# Properties

 Label 880.2.a.i Level $880$ Weight $2$ Character orbit 880.a Self dual yes Analytic conductor $7.027$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$880 = 2^{4} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 880.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.02683537787$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} - 3 q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + q^5 - 3 * q^7 - 2 * q^9 $$q + q^{3} + q^{5} - 3 q^{7} - 2 q^{9} - q^{11} - 6 q^{13} + q^{15} - 7 q^{17} - 5 q^{19} - 3 q^{21} + 6 q^{23} + q^{25} - 5 q^{27} + 5 q^{29} + 3 q^{31} - q^{33} - 3 q^{35} + 3 q^{37} - 6 q^{39} + 2 q^{41} - 4 q^{43} - 2 q^{45} + 2 q^{47} + 2 q^{49} - 7 q^{51} - q^{53} - q^{55} - 5 q^{57} + 10 q^{59} + 7 q^{61} + 6 q^{63} - 6 q^{65} - 8 q^{67} + 6 q^{69} - 7 q^{71} + 14 q^{73} + q^{75} + 3 q^{77} - 10 q^{79} + q^{81} + 6 q^{83} - 7 q^{85} + 5 q^{87} - 15 q^{89} + 18 q^{91} + 3 q^{93} - 5 q^{95} - 12 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^3 + q^5 - 3 * q^7 - 2 * q^9 - q^11 - 6 * q^13 + q^15 - 7 * q^17 - 5 * q^19 - 3 * q^21 + 6 * q^23 + q^25 - 5 * q^27 + 5 * q^29 + 3 * q^31 - q^33 - 3 * q^35 + 3 * q^37 - 6 * q^39 + 2 * q^41 - 4 * q^43 - 2 * q^45 + 2 * q^47 + 2 * q^49 - 7 * q^51 - q^53 - q^55 - 5 * q^57 + 10 * q^59 + 7 * q^61 + 6 * q^63 - 6 * q^65 - 8 * q^67 + 6 * q^69 - 7 * q^71 + 14 * q^73 + q^75 + 3 * q^77 - 10 * q^79 + q^81 + 6 * q^83 - 7 * q^85 + 5 * q^87 - 15 * q^89 + 18 * q^91 + 3 * q^93 - 5 * q^95 - 12 * q^97 + 2 * 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
0 1.00000 0 1.00000 0 −3.00000 0 −2.00000 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 880.2.a.i 1
3.b odd 2 1 7920.2.a.d 1
4.b odd 2 1 110.2.a.b 1
5.b even 2 1 4400.2.a.l 1
5.c odd 4 2 4400.2.b.i 2
8.b even 2 1 3520.2.a.h 1
8.d odd 2 1 3520.2.a.y 1
11.b odd 2 1 9680.2.a.x 1
12.b even 2 1 990.2.a.d 1
20.d odd 2 1 550.2.a.f 1
20.e even 4 2 550.2.b.a 2
28.d even 2 1 5390.2.a.bf 1
44.c even 2 1 1210.2.a.b 1
60.h even 2 1 4950.2.a.bc 1
60.l odd 4 2 4950.2.c.m 2
220.g even 2 1 6050.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.b 1 4.b odd 2 1
550.2.a.f 1 20.d odd 2 1
550.2.b.a 2 20.e even 4 2
880.2.a.i 1 1.a even 1 1 trivial
990.2.a.d 1 12.b even 2 1
1210.2.a.b 1 44.c even 2 1
3520.2.a.h 1 8.b even 2 1
3520.2.a.y 1 8.d odd 2 1
4400.2.a.l 1 5.b even 2 1
4400.2.b.i 2 5.c odd 4 2
4950.2.a.bc 1 60.h even 2 1
4950.2.c.m 2 60.l odd 4 2
5390.2.a.bf 1 28.d even 2 1
6050.2.a.bj 1 220.g even 2 1
7920.2.a.d 1 3.b odd 2 1
9680.2.a.x 1 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(880))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} + 3$$ T7 + 3 $$T_{13} + 6$$ T13 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 3$$
$11$ $$T + 1$$
$13$ $$T + 6$$
$17$ $$T + 7$$
$19$ $$T + 5$$
$23$ $$T - 6$$
$29$ $$T - 5$$
$31$ $$T - 3$$
$37$ $$T - 3$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T - 2$$
$53$ $$T + 1$$
$59$ $$T - 10$$
$61$ $$T - 7$$
$67$ $$T + 8$$
$71$ $$T + 7$$
$73$ $$T - 14$$
$79$ $$T + 10$$
$83$ $$T - 6$$
$89$ $$T + 15$$
$97$ $$T + 12$$
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