# Properties

 Label 16.28.a.b Level $16$ Weight $28$ Character orbit 16.a Self dual yes Analytic conductor $73.897$ 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] = [16,28,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 28, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$73.8968919741$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2) 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)

 $$f(q)$$ $$=$$ $$q + 1016388 q^{3} - 3341197410 q^{5} + 51021361384 q^{7} - 6592552918443 q^{9}+O(q^{10})$$ q + 1016388 * q^3 - 3341197410 * q^5 + 51021361384 * q^7 - 6592552918443 * q^9 $$q + 1016388 q^{3} - 3341197410 q^{5} + 51021361384 q^{7} - 6592552918443 q^{9} + 177413845094508 q^{11} - 264386643393418 q^{13} - 33\!\cdots\!80 q^{15}+ \cdots - 11\!\cdots\!44 q^{99}+O(q^{100})$$ q + 1016388 * q^3 - 3341197410 * q^5 + 51021361384 * q^7 - 6592552918443 * q^9 + 177413845094508 * q^11 - 264386643393418 * q^13 - 3395952953155080 * q^15 + 76811888571465906 * q^17 + 147764402234885140 * q^19 + 51857499454360992 * q^21 - 3583628816727527112 * q^23 + 3713019535666879975 * q^25 - 14451157452241410840 * q^27 - 75208679254256666970 * q^29 + 131071363974700812448 * q^31 + 180321303187916797104 * q^33 - 170472440510894815440 * q^35 - 662288183049353372674 * q^37 - 268719411705349334184 * q^39 + 638680458671096629242 * q^41 - 6917115499338769700852 * q^43 + 22027020736389692832630 * q^45 + 30908908867583003138544 * q^47 - 63109183046057553744087 * q^49 + 78070681801375089267528 * q^51 - 72917877175183391659218 * q^53 - 592774679727911334824280 * q^55 + 150185965258710437674320 * q^57 + 428694162599071555282140 * q^59 - 308035309306328670786298 * q^61 - 336361024895024181605112 * q^63 + 883367968144681832647380 * q^65 + 2019363317538472735190404 * q^67 - 3642357325776057826311456 * q^69 - 12815663453552821957376472 * q^71 - 22881210928165048507710118 * q^73 + 3773868499817388804030300 * q^75 + 9051895905091888301679072 * q^77 - 53028364881871099261766000 * q^79 + 35584171933953904647069321 * q^81 - 52517451913599957686712492 * q^83 - 256643683152190485030503460 * q^85 - 76441199089875425228304360 * q^87 - 215234163567542209160339190 * q^89 - 13489366477678315864970512 * q^91 + 133219361487518209362397824 * q^93 - 493710038037396441415487400 * q^95 - 356303023225402052577970654 * q^97 - 1169610162249993034551211044 * 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 1.01639e6 0 −3.34120e9 0 5.10214e10 0 −6.59255e12 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
$$2$$ $$-1$$

## 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 16.28.a.b 1
4.b odd 2 1 2.28.a.b 1
12.b even 2 1 18.28.a.c 1
20.d odd 2 1 50.28.a.a 1
20.e even 4 2 50.28.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.28.a.b 1 4.b odd 2 1
16.28.a.b 1 1.a even 1 1 trivial
18.28.a.c 1 12.b even 2 1
50.28.a.a 1 20.d odd 2 1
50.28.b.b 2 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1016388$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(16))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1016388$$
$5$ $$T + 3341197410$$
$7$ $$T - 51021361384$$
$11$ $$T - 177413845094508$$
$13$ $$T + 264386643393418$$
$17$ $$T - 76\!\cdots\!06$$
$19$ $$T - 14\!\cdots\!40$$
$23$ $$T + 35\!\cdots\!12$$
$29$ $$T + 75\!\cdots\!70$$
$31$ $$T - 13\!\cdots\!48$$
$37$ $$T + 66\!\cdots\!74$$
$41$ $$T - 63\!\cdots\!42$$
$43$ $$T + 69\!\cdots\!52$$
$47$ $$T - 30\!\cdots\!44$$
$53$ $$T + 72\!\cdots\!18$$
$59$ $$T - 42\!\cdots\!40$$
$61$ $$T + 30\!\cdots\!98$$
$67$ $$T - 20\!\cdots\!04$$
$71$ $$T + 12\!\cdots\!72$$
$73$ $$T + 22\!\cdots\!18$$
$79$ $$T + 53\!\cdots\!00$$
$83$ $$T + 52\!\cdots\!92$$
$89$ $$T + 21\!\cdots\!90$$
$97$ $$T + 35\!\cdots\!54$$