Properties

Label 126.10.a.e
Level $126$
Weight $10$
Character orbit 126.a
Self dual yes
Analytic conductor $64.895$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,10,Mod(1,126)]
 
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, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)

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: yes
Analytic conductor: \(64.8945153566\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 16 q^{2} + 256 q^{4} - 560 q^{5} - 2401 q^{7} + 4096 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{2} + 256 q^{4} - 560 q^{5} - 2401 q^{7} + 4096 q^{8} - 8960 q^{10} + 54152 q^{11} - 113172 q^{13} - 38416 q^{14} + 65536 q^{16} - 6262 q^{17} + 257078 q^{19} - 143360 q^{20} + 866432 q^{22} + 266000 q^{23} - 1639525 q^{25} - 1810752 q^{26} - 614656 q^{28} - 1574714 q^{29} - 4637484 q^{31} + 1048576 q^{32} - 100192 q^{34} + 1344560 q^{35} - 11946238 q^{37} + 4113248 q^{38} - 2293760 q^{40} - 21909126 q^{41} + 27520592 q^{43} + 13862912 q^{44} + 4256000 q^{46} - 52927836 q^{47} + 5764801 q^{49} - 26232400 q^{50} - 28972032 q^{52} - 16221222 q^{53} - 30325120 q^{55} - 9834496 q^{56} - 25195424 q^{58} + 140509618 q^{59} - 202963560 q^{61} - 74199744 q^{62} + 16777216 q^{64} + 63376320 q^{65} + 153734572 q^{67} - 1603072 q^{68} + 21512960 q^{70} - 279655936 q^{71} - 404022830 q^{73} - 191139808 q^{74} + 65811968 q^{76} - 130018952 q^{77} - 130689816 q^{79} - 36700160 q^{80} - 350546016 q^{82} - 420134014 q^{83} + 3506720 q^{85} + 440329472 q^{86} + 221806592 q^{88} + 469542390 q^{89} + 271725972 q^{91} + 68096000 q^{92} - 846845376 q^{94} - 143963680 q^{95} - 872501690 q^{97} + 92236816 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
16.0000 0 256.000 −560.000 0 −2401.00 4096.00 0 −8960.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.10.a.e 1
3.b odd 2 1 14.10.a.a 1
12.b even 2 1 112.10.a.b 1
15.d odd 2 1 350.10.a.c 1
15.e even 4 2 350.10.c.b 2
21.c even 2 1 98.10.a.a 1
21.g even 6 2 98.10.c.e 2
21.h odd 6 2 98.10.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.a 1 3.b odd 2 1
98.10.a.a 1 21.c even 2 1
98.10.c.e 2 21.g even 6 2
98.10.c.f 2 21.h odd 6 2
112.10.a.b 1 12.b even 2 1
126.10.a.e 1 1.a even 1 1 trivial
350.10.a.c 1 15.d odd 2 1
350.10.c.b 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 560 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(126))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 16 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 560 \) Copy content Toggle raw display
$7$ \( T + 2401 \) Copy content Toggle raw display
$11$ \( T - 54152 \) Copy content Toggle raw display
$13$ \( T + 113172 \) Copy content Toggle raw display
$17$ \( T + 6262 \) Copy content Toggle raw display
$19$ \( T - 257078 \) Copy content Toggle raw display
$23$ \( T - 266000 \) Copy content Toggle raw display
$29$ \( T + 1574714 \) Copy content Toggle raw display
$31$ \( T + 4637484 \) Copy content Toggle raw display
$37$ \( T + 11946238 \) Copy content Toggle raw display
$41$ \( T + 21909126 \) Copy content Toggle raw display
$43$ \( T - 27520592 \) Copy content Toggle raw display
$47$ \( T + 52927836 \) Copy content Toggle raw display
$53$ \( T + 16221222 \) Copy content Toggle raw display
$59$ \( T - 140509618 \) Copy content Toggle raw display
$61$ \( T + 202963560 \) Copy content Toggle raw display
$67$ \( T - 153734572 \) Copy content Toggle raw display
$71$ \( T + 279655936 \) Copy content Toggle raw display
$73$ \( T + 404022830 \) Copy content Toggle raw display
$79$ \( T + 130689816 \) Copy content Toggle raw display
$83$ \( T + 420134014 \) Copy content Toggle raw display
$89$ \( T - 469542390 \) Copy content Toggle raw display
$97$ \( T + 872501690 \) Copy content Toggle raw display
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