# Properties

 Label 1764.2.e.g Level $1764$ Weight $2$ Character orbit 1764.e Analytic conductor $14.086$ Analytic rank $0$ Dimension $12$ 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.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.653473922154496.1 Defining polynomial: $$x^{12} - 4 x^{10} + 13 x^{8} - 28 x^{6} + 52 x^{4} - 64 x^{2} + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} + ( 1 + \beta_{10} ) q^{4} -\beta_{4} q^{5} + ( \beta_{6} - \beta_{9} ) q^{8} +O(q^{10})$$ $$q + \beta_{6} q^{2} + ( 1 + \beta_{10} ) q^{4} -\beta_{4} q^{5} + ( \beta_{6} - \beta_{9} ) q^{8} + ( 1 - \beta_{2} - \beta_{5} + \beta_{8} - \beta_{10} ) q^{10} + ( \beta_{3} + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{2} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{13} + ( -1 + \beta_{8} + \beta_{10} ) q^{16} + ( 2 \beta_{1} - \beta_{4} ) q^{17} + ( \beta_{2} - \beta_{8} + 2 \beta_{10} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{20} + ( 2 + \beta_{8} + \beta_{10} + \beta_{11} ) q^{22} + ( -2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{23} + ( -1 - \beta_{2} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{25} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{7} ) q^{26} + \beta_{1} q^{29} + ( -1 + 3 \beta_{2} + \beta_{5} - \beta_{8} - \beta_{11} ) q^{31} + ( -\beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{32} + ( 1 + \beta_{2} - \beta_{5} + \beta_{8} - \beta_{10} ) q^{34} + ( \beta_{2} - 2 \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} ) q^{37} + ( 2 \beta_{4} - 2 \beta_{9} ) q^{38} + ( -2 \beta_{2} - 2 \beta_{5} ) q^{40} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{9} ) q^{41} + ( -1 + 2 \beta_{2} + \beta_{5} - 2 \beta_{10} - \beta_{11} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{44} + ( -4 - 2 \beta_{5} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{46} + ( -2 \beta_{1} + 2 \beta_{4} + 4 \beta_{7} ) q^{47} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{50} + ( 2 - 2 \beta_{2} + 2 \beta_{8} + 2 \beta_{11} ) q^{52} + ( -\beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} ) q^{53} + ( 2 - 3 \beta_{2} - 2 \beta_{5} + 3 \beta_{8} ) q^{55} + \beta_{2} q^{58} + ( 2 \beta_{1} - 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{59} + ( -5 - \beta_{2} - \beta_{5} - \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{61} + ( 4 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{62} + ( -1 - \beta_{2} - 2 \beta_{5} + 3 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{64} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{9} ) q^{65} + ( 3 \beta_{2} - \beta_{8} + 3 \beta_{10} - \beta_{11} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{68} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} ) q^{71} + ( 1 - 2 \beta_{2} + 3 \beta_{5} - 2 \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{73} + ( -6 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{74} + ( -6 + 2 \beta_{2} + 2 \beta_{5} + 2 \beta_{10} ) q^{76} + ( 2 - 3 \beta_{2} - 2 \beta_{5} + \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{79} + ( -4 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} ) q^{80} + ( -3 - \beta_{2} - \beta_{5} + \beta_{8} + 3 \beta_{10} ) q^{82} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( -4 - \beta_{2} + 2 \beta_{5} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{85} + ( 4 \beta_{1} + 2 \beta_{7} + 2 \beta_{9} ) q^{86} + ( 4 - 3 \beta_{2} + 2 \beta_{8} + \beta_{10} + \beta_{11} ) q^{88} + ( -2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} ) q^{89} + ( -6 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{92} + ( 2 - 2 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} + 4 \beta_{11} ) q^{94} + ( 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{9} ) q^{95} + ( -1 - 3 \beta_{5} - 3 \beta_{10} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 8q^{4} + O(q^{10})$$ $$12q + 8q^{4} + 8q^{10} - 20q^{16} + 20q^{22} - 12q^{25} + 16q^{34} + 8q^{37} - 8q^{40} - 36q^{46} + 16q^{52} + 4q^{58} - 56q^{61} - 16q^{64} - 72q^{76} - 56q^{82} - 56q^{85} + 28q^{88} + 24q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{10} + 13 x^{8} - 28 x^{6} + 52 x^{4} - 64 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{9} - 5 \nu^{7} + 12 \nu^{5} - 12 \nu^{3} + 16 \nu$$$$)/32$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{10} + 4 \nu^{8} - 5 \nu^{6} + 12 \nu^{4} - 12 \nu^{2} + 16$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{9} - 13 \nu^{7} + 28 \nu^{5} - 20 \nu^{3} + 32 \nu$$$$)/32$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{9} - 2 \nu^{7} + 9 \nu^{5} - 10 \nu^{3} + 16 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{10} + 4 \nu^{8} - 13 \nu^{6} + 28 \nu^{4} - 20 \nu^{2} + 32$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{11} + 4 \nu^{9} - 13 \nu^{7} + 28 \nu^{5} - 52 \nu^{3} + 64 \nu$$$$)/32$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{11} - 4 \nu^{9} + 11 \nu^{7} - 28 \nu^{5} + 36 \nu^{3} - 48 \nu$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{10} + 8 \nu^{8} - 23 \nu^{6} + 32 \nu^{4} - 44 \nu^{2} + 16$$$$)/16$$ $$\beta_{9}$$ $$=$$ $$($$$$3 \nu^{11} - 8 \nu^{9} + 23 \nu^{7} - 32 \nu^{5} + 44 \nu^{3} + 16 \nu$$$$)/32$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{10} + 4 \nu^{8} - 13 \nu^{6} + 28 \nu^{4} - 52 \nu^{2} + 48$$$$)/16$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{10} - 8 \nu^{8} + 21 \nu^{6} - 32 \nu^{4} + 60 \nu^{2} - 48$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} + \beta_{5} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{9} - \beta_{6} - \beta_{4} + 3 \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} - \beta_{8} + 2 \beta_{5} + \beta_{2} - 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{9} + 2 \beta_{7} - 3 \beta_{6} + 5 \beta_{4} + \beta_{3} - 3 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$2 \beta_{11} + 3 \beta_{10} - 2 \beta_{8} - \beta_{5} + 6 \beta_{2} - 5$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{9} + 4 \beta_{7} - 3 \beta_{6} + 9 \beta_{4} - 7 \beta_{3} + 3 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-5 \beta_{11} - 5 \beta_{10} - 3 \beta_{8} - 2 \beta_{5} + 11 \beta_{2} - 4$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$\beta_{9} - 10 \beta_{7} - 5 \beta_{6} - 5 \beta_{4} - 9 \beta_{3} + 27 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-18 \beta_{11} - 11 \beta_{10} - 14 \beta_{8} + 9 \beta_{5} - 6 \beta_{2} - 19$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$\beta_{9} - 36 \beta_{7} - 13 \beta_{6} - 9 \beta_{4} - 9 \beta_{3} - 3 \beta_{1}$$$$)/2$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1079.1
 −1.35489 + 0.405301i −1.35489 − 0.405301i −1.16947 + 0.795191i −1.16947 − 0.795191i −0.892524 + 1.09700i −0.892524 − 1.09700i 0.892524 + 1.09700i 0.892524 − 1.09700i 1.16947 + 0.795191i 1.16947 − 0.795191i 1.35489 + 0.405301i 1.35489 − 0.405301i
−1.35489 0.405301i 0 1.67146 + 1.09828i 3.31339i 0 0 −1.81951 2.16549i 0 −1.34292 + 4.48929i
