# Properties

 Label 1904.2.a.i Level $1904$ Weight $2$ Character orbit 1904.a Self dual yes Analytic conductor $15.204$ Analytic rank $0$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 952) 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - \beta - 1) q^{5} - q^{7} + (\beta - 2) q^{9}+O(q^{10})$$ q + b * q^3 + (-b - 1) * q^5 - q^7 + (b - 2) * q^9 $$q + \beta q^{3} + ( - \beta - 1) q^{5} - q^{7} + (\beta - 2) q^{9} + ( - 4 \beta + 2) q^{11} + (4 \beta - 2) q^{13} + ( - 2 \beta - 1) q^{15} + q^{17} + 4 \beta q^{19} - \beta q^{21} + ( - 2 \beta + 8) q^{23} + (3 \beta - 3) q^{25} + ( - 4 \beta + 1) q^{27} + 2 q^{29} + (5 \beta + 1) q^{31} + ( - 2 \beta - 4) q^{33} + (\beta + 1) q^{35} + 4 q^{37} + (2 \beta + 4) q^{39} + ( - 5 \beta - 2) q^{41} + (5 \beta + 2) q^{43} + q^{45} + (2 \beta + 8) q^{47} + q^{49} + \beta q^{51} + ( - 5 \beta + 5) q^{53} + (6 \beta + 2) q^{55} + (4 \beta + 4) q^{57} + 4 q^{59} + (\beta + 6) q^{61} + ( - \beta + 2) q^{63} + ( - 6 \beta - 2) q^{65} + ( - 5 \beta - 3) q^{67} + (6 \beta - 2) q^{69} + (9 \beta - 6) q^{73} + 3 q^{75} + (4 \beta - 2) q^{77} + ( - 10 \beta + 8) q^{79} + ( - 6 \beta + 2) q^{81} + (2 \beta + 8) q^{83} + ( - \beta - 1) q^{85} + 2 \beta q^{87} + (6 \beta - 8) q^{89} + ( - 4 \beta + 2) q^{91} + (6 \beta + 5) q^{93} + ( - 8 \beta - 4) q^{95} + ( - 13 \beta + 7) q^{97} + (6 \beta - 8) q^{99} +O(q^{100})$$ q + b * q^3 + (-b - 1) * q^5 - q^7 + (b - 2) * q^9 + (-4*b + 2) * q^11 + (4*b - 2) * q^13 + (-2*b - 1) * q^15 + q^17 + 4*b * q^19 - b * q^21 + (-2*b + 8) * q^23 + (3*b - 3) * q^25 + (-4*b + 1) * q^27 + 2 * q^29 + (5*b + 1) * q^31 + (-2*b - 4) * q^33 + (b + 1) * q^35 + 4 * q^37 + (2*b + 4) * q^39 + (-5*b - 2) * q^41 + (5*b + 2) * q^43 + q^45 + (2*b + 8) * q^47 + q^49 + b * q^51 + (-5*b + 5) * q^53 + (6*b + 2) * q^55 + (4*b + 4) * q^57 + 4 * q^59 + (b + 6) * q^61 + (-b + 2) * q^63 + (-6*b - 2) * q^65 + (-5*b - 3) * q^67 + (6*b - 2) * q^69 + (9*b - 6) * q^73 + 3 * q^75 + (4*b - 2) * q^77 + (-10*b + 8) * q^79 + (-6*b + 2) * q^81 + (2*b + 8) * q^83 + (-b - 1) * q^85 + 2*b * q^87 + (6*b - 8) * q^89 + (-4*b + 2) * q^91 + (6*b + 5) * q^93 + (-8*b - 4) * q^95 + (-13*b + 7) * q^97 + (6*b - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q + q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9 $$2 q + q^{3} - 3 q^{5} - 2 q^{7} - 3 q^{9} - 4 q^{15} + 2 q^{17} + 4 q^{19} - q^{21} + 14 q^{23} - 3 q^{25} - 2 q^{27} + 4 q^{29} + 7 q^{31} - 10 q^{33} + 3 q^{35} + 8 q^{37} + 10 q^{39} - 9 q^{41} + 9 q^{43} + 2 q^{45} + 18 q^{47} + 2 q^{49} + q^{51} + 5 q^{53} + 10 q^{55} + 12 q^{57} + 8 q^{59} + 13 q^{61} + 3 q^{63} - 10 q^{65} - 11 q^{67} + 2 q^{69} - 3 q^{73} + 6 q^{75} + 6 q^{79} - 2 q^{81} + 18 q^{83} - 3 q^{85} + 2 q^{87} - 10 q^{89} + 16 q^{93} - 16 q^{95} + q^{97} - 10 q^{99}+O(q^{100})$$ 2 * q + q^3 - 3 * q^5 - 2 * q^7 - 3 * q^9 - 4 * q^15 + 2 * q^17 + 4 * q^19 - q^21 + 14 * q^23 - 3 * q^25 - 2 * q^27 + 4 * q^29 + 7 * q^31 - 10 * q^33 + 3 * q^35 + 8 * q^37 + 10 * q^39 - 9 * q^41 + 9 * q^43 + 2 * q^45 + 18 * q^47 + 2 * q^49 + q^51 + 5 * q^53 + 10 * q^55 + 12 * q^57 + 8 * q^59 + 13 * q^61 + 3 * q^63 - 10 * q^65 - 11 * q^67 + 2 * q^69 - 3 * q^73 + 6 * q^75 + 6 * q^79 - 2 * q^81 + 18 * q^83 - 3 * q^85 + 2 * q^87 - 10 * q^89 + 16 * q^93 - 16 * q^95 + q^97 - 10 * 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
 −0.618034 1.61803
0 −0.618034 0 −0.381966 0 −1.00000 0 −2.61803 0
1.2 0 1.61803 0 −2.61803 0 −1.00000 0 −0.381966 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.i 2
4.b odd 2 1 952.2.a.a 2
8.b even 2 1 7616.2.a.r 2
8.d odd 2 1 7616.2.a.w 2
12.b even 2 1 8568.2.a.v 2
28.d even 2 1 6664.2.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.a.a 2 4.b odd 2 1
1904.2.a.i 2 1.a even 1 1 trivial
6664.2.a.h 2 28.d even 2 1
7616.2.a.r 2 8.b even 2 1
7616.2.a.w 2 8.d odd 2 1
8568.2.a.v 2 12.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1904))$$:

 $$T_{3}^{2} - T_{3} - 1$$ T3^2 - T3 - 1 $$T_{5}^{2} + 3T_{5} + 1$$ T5^2 + 3*T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 1$$
$5$ $$T^{2} + 3T + 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 20$$
$13$ $$T^{2} - 20$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2} - 4T - 16$$
$23$ $$T^{2} - 14T + 44$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} - 7T - 19$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} + 9T - 11$$
$43$ $$T^{2} - 9T - 11$$
$47$ $$T^{2} - 18T + 76$$
$53$ $$T^{2} - 5T - 25$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} - 13T + 41$$
$67$ $$T^{2} + 11T - 1$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 3T - 99$$
$79$ $$T^{2} - 6T - 116$$
$83$ $$T^{2} - 18T + 76$$
$89$ $$T^{2} + 10T - 20$$
$97$ $$T^{2} - T - 211$$