# Properties

 Label 4400.2.b.s Level $4400$ Weight $2$ Character orbit 4400.b Analytic conductor $35.134$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1100) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (3 \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{9}+O(q^{10})$$ q + b1 * q^3 + (3*b2 + b1) * q^7 + (b3 - 3) * q^9 $$q + \beta_1 q^{3} + (3 \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{9} + q^{11} + \beta_{2} q^{13} + ( - 2 \beta_{2} - \beta_1) q^{17} + ( - 2 \beta_{3} + 3) q^{19} + ( - 2 \beta_{3} - 3) q^{21} + ( - \beta_{2} + \beta_1) q^{23} + 5 \beta_{2} q^{27} + (\beta_{3} - 5) q^{29} + ( - 2 \beta_{3} + 5) q^{31} + \beta_1 q^{33} + (3 \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{3} + 1) q^{39} + ( - 2 \beta_{3} - 5) q^{41} - 10 \beta_{2} q^{43} + (7 \beta_{2} + 2 \beta_1) q^{47} + ( - 5 \beta_{3} - 2) q^{49} + (\beta_{3} + 4) q^{51} + ( - 3 \beta_{2} + 3 \beta_1) q^{53} + ( - 10 \beta_{2} + 3 \beta_1) q^{57} + (2 \beta_{3} - 7) q^{59} + (\beta_{3} + 6) q^{61} - \beta_{2} q^{63} + 4 \beta_{2} q^{67} + (2 \beta_{3} - 7) q^{69} - 6 \beta_{3} q^{71} + (5 \beta_{2} - \beta_1) q^{73} + (3 \beta_{2} + \beta_1) q^{77} + (7 \beta_{3} - 3) q^{79} + ( - 2 \beta_{3} - 4) q^{81} + (4 \beta_{2} + 5 \beta_1) q^{83} + (5 \beta_{2} - 5 \beta_1) q^{87} + ( - \beta_{3} - 1) q^{89} + ( - \beta_{3} - 2) q^{91} + ( - 10 \beta_{2} + 5 \beta_1) q^{93} + ( - 9 \beta_{2} - \beta_1) q^{97} + (\beta_{3} - 3) q^{99}+O(q^{100})$$ q + b1 * q^3 + (3*b2 + b1) * q^7 + (b3 - 3) * q^9 + q^11 + b2 * q^13 + (-2*b2 - b1) * q^17 + (-2*b3 + 3) * q^19 + (-2*b3 - 3) * q^21 + (-b2 + b1) * q^23 + 5*b2 * q^27 + (b3 - 5) * q^29 + (-2*b3 + 5) * q^31 + b1 * q^33 + (3*b2 + 2*b1) * q^37 + (-b3 + 1) * q^39 + (-2*b3 - 5) * q^41 - 10*b2 * q^43 + (7*b2 + 2*b1) * q^47 + (-5*b3 - 2) * q^49 + (b3 + 4) * q^51 + (-3*b2 + 3*b1) * q^53 + (-10*b2 + 3*b1) * q^57 + (2*b3 - 7) * q^59 + (b3 + 6) * q^61 - b2 * q^63 + 4*b2 * q^67 + (2*b3 - 7) * q^69 - 6*b3 * q^71 + (5*b2 - b1) * q^73 + (3*b2 + b1) * q^77 + (7*b3 - 3) * q^79 + (-2*b3 - 4) * q^81 + (4*b2 + 5*b1) * q^83 + (5*b2 - 5*b1) * q^87 + (-b3 - 1) * q^89 + (-b3 - 2) * q^91 + (-10*b2 + 5*b1) * q^93 + (-9*b2 - b1) * q^97 + (b3 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{9}+O(q^{10})$$ 4 * q - 10 * q^9 $$4 q - 10 q^{9} + 4 q^{11} + 8 q^{19} - 16 q^{21} - 18 q^{29} + 16 q^{31} + 2 q^{39} - 24 q^{41} - 18 q^{49} + 18 q^{51} - 24 q^{59} + 26 q^{61} - 24 q^{69} - 12 q^{71} + 2 q^{79} - 20 q^{81} - 6 q^{89} - 10 q^{91} - 10 q^{99}+O(q^{100})$$ 4 * q - 10 * q^9 + 4 * q^11 + 8 * q^19 - 16 * q^21 - 18 * q^29 + 16 * q^31 + 2 * q^39 - 24 * q^41 - 18 * q^49 + 18 * q^51 - 24 * q^59 + 26 * q^61 - 24 * q^69 - 12 * q^71 + 2 * q^79 - 20 * q^81 - 6 * q^89 - 10 * q^91 - 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 5$$ (v^3 + 6*v) / 5 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 6$$ v^2 + 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 6$$ b3 - 6 $$\nu^{3}$$ $$=$$ $$5\beta_{2} - 6\beta_1$$ 5*b2 - 6*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4049.1
 − 2.79129i − 1.79129i 1.79129i 2.79129i
0 2.79129i 0 0 0 0.208712i 0 −4.79129 0
4049.2 0 1.79129i 0 0 0 4.79129i 0 −0.208712 0
4049.3 0 1.79129i 0 0 0 4.79129i 0 −0.208712 0
4049.4 0 2.79129i 0 0 0 0.208712i 0 −4.79129 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{4} + 11T_{3}^{2} + 25$$ T3^4 + 11*T3^2 + 25 $$T_{7}^{4} + 23T_{7}^{2} + 1$$ T7^4 + 23*T7^2 + 1 $$T_{13}^{2} + 1$$ T13^2 + 1 $$T_{17}^{4} + 15T_{17}^{2} + 9$$ T17^4 + 15*T17^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 11T^{2} + 25$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 23T^{2} + 1$$
$11$ $$(T - 1)^{4}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$T^{4} + 15T^{2} + 9$$
$19$ $$(T^{2} - 4 T - 17)^{2}$$
$23$ $$T^{4} + 15T^{2} + 9$$
$29$ $$(T^{2} + 9 T + 15)^{2}$$
$31$ $$(T^{2} - 8 T - 5)^{2}$$
$37$ $$T^{4} + 50T^{2} + 289$$
$41$ $$(T^{2} + 12 T + 15)^{2}$$
$43$ $$(T^{2} + 100)^{2}$$
$47$ $$T^{4} + 114T^{2} + 225$$
$53$ $$T^{4} + 135T^{2} + 729$$
$59$ $$(T^{2} + 12 T + 15)^{2}$$
$61$ $$(T^{2} - 13 T + 37)^{2}$$
$67$ $$(T^{2} + 16)^{2}$$
$71$ $$(T^{2} + 6 T - 180)^{2}$$
$73$ $$T^{4} + 71T^{2} + 625$$
$79$ $$(T^{2} - T - 257)^{2}$$
$83$ $$T^{4} + 267 T^{2} + 16641$$
$89$ $$(T^{2} + 3 T - 3)^{2}$$
$97$ $$T^{4} + 155T^{2} + 4489$$