# Properties

 Label 3420.2.a.i Level $3420$ Weight $2$ Character orbit 3420.a Self dual yes Analytic conductor $27.309$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3420.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.3088374913$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1140) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{5} + (\beta + 1) q^{7}+O(q^{10})$$ q + q^5 + (b + 1) * q^7 $$q + q^{5} + (\beta + 1) q^{7} + ( - \beta + 1) q^{11} + ( - \beta + 1) q^{13} - 2 q^{17} - q^{19} + 2 q^{23} + q^{25} + (\beta + 1) q^{29} + 4 q^{31} + (\beta + 1) q^{35} + (\beta + 7) q^{37} + (\beta - 3) q^{41} + ( - \beta + 7) q^{43} - 6 q^{47} + (2 \beta + 7) q^{49} + ( - \beta + 1) q^{55} + (2 \beta + 2) q^{59} + 2 \beta q^{61} + ( - \beta + 1) q^{65} + 4 q^{67} + ( - 2 \beta + 2) q^{71} + 6 q^{73} - 12 q^{77} + 8 q^{79} + ( - 2 \beta + 4) q^{83} - 2 q^{85} + (\beta - 3) q^{89} - 12 q^{91} - q^{95} + ( - \beta + 13) q^{97}+O(q^{100})$$ q + q^5 + (b + 1) * q^7 + (-b + 1) * q^11 + (-b + 1) * q^13 - 2 * q^17 - q^19 + 2 * q^23 + q^25 + (b + 1) * q^29 + 4 * q^31 + (b + 1) * q^35 + (b + 7) * q^37 + (b - 3) * q^41 + (-b + 7) * q^43 - 6 * q^47 + (2*b + 7) * q^49 + (-b + 1) * q^55 + (2*b + 2) * q^59 + 2*b * q^61 + (-b + 1) * q^65 + 4 * q^67 + (-2*b + 2) * q^71 + 6 * q^73 - 12 * q^77 + 8 * q^79 + (-2*b + 4) * q^83 - 2 * q^85 + (b - 3) * q^89 - 12 * q^91 - q^95 + (-b + 13) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + 2 * q^7 $$2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} - 4 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{35} + 14 q^{37} - 6 q^{41} + 14 q^{43} - 12 q^{47} + 14 q^{49} + 2 q^{55} + 4 q^{59} + 2 q^{65} + 8 q^{67} + 4 q^{71} + 12 q^{73} - 24 q^{77} + 16 q^{79} + 8 q^{83} - 4 q^{85} - 6 q^{89} - 24 q^{91} - 2 q^{95} + 26 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 + 2 * q^13 - 4 * q^17 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 2 * q^29 + 8 * q^31 + 2 * q^35 + 14 * q^37 - 6 * q^41 + 14 * q^43 - 12 * q^47 + 14 * q^49 + 2 * q^55 + 4 * q^59 + 2 * q^65 + 8 * q^67 + 4 * q^71 + 12 * q^73 - 24 * q^77 + 16 * q^79 + 8 * q^83 - 4 * q^85 - 6 * q^89 - 24 * q^91 - 2 * q^95 + 26 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.30278 2.30278
0 0 0 1.00000 0 −2.60555 0 0 0
1.2 0 0 0 1.00000 0 4.60555 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.2.a.i 2
3.b odd 2 1 1140.2.a.e 2
12.b even 2 1 4560.2.a.bl 2
15.d odd 2 1 5700.2.a.u 2
15.e even 4 2 5700.2.f.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.e 2 3.b odd 2 1
3420.2.a.i 2 1.a even 1 1 trivial
4560.2.a.bl 2 12.b even 2 1
5700.2.a.u 2 15.d odd 2 1
5700.2.f.n 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3420))$$:

 $$T_{7}^{2} - 2T_{7} - 12$$ T7^2 - 2*T7 - 12 $$T_{11}^{2} - 2T_{11} - 12$$ T11^2 - 2*T11 - 12 $$T_{13}^{2} - 2T_{13} - 12$$ T13^2 - 2*T13 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 2T - 12$$
$11$ $$T^{2} - 2T - 12$$
$13$ $$T^{2} - 2T - 12$$
$17$ $$(T + 2)^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$(T - 2)^{2}$$
$29$ $$T^{2} - 2T - 12$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} - 14T + 36$$
$41$ $$T^{2} + 6T - 4$$
$43$ $$T^{2} - 14T + 36$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 4T - 48$$
$61$ $$T^{2} - 52$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} - 4T - 48$$
$73$ $$(T - 6)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 8T - 36$$
$89$ $$T^{2} + 6T - 4$$
$97$ $$T^{2} - 26T + 156$$