↑

Properties

Label 208.4.k.b
Level $208$
Weight $4$
Character orbit 208.k
Analytic conductor $12.272$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,4,Mod(31,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.k (of order \(4\), degree \(2\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 12x^{10} + 120x^{8} - 15654x^{6} + 63480x^{4} + 3358092x^{2} + 148035889 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 2) q^{5}+ \cdots + (3 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 2) q^{5}+ \cdots + (6 \beta_{11} - 25 \beta_{9} + \cdots - 239 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 28 q^{5} - 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 28 q^{5} - 180 q^{9} + 12 q^{13} - 296 q^{21} - 1432 q^{29} - 928 q^{33} + 244 q^{37} - 92 q^{41} - 1892 q^{45} + 2648 q^{53} + 424 q^{57} - 344 q^{61} + 1284 q^{65} - 2236 q^{73} + 1212 q^{81} + 5152 q^{85} + 5140 q^{89} + 3096 q^{93} + 316 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 12x^{10} + 120x^{8} - 15654x^{6} + 63480x^{4} + 3358092x^{2} + 148035889 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 250\nu^{10} + 12522\nu^{8} - 135577\nu^{6} - 1091814\nu^{4} - 36362402\nu^{2} + 3634295067 ) / 206522658 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 857\nu^{11} - 20927\nu^{9} + 8149\nu^{7} - 7366363\nu^{5} + 274052153\nu^{3} - 1752644183\nu ) / 9500042268 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 12\nu^{9} + 120\nu^{7} - 15654\nu^{5} + 63480\nu^{3} - 3078251\nu ) / 6436343 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27\nu^{10} - 205\nu^{8} - 3108\nu^{6} - 206297\nu^{4} + 5797311\nu^{2} + 50931062 ) / 5037138 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2333\nu^{11} - 3215\nu^{9} + 185269\nu^{7} - 30471667\nu^{5} + 367748633\nu^{3} + 12703941877\nu ) / 9500042268 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2563 \nu^{11} - 19803 \nu^{10} + 60232 \nu^{9} + 261211 \nu^{8} - 614909 \nu^{7} + \cdots - 33777928064 ) / 9500042268 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2563 \nu^{11} - 19803 \nu^{10} - 60232 \nu^{9} + 261211 \nu^{8} + 614909 \nu^{7} + \cdots - 33777928064 ) / 9500042268 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2460 \nu^{11} + 66815 \nu^{10} - 100614 \nu^{9} + 3466353 \nu^{8} - 1266408 \nu^{7} + \cdots + 1350685887579 ) / 9500042268 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2460 \nu^{11} - 66815 \nu^{10} - 100614 \nu^{9} - 3466353 \nu^{8} - 1266408 \nu^{7} + \cdots - 1350685887579 ) / 9500042268 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17999 \nu^{11} + 11224 \nu^{10} + 302744 \nu^{9} - 84318 \nu^{8} - 8272529 \nu^{7} + \cdots - 26588532933 ) / 9500042268 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17999 \nu^{11} + 11224 \nu^{10} - 302744 \nu^{9} - 84318 \nu^{8} + 8272529 \nu^{7} + \cdots - 26588532933 ) / 9500042268 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} - 3\beta_{7} - 3\beta_{6} - 3\beta_{4} + \beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{9} + 3\beta_{8} + 6\beta_{7} - 6\beta_{6} + 2\beta_{5} + 2\beta_{3} - 62\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -27\beta_{11} - 27\beta_{10} + 6\beta_{9} - 6\beta_{8} - 45\beta_{7} - 45\beta_{6} - 27\beta_{4} + 87\beta _1 - 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18 \beta_{11} + 18 \beta_{10} - 24 \beta_{9} - 24 \beta_{8} + 282 \beta_{7} - 282 \beta_{6} + \cdots - 1169 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 82 \beta_{11} + 82 \beta_{10} + 72 \beta_{9} - 72 \beta_{8} + 450 \beta_{7} + 450 \beta_{6} + \cdots + 8145 