Properties

Label 1800.2.k.o
Level $1800$
Weight $2$
Character orbit 1800.k
Analytic conductor $14.373$
Analytic rank $0$
Dimension $4$
CM discriminant -120
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: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 q^{4} + 2 \beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 q^{4} + 2 \beta_{1} q^{8} + \beta_{3} q^{11} + \beta_{2} q^{13} + 4 q^{16} + 2 \beta_{1} q^{17} -\beta_{2} q^{22} + 4 \beta_{1} q^{23} -2 \beta_{3} q^{26} -\beta_{3} q^{29} + 2 q^{31} + 4 \beta_{1} q^{32} + 4 q^{34} -\beta_{2} q^{37} -2 \beta_{2} q^{43} + 2 \beta_{3} q^{44} + 8 q^{46} + 8 \beta_{1} q^{47} -7 q^{49} + 2 \beta_{2} q^{52} + \beta_{2} q^{58} -\beta_{3} q^{59} + 2 \beta_{1} q^{62} + 8 q^{64} + 2 \beta_{2} q^{67} + 4 \beta_{1} q^{68} + 2 \beta_{3} q^{74} -14 q^{79} + 4 \beta_{3} q^{86} -2 \beta_{2} q^{88} + 8 \beta_{1} q^{92} + 16 q^{94} -7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + O(q^{10}) \) \( 4 q + 8 q^{4} + 16 q^{16} + 8 q^{31} + 16 q^{34} + 32 q^{46} - 28 q^{49} + 32 q^{64} - 56 q^{79} + 64 q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{3} + 14 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 14 \beta_{1}\)\()/4\)

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
0.707107 1.58114i
0.707107 + 1.58114i
−0.707107 + 1.58114i
−0.707107 1.58114i
−1.41421 0 2.00000 0 0 0 −2.82843 0 0
901.2 −1.41421 0 2.00000 0 0 0 −2.82843 0 0
901.3 1.41421 0 2.00000 0 0 0 2.82843 0 0
901.4 1.41421 0 2.00000 0 0 0 2.82843 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
120.i odd 2 1 CM by \(\Q(\sqrt{-30}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
24.h odd 2 1 inner
40.f 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.o 4
3.b odd 2 1 inner 1800.2.k.o 4
4.b odd 2 1 7200.2.k.m 4
5.b even 2 1 inner 1800.2.k.o 4
5.c odd 4 2 360.2.d.a 4
8.b even 2 1 inner 1800.2.k.o 4
8.d odd 2 1 7200.2.k.m 4
12.b even 2 1 7200.2.k.m 4
15.d odd 2 1 inner 1800.2.k.o 4
15.e even 4 2 360.2.d.a 4
20.d odd 2 1 7200.2.k.m 4
20.e even 4 2 1440.2.d.a 4
24.f even 2 1 7200.2.k.m 4
24.h odd 2 1 inner 1800.2.k.o 4
40.e odd 2 1 7200.2.k.m 4
40.f even 2 1 inner 1800.2.k.o 4
40.i odd 4 2 360.2.d.a 4
40.k even 4 2 1440.2.d.a 4
60.h even 2 1 7200.2.k.m 4
60.l odd 4 2 1440.2.d.a 4
120.i odd 2 1 CM 1800.2.k.o 4
120.m even 2 1 7200.2.k.m 4
120.q odd 4 2 1440.2.d.a 4
120.w even 4 2 360.2.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.d.a 4 5.c odd 4 2
360.2.d.a 4 15.e even 4 2
360.2.d.a 4 40.i odd 4 2
360.2.d.a 4 120.w even 4 2
1440.2.d.a 4 20.e even 4 2
1440.2.d.a 4 40.k even 4 2
1440.2.d.a 4 60.l odd 4 2
1440.2.d.a 4 120.q odd 4 2
1800.2.k.o 4 1.a even 1 1 trivial
1800.2.k.o 4 3.b odd 2 1 inner
1800.2.k.o 4 5.b even 2 1 inner
1800.2.k.o 4 8.b even 2 1 inner
1800.2.k.o 4 15.d odd 2 1 inner
1800.2.k.o 4 24.h odd 2 1 inner
1800.2.k.o 4 40.f even 2 1 inner
1800.2.k.o 4 120.i odd 2 1 CM
7200.2.k.m 4 4.b odd 2 1
7200.2.k.m 4 8.d odd 2 1
7200.2.k.m 4 12.b even 2 1
7200.2.k.m 4 20.d odd 2 1
7200.2.k.m 4 24.f even 2 1
7200.2.k.m 4 40.e odd 2 1
7200.2.k.m 4 60.h even 2 1
7200.2.k.m 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 \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7} \)
\( T_{11}^{2} + 20 \)
\( T_{17}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 20 + T^{2} )^{2} \)
$13$ \( ( 40 + T^{2} )^{2} \)
$17$ \( ( -8 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( -32 + T^{2} )^{2} \)
$29$ \( ( 20 + T^{2} )^{2} \)
$31$ \( ( -2 + T )^{4} \)
$37$ \( ( 40 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( 160 + T^{2} )^{2} \)
$47$ \( ( -128 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( ( 20 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 160 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 14 + T )^{4} \)
$83$ \( T^{4} \)
$89$ \( T^{4} \)
$97$ \( T^{4} \)
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