# Properties

 Label 841.2.a.g Level $841$ Weight $2$ Character orbit 841.a Self dual yes Analytic conductor $6.715$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: 6.6.11973625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 8x^{4} + 15x^{3} + 13x^{2} - 27x + 9$$ x^6 - 2*x^5 - 8*x^4 + 15*x^3 + 13*x^2 - 27*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{4} - \beta_{3} + 2) q^{4} + ( - \beta_{5} + \beta_1 - 1) q^{5} + ( - \beta_{5} - \beta_{2} - 1) q^{6} + (\beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{4} + \beta_{3} + \beta_1) q^{9}+O(q^{10})$$ q - b1 * q^2 + b5 * q^3 + (b4 - b3 + 2) * q^4 + (-b5 + b1 - 1) * q^5 + (-b5 - b2 - 1) * q^6 + (b3 + b2 - 1) * q^7 + (-b4 + b3 - b2 - b1 - 1) * q^8 + (-b4 + b3 + b1) * q^9 $$q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{4} - \beta_{3} + 2) q^{4} + ( - \beta_{5} + \beta_1 - 1) q^{5} + ( - \beta_{5} - \beta_{2} - 1) q^{6} + (\beta_{3} + \beta_{2} - 1) q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{8}+ \cdots + (3 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + \cdots - 1) q^{99}+O(q^{100})$$ q - b1 * q^2 + b5 * q^3 + (b4 - b3 + 2) * q^4 + (-b5 + b1 - 1) * q^5 + (-b5 - b2 - 1) * q^6 + (b3 + b2 - 1) * q^7 + (-b4 + b3 - b2 - b1 - 1) * q^8 + (-b4 + b3 + b1) * q^9 + (b5 - b4 + b3 + b2 + b1 - 3) * q^10 + (b5 - b2 - 2) * q^11 + (-3*b3 + b2 + b1 + 2) * q^12 + (-b5 - b3 + b2 + b1 - 1) * q^13 + (-b5 + 3*b3 + b2 + b1 - 1) * q^14 + (b4 - b3 + b2 - b1 - 2) * q^15 + (b5 - 3*b3 + b2 + 2*b1 + 2) * q^16 + (-b5 - b2 - 1) * q^17 + (b2 + b1 - 3) * q^18 + (-b4 - b3 - 2*b1 + 2) * q^19 + (3*b3 + 2*b1 - 3) * q^20 + (-b5 - b3 - b2 + b1) * q^21 + (-3*b3 - b2 + 2*b1) * q^22 + (-b5 - b2 - b1 - 1) * q^23 + (b5 - b4 + 4*b3 - b2 - 2*b1 - 3) * q^24 + (-2*b2 - b1 + 1) * q^25 + (-b4 + 4*b3 + b1 - 4) * q^26 + (3*b3 - b1 - 1) * q^27 + (-b4 + 2*b3 + 2*b2 + b1 - 2) * q^28 + (-b5 + 3*b3 - b2 + b1 + 2) * q^30 + (-b5 + b4 - 2*b3 - b2 - 2) * q^31 + (-2*b5 + 3*b3 - 2*b2 - 8) * q^32 + (-b5 - 2*b4 + 2*b3 + b2 + 3) * q^33 + (2*b5 - 3*b3 + b2 + b1 + 2) * q^34 + (2*b5 - 3*b3 - b2 - 2*b1 + 2) * q^35 + (-b5 + b4 + 2*b3 + b1 - 5) * q^36 + (2*b4 - b3 - b2 + b1 + 2) * q^37 + (3*b4 - 3*b3 - b2 - b1 + 9) * q^38 + (-2*b5 + 3*b4 - 2*b3 - 2) * q^39 + (-2*b5 + b2 + b1 - 2) * q^40 + (-b4 - 2*b3 + b2 - 4) * q^41 + (2*b5 - b4 - 2*b3 - 2) * q^42 + (2*b5 + b4 + b3 - 2*b2 - 2*b1 + 1) * q^43 + (-b5 - 2*b4 - b3 - b2 - 3) * q^44 + (-3*b5 + b4 - 4*b3 - b2 - b1 + 4) * q^45 + (2*b5 + b4 - 4*b3 + b2 + b1 + 6) * q^46 + (-3*b3 + 2*b1) * q^47 + (3*b4 + b2 + 2*b1 + 5) * q^48 + (-b5 + b4 + b3 - 2*b1 - 2) * q^49 + (2*b5 + b4 - 7*b3 - b1 + 6) * q^50 + (b2 - 2*b1 - 3) * q^51 + (2*b5 + 2*b3 + 2*b2 + 3*b1 - 1) * q^52 + (-2*b5 + 2*b4 + b3 + 2*b2 + b1 - 3) * q^53 + (b4 - b3 + 3*b2 + b1 + 4) * q^54 + (2*b4 + b3 + b2 - 2*b1 - 1) * q^55 + (b1 - 3) * q^56 + (2*b4 + 3*b3 - 3*b2 - b1 - 4) * q^57 + (2*b5 - b4 + b3 + b2 - 2*b1 - 2) * q^59 + (2*b5 - 3*b4 + 2*b2 + 2) * q^60 + (-2*b5 + 2*b3 + 4*b2 + 2*b1 - 1) * q^61 + (2*b5 - b4 - 2*b3 - b2 + b1 + 1) * q^62 + (b5 + b4 - 3*b3 - b2 - 2*b1 + 1) * q^63 + (2*b5 + 3*b2 + 4*b1) * q^64 + (3*b5 - 2*b4 - b3 - b2 - 2*b1 + 7) * q^65 + (2*b4 + b3 + 3*b2 - b1 + 2) * q^66 + (b5 - b4 + 3*b3 + b2 + b1 - 3) * q^67 + (-b5 - b4 + 4*b3 - 3*b2 - 2*b1 - 5) * q^68 + (-b5 - 2*b1 - 4) * q^69 + (-b5 + 2*b4 - 5*b3 - 5*b2 - 2*b1 + 7) * q^70 + (b5 + 3*b4 - b2 - b1 + 1) * q^71 + (b5 - 2*b4 + 2*b3 + b2 + 2*b1 + 2) * q^72 + (3*b5 - b4 - b3 - b2 + 2*b1 - 1) * q^73 + (b5 - 3*b4 - b2 - 4*b1 - 5) * q^74 + (2*b5 - 2*b4 + 2*b3 + b2 - 2*b1 - 1) * q^75 + (b5 + b3 - 3*b2 - 8*b1 - 2) * q^76 + (-b4 - 5*b3 - 3*b2 + 2*b1 - 1) * q^77 + (2*b5 - 3*b4 + 3*b3 - b1 - 1) * q^78 + (2*b5 - 2*b4 + 6*b3 + 4*b2 - 3) * q^79 + (b5 - b4 - 2*b3 + 2*b2 - 2*b1 + 3) * q^80 + (b5 - 3*b3 - b2 - 3*b1 - 1) * q^81 + (-b5 + b4 + 2*b3 - 2*b2 + 5*b1) * q^82 + (-3*b4 + 3*b3 + 2*b2 + b1 - 6) * q^83 + (b4 + b3 - 2*b2 + b1 - 1) * q^84 + (-b5 + 3*b3 - b2 + b1 + 2) * q^85 + (b4 - 7*b3 - b2 - 2*b1 + 7) * q^86 + (2*b5 + 2*b4 + b3 + 2*b2 + b1 + 4) * q^88 + (2*b5 + b4 + 2*b3 + 4*b1 - 3) * q^89 + (4*b5 - 3*b3 - b2 - 5*b1 + 7) * q^90 + (b5 + b4 - b3 - 2*b2 - 2*b1 + 4) * q^91 + (-b5 - 2*b4 + 5*b3 - 4*b2 - 5*b1 - 6) * q^92 + (-4*b5 + b4 - 3*b3 + 2*b2 - b1 - 1) * q^93 + (-2*b4 + 2*b3 - 3*b2 - 8) * q^94 + (-4*b4 + b3 + 4*b2 + 4*b1 - 7) * q^95 + (-3*b5 - 3*b4 + 2*b2 - 4*b1 - 6) * q^96 + (2*b5 - 3*b4 - b3 + 2*b2 - b1 + 3) * q^97 + (b5 + b4 - b3 + 2*b2 + b1 + 8) * q^98 + (3*b5 + 2*b4 + 4*b3 - 2*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} - 3 q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9}+O(q^{10})$$ 6 * q - 2 * q^2 - 2 * q^3 + 8 * q^4 - 2 * q^5 - 3 * q^6 - 4 * q^7 - 3 * q^8 + 6 * q^9 $$6 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{5} - 3 q^{6} - 4 q^{7} - 3 q^{8} + 6 q^{9} - 15 q^{10} - 13 q^{11} + 4 q^{12} - 6 q^{13} + 6 q^{14} - 19 q^{15} + 4 q^{16} - 3 q^{17} - 17 q^{18} + 6 q^{19} - 5 q^{20} + 2 q^{21} - 4 q^{22} - 5 q^{23} - 10 q^{24} + 6 q^{25} - 9 q^{26} + q^{27} - 5 q^{28} + 26 q^{30} - 16 q^{31} - 33 q^{32} + 27 q^{33} - 