# Properties

 Label 950.2.bi.a Level $950$ Weight $2$ Character orbit 950.bi Analytic conductor $7.586$ Analytic rank $0$ Dimension $2400$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.bi (of order $$180$$, degree $$48$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.58578819202$$ Analytic rank: $$0$$ Dimension: $$2400$$ Relative dimension: $$50$$ over $$\Q(\zeta_{180})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{180}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$2400q + 12q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$2400q + 12q^{7} + 36q^{15} - 36q^{17} + 96q^{22} + 120q^{23} + 24q^{25} - 120q^{29} - 24q^{33} - 36q^{35} - 168q^{43} + 36q^{45} - 96q^{47} + 144q^{50} - 36q^{53} + 72q^{55} + 156q^{57} - 120q^{59} + 24q^{60} + 24q^{62} - 36q^{63} - 720q^{65} + 96q^{67} + 12q^{68} + 48q^{70} + 36q^{73} - 96q^{78} - 48q^{82} - 84q^{83} - 540q^{84} - 288q^{85} - 192q^{87} - 168q^{90} - 72q^{92} - 156q^{93} - 24q^{95} - 120q^{97} - 24q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −0.974370 0.224951i −2.97099 + 1.64685i 0.898794 + 0.438371i −0.797955 2.08884i 3.26530 0.936311i −0.389434 0.104349i −0.777146 0.629320i 4.52492 7.24138i 0.307615 + 2.21481i
3.2 −0.974370 0.224951i −2.56787 + 1.42339i 0.898794 + 0.438371i −1.74255 + 1.40125i 2.82225 0.809267i 3.37715 + 0.904904i −0.777146 0.629320i 2.97815 4.76603i 2.01310 0.973350i
3.3 −0.974370 0.224951i −2.43284 + 1.34854i 0.898794 + 0.438371i 2.14980 0.615105i 2.67384 0.766712i −0.384955 0.103148i −0.777146 0.629320i 2.51038 4.01744i −2.23307 + 0.115740i
3.4 −0.974370 0.224951i −2.06215 + 1.14307i 0.898794 + 0.438371i −1.23872 + 1.86160i 2.26643 0.649887i −4.76696 1.27730i −0.777146 0.629320i 1.35609 2.17019i 1.62575 1.53524i
3.5 −0.974370 0.224951i −1.69565 + 0.939914i 0.898794 + 0.438371i 1.96376 1.06941i 1.86363 0.534386i 4.14562 + 1.11082i −0.777146 0.629320i 0.402034 0.643389i −2.15400 + 0.600253i
3.6 −0.974370 0.224951i −1.63632 + 0.907028i 0.898794 + 0.438371i −2.13706 0.658015i 1.79842 0.515689i 2.60059 + 0.696825i −0.777146 0.629320i 0.265093 0.424238i 1.93426 + 1.12188i
3.7 −0.974370 0.224951i −1.33092 + 0.737740i 0.898794 + 0.438371i −1.85939 1.24204i 1.46276 0.419440i −2.53826 0.680125i −0.777146 0.629320i −0.362675 + 0.580401i 1.53234 + 1.62848i
3.8 −0.974370 0.224951i −1.15312 + 0.639184i 0.898794 + 0.438371i 1.05144 1.97344i 1.26735 0.363406i −4.51015 1.20849i −0.777146 0.629320i −0.668633 + 1.07004i −1.46842 + 1.68634i
3.9 −0.974370 0.224951i −0.957340 + 0.530662i 0.898794 + 0.438371i 1.21737 + 1.87564i 1.05218 0.301707i 1.58359 + 0.424321i −0.777146 0.629320i −0.954860 + 1.52810i −0.764238 2.10141i
3.10 −0.974370 0.224951i −0.697754 + 0.386771i 0.898794 + 0.438371i −1.12307 + 1.93357i 0.766875 0.219898i −1.17316 0.314348i −0.777146 0.629320i −1.25249 + 2.00440i 1.52925 1.63138i
3.11 −0.974370 0.224951i −0.607413 + 0.336695i 0.898794 + 0.438371i 2.22562 0.215946i 0.667585 0.191427i −0.710217 0.190302i −0.777146 0.629320i −1.33417 + 2.13512i −2.21715 0.290244i
3.12 −0.974370 0.224951i −0.195966 + 0.108626i 0.898794 + 0.438371i 0.0267602 2.23591i 0.215379 0.0617590i 2.91467 + 0.780983i −0.777146 0.629320i −1.56315 + 2.50157i −0.529044 + 2.17258i
3.13 −0.974370 0.224951i −0.00117903 0.000653545i 0.898794 + 0.438371i 1.22275 + 1.87213i 0.00129582 0.000371572i −2.92963 0.784993i −0.777146 0.629320i −1.58976 + 2.54414i −0.770274 2.09921i
3.14 −0.974370 0.224951i 0.344356 0.190880i 0.898794 + 0.438371i −0.109124 2.23340i −0.378469 + 0.108524i 2.03298 + 0.544736i −0.777146 0.629320i −1.50761 + 2.41268i −0.396079 + 2.20071i
3.15 −0.974370 0.224951i 0.407545 0.225906i 0.898794 + 0.438371i 2.17316 + 0.526646i −0.447918 + 0.128438i −2.14541 0.574860i −0.777146 0.629320i −1.47470 + 2.36001i −1.99900 1.00200i
3.16 −0.974370 0.224951i 0.979229 0.542796i 0.898794 + 0.438371i −2.02665 + 0.944813i −1.07623 + 0.308605i 4.34720 + 1.16483i −0.777146 0.629320i −0.925495 + 1.48110i 2.18725 0.464699i
3.17 −0.974370 0.224951i 1.03787 0.575301i 0.898794 + 0.438371i 0.253929 + 2.22160i −1.14069 + 0.327086i 4.09315 + 1.09676i −0.777146 0.629320i −0.843553 + 1.34997i 0.252331 2.22179i
3.18 −0.974370 0.224951i 1.07952 0.598386i 0.898794 + 0.438371i −0.707459 2.12120i −1.18646 + 0.340211i −1.79990 0.482282i −0.777146 0.629320i −0.782466 + 1.25221i 0.212160 + 2.22598i
3.19 −0.974370 0.224951i 1.12268 0.622309i 0.898794 + 0.438371i −2.20793 + 0.353612i −1.23389 + 0.353812i −2.85847 0.765924i −0.777146 0.629320i −0.716626 + 1.14684i 2.23089 + 0.152128i
3.20 −0.974370 0.224951i 1.72849 0.958117i 0.898794 + 0.438371i 1.89512 1.18681i −1.89972 + 0.544735i 0.768108 + 0.205814i −0.777146 0.629320i 0.479928 0.768045i −2.11352 + 0.730084i
See next 80 embeddings (of 2400 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 933.50 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner
25.f odd 20 1 inner
475.bi even 180 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.bi.a 2400
19.f odd 18 1 inner 950.2.bi.a 2400
25.f odd 20 1 inner 950.2.bi.a 2400
475.bi even 180 1 inner 950.2.bi.a 2400

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.bi.a 2400 1.a even 1 1 trivial
950.2.bi.a 2400 19.f odd 18 1 inner
950.2.bi.a 2400 25.f odd 20 1 inner
950.2.bi.a 2400 475.bi even 180 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(950, [\chi])$$.