Properties

Label 76.4.k.a
Level $76$
Weight $4$
Character orbit 76.k
Analytic conductor $4.484$
Analytic rank $0$
Dimension $168$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.k (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.48414516044\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 168q - 6q^{2} - 24q^{4} - 12q^{5} - 24q^{6} - 9q^{8} + 18q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 168q - 6q^{2} - 24q^{4} - 12q^{5} - 24q^{6} - 9q^{8} + 18q^{9} - 105q^{10} - 9q^{12} - 120q^{13} + 69q^{14} + 192q^{16} - 12q^{17} + 558q^{20} + 6q^{21} - 30q^{22} + 96q^{24} - 12q^{25} - 411q^{26} + 756q^{28} - 12q^{29} + 276q^{30} - 471q^{32} - 576q^{33} + 36q^{34} - 2673q^{36} - 648q^{38} - 2298q^{40} - 606q^{41} - 321q^{42} - 1203q^{44} - 6q^{45} + 1566q^{46} + 3237q^{48} + 2346q^{49} + 3204q^{50} + 1077q^{52} + 576q^{53} - 627q^{54} - 12q^{57} - 4116q^{58} + 90q^{60} + 3528q^{61} - 3300q^{62} - 381q^{64} + 1242q^{65} + 276q^{66} + 1170q^{68} - 4770q^{69} + 1449q^{70} + 1146q^{72} - 3468q^{73} + 3105q^{74} + 4386q^{76} - 9396q^{77} + 6939q^{78} + 2133q^{80} + 1980q^{81} + 7299q^{82} + 315q^{84} - 516q^{85} - 3804q^{86} - 5841q^{88} + 3576q^{89} - 8898q^{90} - 7668q^{92} + 5694q^{93} + 18942q^{96} + 774q^{97} + 8745q^{98} + O(q^{100}) \)

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 −2.82429 + 0.152912i 5.54644 2.01874i 7.95324 0.863738i 2.56623 14.5538i −15.3561 + 6.54962i −1.22017 + 0.704465i −22.3302 + 3.65560i 6.00448 5.03835i −5.02232 + 41.4966i
3.2 −2.77864 + 0.528343i −2.57091 + 0.935735i 7.44171 2.93615i −1.56986 + 8.90313i 6.64925 3.95839i 16.8987 9.75650i −19.1266 + 12.0903i −14.9492 + 12.5439i −0.341819 25.5680i
3.3 −2.67496 + 0.919012i −7.61194 + 2.77052i 6.31083 4.91664i 2.53263 14.3633i 17.8155 14.4065i −10.1706 + 5.87199i −12.3628 + 18.9516i 29.5827 24.8228i 6.42533 + 40.7487i
3.4 −2.66917 0.935695i 2.28426 0.831401i 6.24895 + 4.99506i −1.38264 + 7.84134i −6.87501 + 0.0817851i −8.80596 + 5.08413i −12.0057 19.1798i −16.1566 + 13.5570i 11.0276 19.6361i
3.5 −2.42254 1.45990i −6.72563 + 2.44793i 3.73740 + 7.07332i −0.791289 + 4.48762i 19.8668 + 3.88852i −1.86219 + 1.07514i 1.27231 22.5916i 18.5585 15.5724i 8.46839 9.71624i
3.6 −2.38811 + 1.51557i 9.08416 3.30637i 3.40609 7.23868i −3.54781 + 20.1206i −16.6829 + 21.6636i 4.44622 2.56703i 2.83662 + 22.4489i 50.9068 42.7159i −22.0217 53.4271i
3.7 −1.90584 + 2.08992i 1.16156 0.422774i −0.735541 7.96611i 0.336107 1.90616i −1.33019 + 3.23331i −25.4625 + 14.7008i 18.0504 + 13.6449i −19.5127 + 16.3731i 3.34316 + 4.33527i
