# Properties

 Label 448.6.a.x 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: $$\Q(\sqrt{193})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 48$$ x^2 - x - 48 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 $$\beta = \sqrt{193}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 7) q^{3} + (5 \beta - 21) q^{5} - 49 q^{7} + ( - 14 \beta - 1) q^{9}+O(q^{10})$$ q + (-b + 7) * q^3 + (5*b - 21) * q^5 - 49 * q^7 + (-14*b - 1) * q^9 $$q + ( - \beta + 7) q^{3} + (5 \beta - 21) q^{5} - 49 q^{7} + ( - 14 \beta - 1) q^{9} + ( - 14 \beta + 358) q^{11} + ( - 17 \beta + 357) q^{13} + (56 \beta - 1112) q^{15} + (54 \beta - 672) q^{17} + (93 \beta + 973) q^{19} + (49 \beta - 343) q^{21} + ( - 112 \beta - 896) q^{23} + ( - 210 \beta + 2141) q^{25} + (146 \beta + 994) q^{27} + ( - 546 \beta + 600) q^{29} + (446 \beta - 3402) q^{31} + ( - 456 \beta + 5208) q^{33} + ( - 245 \beta + 1029) q^{35} + ( - 154 \beta - 7320) q^{37} + ( - 476 \beta + 5780) q^{39} + (442 \beta + 3948) q^{41} + (518 \beta - 262) q^{43} + (289 \beta - 13489) q^{45} + (746 \beta - 9198) q^{47} + 2401 q^{49} + (1050 \beta - 15126) q^{51} + (1008 \beta - 22566) q^{53} + (2084 \beta - 21028) q^{55} + ( - 322 \beta - 11138) q^{57} + (365 \beta - 11291) q^{59} + (1117 \beta + 26411) q^{61} + (686 \beta + 49) q^{63} + (2142 \beta - 23902) q^{65} + (224 \beta - 4924) q^{67} + (112 \beta + 15344) q^{69} + ( - 4060 \beta - 420) q^{71} + (1204 \beta - 61026) q^{73} + ( - 3611 \beta + 55517) q^{75} + (686 \beta - 17542) q^{77} + (1372 \beta + 15852) q^{79} + (3430 \beta - 20977) q^{81} + (4185 \beta - 18487) q^{83} + ( - 4494 \beta + 66222) q^{85} + ( - 4422 \beta + 109578) q^{87} + ( - 1512 \beta - 105294) q^{89} + (833 \beta - 17493) q^{91} + (6524 \beta - 109892) q^{93} + (2912 \beta + 69312) q^{95} + ( - 8882 \beta - 22120) q^{97} + ( - 4998 \beta + 37470) q^{99}+O(q^{100})$$ q + (-b + 7) * q^3 + (5*b - 21) * q^5 - 49 * q^7 + (-14*b - 1) * q^9 + (-14*b + 358) * q^11 + (-17*b + 357) * q^13 + (56*b - 1112) * q^15 + (54*b - 672) * q^17 + (93*b + 973) * q^19 + (49*b - 343) * q^21 + (-112*b - 896) * q^23 + (-210*b + 2141) * q^25 + (146*b + 994) * q^27 + (-546*b + 600) * q^29 + (446*b - 3402) * q^31 + (-456*b + 5208) * q^33 + (-245*b + 1029) * q^35 + (-154*b - 7320) * q^37 + (-476*b + 5780) * q^39 + (442*b + 3948) * q^41 + (518*b - 262) * q^43 + (289*b - 13489) * q^45 + (746*b - 9198) * q^47 + 2401 * q^49 + (1050*b - 15126) * q^51 + (1008*b - 22566) * q^53 + (2084*b - 21028) * q^55 + (-322*b - 11138) * q^57 + (365*b - 11291) * q^59 + (1117*b + 26411) * q^61 + (686*b + 49) * q^63 + (2142*b - 23902) * q^65 + (224*b - 4924) * q^67 + (112*b + 15344) * q^69 + (-4060*b - 420) * q^71 + (1204*b - 61026) * q^73 + (-3611*b + 55517) * q^75 + (686*b - 17542) * q^77 + (1372*b + 15852) * q^79 + (3430*b - 20977) * q^81 + (4185*b - 18487) * q^83 + (-4494*b + 66222) * q^85 + (-4422*b + 109578) * q^87 + (-1512*b - 105294) * q^89 + (833*b - 17493) * q^91 + (6524*b - 109892) * q^93 + (2912*b + 69312) * q^95 + (-8882*b - 22120) * q^97 + (-4998*b + 37470) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{3} - 42 q^{5} - 98 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 14 * q^3 - 42 * q^5 - 98 * q^7 - 2 * q^9 $$2 q + 14 q^{3} - 42 q^{5} - 98 q^{7} - 2 q^{9} + 716 q^{11} + 714 q^{13} - 2224 q^{15} - 1344 q^{17} + 1946 q^{19} - 686 q^{21} - 1792 q^{23} + 4282 q^{25} + 1988 q^{27} + 1200 q^{29} - 6804 q^{31} + 10416 q^{33} + 2058 q^{35} - 14640 q^{37} + 11560 q^{39} + 7896 q^{41} - 524 q^{43} - 26978 q^{45} - 18396 q^{47} + 4802 q^{49} - 30252 q^{51} - 45132 q^{53} - 42056 q^{55} - 22276 q^{57} - 22582 q^{59} + 52822 q^{61} + 98 q^{63} - 47804 q^{65} - 9848 q^{67} + 30688 q^{69} - 840 q^{71} - 122052 q^{73} + 111034 q^{75} - 35084 q^{77} + 31704 q^{79} - 41954 q^{81} - 36974 q^{83} + 132444 q^{85} + 219156 q^{87} - 210588 q^{89} - 34986 q^{91} - 219784 q^{93} + 138624 q^{95} - 44240 q^{97} + 74940 q^{99}+O(q^{100})$$ 2 * q + 14 * q^3 - 42 * q^5 - 98 * q^7 - 2 * q^9 + 716 * q^11 + 714 * q^13 - 2224 * q^15 - 1344 * q^17 + 1946 * q^19 - 686 * q^21 - 1792 * q^23 + 4282 * q^25 + 1988 * q^27 + 1200 * q^29 - 6804 * q^31 + 10416 * q^33 + 2058 * q^35 - 14640 * q^37 + 11560 * q^39 + 7896 * q^41 - 524 * q^43 - 26978 * q^45 - 18396 * q^47 + 4802 * q^49 - 30252 * q^51 - 45132 * q^53 - 42056 * q^55 - 22276 * q^57 - 22582 * q^59 + 52822 * q^61 + 98 * q^63 - 47804 * q^65 - 9848 * q^67 + 30688 * q^69 - 840 * q^71 - 122052 * q^73 + 111034 * q^75 - 35084 * q^77 + 31704 * q^79 - 41954 * q^81 - 36974 * q^83 + 132444 * q^85 + 219156 * q^87 - 210588 * q^89 - 34986 * q^91 - 219784 * q^93 + 138624 * q^95 - 44240 * q^97 + 74940 * 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
 7.44622 −6.44622
0 −6.89244 0 48.4622 0 −49.0000 0 −195.494 0
1.2 0 20.8924 0 −90.4622 0 −49.0000 0 193.494 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.x 2
4.b odd 2 1 448.6.a.r 2
8.b even 2 1 56.6.a.d 2
8.d odd 2 1 112.6.a.j 2
24.f even 2 1 1008.6.a.bi 2
24.h odd 2 1 504.6.a.m 2
56.e even 2 1 784.6.a.q 2
56.h odd 2 1 392.6.a.e 2
56.j odd 6 2 392.6.i.h 4
56.p even 6 2 392.6.i.k 4