4 q^{35} - 21 q^{36} + 10 q^{37} + 41 q^{38} - 17 q^{39} - 7 q^{40} - 30 q^{41} - 21 q^{42} + 2 q^{43} - 16 q^{44} + 16 q^{45} + 20 q^{46} - 5 q^{47} + 30 q^{48} - 12 q^{49} + 8 q^{50} - 23 q^{51} - 13 q^{53} + 19 q^{54} - 10 q^{55} - 16 q^{56} - 16 q^{57} - 17 q^{59} + 9 q^{60} + 4 q^{61} - 9 q^{63} + q^{64} + 32 q^{65} + 8 q^{66} - 9 q^{67} - 16 q^{68} - 26 q^{69} + 28 q^{70} + 21 q^{72} - 9 q^{73} - 36 q^{74} - 7 q^{75} - 24 q^{76} - 13 q^{77} - 6 q^{79} + 5 q^{80} - 22 q^{81} + 19 q^{82} - 24 q^{83} + 26 q^{85} + 17 q^{86} + 21 q^{88} - 9 q^{89} + 16 q^{90} + 16 q^{91} - 23 q^{92} - 12 q^{93} - 37 q^{94} - 31 q^{95} - 37 q^{96} + 10 q^{97} + 42 q^{98} - 6 q^{99}+O(q^{100})$$ 6 * q - 2 * q^2 - 2 * q^3 + 8 * q^4 - 2 * q^5 - 3 * q^6 - 4 * q^7 - 3 * q^8 + 6 * q^9 - 15 * q^10 - 13 * q^11 + 4 * q^12 - 6 * q^13 + 6 * q^14 - 19 * q^15 + 4 * q^16 - 3 * q^17 - 17 * q^18 + 6 * q^19 - 5 * q^20 + 2 * q^21 - 4 * q^22 - 5 * q^23 - 10 * q^24 + 6 * q^25 - 9 * q^26 + q^27 - 5 * q^28 + 26 * q^30 - 16 * q^31 - 33 * q^32 + 27 * q^33 - 4 * q^35 - 21 * q^36 + 10 * q^37 + 41 * q^38 - 17 * q^39 - 7 * q^40 - 30 * q^41 - 21 * q^42 + 2 * q^43 - 16 * q^44 + 16 * q^45 + 20 * q^46 - 5 * q^47 + 30 * q^48 - 12 * q^49 + 8 * q^50 - 23 * q^51 - 13 * q^53 + 19 * q^54 - 10 * q^55 - 16 * q^56 - 16 * q^57 - 17 * q^59 + 9 * q^60 + 4 * q^61 - 9 * q^63 + q^64 + 32 * q^65 + 8 * q^66 - 9 * q^67 - 16 * q^68 - 26 * q^69 + 28 * q^70 + 21 * q^72 - 9 * q^73 - 36 * q^74 - 7 * q^75 - 24 * q^76 - 13 * q^77 - 6 * q^79 + 5 * q^80 - 22 * q^81 + 19 * q^82 - 24 * q^83 + 26 * q^85 + 17 * q^86 + 21 * q^88 - 9 * q^89 + 16 * q^90 + 16 * q^91 - 23 * q^92 - 12 * q^93 - 37 * q^94 - 31 * q^95 - 37 * q^96 + 10 * q^97 + 42 * q^98 - 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 8x^{4} + 15x^{3} + 13x^{2} - 27x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5\nu + 3$$ v^3 - v^2 - 5*v + 3 $$\beta_{3}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 12\nu^{2} + 16\nu - 12 ) / 3$$ (v^5 - 2*v^4 - 8*v^3 + 12*v^2 + 16*v - 12) / 3 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 15\nu^{2} + 16\nu - 24 ) / 3$$ (v^5 - 2*v^4 - 8*v^3 + 15*v^2 + 16*v - 24) / 3 $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 9\nu^{3} + 7\nu^{2} + 19\nu - 13$$ v^5 - v^4 - 9*v^3 + 7*v^2 + 19*v - 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + 4$$ b4 - b3 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1$$ b4 - b3 + b2 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{5} + 6\beta_{4} - 9\beta_{3} + \beta_{2} + 2\beta _1 + 22$$ b5 + 6*b4 - 9*b3 + b2 + 2*b1 + 22 $$\nu^{5}$$ $$=$$ $$2\beta_{5} + 8\beta_{4} - 11\beta_{3} + 10\beta_{2} + 28\beta _1 + 16$$ 2*b5 + 8*b4 - 11*b3 + 10*b2 + 28*b1 + 16

