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Properties

Label 76.8.f.a
Level $76$
Weight $8$
Character orbit 76.f
Analytic conductor $23.741$
Analytic rank $0$
Dimension $136$
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(27,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.27");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.f (of order \(6\), 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: \(23.7412619368\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(68\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 3 q^{2} - 27 q^{4} - 2 q^{5} + 309 q^{6} - 45540 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 136 q - 3 q^{2} - 27 q^{4} - 2 q^{5} + 309 q^{6} - 45540 q^{9} - 19464 q^{10} - 10602 q^{13} + 13602 q^{14} + 6525 q^{16} - 23974 q^{17} - 91316 q^{20} + 54264 q^{21} + 381 q^{22} - 51631 q^{24} - 937502 q^{25} + 172188 q^{26} - 165428 q^{28} - 6 q^{29} - 454352 q^{30} + 573507 q^{32} - 182106 q^{33} - 629706 q^{34} - 277130 q^{36} - 759906 q^{38} - 627504 q^{40} - 1873020 q^{41} + 332080 q^{42} - 358069 q^{44} + 321240 q^{45} - 7221687 q^{48} - 15750664 q^{49} + 2848050 q^{52} - 1027650 q^{53} + 4352085 q^{54} - 137426 q^{57} + 10387724 q^{58} + 1607094 q^{60} + 5318582 q^{61} + 3394376 q^{62} - 1319538 q^{64} + 2439607 q^{66} - 87756 q^{68} - 17825238 q^{70} + 26155014 q^{72} + 299212 q^{73} + 1895842 q^{74} + 29672379 q^{76} + 907280 q^{77} + 7456734 q^{78} + 5489562 q^{80} - 11065208 q^{81} + 1288975 q^{82} + 6118214 q^{85} + 16831518 q^{86} - 17728542 q^{89} + 52167894 q^{90} + 10184840 q^{92} - 3299132 q^{93} - 68841730 q^{96} - 9435888 q^{97} - 35100021 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −11.2888 + 0.750740i −22.0784 + 38.2409i 126.873 − 16.9499i −25.0418 + 43.3737i 220.529 − 448.268i − 1264.81i −1419.51 + 286.592i 118.588 + 205.400i 250.129 − 508.436i
27.2 −11.2846 − 0.811527i −19.0858 + 33.0575i 126.683 + 18.3155i 115.282 − 199.674i 242.202 − 357.551i 1657.40i −1414.70 − 309.489i 364.967 + 632.141i −1462.95 + 2159.68i
27.3 −11.2351 − 1.33139i −40.8281 + 70.7163i 124.455 + 29.9167i −132.258 + 229.077i 552.859 − 740.147i 660.351i −1358.43 − 501.815i −2240.37 − 3880.43i 1790.92 − 2397.62i
27.4 −11.1144 + 2.11428i 22.4078 − 38.8115i 119.060 − 46.9979i 87.9976 − 152.416i −166.991 + 478.742i 1.37975i −1223.91 + 774.079i 89.2804 + 154.638i −655.790 + 1880.07i
27.5 −10.9494 − 2.84804i 35.9354 − 62.2419i 111.777 + 62.3685i −104.293 + 180.641i −570.738 + 579.164i 565.242i −1046.26 − 1001.24i −1489.21 − 2579.38i 1656.42 − 1680.88i
27.6 −10.9379 − 2.89184i 32.1577 − 55.6988i 111.275 + 63.2611i 200.122 − 346.621i −512.809 + 516.232i 134.782i −1034.17 − 1013.73i −974.738 − 1688.30i −3191.28 + 3212.58i
27.7 −10.9017 + 3.02556i 10.5842 − 18.3323i 109.692 − 65.9672i −226.342 + 392.035i −59.9194 + 231.876i 671.124i −996.236 + 1051.03i 869.451 + 1505.93i 1281.37 − 4958.64i
