# Properties

 Label 950.2.q.a Level $950$ Weight $2$ Character orbit 950.q Analytic conductor $7.586$ Analytic rank $0$ Dimension $8$ 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.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: no (minimal twist has level 190) 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{24} q^{2} + ( -2 + \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{3} + \zeta_{24}^{2} q^{4} + ( 2 \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{6} + ( 2 - \zeta_{24} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{7} -\zeta_{24}^{3} q^{8} -3 \zeta_{24}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{24} q^{2} + ( -2 + \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{3} + \zeta_{24}^{2} q^{4} + ( 2 \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{6} + ( 2 - \zeta_{24} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{7} -\zeta_{24}^{3} q^{8} -3 \zeta_{24}^{2} q^{9} + ( 1 - \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{11} + ( -1 - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{12} + ( 2 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{13} + ( -2 \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{14} + \zeta_{24}^{4} q^{16} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{17} + 3 \zeta_{24}^{3} q^{18} + ( 3 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{19} + ( -8 + 3 \zeta_{24} + 4 \zeta_{24}^{4} - 3 \zeta_{24}^{7} ) q^{21} + ( 2 - \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{22} + ( 3 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{23} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{24} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{26} + ( -2 + 2 \zeta_{24}^{2} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{7} ) q^{28} + ( -2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 4 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{29} + ( 2 - 3 \zeta_{24} + 3 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{31} -\zeta_{24}^{5} q^{32} + ( -2 + \zeta_{24}^{2} + \zeta_{24}^{4} + \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{33} + ( 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{34} -3 \zeta_{24}^{4} q^{36} + ( 1 + 3 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{37} + ( 3 - 3 \zeta_{24}^{2} - \zeta_{24}^{3} - 3 \zeta_{24}^{4} + \zeta_{24}^{7} ) q^{38} + 12 \zeta_{24}^{6} q^{39} + ( -2 + \zeta_{24}^{4} ) q^{41} + ( -3 + 8 \zeta_{24} - 3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 4 \zeta_{24}^{5} ) q^{42} + ( -5 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 5 \zeta_{24}^{4} + 5 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{43} + ( -2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{44} + ( 3 - 6 \zeta_{24}^{4} ) q^{46} + ( 8 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{47} + ( -1 - \zeta_{24}^{2} - \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{48} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{49} + ( -4 - 4 \zeta_{24}^{4} ) q^{51} + ( 4 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{52} + ( -3 - 3 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{4} + 6 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{53} + ( -1 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{56} + ( 1 - 12 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{57} + ( 3 + 2 \zeta_{24}^{3} - 3 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{58} + ( -2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{59} + ( 8 - 8 \zeta_{24}^{4} ) q^{61} + ( -2 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{62} + ( 6 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{7} ) q^{63} + \zeta_{24}^{6} q^{64} + ( -6 + 2 \zeta_{24} - \zeta_{24}^{3} + 6 \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{66} + ( 1 - \zeta_{24}^{2} + \zeta_{24}^{4} + 12 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{67} + ( -2 + 2 \zeta_{24}^{6} ) q^{68} + ( -9 \zeta_{24} - 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} ) q^{69} + ( -4 + 2 \zeta_{24}^{4} ) q^{71} + 3 \zeta_{24}^{5} q^{72} + ( -4 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{73} + ( -\zeta_{24} - 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{74} + ( 1 - 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{76} + ( 5 - 5 \zeta_{24} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{77} -12 \zeta_{24}^{7} q^{78} + ( 6 \zeta_{24} - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{79} -9 \zeta_{24}^{4} q^{81} + ( 2 \zeta_{24} - \zeta_{24}^{5} ) q^{82} + ( -5 - 6 \zeta_{24}^{3} + 5 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{83} + ( 3 \zeta_{24} - 8 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{84} + ( 2 + 5 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{86} + ( -6 - 6 \zeta_{24} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{87} + ( -1 + 2 \zeta_{24}^{2} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{88} + ( -3 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{89} + ( 8 - 6 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{91} + ( -3 \zeta_{24} + 6 \zeta_{24}^{5} ) q^{92} + ( 6 \zeta_{24}^{2} - 12 \zeta_{24}^{3} + 6 \zeta_{24}^{4} - 6 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{93} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{94} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{96} + ( -2 - 12 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{97} + ( -8 + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{98} + ( 6 \zeta_{24} - 3 \zeta_{24}^{2} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 12q^{3} + 16q^{7} + O(q^{10})$$ $$8q - 12q^{3} + 16q^{7} + 8q^{11} + 24q^{13} + 4q^{16} + 8q^{17} - 48q^{21} + 12q^{22} - 8q^{28} - 12q^{33} - 12q^{36} + 12q^{38} - 12q^{41} - 12q^{42} - 20q^{43} - 12q^{48} - 48q^{51} + 24q^{52} - 36q^{53} + 12q^{57} + 24q^{58} + 32q^{61} - 12q^{62} + 24q^{63} - 24q^{66} + 12q^{67} - 16q^{68} - 24q^{71} - 16q^{73} + 8q^{76} + 40q^{77} - 36q^{81} - 40q^{83} + 24q^{86} - 48q^{87} + 96q^{91} + 24q^{93} - 12q^{97} - 48q^{98} + O(q^{100})$$