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

 $$T_{3}^{2} - 14T_{3} - 144$$ T3^2 - 14*T3 - 144 $$T_{5}^{2} + 42T_{5} - 4384$$ T5^2 + 42*T5 - 4384

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 14T - 144$$
$5$ $$T^{2} + 42T - 4384$$
$7$ $$(T + 49)^{2}$$
$11$ $$T^{2} - 716T + 90336$$
$13$ $$T^{2} - 714T + 71672$$
$17$ $$T^{2} + 1344 T - 111204$$
$19$ $$T^{2} - 1946 T - 722528$$
$23$ $$T^{2} + 1792 T - 1618176$$
$29$ $$T^{2} - 1200 T - 57176388$$
$31$ $$T^{2} + 6804 T - 26817184$$
$37$ $$T^{2} + 14640 T + 49005212$$
$41$ $$T^{2} - 7896 T - 22118548$$
$43$ $$T^{2} + 524 T - 51717888$$
$47$ $$T^{2} + 18396 T - 22804384$$
$53$ $$T^{2} + 45132 T + 313124004$$
$59$ $$T^{2} + 22582 T + 101774256$$
$61$ $$T^{2} - 52822 T + 456736944$$
$67$ $$T^{2} + 9848 T + 14561808$$
$71$ $$T^{2} + \cdots - 3181158400$$
$73$ $$T^{2} + \cdots + 3444396788$$
$79$ $$T^{2} - 31704 T - 112014208$$
$83$ $$T^{2} + \cdots - 3038476256$$
$89$ $$T^{2} + \cdots + 10645600644$$
$97$ $$T^{2} + \cdots - 14736460932$$