# Properties

 Label 354.2.e.d Level 354 Weight 2 Character orbit 354.e Analytic conductor 2.827 Analytic rank 0 Dimension 84 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: $$84$$ Relative dimension: $$3$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$84q + 3q^{2} - 3q^{3} - 3q^{4} - 2q^{5} + 3q^{6} - 7q^{7} + 3q^{8} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$84q + 3q^{2} - 3q^{3} - 3q^{4} - 2q^{5} + 3q^{6} - 7q^{7} + 3q^{8} - 3q^{9} + 2q^{10} + 30q^{11} - 3q^{12} - 3q^{13} + 7q^{14} - 2q^{15} - 3q^{16} + 3q^{17} + 3q^{18} - 4q^{19} - 2q^{20} - 7q^{21} - q^{22} + 2q^{23} + 3q^{24} - 67q^{25} + 32q^{26} - 3q^{27} - 7q^{28} + 4q^{29} + 2q^{30} - 6q^{31} + 3q^{32} + q^{33} + 26q^{34} + 79q^{35} - 3q^{36} + 55q^{37} + 4q^{38} - 3q^{39} + 2q^{40} + q^{41} + 7q^{42} + 51q^{43} + q^{44} - 2q^{45} - 31q^{46} - 62q^{47} - 3q^{48} - 70q^{49} + 9q^{50} + 3q^{51} - 32q^{52} - 27q^{53} + 3q^{54} - 83q^{55} + 7q^{56} - 4q^{57} - 120q^{58} - 55q^{59} + 56q^{60} - 46q^{61} - 23q^{62} - 7q^{63} - 3q^{64} - 121q^{65} - q^{66} + 8q^{67} - 26q^{68} - 27q^{69} - 50q^{70} - 61q^{71} + 3q^{72} + 49q^{73} - 26q^{74} - 9q^{75} + 25q^{76} + 77q^{77} + 3q^{78} - 5q^{79} - 2q^{80} - 3q^{81} - q^{82} + 75q^{83} - 7q^{84} + 189q^{85} + 65q^{86} - 25q^{87} - 30q^{88} + 54q^{89} + 2q^{90} - 161q^{91} + 2q^{92} + 23q^{93} + 33q^{94} - 54q^{95} + 3q^{96} + 28q^{97} + 12q^{98} + q^{99} + O(q^{100})$$

