Properties

 Label 3200.2.f.o Level $3200$ Weight $2$ Character orbit 3200.f Analytic conductor $25.552$ Analytic rank $0$ Dimension $4$ CM discriminant -8 Inner twists $8$

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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 128) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{3} + 5 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{3} + 5 q^{9} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} -6 \zeta_{8}^{2} q^{17} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{19} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} -8 \zeta_{8}^{2} q^{33} -6 q^{41} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{43} + 7 q^{49} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{51} -24 \zeta_{8}^{2} q^{57} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{59} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{67} -2 \zeta_{8}^{2} q^{73} + q^{81} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{83} + 18 q^{89} + 10 \zeta_{8}^{2} q^{97} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 20 q^{9} + O(q^{10})$$ $$4 q + 20 q^{9} - 24 q^{41} + 28 q^{49} + 4 q^{81} + 72 q^{89} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0 −2.82843 0 0 0 0 0 5.00000 0
449.2 0 −2.82843 0 0 0 0 0 5.00000 0
449.3 0 2.82843 0 0 0 0 0 5.00000 0
449.4 0 2.82843 0 0 0 0 0 5.00000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
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.o 4
4.b odd 2 1 inner 3200.2.f.o 4
5.b even 2 1 inner 3200.2.f.o 4
5.c odd 4 1 128.2.b.a 2
5.c odd 4 1 3200.2.d.c 2
8.b even 2 1 inner 3200.2.f.o 4
8.d odd 2 1 CM 3200.2.f.o 4
15.e even 4 1 1152.2.d.c 2
20.d odd 2 1 inner 3200.2.f.o 4
20.e even 4 1 128.2.b.a 2
20.e even 4 1 3200.2.d.c 2
40.e odd 2 1 inner 3200.2.f.o 4
40.f even 2 1 inner 3200.2.f.o 4
40.i odd 4 1 128.2.b.a 2
40.i odd 4 1 3200.2.d.c 2
40.k even 4 1 128.2.b.a 2
40.k even 4 1 3200.2.d.c 2
60.l odd 4 1 1152.2.d.c 2
80.i odd 4 1 256.2.a.e 2
80.i odd 4 1 6400.2.a.by 2
80.j even 4 1 256.2.a.e 2
80.j even 4 1 6400.2.a.by 2
80.s even 4 1 256.2.a.e 2
80.s even 4 1 6400.2.a.by 2
80.t odd 4 1 256.2.a.e 2
80.t odd 4 1 6400.2.a.by 2
120.q odd 4 1 1152.2.d.c 2
120.w even 4 1 1152.2.d.c 2
160.u even 8 1 1024.2.e.a 2
160.u even 8 1 1024.2.e.f 2
160.v odd 8 1 1024.2.e.a 2
160.v odd 8 1 1024.2.e.f 2
160.ba even 8 1 1024.2.e.a 2
160.ba even 8 1 1024.2.e.f 2
160.bb odd 8 1 1024.2.e.a 2
160.bb odd 8 1 1024.2.e.f 2
240.z odd 4 1 2304.2.a.t 2
240.bb even 4 1 2304.2.a.t 2
240.bd odd 4 1 2304.2.a.t 2
240.bf even 4 1 2304.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 5.c odd 4 1
128.2.b.a 2 20.e even 4 1
128.2.b.a 2 40.i odd 4 1
128.2.b.a 2 40.k even 4 1
256.2.a.e 2 80.i odd 4 1
256.2.a.e 2 80.j even 4 1
256.2.a.e 2 80.s even 4 1
256.2.a.e 2 80.t odd 4 1
1024.2.e.a 2 160.u even 8 1
1024.2.e.a 2 160.v odd 8 1
1024.2.e.a 2 160.ba even 8 1
1024.2.e.a 2 160.bb odd 8 1
1024.2.e.f 2 160.u even 8 1
1024.2.e.f 2 160.v odd 8 1
1024.2.e.f 2 160.ba even 8 1
1024.2.e.f 2 160.bb odd 8 1
1152.2.d.c 2 15.e even 4 1
1152.2.d.c 2 60.l odd 4 1
1152.2.d.c 2 120.q odd 4 1
1152.2.d.c 2 120.w even 4 1
2304.2.a.t 2 240.z odd 4 1
2304.2.a.t 2 240.bb even 4 1
2304.2.a.t 2 240.bd odd 4 1
2304.2.a.t 2 240.bf even 4 1
3200.2.d.c 2 5.c odd 4 1
3200.2.d.c 2 20.e even 4 1
3200.2.d.c 2 40.i odd 4 1
3200.2.d.c 2 40.k even 4 1
3200.2.f.o 4 1.a even 1 1 trivial
3200.2.f.o 4 4.b odd 2 1 inner
3200.2.f.o 4 5.b even 2 1 inner
3200.2.f.o 4 8.b even 2 1 inner
3200.2.f.o 4 8.d odd 2 1 CM
3200.2.f.o 4 20.d odd 2 1 inner
3200.2.f.o 4 40.e odd 2 1 inner
3200.2.f.o 4 40.f even 2 1 inner
6400.2.a.by 2 80.i odd 4 1
6400.2.a.by 2 80.j even 4 1
6400.2.a.by 2 80.s even 4 1
6400.2.a.by 2 80.t odd 4 1

Hecke kernels

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

 $$T_{3}^{2} - 8$$ $$T_{7}$$ $$T_{11}^{2} + 8$$ $$T_{13}$$ $$T_{31}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -8 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 8 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 36 + T^{2} )^{2}$$
$19$ $$( 72 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 6 + T )^{4}$$
$43$ $$( -72 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$( 200 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( -72 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 4 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$( -8 + T^{2} )^{2}$$
$89$ $$( -18 + T )^{4}$$
$97$ $$( 100 + T^{2} )^{2}$$