# Properties

 Label 840.2.t.d Level $840$ Weight $2$ Character orbit 840.t Analytic conductor $6.707$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$840 = 2^{3} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 840.t (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.70743376979$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{2} q^{5} + \beta_{1} q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{2} q^{5} + \beta_{1} q^{7} - q^{9} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( 4 \beta_{1} + \beta_{3} - \beta_{4} ) q^{13} + \beta_{4} q^{15} + ( -4 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{17} + ( \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{19} + q^{21} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{23} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{25} + \beta_{1} q^{27} + 2 q^{29} + ( -4 - \beta_{2} + \beta_{5} ) q^{31} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{33} -\beta_{4} q^{35} + ( -4 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{37} + ( 4 - \beta_{2} + \beta_{5} ) q^{39} + ( -4 + \beta_{2} - \beta_{5} ) q^{41} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{43} + \beta_{2} q^{45} - q^{49} + ( -4 + 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{51} + ( 4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{53} + ( 6 - 8 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{55} + ( -2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{57} + ( 4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{59} + ( -2 + 4 \beta_{2} - 4 \beta_{5} ) q^{61} -\beta_{1} q^{63} + ( 2 + 4 \beta_{1} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{65} + ( 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{67} + ( 2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{69} + ( -4 - 3 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{71} + ( 8 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{73} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{75} + ( \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{77} + ( -2 \beta_{2} + 2 \beta_{5} ) q^{79} + q^{81} + ( 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( 2 - 6 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{85} -2 \beta_{1} q^{87} + ( 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{89} + ( -4 + \beta_{2} - \beta_{5} ) q^{91} + ( 4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{93} + ( -10 \beta_{1} - \beta_{2} + 2 \beta_{4} + 5 \beta_{5} ) q^{95} + ( -2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{97} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{5} - 6q^{9} + O(q^{10})$$ $$6q - 2q^{5} - 6q^{9} - 4q^{11} + 4q^{19} + 6q^{21} - 2q^{25} + 12q^{29} - 28q^{31} + 20q^{39} - 20q^{41} + 2q^{45} - 6q^{49} - 16q^{51} + 36q^{55} + 16q^{59} + 4q^{61} + 8q^{65} + 16q^{69} - 36q^{71} - 8q^{75} - 8q^{79} + 6q^{81} + 16q^{85} + 12q^{89} - 20q^{91} - 12q^{95} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-7 \nu^{5} + 10 \nu^{4} - 5 \nu^{3} - 30 \nu^{2} - 32 \nu + 13$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-9 \nu^{5} + 3 \nu^{4} + 10 \nu^{3} - 32 \nu^{2} - 74 \nu - 3$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$-10 \nu^{5} + 11 \nu^{4} - 17 \nu^{3} - 10 \nu^{2} - 72 \nu - 11$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$12 \nu^{5} - 27 \nu^{4} + 25 \nu^{3} + 12 \nu^{2} + 68 \nu - 65$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-19 \nu^{5} + 37 \nu^{4} - 30 \nu^{3} - 42 \nu^{2} - 54 \nu + 55$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{2} - 4 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 4 \beta_{1} - 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} - 5 \beta_{4} - 5 \beta_{3} + \beta_{2} - 14$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{5} - 11 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} + 18 \beta_{1} - 18$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$281$$ $$337$$ $$421$$ $$631$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
169.1
 −0.854638 + 0.854638i 1.45161 − 1.45161i 0.403032 − 0.403032i −0.854638 − 0.854638i 1.45161 + 1.45161i 0.403032 + 0.403032i
0 1.00000i 0 −2.17009 0.539189i 0 1.00000i 0 −1.00000 0
169.2 0 1.00000i 0 −0.311108 + 2.21432i 0 1.00000i 0 −1.00000 0
169.3 0 1.00000i 0 1.48119 1.67513i 0 1.00000i 0 −1.00000 0
169.4 0 1.00000i 0 −2.17009 + 0.539189i 0 1.00000i 0 −1.00000 0
169.5 0 1.00000i 0 −0.311108 2.21432i 0 1.00000i 0 −1.00000 0
169.6 0 1.00000i 0 1.48119 + 1.67513i 0 1.00000i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 169.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.t.d 6
3.b odd 2 1 2520.2.t.k 6
4.b odd 2 1 1680.2.t.j 6
5.b even 2 1 inner 840.2.t.d 6
5.c odd 4 1 4200.2.a.bn 3
5.c odd 4 1 4200.2.a.bp 3
12.b even 2 1 5040.2.t.z 6
15.d odd 2 1 2520.2.t.k 6
20.d odd 2 1 1680.2.t.j 6
20.e even 4 1 8400.2.a.di 3
20.e even 4 1 8400.2.a.dl 3
60.h even 2 1 5040.2.t.z 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.d 6 1.a even 1 1 trivial
840.2.t.d 6 5.b even 2 1 inner
1680.2.t.j 6 4.b odd 2 1
1680.2.t.j 6 20.d odd 2 1
2520.2.t.k 6 3.b odd 2 1
2520.2.t.k 6 15.d odd 2 1
4200.2.a.bn 3 5.c odd 4 1
4200.2.a.bp 3 5.c odd 4 1
5040.2.t.z 6 12.b even 2 1
5040.2.t.z 6 60.h even 2 1
8400.2.a.di 3 20.e even 4 1
8400.2.a.dl 3 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(840, [\chi])$$:

 $$T_{11}^{3} + 2 T_{11}^{2} - 36 T_{11} - 104$$ $$T_{19}^{3} - 2 T_{19}^{2} - 60 T_{19} + 200$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$125 + 50 T + 15 T^{2} + 12 T^{3} + 3 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$( 1 + T^{2} )^{3}$$
$11$ $$( -104 - 36 T + 2 T^{2} + T^{3} )^{2}$$
$13$ $$64 + 560 T^{2} + 60 T^{4} + T^{6}$$
$17$ $$73984 + 5376 T^{2} + 128 T^{4} + T^{6}$$
$19$ $$( 200 - 60 T - 2 T^{2} + T^{3} )^{2}$$
$23$ $$92416 + 6464 T^{2} + 144 T^{4} + T^{6}$$
$29$ $$( -2 + T )^{6}$$
$31$ $$( 40 + 52 T + 14 T^{2} + T^{3} )^{2}$$
$37$ $$102400 + 7936 T^{2} + 176 T^{4} + T^{6}$$
$41$ $$( -8 + 20 T + 10 T^{2} + T^{3} )^{2}$$
$43$ $$4096 + 2816 T^{2} + 112 T^{4} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$1600 + 1584 T^{2} + 92 T^{4} + T^{6}$$
$59$ $$( 256 - 64 T - 8 T^{2} + T^{3} )^{2}$$
$61$ $$( 104 - 212 T - 2 T^{2} + T^{3} )^{2}$$
$67$ $$65536 + 8192 T^{2} + 256 T^{4} + T^{6}$$
$71$ $$( -1352 - 52 T + 18 T^{2} + T^{3} )^{2}$$
$73$ $$10816 + 5424 T^{2} + 284 T^{4} + T^{6}$$
$79$ $$( -64 - 48 T + 4 T^{2} + T^{3} )^{2}$$
$83$ $$65536 + 8192 T^{2} + 192 T^{4} + T^{6}$$
$89$ $$( -232 - 124 T - 6 T^{2} + T^{3} )^{2}$$
$97$ $$40000 + 4400 T^{2} + 124 T^{4} + T^{6}$$