↑

Properties

Label 2736.3.h.e
Level $2736$
Weight $3$
Character orbit 2736.h
Analytic conductor $74.551$
Analytic rank $0$
Dimension $16$
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] = [2736,3,Mod(305,2736)]
 
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, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.305");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.h (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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 208 x^{14} + 15922 x^{12} + 569560 x^{10} + 10305505 x^{8} + 93221832 x^{6} + 369937408 x^{4} + \cdots + 83612736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{9} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_1) q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} + \beta_1) q^{5} + \beta_{3} q^{7} + (\beta_{10} + \beta_{9}) q^{11} + (\beta_{6} + \beta_{4}) q^{13} + (\beta_{14} - \beta_{13} + \cdots + \beta_1) q^{17}+ \cdots + ( - 2 \beta_{8} - 3 \beta_{7} + \cdots + 14) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} + 8 q^{13} - 64 q^{25} + 16 q^{31} + 48 q^{37} - 56 q^{43} + 24 q^{49} + 120 q^{55} - 176 q^{61} + 8 q^{67} + 312 q^{73} - 88 q^{79} - 136 q^{85} - 280 q^{91} + 200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 208 x^{14} + 15922 x^{12} + 569560 x^{10} + 10305505 x^{8} + 93221832 x^{6} + 369937408 x^{4} + \cdots + 83612736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 72722245 \nu^{14} + 14392450960 \nu^{12} + 1021895658322 \nu^{10} + 32747787932008 \nu^{8} + \cdots - 962971852921632 ) / 40\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 432911832382759 \nu^{14} + \cdots - 22\!\cdots\!88 ) / 19\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!85 \nu^{14} + \cdots - 10\!\cdots\!12 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!23 \nu^{14} + \cdots + 17\!\cdots\!00 ) / 97\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18958936 \nu^{14} + 3787319668 \nu^{12} + 271336520083 \nu^{10} + 8692345068082 \nu^{8} + \cdots + 12\!\cdots\!80 ) / 150616166499216 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 817318524952667 \nu^{14} + \cdots + 12\!\cdots\!44 ) / 53\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 34\!\cdots\!97 \nu^{14} + \cdots - 21\!\cdots\!92 ) / 19\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 95\!\cdots\!17 \nu^{15} + \cdots + 98\!\cdots\!00 \nu ) / 44\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26327970607 \nu^{15} - 5642460938326 \nu^{13} - 452095090899214 \nu^{11} + \cdots - 29\!\cdots\!88 \nu ) / 91\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 41\!\cdots\!17 \nu^{15} + \cdots - 91\!\cdots\!64 \nu ) / 16\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2279711 \nu^{15} - 490720241 \nu^{13} - 39482752070 \nu^{11} + \cdots - 823598510539320 \nu ) / 45056278517184 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23\!\cdots\!93 \nu^{15} + \cdots + 23\!\cdots\!24 \nu ) / 44\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!29 \nu^{15} + \cdots - 13\!\cdots\!28 \nu ) / 49\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 13\!\cdots\!25 \nu^{15} + \cdots - 32\!\cdots\!68 \nu ) / 55\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 25 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} - 3\beta_{14} + 5\beta_{13} - 2\beta_{12} - 4\beta_{11} + 37\beta_{10} + 7\beta_{9} - 57\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -102\beta_{8} - 73\beta_{7} - 194\beta_{6} + 49\beta_{5} + 46\beta_{4} + 69\beta_{3} - 126\beta_{2} + 1394 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 201 \beta_{15} + 272 \beta_{14} - 543 \beta_{13} + 369 \beta_{12} + 427 \beta_{11} + \cdots + 4016 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7749 \beta_{8} + 5943 \beta_{7} + 15808 \beta_{6} - 2449 \beta_{5} - 2275 \beta_{4} - 5335 \beta_{3} + \cdots - 96443 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14882 \beta_{15} - 19621 \beta_{14} + 50217 \beta_{13} - 