# Properties

 Label 4400.2.a.bv Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) 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.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + ( - 3 \beta + 2) q^{7} + (3 \beta - 1) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + (-3*b + 2) * q^7 + (3*b - 1) * q^9 $$q + (\beta + 1) q^{3} + ( - 3 \beta + 2) q^{7} + (3 \beta - 1) q^{9} - q^{11} + ( - 2 \beta - 3) q^{13} + (\beta - 1) q^{17} + (6 \beta - 3) q^{19} + ( - 4 \beta - 1) q^{21} + ( - 5 \beta + 4) q^{23} + (2 \beta - 1) q^{27} + (\beta - 3) q^{29} + 3 q^{31} + ( - \beta - 1) q^{33} + ( - 2 \beta - 7) q^{37} + ( - 7 \beta - 5) q^{39} - 3 q^{41} - 6 q^{43} + (8 \beta - 1) q^{47} + ( - 3 \beta + 6) q^{49} + \beta q^{51} + ( - 7 \beta + 2) q^{53} + (9 \beta + 3) q^{57} + (4 \beta - 7) q^{59} + (5 \beta - 8) q^{61} - 11 q^{63} + 8 q^{67} + ( - 6 \beta - 1) q^{69} + ( - 10 \beta + 8) q^{71} + (\beta - 12) q^{73} + (3 \beta - 2) q^{77} + ( - 3 \beta - 1) q^{79} + ( - 6 \beta + 4) q^{81} + (3 \beta - 15) q^{83} + ( - \beta - 2) q^{87} + (5 \beta - 15) q^{89} + 11 \beta q^{91} + (3 \beta + 3) q^{93} - \beta q^{97} + ( - 3 \beta + 1) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + (-3*b + 2) * q^7 + (3*b - 1) * q^9 - q^11 + (-2*b - 3) * q^13 + (b - 1) * q^17 + (6*b - 3) * q^19 + (-4*b - 1) * q^21 + (-5*b + 4) * q^23 + (2*b - 1) * q^27 + (b - 3) * q^29 + 3 * q^31 + (-b - 1) * q^33 + (-2*b - 7) * q^37 + (-7*b - 5) * q^39 - 3 * q^41 - 6 * q^43 + (8*b - 1) * q^47 + (-3*b + 6) * q^49 + b * q^51 + (-7*b + 2) * q^53 + (9*b + 3) * q^57 + (4*b - 7) * q^59 + (5*b - 8) * q^61 - 11 * q^63 + 8 * q^67 + (-6*b - 1) * q^69 + (-10*b + 8) * q^71 + (b - 12) * q^73 + (3*b - 2) * q^77 + (-3*b - 1) * q^79 + (-6*b + 4) * q^81 + (3*b - 15) * q^83 + (-b - 2) * q^87 + (5*b - 15) * q^89 + 11*b * q^91 + (3*b + 3) * q^93 - b * q^97 + (-3*b + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + q^{7} + q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + q^7 + q^9 $$2 q + 3 q^{3} + q^{7} + q^{9} - 2 q^{11} - 8 q^{13} - q^{17} - 6 q^{21} + 3 q^{23} - 5 q^{29} + 6 q^{31} - 3 q^{33} - 16 q^{37} - 17 q^{39} - 6 q^{41} - 12 q^{43} + 6 q^{47} + 9 q^{49} + q^{51} - 3 q^{53} + 15 q^{57} - 10 q^{59} - 11 q^{61} - 22 q^{63} + 16 q^{67} - 8 q^{69} + 6 q^{71} - 23 q^{73} - q^{77} - 5 q^{79} + 2 q^{81} - 27 q^{83} - 5 q^{87} - 25 q^{89} + 11 q^{91} + 9 q^{93} - q^{97} - q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + q^7 + q^9 - 2 * q^11 - 8 * q^13 - q^17 - 6 * q^21 + 3 * q^23 - 5 * q^29 + 6 * q^31 - 3 * q^33 - 16 * q^37 - 17 * q^39 - 6 * q^41 - 12 * q^43 + 6 * q^47 + 9 * q^49 + q^51 - 3 * q^53 + 15 * q^57 - 10 * q^59 - 11 * q^61 - 22 * q^63 + 16 * q^67 - 8 * q^69 + 6 * q^71 - 23 * q^73 - q^77 - 5 * q^79 + 2 * q^81 - 27 * q^83 - 5 * q^87 - 25 * q^89 + 11 * q^91 + 9 * q^93 - q^97 - 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 0.381966 0 0 0 3.85410 0 −2.85410 0
1.2 0 2.61803 0 0 0 −2.85410 0 3.85410 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.bv 2
4.b odd 2 1 275.2.a.d 2
5.b even 2 1 4400.2.a.bg 2
5.c odd 4 2 4400.2.b.x 4
12.b even 2 1 2475.2.a.s 2
20.d odd 2 1 275.2.a.g yes 2
20.e even 4 2 275.2.b.e 4
44.c even 2 1 3025.2.a.m 2
60.h even 2 1 2475.2.a.n 2
60.l odd 4 2 2475.2.c.p 4
220.g even 2 1 3025.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 4.b odd 2 1
275.2.a.g yes 2 20.d odd 2 1
275.2.b.e 4 20.e even 4 2
2475.2.a.n 2 60.h even 2 1
2475.2.a.s 2 12.b even 2 1
2475.2.c.p 4 60.l odd 4 2
3025.2.a.i 2 220.g even 2 1
3025.2.a.m 2 44.c even 2 1
4400.2.a.bg 2 5.b even 2 1
4400.2.a.bv 2 1.a even 1 1 trivial
4400.2.b.x 4 5.c odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 3T_{3} + 1$$ T3^2 - 3*T3 + 1 $$T_{7}^{2} - T_{7} - 11$$ T7^2 - T7 - 11 $$T_{13}^{2} + 8T_{13} + 11$$ T13^2 + 8*T13 + 11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T - 11$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 8T + 11$$
$17$ $$T^{2} + T - 1$$
$19$ $$T^{2} - 45$$
$23$ $$T^{2} - 3T - 29$$
$29$ $$T^{2} + 5T + 5$$
$31$ $$(T - 3)^{2}$$
$37$ $$T^{2} + 16T + 59$$
$41$ $$(T + 3)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} - 6T - 71$$
$53$ $$T^{2} + 3T - 59$$
$59$ $$T^{2} + 10T + 5$$
$61$ $$T^{2} + 11T - 1$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} - 6T - 116$$
$73$ $$T^{2} + 23T + 131$$
$79$ $$T^{2} + 5T - 5$$
$83$ $$T^{2} + 27T + 171$$
$89$ $$T^{2} + 25T + 125$$
$97$ $$T^{2} + T - 1$$