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Properties

Label 150.10.a.n
Level $150$
Weight $10$
Character orbit 150.a
Self dual yes
Analytic conductor $77.255$
Analytic rank $1$
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] = [150,10,Mod(1,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-32,162,512,0,-2592,8442] 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: yes
Analytic conductor: \(77.2553754246\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{889}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 222 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 30)
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 \(\beta = 5\sqrt{889}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 16 q^{2} + 81 q^{3} + 256 q^{4} - 1296 q^{6} + ( - 49 \beta + 4221) q^{7} - 4096 q^{8} + 6561 q^{9} + ( - 157 \beta + 225) q^{11} + 20736 q^{12} + (741 \beta - 69453) q^{13} + (784 \beta - 67536) q^{14}+ \cdots + ( - 1030077 \beta + 1476225) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{2} + 162 q^{3} + 512 q^{4} - 2592 q^{6} + 8442 q^{7} - 8192 q^{8} + 13122 q^{9} + 450 q^{11} + 41472 q^{12} - 138906 q^{13} - 135072 q^{14} + 131072 q^{16} + 502 q^{17} - 209952 q^{18} - 420168 q^{19}+ \cdots + 2952450 q^{99}+O(q^{100}) \) 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
15.4081
−14.4081
−16.0000 81.0000 256.000 0 −1296.00 −3083.95 −4096.00 6561.00 0
1.2 −16.0000 81.0000 256.000 0 −1296.00 11525.9 −4096.00 6561.00 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 150.10.a.n 2
5.b even 2 1 150.10.a.o 2
5.c odd 4 2 30.10.c.a ✓ 4
15.e even 4 2 90.10.c.a 4
20.e even 4 2 240.10.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.10.c.a ✓ 4 5.c odd 4 2
90.10.c.a 4 15.e even 4 2
150.10.a.n 2 1.a even 1 1 trivial
150.10.a.o 2 5.b even 2 1
240.10.f.a 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 8442T_{7} - 35545384 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(150))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T - 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8442 T - 35545384 \) Copy content Toggle raw display
$11$ \( T^{2} - 450 T - 547773400 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 7379606016 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 142147503224 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 26133037056 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 610163350000 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 9353315299104 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 2406245504224 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 9974707756816 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 558212189313356 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 57874817302384 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 873562225097600 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 27\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 91\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 10\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 99319265306704 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 64\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 63\!\cdots\!44 \) Copy content Toggle raw display
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