# Properties

 Label 91.2.a.a Level $91$ Weight $2$ Character orbit 91.a Self dual yes Analytic conductor $0.727$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} - 3 q^{5} - q^{7} - 3 q^{9}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 - 3 * q^5 - q^7 - 3 * q^9 $$q - 2 q^{2} + 2 q^{4} - 3 q^{5} - q^{7} - 3 q^{9} + 6 q^{10} - 6 q^{11} - q^{13} + 2 q^{14} - 4 q^{16} + 4 q^{17} + 6 q^{18} + 5 q^{19} - 6 q^{20} + 12 q^{22} + 3 q^{23} + 4 q^{25} + 2 q^{26} - 2 q^{28} - 5 q^{29} - 3 q^{31} + 8 q^{32} - 8 q^{34} + 3 q^{35} - 6 q^{36} - 4 q^{37} - 10 q^{38} - 6 q^{41} - q^{43} - 12 q^{44} + 9 q^{45} - 6 q^{46} + 7 q^{47} + q^{49} - 8 q^{50} - 2 q^{52} - 9 q^{53} + 18 q^{55} + 10 q^{58} + 8 q^{59} - 10 q^{61} + 6 q^{62} + 3 q^{63} - 8 q^{64} + 3 q^{65} - 6 q^{67} + 8 q^{68} - 6 q^{70} - 8 q^{71} - 13 q^{73} + 8 q^{74} + 10 q^{76} + 6 q^{77} + 3 q^{79} + 12 q^{80} + 9 q^{81} + 12 q^{82} + 15 q^{83} - 12 q^{85} + 2 q^{86} + 3 q^{89} - 18 q^{90} + q^{91} + 6 q^{92} - 14 q^{94} - 15 q^{95} + 7 q^{97} - 2 q^{98} + 18 q^{99}+O(q^{100})$$ q - 2 * q^2 + 2 * q^4 - 3 * q^5 - q^7 - 3 * q^9 + 6 * q^10 - 6 * q^11 - q^13 + 2 * q^14 - 4 * q^16 + 4 * q^17 + 6 * q^18 + 5 * q^19 - 6 * q^20 + 12 * q^22 + 3 * q^23 + 4 * q^25 + 2 * q^26 - 2 * q^28 - 5 * q^29 - 3 * q^31 + 8 * q^32 - 8 * q^34 + 3 * q^35 - 6 * q^36 - 4 * q^37 - 10 * q^38 - 6 * q^41 - q^43 - 12 * q^44 + 9 * q^45 - 6 * q^46 + 7 * q^47 + q^49 - 8 * q^50 - 2 * q^52 - 9 * q^53 + 18 * q^55 + 10 * q^58 + 8 * q^59 - 10 * q^61 + 6 * q^62 + 3 * q^63 - 8 * q^64 + 3 * q^65 - 6 * q^67 + 8 * q^68 - 6 * q^70 - 8 * q^71 - 13 * q^73 + 8 * q^74 + 10 * q^76 + 6 * q^77 + 3 * q^79 + 12 * q^80 + 9 * q^81 + 12 * q^82 + 15 * q^83 - 12 * q^85 + 2 * q^86 + 3 * q^89 - 18 * q^90 + q^91 + 6 * q^92 - 14 * q^94 - 15 * q^95 + 7 * q^97 - 2 * q^98 + 18 * q^99

## 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
 0
−2.00000 0 2.00000 −3.00000 0 −1.00000 0 −3.00000 6.00000
 $$n$$: 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.a 1
3.b odd 2 1 819.2.a.f 1
4.b odd 2 1 1456.2.a.g 1
5.b even 2 1 2275.2.a.h 1
7.b odd 2 1 637.2.a.a 1
7.c even 3 2 637.2.e.e 2
7.d odd 6 2 637.2.e.d 2
8.b even 2 1 5824.2.a.s 1
8.d odd 2 1 5824.2.a.t 1
13.b even 2 1 1183.2.a.b 1
13.d odd 4 2 1183.2.c.b 2
21.c even 2 1 5733.2.a.l 1
91.b odd 2 1 8281.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.a 1 1.a even 1 1 trivial
637.2.a.a 1 7.b odd 2 1
637.2.e.d 2 7.d odd 6 2
637.2.e.e 2 7.c even 3 2
819.2.a.f 1 3.b odd 2 1
1183.2.a.b 1 13.b even 2 1
1183.2.c.b 2 13.d odd 4 2
1456.2.a.g 1 4.b odd 2 1
2275.2.a.h 1 5.b even 2 1
5733.2.a.l 1 21.c even 2 1
5824.2.a.s 1 8.b even 2 1
5824.2.a.t 1 8.d odd 2 1
8281.2.a.l 1 91.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T + 3$$
$7$ $$T + 1$$
$11$ $$T + 6$$
$13$ $$T + 1$$
$17$ $$T - 4$$
$19$ $$T - 5$$
$23$ $$T - 3$$
$29$ $$T + 5$$
$31$ $$T + 3$$
$37$ $$T + 4$$
$41$ $$T + 6$$
$43$ $$T + 1$$
$47$ $$T - 7$$
$53$ $$T + 9$$
$59$ $$T - 8$$
$61$ $$T + 10$$
$67$ $$T + 6$$
$71$ $$T + 8$$
$73$ $$T + 13$$
$79$ $$T - 3$$
$83$ $$T - 15$$
$89$ $$T - 3$$
$97$ $$T - 7$$