# Properties

 Label 1764.2.b.a Level $1764$ Weight $2$ Character orbit 1764.b Analytic conductor $14.086$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} - 2 \beta_1 q^{4} - \beta_{2} q^{5} + (2 \beta_1 + 2) q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 - 2*b1 * q^4 - b2 * q^5 + (2*b1 + 2) * q^8 $$q + (\beta_1 - 1) q^{2} - 2 \beta_1 q^{4} - \beta_{2} q^{5} + (2 \beta_1 + 2) q^{8} + (\beta_{3} + \beta_{2}) q^{10} + \beta_1 q^{11} + 2 \beta_{2} q^{13} - 4 q^{16} + \beta_{2} q^{17} - 3 \beta_{3} q^{19} - 2 \beta_{3} q^{20} + ( - \beta_1 - 1) q^{22} - \beta_1 q^{23} + 2 q^{25} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{26} - 4 q^{29} - \beta_{3} q^{31} + ( - 4 \beta_1 + 4) q^{32} + ( - \beta_{3} - \beta_{2}) q^{34} + 3 q^{37} + (3 \beta_{3} - 3 \beta_{2}) q^{38} + (2 \beta_{3} - 2 \beta_{2}) q^{40} + 2 \beta_{2} q^{41} + 2 \beta_1 q^{43} + 2 q^{44} + (\beta_1 + 1) q^{46} - 5 \beta_{3} q^{47} + (2 \beta_1 - 2) q^{50} + 4 \beta_{3} q^{52} + q^{53} + \beta_{3} q^{55} + ( - 4 \beta_1 + 4) q^{58} + 3 \beta_{3} q^{59} + 3 \beta_{2} q^{61} + (\beta_{3} - \beta_{2}) q^{62} + 8 \beta_1 q^{64} + 6 q^{65} - 3 \beta_1 q^{67} + 2 \beta_{3} q^{68} + 14 \beta_1 q^{71} + 5 \beta_{2} q^{73} + (3 \beta_1 - 3) q^{74} + 6 \beta_{2} q^{76} + 9 \beta_1 q^{79} + 4 \beta_{2} q^{80} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{82} - 8 \beta_{3} q^{83} + 3 q^{85} + ( - 2 \beta_1 - 2) q^{86} + (2 \beta_1 - 2) q^{88} + 9 \beta_{2} q^{89} - 2 q^{92} + (5 \beta_{3} - 5 \beta_{2}) q^{94} + 9 \beta_1 q^{95} - 10 \beta_{2} q^{97}+O(q^{100})$$ q + (b1 - 1) * q^2 - 2*b1 * q^4 - b2 * q^5 + (2*b1 + 2) * q^8 + (b3 + b2) * q^10 + b1 * q^11 + 2*b2 * q^13 - 4 * q^16 + b2 * q^17 - 3*b3 * q^19 - 2*b3 * q^20 + (-b1 - 1) * q^22 - b1 * q^23 + 2 * q^25 + (-2*b3 - 2*b2) * q^26 - 4 * q^29 - b3 * q^31 + (-4*b1 + 4) * q^32 + (-b3 - b2) * q^34 + 3 * q^37 + (3*b3 - 3*b2) * q^38 + (2*b3 - 2*b2) * q^40 + 2*b2 * q^41 + 2*b1 * q^43 + 2 * q^44 + (b1 + 1) * q^46 - 5*b3 * q^47 + (2*b1 - 2) * q^50 + 4*b3 * q^52 + q^53 + b3 * q^55 + (-4*b1 + 4) * q^58 + 3*b3 * q^59 + 3*b2 * q^61 + (b3 - b2) * q^62 + 8*b1 * q^64 + 6 * q^65 - 3*b1 * q^67 + 2*b3 * q^68 + 14*b1 * q^71 + 5*b2 * q^73 + (3*b1 - 3) * q^74 + 6*b2 * q^76 + 9*b1 * q^79 + 4*b2 * q^80 + (-2*b3 - 2*b2) * q^82 - 8*b3 * q^83 + 3 * q^85 + (-2*b1 - 2) * q^86 + (2*b1 - 2) * q^88 + 9*b2 * q^89 - 2 * q^92 + (5*b3 - 5*b2) * q^94 + 9*b1 * q^95 - 10*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + 8 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 + 8 * q^8 $$4 q - 4 q^{2} + 8 q^{8} - 16 q^{16} - 4 q^{22} + 8 q^{25} - 16 q^{29} + 16 q^{32} + 12 q^{37} + 8 q^{44} + 4 q^{46} - 8 q^{50} + 4 q^{53} + 16 q^{58} + 24 q^{65} - 12 q^{74} + 12 q^{85} - 8 q^{86} - 8 q^{88} - 8 q^{92}+O(q^{100})$$ 4 * q - 4 * q^2 + 8 * q^8 - 16 * q^16 - 4 * q^22 + 8 * q^25 - 16 * q^29 + 16 * q^32 + 12 * q^37 + 8 * q^44 + 4 * q^46 - 8 * q^50 + 4 * q^53 + 16 * q^58 + 24 * q^65 - 12 * q^74 + 12 * q^85 - 8 * q^86 - 8 * q^88 - 8 * q^92

