# Properties

 Label 4400.2.b.y Level $4400$ Weight $2$ Character orbit 4400.b Analytic conductor $35.134$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (3 \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + (3*b2 + b1) * q^7 + (b3 - 1) * q^9 $$q + \beta_1 q^{3} + (3 \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 1) q^{9} + q^{11} + 5 \beta_{2} q^{13} + 3 \beta_1 q^{17} - q^{19} + ( - 2 \beta_{3} - 1) q^{21} + (5 \beta_{2} - \beta_1) q^{23} + (3 \beta_{2} + 2 \beta_1) q^{27} + (3 \beta_{3} + 3) q^{29} + (4 \beta_{3} - 5) q^{31} + \beta_1 q^{33} + ( - 7 \beta_{2} - 2 \beta_1) q^{37} + ( - 5 \beta_{3} + 5) q^{39} + (2 \beta_{3} - 3) q^{41} + (2 \beta_{2} + 4 \beta_1) q^{43} - 3 \beta_{2} q^{47} - 5 \beta_{3} q^{49} + (3 \beta_{3} - 12) q^{51} + (\beta_{2} + \beta_1) q^{53} - \beta_1 q^{57} + (4 \beta_{3} - 9) q^{59} + (3 \beta_{3} - 4) q^{61} + (3 \beta_{2} + 2 \beta_1) q^{63} - 4 \beta_{2} q^{67} + ( - 6 \beta_{3} + 9) q^{69} - 2 \beta_{3} q^{71} + ( - \beta_{2} + 3 \beta_1) q^{73} + (3 \beta_{2} + \beta_1) q^{77} + (3 \beta_{3} - 7) q^{79} + (2 \beta_{3} - 8) q^{81} + ( - 8 \beta_{2} - 5 \beta_1) q^{83} + (9 \beta_{2} + 3 \beta_1) q^{87} + ( - \beta_{3} - 3) q^{89} + ( - 5 \beta_{3} - 10) q^{91} + (12 \beta_{2} - 5 \beta_1) q^{93} + ( - 13 \beta_{2} + \beta_1) q^{97} + (\beta_{3} - 1) q^{99}+O(q^{100})$$ q + b1 * q^3 + (3*b2 + b1) * q^7 + (b3 - 1) * q^9 + q^11 + 5*b2 * q^13 + 3*b1 * q^17 - q^19 + (-2*b3 - 1) * q^21 + (5*b2 - b1) * q^23 + (3*b2 + 2*b1) * q^27 + (3*b3 + 3) * q^29 + (4*b3 - 5) * q^31 + b1 * q^33 + (-7*b2 - 2*b1) * q^37 + (-5*b3 + 5) * q^39 + (2*b3 - 3) * q^41 + (2*b2 + 4*b1) * q^43 - 3*b2 * q^47 - 5*b3 * q^49 + (3*b3 - 12) * q^51 + (b2 + b1) * q^53 - b1 * q^57 + (4*b3 - 9) * q^59 + (3*b3 - 4) * q^61 + (3*b2 + 2*b1) * q^63 - 4*b2 * q^67 + (-6*b3 + 9) * q^69 - 2*b3 * q^71 + (-b2 + 3*b1) * q^73 + (3*b2 + b1) * q^77 + (3*b3 - 7) * q^79 + (2*b3 - 8) * q^81 + (-8*b2 - 5*b1) * q^83 + (9*b2 + 3*b1) * q^87 + (-b3 - 3) * q^89 + (-5*b3 - 10) * q^91 + (12*b2 - 5*b1) * q^93 + (-13*b2 + b1) * q^97 + (b3 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^9 $$4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91} - 2 q^{99}+O(q^{100})$$ 4 * q - 2 * q^9 + 4 * q^11 - 4 * q^19 - 8 * q^21 + 18 * q^29 - 12 * q^31 + 10 * q^39 - 8 * q^41 - 10 * q^49 - 42 * q^51 - 28 * q^59 - 10 * q^61 + 24 * q^69 - 4 * q^71 - 22 * q^79 - 28 * q^81 - 14 * q^89 - 50 * q^91 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 3$$ (v^3 + 4*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ b3 - 4 $$\nu^{3}$$ $$=$$ $$3\beta_{2} - 4\beta_1$$ 3*b2 - 4*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
 − 2.30278i − 1.30278i 1.30278i 2.30278i
0 2.30278i 0 0 0 0.697224i 0 −2.30278 0
4049.2 0 1.30278i 0 0 0 4.30278i 0 1.30278 0
4049.3 0 1.30278i 0 0 0 4.30278i 0 1.30278 0
4049.4 0 2.30278i 0 0 0 0.697224i 0 −2.30278 0
 $$n$$: e.g. 2-40 or 990-1000 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.y 4
4.b odd 2 1 275.2.b.c 4
5.b even 2 1 inner 4400.2.b.y 4
5.c odd 4 1 4400.2.a.bh 2
5.c odd 4 1 4400.2.a.bs 2
12.b even 2 1 2475.2.c.k 4
20.d odd 2 1 275.2.b.c 4
20.e even 4 1 275.2.a.e 2
20.e even 4 1 275.2.a.f yes 2
60.h even 2 1 2475.2.c.k 4
60.l odd 4 1 2475.2.a.o 2
60.l odd 4 1 2475.2.a.t 2
220.i odd 4 1 3025.2.a.h 2
220.i odd 4 1 3025.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 20.e even 4 1
275.2.a.f yes 2 20.e even 4 1
275.2.b.c 4 4.b odd 2 1
275.2.b.c 4 20.d odd 2 1
2475.2.a.o 2 60.l odd 4 1
2475.2.a.t 2 60.l odd 4 1
2475.2.c.k 4 12.b even 2 1
2475.2.c.k 4 60.h even 2 1
3025.2.a.h 2 220.i odd 4 1
3025.2.a.n 2 220.i odd 4 1
4400.2.a.bh 2 5.c odd 4 1
4400.2.a.bs 2 5.c odd 4 1
4400.2.b.y 4 1.a even 1 1 trivial
4400.2.b.y 4 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 $$S_{2}^{\mathrm{new}}(4400, [\chi])$$:

