# Properties

 Label 418.2.m.b Level $418$ Weight $2$ Character orbit 418.m Analytic conductor $3.338$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.m (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 10 q^{2} - 10 q^{4} + 2 q^{5} + 5 q^{6} - 5 q^{7} + 10 q^{8} + 8 q^{9}+O(q^{10})$$ 40 * q + 10 * q^2 - 10 * q^4 + 2 * q^5 + 5 * q^6 - 5 * q^7 + 10 * q^8 + 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q + 10 q^{2} - 10 q^{4} + 2 q^{5} + 5 q^{6} - 5 q^{7} + 10 q^{8} + 8 q^{9} - 2 q^{10} + 4 q^{11} - 8 q^{13} - 5 q^{14} - 30 q^{15} - 10 q^{16} - 15 q^{17} - 13 q^{18} + 11 q^{19} + 2 q^{20} - 4 q^{22} + 6 q^{23} - 5 q^{24} - 36 q^{25} - 2 q^{26} + 45 q^{27} + 2 q^{29} - 30 q^{30} - 40 q^{32} - 27 q^{33} - 5 q^{35} + 13 q^{36} + 14 q^{38} + 30 q^{39} + 3 q^{40} + 8 q^{41} + 20 q^{42} - 6 q^{44} + 18 q^{45} - q^{46} - 8 q^{47} + 31 q^{49} - 9 q^{50} - 41 q^{51} + 2 q^{52} + 40 q^{53} - 31 q^{55} - 10 q^{57} - 2 q^{58} - 35 q^{59} - 20 q^{60} + 5 q^{61} + 30 q^{62} - 25 q^{63} - 10 q^{64} - 8 q^{65} - 48 q^{66} + 15 q^{68} + 60 q^{69} + 10 q^{70} - 50 q^{71} - 8 q^{72} + 10 q^{73} + 35 q^{75} + 11 q^{76} - 64 q^{77} + 42 q^{79} - 3 q^{80} + 11 q^{81} + 7 q^{82} + 25 q^{83} + 20 q^{84} - 45 q^{85} + 40 q^{86} + 6 q^{88} + 22 q^{90} + 70 q^{91} - 4 q^{92} - 18 q^{93} - 7 q^{94} - 5 q^{95} + 15 q^{97} + 74 q^{98} + 50 q^{99}+O(q^{100})$$ 40 * q + 10 * q^2 - 10 * q^4 + 2 * q^5 + 5 * q^6 - 5 * q^7 + 10 * q^8 + 8 * q^9 - 2 * q^10 + 4 * q^11 - 8 * q^13 - 5 * q^14 - 30 * q^15 - 10 * q^16 - 15 * q^17 - 13 * q^18 + 11 * q^19 + 2 * q^20 - 4 * q^22 + 6 * q^23 - 5 * q^24 - 36 * q^25 - 2 * q^26 + 45 * q^27 + 2 * q^29 - 30 * q^30 - 40 * q^32 - 27 * q^33 - 5 * q^35 + 13 * q^36 + 14 * q^38 + 30 * q^39 + 3 * q^40 + 8 * q^41 + 20 * q^42 - 6 * q^44 + 18 * q^45 - q^46 - 8 * q^47 + 31 * q^49 - 9 * q^50 - 41 * q^51 + 2 * q^52 + 40 * q^53 - 31 * q^55 - 10 * q^57 - 2 * q^58 - 35 * q^59 - 20 * q^60 + 5 * q^61 + 30 * q^62 - 25 * q^63 - 10 * q^64 - 8 * q^65 - 48 * q^66 + 15 * q^68 + 60 * q^69 + 10 * q^70 - 50 * q^71 - 8 * q^72 + 10 * q^73 + 35 * q^75 + 11 * q^76 - 64 * q^77 + 42 * q^79 - 3 * q^80 + 11 * q^81 + 7 * q^82 + 25 * q^83 + 20 * q^84 - 45 * q^85 + 40 * q^86 + 6 * q^88 + 22 * q^90 + 70 * q^91 - 4 * q^92 - 18 * q^93 - 7 * q^94 - 5 * q^95 + 15 * q^97 + 74 * q^98 + 50 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1 0.809017 0.587785i −3.06452 + 0.995722i 0.309017 0.951057i 2.54228 + 1.84707i −1.89398 + 2.60683i −2.55384 0.829793i −0.309017 0.951057i 5.97275 4.33946i 3.14243
151.2 0.809017 0.587785i −2.13440 + 0.693509i 0.309017 0.951057i −0.363676 0.264226i −1.31913 + 1.81563i 1.26293 + 0.410350i −0.309017 0.951057i 1.64766 1.19709i −0.449528
151.3 0.809017 0.587785i −1.02923 + 0.334417i 0.309017 0.951057i −0.0854069 0.0620518i −0.636098 + 0.875514i −4.39704 1.42869i −0.309017 0.951057i −1.47957 + 1.07497i −0.105569
151.4 0.809017 0.587785i −0.871743 + 0.283246i 0.309017 0.951057i −1.71497 1.24600i −0.538767 + 0.741549i −1.29565 0.420982i −0.309017 0.951057i −1.74734 + 1.26952i −2.11982
151.5 0.809017 0.587785i −0.819072 + 0.266133i 0.309017 0.951057i 2.09583 + 1.52271i −0.506214 + 0.696744i 3.24582 + 1.05463i −0.309017 0.951057i −1.82700 + 1.32739i 2.59059
