# Properties

 Label 418.2.j.c 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 12 q^{7} + 15 q^{8}+O(q^{10})$$ 30 * q - 12 * q^7 + 15 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 12 q^{7} + 15 q^{8} - 15 q^{11} + 3 q^{12} - 3 q^{13} + 9 q^{14} + 27 q^{15} - 36 q^{18} - 9 q^{19} + 18 q^{20} - 27 q^{21} - 3 q^{23} - 12 q^{25} + 3 q^{27} + 9 q^{28} + 3 q^{29} + 9 q^{30} + 30 q^{31} - 9 q^{34} + 15 q^{35} + 18 q^{37} + 6 q^{38} - 6 q^{41} - 45 q^{42} + 39 q^{43} - 18 q^{45} + 21 q^{46} + 45 q^{47} - 33 q^{49} + 36 q^{50} - 36 q^{51} + 6 q^{52} - 24 q^{53} + 45 q^{54} - 24 q^{56} - 24 q^{57} - 30 q^{58} + 3 q^{59} - 9 q^{60} - 27 q^{61} + 15 q^{62} - 93 q^{63} - 15 q^{64} + 18 q^{65} - 9 q^{67} - 21 q^{68} + 48 q^{69} - 15 q^{70} + 39 q^{73} + 3 q^{74} - 42 q^{75} - 15 q^{76} + 24 q^{77} + 6 q^{78} + 21 q^{79} + 84 q^{81} + 6 q^{82} - 36 q^{83} - 27 q^{84} + 63 q^{85} + 6 q^{86} - 21 q^{87} + 15 q^{88} + 54 q^{89} + 12 q^{90} + 3 q^{91} - 3 q^{92} + 51 q^{93} - 78 q^{94} + 6 q^{95} + 6 q^{96} - 18 q^{97} + 3 q^{98}+O(q^{100})$$ 30 * q - 12 * q^7 + 15 * q^8 - 15 * q^11 + 3 * q^12 - 3 * q^13 + 9 * q^14 + 27 * q^15 - 36 * q^18 - 9 * q^19 + 18 * q^20 - 27 * q^21 - 3 * q^23 - 12 * q^25 + 3 * q^27 + 9 * q^28 + 3 * q^29 + 9 * q^30 + 30 * q^31 - 9 * q^34 + 15 * q^35 + 18 * q^37 + 6 * q^38 - 6 * q^41 - 45 * q^42 + 39 * q^43 - 18 * q^45 + 21 * q^46 + 45 * q^47 - 33 * q^49 + 36 * q^50 - 36 * q^51 + 6 * q^52 - 24 * q^53 + 45 * q^54 - 24 * q^56 - 24 * q^57 - 30 * q^58 + 3 * q^59 - 9 * q^60 - 27 * q^61 + 15 * q^62 - 93 * q^63 - 15 * q^64 + 18 * q^65 - 9 * q^67 - 21 * q^68 + 48 * q^69 - 15 * q^70 + 39 * q^73 + 3 * q^74 - 42 * q^75 - 15 * q^76 + 24 * q^77 + 6 * q^78 + 21 * q^79 + 84 * q^81 + 6 * q^82 - 36 * q^83 - 27 * q^84 + 63 * q^85 + 6 * q^86 - 21 * q^87 + 15 * q^88 + 54 * q^89 + 12 * q^90 + 3 * q^91 - 3 * q^92 + 51 * q^93 - 78 * q^94 + 6 * q^95 + 6 * q^96 - 18 * q^97 + 3 * 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 −0.506748 2.87391i 0.766044 + 0.642788i −1.93343 + 1.62234i 0.506748 2.87391i 2.38935 4.13848i 0.500000 + 0.866025i −5.18349 + 1.88664i −2.37170 + 0.863229i
23.2 0.939693 + 0.342020i −0.315286 1.78808i 0.766044 + 0.642788i 2.76685 2.32166i 0.315286 1.78808i −0.752408 + 1.30321i 0.500000 + 0.866025i −0.278741 + 0.101453i 3.39404 1.23533i
23.3 0.939693 + 0.342020i 0.0304860 + 0.172895i 0.766044 + 0.642788i −0.190742 + 0.160051i −0.0304860 + 0.172895i 1.48347 2.56944i 0.500000 + 0.866025i 2.79011 1.01552i −0.233979 + 0.0851616i
23.4 0.939693 + 0.342020i 0.102554 + 0.581612i 0.766044 + 0.642788i −1.23439 + 1.03577i −0.102554 + 0.581612i −1.96792 + 3.40854i 0.500000 + 0.866025i 2.49132 0.906767i −1.51420 + 0.551123i
23.5 0.939693 + 0.342020i 0.515346 + 2.92267i 0.766044 + 0.642788i 2.88984 2.42486i −0.515346 + 2.92267i −1.44675 + 2.50585i 0.500000 + 0.866025i −5.45736 + 1.98632i 3.54491 1.29024i
111.1 −0.766044 0.642788i −3.00494 + 1.09371i 0.173648 + 0.984808i −0.0619925 + 0.351577i 3.00494 + 1.09371i −2.21848 3.84252i 0.500000 0.866025i 5.53533 4.64469i 0.273478 0.229475i
