# Properties

 Label 1050.2.b.b Level 1050 Weight 2 Character orbit 1050.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -\beta_{3} q^{3} - q^{4} -\beta_{1} q^{6} + ( 1 + \beta_{1} + \beta_{3} ) q^{7} + \beta_{2} q^{8} -3 \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} -\beta_{3} q^{3} - q^{4} -\beta_{1} q^{6} + ( 1 + \beta_{1} + \beta_{3} ) q^{7} + \beta_{2} q^{8} -3 \beta_{2} q^{9} + \beta_{3} q^{12} + ( -\beta_{1} - \beta_{3} ) q^{13} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{14} + q^{16} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{17} -3 q^{18} + ( -\beta_{1} - \beta_{3} ) q^{19} + ( 3 + 3 \beta_{2} - \beta_{3} ) q^{21} -6 \beta_{2} q^{23} + \beta_{1} q^{24} + ( -\beta_{1} + \beta_{3} ) q^{26} -3 \beta_{1} q^{27} + ( -1 - \beta_{1} - \beta_{3} ) q^{28} -6 \beta_{2} q^{29} -\beta_{2} q^{32} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{34} + 3 \beta_{2} q^{36} + 2 q^{37} + ( -\beta_{1} + \beta_{3} ) q^{38} + ( -3 - 3 \beta_{2} ) q^{39} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{42} -4 q^{43} -6 q^{46} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{47} -\beta_{3} q^{48} + ( -5 + 2 \beta_{1} + 2 \beta_{3} ) q^{49} + ( 6 - 6 \beta_{2} ) q^{51} + ( \beta_{1} + \beta_{3} ) q^{52} -6 \beta_{2} q^{53} + 3 \beta_{3} q^{54} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{56} + ( -3 - 3 \beta_{2} ) q^{57} -6 q^{58} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{59} + ( 5 \beta_{1} + 5 \beta_{3} ) q^{61} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{63} - q^{64} -8 q^{67} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{68} -6 \beta_{1} q^{69} + 3 q^{72} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{73} -2 \beta_{2} q^{74} + ( \beta_{1} + \beta_{3} ) q^{76} + ( -3 + 3 \beta_{2} ) q^{78} -10 q^{79} -9 q^{81} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{82} + ( \beta_{1} - \beta_{3} ) q^{83} + ( -3 - 3 \beta_{2} + \beta_{3} ) q^{84} + 4 \beta_{2} q^{86} -6 \beta_{1} q^{87} + ( 6 - \beta_{1} - \beta_{3} ) q^{91} + 6 \beta_{2} q^{92} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{94} -\beta_{1} q^{96} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{97} + ( 2 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{7} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{7} + 4q^{16} - 12q^{18} + 12q^{21} - 4q^{28} + 8q^{37} - 12q^{39} + 12q^{42} - 16q^{43} - 24q^{46} - 20q^{49} + 24q^{51} - 12q^{57} - 24q^{58} - 4q^{64} - 32q^{67} + 12q^{72} - 12q^{78} - 40q^{79} - 36q^{81} - 12q^{84} + 24q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

