↑

Properties

Label 2664.2.r.o
Level $2664$
Weight $2$
Character orbit 2664.r
Analytic conductor $21.272$
Analytic rank $0$
Dimension $16$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2664,2,Mod(433,2664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2664.433"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2664.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,-2,0,-4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.2721470985\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 24 x^{14} - 8 x^{13} + 344 x^{12} - 88 x^{11} + 2262 x^{10} + 1768 x^{9} + \cdots + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + (\beta_{9} - \beta_{3} + \beta_{2} + \beta_1) q^{7} + ( - \beta_{10} + \beta_{4} - \beta_{2}) q^{11} + (\beta_{7} + \beta_{6} + \beta_1) q^{13} + (\beta_{15} + \beta_{13} + \beta_{12} + \cdots - 1) q^{17}+ \cdots + ( - 2 \beta_{15} + \beta_{14} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{5} - 4 q^{7} - 4 q^{11} - 8 q^{13} - 4 q^{17} + 4 q^{19} - 24 q^{23} - 8 q^{25} + 12 q^{29} + 24 q^{31} - 10 q^{35} + 12 q^{37} + 4 q^{41} - 4 q^{43} - 4 q^{47} - 14 q^{49} + 16 q^{53} + 10 q^{55}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 24 x^{14} - 8 x^{13} + 344 x^{12} - 88 x^{11} + 2262 x^{10} + 1768 x^{9} + \cdots + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 39\!\cdots\!47 \nu^{15} + \cdots - 91\!\cdots\!40 ) / 60\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 84\!\cdots\!66 \nu^{15} + \cdots + 40\!\cdots\!80 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\!\cdots\!98 \nu^{15} + \cdots + 23\!\cdots\!52 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 60\!\cdots\!17 \nu^{15} + \cdots + 35\!\cdots\!58 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 66\!\cdots\!21 \nu^{15} + \cdots - 26\!\cdots\!16 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 86\!\cdots\!83 \nu^{15} + \cdots - 35\!\cdots\!28 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 46\!\cdots\!24 \nu^{15} + \cdots - 19\!\cdots\!60 ) / 60\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 71\!\cdots\!78 \nu^{15} + \cdots - 28\!\cdots\!20 ) / 60\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!18 \nu^{15} + \cdots + 76\!\cdots\!56 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23\!\cdots\!03 \nu^{15} + \cdots - 41\!\cdots\!82 ) / 12\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!80 \nu^{15} + \cdots + 47\!\cdots\!06 ) / 12\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 48\!\cdots\!44 \nu^{15} + \cdots + 29\!\cdots\!96 ) / 13\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26\!\cdots\!69 \nu^{15} + \cdots + 15\!\cdots\!86 ) / 60\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27\!\cdots\!96 \nu^{15} + \cdots + 49\!\cdots\!00 ) / 55\!\cdots\!26 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 76\!\cdots\!07 \nu^{15} + \cdots + 12\!\cdots\!18 ) / 12\!\cdots\!98 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{13} - \beta_{12} + \beta_{9} + \beta_{4} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 2\beta_{6} - \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13 \beta_{15} - 4 \beta_{14} + 13 \beta_{13} + 9 \beta_{12} - 6 \beta_{10} + 13 \beta_{7} + 9 \beta_{6} + \cdots - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 20 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} + 28 \beta_{12} - 5 \beta_{11} - 4 \beta_{10} - 20 \beta_{9} + \cdots - 71 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -217\beta_{9} - 80\beta_{8} - 191\beta_{7} - 149\beta_{6} - 4\beta_{5} - 102\beta_{3} - 49\beta_{2} - 293\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 382 \beta_{15} + 86 \beta_{14} - 151 \beta_{13} - 440 \beta_{12} + 89 \beta_{11} + 96 \beta_{10} + \cdots + 1043 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3783 \beta_{15} + 1380 \beta_{14} - 3007 \beta_{13} - 2749 \beta_{12} + 186 \beta_{11} + 1602 \beta_{10} + \cdots + 5525 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7125 \beta_{9} + 1876 \beta_{8} + 3428 \beta_{7} + 7360 \beta_{6} + 1398 \beta_{5} + 1954 \beta_{3} + \cdots + 16806 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 66715 \beta_{15} - 23492 \beta_{14} + 49251 \beta_{13} + 51183 \beta_{12} - 4864 \beta_{11} + \cdots - 104927 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 130918 \beta_{15} - 37215 \beta_{14} + 69921 \beta_{13} + 126824 \beta_{12} - 22249 \beta_{11} + \cdots - 283175 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1183263 \beta_{9} - 403872 \beta_{8} - 829149 \beta_{7} - 945259 \beta_{6} - 105932 \beta_{5} + \cdots - 1967267 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2384334 \beta_{15} + 707228 \beta_{14} - 1353249 \beta_{13} - 2220320 \beta_{12} + 365119 \beta_{11} + \cdots + 4888821 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 21068501 \beta_{15} + 7027020 \beta_{14} - 14241049 \beta_{13} - 17308331 \beta_{12} + \cdots + 36402579 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 43197793 \beta_{9} + 13141760 \beta_{8} + 25443234 \beta_{7} + 39235776 \beta_{6} + \cdots + 85626028 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 376187293 \beta_{15} - 123466332 \beta_{14} + 248130837 \beta_{13} + 314908197 \beta_{12} + \cdots - 666994605 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1333\) \(1999\) \(2369\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
−1.71191 − 2.96512i
2.11910 + 3.67039i
−0.213073 − 0.369053i
−0.327652 − 0.567510i
1.29882 + 2.24962i
−0.324940 − 0.562812i
0.980040 + 1.69748i
−0.820377 − 1.42093i
−1.71191 + 2.96512i
2.11910 − 3.67039i
−0.213073 + 0.369053i
−0.327652 + 0.567510i
1.29882 − 2.24962i
−0.324940 + 0.562812i
0.980040 − 1.69748i
−0.820377 + 1.42093i
0 0 0 −1.59849 + 2.76867i 0 0.442709 − 0.766794i 0 0 0
433.2 0 0 0 −1.41573 + 2.45212i 0 −0.887981 + 1.53803i 0 0 0
433.3 0 0 0 −1.38223 + 2.39409i 0 −1.64940 + 2.85685i 0 0 0
433.4 0 0 0 −0.622085 + 1.07748i 0 1.83661 − 3.18110i 0 0 0
433.5 0 0 0 0.654030 − 1.13281i 0 0.166501 − 0.288388i 0 0 0
433.6 0 0 0 0.717102 − 1.24206i 0 −1.31848 + 2.28368i 0 0 0
433.7 0 0 0 0.733536 − 1.27052i 0 −2.35431 + 4.07778i 0 0 0
433.8 0 0 0 1.91387 − 3.31492i 0 1.76436 − 3.05596i 0 0 0
1009.1 0 0 0 −1.59849 − 2.76867i 0 0.442709 + 0.766794i 0 0 0
1009.2 0 0 0 −1.41573 − 2.45212i 0 −0.887981 − 1.53803i 0 0 0
1009.3 0 0 0 −1.38223 − 2.39409i 0 −1.64940 − 2.85685i 0 0 0
1009.4 0 0 0 −0.622085 − 1.07748i 0 1.83661 + 3.18110i 0 0 0
1009.5 0 0 0 0.654030 + 1.13281i 0 0.166501 + 0.288388i 0 0 0
1009.6 0 0 0 0.717102 + 1.24206i 0 −1.31848 − 2.28368i 0 0 0
1009.7 0 0 0 0.733536 + 1.27052i 0 −2.35431 − 4.07778i 0 0 0
1009.8 0 0 0 1.91387 + 3.31492i 0 1.76436 + 3.05596i 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 433.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2664.2.r.o ✓ 16
3.b odd 2 1 2664.2.r.p yes 16
37.c even 3 1 inner 2664.2.r.o ✓ 16
111.i odd 6 1 2664.2.r.p yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2664.2.r.o ✓ 16 1.a even 1 1 trivial
2664.2.r.o ✓ 16 37.c even 3 1 inner
2664.2.r.p yes 16 3.b odd 2 1
2664.2.r.p yes 16 111.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2664, [\chi])\):

