# Properties

 Label 114.2.h.f Level $114$ Weight $2$ Character orbit 114.h Analytic conductor $0.910$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{2} + (\beta_{3} + \beta_{2} - \beta_1) q^{3} - \beta_{2} q^{4} + \beta_1 q^{5} + (\beta_{3} + 1) q^{6} + ( - \beta_{3} + 2 \beta_1 + 2) q^{7} - q^{8} + ( - \beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b2 + 1) * q^2 + (b3 + b2 - b1) * q^3 - b2 * q^4 + b1 * q^5 + (b3 + 1) * q^6 + (-b3 + 2*b1 + 2) * q^7 - q^8 + (-b2 - 2*b1 + 1) * q^9 $$q + ( - \beta_{2} + 1) q^{2} + (\beta_{3} + \beta_{2} - \beta_1) q^{3} - \beta_{2} q^{4} + \beta_1 q^{5} + (\beta_{3} + 1) q^{6} + ( - \beta_{3} + 2 \beta_1 + 2) q^{7} - q^{8} + ( - \beta_{2} - 2 \beta_1 + 1) q^{9} + ( - \beta_{3} + \beta_1) q^{10} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{11} + ( - \beta_{2} + \beta_1 + 1) q^{12} + (2 \beta_{2} - 4) q^{13} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{14} + (\beta_{3} - 2) q^{15} + (\beta_{2} - 1) q^{16} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{17} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{18} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{19} - \beta_{3} q^{20} + (3 \beta_{3} + 4 \beta_{2} - \beta_1 - 4) q^{21} + (\beta_{2} - \beta_1 + 1) q^{22} + ( - 5 \beta_{3} + 5 \beta_1) q^{23} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{24} - 3 \beta_{2} q^{25} + (4 \beta_{2} - 2) q^{26} + ( - \beta_{3} + 5) q^{27} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{28} + ( - \beta_{3} - \beta_1) q^{29} + (2 \beta_{2} + \beta_1 - 2) q^{30} + 3 \beta_{3} q^{31} + \beta_{2} q^{32} + ( - 2 \beta_{3} + 3 \beta_{2} - 2) q^{33} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{34} + (2 \beta_{2} + 2 \beta_1 + 2) q^{35} + (2 \beta_{3} - 1) q^{36} + ( - 3 \beta_{3} + 4 \beta_{2} - 2) q^{37} + (3 \beta_{3} - 1) q^{38} + ( - 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{39} - \beta_1 q^{40} + (3 \beta_{2} - 3) q^{41} + (\beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{42} + (4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{43} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{44} + ( - \beta_{3} - 4 \beta_{2} + \beta_1) q^{45} - 5 \beta_{3} q^{46} + (7 \beta_{3} - 2 \beta_{2} - 7 \beta_1 + 4) q^{47} + ( - \beta_{3} - 1) q^{48} + ( - 4 \beta_{3} + 8 \beta_1 + 3) q^{49} - 3 q^{50} + ( - 4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 6) q^{51} + (2 \beta_{2} + 2) q^{52} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{53} + ( - 5 \beta_{2} - \beta_1 + 5) q^{54} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{55} + (\beta_{3} - 2 \beta_1 - 2) q^{56} + ( - 7 \beta_{2} - 2 \beta_1 + 7) q^{57} + (\beta_{3} - 2 \beta_1) q^{58} + (2 \beta_{3} - 9 \beta_{2} - \beta_1 + 9) q^{59} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{60} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{61} + 