# Properties

 Label 1764.2.l.f Level 1764 Weight 2 Character orbit 1764.l Analytic conductor 14.086 Analytic rank 0 Dimension 6 CM no Inner twists 2

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Defining polynomial: $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} + \beta_{4} + \beta_{5} ) q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -1 - 2 \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{3} + \beta_{4} + \beta_{5} ) q^{3} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -1 - 2 \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{9} + ( -1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} ) q^{11} + ( -2 + 4 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{13} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{15} + ( 1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{17} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} ) q^{19} + ( 5 - \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{23} + ( -3 + \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{25} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{27} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{29} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{31} + ( 5 - 5 \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{33} + ( 2 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{37} + ( -\beta_{2} - 5 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{39} + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{41} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{43} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{45} + ( -1 - \beta_{1} - \beta_{3} - 7 \beta_{4} + 3 \beta_{5} ) q^{47} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{51} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{53} + ( -\beta_{1} + 3 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{55} + ( -6 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{57} + ( 11 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 11 \beta_{4} - 2 \beta_{5} ) q^{59} + ( 1 - 8 \beta_{1} + \beta_{3} - 5 \beta_{4} + 6 \beta_{5} ) q^{61} + ( 2 - 7 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} ) q^{65} + ( -3 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{67} + ( -1 + \beta_{1} - \beta_{2} - 6 \beta_{3} + 9 \beta_{4} + 4 \beta_{5} ) q^{69} + ( 6 - 3 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{71} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{73} + ( 3 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{75} + ( 1 - 5 \beta_{1} + \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{79} + ( -10 + 4 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{81} + ( 7 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 7 \beta_{4} - \beta_{5} ) q^{83} + ( -2 + 4 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} ) q^{85} + ( 1 - 7 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{87} + ( 5 - \beta_{1} + 3 \beta_{2} - \beta_{3} + 5 \beta_{4} - 3 \beta_{5} ) q^{89} + ( 3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{5} ) q^{93} + ( -7 + \beta_{1} - \beta_{2} + \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{95} + ( -1 + \beta_{1} + 6 \beta_{2} + \beta_{3} - \beta_{4} - 6 \beta_{5} ) q^{97} + ( 1 - 4 \beta_{1} - 3 \beta_{2} + 5 \beta_{4} + 8 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 2q^{3} - 2q^{5} - 4q^{9} + O(q^{10})$$ $$6q + 2q^{3} - 2q^{5} - 4q^{9} + 4q^{11} + 3q^{13} + q^{15} + 2q^{17} + 3q^{19} + 28q^{23} - 12q^{25} - 7q^{27} - q^{29} - 3q^{31} + 25q^{33} + 