# Properties

 Label 414.3.k.a Level $414$ Weight $3$ Character orbit 414.k Analytic conductor $11.281$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$11.2806829445$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 16 q^{4} - 16 q^{7}+O(q^{10})$$ 80 * q + 16 * q^4 - 16 * q^7 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 16 q^{4} - 16 q^{7} + 8 q^{10} + 8 q^{13} - 32 q^{16} - 128 q^{19} - 32 q^{22} - 352 q^{25} + 32 q^{28} + 32 q^{31} - 300 q^{34} - 384 q^{37} - 16 q^{40} + 540 q^{43} - 80 q^{49} - 16 q^{52} + 1244 q^{55} + 424 q^{58} + 568 q^{61} + 64 q^{64} + 60 q^{67} + 296 q^{70} + 36 q^{73} - 96 q^{76} - 1476 q^{79} + 12 q^{82} - 276 q^{85} - 112 q^{88} - 368 q^{91} - 304 q^{94} + 712 q^{97}+O(q^{100})$$ 80 * q + 16 * q^4 - 16 * q^7 + 8 * q^10 + 8 * q^13 - 32 * q^16 - 128 * q^19 - 32 * q^22 - 352 * q^25 + 32 * q^28 + 32 * q^31 - 300 * q^34 - 384 * q^37 - 16 * q^40 + 540 * q^43 - 80 * q^49 - 16 * q^52 + 1244 * q^55 + 424 * q^58 + 568 * q^61 + 64 * q^64 + 60 * q^67 + 296 * q^70 + 36 * q^73 - 96 * q^76 - 1476 * q^79 + 12 * q^82 - 276 * q^85 - 112 * q^88 - 368 * q^91 - 304 * q^94 + 712 * q^97

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −0.764582 1.18971i 0 −0.830830 + 1.81926i −1.90263 6.47977i 0 −7.74127 8.93390i 2.79964 0.402527i 0 −6.25434 + 7.21790i
35.2 −0.764582 1.18971i 0 −0.830830 + 1.81926i −1.27049 4.32689i 0 6.32390 + 7.29817i 2.79964 0.402527i 0 −4.17636 + 4.81978i
35.3 −0.764582 1.18971i 0 −0.830830 + 1.81926i 0.466986 + 1.59041i 0 −2.89002 3.33526i 2.79964 0.402527i 0 1.53508 1.77157i
35.4 −0.764582 1.18971i 0 −0.830830 + 1.81926i 2.59273 + 8.83003i 0 −0.0998440 0.115226i 2.79964 0.402527i 0 8.52284 9.83588i
35.5 0.764582 + 1.18971i 0 −0.830830 + 1.81926i −2.59273 8.83003i 0 −0.0998440 0.115226i −2.79964 + 0.402527i 0 8.52284 9.83588i
35.6 0.764582 + 1.18971i 0 −0.830830 + 1.81926i −0.466986 1.59041i 0 −2.89002 3.33526i −2.79964 + 0.402527i 0 1.53508 1.77157i
35.7 0.764582 + 1.18971i 0 −0.830830 + 1.81926i 1.27049 + 4.32689i 0 6.32390 + 7.29817i −2.79964 + 0.402527i 0 −4.17636 + 4.81978i
35.8 0.764582 + 1.18971i 0 −0.830830 + 1.81926i 1.90263 + 6.47977i 0 −7.74127 8.93390i −2.79964 + 0.402527i 0 −6.25434 + 7.21790i
71.1 −0.764582 + 1.18971i 0 −0.830830 1.81926i −1.90263 + 6.47977i 0 −7.74127 + 8.93390i 2.79964 + 0.402527i 0 −6.25434 7.21790i
71.2 −0.764582 + 1.18971i 0 −0.830830 1.81926i −1.27049 + 4.32689i 0 6.32390 7.29817i 2.79964 + 0.402527i 0 −4.17636 4.81978i
71.3 −0.764582 + 1.18971i 0 −0.830830 1.81926i 0.466986 1.59041i 0 −2.89002 + 3.33526i 2.79964 + 0.402527i 0 1.53508 + 1.77157i
71.4 −0.764582 + 1.18971i 0 −0.830830 1.81926i 2.59273 8.83003i 0 −0.0998440 + 0.115226i 2.79964 + 0.402527i 0 8.52284 + 9.83588i
71.5 0.764582 1.18971i 0 −0.830830 1.81926i −2.59273 + 8.83003i 0 −0.0998440 + 0.115226i −2.79964 0.402527i 0 8.52284 + 9.83588i
71.6 0.764582 1.18971i 0 −0.830830 1.81926i −0.466986 + 1.59041i 0 −2.89002 + 3.33526i −2.79964 0.402527i 0 1.53508 + 1.77157i
71.7 0.764582 1.18971i 0 −0.830830 1.81926i 1.27049 4.32689i 0 6.32390 7.29817i −2.79964 0.402527i 0 −4.17636 4.81978i
71.8 0.764582 1.18971i 0 −0.830830 1.81926i 1.90263 6.47977i 0 −7.74127 + 8.93390i −2.79964 0.402527i 0 −6.25434 7.21790i
179.1 −0.398430 + 1.35693i 0 −1.68251 1.08128i −5.13916 0.738899i 0 −5.58919 12.2386i 2.13758 1.85223i 0 3.05023 6.67907i
179.2 −0.398430 + 1.35693i 0 −1.68251 1.08128i −4.66712 0.671030i 0 3.18481 + 6.97376i 2.13758 1.85223i 0 2.77006 6.06558i
179.3 −0.398430 + 1.35693i 0 −1.68251 1.08128i 1.62786 + 0.234051i 0 2.08299 + 4.56111i 2.13758 1.85223i 0 −0.966179 + 2.11564i
179.4 −0.398430 + 1.35693i 0 −1.68251 1.08128i 6.34504 + 0.912279i 0 −2.86731 6.27853i 2.13758 1.85223i 0 −3.76595 + 8.24628i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 395.8 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.c even 11 1 inner
69.h odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.3.k.a 80
3.b odd 2 1 inner 414.3.k.a 80
23.c even 11 1 inner 414.3.k.a 80
69.h odd 22 1 inner 414.3.k.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.3.k.a 80 1.a even 1 1 trivial
414.3.k.a 80 3.b odd 2 1 inner
414.3.k.a 80 23.c even 11 1 inner
414.3.k.a 80 69.h odd 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{80} + 76 T_{5}^{78} + 7354 T_{5}^{76} - 308566 T_{5}^{74} - 6443281 T_{5}^{72} - 1226956596 T_{5}^{70} + 171739502288 T_{5}^{68} - 11782301627970 T_{5}^{66} + \cdots + 26\!\cdots\!01$$ acting on $$S_{3}^{\mathrm{new}}(414, [\chi])$$.