# Properties

 Label 3600.2.f.p Level $3600$ Weight $2$ Character orbit 3600.f Analytic conductor $28.746$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3600.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.7461447277$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{7}+O(q^{10})$$ q + 3*i * q^7 $$q + 3 i q^{7} + 2 q^{11} - i q^{13} - 2 i q^{17} - 5 q^{19} - 6 i q^{23} + 10 q^{29} + 3 q^{31} + 2 i q^{37} + 8 q^{41} + i q^{43} + 2 i q^{47} - 2 q^{49} - 4 i q^{53} + 10 q^{59} + 7 q^{61} + 3 i q^{67} - 8 q^{71} + 14 i q^{73} + 6 i q^{77} - 6 i q^{83} + 3 q^{91} + 17 i q^{97} +O(q^{100})$$ q + 3*i * q^7 + 2 * q^11 - i * q^13 - 2*i * q^17 - 5 * q^19 - 6*i * q^23 + 10 * q^29 + 3 * q^31 + 2*i * q^37 + 8 * q^41 + i * q^43 + 2*i * q^47 - 2 * q^49 - 4*i * q^53 + 10 * q^59 + 7 * q^61 + 3*i * q^67 - 8 * q^71 + 14*i * q^73 + 6*i * q^77 - 6*i * q^83 + 3 * q^91 + 17*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 4 q^{11} - 10 q^{19} + 20 q^{29} + 6 q^{31} + 16 q^{41} - 4 q^{49} + 20 q^{59} + 14 q^{61} - 16 q^{71} + 6 q^{91}+O(q^{100})$$ 2 * q + 4 * q^11 - 10 * q^19 + 20 * q^29 + 6 * q^31 + 16 * q^41 - 4 * q^49 + 20 * q^59 + 14 * q^61 - 16 * q^71 + 6 * q^91

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2449.1
 − 1.00000i 1.00000i
0 0 0 0 0 3.00000i 0 0 0
2449.2 0 0 0 0 0 3.00000i 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.f.p 2
3.b odd 2 1 1200.2.f.d 2
4.b odd 2 1 225.2.b.a 2
5.b even 2 1 inner 3600.2.f.p 2
5.c odd 4 1 3600.2.a.j 1
5.c odd 4 1 3600.2.a.bk 1
12.b even 2 1 75.2.b.a 2
15.d odd 2 1 1200.2.f.d 2
15.e even 4 1 1200.2.a.c 1
15.e even 4 1 1200.2.a.p 1
20.d odd 2 1 225.2.b.a 2
20.e even 4 1 225.2.a.a 1
20.e even 4 1 225.2.a.e 1
24.f even 2 1 4800.2.f.l 2
24.h odd 2 1 4800.2.f.y 2
60.h even 2 1 75.2.b.a 2
60.l odd 4 1 75.2.a.a 1
60.l odd 4 1 75.2.a.c yes 1
120.i odd 2 1 4800.2.f.y 2
120.m even 2 1 4800.2.f.l 2
120.q odd 4 1 4800.2.a.bb 1
120.q odd 4 1 4800.2.a.bq 1
120.w even 4 1 4800.2.a.be 1
120.w even 4 1 4800.2.a.br 1
420.w even 4 1 3675.2.a.b 1
420.w even 4 1 3675.2.a.q 1
660.q even 4 1 9075.2.a.a 1
660.q even 4 1 9075.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 60.l odd 4 1
75.2.a.c yes 1 60.l odd 4 1
75.2.b.a 2 12.b even 2 1
75.2.b.a 2 60.h even 2 1
225.2.a.a 1 20.e even 4 1
225.2.a.e 1 20.e even 4 1
225.2.b.a 2 4.b odd 2 1
225.2.b.a 2 20.d odd 2 1
1200.2.a.c 1 15.e even 4 1
1200.2.a.p 1 15.e even 4 1
1200.2.f.d 2 3.b odd 2 1
1200.2.f.d 2 15.d odd 2 1
3600.2.a.j 1 5.c odd 4 1
3600.2.a.bk 1 5.c odd 4 1
3600.2.f.p 2 1.a even 1 1 trivial
3600.2.f.p 2 5.b even 2 1 inner
3675.2.a.b 1 420.w even 4 1
3675.2.a.q 1 420.w even 4 1
4800.2.a.bb 1 120.q odd 4 1
4800.2.a.be 1 120.w even 4 1
4800.2.a.bq 1 120.q odd 4 1
4800.2.a.br 1 120.w even 4 1
4800.2.f.l 2 24.f even 2 1
4800.2.f.l 2 120.m even 2 1
4800.2.f.y 2 24.h odd 2 1
4800.2.f.y 2 120.i odd 2 1
9075.2.a.a 1 660.q even 4 1
9075.2.a.s 1 660.q even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{11} - 2$$ T11 - 2 $$T_{13}^{2} + 1$$ T13^2 + 1 $$T_{17}^{2} + 4$$ T17^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 4$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T - 3)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 16$$
$59$ $$(T - 10)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$T^{2} + 9$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 289$$