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Properties

Label 108.5.d.a
Level $108$
Weight $5$
Character orbit 108.d
Analytic conductor $11.164$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,5,Mod(55,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.55");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 108.d (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 6 x^{14} - 22 x^{13} + 19 x^{12} + 18 x^{11} + 1423 x^{10} + 660 x^{9} + \cdots + 2924100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{26} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{7} - \beta_1) q^{5} + \beta_{4} q^{7} + (\beta_{11} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{7} - \beta_1) q^{5} + \beta_{4} q^{7} + (\beta_{11} - \beta_1) q^{8} + (\beta_{12} + \beta_{2} - 12) q^{10} + ( - \beta_{11} - \beta_{3} - 3 \beta_1) q^{11} + (\beta_{12} + \beta_{10} - 22) q^{13} + ( - \beta_{15} + \beta_{7} + \cdots + \beta_1) q^{14}+ \cdots + (24 \beta_{14} + 64 \beta_{11} + \cdots + 58 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 14 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 14 q^{4} - 202 q^{10} - 352 q^{13} - 206 q^{16} + 738 q^{22} + 1632 q^{25} + 342 q^{28} - 2536 q^{34} + 3200 q^{37} - 2854 q^{40} + 36 q^{46} - 896 q^{49} + 2288 q^{52} + 2492 q^{58} - 2752 q^{61} + 682 q^{64} - 14166 q^{70} + 8240 q^{73} - 33084 q^{76} + 68 q^{82} + 8800 q^{85} + 48294 q^{88} + 52596 q^{94} - 6928 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 6 x^{14} - 22 x^{13} + 19 x^{12} + 18 x^{11} + 1423 x^{10} + 660 x^{9} + \cdots + 2924100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 76\!\cdots\!53 \nu^{15} + \cdots + 10\!\cdots\!60 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 50\!\cdots\!47 \nu^{15} + \cdots + 65\!\cdots\!28 ) / 53\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 64\!\cdots\!77 \nu^{15} + \cdots - 55\!\cdots\!40 ) / 67\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 24\!\cdots\!37 \nu^{15} + \cdots - 91\!\cdots\!84 ) / 21\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 37\!\cdots\!03 \nu^{15} + \cdots - 81\!\cdots\!32 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!77 \nu^{15} + \cdots + 22\!\cdots\!40 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!49 \nu^{15} + \cdots - 33\!\cdots\!80 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!85 \nu^{15} + \cdots + 54\!\cdots\!00 ) / 42\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 58\!\cdots\!19 \nu^{15} + \cdots + 82\!\cdots\!80 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39\!\cdots\!07 \nu^{15} + \cdots - 53\!\cdots\!00 ) / 42\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 61\!\cdots\!69 \nu^{15} + \cdots + 97\!\cdots\!80 ) / 40\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 89\!\cdots\!71 \nu^{15} + \cdots + 41\!\cdots\!28 ) / 42\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 98\!\cdots\!35 \nu^{15} + \cdots + 12\!\cdots\!56 ) / 42\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16\!\cdots\!73 \nu^{15} + \cdots + 14\!\cdots\!60 ) / 67\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 15\!\cdots\!53 \nu^{15} + \cdots - 25\!\cdots\!60 ) / 56\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} + 3 \beta_{11} - \beta_{10} - 3 \beta_{9} + 6 \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} + \cdots + 29 ) / 216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{15} + \beta_{12} + 12 \beta_{11} + 2 \beta_{10} - 3 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} + \cdots - 109 ) / 216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{13} + 11\beta_{12} + 7\beta_{10} - \beta_{8} + 23\beta_{5} - 90\beta_{4} - 104\beta_{2} + 499 ) / 216 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 78 \beta_{15} - 24 \beta_{14} + 4 \beta_{13} + 21 \beta_{12} - 3 \beta_{11} - 39 \beta_{10} + \cdots + 1239 ) / 216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 270 \beta_{15} + 45 \beta_{14} + 53 \beta_{13} - 22 \beta_{12} - 66 \beta_{11} + 145 \beta_{10} + \cdots - 4786 ) / 216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 629 \beta_{13} + 101 \beta_{12} + 685 \beta_{10} - 355 \beta_{8} - 427 \beta_{5} + 2118 \beta_{4} + \cdots - 120359 ) / 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1011 \beta_{15} - 1761 \beta_{14} + 885 \beta_{13} - 50 \beta_{12} - 813 \beta_{11} + 884 \beta_{10} + \cdots - 294398 ) / 216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2823 \beta_{15} - 4122 \beta_{14} - 1754 \beta_{13} + 615 \beta_{12} - 14604 \beta_{11} + \cdots + 935853 ) / 216 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7989 \beta_{13} - 3365 \beta_{12} - 18649 \beta_{10} + 20047 \beta_{8} - 7217 \beta_{5} + \cdots + 3424123 ) / 216 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 179142 \beta_{15} + 27996 \beta_{14} - 8616 \beta_{13} - 26495 \beta_{12} + 256005 \beta_{11} + \cdots - 206529 ) / 216 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 734322 \beta_{15} + 29925 \beta_{14} - 23563 \beta_{13} + 124586 \beta_{12} + 569910 \beta_{11} + \cdots + 17935234 ) / 216 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 761411 \beta_{13} + 660669 \beta_{12} - 2164587 \beta_{10} + 613557 \beta_{8} + 1843941 \beta_{5} + \cdots + 232789369 ) / 216 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 5858433 \beta_{15} + 3139371 \beta_{14} - 2150335 \beta_{13} + 526994 \beta_{12} + 1480539 \beta_{11} + \cdots + 579594810 ) / 216 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 6197223 \beta_{15} + 14141502 \beta_{14} + 8016662 \beta_{13} - 183529 \beta_{12} + \cdots - 2350843623 ) / 216 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 44334675 \beta_{13} + 8238067 \beta_{12} + 55447631 \beta_{10} - 55186649 \beta_{8} + \cdots - 14690631509 ) / 216 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(55\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−1.64756 − 0.974628i
−1.64756 + 0.974628i
−1.81633 + 0.757118i
−1.81633 − 0.757118i
2.91300 + 0.609585i
2.91300 − 0.609585i
3.07345 + 2.03963i
3.07345 − 2.03963i
0.229644 + 3.68150i
0.229644 − 3.68150i
−1.98442 − 2.21794i
−1.98442 + 2.21794i
0.252484 + 1.95155i
0.252484 − 1.95155i
−0.0202748 − 1.91414i
−0.0202748 + 1.91414i
−3.81409 − 1.20528i 0 13.0946 + 9.19411i −13.3855 0 25.4062i −38.8625 − 50.8499i 0 51.0536 + 16.1333i
55.2 −3.81409 + 1.20528i 0 13.0946 − 9.19411i −13.3855 0 − 25.4062i −38.8625 + 50.8499i 0 51.0536 − 16.1333i
55.3 −3.36416 − 2.16389i 0 6.63513 + 14.5594i 43.0579 0 14.1139i 9.18328 − 63.3377i 0 −144.853 − 93.1726i
55.4 −3.36416 + 2.16389i 0 6.63513 − 14.5594i 43.0579 0 − 14.1139i 9.18328 + 63.3377i 0 −144.853 + 93.1726i
55.5 −1.59266 − 3.66925i 0 −10.9268 + 11.6878i −29.4580 0 32.9098i 60.2882 + 21.4787i 0 46.9167 + 108.089i
55.6 −1.59266 + 3.66925i 0 −10.9268 − 11.6878i −29.4580 0 − 32.9098i 60.2882 − 21.4787i 0 46.9167 − 108.089i
55.7 −1.35962 − 3.76184i 0 −12.3029 + 10.2293i 2.66014 0 − 88.8835i 55.2083 + 32.3735i 0 −3.61678 − 10.0070i
55.8 −1.35962 + 3.76184i 0 −12.3029 − 10.2293i 2.66014 0 88.8835i 55.2083 − 32.3735i 0 −3.61678 + 10.0070i
55.9 1.35962 − 3.76184i 0 −12.3029 − 10.2293i −2.66014 0 88.8835i −55.2083 + 32.3735i 0 −3.61678 + 10.0070i
55.10 1.35962 + 3.76184i 0 −12.3029 + 10.2293i −2.66014 0 − 88.8835i −55.2083 − 32.3735i 0 −3.61678 − 10.0070i
55.11 1.59266 − 3.66925i 0 −10.9268 − 11.6878i 29.4580 0 − 32.9098i −60.2882 + 21.4787i 0 46.9167 − 108.089i
55.12 1.59266 + 3.66925i 0 −10.9268 + 11.6878i 29.4580 0 32.9098i −60.2882 − 21.4787i 0 46.9167 + 108.089i
55.13 3.36416 − 2.16389i 0 6.63513 − 14.5594i −43.0579 0 − 14.1139i −9.18328 − 63.3377i 0 −144.853 + 93.1726i
55.14 3.36416 + 2.16389i 0 6.63513 + 14.5594i −43.0579 0 14.1139i −9.18328 + 63.3377i 0 −144.853 − 93.1726i
55.15 3.81409 − 1.20528i 0 13.0946 − 9.19411i 13.3855 0 − 25.4062i 38.8625 − 50.8499i 0 51.0536 − 16.1333i
55.16 3.81409 + 1.20528i 0 13.0946 + 9.19411i 13.3855 0 25.4062i 38.8625 + 50.8499i 0 51.0536 + 16.1333i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.5.d.a ✓ 16
3.b odd 2 1 inner 108.5.d.a ✓ 16
4.b odd 2 1 inner 108.5.d.a ✓ 16
12.b even 2 1 inner 108.5.d.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.d.a ✓ 16 1.a even 1 1 trivial
108.5.d.a ✓ 16 3.b odd 2 1 inner
108.5.d.a ✓ 16 4.b odd 2 1 inner
108.5.d.a ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2908T_{5}^{6} + 2117022T_{5}^{4} - 303093292T_{5}^{2} + 2039817193 \) acting on \(S_{5}^{\mathrm{new}}(108, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 2908 T^{6} + \cdots + 2039817193)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 1100177393193)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 52\!\cdots\!49)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 88 T^{3} + \cdots + 653937664)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 28\!\cdots\!12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 26\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 35\!\cdots\!08)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 23\!\cdots\!13)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 1796371513360)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 69\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 45\!\cdots\!37)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 37518169784320)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 36\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 30\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 675706016782681)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 74\!\cdots\!89)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 82\!\cdots\!92)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 27634716214849)^{4} \) Copy content Toggle raw display
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