Properties

Label 8.22.a.a
Level $8$
Weight $22$
Character orbit 8.a
Self dual yes
Analytic conductor $22.358$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,22,Mod(1,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

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: yes
Analytic conductor: \(22.3581875430\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{358549}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 89637 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 192\sqrt{358549}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 52716) q^{3} + (20 \beta + 1054070) q^{5} + (4222 \beta + 222385896) q^{7} + (105432 \beta + 5536173789) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 52716) q^{3} + (20 \beta + 1054070) q^{5} + (4222 \beta + 222385896) q^{7} + (105432 \beta + 5536173789) q^{9} + (527181 \beta + 26903201660) q^{11} + ( - 4314508 \beta - 245183338466) q^{13} + ( - 2108390 \beta - 319917360840) q^{15} + ( - 30718184 \beta - 3296932336046) q^{17} + (366379451 \beta + 9651198962660) q^{19} + ( - 444952848 \beta - 67527792412128) q^{21} + ( - 944258822 \beta - 204868932888136) q^{23} + (42162800 \beta - 470439074503825) q^{25} + ( - 633773898 \beta - 11\!\cdots\!28) q^{27}+ \cdots + (57\!\cdots\!29 \beta + 88\!\cdots\!52) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 105432 q^{3} + 2108140 q^{5} + 444771792 q^{7} + 11072347578 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 105432 q^{3} + 2108140 q^{5} + 444771792 q^{7} + 11072347578 q^{9} + 53806403320 q^{11} - 490366676932 q^{13} - 639834721680 q^{15} - 6593864672092 q^{17} + 19302397925320 q^{19} - 135055584824256 q^{21} - 409737865776272 q^{23} - 940878149007650 q^{25} - 22\!\cdots\!56 q^{27}+ \cdots + 17\!\cdots\!04 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
299.895
−298.895
0 −167684. 0 3.35342e6 0 7.07779e8 0 1.76574e10 0
1.2 0 62251.6 0 −1.24528e6 0 −2.63007e8 0 −6.58509e9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.22.a.a 2
3.b odd 2 1 72.22.a.b 2
4.b odd 2 1 16.22.a.e 2
8.b even 2 1 64.22.a.k 2
8.d odd 2 1 64.22.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.22.a.a 2 1.a even 1 1 trivial
16.22.a.e 2 4.b odd 2 1
64.22.a.h 2 8.d odd 2 1
64.22.a.k 2 8.b even 2 1
72.22.a.b 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 105432T_{3} - 10438573680 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + \cdots - 10438573680 \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 4175956569500 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 18\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 29\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 18\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 16\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 30\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 56\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 14\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 32\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 96\!\cdots\!48 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 25\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 55\!\cdots\!12 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 56\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 10\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 68\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 36\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 31\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 21\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
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