Properties

Label 4.22.a.a
Level $4$
Weight $22$
Character orbit 4.a
Self dual yes
Analytic conductor $11.179$
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] = [4,22,Mod(1,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 4.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: \(11.1790937715\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2161}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 540 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 7 \)
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 \(\beta = 2688\sqrt{2161}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 32820) q^{3} + ( - 204 \beta + 6844662) q^{5} + (5886 \beta - 130254040) q^{7} + ( - 65640 \beta + 6230767581) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 32820) q^{3} + ( - 204 \beta + 6844662) q^{5} + (5886 \beta - 130254040) q^{7} + ( - 65640 \beta + 6230767581) q^{9} + (314445 \beta + 72717981660) q^{11} + (324756 \beta + 714450208670) q^{13} + ( - 13539942 \beta + 3409891357176) q^{15} + (84858072 \beta + 920310288210) q^{17} + ( - 265449285 \beta - 8390371964284) q^{19} + (323432560 \beta - 96178755501024) q^{21} + (627682938 \beta - 159845962713480) q^{23} + ( - 2792622096 \beta + 219803147959663) q^{25} + (2075280822 \beta + 886085884611720) q^{27} + (4723712580 \beta + 18\!\cdots\!46) q^{29}+ \cdots + ( - 28\!\cdots\!55 \beta + 13\!\cdots\!60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 65640 q^{3} + 13689324 q^{5} - 260508080 q^{7} + 12461535162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 65640 q^{3} + 13689324 q^{5} - 260508080 q^{7} + 12461535162 q^{9} + 145435963320 q^{11} + 1428900417340 q^{13} + 6819782714352 q^{15} + 1840620576420 q^{17} - 16780743928568 q^{19} - 192357511002048 q^{21} - 319691925426960 q^{23} + 439606295919326 q^{25} + 17\!\cdots\!40 q^{27}+ \cdots + 26\!\cdots\!20 q^{99}+O(q^{100}) \) 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
23.7433
−22.7433
0 −92135.9 0 −1.86463e7 0 6.05236e8 0 −1.97134e9 0
1.2 0 157776. 0 3.23357e7 0 −8.65744e8 0 1.44329e10 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 4.22.a.a 2
3.b odd 2 1 36.22.a.c 2
4.b odd 2 1 16.22.a.d 2
8.b even 2 1 64.22.a.i 2
8.d odd 2 1 64.22.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.22.a.a 2 1.a even 1 1 trivial
16.22.a.d 2 4.b odd 2 1
36.22.a.c 2 3.b odd 2 1
64.22.a.i 2 8.b even 2 1
64.22.a.j 2 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(\Gamma_0(4))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 65640 T - 14536815984 \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 602941510374300 \) Copy content Toggle raw display
$7$ \( T^{2} + 260508080 T - 52\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{2} - 145435963320 T + 37\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{2} - 1428900417340 T + 50\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{2} - 1840620576420 T - 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{2} + 16780743928568 T - 10\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{2} + 319691925426960 T + 19\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( T^{2} - 112042353462592 T - 32\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 66\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 65\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 70\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 56\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 19\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 63\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 35\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 88\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 28\!\cdots\!24 \) Copy content Toggle raw display
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