↑

Properties

Label 2016.3.f.e
Level $2016$
Weight $3$
Character orbit 2016.f
Analytic conductor $54.932$
Analytic rank $0$
Dimension $16$
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(1441,2016)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.1441");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.f (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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 50 x^{14} + 1746 x^{12} - 28928 x^{10} + 340567 x^{8} - 2442144 x^{6} + 12715650 x^{4} + \cdots + 74805201 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{32} \)
Twist minimal: no (minimal twist has level 672)
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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{5} + \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{5} + \beta_{7} q^{7} + ( - \beta_{11} + \beta_{9} + \cdots + \beta_{7}) q^{11}+ \cdots + (2 \beta_{15} + 4 \beta_{14} + \cdots - 6 \beta_{12}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 128 q^{25} - 128 q^{29} - 208 q^{37} + 64 q^{49} - 96 q^{53} - 608 q^{65} + 384 q^{77} - 528 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 50 x^{14} + 1746 x^{12} - 28928 x^{10} + 340567 x^{8} - 2442144 x^{6} + 12715650 x^{4} + \cdots + 74805201 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1897758509552 \nu^{14} + 88477685823184 \nu^{12} + \cdots + 40\!\cdots\!84 ) / 41\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5966104924 \nu^{14} - 260609201615 \nu^{12} + 8750020830798 \nu^{10} + \cdots - 64\!\cdots\!32 ) / 42\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18780606476168 \nu^{14} + 786875017390903 \nu^{12} + \cdots + 30\!\cdots\!43 ) / 82\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 51868870002584 \nu^{14} + \cdots - 14\!\cdots\!25 ) / 82\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 66417653042 \nu^{14} - 2944800285091 \nu^{12} + 97409592196809 \nu^{10} + \cdots + 16\!\cdots\!49 ) / 85\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5312784282 \nu^{14} - 237071877917 \nu^{12} + 7791846393789 \nu^{10} + \cdots - 10\!\cdots\!95 ) / 474198951142413 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 162255083710115 \nu^{15} + 836822402685160 \nu^{14} + \cdots - 21\!\cdots\!35 ) / 50\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 162255083710115 \nu^{15} - 836822402685160 \nu^{14} + \cdots + 21\!\cdots\!35 ) / 50\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 44845750008008 \nu^{15} + \cdots + 17\!\cdots\!06 \nu ) / 12\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 354767846821991 \nu^{15} + \cdots + 23\!\cdots\!04 \nu ) / 76\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13219477 \nu^{15} - 509267507 \nu^{13} + 15873401244 \nu^{11} - 139896220973 \nu^{9} + \cdots + 209951734720032 \nu ) / 19231628102883 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 272299622921596 \nu^{15} + \cdots + 35\!\cdots\!60 \nu ) / 38\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 220971442739801 \nu^{15} + \cdots - 29\!\cdots\!90 \nu ) / 10\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 34535041 \nu^{15} - 1731621437 \nu^{13} + 59396067822 \nu^{11} - 956347966769 \nu^{9} + \cdots - 463652460214398 \nu ) / 15952994970039 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 410750404365620 \nu^{15} + \cdots - 56\!\cdots\!20 \nu ) / 12\!\cdots\!47 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} + 2\beta_{13} + 2\beta_{12} - \beta_{11} + 2\beta_{10} - 2\beta_{9} ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11\beta_{8} - 11\beta_{7} + \beta_{6} - 2\beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_{2} - 22\beta _1 + 100 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{15} - 35\beta_{14} + 38\beta_{13} + 38\beta_{12} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 178 \beta_{8} - 178 \beta_{7} - 14 \beta_{6} + 28 \beta_{5} + 68 \beta_{4} - 26 \beta_{3} - 40 \beta_{2} + \cdots - 992 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 284 \beta_{15} - 1079 \beta_{14} + 974 \beta_{13} + 794 \beta_{12} + 1411 \beta_{11} + \cdots + 1704 \beta_{7} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -727\beta_{6} + 1502\beta_{5} - 2534\beta_{2} - 50116 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 8540 \beta_{15} + 31853 \beta_{14} - 27146 \beta_{13} - 19262 \beta_{12} + 42793 \beta_{11} + \cdots + 51240 \beta_{7} ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 147436 \beta_{8} + 147436 \beta_{7} - 9812 \beta_{6} + 20824 \beta_{5} - 57200 \beta_{4} + \cdots - 689636 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -247952\beta_{15} + 923945\beta_{14} - 772274\beta_{13} - 513794\beta_{12} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 8442923 \beta_{8} + 8442923 \beta_{7} + 547201 \beta_{6} - 