# Properties

 Label 403.2.bi.a Level 403 Weight 2 Character orbit 403.bi Analytic conductor 3.218 Analytic rank 0 Dimension 120 CM no Inner twists 2

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## 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: $$120$$ Relative dimension: $$15$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$120q + 6q^{2} + 2q^{3} - 22q^{4} + 13q^{5} - 10q^{6} - 16q^{7} - 6q^{8} + 13q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$120q + 6q^{2} + 2q^{3} - 22q^{4} + 13q^{5} - 10q^{6} - 16q^{7} - 6q^{8} + 13q^{9} + 3q^{10} + 3q^{11} - 12q^{12} - 15q^{13} - 3q^{14} - 39q^{15} - 2q^{16} + 6q^{17} - 34q^{18} - 25q^{19} - 10q^{20} + 40q^{21} + 18q^{22} + q^{23} + 46q^{24} - 45q^{25} - 12q^{26} + 26q^{27} - 13q^{28} + 14q^{29} - 28q^{30} - 14q^{31} - 76q^{32} + 63q^{33} + 36q^{34} + 50q^{35} - 42q^{36} + 14q^{37} + 44q^{38} - 6q^{39} + 27q^{40} - 17q^{41} - 159q^{42} - 15q^{43} + 70q^{44} + 40q^{45} - 62q^{46} + 9q^{47} + 19q^{48} - 151q^{49} - 22q^{50} - 25q^{51} - q^{52} - 69q^{53} + 51q^{54} + 27q^{55} + 51q^{56} - 28q^{57} + 156q^{58} - 27q^{59} + 87q^{60} - 28q^{61} - 25q^{62} - 88q^{63} + 10q^{64} + 7q^{65} + 30q^{66} + 61q^{67} + 124q^{68} + 3q^{69} - 263q^{70} + 31q^{71} + 31q^{72} - 50q^{73} - 90q^{74} + 78q^{75} + 66q^{76} - 38q^{77} + 31q^{79} - 145q^{80} + 36q^{81} + 3q^{82} + 23q^{83} - 228q^{84} - 58q^{85} - 12q^{86} + 68q^{87} + 28q^{88} + 39q^{89} - 7q^{90} - 12q^{91} - 32q^{92} - 17q^{93} + 14q^{94} + 123q^{95} + 165q^{96} - 51q^{97} + 45q^{98} - 45q^{99} + O(q^{100})$$