## 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 −0.633975 + 2.36603i 0.866025 + 0.500000i 0 1.22474 2.12132i 0.775255 + 0.775255i −0.707107 0.707107i −2.59808 1.50000i 0
107.2 0.965926 + 0.258819i −0.633975 + 2.36603i 0.866025 + 0.500000i 0 −1.22474 + 2.12132i 3.22474 + 3.22474i 0.707107 + 0.707107i −2.59808 1.50000i 0
293.1 −0.965926 + 0.258819i −0.633975 2.36603i 0.866025 0.500000i 0 1.22474 + 2.12132i 0.775255 0.775255i −0.707107 + 0.707107i −2.59808 + 1.50000i 0
293.2 0.965926 0.258819i −0.633975 2.36603i 0.866025 0.500000i 0 −1.22474 2.12132i 3.22474 3.22474i 0.707107 0.707107i −2.59808 + 1.50000i 0
407.1 −0.258819 0.965926i −2.36603 + 0.633975i −0.866025 + 0.500000i 0 1.22474 + 2.12132i 0.775255 + 0.775255i 0.707107 + 0.707107i 2.59808 1.50000i 0
407.2 0.258819 + 0.965926i −2.36603 + 0.633975i −0.866025 + 0.500000i 0 −1.22474 2.12132i 3.22474 + 3.22474i −0.707107 0.707107i 2.59808 1.50000i 0
943.1 −0.258819 + 0.965926i −2.36603 0.633975i −0.866025 0.500000i 0 1.22474 2.12132i 0.775255 0.775255i 0.707107 0.707107i 2.59808 + 1.50000i 0
943.2 0.258819 0.965926i −2.36603 0.633975i −0.866025 0.500000i 0 −1.22474 + 2.12132i 3.22474 3.22474i −0.707107 + 0.707107i 2.59808 + 1.50000i 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.c odd 4 1 inner
19.d odd 6 1 inner
95.l even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.q.a 8
5.b even 2 1 190.2.m.a 8
5.c odd 4 1 190.2.m.a 8
5.c odd 4 1 inner 950.2.q.a 8
19.d odd 6 1 inner 950.2.q.a 8
95.h odd 6 1 190.2.m.a 8
95.l even 12 1 190.2.m.a 8
95.l even 12 1 inner 950.2.q.a 8

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

## Hecke kernels

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

 $$T_{3}^{4} + 6 T_{3}^{3} + 18 T_{3}^{2} + 36 T_{3} + 36$$ $$T_{7}^{4} - 8 T_{7}^{3} + 32 T_{7}^{2} - 40 T_{7} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$( 36 + 36 T + 18 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 25 - 40 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$11$ $$( -5 - 2 T + T^{2} )^{4}$$
$13$ $$( 576 - 288 T + 72 T^{2} - 12 T^{3} + T^{4} )^{2}$$
$17$ $$( 64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$19$ $$130321 + 12274 T^{2} + 795 T^{4} + 34 T^{6} + T^{8}$$
$23$ $$531441 - 729 T^{4} + T^{8}$$
$29$ $$1296 + 2160 T^{2} + 3564 T^{4} + 60 T^{6} + T^{8}$$
$31$ $$( 36 + 60 T^{2} + T^{4} )^{2}$$
$37$ $$81 + 882 T^{4} + T^{8}$$
$41$ $$( 3 + 3 T + T^{2} )^{4}$$
$43$ $$2085136 + 1097440 T + 288800 T^{2} + 94240 T^{3} + 23356 T^{4} + 2480 T^{5} + 200 T^{6} + 20 T^{7} + T^{8}$$
$47$ $$5308416 - 2304 T^{4} + T^{8}$$
$53$ $$4100625 + 3280500 T + 1312200 T^{2} + 349920 T^{3} + 64071 T^{4} + 7776 T^{5} + 648 T^{6} + 36 T^{7} + T^{8}$$
$59$ $$( 144 + 12 T^{2} + T^{4} )^{2}$$
$61$ $$( 64 - 8 T + T^{2} )^{4}$$
$67$ $$362673936 + 31536864 T + 1371168 T^{2} + 39744 T^{3} - 25092 T^{4} - 288 T^{5} + 72 T^{6} - 12 T^{7} + T^{8}$$
$71$ $$( 12 + 6 T + T^{2} )^{4}$$
$73$ $$160000 + 128000 T + 51200 T^{2} + 28160 T^{3} + 10864 T^{4} + 1408 T^{5} + 128 T^{6} + 16 T^{7} + T^{8}$$
$79$ $$12960000 + 604800 T^{2} + 24624 T^{4} + 168 T^{6} + T^{8}$$
$83$ $$( 3364 - 1160 T + 200 T^{2} + 20 T^{3} + T^{4} )^{2}$$
$89$ $$4100625 + 400950 T^{2} + 37179 T^{4} + 198 T^{6} + T^{8}$$
$97$ $$362673936 - 31536864 T + 1371168 T^{2} - 39744 T^{3} - 25092 T^{4} + 288 T^{5} + 72 T^{6} + 12 T^{7} + T^{8}$$