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Properties

Label 2016.2.cs.a
Level $2016$
Weight $2$
Character orbit 2016.cs
Analytic conductor $16.098$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,2,Mod(703,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.cs (of order \(6\), degree \(2\), 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: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{5} + ( - \beta_{13} - \beta_{6} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{5} + ( - \beta_{13} - \beta_{6} - \beta_1) q^{7} + (\beta_{13} + \beta_{11} + \cdots - \beta_1) q^{11}+ \cdots + ( - \beta_{15} + \beta_{11} - \beta_{10} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} - 12 q^{11} + 4 q^{19} + 4 q^{25} + 4 q^{31} - 8 q^{35} + 4 q^{37} - 8 q^{47} + 24 q^{49} - 8 q^{53} - 16 q^{55} - 4 q^{59} + 24 q^{61} - 8 q^{65} - 12 q^{67} + 12 q^{73} + 32 q^{77} + 12 q^{79} - 8 q^{83} + 32 q^{85} - 4 q^{91} + 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 22\nu^{13} + 174\nu^{11} + 612\nu^{9} + 865\nu^{7} - 26\nu^{5} - 1020\nu^{3} - 488\nu + 64 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} + 14 \nu^{14} - 22 \nu^{13} + 320 \nu^{12} - 166 \nu^{11} + 2692 \nu^{10} - 476 \nu^{9} + \cdots + 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{15} - 22 \nu^{14} - 242 \nu^{13} - 496 \nu^{12} - 1898 \nu^{11} - 4084 \nu^{10} + \cdots - 704 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{15} + 34 \nu^{14} - 110 \nu^{13} + 776 \nu^{12} - 862 \nu^{11} + 6508 \nu^{10} - 2940 \nu^{9} + \cdots + 928 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6 \nu^{15} + 21 \nu^{14} - 136 \nu^{13} + 480 \nu^{12} - 1124 \nu^{11} + 4030 \nu^{10} - 4248 \nu^{9} + \cdots + 352 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11 \nu^{15} + 42 \nu^{14} + 250 \nu^{13} + 960 \nu^{12} + 2074 \nu^{11} + 8060 \nu^{10} + 7884 \nu^{9} + \cdots + 768 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{15} + 46 \nu^{14} - 242 \nu^{13} + 1056 \nu^{12} - 1898 \nu^{11} + 8932 \nu^{10} + \cdots + 1152 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11 \nu^{15} - 46 \nu^{14} - 242 \nu^{13} - 1056 \nu^{12} - 1898 \nu^{11} - 8932 \nu^{10} + \cdots - 1152 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23 \nu^{15} - 10 \nu^{14} - 530 \nu^{13} - 224 \nu^{12} - 4506 \nu^{11} - 1820 \nu^{10} + \cdots + 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11 \nu^{15} - 12 \nu^{14} + 254 \nu^{13} - 272 \nu^{12} + 2170 \nu^{11} - 2248 \nu^{10} + 8740 \nu^{9} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11 \nu^{15} - 12 \nu^{14} - 254 \nu^{13} - 272 \nu^{12} - 2170 \nu^{11} - 2248 \nu^{10} - 8740 \nu^{9} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16 \nu^{15} - 11 \nu^{14} + 360 \nu^{13} - 248 \nu^{12} + 2944 \nu^{11} - 2042 \nu^{10} + \cdots + 112 \nu ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17 \nu^{15} - 6 \nu^{14} - 386 \nu^{13} - 136 \nu^{12} - 3202 \nu^{11} - 1132 \nu^{10} - 12216 \nu^{9} + \cdots - 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17 \nu^{15} - 6 \nu^{14} + 386 \nu^{13} - 136 \nu^{12} + 3202 \nu^{11} - 1132 \nu^{10} + 12216 \nu^{9} + \cdots + 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 45 \nu^{15} - 14 \nu^{14} + 1022 \nu^{13} - 320 \nu^{12} + 8478 \nu^{11} - 2692 \nu^{10} + \cdots - 1288 \nu ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{9} - \beta_{8} - \beta_{6} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{14} - \beta_{12} + \beta_{9} - \beta_{6} + \beta_{5} + \beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{15} + 13 \beta_{14} - 11 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + 7 \beta_{9} + \cdots - 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{15} - 4 \beta_{14} + 4 \beta_{12} + \beta_{11} + \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 53 \beta_{15} - 123 \beta_{14} + 103 \beta_{13} + 20 \beta_{12} - 26 \beta_{11} + 26 \beta_{10} + \cdots + 147 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 33 \beta_{15} + 73 \beta_{14} + 8 \beta_{13} - 65 \beta_{12} - 26 \beta_{11} - 26 \beta_{10} + \cdots - 214 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 433 \beta_{15} + 1135 \beta_{14} - 961 \beta_{13} - 174 \beta_{12} + 282 \beta_{11} - 282 \beta_{10} + \cdots - 1365 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 95 \beta_{15} - 342 \beta_{14} - 62 \beta_{13} + 280 \beta_{12} + 135 \beta_{11} + 135 \beta_{10} + \cdots + 822 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3735 \beta_{15} - 10481 \beta_{14} + 8965 \beta_{13} + 1516 \beta_{12} - 2838 \beta_{11} + 2838 \beta_{10} + \cdots + 12633 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1203 \beta_{15} + 6411 \beta_{14} + 1400 \beta_{13} - 5011 \beta_{12} - 2622 \beta_{11} - 2622 \beta_{10} + \cdots - 13762 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 33339 \beta_{15} + 97045 \beta_{14} - 83563 \beta_{13} - 13482 \beta_{12} + 27486 \beta_{11} + \cdots - 117159 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4259 \beta_{15} - 29960 \beta_{14} - 7084 \beta_{13} + 22876 \beta_{12} + 12421 \beta_{11} + \cdots + 60536 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 303493 \beta_{15} - 900371 \beta_{14} + 778351 \beta_{13} + 122020 \beta_{12} - 261010 \beta_{11} + \cdots + 1088363 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 66673 \beta_{15} + 558905 \beta_{14} + 136968 \beta_{13} - 421937 \beta_{12} - 233026 \beta_{11} + \cdots - 1094422 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2792409 \beta_{15} + 8364503 \beta_{14} - 7247129 \beta_{13} - 1117374 \beta_{12} + 2453946 \beta_{11} + \cdots - 10120237 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
1.06676i
1.60698i
− 0.266470i
− 2.12462i
− 3.05093i
2.19832i
1.11907i
− 0.549125i
− 1.06676i
− 1.60698i
0.266470i
2.12462i
3.05093i
− 2.19832i
− 1.11907i
0.549125i
0 0 0 −3.08397 + 1.78053i 0 −2.27807 + 1.34552i 0 0 0
703.2 0 0 0 −2.33837 + 1.35006i 0 2.63871 + 0.192843i 0 0 0
703.3 0 0 0 −1.51314 + 0.873609i 0 −2.46664 − 0.956913i 0 0 0
703.4 0 0 0 0.247599 − 0.142951i 0 −1.36578 − 2.26597i 0 0 0
703.5 0 0 0 0.658228 − 0.380028i 0 −1.09610 + 2.40802i 0 0 0
703.6 0 0 0 1.15442 − 0.666502i 0 2.64320 − 0.116219i 0 0 0
703.7 0 0 0 1.43254 − 0.827079i 0 1.65159 − 2.06694i 0 0 0
703.8 0 0 0 3.44269 − 1.98764i 0 −1.72692 − 2.00443i 0 0 0
1279.1 0 0 0 −3.08397 − 1.78053i 0 −2.27807 − 1.34552i 0 0 0
1279.2 0 0 0 −2.33837 − 1.35006i 0 2.63871 − 0.192843i 0 0 0
1279.3 0 0 0 −1.51314 − 0.873609i 0 −2.46664 + 0.956913i 0 0 0
1279.4 0 0 0 0.247599 + 0.142951i 0 −1.36578 + 2.26597i 0 0 0
