Properties

Label 11.16.a.b
Level $11$
Weight $16$
Character orbit 11.a
Self dual yes
Analytic conductor $15.696$
Analytic rank $0$
Dimension $7$
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] = [11,16,Mod(1,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 11.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: \(15.6962855610\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 178974 x^{5} - 2029324 x^{4} + 8013742752 x^{3} + 65268605952 x^{2} - 36793730914304 x + 872452335304704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - 5 \beta_1 + 877) q^{3} + (\beta_{3} + 3 \beta_{2} + 17 \beta_1 + 18366) q^{4} + (3 \beta_{6} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 119 \beta_1 + 28956) q^{5} + ( - 2 \beta_{6} - 6 \beta_{4} + 2 \beta_{3} + 102 \beta_{2} + \cdots + 248494) q^{6}+ \cdots + (113 \beta_{6} + 253 \beta_{5} - 131 \beta_{4} - 13 \beta_{3} + \cdots + 4281359) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - 5 \beta_1 + 877) q^{3} + (\beta_{3} + 3 \beta_{2} + 17 \beta_1 + 18366) q^{4} + (3 \beta_{6} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 119 \beta_1 + 28956) q^{5} + ( - 2 \beta_{6} - 6 \beta_{4} + 2 \beta_{3} + 102 \beta_{2} + \cdots + 248494) q^{6}+ \cdots + ( - 2202050323 \beta_{6} + \cdots - 83431574945389) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 6142 q^{3} + 128572 q^{4} + 202692 q^{5} + 1739782 q^{6} - 475420 q^{7} - 6087972 q^{8} + 29969225 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 6142 q^{3} + 128572 q^{4} + 202692 q^{5} + 1739782 q^{6} - 475420 q^{7} - 6087972 q^{8} + 29969225 q^{9} - 42051570 q^{10} - 136410197 q^{11} + 483668840 q^{12} + 744000686 q^{13} + 62981628 q^{14} + 84801586 q^{15} + 4336886920 q^{16} + 1973544606 q^{17} + 16748380154 q^{18} + 6476441180 q^{19} + 20195961648 q^{20} + 56954353588 q^{21} + 28726414890 q^{23} + 62716246956 q^{24} + 57324660271 q^{25} + 63012786252 q^{26} - 8874384698 q^{27} + 245894842688 q^{28} + 16211787966 q^{29} - 384944091026 q^{30} - 398145552430 q^{31} - 691675472136 q^{32} - 119690204282 q^{33} - 1457160703992 q^{34} - 1295921304612 q^{35} - 1640397143132 q^{36} + 825264084752 q^{37} - 1196804313144 q^{38} - 922894286600 q^{39} - 1079524381452 q^{40} + 478709387970 q^{41} + 5635806608740 q^{42} + 3288874902632 q^{43} - 2505504549812 q^{44} - 6028293522982 q^{45} + 4022813399022 q^{46} + 8702262542568 q^{47} + 2248331000264 q^{48} + 25095993399927 q^{49} + 26681725341282 q^{50} + 11311557006508 q^{51} + 33613883776616 q^{52} - 11672851418694 q^{53} + 8695296319930 q^{54} - 3949893664332 q^{55} - 43126650742440 q^{56} - 3609250568088 q^{57} - 72114160054860 q^{58} - 5175319709862 q^{59} - 70667564795368 q^{60} + 63377381302070 q^{61} - 186937585184130 q^{62} + 22790425932032 q^{63} - 91199168852960 q^{64} + 73870271088900 q^{65} - 33903429336722 q^{66} - 18138459048406 q^{67} - 173306548194648 q^{68} - 5610571633594 q^{69} - 430773415625556 q^{70} + 126010547432334 q^{71} + 61683074386032 q^{72} + 208521291244442 q^{73} + 315125345221170 q^{74} + 539260571374820 q^{75} + 35647303012640 q^{76} + 9264590836820 q^{77} - 81697585097960 q^{78} + 499794357421700 q^{79} + 662893611377496 q^{80} + 11\!\cdots\!67 q^{81}+ \cdots - 584015412312475 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 178974 x^{5} - 2029324 x^{4} + 8013742752 x^{3} + 65268605952 x^{2} - 36793730914304 x + 872452335304704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 368359203 \nu^{6} - 7109508560 \nu^{5} + 64735650385402 \nu^{4} + \cdots + 78\!\cdots\!36 ) / 30\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 368359203 \nu^{6} + 7109508560 \nu^{5} - 64735650385402 \nu^{4} + \cdots - 13\!\cdots\!40 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4514709711 \nu^{6} - 198985994144 \nu^{5} + 822029412501442 \nu^{4} + \cdots + 12\!\cdots\!68 ) / 30\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4662510411 \nu^{6} - 128938123664 \nu^{5} + 817533865660906 \nu^{4} + \cdots + 11\!\cdots\!60 ) / 30\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7691866891 \nu^{6} - 339833234608 \nu^{5} + \cdots + 18\!\cdots\!64 ) / 30\!\cdots\!68 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + 17\beta _1 + 51134 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -27\beta_{6} + 58\beta_{5} + 26\beta_{4} + 82\beta_{3} - 243\beta_{2} + 89456\beta _1 + 869826 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12042\beta_{6} + 20604\beta_{4} + 105750\beta_{3} + 316176\beta_{2} + 4151066\beta _1 + 4572490560 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 6718410 \beta_{6} + 7814964 \beta_{5} + 6039804 \beta_{4} + 12326936 \beta_{3} + 4327362 \beta_{2} + 8640168988 \beta _1 + 212806497412 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2149008624 \beta_{6} + 198046904 \beta_{5} + 3660781504 \beta_{4} + 11327728844 \beta_{3} + 30653070108 \beta_{2} + 706215850924 \beta _1 + 441885379121112 \) Copy content Toggle raw display

