# Properties

 Label 420.2.q.c Level $420$ Weight $2$ Character orbit 420.q Analytic conductor $3.354$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_1 q^{7} + ( - \beta_{2} - 1) q^{9}+O(q^{10})$$ q + b2 * q^3 + (-b2 - 1) * q^5 + b1 * q^7 + (-b2 - 1) * q^9 $$q + \beta_{2} q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_1 q^{7} + ( - \beta_{2} - 1) q^{9} + (\beta_{3} + \beta_{2} + \beta_1) q^{11} - \beta_{3} q^{13} + q^{15} + (\beta_{3} + \beta_{2} + \beta_1) q^{17} + ( - 3 \beta_{2} + 2 \beta_1 - 3) q^{19} + \beta_{3} q^{21} + (\beta_{2} + \beta_1 + 1) q^{23} + \beta_{2} q^{25} + q^{27} + (\beta_{3} + 5) q^{29} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{31} + ( - \beta_{2} - \beta_1 - 1) q^{33} + ( - \beta_{3} - \beta_1) q^{35} + (2 \beta_{2} + \beta_1 + 2) q^{37} + (\beta_{3} + \beta_1) q^{39} + (3 \beta_{3} - 3) q^{41} + ( - 3 \beta_{3} + 2) q^{43} + \beta_{2} q^{45} + ( - 6 \beta_{2} - 6) q^{47} + 7 \beta_{2} q^{49} + ( - \beta_{2} - \beta_1 - 1) q^{51} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{53} + ( - \beta_{3} + 1) q^{55} + (2 \beta_{3} + 3) q^{57} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{59} + ( - 8 \beta_{2} - 8) q^{61} + ( - \beta_{3} - \beta_1) q^{63} - \beta_1 q^{65} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{67} + (\beta_{3} - 1) q^{69} + (\beta_{3} + 11) q^{71} + ( - 5 \beta_{3} - 5 \beta_1) q^{73} + ( - \beta_{2} - 1) q^{75} + (\beta_{3} - 7) q^{77} + (3 \beta_{2} + 2 \beta_1 + 3) q^{79} + \beta_{2} q^{81} + ( - 3 \beta_{3} - 3) q^{83} + ( - \beta_{3} + 1) q^{85} + ( - \beta_{3} + 5 \beta_{2} - \beta_1) q^{87} + (\beta_{2} - 5 \beta_1 + 1) q^{89} + (7 \beta_{2} + 7) q^{91} + ( - \beta_{2} - 2 \beta_1 - 1) q^{93} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{95} + 8 q^{97} + ( - \beta_{3} + 1) q^{99}+O(q^{100})$$ q + b2 * q^3 + (-b2 - 1) * q^5 + b1 * q^7 + (-b2 - 1) * q^9 + (b3 + b2 + b1) * q^11 - b3 * q^13 + q^15 + (b3 + b2 + b1) * q^17 + (-3*b2 + 2*b1 - 3) * q^19 + b3 * q^21 + (b2 + b1 + 1) * q^23 + b2 * q^25 + q^27 + (b3 + 5) * q^29 + (2*b3 + b2 + 2*b1) * q^31 + (-b2 - b1 - 1) * q^33 + (-b3 - b1) * q^35 + (2*b2 + b1 + 2) * q^37 + (b3 + b1) * q^39 + (3*b3 - 3) * q^41 + (-3*b3 + 2) * q^43 + b2 * q^45 + (-6*b2 - 6) * q^47 + 7*b2 * q^49 + (-b2 - b1 - 1) * q^51 + (-2*b3 - 2*b2 - 2*b1) * q^53 + (-b3 + 1) * q^55 + (2*b3 + 3) * q^57 + (-3*b3 + 3*b2 - 3*b1) * q^59 + (-8*b2 - 8) * q^61 + (-b3 - b1) * q^63 - b1 * q^65 + (-b3 - 2*b2 - b1) * q^67 + (b3 - 1) * q^69 + (b3 + 11) * q^71 + (-5*b3 - 5*b1) * q^73 + (-b2 - 1) * q^75 + (b3 - 7) * q^77 + (3*b2 + 2*b1 + 3) * q^79 + b2 * q^81 + (-3*b3 - 3) * q^83 + (-b3 + 1) * q^85 + (-b3 + 5*b2 - b1) * q^87 + (b2 - 5*b1 + 1) * q^89 + (7*b2 + 7) * q^91 + (-b2 - 2*b1 - 1) * q^93 + (-2*b3 + 3*b2 - 2*b1) * q^95 + 8 * q^97 + (-b3 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 2 * q^5 - 2 * q^9 $$4 q - 2 q^{3} - 2 q^{5} - 2 q^{9} - 2 q^{11} + 4 q^{15} - 2 q^{17} - 6 q^{19} + 2 q^{23} - 2 q^{25} + 4 q^{27} + 20 q^{29} - 2 q^{31} - 2 q^{33} + 4 q^{37} - 12 q^{41} + 8 q^{43} - 2 q^{45} - 12 q^{47} - 14 q^{49} - 2 q^{51} + 4 q^{53} + 4 q^{55} + 12 q^{57} - 6 q^{59} - 16 q^{61} + 