# Properties

 Label 403.2.bj.a Level 403 Weight 2 Character orbit 403.bj 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.bj (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.

 $$\operatorname{Tr}(f)(q) =$$ $$280q - 3q^{2} + 3q^{3} + 29q^{4} - 8q^{5} + 5q^{6} - q^{7} - 29q^{8} - 63q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$280q - 3q^{2} + 3q^{3} + 29q^{4} - 8q^{5} + 5q^{6} - q^{7} - 29q^{8} - 63q^{9} + 12q^{10} + 3q^{11} - 50q^{12} - 24q^{13} + 24q^{15} + 23q^{16} - 9q^{17} - 15q^{18} - 3q^{19} - 75q^{20} - 34q^{21} - 22q^{22} - 12q^{23} + 43q^{24} - 112q^{25} - 21q^{26} + 48q^{27} + 33q^{28} - 11q^{29} + 24q^{30} - 22q^{31} + q^{32} + 53q^{33} - 49q^{34} - 34q^{35} - 67q^{36} - 34q^{37} - 23q^{38} - 25q^{39} + 31q^{40} + 7q^{41} + 28q^{42} + 41q^{43} - 42q^{44} - q^{45} - 39q^{46} - 20q^{47} + 84q^{48} + 42q^{49} + 24q^{50} - 71q^{51} + 109q^{52} - 8q^{53} + 31q^{54} + 35q^{55} + 63q^{56} + 21q^{57} - 191q^{58} - 46q^{59} + 2q^{60} + 53q^{61} + 18q^{62} + 17q^{63} - 91q^{64} + 47q^{65} - 62q^{66} + 7q^{67} - 166q^{68} + 53q^{69} + 134q^{70} - 79q^{71} + 65q^{72} - 31q^{73} + 12q^{74} + 73q^{75} - 46q^{76} - 80q^{77} + 114q^{78} - 142q^{79} + 87q^{80} - 27q^{81} - 9q^{82} + 94q^{83} + 85q^{84} - 45q^{85} - 34q^{86} - 11q^{87} + 61q^{88} + 75q^{89} + 43q^{90} - 104q^{91} - 294q^{92} + 26q^{93} - 57q^{94} - 101q^{95} + 24q^{96} + 46q^{97} - 146q^{98} + 12q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1 −2.71366 0.576807i 0.313803 0.965786i 5.20416 + 2.31704i −0.254988 + 0.441652i −1.40863 + 2.43981i 0.0372303 + 0.354222i −8.29696 6.02809i 1.59278 + 1.15722i 0.946699 1.05142i
100.2 −2.57899 0.548181i −0.805625 + 2.47946i 4.52359 + 2.01403i −1.73519 + 3.00543i 3.43689 5.95286i −0.217284 2.06732i −6.29611 4.57439i −3.07163 2.23167i 6.12254 6.79977i
100.3 −2.28137 0.484920i −0.170532 + 0.524843i 3.14241 + 1.39909i 0.529585 0.917267i 0.643553 1.11467i −0.0507599 0.482948i −2.71676 1.97384i 2.18067 + 1.58435i −1.65298 + 1.83582i
100.4 −2.24686 0.477586i −0.679969 + 2.09273i 2.99322 + 1.33267i 0.449810 0.779094i 2.52726 4.37734i 0.399570 + 3.80166i −2.37217 1.72348i −1.49011 1.08263i −1.38275 + 1.53570i
100.5 −2.23294 0.474625i 0.848869 2.61255i 2.93364 + 1.30614i 1.01379 1.75593i −3.13545 + 5.43076i −0.161425 1.53586i −2.23702 1.62529i −3.67779 2.67207i −3.09713 + 3.43971i
100.6 −2.14008 0.454887i −0.792588 + 2.43933i 2.54591 + 1.13351i 1.99812 3.46084i 2.80582 4.85982i −0.425169 4.04521i −1.39275 1.01189i −2.89510 2.10342i −5.85041 + 6.49754i
100.7 −2.08345 0.442850i 0.752630 2.31636i 2.31754 + 1.03184i −1.53370 + 2.65645i −2.59386 + 4.49271i 0.0462700 + 0.440229i −0.925131 0.672147i −2.37201 1.72337i 4.37179 4.85536i
100.8 −1.52609 0.324380i −0.102259 + 0.314720i 0.396636 + 0.176594i −0.0573847 + 0.0993931i 0.258145 0.447120i −0.116771 1.11100i 1.97641 + 1.43595i 2.33846 + 1.69899i 0.119815 0.133068i