3 \beta_1 q^{62} + ( - 2 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 2) q^{63} + q^{64} + (2 \beta_{3} - 4 \beta_1) q^{65} + (2 \beta_{2} - 2 \beta_1 + 1) q^{66} + (3 \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{67} + (2 \beta_{3} + 4 \beta_{2} - 2) q^{68} + (10 \beta_{2} + 5 \beta_1 - 10) q^{69} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{70} + ( - 6 \beta_{2} + 6) q^{71} + (\beta_{2} + 2 \beta_1 - 1) q^{72} + ( - 8 \beta_{3} - \beta_{2} + 4 \beta_1 + 1) q^{73} + (2 \beta_{2} - 3 \beta_1 + 2) q^{74} + ( - 3 \beta_{2} + 3 \beta_1 + 3) q^{75} + (\beta_{2} + 3 \beta_1 - 1) q^{76} + \beta_{3} q^{77} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{78} + 6 \beta_1 q^{79} + (\beta_{3} - \beta_1) q^{80} + (4 \beta_{3} + 7 \beta_{2} - 4 \beta_1) q^{81} + 3 \beta_{2} q^{82} + ( - \beta_{3} + 18 \beta_{2} - 9) q^{83} + ( - 2 \beta_{3} + 3 \beta_1 + 4) q^{84} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{85} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{86} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{87} + (\beta_{3} - 2 \beta_{2} + 1) q^{88} + (2 \beta_{3} + 12 \beta_{2} + 2 \beta_1) q^{89} + ( - \beta_{3} - 4) q^{90} + (6 \beta_{3} + 4 \beta_{2} - 6 \beta_1 - 8) q^{91} - 5 \beta_1 q^{92} + (3 \beta_{3} - 6 \beta_{2} - 3 \beta_1) q^{93} + (7 \beta_{3} - 4 \beta_{2} + 2) q^{94} + ( - \beta_{3} - 6) q^{95} + (\beta_{2} - \beta_1 - 1) q^{96} + ( - 3 \beta_{2} - 6 \beta_1 - 3) q^{97} + ( - 8 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 3) q^{98} + ( - 4 \beta_{3} + 5 \beta_{2} + \beta_1 - 3) q^{99}+O(q^{100})$$ q + (-b2 + 1) * q^2 + (b3 + b2 - b1) * q^3 - b2 * q^4 + b1 * q^5 + (b3 + 1) * q^6 + (-b3 + 2*b1 + 2) * q^7 - q^8 + (-b2 - 2*b1 + 1) * q^9 + (-b3 + b1) * q^10 + (-b3 + 2*b2 - 1) * q^11 + (-b2 + b1 + 1) * q^12 + (2*b2 - 4) * q^13 + (-2*b3 - 2*b2 + b1 + 2) * q^14 + (b3 - 2) * q^15 + (b2 - 1) * q^16 + (-2*b2 - 2*b1 - 2) * q^17 + (2*b3 - b2 - 2*b1) * q^18 + (3*b3 - b2 - 3*b1) * q^19 - b3 * q^20 + (3*b3 + 4*b2 - b1 - 4) * q^21 + (b2 - b1 + 1) * q^22 + (-5*b3 + 5*b1) * q^23 + (-b3 - b2 + b1) * q^24 - 3*b2 * q^25 + (4*b2 - 2) * q^26 + (-b3 + 5) * q^27 + (-b3 - 2*b2 - b1) * q^28 + (-b3 - b1) * q^29 + (2*b2 + b1 - 2) * q^30 + 3*b3 * q^31 + b2 * q^32 + (-2*b3 + 3*b2 - 2) * q^33 + (2*b3 + 2*b2 - 2*b1 - 4) * q^34 + (2*b2 + 2*b1 + 2) * q^35 + (2*b3 - 1) * q^36 + (-3*b3 + 4*b2 - 2) * q^37 + (3*b3 - 1) * q^38 + (-4*b3 - 2*b2 + 2*b1 - 2) * q^39 - b1 * q^40 + (3*b2 - 3) * q^41 + (b3 + 4*b2 + 2*b1) * q^42 + (4*b3 - 4*b2 - 2*b1 + 4) * q^43 + (b3 - b2 - b1 + 2) * q^44 + (-b3 - 4*b2 + b1) * q^45 - 5*b3 * q^46 + (7*b3 - 2*b2 - 7*b1 + 4) * q^47 + (-b3 - 1) * q^48 + (-4*b3 + 8*b1 + 3) * q^49 - 3 * q^50 + (-4*b3 - 4*b2 + 4*b1 + 6) * q^51 + (2*b2 + 2) * q^52 + (2*b3 - 6*b2 + 2*b1) * q^53 + (-5*b2 - b1 + 5) * q^54 + (2*b3 - 2*b2 - b1 + 