Properties

 Label 1050.3.q.a Level $1050$ Weight $3$ Character orbit 1050.q Analytic conductor $28.610$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.q (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} - \zeta_{24}^{7} ) q^{2} + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{3} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{6} + ( 2 \zeta_{24} + 5 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 5 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} -3 \zeta_{24}^{4} q^{9} +O(q^{10})$$ $$q + ( \zeta_{24} - \zeta_{24}^{7} ) q^{2} + ( \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{3} + ( 2 - 2 \zeta_{24}^{4} ) q^{4} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{6} + ( 2 \zeta_{24} + 5 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 5 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{8} -3 \zeta_{24}^{4} q^{9} + ( -6 + 6 \zeta_{24}^{4} ) q^{11} + ( -2 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{12} + ( 8 \zeta_{24} - 2 \zeta_{24}^{2} + 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{13} + ( -4 + 5 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{14} -4 \zeta_{24}^{4} q^{16} + ( 4 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 16 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{17} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{18} + ( 14 + 2 \zeta_{24} + 4 \zeta_{24}^{3} - 7 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{19} + ( 5 - 6 \zeta_{24} - 6 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 6 \zeta_{24}^{5} ) q^{21} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{22} + ( -18 \zeta_{24} + 12 \zeta_{24}^{2} - 12 \zeta_{24}^{6} + 18 \zeta_{24}^{7} ) q^{23} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{24} + ( 32 - \zeta_{24} - 2 \zeta_{24}^{3} - 16 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{26} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{27} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 10 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{28} + ( -24 \zeta_{24} - 24 \zeta_{24}^{3} + 24 \zeta_{24}^{5} ) q^{29} + ( 17 - 12 \zeta_{24} - 6 \zeta_{24}^{3} + 17 \zeta_{24}^{4} + 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{32} + ( 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{33} + ( 4 - 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{4} - 8 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{34} -6 q^{36} + ( 12 \zeta_{24} - 11 \zeta_{24}^{2} + 11 \zeta_{24}^{6} - 12 \zeta_{24}^{7} ) q^{37} + ( 14 \zeta_{24} + 4 \zeta_{24}^{2} - 7 \zeta_{24}^{3} - 7 \zeta_{24}^{5} - 8 \zeta_{24}^{6} - 7 \zeta_{24}^{7} ) q^{38} + ( -3 + 24 \zeta_{24}^{3} + 3 \zeta_{24}^{4} - 24 \zeta_{24}^{5} - 24 \zeta_{24}^{7} ) q^{39} + ( 26 + 4 \zeta_{24} - 4 \zeta_{24}^{3} - 52 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{41} + ( 5 \zeta_{24} - 12 \zeta_{24}^{2} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} + 12 \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{42} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 7 \zeta_{24}^{6} ) q^{43} + 12 \zeta_{24}^{4} q^{44} + ( -36 + 12 \zeta_{24}^{3} + 36 \zeta_{24}^{4} - 12 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{46} + ( -2 \zeta_{24} - 22 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 22 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{47} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{48} + ( 1 - 40 \zeta_{24} - 20 \zeta_{24}^{3} - \zeta_{24}^{4} + 20 \zeta_{24}^{5} - 20 \zeta_{24}^{7} ) q^{49} + ( -6 \zeta_{24} - 24 \zeta_{24}^{4} - 6 \zeta_{24}^{7} ) q^{51} + ( 32 \zeta_{24} - 2 \zeta_{24}^{2} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} + 4 \zeta_{24}^{6} - 16 \zeta_{24}^{7} ) q^{52} + ( -60 \zeta_{24}^{2} + 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} - 18 \zeta_{24}^{7} ) q^{53} + ( -3 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{54} + ( -16 - 10 \zeta_{24} + 8 \zeta_{24}^{4} - 10 \zeta_{24}^{7} ) q^{56} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 21 \zeta_{24}^{6} ) q^{57} + ( -48 \zeta_{24}^{2} + 48 \zeta_{24}^{6} ) q^{58} + ( 4 - 28 \zeta_{24} - 14 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 14 \zeta_{24}^{5} - 14 \zeta_{24}^{7} ) q^{59} + ( -24 - 8 \zeta_{24} - 16 \zeta_{24}^{3} + 12 \zeta_{24}^{4} + 16 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{61} + ( 17 \zeta_{24} - 24 \zeta_{24}^{2} + 17 \zeta_{24}^{3} + 17 \zeta_{24}^{5} + 12 \zeta_{24}^{6} - 34 \zeta_{24}^{7} ) q^{62} + ( -12 \zeta_{24} - 15 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{63} -8 q^{64} + ( 12 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{66} + ( 55 \zeta_{24}^{2} + 42 \zeta_{24}^{3} + 42 \zeta_{24}^{5} - 42 \zeta_{24}^{7} ) q^{67} + ( 4 \zeta_{24} - 16 \zeta_{24}^{2} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 16 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{68} + ( 12 + 18 \zeta_{24} - 18 \zeta_{24}^{3} - 24 \zeta_{24}^{4} + 18 \zeta_{24}^{5} + 36 \zeta_{24}^{7} ) q^{69} + ( 78 - 42 \zeta_{24} - 42 \zeta_{24}^{3} + 42 \zeta_{24}^{5} ) q^{71} + ( -6 \zeta_{24} + 6 \zeta_{24}^{7} ) q^{72} + ( -80 \zeta_{24} - 11 \zeta_{24}^{2} + 40 \zeta_{24}^{3} + 40 \zeta_{24}^{5} + 22 \zeta_{24}^{6} + 40 \zeta_{24}^{7} ) q^{73} + ( 24 - 11 \zeta_{24}^{3} - 24 \zeta_{24}^{4} + 11 \zeta_{24}^{5} + 11 \zeta_{24}^{7} ) q^{74} + ( 14 - 4 \zeta_{24} + 4 \zeta_{24}^{3} - 28 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{76} + ( 12 \zeta_{24} + 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} + 30 \zeta_{24}^{6} - 24 \zeta_{24}^{7} ) q^{77} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 48 \zeta_{24}^{6} ) q^{78} + ( -66 \zeta_{24} + 5 \zeta_{24}^{4} - 66 \zeta_{24}^{7} ) q^{79} + ( -9 + 9 \zeta_{24}^{4} ) q^{81} + ( 26 \zeta_{24} + 8 \zeta_{24}^{2} - 52 \zeta_{24}^{3} - 52 \zeta_{24}^{5} + 8 \zeta_{24}^{6} + 26 \zeta_{24}^{7} ) q^{82} + ( -38 \zeta_{24} + 20 \zeta_{24}^{2} - 38 \zeta_{24}^{3} - 38 \zeta_{24}^{5} - 10 \zeta_{24}^{6} + 76 \zeta_{24}^{7} ) q^{83} + ( -10 - 12 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 12 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{84} + ( -7 \zeta_{24} - 12 \zeta_{24}^{4} - 7 \zeta_{24}^{7} ) q^{86} + ( -48 \zeta_{24} + 24 \zeta_{24}^{3} + 24 \zeta_{24}^{5} + 24 \zeta_{24}^{7} ) q^{87} + ( 12 \zeta_{24}^{3} + 12 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{88} + ( 24 - 12 \zeta_{24}^{4} ) q^{89} + ( -10 + 46 \zeta_{24} + 80 \zeta_{24}^{3} - 91 \zeta_{24}^{4} - 80 \zeta_{24}^{5} - 34 \zeta_{24}^{7} ) q^{91} + ( -36 \zeta_{24} + 36 \zeta_{24}^{3} + 36 \zeta_{24}^{5} - 24 \zeta_{24}^{6} ) q^{92} + ( -18 \zeta_{24} + 51 \zeta_{24}^{2} - 51 \zeta_{24}^{6} + 18 \zeta_{24}^{7} ) q^{93} + ( 4 - 44 \zeta_{24} - 22 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 22 \zeta_{24}^{5} - 22 \zeta_{24}^{7} ) q^{94} + ( -4 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{96} + ( 4 \zeta_{24} + 24 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 12 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{97} + ( \zeta_{24} - 80 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 40 \zeta_{24}^{6} ) q^{98} + 18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{4} - 12 q^{9} + O(q^{10})$$ $$8 q + 8 q^{4} - 12 q^{9} - 24 q^{11} - 48 q^{14} - 16 q^{16} + 84 q^{19} + 192 q^{26} + 204 q^{31} - 48 q^{36} - 12 q^{39} + 48 q^{44} - 144 q^{46} + 4 q^{49} - 96 q^{51} - 96 q^{56} + 48 q^{59} - 144 q^{61} - 64 q^{64} + 624 q^{71} + 96 q^{74} + 20 q^{79} - 36 q^{81} - 120 q^{84} - 48 q^{86} + 144 q^{89} - 444 q^{91} + 48 q^{94} + 144 q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$\zeta_{24}^{2}$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −0.258819 + 0.965926i −0.965926 − 0.258819i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.258819 − 0.965926i −0.965926 + 0.258819i 0.258819 + 0.965926i 0.965926 − 0.258819i
