# Properties

 Label 320.2.bd.a Level $320$ Weight $2$ Character orbit 320.bd Analytic conductor $2.555$ Analytic rank $0$ Dimension $368$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 320.bd (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.55521286468$$ Analytic rank: $$0$$ Dimension: $$368$$ Relative dimension: $$46$$ over $$\Q(\zeta_{16})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$368q - 8q^{2} - 8q^{3} - 8q^{5} - 16q^{6} - 8q^{7} - 8q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$368q - 8q^{2} - 8q^{3} - 8q^{5} - 16q^{6} - 8q^{7} - 8q^{8} - 8q^{10} - 16q^{11} + 24q^{12} - 8q^{13} + 32q^{14} - 8q^{15} - 16q^{16} - 8q^{18} - 8q^{20} - 16q^{21} - 40q^{22} - 8q^{23} - 16q^{24} - 8q^{25} - 16q^{26} - 8q^{27} - 8q^{28} - 72q^{30} - 32q^{31} - 8q^{32} + 32q^{34} - 8q^{35} - 16q^{36} - 8q^{37} - 64q^{38} - 112q^{40} - 16q^{41} - 8q^{42} - 8q^{43} - 32q^{45} - 16q^{46} - 16q^{47} + 96q^{48} + 96q^{50} - 48q^{51} - 8q^{52} - 8q^{53} - 8q^{55} + 80q^{56} - 8q^{57} - 72q^{58} - 64q^{60} - 16q^{61} - 24q^{62} - 16q^{65} + 80q^{66} - 8q^{67} + 80q^{68} - 64q^{69} - 8q^{70} - 80q^{71} - 128q^{72} - 8q^{73} - 8q^{75} + 48q^{76} - 8q^{77} - 160q^{78} + 32q^{79} - 8q^{80} - 16q^{81} - 8q^{82} - 8q^{83} + 32q^{85} - 16q^{86} - 120q^{87} + 80q^{88} - 8q^{90} - 16q^{91} - 232q^{92} - 32q^{93} - 32q^{94} - 16q^{95} - 16q^{96} - 48q^{98} - 128q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −1.41418 0.00906921i −2.51007 1.67717i 1.99984 + 0.0256511i 2.18343 + 0.482341i 3.53449 + 2.39460i −0.174976 0.422430i −2.82790 0.0544123i 2.33948 + 5.64800i −3.08339 0.701921i
43.2 −1.41324 0.0525141i −0.558594 0.373241i 1.99448 + 0.148430i −1.14844 + 1.91862i 0.769826 + 0.556812i −0.384293 0.927767i −2.81089 0.314505i −0.975332 2.35466i 1.72377 2.65115i
43.3 −1.40925 0.118345i 2.70757 + 1.80914i 1.97199 + 0.333555i 0.594278 2.15565i −3.60155 2.86996i 1.00795 + 2.43340i −2.73956 0.703438i 2.90990 + 7.02511i −1.09260 + 2.96753i
43.4 −1.38824 0.269779i 0.618789 + 0.413462i 1.85444 + 0.749037i 2.19748 0.413640i −0.747487 0.740922i −1.02179 2.46682i −2.37234 1.54013i −0.936101 2.25995i −3.16222 0.0185990i
43.5 −1.34845 0.426245i −0.603776 0.403430i 1.63663 + 1.14954i −1.74032 1.40403i 0.642201 + 0.801362i 1.76831 + 4.26909i −1.71693 2.24770i −0.946261 2.28448i 1.74827 + 2.63506i
43.6 −1.28421 + 0.592295i 0.985871 + 0.658738i 1.29837 1.52126i −2.04358 0.907628i −1.65623 0.262029i −0.333079 0.804125i −0.766346 + 2.72263i −0.610044 1.47278i 3.16196 0.0448198i
43.7 −1.23347 + 0.691783i 2.58768 + 1.72903i 1.04287 1.70658i 0.659331 + 2.13665i −4.38793 0.342588i −1.62653 3.92678i −0.105762 + 2.82645i 2.55849 + 6.17673i −2.29136 2.17937i
43.8 −1.18016 0.779237i 1.62297 + 1.08444i 0.785578 + 1.83926i −0.183333 + 2.22854i −1.07034 2.54449i 0.479255 + 1.15702i 0.506107 2.78278i 0.309989 + 0.748380i 1.95292 2.48718i