2) * q^55 + (b3 - 2*b1 - 2) * q^56 + (-7*b2 - 2*b1 + 7) * q^57 + (b3 - 2*b1) * q^58 + (2*b3 - 9*b2 - b1 + 9) * q^59 + (-b3 + 2*b2 + b1) * q^60 + (b3 + 4*b2 + b1) * q^61 + 3*b1 * q^62 + (-2*b3 - 6*b2 - 3*b1 - 2) * q^63 + q^64 + (2*b3 - 4*b1) * q^65 + (2*b2 - 2*b1 + 1) * q^66 + (3*b3 + b2 - 3*b1 - 2) * q^67 + (2*b3 + 4*b2 - 2) * q^68 + (10*b2 + 5*b1 - 10) * q^69 + (-2*b3 - 2*b2 + 2*b1 + 4) * q^70 + (-6*b2 + 6) * q^71 + (b2 + 2*b1 - 1) * q^72 + (-8*b3 - b2 + 4*b1 + 1) * q^73 + (2*b2 - 3*b1 + 2) * q^74 + (-3*b2 + 3*b1 + 3) * q^75 + (b2 + 3*b1 - 1) * q^76 + b3 * q^77 + (-2*b3 + 2*b2 - 2*b1 - 4) * q^78 + 6*b1 * q^79 + (b3 - b1) * q^80 + (4*b3 + 7*b2 - 4*b1) * q^81 + 3*b2 * q^82 + (-b3 + 18*b2 - 9) * q^83 + (-2*b3 + 3*b1 + 4) * q^84 + (-2*b3 - 4*b2 - 2*b1) * q^85 + (2*b3 - 4*b2 + 2*b1) * q^86 + (-2*b3 + 2*b2 + b1 + 2) * q^87 + (b3 - 2*b2 + 1) * q^88 + (2*b3 + 12*b2 + 2*b1) * q^89 + (-b3 - 4) * q^90 + (6*b3 + 4*b2 - 6*b1 - 8) * q^91 - 5*b1 * q^92 + (3*b3 - 6*b2 - 3*b1) * q^93 + (7*b3 - 4*b2 + 2) * q^94 + (-b3 - 6) * q^95 + (b2 - b1 - 1) * q^96 + (-3*b2 - 6*b1 - 3) * q^97 + (-8*b3 - 3*b2 + 4*b1 + 3) * q^98 + (-4*b3 + 5*b2 + b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{6} + 8 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^6 + 8 * q^7 - 4 * q^8 + 2 * q^9 $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{6} + 8 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{12} - 12 q^{13} + 4 q^{14} - 8 q^{15} - 2 q^{16} - 12 q^{17} - 2 q^{18} - 2 q^{19} - 8 q^{21} + 6 q^{22} - 2 q^{24} - 6 q^{25} + 20 q^{27} - 4 q^{28} - 4 q^{30} + 2 q^{32} - 2 q^{33} - 12 q^{34} + 12 q^{35} - 4 q^{36} - 4 q^{38} - 12 q^{39} - 6 q^{41} + 8 q^{42} + 8 q^{43} + 6 q^{44} - 8 q^{45} + 12 q^{47} - 4 q^{48} + 12 q^{49} - 12 q^{50} + 16 q^{51} + 12 q^{52} - 12 q^{53} + 10 q^{54} + 4 q^{55} - 8 q^{56} + 14 q^{57} + 18 q^{59} + 4 q^{60} + 8 q^{61} - 20 q^{63} + 4 q^{64} + 8 q^{66} - 6 q^{67} - 20 q^{69} + 12 q^{70} + 12 q^{71} - 2 q^{72} + 2 q^{73} + 12 q^{74} + 6 q^{75} - 2 q^{76} - 12 q^{78} + 14 q^{81} + 6 q^{82} + 16 q^{84} - 8 q^{85} - 8 q^{86} + 12 q^{87} + 24 q^{89} - 16 q^{90} - 24 q^{91} - 12 q^{93} - 24 q^{95} - 2 q^{96} - 18 q^{97} + 6 q^{98} - 2 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^6 + 8 * q^7 - 4 * q^8 + 2 * q^9 + 2 * q^12 - 12 * q^13 + 4 * q^14 - 8 * q^15 - 2 * q^16 - 12 * q^17 - 2 * q^18 - 2 * q^19 - 8 * q^21 + 6 * q^22 - 2 * q^24 - 6 * q^25 + 20 * q^27 - 4 * q^28 - 4 * q^30 + 2 * q^32 - 2 * q^33 - 12 * q^34 + 12 * q^35 - 4 * q^36 - 4 * q^38 - 12 * q^39 - 6 * q^41 + 8 * q^42 + 8 * q^43 + 6 * q^44 - 8 * q^45 + 12 * q^47 - 4 * q^48 + 12 * q^49 - 12 * q^50 + 16 * q^51 + 12 * q^52 - 12 * q^53 + 10 * q^54 + 4 * q^55 - 8 * q^56 + 14 * q^57 + 18 * q^59 + 4 * q^60 + 8 * q^61 - 20 * q^63 + 4 * q^64 + 8 * q^66 - 6 * q^67 - 20 * q^69 + 12 * q^70 + 12 * q^71 - 2 * q^72 + 2 * q^73 + 12 * q^74 + 6 * q^75 - 2 * q^76 - 12 * q^78 + 14 * q^81 + 6 * q^82 + 16 * q^84 - 8 * q^85 - 8 * q^86 + 12 * q^87 + 24 * q^89 - 16 * q^90 - 24 * q^91 - 12 * q^93 - 24 * q^95 - 2 * q^96 - 18 * q^97 + 6 * q^98 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$-1$$ $$\beta_{2}$$

