# Properties

 Label 96.4.a.d Level $96$ Weight $4$ Character orbit 96.a Self dual yes Analytic conductor $5.664$ 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] = [96,4,Mod(1,96)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(96, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("96.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 96.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: $$5.66418336055$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 14 q^{5} - 36 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 14 * q^5 - 36 * q^7 + 9 * q^9 $$q + 3 q^{3} - 14 q^{5} - 36 q^{7} + 9 q^{9} - 36 q^{11} + 54 q^{13} - 42 q^{15} - 22 q^{17} + 36 q^{19} - 108 q^{21} - 144 q^{23} + 71 q^{25} + 27 q^{27} + 50 q^{29} - 108 q^{31} - 108 q^{33} + 504 q^{35} + 214 q^{37} + 162 q^{39} - 446 q^{41} + 252 q^{43} - 126 q^{45} + 72 q^{47} + 953 q^{49} - 66 q^{51} - 22 q^{53} + 504 q^{55} + 108 q^{57} - 684 q^{59} - 466 q^{61} - 324 q^{63} - 756 q^{65} - 180 q^{67} - 432 q^{69} + 576 q^{71} - 54 q^{73} + 213 q^{75} + 1296 q^{77} - 972 q^{79} + 81 q^{81} - 684 q^{83} + 308 q^{85} + 150 q^{87} + 346 q^{89} - 1944 q^{91} - 324 q^{93} - 504 q^{95} - 1134 q^{97} - 324 q^{99}+O(q^{100})$$ q + 3 * q^3 - 14 * q^5 - 36 * q^7 + 9 * q^9 - 36 * q^11 + 54 * q^13 - 42 * q^15 - 22 * q^17 + 36 * q^19 - 108 * q^21 - 144 * q^23 + 71 * q^25 + 27 * q^27 + 50 * q^29 - 108 * q^31 - 108 * q^33 + 504 * q^35 + 214 * q^37 + 162 * q^39 - 446 * q^41 + 252 * q^43 - 126 * q^45 + 72 * q^47 + 953 * q^49 - 66 * q^51 - 22 * q^53 + 504 * q^55 + 108 * q^57 - 684 * q^59 - 466 * q^61 - 324 * q^63 - 756 * q^65 - 180 * q^67 - 432 * q^69 + 576 * q^71 - 54 * q^73 + 213 * q^75 + 1296 * q^77 - 972 * q^79 + 81 * q^81 - 684 * q^83 + 308 * q^85 + 150 * q^87 + 346 * q^89 - 1944 * q^91 - 324 * q^93 - 504 * q^95 - 1134 * q^97 - 324 * 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 −14.0000 0 −36.0000 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
$$2$$ $$1$$
$$3$$ $$-1$$

## 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 96.4.a.d yes 1
3.b odd 2 1 288.4.a.j 1
4.b odd 2 1 96.4.a.a 1
5.b even 2 1 2400.4.a.k 1
8.b even 2 1 192.4.a.e 1
8.d odd 2 1 192.4.a.k 1
12.b even 2 1 288.4.a.k 1
16.e even 4 2 768.4.d.p 2
16.f odd 4 2 768.4.d.a 2
20.d odd 2 1 2400.4.a.l 1
24.f even 2 1 576.4.a.f 1
24.h odd 2 1 576.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.4.a.a 1 4.b odd 2 1
96.4.a.d yes 1 1.a even 1 1 trivial
192.4.a.e 1 8.b even 2 1
192.4.a.k 1 8.d odd 2 1
288.4.a.j 1 3.b odd 2 1
288.4.a.k 1 12.b even 2 1
576.4.a.e 1 24.h odd 2 1
576.4.a.f 1 24.f even 2 1
768.4.d.a 2 16.f odd 4 2
768.4.d.p 2 16.e even 4 2
2400.4.a.k 1 5.b even 2 1
2400.4.a.l 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(96))$$:

 $$T_{5} + 14$$ T5 + 14 $$T_{7} + 36$$ T7 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 14$$
$7$ $$T + 36$$
$11$ $$T + 36$$
$13$ $$T - 54$$
$17$ $$T + 22$$
$19$ $$T - 36$$
$23$ $$T + 144$$
$29$ $$T - 50$$
$31$ $$T + 108$$
$37$ $$T - 214$$
$41$ $$T + 446$$
$43$ $$T - 252$$
$47$ $$T - 72$$
$53$ $$T + 22$$
$59$ $$T + 684$$
$61$ $$T + 466$$
$67$ $$T + 180$$
$71$ $$T - 576$$
$73$ $$T + 54$$
$79$ $$T + 972$$
$83$ $$T + 684$$
$89$ $$T - 346$$
$97$ $$T + 1134$$