# Properties

 Label 114.2.e.b Level $114$ Weight $2$ Character orbit 114.e Analytic conductor $0.910$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.910294583043$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} + q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{2} + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -\zeta_{6} q^{6} + q^{7} - q^{8} -\zeta_{6} q^{9} -2 q^{11} - q^{12} + 3 \zeta_{6} q^{13} + ( 1 - \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -4 + 4 \zeta_{6} ) q^{17} - q^{18} + ( 3 + 2 \zeta_{6} ) q^{19} + ( 1 - \zeta_{6} ) q^{21} + ( -2 + 2 \zeta_{6} ) q^{22} -4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{24} + 5 \zeta_{6} q^{25} + 3 q^{26} - q^{27} -\zeta_{6} q^{28} -3 q^{31} + \zeta_{6} q^{32} + ( -2 + 2 \zeta_{6} ) q^{33} + 4 \zeta_{6} q^{34} + ( -1 + \zeta_{6} ) q^{36} -5 q^{37} + ( 5 - 3 \zeta_{6} ) q^{38} + 3 q^{39} + ( -4 + 4 \zeta_{6} ) q^{41} -\zeta_{6} q^{42} + ( 9 - 9 \zeta_{6} ) q^{43} + 2 \zeta_{6} q^{44} -4 q^{46} -10 \zeta_{6} q^{47} + \zeta_{6} q^{48} -6 q^{49} + 5 q^{50} + 4 \zeta_{6} q^{51} + ( 3 - 3 \zeta_{6} ) q^{52} + 4 \zeta_{6} q^{53} + ( -1 + \zeta_{6} ) q^{54} - q^{56} + ( 5 - 3 \zeta_{6} ) q^{57} + ( 14 - 14 \zeta_{6} ) q^{59} -11 \zeta_{6} q^{61} + ( -3 + 3 \zeta_{6} ) q^{62} -\zeta_{6} q^{63} + q^{64} + 2 \zeta_{6} q^{66} -3 \zeta_{6} q^{67} + 4 q^{68} -4 q^{69} + ( -14 + 14 \zeta_{6} ) q^{71} + \zeta_{6} q^{72} + ( 11 - 11 \zeta_{6} ) q^{73} + ( -5 + 5 \zeta_{6} ) q^{74} + 5 q^{75} + ( 2 - 5 \zeta_{6} ) q^{76} -2 q^{77} + ( 3 - 3 \zeta_{6} ) q^{78} + ( -1 + \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 4 \zeta_{6} q^{82} + 8 q^{83} - q^{84} -9 \zeta_{6} q^{86} + 2 q^{88} + 14 \zeta_{6} q^{89} + 3 \zeta_{6} q^{91} + ( -4 + 4 \zeta_{6} ) q^{92} + ( -3 + 3 \zeta_{6} ) q^{93} -10 q^{94} + q^{96} + ( -2 + 2 \zeta_{6} ) q^{97} + ( -6 + 6 \zeta_{6} ) q^{98} + 2 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + q^{3} - q^{4} - q^{6} + 2q^{7} - 2q^{8} - q^{9} + O(q^{10})$$ $$2q + q^{2} + q^{3} - q^{4} - q^{6} + 2q^{7} - 2q^{8} - q^{9} - 4q^{11} - 2q^{12} + 3q^{13} + q^{14} - q^{16} - 4q^{17} - 2q^{18} + 8q^{19} + q^{21} - 2q^{22} - 4q^{23} - q^{24} + 5q^{25} + 6q^{26} - 2q^{27} - q^{28} - 6q^{31} + q^{32} - 2q^{33} + 4q^{34} - q^{36} - 10q^{37} + 7q^{38} + 6q^{39} - 4q^{41} - q^{42} + 9q^{43} + 2q^{44} - 8q^{46} - 10q^{47} + q^{48} - 12q^{49} + 10q^{50} + 4q^{51} + 3q^{52} + 4q^{53} - q^{54} - 2q^{56} + 7q^{57} + 14q^{59} - 11q^{61} - 3q^{62} - q^{63} + 2q^{64} + 2q^{66} - 3q^{67} + 8q^{68} - 8q^{69} - 14q^{71} + q^{72} + 11q^{73} - 5q^{74} + 10q^{75} - q^{76} - 4q^{77} + 3q^{78} - q^{79} - q^{81} + 4q^{82} + 16q^{83} - 2q^{84} - 9q^{86} + 4q^{88} + 14q^{89} + 3q^{91} - 4q^{92} - 3q^{93} - 20q^{94} + 2q^{96} - 2q^{97} - 6q^{98} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 −1.00000 −0.500000 + 0.866025i 0
49.1 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 −1.00000 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.e.b 2
3.b odd 2 1 342.2.g.c 2
4.b odd 2 1 912.2.q.b 2
12.b even 2 1 2736.2.s.k 2
19.c even 3 1 inner 114.2.e.b 2
19.c even 3 1 2166.2.a.b 1
19.d odd 6 1 2166.2.a.h 1
57.f even 6 1 6498.2.a.g 1
57.h odd 6 1 342.2.g.c 2
57.h odd 6 1 6498.2.a.u 1
76.g odd 6 1 912.2.q.b 2
228.m even 6 1 2736.2.s.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.b 2 1.a even 1 1 trivial
114.2.e.b 2 19.c even 3 1 inner
342.2.g.c 2 3.b odd 2 1
342.2.g.c 2 57.h odd 6 1
912.2.q.b 2 4.b odd 2 1
912.2.q.b 2 76.g odd 6 1
2166.2.a.b 1 19.c even 3 1
2166.2.a.h 1 19.d odd 6 1
2736.2.s.k 2 12.b even 2 1
2736.2.s.k 2 228.m even 6 1
6498.2.a.g 1 57.f even 6 1
6498.2.a.u 1 57.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(114, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$9 - 3 T + T^{2}$$
$17$ $$16 + 4 T + T^{2}$$
$19$ $$19 - 8 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 3 + T )^{2}$$
$37$ $$( 5 + T )^{2}$$
$41$ $$16 + 4 T + T^{2}$$
$43$ $$81 - 9 T + T^{2}$$
$47$ $$100 + 10 T + T^{2}$$
$53$ $$16 - 4 T + T^{2}$$
$59$ $$196 - 14 T + T^{2}$$
$61$ $$121 + 11 T + T^{2}$$
$67$ $$9 + 3 T + T^{2}$$
$71$ $$196 + 14 T + T^{2}$$
$73$ $$121 - 11 T + T^{2}$$
$79$ $$1 + T + T^{2}$$
$83$ $$( -8 + T )^{2}$$
$89$ $$196 - 14 T + T^{2}$$
$97$ $$4 + 2 T + T^{2}$$