# Properties

 Label 448.6.a.z Level $448$ Weight $6$ Character orbit 448.a Self dual yes Analytic conductor $71.852$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 44$$ x^2 - x - 44 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 $$\beta = \sqrt{177}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 13) q^{3} + ( - 5 \beta + 31) q^{5} + 49 q^{7} + ( - 26 \beta + 103) q^{9}+O(q^{10})$$ q + (-b + 13) * q^3 + (-5*b + 31) * q^5 + 49 * q^7 + (-26*b + 103) * q^9 $$q + ( - \beta + 13) q^{3} + ( - 5 \beta + 31) q^{5} + 49 q^{7} + ( - 26 \beta + 103) q^{9} + ( - 6 \beta - 486) q^{11} + ( - 71 \beta - 39) q^{13} + ( - 96 \beta + 1288) q^{15} + ( - 6 \beta + 280) q^{17} + (37 \beta - 1321) q^{19} + ( - 49 \beta + 637) q^{21} + (248 \beta + 1136) q^{23} + ( - 310 \beta + 2261) q^{25} + ( - 198 \beta + 2782) q^{27} + ( - 14 \beta + 3904) q^{29} + ( - 130 \beta + 2722) q^{31} + (408 \beta - 5256) q^{33} + ( - 245 \beta + 1519) q^{35} + (650 \beta - 288) q^{37} + ( - 884 \beta + 12060) q^{39} + ( - 378 \beta - 8444) q^{41} + ( - 338 \beta + 4198) q^{43} + ( - 1321 \beta + 26203) q^{45} + ( - 1238 \beta - 2266) q^{47} + 2401 q^{49} + ( - 358 \beta + 4702) q^{51} + ( - 1152 \beta - 710) q^{53} + (2244 \beta - 9756) q^{55} + (1802 \beta - 23722) q^{57} + (325 \beta - 17073) q^{59} + (3347 \beta - 9553) q^{61} + ( - 1274 \beta + 5047) q^{63} + ( - 2006 \beta + 61626) q^{65} + (1240 \beta - 28476) q^{67} + (2088 \beta - 29128) q^{69} + ( - 700 \beta - 3612) q^{71} + (1260 \beta + 64414) q^{73} + ( - 6291 \beta + 84263) q^{75} + ( - 294 \beta - 23814) q^{77} + (1548 \beta + 26404) q^{79} + (962 \beta + 46183) q^{81} + (5209 \beta + 42243) q^{83} + ( - 1586 \beta + 13990) q^{85} + ( - 4086 \beta + 53230) q^{87} + (2312 \beta - 65486) q^{89} + ( - 3479 \beta - 1911) q^{91} + ( - 4412 \beta + 58396) q^{93} + (7752 \beta - 73696) q^{95} + ( - 350 \beta + 97312) q^{97} + (12018 \beta - 22446) q^{99}+O(q^{100})$$ q + (-b + 13) * q^3 + (-5*b + 31) * q^5 + 49 * q^7 + (-26*b + 103) * q^9 + (-6*b - 486) * q^11 + (-71*b - 39) * q^13 + (-96*b + 1288) * q^15 + (-6*b + 280) * q^17 + (37*b - 1321) * q^19 + (-49*b + 637) * q^21 + (248*b + 1136) * q^23 + (-310*b + 2261) * q^25 + (-198*b + 2782) * q^27 + (-14*b + 3904) * q^29 + (-130*b + 2722) * q^31 + (408*b - 5256) * q^33 + (-245*b + 1519) * q^35 + (650*b - 288) * q^37 + (-884*b + 12060) * q^39 + (-378*b - 8444) * q^41 + (-338*b + 4198) * q^43 + (-1321*b + 26203) * q^45 + (-1238*b - 2266) * q^47 + 2401 * q^49 + (-358*b + 4702) * q^51 + (-1152*b - 710) * q^53 + (2244*b - 9756) * q^55 + (1802*b - 23722) * q^57 + (325*b - 17073) * q^59 + (3347*b - 9553) * q^61 + (-1274*b + 5047) * q^63 + (-2006*b + 61626) * q^65 + (1240*b - 28476) * q^67 + (2088*b - 29128) * q^69 + (-700*b - 3612) * q^71 + (1260*b + 64414) * q^73 + (-6291*b + 84263) * q^75 + (-294*b - 23814) * q^77 + (1548*b + 26404) * q^79 + (962*b + 46183) * q^81 + (5209*b + 42243) * q^83 + (-1586*b + 13990) * q^85 + (-4086*b + 53230) * q^87 + (2312*b - 65486) * q^89 + (-3479*b - 1911) * q^91 + (-4412*b + 58396) * q^93 + (7752*b - 73696) * q^95 + (-350*b + 97312) * q^97 + (12018*b - 22446) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 26 q^{3} + 62 q^{5} + 98 q^{7} + 206 q^{9}+O(q^{10})$$ 2 * q + 26 * q^3 + 62 * q^5 + 98 * q^7 + 206 * q^9 $$2 q + 26 q^{3} + 62 q^{5} + 98 q^{7} + 206 q^{9} - 972 q^{11} - 78 q^{13} + 2576 q^{15} + 560 q^{17} - 2642 q^{19} + 1274 q^{21} + 2272 q^{23} + 4522 q^{25} + 5564 q^{27} + 7808 q^{29} + 5444 q^{31} - 10512 q^{33} + 3038 q^{35} - 576 q^{37} + 24120 q^{39} - 16888 q^{41} + 8396 q^{43} + 52406 q^{45} - 4532 q^{47} + 4802 q^{49} + 9404 q^{51} - 1420 q^{53} - 19512 q^{55} - 47444 q^{57} - 34146 q^{59} - 19106 q^{61} + 10094 q^{63} + 123252 q^{65} - 56952 q^{67} - 58256 q^{69} - 7224 q^{71} + 128828 q^{73} + 168526 q^{75} - 47628 q^{77} + 52808 q^{79} + 92366 q^{81} + 84486 q^{83} + 27980 q^{85} + 106460 q^{87} - 130972 q^{89} - 3822 q^{91} + 116792 q^{93} - 147392 q^{95} + 194624 q^{97} - 44892 q^{99}+O(q^{100})$$ 2 * q + 26 * q^3 + 62 * q^5 + 98 * q^7 + 206 * q^9 - 972 * q^11 - 78 * q^13 + 2576 * q^15 + 560 * q^17 - 2642 * q^19 + 1274 * q^21 + 2272 * q^23 + 4522 * q^25 + 5564 * q^27 + 7808 * q^29 + 5444 * q^31 - 10512 * q^33 + 3038 * q^35 - 576 * q^37 + 24120 * q^39 - 16888 * q^41 + 8396 * q^43 + 52406 * q^45 - 4532 * q^47 + 4802 * q^49 + 9404 * q^51 - 1420 * q^53 - 19512 * q^55 - 47444 * q^57 - 34146 * q^59 - 19106 * q^61 + 10094 * q^63 + 123252 * q^65 - 56952 * q^67 - 58256 * q^69 - 7224 * q^71 + 128828 * q^73 + 168526 * q^75 - 47628 * q^77 + 52808 * q^79 + 92366 * q^81 + 84486 * q^83 + 27980 * q^85 + 106460 * q^87 - 130972 * q^89 - 3822 * q^91 + 116792 * q^93 - 147392 * q^95 + 194624 * q^97 - 44892 * 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
 7.15207 −6.15207
0 −0.304135 0 −35.5207 0 49.0000 0 −242.908 0
1.2 0 26.3041 0 97.5207 0 49.0000 0 448.908 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.z 2
4.b odd 2 1 448.6.a.q 2
8.b even 2 1 56.6.a.c 2
8.d odd 2 1 112.6.a.k 2
24.f even 2 1 1008.6.a.bt 2
24.h odd 2 1 504.6.a.s 2
56.e even 2 1 784.6.a.p 2
56.h odd 2 1 392.6.a.f 2
56.j odd 6 2 392.6.i.g 4
56.p even 6 2 392.6.i.l 4

