Properties

Label 1200.3.l.s
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: \(\Q(\sqrt{-11}) \)
Defining polynomial: \(x^{2} - x + 3\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta ) q^{3} + ( 6 - 5 \beta ) q^{9} +O(q^{10})\) \( q + ( 3 - \beta ) q^{3} + ( 6 - 5 \beta ) q^{9} + ( 5 - 10 \beta ) q^{11} -10 q^{13} + ( -1 + 2 \beta ) q^{17} -7 q^{19} + ( 6 - 12 \beta ) q^{23} + ( 3 - 16 \beta ) q^{27} + ( -10 + 20 \beta ) q^{29} -42 q^{31} + ( -15 - 25 \beta ) q^{33} + 40 q^{37} + ( -30 + 10 \beta ) q^{39} + ( 5 - 10 \beta ) q^{41} -50 q^{43} + ( -14 + 28 \beta ) q^{47} -49 q^{49} + ( 3 + 5 \beta ) q^{51} + ( 14 - 28 \beta ) q^{53} + ( -21 + 7 \beta ) q^{57} + ( 20 - 40 \beta ) q^{59} -8 q^{61} + 45 q^{67} + ( -18 - 30 \beta ) q^{69} + ( 10 - 20 \beta ) q^{71} + 35 q^{73} -12 q^{79} + ( -39 - 35 \beta ) q^{81} + ( 21 - 42 \beta ) q^{83} + ( 30 + 50 \beta ) q^{87} + ( 45 - 90 \beta ) q^{89} + ( -126 + 42 \beta ) q^{93} + 70 q^{97} + ( -120 - 35 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 5q^{3} + 7q^{9} + O(q^{10}) \) \( 2q + 5q^{3} + 7q^{9} - 20q^{13} - 14q^{19} - 10q^{27} - 84q^{31} - 55q^{33} + 80q^{37} - 50q^{39} - 100q^{43} - 98q^{49} + 11q^{51} - 35q^{57} - 16q^{61} + 90q^{67} - 66q^{69} + 70q^{73} - 24q^{79} - 113q^{81} + 110q^{87} - 210q^{93} + 140q^{97} - 275q^{99} + O(q^{100}) \)

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
0.500000 + 1.65831i
0.500000 1.65831i
0 2.50000 1.65831i 0 0 0 0 0 3.50000 8.29156i 0
401.2 0 2.50000 + 1.65831i 0 0 0 0 0 3.50000 + 8.29156i 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.s 2
3.b odd 2 1 inner 1200.3.l.s 2
4.b odd 2 1 75.3.c.c 2
5.b even 2 1 1200.3.l.f 2
5.c odd 4 2 1200.3.c.d 4
12.b even 2 1 75.3.c.c 2
15.d odd 2 1 1200.3.l.f 2
15.e even 4 2 1200.3.c.d 4
20.d odd 2 1 75.3.c.f yes 2
20.e even 4 2 75.3.d.c 4
60.h even 2 1 75.3.c.f yes 2
60.l odd 4 2 75.3.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.c 2 4.b odd 2 1
75.3.c.c 2 12.b even 2 1
75.3.c.f yes 2 20.d odd 2 1
75.3.c.f yes 2 60.h even 2 1
75.3.d.c 4 20.e even 4 2
75.3.d.c 4 60.l odd 4 2
1200.3.c.d 4 5.c odd 4 2
1200.3.c.d 4 15.e even 4 2
1200.3.l.f 2 5.b even 2 1
1200.3.l.f 2 15.d odd 2 1
1200.3.l.s 2 1.a even 1 1 trivial
1200.3.l.s 2 3.b odd 2 1 inner

Hecke kernels

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

\( T_{7} \)
\( T_{11}^{2} + 275 \)
\( T_{13} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 - 5 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 275 + T^{2} \)
$13$ \( ( 10 + T )^{2} \)
$17$ \( 11 + T^{2} \)
$19$ \( ( 7 + T )^{2} \)
$23$ \( 396 + T^{2} \)
$29$ \( 1100 + T^{2} \)
$31$ \( ( 42 + T )^{2} \)
$37$ \( ( -40 + T )^{2} \)
$41$ \( 275 + T^{2} \)
$43$ \( ( 50 + T )^{2} \)
$47$ \( 2156 + T^{2} \)
$53$ \( 2156 + T^{2} \)
$59$ \( 4400 + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( ( -45 + T )^{2} \)
$71$ \( 1100 + T^{2} \)
$73$ \( ( -35 + T )^{2} \)
$79$ \( ( 12 + T )^{2} \)
$83$ \( 4851 + T^{2} \)
$89$ \( 22275 + T^{2} \)
$97$ \( ( -70 + T )^{2} \)
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