↑

Properties

Label 100.6.e.d
Level $100$
Weight $6$
Character orbit 100.e
Analytic conductor $16.038$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,6,Mod(7,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 100.e (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(16.0383819813\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 30 x^{14} + 431 x^{12} - 12720 x^{10} + 443951 x^{8} - 7845990 x^{6} + 82371841 x^{4} + \cdots + 4975327296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{32} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - \beta_{10} q^{3} + (\beta_{11} + 9 \beta_{6}) q^{4} + ( - \beta_{3} + \beta_1 - 12) q^{6} + ( - 7 \beta_{9} - 9 \beta_{8} + 3 \beta_{4}) q^{7} + ( - \beta_{12} - 2 \beta_{10} - 10 \beta_{7}) q^{8} + (\beta_{14} + 4 \beta_{11} - 15 \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} - \beta_{10} q^{3} + (\beta_{11} + 9 \beta_{6}) q^{4} + ( - \beta_{3} + \beta_1 - 12) q^{6} + ( - 7 \beta_{9} - 9 \beta_{8} + 3 \beta_{4}) q^{7} + ( - \beta_{12} - 2 \beta_{10} - 10 \beta_{7}) q^{8} + (\beta_{14} + 4 \beta_{11} - 15 \beta_{6}) q^{9} + (4 \beta_{3} - \beta_{2} + 4 \beta_1 + 2) q^{11} + (\beta_{13} - 30 \beta_{9} + \cdots + 8 \beta_{4}) q^{12}+ \cdots + ( - 796 \beta_{15} + \cdots + 466 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 200 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 200 q^{6} - 2064 q^{16} - 20960 q^{21} - 5888 q^{26} - 60264 q^{36} - 30208 q^{41} + 24440 q^{46} - 71120 q^{56} + 131072 q^{61} + 423680 q^{66} - 329600 q^{76} + 393616 q^{81} + 643720 q^{86} - 711200 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 30 x^{14} + 431 x^{12} - 12720 x^{10} + 443951 x^{8} - 7845990 x^{6} + 82371841 x^{4} + \cdots + 4975327296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13333283 \nu^{14} + 612525077 \nu^{12} - 7832917034 \nu^{10} + 182565794058 \nu^{8} + \cdots + 10\!\cdots\!04 ) / 284283094892544 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1147765 \nu^{14} + 20066539 \nu^{12} - 162379750 \nu^{10} + 9407609814 \nu^{8} + \cdots - 17277045087552 ) / 4441923357696 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4830357 \nu^{14} + 24121921 \nu^{12} + 869553094 \nu^{10} - 35140881678 \nu^{8} + \cdots + 154943270021184 ) / 17767693430784 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7387310251 \nu^{15} + 1075804112469 \nu^{14} - 1736495260341 \nu^{13} + \cdots - 57\!\cdots\!44 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7387310251 \nu^{15} - 1075804112469 \nu^{14} - 1736495260341 \nu^{13} + \cdots + 57\!\cdots\!44 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 331482203 \nu^{15} + 175377915 \nu^{13} - 34057844870 \nu^{11} - 2784751366410 \nu^{9} + \cdots - 68\!\cdots\!40 \nu ) / 62\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49817032235 \nu^{15} - 219535920171 \nu^{14} + 1714046887221 \nu^{13} + \cdots + 72\!\cdots\!84 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49817032235 \nu^{15} + 219535920171 \nu^{14} + 1714046887221 \nu^{13} + \cdots - 72\!\cdots\!84 ) / 20\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35202085105 \nu^{15} + 602447590991 \nu^{14} + 239065862673 \nu^{13} + \cdots - 15\!\cdots\!52 ) / 66\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 35202085105 \nu^{15} - 602447590991 \nu^{14} + 239065862673 \nu^{13} + \cdots + 15\!\cdots\!52 ) / 66\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 69775213421 \nu^{15} - 1461867549915 \nu^{13} + 24544830078646 \nu^{11} + \cdots - 35\!\cdots\!28 \nu ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 267418062063 \nu^{15} - 1090019790959 \nu^{14} + 5911943628849 \nu^{13} + \cdots + 59\!