Properties

 Label 1050.2.o.h 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: yes 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.

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

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 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.h 4
5.b even 2 1 inner 1050.2.o.h 4
5.c odd 4 1 1050.2.i.c 2
5.c odd 4 1 1050.2.i.q yes 2
7.c even 3 1 inner 1050.2.o.h 4
35.j even 6 1 inner 1050.2.o.h 4
35.k even 12 1 7350.2.a.bm 1
35.k even 12 1 7350.2.a.cg 1
35.l odd 12 1 1050.2.i.c 2
35.l odd 12 1 1050.2.i.q yes 2
35.l odd 12 1 7350.2.a.s 1
35.l odd 12 1 7350.2.a.cy 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.i.c 2 5.c odd 4 1
1050.2.i.c 2 35.l odd 12 1
1050.2.i.q yes 2 5.c odd 4 1
1050.2.i.q yes 2 35.l odd 12 1
1050.2.o.h 4 1.a even 1 1 trivial
1050.2.o.h 4 5.b even 2 1 inner
1050.2.o.h 4 7.c even 3 1 inner
1050.2.o.h 4 35.j even 6 1 inner
7350.2.a.s 1 35.l odd 12 1
7350.2.a.bm 1 35.k even 12 1
7350.2.a.cg 1 35.k even 12 1
7350.2.a.cy 1 35.l odd 12 1

Hecke kernels

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

 $$T_{11}^{2} + 6 T_{11} + 36$$ $$T_{13}^{2} + 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$49 + 11 T^{2} + T^{4}$$
$11$ $$( 36 + 6 T + T^{2} )^{2}$$
$13$ $$( 16 + T^{2} )^{2}$$
$17$ $$81 - 9 T^{2} + T^{4}$$
$19$ $$( 16 + 4 T + T^{2} )^{2}$$
$23$ $$81 - 9 T^{2} + T^{4}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 25 + 5 T + T^{2} )^{2}$$
$37$ $$4096 - 64 T^{2} + T^{4}$$
$41$ $$( 3 + T )^{4}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$6561 - 81 T^{2} + T^{4}$$
$53$ $$20736 - 144 T^{2} + T^{4}$$
$59$ $$( 36 - 6 T + T^{2} )^{2}$$
$61$ $$( 4 + 2 T + T^{2} )^{2}$$
$67$ $$4096 - 64 T^{2} + T^{4}$$
$71$ $$( 9 + T )^{4}$$
$73$ $$38416 - 196 T^{2} + T^{4}$$
$79$ $$( 49 + 7 T + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 9 - 3 T + T^{2} )^{2}$$
$97$ $$( 289 + T^{2} )^{2}$$