# Properties

 Label 950.2.e.i 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: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{2} + \beta_{3} q^{3} + \beta_{1} q^{4} + ( \beta_{2} - \beta_{3} ) q^{6} - q^{7} + q^{8} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{2} + \beta_{3} q^{3} + \beta_{1} q^{4} + ( \beta_{2} - \beta_{3} ) q^{6} - q^{7} + q^{8} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( -1 - \beta_{2} ) q^{11} -\beta_{2} q^{12} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( 1 + \beta_{1} ) q^{14} + ( -1 - \beta_{1} ) q^{16} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{17} + ( 2 + \beta_{2} ) q^{18} + ( -2 + 3 \beta_{1} ) q^{19} -\beta_{3} q^{21} + ( 1 + \beta_{1} + \beta_{3} ) q^{22} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{23} + \beta_{3} q^{24} + ( -3 + \beta_{2} ) q^{26} -5 q^{27} -\beta_{1} q^{28} + ( 4 \beta_{1} - \beta_{2} + \beta_{3} ) q^{29} + ( -2 - \beta_{2} ) q^{31} + \beta_{1} q^{32} + ( -5 - 5 \beta_{1} - 2 \beta_{3} ) q^{33} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{34} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{36} + ( -2 + 2 \beta_{2} ) q^{37} + ( 5 + 2 \beta_{1} ) q^{38} + ( -5 + 2 \beta_{2} ) q^{39} + ( 3 + 3 \beta_{1} + 3 \beta_{3} ) q^{41} + ( -\beta_{2} + \beta_{3} ) q^{42} + ( -3 - 3 \beta_{1} - \beta_{3} ) q^{43} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{44} + ( -1 + 2 \beta_{2} ) q^{46} + ( -\beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{47} + ( \beta_{2} - \beta_{3} ) q^{48} -6 q^{49} + ( 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( 3 + 3 \beta_{1} - \beta_{3} ) q^{52} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 5 + 5 \beta_{1} ) q^{54} - q^{56} + ( -3 \beta_{2} - 2 \beta_{3} ) q^{57} + ( 4 + \beta_{2} ) q^{58} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{59} + ( -7 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{61} + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{62} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{63} + q^{64} + ( 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{66} + ( 10 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( 2 - \beta_{2} ) q^{68} + ( -10 - \beta_{2} ) q^{69} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{71} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{72} -\beta_{3} q^{73} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{74} + ( -3 - 5 \beta_{1} ) q^{76} + ( 1 + \beta_{2} ) q^{77} + ( 5 + 5 \beta_{1} - 2 \beta_{3} ) q^{78} + ( -1 - \beta_{1} - 5 \beta_{3} ) q^{79} + ( 6 + 6 \beta_{1} - 2 \beta_{3} ) q^{81} + ( -3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{82} + ( 1 + \beta_{2} ) q^{83} + \beta_{2} q^{84} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{86} + ( -5 - 5 \beta_{2} ) q^{87} + ( -1 - \beta_{2} ) q^{88} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{91} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{92} + ( -5 - 5 \beta_{1} - 3 \beta_{3} ) q^{93} + ( -1 - 4 \beta_{2} ) q^{94} -\beta_{2} q^{96} + ( 4 + 4 \beta_{1} + 3 \beta_{3} ) q^{97} + ( 6 + 6 \beta_{1} ) q^{98} + ( -7 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} + q^{3} - 2q^{4} + q^{6} - 4q^{7} + 4q^{8} - 5q^{9} + O(q^{10})$$ $$4q - 2q^{2} + q^{3} - 2q^{4} + q^{6} - 4q^{7} + 4q^{8} - 5q^{9} - 6q^{11} - 2q^{12} + 5q^{13} + 2q^{14} - 2q^{16} - 3q^{17} + 10q^{18} - 14q^{19} - q^{21} + 3q^{22} + q^{24} - 10q^{26} - 20q^{27} + 2q^{28} - 9q^{29} - 10q^{31} - 2q^{32} - 12q^{33} - 3q^{34} - 5q^{36} - 4q^{37} + 16q^{38} - 16q^{39} + 9q^{41} - q^{42} - 7q^{43} + 3q^{44} + 6q^{47} + q^{48} - 24q^{49} - 9q^{51} + 5q^{52} + 10q^{54} - 4q^{56} - 8q^{57} + 18q^{58} + 11q^{61} + 5q^{62} + 5q^{63} + 4q^{64} - 12q^{66} - 22q^{67} + 6q^{68} - 42q^{69} - 5q^{72} - q^{73} + 2q^{74} - 2q^{76} + 6q^{77} + 8q^{78} - 7q^{79} + 10q^{81} + 9q^{82} + 6q^{83} + 2q^{84} - 7q^{86} - 30q^{87} - 6q^{88} - 5q^{91} - 13q^{93} - 12q^{94} - 2q^{96} + 11q^{97} + 12q^{98} + 18q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + \nu^{2} - \nu - 4$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 3 \nu + 2$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{3} + \nu^{2} + \nu - 8$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} - 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 4$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{3} - \beta_{2} - \beta_{1} + 7$$$$)/3$$

