Properties

Label 20.11.f.a
Level $20$
Weight $11$
Character orbit 20.f
Analytic conductor $12.707$
Analytic rank $0$
Dimension $10$
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] = [20,11,Mod(13,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.13");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 20.f (of order \(4\), degree \(2\), 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: \(12.7071450535\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 75402 x^{8} + 1918432665 x^{6} + 20025190470928 x^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{2}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 6 \beta_1 + 6) q^{3} + (\beta_{5} + 2 \beta_{2} - 174 \beta_1 + 90) q^{5} + (\beta_{8} + \beta_{5} - 7 \beta_{3} - 2227 \beta_1 + 2227) q^{7} + (\beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} - 3 \beta_{4} + 16 \beta_{3} - 14 \beta_{2} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 6 \beta_1 + 6) q^{3} + (\beta_{5} + 2 \beta_{2} - 174 \beta_1 + 90) q^{5} + (\beta_{8} + \beta_{5} - 7 \beta_{3} - 2227 \beta_1 + 2227) q^{7} + (\beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{5} - 3 \beta_{4} + 16 \beta_{3} - 14 \beta_{2} + \cdots + 3) q^{9}+ \cdots + (229207 \beta_{9} - 76522 \beta_{8} + 76522 \beta_{7} + \cdots + 687621) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 62 q^{3} + 894 q^{5} + 22286 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 62 q^{3} + 894 q^{5} + 22286 q^{7} - 201700 q^{11} + 239298 q^{13} + 213662 q^{15} + 1045442 q^{17} + 4578860 q^{21} - 4097986 q^{23} - 4233934 q^{25} - 4817488 q^{27} + 23221660 q^{31} + 31816220 q^{33} - 55388242 q^{35} - 87811974 q^{37} + 29776460 q^{41} + 156325470 q^{43} - 144135236 q^{45} - 450750018 q^{47} + 1632585820 q^{51} + 701393866 q^{53} - 1301185140 q^{55} - 2564330416 q^{57} + 2991488220 q^{61} + 3352397678 q^{63} - 2867494182 q^{65} - 6990333394 q^{67} + 9915200380 q^{71} + 8401915018 q^{73} - 10170758642 q^{75} - 19825815140 q^{77} + 26071184290 q^{81} + 16998617454 q^{83} - 24280829854 q^{85} - 36065578576 q^{87} + 52347612540 q^{91} + 26277966572 q^{93} - 23431125296 q^{95} - 48945511254 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 75402 x^{8} + 1918432665 x^{6} + 20025190470928 x^{4} + \cdots + 13\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4541161 \nu^{9} - 296647947642 \nu^{7} + \cdots - 25\!\cdots\!24 \nu ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 90309214939079 \nu^{9} + 904047821228016 \nu^{8} + \cdots + 52\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 90309214939079 \nu^{9} + 904047821228016 \nu^{8} + \cdots + 52\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 564930968079603 \nu^{9} + \cdots + 35\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 914897678021365 \nu^{9} + \cdots + 69\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22\!\cdots\!81 \nu^{9} + \cdots - 10\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 24\!\cdots\!59 \nu^{9} + \cdots - 84\!\cdots\!00 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 39\!\cdots\!29 \nu^{9} + \cdots + 14\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30\!\cdots\!15 \nu^{9} + \cdots + 55\!\cdots\!00 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{9} + 2\beta_{6} + 5\beta_{5} + 10\beta_{4} - \beta_{3} + 3\beta_{2} + 4\beta _1 - 3 ) / 400 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 56 \beta_{9} - 50 \beta_{8} - 50 \beta_{7} - 113 \beta_{6} + 570 \beta_{5} + 65 \beta_{4} - 10591 \beta_{3} - 10477 \beta_{2} + 339 \beta _1 - 6036298 ) / 400 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 31877 \beta_{9} + 13300 \beta_{8} - 13300 \beta_{7} - 46694 \beta_{6} - 77845 \beta_{5} - 280890 \beta_{4} - 595543 \beta_{3} + 565909 \beta_{2} + 272252392 \beta _1 + 95631 ) / 400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 603088 \beta_{9} + 136210 \beta_{8} + 136210 \beta_{7} + 896413 \beta_{6} - 5258706 \beta_{5} - 140677 \beta_{4} + 80000603 \beta_{3} + 79413953 \beta_{2} - 2689239 \beta _1 + 29608641818 ) / 80 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1081368341 \beta_{9} - 444738700 \beta_{8} + 444738700 \beta_{7} + 1326829982 \beta_{6} + 1618476205 \beta_{5} + 8750702010 \beta_{4} + 34578791359 \beta_{3} + \cdots - 3244105023 ) / 400 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 129853710776 \beta_{9} + 1828519150 \beta_{8} + 1828519150 \beta_{7} - 165607670273 \beta_{6} + 1053820332570 \beta_{5} - 20763313135 \beta_{4} + \cdots - 43\!