# Properties

 Label 3200.2.c.ba Level $3200$ Weight $2$ Character orbit 3200.c Analytic conductor $25.552$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{3} + (2 \beta_{2} - 2 \beta_1) q^{7} - 2 \beta_{3} q^{9}+O(q^{10})$$ q + (b2 + b1) * q^3 + (2*b2 - 2*b1) * q^7 - 2*b3 * q^9 $$q + (\beta_{2} + \beta_1) q^{3} + (2 \beta_{2} - 2 \beta_1) q^{7} - 2 \beta_{3} q^{9} + (3 \beta_{3} + 1) q^{11} - 4 \beta_{2} q^{13} + ( - 2 \beta_{2} + 3 \beta_1) q^{17} + (\beta_{3} - 3) q^{19} - 2 q^{21} + (2 \beta_{2} + 2 \beta_1) q^{23} + (\beta_{2} - \beta_1) q^{27} - 8 q^{29} + (2 \beta_{3} - 2) q^{31} + (4 \beta_{2} + 7 \beta_1) q^{33} + (4 \beta_{2} + 2 \beta_1) q^{37} + (4 \beta_{3} + 8) q^{39} + (4 \beta_{3} + 5) q^{41} + 10 \beta_1 q^{43} + ( - 4 \beta_{2} - 4 \beta_1) q^{47} + (8 \beta_{3} - 5) q^{49} + ( - \beta_{3} + 1) q^{51} + (4 \beta_{2} + 2 \beta_1) q^{53} + ( - 2 \beta_{2} - \beta_1) q^{57} + (4 \beta_{3} - 2) q^{59} + 6 q^{61} + (4 \beta_{2} - 8 \beta_1) q^{63} + ( - 3 \beta_{2} + 7 \beta_1) q^{67} + ( - 4 \beta_{3} - 6) q^{69} + (4 \beta_{3} + 4) q^{71} + (2 \beta_{2} + 3 \beta_1) q^{73} + ( - 4 \beta_{2} + 10 \beta_1) q^{77} + (2 \beta_{3} + 10) q^{79} + ( - 6 \beta_{3} - 1) q^{81} + (3 \beta_{2} - 3 \beta_1) q^{83} + ( - 8 \beta_{2} - 8 \beta_1) q^{87} + (2 \beta_{3} - 9) q^{89} + ( - 8 \beta_{3} + 16) q^{91} + 2 \beta_1 q^{93} + (8 \beta_{2} + 6 \beta_1) q^{97} + ( - 2 \beta_{3} - 12) q^{99}+O(q^{100})$$ q + (b2 + b1) * q^3 + (2*b2 - 2*b1) * q^7 - 2*b3 * q^9 + (3*b3 + 1) * q^11 - 4*b2 * q^13 + (-2*b2 + 3*b1) * q^17 + (b3 - 3) * q^19 - 2 * q^21 + (2*b2 + 2*b1) * q^23 + (b2 - b1) * q^27 - 8 * q^29 + (2*b3 - 2) * q^31 + (4*b2 + 7*b1) * q^33 + (4*b2 + 2*b1) * q^37 + (4*b3 + 8) * q^39 + (4*b3 + 5) * q^41 + 10*b1 * q^43 + (-4*b2 - 4*b1) * q^47 + (8*b3 - 5) * q^49 + (-b3 + 1) * q^51 + (4*b2 + 2*b1) * q^53 + (-2*b2 - b1) * q^57 + (4*b3 - 2) * q^59 + 6 * q^61 + (4*b2 - 8*b1) * q^63 + (-3*b2 + 7*b1) * q^67 + (-4*b3 - 6) * q^69 + (4*b3 + 4) * q^71 + (2*b2 + 3*b1) * q^73 + (-4*b2 + 10*b1) * q^77 + (2*b3 + 10) * q^79 + (-6*b3 - 1) * q^81 + (3*b2 - 3*b1) * q^83 + (-8*b2 - 8*b1) * q^87 + (2*b3 - 9) * q^89 + (-8*b3 + 16) * q^91 + 2*b1 * q^93 + (8*b2 + 6*b1) * q^97 + (-2*b3 - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{11} - 12 q^{19} - 8 q^{21} - 32 q^{29} - 8 q^{31} + 32 q^{39} + 20 q^{41} - 20 q^{49} + 4 q^{51} - 8 q^{59} + 24 q^{61} - 24 q^{69} + 16 q^{71} + 40 q^{79} - 4 q^{81} - 36 q^{89} + 64 q^{91} - 48 q^{99}+O(q^{100})$$ 4 * q + 4 * q^11 - 12 * q^19 - 8 * q^21 - 32 * q^29 - 8 * q^31 + 32 * q^39 + 20 * q^41 - 20 * q^49 + 4 * q^51 - 8 * q^59 + 24 * q^61 - 24 * q^69 + 16 * q^71 + 40 * q^79 - 4 * q^81 - 36 * q^89 + 64 * q^91 - 48 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 2.41421i 0 0 0 0.828427i 0 −2.82843 0
2049.2 0 0.414214i 0 0 0 4.82843i 0 2.82843 0
2049.3 0 0.414214i 0 0 0 4.82843i 0 2.82843 0
2049.4 0 2.41421i 0 0 0 0.828427i 0 −2.82843 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.ba 4
4.b odd 2 1 3200.2.c.y 4
5.b even 2 1 inner 3200.2.c.ba 4
5.c odd 4 1 3200.2.a.bd yes 2
5.c odd 4 1 3200.2.a.bn yes 2
8.b even 2 1 3200.2.c.z 4
8.d odd 2 1 3200.2.c.bb 4
20.d odd 2 1 3200.2.c.y 4
20.e even 4 1 3200.2.a.bc 2
20.e even 4 1 3200.2.a.bm yes 2
40.e odd 2 1 3200.2.c.bb 4
40.f even 2 1 3200.2.c.z 4
40.i odd 4 1 3200.2.a.bg yes 2
40.i odd 4 1 3200.2.a.bi yes 2
40.k even 4 1 3200.2.a.bh yes 2
40.k even 4 1 3200.2.a.bj yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.a.bc 2 20.e even 4 1
3200.2.a.bd yes 2 5.c odd 4 1
3200.2.a.bg yes 2 40.i odd 4 1
3200.2.a.bh yes 2 40.k even 4 1
3200.2.a.bi yes 2 40.i odd 4 1
3200.2.a.bj yes 2 40.k even 4 1
3200.2.a.bm yes 2 20.e even 4 1
3200.2.a.bn yes 2 5.c odd 4 1
3200.2.c.y 4 4.b odd 2 1
3200.2.c.y 4 20.d odd 2 1
3200.2.c.z 4 8.b even 2 1
3200.2.c.z 4 40.f even 2 1
3200.2.c.ba 4 1.a even 1 1 trivial
3200.2.c.ba 4 5.b even 2 1 inner
3200.2.c.bb 4 8.d odd 2 1
3200.2.c.bb 4 40.e odd 2 1

