# Properties

 Label 403.2.bl.b Level 403 Weight 2 Character orbit 403.bl Analytic conductor 3.218 Analytic rank 0 Dimension 280 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.bl (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$280$$ Relative dimension: $$35$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$280q + q^{2} + 39q^{4} - 8q^{5} - 2q^{7} + 12q^{8} + 29q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$280q + q^{2} + 39q^{4} - 8q^{5} - 2q^{7} + 12q^{8} + 29q^{9} - 21q^{10} - q^{11} - 16q^{12} - 54q^{14} - 27q^{15} + 31q^{16} + 2q^{17} - 10q^{18} + 5q^{19} - 3q^{20} + 68q^{21} - 39q^{22} - 7q^{23} + 48q^{24} + 200q^{25} + 6q^{26} - 78q^{27} + 30q^{28} - 16q^{29} - 66q^{30} - 62q^{31} - 56q^{32} - 20q^{33} - 126q^{34} - 37q^{35} - 140q^{36} - 36q^{37} + 4q^{38} + 28q^{39} - 158q^{40} - 4q^{41} + 16q^{42} - 16q^{43} + 42q^{44} - 46q^{45} - 29q^{46} + 8q^{47} - 36q^{48} + 43q^{49} - 5q^{50} - 134q^{51} - q^{52} + 8q^{53} + 44q^{54} - 55q^{55} - 42q^{56} + 140q^{57} + 38q^{58} - 23q^{59} + 38q^{60} + 40q^{61} + 19q^{62} - 146q^{63} - 68q^{64} + 2q^{65} + 6q^{66} + 46q^{67} + 86q^{68} - 32q^{69} - 4q^{70} + 60q^{71} + 73q^{72} - 12q^{73} - 44q^{74} + 16q^{75} - 70q^{76} + 10q^{77} - 142q^{78} + 134q^{79} - 72q^{80} - 18q^{81} + 28q^{82} - 88q^{83} + 81q^{84} - 69q^{85} + 188q^{86} - 28q^{87} + 42q^{88} + 12q^{89} + 22q^{90} + 67q^{91} - 324q^{92} - 25q^{93} - 62q^{94} + 16q^{95} + 276q^{96} + 16q^{97} + 76q^{98} - 60q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −2.44344 1.08789i −0.216774 + 2.06246i 3.44862 + 3.83009i −2.87099 2.77340 4.80367i 2.21936 + 2.46485i −2.60675 8.02276i −1.27232 0.270439i 7.01509 + 3.12332i
16.2 −2.39574 1.06665i −0.206334 + 1.96314i 3.26354 + 3.62453i 1.57328 2.58831 4.48308i −2.63168 2.92278i −2.33171 7.17627i −0.876907 0.186392i −3.76916 1.67814i
16.3 −2.24423 0.999195i 0.330998 3.14923i 2.69991 + 2.99855i 2.52374 −3.88953 + 6.73687i 2.19826 + 2.44142i −1.54481 4.75442i −6.87367 1.46104i −5.66385 2.52171i
16.4 −2.07437 0.923570i 0.0811184 0.771790i 2.11178 + 2.34536i 2.89709 −0.881072 + 1.52606i −1.22242 1.35763i −0.811140 2.49643i 2.34536 + 0.498522i −6.00965 2.67567i
16.5 −2.04851 0.912054i −0.0396616 + 0.377355i 2.02628 + 2.25041i 0.301769 0.425415 0.736841i 2.74290 + 3.04630i −0.712488 2.19281i 2.79362 + 0.593802i −0.618177 0.275230i
16.6 −1.74019 0.774780i −0.239952 + 2.28299i 1.08970 + 1.21023i 3.51072 2.18638 3.78692i 1.97153 + 2.18961i 0.218662 + 0.672972i −2.22004 0.471884i −6.10930 2.72004i
16.7 −1.58921 0.707561i 0.119963 1.14138i 0.686679 + 0.762634i −0.807451 −0.998239 + 1.72900i 0.398118 + 0.442154i 0.523470 + 1.61107i 1.64610 + 0.349889i 1.28321 + 0.571321i
16.8 −1.58448 0.705458i −0.208907 + 1.98762i 0.674658 + 0.749284i −0.120273 1.73319 3.00197i −1.90609 2.11693i 0.531542 + 1.63592i −0.972535 0.206719i 0.190570 + 0.0848472i
