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Properties

Label 108.6.e.a
Level $108$
Weight $6$
Character orbit 108.e
Analytic conductor $17.321$
Analytic rank $0$
Dimension $10$
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] = [108,6,Mod(37,108)]
 
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, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.37");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 108.e (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: \(17.3214525398\)
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: \( x^{10} + 175x^{8} + 8800x^{6} + 124623x^{4} + 498609x^{2} + 442368 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} - 4 \beta_1 + 4) q^{5} + (\beta_{8} + \beta_{7} + \cdots + 6 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - 4 \beta_1 + 4) q^{5} + (\beta_{8} + \beta_{7} + \cdots + 6 \beta_1) q^{7}+ \cdots + ( - 43 \beta_{9} + 605 \beta_{8} + \cdots + 8219 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 21 q^{5} + 29 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 21 q^{5} + 29 q^{7} - 177 q^{11} - 181 q^{13} - 2280 q^{17} - 832 q^{19} - 399 q^{23} - 4778 q^{25} + 6033 q^{29} + 2759 q^{31} - 37146 q^{35} - 15172 q^{37} + 18435 q^{41} + 1469 q^{43} + 25155 q^{47} - 4056 q^{49} - 116844 q^{53} + 14778 q^{55} + 90537 q^{59} + 1403 q^{61} + 148407 q^{65} + 13907 q^{67} - 229368 q^{71} + 15200 q^{73} + 211983 q^{77} + 29993 q^{79} + 228951 q^{83} - 49662 q^{85} - 598332 q^{89} + 124930 q^{91} + 394764 q^{95} + 40541 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 175x^{8} + 8800x^{6} + 124623x^{4} + 498609x^{2} + 442368 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -99\nu^{9} - 23021\nu^{7} - 1847072\nu^{5} - 56550029\nu^{3} - 389674035\nu + 207097728 ) / 414195456 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -47225\nu^{8} - 9472484\nu^{6} - 609253100\nu^{4} - 13835630451\nu^{2} - 61649294112 ) / 427139064 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1089 \nu^{9} + 102528 \nu^{8} + 253231 \nu^{7} + 17565696 \nu^{6} + 20317792 \nu^{5} + \cdots + 1489775232 ) / 414195456 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 267\nu^{8} + 45744\nu^{6} + 2072454\nu^{4} + 15952113\nu^{2} + 3879623 ) / 539317 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -497\nu^{8} - 86492\nu^{6} - 4241924\nu^{4} - 51195963\nu^{2} - 57765168 ) / 563508 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 136611\nu^{8} + 23186844\nu^{6} + 1089497700\nu^{4} + 12357836001\nu^{2} + 24330997448 ) / 47459896 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1172161 \nu^{9} + 188900 \nu^{8} + 202512139 \nu^{7} + 37889936 \nu^{6} + 9878166064 \nu^{5} + \cdots + 246597176448 ) / 3417112512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1720421 \nu^{9} - 1506904 \nu^{8} + 291062195 \nu^{7} - 262243744 \nu^{6} + 13499523392 \nu^{5} + \cdots - 175143989376 ) / 3417112512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6094287 \nu^{9} - 6557328 \nu^{8} + 1053931697 \nu^{7} - 1112968512 \nu^{6} + \cdots - 1167887877504 ) / 4556150016 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{3} - 22\beta _1 + 11 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{6} + 27\beta_{5} + 13\beta_{4} - 27\beta_{2} - 3786 ) / 108 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 18 \beta_{9} - 36 \beta_{8} + 126 \beta_{7} - 9 \beta_{6} + 18 \beta_{5} - 74 \beta_{4} + \cdots - 2947 ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -182\beta_{6} - 1188\beta_{5} - 805\beta_{4} + 1134\beta_{2} + 140985 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2338 \beta_{9} + 11106 \beta_{8} - 26370 \beta_{7} + 1169 \beta_{6} - 5553 \beta_{5} + 12955 \beta_{4} + \cdots + 826042 ) / 108 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 12040\beta_{6} + 101061\beta_{5} + 85859\beta_{4} - 108270\beta_{2} - 12057045 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 60644 \beta_{9} - 671580 \beta_{8} + 1278036 \beta_{7} - 30322 \beta_{6} + 335790 \beta_{5} + \cdots - 46631401 ) / 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1598887\beta_{6} - 17799345\beta_{5} - 17481455\beta_{4} + 21107817\beta_{2} + 2167343742 ) / 108 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2573322 \beta_{9} + 73125792 \beta_{8} - 123164910 \beta_{7} + 1286661 \beta_{6} - 36562896 \beta_{5} + \cdots + 4890623801 ) / 54 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
3.71922i
− 9.84603i
7.64342i
− 2.13639i
− 1.11227i
− 3.71922i
9.84603i
− 7.64342i
2.13639i
1.11227i
0 0 0 −40.7270 + 70.5412i 0 89.6312 + 155.246i 0 0 0
37.2 0 0 0 −13.1603 + 22.7942i 0 −31.6287 − 54.7826i 0 0 0
37.3 0 0 0 −4.88422 + 8.45972i 0 −68.3340 − 118.358i 0 0 0
37.4 0 0 0 14.0718 − 24.3731i 0 75.7039 + 131.123i 0 0 0
37.5 0 0 0 55.1996 − 95.6086i 0 −50.8724 − 88.1135i 0 0 0
73.1 0 0 0 −40.7270 − 70.5412i 0 89.6312 − 155.246i 0 0 0
73.2 0 0 0 −13.1603 − 22.7942i 0 −31.6287 + 54.7826i 0 0 0
73.3 0 0 0 −4.88422 − 8.45972i 0 −68.3340 + 118.358i 0 0 0
73.4 0 0 0 14.0718 + 24.3731i 0 75.7039 − 131.123i 0 0 0
73.5 0 0 0 55.1996 + 95.6086i 0 −50.8724 + 88.1135i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
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 108.6.e.a 10
3.b odd 2 1 36.6.e.a ✓ 10
4.b odd 2 1 432.6.i.d 10
9.c even 3 1 inner 108.6.e.a 10
9.c even 3 1 324.6.a.d 5
9.d odd 6 1 36.6.e.a ✓ 10
9.d odd 6 1 324.6.a.e 5
12.b even 2 1 144.6.i.d 10
36.f odd 6 1 432.6.i.d 10
36.h even 6 1 144.6.i.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.6.e.a ✓ 10 3.b odd 2 1
36.6.e.a ✓ 10 9.d odd 6 1
108.6.e.a 10 1.a even 1 1 trivial
108.6.e.a 10 9.c even 3 1 inner
144.6.i.d 10 12.b even 2 1
144.6.i.d 10 36.h even 6 1
324.6.a.d 5 9.c even 3 1
324.6.a.e 5 9.d odd 6 1
432.6.i.d 10 4.b odd 2 1
432.6.i.d 10 36.f odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(108, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 42\!\cdots\!24 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 113704762586184)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 97\!\cdots\!92)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 96\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 85\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 66\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 86\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 48\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 18\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 18\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
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