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Properties

Label 108.2.i.a
Level $108$
Weight $2$
Character orbit 108.i
Analytic conductor $0.862$
Analytic rank $0$
Dimension $18$
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] = [108,2,Mod(13,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 108.i (of order \(9\), degree \(6\), 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: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{3} + ( - \beta_{16} + \beta_{11} + \beta_{8} + \cdots - 1) q^{5}+ \cdots + (\beta_{17} + \beta_{15} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{3} + ( - \beta_{16} + \beta_{11} + \beta_{8} + \cdots - 1) q^{5}+ \cdots + (\beta_{17} + 3 \beta_{16} - \beta_{15} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 3 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 3 q^{5} + 6 q^{9} + 3 q^{11} - 9 q^{15} - 12 q^{17} - 30 q^{21} - 30 q^{23} + 9 q^{25} - 27 q^{27} - 24 q^{29} + 9 q^{31} - 18 q^{33} - 21 q^{35} + 3 q^{39} + 21 q^{41} - 9 q^{43} + 45 q^{45} + 45 q^{47} - 18 q^{49} + 63 q^{51} + 66 q^{53} + 54 q^{57} + 60 q^{59} - 18 q^{61} + 57 q^{63} + 33 q^{65} - 27 q^{67} - 9 q^{69} - 12 q^{71} + 9 q^{73} - 33 q^{75} - 75 q^{77} - 36 q^{79} - 54 q^{81} - 45 q^{83} - 36 q^{85} - 63 q^{87} - 48 q^{89} + 9 q^{91} - 33 q^{93} + 6 q^{95} - 27 q^{97} + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} + 24 x^{16} - 66 x^{15} + 153 x^{14} - 315 x^{13} + 651 x^{12} - 1350 x^{11} + \cdots + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 157 \nu^{17} - 5979 \nu^{16} + 22788 \nu^{15} - 73797 \nu^{14} + 149931 \nu^{13} + \cdots - 32102973 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 256 \nu^{17} - 252 \nu^{16} + 819 \nu^{15} - 2946 \nu^{14} + 8433 \nu^{13} - 18684 \nu^{12} + \cdots - 3726648 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 491 \nu^{17} - 3462 \nu^{16} + 16353 \nu^{15} - 43899 \nu^{14} + 97020 \nu^{13} - 191322 \nu^{12} + \cdots - 11881971 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 463 \nu^{17} - 1365 \nu^{16} + 5913 \nu^{15} - 26250 \nu^{14} + 52794 \nu^{13} + \cdots - 17445699 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 160 \nu^{17} + 1006 \nu^{16} - 2643 \nu^{15} + 13329 \nu^{14} - 26211 \nu^{13} + 61623 \nu^{12} + \cdots + 11763873 ) / 391473 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 428 \nu^{17} - 1775 \nu^{16} + 4650 \nu^{15} - 10245 \nu^{14} + 20652 \nu^{13} - 44226 \nu^{12} + \cdots - 1679616 ) / 391473 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1330 \nu^{17} + 5085 \nu^{16} - 18540 \nu^{15} + 38388 \nu^{14} - 80928 \nu^{13} + \cdots - 4494285 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1537 \nu^{17} + 10986 \nu^{16} - 39195 \nu^{15} + 107463 \nu^{14} - 225909 \nu^{13} + \cdots + 36577575 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 965 \nu^{17} + 4460 \nu^{16} - 16464 \nu^{15} + 40854 \nu^{14} - 88314 \nu^{13} + 176328 \nu^{12} + \cdots + 8726130 ) / 391473 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1334 \nu^{17} - 5391 \nu^{16} + 17655 \nu^{15} - 38283 \nu^{14} + 79776 \nu^{13} - 161046 \nu^{12} + \cdots - 2267919 ) / 391473 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1381 \nu^{17} - 5675 \nu^{16} + 18936 \nu^{15} - 41211 \nu^{14} + 86109 \nu^{13} - 172053 \nu^{12} + \cdots - 3037743 ) / 391473 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1631 \nu^{17} - 6363 \nu^{16} + 21888 \nu^{15} - 45120 \nu^{14} + 94122 \nu^{13} - 183402 \nu^{12} + \cdots + 1421550 ) / 391473 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5449 \nu^{17} - 26718 \nu^{16} + 90324 \nu^{15} - 219036 \nu^{14} + 462159 \nu^{13} + \cdots - 43906212 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 6344 \nu^{17} + 41754 \nu^{16} - 141876 \nu^{15} + 376377 \nu^{14} - 781137 \nu^{13} + \cdots + 104976000 