↑

Properties

Label 9464.2.a.bk
Level $9464$
Weight $2$
Character orbit 9464.a
Self dual yes
Analytic conductor $75.570$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9464,2,Mod(1,9464)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9464, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("9464.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9464.a (trivial)

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: yes
Analytic conductor: \(75.5704204729\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 20 x^{10} + 33 x^{9} + 154 x^{8} - 182 x^{7} - 582 x^{6} + 382 x^{5} + 1056 x^{4} + \cdots + 104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{7} q^{5} - q^{7} + (\beta_{6} + \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{7} q^{5} - q^{7} + (\beta_{6} + \beta_{5}) q^{9} + ( - \beta_{10} - \beta_{8} + \cdots - \beta_1) q^{11}+ \cdots + ( - \beta_{10} - \beta_{8} - 2 \beta_{7} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 4 q^{5} - 12 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} + 4 q^{5} - 12 q^{7} + 8 q^{9} - 4 q^{11} + 14 q^{15} - 3 q^{17} + 16 q^{19} + 2 q^{21} - 14 q^{23} + 6 q^{25} - 17 q^{27} + 15 q^{29} + 17 q^{31} - 4 q^{35} - 2 q^{37} + 13 q^{41} - 24 q^{43} - 6 q^{45} + 21 q^{47} + 12 q^{49} + 3 q^{51} + 6 q^{53} - 3 q^{55} + 26 q^{57} - 5 q^{59} + 23 q^{61} - 8 q^{63} + 18 q^{67} - 14 q^{69} + 16 q^{71} + 18 q^{73} + 11 q^{75} + 4 q^{77} - 47 q^{79} - 32 q^{81} + 22 q^{83} + 56 q^{85} - 4 q^{87} + 24 q^{89} + 26 q^{93} - 42 q^{95} + 18 q^{97} + 29 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 20 x^{10} + 33 x^{9} + 154 x^{8} - 182 x^{7} - 582 x^{6} + 382 x^{5} + 1056 x^{4} + \cdots + 104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11 \nu^{11} - 7808 \nu^{10} + 4312 \nu^{9} + 130397 \nu^{8} - 15784 \nu^{7} - 723834 \nu^{6} + \cdots + 190564 ) / 58772 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 101 \nu^{11} + 8452 \nu^{10} - 19180 \nu^{9} - 122417 \nu^{8} + 277164 \nu^{7} + 572158 \nu^{6} + \cdots - 130980 ) / 58772 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1539 \nu^{11} + 13572 \nu^{10} + 15568 \nu^{9} - 226711 \nu^{8} - 39104 \nu^{7} + 1291526 \nu^{6} + \cdots - 373484 ) / 58772 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 487 \nu^{11} - 5071 \nu^{10} + 14588 \nu^{9} + 90847 \nu^{8} - 124438 \nu^{7} - 560342 \nu^{6} + \cdots + 179322 ) / 14693 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 487 \nu^{11} + 5071 \nu^{10} - 14588 \nu^{9} - 90847 \nu^{8} + 124438 \nu^{7} + 560342 \nu^{6} + \cdots - 223401 ) / 14693 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 561 \nu^{11} - 1497 \nu^{10} + 14210 \nu^{9} + 23704 \nu^{8} - 114413 \nu^{7} - 124262 \nu^{6} + \cdots - 8002 ) / 14693 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2679 \nu^{11} + 863 \nu^{10} - 51044 \nu^{9} - 28731 \nu^{8} + 329823 \nu^{7} + 267012 \nu^{6} + \cdots - 128046 ) / 29386 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4071 \nu^{11} - 844 \nu^{10} - 67760 \nu^{9} - 22961 \nu^{8} + 370376 \nu^{7} + 388968 \nu^{6} + \cdots - 234692 ) / 29386 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5247 \nu^{11} - 7606 \nu^{10} - 87962 \nu^{9} + 98951 \nu^{8} + 505714 \nu^{7} - 369314 \nu^{6} + \cdots + 80028 ) / 29386 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 6941 \nu^{11} - 4693 \nu^{10} + 134904 \nu^{9} + 117251 \nu^{8} - 894123 \nu^{7} - 933008 \nu^{6} + \cdots + 174688 ) / 29386 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + 5\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{8} + 8\beta_{6} + 10\beta_{5} + 3\beta_{4} + \beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11 \beta_{11} + 14 \beta_{10} + 8 \beta_{9} - 11 \beta_{8} - 9 \beta_{7} - \beta_{5} + 14 \beta_{4} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9 \beta_{11} + 4 \beta_{10} + \beta_{9} - 11 \beta_{8} + \beta_{7} + 61 \beta_{6} + 86 \beta_{5} + \cdots + 118 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 97 \beta_{11} + 143 \beta_{10} + 59 \beta_{9} - 105 \beta_{8} - 68 \beta_{7} + 3 \beta_{6} - 5 \beta_{5} + \cdots + 270 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 63 \beta_{11} + 80 \beta_{10} + 12 \beta_{9} - 113 \beta_{8} + 19 \beta_{7} + 470 \beta_{6} + \cdots + 901 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 805 \beta_{11} + 1320 \beta_{10} + 439 \beta_{9} - 945 \beta_{8} - 488 \beta_{7} + 60 \beta_{6} + \cdots + 2279 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 391 \beta_{11} + 1071 \beta_{10} + 109 \beta_{9} - 1143 \beta_{8} + 248 \beta_{7} + 3679 \beta_{6} + \cdots + 7244 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6553 \beta_{11} + 11696 \beta_{10} + 3322 \beta_{9} - 8269 \beta_{8} - 3431 \beta_{7} + 851 \beta_{6} + \cdots + 19240 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.96153
2.44693
2.42144
2.18667
0.919747
0.427363
−0.508489
−1.07845
−1.38057
−1.75548
−1.87813
−2.76256
0 −2.96153 0 −0.0896349 0 −1.00000 0 5.77065 0
1.2 0 −2.44693 0 −1.96160 0 −1.00000 0 2.98747 0
1.3 0 −2.42144 0 −3.26196 0 −1.00000 0 2.86336 0
1.4 0 −2.18667 0 2.75504 0 −1.00000 0 1.78154 0
1.5 0 −0.919747 0 2.82158 0 −1.00000 0 −2.15407 0
1.6 0 −0.427363 0 −0.444709 0 −1.00000 0 −2.81736 0
1.7 0 0.508489 0 −1.22080 0 −1.00000 0 −2.74144 0
1.8 0 1.07845 0 3.20179 0 −1.00000 0 −1.83695 0
1.9 0 1.38057 0 −0.344327 0 −1.00000 0 −1.09404 0
1.10 0 1.75548 0 −2.91442 0 −1.00000 0 0.0817252 0
1.11 0 1.87813 0 3.22930 0 −1.00000 0 0.527390 0
1.12 0 2.76256 0 2.22974 0 −1.00000 0 4.63172 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9464.2.a.bk yes 12
13.b even 2 1 9464.2.a.bj ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9464.2.a.bj ✓ 12 13.b even 2 1
9464.2.a.bk yes 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9464))\):

