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Properties

Label 8281.2.a.cs
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
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] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
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: \( x^{16} - 42x^{14} + 641x^{12} - 4448x^{10} + 14076x^{8} - 17900x^{6} + 6960x^{4} - 416x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 637)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - \beta_{11} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{10} q^{5} - \beta_{4} q^{6} + (\beta_{14} + \beta_{12} + \beta_{7}) q^{8} + (\beta_{5} - \beta_{3} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - \beta_{11} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{10} q^{5} - \beta_{4} q^{6} + (\beta_{14} + \beta_{12} + \beta_{7}) q^{8} + (\beta_{5} - \beta_{3} - \beta_{2} + 1) q^{9} + (\beta_{15} - \beta_{6}) q^{10} - \beta_{12} q^{11} - \beta_{9} q^{12} + (\beta_{14} - \beta_{12}) q^{15} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 1) q^{16} + ( - \beta_{11} - \beta_{9} + \beta_{6}) q^{17} + ( - \beta_{14} - \beta_{13}) q^{18} + (\beta_{10} + \beta_{4} - \beta_1) q^{19} + (\beta_{8} - \beta_{4} + \beta_1) q^{20} + ( - \beta_{5} + \beta_{3} - \beta_{2}) q^{22} + (2 \beta_{5} - \beta_{2} + 2) q^{23} + (\beta_{8} + 2 \beta_1) q^{24} + ( - \beta_{2} + 3) q^{25} + ( - \beta_{15} - \beta_{11} + 2 \beta_{6}) q^{27} + ( - \beta_{5} + 2 \beta_{2} + 2) q^{29} + ( - \beta_{5} + 3 \beta_{3} + \beta_{2}) q^{30} + (\beta_{10} - 2 \beta_{4} - \beta_1) q^{31} + (2 \beta_{14} - \beta_{13} + \cdots + 4 \beta_{7}) q^{32}+ \cdots + (2 \beta_{14} - \beta_{13} + \cdots - \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 20 q^{4} + 16 q^{9} + 28 q^{16} - 8 q^{22} + 36 q^{23} + 44 q^{25} + 36 q^{29} - 52 q^{36} + 36 q^{43} + 72 q^{51} + 12 q^{53} + 164 q^{64} + 96 q^{74} + 36 q^{79} + 16 q^{81} - 136 q^{88} + 24 q^{92} + 84 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 42x^{14} + 641x^{12} - 4448x^{10} + 14076x^{8} - 17900x^{6} + 6960x^{4} - 416x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 953676 \nu^{14} + 39533101 \nu^{12} - 590131598 \nu^{10} + 3936732237 \nu^{8} + \cdots - 1018141856 ) / 487134164 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 953676 \nu^{14} - 39533101 \nu^{12} + 590131598 \nu^{10} - 3936732237 \nu^{8} + 11514360568 \nu^{6} + \cdots - 199693554 ) / 243567082 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1349387 \nu^{14} - 56847540 \nu^{12} + 871802467 \nu^{10} - 6097919603 \nu^{8} + \cdots - 643641148 ) / 121783541 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6524510 \nu^{14} - 273768661 \nu^{12} + 4173187693 \nu^{10} - 28922548559 \nu^{8} + \cdots - 2768860598 ) / 487134164 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2735448 \nu^{14} + 114739024 \nu^{12} - 1746847052 \nu^{10} + 12061801269 \nu^{8} + \cdots + 186728765 ) / 121783541 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10968382 \nu^{15} + 461625720 \nu^{13} - 7070265963 \nu^{11} + 49377494734 \nu^{9} + \cdots + 7335835374 \nu ) / 487134164 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10968382 \nu^{15} + 461625720 \nu^{13} - 7070265963 \nu^{11} + 49377494734 \nu^{9} + \cdots + 7822969538 \nu ) / 487134164 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11574053 \nu^{14} - 486829503 \nu^{12} + 7451025177 \nu^{10} - 51998782073 \nu^{8} + \cdots - 2654975036 ) / 487134164 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11922058 \nu^{15} - 501158821 \nu^{13} + 7660397561 \nu^{11} - 53314226971 \nu^{9} + \cdots - 5343425190 \nu ) / 487134164 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25668835 \nu^{14} - 1076348524 \nu^{12} + 16382483065 \nu^{10} - 