# Properties

 Label 950.2.l.f Level $950$ Weight $2$ Character orbit 950.l Analytic conductor $7.586$ Analytic rank $0$ Dimension $6$ 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.l (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{2} + ( 1 + \zeta_{18}^{2} ) q^{3} -\zeta_{18}^{5} q^{4} + ( -1 - \zeta_{18} + \zeta_{18}^{4} ) q^{6} + ( \zeta_{18} - \zeta_{18}^{2} ) q^{7} + \zeta_{18}^{3} q^{8} + ( 1 - \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{2} + ( 1 + \zeta_{18}^{2} ) q^{3} -\zeta_{18}^{5} q^{4} + ( -1 - \zeta_{18} + \zeta_{18}^{4} ) q^{6} + ( \zeta_{18} - \zeta_{18}^{2} ) q^{7} + \zeta_{18}^{3} q^{8} + ( 1 - \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{9} + ( 2 \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{11} + ( \zeta_{18} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{12} + ( 3 + 3 \zeta_{18} - 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{13} + ( 1 - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{14} -\zeta_{18} q^{16} + ( 1 + 4 \zeta_{18} - 4 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{17} + ( 1 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{18} + ( 2 - 4 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{19} + ( \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{21} + ( -1 - \zeta_{18} - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{22} + ( 2 - 3 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{23} + ( \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{24} + ( 2 - 2 \zeta_{18} - \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{26} + ( -3 \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{27} + ( 1 - \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{28} + ( 4 + \zeta_{18}^{2} + 4 \zeta_{18}^{4} ) q^{29} + ( -2 - 3 \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} ) q^{31} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{32} + ( -1 + \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{33} + ( -\zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{34} + ( 1 - \zeta_{18} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{36} + ( -5 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{37} + ( 2 + 2 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{38} + ( 5 + \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{39} + ( 2 + 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{41} + ( 1 - \zeta_{18} + \zeta_{18}^{5} ) q^{42} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{43} + ( 3 + 3 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{44} + ( -3 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{46} + ( -1 + \zeta_{18} - 5 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{47} + ( -\zeta_{18} - \zeta_{18}^{3} ) q^{48} + ( \zeta_{18}^{2} + 5 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{49} + ( 5 + 5 \zeta_{18} + \zeta_{18}^{2} - 5 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{51} + ( 1 - \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{52} + ( 3 - \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{53} + ( 3 - \zeta_{18} + 3 \zeta_{18}^{2} ) q^{54} + ( \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{56} + ( -5 \zeta_{18} - 4 \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{57} + ( -1 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 4 \zeta_{18}^{4} ) q^{58} + ( 3 - 7 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 7 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{59} + ( -5 + 2 \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{61} + ( -3 + 3 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{62} + ( 1 + \zeta_{18} - \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{63} + ( -1 + \zeta_{18}^{3} ) q^{64} + ( -1 - 3 \zeta_{18} - 4 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{66} + ( -3 + 3 \zeta_{18} + 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{67} + ( -\zeta_{18} - 4 \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{68} + ( 1 - 5 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{69} + ( 6 - 6 \zeta_{18}^{2} - 7 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{71} + ( -\zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{72} + ( -5 + 7 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{73} + ( -4 + 5 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 5 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{74} + ( -2 - \zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{76} + ( -1 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{4} ) q^{77} + ( -1 - 5 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 5 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{78} + \zeta_{18} q^{79} + ( 6 - 6 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{81} + ( -4 - 4 \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{82} + ( 1 + 2 \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{83} + ( -\zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{84} + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{4} ) q^{86} + ( 5 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 