1279.5 0 0 0 0.658228 + 0.380028i 0 −1.09610 − 2.40802i 0 0 0
1279.6 0 0 0 1.15442 + 0.666502i 0 2.64320 + 0.116219i 0 0 0
1279.7 0 0 0 1.43254 + 0.827079i 0 1.65159 + 2.06694i 0 0 0
1279.8 0 0 0 3.44269 + 1.98764i 0 −1.72692 + 2.00443i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 703.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.cs.a 16
3.b odd 2 1 672.2.bl.a ✓ 16
4.b odd 2 1 2016.2.cs.c 16
7.d odd 6 1 2016.2.cs.c 16
12.b even 2 1 672.2.bl.b yes 16
21.g even 6 1 672.2.bl.b yes 16
21.g even 6 1 4704.2.b.d 16
21.h odd 6 1 4704.2.b.e 16
24.f even 2 1 1344.2.bl.k 16
24.h odd 2 1 1344.2.bl.l 16
28.f even 6 1 inner 2016.2.cs.a 16
84.j odd 6 1 672.2.bl.a ✓ 16
84.j odd 6 1 4704.2.b.e 16
84.n even 6 1 4704.2.b.d 16
168.ba even 6 1 1344.2.bl.k 16
168.be odd 6 1 1344.2.bl.l 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bl.a ✓ 16 3.b odd 2 1
672.2.bl.a ✓ 16 84.j odd 6 1
672.2.bl.b yes 16 12.b even 2 1
672.2.bl.b yes 16 21.g even 6 1
1344.2.bl.k 16 24.f even 2 1
1344.2.bl.k 16 168.ba even 6 1
1344.2.bl.l 16 24.h odd 2 1
1344.2.bl.l 16 168.be odd 6 1
2016.2.cs.a 16 1.a even 1 1 trivial
2016.2.cs.a 16 28.f even 6 1 inner
2016.2.cs.c 16 4.b odd 2 1
2016.2.cs.c 16 7.d odd 6 1
4704.2.b.d 16 21.g even 6 1
4704.2.b.d 16 84.n even 6 1
4704.2.b.e 16 21.h odd 6 1
4704.2.b.e 16 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\):

\( T_{5}^{16} - 22 T_{5}^{14} + 363 T_{5}^{12} + 96 T_{5}^{11} - 2294 T_{5}^{10} - 288 T_{5}^{9} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{11}^{16} + 12 T_{11}^{15} + 34 T_{11}^{14} - 168 T_{11}^{13} - 813 T_{11}^{12} + 2448 T_{11}^{11} + \cdots + 135424 \) 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
$5$ \( T^{16} - 22 T^{14} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 12 T^{15} + \cdots + 135424 \) Copy content Toggle raw display
$13$ \( T^{16} + 124 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 3743481856 \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} + \cdots + 29073664 \) Copy content Toggle raw display
$23$ \( T^{16} - 96 T^{14} + \cdots + 4194304 \) Copy content Toggle raw display
$29$ \( (T^{8} - 106 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 20535749809 \) Copy content Toggle raw display
$37$ \( T^{16} - 4 T^{15} + \cdots + 135424 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 57538576384 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 17573803020544 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 39464206336 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1630259097856 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 47901450496 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 644513529856 \) Copy content Toggle raw display
$67$ \( T^{16} + 12 T^{15} + \cdots + 21827584 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 2218786816 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 362073308594176 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 157558981969 \) Copy content Toggle raw display
$83$ \( (T^{8} + 4 T^{7} + \cdots - 314144)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 16211639271424 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 135706771456 \) Copy content Toggle raw display
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