# Properties

 Label 572.2.e.b Level $572$ Weight $2$ Character orbit 572.e Analytic conductor $4.567$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$64$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q + 6q^{4} + 12q^{5} - 104q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q + 6q^{4} + 12q^{5} - 104q^{9} - 8q^{12} - 34q^{14} + 26q^{16} + 14q^{20} - 8q^{22} + 76q^{25} + 2q^{26} - 16q^{33} + 32q^{34} - 24q^{36} - 32q^{37} - 30q^{38} + 54q^{42} + 10q^{44} + 12q^{45} + 34q^{48} - 36q^{49} - 32q^{53} - 36q^{56} + 62q^{58} + 4q^{60} - 18q^{64} - 30q^{66} - 24q^{69} - 30q^{70} + 88q^{77} - 10q^{78} - 46q^{80} + 64q^{81} - 46q^{82} + 98q^{86} + 16q^{88} + 24q^{89} + 84q^{92} + 8q^{93} - 88q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1 −1.41398 0.0257972i 0.803304i 1.99867 + 0.0729533i −1.80361 0.0207230 1.13585i 0.264488 −2.82419 0.154714i 2.35470 2.55027 + 0.0465281i
131.2 −1.41398 + 0.0257972i 0.803304i 1.99867 0.0729533i −1.80361 0.0207230 + 1.13585i 0.264488 −2.82419 + 0.154714i 2.35470 2.55027 0.0465281i
131.3 −1.40623 0.150014i 1.91998i 1.95499 + 0.421910i 1.37022 0.288024 2.69995i −2.10998 −2.68588 0.886580i −0.686333 −1.92686 0.205553i
131.4 −1.40623 + 0.150014i 1.91998i 1.95499 0.421910i 1.37022 0.288024 + 2.69995i −2.10998 −2.68588 + 0.886580i −0.686333 −1.92686 + 0.205553i
131.5 −1.39939 0.204204i 3.30990i 1.91660 + 0.571522i 3.38816 −0.675895 + 4.63186i 3.64850 −2.56537 1.19116i −7.95546 −4.74137 0.691875i
131.6 −1.39939 + 0.204204i 3.30990i 1.91660 0.571522i 3.38816 −0.675895 4.63186i 3.64850 −2.56537 + 1.19116i −7.95546 −4.74137 + 0.691875i
131.7 −1.33395 0.469670i 3.04863i 1.55882 + 1.25303i −2.77923 1.43185 4.06671i −0.398342 −1.49087 2.40360i −6.29417 3.70734 + 1.30532i
131.8 −1.33395 + 0.469670i 3.04863i 1.55882 1.25303i −2.77923 1.43185 + 4.06671i −0.398342 −1.49087 + 2.40360i −6.29417 3.70734 1.30532i
131.9 −1.29936 0.558274i 0.726543i 1.37666 + 1.45079i 2.29501 0.405610 0.944039i 3.79351 −0.978833 2.65366i 2.47214 −2.98204 1.28125i
131.10 −1.29936 + 0.558274i 0.726543i 1.37666 1.45079i 2.29501 0.405610 + 0.944039i 3.79351 −0.978833 + 2.65366i 2.47214 −2.98204 + 1.28125i
131.11 −1.25608 0.649821i 1.96816i 1.15547 + 1.63245i −3.24710 −1.27895 + 2.47217i 4.43351 −0.390556 2.80133i −0.873660 4.07861 + 2.11003i
131.12 −1.25608 + 0.649821i 1.96816i 1.15547 1.63245i −3.24710 −1.27895 2.47217i 4.43351 −0.390556 + 2.80133i −0.873660 4.07861 2.11003i
131.13 −1.24299 0.674518i 2.42300i 1.09005 + 1.67684i −0.123164 −1.63435 + 3.01176i −2.37508 −0.223868 2.81955i −2.87092 0.153091 + 0.0830761i
131.14 −1.24299 + 0.674518i 2.42300i 1.09005 1.67684i −0.123164 −1.63435 3.01176i −2.37508 −0.223868 + 2.81955i −2.87092 0.153091 0.0830761i
131.15 −1.16339 0.804061i 0.360913i 0.706971 + 1.87088i 2.06255 0.290196 0.419884i 0.974044 0.681817 2.74502i 2.86974 −2.39956 1.65842i
131.16 −1.16339 + 0.804061i 0.360913i 0.706971 1.87088i 2.06255 0.290196 + 0.419884i 0.974044 0.681817 + 2.74502i 2.86974 −2.39956 + 1.65842i
131.17 −0.936395 1.05979i 2.78715i −0.246329 + 1.98477i 1.09997 2.95381 2.60987i −3.18482 2.33411 1.59747i −4.76820 −1.03001 1.16575i
131.18 −0.936395 + 1.05979i 2.78715i −0.246329 1.98477i 1.09997 2.95381 + 2.60987i −3.18482 2.33411 + 1.59747i −4.76820 −1.03001 + 1.16575i
131.19 −0.847391 1.13222i 1.51262i −0.563856 + 1.91887i 4.09054 −1.71262 + 1.28178i −1.07796 2.65040 0.987625i 0.711990 −3.46629 4.63140i
131.20 −0.847391 + 1.13222i 1.51262i −0.563856 1.91887i 4.09054 −1.71262 1.28178i −1.07796 2.65040 + 0.987625i 0.711990 −3.46629 + 4.63140i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.64 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.e.b 64
4.b odd 2 1 inner 572.2.e.b 64
11.b odd 2 1 inner 572.2.e.b 64
44.c even 2 1 inner 572.2.e.b 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.e.b 64 1.a even 1 1 trivial
572.2.e.b 64 4.b odd 2 1 inner
572.2.e.b 64 11.b odd 2 1 inner
572.2.e.b 64 44.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{32} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(572, [\chi])$$.