# Properties

 Label 196.6.e.a Level $196$ Weight $6$ Character orbit 196.e Analytic conductor $31.435$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,6,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, 6, names="a")

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

chi := DirichletCharacter("196.165");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$31.4352286833$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (26 \zeta_{6} - 26) q^{3} - 16 \zeta_{6} q^{5} - 433 \zeta_{6} q^{9} +O(q^{10})$$ q + (26*z - 26) * q^3 - 16*z * q^5 - 433*z * q^9 $$q + (26 \zeta_{6} - 26) q^{3} - 16 \zeta_{6} q^{5} - 433 \zeta_{6} q^{9} + (8 \zeta_{6} - 8) q^{11} + 684 q^{13} + 416 q^{15} + ( - 2218 \zeta_{6} + 2218) q^{17} + 2698 \zeta_{6} q^{19} - 3344 \zeta_{6} q^{23} + ( - 2869 \zeta_{6} + 2869) q^{25} + 4940 q^{27} - 3254 q^{29} + (4788 \zeta_{6} - 4788) q^{31} - 208 \zeta_{6} q^{33} + 11470 \zeta_{6} q^{37} + (17784 \zeta_{6} - 17784) q^{39} + 13350 q^{41} - 928 q^{43} + (6928 \zeta_{6} - 6928) q^{45} - 1212 \zeta_{6} q^{47} + 57668 \zeta_{6} q^{51} + (13110 \zeta_{6} - 13110) q^{53} + 128 q^{55} - 70148 q^{57} + (34702 \zeta_{6} - 34702) q^{59} + 1032 \zeta_{6} q^{61} - 10944 \zeta_{6} q^{65} + (10108 \zeta_{6} - 10108) q^{67} + 86944 q^{69} + 62720 q^{71} + ( - 18926 \zeta_{6} + 18926) q^{73} + 74594 \zeta_{6} q^{75} - 11400 \zeta_{6} q^{79} + (23221 \zeta_{6} - 23221) q^{81} + 88958 q^{83} - 35488 q^{85} + ( - 84604 \zeta_{6} + 84604) q^{87} - 19722 \zeta_{6} q^{89} - 124488 \zeta_{6} q^{93} + ( - 43168 \zeta_{6} + 43168) q^{95} + 17062 q^{97} + 3464 q^{99} +O(q^{100})$$ q + (26*z - 26) * q^3 - 16*z * q^5 - 433*z * q^9 + (8*z - 8) * q^11 + 684 * q^13 + 416 * q^15 + (-2218*z + 2218) * q^17 + 2698*z * q^19 - 3344*z * q^23 + (-2869*z + 2869) * q^25 + 4940 * q^27 - 3254 * q^29 + (4788*z - 4788) * q^31 - 208*z * q^33 + 11470*z * q^37 + (17784*z - 17784) * q^39 + 13350 * q^41 - 928 * q^43 + (6928*z - 6928) * q^45 - 1212*z * q^47 + 57668*z * q^51 + (13110*z - 13110) * q^53 + 128 * q^55 - 70148 * q^57 + (34702*z - 34702) * q^59 + 1032*z * q^61 - 10944*z * q^65 + (10108*z - 10108) * q^67 + 86944 * q^69 + 62720 * q^71 + (-18926*z + 18926) * q^73 + 74594*z * q^75 - 11400*z * q^79 + (23221*z - 23221) * q^81 + 88958 * q^83 - 35488 * q^85 + (-84604*z + 84604) * q^87 - 19722*z * q^89 - 124488*z * q^93 + (-43168*z + 43168) * q^95 + 17062 * q^97 + 3464 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 26 q^{3} - 16 q^{5} - 433 q^{9}+O(q^{10})$$ 2 * q - 26 * q^3 - 16 * q^5 - 433 * q^9 $$2 q - 26 q^{3} - 16 q^{5} - 433 q^{9} - 8 q^{11} + 1368 q^{13} + 832 q^{15} + 2218 q^{17} + 2698 q^{19} - 3344 q^{23} + 2869 q^{25} + 9880 q^{27} - 6508 q^{29} - 4788 q^{31} - 208 q^{33} + 11470 q^{37} - 17784 q^{39} + 26700 q^{41} - 1856 q^{43} - 6928 q^{45} - 1212 q^{47} + 57668 q^{51} - 13110 q^{53} + 256 q^{55} - 140296 q^{57} - 34702 q^{59} + 1032 q^{61} - 10944 q^{65} - 10108 q^{67} + 173888 q^{69} + 125440 q^{71} + 18926 q^{73} + 74594 q^{75} - 11400 q^{79} - 23221 q^{81} + 177916 q^{83} - 70976 q^{85} + 84604 q^{87} - 19722 q^{89} - 124488 q^{93} + 43168 q^{95} + 34124 q^{97} + 6928 q^{99}+O(q^{100})$$ 2 * q - 26 * q^3 - 16 * q^5 - 433 * q^9 - 8 * q^11 + 1368 * q^13 + 832 * q^15 + 2218 * q^17 + 2698 * q^19 - 3344 * q^23 + 2869 * q^25 + 9880 * q^27 - 6508 * q^29 - 4788 * q^31 - 208 * q^33 + 11470 * q^37 - 17784 * q^39 + 26700 * q^41 - 1856 * q^43 - 6928 * q^45 - 1212 * q^47 + 57668 * q^51 - 13110 * q^53 + 256 * q^55 - 140296 * q^57 - 34702 * q^59 + 1032 * q^61 - 10944 * q^65 - 10108 * q^67 + 173888 * q^69 + 125440 * q^71 + 18926 * q^73 + 74594 * q^75 - 11400 * q^79 - 23221 * q^81 + 177916 * q^83 - 70976 * q^85 + 84604 * q^87 - 19722 * q^89 - 124488 * q^93 + 43168 * q^95 + 34124 * q^97 + 6928 * q^99

