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Properties

Label 252.8.k.b
Level $252$
Weight $8$
Character orbit 252.k
Analytic conductor $78.721$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 659x^{6} + 12718x^{5} + 417701x^{4} + 3735784x^{3} + 32480596x^{2} + 479136x + 7056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 84)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + 49 \beta_{2} + \beta_1) q^{5} + ( - 3 \beta_{7} + 2 \beta_{5} + \cdots + 47) q^{7} + ( - 7 \beta_{7} + 6 \beta_{6} + \cdots - 99) q^{11} + (13 \beta_{7} - 13 \beta_{6} + \cdots + 500) q^{13}+ \cdots + (17755 \beta_{7} - 17755 \beta_{6} + \cdots + 2967567) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 196 q^{5} - 434 q^{7} - 406 q^{11} + 3948 q^{13} + 7436 q^{17} + 15874 q^{19} + 6788 q^{23} + 69898 q^{25} + 189088 q^{29} - 55890 q^{31} + 750596 q^{35} + 93742 q^{37} - 13944 q^{41} - 487844 q^{43}+ \cdots + 23722580 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 659x^{6} + 12718x^{5} + 417701x^{4} + 3735784x^{3} + 32480596x^{2} + 479136x + 7056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 37941104905 \nu^{7} + 1267919624362 \nu^{6} - 25668866266087 \nu^{5} + \cdots - 80\!\cdots\!92 ) / 38\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 134367782106 \nu^{7} - 270767347357 \nu^{6} + 88569667465984 \nu^{5} + \cdots + 64\!\cdots\!64 ) / 64\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 72440898568484 \nu^{7} + 142292235017555 \nu^{6} + \cdots - 89\!\cdots\!56 ) / 19\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 145815130640753 \nu^{7} + 292628177231816 \nu^{6} + \cdots + 11\!\cdots\!16 ) / 38\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 84300389908064 \nu^{7} - 169899669742907 \nu^{6} + \cdots + 68\!\cdots\!80 ) / 19\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 375763963350301 \nu^{7} - 749083617441388 \nu^{6} + \cdots + 69\!\cdots\!76 ) / 38\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 376398131139491 \nu^{7} - 759212300883416 \nu^{6} + \cdots + 11\!\cdots\!92 ) / 38\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} - 3\beta_{6} + 8\beta_{5} - \beta_{4} - 3\beta_{3} - 61\beta_{2} + 59 ) / 126 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + 5\beta_{6} + 85\beta_{5} - 3\beta_{4} - 8\beta_{3} - 20690\beta_{2} - 85\beta _1 - 3 ) / 63 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -1159\beta_{7} + 1159\beta_{6} - 554\beta_{4} + 554\beta_{3} - 5744\beta _1 - 662798 ) / 126 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2983 \beta_{7} + 12234 \beta_{6} - 84235 \beta_{5} + 9251 \beta_{4} + 12234 \beta_{3} + 14392853 \beta_{2} - 14389870 ) / 63 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1205427 \beta_{7} + 317930 \beta_{6} - 5069612 \beta_{5} + 1205427 \beta_{4} + 887497 \beta_{3} + \cdots + 1205427 ) / 126 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10253042 \beta_{7} - 10253042 \beta_{6} + 3851383 \beta_{4} - 3851383 \beta_{3} + 76632565 \beta _1 + 12053285746 ) / 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 257726282 \beta_{7} - 985681749 \beta_{6} + 4588561496 \beta_{5} - 727955467 \beta_{4} + \cdots + 688336597889 ) / 126 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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
−0.00737575 − 0.0127752i
−5.63535 − 9.76071i
−8.39785 − 14.5455i
15.0406 + 26.0510i
−0.00737575 + 0.0127752i
−5.63535 + 9.76071i
−8.39785 + 14.5455i
15.0406 − 26.0510i
0 0 0 −80.0854 + 138.712i 0 −254.682 + 871.022i 0 0 0
37.2 0 0 0 −64.7849 + 112.211i 0 −816.586 − 395.892i 0 0 0
37.3 0 0 0 20.0775 − 34.7753i 0 641.417 − 641.971i 0 0 0
37.4 0 0 0 222.793 − 385.888i 0 212.851 + 882.178i 0 0 0
109.1 0 0 0 −80.0854 − 138.712i 0 −254.682 − 871.022i 0 0 0
109.2 0 0 0 −64.7849 − 112.211i 0 −816.586 + 395.892i 0 0 0
109.3 0 0 0 20.0775 + 34.7753i 0 641.417 + 641.971i 0 0 0
109.4 0 0 0 222.793 + 385.888i 0 212.851 − 882.178i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.k.b 8
3.b odd 2 1 84.8.i.a ✓ 8
7.c even 3 1 inner 252.8.k.b 8
21.c even 2 1 588.8.i.o 8
21.g even 6 1 588.8.a.j 4
21.g even 6 1 588.8.i.o 8
21.h odd 6 1 84.8.i.a ✓ 8
21.h odd 6 1 588.8.a.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.8.i.a ✓ 8 3.b odd 2 1
84.8.i.a ✓ 8 21.h odd 6 1
252.8.k.b 8 1.a even 1 1 trivial
252.8.k.b 8 7.c even 3 1 inner
588.8.a.i 4 21.h odd 6 1
588.8.a.j 4 21.g even 6 1
588.8.i.o 8 21.c even 2 1
588.8.i.o 8 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 196 T_{5}^{7} + 140509 T_{5}^{6} + 29803308 T_{5}^{5} + 9091930509 T_{5}^{4} + \cdots + 13\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 45\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 31670239466400)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 97\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 28\!\cdots\!32)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 81\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 74\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 33\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 96\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 28\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 28\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 95\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 45\!\cdots\!56)^{2} \) Copy content Toggle raw display
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