# Properties

 Label 403.2.bn.a Level 403 Weight 2 Character orbit 403.bn Analytic conductor 3.218 Analytic rank 0 Dimension 272 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.bn (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$272$$ Relative dimension: $$34$$ over $$\Q(\zeta_{20})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$272q - 6q^{2} - 20q^{3} - 16q^{5} - 18q^{7} + 4q^{8} + 40q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$272q - 6q^{2} - 20q^{3} - 16q^{5} - 18q^{7} + 4q^{8} + 40q^{9} + 8q^{14} - 40q^{15} + 20q^{16} + 16q^{18} + 14q^{19} - 4q^{20} + 50q^{21} - 120q^{22} - 50q^{24} + 40q^{27} + 46q^{28} - 20q^{29} - 14q^{31} + 52q^{32} - 70q^{33} - 10q^{34} + 20q^{35} - 46q^{39} - 44q^{40} - 20q^{41} - 20q^{42} + 30q^{44} - 36q^{45} - 50q^{46} + 26q^{47} - 20q^{48} + 30q^{50} - 50q^{52} + 50q^{54} - 20q^{55} - 100q^{58} + 2q^{59} - 40q^{60} - 4q^{63} - 172q^{66} + 12q^{67} - 102q^{70} + 50q^{71} - 18q^{72} - 10q^{73} - 20q^{74} + 148q^{76} + 66q^{78} - 100q^{79} - 120q^{80} + 88q^{81} - 40q^{83} - 70q^{84} - 10q^{86} - 88q^{87} - 20q^{89} + 150q^{91} + 48q^{93} - 24q^{94} + 40q^{96} - 76q^{97} - 8q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
60.1 −0.399110 2.51988i 0.464849 + 0.639810i −4.28839 + 1.39338i −1.89475 1.89475i 1.42672 1.42672i −1.39495 0.710764i 2.90617 + 5.70368i 0.733779 2.25834i −4.01833 + 5.53075i
60.2 −0.393738 2.48596i 1.63498 + 2.25036i −4.12287 + 1.33960i 1.33236 + 1.33236i 4.95055 4.95055i −0.882842 0.449831i 2.66818 + 5.23661i −1.46390 + 4.50541i 2.78760 3.83680i
60.3 −0.390481 2.46540i −1.53915 2.11846i −4.02361 + 1.30735i −1.32495 1.32495i −4.62184 + 4.62184i −2.80514 1.42929i 2.52785 + 4.96118i −1.19183 + 3.66808i −2.74916 + 3.78390i
60.4 −0.388669 2.45396i 0.0803757 + 0.110628i −3.96875 + 1.28952i 0.523296 + 0.523296i 0.240236 0.240236i 3.99187 + 2.03396i 2.45104 + 4.81045i 0.921273 2.83539i 1.08076 1.48754i
60.5 −0.362855 2.29098i −0.115498 0.158970i −3.21480 + 1.04455i 2.00673 + 2.00673i −0.322287 + 0.322287i −1.28505 0.654764i 1.45346 + 2.85258i 0.915119 2.81645i 3.86922 5.32553i
60.6 −0.327123 2.06537i −1.61302 2.22013i −2.25663 + 0.733225i 0.152237 + 0.152237i −4.05774 + 4.05774i 0.794879 + 0.405011i 0.353884 + 0.694537i −1.40010 + 4.30907i 0.264626 0.364226i
60.7 −0.279767 1.76638i 1.47487 + 2.02999i −1.13970 + 0.370312i −2.62399 2.62399i 3.17310 3.17310i 2.14712 + 1.09401i −0.650866 1.27740i −1.01855 + 3.13478i −3.90085 + 5.36906i
60.8 −0.242952 1.53394i −0.916843 1.26193i −0.391834 + 0.127315i −1.09885 1.09885i −1.71297 + 1.71297i −1.58544 0.807824i −1.11966 2.19746i 0.175195 0.539194i −1.41860 + 1.95254i
