Properties

 Label 3600.2.a.bh 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 360) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{7}+O(q^{10})$$ q + 2 * q^7 $$q + 2 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{17} - 4 q^{19} - 8 q^{23} + 10 q^{29} - 4 q^{31} - 8 q^{43} - 8 q^{47} - 3 q^{49} + 6 q^{53} - 14 q^{59} - 14 q^{61} - 4 q^{67} + 12 q^{71} - 6 q^{73} + 4 q^{77} + 12 q^{79} - 4 q^{83} + 12 q^{89} - 8 q^{91} + 14 q^{97}+O(q^{100})$$ q + 2 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^17 - 4 * q^19 - 8 * q^23 + 10 * q^29 - 4 * q^31 - 8 * q^43 - 8 * q^47 - 3 * q^49 + 6 * q^53 - 14 * q^59 - 14 * q^61 - 4 * q^67 + 12 * q^71 - 6 * q^73 + 4 * q^77 + 12 * q^79 - 4 * q^83 + 12 * q^89 - 8 * q^91 + 14 * q^97

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.bh 1
3.b odd 2 1 3600.2.a.bd 1
4.b odd 2 1 1800.2.a.f 1
5.b even 2 1 720.2.a.i 1
5.c odd 4 2 3600.2.f.q 2
12.b even 2 1 1800.2.a.i 1
15.d odd 2 1 720.2.a.a 1
15.e even 4 2 3600.2.f.g 2
20.d odd 2 1 360.2.a.d yes 1
20.e even 4 2 1800.2.f.d 2
40.e odd 2 1 2880.2.a.n 1
40.f even 2 1 2880.2.a.e 1
60.h even 2 1 360.2.a.c 1
60.l odd 4 2 1800.2.f.h 2
120.i odd 2 1 2880.2.a.w 1
120.m even 2 1 2880.2.a.bd 1
180.n even 6 2 3240.2.q.n 2
180.p odd 6 2 3240.2.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.a.c 1 60.h even 2 1
360.2.a.d yes 1 20.d odd 2 1
720.2.a.a 1 15.d odd 2 1
720.2.a.i 1 5.b even 2 1
1800.2.a.f 1 4.b odd 2 1
1800.2.a.i 1 12.b even 2 1
1800.2.f.d 2 20.e even 4 2
1800.2.f.h 2 60.l odd 4 2
2880.2.a.e 1 40.f even 2 1
2880.2.a.n 1 40.e odd 2 1
2880.2.a.w 1 120.i odd 2 1
2880.2.a.bd 1 120.m even 2 1
3240.2.q.d 2 180.p odd 6 2
3240.2.q.n 2 180.n even 6 2
3600.2.a.bd 1 3.b odd 2 1
3600.2.a.bh 1 1.a even 1 1 trivial
3600.2.f.g 2 15.e even 4 2
3600.2.f.q 2 5.c odd 4 2

Hecke kernels

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

 $$T_{7} - 2$$ T7 - 2 $$T_{11} - 2$$ T11 - 2 $$T_{13} + 4$$ T13 + 4 $$T_{17} + 2$$ T17 + 2

Hecke characteristic polynomials

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