# Properties

 Label 354.2.e.b Level 354 Weight 2 Character orbit 354.e Analytic conductor 2.827 Analytic rank 0 Dimension 56 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 354.e (of order $$29$$, degree $$28$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.82670423155$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$2$$ over $$\Q(\zeta_{29})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} - 2q^{6} + q^{7} + 2q^{8} - 2q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} - 2q^{6} + q^{7} + 2q^{8} - 2q^{9} + 2q^{10} + 30q^{11} + 2q^{12} + 5q^{13} + 28q^{14} + 2q^{15} - 2q^{16} - 3q^{17} + 2q^{18} - 4q^{19} - 2q^{20} + 28q^{21} - q^{22} - 2q^{23} - 2q^{24} + 64q^{25} - 34q^{26} + 2q^{27} + q^{28} + 8q^{29} - 2q^{30} + 2q^{32} - q^{33} + 32q^{34} - 87q^{35} - 2q^{36} + 47q^{37} + 4q^{38} - 5q^{39} + 2q^{40} + 3q^{41} + q^{42} - 61q^{43} + q^{44} - 2q^{45} - 27q^{46} + 50q^{47} + 2q^{48} - 45q^{49} - 6q^{50} + 3q^{51} - 24q^{52} - 27q^{53} - 2q^{54} - 95q^{55} - q^{56} + 4q^{57} + 50q^{58} - 58q^{59} + 2q^{60} - 74q^{61} - 29q^{62} + q^{63} - 2q^{64} - 17q^{65} + q^{66} + 4q^{67} - 32q^{68} + 31q^{69} - 58q^{70} - 47q^{71} + 2q^{72} + 43q^{73} - 18q^{74} - 6q^{75} - 33q^{76} - 5q^{77} + 5q^{78} + 19q^{79} - 2q^{80} - 2q^{81} - 3q^{82} + 47q^{83} - q^{84} - 149q^{85} - 55q^{86} + 21q^{87} + 28q^{88} + 36q^{89} + 2q^{90} + 175q^{91} - 2q^{92} - 29q^{93} - 21q^{94} + 50q^{95} - 2q^{96} + 10q^{97} - 13q^{98} + q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 0.561187 + 0.827689i −0.0541389 + 0.998533i −0.370138 + 0.928977i −2.31715 + 1.07203i −0.856857 + 0.515554i 0.0770120 0.277372i −0.976621 + 0.214970i −0.994138 0.108119i −2.18766 1.31627i
7.2 0.561187 + 0.827689i −0.0541389 + 0.998533i −0.370138 + 0.928977i 2.80275 1.29669i −0.856857 + 0.515554i −0.846851 + 3.05008i −0.976621 + 0.214970i −0.994138 0.108119i 2.64613 + 1.59212i
19.1 −0.468408 + 0.883512i 0.725995 0.687699i −0.561187 0.827689i −2.36756 + 0.521140i 0.267528 + 0.963550i 2.60217 1.97812i 0.994138 0.108119i 0.0541389 0.998533i 0.648553 2.33588i
19.2 −0.468408 + 0.883512i 0.725995 0.687699i −0.561187 0.827689i 3.92252 0.863413i 0.267528 + 0.963550i −0.604650 + 0.459643i 0.994138 0.108119i 0.0541389 0.998533i −1.07451 + 3.87003i
25.1 −0.796093 0.605174i −0.468408 0.883512i 0.267528 + 0.963550i −1.39125 1.31786i −0.161782 + 0.986827i 1.52711 1.79785i 0.370138 0.928977i −0.561187 + 0.827689i 0.310029 + 1.89109i
25.2 −0.796093 0.605174i −0.468408 0.883512i 0.267528 + 0.963550i 0.451252 + 0.427449i −0.161782 + 0.986827i −1.39072 + 1.63728i 0.370138 0.928977i −0.561187 + 0.827689i −0.100558 0.613375i
49.1 0.370138 0.928977i 0.994138 + 0.108119i −0.725995 0.687699i −2.09789 + 2.46983i 0.468408 0.883512i 3.22015 + 1.93750i −0.907575 + 0.419889i 0.976621 + 0.214970i 1.51790 + 2.86307i
