Properties

Label 74.4.h.a
Level $74$
Weight $4$
Character orbit 74.h
Analytic conductor $4.366$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 74.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.36614134042\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60q - 12q^{3} + 18q^{5} + 150q^{7} - 96q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q - 12q^{3} + 18q^{5} + 150q^{7} - 96q^{9} - 60q^{10} + 66q^{11} + 48q^{12} + 204q^{13} - 36q^{14} - 198q^{15} - 90q^{17} + 18q^{19} + 72q^{20} - 18q^{21} + 492q^{25} - 192q^{26} + 426q^{27} + 192q^{28} + 360q^{29} + 144q^{30} - 624q^{33} - 24q^{34} - 1494q^{35} - 2592q^{36} - 1482q^{37} + 960q^{38} - 2298q^{39} - 672q^{40} + 828q^{41} - 96q^{42} - 168q^{44} + 3384q^{45} + 1884q^{46} + 444q^{47} + 288q^{48} - 126q^{49} + 1512q^{50} - 552q^{52} + 834q^{53} - 1080q^{54} - 864q^{55} + 3318q^{57} - 1332q^{58} - 2112q^{59} + 2532q^{61} + 2520q^{62} + 2082q^{63} + 1920q^{64} - 540q^{65} - 4002q^{67} + 1596q^{69} - 1512q^{70} - 4302q^{71} - 5460q^{73} + 2328q^{74} + 9144q^{75} + 72q^{76} - 4392q^{77} + 732q^{78} - 1854q^{79} - 2856q^{81} - 1320q^{83} - 1008q^{84} + 888q^{85} + 1512q^{86} + 3936q^{87} + 2592q^{88} + 3198q^{89} - 8868q^{90} - 2088q^{91} + 2832q^{92} + 15408q^{93} + 5568q^{94} + 2166q^{95} - 540q^{97} + 4056q^{98} - 840q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −1.28558 + 1.53209i −7.78411 + 6.53164i −0.694593 3.93923i −6.20210 + 17.0401i 20.3229i 7.25695 + 2.64131i 6.92820 + 4.00000i 13.2415 75.0961i −18.1337 31.4085i
3.2 −1.28558 + 1.53209i −3.90885 + 3.27991i −0.694593 3.93923i 3.66320 10.0646i 10.2053i −15.5322 5.65327i 6.92820 + 4.00000i −0.167232 + 0.948422i 10.7105 + 18.5511i
3.3 −1.28558 + 1.53209i −1.02557 + 0.860552i −0.694593 3.93923i 2.03799 5.59934i 2.67756i 27.8721 + 10.1446i 6.92820 + 4.00000i −4.37727 + 24.8247i 5.95870 + 10.3208i
3.4 −1.28558 + 1.53209i 0.613106 0.514457i −0.694593 3.93923i −3.29690 + 9.05815i 1.60071i −17.1096 6.22737i 6.92820 + 4.00000i −4.57727 + 25.9590i −9.63948 16.6961i
3.5 −1.28558 + 1.53209i 6.92790 5.81319i −0.694593 3.93923i −5.26900 + 14.4765i 18.0875i 22.8859 + 8.32979i 6.92820 + 4.00000i 9.51400 53.9566i −15.4055 26.6831i
3.6 1.28558 1.53209i −6.88175 + 5.77448i −0.694593 3.93923i 5.49624 15.1008i 17.9670i 27.7081 + 10.0849i −6.92820 4.00000i 9.32545 52.8873i −16.0699 27.8339i
3.7 1.28558 1.53209i −4.42123 + 3.70986i −0.694593 3.93923i −1.12359 + 3.08705i 11.5430i −33.1337 12.0597i −6.92820 4.00000i 1.09578 6.21447i 3.28516 + 5.69007i
3.8 1.28558 1.53209i −3.00702 + 2.52319i −0.694593 3.93923i −4.82880 + 13.2670i 7.85076i 21.8021 + 7.93531i −6.92820 4.00000i −2.01282 + 11.4153i 14.1185 + 24.4539i
3.9 1.28558 1.53209i 2.61104 2.19092i −0.694593 3.93923i 3.30841 9.08979i 6.81693i −7.17041 2.60982i −6.92820 4.00000i −2.67112 + 15.1487i −9.67315 16.7544i
3.10 1.28558 1.53209i 6.52145 5.47215i −0.694593 3.93923i −1.16279 + 3.19474i 17.0263i 12.9120 + 4.69957i −6.92820 4.00000i 7.89642 44.7828i 3.39977 + 5.88857i
21.1 −0.684040 + 1.87939i −8.01099 + 2.91576i −3.06418 2.57115i −1.36441 0.240582i 17.0502i 0.234093 1.32761i 6.92820 4.00000i 34.9911 29.3610i 1.38546 2.39968i
21.2 −0.684040 + 1.87939i −1.59789 + 0.581585i −3.06418 2.57115i 19.8439 + 3.49901i 3.40088i 1.06400 6.03424i 6.92820 4.00000i −18.4682 + 15.4966i −20.1500 + 34.9008i
21.3 −0.684040 + 1.87939i 1.13456 0.412945i −3.06418 2.57115i −13.1250 2.31430i 2.41474i 5.34436 30.3094i 6.92820 4.00000i −19.5665 + 16.4182i 13.3275 23.0839i
21.4 −0.684040 + 1.87939i 2.51768 0.916361i −3.06418 2.57115i −10.4603 1.84444i 5.35852i −6.29270 + 35.6877i 6.92820 4.00000i −15.1842 + 12.7411i 10.6217 18.3974i
21.5 −0.684040 + 1.87939i 8.12302 2.95654i −3.06418 2.57115i 5.91804 + 1.04351i 17.2887i 0.991911 5.62541i 6.92820 4.00000i 36.5591 30.6767i −6.00933 + 10.4085i
21.6 0.684040 1.87939i −5.71416 + 2.07978i −3.06418 2.57115i 5.14822 + 0.907770i 12.1618i −3.41363 + 19.3596i −6.92820 + 4.00000i 7.64290 6.41315i 5.22764 9.05454i
21.7 0.684040 1.87939i −4.20598 + 1.53085i −3.06418 2.57115i 16.4608 + 2.90248i 8.95182i 5.69922 32.3219i −6.92820 + 4.00000i −5.33644 + 4.47780i 16.7147 28.9507i
21.8 0.684040 1.87939i −1.20040 + 0.436908i −3.06418 2.57115i −14.3629 2.53256i 2.55487i −1.05161 + 5.96397i −6.92820 + 4.00000i −19.4331 + 16.3063i −14.5844 + 25.2610i
21.9 0.684040 1.87939i 6.26608 2.28067i −3.06418 2.57115i 17.1326 + 3.02094i 13.3365i −3.13680 + 17.7897i −6.92820 + 4.00000i 13.3792 11.2264i 17.3969 30.1323i
21.10 0.684040 1.87939i 7.02082 2.55537i −3.06418 2.57115i −7.50303 1.32299i 14.9428i 3.84603 21.8119i −6.92820 + 4.00000i 22.0789 18.5264i −7.61878 + 13.1961i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.4.h.a 60
37.h even 18 1 inner 74.4.h.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.h.a 60 1.a even 1 1 trivial
74.4.h.a 60 37.h even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(74, [\chi])\).