# Properties

 Label 196.4.a.b Level $196$ Weight $4$ Character orbit 196.a Self dual yes Analytic conductor $11.564$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 196.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.5643743611$$ 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

 $$f(q)$$ $$=$$ $$q - 4 q^{3} - 6 q^{5} - 11 q^{9}+O(q^{10})$$ q - 4 * q^3 - 6 * q^5 - 11 * q^9 $$q - 4 q^{3} - 6 q^{5} - 11 q^{9} - 12 q^{11} + 82 q^{13} + 24 q^{15} + 30 q^{17} - 68 q^{19} + 216 q^{23} - 89 q^{25} + 152 q^{27} + 246 q^{29} + 112 q^{31} + 48 q^{33} + 110 q^{37} - 328 q^{39} + 246 q^{41} - 172 q^{43} + 66 q^{45} - 192 q^{47} - 120 q^{51} + 558 q^{53} + 72 q^{55} + 272 q^{57} - 540 q^{59} - 110 q^{61} - 492 q^{65} + 140 q^{67} - 864 q^{69} - 840 q^{71} + 550 q^{73} + 356 q^{75} - 208 q^{79} - 311 q^{81} - 516 q^{83} - 180 q^{85} - 984 q^{87} + 1398 q^{89} - 448 q^{93} + 408 q^{95} - 1586 q^{97} + 132 q^{99}+O(q^{100})$$ q - 4 * q^3 - 6 * q^5 - 11 * q^9 - 12 * q^11 + 82 * q^13 + 24 * q^15 + 30 * q^17 - 68 * q^19 + 216 * q^23 - 89 * q^25 + 152 * q^27 + 246 * q^29 + 112 * q^31 + 48 * q^33 + 110 * q^37 - 328 * q^39 + 246 * q^41 - 172 * q^43 + 66 * q^45 - 192 * q^47 - 120 * q^51 + 558 * q^53 + 72 * q^55 + 272 * q^57 - 540 * q^59 - 110 * q^61 - 492 * q^65 + 140 * q^67 - 864 * q^69 - 840 * q^71 + 550 * q^73 + 356 * q^75 - 208 * q^79 - 311 * q^81 - 516 * q^83 - 180 * q^85 - 984 * q^87 + 1398 * q^89 - 448 * q^93 + 408 * q^95 - 1586 * q^97 + 132 * 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.

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 −4.00000 0 −6.00000 0 0 0 −11.0000 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 196.4.a.b 1
3.b odd 2 1 1764.4.a.k 1
4.b odd 2 1 784.4.a.n 1
7.b odd 2 1 28.4.a.b 1
7.c even 3 2 196.4.e.d 2
7.d odd 6 2 196.4.e.c 2
21.c even 2 1 252.4.a.c 1
21.g even 6 2 1764.4.k.k 2
21.h odd 6 2 1764.4.k.e 2
28.d even 2 1 112.4.a.c 1
35.c odd 2 1 700.4.a.e 1
35.f even 4 2 700.4.e.f 2
56.e even 2 1 448.4.a.m 1
56.h odd 2 1 448.4.a.d 1
84.h odd 2 1 1008.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 7.b odd 2 1
112.4.a.c 1 28.d even 2 1
196.4.a.b 1 1.a even 1 1 trivial
196.4.e.c 2 7.d odd 6 2
196.4.e.d 2 7.c even 3 2
252.4.a.c 1 21.c even 2 1
448.4.a.d 1 56.h odd 2 1
448.4.a.m 1 56.e even 2 1
700.4.a.e 1 35.c odd 2 1
700.4.e.f 2 35.f even 4 2
784.4.a.n 1 4.b odd 2 1
1008.4.a.f 1 84.h odd 2 1
1764.4.a.k 1 3.b odd 2 1
1764.4.k.e 2 21.h odd 6 2
1764.4.k.k 2 21.g even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(196))$$:

 $$T_{3} + 4$$ T3 + 4 $$T_{5} + 6$$ T5 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 4$$
$5$ $$T + 6$$
$7$ $$T$$
$11$ $$T + 12$$
$13$ $$T - 82$$
$17$ $$T - 30$$
$19$ $$T + 68$$
$23$ $$T - 216$$
$29$ $$T - 246$$
$31$ $$T - 112$$
$37$ $$T - 110$$
$41$ $$T - 246$$
$43$ $$T + 172$$
$47$ $$T + 192$$
$53$ $$T - 558$$
$59$ $$T + 540$$
$61$ $$T + 110$$
$67$ $$T - 140$$
$71$ $$T + 840$$
$73$ $$T - 550$$
$79$ $$T + 208$$
$83$ $$T + 516$$
$89$ $$T - 1398$$
$97$ $$T + 1586$$