# Properties

 Label 2850.2.d.f Level $2850$ Weight $2$ Character orbit 2850.d Analytic conductor $22.757$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2850.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.7573645761$$ 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 570) 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} -i q^{3} - q^{4} - q^{6} + 4 i q^{7} + i q^{8} - q^{9} +O(q^{10})$$ $$q -i q^{2} -i q^{3} - q^{4} - q^{6} + 4 i q^{7} + i q^{8} - q^{9} + i q^{12} -2 i q^{13} + 4 q^{14} + q^{16} -2 i q^{17} + i q^{18} + q^{19} + 4 q^{21} + q^{24} -2 q^{26} + i q^{27} -4 i q^{28} -10 q^{29} -i q^{32} -2 q^{34} + q^{36} + 2 i q^{37} -i q^{38} -2 q^{39} + 2 q^{41} -4 i q^{42} + 4 i q^{43} -i q^{48} -9 q^{49} -2 q^{51} + 2 i q^{52} + 6 i q^{53} + q^{54} -4 q^{56} -i q^{57} + 10 i q^{58} -8 q^{59} + 6 q^{61} -4 i q^{63} - q^{64} + 12 i q^{67} + 2 i q^{68} -i q^{72} + 14 i q^{73} + 2 q^{74} - q^{76} + 2 i q^{78} + q^{81} -2 i q^{82} + 12 i q^{83} -4 q^{84} + 4 q^{86} + 10 i q^{87} -10 q^{89} + 8 q^{91} - q^{96} + 2 i q^{97} + 9 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + 8q^{14} + 2q^{16} + 2q^{19} + 8q^{21} + 2q^{24} - 4q^{26} - 20q^{29} - 4q^{34} + 2q^{36} - 4q^{39} + 4q^{41} - 18q^{49} - 4q^{51} + 2q^{54} - 8q^{56} - 16q^{59} + 12q^{61} - 2q^{64} + 4q^{74} - 2q^{76} + 2q^{81} - 8q^{84} + 8q^{86} - 20q^{89} + 16q^{91} - 2q^{96} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1027$$ $$1351$$ $$1901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 4.00000i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 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 2850.2.d.f 2
5.b even 2 1 inner 2850.2.d.f 2
5.c odd 4 1 570.2.a.d 1
5.c odd 4 1 2850.2.a.p 1
15.e even 4 1 1710.2.a.t 1
15.e even 4 1 8550.2.a.b 1
20.e even 4 1 4560.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.d 1 5.c odd 4 1
1710.2.a.t 1 15.e even 4 1
2850.2.a.p 1 5.c odd 4 1
2850.2.d.f 2 1.a even 1 1 trivial
2850.2.d.f 2 5.b even 2 1 inner
4560.2.a.a 1 20.e even 4 1
8550.2.a.b 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2850, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( 8 + T )^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$144 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$196 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$4 + T^{2}$$