# Properties

 Label 196.2.f.a Level $196$ Weight $2$ Character orbit 196.f Analytic conductor $1.565$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$196 = 2^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 196.f (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.56506787962$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} + 2) q^{5} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{6} + ( - 2 \zeta_{12}^{3} - 2) q^{8}+O(q^{10})$$ q + (z^3 - z^2 - z) * q^2 + (2*z^3 - z) * q^3 + 2*z * q^4 + (-z^2 + 2) * q^5 + (-z^3 - 2*z^2 + 2*z + 1) * q^6 + (-2*z^3 - 2) * q^8 $$q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} + 2) q^{5} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{6} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (2 \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} - 1) q^{10} + \zeta_{12} q^{11} + (2 \zeta_{12}^{2} - 4) q^{12} + (4 \zeta_{12}^{2} - 2) q^{13} + 3 \zeta_{12}^{3} q^{15} + 4 \zeta_{12}^{2} q^{16} + (\zeta_{12}^{2} + 1) q^{17} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{19} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{20} + ( - \zeta_{12}^{3} - 1) q^{22} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{23} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{24} + (2 \zeta_{12}^{2} - 2) q^{25} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + 4 q^{29} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{30} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + (\zeta_{12}^{2} - 2) q^{33} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 1) q^{34} - 3 \zeta_{12}^{2} q^{37} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{38} - 6 \zeta_{12} q^{39} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{40} + ( - 4 \zeta_{12}^{2} + 2) q^{41} + 2 \zeta_{12}^{3} q^{43} + 2 \zeta_{12}^{2} q^{44} + (\zeta_{12}^{2} - \zeta_{12} - 1) q^{46} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{47} + (4 \zeta_{12}^{3} - 8 \zeta_{12}) q^{48} + ( - 2 \zeta_{12}^{3} + 2) q^{50} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{51} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{52} + ( - \zeta_{12}^{2} + 1) q^{53} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 6) q^{54} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{55} - 9 q^{57} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{58} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{59} + (6 \zeta_{12}^{2} - 6) q^{60} + ( - 3 \zeta_{12}^{2} + 6) q^{61} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{62} + 8 \zeta_{12}^{3} q^{64} + 6 \zeta_{12}^{2} q^{65} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{66} + 3 \zeta_{12} q^{67} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{68} + (2 \zeta_{12}^{2} - 1) q^{69} - 14 \zeta_{12}^{3} q^{71} + ( - 5 \zeta_{12}^{2} - 5) q^{73} + (3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{74} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{75} + (12 \zeta_{12}^{2} - 6) q^{76} + (6 \zeta_{12}^{3} + 6) q^{78} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{79} + (4 \zeta_{12}^{2} + 4) q^{80} + ( - 9 \zeta_{12}^{2} + 9) q^{81} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{82} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}) q^{83} + 3 q^{85} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{86} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{87} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{88} + (9 \zeta_{12}^{2} - 18) q^{89} + 2 q^{92} + 3 \zeta_{12}^{2} q^{93} + (10 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 5 \zeta_{12} + 5) q^{94} + 9 \zeta_{12} q^{95} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 8) q^{96} + ( - 20 \zeta_{12}^{2} + 10) q^{97} +O(q^{100})$$ q + (z^3 - z^2 - z) * q^2 + (2*z^3 - z) * q^3 + 2*z * q^4 + (-z^2 + 2) * q^5 + (-z^3 - 2*z^2 + 2*z + 1) * q^6 + (-2*z^3 - 2) * q^8 + (2*z^3 - z^2 - z - 1) * q^10 + z * q^11 + (2*z^2 - 4) * q^12 + (4*z^2 - 2) * q^13 + 3*z^3 * q^15 + 4*z^2 * q^16 + (z^2 + 1) * q^17 + (3*z^3 + 3*z) * q^19 + (-2*z^3 + 4*z) * q^20 + (-z^3 - 1) * q^22 + (-z^3 + z) * q^23 + (-4*z^3 + 2*z^2 + 2*z + 2) * q^24 + (2*z^2 - 2) * q^25 + (-2*z^3 - 2*z^2 - 2*z + 4) * q^26 + (3*z^3 - 6*z) * q^27 + 4 * q^29 + (-3*z^3 - 3*z^2 + 3*z) * q^30 + (-2*z^3 + z) * q^31 + (-4*z^2 - 4*z + 4) * q^32 + (z^2 - 2) * q^33 + (z^3 - 2*z^2 - 2*z + 1) * q^34 - 3*z^2 * q^37 + (-6*z^3 - 3*z^2 + 3*z - 3) * q^38 - 6*z * q^39 + (-2*z^3 + 2*z^2 - 2*z - 4) * q^40 + (-4*z^2 + 2) * q^41 + 2*z^3 * q^43 + 2*z^2 * q^44 + (z^2 - z - 1) * q^46 + (-5*z^3 - 5*z) * q^47 + (4*z^3 - 8*z) * q^48 + (-2*z^3 + 2) * q^50 + (3*z^3 - 3*z) * q^51 + (8*z^3 - 4*z) * q^52 + (-z^2 + 1) * q^53 + (3*z^3 - 3*z^2 + 3*z + 6) * q^54 + (-z^3 + 2*z) * q^55 - 9 * q^57 + (4*z^3 - 4*z^2 - 4*z) * q^58 + (-6*z^3 + 3*z) * q^59 + (6*z^2 - 6) * q^60 + (-3*z^2 + 6) * q^61 + (z^3 + 2*z^2 - 2*z - 1) * q^62 + 8*z^3 * q^64 + 6*z^2 * q^65 + (-2*z^3 + z^2 + z + 1) * q^66 + 3*z * q^67 + (2*z^3 + 2*z) * q^68 + (2*z^2 - 1) * q^69 - 14*z^3 * q^71 + (-5*z^2 - 5) * q^73 + (3*z^2 + 3*z - 3) * q^74 + (-2*z^3 - 2*z) * q^75 + (12*z^2 - 6) * q^76 + (6*z^3 + 6) * q^78 + (-9*z^3 + 9*z) * q^79 + (4*z^2 + 4) * q^80 + (-9*z^2 + 9) * q^81 + (2*z^3 + 2*z^2 + 2*z - 4) * q^82 + (-8*z^3 + 16*z) * q^83 + 3 * q^85 + (-2*z^3 - 2*z^2 + 2*z) * q^86 + (8*z^3 - 4*z) * q^87 + (-2*z^2 - 2*z + 2) * q^88 + (9*z^2 - 18) * q^89 + 2 * q^92 + 3*z^2 * q^93 + (10*z^3 + 5*z^2 - 5*z + 5) * q^94 + 9*z * q^95 + (4*z^3 - 4*z^2 + 4*z + 8) * q^96 + (-20*z^2 + 10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + 6 q^{5} - 8 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 + 6 * q^5 - 8 * q^8 $$4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} - 6 q^{10} - 12 q^{12} + 8 q^{16} + 6 q^{17} - 4 q^{22} + 12 q^{24} - 4 q^{25} + 12 q^{26} + 16 q^{29} - 6 q^{30} + 8 q^{32} - 6 q^{33} - 6 q^{37} - 18 q^{38} - 12 q^{40} + 4 q^{44} - 2 q^{46} + 8 q^{50} + 2 q^{53} + 18 q^{54} - 36 q^{57} - 8 q^{58} - 12 q^{60} + 18 q^{61} + 12 q^{65} + 6 q^{66} - 30 q^{73} - 6 q^{74} + 24 q^{78} + 24 q^{80} + 18 q^{81} - 12 q^{82} + 12 q^{85} - 4 q^{86} + 4 q^{88} - 54 q^{89} + 8 q^{92} + 6 q^{93} + 30 q^{94} + 24 q^{96}+O(q^{100})$$ 4 * q - 2 * q^2 + 6 * q^5 - 8 * q^8 - 6 * q^10 - 12 * q^12 + 8 * q^16 + 6 * q^17 - 4 * q^22 + 12 * q^24 - 4 * q^25 + 12 * q^26 + 16 * q^29 - 6 * q^30 + 8 * q^32 - 6 * q^33 - 6 * q^37 - 18 * q^38 - 12 * q^40 + 4 * q^44 - 2 * q^46 + 8 * q^50 + 2 * q^53 + 18 * q^54 - 36 * q^57 - 8 * q^58 - 12 * q^60 + 18 * q^61 + 12 * q^65 + 6 * q^66 - 30 * q^73 - 6 * q^74 + 24 * q^78 + 24 * q^80 + 18 * q^81 - 12 * q^82 + 12 * q^85 - 4 * q^86 + 4 * q^88 - 54 * q^89 + 8 * q^92 + 6 * q^93 + 30 * q^94 + 24 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/196\mathbb{Z}\right)^\times$$.