 $$T_{3}^{4} + 7T_{3}^{2} + 9$$ T3^4 + 7*T3^2 + 9 $$T_{7}^{4} + 19T_{7}^{2} + 9$$ T7^4 + 19*T7^2 + 9 $$T_{13}^{2} + 25$$ T13^2 + 25 $$T_{17}^{4} + 63T_{17}^{2} + 729$$ T17^4 + 63*T17^2 + 729

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 7T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 19T^{2} + 9$$
$11$ $$(T - 1)^{4}$$
$13$ $$(T^{2} + 25)^{2}$$
$17$ $$T^{4} + 63T^{2} + 729$$
$19$ $$(T + 1)^{4}$$
$23$ $$T^{4} + 67T^{2} + 729$$
$29$ $$(T^{2} - 9 T - 9)^{2}$$
$31$ $$(T^{2} + 6 T - 43)^{2}$$
$37$ $$T^{4} + 98T^{2} + 529$$
$41$ $$(T^{2} + 4 T - 9)^{2}$$
$43$ $$(T^{2} + 52)^{2}$$
$47$ $$(T^{2} + 9)^{2}$$
$53$ $$T^{4} + 7T^{2} + 9$$
$59$ $$(T^{2} + 14 T - 3)^{2}$$
$61$ $$(T^{2} + 5 T - 23)^{2}$$
$67$ $$(T^{2} + 16)^{2}$$
$71$ $$(T^{2} + 2 T - 12)^{2}$$
$73$ $$T^{4} + 71T^{2} + 529$$
$79$ $$(T^{2} + 11 T + 1)^{2}$$
$83$ $$T^{4} + 223T^{2} + 2601$$
$89$ $$(T^{2} + 7 T + 9)^{2}$$
$97$ $$T^{4} + 371 T^{2} + 32041$$