# Properties

 Label 8954.2.a.j Level $8954$ Weight $2$ Character orbit 8954.a Self dual yes Analytic conductor $71.498$ 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] = [8954,2,Mod(1,8954)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8954, 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("8954.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8954 = 2 \cdot 11^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8954.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: $$71.4980499699$$ 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: $$1$$ Twist minimal: no (minimal twist has level 74) 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 −1.61803 1.00000 3.85410 1.61803 3.23607 −1.00000 −0.381966 −3.85410
1.2 −1.00000 0.618034 1.00000 −2.85410 −0.618034 −1.23607 −1.00000 −2.61803 2.85410
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$11$$ $$-1$$
$$37$$ $$+1$$

## 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 8954.2.a.j 2
11.b odd 2 1 74.2.a.b 2
33.d even 2 1 666.2.a.i 2
44.c even 2 1 592.2.a.g 2
55.d odd 2 1 1850.2.a.t 2
55.e even 4 2 1850.2.b.j 4
77.b even 2 1 3626.2.a.s 2
88.b odd 2 1 2368.2.a.y 2
88.g even 2 1 2368.2.a.u 2
132.d odd 2 1 5328.2.a.bc 2
407.d odd 2 1 2738.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 11.b odd 2 1
592.2.a.g 2 44.c even 2 1
666.2.a.i 2 33.d even 2 1
1850.2.a.t 2 55.d odd 2 1
1850.2.b.j 4 55.e even 4 2
2368.2.a.u 2 88.g even 2 1
2368.2.a.y 2 88.b odd 2 1
2738.2.a.g 2 407.d odd 2 1
3626.2.a.s 2 77.b even 2 1
5328.2.a.bc 2 132.d odd 2 1
8954.2.a.j 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8954))$$:

 $$T_{3}^{2} + T_{3} - 1$$ T3^2 + T3 - 1 $$T_{5}^{2} - T_{5} - 11$$ T5^2 - T5 - 11 $$T_{7}^{2} - 2T_{7} - 4$$ T7^2 - 2*T7 - 4 $$T_{17}^{2} - 20$$ T17^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + T - 1$$
$5$ $$T^{2} - T - 11$$
$7$ $$T^{2} - 2T - 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + T - 11$$
$17$ $$T^{2} - 20$$
$19$ $$T^{2} - 20$$
$23$ $$T^{2} + T - 11$$
$29$ $$T^{2} - 3T - 59$$
$31$ $$T^{2} - 17T + 71$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} + 17T + 71$$
$43$ $$T^{2} - 6T + 4$$
$47$ $$T^{2} - 2T - 4$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} + 14T + 44$$
$61$ $$T^{2} + 19T + 89$$
$67$ $$T^{2} + 9T - 11$$
$71$ $$T^{2} + 12T - 44$$
$73$ $$T^{2} + 3T - 29$$
$79$ $$T^{2} + 3T - 99$$
$83$ $$T^{2} - 20T + 80$$
$89$ $$T^{2} + 12T + 16$$
$97$ $$T^{2} - 8T - 4$$