# Properties

 Label 403.2.k.e Level 403 Weight 2 Character orbit 403.k 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.k (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$68$$ Relative dimension: $$17$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$68q - 3q^{2} - 2q^{3} - 23q^{4} + 12q^{5} + 4q^{6} + 2q^{7} - 3q^{8} - 23q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$68q - 3q^{2} - 2q^{3} - 23q^{4} + 12q^{5} + 4q^{6} + 2q^{7} - 3q^{8} - 23q^{9} - 13q^{10} - 5q^{11} - 28q^{12} - 17q^{13} - 3q^{14} - 14q^{15} + 9q^{16} + 12q^{17} - 19q^{18} - 4q^{19} - 53q^{20} - 13q^{21} - 14q^{22} - 9q^{23} + 2q^{24} + 96q^{25} + 12q^{26} + 25q^{27} - 25q^{28} - 78q^{30} - 2q^{31} + 76q^{32} + 29q^{33} - 15q^{34} - 36q^{35} + 52q^{36} + 24q^{37} - 19q^{38} + 3q^{39} - 12q^{40} - 40q^{41} + 11q^{42} - 22q^{43} + 4q^{44} + 63q^{45} - 24q^{46} + 3q^{47} + 68q^{48} + 33q^{49} - 76q^{50} - 59q^{51} - 13q^{52} - q^{53} + 18q^{54} - 22q^{55} + 78q^{56} - 16q^{57} + 5q^{58} - 18q^{59} + 43q^{60} - 32q^{61} - 39q^{62} + 20q^{63} + 23q^{64} + 2q^{65} + 11q^{66} + 114q^{67} + 98q^{68} - 46q^{69} + 32q^{70} - 2q^{71} + 28q^{72} + 10q^{73} - 43q^{74} - 12q^{75} - 35q^{76} - 3q^{77} - 6q^{78} - 10q^{79} + 68q^{80} - 54q^{81} - 80q^{82} - 22q^{83} - 14q^{84} - 50q^{85} - 66q^{86} + 76q^{87} - 34q^{88} - 10q^{89} - 63q^{90} - 8q^{91} - 64q^{92} - 16q^{93} + 30q^{94} + 15q^{95} + 34q^{96} - 7q^{97} + 138q^{98} - 48q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
66.1 −0.831221 2.55823i −0.796905 + 2.45262i −4.23560 + 3.07735i −0.866460 6.93679 1.08832 0.790714i 7.04097 + 5.11556i −2.95324 2.14566i 0.720220 + 2.21661i
66.2 −0.717286 2.20758i 0.693769 2.13520i −2.74087 + 1.99136i 3.88389 −5.21126 1.00783 0.732228i 2.60631 + 1.89360i −1.65072 1.19932i −2.78586 8.57400i
66.3 −0.651553 2.00527i −0.602423 + 1.85407i −1.97856 + 1.43751i 2.75944 4.11042 −1.61436 + 1.17290i 0.760164 + 0.552291i −0.647600 0.470509i −1.79792 5.53344i
66.4 −0.537701 1.65487i 0.587091 1.80688i −0.831453 + 0.604086i −2.54418 −3.30584 −1.97022 + 1.43145i −1.36868 0.994405i −0.493085 0.358248i 1.36801 + 4.21030i
66.5 −0.471037 1.44970i 0.0410411 0.126311i −0.261732 + 0.190160i 1.10606 −0.202446 0.346937 0.252065i −2.06742 1.50207i 2.41278 + 1.75299i −0.520995 1.60346i
66.6 −0.285917 0.879963i −0.982047 + 3.02243i 0.925448 0.672377i −3.20777 2.94041 3.53175 2.56597i −2.35335 1.70981i −5.74362 4.17298i 0.917156 + 2.82271i
66.7 −0.100028 0.307853i 0.126453 0.389181i 1.53327 1.11398i 0.747578 −0.132459 2.14953 1.56172i −1.02006 0.741120i 2.29158 + 1.66493i −0.0747784 0.230144i
66.8 −0.0877385 0.270031i 0.851349 2.62018i 1.55282 1.12819i 0.880734 −0.782228 −3.90316 + 2.83581i −0.900292 0.654101i −3.71351 2.69802i −0.0772743 0.237826i
66.9 0.139693 + 0.429932i −0.0999964 + 0.307757i 1.45271 1.05545i 4.04679 −0.146284 −3.32806 + 2.41798i 1.38815 + 1.00855i 2.34234 + 1.70181i 0.565310 + 1.73984i
66.10 0.165656 + 0.509838i 0.987660 3.03970i 1.38554 1.00665i −3.75225 1.71337 1.77021 1.28613i 1.61014 + 1.16984i −5.83728 4.24103i −0.621584 1.91304i
66.11 0.288452 + 0.887764i −0.712637 + 2.19327i 0.913113 0.663415i 1.71724 −2.15267 1.85539 1.34802i 2.36270 + 1.71660i −1.87554 1.36266i 0.495341 + 1.52450i
66.12 0.455627 + 1.40228i 0.322084 0.991272i −0.140749 + 0.102260i −2.15722 1.53679 0.0381524 0.0277193i 2.17817 + 1.58253i 1.54817 + 1.12481i −0.982888 3.02502i
66.13 0.545494 + 1.67886i −0.170573 + 0.524968i −0.902965 + 0.656042i −0.444421 −0.974393 3.36537 2.44508i 1.26228 + 0.917100i 2.18055 + 1.58427i −0.242429 0.746119i
66.14 0.614130 + 1.89010i −0.776144 + 2.38873i −1.57728 + 1.14596i 0.0289380 −4.99158 −3.33658 + 2.42417i 0.0810008 + 0.0588505i −2.67656 1.94463i 0.0177717 + 0.0546957i
66.15 0.767174 + 2.36112i 0.517972 1.59415i −3.36829 + 2.44720i 3.20234 4.16136 −2.60176 + 1.89029i −4.34523 3.15699i 0.154019 + 0.111901i 2.45675 + 7.56111i
66.16 0.773013 + 2.37909i 0.384456 1.18324i −3.44449 + 2.50257i −2.86335 3.11221 −1.19324 + 0.866941i −4.56892 3.31951i 1.17481 + 0.853551i −2.21341 6.81217i
66.17 0.860293 + 2.64771i −0.871148 + 2.68112i −4.65223 + 3.38004i 2.69869 −7.84827 2.17586 1.58085i −8.44709 6.13717i −4.00245 2.90795i 2.32167 + 7.14536i
157.1 −2.24707 1.63259i −0.470599 + 0.341910i 1.76594 + 5.43499i 0.224436 1.61567 0.946374 + 2.91264i 3.18833 9.81268i −0.822490 + 2.53136i −0.504323 0.366412i
157.2 −1.85614 1.34856i 0.0303627 0.0220598i 1.00860 + 3.10414i −4.26015 −0.0861064 −0.501866 1.54459i 0.896070 2.75782i −0.926616 + 2.85183i 7.90744 + 5.74509i
157.3 −1.84627 1.34139i 1.79347 1.30303i 0.991339 + 3.05103i 4.23714 −5.05910 −0.0173329 0.0533453i 0.851924 2.62195i 0.591588 1.82072i −7.82289 5.68366i
See all 68 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 326.17 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.k.e 68
31.d even 5 1 inner 403.2.k.e 68

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.k.e 68 1.a even 1 1 trivial
403.2.k.e 68 31.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{68} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database