# Properties

 Label 154.2.e.f Level $154$ Weight $2$ Character orbit 154.e 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(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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 49$$ x^4 + 7*x^2 + 49 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + (\beta_{3} + \beta_1) q^{3} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} + \beta_{3} q^{6} - \beta_1 q^{7} - q^{8} + ( - 4 \beta_{2} - 4) q^{9}+O(q^{10})$$ q + (b2 + 1) * q^2 + (b3 + b1) * q^3 + b2 * q^4 + (-b2 + b1 - 1) * q^5 + b3 * q^6 - b1 * q^7 - q^8 + (-4*b2 - 4) * q^9 $$q + (\beta_{2} + 1) q^{2} + (\beta_{3} + \beta_1) q^{3} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} + \beta_{3} q^{6} - \beta_1 q^{7} - q^{8} + ( - 4 \beta_{2} - 4) q^{9} + (\beta_{3} - \beta_{2} + \beta_1) q^{10} + \beta_{2} q^{11} - \beta_1 q^{12} + 5 q^{13} + ( - \beta_{3} - \beta_1) q^{14} + ( - \beta_{3} - 7) q^{15} + ( - \beta_{2} - 1) q^{16} + 6 \beta_{2} q^{17} - 4 \beta_{2} q^{18} + (3 \beta_{2} + \beta_1 + 3) q^{19} + (\beta_{3} + 1) q^{20} + 7 q^{21} - q^{22} + (\beta_{2} - \beta_1 + 1) q^{23} + ( - \beta_{3} - \beta_1) q^{24} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{25} + (5 \beta_{2} + 5) q^{26} - \beta_{3} q^{27} - \beta_{3} q^{28} + ( - 2 \beta_{3} + 1) q^{29} + ( - 7 \beta_{2} + \beta_1 - 7) q^{30} - 4 \beta_{2} q^{31} - \beta_{2} q^{32} - \beta_1 q^{33} - 6 q^{34} + (\beta_{3} - 7 \beta_{2} + \beta_1) q^{35} + 4 q^{36} + ( - \beta_{2} - \beta_1 - 1) q^{37} + (\beta_{3} + 3 \beta_{2} + \beta_1) q^{38} + (5 \beta_{3} + 5 \beta_1) q^{39} + (\beta_{2} - \beta_1 + 1) q^{40} + ( - 3 \beta_{3} + 3) q^{41} + (7 \beta_{2} + 7) q^{42} - 4 q^{43} + ( - \beta_{2} - 1) q^{44} + ( - 4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{45} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{46} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{47} - \beta_{3} q^{48} + 7 \beta_{2} q^{49} + ( - 2 \beta_{3} - 3) q^{50} - 6 \beta_1 q^{51} + 5 \beta_{2} q^{52} + (\beta_{3} - \beta_{2} + \beta_1) q^{53} + \beta_1 q^{54} + (\beta_{3} + 1) q^{55} + \beta_1 q^{56} + (3 \beta_{3} - 7) q^{57} + (\beta_{2} + 2 \beta_1 + 1) q^{58} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{59} + (\beta_{3} - 7 \beta_{2} + \beta_1) q^{60} + ( - 9 \beta_{2} - 2 \beta_1 - 9) q^{61} + 4 q^{62} + (4 \beta_{3} + 4 \beta_1) q^{63} + q^{64} + ( - 5 \beta_{2} + 5 \beta_1 - 5) q^{65} + ( - \beta_{3} - \beta_1) q^{66} + ( - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1) q^{67} + ( - 6 \beta_{2} - 