# Properties

 Label 1050.2.m.d Level 1050 Weight 2 Character orbit 1050.m Analytic conductor 8.384 Analytic rank 0 Dimension 8 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.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$\zeta_{24}^{3}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i
−0.707107 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000i −2.63896 0.189469i 0.707107 0.707107i 1.00000i 0
307.2 −0.707107 0.707107i −0.707107 0.707107i 1.00000i 0 1.00000i −0.189469 2.63896i 0.707107 0.707107i 1.00000i 0
307.3 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000i 0.189469 + 2.63896i −0.707107 + 0.707107i 1.00000i 0
307.4 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0 1.00000i 2.63896 + 0.189469i −0.707107 + 0.707107i 1.00000i 0
643.1 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000i −2.63896 + 0.189469i 0.707107 + 0.707107i 1.00000i 0
643.2 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0 1.00000i −0.189469 + 2.63896i 0.707107 + 0.707107i 1.00000i 0
643.3 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000i 0.189469 2.63896i −0.707107 0.707107i 1.00000i 0
643.4 0.707107 0.707107i 0.707107 0.707107i 1.00000i 0 1.00000i 2.63896 0.189469i −0.707107 0.707107i 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 643.4 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
35.f even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.m.d yes 8
5.b even 2 1 inner 1050.2.m.d yes 8
5.c odd 4 2 1050.2.m.c 8
7.b odd 2 1 1050.2.m.c 8
35.c odd 2 1 1050.2.m.c 8
35.f even 4 2 inner 1050.2.m.d yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1050.2.m.c 8 5.c odd 4 2
1050.2.m.c 8 7.b odd 2 1
1050.2.m.c 8 35.c odd 2 1
1050.2.m.d yes 8 1.a even 1 1 trivial
1050.2.m.d yes 8 5.b even 2 1 inner
1050.2.m.d yes 8 35.f even 4 2 inner

## Hecke kernels

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

 $$T_{11}^{2} + 4 T_{11} - 8$$ $$T_{13}^{8} + 194 T_{13}^{4} + 1$$ $$T_{19}^{2} - 8 T_{19} + 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$( 1 + T^{4} )^{2}$$
$5$ 1
$7$ $$1 - 94 T^{4} + 2401 T^{8}$$
$11$ $$( 1 + 4 T + 14 T^{2} + 44 T^{3} + 121 T^{4} )^{4}$$
$13$ $$1 + 142 T^{4} - 7149 T^{8} + 4055662 T^{12} + 815730721 T^{16}$$
$17$ $$( 1 - 497 T^{4} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 8 T + 42 T^{2} - 152 T^{3} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 967 T^{4} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 38 T^{2} + 1611 T^{4} - 31958 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 26 T^{2} + 1899 T^{4} - 24986 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 2062 T^{4} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 126 T^{2} + 7139 T^{4} - 211806 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$1 + 5182 T^{4} + 12352611 T^{8} + 17716226782 T^{12} + 11688200277601 T^{16}$$
$47$ $$1 - 4732 T^{4} + 13117830 T^{8} - 23090650492 T^{12} + 23811286661761 T^{16}$$
$53$ $$( 1 - 56 T^{2} + 327 T^{4} - 157304 T^{6} + 7890481 T^{8} )( 1 + 56 T^{2} + 327 T^{4} + 157304 T^{6} + 7890481 T^{8} )$$
$59$ $$( 1 - 4 T + 95 T^{2} - 236 T^{3} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 218 T^{2} + 19275 T^{4} - 811178 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$1 - 3932 T^{4} + 41402598 T^{8} - 79234207772 T^{12} + 406067677556641 T^{16}$$
$71$ $$( 1 + 16 T + 194 T^{2} + 1136 T^{3} + 5041 T^{4} )^{4}$$
$73$ $$1 - 15548 T^{4} + 109241286 T^{8} - 441535851068 T^{12} + 806460091894081 T^{16}$$
$79$ $$( 1 - 92 T^{2} + 12870 T^{4} - 574172 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$1 + 15358 T^{4} + 117552483 T^{8} + 728864893918 T^{12} + 2252292232139041 T^{16}$$
$89$ $$( 1 - 12 T + 89 T^{2} )^{8}$$
$97$ $$1 + 4996 T^{4} + 20789766 T^{8} + 442292287876 T^{12} + 7837433594376961 T^{16}$$