↑

Properties

Label 1728.3.h.j
Level $1728$
Weight $3$
Character orbit 1728.h
Analytic conductor $47.085$
Analytic rank $0$
Dimension $12$
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] = [1728,3,Mod(161,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.h (of order \(2\), degree \(1\), 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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 13x^{10} + 129x^{8} - 512x^{6} + 1548x^{4} - 160x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{5} + ( - \beta_{8} + \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{5} + ( - \beta_{8} + \beta_{3}) q^{7} + (\beta_1 + 1) q^{11} - \beta_{5} q^{13} - \beta_{11} q^{17} + ( - \beta_{10} - \beta_{9}) q^{19} + ( - \beta_{7} + 2 \beta_{4}) q^{23} + ( - \beta_{2} + 6) q^{25} + (2 \beta_{8} + 3 \beta_{6} + 3 \beta_{3}) q^{29} + ( - \beta_{8} + \beta_{6} - 2 \beta_{3}) q^{31} + ( - \beta_{2} - \beta_1 + 21) q^{35} + (\beta_{5} - \beta_{4}) q^{37} + (\beta_{10} + 2 \beta_{9}) q^{41} + ( - 3 \beta_{11} + \beta_{10} - 2 \beta_{9}) q^{43} + (2 \beta_{7} + 5 \beta_{4}) q^{47} + ( - 3 \beta_1 + 14) q^{49} + ( - 5 \beta_{8} + 6 \beta_{6} - 3 \beta_{3}) q^{53} + ( - 3 \beta_{8} - 3 \beta_{6} - 3 \beta_{3}) q^{55} + (\beta_{2} - \beta_1 + 22) q^{59} + ( - 3 \beta_{7} - 3 \beta_{5} - \beta_{4}) q^{61} + (3 \beta_{11} + 9 \beta_{9}) q^{65} + (3 \beta_{11} - \beta_{10} + 3 \beta_{9}) q^{67} + ( - 2 \beta_{7} - 3 \beta_{5} + 10 \beta_{4}) q^{71} + (\beta_{2} - 23) q^{73} + (9 \beta_{8} + 12 \beta_{6} - 6 \beta_{3}) q^{77} + ( - 2 \beta_{8} - 2 \beta_{6} + 5 \beta_{3}) q^{79} + (3 \beta_{2} + 33) q^{83} + (3 \beta_{7} + 6 \beta_{5} + 9 \beta_{4}) q^{85} + (\beta_{11} - 5 \beta_{10} - \beta_{9}) q^{89} + (3 \beta_{11} + 2 \beta_{10} + \beta_{9}) q^{91} + (5 \beta_{7} + 6 \beta_{5} + 14 \beta_{4}) q^{95} + (\beta_{2} + 9 \beta_1 - 33) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{11} + 72 q^{25} + 252 q^{35} + 168 q^{49} + 264 q^{59} - 276 q^{73} + 396 q^{83} - 396 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 13x^{10} + 129x^{8} - 512x^{6} + 1548x^{4} - 160x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -1989\nu^{10} + 19737\nu^{8} - 150249\nu^{6} + 236844\nu^{4} - 24480\nu^{2} + 30104 ) / 395224 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -6045\nu^{10} + 59985\nu^{8} - 549633\nu^{6} + 719820\nu^{4} - 74400\nu^{2} - 20026184 ) / 395224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -1290\nu^{11} + 16601\nu^{9} - 164733\nu^{7} + 643839\nu^{5} - 1976796\nu^{3} + 401932\nu ) / 98806 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2580\nu^{10} + 33202\nu^{8} - 329466\nu^{6} + 1287678\nu^{4} - 3953592\nu^{2} + 211028 ) / 49403 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23759\nu^{10} + 311767\nu^{8} - 3078487\nu^{6} + 12238520\nu^{4} - 35953784\nu^{2} + 1921816 ) / 197612 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\nu^{11} - 514\nu^{9} + 5120\nu^{7} - 20773\nu^{5} + 63218\nu^{3} - 12852\nu ) / 508 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -27165\nu^{10} + 345565\nu^{8} - 3444269\nu^{6} + 13434544\nu^{4} - 41528840\nu^{2} + 2216184 ) / 197612 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -80281\nu^{11} + 1051250\nu^{9} - 10454436\nu^{7} + 42108591\nu^{5} - 128318606\nu^{3} + 26087260\nu ) / 790448 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -130\nu^{11} + 1679\nu^{9} - 16601\nu^{7} + 64883\nu^{5} - 193766\nu^{3} + 676\nu ) / 778 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -47470\nu^{11} + 611657\nu^{9} - 6061919\nu^{7} + 23692277\nu^{5} - 71458550\nu^{3} + 246844\nu ) / 98806 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -95970\nu^{11} + 1237335\nu^{9} - 12255369\nu^{7} + 47898627\nu^{5} - 143803830\nu^{3} + 499044\nu ) / 197612 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{11} - 2\beta_{10} - \beta_{9} + 12\beta_{3} ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + 3\beta_{5} - 15\beta_{4} + \beta_{2} - \beta _1 + 52 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{11} - 3\beta_{10} - 3\beta_{9} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 27\beta_{7} + 15\beta_{5} - 105\beta_{4} - 7\beta_{2} + 15\beta _1 - 356 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 62\beta_{11} - 40\beta_{10} - 65\beta_{9} + 132\beta_{8} + 123\beta_{6} - 306\beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -51\beta_{2} + 