Properties

Label 3216.1.dh.a
Level $3216$
Weight $1$
Character orbit 3216.dh
Analytic conductor $1.605$
Analytic rank $0$
Dimension $20$
Projective image $D_{33}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3216,1,Mod(17,3216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3216, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 0, 33, 64]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3216.17");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3216 = 2^{4} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3216.dh (of order \(66\), degree \(20\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 804)
Projective image: \(D_{33}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q - \zeta_{66}^{12} q^{3} + (\zeta_{66}^{31} - \zeta_{66}^{28}) q^{7} + \zeta_{66}^{24} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{66}^{12} q^{3} + (\zeta_{66}^{31} - \zeta_{66}^{28}) q^{7} + \zeta_{66}^{24} q^{9} + ( - \zeta_{66}^{25} + \zeta_{66}^{18}) q^{13} + ( - \zeta_{66}^{32} - \zeta_{66}^{8}) q^{19} + (\zeta_{66}^{10} - \zeta_{66}^{7}) q^{21} - \zeta_{66}^{27} q^{25} + \zeta_{66}^{3} q^{27} + ( - \zeta_{66}^{14} + \zeta_{66}^{9}) q^{31} + ( - \zeta_{66}^{17} - \zeta_{66}^{5}) q^{37} + ( - \zeta_{66}^{30} - \zeta_{66}^{4}) q^{39} + ( - \zeta_{66}^{2} + \zeta_{66}) q^{43} + ( - \zeta_{66}^{29} + \zeta_{66}^{26} - \zeta_{66}^{23}) q^{49} + (\zeta_{66}^{20} - \zeta_{66}^{11}) q^{57} + ( - \zeta_{66}^{15} - \zeta_{66}^{13}) q^{61} + ( - \zeta_{66}^{22} + \zeta_{66}^{19}) q^{63} - \zeta_{66}^{16} q^{67} + (\zeta_{66}^{22} + \zeta_{66}^{6}) q^{73} - \zeta_{66}^{6} q^{75} + (\zeta_{66}^{21} + \zeta_{66}^{11}) q^{79} - \zeta_{66}^{15} q^{81} + (\zeta_{66}^{23} - \zeta_{66}^{20} - \zeta_{66}^{16} + \zeta_{66}^{13}) q^{91} + (\zeta_{66}^{26} - \zeta_{66}^{21}) q^{93} + ( - \zeta_{66}^{21} - \zeta_{66}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} - 2 q^{7} - 2 q^{9} - q^{13} - 2 q^{19} + 2 q^{21} - 2 q^{25} + 2 q^{27} + q^{31} + 2 q^{37} + q^{39} - 2 q^{43} + 3 q^{49} - 9 q^{57} - q^{61} + 9 q^{63} - q^{67} - 12 q^{73} + 2 q^{75} + 12 q^{79} - 2 q^{81} - 4 q^{91} - q^{93} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3216\mathbb{Z}\right)^\times\).

