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Properties

Label 208.3.bd.c
Level $208$
Weight $3$
Character orbit 208.bd
Analytic conductor $5.668$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,3,Mod(33,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.33");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 208.bd (of order \(12\), degree \(4\), 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: \(5.66758949869\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 7 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{7} + ( - 6 \zeta_{12}^{2} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - 7 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{7} + ( - 6 \zeta_{12}^{2} + 6) q^{9} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 1) q^{11} + ( - 2 \zeta_{12}^{3} + 12 \zeta_{12}^{2} + 8 \zeta_{12} - 9) q^{13} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{15} + (13 \zeta_{12}^{2} - 6 \zeta_{12} + 13) q^{17} + (17 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 12 \zeta_{12} + 12) q^{19} + (10 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 10) q^{21} + (21 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 21 \zeta_{12} - 4) q^{23} + 19 \zeta_{12}^{3} q^{25} + ( - 15 \zeta_{12}^{3} + 30 \zeta_{12}) q^{27} + ( - 16 \zeta_{12}^{3} - 27 \zeta_{12}^{2} - 16 \zeta_{12}) q^{29} + ( - 37 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 10 \zeta_{12} + 37) q^{31} + (9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{33} + (2 \zeta_{12}^{3} + 21 \zeta_{12}^{2} - \zeta_{12} - 21) q^{35} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 13 \zeta_{12} + 7) q^{37} + (15 \zeta_{12}^{3} + 14 \zeta_{12}^{2} - 21 \zeta_{12} - 4) q^{39} + ( - 26 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 13 \zeta_{12} - 13) q^{41} + ( - 20 \zeta_{12}^{2} + 39 \zeta_{12} - 20) q^{43} + (12 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{45} + ( - 33 \zeta_{12}^{3} - 33) q^{47} + ( - 25 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 25 \zeta_{12} + 14) q^{49} + (39 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 6) q^{51} + ( - 22 \zeta_{12}^{3} + 44 \zeta_{12} - 42) q^{53} + ( - 9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 9 \zeta_{12}) q^{55} + (22 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12} - 22) q^{57} + (13 \zeta_{12}^{3} - 23 \zeta_{12}^{2} + 10 \zeta_{12} + 10) q^{59} + ( - 12 \zeta_{12}^{3} - 39 \zeta_{12}^{2} + 6 \zeta_{12} + 39) q^{61} + ( - 24 \zeta_{12}^{3} - 24 \zeta_{12}^{2} - 18 \zeta_{12} + 42) q^{63} + ( - 7 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 2 \zeta_{12} - 25) q^{65} + ( - 23 \zeta_{12}^{3} + 23 \zeta_{12}^{2} - 10 \zeta_{12} - 33) q^{67} + ( - 21 \zeta_{12}^{2} - 6 \zeta_{12} - 21) q^{69} + ( - 25 \zeta_{12}^{3} - 29 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{71} + ( - 7 \zeta_{12}^{3} - 54 \zeta_{12}^{2} - 54 \zeta_{12} - 7) q^{73} + (19 \zeta_{12}^{2} - 38) q^{75} + (15 \zeta_{12}^{3} + 64 \zeta_{12}^{2} - 32) q^{77} + (36 \zeta_{12}^{3} - 72 \zeta_{12} - 48) q^{79} - 9 \zeta_{12}^{2} q^{81} + (25 \zeta_{12}^{3} - 46 \zeta_{12}^{2} + 46 \zeta_{12} - 25) q^{83} + ( - 12 \zeta_{12}^{3} + 45 \zeta_{12}^{2} - 33 \zeta_{12} - 33) q^{85} + ( - 54 \zeta_{12}^{3} - 48 \zeta_{12}^{2} + 27 \zeta_{12} + 48) q^{87} + ( - 56 \zeta_{12}^{3} - 56 \zeta_{12}^{2} + 43 \zeta_{12} + 13) q^{89} + (37 \zeta_{12}^{3} + 25 \zeta_{12}^{2} + 86 \zeta_{12} - 22) q^{91} + (17 \zeta_{12}^{3} - 17 \zeta_{12}^{2} + 47 \zeta_{12} + 64) q^{93} + (7 \zeta_{12}^{2} - 51 \zeta_{12} + 7) q^{95} + (69 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 71 \zeta_{12} + 71) q^{97} + (24 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12} + 24) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 22 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 22 q^{7} + 12 q^{9} + 6 q^{11} - 