# Properties

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

# Related objects

## 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 28 q^{2} + 42 q^{5} - 40 q^{6} + 48 q^{7} - 448 q^{8} + 556 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28 q - 28 q^{2} + 42 q^{5} - 40 q^{6} + 48 q^{7} - 448 q^{8} + 556 q^{9} - 72 q^{10} + 80 q^{12} + 248 q^{13} - 192 q^{14} - 852 q^{15} + 896 q^{16} + 1092 q^{17} + 1112 q^{18} - 320 q^{19} + 336 q^{20} - 858 q^{21} - 112 q^{22} + 2348 q^{23} + 160 q^{24} - 162 q^{25} - 184 q^{26} + 1584 q^{28} - 1308 q^{29} + 2652 q^{30} - 3444 q^{31} + 1792 q^{32} + 1052 q^{33} + 1428 q^{34} - 580 q^{35} - 5694 q^{37} - 824 q^{38} - 6514 q^{39} - 1056 q^{40} - 12186 q^{41} - 1276 q^{42} - 6220 q^{43} + 1280 q^{44} + 12804 q^{45} - 4696 q^{46} + 15928 q^{47} - 4892 q^{49} - 10792 q^{50} - 6218 q^{51} + 1984 q^{52} - 9662 q^{53} - 9836 q^{54} + 1744 q^{55} - 3936 q^{56} + 16046 q^{57} + 13800 q^{58} - 9356 q^{59} - 3024 q^{60} - 6682 q^{61} + 16128 q^{62} + 60268 q^{63} - 28086 q^{65} - 4208 q^{66} + 28086 q^{67} - 5712 q^{68} + 19878 q^{69} - 6028 q^{70} + 680 q^{71} - 8896 q^{72} - 2420 q^{74} + 11076 q^{75} - 2560 q^{76} - 32208 q^{77} + 23436 q^{78} - 15136 q^{79} + 1152 q^{80} - 21274 q^{81} - 2608 q^{82} - 19732 q^{83} + 23936 q^{84} + 12440 q^{86} - 31624 q^{87} - 5120 q^{88} + 45940 q^{89} - 25608 q^{90} - 13106 q^{91} - 6944 q^{92} - 37734 q^{93} - 16960 q^{94} - 37458 q^{95} - 1280 q^{96} + 31792 q^{97} + 5216 q^{98} + 141456 q^{99} + O(q^{100})$$

