# Properties

 Label 5445.2.a.bf Level $5445$ Weight $2$ Character orbit 5445.a Self dual yes Analytic conductor $43.479$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -\beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( 2 - \beta_{2} ) q^{7} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -\beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( 2 - \beta_{2} ) q^{7} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{8} + ( 1 - \beta_{1} ) q^{10} + ( 1 + \beta_{1} + \beta_{3} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{14} + ( 1 + \beta_{1} - 3 \beta_{3} ) q^{16} + ( -1 - 2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 2 - \beta_{1} + \beta_{2} ) q^{19} + ( \beta_{1} - \beta_{2} ) q^{20} + ( -2 + \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{23} + q^{25} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{26} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{28} + ( -3 + 5 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{29} + ( -4 - 5 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{31} + ( 2 \beta_{2} + \beta_{3} ) q^{32} + ( 1 - 3 \beta_{1} - 2 \beta_{3} ) q^{34} + ( -2 + \beta_{2} ) q^{35} + ( -4 + 3 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{37} + ( -4 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{38} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{40} + ( -8 + \beta_{2} + 4 \beta_{3} ) q^{41} + ( 4 - 5 \beta_{2} - 3 \beta_{3} ) q^{43} + ( -1 + 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{46} + ( 2 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{47} + ( -1 - 3 \beta_{2} - \beta_{3} ) q^{49} + ( -1 + \beta_{1} ) q^{50} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( -1 + 3 \beta_{2} - 6 \beta_{3} ) q^{53} + ( 3 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{56} + ( 9 - 7 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{58} + ( 4 - 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 3 + 7 \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{61} + ( \beta_{2} - 6 \beta_{3} ) q^{62} + ( -4 + \beta_{1} - \beta_{2} + 7 \beta_{3} ) q^{64} + ( -1 - \beta_{1} - \beta_{3} ) q^{65} + ( 3 + 3 \beta_{1} - 7 \beta_{3} ) q^{67} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{68} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{70} + ( 4 - 3 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{71} + ( 1 + 5 \beta_{1} - 4 \beta_{3} ) q^{73} + ( 1 + \beta_{1} - 4 \beta_{2} + 7 \beta_{3} ) q^{74} + ( 5 - 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{76} + ( -1 - 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{79} + ( -1 - \beta_{1} + 3 \beta_{3} ) q^{80} + ( 7 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{82} + ( \beta_{1} + 3 \beta_{2} - 5 \beta_{3} ) q^{83} + ( 1 + 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( 1 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{86} + ( 5 + 4 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{89} + ( 3 - \beta_{2} + \beta_{3} ) q^{91} + ( 10 - 2 \beta_{1} - 7 \beta_{3} ) q^{92} + ( -7 + \beta_{1} - 7 \beta_{2} + 3 \beta_{3} ) q^{94} + ( -2 + \beta_{1} - \beta_{2} ) q^{95} + ( -7 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{97} + ( 4 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3q^{2} + q^{4} - 4q^{5} + 6q^{7} - 3q^{8} + O(q^{10})$$ $$4q - 3q^{2} + q^{4} - 4q^{5} + 6q^{7} - 3q^{8} + 3q^{10} + 7q^{13} - 3q^{14} - q^{16} - 10q^{17} + 9q^{19} - q^{20} + 3q^{23} + 4q^{25} + 4q^{26} - 7q^{28} - 15q^{29} - 13q^{31} + 6q^{32} - 3q^{34} - 6q^{35} - 3q^{37} - 15q^{38} + 3q^{40} - 22q^{41} + q^{46} + 2q^{47} - 12q^{49} - 3q^{50} - 9q^{52} - 10q^{53} + 8q^{56} + 39q^{58} + 21q^{59} + 11q^{61} - 10q^{62} - 3q^{64} - 7q^{65} + q^{67} - 3q^{68} + 3q^{70} + 13q^{71} + q^{73} + 11q^{74} + 19q^{76} - 4q^{79} + q^{80} + 25q^{82} - 3q^{83} + 10q^{85} + 10q^{89} + 12q^{91} + 24q^{92} - 35q^{94} - 9q^{95} - 22q^{97} + 11q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 3 x^{2} + x + 1$$:

