Properties

Label 1800.2.k.l
Level $1800$
Weight $2$
Character orbit 1800.k
Analytic conductor $14.373$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a root \(\beta\) of the polynomial \(x^{4} + x^{2} + 4\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} + \beta^{2} q^{4} -\beta^{3} q^{8} +O(q^{10})\) \( q -\beta q^{2} + \beta^{2} q^{4} -\beta^{3} q^{8} + ( -4 - \beta^{2} ) q^{16} + ( 2 \beta - 2 \beta^{3} ) q^{17} + ( 2 + 4 \beta^{2} ) q^{19} + ( -\beta + \beta^{3} ) q^{23} -8 q^{31} + ( 4 \beta + \beta^{3} ) q^{32} + ( -8 - 4 \beta^{2} ) q^{34} + ( -2 \beta - 4 \beta^{3} ) q^{38} + ( 4 + 2 \beta^{2} ) q^{46} + ( 3 \beta - 3 \beta^{3} ) q^{47} -7 q^{49} + ( -3 \beta - \beta^{3} ) q^{53} + ( 4 + 8 \beta^{2} ) q^{61} + 8 \beta q^{62} + ( 4 - 3 \beta^{2} ) q^{64} + ( 8 \beta + 4 \beta^{3} ) q^{68} + ( -16 - 2 \beta^{2} ) q^{76} + 16 q^{79} + ( 12 \beta + 4 \beta^{3} ) q^{83} + ( -4 \beta - 2 \beta^{3} ) q^{92} + ( -12 - 6 \beta^{2} ) q^{94} + 7 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + O(q^{10}) \) \( 4 q - 2 q^{4} - 14 q^{16} - 32 q^{31} - 24 q^{34} + 12 q^{46} - 28 q^{49} + 22 q^{64} - 60 q^{76} + 64 q^{79} - 36 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
0.866025 + 1.11803i
0.866025 1.11803i
−0.866025 + 1.11803i
−0.866025 1.11803i
−0.866025 1.11803i 0 −0.500000 + 1.93649i 0 0 0 2.59808 1.11803i 0 0
901.2 −0.866025 + 1.11803i 0 −0.500000 1.93649i 0 0 0 2.59808 + 1.11803i 0 0
901.3 0.866025 1.11803i 0 −0.500000 1.93649i 0 0 0 −2.59808 1.11803i 0 0
901.4 0.866025 + 1.11803i 0 −0.500000 + 1.93649i 0 0 0 −2.59808 + 1.11803i 0 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
8.b even 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner
120.i odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.k.l 4
3.b odd 2 1 inner 1800.2.k.l 4
4.b odd 2 1 7200.2.k.n 4
5.b even 2 1 inner 1800.2.k.l 4
5.c odd 4 2 360.2.d.c 4
8.b even 2 1 inner 1800.2.k.l 4
8.d odd 2 1 7200.2.k.n 4
12.b even 2 1 7200.2.k.n 4
15.d odd 2 1 CM 1800.2.k.l 4
15.e even 4 2 360.2.d.c 4
20.d odd 2 1 7200.2.k.n 4
20.e even 4 2 1440.2.d.d 4
24.f even 2 1 7200.2.k.n 4
24.h odd 2 1 inner 1800.2.k.l 4
40.e odd 2 1 7200.2.k.n 4
40.f even 2 1 inner 1800.2.k.l 4
40.i odd 4 2 360.2.d.c 4
40.k even 4 2 1440.2.d.d 4
60.h even 2 1 7200.2.k.n 4
60.l odd 4 2 1440.2.d.d 4
120.i odd 2 1 inner 1800.2.k.l 4
120.m even 2 1 7200.2.k.n 4
120.q odd 4 2 1440.2.d.d 4
120.w even 4 2 360.2.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.d.c 4 5.c odd 4 2
360.2.d.c 4 15.e even 4 2
360.2.d.c 4 40.i odd 4 2
360.2.d.c 4 120.w even 4 2
1440.2.d.d 4 20.e even 4 2
1440.2.d.d 4 40.k even 4 2
1440.2.d.d 4 60.l odd 4 2
1440.2.d.d 4 120.q odd 4 2
1800.2.k.l 4 1.a even 1 1 trivial
1800.2.k.l 4 3.b odd 2 1 inner
1800.2.k.l 4 5.b even 2 1 inner
1800.2.k.l 4 8.b even 2 1 inner
1800.2.k.l 4 15.d odd 2 1 CM
1800.2.k.l 4 24.h odd 2 1 inner
1800.2.k.l 4 40.f even 2 1 inner
1800.2.k.l 4 120.i odd 2 1 inner
7200.2.k.n 4 4.b odd 2 1
7200.2.k.n 4 8.d odd 2 1
7200.2.k.n 4 12.b even 2 1
7200.2.k.n 4 20.d odd 2 1
7200.2.k.n 4 24.f even 2 1
7200.2.k.n 4 40.e odd 2 1
7200.2.k.n 4 60.h even 2 1
7200.2.k.n 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7} \)
\( T_{11} \)
\( T_{17}^{2} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -48 + T^{2} )^{2} \)
$19$ \( ( 60 + T^{2} )^{2} \)
$23$ \( ( -12 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 8 + T )^{4} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( -108 + T^{2} )^{2} \)
$53$ \( ( 20 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( 240 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( -16 + T )^{4} \)
$83$ \( ( 320 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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