Properties

Label 1800.2.k.s
Level $1800$
Weight $2$
Character orbit 1800.k
Analytic conductor $14.373$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.212336640000.29
Defining polynomial: \(x^{8} + 2 x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} -\beta_{5} q^{4} + ( 1 + \beta_{2} - \beta_{5} ) q^{7} + \beta_{4} q^{8} +O(q^{10})\) \( q + \beta_{6} q^{2} -\beta_{5} q^{4} + ( 1 + \beta_{2} - \beta_{5} ) q^{7} + \beta_{4} q^{8} + ( \beta_{3} + \beta_{4} ) q^{11} -\beta_{7} q^{13} + ( -\beta_{1} + \beta_{4} + \beta_{6} ) q^{14} + ( -1 - \beta_{7} ) q^{16} + ( \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{17} + ( \beta_{2} + \beta_{5} + \beta_{7} ) q^{19} + ( -1 - 2 \beta_{2} - \beta_{7} ) q^{22} + ( -\beta_{3} + \beta_{4} ) q^{23} + ( 2 \beta_{3} - \beta_{6} ) q^{26} + ( 3 - \beta_{5} - \beta_{7} ) q^{28} + ( -\beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{29} + ( 1 - \beta_{2} + \beta_{5} ) q^{31} + ( 2 \beta_{3} - 2 \beta_{6} ) q^{32} + ( -3 - 2 \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{34} + ( 2 \beta_{2} + 2 \beta_{5} ) q^{37} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{38} + ( -\beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{41} + ( \beta_{2} + \beta_{5} - \beta_{7} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} ) q^{44} + ( -1 + 2 \beta_{2} - \beta_{7} ) q^{46} + ( -2 \beta_{1} + 4 \beta_{6} ) q^{47} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{49} + ( -4 \beta_{2} + \beta_{5} ) q^{52} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{53} + ( 2 \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{56} + ( 3 - 2 \beta_{2} + 2 \beta_{5} - \beta_{7} ) q^{58} + ( 2 \beta_{1} + 4 \beta_{6} ) q^{59} + ( -2 \beta_{2} - 2 \beta_{5} - \beta_{7} ) q^{61} + ( \beta_{1} - \beta_{4} + \beta_{6} ) q^{62} + ( -4 \beta_{2} + 2 \beta_{5} ) q^{64} + ( 3 \beta_{2} + 3 \beta_{5} - \beta_{7} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{68} + ( 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} - 4 \beta_{6} ) q^{71} + ( 2 + 2 \beta_{2} - 2 \beta_{5} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{74} + ( 5 + 4 \beta_{2} - \beta_{5} + \beta_{7} ) q^{76} + ( \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} ) q^{77} + ( -2 + 2 \beta_{2} - 2 \beta_{5} ) q^{79} + ( 5 - 2 \beta_{2} - 2 \beta_{5} + \beta_{7} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} ) q^{83} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{86} + ( -8 - 4 \beta_{2} + 2 \beta_{5} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{6} ) q^{89} + ( -3 \beta_{2} - 3 \beta_{5} - \beta_{7} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} ) q^{92} + ( 8 - 4 \beta_{5} ) q^{94} + ( 1 + 6 \beta_{2} - 6 \beta_{5} ) q^{97} + ( -2 \beta_{1} + 2 \beta_{4} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7} + O(q^{10}) \) \( 8 q + 8 q^{7} - 8 q^{16} - 8 q^{22} + 24 q^{28} + 8 q^{31} - 24 q^{34} - 8 q^{46} + 24 q^{58} + 16 q^{73} + 40 q^{76} - 16 q^{79} + 40 q^{82} - 64 q^{88} + 64 q^{94} + 8 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 2 x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 2 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{2} \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{3} \)\()/8\)
\(\beta_{7}\)\(=\)\( \nu^{4} + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{7} - 1\)
\(\nu^{5}\)\(=\)\(2 \beta_{4} - \beta_{1}\)
\(\nu^{6}\)\(=\)\(4 \beta_{5} - 2 \beta_{2}\)
\(\nu^{7}\)\(=\)\(8 \beta_{6} - 2 \beta_{3}\)

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
