Properties

Label 3600.1.ct.a
Level 3600
Weight 1
Character orbit 3600.ct
Analytic conductor 1.797
Analytic rank 0
Dimension 4
Projective image \(D_{10}\)
CM discriminant -4
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.ct (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 400)
Projective image \(D_{10}\)
Projective field Galois closure of 10.2.195312500000000.4

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{10}^{4} q^{5} +O(q^{10})\) \( q + \zeta_{10}^{4} q^{5} + ( -\zeta_{10} + \zeta_{10}^{3} ) q^{13} + ( -\zeta_{10} - \zeta_{10}^{2} ) q^{17} -\zeta_{10}^{3} q^{25} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{29} + ( -1 + \zeta_{10}^{4} ) q^{37} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{41} - q^{49} + ( 1 - \zeta_{10}^{2} ) q^{53} + ( -\zeta_{10}^{2} - \zeta_{10}^{4} ) q^{61} + ( 1 - \zeta_{10}^{2} ) q^{65} + ( -\zeta_{10}^{2} + \zeta_{10}^{4} ) q^{73} + ( 1 + \zeta_{10} ) q^{85} + ( -1 + \zeta_{10} ) q^{89} + ( -\zeta_{10}^{3} - \zeta_{10}^{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{5} + O(q^{10}) \) \( 4q - q^{5} - q^{25} + 2q^{29} - 5q^{37} - 2q^{41} - 4q^{49} + 5q^{53} + 2q^{61} + 5q^{65} + 5q^{85} - 3q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(\zeta_{10}\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−0.309017 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 + 0.951057i
0 0 0 0.309017 0.951057i 0 0 0 0 0
1279.1 0 0 0 −0.809017 + 0.587785i 0 0 0 0 0
2719.1 0 0 0 −0.809017 0.587785i 0 0 0 0 0
3439.1 0 0 0 0.309017 + 0.951057i 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
25.e even 10 1 inner
100.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.ct.a 4
3.b odd 2 1 400.1.x.a 4
4.b odd 2 1 CM 3600.1.ct.a 4
12.b even 2 1 400.1.x.a 4
15.d odd 2 1 2000.1.x.a 4
15.e even 4 2 2000.1.z.a 8
24.f even 2 1 1600.1.bf.a 4
24.h odd 2 1 1600.1.bf.a 4
25.e even 10 1 inner 3600.1.ct.a 4
60.h even 2 1 2000.1.x.a 4
60.l odd 4 2 2000.1.z.a 8
75.h odd 10 1 400.1.x.a 4
75.j odd 10 1 2000.1.x.a 4
75.l even 20 2 2000.1.z.a 8
100.h odd 10 1 inner 3600.1.ct.a 4
300.n even 10 1 2000.1.x.a 4
300.r even 10 1 400.1.x.a 4
300.u odd 20 2 2000.1.z.a 8
600.z odd 10 1 1600.1.bf.a 4
600.bk even 10 1 1600.1.bf.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.1.x.a 4 3.b odd 2 1
400.1.x.a 4 12.b even 2 1
400.1.x.a 4 75.h odd 10 1
400.1.x.a 4 300.r even 10 1
1600.1.bf.a 4 24.f even 2 1
1600.1.bf.a 4 24.h odd 2 1
1600.1.bf.a 4 600.z odd 10 1
1600.1.bf.a 4 600.bk even 10 1
2000.1.x.a 4 15.d odd 2 1
2000.1.x.a 4 60.h even 2 1
2000.1.x.a 4 75.j odd 10 1
2000.1.x.a 4 300.n even 10 1
2000.1.z.a 8 15.e even 4 2
2000.1.z.a 8 60.l odd 4 2
2000.1.z.a 8 75.l even 20 2
2000.1.z.a 8 300.u odd 20 2
3600.1.ct.a 4 1.a even 1 1 trivial
3600.1.ct.a 4 4.b odd 2 1 CM
3600.1.ct.a 4 25.e even 10 1 inner
3600.1.ct.a 4 100.h odd 10 1 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
$5$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$13$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$17$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$19$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$23$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$29$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$31$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$37$ \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$41$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$43$ \( ( 1 + T^{2} )^{4} \)
$47$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$53$ \( ( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
$59$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$61$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$71$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$73$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$79$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
$83$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$89$ \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \)
$97$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
show more
show less