Properties

Label 76.6.k.a
Level $76$
Weight $6$
Character orbit 76.k
Analytic conductor $12.189$
Analytic rank $0$
Dimension $288$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 76.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1891703058\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(48\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288q - 6q^{2} + 24q^{4} - 12q^{5} + 264q^{6} - 9q^{8} + 54q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 288q - 6q^{2} + 24q^{4} - 12q^{5} + 264q^{6} - 9q^{8} + 54q^{9} - 657q^{10} - 9q^{12} + 336q^{13} - 2163q^{14} + 1824q^{16} - 12q^{17} - 10242q^{20} - 7842q^{21} - 102q^{22} - 11856q^{24} - 12q^{25} + 1317q^{26} + 6660q^{28} - 12q^{29} - 40692q^{30} - 3351q^{32} - 8604q^{33} - 15756q^{34} + 81783q^{36} + 50400q^{38} + 39246q^{40} - 29082q^{41} - 123081q^{42} - 39651q^{44} - 6q^{45} - 112554q^{46} - 68403q^{48} + 259302q^{49} + 154188q^{50} - 36867q^{52} - 26280q^{53} + 172605q^{54} - 12q^{57} + 229740q^{58} - 136062q^{60} + 169296q^{61} + 35076q^{62} + 63795q^{64} - 189918q^{65} + 73644q^{66} + 59706q^{68} - 200898q^{69} - 177831q^{70} - 188502q^{72} + 397428q^{73} - 530703q^{74} - 53022q^{76} + 159804q^{77} - 618237q^{78} + 18045q^{80} - 110160q^{81} + 155859q^{82} + 733851q^{84} - 645108q^{85} + 616836q^{86} + 598383q^{88} - 226320q^{89} + 546366q^{90} - 754308q^{92} + 558942q^{93} - 454530q^{96} - 177198q^{97} - 76791q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −5.65633 0.0770723i −6.62020 + 2.40955i 31.9881 + 0.871892i 8.57723 48.6439i 37.6317 13.1190i 27.9453 16.1342i −180.868 7.39710i −148.128 + 124.294i −52.2648 + 274.485i
3.2 −5.64828 0.311409i −24.0467 + 8.75229i 31.8060 + 3.51784i −13.4502 + 76.2798i 138.548 41.9470i 182.624 105.438i −178.554 29.7744i 315.494 264.731i 99.7246 426.661i
3.3 −5.64633 0.344965i −17.2898 + 6.29296i 31.7620 + 3.89557i 4.98692 28.2822i 99.7945 29.5678i −134.579 + 77.6993i −177.995 32.9524i 73.1861 61.4104i −37.9142 + 157.970i
3.4 −5.60133 0.790627i 26.7700 9.74349i 30.7498 + 8.85713i 3.91029 22.1764i −157.651 + 33.4114i 174.375 100.675i −165.237 73.9233i 435.549 365.469i −39.4361 + 121.126i
3.5 −5.50165 1.31601i 12.1664 4.42820i 28.5362 + 14.4804i −16.8954 + 95.8188i −72.7627 + 8.35134i −45.6283 + 26.3435i −137.940 117.220i −57.7369 + 48.4470i 219.051 504.926i
3.6 −5.42730 + 1.59511i 17.0177 6.19395i 26.9112 17.3143i 1.86638 10.5848i −82.4803 + 60.7617i −201.097 + 116.103i −118.437 + 136.897i 65.0895 54.6166i 6.75451 + 60.4239i
3.7 −5.16416 + 2.30899i 8.70220 3.16734i 21.3371 23.8480i −6.50218 + 36.8757i −37.6262 + 36.4499i 104.175 60.1453i −55.1237 + 172.422i −120.453 + 101.072i −51.5672 205.445i
