# Properties

 Label 1344.4.a.n Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ 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] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.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: $$79.2985670477$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 18 q^{5} - 7 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 18 * q^5 - 7 * q^7 + 9 * q^9 $$q - 3 q^{3} + 18 q^{5} - 7 q^{7} + 9 q^{9} - 36 q^{11} + 34 q^{13} - 54 q^{15} + 42 q^{17} - 124 q^{19} + 21 q^{21} + 199 q^{25} - 27 q^{27} - 102 q^{29} + 160 q^{31} + 108 q^{33} - 126 q^{35} - 398 q^{37} - 102 q^{39} - 318 q^{41} - 268 q^{43} + 162 q^{45} - 240 q^{47} + 49 q^{49} - 126 q^{51} + 498 q^{53} - 648 q^{55} + 372 q^{57} - 132 q^{59} - 398 q^{61} - 63 q^{63} + 612 q^{65} + 92 q^{67} + 720 q^{71} - 502 q^{73} - 597 q^{75} + 252 q^{77} + 1024 q^{79} + 81 q^{81} - 204 q^{83} + 756 q^{85} + 306 q^{87} + 354 q^{89} - 238 q^{91} - 480 q^{93} - 2232 q^{95} - 286 q^{97} - 324 q^{99}+O(q^{100})$$ q - 3 * q^3 + 18 * q^5 - 7 * q^7 + 9 * q^9 - 36 * q^11 + 34 * q^13 - 54 * q^15 + 42 * q^17 - 124 * q^19 + 21 * q^21 + 199 * q^25 - 27 * q^27 - 102 * q^29 + 160 * q^31 + 108 * q^33 - 126 * q^35 - 398 * q^37 - 102 * q^39 - 318 * q^41 - 268 * q^43 + 162 * q^45 - 240 * q^47 + 49 * q^49 - 126 * q^51 + 498 * q^53 - 648 * q^55 + 372 * q^57 - 132 * q^59 - 398 * q^61 - 63 * q^63 + 612 * q^65 + 92 * q^67 + 720 * q^71 - 502 * q^73 - 597 * q^75 + 252 * q^77 + 1024 * q^79 + 81 * q^81 - 204 * q^83 + 756 * q^85 + 306 * q^87 + 354 * q^89 - 238 * q^91 - 480 * q^93 - 2232 * q^95 - 286 * 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 18.0000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$

## 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 1344.4.a.n 1
4.b odd 2 1 1344.4.a.ba 1
8.b even 2 1 336.4.a.f 1
8.d odd 2 1 21.4.a.a 1
24.f even 2 1 63.4.a.c 1
24.h odd 2 1 1008.4.a.v 1
40.e odd 2 1 525.4.a.g 1
40.k even 4 2 525.4.d.c 2
56.e even 2 1 147.4.a.c 1
56.h odd 2 1 2352.4.a.r 1
56.k odd 6 2 147.4.e.i 2
56.m even 6 2 147.4.e.g 2
120.m even 2 1 1575.4.a.b 1
168.e odd 2 1 441.4.a.j 1
168.v even 6 2 441.4.e.b 2
168.be odd 6 2 441.4.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.a 1 8.d odd 2 1
63.4.a.c 1 24.f even 2 1
147.4.a.c 1 56.e even 2 1
147.4.e.g 2 56.m even 6 2
147.4.e.i 2 56.k odd 6 2
336.4.a.f 1 8.b even 2 1
441.4.a.j 1 168.e odd 2 1
441.4.e.b 2 168.v even 6 2
441.4.e.d 2 168.be odd 6 2
525.4.a.g 1 40.e odd 2 1
525.4.d.c 2 40.k even 4 2
1008.4.a.v 1 24.h odd 2 1
1344.4.a.n 1 1.a even 1 1 trivial
1344.4.a.ba 1 4.b odd 2 1
1575.4.a.b 1 120.m even 2 1
2352.4.a.r 1 56.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5} - 18$$ T5 - 18 $$T_{11} + 36$$ T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 18$$
$7$ $$T + 7$$
$11$ $$T + 36$$
$13$ $$T - 34$$
$17$ $$T - 42$$
$19$ $$T + 124$$
$23$ $$T$$
$29$ $$T + 102$$
$31$ $$T - 160$$
$37$ $$T + 398$$
$41$ $$T + 318$$
$43$ $$T + 268$$
$47$ $$T + 240$$
$53$ $$T - 498$$
$59$ $$T + 132$$
$61$ $$T + 398$$
$67$ $$T - 92$$
$71$ $$T - 720$$
$73$ $$T + 502$$
$79$ $$T - 1024$$
$83$ $$T + 204$$
$89$ $$T - 354$$
$97$ $$T + 286$$