# Properties

 Label 1764.3.z.m Level $1764$ Weight $3$ Character orbit 1764.z Analytic conductor $48.066$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$48.0655186332$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$7^{2}$$ Twist minimal: no (minimal twist has level 588) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{2} + \beta_{5} + \beta_{6} ) q^{5} +O(q^{10})$$ $$q + ( 2 \beta_{2} + \beta_{5} + \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{5} + 2 \beta_{7} ) q^{11} + ( 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - 8 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{13} + ( 4 + 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} ) q^{17} + ( 16 - 4 \beta_{2} + 8 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{19} + ( 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{23} + ( -9 - 2 \beta_{1} - \beta_{2} - 9 \beta_{4} - \beta_{5} - 4 \beta_{7} ) q^{25} + ( -10 + \beta_{1} - 5 \beta_{2} + 6 \beta_{3} - 6 \beta_{6} - \beta_{7} ) q^{29} + ( -4 + 3 \beta_{1} - 14 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 14 \beta_{5} ) q^{31} + ( -12 \beta_{1} + 8 \beta_{3} + 16 \beta_{4} - 9 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{37} + ( -14 + 5 \beta_{1} + 13 \beta_{2} - 7 \beta_{3} - 28 \beta_{4} + 26 \beta_{5} - 7 \beta_{6} + 5 \beta_{7} ) q^{41} + ( -14 - 2 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{6} + 2 \beta_{7} ) q^{43} + ( -44 + 4 \beta_{2} - 22 \beta_{4} + 2 \beta_{5} - 8 \beta_{6} + 7 \beta_{7} ) q^{47} + ( -18 + 6 \beta_{1} + 26 \beta_{2} - 4 \beta_{3} - 18 \beta_{4} + 26 \beta_{5} - 8 \beta_{6} + 12 \beta_{7} ) q^{53} + ( \beta_{1} - 10 \beta_{2} - 6 \beta_{3} - 20 \beta_{5} - 6 \beta_{6} + \beta_{7} ) q^{55} + ( 14 + 5 \beta_{1} + 22 \beta_{2} + 12 \beta_{3} - 14 \beta_{4} - 22 \beta_{5} ) q^{59} + ( 24 - 28 \beta_{2} + 12 \beta_{4} - 14 \beta_{5} + 5 \beta_{6} + 8 \beta_{7} ) q^{61} + ( 10 \beta_{1} - 8 \beta_{3} - 30 \beta_{4} - 19 \beta_{5} - 4 \beta_{6} + 5 \beta_{7} ) q^{65} + ( 8 - 6 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 12 \beta_{5} - 8 \beta_{6} - 12 \beta_{7} ) q^{67} + ( -28 - \beta_{1} + 43 \beta_{2} + 8 \beta_{3} - 8 \beta_{6} + \beta_{7} ) q^{71} + ( -28 - 8 \beta_{1} - 6 \beta_{2} + 15 \beta_{3} + 28 \beta_{4} + 6 \beta_{5} ) q^{73} + ( 8 \beta_{1} - 54 \beta_{4} - 28 \beta_{5} + 4 \beta_{7} ) q^{79} + ( 16 + 14 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + 32 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} + 14 \beta_{7} ) q^{83} + ( -12 + 15 \beta_{2} - 4 \beta_{3} + 4 \beta_{6} ) q^{85} + ( 16 + 46 \beta_{2} + 8 \beta_{4} + 23 \beta_{5} + 11 \beta_{6} + 16 \beta_{7} ) q^{89} + ( -34 + 4 \beta_{1} + 36 \beta_{2} + 10 \beta_{3} - 34 \beta_{4} + 36 \beta_{5} + 20 \beta_{6} + 8 \beta_{7} ) q^{95} + ( 28 + 4 \beta_{1} - 10 \beta_{2} + 3 \beta_{3} + 56 \beta_{4} - 20 