# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.bs (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.

 $$\operatorname{Tr}(f)(q) =$$ $$288q - 9q^{2} - q^{3} - 39q^{4} - 42q^{6} - 15q^{7} + 31q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$288q - 9q^{2} - q^{3} - 39q^{4} - 42q^{6} - 15q^{7} + 31q^{9} + 3q^{10} - 18q^{11} - 46q^{12} - q^{13} - 32q^{14} + 18q^{15} + 21q^{16} - 15q^{17} - 15q^{19} + 51q^{20} - 10q^{22} - 4q^{23} - 51q^{24} - 296q^{25} - 6q^{26} - 52q^{27} + 21q^{28} + q^{29} + 60q^{30} + 138q^{32} + 69q^{33} - 10q^{35} + 128q^{36} - 18q^{37} + 32q^{38} - 14q^{39} + 60q^{40} - 15q^{41} - 49q^{42} - 36q^{43} + 6q^{45} - 69q^{46} + 21q^{48} - 23q^{49} + 117q^{50} + 8q^{51} + 26q^{52} - 48q^{53} + 75q^{54} + 46q^{55} - 98q^{56} - 21q^{58} - 105q^{59} - 74q^{61} - 3q^{62} - 90q^{63} + 90q^{64} + 89q^{65} - 8q^{66} + 6q^{67} - 182q^{68} + 29q^{69} + 3q^{71} - 183q^{72} - 53q^{74} - 38q^{75} + 144q^{76} - 128q^{78} - 72q^{79} - 72q^{80} + 11q^{81} - 11q^{82} - 33q^{84} + 72q^{85} - 18q^{87} - 14q^{88} + 81q^{89} - 34q^{90} - 48q^{91} + 8q^{92} + 72q^{93} - 6q^{94} + 141q^{97} + 96q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −0.552779 + 2.60062i −0.533122 0.592092i −4.63058 2.06167i 1.27621i 1.83451 1.05915i 1.08933 2.44667i 4.79578 6.60083i 0.247231 2.35225i −3.31895 0.705465i
4.2 −0.525326 + 2.47147i 0.803900 + 0.892821i −4.00509 1.78318i 0.793316i −2.62889 + 1.51779i −0.479028 + 1.07591i 3.54075 4.87343i 0.162711 1.54809i −1.96065 0.416750i
4.3 −0.518746 + 2.44051i −1.86014 2.06590i −3.85989 1.71854i 2.00004i 6.00678 3.46802i −1.59632 + 3.58539i 3.26332 4.49158i −0.494216 + 4.70215i −4.88112 1.03751i
4.4 −0.502235 + 2.36283i 0.181860 + 0.201976i −3.50363 1.55992i 2.70808i −0.568571 + 0.328265i −0.369305 + 0.829472i 2.60573 3.58648i 0.305864 2.91010i 6.39874 + 1.36009i
4.5 −0.490829 + 2.30917i 2.17363 + 2.41407i −3.26425 1.45334i 1.85570i −6.64136 + 3.83439i −1.05002 + 2.35838i 2.18296 3.00459i −0.789440 + 7.51102i 4.28511 + 0.910829i
4.6 −0.431958 + 2.03220i 1.72231 + 1.91282i −2.11617 0.942181i 3.56131i −4.63120 + 2.67382i 1.47626 3.31574i 0.386433 0.531879i −0.378938 + 3.60536i −7.23732 1.53834i
4.7 −0.364510 + 1.71488i −0.797644 0.885874i −0.980870 0.436711i 3.57659i 1.80992 1.04496i 0.0329582 0.0740253i −0.954559 + 1.31384i 0.165050 1.57034i −6.13344 1.30370i
4.8 −0.360928 + 1.69803i 0.922989 + 1.02508i −0.925954 0.412261i 3.21206i −2.07376 + 1.19728i 2.07463 4.65970i −1.00652 + 1.38535i 0.114699 1.09128i 5.45418 + 1.15932i
4.9 −0.350844 + 1.65059i −1.46122 1.62285i −0.774264 0.344725i 3.64318i 3.19131 1.84250i −1.03720 + 2.32958i −1.14309 + 1.57333i −0.184888 + 1.75909i 6.01339 + 1.27819i
