# Properties

 Label 210.2.t.d Level $210$ Weight $2$ Character orbit 210.t Analytic conductor $1.677$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 210.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.67685844245$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 1 - \beta_{1} + \beta_{3} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{5} + ( \beta_{2} + \beta_{3} ) q^{6} + ( 3 - \beta_{2} ) q^{7} - q^{8} + ( -2 - \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 1 - \beta_{1} + \beta_{3} ) q^{3} + ( -1 + \beta_{2} ) q^{4} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{5} + ( \beta_{2} + \beta_{3} ) q^{6} + ( 3 - \beta_{2} ) q^{7} - q^{8} + ( -2 - \beta_{1} + \beta_{3} ) q^{9} + ( 2 - \beta_{1} + \beta_{3} ) q^{10} + ( -2 + \beta_{1} - \beta_{2} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} ) q^{12} + 2 q^{13} + ( 1 + 2 \beta_{2} ) q^{14} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{15} -\beta_{2} q^{16} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{17} + ( -2 \beta_{2} + \beta_{3} ) q^{18} + ( -4 + 2 \beta_{2} ) q^{19} + ( 2 \beta_{2} + \beta_{3} ) q^{20} + ( 3 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{21} + ( 1 + \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{22} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{23} + ( -1 + \beta_{1} - \beta_{3} ) q^{24} + ( -\beta_{2} - 3 \beta_{3} ) q^{25} + 2 \beta_{2} q^{26} + ( -5 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} + ( -2 + 3 \beta_{2} ) q^{28} + ( -1 + 2 \beta_{1} - 2 \beta_{3} ) q^{29} + ( -1 - 2 \beta_{1} + 2 \beta_{3} ) q^{30} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{31} + ( 1 - \beta_{2} ) q^{32} + ( 1 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{33} + ( -2 + 4 \beta_{1} - 4 \beta_{3} ) q^{34} + ( 4 - 2 \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{35} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{36} + ( 2 - 4 \beta_{2} - 6 \beta_{3} ) q^{37} + ( -2 - 2 \beta_{2} ) q^{38} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{39} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{40} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( 1 - \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{42} + ( -2 - 3 \beta_{1} + 7 \beta_{2} + 3 \beta_{3} ) q^{43} + ( 3 - 2 \beta_{2} - \beta_{3} ) q^{44} + ( -7 + \beta_{1} + 7 \beta_{2} ) q^{45} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{46} + ( 6 - 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -\beta_{2} - \beta_{3} ) q^{48} + ( 8 - 5 \beta_{2} ) q^{49} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{50} + ( 10 + 2 \beta_{1} - 10 \beta_{2} ) q^{51} + ( -2 + 2 \beta_{2} ) q^{52} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{53} + ( -5 \beta_{2} - 2 \beta_{3} ) q^{54} + ( -6 + 3 \beta_{1} + \beta_{2} ) q^{55} + ( -3 + \beta_{2} ) q^{56} + ( -4 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{57} + ( -\beta_{2} - 2 \beta_{3} ) q^{58} + ( 6 - 3 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{59} + ( -\beta_{2} + 2 \beta_{3} ) q^{60} + ( -9 + 3 \beta_{2} - 3 \beta_{3} ) q^{61} + ( -3 