# Properties

 Label 1050.3.f.a Level $1050$ Weight $3$ Character orbit 1050.f Analytic conductor $28.610$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{3} + 2 q^{4} + \beta_{3} q^{6} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{7} -2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{2} q^{3} + 2 q^{4} + \beta_{3} q^{6} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{7} -2 \beta_{1} q^{8} -3 q^{9} + ( -6 - 3 \beta_{1} ) q^{11} + 2 \beta_{2} q^{12} + ( -8 \beta_{2} - 2 \beta_{3} ) q^{13} + ( -6 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{14} + 4 q^{16} + ( 2 \beta_{2} - 11 \beta_{3} ) q^{17} + 3 \beta_{1} q^{18} + ( -2 \beta_{2} - 8 \beta_{3} ) q^{19} + ( 3 + 6 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{21} + ( 6 + 6 \beta_{1} ) q^{22} + ( -6 + 9 \beta_{1} ) q^{23} + 2 \beta_{3} q^{24} + ( -4 \beta_{2} - 8 \beta_{3} ) q^{26} -3 \beta_{2} q^{27} + ( -4 + 6 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{28} + 30 q^{29} + ( -12 \beta_{2} + 12 \beta_{3} ) q^{31} -4 \beta_{1} q^{32} + ( -6 \beta_{2} + 3 \beta_{3} ) q^{33} + ( -22 \beta_{2} + 2 \beta_{3} ) q^{34} -6 q^{36} + ( 20 - 36 \beta_{1} ) q^{37} + ( -16 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 24 - 6 \beta_{1} ) q^{39} + ( 14 \beta_{2} + 7 \beta_{3} ) q^{41} + ( -12 - 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{42} + ( 32 + 30 \beta_{1} ) q^{43} + ( -12 - 6 \beta_{1} ) q^{44} + ( -18 + 6 \beta_{1} ) q^{46} + ( -28 \beta_{2} + 4 \beta_{3} ) q^{47} + 4 \beta_{2} q^{48} + ( -5 - 24 \beta_{1} - 20 \beta_{2} - 2 \beta_{3} ) q^{49} + ( -6 - 33 \beta_{1} ) q^{51} + ( -16 \beta_{2} - 4 \beta_{3} ) q^{52} + ( 54 + 12 \beta_{1} ) q^{53} -3 \beta_{3} q^{54} + ( -12 + 4 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{56} + ( 6 - 24 \beta_{1} ) q^{57} -30 \beta_{1} q^{58} + ( -28 \beta_{2} - 20 \beta_{3} ) q^{59} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{61} + ( 24 \beta_{2} - 12 \beta_{3} ) q^{62} + ( 6 - 9 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{63} + 8 q^{64} + ( 6 \beta_{2} - 6 \beta_{3} ) q^{66} + ( -44 + 12 \beta_{1} ) q^{67} + ( 4 \beta_{2} - 22 \beta_{3} ) q^{68} + ( -6 \beta_{2} - 9 \beta_{3} ) q^{69} + ( -30 + 57 \beta_{1} ) q^{71} + 6 \beta_{1} q^{72} + ( -4 \beta_{2} + 26 \beta_{3} ) q^{73} + ( 72 - 20 \beta_{1} ) q^{74} + ( -4 \beta_{2} - 16 \beta_{3} ) q^{76} + ( -6 - 12 \beta_{1} + 18 \beta_{2} - 15 \beta_{3} ) q^{77} + ( 12 - 24 \beta_{1} ) q^{78} + ( 32 + 72 \beta_{1} ) q^{79} + 9 q^{81} + ( 14 \beta_{2} + 14 \beta_{3} ) q^{82} + ( -32 \beta_{2} + 20 \beta_{3} ) q^{83} + ( 6 + 12 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{84} + ( -60 - 32 \beta_{1} ) q^{86} + 30 \beta_{2} q^{87} + ( 12 + 12 \beta_{1} ) q^{88} + ( -54 \beta_{2} + 21 \beta_{3} ) q^{89} + ( -42 \beta_{1} + 28 \beta_{2} + 28 \beta_{3} ) q^{91} + ( -12 + 18 \beta_{1} ) q^{92} + ( 36 + 36 \beta_{1} ) q^{93} + ( 8 \beta_{2} - 28 \beta_{3} ) q^{94} + 4 \beta_{3} q^{96} + ( -44 \beta_{2} + 10 \beta_{3} ) q^{97} + ( 48 + 5 \beta_{1} - 4 \beta_{2} - 20 \beta_{3} ) q^{98} + ( 18 + 9 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} - 8 q^{7} - 12 q^{9} + O(q^{10})$$ $$4 q + 8 q^{4} - 8 q^{7} - 12 q^{9} - 