# Properties

 Label 3584.2.m.d Level $3584$ Weight $2$ Character orbit 3584.m Analytic conductor $28.618$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3584 = 2^{9} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3584.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6183840844$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (i - 1) q^{3} + ( - 2 i - 2) q^{5} - i q^{7} + i q^{9} +O(q^{10})$$ q + (i - 1) * q^3 + (-2*i - 2) * q^5 - i * q^7 + i * q^9 $$q + (i - 1) q^{3} + ( - 2 i - 2) q^{5} - i q^{7} + i q^{9} + ( - 2 i - 2) q^{11} + 4 q^{15} - 2 q^{17} + ( - 3 i + 3) q^{19} + (i + 1) q^{21} + 4 i q^{23} + 3 i q^{25} + ( - 4 i - 4) q^{27} + ( - 7 i + 7) q^{29} + 6 q^{31} + 4 q^{33} + (2 i - 2) q^{35} + (3 i + 3) q^{37} - 2 i q^{41} + ( - 6 i - 6) q^{43} + ( - 2 i + 2) q^{45} - 10 q^{47} - q^{49} + ( - 2 i + 2) q^{51} + (5 i + 5) q^{53} + 8 i q^{55} + 6 i q^{57} + ( - 7 i - 7) q^{59} + ( - 2 i + 2) q^{61} + q^{63} + (8 i - 8) q^{67} + ( - 4 i - 4) q^{69} - 4 i q^{71} + 14 i q^{73} + ( - 3 i - 3) q^{75} + (2 i - 2) q^{77} + 4 q^{79} + 5 q^{81} + (3 i - 3) q^{83} + (4 i + 4) q^{85} + 14 i q^{87} - 10 i q^{89} + (6 i - 6) q^{93} - 12 q^{95} - 6 q^{97} + ( - 2 i + 2) q^{99} +O(q^{100})$$ q + (i - 1) * q^3 + (-2*i - 2) * q^5 - i * q^7 + i * q^9 + (-2*i - 2) * q^11 + 4 * q^15 - 2 * q^17 + (-3*i + 3) * q^19 + (i + 1) * q^21 + 4*i * q^23 + 3*i * q^25 + (-4*i - 4) * q^27 + (-7*i + 7) * q^29 + 6 * q^31 + 4 * q^33 + (2*i - 2) * q^35 + (3*i + 3) * q^37 - 2*i * q^41 + (-6*i - 6) * q^43 + (-2*i + 2) * q^45 - 10 * q^47 - q^49 + (-2*i + 2) * q^51 + (5*i + 5) * q^53 + 8*i * q^55 + 6*i * q^57 + (-7*i - 7) * q^59 + (-2*i + 2) * q^61 + q^63 + (8*i - 8) * q^67 + (-4*i - 4) * q^69 - 4*i * q^71 + 14*i * q^73 + (-3*i - 3) * q^75 + (2*i - 2) * q^77 + 4 * q^79 + 5 * q^81 + (3*i - 3) * q^83 + (4*i + 4) * q^85 + 14*i * q^87 - 10*i * q^89 + (6*i - 6) * q^93 - 12 * q^95 - 6 * q^97 + (-2*i + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^5 $$2 q - 2 q^{3} - 4 q^{5} - 4 q^{11} + 8 q^{15} - 4 q^{17} + 6 q^{19} + 2 q^{21} - 8 q^{27} + 14 q^{29} + 12 q^{31} + 8 q^{33} - 4 q^{35} + 6 q^{37} - 12 q^{43} + 4 q^{45} - 20 q^{47} - 2 q^{49} + 4 q^{51} + 10 q^{53} - 14 q^{59} + 4 q^{61} + 2 q^{63} - 16 q^{67} - 8 q^{69} - 6 q^{75} - 4 q^{77} + 8 q^{79} + 10 q^{81} - 6 q^{83} + 8 q^{85} - 12 q^{93} - 24 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^5 - 4 * q^11 + 8 * q^15 - 4 * q^17 + 6 * q^19 + 2 * q^21 - 8 * q^27 + 14 * q^29 + 12 * q^31 + 8 * q^33 - 4 * q^35 + 6 * q^37 - 12 * q^43 + 4 * q^45 - 20 * q^47 - 2 * q^49 + 4 * q^51 + 10 * q^53 - 14 * q^59 + 4 * q^61 + 2 * q^63 - 16 * q^67 - 8 * q^69 - 6 * q^75 - 4 * q^77 + 8 * q^79 + 10 * q^81 - 6 * q^83 + 8 * q^85 - 12 * q^93 - 24 * q^95 - 12 * q^97 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
897.1
 1.00000i − 1.00000i
0 −1.00000 + 1.00000i 0 −2.00000 2.00000i 0 1.00000i 0 1.00000i 0
2689.1 0 −1.00000 1.00000i 0 −2.00000 + 2.00000i 0 1.00000i 0 1.00000i 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
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.m.d 2
4.b odd 2 1 3584.2.m.s yes 2
8.b even 2 1 3584.2.m.y yes 2
8.d odd 2 1 3584.2.m.j yes 2
16.e even 4 1 inner 3584.2.m.d 2
16.e even 4 1 3584.2.m.y yes 2
16.f odd 4 1 3584.2.m.j yes 2
16.f odd 4 1 3584.2.m.s yes 2
32.g even 8 2 7168.2.a.d 2
32.h odd 8 2 7168.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.m.d 2 1.a even 1 1 trivial
3584.2.m.d 2 16.e even 4 1 inner
3584.2.m.j yes 2 8.d odd 2 1
3584.2.m.j yes 2 16.f odd 4 1
3584.2.m.s yes 2 4.b odd 2 1
3584.2.m.s yes 2 16.f odd 4 1
3584.2.m.y yes 2 8.b even 2 1
3584.2.m.y yes 2 16.e even 4 1
7168.2.a.d 2 32.g even 8 2
7168.2.a.p 2 32.h odd 8 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3584, [\chi])$$:

 $$T_{3}^{2} + 2T_{3} + 2$$ T3^2 + 2*T3 + 2 $$T_{5}^{2} + 4T_{5} + 8$$ T5^2 + 4*T5 + 8 $$T_{11}^{2} + 4T_{11} + 8$$ T11^2 + 4*T11 + 8 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2} + 4T + 8$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 4T + 8$$
$13$ $$T^{2}$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} - 6T + 18$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2} - 14T + 98$$
$31$ $$(T - 6)^{2}$$
$37$ $$T^{2} - 6T + 18$$
$41$ $$T^{2} + 4$$
$43$ $$T^{2} + 12T + 72$$
$47$ $$(T + 10)^{2}$$
$53$ $$T^{2} - 10T + 50$$
$59$ $$T^{2} + 14T + 98$$
$61$ $$T^{2} - 4T + 8$$
$67$ $$T^{2} + 16T + 128$$
$71$ $$T^{2} + 16$$
$73$ $$T^{2} + 196$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 6T + 18$$
$89$ $$T^{2} + 100$$
$97$ $$(T + 6)^{2}$$