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Properties

Label 72.8.i.a
Level $72$
Weight $8$
Character orbit 72.i
Analytic conductor $22.492$
Analytic rank $0$
Dimension $20$
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] = [72,8,Mod(25,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.25");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.i (of order \(3\), degree \(2\), 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: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} + 332929 x^{18} - 2996076 x^{17} + 44578211685 x^{16} - 356557783716 x^{15} + \cdots + 79\!\cdots\!67 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{30} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + 3) q^{3} + ( - \beta_{5} - \beta_{4} + \cdots + 13 \beta_{2}) q^{5}+ \cdots + (\beta_{11} - \beta_{10} + \beta_{9} + \cdots + 136) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + 3) q^{3} + ( - \beta_{5} - \beta_{4} + \cdots + 13 \beta_{2}) q^{5}+ \cdots + (240 \beta_{19} + 81 \beta_{18} + \cdots - 820773) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 57 q^{3} - 125 q^{5} + 1245 q^{7} + 1335 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 57 q^{3} - 125 q^{5} + 1245 q^{7} + 1335 q^{9} + 6106 q^{11} + 4937 q^{13} - 6489 q^{15} + 48722 q^{17} - 26882 q^{19} - 20961 q^{21} + 19387 q^{23} - 218957 q^{25} - 440208 q^{27} - 46791 q^{29} - 185039 q^{31} + 202926 q^{33} + 83094 q^{35} + 108420 q^{37} + 51147 q^{39} - 638112 q^{41} + 892628 q^{43} + 85839 q^{45} - 230883 q^{47} - 1034741 q^{49} + 529941 q^{51} - 2872940 q^{53} + 1089998 q^{55} + 622545 q^{57} + 2172454 q^{59} - 1878325 q^{61} - 3873537 q^{63} + 1239133 q^{65} + 531496 q^{67} - 2501703 q^{69} + 3723056 q^{71} - 1804522 q^{73} + 223545 q^{75} + 6276543 q^{77} + 3607847 q^{79} + 309843 q^{81} + 10794491 q^{83} + 3597658 q^{85} + 13198923 q^{87} + 32214888 q^{89} - 16117530 q^{91} - 27056079 q^{93} - 756868 q^{95} + 17951260 q^{97} - 497775 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} + 332929 x^{18} - 2996076 x^{17} + 44578211685 x^{16} - 356557783716 x^{15} + \cdots + 79\!\cdots\!67 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12\!\cdots\!47 \nu^{18} + \cdots - 67\!\cdots\!71 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 74\!\cdots\!92 \nu^{19} + \cdots - 17\!\cdots\!53 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!82 \nu^{19} + \cdots + 23\!\cdots\!53 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 35\!\cdots\!77 \nu^{19} + \cdots - 66\!\cdots\!01 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!53 \nu^{19} + \cdots + 38\!\cdots\!08 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 45\!\cdots\!33 \nu^{19} + \cdots + 12\!\cdots\!57 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!11 \nu^{19} + \cdots + 33\!\cdots\!89 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 78\!\cdots\!88 \nu^{19} + \cdots + 13\!\cdots\!97 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 32\!\cdots\!80 \nu^{19} + \cdots + 12\!\cdots\!93 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 69\!\cdots\!48 \nu^{19} + \cdots + 38\!\cdots\!35 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 71\!\cdots\!53 \nu^{19} + \cdots + 23\!\cdots\!93 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!33 \nu^{19} + \cdots - 24\!\cdots\!42 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 26\!\cdots\!47 \nu^{19} + \cdots + 78\!\cdots\!73 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 61\!\cdots\!99 \nu^{19} + \cdots - 14\!\cdots\!26 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 34\!\cdots\!39 \nu^{19} + \cdots + 20\!\cdots\!79 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 47\!\cdots\!48 \nu^{19} + \cdots + 14\!\cdots\!07 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 24\!\cdots\!29 \nu^{19} + \cdots - 92\!\cdots\!91 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 11\!\cdots\!42 \nu^{19} + \cdots - 47\!\cdots\!93 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 57\!\cdots\!73 \nu^{19} + \cdots - 15\!\cdots\!42 ) / 22\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} - 2\beta_{3} + \beta_{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{18} + 6 \beta_{15} - 4 \beta_{14} + 8 \beta_{13} + 5 \beta_{12} - 9 \beta_{11} + 18 \beta_{10} + \cdots - 99837 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 468 \beta_{19} - 1273 \beta_{18} - 179 \beta_{17} - 1889 \beta_{16} + 817 \beta_{15} + 6478 \beta_{14} + \cdots + 554155 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 936 \beta_{19} - 764318 \beta_{18} - 358 \beta_{17} - 3778 \beta_{16} - 1438310 \beta_{15} + \cdots + 19494169764 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 31298978 \beta_{19} + 179386409 \beta_{18} + 17067271 \beta_{17} + 286895225 \beta_{16} + \cdots - 448422198287 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 93899274 \beta_{19} + 61142745100 \beta_{18} + 51202708 \beta_{17} + 860695120 \beta_{16} + \cdots - 15\!