# Properties

 Label 850.2.c.b Level $850$ Weight $2$ Character orbit 850.c Analytic conductor $6.787$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$850 = 2 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.78728417181$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 34) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/850\mathbb{Z}\right)^\times$$.

 $$n$$ $$477$$ $$751$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
749.1
 − 1.00000i 1.00000i
1.00000i 2.00000i −1.00000 0 −2.00000 4.00000i 1.00000i −1.00000 0
749.2 1.00000i 2.00000i −1.00000 0 −2.00000 4.00000i 1.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ $$T_{7}^{2} + 16$$ $$T_{11} - 6$$ $$T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$1 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 4 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$196 + T^{2}$$