# Properties

 Label 414.2.j.a Level $414$ Weight $2$ Character orbit 414.j Analytic conductor $3.306$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 414.j (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.30580664368$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$8$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 8 q^{4}+O(q^{10})$$ 80 * q + 8 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 8 q^{4} - 16 q^{13} - 8 q^{16} + 24 q^{25} - 16 q^{31} + 88 q^{37} + 88 q^{43} + 8 q^{46} + 8 q^{49} + 16 q^{52} - 32 q^{55} - 72 q^{58} - 176 q^{61} + 8 q^{64} - 88 q^{67} - 176 q^{70} - 56 q^{73} - 176 q^{79} - 88 q^{82} - 88 q^{85} + 16 q^{94}+O(q^{100})$$ 80 * q + 8 * q^4 - 16 * q^13 - 8 * q^16 + 24 * q^25 - 16 * q^31 + 88 * q^37 + 88 * q^43 + 8 * q^46 + 8 * q^49 + 16 * q^52 - 32 * q^55 - 72 * q^58 - 176 * q^61 + 8 * q^64 - 88 * q^67 - 176 * q^70 - 56 * q^73 - 176 * q^79 - 88 * q^82 - 88 * q^85 + 16 * q^94

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −0.909632 0.415415i 0 0.654861 + 0.755750i −2.75649 + 1.77149i 0 −0.318197 0.0457498i −0.281733 0.959493i 0 3.24329 0.466315i
17.2 −0.909632 0.415415i 0 0.654861 + 0.755750i −2.01516 + 1.29506i 0 0.691369 + 0.0994038i −0.281733 0.959493i 0 2.37104 0.340905i
17.3 −0.909632 0.415415i 0 0.654861 + 0.755750i 2.08613 1.34067i 0 2.86484 + 0.411901i −0.281733 0.959493i 0 −2.45454 + 0.352910i
17.4 −0.909632 0.415415i 0 0.654861 + 0.755750i 2.68552 1.72588i 0 −3.23801 0.465555i −0.281733 0.959493i 0 −3.15979 + 0.454310i
17.5 0.909632 + 0.415415i 0 0.654861 + 0.755750i −2.68552 + 1.72588i 0 −3.23801 0.465555i 0.281733 + 0.959493i 0 −3.15979 + 0.454310i
17.6 0.909632 + 0.415415i 0 0.654861 + 0.755750i −2.08613 + 1.34067i 0 2.86484 + 0.411901i 0.281733 + 0.959493i 0 −2.45454 + 0.352910i
17.7 0.909632 + 0.415415i 0 0.654861 + 0.755750i 2.01516 1.29506i 0 0.691369 + 0.0994038i 0.281733 + 0.959493i 0 2.37104 0.340905i
17.8 0.909632 + 0.415415i 0 0.654861 + 0.755750i 2.75649 1.77149i 0 −0.318197 0.0457498i 0.281733 + 0.959493i 0 3.24329 0.466315i
53.1 −0.755750 0.654861i 0 0.142315 + 0.989821i −0.959224 2.10041i 0 0.976881 + 3.32695i 0.540641 0.841254i 0 −0.650541 + 2.21554i
53.2 −0.755750 0.654861i 0 0.142315 + 0.989821i −0.922959 2.02100i 0 0.0130977 + 0.0446065i 0.540641 0.841254i 0 −0.625947 + 2.13178i
53.3 −0.755750 0.654861i 0 0.142315 + 0.989821i 0.323986 + 0.709431i 0 −0.374655 1.27596i 0.540641 0.841254i 0 0.219726 0.748318i
53.4 −0.755750 0.654861i 0 0.142315 + 0.989821i 1.55820 + 3.41197i 0 −0.615324 2.09560i 0.540641 0.841254i 0 1.05676 3.59900i
53.5 0.755750 + 0.654861i 0 0.142315 + 0.989821i −1.55820 3.41197i 0 −0.615324 2.09560i −0.540641 + 0.841254i 0 1.05676 3.59900i
53.6 0.755750 + 0.654861i 0 0.142315 + 0.989821i −0.323986 0.709431i 0 −0.374655 1.27596i −0.540641 + 0.841254i 0 0.219726 0.748318i
53.7 0.755750 + 0.654861i 0 0.142315 + 0.989821i 0.922959 + 2.02100i 0 0.0130977 + 0.0446065i −0.540641 + 0.841254i 0 −0.625947 + 2.13178i
53.8 0.755750 + 0.654861i 0 0.142315 + 0.989821i 0.959224 + 2.10041i 0 0.976881 + 3.32695i −0.540641 + 0.841254i 0 −0.650541 + 2.21554i
89.1 −0.989821 + 0.142315i 0 0.959493 0.281733i −0.922790 + 1.06496i 0 2.78807 + 4.33833i −0.909632 + 0.415415i 0 0.761838 1.18544i
89.2 −0.989821 + 0.142315i 0 0.959493 0.281733i −0.762054 + 0.879457i 0 −1.27448 1.98312i −0.909632 + 0.415415i 0 0.629138 0.978957i
89.3 −0.989821 + 0.142315i 0 0.959493 0.281733i −0.408438 + 0.471363i 0 −2.27479 3.53964i −0.909632 + 0.415415i 0 0.337199 0.524692i
89.4 −0.989821 + 0.142315i 0 0.959493 0.281733i 2.09328 2.41578i 0 0.761188 + 1.18443i −0.909632 + 0.415415i 0 −1.72817 + 2.68909i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 359.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.j.a 80
3.b odd 2 1 inner 414.2.j.a 80
23.d odd 22 1 inner 414.2.j.a 80
69.g even 22 1 inner 414.2.j.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.j.a 80 1.a even 1 1 trivial
414.2.j.a 80 3.b odd 2 1 inner
414.2.j.a 80 23.d odd 22 1 inner
414.2.j.a 80 69.g even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(414, [\chi])$$.