Properties

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

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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: \( x^{2} - x - 1 \) 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 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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{3} + ( - \beta + 1) q^{5} + q^{7} + (2 \beta + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{3} + ( - \beta + 1) q^{5} + q^{7} + (2 \beta + 3) q^{9} + ( - \beta - 3) q^{11} + ( - 2 \beta + 2) q^{13} + 4 q^{15} + q^{17} + (2 \beta + 4) q^{19} + ( - \beta - 1) q^{21} - 8 q^{23} + ( - 2 \beta + 1) q^{25} + ( - 2 \beta - 10) q^{27} + ( - 3 \beta + 1) q^{29} + ( - 2 \beta + 6) q^{31} + (4 \beta + 8) q^{33} + ( - \beta + 1) q^{35} + ( - \beta - 1) q^{37} + 8 q^{39} + (2 \beta - 8) q^{41} + ( - 2 \beta + 2) q^{43} + ( - \beta - 7) q^{45} + (2 \beta - 2) q^{47} + q^{49} + ( - \beta - 1) q^{51} - 2 \beta q^{53} + (2 \beta + 2) q^{55} + ( - 6 \beta - 14) q^{57} + 6 q^{59} + (\beta - 1) q^{61} + (2 \beta + 3) q^{63} + ( - 4 \beta + 12) q^{65} + ( - 2 \beta + 6) q^{67} + (8 \beta + 8) q^{69} + (2 \beta - 2) q^{71} + 6 \beta q^{73} + (\beta + 9) q^{75} + ( - \beta - 3) q^{77} + (2 \beta + 6) q^{79} + (6 \beta + 11) q^{81} - 2 q^{83} + ( - \beta + 1) q^{85} + (2 \beta + 14) q^{87} + 2 q^{89} + ( - 2 \beta + 2) q^{91} + ( - 4 \beta + 4) q^{93} + ( - 2 \beta - 6) q^{95} + ( - 2 \beta - 4) q^{97} + ( - 9 \beta - 19) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 6 q^{9} - 6 q^{11} + 4 q^{13} + 8 q^{15} + 2 q^{17} + 8 q^{19} - 2 q^{21} - 16 q^{23} + 2 q^{25} - 20 q^{27} + 2 q^{29} + 12 q^{31} + 16 q^{33} + 2 q^{35} - 2 q^{37} + 16 q^{39} - 16 q^{41} + 4 q^{43} - 14 q^{45} - 4 q^{47} + 2 q^{49} - 2 q^{51} + 4 q^{55} - 28 q^{57} + 12 q^{59} - 2 q^{61} + 6 q^{63} + 24 q^{65} + 12 q^{67} + 16 q^{69} - 4 q^{71} + 18 q^{75} - 6 q^{77} + 12 q^{79} + 22 q^{81} - 4 q^{83} + 2 q^{85} + 28 q^{87} + 4 q^{89} + 4 q^{91} + 8 q^{93} - 12 q^{95} - 8 q^{97} - 38 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
0 −3.23607 0 −1.23607 0 1.00000 0 7.47214 0
1.2 0 1.23607 0 3.23607 0 1.00000 0 −1.47214 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.f 2
4.b odd 2 1 238.2.a.f 2
8.b even 2 1 7616.2.a.y 2
8.d odd 2 1 7616.2.a.n 2
12.b even 2 1 2142.2.a.x 2
20.d odd 2 1 5950.2.a.x 2
28.d even 2 1 1666.2.a.o 2
68.d odd 2 1 4046.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.2.a.f 2 4.b odd 2 1
1666.2.a.o 2 28.d even 2 1
1904.2.a.f 2 1.a even 1 1 trivial
2142.2.a.x 2 12.b even 2 1
4046.2.a.v 2 68.d odd 2 1
5950.2.a.x 2 20.d odd 2 1
7616.2.a.n 2 8.d odd 2 1
7616.2.a.y 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1904))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$17$ \( (T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$23$ \( (T + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 44 \) Copy content Toggle raw display
$31$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 16T + 44 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} - 20 \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} - 180 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$83$ \( (T + 2)^{2} \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
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