# Properties

 Label 403.2.g.a Level 403 Weight 2 Character orbit 403.g Analytic conductor 3.218 Analytic rank 0 Dimension 70 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$70$$ Relative dimension: $$35$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

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

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
87.1 −1.34742 + 2.33379i 1.77488 −2.63106 4.55713i 0.115559 0.200154i −2.39150 + 4.14220i −0.966983 + 1.67486i 8.79087 0.150199 0.311412 + 0.539381i
87.2 −1.27542 + 2.20909i −1.13859 −2.25340 3.90300i −0.675176 + 1.16944i 1.45218 2.51524i −0.952417 + 1.64964i 6.39445 −1.70362 −1.72227 2.98305i
87.3 −1.25452 + 2.17290i −3.37142 −2.14766 3.71985i 0.669409 1.15945i 4.22952 7.32574i −0.576214 + 0.998031i 5.75904 8.36645 1.67958 + 2.90912i
87.4 −1.18452 + 2.05166i 0.601104 −1.80620 3.12843i 2.14607 3.71711i −0.712022 + 1.23326i −0.998859 + 1.73007i 3.81984 −2.63867 5.08415 + 8.80601i
87.5 −1.06665 + 1.84749i −1.21169 −1.27547 2.20918i −1.78766 + 3.09632i 1.29245 2.23859i −0.455966 + 0.789757i 1.17532 −1.53180 −3.81360 6.60535i
87.6 −1.05012 + 1.81887i 2.74399 −1.20552 2.08802i 0.498066 0.862675i −2.88153 + 4.99096i 1.76422 3.05572i 0.863293 4.52949 1.04606 + 1.81183i
87.7 −0.948089 + 1.64214i 0.407364 −0.797747 1.38174i 0.840173 1.45522i −0.386218 + 0.668949i 1.87077 3.24028i −0.767016 −2.83405 1.59312 + 2.75936i
87.8 −0.937381 + 1.62359i −2.30722 −0.757366 1.31180i −1.23736 + 2.14316i 2.16274 3.74598i 2.24919 3.89572i −0.909763 2.32325 −2.31975 4.01792i
87.9 −0.925450 + 1.60293i 3.11839 −0.712914 1.23480i −0.0505045 + 0.0874763i −2.88591 + 4.99854i −2.21347 + 3.83384i −1.06274 6.72433 −0.0934787 0.161910i
87.10 −0.852641 + 1.47682i −1.87577 −0.453992 0.786337i 1.48433 2.57094i 1.59936 2.77017i −0.0391156 + 0.0677502i −1.86219 0.518520 2.53120 + 4.38417i
87.11 −0.770644 + 1.33479i 0.658984 −0.187783 0.325250i −0.614094 + 1.06364i −0.507842 + 0.879608i −0.291650 + 0.505152i −2.50372 −2.56574 −0.946495 1.63938i
87.12 −0.447963 + 0.775894i −0.172885 0.598659 + 1.03691i −0.364951 + 0.632113i 0.0774459 0.134140i −1.61455 + 2.79649i −2.86456 −2.97011 −0.326969 0.566326i
87.13 −0.381984 + 0.661616i −2.36827 0.708176 + 1.22660i 0.878528 1.52165i 0.904643 1.56689i 0.396513 0.686780i −2.60998 2.60872 0.671167 + 1.16250i
87.14 −0.356919 + 0.618202i 1.24248 0.745217 + 1.29075i −0.797665 + 1.38160i −0.443464 + 0.768103i 0.782978 1.35616i −2.49161 −1.45625 −0.569404 0.986237i
87.15 −0.351546 + 0.608895i −3.24498 0.752831 + 1.30394i −1.57412 + 2.72646i 1.14076 1.97585i −2.53334 + 4.38787i −2.46480 7.52991 −1.10675 1.91695i
87.16 −0.333798 + 0.578156i 2.80149 0.777157 + 1.34608i −1.97182 + 3.41529i −0.935132 + 1.61970i 0.768201 1.33056i −2.37285 4.84832 −1.31638 2.28004i
87.17 −0.254102 + 0.440118i 1.55532 0.870864 + 1.50838i 1.96757 3.40793i −0.395209 + 0.684522i −1.78085 + 3.08453i −1.90156 −0.580989 0.999926 + 1.73192i
87.18 −0.0861040 + 0.149137i −2.47814 0.985172 + 1.70637i −0.342759 + 0.593675i 0.213378 0.369581i 1.94989 3.37730i −0.683725 3.14119 −0.0590258 0.102236i
87.19 0.0840732 0.145619i −0.346311 0.985863 + 1.70757i 1.49511 2.58961i −0.0291155 + 0.0504294i 1.68076 2.91117i 0.667832 −2.88007 −0.251398 0.435434i
87.20 0.170223 0.294835i 2.25499 0.942048 + 1.63168i 0.624468 1.08161i 0.383851 0.664850i 0.451761 0.782473i 1.32232 2.08499 −0.212598 0.368230i
See all 70 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 315.35 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.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.g.a yes 70
13.c even 3 1 403.2.e.a 70
31.c even 3 1 403.2.e.a 70
403.g even 3 1 inner 403.2.g.a yes 70

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.e.a 70 13.c even 3 1
403.2.e.a 70 31.c even 3 1
403.2.g.a yes 70 1.a even 1 1 trivial
403.2.g.a yes 70 403.g even 3 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