# Properties

 Label 4600.2.e.m Level $4600$ Weight $2$ Character orbit 4600.e Analytic conductor $36.731$ 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] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (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: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - 2 \beta_1 q^{7} + (\beta_{3} - 2) q^{9}+O(q^{10})$$ q + b1 * q^3 - 2*b1 * q^7 + (b3 - 2) * q^9 $$q + \beta_1 q^{3} - 2 \beta_1 q^{7} + (\beta_{3} - 2) q^{9} - 4 q^{11} + ( - 2 \beta_{2} - \beta_1) q^{13} + (2 \beta_{2} + 2 \beta_1) q^{17} - 4 q^{19} + ( - 2 \beta_{3} + 10) q^{21} - \beta_{2} q^{23} + (4 \beta_{2} + \beta_1) q^{27} + ( - \beta_{3} + 7) q^{29} + (\beta_{3} + 3) q^{31} - 4 \beta_1 q^{33} + ( - 2 \beta_{2} + 2 \beta_1) q^{37} + (\beta_{3} + 3) q^{39} + ( - \beta_{3} - 1) q^{41} + (4 \beta_{2} - 2 \beta_1) q^{43} + (4 \beta_{2} + 3 \beta_1) q^{47} + (4 \beta_{3} - 13) q^{49} - 8 q^{51} + ( - 6 \beta_{2} - 4 \beta_1) q^{53} - 4 \beta_1 q^{57} - 4 \beta_{3} q^{59} + ( - 2 \beta_{3} + 8) q^{61} + ( - 8 \beta_{2} + 4 \beta_1) q^{63} + ( - 4 \beta_{2} - 4 \beta_1) q^{67} + (\beta_{3} - 1) q^{69} + ( - 3 \beta_{3} - 1) q^{71} + ( - 14 \beta_{2} + \beta_1) q^{73} + 8 \beta_1 q^{77} + ( - 4 \beta_{3} + 4) q^{79} - 7 q^{81} - 12 \beta_{2} q^{83} + ( - 4 \beta_{2} + 7 \beta_1) q^{87} - 10 q^{89} + ( - 2 \beta_{3} - 6) q^{91} + (4 \beta_{2} + 3 \beta_1) q^{93} + ( - 6 \beta_{2} + 4 \beta_1) q^{97} + ( - 4 \beta_{3} + 8) q^{99}+O(q^{100})$$ q + b1 * q^3 - 2*b1 * q^7 + (b3 - 2) * q^9 - 4 * q^11 + (-2*b2 - b1) * q^13 + (2*b2 + 2*b1) * q^17 - 4 * q^19 + (-2*b3 + 10) * q^21 - b2 * q^23 + (4*b2 + b1) * q^27 + (-b3 + 7) * q^29 + (b3 + 3) * q^31 - 4*b1 * q^33 + (-2*b2 + 2*b1) * q^37 + (b3 + 3) * q^39 + (-b3 - 1) * q^41 + (4*b2 - 2*b1) * q^43 + (4*b2 + 3*b1) * q^47 + (4*b3 - 13) * q^49 - 8 * q^51 + (-6*b2 - 4*b1) * q^53 - 4*b1 * q^57 - 4*b3 * q^59 + (-2*b3 + 8) * q^61 + (-8*b2 + 4*b1) * q^63 + (-4*b2 - 4*b1) * q^67 + (b3 - 1) * q^69 + (-3*b3 - 1) * q^71 + (-14*b2 + b1) * q^73 + 8*b1 * q^77 + (-4*b3 + 4) * q^79 - 7 * q^81 - 12*b2 * q^83 + (-4*b2 + 7*b1) * q^87 - 10 * q^89 + (-2*b3 - 6) * q^91 + (4*b2 + 3*b1) * q^93 + (-6*b2 + 4*b1) * q^97 + (-4*b3 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{9}+O(q^{10})$$ 4 * q - 6 * q^9 $$4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89} - 28 q^{91} + 24 q^{99}+O(q^{100})$$ 4 * q - 6 * q^9 - 16 * q^11 - 16 * q^19 + 36 * q^21 + 26 * q^29 + 14 * q^31 + 14 * q^39 - 6 * q^41 - 44 * q^49 - 32 * q^51 - 8 * q^59 + 28 * q^61 - 2 * q^69 - 10 * q^71 + 8 * q^79 - 28 * q^81 - 40 * q^89 - 28 * q^91 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
 − 2.56155i − 1.56155i 1.56155i 2.56155i
0 2.56155i 0 0 0 5.12311i 0 −3.56155 0
4049.2 0 1.56155i 0 0 0 3.12311i 0 0.561553 0
4049.3 0 1.56155i 0 0 0 3.12311i 0 0.561553 0
4049.4 0 2.56155i 0 0 0 5.12311i 0 −3.56155 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 4600.2.e.m 4
5.b even 2 1 inner 4600.2.e.m 4
5.c odd 4 1 920.2.a.f 2
5.c odd 4 1 4600.2.a.r 2
15.e even 4 1 8280.2.a.bb 2
20.e even 4 1 1840.2.a.k 2
20.e even 4 1 9200.2.a.bx 2
40.i odd 4 1 7360.2.a.bj 2
40.k even 4 1 7360.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.f 2 5.c odd 4 1
1840.2.a.k 2 20.e even 4 1
4600.2.a.r 2 5.c odd 4 1
4600.2.e.m 4 1.a even 1 1 trivial
4600.2.e.m 4 5.b even 2 1 inner
7360.2.a.bj 2 40.i odd 4 1
7360.2.a.bm 2 40.k even 4 1
8280.2.a.bb 2 15.e even 4 1
9200.2.a.bx 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4600, [\chi])$$:

 $$T_{3}^{4} + 9T_{3}^{2} + 16$$ T3^4 + 9*T3^2 + 16 $$T_{7}^{4} + 36T_{7}^{2} + 256$$ T7^4 + 36*T7^2 + 256 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{4} + 13T_{13}^{2} + 4$$ T13^4 + 13*T13^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 9T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 36T^{2} + 256$$
$11$ $$(T + 4)^{4}$$
$13$ $$T^{4} + 13T^{2} + 4$$
$17$ $$T^{4} + 36T^{2} + 256$$
$19$ $$(T + 4)^{4}$$
$23$ $$(T^{2} + 1)^{2}$$
$29$ $$(T^{2} - 13 T + 38)^{2}$$
$31$ $$(T^{2} - 7 T + 8)^{2}$$
$37$ $$T^{4} + 52T^{2} + 64$$
$41$ $$(T^{2} + 3 T - 2)^{2}$$
$43$ $$T^{4} + 84T^{2} + 64$$
$47$ $$T^{4} + 89T^{2} + 1024$$
$53$ $$T^{4} + 168T^{2} + 2704$$
$59$ $$(T^{2} + 4 T - 64)^{2}$$
$61$ $$(T^{2} - 14 T + 32)^{2}$$
$67$ $$T^{4} + 144T^{2} + 4096$$
$71$ $$(T^{2} + 5 T - 32)^{2}$$
$73$ $$T^{4} + 429 T^{2} + 42436$$
$79$ $$(T^{2} - 4 T - 64)^{2}$$
$83$ $$(T^{2} + 144)^{2}$$
$89$ $$(T + 10)^{4}$$
$97$ $$T^{4} + 264T^{2} + 16$$