Properties

Label 2016.3.d.e
Level $2016$
Weight $3$
Character orbit 2016.d
Analytic conductor $54.932$
Analytic rank $0$
Dimension $12$
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] = [2016,3,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: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 44x^{10} + 719x^{8} + 5356x^{6} + 17809x^{4} + 20000x^{2} + 144 \) 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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{5} + \beta_{4} q^{7} + \beta_{6} q^{11} + (\beta_{9} + 1) q^{13} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{17} + (\beta_{11} - \beta_{4} - 5) q^{19} + ( - \beta_{8} - \beta_{6} - \beta_{5} - 7 \beta_{2}) q^{23} + ( - 2 \beta_{10} - \beta_{9} - 6 \beta_{4} - 10) q^{25} + ( - \beta_{8} - 2 \beta_{7} - \beta_{5} + 2 \beta_{3} - 6 \beta_{2}) q^{29} + ( - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 5 \beta_{4} - 13) q^{31} + (\beta_{8} - \beta_{7}) q^{35} + ( - 2 \beta_{11} - \beta_{9} - 6 \beta_{4} - 2 \beta_1 + 5) q^{37} + (3 \beta_{8} - \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{3} + 15 \beta_{2}) q^{41} + (2 \beta_{10} + 2 \beta_{9} + 6 \beta_{4} + 2 \beta_1 + 4) q^{43} + (5 \beta_{8} + 3 \beta_{7} + \beta_{6} + 2 \beta_{5} - 3 \beta_{3} + 8 \beta_{2}) q^{47} + 7 q^{49} + (2 \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + 8 \beta_{3} - 3 \beta_{2}) q^{53} + ( - 3 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{4} - 4 \beta_1 + 13) q^{55} + ( - 3 \beta_{8} - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 9 \beta_{3} - 12 \beta_{2}) q^{59} + ( - 4 \beta_{11} - \beta_{9} - 12 \beta_{4} - \beta_1 + 8) q^{61} + (\beta_{8} - 4 \beta_{6} - \beta_{5} + 8 \beta_{3} - 9 \beta_{2}) q^{65} + (3 \beta_{11} + \beta_{10} - 2 \beta_{9} + 4 \beta_{4} + 3 \beta_1 + 6) q^{67} + ( - 11 \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 6 \beta_{3} + 13 \beta_{2}) q^{71} + ( - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} - 18 \beta_{4} - 5 \beta_1 - 5) q^{73} + (2 \beta_{6} - \beta_{5}) q^{77} + (\beta_{11} - 3 \beta_{10} + 6 \beta_{9} - 6 \beta_{4} - \beta_1 - 4) q^{79} + (7 \beta_{8} - 3 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + 12 \beta_{2}) q^{83} + (8 \beta_{11} - 2 \beta_{10} - \beta_{9} - 30 \beta_{4} + 2 \beta_1 - 13) q^{85} + ( - 5 \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_{5} + 15 \beta_{3} - 3 \beta_{2}) q^{89} + ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} + \beta_{4} - 3 \beta_1 + 1) q^{91} + ( - 4 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} - 12 \beta_{3} - 8 \beta_{2}) q^{95} + (2 \beta_{11} - 4 \beta_{10} - 30 \beta_{4} + 5 \beta_1 - 9) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 16 q^{13} - 64 q^{19} - 124 q^{25} - 160 q^{31} + 56 q^{37} + 64 q^{43} + 84 q^{49} + 160 q^{55} + 104 q^{61} + 64 q^{67} - 64 q^{73} - 32 q^{79} - 184 q^{85} - 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 44x^{10} + 719x^{8} + 5356x^{6} + 17809x^{4} + 20000x^{2} + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -30\nu^{10} - 1330\nu^{8} - 20306\nu^{6} - 123058\nu^{4} - 245774\nu^{2} - 25455 ) / 7683 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 40\nu^{9} + 559\nu^{7} + 3224\nu^{5} + 7305\nu^{3} + 5236\nu ) / 312 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -17\nu^{11} - 688\nu^{9} - 10351\nu^{7} - 70928\nu^{5} - 213449\nu^{3} - 209356\nu ) / 4728 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 363\nu^{10} + 13532\nu^{8} + 169897\nu^{6} + 798044\nu^{4} + 1072579\nu^{2} + 89040 ) / 30732 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{11} + 133\nu^{9} + 2267\nu^{7} + 18334\nu^{5} + 67484\nu^{3} + 81444\nu ) / 591 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44\nu^{11} + 1097\nu^{9} + 416\nu^{7} - 156884\nu^{5} - 1075911\nu^{3} - 1637560\nu ) / 7683 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -86\nu^{11} - 2959\nu^{9} - 32942\nu^{7} - 117325\nu^{5} + 31650\nu^{3} + 428985\nu ) / 7683 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -875\nu^{11} - 32816\nu^{9} - 421525\nu^{7} - 2128568\nu^{5} - 3760571\nu^{3} - 2111292\nu ) / 61464 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 362\nu^{10} + 12634\nu^{8} + 150098\nu^{6} + 684502\nu^{4} + 905946\nu^{2} - 41139 ) / 7683 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -113\nu^{10} - 4156\nu^{8} - 52163\nu^{6} - 253372\nu^{4} - 387177\nu^{2} - 45744 ) / 2364 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -2541\nu^{10} - 94724\nu^{8} - 1189279\nu^{6} - 5586308\nu^{4} - 7385125\nu^{2} + 267948 ) / 30732 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 7\beta_{4} - 29 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{8} - 