# Properties

 Label 1904.2.a.t Level $1904$ Weight $2$ Character orbit 1904.a Self dual yes Analytic conductor $15.204$ Analytic rank $0$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.453749.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3$$ x^5 - 2*x^4 - 5*x^3 + 10*x^2 + x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 119) 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + \beta_{3} q^{5} + q^{7} + ( - \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10})$$ q - b1 * q^3 + b3 * q^5 + q^7 + (-b2 - b1 + 2) * q^9 $$q - \beta_1 q^{3} + \beta_{3} q^{5} + q^{7} + ( - \beta_{2} - \beta_1 + 2) q^{9} + (\beta_{4} - \beta_{2} + 1) q^{11} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{13} + (2 \beta_{4} + \beta_{2} - 1) q^{15} + q^{17} + ( - \beta_{4} + \beta_{3} - \beta_1 - 2) q^{19} - \beta_1 q^{21} + ( - \beta_{4} + \beta_{3} + \beta_1 + 2) q^{23} + ( - \beta_{3} + \beta_{2} + 4) q^{25} + ( - \beta_{4} - 2 \beta_1 + 4) q^{27} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} + \cdots - 3) q^{29}+ \cdots + (3 \beta_{4} + \beta_{3} + \beta_1 + 14) q^{99}+O(q^{100})$$ q - b1 * q^3 + b3 * q^5 + q^7 + (-b2 - b1 + 2) * q^9 + (b4 - b2 + 1) * q^11 + (-b3 - b2 + b1 + 1) * q^13 + (2*b4 + b2 - 1) * q^15 + q^17 + (-b4 + b3 - b1 - 2) * q^19 - b1 * q^21 + (-b4 + b3 + b1 + 2) * q^23 + (-b3 + b2 + 4) * q^25 + (-b4 - 2*b1 + 4) * q^27 + (-b4 + 2*b3 + b2 - 2*b1 - 3) * q^29 + (-b4 + b1) * q^31 + (b4 + b3 + 2*b2 - 3*b1) * q^33 + b3 * q^35 + (-3*b4 + b3 - b1) * q^37 + (-3*b4 + b2 - 2*b1 - 5) * q^39 + (-b4 - b2 - b1 + 3) * q^41 + (-b4 - b3 - 2) * q^43 + (5*b4 - b3 + b2 + 3*b1 + 3) * q^45 + (b4 + b2 + 2*b1 + 3) * q^47 + q^49 - b1 * q^51 + (b2 + b1 + 1) * q^53 + (4*b4 - 3*b3 - b2 + b1 + 7) * q^55 + (-b3 - b2 + b1 + 3) * q^57 + (2*b4 - 2*b3 + 2*b1) * q^59 + (b4 + b2 + b1 + 5) * q^61 + (-b2 - b1 + 2) * q^63 + (b4 - b3 - 2*b2 + 3*b1 - 4) * q^65 + (b4 - b3 + b2 - 3) * q^67 + (-b3 + b2 - b1 - 7) * q^69 + (-3*b4 + 3*b3 - 3*b1 - 2) * q^71 + (-3*b4 + b2 + b1 + 1) * q^73 + (-b4 - 2*b2 - 2*b1 + 2) * q^75 + (b4 - b2 + 1) * q^77 + (-b4 - b2 + 2*b1 - 3) * q^79 + (-2*b4 - b3 - 3*b1 + 3) * q^81 + (b3 + b2 - 3*b1 + 1) * q^83 + b3 * q^85 + (3*b4 - b3 - 2*b2 + 3*b1 + 8) * q^87 + (b4 - 2*b3 - 3*b2 + 5) * q^89 + (-b3 - b2 + b1 + 1) * q^91 + (-2*b4 - b3 + b1 - 6) * q^93 + (b4 - 2*b3 + 3*b2 + 2*b1 + 5) * q^95 + (4*b4 - 3*b3 + 4) * q^97 + (3*b4 + b3 + b1 + 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 2 q^{3} + 5 q^{7} + 11 q^{9}+O(q^{10})$$ 5 * q + 2 * q^3 + 5 * q^7 + 11 * q^9 $$5 q + 2 q^{3} + 5 q^{7} + 11 q^{9} + 2 q^{11} + 2 q^{13} - 8 q^{15} + 5 q^{17} - 6 q^{19} + 2 q^{21} + 10 q^{23} + 21 q^{25} + 26 q^{27} - 8 q^{29} + 6 q^{33} + 8 q^{37} - 14 q^{39} + 18 q^{41} - 8 q^{43} + 10 q^{47} + 5 q^{49} + 2 q^{51} + 4 q^{53} + 24 q^{55} + 12 q^{57} - 8 q^{59} + 22 q^{61} + 11 q^{63} - 30 q^{65} - 16 q^{67} - 32 q^{69} + 2 q^{71} + 10 q^{73} + 14 q^{75} + 2 q^{77} - 18 q^{79} + 25 q^{81} + 12 q^{83} + 26 q^{87} + 20 q^{89} + 2 q^{91} - 28 q^{93} + 22 q^{95} + 12 q^{97} + 62 q^{99}+O(q^{100})$$ 5 * q + 2 * q^3 + 5 * q^7 + 11 * q^9 + 2 * q^11 + 2 * q^13 - 8 * q^15 + 5 * q^17 - 6 * q^19 + 2 * q^21 + 10 * q^23 + 21 * q^25 + 26 * q^27 - 8 * q^29 + 6 * q^33 + 8 * q^37 - 14 * q^39 + 18 * q^41 - 8 * q^43 + 10 * q^47 + 5 * q^49 + 2 * q^51 + 4 * q^53 + 24 * q^55 + 12 * q^57 - 8 * q^59 + 22 * q^61 + 11 * q^63 - 30 * q^65 - 16 * q^67 - 32 * q^69 + 2 * q^71 + 10 * q^73 + 14 * q^75 + 2 * q^77 - 18 * q^79 + 25 * q^81 + 12 * q^83 + 26 * q^87 + 20 * q^89 + 2 * q^91 - 28 * q^93 + 22 * q^95 + 12 * q^97 + 62 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} - 5\nu$$ v^3 - 5*v $$\beta_{2}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4\nu^{2} + 4\nu - 1$$ v^4 - v^3 - 4*v^2 + 4*v - 1 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 6\nu^{2} + \nu + 4$$ v^4 - 6*v^2 + v + 4 $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 4$$ v^4 - v^3 - 6*v^2 + 4*v + 4
 $$\nu$$ $$=$$ $$( -\beta_{4} + \beta_{3} - \beta_1 ) / 2$$ (-b4 + b3 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{4} + \beta_{2} + 5 ) / 2$$ (-b4 + b2 + 5) / 2 $$\nu^{3}$$ $$=$$ $$( -5\beta_{4} + 5\beta_{3} - 3\beta_1 ) / 2$$ (-5*b4 + 5*b3 - 3*b1) / 2 $$\nu^{4}$$ $$=$$ $$( -5\beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 22 ) / 2$$ (-5*b4 + b3 + 6*b2 + b1 + 22) / 2

