↑

Properties

Label 2016.3.m.e
Level $2016$
Weight $3$
Character orbit 2016.m
Analytic conductor $54.932$
Analytic rank $0$
Dimension $12$
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] = [2016,3,Mod(127,2016)]
 
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([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.m (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: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 2 x^{9} + 30 x^{8} - 110 x^{7} + 202 x^{6} + 10 x^{5} + 21 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{5} - \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{5} - \beta_{6} q^{7} + ( - \beta_{6} + \beta_{3}) q^{11} + ( - \beta_{5} + 2) q^{13} + ( - \beta_{7} + \beta_1 + 7) q^{17} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_{2}) q^{19}+ \cdots + ( - 4 \beta_{9} - 2 \beta_{5} + \cdots - 26) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 16 q^{5} + 24 q^{13} + 80 q^{17} + 132 q^{25} + 8 q^{37} + 48 q^{41} - 84 q^{49} - 256 q^{53} + 40 q^{61} + 128 q^{65} + 72 q^{73} - 112 q^{77} - 112 q^{85} + 48 q^{89} - 312 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 2 x^{9} + 30 x^{8} - 110 x^{7} + 202 x^{6} + 10 x^{5} + 21 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5563541 \nu^{11} - 14766376 \nu^{10} + 15008039 \nu^{9} + 68620133 \nu^{8} + 184983481 \nu^{7} + \cdots - 14626063 ) / 82928421 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 26850722 \nu^{11} - 106080670 \nu^{10} + 208595657 \nu^{9} + 72797216 \nu^{8} + 779669107 \nu^{7} + \cdots - 98711359 ) / 82928421 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59297183 \nu^{11} - 221893864 \nu^{10} + 432664322 \nu^{9} + 175163543 \nu^{8} + 1922643130 \nu^{7} + \cdots - 426743275 ) / 82928421 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 63224737 \nu^{11} + 210062228 \nu^{10} - 352457317 \nu^{9} - 395973241 \nu^{8} + \cdots - 879040981 ) / 82928421 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 64638478 \nu^{11} + 184090808 \nu^{10} - 227798536 \nu^{9} - 692929198 \nu^{8} + \cdots - 1256309122 ) / 82928421 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7892 \nu^{11} + 30994 \nu^{10} - 60413 \nu^{9} - 21878 \nu^{8} - 235147 \nu^{7} + 852560 \nu^{6} + \cdots + 36337 ) / 9909 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 76293862 \nu^{11} - 224570066 \nu^{10} + 300126502 \nu^{9} + 751754500 \nu^{8} + \cdots + 2496503620 ) / 82928421 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 77805118 \nu^{11} + 302365352 \nu^{10} - 589820728 \nu^{9} - 218592154 \nu^{8} + \cdots + 399152426 ) / 82928421 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 78134713 \nu^{11} + 267443618 \nu^{10} - 469075105 \nu^{9} - 418316803 \nu^{8} + \cdots - 68858497 ) / 82928421 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 407888668 \nu^{11} + 1593221348 \nu^{10} - 3101056285 \nu^{9} - 1141689160 \nu^{8} + \cdots + 2036593481 ) / 82928421 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 518860541 \nu^{11} + 2029391104 \nu^{10} - 3953500163 \nu^{9} - 1447643981 \nu^{8} + \cdots + 2520446086 ) / 82928421 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{11} - 2\beta_{10} + \beta_{9} - 2\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + 8 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} - 10\beta_{8} + 7\beta_{6} - 3\beta_{3} - \beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{11} - 4 \beta_{10} - \beta_{9} - 7 \beta_{8} + 3 \beta_{7} + \beta_{6} + 3 \beta_{5} + \cdots - 24 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{9} + 17\beta_{7} + 20\beta_{5} - 7\beta_{4} + 18\beta _1 - 274 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 85 \beta_{11} + 94 \beta_{10} - 8 \beta_{9} + 217 \beta_{8} + 