Properties

Label 1200.3.l.b
Level $1200$
Weight $3$
Character orbit 1200.l
Self dual yes
Analytic conductor $32.698$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{3} + 2q^{7} + 9q^{9} + O(q^{10}) \) \( q - 3q^{3} + 2q^{7} + 9q^{9} + 22q^{13} - 26q^{19} - 6q^{21} - 27q^{27} + 46q^{31} - 26q^{37} - 66q^{39} - 22q^{43} - 45q^{49} + 78q^{57} + 74q^{61} + 18q^{63} + 122q^{67} + 46q^{73} + 142q^{79} + 81q^{81} + 44q^{91} - 138q^{93} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
0
0 −3.00000 0 0 0 2.00000 0 9.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.l.b 1
3.b odd 2 1 CM 1200.3.l.b 1
4.b odd 2 1 300.3.g.b 1
5.b even 2 1 48.3.e.a 1
5.c odd 4 2 1200.3.c.c 2
12.b even 2 1 300.3.g.b 1
15.d odd 2 1 48.3.e.a 1
15.e even 4 2 1200.3.c.c 2
20.d odd 2 1 12.3.c.a 1
20.e even 4 2 300.3.b.a 2
40.e odd 2 1 192.3.e.b 1
40.f even 2 1 192.3.e.a 1
45.h odd 6 2 1296.3.q.b 2
45.j even 6 2 1296.3.q.b 2
60.h even 2 1 12.3.c.a 1
60.l odd 4 2 300.3.b.a 2
80.k odd 4 2 768.3.h.a 2
80.q even 4 2 768.3.h.b 2
120.i odd 2 1 192.3.e.a 1
120.m even 2 1 192.3.e.b 1
140.c even 2 1 588.3.c.c 1
140.p odd 6 2 588.3.p.c 2
140.s even 6 2 588.3.p.b 2
180.n even 6 2 324.3.g.b 2
180.p odd 6 2 324.3.g.b 2
220.g even 2 1 1452.3.e.b 1
240.t even 4 2 768.3.h.a 2
240.bm odd 4 2 768.3.h.b 2
420.o odd 2 1 588.3.c.c 1
420.ba even 6 2 588.3.p.c 2
420.be odd 6 2 588.3.p.b 2
660.g odd 2 1 1452.3.e.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.3.c.a 1 20.d odd 2 1
12.3.c.a 1 60.h even 2 1
48.3.e.a 1 5.b even 2 1
48.3.e.a 1 15.d odd 2 1
192.3.e.a 1 40.f even 2 1
192.3.e.a 1 120.i odd 2 1
192.3.e.b 1 40.e odd 2 1
192.3.e.b 1 120.m even 2 1
300.3.b.a 2 20.e even 4 2
300.3.b.a 2 60.l odd 4 2
300.3.g.b 1 4.b odd 2 1
300.3.g.b 1 12.b even 2 1
324.3.g.b 2 180.n even 6 2
324.3.g.b 2 180.p odd 6 2
588.3.c.c 1 140.c even 2 1
588.3.c.c 1 420.o odd 2 1
588.3.p.b 2 140.s even 6 2
588.3.p.b 2 420.be odd 6 2
588.3.p.c 2 140.p odd 6 2
588.3.p.c 2 420.ba even 6 2
768.3.h.a 2 80.k odd 4 2
768.3.h.a 2 240.t even 4 2
768.3.h.b 2 80.q even 4 2
768.3.h.b 2 240.bm odd 4 2
1200.3.c.c 2 5.c odd 4 2
1200.3.c.c 2 15.e even 4 2
1200.3.l.b 1 1.a even 1 1 trivial
1200.3.l.b 1 3.b odd 2 1 CM
1296.3.q.b 2 45.h odd 6 2
1296.3.q.b 2 45.j even 6 2
1452.3.e.b 1 220.g even 2 1
1452.3.e.b 1 660.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7} - 2 \)
\( T_{11} \)
\( T_{13} - 22 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( T \)
$7$ \( -2 + T \)
$11$ \( T \)
$13$ \( -22 + T \)
$17$ \( T \)
$19$ \( 26 + T \)
$23$ \( T \)
$29$ \( T \)
$31$ \( -46 + T \)
$37$ \( 26 + T \)
$41$ \( T \)
$43$ \( 22 + T \)
$47$ \( T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( -74 + T \)
$67$ \( -122 + T \)
$71$ \( T \)
$73$ \( -46 + T \)
$79$ \( -142 + T \)
$83$ \( T \)
$89$ \( T \)
$97$ \( 2 + T \)
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