155\beta _1 - 2596 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -490\beta_{11} + 284\beta_{10} + 607\beta_{9} + 1236\beta_{8} + 1239\beta_{6} - 2322\beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -645\beta_{7} - 129\beta_{5} + 1987\beta_{4} - 129\beta_{2} + 473\beta _1 - 6572 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -3922\beta_{11} + 2116\beta_{10} + 5315\beta_{9} ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -16023\beta_{7} - 2091\beta_{5} + 46833\beta_{4} + 3019\beta_{2} - 12307\beta _1 + 153892 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -31634\beta_{11} + 16308\beta_{10} + 45075\beta_{9} - 91956\beta_{8} - 96831\beta_{6} + 143826\beta_{3} ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
0.278546 − 0.160819i
0.278546 + 0.160819i
1.88569 + 1.08870i
1.88569 − 1.08870i
−2.47317 + 1.42789i
−2.47317 − 1.42789i
2.47317 − 1.42789i
2.47317 + 1.42789i
−1.88569 − 1.08870i
−1.88569 + 1.08870i
−0.278546 + 0.160819i
−0.278546 − 0.160819i
0 0 0 −9.03816 0 −7.92398 0 0 0
161.2 0 0 0 −9.03816 0 −7.92398 0 0 0
161.3 0 0 0 −3.32298 0 4.21979 0 0 0
161.4 0 0 0 −3.32298 0 4.21979 0 0 0
161.5 0 0 0 −0.519033 0 −10.4117 0 0 0
161.6 0 0 0 −0.519033 0 −10.4117 0 0 0
161.7 0 0 0 0.519033 0 10.4117 0 0 0
161.8 0 0 0 0.519033 0 10.4117 0 0 0
161.9 0 0 0 3.32298 0 −4.21979 0 0 0
161.10 0 0 0 3.32298 0 −4.21979 0 0 0
161.11 0 0 0 9.03816 0 7.92398 0 0 0
161.12 0 0 0 9.03816 0 7.92398 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
12.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.3.h.j yes 12
3.b odd 2 1 1728.3.h.i ✓ 12
4.b odd 2 1 1728.3.h.i ✓ 12
8.b even 2 1 1728.3.h.i ✓ 12
8.d odd 2 1 inner 1728.3.h.j yes 12
12.b even 2 1 inner 1728.3.h.j yes 12
24.f even 2 1 1728.3.h.i ✓ 12
24.h odd 2 1 inner 1728.3.h.j yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.3.h.i ✓ 12 3.b odd 2 1
1728.3.h.i ✓ 12 4.b odd 2 1
1728.3.h.i ✓ 12 8.b even 2 1
1728.3.h.i ✓ 12 24.f even 2 1
1728.3.h.j yes 12 1.a even 1 1 trivial
1728.3.h.j yes 12 8.d odd 2 1 inner
1728.3.h.j yes 12 12.b even 2 1 inner
1728.3.h.j yes 12 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{6} - 93T_{5}^{4} + 927T_{5}^{2} - 243 \) Copy content Toggle raw display
\( T_{7}^{6} - 189T_{7}^{4} + 9855T_{7}^{2} - 121203 \) Copy content Toggle raw display
\( T_{11}^{3} - 3T_{11}^{2} - 225T_{11} + 243 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 93 T^{4} + \cdots - 243)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 189 T^{4} + \cdots - 121203)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} - 3 T^{2} + \cdots + 243)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} + 588 T^{4} + \cdots + 2239488)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 1188 T^{4} + \cdots + 6718464)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 1860 T^{4} + \cdots + 122589184)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 1740 T^{4} + \cdots + 995328)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 2568 T^{4} + \cdots - 62208)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 1149 T^{4} + \cdots - 29937843)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 684 T^{4} + \cdots + 110592)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 2340 T^{4} + \cdots + 241864704)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 9456 T^{4} + \cdots + 22562443264)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 10848 T^{4} + \cdots + 10765467648)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 12837 T^{4} + \cdots - 207018747)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 66 T^{2} + \cdots + 52488)^{4} \) Copy content Toggle raw display
$61$ \( (T^{6} + 15408 T^{4} + \cdots + 6159532032)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 10224 T^{4} + \cdots + 32411521024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 18732 T^{4} + \cdots + 16435104768)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 69 T^{2} + \cdots - 1733)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 4440 T^{4} + \cdots - 2239488)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 99 T^{2} + \cdots + 1384371)^{4} \) Copy content Toggle raw display
$89$ \( (T^{6} + 39456 T^{4} + \cdots + 929727922176)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 99 T^{2} + \cdots - 1409407)^{4} \) Copy content Toggle raw display
show more
show less