# Properties

 Label 1764.4.k.n Level $1764$ Weight $4$ Character orbit 1764.k Analytic conductor $104.079$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 14 \zeta_{6} q^{5} +O(q^{10})$$ $$q + 14 \zeta_{6} q^{5} + ( 4 - 4 \zeta_{6} ) q^{11} + 54 q^{13} + ( -14 + 14 \zeta_{6} ) q^{17} -92 \zeta_{6} q^{19} -152 \zeta_{6} q^{23} + ( -71 + 71 \zeta_{6} ) q^{25} + 106 q^{29} + ( 144 - 144 \zeta_{6} ) q^{31} -158 \zeta_{6} q^{37} + 390 q^{41} -508 q^{43} -528 \zeta_{6} q^{47} + ( 606 - 606 \zeta_{6} ) q^{53} + 56 q^{55} + ( -364 + 364 \zeta_{6} ) q^{59} -678 \zeta_{6} q^{61} + 756 \zeta_{6} q^{65} + ( -844 + 844 \zeta_{6} ) q^{67} + 8 q^{71} + ( 422 - 422 \zeta_{6} ) q^{73} -384 \zeta_{6} q^{79} + 548 q^{83} -196 q^{85} + 1194 \zeta_{6} q^{89} + ( 1288 - 1288 \zeta_{6} ) q^{95} -1502 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{5} + O(q^{10})$$ $$2q + 14q^{5} + 4q^{11} + 108q^{13} - 14q^{17} - 92q^{19} - 152q^{23} - 71q^{25} + 212q^{29} + 144q^{31} - 158q^{37} + 780q^{41} - 1016q^{43} - 528q^{47} + 606q^{53} + 112q^{55} - 364q^{59} - 678q^{61} + 756q^{65} - 844q^{67} + 16q^{71} + 422q^{73} - 384q^{79} + 1096q^{83} - 392q^{85} + 1194q^{89} + 1288q^{95} - 3004q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 7.00000 + 12.1244i 0 0 0 0 0
1549.1 0 0 0 7.00000 12.1244i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.k.n 2
3.b odd 2 1 588.4.i.a 2
7.b odd 2 1 1764.4.k.c 2
7.c even 3 1 252.4.a.a 1
7.c even 3 1 inner 1764.4.k.n 2
7.d odd 6 1 1764.4.a.l 1
7.d odd 6 1 1764.4.k.c 2
21.c even 2 1 588.4.i.h 2
21.g even 6 1 588.4.a.a 1
21.g even 6 1 588.4.i.h 2
21.h odd 6 1 84.4.a.b 1
21.h odd 6 1 588.4.i.a 2
28.g odd 6 1 1008.4.a.d 1
84.j odd 6 1 2352.4.a.v 1
84.n even 6 1 336.4.a.e 1
105.o odd 6 1 2100.4.a.g 1
105.x even 12 2 2100.4.k.g 2
168.s odd 6 1 1344.4.a.b 1
168.v even 6 1 1344.4.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 21.h odd 6 1
252.4.a.a 1 7.c even 3 1
336.4.a.e 1 84.n even 6 1
588.4.a.a 1 21.g even 6 1
588.4.i.a 2 3.b odd 2 1
588.4.i.a 2 21.h odd 6 1
588.4.i.h 2 21.c even 2 1
588.4.i.h 2 21.g even 6 1
1008.4.a.d 1 28.g odd 6 1
1344.4.a.b 1 168.s odd 6 1
1344.4.a.p 1 168.v even 6 1
1764.4.a.l 1 7.d odd 6 1
1764.4.k.c 2 7.b odd 2 1
1764.4.k.c 2 7.d odd 6 1
1764.4.k.n 2 1.a even 1 1 trivial
1764.4.k.n 2 7.c even 3 1 inner
2100.4.a.g 1 105.o odd 6 1
2100.4.k.g 2 105.x even 12 2
2352.4.a.v 1 84.j odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 14 T_{5} + 196$$ $$T_{11}^{2} - 4 T_{11} + 16$$ $$T_{13} - 54$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$196 - 14 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$( -54 + T )^{2}$$
$17$ $$196 + 14 T + T^{2}$$
$19$ $$8464 + 92 T + T^{2}$$
$23$ $$23104 + 152 T + T^{2}$$
$29$ $$( -106 + T )^{2}$$
$31$ $$20736 - 144 T + T^{2}$$
$37$ $$24964 + 158 T + T^{2}$$
$41$ $$( -390 + T )^{2}$$
$43$ $$( 508 + T )^{2}$$
$47$ $$278784 + 528 T + T^{2}$$
$53$ $$367236 - 606 T + T^{2}$$
$59$ $$132496 + 364 T + T^{2}$$
$61$ $$459684 + 678 T + T^{2}$$
$67$ $$712336 + 844 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$178084 - 422 T + T^{2}$$
$79$ $$147456 + 384 T + T^{2}$$
$83$ $$( -548 + T )^{2}$$
$89$ $$1425636 - 1194 T + T^{2}$$
$97$ $$( 1502 + T )^{2}$$
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