↑

Properties

Label 9464.2.a.br
Level $9464$
Weight $2$
Character orbit 9464.a
Self dual yes
Analytic conductor $75.570$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [15,0,-3,0,-4,0,-15,0,16,0,-15,0,0,0,-8,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(75.5704204729\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 26 x^{13} + 78 x^{12} + 253 x^{11} - 782 x^{10} - 1087 x^{9} + 3776 x^{8} + \cdots - 344 \) 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

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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{6} q^{5} - q^{7} + (\beta_{2} + 1) q^{9} + ( - \beta_{14} + \beta_{9} - \beta_1 - 1) q^{11} + ( - \beta_{12} - \beta_{10} + \beta_{9} + \cdots - 1) q^{15} + ( - \beta_{10} + 2 \beta_{8} + \cdots - 1) q^{17}+ \cdots + ( - \beta_{14} + \beta_{13} + \beta_{12} + \cdots - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 26 x^{13} + 78 x^{12} + 253 x^{11} - 782 x^{10} - 1087 x^{9} + 3776 x^{8} + \cdots - 344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23023337 \nu^{14} - 64046675 \nu^{13} - 620688986 \nu^{12} + 1671623546 \nu^{11} + \cdots + 18620660252 ) / 218221572 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 92738042 \nu^{14} - 219729253 \nu^{13} - 2559280702 \nu^{12} + 5649666411 \nu^{11} + \cdots + 50372600862 ) / 709220109 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 585727917 \nu^{14} - 1386231583 \nu^{13} - 16107842854 \nu^{12} + 35449654718 \nu^{11} + \cdots + 304679107652 ) / 2836880436 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 254629997 \nu^{14} + 600175837 \nu^{13} + 7017767884 \nu^{12} - 15368643099 \nu^{11} + \cdots - 125788851738 ) / 709220109 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2206407839 \nu^{14} - 4834284839 \nu^{13} - 61250796872 \nu^{12} + 122455303622 \nu^{11} + \cdots + 934704272036 ) / 4255320654 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3595755457 \nu^{14} + 8024430280 \nu^{13} + 99642302239 \nu^{12} - 203862444562 \nu^{11} + \cdots - 1619152665946 ) / 4255320654 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8640111647 \nu^{14} + 18930475097 \nu^{13} + 239824869866 \nu^{12} - 479542458386 \nu^{11} + \cdots - 3713816835356 ) / 8510641308 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5111143377 \nu^{14} - 11087150411 \nu^{13} - 142104787334 \nu^{12} + 280575087022 \nu^{11} + \cdots + 2085327722656 ) / 2836880436 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8398542223 \nu^{14} - 19014425479 \nu^{13} - 232440590170 \nu^{12} + 484100557006 \nu^{11} + \cdots + 3896861605300 ) / 4255320654 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5166505585 \nu^{14} + 11407784443 \nu^{13} + 143357735005 \nu^{12} - 289422601090 \nu^{11} + \cdots - 2243045788762 ) / 2127660327 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12545964577 \nu^{14} - 27298336993 \nu^{13} - 348659131156 \nu^{12} + 691176902722 \nu^{11} + \cdots + 5233494803908 ) / 4255320654 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14181642859 \nu^{14} + 31126121809 \nu^{13} + 393696175786 \nu^{12} - 789078448300 \nu^{11} + \cdots - 6111169700050 ) / 4255320654 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{11} - 6\beta_{8} + \beta_{7} - 2\beta_{5} + \beta_{4} + 9\beta_{2} + 2\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{14} - 2 \beta_{12} - 2 \beta_{11} - \beta_{10} + 10 \beta_{9} - 4 \beta_{8} + 27 \beta_{7} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{14} - 4 \beta_{13} - 7 \beta_{12} - 20 \beta_{11} + \beta_{10} + 2 \beta_{9} - 92 \beta_{8} + \cdots + 239 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 37 \beta_{14} - 13 \beta_{13} - 38 \beta_{12} - 43 \beta_{11} - 9 \beta_{10} + 83 \beta_{9} - 110 \beta_{8} + \cdots + 180 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 201 \beta_{14} - 92 \beta_{13} - 148 \beta_{12} - 280 \beta_{11} + 19 \beta_{10} + 35 \beta_{9} + \cdots + 2124 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 510 \beta_{14} - 305 \beta_{13} - 545 \beta_{12} - 660 \beta_{11} - 36 \beta_{10} + 653 \beta_{9} + \cdots + 2622 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2332 \beta_{14} - 1466 \beta_{13} - 2201 \beta_{12} - 3460 \beta_{11} + 274 \beta_{10} + 427 \beta_{9} + \cdots + 20004 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6298 \beta_{14} - 4897 \beta_{13} - 7071 \beta_{12} - 8836 \beta_{11} + 263 \beta_{10} + 5072 \beta_{9} + \cdots + 33781 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 26111 \beta_{14} - 20057 \beta_{13} - 28442 \beta_{12} - 40434 \beta_{11} + 3570 \beta_{10} + \cdots + 196817 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 73775 \beta_{14} - 67149 \beta_{13} - 87073 \beta_{12} - 110054 \beta_{11} + 8347 \beta_{10} + \cdots + 409072 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 287269 \beta_{14} - 253139 \beta_{13} - 342536 \beta_{12} - 459349 \beta_{11} + 44033 \beta_{10} + \cdots + 1998854 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.30816
2.81891
2.66171
2.23382
1.05916
0.899537
0.790243
0.739098
0.354884
−0.619176
−1.45081
−2.05925
−2.32680
−2.57537
−2.83411
0 −3.30816 0 −1.62495 0 −1.00000 0 7.94392 0
1.2 0 −2.81891 0 3.47162 0 −1.00000 0 4.94627 0
1.3 0 −2.66171 0 −1.73025 0 −1.00000 0 4.08469 0
1.4 0 −2.23382 0 3.65424 0 −1.00000 0 1.98994 0
1.5 0 −1.05916 0 1.23771 0 −1.00000 0 −1.87817 0
1.6 0 −0.899537 0 −4.33228 0 −1.00000 0 −2.19083 0
1.7 0 −0.790243 0 −3.98493 0 −1.00000 0 −2.37552 0
1.8 0 −0.739098 0 −1.43777 0 −1.00000 0 −2.45373 0
1.9 0 −0.354884 0 1.29454 0 −1.00000 0 −2.87406 0
1.10 0 0.619176 0 2.60169 0 −1.00000 0 −2.61662 0
1.11 0 1.45081 0 −1.40254 0 −1.00000 0 −0.895145 0
1.12 0 2.05925 0 1.64516 0 −1.00000 0 1.24052 0
1.13 0 2.32680 0 1.27247 0 −1.00000 0 2.41400 0
1.14 0 2.57537 0 −3.57643 0 −1.00000 0 3.63254 0
1.15 0 2.83411 0 −1.08830 0 −1.00000 0 5.03219 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.br ✓ 15
13.b even 2 1 9464.2.a.bs yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9464.2.a.br ✓ 15 1.a even 1 1 trivial
9464.2.a.bs yes 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9464))\):

