# Properties

 Label 196.2.a.b Level $196$ Weight $2$ Character orbit 196.a Self dual yes Analytic conductor $1.565$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,2,Mod(1,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, 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("196.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 196.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: $$1.56506787962$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + 3 q^{5} - 2 q^{9}+O(q^{10})$$ q + q^3 + 3 * q^5 - 2 * q^9 $$q + q^{3} + 3 q^{5} - 2 q^{9} - 3 q^{11} + 2 q^{13} + 3 q^{15} + 3 q^{17} - q^{19} + 3 q^{23} + 4 q^{25} - 5 q^{27} - 6 q^{29} - 7 q^{31} - 3 q^{33} - q^{37} + 2 q^{39} + 6 q^{41} - 4 q^{43} - 6 q^{45} - 9 q^{47} + 3 q^{51} + 3 q^{53} - 9 q^{55} - q^{57} + 9 q^{59} - q^{61} + 6 q^{65} - 7 q^{67} + 3 q^{69} - q^{73} + 4 q^{75} - 13 q^{79} + q^{81} + 12 q^{83} + 9 q^{85} - 6 q^{87} + 15 q^{89} - 7 q^{93} - 3 q^{95} - 10 q^{97} + 6 q^{99}+O(q^{100})$$ q + q^3 + 3 * q^5 - 2 * q^9 - 3 * q^11 + 2 * q^13 + 3 * q^15 + 3 * q^17 - q^19 + 3 * q^23 + 4 * q^25 - 5 * q^27 - 6 * q^29 - 7 * q^31 - 3 * q^33 - q^37 + 2 * q^39 + 6 * q^41 - 4 * q^43 - 6 * q^45 - 9 * q^47 + 3 * q^51 + 3 * q^53 - 9 * q^55 - q^57 + 9 * q^59 - q^61 + 6 * q^65 - 7 * q^67 + 3 * q^69 - q^73 + 4 * q^75 - 13 * q^79 + q^81 + 12 * q^83 + 9 * q^85 - 6 * q^87 + 15 * q^89 - 7 * q^93 - 3 * q^95 - 10 * q^97 + 6 * 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
0 1.00000 0 3.00000 0 0 0 −2.00000 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
$$2$$ $$-1$$
$$7$$ $$+1$$

## 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 196.2.a.b 1
3.b odd 2 1 1764.2.a.a 1
4.b odd 2 1 784.2.a.d 1
5.b even 2 1 4900.2.a.g 1
5.c odd 4 2 4900.2.e.i 2
7.b odd 2 1 196.2.a.a 1
7.c even 3 2 28.2.e.a 2
7.d odd 6 2 196.2.e.a 2
8.b even 2 1 3136.2.a.h 1
8.d odd 2 1 3136.2.a.s 1
12.b even 2 1 7056.2.a.f 1
21.c even 2 1 1764.2.a.j 1
21.g even 6 2 1764.2.k.b 2
21.h odd 6 2 252.2.k.c 2
28.d even 2 1 784.2.a.g 1
28.f even 6 2 784.2.i.d 2
28.g odd 6 2 112.2.i.b 2
35.c odd 2 1 4900.2.a.n 1
35.f even 4 2 4900.2.e.h 2
35.j even 6 2 700.2.i.c 2
35.l odd 12 4 700.2.r.b 4
56.e even 2 1 3136.2.a.k 1
56.h odd 2 1 3136.2.a.v 1
56.k odd 6 2 448.2.i.c 2
56.p even 6 2 448.2.i.e 2
63.g even 3 2 2268.2.l.h 2
63.h even 3 2 2268.2.i.a 2
63.j odd 6 2 2268.2.i.h 2
63.n odd 6 2 2268.2.l.a 2
84.h odd 2 1 7056.2.a.bw 1
84.n even 6 2 1008.2.s.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.e.a 2 7.c even 3 2
112.2.i.b 2 28.g odd 6 2
196.2.a.a 1 7.b odd 2 1
196.2.a.b 1 1.a even 1 1 trivial
196.2.e.a 2 7.d odd 6 2
252.2.k.c 2 21.h odd 6 2
448.2.i.c 2 56.k odd 6 2
448.2.i.e 2 56.p even 6 2
700.2.i.c 2 35.j even 6 2
700.2.r.b 4 35.l odd 12 4
784.2.a.d 1 4.b odd 2 1
784.2.a.g 1 28.d even 2 1
784.2.i.d 2 28.f even 6 2
1008.2.s.p 2 84.n even 6 2
1764.2.a.a 1 3.b odd 2 1
1764.2.a.j 1 21.c even 2 1
1764.2.k.b 2 21.g even 6 2
2268.2.i.a 2 63.h even 3 2
2268.2.i.h 2 63.j odd 6 2
2268.2.l.a 2 63.n odd 6 2
2268.2.l.h 2 63.g even 3 2
3136.2.a.h 1 8.b even 2 1
3136.2.a.k 1 56.e even 2 1
3136.2.a.s 1 8.d odd 2 1
3136.2.a.v 1 56.h odd 2 1
4900.2.a.g 1 5.b even 2 1
4900.2.a.n 1 35.c odd 2 1
4900.2.e.h 2 35.f even 4 2
4900.2.e.i 2 5.c odd 4 2
7056.2.a.f 1 12.b even 2 1
7056.2.a.bw 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(196))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 3$$
$7$ $$T$$
$11$ $$T + 3$$
$13$ $$T - 2$$
$17$ $$T - 3$$
$19$ $$T + 1$$
$23$ $$T - 3$$
$29$ $$T + 6$$
$31$ $$T + 7$$
$37$ $$T + 1$$
$41$ $$T - 6$$
$43$ $$T + 4$$
$47$ $$T + 9$$
$53$ $$T - 3$$
$59$ $$T - 9$$
$61$ $$T + 1$$
$67$ $$T + 7$$
$71$ $$T$$
$73$ $$T + 1$$
$79$ $$T + 13$$
$83$ $$T - 12$$
$89$ $$T - 15$$
$97$ $$T + 10$$
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