# Properties

 Label 1280.4.d.bc Level $1280$ Weight $4$ Character orbit 1280.d Analytic conductor $75.522$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1280.d (of order $$2$$, degree $$1$$, not 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: $$75.5224448073$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 2x^{7} + 2x^{6} + 52x^{5} + 231x^{4} + 324x^{3} + 242x^{2} + 66x + 9$$ x^8 - 2*x^7 + 2*x^6 + 52*x^5 + 231*x^4 + 324*x^3 + 242*x^2 + 66*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{16}$$ Twist minimal: no (minimal twist has level 640) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_1) q^{3} - 5 \beta_1 q^{5} + ( - \beta_{2} + 5) q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 27) q^{9}+O(q^{10})$$ q + (-b5 + b1) * q^3 - 5*b1 * q^5 + (-b2 + 5) * q^7 + (-b4 + b3 + b2 - 27) * q^9 $$q + ( - \beta_{5} + \beta_1) q^{3} - 5 \beta_1 q^{5} + ( - \beta_{2} + 5) q^{7} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 27) q^{9} + (2 \beta_{7} - \beta_{6} - \beta_{5} - 18 \beta_1) q^{11} + (\beta_{7} - \beta_{6} - \beta_{5} + 8 \beta_1) q^{13} + (5 \beta_{4} + 5) q^{15} + (9 \beta_{4} + \beta_{3} - \beta_{2} + 24) q^{17} + (2 \beta_{7} + \beta_{6} - 5 \beta_{5} - 32 \beta_1) q^{19} + (4 \beta_{7} - 6 \beta_{6} - 22 \beta_{5} - 12 \beta_1) q^{21} + ( - 10 \beta_{4} - 2 \beta_{3} - \beta_{2} + 11) q^{23} - 25 q^{25} + ( - 6 \beta_{7} + 6 \beta_{6} + 28 \beta_{5} - 34 \beta_1) q^{27} + ( - 2 \beta_{6} + 6 \beta_{5} + 50 \beta_1) q^{29} + (8 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 6) q^{31} + (23 \beta_{4} + 3 \beta_{3} + 9 \beta_{2} - 22) q^{33} + (5 \beta_{6} - 25 \beta_1) q^{35} + ( - 10 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} + 6 \beta_1) q^{37} + ( - 14 \beta_{4} + 8 \beta_{2} - 46) q^{39} + ( - 3 \beta_{4} - 9 \beta_{3} + 3 \beta_{2} + 8) q^{41} + (10 \beta_{7} - 4 \beta_{6} - 3 \beta_{5} - 209 \beta_1) q^{43} + ( - 5 \beta_{7} - 5 \beta_{6} - 5 \beta_{5} + 135 \beta_1) q^{45} + (34 \beta_{4} + 4 \beta_{3} - 9 \beta_{2} - 241) q^{47} + (47 \beta_{4} - 15 \beta_{3} + \beta_{2} + 367) q^{49} + ( - 8 \beta_{7} - 16 \beta_{6} - 30 \beta_{5} + 486 \beta_1) q^{51} + ( - 3 \beta_{7} + 3 \beta_{6} - 53 \beta_{5} + 4 \beta_1) q^{53} + (5 \beta_{4} + 10 \beta_{3} - 5 \beta_{2} - 90) q^{55} + (71 \beta_{4} + 15 \beta_{3} + \beta_{2} - 254) q^{57} + ( - 10 \beta_{7} - 7 \beta_{6} - 41 \beta_{5} + 44 \beta_1) q^{59} + (4 \beta_{7} + 4 \beta_{6} - 52 \beta_{5} + 222 \beta_1) q^{61} + ( - 46 \beta_{4} + 10 \beta_{3} + 35 \beta_{2} - 925) q^{63} + (5 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} + 40) q^{65} + ( - 6 \beta_{7} - 4 \beta_{6} + 5 \beta_{5} + 615 \beta_1) q^{67} + (20 \beta_{7} + 6 \beta_{6} - 50 \beta_{5} - 540 \beta_1) q^{69} + (50 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 246) q^{71} + (35 \beta_{4} - 5 \beta_{3} - 19 \beta_{2} - 40) q^{73} + (25 \beta_{5} - 25 \beta_1) q^{75} + ( - 13 \beta_{7} + 37 \beta_{6} - 115 \beta_{5} + 334 \beta_1) q^{77} + ( - 84 \beta_{4} - 8 \beta_{3} - 16 \beta_{2} - 444) q^{79} + (43 \beta_{4} + 5 \beta_{3} - 43 \beta_{2} + 699) q^{81} + (22 \beta_{7} + 22 \beta_{6} + 21 \beta_{5} + 309 \beta_1) q^{83} + ( - 5 \beta_{7} + 5 \beta_{6} + 45 \beta_{5} - 120 \beta_1) q^{85} + ( - 84 \beta_{4} - 14 \beta_{3} + 6 \beta_{2} + 302) q^{87} + (22 \beta_{4} - 22 \beta_{3} + 26 \beta_{2} - 330) q^{89} + ( - 12 \beta_{7} + 2 \beta_{6} - 92 \beta_{5} + 586 \beta_1) q^{91} + (12 \beta_{7} - 16 \beta_{6} - 72 \beta_{5} + 376 \beta_1) q^{93} + (25 \beta_{4} + 10 \beta_{3} + 5 \beta_{2} - 160) q^{95} + ( - 121 \beta_{4} + 27 \beta_{3} + 9 \beta_{2} - 164) q^{97} + ( - 14 \beta_{7} + \beta_{6} + 181 \beta_{5} + 870 \beta_1) q^{99}+O(q^{100})$$ q + (-b5 + b1) * q^3 - 5*b1 * q^5 + (-b2 + 5) * q^7 + (-b4 + b3 + b2 - 27) * q^9 + (2*b7 - b6 - b5 - 18*b1) * q^11 + (b7 - b6 - b5 + 8*b1) * q^13 + (5*b4 + 5) * q^15 + (9*b4 + b3 - b2 + 24) * q^17 + (2*b7 + b6 - 5*b5 - 32*b1) * q^19 + (4*b7 - 6*b6 - 22*b5 - 12*b1) * q^21 + (-10*b4 - 2*b3 - b2 + 11) * q^23 - 25 * q^25 + (-6*b7 + 6*b6 + 28*b5 - 34*b1) * q^27 + (-2*b6 + 6*b5 + 50*b1) * q^29 + (8*b4 - 4*b3 - 2*b2 - 6) * q^31 + (23*b4 + 3*b3 + 9*b2 - 22) * q^33 + (5*b6 - 25*b1) * q^35 + (-10*b7 + 6*b6 - 2*b5 + 6*b1) * q^37 + (-14*b4 + 8*b2 - 46) * q^39 + (-3*b4 - 9*b3 + 3*b2 + 8) * q^41 + (10*b7 - 4*b6 - 3*b5 - 209*b1) * q^43 + (-5*b7 - 5*b6 - 5*b5 + 135*b1) * q^45 + (34*b4 + 4*b3 - 9*b2 - 241) * q^47 + (47*b4 - 15*b3 + b2 + 367) * q^49 + (-8*b7 - 16*b6 - 30*b5 + 486*b1) * q^51 + (-3*b7 + 3*b6 - 53*b5 + 4*b1) * q^53 + (5*b4 + 10*b3 - 5*b2 - 90) * q^55 + (71*b4 + 15*b3 + b2 - 254) * q^57 + (-10*b7 - 7*b6 - 41*b5 + 44*b1) * q^59 + (4*b7 + 4*b6 - 52*b5 + 222*b1) * q^61 + (-46*b4 + 10*b3 + 35*b2 - 925) * q^63 + (5*b4 + 5*b3 - 5*b2 + 40) * q^65 + (-6*b7 - 4*b6 + 5*b5 + 615*b1) * q^67 + (20*b7 + 6*b6 - 50*b5 - 540*b1) * q^69 + (50*b4 - 4*b3 - 4*b2 + 246) * q^71 + (35*b4 - 5*b3 - 19*b2 - 40) * q^73 + (25*b5 - 