# Properties

 Label 2850.2.d.x 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
1.00000i 1.00000i −1.00000 0 1.00000 2.73205i 1.00000i −1.00000 0
799.2 1.00000i 1.00000i −1.00000 0 1.00000 0.732051i 1.00000i −1.00000 0
799.3 1.00000i 1.00000i −1.00000 0 1.00000 0.732051i 1.00000i −1.00000 0
799.4 1.00000i 1.00000i −1.00000 0 1.00000 2.73205i 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.x 4
5.b even 2 1 inner 2850.2.d.x 4
5.c odd 4 1 2850.2.a.bd 2
5.c odd 4 1 2850.2.a.bi yes 2
15.e even 4 1 8550.2.a.bs 2
15.e even 4 1 8550.2.a.by 2

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

 $$T_{7}^{4} + 8 T_{7}^{2} + 4$$ $$T_{11}^{2} - 4 T_{11} + 1$$ $$T_{13}^{4} + 8 T_{13}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$4 + 8 T^{2} + T^{4}$$
$11$ $$( 1 - 4 T + T^{2} )^{2}$$
$13$ $$4 + 8 T^{2} + T^{4}$$
$17$ $$676 + 56 T^{2} + T^{4}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$2209 + 98 T^{2} + T^{4}$$
$29$ $$( -3 + T^{2} )^{2}$$
$31$ $$( -11 - 2 T + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( -32 - 8 T + T^{2} )^{2}$$
$43$ $$324 + 72 T^{2} + T^{4}$$
$47$ $$( 12 + T^{2} )^{2}$$
$53$ $$( 3 + T^{2} )^{2}$$
$59$ $$( -18 - 6 T + T^{2} )^{2}$$
$61$ $$( -71 - 4 T + T^{2} )^{2}$$
$67$ $$1 + 14 T^{2} + T^{4}$$
$71$ $$( 22 - 14 T + T^{2} )^{2}$$
$73$ $$121 + 26 T^{2} + T^{4}$$
$79$ $$( 69 - 18 T + T^{2} )^{2}$$
$83$ $$1 + 14 T^{2} + T^{4}$$
$89$ $$( -183 - 6 T + T^{2} )^{2}$$
$97$ $$19044 + 312 T^{2} + T^{4}$$