# Properties

 Label 850.2.b.c Level 850 Weight 2 Character orbit 850.b 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.b (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 170) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + i q^{3} + q^{4} -i q^{6} -2 i q^{7} - q^{8} + 2 q^{9} +O(q^{10})$$ $$q - q^{2} + i q^{3} + q^{4} -i q^{6} -2 i q^{7} - q^{8} + 2 q^{9} + i q^{12} + q^{13} + 2 i q^{14} + q^{16} + ( -4 - i ) q^{17} -2 q^{18} + 5 q^{19} + 2 q^{21} -4 i q^{23} -i q^{24} - q^{26} + 5 i q^{27} -2 i q^{28} -9 i q^{29} -5 i q^{31} - q^{32} + ( 4 + i ) q^{34} + 2 q^{36} -2 i q^{37} -5 q^{38} + i q^{39} + 10 i q^{41} -2 q^{42} + 6 q^{43} + 4 i q^{46} + 7 q^{47} + i q^{48} + 3 q^{49} + ( 1 - 4 i ) q^{51} + q^{52} + q^{53} -5 i q^{54} + 2 i q^{56} + 5 i q^{57} + 9 i q^{58} + 5 q^{59} -5 i q^{61} + 5 i q^{62} -4 i q^{63} + q^{64} + 2 q^{67} + ( -4 - i ) q^{68} + 4 q^{69} -5 i q^{71} -2 q^{72} + 11 i q^{73} + 2 i q^{74} + 5 q^{76} -i q^{78} + 16 i q^{79} + q^{81} -10 i q^{82} + 6 q^{83} + 2 q^{84} -6 q^{86} + 9 q^{87} -5 q^{89} -2 i q^{91} -4 i q^{92} + 5 q^{93} -7 q^{94} -i q^{96} -7 i q^{97} -3 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} + 4q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{8} + 4q^{9} + 2q^{13} + 2q^{16} - 8q^{17} - 4q^{18} + 10q^{19} + 4q^{21} - 2q^{26} - 2q^{32} + 8q^{34} + 4q^{36} - 10q^{38} - 4q^{42} + 12q^{43} + 14q^{47} + 6q^{49} + 2q^{51} + 2q^{52} + 2q^{53} + 10q^{59} + 2q^{64} + 4q^{67} - 8q^{68} + 8q^{69} - 4q^{72} + 10q^{76} + 2q^{81} + 12q^{83} + 4q^{84} - 12q^{86} + 18q^{87} - 10q^{89} + 10q^{93} - 14q^{94} - 6q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 − 1.00000i 1.00000i
−1.00000 1.00000i 1.00000 0 1.00000i 2.00000i −1.00000 2.00000 0
101.2 −1.00000 1.00000i 1.00000 0 1.00000i 2.00000i −1.00000 2.00000 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
17.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.b.c 2
5.b even 2 1 850.2.b.i 2
5.c odd 4 1 170.2.d.a 2
5.c odd 4 1 170.2.d.b yes 2
15.e even 4 1 1530.2.f.b 2
15.e even 4 1 1530.2.f.e 2
17.b even 2 1 inner 850.2.b.c 2
20.e even 4 1 1360.2.o.a 2
20.e even 4 1 1360.2.o.b 2
85.c even 2 1 850.2.b.i 2
85.g odd 4 1 170.2.d.a 2
85.g odd 4 1 170.2.d.b yes 2
255.o even 4 1 1530.2.f.b 2
255.o even 4 1 1530.2.f.e 2
340.r even 4 1 1360.2.o.a 2
340.r even 4 1 1360.2.o.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.d.a 2 5.c odd 4 1
170.2.d.a 2 85.g odd 4 1
170.2.d.b yes 2 5.c odd 4 1
170.2.d.b yes 2 85.g odd 4 1
850.2.b.c 2 1.a even 1 1 trivial
850.2.b.c 2 17.b even 2 1 inner
850.2.b.i 2 5.b even 2 1
850.2.b.i 2 85.c even 2 1
1360.2.o.a 2 20.e even 4 1
1360.2.o.a 2 340.r even 4 1
1360.2.o.b 2 20.e even 4 1
1360.2.o.b 2 340.r even 4 1
1530.2.f.b 2 15.e even 4 1
1530.2.f.b 2 255.o even 4 1
1530.2.f.e 2 15.e even 4 1
1530.2.f.e 2 255.o even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ $$T_{7}^{2} + 4$$ $$T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 - 5 T^{2} + 9 T^{4}$$
$5$ 1
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ $$( 1 - T + 13 T^{2} )^{2}$$
$17$ $$1 + 8 T + 17 T^{2}$$
$19$ $$( 1 - 5 T + 19 T^{2} )^{2}$$
$23$ $$1 - 30 T^{2} + 529 T^{4}$$
$29$ $$1 + 23 T^{2} + 841 T^{4}$$
$31$ $$1 - 37 T^{2} + 961 T^{4}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 - 8 T + 41 T^{2} )( 1 + 8 T + 41 T^{2} )$$
$43$ $$( 1 - 6 T + 43 T^{2} )^{2}$$
$47$ $$( 1 - 7 T + 47 T^{2} )^{2}$$
$53$ $$( 1 - T + 53 T^{2} )^{2}$$
$59$ $$( 1 - 5 T + 59 T^{2} )^{2}$$
$61$ $$1 - 97 T^{2} + 3721 T^{4}$$
$67$ $$( 1 - 2 T + 67 T^{2} )^{2}$$
$71$ $$1 - 117 T^{2} + 5041 T^{4}$$
$73$ $$1 - 25 T^{2} + 5329 T^{4}$$
$79$ $$1 + 98 T^{2} + 6241 T^{4}$$
$83$ $$( 1 - 6 T + 83 T^{2} )^{2}$$
$89$ $$( 1 + 5 T + 89 T^{2} )^{2}$$
$97$ $$1 - 145 T^{2} + 9409 T^{4}$$