25*b1) * q^75 + (-13*b7 + 37*b6 - 115*b5 + 334*b1) * q^77 + (-84*b4 - 8*b3 - 16*b2 - 444) * q^79 + (43*b4 + 5*b3 - 43*b2 + 699) * q^81 + (22*b7 + 22*b6 + 21*b5 + 309*b1) * q^83 + (-5*b7 + 5*b6 + 45*b5 - 120*b1) * q^85 + (-84*b4 - 14*b3 + 6*b2 + 302) * q^87 + (22*b4 - 22*b3 + 26*b2 - 330) * q^89 + (-12*b7 + 2*b6 - 92*b5 + 586*b1) * q^91 + (12*b7 - 16*b6 - 72*b5 + 376*b1) * q^93 + (25*b4 + 10*b3 + 5*b2 - 160) * q^95 + (-121*b4 + 27*b3 + 9*b2 - 164) * q^97 + (-14*b7 + b6 + 181*b5 + 870*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 40 q^{7} - 216 q^{9}+O(q^{10})$$ 8 * q + 40 * q^7 - 216 * q^9 $$8 q + 40 q^{7} - 216 q^{9} + 40 q^{15} + 192 q^{17} + 88 q^{23} - 200 q^{25} - 48 q^{31} - 176 q^{33} - 368 q^{39} + 64 q^{41} - 1928 q^{47} + 2936 q^{49} - 720 q^{55} - 2032 q^{57} - 7400 q^{63} + 320 q^{65} + 1968 q^{71} - 320 q^{73} - 3552 q^{79} + 5592 q^{81} + 2416 q^{87} - 2640 q^{89} - 1280 q^{95} - 1312 q^{97}+O(q^{100})$$ 8 * q + 40 * q^7 - 216 * q^9 + 40 * q^15 + 192 * q^17 + 88 * q^23 - 200 * q^25 - 48 * q^31 - 176 * q^33 - 368 * q^39 + 64 * q^41 - 1928 * q^47 + 2936 * q^49 - 720 * q^55 - 2032 * q^57 - 7400 * q^63 + 320 * q^65 + 1968 * q^71 - 320 * q^73 - 3552 * q^79 + 5592 * q^81 + 2416 * q^87 - 2640 * q^89 - 1280 * q^95 - 1312 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 2x^{6} + 52x^{5} + 231x^{4} + 324x^{3} + 242x^{2} + 66x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( - 2601 \nu^{7} + 6014 \nu^{6} - 7455 \nu^{5} - 131867 \nu^{4} - 561497 \nu^{3} - 681745 \nu^{2} - 494252 \nu - 82437 ) / 51180$$ (-2601*v^7 + 6014*v^6 - 7455*v^5 - 131867*v^4 - 561497*v^3 - 681745*v^2 - 494252*v - 82437) / 51180 $$\beta_{2}$$ $$=$$ $$( - 2642 \nu^{7} + 7173 \nu^{6} - 8335 \nu^{5} - 141154 \nu^{4} - 491789 \nu^{3} - 414340 \nu^{2} - 116709 \nu - 248394 ) / 12795$$ (-2642*v^7 + 7173*v^6 - 8335*v^5 - 141154*v^4 - 491789*v^3 - 414340*v^2 - 116709*v - 248394) / 12795 $$\beta_{3}$$ $$=$$ $$( 613\nu^{7} - 1704\nu^{6} + 2609\nu^{5} + 28919\nu^{4} + 122167\nu^{3} + 102569\nu^{2} + 28866\nu - 51027 ) / 2559$$ (613*v^7 - 1704*v^6 + 2609*v^5 + 28919*v^4 + 122167*v^3 + 102569*v^2 + 28866*v - 51027) / 2559 $$\beta_{4}$$ $$=$$ $$( 1361\nu^{7} - 3729\nu^{6} + 4870\nu^{5} + 69027\nu^{4} + 260222\nu^{3} + 218935\nu^{2} + 61647\nu - 34913 ) / 4265$$ (1361*v^7 - 3729*v^6 + 4870*v^5 + 69027*v^4 + 260222*v^3 + 218935*v^2 + 61647*v - 34913) / 4265 $$\beta_{5}$$ $$=$$ $$( - 27301 \nu^{7} + 52424 \nu^{6} - 41405 \nu^{5} - 1447987 \nu^{4} - 