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Properties

Label 76.4.d.a
Level $76$
Weight $4$
Character orbit 76.d
Analytic conductor $4.484$
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] = [76,4,Mod(75,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.75");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.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: \(4.48414516044\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 10 q^{4} - 4 q^{5} - 6 q^{6} + 192 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 10 q^{4} - 4 q^{5} - 6 q^{6} + 192 q^{9} - 134 q^{16} - 80 q^{17} - 300 q^{20} - 26 q^{24} + 496 q^{25} - 90 q^{26} + 254 q^{28} - 16 q^{30} - 556 q^{36} - 626 q^{38} - 850 q^{42} + 976 q^{44} - 612 q^{45} + 188 q^{49} + 354 q^{54} - 580 q^{57} + 2534 q^{58} - 948 q^{61} - 1068 q^{62} - 1634 q^{64} + 1244 q^{66} + 1630 q^{68} - 184 q^{73} + 2276 q^{74} + 1688 q^{76} + 308 q^{77} + 3376 q^{80} - 2284 q^{81} - 740 q^{82} + 684 q^{85} + 1810 q^{92} + 824 q^{93} - 5222 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −2.78099 − 0.515856i −3.18171 7.46778 + 2.86918i 2.55979 8.84829 + 1.64130i − 16.6641i −19.2877 − 11.8315i −16.8767 −7.11874 − 1.32048i
75.2 −2.78099 + 0.515856i −3.18171 7.46778 − 2.86918i 2.55979 8.84829 − 1.64130i 16.6641i −19.2877 + 11.8315i −16.8767 −7.11874 + 1.32048i
75.3 −2.63841 − 1.01920i 6.23785 5.92246 + 5.37815i 8.54022 −16.4580 − 6.35762i 28.2101i −10.1445 − 20.2260i 11.9108 −22.5326 − 8.70420i
75.4 −2.63841 + 1.01920i 6.23785 5.92246 − 5.37815i 8.54022 −16.4580 + 6.35762i − 28.2101i −10.1445 + 20.2260i 11.9108 −22.5326 + 8.70420i
75.5 −2.38430 − 1.52155i 5.25020 3.36979 + 7.25565i −19.3166 −12.5181 − 7.98843i − 22.0985i 3.00522 − 22.4270i 0.564610 46.0565 + 29.3911i
75.6 −2.38430 + 1.52155i 5.25020 3.36979 − 7.25565i −19.3166 −12.5181 + 7.98843i 22.0985i 3.00522 + 22.4270i 0.564610 46.0565 − 29.3911i
75.7 −2.25296 − 1.71002i −8.24449 2.15167 + 7.70521i −10.8316 18.5745 + 14.0982i 8.32669i 8.32842 − 21.0389i 40.9716 24.4032 + 18.5223i
75.8 −2.25296 + 1.71002i −8.24449 2.15167 − 7.70521i −10.8316 18.5745 − 14.0982i − 8.32669i 8.32842 + 21.0389i 40.9716 24.4032 − 18.5223i
75.9 −1.60147 − 2.33137i −3.54475 −2.87062 + 7.46723i 18.5634 5.67679 + 8.26413i − 16.5701i 22.0061 − 5.26604i −14.4348 −29.7286 − 43.2782i
75.10 −1.60147 + 2.33137i −3.54475 −2.87062 − 7.46723i 18.5634 5.67679 − 8.26413i 16.5701i 22.0061 + 5.26604i −14.4348 −29.7286 + 43.2782i
75.11 −0.929967 − 2.67117i 0.246541 −6.27032 + 4.96820i −6.75093 −0.229275 − 0.658553i 18.0267i 19.1021 + 12.1288i −26.9392 6.27814 + 18.0329i
75.12 −0.929967 + 2.67117i 0.246541 −6.27032 − 4.96820i −6.75093 −0.229275 + 0.658553i − 18.0267i 19.1021 − 12.1288i −26.9392 6.27814 − 18.0329i
75.13 −0.603834 − 2.76322i 8.93329 −7.27077 + 3.33705i 6.23571 −5.39422 − 24.6847i − 11.1035i 13.6113 + 18.0757i 52.8037 −3.76533 − 17.2306i
75.14 −0.603834 + 2.76322i 8.93329 −7.27077 − 3.33705i 6.23571 −5.39422 + 24.6847i 11.1035i 13.6113 − 18.0757i 52.8037 −3.76533 + 17.2306i
75.15 0.603834 − 2.76322i −8.93329 −7.27077 − 3.33705i 6.23571 −5.39422 + 24.6847i 11.1035i −13.6113 + 18.0757i 52.8037 3.76533 − 17.2306i
75.16 0.603834 + 2.76322i −8.93329 −7.27077 + 3.33705i 6.23571 −5.39422 − 24.6847i − 11.1035i −13.6113 − 18.0757i 52.8037 3.76533 + 17.2306i
75.17 0.929967 − 2.67117i −0.246541 −6.27032 − 4.96820i −6.75093 −0.229275 + 0.658553i − 18.0267i −19.1021 + 12.1288i −26.9392 −6.27814 + 18.0329i
75.18 0.929967 + 2.67117i −0.246541 −6.27032 + 4.96820i −6.75093 −0.229275 − 0.658553i 18.0267i −19.1021 − 12.1288i −26.9392 −6.27814 − 18.0329i
75.19 1.60147 − 2.33137i 3.54475 −2.87062 − 7.46723i 18.5634 5.67679 − 8.26413i 16.5701i −22.0061 − 5.26604i −14.4348 29.7286 − 43.2782i
75.20 1.60147 + 2.33137i 3.54475 −2.87062 + 7.46723i 18.5634 5.67679 + 8.26413i − 16.5701i −22.0061 + 5.26604i −14.4348 29.7286 + 43.2782i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.4.d.a ✓ 28
4.b odd 2 1 inner 76.4.d.a ✓ 28
19.b odd 2 1 inner 76.4.d.a ✓ 28
76.d even 2 1 inner 76.4.d.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.d.a ✓ 28 1.a even 1 1 trivial
76.4.d.a ✓ 28 4.b odd 2 1 inner
76.4.d.a ✓ 28 19.b odd 2 1 inner
76.4.d.a ✓ 28 76.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(76, [\chi])\).