# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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
$5$ $$1 + T + T^{2} + T^{3} + T^{4}$$
$7$ $$( 1 + T^{2} )^{4}$$
$11$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$13$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$17$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$19$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$23$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$29$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$31$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$37$ $$( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )$$
$41$ $$( 1 + T + T^{2} + T^{3} + T^{4} )^{2}$$
$43$ $$( 1 + T^{2} )^{4}$$
$47$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$53$ $$( 1 - T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
$59$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$61$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$67$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$71$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$73$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$79$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$
$83$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$89$ $$( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} )$$
$97$ $$( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} )$$