Properties

Label 201.4.j.a
Level $201$
Weight $4$
Character orbit 201.j
Analytic conductor $11.859$
Analytic rank $0$
Dimension $20$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,4,Mod(5,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.j (of order \(22\), degree \(10\), 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: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{18} q^{3} + ( - 8 \beta_{9} + 8 \beta_{8} - 8 \beta_{7} + 8 \beta_{6} - 8 \beta_{5} + 8 \beta_{4} + \cdots + 8) q^{4}+ \cdots + 27 \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{18} q^{3} + ( - 8 \beta_{9} + 8 \beta_{8} - 8 \beta_{7} + 8 \beta_{6} - 8 \beta_{5} + 8 \beta_{4} + \cdots + 8) q^{4}+ \cdots + (132 \beta_{18} + 132 \beta_{13} + 665 \beta_{9} + 665 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{33}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{33}^{6} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{33}^{9} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{33}^{12} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{33}^{13} + \zeta_{33}^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{33}^{15} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{33}^{16} + \zeta_{33}^{5} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \zeta_{33}^{18} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( \zeta_{33}^{19} + \zeta_{33}^{8} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( 6\zeta_{33}^{11} + 3 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 3\zeta_{33}^{12} + 6\zeta_{33} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( -3\zeta_{33}^{13} + 3\zeta_{33}^{2} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( 6\zeta_{33}^{14} + 3\zeta_{33}^{3} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( 3\zeta_{33}^{15} + 6\zeta_{33}^{4} \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( -3\zeta_{33}^{16} + 3\zeta_{33}^{5} \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( 6\zeta_{33}^{17} + 3\zeta_{33}^{6} \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( 3\zeta_{33}^{18} + 6\zeta_{33}^{7} \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( -3\zeta_{33}^{19} + 3\zeta_{33}^{8} \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( 3 \zeta_{33}^{19} - 3 \zeta_{33}^{18} + 3 \zeta_{33}^{16} - 3 \zeta_{33}^{15} + 3 \zeta_{33}^{13} - 3 \zeta_{33}^{12} + 6 \zeta_{33}^{10} - 3 \zeta_{33}^{9} + 3 \zeta_{33}^{8} - 3 \zeta_{33}^{6} + 3 \zeta_{33}^{5} - 3 \zeta_{33}^{3} + 3 \zeta_{33}^{2} - 3 \) Copy content Toggle raw display
\(\zeta_{33}\)\(=\) \( ( \beta_{11} - 3\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{2}\)\(=\) \( ( \beta_{12} + 3\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{33}^{4}\)\(=\) \( ( \beta_{14} - 3\beta_{6} ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{5}\)\(=\) \( ( \beta_{15} + 3\beta_{7} ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{6}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{33}^{7}\)\(=\) \( ( \beta_{17} - 3\beta_{8} ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{8}\)\(=\) \( ( \beta_{18} + 3\beta_{9} ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{9}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{33}^{10}\)\(=\) \( ( \beta_{19} - 3 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{11}\)\(=\) \( ( \beta_{10} - 3 ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{12}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{33}^{13}\)\(=\) \( ( -\beta_{12} + 3\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{14}\)\(=\) \( ( \beta_{13} - 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{15}\)\(=\) \( \beta_{6} \) Copy content Toggle raw display
\(\zeta_{33}^{16}\)\(=\) \( ( -\beta_{15} + 3\beta_{7} ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{17}\)\(=\) \( ( \beta_{16} - 3\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{33}^{18}\)\(=\) \( \beta_{8} \) Copy content Toggle raw display
\(\zeta_{33}^{19}\)\(=\) \( ( -\beta_{18} + 3\beta_{9} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(-1\) \(1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.928368 0.371662i
