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Properties

Label 76.6.d.a
Level $76$
Weight $6$
Character orbit 76.d
Analytic conductor $12.189$
Analytic rank $0$
Dimension $48$
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] = [76,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.75");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(12.1891703058\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 22 q^{4} - 4 q^{5} - 246 q^{6} + 3516 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 22 q^{4} - 4 q^{5} - 246 q^{6} + 3516 q^{9} - 1222 q^{16} + 1000 q^{17} - 300 q^{20} + 17158 q^{24} + 24996 q^{25} + 12102 q^{26} + 6382 q^{28} - 16528 q^{30} + 27524 q^{36} + 3598 q^{38} + 35966 q^{42} - 65408 q^{44} - 13476 q^{45} - 143096 q^{49} - 148446 q^{54} - 53980 q^{57} - 117018 q^{58} + 133740 q^{61} + 28068 q^{62} - 78130 q^{64} + 128204 q^{66} - 48338 q^{68} - 104080 q^{73} + 48932 q^{74} - 251688 q^{76} + 184844 q^{77} + 80080 q^{80} + 375128 q^{81} + 175628 q^{82} - 384748 q^{85} + 139810 q^{92} + 183272 q^{93} + 601258 q^{96}+O(q^{100}) \) 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 −5.61709 − 0.669578i 15.7189 31.1033 + 7.52215i −23.2669 −88.2945 − 10.5250i − 110.037i −169.673 − 63.0787i 4.08426 130.692 + 15.5790i
75.2 −5.61709 + 0.669578i 15.7189 31.1033 − 7.52215i −23.2669 −88.2945 + 10.5250i 110.037i −169.673 + 63.0787i 4.08426 130.692 − 15.5790i
75.3 −5.48835 − 1.37043i −24.7554 28.2439 + 15.0427i −13.4021 135.866 + 33.9254i − 215.359i −134.397 − 121.266i 369.829 73.5554 + 18.3666i
75.4 −5.48835 + 1.37043i −24.7554 28.2439 − 15.0427i −13.4021 135.866 − 33.9254i 215.359i −134.397 + 121.266i 369.829 73.5554 − 18.3666i
75.5 −5.30781 − 1.95631i −7.37371 24.3457 + 20.7674i 74.1521 39.1383 + 14.4252i 80.5452i −88.5950 − 157.857i −188.628 −393.585 − 145.064i
75.6 −5.30781 + 1.95631i −7.37371 24.3457 − 20.7674i 74.1521 39.1383 − 14.4252i − 80.5452i −88.5950 + 157.857i −188.628 −393.585 + 145.064i
75.7 −5.06482 − 2.51944i −9.05324 19.3048 + 25.5211i −89.6042 45.8530 + 22.8091i 100.840i −33.4765 − 177.897i −161.039 453.829 + 225.753i
75.8 −5.06482 + 2.51944i −9.05324 19.3048 − 25.5211i −89.6042 45.8530 − 22.8091i − 100.840i −33.4765 + 177.897i −161.039 453.829 − 225.753i
75.9 −4.77452 − 3.03380i 25.5646 13.5921 + 28.9699i 94.0869 −122.059 − 77.5580i − 166.433i 22.9933 − 179.553i 410.550 −449.220 − 285.441i
75.10 −4.77452 + 3.03380i 25.5646 13.5921 − 28.9699i 94.0869 −122.059 + 77.5580i 166.433i 22.9933 + 179.553i 410.550 −449.220 + 285.441i
75.11 −3.92565 − 4.07299i 5.43850 −1.17856 + 31.9783i 4.82210 −21.3497 − 22.1510i 68.2624i 134.874 − 120.735i −213.423 −18.9299 − 19.6404i
75.12 −3.92565 + 4.07299i 5.43850 −1.17856 − 31.9783i 4.82210 −21.3497 + 22.1510i − 68.2624i 134.874 + 120.735i −213.423 −18.9299 + 19.6404i
75.13 −3.73141 − 4.25166i 29.4761 −4.15316 + 31.7293i −71.3638 −109.988 − 125.322i 160.370i 150.399 − 100.737i 625.843 266.288 + 303.414i
75.14 −3.73141 + 4.25166i 29.4761 −4.15316 − 31.7293i −71.3638 −109.988 + 125.322i − 160.370i 150.399 + 100.737i 625.843 266.288 − 303.414i
75.15 −3.09334 − 4.73617i −0.774530 −12.8625 + 29.3011i −32.1016 2.39588 + 3.66830i − 237.436i 178.563 − 29.7192i −242.400 99.3011 + 152.038i
75.16 −3.09334 + 4.73617i −0.774530 −12.8625 − 29.3011i −32.1016 2.39588 − 3.66830i 237.436i 178.563 + 29.7192i −242.400 99.3011 − 152.038i
75.17 −2.99929 − 4.79628i −24.6139 −14.0085 + 28.7708i 37.3653 73.8243 + 118.055i 35.5589i 180.009 − 19.1033i 362.846 −112.069 − 179.214i
75.18 −2.99929 + 4.79628i −24.6139 −14.0085 − 28.7708i 37.3653 73.8243 − 118.055i − 35.5589i 180.009 + 19.1033i 362.846 −112.069 + 179.214i
75.19 −1.28503 − 5.50897i 16.4423 −28.6974 + 14.1583i 16.6706 −21.1288 − 90.5803i − 54.8970i 114.875 + 139.899i 27.3504 −21.4221 − 91.8375i
75.20 −1.28503 + 5.50897i 16.4423 −28.6974 − 14.1583i 16.6706 −21.1288 + 90.5803i 54.8970i 114.875 − 139.899i 27.3504 −21.4221 + 91.8375i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.48
Significant digits:
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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 76.6.d.a ✓ 48
4.b odd 2 1 inner 76.6.d.a ✓ 48
19.b odd 2 1 inner 76.6.d.a ✓ 48
76.d even 2 1 inner 76.6.d.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.d.a ✓ 48 1.a even 1 1 trivial
76.6.d.a ✓ 48 4.b odd 2 1 inner
76.6.d.a ✓ 48 19.b odd 2 1 inner
76.6.d.a ✓ 48 76.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(76, [\chi])\).