# Properties

 Label 196.2.e.b Level $196$ Weight $2$ Character orbit 196.e Analytic conductor $1.565$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,2,Mod(165,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("196.165");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 196.e (of order $$3$$, degree $$2$$, not 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: $$1.56506787962$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{5} + 5 \beta_{2} q^{9}+O(q^{10})$$ q + 2*b1 * q^3 + (b3 + b1) * q^5 + 5*b2 * q^9 $$q + 2 \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{5} + 5 \beta_{2} q^{9} + ( - 4 \beta_{2} - 4) q^{11} - 3 \beta_{3} q^{13} - 4 q^{15} - \beta_1 q^{17} + (2 \beta_{3} + 2 \beta_1) q^{19} - 4 \beta_{2} q^{23} + (3 \beta_{2} + 3) q^{25} + 4 \beta_{3} q^{27} + 8 q^{29} + ( - 8 \beta_{3} - 8 \beta_1) q^{33} - 8 \beta_{2} q^{37} + (12 \beta_{2} + 12) q^{39} + 5 \beta_{3} q^{41} - 4 q^{43} - 5 \beta_1 q^{45} + (4 \beta_{3} + 4 \beta_1) q^{47} - 4 \beta_{2} q^{51} + ( - 10 \beta_{2} - 10) q^{53} - 4 \beta_{3} q^{55} - 8 q^{57} - 10 \beta_1 q^{59} + ( - 5 \beta_{3} - 5 \beta_1) q^{61} + 6 \beta_{2} q^{65} - 8 \beta_{3} q^{69} + 5 \beta_1 q^{73} + (6 \beta_{3} + 6 \beta_1) q^{75} + 8 \beta_{2} q^{79} + ( - \beta_{2} - 1) q^{81} + 10 \beta_{3} q^{83} + 2 q^{85} + 16 \beta_1 q^{87} + (5 \beta_{3} + 5 \beta_1) q^{89} + ( - 4 \beta_{2} - 4) q^{95} + \beta_{3} q^{97} + 20 q^{99}+O(q^{100})$$ q + 2*b1 * q^3 + (b3 + b1) * q^5 + 5*b2 * q^9 + (-4*b2 - 4) * q^11 - 3*b3 * q^13 - 4 * q^15 - b1 * q^17 + (2*b3 + 2*b1) * q^19 - 4*b2 * q^23 + (3*b2 + 3) * q^25 + 4*b3 * q^27 + 8 * q^29 + (-8*b3 - 8*b1) * q^33 - 8*b2 * q^37 + (12*b2 + 12) * q^39 + 5*b3 * q^41 - 4 * q^43 - 5*b1 * q^45 + (4*b3 + 4*b1) * q^47 - 4*b2 * q^51 + (-10*b2 - 10) * q^53 - 4*b3 * q^55 - 8 * q^57 - 10*b1 * q^59 + (-5*b3 - 5*b1) * q^61 + 6*b2 * q^65 - 8*b3 * q^69 + 5*b1 * q^73 + (6*b3 + 6*b1) * q^75 + 8*b2 * q^79 + (-b2 - 1) * q^81 + 10*b3 * q^83 + 2 * q^85 + 16*b1 * q^87 + (5*b3 + 5*b1) * q^89 + (-4*b2 - 4) * q^95 + b3 * q^97 + 20 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{9}+O(q^{10})$$ 4 * q - 10 * q^9 $$4 q - 10 q^{9} - 8 q^{11} - 16 q^{15} + 8 q^{23} + 6 q^{25} + 32 q^{29} + 16 q^{37} + 24 q^{39} - 16 q^{43} + 8 q^{51} - 20 q^{53} - 32 q^{57} - 12 q^{65} - 16 q^{79} - 2 q^{81} + 8 q^{85} - 8 q^{95} + 80 q^{99}+O(q^{100})$$ 4 * q - 10 * q^9 - 8 * q^11 - 16 * q^15 + 8 * q^23 + 6 * q^25 + 32 * q^29 + 16 * q^37 + 24 * q^39 - 16 * q^43 + 8 * q^51 - 20 * q^53 - 32 * q^57 - 12 * q^65 - 16 * q^79 - 2 * q^81 + 8 * q^85 - 8 * q^95 + 80 * q^99

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/196\mathbb{Z}\right)^\times$$.

