# Properties

 Label 1932.2.a.h Level $1932$ Weight $2$ Character orbit 1932.a Self dual yes Analytic conductor $15.427$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + (3 \beta - 1) q^{5} + q^{7} + q^{9}+O(q^{10})$$ q + q^3 + (3*b - 1) * q^5 + q^7 + q^9 $$q + q^{3} + (3 \beta - 1) q^{5} + q^{7} + q^{9} + 3 q^{11} + (\beta - 2) q^{13} + (3 \beta - 1) q^{15} + ( - 4 \beta + 5) q^{17} + (2 \beta - 3) q^{19} + q^{21} - q^{23} + (3 \beta + 5) q^{25} + q^{27} - 5 q^{29} + ( - 2 \beta + 9) q^{31} + 3 q^{33} + (3 \beta - 1) q^{35} + ( - 2 \beta - 1) q^{37} + (\beta - 2) q^{39} + 3 q^{41} - \beta q^{43} + (3 \beta - 1) q^{45} - 2 \beta q^{47} + q^{49} + ( - 4 \beta + 5) q^{51} + ( - 7 \beta + 9) q^{53} + (9 \beta - 3) q^{55} + (2 \beta - 3) q^{57} + ( - 9 \beta + 1) q^{59} - 3 \beta q^{61} + q^{63} + ( - 4 \beta + 5) q^{65} + (7 \beta - 9) q^{67} - q^{69} + (\beta + 6) q^{71} + ( - 8 \beta + 7) q^{73} + (3 \beta + 5) q^{75} + 3 q^{77} + ( - 4 \beta - 3) q^{79} + q^{81} + (2 \beta + 9) q^{83} + (7 \beta - 17) q^{85} - 5 q^{87} + ( - 9 \beta + 8) q^{89} + (\beta - 2) q^{91} + ( - 2 \beta + 9) q^{93} + ( - 5 \beta + 9) q^{95} + ( - 8 \beta + 11) q^{97} + 3 q^{99} +O(q^{100})$$ q + q^3 + (3*b - 1) * q^5 + q^7 + q^9 + 3 * q^11 + (b - 2) * q^13 + (3*b - 1) * q^15 + (-4*b + 5) * q^17 + (2*b - 3) * q^19 + q^21 - q^23 + (3*b + 5) * q^25 + q^27 - 5 * q^29 + (-2*b + 9) * q^31 + 3 * q^33 + (3*b - 1) * q^35 + (-2*b - 1) * q^37 + (b - 2) * q^39 + 3 * q^41 - b * q^43 + (3*b - 1) * q^45 - 2*b * q^47 + q^49 + (-4*b + 5) * q^51 + (-7*b + 9) * q^53 + (9*b - 3) * q^55 + (2*b - 3) * q^57 + (-9*b + 1) * q^59 - 3*b * q^61 + q^63 + (-4*b + 5) * q^65 + (7*b - 9) * q^67 - q^69 + (b + 6) * q^71 + (-8*b + 7) * q^73 + (3*b + 5) * q^75 + 3 * q^77 + (-4*b - 3) * q^79 + q^81 + (2*b + 9) * q^83 + (7*b - 17) * q^85 - 5 * q^87 + (-9*b + 8) * q^89 + (b - 2) * q^91 + (-2*b + 9) * q^93 + (-5*b + 9) * q^95 + (-8*b + 11) * q^97 + 3 * 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 $$2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} - 2 q^{23} + 13 q^{25} + 2 q^{27} - 10 q^{29} + 16 q^{31} + 6 q^{33} + q^{35} - 4 q^{37} - 3 q^{39} + 6 q^{41} - q^{43} + q^{45} - 2 q^{47} + 2 q^{49} + 6 q^{51} + 11 q^{53} + 3 q^{55} - 4 q^{57} - 7 q^{59} - 3 q^{61} + 2 q^{63} + 6 q^{65} - 11 q^{67} - 2 q^{69} + 13 q^{71} + 6 q^{73} + 13 q^{75} + 6 q^{77} - 10 q^{79} + 2 q^{81} + 20 q^{83} - 27 q^{85} - 10 q^{87} + 7 q^{89} - 3 q^{91} + 16 q^{93} + 13 q^{95} + 14 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 3 * q^13 + q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 - 2 * q^23 + 13 * q^25 + 2 * q^27 - 10 * q^29 + 16 * q^31 + 6 * q^33 + q^35 - 4 * q^37 - 3 * q^39 + 6 * q^41 - q^43 + q^45 - 2 * q^47 + 2 * q^49 + 6 * q^51 + 11 * q^53 + 3 * q^55 - 4 * q^57 - 7 * q^59 - 3 * q^61 + 2 * q^63 + 6 * q^65 - 11 * q^67 - 2 * q^69 + 13 * q^71 + 6 * q^73 + 13 * q^75 + 6 * q^77 - 10 * q^79 + 2 * q^81 + 20 * q^83 - 27 * q^85 - 10 * q^87 + 7 * q^89 - 3 * q^91 + 16 * q^93 + 13 * q^95 + 14 * q^97 + 6 * 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 1.00000 0 −2.85410 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 3.85410 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.h 2
3.b odd 2 1 5796.2.a.j 2
4.b odd 2 1 7728.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.h 2 1.a even 1 1 trivial
5796.2.a.j 2 3.b odd 2 1
7728.2.a.bb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1932))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - T - 11$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 3T + 1$$
$17$ $$T^{2} - 6T - 11$$
$19$ $$T^{2} + 4T - 1$$
$23$ $$(T + 1)^{2}$$
$29$ $$(T + 5)^{2}$$
$31$ $$T^{2} - 16T + 59$$
$37$ $$T^{2} + 4T - 1$$
$41$ $$(T - 3)^{2}$$
$43$ $$T^{2} + T - 1$$
$47$ $$T^{2} + 2T - 4$$
$53$ $$T^{2} - 11T - 31$$
$59$ $$T^{2} + 7T - 89$$
$61$ $$T^{2} + 3T - 9$$
$67$ $$T^{2} + 11T - 31$$
$71$ $$T^{2} - 13T + 41$$
$73$ $$T^{2} - 6T - 71$$
$79$ $$T^{2} + 10T + 5$$
$83$ $$T^{2} - 20T + 95$$
$89$ $$T^{2} - 7T - 89$$
$97$ $$T^{2} - 14T - 31$$