Properties

Label 1800.2.s.b
Level $1800$
Weight $2$
Character orbit 1800.s
Analytic conductor $14.373$
Analytic rank $0$
Dimension $4$
CM no
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.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} + ( -3 + 3 \zeta_{8}^{2} ) q^{13} -4 \zeta_{8}^{2} q^{19} -4 \zeta_{8} q^{23} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{29} -8 q^{31} + ( -7 - 7 \zeta_{8}^{2} ) q^{37} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} + ( -4 + 4 \zeta_{8}^{2} ) q^{43} + 4 \zeta_{8}^{3} q^{47} -7 \zeta_{8}^{2} q^{49} + 12 \zeta_{8} q^{53} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} -12 q^{61} + ( 8 + 8 \zeta_{8}^{2} ) q^{67} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{71} + ( 3 - 3 \zeta_{8}^{2} ) q^{73} + 8 \zeta_{8}^{2} q^{79} + 16 \zeta_{8} q^{83} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{89} + ( -5 - 5 \zeta_{8}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 12q^{13} - 32q^{31} - 28q^{37} - 16q^{43} - 48q^{61} + 32q^{67} + 12q^{73} - 20q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(\zeta_{8}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 0 0 0 0 0 0
593.2 0 0 0 0 0 0 0 0 0
1457.1 0 0 0 0 0 0 0 0 0
1457.2 0 0 0 0 0 0 0 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.s.b 4
3.b odd 2 1 inner 1800.2.s.b 4
4.b odd 2 1 3600.2.w.d 4
5.b even 2 1 360.2.s.a 4
5.c odd 4 1 360.2.s.a 4
5.c odd 4 1 inner 1800.2.s.b 4
12.b even 2 1 3600.2.w.d 4
15.d odd 2 1 360.2.s.a 4
15.e even 4 1 360.2.s.a 4
15.e even 4 1 inner 1800.2.s.b 4
20.d odd 2 1 720.2.w.c 4
20.e even 4 1 720.2.w.c 4
20.e even 4 1 3600.2.w.d 4
40.e odd 2 1 2880.2.w.f 4
40.f even 2 1 2880.2.w.e 4
40.i odd 4 1 2880.2.w.e 4
40.k even 4 1 2880.2.w.f 4
60.h even 2 1 720.2.w.c 4
60.l odd 4 1 720.2.w.c 4
60.l odd 4 1 3600.2.w.d 4
120.i odd 2 1 2880.2.w.e 4
120.m even 2 1 2880.2.w.f 4
120.q odd 4 1 2880.2.w.f 4
120.w even 4 1 2880.2.w.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.s.a 4 5.b even 2 1
360.2.s.a 4 5.c odd 4 1
360.2.s.a 4 15.d odd 2 1
360.2.s.a 4 15.e even 4 1
720.2.w.c 4 20.d odd 2 1
720.2.w.c 4 20.e even 4 1
720.2.w.c 4 60.h even 2 1
720.2.w.c 4 60.l odd 4 1
1800.2.s.b 4 1.a even 1 1 trivial
1800.2.s.b 4 3.b odd 2 1 inner
1800.2.s.b 4 5.c odd 4 1 inner
1800.2.s.b 4 15.e even 4 1 inner
2880.2.w.e 4 40.f even 2 1
2880.2.w.e 4 40.i odd 4 1
2880.2.w.e 4 120.i odd 2 1
2880.2.w.e 4 120.w even 4 1
2880.2.w.f 4 40.e odd 2 1
2880.2.w.f 4 40.k even 4 1
2880.2.w.f 4 120.m even 2 1
2880.2.w.f 4 120.q odd 4 1
3600.2.w.d 4 4.b odd 2 1
3600.2.w.d 4 12.b even 2 1
3600.2.w.d 4 20.e even 4 1
3600.2.w.d 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(1800, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 32 + T^{2} )^{2} \)
$13$ \( ( 18 + 6 T + T^{2} )^{2} \)
$17$ \( T^{4} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( 256 + T^{4} \)
$29$ \( ( -2 + T^{2} )^{2} \)
$31$ \( ( 8 + T )^{4} \)
$37$ \( ( 98 + 14 T + T^{2} )^{2} \)
$41$ \( ( 2 + T^{2} )^{2} \)
$43$ \( ( 32 + 8 T + T^{2} )^{2} \)
$47$ \( 256 + T^{4} \)
$53$ \( 20736 + T^{4} \)
$59$ \( ( -128 + T^{2} )^{2} \)
$61$ \( ( 12 + T )^{4} \)
$67$ \( ( 128 - 16 T + T^{2} )^{2} \)
$71$ \( ( 32 + T^{2} )^{2} \)
$73$ \( ( 18 - 6 T + T^{2} )^{2} \)
$79$ \( ( 64 + T^{2} )^{2} \)
$83$ \( 65536 + T^{4} \)
$89$ \( ( -50 + T^{2} )^{2} \)
$97$ \( ( 50 + 10 T + T^{2} )^{2} \)
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