Properties

Label 3600.1.da.a
Level 3600
Weight 1
Character orbit 3600.da
Analytic conductor 1.797
Analytic rank 0
Dimension 8
Projective image \(D_{6}\)
CM discriminant -20
Inner twists 16

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.da (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.157464000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{7} q^{3} + \zeta_{24}^{5} q^{7} -\zeta_{24}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{7} q^{3} + \zeta_{24}^{5} q^{7} -\zeta_{24}^{2} q^{9} + q^{21} + ( \zeta_{24}^{3} - \zeta_{24}^{11} ) q^{23} + \zeta_{24}^{9} q^{27} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{29} + ( 1 - \zeta_{24}^{8} ) q^{41} + 2 \zeta_{24}^{11} q^{43} + ( \zeta_{24} - \zeta_{24}^{9} ) q^{47} -\zeta_{24}^{8} q^{61} -\zeta_{24}^{7} q^{63} + \zeta_{24} q^{67} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{69} + \zeta_{24}^{4} q^{81} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{83} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{87} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{21} + 12q^{41} + 4q^{61} + 4q^{81} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(1\) \(\zeta_{24}^{4}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1343.1
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0 −0.965926 0.258819i 0 0 0 −0.965926 + 0.258819i 0 0.866025 + 0.500000i 0
1343.2 0 0.965926 + 0.258819i 0 0 0 0.965926 0.258819i 0 0.866025 + 0.500000i 0
2207.1 0 −0.258819 + 0.965926i 0 0 0 −0.258819 0.965926i 0 −0.866025 0.500000i 0
2207.2 0 0.258819 0.965926i 0 0 0 0.258819 + 0.965926i 0 −0.866025 0.500000i 0
2543.1 0 −0.258819 0.965926i 0 0 0 −0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0
2543.2 0 0.258819 + 0.965926i 0 0 0 0.258819 0.965926i 0 −0.866025 + 0.500000i 0
3407.1 0 −0.965926 + 0.258819i 0 0 0 −0.965926 0.258819i 0 0.866025 0.500000i 0
3407.2 0 0.965926 0.258819i 0 0 0 0.965926 + 0.258819i 0 0.866025 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3407.2
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
5.c odd 4 2 inner
9.d odd 6 1 inner
20.e even 4 2 inner
36.h even 6 1 inner
45.h odd 6 1 inner
45.l even 12 2 inner
180.n even 6 1 inner
180.v odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.da.a 8
4.b odd 2 1 inner 3600.1.da.a 8
5.b even 2 1 inner 3600.1.da.a 8
5.c odd 4 2 inner 3600.1.da.a 8
9.d odd 6 1 inner 3600.1.da.a 8
20.d odd 2 1 CM 3600.1.da.a 8
20.e even 4 2 inner 3600.1.da.a 8
36.h even 6 1 inner 3600.1.da.a 8
45.h odd 6 1 inner 3600.1.da.a 8
45.l even 12 2 inner 3600.1.da.a 8
180.n even 6 1 inner 3600.1.da.a 8
180.v odd 12 2 inner 3600.1.da.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.da.a 8 1.a even 1 1 trivial
3600.1.da.a 8 4.b odd 2 1 inner
3600.1.da.a 8 5.b even 2 1 inner
3600.1.da.a 8 5.c odd 4 2 inner
3600.1.da.a 8 9.d odd 6 1 inner
3600.1.da.a 8 20.d odd 2 1 CM
3600.1.da.a 8 20.e even 4 2 inner
3600.1.da.a 8 36.h even 6 1 inner
3600.1.da.a 8 45.h odd 6 1 inner
3600.1.da.a 8 45.l even 12 2 inner
3600.1.da.a 8 180.n even 6 1 inner
3600.1.da.a 8 180.v odd 12 2 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3600, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T^{4} + T^{8} \)
$5$ 1
$7$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$11$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$13$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$17$ \( ( 1 + T^{4} )^{4} \)
$19$ \( ( 1 + T^{2} )^{8} \)
$23$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$29$ \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \)
$31$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$37$ \( ( 1 + T^{4} )^{4} \)
$41$ \( ( 1 - T )^{8}( 1 - T + T^{2} )^{4} \)
$43$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$47$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$53$ \( ( 1 + T^{4} )^{4} \)
$59$ \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
$61$ \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \)
$67$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$71$ \( ( 1 + T^{2} )^{8} \)
$73$ \( ( 1 + T^{4} )^{4} \)
$79$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$83$ \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \)
$89$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$97$ \( ( 1 - T^{4} + T^{8} )^{2} \)
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