Properties

Label 76.5.j.a
Level $76$
Weight $5$
Character orbit 76.j
Analytic conductor $7.856$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 76.j (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.85611719437\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 42q + 12q^{3} - 45q^{7} - 84q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 42q + 12q^{3} - 45q^{7} - 84q^{9} - 45q^{11} + 33q^{13} - 393q^{15} + 909q^{17} + 1242q^{19} + 1107q^{21} - 360q^{23} - 810q^{25} - 7056q^{27} - 2889q^{29} + 2808q^{31} + 10875q^{33} + 6741q^{35} - 3480q^{39} - 3060q^{41} - 8079q^{43} - 4320q^{45} - 2655q^{47} - 474q^{49} - 12222q^{51} - 6705q^{53} + 4623q^{55} - 8022q^{57} + 24309q^{59} + 7104q^{61} + 12063q^{63} + 25245q^{65} + 15573q^{67} - 10881q^{69} - 25506q^{71} + 3036q^{73} + 12924q^{77} - 16839q^{79} - 2208q^{81} - 6363q^{83} - 37890q^{85} - 21924q^{87} - 22644q^{89} + 17418q^{91} + 8184q^{93} - 82413q^{95} + 13383q^{97} + 23565q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −4.65644 + 12.7935i 0 6.59099 + 37.3793i 0 5.00284 8.66518i 0 −79.9406 67.0781i 0
13.2 0 −4.13985 + 11.3741i 0 −6.25205 35.4571i 0 −25.8702 + 44.8086i 0 −50.1832 42.1087i 0
13.3 0 −2.28730 + 6.28431i 0 −2.03045 11.5152i 0 38.6373 66.9218i 0 27.7888 + 23.3175i 0
13.4 0 −0.180467 + 0.495829i 0 −0.550159 3.12011i 0 −25.9990 + 45.0316i 0 61.8363 + 51.8868i 0
13.5 0 2.65366 7.29088i 0 0.405227 + 2.29816i 0 23.2726 40.3094i 0 15.9346 + 13.3707i 0
13.6 0 3.28951 9.03787i 0 7.51461 + 42.6175i 0 −18.8739 + 32.6905i 0 −8.81253 7.39459i 0
13.7 0 5.33930 14.6696i 0 −7.24101 41.0658i 0 −15.7120 + 27.2140i 0 −124.640 104.585i 0
21.1 0 −8.26669 + 9.85185i 0 −19.2135 + 6.99315i 0 1.56085 + 2.70347i 0 −14.6554 83.1149i 0
21.2 0 −7.77177 + 9.26203i 0 42.9478 15.6317i 0 2.15264 + 3.72849i 0 −11.3194 64.1954i 0
21.3 0 −0.504186 + 0.600866i 0 −8.75235 + 3.18559i 0 −3.87876 6.71820i 0 13.9587 + 79.1635i 0
21.4 0 1.91062 2.27699i 0 −16.1447 + 5.87619i 0 23.9804 + 41.5352i 0 12.5313 + 71.0685i 0
21.5 0 2.92113 3.48127i 0 20.0929 7.31321i 0 −36.0108 62.3725i 0 10.4793 + 59.4309i 0
21.6 0 9.11558 10.8635i 0 31.4208 11.4362i 0 39.9854 + 69.2568i 0 −20.8569 118.285i 0
21.7 0 10.9995 13.1087i 0 −41.8937 + 15.2481i 0 −26.2709 45.5025i 0 −36.7834 208.609i 0
29.1 0 −8.26669 9.85185i 0 −19.2135 6.99315i 0 1.56085 2.70347i 0 −14.6554 + 83.1149i 0
29.2 0 −7.77177 9.26203i 0 42.9478 + 15.6317i 0 2.15264 3.72849i 0 −11.3194 + 64.1954i 0
29.3 0 −0.504186 0.600866i 0 −8.75235 3.18559i 0 −3.87876 + 6.71820i 0 13.9587 79.1635i 0
29.4 0 1.91062 + 2.27699i 0 −16.1447 5.87619i 0 23.9804 41.5352i 0 12.5313 71.0685i 0
29.5 0 2.92113 + 3.48127i 0 20.0929 + 7.31321i 0 −36.0108 + 62.3725i 0 10.4793 59.4309i 0
29.6 0 9.11558 + 10.8635i 0 31.4208 + 11.4362i 0 39.9854 69.2568i 0 −20.8569 + 118.285i 0
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.5.j.a 42
19.f odd 18 1 inner 76.5.j.a 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.j.a 42 1.a even 1 1 trivial
76.5.j.a 42 19.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(76, [\chi])\).