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Properties

Label 11.19.b.b
Level $11$
Weight $19$
Character orbit 11.b
Analytic conductor $22.592$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [11,19,Mod(10,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 11.b (of order \(2\), degree \(1\), 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: \(22.5924751481\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 3191886 x^{14} + 4122245997720 x^{12} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{41}\cdot 3^{13}\cdot 11^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 2409) q^{3} + (\beta_{3} + \beta_{2} - 136842) q^{4} + (\beta_{4} + 13 \beta_{2} + 326104) q^{5} + (\beta_{7} + 3756 \beta_1) q^{6} + ( - \beta_{11} - \beta_{7} - 9665 \beta_1) q^{7} + ( - \beta_{11} + \beta_{10} + \cdots - 84645 \beta_1) q^{8}+ \cdots + (\beta_{5} - 23 \beta_{4} + \cdots + 100603800) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 2409) q^{3} + (\beta_{3} + \beta_{2} - 136842) q^{4} + (\beta_{4} + 13 \beta_{2} + 326104) q^{5} + (\beta_{7} + 3756 \beta_1) q^{6} + ( - \beta_{11} - \beta_{7} - 9665 \beta_1) q^{7} + ( - \beta_{11} + \beta_{10} + \cdots - 84645 \beta_1) q^{8}+ \cdots + (419543 \beta_{15} + \cdots + 15\!\cdots\!05) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 38544 q^{3} - 2189468 q^{4} + 5217660 q^{5} + 1609661148 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 38544 q^{3} - 2189468 q^{4} + 5217660 q^{5} + 1609661148 q^{9} + 2421398892 q^{11} - 13872148524 q^{12} + 61698809208 q^{14} - 88588731000 q^{15} - 33602470616 q^{16} + 51092984700 q^{20} - 339227654040 q^{22} - 1850264563776 q^{23} + 12501851237580 q^{25} - 26205945279672 q^{26} - 14625827701128 q^{27} - 15196014526576 q^{31} - 89105137275396 q^{33} + 300902830051608 q^{34} - 145752168167040 q^{36} - 204410474532724 q^{37} - 450785229418200 q^{38} + 17\!\cdots\!60 q^{42}+ \cdots + 24\!\cdots\!64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 3191886 x^{14} + 4122245997720 x^{12} + \cdots + 35\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 18\!\cdots\!59 \nu^{14} + \cdots + 50\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18\!\cdots\!59 \nu^{14} + \cdots - 11\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\!\cdots\!21 \nu^{14} + \cdots + 64\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 64\!\cdots\!67 \nu^{14} + \cdots + 27\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!51 \nu^{14} + \cdots + 13\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!59 \nu^{15} + \cdots - 50\!\cdots\!00 \nu ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!57 \nu^{14} + \cdots + 10\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 28\!\cdots\!07 \nu^{14} + \cdots + 92\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 80\!\cdots\!09 \nu^{15} + \cdots + 19\!\cdots\!00 \nu ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!03 \nu^{15} + \cdots + 23\!\cdots\!00 \nu ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 67\!\cdots\!57 \nu^{15} + \cdots + 22\!\cdots\!00 \nu ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 69\!\cdots\!59 \nu^{15} + \cdots - 21\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!39 \nu^{15} + \cdots + 47\!\cdots\!00 \nu ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28\!\cdots\!17 \nu^{15} + \cdots + 53\!