Properties

Label 210.3.c.a
Level 210
Weight 3
Character orbit 210.c
Analytic conductor 5.722
Analytic rank 0
Dimension 24
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 210.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.72208555157\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24q + 48q^{4} + 44q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 48q^{4} + 44q^{9} + 16q^{10} + 4q^{15} + 96q^{16} - 80q^{19} - 28q^{21} + 48q^{25} - 56q^{30} + 224q^{31} + 128q^{34} + 88q^{36} - 92q^{39} + 32q^{40} - 72q^{45} - 144q^{46} - 168q^{49} - 284q^{51} - 144q^{54} - 320q^{55} + 8q^{60} + 192q^{64} + 224q^{66} - 152q^{69} - 56q^{70} + 48q^{75} - 160q^{76} - 72q^{79} - 212q^{81} - 56q^{84} - 64q^{85} - 240q^{90} + 168q^{91} - 128q^{94} - 876q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.41421 −2.94835 0.554262i 2.00000 −0.0300576 4.99991i 4.16960 + 0.783844i 2.64575i −2.82843 8.38559 + 3.26832i 0.0425079 + 7.07094i
29.2 −1.41421 −2.94835 + 0.554262i 2.00000 −0.0300576 + 4.99991i 4.16960 0.783844i 2.64575i −2.82843 8.38559 3.26832i 0.0425079 7.07094i
29.3 −1.41421 −2.01468 2.22285i 2.00000 3.91845 + 3.10576i 2.84919 + 3.14358i 2.64575i −2.82843 −0.882103 + 8.95667i −5.54153 4.39221i
29.4 −1.41421 −2.01468 + 2.22285i 2.00000 3.91845 3.10576i 2.84919 3.14358i 2.64575i −2.82843 −0.882103 8.95667i −5.54153 + 4.39221i
29.5 −1.41421 −1.70910 2.46556i 2.00000 −4.99890 0.104764i 2.41703 + 3.48683i 2.64575i −2.82843 −3.15796 + 8.42777i 7.06952 + 0.148158i
29.6 −1.41421 −1.70910 + 2.46556i 2.00000 −4.99890 + 0.104764i 2.41703 3.48683i 2.64575i −2.82843 −3.15796 8.42777i 7.06952 0.148158i
29.7 −1.41421 1.07726 2.79991i 2.00000 −2.52563 + 4.31523i −1.52347 + 3.95968i 2.64575i −2.82843 −6.67904 6.03245i 3.57177 6.10266i
29.8 −1.41421 1.07726 + 2.79991i 2.00000 −2.52563 4.31523i −1.52347 3.95968i 2.64575i −2.82843 −6.67904 + 6.03245i 3.57177 + 6.10266i
29.9 −1.41421 2.70965 1.28756i 2.00000 −3.71627 3.34505i −3.83202 + 1.82088i 2.64575i −2.82843 5.68438 6.97766i 5.25560 + 4.73061i
29.10 −1.41421 2.70965 + 1.28756i 2.00000 −3.71627 + 3.34505i −3.83202 1.82088i 2.64575i −2.82843 5.68438 + 6.97766i 5.25560 4.73061i
29.11 −1.41421 2.88523 0.821846i 2.00000 4.52398 2.12923i −4.08034 + 1.16227i 2.64575i −2.82843 7.64914 4.74244i −6.39787 + 3.01119i
29.12 −1.41421 2.88523 + 0.821846i 2.00000 4.52398 + 2.12923i −4.08034 1.16227i 2.64575i −2.82843 7.64914 + 4.74244i −6.39787 3.01119i
29.13 1.41421 −2.88523 0.821846i 2.00000 −4.52398 + 2.12923i −4.08034 1.16227i 2.64575i 2.82843 7.64914 + 4.74244i −6.39787 + 3.01119i
29.14 1.41421 −2.88523 + 0.821846i 2.00000 −4.52398 2.12923i −4.08034 + 1.16227i 2.64575i 2.82843 7.64914 4.74244i −6.39787 3.01119i
29.15 1.41421 −2.70965 1.28756i 2.00000 3.71627 + 3.34505i −3.83202 1.82088i 2.64575i 2.82843 5.68438 + 6.97766i 5.25560 + 4.73061i
29.16 1.41421 −2.70965 + 1.28756i 2.00000 3.71627 3.34505i −3.83202 + 1.82088i 2.64575i 2.82843 5.68438 6.97766i 5.25560 4.73061i
29.17 1.41421 −1.07726 2.79991i 2.00000 2.52563 4.31523i −1.52347 3.95968i 2.64575i 2.82843 −6.67904 + 6.03245i 3.57177 6.10266i
29.18 1.41421 −1.07726 + 2.79991i 2.00000 2.52563 + 4.31523i −1.52347 + 3.95968i 2.64575i 2.82843 −6.67904 6.03245i 3.57177 + 6.10266i
29.19 1.41421 1.70910 2.46556i 2.00000 4.99890 + 0.104764i 2.41703 3.48683i 2.64575i 2.82843 −3.15796 8.42777i 7.06952 + 0.148158i
29.20 1.41421 1.70910 + 2.46556i 2.00000 4.99890 0.104764i 2.41703 + 3.48683i 2.64575i 2.82843 −3.15796 + 8.42777i 7.06952 0.148158i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.3.c.a 24
3.b odd 2 1 inner 210.3.c.a 24
5.b even 2 1 inner 210.3.c.a 24
5.c odd 4 2 1050.3.e.e 24
15.d odd 2 1 inner 210.3.c.a 24
15.e even 4 2 1050.3.e.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.c.a 24 1.a even 1 1 trivial
210.3.c.a 24 3.b odd 2 1 inner
210.3.c.a 24 5.b even 2 1 inner
210.3.c.a 24 15.d odd 2 1 inner
1050.3.e.e 24 5.c odd 4 2
1050.3.e.e 24 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(210, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database