# Properties

 Label 3600.1.bh.b Level 3600 Weight 1 Character orbit 3600.bh Analytic conductor 1.797 Analytic rank 0 Dimension 4 Projective image $$D_{6}$$ CM discriminant -3 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3600.bh (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 900) Projective image $$D_{6}$$ Projective field Galois closure of 6.2.450000.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{7} + \beta_{3} q^{13} + \beta_{2} q^{19} - q^{31} + \beta_{3} q^{43} + 2 \beta_{2} q^{49} - q^{61} + \beta_{1} q^{67} + 2 \beta_{2} q^{79} + 3 q^{91} -\beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{31} - 4q^{61} + 12q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2593.1
 1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i −1.22474 − 1.22474i
0 0 0 0 0 −1.22474 + 1.22474i 0 0 0
2593.2 0 0 0 0 0 1.22474 1.22474i 0 0 0
3457.1 0 0 0 0 0 −1.22474 1.22474i 0 0 0
3457.2 0 0 0 0 0 1.22474 + 1.22474i 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 $$\Q(\sqrt{-3})$$
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.bh.b 4
3.b odd 2 1 CM 3600.1.bh.b 4
4.b odd 2 1 900.1.l.a 4
5.b even 2 1 inner 3600.1.bh.b 4
5.c odd 4 2 inner 3600.1.bh.b 4
12.b even 2 1 900.1.l.a 4
15.d odd 2 1 inner 3600.1.bh.b 4
15.e even 4 2 inner 3600.1.bh.b 4
20.d odd 2 1 900.1.l.a 4
20.e even 4 2 900.1.l.a 4
60.h even 2 1 900.1.l.a 4
60.l odd 4 2 900.1.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.1.l.a 4 4.b odd 2 1
900.1.l.a 4 12.b even 2 1
900.1.l.a 4 20.d odd 2 1
900.1.l.a 4 20.e even 4 2
900.1.l.a 4 60.h even 2 1
900.1.l.a 4 60.l odd 4 2
3600.1.bh.b 4 1.a even 1 1 trivial
3600.1.bh.b 4 3.b odd 2 1 CM
3600.1.bh.b 4 5.b even 2 1 inner
3600.1.bh.b 4 5.c odd 4 2 inner
3600.1.bh.b 4 15.d odd 2 1 inner
3600.1.bh.b 4 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 9$$ acting on $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.

## Hecke characteristic polynomials

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