6392307 \nu^{3} - 8887065 \nu^{2} - 5671742 \nu - 946347 ) / 51180$$ (-27301*v^7 + 52424*v^6 - 41405*v^5 - 1447987*v^4 - 6392307*v^3 - 8887065*v^2 - 5671742*v - 946347) / 51180 $$\beta_{6}$$ $$=$$ $$( 47481 \nu^{7} - 118984 \nu^{6} + 168705 \nu^{5} + 2350687 \nu^{4} + 9808687 \nu^{3} + 11229245 \nu^{2} + 8594182 \nu + 1433127 ) / 51180$$ (47481*v^7 - 118984*v^6 + 168705*v^5 + 2350687*v^4 + 9808687*v^3 + 11229245*v^2 + 8594182*v + 1433127) / 51180 $$\beta_{7}$$ $$=$$ $$( 28941 \nu^{7} - 64664 \nu^{6} + 76605 \nu^{5} + 1478267 \nu^{4} + 6333587 \nu^{3} + 8153905 \nu^{2} + 5582822 \nu + 931227 ) / 25590$$ (28941*v^7 - 64664*v^6 + 76605*v^5 + 1478267*v^4 + 6333587*v^3 + 8153905*v^2 + 5582822*v + 931227) / 25590
 $$\nu$$ $$=$$ $$( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 + 4 ) / 16$$ (-b7 + b6 - b5 + b4 - b3 + b2 + 4*b1 + 4) / 16 $$\nu^{2}$$ $$=$$ $$( -\beta_{7} + 3\beta_{6} - 3\beta_{5} + 64\beta_1 ) / 8$$ (-b7 + 3*b6 - 3*b5 + 64*b1) / 8 $$\nu^{3}$$ $$=$$ $$( -8\beta_{7} + 11\beta_{6} - 13\beta_{5} - 13\beta_{4} + 8\beta_{3} - 11\beta_{2} + 160\beta _1 - 160 ) / 8$$ (-8*b7 + 11*b6 - 13*b5 - 13*b4 + 8*b3 - 11*b2 + 160*b1 - 160) / 8 $$\nu^{4}$$ $$=$$ $$( -41\beta_{4} + 21\beta_{3} - 39\beta_{2} - 674 ) / 4$$ (-41*b4 + 21*b3 - 39*b2 - 674) / 4 $$\nu^{5}$$ $$=$$ $$( 349\beta_{7} - 555\beta_{6} + 623\beta_{5} - 623\beta_{4} + 349\beta_{3} - 555\beta_{2} - 8916\beta _1 - 8916 ) / 16$$ (349*b7 - 555*b6 + 623*b5 - 623*b4 + 349*b3 - 555*b2 - 8916*b1 - 8916) / 16 $$\nu^{6}$$ $$=$$ $$( 579\beta_{7} - 988\beta_{6} + 1074\beta_{5} - 16424\beta_1 ) / 4$$ (579*b7 - 988*b6 + 1074*b5 - 16424*b1) / 4 $$\nu^{7}$$ $$=$$ $$( 8523 \beta_{7} - 14059 \beta_{6} + 15523 \beta_{5} + 15523 \beta_{4} - 8523 \beta_{3} + 14059 \beta_{2} - 230052 \beta _1 + 230052 ) / 16$$ (8523*b7 - 14059*b6 + 15523*b5 + 15523*b4 - 8523*b3 + 14059*b2 - 230052*b1 + 230052) / 16

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 −0.703738 + 0.703738i −0.178675 − 0.178675i 3.55856 − 3.55856i −1.67615 − 1.67615i −1.67615 + 1.67615i 3.55856 + 3.55856i −0.178675 + 0.178675i −0.703738 − 0.703738i
0 9.86321i 0 5.00000i 0 34.2713 0 −70.2829 0
641.2 0 9.13763i 0 5.00000i 0 23.1776 0 −56.4963 0
641.3 0 5.50114i 0 5.00000i 0 −33.3372 0 −3.26254 0
641.4 0 2.22671i 0 5.00000i 0 −4.11169 0 22.0417 0
641.5 0 2.22671i 0 5.00000i 0 −4.11169 0 22.0417 0
641.6 0 5.50114i 0 5.00000i 0 −33.3372 0 −3.26254 0
641.7 0 9.13763i 0 5.00000i 0 23.1776 0 −56.4963 0
