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Properties

Label 76.8.d.a
Level $76$
Weight $8$
Character orbit 76.d
Analytic conductor $23.741$
Analytic rank $0$
Dimension $68$
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] = [76,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.75");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(23.7412619368\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 54 q^{4} - 4 q^{5} - 150 q^{6} + 48888 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 54 q^{4} - 4 q^{5} - 150 q^{6} + 48888 q^{9} + 13050 q^{16} + 23968 q^{17} - 79660 q^{20} - 232154 q^{24} + 937496 q^{25} - 391242 q^{26} - 37666 q^{28} - 468496 q^{30} - 1128652 q^{36} - 530466 q^{38} + 651806 q^{42} + 1433776 q^{44} - 321252 q^{45} - 4014380 q^{49} + 1697586 q^{54} + 137420 q^{57} + 2227894 q^{58} - 3042164 q^{61} + 2261524 q^{62} + 9825150 q^{64} - 749476 q^{66} - 11176962 q^{68} + 598424 q^{73} - 6195964 q^{74} - 1714536 q^{76} - 2556764 q^{77} + 16623216 q^{80} + 43726748 q^{81} - 20460388 q^{82} + 27222844 q^{85} - 61733582 q^{92} + 17778200 q^{93} - 36370886 q^{96}+O(q^{100}) \) 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 −11.3050 − 0.442790i 16.6522 127.608 + 10.0115i 341.116 −188.254 − 7.37345i − 1382.96i −1438.18 − 169.684i −1909.70 −3856.33 − 151.043i
75.2 −11.3050 + 0.442790i 16.6522 127.608 − 10.0115i 341.116 −188.254 + 7.37345i 1382.96i −1438.18 + 169.684i −1909.70 −3856.33 + 151.043i
75.3 −11.1927 − 1.65015i −65.7104 122.554 + 36.9394i −95.9690 735.478 + 108.432i 851.669i −1310.76 − 615.686i 2130.86 1074.15 + 158.364i
75.4 −11.1927 + 1.65015i −65.7104 122.554 − 36.9394i −95.9690 735.478 − 108.432i − 851.669i −1310.76 + 615.686i 2130.86 1074.15 − 158.364i
75.5 −10.9753 − 2.74630i −8.30507 112.916 + 60.2831i −360.155 91.1509 + 22.8082i 965.147i −1073.73 − 971.727i −2118.03 3952.82 + 989.094i
75.6 −10.9753 + 2.74630i −8.30507 112.916 − 60.2831i −360.155 91.1509 − 22.8082i − 965.147i −1073.73 + 971.727i −2118.03 3952.82 − 989.094i
75.7 −10.7294 − 3.58884i 80.7260 102.240 + 77.0123i 174.727 −866.143 − 289.713i 433.835i −820.594 − 1193.22i 4329.69 −1874.71 − 627.067i
75.8 −10.7294 + 3.58884i 80.7260 102.240 − 77.0123i 174.727 −866.143 + 289.713i − 433.835i −820.594 + 1193.22i 4329.69 −1874.71 + 627.067i
75.9 −10.6582 − 3.79499i 60.8414 99.1961 + 80.8959i −369.508 −648.462 − 230.893i − 900.474i −750.256 − 1238.66i 1514.68 3938.30 + 1402.28i
75.10 −10.6582 + 3.79499i 60.8414 99.1961 − 80.8959i −369.508 −648.462 + 230.893i 900.474i −750.256 + 1238.66i 1514.68 3938.30 − 1402.28i
75.11 −9.80435 − 5.64577i −65.3605 64.2506 + 110.706i 399.628 640.817 + 369.010i − 321.615i −4.91377 − 1448.15i 2084.99 −3918.10 − 2256.21i
75.12 −9.80435 + 5.64577i −65.3605 64.2506 − 110.706i 399.628 640.817 − 369.010i 321.615i −4.91377 + 1448.15i 2084.99 −3918.10 + 2256.21i
75.13 −9.59747 − 5.99070i −1.36022 56.2230 + 114.991i 59.9018 13.0547 + 8.14867i − 254.899i 149.279 − 1440.44i −2185.15 −574.906 − 358.854i
75.14 −9.59747 + 5.99070i −1.36022 56.2230 − 114.991i 59.9018 13.0547 − 8.14867i 254.899i 149.279 + 1440.44i −2185.15 −574.906 + 358.854i
75.15 −8.06300 − 7.93650i 49.8474 2.02391 + 127.984i 33.4044 −401.919 − 395.614i 865.446i 999.426 − 1048.00i 297.759 −269.340 − 265.114i
75.16 −8.06300 + 7.93650i 49.8474 2.02391 − 127.984i 33.4044 −401.919 + 395.614i − 865.446i 999.426 + 1048.00i 297.759 −269.340 + 265.114i
75.17 −7.73771 − 8.25396i −29.6173 −8.25557 + 127.733i −224.326 229.170 + 244.460i − 1196.17i 1118.19 − 920.224i −1309.82 1735.77 + 1851.58i
75.18 −7.73771 + 8.25396i −29.6173 −8.25557 − 127.733i −224.326 229.170 − 244.460i 1196.17i 1118.19 + 920.224i −1309.82 1735.77 − 1851.58i
75.19 −7.35548 − 8.59633i −83.0232 −19.7938 + 126.460i −447.252 610.676 + 713.695i 106.200i 1232.69 − 760.022i 4705.86 3289.75 + 3844.72i
75.20 −7.35548 + 8.59633i −83.0232 −19.7938 − 126.460i −447.252 610.676 − 713.695i − 106.200i 1232.69 + 760.022i 4705.86 3289.75 − 3844.72i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.68
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 76.8.d.a ✓ 68
4.b odd 2 1 inner 76.8.d.a ✓ 68
19.b odd 2 1 inner 76.8.d.a ✓ 68
76.d even 2 1 inner 76.8.d.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.8.d.a ✓ 68 1.a even 1 1 trivial
76.8.d.a ✓ 68 4.b odd 2 1 inner
76.8.d.a ✓ 68 19.b odd 2 1 inner
76.8.d.a ✓ 68 76.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(76, [\chi])\).