# Properties

 Label 403.2.bt.a Level 403 Weight 2 Character orbit 403.bt Analytic conductor 3.218 Analytic rank 0 Dimension 280 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$280$$ Relative dimension: $$35$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$280q - 9q^{2} - 3q^{3} - 35q^{4} - 15q^{7} + 45q^{8} + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$280q - 9q^{2} - 3q^{3} - 35q^{4} - 15q^{7} + 45q^{8} + 24q^{9} + 3q^{10} - 8q^{12} - 6q^{13} + 4q^{14} - 45q^{15} + 23q^{16} + 27q^{17} + 45q^{18} - 15q^{19} - 12q^{20} - 76q^{21} + 41q^{22} - 10q^{23} - 33q^{24} + 96q^{25} + 9q^{26} - 24q^{27} - 32q^{28} + 13q^{29} + 36q^{30} + 2q^{31} - 141q^{32} - 93q^{33} - 9q^{34} - 43q^{35} - 194q^{36} + 3q^{37} - 49q^{38} + 50q^{39} - 75q^{40} - 15q^{41} + 17q^{42} + 33q^{43} + 18q^{44} - 15q^{45} - 9q^{46} - 59q^{48} + 3q^{49} + 36q^{50} + 47q^{51} - 56q^{52} + 12q^{53} - 33q^{54} - 5q^{55} - 50q^{56} - 105q^{57} - 3q^{58} - 15q^{59} + 90q^{60} - 57q^{61} - 72q^{62} + 201q^{63} + 13q^{64} - 43q^{65} + 22q^{66} - 71q^{68} - 7q^{69} - 42q^{71} + 90q^{72} + 9q^{73} - 113q^{74} + 45q^{75} + 14q^{76} - 24q^{77} + 61q^{78} + 54q^{79} + 30q^{80} + 106q^{81} + 16q^{82} + 54q^{83} + 60q^{84} + 18q^{85} + 84q^{86} + 42q^{87} - 98q^{88} - 99q^{89} + 11q^{90} + 60q^{91} + 266q^{92} - 104q^{93} + 33q^{94} - 120q^{95} + 204q^{96} - 50q^{97} - 15q^{98} - 168q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1 −0.558017 + 2.62526i −2.44380 + 0.519446i −4.75353 2.11641i −1.49782 + 0.864767i 6.70549i −0.203294 0.279810i 5.05356 6.95562i 2.96171 1.31864i −1.43443 4.41473i
82.2 −0.554356 + 2.60804i 0.790982 0.168128i −4.66747 2.07809i 2.97528 1.71778i 2.15612i −2.45564 3.37990i 4.87275 6.70677i −2.14325 + 0.954237i 2.83067 + 8.71192i
82.3 −0.526303 + 2.47606i 1.87677 0.398919i −4.02678 1.79284i −3.24110 + 1.87125i 4.85693i −1.04985 1.44500i 3.58267 4.93112i 0.622476 0.277144i −2.92753 9.01000i
82.4 −0.510705 + 2.40268i −1.82246 + 0.387376i −3.68496 1.64065i 3.06150 1.76756i 4.57663i 2.12591 + 2.92607i 2.93626 4.04141i 0.430673 0.191748i 2.68335 + 8.25849i
82.5 −0.475411 + 2.23663i 0.254115 0.0540139i −2.94943 1.31317i −0.204817 + 0.118251i 0.594042i 1.78231 + 2.45313i 1.65121 2.27270i −2.67898 + 1.19276i −0.167112 0.514319i
82.6 −0.448118 + 2.10823i 3.13762 0.666922i −2.41674 1.07600i 0.330706 0.190933i 6.91369i 0.412693 + 0.568023i 0.817700 1.12547i 6.65924 2.96488i 0.254336 + 0.782765i
82.7 −0.387542 + 1.82324i −0.271363 + 0.0576799i −1.34692 0.599689i 0.212568 0.122726i 0.517113i −2.14262 2.94906i −0.575865 + 0.792611i −2.67033 + 1.18891i 0.141380 + 0.435123i
82.8 −0.384687 + 1.80981i −2.40054 + 0.510251i −1.30034 0.578951i −0.272473 + 0.157312i 4.54082i −0.636632 0.876249i −0.627074 + 0.863094i 2.76162 1.22955i −0.179889 0.553641i
