# Properties

 Label 3700.2.d.c Level $3700$ Weight $2$ Character orbit 3700.d Analytic conductor $29.545$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,2,Mod(149,3700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3700.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3700 = 2^{2} \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3700.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: $$29.5446487479$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 740) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + i q^{7} + 2 q^{9}+O(q^{10})$$ q + i * q^3 + i * q^7 + 2 * q^9 $$q + i q^{3} + i q^{7} + 2 q^{9} - 3 q^{11} + 6 i q^{13} - q^{21} - 2 i q^{23} + 5 i q^{27} + 6 q^{29} - 3 i q^{33} - i q^{37} - 6 q^{39} - 9 q^{41} + 10 i q^{43} + i q^{47} + 6 q^{49} - i q^{53} - 12 q^{61} + 2 i q^{63} + 2 q^{69} - 5 q^{71} - 3 i q^{73} - 3 i q^{77} - 16 q^{79} + q^{81} - 11 i q^{83} + 6 i q^{87} - 6 q^{91} + 8 i q^{97} - 6 q^{99} +O(q^{100})$$ q + i * q^3 + i * q^7 + 2 * q^9 - 3 * q^11 + 6*i * q^13 - q^21 - 2*i * q^23 + 5*i * q^27 + 6 * q^29 - 3*i * q^33 - i * q^37 - 6 * q^39 - 9 * q^41 + 10*i * q^43 + i * q^47 + 6 * q^49 - i * q^53 - 12 * q^61 + 2*i * q^63 + 2 * q^69 - 5 * q^71 - 3*i * q^73 - 3*i * q^77 - 16 * q^79 + q^81 - 11*i * q^83 + 6*i * q^87 - 6 * q^91 + 8*i * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} - 6 q^{11} - 2 q^{21} + 12 q^{29} - 12 q^{39} - 18 q^{41} + 12 q^{49} - 24 q^{61} + 4 q^{69} - 10 q^{71} - 32 q^{79} + 2 q^{81} - 12 q^{91} - 12 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 - 6 * q^11 - 2 * q^21 + 12 * q^29 - 12 * q^39 - 18 * q^41 + 12 * q^49 - 24 * q^61 + 4 * q^69 - 10 * q^71 - 32 * q^79 + 2 * q^81 - 12 * q^91 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$1851$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 − 1.00000i 1.00000i
0 1.00000i 0 0 0 1.00000i 0 2.00000 0
149.2 0 1.00000i 0 0 0 1.00000i 0 2.00000 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 3700.2.d.c 2
5.b even 2 1 inner 3700.2.d.c 2
5.c odd 4 1 740.2.a.a 1
5.c odd 4 1 3700.2.a.d 1
15.e even 4 1 6660.2.a.a 1
20.e even 4 1 2960.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.a 1 5.c odd 4 1
2960.2.a.h 1 20.e even 4 1
3700.2.a.d 1 5.c odd 4 1
3700.2.d.c 2 1.a even 1 1 trivial
3700.2.d.c 2 5.b even 2 1 inner
6660.2.a.a 1 15.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 $$S_{2}^{\mathrm{new}}(3700, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{13}^{2} + 36$$ T13^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 4$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 9)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 1$$
$53$ $$T^{2} + 1$$
$59$ $$T^{2}$$
$61$ $$(T + 12)^{2}$$
$67$ $$T^{2}$$
$71$ $$(T + 5)^{2}$$
$73$ $$T^{2} + 9$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} + 121$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 64$$