# Properties

 Label 1859.2.a.b Level $1859$ Weight $2$ Character orbit 1859.a Self dual yes Analytic conductor $14.844$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1859.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.8441897358$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9}+O(q^{10})$$ q + 2 * q^2 - q^3 + 2 * q^4 - q^5 - 2 * q^6 + 2 * q^7 - 2 * q^9 $$q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9} - 2 q^{10} - q^{11} - 2 q^{12} + 4 q^{14} + q^{15} - 4 q^{16} - 2 q^{17} - 4 q^{18} - 2 q^{20} - 2 q^{21} - 2 q^{22} - q^{23} - 4 q^{25} + 5 q^{27} + 4 q^{28} + 2 q^{30} - 7 q^{31} - 8 q^{32} + q^{33} - 4 q^{34} - 2 q^{35} - 4 q^{36} - 3 q^{37} + 8 q^{41} - 4 q^{42} - 6 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{46} - 8 q^{47} + 4 q^{48} - 3 q^{49} - 8 q^{50} + 2 q^{51} - 6 q^{53} + 10 q^{54} + q^{55} - 5 q^{59} + 2 q^{60} + 12 q^{61} - 14 q^{62} - 4 q^{63} - 8 q^{64} + 2 q^{66} + 7 q^{67} - 4 q^{68} + q^{69} - 4 q^{70} + 3 q^{71} - 4 q^{73} - 6 q^{74} + 4 q^{75} - 2 q^{77} - 10 q^{79} + 4 q^{80} + q^{81} + 16 q^{82} + 6 q^{83} - 4 q^{84} + 2 q^{85} - 12 q^{86} - 15 q^{89} + 4 q^{90} - 2 q^{92} + 7 q^{93} - 16 q^{94} + 8 q^{96} + 7 q^{97} - 6 q^{98} + 2 q^{99}+O(q^{100})$$ q + 2 * q^2 - q^3 + 2 * q^4 - q^5 - 2 * q^6 + 2 * q^7 - 2 * q^9 - 2 * q^10 - q^11 - 2 * q^12 + 4 * q^14 + q^15 - 4 * q^16 - 2 * q^17 - 4 * q^18 - 2 * q^20 - 2 * q^21 - 2 * q^22 - q^23 - 4 * q^25 + 5 * q^27 + 4 * q^28 + 2 * q^30 - 7 * q^31 - 8 * q^32 + q^33 - 4 * q^34 - 2 * q^35 - 4 * q^36 - 3 * q^37 + 8 * q^41 - 4 * q^42 - 6 * q^43 - 2 * q^44 + 2 * q^45 - 2 * q^46 - 8 * q^47 + 4 * q^48 - 3 * q^49 - 8 * q^50 + 2 * q^51 - 6 * q^53 + 10 * q^54 + q^55 - 5 * q^59 + 2 * q^60 + 12 * q^61 - 14 * q^62 - 4 * q^63 - 8 * q^64 + 2 * q^66 + 7 * q^67 - 4 * q^68 + q^69 - 4 * q^70 + 3 * q^71 - 4 * q^73 - 6 * q^74 + 4 * q^75 - 2 * q^77 - 10 * q^79 + 4 * q^80 + q^81 + 16 * q^82 + 6 * q^83 - 4 * q^84 + 2 * q^85 - 12 * q^86 - 15 * q^89 + 4 * q^90 - 2 * q^92 + 7 * q^93 - 16 * q^94 + 8 * q^96 + 7 * q^97 - 6 * 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
2.00000 −1.00000 2.00000 −1.00000 −2.00000 2.00000 0 −2.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$
$$13$$ $$1$$

## 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 1859.2.a.b 1
13.b even 2 1 11.2.a.a 1
39.d odd 2 1 99.2.a.d 1
52.b odd 2 1 176.2.a.b 1
65.d even 2 1 275.2.a.b 1
65.h odd 4 2 275.2.b.a 2
91.b odd 2 1 539.2.a.a 1
91.r even 6 2 539.2.e.h 2
91.s odd 6 2 539.2.e.g 2
104.e even 2 1 704.2.a.h 1
104.h odd 2 1 704.2.a.c 1
117.n odd 6 2 891.2.e.b 2
117.t even 6 2 891.2.e.k 2
143.d odd 2 1 121.2.a.d 1
143.l odd 10 4 121.2.c.a 4
143.n even 10 4 121.2.c.e 4
156.h even 2 1 1584.2.a.g 1
195.e odd 2 1 2475.2.a.a 1
195.s even 4 2 2475.2.c.a 2
208.o odd 4 2 2816.2.c.f 2
208.p even 4 2 2816.2.c.j 2
221.b even 2 1 3179.2.a.a 1
247.d odd 2 1 3971.2.a.b 1
260.g odd 2 1 4400.2.a.i 1
260.p even 4 2 4400.2.b.h 2
273.g even 2 1 4851.2.a.t 1
299.c odd 2 1 5819.2.a.a 1
312.b odd 2 1 6336.2.a.br 1
312.h even 2 1 6336.2.a.bu 1
364.h even 2 1 8624.2.a.j 1
377.d even 2 1 9251.2.a.d 1
429.e even 2 1 1089.2.a.b 1
572.b even 2 1 1936.2.a.i 1
715.c odd 2 1 3025.2.a.a 1
1001.g even 2 1 5929.2.a.h 1
1144.h odd 2 1 7744.2.a.x 1
1144.o even 2 1 7744.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 13.b even 2 1
99.2.a.d 1 39.d odd 2 1
121.2.a.d 1 143.d odd 2 1
121.2.c.a 4 143.l odd 10 4
121.2.c.e 4 143.n even 10 4
176.2.a.b 1 52.b odd 2 1
275.2.a.b 1 65.d even 2 1
275.2.b.a 2 65.h odd 4 2
539.2.a.a 1 91.b odd 2 1
539.2.e.g 2 91.s odd 6 2
539.2.e.h 2 91.r even 6 2
704.2.a.c 1 104.h odd 2 1
704.2.a.h 1 104.e even 2 1
891.2.e.b 2 117.n odd 6 2
891.2.e.k 2 117.t even 6 2
1089.2.a.b 1 429.e even 2 1
1584.2.a.g 1 156.h even 2 1
1859.2.a.b 1 1.a even 1 1 trivial
1936.2.a.i 1 572.b even 2 1
2475.2.a.a 1 195.e odd 2 1
2475.2.c.a 2 195.s even 4 2
2816.2.c.f 2 208.o odd 4 2
2816.2.c.j 2 208.p even 4 2
3025.2.a.a 1 715.c odd 2 1
3179.2.a.a 1 221.b even 2 1
3971.2.a.b 1 247.d odd 2 1
4400.2.a.i 1 260.g odd 2 1
4400.2.b.h 2 260.p even 4 2
4851.2.a.t 1 273.g even 2 1
5819.2.a.a 1 299.c odd 2 1
5929.2.a.h 1 1001.g even 2 1
6336.2.a.br 1 312.b odd 2 1
6336.2.a.bu 1 312.h even 2 1
7744.2.a.k 1 1144.o even 2 1
7744.2.a.x 1 1144.h odd 2 1
8624.2.a.j 1 364.h even 2 1
9251.2.a.d 1 377.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1859))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

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