# Properties

 Label 1932.2.a.e Level $1932$ Weight $2$ Character orbit 1932.a Self dual yes Analytic conductor $15.427$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1932.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.4270976705$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} + q^{7} + q^{9}+O(q^{10})$$ q - q^3 + b * q^5 + q^7 + q^9 $$q - q^{3} + \beta q^{5} + q^{7} + q^{9} + ( - 2 \beta + 1) q^{11} + (\beta - 3) q^{13} - \beta q^{15} - 3 q^{17} + ( - 2 \beta - 3) q^{19} - q^{21} - q^{23} + (\beta - 2) q^{25} - q^{27} + q^{29} + q^{31} + (2 \beta - 1) q^{33} + \beta q^{35} + ( - 4 \beta + 1) q^{37} + ( - \beta + 3) q^{39} + ( - 2 \beta - 1) q^{41} + ( - 3 \beta - 1) q^{43} + \beta q^{45} + ( - 2 \beta - 6) q^{47} + q^{49} + 3 q^{51} + (3 \beta + 6) q^{53} + ( - \beta - 6) q^{55} + (2 \beta + 3) q^{57} + (3 \beta + 4) q^{59} + (5 \beta - 3) q^{61} + q^{63} + ( - 2 \beta + 3) q^{65} + (7 \beta - 6) q^{67} + q^{69} + ( - 3 \beta - 5) q^{71} + (6 \beta - 3) q^{73} + ( - \beta + 2) q^{75} + ( - 2 \beta + 1) q^{77} + (2 \beta - 11) q^{79} + q^{81} + (2 \beta - 5) q^{83} - 3 \beta q^{85} - q^{87} + (\beta + 1) q^{89} + (\beta - 3) q^{91} - q^{93} + ( - 5 \beta - 6) q^{95} - 11 q^{97} + ( - 2 \beta + 1) q^{99} +O(q^{100})$$ q - q^3 + b * q^5 + q^7 + q^9 + (-2*b + 1) * q^11 + (b - 3) * q^13 - b * q^15 - 3 * q^17 + (-2*b - 3) * q^19 - q^21 - q^23 + (b - 2) * q^25 - q^27 + q^29 + q^31 + (2*b - 1) * q^33 + b * q^35 + (-4*b + 1) * q^37 + (-b + 3) * q^39 + (-2*b - 1) * q^41 + (-3*b - 1) * q^43 + b * q^45 + (-2*b - 6) * q^47 + q^49 + 3 * q^51 + (3*b + 6) * q^53 + (-b - 6) * q^55 + (2*b + 3) * q^57 + (3*b + 4) * q^59 + (5*b - 3) * q^61 + q^63 + (-2*b + 3) * q^65 + (7*b - 6) * q^67 + q^69 + (-3*b - 5) * q^71 + (6*b - 3) * q^73 + (-b + 2) * q^75 + (-2*b + 1) * q^77 + (2*b - 11) * q^79 + q^81 + (2*b - 5) * q^83 - 3*b * q^85 - q^87 + (b + 1) * q^89 + (b - 3) * q^91 - q^93 + (-5*b - 6) * q^95 - 11 * q^97 + (-2*b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 5 q^{13} - q^{15} - 6 q^{17} - 8 q^{19} - 2 q^{21} - 2 q^{23} - 3 q^{25} - 2 q^{27} + 2 q^{29} + 2 q^{31} + q^{35} - 2 q^{37} + 5 q^{39} - 4 q^{41} - 5 q^{43} + q^{45} - 14 q^{47} + 2 q^{49} + 6 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} - q^{61} + 2 q^{63} + 4 q^{65} - 5 q^{67} + 2 q^{69} - 13 q^{71} + 3 q^{75} - 20 q^{79} + 2 q^{81} - 8 q^{83} - 3 q^{85} - 2 q^{87} + 3 q^{89} - 5 q^{91} - 2 q^{93} - 17 q^{95} - 22 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 - 5 * q^13 - q^15 - 6 * q^17 - 8 * q^19 - 2 * q^21 - 2 * q^23 - 3 * q^25 - 2 * q^27 + 2 * q^29 + 2 * q^31 + q^35 - 2 * q^37 + 5 * q^39 - 4 * q^41 - 5 * q^43 + q^45 - 14 * q^47 + 2 * q^49 + 6 * q^51 + 15 * q^53 - 13 * q^55 + 8 * q^57 + 11 * q^59 - q^61 + 2 * q^63 + 4 * q^65 - 5 * q^67 + 2 * q^69 - 13 * q^71 + 3 * q^75 - 20 * q^79 + 2 * q^81 - 8 * q^83 - 3 * q^85 - 2 * q^87 + 3 * q^89 - 5 * q^91 - 2 * q^93 - 17 * q^95 - 22 * q^97

## 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
 −1.30278 2.30278
0 −1.00000 0 −1.30278 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 2.30278 0 1.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$1$$

## 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 1932.2.a.e 2
3.b odd 2 1 5796.2.a.k 2
4.b odd 2 1 7728.2.a.bk 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.e 2 1.a even 1 1 trivial
5796.2.a.k 2 3.b odd 2 1
7728.2.a.bk 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1932))$$:

 $$T_{5}^{2} - T_{5} - 3$$ T5^2 - T5 - 3 $$T_{11}^{2} - 13$$ T11^2 - 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - T - 3$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 13$$
$13$ $$T^{2} + 5T + 3$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 8T + 3$$
$23$ $$(T + 1)^{2}$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 2T - 51$$
$41$ $$T^{2} + 4T - 9$$
$43$ $$T^{2} + 5T - 23$$
$47$ $$T^{2} + 14T + 36$$
$53$ $$T^{2} - 15T + 27$$
$59$ $$T^{2} - 11T + 1$$
$61$ $$T^{2} + T - 81$$
$67$ $$T^{2} + 5T - 153$$
$71$ $$T^{2} + 13T + 13$$
$73$ $$T^{2} - 117$$
$79$ $$T^{2} + 20T + 87$$
$83$ $$T^{2} + 8T + 3$$
$89$ $$T^{2} - 3T - 1$$
$97$ $$(T + 11)^{2}$$
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