Properties

Label 1200.3.l.n
Level $1200$
Weight $3$
Character orbit 1200.l
Analytic conductor $32.698$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{3} -6 q^{7} + ( -7 + 2 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{3} -6 q^{7} + ( -7 + 2 \beta ) q^{9} + 2 \beta q^{11} -10 q^{13} -8 \beta q^{17} -2 q^{19} + ( -6 - 6 \beta ) q^{21} + 4 \beta q^{23} + ( -23 - 5 \beta ) q^{27} -6 \beta q^{29} + 22 q^{31} + ( -16 + 2 \beta ) q^{33} + 6 q^{37} + ( -10 - 10 \beta ) q^{39} -12 \beta q^{41} + 82 q^{43} -24 \beta q^{47} -13 q^{49} + ( 64 - 8 \beta ) q^{51} -22 \beta q^{53} + ( -2 - 2 \beta ) q^{57} + 26 \beta q^{59} -86 q^{61} + ( 42 - 12 \beta ) q^{63} + 2 q^{67} + ( -32 + 4 \beta ) q^{69} -44 \beta q^{71} -82 q^{73} -12 \beta q^{77} -10 q^{79} + ( 17 - 28 \beta ) q^{81} + 26 \beta q^{83} + ( 48 - 6 \beta ) q^{87} + 12 \beta q^{89} + 60 q^{91} + ( 22 + 22 \beta ) q^{93} + 94 q^{97} + ( -32 - 14 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 12q^{7} - 14q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 12q^{7} - 14q^{9} - 20q^{13} - 4q^{19} - 12q^{21} - 46q^{27} + 44q^{31} - 32q^{33} + 12q^{37} - 20q^{39} + 164q^{43} - 26q^{49} + 128q^{51} - 4q^{57} - 172q^{61} + 84q^{63} + 4q^{67} - 64q^{69} - 164q^{73} - 20q^{79} + 34q^{81} + 96q^{87} + 120q^{91} + 44q^{93} + 188q^{97} - 64q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.41421i
1.41421i
0 1.00000 2.82843i 0 0 0 −6.00000 0 −7.00000 5.65685i 0
401.2 0 1.00000 + 2.82843i 0 0 0 −6.00000 0 −7.00000 + 5.65685i 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.l.n 2
3.b odd 2 1 inner 1200.3.l.n 2
4.b odd 2 1 600.3.l.b 2
5.b even 2 1 48.3.e.b 2
5.c odd 4 2 1200.3.c.i 4
12.b even 2 1 600.3.l.b 2
15.d odd 2 1 48.3.e.b 2
15.e even 4 2 1200.3.c.i 4
20.d odd 2 1 24.3.e.a 2
20.e even 4 2 600.3.c.a 4
40.e odd 2 1 192.3.e.c 2
40.f even 2 1 192.3.e.d 2
45.h odd 6 2 1296.3.q.e 4
45.j even 6 2 1296.3.q.e 4
60.h even 2 1 24.3.e.a 2
60.l odd 4 2 600.3.c.a 4
80.k odd 4 2 768.3.h.d 4
80.q even 4 2 768.3.h.c 4
120.i odd 2 1 192.3.e.d 2
120.m even 2 1 192.3.e.c 2
140.c even 2 1 1176.3.d.a 2
180.n even 6 2 648.3.m.d 4
180.p odd 6 2 648.3.m.d 4
240.t even 4 2 768.3.h.d 4
240.bm odd 4 2 768.3.h.c 4
420.o odd 2 1 1176.3.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.e.a 2 20.d odd 2 1
24.3.e.a 2 60.h even 2 1
48.3.e.b 2 5.b even 2 1
48.3.e.b 2 15.d odd 2 1
192.3.e.c 2 40.e odd 2 1
192.3.e.c 2 120.m even 2 1
192.3.e.d 2 40.f even 2 1
192.3.e.d 2 120.i odd 2 1
600.3.c.a 4 20.e even 4 2
600.3.c.a 4 60.l odd 4 2
600.3.l.b 2 4.b odd 2 1
600.3.l.b 2 12.b even 2 1
648.3.m.d 4 180.n even 6 2
648.3.m.d 4 180.p odd 6 2
768.3.h.c 4 80.q even 4 2
768.3.h.c 4 240.bm odd 4 2
768.3.h.d 4 80.k odd 4 2
768.3.h.d 4 240.t even 4 2
1176.3.d.a 2 140.c even 2 1
1176.3.d.a 2 420.o odd 2 1
1200.3.c.i 4 5.c odd 4 2
1200.3.c.i 4 15.e even 4 2
1200.3.l.n 2 1.a even 1 1 trivial
1200.3.l.n 2 3.b odd 2 1 inner
1296.3.q.e 4 45.h odd 6 2
1296.3.q.e 4 45.j even 6 2

Hecke kernels

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

\( T_{7} + 6 \)
\( T_{11}^{2} + 32 \)
\( T_{13} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 6 + T )^{2} \)
$11$ \( 32 + T^{2} \)
$13$ \( ( 10 + T )^{2} \)
$17$ \( 512 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 128 + T^{2} \)
$29$ \( 288 + T^{2} \)
$31$ \( ( -22 + T )^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( 1152 + T^{2} \)
$43$ \( ( -82 + T )^{2} \)
$47$ \( 4608 + T^{2} \)
$53$ \( 3872 + T^{2} \)
$59$ \( 5408 + T^{2} \)
$61$ \( ( 86 + T )^{2} \)
$67$ \( ( -2 + T )^{2} \)
$71$ \( 15488 + T^{2} \)
$73$ \( ( 82 + T )^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( 5408 + T^{2} \)
$89$ \( 1152 + T^{2} \)
$97$ \( ( -94 + T )^{2} \)
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