# Properties

 Label 3648.2.a.br Level $3648$ Weight $2$ Character orbit 3648.a Self dual yes Analytic conductor $29.129$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3648 = 2^{6} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3648.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.1294266574$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 456) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta q^{5} - \beta q^{7} + q^{9} +O(q^{10})$$ q + q^3 - b * q^5 - b * q^7 + q^9 $$q + q^{3} - \beta q^{5} - \beta q^{7} + q^{9} + ( - \beta + 4) q^{11} + 2 \beta q^{13} - \beta q^{15} + ( - 3 \beta + 2) q^{17} + q^{19} - \beta q^{21} + (2 \beta - 6) q^{23} + (\beta - 1) q^{25} + q^{27} + ( - 4 \beta + 2) q^{29} + 2 q^{31} + ( - \beta + 4) q^{33} + (\beta + 4) q^{35} + 8 q^{37} + 2 \beta q^{39} + (2 \beta - 2) q^{41} + \beta q^{43} - \beta q^{45} + ( - 3 \beta + 2) q^{47} + (\beta - 3) q^{49} + ( - 3 \beta + 2) q^{51} + (4 \beta + 2) q^{53} + ( - 3 \beta + 4) q^{55} + q^{57} + 12 q^{59} + (3 \beta - 2) q^{61} - \beta q^{63} + ( - 2 \beta - 8) q^{65} - 4 \beta q^{67} + (2 \beta - 6) q^{69} + (7 \beta - 6) q^{73} + (\beta - 1) q^{75} + ( - 3 \beta + 4) q^{77} + (6 \beta - 2) q^{79} + q^{81} + 4 q^{83} + (\beta + 12) q^{85} + ( - 4 \beta + 2) q^{87} - 6 q^{89} + ( - 2 \beta - 8) q^{91} + 2 q^{93} - \beta q^{95} + ( - 4 \beta - 2) q^{97} + ( - \beta + 4) q^{99} +O(q^{100})$$ q + q^3 - b * q^5 - b * q^7 + q^9 + (-b + 4) * q^11 + 2*b * q^13 - b * q^15 + (-3*b + 2) * q^17 + q^19 - b * q^21 + (2*b - 6) * q^23 + (b - 1) * q^25 + q^27 + (-4*b + 2) * q^29 + 2 * q^31 + (-b + 4) * q^33 + (b + 4) * q^35 + 8 * q^37 + 2*b * q^39 + (2*b - 2) * q^41 + b * q^43 - b * q^45 + (-3*b + 2) * q^47 + (b - 3) * q^49 + (-3*b + 2) * q^51 + (4*b + 2) * q^53 + (-3*b + 4) * q^55 + q^57 + 12 * q^59 + (3*b - 2) * q^61 - b * q^63 + (-2*b - 8) * q^65 - 4*b * q^67 + (2*b - 6) * q^69 + (7*b - 6) * q^73 + (b - 1) * q^75 + (-3*b + 4) * q^77 + (6*b - 2) * q^79 + q^81 + 4 * q^83 + (b + 12) * q^85 + (-4*b + 2) * q^87 - 6 * q^89 + (-2*b - 8) * q^91 + 2 * q^93 - b * q^95 + (-4*b - 2) * q^97 + (-b + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - q^{5} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - q^5 - q^7 + 2 * q^9 $$2 q + 2 q^{3} - q^{5} - q^{7} + 2 q^{9} + 7 q^{11} + 2 q^{13} - q^{15} + q^{17} + 2 q^{19} - q^{21} - 10 q^{23} - q^{25} + 2 q^{27} + 4 q^{31} + 7 q^{33} + 9 q^{35} + 16 q^{37} + 2 q^{39} - 2 q^{41} + q^{43} - q^{45} + q^{47} - 5 q^{49} + q^{51} + 8 q^{53} + 5 q^{55} + 2 q^{57} + 24 q^{59} - q^{61} - q^{63} - 18 q^{65} - 4 q^{67} - 10 q^{69} - 5 q^{73} - q^{75} + 5 q^{77} + 2 q^{79} + 2 q^{81} + 8 q^{83} + 25 q^{85} - 12 q^{89} - 18 q^{91} + 4 q^{93} - q^{95} - 8 q^{97} + 7 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - q^5 - q^7 + 2 * q^9 + 7 * q^11 + 2 * q^13 - q^15 + q^17 + 2 * q^19 - q^21 - 10 * q^23 - q^25 + 2 * q^27 + 4 * q^31 + 7 * q^33 + 9 * q^35 + 16 * q^37 + 2 * q^39 - 2 * q^41 + q^43 - q^45 + q^47 - 5 * q^49 + q^51 + 8 * q^53 + 5 * q^55 + 2 * q^57 + 24 * q^59 - q^61 - q^63 - 18 * q^65 - 4 * q^67 - 10 * q^69 - 5 * q^73 - q^75 + 5 * q^77 + 2 * q^79 + 2 * q^81 + 8 * q^83 + 25 * q^85 - 12 * q^89 - 18 * q^91 + 4 * q^93 - q^95 - 8 * q^97 + 7 * 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
 2.56155 −1.56155
0 1.00000 0 −2.56155 0 −2.56155 0 1.00000 0
1.2 0 1.00000 0 1.56155 0 1.56155 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

## 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 3648.2.a.br 2
4.b odd 2 1 3648.2.a.bl 2
8.b even 2 1 912.2.a.m 2
8.d odd 2 1 456.2.a.f 2
24.f even 2 1 1368.2.a.k 2
24.h odd 2 1 2736.2.a.z 2
152.b even 2 1 8664.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.a.f 2 8.d odd 2 1
912.2.a.m 2 8.b even 2 1
1368.2.a.k 2 24.f even 2 1
2736.2.a.z 2 24.h odd 2 1
3648.2.a.bl 2 4.b odd 2 1
3648.2.a.br 2 1.a even 1 1 trivial
8664.2.a.r 2 152.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3648))$$:

 $$T_{5}^{2} + T_{5} - 4$$ T5^2 + T5 - 4 $$T_{7}^{2} + T_{7} - 4$$ T7^2 + T7 - 4 $$T_{11}^{2} - 7T_{11} + 8$$ T11^2 - 7*T11 + 8 $$T_{23}^{2} + 10T_{23} + 8$$ T23^2 + 10*T23 + 8 $$T_{31} - 2$$ T31 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + T - 4$$
$7$ $$T^{2} + T - 4$$
$11$ $$T^{2} - 7T + 8$$
$13$ $$T^{2} - 2T - 16$$
$17$ $$T^{2} - T - 38$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 10T + 8$$
$29$ $$T^{2} - 68$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} + 2T - 16$$
$43$ $$T^{2} - T - 4$$
$47$ $$T^{2} - T - 38$$
$53$ $$T^{2} - 8T - 52$$
$59$ $$(T - 12)^{2}$$
$61$ $$T^{2} + T - 38$$
$67$ $$T^{2} + 4T - 64$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 5T - 202$$
$79$ $$T^{2} - 2T - 152$$
$83$ $$(T - 4)^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 8T - 52$$