# Properties

 Label 2850.2.d.w 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{1} q^{3} - q^{4} + q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{7} -\beta_{1} q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{1} q^{3} - q^{4} + q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{7} -\beta_{1} q^{8} - q^{9} + ( 1 - \beta_{3} ) q^{11} + \beta_{1} q^{12} -\beta_{2} q^{13} + ( -2 + \beta_{3} ) q^{14} + q^{16} + ( 2 \beta_{1} - \beta_{2} ) q^{17} -\beta_{1} q^{18} - q^{19} + ( 2 - \beta_{3} ) q^{21} + ( \beta_{1} - \beta_{2} ) q^{22} + \beta_{1} q^{23} - q^{24} + \beta_{3} q^{26} + \beta_{1} q^{27} + ( -2 \beta_{1} + \beta_{2} ) q^{28} + ( -3 - 3 \beta_{3} ) q^{29} -3 q^{31} + \beta_{1} q^{32} + ( -\beta_{1} + \beta_{2} ) q^{33} + ( -2 + \beta_{3} ) q^{34} + q^{36} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{37} -\beta_{1} q^{38} -\beta_{3} q^{39} + ( 4 + 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} - \beta_{2} ) q^{42} -\beta_{2} q^{43} + ( -1 + \beta_{3} ) q^{44} - q^{46} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{47} -\beta_{1} q^{48} + ( -3 + 4 \beta_{3} ) q^{49} + ( 2 - \beta_{3} ) q^{51} + \beta_{2} q^{52} + ( 5 \beta_{1} - \beta_{2} ) q^{53} - q^{54} + ( 2 - \beta_{3} ) q^{56} + \beta_{1} q^{57} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{58} + ( -4 + \beta_{3} ) q^{59} + ( -7 + \beta_{3} ) q^{61} -3 \beta_{1} q^{62} + ( -2 \beta_{1} + \beta_{2} ) q^{63} - q^{64} + ( 1 - \beta_{3} ) q^{66} + ( 3 \beta_{1} - 5 \beta_{2} ) q^{67} + ( -2 \beta_{1} + \beta_{2} ) q^{68} + q^{69} + ( -4 - \beta_{3} ) q^{71} + \beta_{1} q^{72} + \beta_{1} q^{73} + ( -2 - 4 \beta_{3} ) q^{74} + q^{76} + ( 8 \beta_{1} - 3 \beta_{2} ) q^{77} -\beta_{2} q^{78} + 5 q^{79} + q^{81} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{82} + ( -\beta_{1} + 3 \beta_{2} ) q^{83} + ( -2 + \beta_{3} ) q^{84} + \beta_{3} q^{86} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{87} + ( -\beta_{1} + \beta_{2} ) q^{88} + ( -7 - 2 \beta_{3} ) q^{89} + ( -6 + 2 \beta_{3} ) q^{91} -\beta_{1} q^{92} + 3 \beta_{1} q^{93} + ( -2 - 4 \beta_{3} ) q^{94} + q^{96} + ( 4 \beta_{1} + \beta_{2} ) q^{97} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{98} + ( -1 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + 4q^{11} - 8q^{14} + 4q^{16} - 4q^{19} + 8q^{21} - 4q^{24} - 12q^{29} - 12q^{31} - 8q^{34} + 4q^{36} + 16q^{41} - 4q^{44} - 4q^{46} - 12q^{49} + 8q^{51} - 4q^{54} + 8q^{56} - 16q^{59} - 28q^{61} - 4q^{64} + 4q^{66} + 4q^{69} - 16q^{71} - 8q^{74} + 4q^{76} + 20q^{79} + 4q^{81} - 8q^{84} - 28q^{89} - 24q^{91} - 8q^{94} + 4q^{96} - 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

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

## 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
 −1.22474 + 1.22474i 1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 − 1.22474i
1.00000i 1.00000i −1.00000 0 1.00000 4.44949i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 1.00000 0.449490i 1.00000i −1.00000 0
799.3 1.00000i 1.00000i −1.00000 0 1.00000 0.449490i 1.00000i −1.00000 0
799.4 1.00000i 1.00000i −1.00000 0 1.00000 4.44949i 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.w 4
5.b even 2 1 inner 2850.2.d.w 4
5.c odd 4 1 2850.2.a.bc 2
5.c odd 4 1 2850.2.a.bj yes 2
15.e even 4 1 8550.2.a.bu 2
15.e even 4 1 8550.2.a.bv 2

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

 $$T_{7}^{4} + 20 T_{7}^{2} + 4$$ $$T_{11}^{2} - 2 T_{11} - 5$$ $$T_{13}^{2} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$4 + 20 T^{2} + T^{4}$$
$11$ $$( -5 - 2 T + T^{2} )^{2}$$
$13$ $$( 6 + T^{2} )^{2}$$
$17$ $$4 + 20 T^{2} + T^{4}$$
$19$ $$( 1 + T )^{4}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( -45 + 6 T + T^{2} )^{2}$$
$31$ $$( 3 + T )^{4}$$
$37$ $$8464 + 200 T^{2} + T^{4}$$
$41$ $$( -8 - 8 T + T^{2} )^{2}$$
$43$ $$( 6 + T^{2} )^{2}$$
$47$ $$8464 + 200 T^{2} + T^{4}$$
$53$ $$361 + 62 T^{2} + T^{4}$$
$59$ $$( 10 + 8 T + T^{2} )^{2}$$
$61$ $$( 43 + 14 T + T^{2} )^{2}$$
$67$ $$19881 + 318 T^{2} + T^{4}$$
$71$ $$( 10 + 8 T + T^{2} )^{2}$$
$73$ $$( 1 + T^{2} )^{2}$$
$79$ $$( -5 + T )^{4}$$
$83$ $$2809 + 110 T^{2} + T^{4}$$
$89$ $$( 25 + 14 T + T^{2} )^{2}$$
$97$ $$100 + 44 T^{2} + T^{4}$$