# Properties

 Label 644.2.a.d Level $644$ Weight $2$ Character orbit 644.a Self dual yes Analytic conductor $5.142$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$644 = 2^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 644.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.14236589017$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.6963152.1 Defining polynomial: $$x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10$$ x^5 - 2*x^4 - 10*x^3 + 10*x^2 + 29*x + 10 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2} + 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^3 + (b3 - b2 + 1) * q^5 - q^7 + (b2 - b1 + 2) * q^9 $$q + ( - \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2} + 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + (\beta_{4} - \beta_1 + 1) q^{11} + ( - \beta_{4} + \beta_{2} + 2) q^{13} + ( - \beta_{4} - 2 \beta_{2} + \beta_1 + 1) q^{15} + (\beta_{4} - \beta_{2} + \beta_1 + 1) q^{17} + ( - 2 \beta_{3} + 2) q^{19} + (\beta_1 - 1) q^{21} - q^{23} + ( - \beta_{4} - \beta_1 + 4) q^{25} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 + 2) q^{27} + (2 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{29} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - 1) q^{31} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{33} + ( - \beta_{3} + \beta_{2} - 1) q^{35} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{37} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 2) q^{39} + (\beta_{4} - \beta_{2} + 2 \beta_1) q^{41} + (2 \beta_{3} + 2 \beta_1 - 2) q^{43} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{45} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{47} + q^{49} + (\beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{51} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{53} + (\beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 2) q^{55} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_1 + 4) q^{57} + ( - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{59} + (\beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{61} + ( - \beta_{2} + \beta_1 - 2) q^{63} + ( - 2 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{65} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{67} + (\beta_1 - 1) q^{69} + ( - \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 2) q^{71} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{73} + ( - \beta_{4} + 3 \beta_{3} - 4 \beta_1 + 9) q^{75} + ( - \beta_{4} + \beta_1 - 1) q^{77} + (\beta_{4} + 2 \beta_{2} - 3 \beta_1 - 1) q^{79} + (\beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 5 \beta_1 - 2) q^{81} + (2 \beta_{3} + 2 \beta_{2} - 6) q^{83} + (3 \beta_{4} - 2 \beta_{3} + \beta_1 + 3) q^{85} + ( - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{87} + (\beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{89} + (\beta_{4} - \beta_{2} - 2) q^{91} + ( - 6 \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 5) q^{93} + (2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 4) q^{95} + (\beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{97} + (\beta_{4} - 2 \beta_{3} + 6 \beta_{2} - 5 \beta_1 + 5) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^3 + (b3 - b2 + 1) * q^5 - q^7 + (b2 - b1 + 2) * q^9 + (b4 - b1 + 1) * q^11 + (-b4 + b2 + 2) * q^13 + (-b4 - 2*b2 + b1 + 1) * q^15 + (b4 - b2 + b1 + 1) * q^17 + (-2*b3 + 2) * q^19 + (b1 - 1) * q^21 - q^23 + (-b4 - b1 + 4) * q^25 + (-b3 + 3*b2 - b1 + 2) * q^27 + (2*b3 - b2 + b1 + 3) * q^29 + (b4 + b3 + 2*b2 - 1) * q^31 + (b4 - 3*b3 + 2*b2 - b1 + 4) * q^33 + (-b3 + b2 - 1) * q^35 + (-2*b3 + 2*b1 - 2) * q^37 + (-b4 + 2*b3 + b2 - 4*b1 + 2) * q^39 + (b4 - b2 + 2*b1) * q^41 + (2*b3 + 2*b1 - 2) * q^43 + (-b4 + 2*b3 - 3*b2 + 3*b1 - 3) * q^45 + (-b4 + b3 + 2*b2 - 2*b1 + 1) * q^47 + q^49 + (b4 - 2*b3 - 2*b2 + b1 - 3) * q^51 + (-2*b3 - 2*b2 + 2*b1 - 2) * q^53 + (b4 - 3*b3 - 2*b2 + 5*b1 - 2) * q^55 + (2*b4 + 2*b3 - 2*b1 + 4) * q^57 + (-b4 - 2*b3 + b2 + b1 + 1) * q^59 + (b3 + b2 + 2*b1 + 3) * q^61 + (-b2 + b1 - 2) * q^63 + (-2*b4 + 6*b3 - 2*b2 - 2*b1) * q^65 + (-b4 + 2*b3 - 2*b2 - b1 - 1) * q^67 + (b1 - 1) * q^69 + (-b4 - 3*b2 + 2*b1 + 2) * q^71 + (-b4 + 2*b3 - b2 + 2*b1 - 2) * q^73 + (-b4 + 3*b3 - 4*b1 + 9) * q^75 + (-b4 + b1 - 1) * q^77 + (b4 + 2*b2 - 3*b1 - 1) * q^79 + (b4 - 2*b3 + 4*b2 - 5*b1 - 2) * q^81 + (2*b3 + 2*b2 - 6) * q^83 + (3*b4 - 2*b3 + b1 + 3) * q^85 + (-2*b4 - b3 - 3*b2 - b1 - 2) * q^87 + (b4 - 4*b3 + b2 + b1 + 5) * q^89 + (b4 - b2 - 2) * q^91 + (-6*b3 + 5*b2 - 3*b1 - 5) * q^93 + (2*b4 + 2*b3 - 4*b2 - 2*b1 - 4) * q^95 + (b3 - 3*b2 - 2*b1 + 3) * q^97 + (b4 - 2*b3 + 6*b2 - 5*b1 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10})$$ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9 $$5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9} + 2 q^{11} + 13 q^{13} + 4 q^{15} + 4 q^{17} + 12 q^{19} - 3 q^{21} - 5 q^{23} + 19 q^{25} + 15 q^{27} + 13 q^{29} - 3 q^{31} + 24 q^{33} - 2 q^{35} - 4 q^{37} + 3 q^{39} + q^{41} - 8 q^{43} - 16 q^{45} + 5 q^{47} + 5 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 12 q^{59} + 20 q^{61} - 10 q^{63} - 12 q^{65} - 12 q^{67} - 3 q^{69} + 9 q^{71} - 9 q^{73} + 35 q^{75} - 2 q^{77} - 8 q^{79} - 11 q^{81} - 28 q^{83} + 16 q^{85} - 15 q^{87} + 32 q^{89} - 13 q^{91} - 15 q^{93} - 36 q^{95} + 4 q^{97} + 28 q^{99}+O(q^{100})$$ 5 * q + 3 * q^3 + 2 * q^5 - 5 * q^7 + 10 * q^9 + 2 * q^11 + 13 * q^13 + 4 * q^15 + 4 * q^17 + 12 * q^19 - 3 * q^21 - 5 * q^23 + 19 * q^25 + 15 * q^27 + 13 * q^29 - 3 * q^31 + 24 * q^33 - 2 * q^35 - 4 * q^37 + 3 * q^39 + q^41 - 8 * q^43 - 16 * q^45 + 5 * q^47 + 5 * q^49 - 16 * q^51 - 8 * q^53 - 2 * q^55 + 12 * q^57 + 12 * q^59 + 20 * q^61 - 10 * q^63 - 12 * q^65 - 12 * q^67 - 3 * q^69 + 9 * q^71 - 9 * q^73 + 35 * q^75 - 2 * q^77 - 8 * q^79 - 11 * q^81 - 28 * q^83 + 16 * q^85 - 15 * q^87 + 32 * q^89 - 13 * q^91 - 15 * q^93 - 36 * q^95 + 4 * q^97 + 28 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 7\nu - 5$$ v^3 - 7*v - 5 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 2\nu^{3} - 7\nu^{2} + 9\nu + 9$$ v^4 - 2*v^3 - 7*v^2 + 9*v + 9
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 7\beta _1 + 5$$ b3 + 7*b1 + 5 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 2\beta_{3} + 7\beta_{2} + 12\beta _1 + 29$$ b4 + 2*b3 + 7*b2 + 12*b1 + 29

