# Properties

 Label 403.2.bz.a Level 403 Weight 2 Character orbit 403.bz Analytic conductor 3.218 Analytic rank 0 Dimension 288 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.bz (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$288$$ Relative dimension: $$36$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$288q - 16q^{3} + 60q^{4} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$288q - 16q^{3} + 60q^{4} + 16q^{9} - 42q^{10} - 58q^{12} - 7q^{13} - 26q^{14} - 84q^{16} - 30q^{17} + 44q^{22} + 44q^{23} + 124q^{25} - 21q^{26} + 2q^{27} - 44q^{29} - 204q^{30} - 34q^{35} + 134q^{36} + 38q^{38} - 11q^{39} - 84q^{40} - 64q^{42} + 60q^{43} - 42q^{48} - 50q^{49} + 14q^{51} - 13q^{52} - 12q^{53} - 92q^{55} - 56q^{56} + 52q^{61} + 18q^{62} + 174q^{64} - 46q^{65} - 128q^{66} - 140q^{68} - 226q^{69} + 82q^{74} + 46q^{75} + 144q^{77} - 95q^{78} + 30q^{79} + 104q^{81} - 2q^{82} + 30q^{87} + 52q^{88} - 328q^{90} + 78q^{91} + 152q^{92} - 60q^{94} - 150q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
38.1 −2.60476 + 0.846338i −2.83870 + 0.603383i 4.45046 3.23345i 2.15413 + 1.24368i 6.88346 3.97417i −0.647070 + 1.45334i −5.63613 + 7.75747i 4.95348 2.20543i −6.66356 1.41638i
38.2 −2.44210 + 0.793486i 0.151621 0.0322281i 3.71619 2.69997i −1.21524 0.701620i −0.344702 + 0.199014i 0.456028 1.02426i −3.91432 + 5.38760i −2.71869 + 1.21044i 3.52447 + 0.749148i
38.3 −2.39524 + 0.778262i 1.87376 0.398279i 3.51347 2.55268i 1.54681 + 0.893054i −4.17813 + 2.41225i 1.14729 2.57686i −3.46827 + 4.77366i 0.611696 0.272345i −4.40003 0.935255i
38.4 −2.29939 + 0.747117i 0.581957 0.123699i 3.11098 2.26026i 2.81265 + 1.62389i −1.24573 + 0.719222i −0.764399 + 1.71687i −2.62248 + 3.60953i −2.41726 + 1.07624i −7.68063 1.63257i
38.5 −2.19756 + 0.714031i −2.27192 + 0.482911i 2.70141 1.96269i −2.61081 1.50735i 4.64786 2.68345i 1.79222 4.02540i −1.81875 + 2.50330i 2.18776 0.974055i 6.81372 + 1.44830i
38.6 −2.08918 + 0.678815i 0.327283 0.0695662i 2.28584 1.66076i −2.01582 1.16383i −0.636531 + 0.367501i −1.69303 + 3.80261i −1.06581 + 1.46696i −2.63836 + 1.17467i 5.00143 + 1.06309i
38.7 −1.66585 + 0.541266i 3.00668 0.639090i 0.864039 0.627761i 1.65657 + 0.956423i −4.66275 + 2.69204i −0.104310 + 0.234285i 0.959529 1.32068i 5.89106 2.62287i −3.27727 0.696606i
38.8 −1.65648 + 0.538222i −2.26380 + 0.481186i 0.836200 0.607535i −0.549117 0.317033i 3.49095 2.01550i −1.65725 + 3.72223i 0.989361 1.36174i 2.15263 0.958411i 1.08023 + 0.229611i
38.9 −1.60645 + 0.521968i 1.69082 0.359396i 0.690208 0.501465i −1.67367 0.966293i −2.52864 + 1.45991i 0.582041 1.30729i 1.13865 1.56721i −0.0109161 + 0.00486016i 3.19305 + 0.678703i
38.10 −1.52508 + 0.495529i −2.41071 + 0.512413i 0.462291 0.335874i −0.561077 0.323938i 3.42262 1.97605i −0.0399925 + 0.0898246i 1.34651 1.85331i 2.80834 1.25035i 1.01621 + 0.216002i
38.11 −1.09973 + 0.357324i 0.0227929 0.00484479i −0.536308 + 0.389651i 2.94283 + 1.69904i −0.0233349 + 0.0134724i −1.47906 + 3.32203i 1.80990 2.49112i −2.74014 + 1.21999i −3.84343 0.816946i
38.12 −1.08301 + 0.351891i −0.146913 + 0.0312274i −0.568950 + 0.413366i 1.21706 + 0.702668i 0.148120 0.0855171i 1.11912 2.51358i 1.80939 2.49042i −2.72003 + 1.21103i −1.56535 0.332725i
38.13 −0.802861 + 0.260866i −0.963024 + 0.204697i −1.04150 + 0.756693i −3.54704 2.04789i 0.719776 0.415563i 0.345954 0.777024i 1.63118 2.24512i −1.85512 + 0.825954i 3.38201 + 0.718868i
38.14 −0.697365 + 0.226588i 0.705648 0.149990i −1.18306 + 0.859542i −0.658335 0.380090i −0.458108 + 0.264489i 0.753519 1.69243i 1.49225 2.05391i −2.26519 + 1.00853i 0.545224 + 0.115891i
38.15 −0.614391 + 0.199628i −3.29662 + 0.700717i −1.28041 + 0.930272i 3.41239 + 1.97014i 1.88553 1.08861i 0.675482 1.51716i 1.36039 1.87242i 7.63604 3.39978i −2.48984 0.529231i
38.16 −0.245789 + 0.0798617i −1.67915 + 0.356913i −1.56400 + 1.13631i −0.442541 0.255501i 0.384212 0.221825i −0.641731 + 1.44135i 0.597478 0.822358i −0.0484946 + 0.0215912i 0.129176 + 0.0274573i
38.17 −0.181578 + 0.0589982i 2.72617 0.579466i −1.58854 + 1.15415i −2.54353 1.46851i −0.460825 + 0.266057i 0.826174 1.85562i 0.444794 0.612207i 4.35561 1.93924i 0.548488 + 0.116585i
38.18 −0.0579861 + 0.0188408i 1.89240 0.402242i −1.61503 + 1.17339i −2.24387 1.29550i −0.102154 + 0.0589787i −2.03436 + 4.56926i 0.143216 0.197120i 0.678735 0.302192i 0.154522 + 0.0328446i
38.19 0.0579861 0.0188408i 1.89240 0.402242i −1.61503 + 1.17339i 2.24387 + 1.29550i 0.102154 0.0589787i 2.03436 4.56926i −0.143216 + 0.197120i 0.678735 0.302192i 0.154522 + 0.0328446i
38.20 0.181578 0.0589982i 2.72617 0.579466i −1.58854 + 1.15415i 2.54353 + 1.46851i 0.460825 0.266057i −0.826174 + 1.85562i −0.444794 + 0.612207i 4.35561 1.93924i 0.548488 + 0.116585i
See next 80 embeddings (of 288 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 350.36 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.b even 2 1 inner
31.g even 15 1 inner
403.bz even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bz.a 288
13.b even 2 1 inner 403.2.bz.a 288
31.g even 15 1 inner 403.2.bz.a 288
403.bz even 30 1 inner 403.2.bz.a 288

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bz.a 288 1.a even 1 1 trivial
403.2.bz.a 288 13.b even 2 1 inner
403.2.bz.a 288 31.g even 15 1 inner
403.2.bz.a 288 403.bz even 30 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