# Properties

 Label 64.8.b.c Level $64$ Weight $8$ Character orbit 64.b Analytic conductor $19.993$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,8,Mod(33,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("64.33");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 64.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$19.9926416310$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2705346343547136.10 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 301x^{6} + 68101x^{4} - 6772500x^{2} + 506250000$$ x^8 - 301*x^6 + 68101*x^4 - 6772500*x^2 + 506250000 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{42}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + \beta_{4}) q^{3} + \beta_1 q^{5} - \beta_{7} q^{7} + (5 \beta_{2} - 617) q^{9}+O(q^{10})$$ q + (b6 + b4) * q^3 + b1 * q^5 - b7 * q^7 + (5*b2 - 617) * q^9 $$q + (\beta_{6} + \beta_{4}) q^{3} + \beta_1 q^{5} - \beta_{7} q^{7} + (5 \beta_{2} - 617) q^{9} + ( - 96 \beta_{6} - 75 \beta_{4}) q^{11} + (3 \beta_{3} + 45 \beta_1) q^{13} + ( - 10 \beta_{7} + 11 \beta_{5}) q^{15} + ( - 21 \beta_{2} + 11370) q^{17} + ( - 208 \beta_{6} + 225 \beta_{4}) q^{19} + (49 \beta_{3} - 358 \beta_1) q^{21} + (20 \beta_{7} + 115 \beta_{5}) q^{23} + (54 \beta_{2} + 45389) q^{25} + ( - 1735 \beta_{6} + 770 \beta_{4}) q^{27} + (192 \beta_{3} - 449 \beta_1) q^{29} + ( - 179 \beta_{7} + 357 \beta_{5}) q^{31} + ( - 459 \beta_{2} + 223740) q^{33} + ( - 7470 \beta_{6} - 2976 \beta_{4}) q^{35} + (361 \beta_{3} + 1135 \beta_1) q^{37} + ( - 387 \beta_{7} + 576 \beta_{5}) q^{39} + (1890 \beta_{2} - 104694) q^{41} + ( - 11429 \beta_{6} - 2259 \beta_{4}) q^{43} + (325 \beta_{3} - 1987 \beta_1) q^{45} + ( - 43 \beta_{7} + 295 \beta_{5}) q^{47} + ( - 2520 \beta_{2} + 450185) q^{49} + (25251 \beta_{6} + 14730 \beta_{4}) q^{51} + ( - 345 \beta_{3} - 2665 \beta_1) q^{53} + (855 \beta_{7} - 804 \beta_{5}) q^{55} + ( - 607 \beta_{2} - 353780) q^{57} + ( - 3219 \beta_{6} - 3435 \beta_{4}) q^{59} + ( - 1541 \beta_{3} + 4537 \beta_1) q^{61} + (2422 \beta_{7} - 2615 \beta_{5}) q^{63} + (1602 \beta_{2} - 1503648) q^{65} + (52624 \beta_{6} - 28791 \beta_{4}) q^{67} + ( - 2705 \beta_{3} + 950 \beta_1) q^{69} + ( - 1856 \beta_{7} - 5347 \beta_{5}) q^{71} + (2349 \beta_{2} - 1912030) q^{73} + (9695 \beta_{6} + 36749 \beta_{4}) q^{75} + ( - 3759 \beta_{3} + 31050 \beta_1) q^{77} + ( - 2406 \beta_{7} - 4572 \beta_{5}) q^{79} + (4765 \beta_{2} - 1905259) q^{81} + (159705 \beta_{6} - 13095 \beta_{4}) q^{83} + ( - 