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
−1.00000 1.00000i 0 2.00000i 1.73205i 0 0 2.00000 2.00000i 0 −1.73205 + 1.73205i
1567.2 −1.00000 1.00000i 0 2.00000i 1.73205i 0 0 2.00000 2.00000i 0 1.73205 1.73205i
1567.3 −1.00000 + 1.00000i 0 2.00000i 1.73205i 0 0 2.00000 + 2.00000i 0 1.73205 + 1.73205i
1567.4 −1.00000 + 1.00000i 0 2.00000i 1.73205i 0 0 2.00000 + 2.00000i 0 −1.73205 1.73205i
 $$n$$: 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.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.b.a 4
3.b odd 2 1 196.2.d.b 4
4.b odd 2 1 inner 1764.2.b.a 4
7.b odd 2 1 inner 1764.2.b.a 4
7.c even 3 1 252.2.bf.e 4
7.d odd 6 1 252.2.bf.e 4
12.b even 2 1 196.2.d.b 4
21.c even 2 1 196.2.d.b 4
21.g even 6 1 28.2.f.a 4
21.g even 6 1 196.2.f.a 4
21.h odd 6 1 28.2.f.a 4
21.h odd 6 1 196.2.f.a 4
24.f even 2 1 3136.2.f.e 4
24.h odd 2 1 3136.2.f.e 4
28.d even 2 1 inner 1764.2.b.a 4
28.f even 6 1 252.2.bf.e 4
28.g odd 6 1 252.2.bf.e 4
84.h odd 2 1 196.2.d.b 4
84.j odd 6 1 28.2.f.a 4
84.j odd 6 1 196.2.f.a 4
84.n even 6 1 28.2.f.a 4
84.n even 6 1 196.2.f.a 4
105.o odd 6 1 700.2.p.a 4
105.p even 6 1 700.2.p.a 4
105.w odd 12 1 700.2.t.a 4
105.w odd 12 1 700.2.t.b 4
105.x even 12 1 700.2.t.a 4
105.x even 12 1 700.2.t.b 4
168.e odd 2 1 3136.2.f.e 4
168.i even 2 1 3136.2.f.e 4
168.s odd 6 1 448.2.p.d 4
168.v even 6 1 448.2.p.d 4
168.ba even 6 1 448.2.p.d 4
168.be odd 6 1 448.2.p.d 4
420.ba even 6 1 700.2.p.a 4
420.be odd 6 1 700.2.p.a 4
420.bp odd 12 1 700.2.t.a 4
420.bp odd 12 1 700.2.t.b 4
420.br even 12 1 700.2.t.a 4
420.br 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 21.g even 6 1
28.2.f.a 4 21.h odd 6 1
28.2.f.a 4 84.j odd 6 1
28.2.f.a 4 84.n even 6 1
196.2.d.b 4 3.b odd 2 1
196.2.d.b 4 12.b even 2 1
196.2.d.b 4 21.c even 2 1
196.2.d.b 4 84.h odd 2 1
196.2.f.a 4 21.g even 6 1
196.2.f.a 4 21.h odd 6 1
196.2.f.a 4 84.j odd 6 1
196.2.f.a 4 84.n even 6 1
252.2.bf.e 4 7.c even 3 1
252.2.bf.e 4 7.d odd 6 1
252.2.bf.e 4 28.f even 6 1
252.2.bf.e 4 28.g odd 6 1
448.2.p.d 4 168.s odd 6 1
448.2.p.d 4 168.v even 6 1
448.2.p.d 4 168.ba even 6 1
448.2.p.d 4 168.be odd 6 1
700.2.p.a 4 105.o odd 6 1
700.2.p.a 4 105.p even 6 1
700.2.p.a 4 420.ba even 6 1
700.2.p.a 4 420.be odd 6 1
700.2.t.a 4 105.w odd 12 1
700.2.t.a 4 105.x even 12 1
700.2.t.a 4 420.bp odd 12 1
700.2.t.a 4 420.br even 12 1
700.2.t.b 4 105.w odd 12 1
700.2.t.b 4 105.x even 12 1
700.2.t.b 4 420.bp odd 12 1
700.2.t.b 4 420.br even 12 1
1764.2.b.a 4 1.a even 1 1 trivial
1764.2.b.a 4 4.b odd 2 1 inner
1764.2.b.a 4 7.b odd 2 1 inner
1764.2.b.a 4 28.d even 2 1 inner
3136.2.f.e 4 24.f even 2 1
3136.2.f.e 4 24.h odd 2 1
3136.2.f.e 4 168.e odd 2 1
3136.2.f.e 4 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{2} + 3$$ T5^2 + 3 $$T_{11}^{2} + 1$$ T11^2 + 1 $$T_{19}^{2} - 27$$ T19^2 - 27 $$T_{29} + 4$$ T29 + 4 $$T_{53} - 1$$ T53 - 1

## Hecke characteristic polynomials

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