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Properties

Label 4732.2.g.k
Level $4732$
Weight $2$
Character orbit 4732.g
Analytic conductor $37.785$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4732,2,Mod(337,4732)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4732, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4732.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4732.g (of order \(2\), degree \(1\), not 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: \(37.7852102365\)
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: \( x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 364)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{13} - \beta_{3}) q^{5} + \beta_{3} q^{7} + (\beta_{8} + \beta_{6} - \beta_{5} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{13} - \beta_{3}) q^{5} + \beta_{3} q^{7} + (\beta_{8} + \beta_{6} - \beta_{5} + \cdots + 2) q^{9}+ \cdots + ( - \beta_{15} + 2 \beta_{14} + \cdots - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 28 q^{9} - 4 q^{17} - 44 q^{25} - 12 q^{27} + 44 q^{29} + 12 q^{35} - 12 q^{43} - 16 q^{49} - 4 q^{51} + 8 q^{53} - 4 q^{55} - 8 q^{61} + 104 q^{69} + 20 q^{75} - 24 q^{77} + 8 q^{79} - 52 q^{87} + 60 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 125 \nu^{14} + 4204 \nu^{12} + 55047 \nu^{10} + 355436 \nu^{8} + 1184157 \nu^{6} + 1944230 \nu^{4} + \cdots + 104520 ) / 70144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1005 \nu^{15} - 34940 \nu^{13} - 480631 \nu^{11} - 3354588 \nu^{9} - 12716909 \nu^{7} + \cdots - 8550920 \nu ) / 1823744 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1899 \nu^{14} - 66708 \nu^{12} - 922481 \nu^{10} - 6377556 \nu^{8} - 23040011 \nu^{6} + \cdots - 2071160 ) / 70144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1037 \nu^{14} - 35968 \nu^{12} - 489471 \nu^{10} - 3315160 \nu^{8} - 11656901 \nu^{6} + \cdots - 646232 ) / 35072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 543 \nu^{14} + 18968 \nu^{12} + 260597 \nu^{10} + 1787760 \nu^{8} + 6396215 \nu^{6} + 11178918 \nu^{4} + \cdots + 516200 ) / 17536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2921 \nu^{14} - 102588 \nu^{12} - 1417227 \nu^{10} - 9768892 \nu^{8} - 35032713 \nu^{6} + \cdots - 2586920 ) / 70144 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4371 \nu^{14} - 152012 \nu^{12} - 2076377 \nu^{10} - 14136796 \nu^{8} - 50082819 \nu^{6} + \cdots - 3812504 ) / 70144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2547 \nu^{14} - 88484 \nu^{12} - 1207529 \nu^{10} - 8217348 \nu^{8} - 29119795 \nu^{6} + \cdots - 1922424 ) / 35072 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2195 \nu^{15} - 76628 \nu^{13} - 1052201 \nu^{11} - 7213652 \nu^{9} - 25761395 \nu^{7} + \cdots + 44488 \nu ) / 140288 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14471 \nu^{15} - 505724 \nu^{13} - 6952053 \nu^{11} - 47752076 \nu^{9} - 171512567 \nu^{7} + \cdots - 31407032 \nu ) / 911872 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 241 \nu^{15} - 8404 \nu^{13} - 115155 \nu^{11} - 786788 \nu^{9} - 2796961 \nu^{7} + \cdots - 116680 \nu ) / 13312 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12863 \nu^{15} - 446258 \nu^{13} - 6079033 \nu^{11} - 41271254 \nu^{9} - 145826787 \nu^{7} + \cdots - 11271216 \nu ) / 455936 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 7523 \nu^{15} - 262175 \nu^{13} - 3588551 \nu^{11} - 24470313 \nu^{9} - 86676541 \nu^{7} + \cdots - 4617300 \nu ) / 227968 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4249 \nu^{15} + 149086 \nu^{13} + 2059183 \nu^{11} + 14218170 \nu^{9} + 51301557 \nu^{7} + \cdots + 6680784 \nu ) / 113984 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - \beta_{6} + \beta_{5} - \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{13} + \beta_{12} + 2\beta_{11} + \beta_{10} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + 9\beta_{8} + \beta_{7} + 14\beta_{6} - 11\beta_{5} + \beta_{4} + 9\beta_{2} + 37 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{15} + \beta_{14} + 12\beta_{13} - 11\beta_{12} - 29\beta_{11} - 13\beta_{10} + 12\beta_{3} + 44\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -28\beta_{9} - 82\beta_{8} - 15\beta_{7} - 153\beta_{6} + 112\beta_{5} - 15\beta_{4} - 70\beta_{2} - 316 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 142 \beta_{15} - 27 \beta_{14} - 123 \beta_{13} + 124 \beta_{12} + 337 \beta_{11} + 140 \beta_{10} + \cdots - 358 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 320\beta_{9} + 771\beta_{8} + 188\beta_{7} + 1575\beta_{6} - 1139\beta_{5} + 158\beta_{4} + 529\beta_{2} + 2907 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1485 \beta_{15} + 460 \beta_{14} + 1223 \beta_{13} - 1463 \beta_{12} - 3670 \beta_{11} + \cdots + 3086 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3480 \beta_{9} - 7405 \beta_{8} - 2207 \beta_{7} - 15888 \beta_{6} + 11699 \beta_{5} - 1435 \beta_{4} + \cdots - 27861 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 15341 \beta_{15} - 6435 \beta_{14} - 12158 \beta_{13} + 17351 \beta_{12} + 38979 \beta_{11} + \cdots - 27558 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 37294 \beta_{9} + 72152 \beta_{8} + 24959 \beta_{7} + 159203 \beta_{6} - 121182 \beta_{5} + \cdots + 273468 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 157974 \beta_{15} + 81375 \beta_{14} + 121781 \beta_{13} - 202962 \beta_{12} - 409497 \beta_{11} + \cdots + 251910 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 397536 \beta_{9} - 710589 \beta_{8} - 275542 \beta_{7} - 1593911 \beta_{6} + 1261905 \beta_{5} + \cdots - 2724011 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1626201 \beta_{15} - 971326 \beta_{14} - 1230119 \beta_{13} + 2331497 \beta_{12} + 4280424 \beta_{11} + \cdots - 2341280 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times\).

\(n\) \(2367\) \(2705\) \(4565\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
2.98100i
− 2.98100i
2.42977i
− 2.42977i
− 2.25607i
2.25607i
− 0.268953i
0.268953i
− 1.17707i
1.17707i
− 1.75101i
1.75101i
1.77673i
− 1.77673i
3.23100i
− 3.23100i
0 −2.98100 0 − 0.118179i 0 − 1.00000i 0 5.88638 0
337.2 0 −2.98100 0 0.118179i 0 1.00000i 0 5.88638 0
337.3 0 −2.42977 0 − 3.63781i 0 − 1.00000i 0 2.90380 0
337.4 0 −2.42977 0 3.63781i 0 1.00000i 0 2.90380 0
337.5 0 −2.25607 0 − 4.27591i 0 1.00000i 0 2.08985 0
337.6 0 −2.25607 0 4.27591i 0 − 1.00000i 0 2.08985 0
337.7 0 −0.268953 0 − 1.35585i 0 1.00000i 0 −2.92766 0
337.8 0 −0.268953 0 1.35585i 0 − 1.00000i 0 −2.92766 0
337.9 0 1.17707 0 − 1.46614i 0 − 1.00000i 0 −1.61452 0
337.10 0 1.17707 0 1.46614i 0 1.00000i 0 −1.61452 0
337.11 0 1.75101 0 − 1.38536i 0 − 1.00000i 0 0.0660200 0
337.12 0 1.75101 0 1.38536i 0 1.00000i 0 0.0660200 0
337.13 0 1.77673 0 − 3.82804i 0 1.00000i 0 0.156761 0
337.14 0 1.77673 0 3.82804i 0 − 1.00000i 0 0.156761 0
337.15 0 3.23100 0 − 3.14769i 0 1.00000i 0 7.43937 0
