# Properties

 Label 547.2.f.a Level 547 Weight 2 Character orbit 547.f Analytic conductor 4.368 Analytic rank 0 Dimension 528 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$547$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 547.f (of order $$13$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.36781699056$$ Analytic rank: $$0$$ Dimension: $$528$$ Relative dimension: $$44$$ over $$\Q(\zeta_{13})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$528q - 8q^{2} - 46q^{3} - 50q^{4} - 9q^{5} - 21q^{6} - 3q^{7} + 14q^{8} + 466q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$528q - 8q^{2} - 46q^{3} - 50q^{4} - 9q^{5} - 21q^{6} - 3q^{7} + 14q^{8} + 466q^{9} + 7q^{10} - 16q^{11} - 53q^{12} - 12q^{13} - 38q^{15} - 38q^{16} + 7q^{17} - 23q^{18} + 5q^{19} + 45q^{20} + 4q^{21} + 43q^{22} + 2q^{23} - 51q^{24} - 69q^{25} - 4q^{26} - 142q^{27} - 2q^{28} - 72q^{29} - 50q^{30} + 25q^{31} + 66q^{32} - 4q^{34} + 33q^{35} - 55q^{36} + 21q^{37} + 40q^{38} - 142q^{39} + 78q^{40} - 140q^{41} - 248q^{42} + 23q^{43} + 64q^{44} - 163q^{45} + 14q^{46} + 37q^{47} - 35q^{48} + q^{49} + 70q^{50} - 72q^{51} - 74q^{52} - 17q^{53} - 76q^{54} + 6q^{55} - 38q^{56} + 5q^{57} + 29q^{58} + 30q^{59} - 108q^{60} - 94q^{61} - 75q^{62} - 77q^{63} - 252q^{64} - 122q^{65} + 79q^{66} + 43q^{67} - 193q^{68} + 14q^{69} - 267q^{70} + 61q^{71} + 49q^{72} + 11q^{73} + 85q^{74} + 120q^{75} - 23q^{76} + 31q^{77} - 90q^{78} + 65q^{79} + 58q^{80} + 280q^{81} + 10q^{82} - 8q^{83} - 136q^{84} + 33q^{85} + 91q^{86} - 70q^{87} - 198q^{88} + 41q^{89} - 68q^{90} + 38q^{91} + 43q^{92} + 132q^{93} + 8q^{94} - 49q^{95} + 262q^{96} - 14q^{97} + 41q^{98} - 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1 −0.335888 + 2.76629i 1.13158 −5.59764 1.37970i −0.153172 0.403882i −0.380085 + 3.13028i −0.393363 3.23964i 3.72053 9.81022i −1.71952 1.16870 0.288060i
46.2 −0.311246 + 2.56334i −2.48877 −4.53197 1.11703i 0.771833 + 2.03516i 0.774620 6.37957i −0.224653 1.85018i 2.44259 6.44059i 3.19397 −5.45703 + 1.34504i
46.3 −0.305966 + 2.51985i 0.544519 −4.31417 1.06335i 0.299331 + 0.789270i −0.166604 + 1.37211i 0.466827 + 3.84467i 2.19924 5.79891i −2.70350 −2.08043 + 0.512780i
46.4 −0.304286 + 2.50602i −1.95070 −4.24565 1.04646i −1.15766 3.05250i 0.593571 4.88850i 0.109988 + 0.905836i 2.12398 5.60049i 0.805247 8.00187 1.97228i
46.5 −0.268734 + 2.21322i 2.12681 −2.88424 0.710902i −0.947553 2.49849i −0.571546 + 4.70710i −0.292661 2.41028i 0.767309 2.02323i 1.52333 5.78435 1.42572i
46.6 −0.255736 + 2.10617i 1.99966 −2.42868 0.598615i 0.984637 + 2.59628i −0.511385 + 4.21164i 0.216641 + 1.78420i 0.377195 0.994582i 0.998658 −5.72001 + 1.40986i
