# Properties

 Label 1045.2.b.d Level $1045$ Weight $2$ Character orbit 1045.b Analytic conductor $8.344$ Analytic rank $0$ Dimension $22$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.34436701122$$ Analytic rank: $$0$$ Dimension: $$22$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 32 q^{4} + 7 q^{5} - 12 q^{6} - 34 q^{9}+O(q^{10})$$ 22 * q - 32 * q^4 + 7 * q^5 - 12 * q^6 - 34 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$22 q - 32 q^{4} + 7 q^{5} - 12 q^{6} - 34 q^{9} + 2 q^{10} - 22 q^{11} + 8 q^{14} - 23 q^{15} + 40 q^{16} + 22 q^{19} - 22 q^{20} - 22 q^{21} + 22 q^{24} + 13 q^{25} + 16 q^{26} + 10 q^{29} - 22 q^{30} + 76 q^{31} - 56 q^{34} - 2 q^{35} + 104 q^{36} + 8 q^{39} - 20 q^{40} + 6 q^{41} + 32 q^{44} - 12 q^{45} + 88 q^{46} - 28 q^{49} - 20 q^{50} + 8 q^{51} - 38 q^{54} - 7 q^{55} + 44 q^{56} - 40 q^{59} + 78 q^{60} - 6 q^{61} - 140 q^{64} - 22 q^{65} + 12 q^{66} - 74 q^{69} - 24 q^{70} + 62 q^{71} + 26 q^{74} + 13 q^{75} - 32 q^{76} - 102 q^{79} + 142 q^{80} + 94 q^{81} + 38 q^{84} + 26 q^{85} + 28 q^{86} - 54 q^{89} + 118 q^{90} + 88 q^{91} - 36 q^{94} + 7 q^{95} + 2 q^{96} + 34 q^{99}+O(q^{100})$$ 22 * q - 32 * q^4 + 7 * q^5 - 12 * q^6 - 34 * q^9 + 2 * q^10 - 22 * q^11 + 8 * q^14 - 23 * q^15 + 40 * q^16 + 22 * q^19 - 22 * q^20 - 22 * q^21 + 22 * q^24 + 13 * q^25 + 16 * q^26 + 10 * q^29 - 22 * q^30 + 76 * q^31 - 56 * q^34 - 2 * q^35 + 104 * q^36 + 8 * q^39 - 20 * q^40 + 6 * q^41 + 32 * q^44 - 12 * q^45 + 88 * q^46 - 28 * q^49 - 20 * q^50 + 8 * q^51 - 38 * q^54 - 7 * q^55 + 44 * q^56 - 40 * q^59 + 78 * q^60 - 6 * q^61 - 140 * q^64 - 22 * q^65 + 12 * q^66 - 74 * q^69 - 24 * q^70 + 62 * q^71 + 26 * q^74 + 13 * q^75 - 32 * q^76 - 102 * q^79 + 142 * q^80 + 94 * q^81 + 38 * q^84 + 26 * q^85 + 28 * q^86 - 54 * q^89 + 118 * q^90 + 88 * q^91 - 36 * q^94 + 7 * q^95 + 2 * q^96 + 34 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
419.1 2.74370i 2.61424i −5.52788 1.89425 1.18821i −7.17269 3.68386i 9.67944i −3.83426 −3.26008 5.19724i
419.2 2.71685i 2.46712i −5.38125 2.08881 + 0.798049i 6.70278 0.267393i 9.18634i −3.08666 2.16818 5.67497i
419.3 2.58163i 0.787674i −4.66484 −1.04139 + 1.97876i −2.03349 0.906789i 6.87964i 2.37957 5.10844 + 2.68850i
419.4 2.06343i 3.28666i −2.25776 0.951609 2.02347i −6.78181 3.51972i 0.531875i −7.80213 −4.17530 1.96358i
419.5 1.84475i 3.33093i −1.40312 −2.19819 + 0.409812i 6.14475 3.04962i 1.10110i −8.09513 0.756002 + 4.05513i
419.6 1.66420i 1.15649i −0.769547 1.63516 + 1.52521i −1.92462 3.92678i 2.04771i 1.66254 2.53825 2.72122i
419.7 1.60597i 0.776012i −0.579148 −1.81070 1.31201i −1.24625 0.953282i 2.28185i 2.39780 −2.10704 + 2.90794i
419.8 1.28593i 0.496540i 0.346393 −0.328055 + 2.21187i 0.638514 3.51780i 3.01729i 2.75345 2.84431 + 0.421855i
419.9 1.06190i 2.29282i 0.872373 −1.55246 1.60931i −2.43474 3.97400i 3.05017i −2.25702 −1.70892 + 1.64855i
419.10 0.790175i 2.57534i 1.37562 1.76786 1.36919i 2.03497 0.0669205i 2.66733i −3.63236 −1.08190 1.39692i
419.11 0.104144i 0.696995i 1.98915 2.09312 0.786662i 0.0725876 3.21657i 0.415445i 2.51420 −0.0819259 0.217986i
419.12 0.104144i 0.696995i 1.98915 2.09312 + 0.786662i 0.0725876 3.21657i 0.415445i 2.51420 −0.0819259 + 0.217986i
419.13 0.790175i 2.57534i 1.37562 1.76786 + 1.36919i 2.03497 0.0669205i 2.66733i −3.63236 −1.08190 + 1.39692i
419.14 1.06190i 2.29282i 0.872373 −1.55246 + 1.60931i −2.43474 3.97400i 3.05017i −2.25702 −1.70892 1.64855i
419.15 1.28593i 0.496540i 0.346393 −0.328055 2.21187i 0.638514 3.51780i 3.01729i 2.75345 2.84431 0.421855i
419.16 1.60597i 0.776012i −0.579148 −1.81070 + 1.31201i −1.24625 0.953282i 2.28185i 2.39780 −2.10704 2.90794i
419.17 1.66420i 1.15649i −0.769547 1.63516 1.52521i −1.92462 3.92678i 2.04771i 1.66254 2.53825 + 2.72122i
419.18 1.84475i 3.33093i −1.40312 −2.19819 0.409812i 6.14475 3.04962i 1.10110i −8.09513 0.756002 4.05513i
419.19 2.06343i 3.28666i −2.25776 0.951609 + 2.02347i −6.78181 3.51972i 0.531875i −7.80213 −4.17530 + 1.96358i
419.20 2.58163i 0.787674i −4.66484 −1.04139 1.97876i −2.03349 0.906789i 6.87964i 2.37957 5.10844 2.68850i
See all 22 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.d 22
5.b even 2 1 inner 1045.2.b.d 22
5.c odd 4 2 5225.2.a.bb 22

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.d 22 1.a even 1 1 trivial
1045.2.b.d 22 5.b even 2 1 inner
5225.2.a.bb 22 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{22} + 38 T_{2}^{20} + 620 T_{2}^{18} + 5698 T_{2}^{16} + 32546 T_{2}^{14} + 120279 T_{2}^{12} + 290067 T_{2}^{10} + 448488 T_{2}^{8} + 422895 T_{2}^{6} + 218245 T_{2}^{4} + 46943 T_{2}^{2} + 484$$ acting on $$S_{2}^{\mathrm{new}}(1045, [\chi])$$.