# Properties

 Label 1050.2.s.b Level 1050 Weight 2 Character orbit 1050.s 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.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ 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.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 + 0.500000i 0.866025 1.50000i 0.500000 0.866025i 0 1.73205i 2.50000 0.866025i 1.00000i −1.50000 2.59808i 0
101.2 0.866025 0.500000i −0.866025 + 1.50000i 0.500000 0.866025i 0 1.73205i 2.50000 0.866025i 1.00000i −1.50000 2.59808i 0
551.1 −0.866025 0.500000i 0.866025 + 1.50000i 0.500000 + 0.866025i 0 1.73205i 2.50000 + 0.866025i 1.00000i −1.50000 + 2.59808i 0
551.2 0.866025 + 0.500000i −0.866025 1.50000i 0.500000 + 0.866025i 0 1.73205i 2.50000 + 0.866025i 1.00000i −1.50000 + 2.59808i 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g 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.s.b 4
3.b odd 2 1 inner 1050.2.s.b 4
5.b even 2 1 42.2.f.a 4
5.c odd 4 1 1050.2.u.a 4
5.c odd 4 1 1050.2.u.d 4
7.d odd 6 1 inner 1050.2.s.b 4
15.d odd 2 1 42.2.f.a 4
15.e even 4 1 1050.2.u.a 4
15.e even 4 1 1050.2.u.d 4
20.d odd 2 1 336.2.bc.e 4
21.g even 6 1 inner 1050.2.s.b 4
35.c odd 2 1 294.2.f.a 4
35.i odd 6 1 42.2.f.a 4
35.i odd 6 1 294.2.d.a 4
35.j even 6 1 294.2.d.a 4
35.j even 6 1 294.2.f.a 4
35.k even 12 1 1050.2.u.a 4
35.k even 12 1 1050.2.u.d 4
45.h odd 6 1 1134.2.l.c 4
45.h odd 6 1 1134.2.t.d 4
45.j even 6 1 1134.2.l.c 4
45.j even 6 1 1134.2.t.d 4
60.h even 2 1 336.2.bc.e 4
105.g even 2 1 294.2.f.a 4
105.o odd 6 1 294.2.d.a 4
105.o odd 6 1 294.2.f.a 4
105.p even 6 1 42.2.f.a 4
105.p even 6 1 294.2.d.a 4
105.w odd 12 1 1050.2.u.a 4
105.w odd 12 1 1050.2.u.d 4
140.p odd 6 1 2352.2.k.e 4
140.s even 6 1 336.2.bc.e 4
140.s even 6 1 2352.2.k.e 4
315.q odd 6 1 1134.2.t.d 4
315.u even 6 1 1134.2.l.c 4
315.bn odd 6 1 1134.2.l.c 4
315.bq even 6 1 1134.2.t.d 4
420.ba even 6 1 2352.2.k.e 4
420.be odd 6 1 336.2.bc.e 4
420.be odd 6 1 2352.2.k.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.f.a 4 5.b even 2 1
42.2.f.a 4 15.d odd 2 1
42.2.f.a 4 35.i odd 6 1
42.2.f.a 4 105.p even 6 1
294.2.d.a 4 35.i odd 6 1
294.2.d.a 4 35.j even 6 1
294.2.d.a 4 105.o odd 6 1
294.2.d.a 4 105.p even 6 1
294.2.f.a 4 35.c odd 2 1
294.2.f.a 4 35.j even 6 1
294.2.f.a 4 105.g even 2 1
294.2.f.a 4 105.o odd 6 1
336.2.bc.e 4 20.d odd 2 1
336.2.bc.e 4 60.h even 2 1
336.2.bc.e 4 140.s even 6 1
336.2.bc.e 4 420.be odd 6 1
1050.2.s.b 4 1.a even 1 1 trivial
1050.2.s.b 4 3.b odd 2 1 inner
1050.2.s.b 4 7.d odd 6 1 inner
1050.2.s.b 4 21.g even 6 1 inner
1050.2.u.a 4 5.c odd 4 1
1050.2.u.a 4 15.e even 4 1
1050.2.u.a 4 35.k even 12 1
1050.2.u.a 4 105.w odd 12 1
1050.2.u.d 4 5.c odd 4 1
1050.2.u.d 4 15.e even 4 1
1050.2.u.d 4 35.k even 12 1
1050.2.u.d 4 105.w odd 12 1
1134.2.l.c 4 45.h odd 6 1
1134.2.l.c 4 45.j even 6 1
1134.2.l.c 4 315.u even 6 1
1134.2.l.c 4 315.bn odd 6 1
1134.2.t.d 4 45.h odd 6 1
1134.2.t.d 4 45.j even 6 1
1134.2.t.d 4 315.q odd 6 1
1134.2.t.d 4 315.bq even 6 1
2352.2.k.e 4 140.p odd 6 1
2352.2.k.e 4 140.s even 6 1
2352.2.k.e 4 420.ba even 6 1
2352.2.k.e 4 420.be odd 6 1

## Hecke kernels

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

 $$T_{11}^{4} - 9 T_{11}^{2} + 81$$ $$T_{37}^{2} + 2 T_{37} + 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4}$$
$3$ $$1 + 3 T^{2} + 9 T^{4}$$
$5$ 1
$7$ $$( 1 - 5 T + 7 T^{2} )^{2}$$
$11$ $$1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2}$$
$23$ $$1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 49 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 34 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{4}$$
$47$ $$1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8}$$
$61$ $$( 1 + 61 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 2 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 12 T + 121 T^{2} + 876 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 91 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2}$$