# Properties

 Label 154.2.e.c Level $154$ Weight $2$ Character orbit 154.e Analytic conductor $1.230$ 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] = [154,2,Mod(23,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("154.23");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.e (of order $$3$$, degree $$2$$, 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: $$1.22969619113$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/154\mathbb{Z}\right)^\times$$.

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0 −1.00000 −0.500000 2.59808i −1.00000 1.00000 1.73205i 0
67.1 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0 −1.00000 −0.500000 + 2.59808i −1.00000 1.00000 + 1.73205i 0
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.e.c 2
3.b odd 2 1 1386.2.k.e 2
4.b odd 2 1 1232.2.q.d 2
7.b odd 2 1 1078.2.e.k 2
7.c even 3 1 inner 154.2.e.c 2
7.c even 3 1 1078.2.a.e 1
7.d odd 6 1 1078.2.a.c 1
7.d odd 6 1 1078.2.e.k 2
21.g even 6 1 9702.2.a.bs 1
21.h odd 6 1 1386.2.k.e 2
21.h odd 6 1 9702.2.a.br 1
28.f even 6 1 8624.2.a.u 1
28.g odd 6 1 1232.2.q.d 2
28.g odd 6 1 8624.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.c 2 1.a even 1 1 trivial
154.2.e.c 2 7.c even 3 1 inner
1078.2.a.c 1 7.d odd 6 1
1078.2.a.e 1 7.c even 3 1
1078.2.e.k 2 7.b odd 2 1
1078.2.e.k 2 7.d odd 6 1
1232.2.q.d 2 4.b odd 2 1
1232.2.q.d 2 28.g odd 6 1
1386.2.k.e 2 3.b odd 2 1
1386.2.k.e 2 21.h odd 6 1
8624.2.a.k 1 28.g odd 6 1
8624.2.a.u 1 28.f even 6 1
9702.2.a.br 1 21.h odd 6 1
9702.2.a.bs 1 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(154, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 7$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2}$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + 11T + 121$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 18T + 324$$
$97$ $$(T + 13)^{2}$$