Properties

Label 2016.3.d.d
Level $2016$
Weight $3$
Character orbit 2016.d
Analytic conductor $54.932$
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] = [2016,3,Mod(449,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, 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("2016.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.d (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: \(54.9320212950\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.204004793232640142475264.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 84x^{10} + 2142x^{8} + 22400x^{6} + 102753x^{4} + 174636x^{2} + 15876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{5}) q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{5}) q^{5} - \beta_1 q^{7} + ( - \beta_{10} + \beta_{8}) q^{11} + (\beta_{2} - 4) q^{13} + ( - \beta_{11} + \beta_{7} - 5 \beta_{5}) q^{17} + ( - \beta_{6} + \beta_{4} + 4 \beta_1) q^{19} + ( - \beta_{10} - 3 \beta_{8}) q^{23} + ( - \beta_{2} - 5) q^{25} + (2 \beta_{7} - \beta_{5}) q^{29} + ( - \beta_{6} - \beta_{4} + 4 \beta_1) q^{31} + ( - \beta_{10} - \beta_{9} - \beta_{8}) q^{35} + (\beta_{2} - 14) q^{37} + ( - \beta_{7} - 7 \beta_{5}) q^{41} + ( - 2 \beta_{6} + 6 \beta_{4} + 16 \beta_1) q^{43} + ( - \beta_{10} + 4 \beta_{9} - 4 \beta_{8}) q^{47} + 7 q^{49} + (3 \beta_{11} + 11 \beta_{5}) q^{53} + (5 \beta_{6} + \beta_{4} + 4 \beta_1) q^{55} + (5 \beta_{10} + 4 \beta_{9} - 4 \beta_{8}) q^{59} + ( - 5 \beta_{3} - 5 \beta_{2} + 6) q^{61} + (3 \beta_{11} - 2 \beta_{7} - 4 \beta_{5}) q^{65} + (\beta_{6} + 2 \beta_{4} + 24 \beta_1) q^{67} + ( - 5 \beta_{10} + 16 \beta_{9} + 3 \beta_{8}) q^{71} + (5 \beta_{3} - 4 \beta_{2} + 16) q^{73} + ( - 2 \beta_{11} - \beta_{7}) q^{77} + ( - 5 \beta_{6} - 4 \beta_{4} + 16 \beta_1) q^{79} + ( - 3 \beta_{10} + 4 \beta_{9} + 10 \beta_{8}) q^{83} + (10 \beta_{3} + 7 \beta_{2} + 38) q^{85} + ( - 13 \beta_{7} + 49 \beta_{5}) q^{89} + (2 \beta_{6} + 3 \beta_{4} + 4 \beta_1) q^{91} + (24 \beta_{9} + 8 \beta_{8}) q^{95} + ( - 5 \beta_{3} + 6 \beta_{2} + 24) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 48 q^{13} - 60 q^{25} - 168 q^{37} + 84 q^{49} + 72 q^{61} + 192 q^{73} + 456 q^{85} + 288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 84x^{10} + 2142x^{8} + 22400x^{6} + 102753x^{4} + 174636x^{2} + 15876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -96\nu^{10} - 7353\nu^{8} - 150983\nu^{6} - 1020159\nu^{4} - 2200527\nu^{2} - 1107302 ) / 341992 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2831\nu^{10} - 227715\nu^{8} - 5251743\nu^{6} - 44801743\nu^{4} - 138197808\nu^{2} - 67023180 ) / 3077928 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4451\nu^{10} - 354087\nu^{8} - 7943859\nu^{6} - 63315427\nu^{4} - 159445104\nu^{2} - 5586588 ) / 3077928 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5437\nu^{10} + 421593\nu^{8} + 8920395\nu^{6} + 63947681\nu^{4} + 139412238\nu^{2} - 14566104 ) / 3077928 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{11} - 700\nu^{9} - 14826\nu^{7} - 101682\nu^{5} - 121961\nu^{3} + 487746\nu ) / 109368 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -10121\nu^{10} - 796389\nu^{8} - 17476191\nu^{6} - 135808141\nu^{4} - 350002422\nu^{2} - 62955144 ) / 3077928 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -830\nu^{11} - 67008\nu^{9} - 1556397\nu^{7} - 13418314\nu^{5} - 44314095\nu^{3} - 66912426\nu ) / 9233784 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 244\nu^{11} + 20979\nu^{9} + 554490\nu^{7} + 5857859\nu^{5} + 23948358\nu^{3} + 25432176\nu ) / 1025976 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\nu^{11} + 1391\nu^{9} + 29085\nu^{7} + 198261\nu^{5} + 310639\nu^{3} - 570654\nu ) / 49644 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -5720\nu^{11} - 451857\nu^{9} - 10030698\nu^{7} - 80972689\nu^{5} - 245735490\nu^{3} - 210261744\nu ) / 4616892 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -2362\nu^{11} - 191220\nu^{9} - 4488237\nu^{7} - 40091702\nu^{5} - 138985539\nu^{3} - 136391346\nu ) / 1538964 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} - 2\beta_{8} - 2\beta_{7} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - 5\beta_{4} - 2\beta_{3} + 2\beta_{2} - 16\beta _1 - 56 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 39\beta_{11} - 51\beta_{10} - 88\beta_{9} + 54\beta_{8} + 30\beta_{7} - 224\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 72\beta_{6} + 248\beta_{4} + 101\beta_{3} - 145\beta_{2} + 672\beta _1 + 1848 