Properties

Label 1800.2.k.e
Level $1800$
Weight $2$
Character orbit 1800.k
Analytic conductor $14.373$
Analytic rank $1$
Dimension $2$
CM discriminant -24
Inner twists $4$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta q^{2} -2 q^{4} -2 q^{7} + 2 \beta q^{8} +O(q^{10})\) \( q -\beta q^{2} -2 q^{4} -2 q^{7} + 2 \beta q^{8} -4 \beta q^{11} + 2 \beta q^{14} + 4 q^{16} -8 q^{22} + 4 q^{28} + 2 \beta q^{29} -10 q^{31} -4 \beta q^{32} + 8 \beta q^{44} -3 q^{49} + 10 \beta q^{53} -4 \beta q^{56} + 4 q^{58} + 8 \beta q^{59} + 10 \beta q^{62} -8 q^{64} -14 q^{73} + 8 \beta q^{77} -10 q^{79} + 4 \beta q^{83} + 16 q^{88} -2 q^{97} + 3 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 4 q^{7} + O(q^{10}) \) \( 2 q - 4 q^{4} - 4 q^{7} + 8 q^{16} - 16 q^{22} + 8 q^{28} - 20 q^{31} - 6 q^{49} + 8 q^{58} - 16 q^{64} - 28 q^{73} - 20 q^{79} + 32 q^{88} - 4 q^{97} + O(q^{100}) \)

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
1.41421i
1.41421i
1.41421i 0 −2.00000 0 0 −2.00000 2.82843i 0 0
901.2 1.41421i 0 −2.00000 0 0 −2.00000 2.82843i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 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.e 2
3.b odd 2 1 inner 1800.2.k.e 2
4.b odd 2 1 7200.2.k.h 2
5.b even 2 1 72.2.d.a 2
5.c odd 4 2 1800.2.d.n 4
8.b even 2 1 inner 1800.2.k.e 2
8.d odd 2 1 7200.2.k.h 2
12.b even 2 1 7200.2.k.h 2
15.d odd 2 1 72.2.d.a 2
15.e even 4 2 1800.2.d.n 4
20.d odd 2 1 288.2.d.a 2
20.e even 4 2 7200.2.d.p 4
24.f even 2 1 7200.2.k.h 2
24.h odd 2 1 CM 1800.2.k.e 2
40.e odd 2 1 288.2.d.a 2
40.f even 2 1 72.2.d.a 2
40.i odd 4 2 1800.2.d.n 4
40.k even 4 2 7200.2.d.p 4
45.h odd 6 2 648.2.n.h 4
45.j even 6 2 648.2.n.h 4
60.h even 2 1 288.2.d.a 2
60.l odd 4 2 7200.2.d.p 4
80.k odd 4 2 2304.2.a.y 2
80.q even 4 2 2304.2.a.q 2
120.i odd 2 1 72.2.d.a 2
120.m even 2 1 288.2.d.a 2
120.q odd 4 2 7200.2.d.p 4
120.w even 4 2 1800.2.d.n 4
180.n even 6 2 2592.2.r.i 4
180.p odd 6 2 2592.2.r.i 4
240.t even 4 2 2304.2.a.y 2
240.bm odd 4 2 2304.2.a.q 2
360.z odd 6 2 2592.2.r.i 4
360.bd even 6 2 2592.2.r.i 4
360.bh odd 6 2 648.2.n.h 4
360.bk even 6 2 648.2.n.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.d.a 2 5.b even 2 1
72.2.d.a 2 15.d odd 2 1
72.2.d.a 2 40.f even 2 1
72.2.d.a 2 120.i odd 2 1
288.2.d.a 2 20.d odd 2 1
288.2.d.a 2 40.e odd 2 1
288.2.d.a 2 60.h even 2 1
288.2.d.a 2 120.m even 2 1
648.2.n.h 4 45.h odd 6 2
648.2.n.h 4 45.j even 6 2
648.2.n.h 4 360.bh odd 6 2
648.2.n.h 4 360.bk even 6 2
1800.2.d.n 4 5.c odd 4 2
1800.2.d.n 4 15.e even 4 2
1800.2.d.n 4 40.i odd 4 2
1800.2.d.n 4 120.w even 4 2
1800.2.k.e 2 1.a even 1 1 trivial
1800.2.k.e 2 3.b odd 2 1 inner
1800.2.k.e 2 8.b even 2 1 inner
1800.2.k.e 2 24.h odd 2 1 CM
2304.2.a.q 2 80.q even 4 2
2304.2.a.q 2 240.bm odd 4 2
2304.2.a.y 2 80.k odd 4 2
2304.2.a.y 2 240.t even 4 2
2592.2.r.i 4 180.n even 6 2
2592.2.r.i 4 180.p odd 6 2
2592.2.r.i 4 360.z odd 6 2
2592.2.r.i 4 360.bd even 6 2
7200.2.d.p 4 20.e even 4 2
7200.2.d.p 4 40.k even 4 2
7200.2.d.p 4 60.l odd 4 2
7200.2.d.p 4 120.q odd 4 2
7200.2.k.h 2 4.b odd 2 1
7200.2.k.h 2 8.d odd 2 1
7200.2.k.h 2 12.b even 2 1
7200.2.k.h 2 24.f even 2 1

Hecke kernels

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

\( T_{7} + 2 \)
\( T_{11}^{2} + 32 \)
\( T_{17} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( 32 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 8 + T^{2} \)
$31$ \( ( 10 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 200 + T^{2} \)
$59$ \( 128 + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 14 + T )^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( 32 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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