# Properties

 Label 1050.2.d.f Level 1050 Weight 2 Character orbit 1050.d Analytic conductor 8.384 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.38429221223$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 210) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1049.1
 − 0.618034i 0.618034i − 1.61803i 1.61803i
1.00000 −0.618034 1.61803i 1.00000 0 −0.618034 1.61803i −0.381966 + 2.61803i 1.00000 −2.23607 + 2.00000i 0
1049.2 1.00000 −0.618034 + 1.61803i 1.00000 0 −0.618034 + 1.61803i −0.381966 2.61803i 1.00000 −2.23607 2.00000i 0
1049.3 1.00000 1.61803 0.618034i 1.00000 0 1.61803 0.618034i −2.61803 0.381966i 1.00000 2.23607 2.00000i 0
1049.4 1.00000 1.61803 + 0.618034i 1.00000 0 1.61803 + 0.618034i −2.61803 + 0.381966i 1.00000 2.23607 + 2.00000i 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
105.g 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.d.f 4
3.b odd 2 1 1050.2.d.c 4
5.b even 2 1 1050.2.d.a 4
5.c odd 4 1 210.2.b.b yes 4
5.c odd 4 1 1050.2.b.a 4
7.b odd 2 1 1050.2.d.d 4
15.d odd 2 1 1050.2.d.d 4
15.e even 4 1 210.2.b.a 4
15.e even 4 1 1050.2.b.c 4
20.e even 4 1 1680.2.f.e 4
21.c even 2 1 1050.2.d.a 4
35.c odd 2 1 1050.2.d.c 4
35.f even 4 1 210.2.b.a 4
35.f even 4 1 1050.2.b.c 4
60.l odd 4 1 1680.2.f.i 4
105.g even 2 1 inner 1050.2.d.f 4
105.k odd 4 1 210.2.b.b yes 4
105.k odd 4 1 1050.2.b.a 4
140.j odd 4 1 1680.2.f.i 4
420.w even 4 1 1680.2.f.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.b.a 4 15.e even 4 1
210.2.b.a 4 35.f even 4 1
210.2.b.b yes 4 5.c odd 4 1
210.2.b.b yes 4 105.k odd 4 1
1050.2.b.a 4 5.c odd 4 1
1050.2.b.a 4 105.k odd 4 1
1050.2.b.c 4 15.e even 4 1
1050.2.b.c 4 35.f even 4 1
1050.2.d.a 4 5.b even 2 1
1050.2.d.a 4 21.c even 2 1
1050.2.d.c 4 3.b odd 2 1
1050.2.d.c 4 35.c odd 2 1
1050.2.d.d 4 7.b odd 2 1
1050.2.d.d 4 15.d odd 2 1
1050.2.d.f 4 1.a even 1 1 trivial
1050.2.d.f 4 105.g even 2 1 inner
1680.2.f.e 4 20.e even 4 1
1680.2.f.e 4 420.w even 4 1
1680.2.f.i 4 60.l odd 4 1
1680.2.f.i 4 140.j odd 4 1

## Hecke kernels

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

 $$T_{11}^{2} + 20$$ $$T_{13}^{2} - 2 T_{13} - 4$$ $$T_{23} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{4}$$
$3$ $$1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4}$$
$5$ 1
$7$ $$1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4}$$
$11$ $$( 1 - 2 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} )^{2}$$
$17$ $$1 - 40 T^{2} + 798 T^{4} - 11560 T^{6} + 83521 T^{8}$$
$19$ $$1 - 4 T^{2} - 554 T^{4} - 1444 T^{6} + 130321 T^{8}$$
$23$ $$( 1 - 4 T + 23 T^{2} )^{4}$$
$29$ $$1 - 24 T^{2} + 1646 T^{4} - 20184 T^{6} + 707281 T^{8}$$
$31$ $$1 - 64 T^{2} + 2446 T^{4} - 61504 T^{6} + 923521 T^{8}$$
$37$ $$1 - 120 T^{2} + 6158 T^{4} - 164280 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 20 T^{2} - 1322 T^{4} + 36980 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 140 T^{2} + 8998 T^{4} - 309260 T^{6} + 4879681 T^{8}$$
$53$ $$( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 98 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$1 - 184 T^{2} + 15406 T^{4} - 684664 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 + 10 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$1 - 224 T^{2} + 22126 T^{4} - 1129184 T^{6} + 25411681 T^{8}$$
$73$ $$( 1 + 18 T + 222 T^{2} + 1314 T^{3} + 5329 T^{4} )^{2}$$
$79$ $$( 1 + 78 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 + 160 T^{2} + 19198 T^{4} + 1102240 T^{6} + 47458321 T^{8}$$
$89$ $$( 1 + 20 T + 258 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 6 T + 198 T^{2} + 582 T^{3} + 9409 T^{4} )^{2}$$
show more
show less