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Properties

Label 450.5.d.d
Level $450$
Weight $5$
Character orbit 450.d
Analytic conductor $46.516$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,5,Mod(251,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("450.251"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 450.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-16,0,0,166] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.5164833877\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{2} - 8 q^{4} + 83 q^{7} - 16 \beta q^{8} - 57 \beta q^{11} + 41 q^{13} + 166 \beta q^{14} + 64 q^{16} - 363 \beta q^{17} - 139 q^{19} + 228 q^{22} + 159 \beta q^{23} + 82 \beta q^{26} - 664 q^{28} + \cdots + 8976 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} + 166 q^{7} + 82 q^{13} + 128 q^{16} - 278 q^{19} + 456 q^{22} - 1328 q^{28} - 2102 q^{31} + 2904 q^{34} - 3344 q^{37} - 5030 q^{43} - 1272 q^{46} + 8976 q^{49} - 656 q^{52} + 3816 q^{58}+ \cdots + 3586 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
− 1.41421i
1.41421i
− 2.82843i 0 −8.00000 0 0 83.0000 22.6274i 0 0
251.2 2.82843i 0 −8.00000 0 0 83.0000 − 22.6274i 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.5.d.d yes 2
3.b odd 2 1 inner 450.5.d.d yes 2
5.b even 2 1 450.5.d.a ✓ 2
5.c odd 4 2 450.5.b.b 4
15.d odd 2 1 450.5.d.a ✓ 2
15.e even 4 2 450.5.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.5.b.b 4 5.c odd 4 2
450.5.b.b 4 15.e even 4 2
450.5.d.a ✓ 2 5.b even 2 1
450.5.d.a ✓ 2 15.d odd 2 1
450.5.d.d yes 2 1.a even 1 1 trivial
450.5.d.d yes 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 83 \) acting on \(S_{5}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 83)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6498 \) Copy content Toggle raw display
$13$ \( (T - 41)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 263538 \) Copy content Toggle raw display
$19$ \( (T + 139)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 50562 \) Copy content Toggle raw display
$29$ \( T^{2} + 455058 \) Copy content Toggle raw display
$31$ \( (T + 1051)^{2} \) Copy content Toggle raw display
$37$ \( (T + 1672)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 691488 \) Copy content Toggle raw display
$43$ \( (T + 2515)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 8694450 \) Copy content Toggle raw display
$53$ \( T^{2} + 152352 \) Copy content Toggle raw display
$59$ \( T^{2} + 563922 \) Copy content Toggle raw display
$61$ \( (T - 5825)^{2} \) Copy content Toggle raw display
$67$ \( (T - 7259)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 47706912 \) Copy content Toggle raw display
$73$ \( (T + 4552)^{2} \) Copy content Toggle raw display
$79$ \( (T - 9296)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 63686898 \) Copy content Toggle raw display
$89$ \( T^{2} + 36911232 \) Copy content Toggle raw display
$97$ \( (T - 1793)^{2} \) Copy content Toggle raw display
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