\cdots\!05 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1291884703206 \beta_{19} - 21248083420090 \beta_{18} - 2740152393022 \beta_{17} + \cdots + 76\!\cdots\!94 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5167100614028 \beta_{19} + \cdots + 12\!\cdots\!87 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 58\!\cdots\!82 \beta_{19} + \cdots - 98\!\cdots\!27 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 29\!\cdots\!22 \beta_{19} + \cdots - 11\!\cdots\!60 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 97\!\cdots\!06 \beta_{19} + \cdots + 11\!\cdots\!95 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 58\!\cdots\!68 \beta_{19} + \cdots + 10\!\cdots\!33 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 12\!\cdots\!68 \beta_{19} + \cdots - 12\!\cdots\!50 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 89\!\cdots\!36 \beta_{19} + \cdots - 99\!\cdots\!83 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 15\!\cdots\!64 \beta_{19} + \cdots + 12\!\cdots\!23 ) / 9 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 12\!\cdots\!20 \beta_{19} + \cdots + 95\!\cdots\!04 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 16\!\cdots\!22 \beta_{19} + \cdots - 13\!\cdots\!51 ) / 9 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 15\!\cdots\!54 \beta_{19} + \cdots - 92\!\cdots\!13 ) / 9 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 17\!\cdots\!42 \beta_{19} + \cdots + 13\!\cdots\!46 ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 + 315.199i
0.500000 − 186.740i
0.500000 − 46.3348i
0.500000 + 10.7476i
0.500000 + 73.9766i
0.500000 + 187.726i
0.500000 − 271.860i
0.500000 − 2.08484i
0.500000 − 238.457i
0.500000 + 157.828i
0.500000 − 315.199i
0.500000 + 186.740i
0.500000 + 46.3348i
0.500000 − 10.7476i
0.500000 − 73.9766i
0.500000 − 187.726i
0.500000 + 271.860i
0.500000 + 2.08484i
0.500000 + 238.457i
0.500000 − 157.828i
0 −46.7467 − 1.32052i 0 266.720 + 461.973i 0 −66.5594 + 115.284i 0 2183.51 + 123.460i 0
25.2 0 −46.6129 + 3.77327i 0 −167.971 − 290.935i 0 410.729 − 711.403i 0 2158.52 − 351.766i 0
25.3 0 −23.6208 − 40.3616i 0 −46.3771 − 80.3275i 0 −227.962 + 394.841i 0 −1071.12 + 1906.75i 0
25.4 0 −22.7249 + 40.8727i 0 3.05772 + 5.29612i 0 −422.475 + 731.748i 0 −1154.16 − 1857.66i 0
25.5 0 7.11672 + 46.2207i 0 57.8156 + 100.140i 0 783.840 − 1357.65i 0 −2085.70 + 657.879i 0
25.6 0 12.8026 − 44.9788i 0 156.325 + 270.763i 0 626.107 − 1084.45i 0 −1859.19 − 1151.69i 0
25.7 0 29.8220 + 36.0229i 0 −241.687 − 418.615i 0 −351.089 + 608.105i 0 −408.300 + 2148.55i 0
25.8 0 33.7485 − 32.3735i 0 −8.05553 − 13.9526i 0 −629.010 + 1089.48i 0 90.9162 − 2185.11i 0
25.9 0 40.7537 − 22.9377i 0 −212.760 − 368.511i 0 612.894 − 1061.56i 0 1134.72 − 1869.59i 0
25.10 0 43.9619 + 15.9485i 0 130.433 + 225.916i 0 −113.976 + 197.412i 0 1678.29 + 1402.25i 0
49.1 0 −46.7467 + 1.32052i 0 266.720 − 461.973i 0 −66.5594 − 115.284i 0 2183.51 − 123.460i 0
49.2 0 −46.6129 − 3.77327i 0 −167.971 + 290.935i 0 410.729 + 711.403i 0 2158.52 + 351.766i 0
49.3 0 −23.6208 + 40.3616i 0 −46.3771 + 80.3275i 0 −227.962 − 394.841i 0 −1071.12 − 1906.75i 0
49.4 0 −22.7249 − 40.8727i 0 3.05772 − 5.29612i 0 −422.475 − 731.748i 0 −1154.16 + 1857.66i 0
49.5 0 7.11672 − 46.2207i 0 57.8156 − 100.140i 0 783.840 + 1357.65i 0 −2085.70 − 657.879i 0
49.6 0 12.8026 + 44.9788i 0 156.325 − 270.763i 0 626.107 + 1084.45i 0 −1859.19 + 1151.69i 0
49.7 0 29.8220 − 36.0229i 0 −241.687 + 418.615i 0 −351.089 − 608.105i 0 −408.300 − 2148.55i 0
49.8 0 33.7485 + 32.3735i 0 −8.05553 + 13.9526i 0 −629.010 − 1089.48i 0 90.9162 + 2185.11i 0
49.9 0 40.7537 + 22.9377i 0 −212.760 + 368.511i 0 612.894 + 1061.56i 0 1134.72 + 1869.59i 0
49.10 0 43.9619 − 15.9485i 0 130.433 − 225.916i 0 −113.976 − 197.412i 0 1678.29 − 1402.25i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.10
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 72.8.i.a ✓ 20
3.b odd 2 1 216.8.i.a 20
4.b odd 2 1 144.8.i.e 20
9.c even 3 1 inner 72.8.i.a ✓ 20
9.d odd 6 1 216.8.i.a 20
12.b even 2 1 432.8.i.e 20
36.f odd 6 1 144.8.i.e 20
36.h even 6 1 432.8.i.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.i.a ✓ 20 1.a even 1 1 trivial
72.8.i.a ✓ 20 9.c even 3 1 inner
144.8.i.e 20 4.b odd 2 1
144.8.i.e 20 36.f odd 6 1
216.8.i.a 20 3.b odd 2 1
216.8.i.a 20 9.d odd 6 1
432.8.i.e 20 12.b even 2 1
432.8.i.e 20 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 125 T_{5}^{19} + 507916 T_{5}^{18} + 44805399 T_{5}^{17} + 175596788865 T_{5}^{16} + \cdots + 10\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 25\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 76\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 99\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 49\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 65\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 15\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 18\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 57\!\cdots\!41 \) Copy content Toggle raw display
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