# Properties

 Label 4400.2.b.bb Level $4400$ Weight $2$ Character orbit 4400.b Analytic conductor $35.134$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (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: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.96668224.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 15x^{4} + 61x^{2} + 36$$ x^6 + 15*x^4 + 61*x^2 + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2200) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{3} - 2) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b4 - b2) * q^7 + (b3 - 2) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{3} - 2) q^{9} - q^{11} - \beta_{2} q^{13} + ( - 2 \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{5} + 3) q^{19} + (3 \beta_{5} - \beta_{3} - 1) q^{21} + (3 \beta_{2} + \beta_1) q^{23} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{27} + (2 \beta_{5} - \beta_{3} - 4) q^{29} + ( - 2 \beta_{3} + 1) q^{31} - \beta_1 q^{33} + ( - \beta_{4} - 3 \beta_{2} + \beta_1) q^{37} + \beta_{5} q^{39} + (\beta_{5} + \beta_{3} + 1) q^{41} + ( - \beta_{4} + 2 \beta_{2} + 3 \beta_1) q^{43} + (\beta_{4} - \beta_{2} + 3 \beta_1) q^{47} + ( - \beta_{5} - 3) q^{49} + (2 \beta_{5} + \beta_{3} - 5) q^{51} + ( - 3 \beta_{4} - \beta_{2} - 2 \beta_1) q^{53} + ( - 2 \beta_{4} - 10 \beta_{2} + 3 \beta_1) q^{57} + ( - 3 \beta_{5} + \beta_{3} + 1) q^{59} + (2 \beta_{5} - \beta_{3} - 1) q^{61} + (\beta_{4} + 11 \beta_{2} + \beta_1) q^{63} + (2 \beta_{4} + 4 \beta_{2} + 2 \beta_1) q^{67} + ( - 3 \beta_{5} + \beta_{3} - 5) q^{69} + (\beta_{5} + \beta_{3} - 6) q^{71} + (3 \beta_{4} + 7 \beta_{2}) q^{73} + (\beta_{4} + \beta_{2}) q^{77} + ( - \beta_{5} + 2 \beta_{3} + 4) q^{79} + (\beta_{5} + \beta_{3} - 2) q^{81} + ( - \beta_{4} + 8 \beta_{2} - 4 \beta_1) q^{83} + (3 \beta_{4} + 9 \beta_{2} - 2 \beta_1) q^{87} + ( - \beta_{3} - 10) q^{89} + (\beta_{3} - 1) q^{91} + (2 \beta_{4} - 2 \beta_{2} + 5 \beta_1) q^{93} + (2 \beta_{4} - 3 \beta_{2} - \beta_1) q^{97} + ( - \beta_{3} + 2) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b4 - b2) * q^7 + (b3 - 2) * q^9 - q^11 - b2 * q^13 + (-2*b2 + b1) * q^17 + (-2*b5 + 3) * q^19 + (3*b5 - b3 - 1) * q^21 + (3*b2 + b1) * q^23 + (-b4 + b2 - b1) * q^27 + (2*b5 - b3 - 4) * q^29 + (-2*b3 + 1) * q^31 - b1 * q^33 + (-b4 - 3*b2 + b1) * q^37 + b5 * q^39 + (b5 + b3 + 1) * q^41 + (-b4 + 2*b2 + 3*b1) * q^43 + (b4 - b2 + 3*b1) * q^47 + (-b5 - 3) * q^49 + (2*b5 + b3 - 5) * q^51 + (-3*b4 - b2 - 2*b1) * q^53 + (-2*b4 - 10*b2 + 3*b1) * q^57 + (-3*b5 + b3 + 1) * q^59 + (2*b5 - b3 - 1) * q^61 + (b4 + 11*b2 + b1) * q^63 + (2*b4 + 4*b2 + 2*b1) * q^67 + (-3*b5 + b3 - 5) * q^69 + (b5 + b3 - 6) * q^71 + (3*b4 + 7*b2) * q^73 + (b4 + b2) * q^77 + (-b5 + 2*b3 + 4) * q^79 + (b5 + b3 - 2) * q^81 + (-b4 + 8*b2 - 4*b1) * q^83 + (3*b4 + 9*b2 - 2*b1) * q^87 + (-b3 - 10) * q^89 + (b3 - 1) * q^91 + (2*b4 - 2*b2 + 5*b1) * q^93 + (2*b4 - 3*b2 - b1) * q^97 + (-b3 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{9}+O(q^{10})$$ 6 * q - 12 * q^9 $$6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99}+O(q^{100})$$ 6 * q - 12 * q^9 - 