# Properties

 Label 1764.2.t.c Level $1764$ Weight $2$ Character orbit 1764.t Analytic conductor $14.086$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ x^16 - x^8 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{15} + \beta_{4}) q^{5}+O(q^{10})$$ q + (-b15 + b4) * q^5 $$q + ( - \beta_{15} + \beta_{4}) q^{5} + ( - \beta_{5} + \beta_1) q^{11} + (\beta_{13} - \beta_{10} + 2 \beta_{7}) q^{13} + (3 \beta_{11} - 3 \beta_{8}) q^{17} + 2 \beta_{13} q^{19} + ( - 2 \beta_{12} - 3 \beta_1) q^{23} + ( - \beta_{6} + 3 \beta_{3}) q^{25} + (2 \beta_{12} - 2 \beta_{9} - 2 \beta_{5}) q^{29} + ( - 2 \beta_{10} - 4 \beta_{7} + 4 \beta_{2}) q^{31} + ( - 3 \beta_{14} + 3 \beta_{6} + 4 \beta_{3} - 4) q^{37} + ( - 6 \beta_{15} - \beta_{11}) q^{41} + ( - 2 \beta_{14} + 4) q^{43} + ( - 6 \beta_{15} + 4 \beta_{8} + 6 \beta_{4}) q^{47} + ( - 3 \beta_{9} - 4 \beta_{5} + 4 \beta_1) q^{53} + 2 \beta_{7} q^{55} + (8 \beta_{11} - 8 \beta_{8} + 2 \beta_{4}) q^{59} + (4 \beta_{13} + 5 \beta_{2}) q^{61} + ( - 3 \beta_{12} - 2 \beta_1) q^{65} + ( - 6 \beta_{6} - 4 \beta_{3}) q^{67} + (8 \beta_{12} - 8 \beta_{9} + \beta_{5}) q^{71} + ( - 6 \beta_{10} - \beta_{7} + \beta_{2}) q^{73} + ( - 8 \beta_{14} + 8 \beta_{6}) q^{79} + ( - 4 \beta_{15} - 4 \beta_{11}) q^{83} - 3 \beta_{14} q^{85} + ( - 7 \beta_{15} - 2 \beta_{8} + 7 \beta_{4}) q^{89} - 2 \beta_{9} q^{95} + (6 \beta_{13} - 6 \beta_{10} - \beta_{7}) q^{97}+O(q^{100})$$ q + (-b15 + b4) * q^5 + (-b5 + b1) * q^11 + (b13 - b10 + 2*b7) * q^13 + (3*b11 - 3*b8) * q^17 + 2*b13 * q^19 + (-2*b12 - 3*b1) * q^23 + (-b6 + 3*b3) * q^25 + (2*b12 - 2*b9 - 2*b5) * q^29 + (-2*b10 - 4*b7 + 4*b2) * q^31 + (-3*b14 + 3*b6 + 4*b3 - 4) * q^37 + (-6*b15 - b11) * q^41 + (-2*b14 + 4) * q^43 + (-6*b15 + 4*b8 + 6*b4) * q^47 + (-3*b9 - 4*b5 + 4*b1) * q^53 + 2*b7 * q^55 + (8*b11 - 8*b8 + 2*b4) * q^59 + (4*b13 + 5*b2) * q^61 + (-3*b12 - 2*b1) * q^65 + (-6*b6 - 4*b3) * q^67 + (8*b12 - 8*b9 + b5) * q^71 + (-6*b10 - b7 + b2) * q^73 + (-8*b14 + 8*b6) * q^79 + (-4*b15 - 4*b11) * q^83 - 3*b14 * q^85 + (-7*b15 - 2*b8 + 7*b4) * q^89 - 2*b9 * q^95 + (6*b13 - 6*b10 - b7) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q+O(q^{10})$$ 16 * q $$16 q + 24 q^{25} - 32 q^{37} + 64 q^{43} - 32 q^{67}+O(q^{100})$$ 16 * q + 24 * q^25 - 32 * q^37 + 64 * q^43 - 32 * q^67

