# Properties

 Label 403.2.ch.a Level 403 Weight 2 Character orbit 403.ch Analytic conductor 3.218 Analytic rank 0 Dimension 560 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.ch (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$560$$ Relative dimension: $$35$$ over $$\Q(\zeta_{60})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$560q - 12q^{2} - 10q^{3} - 18q^{4} - 8q^{5} - 36q^{6} - 32q^{7} + 50q^{8} + 114q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$560q - 12q^{2} - 10q^{3} - 18q^{4} - 8q^{5} - 36q^{6} - 32q^{7} + 50q^{8} + 114q^{9} - 24q^{10} - 18q^{11} - 26q^{12} + 6q^{13} + 4q^{14} - 8q^{15} - 58q^{16} - 30q^{17} + 44q^{18} - 30q^{19} + 10q^{20} - 42q^{21} - 60q^{22} - 30q^{23} + 32q^{24} + 24q^{26} - 100q^{27} - 66q^{28} - 10q^{29} - 12q^{31} + 20q^{32} + 70q^{33} + 28q^{34} - 2q^{35} - 78q^{36} - 38q^{37} + 108q^{38} - 22q^{39} - 28q^{40} - 10q^{41} - 40q^{42} - 126q^{43} + 60q^{44} - 120q^{45} - 4q^{46} - 32q^{47} - 64q^{48} + 78q^{49} - 150q^{50} - 36q^{51} - 26q^{52} - 84q^{53} - 62q^{54} - 10q^{55} + 216q^{56} + 22q^{57} - 158q^{58} - 50q^{59} - 152q^{60} - 36q^{61} - 42q^{62} + 78q^{63} - 270q^{64} + 54q^{65} - 56q^{66} - 46q^{67} + 162q^{69} + 90q^{70} + 34q^{71} - 42q^{72} - 40q^{73} + 80q^{74} - 38q^{75} + 10q^{76} - 72q^{77} + 6q^{78} - 32q^{79} + 78q^{80} - 62q^{81} - 30q^{82} + 16q^{83} + 126q^{84} - 42q^{85} + 154q^{86} - 56q^{87} - 54q^{88} - 46q^{89} + 90q^{90} + 134q^{91} - 12q^{93} - 24q^{94} - 78q^{95} - 136q^{96} + 204q^{97} - 130q^{98} - 12q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −2.57671 + 0.135040i −2.14204 0.695992i 4.63216 0.486860i −0.765094 + 2.85537i 5.61341 + 1.50411i 2.68803 1.03184i −6.77303 + 1.07274i 1.67689 + 1.21833i 1.58584 7.46078i
11.2 −2.52766 + 0.132469i 2.14081 + 0.695592i 4.38247 0.460616i −0.253940 + 0.947716i −5.50339 1.47463i 0.297869 0.114341i −6.01643 + 0.952909i 1.67218 + 1.21491i 0.516330 2.42914i
11.3 −2.38428 + 0.124955i −0.874415 0.284115i 3.68013 0.386797i 0.207198 0.773272i 2.12035 + 0.568146i −0.517083 + 0.198489i −4.00980 + 0.635090i −1.74317 1.26649i −0.397393 + 1.86959i
11.4 −2.34602 + 0.122950i −2.92390 0.950033i 3.49967 0.367830i 0.583120 2.17623i 6.97635 + 1.86931i −4.00906 + 1.53893i −3.52444 + 0.558217i 5.21959 + 3.79225i −1.10045 + 5.17719i
11.5 −2.15166 + 0.112764i 2.54241 + 0.826080i 2.62788 0.276201i 0.835228 3.11711i −5.56356 1.49075i −3.32428 + 1.27607i −1.36698 + 0.216509i 3.35441 + 2.43712i −1.44563 + 6.80115i
11.6 −1.98259 + 0.103903i 1.93799 + 0.629693i 1.93081 0.202936i −0.925407 + 3.45367i −3.90767 1.04706i 3.57658 1.37292i 0.114823 0.0181862i 0.932260 + 0.677327i 1.47585 6.94334i
11.7 −1.96385 + 0.102921i 1.22501 + 0.398029i 1.85706 0.195185i 0.861644 3.21570i −2.44669 0.655589i 4.08147 1.56673i 0.257774 0.0408275i −1.08484 0.788180i −1.36117 + 6.40382i
