Properties

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

 $$\operatorname{Tr}(f)(q) =$$ $$280q - 3q^{2} - 9q^{3} + 29q^{4} - 8q^{5} - 10q^{6} - 13q^{7} - 29q^{8} + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$280q - 3q^{2} - 9q^{3} + 29q^{4} - 8q^{5} - 10q^{6} - 13q^{7} - 29q^{8} + 24q^{9} - 9q^{10} - 6q^{11} - 50q^{12} - 6q^{13} - 21q^{15} + 23q^{16} + 3q^{17} - 15q^{18} - 9q^{19} + 60q^{20} - 34q^{21} + 41q^{22} - 12q^{23} - 89q^{24} - 112q^{25} + 39q^{26} + 48q^{27} + 66q^{28} - 11q^{29} + 24q^{30} - 22q^{31} + q^{32} - 37q^{33} - 49q^{34} + 41q^{35} + 134q^{36} + 17q^{37} - 23q^{38} + 10q^{39} + 31q^{40} + q^{41} - 71q^{42} - 7q^{43} - 42q^{44} - 13q^{45} + 21q^{46} - 20q^{47} - 3q^{48} - 69q^{49} - 42q^{50} - 71q^{51} - 140q^{52} - 8q^{53} - 149q^{54} + 5q^{55} - 126q^{56} + 21q^{57} + 79q^{58} + 77q^{59} + 2q^{60} + 53q^{61} + 72q^{62} + 17q^{63} - 91q^{64} - 25q^{65} - 62q^{66} - 14q^{67} + 83q^{68} - 31q^{69} + 134q^{70} + 32q^{71} - 70q^{72} - 31q^{73} + 51q^{74} + 79q^{75} + 98q^{76} - 80q^{77} - 141q^{78} - 142q^{79} + 6q^{80} + 66q^{81} - 48q^{82} + 94q^{83} + 130q^{84} + 90q^{85} - 34q^{86} + 22q^{87} - 122q^{88} - 15q^{89} + 43q^{90} + 56q^{91} - 294q^{92} - 52q^{93} - 57q^{94} + 94q^{95} + 24q^{96} + 46q^{97} + 73q^{98} + 12q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −1.83126 2.03382i −0.500677 + 0.556058i −0.573858 + 5.45990i −0.330308 0.572109i 2.04780 −3.67674 2.67131i 7.72715 5.61410i 0.255062 + 2.42676i −0.558690 + 1.71947i
9.2 −1.65225 1.83501i 2.14947 2.38723i −0.428272 + 4.07474i −1.22323 2.11870i −7.93205 −1.37387 0.998173i 4.18946 3.04382i −0.765054 7.27901i −1.86675 + 5.74525i
9.3 −1.59232 1.76846i −1.58986 + 1.76572i −0.382880 + 3.64286i 1.32032 + 2.28686i 5.65416 2.08439 + 1.51440i 3.20149 2.32602i −0.276520 2.63091i 1.94183 5.97634i
9.4 −1.55327 1.72508i 1.03676 1.15144i −0.354201 + 3.36999i 0.252831 + 0.437917i −3.59669 1.69707 + 1.23299i 2.60770 1.89460i 0.0626468 + 0.596045i 0.362727 1.11636i
9.5 −1.46781 1.63017i −2.12808 + 2.36348i −0.293925 + 2.79651i −0.951364 1.64781i 6.97649 −1.54320 1.12120i 1.44087 1.04685i −0.743696 7.07579i −1.28979 + 3.96956i
9.6 −1.44960 1.60995i 0.384158 0.426651i −0.281525 + 2.67853i −1.83417 3.17687i −1.24376 0.633388 + 0.460183i 1.21509 0.882816i 0.279132 + 2.65576i −2.45578 + 7.55812i
9.7 −1.32661 1.47335i 0.650439 0.722386i −0.201810 + 1.92010i 1.17248 + 2.03079i −1.92721 0.184244 + 0.133861i −0.111198 + 0.0807903i 0.214815 + 2.04383i 1.43665 4.42155i
