Properties

Label 600.3.l.b
Level $600$
Weight $3$
Character orbit 600.l
Analytic conductor $16.349$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.3488158616\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{3} + 6 q^{7} + ( -7 - 2 \beta ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta ) q^{3} + 6 q^{7} + ( -7 - 2 \beta ) q^{9} + 2 \beta q^{11} -10 q^{13} + 8 \beta q^{17} + 2 q^{19} + ( -6 + 6 \beta ) q^{21} + 4 \beta q^{23} + ( 23 - 5 \beta ) q^{27} + 6 \beta q^{29} -22 q^{31} + ( -16 - 2 \beta ) q^{33} + 6 q^{37} + ( 10 - 10 \beta ) q^{39} + 12 \beta q^{41} -82 q^{43} -24 \beta q^{47} -13 q^{49} + ( -64 - 8 \beta ) q^{51} + 22 \beta q^{53} + ( -2 + 2 \beta ) q^{57} + 26 \beta q^{59} -86 q^{61} + ( -42 - 12 \beta ) q^{63} -2 q^{67} + ( -32 - 4 \beta ) q^{69} -44 \beta q^{71} -82 q^{73} + 12 \beta q^{77} + 10 q^{79} + ( 17 + 28 \beta ) q^{81} + 26 \beta q^{83} + ( -48 - 6 \beta ) q^{87} -12 \beta q^{89} -60 q^{91} + ( 22 - 22 \beta ) q^{93} + 94 q^{97} + ( 32 - 14 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 12q^{7} - 14q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 12q^{7} - 14q^{9} - 20q^{13} + 4q^{19} - 12q^{21} + 46q^{27} - 44q^{31} - 32q^{33} + 12q^{37} + 20q^{39} - 164q^{43} - 26q^{49} - 128q^{51} - 4q^{57} - 172q^{61} - 84q^{63} - 4q^{67} - 64q^{69} - 164q^{73} + 20q^{79} + 34q^{81} - 96q^{87} - 120q^{91} + 44q^{93} + 188q^{97} + 64q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.41421i
1.41421i
0 −1.00000 2.82843i 0 0 0 6.00000 0 −7.00000 + 5.65685i 0
401.2 0 −1.00000 + 2.82843i 0 0 0 6.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.3.l.b 2
3.b odd 2 1 inner 600.3.l.b 2
4.b odd 2 1 1200.3.l.n 2
5.b even 2 1 24.3.e.a 2
5.c odd 4 2 600.3.c.a 4
12.b even 2 1 1200.3.l.n 2
15.d odd 2 1 24.3.e.a 2
15.e even 4 2 600.3.c.a 4
20.d odd 2 1 48.3.e.b 2
20.e even 4 2 1200.3.c.i 4
35.c odd 2 1 1176.3.d.a 2
40.e odd 2 1 192.3.e.d 2
40.f even 2 1 192.3.e.c 2
45.h odd 6 2 648.3.m.d 4
45.j even 6 2 648.3.m.d 4
60.h even 2 1 48.3.e.b 2
60.l odd 4 2 1200.3.c.i 4
80.k odd 4 2 768.3.h.c 4
80.q even 4 2 768.3.h.d 4
105.g even 2 1 1176.3.d.a 2
120.i odd 2 1 192.3.e.c 2
120.m even 2 1 192.3.e.d 2
180.n even 6 2 1296.3.q.e 4
180.p odd 6 2 1296.3.q.e 4
240.t even 4 2 768.3.h.c 4
240.bm odd 4 2 768.3.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 5.b even 2 1
24.3.e.a 2 15.d odd 2 1
48.3.e.b 2 20.d odd 2 1
48.3.e.b 2 60.h even 2 1
192.3.e.c 2 40.f even 2 1
192.3.e.c 2 120.i odd 2 1
192.3.e.d 2 40.e odd 2 1
192.3.e.d 2 120.m even 2 1
600.3.c.a 4 5.c odd 4 2
600.3.c.a 4 15.e even 4 2
600.3.l.b 2 1.a even 1 1 trivial
600.3.l.b 2 3.b odd 2 1 inner
648.3.m.d 4 45.h odd 6 2
648.3.m.d 4 45.j even 6 2
768.3.h.c 4 80.k odd 4 2
768.3.h.c 4 240.t even 4 2
768.3.h.d 4 80.q even 4 2
768.3.h.d 4 240.bm odd 4 2
1176.3.d.a 2 35.c odd 2 1
1176.3.d.a 2 105.g even 2 1
1200.3.c.i 4 20.e even 4 2
1200.3.c.i 4 60.l odd 4 2
1200.3.l.n 2 4.b odd 2 1
1200.3.l.n 2 12.b even 2 1
1296.3.q.e 4 180.n even 6 2
1296.3.q.e 4 180.p odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 6 \) acting on \(S_{3}^{\mathrm{new}}(600, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -6 + T )^{2} \)
$11$ \( 32 + T^{2} \)
$13$ \( ( 10 + T )^{2} \)
$17$ \( 512 + T^{2} \)
$19$ \( ( -2 + T )^{2} \)
$23$ \( 128 + T^{2} \)
$29$ \( 288 + T^{2} \)
$31$ \( ( 22 + T )^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( 1152 + T^{2} \)
$43$ \( ( 82 + T )^{2} \)
$47$ \( 4608 + T^{2} \)
$53$ \( 3872 + T^{2} \)
$59$ \( 5408 + T^{2} \)
$61$ \( ( 86 + T )^{2} \)
$67$ \( ( 2 + T )^{2} \)
$71$ \( 15488 + T^{2} \)
$73$ \( ( 82 + T )^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( 5408 + T^{2} \)
$89$ \( 1152 + T^{2} \)
$97$ \( ( -94 + T )^{2} \)
show more
show less