4\beta_{7} - 6\beta_{6} - 7\beta_{5} + 4\beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -14\beta_{11} - 6\beta_{10} - 3\beta_{9} - 108\beta_{4} + 7\beta _1 + 326 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 94\beta_{8} + 81\beta_{7} + 149\beta_{6} + 199\beta_{5} - 41\beta_{3} + 73\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 99\beta_{11} + 59\beta_{10} + 32\beta_{9} + 778\beta_{4} - 79\beta _1 - 2066 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1334\beta_{8} - 957\beta_{7} - 1965\beta_{6} - 2831\beta_{5} - 11\beta_{3} - 1285\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -2826\beta_{11} - 1874\beta_{10} - 1085\beta_{9} - 22164\beta_{4} + 2643\beta _1 + 55540 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9471\beta_{8} + 6224\beta_{7} + 13454\beta_{6} + 20135\beta_{5} + 2416\beta_{3} + 10585\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 40501\beta_{11} + 27822\beta_{10} + 17088\beta_{9} + 315061\beta_{4} - 39966\beta _1 - 768503 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -271200\beta_{8} - 170897\beta_{7} - 375837\beta_{6} - 572813\beta_{5} - 108151\beta_{3} - 341495\beta_{2} ) / 8 \) 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
1.48336i
2.56196i
2.61146i
3.75209i
0.0851273i
3.78561i
3.78561i
0.0851273i
3.75209i
2.61146i
2.56196i
1.48336i
0 0 0 9.07031i 0 2.64575 0 0 0
449.2 0 0 0 7.59798i 0 2.64575 0 0 0
449.3 0 0 0 6.81315i 0 −2.64575 0 0 0
449.4 0 0 0 3.85589i 0 −2.64575 0 0 0
449.5 0 0 0 2.88654i 0 2.64575 0 0 0
449.6 0 0 0 1.54305i 0 −2.64575 0 0 0
449.7 0 0 0 1.54305i 0 −2.64575 0 0 0
449.8 0 0 0 2.88654i 0 2.64575 0 0 0
449.9 0 0 0 3.85589i 0 −2.64575 0 0 0
449.10 0 0 0 6.81315i 0 −2.64575 0 0 0
449.11 0 0 0 7.59798i 0 2.64575 0 0 0
449.12 0 0 0 9.07031i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2016.3.d.e 12
3.b odd 2 1 inner 2016.3.d.e 12
4.b odd 2 1 2016.3.d.f yes 12
8.b even 2 1 4032.3.d.o 12
8.d odd 2 1 4032.3.d.n 12
12.b even 2 1 2016.3.d.f yes 12
24.f even 2 1 4032.3.d.n 12
24.h odd 2 1 4032.3.d.o 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.3.d.e 12 1.a even 1 1 trivial
2016.3.d.e 12 3.b odd 2 1 inner
2016.3.d.f yes 12 4.b odd 2 1
2016.3.d.f yes 12 12.b even 2 1
4032.3.d.n 12 8.d odd 2 1
4032.3.d.n 12 24.f even 2 1
4032.3.d.o 12 8.b even 2 1
4032.3.d.o 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 \(S_{3}^{\mathrm{new}}(2016, [\chi])\):

\( T_{5}^{12} + 212T_{5}^{10} + 16196T_{5}^{8} + 541888T_{5}^{6} + 7709440T_{5}^{4} + 42807296T_{5}^{2} + 65028096 \) Copy content Toggle raw display
\( T_{19}^{6} + 32T_{19}^{5} - 80T_{19}^{4} - 9984T_{19}^{3} - 82688T_{19}^{2} - 28672T_{19} + 950272 \) 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
$5$ \( T^{12} + 212 T^{10} + \cdots + 65028096 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 664 T^{10} + \cdots + 1230045184 \) Copy content Toggle raw display
$13$ \( (T^{6} - 8 T^{5} - 456 T^{4} + 1760 T^{3} + \cdots - 60416)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 2004 T^{10} + \cdots + 3455672795136 \) Copy content Toggle raw display
$19$ \( (T^{6} + 32 T^{5} - 80 T^{4} + \cdots + 950272)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 2936 T^{10} + \cdots + 1496766283776 \) Copy content Toggle raw display
$29$ \( T^{12} + 5516 T^{10} + \cdots + 71780341739584 \) Copy content Toggle raw display
$31$ \( (T^{6} + 80 T^{5} - 912 T^{4} + \cdots + 273825792)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 28 T^{5} - 2236 T^{4} + \cdots - 10693632)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 12820 T^{10} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{6} - 32 T^{5} - 3968 T^{4} + \cdots - 81133568)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 16352 T^{10} + \cdots + 54\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + 15148 T^{10} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{12} + 27488 T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} - 52 T^{5} - 7780 T^{4} + \cdots - 11681644992)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 32 T^{5} - 11320 T^{4} + \cdots + 10217426944)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 32056 T^{10} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{6} + 32 T^{5} - 17800 T^{4} + \cdots + 9192474624)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 16 T^{5} - 27032 T^{4} + \cdots - 5097442304)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 37440 T^{10} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{12} + 30356 T^{10} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{6} + 48 T^{5} - 26216 T^{4} + \cdots - 14709774336)^{2} \) Copy content Toggle raw display
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