# Properties

 Label 418.2.u.b Level $418$ Weight $2$ Character orbit 418.u Analytic conductor $3.338$ Analytic rank $0$ Dimension $264$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.u (of order $$45$$, degree $$24$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$264$$ Relative dimension: $$11$$ over $$\Q(\zeta_{45})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{45}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$264 q + 6 q^{3} - 9 q^{6} - 15 q^{7} + 33 q^{8} + 6 q^{9}+O(q^{10})$$ 264 * q + 6 * q^3 - 9 * q^6 - 15 * q^7 + 33 * q^8 + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$264 q + 6 q^{3} - 9 q^{6} - 15 q^{7} + 33 q^{8} + 6 q^{9} + 3 q^{11} - 6 q^{13} + 18 q^{14} - 39 q^{15} - 3 q^{17} - 78 q^{18} - 45 q^{19} - 24 q^{20} + 48 q^{21} + 6 q^{23} - 9 q^{24} + 30 q^{25} + 18 q^{26} - 24 q^{27} + 6 q^{28} - 3 q^{31} - 63 q^{33} - 36 q^{34} + 42 q^{35} - 9 q^{36} + 60 q^{37} - 3 q^{38} + 36 q^{39} + 39 q^{41} + 6 q^{42} - 60 q^{43} + 60 q^{44} - 108 q^{45} - 12 q^{46} - 24 q^{47} - 12 q^{48} + 6 q^{49} + 18 q^{50} + 96 q^{51} + 3 q^{52} - 117 q^{53} + 54 q^{54} + 102 q^{55} - 96 q^{57} - 60 q^{58} - 141 q^{59} + 36 q^{60} + 24 q^{61} - 27 q^{62} - 81 q^{63} + 33 q^{64} - 102 q^{65} + 72 q^{66} + 102 q^{67} - 21 q^{68} - 6 q^{69} - 33 q^{70} - 66 q^{71} - 12 q^{72} + 36 q^{73} + 18 q^{74} + 6 q^{76} - 174 q^{77} + 18 q^{78} + 36 q^{79} + 60 q^{81} - 36 q^{82} - 24 q^{83} + 48 q^{84} + 174 q^{85} - 21 q^{86} + 12 q^{87} + 3 q^{88} + 30 q^{89} - 48 q^{90} - 18 q^{91} + 18 q^{92} - 123 q^{93} - 120 q^{94} - 18 q^{95} - 24 q^{97} - 84 q^{98} - 141 q^{99}+O(q^{100})$$ 264 * q + 6 * q^3 - 9 * q^6 - 15 * q^7 + 33 * q^8 + 6 * q^9 + 3 * q^11 - 6 * q^13 + 18 * q^14 - 39 * q^15 - 3 * q^17 - 78 * q^18 - 45 * q^19 - 24 * q^20 + 48 * q^21 + 6 * q^23 - 9 * q^24 + 30 * q^25 + 18 * q^26 - 24 * q^27 + 6 * q^28 - 3 * q^31 - 63 * q^33 - 36 * q^34 + 42 * q^35 - 9 * q^36 + 60 * q^37 - 3 * q^38 + 36 * q^39 + 39 * q^41 + 6 * q^42 - 60 * q^43 + 60 * q^44 - 108 * q^45 - 12 * q^46 - 24 * q^47 - 12 * q^48 + 6 * q^49 + 18 * q^50 + 96 * q^51 + 3 * q^52 - 117 * q^53 + 54 * q^54 + 102 * q^55 - 96 * q^57 - 60 * q^58 - 141 * q^59 + 36 * q^60 + 24 * q^61 - 27 * q^62 - 81 * q^63 + 33 * q^64 - 102 * q^65 + 72 * q^66 + 102 * q^67 - 21 * q^68 - 6 * q^69 - 33 * q^70 - 66 * q^71 - 12 * q^72 + 36 * q^73 + 18 * q^74 + 6 * q^76 - 174 * q^77 + 18 * q^78 + 36 * q^79 + 60 * q^81 - 36 * q^82 - 24 * q^83 + 48 * q^84 + 174 * q^85 - 21 * q^86 + 12 * q^87 + 3 * q^88 + 30 * q^89 - 48 * q^90 - 18 * q^91 + 18 * q^92 - 123 * q^93 - 120 * q^94 - 18 * q^95 - 24 * q^97 - 84 * q^98 - 141 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −0.615661 0.788011i −2.04010 + 1.97011i −0.241922 + 0.970296i −0.174672 + 0.109147i 2.80848 + 0.394706i 3.62878 + 1.61564i 0.913545 0.406737i 0.176010 5.04027i 0.193548 + 0.0704456i
5.2 −0.615661 0.788011i −2.01318 + 1.94411i −0.241922 + 0.970296i 3.22722 2.01659i 2.77142 + 0.389497i −3.27079 1.45625i 0.913545 0.406737i 0.168650 4.82951i −3.57597 1.30155i
5.3 −0.615661 0.788011i −1.27054 + 1.22694i −0.241922 + 0.970296i −3.54395 + 2.21450i 1.74906 + 0.245815i −1.43392 0.638421i 0.913545 0.406737i 0.00417577 0.119578i 3.92692 + 1.42928i
5.4 −0.615661 0.788011i −0.706965 + 0.682708i −0.241922 + 0.970296i 3.26123 2.03784i 0.973232 + 0.136779i 1.62272 + 0.722480i 0.913545 0.406737i −0.0709895 + 2.03287i −3.61366 1.31526i
5.5 −0.615661 0.788011i −0.215741 + 0.208339i −0.241922 + 0.970296i −0.160523 + 0.100306i 0.296996 + 0.0417401i −1.81207 0.806787i 0.913545 0.406737i −0.101559 + 2.90828i 0.177870 + 0.0647394i
