↑

Properties

Label 252.8.f.a
Level $252$
Weight $8$
Character orbit 252.f
Analytic conductor $78.721$
Analytic rank $0$
Dimension $20$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(125,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.125");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.f (of order \(2\), degree \(1\), 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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 71480 x^{18} + 1912507236 x^{16} + 23093807115120 x^{14} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{52}\cdot 3^{47}\cdot 7^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{6} - 123) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (\beta_{6} - 123) q^{7} - \beta_{9} q^{11} + (\beta_{13} + \beta_{6}) q^{13} + (\beta_{2} + 6 \beta_1) q^{17} + ( - \beta_{13} + \beta_{8} + \beta_{6} + 1) q^{19} + (\beta_{17} - 2 \beta_{9} - 14 \beta_{4}) q^{23} + ( - \beta_{10} + 8 \beta_{6} + \cdots + 27442) q^{25}+ \cdots + (75 \beta_{18} - 110 \beta_{13} + \cdots - 971) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2468 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2468 q^{7} + 548780 q^{25} - 257344 q^{37} - 1589224 q^{43} - 291556 q^{49} - 11860256 q^{67} - 1991600 q^{79} + 11742576 q^{85} - 11268816 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 71480 x^{18} + 1912507236 x^{16} + 23093807115120 x^{14} + \cdots + 18\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 56\!\cdots\!57 \nu^{18} + \cdots - 18\!\cdots\!56 ) / 11\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!03 \nu^{18} + \cdots - 26\!\cdots\!64 ) / 16\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 56\!\cdots\!95 \nu^{18} + \cdots + 34\!\cdots\!80 ) / 17\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!63 \nu^{19} + \cdots - 10\!\cdots\!56 \nu ) / 19\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!79 \nu^{19} + \cdots - 52\!\cdots\!40 ) / 62\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29\!\cdots\!79 \nu^{19} + \cdots - 24\!\cdots\!80 ) / 62\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29\!\cdots\!79 \nu^{19} + \cdots + 57\!\cdots\!60 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 45\!\cdots\!91 \nu^{19} + \cdots + 24\!\cdots\!20 ) / 31\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!84 \nu^{19} + \cdots - 68\!\cdots\!28 \nu ) / 71\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29\!\cdots\!79 \nu^{19} + \cdots + 24\!\cdots\!00 ) / 89\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 41\!\cdots\!75 \nu^{19} + \cdots - 10\!\cdots\!44 ) / 62\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 22\!\cdots\!77 \nu^{19} + \cdots + 56\!\cdots\!56 \nu ) / 31\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17\!\cdots\!81 \nu^{19} + \cdots + 81\!\cdots\!60 ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41\!\cdots\!75 \nu^{19} + \cdots + 17\!\cdots\!44 ) / 31\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 13\!\cdots\!85 \nu^{19} + \cdots + 75\!\cdots\!44 ) / 62\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 35\!\cdots\!59 \nu^{19} + \cdots - 19\!\cdots\!24 \nu ) / 14\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 21\!\cdots\!90 \nu^{19} + \cdots - 11\!\cdots\!84 \nu ) / 79\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 63\!\cdots\!93 \nu^{19} + \cdots + 16\!\cdots\!44 \nu ) / 20\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 88\!\cdots\!30 \nu^{19} + \cdots + 39\!\cdots\!08 ) / 15\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 18\beta_{18} + 33\beta_{13} + 3\beta_{12} + 48\beta_{10} + 45\beta_{8} + 459\beta_{6} - 112\beta_{4} + 189 ) / 27216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 3 \beta_{11} - 1566 \beta_{10} - \beta_{9} + 60 \beta_{7} + 10467 \beta_{6} + \cdots - 32419458 ) / 4536 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 189 \beta_{19} - 78651 \beta_{18} + 243 \beta_{17} - 1458 \beta_{16} + 378 \beta_{15} + \cdots - 868833 ) / 6804 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2352 \beta_{19} - 29192 \beta_{15} + 7644 \beta_{14} + 53472 \beta_{11} + 21287988 \beta_{10} + \cdots + 291243232584 ) / 2268 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16502220 \beta_{19} + 2947030929 \beta_{18} - 34868880 \beta_{17} + 85805730 \beta_{16} + \cdots + 23137487730 ) / 13608 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7687106 \beta_{19} + 219581346 \beta_{15} - 86708818 \beta_{14} - 546823250 \beta_{11} + \cdots - 18\!