Properties

Label 600.1.n.b
Level 600
Weight 1
Character orbit 600.n
Self dual yes
Analytic conductor 0.299
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -15, -24, 40
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.299439007580\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-6}, \sqrt{10})\)
Artin image $D_4$
Artin field Galois closure of 4.0.9000.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} - q^{12} + q^{16} + q^{18} - q^{24} - q^{27} - 2q^{31} + q^{32} + q^{36} - q^{48} - q^{49} - 2q^{53} - q^{54} - 2q^{62} + q^{64} + q^{72} - 2q^{79} + q^{81} + 2q^{83} + 2q^{93} - q^{96} - q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0
1.00000 −1.00000 1.00000 0 −1.00000 0 1.00000 1.00000 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.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
40.f even 2 1 RM by \(\Q(\sqrt{10}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.1.n.b 1
3.b odd 2 1 600.1.n.a 1
4.b odd 2 1 2400.1.n.b 1
5.b even 2 1 600.1.n.a 1
5.c odd 4 2 120.1.i.a 2
8.b even 2 1 600.1.n.a 1
8.d odd 2 1 2400.1.n.a 1
12.b even 2 1 2400.1.n.a 1
15.d odd 2 1 CM 600.1.n.b 1
15.e even 4 2 120.1.i.a 2
20.d odd 2 1 2400.1.n.a 1
20.e even 4 2 480.1.i.a 2
24.f even 2 1 2400.1.n.b 1
24.h odd 2 1 CM 600.1.n.b 1
40.e odd 2 1 2400.1.n.b 1
40.f even 2 1 RM 600.1.n.b 1
40.i odd 4 2 120.1.i.a 2
40.k even 4 2 480.1.i.a 2
45.k odd 12 4 3240.1.bh.h 4
45.l even 12 4 3240.1.bh.h 4
60.h even 2 1 2400.1.n.b 1
60.l odd 4 2 480.1.i.a 2
80.i odd 4 2 3840.1.c.d 1
80.j even 4 2 3840.1.c.c 1
80.s even 4 2 3840.1.c.b 1
80.t odd 4 2 3840.1.c.a 1
120.i odd 2 1 600.1.n.a 1
120.m even 2 1 2400.1.n.a 1
120.q odd 4 2 480.1.i.a 2
120.w even 4 2 120.1.i.a 2
240.z odd 4 2 3840.1.c.c 1
240.bb even 4 2 3840.1.c.a 1
240.bd odd 4 2 3840.1.c.b 1
240.bf even 4 2 3840.1.c.d 1
360.br even 12 4 3240.1.bh.h 4
360.bu odd 12 4 3240.1.bh.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.1.i.a 2 5.c odd 4 2
120.1.i.a 2 15.e even 4 2
120.1.i.a 2 40.i odd 4 2
120.1.i.a 2 120.w even 4 2
480.1.i.a 2 20.e even 4 2
480.1.i.a 2 40.k even 4 2
480.1.i.a 2 60.l odd 4 2
480.1.i.a 2 120.q odd 4 2
600.1.n.a 1 3.b odd 2 1
600.1.n.a 1 5.b even 2 1
600.1.n.a 1 8.b even 2 1
600.1.n.a 1 120.i odd 2 1
600.1.n.b 1 1.a even 1 1 trivial
600.1.n.b 1 15.d odd 2 1 CM
600.1.n.b 1 24.h odd 2 1 CM
600.1.n.b 1 40.f even 2 1 RM
2400.1.n.a 1 8.d odd 2 1
2400.1.n.a 1 12.b even 2 1
2400.1.n.a 1 20.d odd 2 1
2400.1.n.a 1 120.m even 2 1
2400.1.n.b 1 4.b odd 2 1
2400.1.n.b 1 24.f even 2 1
2400.1.n.b 1 40.e odd 2 1
2400.1.n.b 1 60.h even 2 1
3240.1.bh.h 4 45.k odd 12 4
3240.1.bh.h 4 45.l even 12 4
3240.1.bh.h 4 360.br even 12 4
3240.1.bh.h 4 360.bu odd 12 4
3840.1.c.a 1 80.t odd 4 2
3840.1.c.a 1 240.bb even 4 2
3840.1.c.b 1 80.s even 4 2
3840.1.c.b 1 240.bd odd 4 2
3840.1.c.c 1 80.j even 4 2
3840.1.c.c 1 240.z odd 4 2
3840.1.c.d 1 80.i odd 4 2
3840.1.c.d 1 240.bf even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{53} + 2 \) acting on \(S_{1}^{\mathrm{new}}(600, [\chi])\).

Hecke Characteristic Polynomials

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