# Properties

 Label 91.2.g.a Level $91$ Weight $2$ Character orbit 91.g Analytic conductor $0.727$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,2,Mod(9,91)]

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([2, 4]))

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

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

chi := DirichletCharacter("91.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.g (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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-1 + \zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i −3.00000 0.500000 0.866025i −1.50000 + 2.59808i 1.50000 + 2.59808i −2.50000 + 0.866025i −3.00000 6.00000 3.00000
81.1 −0.500000 + 0.866025i −3.00000 0.500000 + 0.866025i −1.50000 2.59808i 1.50000 2.59808i −2.50000 0.866025i −3.00000 6.00000 3.00000
 $$n$$: 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
91.g 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.g.a 2
3.b odd 2 1 819.2.n.c 2
7.b odd 2 1 637.2.g.a 2
7.c even 3 1 91.2.h.a yes 2
7.c even 3 1 637.2.f.b 2
7.d odd 6 1 637.2.f.a 2
7.d odd 6 1 637.2.h.a 2
13.c even 3 1 91.2.h.a yes 2
13.c even 3 1 1183.2.e.a 2
13.e even 6 1 1183.2.e.c 2
21.h odd 6 1 819.2.s.a 2
39.i odd 6 1 819.2.s.a 2
91.g even 3 1 inner 91.2.g.a 2
91.g even 3 1 8281.2.a.i 1
91.h even 3 1 637.2.f.b 2
91.h even 3 1 1183.2.e.a 2
91.k even 6 1 1183.2.e.c 2
91.m odd 6 1 637.2.g.a 2
91.m odd 6 1 8281.2.a.j 1
91.n odd 6 1 637.2.h.a 2
91.p odd 6 1 8281.2.a.g 1
91.u even 6 1 8281.2.a.c 1
91.v odd 6 1 637.2.f.a 2
273.bm odd 6 1 819.2.n.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 1.a even 1 1 trivial
91.2.g.a 2 91.g even 3 1 inner
91.2.h.a yes 2 7.c even 3 1
91.2.h.a yes 2 13.c even 3 1
637.2.f.a 2 7.d odd 6 1
637.2.f.a 2 91.v odd 6 1
637.2.f.b 2 7.c even 3 1
637.2.f.b 2 91.h even 3 1
637.2.g.a 2 7.b odd 2 1
637.2.g.a 2 91.m odd 6 1
637.2.h.a 2 7.d odd 6 1
637.2.h.a 2 91.n odd 6 1
819.2.n.c 2 3.b odd 2 1
819.2.n.c 2 273.bm odd 6 1
819.2.s.a 2 21.h odd 6 1
819.2.s.a 2 39.i odd 6 1
1183.2.e.a 2 13.c even 3 1
1183.2.e.a 2 91.h even 3 1
1183.2.e.c 2 13.e even 6 1
1183.2.e.c 2 91.k even 6 1
8281.2.a.c 1 91.u even 6 1
8281.2.a.g 1 91.p odd 6 1
8281.2.a.i 1 91.g even 3 1
8281.2.a.j 1 91.m odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$T^{2} - 2T + 4$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 7T + 49$$
$31$ $$T^{2} + 3T + 9$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - 7T + 49$$
$47$ $$T^{2} + T + 1$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$(T + 13)^{2}$$
$67$ $$(T + 3)^{2}$$
$71$ $$T^{2} + 13T + 169$$
$73$ $$T^{2} - 13T + 169$$
$79$ $$T^{2} - 3T + 9$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$T^{2} - 5T + 25$$