↑

Properties

Label 1764.2.j.h
Level $1764$
Weight $2$
Character orbit 1764.j
Analytic conductor $14.086$
Analytic rank $0$
Dimension $14$
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] = [1764,2,Mod(589,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.589");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.j (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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 252)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{5} q^{5} + (\beta_{11} - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{5} q^{5} + (\beta_{11} - \beta_{2} - \beta_1) q^{9} + (\beta_{13} - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{11}+ \cdots + (2 \beta_{12} - 2 \beta_{11} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{3} + 2 q^{5} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 3 q^{3} + 2 q^{5} - 5 q^{9} + 2 q^{11} - 2 q^{13} + 7 q^{15} + 4 q^{17} + 14 q^{19} + 11 q^{23} - 9 q^{25} - 9 q^{27} + q^{29} + q^{31} + q^{33} - 20 q^{37} + 22 q^{39} + 33 q^{41} + 7 q^{43} - 5 q^{45} + 3 q^{47} - 7 q^{51} + 30 q^{53} + 28 q^{55} - 18 q^{57} + 14 q^{59} + 10 q^{61} + 15 q^{65} + 6 q^{67} + 43 q^{69} + 2 q^{71} + 42 q^{73} - 43 q^{75} - 10 q^{79} + 7 q^{81} + 25 q^{83} + 8 q^{85} + 26 q^{87} - 12 q^{89} - 38 q^{93} - 28 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 35 \nu^{13} + 72 \nu^{12} + 157 \nu^{11} + 312 \nu^{10} - 290 \nu^{9} - 1383 \nu^{8} + \cdots - 3645 ) / 43011 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8 \nu^{13} + 2 \nu^{12} - 23 \nu^{11} + 5 \nu^{10} + 37 \nu^{9} - 127 \nu^{8} + 78 \nu^{7} + \cdots - 8505 ) / 4779 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8 \nu^{13} + 2 \nu^{12} - 23 \nu^{11} + 5 \nu^{10} + 37 \nu^{9} - 127 \nu^{8} + 78 \nu^{7} + \cdots - 8505 ) / 4779 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26 \nu^{13} + 7 \nu^{12} - 70 \nu^{11} - 266 \nu^{10} - 301 \nu^{9} + 955 \nu^{8} + 846 \nu^{7} + \cdots - 30618 ) / 14337 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{13} + 3 \nu^{12} + 5 \nu^{11} - 12 \nu^{10} - 16 \nu^{9} - 9 \nu^{8} + 90 \nu^{7} + \cdots - 2916 ) / 729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23 \nu^{13} + 70 \nu^{12} - 29 \nu^{11} - 245 \nu^{10} - 785 \nu^{9} - 110 \nu^{8} + \cdots - 126360 ) / 14337 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 86 \nu^{13} - 159 \nu^{12} - 763 \nu^{11} - 165 \nu^{10} + 2420 \nu^{9} + 3060 \nu^{8} + \cdots + 116640 ) / 43011 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 166 \nu^{13} - 120 \nu^{12} - 533 \nu^{11} + 21 \nu^{10} + 280 \nu^{9} + 495 \nu^{8} - 2007 \nu^{7} + \cdots - 27702 ) / 43011 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 73 \nu^{13} - 360 \nu^{12} - 40 \nu^{11} + 804 \nu^{10} + 1703 \nu^{9} - 417 \nu^{8} + \cdots + 239841 ) / 43011 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17 \nu^{13} - 495 \nu^{12} - 14 \nu^{11} + 1149 \nu^{10} + 727 \nu^{9} - 681 \nu^{8} + \cdots + 110808 ) / 43011 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 163 \nu^{13} + \nu^{12} - 374 \nu^{11} - 476 \nu^{10} + 13 \nu^{9} + 2647 \nu^{8} - 231 \nu^{7} + \cdots - 13122 ) / 14337 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 520 \nu^{13} - 246 \nu^{12} + 701 \nu^{11} + 1440 \nu^{10} + 923 \nu^{9} - 7467 \nu^{8} + \cdots - 67068 ) / 43011 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{13} - \beta_{12} + 2 \beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \cdots + 3 \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 2\beta_{7} - 3\beta_{5} - \beta_{4} - 2\beta_{3} - 2\beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{13} - 4 \beta_{12} + \beta_{11} + 2 \beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{13} + \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 5 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 7 \beta_{13} - 10 \beta_{12} + 8 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} + 14 \beta_{8} + 5 \beta_{7} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5 \beta_{13} + 8 \beta_{12} + 9 \beta_{11} - 3 \beta_{10} - 31 \beta_{9} - 8 \beta_{8} - 4 \beta_{7} + \cdots - 21 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 10 \beta_{13} - 19 \beta_{12} + 6 \beta_{11} + 2 \beta_{10} + 16 \beta_{9} + 38 \beta_{8} - 7 \beta_{7} + \cdots - 60 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 15 \beta_{13} + 6 \beta_{12} + 18 \beta_{11} + 45 \beta_{10} - 21 \beta_{9} + 64 \beta_{8} + \cdots - 80 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 23 \beta_{13} - 31 \beta_{12} - 8 \beta_{11} - 61 \beta_{10} + 28 \beta_{9} + 29 \beta_{8} - 16 \beta_{7} + \cdots - 202 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 58 \beta_{13} + 19 \beta_{12} - 78 \beta_{11} + 66 \beta_{10} - 113 \beta_{9} - 74 \beta_{8} + \cdots - 328 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11 \beta_{13} + 62 \beta_{12} + 53 \beta_{11} - 121 \beta_{10} + 121 \beta_{9} - 85 \beta_{8} + \cdots - 452 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
−0.473632 + 1.66604i
−1.73040 + 0.0755709i
−1.58203 − 0.705117i
1.64515 + 0.541745i
−0.674693 − 1.59524i
1.68442 − 0.403398i
1.13119 − 1.31165i
−0.473632 − 1.66604i
−1.73040 − 0.0755709i
−1.58203 + 0.705117i
1.64515 − 0.541745i
−0.674693 + 1.59524i
1.68442 + 0.403398i
1.13119 + 1.31165i
0 −1.67964 + 0.422841i 0 −0.951504 + 1.64805i 0 0 0 2.64241 − 1.42044i 0
589.2 0 −0.930647 − 1.46079i 0 0.483929 − 0.838189i 0 0 0 −1.26779 + 2.71895i 0
589.3 0 −0.180364 − 1.72263i 0 −1.26013 + 2.18261i 0 0 0 −2.93494 + 0.621403i 0
589.4 0 0.353409 + 1.69561i 0 −0.381918 + 0.661502i 0 0 0 −2.75020 + 1.19849i 0
589.5 0 1.04417 − 1.38192i 0 2.07260 − 3.58985i 0 0 0 −0.819413 − 2.88592i 0
589.6 0 1.19156 + 1.25705i 0 1.80173 − 3.12069i 0 0 0 −0.160357 + 2.99571i 0
589.7 0 1.70151 + 0.323812i 0 −0.764702 + 1.32450i 0 0 0 2.79029 + 1.10194i 0
1177.1 0 −1.67964 − 0.422841i 0 −0.951504 − 1.64805i 0 0 0 2.64241 + 1.42044i 0
1177.2 0 −0.930647 + 1.46079i 0 0.483929 + 0.838189i 0 0 0 −1.26779 − 2.71895i 0
1177.3 0 −0.180364 + 1.72263i 0 −1.26013 − 2.18261i 0 0 0 −2.93494 − 0.621403i 0
1177.4 0 0.353409 − 1.69561i 0 −0.381918 − 0.661502i 0 0 0 −2.75020 − 1.19849i 0
1177.5 0 1.04417 + 1.38192i 0 2.07260 + 3.58985i 0 0 0 −0.819413 + 2.88592i 0
1177.6 0 1.19156 − 1.25705i 0 1.80173 + 3.12069i 0 0 0 −0.160357 − 2.99571i 0
1177.7 0 1.70151 − 0.323812i 0 −0.764702 − 1.32450i 0 0 0 2.79029 − 1.10194i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.j.h 14
3.b odd 2 1 5292.2.j.g 14
7.b odd 2 1 1764.2.j.g 14
7.c even 3 1 1764.2.i.i 14
7.c even 3 1 1764.2.l.i 14
7.d odd 6 1 252.2.i.b ✓ 14
7.d odd 6 1 252.2.l.b yes 14
9.c even 3 1 inner 1764.2.j.h 14
9.d odd 6 1 5292.2.j.g 14
21.c even 2 1 5292.2.j.h 14
21.g even 6 1 756.2.i.b 14
21.g even 6 1 756.2.l.b 14
21.h odd 6 1 5292.2.i.i 14
21.h odd 6 1 5292.2.l.i 14
