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Properties

Label 234.2.h.c
Level $234$
Weight $2$
Character orbit 234.h
Analytic conductor $1.868$
Analytic rank $0$
Dimension $2$
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] = [234,2,Mod(55,234)] 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("234.55"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,0,-1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} + q^{5} - 4 \zeta_{6} q^{7} - q^{8} + ( - \zeta_{6} + 1) q^{10} + ( - 4 \zeta_{6} + 4) q^{11} + (\zeta_{6} + 3) q^{13} - 4 q^{14} + (\zeta_{6} - 1) q^{16} + \cdots + 9 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 2 q^{5} - 4 q^{7} - 2 q^{8} + q^{10} + 4 q^{11} + 7 q^{13} - 8 q^{14} - q^{16} + 3 q^{17} - q^{20} - 4 q^{22} - 4 q^{23} - 8 q^{25} + 5 q^{26} - 4 q^{28} - q^{29} + 8 q^{31}+ \cdots + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 −2.00000 + 3.46410i −1.00000 0 0.500000 + 0.866025i
217.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.00000 0 −2.00000 − 3.46410i −1.00000 0 0.500000 − 0.866025i
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.2.h.c 2
3.b odd 2 1 26.2.c.a ✓ 2
4.b odd 2 1 1872.2.t.k 2
12.b even 2 1 208.2.i.b 2
13.c even 3 1 inner 234.2.h.c 2
13.c even 3 1 3042.2.a.e 1
13.e even 6 1 3042.2.a.k 1
13.f odd 12 2 3042.2.b.e 2
15.d odd 2 1 650.2.e.c 2
15.e even 4 2 650.2.o.c 4
21.c even 2 1 1274.2.g.a 2
21.g even 6 1 1274.2.e.m 2
21.g even 6 1 1274.2.h.a 2
21.h odd 6 1 1274.2.e.n 2
21.h odd 6 1 1274.2.h.b 2
24.f even 2 1 832.2.i.f 2
24.h odd 2 1 832.2.i.e 2
39.d odd 2 1 338.2.c.e 2
39.f even 4 2 338.2.e.b 4
39.h odd 6 1 338.2.a.c 1
39.h odd 6 1 338.2.c.e 2
39.i odd 6 1 26.2.c.a ✓ 2
39.i odd 6 1 338.2.a.e 1
39.k even 12 2 338.2.b.b 2
39.k even 12 2 338.2.e.b 4
52.j odd 6 1 1872.2.t.k 2
156.p even 6 1 208.2.i.b 2
156.p even 6 1 2704.2.a.h 1
156.r even 6 1 2704.2.a.i 1
156.v odd 12 2 2704.2.f.g 2
195.x odd 6 1 650.2.e.c 2
195.x odd 6 1 8450.2.a.f 1
195.y odd 6 1 8450.2.a.s 1
195.bl even 12 2 650.2.o.c 4
273.r even 6 1 1274.2.h.a 2
273.s odd 6 1 1274.2.h.b 2
273.bf even 6 1 1274.2.e.m 2
273.bm odd 6 1 1274.2.e.n 2
273.bn even 6 1 1274.2.g.a 2
312.bh odd 6 1 832.2.i.e 2
312.bn even 6 1 832.2.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.c.a ✓ 2 3.b odd 2 1
26.2.c.a ✓ 2 39.i odd 6 1
208.2.i.b 2 12.b even 2 1
208.2.i.b 2 156.p even 6 1
234.2.h.c 2 1.a even 1 1 trivial
234.2.h.c 2 13.c even 3 1 inner
338.2.a.c 1 39.h odd 6 1
338.2.a.e 1 39.i odd 6 1
338.2.b.b 2 39.k even 12 2
338.2.c.e 2 39.d odd 2 1
338.2.c.e 2 39.h odd 6 1
338.2.e.b 4 39.f even 4 2
338.2.e.b 4 39.k even 12 2
650.2.e.c 2 15.d odd 2 1
650.2.e.c 2 195.x odd 6 1
650.2.o.c 4 15.e even 4 2
650.2.o.c 4 195.bl even 12 2
832.2.i.e 2 24.h odd 2 1
832.2.i.e 2 312.bh odd 6 1
832.2.i.f 2 24.f even 2 1
832.2.i.f 2 312.bn even 6 1
1274.2.e.m 2 21.g even 6 1
1274.2.e.m 2 273.bf even 6 1
1274.2.e.n 2 21.h odd 6 1
1274.2.e.n 2 273.bm odd 6 1
1274.2.g.a 2 21.c even 2 1
1274.2.g.a 2 273.bn even 6 1
1274.2.h.a 2 21.g even 6 1
1274.2.h.a 2 273.r even 6 1
1274.2.h.b 2 21.h odd 6 1
1274.2.h.b 2 273.s odd 6 1
1872.2.t.k 2 4.b odd 2 1
1872.2.t.k 2 52.j odd 6 1
2704.2.a.h 1 156.p even 6 1
2704.2.a.i 1 156.r even 6 1
2704.2.f.g 2 156.v odd 12 2
3042.2.a.e 1 13.c even 3 1
3042.2.a.k 1 13.e even 6 1
3042.2.b.e 2 13.f odd 12 2
8450.2.a.f 1 195.x odd 6 1
8450.2.a.s 1 195.y odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(234, [\chi])\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$73$ \( (T - 11)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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