# Properties

 Label 4056.2.c.m Level $4056$ Weight $2$ Character orbit 4056.c Analytic conductor $32.387$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(337,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4056.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.c (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: $$32.3873230598$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.44836416.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 12x^{4} + 36x^{2} + 1$$ x^6 + 12*x^4 + 36*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 312) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} - \beta_{4} q^{5} + (\beta_{5} + \beta_1) q^{7} + q^{9}+O(q^{10})$$ q - q^3 - b4 * q^5 + (b5 + b1) * q^7 + q^9 $$q - q^{3} - \beta_{4} q^{5} + (\beta_{5} + \beta_1) q^{7} + q^{9} + (\beta_{5} - \beta_{4}) q^{11} + \beta_{4} q^{15} - \beta_{2} q^{17} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{19} + ( - \beta_{5} - \beta_1) q^{21} + (\beta_{3} - \beta_{2}) q^{23} + (\beta_{3} - 2 \beta_{2} - 5) q^{25} - q^{27} + (\beta_{3} - 4) q^{29} + ( - \beta_{4} - \beta_1) q^{31} + ( - \beta_{5} + \beta_{4}) q^{33} + (3 \beta_{3} + \beta_{2} + 4) q^{35} + ( - \beta_{5} + 2 \beta_1) q^{37} + (\beta_{5} - 2 \beta_{4}) q^{41} + ( - \beta_{2} + 7) q^{43} - \beta_{4} q^{45} + (\beta_{5} - \beta_{4} - 4 \beta_1) q^{47} + ( - 3 \beta_{3} - 8) q^{49} + \beta_{2} q^{51} + (3 \beta_{3} - 2) q^{53} + (3 \beta_{3} - \beta_{2} - 6) q^{55} + ( - \beta_{5} - \beta_{4} + 2 \beta_1) q^{57} + ( - 2 \beta_{5} + 2 \beta_1) q^{59} + ( - \beta_{3} - \beta_{2} + 1) q^{61} + (\beta_{5} + \beta_1) q^{63} + ( - \beta_{5} - 9 \beta_1) q^{67} + ( - \beta_{3} + \beta_{2}) q^{69} + ( - \beta_{5} + \beta_{4} - 4 \beta_1) q^{71} + ( - 2 \beta_{4} - 3 \beta_1) q^{73} + ( - \beta_{3} + 2 \beta_{2} + 5) q^{75} + (2 \beta_{2} - 10) q^{77} + ( - 3 \beta_{3} + 2 \beta_{2} + 3) q^{79} + q^{81} + ( - \beta_{5} + 3 \beta_{4} + 6 \beta_1) q^{83} + (\beta_{5} + 2 \beta_{4} + 4 \beta_1) q^{85} + ( - \beta_{3} + 4) q^{87} + ( - 2 \beta_{4} + 8 \beta_1) q^{89} + (\beta_{4} + \beta_1) q^{93} + ( - \beta_{3} + 3 \beta_{2} + 14) q^{95} + ( - \beta_{4} - 7 \beta_1) q^{97} + (\beta_{5} - \beta_{4}) q^{99}+O(q^{100})$$ q - q^3 - b4 * q^5 + (b5 + b1) * q^7 + q^9 + (b5 - b4) * q^11 + b4 * q^15 - b2 * q^17 + (b5 + b4 - 2*b1) * q^19 + (-b5 - b1) * q^21 + (b3 - b2) * q^23 + (b3 - 2*b2 - 5) * q^25 - q^27 + (b3 - 4) * q^29 + (-b4 - b1) * q^31 + (-b5 + b4) * q^33 + (3*b3 + b2 + 4) * q^35 + (-b5 + 2*b1) * q^37 + (b5 - 2*b4) * q^41 + (-b2 + 7) * q^43 - b4 * q^45 + (b5 - b4 - 4*b1) * q^47 + (-3*b3 - 8) * q^49 + b2 * q^51 + (3*b3 - 2) * q^53 + (3*b3 - b2 - 6) * q^55 + (-b5 - b4 + 2*b1) * q^57 + (-2*b5 + 2*b1) * q^59 + (-b3 - b2 + 1) * q^61 + (b5 + b1) * q^63 + (-b5 - 9*b1) * q^67 + (-b3 + b2) * q^69 + (-b5 + b4 - 4*b1) * q^71 + (-2*b4 - 3*b1) * q^73 + (-b3 + 2*b2 + 5) * q^75 + (2*b2 - 10) * q^77 + (-3*b3 + 2*b2 + 3) * q^79 + q^81 + (-b5 + 3*b4 + 6*b1) * q^83 + (b5 + 2*b4 + 4*b1) * q^85 + (-b3 + 4) * q^87 + (-2*b4 + 8*b1) * q^89 + (b4 + b1) * q^93 + (-b3 + 3*b2 + 14) * q^95 + (-b4 - 7*b1) * q^97 + (b5 - b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{3} + 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^3 + 6 * q^9 $$6 q - 6 q^{3} + 6 q^{9} - 30 q^{25} - 6 q^{27} - 24 q^{29} + 24 q^{35} + 42 q^{43} - 48 q^{49} - 12 q^{53} - 36 q^{55} + 6 q^{61} + 30 q^{75} - 60 q^{77} + 18 q^{79} + 6 q^{81} + 24 q^{87} + 84 q^{95}+O(q^{100})$$ 6 * q - 6 * q^3 + 6 * q^9 - 30 * q^25 - 6 * q^27 - 24 * q^29 + 24 * q^35 + 42 * q^43 - 48 * q^49 - 12 * q^53 - 36 * q^55 + 6 * q^61 + 30 * q^75 - 60 * q^77 + 18 * q^79 + 6 * q^81 + 24 * q^87 + 84 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 12x^{4} + 36x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 6\nu$$ v^3 + 6*v $$\beta_{2}$$ $$=$$ $$-\nu^{4} - 5\nu^{2} + 4$$ -v^4 - 5*v^2 + 4 $$\beta_{3}$$ $$=$$ $$\nu^{4} + 7\nu^{2} + 4$$ v^4 + 7*v^2 + 4 $$\beta_{4}$$ $$=$$ $$\nu^{5} + 10\nu^{3} + 23\nu$$ v^5 + 10*v^3 + 23*v $$\beta_{5}$$ $$=$$ $$\nu^{5} + 10\nu^{3} + 25\nu$$ v^5 + 10*v^3 + 25*v
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} ) / 2$$ (b5 - b4) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} - 8 ) / 2$$ (b3 + b2 - 8) / 2 $$\nu^{3}$$ $$=$$ $$-3\beta_{5} + 3\beta_{4} + \beta_1$$ -3*b5 + 3*b4 + b1 $$\nu^{4}$$ $$=$$ $$( -5\beta_{3} - 7\beta_{2} + 48 ) / 2$$ (-5*b3 - 7*b2 + 48) / 2 $$\nu^{5}$$ $$=$$ $$( 37\beta_{5} - 35\beta_{4} - 20\beta_1 ) / 2$$ (37*b5 - 35*b4 - 20*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$$.

