# Properties

 Label 5390.2.a.bf Level $5390$ Weight $2$ Character orbit 5390.a Self dual yes Analytic conductor $43.039$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5390.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.0393666895$$ Analytic rank: $$0$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + q^8 - 2 * q^9 $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} - 2 q^{9} - q^{10} + q^{11} + q^{12} + 6 q^{13} - q^{15} + q^{16} + 7 q^{17} - 2 q^{18} - 5 q^{19} - q^{20} + q^{22} - 6 q^{23} + q^{24} + q^{25} + 6 q^{26} - 5 q^{27} + 5 q^{29} - q^{30} + 3 q^{31} + q^{32} + q^{33} + 7 q^{34} - 2 q^{36} + 3 q^{37} - 5 q^{38} + 6 q^{39} - q^{40} - 2 q^{41} + 4 q^{43} + q^{44} + 2 q^{45} - 6 q^{46} + 2 q^{47} + q^{48} + q^{50} + 7 q^{51} + 6 q^{52} - q^{53} - 5 q^{54} - q^{55} - 5 q^{57} + 5 q^{58} + 10 q^{59} - q^{60} - 7 q^{61} + 3 q^{62} + q^{64} - 6 q^{65} + q^{66} + 8 q^{67} + 7 q^{68} - 6 q^{69} + 7 q^{71} - 2 q^{72} - 14 q^{73} + 3 q^{74} + q^{75} - 5 q^{76} + 6 q^{78} + 10 q^{79} - q^{80} + q^{81} - 2 q^{82} + 6 q^{83} - 7 q^{85} + 4 q^{86} + 5 q^{87} + q^{88} + 15 q^{89} + 2 q^{90} - 6 q^{92} + 3 q^{93} + 2 q^{94} + 5 q^{95} + q^{96} + 12 q^{97} - 2 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + q^8 - 2 * q^9 - q^10 + q^11 + q^12 + 6 * q^13 - q^15 + q^16 + 7 * q^17 - 2 * q^18 - 5 * q^19 - q^20 + q^22 - 6 * q^23 + q^24 + q^25 + 6 * q^26 - 5 * q^27 + 5 * q^29 - q^30 + 3 * q^31 + q^32 + q^33 + 7 * q^34 - 2 * q^36 + 3 * q^37 - 5 * q^38 + 6 * q^39 - q^40 - 2 * q^41 + 4 * q^43 + q^44 + 2 * q^45 - 6 * q^46 + 2 * q^47 + q^48 + q^50 + 7 * q^51 + 6 * q^52 - q^53 - 5 * q^54 - q^55 - 5 * q^57 + 5 * q^58 + 10 * q^59 - q^60 - 7 * q^61 + 3 * q^62 + q^64 - 6 * q^65 + q^66 + 8 * q^67 + 7 * q^68 - 6 * q^69 + 7 * q^71 - 2 * q^72 - 14 * q^73 + 3 * q^74 + q^75 - 5 * q^76 + 6 * q^78 + 10 * q^79 - q^80 + q^81 - 2 * q^82 + 6 * q^83 - 7 * q^85 + 4 * q^86 + 5 * q^87 + q^88 + 15 * q^89 + 2 * q^90 - 6 * q^92 + 3 * q^93 + 2 * q^94 + 5 * q^95 + q^96 + 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
1.00000 1.00000 1.00000 −1.00000 1.00000 0 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 5390.2.a.bf 1
7.b odd 2 1 110.2.a.b 1
21.c even 2 1 990.2.a.d 1
28.d even 2 1 880.2.a.i 1
35.c odd 2 1 550.2.a.f 1
35.f even 4 2 550.2.b.a 2
56.e even 2 1 3520.2.a.h 1
56.h odd 2 1 3520.2.a.y 1
77.b even 2 1 1210.2.a.b 1
84.h odd 2 1 7920.2.a.d 1
105.g even 2 1 4950.2.a.bc 1
105.k odd 4 2 4950.2.c.m 2
140.c even 2 1 4400.2.a.l 1
140.j odd 4 2 4400.2.b.i 2
308.g odd 2 1 9680.2.a.x 1
385.h 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 7.b odd 2 1
550.2.a.f 1 35.c odd 2 1
550.2.b.a 2 35.f even 4 2
880.2.a.i 1 28.d even 2 1
990.2.a.d 1 21.c even 2 1
1210.2.a.b 1 77.b even 2 1
3520.2.a.h 1 56.e even 2 1
3520.2.a.y 1 56.h odd 2 1
4400.2.a.l 1 140.c even 2 1
4400.2.b.i 2 140.j odd 4 2
4950.2.a.bc 1 105.g even 2 1
4950.2.c.m 2 105.k odd 4 2
5390.2.a.bf 1 1.a even 1 1 trivial
6050.2.a.bj 1 385.h even 2 1
7920.2.a.d 1 84.h odd 2 1
9680.2.a.x 1 308.g odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5390))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{13} - 6$$ T13 - 6 $$T_{17} - 7$$ T17 - 7 $$T_{19} + 5$$ T19 + 5 $$T_{31} - 3$$ T31 - 3

## Hecke characteristic polynomials

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