# Properties

 Label 3549.2.a.e Level $3549$ Weight $2$ Character orbit 3549.a Self dual yes Analytic conductor $28.339$ Analytic rank $0$ Dimension $1$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} + q^{3} + 2q^{4} + 3q^{5} + 2q^{6} + q^{7} + q^{9} + O(q^{10})$$ $$q + 2q^{2} + q^{3} + 2q^{4} + 3q^{5} + 2q^{6} + q^{7} + q^{9} + 6q^{10} + 2q^{12} + 2q^{14} + 3q^{15} - 4q^{16} + 2q^{17} + 2q^{18} + q^{19} + 6q^{20} + q^{21} + q^{23} + 4q^{25} + q^{27} + 2q^{28} + 5q^{29} + 6q^{30} + 5q^{31} - 8q^{32} + 4q^{34} + 3q^{35} + 2q^{36} + 8q^{37} + 2q^{38} - 10q^{41} + 2q^{42} - 9q^{43} + 3q^{45} + 2q^{46} - 7q^{47} - 4q^{48} + q^{49} + 8q^{50} + 2q^{51} + 9q^{53} + 2q^{54} + q^{57} + 10q^{58} + 4q^{59} + 6q^{60} - 8q^{61} + 10q^{62} + q^{63} - 8q^{64} + 2q^{67} + 4q^{68} + q^{69} + 6q^{70} - 9q^{73} + 16q^{74} + 4q^{75} + 2q^{76} + 15q^{79} - 12q^{80} + q^{81} - 20q^{82} + 9q^{83} + 2q^{84} + 6q^{85} - 18q^{86} + 5q^{87} + 9q^{89} + 6q^{90} + 2q^{92} + 5q^{93} - 14q^{94} + 3q^{95} - 8q^{96} - 13q^{97} + 2q^{98} + O(q^{100})$$

## 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
 0
2.00000 1.00000 2.00000 3.00000 2.00000 1.00000 0 1.00000 6.00000
 $$n$$: 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.e 1
13.b even 2 1 3549.2.a.a 1
13.d odd 4 2 273.2.c.a 2
39.f even 4 2 819.2.c.a 2
52.f even 4 2 4368.2.h.e 2
91.i even 4 2 1911.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.a 2 13.d odd 4 2
819.2.c.a 2 39.f even 4 2
1911.2.c.a 2 91.i even 4 2
3549.2.a.a 1 13.b even 2 1
3549.2.a.e 1 1.a even 1 1 trivial
4368.2.h.e 2 52.f even 4 2

## Hecke kernels

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

 $$T_{2} - 2$$ $$T_{5} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$-1 + T$$
$5$ $$-3 + T$$
$7$ $$-1 + T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$-2 + T$$
$19$ $$-1 + T$$
$23$ $$-1 + T$$
$29$ $$-5 + T$$
$31$ $$-5 + T$$
$37$ $$-8 + T$$
$41$ $$10 + T$$
$43$ $$9 + T$$
$47$ $$7 + T$$
$53$ $$-9 + T$$
$59$ $$-4 + T$$
$61$ $$8 + T$$
$67$ $$-2 + T$$
$71$ $$T$$
$73$ $$9 + T$$
$79$ $$-15 + T$$
$83$ $$-9 + T$$
$89$ $$-9 + T$$
$97$ $$13 + T$$