28.f even 6 1 1008.2.q.j 14
28.f even 6 1 1008.2.t.j 14
63.g even 3 1 1764.2.i.i 14
63.h even 3 1 1764.2.l.i 14
63.i even 6 1 756.2.l.b 14
63.i even 6 1 2268.2.k.f 14
63.j odd 6 1 5292.2.l.i 14
63.k odd 6 1 252.2.i.b ✓ 14
63.k odd 6 1 2268.2.k.e 14
63.l odd 6 1 1764.2.j.g 14
63.n odd 6 1 5292.2.i.i 14
63.o even 6 1 5292.2.j.h 14
63.s even 6 1 756.2.i.b 14
63.s even 6 1 2268.2.k.f 14
63.t odd 6 1 252.2.l.b yes 14
63.t odd 6 1 2268.2.k.e 14
84.j odd 6 1 3024.2.q.j 14
84.j odd 6 1 3024.2.t.j 14
252.n even 6 1 1008.2.q.j 14
252.r odd 6 1 3024.2.t.j 14
252.bj even 6 1 1008.2.t.j 14
252.bn odd 6 1 3024.2.q.j 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.b ✓ 14 7.d odd 6 1
252.2.i.b ✓ 14 63.k odd 6 1
252.2.l.b yes 14 7.d odd 6 1
252.2.l.b yes 14 63.t odd 6 1
756.2.i.b 14 21.g even 6 1
756.2.i.b 14 63.s even 6 1
756.2.l.b 14 21.g even 6 1
756.2.l.b 14 63.i even 6 1
1008.2.q.j 14 28.f even 6 1
1008.2.q.j 14 252.n even 6 1
1008.2.t.j 14 28.f even 6 1
1008.2.t.j 14 252.bj even 6 1
1764.2.i.i 14 7.c even 3 1
1764.2.i.i 14 63.g even 3 1
1764.2.j.g 14 7.b odd 2 1
1764.2.j.g 14 63.l odd 6 1
1764.2.j.h 14 1.a even 1 1 trivial
1764.2.j.h 14 9.c even 3 1 inner
1764.2.l.i 14 7.c even 3 1
1764.2.l.i 14 63.h even 3 1
2268.2.k.e 14 63.k odd 6 1
2268.2.k.e 14 63.t odd 6 1
2268.2.k.f 14 63.i even 6 1
2268.2.k.f 14 63.s even 6 1
3024.2.q.j 14 84.j odd 6 1
3024.2.q.j 14 252.bn odd 6 1
3024.2.t.j 14 84.j odd 6 1
3024.2.t.j 14 252.r odd 6 1
5292.2.i.i 14 21.h odd 6 1
5292.2.i.i 14 63.n odd 6 1
5292.2.j.g 14 3.b odd 2 1
5292.2.j.g 14 9.d odd 6 1
5292.2.j.h 14 21.c even 2 1
5292.2.j.h 14 63.o even 6 1
5292.2.l.i 14 21.h odd 6 1
5292.2.l.i 14 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} - 2 T_{5}^{13} + 24 T_{5}^{12} + 16 T_{5}^{11} + 295 T_{5}^{10} + 357 T_{5}^{9} + 2670 T_{5}^{8} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 3 T^{13} + \cdots + 2187 \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} - 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 150626529 \) Copy content Toggle raw display
$17$ \( (T^{7} - 2 T^{6} - 50 T^{5} + \cdots + 81)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} - 7 T^{6} + \cdots + 2021)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 105822369 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 145660761 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 13807190016 \) Copy content Toggle raw display
$37$ \( (T^{7} + 10 T^{6} + \cdots - 39584)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 1108290681 \) Copy content Toggle raw display
$43$ \( T^{14} - 7 T^{13} + \cdots + 4084441 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 136048896 \) Copy content Toggle raw display
$53$ \( (T^{7} - 15 T^{6} + \cdots - 30861)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 688747536 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 148644864 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 116985856 \) Copy content Toggle raw display
$71$ \( (T^{7} - T^{6} - 116 T^{5} + \cdots - 972)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} - 21 T^{6} + \cdots + 52427)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 54397165824 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 901054679121 \) Copy content Toggle raw display
$89$ \( (T^{7} + 6 T^{6} + \cdots - 128547)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 767677849 \) Copy content Toggle raw display
show more
show less