4 q^{67} - 4 q^{69} + 44 q^{71} - 2 q^{75} - 28 q^{77} + 6 q^{79} - 2 q^{81} - 12 q^{83} + 4 q^{85} - 10 q^{87} + 2 q^{89} + 14 q^{91} - 2 q^{93} - 6 q^{95} + 32 q^{97} + 4 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 2 * q^5 - 2 * q^9 - 2 * q^11 + 4 * q^15 - 2 * q^17 - 6 * q^19 + 2 * q^23 - 2 * q^25 + 4 * q^27 + 20 * q^29 - 2 * q^31 - 2 * q^33 + 4 * q^37 - 12 * q^41 + 8 * q^43 - 2 * q^45 - 12 * q^47 - 14 * q^49 - 2 * q^51 + 4 * q^53 + 4 * q^55 + 12 * q^57 - 6 * q^59 - 16 * q^61 + 4 * q^67 - 4 * q^69 + 44 * q^71 - 2 * q^75 - 28 * q^77 + 6 * q^79 - 2 * q^81 - 12 * q^83 + 4 * q^85 - 10 * q^87 + 2 * q^89 + 14 * q^91 - 2 * q^93 - 6 * q^95 + 32 * q^97 + 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −1.32288 + 2.29129i 1.32288 − 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −1.32288 + 2.29129i 0 −0.500000 + 0.866025i 0
121.2 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.32288 2.29129i 0 −0.500000 + 0.866025i 0
361.1 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −1.32288 2.29129i 0 −0.500000 0.866025i 0
361.2 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.32288 + 2.29129i 0 −0.500000 0.866025i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.q.c 4
3.b odd 2 1 1260.2.s.f 4
4.b odd 2 1 1680.2.bg.q 4
5.b even 2 1 2100.2.q.h 4
5.c odd 4 2 2100.2.bc.e 8
7.b odd 2 1 2940.2.q.t 4
7.c even 3 1 inner 420.2.q.c 4
7.c even 3 1 2940.2.a.s 2
7.d odd 6 1 2940.2.a.m 2
7.d odd 6 1 2940.2.q.t 4
21.g even 6 1 8820.2.a.bj 2
21.h odd 6 1 1260.2.s.f 4
21.h odd 6 1 8820.2.a.be 2
28.g odd 6 1 1680.2.bg.q 4
35.j even 6 1 2100.2.q.h 4
35.l odd 12 2 2100.2.bc.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.c 4 1.a even 1 1 trivial
420.2.q.c 4 7.c even 3 1 inner
1260.2.s.f 4 3.b odd 2 1
1260.2.s.f 4 21.h odd 6 1
1680.2.bg.q 4 4.b odd 2 1
1680.2.bg.q 4 28.g odd 6 1
2100.2.q.h 4 5.b even 2 1
2100.2.q.h 4 35.j even 6 1
2100.2.bc.e 8 5.c odd 4 2
2100.2.bc.e 8 35.l odd 12 2
2940.2.a.m 2 7.d odd 6 1
2940.2.a.s 2 7.c even 3 1
2940.2.q.t 4 7.b odd 2 1
2940.2.q.t 4 7.d odd 6 1
8820.2.a.be 2 21.h odd 6 1
8820.2.a.bj 2 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{4} + 2T_{11}^{3} + 10T_{11}^{2} - 12T_{11} + 36$$ acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + 7T^{2} + 49$$
$11$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$
$13$ $$(T^{2} - 7)^{2}$$
$17$ $$T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36$$
$19$ $$T^{4} + 6 T^{3} + 55 T^{2} - 114 T + 361$$
$23$ $$T^{4} - 2 T^{3} + 10 T^{2} + 12 T + 36$$
$29$ $$(T^{2} - 10 T + 18)^{2}$$
$31$ $$T^{4} + 2 T^{3} + 31 T^{2} - 54 T + 729$$
$37$ $$T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9$$
$41$ $$(T^{2} + 6 T - 54)^{2}$$
$43$ $$(T^{2} - 4 T - 59)^{2}$$
$47$ $$(T^{2} + 6 T + 36)^{2}$$
$53$ $$T^{4} - 4 T^{3} + 40 T^{2} + 96 T + 576$$
$59$ $$T^{4} + 6 T^{3} + 90 T^{2} + \cdots + 2916$$
$61$ $$(T^{2} + 8 T + 64)^{2}$$
$67$ $$T^{4} - 4 T^{3} + 19 T^{2} + 12 T + 9$$
$71$ $$(T^{2} - 22 T + 114)^{2}$$
$73$ $$T^{4} + 175 T^{2} + 30625$$
$79$ $$T^{4} - 6 T^{3} + 55 T^{2} + 114 T + 361$$
$83$ $$(T^{2} + 6 T - 54)^{2}$$
$89$ $$T^{4} - 2 T^{3} + 178 T^{2} + \cdots + 30276$$
$97$ $$(T - 8)^{4}$$