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Properties

Label 405.4.a.e
Level $405$
Weight $4$
Character orbit 405.a
Self dual yes
Analytic conductor $23.896$
Analytic rank $1$
Dimension $2$
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] = [405,4,Mod(1,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,1] 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: \(23.8957735523\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + \beta q^{4} - 5 q^{5} + ( - \beta + 5) q^{7} + ( - 7 \beta + 8) q^{8} - 5 \beta q^{10} + (5 \beta + 16) q^{11} + (8 \beta - 60) q^{13} + (4 \beta - 8) q^{14} + ( - 7 \beta - 56) q^{16} + \cdots + ( - 319 \beta - 72) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} - 10 q^{5} + 9 q^{7} + 9 q^{8} - 5 q^{10} + 37 q^{11} - 112 q^{13} - 12 q^{14} - 119 q^{16} - 77 q^{17} + 35 q^{19} - 5 q^{20} + 101 q^{22} + 267 q^{23} + 50 q^{25} + 76 q^{26} - 12 q^{28}+ \cdots - 463 q^{98}+O(q^{100}) \) 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.

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
−2.37228
3.37228
−2.37228 0 −2.37228 −5.00000 0 7.37228 24.6060 0 11.8614
1.2 3.37228 0 3.37228 −5.00000 0 1.62772 −15.6060 0 −16.8614
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)

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 405.4.a.e 2
3.b odd 2 1 405.4.a.d 2
5.b even 2 1 2025.4.a.j 2
9.c even 3 2 135.4.e.a 4
9.d odd 6 2 45.4.e.a ✓ 4
15.d odd 2 1 2025.4.a.l 2
45.h odd 6 2 225.4.e.a 4
45.l even 12 4 225.4.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.a ✓ 4 9.d odd 6 2
135.4.e.a 4 9.c even 3 2
225.4.e.a 4 45.h odd 6 2
225.4.k.a 8 45.l even 12 4
405.4.a.d 2 3.b odd 2 1
405.4.a.e 2 1.a even 1 1 trivial
2025.4.a.j 2 5.b even 2 1
2025.4.a.l 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(405))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 9T + 12 \) Copy content Toggle raw display
$11$ \( T^{2} - 37T + 136 \) Copy content Toggle raw display
$13$ \( T^{2} + 112T + 2608 \) Copy content Toggle raw display
$17$ \( T^{2} + 77T - 3674 \) Copy content Toggle raw display
$19$ \( T^{2} - 35T - 4850 \) Copy content Toggle raw display
$23$ \( T^{2} - 267T + 13458 \) Copy content Toggle raw display
$29$ \( T^{2} + 325T + 13858 \) Copy content Toggle raw display
$31$ \( (T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 638T + 100936 \) Copy content Toggle raw display
$41$ \( T^{2} + 238T - 15539 \) Copy content Toggle raw display
$43$ \( T^{2} + 97T - 60092 \) Copy content Toggle raw display
$47$ \( T^{2} - 901T + 201952 \) Copy content Toggle raw display
$53$ \( T^{2} + 224T - 69956 \) Copy content Toggle raw display
$59$ \( T^{2} - 85T - 18002 \) Copy content Toggle raw display
$61$ \( T^{2} + 247T - 298454 \) Copy content Toggle raw display
$67$ \( T^{2} + 606T - 187503 \) Copy content Toggle raw display
$71$ \( T^{2} + 394T - 61016 \) Copy content Toggle raw display
$73$ \( T^{2} + 811T - 182276 \) Copy content Toggle raw display
$79$ \( T^{2} + 840T + 128748 \) Copy content Toggle raw display
$83$ \( T^{2} - 387 T - 1655532 \) Copy content Toggle raw display
$89$ \( T^{2} - 1065 T - 535050 \) Copy content Toggle raw display
$97$ \( T^{2} + 1031 T + 124162 \) Copy content Toggle raw display
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