# Properties

 Label 950.2.e.k 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: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## Character values

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

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

## 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
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0.500000 0.866025i −1.32288 + 2.29129i −0.500000 0.866025i 0 1.32288 + 2.29129i 1.64575 −1.00000 −2.00000 3.46410i 0
201.2 0.500000 0.866025i 1.32288 2.29129i −0.500000 0.866025i 0 −1.32288 2.29129i −3.64575 −1.00000 −2.00000 3.46410i 0
501.1 0.500000 + 0.866025i −1.32288 2.29129i −0.500000 + 0.866025i 0 1.32288 2.29129i 1.64575 −1.00000 −2.00000 + 3.46410i 0
501.2 0.500000 + 0.866025i 1.32288 + 2.29129i −0.500000 + 0.866025i 0 −1.32288 + 2.29129i −3.64575 −1.00000 −2.00000 + 3.46410i 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.k 4
5.b even 2 1 38.2.c.b 4
5.c odd 4 2 950.2.j.g 8
15.d odd 2 1 342.2.g.f 4
19.c even 3 1 inner 950.2.e.k 4
20.d odd 2 1 304.2.i.e 4
40.e odd 2 1 1216.2.i.k 4
40.f even 2 1 1216.2.i.l 4
60.h even 2 1 2736.2.s.v 4
95.d odd 2 1 722.2.c.j 4
95.h odd 6 1 722.2.a.g 2
95.h odd 6 1 722.2.c.j 4
95.i even 6 1 38.2.c.b 4
95.i even 6 1 722.2.a.j 2
95.m odd 12 2 950.2.j.g 8
95.o odd 18 6 722.2.e.o 12
95.p even 18 6 722.2.e.n 12
285.n odd 6 1 342.2.g.f 4
285.n odd 6 1 6498.2.a.ba 2
285.q even 6 1 6498.2.a.bg 2
380.p odd 6 1 304.2.i.e 4
380.p odd 6 1 5776.2.a.ba 2
380.s even 6 1 5776.2.a.z 2
760.z even 6 1 1216.2.i.l 4
760.bm odd 6 1 1216.2.i.k 4
1140.bn even 6 1 2736.2.s.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b 4 5.b even 2 1
38.2.c.b 4 95.i even 6 1
304.2.i.e 4 20.d odd 2 1
304.2.i.e 4 380.p odd 6 1
342.2.g.f 4 15.d odd 2 1
342.2.g.f 4 285.n odd 6 1
722.2.a.g 2 95.h odd 6 1
722.2.a.j 2 95.i even 6 1
722.2.c.j 4 95.d odd 2 1
722.2.c.j 4 95.h odd 6 1
722.2.e.n 12 95.p even 18 6
722.2.e.o 12 95.o odd 18 6
950.2.e.k 4 1.a even 1 1 trivial
950.2.e.k 4 19.c even 3 1 inner
950.2.j.g 8 5.c odd 4 2
950.2.j.g 8 95.m odd 12 2
1216.2.i.k 4 40.e odd 2 1
1216.2.i.k 4 760.bm odd 6 1
1216.2.i.l 4 40.f even 2 1
1216.2.i.l 4 760.z even 6 1
2736.2.s.v 4 60.h even 2 1
2736.2.s.v 4 1140.bn even 6 1
5776.2.a.z 2 380.s even 6 1
5776.2.a.ba 2 380.p odd 6 1
6498.2.a.ba 2 285.n odd 6 1
6498.2.a.bg 2 285.q even 6 1

## Hecke kernels

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

 $$T_{3}^{4} + 7 T_{3}^{2} + 49$$ $$T_{7}^{2} + 2 T_{7} - 6$$ $$T_{11}^{2} + 4 T_{11} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$49 + 7 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -6 + 2 T + T^{2} )^{2}$$
$11$ $$( -3 + 4 T + T^{2} )^{2}$$
$13$ $$( 4 - 2 T + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$361 - 228 T + 67 T^{2} - 12 T^{3} + T^{4}$$
$23$ $$36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$36 + 12 T + 10 T^{2} - 2 T^{3} + T^{4}$$
$31$ $$( 2 + 6 T + T^{2} )^{2}$$
$37$ $$( 2 + 6 T + T^{2} )^{2}$$
$41$ $$9 - 30 T + 103 T^{2} + 10 T^{3} + T^{4}$$
$43$ $$64 - 96 T + 136 T^{2} - 12 T^{3} + T^{4}$$
$47$ $$1764 - 588 T + 154 T^{2} - 14 T^{3} + T^{4}$$
$53$ $$11664 - 432 T + 124 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$3969 + 63 T^{2} + T^{4}$$
$61$ $$196 + 196 T + 210 T^{2} - 14 T^{3} + T^{4}$$
$67$ $$9 - 12 T + 19 T^{2} + 4 T^{3} + T^{4}$$
$71$ $$1296 - 576 T + 220 T^{2} - 16 T^{3} + T^{4}$$
$73$ $$441 - 294 T + 175 T^{2} - 14 T^{3} + T^{4}$$
$79$ $$( 16 - 4 T + T^{2} )^{2}$$
$83$ $$( -63 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$2809 + 954 T + 271 T^{2} + 18 T^{3} + T^{4}$$