# 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: $$24$$ 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.

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

## 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: 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
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 $$S_{3}^{\mathrm{new}}(210, [\chi])$$.