Properties

Label 74.5.g.b
Level $74$
Weight $5$
Character orbit 74.g
Analytic conductor $7.649$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 74.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.64937726820\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28q + 28q^{2} - 102q^{5} + 40q^{6} - 48q^{7} + 448q^{8} + 556q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 28q + 28q^{2} - 102q^{5} + 40q^{6} - 48q^{7} + 448q^{8} + 556q^{9} - 120q^{10} + 80q^{12} - 88q^{13} - 192q^{14} + 12q^{15} + 896q^{16} - 540q^{17} - 1112q^{18} + 976q^{19} - 816q^{20} - 618q^{21} + 752q^{22} + 1996q^{23} - 160q^{24} + 414q^{25} - 2504q^{26} + 1584q^{28} + 852q^{29} + 804q^{30} + 300q^{31} - 1792q^{32} + 316q^{33} + 300q^{34} + 796q^{35} - 3678q^{37} - 328q^{38} + 3630q^{39} - 2784q^{40} - 6426q^{41} - 1348q^{42} - 5548q^{43} - 1280q^{44} + 10164q^{45} + 3992q^{46} - 7624q^{47} - 7388q^{49} + 1768q^{50} + 15382q^{51} - 704q^{52} + 13538q^{53} - 1108q^{54} - 10352q^{55} + 2400q^{56} - 12946q^{57} + 3000q^{58} + 5540q^{59} - 3024q^{60} - 26794q^{61} + 2880q^{62} - 61204q^{63} + 39498q^{65} + 1264q^{66} + 23910q^{67} + 1200q^{68} + 19510q^{69} + 524q^{70} - 15992q^{71} + 8896q^{72} - 16780q^{74} + 56644q^{75} + 7808q^{76} + 18960q^{77} + 11892q^{78} - 1792q^{79} - 1920q^{80} - 62170q^{81} - 2128q^{82} - 2516q^{83} - 896q^{84} - 11096q^{86} + 39608q^{87} - 5120q^{88} + 25460q^{89} + 20328q^{90} - 20162q^{91} - 20512q^{92} + 8394q^{93} - 31552q^{94} + 7470q^{95} + 1280q^{96} - 12272q^{97} + 14752q^{98} + 23376q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.732051 2.73205i −13.9920 + 8.07829i −6.92820 + 4.00000i −9.30753 + 34.7362i 32.3132 + 32.3132i −30.9246 53.5629i 16.0000 + 16.0000i 90.0175 155.915i 101.715
23.2 −0.732051 2.73205i −9.27942 + 5.35748i −6.92820 + 4.00000i 1.47068 5.48866i 21.4299 + 21.4299i 20.7504 + 35.9408i 16.0000 + 16.0000i 16.9051 29.2805i −16.0719
23.3 −0.732051 2.73205i −7.91112 + 4.56749i −6.92820 + 4.00000i 9.46674 35.3304i 18.2700 + 18.2700i −10.9589 18.9814i 16.0000 + 16.0000i 1.22390 2.11986i −103.454
23.4 −0.732051 2.73205i 0.515191 0.297445i −6.92820 + 4.00000i −2.76875 + 10.3331i −1.18978 1.18978i −5.68857 9.85290i 16.0000 + 16.0000i −40.3231 + 69.8416i 30.2575
23.5 −0.732051 2.73205i 4.65870 2.68970i −6.92820 + 4.00000i 6.50685 24.2839i −10.7588 10.7588i −14.6009 25.2895i 16.0000 + 16.0000i −26.0310 + 45.0870i −71.1082
23.6 −0.732051 2.73205i 6.81857 3.93670i −6.92820 + 4.00000i −7.04485 + 26.2917i −15.7468 15.7468i 41.4796 + 71.8448i 16.0000 + 16.0000i −9.50474 + 16.4627i 76.9875
23.7 −0.732051 2.73205i 14.8600 8.57940i −6.92820 + 4.00000i −4.77059 + 17.8041i −34.3176 34.3176i −40.6359 70.3834i 16.0000 + 16.0000i 106.712 184.831i 52.1339
29.1 −0.732051 + 2.73205i −13.9920 8.07829i −6.92820 4.00000i −9.30753 34.7362i 32.3132 32.3132i −30.9246 + 53.5629i 16.0000 16.0000i 90.0175 + 155.915i 101.715