\( T_{3}^{12} + 2 T_{3}^{11} - 20 T_{3}^{10} - 33 T_{3}^{9} + 154 T_{3}^{8} + 182 T_{3}^{7} - 582 T_{3}^{6} + \cdots + 104 \) Copy content Toggle raw display
\( T_{5}^{12} - 4 T_{5}^{11} - 25 T_{5}^{10} + 104 T_{5}^{9} + 220 T_{5}^{8} - 933 T_{5}^{7} - 904 T_{5}^{6} + \cdots - 56 \) Copy content Toggle raw display
\( T_{11}^{12} + 4 T_{11}^{11} - 47 T_{11}^{10} - 153 T_{11}^{9} + 872 T_{11}^{8} + 2207 T_{11}^{7} + \cdots + 94871 \) Copy content Toggle raw display
\( T_{17}^{12} + 3 T_{17}^{11} - 103 T_{17}^{10} - 337 T_{17}^{9} + 3680 T_{17}^{8} + 13559 T_{17}^{7} + \cdots - 978104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 104 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + \cdots - 56 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 94871 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 3 T^{11} + \cdots - 978104 \) Copy content Toggle raw display
$19$ \( T^{12} - 16 T^{11} + \cdots - 791288 \) Copy content Toggle raw display
$23$ \( T^{12} + 14 T^{11} + \cdots - 42601 \) Copy content Toggle raw display
$29$ \( T^{12} - 15 T^{11} + \cdots - 4336919 \) Copy content Toggle raw display
$31$ \( T^{12} - 17 T^{11} + \cdots + 15784 \) Copy content Toggle raw display
$37$ \( T^{12} + 2 T^{11} + \cdots + 935719 \) Copy content Toggle raw display
$41$ \( T^{12} - 13 T^{11} + \cdots - 2030168 \) Copy content Toggle raw display
$43$ \( T^{12} + 24 T^{11} + \cdots + 6020287 \) Copy content Toggle raw display
$47$ \( T^{12} - 21 T^{11} + \cdots + 41520296 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 1712374047 \) Copy content Toggle raw display
$59$ \( T^{12} + 5 T^{11} + \cdots + 10598848 \) Copy content Toggle raw display
$61$ \( T^{12} - 23 T^{11} + \cdots - 9976 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 1727096231 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 2501809121 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 356158712 \) Copy content Toggle raw display
$79$ \( T^{12} + 47 T^{11} + \cdots + 11076561 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 2742153728 \) Copy content Toggle raw display
$89$ \( T^{12} - 24 T^{11} + \cdots + 57683128 \) Copy content Toggle raw display
$97$ \( T^{12} - 18 T^{11} + \cdots - 4423384 \) Copy content Toggle raw display
show more
show less