# Properties

 Label 3200.2.c.f Level $3200$ Weight $2$ Character orbit 3200.c Analytic conductor $25.552$ Analytic rank $1$ 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.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.5521286468$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 128) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} -4 i q^{7} - q^{9} +O(q^{10})$$ $$q + 2 i q^{3} -4 i q^{7} - q^{9} -2 q^{11} -2 i q^{13} + 2 i q^{17} -2 q^{19} + 8 q^{21} -4 i q^{23} + 4 i q^{27} -6 q^{29} -4 i q^{33} + 10 i q^{37} + 4 q^{39} -6 q^{41} + 6 i q^{43} -8 i q^{47} -9 q^{49} -4 q^{51} + 6 i q^{53} -4 i q^{57} -14 q^{59} -2 q^{61} + 4 i q^{63} -10 i q^{67} + 8 q^{69} -12 q^{71} + 14 i q^{73} + 8 i q^{77} -8 q^{79} -11 q^{81} -6 i q^{83} -12 i q^{87} + 2 q^{89} -8 q^{91} + 2 i q^{97} + 2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 4q^{11} - 4q^{19} + 16q^{21} - 12q^{29} + 8q^{39} - 12q^{41} - 18q^{49} - 8q^{51} - 28q^{59} - 4q^{61} + 16q^{69} - 24q^{71} - 16q^{79} - 22q^{81} + 4q^{89} - 16q^{91} + 4q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2049.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 4.00000i 0 −1.00000 0
2049.2 0 2.00000i 0 0 0 4.00000i 0 −1.00000 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
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 3200.2.c.f 2
4.b odd 2 1 3200.2.c.l 2
5.b even 2 1 inner 3200.2.c.f 2
5.c odd 4 1 128.2.a.c yes 1
5.c odd 4 1 3200.2.a.e 1
8.b even 2 1 3200.2.c.k 2
8.d odd 2 1 3200.2.c.e 2
15.e even 4 1 1152.2.a.r 1
20.d odd 2 1 3200.2.c.l 2
20.e even 4 1 128.2.a.a 1
20.e even 4 1 3200.2.a.x 1
35.f even 4 1 6272.2.a.b 1
40.e odd 2 1 3200.2.c.e 2
40.f even 2 1 3200.2.c.k 2
40.i odd 4 1 128.2.a.b yes 1
40.i odd 4 1 3200.2.a.u 1
40.k even 4 1 128.2.a.d yes 1
40.k even 4 1 3200.2.a.h 1
60.l odd 4 1 1152.2.a.m 1
80.i odd 4 1 256.2.b.a 2
80.j even 4 1 256.2.b.c 2
80.s even 4 1 256.2.b.c 2
80.t odd 4 1 256.2.b.a 2
120.q odd 4 1 1152.2.a.c 1
120.w even 4 1 1152.2.a.h 1
140.j odd 4 1 6272.2.a.h 1
160.u even 8 2 1024.2.e.i 4
160.v odd 8 2 1024.2.e.m 4
160.ba even 8 2 1024.2.e.i 4
160.bb odd 8 2 1024.2.e.m 4
240.z odd 4 1 2304.2.d.r 2
240.bb even 4 1 2304.2.d.b 2
240.bd odd 4 1 2304.2.d.r 2
240.bf even 4 1 2304.2.d.b 2
280.s even 4 1 6272.2.a.g 1
280.y odd 4 1 6272.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 20.e even 4 1
128.2.a.b yes 1 40.i odd 4 1
128.2.a.c yes 1 5.c odd 4 1
128.2.a.d yes 1 40.k even 4 1
256.2.b.a 2 80.i odd 4 1
256.2.b.a 2 80.t odd 4 1
256.2.b.c 2 80.j even 4 1
256.2.b.c 2 80.s even 4 1
1024.2.e.i 4 160.u even 8 2
1024.2.e.i 4 160.ba even 8 2
1024.2.e.m 4 160.v odd 8 2
1024.2.e.m 4 160.bb odd 8 2
1152.2.a.c 1 120.q odd 4 1
1152.2.a.h 1 120.w even 4 1
1152.2.a.m 1 60.l odd 4 1
1152.2.a.r 1 15.e even 4 1
2304.2.d.b 2 240.bb even 4 1
2304.2.d.b 2 240.bf even 4 1
2304.2.d.r 2 240.z odd 4 1
2304.2.d.r 2 240.bd odd 4 1
3200.2.a.e 1 5.c odd 4 1
3200.2.a.h 1 40.k even 4 1
3200.2.a.u 1 40.i odd 4 1
3200.2.a.x 1 20.e even 4 1
3200.2.c.e 2 8.d odd 2 1
3200.2.c.e 2 40.e odd 2 1
3200.2.c.f 2 1.a even 1 1 trivial
3200.2.c.f 2 5.b even 2 1 inner
3200.2.c.k 2 8.b even 2 1
3200.2.c.k 2 40.f even 2 1
3200.2.c.l 2 4.b odd 2 1
3200.2.c.l 2 20.d odd 2 1
6272.2.a.a 1 280.y odd 4 1
6272.2.a.b 1 35.f even 4 1
6272.2.a.g 1 280.s even 4 1
6272.2.a.h 1 140.j odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{7}^{2} + 16$$ $$T_{11} + 2$$ $$T_{29} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$16 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 14 + T )^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$100 + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$196 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$36 + T^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$4 + T^{2}$$