## Hecke kernels

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

 $$T_{3}^{4} + 6T_{3}^{2} + 1$$ T3^4 + 6*T3^2 + 1 $$T_{7}^{4} + 24T_{7}^{2} + 16$$ T7^4 + 24*T7^2 + 16 $$T_{11}^{2} - 2T_{11} - 17$$ T11^2 - 2*T11 - 17 $$T_{29} + 8$$ T29 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 6T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 24T^{2} + 16$$
$11$ $$(T^{2} - 2 T - 17)^{2}$$
$13$ $$(T^{2} + 32)^{2}$$
$17$ $$T^{4} + 34T^{2} + 1$$
$19$ $$(T^{2} + 6 T + 7)^{2}$$
$23$ $$T^{4} + 24T^{2} + 16$$
$29$ $$(T + 8)^{4}$$
$31$ $$(T^{2} + 4 T - 4)^{2}$$
$37$ $$T^{4} + 72T^{2} + 784$$
$41$ $$(T^{2} - 10 T - 7)^{2}$$
$43$ $$(T^{2} + 100)^{2}$$
$47$ $$T^{4} + 96T^{2} + 256$$
$53$ $$T^{4} + 72T^{2} + 784$$
$59$ $$(T^{2} + 4 T - 28)^{2}$$
$61$ $$(T - 6)^{4}$$
$67$ $$T^{4} + 134T^{2} + 961$$
$71$ $$(T^{2} - 8 T - 16)^{2}$$
$73$ $$T^{4} + 34T^{2} + 1$$
$79$ $$(T^{2} - 20 T + 92)^{2}$$
$83$ $$T^{4} + 54T^{2} + 81$$
$89$ $$(T^{2} + 18 T + 73)^{2}$$
$97$ $$T^{4} + 328T^{2} + 8464$$