# Properties

 Label 950.2.e.h Level $950$ Weight $2$ Character orbit 950.e Analytic conductor $7.586$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -\beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{3} ) q^{6} + ( -3 - \beta_{3} ) q^{7} + q^{8} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} -\beta_{1} q^{3} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{3} ) q^{6} + ( -3 - \beta_{3} ) q^{7} + q^{8} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + q^{11} -\beta_{3} q^{12} + ( -2 + 2 \beta_{2} ) q^{13} + ( -\beta_{1} + 3 \beta_{2} ) q^{14} -\beta_{2} q^{16} + 2 \beta_{1} q^{17} + ( 1 - \beta_{3} ) q^{18} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{21} -\beta_{2} q^{22} + ( 3 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{23} -\beta_{1} q^{24} + 2 q^{26} + ( 4 + \beta_{3} ) q^{27} + ( 3 + \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{28} + ( -2 + 2 \beta_{2} ) q^{29} -4 \beta_{3} q^{31} + ( -1 + \beta_{2} ) q^{32} -\beta_{1} q^{33} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{34} + ( -\beta_{1} - \beta_{2} ) q^{36} + ( -3 - 3 \beta_{3} ) q^{37} + ( -1 - \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{38} -2 \beta_{3} q^{39} + ( 2 \beta_{1} + \beta_{2} ) q^{41} + ( -4 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{42} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -1 + \beta_{2} ) q^{44} + ( -3 - 3 \beta_{3} ) q^{46} + ( 8 + 2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{47} + ( \beta_{1} + \beta_{3} ) q^{48} + ( 6 + 5 \beta_{3} ) q^{49} + ( 8 - 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{51} -2 \beta_{2} q^{52} + ( -5 + \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{53} + ( \beta_{1} - 4 \beta_{2} ) q^{54} + ( -3 - \beta_{3} ) q^{56} + ( 8 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{57} + 2 q^{58} + ( -\beta_{1} - 12 \beta_{2} ) q^{59} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{61} -4 \beta_{1} q^{62} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + q^{64} + ( \beta_{1} + \beta_{3} ) q^{66} + ( 12 + \beta_{1} - 12 \beta_{2} + \beta_{3} ) q^{67} + 2 \beta_{3} q^{68} + 12 q^{69} + ( -4 \beta_{1} - 6 \beta_{2} ) q^{71} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{72} + ( -3 \beta_{1} + 6 \beta_{2} ) q^{73} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{74} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{76} + ( -3 - \beta_{3} ) q^{77} -2 \beta_{1} q^{78} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{79} + 7 \beta_{2} q^{81} + ( 1 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{82} + ( -2 - 5 \beta_{3} ) q^{83} + ( 4 + 2 \beta_{3} ) q^{84} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{86} -2 \beta_{3} q^{87} + q^{88} + ( -5 - 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{89} + ( 6 + 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{92} + ( -4 \beta_{1} - 16 \beta_{2} ) q^{93} + ( -8 - 2 \beta_{3} ) q^{94} -\beta_{3} q^{96} + ( 3 \beta_{1} - 6 \beta_{2} ) q^{97} + ( 5 \beta_{1} - 6 \beta_{2} ) q^{98} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - q^{3} - 2q^{4} - q^{6} - 10q^{7} + 4q^{8} - 3q^{9} + O(q^{10})$$ $$4q - 2q^{2} - q^{3} - 2q^{4} - q^{6} - 10q^{7} + 4q^{8} - 3q^{9} + 4q^{11} + 2q^{12} - 4q^{13} + 5q^{14} - 2q^{16} + 2q^{17} + 6q^{18} - 10q^{19} - 