# Properties

 Label 1050.2.o.a Level $1050$ Weight $2$ Character orbit 1050.o Analytic conductor $8.384$ Analytic rank $0$ 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.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
499.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i 0 −1.00000 2.59808 + 0.500000i 1.00000i 0.500000 0.866025i 0
499.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 −1.00000 −2.59808 0.500000i 1.00000i 0.500000 0.866025i 0
949.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0 −1.00000 2.59808 0.500000i 1.00000i 0.500000 + 0.866025i 0
949.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0 −1.00000 −2.59808 + 0.500000i 1.00000i 0.500000 + 0.866025i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j 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.o.a 4
5.b even 2 1 inner 1050.2.o.a 4
5.c odd 4 1 42.2.e.a 2
5.c odd 4 1 1050.2.i.l 2
7.c even 3 1 inner 1050.2.o.a 4
15.e even 4 1 126.2.g.c 2
20.e even 4 1 336.2.q.b 2
35.f even 4 1 294.2.e.b 2
35.j even 6 1 inner 1050.2.o.a 4
35.k even 12 1 294.2.a.f 1
35.k even 12 1 294.2.e.b 2
35.k even 12 1 7350.2.a.q 1
35.l odd 12 1 42.2.e.a 2
35.l odd 12 1 294.2.a.e 1
35.l odd 12 1 1050.2.i.l 2
35.l odd 12 1 7350.2.a.bl 1
40.i odd 4 1 1344.2.q.g 2
40.k even 4 1 1344.2.q.s 2
45.k odd 12 1 1134.2.e.l 2
45.k odd 12 1 1134.2.h.e 2
45.l even 12 1 1134.2.e.e 2
45.l even 12 1 1134.2.h.l 2
60.l odd 4 1 1008.2.s.k 2
105.k odd 4 1 882.2.g.i 2
105.w odd 12 1 882.2.a.d 1
105.w odd 12 1 882.2.g.i 2
105.x even 12 1 126.2.g.c 2
105.x even 12 1 882.2.a.c 1
140.j odd 4 1 2352.2.q.u 2
140.w even 12 1 336.2.q.b 2
140.w even 12 1 2352.2.a.t 1
140.x odd 12 1 2352.2.a.f 1
140.x odd 12 1 2352.2.q.u 2
280.bp odd 12 1 9408.2.a.cr 1
280.br even 12 1 1344.2.q.s 2
280.br even 12 1 9408.2.a.q 1
280.bt odd 12 1 1344.2.q.g 2
280.bt odd 12 1 9408.2.a.ce 1
280.bv even 12 1 9408.2.a.z 1
315.bt odd 12 1 1134.2.h.e 2
315.bv even 12 1 1134.2.h.l 2
315.bx even 12 1 1134.2.e.e 2
315.ch odd 12 1 1134.2.e.l 2
420.bp odd 12 1 1008.2.s.k 2
420.bp odd 12 1 7056.2.a.w 1
420.br even 12 1 7056.2.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.a 2 5.c odd 4 1
42.2.e.a 2 35.l odd 12 1
126.2.g.c 2 15.e even 4 1
126.2.g.c 2 105.x even 12 1
294.2.a.e 1 35.l odd 12 1
294.2.a.f 1 35.k even 12 1
294.2.e.b 2 35.f even 4 1
294.2.e.b 2 35.k even 12 1
336.2.q.b 2 20.e even 4 1
336.2.q.b 2 140.w even 12 1
882.2.a.c 1 105.x even 12 1
882.2.a.d 1 105.w odd 12 1
882.2.g.i 2 105.k odd 4 1
882.2.g.i 2 105.w odd 12 1
1008.2.s.k 2 60.l odd 4 1
1008.2.s.k 2 420.bp odd 12 1
1050.2.i.l 2 5.c odd 4 1
1050.2.i.l 2 35.l odd 12 1
1050.2.o.a 4 1.a even 1 1 trivial
1050.2.o.a 4 5.b even 2 1 inner
1050.2.o.a 4 7.c even 3 1 inner
1050.2.o.a 4 35.j even 6 1 inner
1134.2.e.e 2 45.l even 12 1
1134.2.e.e 2 315.bx even 12 1
1134.2.e.l 2 45.k odd 12 1
1134.2.e.l 2 315.ch odd 12 1
1134.2.h.e 2 45.k odd 12 1
1134.2.h.e 2 315.bt odd 12 1
1134.2.h.l 2 45.l even 12 1
1134.2.h.l 2 315.bv even 12 1
1344.2.q.g 2 40.i odd 4 1
1344.2.q.g 2 280.bt odd 12 1
1344.2.q.s 2 40.k even 4 1
1344.2.q.s 2 280.br even 12 1
2352.2.a.f 1 140.x odd 12 1
2352.2.a.t 1 140.w even 12 1
2352.2.q.u 2 140.j odd 4 1
2352.2.q.u 2 140.x odd 12 1
7056.2.a.w 1 420.bp odd 12 1
7056.2.a.bl 1 420.br even 12 1
7350.2.a.q 1 35.k even 12 1
7350.2.a.bl 1 35.l odd 12 1
9408.2.a.q 1 280.br even 12 1
9408.2.a.z 1 280.bv even 12 1
9408.2.a.ce 1 280.bt odd 12 1
9408.2.a.cr 1 280.bp odd 12 1

## Hecke kernels

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

 $$T_{11}^{2} + 5 T_{11} + 25$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 - 13 T^{2} + T^{4}$$
$11$ $$( 25 + 5 T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$256 - 16 T^{2} + T^{4}$$
$19$ $$( 64 - 8 T + T^{2} )^{2}$$
$23$ $$256 - 16 T^{2} + T^{4}$$
$29$ $$( -5 + T )^{4}$$
$31$ $$( 9 + 3 T + T^{2} )^{2}$$
$37$ $$256 - 16 T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 4 + T^{2} )^{2}$$
$47$ $$1296 - 36 T^{2} + T^{4}$$
$53$ $$6561 - 81 T^{2} + T^{4}$$
$59$ $$( 121 + 11 T + T^{2} )^{2}$$
$61$ $$( 36 - 6 T + T^{2} )^{2}$$
$67$ $$16 - 4 T^{2} + T^{4}$$
$71$ $$( -2 + T )^{4}$$
$73$ $$10000 - 100 T^{2} + T^{4}$$
$79$ $$( 9 - 3 T + T^{2} )^{2}$$
$83$ $$( 49 + T^{2} )^{2}$$
$89$ $$( 36 + 6 T + T^{2} )^{2}$$
$97$ $$( 49 + T^{2} )^{2}$$