1.26979 0.622597i
1.26979 + 0.622597i
0.622597 1.26979i
0.622597 + 1.26979i
−0.622597 1.26979i
−0.622597 + 1.26979i
−1.26979 0.622597i
−1.26979 + 0.622597i
−1.26979 0.622597i 0 1.22474 + 1.58114i 0 0 3.44949 −0.570759 2.77024i 0 0
901.2 −1.26979 + 0.622597i 0 1.22474 1.58114i 0 0 3.44949 −0.570759 + 2.77024i 0 0
901.3 −0.622597 1.26979i 0 −1.22474 + 1.58114i 0 0 −1.44949 2.77024 + 0.570759i 0 0
901.4 −0.622597 + 1.26979i 0 −1.22474 1.58114i 0 0 −1.44949 2.77024 0.570759i 0 0
901.5 0.622597 1.26979i 0 −1.22474 1.58114i 0 0 −1.44949 −2.77024 + 0.570759i 0 0
901.6 0.622597 + 1.26979i 0 −1.22474 + 1.58114i 0 0 −1.44949 −2.77024 0.570759i 0 0
901.7 1.26979 0.622597i 0 1.22474 1.58114i 0 0 3.44949 0.570759 2.77024i 0 0
901.8 1.26979 + 0.622597i 0 1.22474 + 1.58114i 0 0 3.44949 0.570759 + 2.77024i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h 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.s yes 8
3.b odd 2 1 inner 1800.2.k.s yes 8
4.b odd 2 1 7200.2.k.q 8
5.b even 2 1 1800.2.k.r 8
5.c odd 4 2 1800.2.d.u 16
8.b even 2 1 inner 1800.2.k.s yes 8
8.d odd 2 1 7200.2.k.q 8
12.b even 2 1 7200.2.k.q 8
15.d odd 2 1 1800.2.k.r 8
15.e even 4 2 1800.2.d.u 16
20.d odd 2 1 7200.2.k.t 8
20.e even 4 2 7200.2.d.u 16
24.f even 2 1 7200.2.k.q 8
24.h odd 2 1 inner 1800.2.k.s yes 8
40.e odd 2 1 7200.2.k.t 8
40.f even 2 1 1800.2.k.r 8
40.i odd 4 2 1800.2.d.u 16
40.k even 4 2 7200.2.d.u 16
60.h even 2 1 7200.2.k.t 8
60.l odd 4 2 7200.2.d.u 16
120.i odd 2 1 1800.2.k.r 8
120.m even 2 1 7200.2.k.t 8
120.q odd 4 2 7200.2.d.u 16
120.w even 4 2 1800.2.d.u 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.2.d.u 16 5.c odd 4 2
1800.2.d.u 16 15.e even 4 2
1800.2.d.u 16 40.i odd 4 2
1800.2.d.u 16 120.w even 4 2
1800.2.k.r 8 5.b even 2 1
1800.2.k.r 8 15.d odd 2 1
1800.2.k.r 8 40.f even 2 1
1800.2.k.r 8 120.i odd 2 1
1800.2.k.s yes 8 1.a even 1 1 trivial
1800.2.k.s yes 8 3.b odd 2 1 inner
1800.2.k.s yes 8 8.b even 2 1 inner
1800.2.k.s yes 8 24.h odd 2 1 inner
7200.2.d.u 16 20.e even 4 2
7200.2.d.u 16 40.k even 4 2
7200.2.d.u 16 60.l odd 4 2
7200.2.d.u 16 120.q odd 4 2
7200.2.k.q 8 4.b odd 2 1
7200.2.k.q 8 8.d odd 2 1
7200.2.k.q 8 12.b even 2 1
7200.2.k.q 8 24.f even 2 1
7200.2.k.t 8 20.d odd 2 1
7200.2.k.t 8 40.e odd 2 1
7200.2.k.t 8 60.h even 2 1
7200.2.k.t 8 120.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 \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{2} - 2 T_{7} - 5 \)
\( T_{11}^{4} + 32 T_{11}^{2} + 40 \)
\( T_{17}^{4} - 48 T_{17}^{2} + 360 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 2 T^{4} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( -5 - 2 T + T^{2} )^{4} \)
$11$ \( ( 40 + 32 T^{2} + T^{4} )^{2} \)
$13$ \( ( 15 + T^{2} )^{4} \)
$17$ \( ( 360 - 48 T^{2} + T^{4} )^{2} \)
$19$ \( ( 25 + 50 T^{2} + T^{4} )^{2} \)
$23$ \( ( 40 - 32 T^{2} + T^{4} )^{2} \)
$29$ \( ( 360 + 48 T^{2} + T^{4} )^{2} \)
$31$ \( ( -5 - 2 T + T^{2} )^{4} \)
$37$ \( ( 40 + T^{2} )^{4} \)
$41$ \( ( 1000 - 80 T^{2} + T^{4} )^{2} \)
$43$ \( ( 25 + 50 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2560 - 128 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1000 + 80 T^{2} + T^{4} )^{2} \)
$59$ \( ( 2560 + 128 T^{2} + T^{4} )^{2} \)
$61$ \( ( 625 + 110 T^{2} + T^{4} )^{2} \)
$67$ \( ( 5625 + 210 T^{2} + T^{4} )^{2} \)
$71$ \( ( 25000 - 320 T^{2} + T^{4} )^{2} \)
$73$ \( ( -20 - 4 T + T^{2} )^{4} \)
$79$ \( ( -20 + 4 T + T^{2} )^{4} \)
$83$ \( ( 5760 + 192 T^{2} + T^{4} )^{2} \)
$89$ \( ( 16000 - 320 T^{2} + T^{4} )^{2} \)
$97$ \( ( -215 - 2 T + T^{2} )^{4} \)
show more
show less