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Properties

Label 3264.2.c.q
Level $3264$
Weight $2$
Character orbit 3264.c
Analytic conductor $26.063$
Analytic rank $0$
Dimension $10$
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] = [3264,2,Mod(577,3264)]
 
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, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3264.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3264.c (of order \(2\), degree \(1\), 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: \(26.0631712197\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 12x^{8} + 38x^{6} + 44x^{4} + 17x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 1632)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{9} q^{5} + \beta_{3} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_{9} q^{5} + \beta_{3} q^{7} - q^{9} + (\beta_{9} - \beta_{7} + \beta_{6}) q^{11} + ( - \beta_{4} - \beta_1 + 1) q^{13} - \beta_1 q^{15} + (\beta_{7} + \beta_{6} - \beta_{5} - 1) q^{17} + (\beta_{2} - \beta_1) q^{19} - \beta_{2} q^{21} + (\beta_{9} + \beta_{8} + \cdots - \beta_{3}) q^{23}+ \cdots + ( - \beta_{9} + \beta_{7} - \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{9} + 8 q^{13} - 4 q^{15} - 6 q^{17} - 8 q^{19} + 4 q^{21} - 10 q^{25} + 4 q^{33} - 16 q^{35} - 16 q^{43} + 24 q^{47} - 34 q^{49} + 8 q^{51} - 12 q^{53} - 56 q^{55} + 16 q^{59} + 16 q^{69} + 8 q^{77} + 10 q^{81} + 8 q^{83} + 4 q^{85} + 12 q^{87} - 12 q^{89} - 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 12x^{8} + 38x^{6} + 44x^{4} + 17x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 9\nu^{6} + 4\nu^{4} - 45\nu^{2} - 30 ) / 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{8} + 36\nu^{6} + 30\nu^{4} - 54\nu^{2} - 22 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{9} + 18\nu^{7} + 8\nu^{5} - 90\nu^{3} - 74\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{8} + 86\nu^{6} + 200\nu^{4} + 130\nu^{2} + 5 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{9} + 13\nu^{8} - 34\nu^{7} + 145\nu^{6} - 89\nu^{5} + 374\nu^{4} - 47\nu^{3} + 283\nu^{2} + 41\nu + 30 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -6\nu^{9} - 68\nu^{7} - 185\nu^{5} - 164\nu^{3} - 37\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{9} - 68\nu^{7} - 185\nu^{5} - 164\nu^{3} - 23\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} + 13\nu^{8} + 34\nu^{7} + 145\nu^{6} + 89\nu^{5} + 374\nu^{4} + 47\nu^{3} + 283\nu^{2} - 41\nu + 30 ) / 7 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{9} - 66\nu^{7} - 251\nu^{5} - 335\nu^{3} - 123\nu ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{5} - 4\beta_{4} + 2\beta_{2} - 2\beta _1 - 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{9} - 3\beta_{8} - 14\beta_{7} + 14\beta_{6} + 3\beta_{5} + 4\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{8} - 9\beta_{5} + 36\beta_{4} - 16\beta_{2} + 10\beta _1 + 44 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10\beta_{9} + 13\beta_{8} + 53\beta_{7} - 55\beta_{6} - 13\beta_{5} - 20\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 73\beta_{8} + 73\beta_{5} - 290\beta_{4} + 122\beta_{2} - 66\beta _1 - 318 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -84\beta_{9} - 103\beta_{8} - 411\beta_{7} + 433\beta_{6} + 103\beta_{5} + 165\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -288\beta_{8} - 288\beta_{5} + 1143\beta_{4} - 472\beta_{2} + 246\beta _1 + 1223 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1342\beta_{9} + 1615\beta_{8} + 6418\beta_{7} - 6798\beta_{6} - 1615\beta_{5} - 2616\beta_{3} ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
− 1.17316i
1.45030i
− 2.79802i
0.787589i
− 0.266708i
0.266708i
− 0.787589i
2.79802i
− 1.45030i
1.17316i
0 − 1.00000i 0 − 2.80506i 0 − 4.23523i 0 −1.00000 0
577.2 0 − 1.00000i 0 − 2.59584i 0 4.62887i 0 −1.00000 0
577.3 0 − 1.00000i 0 − 0.789041i 0 1.18054i 0 −1.00000 0
577.4 0 − 1.00000i 0 0.363938i 0 − 2.14845i 0 −1.00000 0
577.5 0 − 1.00000i 0 3.82600i 0 2.57427i 0 −1.00000 0
577.6 0 1.00000i 0 − 3.82600i 0 − 2.57427i 0 −1.00000 0
577.7 0 1.00000i 0 − 0.363938i 0 2.14845i 0 −1.00000 0
577.8 0 1.00000i 0 0.789041i 0 − 1.18054i 0 −1.00000 0
577.9 0 1.00000i 0 2.59584i 0 − 4.62887i 0 −1.00000 0
577.10 0 1.00000i 0 2.80506i 0 4.23523i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3264.2.c.q 10
4.b odd 2 1 3264.2.c.r 10
8.b even 2 1 1632.2.c.e ✓ 10
8.d odd 2 1 1632.2.c.f yes 10
17.b even 2 1 inner 3264.2.c.q 10
24.f even 2 1 4896.2.c.r 10
24.h odd 2 1 4896.2.c.s 10
68.d odd 2 1 3264.2.c.r 10
136.e odd 2 1 1632.2.c.f yes 10
136.h even 2 1 1632.2.c.e ✓ 10
408.b odd 2 1 4896.2.c.s 10
408.h even 2 1 4896.2.c.r 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1632.2.c.e ✓ 10 8.b even 2 1
1632.2.c.e ✓ 10 136.h even 2 1
1632.2.c.f yes 10 8.d odd 2 1
1632.2.c.f yes 10 136.e odd 2 1
3264.2.c.q 10 1.a even 1 1 trivial
3264.2.c.q 10 17.b even 2 1 inner
3264.2.c.r 10 4.b odd 2 1
3264.2.c.r 10 68.d odd 2 1
4896.2.c.r 10 24.f even 2 1
4896.2.c.r 10 408.h even 2 1
4896.2.c.s 10 24.h odd 2 1
4896.2.c.s 10 408.b odd 2 1

