# Properties

 Label 1050.3.e.a Level $1050$ Weight $3$ Character orbit 1050.e 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} -2 q^{4} + ( -2 + \beta_{2} ) q^{6} -\beta_{3} q^{7} -2 \beta_{1} q^{8} + ( 5 + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} -2 q^{4} + ( -2 + \beta_{2} ) q^{6} -\beta_{3} q^{7} -2 \beta_{1} q^{8} + ( 5 + 2 \beta_{2} ) q^{9} + ( 5 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{12} + ( -10 - 4 \beta_{3} ) q^{13} -\beta_{2} q^{14} + 4 q^{16} + ( -2 \beta_{1} + 5 \beta_{2} ) q^{17} + ( 5 \beta_{1} - 4 \beta_{3} ) q^{18} -16 q^{19} + ( -7 - \beta_{2} ) q^{21} + ( -10 + 4 \beta_{3} ) q^{22} + ( \beta_{1} - 10 \beta_{2} ) q^{23} + ( 4 - 2 \beta_{2} ) q^{24} + ( -10 \beta_{1} - 4 \beta_{2} ) q^{26} + ( 19 \beta_{1} + \beta_{3} ) q^{27} + 2 \beta_{3} q^{28} + ( -20 \beta_{1} + 2 \beta_{2} ) q^{29} + ( -32 + 10 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( -10 - 14 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{33} + ( 4 - 10 \beta_{3} ) q^{34} + ( -10 - 4 \beta_{2} ) q^{36} -20 q^{37} -16 \beta_{1} q^{38} + ( -28 - 10 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} ) q^{39} + ( 30 \beta_{1} + 9 \beta_{2} ) q^{41} + ( -7 \beta_{1} + 2 \beta_{3} ) q^{42} + ( -20 - 12 \beta_{3} ) q^{43} + ( -10 \beta_{1} + 4 \beta_{2} ) q^{44} + ( -2 + 20 \beta_{3} ) q^{46} -6 \beta_{1} q^{47} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{48} + 7 q^{49} + ( 4 + 35 \beta_{1} - 2 \beta_{2} - 10 \beta_{3} ) q^{51} + ( 20 + 8 \beta_{3} ) q^{52} -36 \beta_{1} q^{53} + ( -38 + \beta_{2} ) q^{54} + 2 \beta_{2} q^{56} + ( -16 \beta_{1} - 16 \beta_{3} ) q^{57} + ( 40 - 4 \beta_{3} ) q^{58} + ( -20 \beta_{1} + 8 \beta_{2} ) q^{59} + ( -14 - 20 \beta_{3} ) q^{61} + ( -32 \beta_{1} + 10 \beta_{2} ) q^{62} + ( -14 \beta_{1} - 5 \beta_{3} ) q^{63} -8 q^{64} + ( 28 - 10 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} ) q^{66} + ( -60 + 4 \beta_{3} ) q^{67} + ( 4 \beta_{1} - 10 \beta_{2} ) q^{68} + ( -2 - 70 \beta_{1} + \beta_{2} + 20 \beta_{3} ) q^{69} + ( -25 \beta_{1} - 14 \beta_{2} ) q^{71} + ( -10 \beta_{1} + 8 \beta_{3} ) q^{72} + ( -30 + 16 \beta_{3} ) q^{73} -20 \beta_{1} q^{74} + 32 q^{76} + ( 14 \beta_{1} - 5 \beta_{2} ) q^{77} + ( 20 - 28 \beta_{1} - 10 \beta_{2} + 8 \beta_{3} ) q^{78} + ( -32 - 20 \beta_{3} ) q^{79} + ( -31 + 20 \beta_{2} ) q^{81} + ( -60 - 18 \beta_{3} ) q^{82} + ( -50 \beta_{1} + 20 \beta_{2} ) q^{83} + ( 14 + 2 \beta_{2} ) q^{84} + ( -20 \beta_{1} - 12 \beta_{2} ) q^{86} + ( 40 + 14 \beta_{1} - 20 \beta_{2} - 4 \beta_{3} ) q^{87} + ( 20 - 8 \beta_{3} ) q^{88} + ( -30 \beta_{1} + 3 \beta_{2} ) q^{89} + ( 28 + 10 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 20 \beta_{2} ) q^{92} + ( 70 - 32 \beta_{1} + 10 \beta_{2} - 32 \beta_{3} ) q^{93} + 12 q^{94} + ( -8 + 4 \beta_{2} ) q^{96} + ( 90 - 8 \beta_{3} ) q^{97} + 7 \beta_{1} q^{98} + ( 56 + 25 \beta_{1} - 10 \beta_{2} - 20 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} - 8 q^{6} + 20 q^{9} + O(q^{10})$$ $$4 q - 8 q^{4} - 8 q^{6} + 20 q^{9} - 40 q^{13} + 16 q^{16} - 64 q^{19} - 28 q^{21} - 40 q^{22} + 16 q^{24} - 128 q^{31} - 40 q^{33} + 16 q^{34} - 40 q^{36} - 80 q^{37} - 112 q^{39} - 80 q^{43} - 8 q^{46} + 28 q^{49} + 16 q^{51} + 80 q^{52} - 152 q^{54} + 160 q^{58} - 56 q^{61} - 32 q^{64} + 112 q^{66} - 240 q^{67} - 8 q^{69} - 120 q^{73} + 128 q^{76} + 80 q^{78} - 128 q^{79} - 124 q^{81} - 240 q^{82} + 56 q^{84} + 160 q^{87} + 80 q^{88} + 112 q^{91} + 280 q^{93} + 48 q^{94} - 32 q^{96} + 360 q^{97} + 224 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 11 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
701.1
 2.57794i − 1.16372i − 2.57794i 1.16372i
1.41421i −2.64575 1.41421i −2.00000 0 −2.00000 + 3.74166i 2.64575 2.82843i 5.00000 + 7.48331i 0
701.2 1.41421i 2.64575 1.41421i −2.00000 0 −2.00000 3.74166i −2.64575 2.82843i 5.00000 7.48331i 0
701.3 1.41421i −2.64575 + 1.41421i −2.00000 0 −2.00000 3.74166i 2.64575 2.82843i 5.00000 7.48331i 0
701.4 1.41421i 2.64575 + 1.41421i −2.00000 0 −2.00000 + 3.74166i −2.64575 2.82843i 5.00000 + 7.48331i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.e.a 4
3.b odd 2 1 inner 1050.3.e.a 4
5.b even 2 1 42.3.b.a 4
5.c odd 4 2 1050.3.c.a 8
15.d odd 2 1 42.3.b.a 4
15.e even 4 2 1050.3.c.a 8
20.d odd 2 1 336.3.d.b 4
35.c odd 2 1 294.3.b.h 4
35.i odd 6 2 294.3.h.g 8
35.j even 6 2 294.3.h.d 8
40.e odd 2 1 1344.3.d.e 4
40.f even 2 1 1344.3.d.c 4
45.h odd 6 2 1134.3.q.a 8
45.j even 6 2 1134.3.q.a 8
60.h even 2 1 336.3.d.b 4
105.g even 2 1 294.3.b.h 4
105.o odd 6 2 294.3.h.d 8
105.p even 6 2 294.3.h.g 8
120.i odd 2 1 1344.3.d.c 4
120.m even 2 1 1344.3.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.b.a 4 5.b even 2 1
42.3.b.a 4 15.d odd 2 1
294.3.b.h 4 35.c odd 2 1
294.3.b.h 4 105.g even 2 1
294.3.h.d 8 35.j even 6 2
294.3.h.d 8 105.o odd 6 2
294.3.h.g 8 35.i odd 6 2
294.3.h.g 8 105.p even 6 2
336.3.d.b 4 20.d odd 2 1
336.3.d.b 4 60.h even 2 1
1050.3.c.a 8 5.c odd 4 2
1050.3.c.a 8 15.e even 4 2
1050.3.e.a 4 1.a even 1 1 trivial
1050.3.e.a 4 3.b odd 2 1 inner
1134.3.q.a 8 45.h odd 6 2
1134.3.q.a 8 45.j even 6 2
1344.3.d.c 4 40.f even 2 1
1344.3.d.c 4 120.i odd 2 1
1344.3.d.e 4 40.e odd 2 1
1344.3.d.e 4 120.m even 2 1

