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Properties

Label 252.9.c.a
Level $252$
Weight $9$
Character orbit 252.c
Analytic conductor $102.659$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,9,Mod(197,252)]
 
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, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.197");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.c (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: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 4002260 x^{14} + 6534459751956 x^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{37}\cdot 7^{12} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} - \beta_{2} q^{7} + (\beta_{10} + 5 \beta_{9} + 3 \beta_1) q^{11} + ( - \beta_{4} + 2 \beta_{2} - 5968) q^{13} + ( - \beta_{12} + 13 \beta_{9} + 8 \beta_1) q^{17} + (\beta_{6} + \beta_{4} - 7 \beta_{2} - 17972) q^{19} + ( - \beta_{15} + 2 \beta_{13} + \cdots - 25 \beta_1) q^{23}+ \cdots + ( - 34 \beta_{8} + 107 \beta_{7} + \cdots + 8376986) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 95480 q^{13} - 287560 q^{19} - 1754520 q^{25} - 3554264 q^{31} - 182920 q^{37} + 8472416 q^{43} + 13176688 q^{49} - 18692072 q^{55} + 34224568 q^{61} + 22683096 q^{67} + 2137296 q^{73} - 90245624 q^{79} - 56204456 q^{85} - 25661888 q^{91} + 134041152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 4002260 x^{14} + 6534459751956 x^{12} + \cdots + 19\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 71\!\cdots\!41 \nu^{14} + \cdots + 13\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\!\cdots\!67 \nu^{14} + \cdots - 54\!\cdots\!00 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 45\!\cdots\!33 \nu^{14} + \cdots + 28\!\cdots\!00 ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 73\!\cdots\!19 \nu^{14} + \cdots + 21\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43\!\cdots\!59 \nu^{14} + \cdots + 16\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\!\cdots\!27 \nu^{14} + \cdots + 24\!\cdots\!00 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!93 \nu^{14} + \cdots + 90\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!07 \nu^{15} + \cdots + 63\!\cdots\!00 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!53 \nu^{15} + \cdots + 18\!\cdots\!00 \nu ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 89\!\cdots\!81 \nu^{15} + \cdots + 36\!\cdots\!00 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!09 \nu^{15} + \cdots + 32\!\cdots\!00 \nu ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14\!\cdots\!61 \nu^{15} + \cdots + 43\!\cdots\!00 \nu ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 47\!\cdots\!07 \nu^{15} + \cdots - 11\!\cdots\!00 \nu ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 84\!\cdots\!87 \nu^{15} + \cdots - 58\!\cdots\!00 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 75\beta_{2} - 500282 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 502 \beta_{15} - 31 \beta_{14} + 460 \beta_{13} + 1454 \beta_{12} - 1513 \beta_{11} + \cdots - 735213 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2788 \beta_{8} - 27244 \beta_{7} - 545604 \beta_{6} + 558072 \beta_{5} - 175942 \beta_{4} + \cdots + 368645611096 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 454235776 \beta_{15} - 131527410 \beta_{14} - 890878772 \beta_{13} - 1720652240 \beta_{12} + \cdots + 625133110974 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12957774648 \beta_{8} + 76508573832 \beta_{7} + 755360666664 \beta_{6} - 996846903360 \beta_{5} + \cdots - 31\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 419814568152048 \beta_{15} + 242553119836164 \beta_{14} + \cdots - 56\!\cdots\!28 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 24\!\cdots\!20 \beta_{8} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 41\!\cdots\!76 \beta_{15} + \cdots + 52\!\cdots\!28 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 34\!\cdots\!00 \beta_{8} + \cdots - 25\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 43\!\cdots\!16 \beta_{15} + \cdots - 50\!\cdots\!28 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 41\!\cdots\!20 \beta_{8} + \cdots + 24\!\cdots\!88 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 46\!\cdots\!36 \beta_{15} + \cdots + 48\!\cdots\!56 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 46\!\cdots\!52 \beta_{8} + \cdots - 23\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 50\!\cdots\!72 \beta_{15} + \cdots - 47\!\cdots\!72 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
− 1013.87i
− 941.588i
− 899.394i
− 675.810i
− 640.282i
− 491.437i
− 348.567i
− 221.696i
221.696i
348.567i
491.437i
640.282i
675.810i
899.394i
941.588i
1013.87i
0 0 0 − 1013.87i 0 −907.493 0 0 0
197.2 0 0 0 − 941.588i 0 907.493 0 0 0
197.3 0 0 0 − 899.394i 0 907.493 0 0 0
197.4 0 0 0 − 675.810i 0 907.493 0 0 0
197.5 0 0 0 − 640.282i 0 −907.493 0 0 0
197.6 0 0 0 − 491.437i 0 −907.493 0 0 0
197.7 0 0 0 − 348.567i 0 907.493 0 0 0
197.8 0 0 0 − 221.696i 0 −907.493 0 0 0
197.9 0 0 0 221.696i 0 −907.493 0 0 0
197.10 0 0 0 348.567i 0 907.493 0 0 0
197.11 0 0 0 491.437i 0 −907.493 0 0 0
197.12 0 0 0 640.282i 0 −907.493 0 0 0
197.13 0 0 0 675.810i 0 907.493 0 0 0
197.14 0 0 0 899.394i 0 907.493 0 0 0
197.15 0 0 0 941.588i 0 907.493 0 0 0
197.16 0 0 0 1013.87i 0 −907.493 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.c.a ✓ 16
3.b odd 2 1 inner 252.9.c.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.9.c.a ✓ 16 1.a even 1 1 trivial
252.9.c.a ✓ 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} - 823543)^{8} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 25\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 33\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 23\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 16\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 30\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 37\!\cdots\!08)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 26\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 21\!\cdots\!04)^{2} \) Copy content Toggle raw display
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