# Properties

 Label 3800.2.d.l Level $3800$ Weight $2$ Character orbit 3800.d Analytic conductor $30.343$ 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] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.d (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: $$30.3431527681$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.3356224.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 8x^{4} + 16x^{2} + 1$$ x^6 + 8*x^4 + 16*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 760) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{3} + \beta_{5} q^{7} + (\beta_1 - 1) q^{9}+O(q^{10})$$ q - b4 * q^3 + b5 * q^7 + (b1 - 1) * q^9 $$q - \beta_{4} q^{3} + \beta_{5} q^{7} + (\beta_1 - 1) q^{9} + ( - \beta_{4} + \beta_{2}) q^{13} + (\beta_{5} - \beta_{2}) q^{17} - q^{19} + (\beta_{3} - \beta_1 + 1) q^{21} + (\beta_{5} - 2 \beta_{4}) q^{23} + \beta_{5} q^{27} + ( - \beta_1 - 6) q^{29} - 2 \beta_1 q^{31} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{37} + (\beta_{3} + \beta_1 - 3) q^{39} + (\beta_{3} + 2 \beta_1 + 3) q^{41} + (2 \beta_{4} - 2 \beta_{2}) q^{43} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{47} + ( - \beta_{3} - \beta_1 - 2) q^{49} - \beta_1 q^{51} + ( - 2 \beta_{5} + \beta_{4} + \beta_{2}) q^{53} + \beta_{4} q^{57} + ( - 2 \beta_{3} - \beta_1 - 6) q^{59} + (\beta_{3} + 3) q^{61} + ( - 2 \beta_{4} - 4 \beta_{2}) q^{63} + (2 \beta_{5} - 5 \beta_{4}) q^{67} + (\beta_{3} + \beta_1 - 7) q^{69} + 4 \beta_1 q^{71} + (\beta_{5} - 4 \beta_{4} + \beta_{2}) q^{73} + ( - \beta_{3} - 1) q^{79} + (\beta_{3} + 2 \beta_1 - 2) q^{81} + (2 \beta_{5} - 2 \beta_{2}) q^{83} + ( - \beta_{5} + 4 \beta_{4}) q^{87} + (\beta_{3} + 2 \beta_1 - 1) q^{89} + (\beta_{3} + \beta_1 + 1) q^{91} + ( - 2 \beta_{5} - 4 \beta_{4}) q^{93} + ( - 3 \beta_{5} + \beta_{4} + \beta_{2}) q^{97}+O(q^{100})$$ q - b4 * q^3 + b5 * q^7 + (b1 - 1) * q^9 + (-b4 + b2) * q^13 + (b5 - b2) * q^17 - q^19 + (b3 - b1 + 1) * q^21 + (b5 - 2*b4) * q^23 + b5 * q^27 + (-b1 - 6) * q^29 - 2*b1 * q^31 + (-b5 - b4 - b2) * q^37 + (b3 + b1 - 3) * q^39 + (b3 + 2*b1 + 3) * q^41 + (2*b4 - 2*b2) * q^43 + (-2*b5 - 2*b4) * q^47 + (-b3 - b1 - 2) * q^49 - b1 * q^51 + (-2*b5 + b4 + b2) * q^53 + b4 * q^57 + (-2*b3 - b1 - 6) * q^59 + (b3 + 3) * q^61 + (-2*b4 - 4*b2) * q^63 + (2*b5 - 5*b4) * q^67 + (b3 + b1 - 7) * q^69 + 4*b1 * q^71 + (b5 - 4*b4 + b2) * q^73 + (-b3 - 1) * q^79 + (b3 + 2*b1 - 2) * q^81 + (2*b5 - 2*b2) * q^83 + (-b5 + 4*b4) * q^87 + (b3 + 2*b1 - 1) * q^89 + (b3 + b1 + 1) * q^91 + (-2*b5 - 4*b4) * q^93 + (-3*b5 + b4 + b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{9}+O(q^{10})$$ 6 * q - 8 * q^9 $$6 q - 8 q^{9} - 6 q^{19} + 6 q^{21} - 34 q^{29} + 4 q^{31} - 22 q^{39} + 12 q^{41} - 8 q^{49} + 2 q^{51} - 30 q^{59} + 16 q^{61} - 46 q^{69} - 8 q^{71} - 4 q^{79} - 18 q^{81} - 12 q^{89} + 2 q^{91}+O(q^{100})$$ 6 * q - 8 * q^9 - 6 * q^19 + 6 * q^21 - 34 * q^29 + 4 * q^31 - 22 * q^39 + 12 * q^41 - 8 * q^49 + 2 * q^51 - 30 * q^59 + 16 * q^61 - 46 * q^69 - 8 * q^71 - 4 * q^79 - 18 * q^81 - 12 * q^89 + 2 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 8x^{4} + 16x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{4} + 3\nu^{2} - 3$$ v^4 + 3*v^2 - 3 $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 8\nu$$ 2*v^3 + 8*v $$\beta_{3}$$ $$=$$ $$2\nu^{4} + 10\nu^{2} + 5$$ 2*v^4 + 10*v^2 + 5 $$\beta_{4}$$ $$=$$ $$\nu^{5} + 7\nu^{3} + 11\nu$$ v^5 + 7*v^3 + 11*v $$\beta_{5}$$ $$=$$ $$\nu^{5} + 7\nu^{3} + 13\nu$$ v^5 + 7*v^3 + 13*v
