# Properties

 Label 403.2.e.a Level 403 Weight 2 Character orbit 403.e 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.e (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.

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

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1 −1.34742 + 2.33379i −0.887440 1.53709i −2.63106 4.55713i 0.115559 + 0.200154i 4.78300 1.93397 8.79087 −0.0750997 + 0.130077i −0.622824
191.2 −1.27542 + 2.20909i 0.569293 + 0.986044i −2.25340 3.90300i −0.675176 1.16944i −2.90435 1.90483 6.39445 0.851812 1.47538i 3.44453
191.3 −1.25452 + 2.17290i 1.68571 + 2.91973i −2.14766 3.71985i 0.669409 + 1.15945i −8.45904 1.15243 5.75904 −4.18323 + 7.24556i −3.35916
191.4 −1.18452 + 2.05166i −0.300552 0.520571i −1.80620 3.12843i 2.14607 + 3.71711i 1.42404 1.99772 3.81984 1.31934 2.28516i −10.1683
191.5 −1.06665 + 1.84749i 0.605846 + 1.04936i −1.27547 2.20918i −1.78766 3.09632i −2.58490 0.911932 1.17532 0.765900 1.32658i 7.62720
191.6 −1.05012 + 1.81887i −1.37200 2.37637i −1.20552 2.08802i 0.498066 + 0.862675i 5.76307 −3.52844 0.863293 −2.26475 + 3.92266i −2.09212
191.7 −0.948089 + 1.64214i −0.203682 0.352788i −0.797747 1.38174i 0.840173 + 1.45522i 0.772435 −3.74155 −0.767016 1.41703 2.45436i −3.18624
191.8 −0.937381 + 1.62359i 1.15361 + 1.99811i −0.757366 1.31180i −1.23736 2.14316i −4.32548 −4.49839 −0.909763 −1.16162 + 2.01199i 4.63950
191.9 −0.925450 + 1.60293i −1.55919 2.70060i −0.712914 1.23480i −0.0505045 0.0874763i 5.77182 4.42694 −1.06274 −3.36217 + 5.82344i 0.186957
191.10 −0.852641 + 1.47682i 0.937886 + 1.62447i −0.453992 0.786337i 1.48433 + 2.57094i −3.19872 0.0782312 −1.86219 −0.259260 + 0.449051i −5.06240
191.11 −0.770644 + 1.33479i −0.329492 0.570697i −0.187783 0.325250i −0.614094 1.06364i 1.01568 0.583300 −2.50372 1.28287 2.22200i 1.89299
191.12 −0.447963 + 0.775894i 0.0864423 + 0.149722i 0.598659 + 1.03691i −0.364951 0.632113i −0.154892 3.22911 −2.86456 1.48506 2.57219i 0.653937
191.13 −0.381984 + 0.661616i 1.18414 + 2.05098i 0.708176 + 1.22660i 0.878528 + 1.52165i −1.80929 −0.793025 −2.60998 −1.30436 + 2.25922i −1.34233
191.14 −0.356919 + 0.618202i −0.621239 1.07602i 0.745217 + 1.29075i −0.797665 1.38160i 0.886928 −1.56596 −2.49161 0.728125 1.26115i 1.13881
191.15 −0.351546 + 0.608895i 1.62249 + 2.81024i 0.752831 + 1.30394i −1.57412 2.72646i −2.28152 5.06668 −2.46480 −3.76495 + 6.52109i 2.21350
191.16 −0.333798 + 0.578156i −1.40074 2.42616i 0.777157 + 1.34608i −1.97182 3.41529i 1.87026 −1.53640 −2.37285 −2.42416 + 4.19877i 2.63276
191.17 −0.254102 + 0.440118i −0.777658 1.34694i 0.870864 + 1.50838i 1.96757 + 3.40793i 0.790418 3.56171 −1.90156 0.290495 0.503151i −1.99985
191.18 −0.0861040 + 0.149137i 1.23907 + 2.14613i 0.985172 + 1.70637i −0.342759 0.593675i −0.426756 −3.89977 −0.683725 −1.57059 + 2.72035i 0.118052
191.19 0.0840732 0.145619i 0.173155 + 0.299914i 0.985863 + 1.70757i 1.49511 + 2.58961i 0.0582309 −3.36152 0.667832 1.44003 2.49421i 0.502795
191.20 0.170223 0.294835i −1.12750 1.95288i 0.942048 + 1.63168i 0.624468 + 1.08161i −0.767702 −0.903522 1.32232 −1.04250 + 1.80566i 0.425195
See all 70 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 211.35 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.e 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.e.a 70
13.c even 3 1 403.2.g.a yes 70
31.c even 3 1 403.2.g.a yes 70
403.e even 3 1 inner 403.2.e.a 70

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