9.8 −1.24715 1.38510i −0.607940 + 0.675186i −0.154065 + 1.46583i 1.19707 + 2.07339i 1.69340 −2.18489 1.58741i −0.793286 + 0.576356i 0.227301 + 2.16262i 1.37893 4.24390i
9.9 −0.918709 1.02033i −0.334947 + 0.371997i 0.0120106 0.114273i −0.585861 1.01474i 0.687278 3.13175 + 2.27535i −2.34917 + 1.70677i 0.287394 + 2.73437i −0.497134 + 1.53002i
9.10 −0.769629 0.854759i 1.59325 1.76948i 0.0707719 0.673349i −0.220512 0.381938i −2.73870 −1.17517 0.853812i −2.49107 + 1.80987i −0.279042 2.65490i −0.156753 + 0.482435i
9.11 −0.736014 0.817427i 2.06902 2.29788i 0.0825875 0.785768i 1.51346 + 2.62138i −3.40117 2.69694 + 1.95944i −2.48286 + 1.80390i −0.685820 6.52514i 1.02886 3.16652i
9.12 −0.718707 0.798205i −0.717726 + 0.797115i 0.0884655 0.841693i 1.01665 + 1.76088i 1.15210 −3.07573 2.23465i −2.47334 + 1.79699i 0.193323 + 1.83935i 0.674874 2.07705i
9.13 −0.669055 0.743061i −1.14502 + 1.27167i 0.104552 0.994745i −1.77181 3.06886i 1.71101 −2.25690 1.63973i −2.42696 + 1.76329i 0.00750493 + 0.0714046i −1.09491 + 3.36979i
9.14 −0.628504 0.698025i −1.76109 + 1.95588i 0.116836 1.11162i −0.604953 1.04781i 2.47210 2.24460 + 1.63080i −2.36917 + 1.72130i −0.410474 3.90540i −0.351181 + 1.08082i
9.15 −0.165338 0.183626i 1.65099 1.83361i 0.202675 1.92832i −1.68082 2.91126i −0.609670 0.0202552 + 0.0147163i −0.787407 + 0.572085i −0.322771 3.07096i −0.256681 + 0.789984i
9.16 −0.142826 0.158624i 0.434282 0.482319i 0.204295 1.94373i 1.97676 + 3.42385i −0.138534 0.326262 + 0.237043i −0.682872 + 0.496136i 0.269555 + 2.56464i 0.260774 0.802579i
9.17 −0.133023 0.147737i 0.0833490 0.0925684i 0.204926 1.94974i −0.998333 1.72916i −0.0247631 −1.37062 0.995817i −0.636973 + 0.462788i 0.311964 + 2.96813i −0.122660 + 0.377509i
9.18 −0.0525666 0.0583811i −1.70784 + 1.89675i 0.208412 1.98291i 1.96469 + 3.40295i 0.200510 2.32100 + 1.68631i −0.253832 + 0.184420i −0.367355 3.49515i 0.0953908 0.293583i
9.19 0.117605 + 0.130614i 0.508986 0.565287i 0.205828 1.95832i 0.00796452 + 0.0137949i 0.133693 3.75325 + 2.72689i 0.564372 0.410040i 0.253104 + 2.40812i −0.000865141 0.00266263i
9.20 0.169063 + 0.187763i −1.38560 + 1.53887i 0.202384 1.92556i 0.388050 + 0.672122i −0.523196 −2.21450 1.60893i 0.804576 0.584559i −0.134633 1.28095i −0.0605950 + 0.186492i
See next 80 embeddings (of 280 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 276.35 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.bk 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.bk.a yes 280
13.c even 3 1 403.2.bj.a 280
31.g even 15 1 403.2.bj.a 280
403.bk even 15 1 inner 403.2.bk.a yes 280

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