# Properties

 Label 448.6.a.bf Level $448$ Weight $6$ Character orbit 448.a Self dual yes Analytic conductor $71.852$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998$$ x^5 - 229*x^3 - 272*x^2 + 7973*x - 13998 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: no (minimal twist has level 224) 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 a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 2) q^{3} + (\beta_{3} - 7) q^{5} + 49 q^{7} + ( - 2 \beta_{3} + \beta_{2} + \cdots + 127) q^{9}+O(q^{10})$$ q + (-b1 + 2) * q^3 + (b3 - 7) * q^5 + 49 * q^7 + (-2*b3 + b2 - b1 + 127) * q^9 $$q + ( - \beta_1 + 2) q^{3} + (\beta_{3} - 7) q^{5} + 49 q^{7} + ( - 2 \beta_{3} + \beta_{2} + \cdots + 127) q^{9}+ \cdots + ( - 183 \beta_{4} + 1212 \beta_{3} + \cdots + 52253) q^{99}+O(q^{100})$$ q + (-b1 + 2) * q^3 + (b3 - 7) * q^5 + 49 * q^7 + (-2*b3 + b2 - b1 + 127) * q^9 + (-b4 + 2*b3 + b2 - 9*b1 - 23) * q^11 + (-b4 + 6*b3 + b2 - 6*b1 - 7) * q^13 + (-2*b4 + 6*b3 + 3*b2 + 31*b1 + 4) * q^15 + (-4*b4 - 4*b3 - 2*b2 - 24*b1 - 82) * q^17 + (5*b4 + 3*b3 + b2 - 57*b1 + 718) * q^19 + (-49*b1 + 98) * q^21 + (10*b4 + 8*b3 - b2 + 117*b1 + 98) * q^23 + (16*b4 - 6*b3 - 7*b2 - 57*b1 + 1925) * q^25 + (-7*b4 + 33*b3 - 3*b2 - 138*b1 - 66) * q^27 + (10*b4 - 36*b3 - 10*b2 - 96*b1 - 956) * q^29 + (-10*b4 + 30*b3 - 20*b2 + 48*b1 - 2096) * q^31 + (-16*b4 + 62*b3 + 26*b2 - 116*b1 + 3182) * q^33 + (49*b3 - 343) * q^35 + (-38*b3 + 6*b2 - 114*b1 - 3936) * q^37 + (-24*b4 + 92*b3 + 35*b2 - 39*b1 + 2188) * q^39 + (-2*b4 + 4*b3 - 28*b2 - 450*b1 + 4680) * q^41 + (-3*b4 + 32*b3 - 39*b2 + 249*b1 - 4403) * q^43 + (-47*b4 + 38*b3 + 11*b2 - 490*b1 - 9897) * q^45 + (48*b4 + 28*b3 - 48*b2 - 354*b1 + 3216) * q^47 + 2401 * q^49 + (26*b4 - 82*b3 + 44*b2 + 468*b1 + 9124) * q^51 + (32*b4 + 124*b3 + 22*b2 - 546*b1 - 10818) * q^53 + (-6*b4 - 290*b3 + 24*b2 - 798*b1 + 10632) * q^55 + (-12*b4 - 154*b3 + 23*b2 - 851*b1 + 21878) * q^57 + (-5*b4 - 41*b3 + 59*b2 + 537*b1 + 14864) * q^59 + (38*b4 + 83*b3 + 22*b2 - 540*b1 - 13639) * q^61 + (-98*b3 + 49*b2 - 49*b1 + 6223) * q^63 + (64*b4 - 282*b3 - 13*b2 - 1119*b1 + 30470) * q^65 + (-93*b4 - 402*b3 + 3*b2 + 615*b1 + 5213) * q^67 + (5*b4 + 25*b3 - 185*b2 + 74*b1 - 42938) * q^69 + (-50*b4 - 522*b3 + 44*b2 - 522*b1 + 18500) * q^71 + (-58*b4 + 140*b3 + 4*b2 + 372*b1 + 27236) * q^73 + (105*b4 - 815*b3 - 119*b2 - 739*b1 + 24760) * q^75 + (-49*b4 + 98*b3 + 49*b2 - 441*b1 - 1127) * q^77 + (-56*b4 + 406*b3 - 58*b2 + 204*b1 + 19286) * q^79 + (-40*b4 + 414*b3 + 51*b2 + 1937*b1 + 21035) * q^81 + (68*b4 + 678*b3 - 8*b2 - 945*b1 + 29328) * q^83 + (-114*b4 - 664*b3 + 180*b2 + 930*b1 - 23626) * q^85 + (192*b4 - 1088*b3 - 122*b2 + 2148*b1 + 33596) * q^87 + (140*b4 + 46*b3 - 32*b2 + 2094*b1 + 37748) * q^89 + (-49*b4 + 294*b3 + 49*b2 - 294*b1 - 343) * q^91 + (150*b4 - 454*b3 + 92*b2 + 7018*b1 - 17320) * q^93 + (-44*b4 + 1928*b3 + 59*b2 + 2601*b1 + 15324) * q^95 + (-54*b4 + 910*b3 - 270*b2 + 822*b1 - 17458) * q^97 + (-183*b4 + 1212*b3 + 255*b2 - 4597*b1 + 52253) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 10 q^{3} - 36 q^{5} + 245 q^{7} + 637 q^{9}+O(q^{10})$$ 5 * q + 10 * q^3 - 36 * q^5 + 245 * q^7 + 637 * q^9 $$5 q + 10 q^{3} - 36 q^{5} + 245 q^{7} + 637 q^{9} - 116 q^{11} - 40 q^{13} + 16 q^{15} - 402 q^{17} + 3582 q^{19} + 490 q^{21} + 472 q^{23} + 9615 q^{25} - 356 q^{27} - 4754 q^{29} - 10500 q^{31} + 15864 q^{33} - 1764 q^{35} - 19642 q^{37} + 10872 q^{39} + 23398 q^{41} - 22044 q^{43} - 49476 q^{45} + 16004 q^{47} + 12005 q^{49} + 45676 q^{51} - 54246 q^{53} + 53456 q^{55} + 109556 q^{57} + 74366 q^{59} - 68316 q^{61} + 31213 q^{63} + 152568 q^{65} + 26560 q^{67} - 214720 q^{69} + 93072 q^{71} + 136098 q^{73} + 124510 q^{75} - 5684 q^{77} + 96080 q^{79} + 104801 q^{81} + 145894 q^{83} - 117352 q^{85} + 168876 q^{87} + 188554 q^{89} - 1960 q^{91} - 86296 q^{93} + 74736 q^{95} - 88146 q^{97} + 260236 q^{99}+O(q^{100})$$ 5 * q + 10 * q^3 - 36 * q^5 + 245 * q^7 + 637 * q^9 - 116 * q^11 - 40 * q^13 + 16 * q^15 - 402 * q^17 + 3582 * q^19 + 490 * q^21 + 472 * q^23 + 9615 * q^25 - 356 * q^27 - 4754 * q^29 - 10500 * q^31 + 15864 * q^33 - 1764 * q^35 - 19642 * q^37 + 10872 * q^39 + 23398 * q^41 - 22044 * q^43 - 49476 * q^45 + 16004 * q^47 + 12005 * q^49 + 45676 * q^51 - 54246 * q^53 + 53456 * q^55 + 109556 * q^57 + 74366 * q^59 - 68316 * q^61 + 31213 * q^63 + 152568 * q^65 + 26560 * q^67 - 214720 * q^69 + 93072 * q^71 + 136098 * q^73 + 124510 * q^75 - 5684 * q^77 + 96080 * q^79 + 104801 * q^81 + 145894 * q^83 - 117352 * q^85 + 168876 * q^87 + 188554 * q^89 - 1960 * q^91 - 86296 * q^93 + 74736 * q^95 - 88146 * q^97 + 260236 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( 56\nu^{4} - 64\nu^{3} - 8772\nu^{2} - 20218\nu - 3528 ) / 633$$ (56*v^4 - 64*v^3 - 8772*v^2 - 20218*v - 3528) / 633 $$\beta_{3}$$ $$=$$ $$( 28\nu^{4} - 32\nu^{3} - 5652\nu^{2} - 8210\nu + 114075 ) / 633$$ (28*v^4 - 32*v^3 - 5652*v^2 - 8210*v + 114075) / 633 $$\beta_{4}$$ $$=$$ $$( 36\nu^{4} + 200\nu^{3} - 8352\nu^{2} - 46908\nu + 206833 ) / 211$$ (36*v^4 + 200*v^3 - 8352*v^2 - 46908*v + 206833) / 211