\cdots\!64 ) / 33\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 267418062063 \nu^{15} + 1090019790959 \nu^{14} + 5911943628849 \nu^{13} + \cdots - 59\!\cdots\!64 ) / 33\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 32855285351 \nu^{15} - 711740389497 \nu^{13} + 6510139455010 \nu^{11} + \cdots - 18\!\cdots\!36 \nu ) / 31\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 541793935 \nu^{15} + 8001873129 \nu^{13} - 80085262610 \nu^{11} + \cdots + 15\!\cdots\!76 \nu ) / 31\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + 8\beta_{6} + \beta_{5} + \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{8} - 2\beta_{7} - 2\beta_{5} + 2\beta_{4} + \beta_{2} + 30 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 4 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + \cdots + 2 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8 \beta_{13} - 8 \beta_{12} + 16 \beta_{10} - 16 \beta_{9} + 16 \beta_{8} - 16 \beta_{7} - 16 \beta_{5} + \cdots + 70 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 40 \beta_{15} + 5 \beta_{14} - 14 \beta_{13} - 14 \beta_{12} + 320 \beta_{11} - 100 \beta_{10} + \cdots - 66 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4 \beta_{13} + 4 \beta_{12} + 760 \beta_{10} - 760 \beta_{9} + 5156 \beta_{8} - 5156 \beta_{7} + \cdots + 25986 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 784 \beta_{15} - 497 \beta_{14} - 174 \beta_{13} - 174 \beta_{12} - 896 \beta_{11} + \cdots + 4494 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1168 \beta_{13} + 1168 \beta_{12} + 9952 \beta_{10} - 9952 \beta_{9} - 12528 \beta_{8} + \cdots - 621818 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 5880 \beta_{15} + 35853 \beta_{14} + 11626 \beta_{13} + 11626 \beta_{12} - 42048 \beta_{11} + \cdots - 107882 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 83332 \beta_{13} - 83332 \beta_{12} + 220680 \beta_{10} - 220680 \beta_{9} - 890500 \beta_{8} + \cdots - 3348110 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 387904 \beta_{15} - 363913 \beta_{14} - 374022 \beta_{13} - 374022 \beta_{12} + 3080704 \beta_{11} + \cdots - 849178 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1101848 \beta_{13} + 1101848 \beta_{12} + 3957712 \beta_{10} - 3957712 \beta_{9} + 43279000 \beta_{8} + \cdots + 212522454 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 175032 \beta_{15} - 7729371 \beta_{14} + 901890 \beta_{13} + 901890 \beta_{12} + \cdots + 47238078 \beta_{4} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 14556852 \beta_{13} + 14556852 \beta_{12} - 68940392 \beta_{10} + 68940392 \beta_{9} + \cdots - 8531576798 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 131297680 \beta_{15} + 392574495 \beta_{14} + 141201058 \beta_{13} + 141201058 \beta_{12} + \cdots - 1545736578 \beta_{4} ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−2.20505 − 2.33733i
−3.98595 + 0.664991i
−3.98595 + 1.33501i
2.20505 − 2.33733i
−2.20505 + 4.33733i
3.98595 + 0.664991i
3.98595 + 1.33501i
2.20505 + 4.33733i
−2.20505 + 2.33733i
−3.98595 − 0.664991i
−3.98595 − 1.33501i
2.20505 + 2.33733i
−2.20505 − 4.33733i
3.98595 − 0.664991i
3.98595 − 1.33501i
2.20505 − 4.33733i
−5.54238 + 1.13228i 5.17646 + 5.17646i 29.4359 − 12.5511i 0 −34.5511 − 22.8287i 43.8607 − 43.8607i −148.933 + 102.893i − 189.409i 0
7.2 −4.32096 − 3.65094i −14.2550 − 14.2550i 5.34130 + 31.5511i 0 9.55108 + 113.639i 107.825 − 107.825i 92.1115 − 155.832i 163.409i 0
7.3 −3.65094 − 4.32096i 14.2550 + 14.2550i −5.34130 + 31.5511i 0 9.55108 − 113.639i −107.825 + 107.825i 155.832 − 92.1115i 163.409i 0