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 48q^{17} + 96q^{19} - 8q^{23} - 36q^{25} - 80q^{29} - 48q^{31} - 64q^{37} - 112q^{43} - 264q^{47} - 72q^{53} + 168q^{59} + 144q^{61} + 120q^{65} + 32q^{67} - 224q^{71} - 336q^{73} + 216q^{79} - 96q^{85} + 96q^{89} - 136q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 76 \nu$$$$)/14$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 20$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 27 \nu$$$$)/7$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 2$$$$)/14$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{6} + 7 \nu^{4} - 21 \nu^{2} + 2$$$$)/7$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{7} + 35 \nu^{5} - 112 \nu^{3} + 64 \nu$$$$)/14$$ $$\beta_{7}$$ $$=$$ $$($$$$11 \nu^{7} - 42 \nu^{5} + 154 \nu^{3} - 88 \nu$$$$)/14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{1}$$$$)/7$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - 2 \beta_{4}$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{7} + 6 \beta_{6} + 6 \beta_{3} + 5 \beta_{1}$$$$)/7$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{5} - 6 \beta_{4} + 4 \beta_{2} - 6$$ $$\nu^{5}$$ $$=$$ $$($$$$16 \beta_{7} + 22 \beta_{6}$$$$)/7$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{2} - 20$$ $$\nu^{7}$$ $$=$$ $$($$$$-76 \beta_{3} - 54 \beta_{1}$$$$)/7$$

## Character values

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

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
325.1
 1.60021 + 0.923880i −0.662827 − 0.382683i −1.60021 − 0.923880i 0.662827 + 0.382683i 1.60021 − 0.923880i −0.662827 + 0.382683i −1.60021 + 0.923880i 0.662827 − 0.382683i
0 0 0 −5.04718 + 2.91399i 0 0 0 0 0
325.2 0 0 0 −0.416265 + 0.240331i 0 0 0 0 0
325.3 0 0 0 0.804540 0.464502i 0 0 0 0 0
325.4 0 0 0 4.65891 2.68982i 0 0 0 0 0
901.1 0 0 0 −5.04718 2.91399i 0 0 0 0 0
901.2 0 0 0 −0.416265 0.240331i 0 0 0 0 0
901.3 0 0 0 0.804540 + 0.464502i 0 0 0 0 0
901.4 0 0 0 4.65891 + 2.68982i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.z.m 8
3.b odd 2 1 588.3.m.e 8
7.b odd 2 1 1764.3.z.l 8
7.c even 3 1 1764.3.d.h 8
7.c even 3 1 1764.3.z.l 8
7.d odd 6 1 1764.3.d.h 8
7.d odd 6 1 inner 1764.3.z.m 8
21.c even 2 1 588.3.m.f 8
21.g even 6 1 588.3.d.c 8
21.g even 6 1 588.3.m.e 8
21.h odd 6 1 588.3.d.c 8
21.h odd 6 1 588.3.m.f 8
84.j odd 6 1 2352.3.f.j 8
84.n even 6 1 2352.3.f.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.d.c 8 21.g even 6 1
588.3.d.c 8 21.h odd 6 1
588.3.m.e 8 3.b odd 2 1
588.3.m.e 8 21.g even 6 1
588.3.m.f 8 21.c even 2 1
588.3.m.f 8 21.h odd 6 1
1764.3.d.h 8 7.c even 3 1
1764.3.d.h 8 7.d odd 6 1
1764.3.z.l 8 7.b odd 2 1
1764.3.z.l 8 7.c even 3 1
1764.3.z.m 8 1.a even 1 1 trivial
1764.3.z.m 8 7.d odd 6 1 inner
2352.3.f.j 8 84.j odd 6 1
