# Properties

 Label 403.2.r.a Level 403 Weight 2 Character orbit 403.r Analytic conductor 3.218 Analytic rank 0 Dimension 68 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.r (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$68$$ Relative dimension: $$34$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$68q + 32q^{4} - 34q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$68q + 32q^{4} - 34q^{9} + 8q^{10} - 12q^{11} - 16q^{12} + 6q^{13} - 8q^{14} - 36q^{16} - 6q^{17} + 12q^{19} - 12q^{20} - 20q^{22} - 8q^{23} + 48q^{24} - 72q^{25} - 12q^{27} - 6q^{28} + 32q^{30} + 6q^{33} + 30q^{35} + 40q^{36} - 42q^{37} - 36q^{38} - 14q^{39} + 8q^{40} + 18q^{41} - 16q^{42} + 12q^{43} + 60q^{45} + 30q^{46} - 46q^{48} + 22q^{49} + 56q^{51} + 20q^{53} - 114q^{54} - 6q^{55} - 2q^{56} - 12q^{58} + 6q^{59} + 6q^{61} - 8q^{62} - 30q^{63} + 24q^{64} + 24q^{65} + 8q^{66} - 48q^{67} + 58q^{68} - 28q^{69} - 30q^{71} + 72q^{72} + 8q^{74} - 4q^{75} - 12q^{76} - 20q^{77} + 26q^{78} + 16q^{79} + 42q^{80} - 58q^{81} - 42q^{82} - 72q^{84} + 30q^{85} - 20q^{87} + 64q^{88} + 18q^{89} + 52q^{90} - 22q^{91} + 48q^{92} + 8q^{94} - 32q^{95} - 168q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
218.1 −2.26368 + 1.30694i −0.194798 0.337400i 2.41618 4.18495i 3.31493i 0.881922 + 0.509178i −1.89144 1.09203i 7.40344i 1.42411 2.46663i 4.33241 + 7.50395i
218.2 −2.19517 + 1.26738i −1.69135 2.92950i 2.21251 3.83219i 0.452856i 7.42559 + 4.28717i 1.73117 + 0.999489i 6.14687i −4.22131 + 7.31153i 0.573941 + 0.994095i
218.3 −2.18149 + 1.25949i −0.435889 0.754983i 2.17261 3.76307i 3.48615i 1.90178 + 1.09799i −2.02718 1.17039i 5.90753i 1.12000 1.93990i −4.39076 7.60502i
218.4 −2.14952 + 1.24103i −0.331917 0.574896i 2.08030 3.60319i 0.685218i 1.42693 + 0.823836i 3.86933 + 2.23396i 5.36274i 1.27966 2.21644i 0.850375 + 1.47289i
218.5 −1.98176 + 1.14417i 0.827863 + 1.43390i 1.61825 2.80289i 1.16564i −3.28125 1.89443i −2.58921 1.49488i 2.82952i 0.129286 0.223931i 1.33368 + 2.31001i
218.6 −1.86490 + 1.07670i 1.32944 + 2.30267i 1.31857 2.28383i 3.07578i −4.95857 2.86283i −0.437880 0.252810i 1.37202i −2.03485 + 3.52446i −3.31169 5.73602i
218.7 −1.60943 + 0.929207i 1.27215 + 2.20343i 0.726853 1.25895i 3.22618i −4.09489 2.36418i 2.41260 + 1.39291i 1.01524i −1.73673 + 3.00811i 2.99779 + 5.19232i
218.8 −1.56202 + 0.901835i 0.553992 + 0.959542i 0.626612 1.08532i 0.771735i −1.73070 0.999218i 3.80672 + 2.19781i 1.34694i 0.886187 1.53492i −0.695977 1.20547i
218.9 −1.42642 + 0.823544i −1.30685 2.26354i 0.356451 0.617391i 2.48483i 3.72825 + 2.15250i −4.08112 2.35624i 2.11996i −1.91573 + 3.31814i 2.04637 + 3.54442i
218.10 −1.25179 + 0.722722i −0.0126133 0.0218468i 0.0446555 0.0773455i 1.27994i 0.0315784 + 0.0182318i −1.82792 1.05535i 2.76180i 1.49968 2.59753i −0.925042 1.60222i
218.11 −0.697346 + 0.402613i 0.727281 + 1.25969i −0.675806 + 1.17053i 3.57617i −1.01433 0.585626i −1.02623 0.592496i 2.69880i 0.442123 0.765780i 1.43981 + 2.49383i
218.12 −0.624164 + 0.360361i −1.13079 1.95859i −0.740280 + 1.28220i 4.09116i 1.41160 + 0.814986i 3.77469 + 2.17932i 2.50852i −1.05738 + 1.83143i 1.47430 + 2.55356i
218.13 −0.565608 + 0.326554i −0.481134 0.833348i −0.786725 + 1.36265i 0.172354i 0.544266 + 0.314232i 2.14068 + 1.23592i 2.33385i 1.03702 1.79617i 0.0562829 + 0.0974848i
218.14 −0.544301 + 0.314252i 1.36076 + 2.35690i −0.802491 + 1.38996i 1.23905i −1.48132 0.855242i −3.53624 2.04165i 2.26575i −2.20333 + 3.81627i −0.389373 0.674413i
218.15 −0.453928 + 0.262075i 0.409673 + 0.709575i −0.862633 + 1.49412i 4.38147i −0.371924 0.214730i 2.35938 + 1.36219i 1.95260i 1.16434 2.01669i −1.14828 1.98887i
218.16 −0.441172 + 0.254711i −1.38443 2.39789i −0.870245 + 1.50731i 1.59074i 1.22154 + 0.705256i 0.253842 + 0.146556i 1.90549i −2.33327 + 4.04134i −0.405179 0.701791i
218.17 −0.244810 + 0.141341i 1.61186 + 2.79182i −0.960045 + 1.66285i 0.0582096i −0.789198 0.455643i 2.44759 + 1.41312i 1.10814i −3.69618 + 6.40198i 0.00822740 + 0.0142503i
218.18 −0.117721 + 0.0679661i 0.0351399 + 0.0608641i −0.990761 + 1.71605i 1.52124i −0.00827339 0.00477665i −2.01496 1.16334i 0.541217i 1.49753 2.59380i 0.103393 + 0.179081i
218.19 0.228638 0.132004i −0.254003 0.439946i −0.965150 + 1.67169i 2.92638i −0.116149 0.0670588i −4.12569 2.38197i 1.03763i 1.37097 2.37458i 0.386294 + 0.669081i
218.20 0.514518 0.297057i −0.848877 1.47030i −0.823514 + 1.42637i 2.73915i −0.873525 0.504330i −2.55168 1.47321i 2.16675i 0.0588173 0.101874i −0.813684 1.40934i
See all 68 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 342.34 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.r.a 68
13.e even 6 1 inner 403.2.r.a 68
13.f odd 12 1 5239.2.a.q 34
13.f odd 12 1 5239.2.a.r 34

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.r.a 68 1.a even 1 1 trivial
403.2.r.a 68 13.e even 6 1 inner
5239.2.a.q 34 13.f odd 12 1
5239.2.a.r 34 13.f odd 12 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