# Properties

 Label 3200.2.a.bg Level $3200$ Weight $2$ Character orbit 3200.a Self dual yes Analytic conductor $25.552$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} + (2 \beta + 2) q^{7} - 2 \beta q^{9} +O(q^{10})$$ q + (b - 1) * q^3 + (2*b + 2) * q^7 - 2*b * q^9 $$q + (\beta - 1) q^{3} + (2 \beta + 2) q^{7} - 2 \beta q^{9} + (3 \beta - 1) q^{11} - 4 \beta q^{13} + ( - 2 \beta - 3) q^{17} + ( - \beta - 3) q^{19} + 2 q^{21} + ( - 2 \beta + 2) q^{23} + ( - \beta - 1) q^{27} - 8 q^{29} + ( - 2 \beta - 2) q^{31} + ( - 4 \beta + 7) q^{33} + ( - 4 \beta + 2) q^{37} + (4 \beta - 8) q^{39} + ( - 4 \beta + 5) q^{41} - 10 q^{43} + ( - 4 \beta + 4) q^{47} + (8 \beta + 5) q^{49} + ( - \beta - 1) q^{51} + (4 \beta - 2) q^{53} + ( - 2 \beta + 1) q^{57} + ( - 4 \beta - 2) q^{59} - 6 q^{61} + ( - 4 \beta - 8) q^{63} + (3 \beta + 7) q^{67} + (4 \beta - 6) q^{69} + ( - 4 \beta + 4) q^{71} + ( - 2 \beta + 3) q^{73} + (4 \beta + 10) q^{77} + (2 \beta - 10) q^{79} + (6 \beta - 1) q^{81} + (3 \beta + 3) q^{83} + ( - 8 \beta + 8) q^{87} + (2 \beta + 9) q^{89} + ( - 8 \beta - 16) q^{91} - 2 q^{93} + (8 \beta - 6) q^{97} + (2 \beta - 12) q^{99} +O(q^{100})$$ q + (b - 1) * q^3 + (2*b + 2) * q^7 - 2*b * q^9 + (3*b - 1) * q^11 - 4*b * q^13 + (-2*b - 3) * q^17 + (-b - 3) * q^19 + 2 * q^21 + (-2*b + 2) * q^23 + (-b - 1) * q^27 - 8 * q^29 + (-2*b - 2) * q^31 + (-4*b + 7) * q^33 + (-4*b + 2) * q^37 + (4*b - 8) * q^39 + (-4*b + 5) * q^41 - 10 * q^43 + (-4*b + 4) * q^47 + (8*b + 5) * q^49 + (-b - 1) * q^51 + (4*b - 2) * q^53 + (-2*b + 1) * q^57 + (-4*b - 2) * q^59 - 6 * q^61 + (-4*b - 8) * q^63 + (3*b + 7) * q^67 + (4*b - 6) * q^69 + (-4*b + 4) * q^71 + (-2*b + 3) * q^73 + (4*b + 10) * q^77 + (2*b - 10) * q^79 + (6*b - 1) * q^81 + (3*b + 3) * q^83 + (-8*b + 8) * q^87 + (2*b + 9) * q^89 + (-8*b - 16) * q^91 - 2 * q^93 + (8*b - 6) * q^97 + (2*b - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 4 q^{7}+O(q^{10})$$ 2 * q - 2 * q^3 + 4 * q^7 $$2 q - 2 q^{3} + 4 q^{7} - 2 q^{11} - 6 q^{17} - 6 q^{19} + 4 q^{21} + 4 q^{23} - 2 q^{27} - 16 q^{29} - 4 q^{31} + 14 q^{33} + 4 q^{37} - 16 q^{39} + 10 q^{41} - 20 q^{43} + 8 q^{47} + 10 q^{49} - 2 q^{51} - 4 q^{53} + 2 q^{57} - 4 q^{59} - 12 q^{61} - 16 q^{63} + 14 q^{67} - 12 q^{69} + 8 q^{71} + 6 q^{73} + 20 q^{77} - 20 q^{79} - 2 q^{81} + 6 q^{83} + 16 q^{87} + 18 q^{89} - 32 q^{91} - 4 q^{93} - 12 q^{97} - 24 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 4 * q^7 - 2 * q^11 - 6 * q^17 - 6 * q^19 + 4 * q^21 + 4 * q^23 - 2 * q^27 - 16 * q^29 - 4 * q^31 + 14 * q^33 + 4 * q^37 - 16 * q^39 + 10 * q^41 - 20 * q^43 + 8 * q^47 + 10 * q^49 - 2 * q^51 - 4 * q^53 + 2 * q^57 - 4 * q^59 - 12 * q^61 - 16 * q^63 + 14 * q^67 - 12 * q^69 + 8 * q^71 + 6 * q^73 + 20 * q^77 - 20 * q^79 - 2 * q^81 + 6 * q^83 + 16 * q^87 + 18 * q^89 - 32 * q^91 - 4 * q^93 - 12 * q^97 - 24 * 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
 −1.41421 1.41421
0 −2.41421 0 0 0 −0.828427 0 2.82843 0
1.2 0 0.414214 0 0 0 4.82843 0 −2.82843 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.bg yes 2
4.b odd 2 1 3200.2.a.bj yes 2
5.b even 2 1 3200.2.a.bi yes 2
5.c odd 4 2 3200.2.c.z 4
8.b even 2 1 3200.2.a.bn yes 2
8.d odd 2 1 3200.2.a.bc 2
20.d odd 2 1 3200.2.a.bh yes 2
20.e even 4 2 3200.2.c.bb 4
40.e odd 2 1 3200.2.a.bm yes 2
40.f even 2 1 3200.2.a.bd yes 2
40.i odd 4 2 3200.2.c.ba 4
40.k even 4 2 3200.2.c.y 4

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

## Hecke kernels

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

 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1 $$T_{7}^{2} - 4T_{7} - 4$$ T7^2 - 4*T7 - 4 $$T_{11}^{2} + 2T_{11} - 17$$ T11^2 + 2*T11 - 17 $$T_{13}^{2} - 32$$ T13^2 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T - 4$$
$11$ $$T^{2} + 2T - 17$$
$13$ $$T^{2} - 32$$
$17$ $$T^{2} + 6T + 1$$
$19$ $$T^{2} + 6T + 7$$
$23$ $$T^{2} - 4T - 4$$
$29$ $$(T + 8)^{2}$$
$31$ $$T^{2} + 4T - 4$$
$37$ $$T^{2} - 4T - 28$$
$41$ $$T^{2} - 10T - 7$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} - 8T - 16$$
$53$ $$T^{2} + 4T - 28$$
$59$ $$T^{2} + 4T - 28$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} - 14T + 31$$
$71$ $$T^{2} - 8T - 16$$
$73$ $$T^{2} - 6T + 1$$
$79$ $$T^{2} + 20T + 92$$
$83$ $$T^{2} - 6T - 9$$
$89$ $$T^{2} - 18T + 73$$
$97$ $$T^{2} + 12T - 92$$