Properties

 Label 950.2.l.c 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: no (minimal twist has level 190) 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.

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

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 −1.43969 0.524005i 0.173648 0.984808i 0 1.43969 0.524005i −0.347296 + 0.601535i 0.500000 + 0.866025i −0.500000 0.419550i 0
251.1 0.939693 + 0.342020i −0.326352 1.85083i 0.766044 + 0.642788i 0 0.326352 1.85083i −1.53209 + 2.65366i 0.500000 + 0.866025i −0.500000 + 0.181985i 0
301.1 −0.766044 0.642788i −1.43969 + 0.524005i 0.173648 + 0.984808i 0 1.43969 + 0.524005i −0.347296 0.601535i 0.500000 0.866025i −0.500000 + 0.419550i 0
351.1 −0.173648 0.984808i 0.266044 0.223238i −0.939693 + 0.342020i 0 −0.266044 0.223238i 1.87939 3.25519i 0.500000 + 0.866025i −0.500000 + 2.83564i 0
651.1 0.939693 0.342020i −0.326352 + 1.85083i 0.766044 0.642788i 0 0.326352 + 1.85083i −1.53209 2.65366i 0.500000 0.866025i −0.500000 0.181985i 0
701.1 −0.173648 + 0.984808i 0.266044 + 0.223238i −0.939693 0.342020i 0 −0.266044 + 0.223238i 1.87939 + 3.25519i 0.500000 0.866025i −0.500000 2.83564i 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.c 6
5.b even 2 1 190.2.k.a 6
5.c odd 4 2 950.2.u.c 12
19.e even 9 1 inner 950.2.l.c 6
95.o odd 18 1 3610.2.a.w 3
95.p even 18 1 190.2.k.a 6
95.p even 18 1 3610.2.a.x 3
95.q odd 36 2 950.2.u.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.a 6 5.b even 2 1
190.2.k.a 6 95.p even 18 1
950.2.l.c 6 1.a even 1 1 trivial
950.2.l.c 6 19.e even 9 1 inner
950.2.u.c 12 5.c odd 4 2
950.2.u.c 12 95.q odd 36 2
3610.2.a.w 3 95.o odd 18 1
3610.2.a.x 3 95.p even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 3 T_{3}^{5} + 6 T_{3}^{4} + 8 T_{3}^{3} + 3 T_{3}^{2} - 3 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{3} + T^{6}$$
$3$ $$1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$64 + 96 T + 144 T^{2} + 16 T^{3} + 12 T^{4} + T^{6}$$
$11$ $$2601 - 2295 T + 1413 T^{2} - 438 T^{3} + 99 T^{4} - 12 T^{5} + T^{6}$$
$13$ $$64 - 192 T + 240 T^{2} - 152 T^{3} + 48 T^{4} - 6 T^{5} + T^{6}$$
$17$ $$9 - 54 T + 144 T^{2} - 111 T^{3} + 36 T^{4} - 6 T^{5} + T^{6}$$
$19$ $$6859 - 2166 T - 228 T^{2} + 169 T^{3} - 12 T^{4} - 6 T^{5} + T^{6}$$
$23$ $$576 + 576 T^{2} + 240 T^{3} - 6 T^{5} + T^{6}$$
$29$ $$576 + 1728 T + 2304 T^{2} + 888 T^{3} + 144 T^{4} + 12 T^{5} + T^{6}$$
$31$ $$23104 + 1824 T + 1968 T^{2} - 448 T^{3} + 132 T^{4} - 12 T^{5} + T^{6}$$
$37$ $$( -4 + T )^{6}$$
$41$ $$81 + 162 T + 3888 T^{2} - 1692 T^{3} + 306 T^{4} - 27 T^{5} + T^{6}$$
$43$ $$2809 - 318 T + 5766 T^{2} + 2395 T^{3} + 390 T^{4} + 30 T^{5} + T^{6}$$
$47$ $$166464 - 102816 T + 28224 T^{2} - 4296 T^{3} + 396 T^{4} - 24 T^{5} + T^{6}$$
$53$ $$576 - 864 T + 576 T^{2} - 240 T^{3} + 72 T^{4} - 12 T^{5} + T^{6}$$
$59$ $$239121 + 8802 T + 1980 T^{2} + 624 T^{3} - 18 T^{4} - 3 T^{5} + T^{6}$$
$61$ $$18496 - 11424 T + 1056 T^{2} + 224 T^{3} + 96 T^{4} + 12 T^{5} + T^{6}$$
$67$ $$982081 - 383517 T + 60957 T^{2} - 5624 T^{3} + 450 T^{4} - 27 T^{5} + T^{6}$$
$71$ $$166464 - 102816 T + 28224 T^{2} - 4296 T^{3} + 396 T^{4} - 24 T^{5} + T^{6}$$
$73$ $$289 - 255 T + 132 T^{2} + T^{3} + 39 T^{4} + 12 T^{5} + T^{6}$$
$79$ $$64 + 432 T^{2} - 424 T^{3} + 144 T^{4} - 18 T^{5} + T^{6}$$
$83$ $$25281 + 32913 T + 43803 T^{2} - 924 T^{3} + 243 T^{4} + 6 T^{5} + T^{6}$$
$89$ $$103041 - 46224 T + 40536 T^{2} - 8961 T^{3} + 936 T^{4} - 48 T^{5} + T^{6}$$
$97$ $$253009 + 95067 T + 19197 T^{2} + 2656 T^{3} + 324 T^{4} + 27 T^{5} + T^{6}$$