6q^{21} - 2q^{22} + 3q^{23} - q^{24} + 8q^{26} + 14q^{27} + 5q^{28} - 4q^{29} + 8q^{31} - 2q^{32} - q^{33} + 2q^{34} - 3q^{36} - 6q^{37} + 5q^{38} + 4q^{39} + 4q^{41} - 6q^{42} - 2q^{43} - 2q^{44} - 6q^{46} + 14q^{47} - q^{48} + 14q^{49} + 18q^{51} - 4q^{52} - 11q^{53} - 7q^{54} - 10q^{56} + 28q^{57} + 8q^{58} - 25q^{59} - 2q^{61} - 4q^{62} - q^{63} + 4q^{64} - q^{66} + 23q^{67} - 4q^{68} + 48q^{69} - 16q^{71} - 3q^{72} + 9q^{73} + 3q^{74} + 5q^{76} - 10q^{77} - 2q^{78} + 14q^{79} + 14q^{81} + 4q^{82} + 2q^{83} + 12q^{84} - 2q^{86} + 4q^{87} + 4q^{88} - 7q^{89} + 10q^{91} + 3q^{92} - 36q^{93} - 28q^{94} + 2q^{96} - 9q^{97} - 7q^{98} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 4$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2}$$

## 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
 1.28078 − 2.21837i −0.780776 + 1.35234i 1.28078 + 2.21837i −0.780776 − 1.35234i
−0.500000 + 0.866025i −1.28078 + 2.21837i −0.500000 0.866025i 0 −1.28078 2.21837i −0.438447 1.00000 −1.78078 3.08440i 0
201.2 −0.500000 + 0.866025i 0.780776 1.35234i −0.500000 0.866025i 0 0.780776 + 1.35234i −4.56155 1.00000 0.280776 + 0.486319i 0
501.1 −0.500000 0.866025i −1.28078 2.21837i −0.500000 + 0.866025i 0 −1.28078 + 2.21837i −0.438447 1.00000 −1.78078 + 3.08440i 0
501.2 −0.500000 0.866025i 0.780776 + 1.35234i −0.500000 + 0.866025i 0 0.780776 1.35234i −4.56155 1.00000 0.280776 0.486319i 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.h 4
5.b even 2 1 190.2.e.c 4
5.c odd 4 2 950.2.j.f 8
15.d odd 2 1 1710.2.l.m 4
19.c even 3 1 inner 950.2.e.h 4
20.d odd 2 1 1520.2.q.h 4
95.h odd 6 1 3610.2.a.u 2
95.i even 6 1 190.2.e.c 4
95.i even 6 1 3610.2.a.k 2
95.m odd 12 2 950.2.j.f 8
285.n odd 6 1 1710.2.l.m 4
380.p odd 6 1 1520.2.q.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.e.c 4 5.b even 2 1
190.2.e.c 4 95.i even 6 1
950.2.e.h 4 1.a even 1 1 trivial
950.2.e.h 4 19.c even 3 1 inner
950.2.j.f 8 5.c odd 4 2
950.2.j.f 8 95.m odd 12 2
1520.2.q.h 4 20.d odd 2 1
1520.2.q.h 4 380.p odd 6 1
1710.2.l.m 4 15.d odd 2 1
1710.2.l.m 4 285.n odd 6 1
3610.2.a.k 2 95.i even 6 1
3610.2.a.u 2 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 $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{4} + T_{3}^{3} + 5 T_{3}^{2} - 4 T_{3} + 16$$ $$T_{7}^{2} + 5 T_{7} + 2$$ $$T_{11} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 2 + 5 T + T^{2} )^{2}$$
$11$ $$( -1 + T )^{4}$$
$13$ $$( 4 + 2 T + T^{2} )^{2}$$
$17$ $$256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$( 19 + 5 T + T^{2} )^{2}$$
$23$ $$1296 + 108 T + 45 T^{2} - 3 T^{3} + T^{4}$$
$29$ $$( 4 + 2 T + T^{2} )^{2}$$
$31$ $$( -64 - 4 T + T^{2} )^{2}$$
$37$ $$( -36 + 3 T + T^{2} )^{2}$$
$41$ $$169 + 52 T + 29 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4}$$
$47$ $$1024 - 448 T + 164 T^{2} - 14 T^{3} + T^{4}$$
$53$ $$676 + 286 T + 95 T^{2} + 11 T^{3} + T^{4}$$
$59$ $$23104 + 3800 T + 473 T^{2} + 25 T^{3} + T^{4}$$
$61$ $$256 - 32 T + 20 T^{2} + 2 T^{3} + T^{4}$$
$67$ $$16384 - 2944 T + 401 T^{2} - 23 T^{3} + T^{4}$$
$71$ $$16 - 64 T + 260 T^{2} + 16 T^{3} + T^{4}$$
$73$ $$324 + 162 T + 99 T^{2} - 9 T^{3} + T^{4}$$
$79$ $$1024 - 448 T + 164 T^{2} - 14 T^{3} + T^{4}$$
$83$ $$( -106 - T + T^{2} )^{2}$$
$89$ $$676 - 182 T + 75 T^{2} + 7 T^{3} + T^{4}$$
$97$ $$324 - 162 T + 99 T^{2} + 9 T^{3} + T^{4}$$