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Properties

Label 126.16.d.a
Level $126$
Weight $16$
Character orbit 126.d
Analytic conductor $179.794$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [126,16,Mod(125,126)]
 
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([1, 1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.125");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 126.d (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: \(179.793816426\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 655360 q^{4} - 577096 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 655360 q^{4} - 577096 q^{7} + 10737418240 q^{16} + 15012741120 q^{22} + 240658578520 q^{25} + 9455140864 q^{28} + 19750837888 q^{37} + 1178793607984 q^{43} + 5919750832128 q^{46} - 5977420874696 q^{49} - 9318748028928 q^{58} - 175921860444160 q^{64} + 7832586895808 q^{67} - 4428743147520 q^{70} + 11\!\cdots\!28 q^{79}+ \cdots + 642048330852192 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 − 128.000i 0 −16384.0 116893. 0 1.00746e6 − 1.93199e6i 2.09715e6i 0 − 1.49623e7i
125.2 128.000i 0 −16384.0 116893. 0 1.00746e6 + 1.93199e6i − 2.09715e6i 0 1.49623e7i
125.3 − 128.000i 0 −16384.0 12817.6 0 −769584. + 2.03846e6i 2.09715e6i 0 − 1.64065e6i
125.4 128.000i 0 −16384.0 12817.6 0 −769584. − 2.03846e6i − 2.09715e6i 0 1.64065e6i
125.5 − 128.000i 0 −16384.0 −175905. 0 2.11091e6 + 540010.i 2.09715e6i 0 2.25158e7i
125.6 128.000i 0 −16384.0 −175905. 0 2.11091e6 − 540010.i − 2.09715e6i 0 − 2.25158e7i
125.7 − 128.000i 0 −16384.0 −284030. 0 1.99830e6 − 868547.i 2.09715e6i 0 3.63558e7i
125.8 128.000i 0 −16384.0 −284030. 0 1.99830e6 + 868547.i − 2.09715e6i 0 − 3.63558e7i
125.9 − 128.000i 0 −16384.0 299119. 0 −2.10066e6 + 578619.i 2.09715e6i 0 − 3.82872e7i
125.10 128.000i 0 −16384.0 299119. 0 −2.10066e6 − 578619.i − 2.09715e6i 0 3.82872e7i
125.11 − 128.000i 0 −16384.0 −221953. 0 879388. − 1.99355e6i 2.09715e6i 0 2.84100e7i
125.12 128.000i 0 −16384.0 −221953. 0 879388. + 1.99355e6i − 2.09715e6i 0 − 2.84100e7i
125.13 − 128.000i 0 −16384.0 −79801.3 0 −1.71695e6 − 1.34150e6i 2.09715e6i 0 1.02146e7i
125.14 128.000i 0 −16384.0 −79801.3 0 −1.71695e6 + 1.34150e6i − 2.09715e6i 0 − 1.02146e7i
125.15 − 128.000i 0 −16384.0 −72071.1 0 −2.03378e6 − 781846.i 2.09715e6i 0 9.22510e6i
125.16 128.000i 0 −16384.0 −72071.1 0 −2.03378e6 + 781846.i − 2.09715e6i 0 − 9.22510e6i
125.17 − 128.000i 0 −16384.0 49990.7 0 763982. − 2.04056e6i 2.09715e6i 0 − 6.39881e6i
125.18 128.000i 0 −16384.0 49990.7 0 763982. + 2.04056e6i − 2.09715e6i 0 6.39881e6i
125.19 − 128.000i 0 −16384.0 −295127. 0 −283337. + 2.16039e6i 2.09715e6i 0 3.77762e7i
125.20 128.000i 0 −16384.0 −295127. 0 −283337. − 2.16039e6i − 2.09715e6i 0 − 3.77762e7i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.40
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 126.16.d.a ✓ 40
3.b odd 2 1 inner 126.16.d.a ✓ 40
7.b odd 2 1 inner 126.16.d.a ✓ 40
21.c even 2 1 inner 126.16.d.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.16.d.a ✓ 40 1.a even 1 1 trivial
126.16.d.a ✓ 40 3.b odd 2 1 inner
126.16.d.a ✓ 40 7.b odd 2 1 inner
126.16.d.a ✓ 40 21.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(126, [\chi])\).