3q^{37} + 18q^{39} - 3q^{43} + 10q^{45} + 21q^{47} - 13q^{51} - 6q^{53} - 12q^{55} - 37q^{57} + 31q^{59} + 6q^{61} - 15q^{65} - 6q^{67} - 5q^{69} + 34q^{71} - 3q^{73} + 7q^{75} + 9q^{79} - 40q^{81} + 20q^{83} + 15q^{85} - 16q^{87} + 12q^{89} + 33q^{93} - 20q^{95} - 9q^{97} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + 5 \nu^{3} + \nu^{2} + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 5 \nu^{3} + 2 \nu^{2} - 3 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 33 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 6 \beta_{1} + 6$$ $$\nu^{5}$$ $$=$$ $$-3 \beta_{5} - 2 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} + 7$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 0.5 + 1.41036i 0.5 − 2.05195i 0.5 − 0.224437i 0.5 − 1.41036i 0.5 + 2.05195i 0.5 + 0.224437i
0 −1.09097 + 1.34528i 0 −0.239123 0 0 0 −0.619562 2.93533i 0
949.2 0 0.796790 + 1.53790i 0 −2.46050 0 0 0 −1.73025 + 2.45076i 0
949.3 0 1.29418 1.15113i 0 1.69963 0 0 0 0.349814 2.97954i 0
961.1 0 −1.09097 1.34528i 0 −0.239123 0 0 0 −0.619562 + 2.93533i 0
961.2 0 0.796790 1.53790i 0 −2.46050 0 0 0 −1.73025 2.45076i 0
961.3 0 1.29418 + 1.15113i 0 1.69963 0 0 0 0.349814 + 2.97954i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.l.f 6
3.b odd 2 1 5292.2.l.f 6
7.b odd 2 1 1764.2.l.e 6
7.c even 3 1 1764.2.i.d 6
7.c even 3 1 1764.2.j.e 6
7.d odd 6 1 252.2.j.a 6
7.d odd 6 1 1764.2.i.g 6
9.c even 3 1 1764.2.i.d 6
9.d odd 6 1 5292.2.i.e 6
21.c even 2 1 5292.2.l.e 6
21.g even 6 1 756.2.j.b 6
21.g even 6 1 5292.2.i.f 6
21.h odd 6 1 5292.2.i.e 6
21.h odd 6 1 5292.2.j.d 6
28.f even 6 1 1008.2.r.j 6
63.g even 3 1 inner 1764.2.l.f 6
63.h even 3 1 1764.2.j.e 6
63.i even 6 1 756.2.j.b 6
63.j odd 6 1 5292.2.j.d 6
63.k odd 6 1 1764.2.l.e 6
63.k odd 6 1 2268.2.a.i 3
63.l odd 6 1 1764.2.i.g 6
63.n odd 6 1 5292.2.l.f 6
63.o even 6 1 5292.2.i.f 6
63.s even 6 1 2268.2.a.h 3
63.s even 6 1 5292.2.l.e 6
63.t odd 6 1 252.2.j.a 6
84.j odd 6 1 3024.2.r.j 6
252.n even 6 1 9072.2.a.by 3
252.r odd 6 1 3024.2.r.j 6
252.bj even 6 1 1008.2.r.j 6
252.bn odd 6 1 9072.2.a.bv 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.a 6 7.d odd 6 1
252.2.j.a 6 63.t odd 6 1
756.2.j.b 6 21.g even 6 1
756.2.j.b 6 63.i even 6 1
1008.2.r.j 6 28.f even 6 1
1008.2.r.j 6 252.bj even 6 1
1764.2.i.d 6 7.c even 3 1
1764.2.i.d 6 9.c even 3 1
1764.2.i.g 6 7.d odd 6 1
1764.2.i.g 6 63.l odd 6 1
1764.2.j.e 6 7.c even 3 1
1764.2.j.e 6 63.h even 3 1
1764.2.l.e 6 7.b odd 2 1
1764.2.l.e 6 63.k odd 6 1
1764.2.l.f 6 1.a even 1 1 trivial
1764.2.l.f 6 63.g even 3 1 inner
2268.2.a.h 3 63.s even 6 1
2268.2.a.i 3 63.k odd 6 1
3024.2.r.j 6 84.j odd 6 1
3024.2.r.j 6 252.r odd 6 1
5292.2.i.e 6 9.d odd 6 1
5292.2.i.e 6 21.h odd 6 1
5292.2.i.f 6 21.g even 6 1
5292.2.i.f 6 63.o even 6 1
5292.2.j.d 6 21.h odd 6 1
5292.2.j.d 6 63.j odd 6 1
5292.2.l.e 6 21.c even 2 1
5292.2.l.e 6 63.s even 6 1
5292.2.l.f 6 3.b odd 2 1
5292.2.l.f 6 63.n odd 6 1
9072.2.a.bv 3 252.bn odd 6 1
9072.2.a.by 3 252.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} + T_{5}^{2} - 4 T_{5} - 1$$ acting on $$S_{2}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T + 4 T^{2} - 3 T^{3} + 12 T^{4} - 18 T^{5} + 27 T^{6}$$
$5$ $$( 1 + T + 11 T^{2} + 9 T^{3} + 55 T^{4} + 25 T^{5} + 125 T^{6} )^{2}$$
$7$ 1
$11$ $$( 1 - 2 T + 8 T^{2} + 15 T^{3} + 88 T^{4} - 242 T^{5} + 1331 T^{6} )^{2}$$
$13$ $$1 - 3 T + 3 T^{2} + 84 T^{3} - 195 T^{4} - 345 T^{5} + 5006 T^{6} - 4485 T^{7} - 32955 T^{8} + 184548 T^{9} + 85683 T^{10} - 1113879 T^{11} + 4826809 T^{12}$$
