Properties

Label 1800.1.l.a
Level 1800
Weight 1
Character orbit 1800.l
Analytic conductor 0.898
Analytic rank 0
Dimension 2
Projective image \(S_{4}\)
CM/RM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 1800.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.10800.2
Artin image $\GL(2,3)$
Artin field Galois closure of 8.2.1399680000.1

$q$-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.

\(f(q)\) \(=\) \( q - q^{7} +O(q^{10})\) \( q - q^{7} -\beta q^{11} + q^{13} + \beta q^{17} + q^{19} -\beta q^{23} -\beta q^{29} + q^{31} + q^{43} -\beta q^{47} + \beta q^{59} + q^{61} - q^{67} + \beta q^{77} + \beta q^{83} - q^{91} + q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} + 2q^{13} + 2q^{19} + 2q^{31} + 2q^{43} + 2q^{61} - 2q^{67} - 2q^{91} + 2q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
1.41421i
1.41421i
0 0 0 0 0 −1.00000 0 0 0
1601.2 0 0 0 0 0 −1.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.1.l.a 2
3.b odd 2 1 inner 1800.1.l.a 2
4.b odd 2 1 3600.1.l.b 2
5.b even 2 1 1800.1.l.b yes 2
5.c odd 4 2 1800.1.c.a 4
12.b even 2 1 3600.1.l.b 2
15.d odd 2 1 1800.1.l.b yes 2
15.e even 4 2 1800.1.c.a 4
20.d odd 2 1 3600.1.l.a 2
20.e even 4 2 3600.1.c.a 4
60.h even 2 1 3600.1.l.a 2
60.l odd 4 2 3600.1.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.1.c.a 4 5.c odd 4 2
1800.1.c.a 4 15.e even 4 2
1800.1.l.a 2 1.a even 1 1 trivial
1800.1.l.a 2 3.b odd 2 1 inner
1800.1.l.b yes 2 5.b even 2 1
1800.1.l.b yes 2 15.d odd 2 1
3600.1.c.a 4 20.e even 4 2
3600.1.c.a 4 60.l odd 4 2
3600.1.l.a 2 20.d odd 2 1
3600.1.l.a 2 60.h even 2 1
3600.1.l.b 2 4.b odd 2 1
3600.1.l.b 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\).