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{48}^{4}$$ 2*v^4 $$\beta_{2}$$ $$=$$ $$\zeta_{48}^{7} + \zeta_{48}$$ v^7 + v $$\beta_{3}$$ $$=$$ $$\zeta_{48}^{8}$$ v^8 $$\beta_{4}$$ $$=$$ $$\zeta_{48}^{11} + \zeta_{48}^{5}$$ v^11 + v^5 $$\beta_{5}$$ $$=$$ $$2\zeta_{48}^{12}$$ 2*v^12 $$\beta_{6}$$ $$=$$ $$\zeta_{48}^{14} + \zeta_{48}^{2}$$ v^14 + v^2 $$\beta_{7}$$ $$=$$ $$\zeta_{48}^{15} + \zeta_{48}^{9}$$ v^15 + v^9 $$\beta_{8}$$ $$=$$ $$-\zeta_{48}^{7} + \zeta_{48}$$ -v^7 + v $$\beta_{9}$$ $$=$$ $$-\zeta_{48}^{14} + \zeta_{48}^{2}$$ -v^14 + v^2 $$\beta_{10}$$ $$=$$ $$-\zeta_{48}^{11} + \zeta_{48}^{5}$$ -v^11 + v^5 $$\beta_{11}$$ $$=$$ $$-\zeta_{48}^{15} + \zeta_{48}^{9}$$ -v^15 + v^9 $$\beta_{12}$$ $$=$$ $$-\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6}$$ -v^14 + v^10 + v^6 $$\beta_{13}$$ $$=$$ $$\zeta_{48}^{13} - \zeta_{48}^{11} + \zeta_{48}^{3}$$ v^13 - v^11 + v^3 $$\beta_{14}$$ $$=$$ $$-\zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2}$$ -v^10 + v^6 + v^2 $$\beta_{15}$$ $$=$$ $$-\zeta_{48}^{13} + \zeta_{48}^{5} + \zeta_{48}^{3}$$ -v^13 + v^5 + v^3
 $$\zeta_{48}$$ $$=$$ $$( \beta_{8} + \beta_{2} ) / 2$$ (b8 + b2) / 2 $$\zeta_{48}^{2}$$ $$=$$ $$( \beta_{9} + \beta_{6} ) / 2$$ (b9 + b6) / 2 $$\zeta_{48}^{3}$$ $$=$$ $$( \beta_{15} + \beta_{13} - \beta_{10} ) / 2$$ (b15 + b13 - b10) / 2 $$\zeta_{48}^{4}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{48}^{5}$$ $$=$$ $$( \beta_{10} + \beta_{4} ) / 2$$ (b10 + b4) / 2 $$\zeta_{48}^{6}$$ $$=$$ $$( \beta_{14} + \beta_{12} - \beta_{9} ) / 2$$ (b14 + b12 - b9) / 2 $$\zeta_{48}^{7}$$ $$=$$ $$( -\beta_{8} + \beta_{2} ) / 2$$ (-b8 + b2) / 2 $$\zeta_{48}^{8}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{48}^{9}$$ $$=$$ $$( \beta_{11} + \beta_{7} ) / 2$$ (b11 + b7) / 2 $$\zeta_{48}^{10}$$ $$=$$ $$( -\beta_{14} + \beta_{12} + \beta_{6} ) / 2$$ (-b14 + b12 + b6) / 2 $$\zeta_{48}^{11}$$ $$=$$ $$( -\beta_{10} + \beta_{4} ) / 2$$ (-b10 + b4) / 2 $$\zeta_{48}^{12}$$ $$=$$ $$( \beta_{5} ) / 2$$ (b5) / 2 $$\zeta_{48}^{13}$$ $$=$$ $$( -\beta_{15} + \beta_{13} + \beta_{4} ) / 2$$ (-b15 + b13 + b4) / 2 $$\zeta_{48}^{14}$$ $$=$$ $$( -\beta_{9} + \beta_{6} ) / 2$$ (-b9 + b6) / 2 $$\zeta_{48}^{15}$$ $$=$$ $$( -\beta_{11} + \beta_{7} ) / 2$$ (-b11 + b7) / 2

