Properties

Label 3600.1.cc.b
Level 3600
Weight 1
Character orbit 3600.cc
Analytic conductor 1.797
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM discriminant -20
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.cc (of order \(6\), degree \(2\), 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 720)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.10497600.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} + ( 1 + \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{3} + ( 1 + \zeta_{6} ) q^{7} + \zeta_{6}^{2} q^{9} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{21} + ( 1 - \zeta_{6}^{2} ) q^{23} - q^{27} + \zeta_{6}^{2} q^{29} -\zeta_{6} q^{41} + ( -1 - \zeta_{6} ) q^{47} + ( 1 + \zeta_{6} + \zeta_{6}^{2} ) q^{49} + \zeta_{6}^{2} q^{61} + ( -1 + \zeta_{6}^{2} ) q^{63} + ( 1 - \zeta_{6}^{2} ) q^{67} + ( 1 + \zeta_{6} ) q^{69} -\zeta_{6} q^{81} + ( -1 - \zeta_{6} ) q^{83} - q^{87} + q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{3} + 3q^{7} - q^{9} + O(q^{10}) \) \( 2q + q^{3} + 3q^{7} - q^{9} + 3q^{23} - 2q^{27} - q^{29} - q^{41} - 3q^{47} + 2q^{49} - q^{61} - 3q^{63} + 3q^{67} + 3q^{69} - q^{81} - 3q^{83} - 2q^{87} + 2q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 0.866025i 0 0 0 1.50000 0.866025i 0 −0.500000 0.866025i 0
1951.1 0 0.500000 + 0.866025i 0 0 0 1.50000 + 0.866025i 0 −0.500000 + 0.866025i 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 \(\Q(\sqrt{-5}) \)
36.f odd 6 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.cc.b 2
4.b odd 2 1 3600.1.cc.a 2
5.b even 2 1 3600.1.cc.a 2
5.c odd 4 2 720.1.bu.a 4
9.c even 3 1 3600.1.cc.a 2
15.e even 4 2 2160.1.bu.a 4
20.d odd 2 1 CM 3600.1.cc.b 2
20.e even 4 2 720.1.bu.a 4
36.f odd 6 1 inner 3600.1.cc.b 2
40.i odd 4 2 2880.1.bu.c 4
40.k even 4 2 2880.1.bu.c 4
45.j even 6 1 inner 3600.1.cc.b 2
45.k odd 12 2 720.1.bu.a 4
45.l even 12 2 2160.1.bu.a 4
60.l odd 4 2 2160.1.bu.a 4
180.p odd 6 1 3600.1.cc.a 2
180.v odd 12 2 2160.1.bu.a 4
180.x even 12 2 720.1.bu.a 4
360.bo even 12 2 2880.1.bu.c 4
360.bu odd 12 2 2880.1.bu.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bu.a 4 5.c odd 4 2
720.1.bu.a 4 20.e even 4 2
720.1.bu.a 4 45.k odd 12 2
720.1.bu.a 4 180.x even 12 2
2160.1.bu.a 4 15.e even 4 2
2160.1.bu.a 4 45.l even 12 2
2160.1.bu.a 4 60.l odd 4 2
2160.1.bu.a 4 180.v odd 12 2
2880.1.bu.c 4 40.i odd 4 2
2880.1.bu.c 4 40.k even 4 2
2880.1.bu.c 4 360.bo even 12 2
2880.1.bu.c 4 360.bu odd 12 2
3600.1.cc.a 2 4.b odd 2 1
3600.1.cc.a 2 5.b even 2 1
3600.1.cc.a 2 9.c even 3 1
3600.1.cc.a 2 180.p odd 6 1
3600.1.cc.b 2 1.a even 1 1 trivial
3600.1.cc.b 2 20.d odd 2 1 CM
3600.1.cc.b 2 36.f odd 6 1 inner
3600.1.cc.b 2 45.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 3 T_{7} + 3 \) acting on \(S_{1}^{\mathrm{new}}(3600, [\chi])\).

Hecke characteristic polynomials

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