# Properties

 Label 3200.2.f.a Level $3200$ Weight $2$ Character orbit 3200.f Analytic conductor $25.552$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + 4 i q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + 4*i * q^7 - 2 * q^9 $$q - q^{3} + 4 i q^{7} - 2 q^{9} + 3 i q^{11} - i q^{17} + 7 i q^{19} - 4 i q^{21} - 4 i q^{23} + 5 q^{27} + 8 i q^{29} - 4 q^{31} - 3 i q^{33} - 4 q^{37} + 3 q^{41} + 8 q^{43} - 9 q^{49} + i q^{51} - 12 q^{53} - 7 i q^{57} - 8 i q^{59} + 4 i q^{61} - 8 i q^{63} - 9 q^{67} + 4 i q^{69} + 16 q^{71} - 11 i q^{73} - 12 q^{77} + 4 q^{79} + q^{81} + q^{83} - 8 i q^{87} - 13 q^{89} + 4 q^{93} - 14 i q^{97} - 6 i q^{99} +O(q^{100})$$ q - q^3 + 4*i * q^7 - 2 * q^9 + 3*i * q^11 - i * q^17 + 7*i * q^19 - 4*i * q^21 - 4*i * q^23 + 5 * q^27 + 8*i * q^29 - 4 * q^31 - 3*i * q^33 - 4 * q^37 + 3 * q^41 + 8 * q^43 - 9 * q^49 + i * q^51 - 12 * q^53 - 7*i * q^57 - 8*i * q^59 + 4*i * q^61 - 8*i * q^63 - 9 * q^67 + 4*i * q^69 + 16 * q^71 - 11*i * q^73 - 12 * q^77 + 4 * q^79 + q^81 + q^83 - 8*i * q^87 - 13 * q^89 + 4 * q^93 - 14*i * q^97 - 6*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^9 $$2 q - 2 q^{3} - 4 q^{9} + 10 q^{27} - 8 q^{31} - 8 q^{37} + 6 q^{41} + 16 q^{43} - 18 q^{49} - 24 q^{53} - 18 q^{67} + 32 q^{71} - 24 q^{77} + 8 q^{79} + 2 q^{81} + 2 q^{83} - 26 q^{89} + 8 q^{93}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^9 + 10 * q^27 - 8 * q^31 - 8 * q^37 + 6 * q^41 + 16 * q^43 - 18 * q^49 - 24 * q^53 - 18 * q^67 + 32 * q^71 - 24 * q^77 + 8 * q^79 + 2 * q^81 + 2 * q^83 - 26 * q^89 + 8 * q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1151$$ $$2177$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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
 − 1.00000i 1.00000i
0 −1.00000 0 0 0 4.00000i 0 −2.00000 0
449.2 0 −1.00000 0 0 0 4.00000i 0 −2.00000 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
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.f.a 2
4.b odd 2 1 3200.2.f.f 2
5.b even 2 1 3200.2.f.e 2
5.c odd 4 1 3200.2.d.b yes 2
5.c odd 4 1 3200.2.d.g yes 2
8.b even 2 1 3200.2.f.e 2
8.d odd 2 1 3200.2.f.b 2
20.d odd 2 1 3200.2.f.b 2
20.e even 4 1 3200.2.d.a 2
20.e even 4 1 3200.2.d.h yes 2
40.e odd 2 1 3200.2.f.f 2
40.f even 2 1 inner 3200.2.f.a 2
40.i odd 4 1 3200.2.d.b yes 2
40.i odd 4 1 3200.2.d.g yes 2
40.k even 4 1 3200.2.d.a 2
40.k even 4 1 3200.2.d.h yes 2
80.i odd 4 1 6400.2.a.q 1
80.i odd 4 1 6400.2.a.v 1
80.j even 4 1 6400.2.a.p 1
80.j even 4 1 6400.2.a.w 1
80.s even 4 1 6400.2.a.c 1
80.s even 4 1 6400.2.a.h 1
80.t odd 4 1 6400.2.a.b 1
80.t odd 4 1 6400.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.a 2 20.e even 4 1
3200.2.d.a 2 40.k even 4 1
3200.2.d.b yes 2 5.c odd 4 1
3200.2.d.b yes 2 40.i odd 4 1
3200.2.d.g yes 2 5.c odd 4 1
3200.2.d.g yes 2 40.i odd 4 1
3200.2.d.h yes 2 20.e even 4 1
3200.2.d.h yes 2 40.k even 4 1
3200.2.f.a 2 1.a even 1 1 trivial
3200.2.f.a 2 40.f even 2 1 inner
3200.2.f.b 2 8.d odd 2 1
3200.2.f.b 2 20.d odd 2 1
3200.2.f.e 2 5.b even 2 1
3200.2.f.e 2 8.b even 2 1
3200.2.f.f 2 4.b odd 2 1
3200.2.f.f 2 40.e odd 2 1
6400.2.a.b 1 80.t odd 4 1
6400.2.a.c 1 80.s even 4 1
6400.2.a.h 1 80.s even 4 1
6400.2.a.i 1 80.t odd 4 1
6400.2.a.p 1 80.j even 4 1
6400.2.a.q 1 80.i odd 4 1
6400.2.a.v 1 80.i odd 4 1
6400.2.a.w 1 80.j even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3200, [\chi])$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11}^{2} + 9$$ T11^2 + 9 $$T_{13}$$ T13 $$T_{31} + 4$$ T31 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 9$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 1$$
$19$ $$T^{2} + 49$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2} + 64$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T + 4)^{2}$$
$41$ $$(T - 3)^{2}$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} + 64$$
$61$ $$T^{2} + 16$$
$67$ $$(T + 9)^{2}$$
$71$ $$(T - 16)^{2}$$
$73$ $$T^{2} + 121$$
$79$ $$(T - 4)^{2}$$
$83$ $$(T - 1)^{2}$$
$89$ $$(T + 13)^{2}$$
$97$ $$T^{2} + 196$$