# Properties

 Label 950.2.j.a Level $950$ Weight $2$ Character orbit 950.j Analytic conductor $7.586$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.j (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{2} + \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{2} q^{6} -2 \zeta_{12}^{3} q^{7} -\zeta_{12}^{3} q^{8} -2 \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{12} q^{2} + \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} -\zeta_{12}^{2} q^{6} -2 \zeta_{12}^{3} q^{7} -\zeta_{12}^{3} q^{8} -2 \zeta_{12}^{2} q^{9} -3 q^{11} + \zeta_{12}^{3} q^{12} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{13} + ( -2 + 2 \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + 2 \zeta_{12} q^{17} + 2 \zeta_{12}^{3} q^{18} + ( -2 - 3 \zeta_{12}^{2} ) q^{19} + ( 2 - 2 \zeta_{12}^{2} ) q^{21} + 3 \zeta_{12} q^{22} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{23} + ( 1 - \zeta_{12}^{2} ) q^{24} -6 q^{26} -5 \zeta_{12}^{3} q^{27} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} -2 \zeta_{12}^{2} q^{29} -8 q^{31} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{32} -3 \zeta_{12} q^{33} -2 \zeta_{12}^{2} q^{34} + ( 2 - 2 \zeta_{12}^{2} ) q^{36} -8 \zeta_{12}^{3} q^{37} + ( 2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{38} + 6 q^{39} + ( -5 + 5 \zeta_{12}^{2} ) q^{41} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{42} -3 \zeta_{12}^{2} q^{44} + 8 q^{46} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{47} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{48} + 3 q^{49} + 2 \zeta_{12}^{2} q^{51} + 6 \zeta_{12} q^{52} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{53} + ( -5 + 5 \zeta_{12}^{2} ) q^{54} -2 q^{56} + ( -2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{57} + 2 \zeta_{12}^{3} q^{58} + ( 5 - 5 \zeta_{12}^{2} ) q^{59} -14 \zeta_{12}^{2} q^{61} + 8 \zeta_{12} q^{62} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{63} - q^{64} + 3 \zeta_{12}^{2} q^{66} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{67} + 2 \zeta_{12}^{3} q^{68} -8 q^{69} + ( 6 - 6 \zeta_{12}^{2} ) q^{71} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{72} + 9 \zeta_{12} q^{73} + ( -8 + 8 \zeta_{12}^{2} ) q^{74} + ( 3 - 5 \zeta_{12}^{2} ) q^{76} + 6 \zeta_{12}^{3} q^{77} -6 \zeta_{12} q^{78} + ( 8 - 8 \zeta_{12}^{2} ) q^{79} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{82} -11 \zeta_{12}^{3} q^{83} + 2 q^{84} -2 \zeta_{12}^{3} q^{87} + 3 \zeta_{12}^{3} q^{88} + 14 \zeta_{12}^{2} q^{89} -12 \zeta_{12}^{2} q^{91} -8 \zeta_{12} q^{92} -8 \zeta_{12} q^{93} -6 q^{94} + q^{96} -15 \zeta_{12} q^{97} -3 \zeta_{12} q^{98} + 6 \zeta_{12}^{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{4} - 2q^{6} - 4q^{9} + O(q^{10})$$ $$4q + 2q^{4} - 2q^{6} - 4q^{9} - 12q^{11} - 4q^{14} - 2q^{16} - 14q^{19} + 4q^{21} + 2q^{24} - 24q^{26} - 4q^{29} - 32q^{31} - 4q^{34} + 4q^{36} + 24q^{39} - 10q^{41} - 6q^{44} + 32q^{46} + 12q^{49} + 4q^{51} - 10q^{54} - 8q^{56} + 10q^{59} - 28q^{61} - 4q^{64} + 6q^{66} - 32q^{69} + 12q^{71} - 16q^{74} + 2q^{76} + 16q^{79} - 2q^{81} + 8q^{84} + 28q^{89} - 24q^{91} - 24q^{94} + 4q^{96} + 12q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 2.00000i 1.00000i −1.00000 1.73205i 0
49.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 2.00000i 1.00000i −1.00000 1.73205i 0
349.1 −0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 2.00000i 1.00000i −1.00000 + 1.73205i 0
349.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 2.00000i 1.00000i −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
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.j.a 4
5.b even 2 1 inner 950.2.j.a 4
5.c odd 4 1 190.2.e.b 2
5.c odd 4 1 950.2.e.a 2
15.e even 4 1 1710.2.l.b 2
19.c even 3 1 inner 950.2.j.a 4
20.e even 4 1 1520.2.q.e 2
95.i even 6 1 inner 950.2.j.a 4
95.l even 12 1 3610.2.a.i 1
95.m odd 12 1 190.2.e.b 2
95.m odd 12 1 950.2.e.a 2
95.m odd 12 1 3610.2.a.a 1
285.v even 12 1 1710.2.l.b 2
380.v even 12 1 1520.2.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.e.b 2 5.c odd 4 1
190.2.e.b 2 95.m odd 12 1
950.2.e.a 2 5.c odd 4 1
950.2.e.a 2 95.m odd 12 1
950.2.j.a 4 1.a even 1 1 trivial
950.2.j.a 4 5.b even 2 1 inner
950.2.j.a 4 19.c even 3 1 inner
950.2.j.a 4 95.i even 6 1 inner
1520.2.q.e 2 20.e even 4 1
1520.2.q.e 2 380.v even 12 1
1710.2.l.b 2 15.e even 4 1
1710.2.l.b 2 285.v even 12 1
3610.2.a.a 1 95.m odd 12 1
3610.2.a.i 1 95.l even 12 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( 3 + T )^{4}$$
$13$ $$1296 - 36 T^{2} + T^{4}$$
$17$ $$16 - 4 T^{2} + T^{4}$$
$19$ $$( 19 + 7 T + T^{2} )^{2}$$
$23$ $$4096 - 64 T^{2} + T^{4}$$
$29$ $$( 4 + 2 T + T^{2} )^{2}$$
$31$ $$( 8 + T )^{4}$$
$37$ $$( 64 + T^{2} )^{2}$$
$41$ $$( 25 + 5 T + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$1296 - 36 T^{2} + T^{4}$$
$53$ $$1296 - 36 T^{2} + T^{4}$$
$59$ $$( 25 - 5 T + T^{2} )^{2}$$
$61$ $$( 196 + 14 T + T^{2} )^{2}$$
$67$ $$625 - 25 T^{2} + T^{4}$$
$71$ $$( 36 - 6 T + T^{2} )^{2}$$
$73$ $$6561 - 81 T^{2} + T^{4}$$
$79$ $$( 64 - 8 T + T^{2} )^{2}$$
$83$ $$( 121 + T^{2} )^{2}$$
$89$ $$( 196 - 14 T + T^{2} )^{2}$$
$97$ $$50625 - 225 T^{2} + T^{4}$$