# Properties

 Label 110.2.a.b Level $110$ Weight $2$ Character orbit 110.a Self dual yes Analytic conductor $0.878$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$110 = 2 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 110.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.878354422234$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3 q^{7} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + 3 * q^7 + q^8 - 2 * q^9 $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3 q^{7} + q^{8} - 2 q^{9} + q^{10} + q^{11} - q^{12} - 6 q^{13} + 3 q^{14} - q^{15} + q^{16} - 7 q^{17} - 2 q^{18} + 5 q^{19} + q^{20} - 3 q^{21} + q^{22} - 6 q^{23} - q^{24} + q^{25} - 6 q^{26} + 5 q^{27} + 3 q^{28} + 5 q^{29} - q^{30} - 3 q^{31} + q^{32} - q^{33} - 7 q^{34} + 3 q^{35} - 2 q^{36} + 3 q^{37} + 5 q^{38} + 6 q^{39} + q^{40} + 2 q^{41} - 3 q^{42} + 4 q^{43} + q^{44} - 2 q^{45} - 6 q^{46} - 2 q^{47} - q^{48} + 2 q^{49} + q^{50} + 7 q^{51} - 6 q^{52} - q^{53} + 5 q^{54} + q^{55} + 3 q^{56} - 5 q^{57} + 5 q^{58} - 10 q^{59} - q^{60} + 7 q^{61} - 3 q^{62} - 6 q^{63} + q^{64} - 6 q^{65} - q^{66} + 8 q^{67} - 7 q^{68} + 6 q^{69} + 3 q^{70} + 7 q^{71} - 2 q^{72} + 14 q^{73} + 3 q^{74} - q^{75} + 5 q^{76} + 3 q^{77} + 6 q^{78} + 10 q^{79} + q^{80} + q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} - 7 q^{85} + 4 q^{86} - 5 q^{87} + q^{88} - 15 q^{89} - 2 q^{90} - 18 q^{91} - 6 q^{92} + 3 q^{93} - 2 q^{94} + 5 q^{95} - q^{96} - 12 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 + 3 * q^7 + q^8 - 2 * q^9 + q^10 + q^11 - q^12 - 6 * q^13 + 3 * q^14 - q^15 + q^16 - 7 * q^17 - 2 * q^18 + 5 * q^19 + q^20 - 3 * q^21 + q^22 - 6 * q^23 - q^24 + q^25 - 6 * q^26 + 5 * q^27 + 3 * q^28 + 5 * q^29 - q^30 - 3 * q^31 + q^32 - q^33 - 7 * q^34 + 3 * q^35 - 2 * q^36 + 3 * q^37 + 5 * q^38 + 6 * q^39 + q^40 + 2 * q^41 - 3 * q^42 + 4 * q^43 + q^44 - 2 * q^45 - 6 * q^46 - 2 * q^47 - q^48 + 2 * q^49 + q^50 + 7 * q^51 - 6 * q^52 - q^53 + 5 * q^54 + q^55 + 3 * q^56 - 5 * q^57 + 5 * q^58 - 10 * q^59 - q^60 + 7 * q^61 - 3 * q^62 - 6 * q^63 + q^64 - 6 * q^65 - q^66 + 8 * q^67 - 7 * q^68 + 6 * q^69 + 3 * q^70 + 7 * q^71 - 2 * q^72 + 14 * q^73 + 3 * q^74 - q^75 + 5 * q^76 + 3 * q^77 + 6 * q^78 + 10 * q^79 + q^80 + q^81 + 2 * q^82 - 6 * q^83 - 3 * q^84 - 7 * q^85 + 4 * q^86 - 5 * q^87 + q^88 - 15 * q^89 - 2 * q^90 - 18 * q^91 - 6 * q^92 + 3 * q^93 - 2 * q^94 + 5 * q^95 - q^96 - 12 * q^97 + 2 * q^98 - 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
1.00000 −1.00000 1.00000 1.00000 −1.00000 3.00000 1.00000 −2.00000 1.00000
 $$n$$: 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 110.2.a.b 1
3.b odd 2 1 990.2.a.d 1
4.b odd 2 1 880.2.a.i 1
5.b even 2 1 550.2.a.f 1
5.c odd 4 2 550.2.b.a 2
7.b odd 2 1 5390.2.a.bf 1
8.b even 2 1 3520.2.a.y 1
8.d odd 2 1 3520.2.a.h 1
11.b odd 2 1 1210.2.a.b 1
12.b even 2 1 7920.2.a.d 1
15.d odd 2 1 4950.2.a.bc 1
15.e even 4 2 4950.2.c.m 2
20.d odd 2 1 4400.2.a.l 1
20.e even 4 2 4400.2.b.i 2
44.c even 2 1 9680.2.a.x 1
55.d odd 2 1 6050.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.b 1 1.a even 1 1 trivial
550.2.a.f 1 5.b even 2 1
550.2.b.a 2 5.c odd 4 2
880.2.a.i 1 4.b odd 2 1
990.2.a.d 1 3.b odd 2 1
1210.2.a.b 1 11.b odd 2 1
3520.2.a.h 1 8.d odd 2 1
3520.2.a.y 1 8.b even 2 1
4400.2.a.l 1 20.d odd 2 1
4400.2.b.i 2 20.e even 4 2
4950.2.a.bc 1 15.d odd 2 1
4950.2.c.m 2 15.e even 4 2
5390.2.a.bf 1 7.b odd 2 1
6050.2.a.bj 1 55.d odd 2 1
7920.2.a.d 1 12.b even 2 1
9680.2.a.x 1 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(110))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7} - 3$$ T7 - 3

## Hecke characteristic polynomials

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