# Properties

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

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## 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 38) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} + 4 q^{7} + q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{6} + 4 q^{7} + q^{8} + 2 \zeta_{6} q^{9} + 3 q^{11} - q^{12} + 2 \zeta_{6} q^{13} + ( -4 + 4 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} -2 q^{18} + ( -2 - 3 \zeta_{6} ) q^{19} + ( 4 - 4 \zeta_{6} ) q^{21} + ( -3 + 3 \zeta_{6} ) q^{22} -6 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} -2 q^{26} + 5 q^{27} -4 \zeta_{6} q^{28} + 2 q^{31} -\zeta_{6} q^{32} + ( 3 - 3 \zeta_{6} ) q^{33} -6 \zeta_{6} q^{34} + ( 2 - 2 \zeta_{6} ) q^{36} + 10 q^{37} + ( 5 - 2 \zeta_{6} ) q^{38} + 2 q^{39} + ( -9 + 9 \zeta_{6} ) q^{41} + 4 \zeta_{6} q^{42} + ( -4 + 4 \zeta_{6} ) q^{43} -3 \zeta_{6} q^{44} + 6 q^{46} + \zeta_{6} q^{48} + 9 q^{49} + 6 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{52} + 6 \zeta_{6} q^{53} + ( -5 + 5 \zeta_{6} ) q^{54} + 4 q^{56} + ( -5 + 2 \zeta_{6} ) q^{57} + ( 9 - 9 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} + ( -2 + 2 \zeta_{6} ) q^{62} + 8 \zeta_{6} q^{63} + q^{64} + 3 \zeta_{6} q^{66} -7 \zeta_{6} q^{67} + 6 q^{68} -6 q^{69} + ( 6 - 6 \zeta_{6} ) q^{71} + 2 \zeta_{6} q^{72} + ( -1 + \zeta_{6} ) q^{73} + ( -10 + 10 \zeta_{6} ) q^{74} + ( -3 + 5 \zeta_{6} ) q^{76} + 12 q^{77} + ( -2 + 2 \zeta_{6} ) q^{78} + ( 4 - 4 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} -9 \zeta_{6} q^{82} -3 q^{83} -4 q^{84} -4 \zeta_{6} q^{86} + 3 q^{88} -6 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( -6 + 6 \zeta_{6} ) q^{92} + ( 2 - 2 \zeta_{6} ) q^{93} - q^{96} + ( 17 - 17 \zeta_{6} ) q^{97} + ( -9 + 9 \zeta_{6} ) q^{98} + 6 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + q^{3} - q^{4} + q^{6} + 8q^{7} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - q^{2} + q^{3} - q^{4} + q^{6} + 8q^{7} + 2q^{8} + 2q^{9} + 6q^{11} - 2q^{12} + 2q^{13} - 4q^{14} - q^{16} - 6q^{17} - 4q^{18} - 7q^{19} + 4q^{21} - 3q^{22} - 6q^{23} + q^{24} - 4q^{26} + 10q^{27} - 4q^{28} + 4q^{31} - q^{32} + 3q^{33} - 6q^{34} + 2q^{36} + 20q^{37} + 8q^{38} + 4q^{39} - 9q^{41} + 4q^{42} - 4q^{43} - 3q^{44} + 12q^{46} + q^{48} + 18q^{49} + 6q^{51} + 2q^{52} + 6q^{53} - 5q^{54} + 8q^{56} - 8q^{57} + 9q^{59} + 4q^{61} - 2q^{62} + 8q^{63} + 2q^{64} + 3q^{66} - 7q^{67} + 12q^{68} - 12q^{69} + 6q^{71} + 2q^{72} - q^{73} - 10q^{74} - q^{76} + 24q^{77} - 2q^{78} + 4q^{79} - q^{81} - 9q^{82} - 6q^{83} - 8q^{84} - 4q^{86} + 6q^{88} - 6q^{89} + 8q^{91} - 6q^{92} + 2q^{93} - 2q^{96} + 17q^{97} - 9q^{98} + 6q^{99} + O(q^{100})$$

## 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 4.00000 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 4.00000 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.d 2
5.b even 2 1 38.2.c.a 2
5.c odd 4 2 950.2.j.e 4
15.d odd 2 1 342.2.g.b 2
19.c even 3 1 inner 950.2.e.d 2
20.d odd 2 1 304.2.i.c 2
40.e odd 2 1 1216.2.i.d 2
40.f even 2 1 1216.2.i.h 2
60.h even 2 1 2736.2.s.m 2
95.d odd 2 1 722.2.c.b 2
95.h odd 6 1 722.2.a.d 1
95.h odd 6 1 722.2.c.b 2
95.i even 6 1 38.2.c.a 2
95.i even 6 1 722.2.a.c 1
95.m odd 12 2 950.2.j.e 4
95.o odd 18 6 722.2.e.i 6
95.p even 18 6 722.2.e.j 6
285.n odd 6 1 342.2.g.b 2
285.n odd 6 1 6498.2.a.s 1
285.q even 6 1 6498.2.a.e 1
380.p odd 6 1 304.2.i.c 2
380.p odd 6 1 5776.2.a.g 1
380.s even 6 1 5776.2.a.n 1
760.z even 6 1 1216.2.i.h 2
760.bm odd 6 1 1216.2.i.d 2
1140.bn even 6 1 2736.2.s.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a 2 5.b even 2 1
38.2.c.a 2 95.i even 6 1
304.2.i.c 2 20.d odd 2 1
304.2.i.c 2 380.p odd 6 1
342.2.g.b 2 15.d odd 2 1
342.2.g.b 2 285.n odd 6 1
722.2.a.c 1 95.i even 6 1
722.2.a.d 1 95.h odd 6 1
722.2.c.b 2 95.d odd 2 1
722.2.c.b 2 95.h odd 6 1
722.2.e.i 6 95.o odd 18 6
722.2.e.j 6 95.p even 18 6
950.2.e.d 2 1.a even 1 1 trivial
950.2.e.d 2 19.c even 3 1 inner
950.2.j.e 4 5.c odd 4 2
950.2.j.e 4 95.m odd 12 2
1216.2.i.d 2 40.e odd 2 1
1216.2.i.d 2 760.bm odd 6 1
1216.2.i.h 2 40.f even 2 1
1216.2.i.h 2 760.z even 6 1
2736.2.s.m 2 60.h even 2 1
2736.2.s.m 2 1140.bn even 6 1
5776.2.a.g 1 380.p odd 6 1
5776.2.a.n 1 380.s even 6 1
6498.2.a.e 1 285.q even 6 1
6498.2.a.s 1 285.n odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{7} - 4$$ $$T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$19 + 7 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$81 + 9 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$16 - 4 T + T^{2}$$
$67$ $$49 + 7 T + T^{2}$$
$71$ $$36 - 6 T + T^{2}$$
$73$ $$1 + T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$( 3 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$289 - 17 T + T^{2}$$
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