43.9 −1.17265 + 0.790504i −1.40177 0.936632i 0.750206 1.85397i −0.231774 + 2.22402i 2.38419 0.00976493i 0.114006 + 0.275234i 0.585841 + 2.76709i −0.0603745 0.145757i −1.48631 2.79121i
43.10 −1.15017 + 0.822872i −1.52991 1.02225i 0.645765 1.89288i 1.06128 1.96817i 2.60083 0.0831583i 1.08505 + 2.61953i 0.814859 + 2.70851i 0.147570 + 0.356266i 0.398902 + 3.13702i
43.11 −1.04815 0.949416i 1.57122 + 1.04985i 0.197220 + 1.99025i −1.85697 1.24566i −0.650118 2.59214i −1.56668 3.78229i 1.68286 2.27332i 0.218482 + 0.527463i 0.763728 + 3.06867i
43.12 −1.00711 0.992839i −1.39071 0.929242i 0.0285426 + 1.99980i 0.728044 2.11423i 0.478010 + 2.31660i −0.473507 1.14315i 1.95673 2.04235i −0.0774701 0.187029i −2.83231 + 1.40643i
43.13 −0.988622 1.01125i −2.72182 1.81866i −0.0452538 + 1.99949i −1.63219 + 1.52838i 0.851727 + 4.55040i 0.160177 + 0.386702i 2.06672 1.93097i 2.95271 + 7.12847i 3.15920 + 0.139569i
43.14 −0.850415 + 1.12995i −2.07904 1.38917i −0.553590 1.92186i −2.12829 0.685857i 3.33774 1.16785i −1.73532 4.18944i 2.64239 + 1.00885i 1.24456 + 3.00463i 2.58491 1.82160i
43.15 −0.828734 + 1.14595i 1.54280 + 1.03087i −0.626401 1.89937i −2.19943 + 0.403103i −2.45989 + 0.913659i 1.23320 + 2.97720i 2.69571 + 0.856252i 0.169501 + 0.409211i 1.36081 2.85451i
43.16 −0.759115 + 1.19321i 0.0705545 + 0.0471430i −0.847488 1.81156i 2.23184 + 0.137441i −0.109810 + 0.0483992i −1.17225 2.83007i 2.80491 + 0.363957i −1.14529 2.76499i −1.85822 + 2.55872i
43.17 −0.643040 1.25956i 1.22104 + 0.815871i −1.17300 + 1.61990i 2.16782 0.548217i 0.242466 2.06261i 0.908242 + 2.19269i 2.79465 + 0.435808i −0.322763 0.779219i −2.08451 2.37799i
43.18 −0.627366 + 1.26744i 1.84918 + 1.23558i −1.21282 1.59030i 1.19255 1.89151i −2.72615 + 1.56857i 0.503585 + 1.21576i 2.77650 0.539487i 0.744758 + 1.79801i 1.64921 + 2.69816i
43.19 −0.497102 1.32397i −0.705925 0.471684i −1.50578 + 1.31629i 1.24689 + 1.85614i −0.273577 + 1.16910i −0.381149 0.920174i 2.49126 + 1.33927i −0.872206 2.10569i 1.83764 2.57354i
43.20 −0.317156 1.37819i −0.627981 0.419603i −1.79882 + 0.874203i −2.23256 0.125169i −0.379126 + 0.998558i −0.150193 0.362597i 1.77533 + 2.20187i −0.929757 2.24463i 0.535564 + 3.11660i
See next 80 embeddings (of 368 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
320.bd even 16 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.2.bd.a 368
5.c odd 4 1 320.2.bj.a yes 368
64.j odd 16 1 320.2.bj.a yes 368
320.bd even 16 1 inner 320.2.bd.a 368

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.bd.a 368 1.a even 1 1 trivial
320.2.bd.a 368 320.bd even 16 1 inner
320.2.bj.a yes 368 5.c odd 4 1
320.2.bj.a yes 368 64.j odd 16 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(320, [\chi])$$.