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 871 \nu^{17} - 4787 \nu^{16} + 16587 \nu^{15} - 41442 \nu^{14} + 86388 \nu^{13} - 175257 \nu^{12} + \cdots - 8752374 ) / 130491 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 8860 \nu^{17} - 46278 \nu^{16} + 162486 \nu^{15} - 411825 \nu^{14} + 874917 \nu^{13} + \cdots - 103873752 ) / 1174419 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} + \beta_{16} + \beta_{14} - \beta_{11} + \beta_{8} + \beta_{4} + \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{16} + 2 \beta_{14} + \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{11} - 3 \beta_{10} - \beta_{9} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} - 2 \beta_{16} + \beta_{15} + 2 \beta_{14} + 4 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{17} - \beta_{16} - 3 \beta_{14} - \beta_{12} - 3 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{17} + 9 \beta_{16} - 6 \beta_{15} - 9 \beta_{14} - 12 \beta_{13} + 27 \beta_{12} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{17} - 18 \beta_{16} - 18 \beta_{15} + 6 \beta_{13} + 33 \beta_{12} - 9 \beta_{11} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 18 \beta_{17} - 18 \beta_{16} - 36 \beta_{15} - 24 \beta_{14} + 15 \beta_{13} + 6 \beta_{12} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 27 \beta_{17} + 36 \beta_{16} + 39 \beta_{15} + 66 \beta_{14} + 51 \beta_{13} - 90 \beta_{12} + \cdots + 108 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 69 \beta_{17} - 66 \beta_{16} + 69 \beta_{15} + 48 \beta_{14} + 6 \beta_{13} - 45 \beta_{12} + \cdots - 114 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 81 \beta_{17} + 39 \beta_{16} - 36 \beta_{15} - 90 \beta_{14} - 144 \beta_{13} + 210 \beta_{12} + \cdots - 306 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 243 \beta_{17} - 207 \beta_{16} - 117 \beta_{15} + 72 \beta_{14} + 180 \beta_{13} - 9 \beta_{12} + \cdots - 45 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 513 \beta_{17} - 621 \beta_{16} - 306 \beta_{15} - 720 \beta_{14} - 27 \beta_{13} + 234 \beta_{12} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 54 \beta_{17} + 441 \beta_{16} + 513 \beta_{15} - 495 \beta_{14} - 99 \beta_{13} + 315 \beta_{12} + \cdots + 2151 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 270 \beta_{17} + 675 \beta_{16} + 261 \beta_{15} - 603 \beta_{14} - 423 \beta_{13} - 54 \beta_{12} + \cdots - 1782 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 3366 \beta_{17} + 4329 \beta_{16} - 693 \beta_{15} - 2061 \beta_{14} - 3339 \beta_{13} + 2160 \beta_{12} + \cdots - 3276 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-\beta_{5} - \beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.381933 + 1.68942i
1.20201 − 1.24706i
−1.34999 − 1.08514i
0.381933 − 1.68942i
1.20201 + 1.24706i
−1.34999 + 1.08514i
1.68668 − 0.393823i
0.472963 − 1.66622i
−0.219955 + 1.71803i
−1.29960 − 1.14501i
0.960398 + 1.44140i
1.16555 − 1.28120i
−1.29960 + 1.14501i
0.960398 − 1.44140i
1.16555 + 1.28120i
1.68668 + 0.393823i
0.472963 + 1.66622i
−0.219955 − 1.71803i
0 −1.73007 + 0.0827666i 0 2.26400 + 1.89972i 0 2.50885 + 0.913148i 0 2.98630 − 0.286384i 0
13.2 0 1.01939 + 1.40030i 0 1.46957 + 1.23312i 0 −3.86125 − 1.40538i 0 −0.921685 + 2.85491i 0
13.3 0 1.30308 − 1.14105i 0 −0.761786 − 0.639214i 0 1.35240 + 0.492232i 0 0.396022 − 2.97375i 0
25.1 0 −1.73007 − 0.0827666i 0 2.26400 − 1.89972i 0 2.50885 − 0.913148i 0 2.98630 + 0.286384i 0
25.2 0 1.01939 − 1.40030i 0 1.46957 − 1.23312i 0 −3.86125 + 1.40538i 0 −0.921685 − 2.85491i 0
25.3 0 1.30308 + 1.14105i 0 −0.761786 + 0.639214i 0 1.35240 − 0.492232i 0 0.396022 + 2.97375i 0
49.1 0 −1.54522 − 0.782494i 0 2.29878 − 0.836687i 0 −0.775345 − 4.39720i 0 1.77541 + 2.41825i 0
49.2 0 −1.43334 + 0.972387i 0 −3.94709 + 1.43662i 0 0.610312 + 3.46125i 0 1.10893 − 2.78752i 0
