Properties

Label 2016.2.k.a
Level $2016$
Weight $2$
Character orbit 2016.k
Analytic conductor $16.098$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,2,Mod(1889,2016)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.1889");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.k (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: \(16.0978410475\)
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: \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{25} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} - \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} - \beta_{6} q^{7} + \beta_{3} q^{11} - \beta_{13} q^{13} + ( - \beta_{2} + \beta_1) q^{17} + (\beta_{15} + \beta_{7}) q^{19} + (\beta_{10} - \beta_{9} + \beta_{3}) q^{23} + ( - \beta_{5} - 1) q^{25} + \beta_{8} q^{29} + (\beta_{15} - \beta_{6}) q^{31} + (\beta_{9} + \beta_{3}) q^{35} + \beta_{5} q^{37} + ( - \beta_{2} - 2 \beta_1) q^{41} + (\beta_{7} - \beta_{6} + 2 \beta_{4}) q^{43} + ( - \beta_{11} - \beta_{10}) q^{47} + (\beta_{13} - 2 \beta_{12} + \beta_{5} - 1) q^{49} + ( - \beta_{14} - \beta_{8}) q^{53} + (\beta_{15} - \beta_{7} - 2 \beta_{6}) q^{55} + ( - \beta_{11} + \beta_{9}) q^{59} + (3 \beta_{13} + \beta_{12}) q^{61} + \beta_{14} q^{65} + (3 \beta_{7} - 3 \beta_{6} + \beta_{4}) q^{67} + (2 \beta_{10} - 2 \beta_{9} - \beta_{3}) q^{71} + 3 \beta_{12} q^{73} + (\beta_{14} - \beta_{8} - 3 \beta_{2} - \beta_1) q^{77} + (\beta_{7} - \beta_{6} + 3 \beta_{4}) q^{79} + ( - \beta_{11} + \beta_{10} + 2 \beta_{9}) q^{83} + ( - \beta_{5} - 4) q^{85} + (5 \beta_{2} + 2 \beta_1) q^{89} + (\beta_{15} + 2 \beta_{7} - \beta_{6} + \beta_{4}) q^{91} + (\beta_{10} - \beta_{9} + 2 \beta_{3}) q^{95} + ( - 2 \beta_{13} - \beta_{12}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{25} - 16 q^{49} - 64 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} - 20\nu^{12} - 100\nu^{10} + 4\nu^{8} + 918\nu^{6} + 1440\nu^{4} + 664\nu^{2} + 72 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{14} - 164\nu^{12} - 1250\nu^{10} - 3984\nu^{8} - 5334\nu^{6} - 3024\nu^{4} - 596\nu^{2} - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{15} - 110\nu^{13} - 728\nu^{11} - 1625\nu^{9} - 118\nu^{7} + 2276\nu^{5} + 1648\nu^{3} + 270\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{14} - 261\nu^{12} - 2040\nu^{10} - 6818\nu^{8} - 10038\nu^{6} - 6666\nu^{4} - 1856\nu^{2} - 172 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{14} - 141\nu^{12} - 1082\nu^{10} - 3502\nu^{8} - 4876\nu^{6} - 3046\nu^{4} - 804\nu^{2} - 68 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{15} - 6 \nu^{14} - 260 \nu^{13} - 138 \nu^{12} - 2018 \nu^{11} - 1012 \nu^{10} - 6672 \nu^{9} - 2976 \nu^{8} - 9702 \nu^{7} - 3276 \nu^{6} - 6544 \nu^{5} - 1188 \nu^{4} - 2004 \nu^{3} + \cdots - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{15} + 6 \nu^{14} - 260 \nu^{13} + 138 \nu^{12} - 2018 \nu^{11} + 1012 \nu^{10} - 6672 \nu^{9} + 2976 \nu^{8} - 9702 \nu^{7} + 3276 \nu^{6} - 6544 \nu^{5} + 1188 \nu^{4} - 2004 \nu^{3} + \cdots + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\nu^{15} + 258\nu^{13} + 1971\nu^{11} + 6311\nu^{9} + 8530\nu^{7} + 4916\nu^{5} + 1046\nu^{3} + 38\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{15} - 15 \nu^{14} - 261 \nu^{13} - 352 \nu^{12} - 2040 \nu^{11} - 2692 \nu^{10} - 6817 \nu^{9} - 8638 \nu^{8} - 10018 \nu^{7} - 11726 \nu^{6} - 6554 \nu^{5} - 6808 \nu^{4} - 1640 \nu^{3} + \cdots - 76 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11 \nu^{15} - 15 \nu^{14} + 261 \nu^{13} - 352 \nu^{12} + 2040 \nu^{11} - 2692 \nu^{10} + 6817 \nu^{9} - 8638 \nu^{8} + 10018 \nu^{7} - 11726 \nu^{6} + 6554 \nu^{5} - 6808 \nu^{4} + 1640 \nu^{3} + \cdots - 76 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11 \nu^{15} - 27 \nu^{14} - 261 \nu^{13} - 636 \nu^{12} - 2040 \nu^{11} - 4904 \nu^{10} - 