Properties

 Label 3600.1.e.b Level 3600 Weight 1 Character orbit 3600.e Self dual yes Analytic conductor 1.797 Analytic rank 0 Dimension 1 Projective image $$D_{2}$$ CM/RM discs -3, -4, 12 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: yes Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 144) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\zeta_{12})$$ Artin image $D_4$ Artin field Galois closure of 4.0.10800.2

$q$-expansion

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

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
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 $$\Q(\sqrt{3})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.e.b 1
3.b odd 2 1 CM 3600.1.e.b 1
4.b odd 2 1 CM 3600.1.e.b 1
5.b even 2 1 144.1.g.a 1
5.c odd 4 2 3600.1.j.a 2
12.b even 2 1 RM 3600.1.e.b 1
15.d odd 2 1 144.1.g.a 1
15.e even 4 2 3600.1.j.a 2
20.d odd 2 1 144.1.g.a 1
20.e even 4 2 3600.1.j.a 2
40.e odd 2 1 576.1.g.a 1
40.f even 2 1 576.1.g.a 1
45.h odd 6 2 1296.1.o.b 2
45.j even 6 2 1296.1.o.b 2
60.h even 2 1 144.1.g.a 1
60.l odd 4 2 3600.1.j.a 2
80.k odd 4 2 2304.1.b.a 2
80.q even 4 2 2304.1.b.a 2
120.i odd 2 1 576.1.g.a 1
120.m even 2 1 576.1.g.a 1
180.n even 6 2 1296.1.o.b 2
180.p odd 6 2 1296.1.o.b 2
240.t even 4 2 2304.1.b.a 2
240.bm odd 4 2 2304.1.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.1.g.a 1 5.b even 2 1
144.1.g.a 1 15.d odd 2 1
144.1.g.a 1 20.d odd 2 1
144.1.g.a 1 60.h even 2 1
576.1.g.a 1 40.e odd 2 1
576.1.g.a 1 40.f even 2 1
576.1.g.a 1 120.i odd 2 1
576.1.g.a 1 120.m even 2 1
1296.1.o.b 2 45.h odd 6 2
1296.1.o.b 2 45.j even 6 2
1296.1.o.b 2 180.n even 6 2
1296.1.o.b 2 180.p odd 6 2
2304.1.b.a 2 80.k odd 4 2
2304.1.b.a 2 80.q even 4 2
2304.1.b.a 2 240.t even 4 2
2304.1.b.a 2 240.bm odd 4 2
3600.1.e.b 1 1.a even 1 1 trivial
3600.1.e.b 1 3.b odd 2 1 CM
3600.1.e.b 1 4.b odd 2 1 CM
3600.1.e.b 1 12.b even 2 1 RM
3600.1.j.a 2 5.c odd 4 2
3600.1.j.a 2 15.e even 4 2
3600.1.j.a 2 20.e even 4 2
3600.1.j.a 2 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 $$S_{1}^{\mathrm{new}}(3600, [\chi])$$:

 $$T_{7}$$ $$T_{13} - 2$$

Hecke characteristic polynomials

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