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Properties

Label 1716.2.q.b
Level $1716$
Weight $2$
Character orbit 1716.q
Analytic conductor $13.702$
Analytic rank $0$
Dimension $12$
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(133,1716)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1716.133");
 
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.q (of order \(3\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 19 x^{10} - 28 x^{9} + 301 x^{8} - 319 x^{7} + 1300 x^{6} - 1174 x^{5} + 4000 x^{4} + \cdots + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} - \beta_{3} q^{5} - \beta_{4} q^{7} + ( - \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} - \beta_{3} q^{5} - \beta_{4} q^{7} + ( - \beta_{5} - 1) q^{9} + \beta_{5} q^{11} + ( - \beta_{10} - \beta_{5} - 1) q^{13} + (\beta_{3} - \beta_1) q^{15} + ( - \beta_{11} + \beta_{5} + 1) q^{17} + ( - \beta_{11} - \beta_{5} - 1) q^{19} + (\beta_{6} + \beta_{4}) q^{21} + (\beta_{11} - \beta_{5} + \beta_{3} + \cdots - \beta_1) q^{23}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} + 2 q^{7} - 6 q^{9} - 6 q^{11} - 4 q^{13} + 5 q^{17} - 7 q^{19} - 4 q^{21} + 5 q^{23} + 16 q^{25} + 12 q^{27} - 22 q^{31} - 6 q^{33} - 33 q^{37} + 8 q^{39} - 14 q^{41} - 4 q^{47} - 10 q^{51} + 10 q^{53} + 14 q^{57} - 30 q^{59} - 9 q^{61} + 2 q^{63} + 4 q^{65} + 14 q^{67} + 5 q^{69} + 3 q^{71} - 6 q^{73} - 8 q^{75} - 4 q^{77} + 2 q^{79} - 6 q^{81} + 8 q^{83} + q^{85} - 32 q^{89} + 16 q^{91} + 11 q^{93} + q^{95} + 10 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 19 x^{10} - 28 x^{9} + 301 x^{8} - 319 x^{7} + 1300 x^{6} - 1174 x^{5} + 4000 x^{4} + \cdots + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1492313480 \nu^{11} - 22603279200 \nu^{10} - 49428652069 \nu^{9} - 320461520120 \nu^{8} + \cdots + 56658975561609 ) / 8240323363467 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 130063460 \nu^{11} - 175478800 \nu^{10} - 2436386123 \nu^{9} + 499014660 \nu^{8} + \cdots - 171614988738 ) / 633871027959 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 40728356681 \nu^{11} - 82794842220 \nu^{10} - 529698613619 \nu^{9} + \cdots - 13418256443760 ) / 49441940180802 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3178055347 \nu^{11} + 780380760 \nu^{10} - 59330178793 \nu^{9} + 103603866454 \nu^{8} + \cdots - 2759831620794 ) / 3803226167754 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 76605577039 \nu^{11} - 123418072080 \nu^{10} - 1769512501291 \nu^{9} + \cdots - 123509794771458 ) / 49441940180802 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14293743497 \nu^{11} - 39590777857 \nu^{10} - 256822681567 \nu^{9} + \cdots + 27639648665856 ) / 8240323363467 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15483128708 \nu^{11} - 57257413998 \nu^{10} + 229681967285 \nu^{9} - 1486465057349 \nu^{8} + \cdots - 2206154941164 ) / 8240323363467 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23550366403 \nu^{11} - 47565119139 \nu^{10} - 500754448164 \nu^{9} + \cdots + 10810146875796 ) / 8240323363467 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 33748921585 \nu^{11} - 15838066762 \nu^{10} + 559033577311 \nu^{9} - 1335267827910 \nu^{8} + \cdots + 21279304133856 ) / 8240323363467 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 329092289185 \nu^{11} + 14500435680 \nu^{10} - 6026397664435 \nu^{9} + \cdots + 94417124750640 ) / 49441940180802 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{7} + 6\beta_{5} + \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{10} + 2\beta_{9} + 2\beta_{8} - 3\beta_{6} - 3\beta_{4} - 13\beta_{3} - 2\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{10} + 15\beta_{9} - 15\beta_{8} - 3\beta_{7} - 71\beta_{5} + 7\beta_{4} + 27\beta _1 - 71 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 33 \beta_{11} + 12 \beta_{10} - 49 \beta_{9} - 37 \beta_{8} + 49 \beta_{7} + 61 \beta_{6} + \cdots - 204 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 310 \beta_{10} + 85 \beta_{9} + 310 \beta_{8} - 225 \beta_{7} - 175 \beta_{6} - 175 \beta_{4} + \cdots + 1077 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 507 \beta_{11} + 625 \beta_{10} + 385 \beta_{9} - 385 \beta_{8} - 625 \beta_{7} - 2922 \beta_{5} + \cdots - 2922 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 888 \beta_{11} + 3561 \beta_{10} - 5408 \beta_{9} - 1847 \beta_{8} + 5408 \beta_{7} + \cdots - 11623 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 19672 \beta_{10} + 10722 \beta_{9} + 19672 \beta_{8} - 8950 \beta_{7} - 18936 \beta_{6} + \cdots + 58883 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20541 \beta_{11} + 36836 \beta_{10} + 59183 \beta_{9} - 59183 \beta_{8} - 36836 \beta_{7} + \cdots - 310982 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 134661 \beta_{11} + 185095 \beta_{10} - 373332 \beta_{9} - 188237 \beta_{8} + 373332 \beta_{7} + \cdots - 1070196 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(859\) \(925\) \(937\) \(1145\)
\(\chi(n)\) \(1\) \(-1 - \beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1
1.64778 − 2.85404i
0.832261 − 1.44152i
0.752276 − 1.30298i
−0.133093 + 0.230523i
−0.955509 + 1.65499i
−2.14372 + 3.71303i
1.64778 + 2.85404i
0.832261 + 1.44152i
0.752276 + 1.30298i
−0.133093 − 0.230523i
−0.955509 − 1.65499i
−2.14372 − 3.71303i
0 −0.500000 − 0.866025i 0 −3.29557 0 −0.0598587 + 0.103678i 0 −0.500000 + 0.866025i 0
133.2 0 −0.500000 − 0.866025i 0 −1.66452 0 1.46059 − 2.52982i 0 −0.500000 + 0.866025i 0
133.3 0 −0.500000 − 0.866025i 0 −1.50455 0 −0.691197 + 1.19719i 0 −0.500000 + 0.866025i 0
133.4 0 −0.500000 − 0.866025i 0 0.266185 0 1.30732 − 2.26435i 0 −0.500000 + 0.866025i 0
133.5 0 −0.500000 − 0.866025i 0 1.91102 0 −2.19058 + 3.79419i 0 −0.500000 + 0.866025i 0
133.6 0 −0.500000 − 0.866025i 0 4.28743 0 1.17371 − 2.03293i 0 −0.500000 + 0.866025i 0
529.1 0 −0.500000 + 0.866025i 0 −3.29557 0 −0.0598587 − 0.103678i 0 −0.500000 − 0.866025i 0
529.2 0 −0.500000 + 0.866025i 0 −1.66452 0 1.46059 + 2.52982i 0 −0.500000 − 0.866025i 0
529.3 0 −0.500000 + 0.866025i 0 −1.50455 0 −0.691197 − 1.19719i 0 −0.500000 − 0.866025i 0
529.4 0 −0.500000 + 0.866025i 0 0.266185 0 1.30732 + 2.26435i 0 −0.500000 − 0.866025i 0
529.5 0 −0.500000 + 0.866025i 0 1.91102 0 −2.19058 − 3.79419i 0 −0.500000 − 0.866025i 0
529.6 0 −0.500000 + 0.866025i 0 4.28743 0 1.17371 + 2.03293i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1716.2.q.b ✓ 12
13.c even 3 1 inner 1716.2.q.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1716.2.q.b ✓ 12 1.a even 1 1 trivial
1716.2.q.b ✓ 12 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 19T_{5}^{4} - 14T_{5}^{3} + 60T_{5}^{2} + 53T_{5} - 18 \) acting on \(S_{2}^{\mathrm{new}}(1716, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 19 T^{4} + \cdots - 18)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{11} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 5 T^{11} + \cdots + 492804 \) Copy content Toggle raw display
$19$ \( T^{12} + 7 T^{11} + \cdots + 144 \) Copy content Toggle raw display
$23$ \( T^{12} - 5 T^{11} + \cdots + 104976 \) Copy content Toggle raw display
$29$ \( T^{12} + 88 T^{10} + \cdots + 1166400 \) Copy content Toggle raw display
$31$ \( (T^{6} + 11 T^{5} + \cdots + 10349)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 281967372036 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 48636127296 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 1287589689 \) Copy content Toggle raw display
$47$ \( (T^{6} + 2 T^{5} + \cdots - 15192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 5 T^{5} + \cdots - 1614)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 2489610816 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 13845993561 \) Copy content Toggle raw display
$67$ \( T^{12} - 14 T^{11} + \cdots + 91298025 \) Copy content Toggle raw display
$71$ \( T^{12} - 3 T^{11} + \cdots + 1602756 \) Copy content Toggle raw display
$73$ \( (T^{6} + 3 T^{5} + \cdots + 549741)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - T^{5} - 112 T^{4} + \cdots + 7727)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 4 T^{5} + \cdots + 3438)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 275725809216 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 18103971601 \) Copy content Toggle raw display
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