# Properties

 Label 114.2.h.a Level $114$ Weight $2$ Character orbit 114.h Analytic conductor $0.910$ Analytic rank $1$ 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.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.910294583043$$ Analytic rank: $$1$$ 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -2 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -2 - 2 \zeta_{6} ) q^{5} + ( 1 - 2 \zeta_{6} ) q^{6} -2 q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( -2 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -2 - 2 \zeta_{6} ) q^{5} + ( 1 - 2 \zeta_{6} ) q^{6} -2 q^{7} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + ( 4 - 2 \zeta_{6} ) q^{10} + ( -1 + 2 \zeta_{6} ) q^{11} + ( 1 + \zeta_{6} ) q^{12} + ( -4 + 2 \zeta_{6} ) q^{13} + ( 2 - 2 \zeta_{6} ) q^{14} + 6 q^{15} + ( -1 + \zeta_{6} ) q^{16} + ( -4 - 4 \zeta_{6} ) q^{17} + 3 \zeta_{6} q^{18} + ( -2 + 5 \zeta_{6} ) q^{19} + ( -2 + 4 \zeta_{6} ) q^{20} + ( 4 - 2 \zeta_{6} ) q^{21} + ( -1 - \zeta_{6} ) q^{22} + ( -2 + \zeta_{6} ) q^{24} + 7 \zeta_{6} q^{25} + ( 2 - 4 \zeta_{6} ) q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + 2 \zeta_{6} q^{28} + 6 \zeta_{6} q^{29} + ( -6 + 6 \zeta_{6} ) q^{30} + ( 4 - 8 \zeta_{6} ) q^{31} -\zeta_{6} q^{32} -3 \zeta_{6} q^{33} + ( 8 - 4 \zeta_{6} ) q^{34} + ( 4 + 4 \zeta_{6} ) q^{35} -3 q^{36} + ( 4 - 8 \zeta_{6} ) q^{37} + ( -3 - 2 \zeta_{6} ) q^{38} + ( 6 - 6 \zeta_{6} ) q^{39} + ( -2 - 2 \zeta_{6} ) q^{40} + ( 3 - 3 \zeta_{6} ) q^{41} + ( -2 + 4 \zeta_{6} ) q^{42} + ( -8 + 8 \zeta_{6} ) q^{43} + ( 2 - \zeta_{6} ) q^{44} + ( -12 + 6 \zeta_{6} ) q^{45} + ( 4 - 2 \zeta_{6} ) q^{47} + ( 1 - 2 \zeta_{6} ) q^{48} -3 q^{49} -7 q^{50} + 12 q^{51} + ( 2 + 2 \zeta_{6} ) q^{52} -6 \zeta_{6} q^{53} + ( -3 - 3 \zeta_{6} ) q^{54} + ( 6 - 6 \zeta_{6} ) q^{55} -2 q^{56} + ( -1 - 7 \zeta_{6} ) q^{57} -6 q^{58} + ( -3 + 3 \zeta_{6} ) q^{59} -6 \zeta_{6} q^{60} -10 \zeta_{6} q^{61} + ( 4 + 4 \zeta_{6} ) q^{62} + ( -6 + 6 \zeta_{6} ) q^{63} + q^{64} + 12 q^{65} + 3 q^{66} + ( 6 - 3 \zeta_{6} ) q^{67} + ( -4 + 8 \zeta_{6} ) q^{68} + ( -8 + 4 \zeta_{6} ) q^{70} + ( -6 + 6 \zeta_{6} ) q^{71} + ( 3 - 3 \zeta_{6} ) q^{72} + ( -5 + 5 \zeta_{6} ) q^{73} + ( 4 + 4 \zeta_{6} ) q^{74} + ( -7 - 7 \zeta_{6} ) q^{75} + ( 5 - 3 \zeta_{6} ) q^{76} + ( 2 - 4 \zeta_{6} ) q^{77} + 6 \zeta_{6} q^{78} + ( -8 - 8 \zeta_{6} ) q^{79} + ( 4 - 2 \zeta_{6} ) q^{80} -9 \zeta_{6} q^{81} + 3 \zeta_{6} q^{82} + ( -3 + 6 \zeta_{6} ) q^{83} + ( -2 - 2 \zeta_{6} ) q^{84} + 24 \zeta_{6} q^{85} -8 \zeta_{6} q^{86} + ( -6 - 6 \zeta_{6} ) q^{87} + ( -1 + 2 \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} + ( 6 - 12 \zeta_{6} ) q^{90} + ( 8 - 4 \zeta_{6} ) q^{91} + 12 \zeta_{6} q^{93} + ( -2 + 4 \zeta_{6} ) q^{94} + ( 14 - 16 \zeta_{6} ) q^{95} + ( 1 + \zeta_{6} ) q^{96} + ( 5 + 5 \zeta_{6} ) q^{97} + ( 3 - 3 \zeta_{6} ) q^{98} + ( 3 + 3 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - 3q^{3} - q^{4} - 6q^{5} - 4q^{7} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} - 3q^{3} - q^{4} - 6q^{5} - 4q^{7} + 2q^{8} + 3q^{9} + 6q^{10} + 3q^{12} - 6q^{13} + 2q^{14} + 12q^{15} - q^{16} - 12q^{17} + 3q^{18} + q^{19} + 6q^{21} - 3q^{22} - 3q^{24} + 7q^{25} + 2q^{28} + 6q^{29} - 6q^{30} - q^{32} - 3q^{33} + 12q^{34} + 12q^{35} - 6q^{36} - 8q^{38} + 6q^{39} - 6q^{40} + 3q^{41} - 8q^{43} + 3q^{44} - 18q^{45} + 6q^{47} - 6q^{49} - 14q^{50} + 24q^{51} + 6q^{52} - 6q^{53} - 9q^{54} + 6q^{55} - 4q^{56} - 9q^{57} - 12q^{58} - 3q^{59} - 6q^{60} - 10q^{61} + 12q^{62} - 6q^{63} + 2q^{64} + 24q^{65} + 6q^{66} + 9q^{67} - 12q^{70} - 6q^{71} + 3q^{72} - 5q^{73} + 12q^{74} - 21q^{75} + 7q^{76} + 6q^{78} - 24q^{79} + 6q^{80} - 9q^{81} + 3q^{82} - 6q^{84} + 24q^{85} - 8q^{86} - 18q^{87} + 6q^{89} + 12q^{91} + 12q^{93} + 12q^{95} + 3q^{96} + 15q^{97} + 3q^{98} + 9q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i −3.00000 1.73205i 1.73205i −2.00000 1.00000 1.50000 2.59808i 3.00000 1.73205i
107.1 −0.500000 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i −3.00000 + 1.73205i 1.73205i −2.00000 1.00000 1.50000 + 2.59808i 3.00000 + 1.73205i
 $$n$$: 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
57.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.2.h.a 2
3.b odd 2 1 114.2.h.d yes 2
4.b odd 2 1 912.2.bn.d 2
12.b even 2 1 912.2.bn.b 2
19.d odd 6 1 114.2.h.d yes 2
57.f even 6 1 inner 114.2.h.a 2
76.f even 6 1 912.2.bn.b 2
228.n odd 6 1 912.2.bn.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.h.a 2 1.a even 1 1 trivial
114.2.h.a 2 57.f even 6 1 inner
114.2.h.d yes 2 3.b odd 2 1
114.2.h.d yes 2 19.d odd 6 1
912.2.bn.b 2 12.b even 2 1
912.2.bn.b 2 76.f even 6 1
912.2.bn.d 2 4.b odd 2 1
912.2.bn.d 2 228.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(114, [\chi])$$:

 $$T_{5}^{2} + 6 T_{5} + 12$$ $$T_{7} + 2$$ $$T_{17}^{2} + 12 T_{17} + 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$12 + 6 T + T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$3 + T^{2}$$
$13$ $$12 + 6 T + T^{2}$$
$17$ $$48 + 12 T + T^{2}$$
$19$ $$19 - T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$48 + T^{2}$$
$37$ $$48 + T^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$64 + 8 T + T^{2}$$
$47$ $$12 - 6 T + T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$100 + 10 T + T^{2}$$
$67$ $$27 - 9 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$25 + 5 T + T^{2}$$
$79$ $$192 + 24 T + T^{2}$$
$83$ $$27 + T^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$75 - 15 T + T^{2}$$