3.8 −1.72722 + 2.23980i −1.16156 + 0.422774i −2.03339 7.73727i 0.336107 1.90616i 1.05935 3.33189i 25.4625 14.7008i 20.8420 + 8.80962i −19.5127 + 16.3731i 3.68888 + 4.04518i
3.9 −1.70042 2.26022i 5.85727 2.13187i −2.21716 + 7.68663i 0.184577 1.04679i −14.7783 9.61362i 24.9926 14.4295i 21.1435 8.05922i 9.07949 7.61860i −2.67983 + 1.36279i
3.10 −1.58740 2.34097i −1.55643 + 0.566493i −2.96030 + 7.43213i 2.88655 16.3705i 3.79682 + 2.74430i −17.3301 + 10.0055i 22.0976 4.86782i −18.5816 + 15.5919i −42.9049 + 19.2292i
3.11 −1.07786 + 2.61500i −9.08416 + 3.30637i −5.67646 5.63718i −3.54781 + 20.1206i 1.14527 27.3189i −4.44622 + 2.56703i 20.8596 8.76787i 50.9068 42.7159i −48.7914 30.9646i
3.12 −0.518452 2.78050i 2.06434 0.751360i −7.46241 + 2.88312i −3.33172 + 18.8951i −3.15942 5.35037i −15.6472 + 9.03393i 11.8854 + 19.2545i −16.9862 + 14.2531i 54.2653 0.532358i
3.13 −0.440548 + 2.79391i 7.61194 2.77052i −7.61183 2.46170i 2.53263 14.3633i 4.38715 + 22.4876i 10.1706 5.87199i 10.2311 20.1823i 29.5827 24.8228i 39.0139 + 13.4036i
3.14 −0.428429 2.79579i −6.50066 + 2.36605i −7.63290 + 2.39560i −0.324438 + 1.83998i 9.40005 + 17.1608i 19.7617 11.4094i 9.96774 + 20.3136i 15.9773 13.4065i 5.28320 + 0.118761i
3.15 −0.0378098 + 2.82817i 2.57091 0.935735i −7.99714 0.213866i −1.56986 + 8.90313i 2.54921 + 7.30636i −16.8987 + 9.75650i 0.907220 22.6092i −14.9492 + 12.5439i −25.1202 4.77647i
3.16 0.311236 2.81125i 8.99625 3.27437i −7.80626 1.74992i 1.82627 10.3573i −6.40511 26.3098i −21.9414 + 12.6679i −7.34906 + 21.4007i 49.5278 41.5588i −28.5485 8.35765i
3.17 0.339844 + 2.80794i −5.54644 + 2.01874i −7.76901 + 1.90852i 2.56623 14.5538i −7.55341 14.8880i 1.22017 0.704465i −7.99925 21.1663i 6.00448 5.03835i 41.7383 + 2.25979i
3.18 0.844697 2.69935i −1.12992 + 0.411256i −6.57297 4.56026i 2.09426 11.8771i 0.155687 + 3.39743i 6.44819 3.72286i −17.8619 + 13.8907i −19.5756 + 16.4259i −30.2915 15.6857i
3.19 1.38498 + 2.46614i −2.28426 + 0.831401i −4.16368 + 6.83109i −1.38264 + 7.84134i −5.21399 4.48182i 8.80596 5.08413i −22.6130 0.807318i −16.1566 + 13.5570i −21.2527 + 7.45029i
3.20 1.71026 2.25278i −5.58264 + 2.03191i −2.15002 7.70567i −1.53918 + 8.72914i −4.97031 + 16.0515i −18.0252 + 10.4069i −21.0363 8.33517i 6.35395 5.33160i 17.0324 + 18.3965i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.28
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 76.4.k.a 168
4.b odd 2 1 inner 76.4.k.a 168
19.f odd 18 1 inner 76.4.k.a 168
76.k even 18 1 inner 76.4.k.a 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.k.a 168 1.a even 1 1 trivial
76.4.k.a 168 4.b odd 2 1 inner
76.4.k.a 168 19.f odd 18 1 inner
76.4.k.a 168 76.k even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(76, [\chi])\).