## Character values

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

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1 - \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
521.1
 −0.608761 + 0.793353i 0.991445 − 0.130526i −0.793353 − 0.608761i −0.130526 − 0.991445i 0.793353 + 0.608761i 0.130526 + 0.991445i 0.608761 − 0.793353i −0.991445 + 0.130526i −0.608761 − 0.793353i 0.991445 + 0.130526i −0.793353 + 0.608761i −0.130526 + 0.991445i 0.793353 − 0.608761i 0.130526 − 0.991445i 0.608761 + 0.793353i −0.991445 − 0.130526i
0 0 0 −0.923880 1.60021i 0 0 0 0 0
521.2 0 0 0 −0.923880 1.60021i 0 0 0 0 0
521.3 0 0 0 −0.382683 0.662827i 0 0 0 0 0
521.4 0 0 0 −0.382683 0.662827i 0 0 0 0 0
521.5 0 0 0 0.382683 + 0.662827i 0 0 0 0 0
521.6 0 0 0 0.382683 + 0.662827i 0 0 0 0 0
521.7 0 0 0 0.923880 + 1.60021i 0 0 0 0 0
521.8 0 0 0 0.923880 + 1.60021i 0 0 0 0 0
1097.1 0 0 0 −0.923880 + 1.60021i 0 0 0 0 0
1097.2 0 0 0 −0.923880 + 1.60021i 0 0 0 0 0
1097.3 0 0 0 −0.382683 + 0.662827i 0 0 0 0 0
1097.4 0 0 0 −0.382683 + 0.662827i 0 0 0 0 0
1097.5 0 0 0 0.382683 0.662827i 0 0 0 0 0
1097.6 0 0 0 0.382683 0.662827i 0 0 0 0 0
1097.7 0 0 0 0.923880 1.60021i 0 0 0 0 0
1097.8 0 0 0 0.923880 1.60021i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1097.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.t.c 16
3.b odd 2 1 inner 1764.2.t.c 16
7.b odd 2 1 inner 1764.2.t.c 16
7.c even 3 1 1764.2.f.b 8
7.c even 3 1 inner 1764.2.t.c 16
7.d odd 6 1 1764.2.f.b 8
7.d odd 6 1 inner 1764.2.t.c 16
21.c even 2 1 inner 1764.2.t.c 16
21.g even 6 1 1764.2.f.b 8
21.g even 6 1 inner 1764.2.t.c 16
21.h odd 6 1 1764.2.f.b 8
21.h odd 6 1 inner 1764.2.t.c 16
28.f even 6 1 7056.2.k.e 8
28.g odd 6 1 7056.2.k.e 8
84.j odd 6 1 7056.2.k.e 8
84.n even 6 1 7056.2.k.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.f.b 8 7.c even 3 1
1764.2.f.b 8 7.d odd 6 1
1764.2.f.b 8 21.g even 6 1
1764.2.f.b 8 21.h odd 6 1
1764.2.t.c 16 1.a even 1 1 trivial
1764.2.t.c 16 3.b odd 2 1 inner
1764.2.t.c 16 7.b odd 2 1 inner
1764.2.t.c 16 7.c even 3 1 inner
1764.2.t.c 16 7.d odd 6 1 inner
1764.2.t.c 16 21.c even 2 1 inner
1764.2.t.c 16 21.g even 6 1 inner
1764.2.t.c 16 21.h odd 6 1 inner
7056.2.k.e 8 28.f even 6 1
7056.2.k.e 8 28.g odd 6 1
7056.2.k.e 8 84.j odd 6 1
7056.2.k.e 8 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 4T_{5}^{6} + 14T_{5}^{4} + 8T_{5}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$(T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2}$$
$7$ $$T^{16}$$
$11$ $$(T^{4} - 4 T^{2} + 16)^{4}$$
$13$ $$(T^{4} + 20 T^{2} + 2)^{4}$$
$17$ $$(T^{8} + 36 T^{6} + 1134 T^{4} + \cdots + 26244)^{2}$$
$19$ $$(T^{8} - 16 T^{6} + 224 T^{4} - 512 T^{2} + \cdots + 1024)^{2}$$
$23$ $$(T^{8} - 88 T^{6} + 6960 T^{4} + \cdots + 614656)^{2}$$
$29$ $$(T^{4} + 48 T^{2} + 64)^{4}$$
$31$ $$(T^{8} - 80 T^{6} + 4832 T^{4} + \cdots + 2458624)^{2}$$
$37$ $$(T^{4} + 8 T^{3} + 66 T^{2} - 16 T + 4)^{4}$$
$41$ $$(T^{4} - 148 T^{2} + 1058)^{4}$$
$43$ $$(T^{2} - 8 T + 8)^{8}$$
$47$ $$(T^{8} + 208 T^{6} + 34016 T^{4} + \cdots + 85525504)^{2}$$
$53$ $$(T^{8} - 164 T^{6} + 24780 T^{4} + \cdots + 4477456)^{2}$$
$59$ $$(T^{8} + 272 T^{6} + 57056 T^{4} + \cdots + 286557184)^{2}$$
$61$ $$(T^{8} - 164 T^{6} + 24974 T^{4} + \cdots + 3694084)^{2}$$
$67$ $$(T^{4} + 8 T^{3} + 120 T^{2} - 448 T + 3136)^{4}$$
$71$ $$(T^{4} + 264 T^{2} + 15376)^{4}$$
$73$ $$(T^{8} - 148 T^{6} + 20846 T^{4} + \cdots + 1119364)^{2}$$
$79$ $$(T^{4} + 128 T^{2} + 16384)^{4}$$
$83$ $$(T^{4} - 128 T^{2} + 2048)^{4}$$
$89$ $$(T^{8} + 212 T^{6} + 44366 T^{4} + \cdots + 334084)^{2}$$
$97$ $$(T^{4} + 148 T^{2} + 1058)^{4}$$