151.6 0.809017 0.587785i 0.809704 0.263089i 0.309017 0.951057i 2.31139 + 1.67932i 0.500425 0.688776i 1.24718 + 0.405232i −0.309017 0.951057i −1.84065 + 1.33731i 2.85703
151.7 0.809017 0.587785i 1.03661 0.336815i 0.309017 0.951057i −1.83162 1.33075i 0.640660 0.881793i 3.63997 + 1.18270i −0.309017 0.951057i −1.46594 + 1.06506i −2.26401
151.8 0.809017 0.587785i 2.05378 0.667313i 0.309017 0.951057i −2.88627 2.09700i 1.26930 1.74705i −3.81661 1.24009i −0.309017 0.951057i 1.34564 0.977666i −3.56763
151.9 0.809017 0.587785i 2.42317 0.787336i 0.309017 0.951057i 1.61580 + 1.17395i 1.49760 2.06127i −2.37273 0.770947i −0.309017 0.951057i 2.82481 2.05234i 1.99724
151.10 0.809017 0.587785i 2.71373 0.881745i 0.309017 0.951057i −1.18335 0.859758i 1.67718 2.30844i 3.23095 + 1.04980i −0.309017 0.951057i 4.15981 3.02228i −1.46271
189.1 −0.309017 + 0.951057i −1.97478 2.71805i −0.809017 0.587785i 0.505434 + 1.55557i 3.19526 1.03820i 2.28733 3.14824i 0.809017 0.587785i −2.56099 + 7.88192i −1.63562
189.2 −0.309017 + 0.951057i −1.43308 1.97246i −0.809017 0.587785i 0.0275048 + 0.0846511i 2.31877 0.753413i −2.13510 + 2.93871i 0.809017 0.587785i −0.909840 + 2.80020i −0.0890075
189.3 −0.309017 + 0.951057i −1.43219 1.97124i −0.809017 0.587785i −1.36407 4.19816i 2.31733 0.752945i 0.381062 0.524486i 0.809017 0.587785i −0.907563 + 2.79319i 4.41421
189.4 −0.309017 + 0.951057i −0.897714 1.23560i −0.809017 0.587785i 1.27104 + 3.91184i 1.45253 0.471956i −0.757296 + 1.04233i 0.809017 0.587785i 0.206241 0.634743i −4.11316
189.5 −0.309017 + 0.951057i −0.238794 0.328671i −0.809017 0.587785i 0.255338 + 0.785850i 0.386376 0.125541i 0.299067 0.411631i 0.809017 0.587785i 0.876049 2.69620i −0.826292
189.6 −0.309017 + 0.951057i 0.366195 + 0.504024i −0.809017 0.587785i −0.493630 1.51924i −0.592516 + 0.192520i 0.534358 0.735481i 0.809017 0.587785i 0.807109 2.48403i 1.59742
189.7 −0.309017 + 0.951057i 0.775778 + 1.06777i −0.809017 0.587785i 1.01291 + 3.11743i −1.25524 + 0.407851i 1.17374 1.61551i 0.809017 0.587785i 0.388756 1.19647i −3.27786
189.8 −0.309017 + 0.951057i 0.943373 + 1.29844i −0.809017 0.587785i −0.757060 2.32999i −1.52641 + 0.495961i −2.75444 + 3.79116i 0.809017 0.587785i 0.131053 0.403339i 2.44990
189.9 −0.309017 + 0.951057i 1.24511 + 1.71374i −0.809017 0.587785i −0.810373 2.49407i −2.01463 + 0.654592i 2.23258 3.07288i 0.809017 0.587785i −0.459575 + 1.41443i 2.62242
189.10 −0.309017 + 0.951057i 1.52806 + 2.10319i −0.809017 0.587785i 0.852902 + 2.62496i −2.47245 + 0.803349i −1.95228 + 2.68709i 0.809017 0.587785i −1.16141 + 3.57445i −2.76005
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 303.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.k even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.m.b yes 40
11.d odd 10 1 418.2.m.a 40
19.b odd 2 1 418.2.m.a 40
209.k even 10 1 inner 418.2.m.b yes 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.m.a 40 11.d odd 10 1
418.2.m.a 40 19.b odd 2 1
418.2.m.b yes 40 1.a even 1 1 trivial
418.2.m.b yes 40 209.k even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} - 19 T_{3}^{38} - 25 T_{3}^{37} + 252 T_{3}^{36} + 475 T_{3}^{35} - 2584 T_{3}^{34} - 5225 T_{3}^{33} + 24636 T_{3}^{32} + 37895 T_{3}^{31} - 162463 T_{3}^{30} - 335915 T_{3}^{29} + 1132571 T_{3}^{28} + \cdots + 65610000$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.