# Properties

 Label 5550.2.a.cj Level $5550$ Weight $2$ Character orbit 5550.a Self dual yes Analytic conductor $44.317$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.3169731218$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.54764.1 Defining polynomial: $$x^{4} - x^{3} - 9 x^{2} + 3 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1110) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 - \beta_{2} ) q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 - \beta_{2} ) q^{7} - q^{8} + q^{9} + ( 1 - \beta_{1} ) q^{11} - q^{12} + ( 1 - \beta_{1} + \beta_{3} ) q^{13} + ( 1 + \beta_{2} ) q^{14} + q^{16} + ( -1 + \beta_{2} - 2 \beta_{3} ) q^{17} - q^{18} + ( 1 - \beta_{2} - \beta_{3} ) q^{19} + ( 1 + \beta_{2} ) q^{21} + ( -1 + \beta_{1} ) q^{22} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{23} + q^{24} + ( -1 + \beta_{1} - \beta_{3} ) q^{26} - q^{27} + ( -1 - \beta_{2} ) q^{28} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{29} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} - q^{32} + ( -1 + \beta_{1} ) q^{33} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{34} + q^{36} + q^{37} + ( -1 + \beta_{2} + \beta_{3} ) q^{38} + ( -1 + \beta_{1} - \beta_{3} ) q^{39} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{41} + ( -1 - \beta_{2} ) q^{42} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( 1 - \beta_{1} ) q^{44} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{46} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{47} - q^{48} + ( 4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{49} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{51} + ( 1 - \beta_{1} + \beta_{3} ) q^{52} + ( 5 - \beta_{2} + 2 \beta_{3} ) q^{53} + q^{54} + ( 1 + \beta_{2} ) q^{56} + ( -1 + \beta_{2} + \beta_{3} ) q^{57} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{58} + ( -2 - 2 \beta_{2} ) q^{59} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{61} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{62} + ( -1 - \beta_{2} ) q^{63} + q^{64} + ( 1 - \beta_{1} ) q^{66} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{67} + ( -1 + \beta_{2} - 2 \beta_{3} ) q^{68} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{69} + ( -6 + 2 \beta_{1} + 2 \beta_{3} ) q^{71} - q^{72} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{73} - q^{74} + ( 1 - \beta_{2} - \beta_{3} ) q^{76} + ( 1 - 3 \beta_{2} ) q^{77} + ( 1 - \beta_{1} + \beta_{3} ) q^{78} + 4 q^{79} + q^{81} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{82} + ( -1 + 3 \beta_{1} - \beta_{3} ) q^{83} + ( 1 + \beta_{2} ) q^{84} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{86} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{87} + ( -1 + \beta_{1} ) q^{88} + ( -3 + 3 \beta_{1} + \beta_{3} ) q^{89} + ( -3 + \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{91} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{92} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{94} + q^{96} + ( -2 + 2 \beta_{1} - 3 \beta_{3} ) q^{97} + ( -4 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{98} + ( 1 - \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{7} - 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 4q^{3} + 4q^{4} + 4q^{6} - 4q^{7} - 4q^{8} + 4q^{9} + 2q^{11} - 4q^{12} + 3q^{13} + 4q^{14} + 4q^{16} - 6q^{17} - 4q^{18} + 3q^{19} + 4q^{21} - 2q^{22} + q^{23} + 4q^{24} - 3q^{26} - 4q^{27} - 4q^{28} - 3q^{29} + 3q^{31} - 4q^{32} - 2q^{33} + 6q^{34} + 4q^{36} + 4q^{37} - 3q^{38} - 3q^{39} - 11q^{41} - 4q^{42} + 5q^{43} + 2q^{44} - q^{46} - 4q^{48} + 14q^{49} + 6q^{51} + 3q^{52} + 22q^{53} + 4q^{54} + 4q^{56} - 3q^{57} + 3q^{58} - 8q^{59} - 5q^{61} - 3q^{62} - 4q^{63} + 4q^{64} + 2q^{66} - 6q^{67} - 6q^{68} - q^{69} - 18q^{71} - 4q^{72} + 5q^{73} - 4q^{74} + 3q^{76} + 4q^{77} + 3q^{78} + 16q^{79} + 4q^{81} + 11q^{82} + q^{83} + 4q^{84} - 5q^{86} + 3q^{87} - 2q^{88} - 5q^{89} - 13q^{91} + q^{92} - 3q^{93} + 4q^{96} - 7q^{97} - 14q^{98} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 9 x^{2} + 3 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 7 \nu - 2$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 11 \nu - 2$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} + 7 \nu - 6$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{2} + 11 \beta_{1} + 4$$$$)/2$$

