Properties

Label 76.6.i.a
Level $76$
Weight $6$
Character orbit 76.i
Analytic conductor $12.189$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 76.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1891703058\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48q + 33q^{3} - 177q^{7} + 33q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q + 33q^{3} - 177q^{7} + 33q^{9} - 237q^{11} + 2049q^{13} + 2085q^{15} - 609q^{17} - 6012q^{19} + 3591q^{21} + 14100q^{23} + 11802q^{25} - 861q^{27} - 10575q^{29} - 6546q^{31} - 21234q^{33} - 231q^{35} + 20052q^{37} + 72204q^{39} - 3249q^{41} - 34677q^{43} - 34956q^{45} + 4461q^{47} - 41139q^{49} + 12099q^{51} - 24291q^{53} - 61767q^{55} - 64470q^{57} - 20100q^{59} + 95490q^{61} + 86403q^{63} - 57915q^{65} - 64452q^{67} - 99315q^{69} - 115536q^{71} - 16362q^{73} + 236250q^{75} + 26688q^{77} + 29799q^{79} + 180327q^{81} + 52347q^{83} + 204618q^{85} + 69414q^{87} + 47394q^{89} - 249384q^{91} - 462126q^{93} + 412869q^{95} - 229974q^{97} - 692274q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0 −4.36641 + 24.7631i 0 −45.0243 37.7798i 0 −12.6797 21.9619i 0 −365.802 133.141i 0
5.2 0 −2.77836 + 15.7569i 0 40.2412 + 33.7663i 0 35.1538 + 60.8882i 0 −12.2145 4.44570i 0
5.3 0 −0.700721 + 3.97399i 0 26.7333 + 22.4319i 0 −120.347 208.448i 0 213.044 + 77.5416i 0
5.4 0 0.149131 0.845767i 0 −73.5089 61.6813i 0 18.6370 + 32.2802i 0 227.652 + 82.8586i 0
5.5 0 1.50126 8.51408i 0 −8.90022 7.46817i 0 70.8987 + 122.800i 0 158.110 + 57.5472i 0
5.6 0 2.69006 15.2561i 0 0.287480 + 0.241225i 0 −83.3571 144.379i 0 2.83426 + 1.03159i 0
5.7 0 3.64287 20.6597i 0 83.5459 + 70.1033i 0 58.2786 + 100.941i 0 −185.208 67.4103i 0
5.8 0 5.01488 28.4408i 0 −31.8009 26.6842i 0 −9.54633 16.5347i 0 −555.385 202.143i 0
9.1 0 −16.5327 + 13.8726i 0 17.0893 + 6.21998i 0 −92.6180 + 160.419i 0 38.6850 219.393i 0
9.2 0 −15.3348 + 12.8674i 0 −87.8167 31.9627i 0 31.3213 54.2501i 0 27.3887 155.329i 0
9.3 0 −13.4877 + 11.3175i 0 63.2325 + 23.0148i 0 73.8179 127.856i 0 11.6351 65.9857i 0
9.4 0 −0.670817 + 0.562882i 0 −12.4943 4.54757i 0 41.6708 72.1759i 0 −42.0633 + 238.553i 0
9.5 0 6.23353 5.23055i 0 −51.5730 18.7710i 0 −28.9937 + 50.2186i 0 −30.6983 + 174.099i 0
9.6 0 7.22070 6.05889i 0 72.9012 + 26.5339i 0 −122.846 + 212.775i 0 −26.7681 + 151.809i 0
9.7 0 14.0139 11.7590i 0 57.8858 + 21.0687i 0 68.1093 117.969i 0 15.9173 90.2714i 0
9.8 0 22.5258 18.9014i 0 −48.8881 17.7938i 0 −11.8403 + 20.5081i 0 107.952 612.226i 0
17.1 0 −16.5327 13.8726i 0 17.0893 6.21998i 0 −92.6180 160.419i 0 38.6850 + 219.393i 0
17.2 0 −15.3348 12.8674i 0 −87.8167 + 31.9627i 0 31.3213 + 54.2501i 0 27.3887 + 155.329i 0
17.3 0 −13.4877 11.3175i 0 63.2325 23.0148i 0 73.8179 + 127.856i 0 11.6351 + 65.9857i 0
17.4 0 −0.670817 0.562882i 0 −12.4943 + 4.54757i 0 41.6708 + 72.1759i 0 −42.0633 238.553i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.6.i.a 48
19.e even 9 1 inner 76.6.i.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.i.a 48 1.a even 1 1 trivial
76.6.i.a 48 19.e even 9 1 inner

Hecke kernels

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