−1.22474 + 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −1.88064 + 6.74264i 2.82843i −1.50000 2.59808i 0
199.2 −1.22474 + 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 6.77962 + 1.74264i 2.82843i −1.50000 2.59808i 0
199.3 1.22474 0.707107i −0.866025 + 1.50000i 1.00000 1.73205i 0 2.44949i −6.77962 1.74264i 2.82843i −1.50000 2.59808i 0
199.4 1.22474 0.707107i 0.866025 1.50000i 1.00000 1.73205i 0 2.44949i 1.88064 6.74264i 2.82843i −1.50000 2.59808i 0
649.1 −1.22474 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −1.88064 6.74264i 2.82843i −1.50000 + 2.59808i 0
649.2 −1.22474 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 6.77962 1.74264i 2.82843i −1.50000 + 2.59808i 0
649.3 1.22474 + 0.707107i −0.866025 1.50000i 1.00000 + 1.73205i 0 2.44949i −6.77962 + 1.74264i 2.82843i −1.50000 + 2.59808i 0
649.4 1.22474 + 0.707107i 0.866025 + 1.50000i 1.00000 + 1.73205i 0 2.44949i 1.88064 + 6.74264i 2.82843i −1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.4 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.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.q.a 8
5.b even 2 1 inner 1050.3.q.a 8
5.c odd 4 1 42.3.g.a 4
5.c odd 4 1 1050.3.p.a 4
7.d odd 6 1 inner 1050.3.q.a 8
15.e even 4 1 126.3.n.a 4
20.e even 4 1 336.3.bh.e 4
35.f even 4 1 294.3.g.a 4
35.i odd 6 1 inner 1050.3.q.a 8
35.k even 12 1 42.3.g.a 4
35.k even 12 1 294.3.c.a 4
35.k even 12 1 1050.3.p.a 4
35.l odd 12 1 294.3.c.a 4
35.l odd 12 1 294.3.g.a 4
60.l odd 4 1 1008.3.cg.h 4
105.k odd 4 1 882.3.n.e 4
105.w odd 12 1 126.3.n.a 4
105.w odd 12 1 882.3.c.b 4
105.x even 12 1 882.3.c.b 4
105.x even 12 1 882.3.n.e 4
140.w even 12 1 2352.3.f.e 4
140.x odd 12 1 336.3.bh.e 4
140.x odd 12 1 2352.3.f.e 4
420.br even 12 1 1008.3.cg.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 5.c odd 4 1
42.3.g.a 4 35.k even 12 1
126.3.n.a 4 15.e even 4 1
126.3.n.a 4 105.w odd 12 1
294.3.c.a 4 35.k even 12 1
294.3.c.a 4 35.l odd 12 1
294.3.g.a 4 35.f even 4 1
294.3.g.a 4 35.l odd 12 1
336.3.bh.e 4 20.e even 4 1
336.3.bh.e 4 140.x odd 12 1
882.3.c.b 4 105.w odd 12 1
882.3.c.b 4 105.x even 12 1
882.3.n.e 4 105.k odd 4 1
882.3.n.e 4 105.x even 12 1
1008.3.cg.h 4 60.l odd 4 1
1008.3.cg.h 4 420.br even 12 1
1050.3.p.a 4 5.c odd 4 1
1050.3.p.a 4 35.k even 12 1
1050.3.q.a 8 1.a even 1 1 trivial
1050.3.q.a 8 5.b even 2 1 inner
1050.3.q.a 8 7.d odd 6 1 inner
1050.3.q.a 8 35.i odd 6 1 inner
2352.3.f.e 4 140.w even 12 1
2352.3.f.e 4 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} + 6 T_{11} + 36$$ acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$3$ $$( 9 + 3 T^{2} + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$5764801 - 4802 T^{2} - 2397 T^{4} - 2 T^{6} + T^{8}$$
$11$ $$( 36 + 6 T + T^{2} )^{4}$$
$13$ $$( 145161 - 774 T^{2} + T^{4} )^{2}$$
$17$ $$796594176 + 12192768 T^{2} + 158400 T^{4} + 432 T^{6} + T^{8}$$
$19$ $$( 15129 - 5166 T + 711 T^{2} - 42 T^{3} + T^{4} )^{2}$$
$23$ $$64524128256 - 402361344 T^{2} + 2255040 T^{4} - 1584 T^{6} + T^{8}$$
$29$ $$( -1152 + T^{2} )^{4}$$
$31$ $$( 423801 - 66402 T + 4119 T^{2} - 102 T^{3} + T^{4} )^{2}$$
$37$ $$777796321 - 22813202 T^{2} + 641235 T^{4} - 818 T^{6} + T^{8}$$
$41$ $$( 3732624 + 4248 T^{2} + T^{4} )^{2}$$
$43$ $$( 529 + 242 T^{2} + T^{4} )^{2}$$
$47$ $$4158271385856 + 6019671168 T^{2} + 6675120 T^{4} + 2952 T^{6} + T^{8}$$
$53$ $$75939094204416 - 74036726784 T^{2} + 63467712 T^{4} - 8496 T^{6} + T^{8}$$
$59$ $$( 1272384 + 27072 T - 936 T^{2} - 24 T^{3} + T^{4} )^{2}$$
$61$ $$( 2304 + 3456 T + 1776 T^{2} + 72 T^{3} + T^{4} )^{2}$$
$67$ $$64013554081 - 3315935954 T^{2} + 171514227 T^{4} - 13106 T^{6} + T^{8}$$
$71$ $$( 2556 - 156 T + T^{2} )^{4}$$
$73$ $$7279872522864561 + 1700129539494 T^{2} + 311723307 T^{4} + 19926 T^{6} + T^{8}$$
$79$ $$( 75463969 + 86870 T + 8787 T^{2} - 10 T^{3} + T^{4} )^{2}$$
$83$ $$( 69956496 - 17928 T^{2} + T^{4} )^{2}$$
$89$ $$( 432 - 36 T + T^{2} )^{4}$$
$97$ $$( 112896 - 1056 T^{2} + T^{4} )^{2}$$
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