## 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
 −0.544198 2.32183 −2.17679 0.609440 1.78972
0 −2.55982 0 1.76660 0 1.00000 0 3.55270 0
1.2 0 −0.907578 0 3.03818 0 1.00000 0 −2.17630 0
1.3 0 −0.569378 0 −4.15465 0 1.00000 0 −2.67581 0
1.4 0 2.82084 0 2.51889 0 1.00000 0 4.95716 0
1.5 0 3.21594 0 −3.16902 0 1.00000 0 7.34225 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.t 5
4.b odd 2 1 119.2.a.b 5
8.b even 2 1 7616.2.a.bq 5
8.d odd 2 1 7616.2.a.bt 5
12.b even 2 1 1071.2.a.m 5
20.d odd 2 1 2975.2.a.m 5
28.d even 2 1 833.2.a.g 5
28.f even 6 2 833.2.e.h 10
28.g odd 6 2 833.2.e.i 10
68.d odd 2 1 2023.2.a.j 5
84.h odd 2 1 7497.2.a.br 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
119.2.a.b 5 4.b odd 2 1
833.2.a.g 5 28.d even 2 1
833.2.e.h 10 28.f even 6 2
833.2.e.i 10 28.g odd 6 2
1071.2.a.m 5 12.b even 2 1
1904.2.a.t 5 1.a even 1 1 trivial
2023.2.a.j 5 68.d odd 2 1
2975.2.a.m 5 20.d odd 2 1
7497.2.a.br 5 84.h odd 2 1
7616.2.a.bq 5 8.b even 2 1
7616.2.a.bt 5 8.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{5} - 2T_{3}^{4} - 11T_{3}^{3} + 12T_{3}^{2} + 31T_{3} + 12$$ T3^5 - 2*T3^4 - 11*T3^3 + 12*T3^2 + 31*T3 + 12 $$T_{5}^{5} - 23T_{5}^{3} + 18T_{5}^{2} + 131T_{5} - 178$$ T5^5 - 23*T5^3 + 18*T5^2 + 131*T5 - 178

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 2 T^{4} + \cdots + 12$$
$5$ $$T^{5} - 23 T^{3} + \cdots - 178$$
$7$ $$(T - 1)^{5}$$
$11$ $$T^{5} - 2 T^{4} + \cdots + 192$$
$13$ $$T^{5} - 2 T^{4} + \cdots - 544$$
$17$ $$(T - 1)^{5}$$
$19$ $$T^{5} + 6 T^{4} + \cdots + 64$$
$23$ $$T^{5} - 10 T^{4} + \cdots + 128$$
$29$ $$T^{5} + 8 T^{4} + \cdots + 2592$$
$31$ $$T^{5} - 33 T^{3} + \cdots + 16$$
$37$ $$T^{5} - 8 T^{4} + \cdots + 4384$$
$41$ $$T^{5} - 18 T^{4} + \cdots + 162$$
$43$ $$T^{5} + 8 T^{4} + \cdots + 1052$$
$47$ $$T^{5} - 10 T^{4} + \cdots + 2304$$
$53$ $$T^{5} - 4 T^{4} + \cdots + 138$$
$59$ $$T^{5} + 8 T^{4} + \cdots + 3072$$
$61$ $$T^{5} - 22 T^{4} + \cdots + 5542$$
$67$ $$T^{5} + 16 T^{4} + \cdots - 1868$$
$71$ $$T^{5} - 2 T^{4} + \cdots - 13696$$
$73$ $$T^{5} - 10 T^{4} + \cdots - 11118$$
$79$ $$T^{5} + 18 T^{4} + \cdots - 3072$$
$83$ $$T^{5} - 12 T^{4} + \cdots - 1984$$
$89$ $$T^{5} - 20 T^{4} + \cdots + 7456$$
$97$ $$T^{5} - 12 T^{4} + \cdots + 218$$