# 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: $$x^{4} - 2 x^{2} + 4$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## 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 $$S_{2}^{\mathrm{new}}(114, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$9 - 6 T + T^{2} - 2 T^{3} + T^{4}$$
$5$ $$4 - 2 T^{2} + T^{4}$$
$7$ $$( -2 - 4 T + T^{2} )^{2}$$
$11$ $$1 + 10 T^{2} + T^{4}$$
$13$ $$( 12 + 6 T + T^{2} )^{2}$$
$17$ $$16 + 48 T + 52 T^{2} + 12 T^{3} + T^{4}$$
$19$ $$361 + 38 T - 15 T^{2} + 2 T^{3} + T^{4}$$
$23$ $$2500 - 50 T^{2} + T^{4}$$
$29$ $$36 + 6 T^{2} + T^{4}$$
$31$ $$( 18 + T^{2} )^{2}$$
$37$ $$36 + 60 T^{2} + T^{4}$$
$41$ $$( 9 + 3 T + T^{2} )^{2}$$
$43$ $$64 + 64 T + 72 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$7396 + 1032 T - 38 T^{2} - 12 T^{3} + T^{4}$$
$53$ $$144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4}$$
$59$ $$5625 - 1350 T + 249 T^{2} - 18 T^{3} + T^{4}$$
$61$ $$100 - 80 T + 54 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$225 - 90 T - 3 T^{2} + 6 T^{3} + T^{4}$$
$71$ $$( 36 - 6 T + T^{2} )^{2}$$
$73$ $$9025 + 190 T + 99 T^{2} - 2 T^{3} + T^{4}$$
$79$ $$5184 - 72 T^{2} + T^{4}$$
$83$ $$58081 + 490 T^{2} + T^{4}$$
$89$ $$14400 - 2880 T + 456 T^{2} - 24 T^{3} + T^{4}$$
$97$ $$2025 - 810 T + 63 T^{2} + 18 T^{3} + T^{4}$$