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

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
1.00000i −1.22474 + 1.22474i −1.00000 0 1.22474 + 1.22474i 1.00000 2.44949i 1.00000i 3.00000i 0
251.2 1.00000i 1.22474 1.22474i −1.00000 0 −1.22474 1.22474i 1.00000 + 2.44949i 1.00000i 3.00000i 0
251.3 1.00000i −1.22474 1.22474i −1.00000 0 1.22474 1.22474i 1.00000 + 2.44949i 1.00000i 3.00000i 0
251.4 1.00000i 1.22474 + 1.22474i −1.00000 0 −1.22474 + 1.22474i 1.00000 2.44949i 1.00000i 3.00000i 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.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.b.b 4
3.b odd 2 1 inner 1050.2.b.b 4
5.b even 2 1 42.2.d.a 4
5.c odd 4 1 1050.2.d.b 4
5.c odd 4 1 1050.2.d.e 4
7.b odd 2 1 inner 1050.2.b.b 4
15.d odd 2 1 42.2.d.a 4
15.e even 4 1 1050.2.d.b 4
15.e even 4 1 1050.2.d.e 4
20.d odd 2 1 336.2.k.b 4
21.c even 2 1 inner 1050.2.b.b 4
35.c odd 2 1 42.2.d.a 4
35.f even 4 1 1050.2.d.b 4
35.f even 4 1 1050.2.d.e 4
35.i odd 6 2 294.2.f.b 8
35.j even 6 2 294.2.f.b 8
40.e odd 2 1 1344.2.k.d 4
40.f even 2 1 1344.2.k.c 4
45.h odd 6 2 1134.2.m.g 8
45.j even 6 2 1134.2.m.g 8
60.h even 2 1 336.2.k.b 4
105.g even 2 1 42.2.d.a 4
105.k odd 4 1 1050.2.d.b 4
105.k odd 4 1 1050.2.d.e 4
105.o odd 6 2 294.2.f.b 8
105.p even 6 2 294.2.f.b 8
120.i odd 2 1 1344.2.k.c 4
120.m even 2 1 1344.2.k.d 4
140.c even 2 1 336.2.k.b 4
280.c odd 2 1 1344.2.k.c 4
280.n even 2 1 1344.2.k.d 4
315.z even 6 2 1134.2.m.g 8
315.bg odd 6 2 1134.2.m.g 8
420.o odd 2 1 336.2.k.b 4
840.b odd 2 1 1344.2.k.d 4
840.u even 2 1 1344.2.k.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.d.a 4 5.b even 2 1
42.2.d.a 4 15.d odd 2 1
42.2.d.a 4 35.c odd 2 1
42.2.d.a 4 105.g even 2 1
294.2.f.b 8 35.i odd 6 2
294.2.f.b 8 35.j even 6 2
294.2.f.b 8 105.o odd 6 2
294.2.f.b 8 105.p even 6 2
336.2.k.b 4 20.d odd 2 1
336.2.k.b 4 60.h even 2 1
336.2.k.b 4 140.c even 2 1
336.2.k.b 4 420.o odd 2 1
1050.2.b.b 4 1.a even 1 1 trivial
1050.2.b.b 4 3.b odd 2 1 inner
1050.2.b.b 4 7.b odd 2 1 inner
1050.2.b.b 4 21.c even 2 1 inner
1050.2.d.b 4 5.c odd 4 1
1050.2.d.b 4 15.e even 4 1
1050.2.d.b 4 35.f even 4 1
1050.2.d.b 4 105.k odd 4 1
1050.2.d.e 4 5.c odd 4 1
1050.2.d.e 4 15.e even 4 1
1050.2.d.e 4 35.f even 4 1
1050.2.d.e 4 105.k odd 4 1
1134.2.m.g 8 45.h odd 6 2
1134.2.m.g 8 45.j even 6 2
1134.2.m.g 8 315.z even 6 2
1134.2.m.g 8 315.bg odd 6 2
1344.2.k.c 4 40.f even 2 1
1344.2.k.c 4 120.i odd 2 1
1344.2.k.c 4 280.c odd 2 1
1344.2.k.c 4 840.u even 2 1
1344.2.k.d 4 40.e odd 2 1
1344.2.k.d 4 120.m even 2 1
1344.2.k.d 4 280.n even 2 1
1344.2.k.d 4 840.b odd 2 1

## Hecke kernels

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

 $$T_{11}$$ $$T_{17}^{2} - 24$$ $$T_{37} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$1 + 9 T^{4}$$
$5$ 1
$7$ $$( 1 - 2 T + 7 T^{2} )^{2}$$
$11$ $$( 1 - 11 T^{2} )^{4}$$
$13$ $$( 1 - 20 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 10 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 32 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 10 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 22 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 58 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 70 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 32 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 28 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 8 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 71 T^{2} )^{4}$$
$73$ $$( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 10 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 160 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{4}$$
$97$ $$( 1 - 170 T^{2} + 9409 T^{4} )^{2}$$