Properties

Label 8.18.a.a
Level $8$
Weight $18$
Character orbit 8.a
Self dual yes
Analytic conductor $14.658$
Analytic rank $0$
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] = [8,18,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, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 18 \)
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: \(14.6577669876\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2146}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2146 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
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 \(\beta = 256\sqrt{2146}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 476) q^{3} + (60 \beta - 26810) q^{5} + (978 \beta - 166584) q^{7} + ( - 952 \beta + 11726669) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 476) q^{3} + (60 \beta - 26810) q^{5} + (978 \beta - 166584) q^{7} + ( - 952 \beta + 11726669) q^{9} + ( - 74781 \beta + 215487340) q^{11} + (329628 \beta + 1333760974) q^{13} + ( - 55370 \beta + 8451176920) q^{15} + ( - 1629816 \beta + 30336751634) q^{17} + (1134309 \beta + 89314980020) q^{19} + ( - 632112 \beta + 137625464352) q^{21} + (18242742 \beta + 264378297304) q^{23} + ( - 3217200 \beta - 255915755425) q^{25} + ( - 116960342 \beta - 78000700568) q^{27} + (71618988 \beta - 3620330045730) q^{29} + (165125256 \beta - 939175570144) q^{31} + (251083096 \beta - 10619790957776) q^{33} + ( - 36215220 \beta + 8257236339120) q^{35} + ( - 2527244916 \beta - 10166232283290) q^{37} + (1176858046 \beta + 45724096081144) q^{39} + (4750902096 \beta - 881520952662) q^{41} + ( - 1772936949 \beta + 96697262984332) q^{43} + (729123260 \beta - 8347763418610) q^{45} + ( - 12557431380 \beta + 50381918882736) q^{47} + ( - 325838304 \beta - 98082609138247) q^{49} + (31112544050 \beta - 243658033250680) q^{51} + ( - 40341828660 \beta - 158909073030026) q^{53} + (14934119010 \beta - 636810354621560) q^{55} + (88775048936 \beta + 117015577653584) q^{57} + ( - 86330682801 \beta - 631025394160868) q^{59} + ( - 97871720196 \beta + 13\!\cdots\!54) q^{61}+ \cdots + ( - 1082075982169 \beta + 12\!\cdots\!32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 952 q^{3} - 53620 q^{5} - 333168 q^{7} + 23453338 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 952 q^{3} - 53620 q^{5} - 333168 q^{7} + 23453338 q^{9} + 430974680 q^{11} + 2667521948 q^{13} + 16902353840 q^{15} + 60673503268 q^{17} + 178629960040 q^{19} + 275250928704 q^{21} + 528756594608 q^{23} - 511831510850 q^{25} - 156001401136 q^{27} - 7240660091460 q^{29} - 1878351140288 q^{31} - 21239581915552 q^{33} + 16514472678240 q^{35} - 20332464566580 q^{37} + 91448192162288 q^{39} - 1763041905324 q^{41} + 193394525968664 q^{43} - 16695526837220 q^{45} + 100763837765472 q^{47} - 196165218276494 q^{49} - 487316066501360 q^{51} - 317818146060052 q^{53} - 12\!\cdots\!20 q^{55}+ \cdots + 25\!\cdots\!64 q^{99}+O(q^{100}) \) 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
−46.3249
46.3249
0 −12335.2 0 −738361. 0 −1.17649e7 0 2.30166e7 0
1.2 0 11383.2 0 684741. 0 1.14317e7 0 436725. 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 8.18.a.a 2
3.b odd 2 1 72.18.a.c 2
4.b odd 2 1 16.18.a.d 2
8.b even 2 1 64.18.a.k 2
8.d odd 2 1 64.18.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.18.a.a 2 1.a even 1 1 trivial
16.18.a.d 2 4.b odd 2 1
64.18.a.h 2 8.d odd 2 1
64.18.a.k 2 8.b even 2 1
72.18.a.c 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 952T_{3} - 140413680 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 952 T - 140413680 \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 505586145500 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 134492404390848 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots - 74\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 13\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 54\!\cdots\!20 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 77\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 23\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 29\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 79\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 31\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 89\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 19\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 20\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 64\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 56\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 40\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 89\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 76\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 23\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 31\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 41\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
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