# Properties

 Label 950.2.q.b Level $950$ Weight $2$ Character orbit 950.q Analytic conductor $7.586$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.q (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$\zeta_{24}^{6}$$ $$1 - \zeta_{24}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i −0.258819 + 0.965926i 0.258819 − 0.965926i
−0.965926 0.258819i 0.258819 0.965926i 0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 −0.707107 0.707107i 1.73205 + 1.00000i 0
107.2 0.965926 + 0.258819i −0.258819 + 0.965926i 0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 0.707107 + 0.707107i 1.73205 + 1.00000i 0
293.1 −0.965926 + 0.258819i 0.258819 + 0.965926i 0.866025 0.500000i 0 −0.500000 0.866025i 0 −0.707107 + 0.707107i 1.73205 1.00000i 0
293.2 0.965926 0.258819i −0.258819 0.965926i 0.866025 0.500000i 0 −0.500000 0.866025i 0 0.707107 0.707107i 1.73205 1.00000i 0
407.1 −0.258819 0.965926i 0.965926 0.258819i −0.866025 + 0.500000i 0 −0.500000 0.866025i 0 0.707107 + 0.707107i −1.73205 + 1.00000i 0
407.2 0.258819 + 0.965926i −0.965926 + 0.258819i −0.866025 + 0.500000i 0 −0.500000 0.866025i 0 −0.707107 0.707107i −1.73205 + 1.00000i 0
943.1 −0.258819 + 0.965926i 0.965926 + 0.258819i −0.866025 0.500000i 0 −0.500000 + 0.866025i 0 0.707107 0.707107i −1.73205 1.00000i 0
943.2 0.258819 0.965926i −0.965926 0.258819i −0.866025 0.500000i 0 −0.500000 + 0.866025i 0 −0.707107 + 0.707107i −1.73205 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 943.2 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
5.c odd 4 2 inner
19.d odd 6 1 inner
95.h odd 6 1 inner
95.l even 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.q.b 8
5.b even 2 1 inner 950.2.q.b 8
5.c odd 4 2 inner 950.2.q.b 8
19.d odd 6 1 inner 950.2.q.b 8
95.h odd 6 1 inner 950.2.q.b 8
95.l even 12 2 inner 950.2.q.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.q.b 8 1.a even 1 1 trivial
950.2.q.b 8 5.b even 2 1 inner
950.2.q.b 8 5.c odd 4 2 inner
950.2.q.b 8 19.d odd 6 1 inner
950.2.q.b 8 95.h odd 6 1 inner
950.2.q.b 8 95.l even 12 2 inner

## Hecke kernels

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

 $$T_{3}^{8} - T_{3}^{4} + 1$$ $$T_{7}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$1 - T^{4} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 3 + T )^{8}$$
$13$ $$256 - 16 T^{4} + T^{8}$$
$17$ $$5308416 - 2304 T^{4} + T^{8}$$
$19$ $$( 361 - 37 T^{2} + T^{4} )^{2}$$
$23$ $$20736 - 144 T^{4} + T^{8}$$
$29$ $$( 2304 + 48 T^{2} + T^{4} )^{2}$$
$31$ $$( 12 + T^{2} )^{4}$$
$37$ $$( 16 + T^{4} )^{2}$$
$41$ $$( 3 + 3 T + T^{2} )^{4}$$
$43$ $$20736 - 144 T^{4} + T^{8}$$
$47$ $$5308416 - 2304 T^{4} + T^{8}$$
$53$ $$1679616 - 1296 T^{4} + T^{8}$$
$59$ $$( 729 + 27 T^{2} + T^{4} )^{2}$$
$61$ $$( 64 - 8 T + T^{2} )^{4}$$
$67$ $$1 - T^{4} + T^{8}$$
$71$ $$( 12 + 6 T + T^{2} )^{4}$$
$73$ $$3486784401 - 59049 T^{4} + T^{8}$$
$79$ $$( 2304 + 48 T^{2} + T^{4} )^{2}$$
$83$ $$( 9 + T^{4} )^{2}$$
$89$ $$( 2304 + 48 T^{2} + T^{4} )^{2}$$
$97$ $$1 - T^{4} + T^{8}$$