Properties

Label 43.4.g.a
Level 43
Weight 4
Character orbit 43.g
Analytic conductor 2.537
Analytic rank 0
Dimension 120
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.53708213025\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{21})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120q - 12q^{2} - 9q^{3} - 92q^{4} + 5q^{5} - 22q^{6} - 54q^{7} + 2q^{8} + 201q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 120q - 12q^{2} - 9q^{3} - 92q^{4} + 5q^{5} - 22q^{6} - 54q^{7} + 2q^{8} + 201q^{9} - 41q^{10} - 68q^{11} + 114q^{12} - 167q^{13} + 254q^{14} - 163q^{15} - 344q^{16} + 68q^{17} - 72q^{18} - 407q^{19} + 621q^{20} + 193q^{21} - 520q^{22} - 219q^{23} + 1072q^{24} - 87q^{25} - 133q^{26} + 180q^{27} + 1228q^{28} - 17q^{29} - 1796q^{30} - 953q^{31} - 2730q^{32} + 473q^{33} - 1043q^{34} - 241q^{35} - 175q^{36} - 228q^{37} + 1512q^{38} + 1250q^{39} + 2673q^{40} - 236q^{41} + 5286q^{42} + 1789q^{43} - 2756q^{44} + 856q^{45} + 4331q^{46} + 962q^{47} + 5243q^{48} - 1264q^{49} - 3273q^{50} - 4803q^{51} - 3538q^{52} - 1375q^{53} - 2646q^{54} - 1460q^{55} - 3305q^{56} - 719q^{57} + 142q^{58} + 1202q^{59} + 4043q^{60} + 837q^{61} - 3959q^{62} + 3279q^{63} + 5718q^{64} + 54q^{65} - 3457q^{66} + 1384q^{67} - 747q^{68} - 4715q^{69} - 2553q^{70} - 1619q^{71} - 20137q^{72} - 3630q^{73} - 5006q^{74} - 1186q^{75} + 1092q^{76} + 3515q^{77} + 2980q^{78} + 4422q^{79} - 1610q^{80} - 4089q^{81} + 5292q^{82} + 10398q^{83} + 17399q^{84} + 8666q^{85} + 1858q^{86} + 2754q^{87} + 7290q^{88} + 11478q^{89} + 29113q^{90} + 11920q^{91} + 3286q^{92} - 4q^{93} - 14736q^{94} - 1741q^{95} - 6200q^{96} + 2236q^{97} - 5254q^{98} - 7417q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −3.45964 4.33825i −3.12542 0.471081i −5.07115 + 22.2182i −0.150276 0.102456i 8.76916 + 15.1886i −0.220734 + 0.382322i 73.9379 35.6066i −16.2541 5.01374i 0.0754189 + 1.00640i
9.2 −2.34805 2.94437i 6.79273 + 1.02384i −1.37577 + 6.02765i 14.9228 + 10.1742i −12.9351 22.4043i 3.09950 5.36848i −6.16631 + 2.96954i 19.2924 + 5.95092i −5.08298 67.8276i
9.3 −1.84616 2.31501i 0.851997 + 0.128418i −0.170806 + 0.748350i −8.91021 6.07488i −1.27563 2.20946i 4.30159 7.45057i −19.2945 + 9.29173i −25.0911 7.73957i 2.38626 + 31.8425i
9.4 −1.62068 2.03226i −6.93204 1.04484i 0.276661 1.21213i 6.69761 + 4.56636i 9.11121 + 15.7811i −16.5418 + 28.6512i −21.6473 + 10.4248i 21.1610 + 6.52730i −1.57463 21.0119i
9.5 0.243093 + 0.304829i 7.00851 + 1.05636i 1.74634 7.65122i −2.70479 1.84409i 1.38171 + 2.39319i −4.58307 + 7.93810i 5.56708 2.68096i 22.2028 + 6.84867i −0.0953820 1.27278i
9.6 0.277412 + 0.347863i −3.67378 0.553733i 1.73612 7.60642i 14.5525 + 9.92173i −0.826526 1.43158i 18.2723 31.6486i 6.33459 3.05058i −12.6104 3.88980i 0.585630 + 7.81469i
