# Properties

 Label 418.2.n.d Level $418$ Weight $2$ Character orbit 418.n Analytic conductor $3.338$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.n (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$8$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 8 q^{2} - 6 q^{3} + 8 q^{4} - 7 q^{5} - 4 q^{6} + 22 q^{7} + 16 q^{8} + 14 q^{9}+O(q^{10})$$ 64 * q - 8 * q^2 - 6 * q^3 + 8 * q^4 - 7 * q^5 - 4 * q^6 + 22 * q^7 + 16 * q^8 + 14 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$64 q - 8 q^{2} - 6 q^{3} + 8 q^{4} - 7 q^{5} - 4 q^{6} + 22 q^{7} + 16 q^{8} + 14 q^{9} - 8 q^{10} - 6 q^{11} - 8 q^{12} + 9 q^{13} + 11 q^{14} + 9 q^{15} + 8 q^{16} - 2 q^{17} - 12 q^{18} + 4 q^{19} + 14 q^{20} - 36 q^{21} + 7 q^{22} + 8 q^{23} - 4 q^{24} + 31 q^{25} - 12 q^{26} + 54 q^{27} + 9 q^{28} + 18 q^{29} + 18 q^{30} + 20 q^{31} + 32 q^{32} + 10 q^{33} + 2 q^{34} - 16 q^{35} - 6 q^{36} + 18 q^{37} - 31 q^{38} + 2 q^{39} - 3 q^{40} + 16 q^{41} + 6 q^{42} + 42 q^{43} - 2 q^{44} - 8 q^{45} - 24 q^{46} - 34 q^{47} - 6 q^{48} - 10 q^{49} - 58 q^{50} - 40 q^{51} - 6 q^{52} + 15 q^{53} - 28 q^{54} + 49 q^{55} + 8 q^{56} + 8 q^{57} + 36 q^{58} - 7 q^{59} + 4 q^{60} - 15 q^{61} - 37 q^{63} - 16 q^{64} - 48 q^{65} - 10 q^{66} - 14 q^{67} + 4 q^{68} - 30 q^{69} - 19 q^{70} - 4 q^{71} - 14 q^{72} + 8 q^{73} + 9 q^{74} - 96 q^{75} - 10 q^{76} - 58 q^{77} + 46 q^{78} + 12 q^{79} + 3 q^{80} - 8 q^{81} + 4 q^{82} - 6 q^{83} - 48 q^{84} + 18 q^{85} + 3 q^{86} - 244 q^{87} + 6 q^{88} - 4 q^{89} - 9 q^{90} - 33 q^{91} + 8 q^{92} + 3 q^{93} + 62 q^{94} - 49 q^{95} - 12 q^{96} - 15 q^{97} - 4 q^{99}+O(q^{100})$$ 64 * q - 8 * q^2 - 6 * q^3 + 8 * q^4 - 7 * q^5 - 4 * q^6 + 22 * q^7 + 16 * q^8 + 14 * q^9 - 8 * q^10 - 6 * q^11 - 8 * q^12 + 9 * q^13 + 11 * q^14 + 9 * q^15 + 8 * q^16 - 2 * q^17 - 12 * q^18 + 4 * q^19 + 14 * q^20 - 36 * q^21 + 7 * q^22 + 8 * q^23 - 4 * q^24 + 31 * q^25 - 12 * q^26 + 54 * q^27 + 9 * q^28 + 18 * q^29 + 18 * q^30 + 20 * q^31 + 32 * q^32 + 10 * q^33 + 2 * q^34 - 16 * q^35 - 6 * q^36 + 18 * q^37 - 31 * q^38 + 2 * q^39 - 3 * q^40 + 16 * q^41 + 6 * q^42 + 42 * q^43 - 2 * q^44 - 8 * q^45 - 24 * q^46 - 34 * q^47 - 6 * q^48 - 10 * q^49 - 58 * q^50 - 40 * q^51 - 6 * q^52 + 15 * q^53 - 28 * q^54 + 49 * q^55 + 8 * q^56 + 8 * q^57 + 36 * q^58 - 7 * q^59 + 4 * q^60 - 15 * q^61 - 37 * q^63 - 16 * q^64 - 48 * q^65 - 10 * q^66 - 14 * q^67 + 4 * q^68 - 30 * q^69 - 19 * q^70 - 4 * q^71 - 14 * q^72 + 8 * q^73 + 9 * q^74 - 96 * q^75 - 10 * q^76 - 58 * q^77 + 46 * q^78 + 12 * q^79 + 3 * q^80 - 8 * q^81 + 4 * q^82 - 6 * q^83 - 48 * q^84 + 18 * q^85 + 3 * q^86 - 244 * q^87 + 6 * q^88 - 4 * q^89 - 9 * q^90 - 33 * q^91 + 8 * q^92 + 3 * q^93 + 62 * q^94 - 49 * q^95 - 12 * q^96 - 15 * q^97 - 4 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 0.978148 + 0.207912i −2.78374 + 1.23940i 0.913545 + 0.406737i −1.23365 1.37011i −2.98059 + 0.633544i 2.74006 1.99077i 0.809017 + 0.587785i 4.20569 4.67089i −0.921833 1.59666i
49.2 0.978148 + 0.207912i −1.41989 + 0.632177i 0.913545 + 0.406737i 1.85966 + 2.06536i −1.52030 + 0.323150i 3.87852 2.81791i 0.809017 + 0.587785i −0.390947 + 0.434190i 1.38961 + 2.40687i
49.3 0.978148 + 0.207912i −1.22145 + 0.543824i 0.913545 + 0.406737i −1.54111 1.71158i −1.30782 + 0.277987i −1.94898 + 1.41602i 0.809017 + 0.587785i −0.811200 + 0.900929i −1.15158 1.99459i
49.4 0.978148 + 0.207912i −0.876283 + 0.390146i 0.913545 + 0.406737i 2.35086 + 2.61090i −0.938250 + 0.199431i −1.45336 + 1.05593i 0.809017 + 0.587785i −1.39173 + 1.54568i 1.75666 + 3.04262i
49.5 0.978148 + 0.207912i −0.690099 + 0.307252i 0.913545 + 0.406737i −2.78499 3.09304i −0.738900 + 0.157058i 1.79660 1.30531i 0.809017 + 0.587785i −1.62556 + 1.80537i −2.08105 3.60448i
49.6 0.978148 + 0.207912i 0.375886 0.167355i 0.913545 + 0.406737i 1.03755 + 1.15232i 0.402467 0.0855471i −0.764726 + 0.555606i 0.809017 + 0.587785i −1.89411 + 2.10362i 0.775298 + 1.34285i
49.7 0.978148 + 0.207912i 1.44379 0.642816i 0.913545 + 0.406737i −0.891991 0.990657i 1.54589 0.328588i 1.29335 0.939674i 0.809017 + 0.587785i −0.336079 + 0.373254i −0.666530 1.15446i
49.8 0.978148 + 0.207912i 2.21549 0.986399i 0.913545 + 0.406737i 1.36163 + 1.51224i 2.37216 0.504218i −1.11442 + 0.809676i 0.809017 + 0.587785i 1.92802 2.14128i 1.01746 + 1.76229i
125.1 −0.913545 + 0.406737i −1.96426 2.18153i 0.669131 0.743145i −0.0453791 0.431753i 2.68175 + 1.19399i 0.873693 + 2.68895i −0.309017 + 0.951057i −0.587180 + 5.58664i 0.217066 + 0.375969i
125.2 −0.913545 + 0.406737i −1.84910 2.05364i 0.669131 0.743145i 0.213696 + 2.03318i 2.52453 + 1.12399i −0.473002 1.45575i −0.309017 + 0.951057i −0.484658 + 4.61121i −1.02219 1.77049i
125.3 −0.913545 + 0.406737i −0.311198 0.345621i 0.669131 0.743145i 0.187144 + 1.78056i 0.424871 + 0.189165i −0.928913 2.85890i −0.309017 + 0.951057i 0.290976 2.76845i −0.895184 1.55050i
