Properties

 Label 3960.2.d.i Level $3960$ Weight $2$ Character orbit 3960.d Analytic conductor $31.621$ Analytic rank $0$ Dimension $14$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3960,2,Mod(3169,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3960.3169");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3960.d (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: $$31.6207592004$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{14} + 24x^{12} + 226x^{10} + 1052x^{8} + 2497x^{6} + 2788x^{4} + 1156x^{2} + 64$$ x^14 + 24*x^12 + 226*x^10 + 1052*x^8 + 2497*x^6 + 2788*x^4 + 1156*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{13}$$ 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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{10} q^{5} + \beta_{5} q^{7}+O(q^{10})$$ q - b10 * q^5 + b5 * q^7 $$q - \beta_{10} q^{5} + \beta_{5} q^{7} - q^{11} + ( - \beta_{5} + \beta_1) q^{13} + (\beta_{5} - \beta_{3} + \beta_{2}) q^{17} + ( - \beta_{11} - \beta_{4}) q^{19} + (\beta_{2} + \beta_1) q^{23} + (\beta_{12} - \beta_{11}) q^{25} + (\beta_{13} - \beta_{12} + \cdots + \beta_{7}) q^{29}+ \cdots + ( - \beta_{13} - \beta_{12} + \cdots - \beta_1) q^{97}+O(q^{100})$$ q - b10 * q^5 + b5 * q^7 - q^11 + (-b5 + b1) * q^13 + (b5 - b3 + b2) * q^17 + (-b11 - b4) * q^19 + (b2 + b1) * q^23 + (b12 - b11) * q^25 + (b13 - b12 + b11 + b10 - b9 + b7) * q^29 + (-b13 + b12 - b10 + b9 - b7 - b4) * q^31 + (-b12 + b7 + b5 + b4 - b1 + 1) * q^35 + (b13 + b12 - b6 - b5 + 2*b3 + b1) * q^37 + (b11 + b10 - b9 + b7 - 2) * q^41 + (b6 - b2) * q^43 + (-b6 - b5 + b2) * q^47 + (-b11 + 2*b10 - 2*b9 + b8 + b7 - b4) * q^49 + (b13 + b12 + b3 + b2) * q^53 + b10 * q^55 + (-b13 + b12 - b11 + b8 + b7 - b4 - 1) * q^59 + (b13 - b12 + 2*b11 - 2*b8 + b7 + b4) * q^61 + (b12 - b11 + b10 + b9 + b8 - b6 - 2*b5 - b4 + b3 + b2 + b1) * q^65 + (b10 + b9 + b3 + b2) * q^67 + (-b11 - b8 + 2*b7 + 1) * q^71 + (-2*b10 - 2*b9 + b6 + 2*b5 - 2*b3 + b2 + b1) * q^73 - b5 * q^77 + (-b13 + b12 - b11 + 2*b8 - b7 - 2*b4 - 2) * q^79 + (-b13 - b12 - b5 + b2) * q^83 + (b13 - b12 + b11 + b5 + 2*b4 - 2*b3 + b2) * q^85 + (-b13 + b12 - 2*b11 - b10 + b9 + b7 - b4 + 4) * q^89 + (-3*b10 + 3*b9 - 2*b8 + b7 + b4 + 4) * q^91 + (-b13 + b12 + b8 + b6 + b5 + b2 + 1) * q^95 + (-b13 - b12 - 2*b10 - 2*b9 + b6 - b5 + 2*b3 - 2*b2 - b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q+O(q^{10})$$ 14 * q $$14 q - 14 q^{11} - 2 q^{25} + 12 q^{35} - 32 q^{41} - 6 q^{49} - 