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Properties

Label 252.8.k.e
Level $252$
Weight $8$
Character orbit 252.k
Analytic conductor $78.721$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(37,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), 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: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 89566 x^{14} - 207320 x^{13} + 5161603375 x^{12} - 17143558340 x^{11} + 178819626045814 x^{10} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{13}\cdot 7^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{2}) q^{5} + (2 \beta_{9} - 3 \beta_{3} + 140 \beta_1 + 35) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{2}) q^{5} + (2 \beta_{9} - 3 \beta_{3} + 140 \beta_1 + 35) q^{7} - \beta_{10} q^{11} + (7 \beta_{9} + 5 \beta_{8} + \cdots - 1765) q^{13}+ \cdots + (515 \beta_{9} + 17595 \beta_{8} + \cdots - 1697780) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 1680 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 1680 q^{7} - 28280 q^{13} + 42224 q^{19} - 80460 q^{25} + 164752 q^{31} - 647980 q^{37} + 1341440 q^{43} + 230104 q^{49} - 323120 q^{55} - 4319336 q^{61} - 3905760 q^{67} + 6471780 q^{73} - 6093104 q^{79} + 456400 q^{85} + 15969856 q^{91} - 27141240 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 89566 x^{14} - 207320 x^{13} + 5161603375 x^{12} - 17143558340 x^{11} + 178819626045814 x^{10} + \cdots + 34\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 63\!\cdots\!15 \nu^{15} + \cdots - 15\!\cdots\!72 ) / 18\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 75\!\cdots\!99 \nu^{15} + \cdots + 51\!\cdots\!36 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 85\!\cdots\!63 \nu^{15} + \cdots - 33\!\cdots\!64 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 77\!\cdots\!57 \nu^{15} + \cdots - 11\!\cdots\!84 ) / 68\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30\!\cdots\!77 \nu^{15} + \cdots + 29\!\cdots\!20 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 82\!\cdots\!73 \nu^{15} + \cdots + 77\!\cdots\!20 ) / 30\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!13 \nu^{15} + \cdots - 16\!\cdots\!44 ) / 24\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38\!\cdots\!90 \nu^{15} + \cdots + 12\!\cdots\!32 ) / 38\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33\!\cdots\!33 \nu^{15} + \cdots - 82\!\cdots\!00 ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 48\!\cdots\!03 \nu^{15} + \cdots + 36\!\cdots\!52 ) / 72\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 91\!\cdots\!09 \nu^{15} + \cdots - 14\!\cdots\!40 ) / 80\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!97 \nu^{15} + \cdots + 64\!\cdots\!88 ) / 36\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 81\!\cdots\!09 \nu^{15} + \cdots - 14\!\cdots\!24 ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 46\!\cdots\!07 \nu^{15} + \cdots - 20\!\cdots\!08 ) / 72\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 59\!\cdots\!53 \nu^{15} + \cdots + 92\!\cdots\!88 ) / 80\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5\beta_{9} - 5\beta_{8} + \beta_{5} + 63\beta_{4} - 4\beta_{3} + 63\beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{14} - 33 \beta_{12} - 57 \beta_{10} + 1052 \beta_{9} + 2089 \beta_{8} + \cdots - 2821224 \beta_1 ) / 126 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3249 \beta_{15} - 864 \beta_{13} - 16263 \beta_{11} - 37634 \beta_{9} + 619508 \beta_{8} + \cdots + 9763992 ) / 252 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 194508 \beta_{15} + 194508 \beta_{14} + 1807515 \beta_{13} + 1807515 \beta_{12} + 1523958 \beta_{11} + \cdots - 72489863670 ) / 126 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 163253895 \beta_{14} - 34799100 \beta_{12} - 641970105 \beta_{10} - 28291292084 \beta_{9} + \cdots - 226848463380 \beta_1 ) / 252 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9066555951 \beta_{15} - 77238171597 \beta_{13} - 25403468157 \beta_{11} - 3847036593817 \beta_{9} + \cdots + 20\!