6) q^{68} + (\beta_{3} + 7) q^{69} + (\beta_{3} + 7) q^{70} + (\beta_{3} + 7) q^{71} + (4 \beta_{2} + 4) q^{72} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{73} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{74} + (14 \beta_{2} - 3 \beta_1 + 14) q^{75} + (\beta_{3} - 3) q^{76} - \beta_{3} q^{77} + 5 \beta_{3} q^{78} + \beta_1 q^{79} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{80} - 5 \beta_{2} q^{81} + (3 \beta_{2} + 3 \beta_1 + 3) q^{82} + (2 \beta_{3} + 8) q^{83} + 7 \beta_{2} q^{84} + (6 \beta_{3} + 6) q^{85} + ( - 4 \beta_{2} - 4) q^{86} + (\beta_{3} + 14 \beta_{2} + \beta_1) q^{87} - \beta_{2} q^{88} + (4 \beta_{2} - 4 \beta_1 + 4) q^{89} + ( - 4 \beta_{3} - 4) q^{90} - 5 \beta_1 q^{91} + ( - \beta_{3} - 1) q^{92} + 4 \beta_1 q^{93} + (2 \beta_{3} - 8 \beta_{2} + 2 \beta_1) q^{94} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{95} + \beta_1 q^{96} + (2 \beta_{3} - 11) q^{97} - 7 q^{98} + 4 q^{99}+O(q^{100})$$ q + (b2 + 1) * q^2 + (b3 + b1) * q^3 + b2 * q^4 + (-b2 + b1 - 1) * q^5 + b3 * q^6 - b1 * q^7 - q^8 + (-4*b2 - 4) * q^9 + (b3 - b2 + b1) * q^10 + b2 * q^11 - b1 * q^12 + 5 * q^13 + (-b3 - b1) * q^14 + (-b3 - 7) * q^15 + (-b2 - 1) * q^16 + 6*b2 * q^17 - 4*b2 * q^18 + (3*b2 + b1 + 3) * q^19 + (b3 + 1) * q^20 + 7 * q^21 - q^22 + (b2 - b1 + 1) * q^23 + (-b3 - b1) * q^24 + (-2*b3 + 3*b2 - 2*b1) * q^25 + (5*b2 + 5) * q^26 - b3 * q^27 - b3 * q^28 + (-2*b3 + 1) * q^29 + (-7*b2 + b1 - 7) * q^30 - 4*b2 * q^31 - b2 * q^32 - b1 * q^33 - 6 * q^34 + (b3 - 7*b2 + b1) * q^35 + 4 * q^36 + (-b2 - b1 - 1) * q^37 + (b3 + 3*b2 + b1) * q^38 + (5*b3 + 5*b1) * q^39 + (b2 - b1 + 1) * q^40 + (-3*b3 + 3) * q^41 + (7*b2 + 7) * q^42 - 4 * q^43 + (-b2 - 1) * q^44 + (-4*b3 + 4*b2 - 4*b1) * q^45 + (-b3 + b2 - b1) * q^46 + (-8*b2 + 2*b1 - 8) * q^47 - b3 * q^48 + 7*b2 * q^49 + (-2*b3 - 3) * q^50 - 6*b1 * q^51 + 5*b2 * q^52 + (b3 - b2 + b1) * q^53 + b1 * q^54 + (b3 + 1) * q^55 + b1 * q^56 + (3*b3 - 7) * q^57 + (b2 + 2*b1 + 1) * q^58 + (b3 + 2*b2 + b1) * q^59 + (b3 - 7*b2 + b1) * q^60 + (-9*b2 - 2*b1 - 9) * q^61 + 4 * q^62 + (4*b3 + 4*b1) * q^63 + q^64 + (-5*b2 + 5*b1 - 5) * q^65 + (-b3 - b1) * q^66 + (-3*b3 - 4*b2 - 3*b1) * q^67 + (-6*b2 - 6) * q^68 + (b3 + 7) * q^69 + (b3 + 7) * q^70 + (b3 + 7) * q^71 + (4*b2 + 4) * q^72 + (-b3 + 3*b2 - b1) * q^73 + (-b3 - b2 - b1) * q^74 + (14*b2 - 3*b1 + 14) * q^75 + (b3 - 3) * q^76 - b3 * q^77 + 5*b3 * q^78 + b1 * q^79 + (-b3 + b2 - b1) * q^80 - 5*b2 * q^81 + (3*b2 + 3*b1 + 3) * q^82 + (2*b3 + 8) * q^83 + 7*b2 * q^84 + (6*b3 + 6) * q^85 + (-4*b2 - 4) * q^86 + (b3 + 14*b2 + b1) * q^87 - b2 * q^88 + (4*b2 - 4*b1 + 4) * q^89 + (-4*b3 - 4) * q^90 - 5*b1 * q^91 + (-b3 - 1) * q^92 + 4*b1 * q^93 + (2*b3 - 8*b2 + 2*b1) * q^94 + (2*b3 + 4*b2 + 2*b1) * q^95 + b1 * q^96 + (2*b3 - 11) * q^97 - 7 * q^98 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{8} - 8 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 - 4 * q^8 - 8 * q^9 $$4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{8} - 8 q^{9} + 2 q^{10} - 2 q^{11} + 20 q^{13} - 28 q^{15} - 2 q^{16} - 12 q^{17} + 8 q^{18} + 6 q^{19} + 4 q^{20} + 28 q^{21} - 4 q^{22} + 2 q^{23} - 6 q^{25} + 10 q^{26} + 4 q^{29} - 14 q^{30} + 8 q^{31} + 2 q^{32} - 24 q^{34} + 14 q^{35} + 16 q^{36} - 2 q^{37} - 6 q^{38} + 2 q^{40} + 12 q^{41} + 14 q^{42} - 16 q^{43} - 2 q^{44} - 8 q^{45} - 2 q^{46} - 16 q^{47} - 14 q^{49} - 12 q^{50} - 10 q^{52} + 2 q^{53} + 4 q^{55} - 28 q^{57} + 2 q^{58} - 4 q^{59} + 14 q^{60} - 18 q^{61} + 16 q^{62} + 4 q^{64} - 10 q^{65} + 8 q^{67} - 12 q^{68} + 28 q^{69} + 28 q^{70} + 28 q^{71} + 8 q^{72} - 6 q^{73} + 2 q^{74} + 28 q^{75} - 12 q^{76} - 2 q^{80} + 10 q^{81} + 6 q^{82} + 32 q^{83} - 14 q^{84} + 24 q^{85} - 8 q^{86} - 28 q^{87} + 2 q^{88} + 8 q^{89} - 16 q^{90} - 4 q^{92} + 16 q^{94} - 8 q^{95} - 44 q^{97} - 28 q^{98} + 16 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 - 4 * q^8 - 8 * q^9 + 2 * q^10 - 2 * q^11 + 20 * q^13 - 28 * q^15 - 2 * q^16 - 12 * q^17 + 8 * q^18 + 6 * q^19 + 4 * q^20 + 28 * q^21 - 4 * q^22 + 2 * q^23 - 6 * q^25 + 10 * q^26 + 4 * q^29 - 14 * q^30 + 8 * q^31 + 2 * q^32 - 24 * q^34 + 14 * q^35 + 16 * q^36 - 2 * q^37 - 6 * q^38 + 2 * q^40 + 12 * q^41 + 14 * q^42 - 16 * q^43 - 2 * q^44 - 8 * q^45 - 2 * q^46 - 16 * q^47 - 14 * q^49 - 12 * q^50 - 10 * q^52 + 2 * q^53 + 4 * q^55 - 28 * q^57 + 2 * q^58 - 4 * q^59 + 14 * q^60 - 18 * q^61 + 16 * q^62 + 4 * q^64 - 10 * q^65 + 8 * q^67 - 12 * q^68 + 28 * q^69 + 28 * q^70 + 28 * q^71 + 8 * q^72 - 6 * q^73 + 2 * q^74 + 28 * q^75 - 12 * q^76 - 2 * q^80 + 10 * q^81 + 6 * q^82 + 32 * q^83 - 14 * q^84 + 24 * q^85 - 8 * q^86 - 28 * q^87 + 2 * q^88 + 8 * q^89 - 16 * q^90 - 4 * q^92 + 16 * q^94 - 8 * q^95 - 44 * q^97 - 28 * q^98 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 49$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 7$$ (v^3) / 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$7\beta_{3}$$ 7*b3

## Character values

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

 $$n$$ $$45$$ $$57$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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