113117579184 \nu^{8} + \cdots - 4616203728 ) / 974268328 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 97148003 \nu^{15} - 4077962202 \nu^{13} + 62179112921 \nu^{11} - 430742361694 \nu^{9} + \cdots - 34215865848 \nu ) / 974268328 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66779377 \nu^{15} + 2801875899 \nu^{13} - 42685130106 \nu^{11} + 295191359193 \nu^{9} + \cdots + 15401679066 \nu ) / 487134164 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 95646446 \nu^{15} - 4014584759 \nu^{13} + 61200884957 \nu^{11} - 423765756991 \nu^{9} + \cdots - 18898589406 \nu ) / 487134164 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 102545551 \nu^{15} - 4305352362 \nu^{13} + 65666322789 \nu^{11} - 455134040106 \nu^{9} + \cdots - 36790430440 \nu ) / 487134164 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 216961727 \nu^{15} - 9105518042 \nu^{13} + 138784252979 \nu^{11} - 960649093166 \nu^{9} + \cdots - 43387650868 \nu ) / 974268328 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{7} - \beta_{6} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{12} - 3\beta_{9} + 11\beta_{7} - 11\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{8} + \beta_{5} - 8\beta_{4} + \beta_{3} + 20\beta_{2} + 28\beta _1 + 55 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{15} + 30\beta_{14} - \beta_{13} + 29\beta_{12} - 15\beta_{11} - 60\beta_{9} + 152\beta_{7} - 139\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{10} + 94\beta_{8} + 41\beta_{5} - 200\beta_{4} + 40\beta_{3} + 360\beta_{2} + 410\beta _1 + 733 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 147 \beta_{15} + 641 \beta_{14} - 53 \beta_{13} + 589 \beta_{12} - 427 \beta_{11} + \cdots - 1975 \beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 400\beta_{10} + 1800\beta_{8} + 989\beta_{5} - 3984\beta_{4} + 920\beta_{3} + 6272\beta_{2} + 6376\beta _1 + 10875 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3189 \beta_{15} + 12165 \beta_{14} - 1389 \beta_{13} + 10861 \beta_{12} - 8997 \beta_{11} + \cdots - 30403 \beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9156 \beta_{10} + 32482 \beta_{8} + 20017 \beta_{5} - 73380 \beta_{4} + 17888 \beta_{3} + 108226 \beta_{2} + \cdots + 172097 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 61655 \beta_{15} + 219511 \beta_{14} - 29173 \beta_{13} + 193207 \beta_{12} - 170071 \beta_{11} + \cdots - 491567 \beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 181656 \beta_{10} + 572388 \beta_{8} + 374863 \beta_{5} - 1305800 \beta_{4} + 325058 \beta_{3} + \cdots + 2827755 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1128907 \beta_{15} + 3868427 \beta_{14} - 556519 \beta_{13} + 3382345 \beta_{12} + \cdots - 8175543 \beta_{6} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3370852 \beta_{10} + 9974190 \beta_{8} + 6753197 \beta_{5} - 22855828 \beta_{4} + 5733274 \beta_{3} + \cdots + 47455889 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 20098239 \beta_{15} + 67384629 \beta_{14} - 10124049 \beta_{13} + 58743907 \beta_{12} + \cdots - 138122763 \beta_{6} \) 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
−4.14842
−1.31999
−0.241304
−3.06973
−0.109182
−2.93761
−2.09441
0.734017
−0.734017
2.09441
2.93761
0.109182
3.06973
0.241304
1.31999
4.14842
−2.73420 −1.15595 5.47586 1.87727 3.16060 0 −9.50370 −1.66378 −5.13283
1.2 −2.73420 1.15595 5.47586 −1.87727 −3.16060 0 −9.50370 −1.66378 5.13283
1.3 −1.65552 −0.494977 0.740740 2.87389 0.819443 0 2.08473 −2.75500 −4.75778
1.4 −1.65552 0.494977 0.740740 −2.87389 −0.819443 0 2.08473 −2.75500 4.75778
1.5 −1.52340 −2.99510 0.320733 −2.94606 4.56272 0 2.55819 5.97063 4.48801
1.6 −1.52340 2.99510 0.320733 2.94606 −4.56272 0 2.55819 5.97063 −4.48801
1.7 −0.680196 −2.33413 −1.53733 3.24613 1.58766 0 2.40608 2.44815 −2.20800
