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Properties

Label 972.2.i.b
Level $972$
Weight $2$
Character orbit 972.i
Analytic conductor $7.761$
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] = [972,2,Mod(109,972)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(972, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("972.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 972 = 2^{2} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 972.i (of order \(9\), degree \(6\), 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: \(7.76145907647\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 108)
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.

\(f(q)\) \(=\) \( q + (\beta_{12} - \beta_{7}) q^{5} + (\beta_{17} - \beta_{16} + \beta_{15} + \cdots - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} - \beta_{7}) q^{5} + (\beta_{17} - \beta_{16} + \beta_{15} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{17} + \beta_{16} - 2 \beta_{15} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{5} + 15 q^{11} + 12 q^{17} - 33 q^{23} + 9 q^{25} - 21 q^{29} + 9 q^{31} + 21 q^{35} + 33 q^{41} + 18 q^{43} - 9 q^{47} + 36 q^{49} - 66 q^{53} + 12 q^{59} + 36 q^{61} + 66 q^{65} + 27 q^{67} + 12 q^{71} + 9 q^{73} - 33 q^{77} + 18 q^{79} - 81 q^{83} + 18 q^{85} + 48 q^{89} + 9 q^{91} + 84 q^{95}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 256 \nu^{17} - 252 \nu^{16} + 819 \nu^{15} - 2946 \nu^{14} + 8433 \nu^{13} - 18684 \nu^{12} + \cdots - 3726648 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 463 \nu^{17} + 1365 \nu^{16} - 5913 \nu^{15} + 26250 \nu^{14} - 52794 \nu^{13} + 112482 \nu^{12} + \cdots + 17445699 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1259 \nu^{17} - 3681 \nu^{16} + 13977 \nu^{15} - 36627 \nu^{14} + 86877 \nu^{13} - 179712 \nu^{12} + \cdots - 16015401 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1330 \nu^{17} - 5085 \nu^{16} + 18540 \nu^{15} - 38388 \nu^{14} + 80928 \nu^{13} - 154008 \nu^{12} + \cdots + 4494285 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 155 \nu^{17} - 404 \nu^{16} + 1671 \nu^{15} - 7425 \nu^{14} + 14568 \nu^{13} - 30951 \nu^{12} + \cdots - 4166235 ) / 130491 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1495 \nu^{17} - 1329 \nu^{16} + 7407 \nu^{15} + 3399 \nu^{14} - 3645 \nu^{13} + 36558 \nu^{12} + \cdots + 26204634 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1537 \nu^{17} - 10986 \nu^{16} + 39195 \nu^{15} - 107463 \nu^{14} + 225909 \nu^{13} + \cdots - 36577575 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2093 \nu^{17} + 3042 \nu^{16} - 4248 \nu^{15} - 21513 \nu^{14} + 46989 \nu^{13} + \cdots - 38565558 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4 \nu^{17} - 25 \nu^{16} + 84 \nu^{15} - 225 \nu^{14} + 471 \nu^{13} - 981 \nu^{12} + 2001 \nu^{11} + \cdots - 69984 ) / 2187 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2600 \nu^{17} - 14172 \nu^{16} + 50355 \nu^{15} - 126681 \nu^{14} + 272574 \nu^{13} + \cdots - 32857488 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2 \nu^{17} + 12 \nu^{16} - 42 \nu^{15} + 111 \nu^{14} - 237 \nu^{13} + 486 \nu^{12} - 996 \nu^{11} + \cdots + 33534 ) / 729 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3458 \nu^{17} + 25572 \nu^{16} - 91701 \nu^{15} + 251565 \nu^{14} - 529713 \nu^{13} + \cdots + 