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Properties

Label 3264.2.f.j
Level $3264$
Weight $2$
Character orbit 3264.f
Analytic conductor $26.063$
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] = [3264,2,Mod(1633,3264)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3264, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3264.1633");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3264.f (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: \(26.0631712197\)
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: \( x^{16} + 11x^{14} + 85x^{12} + 326x^{10} + 910x^{8} + 1238x^{6} + 1189x^{4} + 35x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{22} \)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{3} - \beta_{9} q^{5} + \beta_{6} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{3} - \beta_{9} q^{5} + \beta_{6} q^{7} - q^{9} + \beta_{8} q^{11} + (\beta_{9} - \beta_{3}) q^{13} - \beta_{14} q^{15} + q^{17} + (\beta_{12} - 2 \beta_{10} - \beta_{8}) q^{19} + \beta_{3} q^{21} + ( - \beta_{14} + \beta_{13}) q^{23} + (\beta_{4} - 1) q^{25} - \beta_{10} q^{27} + (2 \beta_{9} - \beta_{7} + \cdots - \beta_{3}) q^{29}+ \cdots - \beta_{8} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} + 16 q^{17} - 24 q^{25} + 8 q^{33} - 24 q^{41} + 64 q^{49} + 24 q^{57} + 72 q^{65} + 16 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 11x^{14} + 85x^{12} + 326x^{10} + 910x^{8} + 1238x^{6} + 1189x^{4} + 35x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 525 \nu^{14} - 5582 \nu^{12} - 39025 \nu^{10} - 127925 \nu^{8} - 243011 \nu^{6} - 190225 \nu^{4} + \cdots - 435057 ) / 129954 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1302 \nu^{14} - 12977 \nu^{12} - 96782 \nu^{10} - 317254 \nu^{8} - 777672 \nu^{6} + \cdots + 481373 ) / 259908 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8879 \nu^{14} - 102986 \nu^{12} - 804399 \nu^{10} - 3232027 \nu^{8} - 9000043 \nu^{6} + \cdots - 165329 ) / 779724 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1230 \nu^{14} + 12459 \nu^{12} + 91430 \nu^{10} + 299710 \nu^{8} + 706720 \nu^{6} + 445670 \nu^{4} + \cdots - 813075 ) / 86636 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18242 \nu^{14} - 204641 \nu^{12} - 1587018 \nu^{10} - 6206566 \nu^{8} - 17302642 \nu^{6} + \cdots - 327149 ) / 389862 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15511 \nu^{15} - 169952 \nu^{13} - 1311817 \nu^{11} - 5006857 \nu^{9} - 13951997 \nu^{7} + \cdots - 15933 \nu ) / 259908 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15511 \nu^{14} - 169952 \nu^{12} - 1311817 \nu^{10} - 5006857 \nu^{8} - 13951997 \nu^{6} + \cdots - 275841 ) / 129954 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 33431 \nu^{15} - 366260 \nu^{13} - 2824362 \nu^{11} - 10773979 \nu^{9} - 29967484 \nu^{7} + \cdots - 2273336 \nu ) / 389862 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 105865 \nu^{14} + 1162933 \nu^{12} + 8981127 \nu^{10} + 34387175 \nu^{8} + 95872253 \nu^{6} + \cdots + 1876264 ) / 779724 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 613 \nu^{15} - 6772 \nu^{13} - 52428 \nu^{11} - 202292 \nu^{9} - 567104 \nu^{7} - 783156 \nu^{5} + \cdots - 44020 \nu ) / 6444 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15511 \nu^{15} - 169952 \nu^{13} - 1311817 \nu^{11} - 5006857 \nu^{9} - 13951997 \nu^{7} + \cdots - 15933 \nu ) / 64977 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 41383 \nu^{15} - 457162 \nu^{13} - 3538195 \nu^{11} - 13650173 \nu^{9} - 38248951 \nu^{7} + \cdots - 2970885 \nu ) / 129954 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 446 \nu^{15} + 4896 \nu^{13} + 37791 \nu^{11} + 144451 \nu^{9} + 401931 \nu^{7} + 540855 \nu^{5} + \cdots + 459 \nu ) / 726 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 60095 \nu^{15} - 659168 \nu^{13} - 5087953 \nu^{11} - 19437007 \nu^{9} - 54113573 \nu^{7} + \cdots - 61797 \nu ) / 86636 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 630745 \nu^{15} + 6953440 \nu^{13} + 53780160 \nu^{11} + 206922134 \nu^{9} + 578999408 \nu^{7} + \cdots + 44684632 \nu ) / 779724 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{12} + 3\beta_{10} + \beta_{8} + 2\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{9} - 4\beta_{7} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} - \beta _1 - 11 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 4\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{9} + 14\beta_{7} + 14\beta_{5} - 5\beta_{4} - 14\beta_{3} - 9\beta_{2} - 5\beta _1 - 47 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -21\beta_{15} + 4\beta_{14} + 2\beta_{13} - \beta_{12} - 16\beta_{11} - 131\beta_{10} - 49\beta_{8} + 38\beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{4} + 15\beta_{2} + 6\beta _1 + 58 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 119 \beta_{15} + 40 \beta_{14} + 24 \beta_{13} - 25 \beta_{12} - 104 \beta_{11} + 837 \beta_{10} + \cdots + 198 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -356\beta_{9} - 