# Properties

 Label 354.2.g.a Level 354 Weight 2 Character orbit 354.g Analytic conductor 2.827 Analytic rank 0 Dimension 280 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$354 = 2 \cdot 3 \cdot 59$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 354.g (of order $$58$$, degree $$28$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.82670423155$$ Analytic rank: $$0$$ Dimension: $$280$$ Relative dimension: $$10$$ over $$\Q(\zeta_{58})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{58}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$280q - 10q^{2} + q^{3} - 10q^{4} + q^{6} + 2q^{7} - 10q^{8} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$280q - 10q^{2} + q^{3} - 10q^{4} + q^{6} + 2q^{7} - 10q^{8} - 3q^{9} + 4q^{11} + q^{12} - 27q^{14} + 7q^{15} - 10q^{16} - 3q^{18} + 6q^{19} + 3q^{21} + 4q^{22} - 8q^{23} + q^{24} - 4q^{25} - 77q^{27} + 2q^{28} + 7q^{30} - 10q^{32} - 3q^{36} + 6q^{38} - 4q^{39} + 3q^{42} + 4q^{44} - 18q^{45} - 8q^{46} + 232q^{47} + q^{48} - 8q^{49} - 4q^{50} - 118q^{51} - 58q^{53} - 77q^{54} + 2q^{56} - 316q^{57} + 20q^{59} - 80q^{60} + 77q^{63} - 10q^{64} + 16q^{65} - 116q^{66} - 102q^{69} - 3q^{72} - 25q^{75} + 6q^{76} + 106q^{77} - 4q^{78} - 6q^{79} - 7q^{81} - 104q^{83} + 3q^{84} + 4q^{85} + 15q^{87} + 4q^{88} - 16q^{89} + 11q^{90} - 8q^{92} + 122q^{93} + q^{96} - 8q^{98} + 2q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 0.976621 0.214970i −1.68205 + 0.413161i 0.907575 0.419889i −2.23908 0.621678i −1.55391 + 0.765093i 1.22747 1.16272i 0.796093 0.605174i 2.65860 1.38992i −2.32038 0.125807i
11.2 0.976621 0.214970i −1.66116 0.490442i 0.907575 0.419889i 3.54823 + 0.985161i −1.72776 0.121874i −3.12490 + 2.96006i 0.796093 0.605174i 2.51893 + 1.62941i 3.67705 + 0.199364i
11.3 0.976621 0.214970i −1.36107 + 1.07121i 0.907575 0.419889i 1.88995 + 0.524743i −1.09897 + 1.33876i 0.743378 0.704165i 0.796093 0.605174i 0.705020 2.91598i 1.95857 + 0.106191i
11.4 0.976621 0.214970i −0.326586 + 1.70098i 0.907575 0.419889i −2.85285 0.792090i 0.0467107 + 1.73142i −3.45977 + 3.27726i 0.796093 0.605174i −2.78668 1.11103i −2.95643 0.160293i
11.5 0.976621 0.214970i −0.207186 1.71961i 0.907575 0.419889i 1.05087 + 0.291772i −0.572009 1.63487i 1.62439 1.53870i 0.796093 0.605174i −2.91415 + 0.712562i 1.08902 + 0.0590451i
11.6 0.976621 0.214970i 0.308899 + 1.70428i 0.907575 0.419889i −0.897171 0.249098i 0.668047 + 1.59803i 3.72971 3.53297i 0.796093 0.605174i −2.80916 + 1.05290i −0.929745 0.0504093i
11.7 0.976621 0.214970i 0.566341 + 1.63684i 0.907575 0.419889i 3.87317 + 1.07538i 0.904973 + 1.47683i −0.391370 + 0.370725i 0.796093 0.605174i −2.35852 + 1.85402i 4.01379 + 0.217621i
11.8 0.976621 0.214970i 1.28026 1.16660i 0.907575 0.419889i −4.02621 1.11787i 0.999541 1.41454i 1.69268 1.60339i 0.796093 0.605174i 0.278110 2.98708i −4.17238 0.226220i