\cdots\!00 ) / 94\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 398986 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} - 2\beta_{7} - 608933\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} - 21\beta_{8} + 13\beta_{6} + 15099\beta_{4} - 872105\beta_{3} - 1175472\beta_{2} + 242955752002 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} - 52 \beta_{14} - 11469 \beta_{13} + 13744 \beta_{12} + 1219545 \beta_{11} - 965948 \beta_{10} + \cdots - 5735 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4356338 \beta_{9} + 30981678 \beta_{8} - 15275142 \beta_{6} - 12630032 \beta_{5} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9992510 \beta_{15} + 403115672 \beta_{14} + 15978379430 \beta_{13} - 21233788064 \beta_{12} + \cdots + 7994185970 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4591289326068 \beta_{9} - 33177551821452 \beta_{8} + 13557771575580 \beta_{6} + 23269300402656 \beta_{5} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 17423388134988 \beta_{15} - 675064790731632 \beta_{14} + \cdots - 82\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 42\!\cdots\!88 \beta_{9} + \cdots - 93\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 20\!\cdots\!48 \beta_{15} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 37\!\cdots\!48 \beta_{9} + \cdots + 72\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 21\!\cdots\!88 \beta_{15} + \cdots - 65\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 30\!\cdots\!68 \beta_{9} + \cdots - 56\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 20\!\cdots\!88 \beta_{15} + \cdots + 55\!\cdots\!80 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
− 908.996i
− 874.327i
− 767.789i
− 709.740i
− 461.076i
− 341.817i
− 324.216i
− 270.903i
270.903i
324.216i
341.817i
461.076i
709.740i
767.789i
874.327i
908.996i
− 908.996i 23276.1 −564129. 3.28580e6 − 2.11579e7i 3.22343e7i 2.74503e8i 1.54356e8 − 2.98678e9i
10.2 − 874.327i −21222.2 −502304. −1.08295e6 1.85552e7i 3.16631e7i 2.09978e8i 6.29623e7 9.46857e8i
10.3 − 767.789i 1141.01 −327356. 125109. − 876053.i − 6.42132e7i 5.00689e7i −3.86119e8 − 9.60571e7i
10.4 − 709.740i 28368.2 −241587. −3.59402e6 − 2.01340e7i 3.70715e7i − 1.45900e7i 4.17333e8 2.55082e9i
10.5 − 461.076i −34095.8 49553.3 875875. 1.57208e7i − 3.92157e7i − 1.43716e8i 7.75106e8 − 4.03845e8i
10.6 − 341.817i −11062.2 145305. 3.04947e6 3.78124e6i 6.01319e7i − 1.39273e8i −2.65049e8 − 1.04236e9i
10.7 − 324.216i 4580.99 157028. −973631. − 1.48523e6i 1.60687e7i − 1.35902e8i −3.66435e8 3.15667e8i
10.8 − 270.903i 28286.0 188756. 923178. − 7.66276e6i − 3.99654e7i − 1.22150e8i 4.12677e8 − 2.50092e8i
10.9 270.903i 28286.0 188756. 923178. 7.66276e6i 3.99654e7i 1.22150e8i 4.12677e8 2.50092e8i
10.10 324.216i 4580.99 157028. −973631. 1.48523e6i − 1.60687e7i 1.35902e8i −3.66435e8 − 3.15667e8i
10.11 341.817i −11062.2 145305. 3.04947e6 − 3.78124e6i − 6.01319e7i 1.39273e8i −2.65049e8 1.04236e9i
10.12 461.076i −34095.8 49553.3 875875. − 1.57208e7i 3.92157e7i 1.43716e8i 7.75106e8 4.03845e8i
10.13 709.740i 28368.2 −241587. −3.59402e6 2.01340e7i − 3.70715e7i 1.45900e7i 4.17333e8 − 2.55082e9i
10.14 767.789i 1141.01 −327356. 125109. 876053.i 6.42132e7i − 5.00689e7i −3.86119e8 9.60571e7i
10.15 874.327i −21222.2 −502304. −1.08295e6 − 1.85552e7i − 3.16631e7i − 2.09978e8i 6.29623e7 − 9.46857e8i
10.16 908.996i 23276.1 −564129. 3.28580e6 2.11579e7i − 3.22343e7i − 2.74503e8i 1.54356e8 2.98678e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.19.b.b ✓ 16
3.b odd 2 1 99.19.c.b 16
11.b odd 2 1 inner 11.19.b.b ✓ 16
33.d even 2 1 99.19.c.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.19.b.b ✓ 16 1.a even 1 1 trivial
11.19.b.b ✓ 16 11.b odd 2 1 inner
99.19.c.b 16 3.b odd 2 1
99.19.c.b 16 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 3191886 T_{2}^{14} + 4122245997720 T_{2}^{12} + \cdots + 35\!\cdots\!00 \) acting on \(S_{19}^{\mathrm{new}}(11, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( (T^{8} + \cdots - 78\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 91\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 77\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 19\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 24\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 20\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
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