Properties

Label 208.4.i.a
Level $208$
Weight $4$
Character orbit 208.i
Analytic conductor $12.272$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,4,Mod(81,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.i (of order \(3\), degree \(2\), 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: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 3) q^{3} + 2 q^{5} - 5 \zeta_{6} q^{7} + 18 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 3) q^{3} + 2 q^{5} - 5 \zeta_{6} q^{7} + 18 \zeta_{6} q^{9} + ( - 13 \zeta_{6} + 13) q^{11} + (52 \zeta_{6} - 39) q^{13} + (6 \zeta_{6} - 6) q^{15} - 27 \zeta_{6} q^{17} + 75 \zeta_{6} q^{19} + 15 q^{21} + (187 \zeta_{6} - 187) q^{23} - 121 q^{25} - 135 q^{27} + ( - 13 \zeta_{6} + 13) q^{29} + 104 q^{31} + 39 \zeta_{6} q^{33} - 10 \zeta_{6} q^{35} + (423 \zeta_{6} - 423) q^{37} + ( - 117 \zeta_{6} - 39) q^{39} + (195 \zeta_{6} - 195) q^{41} + 199 \zeta_{6} q^{43} + 36 \zeta_{6} q^{45} - 388 q^{47} + ( - 318 \zeta_{6} + 318) q^{49} + 81 q^{51} + 618 q^{53} + ( - 26 \zeta_{6} + 26) q^{55} - 225 q^{57} + 491 \zeta_{6} q^{59} - 175 \zeta_{6} q^{61} + ( - 90 \zeta_{6} + 90) q^{63} + (104 \zeta_{6} - 78) q^{65} + ( - 817 \zeta_{6} + 817) q^{67} - 561 \zeta_{6} q^{69} + 79 \zeta_{6} q^{71} + 230 q^{73} + ( - 363 \zeta_{6} + 363) q^{75} - 65 q^{77} - 764 q^{79} + (81 \zeta_{6} - 81) q^{81} + 732 q^{83} - 54 \zeta_{6} q^{85} + 39 \zeta_{6} q^{87} + ( - 1041 \zeta_{6} + 1041) q^{89} + ( - 65 \zeta_{6} + 260) q^{91} + (312 \zeta_{6} - 312) q^{93} + 150 \zeta_{6} q^{95} + 97 \zeta_{6} q^{97} + 234 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 4 q^{5} - 5 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 4 q^{5} - 5 q^{7} + 18 q^{9} + 13 q^{11} - 26 q^{13} - 6 q^{15} - 27 q^{17} + 75 q^{19} + 30 q^{21} - 187 q^{23} - 242 q^{25} - 270 q^{27} + 13 q^{29} + 208 q^{31} + 39 q^{33} - 10 q^{35} - 423 q^{37} - 195 q^{39} - 195 q^{41} + 199 q^{43} + 36 q^{45} - 776 q^{47} + 318 q^{49} + 162 q^{51} + 1236 q^{53} + 26 q^{55} - 450 q^{57} + 491 q^{59} - 175 q^{61} + 90 q^{63} - 52 q^{65} + 817 q^{67} - 561 q^{69} + 79 q^{71} + 460 q^{73} + 363 q^{75} - 130 q^{77} - 1528 q^{79} - 81 q^{81} + 1464 q^{83} - 54 q^{85} + 39 q^{87} + 1041 q^{89} + 455 q^{91} - 312 q^{93} + 150 q^{95} + 97 q^{97} + 468 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.50000 2.59808i 0 2.00000 0 −2.50000 + 4.33013i 0 9.00000 15.5885i 0
113.1 0 −1.50000 + 2.59808i 0 2.00000 0 −2.50000 4.33013i 0 9.00000 + 15.5885i 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 208.4.i.a 2
4.b odd 2 1 26.4.c.a 2
12.b even 2 1 234.4.h.b 2
13.c even 3 1 inner 208.4.i.a 2
52.b odd 2 1 338.4.c.d 2
52.f even 4 2 338.4.e.d 4
52.i odd 6 1 338.4.a.d 1
52.i odd 6 1 338.4.c.d 2
52.j odd 6 1 26.4.c.a 2
52.j odd 6 1 338.4.a.a 1
52.l even 12 2 338.4.b.a 2
52.l even 12 2 338.4.e.d 4
156.p even 6 1 234.4.h.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.c.a 2 4.b odd 2 1
26.4.c.a 2 52.j odd 6 1
208.4.i.a 2 1.a even 1 1 trivial
208.4.i.a 2 13.c even 3 1 inner
234.4.h.b 2 12.b even 2 1
234.4.h.b 2 156.p even 6 1
338.4.a.a 1 52.j odd 6 1
338.4.a.d 1 52.i odd 6 1
338.4.b.a 2 52.l even 12 2
338.4.c.d 2 52.b odd 2 1
338.4.c.d 2 52.i odd 6 1
338.4.e.d 4 52.f even 4 2
338.4.e.d 4 52.l even 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$11$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$13$ \( T^{2} + 26T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 27T + 729 \) Copy content Toggle raw display
$19$ \( T^{2} - 75T + 5625 \) Copy content Toggle raw display
$23$ \( T^{2} + 187T + 34969 \) Copy content Toggle raw display
$29$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$31$ \( (T - 104)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 423T + 178929 \) Copy content Toggle raw display
$41$ \( T^{2} + 195T + 38025 \) Copy content Toggle raw display
$43$ \( T^{2} - 199T + 39601 \) Copy content Toggle raw display
$47$ \( (T + 388)^{2} \) Copy content Toggle raw display
$53$ \( (T - 618)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 491T + 241081 \) Copy content Toggle raw display
$61$ \( T^{2} + 175T + 30625 \) Copy content Toggle raw display
$67$ \( T^{2} - 817T + 667489 \) Copy content Toggle raw display
$71$ \( T^{2} - 79T + 6241 \) Copy content Toggle raw display
$73$ \( (T - 230)^{2} \) Copy content Toggle raw display
$79$ \( (T + 764)^{2} \) Copy content Toggle raw display
$83$ \( (T - 732)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1041 T + 1083681 \) Copy content Toggle raw display
$97$ \( T^{2} - 97T + 9409 \) Copy content Toggle raw display
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