## 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 −0.745649 + 2.29487i −1.56275 + 1.73561i −3.09241 2.24677i −1.53204 + 2.65358i −2.81775 4.88048i 0.477992 4.54779i 3.55762 2.58476i −0.256571 2.44111i −4.94725 5.49448i
14.2 −0.717728 + 2.20894i 1.96095 2.17786i −2.74625 1.99527i 1.43599 2.48720i 3.40333 + 5.89474i −0.518982 + 4.93778i 2.62042 1.90385i −0.584148 5.55780i 4.46343 + 4.95714i
14.3 −0.629077 + 1.93610i −0.654890 + 0.727329i −1.73471 1.26034i 0.778108 1.34772i −0.996205 1.72548i −0.152927 + 1.45500i 0.237526 0.172573i 0.213459 + 2.03093i 2.11984 + 2.35432i
14.4 −0.540256 + 1.66274i −0.217770 + 0.241858i −0.854781 0.621035i 1.00253 1.73643i −0.284495 0.492760i 0.416075 3.95868i −1.33440 + 0.969498i 0.302514 + 2.87823i 2.34560 + 2.60506i
14.5 −0.527412 + 1.62321i 1.32464 1.47116i −0.738608 0.536630i −1.29201 + 2.23783i 1.68936 + 2.92607i −0.115558 + 1.09946i −1.50095 + 1.09051i −0.0960576 0.913927i −2.95105 3.27747i
14.6 −0.269588 + 0.829708i −1.42637 + 1.58415i 1.00230 + 0.728211i −1.58260 + 2.74115i −0.929847 1.61054i −0.112042 + 1.06601i −2.28599 + 1.66087i −0.161400 1.53561i −1.84770 2.05208i
14.7 −0.162742 + 0.500869i −1.66779 + 1.85227i 1.39365 + 1.01255i 1.93093 3.34447i −0.656324 1.13679i −0.465155 + 4.42565i −1.58609 + 1.15236i −0.335789 3.19482i 1.36090 + 1.51143i
14.8 −0.0908434 + 0.279587i 1.96940 2.18724i 1.54812 + 1.12477i 0.739803 1.28138i 0.432618 + 0.749317i 0.0548395 0.521763i −0.930771 + 0.676245i −0.591901 5.63156i 0.291050 + 0.323244i
14.9 −0.0727036 + 0.223759i 0.337206 0.374505i 1.57325 + 1.14303i 0.254802 0.441330i 0.0592827 + 0.102681i −0.0906697 + 0.862665i −0.750826 + 0.545507i 0.287039 + 2.73099i 0.0802263 + 0.0891004i
14.10 0.138780 0.427121i −2.06376 + 2.29204i 1.45486 + 1.05702i 0.873401 1.51278i 0.692568 + 1.19956i 0.471332 4.48443i 1.38004 1.00266i −0.680746 6.47687i −0.524927 0.582991i
14.11 0.362743 1.11641i −0.166092 + 0.184464i 0.503249 + 0.365632i 2.00419 3.47136i 0.145688 + 0.252339i 0.284216 2.70413i 2.49009 1.80916i 0.307145 + 2.92229i −3.14845 3.49671i
14.12 0.395429 1.21700i 0.354263 0.393449i 0.293298 + 0.213093i −1.48333 + 2.56921i −0.338744 0.586721i −0.222430 + 2.11628i 2.44581 1.77698i 0.284286 + 2.70480i 2.54018 + 2.82116i
14.13 0.523105 1.60995i 0.991294 1.10094i −0.700275 0.508780i −1.03169 + 1.78693i −1.25392 2.17185i 0.334356 3.18118i 1.55359 1.12875i 0.0841724 + 0.800847i 2.33720 + 2.59572i
14.14 0.635252 1.95510i −1.26940 + 1.40981i −1.80085 1.30840i 0.148342 0.256935i 1.94994 + 3.37739i −0.266274 + 2.53343i −0.375822 + 0.273051i −0.0626061 0.595658i −0.408101 0.453242i
14.15 0.773640 2.38102i 1.00840 1.11994i −3.45270 2.50854i 1.04075 1.80263i −1.88646 3.26744i −0.0586254 + 0.557783i −4.59319 + 3.33715i 0.0761884 + 0.724884i −3.48694 3.87264i
40.1 −0.844756 + 2.59989i 1.33170 + 0.283062i −4.42779 3.21698i 1.99341 + 3.45269i −1.86090 + 3.22317i 3.32958 1.48242i 7.68099 5.58056i −1.04733 0.466300i −10.6606 + 2.26597i
40.2 −0.820567 + 2.52545i −1.82037 0.386931i −4.08652 2.96903i −1.79431 3.10783i 2.47091 4.27974i −1.93457 + 0.861327i 6.55484 4.76237i 0.423384 + 0.188503i 9.32102 1.98124i
40.3 −0.608805 + 1.87371i −1.89076 0.401893i −1.52211 1.10588i 0.912174 + 1.57993i 1.90413 3.29806i −0.756252 + 0.336705i −0.188985 + 0.137305i 0.672810 + 0.299554i −3.51567 + 0.747279i
40.4 −0.584417 + 1.79865i 3.21845 + 0.684102i −1.27557 0.926754i 1.13209 + 1.96083i −3.11138 + 5.38906i −1.46764 + 0.653435i −0.647676 + 0.470565i 7.14976 + 3.18328i −4.18846 + 0.890285i
40.5 −0.395989 + 1.21873i 2.04438 + 0.434547i 0.289543 + 0.210365i −0.128714 0.222939i −1.33915 + 2.31947i 1.64432 0.732098i −2.44446 + 1.77600i 1.25003 + 0.556550i 0.322672 0.0685860i
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 391.15 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.a 120
31.g even 15 1 inner 403.2.bi.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bi.a 120 1.a even 1 1 trivial
403.2.bi.a 120 31.g even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{120} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database