# Properties

 Label 448.6.a.v Level $448$ Weight $6$ Character orbit 448.a Self dual yes Analytic conductor $71.852$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{345})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 86$$ x^2 - x - 86 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{345}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 3) q^{3} + (3 \beta - 41) q^{5} - 49 q^{7} + (6 \beta + 111) q^{9}+O(q^{10})$$ q + (b + 3) * q^3 + (3*b - 41) * q^5 - 49 * q^7 + (6*b + 111) * q^9 $$q + (\beta + 3) q^{3} + (3 \beta - 41) q^{5} - 49 q^{7} + (6 \beta + 111) q^{9} + (6 \beta - 170) q^{11} + ( - 39 \beta - 455) q^{13} + ( - 32 \beta + 912) q^{15} + ( - 6 \beta + 1608) q^{17} + ( - 45 \beta + 337) q^{19} + ( - 49 \beta - 147) q^{21} + ( - 72 \beta - 552) q^{23} + ( - 246 \beta + 1661) q^{25} + ( - 114 \beta + 1674) q^{27} + (114 \beta - 4032) q^{29} + (210 \beta - 3106) q^{31} + ( - 152 \beta + 1560) q^{33} + ( - 147 \beta + 2009) q^{35} + ( - 390 \beta + 4256) q^{37} + ( - 572 \beta - 14820) q^{39} + (678 \beta - 652) q^{41} + ( - 798 \beta + 5002) q^{43} + (87 \beta + 1659) q^{45} + ( - 714 \beta - 6374) q^{47} + 2401 q^{49} + (1590 \beta + 2754) q^{51} + (768 \beta + 5610) q^{53} + ( - 756 \beta + 13180) q^{55} + (202 \beta - 14514) q^{57} + ( - 2253 \beta + 6009) q^{59} + (75 \beta - 51369) q^{61} + ( - 294 \beta - 5439) q^{63} + (234 \beta - 21710) q^{65} + ( - 1944 \beta - 12068) q^{67} + ( - 768 \beta - 26496) q^{69} + (1740 \beta + 44860) q^{71} + ( - 1716 \beta - 27794) q^{73} + (923 \beta - 79887) q^{75} + ( - 294 \beta + 8330) q^{77} + (1956 \beta + 24412) q^{79} + ( - 126 \beta - 61281) q^{81} + (3447 \beta - 17891) q^{83} + (5070 \beta - 72138) q^{85} + ( - 3690 \beta + 27234) q^{87} + (4728 \beta - 9150) q^{89} + (1911 \beta + 22295) q^{91} + ( - 2476 \beta + 63132) q^{93} + (2856 \beta - 60392) q^{95} + ( - 462 \beta - 34992) q^{97} + ( - 354 \beta - 6450) q^{99}+O(q^{100})$$ q + (b + 3) * q^3 + (3*b - 41) * q^5 - 49 * q^7 + (6*b + 111) * q^9 + (6*b - 170) * q^11 + (-39*b - 455) * q^13 + (-32*b + 912) * q^15 + (-6*b + 1608) * q^17 + (-45*b + 337) * q^19 + (-49*b - 147) * q^21 + (-72*b - 552) * q^23 + (-246*b + 1661) * q^25 + (-114*b + 1674) * q^27 + (114*b - 4032) * q^29 + (210*b - 3106) * q^31 + (-152*b + 1560) * q^33 + (-147*b + 2009) * q^35 + (-390*b + 4256) * q^37 + (-572*b - 14820) * q^39 + (678*b - 652) * q^41 + (-798*b + 5002) * q^43 + (87*b + 1659) * q^45 + (-714*b - 6374) * q^47 + 2401 * q^49 + (1590*b + 2754) * q^51 + (768*b + 5610) * q^53 + (-756*b + 13180) * q^55 + (202*b - 14514) * q^57 + (-2253*b + 6009) * q^59 + (75*b - 51369) * q^61 + (-294*b - 5439) * q^63 + (234*b - 21710) * q^65 + (-1944*b - 12068) * q^67 + (-768*b - 26496) * q^69 + (1740*b + 44860) * q^71 + (-1716*b - 27794) * q^73 + (923*b - 79887) * q^75 + (-294*b + 8330) * q^77 + (1956*b + 24412) * q^79 + (-126*b - 61281) * q^81 + (3447*b - 17891) * q^83 + (5070*b - 72138) * q^85 + (-3690*b + 27234) * q^87 + (4728*b - 9150) * q^89 + (1911*b + 22295) * q^91 + (-2476*b + 63132) * q^93 + (2856*b - 60392) * q^95 + (-462*b - 34992) * q^97 + (-354*b - 6450) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 82 q^{5} - 98 q^{7} + 222 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 82 * q^5 - 98 * q^7 + 222 * q^9 $$2 q + 6 q^{3} - 82 q^{5} - 98 q^{7} + 222 q^{9} - 340 q^{11} - 910 q^{13} + 1824 q^{15} + 3216 q^{17} + 674 q^{19} - 294 q^{21} - 1104 q^{23} + 3322 q^{25} + 3348 q^{27} - 8064 q^{29} - 6212 q^{31} + 3120 q^{33} + 4018 q^{35} + 8512 q^{37} - 29640 q^{39} - 1304 q^{41} + 10004 q^{43} + 3318 q^{45} - 12748 q^{47} + 4802 q^{49} + 5508 q^{51} + 11220 q^{53} + 26360 q^{55} - 29028 q^{57} + 12018 q^{59} - 102738 q^{61} - 10878 q^{63} - 43420 q^{65} - 24136 q^{67} - 52992 q^{69} + 89720 q^{71} - 55588 q^{73} - 159774 q^{75} + 16660 q^{77} + 48824 q^{79} - 122562 q^{81} - 35782 q^{83} - 144276 q^{85} + 54468 q^{87} - 18300 q^{89} + 44590 q^{91} + 126264 q^{93} - 120784 q^{95} - 69984 q^{97} - 12900 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 82 * q^5 - 98 * q^7 + 222 * q^9 - 340 * q^11 - 910 * q^13 + 1824 * q^15 + 3216 * q^17 + 674 * q^19 - 294 * q^21 - 1104 * q^23 + 3322 * q^25 + 3348 * q^27 - 8064 * q^29 - 6212 * q^31 + 3120 * q^33 + 4018 * q^35 + 8512 * q^37 - 29640 * q^39 - 1304 * q^41 + 10004 * q^43 + 3318 * q^45 - 12748 * q^47 + 4802 * q^49 + 5508 * q^51 + 11220 * q^53 + 26360 * q^55 - 29028 * q^57 + 12018 * q^59 - 102738 * q^61 - 10878 * q^63 - 43420 * q^65 - 24136 * q^67 - 52992 * q^69 + 89720 * q^71 - 55588 * q^73 - 159774 * q^75 + 16660 * q^77 + 48824 * q^79 - 122562 * q^81 - 35782 * q^83 - 144276 * q^85 + 54468 * q^87 - 18300 * q^89 + 44590 * q^91 + 126264 * q^93 - 120784 * q^95 - 69984 * q^97 - 12900 * 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
 −8.78709 9.78709
0 −15.5742 0 −96.7225 0 −49.0000 0 −0.445054 0
1.2 0 21.5742 0 14.7225 0 −49.0000 0 222.445 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.v 2
4.b odd 2 1 448.6.a.t 2
8.b even 2 1 56.6.a.e 2
8.d odd 2 1 112.6.a.i 2
24.f even 2 1 1008.6.a.bd 2
24.h odd 2 1 504.6.a.i 2
56.e even 2 1 784.6.a.u 2
56.h odd 2 1 392.6.a.d 2
56.j odd 6 2 392.6.i.i 4
56.p even 6 2 392.6.i.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.e 2 8.b even 2 1
112.6.a.i 2 8.d odd 2 1
392.6.a.d 2 56.h odd 2 1
392.6.i.i 4 56.j odd 6 2
392.6.i.j 4 56.p even 6 2
448.6.a.t 2 4.b odd 2 1
448.6.a.v 2 1.a even 1 1 trivial
504.6.a.i 2 24.h odd 2 1
784.6.a.u 2 56.e even 2 1
1008.6.a.bd 2 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(448))$$:

