# Properties

 Label 252.4.e.a Level $252$ Weight $4$ Character orbit 252.e Analytic conductor $14.868$ Analytic rank $0$ Dimension $36$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$36$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 24q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 24q^{4} + 264q^{10} - 468q^{16} + 444q^{22} - 900q^{25} - 84q^{28} - 432q^{34} - 264q^{37} + 1416q^{40} + 180q^{46} - 1764q^{49} + 2736q^{52} + 636q^{58} - 3960q^{61} + 1392q^{64} - 504q^{70} - 2520q^{76} + 1032q^{82} - 3144q^{85} + 2748q^{88} + 5496q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1 −2.82720 0.0833946i 0 7.98609 + 0.471546i 8.62666i 0 7.00000i −22.5389 1.99915i 0 −0.719417 + 24.3893i
71.2 −2.82720 + 0.0833946i 0 7.98609 0.471546i 8.62666i 0 7.00000i −22.5389 + 1.99915i 0 −0.719417 24.3893i
71.3 −2.45159 1.41057i 0 4.02056 + 6.91629i 16.9233i 0 7.00000i −0.100815 22.6272i 0 23.8716 41.4889i
71.4 −2.45159 + 1.41057i 0 4.02056 6.91629i 16.9233i 0 7.00000i −0.100815 + 22.6272i 0 23.8716 + 41.4889i
71.5 −2.36083 1.55771i 0 3.14707 + 7.35499i 8.56325i 0 7.00000i 4.02725 22.2661i 0 −13.3391 + 20.2164i
71.6 −2.36083 + 1.55771i 0 3.14707 7.35499i 8.56325i 0 7.00000i 4.02725 + 22.2661i 0 −13.3391 20.2164i
71.7 −2.27330 1.68288i 0 2.33582 + 7.65140i 4.43347i 0 7.00000i 7.56635 21.3249i 0 7.46100 10.0786i
71.8 −2.27330 + 1.68288i 0 2.33582 7.65140i 4.43347i 0 7.00000i 7.56635 + 21.3249i 0 7.46100 + 10.0786i
71.9 −1.88294 2.11058i 0 −0.909099 + 7.94818i 21.8666i 0 7.00000i 18.4870 13.0472i 0 46.1512 41.1734i
71.10 −1.88294 + 2.11058i 0 −0.909099 7.94818i 21.8666i 0 7.00000i 18.4870 + 13.0472i 0 46.1512 + 41.1734i
71.11 −1.42938 2.44067i 0 −3.91377 + 6.97728i 15.7775i 0 7.00000i 22.6235 0.420902i 0 −38.5077 + 22.5520i
71.12 −1.42938 + 2.44067i 0 −3.91377 6.97728i 15.7775i 0 7.00000i 22.6235 + 0.420902i 0 −38.5077 22.5520i
71.13 −1.16208 2.57868i 0 −5.29916 + 5.99324i 0.300247i 0 7.00000i 21.6127 + 6.70021i 0 −0.774240 + 0.348910i
71.14 −1.16208 + 2.57868i 0 −5.29916 5.99324i 0.300247i 0 7.00000i 21.6127 6.70021i 0 −0.774240 0.348910i
71.15 −0.968530 2.65743i 0 −6.12390 + 5.14761i 12.6551i 0 7.00000i 19.6106 + 11.2882i 0 33.6300 12.2568i
71.16 −0.968530 + 2.65743i 0 −6.12390 5.14761i 12.6551i 0 7.00000i 19.6106 11.2882i 0 33.6300 + 12.2568i
71.17 −0.614971 2.76076i 0 −7.24362 + 3.39558i 2.97985i 0 7.00000i 13.8290 + 17.9097i 0 8.22665 1.83252i
71.18 −0.614971 + 2.76076i 0 −7.24362 3.39558i 2.97985i 0 7.00000i 13.8290 17.9097i 0 8.22665 + 1.83252i
71.19 0.614971 2.76076i 0 −7.24362 3.39558i 2.97985i 0 7.00000i −13.8290 + 17.9097i 0 8.22665 + 1.83252i
71.20 0.614971 + 2.76076i 0 −7.24362 + 3.39558i 2.97985i 0 7.00000i −13.8290 17.9097i 0 8.22665 1.83252i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.36 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
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.e.a 36
3.b odd 2 1 inner 252.4.e.a 36
4.b odd 2 1 inner 252.4.e.a 36
12.b even 2 1 inner 252.4.e.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.e.a 36 1.a even 1 1 trivial
252.4.e.a 36 3.b odd 2 1 inner
252.4.e.a 36 4.b odd 2 1 inner
252.4.e.a 36 12.b even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(252, [\chi])$$.