Properties

 Label 418.2.j.d Level $418$ Weight $2$ Character orbit 418.j Analytic conductor $3.338$ Analytic rank $0$ Dimension $30$ 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: $$30$$ Relative dimension: $$5$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 15 q^{8}+O(q^{10})$$ 30 * q - 15 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 15 q^{8} + 15 q^{11} - 3 q^{12} - 21 q^{13} - 9 q^{14} + 3 q^{15} + 6 q^{17} + 60 q^{18} - 9 q^{19} + 18 q^{20} - 39 q^{21} + 15 q^{23} + 24 q^{25} - 3 q^{27} + 9 q^{28} - 3 q^{29} - 21 q^{30} - 18 q^{31} + 15 q^{34} + 51 q^{35} - 18 q^{37} - 6 q^{38} - 6 q^{41} + 51 q^{42} + 39 q^{43} - 54 q^{45} - 21 q^{46} - 3 q^{47} - 33 q^{49} - 24 q^{50} - 48 q^{51} - 12 q^{52} - 24 q^{53} - 9 q^{54} - 6 q^{57} - 18 q^{58} + 21 q^{59} + 3 q^{60} + 63 q^{61} - 27 q^{62} + 57 q^{63} - 15 q^{64} - 6 q^{65} - 45 q^{67} - 21 q^{68} + 42 q^{69} + 51 q^{70} - 48 q^{71} + 87 q^{73} + 9 q^{74} + 42 q^{75} - 9 q^{76} - 36 q^{78} - 57 q^{79} + 36 q^{81} - 6 q^{82} - 30 q^{83} + 9 q^{84} + 81 q^{85} - 24 q^{86} - 9 q^{87} + 15 q^{88} - 6 q^{89} - 114 q^{90} - 51 q^{91} - 3 q^{92} + 33 q^{93} + 78 q^{94} - 132 q^{95} + 6 q^{96} - 66 q^{97} + 9 q^{98}+O(q^{100})$$ 30 * q - 15 * q^8 + 15 * q^11 - 3 * q^12 - 21 * q^13 - 9 * q^14 + 3 * q^15 + 6 * q^17 + 60 * q^18 - 9 * q^19 + 18 * q^20 - 39 * q^21 + 15 * q^23 + 24 * q^25 - 3 * q^27 + 9 * q^28 - 3 * q^29 - 21 * q^30 - 18 * q^31 + 15 * q^34 + 51 * q^35 - 18 * q^37 - 6 * q^38 - 6 * q^41 + 51 * q^42 + 39 * q^43 - 54 * q^45 - 21 * q^46 - 3 * q^47 - 33 * q^49 - 24 * q^50 - 48 * q^51 - 12 * q^52 - 24 * q^53 - 9 * q^54 - 6 * q^57 - 18 * q^58 + 21 * q^59 + 3 * q^60 + 63 * q^61 - 27 * q^62 + 57 * q^63 - 15 * q^64 - 6 * q^65 - 45 * q^67 - 21 * q^68 + 42 * q^69 + 51 * q^70 - 48 * q^71 + 87 * q^73 + 9 * q^74 + 42 * q^75 - 9 * q^76 - 36 * q^78 - 57 * q^79 + 36 * q^81 - 6 * q^82 - 30 * q^83 + 9 * q^84 + 81 * q^85 - 24 * q^86 - 9 * q^87 + 15 * q^88 - 6 * q^89 - 114 * q^90 - 51 * q^91 - 3 * q^92 + 33 * q^93 + 78 * q^94 - 132 * q^95 + 6 * q^96 - 66 * q^97 + 9 * 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 −0.530846 3.01058i 0.766044 + 0.642788i −1.91059 + 1.60317i −0.530846 + 3.01058i 0.304262 0.526998i −0.500000 0.866025i −5.96271 + 2.17025i 2.34368 0.853031i
23.2 −0.939693 0.342020i −0.235710 1.33678i 0.766044 + 0.642788i 2.14622 1.80090i −0.235710 + 1.33678i 2.28653 3.96039i −0.500000 0.866025i 1.08766 0.395875i −2.63273 + 0.958237i
23.3 −0.939693 0.342020i 0.146051 + 0.828296i 0.766044 + 0.642788i −1.03325 + 0.867001i 0.146051 0.828296i 0.273610 0.473907i −0.500000 0.866025i 2.15433 0.784114i 1.26747 0.461322i
23.4 −0.939693 0.342020i 0.197309 + 1.11900i 0.766044 + 0.642788i 2.56288 2.15051i 0.197309 1.11900i −0.480180 + 0.831696i −0.500000 0.866025i 1.60586 0.584483i −3.14383 + 1.14426i
23.5 −0.939693 0.342020i 0.596844 + 3.38487i 0.766044 + 0.642788i 0.532872 0.447132i 0.596844 3.38487i −0.678489 + 1.17518i −0.500000 0.866025i −8.28207 + 3.01443i −0.653664 + 0.237914i
111.1 0.766044 + 0.642788i −2.46726 + 0.898008i 0.173648 + 0.984808i 0.624693 3.54281i −2.46726 0.898008i 2.28982 + 3.96608i −0.500000 + 0.866025i 2.98280 2.50287i 2.75582 2.31240i
