# Properties

 Label 418.2.q.a Level $418$ Weight $2$ Character orbit 418.q Analytic conductor $3.338$ Analytic rank $0$ Dimension $60$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.q (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$10$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 3 q^{3} - 3 q^{6} - 18 q^{7} + 30 q^{8} - 3 q^{9}+O(q^{10})$$ 60 * q - 3 * q^3 - 3 * q^6 - 18 * q^7 + 30 * q^8 - 3 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$60 q - 3 q^{3} - 3 q^{6} - 18 q^{7} + 30 q^{8} - 3 q^{9} + 3 q^{11} + 6 q^{13} - 12 q^{14} + 24 q^{15} - 6 q^{17} + 60 q^{18} - 30 q^{19} - 12 q^{20} + 12 q^{21} + 12 q^{22} + 3 q^{24} - 12 q^{25} + 6 q^{26} + 9 q^{27} + 6 q^{28} - 3 q^{29} - 9 q^{31} - 42 q^{33} - 6 q^{34} - 24 q^{35} - 3 q^{36} + 6 q^{38} + 15 q^{41} + 6 q^{42} - 3 q^{43} + 12 q^{44} - 48 q^{45} + 3 q^{46} + 54 q^{47} - 6 q^{48} + 6 q^{49} + 36 q^{50} - 45 q^{51} - 3 q^{52} + 24 q^{53} - 27 q^{54} + 6 q^{55} + 30 q^{57} + 24 q^{58} - 39 q^{59} + 12 q^{60} + 54 q^{61} - 66 q^{63} - 30 q^{64} - 63 q^{66} + 9 q^{67} - 27 q^{68} + 54 q^{69} - 24 q^{70} - 33 q^{71} - 6 q^{72} + 12 q^{74} + 18 q^{77} + 36 q^{79} - 93 q^{81} - 15 q^{82} - 36 q^{83} + 24 q^{84} - 60 q^{85} - 3 q^{86} + 54 q^{87} - 3 q^{88} - 3 q^{89} - 24 q^{90} - 12 q^{91} - 102 q^{93} - 12 q^{94} + 24 q^{95} - 6 q^{97} - 18 q^{98} - 78 q^{99}+O(q^{100})$$ 60 * q - 3 * q^3 - 3 * q^6 - 18 * q^7 + 30 * q^8 - 3 * q^9 + 3 * q^11 + 6 * q^13 - 12 * q^14 + 24 * q^15 - 6 * q^17 + 60 * q^18 - 30 * q^19 - 12 * q^20 + 12 * q^21 + 12 * q^22 + 3 * q^24 - 12 * q^25 + 6 * q^26 + 9 * q^27 + 6 * q^28 - 3 * q^29 - 9 * q^31 - 42 * q^33 - 6 * q^34 - 24 * q^35 - 3 * q^36 + 6 * q^38 + 15 * q^41 + 6 * q^42 - 3 * q^43 + 12 * q^44 - 48 * q^45 + 3 * q^46 + 54 * q^47 - 6 * q^48 + 6 * q^49 + 36 * q^50 - 45 * q^51 - 3 * q^52 + 24 * q^53 - 27 * q^54 + 6 * q^55 + 30 * q^57 + 24 * q^58 - 39 * q^59 + 12 * q^60 + 54 * q^61 - 66 * q^63 - 30 * q^64 - 63 * q^66 + 9 * q^67 - 27 * q^68 + 54 * q^69 - 24 * q^70 - 33 * q^71 - 6 * q^72 + 12 * q^74 + 18 * q^77 + 36 * q^79 - 93 * q^81 - 15 * q^82 - 36 * q^83 + 24 * q^84 - 60 * q^85 - 3 * q^86 + 54 * q^87 - 3 * q^88 - 3 * q^89 - 24 * q^90 - 12 * q^91 - 102 * q^93 - 12 * q^94 + 24 * q^95 - 6 * q^97 - 18 * q^98 - 78 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1 −0.173648 + 0.984808i −2.10863 + 2.51297i −0.939693 0.342020i 1.20713 0.439360i −2.10863 2.51297i −2.67427 + 1.54399i 0.500000 0.866025i −1.34774 7.64344i 0.223069 + 1.26509i
21.2 −0.173648 + 0.984808i −1.64579 + 1.96137i −0.939693 0.342020i −3.44031 + 1.25217i −1.64579 1.96137i 0.237067 0.136871i 0.500000 0.866025i −0.617423 3.50158i −0.635743 3.60548i
21.3 −0.173648 + 0.984808i −1.49082 + 1.77669i −0.939693 0.342020i 1.76584 0.642714i −1.49082 1.77669i 3.47026 2.00356i 0.500000 0.866025i −0.413143 2.34305i 0.326315 + 1.85062i
21.4 −0.173648 + 0.984808i −0.596920 + 0.711382i −0.939693 0.342020i 0.587101 0.213687i −0.596920 0.711382i −1.73498 + 1.00169i 0.500000 0.866025i 0.371194 + 2.10515i 0.108492 + 0.615288i
21.5 −0.173648 + 0.984808i −0.138226 + 0.164732i −0.939693 0.342020i 3.86845 1.40800i −0.138226 0.164732i −1.22696 + 0.708384i 0.500000 0.866025i 0.512915 + 2.90888i 0.714861 + 4.05418i
21.6 −0.173648 + 0.984808i 0.0268646 0.0320160i −0.939693 0.342020i −0.843173 + 0.306890i 0.0268646 + 0.0320160i 2.35018 1.35688i 0.500000 0.866025i 0.520641 + 2.95270i −0.155812 0.883655i