78745122 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1334 \nu^{17} - 5391 \nu^{16} + 17655 \nu^{15} - 38283 \nu^{14} + 79776 \nu^{13} - 161046 \nu^{12} + \cdots - 2659392 ) / 391473 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 4597 \nu^{17} - 31350 \nu^{16} + 111285 \nu^{15} - 296175 \nu^{14} + 622152 \nu^{13} + \cdots - 78627024 ) / 1174419 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2059 \nu^{17} - 8138 \nu^{16} + 26538 \nu^{15} - 55365 \nu^{14} + 114774 \nu^{13} - 227628 \nu^{12} + \cdots - 258066 ) / 391473 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2492 \nu^{17} + 10743 \nu^{16} - 37734 \nu^{15} + 90852 \nu^{14} - 197352 \nu^{13} + \cdots + 20177262 ) / 391473 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 1499 \nu^{17} + 7184 \nu^{16} - 25138 \nu^{15} + 60936 \nu^{14} - 129453 \nu^{13} + \cdots + 11744919 ) / 130491 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{17} + \beta_{15} + \beta_{13} - \beta_{12} + 2 \beta_{10} + \beta_{9} + \beta_{6} + \cdots + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{16} + \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} - 2\beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} - \beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{16} - 3 \beta_{15} - 3 \beta_{12} + \beta_{11} - 6 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{17} - 3 \beta_{16} + 4 \beta_{15} - 6 \beta_{14} - 5 \beta_{13} - 5 \beta_{12} - \beta_{11} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{17} - 3 \beta_{16} + 11 \beta_{15} + 9 \beta_{14} + 2 \beta_{13} + 19 \beta_{12} - 15 \beta_{11} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 33 \beta_{17} - 26 \beta_{16} + 21 \beta_{15} + 41 \beta_{14} + 13 \beta_{13} + 52 \beta_{12} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18 \beta_{17} - 3 \beta_{16} + 9 \beta_{15} + 30 \beta_{14} + 21 \beta_{13} + 9 \beta_{12} + 18 \beta_{11} + \cdots - 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 78 \beta_{17} + 87 \beta_{16} - 27 \beta_{15} - 21 \beta_{14} - 84 \beta_{13} - 78 \beta_{12} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 36 \beta_{17} + 45 \beta_{16} + 27 \beta_{15} - 57 \beta_{14} - 33 \beta_{13} - 51 \beta_{12} + \cdots - 96 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 204 \beta_{17} - 81 \beta_{16} - 30 \beta_{15} + 126 \beta_{14} + 240 \beta_{13} + 123 \beta_{12} + \cdots - 279 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 102 \beta_{17} - 333 \beta_{16} - 51 \beta_{15} + 126 \beta_{14} - 303 \beta_{13} + 177 \beta_{12} + \cdots - 288 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 261 \beta_{17} - 219 \beta_{16} + 108 \beta_{15} - 240 \beta_{14} - 345 \beta_{13} + 177 \beta_{12} + \cdots - 144 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 387 \beta_{17} + 126 \beta_{16} + 297 \beta_{15} + 711 \beta_{14} + 540 \beta_{13} + 720 \beta_{12} + \cdots + 2385 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1980 \beta_{17} - 459 \beta_{16} + 1062 \beta_{15} + 1611 \beta_{14} + 1818 \beta_{13} + 891 \beta_{12} + \cdots - 630 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1377 \beta_{17} + 2556 \beta_{16} - 621 \beta_{15} - 63 \beta_{14} + 2142 \beta_{13} - 180 \beta_{12} + \cdots - 3411 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/972\mathbb{Z}\right)^\times\).

