# Properties

 Label 91.2.a.d Level $91$ Weight $2$ Character orbit 91.a Self dual yes Analytic conductor $0.727$ Analytic rank $0$ 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] = [91,2,Mod(1,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("91.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.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: $$0.726638658394$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x + 2$$ x^3 - x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

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

## 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
 −1.81361 0.470683 2.34292
−1.81361 −3.10278 1.28917 2.81361 5.62721 −1.00000 1.28917 6.62721 −5.10278
1.2 0.470683 2.24914 −1.77846 0.529317 1.05863 −1.00000 −1.77846 2.05863 0.249141
1.3 2.34292 −1.14637 3.48929 −1.34292 −2.68585 −1.00000 3.48929 −1.68585 −3.14637
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$+1$$
$$13$$ $$-1$$

## 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 91.2.a.d 3
3.b odd 2 1 819.2.a.i 3
4.b odd 2 1 1456.2.a.t 3
5.b even 2 1 2275.2.a.m 3
7.b odd 2 1 637.2.a.j 3
7.c even 3 2 637.2.e.j 6
7.d odd 6 2 637.2.e.i 6
8.b even 2 1 5824.2.a.by 3
8.d odd 2 1 5824.2.a.bs 3
13.b even 2 1 1183.2.a.i 3
13.d odd 4 2 1183.2.c.f 6
21.c even 2 1 5733.2.a.x 3
91.b odd 2 1 8281.2.a.bg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 1.a even 1 1 trivial
637.2.a.j 3 7.b odd 2 1
637.2.e.i 6 7.d odd 6 2
637.2.e.j 6 7.c even 3 2
819.2.a.i 3 3.b odd 2 1
1183.2.a.i 3 13.b even 2 1
1183.2.c.f 6 13.d odd 4 2
1456.2.a.t 3 4.b odd 2 1
2275.2.a.m 3 5.b even 2 1
5733.2.a.x 3 21.c even 2 1
5824.2.a.bs 3 8.d odd 2 1
5824.2.a.by 3 8.b even 2 1
8281.2.a.bg 3 91.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - T_{2}^{2} - 4T_{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(91))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - T^{2} - 4T + 2$$
$3$ $$T^{3} + 2 T^{2} + \cdots - 8$$
$5$ $$T^{3} - 2 T^{2} + \cdots + 2$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} - 2 T^{2} + \cdots + 8$$
$13$ $$(T - 1)^{3}$$
$17$ $$T^{3} - 4 T^{2} + \cdots - 4$$
$19$ $$T^{3} + 4T^{2} + T - 4$$
$23$ $$T^{3} - 10 T^{2} + \cdots + 136$$
$29$ $$T^{3} - 24 T^{2} + \cdots - 454$$
$31$ $$T^{3} + 4 T^{2} + \cdots + 16$$
$37$ $$T^{3} - 58T - 124$$
$41$ $$T^{3} - 2 T^{2} + \cdots - 8$$
$43$ $$T^{3} - 10 T^{2} + \cdots + 628$$
$47$ $$T^{3} + 8 T^{2} + \cdots - 544$$
$53$ $$T^{3} - 8 T^{2} + \cdots - 22$$
$59$ $$T^{3} + 4 T^{2} + \cdots - 688$$
$61$ $$(T + 2)^{3}$$
$67$ $$T^{3} + 12 T^{2} + \cdots - 976$$
$71$ $$T^{3} + 6 T^{2} + \cdots + 16$$
$73$ $$T^{3} + 10 T^{2} + \cdots - 274$$
$79$ $$T^{3} + 14 T^{2} + \cdots - 16$$
$83$ $$T^{3} + 12 T^{2} + \cdots - 3268$$
$89$ $$T^{3} - 2 T^{2} + \cdots + 422$$
$97$ $$T^{3} + 10 T^{2} + \cdots + 22$$