# Properties

 Label 6050.2.a.bj Level $6050$ Weight $2$ Character orbit 6050.a Self dual yes Analytic conductor $48.309$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6050 = 2 \cdot 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.3094932229$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + 3 q^{7} + q^{8} - 2 q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 + 3 * q^7 + q^8 - 2 * q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} + 3 q^{7} + q^{8} - 2 q^{9} + q^{12} - 6 q^{13} + 3 q^{14} + q^{16} - 7 q^{17} - 2 q^{18} - 5 q^{19} + 3 q^{21} + 6 q^{23} + q^{24} - 6 q^{26} - 5 q^{27} + 3 q^{28} - 5 q^{29} - 3 q^{31} + q^{32} - 7 q^{34} - 2 q^{36} - 3 q^{37} - 5 q^{38} - 6 q^{39} - 2 q^{41} + 3 q^{42} + 4 q^{43} + 6 q^{46} + 2 q^{47} + q^{48} + 2 q^{49} - 7 q^{51} - 6 q^{52} + q^{53} - 5 q^{54} + 3 q^{56} - 5 q^{57} - 5 q^{58} - 10 q^{59} - 7 q^{61} - 3 q^{62} - 6 q^{63} + q^{64} - 8 q^{67} - 7 q^{68} + 6 q^{69} + 7 q^{71} - 2 q^{72} + 14 q^{73} - 3 q^{74} - 5 q^{76} - 6 q^{78} - 10 q^{79} + q^{81} - 2 q^{82} - 6 q^{83} + 3 q^{84} + 4 q^{86} - 5 q^{87} - 15 q^{89} - 18 q^{91} + 6 q^{92} - 3 q^{93} + 2 q^{94} + q^{96} + 12 q^{97} + 2 q^{98}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 + 3 * q^7 + q^8 - 2 * q^9 + q^12 - 6 * q^13 + 3 * q^14 + q^16 - 7 * q^17 - 2 * q^18 - 5 * q^19 + 3 * q^21 + 6 * q^23 + q^24 - 6 * q^26 - 5 * q^27 + 3 * q^28 - 5 * q^29 - 3 * q^31 + q^32 - 7 * q^34 - 2 * q^36 - 3 * q^37 - 5 * q^38 - 6 * q^39 - 2 * q^41 + 3 * q^42 + 4 * q^43 + 6 * q^46 + 2 * q^47 + q^48 + 2 * q^49 - 7 * q^51 - 6 * q^52 + q^53 - 5 * q^54 + 3 * q^56 - 5 * q^57 - 5 * q^58 - 10 * q^59 - 7 * q^61 - 3 * q^62 - 6 * q^63 + q^64 - 8 * q^67 - 7 * q^68 + 6 * q^69 + 7 * q^71 - 2 * q^72 + 14 * q^73 - 3 * q^74 - 5 * q^76 - 6 * q^78 - 10 * q^79 + q^81 - 2 * q^82 - 6 * q^83 + 3 * q^84 + 4 * q^86 - 5 * q^87 - 15 * q^89 - 18 * q^91 + 6 * q^92 - 3 * q^93 + 2 * q^94 + q^96 + 12 * q^97 + 2 * q^98

## 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 0 1.00000 3.00000 1.00000 −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 6050.2.a.bj 1
5.b even 2 1 1210.2.a.b 1
11.b odd 2 1 550.2.a.f 1
20.d odd 2 1 9680.2.a.x 1
33.d even 2 1 4950.2.a.bc 1
44.c even 2 1 4400.2.a.l 1
55.d odd 2 1 110.2.a.b 1
55.e even 4 2 550.2.b.a 2
165.d even 2 1 990.2.a.d 1
165.l odd 4 2 4950.2.c.m 2
220.g even 2 1 880.2.a.i 1
220.i odd 4 2 4400.2.b.i 2
385.h even 2 1 5390.2.a.bf 1
440.c even 2 1 3520.2.a.h 1
440.o odd 2 1 3520.2.a.y 1
660.g odd 2 1 7920.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.b 1 55.d odd 2 1
550.2.a.f 1 11.b odd 2 1
550.2.b.a 2 55.e even 4 2
880.2.a.i 1 220.g even 2 1
990.2.a.d 1 165.d even 2 1
1210.2.a.b 1 5.b even 2 1
3520.2.a.h 1 440.c even 2 1
3520.2.a.y 1 440.o odd 2 1
4400.2.a.l 1 44.c even 2 1
4400.2.b.i 2 220.i odd 4 2
4950.2.a.bc 1 33.d even 2 1
4950.2.c.m 2 165.l odd 4 2
5390.2.a.bf 1 385.h even 2 1
6050.2.a.bj 1 1.a even 1 1 trivial
7920.2.a.d 1 660.g odd 2 1
9680.2.a.x 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6050))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} - 3$$ T7 - 3 $$T_{13} + 6$$ T13 + 6 $$T_{17} + 7$$ T17 + 7 $$T_{19} + 5$$ T19 + 5

## Hecke characteristic polynomials

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