# Properties

 Label 950.2.l.e Level $950$ Weight $2$ Character orbit 950.l Analytic conductor $7.586$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{18} - \zeta_{18}^{4} ) q^{2} + ( 1 + \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{3} -\zeta_{18}^{5} q^{4} + ( \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{6} + ( 1 + \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} -\zeta_{18}^{3} q^{8} + ( -2 + \zeta_{18} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{18} - \zeta_{18}^{4} ) q^{2} + ( 1 + \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{3} -\zeta_{18}^{5} q^{4} + ( \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{6} + ( 1 + \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} -\zeta_{18}^{3} q^{8} + ( -2 + \zeta_{18} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{9} + \zeta_{18}^{3} q^{11} + ( 1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{12} + ( 1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{13} + ( -\zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{14} -\zeta_{18} q^{16} + ( 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{17} + ( -\zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{18} + ( -1 + 4 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{19} + ( 1 + 2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{21} + \zeta_{18} q^{22} + ( 1 - \zeta_{18}^{3} - \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{23} + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{24} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{26} + ( -3 \zeta_{18}^{2} - 3 \zeta_{18}^{4} ) q^{27} + ( -\zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{28} + ( -4 + 2 \zeta_{18} + 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{29} + ( 1 + 2 \zeta_{18} - 2 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{31} + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{32} + ( 1 - \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{33} + ( -1 + \zeta_{18} - 2 \zeta_{18}^{5} ) q^{34} + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{36} + ( -2 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{4} ) q^{37} + ( 2 - 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{38} + ( -2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{39} + ( -3 + 2 \zeta_{18} - 4 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{41} + ( 1 - \zeta_{18}^{3} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{42} + ( 1 - \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{43} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{44} + ( -\zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{46} + ( -2 + \zeta_{18} + 3 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{47} + ( 1 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{48} + ( -\zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{49} + ( 1 + \zeta_{18} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{51} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{52} + ( -1 - 4 \zeta_{18} + 5 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{53} + ( -3 - 3 \zeta_{18}^{2} ) q^{54} + ( -1 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{56} + ( -2 + \zeta_{18} + 5 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{57} + ( -2 \zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{58} + ( 4 + 3 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{59} + ( -1 + 4 \zeta_{18} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{61} + ( -2 + 2 \zeta_{18}^{2} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{62} + ( -2 - 2 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{63} + ( -1 + \zeta_{18}^{3} ) q^{64} + ( -1 + \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{66} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{67} + ( -\zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{68} + ( -2 - 3 \zeta_{18} + 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{69} + ( 4 - 4 \zeta_{18}^{2} - 6 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{71} + ( 1 - 2 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{72} + ( 4 - \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{73} + ( 3 - 2 \zeta_{18} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{74} + ( 2 - \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{76} + ( 1 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{77} + ( 2 - 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{78} + ( 3 + 6 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{79} + ( -3 + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{81} + ( -4 - 4 \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{82} + ( -6 - 9 \zeta_{18} + 8 \zeta_{18}^{2} + 6 