111.2 −0.766044 0.642788i −0.305857 + 0.111323i 0.173648 + 0.984808i 0.749682 4.25166i 0.305857 + 0.111323i −1.19826 2.07545i 0.500000 0.866025i −2.21698 + 1.86027i −3.30720 + 2.77507i
111.3 −0.766044 0.642788i 0.0916892 0.0333721i 0.173648 + 0.984808i 0.327930 1.85978i −0.0916892 0.0333721i 1.90575 + 3.30086i 0.500000 0.866025i −2.29084 + 1.92224i −1.44665 + 1.21389i
111.4 −0.766044 0.642788i 1.46132 0.531876i 0.173648 + 0.984808i −0.712779 + 4.04237i −1.46132 0.531876i −0.291586 0.505042i 0.500000 0.866025i −0.445574 + 0.373881i 3.14441 2.63847i
111.5 −0.766044 0.642788i 2.69748 0.981803i 0.173648 + 0.984808i 0.218104 1.23693i −2.69748 0.981803i −0.789822 1.36801i 0.500000 0.866025i 4.01433 3.36842i −0.962162 + 0.807349i
177.1 −0.766044 + 0.642788i −3.00494 1.09371i 0.173648 0.984808i −0.0619925 0.351577i 3.00494 1.09371i −2.21848 + 3.84252i 0.500000 + 0.866025i 5.53533 + 4.64469i 0.273478 + 0.229475i
177.2 −0.766044 + 0.642788i −0.305857 0.111323i 0.173648 0.984808i 0.749682 + 4.25166i 0.305857 0.111323i −1.19826 + 2.07545i 0.500000 + 0.866025i −2.21698 1.86027i −3.30720 2.77507i
177.3 −0.766044 + 0.642788i 0.0916892 + 0.0333721i 0.173648 0.984808i 0.327930 + 1.85978i −0.0916892 + 0.0333721i 1.90575 3.30086i 0.500000 + 0.866025i −2.29084 1.92224i −1.44665 1.21389i
177.4 −0.766044 + 0.642788i 1.46132 + 0.531876i 0.173648 0.984808i −0.712779 4.04237i −1.46132 + 0.531876i −0.291586 + 0.505042i 0.500000 + 0.866025i −0.445574 0.373881i 3.14441 + 2.63847i
177.5 −0.766044 + 0.642788i 2.69748 + 0.981803i 0.173648 0.984808i 0.218104 + 1.23693i −2.69748 + 0.981803i −0.789822 + 1.36801i 0.500000 + 0.866025i 4.01433 + 3.36842i −0.962162 0.807349i
199.1 −0.173648 0.984808i −2.04235 + 1.71374i −0.939693 + 0.342020i −3.54102 1.28883i 2.04235 + 1.71374i 0.168894 0.292534i 0.500000 + 0.866025i 0.713363 4.04568i −0.654355 + 3.71103i
199.2 −0.173648 0.984808i −1.95711 + 1.64221i −0.939693 + 0.342020i 0.968941 + 0.352666i 1.95711 + 1.64221i −1.38692 + 2.40222i 0.500000 + 0.866025i 0.612477 3.47353i 0.179053 1.01546i
199.3 −0.173648 0.984808i 0.345094 0.289568i −0.939693 + 0.342020i 1.87337 + 0.681850i −0.345094 0.289568i −1.54380 + 2.67395i 0.500000 + 0.866025i −0.485704 + 2.75457i 0.346185 1.96331i
199.4 −0.173648 0.984808i 1.31855 1.10640i −0.939693 + 0.342020i −3.37653 1.22895i −1.31855 1.10640i −1.75103 + 3.03286i 0.500000 + 0.866025i −0.00647669 + 0.0367312i −0.623957 + 3.53863i
199.5 −0.173648 0.984808i 1.56977 1.31719i −0.939693 + 0.342020i 1.25616 + 0.457206i −1.56977 1.31719i 1.39952 2.42403i 0.500000 + 0.866025i 0.208231 1.18093i 0.232130 1.31647i
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.c 30
19.e even 9 1 inner 418.2.j.c 30
19.e even 9 1 7942.2.a.bz 15
19.f odd 18 1 7942.2.a.cb 15

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{30} - 7 T_{3}^{27} - 21 T_{3}^{26} - 81 T_{3}^{25} + 644 T_{3}^{24} - 129 T_{3}^{23} + 3042 T_{3}^{22} - 5648 T_{3}^{21} + 11619 T_{3}^{20} - 91479 T_{3}^{19} + 399310 T_{3}^{18} - 568311 T_{3}^{17} + 927954 T_{3}^{16} + \cdots + 64$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.