↑

Properties

Label 2736.3.o.q
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,3,Mod(721,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 264 x^{18} + 28274 x^{16} - 1545308 x^{14} + 45358441 x^{12} - 637328868 x^{10} + \cdots + 194396337216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{32} \)
Twist minimal: no (minimal twist has level 1368)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{5} + ( - \beta_{3} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{5} + ( - \beta_{3} + 1) q^{7} + \beta_{14} q^{11} + \beta_{2} q^{13} - \beta_{17} q^{17} + \beta_{6} q^{19} - \beta_{12} q^{23} + (\beta_1 + 3) q^{25} + (\beta_{13} - \beta_{11}) q^{29} + ( - \beta_{5} + \beta_{2}) q^{31} + (\beta_{15} - \beta_{14}) q^{35} + ( - \beta_{9} - \beta_{6} - \beta_{4}) q^{37} - \beta_{19} q^{41} + ( - \beta_{7} - \beta_{4} + 6) q^{43} + (\beta_{15} - \beta_{14} - \beta_{12}) q^{47} + ( - \beta_{7} - \beta_{4} - \beta_{3} + \cdots + 6) q^{49}+ \cdots + (3 \beta_{9} + \beta_{6} + \cdots + \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 16 q^{7} + 8 q^{19} + 68 q^{25} + 128 q^{43} + 116 q^{49} - 144 q^{55} - 104 q^{61} - 88 q^{73} - 280 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 264 x^{18} + 28274 x^{16} - 1545308 x^{14} + 45358441 x^{12} - 637328868 x^{10} + \cdots + 194396337216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12\!\cdots\!73 \nu^{18} + \cdots - 63\!\cdots\!76 ) / 22\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!19 \nu^{18} + \cdots + 91\!\cdots\!12 ) / 46\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!12 \nu^{18} + \cdots + 17\!\cdots\!48 ) / 29\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 72\!\cdots\!19 \nu^{18} + \cdots - 29\!\cdots\!36 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 98\!\cdots\!57 \nu^{18} + \cdots + 60\!\cdots\!32 ) / 93\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\!\cdots\!79 \nu^{18} + \cdots + 23\!\cdots\!40 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42\!\cdots\!55 \nu^{18} + \cdots - 51\!\cdots\!48 ) / 18\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!87 \nu^{18} + \cdots - 43\!\cdots\!20 ) / 46\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11\!\cdots\!55 \nu^{18} + \cdots - 46\!\cdots\!72 ) / 31\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 65\!\cdots\!03 \nu^{19} + \cdots - 12\!\cdots\!68 \nu ) / 16\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 65\!\cdots\!03 \nu^{19} + \cdots + 28\!\cdots\!44 \nu ) / 20\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 79\!\cdots\!61 \nu^{19} + \cdots - 38\!\cdots\!12 \nu ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17\!\cdots\!95 \nu^{19} + \cdots - 16\!\cdots\!92 \nu ) / 42\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15\!\cdots\!09 \nu^{19} + \cdots + 18\!\cdots\!00 \nu ) / 34\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 21\!\cdots\!17 \nu^{19} + \cdots + 29\!\cdots\!60 \nu ) / 34\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 92\!\cdots\!21 \nu^{19} + \cdots + 26\!\cdots\!00 \nu ) / 13\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 77\!\cdots\!23 \nu^{19} + \cdots + 21\!\cdots\!76 \nu ) / 85\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 22\!\cdots\!95 \nu^{19} + \cdots + 27\!\cdots\!52 \nu ) / 10\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 10\!\cdots\!11 \nu^{19} + \cdots - 12\!\cdots\!12 \nu ) / 42\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 8\beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} - \beta_{5} + \beta_{2} + 4\beta _1 + 104 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{19} + 6 \beta_{18} + 24 \beta_{17} + 12 \beta_{16} - 32 \beta_{15} - 12 \beta_{13} + \cdots + 400 \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 45 \beta_{9} + 57 \beta_{8} - 6 \beta_{7} - 12 \beta_{6} - 69 \beta_{5} + 70 \beta_{4} + 90 \beta_{3} + \cdots + 2594 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1800 \beta_{19} + 1190 \beta_{18} + 2168 \beta_{17} + 1040 \beta_{16} - 2800 \beta_{15} + \cdots + 20784 \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 6983 \beta_{9} + 10787 \beta_{8} - 1340 \beta_{7} + 3416 \beta_{6} - 15647 \beta_{5} + 17004 \beta_{4} + \cdots + 254132 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 193816 \beta_{19} + 135058 \beta_{18} + 131352 \beta_{17} + 86800 \beta_{16} - 158928 \beta_{15} + \cdots + 963664 \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 139277 \beta_{9} + 237009 \beta_{8} - 19579 \beta_{7} + 200770 \beta_{6} - 383701 \beta_{5} + \cdots + 2422221 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 17687544 \beta_{19} + 12474966 \beta_{18} + 4163672 \beta_{17} + 7095456 \beta_{16} + \cdots + 23106448 \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 44083593 \beta_{9} + 77897725 \beta_{8} - 1068972 \beta_{7} + 101882264 \beta_{6} - 133205905 \beta_{5} + \cdots - 73038332 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1438379976 \beta_{19} + 1016671194 \beta_{18} - 270794024 \beta_{17} + 552352064 \beta_{16} + \cdots - 2545983280 \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1650743743 \beta_{9} + 2953625547 \beta_{8} + 203506234 \beta_{7} + 5149511828 \beta_{6} + \cdots - 36917990150 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 105024498904 \beta_{19} + 74232781038 \beta_{18} - 69629386600 \beta_{17} + \cdots - 544008572912 \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 224805993819 \beta_{9} + 403674825655 \beta_{8} + 69837133684 \beta_{7} + 908375012408 \beta_{6} + \cdots - 10956535197500 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 6723822781032 \beta_{19} + 4751208947538 \beta_{18} - 8911789447912 \beta_{17} + \cdots - 67405260572720 \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 3277009021882 \beta_{9} + 5889328683906 \beta_{8} + 1999352894565 \beta_{7} + 17962554953202 \beta_{6} + \cdots - 302143772553619 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 345929797303224 \beta_{19} + 244394114338790 \beta_{18} - 912359080292776 \beta_{17} + \cdots - 68\!\cdots\!36 \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 545831763358445 \beta_{9} + 981473797990129 \beta_{8} + 767114926063572 \beta_{7} + \cdots - 11\!\cdots\!40 ) / 4 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 91\!\cdots\!96 \beta_{19} + \cdots - 61\!\cdots\!44 \beta_{10} ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
−8.89263 − 1.41421i
−8.89263 + 1.41421i
−6.04300 + 1.41421i
−6.04300 − 1.41421i
−5.08927 + 1.41421i
−5.08927 − 1.41421i
−0.585690 + 1.41421i
−0.585690 − 1.41421i
−0.399553 + 1.41421i
−0.399553 − 1.41421i
0.399553 − 1.41421i
0.399553 + 1.41421i
0.585690 − 1.41421i
0.585690 + 1.41421i
5.08927 − 1.41421i
5.08927 + 1.41421i
6.04300 − 1.41421i
6.04300 + 1.41421i
8.89263 + 1.41421i
8.89263 − 1.41421i
0 0 0 −8.89263 0 −4.08883 0 0 0
721.2 0 0 0 −8.89263 0 −4.08883 0 0 0
721.3 0 0 0 −6.04300 0 0.880524 0 0 0
721.4 0 0 0 −6.04300 0 0.880524 0 0 0
721.5 0 0 0 −5.08927 0 11.6968 0 0 0
721.6 0 0 0 −5.08927 0 11.6968 0 0 0
721.7 0 0 0 −0.585690 0 5.15901 0 0 0
721.8 0 0 0 −0.585690 0 5.15901 0 0 0
721.9 0 0 0 −0.399553 0 −9.64754 0 0 0
721.10 0 0 0 −0.399553 0 −9.64754 0 0 0
721.11 0 0 0 0.399553 0 −9.64754 0 0 0
721.12 0 0 0 0.399553 0 −9.64754 0 0 0
721.13 0 0 0 0.585690 0 5.15901 0 0 0
721.14 0 0 0 0.585690 0 5.15901 0 0 0
721.15 0 0 0 5.08927 0 11.6968 0 0 0
721.16 0 0 0 5.08927 0 11.6968 0 0 0
721.17 0 0 0 6.04300 0 0.880524 0 0 0
721.18 0 0 0 6.04300 0 0.880524 0 0 0
721.19 0 0 0 8.89263 0 −4.08883 0 0 0
721.20 0 0 0 8.89263 0 −4.08883 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.q 20
3.b odd 2 1 inner 2736.3.o.q 20
4.b odd 2 1 1368.3.o.d ✓ 20
12.b even 2 1 1368.3.o.d ✓ 20
19.b odd 2 1 inner 2736.3.o.q 20
57.d even 2 1 inner 2736.3.o.q 20
76.d even 2 1 1368.3.o.d ✓ 20
228.b odd 2 1 1368.3.o.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.3.o.d ✓ 20 4.b odd 2 1
1368.3.o.d ✓ 20 12.b even 2 1
1368.3.o.d ✓ 20 76.d even 2 1
1368.3.o.d ✓ 20 228.b odd 2 1
2736.3.o.q 20 1.a even 1 1 trivial
2736.3.o.q 20 3.b odd 2 1 inner
2736.3.o.q 20 19.b odd 2 1 inner
2736.3.o.q 20 57.d 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 \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{10} - 142T_{5}^{8} + 5953T_{5}^{6} - 77760T_{5}^{4} + 37920T_{5}^{2} - 4096 \) Copy content Toggle raw display
\( T_{7}^{5} - 4T_{7}^{4} - 129T_{7}^{3} + 280T_{7}^{2} + 2236T_{7} - 2096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} - 142 T^{8} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$7$ \( (T^{5} - 4 T^{4} + \cdots - 2096)^{4} \) Copy content Toggle raw display
$11$ \( (T^{10} - 616 T^{8} + \cdots - 7953785856)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 840 T^{8} + \cdots + 55207657472)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 194784526336)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 6131066257801)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 218994049024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 34662336954368)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 7860327022592)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 549755813888)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 31\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 32 T^{4} + \cdots - 103358208)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 72\!\cdots\!48)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 52\!\cdots\!52)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 26 T^{4} + \cdots + 2358752)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 15\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 28\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 22 T^{4} + \cdots - 134016)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 44\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 31\!\cdots\!08)^{2} \) Copy content Toggle raw display
show more
show less