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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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.d (of order \(2\), degree \(1\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 28q + 10q^{4} - 4q^{5} - 6q^{6} + 192q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q + 10q^{4} - 4q^{5} - 6q^{6} + 192q^{9} - 134q^{16} - 80q^{17} - 300q^{20} - 26q^{24} + 496q^{25} - 90q^{26} + 254q^{28} - 16q^{30} - 556q^{36} - 626q^{38} - 850q^{42} + 976q^{44} - 612q^{45} + 188q^{49} + 354q^{54} - 580q^{57} + 2534q^{58} - 948q^{61} - 1068q^{62} - 1634q^{64} + 1244q^{66} + 1630q^{68} - 184q^{73} + 2276q^{74} + 1688q^{76} + 308q^{77} + 3376q^{80} - 2284q^{81} - 740q^{82} + 684q^{85} + 1810q^{92} + 824q^{93} - 5222q^{96} + O(q^{100}) \)

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.

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 \(S_{4}^{\mathrm{new}}(76, [\chi])\).