5.6 −0.615661 0.788011i −0.149553 + 0.144422i −0.241922 + 0.970296i −0.227007 + 0.141850i 0.205880 + 0.0289345i −4.29887 1.91398i 0.913545 0.406737i −0.103190 + 2.95498i 0.251538 + 0.0915525i
5.7 −0.615661 0.788011i 0.555466 0.536408i −0.241922 + 0.970296i −0.977067 + 0.610539i −0.764674 0.107468i 2.99213 + 1.33218i 0.913545 0.406737i −0.0838888 + 2.40226i 1.08265 + 0.394054i
5.8 −0.615661 0.788011i 1.20057 1.15938i −0.241922 + 0.970296i −2.39453 + 1.49627i −1.65275 0.232278i 2.61504 + 1.16429i 0.913545 0.406737i −0.00748509 + 0.214345i 2.65330 + 0.965720i
5.9 −0.615661 0.788011i 1.44037 1.39095i −0.241922 + 0.970296i 1.24695 0.779183i −1.98287 0.278674i 0.930267 + 0.414181i 0.913545 0.406737i 0.0352282 1.00880i −1.38171 0.502900i
5.10 −0.615661 0.788011i 1.73828 1.67864i −0.241922 + 0.970296i 2.93245 1.83240i −2.39298 0.336312i −1.22803 0.546756i 0.913545 0.406737i 0.0990960 2.83774i −3.24934 1.18266i
5.11 −0.615661 0.788011i 2.29692 2.21811i −0.241922 + 0.970296i −2.86618 + 1.79099i −3.16202 0.444392i −4.05265 1.80436i 0.913545 0.406737i 0.251131 7.19145i 3.17592 + 1.15594i
9.1 0.990268 + 0.139173i −0.772549 + 3.09852i 0.961262 + 0.275637i 0.0332390 0.951842i −1.19626 + 2.96085i −2.96764 + 1.32128i 0.913545 + 0.406737i −6.35518 3.37911i 0.165386 0.937953i
9.2 0.990268 + 0.139173i −0.689545 + 2.76561i 0.961262 + 0.275637i −0.144956 + 4.15100i −1.06773 + 2.64273i 1.74927 0.778823i 0.913545 + 0.406737i −4.52431 2.40562i −0.721253 + 4.09043i
9.3 0.990268 + 0.139173i −0.585803 + 2.34953i 0.961262 + 0.275637i 0.126163 3.61284i −0.907093 + 2.24513i 4.27441 1.90309i 0.913545 + 0.406737i −2.52827 1.34431i 0.627745 3.56012i
9.4 0.990268 + 0.139173i −0.349946 + 1.40356i 0.961262 + 0.275637i 0.0105893 0.303239i −0.541878 + 1.34120i 1.01844 0.453439i 0.913545 + 0.406737i 0.801329 + 0.426074i 0.0526890 0.298814i
9.5 0.990268 + 0.139173i −0.00819141 + 0.0328539i 0.961262 + 0.275637i 0.0160118 0.458517i −0.0126841 + 0.0313942i −0.614926 + 0.273783i 0.913545 + 0.406737i 2.64783 + 1.40788i 0.0796692 0.451826i
9.6 0.990268 + 0.139173i 0.0134323 0.0538742i 0.961262 + 0.275637i −0.0826337 + 2.36632i 0.0207995 0.0514805i 1.41811 0.631382i 0.913545 + 0.406737i 2.64612 + 1.40697i −0.411158 + 2.33179i
9.7 0.990268 + 0.139173i 0.161628 0.648256i 0.961262 + 0.275637i 0.0866869 2.48239i 0.250275 0.619453i −0.941853 + 0.419340i 0.913545 + 0.406737i 2.25473 + 1.19886i 0.431325 2.44617i
9.8 0.990268 + 0.139173i 0.276037 1.10712i 0.961262 + 0.275637i −0.120793 + 3.45905i 0.427432 1.05793i −4.47274 + 1.99139i 0.913545 + 0.406737i 1.49932 + 0.797201i −0.601023 + 3.40857i
9.9 0.990268 + 0.139173i 0.530296 2.12690i 0.961262 + 0.275637i 0.0547010 1.56643i 0.821143 2.03240i 2.87167 1.27855i 0.913545 + 0.406737i −1.59366 0.847362i 0.272174 1.54358i
See next 80 embeddings (of 264 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 405.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
19.e even 9 1 inner
209.u even 45 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.u.b 264
11.c even 5 1 inner 418.2.u.b 264
19.e even 9 1 inner 418.2.u.b 264
209.u even 45 1 inner 418.2.u.b 264

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.u.b 264 1.a even 1 1 trivial
418.2.u.b 264 11.c even 5 1 inner
418.2.u.b 264 19.e even 9 1 inner
418.2.u.b 264 209.u even 45 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{264} - 6 T_{3}^{263} + 15 T_{3}^{262} - 10 T_{3}^{261} - 99 T_{3}^{260} + 618 T_{3}^{259} - 3236 T_{3}^{258} + 8898 T_{3}^{257} - 1707 T_{3}^{256} - 61104 T_{3}^{255} + 240666 T_{3}^{254} - 471663 T_{3}^{253} + \cdots + 20\!\cdots\!81$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.