2352.3.f.j 8 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 $$S_{3}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{8} - 32 T_{5}^{6} + 1010 T_{5}^{4} - 768 T_{5}^{3} - 256 T_{5}^{2} + 336 T_{5} + 196$$ $$T_{11}^{8} + 124 T_{11}^{6} + 96 T_{11}^{5} + 12084 T_{11}^{4} + 5952 T_{11}^{3} + 410512 T_{11}^{2} - 158016 T_{11} + 10837264$$ $$T_{13}^{8} + 712 T_{13}^{6} + 123284 T_{13}^{4} + 1723280 T_{13}^{2} + 2979076$$ $$T_{19}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$196 + 336 T - 256 T^{2} - 768 T^{3} + 1010 T^{4} - 32 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$10837264 - 158016 T + 410512 T^{2} + 5952 T^{3} + 12084 T^{4} + 96 T^{5} + 124 T^{6} + T^{8}$$
$13$ $$2979076 + 1723280 T^{2} + 123284 T^{4} + 712 T^{6} + T^{8}$$
$17$ $$168428484 - 68212368 T + 9208512 T^{2} - 71118 T^{4} + 768 T^{6} - 48 T^{7} + T^{8}$$
$19$ $$285745216 - 199602432 T + 62839360 T^{2} - 11430144 T^{3} + 1297976 T^{4} - 92928 T^{5} + 4040 T^{6} - 96 T^{7} + T^{8}$$
$23$ $$1443088144 + 374409728 T + 73892080 T^{2} + 6639680 T^{3} + 491380 T^{4} + 14816 T^{5} + 676 T^{6} + 8 T^{7} + T^{8}$$
$29$ $$( 468892 - 65440 T - 1924 T^{2} + 40 T^{3} + T^{4} )^{2}$$
$31$ $$2052452416 + 7467548928 T + 9185192768 T^{2} + 468122880 T^{3} + 5473592 T^{4} - 136320 T^{5} - 2072 T^{6} + 48 T^{7} + T^{8}$$
$37$ $$298373767696 + 125188551424 T + 50617849744 T^{2} + 870228736 T^{3} + 27408076 T^{4} + 234880 T^{5} + 7588 T^{6} + 64 T^{7} + T^{8}$$
$41$ $$26697826996036 + 51769047200 T^{2} + 34716572 T^{4} + 9808 T^{6} + T^{8}$$
$43$ $$( 192784 - 203168 T - 3784 T^{2} + 56 T^{3} + T^{4} )^{2}$$
$47$ $$8881401308224 + 111863586048 T - 17029896064 T^{2} - 220411392 T^{3} + 34157384 T^{4} + 1550208 T^{5} + 29104 T^{6} + 264 T^{7} + T^{8}$$
$53$ $$71641191948544 + 4130215804416 T + 191864861056 T^{2} + 3885089280 T^{3} + 73453104 T^{4} + 582528 T^{5} + 10648 T^{6} + 72 T^{7} + T^{8}$$
$59$ $$372481662446656 - 28782925866240 T + 869535448960 T^{2} - 9902630400 T^{3} - 20126776 T^{4} + 1115520 T^{5} + 2768 T^{6} - 168 T^{7} + T^{8}$$
$61$ $$454276 + 20414112 T + 303671288 T^{2} - 95104320 T^{3} + 8405102 T^{4} + 452160 T^{5} + 3772 T^{6} - 144 T^{7} + T^{8}$$
$67$ $$306756468736 - 146466111488 T + 66583023616 T^{2} - 1634828288 T^{3} + 45594496 T^{4} - 335360 T^{5} + 7072 T^{6} - 32 T^{7} + T^{8}$$
$71$ $$( -7722596 - 634384 T - 6460 T^{2} + 112 T^{3} + T^{4} )^{2}$$
$73$ $$3712242251524 + 23767993248 T - 14553796808 T^{2} - 93506880 T^{3} + 58001486 T^{4} + 2546880 T^{5} + 45212 T^{6} + 336 T^{7} + T^{8}$$
$79$ $$49705658450176 + 439257156096 T + 91586574976 T^{2} - 3820758528 T^{3} + 148346160 T^{4} - 2562432 T^{5} + 34216 T^{6} - 216 T^{7} + T^{8}$$
$83$ $$61585579131904 + 132200402944 T^{2} + 89759168 T^{4} + 19712 T^{6} + T^{8}$$
$89$ $$1748734585879876 - 136075027453008 T + 4335067503424 T^{2} - 62684901888 T^{3} + 308791826 T^{4} + 1849344 T^{5} - 16192 T^{6} - 96 T^{7} + T^{8}$$
$97$ $$5315948141956 + 28759544464 T^{2} + 36017108 T^{4} + 13640 T^{6} + T^{8}$$