641.8 0 9.86321i 0 5.00000i 0 34.2713 0 −70.2829 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 641.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.4.d.bc 8
4.b odd 2 1 1280.4.d.bb 8
8.b even 2 1 inner 1280.4.d.bc 8
8.d odd 2 1 1280.4.d.bb 8
16.e even 4 1 640.4.a.q 4
16.e even 4 1 640.4.a.t yes 4
16.f odd 4 1 640.4.a.r yes 4
16.f odd 4 1 640.4.a.s yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.a.q 4 16.e even 4 1
640.4.a.r yes 4 16.f odd 4 1
640.4.a.s yes 4 16.f odd 4 1
640.4.a.t yes 4 16.e even 4 1
1280.4.d.bb 8 4.b odd 2 1
1280.4.d.bb 8 8.d odd 2 1
1280.4.d.bc 8 1.a even 1 1 trivial
1280.4.d.bc 8 8.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{8} + 216T_{3}^{6} + 14640T_{3}^{4} + 313216T_{3}^{2} + 1218816$$ T3^8 + 216*T3^6 + 14640*T3^4 + 313216*T3^2 + 1218816 $$T_{7}^{4} - 20T_{7}^{3} - 1220T_{7}^{2} + 21872T_{7} + 108880$$ T7^4 - 20*T7^3 - 1220*T7^2 + 21872*T7 + 108880 $$T_{11}^{8} + 9936T_{11}^{6} + 30267744T_{11}^{4} + 29743187200T_{11}^{2} + 2604066418944$$ T11^8 + 9936*T11^6 + 30267744*T11^4 + 29743187200*T11^2 + 2604066418944

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 216 T^{6} + 14640 T^{4} + \cdots + 1218816$$
$5$ $$(T^{2} + 25)^{4}$$
$7$ $$(T^{4} - 20 T^{3} - 1220 T^{2} + \cdots + 108880)^{2}$$
$11$ $$T^{8} + 9936 T^{6} + \cdots + 2604066418944$$
$13$ $$T^{8} + 4048 T^{6} + \cdots + 66663108864$$
$17$ $$(T^{4} - 96 T^{3} - 6440 T^{2} + \cdots - 3269232)^{2}$$
$19$ $$T^{8} + 22672 T^{6} + \cdots + 1356796313856$$
$23$ $$(T^{4} - 44 T^{3} - 16932 T^{2} + \cdots - 13516080)^{2}$$
$29$ $$T^{8} + \cdots + 500338160697600$$
$31$ $$(T^{4} + 24 T^{3} - 30096 T^{2} + \cdots + 6237440)^{2}$$
$37$ $$T^{8} + 235984 T^{6} + \cdots + 98\!\cdots\!00$$
$41$ $$(T^{4} - 32 T^{3} - 77736 T^{2} + \cdots + 607861648)^{2}$$
$43$ $$T^{8} + 373880 T^{6} + \cdots + 50\!\cdots\!00$$
$47$ $$(T^{4} + 964 T^{3} + \cdots - 20842229424)^{2}$$
$53$ $$T^{8} + 647952 T^{6} + \cdots + 41\!\cdots\!00$$
$59$ $$T^{8} + 714704 T^{6} + \cdots + 39\!\cdots\!24$$
$61$ $$T^{8} + 892432 T^{6} + \cdots + 11\!\cdots\!44$$
$67$ $$T^{8} + 1658360 T^{6} + \cdots + 13\!\cdots\!00$$
$71$ $$(T^{4} - 984 T^{3} + 68080 T^{2} + \cdots + 2987836672)^{2}$$
$73$ $$(T^{4} + 160 T^{3} + \cdots + 14073390480)^{2}$$
$79$ $$(T^{4} + 1776 T^{3} + \cdots - 490073812992)^{2}$$
$83$ $$T^{8} + 3140760 T^{6} + \cdots + 66\!\cdots\!00$$
$89$ $$(T^{4} + 1320 T^{3} + \cdots - 415967432304)^{2}$$
$97$ $$(T^{4} + 656 T^{3} + \cdots + 433240091664)^{2}$$