$17$ $$1 - 2 T - 28 T^{2} - 22 T^{3} + 438 T^{4} + 926 T^{5} - 8297 T^{6} + 15742 T^{7} + 126582 T^{8} - 108086 T^{9} - 2338588 T^{10} - 2839714 T^{11} + 24137569 T^{12}$$
$19$ $$1 - 3 T - 24 T^{2} - 29 T^{3} + 357 T^{4} + 1524 T^{5} - 8997 T^{6} + 28956 T^{7} + 128877 T^{8} - 198911 T^{9} - 3127704 T^{10} - 7428297 T^{11} + 47045881 T^{12}$$
$23$ $$( 1 - 14 T + 122 T^{2} - 675 T^{3} + 2806 T^{4} - 7406 T^{5} + 12167 T^{6} )^{2}$$
$29$ $$1 + T - 46 T^{2} + 149 T^{3} + 897 T^{4} - 4282 T^{5} - 13523 T^{6} - 124178 T^{7} + 754377 T^{8} + 3633961 T^{9} - 32534926 T^{10} + 20511149 T^{11} + 594823321 T^{12}$$
$31$ $$1 + 3 T - 48 T^{2} - 147 T^{3} + 1005 T^{4} + 1344 T^{5} - 24505 T^{6} + 41664 T^{7} + 965805 T^{8} - 4379277 T^{9} - 44329008 T^{10} + 85887453 T^{11} + 887503681 T^{12}$$
$37$ $$1 - 3 T - 72 T^{2} + 155 T^{3} + 2967 T^{4} - 2244 T^{5} - 114171 T^{6} - 83028 T^{7} + 4061823 T^{8} + 7851215 T^{9} - 134939592 T^{10} - 208031871 T^{11} + 2565726409 T^{12}$$
$41$ $$1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 33210 T^{7} + 7413210 T^{8} - 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12}$$
$43$ $$1 + 3 T - 24 T^{2} - 979 T^{3} - 1947 T^{4} + 14820 T^{5} + 386067 T^{6} + 637260 T^{7} - 3600003 T^{8} - 77837353 T^{9} - 82051224 T^{10} + 441025329 T^{11} + 6321363049 T^{12}$$
$47$ $$1 - 21 T + 180 T^{2} - 1119 T^{3} + 10053 T^{4} - 100416 T^{5} + 788551 T^{6} - 4719552 T^{7} + 22207077 T^{8} - 116177937 T^{9} + 878342580 T^{10} - 4816245147 T^{11} + 10779215329 T^{12}$$
$53$ $$1 + 6 T - 126 T^{2} - 282 T^{3} + 13896 T^{4} + 15396 T^{5} - 801173 T^{6} + 815988 T^{7} + 39033864 T^{8} - 41983314 T^{9} - 994200606 T^{10} + 2509172958 T^{11} + 22164361129 T^{12}$$
$59$ $$1 - 31 T + 476 T^{2} - 5741 T^{3} + 62553 T^{4} - 587576 T^{5} + 4781851 T^{6} - 34666984 T^{7} + 217746993 T^{8} - 1179080839 T^{9} + 5767863836 T^{10} - 22162653269 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 6 T + 48 T^{2} - 642 T^{3} + 3018 T^{4} - 35394 T^{5} + 438671 T^{6} - 2159034 T^{7} + 11229978 T^{8} - 145721802 T^{9} + 664600368 T^{10} - 5067577806 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + 6 T - 150 T^{2} - 506 T^{3} + 17268 T^{4} + 28236 T^{5} - 1220289 T^{6} + 1891812 T^{7} + 77516052 T^{8} - 152186078 T^{9} - 3022668150 T^{10} + 8100750642 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 - 17 T + 119 T^{2} - 507 T^{3} + 8449 T^{4} - 85697 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 + 3 T - 186 T^{2} - 133 T^{3} + 22713 T^{4} + 582 T^{5} - 1916871 T^{6} + 42486 T^{7} + 121037577 T^{8} - 51739261 T^{9} - 5282072826 T^{10} + 6219214779 T^{11} + 151334226289 T^{12}$$
$79$ $$1 - 9 T - 114 T^{2} + 351 T^{3} + 13143 T^{4} + 15786 T^{5} - 1414609 T^{6} + 1247094 T^{7} + 82025463 T^{8} + 173056689 T^{9} - 4440309234 T^{10} - 27693507591 T^{11} + 243087455521 T^{12}$$
$83$ $$1 - 20 T + 38 T^{2} - 346 T^{3} + 32058 T^{4} - 183754 T^{5} - 606869 T^{6} - 15251582 T^{7} + 220847562 T^{8} - 197838302 T^{9} + 1803416198 T^{10} - 78780812860 T^{11} + 326940373369 T^{12}$$
$89$ $$1 - 12 T - 72 T^{2} + 258 T^{3} + 10332 T^{4} + 58524 T^{5} - 1852445 T^{6} + 5208636 T^{7} + 81839772 T^{8} + 181882002 T^{9} - 4517441352 T^{10} - 67008713388 T^{11} + 496981290961 T^{12}$$
$97$ $$1 + 9 T - 66 T^{2} - 2023 T^{3} - 7707 T^{4} + 73950 T^{5} + 1766073 T^{6} + 7173150 T^{7} - 72515163 T^{8} - 1846337479 T^{9} - 5842932546 T^{10} + 77286062313 T^{11} + 832972004929 T^{12}$$
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