd 2 1
1200.2.v.d 4 60.l odd 4 1
1200.2.v.j 4 12.b even 2 1
1200.2.v.j 4 20.e even 4 1

Hecke kernels

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

\( T_{7}^{4} - 4T_{7}^{3} + 8T_{7}^{2} + 8T_{7} + 4 \) Copy content Toggle raw display
\( T_{17}^{4} + 4T_{17}^{3} + 8T_{17}^{2} - 56T_{17} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + 8 T^{2} + 8 T + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 4 T^{3} + 8 T^{2} - 56 T + 196 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + 8 T^{2} - 56 T + 196 \) Copy content Toggle raw display
$19$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 12 T^{3} + 72 T^{2} - 24 T + 4 \) Copy content Toggle raw display
$41$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + 8 T^{2} - 8 T + 4 \) Copy content Toggle raw display
$47$ \( T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$53$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$59$ \( (T + 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 6724 \) Copy content Toggle raw display
$71$ \( T^{4} + 152T^{2} + 4624 \) Copy content Toggle raw display
$73$ \( T^{4} + 4 T^{3} + 8 T^{2} - 248 T + 3844 \) Copy content Toggle raw display
$79$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$83$ \( T^{4} - 4 T^{3} + 8 T^{2} + 136 T + 1156 \) Copy content Toggle raw display
$89$ \( (T^{2} - 20 T + 68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
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