# Properties

 Label 403.2.ba.a Level 403 Weight 2 Character orbit 403.ba Analytic conductor 3.218 Analytic rank 0 Dimension 140 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.ba (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$140$$ Relative dimension: $$35$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$140q - 8q^{2} - 12q^{4} - 2q^{5} + 6q^{6} + 12q^{7} - 10q^{8} - 124q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$140q - 8q^{2} - 12q^{4} - 2q^{5} + 6q^{6} + 12q^{7} - 10q^{8} - 124q^{9} - 6q^{10} - 12q^{11} + 26q^{12} - 6q^{13} - 24q^{14} + 18q^{15} + 48q^{16} - 4q^{18} + 10q^{19} - 50q^{20} - 28q^{21} - 12q^{24} + 6q^{26} - 54q^{28} - 28q^{31} - 10q^{32} - 30q^{33} + 72q^{34} - 8q^{35} + 48q^{36} + 8q^{37} + 72q^{38} - 8q^{39} - 12q^{40} - 20q^{41} + 30q^{42} + 26q^{43} + 24q^{46} + 12q^{47} + 54q^{48} - 108q^{49} + 10q^{50} + 36q^{51} + 46q^{52} + 24q^{53} - 18q^{54} + 24q^{56} - 52q^{57} - 42q^{58} - 10q^{59} - 18q^{60} + 36q^{61} + 12q^{62} - 58q^{63} - 84q^{65} + 16q^{66} + 36q^{67} - 12q^{69} + 30q^{70} + 106q^{71} + 62q^{72} + 20q^{73} - 90q^{74} - 82q^{75} + 20q^{76} - 48q^{77} - 6q^{78} - 48q^{79} + 32q^{80} + 132q^{81} - 6q^{83} - 86q^{84} + 42q^{85} + 6q^{86} - 14q^{87} + 24q^{88} + 36q^{89} - 90q^{90} + 46q^{91} - 58q^{93} + 4q^{94} + 48q^{95} - 54q^{96} + 26q^{97} - 40q^{98} - 18q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1 −0.725626 2.70807i 1.15921i −5.07507 + 2.93009i 0.758661 + 2.83136i −3.13923 + 0.841153i 2.03109 0.544229i 7.65262 + 7.65262i 1.65623 7.11703 4.10902i
6.2 −0.648505 2.42025i 0.854246i −3.70501 + 2.13909i −0.846300 3.15843i 2.06749 0.553983i 4.17770 1.11941i 4.03636 + 4.03636i 2.27026 −7.09538 + 4.09652i
6.3 −0.626284 2.33732i 1.39283i −3.33879 + 1.92765i −0.0315189 0.117630i 3.25550 0.872308i −1.74827 + 0.468447i 3.17450 + 3.17450i 1.06002 −0.255199 + 0.147339i
6.4 −0.616245 2.29986i 2.87995i −3.17753 + 1.83455i −0.235608 0.879300i −6.62347 + 1.77475i −1.87834 + 0.503300i 2.81012 + 2.81012i −5.29412 −1.87707 + 1.08373i
6.5 −0.610803 2.27955i 2.86872i −3.09121 + 1.78471i −0.322912 1.20512i 6.53938 1.75222i 0.235099 0.0629946i 2.61896 + 2.61896i −5.22953 −2.54990 + 1.47219i
6.6 −0.529121 1.97471i 0.0635184i −1.88744 + 1.08972i 0.363448 + 1.35641i 0.125430 0.0336089i −3.84966 + 1.03151i 0.259387 + 0.259387i 2.99597 2.48620 1.43541i
6.7 −0.505205 1.88545i 2.78647i −1.56764 + 0.905077i −0.0340828 0.127199i −5.25374 + 1.40774i 4.21322 1.12893i −0.262034 0.262034i −4.76440 −0.222608 + 0.128523i
6.8 −0.504814 1.88399i 2.73123i −1.56254 + 0.902134i 1.05873 + 3.95125i 5.14562 1.37876i 1.60020 0.428773i −0.269949 0.269949i −4.45962 6.90966 3.98929i
6.9 −0.495867 1.85060i 0.831703i −1.44679 + 0.835304i 0.740052 + 2.76191i −1.53915 + 0.412414i 1.04383 0.279694i −0.446239 0.446239i 2.30827 4.74423 2.73908i
6.10 −0.379919 1.41788i 0.797458i −0.133982 + 0.0773543i −0.876737 3.27203i −1.13070 + 0.302969i 2.28277 0.611667i −1.91533 1.91533i 2.36406 −4.30624 + 2.48621i
6.11 −0.361997 1.35099i 2.34712i 0.0379121 0.0218885i −0.843165 3.14674i 3.17094 0.849651i −1.42675 + 0.382295i −2.02129 2.02129i −2.50897 −3.94599 + 2.27822i
6.12 −0.330760 1.23441i 1.61222i 0.317680 0.183412i −0.646845 2.41406i −1.99014 + 0.533256i −2.14247 + 0.574074i −2.13879 2.13879i 0.400758 −2.76599 + 1.59695i
6.13 −0.216378 0.807533i 1.94866i 1.12676 0.650535i −0.0513533 0.191653i 1.57361 0.421647i 1.59671 0.427837i −1.95145 1.95145i −0.797286 −0.143655 + 0.0829390i
6.14 −0.167077 0.623539i 0.591585i 1.37116 0.791642i 0.549606 + 2.05116i 0.368876 0.0988400i −4.24319 + 1.13696i −1.63563 1.63563i 2.65003 1.18715 0.685401i
6.15 −0.137083 0.511599i 1.84118i 1.48911 0.859737i 0.868165 + 3.24004i −0.941949 + 0.252395i 1.36093 0.364660i −1.39301 1.39301i −0.389962 1.53859 0.888306i
6.16 −0.135765 0.506680i 2.82155i 1.49376 0.862422i 0.266274 + 0.993746i −1.42962 + 0.383067i −2.70478 + 0.724745i −1.38160 1.38160i −4.96115 0.467361 0.269831i
6.17 −0.133986 0.500044i 1.10231i 1.49996 0.866002i 0.299955 + 1.11945i 0.551206 0.147695i 3.94703 1.05760i −1.36613 1.36613i 1.78490 0.519583 0.299981i
6.18 0.0225085 + 0.0840028i 0.498964i 1.72550 0.996218i −0.755041 2.81785i −0.0419144 + 0.0112309i −2.56590 + 0.687531i 0.245512 + 0.245512i 2.75103 0.219712 0.126851i
6.19 0.0759990 + 0.283632i 1.83195i 1.65738 0.956888i −0.350763 1.30906i 0.519599 0.139226i 2.10731 0.564653i 0.812630 + 0.812630i −0.356028 0.344635 0.198975i
6.20 0.107992 + 0.403030i 2.83596i 1.58128 0.912952i −0.961540 3.58852i −1.14298 + 0.306260i 1.51235 0.405234i 1.12879 + 1.12879i −5.04265 1.34244 0.775060i
See next 80 embeddings (of 140 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 336.35 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.ba even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.ba.a 140
13.f odd 12 1 403.2.bf.a yes 140
31.e odd 6 1 403.2.bf.a yes 140
403.ba even 12 1 inner 403.2.ba.a 140

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.ba.a 140 1.a even 1 1 trivial
403.2.ba.a 140 403.ba even 12 1 inner
403.2.bf.a yes 140 13.f odd 12 1
403.2.bf.a yes 140 31.e odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database