# Properties

 Label 1050.2.o.e 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.

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

## 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 −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.e 4
5.b even 2 1 inner 1050.2.o.e 4
5.c odd 4 1 1050.2.i.h 2
5.c odd 4 1 1050.2.i.m yes 2
7.c even 3 1 inner 1050.2.o.e 4
35.j even 6 1 inner 1050.2.o.e 4
35.k even 12 1 7350.2.a.n 1
35.k even 12 1 7350.2.a.cq 1
35.l odd 12 1 1050.2.i.h 2
35.l odd 12 1 1050.2.i.m yes 2
35.l odd 12 1 7350.2.a.bb 1
35.l odd 12 1 7350.2.a.by 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.i.h 2 5.c odd 4 1
1050.2.i.h 2 35.l odd 12 1
1050.2.i.m yes 2 5.c odd 4 1
1050.2.i.m yes 2 35.l odd 12 1
1050.2.o.e 4 1.a even 1 1 trivial
1050.2.o.e 4 5.b even 2 1 inner
1050.2.o.e 4 7.c even 3 1 inner
1050.2.o.e 4 35.j even 6 1 inner
7350.2.a.n 1 35.k even 12 1
7350.2.a.bb 1 35.l odd 12 1
7350.2.a.by 1 35.l odd 12 1
7350.2.a.cq 1 35.k even 12 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ 1
$7$ $$1 - 13 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 11 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )^{2}( 1 + 6 T + 13 T^{2} )^{2}$$
$17$ $$1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 73 T^{2} + 1369 T^{4} )( 1 + 47 T^{2} + 1369 T^{4} )$$
$41$ $$( 1 + 9 T + 41 T^{2} )^{4}$$
$43$ $$( 1 + 14 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 118 T^{2} + 9435 T^{4} + 529702 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 3 T + 71 T^{2} )^{4}$$
$73$ $$1 - 50 T^{2} - 2829 T^{4} - 266450 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 83 T^{2} )^{4}$$
$89$ $$( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 145 T^{2} + 9409 T^{4} )^{2}$$