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 $1$

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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: \(5\)
Coefficient field: 5.5.453749.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 \) Copy content Toggle raw display
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 5x^{3} + 10x^{2} + x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - \nu^{3} - 4\nu^{2} + 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 6\nu^{2} + \nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} + \beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{4} + 5\beta_{3} - 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 22 ) / 2 \) 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
−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:

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1904))\):

\( T_{3}^{5} - 2T_{3}^{4} - 11T_{3}^{3} + 12T_{3}^{2} + 31T_{3} + 12 \) Copy content Toggle raw display
\( T_{5}^{5} - 23T_{5}^{3} + 18T_{5}^{2} + 131T_{5} - 178 \) Copy content Toggle raw display

Hecke characteristic polynomials

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