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Properties

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

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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: \(1\)
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: \( x^{12} - 3 x^{11} - 19 x^{10} + 54 x^{9} + 127 x^{8} - 317 x^{7} - 372 x^{6} + 648 x^{5} + 539 x^{4} + \cdots + 8 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{2} - 1) q^{5} - q^{7} + (\beta_{11} + \beta_{9} + \beta_{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{2} - 1) q^{5} - q^{7} + (\beta_{11} + \beta_{9} + \beta_{6} + \cdots + 1) q^{9}+ \cdots + (\beta_{11} - 3 \beta_{10} + 2 \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 8 q^{5} - 12 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} - 8 q^{5} - 12 q^{7} + 11 q^{9} + 4 q^{11} - 19 q^{15} + 6 q^{17} - 2 q^{19} - 3 q^{21} + 7 q^{23} + 28 q^{25} + 18 q^{27} - 16 q^{29} - 31 q^{31} - 3 q^{33} + 8 q^{35} - 34 q^{37} - 13 q^{41} + 28 q^{43} - 33 q^{45} - 4 q^{47} + 12 q^{49} + 6 q^{51} - 52 q^{53} + 6 q^{55} - 20 q^{57} + 13 q^{59} - 24 q^{61} - 11 q^{63} + 36 q^{67} - 47 q^{69} - 12 q^{71} - 28 q^{73} + 38 q^{75} - 4 q^{77} - q^{79} + 16 q^{81} - 6 q^{83} - 7 q^{85} + 37 q^{87} + 33 q^{89} - 47 q^{93} + 41 q^{95} - 2 q^{97} + 31 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 19 x^{10} + 54 x^{9} + 127 x^{8} - 317 x^{7} - 372 x^{6} + 648 x^{5} + 539 x^{4} + \cdots + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17 \nu^{11} - 71 \nu^{10} - 241 \nu^{9} + 1174 \nu^{8} + 891 \nu^{7} - 6139 \nu^{6} - 56 \nu^{5} + \cdots - 550 ) / 234 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21 \nu^{11} + 25 \nu^{10} + 573 \nu^{9} - 632 \nu^{8} - 5487 \nu^{7} + 4959 \nu^{6} + 22654 \nu^{5} + \cdots + 8 ) / 468 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41 \nu^{11} - 107 \nu^{10} - 907 \nu^{9} + 2446 \nu^{8} + 6003 \nu^{7} - 16609 \nu^{6} - 12872 \nu^{5} + \cdots + 284 ) / 468 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 35 \nu^{11} + 85 \nu^{10} + 877 \nu^{9} - 2414 \nu^{8} - 5817 \nu^{7} + 18223 \nu^{6} + 9616 \nu^{5} + \cdots - 2968 ) / 468 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 49 \nu^{11} + 197 \nu^{10} + 713 \nu^{9} - 3208 \nu^{8} - 3183 \nu^{7} + 17135 \nu^{6} + \cdots - 1004 ) / 468 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59 \nu^{11} + 251 \nu^{10} + 763 \nu^{9} - 3712 \nu^{8} - 3285 \nu^{7} + 18085 \nu^{6} + 7742 \nu^{5} + \cdots - 188 ) / 468 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 95 \nu^{11} - 409 \nu^{10} - 1177 \nu^{9} + 6062 \nu^{8} + 4089 \nu^{7} - 29071 \nu^{6} - 4216 \nu^{5} + \cdots + 256 ) / 468 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 99 \nu^{11} + 415 \nu^{10} + 1275 \nu^{9} - 6248 \nu^{8} - 4785 \nu^{7} + 30621 \nu^{6} + \cdots - 1396 ) / 468 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 83 \nu^{11} - 287 \nu^{10} - 1507 \nu^{9} + 5452 \nu^{8} + 8397 \nu^{7} - 33001 \nu^{6} - 14474 \nu^{5} + \cdots + 1724 ) / 468 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 189 \nu^{11} - 719 \nu^{10} - 2895 \nu^{9} + 11902 \nu^{8} + 13971 \nu^{7} - 64365 \nu^{6} + \cdots + 812 ) / 468 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{6} - \beta_{4} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{11} + 9\beta_{9} + \beta_{8} + 2\beta_{7} + 8\beta_{6} - \beta_{5} - 11\beta_{4} - \beta_{3} + 11\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} + 8 \beta_{10} + 14 \beta_{9} + 13 \beta_{8} + 12 \beta_{7} - 11 \beta_{6} + 8 \beta_{5} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 74 \beta_{11} - 3 \beta_{10} + 82 \beta_{9} + 19 \beta_{8} + 29 \beta_{7} + 49 \beta_{6} - 11 \beta_{5} + \cdots + 220 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 35 \beta_{11} + 48 \beta_{10} + 158 \beta_{9} + 135 \beta_{8} + 126 \beta_{7} - 125 \beta_{6} + \cdots + 125 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 603 \beta_{11} - 67 \beta_{10} + 758 \beta_{9} + 248 \beta_{8} + 332 \beta_{7} + 208 \beta_{6} + \cdots + 1695 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 449 \beta_{11} + 192 \beta_{10} + 1664 \beta_{9} + 1325 \beta_{8} + 1276 \beta_{7} - 1416 \beta_{6} + \cdots + 1245 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4967 \beta_{11} - 1026 \beta_{10} + 7063 \beta_{9} + 2833 \beta_{8} + 3545 \beta_{7} - 155 \beta_{6} + \cdots + 13248 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5116 \beta_{11} - 422 \beta_{10} + 16959 \beta_{9} + 12809 \beta_{8} + 12771 \beta_{7} - 15613 \beta_{6} + \cdots + 12163 \) 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
−2.68481
−2.68370
−1.54534
−0.728025
−0.623009
−0.291796
0.130679
0.934943
1.92739
2.60319
2.81913
3.14135
0 −2.68481 0 0.148512 0 −1.00000 0 4.20820 0
1.2 0 −2.68370 0 0.803744 0 −1.00000 0 4.20224 0
1.3 0 −1.54534 0 −3.16236 0 −1.00000 0 −0.611919 0
1.4 0 −0.728025 0 4.24498 0 −1.00000 0 −2.46998 0
1.5 0 −0.623009 0 1.00440 0 −1.00000 0 −2.61186 0
1.6 0 −0.291796 0 −3.10615 0 −1.00000 0 −2.91486 0
1.7 0 0.130679 0 −3.71767 0 −1.00000 0 −2.98292 0
1.8 0 0.934943 0 2.60102 0 −1.00000 0 −2.12588 0
1.9 0 1.92739 0 1.09214 0 −1.00000 0 0.714836 0
1.10 0 2.60319 0 −3.83415 0 −1.00000 0 3.77658 0
1.11 0 2.81913 0 −0.645434 0 −1.00000 0 4.94750 0
1.12 0 3.14135 0 −3.42902 0 −1.00000 0 6.86806 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.bn ✓ 12
13.b even 2 1 9464.2.a.bo yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9464.2.a.bn ✓ 12 1.a even 1 1 trivial
9464.2.a.bo yes 12 13.b even 2 1