1.8 −0.680196 2.33413 −1.53733 −3.24613 −1.58766 0 2.40608 2.44815 2.20800
1.9 0.680196 −2.33413 −1.53733 −3.24613 −1.58766 0 −2.40608 2.44815 −2.20800
1.10 0.680196 2.33413 −1.53733 3.24613 1.58766 0 −2.40608 2.44815 2.20800
1.11 1.52340 −2.99510 0.320733 2.94606 −4.56272 0 −2.55819 5.97063 4.48801
1.12 1.52340 2.99510 0.320733 −2.94606 4.56272 0 −2.55819 5.97063 −4.48801
1.13 1.65552 −0.494977 0.740740 −2.87389 −0.819443 0 −2.08473 −2.75500 −4.75778
1.14 1.65552 0.494977 0.740740 2.87389 0.819443 0 −2.08473 −2.75500 4.75778
1.15 2.73420 −1.15595 5.47586 −1.87727 −3.16060 0 9.50370 −1.66378 −5.13283
1.16 2.73420 1.15595 5.47586 1.87727 3.16060 0 9.50370 −1.66378 5.13283
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.b even 2 1 inner
91.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.cs 16
7.b odd 2 1 inner 8281.2.a.cs 16
13.b even 2 1 inner 8281.2.a.cs 16
13.d odd 4 2 637.2.c.g ✓ 16
91.b odd 2 1 inner 8281.2.a.cs 16
91.i even 4 2 637.2.c.g ✓ 16
91.z odd 12 4 637.2.r.g 32
91.bb even 12 4 637.2.r.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.c.g ✓ 16 13.d odd 4 2
637.2.c.g ✓ 16 91.i even 4 2
637.2.r.g 32 91.z odd 12 4
637.2.r.g 32 91.bb even 12 4
8281.2.a.cs 16 1.a even 1 1 trivial
8281.2.a.cs 16 7.b odd 2 1 inner
8281.2.a.cs 16 13.b even 2 1 inner
8281.2.a.cs 16 91.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{8} - 13T_{2}^{6} + 50T_{2}^{4} - 68T_{2}^{2} + 22 \) Copy content Toggle raw display
\( T_{3}^{8} - 16T_{3}^{6} + 72T_{3}^{4} - 82T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{8} - 31T_{5}^{6} + 347T_{5}^{4} - 1637T_{5}^{2} + 2662 \) Copy content Toggle raw display
\( T_{11}^{8} - 46T_{11}^{6} + 602T_{11}^{4} - 3110T_{11}^{2} + 5632 \) Copy content Toggle raw display
\( T_{17}^{8} - 78T_{17}^{6} + 1908T_{17}^{4} - 15498T_{17}^{2} + 39204 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 13 T^{6} + \cdots + 22)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - 16 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 31 T^{6} + \cdots + 2662)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 46 T^{6} + \cdots + 5632)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( (T^{8} - 78 T^{6} + \cdots + 39204)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 75 T^{6} + \cdots + 7128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 9 T^{3} + \cdots - 792)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 9 T^{3} + \cdots - 144)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 165 T^{6} + \cdots + 1824768)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 90 T^{6} + \cdots + 7128)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 154 T^{6} + \cdots + 59488)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 9 T^{3} - 21 T^{2} + \cdots + 72)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 115 T^{6} + \cdots + 1408)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 3 T^{3} + \cdots + 2502)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 190 T^{6} + \cdots + 1580128)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 168 T^{6} + \cdots + 219024)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 348 T^{6} + \cdots + 456192)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 124 T^{6} + \cdots + 68992)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 189 T^{6} + \cdots + 514998)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 9 T^{3} + 11 T^{2} + \cdots - 44)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 157 T^{6} + \cdots + 25432)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 385 T^{6} + \cdots + 1888678)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 537 T^{6} + \cdots + 28741878)^{2} \) Copy content Toggle raw display
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