# Properties

 Label 3420.2.f.a Level $3420$ Weight $2$ Character orbit 3420.f Analytic conductor $27.309$ Analytic rank $0$ Dimension $4$ 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.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.3088374913$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ Defining polynomial: $$x^{4} + 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 380) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( \beta_{1} - \beta_{3} ) q^{7} + ( 3 - \beta_{2} ) q^{11} + ( -2 \beta_{1} - \beta_{3} ) q^{13} -2 \beta_{3} q^{17} + q^{19} + ( -3 \beta_{1} + \beta_{3} ) q^{23} + 5 q^{25} -2 \beta_{2} q^{29} -4 q^{31} + ( 3 \beta_{1} + \beta_{3} ) q^{35} + 3 \beta_{1} q^{37} + 6 q^{41} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{43} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{47} - q^{49} -3 \beta_{1} q^{53} + ( 5 - 3 \beta_{2} ) q^{55} + ( -6 + 2 \beta_{2} ) q^{59} + ( -5 - 3 \beta_{2} ) q^{61} + ( -3 \beta_{1} + 4 \beta_{3} ) q^{65} -3 \beta_{3} q^{67} + ( 6 + 2 \beta_{2} ) q^{71} + 6 \beta_{1} q^{73} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{77} + ( -2 - 6 \beta_{2} ) q^{79} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{83} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{85} + ( -6 + 4 \beta_{2} ) q^{89} + ( -2 - 6 \beta_{2} ) q^{91} -\beta_{2} q^{95} + ( \beta_{1} - 4 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 12 q^{11} + 4 q^{19} + 20 q^{25} - 16 q^{31} + 24 q^{41} - 4 q^{49} + 20 q^{55} - 24 q^{59} - 20 q^{61} + 24 q^{71} - 8 q^{79} - 24 q^{89} - 8 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 6 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 5 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 5 \beta_{1}$$

## Character values

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

 $$n$$ $$1711$$ $$1901$$ $$2737$$ $$3061$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1369.1
 0.874032i − 0.874032i − 2.28825i 2.28825i
0 0 0 −2.23607 0 2.82843i 0 0 0
1369.2 0 0 0 −2.23607 0 2.82843i 0 0 0
1369.3 0 0 0 2.23607 0 2.82843i 0 0 0
1369.4 0 0 0 2.23607 0 2.82843i 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
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 3420.2.f.a 4
3.b odd 2 1 380.2.c.a 4
5.b even 2 1 inner 3420.2.f.a 4
12.b even 2 1 1520.2.d.f 4
15.d odd 2 1 380.2.c.a 4
15.e even 4 2 1900.2.a.j 4
60.h even 2 1 1520.2.d.f 4
60.l odd 4 2 7600.2.a.ce 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.a 4 3.b odd 2 1
380.2.c.a 4 15.d odd 2 1
1520.2.d.f 4 12.b even 2 1
1520.2.d.f 4 60.h even 2 1
1900.2.a.j 4 15.e even 4 2
3420.2.f.a 4 1.a even 1 1 trivial
3420.2.f.a 4 5.b even 2 1 inner
7600.2.a.ce 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(3420, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$( 8 + T^{2} )^{2}$$
$11$ $$( 4 - 6 T + T^{2} )^{2}$$
$13$ $$484 + 46 T^{2} + T^{4}$$
$17$ $$64 + 56 T^{2} + T^{4}$$
$19$ $$( -1 + T )^{4}$$
$23$ $$64 + 56 T^{2} + T^{4}$$
$29$ $$( -20 + T^{2} )^{2}$$
$31$ $$( 4 + T )^{4}$$
$37$ $$324 + 54 T^{2} + T^{4}$$
$41$ $$( -6 + T )^{4}$$
$43$ $$( 72 + T^{2} )^{2}$$
$47$ $$( 72 + T^{2} )^{2}$$
$53$ $$324 + 54 T^{2} + T^{4}$$
$59$ $$( 16 + 12 T + T^{2} )^{2}$$
$61$ $$( -20 + 10 T + T^{2} )^{2}$$
$67$ $$324 + 126 T^{2} + T^{4}$$
$71$ $$( 16 - 12 T + T^{2} )^{2}$$
$73$ $$5184 + 216 T^{2} + T^{4}$$
$79$ $$( -176 + 4 T + T^{2} )^{2}$$
$83$ $$5184 + 216 T^{2} + T^{4}$$
$89$ $$( -44 + 12 T + T^{2} )^{2}$$
$97$ $$3844 + 214 T^{2} + T^{4}$$