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Properties

Label 3636.1.o.c
Level $3636$
Weight $1$
Character orbit 3636.o
Analytic conductor $1.815$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -404
Inner twists $4$

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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] = [3636,1,Mod(403,3636)] 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("3636.403"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3636, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3636 = 2^{2} \cdot 3^{2} \cdot 101 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3636.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-6] 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: no
Analytic conductor: \(1.81460038593\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\)

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{42}^{14} q^{2} - \zeta_{42}^{3} q^{3} - \zeta_{42}^{7} q^{4} + ( - \zeta_{42}^{19} + \zeta_{42}^{16}) q^{5} - \zeta_{42}^{17} q^{6} + (\zeta_{42}^{6} - \zeta_{42}) q^{7} + q^{8} + \zeta_{42}^{6} q^{9} + \cdots + (\zeta_{42}^{18} - \zeta_{42}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} - 2 q^{3} - 6 q^{4} + 2 q^{5} + q^{6} - q^{7} + 12 q^{8} - 2 q^{9} - 4 q^{10} - q^{11} + q^{12} - q^{13} - q^{14} + 2 q^{15} - 6 q^{16} - 4 q^{17} + q^{18} + 2 q^{20} - q^{21} - q^{22}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3636\mathbb{Z}\right)^\times\).

\(n\) \(1819\) \(3133\) \(3233\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{42}^{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
403.1
−0.988831 − 0.149042i
0.365341 − 0.930874i
0.0747301 + 0.997204i
0.826239 + 0.563320i
0.955573 − 0.294755i
−0.733052 + 0.680173i
0.365341 + 0.930874i
−0.988831 + 0.149042i
0.826239 − 0.563320i
0.0747301 − 0.997204i
−0.733052 − 0.680173i
0.955573 + 0.294755i
−0.500000 + 0.866025i −0.900969 − 0.433884i −0.500000 − 0.866025i 0.222521 + 0.385418i 0.826239 − 0.563320i −0.365341 + 0.632789i 1.00000 0.623490 + 0.781831i −0.445042
403.2 −0.500000 + 0.866025i −0.900969 + 0.433884i −0.500000 − 0.866025i 0.222521 + 0.385418i 0.0747301 − 0.997204i 0.988831 − 1.71271i 1.00000 0.623490 − 0.781831i −0.445042
403.3 −0.500000 + 0.866025i −0.222521 − 0.974928i −0.500000 − 0.866025i −0.623490 − 1.07992i 0.955573 + 0.294755i −0.826239 + 1.43109i 1.00000 −0.900969 + 0.433884i 1.24698
403.4 −0.500000 + 0.866025i −0.222521 + 0.974928i −0.500000 − 0.866025i −0.623490 − 1.07992i −0.733052 − 0.680173i −0.0747301 + 0.129436i 1.00000 −0.900969 − 0.433884i 1.24698
403.5 −0.500000 + 0.866025i 0.623490 − 0.781831i −0.500000 − 0.866025i 0.900969 + 1.56052i 0.365341 + 0.930874i 0.733052 − 1.26968i 1.00000 −0.222521 − 0.974928i −1.80194
403.6 −0.500000 + 0.866025i 0.623490 + 0.781831i −0.500000 − 0.866025i 0.900969 + 1.56052i −0.988831 + 0.149042i −0.955573 + 1.65510i 1.00000 −0.222521 + 0.974928i −1.80194
1615.1 −0.500000 − 0.866025i −0.900969 − 0.433884i −0.500000 + 0.866025i 0.222521 − 0.385418i 0.0747301 + 0.997204i 0.988831 + 1.71271i 1.00000 0.623490 + 0.781831i −0.445042
1615.2 −0.500000 − 0.866025i −0.900969 + 0.433884i −0.500000 + 0.866025i 0.222521 − 0.385418i 0.826239 + 0.563320i −0.365341 − 0.632789i 1.00000 0.623490 − 0.781831i −0.445042
1615.3 −0.500000 − 0.866025i −0.222521 − 0.974928i −0.500000 + 0.866025i −0.623490 + 1.07992i −0.733052 + 0.680173i −0.0747301 − 0.129436i 1.00000 −0.900969 + 0.433884i 1.24698
1615.4 −0.500000 − 0.866025i −0.222521 + 0.974928i −0.500000 + 0.866025i −0.623490 + 1.07992i 0.955573 − 0.294755i −0.826239 − 1.43109i 1.00000 −0.900969 − 0.433884i 1.24698
1615.5 −0.500000 − 0.866025i 0.623490 − 0.781831i −0.500000 + 0.866025i 0.900969 − 1.56052i −0.988831 − 0.149042i −0.955573 − 1.65510i 1.00000 −0.222521 − 0.974928i −1.80194
1615.6 −0.500000 − 0.866025i 0.623490 + 0.781831i −0.500000 + 0.866025i 0.900969 − 1.56052i 0.365341 − 0.930874i 0.733052 + 1.26968i 1.00000 −0.222521 + 0.974928i −1.80194
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 403.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
404.d odd 2 1 CM by \(\Q(\sqrt{-101}) \)
9.c even 3 1 inner
3636.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3636.1.o.c ✓ 12
4.b odd 2 1 3636.1.o.d yes 12
9.c even 3 1 inner 3636.1.o.c ✓ 12
36.f odd 6 1 3636.1.o.d yes 12
101.b even 2 1 3636.1.o.d yes 12
404.d odd 2 1 CM 3636.1.o.c ✓ 12
909.j even 6 1 3636.1.o.d yes 12
3636.o odd 6 1 inner 3636.1.o.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3636.1.o.c ✓ 12 1.a even 1 1 trivial
3636.1.o.c ✓ 12 9.c even 3 1 inner
3636.1.o.c ✓ 12 404.d odd 2 1 CM
3636.1.o.c ✓ 12 3636.o odd 6 1 inner
3636.1.o.d yes 12 4.b odd 2 1
3636.1.o.d yes 12 36.f odd 6 1
3636.1.o.d yes 12 101.b even 2 1
3636.1.o.d yes 12 909.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3636, [\chi])\):

\( T_{5}^{6} - T_{5}^{5} + 3T_{5}^{4} + 5T_{5}^{2} - 2T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{12} + T_{7}^{11} + 7 T_{7}^{10} + 6 T_{7}^{9} + 34 T_{7}^{8} + 28 T_{7}^{7} + 78 T_{7}^{6} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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