Properties

Label 74.6.h.a
Level $74$
Weight $6$
Character orbit 74.h
Analytic conductor $11.868$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(11.8684026662\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96q + 6q^{3} + 90q^{5} - 228q^{7} - 186q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 96q + 6q^{3} + 90q^{5} - 228q^{7} - 186q^{9} + 600q^{10} - 726q^{11} - 96q^{12} - 246q^{13} + 3960q^{14} - 3960q^{15} - 1530q^{17} + 3222q^{19} + 1440q^{20} - 16110q^{21} + 8232q^{25} + 13632q^{26} + 2154q^{27} - 10464q^{28} - 18378q^{29} - 25056q^{30} + 78198q^{33} + 37296q^{34} + 82170q^{35} - 124416q^{36} - 5586q^{37} - 39936q^{38} + 58686q^{39} - 16128q^{40} - 59598q^{41} - 26304q^{42} + 480q^{44} - 74646q^{45} - 22536q^{46} - 1266q^{47} + 193320q^{49} + 18864q^{50} + 68352q^{52} - 57432q^{53} + 101520q^{54} + 133830q^{55} - 228936q^{57} - 79224q^{58} - 52800q^{59} + 68664q^{61} - 72720q^{62} - 166254q^{63} + 196608q^{64} - 122058q^{65} + 254946q^{67} - 218094q^{69} - 25872q^{70} + 200052q^{71} - 32244q^{73} + 124368q^{74} - 234036q^{75} + 51552q^{76} + 397188q^{77} - 16584q^{78} + 267462q^{79} - 371496q^{81} + 121440q^{83} + 84672q^{84} - 337260q^{85} + 43920q^{86} - 667500q^{87} - 135936q^{88} + 377850q^{89} - 576408q^{90} + 875826q^{91} + 233376q^{92} + 365868q^{93} - 80256q^{94} + 764592q^{95} - 823212q^{97} - 342192q^{98} - 928650q^{99} + O(q^{100}) \)

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 −2.57115 + 3.06418i −20.5579 + 17.2501i −2.77837 15.7569i 15.9101 43.7127i 107.346i 202.247 + 73.6118i 55.4256 + 32.0000i 82.8635 469.942i 93.0361 + 161.143i
3.2 −2.57115 + 3.06418i −16.2007 + 13.5940i −2.77837 15.7569i 0.866934 2.38188i 84.5942i −134.390 48.9139i 55.4256 + 32.0000i 35.4695 201.158i 5.06949 + 8.78062i
3.3 −2.57115 + 3.06418i −5.46669 + 4.58709i −2.77837 15.7569i −35.9379 + 98.7386i 28.5450i 74.9073 + 27.2640i 55.4256 + 32.0000i −33.3533 + 189.156i −210.151 363.992i
3.4 −2.57115 + 3.06418i −4.69686 + 3.94113i −2.77837 15.7569i 9.28305 25.5050i 24.5252i 77.8232 + 28.3253i 55.4256 + 32.0000i −35.6686 + 202.286i 54.2837 + 94.0221i
3.5 −2.57115 + 3.06418i 2.39572 2.01025i −2.77837 15.7569i 36.3778 99.9471i 12.5096i −172.609 62.8245i 55.4256 + 32.0000i −40.4981 + 229.676i 212.723 + 368.447i
3.6 −2.57115 + 3.06418i 7.21134 6.05104i −2.77837 15.7569i −5.26761 + 14.4726i 37.6550i −11.7126 4.26304i 55.4256 + 32.0000i −26.8081 + 152.036i −30.8029 53.3522i
3.7 −2.57115 + 3.06418i 18.1172 15.2021i −2.77837 15.7569i 15.7606 43.3020i 94.6013i 83.7120 + 30.4687i 55.4256 + 32.0000i 54.9315 311.532i 92.1620 + 159.629i
3.8 −2.57115 + 3.06418i 20.6376 17.3170i −2.77837 15.7569i −28.8625 + 79.2990i 107.762i −209.650 76.3065i 55.4256 + 32.0000i 83.8350 475.452i −168.776 292.329i
3.9 2.57115 3.06418i −20.6482 + 17.3259i −2.77837 15.7569i −24.4685 + 67.2268i 107.817i −39.0875 14.2267i −55.4256 32.0000i 83.9648 476.188i 143.082 + 247.826i
3.10 2.57115 3.06418i −16.4794 + 13.8279i −2.77837 15.7569i 14.6671 40.2975i 86.0494i 36.2322 + 13.1874i −55.4256 32.0000i 38.1647 216.443i −85.7674 148.554i
3.11 2.57115 3.06418i −9.90239 + 8.30909i −2.77837 15.7569i 16.4126 45.0933i 51.7066i −20.3554 7.40876i −55.4256 32.0000i −13.1802 + 74.7486i −95.9745 166.233i
3.12 2.57115 3.06418i 1.03890 0.871742i −2.77837 15.7569i −11.0838 + 30.4525i 5.42476i 194.404 + 70.7571i −55.4256 32.0000i −41.8771 + 237.497i 64.8138 + 112.261i
3.13 2.57115 3.06418i 2.44498 2.05158i −2.77837 15.7569i −23.1154 + 63.5089i 12.7668i −54.4150 19.8054i −55.4256 32.0000i −40.4276 + 229.276i 135.170 + 234.121i
3.14 2.57115 3.06418i 10.7924 9.05588i −2.77837 15.7569i 17.1692 47.1719i 56.3538i −205.267 74.7112i −55.4256 32.0000i −7.73005 + 43.8393i −100.399 173.895i
3.15 2.57115 3.06418i 12.3631 10.3739i −2.77837 15.7569i 33.5955 92.3028i 64.5555i 200.133 + 72.8426i −55.4256 32.0000i 3.03242 17.1977i −196.453 340.267i
3.16 2.57115 3.06418i 21.8303 18.3178i −2.77837 15.7569i −10.1389 + 27.8564i 113.990i −22.2808 8.10956i −55.4256 32.0000i 98.8244 560.461i 59.2884 + 102.690i
21.1 −1.36808 + 3.75877i −23.0035 + 8.37259i −12.2567 10.2846i 48.7297 + 8.59237i 97.9193i 23.7455 134.667i 55.4256 32.0000i 272.912 229.000i −98.9629 + 171.409i
21.2 −1.36808 + 3.75877i −17.4563 + 6.35356i −12.2567 10.2846i −104.064 18.3493i 74.3062i 19.0842 108.232i 55.4256 32.0000i 78.2045 65.6213i 211.339 366.050i
21.3 −1.36808 + 3.75877i −10.7367 + 3.90785i −12.2567 10.2846i −56.2384 9.91636i 45.7031i −23.1802 + 131.461i 55.4256 32.0000i −86.1429 + 72.2825i 114.212 197.821i
21.4 −1.36808 + 3.75877i −10.5624 + 3.84441i −12.2567 10.2846i 53.2500 + 9.38941i 44.9612i −24.0440 + 136.360i 55.4256 32.0000i −89.3632 + 74.9846i −108.143 + 187.309i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.16
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 74.6.h.a 96
37.h even 18 1 inner 74.6.h.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.6.h.a 96 1.a even 1 1 trivial
74.6.h.a 96 37.h even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(74, [\chi])\).