# Properties

 Label 1904.2.a.q Level $1904$ Weight $2$ Character orbit 1904.a Self dual yes Analytic conductor $15.204$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.5225.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + x + 11$$ x^4 - x^3 - 8*x^2 + x + 11 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{3} + ( - \beta_{3} - \beta_1) q^{5} + q^{7} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^3 + (-b3 - b1) * q^5 + q^7 + (b3 + b2 - b1 + 2) * q^9 $$q + (\beta_1 - 1) q^{3} + ( - \beta_{3} - \beta_1) q^{5} + q^{7} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{9} + (\beta_{2} - 1) q^{11} + ( - 2 \beta_{2} - 2) q^{13} + (\beta_{3} - 2 \beta_{2} - 3) q^{15} + q^{17} + (2 \beta_1 - 4) q^{19} + (\beta_1 - 1) q^{21} + (2 \beta_{3} - 2) q^{23} + (3 \beta_{2} - \beta_1 + 3) q^{25} + ( - \beta_{3} - 3) q^{27} + (2 \beta_{3} - \beta_{2} + 1) q^{29} + (3 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{31} + (2 \beta_{3} + 2) q^{33} + ( - \beta_{3} - \beta_1) q^{35} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{37} + ( - 4 \beta_{3} - 4 \beta_1) q^{39} + ( - 3 \beta_{2} + \beta_1 - 2) q^{41} + (\beta_{2} - 3 \beta_1 - 4) q^{43} + ( - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{45} + ( - 4 \beta_{3} - 2 \beta_{2} - 2) q^{47} + q^{49} + (\beta_1 - 1) q^{51} + (3 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{53} + ( - 2 \beta_1 + 2) q^{55} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 12) q^{57} + 2 \beta_{3} q^{59} + (2 \beta_{2} - 3 \beta_1 - 5) q^{61} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{63} + (4 \beta_{3} + 8 \beta_1 - 4) q^{65} + (\beta_{3} + 3 \beta_{2} - 3 \beta_1 - 3) q^{67} + ( - 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{69} + (2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{71} + (3 \beta_{2} + 3 \beta_1 - 2) q^{73} + (5 \beta_{3} - \beta_{2} + 6 \beta_1 - 4) q^{75} + (\beta_{2} - 1) q^{77} + ( - 4 \beta_{3} - 2 \beta_{2} + 6) q^{79} + ( - \beta_{3} - 4 \beta_{2} - 2) q^{81} + ( - 4 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{83}+ \cdots + ( - 4 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^3 + (-b3 - b1) * q^5 + q^7 + (b3 + b2 - b1 + 2) * q^9 + (b2 - 1) * q^11 + (-2*b2 - 2) * q^13 + (b3 - 2*b2 - 3) * q^15 + q^17 + (2*b1 - 4) * q^19 + (b1 - 1) * q^21 + (2*b3 - 2) * q^23 + (3*b2 - b1 + 3) * q^25 + (-b3 - 3) * q^27 + (2*b3 - b2 + 1) * q^29 + (3*b3 + b2 - b1 - 1) * q^31 + (2*b3 + 2) * q^33 + (-b3 - b1) * q^35 + (-2*b3 - 3*b2 - 1) * q^37 + (-4*b3 - 4*b1) * q^39 + (-3*b2 + b1 - 2) * q^41 + (b2 - 3*b1 - 4) * q^43 + (-3*b3 + b2 - 2*b1) * q^45 + (-4*b3 - 2*b2 - 2) * q^47 + q^49 + (b1 - 1) * q^51 + (3*b3 + 3*b2 + b1 + 1) * q^53 + (-2*b1 + 2) * q^55 + (2*b3 + 2*b2 - 4*b1 + 12) * q^57 + 2*b3 * q^59 + (2*b2 - 3*b1 - 5) * q^61 + (b3 + b2 - b1 + 2) * q^63 + (4*b3 + 8*b1 - 4) * q^65 + (b3 + 3*b2 - 3*b1 - 3) * q^67 + (-4*b3 + 2*b2 - 2*b1) * q^69 + (2*b3 + 4*b2 - 2*b1) * q^71 + (3*b2 + 3*b1 - 2) * q^73 + (5*b3 - b2 + 6*b1 - 4) * q^75 + (b2 - 1) * q^77 + (-4*b3 - 2*b2 + 6) * q^79 + (-b3 - 4*b2 - 2) * q^81 + (-4*b3 - 2*b2 - 2*b1 - 2) * q^83 + (-b3 - b1) * q^85 + (-6*b3 + 2*b2 - 4) * q^87 + (-6*b3 - 4*b2 - 4) * q^89 + (-2*b2 - 2) * q^91 + (-5*b3 + 2*b2 - 5) * q^93 + (4*b3 - 4*b2 + 2*b1 - 6) * q^95 + (-b3 + 3*b2 - 3*b1 - 3) * q^97 + (-4*b3 - b2 + 2*b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{3} - q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10})$$ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + 7 * q^9 $$4 q - 3 q^{3} - q^{5} + 4 q^{7} + 7 q^{9} - 4 q^{11} - 8 q^{13} - 12 q^{15} + 4 q^{17} - 14 q^{19} - 3 q^{21} - 8 q^{23} + 11 q^{25} - 12 q^{27} + 4 q^{29} - 5 q^{31} + 8 q^{33} - q^{35} - 4 q^{37} - 4 q^{39} - 7 q^{41} - 19 q^{43} - 2 q^{45} - 8 q^{47} + 4 q^{49} - 3 q^{51} + 5 q^{53} + 6 q^{55} + 44 q^{57} - 23 q^{61} + 7 q^{63} - 8 q^{65} - 15 q^{67} - 2 q^{69} - 2 q^{71} - 5 q^{73} - 10 q^{75} - 4 q^{77} + 24 q^{79} - 8 q^{81} - 10 q^{83} - q^{85} - 16 q^{87} - 16 q^{89} - 8 q^{91} - 20 q^{93} - 22 q^{95} - 15 q^{97} - 2 q^{99}+O(q^{100})$$ 4 * q - 3 * q^3 - q^5 + 4 * q^7 + 7 * q^9 - 4 * q^11 - 8 * q^13 - 12 * q^15 + 4 * q^17 - 14 * q^19 - 3 * q^21 - 8 * q^23 + 11 * q^25 - 12 * q^27 + 4 * q^29 - 5 * q^31 + 8 * q^33 - q^35 - 4 * q^37 - 4 * q^39 - 7 * q^41 - 19 * q^43 - 2 * q^45 - 8 * q^47 + 4 * q^49 - 3 * q^51 + 5 * q^53 + 6 * q^55 + 44 * q^57 - 23 * q^61 + 7 * q^63 - 8 * q^65 - 15 * q^67 - 2 * q^69 - 2 * q^71 - 5 * q^73 - 10 * q^75 - 4 * q^77 + 24 * q^79 - 8 * q^81 - 10 * q^83 - q^85 - 16 * q^87 - 16 * q^89 - 8 * q^91 - 20 * q^93 - 22 * q^95 - 15 * q^97 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8x^{2} + x + 11$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 4\nu + 4$$ v^3 - 2*v^2 - 4*v + 4 $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 3\nu^{2} + 3\nu - 8$$ -v^3 + 3*v^2 + 3*v - 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta _1 + 4$$ b3 + b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$2\beta_{3} + 3\beta_{2} + 6\beta _1 + 4$$ 2*b3 + 3*b2 + 6*b1 + 4

