# Properties

 Label 4400.2.b.e Level $4400$ Weight $2$ Character orbit 4400.b Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 550) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} - 4 i q^{7} - q^{9} +O(q^{10})$$ q + 2*i * q^3 - 4*i * q^7 - q^9 $$q + 2 i q^{3} - 4 i q^{7} - q^{9} + q^{11} + 5 i q^{13} - 7 q^{19} + 8 q^{21} - 3 i q^{23} + 4 i q^{27} - 3 q^{29} - 5 q^{31} + 2 i q^{33} + 4 i q^{37} - 10 q^{39} + 12 q^{41} - 5 i q^{43} - 9 q^{49} + 6 i q^{53} - 14 i q^{57} + 12 q^{59} - 10 q^{61} + 4 i q^{63} + 14 i q^{67} + 6 q^{69} - 3 q^{71} + 8 i q^{73} - 4 i q^{77} - 4 q^{79} - 11 q^{81} + 15 i q^{83} - 6 i q^{87} - 3 q^{89} + 20 q^{91} - 10 i q^{93} + 13 i q^{97} - q^{99} +O(q^{100})$$ q + 2*i * q^3 - 4*i * q^7 - q^9 + q^11 + 5*i * q^13 - 7 * q^19 + 8 * q^21 - 3*i * q^23 + 4*i * q^27 - 3 * q^29 - 5 * q^31 + 2*i * q^33 + 4*i * q^37 - 10 * q^39 + 12 * q^41 - 5*i * q^43 - 9 * q^49 + 6*i * q^53 - 14*i * q^57 + 12 * q^59 - 10 * q^61 + 4*i * q^63 + 14*i * q^67 + 6 * q^69 - 3 * q^71 + 8*i * q^73 - 4*i * q^77 - 4 * q^79 - 11 * q^81 + 15*i * q^83 - 6*i * q^87 - 3 * q^89 + 20 * q^91 - 10*i * q^93 + 13*i * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 2 q^{11} - 14 q^{19} + 16 q^{21} - 6 q^{29} - 10 q^{31} - 20 q^{39} + 24 q^{41} - 18 q^{49} + 24 q^{59} - 20 q^{61} + 12 q^{69} - 6 q^{71} - 8 q^{79} - 22 q^{81} - 6 q^{89} + 40 q^{91} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 2 * q^11 - 14 * q^19 + 16 * q^21 - 6 * q^29 - 10 * q^31 - 20 * q^39 + 24 * q^41 - 18 * q^49 + 24 * q^59 - 20 * q^61 + 12 * q^69 - 6 * q^71 - 8 * q^79 - 22 * q^81 - 6 * q^89 + 40 * q^91 - 2 * q^99

## 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
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 4.00000i 0 −1.00000 0
4049.2 0 2.00000i 0 0 0 4.00000i 0 −1.00000 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.e 2
4.b odd 2 1 550.2.b.d 2
5.b even 2 1 inner 4400.2.b.e 2
5.c odd 4 1 4400.2.a.d 1
5.c odd 4 1 4400.2.a.bc 1
12.b even 2 1 4950.2.c.ba 2
20.d odd 2 1 550.2.b.d 2
20.e even 4 1 550.2.a.a 1
20.e even 4 1 550.2.a.m yes 1
60.h even 2 1 4950.2.c.ba 2
60.l odd 4 1 4950.2.a.u 1
60.l odd 4 1 4950.2.a.y 1
220.i odd 4 1 6050.2.a.n 1
220.i odd 4 1 6050.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.a.a 1 20.e even 4 1
550.2.a.m yes 1 20.e even 4 1
550.2.b.d 2 4.b odd 2 1
550.2.b.d 2 20.d odd 2 1
4400.2.a.d 1 5.c odd 4 1
4400.2.a.bc 1 5.c odd 4 1
4400.2.b.e 2 1.a even 1 1 trivial
4400.2.b.e 2 5.b even 2 1 inner
4950.2.a.u 1 60.l odd 4 1
4950.2.a.y 1 60.l odd 4 1
4950.2.c.ba 2 12.b even 2 1
4950.2.c.ba 2 60.h even 2 1
6050.2.a.n 1 220.i odd 4 1
6050.2.a.bb 1 220.i odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{13}^{2} + 25$$ T13^2 + 25 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 25$$
$17$ $$T^{2}$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T + 3)^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T - 12)^{2}$$
$43$ $$T^{2} + 25$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$(T + 3)^{2}$$
$73$ $$T^{2} + 64$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 225$$
$89$ $$(T + 3)^{2}$$
$97$ $$T^{2} + 169$$