1365 \beta_{3} + 17124 \beta_1) q^{85} + (8522 \beta_{7} + 245 \beta_{5}) q^{87} + ( - 15315 \beta_{2} + 1889298) q^{89} + ( - 319914 \beta_{6} - 96480 \beta_{4}) q^{91} + (3416 \beta_{3} - 83360 \beta_1) q^{93} + ( - 85 \beta_{7} + 2908 \beta_{5}) q^{95} + (14067 \beta_{2} + 148570) q^{97} + (317187 \beta_{6} + 133155 \beta_{4}) q^{99}+O(q^{100})$$ q + (b6 + b4) * q^3 + b1 * q^5 - b7 * q^7 + (5*b2 - 617) * q^9 + (-96*b6 - 75*b4) * q^11 + (3*b3 + 45*b1) * q^13 + (-10*b7 + 11*b5) * q^15 + (-21*b2 + 11370) * q^17 + (-208*b6 + 225*b4) * q^19 + (49*b3 - 358*b1) * q^21 + (20*b7 + 115*b5) * q^23 + (54*b2 + 45389) * q^25 + (-1735*b6 + 770*b4) * q^27 + (192*b3 - 449*b1) * q^29 + (-179*b7 + 357*b5) * q^31 + (-459*b2 + 223740) * q^33 + (-7470*b6 - 2976*b4) * q^35 + (361*b3 + 1135*b1) * q^37 + (-387*b7 + 576*b5) * q^39 + (1890*b2 - 104694) * q^41 + (-11429*b6 - 2259*b4) * q^43 + (325*b3 - 1987*b1) * q^45 + (-43*b7 + 295*b5) * q^47 + (-2520*b2 + 450185) * q^49 + (25251*b6 + 14730*b4) * q^51 + (-345*b3 - 2665*b1) * q^53 + (855*b7 - 804*b5) * q^55 + (-607*b2 - 353780) * q^57 + (-3219*b6 - 3435*b4) * q^59 + (-1541*b3 + 4537*b1) * q^61 + (2422*b7 - 2615*b5) * q^63 + (1602*b2 - 1503648) * q^65 + (52624*b6 - 28791*b4) * q^67 + (-2705*b3 + 950*b1) * q^69 + (-1856*b7 - 5347*b5) * q^71 + (2349*b2 - 1912030) * q^73 + (9695*b6 + 36749*b4) * q^75 + (-3759*b3 + 31050*b1) * q^77 + (-2406*b7 - 4572*b5) * q^79 + (4765*b2 - 1905259) * q^81 + (159705*b6 - 13095*b4) * q^83 + (-1365*b3 + 17124*b1) * q^85 + (8522*b7 + 245*b5) * q^87 + (-15315*b2 + 1889298) * q^89 + (-319914*b6 - 96480*b4) * q^91 + (3416*b3 - 83360*b1) * q^93 + (-85*b7 + 2908*b5) * q^95 + (14067*b2 + 148570) * q^97 + (317187*b6 + 133155*b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 4936 q^{9}+O(q^{10})$$ 8 * q - 4936 * q^9 $$8 q - 4936 q^{9} + 90960 q^{17} + 363112 q^{25} + 1789920 q^{33} - 837552 q^{41} + 3601480 q^{49} - 2830240 q^{57} - 12029184 q^{65} - 15296240 q^{73} - 15242072 q^{81} + 15114384 q^{89} + 1188560 q^{97}+O(q^{100})$$ 8 * q - 4936 * q^9 + 90960 * q^17 + 363112 * q^25 + 1789920 * q^33 - 837552 * q^41 + 3601480 * q^49 - 2830240 * q^57 - 12029184 * q^65 - 15296240 * q^73 - 15242072 * q^81 + 15114384 * q^89 + 1188560 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 301x^{6} + 68101x^{4} - 6772500x^{2} + 506250000$$ :

