# Properties

 Label 414.3.k.b Level $414$ Weight $3$ Character orbit 414.k Analytic conductor $11.281$ 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: $$k$$ $$=$$ $$3$$ 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 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 16 q^{4} + 16 q^{7} - 8 q^{10} - 24 q^{13} - 32 q^{16} + 208 q^{19} + 64 q^{22} + 256 q^{25} - 32 q^{28} + 268 q^{34} - 256 q^{37} + 16 q^{40} - 524 q^{43} - 48 q^{46} + 144 q^{49} + 48 q^{52} + 396 q^{55} + 456 q^{58} + 376 q^{61} + 64 q^{64} + 44 q^{67} - 520 q^{70} - 188 q^{73} - 64 q^{76} + 164 q^{79} - 924 q^{82} - 1524 q^{85} + 48 q^{88} + 128 q^{91} - 176 q^{94} - 1144 q^{97}+O(q^{100})$$ 80 * q + 16 * q^4 + 16 * q^7 - 8 * q^10 - 24 * q^13 - 32 * q^16 + 208 * q^19 + 64 * q^22 + 256 * q^25 - 32 * q^28 + 268 * q^34 - 256 * q^37 + 16 * q^40 - 524 * q^43 - 48 * q^46 + 144 * q^49 + 48 * q^52 + 396 * q^55 + 456 * q^58 + 376 * q^61 + 64 * q^64 + 44 * q^67 - 520 * q^70 - 188 * q^73 - 64 * q^76 + 164 * q^79 - 924 * q^82 - 1524 * q^85 + 48 * q^88 + 128 * q^91 - 176 * q^94 - 1144 * 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 −2.17725 7.41503i 0 6.85488 + 7.91095i 2.79964 0.402527i 0 −7.15707 + 8.25969i
35.2 −0.764582 1.18971i 0 −0.830830 + 1.81926i −0.172109 0.586151i 0 −4.94341 5.70500i 2.79964 0.402527i 0 −0.565759 + 0.652921i
35.3 −0.764582 1.18971i 0 −0.830830 + 1.81926i 0.404235 + 1.37670i 0 −1.64257 1.89563i 2.79964 0.402527i 0 1.32880 1.53352i
35.4 −0.764582 1.18971i 0 −0.830830 + 1.81926i 2.05853 + 7.01070i 0 4.13834 + 4.77590i 2.79964 0.402527i 0 6.76681 7.80931i
35.5 0.764582 + 1.18971i 0 −0.830830 + 1.81926i −2.05853 7.01070i 0 4.13834 + 4.77590i −2.79964 + 0.402527i 0 6.76681 7.80931i
35.6 0.764582 + 1.18971i 0 −0.830830 + 1.81926i −0.404235 1.37670i 0 −1.64257 1.89563i −2.79964 + 0.402527i 0 1.32880 1.53352i
35.7 0.764582 + 1.18971i 0 −0.830830 + 1.81926i 0.172109 + 0.586151i 0 −4.94341 5.70500i −2.79964 + 0.402527i 0 −0.565759 + 0.652921i
35.8 0.764582 + 1.18971i 0 −0.830830 + 1.81926i 2.17725 + 7.41503i 0 6.85488 + 7.91095i −2.79964 + 0.402527i 0 −7.15707 + 8.25969i
71.1 −0.764582 + 1.18971i 0 −0.830830 1.81926i −2.17725 + 7.41503i 0 6.85488 7.91095i 2.79964 + 0.402527i 0 −7.15707 8.25969i
71.2 −0.764582 + 1.18971i 0 −0.830830 1.81926i −0.172109 + 0.586151i 0 −4.94341 + 5.70500i 2.79964 + 0.402527i 0 −0.565759 0.652921i
71.3 −0.764582 + 1.18971i 0 −0.830830 1.81926i 0.404235 1.37670i 0 −1.64257 + 1.89563i 2.79964 + 0.402527i 0 1.32880 + 1.53352i
71.4 −0.764582 + 1.18971i 0 −0.830830 1.81926i 2.05853 7.01070i 0 4.13834 4.77590i 2.79964 + 0.402527i 0 6.76681 + 7.80931i
71.5 0.764582 1.18971i 0 −0.830830 1.81926i −2.05853 + 7.01070i 0 4.13834 4.77590i −2.79964 0.402527i 0 6.76681 + 7.80931i
71.6 0.764582 1.18971i 0 −0.830830 1.81926i −0.404235 + 1.37670i 0 −1.64257 + 1.89563i −2.79964 0.402527i 0 1.32880 + 1.53352i
71.7 0.764582 1.18971i 0 −0.830830 1.81926i 0.172109 0.586151i 0 −4.94341 + 5.70500i −2.79964 0.402527i 0 −0.565759 0.652921i
71.8 0.764582 1.18971i 0 −0.830830 1.81926i 2.17725 7.41503i 0 6.85488 7.91095i −2.79964 0.402527i 0 −7.15707 8.25969i
179.1 −0.398430 + 1.35693i 0 −1.68251 1.08128i −8.69286 1.24984i 0 −0.246783 0.540379i 2.13758 1.85223i 0 5.15945 11.2976i
179.2 −0.398430 + 1.35693i 0 −1.68251 1.08128i 0.248013 + 0.0356589i 0 −2.43143 5.32409i 2.13758 1.85223i 0 −0.147203 + 0.322329i
179.3 −0.398430 + 1.35693i 0 −1.68251 1.08128i 0.751975 + 0.108118i 0 0.750840 + 1.64411i 2.13758 1.85223i 0 −0.446317 + 0.977298i
179.4 −0.398430 + 1.35693i 0 −1.68251 1.08128i 9.52625 + 1.36967i 0 5.11608 + 11.2026i 2.13758 1.85223i 0 −5.65408 + 12.3807i
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.b 80
3.b odd 2 1 inner 414.3.k.b 80
23.c even 11 1 inner 414.3.k.b 80
69.h odd 22 1 inner 414.3.k.b 80

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{80} - 228 T_{5}^{78} + 22970 T_{5}^{76} - 1637494 T_{5}^{74} + 104428799 T_{5}^{72} - 6075874580 T_{5}^{70} + 468935035136 T_{5}^{68} - 34569703200050 T_{5}^{66} + \cdots + 15\!\cdots\!21$$ acting on $$S_{3}^{\mathrm{new}}(414, [\chi])$$.