# Properties

 Label 2850.2.d.v Level $2850$ Weight $2$ Character orbit 2850.d Analytic conductor $22.757$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{10})$$ Defining polynomial: $$x^{4} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} - q^{6} + \beta_{2} q^{7} -\beta_{1} q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} - q^{6} + \beta_{2} q^{7} -\beta_{1} q^{8} - q^{9} + ( 1 + \beta_{3} ) q^{11} -\beta_{1} q^{12} + ( -2 \beta_{1} - \beta_{2} ) q^{13} -\beta_{3} q^{14} + q^{16} + ( 4 \beta_{1} - \beta_{2} ) q^{17} -\beta_{1} q^{18} + q^{19} -\beta_{3} q^{21} + ( \beta_{1} + \beta_{2} ) q^{22} + ( -\beta_{1} + 2 \beta_{2} ) q^{23} + q^{24} + ( 2 + \beta_{3} ) q^{26} -\beta_{1} q^{27} -\beta_{2} q^{28} + ( 7 - \beta_{3} ) q^{29} + ( -1 + 2 \beta_{3} ) q^{31} + \beta_{1} q^{32} + ( \beta_{1} + \beta_{2} ) q^{33} + ( -4 + \beta_{3} ) q^{34} + q^{36} + 10 \beta_{1} q^{37} + \beta_{1} q^{38} + ( 2 + \beta_{3} ) q^{39} -2 \beta_{3} q^{41} -\beta_{2} q^{42} + ( 6 \beta_{1} + \beta_{2} ) q^{43} + ( -1 - \beta_{3} ) q^{44} + ( 1 - 2 \beta_{3} ) q^{46} + 6 \beta_{1} q^{47} + \beta_{1} q^{48} -3 q^{49} + ( -4 + \beta_{3} ) q^{51} + ( 2 \beta_{1} + \beta_{2} ) q^{52} + ( -5 \beta_{1} + \beta_{2} ) q^{53} + q^{54} + \beta_{3} q^{56} + \beta_{1} q^{57} + ( 7 \beta_{1} - \beta_{2} ) q^{58} + ( -2 + 3 \beta_{3} ) q^{59} + ( -1 - \beta_{3} ) q^{61} + ( -\beta_{1} + 2 \beta_{2} ) q^{62} -\beta_{2} q^{63} - q^{64} + ( -1 - \beta_{3} ) q^{66} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -4 \beta_{1} + \beta_{2} ) q^{68} + ( 1 - 2 \beta_{3} ) q^{69} + ( -2 + \beta_{3} ) q^{71} + \beta_{1} q^{72} + \beta_{1} q^{73} -10 q^{74} - q^{76} + ( 10 \beta_{1} + \beta_{2} ) q^{77} + ( 2 \beta_{1} + \beta_{2} ) q^{78} + ( -1 + 2 \beta_{3} ) q^{79} + q^{81} -2 \beta_{2} q^{82} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{83} + \beta_{3} q^{84} + ( -6 - \beta_{3} ) q^{86} + ( 7 \beta_{1} - \beta_{2} ) q^{87} + ( -\beta_{1} - \beta_{2} ) q^{88} + ( 1 + 2 \beta_{3} ) q^{89} + ( 10 + 2 \beta_{3} ) q^{91} + ( \beta_{1} - 2 \beta_{2} ) q^{92} + ( -\beta_{1} + 2 \beta_{2} ) q^{93} -6 q^{94} - q^{96} + ( 10 \beta_{1} - 3 \beta_{2} ) q^{97} -3 \beta_{1} q^{98} + ( -1 - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{6} - 4q^{9} + 4q^{11} + 4q^{16} + 4q^{19} + 4q^{24} + 8q^{26} + 28q^{29} - 4q^{31} - 16q^{34} + 4q^{36} + 8q^{39} - 4q^{44} + 4q^{46} - 12q^{49} - 16q^{51} + 4q^{54} - 8q^{59} - 4q^{61} - 4q^{64} - 4q^{66} + 4q^{69} - 8q^{71} - 40q^{74} - 4q^{76} - 4q^{79} + 4q^{81} - 24q^{86} + 4q^{89} + 40q^{91} - 24q^{94} - 4q^{96} - 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{3} + 5 \beta_{2}$$$$)/2$$

## 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
 1.58114 − 1.58114i −1.58114 + 1.58114i −1.58114 − 1.58114i 1.58114 + 1.58114i
1.00000i 1.00000i −1.00000 0 −1.00000 3.16228i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 −1.00000 3.16228i 1.00000i −1.00000 0
799.3 1.00000i 1.00000i −1.00000 0 −1.00000 3.16228i 1.00000i −1.00000 0
799.4 1.00000i 1.00000i −1.00000 0 −1.00000 3.16228i 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.v 4
5.b even 2 1 inner 2850.2.d.v 4
5.c odd 4 1 2850.2.a.bf 2
5.c odd 4 1 2850.2.a.bg yes 2
15.e even 4 1 8550.2.a.bq 2
15.e even 4 1 8550.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bf 2 5.c odd 4 1
2850.2.a.bg yes 2 5.c odd 4 1
2850.2.d.v 4 1.a even 1 1 trivial
2850.2.d.v 4 5.b even 2 1 inner
8550.2.a.bq 2 15.e even 4 1
8550.2.a.ca 2 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 $$S_{2}^{\mathrm{new}}(2850, [\chi])$$:

 $$T_{7}^{2} + 10$$ $$T_{11}^{2} - 2 T_{11} - 9$$ $$T_{13}^{4} + 28 T_{13}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 10 + T^{2} )^{2}$$
$11$ $$( -9 - 2 T + T^{2} )^{2}$$
$13$ $$36 + 28 T^{2} + T^{4}$$
$17$ $$36 + 52 T^{2} + T^{4}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$1521 + 82 T^{2} + T^{4}$$
$29$ $$( 39 - 14 T + T^{2} )^{2}$$
$31$ $$( -39 + 2 T + T^{2} )^{2}$$
$37$ $$( 100 + T^{2} )^{2}$$
$41$ $$( -40 + T^{2} )^{2}$$
$43$ $$676 + 92 T^{2} + T^{4}$$
$47$ $$( 36 + T^{2} )^{2}$$
$53$ $$225 + 70 T^{2} + T^{4}$$
$59$ $$( -86 + 4 T + T^{2} )^{2}$$
$61$ $$( -9 + 2 T + T^{2} )^{2}$$
$67$ $$6561 + 198 T^{2} + T^{4}$$
$71$ $$( -6 + 4 T + T^{2} )^{2}$$
$73$ $$( 1 + T^{2} )^{2}$$
$79$ $$( -39 + 2 T + T^{2} )^{2}$$
$83$ $$6561 + 198 T^{2} + T^{4}$$
$89$ $$( -39 - 2 T + T^{2} )^{2}$$
$97$ $$100 + 380 T^{2} + T^{4}$$