\cdots\!94 ) / 756 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 273871666026 \beta_{19} - 28564879738716 \beta_{18} + 659808112446 \beta_{17} + \cdots - 148021815908295 ) / 6804 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1264706261952 \beta_{19} - 6701981109068 \beta_{15} + 3794390001192 \beta_{14} + \cdots + 54\!\cdots\!76 ) / 1134 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 79\!\cdots\!60 \beta_{19} + \cdots + 17\!\cdots\!32 ) / 6804 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 49\!\cdots\!82 \beta_{19} + \cdots - 10\!\cdots\!10 ) / 1134 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10\!\cdots\!74 \beta_{19} + \cdots - 94\!\cdots\!41 ) / 3402 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 25\!\cdots\!56 \beta_{19} + \cdots + 36\!\cdots\!88 ) / 189 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 27\!\cdots\!24 \beta_{19} + \cdots + 68\!\cdots\!40 ) / 3402 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 21\!\cdots\!94 \beta_{19} + \cdots - 22\!\cdots\!82 ) / 567 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 34\!\cdots\!26 \beta_{19} + \cdots + 19\!\cdots\!79 ) / 1701 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 54\!\cdots\!64 \beta_{19} + \cdots + 47\!\cdots\!16 ) / 567 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 83\!\cdots\!64 \beta_{19} + \cdots - 16\!\cdots\!64 ) / 1701 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 15\!\cdots\!32 \beta_{19} + \cdots - 11\!\cdots\!00 ) / 63 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 19\!\cdots\!12 \beta_{19} + \cdots + 53\!\cdots\!66 ) / 1701 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
− 119.399i
119.399i
144.051i
− 144.051i
16.5035i
− 16.5035i
0.733783i
− 0.733783i
21.9392i
− 21.9392i
19.1107i
− 19.1107i
− 2.09464i
2.09464i
13.6751i
− 13.6751i
146.879i
− 146.879i
− 116.571i
116.571i
0 0 0 −524.015 0 423.305 − 802.718i 0 0 0
125.2 0 0 0 −524.015 0 423.305 + 802.718i 0 0 0
125.3 0 0 0 −438.356 0 −907.259 − 20.5933i 0 0 0
125.4 0 0 0 −438.356 0 −907.259 + 20.5933i 0 0 0
125.5 0 0 0 −209.330 0 775.201 − 471.812i 0 0 0
125.6 0 0 0 −209.330 0 775.201 + 471.812i 0 0 0
125.7 0 0 0 −129.165 0 −512.064 − 749.222i 0 0 0
125.8 0 0 0 −129.165 0 −512.064 + 749.222i 0 0 0
125.9 0 0 0 −23.8524 0 −396.183 − 816.445i 0 0 0
125.10 0 0 0 −23.8524 0 −396.183 + 816.445i 0 0 0
125.11 0 0 0 23.8524 0 −396.183 − 816.445i 0 0 0
125.12 0 0 0 23.8524 0 −396.183 + 816.445i 0 0 0
125.13 0 0 0 129.165 0 −512.064 − 749.222i 0 0 0
125.14 0 0 0 129.165 0 −512.064 + 749.222i 0 0 0
125.15 0 0 0 209.330 0 775.201 − 471.812i 0 0 0
125.16 0 0 0 209.330 0 775.201 + 471.812i 0 0 0
125.17 0 0 0 438.356 0 −907.259 − 20.5933i 0 0 0
125.18 0 0 0 438.356 0 −907.259 + 20.5933i 0 0 0
125.19 0 0 0 524.015 0 423.305 − 802.718i 0 0 0
125.20 0 0 0 524.015 0 423.305 + 802.718i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.f.a ✓ 20
3.b odd 2 1 inner 252.8.f.a ✓ 20
7.b odd 2 1 inner 252.8.f.a ✓ 20
21.c even 2 1 inner 252.8.f.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.8.f.a ✓ 20 1.a even 1 1 trivial
252.8.f.a ✓ 20 3.b odd 2 1 inner
252.8.f.a ✓ 20 7.b odd 2 1 inner
252.8.f.a ✓ 20 21.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 37\!\cdots\!43)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 84\!\cdots\!88)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 22\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 60\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 14\!\cdots\!72)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 40\!\cdots\!28)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 98\!\cdots\!04)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 43\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 12\!\cdots\!56)^{4} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 75\!\cdots\!88)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 37\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 21\!\cdots\!08)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 26\!\cdots\!96)^{4} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 47\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 22\!\cdots\!24)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 53\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 31\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 25\!\cdots\!64)^{2} \) Copy content Toggle raw display
show more
show less