Properties

Label 1.100.a.a
Level $1$
Weight $100$
Character orbit 1.a
Self dual yes
Analytic conductor $62.068$
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] = [1,100,Mod(1,1)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 100, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1.1");
 
S:= CuspForms(chi, 100);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 100 \)
Character orbit: \([\chi]\) \(=\) 1.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: \(62.0676682981\)
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: \( x^{8} - x^{7} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{109}\cdot 3^{44}\cdot 5^{13}\cdot 7^{9}\cdot 11^{3}\cdot 13\cdot 17 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 26005077112815) q^{2} + (\beta_{2} - 10743446 \beta_1 - 35\!\cdots\!40) q^{3}+ \cdots + (48 \beta_{7} - 174420 \beta_{6} + 102829320 \beta_{5} + \cdots + 19\!\cdots\!97) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 26005077112815) q^{2} + (\beta_{2} - 10743446 \beta_1 - 35\!\cdots\!40) q^{3}+ \cdots + ( - 34\!\cdots\!16 \beta_{7} + \cdots - 17\!\cdots\!76) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3}+ \cdots + 15\!\cdots\!76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 208040616902520 q^{2} - 28\!\cdots\!20 q^{3}+ \cdots - 13\!\cdots\!08 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + \cdots + 23\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 72\nu - 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\!\cdots\!03 \nu^{7} + \cdots + 13\!\cdots\!16 ) / 59\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!79 \nu^{7} + \cdots - 21\!\cdots\!28 ) / 29\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 35\!\cdots\!17 \nu^{7} + \cdots + 10\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 65\!\cdots\!89 \nu^{7} + \cdots - 23\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24\!\cdots\!81 \nu^{7} + \cdots - 51\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16\!\cdots\!89 \nu^{7} + \cdots - 33\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 9 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 111186\beta_{2} + 119454850513664\beta _1 + 992470782456187687053696265272 ) / 5184 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 117 \beta_{6} - 237529 \beta_{5} + 3005209639 \beta_{4} - 14859349564627 \beta_{3} + \cdots + 11\!\cdots\!36 ) / 373248 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 10509299761779 \beta_{7} + \cdots + 21\!\cdots\!96 ) / 3359232 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21\!\cdots\!39 \beta_{7} + \cdots + 25\!\cdots\!28 ) / 15116544 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 51\!\cdots\!99 \beta_{7} + \cdots + 13\!\cdots\!04 ) / 5038848 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 20\!\cdots\!51 \beta_{7} + \cdots + 22\!\cdots\!96 ) / 30233088 \) Copy content Toggle raw display

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
−2.10181e13
−1.08597e13
−9.79932e12
−8.46093e12
4.33987e12
8.50040e12
1.56804e13
2.16174e13
−1.53931e15 1.09471e23 1.73565e30 −1.42424e34 −1.68510e38 −4.25157e41 −1.69606e45 −1.59809e47 2.19235e49
1.2 −8.07907e14 −6.25114e23 1.88877e28 −5.34820e34 5.05034e38 1.04094e42 4.96812e44 2.18975e47 4.32085e49
1.3 −7.31556e14 −2.11461e23 −9.86512e28 5.46163e34 1.54696e38 −9.42405e41 5.35848e44 −1.27077e47 −3.99548e49
1.4 −6.35192e14 5.52801e23 −2.30357e29 4.78100e33 −3.51135e38 7.17401e41 5.48921e44 1.33796e47 −3.03685e48
1.5 2.86466e14 2.01536e23 −5.51763e29 −4.79915e34 5.77333e37 −6.92478e41 −3.39630e44 −1.31176e47 −1.37479e49
1.6 5.86024e14 −4.40976e23 −2.90401e29 2.74639e34 −2.58422e38 3.47138e41 −5.41619e44 2.26671e46 1.60945e49
1.7 1.10299e15 5.08811e23 5.82752e29 3.48045e34 5.61211e38 4.01523e41 −5.63335e43 8.70965e46 3.83889e49
1.8 1.53045e15 −3.78025e23 1.70845e30 −5.47796e34 −5.78549e38 −5.03791e41 1.64466e45 −2.88893e46 −8.38375e49
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

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 1.100.a.a 8
3.b odd 2 1 9.100.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.100.a.a 8 1.a even 1 1 trivial
9.100.a.d 8 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{100}^{\mathrm{new}}(\Gamma_0(1))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 208040616902520 T^{7} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 33\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 68\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots - 51\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 29\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 51\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 29\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 62\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 64\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 56\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 85\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 73\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 27\!\cdots\!44 \) Copy content Toggle raw display
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