## Character values

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

 $$n$$ $$99$$ $$101$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 −13.0000 + 22.5167i 0 −8.00000 13.8564i 0 0 0 −216.500 374.989i 0
177.1 0 −13.0000 22.5167i 0 −8.00000 + 13.8564i 0 0 0 −216.500 + 374.989i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.6.e.a 2
7.b odd 2 1 196.6.e.i 2
7.c even 3 1 28.6.a.b 1
7.c even 3 1 inner 196.6.e.a 2
7.d odd 6 1 196.6.a.a 1
7.d odd 6 1 196.6.e.i 2
21.h odd 6 1 252.6.a.a 1
28.f even 6 1 784.6.a.m 1
28.g odd 6 1 112.6.a.b 1
35.j even 6 1 700.6.a.b 1
35.l odd 12 2 700.6.e.b 2
56.k odd 6 1 448.6.a.o 1
56.p even 6 1 448.6.a.b 1
84.n even 6 1 1008.6.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.b 1 7.c even 3 1
112.6.a.b 1 28.g odd 6 1
196.6.a.a 1 7.d odd 6 1
196.6.e.a 2 1.a even 1 1 trivial
196.6.e.a 2 7.c even 3 1 inner
196.6.e.i 2 7.b odd 2 1
196.6.e.i 2 7.d odd 6 1
252.6.a.a 1 21.h odd 6 1
448.6.a.b 1 56.p even 6 1
448.6.a.o 1 56.k odd 6 1
700.6.a.b 1 35.j even 6 1
700.6.e.b 2 35.l odd 12 2
784.6.a.m 1 28.f even 6 1
1008.6.a.l 1 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 26T_{3} + 676$$ acting on $$S_{6}^{\mathrm{new}}(196, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 26T + 676$$
$5$ $$T^{2} + 16T + 256$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 8T + 64$$
$13$ $$(T - 684)^{2}$$
$17$ $$T^{2} - 2218 T + 4919524$$
$19$ $$T^{2} - 2698 T + 7279204$$
$23$ $$T^{2} + 3344 T + 11182336$$
$29$ $$(T + 3254)^{2}$$
$31$ $$T^{2} + 4788 T + 22924944$$
$37$ $$T^{2} - 11470 T + 131560900$$
$41$ $$(T - 13350)^{2}$$
$43$ $$(T + 928)^{2}$$
$47$ $$T^{2} + 1212 T + 1468944$$
$53$ $$T^{2} + 13110 T + 171872100$$
$59$ $$T^{2} + \cdots + 1204228804$$
$61$ $$T^{2} - 1032 T + 1065024$$
$67$ $$T^{2} + 10108 T + 102171664$$
$71$ $$(T - 62720)^{2}$$
$73$ $$T^{2} - 18926 T + 358193476$$
$79$ $$T^{2} + 11400 T + 129960000$$
$83$ $$(T - 88958)^{2}$$
$89$ $$T^{2} + 19722 T + 388957284$$
$97$ $$(T - 17062)^{2}$$