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Properties

Label 2916.2.e.c
Level $2916$
Weight $2$
Character orbit 2916.e
Analytic conductor $23.284$
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] = [2916,2,Mod(973,2916)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2916, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2916.973");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2916 = 2^{2} \cdot 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2916.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: \(23.2843772294\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
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: \( 3^{10} \)
Twist minimal: no (minimal twist has level 108)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{13} - \beta_{10} + \beta_{5} - 1) q^{5} - \beta_{15} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{13} - \beta_{10} + \beta_{5} - 1) q^{5} - \beta_{15} q^{7} + ( - \beta_{15} - \beta_{11} + \cdots + \beta_{2}) q^{11}+ \cdots + (\beta_{16} + 2 \beta_{15} + \cdots - \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 6 q^{5} - 6 q^{11} + 24 q^{17} - 3 q^{23} - 9 q^{25} - 24 q^{29} + 42 q^{35} - 33 q^{41} - 9 q^{47} - 9 q^{49} + 66 q^{53} - 30 q^{59} + 9 q^{61} - 39 q^{65} + 9 q^{67} + 24 q^{71} - 18 q^{73} - 39 q^{77} - 36 q^{83} + 96 q^{89} - 18 q^{91} - 12 q^{95}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 53 \nu^{17} - 3732 \nu^{16} + 14277 \nu^{15} - 47439 \nu^{14} + 99036 \nu^{13} - 207198 \nu^{12} + \cdots - 22362075 ) / 391473 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 173 \nu^{17} + 1712 \nu^{16} - 8607 \nu^{15} + 34479 \nu^{14} - 77748 \nu^{13} + 166824 \nu^{12} + \cdots + 26585172 ) / 391473 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2 \nu^{17} - 5 \nu^{16} + 18 \nu^{15} - 117 \nu^{14} + 249 \nu^{13} - 576 \nu^{12} + 1038 \nu^{11} + \cdots - 131220 ) / 2187 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2 \nu^{17} - 37 \nu^{16} + 135 \nu^{15} - 432 \nu^{14} + 903 \nu^{13} - 1908 \nu^{12} + 3768 \nu^{11} + \cdots - 216513 ) / 2187 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4 \nu^{17} - 25 \nu^{16} + 84 \nu^{15} - 225 \nu^{14} + 471 \nu^{13} - 981 \nu^{12} + 2001 \nu^{11} + \cdots - 69984 ) / 2187 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 821 \nu^{17} + 5258 \nu^{16} - 18252 \nu^{15} + 51078 \nu^{14} - 108843 \nu^{13} + \cdots + 21310128 ) / 391473 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5 \nu^{17} - 37 \nu^{16} + 126 \nu^{15} - 351 \nu^{14} + 732 \nu^{13} - 1530 \nu^{12} + 3102 \nu^{11} + \cdots - 113724 ) / 2187 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{17} - 12 \nu^{16} + 42 \nu^{15} - 111 \nu^{14} + 237 \nu^{13} - 486 \nu^{12} + 996 \nu^{11} + \cdots - 33534 ) / 729 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2 \nu^{17} - 22 \nu^{16} + 78 \nu^{15} - 242 \nu^{14} + 513 \nu^{13} - 1086 \nu^{12} + 2154 \nu^{11} + \cdots - 114453 ) / 729 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1334 \nu^{17} + 5391 \nu^{16} - 17655 \nu^{15} + 38283 \nu^{14} - 79776 \nu^{13} + \cdots + 2267919 ) / 391473 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1435 \nu^{17} - 4615 \nu^{16} + 15849 \nu^{15} - 33081 \nu^{14} + 72705 \nu^{13} - 144630 \nu^{12} + \cdots - 1548396 ) / 391473 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11 \nu^{17} + 49 \nu^{16} - 171 \nu^{15} + 369 \nu^{14} - 768 \nu^{13} + 1476 \nu^{12} + \cdots - 13122 ) / 2187 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2059 \nu^{17} - 8138 \nu^{16} + 26538 \nu^{15} - 55365 \nu^{14} + 114774 \nu^{13} - 227628 \nu^{12} + \cdots - 258066 ) / 391473 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2530 \nu^{17} + 12397 \nu^{16} - 43557 \nu^{15} + 107844 \nu^{14} - 227757 \nu^{13} + \cdots + 22565466 ) / 391473 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 853 \nu^{17} - 5021 \nu^{16} + 17437 \nu^{15} - 44510 \nu^{14} + 92646 \nu^{13} - 188694 \nu^{12} + \cdots - 10292751 ) / 130491 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2964 \nu^{17} + 11051 \nu^{16} - 38019 \nu^{15} + 80073 \nu^{14} - 169890 \nu^{13} + \cdots + 1113183 ) / 391473 \) 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 \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{17} - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{17} - 2 \beta_{16} - \beta_{15} + \beta_{13} + 2 \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{5} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{17} + 2 \beta_{16} + \beta_{15} + 3 \beta_{14} + 2 \beta_{13} + \beta_{12} + 4 \beta_{11} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{17} - 3 \beta_{16} - 2 \beta_{15} + \beta_{14} - 2 \beta_{13} - \beta_{12} - \beta_{11} + \cdots - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{17} - \beta_{16} - 2 \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 3 \beta_{9} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 21 \beta_{16} + 3 \beta_{13} - 12 \beta_{12} + 19 \beta_{11} + 3 \beta_{10} - 9 \beta_{9} - 6 \beta_{8} + \cdots - 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5 \beta_{16} + 4 \beta_{15} + \beta_{14} - 2 \beta_{13} + 2 \beta_{12} - 3 \beta_{11} + 16 \beta_{10} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{17} - \beta_{16} + 19 \beta_{15} + 15 \beta_{14} + 11 \beta_{13} + 3 \beta_{12} + 10 \beta_{11} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 11 \beta_{17} - 14 \beta_{16} - \beta_{15} + 24 \beta_{14} + 19 \beta_{13} - 10 \beta_{12} + 18 \beta_{11} + \cdots - 22 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 25 \beta_{17} - 27 \beta_{16} - 7 \beta_{15} - 16 \beta_{14} + 35 \beta_{13} + 37 \beta_{12} + 25 \beta_{11} + \cdots + 77 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 54 \beta_{17} + 48 \beta_{16} + 3 \beta_{15} - 27 \beta_{14} + 33 \beta_{13} + 33 \beta_{12} + \cdots - 117 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 126 \beta_{17} - 162 \beta_{16} + 3 \beta_{15} - 75 \beta_{14} - 147 \beta_{13} - 51 \beta_{12} + \cdots - 447 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 186 \beta_{17} + 150 \beta_{16} + 210 \beta_{15} + 111 \beta_{14} - 69 \beta_{13} - 129 \beta_{12} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 537 \beta_{17} + 552 \beta_{16} + 114 \beta_{15} + 468 \beta_{14} + 30 \beta_{13} - 198 \beta_{12} + \cdots + 237 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 228 \beta_{17} - 354 \beta_{16} + 309 \beta_{15} - 387 \beta_{14} + 186 \beta_{13} + 642 \beta_{12} + \cdots + 1767 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 609 \beta_{17} + 171 \beta_{16} + 984 \beta_{15} - 645 \beta_{14} + 489 \beta_{13} + 1059 \beta_{12} + \cdots - 1266 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 540 \beta_{17} - 2115 \beta_{16} - 342 \beta_{15} - 630 \beta_{14} - 1620 \beta_{13} + 1215 \beta_{12} + \cdots - 6066 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times\).