## 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 −13.4807 + 7.78308i −6.92820 + 4.00000i −9.88545 + 36.8930i −31.1323 31.1323i 23.7073 + 41.0623i −16.0000 16.0000i 80.6527 139.695i −108.030
23.2 0.732051 + 2.73205i −12.4198 + 7.17059i −6.92820 + 4.00000i 11.3964 42.5318i −28.6824 28.6824i 6.42196 + 11.1232i −16.0000 16.0000i 62.3348 107.967i 124.542
23.3 0.732051 + 2.73205i −5.46714 + 3.15645i −6.92820 + 4.00000i −2.29270 + 8.55648i −12.6258 12.6258i −24.9782 43.2636i −16.0000 16.0000i −20.5736 + 35.6345i −25.0551
23.4 0.732051 + 2.73205i 1.03965 0.600242i −6.92820 + 4.00000i 1.65417 6.17343i 2.40097 + 2.40097i 28.4154 + 49.2169i −16.0000 16.0000i −39.7794 + 68.9000i 18.0771
23.5 0.732051 + 2.73205i 4.95016 2.85797i −6.92820 + 4.00000i 7.81136 29.1524i 11.4319 + 11.4319i −41.9747 72.7023i −16.0000 16.0000i −24.1640 + 41.8532i 85.3641
23.6 0.732051 + 2.73205i 7.29967 4.21447i −6.92820 + 4.00000i −12.4312 + 46.3938i 16.8579 + 16.8579i −18.9531 32.8277i −16.0000 16.0000i −4.97651 + 8.61956i −135.850
23.7 0.732051 + 2.73205i 13.7481 7.93744i −6.92820 + 4.00000i 5.58720 20.8517i 31.7498 + 31.7498i 10.7825 + 18.6759i −16.0000 16.0000i 85.5060 148.101i 61.0580
29.1 0.732051 2.73205i −13.4807 7.78308i −6.92820 4.00000i −9.88545 36.8930i −31.1323 + 31.1323i 23.7073 41.0623i −16.0000 + 16.0000i 80.6527 + 139.695i −108.030
29.2 0.732051 2.73205i −12.4198 7.17059i −6.92820 4.00000i 11.3964 + 42.5318i −28.6824 + 28.6824i 6.42196 11.1232i −16.0000 + 16.0000i 62.3348 + 107.967i 124.542
29.3 0.732051 2.73205i −5.46714 3.15645i −6.92820 4.00000i −2.29270 8.55648i −12.6258 + 12.6258i −24.9782 + 43.2636i −16.0000 + 16.0000i −20.5736 35.6345i −25.0551
29.4 0.732051 2.73205i 1.03965 + 0.600242i −6.92820 4.00000i 1.65417 + 6.17343i 2.40097 2.40097i 28.4154 49.2169i −16.0000 + 16.0000i −39.7794 68.9000i 18.0771
29.5 0.732051 2.73205i 4.95016 + 2.85797i −6.92820 4.00000i 7.81136 + 29.1524i 11.4319 11.4319i −41.9747 + 72.7023i −16.0000 + 16.0000i −24.1640 41.8532i 85.3641
29.6 0.732051 2.73205i 7.29967 + 4.21447i −6.92820 4.00000i −12.4312 46.3938i 16.8579 16.8579i −18.9531 + 32.8277i −16.0000 + 16.0000i −4.97651 8.61956i −135.850
29.7 0.732051 2.73205i 13.7481 + 7.93744i −6.92820 4.00000i 5.58720 + 20.8517i 31.7498 31.7498i 10.7825 18.6759i −16.0000 + 16.0000i 85.5060 + 148.101i 61.0580
45.1 −2.73205 0.732051i −14.4302 8.33128i 6.92820 + 4.00000i 44.0328 11.7986i 33.3251 + 33.3251i 20.5087 35.5220i −16.0000 16.0000i 98.3204 + 170.296i −128.937
45.2 −2.73205 0.732051i −9.68326 5.59063i 6.92820 + 4.00000i −24.8057 + 6.64666i 22.3625 + 22.3625i −19.9156 + 34.4949i −16.0000 16.0000i 22.0103 + 38.1230i 72.6360
45.3 −2.73205 0.732051i −4.11612 2.37644i 6.92820 + 4.00000i 0.920018 0.246518i 9.50578 + 9.50578i 14.7301 25.5133i −16.0000 16.0000i −29.2050 50.5846i −2.69400
45.4 −2.73205 0.732051i 4.17866 + 2.41255i 6.92820 + 4.00000i 35.6476 9.55175i −9.65021 9.65021i −13.4517 + 23.2990i −16.0000 16.0000i −28.8592 49.9856i −104.383
45.5 −2.73205 0.732051i 6.38646 + 3.68723i 6.92820 + 4.00000i −43.6075 + 11.6846i −14.7489 14.7489i 39.7034 68.7683i −16.0000 16.0000i −13.3087 23.0514i 127.692
45.6 −2.73205 0.732051i 8.28780 + 4.78497i 6.92820 + 4.00000i −17.4826 + 4.68445i −19.1399 19.1399i −36.2925 + 62.8605i −16.0000 16.0000i 5.29180 + 9.16567i 51.1927
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 51.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.a 28
37.g odd 12 1 inner 74.5.g.a 28

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$59\!\cdots\!84$$$$T_{3}^{18} +$$$$21\!\cdots\!88$$$$T_{3}^{17} +$$$$75\!\cdots\!16$$$$T_{3}^{16} -$$$$40\!\cdots\!16$$$$T_{3}^{15} -$$$$63\!\cdots\!24$$$$T_{3}^{14} +$$$$46\!\cdots\!76$$$$T_{3}^{13} +$$$$35\!\cdots\!24$$$$T_{3}^{12} -$$$$38\!\cdots\!32$$$$T_{3}^{11} -$$$$75\!\cdots\!64$$$$T_{3}^{10} +$$$$16\!\cdots\!88$$$$T_{3}^{9} -$$$$19\!\cdots\!72$$$$T_{3}^{8} -$$$$52\!\cdots\!52$$$$T_{3}^{7} +$$$$12\!\cdots\!56$$$$T_{3}^{6} +$$$$72\!\cdots\!52$$$$T_{3}^{5} -$$$$27\!\cdots\!00$$$$T_{3}^{4} -$$$$65\!\cdots\!36$$$$T_{3}^{3} +$$$$45\!\cdots\!68$$$$T_{3}^{2} -$$$$69\!\cdots\!56$$$$T_{3} +$$$$38\!\cdots\!76$$">$$T_{3}^{28} - \cdots$$ acting on $$S_{5}^{\mathrm{new}}(74, [\chi])$$.