# Properties

 Label 196.6.e.e 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} - 96 \zeta_{6} q^{5} + 239 \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 - 96*z * q^5 + 239*z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} - 96 \zeta_{6} q^{5} + 239 \zeta_{6} q^{9} + ( - 720 \zeta_{6} + 720) q^{11} - 572 q^{13} + 192 q^{15} + ( - 1254 \zeta_{6} + 1254) q^{17} - 94 \zeta_{6} q^{19} - 96 \zeta_{6} q^{23} + (6091 \zeta_{6} - 6091) q^{25} - 964 q^{27} - 4374 q^{29} + (6244 \zeta_{6} - 6244) q^{31} + 1440 \zeta_{6} q^{33} + 10798 \zeta_{6} q^{37} + ( - 1144 \zeta_{6} + 1144) q^{39} - 12006 q^{41} - 9160 q^{43} + ( - 22944 \zeta_{6} + 22944) q^{45} - 25836 \zeta_{6} q^{47} + 2508 \zeta_{6} q^{51} + (1014 \zeta_{6} - 1014) q^{53} - 69120 q^{55} + 188 q^{57} + ( - 1242 \zeta_{6} + 1242) q^{59} + 7592 \zeta_{6} q^{61} + 54912 \zeta_{6} q^{65} + (41132 \zeta_{6} - 41132) q^{67} + 192 q^{69} - 37632 q^{71} + (13438 \zeta_{6} - 13438) q^{73} - 12182 \zeta_{6} q^{75} - 6248 \zeta_{6} q^{79} + (56149 \zeta_{6} - 56149) q^{81} + 25254 q^{83} - 120384 q^{85} + ( - 8748 \zeta_{6} + 8748) q^{87} - 45126 \zeta_{6} q^{89} - 12488 \zeta_{6} q^{93} + (9024 \zeta_{6} - 9024) q^{95} - 107222 q^{97} + 172080 q^{99} +O(q^{100})$$ q + (2*z - 2) * q^3 - 96*z * q^5 + 239*z * q^9 + (-720*z + 720) * q^11 - 572 * q^13 + 192 * q^15 + (-1254*z + 1254) * q^17 - 94*z * q^19 - 96*z * q^23 + (6091*z - 6091) * q^25 - 964 * q^27 - 4374 * q^29 + (6244*z - 6244) * q^31 + 1440*z * q^33 + 10798*z * q^37 + (-1144*z + 1144) * q^39 - 12006 * q^41 - 9160 * q^43 + (-22944*z + 22944) * q^45 - 25836*z * q^47 + 2508*z * q^51 + (1014*z - 1014) * q^53 - 69120 * q^55 + 188 * q^57 + (-1242*z + 1242) * q^59 + 7592*z * q^61 + 54912*z * q^65 + (41132*z - 41132) * q^67 + 192 * q^69 - 37632 * q^71 + (13438*z - 13438) * q^73 - 12182*z * q^75 - 6248*z * q^79 + (56149*z - 56149) * q^81 + 25254 * q^83 - 120384 * q^85 + (-8748*z + 8748) * q^87 - 45126*z * q^89 - 12488*z * q^93 + (9024*z - 9024) * q^95 - 107222 * q^97 + 172080 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 96 q^{5} + 239 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 96 * q^5 + 239 * q^9 $$2 q - 2 q^{3} - 96 q^{5} + 239 q^{9} + 720 q^{11} - 1144 q^{13} + 384 q^{15} + 1254 q^{17} - 94 q^{19} - 96 q^{23} - 6091 q^{25} - 1928 q^{27} - 8748 q^{29} - 6244 q^{31} + 1440 q^{33} + 10798 q^{37} + 1144 q^{39} - 24012 q^{41} - 18320 q^{43} + 22944 q^{45} - 25836 q^{47} + 2508 q^{51} - 1014 q^{53} - 138240 q^{55} + 376 q^{57} + 1242 q^{59} + 7592 q^{61} + 54912 q^{65} - 41132 q^{67} + 384 q^{69} - 75264 q^{71} - 13438 q^{73} - 12182 q^{75} - 6248 q^{79} - 56149 q^{81} + 50508 q^{83} - 240768 q^{85} + 8748 q^{87} - 45126 q^{89} - 12488 q^{93} - 9024 q^{95} - 214444 q^{97} + 344160 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 96 * q^5 + 239 * q^9 + 720 * q^11 - 1144 * q^13 + 384 * q^15 + 1254 * q^17 - 94 * q^19 - 96 * q^23 - 6091 * q^25 - 1928 * q^27 - 8748 * q^29 - 6244 * q^31 + 1440 * q^33 + 10798 * q^37 + 1144 * q^39 - 24012 * q^41 - 18320 * q^43 + 22944 * q^45 - 25836 * q^47 + 2508 * q^51 - 1014 * q^53 - 138240 * q^55 + 376 * q^57 + 1242 * q^59 + 7592 * q^61 + 54912 * q^65 - 41132 * q^67 + 384 * q^69 - 75264 * q^71 - 13438 * q^73 - 12182 * q^75 - 6248 * q^79 - 56149 * q^81 + 50508 * q^83 - 240768 * q^85 + 8748 * q^87 - 45126 * q^89 - 12488 * q^93 - 9024 * q^95 - 214444 * q^97 + 344160 * 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 −1.00000 + 1.73205i 0 −48.0000 83.1384i 0 0 0 119.500 + 206.980i 0
177.1 0 −1.00000 1.73205i 0 −48.0000 + 83.1384i 0 0 0 119.500 206.980i 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.e 2
7.b odd 2 1 196.6.e.f 2
7.c even 3 1 196.6.a.d 1
7.c even 3 1 inner 196.6.e.e 2
7.d odd 6 1 28.6.a.a 1
7.d odd 6 1 196.6.e.f 2
21.g even 6 1 252.6.a.d 1
28.f even 6 1 112.6.a.e 1
28.g odd 6 1 784.6.a.f 1
35.i odd 6 1 700.6.a.d 1
35.k even 12 2 700.6.e.d 2
56.j odd 6 1 448.6.a.i 1
56.m even 6 1 448.6.a.h 1
84.j odd 6 1 1008.6.a.bb 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 2T_{3} + 4$$ acting on $$S_{6}^{\mathrm{new}}(196, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + 96T + 9216$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 720T + 518400$$
$13$ $$(T + 572)^{2}$$
$17$ $$T^{2} - 1254 T + 1572516$$
$19$ $$T^{2} + 94T + 8836$$
$23$ $$T^{2} + 96T + 9216$$
$29$ $$(T + 4374)^{2}$$
$31$ $$T^{2} + 6244 T + 38987536$$
$37$ $$T^{2} - 10798 T + 116596804$$
$41$ $$(T + 12006)^{2}$$
$43$ $$(T + 9160)^{2}$$
$47$ $$T^{2} + 25836 T + 667498896$$
$53$ $$T^{2} + 1014 T + 1028196$$
$59$ $$T^{2} - 1242 T + 1542564$$
$61$ $$T^{2} - 7592 T + 57638464$$
$67$ $$T^{2} + \cdots + 1691841424$$
$71$ $$(T + 37632)^{2}$$
$73$ $$T^{2} + 13438 T + 180579844$$
$79$ $$T^{2} + 6248 T + 39037504$$
$83$ $$(T - 25254)^{2}$$
$89$ $$T^{2} + \cdots + 2036355876$$
$97$ $$(T + 107222)^{2}$$