# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1050.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6104277578$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ 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: $$1$$ 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 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -2 - \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 2 \beta_{1} + \beta_{3} ) q^{6} + ( 5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{7} -2 \beta_{3} q^{8} + ( 3 + 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -2 - \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 2 \beta_{1} + \beta_{3} ) q^{6} + ( 5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{7} -2 \beta_{3} q^{8} + ( 3 + 3 \beta_{2} ) q^{9} + 6 \beta_{2} q^{11} + ( 2 - 2 \beta_{2} ) q^{12} + ( 1 + 16 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{13} + ( 8 - 5 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{14} + ( -4 - 4 \beta_{2} ) q^{16} + ( 16 - 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{18} + ( -7 - 2 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} ) q^{19} + ( -5 - 10 \beta_{2} - 6 \beta_{3} ) q^{21} -6 \beta_{3} q^{22} + ( -12 - 18 \beta_{1} - 12 \beta_{2} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{24} + ( 16 - \beta_{1} - 16 \beta_{2} - 2 \beta_{3} ) q^{26} + ( -3 - 6 \beta_{2} ) q^{27} + ( -10 - 8 \beta_{1} - 4 \beta_{3} ) q^{28} + 24 \beta_{3} q^{29} + ( 34 + 6 \beta_{1} + 17 \beta_{2} - 6 \beta_{3} ) q^{31} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( 6 - 6 \beta_{2} ) q^{33} + ( 4 - 16 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{34} -6 q^{36} + ( -11 - 12 \beta_{1} - 11 \beta_{2} ) q^{37} + ( -8 + 7 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} ) q^{38} + ( -24 \beta_{1} - 3 \beta_{2} - 24 \beta_{3} ) q^{39} + ( -26 - 8 \beta_{1} - 52 \beta_{2} - 4 \beta_{3} ) q^{41} + ( -12 + 5 \beta_{1} - 12 \beta_{2} + 10 \beta_{3} ) q^{42} + ( -7 - 6 \beta_{3} ) q^{43} + ( -12 - 12 \beta_{2} ) q^{44} + ( 12 \beta_{1} + 36 \beta_{2} + 12 \beta_{3} ) q^{46} + ( 22 - 2 \beta_{1} - 22 \beta_{2} - 4 \beta_{3} ) q^{47} + ( 4 + 8 \beta_{2} ) q^{48} + ( -20 \beta_{1} + \beta_{2} + 20 \beta_{3} ) q^{49} + ( -24 + 6 \beta_{1} - 24 \beta_{2} ) q^{51} + ( -4 - 16 \beta_{1} - 2 \beta_{2} + 16 \beta_{3} ) q^{52} + ( 18 \beta_{1} + 60 \beta_{2} + 18 \beta_{3} ) q^{53} + ( 3 \beta_{1} + 6 \beta_{3} ) q^{54} + ( -8 + 10 \beta_{1} + 8 \beta_{2} ) q^{56} + ( 21 + 6 \beta_{3} ) q^{57} + ( 48 + 48 \beta_{2} ) q^{58} + ( -8 - 14 \beta_{1} - 4 \beta_{2} + 14 \beta_{3} ) q^{59} + ( -12 - 8 \beta_{1} + 12 \beta_{2} - 16 \beta_{3} ) q^{61} + ( -12 - 34 \beta_{1} - 24 \beta_{2} - 17 \beta_{3} ) q^{62} + ( -6 \beta_{1} + 