1079.2 −1.35489 + 0.405301i 0 1.67146 1.09828i 3.31339i 0 0 −1.81951 + 2.16549i 0 −1.34292 4.48929i
1079.3 −1.16947 0.795191i 0 0.735342 + 1.85991i 0.665647i 0 0 0.619022 2.75986i 0 0.529317 0.778457i
1079.4 −1.16947 + 0.795191i 0 0.735342 1.85991i 0.665647i 0 0 0.619022 + 2.75986i 0 0.529317 + 0.778457i
1079.5 −0.892524 1.09700i 0 −0.406803 + 1.95819i 2.56483i 0 0 2.51121 1.30147i 0 2.81361 2.28917i
1079.6 −0.892524 + 1.09700i 0 −0.406803 1.95819i 2.56483i 0 0 2.51121 + 1.30147i 0 2.81361 + 2.28917i
1079.7 0.892524 1.09700i 0 −0.406803 1.95819i 2.56483i 0 0 −2.51121 1.30147i 0 2.81361 + 2.28917i
1079.8 0.892524 + 1.09700i 0 −0.406803 + 1.95819i 2.56483i 0 0 −2.51121 + 1.30147i 0 2.81361 2.28917i
1079.9 1.16947 0.795191i 0 0.735342 1.85991i 0.665647i 0 0 −0.619022 2.75986i 0 0.529317 + 0.778457i
1079.10 1.16947 + 0.795191i 0 0.735342 + 1.85991i 0.665647i 0 0 −0.619022 + 2.75986i 0 0.529317 0.778457i
1079.11 1.35489 0.405301i 0 1.67146 1.09828i 3.31339i 0 0 1.81951 2.16549i 0 −1.34292 4.48929i
1079.12 1.35489 + 0.405301i 0 1.67146 + 1.09828i 3.31339i 0 0 1.81951 + 2.16549i 0 −1.34292 + 4.48929i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1079.12 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
4.b odd 2 1 inner
12.b 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.e.g 12
3.b odd 2 1 inner 1764.2.e.g 12
4.b odd 2 1 inner 1764.2.e.g 12
7.b odd 2 1 252.2.e.a 12
12.b even 2 1 inner 1764.2.e.g 12
21.c even 2 1 252.2.e.a 12
28.d even 2 1 252.2.e.a 12
56.e even 2 1 4032.2.h.h 12
56.h odd 2 1 4032.2.h.h 12
84.h odd 2 1 252.2.e.a 12
168.e odd 2 1 4032.2.h.h 12
168.i even 2 1 4032.2.h.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.e.a 12 7.b odd 2 1
252.2.e.a 12 21.c even 2 1
252.2.e.a 12 28.d even 2 1
252.2.e.a 12 84.h odd 2 1
1764.2.e.g 12 1.a even 1 1 trivial
1764.2.e.g 12 3.b odd 2 1 inner
1764.2.e.g 12 4.b odd 2 1 inner
1764.2.e.g 12 12.b even 2 1 inner
4032.2.h.h 12 56.e even 2 1
4032.2.h.h 12 56.h odd 2 1
4032.2.h.h 12 168.e odd 2 1
4032.2.h.h 12 168.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 $$S_{2}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{6} + 18 T_{5}^{4} + 80 T_{5}^{2} + 32$$ $$T_{13}^{3} - 28 T_{13} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 - 64 T^{2} + 52 T^{4} - 28 T^{6} + 13 T^{8} - 4 T^{10} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$( 32 + 80 T^{2} + 18 T^{4} + T^{6} )^{2}$$
$7$ $$T^{12}$$
$11$ $$( -128 + 132 T^{2} - 28 T^{4} + T^{6} )^{2}$$
$13$ $$( 16 - 28 T + T^{3} )^{4}$$
$17$ $$( 32 + 144 T^{2} + 34 T^{4} + T^{6} )^{2}$$
$19$ $$( 4096 + 1088 T^{2} + 64 T^{4} + T^{6} )^{2}$$
$23$ $$( -67712 + 5924 T^{2} - 140 T^{4} + T^{6} )^{2}$$
$29$ $$( 2 + T^{2} )^{6}$$
$31$ $$( 65536 + 5184 T^{2} + 128 T^{4} + T^{6} )^{2}$$
$37$ $$( -128 - 64 T - 2 T^{2} + T^{3} )^{4}$$
$41$ $$( 53792 + 4304 T^{2} + 114 T^{4} + T^{6} )^{2}$$
$43$ $$( 16384 + 4352 T^{2} + 160 T^{4} + T^{6} )^{2}$$
$47$ $$( -524288 + 22592 T^{2} - 272 T^{4} + T^{6} )^{2}$$
$53$ $$( 2312 + 2444 T^{2} + 134 T^{4} + T^{6} )^{2}$$
$59$ $$( -8192 + 4672 T^{2} - 208 T^{4} + T^{6} )^{2}$$
$61$ $$( -8 + 4 T + 14 T^{2} + T^{3} )^{4}$$
$67$ $$( 16384 + 3728 T^{2} + 216 T^{4} + T^{6} )^{2}$$
$71$ $$( -36992 + 6180 T^{2} - 268 T^{4} + T^{6} )^{2}$$
$73$ $$( 352 - 172 T + T^{3} )^{4}$$
$79$ $$( 16384 + 24336 T^{2} + 312 T^{4} + T^{6} )^{2}$$
$83$ $$( -32768 + 4352 T^{2} - 128 T^{4} + T^{6} )^{2}$$
$89$ $$( 170528 + 11536 T^{2} + 226 T^{4} + T^{6} )^{2}$$
$97$ $$( -128 - 156 T + T^{3} )^{4}$$