272\beta_{7} - 490\beta_{5} - 163\beta_{4} + 622\beta_{3} - 371\beta_{2} - 119\beta _1 - 1241 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -148\beta_{14} - 96\beta_{13} + 319\beta_{11} - 544\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1880 \beta_{9} + 1390 \beta_{7} + 2854 \beta_{5} - 959 \beta_{4} - 3874 \beta_{3} - 2235 \beta_{2} + \cdots - 6933 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4059 \beta_{15} + 1956 \beta_{14} + 1318 \beta_{13} + 1729 \beta_{12} - 3832 \beta_{11} + \cdots + 6162 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1414\beta_{4} + 3330\beta_{2} + 840\beta _1 + 9915 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 23881 \beta_{15} + 12256 \beta_{14} + 8424 \beta_{13} - 11183 \beta_{12} - 22808 \beta_{11} + \cdots + 35506 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 58756 \beta_{9} - 42076 \beta_{7} - 97802 \beta_{5} - 33369 \beta_{4} + 141458 \beta_{3} + \cdots - 229875 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -37360\beta_{14} - 25956\beta_{13} + 67581\beta_{11} - 103308\beta_{6} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3264\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(2177\) \(2245\) \(2689\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1633.1
0.0858135 + 0.148633i
−0.916682 + 1.58774i
0.654400 − 1.13345i
−1.21412 − 2.10292i
1.21412 − 2.10292i
−0.654400 − 1.13345i
0.916682 + 1.58774i
−0.0858135 + 0.148633i
−0.0858135 − 0.148633i
0.916682 − 1.58774i
−0.654400 + 1.13345i
1.21412 + 2.10292i
−1.21412 + 2.10292i
0.654400 + 1.13345i
−0.916682 − 1.58774i
0.0858135 − 0.148633i
0 − 1.00000i 0 − 3.92291i 0 0.343254 0 −1.00000 0
1633.2 0 − 1.00000i 0 − 3.01997i 0 −3.66673 0 −1.00000 0
1633.3 0 − 1.00000i 0 − 1.18731i 0 2.61760 0 −1.00000 0
1633.4 0 − 1.00000i 0 − 0.284371i 0 −4.85648 0 −1.00000 0
1633.5 0 − 1.00000i 0 0.284371i 0 4.85648 0 −1.00000 0
1633.6 0 − 1.00000i 0 1.18731i 0 −2.61760 0 −1.00000 0
1633.7 0 − 1.00000i 0 3.01997i 0 3.66673 0 −1.00000 0
1633.8 0 − 1.00000i 0 3.92291i 0 −0.343254 0 −1.00000 0
1633.9 0 1.00000i 0 − 3.92291i 0 −0.343254 0 −1.00000 0
1633.10 0 1.00000i 0 − 3.01997i 0 3.66673 0 −1.00000 0
1633.11 0 1.00000i 0 − 1.18731i 0 −2.61760 0 −1.00000 0
1633.12 0 1.00000i 0 − 0.284371i 0 4.85648 0 −1.00000 0
1633.13 0 1.00000i 0 0.284371i 0 −4.85648 0 −1.00000 0
1633.14 0 1.00000i 0 1.18731i 0 2.61760 0 −1.00000 0
1633.15 0 1.00000i 0 3.01997i 0 −3.66673 0 −1.00000 0
1633.16 0 1.00000i 0 3.92291i 0 0.343254 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1633.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3264.2.f.j ✓ 16
4.b odd 2 1 inner 3264.2.f.j ✓ 16
8.b even 2 1 inner 3264.2.f.j ✓ 16
8.d odd 2 1 inner 3264.2.f.j ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3264.2.f.j ✓ 16 1.a even 1 1 trivial
3264.2.f.j ✓ 16 4.b odd 2 1 inner
3264.2.f.j ✓ 16 8.b even 2 1 inner
3264.2.f.j ✓ 16 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3264, [\chi])\):

\( T_{5}^{8} + 26T_{5}^{6} + 177T_{5}^{4} + 212T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 44T_{7}^{6} + 576T_{7}^{4} - 2240T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{23}^{8} - 74T_{23}^{6} + 1137T_{23}^{4} - 4196T_{23}^{2} + 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + 26 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 44 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 82 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 54 T^{6} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$17$ \( (T - 1)^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} + 162 T^{6} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 74 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 228 T^{6} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 204 T^{6} + \cdots + 3873024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 132 T^{6} + \cdots + 147456)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 6 T^{3} + \cdots + 156)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 178 T^{6} + \cdots + 246016)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 128 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 296 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 376 T^{6} + \cdots + 11505664)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 204 T^{6} + \cdots + 112896)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 544 T^{6} + \cdots + 126877696)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 380 T^{6} + \cdots + 7661824)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 60 T^{2} + \cdots - 96)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 428 T^{6} + \cdots + 30824704)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 864 T^{6} + \cdots + 1504198656)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 96 T^{2} + \cdots + 48)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 96 T^{2} + \cdots + 48)^{4} \) Copy content Toggle raw display
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