Properties

Label 1200.4.a.o
Level $1200$
Weight $4$
Character orbit 1200.a
Self dual yes
Analytic conductor $70.802$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{3} + 20 q^{7} + 9 q^{9} + O(q^{10}) \) \( q - 3 q^{3} + 20 q^{7} + 9 q^{9} + 24 q^{11} - 74 q^{13} - 54 q^{17} + 124 q^{19} - 60 q^{21} - 120 q^{23} - 27 q^{27} - 78 q^{29} - 200 q^{31} - 72 q^{33} + 70 q^{37} + 222 q^{39} + 330 q^{41} + 92 q^{43} - 24 q^{47} + 57 q^{49} + 162 q^{51} - 450 q^{53} - 372 q^{57} - 24 q^{59} - 322 q^{61} + 180 q^{63} - 196 q^{67} + 360 q^{69} + 288 q^{71} + 430 q^{73} + 480 q^{77} + 520 q^{79} + 81 q^{81} + 156 q^{83} + 234 q^{87} + 1026 q^{89} - 1480 q^{91} + 600 q^{93} + 286 q^{97} + 216 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −3.00000 0 0 0 20.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.a.o 1
4.b odd 2 1 75.4.a.a 1
5.b even 2 1 240.4.a.f 1
5.c odd 4 2 1200.4.f.m 2
12.b even 2 1 225.4.a.g 1
15.d odd 2 1 720.4.a.r 1
20.d odd 2 1 15.4.a.b 1
20.e even 4 2 75.4.b.a 2
40.e odd 2 1 960.4.a.bi 1
40.f even 2 1 960.4.a.l 1
60.h even 2 1 45.4.a.b 1
60.l odd 4 2 225.4.b.d 2
140.c even 2 1 735.4.a.i 1
180.n even 6 2 405.4.e.k 2
180.p odd 6 2 405.4.e.d 2
220.g even 2 1 1815.4.a.a 1
420.o odd 2 1 2205.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 20.d odd 2 1
45.4.a.b 1 60.h even 2 1
75.4.a.a 1 4.b odd 2 1
75.4.b.a 2 20.e even 4 2
225.4.a.g 1 12.b even 2 1
225.4.b.d 2 60.l odd 4 2
240.4.a.f 1 5.b even 2 1
405.4.e.d 2 180.p odd 6 2
405.4.e.k 2 180.n even 6 2
720.4.a.r 1 15.d odd 2 1
735.4.a.i 1 140.c even 2 1
960.4.a.l 1 40.f even 2 1
960.4.a.bi 1 40.e odd 2 1
1200.4.a.o 1 1.a even 1 1 trivial
1200.4.f.m 2 5.c odd 4 2
1815.4.a.a 1 220.g even 2 1
2205.4.a.c 1 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7} - 20 \)
\( T_{11} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( T \)
$7$ \( -20 + T \)
$11$ \( -24 + T \)
$13$ \( 74 + T \)
$17$ \( 54 + T \)
$19$ \( -124 + T \)
$23$ \( 120 + T \)
$29$ \( 78 + T \)
$31$ \( 200 + T \)
$37$ \( -70 + T \)
$41$ \( -330 + T \)
$43$ \( -92 + T \)
$47$ \( 24 + T \)
$53$ \( 450 + T \)
$59$ \( 24 + T \)
$61$ \( 322 + T \)
$67$ \( 196 + T \)
$71$ \( -288 + T \)
$73$ \( -430 + T \)
$79$ \( -520 + T \)
$83$ \( -156 + T \)
$89$ \( -1026 + T \)
$97$ \( -286 + T \)
show more
show less