# Properties

 Label 950.2.e.a 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: no (minimal twist has level 190) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} -2 q^{7} + q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} -2 q^{7} + q^{8} + 2 \zeta_{6} q^{9} -3 q^{11} + q^{12} + 6 \zeta_{6} q^{13} + ( 2 - 2 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} -2 q^{18} + ( 2 + 3 \zeta_{6} ) q^{19} + ( 2 - 2 \zeta_{6} ) q^{21} + ( 3 - 3 \zeta_{6} ) q^{22} -8 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} -6 q^{26} -5 q^{27} + 2 \zeta_{6} q^{28} + 2 \zeta_{6} q^{29} -8 q^{31} -\zeta_{6} q^{32} + ( 3 - 3 \zeta_{6} ) q^{33} + 2 \zeta_{6} q^{34} + ( 2 - 2 \zeta_{6} ) q^{36} -8 q^{37} + ( -5 + 2 \zeta_{6} ) q^{38} -6 q^{39} + ( -5 + 5 \zeta_{6} ) q^{41} + 2 \zeta_{6} q^{42} + 3 \zeta_{6} q^{44} + 8 q^{46} -6 \zeta_{6} q^{47} -\zeta_{6} q^{48} -3 q^{49} + 2 \zeta_{6} q^{51} + ( 6 - 6 \zeta_{6} ) q^{52} + 6 \zeta_{6} q^{53} + ( 5 - 5 \zeta_{6} ) q^{54} -2 q^{56} + ( -5 + 2 \zeta_{6} ) q^{57} -2 q^{58} + ( -5 + 5 \zeta_{6} ) q^{59} -14 \zeta_{6} q^{61} + ( 8 - 8 \zeta_{6} ) q^{62} -4 \zeta_{6} q^{63} + q^{64} + 3 \zeta_{6} q^{66} -5 \zeta_{6} q^{67} -2 q^{68} + 8 q^{69} + ( 6 - 6 \zeta_{6} ) q^{71} + 2 \zeta_{6} q^{72} + ( -9 + 9 \zeta_{6} ) q^{73} + ( 8 - 8 \zeta_{6} ) q^{74} + ( 3 - 5 \zeta_{6} ) q^{76} + 6 q^{77} + ( 6 - 6 \zeta_{6} ) q^{78} + ( -8 + 8 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} -5 \zeta_{6} q^{82} + 11 q^{83} -2 q^{84} -2 q^{87} -3 q^{88} -14 \zeta_{6} q^{89} -12 \zeta_{6} q^{91} + ( -8 + 8 \zeta_{6} ) q^{92} + ( 8 - 8 \zeta_{6} ) q^{93} + 6 q^{94} + q^{96} + ( -15 + 15 \zeta_{6} ) q^{97} + ( 3 - 3 \zeta_{6} ) q^{98} -6 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} - q^{4} - q^{6} - 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} - q^{4} - q^{6} - 4q^{7} + 2q^{8} + 2q^{9} - 6q^{11} + 2q^{12} + 6q^{13} + 2q^{14} - q^{16} + 2q^{17} - 4q^{18} + 7q^{19} + 2q^{21} + 3q^{22} - 8q^{23} - q^{24} - 12q^{26} - 10q^{27} + 2q^{28} + 2q^{29} - 16q^{31} - q^{32} + 3q^{33} + 2q^{34} + 2q^{36} - 16q^{37} - 8q^{38} - 12q^{39} - 5q^{41} + 2q^{42} + 3q^{44} + 16q^{46} - 6q^{47} - q^{48} - 6q^{49} + 2q^{51} + 6q^{52} + 6q^{53} + 5q^{54} - 4q^{56} - 8q^{57} - 4q^{58} - 5q^{59} - 14q^{61} + 8q^{62} - 4q^{63} + 2q^{64} + 3q^{66} - 5q^{67} - 4q^{68} + 16q^{69} + 6q^{71} + 2q^{72} - 9q^{73} + 8q^{74} + q^{76} + 12q^{77} + 6q^{78} - 8q^{79} - q^{81} - 5q^{82} + 22q^{83} - 4q^{84} - 4q^{87} - 6q^{88} - 14q^{89} - 12q^{91} - 8q^{92} + 8q^{93} + 12q^{94} + 2q^{96} - 15q^{97} + 3q^{98} - 6q^{99} + O(q^{100})$$

## 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 −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i −2.00000 1.00000 1.00000 + 1.73205i 0
501.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.00000 1.00000 1.00000 1.73205i 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.a 2
5.b even 2 1 190.2.e.b 2
5.c odd 4 2 950.2.j.a 4
15.d odd 2 1 1710.2.l.b 2
19.c even 3 1 inner 950.2.e.a 2
20.d odd 2 1 1520.2.q.e 2
95.h odd 6 1 3610.2.a.i 1
95.i even 6 1 190.2.e.b 2
95.i even 6 1 3610.2.a.a 1
95.m odd 12 2 950.2.j.a 4
285.n odd 6 1 1710.2.l.b 2
380.p odd 6 1 1520.2.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.e.b 2 5.b even 2 1
190.2.e.b 2 95.i even 6 1
950.2.e.a 2 1.a even 1 1 trivial
950.2.e.a 2 19.c even 3 1 inner
950.2.j.a 4 5.c odd 4 2
950.2.j.a 4 95.m odd 12 2
1520.2.q.e 2 20.d odd 2 1
1520.2.q.e 2 380.p odd 6 1
1710.2.l.b 2 15.d odd 2 1
1710.2.l.b 2 285.n odd 6 1
3610.2.a.a 1 95.i even 6 1
3610.2.a.i 1 95.h odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} + 1$$ $$T_{7} + 2$$ $$T_{11} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$36 - 6 T + T^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$19 - 7 T + T^{2}$$
$23$ $$64 + 8 T + T^{2}$$
$29$ $$4 - 2 T + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$25 + 5 T + T^{2}$$
$43$ $$T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$25 + 5 T + T^{2}$$
$61$ $$196 + 14 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$36 - 6 T + T^{2}$$
$73$ $$81 + 9 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$( -11 + T )^{2}$$
$89$ $$196 + 14 T + T^{2}$$
$97$ $$225 + 15 T + T^{2}$$