# Properties

 Label 3800.2.a.n Level $3800$ Weight $2$ Character orbit 3800.a Self dual yes Analytic conductor $30.343$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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, 0, 0]))

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

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

chi := DirichletCharacter("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.a (trivial)

## 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: yes Analytic conductor: $$30.3431527681$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 2 \beta q^{7} - q^{9}+O(q^{10})$$ q + b * q^3 + 2*b * q^7 - q^9 $$q + \beta q^{3} + 2 \beta q^{7} - q^{9} + ( - 2 \beta + 2) q^{11} + ( - \beta - 2) q^{13} + ( - 2 \beta - 2) q^{17} - q^{19} + 4 q^{21} - 4 q^{23} - 4 \beta q^{27} + ( - 2 \beta + 2) q^{29} + (2 \beta - 4) q^{33} + ( - 3 \beta - 6) q^{37} + ( - 2 \beta - 2) q^{39} + ( - 2 \beta + 2) q^{41} - 2 \beta q^{43} + 6 \beta q^{47} + q^{49} + ( - 2 \beta - 4) q^{51} + ( - 5 \beta - 6) q^{53} - \beta q^{57} + 2 \beta q^{59} + ( - 4 \beta + 4) q^{61} - 2 \beta q^{63} + (\beta + 8) q^{67} - 4 \beta q^{69} + (8 \beta + 4) q^{71} + ( - 10 \beta + 2) q^{73} + (4 \beta - 8) q^{77} + (2 \beta - 12) q^{79} - 5 q^{81} - 8 q^{83} + (2 \beta - 4) q^{87} + (2 \beta - 6) q^{89} + ( - 4 \beta - 4) q^{91} + (3 \beta - 2) q^{97} + (2 \beta - 2) q^{99} +O(q^{100})$$ q + b * q^3 + 2*b * q^7 - q^9 + (-2*b + 2) * q^11 + (-b - 2) * q^13 + (-2*b - 2) * q^17 - q^19 + 4 * q^21 - 4 * q^23 - 4*b * q^27 + (-2*b + 2) * q^29 + (2*b - 4) * q^33 + (-3*b - 6) * q^37 + (-2*b - 2) * q^39 + (-2*b + 2) * q^41 - 2*b * q^43 + 6*b * q^47 + q^49 + (-2*b - 4) * q^51 + (-5*b - 6) * q^53 - b * q^57 + 2*b * q^59 + (-4*b + 4) * q^61 - 2*b * q^63 + (b + 8) * q^67 - 4*b * q^69 + (8*b + 4) * q^71 + (-10*b + 2) * q^73 + (4*b - 8) * q^77 + (2*b - 12) * q^79 - 5 * q^81 - 8 * q^83 + (2*b - 4) * q^87 + (2*b - 6) * q^89 + (-4*b - 4) * q^91 + (3*b - 2) * q^97 + (2*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 4 q^{11} - 4 q^{13} - 4 q^{17} - 2 q^{19} + 8 q^{21} - 8 q^{23} + 4 q^{29} - 8 q^{33} - 12 q^{37} - 4 q^{39} + 4 q^{41} + 2 q^{49} - 8 q^{51} - 12 q^{53} + 8 q^{61} + 16 q^{67} + 8 q^{71} + 4 q^{73} - 16 q^{77} - 24 q^{79} - 10 q^{81} - 16 q^{83} - 8 q^{87} - 12 q^{89} - 8 q^{91} - 4 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 4 * q^11 - 4 * q^13 - 4 * q^17 - 2 * q^19 + 8 * q^21 - 8 * q^23 + 4 * q^29 - 8 * q^33 - 12 * q^37 - 4 * q^39 + 4 * q^41 + 2 * q^49 - 8 * q^51 - 12 * q^53 + 8 * q^61 + 16 * q^67 + 8 * q^71 + 4 * q^73 - 16 * q^77 - 24 * q^79 - 10 * q^81 - 16 * q^83 - 8 * q^87 - 12 * q^89 - 8 * q^91 - 4 * q^97 - 4 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 −1.41421 0 0 0 −2.82843 0 −1.00000 0
1.2 0 1.41421 0 0 0 2.82843 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.2.a.n 2
4.b odd 2 1 7600.2.a.ba 2
5.b even 2 1 760.2.a.f 2
5.c odd 4 2 3800.2.d.i 4
15.d odd 2 1 6840.2.a.z 2
20.d odd 2 1 1520.2.a.m 2
40.e odd 2 1 6080.2.a.bg 2
40.f even 2 1 6080.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.f 2 5.b even 2 1
1520.2.a.m 2 20.d odd 2 1
3800.2.a.n 2 1.a even 1 1 trivial
3800.2.d.i 4 5.c odd 4 2
6080.2.a.bf 2 40.f even 2 1
6080.2.a.bg 2 40.e odd 2 1
6840.2.a.z 2 15.d odd 2 1
7600.2.a.ba 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3800))$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{7}^{2} - 8$$ T7^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2} - 4T - 4$$
$13$ $$T^{2} + 4T + 2$$
$17$ $$T^{2} + 4T - 4$$
$19$ $$(T + 1)^{2}$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} - 4T - 4$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 12T + 18$$
$41$ $$T^{2} - 4T - 4$$
$43$ $$T^{2} - 8$$
$47$ $$T^{2} - 72$$
$53$ $$T^{2} + 12T - 14$$
$59$ $$T^{2} - 8$$
$61$ $$T^{2} - 8T - 16$$
$67$ $$T^{2} - 16T + 62$$
$71$ $$T^{2} - 8T - 112$$
$73$ $$T^{2} - 4T - 196$$
$79$ $$T^{2} + 24T + 136$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} + 12T + 28$$
$97$ $$T^{2} + 4T - 14$$