 $$n$$ $$1015$$ $$2029$$ $$2705$$ $$3889$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 2.36147i 0.167449i − 2.52892i 2.52892i − 0.167449i 2.36147i
0 −1.00000 0 3.93800i 0 1.78493i 0 1.00000 0
337.2 0 −1.00000 0 3.80451i 0 5.13941i 0 1.00000 0
337.3 0 −1.00000 0 0.133492i 0 3.92434i 0 1.00000 0
337.4 0 −1.00000 0 0.133492i 0 3.92434i 0 1.00000 0
337.5 0 −1.00000 0 3.80451i 0 5.13941i 0 1.00000 0
337.6 0 −1.00000 0 3.93800i 0 1.78493i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4056.2.c.m 6
13.b even 2 1 inner 4056.2.c.m 6
13.d odd 4 1 4056.2.a.y 3
13.d odd 4 1 4056.2.a.z 3
13.f odd 12 2 312.2.q.e 6
39.k even 12 2 936.2.t.h 6
52.f even 4 1 8112.2.a.ck 3
52.f even 4 1 8112.2.a.cl 3
52.l even 12 2 624.2.q.j 6
156.v odd 12 2 1872.2.t.u 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.e 6 13.f odd 12 2
624.2.q.j 6 52.l even 12 2
936.2.t.h 6 39.k even 12 2
1872.2.t.u 6 156.v odd 12 2
4056.2.a.y 3 13.d odd 4 1
4056.2.a.z 3 13.d odd 4 1
4056.2.c.m 6 1.a even 1 1 trivial
4056.2.c.m 6 13.b even 2 1 inner
8112.2.a.ck 3 52.f even 4 1
8112.2.a.cl 3 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4056, [\chi])$$:

 $$T_{5}^{6} + 30T_{5}^{4} + 225T_{5}^{2} + 4$$ T5^6 + 30*T5^4 + 225*T5^2 + 4 $$T_{7}^{6} + 45T_{7}^{4} + 540T_{7}^{2} + 1296$$ T7^6 + 45*T7^4 + 540*T7^2 + 1296 $$T_{11}^{6} + 48T_{11}^{4} + 576T_{11}^{2} + 64$$ T11^6 + 48*T11^4 + 576*T11^2 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T + 1)^{6}$$
$5$ $$T^{6} + 30 T^{4} + 225 T^{2} + 4$$
$7$ $$T^{6} + 45 T^{4} + 540 T^{2} + \cdots + 1296$$
$11$ $$T^{6} + 48 T^{4} + 576 T^{2} + \cdots + 64$$
$13$ $$T^{6}$$
$17$ $$(T^{3} - 21 T - 16)^{2}$$
$19$ $$T^{6} + 108 T^{4} + 3792 T^{2} + \cdots + 43264$$
$23$ $$(T^{3} - 24 T - 8)^{2}$$
$29$ $$(T^{3} + 12 T^{2} + 33 T + 6)^{2}$$
$31$ $$T^{6} + 33 T^{4} + 240 T^{2} + \cdots + 256$$
$37$ $$T^{6} + 54 T^{4} + 297 T^{2} + \cdots + 324$$
$41$ $$T^{6} + 114 T^{4} + 3249 T^{2} + \cdots + 24336$$
$43$ $$(T^{3} - 21 T^{2} + 126 T - 212)^{2}$$
$47$ $$T^{6} + 96 T^{4} + 1152 T^{2} + \cdots + 576$$
$53$ $$(T^{3} + 6 T^{2} - 123 T - 208)^{2}$$
$59$ $$T^{6} + 180 T^{4} + 5568 T^{2} + \cdots + 1024$$
$61$ $$(T^{3} - 3 T^{2} - 45 T + 167)^{2}$$
$67$ $$T^{6} + 285 T^{4} + 20988 T^{2} + \cdots + 274576$$
$71$ $$T^{6} + 96 T^{4} + 1536 T^{2} + \cdots + 1600$$
$73$ $$T^{6} + 147 T^{4} + 4131 T^{2} + \cdots + 28561$$
$79$ $$(T^{3} - 9 T^{2} - 120 T - 16)^{2}$$
$83$ $$T^{6} + 348 T^{4} + 35856 T^{2} + \cdots + 984064$$
$89$ $$T^{6} + 312 T^{4} + 15120 T^{2} + \cdots + 2304$$
$97$ $$T^{6} + 177 T^{4} + 7512 T^{2} + \cdots + 55696$$