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Properties

Label 2250.4.a.l
Level $2250$
Weight $4$
Character orbit 2250.a
Self dual yes
Analytic conductor $132.754$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2250,4,Mod(1,2250)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2250, 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("2250.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2250 = 2 \cdot 3^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2250.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4,0,8,0,0,43,16,0,0,66,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(132.754297513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 750)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + ( - 9 \beta + 26) q^{7} + 8 q^{8} + (8 \beta + 29) q^{11} + (38 \beta - 11) q^{13} + ( - 18 \beta + 52) q^{14} + 16 q^{16} + (23 \beta + 2) q^{17} + ( - 14 \beta - 13) q^{19} + (16 \beta + 58) q^{22}+ \cdots + ( - 774 \beta + 828) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 8 q^{4} + 43 q^{7} + 16 q^{8} + 66 q^{11} + 16 q^{13} + 86 q^{14} + 32 q^{16} + 27 q^{17} - 40 q^{19} + 132 q^{22} - 16 q^{23} + 32 q^{26} + 172 q^{28} + 440 q^{29} - 201 q^{31} + 64 q^{32}+ \cdots + 882 q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
2.00000 0 4.00000 0 0 11.4377 8.00000 0 0
1.2 2.00000 0 4.00000 0 0 31.5623 8.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( +1 \)

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 2250.4.a.l 2
3.b odd 2 1 750.4.a.b ✓ 2
5.b even 2 1 2250.4.a.a 2
15.d odd 2 1 750.4.a.g yes 2
15.e even 4 2 750.4.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
750.4.a.b ✓ 2 3.b odd 2 1
750.4.a.g yes 2 15.d odd 2 1
750.4.c.c 4 15.e even 4 2
2250.4.a.a 2 5.b even 2 1
2250.4.a.l 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2250))\):

\( T_{7}^{2} - 43T_{7} + 361 \) Copy content Toggle raw display
\( T_{11}^{2} - 66T_{11} + 1009 \) Copy content Toggle raw display
\( T_{17}^{2} - 27T_{17} - 479 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 43T + 361 \) Copy content Toggle raw display
$11$ \( T^{2} - 66T + 1009 \) Copy content Toggle raw display
$13$ \( T^{2} - 16T - 1741 \) Copy content Toggle raw display
$17$ \( T^{2} - 27T - 479 \) Copy content Toggle raw display
$19$ \( T^{2} + 40T + 155 \) Copy content Toggle raw display
$23$ \( T^{2} + 16T - 4141 \) Copy content Toggle raw display
$29$ \( T^{2} - 440T + 47555 \) Copy content Toggle raw display
$31$ \( T^{2} + 201T - 8811 \) Copy content Toggle raw display
$37$ \( T^{2} - 428T + 45551 \) Copy content Toggle raw display
$41$ \( T^{2} - 171T - 8651 \) Copy content Toggle raw display
$43$ \( T^{2} - 431T - 50861 \) Copy content Toggle raw display
$47$ \( T^{2} + 388T - 56209 \) Copy content Toggle raw display
$53$ \( T^{2} + 281T - 319561 \) Copy content Toggle raw display
$59$ \( T^{2} - 205T - 248275 \) Copy content Toggle raw display
$61$ \( T^{2} - 489T - 99531 \) Copy content Toggle raw display
$67$ \( T^{2} - 188T - 7984 \) Copy content Toggle raw display
$71$ \( T^{2} - 771T + 126499 \) Copy content Toggle raw display
$73$ \( T^{2} + 1169 T + 185879 \) Copy content Toggle raw display
$79$ \( T^{2} + 930T - 10620 \) Copy content Toggle raw display
$83$ \( T^{2} - 39T - 58481 \) Copy content Toggle raw display
$89$ \( T^{2} - 310T - 697975 \) Copy content Toggle raw display
$97$ \( T^{2} + 2177 T + 1073081 \) Copy content Toggle raw display
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