111.2 0.766044 + 0.642788i −2.46577 + 0.897465i 0.173648 + 0.984808i −0.00995446 + 0.0564545i −2.46577 0.897465i −1.80146 3.12022i −0.500000 + 0.866025i 2.97642 2.49751i −0.0439138 + 0.0368481i
111.3 0.766044 + 0.642788i −0.317071 + 0.115404i 0.173648 + 0.984808i −0.456329 + 2.58797i −0.317071 0.115404i 0.0977794 + 0.169359i −0.500000 + 0.866025i −2.21092 + 1.85518i −2.01309 + 1.68918i
111.4 0.766044 + 0.642788i 1.64633 0.599214i 0.173648 + 0.984808i 0.477535 2.70824i 1.64633 + 0.599214i −1.91741 3.32106i −0.500000 + 0.866025i 0.0532024 0.0446421i 2.10664 1.76768i
111.5 0.766044 + 0.642788i 2.66407 0.969643i 0.173648 + 0.984808i −0.115000 + 0.652197i 2.66407 + 0.969643i 0.738881 + 1.27978i −0.500000 + 0.866025i 3.85894 3.23803i −0.507320 + 0.425692i
177.1 0.766044 0.642788i −2.46726 0.898008i 0.173648 0.984808i 0.624693 + 3.54281i −2.46726 + 0.898008i 2.28982 3.96608i −0.500000 0.866025i 2.98280 + 2.50287i 2.75582 + 2.31240i
177.2 0.766044 0.642788i −2.46577 0.897465i 0.173648 0.984808i −0.00995446 0.0564545i −2.46577 + 0.897465i −1.80146 + 3.12022i −0.500000 0.866025i 2.97642 + 2.49751i −0.0439138 0.0368481i
177.3 0.766044 0.642788i −0.317071 0.115404i 0.173648 0.984808i −0.456329 2.58797i −0.317071 + 0.115404i 0.0977794 0.169359i −0.500000 0.866025i −2.21092 1.85518i −2.01309 1.68918i
177.4 0.766044 0.642788i 1.64633 + 0.599214i 0.173648 0.984808i 0.477535 + 2.70824i 1.64633 0.599214i −1.91741 + 3.32106i −0.500000 0.866025i 0.0532024 + 0.0446421i 2.10664 + 1.76768i
177.5 0.766044 0.642788i 2.66407 + 0.969643i 0.173648 0.984808i −0.115000 0.652197i 2.66407 0.969643i 0.738881 1.27978i −0.500000 0.866025i 3.85894 + 3.23803i −0.507320 0.425692i
199.1 0.173648 + 0.984808i −2.38321 + 1.99975i −0.939693 + 0.342020i −2.75425 1.00246i −2.38321 1.99975i −2.12923 + 3.68794i −0.500000 0.866025i 1.15974 6.57722i 0.508964 2.88648i
199.2 0.173648 + 0.984808i −1.16564 + 0.978092i −0.939693 + 0.342020i 2.75856 + 1.00403i −1.16564 0.978092i 0.232407 0.402540i −0.500000 0.866025i −0.118881 + 0.674209i −0.509761 + 2.89100i
199.3 0.173648 + 0.984808i 0.854877 0.717327i −0.939693 + 0.342020i −1.23363 0.449003i 0.854877 + 0.717327i −2.13725 + 3.70183i −0.500000 0.866025i −0.304688 + 1.72797i 0.227965 1.29285i
199.4 0.173648 + 0.984808i 1.05344 0.883939i −0.939693 + 0.342020i 2.40688 + 0.876033i 1.05344 + 0.883939i 2.52622 4.37554i −0.500000 0.866025i −0.192562 + 1.09208i −0.444774 + 2.52244i
199.5 0.173648 + 0.984808i 2.40658 2.01936i −0.939693 + 0.342020i −3.99665 1.45466i 2.40658 + 2.01936i 0.394516 0.683322i −0.500000 0.866025i 1.19287 6.76511i 0.738551 4.18853i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 397.5 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.d 30
19.e even 9 1 inner 418.2.j.d 30
19.e even 9 1 7942.2.a.ca 15
19.f odd 18 1 7942.2.a.by 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.j.d 30 1.a even 1 1 trivial
418.2.j.d 30 19.e even 9 1 inner
7942.2.a.by 15 19.f odd 18 1
7942.2.a.ca 15 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{30} + 7 T_{3}^{27} - 9 T_{3}^{26} - 63 T_{3}^{25} + 952 T_{3}^{24} + 735 T_{3}^{23} - 3024 T_{3}^{22} - 3092 T_{3}^{21} - 26649 T_{3}^{20} + 13665 T_{3}^{19} + 826614 T_{3}^{18} - 15939 T_{3}^{17} - 3048342 T_{3}^{16} + \cdots + 12902464$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.