60.9 −0.241622 1.52554i −1.55970 2.14675i −0.366780 + 0.119174i 2.50940 + 2.50940i −2.89809 + 2.89809i 3.71488 + 1.89282i −1.13200 2.22168i −1.24880 + 3.84340i 3.22186 4.43452i
60.10 −0.217339 1.37222i 0.253284 + 0.348615i 0.0663516 0.0215589i −0.524097 0.524097i 0.423330 0.423330i 0.759489 + 0.386979i −1.30549 2.56217i 0.869671 2.67657i −0.605272 + 0.833086i
60.11 −0.197701 1.24823i 1.55219 + 2.13641i 0.383111 0.124480i 2.03317 + 2.03317i 2.35987 2.35987i 0.0423174 + 0.0215618i −1.37862 2.70570i −1.22789 + 3.77906i 2.13592 2.93984i
60.12 −0.194801 1.22992i 0.902955 + 1.24281i 0.427346 0.138853i −1.41853 1.41853i 1.35267 1.35267i −4.54364 2.31510i −1.38469 2.71762i 0.197800 0.608766i −1.46835 + 2.02102i
60.13 −0.170304 1.07526i 0.498908 + 0.686688i 0.774943 0.251794i 2.61824 + 2.61824i 0.653399 0.653399i −0.113188 0.0576723i −1.39120 2.73038i 0.704420 2.16798i 2.36938 3.26117i
60.14 −0.159192 1.00510i −0.939090 1.29255i 0.917230 0.298026i −2.17200 2.17200i −1.14964 + 1.14964i 3.27405 + 1.66821i −1.36955 2.68789i 0.138265 0.425535i −1.83731 + 2.52884i
60.15 −0.0811295 0.512232i −1.59762 2.19894i 1.64631 0.534920i 1.75447 + 1.75447i −0.996753 + 0.996753i −4.04043 2.05870i −0.878462 1.72408i −1.35589 + 4.17300i 0.756357 1.04104i
60.16 −0.0419805 0.265054i 1.85928 + 2.55908i 1.83362 0.595780i −0.657016 0.657016i 0.600242 0.600242i −0.541111 0.275710i −0.478554 0.939216i −2.16493 + 6.66296i −0.146563 + 0.201727i
60.17 0.000221639 0.00139937i −0.216440 0.297904i 1.90211 0.618033i 1.05381 + 1.05381i 0.000368907 0 0.000368907i 1.63845 + 0.834831i 0.00257288 + 0.00504957i 0.885150 2.72421i −0.00124111 + 0.00170824i
60.18 0.00544638 + 0.0343871i −1.45104 1.99718i 1.90096 0.617659i −2.64841 2.64841i 0.0607744 0.0607744i −1.91884 0.977699i 0.0632049 + 0.124047i −0.956174 + 2.94280i 0.0766470 0.105496i
60.19 0.0137208 + 0.0866298i 1.01677 + 1.39947i 1.89480 0.615657i −0.943639 0.943639i −0.107284 + 0.107284i 2.98091 + 1.51885i 0.158971 + 0.311998i 0.00237083 0.00729668i 0.0687997 0.0946947i
60.20 0.0381722 + 0.241010i 0.264626 + 0.364226i 1.84548 0.599634i −0.0177153 0.0177153i −0.0776807 + 0.0776807i −3.14122 1.60053i 0.436524 + 0.856726i 0.864417 2.66040i 0.00359333 0.00494579i
See next 80 embeddings (of 272 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 395.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
31.f odd 10 1 inner
403.bn even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bn.a 272
13.d odd 4 1 inner 403.2.bn.a 272
31.f odd 10 1 inner 403.2.bn.a 272
403.bn even 20 1 inner 403.2.bn.a 272

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bn.a 272 1.a even 1 1 trivial
403.2.bn.a 272 13.d odd 4 1 inner
403.2.bn.a 272 31.f odd 10 1 inner
403.2.bn.a 272 403.bn even 20 1 inner

## 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