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
342.946
248.290
41.0559
34.1853
−87.3124
−273.868
−305.297
−342.946 828.867 84844.1 134280. −284257. 3.30958e6 −1.78593e7 −1.36619e7 −4.60509e7
1.2 −248.290 −1090.46 28880.1 70033.4 270751. −4.28609e6 965337. −1.31598e7 −1.73886e7
1.3 −41.0559 −1011.27 −31082.4 −266062. 41518.5 1.10351e6 2.62144e6 −1.33262e7 1.09234e7
1.4 −34.1853 5391.11 −31599.4 210501. −184297. 908835. 2.20042e6 1.47151e7 −7.19605e6
1.5 87.3124 −6857.11 −25144.5 29790.5 −598711. −3.58293e6 −5.05648e6 3.26710e7 2.60108e6
1.6 273.868 6889.30 42235.6 −244570. 1.88676e6 3.48309e6 2.59288e6 3.31136e7 −6.69799e7
1.7 305.297 1991.56 60438.5 268719. 608018. −1.41141e6 8.44774e6 −1.03826e7 8.20393e7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)

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 11.16.a.b 7
3.b odd 2 1 99.16.a.f 7
11.b odd 2 1 121.16.a.d 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.16.a.b 7 1.a even 1 1 trivial
99.16.a.f 7 3.b odd 2 1
121.16.a.d 7 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 178974 T_{2}^{5} + 2029324 T_{2}^{4} + 8013742752 T_{2}^{3} - 65268605952 T_{2}^{2} - 36793730914304 T_{2} - 872452335304704 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(11))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + \cdots - 872452335304704 \) Copy content Toggle raw display
$3$ \( T^{7} - 6142 T^{6} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{7} - 202692 T^{6} + \cdots - 10\!\cdots\!50 \) Copy content Toggle raw display
$7$ \( T^{7} + 475420 T^{6} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T + 19487171)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} - 744000686 T^{6} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{7} - 1973544606 T^{6} + \cdots - 23\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{7} - 6476441180 T^{6} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{7} - 28726414890 T^{6} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{7} - 16211787966 T^{6} + \cdots - 93\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{7} + 398145552430 T^{6} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{7} - 825264084752 T^{6} + \cdots + 13\!\cdots\!62 \) Copy content Toggle raw display
$41$ \( T^{7} - 478709387970 T^{6} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{7} - 3288874902632 T^{6} + \cdots + 51\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{7} - 8702262542568 T^{6} + \cdots + 63\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{7} + 11672851418694 T^{6} + \cdots - 44\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{7} + 5175319709862 T^{6} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$61$ \( T^{7} - 63377381302070 T^{6} + \cdots + 54\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{7} + 18138459048406 T^{6} + \cdots - 19\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{7} - 126010547432334 T^{6} + \cdots - 51\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{7} - 208521291244442 T^{6} + \cdots - 28\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{7} - 499794357421700 T^{6} + \cdots + 38\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{7} - 291424819897200 T^{6} + \cdots - 39\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{7} - 870223903132404 T^{6} + \cdots + 23\!\cdots\!90 \) Copy content Toggle raw display
$97$ \( T^{7} + 747442043765704 T^{6} + \cdots - 20\!\cdots\!70 \) Copy content Toggle raw display
show more
show less