↑

Properties

Label 338.8.a.m
Level $338$
Weight $8$
Character orbit 338.a
Self dual yes
Analytic conductor $105.586$
Analytic rank $1$
Dimension $8$
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] = [338,8,Mod(1,338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("338.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-64,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(105.586138614\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11090 x^{6} - 22320 x^{5} + 30742777 x^{4} + 246003120 x^{3} - 12575752488 x^{2} + \cdots + 137733023376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 13^{3} \)
Twist minimal: no (minimal twist has level 26)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{2} + \beta_1 q^{3} + 64 q^{4} + (\beta_{5} + \beta_{2} - \beta_1 - 38) q^{5} - 8 \beta_1 q^{6} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - 269) q^{7} - 512 q^{8} + ( - 3 \beta_{7} - 2 \beta_{6} + \cdots + 584) q^{9}+ \cdots + (8136 \beta_{7} - 7426 \beta_{6} + \cdots - 5526274) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{2} + 512 q^{4} - 304 q^{5} - 2160 q^{7} - 4096 q^{8} + 4684 q^{9} + 2432 q^{10} - 2160 q^{11} + 17280 q^{14} - 21296 q^{15} + 32768 q^{16} + 41520 q^{17} - 37472 q^{18} - 66240 q^{19} - 19456 q^{20}+ \cdots - 44172000 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 11090 x^{6} - 22320 x^{5} + 30742777 x^{4} + 246003120 x^{3} - 12575752488 x^{2} + \cdots + 137733023376 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5417837 \nu^{7} - 83535140 \nu^{6} + 59759020330 \nu^{5} + 1080176685280 \nu^{4} + \cdots + 32\!\cdots\!40 ) / 11\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 139860008450 \nu^{7} + 9509108412882 \nu^{6} + \cdots - 36\!\cdots\!48 ) / 72\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3255367945616 \nu^{7} - 3359453901336 \nu^{6} + \cdots + 16\!\cdots\!32 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3494799382402 \nu^{7} + 33434908548717 \nu^{6} + \cdots - 13\!\cdots\!40 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4437093902711 \nu^{7} - 31393890383310 \nu^{6} + \cdots + 10\!\cdots\!20 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22270541742133 \nu^{7} + 27925075617192 \nu^{6} + \cdots + 89\!\cdots\!48 ) / 86\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -3\beta_{7} - 2\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 25\beta_{2} + 4\beta _1 + 2771 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -9\beta_{7} + 123\beta_{6} + 15\beta_{5} + 120\beta_{4} + 42\beta_{3} - 1641\beta_{2} + 5521\beta _1 + 8487 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 19515 \beta_{7} - 15482 \beta_{6} + 11305 \beta_{5} + 9578 \beta_{4} - 6337 \beta_{3} + \cdots + 15362927 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 81639 \beta_{7} + 1018995 \beta_{6} - 86673 \beta_{5} + 951096 \beta_{4} + 510810 \beta_{3} + \cdots + 1979415 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 130229679 \beta_{7} - 108419906 \beta_{6} + 96934861 \beta_{5} + 50486186 \beta_{4} + \cdots + 94806010619 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1505821419 \beta_{7} + 7682612847 \beta_{6} - 1379622045 \beta_{5} + 6604276440 \beta_{4} + \cdots - 620959764357 \) 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
−84.9959
−50.2253
−32.3527
−1.87085
7.49290
12.8548
69.7232
79.3738
−8.00000 −84.9959 64.0000 238.054 679.967 −1224.79 −512.000 5037.30 −1904.43
1.2 −8.00000 −50.2253 64.0000 −469.361 401.803 74.3301 −512.000 335.583 3754.89
1.3 −8.00000 −32.3527 64.0000 211.408 258.821 734.154 −512.000 −1140.31 −1691.27
1.4 −8.00000 −1.87085 64.0000 218.203 14.9668 417.634 −512.000 −2183.50 −1745.62
1.5 −8.00000 7.49290 64.0000 −522.947 −59.9432 −1108.53 −512.000 −2130.86 4183.58
1.6 −8.00000 12.8548 64.0000 260.039 −102.838 −982.460 −512.000 −2021.75 −2080.31
1.7 −8.00000 69.7232 64.0000 −226.832 −557.786 −178.562 −512.000 2674.32 1814.66
1.8 −8.00000 79.3738 64.0000 −12.5639 −634.991 108.218 −512.000 4113.21 100.511
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

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 338.8.a.m 8
13.b even 2 1 338.8.a.n 8
13.d odd 4 2 338.8.b.i 16
13.f odd 12 2 26.8.e.a ✓ 16
39.k even 12 2 234.8.l.c 16
52.l even 12 2 208.8.w.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.e.a ✓ 16 13.f odd 12 2
208.8.w.b 16 52.l even 12 2
234.8.l.c 16 39.k even 12 2
338.8.a.m 8 1.a even 1 1 trivial
338.8.a.n 8 13.b even 2 1
338.8.b.i 16 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(338))\):

\( T_{3}^{8} - 11090 T_{3}^{6} - 22320 T_{3}^{5} + 30742777 T_{3}^{4} + 246003120 T_{3}^{3} + \cdots + 137733023376 \) Copy content Toggle raw display
\( T_{5}^{8} + 304 T_{5}^{7} - 334782 T_{5}^{6} - 42080720 T_{5}^{5} + 41817423121 T_{5}^{4} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 137733023376 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 58\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 66\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 24\!\cdots\!51 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 40\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 46\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 31\!\cdots\!79 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 40\!\cdots\!91 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 82\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 86\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 77\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 68\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 60\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 52\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 27\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 26\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 56\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 62\!\cdots\!96 \) Copy content Toggle raw display
show more
show less