24 q^{59} + 4 q^{61} - 4 q^{65} + 8 q^{71} - 32 q^{79} + 4 q^{85} + 48 q^{89} + 56 q^{91} + 8 q^{95}+O(q^{100})$$ 14 * q - 14 * q^11 - 2 * q^25 + 12 * q^35 - 32 * q^41 - 6 * q^49 - 24 * q^59 + 4 * q^61 - 4 * q^65 + 8 * q^71 - 32 * q^79 + 4 * q^85 + 48 * q^89 + 56 * q^91 + 8 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 24x^{12} + 226x^{10} + 1052x^{8} + 2497x^{6} + 2788x^{4} + 1156x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 5\nu$$ v^3 + 5*v $$\beta_{3}$$ $$=$$ $$( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 34\nu ) / 4$$ (v^7 + 12*v^5 + 41*v^3 + 34*v) / 4 $$\beta_{4}$$ $$=$$ $$( \nu^{8} + 12\nu^{6} + 41\nu^{4} + 34\nu^{2} ) / 4$$ (v^8 + 12*v^6 + 41*v^4 + 34*v^2) / 4 $$\beta_{5}$$ $$=$$ $$( \nu^{9} + 16\nu^{7} + 85\nu^{5} + 162\nu^{3} + 72\nu ) / 8$$ (v^9 + 16*v^7 + 85*v^5 + 162*v^3 + 72*v) / 8 $$\beta_{6}$$ $$=$$ $$( \nu^{9} + 16\nu^{7} + 93\nu^{5} + 234\nu^{3} + 200\nu ) / 8$$ (v^9 + 16*v^7 + 93*v^5 + 234*v^3 + 200*v) / 8 $$\beta_{7}$$ $$=$$ $$( \nu^{10} + 17\nu^{8} + 101\nu^{6} + 247\nu^{4} + 226\nu^{2} + 48 ) / 8$$ (v^10 + 17*v^8 + 101*v^6 + 247*v^4 + 226*v^2 + 48) / 8 $$\beta_{8}$$ $$=$$ $$( -\nu^{12} - 19\nu^{10} - 129\nu^{8} - 361\nu^{6} - 314\nu^{4} + 88\nu^{2} + 16 ) / 16$$ (-v^12 - 19*v^10 - 129*v^8 - 361*v^6 - 314*v^4 + 88*v^2 + 16) / 16 $$\beta_{9}$$ $$=$$ $$( \nu^{11} + \nu^{10} + 19 \nu^{9} + 17 \nu^{8} + 131 \nu^{7} + 101 \nu^{6} + 393 \nu^{5} + \cdots + 164 \nu ) / 16$$ (v^11 + v^10 + 19*v^9 + 17*v^8 + 131*v^7 + 101*v^6 + 393*v^5 + 239*v^4 + 476*v^3 + 170*v^2 + 164*v) / 16 $$\beta_{10}$$ $$=$$ $$( \nu^{11} - \nu^{10} + 19 \nu^{9} - 17 \nu^{8} + 131 \nu^{7} - 101 \nu^{6} + 393 \nu^{5} + \cdots + 164 \nu ) / 16$$ (v^11 - v^10 + 19*v^9 - 17*v^8 + 131*v^7 - 101*v^6 + 393*v^5 - 239*v^4 + 476*v^3 - 170*v^2 + 164*v) / 16 $$\beta_{11}$$ $$=$$ $$( \nu^{12} + 21\nu^{10} + 167\nu^{8} + 627\nu^{6} + 1132\nu^{4} + 868\nu^{2} + 128 ) / 16$$ (v^12 + 21*v^10 + 167*v^8 + 627*v^6 + 1132*v^4 + 868*v^2 + 128) / 16 $$\beta_{12}$$ $$=$$ $$( \nu^{13} + \nu^{12} + 22 \nu^{11} + 21 \nu^{10} + 186 \nu^{9} + 167 \nu^{8} + 752 \nu^{7} + 619 \nu^{6} + \cdots + 48 ) / 16$$ (v^13 + v^12 + 22*v^11 + 21*v^10 + 186*v^9 + 167*v^8 + 752*v^7 + 619*v^6 + 1461*v^5 + 1052*v^4 + 1178*v^3 + 668*v^2 + 240*v + 48) / 16 $$\beta_{13}$$ $$=$$ $$( \nu^{13} - \nu^{12} + 22 \nu^{11} - 21 \nu^{10} + 186 \nu^{9} - 167 \nu^{8} + 752 \nu^{7} - 619 \nu^{6} + \cdots - 48 ) / 16$$ (v^13 - v^12 + 22*v^11 - 21*v^10 + 186*v^9 - 167*v^8 + 752*v^7 - 619*v^6 + 1461*v^5 - 1052*v^4 + 1178*v^3 - 668*v^2 + 240*v - 48) / 16
