# Properties

 Label 91.2.e.b Level $91$ Weight $2$ Character orbit 91.e Analytic conductor $0.727$ 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] = [91,2,Mod(53,91)]

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([4, 0]))

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

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

chi := DirichletCharacter("91.53");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.e (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: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 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.

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

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

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

## Character values

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

 $$n$$ $$15$$ $$66$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −0.309017 − 0.535233i 0.809017 + 1.40126i −0.309017 + 0.535233i 0.809017 − 1.40126i
0.190983 + 0.330792i −1.11803 + 1.93649i 0.927051 1.60570i 1.11803 + 1.93649i −0.854102 −2.00000 + 1.73205i 1.47214 −1.00000 1.73205i −0.427051 + 0.739674i
53.2 1.30902 + 2.26728i 1.11803 1.93649i −2.42705 + 4.20378i −1.11803 1.93649i 5.85410 −2.00000 + 1.73205i −7.47214 −1.00000 1.73205i 2.92705 5.06980i
79.1 0.190983 0.330792i −1.11803 1.93649i 0.927051 + 1.60570i 1.11803 1.93649i −0.854102 −2.00000 1.73205i 1.47214 −1.00000 + 1.73205i −0.427051 0.739674i
79.2 1.30902 2.26728i 1.11803 + 1.93649i −2.42705 4.20378i −1.11803 + 1.93649i 5.85410 −2.00000 1.73205i −7.47214 −1.00000 + 1.73205i 2.92705 + 5.06980i
 $$n$$: 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
7.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.e.b 4
3.b odd 2 1 819.2.j.c 4
4.b odd 2 1 1456.2.r.j 4
7.b odd 2 1 637.2.e.h 4
7.c even 3 1 inner 91.2.e.b 4
7.c even 3 1 637.2.a.f 2
7.d odd 6 1 637.2.a.e 2
7.d odd 6 1 637.2.e.h 4
13.b even 2 1 1183.2.e.d 4
21.g even 6 1 5733.2.a.w 2
21.h odd 6 1 819.2.j.c 4
21.h odd 6 1 5733.2.a.v 2
28.g odd 6 1 1456.2.r.j 4
91.r even 6 1 1183.2.e.d 4
91.r even 6 1 8281.2.a.z 2
91.s odd 6 1 8281.2.a.ba 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.b 4 1.a even 1 1 trivial
91.2.e.b 4 7.c even 3 1 inner
637.2.a.e 2 7.d odd 6 1
637.2.a.f 2 7.c even 3 1
637.2.e.h 4 7.b odd 2 1
637.2.e.h 4 7.d odd 6 1
819.2.j.c 4 3.b odd 2 1
819.2.j.c 4 21.h odd 6 1
1183.2.e.d 4 13.b even 2 1
1183.2.e.d 4 91.r even 6 1
1456.2.r.j 4 4.b odd 2 1
1456.2.r.j 4 28.g odd 6 1
5733.2.a.v 2 21.h odd 6 1
5733.2.a.w 2 21.g even 6 1
8281.2.a.z 2 91.r even 6 1
8281.2.a.ba 2 91.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 3T_{2}^{3} + 8T_{2}^{2} - 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(91, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 3 T^{3} + \cdots + 1$$
$3$ $$T^{4} + 5T^{2} + 25$$
$5$ $$T^{4} + 5T^{2} + 25$$
$7$ $$(T^{2} + 4 T + 7)^{2}$$
$11$ $$(T^{2} - 3 T + 9)^{2}$$
$13$ $$(T + 1)^{4}$$
$17$ $$T^{4} - 6 T^{3} + \cdots + 121$$
$19$ $$(T^{2} + 3 T + 9)^{2}$$
$23$ $$T^{4} - 12 T^{3} + \cdots + 961$$
$29$ $$(T^{2} - 20)^{2}$$
$31$ $$(T^{2} + 5 T + 25)^{2}$$
$37$ $$T^{4} - 4 T^{3} + \cdots + 1681$$
$41$ $$(T^{2} - 20)^{2}$$
$43$ $$(T + 8)^{4}$$
$47$ $$T^{4} - 6 T^{3} + \cdots + 121$$
$53$ $$T^{4} - 6 T^{3} + \cdots + 121$$
$59$ $$T^{4} + 6 T^{3} + \cdots + 121$$
$61$ $$(T^{2} + 3 T + 9)^{2}$$
$67$ $$(T^{2} - 3 T + 9)^{2}$$
$71$ $$(T^{2} - 80)^{2}$$
$73$ $$T^{4} + 8 T^{3} + \cdots + 841$$
$79$ $$T^{4} + 8 T^{3} + \cdots + 841$$
$83$ $$T^{4}$$
$89$ $$T^{4} + 5T^{2} + 25$$
$97$ $$(T^{2} + 8 T - 164)^{2}$$