Properties

Label 1200.3.l.l
Level $1200$
Weight $3$
Character orbit 1200.l
Analytic conductor $32.698$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 i q^{3} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} - 9 q^{9} + 14 i q^{17} - 22 q^{19} - 34 i q^{23} - 27 i q^{27} - 2 q^{31} - 14 i q^{47} - 49 q^{49} - 42 q^{51} - 86 i q^{53} - 66 i q^{57} - 118 q^{61} + 102 q^{69} + 98 q^{79} + 81 q^{81} - 154 i q^{83} - 6 i q^{93} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 44 q^{19} - 4 q^{31} - 98 q^{49} - 84 q^{51} - 236 q^{61} + 204 q^{69} + 196 q^{79} + 162 q^{81}+O(q^{100}) \) Copy content Toggle raw display

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
1.00000i
1.00000i
0 3.00000i 0 0 0 0 0 −9.00000 0
401.2 0 3.00000i 0 0 0 0 0 −9.00000 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
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 1200.3.l.l 2
3.b odd 2 1 inner 1200.3.l.l 2
4.b odd 2 1 75.3.c.d 2
5.b even 2 1 inner 1200.3.l.l 2
5.c odd 4 1 240.3.c.a 1
5.c odd 4 1 240.3.c.b 1
12.b even 2 1 75.3.c.d 2
15.d odd 2 1 CM 1200.3.l.l 2
15.e even 4 1 240.3.c.a 1
15.e even 4 1 240.3.c.b 1
20.d odd 2 1 75.3.c.d 2
20.e even 4 1 15.3.d.a 1
20.e even 4 1 15.3.d.b yes 1
40.i odd 4 1 960.3.c.a 1
40.i odd 4 1 960.3.c.d 1
40.k even 4 1 960.3.c.b 1
40.k even 4 1 960.3.c.c 1
60.h even 2 1 75.3.c.d 2
60.l odd 4 1 15.3.d.a 1
60.l odd 4 1 15.3.d.b yes 1
120.q odd 4 1 960.3.c.b 1
120.q odd 4 1 960.3.c.c 1
120.w even 4 1 960.3.c.a 1
120.w even 4 1 960.3.c.d 1
180.v odd 12 2 405.3.h.a 2
180.v odd 12 2 405.3.h.b 2
180.x even 12 2 405.3.h.a 2
180.x even 12 2 405.3.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 20.e even 4 1
15.3.d.a 1 60.l odd 4 1
15.3.d.b yes 1 20.e even 4 1
15.3.d.b yes 1 60.l odd 4 1
75.3.c.d 2 4.b odd 2 1
75.3.c.d 2 12.b even 2 1
75.3.c.d 2 20.d odd 2 1
75.3.c.d 2 60.h even 2 1
240.3.c.a 1 5.c odd 4 1
240.3.c.a 1 15.e even 4 1
240.3.c.b 1 5.c odd 4 1
240.3.c.b 1 15.e even 4 1
405.3.h.a 2 180.v odd 12 2
405.3.h.a 2 180.x even 12 2
405.3.h.b 2 180.v odd 12 2
405.3.h.b 2 180.x even 12 2
960.3.c.a 1 40.i odd 4 1
960.3.c.a 1 120.w even 4 1
960.3.c.b 1 40.k even 4 1
960.3.c.b 1 120.q odd 4 1
960.3.c.c 1 40.k even 4 1
960.3.c.c 1 120.q odd 4 1
960.3.c.d 1 40.i odd 4 1
960.3.c.d 1 120.w even 4 1
1200.3.l.l 2 1.a even 1 1 trivial
1200.3.l.l 2 3.b odd 2 1 inner
1200.3.l.l 2 5.b even 2 1 inner
1200.3.l.l 2 15.d odd 2 1 CM

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 196 \) Copy content Toggle raw display
$19$ \( (T + 22)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1156 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 196 \) Copy content Toggle raw display
$53$ \( T^{2} + 7396 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 118)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 98)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 23716 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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