# Properties

 Label 3600.2.a.bb Level $3600$ Weight $2$ Character orbit 3600.a Self dual yes Analytic conductor $28.746$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.7461447277$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{7} + O(q^{10})$$ $$q + 2q^{7} - 4q^{11} - 4q^{13} + 4q^{19} + 2q^{23} - 2q^{29} - 4q^{37} - 2q^{41} - 6q^{43} + 6q^{47} - 3q^{49} - 4q^{53} - 12q^{59} - 10q^{61} + 14q^{67} + 8q^{71} - 8q^{73} - 8q^{77} - 16q^{79} - 2q^{83} - 6q^{89} - 8q^{91} - 16q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.a.bb 1
3.b odd 2 1 400.2.a.g 1
4.b odd 2 1 1800.2.a.j 1
5.b even 2 1 3600.2.a.k 1
5.c odd 4 2 720.2.f.e 2
12.b even 2 1 200.2.a.b 1
15.d odd 2 1 400.2.a.b 1
15.e even 4 2 80.2.c.a 2
20.d odd 2 1 1800.2.a.s 1
20.e even 4 2 360.2.f.c 2
24.f even 2 1 1600.2.a.v 1
24.h odd 2 1 1600.2.a.d 1
40.i odd 4 2 2880.2.f.i 2
40.k even 4 2 2880.2.f.h 2
60.h even 2 1 200.2.a.d 1
60.l odd 4 2 40.2.c.a 2
84.h odd 2 1 9800.2.a.bf 1
120.i odd 2 1 1600.2.a.u 1
120.m even 2 1 1600.2.a.f 1
120.q odd 4 2 320.2.c.c 2
120.w even 4 2 320.2.c.b 2
240.z odd 4 2 1280.2.f.f 2
240.bb even 4 2 1280.2.f.b 2
240.bd odd 4 2 1280.2.f.a 2
240.bf even 4 2 1280.2.f.e 2
420.o odd 2 1 9800.2.a.d 1
420.w even 4 2 1960.2.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 60.l odd 4 2
80.2.c.a 2 15.e even 4 2
200.2.a.b 1 12.b even 2 1
200.2.a.d 1 60.h even 2 1
320.2.c.b 2 120.w even 4 2
320.2.c.c 2 120.q odd 4 2
360.2.f.c 2 20.e even 4 2
400.2.a.b 1 15.d odd 2 1
400.2.a.g 1 3.b odd 2 1
720.2.f.e 2 5.c odd 4 2
1280.2.f.a 2 240.bd odd 4 2
1280.2.f.b 2 240.bb even 4 2
1280.2.f.e 2 240.bf even 4 2
1280.2.f.f 2 240.z odd 4 2
1600.2.a.d 1 24.h odd 2 1
1600.2.a.f 1 120.m even 2 1
1600.2.a.u 1 120.i odd 2 1
1600.2.a.v 1 24.f even 2 1
1800.2.a.j 1 4.b odd 2 1
1800.2.a.s 1 20.d odd 2 1
1960.2.g.b 2 420.w even 4 2
2880.2.f.h 2 40.k even 4 2
2880.2.f.i 2 40.i odd 4 2
3600.2.a.k 1 5.b even 2 1
3600.2.a.bb 1 1.a even 1 1 trivial
9800.2.a.d 1 420.o odd 2 1
9800.2.a.bf 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3600))$$:

 $$T_{7} - 2$$ $$T_{11} + 4$$ $$T_{13} + 4$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-2 + T$$
$11$ $$4 + T$$
$13$ $$4 + T$$
$17$ $$T$$
$19$ $$-4 + T$$
$23$ $$-2 + T$$
$29$ $$2 + T$$
$31$ $$T$$
$37$ $$4 + T$$
$41$ $$2 + T$$
$43$ $$6 + T$$
$47$ $$-6 + T$$
$53$ $$4 + T$$
$59$ $$12 + T$$
$61$ $$10 + T$$
$67$ $$-14 + T$$
$71$ $$-8 + T$$
$73$ $$8 + T$$
$79$ $$16 + T$$
$83$ $$2 + T$$
$89$ $$6 + T$$
$97$ $$16 + T$$