# Properties

 Label 114.2.h.d 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: $$x^{2} - x + 1$$ 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.

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

## 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 1.73205i −0.500000 0.866025i 3.00000 + 1.73205i −1.50000 0.866025i −2.00000 −1.00000 −3.00000 3.00000 1.73205i
107.1 0.500000 + 0.866025i 1.73205i −0.500000 + 0.866025i 3.00000 1.73205i −1.50000 + 0.866025i −2.00000 −1.00000 −3.00000 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.d yes 2
3.b odd 2 1 114.2.h.a 2
4.b odd 2 1 912.2.bn.b 2
12.b even 2 1 912.2.bn.d 2
19.d odd 6 1 114.2.h.a 2
57.f even 6 1 inner 114.2.h.d yes 2
76.f even 6 1 912.2.bn.d 2
228.n odd 6 1 912.2.bn.b 2

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

## Hecke kernels

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

 $$T_{5}^{2} - 6T_{5} + 12$$ T5^2 - 6*T5 + 12 $$T_{7} + 2$$ T7 + 2 $$T_{17}^{2} - 12T_{17} + 48$$ T17^2 - 12*T17 + 48

## Hecke characteristic polynomials

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