Properties

Label 2700.1.c.a
Level $2700$
Weight $1$
Character orbit 2700.c
Analytic conductor $1.347$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -15
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34747553411\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.1166400.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + q^{8} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} + q^{8} + \zeta_{6}^{2} q^{16} + q^{17} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{19} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{23} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} -\zeta_{6} q^{32} + \zeta_{6}^{2} q^{34} + ( 1 + \zeta_{6} ) q^{38} + ( -1 - \zeta_{6} ) q^{46} + q^{49} + q^{53} + q^{61} + ( 1 + \zeta_{6} ) q^{62} + q^{64} -\zeta_{6} q^{68} + ( -1 + \zeta_{6}^{2} ) q^{76} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{79} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{83} + ( 1 - \zeta_{6}^{2} ) q^{92} + \zeta_{6}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} + 2q^{8} + O(q^{10}) \) \( 2q - q^{2} - q^{4} + 2q^{8} - q^{16} + 2q^{17} - q^{32} - q^{34} + 3q^{38} - 3q^{46} + 2q^{49} + 2q^{53} + 2q^{61} + 3q^{62} + 2q^{64} - q^{68} - 3q^{76} + 3q^{92} - q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 0 0
1351.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
4.b odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.1.c.a 2
3.b odd 2 1 2700.1.c.b 2
4.b odd 2 1 inner 2700.1.c.a 2
5.b even 2 1 2700.1.c.b 2
5.c odd 4 2 540.1.f.a 4
12.b even 2 1 2700.1.c.b 2
15.d odd 2 1 CM 2700.1.c.a 2
15.e even 4 2 540.1.f.a 4
20.d odd 2 1 2700.1.c.b 2
20.e even 4 2 540.1.f.a 4
45.k odd 12 2 1620.1.p.a 4
45.k odd 12 2 1620.1.p.d 4
45.l even 12 2 1620.1.p.a 4
45.l even 12 2 1620.1.p.d 4
60.h even 2 1 inner 2700.1.c.a 2
60.l odd 4 2 540.1.f.a 4
180.v odd 12 2 1620.1.p.a 4
180.v odd 12 2 1620.1.p.d 4
180.x even 12 2 1620.1.p.a 4
180.x even 12 2 1620.1.p.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.1.f.a 4 5.c odd 4 2
540.1.f.a 4 15.e even 4 2
540.1.f.a 4 20.e even 4 2
540.1.f.a 4 60.l odd 4 2
1620.1.p.a 4 45.k odd 12 2
1620.1.p.a 4 45.l even 12 2
1620.1.p.a 4 180.v odd 12 2
1620.1.p.a 4 180.x even 12 2
1620.1.p.d 4 45.k odd 12 2
1620.1.p.d 4 45.l even 12 2
1620.1.p.d 4 180.v odd 12 2
1620.1.p.d 4 180.x even 12 2
2700.1.c.a 2 1.a even 1 1 trivial
2700.1.c.a 2 4.b odd 2 1 inner
2700.1.c.a 2 15.d odd 2 1 CM
2700.1.c.a 2 60.h even 2 1 inner
2700.1.c.b 2 3.b odd 2 1
2700.1.c.b 2 5.b even 2 1
2700.1.c.b 2 12.b even 2 1
2700.1.c.b 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17} - 1 \) acting on \(S_{1}^{\mathrm{new}}(2700, [\chi])\).

Hecke characteristic polynomials

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