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Properties

Label 108.5.g.a
Level $108$
Weight $5$
Character orbit 108.g
Analytic conductor $11.164$
Analytic rank $0$
Dimension $8$
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] = [108,5,Mod(17,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.17");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 108.g (of order \(6\), 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: \(11.1639560131\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{4} + \beta_{2} + 2) q^{5} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} + \cdots + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{4} + \beta_{2} + 2) q^{5} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} + \cdots + 3) q^{7}+ \cdots + (24 \beta_{7} + 71 \beta_{6} + \cdots + 1874) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 9 q^{5} + 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 9 q^{5} + 13 q^{7} + 18 q^{11} - 5 q^{13} + 562 q^{19} + 1719 q^{23} + 353 q^{25} - 2115 q^{29} + 187 q^{31} + 16 q^{37} + 7920 q^{41} - 68 q^{43} - 13689 q^{47} - 327 q^{49} - 1818 q^{55} + 20052 q^{59} - 1937 q^{61} - 25965 q^{65} + 154 q^{67} - 7802 q^{73} + 25641 q^{77} - 2195 q^{79} - 37017 q^{83} - 3042 q^{85} + 15830 q^{91} + 37116 q^{95} + 7282 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 6x^{6} + 121x^{5} + 1104x^{4} - 1647x^{3} + 6529x^{2} + 85254x + 440076 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 35 \nu^{7} - 3472 \nu^{6} - 34123 \nu^{5} + 527797 \nu^{4} - 988763 \nu^{3} - 3500708 \nu^{2} + \cdots + 307232328 ) / 2814669 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 115 \nu^{7} - 175 \nu^{6} + 4562 \nu^{5} - 26525 \nu^{4} - 65600 \nu^{3} + 13645 \nu^{2} + \cdots - 10751676 ) / 5629338 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 95 \nu^{7} + 21007 \nu^{6} - 8318 \nu^{5} + 612635 \nu^{4} + 681581 \nu^{3} + 17133464 \nu^{2} + \cdots + 416903721 ) / 2814669 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 574 \nu^{7} - 3659 \nu^{6} + 12583 \nu^{5} - 3580 \nu^{4} + 549521 \nu^{3} - 4310506 \nu^{2} + \cdots - 2489799 ) / 2814669 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\nu^{7} - 164\nu^{6} - 509\nu^{5} + 16298\nu^{4} + 11840\nu^{3} - 102856\nu^{2} + 736731\nu + 9836232 ) / 72171 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{7} - 35\nu^{6} + 43\nu^{5} + 419\nu^{4} + 1460\nu^{3} - 18628\nu^{2} - 20592\nu + 201354 ) / 6561 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5003 \nu^{7} - 30661 \nu^{6} + 137138 \nu^{5} + 631207 \nu^{4} + 837436 \nu^{3} - 4541909 \nu^{2} + \cdots + 323067342 ) / 5629338 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 3\beta_{6} + 2\beta_{5} - \beta_{3} - 7\beta_{2} - \beta _1 + 6 ) / 27 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 3\beta_{6} + 8\beta_{5} - 27\beta_{4} - \beta_{3} + 11\beta_{2} - 10\beta _1 - 3 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -50\beta_{7} + 84\beta_{6} + 32\beta_{5} - 27\beta_{4} + 23\beta_{3} + 602\beta_{2} - 49\beta _1 - 1002 ) / 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -89\beta_{7} + 228\beta_{6} - 64\beta_{5} + 54\beta_{4} + 116\beta_{3} + 2135\beta_{2} + 116\beta _1 - 18885 ) / 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 335 \beta_{7} + 1875 \beta_{6} - 2095 \beta_{5} + 2160 \beta_{4} + 1145 \beta_{3} + 22073 \beta_{2} + \cdots - 22944 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1120 \beta_{7} - 4254 \beta_{6} - 11443 \beta_{5} + 22410 \beta_{4} + 3065 \beta_{3} - 71200 \beta_{2} + \cdots + 33477 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 50779 \beta_{7} - 69108 \beta_{6} - 35029 \beta_{5} + 51327 \beta_{4} - 15841 \beta_{3} + \cdots + 1541283 ) / 27 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−3.05006 − 3.25531i
4.23522 + 4.06612i
3.72537 − 4.42407i
−3.41053 + 2.74723i
−3.05006 + 3.25531i
4.23522 − 4.06612i
3.72537 + 4.42407i
−3.41053 − 2.74723i
0 0 0 −27.4152 − 15.8282i 0 37.6830 + 65.2688i 0 0 0
17.2 0 0 0 −10.6364 − 6.14094i 0 7.14202 + 12.3703i 0 0 0
17.3 0 0 0 7.67992 + 4.43400i 0 −30.9381 − 53.5864i 0 0 0
17.4 0 0 0 34.8718 + 20.1332i 0 −7.38688 − 12.7945i 0 0 0
89.1 0 0 0 −27.4152 + 15.8282i 0 37.6830 − 65.2688i 0 0 0
89.2 0 0 0 −10.6364 + 6.14094i 0 7.14202 − 12.3703i 0 0 0
89.3 0 0 0 7.67992 − 4.43400i 0 −30.9381 + 53.5864i 0 0 0
89.4 0 0 0 34.8718 − 20.1332i 0 −7.38688 + 12.7945i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.5.g.a 8
3.b odd 2 1 36.5.g.a ✓ 8
4.b odd 2 1 432.5.q.c 8
9.c even 3 1 36.5.g.a ✓ 8
9.c even 3 1 324.5.c.a 8
9.d odd 6 1 inner 108.5.g.a 8
9.d odd 6 1 324.5.c.a 8
12.b even 2 1 144.5.q.c 8
36.f odd 6 1 144.5.q.c 8
36.f odd 6 1 1296.5.e.g 8
36.h even 6 1 432.5.q.c 8
36.h even 6 1 1296.5.e.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.5.g.a ✓ 8 3.b odd 2 1
36.5.g.a ✓ 8 9.c even 3 1
108.5.g.a 8 1.a even 1 1 trivial
108.5.g.a 8 9.d odd 6 1 inner
144.5.q.c 8 12.b even 2 1
144.5.q.c 8 36.f odd 6 1
324.5.c.a 8 9.c even 3 1
324.5.c.a 8 9.d odd 6 1
432.5.q.c 8 4.b odd 2 1
432.5.q.c 8 36.h even 6 1
1296.5.e.g 8 36.f odd 6 1
1296.5.e.g 8 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(108, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 19274879556 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 968464619236 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{4} - 281 T^{3} + \cdots - 16594049624)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 59\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + \cdots + 400007987296)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 40\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 42\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 31\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 523952945824816)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 51\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
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