# Properties

 Label 403.2.bi.b Level 403 Weight 2 Character orbit 403.bi Analytic conductor 3.218 Analytic rank 0 Dimension 136 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$136$$ Relative dimension: $$17$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$136q - 6q^{2} + 2q^{3} - 34q^{4} - 15q^{5} - 10q^{6} - 8q^{7} - 6q^{8} + 23q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$136q - 6q^{2} + 2q^{3} - 34q^{4} - 15q^{5} - 10q^{6} - 8q^{7} - 6q^{8} + 23q^{9} + 7q^{10} + 5q^{11} + 28q^{12} + 17q^{13} - 3q^{14} + 17q^{15} - 78q^{16} - 12q^{17} + 34q^{18} - 23q^{19} + 8q^{20} - 38q^{21} + 8q^{22} + 9q^{23} + 46q^{24} - 69q^{25} - 12q^{26} - 58q^{27} + 7q^{28} + 18q^{29} + 60q^{30} + 14q^{31} + 212q^{32} - 41q^{33} - 30q^{35} - 106q^{36} + 6q^{37} - 62q^{38} + 6q^{39} + 39q^{40} - 5q^{41} + 37q^{42} + q^{43} - 88q^{44} - 168q^{45} + 6q^{46} - 81q^{47} - 101q^{48} + 117q^{49} - 20q^{50} + 35q^{51} + 7q^{52} + 25q^{53} + 57q^{54} + 19q^{55} - 63q^{56} - 20q^{57} - 44q^{58} + 9q^{59} + 113q^{60} + 8q^{61} + 57q^{62} + 112q^{63} - 146q^{64} - 5q^{65} + 70q^{66} - 99q^{67} - 56q^{68} - 65q^{69} + 169q^{70} - q^{71} - 43q^{72} + 50q^{73} - 26q^{74} - 18q^{75} - 94q^{76} + 114q^{77} - 17q^{79} + 217q^{80} + 54q^{81} - 25q^{82} - 53q^{83} - 22q^{84} + 2q^{85} - 54q^{86} - 28q^{87} + 70q^{88} - 101q^{89} + 165q^{90} - 4q^{91} + 64q^{92} + 61q^{93} - 90q^{94} - 33q^{95} - 217q^{96} + 73q^{97} + 33q^{98} - 15q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1 −0.778445 + 2.39581i −0.897108 + 0.996339i −3.51588 2.55444i −1.29523 + 2.24341i −1.68869 2.92489i −0.499609 + 4.75346i 4.78087 3.47351i 0.125696 + 1.19592i −4.36650 4.84949i
14.2 −0.775728 + 2.38745i −1.68623 + 1.87274i −3.48011 2.52845i 1.72799 2.99297i −3.16302 5.47852i 0.0507028 0.482404i 4.67440 3.39615i −0.350226 3.33218i 5.80510 + 6.44721i
14.3 −0.577531 + 1.77746i 1.86722 2.07376i −1.20778 0.877505i −0.00232995 + 0.00403559i 2.60765 + 4.51658i 0.383684 3.65051i −0.766732 + 0.557063i −0.500379 4.76079i −0.00582747 0.00647206i
14.4 −0.516329 + 1.58910i −0.00193962 + 0.00215417i −0.640602 0.465424i −0.479120 + 0.829859i −0.00242170 0.00419451i −0.214858 + 2.04424i −1.63317 + 1.18657i 0.313585 + 2.98356i −1.07134 1.18985i
14.5 −0.292848 + 0.901292i −1.55469 + 1.72666i 0.891466 + 0.647688i 0.318197 0.551134i −1.10094 1.90688i 0.111899 1.06465i −2.37819 + 1.72786i −0.250704 2.38529i 0.403549 + 0.448187i
14.6 −0.256563 + 0.789619i 0.840376 0.933332i 1.06036 + 0.770397i 2.12781 3.68547i 0.521367 + 0.903035i 0.110851 1.05468i −2.22375 + 1.61565i 0.148708 + 1.41487i 2.36420 + 2.62571i
14.7 −0.215030 + 0.661795i 0.258564 0.287164i 1.22630 + 0.890959i −1.89053 + 3.27449i 0.134445 + 0.232865i 0.319980 3.04441i −1.97924 + 1.43800i 0.297977 + 2.83506i −1.76052 1.95526i
