# Properties

 Label 114.2.b.a Level $114$ Weight $2$ Character orbit 114.b Analytic conductor $0.910$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,2,Mod(113,114)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("114.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.910294583043$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 − 1.41421i 1.41421i
−1.00000 −1.00000 1.41421i 1.00000 1.41421i 1.00000 + 1.41421i −4.00000 −1.00000 −1.00000 + 2.82843i 1.41421i
113.2 −1.00000 −1.00000 + 1.41421i 1.00000 1.41421i 1.00000 1.41421i −4.00000 −1.00000 −1.00000 2.82843i 1.41421i
 $$n$$: 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.d even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{2} + 2$$ T5^2 + 2 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 2T + 3$$
$5$ $$T^{2} + 2$$
$7$ $$(T + 4)^{2}$$
$11$ $$T^{2} + 32$$
$13$ $$T^{2} + 18$$
$17$ $$T^{2} + 8$$
$19$ $$T^{2} - 2T + 19$$
$23$ $$T^{2} + 2$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 18$$
$37$ $$T^{2} + 18$$
$41$ $$T^{2}$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} + 2$$
$53$ $$(T + 6)^{2}$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2}$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} + 18$$
$83$ $$T^{2} + 8$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 72$$