125.4 −0.913545 + 0.406737i 0.0875155 + 0.0971958i 0.669131 0.743145i 0.00825402 + 0.0785317i −0.119482 0.0531970i 1.37067 + 4.21848i −0.309017 + 0.951057i 0.311797 2.96655i −0.0394821 0.0683851i
125.5 −0.913545 + 0.406737i 0.344284 + 0.382366i 0.669131 0.743145i 0.0740341 + 0.704387i −0.470041 0.209276i −0.424117 1.30530i −0.309017 + 0.951057i 0.285913 2.72028i −0.354134 0.613377i
125.6 −0.913545 + 0.406737i 1.27149 + 1.41213i 0.669131 0.743145i −0.245336 2.33422i −1.73593 0.772884i −0.645619 1.98701i −0.309017 + 0.951057i −0.0638446 + 0.607441i 1.17354 + 2.03263i
125.7 −0.913545 + 0.406737i 1.35422 + 1.50401i 0.669131 0.743145i 0.371716 + 3.53664i −1.84887 0.823172i 1.09191 + 3.36056i −0.309017 + 0.951057i −0.114558 + 1.08994i −1.77806 3.07970i
125.8 −0.913545 + 0.406737i 1.89415 + 2.10367i 0.669131 0.743145i −0.121340 1.15447i −2.58603 1.15138i 0.208330 + 0.641173i −0.309017 + 0.951057i −0.524026 + 4.98578i 0.580415 + 1.00531i
159.1 −0.669131 + 0.743145i −0.253498 + 2.41187i −0.104528 0.994522i −1.99045 + 0.423084i −1.62275 1.80224i −1.11442 + 0.809676i 0.809017 + 0.587785i −2.81841 0.599072i 1.01746 1.76229i
159.2 −0.669131 + 0.743145i −0.165199 + 1.57177i −0.104528 0.994522i 1.30393 0.277159i −1.05751 1.17448i 1.29335 0.939674i 0.809017 + 0.587785i 0.491287 + 0.104426i −0.666530 + 1.15446i
159.3 −0.669131 + 0.743145i −0.0430092 + 0.409205i −0.104528 0.994522i −1.51671 + 0.322387i −0.275320 0.305774i −0.764726 + 0.555606i 0.809017 + 0.587785i 2.76884 + 0.588536i 0.775298 1.34285i
159.4 −0.669131 + 0.743145i 0.0789616 0.751269i −0.104528 0.994522i 4.07114 0.865348i 0.505466 + 0.561377i 1.79660 1.30531i 0.809017 + 0.587785i 2.37627 + 0.505092i −2.08105 + 3.60448i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 311.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
19.c even 3 1 inner
209.n even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.n.d 64
11.c even 5 1 inner 418.2.n.d 64
19.c even 3 1 inner 418.2.n.d 64
209.n even 15 1 inner 418.2.n.d 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.n.d 64 1.a even 1 1 trivial
418.2.n.d 64 11.c even 5 1 inner
418.2.n.d 64 19.c even 3 1 inner
418.2.n.d 64 209.n even 15 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{64} + 6 T_{3}^{63} - T_{3}^{62} - 80 T_{3}^{61} - 140 T_{3}^{60} + 214 T_{3}^{59} + 758 T_{3}^{58} + 1862 T_{3}^{57} + 4123 T_{3}^{56} - 25024 T_{3}^{55} - 68168 T_{3}^{54} + 346291 T_{3}^{53} + 1139555 T_{3}^{52} + \cdots + 160000$$ acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$.