# Properties

 Label 3420.2.a.g Level $3420$ Weight $2$ Character orbit 3420.a Self dual yes Analytic conductor $27.309$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 380) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + (2 \beta - 2) q^{7}+O(q^{10})$$ q - q^5 + (2*b - 2) * q^7 $$q - q^{5} + (2 \beta - 2) q^{7} + 2 q^{11} + ( - 3 \beta - 2) q^{13} + ( - 2 \beta + 2) q^{17} - q^{19} + 6 q^{23} + q^{25} + (6 \beta - 2) q^{29} + ( - 2 \beta - 4) q^{31} + ( - 2 \beta + 2) q^{35} + (3 \beta - 6) q^{37} + ( - 4 \beta + 2) q^{41} + (2 \beta + 2) q^{43} + (2 \beta + 2) q^{47} + ( - 8 \beta + 5) q^{49} + ( - 5 \beta - 2) q^{53} - 2 q^{55} + ( - 4 \beta - 8) q^{59} + ( - 4 \beta - 8) q^{61} + (3 \beta + 2) q^{65} + ( - \beta - 2) q^{67} + (2 \beta - 8) q^{71} + ( - 6 \beta + 6) q^{73} + (4 \beta - 4) q^{77} + (4 \beta - 4) q^{79} + (8 \beta + 2) q^{83} + (2 \beta - 2) q^{85} + (6 \beta - 2) q^{89} + (2 \beta - 8) q^{91} + q^{95} + ( - 3 \beta - 6) q^{97}+O(q^{100})$$ q - q^5 + (2*b - 2) * q^7 + 2 * q^11 + (-3*b - 2) * q^13 + (-2*b + 2) * q^17 - q^19 + 6 * q^23 + q^25 + (6*b - 2) * q^29 + (-2*b - 4) * q^31 + (-2*b + 2) * q^35 + (3*b - 6) * q^37 + (-4*b + 2) * q^41 + (2*b + 2) * q^43 + (2*b + 2) * q^47 + (-8*b + 5) * q^49 + (-5*b - 2) * q^53 - 2 * q^55 + (-4*b - 8) * q^59 + (-4*b - 8) * q^61 + (3*b + 2) * q^65 + (-b - 2) * q^67 + (2*b - 8) * q^71 + (-6*b + 6) * q^73 + (4*b - 4) * q^77 + (4*b - 4) * q^79 + (8*b + 2) * q^83 + (2*b - 2) * q^85 + (6*b - 2) * q^89 + (2*b - 8) * q^91 + q^95 + (-3*b - 6) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 4 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 - 4 * q^7 $$2 q - 2 q^{5} - 4 q^{7} + 4 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 12 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} + 4 q^{35} - 12 q^{37} + 4 q^{41} + 4 q^{43} + 4 q^{47} + 10 q^{49} - 4 q^{53} - 4 q^{55} - 16 q^{59} - 16 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{77} - 8 q^{79} + 4 q^{83} - 4 q^{85} - 4 q^{89} - 16 q^{91} + 2 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 4 * q^7 + 4 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 12 * q^23 + 2 * q^25 - 4 * q^29 - 8 * q^31 + 4 * q^35 - 12 * q^37 + 4 * q^41 + 4 * q^43 + 4 * q^47 + 10 * q^49 - 4 * q^53 - 4 * q^55 - 16 * q^59 - 16 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^77 - 8 * q^79 + 4 * q^83 - 4 * q^85 - 4 * q^89 - 16 * q^91 + 2 * q^95 - 12 * 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
 −1.41421 1.41421
0 0 0 −1.00000 0 −4.82843 0 0 0
1.2 0 0 0 −1.00000 0 0.828427 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.g 2
3.b odd 2 1 380.2.a.c 2
12.b even 2 1 1520.2.a.o 2
15.d odd 2 1 1900.2.a.e 2
15.e even 4 2 1900.2.c.d 4
24.f even 2 1 6080.2.a.y 2
24.h odd 2 1 6080.2.a.bl 2
57.d even 2 1 7220.2.a.m 2
60.h even 2 1 7600.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.c 2 3.b odd 2 1
1520.2.a.o 2 12.b even 2 1
1900.2.a.e 2 15.d odd 2 1
1900.2.c.d 4 15.e even 4 2
3420.2.a.g 2 1.a even 1 1 trivial
6080.2.a.y 2 24.f even 2 1
6080.2.a.bl 2 24.h odd 2 1
7220.2.a.m 2 57.d even 2 1
7600.2.a.u 2 60.h even 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 4T_{7} - 4$$ T7^2 + 4*T7 - 4 $$T_{11} - 2$$ T11 - 2 $$T_{13}^{2} + 4T_{13} - 14$$ T13^2 + 4*T13 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 4T - 4$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 4T - 14$$
$17$ $$T^{2} - 4T - 4$$
$19$ $$(T + 1)^{2}$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} + 4T - 68$$
$31$ $$T^{2} + 8T + 8$$
$37$ $$T^{2} + 12T + 18$$
$41$ $$T^{2} - 4T - 28$$
$43$ $$T^{2} - 4T - 4$$
$47$ $$T^{2} - 4T - 4$$
$53$ $$T^{2} + 4T - 46$$
$59$ $$T^{2} + 16T + 32$$
$61$ $$T^{2} + 16T + 32$$
$67$ $$T^{2} + 4T + 2$$
$71$ $$T^{2} + 16T + 56$$
$73$ $$T^{2} - 12T - 36$$
$79$ $$T^{2} + 8T - 16$$
$83$ $$T^{2} - 4T - 124$$
$89$ $$T^{2} + 4T - 68$$
$97$ $$T^{2} + 12T + 18$$