# Properties

 Label 154.2.f.c 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{2}+ \cdots + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \cdots + 1) q^{9}+O(q^{10})$$ q + (-z^3 + z^2 - z + 1) * q^2 + (-z^3 - z) * q^3 - z^3 * q^4 - 2*z * q^5 + (-z^2 - 1) * q^6 + z^3 * q^7 - z^2 * q^8 + (-z^3 + 2*z^2 - 2*z + 1) * q^9 $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots + 1) q^{2}+ \cdots + ( - 5 \zeta_{10}^{2} + 6 \zeta_{10} - 3) q^{99}+O(q^{100})$$ q + (-z^3 + z^2 - z + 1) * q^2 + (-z^3 - z) * q^3 - z^3 * q^4 - 2*z * q^5 + (-z^2 - 1) * q^6 + z^3 * q^7 - z^2 * q^8 + (-z^3 + 2*z^2 - 2*z + 1) * q^9 - 2 * q^10 + (-4*z^3 + 3*z^2 - 2*z + 2) * q^11 + (z^3 - z^2 - 1) * q^12 + (2*z^3 - 2) * q^13 + z^2 * q^14 + (2*z^3 + 2*z - 2) * q^15 - z * q^16 + (3*z^2 + z + 3) * q^17 + (-z^3 + z - 1) * q^18 + (3*z^3 - 5*z^2 + 3*z) * q^19 + (2*z^3 - 2*z^2 + 2*z - 2) * q^20 + (-z^3 + z^2 + 1) * q^21 + (-2*z^3 - 2*z^2 + z) * q^22 + (-4*z^3 + 4*z^2 + 6) * q^23 + (z^3 - 1) * q^24 - z^2 * q^25 + (2*z^3 + 2*z - 2) * q^26 + (-4*z^2 + z - 4) * q^27 + z * q^28 + 8*z^3 * q^29 + (2*z^3 + 2*z) * q^30 + (-2*z^3 - 2*z^2 + 2*z + 2) * q^31 - q^32 + (z^3 - 4*z^2 - 3) * q^33 + (-3*z^3 + 3*z^2 + 4) * q^34 + (-2*z^3 + 2*z^2 - 2*z + 2) * q^35 + (z^3 - 2*z^2 + z) * q^36 + (4*z^3 + 6*z - 6) * q^37 + (3*z^2 - 5*z + 3) * q^38 + (2*z^2 + 2*z + 2) * q^39 + 2*z^3 * q^40 + (3*z^3 - 4*z^2 + 3*z) * q^41 + (-z^3 + 1) * q^42 + (3*z^3 - 3*z^2 - 3) * q^43 + (-2*z^2 - 2*z + 1) * q^44 + (-2*z^3 + 2*z^2 - 2) * q^45 + (-6*z^3 + 2*z^2 - 2*z + 6) * q^46 + (6*z^3 - 8*z^2 + 6*z) * q^47 + (z^3 + z - 1) * q^48 - z * q^49 - z * q^50 + (-7*z^3 - 4*z + 4) * q^51 + (2*z^3 + 2*z) * q^52 + (8*z^2 - 8*z) * q^53 + (4*z^3 - 4*z^2 - 3) * q^54 + (2*z^3 - 4*z^2 + 4*z - 8) * q^55 + q^56 + (-z^3 + 3*z^2 - 3*z + 1) * q^57 + 8*z^2 * q^58 + (-6*z^3 + 3*z - 3) * q^59 + (2*z^2 + 2) * q^60 + (-4*z^2 - 4) * q^61 + (-2*z^3 - 4*z + 4) * q^62 + (-z^3 + 2*z^2 - z) * q^63 + (z^3 - z^2 + z - 1) * q^64 + (-4*z^3 + 4*z^2 + 4) * q^65 + (3*z^3 - 2*z^2 - z - 3) * q^66 + (-5*z^3 + 5*z^2 - 2) * q^67 + (-4*z^3 + z^2 - z + 4) * q^68 + (-6*z^3 - 4*z^2 - 6*z) * q^69 - 2*z^3 * q^70 + (-4*z^2 - 4*z - 4) * q^71 + (z^2 - 2*z + 1) * q^72 + (3*z^3 - z + 1) * q^73 + (6*z^3 - 2*z^2 + 6*z) * q^74 + (z^3 - 1) * q^75 + (-3*z^3 + 3*z^2 - 2) * q^76 + (2*z^2 + 2*z - 1) * q^77 + (-2*z^3 + 2*z^2 + 4) * q^78 + (-2*z^2 + 2*z) * q^79 + 2*z^2 * q^80 + (4*z^3 + 6*z - 6) * q^81 + (3*z^2 - 4*z + 3) * q^82 + (13*z^2 - 8*z + 13) * q^83 + (-z^3 - z + 1) * q^84 + (-6*z^3 - 2*z^2 - 6*z) * q^85 + (3*z^3 - 3) * q^86 + (-8*z^3 + 8*z^2 + 8) * q^87 + (-z^3 + z^2 - 3*z - 1) * q^88 + (7*z^3 - 7*z^2 - 11) * q^89 + (2*z^3 - 4*z^2 + 4*z - 2) * q^90 + (-2*z^3 - 2*z) * q^91 + (-6*z^3 - 4*z + 4) * q^92 + (-2*z^2 - 4*z - 2) * q^93 + (6*z^2 - 8*z + 6) * q^94 + (4*z^3 - 6*z + 6) * q^95 + (z^3 + z) * q^96 + (-5*z^3 + 8*z^2 - 8*z + 5) * q^97 - q^98 + (-5*z^2 + 6*z - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 3 q^{6} + q^{7} + q^{8} - q^{9}+O(q^{10})$$ 4 * q + q^2 - 2 * q^3 - q^4 - 2 * q^5 - 3 * q^6 + q^7 + q^8 - q^9 $$4 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - 3 q^{6} + q^{7} + q^{8} - q^{9} - 8 q^{10} - q^{11} - 2 q^{12} - 6 q^{13} - q^{14} - 4 q^{15} - q^{16} + 10 q^{17} - 4 q^{18} + 11 q^{19} - 2 q^{20} + 2 q^{21} + q^{22} + 16 q^{23} - 3 q^{24} + q^{25} - 4 q^{26} - 11 q^{27} + q^{28} + 8 q^{29} + 4 q^{30} + 10 q^{31} - 4 q^{32} - 7 q^{33} + 10 q^{34} + 2 q^{35} + 4 q^{36} - 14 q^{37} + 4 q^{38} + 8 q^{39} + 2 q^{40} + 10 q^{41} + 3 q^{42} - 6 q^{43} + 4 q^{44} - 12 q^{45} + 14 q^{46} + 20 q^{47} - 2 q^{48} - q^{49} - q^{50} + 5 q^{51} + 4 q^{52} - 16 q^{53} - 4 q^{54} - 22 q^{55} + 4 q^{56} - 3 q^{57} - 8 q^{58} - 15 q^{59} + 6 q^{60} - 12 q^{61} + 10 q^{62} - 4 q^{63} - q^{64} + 8 q^{65} - 8 q^{66} - 18 q^{67} + 10 q^{68} - 8 q^{69} - 2 q^{70} - 16 q^{71} + q^{72} + 6 q^{73} + 14 q^{74} - 3 q^{75} - 14 q^{76} - 4 q^{77} + 12 q^{78} + 4 q^{79} - 2 q^{80} - 14 q^{81} + 5 q^{82} + 31 q^{83} + 2 q^{84} - 10 q^{85} - 9 q^{86} + 16 q^{87} - 9 q^{88} - 30 q^{89} + 2 q^{90} - 4 q^{91} + 6 q^{92} - 10 q^{93} + 10 q^{94} + 22 q^{95} + 2 q^{96} - q^{97} - 4 q^{98} - q^{99}+O(q^{100})$$ 4 * q + q^2 - 2 * q^3 - q^4 - 2 * q^5 - 3 * q^6 + q^7 + q^8 - q^9 - 8 * q^10 - q^11 - 2 * q^12 - 6 * q^13 - q^14 - 4 * q^15 - q^16 + 10 * q^17 - 4 * q^18 + 11 * q^19 - 2 * q^20 + 2 * q^21 + q^22 + 16 * q^23 - 3 * q^24 + q^25 - 4 * q^26 - 11 * q^27 + q^28 + 8 * q^29 + 4 * q^30 + 10 * q^31 - 4 * q^32 - 7 * q^33 + 10 * q^34 + 2 * q^35 + 4 * q^36 - 14 * q^37 + 4 * q^38 + 8 * q^39 + 2 * q^40 + 10 * q^41 + 3 * q^42 - 6 * q^43 + 4 * q^44 - 12 * q^45 + 14 * q^46 + 20 * q^47 - 2 * q^48 - q^49 - q^50 + 5 * q^51 + 4 * q^52 - 16 * q^53 - 4 * q^54 - 22 * q^55 + 4 * q^56 - 3 * q^57 - 8 * q^58 - 15 * q^59 + 6 * q^60 - 12 * q^61 + 10 * q^62 - 4 * q^63 - q^64 + 8 * q^65 - 8 * q^66 - 18 * q^67 + 10 * q^68 - 8 * q^69 - 2 * q^70 - 16 * q^71 + q^72 + 6 * q^73 + 14 * q^74 - 3 * q^75 - 14 * q^76 - 4 * q^77 + 12 * q^78 + 4 * q^79 - 2 * q^80 - 14 * q^81 + 5 * q^82 + 31 * q^83 + 2 * q^84 - 10 * q^85 - 9 * q^86 + 16 * q^87 - 9 * q^88 - 30 * q^89 + 2 * q^90 - 4 * q^91 + 6 * q^92 - 10 * q^93 + 10 * q^94 + 22 * q^95 + 2 * q^96 - q^97 - 4 * q^98 - 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 −0.500000 + 1.53884i 0.309017 + 0.951057i −1.61803 + 1.17557i −1.30902 + 0.951057i −0.309017 0.951057i −0.309017 + 0.951057i 0.309017 + 0.224514i −2.00000
71.1 −0.309017 0.951057i −0.500000 0.363271i −0.809017 + 0.587785i 0.618034 1.90211i −0.190983 + 0.587785i 0.809017 0.587785i 0.809017 + 0.587785i −0.809017 2.48990i −2.00000
113.1 0.809017 0.587785i −0.500000 1.53884i 0.309017 0.951057i −1.61803 1.17557i −1.30902 0.951057i −0.309017 + 0.951057i −0.309017 0.951057i 0.309017 0.224514i −2.00000
141.1 −0.309017 + 0.951057i −0.500000 + 0.363271i −0.809017 0.587785i 0.618034 + 1.90211i −0.190983 0.587785i 0.809017 + 0.587785i 0.809017 0.587785i −0.809017 + 2.48990i −2.00000
 $$n$$: 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.c 4
11.c even 5 1 inner 154.2.f.c 4
11.c even 5 1 1694.2.a.m 2
11.d odd 10 1 1694.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.f.c 4 1.a even 1 1 trivial
154.2.f.c 4 11.c even 5 1 inner
1694.2.a.m 2 11.c even 5 1
1694.2.a.r 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 $$S_{2}^{\mathrm{new}}(154, [\chi])$$:

 $$T_{3}^{4} + 2T_{3}^{3} + 4T_{3}^{2} + 3T_{3} + 1$$ T3^4 + 2*T3^3 + 4*T3^2 + 3*T3 + 1 $$T_{5}^{4} + 2T_{5}^{3} + 4T_{5}^{2} + 8T_{5} + 16$$ T5^4 + 2*T5^3 + 4*T5^2 + 8*T5 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + T^{2} + \cdots + 1$$
$3$ $$T^{4} + 2 T^{3} + \cdots + 1$$
$5$ $$T^{4} + 2 T^{3} + \cdots + 16$$
$7$ $$T^{4} - T^{3} + T^{2} + \cdots + 1$$
$11$ $$T^{4} + T^{3} + \cdots + 121$$
$13$ $$T^{4} + 6 T^{3} + \cdots + 16$$
$17$ $$T^{4} - 10 T^{3} + \cdots + 25$$
$19$ $$T^{4} - 11 T^{3} + \cdots + 1$$
$23$ $$(T^{2} - 8 T - 4)^{2}$$
$29$ $$T^{4} - 8 T^{3} + \cdots + 4096$$
$31$ $$T^{4} - 10 T^{3} + \cdots + 400$$
$37$ $$T^{4} + 14 T^{3} + \cdots + 1936$$
$41$ $$T^{4} - 10 T^{3} + \cdots + 25$$
$43$ $$(T^{2} + 3 T - 9)^{2}$$
$47$ $$T^{4} - 20 T^{3} + \cdots + 400$$
$53$ $$T^{4} + 16 T^{3} + \cdots + 4096$$
$59$ $$T^{4} + 15 T^{3} + \cdots + 2025$$
$61$ $$T^{4} + 12 T^{3} + \cdots + 256$$
$67$ $$(T^{2} + 9 T - 11)^{2}$$
$71$ $$T^{4} + 16 T^{3} + \cdots + 256$$
$73$ $$T^{4} - 6 T^{3} + \cdots + 121$$
$79$ $$T^{4} - 4 T^{3} + \cdots + 16$$
$83$ $$T^{4} - 31 T^{3} + \cdots + 43681$$
$89$ $$(T^{2} + 15 T - 5)^{2}$$
$97$ $$T^{4} + T^{3} + \cdots + 961$$