16.9 −1.41616 0.630515i −0.113478 + 1.07967i 0.269698 + 0.299530i −3.63191 0.841454 1.45744i −0.969238 1.07645i 0.764987 + 2.35439i 1.78162 + 0.378696i 5.14337 + 2.28997i
16.10 −1.40769 0.626744i 0.286757 2.72831i 0.250522 + 0.278233i −3.15693 −2.11362 + 3.66089i 2.27878 + 2.53085i 0.774057 + 2.38230i −4.42699 0.940986i 4.44398 + 1.97859i
16.11 −1.18678 0.528387i 0.262692 2.49935i −0.209017 0.232137i −0.0704053 −1.63238 + 2.82736i −3.21641 3.57218i 0.928279 + 2.85695i −3.24328 0.689381i 0.0835553 + 0.0372012i
16.12 −1.17029 0.521045i −0.352762 + 3.35630i −0.240178 0.266745i −1.20807 2.16162 3.74403i 1.07211 + 1.19070i 0.933816 + 2.87399i −8.20588 1.74421i 1.41379 + 0.629461i
16.13 −0.657025 0.292526i −0.0382007 + 0.363455i −0.992151 1.10190i −0.138184 0.131419 0.227624i −1.74853 1.94194i 0.774026 + 2.38221i 2.80380 + 0.595967i 0.0907902 + 0.0404224i
16.14 −0.654441 0.291376i 0.0273063 0.259802i −0.994868 1.10491i 4.00393 −0.0935704 + 0.162069i 0.155298 + 0.172476i 0.771881 + 2.37561i 2.86769 + 0.609547i −2.62034 1.16665i
16.15 −0.596810 0.265717i 0.0996818 0.948409i −1.05268 1.16913i 0.954231 −0.311499 + 0.539532i 2.48894 + 2.76425i 0.721351 + 2.22009i 2.04490 + 0.434657i −0.569494 0.253555i
16.16 −0.187032 0.0832719i −0.0175618 + 0.167089i −1.31021 1.45514i −4.19195 0.0171984 0.0297885i 1.71571 + 1.90549i 0.250410 + 0.770684i 2.90683 + 0.617866i 0.784027 + 0.349072i
16.17 −0.0396375 0.0176478i −0.210379 + 2.00162i −1.33700 1.48489i 0.511564 0.0436630 0.0756265i 1.64562 + 1.82764i 0.0536061 + 0.164983i −1.02778 0.218461i −0.0202771 0.00902795i
16.18 0.0464992 + 0.0207028i 0.316586 3.01212i −1.33653 1.48436i 3.26010 0.0770802 0.133507i −0.734123 0.815327i −0.0628748 0.193509i −6.03818 1.28346i 0.151592 + 0.0674932i
16.19 0.192109 + 0.0855325i 0.298908 2.84392i −1.30867 1.45343i −1.10354 0.300670 0.520777i 0.556681 + 0.618257i −0.257059 0.791146i −5.06408 1.07640i −0.212000 0.0943885i
16.20 0.376809 + 0.167766i −0.276755 + 2.63315i −1.22442 1.35986i −2.68659 −0.546037 + 0.945764i −0.664950 0.738502i −0.488154 1.50238i −3.92245 0.833743i −1.01233 0.450718i
See next 80 embeddings (of 280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 380.35 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
31.d even 5 1 inner
403.bl even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bl.b 280
13.c even 3 1 inner 403.2.bl.b 280
31.d even 5 1 inner 403.2.bl.b 280
403.bl even 15 1 inner 403.2.bl.b 280

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bl.b 280 1.a even 1 1 trivial
403.2.bl.b 280 13.c even 3 1 inner
403.2.bl.b 280 31.d even 5 1 inner
403.2.bl.b 280 403.bl even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{280} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database