4.10 −0.323679 + 1.52279i 0.859963 + 0.955086i −0.387033 0.172318i 2.26589i −1.73275 + 1.00040i −1.51411 + 3.40074i −1.44246 + 1.98538i 0.140933 1.34089i −3.45048 0.733422i
4.11 −0.319792 + 1.50450i −0.741178 0.823162i −0.334172 0.148783i 1.09342i 1.47547 0.851865i 1.19871 2.69235i −1.47745 + 2.03354i 0.185335 1.76335i 1.64506 + 0.349668i
4.12 −0.254380 + 1.19677i −1.60706 1.78482i 0.459552 + 0.204606i 0.244018i 2.54482 1.46925i −0.0246816 + 0.0554357i −1.80008 + 2.47760i −0.289361 + 2.75309i −0.292033 0.0620735i
4.13 −0.230096 + 1.08252i 0.418891 + 0.465225i 0.708195 + 0.315309i 1.86982i −0.599999 + 0.346409i −2.06505 + 4.63818i −1.80528 + 2.48476i 0.272620 2.59381i 2.02411 + 0.430238i
4.14 −0.129772 + 0.610527i −0.324888 0.360825i 1.47119 + 0.655015i 0.658726i 0.262455 0.151528i 0.662294 1.48754i −1.32458 + 1.82312i 0.288943 2.74911i 0.402170 + 0.0854839i
4.15 −0.117135 + 0.551075i 1.94238 + 2.15723i 1.53713 + 0.684373i 0.526119i −1.41631 + 0.817710i 0.000659214 0.00148062i −1.21949 + 1.67849i −0.567221 + 5.39675i 0.289931 + 0.0616267i
4.16 −0.0758408 + 0.356803i 1.47267 + 1.63557i 1.70553 + 0.759353i 1.67783i −0.695264 + 0.401411i 0.425932 0.956658i −0.829106 + 1.14117i −0.192734 + 1.83374i 0.598654 + 0.127248i
4.17 −0.0525799 + 0.247369i 0.690263 + 0.766614i 1.76866 + 0.787460i 3.00880i −0.225931 + 0.130441i −0.278018 + 0.624439i −0.585086 + 0.805302i 0.202350 1.92523i −0.744285 0.158203i
4.18 −0.00361138 + 0.0169902i −2.04909 2.27575i 1.82682 + 0.813351i 1.58318i 0.0460655 0.0265959i −0.684412 + 1.53721i −0.0408357 + 0.0562056i −0.666663 + 6.34287i −0.0268985 0.00571745i
4.19 0.00797121 0.0375016i −0.129846 0.144209i 1.82575 + 0.812875i 2.93327i −0.00644308 + 0.00371992i 1.61684 3.63149i 0.0901082 0.124023i 0.309649 2.94612i 0.110002 + 0.0233817i
4.20 0.0550517 0.258998i −1.89412 2.10363i 1.76304 + 0.784957i 4.39411i −0.649111 + 0.374764i 0.553765 1.24378i 0.611633 0.841841i −0.523995 + 4.98548i −1.13807 0.241903i
See next 80 embeddings (of 288 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 374.36 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.e even 6 1 inner
31.d even 5 1 inner
403.bs 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.bs.a 288
13.e even 6 1 inner 403.2.bs.a 288
31.d even 5 1 inner 403.2.bs.a 288
403.bs even 30 1 inner 403.2.bs.a 288

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bs.a 288 1.a even 1 1 trivial
403.2.bs.a 288 13.e even 6 1 inner
403.2.bs.a 288 31.d even 5 1 inner
403.2.bs.a 288 403.bs 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