# Properties

 Label 850.2.a.e Level $850$ Weight $2$ Character orbit 850.a Self dual yes Analytic conductor $6.787$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$850 = 2 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.78728417181$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 34) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} + 4q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} + 4q^{7} - q^{8} + q^{9} + 6q^{11} + 2q^{12} - 2q^{13} - 4q^{14} + q^{16} + q^{17} - q^{18} - 4q^{19} + 8q^{21} - 6q^{22} - 2q^{24} + 2q^{26} - 4q^{27} + 4q^{28} - 4q^{31} - q^{32} + 12q^{33} - q^{34} + q^{36} + 4q^{37} + 4q^{38} - 4q^{39} + 6q^{41} - 8q^{42} - 8q^{43} + 6q^{44} + 2q^{48} + 9q^{49} + 2q^{51} - 2q^{52} + 6q^{53} + 4q^{54} - 4q^{56} - 8q^{57} - 4q^{61} + 4q^{62} + 4q^{63} + q^{64} - 12q^{66} - 8q^{67} + q^{68} - q^{72} - 2q^{73} - 4q^{74} - 4q^{76} + 24q^{77} + 4q^{78} + 8q^{79} - 11q^{81} - 6q^{82} + 8q^{84} + 8q^{86} - 6q^{88} - 6q^{89} - 8q^{91} - 8q^{93} - 2q^{96} - 14q^{97} - 9q^{98} + 6q^{99} + O(q^{100})$$

## 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 2.00000 1.00000 0 −2.00000 4.00000 −1.00000 1.00000 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
$$2$$ $$1$$
$$5$$ $$1$$
$$17$$ $$-1$$

## 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 850.2.a.e 1
3.b odd 2 1 7650.2.a.ci 1
4.b odd 2 1 6800.2.a.b 1
5.b even 2 1 34.2.a.a 1
5.c odd 4 2 850.2.c.b 2
15.d odd 2 1 306.2.a.a 1
20.d odd 2 1 272.2.a.d 1
35.c odd 2 1 1666.2.a.m 1
40.e odd 2 1 1088.2.a.d 1
40.f even 2 1 1088.2.a.l 1
55.d odd 2 1 4114.2.a.a 1
60.h even 2 1 2448.2.a.k 1
65.d even 2 1 5746.2.a.b 1
85.c even 2 1 578.2.a.a 1
85.j even 4 2 578.2.b.a 2
85.m even 8 4 578.2.c.e 4
85.p odd 16 8 578.2.d.e 8
120.i odd 2 1 9792.2.a.y 1
120.m even 2 1 9792.2.a.bj 1
255.h odd 2 1 5202.2.a.d 1
340.d odd 2 1 4624.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 5.b even 2 1
272.2.a.d 1 20.d odd 2 1
306.2.a.a 1 15.d odd 2 1
578.2.a.a 1 85.c even 2 1
578.2.b.a 2 85.j even 4 2
578.2.c.e 4 85.m even 8 4
578.2.d.e 8 85.p odd 16 8
850.2.a.e 1 1.a even 1 1 trivial
850.2.c.b 2 5.c odd 4 2
1088.2.a.d 1 40.e odd 2 1
1088.2.a.l 1 40.f even 2 1
1666.2.a.m 1 35.c odd 2 1
2448.2.a.k 1 60.h even 2 1
4114.2.a.a 1 55.d odd 2 1
4624.2.a.a 1 340.d odd 2 1
5202.2.a.d 1 255.h odd 2 1
5746.2.a.b 1 65.d even 2 1
6800.2.a.b 1 4.b odd 2 1
7650.2.a.ci 1 3.b odd 2 1
9792.2.a.y 1 120.i odd 2 1
9792.2.a.bj 1 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(850))$$:

 $$T_{3} - 2$$ $$T_{7} - 4$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

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