# Properties

 Label 3696.2.a.bl Level $3696$ Weight $2$ Character orbit 3696.a Self dual yes Analytic conductor $29.513$ 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] = [3696,2,Mod(1,3696)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3696, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 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("3696.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3696.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: $$29.5127085871$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 231) 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 $$\beta = \sqrt{21}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + 3 q^{5} - q^{7} + q^{9}+O(q^{10})$$ q + q^3 + 3 * q^5 - q^7 + q^9 $$q + q^{3} + 3 q^{5} - q^{7} + q^{9} + q^{11} + q^{13} + 3 q^{15} + ( - \beta + 3) q^{17} + ( - \beta + 2) q^{19} - q^{21} + ( - \beta + 1) q^{23} + 4 q^{25} + q^{27} + (2 \beta + 1) q^{29} + (\beta + 1) q^{31} + q^{33} - 3 q^{35} + q^{37} + q^{39} + (2 \beta - 2) q^{41} + (\beta + 3) q^{43} + 3 q^{45} + ( - \beta - 6) q^{47} + q^{49} + ( - \beta + 3) q^{51} + (\beta - 5) q^{53} + 3 q^{55} + ( - \beta + 2) q^{57} + \beta q^{59} + 10 q^{61} - q^{63} + 3 q^{65} + (\beta - 4) q^{67} + ( - \beta + 1) q^{69} + (2 \beta - 2) q^{71} + 7 q^{73} + 4 q^{75} - q^{77} + (2 \beta + 2) q^{79} + q^{81} + ( - \beta + 7) q^{83} + ( - 3 \beta + 9) q^{85} + (2 \beta + 1) q^{87} - 2 \beta q^{89} - q^{91} + (\beta + 1) q^{93} + ( - 3 \beta + 6) q^{95} + ( - \beta - 7) q^{97} + q^{99} +O(q^{100})$$ q + q^3 + 3 * q^5 - q^7 + q^9 + q^11 + q^13 + 3 * q^15 + (-b + 3) * q^17 + (-b + 2) * q^19 - q^21 + (-b + 1) * q^23 + 4 * q^25 + q^27 + (2*b + 1) * q^29 + (b + 1) * q^31 + q^33 - 3 * q^35 + q^37 + q^39 + (2*b - 2) * q^41 + (b + 3) * q^43 + 3 * q^45 + (-b - 6) * q^47 + q^49 + (-b + 3) * q^51 + (b - 5) * q^53 + 3 * q^55 + (-b + 2) * q^57 + b * q^59 + 10 * q^61 - q^63 + 3 * q^65 + (b - 4) * q^67 + (-b + 1) * q^69 + (2*b - 2) * q^71 + 7 * q^73 + 4 * q^75 - q^77 + (2*b + 2) * q^79 + q^81 + (-b + 7) * q^83 + (-3*b + 9) * q^85 + (2*b + 1) * q^87 - 2*b * q^89 - q^91 + (b + 1) * q^93 + (-3*b + 6) * q^95 + (-b - 7) * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 6 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{13} + 6 q^{15} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} + 2 q^{27} + 2 q^{29} + 2 q^{31} + 2 q^{33} - 6 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41} + 6 q^{43} + 6 q^{45} - 12 q^{47} + 2 q^{49} + 6 q^{51} - 10 q^{53} + 6 q^{55} + 4 q^{57} + 20 q^{61} - 2 q^{63} + 6 q^{65} - 8 q^{67} + 2 q^{69} - 4 q^{71} + 14 q^{73} + 8 q^{75} - 2 q^{77} + 4 q^{79} + 2 q^{81} + 14 q^{83} + 18 q^{85} + 2 q^{87} - 2 q^{91} + 2 q^{93} + 12 q^{95} - 14 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 6 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^13 + 6 * q^15 + 6 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 + 2 * q^27 + 2 * q^29 + 2 * q^31 + 2 * q^33 - 6 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 + 6 * q^43 + 6 * q^45 - 12 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 6 * q^55 + 4 * q^57 + 20 * q^61 - 2 * q^63 + 6 * q^65 - 8 * q^67 + 2 * q^69 - 4 * q^71 + 14 * q^73 + 8 * q^75 - 2 * q^77 + 4 * q^79 + 2 * q^81 + 14 * q^83 + 18 * q^85 + 2 * q^87 - 2 * q^91 + 2 * q^93 + 12 * q^95 - 14 * q^97 + 2 * 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
 2.79129 −1.79129
0 1.00000 0 3.00000 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 3.00000 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$+1$$
$$11$$ $$-1$$

## 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 3696.2.a.bl 2
4.b odd 2 1 231.2.a.b 2
12.b even 2 1 693.2.a.j 2
20.d odd 2 1 5775.2.a.bn 2
28.d even 2 1 1617.2.a.o 2
44.c even 2 1 2541.2.a.z 2
84.h odd 2 1 4851.2.a.ba 2
132.d odd 2 1 7623.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 4.b odd 2 1
693.2.a.j 2 12.b even 2 1
1617.2.a.o 2 28.d even 2 1
2541.2.a.z 2 44.c even 2 1
3696.2.a.bl 2 1.a even 1 1 trivial
4851.2.a.ba 2 84.h odd 2 1
5775.2.a.bn 2 20.d odd 2 1
7623.2.a.bf 2 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3696))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{13} - 1$$ T13 - 1 $$T_{17}^{2} - 6T_{17} - 12$$ T17^2 - 6*T17 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 3)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 6T - 12$$
$19$ $$T^{2} - 4T - 17$$
$23$ $$T^{2} - 2T - 20$$
$29$ $$T^{2} - 2T - 83$$
$31$ $$T^{2} - 2T - 20$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} + 4T - 80$$
$43$ $$T^{2} - 6T - 12$$
$47$ $$T^{2} + 12T + 15$$
$53$ $$T^{2} + 10T + 4$$
$59$ $$T^{2} - 21$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 8T - 5$$
$71$ $$T^{2} + 4T - 80$$
$73$ $$(T - 7)^{2}$$
$79$ $$T^{2} - 4T - 80$$
$83$ $$T^{2} - 14T + 28$$
$89$ $$T^{2} - 84$$
$97$ $$T^{2} + 14T + 28$$