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{13} - \beta_{12} + \beta_{11} - \beta_{8} + \beta_{7} + \beta_{4} - 7 ) / 2$$ (b13 - b12 + b11 - b8 + b7 + b4 - 7) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{2} - 5\beta_1 ) / 2$$ (2*b2 - 5*b1) / 2 $$\nu^{4}$$ $$=$$ $$( -7\beta_{13} + 7\beta_{12} - 7\beta_{11} + 2\beta_{10} - 2\beta_{9} + 7\beta_{8} - 5\beta_{7} - 7\beta_{4} + 37 ) / 2$$ (-7*b13 + 7*b12 - 7*b11 + 2*b10 - 2*b9 + 7*b8 - 5*b7 - 7*b4 + 37) / 2 $$\nu^{5}$$ $$=$$ $$( 2\beta_{6} - 2\beta_{5} - 18\beta_{2} + 29\beta_1 ) / 2$$ (2*b6 - 2*b5 - 18*b2 + 29*b1) / 2 $$\nu^{6}$$ $$=$$ $$( 47 \beta_{13} - 47 \beta_{12} + 49 \beta_{11} - 20 \beta_{10} + 20 \beta_{9} - 45 \beta_{8} + 25 \beta_{7} + \cdots - 215 ) / 2$$ (47*b13 - 47*b12 + 49*b11 - 20*b10 + 20*b9 - 45*b8 + 25*b7 + 45*b4 - 215) / 2 $$\nu^{7}$$ $$=$$ $$( -24\beta_{6} + 24\beta_{5} + 8\beta_{3} + 134\beta_{2} - 177\beta_1 ) / 2$$ (-24*b6 + 24*b5 + 8*b3 + 134*b2 - 177*b1) / 2 $$\nu^{8}$$ $$=$$ $$( - 311 \beta_{13} + 311 \beta_{12} - 335 \beta_{11} + 158 \beta_{10} - 158 \beta_{9} + 287 \beta_{8} + \cdots + 1301 ) / 2$$ (-311*b13 + 311*b12 - 335*b11 + 158*b10 - 158*b9 + 287*b8 - 129*b7 - 279*b4 + 1301) / 2 $$\nu^{9}$$ $$=$$ $$( 214\beta_{6} - 198\beta_{5} - 128\beta_{3} - 938\beta_{2} + 1105\beta_1 ) / 2$$ (214*b6 - 198*b5 - 128*b3 - 938*b2 + 1105*b1) / 2 $$\nu^{10}$$ $$=$$ $$( 2043 \beta_{13} - 2043 \beta_{12} + 2249 \beta_{11} - 1160 \beta_{10} + 1160 \beta_{9} - 1837 \beta_{8} + \cdots - 8055 ) / 2$$ (2043*b13 - 2043*b12 + 2249*b11 - 1160*b10 + 1160*b9 - 1837*b8 + 693*b7 + 1701*b4 - 8055) / 2 $$\nu^{11}$$ $$=$$ $$( 16\beta_{10} + 16\beta_{9} - 1708\beta_{6} + 1404\beta_{5} + 1384\beta_{3} + 6390\beta_{2} - 6989\beta_1 ) / 2$$ (16*b10 + 16*b9 - 1708*b6 + 1404*b5 + 1384*b3 + 6390*b2 - 6989*b1) / 2 $$\nu^{12}$$ $$=$$ $$( - 13379 \beta_{13} + 13379 \beta_{12} - 14919 \beta_{11} + 8250 \beta_{10} - 8250 \beta_{9} + \cdots + 50629 ) / 2$$ (-13379*b13 + 13379*b12 - 14919*b11 + 8250*b10 - 8250*b9 + 11807*b8 - 3893*b7 - 10287*b4 + 50629) / 2 $$\nu^{13}$$ $$=$$ $$( 16 \beta_{13} + 16 \beta_{12} - 352 \beta_{10} - 352 \beta_{9} + 12898 \beta_{6} - 9186 \beta_{5} + \cdots + 44613 \beta_1 ) / 2$$ (16*b13 + 16*b12 - 352*b10 - 352*b9 + 12898*b6 - 9186*b5 - 12656*b3 - 42938*b2 + 44613*b1) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times$$.

