# Properties

 Label 354.2.g.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$280q + 10q^{2} + q^{3} - 10q^{4} - q^{6} + 2q^{7} + 10q^{8} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$280q + 10q^{2} + q^{3} - 10q^{4} - q^{6} + 2q^{7} + 10q^{8} - 3q^{9} - 4q^{11} + q^{12} + 27q^{14} + 7q^{15} - 10q^{16} + 3q^{18} + 6q^{19} + 3q^{21} + 4q^{22} + 8q^{23} - q^{24} - 4q^{25} + 97q^{27} + 2q^{28} - 7q^{30} + 10q^{32} - 3q^{36} - 6q^{38} + 4q^{39} - 3q^{42} - 4q^{44} - 18q^{45} - 8q^{46} - 232q^{47} + q^{48} - 8q^{49} + 4q^{50} - 118q^{51} + 58q^{53} - 126q^{54} - 2q^{56} + 32q^{57} - 20q^{59} - 80q^{60} - 329q^{63} - 10q^{64} - 16q^{65} - 87q^{66} - 130q^{69} + 3q^{72} - 25q^{75} + 6q^{76} - 106q^{77} - 4q^{78} - 6q^{79} - 7q^{81} + 104q^{83} + 3q^{84} + 4q^{85} + 15q^{87} + 4q^{88} + 16q^{89} - 11q^{90} + 8q^{92} - 122q^{93} - q^{96} + 8q^{98} - 2q^{99} + O(q^{100})$$

## 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 −1.72285 + 0.178276i 0.907575 0.419889i 0.135298 + 0.0375654i 1.64425 0.544470i −1.37447 + 1.30197i −0.796093 + 0.605174i 2.93644 0.614285i −0.140211 0.00760200i
11.2 −0.976621 + 0.214970i −1.07546 1.35772i 0.907575 0.419889i −1.11366 0.309205i 1.34218 + 1.09478i −1.48196 + 1.40379i −0.796093 + 0.605174i −0.686781 + 2.92033i 1.15409 + 0.0625729i
11.3 −0.976621 + 0.214970i −1.05934 + 1.37033i 0.907575 0.419889i −3.87317 1.07538i 0.739995 1.56602i −0.391370 + 0.370725i −0.796093 + 0.605174i −0.755592 2.90329i 4.01379 + 0.217621i
11.4 −0.976621 + 0.214970i −0.840743 1.51432i 0.907575 0.419889i 4.02621 + 1.11787i 1.14662 + 1.29818i 1.69268 1.60339i −0.796093 + 0.605174i −1.58630 + 2.54630i −4.17238 0.226220i
11.5 −0.976621 + 0.214970i −0.836909 + 1.51644i 0.907575 0.419889i 0.897171 + 0.249098i 0.491353 1.66089i 3.72971 3.53297i −0.796093 + 0.605174i −1.59917 2.53824i −0.929745 0.0504093i
11.6 −0.976621 + 0.214970i −0.233636 + 1.71622i 0.907575 0.419889i 2.85285 + 0.792090i −0.140763 1.72632i −3.45977 + 3.27726i −0.796093 + 0.605174i −2.89083 0.801943i −2.95643 0.160293i
11.7 −0.976621 + 0.214970i 0.745416 1.56344i 0.907575 0.419889i −1.05087 0.291772i −0.391895 + 1.68713i 1.62439 1.53870i −0.796093 + 0.605174i −1.88871 2.33083i 1.08902 + 0.0590451i
11.8 −0.976621 + 0.214970i 0.947783 + 1.44973i 0.907575 0.419889i −1.88995 0.524743i −1.23727 1.21209i 0.743378 0.704165i −0.796093 + 0.605174i −1.20342 + 2.74805i 1.95857 + 0.106191i
11.9 −0.976621 + 0.214970i 1.46208 + 0.928615i 0.907575 0.419889i 2.23908 + 0.621678i −1.62752 0.592600i 1.22747 1.16272i −0.796093 + 0.605174i 1.27535 + 2.71542i −2.32038 0.125807i
11.10 −0.976621 + 0.214970i 1.73081 + 0.0656435i 0.907575 0.419889i −3.54823 0.985161i −1.70445 + 0.307963i −3.12490 + 2.96006i −0.796093 + 0.605174i 2.99138 + 0.227232i 3.67705 + 0.199364i
23.1 0.725995 0.687699i −1.58067 0.708165i 0.0541389 0.998533i −1.48491 0.243438i −1.63456 + 0.572898i −2.01420 + 3.79918i −0.647386 0.762162i 1.99700 + 2.23874i −1.24545 + 0.844434i
23.2 0.725995 0.687699i −1.55051 + 0.771956i 0.0541389 0.998533i −0.271078 0.0444409i −0.594791 + 1.62672i −0.468623 + 0.883917i −0.647386 0.762162i 1.80817 2.39385i −0.227363 + 0.154156i
23.3 0.725995 0.687699i −1.30996 1.13314i 0.0541389 0.998533i 1.84131 + 0.301868i −1.73028 + 0.0782045i 1.58049 2.98111i −0.647386 0.762162i 0.431990 + 2.96873i 1.54438 1.04712i
23.4 0.725995 0.687699i −0.244608 + 1.71469i 0.0541389 0.998533i −3.62342 0.594030i 1.00161 + 1.41308i −0.837325 + 1.57936i −0.647386 0.762162i −2.88033 0.838855i −3.03910 + 2.06056i
23.5 0.725995 0.687699i 0.0950413 + 1.72944i 0.0541389 0.998533i 1.86219 + 0.305290i 1.25834 + 1.19021i 1.84960 3.48871i −0.647386 0.762162i −2.98193 + 0.328737i 1.56189 1.05899i
23.6 0.725995 0.687699i 0.399507 1.68535i 0.0541389 0.998533i −2.51062 0.411595i −0.868972 1.49829i 0.276889 0.522268i −0.647386 0.762162i −2.68079 1.34661i −2.10575 + 1.42773i
23.7 0.725995 0.687699i 0.667477 1.59827i 0.0541389 0.998533i 3.03432 + 0.497451i −0.614546 1.61936i −0.0547647 + 0.103297i −0.647386 0.762162i −2.10895 2.13362i 2.54500 1.72555i
23.8 0.725995 0.687699i 1.08874 + 1.34709i 0.0541389 0.998533i 1.70784 + 0.279986i 1.71681 + 0.229256i −2.22532 + 4.19739i −0.647386 0.762162i −0.629299 + 2.93325i 1.43243 0.971213i
23.9 0.725995 0.687699i 1.69343 + 0.363726i 0.0541389 0.998533i −3.46404 0.567901i 1.47956 0.900507i 1.50148 2.83210i −0.647386 0.762162i 2.73541 + 1.23189i −2.90542 + 1.96993i
23.10 0.725995 0.687699i 1.73205 + 0.00243249i 0.0541389 0.998533i 1.16466 + 0.190936i 1.25913 1.18936i 0.141142 0.266222i −0.647386 0.762162i 2.99999 + 0.00842639i 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.b yes 280
3.b odd 2 1 354.2.g.a 280
59.d odd 58 1 354.2.g.a 280
177.f even 58 1 inner 354.2.g.b yes 280

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

## 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