11.9 0.976621 0.214970i 1.45268 0.943248i 0.907575 0.419889i 1.11366 + 0.309205i 1.21595 1.23348i −1.48196 + 1.40379i 0.796093 0.605174i 1.22057 2.74048i 1.15409 + 0.0625729i
11.10 0.976621 0.214970i 1.57574 + 0.719053i 0.907575 0.419889i −0.135298 0.0375654i 1.69348 + 0.363504i −1.37447 + 1.30197i 0.796093 0.605174i 1.96593 + 2.26608i −0.140211 0.00760200i
23.1 −0.725995 + 0.687699i −1.73134 0.0495652i 0.0541389 0.998533i 0.271078 + 0.0444409i 1.29103 1.15466i −0.468623 + 0.883917i 0.647386 + 0.762162i 2.99509 + 0.171629i −0.227363 + 0.154156i
23.2 −0.725995 + 0.687699i −1.13722 + 1.30642i 0.0541389 0.998533i 1.48491 + 0.243438i −0.0728043 1.73052i −2.01420 + 3.79918i 0.647386 + 0.762162i −0.413452 2.97137i −1.24545 + 0.844434i
23.3 −0.725995 + 0.687699i −0.941981 1.45350i 0.0541389 0.998533i 3.62342 + 0.594030i 1.68345 + 0.407438i −0.837325 + 1.57936i 0.647386 + 0.762162i −1.22535 + 2.73834i −3.03910 + 2.06056i
23.4 −0.725995 + 0.687699i −0.713094 + 1.57845i 0.0541389 0.998533i −1.84131 0.301868i −0.567794 1.63634i 1.58049 2.98111i 0.647386 + 0.762162i −1.98299 2.25116i 1.54438 1.04712i
23.5 −0.725995 + 0.687699i −0.639916 1.60951i 0.0541389 0.998533i −1.86219 0.305290i 1.57143 + 0.728423i 1.84960 3.48871i 0.647386 + 0.762162i −2.18101 + 2.05990i 1.56189 1.05899i
23.6 −0.725995 + 0.687699i 0.422484 1.67973i 0.0541389 0.998533i −1.70784 0.279986i 0.848431 + 1.51002i −2.22532 + 4.19739i 0.647386 + 0.762162i −2.64301 1.41932i 1.43243 0.971213i
23.7 −0.725995 + 0.687699i 1.07024 + 1.36183i 0.0541389 0.998533i 2.51062 + 0.411595i −1.71352 0.252679i 0.276889 0.522268i 0.647386 + 0.762162i −0.709168 + 2.91498i −2.10575 + 1.42773i
23.8 −0.725995 + 0.687699i 1.27688 + 1.17029i 0.0541389 0.998533i −3.03432 0.497451i −1.73182 + 0.0284890i −0.0547647 + 0.103297i 0.647386 + 0.762162i 0.260860 + 2.98864i 2.54500 1.72555i
23.9 −0.725995 + 0.687699i 1.38419 1.04116i 0.0541389 0.998533i 3.46404 + 0.567901i −0.288910 + 1.70779i 1.50148 2.83210i 0.647386 + 0.762162i 0.831965 2.88233i −2.90542 + 1.96993i
23.10 −0.725995 + 0.687699i 1.57094 0.729476i 0.0541389 0.998533i −1.16466 0.190936i −0.638838 + 1.60993i 0.141142 0.266222i 0.647386 + 0.762162i 1.93573 2.29193i 0.976845 0.662318i
See next 80 embeddings (of 280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 347.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
177.f even 58 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 354.2.g.a 280
3.b odd 2 1 354.2.g.b yes 280
59.d odd 58 1 354.2.g.b yes 280
177.f even 58 1 inner 354.2.g.a 280

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.2.g.a 280 1.a even 1 1 trivial
354.2.g.a 280 177.f even 58 1 inner
354.2.g.b yes 280 3.b odd 2 1
354.2.g.b yes 280 59.d odd 58 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{280} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(354, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database