# Properties

 Label 950.2.q.d 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.

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

## 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 −0.517638 + 1.93185i 0.866025 + 0.500000i 0 1.00000 1.73205i −1.22474 1.22474i −0.707107 0.707107i −0.866025 0.500000i 0
107.2 0.965926 + 0.258819i 0.517638 1.93185i 0.866025 + 0.500000i 0 1.00000 1.73205i 1.22474 + 1.22474i 0.707107 + 0.707107i −0.866025 0.500000i 0
293.1 −0.965926 + 0.258819i −0.517638 1.93185i 0.866025 0.500000i 0 1.00000 + 1.73205i −1.22474 + 1.22474i −0.707107 + 0.707107i −0.866025 + 0.500000i 0
293.2 0.965926 0.258819i 0.517638 + 1.93185i 0.866025 0.500000i 0 1.00000 + 1.73205i 1.22474 1.22474i 0.707107 0.707107i −0.866025 + 0.500000i 0
407.1 −0.258819 0.965926i −1.93185 + 0.517638i −0.866025 + 0.500000i 0 1.00000 + 1.73205i −1.22474 1.22474i 0.707107 + 0.707107i 0.866025 0.500000i 0
407.2 0.258819 + 0.965926i 1.93185 0.517638i −0.866025 + 0.500000i 0 1.00000 + 1.73205i 1.22474 + 1.22474i −0.707107 0.707107i 0.866025 0.500000i 0
943.1 −0.258819 + 0.965926i −1.93185 0.517638i −0.866025 0.500000i 0 1.00000 1.73205i −1.22474 + 1.22474i 0.707107 0.707107i 0.866025 + 0.500000i 0
943.2 0.258819 0.965926i 1.93185 + 0.517638i −0.866025 0.500000i 0 1.00000 1.73205i 1.22474 1.22474i −0.707107 + 0.707107i 0.866025 + 0.500000i 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.d 8
5.b even 2 1 inner 950.2.q.d 8
5.c odd 4 2 inner 950.2.q.d 8
19.d odd 6 1 inner 950.2.q.d 8
95.h odd 6 1 inner 950.2.q.d 8
95.l even 12 2 inner 950.2.q.d 8

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

 $$T_{3}^{8} - 16 T_{3}^{4} + 256$$ $$T_{7}^{4} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$256 - 16 T^{4} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 9 + T^{4} )^{2}$$
$11$ $$( -3 + T )^{8}$$
$13$ $$256 - 16 T^{4} + T^{8}$$
$17$ $$T^{8}$$
$19$ $$( 361 + 11 T^{2} + T^{4} )^{2}$$
$23$ $$531441 - 729 T^{4} + T^{8}$$
$29$ $$T^{8}$$
$31$ $$( 12 + T^{2} )^{4}$$
$37$ $$( 1 + T^{4} )^{2}$$
$41$ $$( 27 + 9 T + T^{2} )^{4}$$
$43$ $$5308416 - 2304 T^{4} + T^{8}$$
$47$ $$136048896 - 11664 T^{4} + T^{8}$$
$53$ $$6561 - 81 T^{4} + T^{8}$$
$59$ $$( 11664 + 108 T^{2} + T^{4} )^{2}$$
$61$ $$( 16 + 4 T + T^{2} )^{4}$$
$67$ $$256 - 16 T^{4} + T^{8}$$
$71$ $$( 108 + 18 T + T^{2} )^{4}$$
$73$ $$20736 - 144 T^{4} + T^{8}$$
$79$ $$( 144 + 12 T^{2} + T^{4} )^{2}$$
$83$ $$T^{8}$$
$89$ $$( 729 + 27 T^{2} + T^{4} )^{2}$$
$97$ $$16777216 - 4096 T^{4} + T^{8}$$