27.8 −10.7404 − 3.55569i −14.8281 + 25.6830i 102.714 + 76.3794i 165.223 − 286.175i 250.581 − 223.122i − 712.107i −831.613 − 1185.57i 653.757 + 1132.34i −2792.12 + 2486.16i
27.9 −10.6072 − 3.93531i 6.98587 − 12.0999i 97.0267 + 83.4854i −138.667 + 240.178i −121.717 + 100.855i − 1057.42i −700.644 − 1267.38i 995.895 + 1724.94i 2416.05 − 2001.93i
27.10 −10.3342 + 4.60487i 37.3006 − 64.6065i 85.5904 − 95.1750i 66.9534 − 115.967i −87.9663 + 839.419i − 1511.36i −446.237 + 1377.69i −1689.17 − 2925.72i −157.897 + 1506.73i
27.11 −10.2494 + 4.79057i −2.37350 + 4.11103i 82.1008 − 98.2011i 75.5894 − 130.925i 4.63282 − 53.5061i 160.712i −371.045 + 1399.81i 1082.23 + 1874.48i −147.542 + 1704.02i
27.12 −9.80808 + 5.63929i −43.2012 + 74.8267i 64.3967 − 110.621i 243.358 − 421.508i 1.75099 − 977.530i − 413.429i −7.78194 + 1448.13i −2639.19 − 4571.21i −9.86356 + 5506.55i
27.13 −9.21894 + 6.55829i −30.2276 + 52.3558i 41.9776 − 120.921i −183.390 + 317.640i −64.6980 − 680.906i − 568.154i 406.047 + 1390.06i −733.917 − 1271.18i −392.520 − 4131.03i
27.14 −8.71169 − 7.21848i −6.98587 + 12.0999i 23.7871 + 125.770i −138.667 + 240.178i 148.201 − 54.9831i 1057.42i 700.644 − 1267.38i 995.895 + 1724.94i 2941.74 − 1091.39i
27.15 −8.44954 − 7.52365i 14.8281 − 25.6830i 14.7894 + 127.143i 165.223 − 286.175i −318.520 + 105.448i 712.107i 831.613 − 1185.57i 653.757 + 1132.34i −3549.14 + 1174.97i
27.16 −8.08949 + 7.90950i 43.0241 − 74.5200i 2.87975 − 127.968i −29.6094 + 51.2850i 241.372 + 943.128i 1634.91i 988.864 + 1057.97i −2608.65 − 4518.32i −166.114 − 649.066i
27.17 −7.97334 − 8.02657i −32.1577 + 55.6988i −0.851561 + 127.997i 200.122 − 346.621i 703.475 − 185.990i − 134.782i 1034.17 − 1013.73i −974.738 − 1688.30i −4377.82 + 1157.44i
27.18 −7.94116 − 8.05841i −35.9354 + 62.2419i −1.87591 + 127.986i −104.293 + 180.641i 786.940 − 204.691i − 565.242i 1046.26 − 1001.24i −1489.21 − 2579.38i 2283.89 − 594.064i
27.19 −7.31847 + 8.62786i −23.5556 + 40.7996i −20.8799 − 126.286i −56.7551 + 98.3028i −179.622 − 501.825i 1227.46i 1242.38 + 744.068i −16.2362 − 28.1219i −432.782 − 1209.10i
27.20 −7.19272 + 8.73297i 27.5600 − 47.7352i −24.5296 − 125.628i −200.879 + 347.932i 218.640 + 584.026i − 1173.75i 1273.54 + 689.387i −425.602 − 737.165i −1593.62 − 4256.84i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.d odd 6 1 inner
76.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.8.f.a ✓ 136
4.b odd 2 1 inner 76.8.f.a ✓ 136
19.d odd 6 1 inner 76.8.f.a ✓ 136
76.f even 6 1 inner 76.8.f.a ✓ 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.8.f.a ✓ 136 1.a even 1 1 trivial
76.8.f.a ✓ 136 4.b odd 2 1 inner
76.8.f.a ✓ 136 19.d odd 6 1 inner
76.8.f.a ✓ 136 76.f even 6 1 inner

Hecke kernels

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