# Properties

 Label 91.2.f.b Level $91$ Weight $2$ Character orbit 91.f Analytic conductor $0.727$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,2,Mod(22,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("91.22");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.f (of order $$3$$, degree $$2$$, 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: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + (\beta_1 - 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{6} + (\beta_1 - 1) q^{7} - \beta_{3} q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10})$$ q - b2 * q^2 + (-b2 - b1) * q^3 + (b1 - 1) * q^4 + b3 * q^5 + (-b3 + b2 + 3*b1 - 3) * q^6 + (b1 - 1) * q^7 - b3 * q^8 + (-2*b3 + 2*b2 + b1 - 1) * q^9 $$q - \beta_{2} q^{2} + ( - \beta_{2} - \beta_1) q^{3} + (\beta_1 - 1) q^{4} + \beta_{3} q^{5} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{6} + (\beta_1 - 1) q^{7} - \beta_{3} q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{9} - 3 \beta_1 q^{10} + (\beta_{2} - 3 \beta_1) q^{11} + (\beta_{3} + 1) q^{12} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{13} + \beta_{3} q^{14} + ( - \beta_{2} - 3 \beta_1) q^{15} + 5 \beta_1 q^{16} + (\beta_{3} - \beta_{2} - 6 \beta_1 + 6) q^{17} + (\beta_{3} + 6) q^{18} + (2 \beta_1 - 2) q^{19} + ( - \beta_{3} + \beta_{2}) q^{20} + (\beta_{3} + 1) q^{21} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{22} + ( - \beta_{2} - 3 \beta_1) q^{23} + (\beta_{2} + 3 \beta_1) q^{24} - 2 q^{25} + (2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{26} + 4 q^{27} - \beta_1 q^{28} + 3 \beta_1 q^{29} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{30} + (3 \beta_{3} - 1) q^{31} + (3 \beta_{3} - 3 \beta_{2}) q^{32} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{33} + ( - 6 \beta_{3} - 3) q^{34} + ( - \beta_{3} + \beta_{2}) q^{35} + ( - 2 \beta_{2} - \beta_1) q^{36} + 7 \beta_1 q^{37} + 2 \beta_{3} q^{38} + (\beta_{3} - 3 \beta_{2} - 5 \beta_1 - 1) q^{39} - 3 q^{40} + 3 \beta_{2} q^{41} + ( - \beta_{2} - 3 \beta_1) q^{42} + (3 \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 5) q^{43} + ( - \beta_{3} + 3) q^{44} + ( - \beta_{3} + \beta_{2} + 6 \beta_1 - 6) q^{45} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 3) q^{46} + (4 \beta_{3} + 6) q^{47} + (5 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 5) q^{48} - \beta_1 q^{49} + 2 \beta_{2} q^{50} + ( - 7 \beta_{3} - 9) q^{51} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{52} + ( - 4 \beta_{3} - 3) q^{53} - 4 \beta_{2} q^{54} + ( - 3 \beta_{2} + 3 \beta_1) q^{55} + (\beta_{3} - \beta_{2}) q^{56} + (2 \beta_{3} + 2) q^{57} + (3 \beta_{3} - 3 \beta_{2}) q^{58} + (\beta_{3} - \beta_{2} + 9 \beta_1 - 9) q^{59} + (\beta_{3} + 3) q^{60} + ( - 3 \beta_{3} + 3 \beta_{2} - 10 \beta_1 + 