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Properties

Label 126.9.c.a
Level $126$
Weight $9$
Character orbit 126.c
Analytic conductor $51.330$
Analytic rank $0$
Dimension $4$
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] = [126,9,Mod(55,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.55");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 126.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: \(51.3297048677\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.3520512.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 120x^{2} + 3438 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 14)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + 128 q^{4} + ( - 2 \beta_{2} - 7 \beta_1) q^{5} + (147 \beta_{3} + 49 \beta_1 - 1519) q^{7} - 256 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + 128 q^{4} + ( - 2 \beta_{2} - 7 \beta_1) q^{5} + (147 \beta_{3} + 49 \beta_1 - 1519) q^{7} - 256 \beta_{3} q^{8} + ( - 6 \beta_{2} + 98 \beta_1) q^{10} + ( - 1308 \beta_{3} + 3390) q^{11} + (360 \beta_{2} - 229 \beta_1) q^{13} + (3038 \beta_{3} - 98 \beta_{2} + \cdots - 9408) q^{14}+ \cdots + ( - 465794 \beta_{3} + 297724 \beta_{2} + \cdots + 57163008) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 512 q^{4} - 6076 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 512 q^{4} - 6076 q^{7} + 13560 q^{11} - 37632 q^{14} + 65536 q^{16} + 334848 q^{22} + 894072 q^{23} + 1216900 q^{25} - 777728 q^{28} - 317064 q^{29} + 1655808 q^{35} - 2495096 q^{37} + 9186568 q^{43} + 1735680 q^{44} + 3059712 q^{46} + 931588 q^{49} + 2984448 q^{50} + 38727288 q^{53} - 4816896 q^{56} + 34661376 q^{58} + 8388608 q^{64} + 11891712 q^{65} - 12320248 q^{67} - 21901824 q^{70} - 62168712 q^{71} + 22957056 q^{74} - 45208968 q^{77} + 24889736 q^{79} + 89943552 q^{85} - 74680320 q^{86} + 42860544 q^{88} + 38158848 q^{91} + 114441216 q^{92} - 227967744 q^{95} + 228652032 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 120x^{2} + 3438 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 156\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 108\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{2} + 240 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + 3\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{3} - 240 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 39\beta_{2} - 81\beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
6.87547i
− 6.87547i
− 8.52807i
8.52807i
−11.3137 0 128.000 − 390.317i 0 −687.442 + 2300.48i −1448.15 0 4415.94i
55.2 −11.3137 0 128.000 390.317i 0 −687.442 − 2300.48i −1448.15 0 − 4415.94i
55.3 11.3137 0 128.000 − 143.012i 0 −2350.56 − 489.571i 1448.15 0 − 1617.99i
55.4 11.3137 0 128.000 143.012i 0 −2350.56 + 489.571i 1448.15 0 1617.99i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.9.c.a 4
3.b odd 2 1 14.9.b.a ✓ 4
7.b odd 2 1 inner 126.9.c.a 4
12.b even 2 1 112.9.c.c 4
15.d odd 2 1 350.9.b.a 4
15.e even 4 2 350.9.d.a 8
21.c even 2 1 14.9.b.a ✓ 4
21.g even 6 2 98.9.d.a 8
21.h odd 6 2 98.9.d.a 8
84.h odd 2 1 112.9.c.c 4
105.g even 2 1 350.9.b.a 4
105.k odd 4 2 350.9.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.9.b.a ✓ 4 3.b odd 2 1
14.9.b.a ✓ 4 21.c even 2 1
98.9.d.a 8 21.g even 6 2
98.9.d.a 8 21.h odd 6 2
112.9.c.c 4 12.b even 2 1
112.9.c.c 4 84.h odd 2 1
126.9.c.a 4 1.a even 1 1 trivial
126.9.c.a 4 7.b odd 2 1 inner
350.9.b.a 4 15.d odd 2 1
350.9.b.a 4 105.g even 2 1
350.9.d.a 8 15.e even 4 2
350.9.d.a 8 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 172800T_{5}^{2} + 3115873152 \) acting on \(S_{9}^{\mathrm{new}}(126, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 3115873152 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6780 T - 43255548)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 202796647224192 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 23\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T^{2} - 447036 T + 45389086596)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 158532 T - 580343359356)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( (T^{2} + 1247548 T + 131756883844)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 54\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 2551346607364)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 25\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( (T^{2} + \cdots + 90844298538372)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 38\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots - 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 599142107080188)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 50828841800444)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 97\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 95\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 34\!\cdots\!12 \) Copy content Toggle raw display
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