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{62} + ( -6 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{63} + q^{64} + ( 4 - 2 \beta_{1} - 4 \beta_{2} ) q^{65} + ( 4 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{66} + ( -3 + 3 \beta_{1} ) q^{67} + ( -2 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -5 - \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{69} + ( 6 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{70} + ( -4 + 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{71} + ( 2 + \beta_{1} - \beta_{3} ) q^{72} + ( -2 + 2 \beta_{2} ) q^{73} + ( 4 - 6 \beta_{1} - 2 \beta_{2} ) q^{74} + ( 8 \beta_{2} - \beta_{3} ) q^{75} + ( 2 - 4 \beta_{2} ) q^{76} + ( -7 + 2 \beta_{1} + \beta_{3} ) q^{77} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{78} + ( 3 - 6 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{79} + ( -2 + \beta_{1} - \beta_{3} ) q^{80} + ( 1 + 5 \beta_{1} - 5 \beta_{3} ) q^{81} + ( 1 - 2 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{82} + ( 5 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{84} + ( -8 \beta_{2} + 6 \beta_{3} ) q^{85} + ( -7 + 5 \beta_{2} + 3 \beta_{3} ) q^{86} + ( 5 + \beta_{1} - \beta_{3} ) q^{87} + ( 2 - \beta_{1} + \beta_{2} ) q^{88} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{89} + ( -7 + \beta_{1} - \beta_{3} ) q^{90} + ( 6 - 2 \beta_{2} ) q^{91} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{92} + ( 9 - 6 \beta_{2} + 3 \beta_{3} ) q^{93} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{94} + ( -4 + 2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{95} + ( 1 - \beta_{1} - \beta_{2} ) q^{96} + ( 5 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{97} + ( 5 + 3 \beta_{2} ) q^{98} + ( 7 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + 2q^{3} - 2q^{4} + 3q^{5} + q^{6} + 10q^{7} - 4q^{8} - 10q^{9} + O(q^{10})$$ $$4q + 2q^{2} + 2q^{3} - 2q^{4} + 3q^{5} + q^{6} + 10q^{7} - 4q^{8} - 10q^{9} + 6q^{10} - 9q^{11} - q^{12} + 8q^{13} + 8q^{14} - 4q^{15} - 2q^{16} - 5q^{18} - 12q^{19} + 3q^{20} + 5q^{21} + 3q^{23} - 2q^{24} + q^{25} + 4q^{26} - 16q^{27} - 2q^{28} - 8q^{30} + 9q^{31} + 2q^{32} + q^{33} + 3q^{35} + 5q^{36} + 6q^{37} - 12q^{38} + 4q^{39} - 3q^{40} - 18q^{41} + 4q^{42} + 9q^{44} - 13q^{45} - 3q^{46} + 18q^{47} - q^{48} + 22q^{49} - q^{50} + 22q^{51} - 4q^{52} - 3q^{53} - 8q^{54} - 19q^{55} - 10q^{56} - 6q^{57} + 9q^{59} - 4q^{60} - 27q^{61} - 25q^{63} + 4q^{64} + 6q^{65} + 11q^{66} - 9q^{67} - 15q^{69} + 15q^{70} + 10q^{72} - 4q^{73} + 6q^{74} + 17q^{75} - 27q^{77} + 2q^{78} - q^{79} - 6q^{80} + 14q^{81} - 9q^{82} - q^{84} - 22q^{85} - 21q^{86} + 22q^{87} + 9q^{88} - 3q^{89} - 26q^{90} + 20q^{91} - 6q^{92} + 21q^{93} + 18q^{94} + q^{96} + 26q^{97} + 26q^{98} + 28q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/210\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$71$$ $$127$$ $$\chi(n)$$ $$\beta_{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 1.68614 + 0.396143i −1.18614 − 1.26217i −1.18614 + 1.26217i 1.68614 − 0.396143i