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 630 \beta_{11} - 630 \beta_{10} - 348 \beta_{9} - 348 \beta_{8} + 4038 \beta_{7} + \cdots - 40945 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 8991 \beta_{11} + 8991 \beta_{10} + 726 \beta_{9} - 726 \beta_{8} - 9513 \beta_{7} - 9513 \beta_{6} + \cdots - 335431 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5400 \beta_{11} + 5400 \beta_{10} + 17517 \beta_{9} + 17517 \beta_{8} + 84630 \beta_{7} + \cdots - 1175806 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 333869 \beta_{11} - 333869 \beta_{10} + 67932 \beta_{9} - 67932 \beta_{8} + 331689 \beta_{7} + \cdots - 3975891 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 292572 \beta_{11} + 292572 \beta_{10} - 734580 \beta_{9} - 734580 \beta_{8} + \cdots + 1581055 \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
4.66017 − 1.13261i
2.33573 − 4.18860i
2.08018 + 4.32121i
−2.08018 + 4.32121i
−2.33573 − 4.18860i
−4.66017 − 1.13261i
−4.66017 + 1.13261i
−2.33573 + 4.18860i
−2.08018 − 4.32121i
2.08018 − 4.32121i
2.33573 + 4.18860i
4.66017 + 1.13261i
0 − 9.32034i 0 5.94407 − 5.94407i 0 3.42421 − 3.42421i 0 −59.8688 0
31.2 0 − 4.67146i 0 −1.84098 + 1.84098i 0 −12.5597 + 12.5597i 0 5.17744 0
31.3 0 − 4.16037i 0 −11.1031 + 11.1031i 0 24.2184 − 24.2184i 0 9.69135 0
31.4 0 4.16037i 0 −11.1031 + 11.1031i 0 −24.2184 + 24.2184i 0 9.69135 0
31.5 0 4.67146i 0 −1.84098 + 1.84098i 0 12.5597 − 12.5597i 0 5.17744 0
31.6 0 9.32034i 0 5.94407 − 5.94407i 0 −3.42421 + 3.42421i 0 −59.8688 0
47.1 0 − 9.32034i 0 5.94407 + 5.94407i 0 −3.42421 − 3.42421i 0 −59.8688 0
47.2 0 − 4.67146i 0 −1.84098 − 1.84098i 0 12.5597 + 12.5597i 0 5.17744 0
47.3 0 − 4.16037i 0 −11.1031 − 11.1031i 0 −24.2184 − 24.2184i 0 9.69135 0
47.4 0 4.16037i 0 −11.1031 − 11.1031i 0 24.2184 + 24.2184i 0 9.69135 0
47.5 0 4.67146i 0 −1.84098 − 1.84098i 0 −12.5597 − 12.5597i 0 5.17744 0
47.6 0 9.32034i 0 5.94407 + 5.94407i 0 3.42421 + 3.42421i 0 −59.8688 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.d odd 4 1 inner
52.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.k.b ✓ 12
4.b odd 2 1 inner 208.4.k.b ✓ 12
13.d odd 4 1 inner 208.4.k.b ✓ 12
52.f even 4 1 inner 208.4.k.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
208.4.k.b ✓ 12 1.a even 1 1 trivial
208.4.k.b ✓ 12 4.b odd 2 1 inner
208.4.k.b ✓ 12 13.d odd 4 1 inner
208.4.k.b ✓ 12 52.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 126T_{3}^{4} + 3777T_{3}^{2} + 32812 \) acting on \(S_{4}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 126 T^{4} + \cdots + 32812)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 14 T^{5} + \cdots + 118098)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 75321118143076 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 79\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{6} - 6 T^{5} + \cdots + 10604499373)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 15266 T^{4} + \cdots + 102517793856)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} - 15172 T^{4} + \cdots - 8334772992)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 358 T^{2} + \cdots - 3077532)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 9218127356258)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 48065090580000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 4120348394032)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{3} - 662 T^{2} + \cdots - 2540196)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{3} + 86 T^{2} + \cdots + 5586840)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 53\!\cdots\!92)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less