# Properties

 Label 3600.1.c.a Level $3600$ Weight $1$ Character orbit 3600.c Analytic conductor $1.797$ Analytic rank $0$ Dimension $4$ Projective image $S_{4}$ CM/RM no 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1800) Projective image $$S_{4}$$ Projective field Galois closure of 4.2.10800.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{7} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{7} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{11} + \zeta_{8}^{2} q^{13} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{17} + q^{19} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{23} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{29} - q^{31} -\zeta_{8}^{2} q^{43} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{47} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{59} + q^{61} -\zeta_{8}^{2} q^{67} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{77} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{83} + q^{91} -\zeta_{8}^{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{19} - 4q^{31} + 4q^{61} + 4q^{91} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
0 0 0 0 0 1.00000i 0 0 0
449.2 0 0 0 0 0 1.00000i 0 0 0
449.3 0 0 0 0 0 1.00000i 0 0 0
449.4 0 0 0 0 0 1.00000i 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 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.c.a 4
3.b odd 2 1 inner 3600.1.c.a 4
4.b odd 2 1 1800.1.c.a 4
5.b even 2 1 inner 3600.1.c.a 4
5.c odd 4 1 3600.1.l.a 2
5.c odd 4 1 3600.1.l.b 2
12.b even 2 1 1800.1.c.a 4
15.d odd 2 1 inner 3600.1.c.a 4
15.e even 4 1 3600.1.l.a 2
15.e even 4 1 3600.1.l.b 2
20.d odd 2 1 1800.1.c.a 4
20.e even 4 1 1800.1.l.a 2
20.e even 4 1 1800.1.l.b yes 2
60.h even 2 1 1800.1.c.a 4
60.l odd 4 1 1800.1.l.a 2
60.l odd 4 1 1800.1.l.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.1.c.a 4 4.b odd 2 1
1800.1.c.a 4 12.b even 2 1
1800.1.c.a 4 20.d odd 2 1
1800.1.c.a 4 60.h even 2 1
1800.1.l.a 2 20.e even 4 1
1800.1.l.a 2 60.l odd 4 1
1800.1.l.b yes 2 20.e even 4 1
1800.1.l.b yes 2 60.l odd 4 1
3600.1.c.a 4 1.a even 1 1 trivial
3600.1.c.a 4 3.b odd 2 1 inner
3600.1.c.a 4 5.b even 2 1 inner
3600.1.c.a 4 15.d odd 2 1 inner
3600.1.l.a 2 5.c odd 4 1
3600.1.l.a 2 15.e even 4 1
3600.1.l.b 2 5.c odd 4 1
3600.1.l.b 2 15.e even 4 1

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