↑

Properties

Label 4016.2.a.l
Level $4016$
Weight $2$
Character orbit 4016.a
Self dual yes
Analytic conductor $32.068$
Analytic rank $0$
Dimension $19$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4016,2,Mod(1,4016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4016.a (trivial)

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: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 2008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{4} q^{5} + (\beta_{13} + 1) q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{4} q^{5} + (\beta_{13} + 1) q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{12} + 1) q^{11} + (\beta_{13} - \beta_{12} + \cdots + \beta_1) q^{13}+ \cdots + ( - \beta_{18} - \beta_{16} - 2 \beta_{13} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 196521312495034 \nu^{18} + \cdots + 16\!\cdots\!88 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 209544026239557 \nu^{18} + \cdots - 24\!\cdots\!40 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 325315462336008 \nu^{18} + \cdots - 47\!\cdots\!20 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 346551847676251 \nu^{18} + \cdots - 49\!\cdots\!04 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 498188113457669 \nu^{18} + \cdots - 40\!\cdots\!92 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!90 \nu^{18} + \cdots + 10\!\cdots\!56 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!42 \nu^{18} + \cdots - 90\!\cdots\!56 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!57 \nu^{18} + \cdots - 35\!\cdots\!96 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!81 \nu^{18} + \cdots + 31\!\cdots\!92 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!57 \nu^{18} + \cdots - 12\!\cdots\!12 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!26 \nu^{18} + \cdots - 36\!\cdots\!16 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!97 \nu^{18} + \cdots + 26\!\cdots\!24 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 741068081655316 \nu^{18} + \cdots - 18\!\cdots\!72 ) / 38\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 17\!\cdots\!19 \nu^{18} + \cdots - 57\!\cdots\!84 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 24\!\cdots\!49 \nu^{18} + \cdots - 44\!\cdots\!68 ) / 77\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 27\!\cdots\!61 \nu^{18} + \cdots - 49\!\cdots\!36 ) / 77\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} + \beta_{15} + \beta_{14} - \beta_{12} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{18} - \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 4 \beta_{13} - 3 \beta_{12} - 2 \beta_{11} + \cdots + 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14 \beta_{18} - \beta_{17} - \beta_{16} + 12 \beta_{15} + 14 \beta_{14} - 2 \beta_{13} - 10 \beta_{12} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32 \beta_{18} - \beta_{17} - 16 \beta_{16} - 30 \beta_{15} + 47 \beta_{14} - 64 \beta_{13} - 40 \beta_{12} + \cdots + 259 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 166 \beta_{18} - 11 \beta_{17} - 20 \beta_{16} + 120 \beta_{15} + 164 \beta_{14} - 42 \beta_{13} + \cdots + 293 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 406 \beta_{18} - 20 \beta_{17} - 197 \beta_{16} - 358 \beta_{15} + 568 \beta_{14} - 810 \beta_{13} + \cdots + 2390 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1866 \beta_{18} - 79 \beta_{17} - 301 \beta_{16} + 1162 \beta_{15} + 1811 \beta_{14} - 646 \beta_{13} + \cdots + 3163 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4762 \beta_{18} - 277 \beta_{17} - 2224 \beta_{16} - 3940 \beta_{15} + 6298 \beta_{14} - 9446 \beta_{13} + \cdots + 22878 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 20481 \beta_{18} - 392 \beta_{17} - 3957 \beta_{16} + 11201 \beta_{15} + 19469 \beta_{14} - 8838 \beta_{13} + \cdots + 33634 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 53886 \beta_{18} - 3302 \beta_{17} - 24225 \beta_{16} - 41620 \beta_{15} + 67437 \beta_{14} + \cdots + 223931 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 222039 \beta_{18} - 343 \beta_{17} - 48163 \beta_{16} + 108145 \beta_{15} + 206563 \beta_{14} + \cdots + 355806 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 598270 \beta_{18} - 36416 \beta_{17} - 