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{83} + ( -\zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{84} + ( -1 + 5 \zeta_{18} + 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{86} + ( -6 \zeta_{18} + 6 \zeta_{18}^{3} - 6 \zeta_{18}^{5} ) q^{87} + ( 1 - \zeta_{18}^{3} ) q^{88} + ( -4 - 4 \zeta_{18} + 4 \zeta_{18}^{2} + 8 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{89} + ( \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{91} + ( -1 + 3 \zeta_{18} - \zeta_{18}^{2} ) q^{92} + ( -2 + 5 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{93} + ( 3 - \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{94} + ( -1 - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{96} + ( -8 + \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{97} + ( -2 + 4 \zeta_{18} - 3 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{98} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{3} + 3q^{6} + 3q^{7} - 3q^{8} - 9q^{9} + O(q^{10})$$ $$6q + 3q^{3} + 3q^{6} + 3q^{7} - 3q^{8} - 9q^{9} + 3q^{11} + 3q^{12} + 3q^{13} - 3q^{14} + 3q^{17} - 9q^{19} + 3q^{21} + 3q^{23} - 6q^{24} - 3q^{26} - 3q^{28} - 18q^{29} + 3q^{31} + 6q^{33} - 6q^{34} - 9q^{36} - 12q^{37} + 6q^{38} - 12q^{39} - 21q^{41} + 3q^{42} + 18q^{43} + 9q^{46} - 9q^{47} + 3q^{48} + 12q^{49} + 9q^{51} + 3q^{52} + 9q^{53} - 18q^{54} - 6q^{56} + 3q^{57} + 27q^{59} - 15q^{61} - 12q^{62} - 9q^{63} - 3q^{64} - 3q^{66} + 6q^{67} - 6q^{68} - 6q^{69} + 6q^{71} + 9q^{72} + 21q^{73} + 18q^{74} + 6q^{77} + 15q^{78} + 12q^{79} - 18q^{81} - 21q^{82} - 18q^{83} - 6q^{84} + 9q^{86} + 18q^{87} + 3q^{88} - 3q^{91} - 6q^{92} - 15q^{93} + 18q^{94} - 6q^{96} - 45q^{97} - 15q^{98} - 9q^{99} + O(q^{100})$$

## 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 1.26604 + 0.460802i 0.173648 0.984808i 0 1.26604 0.460802i 1.43969 2.49362i −0.500000 0.866025i −0.907604 0.761570i 0
251.1 −0.939693 0.342020i −0.439693 2.49362i 0.766044 + 0.642788i 0 −0.439693 + 2.49362i 0.326352 0.565258i −0.500000 0.866025i −3.20574 + 1.16679i 0
301.1 0.766044 + 0.642788i 1.26604 0.460802i 0.173648 + 0.984808i 0 1.26604 + 0.460802i 1.43969 + 2.49362i −0.500000 + 0.866025i −0.907604 + 0.761570i 0
351.1 0.173648 + 0.984808i 0.673648 0.565258i −0.939693 + 0.342020i 0 0.673648 + 0.565258i −0.266044 + 0.460802i −0.500000 0.866025i −0.386659 + 2.19285i 0
651.1 −0.939693 + 0.342020i −0.439693 + 2.49362i 0.766044 0.642788i 0 −0.439693 2.49362i 0.326352 + 0.565258i −0.500000 + 0.866025i −3.20574 1.16679i 0
701.1 0.173648 0.984808i 0.673648 + 0.565258i −0.939693 0.342020i 0 0.673648 0.565258i −0.266044 0.460802i −0.500000 + 0.866025i −0.386659 2.19285i 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.e yes 6
5.b even 2 1 950.2.l.b 6
5.c odd 4 2 950.2.u.d 12
19.e even 9 1 inner 950.2.l.e yes 6
95.p even 18 1 950.2.l.b 6
95.q odd 36 2 950.2.u.d 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 3 T_{3}^{5} + 9 T_{3}^{4} - 24 T_{3}^{3} + 36 T_{3}^{2} - 27 T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{3} + T^{6}$$
$3$ $$9 - 27 T + 36 T^{2} - 24 T^{3} + 9 T^{4} - 3 T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$1 + 3 T^{2} - 2 T^{3} + 9 T^{4} - 3 T^{5} + T^{6}$$
$11$ $$( 1 - T + T^{2} )^{3}$$
$13$ $$1 - 6 T + 15 T^{2} - 19 T^{3} + 12 T^{4} - 3 T^{5} + T^{6}$$
$17$ $$9 + 27 T + 9 T^{2} - 24 T^{3} + 18 T^{4} - 3 T^{5} + T^{6}$$
$19$ $$6859 + 3249 T - 179 T^{3} + 9 T^{5} + T^{6}$$
$23$ $$289 + 102 T + 219 T^{2} + T^{3} - 12 T^{4} - 3 T^{5} + T^{6}$$
$29$ $$5184 + 5184 T + 2592 T^{2} + 720 T^{3} + 144 T^{4} + 18 T^{5} + T^{6}$$
$31$ $$361 - 171 T + 138 T^{2} - 11 T^{3} + 18 T^{4} - 3 T^{5} + T^{6}$$
$37$ $$( -19 - 15 T + 6 T^{2} + T^{3} )^{2}$$
$41$ $$25281 + 17172 T + 5976 T^{2} + 1374 T^{3} + 216 T^{4} + 21 T^{5} + T^{6}$$
$43$ $$29241 - 26163 T + 10530 T^{2} - 2196 T^{3} + 252 T^{4} - 18 T^{5} + T^{6}$$
$47$ $$81 - 81 T + 162 T^{2} - 72 T^{3} + 9 T^{4} + 9 T^{5} + T^{6}$$
$53$ $$32041 - 27387 T + 9891 T^{2} - 1792 T^{3} + 162 T^{4} - 9 T^{5} + T^{6}$$
$59$ $$110889 - 59940 T + 16281 T^{2} - 2925 T^{3} + 360 T^{4} - 27 T^{5} + T^{6}$$
$61$ $$25281 + 15741 T + 4545 T^{2} + 840 T^{3} + 126 T^{4} + 15 T^{5} + T^{6}$$
$67$ $$9 - 27 T + 36 T^{2} - 30 T^{3} + 18 T^{4} - 6 T^{5} + T^{6}$$
$71$ $$166464 + 44064 T + 6192 T^{2} + 672 T^{3} - 6 T^{5} + T^{6}$$
$73$ $$361 - 456 T + 1581 T^{2} - 847 T^{3} + 192 T^{4} - 21 T^{5} + T^{6}$$
$79$ $$2809 + 5565 T + 4548 T^{2} - 899 T^{3} + 87 T^{4} - 12 T^{5} + T^{6}$$
$83$ $$4068289 + 223887 T + 48627 T^{2} + 2036 T^{3} + 435 T^{4} + 18 T^{5} + T^{6}$$
$89$ $$4096 + 9216 T + 4608 T^{2} - 1792 T^{3} + 288 T^{4} + T^{6}$$
$97$ $$289 - 918 T + 28071 T^{2} + 7381 T^{3} + 828 T^{4} + 45 T^{5} + T^{6}$$
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