15 \beta_{2} + 6 \beta_{3} ) q^{63} + 8 q^{64} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{66} + ( -42 \beta_{1} + 55 \beta_{2} - 42 \beta_{3} ) q^{67} + ( -16 - 4 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{68} + ( 12 + 36 \beta_{1} + 24 \beta_{2} + 18 \beta_{3} ) q^{69} + ( 78 - 42 \beta_{3} ) q^{71} + 6 \beta_{1} q^{72} + ( 22 - 40 \beta_{1} + 11 \beta_{2} + 40 \beta_{3} ) q^{73} + ( 11 \beta_{1} + 24 \beta_{2} + 11 \beta_{3} ) q^{74} + ( -14 + 8 \beta_{1} - 28 \beta_{2} + 4 \beta_{3} ) q^{76} + ( -30 - 24 \beta_{1} - 12 \beta_{3} ) q^{77} + ( -48 + 3 \beta_{3} ) q^{78} + ( -5 - 66 \beta_{1} - 5 \beta_{2} ) q^{79} + 9 \beta_{2} q^{81} + ( -8 + 26 \beta_{1} + 8 \beta_{2} + 52 \beta_{3} ) q^{82} + ( -10 - 76 \beta_{1} - 20 \beta_{2} - 38 \beta_{3} ) q^{83} + ( 20 + 12 \beta_{1} + 10 \beta_{2} + 12 \beta_{3} ) q^{84} + ( -12 + 7 \beta_{1} - 12 \beta_{2} ) q^{86} + ( 24 \beta_{1} - 24 \beta_{3} ) q^{87} + ( 12 \beta_{1} + 12 \beta_{3} ) q^{88} + ( -12 + 12 \beta_{2} ) q^{89} + ( -101 + 34 \beta_{1} - 91 \beta_{2} + 80 \beta_{3} ) q^{91} + ( 24 - 36 \beta_{3} ) q^{92} + ( -51 - 18 \beta_{1} - 51 \beta_{2} ) q^{93} + ( -8 - 22 \beta_{1} - 4 \beta_{2} + 22 \beta_{3} ) q^{94} + ( -4 \beta_{1} - 8 \beta_{3} ) q^{96} + ( 12 - 8 \beta_{1} + 24 \beta_{2} - 4 \beta_{3} ) q^{97} + ( 40 + 80 \beta_{2} - \beta_{3} ) q^{98} -18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{3} - 4 q^{4} + 10 q^{7} + 6 q^{9} + O(q^{10})$$ $$4 q - 6 q^{3} - 4 q^{4} + 10 q^{7} + 6 q^{9} - 12 q^{11} + 12 q^{12} + 24 q^{14} - 8 q^{16} + 48 q^{17} - 42 q^{19} - 24 q^{23} + 96 q^{26} - 40 q^{28} + 102 q^{31} + 36 q^{33} - 24 q^{36} - 22 q^{37} - 24 q^{38} + 6 q^{39} - 24 q^{42} - 28 q^{43} - 24 q^{44} - 72 q^{46} + 132 q^{47} - 2 q^{49} - 48 q^{51} - 12 q^{52} - 120 q^{53} - 48 q^{56} + 84 q^{57} + 96 q^{58} - 24 q^{59} - 72 q^{61} - 30 q^{63} + 32 q^{64} - 110 q^{67} - 96 q^{68} + 312 q^{71} + 66 q^{73} - 48 q^{74} - 120 q^{77} - 192 q^{78} - 10 q^{79} - 18 q^{81} - 48 q^{82} + 60 q^{84} - 24 q^{86} - 72 q^{89} - 222 q^{91} + 96 q^{92} - 102 q^{93} - 24 q^{94} - 72 q^{99} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$127$$ $$451$$ $$701$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.707107 + 1.22474i −0.707107 − 1.22474i 0.707107 − 1.22474i −0.707107 + 1.22474i
−0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i −1.74264 + 6.77962i 2.82843 1.50000 + 2.59808i 0
451.2 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 0 2.44949i 6.74264 + 1.88064i −2.82843 1.50000 + 2.59808i 0
901.1 −0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i −1.74264 6.77962i 2.82843 1.50000 2.59808i 0
901.2 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 0 2.44949i 6.74264 1.88064i −2.82843 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
