# Properties

 Label 3420.2.t.b Level $3420$ Weight $2$ Character orbit 3420.t Analytic conductor $27.309$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3420.t (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.3088374913$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 380) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{5} - 4 q^{7}+O(q^{10})$$ q + (z - 1) * q^5 - 4 * q^7 $$q + (\zeta_{6} - 1) q^{5} - 4 q^{7} + 3 q^{11} - 6 \zeta_{6} q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + (3 \zeta_{6} + 2) q^{19} + 4 \zeta_{6} q^{23} - \zeta_{6} q^{25} + \zeta_{6} q^{29} - 5 q^{31} + ( - 4 \zeta_{6} + 4) q^{35} - 4 q^{37} + ( - 2 \zeta_{6} + 2) q^{41} + 6 \zeta_{6} q^{47} + 9 q^{49} - 6 \zeta_{6} q^{53} + (3 \zeta_{6} - 3) q^{55} + (\zeta_{6} - 1) q^{59} + 7 \zeta_{6} q^{61} + 6 q^{65} + 14 \zeta_{6} q^{67} + ( - 15 \zeta_{6} + 15) q^{71} + (12 \zeta_{6} - 12) q^{73} - 12 q^{77} + ( - \zeta_{6} + 1) q^{79} - 16 q^{83} + 2 \zeta_{6} q^{85} + 17 \zeta_{6} q^{89} + 24 \zeta_{6} q^{91} + (2 \zeta_{6} - 5) q^{95} + (12 \zeta_{6} - 12) q^{97} +O(q^{100})$$ q + (z - 1) * q^5 - 4 * q^7 + 3 * q^11 - 6*z * q^13 + (-2*z + 2) * q^17 + (3*z + 2) * q^19 + 4*z * q^23 - z * q^25 + z * q^29 - 5 * q^31 + (-4*z + 4) * q^35 - 4 * q^37 + (-2*z + 2) * q^41 + 6*z * q^47 + 9 * q^49 - 6*z * q^53 + (3*z - 3) * q^55 + (z - 1) * q^59 + 7*z * q^61 + 6 * q^65 + 14*z * q^67 + (-15*z + 15) * q^71 + (12*z - 12) * q^73 - 12 * q^77 + (-z + 1) * q^79 - 16 * q^83 + 2*z * q^85 + 17*z * q^89 + 24*z * q^91 + (2*z - 5) * q^95 + (12*z - 12) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} - 8 q^{7}+O(q^{10})$$ 2 * q - q^5 - 8 * q^7 $$2 q - q^{5} - 8 q^{7} + 6 q^{11} - 6 q^{13} + 2 q^{17} + 7 q^{19} + 4 q^{23} - q^{25} + q^{29} - 10 q^{31} + 4 q^{35} - 8 q^{37} + 2 q^{41} + 6 q^{47} + 18 q^{49} - 6 q^{53} - 3 q^{55} - q^{59} + 7 q^{61} + 12 q^{65} + 14 q^{67} + 15 q^{71} - 12 q^{73} - 24 q^{77} + q^{79} - 32 q^{83} + 2 q^{85} + 17 q^{89} + 24 q^{91} - 8 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q - q^5 - 8 * q^7 + 6 * q^11 - 6 * q^13 + 2 * q^17 + 7 * q^19 + 4 * q^23 - q^25 + q^29 - 10 * q^31 + 4 * q^35 - 8 * q^37 + 2 * q^41 + 6 * q^47 + 18 * q^49 - 6 * q^53 - 3 * q^55 - q^59 + 7 * q^61 + 12 * q^65 + 14 * q^67 + 15 * q^71 - 12 * q^73 - 24 * q^77 + q^79 - 32 * q^83 + 2 * q^85 + 17 * q^89 + 24 * q^91 - 8 * q^95 - 12 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$$.

 $$n$$ $$1711$$ $$1901$$ $$2737$$ $$3061$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1261.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −0.500000 0.866025i 0 −4.00000 0 0 0
3241.1 0 0 0 −0.500000 + 0.866025i 0 −4.00000 0 0 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.2.t.b 2
3.b odd 2 1 380.2.i.a 2
12.b even 2 1 1520.2.q.g 2
15.d odd 2 1 1900.2.i.b 2
15.e even 4 2 1900.2.s.b 4
19.c even 3 1 inner 3420.2.t.b 2
57.f even 6 1 7220.2.a.a 1
57.h odd 6 1 380.2.i.a 2
57.h odd 6 1 7220.2.a.e 1
228.m even 6 1 1520.2.q.g 2
285.n odd 6 1 1900.2.i.b 2
285.v even 12 2 1900.2.s.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.a 2 3.b odd 2 1
380.2.i.a 2 57.h odd 6 1
1520.2.q.g 2 12.b even 2 1
1520.2.q.g 2 228.m even 6 1
1900.2.i.b 2 15.d odd 2 1
1900.2.i.b 2 285.n odd 6 1
1900.2.s.b 4 15.e even 4 2
1900.2.s.b 4 285.v even 12 2
3420.2.t.b 2 1.a even 1 1 trivial
3420.2.t.b 2 19.c even 3 1 inner
7220.2.a.a 1 57.f even 6 1
7220.2.a.e 1 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3420, [\chi])$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 3$$ T11 - 3 $$T_{13}^{2} + 6T_{13} + 36$$ T13^2 + 6*T13 + 36 $$T_{17}^{2} - 2T_{17} + 4$$ T17^2 - 2*T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$(T + 4)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 6T + 36$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$T^{2} - 7T + 19$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} - T + 1$$
$31$ $$(T + 5)^{2}$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} - 2T + 4$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + T + 1$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} - 14T + 196$$
$71$ $$T^{2} - 15T + 225$$
$73$ $$T^{2} + 12T + 144$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T + 16)^{2}$$
$89$ $$T^{2} - 17T + 289$$
$97$ $$T^{2} + 12T + 144$$