# Properties

 Label 841.2.a.f Level $841$ Weight $2$ Character orbit 841.a Self dual yes Analytic conductor $6.715$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [841,2,Mod(1,841)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(841, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("841.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$841 = 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 841.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: $$6.71541880999$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 29) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 −1.24698 0.445042 1.80194
−1.24698 −1.80194 −0.445042 −4.04892 2.24698 0.692021 3.04892 0.246980 5.04892
1.2 0.445042 1.24698 −1.80194 0.692021 0.554958 0.356896 −1.69202 −1.44504 0.307979
1.3 1.80194 −0.445042 1.24698 0.356896 −0.801938 −4.04892 −1.35690 −2.80194 0.643104
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$29$$ $$+1$$

## 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 841.2.a.f 3
3.b odd 2 1 7569.2.a.p 3
29.b even 2 1 841.2.a.e 3
29.c odd 4 2 841.2.b.c 6
29.d even 7 2 841.2.d.a 6
29.d even 7 2 841.2.d.c 6
29.d even 7 2 841.2.d.d 6
29.e even 14 2 29.2.d.a 6
29.e even 14 2 841.2.d.b 6
29.e even 14 2 841.2.d.e 6
29.f odd 28 4 841.2.e.b 12
29.f odd 28 4 841.2.e.c 12
29.f odd 28 4 841.2.e.d 12
87.d odd 2 1 7569.2.a.r 3
87.h odd 14 2 261.2.k.a 6
116.h odd 14 2 464.2.u.f 6
145.l even 14 2 725.2.l.b 6
145.q odd 28 4 725.2.r.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.d.a 6 29.e even 14 2
261.2.k.a 6 87.h odd 14 2
464.2.u.f 6 116.h odd 14 2
725.2.l.b 6 145.l even 14 2
725.2.r.b 12 145.q odd 28 4
841.2.a.e 3 29.b even 2 1
841.2.a.f 3 1.a even 1 1 trivial
841.2.b.c 6 29.c odd 4 2
841.2.d.a 6 29.d even 7 2
841.2.d.b 6 29.e even 14 2
841.2.d.c 6 29.d even 7 2
841.2.d.d 6 29.d even 7 2
841.2.d.e 6 29.e even 14 2
841.2.e.b 12 29.f odd 28 4
841.2.e.c 12 29.f odd 28 4
841.2.e.d 12 29.f odd 28 4
7569.2.a.p 3 3.b odd 2 1
7569.2.a.r 3 87.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - T_{2}^{2} - 2T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(841))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 2T + 1$$
$3$ $$T^{3} + T^{2} - 2T - 1$$
$5$ $$T^{3} + 3 T^{2} + \cdots + 1$$
$7$ $$T^{3} + 3 T^{2} + \cdots + 1$$
$11$ $$T^{3} + 5 T^{2} + \cdots - 41$$
$13$ $$T^{3} - T^{2} + \cdots + 43$$
$17$ $$T^{3} + 4 T^{2} + \cdots - 8$$
$19$ $$T^{3} - 3 T^{2} + \cdots + 13$$
$23$ $$T^{3} + 7T^{2} - 49$$
$29$ $$T^{3}$$
$31$ $$T^{3} - 15 T^{2} + \cdots - 83$$
$37$ $$T^{3} - 5 T^{2} + \cdots + 41$$
$41$ $$T^{3} + 10 T^{2} + \cdots + 8$$
$43$ $$T^{3} + 3 T^{2} + \cdots - 13$$
$47$ $$T^{3} - 19 T^{2} + \cdots - 239$$
$53$ $$T^{3} + 9 T^{2} + \cdots - 1$$
$59$ $$T^{3} + 28 T^{2} + \cdots + 728$$
$61$ $$T^{3} - 9 T^{2} + \cdots - 13$$
$67$ $$T^{3} - 13 T^{2} + \cdots + 13$$
$71$ $$T^{3} + 21 T^{2} + \cdots + 189$$
$73$ $$T^{3} + 5 T^{2} + \cdots - 419$$
$79$ $$T^{3} - 15 T^{2} + \cdots - 27$$
$83$ $$T^{3} + 9 T^{2} + \cdots - 169$$
$89$ $$T^{3} + 7 T^{2} + \cdots - 91$$
$97$ $$T^{3} + 11 T^{2} + \cdots + 13$$