Properties

Label 20.18.a.a
Level $20$
Weight $18$
Character orbit 20.a
Self dual yes
Analytic conductor $36.644$
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] = [20,18,Mod(1,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, 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("20.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 20.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: \(36.6444174689\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{247521}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 61880 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(\beta = 18\sqrt{247521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 5490) q^{3} + 390625 q^{5} + (581 \beta - 5022430) q^{7} + ( - 10980 \beta - 18803259) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 5490) q^{3} + 390625 q^{5} + (581 \beta - 5022430) q^{7} + ( - 10980 \beta - 18803259) q^{9} + (88790 \beta - 478363560) q^{11} + (164156 \beta - 615740410) q^{13} + ( - 390625 \beta + 2144531250) q^{15} + (4233532 \beta - 13644399030) q^{17} + (167380 \beta - 17763948484) q^{19} + (8212120 \beta - 74167483824) q^{21} + ( - 54923597 \beta + 157415994990) q^{23} + 152587890625 q^{25} + (87663222 \beta + 68351521140) q^{27} + ( - 340643240 \beta + 245675120526) q^{29} + (43538950 \beta - 1920005483764) q^{31} + (965820660 \beta - 9746890171560) q^{33} + (226953125 \beta - 1961886718750) q^{35} + ( - 607859784 \beta - 14189827212970) q^{37} + (1516956850 \beta - 16545201408324) q^{39} + ( - 1064602860 \beta - 35928584276334) q^{41} + ( - 2169228925 \beta - 42237129385630) q^{43} + ( - 4289062500 \beta - 7345023046875) q^{45} + ( - 8557664083 \beta - 134111377304550) q^{47} + ( - 5836063660 \beta - 180334397527263) q^{49} + (36886489710 \beta - 414423486706428) q^{51} + ( - 17406496588 \beta - 203891690553570) q^{53} + (34683593750 \beta - 186860765625000) q^{55} + (18682864684 \beta - 110947418230680) q^{57} + (17259579160 \beta - 734385414779988) q^{59} + ( - 85069163280 \beta + 83687661321182) q^{61} + (44221587921 \beta - 417167835402150) q^{63} + (64123437500 \beta - 240523597656250) q^{65} + ( - 356884486239 \beta + 28\!\cdots\!30) q^{67}+ \cdots + (3582890522190 \beta - 69\!\cdots\!60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10980 q^{3} + 781250 q^{5} - 10044860 q^{7} - 37606518 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10980 q^{3} + 781250 q^{5} - 10044860 q^{7} - 37606518 q^{9} - 956727120 q^{11} - 1231480820 q^{13} + 4289062500 q^{15} - 27288798060 q^{17} - 35527896968 q^{19} - 148334967648 q^{21} + 314831989980 q^{23} + 305175781250 q^{25} + 136703042280 q^{27} + 491350241052 q^{29} - 3840010967528 q^{31} - 19493780343120 q^{33} - 3923773437500 q^{35} - 28379654425940 q^{37} - 33090402816648 q^{39} - 71857168552668 q^{41} - 84474258771260 q^{43} - 14690046093750 q^{45} - 268222754609100 q^{47} - 360668795054526 q^{49} - 828846973412856 q^{51} - 407783381107140 q^{53} - 373721531250000 q^{55} - 221894836461360 q^{57} - 14\!\cdots\!76 q^{59}+ \cdots - 13\!\cdots\!20 q^{99}+O(q^{100}) \) 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
249.257
−248.257
0 −3465.27 0 390625. 0 180580. 0 −1.17132e8 0
1.2 0 14445.3 0 390625. 0 −1.02254e7 0 7.95256e7 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

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 20.18.a.a 2
4.b odd 2 1 80.18.a.c 2
5.b even 2 1 100.18.a.a 2
5.c odd 4 2 100.18.c.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.18.a.a 2 1.a even 1 1 trivial
80.18.a.c 2 4.b odd 2 1
100.18.a.a 2 5.b even 2 1
100.18.c.b 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 10980T_{3} - 50056704 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(20))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 10980 T - 50056704 \) Copy content Toggle raw display
$5$ \( (T - 390625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 10044860 T - 1846510250144 \) Copy content Toggle raw display
$11$ \( T^{2} + 956727120 T - 40\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{2} + 1231480820 T - 17\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{2} + 27288798060 T - 12\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{2} + 35527896968 T + 31\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{2} - 314831989980 T - 21\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{2} - 491350241052 T - 92\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{2} + 3840010967528 T + 35\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{2} + 28379654425940 T + 17\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{2} + 71857168552668 T + 11\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{2} + 84474258771260 T + 14\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{2} + 268222754609100 T + 12\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + 407783381107140 T + 17\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{2} - 167375322642364 T - 57\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 19\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 68\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 25\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 18\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
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