77 \beta_{7} - 139 \beta_{6} + \cdots - 724 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -35\beta_{11} + 47\beta_{10} + 206\beta_{8} - 207\beta_{6} + 69\beta_{3} - 27\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 682 \beta_{11} + 814 \beta_{10} - 22 \beta_{9} + 2203 \beta_{8} - 704 \beta_{7} - 1906 \beta_{6} + \cdots + 7324 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -401\beta_{9} - 1489\beta_{7} - 1861\beta_{5} + 344\beta_{4} - 2925\beta _1 + 19430 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1989 \beta_{11} - 2486 \beta_{10} - 237 \beta_{9} - 7364 \beta_{8} - 2226 \beta_{7} + 7096 \beta_{6} + \cdots + 24528 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11060\beta_{11} - 14756\beta_{10} - 52727\beta_{8} + 55271\beta_{6} - 17832\beta_{3} + 12115\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 55025 \beta_{11} - 70526 \beta_{10} + 9440 \beta_{9} - 219548 \beta_{8} + 64465 \beta_{7} + \cdots - 733052 ) / 24 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.43016 − 1.43016i
1.43016 + 1.43016i
−1.60946 − 1.60946i
−1.60946 + 1.60946i
0.113833 − 0.113833i
0.113833 + 0.113833i
0.396237 + 0.396237i
0.396237 − 0.396237i
−0.543994 + 0.543994i
−0.543994 − 0.543994i
2.21323 + 2.21323i
2.21323 − 2.21323i
0 0 0 −8.42802 0 − 2.64575i 0 0 0
127.2 0 0 0 −8.42802 0 2.64575i 0 0 0
127.3 0 0 0 −3.29755 0 − 2.64575i 0 0 0
127.4 0 0 0 −3.29755 0 2.64575i 0 0 0
127.5 0 0 0 1.16998 0 − 2.64575i 0 0 0
127.6 0 0 0 1.16998 0 2.64575i 0 0 0
127.7 0 0 0 3.30421 0 − 2.64575i 0 0 0
127.8 0 0 0 3.30421 0 2.64575i 0 0 0
127.9 0 0 0 5.96654 0 − 2.64575i 0 0 0
127.10 0 0 0 5.96654 0 2.64575i 0 0 0
127.11 0 0 0 9.28484 0 − 2.64575i 0 0 0
127.12 0 0 0 9.28484 0 2.64575i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.3.m.e yes 12
3.b odd 2 1 2016.3.m.d ✓ 12
4.b odd 2 1 inner 2016.3.m.e yes 12
12.b even 2 1 2016.3.m.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.3.m.d ✓ 12 3.b odd 2 1
2016.3.m.d ✓ 12 12.b even 2 1
2016.3.m.e yes 12 1.a even 1 1 trivial
2016.3.m.e yes 12 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 8T_{5}^{5} - 76T_{5}^{4} + 640T_{5}^{3} + 160T_{5}^{2} - 6016T_{5} + 5952 \) acting on \(S_{3}^{\mathrm{new}}(2016, [\chi])\). 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
$5$ \( (T^{6} - 8 T^{5} + \cdots + 5952)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 19591041024 \) Copy content Toggle raw display
$13$ \( (T^{6} - 12 T^{5} + \cdots + 16704)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 40 T^{5} + \cdots - 993984)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 206712078336 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 105245741056 \) Copy content Toggle raw display
$29$ \( (T^{6} - 1936 T^{4} + \cdots + 24072192)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 6893439287296 \) Copy content Toggle raw display
$37$ \( (T^{6} - 4 T^{5} + \cdots + 524755008)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 24 T^{5} + \cdots + 44981568)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 99802977140736 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{6} + 128 T^{5} + \cdots - 1267642368)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 89\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{6} - 20 T^{5} + \cdots - 3840221888)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 35\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( (T^{6} - 36 T^{5} + \cdots + 3076451904)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 356241767399424 \) Copy content Toggle raw display
$89$ \( (T^{6} - 24 T^{5} + \cdots + 203524416)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 156 T^{5} + \cdots - 707714496)^{2} \) Copy content Toggle raw display
show more
show less