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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 16 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 16 q^{4} + 16 q^{7} - 8 q^{10} - 24 q^{13} - 32 q^{16} + 208 q^{19} + 64 q^{22} + 256 q^{25} - 32 q^{28} + 268 q^{34} - 256 q^{37} + 16 q^{40} - 524 q^{43} - 48 q^{46} + 144 q^{49} + 48 q^{52} + 396 q^{55} + 456 q^{58} + 376 q^{61} + 64 q^{64} + 44 q^{67} - 520 q^{70} - 188 q^{73} - 64 q^{76} + 164 q^{79} - 924 q^{82} - 1524 q^{85} + 48 q^{88} + 128 q^{91} - 176 q^{94} - 1144 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
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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 \(S_{3}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display