Properties

Label 224.2.x.a
Level $224$
Weight $2$
Character orbit 224.x
Analytic conductor $1.789$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,2,Mod(27,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.27");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 224.x (of order \(8\), degree \(4\), 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: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{2} + (\beta_{5} + \beta_{3}) q^{4} + (\beta_{4} + 2 \beta_{2}) q^{7} + (\beta_{7} + 2 \beta_{6}) q^{8} - 3 \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2}) q^{2} + (\beta_{5} + \beta_{3}) q^{4} + (\beta_{4} + 2 \beta_{2}) q^{7} + (\beta_{7} + 2 \beta_{6}) q^{8} - 3 \beta_{6} q^{9} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 3 \beta_{3}) q^{11} + (\beta_{5} + 3 \beta_{3}) q^{14} + (3 \beta_1 - 2) q^{16} - 3 \beta_1 q^{18} + ( - 3 \beta_{7} + 4 \beta_{6} - \beta_1 - 4) q^{22} + (2 \beta_{5} - 5 \beta_{3} - 2 \beta_1 + 5) q^{23} - 5 \beta_{4} q^{25} + (3 \beta_{7} + 2 \beta_{6}) q^{28} + ( - 4 \beta_{5} - 3 \beta_{4} + \beta_{3} - 4 \beta_{2}) q^{29} + ( - 5 \beta_{4} + \beta_{2}) q^{32} + (3 \beta_{4} - 3 \beta_{2}) q^{36} + ( - 4 \beta_{7} - \beta_{6} - 4 \beta_{5} - \beta_{3}) q^{37} + (7 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 7) q^{43} + ( - 3 \beta_{4} - 5 \beta_{2} + \beta_1 + 6) q^{44} + ( - 5 \beta_{7} + 4 \beta_{6} + 7 \beta_{4} + 3 \beta_{2}) q^{46} + 7 \beta_{3} q^{49} + ( - 5 \beta_{5} + 5 \beta_{3}) q^{50} + (3 \beta_{4} - 4 \beta_{2} + 4 \beta_1 + 3) q^{53} + (5 \beta_1 - 6) q^{56} + (\beta_{7} - 8 \beta_{6} - 3 \beta_{5} - 5 \beta_{3}) q^{58} + ( - 6 \beta_1 + 3) q^{63} + ( - 5 \beta_{5} + 7 \beta_{3}) q^{64} + (6 \beta_{7} - 5 \beta_{6} + 6 \beta_1 - 1) q^{67} + 16 \beta_{6} q^{71} + (3 \beta_{5} - 9 \beta_{3}) q^{72} + ( - \beta_{7} - 8 \beta_{6} - 5 \beta_1 + 8) q^{74} + ( - 4 \beta_{7} + 9 \beta_{6} - 4 \beta_1 - 5) q^{77} + ( - 6 \beta_{7} - \beta_{6} - \beta_{4} + 6 \beta_{2}) q^{79} + 9 \beta_{3} q^{81} + (7 \beta_{5} - 9 \beta_{4} - 3 \beta_{3} - 5 \beta_{2}) q^{86} + ( - 3 \beta_{5} + 5 \beta_{4} - 7 \beta_{3} + 7 \beta_{2}) q^{88} + (7 \beta_{5} - \beta_{3} - \beta_1 + 10) q^{92} + 7 \beta_{7} q^{98} + (6 \beta_{5} - 9 \beta_{4} + 3 \beta_{3} - 6 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{16} - 12 q^{18} - 36 q^{22} + 32 q^{23} - 48 q^{43} + 52 q^{44} + 40 q^{53} - 28 q^{56} + 16 q^{67} + 44 q^{74} - 56 q^{77} + 76 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 5\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 5\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 7\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 7\nu^{3} ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 5\nu^{3} ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7\beta_{7} - 10\beta_{6} \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
−1.28897 + 0.581861i
0.581861 1.28897i
−1.28897 0.581861i
0.581861 + 1.28897i
−0.581861 + 1.28897i
1.28897 0.581861i
−0.581861 1.28897i
1.28897 + 0.581861i
−1.28897 + 0.581861i 0 1.32288 1.50000i 0 0 −1.87083 + 1.87083i −0.832353 + 2.70318i 2.12132 2.12132i 0
27.2 0.581861 1.28897i 0 −1.32288 1.50000i 0 0 1.87083 1.87083i −2.70318 + 0.832353i 2.12132 2.12132i 0
83.1 −1.28897 0.581861i 0 1.32288 + 1.50000i 0 0 −1.87083 1.87083i −0.832353 2.70318i 2.12132 + 2.12132i 0
83.2 0.581861 + 1.28897i 0 −1.32288 + 1.50000i 0 0 1.87083 + 1.87083i −2.70318 0.832353i 2.12132 + 2.12132i 0
139.1 −0.581861 + 1.28897i 0 −1.32288 1.50000i 0 0 −1.87083 + 1.87083i 2.70318 0.832353i −2.12132 + 2.12132i 0
139.2 1.28897 0.581861i 0 1.32288 1.50000i 0 0 1.87083 1.87083i 0.832353 2.70318i −2.12132 + 2.12132i 0
195.1 −0.581861 1.28897i 0 −1.32288 + 1.50000i 0 0 −1.87083 1.87083i 2.70318 + 0.832353i −2.12132 2.12132i 0
195.2 1.28897 + 0.581861i 0 1.32288 + 1.50000i 0 0 1.87083 + 1.87083i 0.832353 + 2.70318i −2.12132 2.12132i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
32.h odd 8 1 inner
224.x even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.x.a 8
4.b odd 2 1 896.2.x.a 8
7.b odd 2 1 CM 224.2.x.a 8
28.d even 2 1 896.2.x.a 8
32.g even 8 1 896.2.x.a 8
32.h odd 8 1 inner 224.2.x.a 8
224.v odd 8 1 896.2.x.a 8
224.x even 8 1 inner 224.2.x.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.x.a 8 1.a even 1 1 trivial
224.2.x.a 8 7.b odd 2 1 CM
224.2.x.a 8 32.h odd 8 1 inner
224.2.x.a 8 224.x even 8 1 inner
896.2.x.a 8 4.b odd 2 1
896.2.x.a 8 28.d even 2 1
896.2.x.a 8 32.g even 8 1
896.2.x.a 8 224.v odd 8 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{4} + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{6} - 176 T^{5} + \cdots + 42436 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 16 T^{3} + 128 T^{2} - 288 T + 324)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 108 T^{6} + 232 T^{5} + \cdots + 1522756 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 76 T^{6} - 888 T^{5} + \cdots + 1674436 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} + 48 T^{7} + 1036 T^{6} + \cdots + 111556 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 40 T^{7} + 812 T^{6} + \cdots + 31158724 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} + 364 T^{6} + \cdots + 24462916 \) Copy content Toggle raw display
$71$ \( (T^{4} + 65536)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 316 T^{2} + 8836)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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