Properties

 Label 1764.2.k.m Level 1764 Weight 2 Character orbit 1764.k Analytic conductor 14.086 Analytic rank 0 Dimension 8 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.k (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.12745506816.4 Defining polynomial: $$x^{8} + 8 x^{6} + 55 x^{4} + 72 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{5} +O(q^{10})$$ $$q -\beta_{7} q^{5} -\beta_{5} q^{11} -3 \beta_{6} q^{13} + \beta_{3} q^{17} + ( -2 \beta_{2} - 2 \beta_{6} ) q^{19} + ( -\beta_{4} - \beta_{5} ) q^{23} -9 \beta_{1} q^{25} + \beta_{4} q^{29} + 6 \beta_{2} q^{31} + ( -4 + 4 \beta_{1} ) q^{37} + ( -\beta_{3} + \beta_{7} ) q^{41} + 8 q^{43} + 2 \beta_{7} q^{47} + 2 \beta_{5} q^{53} -14 \beta_{6} q^{55} -2 \beta_{3} q^{59} + ( -7 \beta_{2} - 7 \beta_{6} ) q^{61} + ( 3 \beta_{4} + 3 \beta_{5} ) q^{65} -12 \beta_{1} q^{67} -3 \beta_{4} q^{71} + \beta_{2} q^{73} + ( 4 - 4 \beta_{1} ) q^{79} + ( 4 \beta_{3} - 4 \beta_{7} ) q^{83} + 14 q^{85} + \beta_{7} q^{89} + 2 \beta_{5} q^{95} -7 \beta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 36q^{25} - 16q^{37} + 64q^{43} - 48q^{67} + 16q^{79} + 112q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 8 x^{6} + 55 x^{4} + 72 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$8 \nu^{6} + 55 \nu^{4} + 440 \nu^{2} + 576$$$$)/495$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 203 \nu$$$$)/165$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 533 \nu$$$$)/165$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{6} + 296$$$$)/55$$ $$\beta_{5}$$ $$=$$ $$($$$$-46 \nu^{6} - 440 \nu^{4} - 2530 \nu^{2} - 3312$$$$)/495$$ $$\beta_{6}$$ $$=$$ $$($$$$8 \nu^{7} + 55 \nu^{5} + 341 \nu^{3} + 81 \nu$$$$)/297$$ $$\beta_{7}$$ $$=$$ $$($$$$79 \nu^{7} + 605 \nu^{5} + 4345 \nu^{3} + 5688 \nu$$$$)/1485$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 8 \beta_{1} - 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{7} - 11 \beta_{6} - 5 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-4 \beta_{5} - 23 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$-31 \beta_{7} + 79 \beta_{6} + 79 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-55 \beta_{4} + 296$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$203 \beta_{3} - 533 \beta_{2}$$$$)/2$$

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −1.28897 + 2.23256i −0.581861 + 1.00781i 1.28897 − 2.23256i 0.581861 − 1.00781i −1.28897 − 2.23256i −0.581861 − 1.00781i 1.28897 + 2.23256i 0.581861 + 1.00781i
0 0 0 −1.87083 3.24037i 0 0 0 0 0
361.2 0 0 0 −1.87083 3.24037i 0 0 0 0 0
361.3 0 0 0 1.87083 + 3.24037i 0 0 0 0 0
361.4 0 0 0 1.87083 + 3.24037i 0 0 0 0 0
1549.1 0 0 0 −1.87083 + 3.24037i 0 0 0 0 0
1549.2 0 0 0 −1.87083 + 3.24037i 0 0 0 0 0
1549.3 0 0 0 1.87083 3.24037i 0 0 0 0 0
1549.4 0 0 0 1.87083 3.24037i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1549.4 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.k.m 8
3.b odd 2 1 inner 1764.2.k.m 8
7.b odd 2 1 inner 1764.2.k.m 8
7.c even 3 1 1764.2.a.m 4
7.c even 3 1 inner 1764.2.k.m 8
7.d odd 6 1 1764.2.a.m 4
7.d odd 6 1 inner 1764.2.k.m 8
21.c even 2 1 inner 1764.2.k.m 8
21.g even 6 1 1764.2.a.m 4
21.g even 6 1 inner 1764.2.k.m 8
21.h odd 6 1 1764.2.a.m 4
21.h odd 6 1 inner 1764.2.k.m 8
28.f even 6 1 7056.2.a.cy 4
28.g odd 6 1 7056.2.a.cy 4
84.j odd 6 1 7056.2.a.cy 4
84.n even 6 1 7056.2.a.cy 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.a.m 4 7.c even 3 1
1764.2.a.m 4 7.d odd 6 1
1764.2.a.m 4 21.g even 6 1
1764.2.a.m 4 21.h odd 6 1
1764.2.k.m 8 1.a even 1 1 trivial
1764.2.k.m 8 3.b odd 2 1 inner
1764.2.k.m 8 7.b odd 2 1 inner
1764.2.k.m 8 7.c even 3 1 inner
1764.2.k.m 8 7.d odd 6 1 inner
1764.2.k.m 8 21.c even 2 1 inner
1764.2.k.m 8 21.g even 6 1 inner
1764.2.k.m 8 21.h odd 6 1 inner
7056.2.a.cy 4 28.f even 6 1
7056.2.a.cy 4 28.g odd 6 1
7056.2.a.cy 4 84.j odd 6 1
7056.2.a.cy 4 84.n even 6 1

Hecke kernels

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

 $$T_{5}^{4} + 14 T_{5}^{2} + 196$$ $$T_{11}^{4} + 28 T_{11}^{2} + 784$$ $$T_{13}^{2} - 18$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 4 T^{2} - 9 T^{4} + 100 T^{6} + 625 T^{8} )^{2}$$
$7$ 1
$11$ $$( 1 + 6 T^{2} - 85 T^{4} + 726 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 8 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 - 20 T^{2} + 111 T^{4} - 5780 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 30 T^{2} + 539 T^{4} - 10830 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 18 T^{2} - 205 T^{4} - 9522 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 30 T^{2} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 10 T^{2} - 861 T^{4} + 9610 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 68 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{8}$$
$47$ $$( 1 - 38 T^{2} - 765 T^{4} - 83942 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 6 T^{2} - 2773 T^{4} + 16854 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 - 62 T^{2} + 363 T^{4} - 215822 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 24 T^{2} - 3145 T^{4} - 89304 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 110 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 144 T^{2} + 15407 T^{4} - 767376 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 17 T + 79 T^{2} )^{4}( 1 + 13 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 58 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 - 164 T^{2} + 18975 T^{4} - 1299044 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 + 96 T^{2} + 9409 T^{4} )^{4}$$