−0.786053 0.618159i
0.235759 0.971812i
0.723734 + 0.690079i
−0.327068 + 0.945001i
0.981929 0.189251i
−0.327068 0.945001i
0.981929 + 0.189251i
0.580057 0.814576i
−0.995472 0.0950560i
0.580057 + 0.814576i
−0.995472 + 0.0950560i
−0.888835 0.458227i
0.0475819 + 0.998867i
0.928368 + 0.371662i
−0.786053 + 0.618159i
0.235759 + 0.971812i
0.723734 0.690079i
−0.888835 + 0.458227i
0.0475819 0.998867i
0 −4.72659 + 2.15856i 1.13852 + 7.91857i 0 0 18.3144 + 15.8695i 0 17.6812 20.4052i 0
5.2 0 4.72659 2.15856i 1.13852 + 7.91857i 0 0 11.6079 + 10.0583i 0 17.6812 20.4052i 0
8.1 0 −3.92699 + 3.40276i 7.67594 + 2.25386i 0 0 −24.0323 3.45532i 0 3.84250 26.7252i 0
8.2 0 3.92699 3.40276i 7.67594 + 2.25386i 0 0 35.1868 + 5.05910i 0 3.84250 26.7252i 0
53.1 0 −2.80925 + 4.37128i 5.23889 6.04600i 0 0 4.82246 2.20234i 0 −11.2162 24.5601i 0
53.2 0 2.80925 4.37128i 5.23889 6.04600i 0 0 −32.3206 + 14.7603i 0 −11.2162 24.5601i 0
110.1 0 −2.80925 4.37128i 5.23889 + 6.04600i 0 0 4.82246 + 2.20234i 0 −11.2162 + 24.5601i 0
110.2 0 2.80925 + 4.37128i 5.23889 + 6.04600i 0 0 −32.3206 14.7603i 0 −11.2162 + 24.5601i 0
119.1 0 −1.46393 4.98567i −3.32332 + 7.27706i 0 0 −2.83365 4.40925i 0 −22.7138 + 14.5973i 0
119.2 0 1.46393 + 4.98567i −3.32332 + 7.27706i 0 0 −16.8377 26.2000i 0 −22.7138 + 14.5973i 0
125.1 0 −1.46393 + 4.98567i −3.32332 7.27706i 0 0 −2.83365 + 4.40925i 0 −22.7138 14.5973i 0
125.2 0 1.46393 4.98567i −3.32332 7.27706i 0 0 −16.8377 + 26.2000i 0 −22.7138 14.5973i 0
137.1 0 −5.14326 0.739490i −6.73003 4.32513i 0 0 10.4355 35.5401i 0 25.9063 + 7.60678i 0
137.2 0 5.14326 + 0.739490i −6.73003 4.32513i 0 0 −4.34287 + 14.7905i 0 25.9063 + 7.60678i 0
161.1 0 −4.72659 2.15856i 1.13852 7.91857i 0 0 18.3144 15.8695i 0 17.6812 + 20.4052i 0
161.2 0 4.72659 + 2.15856i 1.13852 7.91857i 0 0 11.6079 10.0583i 0 17.6812 + 20.4052i 0
176.1 0 −3.92699 3.40276i 7.67594 2.25386i 0 0 −24.0323 + 3.45532i 0 3.84250 + 26.7252i 0
176.2 0 3.92699 + 3.40276i 7.67594 2.25386i 0 0 35.1868 5.05910i 0 3.84250 + 26.7252i 0
179.1 0 −5.14326 + 0.739490i −6.73003 + 4.32513i 0 0 10.4355 + 35.5401i 0 25.9063 7.60678i 0
179.2 0 5.14326 0.739490i −6.73003 + 4.32513i 0 0 −4.34287 14.7905i 0 25.9063 7.60678i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
67.f odd 22 1 inner
201.j even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.j.a 20
3.b odd 2 1 CM 201.4.j.a 20
67.f odd 22 1 inner 201.4.j.a 20
201.j even 22 1 inner 201.4.j.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.j.a 20 1.a even 1 1 trivial
201.4.j.a 20 3.b odd 2 1 CM
201.4.j.a 20 67.f odd 22 1 inner
201.4.j.a 20 201.j even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{4}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 205891132094649 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 972 T^{18} + \cdots + 31\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} - 3888 T^{18} + \cdots + 57\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} - 112 T^{19} + \cdots + 74\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} - 24300 T^{18} + \cdots + 22\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( (T^{10} - 110 T^{9} + \cdots - 56\!\cdots\!23)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( T^{20} - 47628 T^{18} + \cdots + 59\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( T^{20} \) Copy content Toggle raw display
$53$ \( T^{20} \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( T^{20} - 874800 T^{18} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{20} + 880 T^{19} + \cdots + 60\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( T^{20} + 10710 T^{19} + \cdots + 83\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{20} + 9724 T^{19} + \cdots + 17\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( T^{20} \) Copy content Toggle raw display
$97$ \( T^{20} + 18197014 T^{18} + \cdots + 25\!\cdots\!49 \) Copy content Toggle raw display
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