6 * q^11 + 14 * q^19 - 20 * q^29 + 6 * q^31 + 2 * q^39 + 8 * q^41 - 20 * q^49 - 26 * q^51 - 2 * q^61 - 36 * q^69 - 34 * q^71 + 22 * q^79 - 10 * q^81 - 60 * q^89 - 6 * q^91 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 15x^{4} + 61x^{2} + 36$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 9\nu^{3} + 13\nu ) / 6$$ (v^5 + 9*v^3 + 13*v) / 6 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 3\nu^{3} - 29\nu ) / 6$$ (v^5 + 3*v^3 - 29*v) / 6 $$\beta_{5}$$ $$=$$ $$\nu^{4} + 8\nu^{2} + 6$$ v^4 + 8*v^2 + 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{2} - 7\beta_1$$ -b4 + b2 - 7*b1 $$\nu^{4}$$ $$=$$ $$\beta_{5} - 8\beta_{3} + 34$$ b5 - 8*b3 + 34 $$\nu^{5}$$ $$=$$ $$9\beta_{4} - 3\beta_{2} + 50\beta_1$$ 9*b4 - 3*b2 + 50*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4049.1
 − 2.75153i − 2.59261i − 0.841083i 0.841083i 2.59261i 2.75153i
0 2.75153i 0 0 0 3.57093i 0 −4.57093 0
4049.2 0 2.59261i 0 0 0 2.72165i 0 −3.72165 0
4049.3 0 0.841083i 0 0 0 3.29258i 0 2.29258 0
4049.4 0 0.841083i 0 0 0 3.29258i 0 2.29258 0
4049.5 0 2.59261i 0 0 0 2.72165i 0 −3.72165 0
4049.6 0 2.75153i 0 0 0 3.57093i 0 −4.57093 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4049.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 4400.2.b.bb 6
4.b odd 2 1 2200.2.b.m 6
5.b even 2 1 inner 4400.2.b.bb 6
5.c odd 4 1 4400.2.a.by 3
5.c odd 4 1 4400.2.a.bz 3
20.d odd 2 1 2200.2.b.m 6
20.e even 4 1 2200.2.a.u 3
20.e even 4 1 2200.2.a.v yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.u 3 20.e even 4 1
2200.2.a.v yes 3 20.e even 4 1
2200.2.b.m 6 4.b odd 2 1
2200.2.b.m 6 20.d odd 2 1
4400.2.a.by 3 5.c odd 4 1
4400.2.a.bz 3 5.c odd 4 1
4400.2.b.bb 6 1.a even 1 1 trivial
4400.2.b.bb 6 5.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 $$S_{2}^{\mathrm{new}}(4400, [\chi])$$:

 $$T_{3}^{6} + 15T_{3}^{4} + 61T_{3}^{2} + 36$$ T3^6 + 15*T3^4 + 61*T3^2 + 36 $$T_{7}^{6} + 31T_{7}^{4} + 313T_{7}^{2} + 1024$$ T7^6 + 31*T7^4 + 313*T7^2 + 1024 $$T_{13}^{2} + 1$$ T13^2 + 1 $$T_{17}^{6} + 23T_{17}^{4} + 41T_{17}^{2} + 16$$ T17^6 + 23*T17^4 + 41*T17^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 15 T^{4} + \cdots + 36$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 31 T^{4} + \cdots + 1024$$
$11$ $$(T + 1)^{6}$$
$13$ $$(T^{2} + 1)^{3}$$
$17$ $$T^{6} + 23 T^{4} + \cdots + 16$$
$19$ $$(T^{3} - 7 T^{2} - 13 T + 27)^{2}$$
$23$ $$T^{6} + 48 T^{4} + \cdots + 81$$
$29$ $$(T^{3} + 10 T^{2} + \cdots - 201)^{2}$$
$31$ $$(T^{3} - 3 T^{2} + \cdots + 207)^{2}$$
$37$ $$T^{6} + 66 T^{4} + \cdots + 1936$$
$41$ $$(T^{3} - 4 T^{2} - 17 T + 24)^{2}$$
$43$ $$T^{6} + 193 T^{4} + \cdots + 258064$$
$47$ $$T^{6} + 154 T^{4} + \cdots + 36864$$
$53$ $$T^{6} + 307 T^{4} + \cdots + 20164$$
$59$ $$(T^{3} - 77 T + 192)^{2}$$
$61$ $$(T^{3} + T^{2} - 41 T - 114)^{2}$$
$67$ $$T^{6} + 228 T^{4} + \cdots + 9216$$
$71$ $$(T^{3} + 17 T^{2} + \cdots + 52)^{2}$$
$73$ $$T^{6} + 399 T^{4} + \cdots + 1106704$$
$79$ $$(T^{3} - 11 T^{2} + \cdots + 144)^{2}$$
$83$ $$T^{6} + 388 T^{4} + \cdots + 687241$$
$89$ $$(T^{3} + 30 T^{2} + \cdots + 879)^{2}$$
$97$ $$T^{6} + 164 T^{4} + \cdots + 529$$