↑

Properties

Label 224.3.g.b
Level $224$
Weight $3$
Character orbit 224.g
Analytic conductor $6.104$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,3,Mod(15,224)]
 
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, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.15");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.g (of order \(2\), degree \(1\), 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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.292213762624.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 56)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{3} + (\beta_{4} + \beta_{2} - \beta_1) q^{5} + \beta_{2} q^{7} + ( - \beta_{7} + \beta_{3} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{3} + (\beta_{4} + \beta_{2} - \beta_1) q^{5} + \beta_{2} q^{7} + ( - \beta_{7} + \beta_{3} + 6) q^{9} + (\beta_{7} + \beta_{6} + 4) q^{11} + (3 \beta_{4} - \beta_{2} + \beta_1) q^{13} + (2 \beta_{5} + 4 \beta_{4} + \cdots - \beta_1) q^{15}+ \cdots + (7 \beta_{7} + 7 \beta_{6} - 12 \beta_{3} - 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 48 q^{9} + 32 q^{11} - 80 q^{17} - 56 q^{19} - 16 q^{25} + 32 q^{27} + 32 q^{33} - 56 q^{35} + 128 q^{41} - 56 q^{49} + 368 q^{51} + 56 q^{57} - 104 q^{59} - 72 q^{65} - 304 q^{67} - 112 q^{73} - 72 q^{75} + 48 q^{81} - 72 q^{83} - 512 q^{89} + 56 q^{91} + 64 q^{97} - 256 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 5\nu^{6} + 4\nu^{5} - 6\nu^{4} + 4\nu^{3} - 160\nu^{2} + 96\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 12\nu^{5} + 6\nu^{4} + 12\nu^{3} - 96\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - \nu^{5} - 2\nu^{4} - 2\nu^{3} + 8\nu^{2} + 8\nu - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 14\nu^{4} - 4\nu^{3} - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - \nu^{6} - 4\nu^{5} - 22\nu^{4} + 84\nu^{3} + 48\nu^{2} + 128\nu - 384 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{5} - 4\nu^{4} + 22\nu^{3} - 88\nu - 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + \nu^{6} + 18\nu^{4} - 20\nu^{3} - 8\nu^{2} + 16\nu + 192 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + 3\beta_{4} - 3\beta_{3} + \beta_{2} - 7\beta _1 + 5 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} + 3\beta_{6} + 5\beta_{5} - \beta_{4} - 3\beta_{3} - 7\beta_{2} + \beta _1 + 13 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} - \beta_{6} + \beta_{5} - 21\beta_{4} - 7\beta_{3} + 5\beta_{2} - 3\beta _1 - 71 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{7} + 3\beta_{6} - 3\beta_{5} - \beta_{4} - 35\beta_{3} - 71\beta_{2} - 15\beta _1 - 19 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12\beta_{7} + 23\beta_{6} - 7\beta_{5} - 61\beta_{4} + 89\beta_{3} - 91\beta_{2} + 29\beta _1 - 55 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -84\beta_{7} - 21\beta_{6} - 27\beta_{5} - 153\beta_{4} + 21\beta_{3} + 49\beta_{2} + 9\beta _1 - 27 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
−1.67467 + 1.09337i
−1.67467 − 1.09337i
1.85837 + 0.739226i
1.85837 − 0.739226i
−1.05468 + 1.69931i
−1.05468 − 1.69931i
1.37098 − 1.45617i
1.37098 + 1.45617i
0 −4.56747 0 − 5.73252i 0 − 2.64575i 0 11.8618 0
15.2 0 −4.56747 0 5.73252i 0 2.64575i 0 11.8618 0
15.3 0 −0.0974366 0 − 3.46547i 0 − 2.64575i 0 −8.99051 0
15.4 0 −0.0974366 0 3.46547i 0 2.64575i 0 −8.99051 0
15.5 0 3.44128 0 − 4.88287i 0 2.64575i 0 2.84239 0
15.6 0 3.44128 0 4.88287i 0 − 2.64575i 0 2.84239 0
15.7 0 5.22363 0 − 6.26788i 0 − 2.64575i 0 18.2863 0
15.8 0 5.22363 0 6.26788i 0 2.64575i 0 18.2863 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d 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.g.b 8
3.b odd 2 1 2016.3.g.b 8
4.b odd 2 1 56.3.g.b ✓ 8
7.b odd 2 1 1568.3.g.m 8
8.b even 2 1 56.3.g.b ✓ 8
8.d odd 2 1 inner 224.3.g.b 8
12.b even 2 1 504.3.g.b 8
16.e even 4 2 1792.3.d.j 16
16.f odd 4 2 1792.3.d.j 16
24.f even 2 1 2016.3.g.b 8
24.h odd 2 1 504.3.g.b 8
28.d even 2 1 392.3.g.m 8
28.f even 6 2 392.3.k.n 16
28.g odd 6 2 392.3.k.o 16
56.e even 2 1 1568.3.g.m 8
56.h odd 2 1 392.3.g.m 8
56.j odd 6 2 392.3.k.n 16
56.p even 6 2 392.3.k.o 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.b ✓ 8 4.b odd 2 1
56.3.g.b ✓ 8 8.b even 2 1
224.3.g.b 8 1.a even 1 1 trivial
224.3.g.b 8 8.d odd 2 1 inner
392.3.g.m 8 28.d even 2 1
392.3.g.m 8 56.h odd 2 1
392.3.k.n 16 28.f even 6 2
392.3.k.n 16 56.j odd 6 2
392.3.k.o 16 28.g odd 6 2
392.3.k.o 16 56.p even 6 2
504.3.g.b 8 12.b even 2 1
504.3.g.b 8 24.h odd 2 1
1568.3.g.m 8 7.b odd 2 1
1568.3.g.m 8 56.e even 2 1
1792.3.d.j 16 16.e even 4 2
1792.3.d.j 16 16.f odd 4 2
2016.3.g.b 8 3.b odd 2 1
2016.3.g.b 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} - 22T_{3}^{2} + 80T_{3} + 8 \) 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
$3$ \( (T^{4} - 4 T^{3} - 22 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 108 T^{6} + \cdots + 369664 \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{3} + \cdots - 864)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 908 T^{6} + \cdots + 133448704 \) Copy content Toggle raw display
$17$ \( (T^{4} + 40 T^{3} + \cdots - 752)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 28 T^{3} + \cdots + 1288)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 15607005184 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 13389266944 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 61917364224 \) Copy content Toggle raw display
$37$ \( T^{8} + 7440 T^{6} + \cdots + 554696704 \) Copy content Toggle raw display
$41$ \( (T^{4} - 64 T^{3} + \cdots + 837776)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2716 T^{2} + \cdots - 217952)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 7542537191424 \) Copy content Toggle raw display
$53$ \( T^{8} + 3552 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
$59$ \( (T^{4} + 52 T^{3} + \cdots - 11252856)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 3223777158144 \) Copy content Toggle raw display
$67$ \( (T^{4} + 152 T^{3} + \cdots - 791808)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 221437256269824 \) Copy content Toggle raw display
$73$ \( (T^{4} + 56 T^{3} + \cdots - 1726704)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 89369947930624 \) Copy content Toggle raw display
$83$ \( (T^{4} + 36 T^{3} + \cdots + 3180872)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 256 T^{3} + \cdots - 618736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 32 T^{3} + \cdots + 9539216)^{2} \) Copy content Toggle raw display
show more
show less