0.500000 + 0.866025i 0.500000 1.65831i −0.500000 + 0.866025i −0.686141 2.12819i 1.68614 0.396143i 2.50000 0.866025i −1.00000 −2.50000 1.65831i 1.50000 1.65831i
59.2 0.500000 + 0.866025i 0.500000 + 1.65831i −0.500000 + 0.866025i 2.18614 0.469882i −1.18614 + 1.26217i 2.50000 0.866025i −1.00000 −2.50000 + 1.65831i 1.50000 + 1.65831i
89.1 0.500000 0.866025i 0.500000 1.65831i −0.500000 0.866025i 2.18614 + 0.469882i −1.18614 1.26217i 2.50000 + 0.866025i −1.00000 −2.50000 1.65831i 1.50000 1.65831i
89.2 0.500000 0.866025i 0.500000 + 1.65831i −0.500000 0.866025i −0.686141 + 2.12819i 1.68614 + 0.396143i 2.50000 + 0.866025i −1.00000 −2.50000 + 1.65831i 1.50000 + 1.65831i
 $$n$$: 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.p even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.t.d yes 4
3.b odd 2 1 210.2.t.b yes 4
5.b even 2 1 210.2.t.a 4
5.c odd 4 2 1050.2.s.e 8
7.c even 3 1 1470.2.d.a 4
7.d odd 6 1 210.2.t.c yes 4
7.d odd 6 1 1470.2.d.b 4
15.d odd 2 1 210.2.t.c yes 4
15.e even 4 2 1050.2.s.d 8
21.g even 6 1 210.2.t.a 4
21.g even 6 1 1470.2.d.d 4
21.h odd 6 1 1470.2.d.c 4
35.i odd 6 1 210.2.t.b yes 4
35.i odd 6 1 1470.2.d.c 4
35.j even 6 1 1470.2.d.d 4
35.k even 12 2 1050.2.s.d 8
105.o odd 6 1 1470.2.d.b 4
105.p even 6 1 inner 210.2.t.d yes 4
105.p even 6 1 1470.2.d.a 4
105.w odd 12 2 1050.2.s.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.a 4 5.b even 2 1
210.2.t.a 4 21.g even 6 1
210.2.t.b yes 4 3.b odd 2 1
210.2.t.b yes 4 35.i odd 6 1
210.2.t.c yes 4 7.d odd 6 1
210.2.t.c yes 4 15.d odd 2 1
210.2.t.d yes 4 1.a even 1 1 trivial
210.2.t.d yes 4 105.p even 6 1 inner
1050.2.s.d 8 15.e even 4 2
1050.2.s.d 8 35.k even 12 2
1050.2.s.e 8 5.c odd 4 2
1050.2.s.e 8 105.w odd 12 2
1470.2.d.a 4 7.c even 3 1
1470.2.d.a 4 105.p even 6 1
1470.2.d.b 4 7.d odd 6 1
1470.2.d.b 4 105.o odd 6 1
1470.2.d.c 4 21.h odd 6 1
1470.2.d.c 4 35.i odd 6 1
1470.2.d.d 4 21.g even 6 1
1470.2.d.d 4 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(210, [\chi])$$:

 $$T_{11}^{4} + 9 T_{11}^{3} + 31 T_{11}^{2} + 36 T_{11} + 16$$ $$T_{13} - 2$$ $$T_{23}^{4} - 3 T_{23}^{3} + 15 T_{23}^{2} + 18 T_{23} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$( 3 - T + T^{2} )^{2}$$
$5$ $$25 - 15 T + 4 T^{2} - 3 T^{3} + T^{4}$$
$7$ $$( 7 - 5 T + T^{2} )^{2}$$
$11$ $$16 + 36 T + 31 T^{2} + 9 T^{3} + T^{4}$$
$13$ $$( -2 + T )^{4}$$
$17$ $$1936 - 44 T^{2} + T^{4}$$
$19$ $$( 12 + 6 T + T^{2} )^{2}$$
$23$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$29$ $$( 11 + T^{2} )^{2}$$
$31$ $$324 + 162 T + 9 T^{2} - 9 T^{3} + T^{4}$$
$37$ $$9216 + 576 T - 84 T^{2} - 6 T^{3} + T^{4}$$
$41$ $$( 12 + 9 T + T^{2} )^{2}$$
$43$ $$144 + 123 T^{2} + T^{4}$$
$47$ $$256 - 288 T + 124 T^{2} - 18 T^{3} + T^{4}$$
$53$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$59$ $$2916 + 486 T + 135 T^{2} - 9 T^{3} + T^{4}$$
$61$ $$1296 + 972 T + 279 T^{2} + 27 T^{3} + T^{4}$$
$67$ $$324 - 162 T + 9 T^{2} + 9 T^{3} + T^{4}$$
$71$ $$256 + 76 T^{2} + T^{4}$$
$73$ $$( 4 + 2 T + T^{2} )^{2}$$
$79$ $$5476 - 74 T + 75 T^{2} + T^{3} + T^{4}$$
$83$ $$289 + 142 T^{2} + T^{4}$$
$89$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$97$ $$( -32 - 13 T + T^{2} )^{2}$$