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Properties

Label 78.4.g.a
Level $78$
Weight $4$
Character orbit 78.g
Analytic conductor $4.602$
Analytic rank $0$
Dimension $28$
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] = [78,4,Mod(5,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 78.g (of order \(4\), degree \(2\), 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: \(4.60214898045\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 20 q^{7} + 120 q^{13} + 168 q^{15} - 448 q^{16} - 96 q^{18} - 124 q^{19} + 264 q^{21} + 900 q^{27} - 80 q^{28} + 364 q^{31} - 1188 q^{33} - 672 q^{34} + 796 q^{37} - 912 q^{39} - 360 q^{42} - 252 q^{45} + 384 q^{46} + 592 q^{52} - 72 q^{54} + 144 q^{55} - 540 q^{57} + 1056 q^{58} + 672 q^{60} - 816 q^{61} + 252 q^{63} + 3072 q^{66} - 124 q^{67} + 192 q^{70} - 384 q^{72} + 1028 q^{73} - 496 q^{76} + 984 q^{78} - 10632 q^{79} - 3888 q^{81} - 1056 q^{84} + 9288 q^{85} + 1632 q^{87} - 844 q^{91} + 1788 q^{93} + 5232 q^{94} + 6628 q^{97} - 2052 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.41421 − 1.41421i −4.49979 + 2.59843i 4.00000i −5.97265 − 5.97265i 10.0384 + 2.68893i 11.8309 + 11.8309i 5.65685 − 5.65685i 13.4963 − 23.3848i 16.8932i
5.2 −1.41421 − 1.41421i −4.00635 − 3.30895i 4.00000i 13.5875 + 13.5875i 0.986279 + 10.3454i −4.43178 − 4.43178i 5.65685 − 5.65685i 5.10173 + 26.5136i − 38.4313i
5.3 −1.41421 − 1.41421i −2.60286 + 4.49723i 4.00000i 5.70189 + 5.70189i 10.0411 − 2.67904i −19.6648 − 19.6648i 5.65685 − 5.65685i −13.4502 − 23.4114i − 16.1274i
5.4 −1.41421 − 1.41421i −0.774086 − 5.13817i 4.00000i −6.13104 − 6.13104i −6.17175 + 8.36119i −3.16264 − 3.16264i 5.65685 − 5.65685i −25.8016 + 7.95477i 17.3412i
5.5 −1.41421 − 1.41421i 2.48497 + 4.56343i 4.00000i −4.29762 − 4.29762i 2.93940 − 9.96795i 13.3337 + 13.3337i 5.65685 − 5.65685i −14.6499 + 22.6800i 12.1555i
5.6 −1.41421 − 1.41421i 4.20440 − 3.05336i 4.00000i 6.74028 + 6.74028i −10.2640 − 1.62781i 22.3719 + 22.3719i 5.65685 − 5.65685i 8.35394 − 25.6751i − 19.0644i
5.7 −1.41421 − 1.41421i 5.19373 − 0.158618i 4.00000i −9.62835 − 9.62835i −7.56936 − 7.12073i −15.2773 − 15.2773i 5.65685 − 5.65685i 26.9497 − 1.64763i 27.2331i
5.8 1.41421 + 1.41421i −4.49979 − 2.59843i 4.00000i 5.97265 + 5.97265i −2.68893 − 10.0384i 11.8309 + 11.8309i −5.65685 + 5.65685i 13.4963 + 23.3848i 16.8932i
5.9 1.41421 + 1.41421i −4.00635 + 3.30895i 4.00000i −13.5875 − 13.5875i −10.3454 − 0.986279i −4.43178 − 4.43178i −5.65685 + 5.65685i 5.10173 − 26.5136i − 38.4313i
5.10 1.41421 + 1.41421i −2.60286 − 4.49723i 4.00000i −5.70189 − 5.70189i 2.67904 − 10.0411i −19.6648 − 19.6648i −5.65685 + 5.65685i −13.4502 + 23.4114i − 16.1274i
5.11 1.41421 + 1.41421i −0.774086 + 5.13817i 4.00000i 6.13104 + 6.13104i −8.36119 + 6.17175i −3.16264 − 3.16264i −5.65685 + 5.65685i −25.8016 − 7.95477i 17.3412i
5.12 1.41421 + 1.41421i 2.48497 − 4.56343i 4.00000i 4.29762 + 4.29762i 9.96795 − 2.93940i 13.3337 + 13.3337i −5.65685 + 5.65685i −14.6499 − 22.6800i 12.1555i
5.13 1.41421 + 1.41421i 4.20440 + 3.05336i 4.00000i −6.74028 − 6.74028i 1.62781 + 10.2640i 22.3719 + 22.3719i −5.65685 + 5.65685i 8.35394 + 25.6751i − 19.0644i
5.14 1.41421 + 1.41421i 5.19373 + 0.158618i 4.00000i 9.62835 + 9.62835i 7.12073 + 7.56936i −15.2773 − 15.2773i −5.65685 + 5.65685i 26.9497 + 1.64763i 27.2331i
47.1 −1.41421 + 1.41421i −4.49979 − 2.59843i − 4.00000i −5.97265 + 5.97265i 10.0384 − 2.68893i 11.8309 − 11.8309i 5.65685 + 5.65685i 13.4963 + 23.3848i − 16.8932i
47.2 −1.41421 + 1.41421i −4.00635 + 3.30895i − 4.00000i 13.5875 − 13.5875i 0.986279 − 10.3454i −4.43178 + 4.43178i 5.65685 + 5.65685i 5.10173 − 26.5136i 38.4313i
47.3 −1.41421 + 1.41421i −2.60286 − 4.49723i − 4.00000i 5.70189 − 5.70189i 10.0411 + 2.67904i −19.6648 + 19.6648i 5.65685 + 5.65685i −13.4502 + 23.4114i 16.1274i
47.4 −1.41421 + 1.41421i −0.774086 + 5.13817i − 4.00000i −6.13104 + 6.13104i −6.17175 − 8.36119i −3.16264 + 3.16264i 5.65685 + 5.65685i −25.8016 − 7.95477i − 17.3412i
47.5 −1.41421 + 1.41421i 2.48497 − 4.56343i − 4.00000i −4.29762 + 4.29762i 2.93940 + 9.96795i 13.3337 − 13.3337i 5.65685 + 5.65685i −14.6499 − 22.6800i − 12.1555i
47.6 −1.41421 + 1.41421i 4.20440 + 3.05336i − 4.00000i 6.74028 − 6.74028i −10.2640 + 1.62781i 22.3719 − 22.3719i 5.65685 + 5.65685i 8.35394 + 25.6751i 19.0644i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.4.g.a ✓ 28
3.b odd 2 1 inner 78.4.g.a ✓ 28
13.d odd 4 1 inner 78.4.g.a ✓ 28
39.f even 4 1 inner 78.4.g.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.g.a ✓ 28 1.a even 1 1 trivial
78.4.g.a ✓ 28 3.b odd 2 1 inner
78.4.g.a ✓ 28 13.d odd 4 1 inner
78.4.g.a ✓ 28 39.f even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(78, [\chi])\).