14.8 0.0129642 0.0398998i 1.94913 2.16473i 1.61661 + 1.17454i −1.81601 + 3.14542i −0.0611034 0.105834i −0.401482 + 3.81985i 0.135703 0.0985943i −0.573359 5.45514i 0.101958 + 0.113236i
14.9 0.124680 0.383725i −0.653315 + 0.725580i 1.48633 + 1.07989i −0.174215 + 0.301749i 0.196968 + 0.341158i 0.256927 2.44450i 1.25253 0.910013i 0.213940 + 2.03550i 0.0940675 + 0.104473i
14.10 0.201597 0.620451i −0.733986 + 0.815174i 1.27372 + 0.925409i 0.554991 0.961272i 0.357806 + 0.619739i −0.401317 + 3.81828i 1.88652 1.37064i 0.187812 + 1.78691i −0.484538 0.538134i
14.11 0.249715 0.768544i 1.39968 1.55451i 1.08973 + 0.791737i −0.342316 + 0.592908i −0.845184 1.46390i 0.346639 3.29805i 2.18813 1.58977i −0.143790 1.36807i 0.370195 + 0.411143i
14.12 0.313343 0.964371i −2.18936 + 2.43153i 0.786207 + 0.571213i −1.31857 + 2.28384i 1.65888 + 2.87326i −0.310865 + 2.95768i 2.43790 1.77124i −0.805458 7.66342i 1.78930 + 1.98722i
14.13 0.477192 1.46864i 1.45858 1.61992i −0.311171 0.226079i 1.14752 1.98757i −1.68306 2.91515i −0.410673 + 3.90729i 2.01809 1.46623i −0.183091 1.74199i −2.37144 2.63375i
14.14 0.563363 1.73385i −0.719072 + 0.798611i −1.07083 0.778005i 1.03814 1.79811i 0.979574 + 1.69667i 0.0320232 0.304681i 0.997593 0.724794i 0.192871 + 1.83505i −2.53281 2.81297i
14.15 0.748518 2.30370i 1.94845 2.16398i −3.12873 2.27316i −1.19465 + 2.06919i −3.52671 6.10844i 0.119417 1.13617i −3.65930 + 2.65864i −0.572740 5.44925i 3.87259 + 4.30095i
14.16 0.798897 2.45875i −0.340489 + 0.378151i −3.78920 2.75301i −1.48464 + 2.57146i 0.657765 + 1.13928i −0.327920 + 3.11995i −5.61307 + 4.07814i 0.286520 + 2.72605i 5.13652 + 5.70469i
14.17 0.849257 2.61374i −2.02850 + 2.25288i −4.49238 3.26391i 1.75209 3.03471i 4.16574 + 7.21527i 0.0345203 0.328439i −7.89944 + 5.73928i −0.647064 6.15640i −6.44398 7.15676i
40.1 −0.779971 + 2.40050i 2.95884 + 0.628922i −3.53603 2.56908i −2.10392 3.64409i −3.81754 + 6.61218i 3.44993 1.53601i 4.84111 3.51727i 5.61858 + 2.50155i 10.3887 2.20818i
40.2 −0.756168 + 2.32725i −0.493906 0.104983i −3.22625 2.34401i 0.105129 + 0.182089i 0.617798 1.07006i 0.113244 0.0504194i 3.93532 2.85917i −2.50771 1.11651i −0.503261 + 0.106971i
40.3 −0.715991 + 2.20359i 1.66971 + 0.354909i −2.72515 1.97993i 0.00534395 + 0.00925600i −1.97758 + 3.42526i −3.74078 + 1.66550i 2.56517 1.86371i −0.0786495 0.0350170i −0.0242227 + 0.00514869i
See next 80 embeddings (of 136 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.17 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g 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.bi.b 136
31.g even 15 1 inner 403.2.bi.b 136

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bi.b 136 1.a even 1 1 trivial
403.2.bi.b 136 31.g even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{136} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database