# Properties

 Label 147.6.c.d Level $147$ Weight $6$ Character orbit 147.c Analytic conductor $23.576$ Analytic rank $0$ Dimension $40$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 147.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.5764215125$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 640 q^{4} + 440 q^{9}+O(q^{10})$$ 40 * q - 640 * q^4 + 440 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 640 q^{4} + 440 q^{9} - 3176 q^{15} + 3552 q^{16} - 4544 q^{18} + 21088 q^{22} + 30104 q^{25} + 44664 q^{30} - 59320 q^{36} - 26224 q^{37} - 16216 q^{39} + 21584 q^{43} - 27680 q^{46} + 95784 q^{51} - 130304 q^{57} - 131376 q^{58} + 329208 q^{60} + 53968 q^{64} - 187632 q^{67} + 606736 q^{72} + 70312 q^{78} - 597424 q^{79} + 755256 q^{81} + 108112 q^{85} - 1673072 q^{88} + 590480 q^{93} - 258480 q^{99}+O(q^{100})$$ 40 * q - 640 * q^4 + 440 * q^9 - 3176 * q^15 + 3552 * q^16 - 4544 * q^18 + 21088 * q^22 + 30104 * q^25 + 44664 * q^30 - 59320 * q^36 - 26224 * q^37 - 16216 * q^39 + 21584 * q^43 - 27680 * q^46 + 95784 * q^51 - 130304 * q^57 - 131376 * q^58 + 329208 * q^60 + 53968 * q^64 - 187632 * q^67 + 606736 * q^72 + 70312 * q^78 - 597424 * q^79 + 755256 * q^81 + 108112 * q^85 - 1673072 * q^88 + 590480 * q^93 - 258480 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1 8.23628i 13.6946 + 7.44699i −35.8363 −95.6857 61.3355 112.793i 0 31.5969i 132.085 + 203.967i 788.094i
146.2 8.23628i 13.6946 7.44699i −35.8363 −95.6857 61.3355 + 112.793i 0 31.5969i 132.085 203.967i 788.094i
146.3 9.64154i 2.14483 15.4402i −60.9593 −95.0200 −148.867 20.6794i 0 279.213i −233.799 66.2331i 916.140i
146.4 9.64154i 2.14483 + 15.4402i −60.9593 −95.0200 −148.867 + 20.6794i 0 279.213i −233.799 + 66.2331i 916.140i
146.5 7.35335i 15.5875 0.174581i −22.0717 89.2354 −1.28375 114.620i 0 73.0063i 242.939 5.44254i 656.179i
146.6 7.35335i 15.5875 + 0.174581i −22.0717 89.2354 −1.28375 + 114.620i 0 73.0063i 242.939 + 5.44254i 656.179i
146.7 4.29037i −1.28755 15.5352i 13.5927 −75.7260 −66.6517 + 5.52407i 0 195.610i −239.684 + 40.0047i 324.892i
146.8 4.29037i −1.28755 + 15.5352i 13.5927 −75.7260 −66.6517 5.52407i 0 195.610i −239.684 40.0047i 324.892i
146.9 3.81342i −6.23210 14.2885i 17.4578 63.5678 −54.4880 + 23.7656i 0 188.603i −165.322 + 178.095i 242.411i
146.10 3.81342i −6.23210 + 14.2885i 17.4578 63.5678 −54.4880 23.7656i 0 188.603i −165.322 178.095i 242.411i
146.11 8.31267i −14.0092 + 6.83685i −37.1006 28.6894 56.8325 + 116.454i 0 42.3992i 149.515 191.558i 238.485i
146.12 8.31267i −14.0092 6.83685i −37.1006 28.6894 56.8325 116.454i 0 42.3992i 149.515 + 191.558i 238.485i
146.13 10.7583i 14.9806 4.31048i −83.7406 −26.3120 −46.3734 161.166i 0 556.640i 205.839 129.148i 283.072i
146.14 10.7583i 14.9806 + 4.31048i −83.7406 −26.3120 −46.3734 + 161.166i 0 556.640i 205.839 + 129.148i 283.072i
146.15 1.52115i −15.4465 2.09880i 29.6861 −32.5134 −3.19260 + 23.4966i 0 93.8341i 234.190 + 64.8384i 49.4578i
146.16 1.52115i −15.4465 + 2.09880i 29.6861 −32.5134 −3.19260 23.4966i 0 93.8341i 234.190 64.8384i 49.4578i
146.17 6.54010i 2.32648 15.4139i −10.7729 13.6838 −100.808 15.2154i 0 138.827i −232.175 71.7202i 89.4937i
146.18 6.54010i 2.32648 + 15.4139i −10.7729 13.6838 −100.808 + 15.2154i 0 138.827i −232.175 + 71.7202i 89.4937i
146.19 1.50172i −11.3889 10.6440i 29.7448 9.66598 −15.9842 + 17.1029i 0 92.7234i 16.4124 + 242.445i 14.5156i
146.20 1.50172i −11.3889 + 10.6440i 29.7448 9.66598 −15.9842 17.1029i 0 92.7234i 16.4124 242.445i 14.5156i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 146.40 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
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.6.c.d 40
3.b odd 2 1 inner 147.6.c.d 40
7.b odd 2 1 inner 147.6.c.d 40
21.c even 2 1 inner 147.6.c.d 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.6.c.d 40 1.a even 1 1 trivial
147.6.c.d 40 3.b odd 2 1 inner
147.6.c.d 40 7.b odd 2 1 inner
147.6.c.d 40 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{20} + 480 T_{2}^{18} + 96836 T_{2}^{16} + 10697796 T_{2}^{14} + 706034311 T_{2}^{12} + 28468661112 T_{2}^{10} + 688232406736 T_{2}^{8} + 9399633712896 T_{2}^{6} + \cdots + 162931412267008$$ acting on $$S_{6}^{\mathrm{new}}(147, [\chi])$$.