# Properties

 Label 1764.4.k.bc Level $1764$ Weight $4$ Character orbit 1764.k Analytic conductor $104.079$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1764.k (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$104.079369250$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.7027860559216896.42 Defining polynomial: $$x^{8} - 4 x^{7} - 88 x^{6} + 278 x^{5} + 3869 x^{4} - 8206 x^{3} - 71054 x^{2} + 75204 x + 945876$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{16}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{5} +O(q^{10})$$ $$q -\beta_{5} q^{5} + ( \beta_{2} + \beta_{3} ) q^{11} + \beta_{6} q^{13} + ( 9 \beta_{4} + 9 \beta_{5} ) q^{17} + ( -\beta_{6} - \beta_{7} ) q^{19} -3 \beta_{2} q^{23} + ( 13 + 13 \beta_{1} ) q^{25} + 2 \beta_{3} q^{29} -\beta_{7} q^{31} + 78 \beta_{1} q^{37} -39 \beta_{4} q^{41} + 148 q^{43} + 44 \beta_{5} q^{47} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{53} -7 \beta_{6} q^{55} + ( 52 \beta_{4} + 52 \beta_{5} ) q^{59} + ( 7 \beta_{6} + 7 \beta_{7} ) q^{61} + 16 \beta_{2} q^{65} + ( 260 + 260 \beta_{1} ) q^{67} -13 \beta_{3} q^{71} + 8 \beta_{7} q^{73} + 664 \beta_{1} q^{79} -12 \beta_{4} q^{83} + 1008 q^{85} -83 \beta_{5} q^{89} + ( -16 \beta_{2} - 16 \beta_{3} ) q^{95} + 14 \beta_{6} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 52q^{25} - 312q^{37} + 1184q^{43} + 1040q^{67} - 2656q^{79} + 8064q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 88 x^{6} + 278 x^{5} + 3869 x^{4} - 8206 x^{3} - 71054 x^{2} + 75204 x + 945876$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{5} - 122 \nu^{4} + 249 \nu^{3} + 4585 \nu^{2} - 4710 \nu - 88614$$$$)/46410$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{5} - 122 \nu^{4} + 249 \nu^{3} + 11215 \nu^{2} - 11340 \nu - 221214$$$$)/3315$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{5} - 57 \nu^{4} + 119 \nu^{3} + 1140 \nu^{2} - 1200 \nu + 8366$$$$)/455$$ $$\beta_{4}$$ $$=$$ $$($$$$-212 \nu^{7} + 742 \nu^{6} + 3310 \nu^{5} - 10130 \nu^{4} + 290614 \nu^{3} - 425420 \nu^{2} - 14728656 \nu + 7434876$$$$)/5610969$$ $$\beta_{5}$$ $$=$$ $$($$$$-172 \nu^{7} + 602 \nu^{6} + 16646 \nu^{5} - 43120 \nu^{4} - 834526 \nu^{3} + 1295210 \nu^{2} + 10396488 \nu - 5415564$$$$)/4007835$$ $$\beta_{6}$$ $$=$$ $$($$$$-688 \nu^{7} + 2408 \nu^{6} + 66584 \nu^{5} - 172480 \nu^{4} - 3338104 \nu^{3} + 5180840 \nu^{2} + 105711312 \nu - 53724936$$$$)/4007835$$ $$\beta_{7}$$ $$=$$ $$($$$$-13912 \nu^{7} + 48692 \nu^{6} + 1231676 \nu^{5} - 3200920 \nu^{4} - 37329676 \nu^{3} + 59219780 \nu^{2} + 90846408 \nu - 55401024$$$$)/28054845$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - 4 \beta_{5} + 8$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 4 \beta_{5} + 8 \beta_{2} - 112 \beta_{1} + 328$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$21 \beta_{7} + 28 \beta_{6} - 330 \beta_{5} - 28 \beta_{4} + 12 \beta_{2} - 168 \beta_{1} + 488$$$$)/16$$ $$\nu^{4}$$ $$=$$ $$($$$$42 \beta_{7} + 55 \beta_{6} - 656 \beta_{5} - 56 \beta_{4} + 112 \beta_{3} + 448 \beta_{2} - 17696 \beta_{1} - 5944$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$1715 \beta_{7} + 566 \beta_{6} - 15474 \beta_{5} - 7504 \beta_{4} + 280 \beta_{3} + 1100 \beta_{2} - 43960 \beta_{1} - 15672$$$$)/16$$ $$\nu^{6}$$ $$=$$ $$($$$$5040 \beta_{7} + 1561 \beta_{6} - 44784 \beta_{5} - 22372 \beta_{4} + 14504 \beta_{3} + 18288 \beta_{2} - 992880 \beta_{1} - 942072$$$$)/16$$ $$\nu^{7}$$ $$=$$ $$($$$$71197 \beta_{7} - 21426 \beta_{6} - 533442 \beta_{5} - 654640 \beta_{4} + 49784 \beta_{3} + 60172 \beta_{2} - 3321416 \beta_{1} - 3241832$$$$)/16$$