## 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
 2.67904 1.94573 0.855835 0.511256 −1.80156 −2.19030
−2.67904 1.58170 5.17728 0.0973457 −4.23744 0.0377061 −8.51206 −0.498231 −0.260793
1.2 −1.94573 −2.27154 1.78585 3.21726 4.41979 −2.53022 0.416673 2.15987 −6.25991
1.3 −0.855835 2.66897 −1.26755 −2.81313 −2.28420 −0.766737 2.79648 4.12338 2.40758
1.4 −0.511256 −2.69256 −1.73862 2.20381 1.37659 −1.30206 1.91139 4.24987 −1.12671
1.5 1.80156 −1.39743 1.24563 −1.40413 −2.51756 3.53302 −1.35905 −1.04719 −2.52963
1.6 2.19030 0.110861 2.79741 −3.30116 0.242818 −2.97171 1.74657 −2.98771 −7.23053
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.g 6
3.b odd 2 1 7569.2.a.bc 6
29.b even 2 1 841.2.a.h yes 6
29.c odd 4 2 841.2.b.d 12
29.d even 7 6 841.2.d.o 36
29.e even 14 6 841.2.d.n 36
29.f odd 28 12 841.2.e.l 72
87.d odd 2 1 7569.2.a.y 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
841.2.a.g 6 1.a even 1 1 trivial
841.2.a.h yes 6 29.b even 2 1
841.2.b.d 12 29.c odd 4 2
841.2.d.n 36 29.e even 14 6
841.2.d.o 36 29.d even 7 6
841.2.e.l 72 29.f odd 28 12
7569.2.a.y 6 87.d odd 2 1
7569.2.a.bc 6 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 2T_{2}^{5} - 8T_{2}^{4} - 15T_{2}^{3} + 13T_{2}^{2} + 27T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(841))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 2 T^{5} + \cdots + 9$$
$3$ $$T^{6} + 2 T^{5} + \cdots - 4$$
$5$ $$T^{6} + 2 T^{5} + \cdots - 9$$
$7$ $$T^{6} + 4 T^{5} + \cdots + 1$$
$11$ $$T^{6} + 13 T^{5} + \cdots - 81$$
$13$ $$T^{6} + 6 T^{5} + \cdots + 29$$
$17$ $$T^{6} + 3 T^{5} + \cdots - 36$$
$19$ $$T^{6} - 6 T^{5} + \cdots + 1055$$
$23$ $$T^{6} + 5 T^{5} + \cdots - 81$$
$29$ $$T^{6}$$
$31$ $$T^{6} + 16 T^{5} + \cdots + 199$$
$37$ $$T^{6} - 10 T^{5} + \cdots - 1616$$
$41$ $$T^{6} + 30 T^{5} + \cdots - 12816$$
$43$ $$T^{6} - 2 T^{5} + \cdots - 4751$$
$47$ $$T^{6} + 5 T^{5} + \cdots + 1341$$
$53$ $$T^{6} + 13 T^{5} + \cdots - 29124$$
$59$ $$T^{6} + 17 T^{5} + \cdots - 1305$$
$61$ $$T^{6} - 4 T^{5} + \cdots - 76421$$
$67$ $$T^{6} + 9 T^{5} + \cdots + 361$$
$71$ $$T^{6} - 162 T^{4} + \cdots + 1611$$
$73$ $$T^{6} + 9 T^{5} + \cdots + 33251$$
$79$ $$T^{6} + 6 T^{5} + \cdots + 95125$$
$83$ $$T^{6} + 24 T^{5} + \cdots - 4311$$
$89$ $$T^{6} + 9 T^{5} + \cdots - 585405$$
$97$ $$T^{6} - 10 T^{5} + \cdots + 3824$$