10) q^{61} + (\beta_{2} - 9 \beta_1) q^{62} + ( - 2 \beta_{2} - \beta_1) q^{63} + q^{64} + (2 \beta_{2} - 3 \beta_1 + 6) q^{65} + 6 q^{66} + (3 \beta_{2} + \beta_1) q^{67} + (\beta_{2} + 6 \beta_1) q^{68} + ( - 4 \beta_{3} + 4 \beta_{2} + 6 \beta_1 - 6) q^{69} + 3 q^{70} + (6 \beta_1 - 6) q^{71} + (\beta_{3} - \beta_{2} - 6 \beta_1 + 6) q^{72} + ( - 3 \beta_{3} + 2) q^{73} + (7 \beta_{3} - 7 \beta_{2}) q^{74} + (2 \beta_{2} + 2 \beta_1) q^{75} - 2 \beta_1 q^{76} + ( - \beta_{3} + 3) q^{77} + ( - 5 \beta_{3} + 6 \beta_{2} + 6 \beta_1 - 9) q^{78} + (3 \beta_{3} + 11) q^{79} + 5 \beta_{2} q^{80} + (2 \beta_{2} - \beta_1) q^{81} + ( - 9 \beta_1 + 9) q^{82} + ( - 3 \beta_{3} + 3) q^{83} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{84} + (6 \beta_{3} - 6 \beta_{2} - 3 \beta_1 + 3) q^{85} + (5 \beta_{3} - 9) q^{86} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{87} + (3 \beta_{2} - 3 \beta_1) q^{88} + ( - 4 \beta_{2} - 6 \beta_1) q^{89} + (6 \beta_{3} + 3) q^{90} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{91} + (\beta_{3} + 3) q^{92} + ( - 2 \beta_{2} - 8 \beta_1) q^{93} + ( - 6 \beta_{2} - 12 \beta_1) q^{94} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{95} + ( - 3 \beta_{3} - 9) q^{96} + ( - 6 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 4) q^{97} + ( - \beta_{3} + \beta_{2}) q^{98} + (5 \beta_{3} - 3) q^{99}+O(q^{100})$$ q - b2 * q^2 + (-b2 - b1) * q^3 + (b1 - 1) * q^4 + b3 * q^5 + (-b3 + b2 + 3*b1 - 3) * q^6 + (b1 - 1) * q^7 - b3 * q^8 + (-2*b3 + 2*b2 + b1 - 1) * q^9 - 3*b1 * q^10 + (b2 - 3*b1) * q^11 + (b3 + 1) * q^12 + (2*b3 - b2 + 2*b1) * q^13 + b3 * q^14 + (-b2 - 3*b1) * q^15 + 5*b1 * q^16 + (b3 - b2 - 6*b1 + 6) * q^17 + (b3 + 6) * q^18 + (2*b1 - 2) * q^19 + (-b3 + b2) * q^20 + (b3 + 1) * q^21 + (-3*b3 + 3*b2 - 3*b1 + 3) * q^22 + (-b2 - 3*b1) * q^23 + (b2 + 3*b1) * q^24 - 2 * q^25 + (2*b3 - 2*b2 - 3*b1 - 3) * q^26 + 4 * q^27 - b1 * q^28 + 3*b1 * q^29 + (-3*b3 + 3*b2 + 3*b1 - 3) * q^30 + (3*b3 - 1) * q^31 + (3*b3 - 3*b2) * q^32 + (-2*b3 + 2*b2) * q^33 + (-6*b3 - 3) * q^34 + (-b3 + b2) * q^35 + (-2*b2 - b1) * q^36 + 7*b1 * q^37 + 2*b3 * q^38 + (b3 - 3*b2 - 5*b1 - 1) * q^39 - 3 * q^40 + 3*b2 * q^41 + (-b2 - 3*b1) * q^42 + (3*b3 - 3*b2 + 5*b1 - 5) * q^43 + (-b3 + 3) * q^44 + (-b3 + b2 + 6*b1 - 6) * q^45 + (-3*b3 + 3*b2 + 3*b1 - 3) * q^46 + (4*b3 + 6) * q^47 + (5*b3 - 5*b2 - 5*b1 + 5) * q^48 - b1 * q^49 + 2*b2 * q^50 + (-7*b3 - 9) * q^51 + (-b3 + 2*b2 - 2) * q^52 + (-4*b3 - 3) * q^53 - 4*b2 * q^54 + (-3*b2 + 3*b1) * q^55 + (b3 - b2) * q^56 + (2*b3 + 2) * q^57 + (3*b3 - 3*b2) * q^58 + (b3 - b2 + 9*b1 - 9) * q^59 + (b3 + 3) * q^60 + (-3*b3 + 3*b2 - 10*b1 + 10) * q^61 + (b2 - 9*b1) * q^62 + (-2*b2 - b1) * q^63 + q^64 + (2*b2 - 3*b1 + 6) * q^65 + 6 * q^66 + (3*b2 + b1) * q^67 + (b2 + 6*b1) * q^68 + (-4*b3 + 