## 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 −1.35846 + 0.628489i 0.856857 0.515554i −1.22687 + 4.41877i −0.976621 + 0.214970i −0.994138 0.108119i −1.28254 0.771679i
7.2 0.561187 + 0.827689i 0.0541389 0.998533i −0.370138 + 0.928977i −0.147621 + 0.0682969i 0.856857 0.515554i 1.13460 4.08646i −0.976621 + 0.214970i −0.994138 0.108119i −0.139372 0.0838572i
7.3 0.561187 + 0.827689i 0.0541389 0.998533i −0.370138 + 0.928977i 3.32123 1.53656i 0.856857 0.515554i −0.148313 + 0.534175i −0.976621 + 0.214970i −0.994138 0.108119i 3.13563 + 1.88664i
19.1 −0.468408 + 0.883512i −0.725995 + 0.687699i −0.561187 0.827689i −0.363688 + 0.0800537i −0.267528 0.963550i 2.14232 1.62855i 0.994138 0.108119i 0.0541389 0.998533i 0.0996260 0.358820i
19.2 −0.468408 + 0.883512i −0.725995 + 0.687699i −0.561187 0.827689i −0.307038 + 0.0675842i −0.267528 0.963550i −2.91202 + 2.21366i 0.994138 0.108119i 0.0541389 0.998533i 0.0841078 0.302929i
19.3 −0.468408 + 0.883512i −0.725995 + 0.687699i −0.561187 0.827689i 2.62397 0.577579i −0.267528 0.963550i 1.37326 1.04393i 0.994138 0.108119i 0.0541389 0.998533i −0.718790 + 2.58885i
25.1 −0.796093 0.605174i 0.468408 + 0.883512i 0.267528 + 0.963550i −3.02590 2.86629i 0.161782 0.986827i −1.62278 + 1.91048i 0.370138 0.928977i −0.561187 + 0.827689i 0.674297 + 4.11303i
25.2 −0.796093 0.605174i 0.468408 + 0.883512i 0.267528 + 0.963550i −0.526198 0.498441i 0.161782 0.986827i 2.82058 3.32064i 0.370138 0.928977i −0.561187 + 0.827689i 0.117259 + 0.715246i
25.3 −0.796093 0.605174i 0.468408 + 0.883512i 0.267528 + 0.963550i 2.10011 + 1.98933i 0.161782 0.986827i −0.956124 + 1.12564i 0.370138 0.928977i −0.561187 + 0.827689i −0.467991 2.85462i
49.1 0.370138 0.928977i −0.994138 0.108119i −0.725995 0.687699i −1.43265 + 1.68664i −0.468408 + 0.883512i −0.808791 0.486633i −0.907575 + 0.419889i 0.976621 + 0.214970i 1.03657 + 1.95519i
49.2 0.370138 0.928977i −0.994138 0.108119i −0.725995 0.687699i 0.645875 0.760383i −0.468408 + 0.883512i 4.24666 + 2.55513i −0.907575 + 0.419889i 0.976621 + 0.214970i −0.467315 0.881449i
49.3 0.370138 0.928977i −0.994138 0.108119i −0.725995 0.687699i 2.08155 2.45058i −0.468408 + 0.883512i −3.50142 2.10673i −0.907575 + 0.419889i 0.976621 + 0.214970i −1.50608 2.84076i
79.1 −0.907575 0.419889i −0.947653 + 0.319302i 0.647386 + 0.762162i −3.13475 + 1.88612i 0.994138 + 0.108119i 0.104895 + 1.93468i −0.267528 0.963550i 0.796093 0.605174i 3.63698 0.395546i
79.2 −0.907575 0.419889i −0.947653 + 0.319302i 0.647386 + 0.762162i −0.971226 + 0.584367i 0.994138 + 0.108119i −0.0967973 1.78532i −0.267528 0.963550i 0.796093 0.605174i 1.12683 0.122550i
79.3 −0.907575 0.419889i −0.947653 + 0.319302i 0.647386 + 0.762162i 2.39226 1.43938i 0.994138 + 0.108119i 0.142415 + 2.62669i −0.267528 0.963550i 0.796093 0.605174i −2.77554 + 0.301858i
85.1 −0.796093 + 0.605174i 0.468408 0.883512i 0.267528 0.963550i −3.02590 + 2.86629i 0.161782 + 0.986827i −1.62278 1.91048i 0.370138 + 0.928977i −0.561187 0.827689i 0.674297 4.11303i
85.2 −0.796093 + 0.605174i 0.468408 0.883512i 0.267528 0.963550i −0.526198 + 0.498441i 0.161782 + 0.986827i 2.82058 + 3.32064i 0.370138 + 0.928977i −0.561187 0.827689i 0.117259 0.715246i
85.3 −0.796093 + 0.605174i 0.468408 0.883512i 0.267528 0.963550i 2.10011 1.98933i 0.161782 + 0.986827i −0.956124 1.12564i 0.370138 + 0.928977i −0.561187 0.827689i −0.467991 + 2.85462i
121.1 −0.907575 + 0.419889i −0.947653 0.319302i 0.647386 0.762162i −3.13475 1.88612i 0.994138 0.108119i 0.104895 1.93468i −0.267528 + 0.963550i 0.796093 + 0.605174i 3.63698 + 0.395546i
121.2 −0.907575 + 0.419889i −0.947653 0.319302i 0.647386 0.762162i −0.971226 0.584367i 0.994138 0.108119i −0.0967973 + 1.78532i −0.267528 + 0.963550i 0.796093 + 0.605174i 1.12683 + 0.122550i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 343.3 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.d 84
59.c even 29 1 inner 354.2.e.d 84

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.2.e.d 84 1.a even 1 1 trivial
354.2.e.d 84 59.c even 29 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{84} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(354, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database