# Properties

 Label 4400.2.a.bu Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1100) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - \beta + 3) q^{7} + (\beta + 2) q^{9}+O(q^{10})$$ q + b * q^3 + (-b + 3) * q^7 + (b + 2) * q^9 $$q + \beta q^{3} + ( - \beta + 3) q^{7} + (\beta + 2) q^{9} + q^{11} - q^{13} + (\beta - 2) q^{17} + ( - 2 \beta - 1) q^{19} + (2 \beta - 5) q^{21} + (\beta + 1) q^{23} + 5 q^{27} + (\beta + 4) q^{29} + (2 \beta + 3) q^{31} + \beta q^{33} + ( - 2 \beta + 3) q^{37} - \beta q^{39} + (2 \beta - 7) q^{41} + 10 q^{43} + ( - 2 \beta + 7) q^{47} + ( - 5 \beta + 7) q^{49} + ( - \beta + 5) q^{51} + (3 \beta + 3) q^{53} + ( - 3 \beta - 10) q^{57} + (2 \beta + 5) q^{59} + ( - \beta + 7) q^{61} + q^{63} + 4 q^{67} + (2 \beta + 5) q^{69} + (6 \beta - 6) q^{71} + ( - \beta - 5) q^{73} + ( - \beta + 3) q^{77} + (7 \beta - 4) q^{79} + (2 \beta - 6) q^{81} + (5 \beta - 4) q^{83} + (5 \beta + 5) q^{87} + ( - \beta + 2) q^{89} + (\beta - 3) q^{91} + (5 \beta + 10) q^{93} + (\beta - 9) q^{97} + (\beta + 2) q^{99} +O(q^{100})$$ q + b * q^3 + (-b + 3) * q^7 + (b + 2) * q^9 + q^11 - q^13 + (b - 2) * q^17 + (-2*b - 1) * q^19 + (2*b - 5) * q^21 + (b + 1) * q^23 + 5 * q^27 + (b + 4) * q^29 + (2*b + 3) * q^31 + b * q^33 + (-2*b + 3) * q^37 - b * q^39 + (2*b - 7) * q^41 + 10 * q^43 + (-2*b + 7) * q^47 + (-5*b + 7) * q^49 + (-b + 5) * q^51 + (3*b + 3) * q^53 + (-3*b - 10) * q^57 + (2*b + 5) * q^59 + (-b + 7) * q^61 + q^63 + 4 * q^67 + (2*b + 5) * q^69 + (6*b - 6) * q^71 + (-b - 5) * q^73 + (-b + 3) * q^77 + (7*b - 4) * q^79 + (2*b - 6) * q^81 + (5*b - 4) * q^83 + (5*b + 5) * q^87 + (-b + 2) * q^89 + (b - 3) * q^91 + (5*b + 10) * q^93 + (b - 9) * q^97 + (b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 5 q^{7} + 5 q^{9}+O(q^{10})$$ 2 * q + q^3 + 5 * q^7 + 5 * q^9 $$2 q + q^{3} + 5 q^{7} + 5 q^{9} + 2 q^{11} - 2 q^{13} - 3 q^{17} - 4 q^{19} - 8 q^{21} + 3 q^{23} + 10 q^{27} + 9 q^{29} + 8 q^{31} + q^{33} + 4 q^{37} - q^{39} - 12 q^{41} + 20 q^{43} + 12 q^{47} + 9 q^{49} + 9 q^{51} + 9 q^{53} - 23 q^{57} + 12 q^{59} + 13 q^{61} + 2 q^{63} + 8 q^{67} + 12 q^{69} - 6 q^{71} - 11 q^{73} + 5 q^{77} - q^{79} - 10 q^{81} - 3 q^{83} + 15 q^{87} + 3 q^{89} - 5 q^{91} + 25 q^{93} - 17 q^{97} + 5 q^{99}+O(q^{100})$$ 2 * q + q^3 + 5 * q^7 + 5 * q^9 + 2 * q^11 - 2 * q^13 - 3 * q^17 - 4 * q^19 - 8 * q^21 + 3 * q^23 + 10 * q^27 + 9 * q^29 + 8 * q^31 + q^33 + 4 * q^37 - q^39 - 12 * q^41 + 20 * q^43 + 12 * q^47 + 9 * q^49 + 9 * q^51 + 9 * q^53 - 23 * q^57 + 12 * q^59 + 13 * q^61 + 2 * q^63 + 8 * q^67 + 12 * q^69 - 6 * q^71 - 11 * q^73 + 5 * q^77 - q^79 - 10 * q^81 - 3 * q^83 + 15 * q^87 + 3 * q^89 - 5 * q^91 + 25 * q^93 - 17 * q^97 + 5 * 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
 −1.79129 2.79129
0 −1.79129 0 0 0 4.79129 0 0.208712 0
1.2 0 2.79129 0 0 0 0.208712 0 4.79129 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 4400.2.a.bu 2
4.b odd 2 1 1100.2.a.g 2
5.b even 2 1 4400.2.a.bi 2
5.c odd 4 2 4400.2.b.s 4
12.b even 2 1 9900.2.a.bh 2
20.d odd 2 1 1100.2.a.h yes 2
20.e even 4 2 1100.2.b.d 4
60.h even 2 1 9900.2.a.bz 2
60.l odd 4 2 9900.2.c.x 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.2.a.g 2 4.b odd 2 1
1100.2.a.h yes 2 20.d odd 2 1
1100.2.b.d 4 20.e even 4 2
4400.2.a.bi 2 5.b even 2 1
4400.2.a.bu 2 1.a even 1 1 trivial
4400.2.b.s 4 5.c odd 4 2
9900.2.a.bh 2 12.b even 2 1
9900.2.a.bz 2 60.h even 2 1
9900.2.c.x 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

 $$T_{3}^{2} - T_{3} - 5$$ T3^2 - T3 - 5 $$T_{7}^{2} - 5T_{7} + 1$$ T7^2 - 5*T7 + 1 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 5$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5T + 1$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 3T - 3$$
$19$ $$T^{2} + 4T - 17$$
$23$ $$T^{2} - 3T - 3$$
$29$ $$T^{2} - 9T + 15$$
$31$ $$T^{2} - 8T - 5$$
$37$ $$T^{2} - 4T - 17$$
$41$ $$T^{2} + 12T + 15$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2} - 12T + 15$$
$53$ $$T^{2} - 9T - 27$$
$59$ $$T^{2} - 12T + 15$$
$61$ $$T^{2} - 13T + 37$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} + 6T - 180$$
$73$ $$T^{2} + 11T + 25$$
$79$ $$T^{2} + T - 257$$
$83$ $$T^{2} + 3T - 129$$
$89$ $$T^{2} - 3T - 3$$
$97$ $$T^{2} + 17T + 67$$