# Properties

 Label 3600.1.j.a Level 3600 Weight 1 Character orbit 3600.j Analytic conductor 1.797 Analytic rank 0 Dimension 2 Projective image $$D_{2}$$ CM/RM discs -3, -4, 12 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.j (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 144) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\zeta_{12})$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.2916000000.5

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q -2 i q^{13} -2 i q^{37} - q^{49} + 2 q^{61} -2 i q^{73} + 2 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 2q^{49} + 4q^{61} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1999.1
 1.00000i − 1.00000i
0 0 0 0 0 0 0 0 0
1999.2 0 0 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
5.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h 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.j.a 2
3.b odd 2 1 CM 3600.1.j.a 2
4.b odd 2 1 CM 3600.1.j.a 2
5.b even 2 1 inner 3600.1.j.a 2
5.c odd 4 1 144.1.g.a 1
5.c odd 4 1 3600.1.e.b 1
12.b even 2 1 RM 3600.1.j.a 2
15.d odd 2 1 inner 3600.1.j.a 2
15.e even 4 1 144.1.g.a 1
15.e even 4 1 3600.1.e.b 1
20.d odd 2 1 inner 3600.1.j.a 2
20.e even 4 1 144.1.g.a 1
20.e even 4 1 3600.1.e.b 1
40.i odd 4 1 576.1.g.a 1
40.k even 4 1 576.1.g.a 1
45.k odd 12 2 1296.1.o.b 2
45.l even 12 2 1296.1.o.b 2
60.h even 2 1 inner 3600.1.j.a 2
60.l odd 4 1 144.1.g.a 1
60.l odd 4 1 3600.1.e.b 1
80.i odd 4 1 2304.1.b.a 2
80.j even 4 1 2304.1.b.a 2
80.s even 4 1 2304.1.b.a 2
80.t odd 4 1 2304.1.b.a 2
120.q odd 4 1 576.1.g.a 1
120.w even 4 1 576.1.g.a 1
180.v odd 12 2 1296.1.o.b 2
180.x even 12 2 1296.1.o.b 2
240.z odd 4 1 2304.1.b.a 2
240.bb even 4 1 2304.1.b.a 2
240.bd odd 4 1 2304.1.b.a 2
240.bf even 4 1 2304.1.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a 1 5.c odd 4 1
144.1.g.a 1 15.e even 4 1
144.1.g.a 1 20.e even 4 1
144.1.g.a 1 60.l odd 4 1
576.1.g.a 1 40.i odd 4 1
576.1.g.a 1 40.k even 4 1
576.1.g.a 1 120.q odd 4 1
576.1.g.a 1 120.w even 4 1
1296.1.o.b 2 45.k odd 12 2
1296.1.o.b 2 45.l even 12 2
1296.1.o.b 2 180.v odd 12 2
1296.1.o.b 2 180.x even 12 2
2304.1.b.a 2 80.i odd 4 1
2304.1.b.a 2 80.j even 4 1
2304.1.b.a 2 80.s even 4 1
2304.1.b.a 2 80.t odd 4 1
2304.1.b.a 2 240.z odd 4 1
2304.1.b.a 2 240.bb even 4 1
2304.1.b.a 2 240.bd odd 4 1
2304.1.b.a 2 240.bf even 4 1
3600.1.e.b 1 5.c odd 4 1
3600.1.e.b 1 15.e even 4 1
3600.1.e.b 1 20.e even 4 1
3600.1.e.b 1 60.l odd 4 1
3600.1.j.a 2 1.a even 1 1 trivial
3600.1.j.a 2 3.b odd 2 1 CM
3600.1.j.a 2 4.b odd 2 1 CM
3600.1.j.a 2 5.b even 2 1 inner
3600.1.j.a 2 12.b even 2 1 RM
3600.1.j.a 2 15.d odd 2 1 inner
3600.1.j.a 2 20.d odd 2 1 inner
3600.1.j.a 2 60.h even 2 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

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