\cdots\!58 ) / 400 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 36636410491877 \beta_{9} + 13260748574500 \beta_{8} - 13260748574500 \beta_{7} - 41196564484694 \beta_{6} - 41616277888645 \beta_{5} + \cdots + 109909231475631 ) / 400 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10\!\cdots\!00 \beta_{9} - 118088843084030 \beta_{8} - 118088843084030 \beta_{7} + \cdots + 27\!\cdots\!54 ) / 80 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 12\!\cdots\!81 \beta_{9} + \cdots - 37\!\cdots\!43 ) / 400 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
149.896i
186.802i
56.3206i
75.9602i
95.3750i
149.896i
186.802i
56.3206i
75.9602i
95.3750i
0 −190.862 190.862i 0 2756.18 + 1472.78i 0 14849.7 14849.7i 0 13807.6i 0
13.2 0 −189.544 189.544i 0 −3114.06 261.220i 0 −3939.73 + 3939.73i 0 12805.1i 0
13.3 0 7.29050 + 7.29050i 0 2223.57 2195.76i 0 −14037.3 + 14037.3i 0 58942.7i 0
13.4 0 186.378 + 186.378i 0 −473.688 + 3088.89i 0 −7250.16 + 7250.16i 0 10424.8i 0
13.5 0 217.737 + 217.737i 0 −944.998 2978.69i 0 21520.5 21520.5i 0 35770.2i 0
17.1 0 −190.862 + 190.862i 0 2756.18 1472.78i 0 14849.7 + 14849.7i 0 13807.6i 0
17.2 0 −189.544 + 189.544i 0 −3114.06 + 261.220i 0 −3939.73 3939.73i 0 12805.1i 0
17.3 0 7.29050 7.29050i 0 2223.57 + 2195.76i 0 −14037.3 14037.3i 0 58942.7i 0
17.4 0 186.378 186.378i 0 −473.688 3088.89i 0 −7250.16 7250.16i 0 10424.8i 0
17.5 0 217.737 217.737i 0 −944.998 + 2978.69i 0 21520.5 + 21520.5i 0 35770.2i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.11.f.a 10
3.b odd 2 1 180.11.l.a 10
4.b odd 2 1 80.11.p.e 10
5.b even 2 1 100.11.f.b 10
5.c odd 4 1 inner 20.11.f.a 10
5.c odd 4 1 100.11.f.b 10
15.e even 4 1 180.11.l.a 10
20.e even 4 1 80.11.p.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.f.a 10 1.a even 1 1 trivial
20.11.f.a 10 5.c odd 4 1 inner
80.11.p.e 10 4.b odd 2 1
80.11.p.e 10 20.e even 4 1
100.11.f.b 10 5.b even 2 1
100.11.f.b 10 5.c odd 4 1
180.11.l.a 10 3.b odd 2 1
180.11.l.a 10 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 62 T^{9} + \cdots + 36\!\cdots\!32 \) Copy content Toggle raw display
$5$ \( T^{10} - 894 T^{9} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} - 22286 T^{9} + \cdots + 52\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( (T^{5} + 100850 T^{4} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - 239298 T^{9} + \cdots + 71\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{10} - 1045442 T^{9} + \cdots + 67\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{10} + 48182311879520 T^{8} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{10} + 4097986 T^{9} + \cdots + 37\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{5} - 11610830 T^{4} + \cdots - 49\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 87811974 T^{9} + \cdots + 29\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( (T^{5} - 14888230 T^{4} + \cdots - 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} - 156325470 T^{9} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + 450750018 T^{9} + \cdots + 90\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{10} - 701393866 T^{9} + \cdots + 72\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 59\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{5} - 1495744110 T^{4} + \cdots + 10\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 6990333394 T^{9} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( (T^{5} - 4957600190 T^{4} + \cdots - 20\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} - 8401915018 T^{9} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 58\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{10} - 16998617454 T^{9} + \cdots + 23\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{10} + 48945511254 T^{9} + \cdots + 57\!\cdots\!68 \) Copy content Toggle raw display
show more
show less