Properties

Label 43.4.e.a
Level 43
Weight 4
Character orbit 43.e
Analytic conductor 2.537
Analytic rank 0
Dimension 60
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.53708213025\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{7})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60q - 9q^{2} - 3q^{3} - 31q^{4} - 23q^{5} + 16q^{6} + 96q^{7} + 61q^{8} - 177q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q - 9q^{2} - 3q^{3} - 31q^{4} - 23q^{5} + 16q^{6} + 96q^{7} + 61q^{8} - 177q^{9} - 61q^{10} + 83q^{11} + 33q^{12} + 107q^{13} - 299q^{14} + 109q^{15} + 41q^{16} + 181q^{17} - 414q^{18} + 284q^{19} - 363q^{20} - 88q^{21} + 421q^{22} + 231q^{23} - 937q^{24} + 213q^{25} + 139q^{26} - 27q^{27} + 29q^{28} - 367q^{29} + 1244q^{30} - 319q^{31} + 435q^{32} - 2594q^{33} - 583q^{34} - 902q^{35} + 1552q^{36} + 1020q^{37} + 1251q^{38} - 1571q^{39} + 1263q^{40} + 293q^{41} - 1830q^{42} + 1661q^{43} + 6512q^{44} + 1019q^{45} - 2786q^{46} - 287q^{47} - 95q^{48} + 772q^{49} - 282q^{50} + 1524q^{51} - 1511q^{52} - 1505q^{53} - 3489q^{54} - 1735q^{55} - 1237q^{56} + 1055q^{57} + 335q^{58} + 571q^{59} - 101q^{60} - 339q^{61} + 923q^{62} - 702q^{63} - 5163q^{64} + 2463q^{65} + 985q^{66} - 241q^{67} + 2904q^{68} + 2711q^{69} - 7698q^{70} - 2431q^{71} - 4340q^{72} - 2157q^{73} - 1294q^{74} - 242q^{75} - 4272q^{76} - 3962q^{77} - 2860q^{78} + 1092q^{79} + 11618q^{80} + 12060q^{81} + 4023q^{82} - 2664q^{83} + 3334q^{84} - 3446q^{85} + 10055q^{86} + 11874q^{87} + 9957q^{88} - 5811q^{89} - 1612q^{90} - 760q^{91} + 2120q^{92} + 3994q^{93} + 6057q^{94} + 379q^{95} - 2044q^{96} - 5509q^{97} - 9041q^{98} - 2012q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −3.08835 + 3.87267i −0.198541 0.248963i −3.67950 16.1209i −11.1889 5.38829i 1.57731 3.53619 38.0923 + 18.3443i 5.98550 26.2242i 55.4224 26.6900i
4.2 −2.32023 + 2.90947i 3.61377 + 4.53152i −1.30141 5.70184i 13.3469 + 6.42753i −21.5691 −12.2090 −7.21369 3.47393i −1.46730 + 6.42866i −49.6685 + 23.9191i
4.3 −2.29664 + 2.87990i −5.14704 6.45419i −1.23909 5.42880i 11.0752 + 5.33352i 30.4083 28.5576 −8.06984 3.88623i −9.15641 + 40.1169i −40.7957 + 19.6462i
4.4 −0.976506 + 1.22450i −3.38192 4.24079i 1.23433 + 5.40796i −7.32010 3.52517i 8.49531 −30.7370 −19.1161 9.20584i −0.538880 + 2.36099i 11.4647 5.52111i
4.5 −0.873702 + 1.09559i 4.80787 + 6.02888i 1.34321 + 5.88499i −14.6785 7.06881i −10.8058 16.8292 −17.7214 8.53416i −7.22369 + 31.6490i 20.5692 9.90558i
4.6 0.208948 0.262012i −0.805279 1.00979i 1.75518 + 7.68993i 8.19351 + 3.94579i −0.432838 17.7083 4.79710 + 2.31016i 5.63687 24.6967i 2.74586 1.32234i