24 q^{11} - 24 q^{14} + 16 q^{16} + 12 q^{21} + 24 q^{22} - 24 q^{23} - 16 q^{28} + 120 q^{29} - 24 q^{36} + 80 q^{37} + 96 q^{39} - 48 q^{42} + 128 q^{43} - 48 q^{44} - 72 q^{46} - 20 q^{49} - 24 q^{51} + 216 q^{53} - 48 q^{56} + 24 q^{57} + 24 q^{63} + 32 q^{64} - 176 q^{67} - 120 q^{71} + 288 q^{74} - 24 q^{77} + 48 q^{78} + 128 q^{79} + 36 q^{81} + 24 q^{84} - 240 q^{86} + 48 q^{88} - 48 q^{92} + 144 q^{93} + 192 q^{98} + 72 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
601.1
 −0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i 0.707107 + 1.22474i
−1.41421 1.73205i 2.00000 0 2.44949i 2.24264 + 6.63103i −2.82843 −3.00000 0
601.2 −1.41421 1.73205i 2.00000 0 2.44949i 2.24264 6.63103i −2.82843 −3.00000 0
601.3 1.41421 1.73205i 2.00000 0 2.44949i −6.24264 3.16693i 2.82843 −3.00000 0
601.4 1.41421 1.73205i 2.00000 0 2.44949i −6.24264 + 3.16693i 2.82843 −3.00000 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.f.a 4
5.b even 2 1 42.3.c.a 4
5.c odd 4 2 1050.3.h.a 8
7.b odd 2 1 inner 1050.3.f.a 4
15.d odd 2 1 126.3.c.b 4
20.d odd 2 1 336.3.f.c 4
35.c odd 2 1 42.3.c.a 4
35.f even 4 2 1050.3.h.a 8
35.i odd 6 1 294.3.g.b 4
35.i odd 6 1 294.3.g.c 4
35.j even 6 1 294.3.g.b 4
35.j even 6 1 294.3.g.c 4
40.e odd 2 1 1344.3.f.e 4
40.f even 2 1 1344.3.f.f 4
60.h even 2 1 1008.3.f.g 4
105.g even 2 1 126.3.c.b 4
105.o odd 6 1 882.3.n.a 4
105.o odd 6 1 882.3.n.d 4
105.p even 6 1 882.3.n.a 4
105.p even 6 1 882.3.n.d 4
140.c even 2 1 336.3.f.c 4
280.c odd 2 1 1344.3.f.f 4
280.n even 2 1 1344.3.f.e 4
420.o odd 2 1 1008.3.f.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 5.b even 2 1
42.3.c.a 4 35.c odd 2 1
126.3.c.b 4 15.d odd 2 1
126.3.c.b 4 105.g even 2 1
294.3.g.b 4 35.i odd 6 1
294.3.g.b 4 35.j even 6 1
294.3.g.c 4 35.i odd 6 1
294.3.g.c 4 35.j even 6 1
336.3.f.c 4 20.d odd 2 1
336.3.f.c 4 140.c even 2 1
882.3.n.a 4 105.o odd 6 1
882.3.n.a 4 105.p even 6 1
882.3.n.d 4 105.o odd 6 1
882.3.n.d 4 105.p even 6 1
1008.3.f.g 4 60.h even 2 1
1008.3.f.g 4 420.o odd 2 1
1050.3.f.a 4 1.a even 1 1 trivial
1050.3.f.a 4 7.b odd 2 1 inner
1050.3.h.a 8 5.c odd 4 2
1050.3.h.a 8 35.f even 4 2
1344.3.f.e 4 40.e odd 2 1
1344.3.f.e 4 280.n even 2 1
1344.3.f.f 4 40.f even 2 1
1344.3.f.f 4 280.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{2} + 12 T_{11} + 18$$ $$T_{23}^{2} + 12 T_{23} - 126$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$2401 + 392 T + 42 T^{2} + 8 T^{3} + T^{4}$$
$11$ $$( 18 + 12 T + T^{2} )^{2}$$
$13$ $$28224 + 432 T^{2} + T^{4}$$
$17$ $$509796 + 1476 T^{2} + T^{4}$$
$19$ $$138384 + 792 T^{2} + T^{4}$$
$23$ $$( -126 + 12 T + T^{2} )^{2}$$
$29$ $$( -30 + T )^{4}$$
$31$ $$186624 + 2592 T^{2} + T^{4}$$
$37$ $$( -2192 - 40 T + T^{2} )^{2}$$
$41$ $$86436 + 1764 T^{2} + T^{4}$$
$43$ $$( -776 - 64 T + T^{2} )^{2}$$
$47$ $$5089536 + 4896 T^{2} + T^{4}$$
$53$ $$( 2628 - 108 T + T^{2} )^{2}$$
$59$ $$2304 + 9504 T^{2} + T^{4}$$
$61$ $$2304 + 288 T^{2} + T^{4}$$
$67$ $$( 1648 + 88 T + T^{2} )^{2}$$
$71$ $$( -5598 + 60 T + T^{2} )^{2}$$
$73$ $$16064064 + 8208 T^{2} + T^{4}$$
$79$ $$( -9344 - 64 T + T^{2} )^{2}$$
$83$ $$451584 + 10944 T^{2} + T^{4}$$
$89$ $$37234404 + 22788 T^{2} + T^{4}$$
$97$ $$27123264 + 12816 T^{2} + T^{4}$$