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Properties

Label 115.4.a.c
Level $115$
Weight $4$
Character orbit 115.a
Self dual yes
Analytic conductor $6.785$
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] = [115,4,Mod(1,115)] 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("115.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.78521965066\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{109}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{109})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{2} + ( - \beta - 1) q^{3} + q^{4} + 5 q^{5} + (3 \beta + 3) q^{6} + (5 \beta - 2) q^{7} + 21 q^{8} + (3 \beta + 1) q^{9} - 15 q^{10} + ( - 5 \beta - 11) q^{11} + ( - \beta - 1) q^{12} + ( - 3 \beta - 6) q^{13}+ \cdots + ( - 53 \beta - 416) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} - 3 q^{3} + 2 q^{4} + 10 q^{5} + 9 q^{6} + q^{7} + 42 q^{8} + 5 q^{9} - 30 q^{10} - 27 q^{11} - 3 q^{12} - 15 q^{13} - 3 q^{14} - 15 q^{15} - 142 q^{16} - 79 q^{17} - 15 q^{18} - 71 q^{19}+ \cdots - 885 q^{99}+O(q^{100}) \) 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
5.72015
−4.72015
−3.00000 −6.72015 1.00000 5.00000 20.1605 26.6008 21.0000 18.1605 −15.0000
1.2 −3.00000 3.72015 1.00000 5.00000 −11.1605 −25.6008 21.0000 −13.1605 −15.0000
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(23\) \( +1 \)

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 115.4.a.c ✓ 2
3.b odd 2 1 1035.4.a.g 2
4.b odd 2 1 1840.4.a.h 2
5.b even 2 1 575.4.a.h 2
5.c odd 4 2 575.4.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.c ✓ 2 1.a even 1 1 trivial
575.4.a.h 2 5.b even 2 1
575.4.b.f 4 5.c odd 4 2
1035.4.a.g 2 3.b odd 2 1
1840.4.a.h 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(115))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T - 25 \) Copy content Toggle raw display
$5$ \( (T - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T - 681 \) Copy content Toggle raw display
$11$ \( T^{2} + 27T - 499 \) Copy content Toggle raw display
$13$ \( T^{2} + 15T - 189 \) Copy content Toggle raw display
$17$ \( T^{2} + 79T + 225 \) Copy content Toggle raw display
$19$ \( T^{2} + 71T - 3345 \) Copy content Toggle raw display
$23$ \( (T + 23)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 430T + 43500 \) Copy content Toggle raw display
$31$ \( T^{2} + 305T + 15381 \) Copy content Toggle raw display
$37$ \( T^{2} + 68T - 20208 \) Copy content Toggle raw display
$41$ \( T^{2} + 593T + 84615 \) Copy content Toggle raw display
$43$ \( T^{2} - 648T + 103232 \) Copy content Toggle raw display
$47$ \( T^{2} - 382T - 112740 \) Copy content Toggle raw display
$53$ \( T^{2} + 464T + 1068 \) Copy content Toggle raw display
$59$ \( T^{2} + 18T - 79380 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T - 88523 \) Copy content Toggle raw display
$67$ \( T^{2} - 60T - 156496 \) Copy content Toggle raw display
$71$ \( T^{2} + 1029T + 8315 \) Copy content Toggle raw display
$73$ \( T^{2} - 74T - 103380 \) Copy content Toggle raw display
$79$ \( T^{2} - 692T - 686448 \) Copy content Toggle raw display
$83$ \( T^{2} + 1460T - 32156 \) Copy content Toggle raw display
$89$ \( T^{2} + 220T - 870800 \) Copy content Toggle raw display
$97$ \( T^{2} - 1339 T - 674715 \) Copy content Toggle raw display
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