 1.32288 − 2.29129i −1.32288 + 2.29129i 1.32288 + 2.29129i −1.32288 − 2.29129i
0.500000 0.866025i −1.32288 2.29129i −0.500000 0.866025i 0.822876 1.42526i −2.64575 −1.32288 + 2.29129i −1.00000 −2.00000 + 3.46410i −0.822876 1.42526i
23.2 0.500000 0.866025i 1.32288 + 2.29129i −0.500000 0.866025i −1.82288 + 3.15731i 2.64575 1.32288 2.29129i −1.00000 −2.00000 + 3.46410i 1.82288 + 3.15731i
67.1 0.500000 + 0.866025i −1.32288 + 2.29129i −0.500000 + 0.866025i 0.822876 + 1.42526i −2.64575 −1.32288 2.29129i −1.00000 −2.00000 3.46410i −0.822876 + 1.42526i
67.2 0.500000 + 0.866025i 1.32288 2.29129i −0.500000 + 0.866025i −1.82288 3.15731i 2.64575 1.32288 + 2.29129i −1.00000 −2.00000 3.46410i 1.82288 3.15731i
 $$n$$: 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.f 4
3.b odd 2 1 1386.2.k.s 4
4.b odd 2 1 1232.2.q.g 4
7.b odd 2 1 1078.2.e.v 4
7.c even 3 1 inner 154.2.e.f 4
7.c even 3 1 1078.2.a.s 2
7.d odd 6 1 1078.2.a.n 2
7.d odd 6 1 1078.2.e.v 4
21.g even 6 1 9702.2.a.dr 2
21.h odd 6 1 1386.2.k.s 4
21.h odd 6 1 9702.2.a.cz 2
28.f even 6 1 8624.2.a.bk 2
28.g odd 6 1 1232.2.q.g 4
28.g odd 6 1 8624.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 1.a even 1 1 trivial
154.2.e.f 4 7.c even 3 1 inner
1078.2.a.n 2 7.d odd 6 1
1078.2.a.s 2 7.c even 3 1
1078.2.e.v 4 7.b odd 2 1
1078.2.e.v 4 7.d odd 6 1
1232.2.q.g 4 4.b odd 2 1
1232.2.q.g 4 28.g odd 6 1
1386.2.k.s 4 3.b odd 2 1
1386.2.k.s 4 21.h odd 6 1
8624.2.a.bk 2 28.f even 6 1
8624.2.a.ca 2 28.g odd 6 1
9702.2.a.cz 2 21.h odd 6 1
9702.2.a.dr 2 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 $$S_{2}^{\mathrm{new}}(154, [\chi])$$:

 $$T_{3}^{4} + 7T_{3}^{2} + 49$$ T3^4 + 7*T3^2 + 49 $$T_{13} - 5$$ T13 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - T + 1)^{2}$$
$3$ $$T^{4} + 7T^{2} + 49$$
$5$ $$T^{4} + 2 T^{3} + \cdots + 36$$
$7$ $$T^{4} + 7T^{2} + 49$$
$11$ $$(T^{2} + T + 1)^{2}$$
$13$ $$(T - 5)^{4}$$
$17$ $$(T^{2} + 6 T + 36)^{2}$$
$19$ $$T^{4} - 6 T^{3} + \cdots + 4$$
$23$ $$T^{4} - 2 T^{3} + \cdots + 36$$
$29$ $$(T^{2} - 2 T - 27)^{2}$$
$31$ $$(T^{2} - 4 T + 16)^{2}$$
$37$ $$T^{4} + 2 T^{3} + \cdots + 36$$
$41$ $$(T^{2} - 6 T - 54)^{2}$$
$43$ $$(T + 4)^{4}$$
$47$ $$T^{4} + 16 T^{3} + \cdots + 1296$$
$53$ $$T^{4} - 2 T^{3} + \cdots + 36$$
$59$ $$T^{4} + 4 T^{3} + \cdots + 9$$
$61$ $$T^{4} + 18 T^{3} + \cdots + 2809$$
$67$ $$T^{4} - 8 T^{3} + \cdots + 2209$$
$71$ $$(T^{2} - 14 T + 42)^{2}$$
$73$ $$T^{4} + 6 T^{3} + \cdots + 4$$
$79$ $$T^{4} + 7T^{2} + 49$$
$83$ $$(T^{2} - 16 T + 36)^{2}$$
$89$ $$T^{4} - 8 T^{3} + \cdots + 9216$$
$97$ $$(T^{2} + 22 T + 93)^{2}$$