# Properties

 Label 1764.4.a.l Level $1764$ Weight $4$ Character orbit 1764.a Self dual yes Analytic conductor $104.079$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1764.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$104.079369250$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 14.0000 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.a.l 1
3.b odd 2 1 588.4.a.a 1
7.b odd 2 1 252.4.a.a 1
7.c even 3 2 1764.4.k.c 2
7.d odd 6 2 1764.4.k.n 2
12.b even 2 1 2352.4.a.v 1
21.c even 2 1 84.4.a.b 1
21.g even 6 2 588.4.i.a 2
21.h odd 6 2 588.4.i.h 2
28.d even 2 1 1008.4.a.d 1
84.h odd 2 1 336.4.a.e 1
105.g even 2 1 2100.4.a.g 1
105.k odd 4 2 2100.4.k.g 2
168.e odd 2 1 1344.4.a.p 1
168.i even 2 1 1344.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 21.c even 2 1
252.4.a.a 1 7.b odd 2 1
336.4.a.e 1 84.h odd 2 1
588.4.a.a 1 3.b odd 2 1
588.4.i.a 2 21.g even 6 2
588.4.i.h 2 21.h odd 6 2
1008.4.a.d 1 28.d even 2 1
1344.4.a.b 1 168.i even 2 1
1344.4.a.p 1 168.e odd 2 1
1764.4.a.l 1 1.a even 1 1 trivial
1764.4.k.c 2 7.c even 3 2
1764.4.k.n 2 7.d odd 6 2
2100.4.a.g 1 105.g even 2 1
2100.4.k.g 2 105.k odd 4 2
2352.4.a.v 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1764))$$:

 $$T_{5} - 14$$ $$T_{11} + 4$$ $$T_{13} + 54$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-14 + T$$
$7$ $$T$$
$11$ $$4 + T$$
$13$ $$54 + T$$
$17$ $$14 + T$$
$19$ $$92 + T$$
$23$ $$-152 + T$$
$29$ $$-106 + T$$
$31$ $$-144 + T$$
$37$ $$-158 + T$$
$41$ $$390 + T$$
$43$ $$508 + T$$
$47$ $$528 + T$$
$53$ $$606 + T$$
$59$ $$364 + T$$
$61$ $$678 + T$$
$67$ $$-844 + T$$
$71$ $$-8 + T$$
$73$ $$-422 + T$$
$79$ $$-384 + T$$
$83$ $$548 + T$$
$89$ $$-1194 + T$$
$97$ $$-1502 + T$$
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