Properties

Label 1904.2.a.r
Level $1904$
Weight $2$
Character orbit 1904.a
Self dual yes
Analytic conductor $15.204$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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: \(4\)
Coefficient field: 4.4.13448.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 2 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{3} + ( - \beta_{2} + 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{3} + ( - \beta_{2} + 1) q^{5} - q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} - \beta_1 q^{11} + 2 q^{13} + ( - \beta_{3} + \beta_{2}) q^{15} + q^{17} + (\beta_{3} + 1) q^{21} + ( - 2 \beta_{3} - 2) q^{23} + (\beta_{3} - 4 \beta_{2} + \beta_1) q^{25} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{29}+ \cdots + (4 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + 5 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} + 5 q^{5} - 4 q^{7} + 7 q^{9} + 8 q^{13} + 4 q^{17} + 3 q^{21} - 6 q^{23} + 3 q^{25} - 6 q^{27} + 8 q^{29} + 7 q^{31} - 6 q^{33} - 5 q^{35} + 8 q^{37} - 6 q^{39} + 15 q^{41} + 9 q^{43} - 2 q^{45} - 6 q^{47} + 4 q^{49} - 3 q^{51} + 17 q^{53} + 6 q^{55} + 5 q^{61} - 7 q^{63} + 10 q^{65} + 9 q^{67} + 38 q^{69} + 12 q^{71} - 11 q^{73} + 2 q^{75} + 6 q^{79} + 16 q^{81} + 6 q^{83} + 5 q^{85} + 14 q^{87} + 2 q^{89} - 8 q^{91} + 9 q^{97} + 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - 6\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 6\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} + 3\beta_1 \) Copy content Toggle raw display

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
−2.58874
2.58874
0.546295
−0.546295
0 −3.25886 0 0.557299 0 −1.00000 0 7.62018 0
1.2 0 −1.44270 0 −1.25886 0 −1.00000 0 −0.918614 0
1.3 0 −0.706585 0 4.40815 0 −1.00000 0 −2.50074 0
1.4 0 2.40815 0 1.29341 0 −1.00000 0 2.79917 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

\( T_{3}^{4} + 3T_{3}^{3} - 5T_{3}^{2} - 16T_{3} - 8 \) Copy content Toggle raw display
\( T_{5}^{4} - 5T_{5}^{3} + T_{5}^{2} + 8T_{5} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{4} - 5 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 28T^{2} + 32 \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T - 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} + \cdots - 80 \) Copy content Toggle raw display
$31$ \( T^{4} - 7 T^{3} + \cdots - 1600 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$41$ \( T^{4} - 15 T^{3} + \cdots - 1132 \) Copy content Toggle raw display
$43$ \( T^{4} - 9 T^{3} + \cdots - 1168 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} + \cdots + 640 \) Copy content Toggle raw display
$53$ \( T^{4} - 17 T^{3} + \cdots - 2500 \) Copy content Toggle raw display
$59$ \( T^{4} - 112T^{2} + 512 \) Copy content Toggle raw display
$61$ \( T^{4} - 5 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$67$ \( T^{4} - 9 T^{3} + \cdots + 80 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + \cdots - 2560 \) Copy content Toggle raw display
$73$ \( T^{4} + 11 T^{3} + \cdots + 3284 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} + \cdots + 640 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + \cdots + 14464 \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} + \cdots + 2000 \) Copy content Toggle raw display
$97$ \( T^{4} - 9 T^{3} + \cdots + 1460 \) Copy content Toggle raw display
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