\cdots\!42 ) / 126 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6559965413589 \beta_{15} - 6559965413589 \beta_{14} + 1037354143224 \beta_{13} + \cdots + 11\!\cdots\!56 ) / 252 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 376064851106226 \beta_{14} + \cdots - 64\!\cdots\!32 \beta_1 ) / 126 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 24\!\cdots\!83 \beta_{15} + \cdots - 13\!\cdots\!84 ) / 252 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 14\!\cdots\!25 \beta_{15} + \cdots - 21\!\cdots\!34 ) / 126 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 90\!\cdots\!85 \beta_{14} + \cdots + 80\!\cdots\!60 \beta_1 ) / 252 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 57\!\cdots\!12 \beta_{15} + \cdots + 72\!\cdots\!46 ) / 126 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 33\!\cdots\!71 \beta_{15} + \cdots + 40\!\cdots\!12 ) / 252 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 21\!\cdots\!27 \beta_{14} + \cdots - 25\!\cdots\!68 \beta_1 ) / 126 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 12\!\cdots\!65 \beta_{15} + \cdots - 18\!\cdots\!40 ) / 252 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−77.2060 + 133.725i
−92.9954 + 161.073i
−73.4015 + 127.135i
−50.1630 + 86.8849i
67.3630 − 116.676i
62.2223 − 107.772i
68.2530 − 118.218i
95.9276 − 166.152i
−77.2060 − 133.725i
−92.9954 − 161.073i
−73.4015 − 127.135i
−50.1630 − 86.8849i
67.3630 + 116.676i
62.2223 + 107.772i
68.2530 + 118.218i
95.9276 + 166.152i
0 0 0 −173.134 + 299.876i 0 334.186 − 843.720i 0 0 0
37.2 0 0 0 −161.248 + 279.290i 0 −882.807 + 210.227i 0 0 0
37.3 0 0 0 −135.624 + 234.907i 0 87.0016 + 903.313i 0 0 0
37.4 0 0 0 −117.526 + 203.561i 0 881.619 + 215.154i 0 0 0
37.5 0 0 0 117.526 − 203.561i 0 881.619 + 215.154i 0 0 0
37.6 0 0 0 135.624 − 234.907i 0 87.0016 + 903.313i 0 0 0
37.7 0 0 0 161.248 − 279.290i 0 −882.807 + 210.227i 0 0 0
37.8 0 0 0 173.134 − 299.876i 0 334.186 − 843.720i 0 0 0
109.1 0 0 0 −173.134 − 299.876i 0 334.186 + 843.720i 0 0 0
109.2 0 0 0 −161.248 − 279.290i 0 −882.807 − 210.227i 0 0 0
109.3 0 0 0 −135.624 − 234.907i 0 87.0016 − 903.313i 0 0 0
109.4 0 0 0 −117.526 − 203.561i 0 881.619 − 215.154i 0 0 0
109.5 0 0 0 117.526 + 203.561i 0 881.619 − 215.154i 0 0 0
109.6 0 0 0 135.624 + 234.907i 0 87.0016 − 903.313i 0 0 0
109.7 0 0 0 161.248 + 279.290i 0 −882.807 − 210.227i 0 0 0
109.8 0 0 0 173.134 + 299.876i 0 334.186 + 843.720i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.k.e ✓ 16
3.b odd 2 1 inner 252.8.k.e ✓ 16
7.c even 3 1 inner 252.8.k.e ✓ 16
21.h odd 6 1 inner 252.8.k.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.8.k.e ✓ 16 1.a even 1 1 trivial
252.8.k.e ✓ 16 3.b odd 2 1 inner
252.8.k.e ✓ 16 7.c even 3 1 inner
252.8.k.e ✓ 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 352730 T_{5}^{14} + 79038704955 T_{5}^{12} + \cdots + 25\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 45\!\cdots\!01)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 247198114185804)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 20\!\cdots\!21)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 38\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 50\!\cdots\!16)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 80\!\cdots\!96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 52\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 19\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 37\!\cdots\!01)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 10\!\cdots\!00)^{4} \) Copy content Toggle raw display
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