Properties

Label 216.4.i.b
Level $216$
Weight $4$
Character orbit 216.i
Analytic conductor $12.744$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [216,4,Mod(73,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.73");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.i (of order \(3\), degree \(2\), not 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: \(12.7444125612\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 20x^{8} + 11x^{7} + 284x^{6} + 98x^{5} + 1567x^{4} + 780x^{3} + 6513x^{2} + 972x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{3} + \beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_1) q^{5} + ( - \beta_{7} - \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{8} + 2 \beta_{6} + \beta_{5} - 2 \beta_{2} + 5 \beta_1 - 5) q^{11} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{4} - 7 \beta_1) q^{13} + (\beta_{5} + \beta_{4} + 3 \beta_{2} - 6) q^{17} + (2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 12) q^{19} + ( - 3 \beta_{9} + \beta_{8} - \beta_{7} - 2 \beta_{6} + 3 \beta_{4} + 17 \beta_1) q^{23} + ( - 4 \beta_{9} + 2 \beta_{7} - \beta_{6} + 2 \beta_{3} + \beta_{2} + 62 \beta_1 - 62) q^{25} + (\beta_{9} + \beta_{8} - 6 \beta_{7} + \beta_{6} + \beta_{5} - 6 \beta_{3} - \beta_{2} - 23 \beta_1 + 23) q^{29} + ( - 3 \beta_{9} - \beta_{8} - \beta_{7} - 6 \beta_{6} + 3 \beta_{4} - 49 \beta_1) q^{31} + (6 \beta_{5} + 8 \beta_{4} - 11 \beta_{3} + 2 \beta_{2} + 41) q^{35} + (5 \beta_{5} - 3 \beta_{4} - 6 \beta_{3} - 12 \beta_{2} + 72) q^{37} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} + 4 \beta_{6} + \beta_{4} + 33 \beta_1) q^{41} + (\beta_{9} + 6 \beta_{8} + 8 \beta_{6} + 6 \beta_{5} - 8 \beta_{2} + 161 \beta_1 - 161) q^{43} + (2 \beta_{9} + 10 \beta_{8} - 7 \beta_{7} - 8 \beta_{6} + 10 \beta_{5} - 7 \beta_{3} + \cdots - 99) q^{47}+ \cdots + (7 \beta_{9} + 11 \beta_{8} + 2 \beta_{7} + 48 \beta_{6} + 11 \beta_{5} + \cdots - 577) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5 q^{5} - 3 q^{7} - 25 q^{11} - 29 q^{13} - 56 q^{17} + 128 q^{19} + 89 q^{23} - 322 q^{25} + 129 q^{29} - 241 q^{31} + 486 q^{35} + 732 q^{37} + 171 q^{41} - 803 q^{43} - 477 q^{47} - 1072 q^{49} - 748 q^{53} + 2938 q^{55} - 607 q^{59} - 1349 q^{61} + 527 q^{65} - 1549 q^{67} + 1624 q^{71} + 3856 q^{73} + 903 q^{77} - 1727 q^{79} - 1025 q^{83} - 2902 q^{85} - 4620 q^{89} + 5970 q^{91} - 2012 q^{95} - 2875 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 20x^{8} + 11x^{7} + 284x^{6} + 98x^{5} + 1567x^{4} + 780x^{3} + 6513x^{2} + 972x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 62716607 \nu^{9} + 60280747 \nu^{8} - 1256375540 \nu^{7} - 685723757 \nu^{6} - 17945929304 \nu^{5} - 6134568086 \nu^{4} + \cdots - 539986752 ) / 60861064212 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22624787 \nu^{9} - 181801400 \nu^{8} + 370019320 \nu^{7} - 5502995321 \nu^{6} + 1037337400 \nu^{5} - 48081123200 \nu^{4} + \cdots - 298449333963 ) / 15215266053 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25689887 \nu^{9} + 99422120 \nu^{8} - 202353256 \nu^{7} + 4246210487 \nu^{6} - 567290920 \nu^{5} + 26294226560 \nu^{4} + \cdots - 388432263123 ) / 15215266053 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1412464 \nu^{9} + 13709650 \nu^{8} - 27903170 \nu^{7} + 405213286 \nu^{6} - 78225650 \nu^{5} + 3625799200 \nu^{4} - 1484068446 \nu^{3} + \cdots + 15200272758 ) / 390135027 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 153965140 \nu^{9} - 773366890 \nu^{8} + 1574029082 \nu^{7} - 26916790990 \nu^{6} + 4412740490 \nu^{5} - 204532796320 \nu^{4} + \cdots - 248448520980 ) / 15215266053 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1444155569 \nu^{9} - 2558869777 \nu^{8} + 30805500200 \nu^{7} - 2347165801 \nu^{6} + 389612167328 \nu^{5} - 34124558314 \nu^{4} + \cdots + 3049271424 ) / 60861064212 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 489227971 \nu^{9} + 177100435 \nu^{8} - 9211161792 \nu^{7} - 9889932261 \nu^{6} - 145903603392 \nu^{5} - 77469872970 \nu^{4} + \cdots - 6561845376 ) / 20287021404 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 579522655 \nu^{9} + 1896169913 \nu^{8} - 14062441882 \nu^{7} + 14408750465 \nu^{6} - 138806560420 \nu^{5} + 120204505310 \nu^{4} + \cdots + 5744630880 ) / 15215266053 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1627759855 \nu^{9} - 2102773259 \nu^{8} + 30712557580 \nu^{7} + 21655671781 \nu^{6} + 393573182356 \nu^{5} + 170034350110 \nu^{4} + \cdots + 1184952854772 ) / 30430532106 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} + 2\beta_{7} + 6\beta_{6} + \beta_{4} + 6\beta_1 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} - 4\beta_{7} + 12\beta_{6} + \beta_{5} - 4\beta_{3} - 12\beta_{2} + 282\beta _1 - 282 