Properties

Label 1200.2.f.d
Level $1200$
Weight $2$
Character orbit 1200.f
Analytic conductor $9.582$
Analytic rank $0$
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: \(0\)
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 75)
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.

\(f(q)\) \(=\) \( q -i q^{3} + 3 i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} + 3 i q^{7} - q^{9} -2 q^{11} -i q^{13} + 2 i q^{17} -5 q^{19} + 3 q^{21} + 6 i q^{23} + i q^{27} -10 q^{29} + 3 q^{31} + 2 i q^{33} + 2 i q^{37} - q^{39} -8 q^{41} + i q^{43} -2 i q^{47} -2 q^{49} + 2 q^{51} + 4 i q^{53} + 5 i q^{57} -10 q^{59} + 7 q^{61} -3 i q^{63} + 3 i q^{67} + 6 q^{69} + 8 q^{71} + 14 i q^{73} -6 i q^{77} + q^{81} + 6 i q^{83} + 10 i q^{87} + 3 q^{91} -3 i q^{93} + 17 i q^{97} + 2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 4q^{11} - 10q^{19} + 6q^{21} - 20q^{29} + 6q^{31} - 2q^{39} - 16q^{41} - 4q^{49} + 4q^{51} - 20q^{59} + 14q^{61} + 12q^{69} + 16q^{71} + 2q^{81} + 6q^{91} + 4q^{99} + O(q^{100}) \)

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 3.00000i 0 −1.00000 0
49.2 0 1.00000i 0 0 0 3.00000i 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.d 2
3.b odd 2 1 3600.2.f.p 2
4.b odd 2 1 75.2.b.a 2
5.b even 2 1 inner 1200.2.f.d 2
5.c odd 4 1 1200.2.a.c 1
5.c odd 4 1 1200.2.a.p 1
8.b even 2 1 4800.2.f.y 2
8.d odd 2 1 4800.2.f.l 2
12.b even 2 1 225.2.b.a 2
15.d odd 2 1 3600.2.f.p 2
15.e even 4 1 3600.2.a.j 1
15.e even 4 1 3600.2.a.bk 1
20.d odd 2 1 75.2.b.a 2
20.e even 4 1 75.2.a.a 1
20.e even 4 1 75.2.a.c yes 1
40.e odd 2 1 4800.2.f.l 2
40.f even 2 1 4800.2.f.y 2
40.i odd 4 1 4800.2.a.be 1
40.i odd 4 1 4800.2.a.br 1
40.k even 4 1 4800.2.a.bb 1
40.k even 4 1 4800.2.a.bq 1
60.h even 2 1 225.2.b.a 2
60.l odd 4 1 225.2.a.a 1
60.l odd 4 1 225.2.a.e 1
140.j odd 4 1 3675.2.a.b 1
140.j odd 4 1 3675.2.a.q 1
220.i odd 4 1 9075.2.a.a 1
220.i odd 4 1 9075.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 20.e even 4 1
75.2.a.c yes 1 20.e even 4 1
75.2.b.a 2 4.b odd 2 1
75.2.b.a 2 20.d odd 2 1
225.2.a.a 1 60.l odd 4 1
225.2.a.e 1 60.l odd 4 1
225.2.b.a 2 12.b even 2 1
225.2.b.a 2 60.h even 2 1
1200.2.a.c 1 5.c odd 4 1
1200.2.a.p 1 5.c odd 4 1
1200.2.f.d 2 1.a even 1 1 trivial
1200.2.f.d 2 5.b even 2 1 inner
3600.2.a.j 1 15.e even 4 1
3600.2.a.bk 1 15.e even 4 1
3600.2.f.p 2 3.b odd 2 1
3600.2.f.p 2 15.d odd 2 1
3675.2.a.b 1 140.j odd 4 1
3675.2.a.q 1 140.j odd 4 1
4800.2.a.bb 1 40.k even 4 1
4800.2.a.be 1 40.i odd 4 1
4800.2.a.bq 1 40.k even 4 1
4800.2.a.br 1 40.i odd 4 1
4800.2.f.l 2 8.d odd 2 1
4800.2.f.l 2 40.e odd 2 1
4800.2.f.y 2 8.b even 2 1
4800.2.f.y 2 40.f even 2 1
9075.2.a.a 1 220.i odd 4 1
9075.2.a.s 1 220.i odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 9 \)
\( T_{11} + 2 \)
\( T_{13}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 9 + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 1 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( ( 5 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 10 + T )^{2} \)
$31$ \( ( -3 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 8 + T )^{2} \)
$43$ \( 1 + T^{2} \)
$47$ \( 4 + T^{2} \)
$53$ \( 16 + T^{2} \)
$59$ \( ( 10 + T )^{2} \)
$61$ \( ( -7 + T )^{2} \)
$67$ \( 9 + T^{2} \)
$71$ \( ( -8 + T )^{2} \)
$73$ \( 196 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 289 + T^{2} \)
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