# Properties

 Label 1680.4.a.u Level $1680$ Weight $4$ Character orbit 1680.a Self dual yes Analytic conductor $99.123$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,4,Mod(1,1680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1680.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$99.1232088096$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 5 q^{5} - 7 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 5 * q^5 - 7 * q^7 + 9 * q^9 $$q + 3 q^{3} + 5 q^{5} - 7 q^{7} + 9 q^{9} - 12 q^{11} + 30 q^{13} + 15 q^{15} - 134 q^{17} + 92 q^{19} - 21 q^{21} - 112 q^{23} + 25 q^{25} + 27 q^{27} - 58 q^{29} + 224 q^{31} - 36 q^{33} - 35 q^{35} - 146 q^{37} + 90 q^{39} + 18 q^{41} - 340 q^{43} + 45 q^{45} - 208 q^{47} + 49 q^{49} - 402 q^{51} - 754 q^{53} - 60 q^{55} + 276 q^{57} - 380 q^{59} + 718 q^{61} - 63 q^{63} + 150 q^{65} - 412 q^{67} - 336 q^{69} + 960 q^{71} + 1066 q^{73} + 75 q^{75} + 84 q^{77} - 896 q^{79} + 81 q^{81} - 436 q^{83} - 670 q^{85} - 174 q^{87} - 1038 q^{89} - 210 q^{91} + 672 q^{93} + 460 q^{95} - 702 q^{97} - 108 q^{99}+O(q^{100})$$ q + 3 * q^3 + 5 * q^5 - 7 * q^7 + 9 * q^9 - 12 * q^11 + 30 * q^13 + 15 * q^15 - 134 * q^17 + 92 * q^19 - 21 * q^21 - 112 * q^23 + 25 * q^25 + 27 * q^27 - 58 * q^29 + 224 * q^31 - 36 * q^33 - 35 * q^35 - 146 * q^37 + 90 * q^39 + 18 * q^41 - 340 * q^43 + 45 * q^45 - 208 * q^47 + 49 * q^49 - 402 * q^51 - 754 * q^53 - 60 * q^55 + 276 * q^57 - 380 * q^59 + 718 * q^61 - 63 * q^63 + 150 * q^65 - 412 * q^67 - 336 * q^69 + 960 * q^71 + 1066 * q^73 + 75 * q^75 + 84 * q^77 - 896 * q^79 + 81 * q^81 - 436 * q^83 - 670 * q^85 - 174 * q^87 - 1038 * q^89 - 210 * q^91 + 672 * q^93 + 460 * q^95 - 702 * q^97 - 108 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 3.00000 0 5.00000 0 −7.00000 0 9.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$+1$$

## 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 1680.4.a.u 1
4.b odd 2 1 105.4.a.b 1
12.b even 2 1 315.4.a.a 1
20.d odd 2 1 525.4.a.a 1
20.e even 4 2 525.4.d.a 2
28.d even 2 1 735.4.a.j 1
60.h even 2 1 1575.4.a.l 1
84.h odd 2 1 2205.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.b 1 4.b odd 2 1
315.4.a.a 1 12.b even 2 1
525.4.a.a 1 20.d odd 2 1
525.4.d.a 2 20.e even 4 2
735.4.a.j 1 28.d even 2 1
1575.4.a.l 1 60.h even 2 1
1680.4.a.u 1 1.a even 1 1 trivial
2205.4.a.b 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1680))$$:

 $$T_{11} + 12$$ T11 + 12 $$T_{13} - 30$$ T13 - 30 $$T_{17} + 134$$ T17 + 134

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 5$$
$7$ $$T + 7$$
$11$ $$T + 12$$
$13$ $$T - 30$$
$17$ $$T + 134$$
$19$ $$T - 92$$
$23$ $$T + 112$$
$29$ $$T + 58$$
$31$ $$T - 224$$
$37$ $$T + 146$$
$41$ $$T - 18$$
$43$ $$T + 340$$
$47$ $$T + 208$$
$53$ $$T + 754$$
$59$ $$T + 380$$
$61$ $$T - 718$$
$67$ $$T + 412$$
$71$ $$T - 960$$
$73$ $$T - 1066$$
$79$ $$T + 896$$
$83$ $$T + 436$$
$89$ $$T + 1038$$
$97$ $$T + 702$$