## 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
 3.11181 2.76321 −0.435854 −1.18229 −2.25688
0 −2.11181 0 1.77860 0 −1.00000 0 1.45975 0
1.2 0 −1.76321 0 −3.11657 0 −1.00000 0 0.108911 0
1.3 0 1.43585 0 2.34236 0 −1.00000 0 −0.938323 0
1.4 0 2.18229 0 4.04332 0 −1.00000 0 1.76237 0
1.5 0 3.25688 0 −3.04771 0 −1.00000 0 7.60729 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$1$$

## 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 644.2.a.d 5
3.b odd 2 1 5796.2.a.t 5
4.b odd 2 1 2576.2.a.bb 5
7.b odd 2 1 4508.2.a.f 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.a.d 5 1.a even 1 1 trivial
2576.2.a.bb 5 4.b odd 2 1
4508.2.a.f 5 7.b odd 2 1
5796.2.a.t 5 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{5} - 3T_{3}^{4} - 8T_{3}^{3} + 22T_{3}^{2} + 16T_{3} - 38$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(644))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 3 T^{4} - 8 T^{3} + 22 T^{2} + \cdots - 38$$
$5$ $$T^{5} - 2 T^{4} - 20 T^{3} + 34 T^{2} + \cdots - 160$$
$7$ $$(T + 1)^{5}$$
$11$ $$T^{5} - 2 T^{4} - 48 T^{3} + 68 T^{2} + \cdots - 360$$
$13$ $$T^{5} - 13 T^{4} + 20 T^{3} + \cdots + 1424$$
$17$ $$T^{5} - 4 T^{4} - 48 T^{3} + 70 T^{2} + \cdots + 256$$
$19$ $$T^{5} - 12 T^{4} + 320 T^{2} + \cdots - 1024$$
$23$ $$(T + 1)^{5}$$
$29$ $$T^{5} - 13 T^{4} + 304 T^{2} + \cdots - 796$$
$31$ $$T^{5} + 3 T^{4} - 142 T^{3} + \cdots + 20810$$
$37$ $$T^{5} + 4 T^{4} - 88 T^{3} + \cdots + 1152$$
$41$ $$T^{5} - T^{4} - 84 T^{3} + 96 T^{2} + \cdots - 2032$$
$43$ $$T^{5} + 8 T^{4} - 88 T^{3} + \cdots - 1984$$
$47$ $$T^{5} - 5 T^{4} - 130 T^{3} + \cdots - 1198$$
$53$ $$T^{5} + 8 T^{4} - 144 T^{3} + \cdots + 28224$$
$59$ $$T^{5} - 12 T^{4} - 44 T^{3} + \cdots - 216$$
$61$ $$T^{5} - 20 T^{4} + 60 T^{3} + \cdots + 292$$
$67$ $$T^{5} + 12 T^{4} - 96 T^{3} + \cdots - 9008$$
$71$ $$T^{5} - 9 T^{4} - 180 T^{3} + \cdots - 19840$$
$73$ $$T^{5} + 9 T^{4} - 112 T^{3} + \cdots + 1296$$
$79$ $$T^{5} + 8 T^{4} - 160 T^{3} + \cdots - 6320$$
$83$ $$T^{5} + 28 T^{4} + 160 T^{3} + \cdots - 4096$$
$89$ $$T^{5} - 32 T^{4} + 148 T^{3} + \cdots + 200$$
$97$ $$T^{5} - 4 T^{4} - 196 T^{3} + \cdots + 27700$$