Properties

 Label 841.2.b.b Level $841$ Weight $2$ Character orbit 841.b Analytic conductor $6.715$ 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] = [841,2,Mod(840,841)]

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([1]))

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

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

chi := DirichletCharacter("841.840");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$841 = 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 841.b (of order $$2$$, degree $$1$$, not 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: $$6.71541880999$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/841\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
840.1
 − 1.61803i − 0.618034i 0.618034i 1.61803i
1.61803i 1.61803i −0.618034 2.85410 2.61803 2.23607 2.23607i 0.381966 4.61803i
840.2 0.618034i 0.618034i 1.61803 −3.85410 0.381966 −2.23607 2.23607i 2.61803 2.38197i
840.3 0.618034i 0.618034i 1.61803 −3.85410 0.381966 −2.23607 2.23607i 2.61803 2.38197i
840.4 1.61803i 1.61803i −0.618034 2.85410 2.61803 2.23607 2.23607i 0.381966 4.61803i
 $$n$$: 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
29.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 841.2.b.b 4
29.b even 2 1 inner 841.2.b.b 4
29.c odd 4 1 841.2.a.a 2
29.c odd 4 1 841.2.a.c yes 2
29.d even 7 6 841.2.e.j 24
29.e even 14 6 841.2.e.j 24
29.f odd 28 6 841.2.d.g 12
29.f odd 28 6 841.2.d.i 12
87.f even 4 1 7569.2.a.d 2
87.f even 4 1 7569.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
841.2.a.a 2 29.c odd 4 1
841.2.a.c yes 2 29.c odd 4 1
841.2.b.b 4 1.a even 1 1 trivial
841.2.b.b 4 29.b even 2 1 inner
841.2.d.g 12 29.f odd 28 6
841.2.d.i 12 29.f odd 28 6
841.2.e.j 24 29.d even 7 6
841.2.e.j 24 29.e even 14 6
7569.2.a.d 2 87.f even 4 1
7569.2.a.l 2 87.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(841, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 1$$
$3$ $$T^{4} + 3T^{2} + 1$$
$5$ $$(T^{2} + T - 11)^{2}$$
$7$ $$(T^{2} - 5)^{2}$$
$11$ $$T^{4} + 15T^{2} + 25$$
$13$ $$(T^{2} + 4 T - 1)^{2}$$
$17$ $$T^{4} + 63T^{2} + 841$$
$19$ $$T^{4} + 27T^{2} + 81$$
$23$ $$(T^{2} - 2 T - 4)^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 103T^{2} + 121$$
$37$ $$T^{4} + 98T^{2} + 1681$$
$41$ $$T^{4} + 23T^{2} + 121$$
$43$ $$T^{4} + 60T^{2} + 400$$
$47$ $$(T^{2} + 49)^{2}$$
$53$ $$(T + 2)^{4}$$
$59$ $$(T^{2} - T - 31)^{2}$$
$61$ $$T^{4} + 3T^{2} + 1$$
$67$ $$(T^{2} - 12 T + 16)^{2}$$
$71$ $$(T^{2} + 12 T + 16)^{2}$$
$73$ $$T^{4} + 188T^{2} + 16$$
$79$ $$T^{4} + 63T^{2} + 961$$
$83$ $$(T^{2} + 2 T - 79)^{2}$$
$89$ $$T^{4} + 98T^{2} + 1681$$
$97$ $$T^{4} + 287T^{2} + 3481$$