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{5} - 19\beta_{4} + 14\beta_{3} - 78\beta_{2} - 318 ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 26\beta_{9} - 5\beta_{8} + 20\beta_{7} - 114\beta_{6} - 26\beta_{4} - 1650\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 164 \beta_{9} - 29 \beta_{8} - 40 \beta_{7} - 588 \beta_{6} - 29 \beta_{5} - 40 \beta_{3} + 588 \beta_{2} - 3648 \beta _1 + 3648 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -127\beta_{5} + 1147\beta_{4} + 436\beta_{3} + 4152\beta_{2} + 45810 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2810\beta_{9} + 305\beta_{8} + 79\beta_{7} + 9561\beta_{6} + 2810\beta_{4} + 71046\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 21661 \beta_{9} + 1897 \beta_{8} - 5374 \beta_{7} + 73902 \beta_{6} + 1897 \beta_{5} - 5374 \beta_{3} - 73902 \beta_{2} + 712182 \beta _1 - 712182 ) / 36 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8161\beta_{5} - 96661\beta_{4} - 6628\beta_{3} - 322620\beta_{2} - 2598114 ) / 36 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
2.08481 + 3.61100i
1.25497 + 2.17367i
−1.35977 2.35519i
−1.40499 2.43351i
−0.0750226 0.129943i
2.08481 3.61100i
1.25497 2.17367i
−1.35977 + 2.35519i
−1.40499 + 2.43351i
−0.0750226 + 0.129943i
0 0 0 −9.15919 15.8642i 0 −7.48933 + 12.9719i 0 0 0
73.2 0 0 0 −3.15936 5.47217i 0 14.7898 25.6167i 0 0 0
73.3 0 0 0 −1.49098 2.58245i 0 −5.85543 + 10.1419i 0 0 0
73.4 0 0 0 6.19843 + 10.7360i 0 −15.3163 + 26.5286i 0 0 0
73.5 0 0 0 10.1111 + 17.5129i 0 12.3712 21.4276i 0 0 0
145.1 0 0 0 −9.15919 + 15.8642i 0 −7.48933 12.9719i 0 0 0
145.2 0 0 0 −3.15936 + 5.47217i 0 14.7898 + 25.6167i 0 0 0
145.3 0 0 0 −1.49098 + 2.58245i 0 −5.85543 10.1419i 0 0 0
145.4 0 0 0 6.19843 10.7360i 0 −15.3163 26.5286i 0 0 0
145.5 0 0 0 10.1111 17.5129i 0 12.3712 + 21.4276i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.4.i.b 10
3.b odd 2 1 72.4.i.b 10
4.b odd 2 1 432.4.i.f 10
9.c even 3 1 inner 216.4.i.b 10
9.c even 3 1 648.4.a.k 5
9.d odd 6 1 72.4.i.b 10
9.d odd 6 1 648.4.a.l 5
12.b even 2 1 144.4.i.f 10
36.f odd 6 1 432.4.i.f 10
36.f odd 6 1 1296.4.a.bc 5
36.h even 6 1 144.4.i.f 10
36.h even 6 1 1296.4.a.bd 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.i.b 10 3.b odd 2 1
72.4.i.b 10 9.d odd 6 1
144.4.i.f 10 12.b even 2 1
144.4.i.f 10 36.h even 6 1
216.4.i.b 10 1.a even 1 1 trivial
216.4.i.b 10 9.c even 3 1 inner
432.4.i.f 10 4.b odd 2 1
432.4.i.f 10 36.f odd 6 1
648.4.a.k 5 9.c even 3 1
648.4.a.l 5 9.d odd 6 1
1296.4.a.bc 5 36.f odd 6 1
1296.4.a.bd 5 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 5 T_{5}^{9} + 486 T_{5}^{8} + 111 T_{5}^{7} + 181830 T_{5}^{6} - 57429 T_{5}^{5} + 18313185 T_{5}^{4} + 119463888 T_{5}^{3} + 1213781760 T_{5}^{2} + 3130236928 T_{5} + 7487094784 \) acting on \(S_{4}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 5 T^{9} + \cdots + 7487094784 \) Copy content Toggle raw display
$7$ \( T^{10} + 3 T^{9} + \cdots + 15465374220816 \) Copy content Toggle raw display
$11$ \( T^{10} + 25 T^{9} + \cdots + 6004376846689 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 137434637455504 \) Copy content Toggle raw display
$17$ \( (T^{5} + 28 T^{4} - 8927 T^{3} + \cdots + 465026264)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 64 T^{4} - 10133 T^{3} + \cdots - 219796928)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 89 T^{9} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{10} - 129 T^{9} + \cdots + 57\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{10} + 241 T^{9} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{5} - 366 T^{4} + \cdots - 525481299072)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 171 T^{9} + \cdots + 66\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{10} + 803 T^{9} + \cdots + 55\!\cdots\!89 \) Copy content Toggle raw display
$47$ \( T^{10} + 477 T^{9} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{5} + 374 T^{4} + \cdots + 758189779072)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 607 T^{9} + \cdots + 64\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{10} + 1349 T^{9} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{10} + 1549 T^{9} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{5} - 812 T^{4} + \cdots - 24420797440)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 1928 T^{4} + \cdots + 28702127905256)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 1727 T^{9} + \cdots + 72\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{10} + 1025 T^{9} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + 2310 T^{4} + \cdots - 887800297007520)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 2875 T^{9} + \cdots + 76\!\cdots\!09 \) Copy content Toggle raw display
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