# Properties

 Label 950.2.e.b 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} -2 q^{7} + q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} -2 q^{7} + q^{8} + 2 \zeta_{6} q^{9} + q^{12} -6 \zeta_{6} q^{13} + ( 2 - 2 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -7 + 7 \zeta_{6} ) q^{17} -2 q^{18} + ( 5 - 3 \zeta_{6} ) q^{19} + ( 2 - 2 \zeta_{6} ) q^{21} -2 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + 6 q^{26} -5 q^{27} + 2 \zeta_{6} q^{28} -10 \zeta_{6} q^{29} -2 q^{31} -\zeta_{6} q^{32} -7 \zeta_{6} q^{34} + ( 2 - 2 \zeta_{6} ) q^{36} + 4 q^{37} + ( -2 + 5 \zeta_{6} ) q^{38} + 6 q^{39} + ( -2 + 2 \zeta_{6} ) q^{41} + 2 \zeta_{6} q^{42} + ( 12 - 12 \zeta_{6} ) q^{43} + 2 q^{46} -\zeta_{6} q^{48} -3 q^{49} -7 \zeta_{6} q^{51} + ( -6 + 6 \zeta_{6} ) q^{52} + ( 5 - 5 \zeta_{6} ) q^{54} -2 q^{56} + ( -2 + 5 \zeta_{6} ) q^{57} + 10 q^{58} + ( 1 - \zeta_{6} ) q^{59} -8 \zeta_{6} q^{61} + ( 2 - 2 \zeta_{6} ) q^{62} -4 \zeta_{6} q^{63} + q^{64} -8 \zeta_{6} q^{67} + 7 q^{68} + 2 q^{69} + ( 12 - 12 \zeta_{6} ) q^{71} + 2 \zeta_{6} q^{72} + ( 3 - 3 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} + ( -3 - 2 \zeta_{6} ) q^{76} + ( -6 + 6 \zeta_{6} ) q^{78} + ( 4 - 4 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} -2 \zeta_{6} q^{82} -13 q^{83} -2 q^{84} + 12 \zeta_{6} q^{86} + 10 q^{87} + 13 \zeta_{6} q^{89} + 12 \zeta_{6} q^{91} + ( -2 + 2 \zeta_{6} ) q^{92} + ( 2 - 2 \zeta_{6} ) q^{93} + q^{96} + ( -15 + 15 \zeta_{6} ) q^{97} + ( 3 - 3 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{3} - q^{4} - q^{6} - 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} - q^{4} - q^{6} - 4q^{7} + 2q^{8} + 2q^{9} + 2q^{12} - 6q^{13} + 2q^{14} - q^{16} - 7q^{17} - 4q^{18} + 7q^{19} + 2q^{21} - 2q^{23} - q^{24} + 12q^{26} - 10q^{27} + 2q^{28} - 10q^{29} - 4q^{31} - q^{32} - 7q^{34} + 2q^{36} + 8q^{37} + q^{38} + 12q^{39} - 2q^{41} + 2q^{42} + 12q^{43} + 4q^{46} - q^{48} - 6q^{49} - 7q^{51} - 6q^{52} + 5q^{54} - 4q^{56} + q^{57} + 20q^{58} + q^{59} - 8q^{61} + 2q^{62} - 4q^{63} + 2q^{64} - 8q^{67} + 14q^{68} + 4q^{69} + 12q^{71} + 2q^{72} + 3q^{73} - 4q^{74} - 8q^{76} - 6q^{78} + 4q^{79} - q^{81} - 2q^{82} - 26q^{83} - 4q^{84} + 12q^{86} + 20q^{87} + 13q^{89} + 12q^{91} - 2q^{92} + 2q^{93} + 2q^{96} - 15q^{97} + 3q^{98} + O(q^{100})$$

## 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.b 2
5.b even 2 1 950.2.e.g yes 2
5.c odd 4 2 950.2.j.b 4
19.c even 3 1 inner 950.2.e.b 2
95.i even 6 1 950.2.e.g yes 2
95.m odd 12 2 950.2.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.b 2 1.a even 1 1 trivial
950.2.e.b 2 19.c even 3 1 inner
950.2.e.g yes 2 5.b even 2 1
950.2.e.g yes 2 95.i even 6 1
950.2.j.b 4 5.c odd 4 2
950.2.j.b 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 $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$36 + 6 T + T^{2}$$
$17$ $$49 + 7 T + T^{2}$$
$19$ $$19 - 7 T + T^{2}$$
$23$ $$4 + 2 T + T^{2}$$
$29$ $$100 + 10 T + T^{2}$$
$31$ $$( 2 + T )^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$4 + 2 T + T^{2}$$
$43$ $$144 - 12 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$1 - T + T^{2}$$
$61$ $$64 + 8 T + T^{2}$$
$67$ $$64 + 8 T + T^{2}$$
$71$ $$144 - 12 T + T^{2}$$
$73$ $$9 - 3 T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$( 13 + T )^{2}$$
$89$ $$169 - 13 T + T^{2}$$
$97$ $$225 + 15 T + T^{2}$$