# Properties

 Label 1050.2.u.a Level 1050 Weight 2 Character orbit 1050.u Analytic conductor 8.384 Analytic rank 1 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$1$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
299.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.500000 + 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i 0 1.73205i −0.866025 + 2.50000i 1.00000 1.50000 2.59808i 0
299.2 −0.500000 + 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i 0 1.73205i 0.866025 2.50000i 1.00000 1.50000 2.59808i 0
899.1 −0.500000 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i 0 1.73205i −0.866025 2.50000i 1.00000 1.50000 + 2.59808i 0
899.2 −0.500000 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i 0 1.73205i 0.866025 + 2.50000i 1.00000 1.50000 + 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.d odd 2 1 inner
105.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.u.a 4
3.b odd 2 1 1050.2.u.d 4
5.b even 2 1 1050.2.u.d 4
5.c odd 4 1 42.2.f.a 4
5.c odd 4 1 1050.2.s.b 4
7.d odd 6 1 inner 1050.2.u.a 4
15.d odd 2 1 inner 1050.2.u.a 4
15.e even 4 1 42.2.f.a 4
15.e even 4 1 1050.2.s.b 4
20.e even 4 1 336.2.bc.e 4
21.g even 6 1 1050.2.u.d 4
35.f even 4 1 294.2.f.a 4
35.i odd 6 1 1050.2.u.d 4
35.k even 12 1 42.2.f.a 4
35.k even 12 1 294.2.d.a 4
35.k even 12 1 1050.2.s.b 4
35.l odd 12 1 294.2.d.a 4
35.l odd 12 1 294.2.f.a 4
45.k odd 12 1 1134.2.l.c 4
45.k odd 12 1 1134.2.t.d 4
45.l even 12 1 1134.2.l.c 4
45.l even 12 1 1134.2.t.d 4
60.l odd 4 1 336.2.bc.e 4
105.k odd 4 1 294.2.f.a 4
105.p even 6 1 inner 1050.2.u.a 4
105.w odd 12 1 42.2.f.a 4
105.w odd 12 1 294.2.d.a 4
105.w odd 12 1 1050.2.s.b 4
105.x even 12 1 294.2.d.a 4
105.x even 12 1 294.2.f.a 4
140.w even 12 1 2352.2.k.e 4
140.x odd 12 1 336.2.bc.e 4
140.x odd 12 1 2352.2.k.e 4
315.bs even 12 1 1134.2.t.d 4
315.bu odd 12 1 1134.2.t.d 4
315.bw odd 12 1 1134.2.l.c 4
315.cg even 12 1 1134.2.l.c 4
420.bp odd 12 1 2352.2.k.e 4
420.br even 12 1 336.2.bc.e 4
420.br even 12 1 2352.2.k.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.f.a 4 5.c odd 4 1
42.2.f.a 4 15.e even 4 1
42.2.f.a 4 35.k even 12 1
42.2.f.a 4 105.w odd 12 1
294.2.d.a 4 35.k even 12 1
294.2.d.a 4 35.l odd 12 1
294.2.d.a 4 105.w odd 12 1
294.2.d.a 4 105.x even 12 1
294.2.f.a 4 35.f even 4 1
294.2.f.a 4 35.l odd 12 1
294.2.f.a 4 105.k odd 4 1
294.2.f.a 4 105.x even 12 1
336.2.bc.e 4 20.e even 4 1
336.2.bc.e 4 60.l odd 4 1
336.2.bc.e 4 140.x odd 12 1
336.2.bc.e 4 420.br even 12 1
1050.2.s.b 4 5.c odd 4 1
1050.2.s.b 4 15.e even 4 1
1050.2.s.b 4 35.k even 12 1
1050.2.s.b 4 105.w odd 12 1
1050.2.u.a 4 1.a even 1 1 trivial
1050.2.u.a 4 7.d odd 6 1 inner
1050.2.u.a 4 15.d odd 2 1 inner
1050.2.u.a 4 105.p even 6 1 inner
1050.2.u.d 4 3.b odd 2 1
1050.2.u.d 4 5.b even 2 1
1050.2.u.d 4 21.g even 6 1
1050.2.u.d 4 35.i odd 6 1
1134.2.l.c 4 45.k odd 12 1
1134.2.l.c 4 45.l even 12 1
1134.2.l.c 4 315.bw odd 12 1
1134.2.l.c 4 315.cg even 12 1
1134.2.t.d 4 45.k odd 12 1
1134.2.t.d 4 45.l even 12 1
1134.2.t.d 4 315.bs even 12 1
1134.2.t.d 4 315.bu odd 12 1
2352.2.k.e 4 140.w even 12 1
2352.2.k.e 4 140.x odd 12 1
2352.2.k.e 4 420.bp odd 12 1
2352.2.k.e 4 420.br even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{4} - 9 T_{11}^{2} + 81$$ $$T_{13}^{2} - 12$$ $$T_{17}^{2} + 6 T_{17} + 12$$ $$T_{23}^{2} + 6 T_{23} + 36$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$ $$( 1 + 3 T + 3 T^{2} )^{2}$$
$5$ 1
$7$ $$1 + 11 T^{2} + 49 T^{4}$$
$11$ $$1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8}$$
$13$ $$( 1 + 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 6 T + 29 T^{2} + 102 T^{3} + 289 T^{4} )^{2}$$
$19$ $$( 1 - T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2}$$
$23$ $$( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 107 T^{2} - 444 T^{3} + 1369 T^{4} )( 1 + 12 T + 107 T^{2} + 444 T^{3} + 1369 T^{4} )$$
$41$ $$( 1 + 34 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 22 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 12 T + 95 T^{2} + 564 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$1 + 130 T^{2} + 12411 T^{4} + 583570 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 + 2 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$1 - 98 T^{2} + 4275 T^{4} - 522242 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 91 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + 167 T^{2} + 9409 T^{4} )^{2}$$