# Properties

 Label 418.2.j.a Level $418$ Weight $2$ Character orbit 418.j Analytic conductor $3.338$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.j (of order $$9$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$4$$ over $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 12 q^{8}+O(q^{10})$$ 24 * q + 12 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 12 q^{8} + 12 q^{11} - 3 q^{12} - 3 q^{13} + 3 q^{14} + 27 q^{15} - 6 q^{18} - 21 q^{19} - 18 q^{20} + 15 q^{21} + 9 q^{23} + 36 q^{25} - 21 q^{27} - 3 q^{28} - 9 q^{30} - 27 q^{31} - 9 q^{34} - 45 q^{35} + 18 q^{37} + 9 q^{38} + 36 q^{39} - 18 q^{41} + 39 q^{42} - 48 q^{43} + 36 q^{45} - 18 q^{46} - 9 q^{47} + 6 q^{49} + 3 q^{50} - 18 q^{51} - 3 q^{52} - 36 q^{53} - 45 q^{54} + 18 q^{58} + 9 q^{59} - 9 q^{60} + 15 q^{61} - 33 q^{62} + 87 q^{63} - 12 q^{64} - 36 q^{65} + 33 q^{67} + 9 q^{68} - 18 q^{69} + 45 q^{70} - 9 q^{71} - 3 q^{73} + 3 q^{74} + 42 q^{75} + 9 q^{76} + 12 q^{78} + 15 q^{79} - 108 q^{81} + 18 q^{82} + 36 q^{83} - 9 q^{84} - 99 q^{85} - 33 q^{86} + 63 q^{87} - 12 q^{88} - 27 q^{89} - 36 q^{90} - 21 q^{91} - 9 q^{92} - 21 q^{93} + 54 q^{94} + 18 q^{95} - 6 q^{96} + 45 q^{97} + 39 q^{98}+O(q^{100})$$ 24 * q + 12 * q^8 + 12 * q^11 - 3 * q^12 - 3 * q^13 + 3 * q^14 + 27 * q^15 - 6 * q^18 - 21 * q^19 - 18 * q^20 + 15 * q^21 + 9 * q^23 + 36 * q^25 - 21 * q^27 - 3 * q^28 - 9 * q^30 - 27 * q^31 - 9 * q^34 - 45 * q^35 + 18 * q^37 + 9 * q^38 + 36 * q^39 - 18 * q^41 + 39 * q^42 - 48 * q^43 + 36 * q^45 - 18 * q^46 - 9 * q^47 + 6 * q^49 + 3 * q^50 - 18 * q^51 - 3 * q^52 - 36 * q^53 - 45 * q^54 + 18 * q^58 + 9 * q^59 - 9 * q^60 + 15 * q^61 - 33 * q^62 + 87 * q^63 - 12 * q^64 - 36 * q^65 + 33 * q^67 + 9 * q^68 - 18 * q^69 + 45 * q^70 - 9 * q^71 - 3 * q^73 + 3 * q^74 + 42 * q^75 + 9 * q^76 + 12 * q^78 + 15 * q^79 - 108 * q^81 + 18 * q^82 + 36 * q^83 - 9 * q^84 - 99 * q^85 - 33 * q^86 + 63 * q^87 - 12 * q^88 - 27 * q^89 - 36 * q^90 - 21 * q^91 - 9 * q^92 - 21 * q^93 + 54 * q^94 + 18 * q^95 - 6 * q^96 + 45 * q^97 + 39 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1 0.939693 + 0.342020i −0.293621 1.66521i 0.766044 + 0.642788i −3.02184 + 2.53563i 0.293621 1.66521i −1.16294 + 2.01427i 0.500000 + 0.866025i 0.132366 0.0481775i −3.70684 + 1.34918i
23.2 0.939693 + 0.342020i −0.232600 1.31914i 0.766044 + 0.642788i 0.527700 0.442792i 0.232600 1.31914i 0.106944 0.185233i 0.500000 + 0.866025i 1.13305 0.412397i 0.647319 0.235605i
23.3 0.939693 + 0.342020i 0.291741 + 1.65454i 0.766044 + 0.642788i 1.53274 1.28612i −0.291741 + 1.65454i 1.45368 2.51785i 0.500000 + 0.866025i 0.166674 0.0606642i 1.88018 0.684329i
23.4 0.939693 + 0.342020i 0.408129 + 2.31461i 0.766044 + 0.642788i −1.33673 + 1.12165i −0.408129 + 2.31461i −0.571336 + 0.989583i 0.500000 + 0.866025i −2.37179 + 0.863259i −1.63974 + 0.596815i
111.1 −0.766044 0.642788i −2.02265 + 0.736184i 0.173648 + 0.984808i −0.505679 + 2.86785i 2.02265 + 0.736184i 1.62478 + 2.81421i 0.500000 0.866025i 1.25101 1.04972i 2.23079 1.87186i
111.2 −0.766044 0.642788i −1.50994 + 0.549572i 0.173648 + 0.984808i 0.142998 0.810982i 1.50994 + 0.549572i −0.0412325 0.0714168i 0.500000 0.866025i −0.320257 + 0.268728i −0.630832 + 0.529331i