259618 \beta_{16} - 428934 \beta_{15} + 711545 \beta_{14} + \cdots + 2224383 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2390667 \beta_{18} + 24909 \beta_{17} - 559761 \beta_{16} + 1047245 \beta_{15} + 2178127 \beta_{14} + \cdots + 3759565 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 6568957 \beta_{18} - 383804 \beta_{17} - 2760886 \beta_{16} - 4349501 \beta_{15} + 7464710 \beta_{14} + \cdots + 22327897 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 25636785 \beta_{18} + 460550 \beta_{17} - 6322641 \beta_{16} + 10172705 \beta_{15} + 22913109 \beta_{14} + \cdots + 39746289 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 71626570 \beta_{18} - 3932010 \beta_{17} - 29251285 \beta_{16} - 43615626 \beta_{15} + 78189603 \beta_{14} + \cdots + 225892133 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.12892
−2.63379
−2.08912
−1.99202
−1.53958
−1.31693
−0.455245
−0.312548
−0.157639
0.00508866
0.749639
1.27063
1.56031
2.01204
2.43794
2.50185
2.79120
3.03361
3.26349
0 −3.12892 0 −3.70734 0 −1.25809 0 6.79011 0
1.2 0 −2.63379 0 −2.05129 0 4.98850 0 3.93685 0
1.3 0 −2.08912 0 −2.23992 0 1.92906 0 1.36442 0
1.4 0 −1.99202 0 2.61608 0 4.58459 0 0.968146 0
1.5 0 −1.53958 0 −0.551255 0 −1.16112 0 −0.629704 0
1.6 0 −1.31693 0 −1.28443 0 1.02036 0 −1.26569 0
1.7 0 −0.455245 0 2.37571 0 1.84768 0 −2.79275 0
1.8 0 −0.312548 0 0.780222 0 −4.05339 0 −2.90231 0
1.9 0 −0.157639 0 −0.360777 0 −0.00399840 0 −2.97515 0
1.10 0 0.00508866 0 −3.22565 0 −4.03772 0 −2.99997 0
1.11 0 0.749639 0 0.796495 0 1.70851 0 −2.43804 0
1.12 0 1.27063 0 −4.07798 0 2.09947 0 −1.38551 0
1.13 0 1.56031 0 3.44545 0 1.77285 0 −0.565441 0
1.14 0 2.01204 0 3.16436 0 2.95084 0 1.04829 0
1.15 0 2.43794 0 −3.80787 0 −4.24792 0 2.94357 0
1.16 0 2.50185 0 0.708265 0 −3.63078 0 3.25924 0
1.17 0 2.79120 0 −3.17175 0 1.46800 0 4.79080 0
1.18 0 3.03361 0 −0.0964405 0 5.16490 0 6.20277 0
1.19 0 3.26349 0 2.68811 0 −0.141739 0 7.65038 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(251\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4016.2.a.l 19
4.b odd 2 1 2008.2.a.c ✓ 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2008.2.a.c ✓ 19 4.b odd 2 1
4016.2.a.l 19 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{19} - 6 T_{3}^{18} - 21 T_{3}^{17} + 179 T_{3}^{16} + 90 T_{3}^{15} - 2109 T_{3}^{14} + 926 T_{3}^{13} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} \) Copy content Toggle raw display
$3$ \( T^{19} - 6 T^{18} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{19} + 8 T^{18} + \cdots + 5345 \) Copy content Toggle raw display
$7$ \( T^{19} - 11 T^{18} + \cdots - 2473 \) Copy content Toggle raw display
$11$ \( T^{19} - 15 T^{18} + \cdots - 1245184 \) Copy content Toggle raw display
$13$ \( T^{19} + \cdots - 414711566 \) Copy content Toggle raw display
$17$ \( T^{19} + \cdots + 55086343879 \) Copy content Toggle raw display
$19$ \( T^{19} + \cdots - 1039107162112 \) Copy content Toggle raw display
$23$ \( T^{19} + \cdots + 2426008741043 \) Copy content Toggle raw display
$29$ \( T^{19} + \cdots + 1577122914304 \) Copy content Toggle raw display
$31$ \( T^{19} + \cdots - 33028519916 \) Copy content Toggle raw display
$37$ \( T^{19} + \cdots - 19419455488 \) Copy content Toggle raw display
$41$ \( T^{19} + \cdots - 827419431304861 \) Copy content Toggle raw display
$43$ \( T^{19} + \cdots + 2528938958848 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots - 73244994928640 \) Copy content Toggle raw display
$53$ \( T^{19} + \cdots + 78211349184512 \) Copy content Toggle raw display
$59$ \( T^{19} + \cdots + 1786764591104 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots - 143052707627008 \) Copy content Toggle raw display
$67$ \( T^{19} + \cdots + 2046401095 \) Copy content Toggle raw display
$71$ \( T^{19} + \cdots - 720774358040576 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 6181963821584 \) Copy content Toggle raw display
$79$ \( T^{19} + \cdots - 36\!\cdots\!67 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots - 52\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots - 63121213508746 \) Copy content Toggle raw display
$97$ \( T^{19} + \cdots + 21933613506560 \) Copy content Toggle raw display
show more
show less