## Hecke kernels

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

 $$T_{11}^{4} + 212 T_{11}^{2} + 36$$ $$T_{13}^{2} + 20 T_{13} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$81 - 10 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( -7 + T^{2} )^{2}$$
$11$ $$36 + 212 T^{2} + T^{4}$$
$13$ $$( -12 + 20 T + T^{2} )^{2}$$
$17$ $$116964 + 716 T^{2} + T^{4}$$
$19$ $$( 16 + T )^{4}$$
$23$ $$1954404 + 2804 T^{2} + T^{4}$$
$29$ $$553536 + 1712 T^{2} + T^{4}$$
$31$ $$( 324 + 64 T + T^{2} )^{2}$$
$37$ $$( 20 + T )^{4}$$
$41$ $$443556 + 5868 T^{2} + T^{4}$$
$43$ $$( -608 + 40 T + T^{2} )^{2}$$
$47$ $$( 72 + T^{2} )^{2}$$
$53$ $$( 2592 + T^{2} )^{2}$$
$59$ $$9216 + 3392 T^{2} + T^{4}$$
$61$ $$( -2604 + 28 T + T^{2} )^{2}$$
$67$ $$( 3488 + 120 T + T^{2} )^{2}$$
$71$ $$2232036 + 7988 T^{2} + T^{4}$$
$73$ $$( -892 + 60 T + T^{2} )^{2}$$
$79$ $$( -1776 + 64 T + T^{2} )^{2}$$
$83$ $$360000 + 21200 T^{2} + T^{4}$$
$89$ $$2802276 + 3852 T^{2} + T^{4}$$
$97$ $$( 7652 - 180 T + T^{2} )^{2}$$