## Character values

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

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

## 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
 −6.04303 − 2.29129i 4.39728 − 2.29129i 7.04303 + 2.29129i −3.39728 + 2.29129i −6.04303 + 2.29129i 4.39728 + 2.29129i 7.04303 − 2.29129i −3.39728 − 2.29129i
0 0 0 −5.29150 9.16515i 0 0 0 0 0
361.2 0 0 0 −5.29150 9.16515i 0 0 0 0 0
361.3 0 0 0 5.29150 + 9.16515i 0 0 0 0 0
361.4 0 0 0 5.29150 + 9.16515i 0 0 0 0 0
1549.1 0 0 0 −5.29150 + 9.16515i 0 0 0 0 0
1549.2 0 0 0 −5.29150 + 9.16515i 0 0 0 0 0
1549.3 0 0 0 5.29150 9.16515i 0 0 0 0 0
1549.4 0 0 0 5.29150 9.16515i 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 1764.4.k.bc 8
3.b odd 2 1 inner 1764.4.k.bc 8
7.b odd 2 1 inner 1764.4.k.bc 8
7.c even 3 1 1764.4.a.bb 4
7.c even 3 1 inner 1764.4.k.bc 8
7.d odd 6 1 1764.4.a.bb 4
7.d odd 6 1 inner 1764.4.k.bc 8
21.c even 2 1 inner 1764.4.k.bc 8
21.g even 6 1 1764.4.a.bb 4
21.g even 6 1 inner 1764.4.k.bc 8
21.h odd 6 1 1764.4.a.bb 4
21.h odd 6 1 inner 1764.4.k.bc 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.4.a.bb 4 7.c even 3 1
1764.4.a.bb 4 7.d odd 6 1
1764.4.a.bb 4 21.g even 6 1
1764.4.a.bb 4 21.h odd 6 1
1764.4.k.bc 8 1.a even 1 1 trivial
1764.4.k.bc 8 3.b odd 2 1 inner
1764.4.k.bc 8 7.b odd 2 1 inner
1764.4.k.bc 8 7.c even 3 1 inner
1764.4.k.bc 8 7.d odd 6 1 inner
1764.4.k.bc 8 21.c even 2 1 inner
1764.4.k.bc 8 21.g even 6 1 inner
1764.4.k.bc 8 21.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{4} + 112 T_{5}^{2} + 12544$$ $$T_{11}^{4} + 3052 T_{11}^{2} + 9314704$$ $$T_{13}^{2} - 6976$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 12544 + 112 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 9314704 + 3052 T^{2} + T^{4} )^{2}$$
$13$ $$( -6976 + T^{2} )^{4}$$
$17$ $$( 82301184 + 9072 T^{2} + T^{4} )^{2}$$
$19$ $$( 48664576 + 6976 T^{2} + T^{4} )^{2}$$
$23$ $$( 754491024 + 27468 T^{2} + T^{4} )^{2}$$
$29$ $$( -12208 + T^{2} )^{4}$$
$31$ $$( 48664576 + 6976 T^{2} + T^{4} )^{2}$$
$37$ $$( 6084 + 78 T + T^{2} )^{4}$$
$41$ $$( -170352 + T^{2} )^{4}$$
$43$ $$( -148 + T )^{8}$$
$47$ $$( 47016116224 + 216832 T^{2} + T^{4} )^{2}$$
$53$ $$( 149035264 + 12208 T^{2} + T^{4} )^{2}$$
$59$ $$( 91716911104 + 302848 T^{2} + T^{4} )^{2}$$
$61$ $$( 116843646976 + 341824 T^{2} + T^{4} )^{2}$$
$67$ $$( 67600 - 260 T + T^{2} )^{4}$$
$71$ $$( -515788 + T^{2} )^{4}$$
$73$ $$( 199330103296 + 446464 T^{2} + T^{4} )^{2}$$
$79$ $$( 440896 + 664 T + T^{2} )^{4}$$
$83$ $$( -16128 + T^{2} )^{4}$$
$89$ $$( 595317178624 + 771568 T^{2} + T^{4} )^{2}$$
$97$ $$( -1367296 + T^{2} )^{4}$$