Properties

Label 600.1.q.a
Level 600
Weight 1
Character orbit 600.q
Analytic conductor 0.299
Analytic rank 0
Dimension 4
Projective image \(D_{2}\)
CM/RM discs -8, -15, 120
Inner twists 16

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 600.q (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.299439007580\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-2}, \sqrt{-15})\)
Artin image $OD_{16}:C_2$
Artin field Galois closure of 16.0.164025000000000000.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{2} -\zeta_{8} q^{3} -\zeta_{8}^{2} q^{4} + q^{6} + \zeta_{8} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{8}^{3} q^{2} -\zeta_{8} q^{3} -\zeta_{8}^{2} q^{4} + q^{6} + \zeta_{8} q^{8} + \zeta_{8}^{2} q^{9} + \zeta_{8}^{3} q^{12} - q^{16} -2 \zeta_{8}^{3} q^{17} -\zeta_{8} q^{18} -2 \zeta_{8}^{2} q^{19} -\zeta_{8}^{2} q^{24} -\zeta_{8}^{3} q^{27} -\zeta_{8}^{3} q^{32} + 2 \zeta_{8}^{2} q^{34} + q^{36} + 2 \zeta_{8} q^{38} + \zeta_{8} q^{48} + \zeta_{8}^{2} q^{49} -2 q^{51} + \zeta_{8}^{2} q^{54} + 2 \zeta_{8}^{3} q^{57} + \zeta_{8}^{2} q^{64} -2 \zeta_{8} q^{68} + \zeta_{8}^{3} q^{72} -2 q^{76} - q^{81} + 2 \zeta_{8} q^{83} - q^{96} -\zeta_{8} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{6} + O(q^{10}) \) \( 4q + 4q^{6} - 4q^{16} + 4q^{36} - 8q^{51} - 8q^{76} - 4q^{81} - 4q^{96} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000 0 0.707107 + 0.707107i 1.00000i 0
107.2 0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000 0 −0.707107 0.707107i 1.00000i 0
443.1 −0.707107 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000 0 0.707107 0.707107i 1.00000i 0
443.2 0.707107 + 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000 0 −0.707107 + 0.707107i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
120.m even 2 1 RM by \(\Q(\sqrt{30}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.e even 4 2 inner
24.f even 2 1 inner
40.e odd 2 1 inner
40.k even 4 2 inner
120.q odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.1.q.a 4
3.b odd 2 1 inner 600.1.q.a 4
4.b odd 2 1 2400.1.u.a 4
5.b even 2 1 inner 600.1.q.a 4
5.c odd 4 2 inner 600.1.q.a 4
8.b even 2 1 2400.1.u.a 4
8.d odd 2 1 CM 600.1.q.a 4
12.b even 2 1 2400.1.u.a 4
15.d odd 2 1 CM 600.1.q.a 4
15.e even 4 2 inner 600.1.q.a 4
20.d odd 2 1 2400.1.u.a 4
20.e even 4 2 2400.1.u.a 4
24.f even 2 1 inner 600.1.q.a 4
24.h odd 2 1 2400.1.u.a 4
40.e odd 2 1 inner 600.1.q.a 4
40.f even 2 1 2400.1.u.a 4
40.i odd 4 2 2400.1.u.a 4
40.k even 4 2 inner 600.1.q.a 4
60.h even 2 1 2400.1.u.a 4
60.l odd 4 2 2400.1.u.a 4
120.i odd 2 1 2400.1.u.a 4
120.m even 2 1 RM 600.1.q.a 4
120.q odd 4 2 inner 600.1.q.a 4
120.w even 4 2 2400.1.u.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.1.q.a 4 1.a even 1 1 trivial
600.1.q.a 4 3.b odd 2 1 inner
600.1.q.a 4 5.b even 2 1 inner
600.1.q.a 4 5.c odd 4 2 inner
600.1.q.a 4 8.d odd 2 1 CM
600.1.q.a 4 15.d odd 2 1 CM
600.1.q.a 4 15.e even 4 2 inner
600.1.q.a 4 24.f even 2 1 inner
600.1.q.a 4 40.e odd 2 1 inner
600.1.q.a 4 40.k even 4 2 inner
600.1.q.a 4 120.m even 2 1 RM
600.1.q.a 4 120.q odd 4 2 inner
2400.1.u.a 4 4.b odd 2 1
2400.1.u.a 4 8.b even 2 1
2400.1.u.a 4 12.b even 2 1
2400.1.u.a 4 20.d odd 2 1
2400.1.u.a 4 20.e even 4 2
2400.1.u.a 4 24.h odd 2 1
2400.1.u.a 4 40.f even 2 1
2400.1.u.a 4 40.i odd 4 2
2400.1.u.a 4 60.h even 2 1
2400.1.u.a 4 60.l odd 4 2
2400.1.u.a 4 120.i odd 2 1
2400.1.u.a 4 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} \) acting on \(S_{1}^{\mathrm{new}}(600, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( 1 + T^{4} \)
$5$ 1
$7$ \( ( 1 + T^{4} )^{2} \)
$11$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$13$ \( ( 1 + T^{4} )^{2} \)
$17$ \( ( 1 + T^{4} )^{2} \)
$19$ \( ( 1 + T^{2} )^{4} \)
$23$ \( ( 1 + T^{4} )^{2} \)
$29$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$31$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$37$ \( ( 1 + T^{4} )^{2} \)
$41$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$43$ \( ( 1 + T^{4} )^{2} \)
$47$ \( ( 1 + T^{4} )^{2} \)
$53$ \( ( 1 + T^{4} )^{2} \)
$59$ \( ( 1 + T^{2} )^{4} \)
$61$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
$67$ \( ( 1 + T^{4} )^{2} \)
$71$ \( ( 1 + T^{2} )^{4} \)
$73$ \( ( 1 + T^{4} )^{2} \)
$79$ \( ( 1 + T^{2} )^{4} \)
$83$ \( ( 1 + T^{4} )^{2} \)
$89$ \( ( 1 + T^{2} )^{4} \)
$97$ \( ( 1 + T^{4} )^{2} \)
show more
show less