7.4 −1.13228 + 5.54238i 5.17646 + 5.17646i −29.4359 − 12.5511i 0 −34.5511 + 22.8287i 43.8607 − 43.8607i 102.893 − 148.933i − 189.409i 0
7.5 1.13228 − 5.54238i −5.17646 − 5.17646i −29.4359 − 12.5511i 0 −34.5511 + 22.8287i −43.8607 + 43.8607i −102.893 + 148.933i − 189.409i 0
7.6 3.65094 + 4.32096i −14.2550 − 14.2550i −5.34130 + 31.5511i 0 9.55108 − 113.639i 107.825 − 107.825i −155.832 + 92.1115i 163.409i 0
7.7 4.32096 + 3.65094i 14.2550 + 14.2550i 5.34130 + 31.5511i 0 9.55108 + 113.639i −107.825 + 107.825i −92.1115 + 155.832i 163.409i 0
7.8 5.54238 − 1.13228i −5.17646 − 5.17646i 29.4359 − 12.5511i 0 −34.5511 − 22.8287i −43.8607 + 43.8607i 148.933 − 102.893i − 189.409i 0
43.1 −5.54238 − 1.13228i 5.17646 − 5.17646i 29.4359 + 12.5511i 0 −34.5511 + 22.8287i 43.8607 + 43.8607i −148.933 − 102.893i 189.409i 0
43.2 −4.32096 + 3.65094i −14.2550 + 14.2550i 5.34130 − 31.5511i 0 9.55108 − 113.639i 107.825 + 107.825i 92.1115 + 155.832i − 163.409i 0
43.3 −3.65094 + 4.32096i 14.2550 − 14.2550i −5.34130 − 31.5511i 0 9.55108 + 113.639i −107.825 − 107.825i 155.832 + 92.1115i − 163.409i 0
43.4 −1.13228 − 5.54238i 5.17646 − 5.17646i −29.4359 + 12.5511i 0 −34.5511 − 22.8287i 43.8607 + 43.8607i 102.893 + 148.933i 189.409i 0
43.5 1.13228 + 5.54238i −5.17646 + 5.17646i −29.4359 + 12.5511i 0 −34.5511 − 22.8287i −43.8607 − 43.8607i −102.893 − 148.933i 189.409i 0
43.6 3.65094 − 4.32096i −14.2550 + 14.2550i −5.34130 − 31.5511i 0 9.55108 + 113.639i 107.825 + 107.825i −155.832 − 92.1115i − 163.409i 0
43.7 4.32096 − 3.65094i 14.2550 − 14.2550i 5.34130 − 31.5511i 0 9.55108 − 113.639i −107.825 − 107.825i −92.1115 − 155.832i − 163.409i 0
43.8 5.54238 + 1.13228i −5.17646 + 5.17646i 29.4359 + 12.5511i 0 −34.5511 + 22.8287i −43.8607 − 43.8607i 148.933 + 102.893i 189.409i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.6.e.d ✓ 16
4.b odd 2 1 inner 100.6.e.d ✓ 16
5.b even 2 1 inner 100.6.e.d ✓ 16
5.c odd 4 2 inner 100.6.e.d ✓ 16
20.d odd 2 1 inner 100.6.e.d ✓ 16
20.e even 4 2 inner 100.6.e.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.6.e.d ✓ 16 1.a even 1 1 trivial
100.6.e.d ✓ 16 4.b odd 2 1 inner
100.6.e.d ✓ 16 5.b even 2 1 inner
100.6.e.d ✓ 16 5.c odd 4 2 inner
100.6.e.d ✓ 16 20.d odd 2 1 inner
100.6.e.d ✓ 16 20.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 168040T_{3}^{4} + 474368400 \) acting on \(S_{6}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 1099511627776 \) Copy content Toggle raw display
$3$ \( (T^{8} + 168040 T^{4} + 474368400)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 80\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 337600 T^{2} + 18453381120)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 39\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 62\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 32494379089920)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 538297723147536)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 16\!\cdots\!20)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 16\!\cdots\!96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3776 T - 104764176)^{8} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 25\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 12\!\cdots\!20)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 16384 T - 596245056)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 96\!\cdots\!20)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 46\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 21\!\cdots\!20)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 15\!\cdots\!16)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 65\!\cdots\!56)^{2} \) Copy content Toggle raw display
show more
show less