\(n\) \(245\) \(487\)
\(\chi(n)\) \(\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.29960 − 1.14501i
0.960398 + 1.44140i
1.16555 − 1.28120i
0.472963 − 1.66622i
−0.219955 + 1.71803i
1.68668 − 0.393823i
0.381933 − 1.68942i
1.20201 + 1.24706i
−1.34999 + 1.08514i
0.381933 + 1.68942i
1.20201 − 1.24706i
−1.34999 − 1.08514i
0.472963 + 1.66622i
−0.219955 − 1.71803i
1.68668 + 0.393823i
−1.29960 + 1.14501i
0.960398 − 1.44140i
1.16555 + 1.28120i
0 0 0 −2.16241 + 0.787054i 0 −0.162117 − 0.919412i 0 0 0
109.2 0 0 0 −0.558096 + 0.203130i 0 0.495934 + 2.81258i 0 0 0
109.3 0 0 0 3.16020 − 1.15022i 0 −0.333817 − 1.89317i 0 0 0
217.1 0 0 0 −3.21770 + 2.69997i 0 −3.30269 + 1.20208i 0 0 0
217.2 0 0 0 0.0776734 − 0.0651757i 0 −0.893073 + 0.325052i 0 0 0
217.3 0 0 0 1.87398 − 1.57246i 0 4.19576 − 1.52713i 0 0 0
433.1 0 0 0 −0.513206 + 2.91054i 0 −2.04524 + 1.71616i 0 0 0
433.2 0 0 0 −0.333125 + 1.88924i 0 3.14772 − 2.64125i 0 0 0
433.3 0 0 0 0.172683 − 0.979333i 0 −1.10248 + 0.925093i 0 0 0
541.1 0 0 0 −0.513206 − 2.91054i 0 −2.04524 − 1.71616i 0 0 0
541.2 0 0 0 −0.333125 − 1.88924i 0 3.14772 + 2.64125i 0 0 0
541.3 0 0 0 0.172683 + 0.979333i 0 −1.10248 − 0.925093i 0 0 0
757.1 0 0 0 −3.21770 − 2.69997i 0 −3.30269 − 1.20208i 0 0 0
757.2 0 0 0 0.0776734 + 0.0651757i 0 −0.893073 − 0.325052i 0 0 0
757.3 0 0 0 1.87398 + 1.57246i 0 4.19576 + 1.52713i 0 0 0
865.1 0 0 0 −2.16241 − 0.787054i 0 −0.162117 + 0.919412i 0 0 0
865.2 0 0 0 −0.558096 − 0.203130i 0 0.495934 − 2.81258i 0 0 0
865.3 0 0 0 3.16020 + 1.15022i 0 −0.333817 + 1.89317i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.3
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 972.2.i.b 18
3.b odd 2 1 972.2.i.c 18
9.c even 3 1 324.2.i.a 18
9.c even 3 1 972.2.i.d 18
9.d odd 6 1 108.2.i.a ✓ 18
9.d odd 6 1 972.2.i.a 18
27.e even 9 1 324.2.i.a 18
27.e even 9 1 inner 972.2.i.b 18
27.e even 9 1 972.2.i.d 18
27.e even 9 1 2916.2.a.c 9
27.e even 9 2 2916.2.e.d 18
27.f odd 18 1 108.2.i.a ✓ 18
27.f odd 18 1 972.2.i.a 18
27.f odd 18 1 972.2.i.c 18
27.f odd 18 1 2916.2.a.d 9
27.f odd 18 2 2916.2.e.c 18
36.h even 6 1 432.2.u.d 18
108.l even 18 1 432.2.u.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.i.a ✓ 18 9.d odd 6 1
108.2.i.a ✓ 18 27.f odd 18 1
324.2.i.a 18 9.c even 3 1
324.2.i.a 18 27.e even 9 1
432.2.u.d 18 36.h even 6 1
432.2.u.d 18 108.l even 18 1
972.2.i.a 18 9.d odd 6 1
972.2.i.a 18 27.f odd 18 1
972.2.i.b 18 1.a even 1 1 trivial
972.2.i.b 18 27.e even 9 1 inner
972.2.i.c 18 3.b odd 2 1
972.2.i.c 18 27.f odd 18 1
972.2.i.d 18 9.c even 3 1
972.2.i.d 18 27.e even 9 1
2916.2.a.c 9 27.e even 9 1
2916.2.a.d 9 27.f odd 18 1
2916.2.e.c 18 27.f odd 18 2
2916.2.e.d 18 27.e even 9 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 3 T_{5}^{17} - 48 T_{5}^{15} - 9 T_{5}^{14} + 1395 T_{5}^{12} + 2592 T_{5}^{11} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(972, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + 3 T^{17} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{18} - 18 T^{16} + \cdots + 1456849 \) Copy content Toggle raw display
$11$ \( T^{18} - 15 T^{17} + \cdots + 23357889 \) Copy content Toggle raw display
$13$ \( T^{18} + 9 T^{16} + \cdots + 4068289 \) 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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