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.p.a 4
5.b even 2 1 42.3.g.a 4
5.c odd 4 2 1050.3.q.a 8
7.d odd 6 1 inner 1050.3.p.a 4
15.d odd 2 1 126.3.n.a 4
20.d odd 2 1 336.3.bh.e 4
35.c odd 2 1 294.3.g.a 4
35.i odd 6 1 42.3.g.a 4
35.i odd 6 1 294.3.c.a 4
35.j even 6 1 294.3.c.a 4
35.j even 6 1 294.3.g.a 4
35.k even 12 2 1050.3.q.a 8
60.h even 2 1 1008.3.cg.h 4
105.g even 2 1 882.3.n.e 4
105.o odd 6 1 882.3.c.b 4
105.o odd 6 1 882.3.n.e 4
105.p even 6 1 126.3.n.a 4
105.p even 6 1 882.3.c.b 4
140.p odd 6 1 2352.3.f.e 4
140.s even 6 1 336.3.bh.e 4
140.s even 6 1 2352.3.f.e 4
420.be odd 6 1 1008.3.cg.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 5.b even 2 1
42.3.g.a 4 35.i odd 6 1
126.3.n.a 4 15.d odd 2 1
126.3.n.a 4 105.p even 6 1
294.3.c.a 4 35.i odd 6 1
294.3.c.a 4 35.j even 6 1
294.3.g.a 4 35.c odd 2 1
294.3.g.a 4 35.j even 6 1
336.3.bh.e 4 20.d odd 2 1
336.3.bh.e 4 140.s even 6 1
882.3.c.b 4 105.o odd 6 1
882.3.c.b 4 105.p even 6 1
882.3.n.e 4 105.g even 2 1
882.3.n.e 4 105.o odd 6 1
1008.3.cg.h 4 60.h even 2 1
1008.3.cg.h 4 420.be odd 6 1
1050.3.p.a 4 1.a even 1 1 trivial
1050.3.p.a 4 7.d odd 6 1 inner
1050.3.q.a 8 5.c odd 4 2
1050.3.q.a 8 35.k even 12 2
2352.3.f.e 4 140.p odd 6 1
2352.3.f.e 4 140.s even 6 1

## Hecke kernels

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

 $$T_{11}^{2} + 6 T_{11} + 36$$ $$T_{17}^{4} - 48 T_{17}^{3} + 936 T_{17}^{2} - 8064 T_{17} + 28224$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T^{2} + T^{4}$$
$3$ $$( 3 + 3 T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$2401 - 490 T + 51 T^{2} - 10 T^{3} + T^{4}$$
$11$ $$( 36 + 6 T + T^{2} )^{2}$$
$13$ $$145161 + 774 T^{2} + T^{4}$$
$17$ $$28224 - 8064 T + 936 T^{2} - 48 T^{3} + T^{4}$$
$19$ $$15129 + 5166 T + 711 T^{2} + 42 T^{3} + T^{4}$$
$23$ $$254016 - 12096 T + 1080 T^{2} + 24 T^{3} + T^{4}$$
$29$ $$( -1152 + T^{2} )^{2}$$
$31$ $$423801 - 66402 T + 4119 T^{2} - 102 T^{3} + T^{4}$$
$37$ $$27889 - 3674 T + 651 T^{2} + 22 T^{3} + T^{4}$$
$41$ $$3732624 + 4248 T^{2} + T^{4}$$
$43$ $$( -23 + 14 T + T^{2} )^{2}$$
$47$ $$2039184 - 188496 T + 7236 T^{2} - 132 T^{3} + T^{4}$$
$53$ $$8714304 + 354240 T + 11448 T^{2} + 120 T^{3} + T^{4}$$
$59$ $$1272384 - 27072 T - 936 T^{2} + 24 T^{3} + T^{4}$$
$61$ $$2304 + 3456 T + 1776 T^{2} + 72 T^{3} + T^{4}$$
$67$ $$253009 - 55330 T + 12603 T^{2} + 110 T^{3} + T^{4}$$
$71$ $$( 2556 - 156 T + T^{2} )^{2}$$
$73$ $$85322169 + 609642 T - 7785 T^{2} - 66 T^{3} + T^{4}$$
$79$ $$75463969 - 86870 T + 8787 T^{2} + 10 T^{3} + T^{4}$$
$83$ $$69956496 + 17928 T^{2} + T^{4}$$
$89$ $$( 432 + 36 T + T^{2} )^{2}$$
$97$ $$112896 + 1056 T^{2} + T^{4}$$
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