\(n\) \(1459\) \(2189\)
\(\chi(n)\) \(1\) \(-1 - \beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
973.1
0.472963 − 1.66622i
1.16555 + 1.28120i
0.381933 + 1.68942i
1.20201 − 1.24706i
−0.219955 + 1.71803i
0.960398 − 1.44140i
−1.34999 − 1.08514i
−1.29960 + 1.14501i
1.68668 − 0.393823i
0.472963 + 1.66622i
1.16555 − 1.28120i
0.381933 − 1.68942i
1.20201 + 1.24706i
−0.219955 − 1.71803i
0.960398 + 1.44140i
−1.34999 + 1.08514i
−1.29960 − 1.14501i
1.68668 + 0.393823i
0 0 0 −2.10020 + 3.63766i 0 −1.75732 − 3.04377i 0 0 0
973.2 0 0 0 −1.68151 + 2.91246i 0 0.961187 + 1.66483i 0 0 0
973.3 0 0 0 −1.47772 + 2.55948i 0 1.33493 + 2.31217i 0 0 0
973.4 0 0 0 −0.959195 + 1.66137i 0 −2.05453 − 3.55855i 0 0 0
973.5 0 0 0 0.0506977 − 0.0878110i 0 −0.475194 − 0.823060i 0 0 0
973.6 0 0 0 0.296957 − 0.514344i 0 −1.42798 − 2.47334i 0 0 0
973.7 0 0 0 0.497220 − 0.861211i 0 0.719594 + 1.24637i 0 0 0
973.8 0 0 0 1.15060 − 1.99289i 0 0.466798 + 0.808518i 0 0 0
973.9 0 0 0 1.22316 − 2.11857i 0 2.23252 + 3.86683i 0 0 0
1945.1 0 0 0 −2.10020 − 3.63766i 0 −1.75732 + 3.04377i 0 0 0
1945.2 0 0 0 −1.68151 − 2.91246i 0 0.961187 − 1.66483i 0 0 0
1945.3 0 0 0 −1.47772 − 2.55948i 0 1.33493 − 2.31217i 0 0 0
1945.4 0 0 0 −0.959195 − 1.66137i 0 −2.05453 + 3.55855i 0 0 0
1945.5 0 0 0 0.0506977 + 0.0878110i 0 −0.475194 + 0.823060i 0 0 0
1945.6 0 0 0 0.296957 + 0.514344i 0 −1.42798 + 2.47334i 0 0 0
1945.7 0 0 0 0.497220 + 0.861211i 0 0.719594 − 1.24637i 0 0 0
1945.8 0 0 0 1.15060 + 1.99289i 0 0.466798 − 0.808518i 0 0 0
1945.9 0 0 0 1.22316 + 2.11857i 0 2.23252 − 3.86683i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 973.9
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 2916.2.e.c 18
3.b odd 2 1 2916.2.e.d 18
9.c even 3 1 2916.2.a.d 9
9.c even 3 1 inner 2916.2.e.c 18
9.d odd 6 1 2916.2.a.c 9
9.d odd 6 1 2916.2.e.d 18
27.e even 9 2 108.2.i.a ✓ 18
27.e even 9 2 972.2.i.a 18
27.e even 9 2 972.2.i.c 18
27.f odd 18 2 324.2.i.a 18
27.f odd 18 2 972.2.i.b 18
27.f odd 18 2 972.2.i.d 18
108.j odd 18 2 432.2.u.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.i.a ✓ 18 27.e even 9 2
324.2.i.a 18 27.f odd 18 2
432.2.u.d 18 108.j odd 18 2
972.2.i.a 18 27.e even 9 2
972.2.i.b 18 27.f odd 18 2
972.2.i.c 18 27.e even 9 2
972.2.i.d 18 27.f odd 18 2
2916.2.a.c 9 9.d odd 6 1
2916.2.a.d 9 9.c even 3 1
2916.2.e.c 18 1.a even 1 1 trivial
2916.2.e.c 18 9.c even 3 1 inner
2916.2.e.d 18 3.b odd 2 1
2916.2.e.d 18 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 6 T_{5}^{17} + 45 T_{5}^{16} + 120 T_{5}^{15} + 585 T_{5}^{14} + 972 T_{5}^{13} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(2916, [\chi])\). 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
$5$ \( T^{18} + 6 T^{17} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{18} + 36 T^{16} + \cdots + 1456849 \) Copy content Toggle raw display
$11$ \( T^{18} + 6 T^{17} + \cdots + 23357889 \) Copy content Toggle raw display
$13$ \( T^{18} + 63 T^{16} + \cdots + 4068289 \) Copy content Toggle raw display
$17$ \( (T^{9} - 12 T^{8} + \cdots - 72441)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} - 90 T^{7} + \cdots - 7001)^{2} \) 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
$37$ \( (T^{9} - 171 T^{7} + \cdots + 97057)^{2} \) 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
$67$ \( T^{18} - 9 T^{17} + \cdots + 3568321 \) Copy content Toggle raw display
$71$ \( (T^{9} - 12 T^{8} + \cdots + 11637)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} + 9 T^{8} + \cdots - 115127)^{2} \) 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
$89$ \( (T^{9} - 48 T^{8} + \cdots + 8774217)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 736742449 \) Copy content Toggle raw display
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