# Properties

 Label 96.4.a.f Level $96$ Weight $4$ Character orbit 96.a Self dual yes Analytic conductor $5.664$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 10 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 10 * q^5 + 4 * q^7 + 9 * q^9 $$q + 3 q^{3} + 10 q^{5} + 4 q^{7} + 9 q^{9} - 20 q^{11} + 70 q^{13} + 30 q^{15} + 90 q^{17} - 140 q^{19} + 12 q^{21} + 192 q^{23} - 25 q^{25} + 27 q^{27} - 134 q^{29} - 100 q^{31} - 60 q^{33} + 40 q^{35} - 170 q^{37} + 210 q^{39} - 110 q^{41} - 532 q^{43} + 90 q^{45} + 56 q^{47} - 327 q^{49} + 270 q^{51} - 430 q^{53} - 200 q^{55} - 420 q^{57} + 20 q^{59} + 270 q^{61} + 36 q^{63} + 700 q^{65} + 524 q^{67} + 576 q^{69} + 80 q^{71} + 330 q^{73} - 75 q^{75} - 80 q^{77} - 1060 q^{79} + 81 q^{81} + 1188 q^{83} + 900 q^{85} - 402 q^{87} + 1274 q^{89} + 280 q^{91} - 300 q^{93} - 1400 q^{95} - 590 q^{97} - 180 q^{99}+O(q^{100})$$ q + 3 * q^3 + 10 * q^5 + 4 * q^7 + 9 * q^9 - 20 * q^11 + 70 * q^13 + 30 * q^15 + 90 * q^17 - 140 * q^19 + 12 * q^21 + 192 * q^23 - 25 * q^25 + 27 * q^27 - 134 * q^29 - 100 * q^31 - 60 * q^33 + 40 * q^35 - 170 * q^37 + 210 * q^39 - 110 * q^41 - 532 * q^43 + 90 * q^45 + 56 * q^47 - 327 * q^49 + 270 * q^51 - 430 * q^53 - 200 * q^55 - 420 * q^57 + 20 * q^59 + 270 * q^61 + 36 * q^63 + 700 * q^65 + 524 * q^67 + 576 * q^69 + 80 * q^71 + 330 * q^73 - 75 * q^75 - 80 * q^77 - 1060 * q^79 + 81 * q^81 + 1188 * q^83 + 900 * q^85 - 402 * q^87 + 1274 * q^89 + 280 * q^91 - 300 * q^93 - 1400 * q^95 - 590 * q^97 - 180 * 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 10.0000 0 4.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$

## 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.f yes 1
3.b odd 2 1 288.4.a.c 1
4.b odd 2 1 96.4.a.c 1
5.b even 2 1 2400.4.a.e 1
8.b even 2 1 192.4.a.b 1
8.d odd 2 1 192.4.a.h 1
12.b even 2 1 288.4.a.b 1
16.e even 4 2 768.4.d.h 2
16.f odd 4 2 768.4.d.i 2
20.d odd 2 1 2400.4.a.r 1
24.f even 2 1 576.4.a.s 1
24.h odd 2 1 576.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.4.a.c 1 4.b odd 2 1
96.4.a.f yes 1 1.a even 1 1 trivial
192.4.a.b 1 8.b even 2 1
192.4.a.h 1 8.d odd 2 1
288.4.a.b 1 12.b even 2 1
288.4.a.c 1 3.b odd 2 1
576.4.a.s 1 24.f even 2 1
576.4.a.t 1 24.h odd 2 1
768.4.d.h 2 16.e even 4 2
768.4.d.i 2 16.f odd 4 2
2400.4.a.e 1 5.b even 2 1
2400.4.a.r 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(96))$$:

 $$T_{5} - 10$$ T5 - 10 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 10$$
$7$ $$T - 4$$
$11$ $$T + 20$$
$13$ $$T - 70$$
$17$ $$T - 90$$
$19$ $$T + 140$$
$23$ $$T - 192$$
$29$ $$T + 134$$
$31$ $$T + 100$$
$37$ $$T + 170$$
$41$ $$T + 110$$
$43$ $$T + 532$$
$47$ $$T - 56$$
$53$ $$T + 430$$
$59$ $$T - 20$$
$61$ $$T - 270$$
$67$ $$T - 524$$
$71$ $$T - 80$$
$73$ $$T - 330$$
$79$ $$T + 1060$$
$83$ $$T - 1188$$
$89$ $$T - 1274$$
$97$ $$T + 590$$