# Properties

 Label 1050.2.o.b 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.

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

## Character values

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

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

## 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 0.866025 2.50000i 1.00000i 0.500000 0.866025i 0
499.2 0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0 −1.00000 −0.866025 + 2.50000i 1.00000i 0.500000 0.866025i 0
949.1 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i 0 −1.00000 0.866025 + 2.50000i 1.00000i 0.500000 + 0.866025i 0
949.2 0.866025 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0 −1.00000 −0.866025 2.50000i 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.b 4
5.b even 2 1 inner 1050.2.o.b 4
5.c odd 4 1 42.2.e.b 2
5.c odd 4 1 1050.2.i.e 2
7.c even 3 1 inner 1050.2.o.b 4
15.e even 4 1 126.2.g.b 2
20.e even 4 1 336.2.q.d 2
35.f even 4 1 294.2.e.f 2
35.j even 6 1 inner 1050.2.o.b 4
35.k even 12 1 294.2.a.a 1
35.k even 12 1 294.2.e.f 2
35.k even 12 1 7350.2.a.cw 1
35.l odd 12 1 42.2.e.b 2
35.l odd 12 1 294.2.a.d 1
35.l odd 12 1 1050.2.i.e 2
35.l odd 12 1 7350.2.a.ce 1
40.i odd 4 1 1344.2.q.v 2
40.k even 4 1 1344.2.q.j 2
45.k odd 12 1 1134.2.e.a 2
45.k odd 12 1 1134.2.h.p 2
45.l even 12 1 1134.2.e.p 2
45.l even 12 1 1134.2.h.a 2
60.l odd 4 1 1008.2.s.n 2
105.k odd 4 1 882.2.g.b 2
105.w odd 12 1 882.2.a.k 1
105.w odd 12 1 882.2.g.b 2
105.x even 12 1 126.2.g.b 2
105.x even 12 1 882.2.a.g 1
140.j odd 4 1 2352.2.q.m 2
140.w even 12 1 336.2.q.d 2
140.w even 12 1 2352.2.a.m 1
140.x odd 12 1 2352.2.a.n 1
140.x odd 12 1 2352.2.q.m 2
280.bp odd 12 1 9408.2.a.bm 1
280.br even 12 1 1344.2.q.j 2
280.br even 12 1 9408.2.a.bu 1
280.bt odd 12 1 1344.2.q.v 2
280.bt odd 12 1 9408.2.a.d 1
280.bv even 12 1 9408.2.a.db 1
315.bt odd 12 1 1134.2.h.p 2
315.bv even 12 1 1134.2.h.a 2
315.bx even 12 1 1134.2.e.p 2
315.ch odd 12 1 1134.2.e.a 2
420.bp odd 12 1 1008.2.s.n 2
420.bp odd 12 1 7056.2.a.g 1
420.br even 12 1 7056.2.a.bz 1

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

## Hecke kernels

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

 $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{13}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 + 11 T^{2} + T^{4}$$
$11$ $$( 9 + 3 T + T^{2} )^{2}$$
$13$ $$( 16 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 16 + 4 T + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( 9 + T )^{4}$$
$31$ $$( 1 - T + T^{2} )^{2}$$
$37$ $$4096 - 64 T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$( 100 + T^{2} )^{2}$$
$47$ $$1296 - 36 T^{2} + T^{4}$$
$53$ $$81 - 9 T^{2} + T^{4}$$
$59$ $$( 9 - 3 T + T^{2} )^{2}$$
$61$ $$( 100 - 10 T + T^{2} )^{2}$$
$67$ $$10000 - 100 T^{2} + T^{4}$$
$71$ $$( 6 + T )^{4}$$
$73$ $$16 - 4 T^{2} + T^{4}$$
$79$ $$( 1 + T + T^{2} )^{2}$$
$83$ $$( 81 + T^{2} )^{2}$$
$89$ $$( 36 - 6 T + T^{2} )^{2}$$
$97$ $$( 1 + T^{2} )^{2}$$