# Properties

 Label 29.16.b.a Level $29$ Weight $16$ Character orbit 29.b Analytic conductor $41.381$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 29.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.3811164790$$ Analytic rank: $$0$$ Dimension: $$36$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 508108q^{4} + 470082q^{5} + 1112016q^{6} - 4820620q^{7} - 167460710q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 508108q^{4} + 470082q^{5} + 1112016q^{6} - 4820620q^{7} - 167460710q^{9} + 133305618q^{13} + 5626041364q^{16} - 30737731548q^{20} - 51638088984q^{22} - 23459433564q^{23} - 13473060100q^{24} + 169887741474q^{25} + 281303298768q^{28} - 85550328684q^{29} - 681215606256q^{30} + 831111242422q^{33} - 449988200584q^{34} + 726838987044q^{35} + 1809260484664q^{36} - 2518300733088q^{38} - 5363921425320q^{42} - 16561773855556q^{45} + 29824615981340q^{49} + 1184881612900q^{51} + 21527128606228q^{52} - 40200435711486q^{53} + 9043904345168q^{54} + 42099004809572q^{57} - 3461494533632q^{58} - 50458797940572q^{59} - 298531808710416q^{62} + 159779590145904q^{63} - 71569159267548q^{64} + 92095395748902q^{65} + 130146715692752q^{67} - 178710878083152q^{71} - 205323946615296q^{74} + 13818320315976q^{78} + 857820862108188q^{80} + 126746036597568q^{81} + 249211917251112q^{82} - 541736282848188q^{83} + 630538772195064q^{86} - 633552108095260q^{87} + 969723837884556q^{88} - 962583563732444q^{91} + 2248076163071664q^{92} - 212103831463414q^{93} - 5356847193577464q^{94} + 4021088569901964q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
28.1 350.198i 3847.09i −89870.4 106507. 1.34724e6 −2.76298e6 1.99971e7i −451190. 3.72984e7i
28.2 327.159i 2163.79i −74264.8 235169. −707903. 2.51260e6 1.35760e7i 9.66691e6 7.69374e7i
28.3 317.689i 3423.11i −68158.0 −176108. −1.08748e6 98456.3 1.12430e7i 2.63125e6 5.59476e7i
28.4 297.362i 5034.99i −55656.3 −122226. 1.49722e6 2.97082e6 6.80612e6i −1.10022e7 3.63455e7i
28.5 285.356i 6609.41i −48660.2 173971. −1.88604e6 −3.97639e6 4.53494e6i −2.93355e7 4.96437e7i
28.6 255.661i 4338.48i −32594.8 −284737. 1.10918e6 −3.50246e6 44288.6i −4.47354e6 7.27962e7i
28.7 240.042i 2267.47i −24852.3 195163. 544290. 168543. 1.90009e6i 9.20747e6 4.68475e7i
28.8 231.586i 885.668i −20863.9 −97507.9 −205108. 415555. 2.75682e6i 1.35645e7 2.25814e7i
28.9 202.124i 5797.90i −8086.11 172967. −1.17189e6 3.88134e6 4.98880e6i −1.92667e7 3.49608e7i
28.10 195.247i 7180.80i −5353.24 274602. 1.40203e6 −964261. 5.35264e6i −3.72150e7 5.36151e7i
28.11 169.905i 817.971i 3900.43 58678.2 −138977. −2.72949e6 6.23013e6i 1.36798e7 9.96969e6i
28.12 150.477i 6169.52i 10124.7 −189529. −928371. 634911. 6.45436e6i −2.37141e7 2.85197e7i
28.13 118.325i 6445.47i 18767.1 −115158. 762661. 493995. 6.09791e6i −2.71951e7 1.36261e7i
28.14 101.334i 71.6384i 22499.4 −295367. 7259.42 1.06773e6 5.60048e6i 1.43438e7 2.99307e7i
28.15 84.1317i 2443.60i 25689.9 22406.3 205584. 3.47198e6 4.91816e6i 8.37775e6 1.88508e6i
28.16 68.3669i 2973.41i 28094.0 302546. −203283. −153099. 4.16094e6i 5.50774e6 2.06841e7i
28.17 14.2479i 3944.08i 32565.0 −125301. 56194.9 −3.40529e6 930860.i −1.20682e6 1.78527e6i
28.18 10.1196i 4604.16i 32665.6 98965.0 −46592.0 −632277. 662159.i −6.84936e6 1.00148e6i
28.19 10.1196i 4604.16i 32665.6 98965.0 −46592.0 −632277. 662159.i −6.84936e6 1.00148e6i
28.20 14.2479i 3944.08i 32565.0 −125301. 56194.9 −3.40529e6 930860.i −1.20682e6 1.78527e6i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 28.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.16.b.a 36
29.b even 2 1 inner 29.16.b.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.16.b.a 36 1.a even 1 1 trivial
29.16.b.a 36 29.b even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{16}^{\mathrm{new}}(29, [\chi])$$.