\(n\) \(337\) \(805\) \(1073\) \(2815\)
\(\chi(n)\) \(\zeta_{66}^{4}\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.0475819 + 0.998867i
0.723734 0.690079i
0.580057 0.814576i
0.928368 0.371662i
0.235759 0.971812i
0.928368 + 0.371662i
0.981929 0.189251i
−0.888835 + 0.458227i
−0.995472 + 0.0950560i
0.235759 + 0.971812i
−0.786053 0.618159i
0.0475819 0.998867i
0.723734 + 0.690079i
0.580057 + 0.814576i
−0.327068 0.945001i
−0.786053 + 0.618159i
−0.995472 0.0950560i
−0.888835 0.458227i
0.981929 + 0.189251i
−0.327068 + 0.945001i
0 −0.841254 + 0.540641i 0 0 0 0.759713 + 1.06687i 0 0.415415 0.909632i 0
65.1 0 0.959493 + 0.281733i 0 0 0 0.738471 0.380708i 0 0.841254 + 0.540641i 0
257.1 0 −0.415415 0.909632i 0 0 0 0.279486 + 0.0538665i 0 −0.654861 + 0.755750i 0
449.1 0 0.142315 0.989821i 0 0 0 −0.396666 1.63508i 0 −0.959493 0.281733i 0
689.1 0 0.959493 0.281733i 0 0 0 −0.0395325 0.829889i 0 0.841254 0.540641i 0
881.1 0 0.142315 + 0.989821i 0 0 0 −0.396666 + 1.63508i 0 −0.959493 + 0.281733i 0
977.1 0 0.654861 + 0.755750i 0 0 0 −1.50842 1.18624i 0 −0.142315 + 0.989821i 0
1121.1 0 −0.841254 0.540641i 0 0 0 −1.30379 0.124497i 0 0.415415 + 0.909632i 0
1361.1 0 −0.415415 + 0.909632i 0 0 0 −0.0930932 + 0.268975i 0 −0.654861 0.755750i 0
1601.1 0 0.959493 + 0.281733i 0 0 0 −0.0395325 + 0.829889i 0 0.841254 + 0.540641i 0
2033.1 0 0.142315 0.989821i 0 0 0 −1.21769 + 1.16106i 0 −0.959493 0.281733i 0
2081.1 0 −0.841254 0.540641i 0 0 0 0.759713 1.06687i 0 0.415415 + 0.909632i 0
2177.1 0 0.959493 0.281733i 0 0 0 0.738471 + 0.380708i 0 0.841254 0.540641i 0
2753.1 0 −0.415415 + 0.909632i 0 0 0 0.279486 0.0538665i 0 −0.654861 0.755750i 0
2801.1 0 0.654861 0.755750i 0 0 0 1.78153 + 0.713215i 0 −0.142315 0.989821i 0
2849.1 0 0.142315 + 0.989821i 0 0 0 −1.21769 1.16106i 0 −0.959493 + 0.281733i 0
2897.1 0 −0.415415 0.909632i 0 0 0 −0.0930932 0.268975i 0 −0.654861 + 0.755750i 0
3041.1 0 −0.841254 + 0.540641i 0 0 0 −1.30379 + 0.124497i 0 0.415415 0.909632i 0
3137.1 0 0.654861 0.755750i 0 0 0 −1.50842 + 1.18624i 0 −0.142315 0.989821i 0
3185.1 0 0.654861 + 0.755750i 0 0 0 1.78153 0.713215i 0 −0.142315 + 0.989821i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
67.g even 33 1 inner
201.o odd 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3216.1.dh.a 20
3.b odd 2 1 CM 3216.1.dh.a 20
4.b odd 2 1 804.1.be.a 20
12.b even 2 1 804.1.be.a 20
67.g even 33 1 inner 3216.1.dh.a 20
201.o odd 66 1 inner 3216.1.dh.a 20
268.o odd 66 1 804.1.be.a 20
804.bd even 66 1 804.1.be.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.1.be.a 20 4.b odd 2 1
804.1.be.a 20 12.b even 2 1
804.1.be.a 20 268.o odd 66 1
804.1.be.a 20 804.bd even 66 1
3216.1.dh.a 20 1.a even 1 1 trivial
3216.1.dh.a 20 3.b odd 2 1 CM
3216.1.dh.a 20 67.g even 33 1 inner
3216.1.dh.a 20 201.o odd 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 2 T^{19} - 8 T^{17} - 5 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + T^{19} - T^{17} - T^{16} + 12 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} + 2 T^{19} - 8 T^{17} - 16 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} - T^{19} + T^{17} - T^{16} - 22 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( (T^{10} - T^{9} + 5 T^{8} - 2 T^{7} + 16 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( T^{20} + 2 T^{19} + 3 T^{18} + 4 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{20} \) Copy content Toggle raw display
$53$ \( T^{20} \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( T^{20} + T^{19} - T^{17} - T^{16} + 12 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{20} + T^{19} - T^{17} - T^{16} + T^{14} + T^{13} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( T^{20} + 12 T^{19} + 77 T^{18} + 340 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{20} - 12 T^{19} + 77 T^{18} - 340 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( T^{20} \) Copy content Toggle raw display
$97$ \( T^{20} + T^{19} + 11 T^{18} + 10 T^{17} + \cdots + 1 \) Copy content Toggle raw display
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