## 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
 −2.67673 −0.339102 3.36007 0.655762
−1.00000 −1.00000 1.00000 0 1.00000 −5.13277 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 −1.84556 −1.00000 1.00000 0
1.3 −1.00000 −1.00000 1.00000 0 1.00000 −0.487359 −1.00000 1.00000 0
1.4 −1.00000 −1.00000 1.00000 0 1.00000 3.46569 −1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$37$$ $$-1$$

## 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 5550.2.a.cj 4
5.b even 2 1 1110.2.a.s 4
15.d odd 2 1 3330.2.a.bj 4
20.d odd 2 1 8880.2.a.cg 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.s 4 5.b even 2 1
3330.2.a.bj 4 15.d odd 2 1
5550.2.a.cj 4 1.a even 1 1 trivial
8880.2.a.cg 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5550))$$:

 $$T_{7}^{4} + 4 T_{7}^{3} - 13 T_{7}^{2} - 40 T_{7} - 16$$ $$T_{11}^{4} - 2 T_{11}^{3} - 23 T_{11}^{2} + 68 T_{11} - 40$$ $$T_{13}^{4} - 3 T_{13}^{3} - 38 T_{13}^{2} + 44 T_{13} + 328$$ $$T_{17}^{4} + 6 T_{17}^{3} - 47 T_{17}^{2} - 412 T_{17} - 764$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$T^{4}$$
$7$ $$-16 - 40 T - 13 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$-40 + 68 T - 23 T^{2} - 2 T^{3} + T^{4}$$
$13$ $$328 + 44 T - 38 T^{2} - 3 T^{3} + T^{4}$$
$17$ $$-764 - 412 T - 47 T^{2} + 6 T^{3} + T^{4}$$
$19$ $$-80 + 208 T - 50 T^{2} - 3 T^{3} + T^{4}$$
$23$ $$1280 + 64 T - 76 T^{2} - T^{3} + T^{4}$$
$29$ $$1208 - 204 T - 82 T^{2} + 3 T^{3} + T^{4}$$
$31$ $$32 - 38 T^{2} - 3 T^{3} + T^{4}$$
$37$ $$( -1 + T )^{4}$$
$41$ $$-80 - 84 T + 4 T^{2} + 11 T^{3} + T^{4}$$
$43$ $$1280 + 128 T - 76 T^{2} - 5 T^{3} + T^{4}$$
$47$ $$1280 + 96 T - 88 T^{2} + T^{4}$$
$53$ $$-2524 + 244 T + 121 T^{2} - 22 T^{3} + T^{4}$$
$59$ $$-256 - 320 T - 52 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$1096 + 196 T - 130 T^{2} + 5 T^{3} + T^{4}$$
$67$ $$128 - 32 T - 72 T^{2} + 6 T^{3} + T^{4}$$
$71$ $$512 - 1536 T - 56 T^{2} + 18 T^{3} + T^{4}$$
$73$ $$-2336 + 1132 T - 128 T^{2} - 5 T^{3} + T^{4}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$5344 - 704 T - 234 T^{2} - T^{3} + T^{4}$$
$89$ $$-3208 - 1956 T - 234 T^{2} + 5 T^{3} + T^{4}$$
$97$ $$9344 - 340 T - 236 T^{2} + 7 T^{3} + T^{4}$$