# Properties

 Label 350.2.r.a Level 350 Weight 2 Character orbit 350.r Analytic conductor 2.795 Analytic rank 0 Dimension 160 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.r (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$160$$ Relative dimension: $$20$$ over $$\Q(\zeta_{20})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$160q + 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$160q + 8q^{7} + 16q^{15} + 40q^{16} - 8q^{18} - 24q^{22} - 32q^{23} + 12q^{28} - 40q^{29} - 56q^{30} + 28q^{35} + 40q^{36} - 32q^{37} + 40q^{39} - 112q^{43} - 32q^{50} + 112q^{53} - 152q^{57} + 16q^{58} + 8q^{60} - 100q^{63} - 8q^{65} - 16q^{67} - 56q^{70} - 8q^{72} - 144q^{77} + 40q^{78} + 40q^{81} - 60q^{84} + 48q^{85} - 16q^{88} + 8q^{92} + 56q^{93} - 104q^{95} - 32q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −0.987688 + 0.156434i −2.24936 1.14611i 0.951057 0.309017i 0.898072 2.04780i 2.40096 + 0.780120i 1.91850 + 1.82192i −0.891007 + 0.453990i 1.98272 + 2.72898i −0.566669 + 2.16307i
13.2 −0.987688 + 0.156434i −2.12064 1.08052i 0.951057 0.309017i −2.23074 0.154254i 2.26356 + 0.735475i −2.39319 + 1.12812i −0.891007 + 0.453990i 1.56623 + 2.15573i 2.22741 0.196609i
13.3 −0.987688 + 0.156434i −2.10001 1.07001i 0.951057 0.309017i 0.797761 + 2.08892i 2.24154 + 0.728320i −1.08017 2.41521i −0.891007 + 0.453990i 1.50176 + 2.06699i −1.11472 1.93840i
13.4 −0.987688 + 0.156434i −1.42631 0.726740i 0.951057 0.309017i 1.81867 + 1.30093i 1.52243 + 0.494669i 1.53413 + 2.15556i −0.891007 + 0.453990i −0.257154 0.353942i −1.99979 1.00041i
13.5 −0.987688 + 0.156434i −0.353713 0.180226i 0.951057 0.309017i −1.95928 + 1.07760i 0.377552 + 0.122674i 2.34032 1.23406i −0.891007 + 0.453990i −1.67072 2.29955i 1.76658 1.37083i
13.6 −0.987688 + 0.156434i 0.353713 + 0.180226i 0.951057 0.309017i 1.95928 1.07760i −0.377552 0.122674i 1.23406 2.34032i −0.891007 + 0.453990i −1.67072 2.29955i −1.76658 + 1.37083i
13.7 −0.987688 + 0.156434i 1.42631 + 0.726740i 0.951057 0.309017i −1.81867 1.30093i −1.52243 0.494669i −2.15556 1.53413i −0.891007 + 0.453990i −0.257154 0.353942i 1.99979 + 1.00041i
13.8 −0.987688 + 0.156434i 2.10001 + 1.07001i 0.951057 0.309017i −0.797761 2.08892i −2.24154 0.728320i 2.41521 + 1.08017i −0.891007 + 0.453990i 1.50176 + 2.06699i 1.11472 + 1.93840i
13.9 −0.987688 + 0.156434i 2.12064 + 1.08052i 0.951057 0.309017i 2.23074 + 0.154254i −2.26356 0.735475i −1.12812 + 2.39319i −0.891007 + 0.453990i 1.56623 + 2.15573i −2.22741 + 0.196609i
13.10 −0.987688 + 0.156434i 2.24936 + 1.14611i 0.951057 0.309017i −0.898072 + 2.04780i −2.40096 0.780120i −1.82192 1.91850i −0.891007 + 0.453990i 1.98272 + 2.72898i 0.566669 2.16307i
13.11 0.987688 0.156434i −2.95619 1.50625i 0.951057 0.309017i 1.79263 + 1.33659i −3.15542 1.02526i 2.61159 0.423809i 0.891007 0.453990i 4.70689 + 6.47848i 1.97965 + 1.03971i
13.12 0.987688 0.156434i −1.66180 0.846728i 0.951057 0.309017i −0.499152 + 2.17964i −1.77380 0.576341i −0.737955 + 2.54075i 0.891007 0.453990i 0.281267 + 0.387131i −0.152036 + 2.23089i
13.13 0.987688 0.156434i −1.63645 0.833810i 0.951057 0.309017i 1.48682 1.67014i −1.74673 0.567548i −2.25994 1.37574i 0.891007 0.453990i 0.219357 + 0.301919i 1.20725 1.88217i
13.14 0.987688 0.156434i −1.12308 0.572235i 0.951057 0.309017i −1.93930 + 1.11316i −1.19877 0.389503i −2.35637 1.20314i 0.891007 0.453990i −0.829511 1.14172i −1.74129 + 1.40283i
13.15 0.987688 0.156434i −0.435431 0.221863i 0.951057 0.309017i −1.65984 1.49831i −0.464777 0.151015i 2.58630 0.557723i 0.891007 0.453990i −1.62298 2.23384i −1.87379 1.22021i
13.16 0.987688 0.156434i 0.435431 + 0.221863i 0.951057 0.309017i 1.65984 + 1.49831i 0.464777 + 0.151015i 0.557723 2.58630i 0.891007 0.453990i −1.62298 2.23384i 1.87379 + 1.22021i
13.17 0.987688 0.156434i 1.12308 + 0.572235i 0.951057 0.309017i 1.93930 1.11316i 1.19877 + 0.389503i 1.20314 + 2.35637i 0.891007 0.453990i −0.829511 1.14172i 1.74129 1.40283i
13.18 0.987688 0.156434i 1.63645 + 0.833810i 0.951057 0.309017i −1.48682 + 1.67014i 1.74673 + 0.567548i 1.37574 + 2.25994i 0.891007 0.453990i 0.219357 + 0.301919i −1.20725 + 1.88217i
13.19 0.987688 0.156434i 1.66180 + 0.846728i 0.951057 0.309017i 0.499152 2.17964i 1.77380 + 0.576341i −2.54075 + 0.737955i 0.891007 0.453990i 0.281267 + 0.387131i 0.152036 2.23089i
13.20 0.987688 0.156434i 2.95619 + 1.50625i 0.951057 0.309017i −1.79263 1.33659i 3.15542 + 1.02526i 0.423809 2.61159i 0.891007 0.453990i 4.70689 + 6.47848i −1.97965 1.03971i
See next 80 embeddings (of 160 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 237.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
25.f odd 20 1 inner
175.s even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.r.a 160
7.b odd 2 1 inner 350.2.r.a 160
25.f odd 20 1 inner 350.2.r.a 160
175.s even 20 1 inner 350.2.r.a 160

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.r.a 160 1.a even 1 1 trivial
350.2.r.a 160 7.b odd 2 1 inner
350.2.r.a 160 25.f odd 20 1 inner
350.2.r.a 160 175.s even 20 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database