337.16 0 3.23100 0 3.14769i 0 − 1.00000i 0 7.43937 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4732.2.g.k 16
13.b even 2 1 inner 4732.2.g.k 16
13.c even 3 1 364.2.u.a ✓ 16
13.d odd 4 1 4732.2.a.s 8
13.d odd 4 1 4732.2.a.t 8
13.e even 6 1 364.2.u.a ✓ 16
39.h odd 6 1 3276.2.cf.c 16
39.i odd 6 1 3276.2.cf.c 16
52.i odd 6 1 1456.2.cc.f 16
52.j odd 6 1 1456.2.cc.f 16
91.g even 3 1 2548.2.bq.e 16
91.h even 3 1 2548.2.bb.d 16
91.k even 6 1 2548.2.bb.d 16
91.l odd 6 1 2548.2.bb.c 16
91.m odd 6 1 2548.2.bq.c 16
91.n odd 6 1 2548.2.u.c 16
91.p odd 6 1 2548.2.bq.c 16
91.t odd 6 1 2548.2.u.c 16
91.u even 6 1 2548.2.bq.e 16
91.v odd 6 1 2548.2.bb.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.u.a ✓ 16 13.c even 3 1
364.2.u.a ✓ 16 13.e even 6 1
1456.2.cc.f 16 52.i odd 6 1
1456.2.cc.f 16 52.j odd 6 1
2548.2.u.c 16 91.n odd 6 1
2548.2.u.c 16 91.t odd 6 1
2548.2.bb.c 16 91.l odd 6 1
2548.2.bb.c 16 91.v odd 6 1
2548.2.bb.d 16 91.h even 3 1
2548.2.bb.d 16 91.k even 6 1
2548.2.bq.c 16 91.m odd 6 1
2548.2.bq.c 16 91.p odd 6 1
2548.2.bq.e 16 91.g even 3 1
2548.2.bq.e 16 91.u even 6 1
3276.2.cf.c 16 39.h odd 6 1
3276.2.cf.c 16 39.i odd 6 1
4732.2.a.s 8 13.d odd 4 1
4732.2.a.t 8 13.d odd 4 1
4732.2.g.k 16 1.a even 1 1 trivial
4732.2.g.k 16 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4732, [\chi])\):

\( T_{3}^{8} - 19T_{3}^{6} + 2T_{3}^{5} + 113T_{3}^{4} - 40T_{3}^{3} - 232T_{3}^{2} + 136T_{3} + 52 \) Copy content Toggle raw display
\( T_{5}^{16} + 62 T_{5}^{14} + 1505 T_{5}^{12} + 18058 T_{5}^{10} + 111420 T_{5}^{8} + 339950 T_{5}^{6} + \cdots + 3721 \) Copy content Toggle raw display
\( T_{17}^{8} + 2T_{17}^{7} - 76T_{17}^{6} - 182T_{17}^{5} + 1478T_{17}^{4} + 4270T_{17}^{3} - 2380T_{17}^{2} - 9498T_{17} - 1727 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 19 T^{6} + \cdots + 52)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 62 T^{14} + \cdots + 3721 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 185722384 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{8} + 2 T^{7} + \cdots - 1727)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 1235663104 \) Copy content Toggle raw display
$23$ \( (T^{8} - 92 T^{6} + \cdots - 9408)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 22 T^{7} + \cdots - 23031)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1643965037584 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 592240896 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 1637621852416 \) Copy content Toggle raw display
$43$ \( (T^{8} + 6 T^{7} + \cdots - 2442908)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 5725040896 \) Copy content Toggle raw display
$53$ \( (T^{8} - 4 T^{7} + \cdots + 117909)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 119508624 \) Copy content Toggle raw display
$61$ \( (T^{8} + 4 T^{7} + \cdots + 1286464)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 62066753424 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 7345574313984 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 89262317824 \) Copy content Toggle raw display
$79$ \( (T^{8} - 4 T^{7} + \cdots - 1070784)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 656 T^{14} + \cdots + 1971216 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 33199755264 \) Copy content Toggle raw display
$97$ \( T^{16} + 798 T^{14} + \cdots + 44302336 \) Copy content Toggle raw display
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