Properties

Label 192.7.i.a
Level $192$
Weight $7$
Character orbit 192.i
Analytic conductor $44.170$
Analytic rank $0$
Dimension $92$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,7,Mod(17,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.17");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 192.i (of order \(4\), degree \(2\), not 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: \(44.1703840550\)
Analytic rank: \(0\)
Dimension: \(92\)
Relative dimension: \(46\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 92 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q + 2 q^{3} - 4 q^{13} + 4 q^{15} + 3940 q^{19} + 1456 q^{21} + 34322 q^{27} + 8 q^{31} - 4 q^{33} - 4 q^{37} + 195268 q^{43} - 31252 q^{45} - 1142884 q^{49} - 385056 q^{51} - 326500 q^{61} - 470592 q^{63} - 1207676 q^{67} - 1460 q^{69} - 1413778 q^{75} - 860920 q^{79} - 4 q^{81} + 434496 q^{85} - 2320992 q^{91} - 1176260 q^{93} - 8 q^{97} + 4048228 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −26.8720 2.62588i 0 141.081 + 141.081i 0 308.364i 0 715.209 + 141.126i 0
17.2 0 −26.7985 3.29233i 0 59.2583 + 59.2583i 0 2.09133i 0 707.321 + 176.459i 0
17.3 0 −26.4520 + 5.41236i 0 54.1549 + 54.1549i 0 639.122i 0 670.413 286.335i 0
17.4 0 −26.4071 + 5.62718i 0 −81.2421 81.2421i 0 51.0073i 0 665.670 297.195i 0
17.5 0 −26.3911 5.70179i 0 −89.3741 89.3741i 0 331.991i 0 663.979 + 300.953i 0
17.6 0 −26.3319 + 5.96897i 0 −73.2326 73.2326i 0 386.845i 0 657.743 314.349i 0
17.7 0 −23.2271 + 13.7659i 0 133.976 + 133.976i 0 25.6967i 0 350.001 639.485i 0
17.8 0 −22.8925 14.3156i 0 −4.45378 4.45378i 0 601.125i 0 319.129 + 655.437i 0
17.9 0 −20.6842 17.3541i 0 −77.8915 77.8915i 0 310.530i 0 126.674 + 717.910i 0
17.10 0 −20.4285 17.6543i 0 −132.844 132.844i 0 91.8613i 0 105.651 + 721.304i 0
17.11 0 −19.9194 + 18.2268i 0 14.1816 + 14.1816i 0 191.419i 0 64.5655 726.135i 0
17.12 0 −18.9466 + 19.2360i 0 −148.490 148.490i 0 263.428i 0 −11.0511 728.916i 0
17.13 0 −17.6543 20.4285i 0 132.844 + 132.844i 0 91.8613i 0 −105.651 + 721.304i 0
17.14 0 −17.3541 20.6842i 0 77.8915 + 77.8915i 0 310.530i 0 −126.674 + 717.910i 0
17.15 0 −15.0602 + 22.4096i 0 13.2524 + 13.2524i 0 437.375i 0 −275.382 674.986i 0
17.16 0 −14.3156 22.8925i 0 4.45378 + 4.45378i 0 601.125i 0 −319.129 + 655.437i 0
17.17 0 −12.2138 + 24.0795i 0 −131.879 131.879i 0 543.561i 0 −430.647 588.204i 0
17.18 0 −9.91420 + 25.1139i 0 41.0809 + 41.0809i 0 200.806i 0 −532.417 497.969i 0
17.19 0 −9.03029 + 25.4451i 0 65.5814 + 65.5814i 0 515.866i 0 −565.908 459.554i 0
17.20 0 −5.70179 26.3911i 0 89.3741 + 89.3741i 0 331.991i 0 −663.979 + 300.953i 0
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.7.i.a 92
3.b odd 2 1 inner 192.7.i.a 92
4.b odd 2 1 48.7.i.a 92
12.b even 2 1 48.7.i.a 92
16.e even 4 1 inner 192.7.i.a 92
16.f odd 4 1 48.7.i.a 92
48.i odd 4 1 inner 192.7.i.a 92
48.k even 4 1 48.7.i.a 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.7.i.a 92 4.b odd 2 1
48.7.i.a 92 12.b even 2 1
48.7.i.a 92 16.f odd 4 1
48.7.i.a 92 48.k even 4 1
192.7.i.a 92 1.a even 1 1 trivial
192.7.i.a 92 3.b odd 2 1 inner
192.7.i.a 92 16.e even 4 1 inner
192.7.i.a 92 48.i odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(192, [\chi])\).