49.3 0 1.27282 − 1.17470i 0 0.0952805 − 0.0346793i 0 0.165033 + 0.935950i 0 0.240153 − 2.99037i 0
61.1 0 −0.829611 − 1.52044i 0 −0.399598 − 2.26623i 0 −0.715176 − 0.600104i 0 −1.62349 + 2.52275i 0
61.2 0 0.409491 + 1.68295i 0 −0.103132 − 0.584890i 0 2.18780 + 1.83578i 0 −2.66463 + 1.37830i 0
61.3 0 1.53346 − 0.805294i 0 0.583982 + 3.31193i 0 −1.47262 − 1.23568i 0 1.70300 − 2.46977i 0
85.1 0 −0.829611 + 1.52044i 0 −0.399598 + 2.26623i 0 −0.715176 + 0.600104i 0 −1.62349 − 2.52275i 0
85.2 0 0.409491 − 1.68295i 0 −0.103132 + 0.584890i 0 2.18780 − 1.83578i 0 −2.66463 − 1.37830i 0
85.3 0 1.53346 + 0.805294i 0 0.583982 − 3.31193i 0 −1.47262 + 1.23568i 0 1.70300 + 2.46977i 0
97.1 0 −1.54522 + 0.782494i 0 2.29878 + 0.836687i 0 −0.775345 + 4.39720i 0 1.77541 − 2.41825i 0
97.2 0 −1.43334 − 0.972387i 0 −3.94709 − 1.43662i 0 0.610312 − 3.46125i 0 1.10893 + 2.78752i 0
97.3 0 1.27282 + 1.17470i 0 0.0952805 + 0.0346793i 0 0.165033 − 0.935950i 0 0.240153 + 2.99037i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.2.i.a ✓ 18
3.b odd 2 1 324.2.i.a 18
4.b odd 2 1 432.2.u.d 18
9.c even 3 1 972.2.i.a 18
9.c even 3 1 972.2.i.c 18
9.d odd 6 1 972.2.i.b 18
9.d odd 6 1 972.2.i.d 18
27.e even 9 1 inner 108.2.i.a ✓ 18
27.e even 9 1 972.2.i.a 18
27.e even 9 1 972.2.i.c 18
27.e even 9 1 2916.2.a.d 9
27.e even 9 2 2916.2.e.c 18
27.f odd 18 1 324.2.i.a 18
27.f odd 18 1 972.2.i.b 18
27.f odd 18 1 972.2.i.d 18
27.f odd 18 1 2916.2.a.c 9
27.f odd 18 2 2916.2.e.d 18
108.j odd 18 1 432.2.u.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.i.a ✓ 18 1.a even 1 1 trivial
108.2.i.a ✓ 18 27.e even 9 1 inner
324.2.i.a 18 3.b odd 2 1
324.2.i.a 18 27.f odd 18 1
432.2.u.d 18 4.b odd 2 1
432.2.u.d 18 108.j odd 18 1
972.2.i.a 18 9.c even 3 1
972.2.i.a 18 27.e even 9 1
972.2.i.b 18 9.d odd 6 1
972.2.i.b 18 27.f odd 18 1
972.2.i.c 18 9.c even 3 1
972.2.i.c 18 27.e even 9 1
972.2.i.d 18 9.d odd 6 1
972.2.i.d 18 27.f odd 18 1
2916.2.a.c 9 27.f odd 18 1
2916.2.a.d 9 27.e even 9 1
2916.2.e.c 18 27.e even 9 2
2916.2.e.d 18 27.f odd 18 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(108, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - 3 T^{16} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{18} - 3 T^{17} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{18} + 9 T^{16} + \cdots + 1456849 \) Copy content Toggle raw display
$11$ \( T^{18} - 3 T^{17} + \cdots + 23357889 \) Copy content Toggle raw display
$13$ \( T^{18} - 45 T^{16} + \cdots + 4068289 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 5247698481 \) Copy content Toggle raw display
$19$ \( T^{18} + 90 T^{16} + \cdots + 49014001 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 5597583489 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 451152679041 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 7983601201 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 9420061249 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 29274867801 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 5055621837841 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 941480149401 \) Copy content Toggle raw display
$53$ \( (T^{9} - 33 T^{8} + \cdots + 9249336)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 14164767759321 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 173474749009 \) Copy content Toggle raw display
$67$ \( T^{18} + 27 T^{17} + \cdots + 3568321 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 135419769 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 13254226129 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 1826490081529 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 289679140521369 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 76986883963089 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 736742449 \) Copy content Toggle raw display
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