 $$\beta_{1}$$ $$=$$ $$( 130784\nu^{6} - 39770984\nu^{4} + 5841976184\nu^{2} - 438311070000 ) / 383068125$$ (130784*v^6 - 39770984*v^4 + 5841976184*v^2 - 438311070000) / 383068125 $$\beta_{2}$$ $$=$$ $$( 32\nu^{6} + 111254416 ) / 68101$$ (32*v^6 + 111254416) / 68101 $$\beta_{3}$$ $$=$$ $$( 30784\nu^{6} - 8815984\nu^{4} + 1539231184\nu^{2} - 109304820000 ) / 34824375$$ (30784*v^6 - 8815984*v^4 + 1539231184*v^2 - 109304820000) / 34824375 $$\beta_{4}$$ $$=$$ $$( 408457\nu^{7} - 92413057\nu^{5} + 17090422657\nu^{3} - 686981250000\nu ) / 114920437500$$ (408457*v^7 - 92413057*v^5 + 17090422657*v^3 - 686981250000*v) / 114920437500 $$\beta_{5}$$ $$=$$ $$( 104732\nu^{7} - 349494332\nu^{5} + 56165073932\nu^{3} - 8572919940000\nu ) / 28730109375$$ (104732*v^7 - 349494332*v^5 + 56165073932*v^3 - 8572919940000*v) / 28730109375 $$\beta_{6}$$ $$=$$ $$( -1204\nu^{7} + 272404\nu^{5} - 41403604\nu^{3} + 2025000000\nu ) / 190265625$$ (-1204*v^7 + 272404*v^5 - 41403604*v^3 + 2025000000*v) / 190265625 $$\beta_{7}$$ $$=$$ $$( 256468\nu^{7} + 267773132\nu^{5} - 31566992732\nu^{3} + 4287965940000\nu ) / 28730109375$$ (256468*v^7 + 267773132*v^5 - 31566992732*v^3 + 4287965940000*v) / 28730109375
 $$\nu$$ $$=$$ $$( -\beta_{7} + 7\beta_{6} - \beta_{5} + 16\beta_{4} ) / 128$$ (-b7 + 7*b6 - b5 + 16*b4) / 128 $$\nu^{2}$$ $$=$$ $$( 73\beta_{3} - 8\beta_{2} - 178\beta _1 + 38528 ) / 512$$ (73*b3 - 8*b2 - 178*b1 + 38528) / 512 $$\nu^{3}$$ $$=$$ $$( 1357\beta_{6} + 2416\beta_{4} ) / 64$$ (1357*b6 + 2416*b4) / 64 $$\nu^{4}$$ $$=$$ $$( 10723\beta_{3} + 2408\beta_{2} - 31078\beta _1 - 5836928 ) / 512$$ (10723*b3 + 2408*b2 - 31078*b1 - 5836928) / 512 $$\nu^{5}$$ $$=$$ $$( 206183\beta_{7} + 2007656\beta_{6} + 115883\beta_{5} + 2937728\beta_{4} ) / 1024$$ (206183*b7 + 2007656*b6 + 115883*b5 + 2937728*b4) / 1024 $$\nu^{6}$$ $$=$$ $$( 68101\beta_{2} - 111254416 ) / 32$$ (68101*b2 - 111254416) / 32 $$\nu^{7}$$ $$=$$ $$( 33193583\beta_{7} - 360044456\beta_{6} + 12763283\beta_{5} - 449376128\beta_{4} ) / 1024$$ (33193583*b7 - 360044456*b6 + 12763283*b5 - 449376128*b4) / 1024

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$63$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 −11.0484 − 6.37883i 11.0484 − 6.37883i 10.1824 − 5.87883i −10.1824 − 5.87883i −10.1824 + 5.87883i 10.1824 + 5.87883i 11.0484 + 6.37883i −11.0484 + 6.37883i
0 69.0306i 0 232.201i 0 −1504.06 0 −2578.22 0
33.2 0 69.0306i 0 232.201i 0 1504.06 0 −2578.22 0
33.3 0 29.0306i 0 107.493i 0 534.108 0 1344.22 0
33.4 0 29.0306i 0 107.493i 0 −534.108 0 1344.22 0
33.5 0 29.0306i 0 107.493i 0 −534.108 0 1344.22 0
33.6 0 29.0306i 0 107.493i 0 534.108 0 1344.22 0
