# Properties

 Label 207.6.c.a Level $207$ Weight $6$ Character orbit 207.c Analytic conductor $33.199$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$207 = 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 207.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.1994507013$$ Analytic rank: $$0$$ Dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 600 q^{4}+O(q^{10})$$ 40 * q - 600 * q^4 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 600 q^{4} - 1048 q^{13} + 9728 q^{16} + 14704 q^{25} + 4640 q^{31} - 91864 q^{46} - 8192 q^{49} + 150360 q^{52} + 134592 q^{55} - 195704 q^{58} - 183416 q^{64} - 257448 q^{70} + 31088 q^{73} - 77096 q^{82} - 368760 q^{85} - 123512 q^{94}+O(q^{100})$$ 40 * q - 600 * q^4 - 1048 * q^13 + 9728 * q^16 + 14704 * q^25 + 4640 * q^31 - 91864 * q^46 - 8192 * q^49 + 150360 * q^52 + 134592 * q^55 - 195704 * q^58 - 183416 * q^64 - 257448 * q^70 + 31088 * q^73 - 77096 * q^82 - 368760 * q^85 - 123512 * q^94

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
206.1 6.22736i 0 −6.78003 −105.915 0 86.6285i 157.054i 0 659.570i
206.2 6.22736i 0 −6.78003 −105.915 0 86.6285i 157.054i 0 659.570i
206.3 1.91117i 0 28.3474 86.4475 0 91.5099i 115.334i 0 165.216i
206.4 1.91117i 0 28.3474 86.4475 0 91.5099i 115.334i 0 165.216i
206.5 10.4484i 0 −77.1695 −72.3402 0 9.31885i 471.949i 0 755.841i
206.6 10.4484i 0 −77.1695 −72.3402 0 9.31885i 471.949i 0 755.841i
206.7 6.00386i 0 −4.04628 −54.7821 0 232.221i 167.830i 0 328.904i
206.8 6.00386i 0 −4.04628 −54.7821 0 232.221i 167.830i 0 328.904i
206.9 3.32542i 0 20.9416 −58.3890 0 22.0963i 176.053i 0 194.168i
206.10 3.32542i 0 20.9416 −58.3890 0 22.0963i 176.053i 0 194.168i
206.11 10.4590i 0 −77.3916 44.1133 0 173.403i 474.753i 0 461.383i
206.12 10.4590i 0 −77.3916 44.1133 0 173.403i 474.753i 0 461.383i
206.13 7.94564i 0 −31.1332 −36.1472 0 129.914i 6.88755i 0 287.213i
206.14 7.94564i 0 −31.1332 −36.1472 0 129.914i 6.88755i 0 287.213i
206.15 8.84990i 0 −46.3207 −32.1196 0 32.0928i 126.737i 0 284.255i
206.16 8.84990i 0 −46.3207 −32.1196 0 32.0928i 126.737i 0 284.255i
206.17 1.13533i 0 30.7110 −11.6464 0 79.1972i 71.1980i 0 13.2226i
206.18 1.13533i 0 30.7110 −11.6464 0 79.1972i 71.1980i 0 13.2226i
206.19 4.37708i 0 12.8412 −13.0959 0 213.282i 196.273i 0 57.3218i
206.20 4.37708i 0 12.8412 −13.0959 0 213.282i 196.273i 0 57.3218i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 206.40 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.b odd 2 1 inner
69.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.6.c.a 40
3.b odd 2 1 inner 207.6.c.a 40
23.b odd 2 1 inner 207.6.c.a 40
69.c even 2 1 inner 207.6.c.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.6.c.a 40 1.a even 1 1 trivial
207.6.c.a 40 3.b odd 2 1 inner
207.6.c.a 40 23.b odd 2 1 inner
207.6.c.a 40 69.c even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(207, [\chi])$$.