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} ) / 2$$ (b5 - b4) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 2\beta _1 - 11 ) / 4$$ (b3 - 2*b1 - 11) / 4 $$\nu^{3}$$ $$=$$ $$( -4\beta_{5} + 4\beta_{4} + \beta_{2} ) / 2$$ (-4*b5 + 4*b4 + b2) / 2 $$\nu^{4}$$ $$=$$ $$( -3\beta_{3} + 10\beta _1 + 45 ) / 4$$ (-3*b3 + 10*b1 + 45) / 4 $$\nu^{5}$$ $$=$$ $$( 17\beta_{5} - 15\beta_{4} - 7\beta_{2} ) / 2$$ (17*b5 - 15*b4 - 7*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3649.1
 0.254102i − 1.86081i − 2.11491i 2.11491i 1.86081i − 0.254102i
0 2.68133i 0 0 0 3.18953i 0 −4.18953 0
3649.2 0 2.32340i 0 0 0 1.39821i 0 −2.39821 0
3649.3 0 0.642074i 0 0 0 3.58774i 0 2.58774 0
3649.4 0 0.642074i 0 0 0 3.58774i 0 2.58774 0
3649.5 0 2.32340i 0 0 0 1.39821i 0 −2.39821 0
3649.6 0 2.68133i 0 0 0 3.18953i 0 −4.18953 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3649.6 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 3800.2.d.l 6
5.b even 2 1 inner 3800.2.d.l 6
5.c odd 4 1 760.2.a.j 3
5.c odd 4 1 3800.2.a.x 3
15.e even 4 1 6840.2.a.bg 3
20.e even 4 1 1520.2.a.s 3
20.e even 4 1 7600.2.a.bq 3
40.i odd 4 1 6080.2.a.bv 3
40.k even 4 1 6080.2.a.bq 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.j 3 5.c odd 4 1
1520.2.a.s 3 20.e even 4 1
3800.2.a.x 3 5.c odd 4 1
3800.2.d.l 6 1.a even 1 1 trivial
3800.2.d.l 6 5.b even 2 1 inner
6080.2.a.bq 3 40.k even 4 1
6080.2.a.bv 3 40.i odd 4 1
6840.2.a.bg 3 15.e even 4 1
7600.2.a.bq 3 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 $$S_{2}^{\mathrm{new}}(3800, [\chi])$$:

 $$T_{3}^{6} + 13T_{3}^{4} + 44T_{3}^{2} + 16$$ T3^6 + 13*T3^4 + 44*T3^2 + 16 $$T_{7}^{6} + 25T_{7}^{4} + 176T_{7}^{2} + 256$$ T7^6 + 25*T7^4 + 176*T7^2 + 256 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 13 T^{4} + \cdots + 16$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 25 T^{4} + \cdots + 256$$
$11$ $$T^{6}$$
$13$ $$T^{6} + 21 T^{4} + \cdots + 16$$
$17$ $$T^{6} + 33 T^{4} + \cdots + 16$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} + 65 T^{4} + \cdots + 4096$$
$29$ $$(T^{3} + 17 T^{2} + \cdots + 124)^{2}$$
$31$ $$(T^{3} - 2 T^{2} + \cdots + 128)^{2}$$
$37$ $$T^{6} + 64 T^{4} + \cdots + 64$$
$41$ $$(T^{3} - 6 T^{2} - 52 T + 56)^{2}$$
$43$ $$T^{6} + 84 T^{4} + \cdots + 1024$$
$47$ $$T^{6} + 176 T^{4} + \cdots + 16384$$
$53$ $$T^{6} + 109 T^{4} + \cdots + 2704$$
$59$ $$(T^{3} + 15 T^{2} + \cdots - 784)^{2}$$
$61$ $$(T^{3} - 8 T^{2} - 4 T + 64)^{2}$$
$67$ $$T^{6} + 365 T^{4} + \cdots + 1106704$$
$71$ $$(T^{3} + 4 T^{2} + \cdots - 1024)^{2}$$
$73$ $$T^{6} + 209 T^{4} + \cdots + 85264$$
$79$ $$(T^{3} + 2 T^{2} - 24 T - 32)^{2}$$
$83$ $$T^{6} + 132 T^{4} + \cdots + 1024$$
$89$ $$(T^{3} + 6 T^{2} + \cdots - 184)^{2}$$
$97$ $$T^{6} + 224 T^{4} + \cdots + 87616$$