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Properties

Label 1716.2.be.a
Level $1716$
Weight $2$
Character orbit 1716.be
Analytic conductor $13.702$
Analytic rank $0$
Dimension $20$
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] = [1716,2,Mod(1057,1716)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.1057");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1716.be (of order \(6\), 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: \(13.7023289869\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{17} + 86 x^{16} - 18 x^{15} + 8 x^{14} - 270 x^{13} + 2493 x^{12} - 1262 x^{11} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} - 1) q^{3} + (\beta_{16} + \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + \beta_{11} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} - 1) q^{3} + (\beta_{16} + \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{10} + \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 10 q^{3} + 6 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 10 q^{3} + 6 q^{7} - 10 q^{9} - 2 q^{13} + 4 q^{17} + 12 q^{19} - 10 q^{23} + 12 q^{25} + 20 q^{27} - 2 q^{29} + 18 q^{35} - 6 q^{37} + 4 q^{39} - 36 q^{41} - 16 q^{43} - 8 q^{49} - 8 q^{51} + 20 q^{53} + 4 q^{55} + 12 q^{59} - 6 q^{63} - 4 q^{65} + 18 q^{67} - 10 q^{69} + 18 q^{71} - 6 q^{75} - 12 q^{79} - 10 q^{81} + 60 q^{85} - 2 q^{87} + 18 q^{91} - 4 q^{95} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{17} + 86 x^{16} - 18 x^{15} + 8 x^{14} - 270 x^{13} + 2493 x^{12} - 1262 x^{11} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 28\!\cdots\!38 \nu^{19} + \cdots - 26\!\cdots\!02 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 29\!\cdots\!68 \nu^{19} + \cdots + 14\!\cdots\!88 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 50\!\cdots\!83 \nu^{19} + \cdots - 17\!\cdots\!29 ) / 95\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 74\!\cdots\!43 \nu^{19} + \cdots - 47\!\cdots\!84 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!63 \nu^{19} + \cdots - 54\!\cdots\!47 ) / 95\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!63 \nu^{19} + \cdots + 16\!\cdots\!36 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!96 \nu^{19} + \cdots + 11\!\cdots\!07 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 134315943465638 \nu^{19} + 143672291022478 \nu^{18} + 63818624715948 \nu^{17} + \cdots + 10\!\cdots\!70 ) / 50\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23\!\cdots\!01 \nu^{19} + \cdots + 85\!\cdots\!57 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!70 \nu^{19} + \cdots - 38\!\cdots\!86 ) / 41\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 30\!\cdots\!36 \nu^{19} + \cdots - 17\!\cdots\!49 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 37\!\cdots\!31 \nu^{19} + \cdots - 20\!\cdots\!99 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 43\!\cdots\!50 \nu^{19} + \cdots + 52\!\cdots\!86 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 71\!\cdots\!21 \nu^{19} + \cdots - 20\!\cdots\!69 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 13\!\cdots\!74 \nu^{19} + \cdots + 17\!\cdots\!47 ) / 95\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 17\!\cdots\!73 \nu^{19} + \cdots - 67\!\cdots\!66 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 19\!\cdots\!46 \nu^{19} + \cdots + 23\!\cdots\!60 ) / 86\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 26\!\cdots\!54 \nu^{19} + \cdots + 34\!\cdots\!58 ) / 95\!