# Properties

 Label 3440.2.a.j Level 3440 Weight 2 Character orbit 3440.a Self dual yes Analytic conductor 27.469 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3440 = 2^{4} \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3440.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.4685382953$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 430) 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.

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

## 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 −1.41421 0 1.00000 0 −1.00000 0 −1.00000 0
1.2 0 1.41421 0 1.00000 0 −1.00000 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 3440.2.a.j 2
4.b odd 2 1 430.2.a.g 2
12.b even 2 1 3870.2.a.bc 2
20.d odd 2 1 2150.2.a.v 2
20.e even 4 2 2150.2.b.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.a.g 2 4.b odd 2 1
2150.2.a.v 2 20.d odd 2 1
2150.2.b.o 4 20.e even 4 2
3440.2.a.j 2 1.a even 1 1 trivial
3870.2.a.bc 2 12.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$43$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3440))$$:

 $$T_{3}^{2} - 2$$ $$T_{7} + 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 4 T^{2} + 9 T^{4}$$
$5$ $$( 1 - T )^{2}$$
$7$ $$( 1 + T + 7 T^{2} )^{2}$$
$11$ $$1 + 4 T + 24 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 - 2 T + 19 T^{2} - 26 T^{3} + 169 T^{4}$$
$17$ $$1 + 32 T^{2} + 289 T^{4}$$
$19$ $$( 1 - T + 19 T^{2} )^{2}$$
$23$ $$1 + 4 T + 92 T^{3} + 529 T^{4}$$
$29$ $$1 + 6 T + 49 T^{2} + 174 T^{3} + 841 T^{4}$$
$31$ $$1 + 2 T + 61 T^{2} + 62 T^{3} + 961 T^{4}$$
$37$ $$1 + 4 T + 60 T^{2} + 148 T^{3} + 1369 T^{4}$$
$41$ $$1 - 2 T + 75 T^{2} - 82 T^{3} + 1681 T^{4}$$
$43$ $$( 1 + T )^{2}$$
$47$ $$1 + 44 T^{2} + 2209 T^{4}$$
$53$ $$1 + 74 T^{2} + 2809 T^{4}$$
$59$ $$1 - 8 T + 126 T^{2} - 472 T^{3} + 3481 T^{4}$$
$61$ $$1 + 14 T + 121 T^{2} + 854 T^{3} + 3721 T^{4}$$
$67$ $$1 + 6 T + 125 T^{2} + 402 T^{3} + 4489 T^{4}$$
$71$ $$1 + 4 T + 48 T^{2} + 284 T^{3} + 5041 T^{4}$$
$73$ $$1 + 10 T + 153 T^{2} + 730 T^{3} + 5329 T^{4}$$
$79$ $$1 - 14 T + 205 T^{2} - 1106 T^{3} + 6241 T^{4}$$
$83$ $$1 - 4 T + 138 T^{2} - 332 T^{3} + 6889 T^{4}$$
$89$ $$1 + 4 T + 132 T^{2} + 356 T^{3} + 7921 T^{4}$$
$97$ $$1 + 4 T - 44 T^{2} + 388 T^{3} + 9409 T^{4}$$