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Properties

Label 207.8.a.f
Level $207$
Weight $8$
Character orbit 207.a
Self dual yes
Analytic conductor $64.664$
Analytic rank $0$
Dimension $8$
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] = [207,8,Mod(1,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 207.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: \(64.6637002752\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 23)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 2 \beta_1 + 80) q^{4} + ( - \beta_{4} - 12 \beta_1 - 55) q^{5} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + 181) q^{7}+ \cdots + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 397) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 2 \beta_1 + 80) q^{4} + ( - \beta_{4} - 12 \beta_1 - 55) q^{5} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + 181) q^{7}+ \cdots + ( - 2058 \beta_{7} + 6230 \beta_{6} + \cdots - 9604) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 640 q^{4} - 444 q^{5} + 1446 q^{7} - 3177 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 640 q^{4} - 444 q^{5} + 1446 q^{7} - 3177 q^{8} + 19502 q^{10} - 7588 q^{11} + 19862 q^{13} - 17544 q^{14} + 64336 q^{16} - 42070 q^{17} + 1050 q^{19} - 3364 q^{20} - 128220 q^{22} + 97336 q^{23} + 49496 q^{25} + 371761 q^{26} + 143050 q^{28} + 102578 q^{29} + 304172 q^{31} + 612824 q^{32} - 524530 q^{34} - 531048 q^{35} + 286472 q^{37} + 762932 q^{38} + 2105286 q^{40} - 1324414 q^{41} + 2052578 q^{43} + 867298 q^{44} - 675556 q^{47} - 55404 q^{49} - 1458528 q^{50} - 1695409 q^{52} - 203654 q^{53} - 1024444 q^{55} + 5766846 q^{56} - 5039991 q^{58} + 748892 q^{59} + 61822 q^{61} + 4939277 q^{62} + 2702267 q^{64} + 1571618 q^{65} + 3235604 q^{67} - 4914980 q^{68} + 10871764 q^{70} + 4951664 q^{71} + 11019370 q^{73} - 356954 q^{74} + 21973240 q^{76} + 5284888 q^{77} + 4202464 q^{79} - 8785886 q^{80} + 32636759 q^{82} - 518568 q^{83} + 9854220 q^{85} + 14681386 q^{86} + 20589740 q^{88} - 4203864 q^{89} + 2488406 q^{91} + 7786880 q^{92} + 12314327 q^{94} + 44485300 q^{95} + 18621134 q^{97} - 35756 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10024043 \nu^{7} + 79134062 \nu^{6} + 8296263188 \nu^{5} - 44420443495 \nu^{4} + \cdots + 100601848164448 ) / 368334844960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2454119 \nu^{7} + 23811754 \nu^{6} + 1978218156 \nu^{5} - 14359538067 \nu^{4} + \cdots + 17165478315440 ) / 36833484496 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27119491 \nu^{7} + 169640374 \nu^{6} + 20153927556 \nu^{5} - 113638316255 \nu^{4} + \cdots - 7207035392544 ) / 368334844960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17612663 \nu^{7} - 158824522 \nu^{6} - 14036094188 \nu^{5} + 101624534195 \nu^{4} + \cdots + 16775798636272 ) / 184167422480 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53358609 \nu^{7} - 327590586 \nu^{6} - 41846805564 \nu^{5} + 207378397285 \nu^{4} + \cdots - 394409070388064 ) / 368334844960 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 208 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} + 4\beta_{3} - 3\beta_{2} + 344\beta _1 + 397 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{7} + 43\beta_{6} + 2\beta_{5} + 53\beta_{4} - 16\beta_{3} + 491\beta_{2} + 670\beta _1 + 71496 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 439 \beta_{7} - 811 \beta_{6} + 228 \beta_{5} - 1421 \beta_{4} + 2349 \beta_{3} - 2125 \beta_{2} + \cdots + 127066 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1523 \beta_{7} + 26139 \beta_{6} + 2243 \beta_{5} + 37064 \beta_{4} - 13062 \beta_{3} + \cdots + 27742919 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 191690 \beta_{7} - 469203 \beta_{6} + 11338 \beta_{5} - 745921 \beta_{4} + 1130328 \beta_{3} + \cdots + 23872964 \) 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
19.9556
19.4241
11.0962
−0.570902
−6.60982
−7.41631
−14.5712
−21.3077
−19.9556 0 270.224 −522.229 0 653.513 −2838.16 0 10421.4
1.2 −19.4241 0 249.296 31.9528 0 −461.175 −2356.08 0 −620.655
1.3 −11.0962 0 −4.87416 −165.526 0 952.148 1474.40 0 1836.71
1.4 0.570902 0 −127.674 −128.909 0 1733.23 −145.965 0 −73.5944
1.5 6.60982 0 −84.3103 124.345 0 −780.885 −1403.33 0 821.899
1.6 7.41631 0 −72.9984 −376.733 0 −902.074 −1490.67 0 −2793.97
1.7 14.5712 0 84.3195 404.860 0 −387.911 −636.476 0 5899.30
1.8 21.3077 0 326.017 188.239 0 639.155 4219.28 0 4010.94
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(-1\)

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 207.8.a.f 8
3.b odd 2 1 23.8.a.b ✓ 8
12.b even 2 1 368.8.a.h 8
15.d odd 2 1 575.8.a.b 8
69.c even 2 1 529.8.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.8.a.b ✓ 8 3.b odd 2 1
207.8.a.f 8 1.a even 1 1 trivial
368.8.a.h 8 12.b even 2 1
529.8.a.c 8 69.c even 2 1
575.8.a.b 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 832T_{2}^{6} + 1059T_{2}^{5} + 203052T_{2}^{4} - 678328T_{2}^{3} - 13424272T_{2}^{2} + 73308944T_{2} - 37372224 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 832 T^{6} + \cdots - 37372224 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 86\!\cdots\!92 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 23\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 39\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T - 12167)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 38\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 15\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 59\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 90\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 16\!\cdots\!60 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 38\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 17\!\cdots\!92 \) Copy content Toggle raw display
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