# Properties

 Label 114.2.h.b Level $114$ Weight $2$ Character orbit 114.h 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.h (of order $$6$$, 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

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

## 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 1.50000 0.866025i 1.00000 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 1.50000 + 0.866025i 1.00000 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.b 2
3.b odd 2 1 114.2.h.c yes 2
4.b odd 2 1 912.2.bn.f 2
12.b even 2 1 912.2.bn.c 2
19.d odd 6 1 114.2.h.c yes 2
57.f even 6 1 inner 114.2.h.b 2
76.f even 6 1 912.2.bn.c 2
228.n odd 6 1 912.2.bn.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.h.b 2 1.a even 1 1 trivial
114.2.h.b 2 57.f even 6 1 inner
114.2.h.c yes 2 3.b odd 2 1
114.2.h.c yes 2 19.d odd 6 1
912.2.bn.c 2 12.b even 2 1
912.2.bn.c 2 76.f even 6 1
912.2.bn.f 2 4.b odd 2 1
912.2.bn.f 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 $$S_{2}^{\mathrm{new}}(114, [\chi])$$:

 $$T_{5}^{2} - 6 T_{5} + 12$$ $$T_{7} - 1$$ $$T_{17}^{2} + 6 T_{17} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$12 - 6 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$12 + T^{2}$$
$13$ $$27 - 9 T + T^{2}$$
$17$ $$12 + 6 T + T^{2}$$
$19$ $$19 + 8 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$3 + T^{2}$$
$37$ $$27 + T^{2}$$
$41$ $$144 + 12 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$48 - 12 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$49 + 7 T + T^{2}$$
$67$ $$75 - 15 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$27 - 9 T + T^{2}$$
$83$ $$108 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$48 - 12 T + T^{2}$$