Properties

Label 1805.2.a.o
Level $1805$
Weight $2$
Character orbit 1805.a
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1805,2,Mod(1,1805)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1805, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1805.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1,3,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7537.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2} + 1) q^{4} + q^{5} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} + ( - \beta_{3} - 1) q^{7} + (2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{8}+ \cdots + (5 \beta_{2} - 5 \beta_1 + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 3 q^{3} + 5 q^{4} + 4 q^{5} + 2 q^{6} - 4 q^{7} + 12 q^{8} + q^{9} + q^{10} - 2 q^{11} + 6 q^{12} + 7 q^{13} - q^{14} + 3 q^{15} + 7 q^{16} - q^{17} - 10 q^{18} + 5 q^{20} - 4 q^{21}+ \cdots + 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

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

Copy content comment:embeddings in the coefficient field
 
Copy content 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
2.15976
−2.04717
1.37933
−0.491918
−1.66454 −1.15976 0.770710 1.00000 1.93047 −2.43525 2.04621 −1.65497 −1.66454
1.2 −1.19091 3.04717 −0.581734 1.00000 −3.62891 −0.609175 3.07461 6.28525 −1.19091
1.3 1.09744 −0.379334 −0.795629 1.00000 −0.416295 1.89307 −3.06803 −2.85611 1.09744
1.4 2.75802 1.49192 5.60665 1.00000 4.11474 −2.84864 9.94721 −0.774179 2.75802
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(19\) \( +1 \)

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 1805.2.a.o 4
5.b even 2 1 9025.2.a.bg 4
19.b odd 2 1 1805.2.a.i 4
19.c even 3 2 95.2.e.c 8
57.h odd 6 2 855.2.k.h 8
76.g odd 6 2 1520.2.q.o 8
95.d odd 2 1 9025.2.a.bp 4
95.i even 6 2 475.2.e.e 8
95.m odd 12 4 475.2.j.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 19.c even 3 2
475.2.e.e 8 95.i even 6 2
475.2.j.c 16 95.m odd 12 4
855.2.k.h 8 57.h odd 6 2
1520.2.q.o 8 76.g odd 6 2
1805.2.a.i 4 19.b odd 2 1
1805.2.a.o 4 1.a even 1 1 trivial
9025.2.a.bg 4 5.b even 2 1
9025.2.a.bp 4 95.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\):

\( T_{2}^{4} - T_{2}^{3} - 6T_{2}^{2} + T_{2} + 6 \) Copy content Toggle raw display
\( T_{3}^{4} - 3T_{3}^{3} - 2T_{3}^{2} + 5T_{3} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} - 6 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$3$ \( T^{4} - 3 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$13$ \( T^{4} - 7 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} + \cdots + 108 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + \cdots + 6 \) Copy content Toggle raw display
$29$ \( T^{4} + T^{3} + \cdots - 141 \) Copy content Toggle raw display
$31$ \( T^{4} - 67 T^{2} + \cdots + 1063 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots - 118 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} + \cdots - 2238 \) Copy content Toggle raw display
$43$ \( T^{4} - T^{3} + \cdots + 794 \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} + \cdots - 2316 \) Copy content Toggle raw display
$53$ \( T^{4} + 5 T^{3} + \cdots - 54 \) Copy content Toggle raw display
$59$ \( T^{4} + 5 T^{3} + \cdots + 1875 \) Copy content Toggle raw display
$61$ \( T^{4} - 130 T^{2} + \cdots + 3049 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + \cdots - 243 \) Copy content Toggle raw display
$73$ \( T^{4} + 20 T^{3} + \cdots - 1726 \) Copy content Toggle raw display
$79$ \( T^{4} - 17 T^{3} + \cdots - 184 \) Copy content Toggle raw display
$83$ \( T^{4} - T^{3} + \cdots + 366 \) Copy content Toggle raw display
$89$ \( T^{4} - 11 T^{3} + \cdots - 3816 \) Copy content Toggle raw display
$97$ \( T^{4} - T^{3} + \cdots + 7442 \) Copy content Toggle raw display
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