Properties

 Label 950.2.l.d 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 38) 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.

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

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 1.34730 2.33359i 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 2.53209 4.38571i 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 1.34730 + 2.33359i 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 −0.879385 + 1.52314i 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 2.53209 + 4.38571i 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 −0.879385 1.52314i 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.d 6
5.b even 2 1 38.2.e.a 6
5.c odd 4 2 950.2.u.b 12
15.d odd 2 1 342.2.u.c 6
19.e even 9 1 inner 950.2.l.d 6
20.d odd 2 1 304.2.u.c 6
95.d odd 2 1 722.2.e.k 6
95.h odd 6 1 722.2.e.a 6
95.h odd 6 1 722.2.e.l 6
95.i even 6 1 722.2.e.b 6
95.i even 6 1 722.2.e.m 6
95.o odd 18 1 722.2.a.k 3
95.o odd 18 2 722.2.c.l 6
95.o odd 18 1 722.2.e.a 6
95.o odd 18 1 722.2.e.k 6
95.o odd 18 1 722.2.e.l 6
95.p even 18 1 38.2.e.a 6
95.p even 18 1 722.2.a.l 3
95.p even 18 2 722.2.c.k 6
95.p even 18 1 722.2.e.b 6
95.p even 18 1 722.2.e.m 6
95.q odd 36 2 950.2.u.b 12
285.bd odd 18 1 342.2.u.c 6
285.bd odd 18 1 6498.2.a.bl 3
285.bf even 18 1 6498.2.a.bq 3
380.ba odd 18 1 304.2.u.c 6
380.ba odd 18 1 5776.2.a.bn 3
380.bb even 18 1 5776.2.a.bo 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.e.a 6 5.b even 2 1
38.2.e.a 6 95.p even 18 1
304.2.u.c 6 20.d odd 2 1
304.2.u.c 6 380.ba odd 18 1
342.2.u.c 6 15.d odd 2 1
342.2.u.c 6 285.bd odd 18 1
722.2.a.k 3 95.o odd 18 1
722.2.a.l 3 95.p even 18 1
722.2.c.k 6 95.p even 18 2
722.2.c.l 6 95.o odd 18 2
722.2.e.a 6 95.h odd 6 1
722.2.e.a 6 95.o odd 18 1
722.2.e.b 6 95.i even 6 1
722.2.e.b 6 95.p even 18 1
722.2.e.k 6 95.d odd 2 1
722.2.e.k 6 95.o odd 18 1
722.2.e.l 6 95.h odd 6 1
722.2.e.l 6 95.o odd 18 1
722.2.e.m 6 95.i even 6 1
722.2.e.m 6 95.p even 18 1
950.2.l.d 6 1.a even 1 1 trivial
950.2.l.d 6 19.e even 9 1 inner
950.2.u.b 12 5.c odd 4 2
950.2.u.b 12 95.q odd 36 2
5776.2.a.bn 3 380.ba odd 18 1
5776.2.a.bo 3 380.bb even 18 1
6498.2.a.bl 3 285.bd odd 18 1
6498.2.a.bq 3 285.bf 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

$p$ $F_p(T)$
$2$ $$1 - T^{3} + T^{6}$$
$3$ $$1 + 3 T + 3 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$576 + 144 T^{2} - 48 T^{3} + 36 T^{4} - 6 T^{5} + T^{6}$$
$11$ $$361 - 57 T + 123 T^{2} + 56 T^{3} + 33 T^{4} + 6 T^{5} + T^{6}$$
$13$ $$64 + 96 T + 96 T^{2} + 64 T^{3} + 48 T^{4} + 12 T^{5} + T^{6}$$
$17$ $$12321 - 11988 T + 4356 T^{2} - 753 T^{3} + 108 T^{4} - 12 T^{5} + T^{6}$$
$19$ $$6859 - 6498 T + 3078 T^{2} - 883 T^{3} + 162 T^{4} - 18 T^{5} + T^{6}$$
$23$ $$64 - 96 T + 192 T^{2} - 152 T^{3} + 60 T^{4} - 12 T^{5} + T^{6}$$
$29$ $$23104 + 16416 T + 5616 T^{2} + 928 T^{3} + 144 T^{4} + 18 T^{5} + T^{6}$$
$31$ $$64 + 192 T + 528 T^{2} + 160 T^{3} + 60 T^{4} - 6 T^{5} + T^{6}$$
$37$ $$( 136 - 24 T - 6 T^{2} + T^{3} )^{2}$$
$41$ $$1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$43$ $$289 - 714 T + 786 T^{2} - 271 T^{3} + 42 T^{4} - 6 T^{5} + T^{6}$$
$47$ $$87616 + 24864 T + 8736 T^{2} + 2368 T^{3} + 372 T^{4} + 30 T^{5} + T^{6}$$
$53$ $$18496 + 21216 T + 11136 T^{2} + 2152 T^{3} + 276 T^{4} + 24 T^{5} + T^{6}$$
$59$ $$9 - 18 T^{2} + 24 T^{3} + 54 T^{4} + 3 T^{5} + T^{6}$$
$61$ $$23104 + 1824 T + 1920 T^{2} + 512 T^{3} - 12 T^{4} - 6 T^{5} + T^{6}$$
$67$ $$6561 + 6561 T + 729 T^{2} - 648 T^{3} + 162 T^{4} - 9 T^{5} + T^{6}$$
$71$ $$23104 + 21888 T + 8352 T^{2} + 1664 T^{3} + 216 T^{4} + 18 T^{5} + T^{6}$$
$73$ $$3249 + 5643 T + 4140 T^{2} - 645 T^{3} + 279 T^{4} - 30 T^{5} + T^{6}$$
$79$ $$18496 + 3264 T + 3504 T^{2} + 8 T^{3} - 48 T^{4} - 6 T^{5} + T^{6}$$
$83$ $$2601 - 1377 T + 1035 T^{2} + 60 T^{3} + 63 T^{4} - 6 T^{5} + T^{6}$$
$89$ $$962361 + 141264 T + 4860 T^{2} + 315 T^{3} + 36 T^{4} + T^{6}$$
$97$ $$1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$