# Properties

 Label 4400.2.b.h 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 2 i q^{7} + 2 q^{9} +O(q^{10})$$ q + i * q^3 - 2*i * q^7 + 2 * q^9 $$q + i q^{3} - 2 i q^{7} + 2 q^{9} - q^{11} + 4 i q^{13} + 2 i q^{17} + 2 q^{21} + i q^{23} + 5 i q^{27} - 7 q^{31} - i q^{33} - 3 i q^{37} - 4 q^{39} - 8 q^{41} + 6 i q^{43} + 8 i q^{47} + 3 q^{49} - 2 q^{51} - 6 i q^{53} + 5 q^{59} + 12 q^{61} - 4 i q^{63} - 7 i q^{67} - q^{69} + 3 q^{71} + 4 i q^{73} + 2 i q^{77} - 10 q^{79} + q^{81} + 6 i q^{83} - 15 q^{89} + 8 q^{91} - 7 i q^{93} + 7 i q^{97} - 2 q^{99} +O(q^{100})$$ q + i * q^3 - 2*i * q^7 + 2 * q^9 - q^11 + 4*i * q^13 + 2*i * q^17 + 2 * q^21 + i * q^23 + 5*i * q^27 - 7 * q^31 - i * q^33 - 3*i * q^37 - 4 * q^39 - 8 * q^41 + 6*i * q^43 + 8*i * q^47 + 3 * q^49 - 2 * q^51 - 6*i * q^53 + 5 * q^59 + 12 * q^61 - 4*i * q^63 - 7*i * q^67 - q^69 + 3 * q^71 + 4*i * q^73 + 2*i * q^77 - 10 * q^79 + q^81 + 6*i * q^83 - 15 * q^89 + 8 * q^91 - 7*i * q^93 + 7*i * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} - 2 q^{11} + 4 q^{21} - 14 q^{31} - 8 q^{39} - 16 q^{41} + 6 q^{49} - 4 q^{51} + 10 q^{59} + 24 q^{61} - 2 q^{69} + 6 q^{71} - 20 q^{79} + 2 q^{81} - 30 q^{89} + 16 q^{91} - 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 - 2 * q^11 + 4 * q^21 - 14 * q^31 - 8 * q^39 - 16 * q^41 + 6 * q^49 - 4 * q^51 + 10 * q^59 + 24 * q^61 - 2 * q^69 + 6 * q^71 - 20 * q^79 + 2 * q^81 - 30 * q^89 + 16 * q^91 - 4 * 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 1.00000i 0 0 0 2.00000i 0 2.00000 0
4049.2 0 1.00000i 0 0 0 2.00000i 0 2.00000 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.h 2
4.b odd 2 1 275.2.b.a 2
5.b even 2 1 inner 4400.2.b.h 2
5.c odd 4 1 176.2.a.b 1
5.c odd 4 1 4400.2.a.i 1
12.b even 2 1 2475.2.c.a 2
15.e even 4 1 1584.2.a.g 1
20.d odd 2 1 275.2.b.a 2
20.e even 4 1 11.2.a.a 1
20.e even 4 1 275.2.a.b 1
35.f even 4 1 8624.2.a.j 1
40.i odd 4 1 704.2.a.c 1
40.k even 4 1 704.2.a.h 1
55.e even 4 1 1936.2.a.i 1
60.h even 2 1 2475.2.c.a 2
60.l odd 4 1 99.2.a.d 1
60.l odd 4 1 2475.2.a.a 1
80.i odd 4 1 2816.2.c.f 2
80.j even 4 1 2816.2.c.j 2
80.s even 4 1 2816.2.c.j 2
80.t odd 4 1 2816.2.c.f 2
120.q odd 4 1 6336.2.a.br 1
120.w even 4 1 6336.2.a.bu 1
140.j odd 4 1 539.2.a.a 1
140.w even 12 2 539.2.e.h 2
140.x odd 12 2 539.2.e.g 2
180.v odd 12 2 891.2.e.b 2
180.x even 12 2 891.2.e.k 2
220.i odd 4 1 121.2.a.d 1
220.i odd 4 1 3025.2.a.a 1
220.v even 20 4 121.2.c.e 4
220.w odd 20 4 121.2.c.a 4
260.p even 4 1 1859.2.a.b 1
340.r even 4 1 3179.2.a.a 1
380.j odd 4 1 3971.2.a.b 1
420.w even 4 1 4851.2.a.t 1
440.t even 4 1 7744.2.a.k 1
440.w odd 4 1 7744.2.a.x 1
460.k odd 4 1 5819.2.a.a 1
580.o even 4 1 9251.2.a.d 1
660.q even 4 1 1089.2.a.b 1
1540.x even 4 1 5929.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 20.e even 4 1
99.2.a.d 1 60.l odd 4 1
121.2.a.d 1 220.i odd 4 1
121.2.c.a 4 220.w odd 20 4
121.2.c.e 4 220.v even 20 4
176.2.a.b 1 5.c odd 4 1
275.2.a.b 1 20.e even 4 1
275.2.b.a 2 4.b odd 2 1
275.2.b.a 2 20.d odd 2 1
539.2.a.a 1 140.j odd 4 1
539.2.e.g 2 140.x odd 12 2
539.2.e.h 2 140.w even 12 2
704.2.a.c 1 40.i odd 4 1
704.2.a.h 1 40.k even 4 1
891.2.e.b 2 180.v odd 12 2
891.2.e.k 2 180.x even 12 2
1089.2.a.b 1 660.q even 4 1
1584.2.a.g 1 15.e even 4 1
1859.2.a.b 1 260.p even 4 1
1936.2.a.i 1 55.e even 4 1
2475.2.a.a 1 60.l odd 4 1
2475.2.c.a 2 12.b even 2 1
2475.2.c.a 2 60.h even 2 1
2816.2.c.f 2 80.i odd 4 1
2816.2.c.f 2 80.t odd 4 1
2816.2.c.j 2 80.j even 4 1
2816.2.c.j 2 80.s even 4 1
3025.2.a.a 1 220.i odd 4 1
3179.2.a.a 1 340.r even 4 1
3971.2.a.b 1 380.j odd 4 1
4400.2.a.i 1 5.c odd 4 1
4400.2.b.h 2 1.a even 1 1 trivial
4400.2.b.h 2 5.b even 2 1 inner
4851.2.a.t 1 420.w even 4 1
5819.2.a.a 1 460.k odd 4 1
5929.2.a.h 1 1540.x even 4 1
6336.2.a.br 1 120.q odd 4 1
6336.2.a.bu 1 120.w even 4 1
7744.2.a.k 1 440.t even 4 1
7744.2.a.x 1 440.w odd 4 1
8624.2.a.j 1 35.f even 4 1
9251.2.a.d 1 580.o even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}^{2} + 4$$ T17^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$T^{2}$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 5)^{2}$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 15)^{2}$$
$97$ $$T^{2} + 49$$