## 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
 −1.88301 −1.48718 1.26498 3.10522
0 −2.88301 0 −1.78180 0 1.00000 0 5.31175 0
1.2 0 −2.48718 0 4.02435 0 1.00000 0 3.18609 0
1.3 0 0.264977 0 0.163765 0 1.00000 0 −2.92979 0
1.4 0 2.10522 0 −3.40632 0 1.00000 0 1.43195 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.q 4
4.b odd 2 1 952.2.a.g 4
8.b even 2 1 7616.2.a.bp 4
8.d odd 2 1 7616.2.a.bj 4
12.b even 2 1 8568.2.a.bj 4
28.d even 2 1 6664.2.a.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.a.g 4 4.b odd 2 1
1904.2.a.q 4 1.a even 1 1 trivial
6664.2.a.o 4 28.d even 2 1
7616.2.a.bj 4 8.d odd 2 1
7616.2.a.bp 4 8.b even 2 1
8568.2.a.bj 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1904))$$:

 $$T_{3}^{4} + 3T_{3}^{3} - 5T_{3}^{2} - 14T_{3} + 4$$ T3^4 + 3*T3^3 - 5*T3^2 - 14*T3 + 4 $$T_{5}^{4} + T_{5}^{3} - 15T_{5}^{2} - 22T_{5} + 4$$ T5^4 + T5^3 - 15*T5^2 - 22*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3 T^{3} + \cdots + 4$$
$5$ $$T^{4} + T^{3} - 15 T^{2} + \cdots + 4$$
$7$ $$(T - 1)^{4}$$
$11$ $$(T^{2} + 2 T - 4)^{2}$$
$13$ $$(T^{2} + 4 T - 16)^{2}$$
$17$ $$(T - 1)^{4}$$
$19$ $$T^{4} + 14 T^{3} + \cdots - 176$$
$23$ $$T^{4} + 8 T^{3} + \cdots - 256$$
$29$ $$T^{4} - 4 T^{3} + \cdots + 16$$
$31$ $$T^{4} + 5 T^{3} + \cdots - 400$$
$37$ $$T^{4} + 4 T^{3} + \cdots - 304$$
$41$ $$T^{4} + 7 T^{3} + \cdots + 964$$
$43$ $$T^{4} + 19 T^{3} + \cdots - 176$$
$47$ $$T^{4} + 8 T^{3} + \cdots + 2816$$
$53$ $$T^{4} - 5 T^{3} + \cdots + 484$$
$59$ $$T^{4} - 44 T^{2} + \cdots + 64$$
$61$ $$T^{4} + 23 T^{3} + \cdots - 1964$$
$67$ $$T^{4} + 15 T^{3} + \cdots - 176$$
$71$ $$T^{4} + 2 T^{3} + \cdots - 704$$
$73$ $$T^{4} + 5 T^{3} + \cdots + 244$$
$79$ $$T^{4} - 24 T^{3} + \cdots - 256$$
$83$ $$T^{4} + 10 T^{3} + \cdots + 4400$$
$89$ $$T^{4} + 16 T^{3} + \cdots + 7744$$
$97$ $$T^{4} + 15 T^{3} + \cdots - 5900$$