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Properties

Label 224.3.d.b
Level $224$
Weight $3$
Character orbit 224.d
Analytic conductor $6.104$
Analytic rank $0$
Dimension $8$
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] = [224,3,Mod(127,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.d (of order \(2\), degree \(1\), 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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1997017344.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{5} - \beta_{3} q^{7} + (\beta_{6} - \beta_{4} + \beta_{2} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{5} - \beta_{3} q^{7} + (\beta_{6} - \beta_{4} + \beta_{2} - 5) q^{9} + (\beta_{7} - \beta_{5} - 2 \beta_{3}) q^{11} + (\beta_{4} + 4) q^{13} + ( - \beta_{7} - 6 \beta_{3} - 2 \beta_1) q^{15} + (2 \beta_{6} - 2) q^{17} + (2 \beta_{5} + 3 \beta_1) q^{19} + (\beta_{4} - \beta_{2}) q^{21} + (\beta_{7} + 2 \beta_{5} + \cdots - 6 \beta_1) q^{23}+ \cdots + (5 \beta_{7} + 19 \beta_{5} + \cdots - 8 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{9} + 32 q^{13} - 16 q^{17} + 104 q^{25} - 80 q^{29} - 176 q^{37} + 144 q^{41} + 256 q^{45} - 56 q^{49} + 48 q^{53} - 400 q^{57} - 192 q^{61} - 304 q^{65} + 576 q^{69} + 272 q^{73} - 112 q^{77} + 504 q^{81} - 160 q^{85} - 80 q^{89} - 608 q^{93} + 528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 24\nu^{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 51\nu^{3} + 42\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 13\nu^{4} + 38\nu^{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 13\nu^{5} - 42\nu^{3} - 32\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{4} + 11\nu^{2} + 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 16\nu^{5} - 77\nu^{3} - 110\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_{2} - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 2\beta_{5} - 6\beta_{3} - 12\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{6} + 11\beta_{4} - 11\beta_{2} + 90 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} + 26\beta_{5} + 70\beta_{3} + 88\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 53\beta_{6} - 97\beta_{4} + 105\beta_{2} - 686 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -75\beta_{7} - 262\beta_{5} - 658\beta_{3} - 704\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
− 2.92812i
− 1.92812i
− 1.27733i
− 0.277334i
0.277334i
1.27733i
1.92812i
2.92812i
0 − 5.85623i 0 −5.78167 0 2.64575i 0 −25.2955 0
127.2 0 − 3.85623i 0 0.490168 0 − 2.64575i 0 −5.87054 0
127.3 0 − 2.55467i 0 9.86836 0 − 2.64575i 0 2.47367 0
127.4 0 − 0.554669i 0 −4.57685 0 2.64575i 0 8.69234 0
127.5 0 0.554669i 0 −4.57685 0 − 2.64575i 0 8.69234 0
127.6 0 2.55467i 0 9.86836 0 2.64575i 0 2.47367 0
127.7 0 3.85623i 0 0.490168 0 2.64575i 0 −5.87054 0
127.8 0 5.85623i 0 −5.78167 0 − 2.64575i 0 −25.2955 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.d.b ✓ 8
3.b odd 2 1 2016.3.m.c 8
4.b odd 2 1 inner 224.3.d.b ✓ 8
7.b odd 2 1 1568.3.d.n 8
8.b even 2 1 448.3.d.e 8
8.d odd 2 1 448.3.d.e 8
12.b even 2 1 2016.3.m.c 8
16.e even 4 1 1792.3.g.d 8
16.e even 4 1 1792.3.g.f 8
16.f odd 4 1 1792.3.g.d 8
16.f odd 4 1 1792.3.g.f 8
28.d even 2 1 1568.3.d.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.d.b ✓ 8 1.a even 1 1 trivial
224.3.d.b ✓ 8 4.b odd 2 1 inner
448.3.d.e 8 8.b even 2 1
448.3.d.e 8 8.d odd 2 1
1568.3.d.n 8 7.b odd 2 1
1568.3.d.n 8 28.d even 2 1
1792.3.g.d 8 16.e even 4 1
1792.3.g.d 8 16.f odd 4 1
1792.3.g.f 8 16.e even 4 1
1792.3.g.f 8 16.f odd 4 1
2016.3.m.c 8 3.b odd 2 1
2016.3.m.c 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 56T_{3}^{6} + 848T_{3}^{4} + 3584T_{3}^{2} + 1024 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 56 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$5$ \( (T^{4} - 76 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + 672 T^{6} + \cdots + 94633984 \) Copy content Toggle raw display
$13$ \( (T^{4} - 16 T^{3} + \cdots - 1728)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 8 T^{3} + \cdots - 4528)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 952 T^{6} + \cdots + 82011136 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 68719476736 \) Copy content Toggle raw display
$29$ \( (T^{4} + 40 T^{3} + \cdots - 432)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 90943258624 \) Copy content Toggle raw display
$37$ \( (T^{4} + 88 T^{3} + \cdots - 5849392)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 72 T^{3} + \cdots - 139824)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 23993382076416 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 40757090320384 \) Copy content Toggle raw display
$53$ \( (T^{4} - 24 T^{3} + \cdots + 396944)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 42455778304 \) Copy content Toggle raw display
$61$ \( (T^{4} + 96 T^{3} + \cdots + 3035776)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 18196435369984 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 28179280429056 \) Copy content Toggle raw display
$73$ \( (T^{4} - 136 T^{3} + \cdots - 3527408)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13710609350656 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 20463101715456 \) Copy content Toggle raw display
$89$ \( (T^{4} + 40 T^{3} + \cdots + 837776)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 264 T^{3} + \cdots + 475344)^{2} \) Copy content Toggle raw display
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