1178210 \beta_{5} - 3274948 \beta_{4} + \cdots + 39013060 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7134872 \beta_{15} + 26609867 \beta_{14} - 22088918 \beta_{13} - 14319974 \beta_{12} + \cdots - 42809232 \beta_{7} ) / 16 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 3877501\beta_{6} - 8402834\beta_{5} + 15696224\beta_{2} + 278332768 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 204787916 \beta_{15} - 764213807 \beta_{14} + 632789534 \beta_{13} + 406213850 \beta_{12} + \cdots - 1228727496 \beta_{7} ) / 16 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 6933941501 \beta_{8} - 6933941501 \beta_{7} + 442693687 \beta_{6} - 961873790 \beta_{5} + \cdots + 31866740068 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5873357324\beta_{15} - 21923776805\beta_{14} + 18136840538\beta_{13} + 11600335550\beta_{12} ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1441.1
−2.80575 − 1.61990i
2.80575 − 1.61990i
−1.86542 + 1.07700i
1.86542 + 1.07700i
4.63702 − 2.67718i
−4.63702 − 2.67718i
−2.15547 − 1.24446i
2.15547 − 1.24446i
−2.15547 + 1.24446i
2.15547 + 1.24446i
4.63702 + 2.67718i
−4.63702 + 2.67718i
−1.86542 − 1.07700i
1.86542 − 1.07700i
−2.80575 + 1.61990i
2.80575 + 1.61990i
0 0 0 − 9.48526i 0 −6.66956 − 2.12532i 0 0 0
1441.2 0 0 0 − 9.48526i 0 6.66956 + 2.12532i 0 0 0
1441.3 0 0 0 − 5.54904i 0 −0.647636 − 6.96998i 0 0 0
1441.4 0 0 0 − 5.54904i 0 0.647636 + 6.96998i 0 0 0
1441.5 0 0 0 − 3.28930i 0 −4.63422 + 5.24633i 0 0 0
1441.6 0 0 0 − 3.28930i 0 4.63422 − 5.24633i 0 0 0
1441.7 0 0 0 − 0.646915i 0 −6.29456 − 3.06243i 0 0 0
1441.8 0 0 0 − 0.646915i 0 6.29456 + 3.06243i 0 0 0
1441.9 0 0 0 0.646915i 0 −6.29456 + 3.06243i 0 0 0
1441.10 0 0 0 0.646915i 0 6.29456 − 3.06243i 0 0 0
1441.11 0 0 0 3.28930i 0 −4.63422 − 5.24633i 0 0 0
1441.12 0 0 0 3.28930i 0 4.63422 + 5.24633i 0 0 0
1441.13 0 0 0 5.54904i 0 −0.647636 + 6.96998i 0 0 0
1441.14 0 0 0 5.54904i 0 0.647636 − 6.96998i 0 0 0
1441.15 0 0 0 9.48526i 0 −6.66956 + 2.12532i 0 0 0
1441.16 0 0 0 9.48526i 0 6.66956 − 2.12532i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1441.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.f.e 16
3.b odd 2 1 672.3.f.a ✓ 16
4.b odd 2 1 inner 2016.3.f.e 16
7.b odd 2 1 inner 2016.3.f.e 16
12.b even 2 1 672.3.f.a ✓ 16
21.c even 2 1 672.3.f.a ✓ 16
24.f even 2 1 1344.3.f.i 16
24.h odd 2 1 1344.3.f.i 16
28.d even 2 1 inner 2016.3.f.e 16
84.h odd 2 1 672.3.f.a ✓ 16
168.e odd 2 1 1344.3.f.i 16
168.i even 2 1 1344.3.f.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.3.f.a ✓ 16 3.b odd 2 1
672.3.f.a ✓ 16 12.b even 2 1
672.3.f.a ✓ 16 21.c even 2 1
672.3.f.a ✓ 16 84.h odd 2 1
1344.3.f.i 16 24.f even 2 1
1344.3.f.i 16 24.h odd 2 1
1344.3.f.i 16 168.e odd 2 1
1344.3.f.i 16 168.i even 2 1
2016.3.f.e 16 1.a even 1 1 trivial
2016.3.f.e 16 4.b odd 2 1 inner
2016.3.f.e 16 7.b odd 2 1 inner
2016.3.f.e 16 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 132T_{5}^{6} + 4132T_{5}^{4} + 31680T_{5}^{2} + 12544 \) acting on \(S_{3}^{\mathrm{new}}(2016, [\chi])\). 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
$5$ \( (T^{8} + 132 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{8} - 484 T^{6} + \cdots + 53582400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 800 T^{6} + \cdots + 406425600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 516 T^{6} + \cdots + 23814400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 2168 T^{6} + \cdots + 45158400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2452 T^{6} + \cdots + 3287416896)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 32 T^{3} + \cdots - 625104)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 4912 T^{6} + \cdots + 71224934400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 52 T^{3} + \cdots - 16320)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 3940 T^{6} + \cdots + 153827115264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 27284492583936)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 1835352981504)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 24 T^{3} + \cdots + 29334960)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 1407924633600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 314817739554816)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 18992303456256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 7043079054400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 15824 T^{6} + \cdots + 331776000000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 369230059929600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 26848 T^{6} + \cdots + 337105649664)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 80794210873600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 764411904000000)^{2} \) Copy content Toggle raw display
show more
show less