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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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.f (of order \(6\), degree \(2\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 136q - 3q^{2} - 27q^{4} - 2q^{5} + 309q^{6} - 45540q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 136q - 3q^{2} - 27q^{4} - 2q^{5} + 309q^{6} - 45540q^{9} - 19464q^{10} - 10602q^{13} + 13602q^{14} + 6525q^{16} - 23974q^{17} - 91316q^{20} + 54264q^{21} + 381q^{22} - 51631q^{24} - 937502q^{25} + 172188q^{26} - 165428q^{28} - 6q^{29} - 454352q^{30} + 573507q^{32} - 182106q^{33} - 629706q^{34} - 277130q^{36} - 759906q^{38} - 627504q^{40} - 1873020q^{41} + 332080q^{42} - 358069q^{44} + 321240q^{45} - 7221687q^{48} - 15750664q^{49} + 2848050q^{52} - 1027650q^{53} + 4352085q^{54} - 137426q^{57} + 10387724q^{58} + 1607094q^{60} + 5318582q^{61} + 3394376q^{62} - 1319538q^{64} + 2439607q^{66} - 87756q^{68} - 17825238q^{70} + 26155014q^{72} + 299212q^{73} + 1895842q^{74} + 29672379q^{76} + 907280q^{77} + 7456734q^{78} + 5489562q^{80} - 11065208q^{81} + 1288975q^{82} + 6118214q^{85} + 16831518q^{86} - 17728542q^{89} + 52167894q^{90} + 10184840q^{92} - 3299132q^{93} - 68841730q^{96} - 9435888q^{97} - 35100021q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 31.68
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 \(S_{8}^{\mathrm{new}}(76, [\chi])\).