# Properties

 Label 950.2.e.c Level $950$ Weight $2$ Character orbit 950.e Analytic conductor $7.586$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 4 q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + 4 q^{7} + q^{8} + 3 \zeta_{6} q^{9} - q^{11} -2 \zeta_{6} q^{13} + ( -4 + 4 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 3 - 3 \zeta_{6} ) q^{17} -3 q^{18} + ( 3 + 2 \zeta_{6} ) q^{19} + ( 1 - \zeta_{6} ) q^{22} -4 \zeta_{6} q^{23} + 2 q^{26} -4 \zeta_{6} q^{28} + 6 \zeta_{6} q^{29} -2 q^{31} -\zeta_{6} q^{32} + 3 \zeta_{6} q^{34} + ( 3 - 3 \zeta_{6} ) q^{36} -2 q^{37} + ( -5 + 3 \zeta_{6} ) q^{38} + ( 3 - 3 \zeta_{6} ) q^{41} + ( 12 - 12 \zeta_{6} ) q^{43} + \zeta_{6} q^{44} + 4 q^{46} + 6 \zeta_{6} q^{47} + 9 q^{49} + ( -2 + 2 \zeta_{6} ) q^{52} + 4 \zeta_{6} q^{53} + 4 q^{56} -6 q^{58} + ( -9 + 9 \zeta_{6} ) q^{59} + 12 \zeta_{6} q^{61} + ( 2 - 2 \zeta_{6} ) q^{62} + 12 \zeta_{6} q^{63} + q^{64} + 15 \zeta_{6} q^{67} -3 q^{68} + ( -6 + 6 \zeta_{6} ) q^{71} + 3 \zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 2 - 2 \zeta_{6} ) q^{74} + ( 2 - 5 \zeta_{6} ) q^{76} -4 q^{77} + ( 14 - 14 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + 3 \zeta_{6} q^{82} -4 q^{83} + 12 \zeta_{6} q^{86} - q^{88} + \zeta_{6} q^{89} -8 \zeta_{6} q^{91} + ( -4 + 4 \zeta_{6} ) q^{92} -6 q^{94} + ( -1 + \zeta_{6} ) q^{97} + ( -9 + 9 \zeta_{6} ) q^{98} -3 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 8q^{7} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 8q^{7} + 2q^{8} + 3q^{9} - 2q^{11} - 2q^{13} - 4q^{14} - q^{16} + 3q^{17} - 6q^{18} + 8q^{19} + q^{22} - 4q^{23} + 4q^{26} - 4q^{28} + 6q^{29} - 4q^{31} - q^{32} + 3q^{34} + 3q^{36} - 4q^{37} - 7q^{38} + 3q^{41} + 12q^{43} + q^{44} + 8q^{46} + 6q^{47} + 18q^{49} - 2q^{52} + 4q^{53} + 8q^{56} - 12q^{58} - 9q^{59} + 12q^{61} + 2q^{62} + 12q^{63} + 2q^{64} + 15q^{67} - 6q^{68} - 6q^{71} + 3q^{72} + 10q^{73} + 2q^{74} - q^{76} - 8q^{77} + 14q^{79} - 9q^{81} + 3q^{82} - 8q^{83} + 12q^{86} - 2q^{88} + q^{89} - 8q^{91} - 4q^{92} - 12q^{94} - q^{97} - 9q^{98} - 3q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
201.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 4.00000 1.00000 1.50000 + 2.59808i 0
501.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 4.00000 1.00000 1.50000 2.59808i 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.e.c 2
5.b even 2 1 950.2.e.e yes 2
5.c odd 4 2 950.2.j.c 4
19.c even 3 1 inner 950.2.e.c 2
95.i even 6 1 950.2.e.e yes 2
95.m odd 12 2 950.2.j.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.c 2 1.a even 1 1 trivial
950.2.e.c 2 19.c even 3 1 inner
950.2.e.e yes 2 5.b even 2 1
950.2.e.e yes 2 95.i even 6 1
950.2.j.c 4 5.c odd 4 2
950.2.j.c 4 95.m odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}$$ $$T_{7} - 4$$ $$T_{11} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$19 - 8 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$144 - 12 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$16 - 4 T + T^{2}$$
$59$ $$81 + 9 T + T^{2}$$
$61$ $$144 - 12 T + T^{2}$$
$67$ $$225 - 15 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$100 - 10 T + T^{2}$$
$79$ $$196 - 14 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$1 - T + T^{2}$$
$97$ $$1 + T + T^{2}$$