46.7 −0.245329 + 2.02046i 0.136449 −2.08020 0.512724i 1.38996 + 3.66502i −0.0334750 + 0.275691i −0.418952 3.45038i 0.102818 0.271108i −2.98138 −7.74605 + 1.90923i
46.8 −0.236626 + 1.94879i −2.82258 −1.79991 0.443638i 0.468139 + 1.23438i 0.667896 5.50062i 0.189324 + 1.55922i −0.101791 + 0.268400i 4.96697 −2.51632 + 0.620218i
46.9 −0.234924 + 1.93478i 0.630029 −1.74629 0.430421i −1.10146 2.90432i −0.148009 + 1.21897i −0.214681 1.76806i −0.139228 + 0.367113i −2.60306 5.87797 1.44879i
46.10 −0.192434 + 1.58483i 2.35170 −0.532783 0.131319i −0.750458 1.97880i −0.452547 + 3.72706i 0.413115 + 3.40231i −0.821590 + 2.16636i 2.53051 3.28047 0.808564i
46.11 −0.188009 + 1.54840i −0.908881 −0.420298 0.103594i −0.0244243 0.0644017i 0.170878 1.40731i −0.367947 3.03032i −0.866778 + 2.28551i −2.17394 0.104311 0.0257104i
46.12 −0.167985 + 1.38348i 2.55202 0.0560731 + 0.0138208i 0.714436 + 1.88381i −0.428702 + 3.53068i −0.0490013 0.403562i −1.01693 + 2.68142i 3.51279 −2.72624 + 0.671958i
46.13 −0.166856 + 1.37418i −1.48276 0.0813480 + 0.0200505i −0.394072 1.03908i 0.247408 2.03759i −0.324019 2.66853i −1.02287 + 2.69708i −0.801408 1.49364 0.368149i
46.14 −0.149053 + 1.22756i 0.296875 0.457198 + 0.112689i −0.175522 0.462814i −0.0442500 + 0.364432i 0.391638 + 3.22543i −1.08347 + 2.85688i −2.91187 0.594294 0.146480i
46.15 −0.142855 + 1.17652i −1.73902 0.578096 + 0.142488i 1.43759 + 3.79062i 0.248428 2.04599i 0.544189 + 4.48180i −1.09075 + 2.87607i 0.0242007 −4.66510 + 1.14984i
46.16 −0.137944 + 1.13607i −2.73564 0.670252 + 0.165202i −1.11149 2.93075i 0.377365 3.10788i 0.373963 + 3.07987i −1.09177 + 2.87876i 4.48371 3.48287 0.858450i
46.17 −0.0945775 + 0.778916i 3.19456 1.34412 + 0.331295i −0.702506 1.85236i −0.302134 + 2.48830i −0.187976 1.54812i −0.941647 + 2.48292i 7.20523 1.50927 0.372002i
46.18 −0.0547827 + 0.451176i 1.64947 1.74132 + 0.429198i 0.558494 + 1.47263i −0.0903624 + 0.744201i −0.347117 2.85877i −0.611367 + 1.61204i −0.279256 −0.695011 + 0.171305i
46.19 −0.0443771 + 0.365478i −0.533473 1.81028 + 0.446194i 0.569048 + 1.50046i 0.0236740 0.194973i 0.225842 + 1.85997i −0.504513 + 1.33029i −2.71541 −0.573637 + 0.141389i
46.20 −0.0420575 + 0.346375i −3.30487 1.82368 + 0.449496i 1.10374 + 2.91033i 0.138995 1.14473i −0.459333 3.78294i −0.479850 + 1.26526i 7.92218 −1.05449 + 0.259908i
See next 80 embeddings (of 528 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 519.44 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
547.f even 13 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.2.f.a 528
547.f even 13 1 inner 547.2.f.a 528

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.f.a 528 1.a even 1 1 trivial
547.2.f.a 528 547.f even 13 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(547, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database