# Properties

 Label 154.2.f.d Level $154$ Weight $2$ Character orbit 154.f Analytic conductor $1.230$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [154,2,Mod(15,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("154.15");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.f (of order $$5$$, degree $$4$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1
 0.809017 − 0.587785i −0.309017 + 0.951057i 0.809017 + 0.587785i −0.309017 − 0.951057i
0.809017 + 0.587785i 0.690983 2.12663i 0.309017 + 0.951057i 0.309017 0.224514i 1.80902 1.31433i −0.309017 0.951057i −0.309017 + 0.951057i −1.61803 1.17557i 0.381966
71.1 −0.309017 0.951057i 1.80902 + 1.31433i −0.809017 + 0.587785i −0.809017 + 2.48990i 0.690983 2.12663i 0.809017 0.587785i 0.809017 + 0.587785i 0.618034 + 1.90211i 2.61803
113.1 0.809017 0.587785i 0.690983 + 2.12663i 0.309017 0.951057i 0.309017 + 0.224514i 1.80902 + 1.31433i −0.309017 + 0.951057i −0.309017 0.951057i −1.61803 + 1.17557i 0.381966
141.1 −0.309017 + 0.951057i 1.80902 1.31433i −0.809017 0.587785i −0.809017 2.48990i 0.690983 + 2.12663i 0.809017 + 0.587785i 0.809017 0.587785i 0.618034 1.90211i 2.61803
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.f.d 4
11.c even 5 1 inner 154.2.f.d 4
11.c even 5 1 1694.2.a.n 2
11.d odd 10 1 1694.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.f.d 4 1.a even 1 1 trivial
154.2.f.d 4 11.c even 5 1 inner
1694.2.a.n 2 11.c even 5 1
1694.2.a.s 2 11.d odd 10 1

## Hecke kernels

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

 $$T_{3}^{4} - 5T_{3}^{3} + 15T_{3}^{2} - 25T_{3} + 25$$ T3^4 - 5*T3^3 + 15*T3^2 - 25*T3 + 25 $$T_{5}^{4} + T_{5}^{3} + 6T_{5}^{2} - 4T_{5} + 1$$ T5^4 + T5^3 + 6*T5^2 - 4*T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + T^{2} + \cdots + 1$$
$3$ $$T^{4} - 5 T^{3} + \cdots + 25$$
$5$ $$T^{4} + T^{3} + 6 T^{2} + \cdots + 1$$
$7$ $$T^{4} - T^{3} + T^{2} + \cdots + 1$$
$11$ $$T^{4} + 4 T^{3} + \cdots + 121$$
$13$ $$T^{4} - 3 T^{3} + \cdots + 81$$
$17$ $$T^{4} - 9 T^{3} + \cdots + 361$$
$19$ $$T^{4} + 9 T^{3} + \cdots + 361$$
$23$ $$(T^{2} + 2 T - 19)^{2}$$
$29$ $$T^{4} + 3 T^{3} + \cdots + 81$$
$31$ $$T^{4} + 13 T^{3} + \cdots + 1681$$
$37$ $$T^{4} + 20 T^{3} + \cdots + 3025$$
$41$ $$T^{4} - 16 T^{3} + \cdots + 256$$
$43$ $$(T^{2} + T - 11)^{2}$$
$47$ $$T^{4} + 12 T^{3} + \cdots + 81$$
$53$ $$T^{4} + 11 T^{3} + \cdots + 961$$
$59$ $$T^{4} + 18 T^{3} + \cdots + 6561$$
$61$ $$T^{4} - 14 T^{3} + \cdots + 961$$
$67$ $$(T - 13)^{4}$$
$71$ $$T^{4} + 7 T^{3} + \cdots + 1$$
$73$ $$T^{4} - 21 T^{3} + \cdots + 9801$$
$79$ $$T^{4} - 15 T^{3} + \cdots + 9025$$
$83$ $$T^{4} - T^{3} + \cdots + 121$$
$89$ $$(T^{2} - 3 T - 9)^{2}$$
$97$ $$T^{4} - 14 T^{3} + \cdots + 16$$