3.8 −4.83109 2.94288i 0.477768 0.173893i 14.6789 + 28.4347i 2.39951 13.6083i −2.81989 0.565922i 49.8634 28.7886i 12.7651 180.569i −185.951 + 156.031i −51.6398 + 58.6813i
3.9 −4.82821 + 2.94761i 1.18382 0.430875i 14.6232 28.4634i 16.6949 94.6815i −4.44567 + 5.56980i 127.080 73.3697i 13.2952 + 180.530i −184.933 + 155.177i 198.478 + 506.352i
3.10 −4.64176 + 3.23327i −10.8783 + 3.95939i 11.0919 30.0162i −11.1052 + 62.9808i 37.6929 53.5512i −71.5832 + 41.3286i 45.5645 + 175.191i −83.4872 + 70.0541i −152.086 328.248i
3.11 −4.39545 3.56091i 16.7807 6.10766i 6.63988 + 31.3035i 16.0499 91.0237i −95.5073 32.9085i −124.031 + 71.6095i 82.2838 161.237i 58.1382 48.7837i −394.673 + 342.937i
3.12 −4.03838 3.96124i −17.5013 + 6.36994i 0.617094 + 31.9940i −13.1913 + 74.8117i 95.9098 + 43.6025i −207.031 + 119.529i 124.244 131.649i 79.5697 66.7669i 349.619 249.864i
3.13 −3.98896 + 4.01101i −23.5224 + 8.56144i −0.176366 31.9995i 4.95687 28.1118i 59.4898 128.500i 12.7729 7.37443i 129.054 + 126.937i 293.855 246.573i 92.9838 + 132.019i
3.14 −3.80702 4.18409i −26.7836 + 9.74843i −3.01324 + 31.8578i 14.4150 81.7515i 142.754 + 74.9526i 78.4330 45.2833i 144.767 108.676i 436.180 365.999i −396.934 + 250.916i
3.15 −3.25740 + 4.62486i 23.5224 8.56144i −10.7787 30.1300i 4.95687 28.1118i −37.0261 + 136.676i −12.7729 + 7.37443i 174.458 + 48.2952i 293.855 246.573i 113.867 + 114.496i
3.16 −3.04783 4.76558i 22.6388 8.23986i −13.4215 + 29.0493i −8.94224 + 50.7139i −108.267 82.7735i −13.7539 + 7.94084i 179.343 24.5760i 258.473 216.884i 268.936 111.952i
3.17 −2.84094 4.89174i −6.78809 + 2.47066i −15.8582 + 27.7942i −7.75313 + 43.9702i 31.3703 + 26.1866i 149.921 86.5570i 181.014 1.38754i −146.175 + 122.655i 237.117 86.9902i
3.18 −2.37812 + 5.13269i 10.8783 3.95939i −20.6891 24.4123i −11.1052 + 62.9808i −5.54764 + 65.2511i 71.5832 41.3286i 174.502 48.1354i −83.4872 + 70.0541i −296.852 206.775i
3.19 −2.06442 + 5.26670i −1.18382 + 0.430875i −23.4763 21.7454i 16.6949 94.6815i 0.174611 7.12434i −127.080 + 73.3697i 162.992 78.7512i −184.933 + 155.177i 464.194 + 283.390i
3.20 −1.37716 + 5.48666i −8.70220 + 3.16734i −28.2068 15.1120i −6.50218 + 36.8757i −5.39378 52.1079i −104.175 + 60.1453i 121.760 133.950i −120.453 + 101.072i −193.370 86.4590i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.f odd 18 1 inner
76.k even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.6.k.a 288
4.b odd 2 1 inner 76.6.k.a 288
19.f odd 18 1 inner 76.6.k.a 288
76.k even 18 1 inner 76.6.k.a 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.k.a 288 1.a even 1 1 trivial
76.6.k.a 288 4.b odd 2 1 inner
76.6.k.a 288 19.f odd 18 1 inner
76.6.k.a 288 76.k even 18 1 inner

Hecke kernels

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