100.9 −1.41577 0.300932i −0.586565 + 1.80526i 0.0867606 + 0.0386283i −2.18816 + 3.79001i 1.37370 2.37932i 0.150521 + 1.43211i 2.23074 + 1.62072i −0.487856 0.354448i 4.23848 4.70731i
100.10 −1.36969 0.291136i 0.233287 0.717984i −0.0358094 0.0159434i 1.52418 2.63996i −0.528561 + 0.915495i 0.0830158 + 0.789842i 2.31012 + 1.67840i 1.96597 + 1.42836i −2.85624 + 3.17218i
100.11 −1.34661 0.286231i 0.272316 0.838103i −0.0956586 0.0425900i −1.29583 + 2.24445i −0.606595 + 1.05065i −0.453578 4.31551i 2.34416 + 1.70313i 1.79879 + 1.30690i 2.38741 2.65149i
100.12 −1.10979 0.235894i −0.532074 + 1.63756i −0.651094 0.289886i 0.103550 0.179354i 0.976783 1.69184i 0.239412 + 2.27785i 2.49000 + 1.80909i 0.0285618 + 0.0207514i −0.157228 + 0.174619i
100.13 −0.881283 0.187322i 0.643229 1.97966i −1.08552 0.483305i −0.866187 + 1.50028i −0.937701 + 1.62414i 0.293789 + 2.79521i 2.32392 + 1.68843i −1.07824 0.783389i 1.04439 1.15991i
100.14 −0.661543 0.140615i −1.01562 + 3.12577i −1.40922 0.627427i −0.398234 + 0.689762i 1.11141 1.92502i −0.202627 1.92787i 1.93835 + 1.40829i −6.31190 4.58586i 0.360441 0.400310i
100.15 −0.528168 0.112266i 0.698235 2.14895i −1.56073 0.694883i 1.10173 1.90824i −0.610039 + 1.05662i −0.519865 4.94618i 1.62000 + 1.17700i −1.70339 1.23759i −0.796127 + 0.884188i
100.16 −0.247139 0.0525310i −0.639897 + 1.96940i −1.76877 0.787508i 0.388050 0.672122i 0.261598 0.453101i −0.286123 2.72228i 0.804576 + 0.584559i −1.04202 0.757070i −0.131209 + 0.145723i
100.17 −0.171917 0.0365421i 0.235059 0.723438i −1.79887 0.800909i 0.00796452 0.0137949i −0.0668467 + 0.115782i 0.484936 + 4.61385i 0.564372 + 0.410040i 1.95894 + 1.42325i −0.00187333 + 0.00208055i
100.18 0.0768429 + 0.0163335i −0.788714 + 2.42741i −1.82145 0.810963i 1.96469 3.40295i −0.100255 + 0.173647i 0.299883 + 2.85320i −0.253832 0.184420i −2.84321 2.06571i 0.206555 0.229402i
100.19 0.194455 + 0.0413328i 0.0384921 0.118467i −1.79099 0.797399i −0.998333 + 1.72916i 0.0123816 0.0214455i −0.177091 1.68490i −0.636973 0.462788i 2.41450 + 1.75424i −0.265602 + 0.294981i
100.20 0.208786 + 0.0443788i 0.200559 0.617258i −1.78547 0.794942i 1.97676 3.42385i 0.0692671 0.119974i 0.0421544 + 0.401072i −0.682872 0.496136i 2.08627 + 1.51576i 0.564667 0.627126i
See next 80 embeddings (of 280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 386.35 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.bj 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.bj.a 280
13.c even 3 1 403.2.bk.a yes 280
31.g even 15 1 403.2.bk.a yes 280
403.bj even 15 1 inner 403.2.bj.a 280

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bj.a 280 1.a even 1 1 trivial
403.2.bj.a 280 403.bj even 15 1 inner
403.2.bk.a yes 280 13.c even 3 1
403.2.bk.a yes 280 31.g even 15 1

## 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