 $$n$$ $$991$$ $$1981$$ $$2377$$ $$2521$$ $$3521$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3169.1
 2.59233i − 2.59233i − 0.876610i 0.876610i 0.254885i − 0.254885i − 2.21557i 2.21557i 1.91990i − 1.91990i 2.47637i − 2.47637i − 1.31120i 1.31120i
0 0 0 −2.22960 0.169899i 0 1.83994i 0 0 0
3169.2 0 0 0 −2.22960 + 0.169899i 0 1.83994i 0 0 0
3169.3 0 0 0 −1.85471 1.24902i 0 1.00883i 0 0 0
3169.4 0 0 0 −1.85471 + 1.24902i 0 1.00883i 0 0 0
3169.5 0 0 0 −0.628932 2.14580i 0 1.96993i 0 0 0
3169.6 0 0 0 −0.628932 + 2.14580i 0 1.96993i 0 0 0
3169.7 0 0 0 −0.101079 2.23378i 0 3.61424i 0 0 0
3169.8 0 0 0 −0.101079 + 2.23378i 0 3.61424i 0 0 0
3169.9 0 0 0 1.26135 1.84634i 0 3.12979i 0 0 0
3169.10 0 0 0 1.26135 + 1.84634i 0 3.12979i 0 0 0
3169.11 0 0 0 1.40211 1.74186i 0 0.169515i 0 0 0
3169.12 0 0 0 1.40211 + 1.74186i 0 0.169515i 0 0 0
3169.13 0 0 0 2.15086 0.611385i 0 4.56390i 0 0 0
3169.14 0 0 0 2.15086 + 0.611385i 0 4.56390i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3169.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3960.2.d.i 14
3.b odd 2 1 3960.2.d.j yes 14
5.b even 2 1 inner 3960.2.d.i 14
15.d odd 2 1 3960.2.d.j yes 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3960.2.d.i 14 1.a even 1 1 trivial
3960.2.d.i 14 5.b even 2 1 inner
3960.2.d.j yes 14 3.b odd 2 1
3960.2.d.j yes 14 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3960, [\chi])$$:

 $$T_{7}^{14} + 52T_{7}^{12} + 988T_{7}^{10} + 8608T_{7}^{8} + 35312T_{7}^{6} + 63808T_{7}^{4} + 37440T_{7}^{2} + 1024$$ T7^14 + 52*T7^12 + 988*T7^10 + 8608*T7^8 + 35312*T7^6 + 63808*T7^4 + 37440*T7^2 + 1024 $$T_{29}^{7} - 116T_{29}^{5} + 80T_{29}^{4} + 2528T_{29}^{3} - 6016T_{29}^{2} + 3456T_{29} - 512$$ T29^7 - 116*T29^5 + 80*T29^4 + 2528*T29^3 - 6016*T29^2 + 3456*T29 - 512

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14}$$
$3$ $$T^{14}$$
$5$ $$T^{14} + T^{12} + \cdots + 78125$$
$7$ $$T^{14} + 52 T^{12} + \cdots + 1024$$
$11$ $$(T + 1)^{14}$$
$13$ $$T^{14} + 100 T^{12} + \cdots + 82944$$
$17$ $$T^{14} + 92 T^{12} + \cdots + 4096$$
$19$ $$(T^{7} - 68 T^{5} + \cdots + 256)^{2}$$
$23$ $$T^{14} + 148 T^{12} + \cdots + 9437184$$
$29$ $$(T^{7} - 116 T^{5} + \cdots - 512)^{2}$$
$31$ $$(T^{7} - 128 T^{5} + \cdots + 65536)^{2}$$
$37$ $$T^{14} + \cdots + 17994612736$$
$41$ $$(T^{7} + 16 T^{6} + \cdots + 6144)^{2}$$
$43$ $$T^{14} + \cdots + 764411904$$
$47$ $$T^{14} + \cdots + 220463104$$
$53$ $$T^{14} + \cdots + 19323224064$$
$59$ $$(T^{7} + 12 T^{6} + \cdots - 52736)^{2}$$
$61$ $$(T^{7} - 2 T^{6} + \cdots + 1157056)^{2}$$
$67$ $$T^{14} + 136 T^{12} + \cdots + 16777216$$
$71$ $$(T^{7} - 4 T^{6} + \cdots + 128768)^{2}$$
$73$ $$T^{14} + \cdots + 212572635136$$
$79$ $$(T^{7} + 16 T^{6} + \cdots + 7424)^{2}$$
$83$ $$T^{14} + \cdots + 140847087616$$
$89$ $$(T^{7} - 24 T^{6} + \cdots + 65536)^{2}$$
$97$ $$T^{14} + 904 T^{12} + \cdots + 16777216$$