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

 $$T_{3}^{2} - 26T_{3} - 8$$ T3^2 - 26*T3 - 8 $$T_{5}^{2} - 62T_{5} - 3464$$ T5^2 - 62*T5 - 3464

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 26T - 8$$
$5$ $$T^{2} - 62T - 3464$$
$7$ $$(T - 49)^{2}$$
$11$ $$T^{2} + 972T + 229824$$
$13$ $$T^{2} + 78T - 890736$$
$17$ $$T^{2} - 560T + 72028$$
$19$ $$T^{2} + 2642 T + 1502728$$
$23$ $$T^{2} - 2272 T - 9595712$$
$29$ $$T^{2} - 7808 T + 15206524$$
$31$ $$T^{2} - 5444 T + 4417984$$
$37$ $$T^{2} + 576 T - 74699556$$
$41$ $$T^{2} + 16888 T + 46010668$$
$43$ $$T^{2} - 8396 T - 2597984$$
$47$ $$T^{2} + 4532 T - 266143232$$
$53$ $$T^{2} + 1420 T - 234393308$$
$59$ $$T^{2} + 34146 T + 272791704$$
$61$ $$T^{2} + \cdots - 1891566584$$
$67$ $$T^{2} + 56952 T + 538727376$$
$71$ $$T^{2} + 7224 T - 73683456$$
$73$ $$T^{2} + \cdots + 3868158196$$
$79$ $$T^{2} - 52808 T + 273025408$$
$83$ $$T^{2} + \cdots - 3018190488$$
$89$ $$T^{2} + \cdots + 3342290308$$
$97$ $$T^{2} + \cdots + 9447942844$$