 $$n$$ $$99$$ $$101$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−1.36603 + 0.366025i −0.866025 1.50000i 1.73205 1.00000i 1.50000 + 0.866025i 1.73205 + 1.73205i 0 −2.00000 + 2.00000i 0 −2.36603 0.633975i
19.2 0.366025 + 1.36603i 0.866025 + 1.50000i −1.73205 + 1.00000i 1.50000 + 0.866025i −1.73205 + 1.73205i 0 −2.00000 2.00000i 0 −0.633975 + 2.36603i
31.1 −1.36603 0.366025i −0.866025 + 1.50000i 1.73205 + 1.00000i 1.50000 0.866025i 1.73205 1.73205i 0 −2.00000 2.00000i 0 −2.36603 + 0.633975i
31.2 0.366025 1.36603i 0.866025 1.50000i −1.73205 1.00000i 1.50000 0.866025i −1.73205 1.73205i 0 −2.00000 + 2.00000i 0 −0.633975 2.36603i
 $$n$$: 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
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.2.f.a 4
4.b odd 2 1 inner 196.2.f.a 4
7.b odd 2 1 28.2.f.a 4
7.c even 3 1 28.2.f.a 4
7.c even 3 1 196.2.d.b 4
7.d odd 6 1 196.2.d.b 4
7.d odd 6 1 inner 196.2.f.a 4
21.c even 2 1 252.2.bf.e 4
21.g even 6 1 1764.2.b.a 4
21.h odd 6 1 252.2.bf.e 4
21.h odd 6 1 1764.2.b.a 4
28.d even 2 1 28.2.f.a 4
28.f even 6 1 196.2.d.b 4
28.f even 6 1 inner 196.2.f.a 4
28.g odd 6 1 28.2.f.a 4
28.g odd 6 1 196.2.d.b 4
35.c odd 2 1 700.2.p.a 4
35.f even 4 1 700.2.t.a 4
35.f even 4 1 700.2.t.b 4
35.j even 6 1 700.2.p.a 4
35.l odd 12 1 700.2.t.a 4
35.l odd 12 1 700.2.t.b 4
56.e even 2 1 448.2.p.d 4
56.h odd 2 1 448.2.p.d 4
56.j odd 6 1 3136.2.f.e 4
56.k odd 6 1 448.2.p.d 4
56.k odd 6 1 3136.2.f.e 4
56.m even 6 1 3136.2.f.e 4
56.p even 6 1 448.2.p.d 4
56.p even 6 1 3136.2.f.e 4
84.h odd 2 1 252.2.bf.e 4
84.j odd 6 1 1764.2.b.a 4
84.n even 6 1 252.2.bf.e 4
84.n even 6 1 1764.2.b.a 4
140.c even 2 1 700.2.p.a 4
140.j odd 4 1 700.2.t.a 4
140.j odd 4 1 700.2.t.b 4
140.p odd 6 1 700.2.p.a 4
140.w even 12 1 700.2.t.a 4
140.w even 12 1 700.2.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 7.b odd 2 1
28.2.f.a 4 7.c even 3 1
28.2.f.a 4 28.d even 2 1
28.2.f.a 4 28.g odd 6 1
196.2.d.b 4 7.c even 3 1
196.2.d.b 4 7.d odd 6 1
196.2.d.b 4 28.f even 6 1
196.2.d.b 4 28.g odd 6 1
196.2.f.a 4 1.a even 1 1 trivial
196.2.f.a 4 4.b odd 2 1 inner
196.2.f.a 4 7.d odd 6 1 inner
196.2.f.a 4 28.f even 6 1 inner
252.2.bf.e 4 21.c even 2 1
252.2.bf.e 4 21.h odd 6 1
252.2.bf.e 4 84.h odd 2 1
252.2.bf.e 4 84.n even 6 1
448.2.p.d 4 56.e even 2 1
448.2.p.d 4 56.h odd 2 1
448.2.p.d 4 56.k odd 6 1
448.2.p.d 4 56.p even 6 1
700.2.p.a 4 35.c odd 2 1
700.2.p.a 4 35.j even 6 1
700.2.p.a 4 140.c even 2 1
700.2.p.a 4 140.p odd 6 1
700.2.t.a 4 35.f even 4 1
700.2.t.a 4 35.l odd 12 1
700.2.t.a 4 140.j odd 4 1
700.2.t.a 4 140.w even 12 1
700.2.t.b 4 35.f even 4 1
700.2.t.b 4 35.l odd 12 1
700.2.t.b 4 140.j odd 4 1
700.2.t.b 4 140.w even 12 1
1764.2.b.a 4 21.g even 6 1
1764.2.b.a 4 21.h odd 6 1
1764.2.b.a 4 84.j odd 6 1
1764.2.b.a 4 84.n even 6 1
3136.2.f.e 4 56.j odd 6 1
3136.2.f.e 4 56.k odd 6 1
3136.2.f.e 4 56.m even 6 1
3136.2.f.e 4 56.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(196, [\chi])$$:

 $$T_{3}^{4} + 3T_{3}^{2} + 9$$ T3^4 + 3*T3^2 + 9 $$T_{5}^{2} - 3T_{5} + 3$$ T5^2 - 3*T5 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$(T^{2} - 3 T + 3)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - T^{2} + 1$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} - 3 T + 3)^{2}$$
$19$ $$T^{4} + 27T^{2} + 729$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T - 4)^{4}$$
$31$ $$T^{4} + 3T^{2} + 9$$
$37$ $$(T^{2} + 3 T + 9)^{2}$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$T^{4} + 75T^{2} + 5625$$
$53$ $$(T^{2} - T + 1)^{2}$$
$59$ $$T^{4} + 27T^{2} + 729$$
$61$ $$(T^{2} - 9 T + 27)^{2}$$
$67$ $$T^{4} - 9T^{2} + 81$$
$71$ $$(T^{2} + 196)^{2}$$
$73$ $$(T^{2} + 15 T + 75)^{2}$$
$79$ $$T^{4} - 81T^{2} + 6561$$
$83$ $$(T^{2} - 192)^{2}$$
$89$ $$(T^{2} + 27 T + 243)^{2}$$
$97$ $$(T^{2} + 300)^{2}$$