# Properties

 Label 91.2.a.c Level $91$ Weight $2$ Character orbit 91.a Self dual yes Analytic conductor $0.727$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−1.41421 1.41421 0 1.58579 −2.00000 1.00000 2.82843 −1.00000 −2.24264
1.2 1.41421 −1.41421 0 4.41421 −2.00000 1.00000 −2.82843 −1.00000 6.24264
 $$n$$: 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.c 2
3.b odd 2 1 819.2.a.h 2
4.b odd 2 1 1456.2.a.q 2
5.b even 2 1 2275.2.a.j 2
7.b odd 2 1 637.2.a.g 2
7.c even 3 2 637.2.e.f 4
7.d odd 6 2 637.2.e.g 4
8.b even 2 1 5824.2.a.bl 2
8.d odd 2 1 5824.2.a.bk 2
13.b even 2 1 1183.2.a.d 2
13.d odd 4 2 1183.2.c.d 4
21.c even 2 1 5733.2.a.s 2
91.b odd 2 1 8281.2.a.v 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2} - 6T + 7$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 18$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 2$$
$19$ $$T^{2} + 6T - 9$$
$23$ $$T^{2} + 6T + 1$$
$29$ $$T^{2} - 6T + 1$$
$31$ $$T^{2} + 2T - 17$$
$37$ $$T^{2} + 4T - 14$$
$41$ $$T^{2} - 12T + 28$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2} - 6T + 7$$
$53$ $$T^{2} + 6T + 1$$
$59$ $$T^{2} - 12T + 4$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2} + 12T - 36$$
$71$ $$T^{2} + 12T - 14$$
$73$ $$T^{2} + 10T + 7$$
$79$ $$T^{2} - 14T - 23$$
$83$ $$T^{2} - 18T + 63$$
$89$ $$T^{2} - 6T + 7$$
$97$ $$T^{2} + 2T - 161$$