# Properties

 Label 840.2.t.c Level $840$ Weight $2$ Character orbit 840.t Analytic conductor $6.707$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} - q^{9}+O(q^{10})$$ q - b1 * q^3 - b3 * q^5 - b1 * q^7 - q^9 $$q - \beta_1 q^{3} - \beta_{3} q^{5} - \beta_1 q^{7} - q^{9} - 2 q^{11} + 2 \beta_{2} q^{13} + \beta_{2} q^{15} + (2 \beta_{2} - 2 \beta_1) q^{17} - 2 q^{19} - q^{21} + 4 \beta_1 q^{23} + 5 q^{25} + \beta_1 q^{27} + ( - 2 \beta_{3} + 4) q^{29} + (2 \beta_{3} + 4) q^{31} + 2 \beta_1 q^{33} + \beta_{2} q^{35} + (2 \beta_{2} + 2 \beta_1) q^{37} + 2 \beta_{3} q^{39} + ( - 2 \beta_{3} - 8) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + \beta_{3} q^{45} + (2 \beta_{2} - 2 \beta_1) q^{47} - q^{49} + (2 \beta_{3} - 2) q^{51} + 2 \beta_1 q^{53} + 2 \beta_{3} q^{55} + 2 \beta_1 q^{57} + ( - 2 \beta_{3} - 8) q^{61} + \beta_1 q^{63} - 10 \beta_1 q^{65} + ( - 2 \beta_{2} - 6 \beta_1) q^{67} + 4 q^{69} + ( - 2 \beta_{3} + 8) q^{71} + (2 \beta_{2} + 12 \beta_1) q^{73} - 5 \beta_1 q^{75} + 2 \beta_1 q^{77} + 4 \beta_{3} q^{79} + q^{81} + (4 \beta_{2} + 4 \beta_1) q^{83} + (2 \beta_{2} - 10 \beta_1) q^{85} + (2 \beta_{2} - 4 \beta_1) q^{87} + ( - 6 \beta_{3} + 4) q^{89} + 2 \beta_{3} q^{91} + ( - 2 \beta_{2} - 4 \beta_1) q^{93} + 2 \beta_{3} q^{95} + ( - 2 \beta_{2} - 8 \beta_1) q^{97} + 2 q^{99}+O(q^{100})$$ q - b1 * q^3 - b3 * q^5 - b1 * q^7 - q^9 - 2 * q^11 + 2*b2 * q^13 + b2 * q^15 + (2*b2 - 2*b1) * q^17 - 2 * q^19 - q^21 + 4*b1 * q^23 + 5 * q^25 + b1 * q^27 + (-2*b3 + 4) * q^29 + (2*b3 + 4) * q^31 + 2*b1 * q^33 + b2 * q^35 + (2*b2 + 2*b1) * q^37 + 2*b3 * q^39 + (-2*b3 - 8) * q^41 + (2*b2 + 2*b1) * q^43 + b3 * q^45 + (2*b2 - 2*b1) * q^47 - q^49 + (2*b3 - 2) * q^51 + 2*b1 * q^53 + 2*b3 * q^55 + 2*b1 * q^57 + (-2*b3 - 8) * q^61 + b1 * q^63 - 10*b1 * q^65 + (-2*b2 - 6*b1) * q^67 + 4 * q^69 + (-2*b3 + 8) * q^71 + (2*b2 + 12*b1) * q^73 - 5*b1 * q^75 + 2*b1 * q^77 + 4*b3 * q^79 + q^81 + (4*b2 + 4*b1) * q^83 + (2*b2 - 10*b1) * q^85 + (2*b2 - 4*b1) * q^87 + (-6*b3 + 4) * q^89 + 2*b3 * q^91 + (-2*b2 - 4*b1) * q^93 + 2*b3 * q^95 + (-2*b2 - 8*b1) * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 8 q^{11} - 8 q^{19} - 4 q^{21} + 20 q^{25} + 16 q^{29} + 16 q^{31} - 32 q^{41} - 4 q^{49} - 8 q^{51} - 32 q^{61} + 16 q^{69} + 32 q^{71} + 4 q^{81} + 16 q^{89} + 8 q^{99}+O(q^{100})$$ 4 * q - 4 * q^9 - 8 * q^11 - 8 * q^19 - 4 * q^21 + 20 * q^25 + 16 * q^29 + 16 * q^31 - 32 * q^41 - 4 * q^49 - 8 * q^51 - 32 * q^61 + 16 * q^69 + 32 * q^71 + 4 * q^81 + 16 * q^89 + 8 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## 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
 0.618034i − 1.61803i − 0.618034i 1.61803i
0 1.00000i 0 −2.23607 0 1.00000i 0 −1.00000 0
169.2 0 1.00000i 0 2.23607 0 1.00000i 0 −1.00000 0
169.3 0 1.00000i 0 −2.23607 0 1.00000i 0 −1.00000 0
169.4 0 1.00000i 0 2.23607 0 1.00000i 0 −1.00000 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
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.c 4
3.b odd 2 1 2520.2.t.f 4
4.b odd 2 1 1680.2.t.h 4
5.b even 2 1 inner 840.2.t.c 4
5.c odd 4 1 4200.2.a.bj 2
5.c odd 4 1 4200.2.a.bk 2
12.b even 2 1 5040.2.t.u 4
15.d odd 2 1 2520.2.t.f 4
20.d odd 2 1 1680.2.t.h 4
20.e even 4 1 8400.2.a.cz 2
20.e even 4 1 8400.2.a.db 2
60.h even 2 1 5040.2.t.u 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.c 4 1.a even 1 1 trivial
840.2.t.c 4 5.b even 2 1 inner
1680.2.t.h 4 4.b odd 2 1
1680.2.t.h 4 20.d odd 2 1
2520.2.t.f 4 3.b odd 2 1
2520.2.t.f 4 15.d odd 2 1
4200.2.a.bj 2 5.c odd 4 1
4200.2.a.bk 2 5.c odd 4 1
5040.2.t.u 4 12.b even 2 1
5040.2.t.u 4 60.h even 2 1
8400.2.a.cz 2 20.e even 4 1
8400.2.a.db 2 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 $$S_{2}^{\mathrm{new}}(840, [\chi])$$:

 $$T_{11} + 2$$ T11 + 2 $$T_{19} + 2$$ T19 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T^{2} - 5)^{2}$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T + 2)^{4}$$
$13$ $$(T^{2} + 20)^{2}$$
$17$ $$T^{4} + 48T^{2} + 256$$
$19$ $$(T + 2)^{4}$$
$23$ $$(T^{2} + 16)^{2}$$
$29$ $$(T^{2} - 8 T - 4)^{2}$$
$31$ $$(T^{2} - 8 T - 4)^{2}$$
$37$ $$T^{4} + 48T^{2} + 256$$
$41$ $$(T^{2} + 16 T + 44)^{2}$$
$43$ $$T^{4} + 48T^{2} + 256$$
$47$ $$T^{4} + 48T^{2} + 256$$
$53$ $$(T^{2} + 4)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 16 T + 44)^{2}$$
$67$ $$T^{4} + 112T^{2} + 256$$
$71$ $$(T^{2} - 16 T + 44)^{2}$$
$73$ $$T^{4} + 328 T^{2} + 15376$$
$79$ $$(T^{2} - 80)^{2}$$
$83$ $$T^{4} + 192T^{2} + 4096$$
$89$ $$(T^{2} - 8 T - 164)^{2}$$
$97$ $$T^{4} + 168T^{2} + 1936$$