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( -2\beta_{3} + \beta_{2} + 3\beta _1 + 366 ) / 4$$ (-2*b3 + b2 + 3*b1 + 366) / 4 $$\nu^{3}$$ $$=$$ $$( 7\beta_{4} - 45\beta_{3} + 9\beta_{2} + 630\beta _1 + 1298 ) / 8$$ (7*b4 - 45*b3 + 9*b2 + 630*b1 + 1298) / 8 $$\nu^{4}$$ $$=$$ $$( 4\beta_{4} - 339\beta_{3} + 207\beta_{2} + 1552\beta _1 + 58325 ) / 4$$ (4*b4 - 339*b3 + 207*b2 + 1552*b1 + 58325) / 4

## 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
 14.6594 3.97668 2.24605 −8.98949 −11.8926
0 −27.3187 0 −52.2232 0 49.0000 0 503.312 0
1.2 0 −5.95336 0 −11.6830 0 49.0000 0 −207.557 0
1.3 0 −2.49210 0 99.5911 0 49.0000 0 −236.789 0
1.4 0 19.9790 0 −106.158 0 49.0000 0 156.159 0
1.5 0 25.7852 0 34.4729 0 49.0000 0 421.876 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.bf 5
4.b odd 2 1 448.6.a.be 5
8.b even 2 1 224.6.a.i 5
8.d odd 2 1 224.6.a.j yes 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.i 5 8.b even 2 1
224.6.a.j yes 5 8.d odd 2 1
448.6.a.be 5 4.b odd 2 1
448.6.a.bf 5 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{5} - 10T_{3}^{4} - 876T_{3}^{3} + 7592T_{3}^{2} + 107952T_{3} + 208800$$ T3^5 - 10*T3^4 - 876*T3^3 + 7592*T3^2 + 107952*T3 + 208800 $$T_{5}^{5} + 36T_{5}^{4} - 11972T_{5}^{3} - 342672T_{5}^{2} + 16702736T_{5} + 222365696$$ T5^5 + 36*T5^4 - 11972*T5^3 - 342672*T5^2 + 16702736*T5 + 222365696

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 10 T^{4} + \cdots + 208800$$
$5$ $$T^{5} + 36 T^{4} + \cdots + 222365696$$
$7$ $$(T - 49)^{5}$$
$11$ $$T^{5} + \cdots - 183286596608$$
$13$ $$T^{5} + \cdots - 247266196544$$
$17$ $$T^{5} + \cdots - 353511151373280$$
$19$ $$T^{5} + \cdots - 71\!\cdots\!08$$
$23$ $$T^{5} + \cdots + 11\!\cdots\!52$$
$29$ $$T^{5} + \cdots - 38\!\cdots\!20$$
$31$ $$T^{5} + \cdots + 34\!\cdots\!20$$
$37$ $$T^{5} + \cdots - 11\!\cdots\!00$$
$41$ $$T^{5} + \cdots + 13\!\cdots\!00$$
$43$ $$T^{5} + \cdots + 77\!\cdots\!32$$
$47$ $$T^{5} + \cdots + 62\!\cdots\!40$$
$53$ $$T^{5} + \cdots + 57\!\cdots\!00$$
$59$ $$T^{5} + \cdots + 14\!\cdots\!20$$
$61$ $$T^{5} + \cdots + 30\!\cdots\!16$$
$67$ $$T^{5} + \cdots - 16\!\cdots\!64$$
$71$ $$T^{5} + \cdots + 10\!\cdots\!36$$
$73$ $$T^{5} + \cdots + 81\!\cdots\!00$$
$79$ $$T^{5} + \cdots - 26\!\cdots\!92$$
$83$ $$T^{5} + \cdots + 86\!\cdots\!00$$
$89$ $$T^{5} + \cdots + 64\!\cdots\!92$$
$97$ $$T^{5} + \cdots + 93\!\cdots\!00$$