21.7 −0.173648 + 0.984808i 0.144347 0.172026i −0.939693 0.342020i −2.86607 + 1.04316i 0.144347 + 0.172026i 0.416271 0.240334i 0.500000 0.866025i 0.512188 + 2.90476i −0.529628 3.00367i
21.8 −0.173648 + 0.984808i 0.904669 1.07814i −0.939693 0.342020i −0.316544 + 0.115212i 0.904669 + 1.07814i −4.29189 + 2.47793i 0.500000 0.866025i 0.176980 + 1.00370i −0.0584949 0.331741i
21.9 −0.173648 + 0.984808i 1.60541 1.91325i −0.939693 0.342020i 2.62065 0.953840i 1.60541 + 1.91325i 0.240088 0.138615i 0.500000 0.866025i −0.562250 3.18868i 0.484277 + 2.74647i
21.10 −0.173648 + 0.984808i 2.03306 2.42290i −0.939693 0.342020i −0.703704 + 0.256127i 2.03306 + 2.42290i −0.970561 + 0.560354i 0.500000 0.866025i −1.21619 6.89736i −0.130039 0.737489i
109.1 0.939693 0.342020i −3.24219 0.571685i 0.766044 0.642788i −2.49158 2.09069i −3.24219 + 0.571685i −3.04403 + 1.75747i 0.500000 0.866025i 7.36587 + 2.68096i −3.05638 1.11243i
109.2 0.939693 0.342020i −1.99937 0.352544i 0.766044 0.642788i 0.590160 + 0.495203i −1.99937 + 0.352544i 0.755972 0.436461i 0.500000 0.866025i 1.05413 + 0.383673i 0.723939 + 0.263492i
109.3 0.939693 0.342020i −1.85347 0.326818i 0.766044 0.642788i 2.99121 + 2.50992i −1.85347 + 0.326818i −4.29605 + 2.48032i 0.500000 0.866025i 0.509479 + 0.185435i 3.66926 + 1.33550i
109.4 0.939693 0.342020i −1.39303 0.245629i 0.766044 0.642788i −1.31092 1.09999i −1.39303 + 0.245629i 1.63419 0.943501i 0.500000 0.866025i −0.938877 0.341723i −1.60808 0.585293i
109.5 0.939693 0.342020i −0.274412 0.0483863i 0.766044 0.642788i 1.21736 + 1.02148i −0.274412 + 0.0483863i 2.59958 1.50087i 0.500000 0.866025i −2.74612 0.999505i 1.49331 + 0.543520i
109.6 0.939693 0.342020i 0.00614027 + 0.00108270i 0.766044 0.642788i −3.03422 2.54601i 0.00614027 0.00108270i −2.10812 + 1.21712i 0.500000 0.866025i −2.81904 1.02605i −3.72202 1.35470i
109.7 0.939693 0.342020i 1.58017 + 0.278627i 0.766044 0.642788i −2.47390 2.07585i 1.58017 0.278627i 2.13981 1.23542i 0.500000 0.866025i −0.399773 0.145505i −3.03469 1.10454i
109.8 0.939693 0.342020i 1.64099 + 0.289351i 0.766044 0.642788i 1.95336 + 1.63906i 1.64099 0.289351i 0.632001 0.364886i 0.500000 0.866025i −0.209943 0.0764129i 2.39615 + 0.872127i
109.9 0.939693 0.342020i 2.01819 + 0.355861i 0.766044 0.642788i 1.40938 + 1.18261i 2.01819 0.355861i −2.85785 + 1.64998i 0.500000 0.866025i 1.12738 + 0.410331i 1.72887 + 0.629256i
109.10 0.939693 0.342020i 2.84334 + 0.501357i 0.766044 0.642788i −0.382944 0.321328i 2.84334 0.501357i −0.682188 + 0.393861i 0.500000 0.866025i 5.01412 + 1.82499i −0.469750 0.170975i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 395.10 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.p even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.q.a 60
11.b odd 2 1 418.2.q.b yes 60
19.f odd 18 1 418.2.q.b yes 60
209.p even 18 1 inner 418.2.q.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.q.a 60 1.a even 1 1 trivial
418.2.q.a 60 209.p even 18 1 inner
418.2.q.b yes 60 11.b odd 2 1
418.2.q.b yes 60 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{60} + 18 T_{7}^{59} + 54 T_{7}^{58} - 972 T_{7}^{57} - 6507 T_{7}^{56} + 28458 T_{7}^{55} + 346668 T_{7}^{54} - 193806 T_{7}^{53} - 10886139 T_{7}^{52} - 13289346 T_{7}^{51} + \cdots + 132936103563264$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.