Properties

Label 2700.1.b.b
Level 2700
Weight 1
Character orbit 2700.b
Analytic conductor 1.347
Analytic rank 0
Dimension 2
Projective image \(D_{3}\)
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2700.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34747553411\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 $C_4\times S_3$
Artin field Galois closure of 12.0.265720500000000.3

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -i q^{7} +O(q^{10})\) \( q -i q^{7} + i q^{13} + q^{19} + 2 q^{31} -i q^{37} -2 i q^{43} - q^{61} -i q^{67} + i q^{73} + q^{79} + q^{91} -i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 2q^{19} + 4q^{31} - 2q^{61} + 2q^{79} + 2q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1349.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 0 0
1349.2 0 0 0 0 0 1.00000i 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 \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.1.b.b 2
3.b odd 2 1 CM 2700.1.b.b 2
5.b even 2 1 inner 2700.1.b.b 2
5.c odd 4 1 108.1.c.a 1
5.c odd 4 1 2700.1.g.b 1
15.d odd 2 1 inner 2700.1.b.b 2
15.e even 4 1 108.1.c.a 1
15.e even 4 1 2700.1.g.b 1
20.e even 4 1 432.1.e.a 1
40.i odd 4 1 1728.1.e.a 1
40.k even 4 1 1728.1.e.b 1
45.k odd 12 2 324.1.g.a 2
45.l even 12 2 324.1.g.a 2
60.l odd 4 1 432.1.e.a 1
120.q odd 4 1 1728.1.e.b 1
120.w even 4 1 1728.1.e.a 1
135.q even 36 6 2916.1.k.c 6
135.r odd 36 6 2916.1.k.c 6
180.v odd 12 2 1296.1.q.a 2
180.x even 12 2 1296.1.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.1.c.a 1 5.c odd 4 1
108.1.c.a 1 15.e even 4 1
324.1.g.a 2 45.k odd 12 2
324.1.g.a 2 45.l even 12 2
432.1.e.a 1 20.e even 4 1
432.1.e.a 1 60.l odd 4 1
1296.1.q.a 2 180.v odd 12 2
1296.1.q.a 2 180.x even 12 2
1728.1.e.a 1 40.i odd 4 1
1728.1.e.a 1 120.w even 4 1
1728.1.e.b 1 40.k even 4 1
1728.1.e.b 1 120.q odd 4 1
2700.1.b.b 2 1.a even 1 1 trivial
2700.1.b.b 2 3.b odd 2 1 CM
2700.1.b.b 2 5.b even 2 1 inner
2700.1.b.b 2 15.d odd 2 1 inner
2700.1.g.b 1 5.c odd 4 1
2700.1.g.b 1 15.e even 4 1
2916.1.k.c 6 135.q even 36 6
2916.1.k.c 6 135.r odd 36 6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2700, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ 1
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$13$ \( 1 - T^{2} + T^{4} \)
$17$ \( ( 1 + T^{2} )^{2} \)
$19$ \( ( 1 - T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$31$ \( ( 1 - T )^{4} \)
$37$ \( 1 - T^{2} + T^{4} \)
$41$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$43$ \( ( 1 + T^{2} )^{2} \)
$47$ \( ( 1 + T^{2} )^{2} \)
$53$ \( ( 1 + T^{2} )^{2} \)
$59$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$61$ \( ( 1 + T + T^{2} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$73$ \( 1 - T^{2} + T^{4} \)
$79$ \( ( 1 - T + T^{2} )^{2} \)
$83$ \( ( 1 + T^{2} )^{2} \)
$89$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$97$ \( 1 - T^{2} + T^{4} \)
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