# Properties

 Label 196.6.e.g Level $196$ Weight $6$ Character orbit 196.e Analytic conductor $31.435$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$31.4352286833$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4) 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.

 $$f(q)$$ $$=$$ $$q + ( 12 - 12 \zeta_{6} ) q^{3} -54 \zeta_{6} q^{5} + 99 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 12 - 12 \zeta_{6} ) q^{3} -54 \zeta_{6} q^{5} + 99 \zeta_{6} q^{9} + ( -540 + 540 \zeta_{6} ) q^{11} -418 q^{13} -648 q^{15} + ( -594 + 594 \zeta_{6} ) q^{17} -836 \zeta_{6} q^{19} + 4104 \zeta_{6} q^{23} + ( 209 - 209 \zeta_{6} ) q^{25} + 4104 q^{27} -594 q^{29} + ( -4256 + 4256 \zeta_{6} ) q^{31} + 6480 \zeta_{6} q^{33} + 298 \zeta_{6} q^{37} + ( -5016 + 5016 \zeta_{6} ) q^{39} + 17226 q^{41} -12100 q^{43} + ( 5346 - 5346 \zeta_{6} ) q^{45} + 1296 \zeta_{6} q^{47} + 7128 \zeta_{6} q^{51} + ( -19494 + 19494 \zeta_{6} ) q^{53} + 29160 q^{55} -10032 q^{57} + ( 7668 - 7668 \zeta_{6} ) q^{59} + 34738 \zeta_{6} q^{61} + 22572 \zeta_{6} q^{65} + ( -21812 + 21812 \zeta_{6} ) q^{67} + 49248 q^{69} -46872 q^{71} + ( -67562 + 67562 \zeta_{6} ) q^{73} -2508 \zeta_{6} q^{75} + 76912 \zeta_{6} q^{79} + ( 25191 - 25191 \zeta_{6} ) q^{81} + 67716 q^{83} + 32076 q^{85} + ( -7128 + 7128 \zeta_{6} ) q^{87} -29754 \zeta_{6} q^{89} + 51072 \zeta_{6} q^{93} + ( -45144 + 45144 \zeta_{6} ) q^{95} -122398 q^{97} -53460 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 12 q^{3} - 54 q^{5} + 99 q^{9} + O(q^{10})$$ $$2 q + 12 q^{3} - 54 q^{5} + 99 q^{9} - 540 q^{11} - 836 q^{13} - 1296 q^{15} - 594 q^{17} - 836 q^{19} + 4104 q^{23} + 209 q^{25} + 8208 q^{27} - 1188 q^{29} - 4256 q^{31} + 6480 q^{33} + 298 q^{37} - 5016 q^{39} + 34452 q^{41} - 24200 q^{43} + 5346 q^{45} + 1296 q^{47} + 7128 q^{51} - 19494 q^{53} + 58320 q^{55} - 20064 q^{57} + 7668 q^{59} + 34738 q^{61} + 22572 q^{65} - 21812 q^{67} + 98496 q^{69} - 93744 q^{71} - 67562 q^{73} - 2508 q^{75} + 76912 q^{79} + 25191 q^{81} + 135432 q^{83} + 64152 q^{85} - 7128 q^{87} - 29754 q^{89} + 51072 q^{93} - 45144 q^{95} - 244796 q^{97} - 106920 q^{99} + O(q^{100})$$

