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Properties

Label 2156.3.h.a
Level $2156$
Weight $3$
Character orbit 2156.h
Self dual yes
Analytic conductor $58.747$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $2$

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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] = [2156,3,Mod(197,2156)] 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("2156.197"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2156.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-5,0,-1,0,0,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(58.7467449961\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 2) q^{3} + (3 \beta - 2) q^{5} + (5 \beta + 3) q^{9} - 11 q^{11} + ( - 7 \beta - 20) q^{15} + (9 \beta - 22) q^{23} + ( - 3 \beta + 51) q^{25} + ( - 9 \beta - 28) q^{27} + (15 \beta - 26) q^{31}+ \cdots + ( - 55 \beta - 33) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{3} - q^{5} + 11 q^{9} - 22 q^{11} - 47 q^{15} - 35 q^{23} + 99 q^{25} - 65 q^{27} - 37 q^{31} + 55 q^{33} + 25 q^{37} + 242 q^{45} - 100 q^{47} - 140 q^{53} + 11 q^{55} + 107 q^{59} - 35 q^{67}+ \cdots - 121 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1079\) \(1277\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
3.37228
−2.37228
0 −5.37228 0 8.11684 0 0 0 19.8614 0
197.2 0 0.372281 0 −9.11684 0 0 0 −8.86141 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2156.3.h.a 2
7.b odd 2 1 44.3.d.a ✓ 2
11.b odd 2 1 CM 2156.3.h.a 2
21.c even 2 1 396.3.f.a 2
28.d even 2 1 176.3.h.b 2
35.c odd 2 1 1100.3.f.a 2
35.f even 4 2 1100.3.e.a 4
56.e even 2 1 704.3.h.f 2
56.h odd 2 1 704.3.h.c 2
77.b even 2 1 44.3.d.a ✓ 2
77.j odd 10 4 484.3.f.b 8
77.l even 10 4 484.3.f.b 8
84.h odd 2 1 1584.3.j.c 2
231.h odd 2 1 396.3.f.a 2
308.g odd 2 1 176.3.h.b 2
385.h even 2 1 1100.3.f.a 2
385.l odd 4 2 1100.3.e.a 4
616.g odd 2 1 704.3.h.f 2
616.o even 2 1 704.3.h.c 2
924.n even 2 1 1584.3.j.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.3.d.a ✓ 2 7.b odd 2 1
44.3.d.a ✓ 2 77.b even 2 1
176.3.h.b 2 28.d even 2 1
176.3.h.b 2 308.g odd 2 1
396.3.f.a 2 21.c even 2 1
396.3.f.a 2 231.h odd 2 1
484.3.f.b 8 77.j odd 10 4
484.3.f.b 8 77.l even 10 4
704.3.h.c 2 56.h odd 2 1
704.3.h.c 2 616.o even 2 1
704.3.h.f 2 56.e even 2 1
704.3.h.f 2 616.g odd 2 1
1100.3.e.a 4 35.f even 4 2
1100.3.e.a 4 385.l odd 4 2
1100.3.f.a 2 35.c odd 2 1
1100.3.f.a 2 385.h even 2 1
1584.3.j.c 2 84.h odd 2 1
1584.3.j.c 2 924.n even 2 1
2156.3.h.a 2 1.a even 1 1 trivial
2156.3.h.a 2 11.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} - 2 \) acting on \(S_{3}^{\mathrm{new}}(2156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 74 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 35T - 362 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 37T - 1514 \) Copy content Toggle raw display
$37$ \( T^{2} - 25T - 3482 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T + 50)^{2} \) Copy content Toggle raw display
$53$ \( (T + 70)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 107T + 1006 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 35T - 12242 \) Copy content Toggle raw display
$71$ \( T^{2} - 133T + 2566 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 97T - 14354 \) Copy content Toggle raw display
$97$ \( T^{2} - 95T - 19202 \) Copy content Toggle raw display
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