 $$T_{3}^{2} - 6T_{3} - 336$$ T3^2 - 6*T3 - 336 $$T_{5}^{2} + 82T_{5} - 1424$$ T5^2 + 82*T5 - 1424

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 6T - 336$$
$5$ $$T^{2} + 82T - 1424$$
$7$ $$(T + 49)^{2}$$
$11$ $$T^{2} + 340T + 16480$$
$13$ $$T^{2} + 910T - 317720$$
$17$ $$T^{2} - 3216 T + 2573244$$
$19$ $$T^{2} - 674T - 585056$$
$23$ $$T^{2} + 1104 T - 1483776$$
$29$ $$T^{2} + 8064 T + 11773404$$
$31$ $$T^{2} + 6212 T - 5567264$$
$37$ $$T^{2} - 8512 T - 34360964$$
$41$ $$T^{2} + 1304 T - 158165876$$
$43$ $$T^{2} - 10004 T - 194677376$$
$47$ $$T^{2} + 12748 T - 135251744$$
$53$ $$T^{2} - 11220 T - 172017180$$
$59$ $$T^{2} + \cdots - 1715115024$$
$61$ $$T^{2} + \cdots + 2636833536$$
$67$ $$T^{2} + \cdots - 1158165296$$
$71$ $$T^{2} - 89720 T + 967897600$$
$73$ $$T^{2} + 55588 T - 243399884$$
$79$ $$T^{2} - 48824 T - 724002176$$
$83$ $$T^{2} + \cdots - 3779136224$$
$89$ $$T^{2} + \cdots - 7628401980$$
$97$ $$T^{2} + \cdots + 1150801884$$