# Properties

 Label 196.4.e.a Level $196$ Weight $4$ Character orbit 196.e Analytic conductor $11.564$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 196.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.5643743611$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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

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.

 $$f(q)$$ $$=$$ $$q + ( -10 + 10 \zeta_{6} ) q^{3} -8 \zeta_{6} q^{5} -73 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -10 + 10 \zeta_{6} ) q^{3} -8 \zeta_{6} q^{5} -73 \zeta_{6} q^{9} + ( 40 - 40 \zeta_{6} ) q^{11} + 12 q^{13} + 80 q^{15} + ( -58 + 58 \zeta_{6} ) q^{17} + 26 \zeta_{6} q^{19} + 64 \zeta_{6} q^{23} + ( 61 - 61 \zeta_{6} ) q^{25} + 460 q^{27} -62 q^{29} + ( 252 - 252 \zeta_{6} ) q^{31} + 400 \zeta_{6} q^{33} -26 \zeta_{6} q^{37} + ( -120 + 120 \zeta_{6} ) q^{39} -6 q^{41} + 416 q^{43} + ( -584 + 584 \zeta_{6} ) q^{45} -396 \zeta_{6} q^{47} -580 \zeta_{6} q^{51} + ( 450 - 450 \zeta_{6} ) q^{53} -320 q^{55} -260 q^{57} + ( 274 - 274 \zeta_{6} ) q^{59} -576 \zeta_{6} q^{61} -96 \zeta_{6} q^{65} + ( 476 - 476 \zeta_{6} ) q^{67} -640 q^{69} -448 q^{71} + ( -158 + 158 \zeta_{6} ) q^{73} + 610 \zeta_{6} q^{75} + 936 \zeta_{6} q^{79} + ( -2629 + 2629 \zeta_{6} ) q^{81} -530 q^{83} + 464 q^{85} + ( 620 - 620 \zeta_{6} ) q^{87} -390 \zeta_{6} q^{89} + 2520 \zeta_{6} q^{93} + ( 208 - 208 \zeta_{6} ) q^{95} -214 q^{97} -2920 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 10q^{3} - 8q^{5} - 73q^{9} + O(q^{10})$$ $$2q - 10q^{3} - 8q^{5} - 73q^{9} + 40q^{11} + 24q^{13} + 160q^{15} - 58q^{17} + 26q^{19} + 64q^{23} + 61q^{25} + 920q^{27} - 124q^{29} + 252q^{31} + 400q^{33} - 26q^{37} - 120q^{39} - 12q^{41} + 832q^{43} - 584q^{45} - 396q^{47} - 580q^{51} + 450q^{53} - 640q^{55} - 520q^{57} + 274q^{59} - 576q^{61} - 96q^{65} + 476q^{67} - 1280q^{69} - 896q^{71} - 158q^{73} + 610q^{75} + 936q^{79} - 2629q^{81} - 1060q^{83} + 928q^{85} + 620q^{87} - 390q^{89} + 2520q^{93} + 208q^{95} - 428q^{97} - 5840q^{99} + O(q^{100})$$

## 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.

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 −5.00000 + 8.66025i 0 −4.00000 6.92820i 0 0 0 −36.5000 63.2199i 0
177.1 0 −5.00000 8.66025i 0 −4.00000 + 6.92820i 0 0 0 −36.5000 + 63.2199i 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 196.4.e.a 2
3.b odd 2 1 1764.4.k.m 2
7.b odd 2 1 196.4.e.f 2
7.c even 3 1 196.4.a.d 1
7.c even 3 1 inner 196.4.e.a 2
7.d odd 6 1 28.4.a.a 1
7.d odd 6 1 196.4.e.f 2
21.c even 2 1 1764.4.k.d 2
21.g even 6 1 252.4.a.d 1
21.g even 6 1 1764.4.k.d 2
21.h odd 6 1 1764.4.a.c 1
21.h odd 6 1 1764.4.k.m 2
28.f even 6 1 112.4.a.g 1
28.g odd 6 1 784.4.a.a 1
35.i odd 6 1 700.4.a.n 1
35.k even 12 2 700.4.e.a 2
56.j odd 6 1 448.4.a.p 1
56.m even 6 1 448.4.a.a 1
84.j odd 6 1 1008.4.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 7.d odd 6 1
112.4.a.g 1 28.f even 6 1
196.4.a.d 1 7.c even 3 1
196.4.e.a 2 1.a even 1 1 trivial
196.4.e.a 2 7.c even 3 1 inner
196.4.e.f 2 7.b odd 2 1
196.4.e.f 2 7.d odd 6 1
252.4.a.d 1 21.g even 6 1
448.4.a.a 1 56.m even 6 1
448.4.a.p 1 56.j odd 6 1
700.4.a.n 1 35.i odd 6 1
700.4.e.a 2 35.k even 12 2
784.4.a.a 1 28.g odd 6 1
1008.4.a.o 1 84.j odd 6 1
1764.4.a.c 1 21.h odd 6 1
1764.4.k.d 2 21.c even 2 1
1764.4.k.d 2 21.g even 6 1
1764.4.k.m 2 3.b odd 2 1
1764.4.k.m 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(196, [\chi])$$:

 $$T_{3}^{2} + 10 T_{3} + 100$$ $$T_{5}^{2} + 8 T_{5} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$100 + 10 T + T^{2}$$
$5$ $$64 + 8 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1600 - 40 T + T^{2}$$
$13$ $$( -12 + T )^{2}$$
$17$ $$3364 + 58 T + T^{2}$$
$19$ $$676 - 26 T + T^{2}$$
$23$ $$4096 - 64 T + T^{2}$$
$29$ $$( 62 + T )^{2}$$
$31$ $$63504 - 252 T + T^{2}$$
$37$ $$676 + 26 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -416 + T )^{2}$$
$47$ $$156816 + 396 T + T^{2}$$
$53$ $$202500 - 450 T + T^{2}$$
$59$ $$75076 - 274 T + T^{2}$$
$61$ $$331776 + 576 T + T^{2}$$
$67$ $$226576 - 476 T + T^{2}$$
$71$ $$( 448 + T )^{2}$$
$73$ $$24964 + 158 T + T^{2}$$
$79$ $$876096 - 936 T + T^{2}$$
$83$ $$( 530 + T )^{2}$$
$89$ $$152100 + 390 T + T^{2}$$
$97$ $$( 214 + T )^{2}$$