6817 \nu^{9} - 16030 \nu^{8} - 10018 \nu^{7} - 22886 \nu^{6} - 6554 \nu^{5} - 15248 \nu^{4} + \cdots - 428 ) / 8 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -19\nu^{15} - 448\nu^{13} - 3460\nu^{11} - 11326\nu^{9} - 16094\nu^{7} - 10328\nu^{5} - 2904\nu^{3} - 316\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -19\nu^{15} - 454\nu^{13} - 3598\nu^{11} - 12338\nu^{9} - 19070\nu^{7} - 13604\nu^{5} - 4092\nu^{3} - 372\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\nu^{15} + 261\nu^{13} + 2042\nu^{11} + 6862\nu^{9} + 10334\nu^{7} + 7410\nu^{5} + 2412\nu^{3} + 252\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 141 \nu^{15} - 6 \nu^{14} + 3324 \nu^{13} - 138 \nu^{12} + 25666 \nu^{11} - 1012 \nu^{10} + 84024 \nu^{9} - 2976 \nu^{8} + 119626 \nu^{7} - 3276 \nu^{6} + 77296 \nu^{5} - 1188 \nu^{4} + \cdots - 16 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} - 2\beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 2\beta_{10} + \beta_{9} + 2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 2\beta_{2} + \beta _1 - 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} + 3 \beta_{14} - 11 \beta_{13} + 7 \beta_{12} - 8 \beta_{10} + 8 \beta_{9} + 16 \beta_{8} - 6 \beta_{7} - 3 \beta_{6} - 4 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{11} - 10 \beta_{10} - 4 \beta_{9} - 14 \beta_{7} + 14 \beta_{6} + 13 \beta_{5} - 4 \beta_{4} + 12 \beta_{2} - 10 \beta _1 + 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 50 \beta_{15} - 55 \beta_{14} + 126 \beta_{13} - 64 \beta_{12} + 83 \beta_{10} - 83 \beta_{9} - 188 \beta_{8} + 80 \beta_{7} + 30 \beta_{6} + 58 \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 72 \beta_{11} + 109 \beta_{10} + 37 \beta_{9} + 181 \beta_{7} - 181 \beta_{6} - 161 \beta_{5} + 67 \beta_{4} - 134 \beta_{2} + 140 \beta _1 - 504 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 336 \beta_{15} + 378 \beta_{14} - 752 \beta_{13} + 339 \beta_{12} - 478 \beta_{10} + 478 \beta_{9} + 1142 \beta_{8} - 497 \beta_{7} - 161 \beta_{6} - 372 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 438 \beta_{11} - 636 \beta_{10} - 198 \beta_{9} - 1130 \beta_{7} + 1130 \beta_{6} + 990 \beta_{5} - 450 \beta_{4} + 784 \beta_{2} - 896 \beta _1 + 2910 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4248 \beta_{15} - 4812 \beta_{14} + 9127 \beta_{13} - 3907 \beta_{12} + 5723 \beta_{10} - 5723 \beta_{9} - 13942 \beta_{8} + 6108 \beta_{7} + 1860 \beta_{6} + 4632 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10707 \beta_{11} + 15294 \beta_{10} + 4587 \beta_{9} + 27862 \beta_{7} - 27862 \beta_{6} - 24294 \beta_{5} + 11376 \beta_{4} - 18846 \beta_{2} + 22287 \beta _1 - 69716 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 52613 \beta_{15} + 59741 \beta_{14} - 111425 \beta_{13} + 46733 \beta_{12} - 69508 \beta_{10} + 69508 \beta_{9} + 170488 \beta_{8} - 74866 \beta_{7} - 22253 \beta_{6} - 57084 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 65522 \beta_{11} - 93010 \beta_{10} - 27488 \beta_{9} - 170986 \beta_{7} + 170986 \beta_{6} + 148879 \beta_{5} - 70448 \beta_{4} + 114548 \beta_{2} - 137214 \beta _1 + 423384 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 646906 \beta_{15} - 735189 \beta_{14} + 1363030 \beta_{13} - 567236 \beta_{12} + 848621 \beta_{10} - 848621 \beta_{9} - 2086356 \beta_{8} + 916968 \beta_{7} + 270062 \beta_{6} + \cdots + 700742 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 802232 \beta_{11} + 1136119 \beta_{10} + 333887 \beta_{9} + 2095431 \beta_{7} - 2095431 \beta_{6} - 1823791 \beta_{5} + 866233 \beta_{4} - 1398722 \beta_{2} + 1683568 \beta _1 - 5168936 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3966632 \beta_{15} + 4509412 \beta_{14} - 8342408 \beta_{13} + 3461601 \beta_{12} - 5190196 \beta_{10} + 5190196 \beta_{9} + 12770386 \beta_{8} - 5614479 \beta_{7} + \cdots - 4294488 \beta_{3} \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
