# Properties

 Label 3200.2.a.bl Level $3200$ Weight $2$ Character orbit 3200.a Self dual yes Analytic conductor $25.552$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 640) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + ( - \beta + 1) q^{7} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (-b + 1) * q^7 + (2*b + 3) * q^9 $$q + (\beta + 1) q^{3} + ( - \beta + 1) q^{7} + (2 \beta + 3) q^{9} - 2 q^{11} - 2 \beta q^{13} + 2 \beta q^{17} + 2 \beta q^{19} - 4 q^{21} + (\beta + 7) q^{23} + (2 \beta + 10) q^{27} - 2 q^{29} + ( - 2 \beta + 2) q^{31} + ( - 2 \beta - 2) q^{33} + (4 \beta + 2) q^{37} + ( - 2 \beta - 10) q^{39} + ( - 2 \beta + 8) q^{41} + ( - 3 \beta + 1) q^{43} + ( - \beta + 5) q^{47} + ( - 2 \beta - 1) q^{49} + (2 \beta + 10) q^{51} + (2 \beta + 4) q^{53} + (2 \beta + 10) q^{57} + (2 \beta - 4) q^{59} - 6 q^{61} + ( - \beta - 7) q^{63} + (3 \beta - 1) q^{67} + (8 \beta + 12) q^{69} + (2 \beta + 2) q^{71} + 2 \beta q^{73} + (2 \beta - 2) q^{77} + ( - 4 \beta + 4) q^{79} + (6 \beta + 11) q^{81} + ( - 3 \beta - 3) q^{83} + ( - 2 \beta - 2) q^{87} + (4 \beta - 6) q^{89} + ( - 2 \beta + 10) q^{91} - 8 q^{93} + ( - 2 \beta + 12) q^{97} + ( - 4 \beta - 6) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + (-b + 1) * q^7 + (2*b + 3) * q^9 - 2 * q^11 - 2*b * q^13 + 2*b * q^17 + 2*b * q^19 - 4 * q^21 + (b + 7) * q^23 + (2*b + 10) * q^27 - 2 * q^29 + (-2*b + 2) * q^31 + (-2*b - 2) * q^33 + (4*b + 2) * q^37 + (-2*b - 10) * q^39 + (-2*b + 8) * q^41 + (-3*b + 1) * q^43 + (-b + 5) * q^47 + (-2*b - 1) * q^49 + (2*b + 10) * q^51 + (2*b + 4) * q^53 + (2*b + 10) * q^57 + (2*b - 4) * q^59 - 6 * q^61 + (-b - 7) * q^63 + (3*b - 1) * q^67 + (8*b + 12) * q^69 + (2*b + 2) * q^71 + 2*b * q^73 + (2*b - 2) * q^77 + (-4*b + 4) * q^79 + (6*b + 11) * q^81 + (-3*b - 3) * q^83 + (-2*b - 2) * q^87 + (4*b - 6) * q^89 + (-2*b + 10) * q^91 - 8 * q^93 + (-2*b + 12) * q^97 + (-4*b - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^7 + 6 * q^9 $$2 q + 2 q^{3} + 2 q^{7} + 6 q^{9} - 4 q^{11} - 8 q^{21} + 14 q^{23} + 20 q^{27} - 4 q^{29} + 4 q^{31} - 4 q^{33} + 4 q^{37} - 20 q^{39} + 16 q^{41} + 2 q^{43} + 10 q^{47} - 2 q^{49} + 20 q^{51} + 8 q^{53} + 20 q^{57} - 8 q^{59} - 12 q^{61} - 14 q^{63} - 2 q^{67} + 24 q^{69} + 4 q^{71} - 4 q^{77} + 8 q^{79} + 22 q^{81} - 6 q^{83} - 4 q^{87} - 12 q^{89} + 20 q^{91} - 16 q^{93} + 24 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^7 + 6 * q^9 - 4 * q^11 - 8 * q^21 + 14 * q^23 + 20 * q^27 - 4 * q^29 + 4 * q^31 - 4 * q^33 + 4 * q^37 - 20 * q^39 + 16 * q^41 + 2 * q^43 + 10 * q^47 - 2 * q^49 + 20 * q^51 + 8 * q^53 + 20 * q^57 - 8 * q^59 - 12 * q^61 - 14 * q^63 - 2 * q^67 + 24 * q^69 + 4 * q^71 - 4 * q^77 + 8 * q^79 + 22 * q^81 - 6 * q^83 - 4 * q^87 - 12 * q^89 + 20 * q^91 - 16 * q^93 + 24 * q^97 - 12 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.618034 1.61803
0 −1.23607 0 0 0 3.23607 0 −1.47214 0
1.2 0 3.23607 0 0 0 −1.23607 0 7.47214 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.a.bl 2
4.b odd 2 1 3200.2.a.be 2
5.b even 2 1 640.2.a.i 2
5.c odd 4 2 3200.2.c.u 4
8.b even 2 1 3200.2.a.bf 2
8.d odd 2 1 3200.2.a.bk 2
15.d odd 2 1 5760.2.a.ch 2
20.d odd 2 1 640.2.a.k yes 2
20.e even 4 2 3200.2.c.w 4
40.e odd 2 1 640.2.a.j yes 2
40.f even 2 1 640.2.a.l yes 2
40.i odd 4 2 3200.2.c.x 4
40.k even 4 2 3200.2.c.v 4
60.h even 2 1 5760.2.a.ci 2
80.k odd 4 2 1280.2.d.k 4
80.q even 4 2 1280.2.d.m 4
120.i odd 2 1 5760.2.a.bw 2
120.m even 2 1 5760.2.a.cd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.a.i 2 5.b even 2 1
640.2.a.j yes 2 40.e odd 2 1
640.2.a.k yes 2 20.d odd 2 1
640.2.a.l yes 2 40.f even 2 1
1280.2.d.k 4 80.k odd 4 2
1280.2.d.m 4 80.q even 4 2
3200.2.a.be 2 4.b odd 2 1
3200.2.a.bf 2 8.b even 2 1
3200.2.a.bk 2 8.d odd 2 1
3200.2.a.bl 2 1.a even 1 1 trivial
3200.2.c.u 4 5.c odd 4 2
3200.2.c.v 4 40.k even 4 2
3200.2.c.w 4 20.e even 4 2
3200.2.c.x 4 40.i odd 4 2
5760.2.a.bw 2 120.i odd 2 1
5760.2.a.cd 2 120.m even 2 1
5760.2.a.ch 2 15.d odd 2 1
5760.2.a.ci 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3200))$$:

 $$T_{3}^{2} - 2T_{3} - 4$$ T3^2 - 2*T3 - 4 $$T_{7}^{2} - 2T_{7} - 4$$ T7^2 - 2*T7 - 4 $$T_{11} + 2$$ T11 + 2 $$T_{13}^{2} - 20$$ T13^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 2T - 4$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} - 20$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} - 20$$
$23$ $$T^{2} - 14T + 44$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} - 4T - 16$$
$37$ $$T^{2} - 4T - 76$$
$41$ $$T^{2} - 16T + 44$$
$43$ $$T^{2} - 2T - 44$$
$47$ $$T^{2} - 10T + 20$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} + 8T - 4$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} + 2T - 44$$
$71$ $$T^{2} - 4T - 16$$
$73$ $$T^{2} - 20$$
$79$ $$T^{2} - 8T - 64$$
$83$ $$T^{2} + 6T - 36$$
$89$ $$T^{2} + 12T - 44$$
$97$ $$T^{2} - 24T + 124$$