5 \zeta_{18}^{4} ) q^{87} + ( -1 - \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{88} + ( -3 - 3 \zeta_{18} - 5 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{89} + ( -4 + 5 \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{91} + ( -2 + 2 \zeta_{18} + \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{92} + ( -2 - 3 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{93} + ( 5 - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{94} + ( \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{96} + ( 3 - 2 \zeta_{18} + 9 \zeta_{18}^{2} - 9 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{97} + ( -1 - 5 \zeta_{18} - \zeta_{18}^{2} ) q^{98} + ( -2 + 3 \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{3} - 6q^{6} + 3q^{8} + 6q^{9} + O(q^{10})$$ $$6q + 6q^{3} - 6q^{6} + 3q^{8} + 6q^{9} + 3q^{11} + 15q^{13} + 6q^{14} + 6q^{17} + 6q^{18} + 6q^{19} + 3q^{21} - 12q^{22} + 15q^{23} + 3q^{24} + 6q^{26} + 3q^{27} + 3q^{28} + 24q^{29} - 6q^{31} + 6q^{33} + 3q^{34} + 6q^{36} - 30q^{37} + 9q^{38} + 30q^{39} + 18q^{41} + 6q^{42} - 3q^{43} + 12q^{44} - 6q^{46} - 3q^{47} - 3q^{48} + 15q^{49} + 30q^{51} - 3q^{52} + 12q^{53} + 18q^{54} - 12q^{57} - 6q^{58} + 21q^{59} - 21q^{61} - 18q^{62} - 3q^{64} - 15q^{66} - 9q^{67} - 12q^{68} + 3q^{69} + 15q^{71} + 3q^{72} - 21q^{73} - 18q^{74} - 6q^{77} - 9q^{78} + 21q^{81} - 18q^{82} + 3q^{83} - 3q^{84} - 6q^{86} + 12q^{87} - 3q^{88} - 24q^{89} - 27q^{91} - 3q^{92} - 15q^{93} + 30q^{94} - 9q^{97} - 6q^{98} - 15q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-\zeta_{18}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.939693 + 0.342020i −0.173648 − 0.984808i 0.939693 − 0.342020i −0.766044 + 0.642788i −0.173648 + 0.984808i −0.766044 − 0.642788i
−0.766044 + 0.642788i 1.76604 + 0.642788i 0.173648 0.984808i 0 −1.76604 + 0.642788i 0.173648 0.300767i 0.500000 + 0.866025i 0.407604 + 0.342020i 0
251.1 0.939693 + 0.342020i 0.0603074 + 0.342020i 0.766044 + 0.642788i 0 −0.0603074 + 0.342020i 0.766044 1.32683i 0.500000 + 0.866025i 2.70574 0.984808i 0
301.1 −0.766044 0.642788i 1.76604 0.642788i 0.173648 + 0.984808i 0 −1.76604 0.642788i 0.173648 + 0.300767i 0.500000 0.866025i 0.407604 0.342020i 0
351.1 −0.173648 0.984808i 1.17365 0.984808i −0.939693 + 0.342020i 0 −1.17365 0.984808i −0.939693 + 1.62760i 0.500000 + 0.866025i −0.113341 + 0.642788i 0
651.1 0.939693 0.342020i 0.0603074 0.342020i 0.766044 0.642788i 0 −0.0603074 0.342020i 0.766044 + 1.32683i 0.500000 0.866025i 2.70574 + 0.984808i 0
701.1 −0.173648 + 0.984808i 1.17365 + 0.984808i −0.939693 0.342020i 0 −1.17365 + 0.984808i −0.939693 1.62760i 0.500000 0.866025i −0.113341 0.642788i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 701.1 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.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.l.f yes 6
5.b even 2 1 950.2.l.a 6
5.c odd 4 2 950.2.u.a 12
19.e even 9 1 inner 950.2.l.f yes 6
95.p even 18 1 950.2.l.a 6
95.q odd 36 2 950.2.u.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.l.a 6 5.b even 2 1
950.2.l.a 6 95.p even 18 1
950.2.l.f yes 6 1.a even 1 1 trivial
950.2.l.f yes 6 19.e even 9 1 inner
950.2.u.a 12 5.c odd 4 2
950.2.u.a 12 95.q odd 36 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 6 T_{3}^{5} + 15 T_{3}^{4} - 19 T_{3}^{3} + 12 T_{3}^{2} - 3 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{3} + T^{6}$$
$3$ $$1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$1 - 3 T + 9 T^{2} - 2 T^{3} + 3 T^{4} + T^{6}$$
$11$ $$9 - 54 T + 333 T^{2} + 48 T^{3} + 27 T^{4} - 3 T^{5} + T^{6}$$
$13$ $$5041 - 4899 T + 2211 T^{2} - 620 T^{3} + 120 T^{4} - 15 T^{5} + T^{6}$$
$17$ $$2601 + 1377 T + 36 T^{2} - 3 T^{3} + 27 T^{4} - 6 T^{5} + T^{6}$$
$19$ $$6859 - 2166 T - 228 T^{2} + 169 T^{3} - 12 T^{4} - 6 T^{5} + T^{6}$$
$23$ $$2601 - 1836 T + 846 T^{2} - 300 T^{3} + 90 T^{4} - 15 T^{5} + T^{6}$$
$29$ $$12321 - 7992 T + 4356 T^{2} - 1407 T^{3} + 252 T^{4} - 24 T^{5} + T^{6}$$
$31$ $$5329 + 1095 T + 663 T^{2} + 56 T^{3} + 51 T^{4} + 6 T^{5} + T^{6}$$
$37$ $$( -127 + 39 T + 15 T^{2} + T^{3} )^{2}$$
$41$ $$5184 - 5184 T + 2592 T^{2} - 720 T^{3} + 144 T^{4} - 18 T^{5} + T^{6}$$
$43$ $$1 - 3 T + 6 T^{2} - 8 T^{3} + 3 T^{4} + 3 T^{5} + T^{6}$$
$47$ $$12321 - 1998 T - 477 T^{2} - 57 T^{3} + 36 T^{4} + 3 T^{5} + T^{6}$$
$53$ $$9 + 81 T + 306 T^{2} - 132 T^{3} + 54 T^{4} - 12 T^{5} + T^{6}$$
$59$ $$25281 + 12879 T + 4032 T^{2} + 408 T^{3} + 81 T^{4} - 21 T^{5} + T^{6}$$
$61$ $$54289 + 55221 T + 20775 T^{2} + 3428 T^{3} + 348 T^{4} + 21 T^{5} + T^{6}$$
$67$ $$5041 + 639 T + 1719 T^{2} - 152 T^{3} - 18 T^{4} + 9 T^{5} + T^{6}$$
$71$ $$103041 - 89559 T + 27171 T^{2} - 3108 T^{3} + 252 T^{4} - 15 T^{5} + T^{6}$$
$73$ $$11449 - 7383 T + 1680 T^{2} - 548 T^{3} + 183 T^{4} + 21 T^{5} + T^{6}$$
$79$ $$1 - T^{3} + T^{6}$$
$83$ $$2601 + 2754 T + 2763 T^{2} + 264 T^{3} + 63 T^{4} - 3 T^{5} + T^{6}$$
$89$ $$71289 + 40851 T + 11538 T^{2} + 1596 T^{3} + 234 T^{4} + 24 T^{5} + T^{6}$$
$97$ $$395641 + 22644 T - 8298 T^{2} + 748 T^{3} + 306 T^{4} + 9 T^{5} + T^{6}$$