## Character values

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

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

## 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
 −0.895644 + 1.09445i 1.39564 − 0.228425i −0.895644 − 1.09445i 1.39564 + 0.228425i
−0.500000 + 0.866025i −0.895644 + 1.55130i −0.500000 0.866025i 0 −0.895644 1.55130i −1.00000 1.00000 −0.104356 0.180750i 0
201.2 −0.500000 + 0.866025i 1.39564 2.41733i −0.500000 0.866025i 0 1.39564 + 2.41733i −1.00000 1.00000 −2.39564 4.14938i 0
501.1 −0.500000 0.866025i −0.895644 1.55130i −0.500000 + 0.866025i 0 −0.895644 + 1.55130i −1.00000 1.00000 −0.104356 + 0.180750i 0
501.2 −0.500000 0.866025i 1.39564 + 2.41733i −0.500000 + 0.866025i 0 1.39564 2.41733i −1.00000 1.00000 −2.39564 + 4.14938i 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.i 4
5.b even 2 1 950.2.e.j yes 4
5.c odd 4 2 950.2.j.h 8
19.c even 3 1 inner 950.2.e.i 4
95.i even 6 1 950.2.e.j yes 4
95.m odd 12 2 950.2.j.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.i 4 1.a even 1 1 trivial
950.2.e.i 4 19.c even 3 1 inner
950.2.e.j yes 4 5.b even 2 1
950.2.e.j yes 4 95.i even 6 1
950.2.j.h 8 5.c odd 4 2
950.2.j.h 8 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 $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{4} - T_{3}^{3} + 6 T_{3}^{2} + 5 T_{3} + 25$$ $$T_{7} + 1$$ $$T_{11}^{2} + 3 T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$25 + 5 T + 6 T^{2} - T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$( -3 + 3 T + T^{2} )^{2}$$
$13$ $$1 - 5 T + 24 T^{2} - 5 T^{3} + T^{4}$$
$17$ $$9 - 9 T + 12 T^{2} + 3 T^{3} + T^{4}$$
$19$ $$( 19 + 7 T + T^{2} )^{2}$$
$23$ $$441 + 21 T^{2} + T^{4}$$
$29$ $$225 + 135 T + 66 T^{2} + 9 T^{3} + T^{4}$$
$31$ $$( 1 + 5 T + T^{2} )^{2}$$
$37$ $$( -20 + 2 T + T^{2} )^{2}$$
$41$ $$729 + 243 T + 108 T^{2} - 9 T^{3} + T^{4}$$
$43$ $$49 + 49 T + 42 T^{2} + 7 T^{3} + T^{4}$$
$47$ $$5625 + 450 T + 111 T^{2} - 6 T^{3} + T^{4}$$
$53$ $$441 + 21 T^{2} + T^{4}$$
$59$ $$441 + 21 T^{2} + T^{4}$$
$61$ $$289 + 187 T + 138 T^{2} - 11 T^{3} + T^{4}$$
$67$ $$10000 + 2200 T + 384 T^{2} + 22 T^{3} + T^{4}$$
$71$ $$441 + 21 T^{2} + T^{4}$$
$73$ $$25 - 5 T + 6 T^{2} + T^{3} + T^{4}$$
$79$ $$14161 - 833 T + 168 T^{2} + 7 T^{3} + T^{4}$$
$83$ $$( -3 - 3 T + T^{2} )^{2}$$
$89$ $$441 + 21 T^{2} + T^{4}$$
$97$ $$289 + 187 T + 138 T^{2} - 11 T^{3} + T^{4}$$