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Properties

Label 361.2.c.a
Level $361$
Weight $2$
Character orbit 361.c
Analytic conductor $2.883$
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] = [361,2,Mod(68,361)] 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("361.68"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(361, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 361.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.88259951297\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{3} + 2 \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - q^{7} - \zeta_{6} q^{9} + 3 q^{11} - 4 q^{12} - 4 \zeta_{6} q^{13} - 6 \zeta_{6} q^{15} + (4 \zeta_{6} - 4) q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + \cdots - 3 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} - 3 q^{5} - 2 q^{7} - q^{9} + 6 q^{11} - 8 q^{12} - 4 q^{13} - 6 q^{15} - 4 q^{16} + 3 q^{17} - 12 q^{20} + 2 q^{21} - 4 q^{25} - 8 q^{27} - 2 q^{28} + 6 q^{29} + 8 q^{31} - 6 q^{33}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −1.00000 + 1.73205i 1.00000 + 1.73205i −1.50000 + 2.59808i 0 −1.00000 0 −0.500000 − 0.866025i 0
292.1 0 −1.00000 − 1.73205i 1.00000 − 1.73205i −1.50000 − 2.59808i 0 −1.00000 0 −0.500000 + 0.866025i 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
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.c.a 2
19.b odd 2 1 361.2.c.c 2
19.c even 3 1 361.2.a.b 1
19.c even 3 1 inner 361.2.c.a 2
19.d odd 6 1 19.2.a.a ✓ 1
19.d odd 6 1 361.2.c.c 2
19.e even 9 6 361.2.e.e 6
19.f odd 18 6 361.2.e.d 6
57.f even 6 1 171.2.a.b 1
57.h odd 6 1 3249.2.a.d 1
76.f even 6 1 304.2.a.f 1
76.g odd 6 1 5776.2.a.c 1
95.h odd 6 1 475.2.a.b 1
95.i even 6 1 9025.2.a.d 1
95.l even 12 2 475.2.b.a 2
133.i even 6 1 931.2.f.b 2
133.j odd 6 1 931.2.f.c 2
133.n odd 6 1 931.2.f.c 2
133.p even 6 1 931.2.a.a 1
133.s even 6 1 931.2.f.b 2
152.l odd 6 1 1216.2.a.o 1
152.o even 6 1 1216.2.a.b 1
209.g even 6 1 2299.2.a.b 1
228.n odd 6 1 2736.2.a.c 1
247.n odd 6 1 3211.2.a.a 1
285.q even 6 1 4275.2.a.i 1
323.i odd 6 1 5491.2.a.b 1
380.s even 6 1 7600.2.a.c 1
399.q odd 6 1 8379.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a ✓ 1 19.d odd 6 1
171.2.a.b 1 57.f even 6 1
304.2.a.f 1 76.f even 6 1
361.2.a.b 1 19.c even 3 1
361.2.c.a 2 1.a even 1 1 trivial
361.2.c.a 2 19.c even 3 1 inner
361.2.c.c 2 19.b odd 2 1
361.2.c.c 2 19.d odd 6 1
361.2.e.d 6 19.f odd 18 6
361.2.e.e 6 19.e even 9 6
475.2.a.b 1 95.h odd 6 1
475.2.b.a 2 95.l even 12 2
931.2.a.a 1 133.p even 6 1
931.2.f.b 2 133.i even 6 1
931.2.f.b 2 133.s even 6 1
931.2.f.c 2 133.j odd 6 1
931.2.f.c 2 133.n odd 6 1
1216.2.a.b 1 152.o even 6 1
1216.2.a.o 1 152.l odd 6 1
2299.2.a.b 1 209.g even 6 1
2736.2.a.c 1 228.n odd 6 1
3211.2.a.a 1 247.n odd 6 1
3249.2.a.d 1 57.h odd 6 1
4275.2.a.i 1 285.q even 6 1
5491.2.a.b 1 323.i odd 6 1
5776.2.a.c 1 76.g odd 6 1
7600.2.a.c 1 380.s even 6 1
8379.2.a.j 1 399.q odd 6 1
9025.2.a.d 1 95.i even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(361, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{3}^{2} + 2T_{3} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
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