41444 \beta_{12} - 36284 \beta_{11} + \cdots - 299903 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 563478 \beta_{8} - 487259 \beta_{7} - 1235702 \beta_{6} + 125827 \beta_{5} + 117054 \beta_{4} + \cdots + 7142690 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1021735 \beta_{15} + 1341300 \beta_{14} - 4380293 \beta_{13} + 4006095 \beta_{12} + \cdots + 22887556 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 41000995 \beta_{8} + 39714965 \beta_{7} + 95880264 \beta_{6} - 6335451 \beta_{5} - 5961253 \beta_{4} + \cdots - 543014861 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 68959466 \beta_{15} - 90833419 \beta_{14} + 371250747 \beta_{13} - 361803288 \beta_{12} + \cdots - 1767359457 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3016540182 \beta_{8} - 3221041033 \beta_{7} - 7457099530 \beta_{6} + 292728153 \beta_{5} + \cdots + 41843655762 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4669513761 \beta_{15} + 6191207688 \beta_{14} - 30944254663 \beta_{13} + 31459823921 \beta_{12} + \cdots + 137580783568 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 224832557173 \beta_{8} + 260351677551 \beta_{7} + 582720420960 \beta_{6} - 10640946521 \beta_{5} + \cdots - 3252951838779 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 319852539850 \beta_{15} - 427514103349 \beta_{14} + 2551450510369 \beta_{13} + \cdots - 10775787334575 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
− 7.84773i
− 8.94764i
− 4.13517i
− 3.65489i
− 2.88923i
− 5.09734i
0.512855i
1.14076i
− 1.14076i
− 0.512855i
5.09734i
2.88923i
3.65489i
4.13517i
8.94764i
7.84773i
0 0 0 − 9.26195i 0 −1.38995 0 0 0
305.2 0 0 0 − 7.53343i 0 6.57918 0 0 0
305.3 0 0 0 − 5.54939i 0 13.4833 0 0 0
305.4 0 0 0 − 5.06910i 0 −5.44479 0 0 0
305.5 0 0 0 − 4.30345i 0 −11.9150 0 0 0
305.6 0 0 0 − 3.68313i 0 0.523256 0 0 0
305.7 0 0 0 − 0.901359i 0 −0.0915171 0 0 0
305.8 0 0 0 − 0.273452i 0 2.25556 0 0 0
305.9 0 0 0 0.273452i 0 2.25556 0 0 0
305.10 0 0 0 0.901359i 0 −0.0915171 0 0 0
305.11 0 0 0 3.68313i 0 0.523256 0 0 0
305.12 0 0 0 4.30345i 0 −11.9150 0 0 0
305.13 0 0 0 5.06910i 0 −5.44479 0 0 0
305.14 0 0 0 5.54939i 0 13.4833 0 0 0
305.15 0 0 0 7.53343i 0 6.57918 0 0 0
305.16 0 0 0 9.26195i 0 −1.38995 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.h.e 16
3.b odd 2 1 inner 2736.3.h.e 16
4.b odd 2 1 1368.3.h.a ✓ 16
12.b even 2 1 1368.3.h.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.3.h.a ✓ 16 4.b odd 2 1
1368.3.h.a ✓ 16 12.b even 2 1
2736.3.h.e 16 1.a even 1 1 trivial
2736.3.h.e 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 232 T_{5}^{14} + 20554 T_{5}^{12} + 895832 T_{5}^{10} + 20520433 T_{5}^{8} + \cdots + 58798224 \) acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 232 T^{14} + \cdots + 58798224 \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} + \cdots + 864)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{8} - 4 T^{7} + \cdots + 2523520)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 52\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{8} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 225528790323456 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 510138777600 \) Copy content Toggle raw display
$31$ \( (T^{8} - 8 T^{7} + \cdots - 120211551360)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 24 T^{7} + \cdots + 7379963904)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{8} + 28 T^{7} + \cdots + 101515853584)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 77\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 41\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 26036641119200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 292666448609280)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 28\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 254667202670448)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 119294647066624)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
show more
show less