## 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
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
0.500000 0.866025i −0.724745 + 1.57313i −0.500000 0.866025i 1.22474 + 0.707107i 1.00000 + 1.41421i 4.44949 −1.00000 −1.94949 2.28024i 1.22474 0.707107i
65.2 0.500000 0.866025i 1.72474 + 0.158919i −0.500000 0.866025i −1.22474 0.707107i 1.00000 1.41421i −0.449490 −1.00000 2.94949 + 0.548188i −1.22474 + 0.707107i
107.1 0.500000 + 0.866025i −0.724745 1.57313i −0.500000 + 0.866025i 1.22474 0.707107i 1.00000 1.41421i 4.44949 −1.00000 −1.94949 + 2.28024i 1.22474 + 0.707107i
107.2 0.500000 + 0.866025i 1.72474 0.158919i −0.500000 + 0.866025i −1.22474 + 0.707107i 1.00000 + 1.41421i −0.449490 −1.00000 2.94949 0.548188i −1.22474 0.707107i
 $$n$$: 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.f yes 4
3.b odd 2 1 114.2.h.e 4
4.b odd 2 1 912.2.bn.h 4
12.b even 2 1 912.2.bn.g 4
19.d odd 6 1 114.2.h.e 4
57.f even 6 1 inner 114.2.h.f yes 4
76.f even 6 1 912.2.bn.g 4
228.n odd 6 1 912.2.bn.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.h.e 4 3.b odd 2 1
114.2.h.e 4 19.d odd 6 1
114.2.h.f yes 4 1.a even 1 1 trivial
114.2.h.f yes 4 57.f even 6 1 inner
912.2.bn.g 4 12.b even 2 1
912.2.bn.g 4 76.f even 6 1
912.2.bn.h 4 4.b odd 2 1
912.2.bn.h 4 228.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 $$S_{2}^{\mathrm{new}}(114, [\chi])$$:

 $$T_{5}^{4} - 2T_{5}^{2} + 4$$ T5^4 - 2*T5^2 + 4 $$T_{7}^{2} - 4T_{7} - 2$$ T7^2 - 4*T7 - 2 $$T_{17}^{4} + 12T_{17}^{3} + 52T_{17}^{2} + 48T_{17} + 16$$ T17^4 + 12*T17^3 + 52*T17^2 + 48*T17 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} - 2 T^{3} + T^{2} - 6 T + 9$$
$5$ $$T^{4} - 2T^{2} + 4$$
$7$ $$(T^{2} - 4 T - 2)^{2}$$
$11$ $$T^{4} + 10T^{2} + 1$$
$13$ $$(T^{2} + 6 T + 12)^{2}$$
$17$ $$T^{4} + 12 T^{3} + 52 T^{2} + 48 T + 16$$
$19$ $$T^{4} + 2 T^{3} - 15 T^{2} + 38 T + 361$$
$23$ $$T^{4} - 50T^{2} + 2500$$
$29$ $$T^{4} + 6T^{2} + 36$$
$31$ $$(T^{2} + 18)^{2}$$
$37$ $$T^{4} + 60T^{2} + 36$$
$41$ $$(T^{2} + 3 T + 9)^{2}$$
$43$ $$T^{4} - 8 T^{3} + 72 T^{2} + 64 T + 64$$
$47$ $$T^{4} - 12 T^{3} - 38 T^{2} + \cdots + 7396$$
$53$ $$T^{4} + 12 T^{3} + 132 T^{2} + \cdots + 144$$
$59$ $$T^{4} - 18 T^{3} + 249 T^{2} + \cdots + 5625$$
$61$ $$T^{4} - 8 T^{3} + 54 T^{2} - 80 T + 100$$
$67$ $$T^{4} + 6 T^{3} - 3 T^{2} - 90 T + 225$$
$71$ $$(T^{2} - 6 T + 36)^{2}$$
$73$ $$T^{4} - 2 T^{3} + 99 T^{2} + \cdots + 9025$$
$79$ $$T^{4} - 72T^{2} + 5184$$
$83$ $$T^{4} + 490 T^{2} + 58081$$
$89$ $$T^{4} - 24 T^{3} + 456 T^{2} + \cdots + 14400$$
$97$ $$T^{4} + 18 T^{3} + 63 T^{2} + \cdots + 2025$$