# Properties

 Label 3549.2.a.f Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3549 = 3 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3549.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.3389076774$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} - q^{3} + ( 1 - 2 \beta ) q^{4} + ( 1 - \beta ) q^{6} - q^{7} + ( -3 + \beta ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} - q^{3} + ( 1 - 2 \beta ) q^{4} + ( 1 - \beta ) q^{6} - q^{7} + ( -3 + \beta ) q^{8} + q^{9} -2 q^{11} + ( -1 + 2 \beta ) q^{12} + ( 1 - \beta ) q^{14} + 3 q^{16} + ( 2 + 2 \beta ) q^{17} + ( -1 + \beta ) q^{18} -4 \beta q^{19} + q^{21} + ( 2 - 2 \beta ) q^{22} + ( 4 + 2 \beta ) q^{23} + ( 3 - \beta ) q^{24} -5 q^{25} - q^{27} + ( -1 + 2 \beta ) q^{28} + ( 2 + 4 \beta ) q^{29} + ( 4 + 4 \beta ) q^{31} + ( 3 + \beta ) q^{32} + 2 q^{33} + 2 q^{34} + ( 1 - 2 \beta ) q^{36} + ( 2 - 4 \beta ) q^{37} + ( -8 + 4 \beta ) q^{38} + 4 \beta q^{41} + ( -1 + \beta ) q^{42} + ( 4 - 4 \beta ) q^{43} + ( -2 + 4 \beta ) q^{44} + 2 \beta q^{46} + ( -6 - 2 \beta ) q^{47} -3 q^{48} + q^{49} + ( 5 - 5 \beta ) q^{50} + ( -2 - 2 \beta ) q^{51} + ( -2 - 4 \beta ) q^{53} + ( 1 - \beta ) q^{54} + ( 3 - \beta ) q^{56} + 4 \beta q^{57} + ( 6 - 2 \beta ) q^{58} + ( 2 - 6 \beta ) q^{59} + ( -6 - 4 \beta ) q^{61} + 4 q^{62} - q^{63} + ( -7 + 2 \beta ) q^{64} + ( -2 + 2 \beta ) q^{66} -4 q^{67} + ( -6 - 2 \beta ) q^{68} + ( -4 - 2 \beta ) q^{69} -14 q^{71} + ( -3 + \beta ) q^{72} + ( 2 - 4 \beta ) q^{73} + ( -10 + 6 \beta ) q^{74} + 5 q^{75} + ( 16 - 4 \beta ) q^{76} + 2 q^{77} + 8 \beta q^{79} + q^{81} + ( 8 - 4 \beta ) q^{82} + ( -2 + 10 \beta ) q^{83} + ( 1 - 2 \beta ) q^{84} + ( -12 + 8 \beta ) q^{86} + ( -2 - 4 \beta ) q^{87} + ( 6 - 2 \beta ) q^{88} + ( -4 + 8 \beta ) q^{89} + ( -4 - 6 \beta ) q^{92} + ( -4 - 4 \beta ) q^{93} + ( 2 - 4 \beta ) q^{94} + ( -3 - \beta ) q^{96} + 2 q^{97} + ( -1 + \beta ) q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} - 6q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} - 6q^{8} + 2q^{9} - 4q^{11} - 2q^{12} + 2q^{14} + 6q^{16} + 4q^{17} - 2q^{18} + 2q^{21} + 4q^{22} + 8q^{23} + 6q^{24} - 10q^{25} - 2q^{27} - 2q^{28} + 4q^{29} + 8q^{31} + 6q^{32} + 4q^{33} + 4q^{34} + 2q^{36} + 4q^{37} - 16q^{38} - 2q^{42} + 8q^{43} - 4q^{44} - 12q^{47} - 6q^{48} + 2q^{49} + 10q^{50} - 4q^{51} - 4q^{53} + 2q^{54} + 6q^{56} + 12q^{58} + 4q^{59} - 12q^{61} + 8q^{62} - 2q^{63} - 14q^{64} - 4q^{66} - 8q^{67} - 12q^{68} - 8q^{69} - 28q^{71} - 6q^{72} + 4q^{73} - 20q^{74} + 10q^{75} + 32q^{76} + 4q^{77} + 2q^{81} + 16q^{82} - 4q^{83} + 2q^{84} - 24q^{86} - 4q^{87} + 12q^{88} - 8q^{89} - 8q^{92} - 8q^{93} + 4q^{94} - 6q^{96} + 4q^{97} - 2q^{98} - 4q^{99} + O(q^{100})$$

## 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
−2.41421 −1.00000 3.82843 0 2.41421 −1.00000 −4.41421 1.00000 0
1.2 0.414214 −1.00000 −1.82843 0 −0.414214 −1.00000 −1.58579 1.00000 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
$$3$$ $$1$$
$$7$$ $$1$$
$$13$$ $$1$$

## 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 3549.2.a.f 2
13.b even 2 1 273.2.a.c 2
39.d odd 2 1 819.2.a.g 2
52.b odd 2 1 4368.2.a.bj 2
65.d even 2 1 6825.2.a.m 2
91.b odd 2 1 1911.2.a.k 2
273.g even 2 1 5733.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.c 2 13.b even 2 1
819.2.a.g 2 39.d odd 2 1
1911.2.a.k 2 91.b odd 2 1
3549.2.a.f 2 1.a even 1 1 trivial
4368.2.a.bj 2 52.b odd 2 1
5733.2.a.n 2 273.g even 2 1
6825.2.a.m 2 65.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3549))$$:

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{5}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$-4 - 4 T + T^{2}$$
$19$ $$-32 + T^{2}$$
$23$ $$8 - 8 T + T^{2}$$
$29$ $$-28 - 4 T + T^{2}$$
$31$ $$-16 - 8 T + T^{2}$$
$37$ $$-28 - 4 T + T^{2}$$
$41$ $$-32 + T^{2}$$
$43$ $$-16 - 8 T + T^{2}$$
$47$ $$28 + 12 T + T^{2}$$
$53$ $$-28 + 4 T + T^{2}$$
$59$ $$-68 - 4 T + T^{2}$$
$61$ $$4 + 12 T + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$( 14 + T )^{2}$$
$73$ $$-28 - 4 T + T^{2}$$
$79$ $$-128 + T^{2}$$
$83$ $$-196 + 4 T + T^{2}$$
$89$ $$-112 + 8 T + T^{2}$$
$97$ $$( -2 + T )^{2}$$