# Properties

 Label 1904.2.a.b Level $1904$ Weight $2$ Character orbit 1904.a Self dual yes Analytic conductor $15.204$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1904,2,Mod(1,1904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1904, 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("1904.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1904 = 2^{4} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1904.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: $$15.2035165449$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 238) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + 4 q^{5} - q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + 4 * q^5 - q^7 + q^9 $$q - 2 q^{3} + 4 q^{5} - q^{7} + q^{9} + 4 q^{11} - 4 q^{13} - 8 q^{15} - q^{17} + 6 q^{19} + 2 q^{21} + 11 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} - 8 q^{33} - 4 q^{35} - 10 q^{37} + 8 q^{39} + 6 q^{41} + 4 q^{45} - 4 q^{47} + q^{49} + 2 q^{51} + 14 q^{53} + 16 q^{55} - 12 q^{57} + 6 q^{59} - 12 q^{61} - q^{63} - 16 q^{65} - 4 q^{67} + 8 q^{71} + 2 q^{73} - 22 q^{75} - 4 q^{77} - 11 q^{81} - 10 q^{83} - 4 q^{85} - 12 q^{87} + 10 q^{89} + 4 q^{91} + 8 q^{93} + 24 q^{95} + 6 q^{97} + 4 q^{99}+O(q^{100})$$ q - 2 * q^3 + 4 * q^5 - q^7 + q^9 + 4 * q^11 - 4 * q^13 - 8 * q^15 - q^17 + 6 * q^19 + 2 * q^21 + 11 * q^25 + 4 * q^27 + 6 * q^29 - 4 * q^31 - 8 * q^33 - 4 * q^35 - 10 * q^37 + 8 * q^39 + 6 * q^41 + 4 * q^45 - 4 * q^47 + q^49 + 2 * q^51 + 14 * q^53 + 16 * q^55 - 12 * q^57 + 6 * q^59 - 12 * q^61 - q^63 - 16 * q^65 - 4 * q^67 + 8 * q^71 + 2 * q^73 - 22 * q^75 - 4 * q^77 - 11 * q^81 - 10 * q^83 - 4 * q^85 - 12 * q^87 + 10 * q^89 + 4 * q^91 + 8 * q^93 + 24 * q^95 + 6 * 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
 0
0 −2.00000 0 4.00000 0 −1.00000 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
$$2$$ $$-1$$
$$7$$ $$1$$
$$17$$ $$1$$

## 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 1904.2.a.b 1
4.b odd 2 1 238.2.a.b 1
8.b even 2 1 7616.2.a.i 1
8.d odd 2 1 7616.2.a.a 1
12.b even 2 1 2142.2.a.l 1
20.d odd 2 1 5950.2.a.k 1
28.d even 2 1 1666.2.a.b 1
68.d odd 2 1 4046.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.2.a.b 1 4.b odd 2 1
1666.2.a.b 1 28.d even 2 1
1904.2.a.b 1 1.a even 1 1 trivial
2142.2.a.l 1 12.b even 2 1
4046.2.a.b 1 68.d odd 2 1
5950.2.a.k 1 20.d odd 2 1
7616.2.a.a 1 8.d odd 2 1
7616.2.a.i 1 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1904))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{5} - 4$$ T5 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T - 4$$
$7$ $$T + 1$$
$11$ $$T - 4$$
$13$ $$T + 4$$
$17$ $$T + 1$$
$19$ $$T - 6$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T + 10$$
$41$ $$T - 6$$
$43$ $$T$$
$47$ $$T + 4$$
$53$ $$T - 14$$
$59$ $$T - 6$$
$61$ $$T + 12$$
$67$ $$T + 4$$
$71$ $$T - 8$$
$73$ $$T - 2$$
$79$ $$T$$
$83$ $$T + 10$$
$89$ $$T - 10$$
$97$ $$T - 6$$