111.3 −0.766044 0.642788i 0.361908 0.131724i 0.173648 + 0.984808i −0.0315845 + 0.179124i −0.361908 0.131724i −2.25138 3.89951i 0.500000 0.866025i −2.18451 + 1.83302i 0.139334 0.116915i
111.4 −0.766044 0.642788i 2.23098 0.812012i 0.173648 + 0.984808i −0.126679 + 0.718430i −2.23098 0.812012i 1.60753 + 2.78432i 0.500000 0.866025i 2.01980 1.69481i 0.558839 0.468922i
177.1 −0.766044 + 0.642788i −2.02265 0.736184i 0.173648 0.984808i −0.505679 2.86785i 2.02265 0.736184i 1.62478 2.81421i 0.500000 + 0.866025i 1.25101 + 1.04972i 2.23079 + 1.87186i
177.2 −0.766044 + 0.642788i −1.50994 0.549572i 0.173648 0.984808i 0.142998 + 0.810982i 1.50994 0.549572i −0.0412325 + 0.0714168i 0.500000 + 0.866025i −0.320257 0.268728i −0.630832 0.529331i
177.3 −0.766044 + 0.642788i 0.361908 + 0.131724i 0.173648 0.984808i −0.0315845 0.179124i −0.361908 + 0.131724i −2.25138 + 3.89951i 0.500000 + 0.866025i −2.18451 1.83302i 0.139334 + 0.116915i
177.4 −0.766044 + 0.642788i 2.23098 + 0.812012i 0.173648 0.984808i −0.126679 0.718430i −2.23098 + 0.812012i 1.60753 2.78432i 0.500000 + 0.866025i 2.01980 + 1.69481i 0.558839 + 0.468922i
199.1 −0.173648 0.984808i −1.23590 + 1.03704i −0.939693 + 0.342020i −0.846117 0.307961i 1.23590 + 1.03704i 1.44710 2.50644i 0.500000 + 0.866025i −0.0689564 + 0.391071i −0.156356 + 0.886739i
199.2 −0.173648 0.984808i −0.404607 + 0.339506i −0.939693 + 0.342020i 2.97940 + 1.08441i 0.404607 + 0.339506i 0.0238226 0.0412620i 0.500000 + 0.866025i −0.472502 + 2.67969i 0.550571 3.12244i
199.3 −0.173648 0.984808i −0.0299784 + 0.0251548i −0.939693 + 0.342020i −2.55820 0.931110i 0.0299784 + 0.0251548i −0.483905 + 0.838147i 0.500000 + 0.866025i −0.520679 + 2.95292i −0.472737 + 2.68103i
199.4 −0.173648 0.984808i 2.43653 2.04449i −0.939693 + 0.342020i 3.24400 + 1.18072i −2.43653 2.04449i −1.75306 + 3.03639i 0.500000 + 0.866025i 1.23578 7.00848i 0.599467 3.39975i
309.1 0.939693 0.342020i −0.293621 + 1.66521i 0.766044 0.642788i −3.02184 2.53563i 0.293621 + 1.66521i −1.16294 2.01427i 0.500000 0.866025i 0.132366 + 0.0481775i −3.70684 1.34918i
309.2 0.939693 0.342020i −0.232600 + 1.31914i 0.766044 0.642788i 0.527700 + 0.442792i 0.232600 + 1.31914i 0.106944 + 0.185233i 0.500000 0.866025i 1.13305 + 0.412397i 0.647319 + 0.235605i
309.3 0.939693 0.342020i 0.291741 1.65454i 0.766044 0.642788i 1.53274 + 1.28612i −0.291741 1.65454i 1.45368 + 2.51785i 0.500000 0.866025i 0.166674 + 0.0606642i 1.88018 + 0.684329i
309.4 0.939693 0.342020i 0.408129 2.31461i 0.766044 0.642788i −1.33673 1.12165i −0.408129 2.31461i −0.571336 0.989583i 0.500000 0.866025i −2.37179 0.863259i −1.63974 0.596815i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 397.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.j.a 24
19.e even 9 1 inner 418.2.j.a 24
19.e even 9 1 7942.2.a.bt 12
19.f odd 18 1 7942.2.a.bx 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.j.a 24 1.a even 1 1 trivial
418.2.j.a 24 19.e even 9 1 inner
7942.2.a.bt 12 19.e even 9 1
7942.2.a.bx 12 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} + 13 T_{3}^{21} + 27 T_{3}^{20} + 99 T_{3}^{19} + 346 T_{3}^{18} + 333 T_{3}^{17} - 54 T_{3}^{16} - 1134 T_{3}^{15} - 945 T_{3}^{14} + 8973 T_{3}^{13} + 44574 T_{3}^{12} + 114003 T_{3}^{11} + 218682 T_{3}^{10} + 317238 T_{3}^{9} + \cdots + 9$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.