Properties

Label 1800.2.k.m
Level $1800$
Weight $2$
Character orbit 1800.k
Analytic conductor $14.373$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{7} -2 \beta_{2} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{7} -2 \beta_{2} q^{8} + 2 \beta_{3} q^{11} + ( 3 + \beta_{3} ) q^{14} + ( -2 + 2 \beta_{3} ) q^{16} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{17} -2 \beta_{3} q^{19} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{22} + ( 2 \beta_{1} - \beta_{2} ) q^{23} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{28} + 4 q^{31} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{32} + ( 6 + 2 \beta_{3} ) q^{34} -6 \beta_{2} q^{37} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{38} -3 \beta_{2} q^{43} + ( -6 + 2 \beta_{3} ) q^{44} + ( -3 - \beta_{3} ) q^{46} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{47} - q^{49} + 4 \beta_{2} q^{53} + 4 \beta_{3} q^{56} -6 \beta_{3} q^{59} + 2 \beta_{3} q^{61} -4 \beta_{1} q^{62} -8 q^{64} -3 \beta_{2} q^{67} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{68} + 12 q^{71} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -6 + 6 \beta_{3} ) q^{74} + ( 6 - 2 \beta_{3} ) q^{76} -6 \beta_{2} q^{77} + 4 q^{79} -7 \beta_{2} q^{83} + ( -3 + 3 \beta_{3} ) q^{86} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{88} + 6 q^{89} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{92} + ( -9 - 3 \beta_{3} ) q^{94} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + O(q^{10}) \) \( 4 q + 4 q^{4} + 12 q^{14} - 8 q^{16} + 16 q^{31} + 24 q^{34} - 24 q^{44} - 12 q^{46} - 4 q^{49} - 32 q^{64} + 48 q^{71} - 24 q^{74} + 24 q^{76} + 16 q^{79} - 12 q^{86} + 24 q^{89} - 36 q^{94} + O(q^{100}) \)

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

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

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
1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
−1.22474 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 0 0 −2.44949 2.82843i 0 0
901.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 0 0 −2.44949 2.82843i 0 0
901.3 1.22474 0.707107i 0 1.00000 1.73205i 0 0 2.44949 2.82843i 0 0
901.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 0 0 2.44949 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
5.b even 2 1 inner
8.b even 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.m 4
3.b odd 2 1 200.2.d.e 4
4.b odd 2 1 7200.2.k.l 4
5.b even 2 1 inner 1800.2.k.m 4
5.c odd 4 2 360.2.d.b 4
8.b even 2 1 inner 1800.2.k.m 4
8.d odd 2 1 7200.2.k.l 4
12.b even 2 1 800.2.d.f 4
15.d odd 2 1 200.2.d.e 4
15.e even 4 2 40.2.f.a 4
20.d odd 2 1 7200.2.k.l 4
20.e even 4 2 1440.2.d.c 4
24.f even 2 1 800.2.d.f 4
24.h odd 2 1 200.2.d.e 4
40.e odd 2 1 7200.2.k.l 4
40.f even 2 1 inner 1800.2.k.m 4
40.i odd 4 2 360.2.d.b 4
40.k even 4 2 1440.2.d.c 4
48.i odd 4 2 6400.2.a.co 4
48.k even 4 2 6400.2.a.cm 4
60.h even 2 1 800.2.d.f 4
60.l odd 4 2 160.2.f.a 4
120.i odd 2 1 200.2.d.e 4
120.m even 2 1 800.2.d.f 4
120.q odd 4 2 160.2.f.a 4
120.w even 4 2 40.2.f.a 4
240.t even 4 2 6400.2.a.cm 4
240.z odd 4 2 1280.2.c.k 4
240.bb even 4 2 1280.2.c.i 4
240.bd odd 4 2 1280.2.c.k 4
240.bf even 4 2 1280.2.c.i 4
240.bm odd 4 2 6400.2.a.co 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 15.e even 4 2
40.2.f.a 4 120.w even 4 2
160.2.f.a 4 60.l odd 4 2
160.2.f.a 4 120.q odd 4 2
200.2.d.e 4 3.b odd 2 1
200.2.d.e 4 15.d odd 2 1
200.2.d.e 4 24.h odd 2 1
200.2.d.e 4 120.i odd 2 1
360.2.d.b 4 5.c odd 4 2
360.2.d.b 4 40.i odd 4 2
800.2.d.f 4 12.b even 2 1
800.2.d.f 4 24.f even 2 1
800.2.d.f 4 60.h even 2 1
800.2.d.f 4 120.m even 2 1
1280.2.c.i 4 240.bb even 4 2
1280.2.c.i 4 240.bf even 4 2
1280.2.c.k 4 240.z odd 4 2
1280.2.c.k 4 240.bd odd 4 2
1440.2.d.c 4 20.e even 4 2
1440.2.d.c 4 40.k even 4 2
1800.2.k.m 4 1.a even 1 1 trivial
1800.2.k.m 4 5.b even 2 1 inner
1800.2.k.m 4 8.b even 2 1 inner
1800.2.k.m 4 40.f even 2 1 inner
6400.2.a.cm 4 48.k even 4 2
6400.2.a.cm 4 240.t even 4 2
6400.2.a.co 4 48.i odd 4 2
6400.2.a.co 4 240.bm odd 4 2
7200.2.k.l 4 4.b odd 2 1
7200.2.k.l 4 8.d odd 2 1
7200.2.k.l 4 20.d odd 2 1
7200.2.k.l 4 40.e odd 2 1

Hecke kernels

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

\( T_{7}^{2} - 6 \)
\( T_{11}^{2} + 12 \)
\( T_{17}^{2} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -6 + T^{2} )^{2} \)
$11$ \( ( 12 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( -24 + T^{2} )^{2} \)
$19$ \( ( 12 + T^{2} )^{2} \)
$23$ \( ( -6 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( -4 + T )^{4} \)
$37$ \( ( 72 + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( 18 + T^{2} )^{2} \)
$47$ \( ( -54 + T^{2} )^{2} \)
$53$ \( ( 32 + T^{2} )^{2} \)
$59$ \( ( 108 + T^{2} )^{2} \)
$61$ \( ( 12 + T^{2} )^{2} \)
$67$ \( ( 18 + T^{2} )^{2} \)
$71$ \( ( -12 + T )^{4} \)
$73$ \( ( -24 + T^{2} )^{2} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( ( 98 + T^{2} )^{2} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( ( -24 + T^{2} )^{2} \)
show more
show less