\cdots\!38 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{16} - 4\beta_{10} + \beta_{7} - \beta_{6} - 4\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{19} + \beta_{18} + \beta_{15} + \beta_{11} + \beta_{8} - 4\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{19} - \beta_{18} + 2 \beta_{17} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{15} - 9 \beta_{13} - 9 \beta_{12} - 8 \beta_{11} - 8 \beta_{10} - 9 \beta_{7} + 10 \beta_{6} + \cdots - 19 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{19} - 11 \beta_{18} - 46 \beta_{16} - 9 \beta_{15} + 18 \beta_{14} - 11 \beta_{13} + 9 \beta_{12} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 64 \beta_{19} - 64 \beta_{18} - 2 \beta_{17} - 23 \beta_{16} - 65 \beta_{15} - \beta_{13} - \beta_{12} + \cdots - 51 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 64 \beta_{19} + 63 \beta_{18} - 128 \beta_{17} - 12 \beta_{16} - \beta_{15} - 64 \beta_{14} - 63 \beta_{13} + \cdots + 609 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 13 \beta_{19} + 14 \beta_{18} - 27 \beta_{17} + 4 \beta_{16} - 409 \beta_{15} - 27 \beta_{14} + \cdots + 48 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 234 \beta_{19} + 657 \beta_{18} + 1872 \beta_{16} + 423 \beta_{15} - 832 \beta_{14} + 657 \beta_{13} + \cdots + 706 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2722 \beta_{19} + 2718 \beta_{18} + 252 \beta_{17} + 1455 \beta_{16} + 2833 \beta_{15} - 4 \beta_{14} + \cdots + 1707 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2800 \beta_{19} - 2503 \beta_{18} + 5436 \beta_{17} + 823 \beta_{16} + 297 \beta_{15} + 2718 \beta_{14} + \cdots - 20815 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 862 \beta_{19} - 1253 \beta_{18} + 2033 \beta_{17} - 886 \beta_{16} + 16094 \beta_{15} + 2115 \beta_{14} + \cdots - 4731 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13119 \beta_{19} - 30384 \beta_{18} - 74644 \beta_{16} - 17265 \beta_{15} + 33441 \beta_{14} + \cdots - 34886 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 110224 \beta_{19} - 109174 \beta_{18} - 15258 \beta_{17} - 70739 \beta_{16} - 115070 \beta_{15} + \cdots - 50779 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 119998 \beta_{19} + 90038 \beta_{18} - 218348 \beta_{17} - 44447 \beta_{16} - 29960 \beta_{15} + \cdots + 751348 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 38084 \beta_{19} + 82719 \beta_{18} - 109979 \beta_{17} + 76849 \beta_{16} - 617439 \beta_{15} + \cdots + 306222 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 613839 \beta_{19} + 1303173 \beta_{18} + 2973266 \beta_{16} + 689334 \beta_{15} - 1317597 \beta_{14} + \cdots + 1562579 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 4450431 \beta_{19} + 4351934 \beta_{18} + 773407 \beta_{17} + 3174331 \beta_{16} + 4588625 \beta_{15} + \cdots + 1370779 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1716\mathbb{Z}\right)^\times\).

\(n\) \(859\) \(925\) \(937\) \(1145\)
\(\chi(n)\) \(1\) \(-\beta_{11}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1057.1
0.852562 + 0.852562i
1.79483 − 1.79483i
0.946067 + 0.946067i
−1.73621 + 1.73621i
1.44831 + 1.44831i
−1.55572 − 1.55572i
1.23380 − 1.23380i
−1.09718 + 1.09718i
−1.69122 − 1.69122i
−0.195231 + 0.195231i
−0.195231 − 0.195231i
−1.69122 + 1.69122i
−1.09718 − 1.09718i
1.23380 + 1.23380i
−1.55572 + 1.55572i
1.44831 − 1.44831i
−1.73621 − 1.73621i
0.946067 − 0.946067i
1.79483 + 1.79483i
0.852562 − 0.852562i
0 −0.500000 + 0.866025i 0 − 2.54628i 0 2.72343 − 1.57237i 0 −0.500000 − 0.866025i 0
1057.2 0 −0.500000 + 0.866025i 0 − 2.44280i 0 −2.06418 + 1.19175i 0 −0.500000 − 0.866025i 0
