Properties

Label 2700.1.g.b
Level 2700
Weight 1
Character orbit 2700.g
Self dual yes
Analytic conductor 1.347
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -3
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2700.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.34747553411\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.108.1
Artin image $D_6$
Artin field Galois closure of 6.2.1458000.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{7} + O(q^{10}) \) \( q + q^{7} + q^{13} - q^{19} + 2q^{31} + q^{37} - 2q^{43} - q^{61} + q^{67} + q^{73} - q^{79} + q^{91} + q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
0
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 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.1.g.b 1
3.b odd 2 1 CM 2700.1.g.b 1
5.b even 2 1 108.1.c.a 1
5.c odd 4 2 2700.1.b.b 2
15.d odd 2 1 108.1.c.a 1
15.e even 4 2 2700.1.b.b 2
20.d odd 2 1 432.1.e.a 1
40.e odd 2 1 1728.1.e.b 1
40.f even 2 1 1728.1.e.a 1
45.h odd 6 2 324.1.g.a 2
45.j even 6 2 324.1.g.a 2
60.h even 2 1 432.1.e.a 1
120.i odd 2 1 1728.1.e.a 1
120.m even 2 1 1728.1.e.b 1
135.n odd 18 6 2916.1.k.c 6
135.p even 18 6 2916.1.k.c 6
180.n even 6 2 1296.1.q.a 2
180.p odd 6 2 1296.1.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 5.b even 2 1
108.1.c.a 1 15.d odd 2 1
324.1.g.a 2 45.h odd 6 2
324.1.g.a 2 45.j even 6 2
432.1.e.a 1 20.d odd 2 1
432.1.e.a 1 60.h even 2 1
1296.1.q.a 2 180.n even 6 2
1296.1.q.a 2 180.p odd 6 2
1728.1.e.a 1 40.f even 2 1
1728.1.e.a 1 120.i odd 2 1
1728.1.e.b 1 40.e odd 2 1
1728.1.e.b 1 120.m even 2 1
2700.1.b.b 2 5.c odd 4 2
2700.1.b.b 2 15.e even 4 2
2700.1.g.b 1 1.a even 1 1 trivial
2700.1.g.b 1 3.b odd 2 1 CM
2916.1.k.c 6 135.n odd 18 6
2916.1.k.c 6 135.p even 18 6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 1 \) acting on \(S_{1}^{\mathrm{new}}(2700, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ \( 1 - T + T^{2} \)
$11$ \( ( 1 - T )( 1 + T ) \)
$13$ \( 1 - T + T^{2} \)
$17$ \( ( 1 - T )( 1 + T ) \)
$19$ \( 1 + T + T^{2} \)
$23$ \( ( 1 - T )( 1 + T ) \)
$29$ \( ( 1 - T )( 1 + T ) \)
$31$ \( ( 1 - T )^{2} \)
$37$ \( 1 - T + T^{2} \)
$41$ \( ( 1 - T )( 1 + T ) \)
$43$ \( ( 1 + T )^{2} \)
$47$ \( ( 1 - T )( 1 + T ) \)
$53$ \( ( 1 - T )( 1 + T ) \)
$59$ \( ( 1 - T )( 1 + T ) \)
$61$ \( 1 + T + T^{2} \)
$67$ \( 1 - T + T^{2} \)
$71$ \( ( 1 - T )( 1 + T ) \)
$73$ \( 1 - T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( ( 1 - T )( 1 + T ) \)
$89$ \( ( 1 - T )( 1 + T ) \)
$97$ \( 1 - T + T^{2} \)
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