Hecke kernels

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

\( T_{5}^{10} + 30T_{5}^{8} + 289T_{5}^{6} + 980T_{5}^{4} + 608T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{5} - 4T_{13}^{4} - 31T_{13}^{3} + 110T_{13}^{2} + 220T_{13} - 664 \) Copy content Toggle raw display
\( T_{19}^{5} + 4T_{19}^{4} - 25T_{19}^{3} + 80T_{19} - 64 \) Copy content Toggle raw display
\( T_{43}^{5} + 8T_{43}^{4} - 89T_{43}^{3} - 524T_{43}^{2} + 592T_{43} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + 30 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{10} + 52 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( T^{10} + 74 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( (T^{5} - 4 T^{4} + \cdots - 664)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 6 T^{9} + \cdots + 1419857 \) Copy content Toggle raw display
$19$ \( (T^{5} + 4 T^{4} - 25 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 158 T^{8} + \cdots + 937024 \) Copy content Toggle raw display
$29$ \( T^{10} + 188 T^{8} + \cdots + 541696 \) Copy content Toggle raw display
$31$ \( T^{10} + 140 T^{8} + \cdots + 295936 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 124724224 \) Copy content Toggle raw display
$41$ \( T^{10} + 202 T^{8} + \cdots + 495616 \) Copy content Toggle raw display
$43$ \( (T^{5} + 8 T^{4} - 89 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} - 12 T^{4} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 6 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 8 T^{4} + \cdots - 23552)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 260 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$67$ \( (T^{5} - 272 T^{3} + \cdots - 65536)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + 276 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$73$ \( T^{10} + 496 T^{8} + \cdots + 67108864 \) Copy content Toggle raw display
$79$ \( T^{10} + 196 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$83$ \( (T^{5} - 4 T^{4} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + 6 T^{4} + \cdots + 36992)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 440 T^{8} + \cdots + 18939904 \) Copy content Toggle raw display
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