# Properties

 Label 196.4.e.c 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + (4 \zeta_{6} - 4) q^{3} - 6 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} +O(q^{10})$$ q + (4*z - 4) * q^3 - 6*z * q^5 + 11*z * q^9 $$q + (4 \zeta_{6} - 4) q^{3} - 6 \zeta_{6} q^{5} + 11 \zeta_{6} q^{9} + ( - 12 \zeta_{6} + 12) q^{11} - 82 q^{13} + 24 q^{15} + ( - 30 \zeta_{6} + 30) q^{17} - 68 \zeta_{6} q^{19} - 216 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} - 152 q^{27} + 246 q^{29} + ( - 112 \zeta_{6} + 112) q^{31} + 48 \zeta_{6} q^{33} - 110 \zeta_{6} q^{37} + ( - 328 \zeta_{6} + 328) q^{39} - 246 q^{41} - 172 q^{43} + ( - 66 \zeta_{6} + 66) q^{45} - 192 \zeta_{6} q^{47} + 120 \zeta_{6} q^{51} + (558 \zeta_{6} - 558) q^{53} - 72 q^{55} + 272 q^{57} + (540 \zeta_{6} - 540) q^{59} - 110 \zeta_{6} q^{61} + 492 \zeta_{6} q^{65} + (140 \zeta_{6} - 140) q^{67} + 864 q^{69} - 840 q^{71} + ( - 550 \zeta_{6} + 550) q^{73} + 356 \zeta_{6} q^{75} + 208 \zeta_{6} q^{79} + ( - 311 \zeta_{6} + 311) q^{81} + 516 q^{83} - 180 q^{85} + (984 \zeta_{6} - 984) q^{87} + 1398 \zeta_{6} q^{89} + 448 \zeta_{6} q^{93} + (408 \zeta_{6} - 408) q^{95} + 1586 q^{97} + 132 q^{99} +O(q^{100})$$ q + (4*z - 4) * q^3 - 6*z * q^5 + 11*z * q^9 + (-12*z + 12) * q^11 - 82 * q^13 + 24 * q^15 + (-30*z + 30) * q^17 - 68*z * q^19 - 216*z * q^23 + (-89*z + 89) * q^25 - 152 * q^27 + 246 * q^29 + (-112*z + 112) * q^31 + 48*z * q^33 - 110*z * q^37 + (-328*z + 328) * q^39 - 246 * q^41 - 172 * q^43 + (-66*z + 66) * q^45 - 192*z * q^47 + 120*z * q^51 + (558*z - 558) * q^53 - 72 * q^55 + 272 * q^57 + (540*z - 540) * q^59 - 110*z * q^61 + 492*z * q^65 + (140*z - 140) * q^67 + 864 * q^69 - 840 * q^71 + (-550*z + 550) * q^73 + 356*z * q^75 + 208*z * q^79 + (-311*z + 311) * q^81 + 516 * q^83 - 180 * q^85 + (984*z - 984) * q^87 + 1398*z * q^89 + 448*z * q^93 + (408*z - 408) * q^95 + 1586 * q^97 + 132 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} - 6 q^{5} + 11 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 - 6 * q^5 + 11 * q^9 $$2 q - 4 q^{3} - 6 q^{5} + 11 q^{9} + 12 q^{11} - 164 q^{13} + 48 q^{15} + 30 q^{17} - 68 q^{19} - 216 q^{23} + 89 q^{25} - 304 q^{27} + 492 q^{29} + 112 q^{31} + 48 q^{33} - 110 q^{37} + 328 q^{39} - 492 q^{41} - 344 q^{43} + 66 q^{45} - 192 q^{47} + 120 q^{51} - 558 q^{53} - 144 q^{55} + 544 q^{57} - 540 q^{59} - 110 q^{61} + 492 q^{65} - 140 q^{67} + 1728 q^{69} - 1680 q^{71} + 550 q^{73} + 356 q^{75} + 208 q^{79} + 311 q^{81} + 1032 q^{83} - 360 q^{85} - 984 q^{87} + 1398 q^{89} + 448 q^{93} - 408 q^{95} + 3172 q^{97} + 264 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 - 6 * q^5 + 11 * q^9 + 12 * q^11 - 164 * q^13 + 48 * q^15 + 30 * q^17 - 68 * q^19 - 216 * q^23 + 89 * q^25 - 304 * q^27 + 492 * q^29 + 112 * q^31 + 48 * q^33 - 110 * q^37 + 328 * q^39 - 492 * q^41 - 344 * q^43 + 66 * q^45 - 192 * q^47 + 120 * q^51 - 558 * q^53 - 144 * q^55 + 544 * q^57 - 540 * q^59 - 110 * q^61 + 492 * q^65 - 140 * q^67 + 1728 * q^69 - 1680 * q^71 + 550 * q^73 + 356 * q^75 + 208 * q^79 + 311 * q^81 + 1032 * q^83 - 360 * q^85 - 984 * q^87 + 1398 * q^89 + 448 * q^93 - 408 * q^95 + 3172 * q^97 + 264 * 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.

