# Properties

 Label 116.2.c.a Level 116 Weight 2 Character orbit 116.c Analytic conductor 0.926 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$116 = 2^{2} \cdot 29$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 116.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.926264663447$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + q^{5} -2 q^{7} -4 q^{9} +O(q^{10})$$ $$q -\beta q^{3} + q^{5} -2 q^{7} -4 q^{9} -\beta q^{11} + 5 q^{13} -\beta q^{15} + 2 \beta q^{17} + 2 \beta q^{19} + 2 \beta q^{21} + 6 q^{23} -4 q^{25} + \beta q^{27} + ( -1 - 2 \beta ) q^{29} + \beta q^{31} -7 q^{33} -2 q^{35} + 4 \beta q^{37} -5 \beta q^{39} -2 \beta q^{41} + 3 \beta q^{43} -4 q^{45} -3 \beta q^{47} -3 q^{49} + 14 q^{51} + 5 q^{53} -\beta q^{55} + 14 q^{57} -14 q^{59} -2 \beta q^{61} + 8 q^{63} + 5 q^{65} -4 q^{67} -6 \beta q^{69} -8 q^{71} -4 \beta q^{73} + 4 \beta q^{75} + 2 \beta q^{77} + \beta q^{79} -5 q^{81} + 2 q^{83} + 2 \beta q^{85} + ( -14 + \beta ) q^{87} -2 \beta q^{89} -10 q^{91} + 7 q^{93} + 2 \beta q^{95} + 2 \beta q^{97} + 4 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 4q^{7} - 8q^{9} + O(q^{10})$$ $$2q + 2q^{5} - 4q^{7} - 8q^{9} + 10q^{13} + 12q^{23} - 8q^{25} - 2q^{29} - 14q^{33} - 4q^{35} - 8q^{45} - 6q^{49} + 28q^{51} + 10q^{53} + 28q^{57} - 28q^{59} + 16q^{63} + 10q^{65} - 8q^{67} - 16q^{71} - 10q^{81} + 4q^{83} - 28q^{87} - 20q^{91} + 14q^{93} + O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/116\mathbb{Z}\right)^\times$$.

 $$n$$ $$59$$ $$89$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 0.5 + 1.32288i 0.5 − 1.32288i
0 2.64575i 0 1.00000 0 −2.00000 0 −4.00000 0
57.2 0 2.64575i 0 1.00000 0 −2.00000 0 −4.00000 0
 $$n$$: 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
29.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(116, [\chi])$$.