9.7 0.789787 + 0.990362i −6.14934 0.926863i 1.42311 6.23507i −11.5199 7.85413i −3.93874 6.82209i −0.966032 + 1.67322i 16.4291 7.91186i 11.1548 + 3.44080i −1.31983 17.6120i
9.8 2.22758 + 2.79329i 1.24096 + 0.187044i −1.06022 + 4.64514i 6.85374 + 4.67280i 2.24186 + 3.88302i −6.94688 + 12.0324i 10.4146 5.01540i −24.2955 7.49416i 2.21473 + 29.5535i
9.9 3.02609 + 3.79460i 5.24807 + 0.791020i −3.46159 + 15.1662i −13.3495 9.10152i 12.8796 + 22.3080i 13.9767 24.2083i −33.0423 + 15.9123i 1.11608 + 0.344266i −5.86013 78.1980i
9.10 3.27326 + 4.10454i −9.59032 1.44551i −4.35283 + 19.0710i 4.62627 + 3.15414i −25.4585 44.0953i 0.410104 0.710321i −54.6855 + 26.3351i 64.0843 + 19.7674i 2.19671 + 29.3130i
10.1 −1.17910 5.16599i 2.10604 + 1.95412i −18.0894 + 8.71140i −16.7285 + 2.52142i 7.61172 13.1839i −11.8225 20.4772i 39.9021 + 50.0356i −1.40089 18.6936i 32.7503 + 83.4464i
10.2 −1.08942 4.77305i −6.48010 6.01265i −14.3874 + 6.92863i 20.4076 3.07594i −21.6392 + 37.4801i −5.26876 9.12575i 24.3248 + 30.5023i 3.82197 + 51.0006i −36.9140 94.0553i
10.3 −0.757202 3.31752i 4.24926 + 3.94274i −3.22482 + 1.55299i 10.1014 1.52254i 9.86256 17.0825i 4.90816 + 8.50119i −9.37914 11.7611i 0.493326 + 6.58297i −12.6998 32.3586i
10.4 −0.582989 2.55424i −4.15286 3.85329i 1.02347 0.492879i −13.6446 + 2.05660i −7.42117 + 12.8538i 9.34797 + 16.1912i −14.9236 18.7136i 0.380687 + 5.07991i 13.2077 + 33.6527i
10.5 −0.294103 1.28855i −1.69703 1.57462i 5.63389 2.71314i 1.82643 0.275290i −1.52987 + 2.64981i −11.2969 19.5668i −11.7454 14.7283i −1.61721 21.5801i −0.891883 2.27248i
10.6 0.113555 + 0.497516i 4.85972 + 4.50917i 6.97312 3.35808i −8.91725 + 1.34406i −1.69154 + 2.92983i 2.21697 + 3.83991i 5.00791 + 6.27973i 1.26664 + 16.9021i −1.68129 4.28385i
10.7 0.399395 + 1.74986i −2.95769 2.74433i 4.30524 2.07330i 12.0373 1.81433i 3.62092 6.27163i 7.86579 + 13.6240i 14.3001 + 17.9318i −0.801160 10.6907i 7.98245 + 20.3390i
10.8 0.747545 + 3.27521i −6.89751 6.39996i −2.96041 + 1.42566i −13.7333 + 2.06996i 15.8050 27.3750i −15.9110 27.5586i 9.87423 + 12.3819i 4.59853 + 61.3631i −17.0458 43.4320i
10.9 0.882546 + 3.86669i 4.09405 + 3.79872i −6.96463 + 3.35399i 7.73032 1.16516i −11.0753 + 19.1829i −13.6399 23.6249i 0.667271 + 0.836731i 0.313230 + 4.17976i 11.3277 + 28.8624i
10.10 1.05311 + 4.61397i 0.0821692 + 0.0762418i −12.9720 + 6.24696i −9.16285 + 1.38108i −0.265245 + 0.459417i 13.3494 + 23.1219i −18.8782 23.6726i −2.01677 26.9120i −16.0217 40.8227i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.4.g.a 120
43.g even 21 1 inner 43.4.g.a 120
43.g even 21 1 1849.4.a.l 60
43.h odd 42 1 1849.4.a.k 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.g.a 120 1.a even 1 1 trivial
43.4.g.a 120 43.g even 21 1 inner
1849.4.a.k 60 43.h odd 42 1
1849.4.a.l 60 43.g even 21 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database