# Properties

 Label 154.2.a.d Level $154$ Weight $2$ Character orbit 154.a Self dual yes Analytic conductor $1.230$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [154,2,Mod(1,154)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(154, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("154.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$154 = 2 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 154.a (trivial)

## 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: yes Analytic conductor: $$1.22969619113$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
1.00000 −3.23607 1.00000 3.23607 −3.23607 1.00000 1.00000 7.47214 3.23607
1.2 1.00000 1.23607 1.00000 −1.23607 1.23607 1.00000 1.00000 −1.47214 −1.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 154.2.a.d 2
3.b odd 2 1 1386.2.a.m 2
4.b odd 2 1 1232.2.a.p 2
5.b even 2 1 3850.2.a.bj 2
5.c odd 4 2 3850.2.c.q 4
7.b odd 2 1 1078.2.a.w 2
7.c even 3 2 1078.2.e.q 4
7.d odd 6 2 1078.2.e.n 4
8.b even 2 1 4928.2.a.bt 2
8.d odd 2 1 4928.2.a.bk 2
11.b odd 2 1 1694.2.a.l 2
21.c even 2 1 9702.2.a.cu 2
28.d even 2 1 8624.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 1.a even 1 1 trivial
1078.2.a.w 2 7.b odd 2 1
1078.2.e.n 4 7.d odd 6 2
1078.2.e.q 4 7.c even 3 2
1232.2.a.p 2 4.b odd 2 1
1386.2.a.m 2 3.b odd 2 1
1694.2.a.l 2 11.b odd 2 1
3850.2.a.bj 2 5.b even 2 1
3850.2.c.q 4 5.c odd 4 2
4928.2.a.bk 2 8.d odd 2 1
4928.2.a.bt 2 8.b even 2 1
8624.2.a.bf 2 28.d even 2 1
9702.2.a.cu 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(154))$$:

 $$T_{3}^{2} + 2T_{3} - 4$$ T3^2 + 2*T3 - 4 $$T_{5}^{2} - 2T_{5} - 4$$ T5^2 - 2*T5 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 2T - 4$$
$5$ $$T^{2} - 2T - 4$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 2T - 4$$
$17$ $$T^{2} + 4T - 16$$
$19$ $$T^{2} + 10T + 20$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} - 4T - 16$$
$43$ $$T^{2} + 12T + 16$$
$47$ $$(T + 2)^{2}$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} - 10T + 20$$
$61$ $$T^{2} + 6T + 4$$
$67$ $$T^{2} + 4T - 176$$
$71$ $$T^{2} - 4T - 16$$
$73$ $$T^{2} - 8T - 64$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 2T - 124$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} - 16T + 44$$