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1715\beta_{11} + 2453\beta_{10} + 5320\beta_{9} - 2090\beta_{8} - 1414\beta_{7} + 14000\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4095\beta_{6} - 12075\beta_{4} - 4452\beta_{3} + 7644\beta_{2} - 30128\beta _1 - 80080 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 80017\beta_{11} - 118769\beta_{10} - 274008\beta_{9} + 92946\beta_{8} + 74578\beta_{7} - 724416\beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 210224\beta_{6} + 585424\beta_{4} + 203959\beta_{3} - 380107\beta_{2} + 1416128\beta _1 + 3748696 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3824541 \beta_{11} + 5758515 \beta_{10} + 13549592 \beta_{9} - 4360230 \beta_{8} - 3764250 \beta_{7} + 35845264 \beta_{5} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -10403673\beta_{6} - 28369789\beta_{4} - 9647596\beta_{3} + 18586820\beta_{2} - 67871664\beta _1 - 179582256 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 184443007 \beta_{11} - 279201055 \beta_{10} - 661387496 \beta_{9} + 208788622 \beta_{8} + 185380622 \beta_{7} - 1749844096 \beta_{5} ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
4.20179i
2.09284i
2.76051i
2.40307i
6.96230i
0.310236i
6.96230i
0.310236i
2.76051i
2.40307i
4.20179i
2.09284i
0 0 0 6.57780i 0 −2.64575 0 0 0
449.2 0 0 0 6.57780i 0 2.64575 0 0 0
449.3 0 0 0 5.85832i 0 −2.64575 0 0 0
449.4 0 0 0 5.85832i 0 2.64575 0 0 0
449.5 0 0 0 3.52316i 0 −2.64575 0 0 0
449.6 0 0 0 3.52316i 0 2.64575 0 0 0
449.7 0 0 0 3.52316i 0 −2.64575 0 0 0
449.8 0 0 0 3.52316i 0 2.64575 0 0 0
449.9 0 0 0 5.85832i 0 −2.64575 0 0 0
449.10 0 0 0 5.85832i 0 2.64575 0 0 0
449.11 0 0 0 6.57780i 0 −2.64575 0 0 0
449.12 0 0 0 6.57780i 0 2.64575 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.3.d.d 12
3.b odd 2 1 inner 2016.3.d.d 12
4.b odd 2 1 inner 2016.3.d.d 12
8.b even 2 1 4032.3.d.p 12
8.d odd 2 1 4032.3.d.p 12
12.b even 2 1 inner 2016.3.d.d 12
24.f even 2 1 4032.3.d.p 12
24.h odd 2 1 4032.3.d.p 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.3.d.d 12 1.a even 1 1 trivial
2016.3.d.d 12 3.b odd 2 1 inner
2016.3.d.d 12 4.b odd 2 1 inner
2016.3.d.d 12 12.b even 2 1 inner
4032.3.d.p 12 8.b even 2 1
4032.3.d.p 12 8.d odd 2 1
4032.3.d.p 12 24.f even 2 1
4032.3.d.p 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2016, [\chi])\):

\( T_{5}^{6} + 90T_{5}^{4} + 2448T_{5}^{2} + 18432 \) Copy content Toggle raw display
\( T_{19}^{6} - 1008T_{19}^{4} + 129024T_{19}^{2} - 4128768 \) 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
$5$ \( (T^{6} + 90 T^{4} + 2448 T^{2} + \cdots + 18432)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} + 588 T^{4} + 86436 T^{2} + \cdots + 3118976)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} + 12 T^{2} - 204 T + 64)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 1242 T^{4} + 278736 T^{2} + \cdots + 8258048)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 1008 T^{4} + 129024 T^{2} + \cdots - 4128768)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 2604 T^{4} + 1695204 T^{2} + \cdots + 107886464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 342 T^{4} + 33612 T^{2} + \cdots + 753992)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 1008 T^{4} + 172032 T^{2} + \cdots - 7340032)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 42 T^{2} + 336 T + 224)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 378 T^{4} + 25872 T^{2} + \cdots + 401408)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 13440 T^{4} + \cdots - 37001101312)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 5040 T^{4} + \cdots + 3526885376)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 9798 T^{4} + \cdots + 20671951112)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 13104 T^{4} + \cdots + 9592561664)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 18 T^{2} - 8292 T + 274184)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 13272 T^{4} + \cdots - 50033815552)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 21420 T^{4} + \cdots + 62008361856)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 48 T^{2} - 5364 T - 2976)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 20664 T^{4} + \cdots - 46026846208)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 28896 T^{4} + \cdots + 3526885376)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 28602 T^{4} + \cdots + 110954414592)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 72 T^{2} - 9444 T + 655712)^{4} \) Copy content Toggle raw display
show more
show less