# Properties

 Label 3600.1.e.d 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

# Related objects

## 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
 0.5 + 0.866025i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.d yes 2
3.b odd 2 1 CM 3600.1.e.d yes 2
4.b odd 2 1 inner 3600.1.e.d yes 2
5.b even 2 1 3600.1.e.c 2
5.c odd 4 2 3600.1.j.b 4
12.b even 2 1 inner 3600.1.e.d yes 2
15.d odd 2 1 3600.1.e.c 2
15.e even 4 2 3600.1.j.b 4
20.d odd 2 1 3600.1.e.c 2
20.e even 4 2 3600.1.j.b 4
60.h even 2 1 3600.1.e.c 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 5.b even 2 1
3600.1.e.c 2 15.d odd 2 1
3600.1.e.c 2 20.d odd 2 1
3600.1.e.c 2 60.h even 2 1
3600.1.e.d yes 2 1.a even 1 1 trivial
3600.1.e.d yes 2 3.b odd 2 1 CM
3600.1.e.d yes 2 4.b odd 2 1 inner
3600.1.e.d yes 2 12.b even 2 1 inner
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
$7$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$11$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$13$ $$( 1 - T + T^{2} )^{2}$$
$17$ $$( 1 + T^{2} )^{2}$$
$19$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$23$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$29$ $$( 1 + T^{2} )^{2}$$
$31$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$37$ $$( 1 - T )^{4}$$
$41$ $$( 1 + T^{2} )^{2}$$
$43$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$47$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$53$ $$( 1 + T^{2} )^{2}$$
$59$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$61$ $$( 1 + T + T^{2} )^{2}$$
$67$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$71$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$73$ $$( 1 + T )^{4}$$
$79$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$83$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$89$ $$( 1 + T^{2} )^{2}$$
$97$ $$( 1 - T + T^{2} )^{2}$$