## 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 6.00000 10.3923i 0 −27.0000 46.7654i 0 0 0 49.5000 + 85.7365i 0
177.1 0 6.00000 + 10.3923i 0 −27.0000 + 46.7654i 0 0 0 49.5000 85.7365i 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.g 2
7.b odd 2 1 196.6.e.d 2
7.c even 3 1 4.6.a.a 1
7.c even 3 1 inner 196.6.e.g 2
7.d odd 6 1 196.6.a.e 1
7.d odd 6 1 196.6.e.d 2
21.h odd 6 1 36.6.a.a 1
28.f even 6 1 784.6.a.d 1
28.g odd 6 1 16.6.a.b 1
35.j even 6 1 100.6.a.b 1
35.l odd 12 2 100.6.c.b 2
56.k odd 6 1 64.6.a.b 1
56.p even 6 1 64.6.a.f 1
63.g even 3 1 324.6.e.a 2
63.h even 3 1 324.6.e.a 2
63.j odd 6 1 324.6.e.d 2
63.n odd 6 1 324.6.e.d 2
77.h odd 6 1 484.6.a.a 1
84.n even 6 1 144.6.a.c 1
91.r even 6 1 676.6.a.a 1
91.z odd 12 2 676.6.d.a 2
105.o odd 6 1 900.6.a.h 1
105.x even 12 2 900.6.d.a 2
112.u odd 12 2 256.6.b.c 2
112.w even 12 2 256.6.b.g 2
140.p odd 6 1 400.6.a.d 1
140.w even 12 2 400.6.c.f 2
168.s odd 6 1 576.6.a.bc 1
168.v even 6 1 576.6.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 7.c even 3 1
16.6.a.b 1 28.g odd 6 1
36.6.a.a 1 21.h odd 6 1
64.6.a.b 1 56.k odd 6 1
64.6.a.f 1 56.p even 6 1
100.6.a.b 1 35.j even 6 1
100.6.c.b 2 35.l odd 12 2
144.6.a.c 1 84.n even 6 1
196.6.a.e 1 7.d odd 6 1
196.6.e.d 2 7.b odd 2 1
196.6.e.d 2 7.d odd 6 1
196.6.e.g 2 1.a even 1 1 trivial
196.6.e.g 2 7.c even 3 1 inner
256.6.b.c 2 112.u odd 12 2
256.6.b.g 2 112.w even 12 2
324.6.e.a 2 63.g even 3 1
324.6.e.a 2 63.h even 3 1
324.6.e.d 2 63.j odd 6 1
324.6.e.d 2 63.n odd 6 1
400.6.a.d 1 140.p odd 6 1
400.6.c.f 2 140.w even 12 2
484.6.a.a 1 77.h odd 6 1
576.6.a.bc 1 168.s odd 6 1
576.6.a.bd 1 168.v even 6 1
676.6.a.a 1 91.r even 6 1
676.6.d.a 2 91.z odd 12 2
784.6.a.d 1 28.f even 6 1
900.6.a.h 1 105.o odd 6 1
900.6.d.a 2 105.x even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 12 T_{3} + 144$$ acting on $$S_{6}^{\mathrm{new}}(196, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$144 - 12 T + T^{2}$$
$5$ $$2916 + 54 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$291600 + 540 T + T^{2}$$
$13$ $$( 418 + T )^{2}$$
$17$ $$352836 + 594 T + T^{2}$$
$19$ $$698896 + 836 T + T^{2}$$
$23$ $$16842816 - 4104 T + T^{2}$$
$29$ $$( 594 + T )^{2}$$
$31$ $$18113536 + 4256 T + T^{2}$$
$37$ $$88804 - 298 T + T^{2}$$
$41$ $$( -17226 + T )^{2}$$
$43$ $$( 12100 + T )^{2}$$
$47$ $$1679616 - 1296 T + T^{2}$$
$53$ $$380016036 + 19494 T + T^{2}$$
$59$ $$58798224 - 7668 T + T^{2}$$
$61$ $$1206728644 - 34738 T + T^{2}$$
$67$ $$475763344 + 21812 T + T^{2}$$
$71$ $$( 46872 + T )^{2}$$
$73$ $$4564623844 + 67562 T + T^{2}$$
$79$ $$5915455744 - 76912 T + T^{2}$$
$83$ $$( -67716 + T )^{2}$$
$89$ $$885300516 + 29754 T + T^{2}$$
$97$ $$( 122398 + T )^{2}$$