# Properties

 Label 55.2.e.a Level 55 Weight 2 Character orbit 55.e Analytic conductor 0.439 Analytic rank 0 Dimension 4 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$55 = 5 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 55.e (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.439177211117$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{10})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{3} + 3 \beta_{2} q^{4} + ( -2 - \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{6} + \beta_{3} q^{8} -\beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 - \beta_{2} ) q^{3} + 3 \beta_{2} q^{4} + ( -2 - \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{3} ) q^{6} + \beta_{3} q^{8} -\beta_{2} q^{9} + ( -2 \beta_{1} - \beta_{3} ) q^{10} + ( 1 + \beta_{1} + \beta_{3} ) q^{11} + ( 3 - 3 \beta_{2} ) q^{12} + 2 \beta_{3} q^{13} + ( 1 + 3 \beta_{2} ) q^{15} + q^{16} -2 \beta_{1} q^{17} -\beta_{3} q^{18} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{19} + ( 3 - 6 \beta_{2} ) q^{20} + ( -5 + \beta_{1} + 5 \beta_{2} ) q^{22} + ( -1 - \beta_{2} ) q^{23} + ( \beta_{1} - \beta_{3} ) q^{24} + ( 3 + 4 \beta_{2} ) q^{25} -10 q^{26} + ( -4 + 4 \beta_{2} ) q^{27} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{29} + ( \beta_{1} + 3 \beta_{3} ) q^{30} + 2 q^{31} + 3 \beta_{1} q^{32} + ( -1 - \beta_{2} - 2 \beta_{3} ) q^{33} -10 \beta_{2} q^{34} + 3 q^{36} + ( 3 - 3 \beta_{2} ) q^{37} + ( 10 + 10 \beta_{2} ) q^{38} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{39} + ( \beta_{1} - 2 \beta_{3} ) q^{40} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{41} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{44} + ( -1 + 2 \beta_{2} ) q^{45} + ( -\beta_{1} - \beta_{3} ) q^{46} + ( 3 - 3 \beta_{2} ) q^{47} + ( -1 - \beta_{2} ) q^{48} -7 \beta_{2} q^{49} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{50} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{51} -6 \beta_{1} q^{52} + ( -1 - \beta_{2} ) q^{53} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{54} + ( -2 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{55} -4 \beta_{1} q^{57} + ( -10 - 10 \beta_{2} ) q^{58} + 6 \beta_{2} q^{59} + ( -9 + 3 \beta_{2} ) q^{60} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{61} + 2 \beta_{1} q^{62} + 13 \beta_{2} q^{64} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{65} + ( 10 - \beta_{1} - \beta_{3} ) q^{66} + ( 3 - 3 \beta_{2} ) q^{67} -6 \beta_{3} q^{68} + 2 \beta_{2} q^{69} -8 q^{71} + \beta_{1} q^{72} -2 \beta_{3} q^{73} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{74} + ( 1 - 7 \beta_{2} ) q^{75} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{76} + ( 10 + 10 \beta_{2} ) q^{78} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -2 - \beta_{2} ) q^{80} + 5 q^{81} + ( 10 - 10 \beta_{2} ) q^{82} + 4 \beta_{3} q^{83} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{85} + 4 \beta_{1} q^{87} + ( -5 - 5 \beta_{2} + \beta_{3} ) q^{88} + 6 \beta_{2} q^{89} + ( -\beta_{1} + 2 \beta_{3} ) q^{90} + ( 3 - 3 \beta_{2} ) q^{92} + ( -2 - 2 \beta_{2} ) q^{93} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{94} + ( -6 \beta_{1} + 2 \beta_{3} ) q^{95} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{96} + ( -7 + 7 \beta_{2} ) q^{97} -7 \beta_{3} q^{98} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 8q^{5} + O(q^{10})$$ $$4q - 4q^{3} - 8q^{5} + 4q^{11} + 12q^{12} + 4q^{15} + 4q^{16} + 12q^{20} - 20q^{22} - 4q^{23} + 12q^{25} - 40q^{26} - 16q^{27} + 8q^{31} - 4q^{33} + 12q^{36} + 12q^{37} + 40q^{38} - 4q^{45} + 12q^{47} - 4q^{48} - 4q^{53} - 8q^{55} - 40q^{58} - 36q^{60} + 40q^{66} + 12q^{67} - 32q^{71} + 4q^{75} + 40q^{78} - 8q^{80} + 20q^{81} + 40q^{82} - 20q^{88} + 12q^{92} - 8q^{93} - 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −1.58114 + 1.58114i 1.58114 − 1.58114i −1.58114 − 1.58114i 1.58114 + 1.58114i
−1.58114 + 1.58114i −1.00000 + 1.00000i 3.00000i −2.00000 + 1.00000i 3.16228i 0 1.58114 + 1.58114i 1.00000i 1.58114 4.74342i
32.2 1.58114 1.58114i −1.00000 + 1.00000i 3.00000i −2.00000 + 1.00000i 3.16228i 0 −1.58114 1.58114i 1.00000i −1.58114 + 4.74342i
43.1 −1.58114 1.58114i −1.00000 1.00000i 3.00000i −2.00000 1.00000i 3.16228i 0 1.58114 1.58114i 1.00000i 1.58114 + 4.74342i
43.2 1.58114 + 1.58114i −1.00000 1.00000i 3.00000i −2.00000 1.00000i 3.16228i 0 −1.58114 + 1.58114i 1.00000i −1.58114 4.74342i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.c Odd 1 yes
11.b Odd 1 yes
55.e Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{4} + 25$$ acting on $$S_{2}^{\mathrm{new}}(55, [\chi])$$.