Properties

Label 3600.1.e.c
Level 3600
Weight 1
Character orbit 3600.e
Analytic conductor 1.797
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.4320000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{7} +O(q^{10})\) \( q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{7} - q^{13} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{31} -2 q^{37} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{43} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{49} - q^{61} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{67} + 2 q^{73} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{91} - q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 2q^{13} - 4q^{37} - 4q^{49} - 2q^{61} + 4q^{73} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3151.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 1.73205i 0 0 0
3151.2 0 0 0 0 0 1.73205i 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 \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{2} + 3 \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

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