# Properties

 Label 950.2.e.f 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.

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

## 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 −1.00000 −1.00000 1.50000 + 2.59808i 0
501.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −1.00000 −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.f 2
5.b even 2 1 190.2.e.a 2
5.c odd 4 2 950.2.j.d 4
15.d odd 2 1 1710.2.l.h 2
19.c even 3 1 inner 950.2.e.f 2
20.d odd 2 1 1520.2.q.f 2
95.h odd 6 1 3610.2.a.c 1
95.i even 6 1 190.2.e.a 2
95.i even 6 1 3610.2.a.g 1
95.m odd 12 2 950.2.j.d 4
285.n odd 6 1 1710.2.l.h 2
380.p odd 6 1 1520.2.q.f 2

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

## Hecke kernels

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

 $$T_{3}$$ $$T_{7} + 1$$ $$T_{11} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -5 + T )^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$19 + T + T^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$( -11 + T )^{2}$$
$41$ $$81 - 9 T + T^{2}$$
$43$ $$36 - 6 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$25 - 5 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$144 - 12 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$196 - 14 T + T^{2}$$
$79$ $$100 + 10 T + T^{2}$$
$83$ $$( 14 + T )^{2}$$
$89$ $$49 - 7 T + T^{2}$$
$97$ $$4 + 2 T + T^{2}$$