# Properties

 Label 403.2.k.d Level 403 Weight 2 Character orbit 403.k Analytic conductor 3.218 Analytic rank 0 Dimension 48 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: $$48$$ Relative dimension: $$12$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 7q^{2} - 2q^{3} - 7q^{4} - 12q^{5} - 10q^{6} + 25q^{7} - 14q^{8} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 7q^{2} - 2q^{3} - 7q^{4} - 12q^{5} - 10q^{6} + 25q^{7} - 14q^{8} - 8q^{9} - 19q^{10} - 9q^{11} + 15q^{12} + 12q^{13} - 25q^{14} - 30q^{15} - 21q^{16} + 11q^{17} + 17q^{18} + 36q^{19} + 30q^{20} + 11q^{21} + 15q^{22} - 7q^{23} - 20q^{24} - 16q^{25} + 8q^{26} - 5q^{27} - 9q^{28} + 12q^{29} + 18q^{30} + 22q^{31} - 76q^{32} - 49q^{33} - 26q^{34} + 8q^{35} + 2q^{36} + 64q^{37} - 27q^{38} - 3q^{39} - 24q^{40} + 46q^{41} + 20q^{42} - 28q^{43} - 23q^{45} + 34q^{46} + 5q^{47} - 20q^{48} - 11q^{49} + 9q^{50} + 59q^{51} + 17q^{52} + 23q^{53} + 41q^{54} - 10q^{55} - 60q^{56} + 24q^{57} - 37q^{58} + 71q^{59} - 72q^{60} + 22q^{61} + 43q^{62} - 106q^{63} - 52q^{64} + 2q^{65} - 21q^{66} - 56q^{67} - 104q^{68} - 12q^{69} - 32q^{70} - 36q^{71} + 147q^{72} - 12q^{73} + 10q^{74} + 34q^{75} - 49q^{76} - 30q^{77} + 5q^{78} - 70q^{79} + q^{81} + 130q^{82} + 11q^{83} + 77q^{84} + 8q^{85} + 11q^{86} - 88q^{87} + 96q^{88} - 40q^{89} - 48q^{90} + 10q^{91} + 112q^{92} + 50q^{93} + 78q^{94} + 41q^{95} - 75q^{96} - 47q^{97} - 46q^{98} + 46q^{99} + O(q^{100})$$

## 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 −0.743865 2.28938i 0.00536078 0.0164988i −3.06990 + 2.23041i 0.430402 −0.0417597 3.00261 2.18153i 3.49493 + 2.53921i 2.42681 + 1.76318i −0.320161 0.985354i
66.2 −0.604780 1.86132i 0.854563 2.63007i −1.48072 + 1.07581i −1.67501 −5.41223 1.22914 0.893026i −0.268732 0.195245i −3.75996 2.73177i 1.01301 + 3.11773i
66.3 −0.599354 1.84462i −0.251540 + 0.774160i −1.42537 + 1.03559i −3.24431 1.57879 1.80822 1.31375i −0.373685 0.271498i 1.89100 + 1.37389i 1.94449 + 5.98453i
66.4 −0.365564 1.12509i −0.709225 + 2.18277i 0.485840 0.352983i 3.83735 2.71508 3.69636 2.68556i −2.48886 1.80827i −1.83443 1.33279i −1.40280 4.31737i
66.5 −0.288777 0.888765i 0.813772 2.50453i 0.911523 0.662261i 1.99869 −2.46094 0.592448 0.430438i −2.36388 1.71746i −3.18340 2.31288i −0.577176 1.77636i
66.6 −0.0757307 0.233075i 0.263551 0.811126i 1.56945 1.14027i −2.48983 −0.209012 2.14015 1.55491i −0.781154 0.567542i 1.83859 + 1.33581i 0.188556 + 0.580316i
66.7 −0.00310087 0.00954350i −0.517074 + 1.59139i 1.61795 1.17551i 1.33726 0.0167908 −0.925697 + 0.672558i −0.0324719 0.0235922i 0.161891 + 0.117620i −0.00414667 0.0127621i
66.8 0.247442 + 0.761548i −0.990504 + 3.04846i 1.09931 0.798693i −0.800250 −2.56664 −0.623500 + 0.452999i 2.17588 + 1.58087i −5.88494 4.27566i −0.198015 0.609429i
66.9 0.376127 + 1.15760i 0.106335 0.327267i 0.419469 0.304762i −2.41222 0.418839 −3.46077 + 2.51440i 2.47999 + 1.80182i 2.33125 + 1.69376i −0.907299 2.79238i
66.10 0.655076 + 2.01612i −0.156598 + 0.481958i −2.01757 + 1.46585i 2.06047 −1.07427 0.413610 0.300506i −0.846966 0.615357i 2.21929 + 1.61241i 1.34976 + 4.15414i
66.11 0.700746 + 2.15667i −0.672668 + 2.07026i −2.54216 + 1.84699i −3.69852 −4.93624 0.888338 0.645416i −2.09561 1.52255i −1.40643 1.02183i −2.59172 7.97650i
66.12 0.774730 + 2.38437i 0.754027 2.32066i −3.46700 + 2.51892i −0.580097 6.11748 3.63826 2.64335i −4.63551 3.36789i −2.38983 1.73632i −0.449419 1.38317i
157.1 −1.95579 1.42096i −0.875508 + 0.636094i 1.18794 + 3.65610i 2.12456 2.61618 −0.805291 2.47843i 1.37774 4.24024i −0.565152 + 1.73936i −4.15519 3.01892i
157.2 −1.78093 1.29392i 1.61180 1.17104i 0.879438 + 2.70663i −1.55702 −4.38572 0.929842 + 2.86176i 0.575440 1.77102i 0.299507 0.921787i 2.77294 + 2.01466i
157.3 −1.43812 1.04485i −1.43056 + 1.03937i 0.358428 + 1.10313i 0.805848 3.14330 −0.358493 1.10333i −0.461478 + 1.42028i 0.0391825 0.120591i −1.15890 0.841992i
157.4 −0.804029 0.584162i 1.86138 1.35237i −0.312815 0.962747i 1.77944 −2.28660 −0.394812 1.21511i −0.925111 + 2.84720i 0.708770 2.18137i −1.43072 1.03948i
157.5 −0.0675977 0.0491126i −2.70745 + 1.96708i −0.615877 1.89547i 0.141769 0.279626 −0.415935 1.28012i −0.103100 + 0.317309i 2.53385 7.79838i −0.00958327 0.00696265i
157.6 0.607778 + 0.441577i 0.0305971 0.0222301i −0.443630 1.36535i 2.26489 0.0284125 1.52318 + 4.68785i 0.797580 2.45470i −0.926609 + 2.85181i 1.37655 + 1.00012i
157.7 0.715177 + 0.519606i 2.11502 1.53665i −0.376547 1.15889i −3.34815 2.31107 −0.00198090 0.00609658i 0.879216 2.70595i 1.18497 3.64695i −2.39452 1.73972i
157.8 1.04202 + 0.757072i −0.137474 + 0.0998807i −0.105385 0.324343i −3.49603 −0.218868 1.29282 + 3.97888i 0.931770 2.86769i −0.918128 + 2.82571i −3.64293 2.64675i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 326.12 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.d 48
31.d even 5 1 inner 403.2.k.d 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.k.d 48 1.a even 1 1 trivial
403.2.k.d 48 31.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{48} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database