12 q^{13} - 6 q^{15} + 78 q^{17} + 58 q^{19} + 42 q^{21} - 12 q^{23} - 54 q^{29} + 128 q^{31} - 30 q^{33} - 42 q^{35} + 40 q^{37} + 12 q^{39} - 120 q^{43} + 36 q^{45} - 132 q^{47} + 42 q^{49} - 168 q^{53} - 6 q^{55} - 102 q^{57} - 6 q^{59} + 78 q^{61} + 120 q^{63} - 120 q^{65} - 86 q^{67} - 126 q^{69} - 42 q^{71} - 136 q^{73} - 114 q^{75} - 192 q^{79} - 18 q^{81} - 192 q^{83} - 42 q^{85} + 96 q^{87} - 60 q^{89} - 38 q^{91} + 222 q^{93} + 42 q^{95} + 280 q^{97} + 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 0.866025 − 1.50000i 0 −1.73205 − 1.73205i 0 8.96410 + 2.40192i 0 3.00000 + 5.19615i 0
97.1 0 −0.866025 + 1.50000i 0 1.73205 − 1.73205i 0 2.03590 − 7.59808i 0 3.00000 + 5.19615i 0
145.1 0 0.866025 + 1.50000i 0 −1.73205 + 1.73205i 0 8.96410 − 2.40192i 0 3.00000 − 5.19615i 0
193.1 0 −0.866025 − 1.50000i 0 1.73205 + 1.73205i 0 2.03590 + 7.59808i 0 3.00000 − 5.19615i 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.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.3.bd.c 4
4.b odd 2 1 26.3.f.a ✓ 4
12.b even 2 1 234.3.bb.b 4
13.f odd 12 1 inner 208.3.bd.c 4
52.b odd 2 1 338.3.f.d 4
52.f even 4 1 338.3.f.c 4
52.f even 4 1 338.3.f.f 4
52.i odd 6 1 338.3.d.e 4
52.i odd 6 1 338.3.f.c 4
52.j odd 6 1 338.3.d.d 4
52.j odd 6 1 338.3.f.f 4
52.l even 12 1 26.3.f.a ✓ 4
52.l even 12 1 338.3.d.d 4
52.l even 12 1 338.3.d.e 4
52.l even 12 1 338.3.f.d 4
156.v odd 12 1 234.3.bb.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.3.f.a ✓ 4 4.b odd 2 1
26.3.f.a ✓ 4 52.l even 12 1
208.3.bd.c 4 1.a even 1 1 trivial
208.3.bd.c 4 13.f odd 12 1 inner
234.3.bb.b 4 12.b even 2 1
234.3.bb.b 4 156.v odd 12 1
338.3.d.d 4 52.j odd 6 1
338.3.d.d 4 52.l even 12 1
338.3.d.e 4 52.i odd 6 1
338.3.d.e 4 52.l even 12 1
338.3.f.c 4 52.f even 4 1
338.3.f.c 4 52.i odd 6 1
338.3.f.d 4 52.b odd 2 1
338.3.f.d 4 52.l even 12 1
338.3.f.f 4 52.f even 4 1
338.3.f.f 4 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 36 \) Copy content Toggle raw display
$7$ \( T^{4} - 22 T^{3} + 221 T^{2} + \cdots + 5329 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 45 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} + 182 T^{2} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{4} - 78 T^{3} + 2499 T^{2} + \cdots + 221841 \) Copy content Toggle raw display
$19$ \( T^{4} - 58 T^{3} + 1325 T^{2} + \cdots + 167281 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} - 381 T^{2} + \cdots + 184041 \) Copy content Toggle raw display
$29$ \( T^{4} + 54 T^{3} + 2955 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$31$ \( T^{4} - 128 T^{3} + 8192 T^{2} + \cdots + 3602404 \) Copy content Toggle raw display
$37$ \( T^{4} - 40 T^{3} + 401 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$41$ \( T^{4} + 1521 T^{2} - 39546 T + 257049 \) Copy content Toggle raw display
$43$ \( T^{4} + 120 T^{3} + 4479 T^{2} + \cdots + 103041 \) Copy content Toggle raw display
$47$ \( (T^{2} + 66 T + 2178)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 84 T + 312)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 1305 T^{2} + \cdots + 84681 \) Copy content Toggle raw display
$61$ \( T^{4} - 78 T^{3} + 4671 T^{2} + \cdots + 1996569 \) Copy content Toggle raw display
$67$ \( T^{4} + 86 T^{3} + 4985 T^{2} + \cdots + 2399401 \) Copy content Toggle raw display
$71$ \( T^{4} + 42 T^{3} + 3357 T^{2} + \cdots + 154449 \) Copy content Toggle raw display
$73$ \( T^{4} + 136 T^{3} + 9248 T^{2} + \cdots + 4251844 \) Copy content Toggle raw display
$79$ \( (T^{2} + 96 T - 1584)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 192 T^{3} + 18432 T^{2} + \cdots + 2056356 \) Copy content Toggle raw display
$89$ \( T^{4} + 60 T^{3} + 5661 T^{2} + \cdots + 21594609 \) Copy content Toggle raw display
$97$ \( T^{4} - 280 T^{3} + \cdots + 20043529 \) Copy content Toggle raw display
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