4*b2 + 6*b1 - 6) * q^69 + 3 * q^70 + (6*b1 - 6) * q^71 + (b3 - b2 - 6*b1 + 6) * q^72 + (-3*b3 + 2) * q^73 + (7*b3 - 7*b2) * q^74 + (2*b2 + 2*b1) * q^75 - 2*b1 * q^76 + (-b3 + 3) * q^77 + (-5*b3 + 6*b2 + 6*b1 - 9) * q^78 + (3*b3 + 11) * q^79 + 5*b2 * q^80 + (2*b2 - b1) * q^81 + (-9*b1 + 9) * q^82 + (-3*b3 + 3) * q^83 + (-b3 + b2 + b1 - 1) * q^84 + (6*b3 - 6*b2 - 3*b1 + 3) * q^85 + (5*b3 - 9) * q^86 + (3*b3 - 3*b2 - 3*b1 + 3) * q^87 + (3*b2 - 3*b1) * q^88 + (-4*b2 - 6*b1) * q^89 + (6*b3 + 3) * q^90 + (-b3 + 2*b2 - 2) * q^91 + (b3 + 3) * q^92 + (-2*b2 - 8*b1) * q^93 + (-6*b2 - 12*b1) * q^94 + (-2*b3 + 2*b2) * q^95 + (-3*b3 - 9) * q^96 + (-6*b3 + 6*b2 - 4*b1 + 4) * q^97 + (-b3 + b2) * q^98 + (5*b3 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 2 q^{4} - 6 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 2 * q^4 - 6 * q^6 - 2 * q^7 - 2 * q^9 $$4 q - 2 q^{3} - 2 q^{4} - 6 q^{6} - 2 q^{7} - 2 q^{9} - 6 q^{10} - 6 q^{11} + 4 q^{12} + 4 q^{13} - 6 q^{15} + 10 q^{16} + 12 q^{17} + 24 q^{18} - 4 q^{19} + 4 q^{21} + 6 q^{22} - 6 q^{23} + 6 q^{24} - 8 q^{25} - 18 q^{26} + 16 q^{27} - 2 q^{28} + 6 q^{29} - 6 q^{30} - 4 q^{31} - 12 q^{34} - 2 q^{36} + 14 q^{37} - 14 q^{39} - 12 q^{40} - 6 q^{42} - 10 q^{43} + 12 q^{44} - 12 q^{45} - 6 q^{46} + 24 q^{47} + 10 q^{48} - 2 q^{49} - 36 q^{51} - 8 q^{52} - 12 q^{53} + 6 q^{55} + 8 q^{57} - 18 q^{59} + 12 q^{60} + 20 q^{61} - 18 q^{62} - 2 q^{63} + 4 q^{64} + 18 q^{65} + 24 q^{66} + 2 q^{67} + 12 q^{68} - 12 q^{69} + 12 q^{70} - 12 q^{71} + 12 q^{72} + 8 q^{73} + 4 q^{75} - 4 q^{76} + 12 q^{77} - 24 q^{78} + 44 q^{79} - 2 q^{81} + 18 q^{82} + 12 q^{83} - 2 q^{84} + 6 q^{85} - 36 q^{86} + 6 q^{87} - 6 q^{88} - 12 q^{89} + 12 q^{90} - 8 q^{91} + 12 q^{92} - 16 q^{93} - 24 q^{94} - 36 q^{96} + 8 q^{97} - 12 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 2 * q^4 - 6 * q^6 - 2 * q^7 - 2 * q^9 - 6 * q^10 - 6 * q^11 + 4 * q^12 + 4 * q^13 - 6 * q^15 + 10 * q^16 + 12 * q^17 + 24 * q^18 - 4 * q^19 + 4 * q^21 + 6 * q^22 - 6 * q^23 + 6 * q^24 - 8 * q^25 - 18 * q^26 + 16 * q^27 - 2 * q^28 + 6 * q^29 - 6 * q^30 - 4 * q^31 - 12 * q^34 - 2 * q^36 + 14 * q^37 - 14 * q^39 - 12 * q^40 - 6 * q^42 - 10 * q^43 + 12 * q^44 - 12 * q^45 - 6 * q^46 + 24 * q^47 + 10 * q^48 - 2 * q^49 - 36 * q^51 - 8 * q^52 - 12 * q^53 + 6 * q^55 + 8 * q^57 - 18 * q^59 + 12 * q^60 + 20 * q^61 - 18 * q^62 - 2 * q^63 + 4 * q^64 + 18 * q^65 + 24 * q^66 + 2 * q^67 + 12 * q^68 - 12 * q^69 + 12 * q^70 - 12 * q^71 + 12 * q^72 + 8 * q^73 + 4 * q^75 - 4 * q^76 + 12 * q^77 - 24 * q^78 + 44 * q^79 - 2 * q^81 + 18 * q^82 + 12 * q^83 - 2 * q^84 + 6 * q^85 - 36 * q^86 + 6 * q^87 - 6 * q^88 - 12 * q^89 + 12 * q^90 - 8 * q^91 + 12 * q^92 - 16 * q^93 - 24 * q^94 - 36 * q^96 + 8 * q^97 - 12 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 + 1.50000i −1.36603 + 2.36603i −0.500000 0.866025i 1.73205 −2.36603 4.09808i −0.500000 0.866025i −1.73205 −2.23205 3.86603i −1.50000 + 2.59808i