82.9 −0.309669 + 1.45688i 2.00373 0.425907i −0.199502 0.0888240i 3.03693 1.75337i 3.05108i 0.544135 + 0.748938i −1.55974 + 2.14679i 1.09292 0.486598i 1.61401 + 4.96740i
82.10 −0.285763 + 1.34441i 1.01586 0.215927i 0.101320 + 0.0451105i −1.34641 + 0.777351i 1.42743i 2.50350 + 3.44577i −1.70535 + 2.34722i −1.75529 + 0.781506i −0.660323 2.03226i
82.11 −0.226015 + 1.06331i −0.562471 + 0.119557i 0.747535 + 0.332824i 1.49545 0.863396i 0.625106i −0.503379 0.692842i −1.80078 + 2.47856i −2.43856 + 1.08572i 0.580069 + 1.78527i
82.12 −0.219621 + 1.03324i 2.09766 0.445872i 0.807749 + 0.359633i −3.00000 + 1.73205i 2.26530i −0.0849215 0.116884i −1.79076 + 2.46477i 1.46075 0.650366i −1.13075 3.48010i
82.13 −0.190379 + 0.895663i −3.22061 + 0.684561i 1.06112 + 0.472442i 0.521441 0.301054i 3.01491i 0.972860 + 1.33903i −1.70160 + 2.34205i 7.16305 3.18920i 0.170372 + 0.524350i
82.14 −0.136295 + 0.641215i −1.88071 + 0.399758i 1.43451 + 0.638685i −1.94312 + 1.12186i 1.26043i 1.23396 + 1.69841i −1.37568 + 1.89347i 0.636641 0.283451i −0.454518 1.39886i
82.15 −0.107110 + 0.503915i 2.39868 0.509854i 1.58463 + 0.705524i 0.193800 0.111891i 1.26334i −2.62876 3.61818i −1.13088 + 1.55652i 2.75306 1.22574i 0.0356253 + 0.109644i
82.16 −0.0617274 + 0.290405i −0.640694 + 0.136184i 1.74657 + 0.777621i −2.50432 + 1.44587i 0.194467i −1.38419 1.90518i −0.682654 + 0.939592i −2.34869 + 1.04571i −0.265302 0.816516i
82.17 −0.0166647 + 0.0784010i −2.19989 + 0.467602i 1.82122 + 0.810860i 3.63891 2.10092i 0.180266i −2.71023 3.73031i −0.188147 + 0.258963i 1.88024 0.837136i 0.104073 + 0.320305i
82.18 0.0316136 0.148730i 1.83195 0.389394i 1.80597 + 0.804069i 1.52776 0.882054i 0.284777i −0.353168 0.486094i 0.355432 0.489210i 0.463790 0.206493i −0.0828901 0.255110i
82.19 0.0633758 0.298160i −0.256430 + 0.0545059i 1.74221 + 0.775681i −2.38402 + 1.37641i 0.0799114i 0.229361 + 0.315689i 0.700029 0.963507i −2.67785 + 1.19226i 0.259302 + 0.798049i
82.20 0.0900306 0.423561i −1.34476 + 0.285838i 1.65579 + 0.737206i 1.67453 0.966790i 0.595322i 3.08797 + 4.25023i 0.970373 1.33560i −1.01396 + 0.451444i −0.258735 0.796306i
See next 80 embeddings (of 280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 400.35 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bt 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.bt.a yes 280
13.e even 6 1 403.2.bp.a 280
31.g even 15 1 403.2.bp.a 280
403.bt even 30 1 inner 403.2.bt.a yes 280

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bp.a 280 13.e even 6 1
403.2.bp.a 280 31.g even 15 1
403.2.bt.a yes 280 1.a even 1 1 trivial
403.2.bt.a yes 280 403.bt 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