# Properties

 Label 2850.2.d.d 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: $$x^{2} + 1$$ 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.

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

## 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 2.00000i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 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.d 2
5.b even 2 1 inner 2850.2.d.d 2
5.c odd 4 1 570.2.a.f 1
5.c odd 4 1 2850.2.a.q 1
15.e even 4 1 1710.2.a.o 1
15.e even 4 1 8550.2.a.e 1
20.e even 4 1 4560.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.a.f 1 5.c odd 4 1
1710.2.a.o 1 15.e even 4 1
2850.2.a.q 1 5.c odd 4 1
2850.2.d.d 2 1.a even 1 1 trivial
2850.2.d.d 2 5.b even 2 1 inner
4560.2.a.m 1 20.e even 4 1
8550.2.a.e 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 $$S_{2}^{\mathrm{new}}(2850, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}$$ T11 $$T_{13}^{2} + 4$$ T13^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T + 2)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 12)^{2}$$
$97$ $$T^{2} + 100$$