Hecke kernels

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

\( T_{3}^{12} - 3 T_{3}^{11} - 19 T_{3}^{10} + 54 T_{3}^{9} + 127 T_{3}^{8} - 317 T_{3}^{7} - 372 T_{3}^{6} + \cdots + 8 \) Copy content Toggle raw display
\( T_{5}^{12} + 8 T_{5}^{11} - 12 T_{5}^{10} - 229 T_{5}^{9} - 246 T_{5}^{8} + 1810 T_{5}^{7} + 2839 T_{5}^{6} + \cdots + 448 \) Copy content Toggle raw display
\( T_{11}^{12} - 4 T_{11}^{11} - 59 T_{11}^{10} + 119 T_{11}^{9} + 1534 T_{11}^{8} + 93 T_{11}^{7} + \cdots + 28769 \) Copy content Toggle raw display
\( T_{17}^{12} - 6 T_{17}^{11} - 85 T_{17}^{10} + 478 T_{17}^{9} + 2247 T_{17}^{8} - 11181 T_{17}^{7} + \cdots - 11752 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{12} + 8 T^{11} + \cdots + 448 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 28769 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 6 T^{11} + \cdots - 11752 \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} + \cdots - 56896 \) Copy content Toggle raw display
$23$ \( T^{12} - 7 T^{11} + \cdots - 1517608 \) Copy content Toggle raw display
$29$ \( T^{12} + 16 T^{11} + \cdots + 15112 \) Copy content Toggle raw display
$31$ \( T^{12} + 31 T^{11} + \cdots - 1341376 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 131241592 \) Copy content Toggle raw display
$41$ \( T^{12} + 13 T^{11} + \cdots + 28348312 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 1758619603 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 833448512 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 10318682744 \) Copy content Toggle raw display
$59$ \( T^{12} - 13 T^{11} + \cdots - 19251784 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 765880256 \) Copy content Toggle raw display
$67$ \( T^{12} - 36 T^{11} + \cdots + 17580991 \) Copy content Toggle raw display
$71$ \( T^{12} + 12 T^{11} + \cdots - 3279016 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 2722680584 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 212540776 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 3870066472 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1920006712 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 104395863608 \) Copy content Toggle raw display
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