# Properties

 Label 448.6.a.b Level $448$ Weight $6$ Character orbit 448.a Self dual yes Analytic conductor $71.852$ 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] = [448,6,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 448.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: $$71.8519512762$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) 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)

 $$f(q)$$ $$=$$ $$q - 26 q^{3} - 16 q^{5} - 49 q^{7} + 433 q^{9}+O(q^{10})$$ q - 26 * q^3 - 16 * q^5 - 49 * q^7 + 433 * q^9 $$q - 26 q^{3} - 16 q^{5} - 49 q^{7} + 433 q^{9} - 8 q^{11} - 684 q^{13} + 416 q^{15} - 2218 q^{17} + 2698 q^{19} + 1274 q^{21} + 3344 q^{23} - 2869 q^{25} - 4940 q^{27} + 3254 q^{29} + 4788 q^{31} + 208 q^{33} + 784 q^{35} + 11470 q^{37} + 17784 q^{39} + 13350 q^{41} + 928 q^{43} - 6928 q^{45} + 1212 q^{47} + 2401 q^{49} + 57668 q^{51} - 13110 q^{53} + 128 q^{55} - 70148 q^{57} - 34702 q^{59} + 1032 q^{61} - 21217 q^{63} + 10944 q^{65} - 10108 q^{67} - 86944 q^{69} + 62720 q^{71} - 18926 q^{73} + 74594 q^{75} + 392 q^{77} + 11400 q^{79} + 23221 q^{81} - 88958 q^{83} + 35488 q^{85} - 84604 q^{87} + 19722 q^{89} + 33516 q^{91} - 124488 q^{93} - 43168 q^{95} + 17062 q^{97} - 3464 q^{99}+O(q^{100})$$ q - 26 * q^3 - 16 * q^5 - 49 * q^7 + 433 * q^9 - 8 * q^11 - 684 * q^13 + 416 * q^15 - 2218 * q^17 + 2698 * q^19 + 1274 * q^21 + 3344 * q^23 - 2869 * q^25 - 4940 * q^27 + 3254 * q^29 + 4788 * q^31 + 208 * q^33 + 784 * q^35 + 11470 * q^37 + 17784 * q^39 + 13350 * q^41 + 928 * q^43 - 6928 * q^45 + 1212 * q^47 + 2401 * q^49 + 57668 * q^51 - 13110 * q^53 + 128 * q^55 - 70148 * q^57 - 34702 * q^59 + 1032 * q^61 - 21217 * q^63 + 10944 * q^65 - 10108 * q^67 - 86944 * q^69 + 62720 * q^71 - 18926 * q^73 + 74594 * q^75 + 392 * q^77 + 11400 * q^79 + 23221 * q^81 - 88958 * q^83 + 35488 * q^85 - 84604 * q^87 + 19722 * q^89 + 33516 * q^91 - 124488 * q^93 - 43168 * q^95 + 17062 * q^97 - 3464 * 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 −26.0000 0 −16.0000 0 −49.0000 0 433.000 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
$$2$$ $$+1$$
$$7$$ $$+1$$

## 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 448.6.a.b 1
4.b odd 2 1 448.6.a.o 1
8.b even 2 1 28.6.a.b 1
8.d odd 2 1 112.6.a.b 1
24.f even 2 1 1008.6.a.l 1
24.h odd 2 1 252.6.a.a 1
40.f even 2 1 700.6.a.b 1
40.i odd 4 2 700.6.e.b 2
56.e even 2 1 784.6.a.m 1
56.h odd 2 1 196.6.a.a 1
56.j odd 6 2 196.6.e.i 2
56.p even 6 2 196.6.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.b 1 8.b even 2 1
112.6.a.b 1 8.d odd 2 1
196.6.a.a 1 56.h odd 2 1
196.6.e.a 2 56.p even 6 2
196.6.e.i 2 56.j odd 6 2
252.6.a.a 1 24.h odd 2 1
448.6.a.b 1 1.a even 1 1 trivial
448.6.a.o 1 4.b odd 2 1
700.6.a.b 1 40.f even 2 1
700.6.e.b 2 40.i odd 4 2
784.6.a.m 1 56.e even 2 1
1008.6.a.l 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(448))$$:

 $$T_{3} + 26$$ T3 + 26 $$T_{5} + 16$$ T5 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 26$$
$5$ $$T + 16$$
$7$ $$T + 49$$
$11$ $$T + 8$$
$13$ $$T + 684$$
$17$ $$T + 2218$$
$19$ $$T - 2698$$
$23$ $$T - 3344$$
$29$ $$T - 3254$$
$31$ $$T - 4788$$
$37$ $$T - 11470$$
$41$ $$T - 13350$$
$43$ $$T - 928$$
$47$ $$T - 1212$$
$53$ $$T + 13110$$
$59$ $$T + 34702$$
$61$ $$T - 1032$$
$67$ $$T + 10108$$
$71$ $$T - 62720$$
$73$ $$T + 18926$$
$79$ $$T - 11400$$
$83$ $$T + 88958$$
$89$ $$T - 19722$$
$97$ $$T - 17062$$