11.8 −1.76182 + 0.0923331i −1.13628 0.369201i 1.10645 0.116292i −0.276623 + 1.03237i 2.03602 + 0.545550i 1.09450 0.420140i 1.54642 0.244928i −1.27222 0.924320i 0.392039 1.84440i
11.9 −1.62361 + 0.0850899i 1.10108 + 0.357762i 0.639831 0.0672490i −0.314520 + 1.17380i −1.81817 0.487176i −2.76717 + 1.06222i 2.17853 0.345045i −1.34267 0.975508i 0.410779 1.93256i
11.10 −1.39534 + 0.0731266i −2.17674 0.707266i −0.0474228 + 0.00498434i 0.992216 3.70300i 3.08901 + 0.827698i 2.67826 1.02809i 2.82591 0.447580i 1.81093 + 1.31572i −1.11369 + 5.23949i
11.11 −1.34882 + 0.0706888i −0.497934 0.161789i −0.174716 + 0.0183634i 0.0990127 0.369520i 0.683062 + 0.183026i −1.81118 + 0.695246i 2.90245 0.459704i −2.20529 1.60224i −0.107430 + 0.505417i
11.12 −0.995730 + 0.0521840i −2.77166 0.900568i −1.00029 + 0.105134i −0.373706 + 1.39469i 2.80682 + 0.752086i 2.77787 1.06632i 2.96017 0.468845i 4.44404 + 3.22878i 0.299330 1.40824i
11.13 −0.754685 + 0.0395514i 2.67759 + 0.870001i −1.42106 + 0.149359i −0.552551 + 2.06215i −2.05515 0.550675i −3.74165 + 1.43629i 2.55938 0.405366i 3.98552 + 2.89565i 0.335441 1.57813i
11.14 −0.636159 + 0.0333397i 3.08296 + 1.00171i −1.58546 + 0.166638i 0.0795633 0.296934i −1.99465 0.534465i 2.99900 1.15121i 2.26143 0.358175i 6.07415 + 4.41312i −0.0407153 + 0.191550i
11.15 −0.596380 + 0.0312549i 0.657945 + 0.213779i −1.63435 + 0.171777i 0.974547 3.63706i −0.399067 0.106930i −2.39734 + 0.920252i 2.14902 0.340371i −2.03986 1.48205i −0.467524 + 2.19953i
11.16 −0.284729 + 0.0149220i −2.30585 0.749216i −1.90820 + 0.200559i 0.150829 0.562901i 0.667721 + 0.178915i −3.98575 + 1.52998i 1.10354 0.174784i 2.32856 + 1.69180i −0.0345457 + 0.162525i
11.17 −0.268345 + 0.0140634i −1.39144 0.452107i −1.91723 + 0.201509i −1.07119 + 3.99775i 0.379744 + 0.101752i −1.20967 + 0.464347i 1.04246 0.165109i −0.695342 0.505196i 0.231227 1.08784i
11.18 −0.173208 + 0.00907747i −0.409936 0.133196i −1.95913 + 0.205912i −0.381329 + 1.42314i 0.0722135 + 0.0193495i 4.38084 1.68165i 0.680089 0.107716i −2.27674 1.65415i 0.0531309 0.249961i
11.19 −0.124958 + 0.00654878i 0.895096 + 0.290834i −1.97347 + 0.207420i 0.216048 0.806301i −0.113754 0.0304803i 1.43092 0.549278i 0.492421 0.0779919i −1.71044 1.24271i −0.0217166 + 0.102169i
11.20 0.0361928 0.00189679i 0.910576 + 0.295864i −1.98774 + 0.208920i 0.464170 1.73230i 0.0335175 + 0.00898099i 0.685669 0.263204i −0.143138 + 0.0226709i −1.68544 1.22454i 0.0135138 0.0635775i
See next 80 embeddings (of 560 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 396.35 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.ch even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.ch.a yes 560
13.f odd 12 1 403.2.cc.a 560
31.h odd 30 1 403.2.cc.a 560
403.ch even 60 1 inner 403.2.ch.a yes 560

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.cc.a 560 13.f odd 12 1
403.2.cc.a 560 31.h odd 30 1
403.2.ch.a yes 560 1.a even 1 1 trivial
403.2.ch.a yes 560 403.ch even 60 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