Properties

Label 1200.2.f.b
Level $1200$
Weight $2$
Character orbit 1200.f
Analytic conductor $9.582$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.58204824255\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -i q^{3} - q^{9} +O(q^{10})\) \( q -i q^{3} - q^{9} -4 q^{11} + 2 i q^{13} + 2 i q^{17} -4 q^{19} -8 i q^{23} + i q^{27} -6 q^{29} -8 q^{31} + 4 i q^{33} + 6 i q^{37} + 2 q^{39} -6 q^{41} + 4 i q^{43} + 7 q^{49} + 2 q^{51} + 2 i q^{53} + 4 i q^{57} + 4 q^{59} -2 q^{61} + 4 i q^{67} -8 q^{69} -8 q^{71} -10 i q^{73} -8 q^{79} + q^{81} -4 i q^{83} + 6 i q^{87} + 6 q^{89} + 8 i q^{93} + 2 i q^{97} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 8q^{11} - 8q^{19} - 12q^{29} - 16q^{31} + 4q^{39} - 12q^{41} + 14q^{49} + 4q^{51} + 8q^{59} - 4q^{61} - 16q^{69} - 16q^{71} - 16q^{79} + 2q^{81} + 12q^{89} + 8q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 1.00000i 0 0 0 0 0 −1.00000 0
49.2 0 1.00000i 0 0 0 0 0 −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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.f.b 2
3.b odd 2 1 3600.2.f.r 2
4.b odd 2 1 600.2.f.e 2
5.b even 2 1 inner 1200.2.f.b 2
5.c odd 4 1 48.2.a.a 1
5.c odd 4 1 1200.2.a.d 1
8.b even 2 1 4800.2.f.bg 2
8.d odd 2 1 4800.2.f.d 2
12.b even 2 1 1800.2.f.c 2
15.d odd 2 1 3600.2.f.r 2
15.e even 4 1 144.2.a.b 1
15.e even 4 1 3600.2.a.v 1
20.d odd 2 1 600.2.f.e 2
20.e even 4 1 24.2.a.a 1
20.e even 4 1 600.2.a.h 1
35.f even 4 1 2352.2.a.i 1
35.k even 12 2 2352.2.q.r 2
35.l odd 12 2 2352.2.q.l 2
40.e odd 2 1 4800.2.f.d 2
40.f even 2 1 4800.2.f.bg 2
40.i odd 4 1 192.2.a.b 1
40.i odd 4 1 4800.2.a.cc 1
40.k even 4 1 192.2.a.d 1
40.k even 4 1 4800.2.a.q 1
45.k odd 12 2 1296.2.i.m 2
45.l even 12 2 1296.2.i.e 2
55.e even 4 1 5808.2.a.s 1
60.h even 2 1 1800.2.f.c 2
60.l odd 4 1 72.2.a.a 1
60.l odd 4 1 1800.2.a.m 1
65.h odd 4 1 8112.2.a.be 1
80.i odd 4 1 768.2.d.d 2
80.j even 4 1 768.2.d.e 2
80.s even 4 1 768.2.d.e 2
80.t odd 4 1 768.2.d.d 2
105.k odd 4 1 7056.2.a.q 1
120.q odd 4 1 576.2.a.d 1
120.w even 4 1 576.2.a.b 1
140.j odd 4 1 1176.2.a.i 1
140.w even 12 2 1176.2.q.i 2
140.x odd 12 2 1176.2.q.a 2
180.v odd 12 2 648.2.i.b 2
180.x even 12 2 648.2.i.g 2
220.i odd 4 1 2904.2.a.c 1
240.z odd 4 1 2304.2.d.i 2
240.bb even 4 1 2304.2.d.k 2
240.bd odd 4 1 2304.2.d.i 2
240.bf even 4 1 2304.2.d.k 2
260.l odd 4 1 4056.2.c.e 2
260.p even 4 1 4056.2.a.i 1
260.s odd 4 1 4056.2.c.e 2
280.s even 4 1 9408.2.a.cc 1
280.y odd 4 1 9408.2.a.h 1
340.r even 4 1 6936.2.a.p 1
380.j odd 4 1 8664.2.a.j 1
420.w even 4 1 3528.2.a.d 1
420.bp odd 12 2 3528.2.s.j 2
420.br even 12 2 3528.2.s.y 2
660.q even 4 1 8712.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 20.e even 4 1
48.2.a.a 1 5.c odd 4 1
72.2.a.a 1 60.l odd 4 1
144.2.a.b 1 15.e even 4 1
192.2.a.b 1 40.i odd 4 1
192.2.a.d 1 40.k even 4 1
576.2.a.b 1 120.w even 4 1
576.2.a.d 1 120.q odd 4 1
600.2.a.h 1 20.e even 4 1
600.2.f.e 2 4.b odd 2 1
600.2.f.e 2 20.d odd 2 1
648.2.i.b 2 180.v odd 12 2
648.2.i.g 2 180.x even 12 2
768.2.d.d 2 80.i odd 4 1
768.2.d.d 2 80.t odd 4 1
768.2.d.e 2 80.j even 4 1
768.2.d.e 2 80.s even 4 1
1176.2.a.i 1 140.j odd 4 1
1176.2.q.a 2 140.x odd 12 2
1176.2.q.i 2 140.w even 12 2
1200.2.a.d 1 5.c odd 4 1
1200.2.f.b 2 1.a even 1 1 trivial
1200.2.f.b 2 5.b even 2 1 inner
1296.2.i.e 2 45.l even 12 2
1296.2.i.m 2 45.k odd 12 2
1800.2.a.m 1 60.l odd 4 1
1800.2.f.c 2 12.b even 2 1
1800.2.f.c 2 60.h even 2 1
2304.2.d.i 2 240.z odd 4 1
2304.2.d.i 2 240.bd odd 4 1
2304.2.d.k 2 240.bb even 4 1
2304.2.d.k 2 240.bf even 4 1
2352.2.a.i 1 35.f even 4 1
2352.2.q.l 2 35.l odd 12 2
2352.2.q.r 2 35.k even 12 2
2904.2.a.c 1 220.i odd 4 1
3528.2.a.d 1 420.w even 4 1
3528.2.s.j 2 420.bp odd 12 2
3528.2.s.y 2 420.br even 12 2
3600.2.a.v 1 15.e even 4 1
3600.2.f.r 2 3.b odd 2 1
3600.2.f.r 2 15.d odd 2 1
4056.2.a.i 1 260.p even 4 1
4056.2.c.e 2 260.l odd 4 1
4056.2.c.e 2 260.s odd 4 1
4800.2.a.q 1 40.k even 4 1
4800.2.a.cc 1 40.i odd 4 1
4800.2.f.d 2 8.d odd 2 1
4800.2.f.d 2 40.e odd 2 1
4800.2.f.bg 2 8.b even 2 1
4800.2.f.bg 2 40.f even 2 1
5808.2.a.s 1 55.e even 4 1
6936.2.a.p 1 340.r even 4 1
7056.2.a.q 1 105.k odd 4 1
8112.2.a.be 1 65.h odd 4 1
8664.2.a.j 1 380.j odd 4 1
8712.2.a.u 1 660.q even 4 1
9408.2.a.h 1 280.y odd 4 1
9408.2.a.cc 1 280.s even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7} \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} \)
$5$ 1
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 + 4 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 18 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( 1 - 102 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 - 4 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 46 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 150 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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