22.2 0.866025 1.50000i 0.366025 0.633975i −0.500000 0.866025i −1.73205 −0.633975 1.09808i −0.500000 0.866025i 1.73205 1.23205 + 2.13397i −1.50000 + 2.59808i
29.1 −0.866025 1.50000i −1.36603 2.36603i −0.500000 + 0.866025i 1.73205 −2.36603 + 4.09808i −0.500000 + 0.866025i −1.73205 −2.23205 + 3.86603i −1.50000 2.59808i
29.2 0.866025 + 1.50000i 0.366025 + 0.633975i −0.500000 + 0.866025i −1.73205 −0.633975 + 1.09808i −0.500000 + 0.866025i 1.73205 1.23205 2.13397i −1.50000 2.59808i
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.f.b 4
3.b odd 2 1 819.2.o.b 4
4.b odd 2 1 1456.2.s.o 4
7.b odd 2 1 637.2.f.d 4
7.c even 3 1 637.2.g.e 4
7.c even 3 1 637.2.h.d 4
7.d odd 6 1 637.2.g.d 4
7.d odd 6 1 637.2.h.e 4
13.c even 3 1 inner 91.2.f.b 4
13.c even 3 1 1183.2.a.f 2
13.e even 6 1 1183.2.a.e 2
13.f odd 12 2 1183.2.c.e 4
39.i odd 6 1 819.2.o.b 4
52.j odd 6 1 1456.2.s.o 4
91.g even 3 1 637.2.h.d 4
91.h even 3 1 637.2.g.e 4
91.m odd 6 1 637.2.h.e 4
91.n odd 6 1 637.2.f.d 4
91.n odd 6 1 8281.2.a.r 2
91.t odd 6 1 8281.2.a.t 2
91.v odd 6 1 637.2.g.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.b 4 1.a even 1 1 trivial
91.2.f.b 4 13.c even 3 1 inner
637.2.f.d 4 7.b odd 2 1
637.2.f.d 4 91.n odd 6 1
637.2.g.d 4 7.d odd 6 1
637.2.g.d 4 91.v odd 6 1
637.2.g.e 4 7.c even 3 1
637.2.g.e 4 91.h even 3 1
637.2.h.d 4 7.c even 3 1
637.2.h.d 4 91.g even 3 1
637.2.h.e 4 7.d odd 6 1
637.2.h.e 4 91.m odd 6 1
819.2.o.b 4 3.b odd 2 1
819.2.o.b 4 39.i odd 6 1
1183.2.a.e 2 13.e even 6 1
1183.2.a.f 2 13.c even 3 1
1183.2.c.e 4 13.f odd 12 2
1456.2.s.o 4 4.b odd 2 1
1456.2.s.o 4 52.j odd 6 1
8281.2.a.r 2 91.n odd 6 1
8281.2.a.t 2 91.t odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$(T^{2} + T + 1)^{2}$$
$11$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$13$ $$T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169$$
$17$ $$T^{4} - 12 T^{3} + 111 T^{2} + \cdots + 1089$$
$19$ $$(T^{2} + 2 T + 4)^{2}$$
$23$ $$T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36$$
$29$ $$(T^{2} - 3 T + 9)^{2}$$
$31$ $$(T^{2} + 2 T - 26)^{2}$$
$37$ $$(T^{2} - 7 T + 49)^{2}$$
$41$ $$T^{4} + 27T^{2} + 729$$
$43$ $$T^{4} + 10 T^{3} + 102 T^{2} - 20 T + 4$$
$47$ $$(T^{2} - 12 T - 12)^{2}$$
$53$ $$(T^{2} + 6 T - 39)^{2}$$
$59$ $$T^{4} + 18 T^{3} + 246 T^{2} + \cdots + 6084$$
$61$ $$T^{4} - 20 T^{3} + 327 T^{2} + \cdots + 5329$$
$67$ $$T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676$$
$71$ $$(T^{2} + 6 T + 36)^{2}$$
$73$ $$(T^{2} - 4 T - 23)^{2}$$
$79$ $$(T^{2} - 22 T + 94)^{2}$$
$83$ $$(T^{2} - 6 T - 18)^{2}$$
$89$ $$T^{4} + 12 T^{3} + 156 T^{2} + \cdots + 144$$
$97$ $$T^{4} - 8 T^{3} + 156 T^{2} + \cdots + 8464$$