49.2 0.370138 0.928977i 0.994138 + 0.108119i −0.725995 0.687699i 0.988453 1.16370i 0.468408 0.883512i −2.12914 1.28106i −0.907575 + 0.419889i 0.976621 + 0.214970i −0.715183 1.34898i
79.1 −0.907575 0.419889i 0.947653 0.319302i 0.647386 + 0.762162i −1.91571 + 1.15265i −0.994138 0.108119i 0.0631741 + 1.16518i −0.267528 0.963550i 0.796093 0.605174i 2.22264 0.241727i
79.2 −0.907575 0.419889i 0.947653 0.319302i 0.647386 + 0.762162i 1.82294 1.09682i −0.994138 0.108119i −0.192354 3.54777i −0.267528 0.963550i 0.796093 0.605174i −2.11500 + 0.230020i
85.1 −0.796093 + 0.605174i −0.468408 + 0.883512i 0.267528 0.963550i −1.39125 + 1.31786i −0.161782 0.986827i 1.52711 + 1.79785i 0.370138 + 0.928977i −0.561187 0.827689i 0.310029 1.89109i
85.2 −0.796093 + 0.605174i −0.468408 + 0.883512i 0.267528 0.963550i 0.451252 0.427449i −0.161782 0.986827i −1.39072 1.63728i 0.370138 + 0.928977i −0.561187 0.827689i −0.100558 + 0.613375i
121.1 −0.907575 + 0.419889i 0.947653 + 0.319302i 0.647386 0.762162i −1.91571 1.15265i −0.994138 + 0.108119i 0.0631741 1.16518i −0.267528 + 0.963550i 0.796093 + 0.605174i 2.22264 + 0.241727i
121.2 −0.907575 + 0.419889i 0.947653 + 0.319302i 0.647386 0.762162i 1.82294 + 1.09682i −0.994138 + 0.108119i −0.192354 + 3.54777i −0.267528 + 0.963550i 0.796093 + 0.605174i −2.11500 0.230020i
127.1 −0.647386 + 0.762162i −0.796093 0.605174i −0.161782 0.986827i −1.07211 2.02222i 0.976621 0.214970i −1.17973 0.128303i 0.856857 + 0.515554i 0.267528 + 0.963550i 2.23533 + 0.492034i
127.2 −0.647386 + 0.762162i −0.796093 0.605174i −0.161782 0.986827i 0.140789 + 0.265557i 0.976621 0.214970i −0.694075 0.0754852i 0.856857 + 0.515554i 0.267528 + 0.963550i −0.293542 0.0646136i
133.1 0.994138 0.108119i −0.647386 0.762162i 0.976621 0.214970i −2.85024 + 2.16670i −0.725995 0.687699i 1.29781 + 3.25725i 0.947653 0.319302i −0.161782 + 0.986827i −2.59927 + 2.46216i
133.2 0.994138 0.108119i −0.647386 0.762162i 0.976621 0.214970i 2.26091 1.71870i −0.725995 0.687699i 0.732533 + 1.83852i 0.947653 0.319302i −0.161782 + 0.986827i 2.06183 1.95307i
139.1 0.856857 + 0.515554i 0.370138 0.928977i 0.468408 + 0.883512i −1.41393 0.153774i 0.796093 0.605174i 2.98026 + 1.00417i −0.0541389 + 0.998533i −0.725995 0.687699i −1.13226 0.860721i
139.2 0.856857 + 0.515554i 0.370138 0.928977i 0.468408 + 0.883512i 3.29813 + 0.358693i 0.796093 0.605174i −0.183400 0.0617948i −0.0541389 + 0.998533i −0.725995 0.687699i 2.64110 + 2.00771i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 343.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.2.e.b 56
59.c even 29 1 inner 354.2.e.b 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.2.e.b 56 1.a even 1 1 trivial
354.2.e.b 56 59.c even 29 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{56} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(354, [\chi])$$.