 $$n$$ $$99$$ $$101$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 −1.41421 + 2.44949i 0 0.707107 + 1.22474i 0 0 0 −2.50000 4.33013i 0
165.2 0 1.41421 2.44949i 0 −0.707107 1.22474i 0 0 0 −2.50000 4.33013i 0
177.1 0 −1.41421 2.44949i 0 0.707107 1.22474i 0 0 0 −2.50000 + 4.33013i 0
177.2 0 1.41421 + 2.44949i 0 −0.707107 + 1.22474i 0 0 0 −2.50000 + 4.33013i 0
 $$n$$: 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.2.e.b 4
3.b odd 2 1 1764.2.k.l 4
4.b odd 2 1 784.2.i.l 4
7.b odd 2 1 inner 196.2.e.b 4
7.c even 3 1 196.2.a.c 2
7.c even 3 1 inner 196.2.e.b 4
7.d odd 6 1 196.2.a.c 2
7.d odd 6 1 inner 196.2.e.b 4
21.c even 2 1 1764.2.k.l 4
21.g even 6 1 1764.2.a.l 2
21.g even 6 1 1764.2.k.l 4
21.h odd 6 1 1764.2.a.l 2
21.h odd 6 1 1764.2.k.l 4
28.d even 2 1 784.2.i.l 4
28.f even 6 1 784.2.a.m 2
28.f even 6 1 784.2.i.l 4
28.g odd 6 1 784.2.a.m 2
28.g odd 6 1 784.2.i.l 4
35.i odd 6 1 4900.2.a.y 2
35.j even 6 1 4900.2.a.y 2
35.k even 12 2 4900.2.e.p 4
35.l odd 12 2 4900.2.e.p 4
56.j odd 6 1 3136.2.a.br 2
56.k odd 6 1 3136.2.a.bs 2
56.m even 6 1 3136.2.a.bs 2
56.p even 6 1 3136.2.a.br 2
84.j odd 6 1 7056.2.a.cr 2
84.n even 6 1 7056.2.a.cr 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.a.c 2 7.c even 3 1
196.2.a.c 2 7.d odd 6 1
196.2.e.b 4 1.a even 1 1 trivial
196.2.e.b 4 7.b odd 2 1 inner
196.2.e.b 4 7.c even 3 1 inner
196.2.e.b 4 7.d odd 6 1 inner
784.2.a.m 2 28.f even 6 1
784.2.a.m 2 28.g odd 6 1
784.2.i.l 4 4.b odd 2 1
784.2.i.l 4 28.d even 2 1
784.2.i.l 4 28.f even 6 1
784.2.i.l 4 28.g odd 6 1
1764.2.a.l 2 21.g even 6 1
1764.2.a.l 2 21.h odd 6 1
1764.2.k.l 4 3.b odd 2 1
1764.2.k.l 4 21.c even 2 1
1764.2.k.l 4 21.g even 6 1
1764.2.k.l 4 21.h odd 6 1
3136.2.a.br 2 56.j odd 6 1
3136.2.a.br 2 56.p even 6 1
3136.2.a.bs 2 56.k odd 6 1
3136.2.a.bs 2 56.m even 6 1
4900.2.a.y 2 35.i odd 6 1
4900.2.a.y 2 35.j even 6 1
4900.2.e.p 4 35.k even 12 2
4900.2.e.p 4 35.l odd 12 2
7056.2.a.cr 2 84.j odd 6 1
7056.2.a.cr 2 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 8T_{3}^{2} + 64$$ acting on $$S_{2}^{\mathrm{new}}(196, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 8T^{2} + 64$$
$5$ $$T^{4} + 2T^{2} + 4$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 4 T + 16)^{2}$$
$13$ $$(T^{2} - 18)^{2}$$
$17$ $$T^{4} + 2T^{2} + 4$$
$19$ $$T^{4} + 8T^{2} + 64$$
$23$ $$(T^{2} - 4 T + 16)^{2}$$
$29$ $$(T - 8)^{4}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} - 8 T + 64)^{2}$$
$41$ $$(T^{2} - 50)^{2}$$
$43$ $$(T + 4)^{4}$$
$47$ $$T^{4} + 32T^{2} + 1024$$
$53$ $$(T^{2} + 10 T + 100)^{2}$$
$59$ $$T^{4} + 200 T^{2} + 40000$$
$61$ $$T^{4} + 50T^{2} + 2500$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 50T^{2} + 2500$$
$79$ $$(T^{2} + 8 T + 64)^{2}$$
$83$ $$(T^{2} - 200)^{2}$$
$89$ $$T^{4} + 50T^{2} + 2500$$
$97$ $$(T^{2} - 2)^{2}$$