33.7 0 69.0306i 0 232.201i 0 1504.06 0 −2578.22 0
33.8 0 69.0306i 0 232.201i 0 −1504.06 0 −2578.22 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 33.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.8.b.c 8
3.b odd 2 1 576.8.d.f 8
4.b odd 2 1 inner 64.8.b.c 8
8.b even 2 1 inner 64.8.b.c 8
8.d odd 2 1 inner 64.8.b.c 8
12.b even 2 1 576.8.d.f 8
16.e even 4 1 256.8.a.j 4
16.e even 4 1 256.8.a.p 4
16.f odd 4 1 256.8.a.j 4
16.f odd 4 1 256.8.a.p 4
24.f even 2 1 576.8.d.f 8
24.h odd 2 1 576.8.d.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.8.b.c 8 1.a even 1 1 trivial
64.8.b.c 8 4.b odd 2 1 inner
64.8.b.c 8 8.b even 2 1 inner
64.8.b.c 8 8.d odd 2 1 inner
256.8.a.j 4 16.e even 4 1
256.8.a.j 4 16.f odd 4 1
256.8.a.p 4 16.e even 4 1
256.8.a.p 4 16.f odd 4 1
576.8.d.f 8 3.b odd 2 1
576.8.d.f 8 12.b even 2 1
576.8.d.f 8 24.f even 2 1
576.8.d.f 8 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 5608T_{3}^{2} + 4016016$$ acting on $$S_{8}^{\mathrm{new}}(64, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 5608 T^{2} + 4016016)^{2}$$
$5$ $$(T^{4} + 65472 T^{2} + \cdots + 623001600)^{2}$$
$7$ $$(T^{4} - 2547456 T^{2} + \cdots + 645335875584)^{2}$$
$11$ $$(T^{4} + 36480168 T^{2} + \cdots + 77526545767056)^{2}$$
$13$ $$(T^{4} + 150607296 T^{2} + \cdots + 55\!\cdots\!04)^{2}$$
$17$ $$(T^{2} - 22740 T + 61426404)^{4}$$
$19$ $$(T^{4} + 418474472 T^{2} + \cdots + 11\!\cdots\!96)^{2}$$
$23$ $$(T^{4} - 10219718400 T^{2} + \cdots + 38\!\cdots\!00)^{2}$$
$29$ $$(T^{4} + 61019054016 T^{2} + \cdots + 91\!\cdots\!64)^{2}$$
$31$ $$(T^{4} - 122830319616 T^{2} + \cdots + 29\!\cdots\!64)^{2}$$
$37$ $$(T^{4} + 282477395904 T^{2} + \cdots + 33\!\cdots\!04)^{2}$$
$41$ $$(T^{2} + 209388 T - 538628183964)^{4}$$
$43$ $$(T^{4} + 253862622248 T^{2} + \cdots + 10\!\cdots\!76)^{2}$$
$47$ $$(T^{4} - 49913723904 T^{2} + \cdots + 34\!\cdots\!04)^{2}$$
$53$ $$(T^{4} + 668149656000 T^{2} + \cdots + 99\!\cdots\!00)^{2}$$
$59$ $$(T^{4} + 64366187688 T^{2} + \cdots + 60\!\cdots\!36)^{2}$$
$61$ $$(T^{4} + 4369584928704 T^{2} + \cdots + 42\!\cdots\!04)^{2}$$
$67$ $$(T^{4} + 12222871225448 T^{2} + \cdots + 45\!\cdots\!76)^{2}$$
$71$ $$(T^{4} - 31396418854656 T^{2} + \cdots + 13\!\cdots\!84)^{2}$$
$73$ $$(T^{2} + 3824060 T + 2806911930244)^{4}$$
$79$ $$(T^{4} - 33350493920256 T^{2} + \cdots + 23\!\cdots\!84)^{2}$$
$83$ $$(T^{4} + 56321797732200 T^{2} + \cdots + 74\!\cdots\!00)^{2}$$
$89$ $$(T^{2} - 3778596 T - 32517358628796)^{4}$$
$97$ $$(T^{2} - 297140 T - 30423027470684)^{4}$$