4.7 1.20289 1.50838i 3.21845 + 4.03581i 0.951908 + 4.17058i 3.04073 + 1.46434i 9.95900 −15.9770 21.3417 + 10.2776i 0.0787180 0.344886i 5.86645 2.82513i
4.8 1.80704 2.26595i −3.86902 4.85159i −0.0889949 0.389912i −19.3746 9.33033i −17.9850 25.5918 19.8456 + 9.55715i −2.56060 + 11.2187i −56.1528 + 27.0418i
4.9 2.72209 3.41339i −2.84196 3.56370i −2.46131 10.7837i 8.30189 + 3.99798i −19.9004 −12.2688 −12.0406 5.79846i 1.38482 6.06730i 36.2452 17.4548i
4.10 2.96047 3.71232i 4.10366 + 5.14583i −3.23672 14.1810i −6.52318 3.14140i 31.2517 6.49418 −28.0025 13.4853i −3.63147 + 15.9105i −30.9735 + 14.9161i
11.1 −3.08835 3.87267i −0.198541 + 0.248963i −3.67950 + 16.1209i −11.1889 + 5.38829i 1.57731 3.53619 38.0923 18.3443i 5.98550 + 26.2242i 55.4224 + 26.6900i
11.2 −2.32023 2.90947i 3.61377 4.53152i −1.30141 + 5.70184i 13.3469 6.42753i −21.5691 −12.2090 −7.21369 + 3.47393i −1.46730 6.42866i −49.6685 23.9191i
11.3 −2.29664 2.87990i −5.14704 + 6.45419i −1.23909 + 5.42880i 11.0752 5.33352i 30.4083 28.5576 −8.06984 + 3.88623i −9.15641 40.1169i −40.7957 19.6462i
11.4 −0.976506 1.22450i −3.38192 + 4.24079i 1.23433 5.40796i −7.32010 + 3.52517i 8.49531 −30.7370 −19.1161 + 9.20584i −0.538880 2.36099i 11.4647 + 5.52111i
11.5 −0.873702 1.09559i 4.80787 6.02888i 1.34321 5.88499i −14.6785 + 7.06881i −10.8058 16.8292 −17.7214 + 8.53416i −7.22369 31.6490i 20.5692 + 9.90558i
11.6 0.208948 + 0.262012i −0.805279 + 1.00979i 1.75518 7.68993i 8.19351 3.94579i −0.432838 17.7083 4.79710 2.31016i 5.63687 + 24.6967i 2.74586 + 1.32234i
11.7 1.20289 + 1.50838i 3.21845 4.03581i 0.951908 4.17058i 3.04073 1.46434i 9.95900 −15.9770 21.3417 10.2776i 0.0787180 + 0.344886i 5.86645 + 2.82513i
11.8 1.80704 + 2.26595i −3.86902 + 4.85159i −0.0889949 + 0.389912i −19.3746 + 9.33033i −17.9850 25.5918 19.8456 9.55715i −2.56060 11.2187i −56.1528 27.0418i
11.9 2.72209 + 3.41339i −2.84196 + 3.56370i −2.46131 + 10.7837i 8.30189 3.99798i −19.9004 −12.2688 −12.0406 + 5.79846i 1.38482 + 6.06730i 36.2452 + 17.4548i
11.10 2.96047 + 3.71232i 4.10366 5.14583i −3.23672 + 14.1810i −6.52318 + 3.14140i 31.2517 6.49418 −28.0025 + 13.4853i −3.63147 15.9105i −30.9735 14.9161i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.4.e.a 60
43.e even 7 1 inner 43.4.e.a 60
43.e even 7 1 1849.4.a.h 30
43.f odd 14 1 1849.4.a.g 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.4.e.a 60 1.a even 1 1 trivial
43.4.e.a 60 43.e even 7 1 inner
1849.4.a.g 30 43.f odd 14 1
1849.4.a.h 30 43.e even 7 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