\( T_{3}^{15} + 3 T_{3}^{14} - 26 T_{3}^{13} - 78 T_{3}^{12} + 253 T_{3}^{11} + 782 T_{3}^{10} - 1087 T_{3}^{9} + \cdots + 344 \) Copy content Toggle raw display
\( T_{5}^{15} + 4 T_{5}^{14} - 41 T_{5}^{13} - 156 T_{5}^{12} + 642 T_{5}^{11} + 2279 T_{5}^{10} + \cdots - 42176 \) Copy content Toggle raw display
\( T_{11}^{15} + 15 T_{11}^{14} + T_{11}^{13} - 921 T_{11}^{12} - 2855 T_{11}^{11} + 19628 T_{11}^{10} + \cdots + 5892307 \) Copy content Toggle raw display
\( T_{17}^{15} - 2 T_{17}^{14} - 122 T_{17}^{13} + 361 T_{17}^{12} + 5070 T_{17}^{11} - 19306 T_{17}^{10} + \cdots + 94744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + 3 T^{14} + \cdots + 344 \) Copy content Toggle raw display
$5$ \( T^{15} + 4 T^{14} + \cdots - 42176 \) Copy content Toggle raw display
$7$ \( (T + 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + 15 T^{14} + \cdots + 5892307 \) Copy content Toggle raw display
$13$ \( T^{15} \) Copy content Toggle raw display
$17$ \( T^{15} - 2 T^{14} + \cdots + 94744 \) Copy content Toggle raw display
$19$ \( T^{15} + 13 T^{14} + \cdots - 2228008 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 198287944 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 3008199032 \) Copy content Toggle raw display
$31$ \( T^{15} + 19 T^{14} + \cdots - 15805888 \) Copy content Toggle raw display
$37$ \( T^{15} - 2 T^{14} + \cdots + 61585784 \) Copy content Toggle raw display
$41$ \( T^{15} + 30 T^{14} + \cdots - 224552 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 3300104227 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 1989448768 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 720963375448 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 3164487872 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 713528768 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 5020516607 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 10240196408 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 31621315928 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 9542065672 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 1887848887616 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 1155157416 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 132670219845304 \) Copy content Toggle raw display
show more
show less