29.2 −0.732051 + 2.73205i −9.27942 5.35748i −6.92820 4.00000i 1.47068 + 5.48866i 21.4299 21.4299i 20.7504 35.9408i 16.0000 16.0000i 16.9051 + 29.2805i −16.0719
29.3 −0.732051 + 2.73205i −7.91112 4.56749i −6.92820 4.00000i 9.46674 + 35.3304i 18.2700 18.2700i −10.9589 + 18.9814i 16.0000 16.0000i 1.22390 + 2.11986i −103.454
29.4 −0.732051 + 2.73205i 0.515191 + 0.297445i −6.92820 4.00000i −2.76875 10.3331i −1.18978 + 1.18978i −5.68857 + 9.85290i 16.0000 16.0000i −40.3231 69.8416i 30.2575
29.5 −0.732051 + 2.73205i 4.65870 + 2.68970i −6.92820 4.00000i 6.50685 + 24.2839i −10.7588 + 10.7588i −14.6009 + 25.2895i 16.0000 16.0000i −26.0310 45.0870i −71.1082
29.6 −0.732051 + 2.73205i 6.81857 + 3.93670i −6.92820 4.00000i −7.04485 26.2917i −15.7468 + 15.7468i 41.4796 71.8448i 16.0000 16.0000i −9.50474 16.4627i 76.9875
29.7 −0.732051 + 2.73205i 14.8600 + 8.57940i −6.92820 4.00000i −4.77059 17.8041i −34.3176 + 34.3176i −40.6359 + 70.3834i 16.0000 16.0000i 106.712 + 184.831i 52.1339
45.1 2.73205 + 0.732051i −15.3929 8.88709i 6.92820 + 4.00000i −1.55696 + 0.417185i −35.5484 35.5484i −4.44075 + 7.69161i 16.0000 + 16.0000i 117.461 + 203.448i −4.55909
45.2 2.73205 + 0.732051i −6.14320 3.54678i 6.92820 + 4.00000i 13.7399 3.68160i −14.1871 14.1871i 17.6478 30.5669i 16.0000 + 16.0000i −15.3407 26.5709i 40.2333
45.3 2.73205 + 0.732051i −3.71174 2.14297i 6.92820 + 4.00000i −38.9095 + 10.4258i −8.57189 8.57189i 32.4545 56.2129i 16.0000 + 16.0000i −31.3153 54.2398i −113.935
45.4 2.73205 + 0.732051i −0.984681 0.568506i 6.92820 + 4.00000i −27.3382 + 7.32524i −2.27402 2.27402i −37.5434 + 65.0271i 16.0000 + 16.0000i −39.8536 69.0285i −80.0517
45.5 2.73205 + 0.732051i 6.81652 + 3.93552i 6.92820 + 4.00000i 31.3386 8.39715i 15.7421 + 15.7421i −18.6320 + 32.2716i 16.0000 + 16.0000i −9.52334 16.4949i 91.7657
45.6 2.73205 + 0.732051i 9.05180 + 5.22606i 6.92820 + 4.00000i 8.32031 2.22942i 20.9042 + 20.9042i 41.9891 72.7273i 16.0000 + 16.0000i 14.1234 + 24.4624i 24.3635
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.g odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.5.g.b 28
37.g odd 12 1 inner 74.5.g.b 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.5.g.b 28 1.a even 1 1 trivial
74.5.g.b 28 37.g odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(54\!\cdots\!28\)\( T_{3}^{18} + 324041868432 T_{3}^{17} + \)\(64\!\cdots\!40\)\( T_{3}^{16} - \)\(10\!\cdots\!40\)\( T_{3}^{15} - \)\(53\!\cdots\!88\)\( T_{3}^{14} + \)\(14\!\cdots\!52\)\( T_{3}^{13} + \)\(33\!\cdots\!32\)\( T_{3}^{12} - \)\(13\!\cdots\!24\)\( T_{3}^{11} - \)\(14\!\cdots\!96\)\( T_{3}^{10} + \)\(55\!\cdots\!12\)\( T_{3}^{9} + \)\(42\!\cdots\!68\)\( T_{3}^{8} - \)\(12\!\cdots\!96\)\( T_{3}^{7} - \)\(73\!\cdots\!48\)\( T_{3}^{6} - \)\(86\!\cdots\!12\)\( T_{3}^{5} + \)\(83\!\cdots\!56\)\( T_{3}^{4} + \)\(72\!\cdots\!76\)\( T_{3}^{3} - \)\(35\!\cdots\!92\)\( T_{3}^{2} - \)\(45\!\cdots\!24\)\( T_{3} + \)\(34\!\cdots\!36\)\( \)">\(T_{3}^{28} - \cdots\) acting on \(S_{5}^{\mathrm{new}}(74, [\chi])\).