# Properties

 Label 147.4.c.b Level $147$ Weight $4$ Character orbit 147.c Analytic conductor $8.673$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 147.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.67328077084$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 96 q^{4} - 64 q^{9}+O(q^{10})$$ 24 * q - 96 * q^4 - 64 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 96 q^{4} - 64 q^{9} + 256 q^{15} + 864 q^{16} - 32 q^{18} - 384 q^{22} + 744 q^{25} - 1704 q^{30} + 584 q^{36} + 432 q^{37} - 2368 q^{39} - 624 q^{43} + 3744 q^{46} - 2160 q^{51} + 2032 q^{57} + 6384 q^{58} - 5832 q^{60} - 3504 q^{64} + 3792 q^{67} - 7472 q^{72} + 2248 q^{78} + 2784 q^{79} - 1968 q^{81} - 3744 q^{85} - 624 q^{88} - 3232 q^{93} + 1320 q^{99}+O(q^{100})$$ 24 * q - 96 * q^4 - 64 * q^9 + 256 * q^15 + 864 * q^16 - 32 * q^18 - 384 * q^22 + 744 * q^25 - 1704 * q^30 + 584 * q^36 + 432 * q^37 - 2368 * q^39 - 624 * q^43 + 3744 * q^46 - 2160 * q^51 + 2032 * q^57 + 6384 * q^58 - 5832 * q^60 - 3504 * q^64 + 3792 * q^67 - 7472 * q^72 + 2248 * q^78 + 2784 * q^79 - 1968 * q^81 - 3744 * q^85 - 624 * q^88 - 3232 * q^93 + 1320 * 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 5.37572i −2.08523 4.75939i −20.8984 2.78815 −25.5852 + 11.2096i 0 69.3382i −18.3036 + 19.8488i 14.9883i
146.2 5.37572i 2.08523 + 4.75939i −20.8984 −2.78815 25.5852 11.2096i 0 69.3382i −18.3036 + 19.8488i 14.9883i
146.3 4.84485i −5.11673 + 0.905040i −15.4725 −17.3012 4.38478 + 24.7898i 0 36.2033i 25.3618 9.26169i 83.8216i
146.4 4.84485i 5.11673 0.905040i −15.4725 17.3012 −4.38478 24.7898i 0 36.2033i 25.3618 9.26169i 83.8216i
146.5 3.20022i −0.930073 5.11224i −2.24142 11.9763 −16.3603 + 2.97644i 0 18.4287i −25.2699 + 9.50951i 38.3267i
146.6 3.20022i 0.930073 + 5.11224i −2.24142 −11.9763 16.3603 2.97644i 0 18.4287i −25.2699 + 9.50951i 38.3267i
146.7 2.86969i −2.76278 + 4.40080i −0.235115 −5.57059 12.6289 + 7.92832i 0 22.2828i −11.7341 24.3169i 15.9859i
146.8 2.86969i 2.76278 4.40080i −0.235115 5.57059 −12.6289 7.92832i 0 22.2828i −11.7341 24.3169i 15.9859i
146.9 1.05015i −4.50851 + 2.58329i 6.89718 −10.2305 2.71285 + 4.73462i 0 15.6443i 13.6533 23.2935i 10.7436i
146.10 1.05015i 4.50851 2.58329i 6.89718 10.2305 −2.71285 4.73462i 0 15.6443i 13.6533 23.2935i 10.7436i
146.11 0.222948i −3.69409 + 3.65427i 7.95029 18.7021 0.814713 + 0.823589i 0 3.55609i 0.292585 26.9984i 4.16960i
146.12 0.222948i 3.69409 3.65427i 7.95029 −18.7021 −0.814713 0.823589i 0 3.55609i 0.292585 26.9984i 4.16960i
146.13 0.222948i −3.69409 3.65427i 7.95029 18.7021 0.814713 0.823589i 0 3.55609i 0.292585 + 26.9984i 4.16960i
146.14 0.222948i 3.69409 + 3.65427i 7.95029 −18.7021 −0.814713 + 0.823589i 0 3.55609i 0.292585 + 26.9984i 4.16960i
146.15 1.05015i −4.50851 2.58329i 6.89718 −10.2305 2.71285 4.73462i 0 15.6443i 13.6533 + 23.2935i 10.7436i
146.16 1.05015i 4.50851 + 2.58329i 6.89718 10.2305 −2.71285 + 4.73462i 0 15.6443i 13.6533 + 23.2935i 10.7436i
146.17 2.86969i −2.76278 4.40080i −0.235115 −5.57059 12.6289 7.92832i 0 22.2828i −11.7341 + 24.3169i 15.9859i
146.18 2.86969i 2.76278 + 4.40080i −0.235115 5.57059 −12.6289 + 7.92832i 0 22.2828i −11.7341 + 24.3169i 15.9859i
146.19 3.20022i −0.930073 + 5.11224i −2.24142 11.9763 −16.3603 2.97644i 0 18.4287i −25.2699 9.50951i 38.3267i
146.20 3.20022i 0.930073 5.11224i −2.24142 −11.9763 16.3603 + 2.97644i 0 18.4287i −25.2699 9.50951i 38.3267i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 146.24 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
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 147.4.c.b 24
3.b odd 2 1 inner 147.4.c.b 24
7.b odd 2 1 inner 147.4.c.b 24
7.c even 3 2 147.4.g.e 48
7.d odd 6 2 147.4.g.e 48
21.c even 2 1 inner 147.4.c.b 24
21.g even 6 2 147.4.g.e 48
21.h odd 6 2 147.4.g.e 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.c.b 24 1.a even 1 1 trivial
147.4.c.b 24 3.b odd 2 1 inner
147.4.c.b 24 7.b odd 2 1 inner
147.4.c.b 24 21.c even 2 1 inner
147.4.g.e 48 7.c even 3 2
147.4.g.e 48 7.d odd 6 2
147.4.g.e 48 21.g even 6 2
147.4.g.e 48 21.h odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 72T_{2}^{10} + 1812T_{2}^{8} + 18948T_{2}^{6} + 76839T_{2}^{4} + 66864T_{2}^{2} + 3136$$ acting on $$S_{4}^{\mathrm{new}}(147, [\chi])$$.