# Properties

 Label 1050.2.i.e Level 1050 Weight 2 Character orbit 1050.i Analytic conductor 8.384 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1050.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 −2.50000 + 0.866025i 1.00000 −0.500000 0.866025i 0
751.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 −2.50000 0.866025i 1.00000 −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
7.c even 3 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{11}^{2} + 3 T_{11} + 9$$ $$T_{13} - 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ 1
$7$ $$1 + 5 T + 7 T^{2}$$
$11$ $$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 4 T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 9 T + 29 T^{2} )^{2}$$
$31$ $$1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4}$$
$37$ $$1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 - 10 T + 43 T^{2} )^{2}$$
$47$ $$1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4}$$
$53$ $$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$
$59$ $$1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4}$$
$61$ $$1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4}$$
$67$ $$1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{2}$$
$73$ $$1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4}$$
$79$ $$1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 9 T + 83 T^{2} )^{2}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - T + 97 T^{2} )^{2}$$