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Properties

Label 1872.2.t.k
Level $1872$
Weight $2$
Character orbit 1872.t
Analytic conductor $14.948$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1872,2,Mod(289,1872)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1872, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1872.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.t (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.9479952584\)
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: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + q^{5} + 4 \zeta_{6} q^{7} + (4 \zeta_{6} - 4) q^{11} + (\zeta_{6} + 3) q^{13} + 3 \zeta_{6} q^{17} + ( - 4 \zeta_{6} + 4) q^{23} - 4 q^{25} + (\zeta_{6} - 1) q^{29} - 4 q^{31} + 4 \zeta_{6} q^{35}+ \cdots - 2 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 4 q^{7} - 4 q^{11} + 7 q^{13} + 3 q^{17} + 4 q^{23} - 8 q^{25} - q^{29} - 8 q^{31} + 4 q^{35} - 3 q^{37} - 9 q^{41} - 8 q^{43} - 16 q^{47} - 9 q^{49} + 18 q^{53} - 4 q^{55} + 4 q^{59} - 7 q^{61}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 1.00000 0 2.00000 − 3.46410i 0 0 0
1153.1 0 0 0 1.00000 0 2.00000 + 3.46410i 0 0 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
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 1872.2.t.k 2
3.b odd 2 1 208.2.i.b 2
4.b odd 2 1 234.2.h.c 2
12.b even 2 1 26.2.c.a ✓ 2
13.c even 3 1 inner 1872.2.t.k 2
24.f even 2 1 832.2.i.e 2
24.h odd 2 1 832.2.i.f 2
39.h odd 6 1 2704.2.a.i 1
39.i odd 6 1 208.2.i.b 2
39.i odd 6 1 2704.2.a.h 1
39.k even 12 2 2704.2.f.g 2
52.i odd 6 1 3042.2.a.k 1
52.j odd 6 1 234.2.h.c 2
52.j odd 6 1 3042.2.a.e 1
52.l even 12 2 3042.2.b.e 2
60.h even 2 1 650.2.e.c 2
60.l odd 4 2 650.2.o.c 4
84.h odd 2 1 1274.2.g.a 2
84.j odd 6 1 1274.2.e.m 2
84.j odd 6 1 1274.2.h.a 2
84.n even 6 1 1274.2.e.n 2
84.n even 6 1 1274.2.h.b 2
156.h even 2 1 338.2.c.e 2
156.l odd 4 2 338.2.e.b 4
156.p even 6 1 26.2.c.a ✓ 2
156.p even 6 1 338.2.a.e 1
156.r even 6 1 338.2.a.c 1
156.r even 6 1 338.2.c.e 2
156.v odd 12 2 338.2.b.b 2
156.v odd 12 2 338.2.e.b 4
312.bh odd 6 1 832.2.i.f 2
312.bn even 6 1 832.2.i.e 2
780.br even 6 1 650.2.e.c 2
780.br even 6 1 8450.2.a.f 1
780.cb even 6 1 8450.2.a.s 1
780.cj odd 12 2 650.2.o.c 4
1092.bt odd 6 1 1274.2.h.a 2
1092.cc odd 6 1 1274.2.e.m 2
1092.ck even 6 1 1274.2.h.b 2
1092.dc even 6 1 1274.2.e.n 2
1092.dd odd 6 1 1274.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.c.a ✓ 2 12.b even 2 1
26.2.c.a ✓ 2 156.p even 6 1
208.2.i.b 2 3.b odd 2 1
208.2.i.b 2 39.i odd 6 1
234.2.h.c 2 4.b odd 2 1
234.2.h.c 2 52.j odd 6 1
338.2.a.c 1 156.r even 6 1
338.2.a.e 1 156.p even 6 1
338.2.b.b 2 156.v odd 12 2
338.2.c.e 2 156.h even 2 1
338.2.c.e 2 156.r even 6 1
338.2.e.b 4 156.l odd 4 2
338.2.e.b 4 156.v odd 12 2
650.2.e.c 2 60.h even 2 1
650.2.e.c 2 780.br even 6 1
650.2.o.c 4 60.l odd 4 2
650.2.o.c 4 780.cj odd 12 2
832.2.i.e 2 24.f even 2 1
832.2.i.e 2 312.bn even 6 1
832.2.i.f 2 24.h odd 2 1
832.2.i.f 2 312.bh odd 6 1
1274.2.e.m 2 84.j odd 6 1
1274.2.e.m 2 1092.cc odd 6 1
1274.2.e.n 2 84.n even 6 1
1274.2.e.n 2 1092.dc even 6 1
1274.2.g.a 2 84.h odd 2 1
1274.2.g.a 2 1092.dd odd 6 1
1274.2.h.a 2 84.j odd 6 1
1274.2.h.a 2 1092.bt odd 6 1
1274.2.h.b 2 84.n even 6 1
1274.2.h.b 2 1092.ck even 6 1
1872.2.t.k 2 1.a even 1 1 trivial
1872.2.t.k 2 13.c even 3 1 inner
2704.2.a.h 1 39.i odd 6 1
2704.2.a.i 1 39.h odd 6 1
2704.2.f.g 2 39.k even 12 2
3042.2.a.e 1 52.j odd 6 1
3042.2.a.k 1 52.i odd 6 1
3042.2.b.e 2 52.l even 12 2
8450.2.a.f 1 780.br even 6 1
8450.2.a.s 1 780.cb even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) 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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