1.05636i
1.05636i
0.724535i
0.724535i
0.886177i
0.886177i
2.08509i
2.08509i
3.49930i
3.49930i
0.528036i
0.528036i
2.13875i
2.13875i
0.357857i
0.357857i
0 0 0 −2.61313 0 −1.25928 2.32685i 0 0 0
1889.2 0 0 0 −2.61313 0 −1.25928 + 2.32685i 0 0 0
1889.3 0 0 0 −2.61313 0 1.25928 2.32685i 0 0 0
1889.4 0 0 0 −2.61313 0 1.25928 + 2.32685i 0 0 0
1889.5 0 0 0 −1.08239 0 −2.10100 1.60804i 0 0 0
1889.6 0 0 0 −1.08239 0 −2.10100 + 1.60804i 0 0 0
1889.7 0 0 0 −1.08239 0 2.10100 1.60804i 0 0 0
1889.8 0 0 0 −1.08239 0 2.10100 + 1.60804i 0 0 0
1889.9 0 0 0 1.08239 0 −2.10100 1.60804i 0 0 0
1889.10 0 0 0 1.08239 0 −2.10100 + 1.60804i 0 0 0
1889.11 0 0 0 1.08239 0 2.10100 1.60804i 0 0 0
1889.12 0 0 0 1.08239 0 2.10100 + 1.60804i 0 0 0
1889.13 0 0 0 2.61313 0 −1.25928 2.32685i 0 0 0
1889.14 0 0 0 2.61313 0 −1.25928 + 2.32685i 0 0 0
1889.15 0 0 0 2.61313 0 1.25928 2.32685i 0 0 0
1889.16 0 0 0 2.61313 0 1.25928 + 2.32685i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.16
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
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.2.k.a 16
3.b odd 2 1 inner 2016.2.k.a 16
4.b odd 2 1 inner 2016.2.k.a 16
7.b odd 2 1 inner 2016.2.k.a 16
8.b even 2 1 4032.2.k.g 16
8.d odd 2 1 4032.2.k.g 16
12.b even 2 1 inner 2016.2.k.a 16
21.c even 2 1 inner 2016.2.k.a 16
24.f even 2 1 4032.2.k.g 16
24.h odd 2 1 4032.2.k.g 16
28.d even 2 1 inner 2016.2.k.a 16
56.e even 2 1 4032.2.k.g 16
56.h odd 2 1 4032.2.k.g 16
84.h odd 2 1 inner 2016.2.k.a 16
168.e odd 2 1 4032.2.k.g 16
168.i even 2 1 4032.2.k.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.k.a 16 1.a even 1 1 trivial
2016.2.k.a 16 3.b odd 2 1 inner
2016.2.k.a 16 4.b odd 2 1 inner
2016.2.k.a 16 7.b odd 2 1 inner
2016.2.k.a 16 12.b even 2 1 inner
2016.2.k.a 16 21.c even 2 1 inner
2016.2.k.a 16 28.d even 2 1 inner
2016.2.k.a 16 84.h odd 2 1 inner
4032.2.k.g 16 8.b even 2 1
4032.2.k.g 16 8.d odd 2 1
4032.2.k.g 16 24.f even 2 1
4032.2.k.g 16 24.h odd 2 1
4032.2.k.g 16 56.e even 2 1
4032.2.k.g 16 56.h odd 2 1
4032.2.k.g 16 168.e odd 2 1
4032.2.k.g 16 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 8T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(2016, [\chi])\). 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
$5$ \( (T^{4} - 8 T^{2} + 8)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{6} + 70 T^{4} + 196 T^{2} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 20 T^{2} + 28)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 16 T^{2} + 32)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 40 T^{2} + 8)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 64 T^{2} + 224)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 52 T^{2} + 28)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 64 T^{2} + 896)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 136 T^{2} + 4232)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 152 T^{2} + 5488)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 128 T^{2} + 896)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 36 T^{2} + 196)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 128 T^{2} + 3584)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 160 T^{2} + 6272)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 208 T^{2} + 448)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 244 T^{2} + 14812)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 144 T^{2} + 2592)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 336 T^{2} + 21952)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 256 T^{2} + 896)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 328 T^{2} + 7688)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 80 T^{2} + 1568)^{4} \) Copy content Toggle raw display
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