Properties

Label 2700.1.t.a
Level $2700$
Weight $1$
Character orbit 2700.t
Analytic conductor $1.347$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2700.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.34747553411\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1620.1
Artin image: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} -\zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} +O(q^{10})\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} -\zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{14} + \zeta_{12}^{4} q^{16} -\zeta_{12}^{5} q^{23} -\zeta_{12}^{3} q^{28} -\zeta_{12}^{4} q^{29} -\zeta_{12}^{5} q^{32} -\zeta_{12}^{2} q^{41} -2 \zeta_{12} q^{43} - q^{46} + \zeta_{12} q^{47} + \zeta_{12}^{4} q^{56} + \zeta_{12}^{5} q^{58} -\zeta_{12}^{4} q^{61} - q^{64} -\zeta_{12}^{5} q^{67} + \zeta_{12}^{3} q^{82} -\zeta_{12} q^{83} + 2 \zeta_{12}^{2} q^{86} - q^{89} + \zeta_{12} q^{92} -\zeta_{12}^{2} q^{94} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} + 2q^{14} - 2q^{16} + 2q^{29} - 2q^{41} - 4q^{46} - 2q^{56} + 2q^{61} - 4q^{64} + 4q^{86} - 4q^{89} - 2q^{94} + O(q^{100}) \)

Character values

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

\(n\) \(1001\) \(1351\) \(2377\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −0.866025 0.500000i 1.00000i 0 0
451.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 0.866025 + 0.500000i 1.00000i 0 0
2251.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −0.866025 + 0.500000i 1.00000i 0 0
2251.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 0.866025 0.500000i 1.00000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
45.j even 6 1 inner
180.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.1.t.a 4
3.b odd 2 1 900.1.t.a 4
4.b odd 2 1 inner 2700.1.t.a 4
5.b even 2 1 inner 2700.1.t.a 4
5.c odd 4 1 540.1.p.a 2
5.c odd 4 1 540.1.p.b 2
9.c even 3 1 inner 2700.1.t.a 4
9.d odd 6 1 900.1.t.a 4
12.b even 2 1 900.1.t.a 4
15.d odd 2 1 900.1.t.a 4
15.e even 4 1 180.1.p.a 2
15.e even 4 1 180.1.p.b yes 2
20.d odd 2 1 CM 2700.1.t.a 4
20.e even 4 1 540.1.p.a 2
20.e even 4 1 540.1.p.b 2
36.f odd 6 1 inner 2700.1.t.a 4
36.h even 6 1 900.1.t.a 4
45.h odd 6 1 900.1.t.a 4
45.j even 6 1 inner 2700.1.t.a 4
45.k odd 12 1 540.1.p.a 2
45.k odd 12 1 540.1.p.b 2
45.k odd 12 1 1620.1.f.a 1
45.k odd 12 1 1620.1.f.c 1
45.l even 12 1 180.1.p.a 2
45.l even 12 1 180.1.p.b yes 2
45.l even 12 1 1620.1.f.b 1
45.l even 12 1 1620.1.f.d 1
60.h even 2 1 900.1.t.a 4
60.l odd 4 1 180.1.p.a 2
60.l odd 4 1 180.1.p.b yes 2
120.q odd 4 1 2880.1.bu.a 2
120.q odd 4 1 2880.1.bu.b 2
120.w even 4 1 2880.1.bu.a 2
120.w even 4 1 2880.1.bu.b 2
180.n even 6 1 900.1.t.a 4
180.p odd 6 1 inner 2700.1.t.a 4
180.v odd 12 1 180.1.p.a 2
180.v odd 12 1 180.1.p.b yes 2
180.v odd 12 1 1620.1.f.b 1
180.v odd 12 1 1620.1.f.d 1
180.x even 12 1 540.1.p.a 2
180.x even 12 1 540.1.p.b 2
180.x even 12 1 1620.1.f.a 1
180.x even 12 1 1620.1.f.c 1
360.br even 12 1 2880.1.bu.a 2
360.br even 12 1 2880.1.bu.b 2
360.bt odd 12 1 2880.1.bu.a 2
360.bt odd 12 1 2880.1.bu.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.1.p.a 2 15.e even 4 1
180.1.p.a 2 45.l even 12 1
180.1.p.a 2 60.l odd 4 1
180.1.p.a 2 180.v odd 12 1
180.1.p.b yes 2 15.e even 4 1
180.1.p.b yes 2 45.l even 12 1
180.1.p.b yes 2 60.l odd 4 1
180.1.p.b yes 2 180.v odd 12 1
540.1.p.a 2 5.c odd 4 1
540.1.p.a 2 20.e even 4 1
540.1.p.a 2 45.k odd 12 1
540.1.p.a 2 180.x even 12 1
540.1.p.b 2 5.c odd 4 1
540.1.p.b 2 20.e even 4 1
540.1.p.b 2 45.k odd 12 1
540.1.p.b 2 180.x even 12 1
900.1.t.a 4 3.b odd 2 1
900.1.t.a 4 9.d odd 6 1
900.1.t.a 4 12.b even 2 1
900.1.t.a 4 15.d odd 2 1
900.1.t.a 4 36.h even 6 1
900.1.t.a 4 45.h odd 6 1
900.1.t.a 4 60.h even 2 1
900.1.t.a 4 180.n even 6 1
1620.1.f.a 1 45.k odd 12 1
1620.1.f.a 1 180.x even 12 1
1620.1.f.b 1 45.l even 12 1
1620.1.f.b 1 180.v odd 12 1
1620.1.f.c 1 45.k odd 12 1
1620.1.f.c 1 180.x even 12 1
1620.1.f.d 1 45.l even 12 1
1620.1.f.d 1 180.v odd 12 1
2700.1.t.a 4 1.a even 1 1 trivial
2700.1.t.a 4 4.b odd 2 1 inner
2700.1.t.a 4 5.b even 2 1 inner
2700.1.t.a 4 9.c even 3 1 inner
2700.1.t.a 4 20.d odd 2 1 CM
2700.1.t.a 4 36.f odd 6 1 inner
2700.1.t.a 4 45.j even 6 1 inner
2700.1.t.a 4 180.p odd 6 1 inner
2880.1.bu.a 2 120.q odd 4 1
2880.1.bu.a 2 120.w even 4 1
2880.1.bu.a 2 360.br even 12 1
2880.1.bu.a 2 360.bt odd 12 1
2880.1.bu.b 2 120.q odd 4 1
2880.1.bu.b 2 120.w even 4 1
2880.1.bu.b 2 360.br even 12 1
2880.1.bu.b 2 360.bt odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2700, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 - T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( 1 - T^{2} + T^{4} \)
$29$ \( ( 1 - T + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 1 + T + T^{2} )^{2} \)
$43$ \( 16 - 4 T^{2} + T^{4} \)
$47$ \( 1 - T^{2} + T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( 1 - T^{2} + T^{4} \)
$89$ \( ( 1 + T )^{4} \)
$97$ \( T^{4} \)
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