1057.3 0 −0.500000 + 0.866025i 0 − 2.20991i 0 −2.02115 + 1.16691i 0 −0.500000 − 0.866025i 0
1057.4 0 −0.500000 + 0.866025i 0 − 2.02886i 0 −1.48391 + 0.856738i 0 −0.500000 − 0.866025i 0
1057.5 0 −0.500000 + 0.866025i 0 0.195209i 0 −2.31552 + 1.33686i 0 −0.500000 − 0.866025i 0
1057.6 0 −0.500000 + 0.866025i 0 0.840526i 0 2.43416 − 1.40536i 0 −0.500000 − 0.866025i 0
1057.7 0 −0.500000 + 0.866025i 0 0.955493i 0 4.01755 − 2.31953i 0 −0.500000 − 0.866025i 0
1057.8 0 −0.500000 + 0.866025i 0 1.59240i 0 0.275000 − 0.158771i 0 −0.500000 − 0.866025i 0
1057.9 0 −0.500000 + 0.866025i 0 1.72045i 0 0.679071 − 0.392062i 0 −0.500000 − 0.866025i 0
1057.10 0 −0.500000 + 0.866025i 0 3.92377i 0 0.755539 − 0.436210i 0 −0.500000 − 0.866025i 0
1453.1 0 −0.500000 − 0.866025i 0 − 3.92377i 0 0.755539 + 0.436210i 0 −0.500000 + 0.866025i 0
1453.2 0 −0.500000 − 0.866025i 0 − 1.72045i 0 0.679071 + 0.392062i 0 −0.500000 + 0.866025i 0
1453.3 0 −0.500000 − 0.866025i 0 − 1.59240i 0 0.275000 + 0.158771i 0 −0.500000 + 0.866025i 0
1453.4 0 −0.500000 − 0.866025i 0 − 0.955493i 0 4.01755 + 2.31953i 0 −0.500000 + 0.866025i 0
1453.5 0 −0.500000 − 0.866025i 0 − 0.840526i 0 2.43416 + 1.40536i 0 −0.500000 + 0.866025i 0
1453.6 0 −0.500000 − 0.866025i 0 − 0.195209i 0 −2.31552 − 1.33686i 0 −0.500000 + 0.866025i 0
1453.7 0 −0.500000 − 0.866025i 0 2.02886i 0 −1.48391 − 0.856738i 0 −0.500000 + 0.866025i 0
1453.8 0 −0.500000 − 0.866025i 0 2.20991i 0 −2.02115 − 1.16691i 0 −0.500000 + 0.866025i 0
1453.9 0 −0.500000 − 0.866025i 0 2.44280i 0 −2.06418 − 1.19175i 0 −0.500000 + 0.866025i 0
1453.10 0 −0.500000 − 0.866025i 0 2.54628i 0 2.72343 + 1.57237i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1057.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.2.be.a ✓ 20
13.e even 6 1 inner 1716.2.be.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.be.a ✓ 20 1.a even 1 1 trivial
1716.2.be.a ✓ 20 13.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 44 T_{5}^{18} + 782 T_{5}^{16} + 7468 T_{5}^{14} + 42377 T_{5}^{12} + 147774 T_{5}^{10} + \cdots + 2209 \) acting on \(S_{2}^{\mathrm{new}}(1716, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + 44 T^{18} + \cdots + 2209 \) Copy content Toggle raw display
$7$ \( T^{20} - 6 T^{19} + \cdots + 51529 \) Copy content Toggle raw display
$11$ \( (T^{4} - T^{2} + 1)^{5} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} - 4 T^{19} + \cdots + 1369 \) Copy content Toggle raw display
$19$ \( T^{20} - 12 T^{19} + \cdots + 59049 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 350818105401 \) Copy content Toggle raw display
$29$ \( T^{20} + 2 T^{19} + \cdots + 4108729 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 10583459887729 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 88222662529 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 132411049 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 621056241 \) Copy content Toggle raw display
$53$ \( (T^{10} - 10 T^{9} + \cdots + 2405169)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 3508311361 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 87\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 110766215849809 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 96\!\cdots\!21 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 12\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( (T^{10} + 6 T^{9} + \cdots + 6690393)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 40\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 19671078873681 \) Copy content Toggle raw display
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