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.35567 −0.477260 0.737640 2.09529
−2.35567 0 3.54920 −1.00000 0 −0.193527 −3.64941 0 2.35567
1.2 −1.47726 0 0.182297 −1.00000 0 2.29496 2.68522 0 1.47726
1.3 −0.262360 0 −1.93117 −1.00000 0 3.19353 1.03138 0 0.262360
1.4 1.09529 0 −0.800331 −1.00000 0 0.705037 −3.06719 0 −1.09529
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 5445.2.a.bf 4
3.b odd 2 1 1815.2.a.w 4
11.b odd 2 1 5445.2.a.bt 4
11.d odd 10 2 495.2.n.a 8
15.d odd 2 1 9075.2.a.cm 4
33.d even 2 1 1815.2.a.p 4
33.f even 10 2 165.2.m.d 8
165.d even 2 1 9075.2.a.di 4
165.r even 10 2 825.2.n.g 8
165.u odd 20 4 825.2.bx.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.d 8 33.f even 10 2
495.2.n.a 8 11.d odd 10 2
825.2.n.g 8 165.r even 10 2
825.2.bx.f 16 165.u odd 20 4
1815.2.a.p 4 33.d even 2 1
1815.2.a.w 4 3.b odd 2 1
5445.2.a.bf 4 1.a even 1 1 trivial
5445.2.a.bt 4 11.b odd 2 1
9075.2.a.cm 4 15.d odd 2 1
9075.2.a.di 4 165.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5445))$$:

 $$T_{2}^{4} + 3 T_{2}^{3} - 4 T_{2} - 1$$ $$T_{7}^{4} - 6 T_{7}^{3} + 10 T_{7}^{2} - 3 T_{7} - 1$$ $$T_{23}^{4} - 3 T_{23}^{3} - 55 T_{23}^{2} + 129 T_{23} + 449$$ $$T_{53}^{4} + 10 T_{53}^{3} - 84 T_{53}^{2} - 815 T_{53} - 641$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 4 T + 3 T^{3} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$-1 - 3 T + 10 T^{2} - 6 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$-11 + 6 T + 10 T^{2} - 7 T^{3} + T^{4}$$
$17$ $$-1 - 10 T + 16 T^{2} + 10 T^{3} + T^{4}$$
$19$ $$1 - 16 T + 22 T^{2} - 9 T^{3} + T^{4}$$
$23$ $$449 + 129 T - 55 T^{2} - 3 T^{3} + T^{4}$$
$29$ $$499 - 410 T + 4 T^{2} + 15 T^{3} + T^{4}$$
$31$ $$-1249 - 607 T - 17 T^{2} + 13 T^{3} + T^{4}$$
$37$ $$3541 - 213 T - 121 T^{2} + 3 T^{3} + T^{4}$$
$41$ $$41 + 207 T + 138 T^{2} + 22 T^{3} + T^{4}$$
$43$ $$275 - 375 T - 110 T^{2} + T^{4}$$
$47$ $$31 - 47 T - 92 T^{2} - 2 T^{3} + T^{4}$$
$53$ $$-641 - 815 T - 84 T^{2} + 10 T^{3} + T^{4}$$
$59$ $$-589 - 29 T + 117 T^{2} - 21 T^{3} + T^{4}$$
$61$ $$1151 + 866 T - 88 T^{2} - 11 T^{3} + T^{4}$$
$67$ $$619 + 112 T - 100 T^{2} - T^{3} + T^{4}$$
$71$ $$281 + 387 T - 57 T^{2} - 13 T^{3} + T^{4}$$
$73$ $$931 + 84 T - 74 T^{2} - T^{3} + T^{4}$$
$79$ $$-169 - 286 T - 89 T^{2} + 4 T^{3} + T^{4}$$
$83$ $$41 - 147 T - 77 T^{2} + 3 T^{3} + T^{4}$$
$89$ $$209 + 310 T - 89 T^{2} - 10 T^{3} + T^{4}$$
$97$ $$-1159 - 223 T + 88 T^{2} + 22 T^{3} + T^{4}$$