# Properties

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

# Learn more about

## 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 12) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 18q^{5} + O(q^{10})$$ $$q - 18q^{5} - 36q^{11} + 10q^{13} + 18q^{17} + 100q^{19} - 72q^{23} + 199q^{25} + 234q^{29} + 16q^{31} - 226q^{37} + 90q^{41} + 452q^{43} + 432q^{47} - 414q^{53} + 648q^{55} - 684q^{59} - 422q^{61} - 180q^{65} + 332q^{67} + 360q^{71} - 26q^{73} + 512q^{79} - 1188q^{83} - 324q^{85} - 630q^{89} - 1800q^{95} + 1054q^{97} + O(q^{100})$$

## 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 −18.0000 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.b 1
3.b odd 2 1 588.4.a.c 1
7.b odd 2 1 36.4.a.a 1
7.c even 3 2 1764.4.k.o 2
7.d odd 6 2 1764.4.k.b 2
12.b even 2 1 2352.4.a.bk 1
21.c even 2 1 12.4.a.a 1
21.g even 6 2 588.4.i.d 2
21.h odd 6 2 588.4.i.e 2
28.d even 2 1 144.4.a.g 1
35.c odd 2 1 900.4.a.g 1
35.f even 4 2 900.4.d.c 2
56.e even 2 1 576.4.a.a 1
56.h odd 2 1 576.4.a.b 1
63.l odd 6 2 324.4.e.a 2
63.o even 6 2 324.4.e.h 2
84.h odd 2 1 48.4.a.a 1
105.g even 2 1 300.4.a.b 1
105.k odd 4 2 300.4.d.e 2
168.e odd 2 1 192.4.a.l 1
168.i even 2 1 192.4.a.f 1
231.h odd 2 1 1452.4.a.d 1
273.g even 2 1 2028.4.a.c 1
273.o odd 4 2 2028.4.b.c 2
336.v odd 4 2 768.4.d.j 2
336.y even 4 2 768.4.d.g 2
420.o odd 2 1 1200.4.a.be 1
420.w even 4 2 1200.4.f.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 21.c even 2 1
36.4.a.a 1 7.b odd 2 1
48.4.a.a 1 84.h odd 2 1
144.4.a.g 1 28.d even 2 1
192.4.a.f 1 168.i even 2 1
192.4.a.l 1 168.e odd 2 1
300.4.a.b 1 105.g even 2 1
300.4.d.e 2 105.k odd 4 2
324.4.e.a 2 63.l odd 6 2
324.4.e.h 2 63.o even 6 2
576.4.a.a 1 56.e even 2 1
576.4.a.b 1 56.h odd 2 1
588.4.a.c 1 3.b odd 2 1
588.4.i.d 2 21.g even 6 2
588.4.i.e 2 21.h odd 6 2
768.4.d.g 2 336.y even 4 2
768.4.d.j 2 336.v odd 4 2
900.4.a.g 1 35.c odd 2 1
900.4.d.c 2 35.f even 4 2
1200.4.a.be 1 420.o odd 2 1
1200.4.f.d 2 420.w even 4 2
1452.4.a.d 1 231.h odd 2 1
1764.4.a.b 1 1.a even 1 1 trivial
1764.4.k.b 2 7.d odd 6 2
1764.4.k.o 2 7.c even 3 2
2028.4.a.c 1 273.g even 2 1
2028.4.b.c 2 273.o odd 4 2
2352.4.a.bk 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1764))$$:

 $$T_{5} + 18$$ $$T_{11} + 36$$ $$T_{13} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$18 + T$$
$7$ $$T$$
$11$ $$36 + T$$
$13$ $$-10 + T$$
$17$ $$-18 + T$$
$19$ $$-100 + T$$
$23$ $$72 + T$$
$29$ $$-234 + T$$
$31$ $$-16 + T$$
$37$ $$226 + T$$
$41$ $$-90 + T$$
$43$ $$-452 + T$$
$47$ $$-432 + T$$
$53$ $$414 + T$$
$59$ $$684 + T$$
$61$ $$422 + T$$
$67$ $$-332 + T$$
$71$ $$-360 + T$$
$73$ $$26 + T$$
$79$ $$-512 + T$$
$83$ $$1188 + T$$
$89$ $$630 + T$$
$97$ $$-1054 + T$$
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