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 −2.00000 + 3.46410i 0 −3.00000 5.19615i 0 0 0 5.50000 + 9.52628i 0
177.1 0 −2.00000 3.46410i 0 −3.00000 + 5.19615i 0 0 0 5.50000 9.52628i 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.c 2
3.b odd 2 1 1764.4.k.k 2
7.b odd 2 1 196.4.e.d 2
7.c even 3 1 28.4.a.b 1
7.c even 3 1 inner 196.4.e.c 2
7.d odd 6 1 196.4.a.b 1
7.d odd 6 1 196.4.e.d 2
21.c even 2 1 1764.4.k.e 2
21.g even 6 1 1764.4.a.k 1
21.g even 6 1 1764.4.k.e 2
21.h odd 6 1 252.4.a.c 1
21.h odd 6 1 1764.4.k.k 2
28.f even 6 1 784.4.a.n 1
28.g odd 6 1 112.4.a.c 1
35.j even 6 1 700.4.a.e 1
35.l odd 12 2 700.4.e.f 2
56.k odd 6 1 448.4.a.m 1
56.p even 6 1 448.4.a.d 1
84.n even 6 1 1008.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 7.c even 3 1
112.4.a.c 1 28.g odd 6 1
196.4.a.b 1 7.d odd 6 1
196.4.e.c 2 1.a even 1 1 trivial
196.4.e.c 2 7.c even 3 1 inner
196.4.e.d 2 7.b odd 2 1
196.4.e.d 2 7.d odd 6 1
252.4.a.c 1 21.h odd 6 1
448.4.a.d 1 56.p even 6 1
448.4.a.m 1 56.k odd 6 1
700.4.a.e 1 35.j even 6 1
700.4.e.f 2 35.l odd 12 2
784.4.a.n 1 28.f even 6 1
1008.4.a.f 1 84.n even 6 1
1764.4.a.k 1 21.g even 6 1
1764.4.k.e 2 21.c even 2 1
1764.4.k.e 2 21.g even 6 1
1764.4.k.k 2 3.b odd 2 1
1764.4.k.k 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 $$S_{4}^{\mathrm{new}}(196, [\chi])$$:

 $$T_{3}^{2} + 4T_{3} + 16$$ T3^2 + 4*T3 + 16 $$T_{5}^{2} + 6T_{5} + 36$$ T5^2 + 6*T5 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4T + 16$$
$5$ $$T^{2} + 6T + 36$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 12T + 144$$
$13$ $$(T + 82)^{2}$$
$17$ $$T^{2} - 30T + 900$$
$19$ $$T^{2} + 68T + 4624$$
$23$ $$T^{2} + 216T + 46656$$
$29$ $$(T - 246)^{2}$$
$31$ $$T^{2} - 112T + 12544$$
$37$ $$T^{2} + 110T + 12100$$
$41$ $$(T + 246)^{2}$$
$43$ $$(T + 172)^{2}$$
$47$ $$T^{2} + 192T + 36864$$
$53$ $$T^{2} + 558T + 311364$$
$59$ $$T^{2} + 540T + 291600$$
$61$ $$T^{2} + 110T + 12100$$
$67$ $$T^{2} + 140T + 19600$$
$71$ $$(T + 840)^{2}$$
$73$ $$T^{2} - 550T + 302500$$
$79$ $$T^{2} - 208T + 43264$$
$83$ $$(T - 516)^{2}$$
$89$ $$T^{2} - 1398 T + 1954404$$
$97$ $$(T - 1586)^{2}$$