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Properties

Label 33.8.a.c
Level $33$
Weight $8$
Character orbit 33.a
Self dual yes
Analytic conductor $10.309$
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] = [33,8,Mod(1,33)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(33, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("33.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1] 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: \(10.3087058410\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
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 \(\beta = \frac{1}{2}(1 + 3\sqrt{97})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 27 q^{3} + (\beta + 90) q^{4} + ( - 14 \beta - 90) q^{5} - 27 \beta q^{6} + ( - 42 \beta - 188) q^{7} + ( - 37 \beta + 218) q^{8} + 729 q^{9} + ( - 104 \beta - 3052) q^{10} + 1331 q^{11} + \cdots + 970299 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 54 q^{3} + 181 q^{4} - 194 q^{5} - 27 q^{6} - 418 q^{7} + 399 q^{8} + 1458 q^{9} - 6208 q^{10} + 2662 q^{11} - 4887 q^{12} - 13246 q^{13} - 18542 q^{14} + 5238 q^{15} - 39119 q^{16} - 10256 q^{17}+ \cdots + 1940598 q^{99}+O(q^{100}) \) 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
−4.42443
5.42443
−14.2733 −27.0000 75.7267 109.826 385.379 411.478 746.112 729.000 −1567.58
1.2 15.2733 −27.0000 105.273 −303.826 −412.379 −829.478 −347.112 729.000 −4640.42
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(11\) \( -1 \)

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 33.8.a.c ✓ 2
3.b odd 2 1 99.8.a.b 2
4.b odd 2 1 528.8.a.h 2
11.b odd 2 1 363.8.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.c ✓ 2 1.a even 1 1 trivial
99.8.a.b 2 3.b odd 2 1
363.8.a.c 2 11.b odd 2 1
528.8.a.h 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 218 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 218 \) Copy content Toggle raw display
$3$ \( (T + 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 194T - 33368 \) Copy content Toggle raw display
$7$ \( T^{2} + 418T - 341312 \) Copy content Toggle raw display
$11$ \( (T - 1331)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 13246 T + 43548976 \) Copy content Toggle raw display
$17$ \( T^{2} + 10256 T + 19225084 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 1177576704 \) Copy content Toggle raw display
$23$ \( T^{2} + 13666 T - 870505736 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 4487554116 \) Copy content Toggle raw display
$31$ \( T^{2} + 48040 T + 569571328 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 18695152476 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 49849727804 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 714621373632 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 47516184584 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 1006243830216 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 1058263475744 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 6224547468744 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 2655573007408 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 7547380719912 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 4341768952708 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 177407563168 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 5354532740304 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 10431675116388 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 26135282253956 \) Copy content Toggle raw display
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