\( T_{5}^{16} + 2 T_{5}^{15} + 26 T_{5}^{14} + 40 T_{5}^{13} + 432 T_{5}^{12} + 562 T_{5}^{11} + \cdots + 107584 \) Copy content Toggle raw display
\( T_{11}^{8} + 2T_{11}^{7} - 35T_{11}^{6} - 102T_{11}^{5} + 197T_{11}^{4} + 938T_{11}^{3} + 846T_{11}^{2} - 72T_{11} - 216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 107584 \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 77284 \) Copy content Toggle raw display
$11$ \( (T^{8} + 2 T^{7} + \cdots - 216)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 8 T^{15} + \cdots + 2108304 \) Copy content Toggle raw display
$17$ \( T^{16} + 4 T^{15} + \cdots + 620944 \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} + \cdots + 22733824 \) Copy content Toggle raw display
$23$ \( (T^{8} + 12 T^{7} + \cdots + 192)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 6 T^{7} + \cdots + 218248)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 12 T^{7} + \cdots + 4192)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 3512479453921 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 3617781904 \) Copy content Toggle raw display
$43$ \( (T^{8} + 2 T^{7} + \cdots + 57934)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 2 T^{7} + \cdots - 1752)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 571401216 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 637461504 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 32480690176 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 530050978116 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 449396583448576 \) Copy content Toggle raw display
$73$ \( (T^{8} + 6 T^{7} + \cdots - 128)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 22211133156 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 170721017856 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 4160033427456 \) Copy content Toggle raw display
$97$ \( (T^{8} + 6 T^{7} + \cdots - 10566851)^{2} \) Copy content Toggle raw display
show more
show less