# Properties

 Label 252.2.be.a Level $252$ Weight $2$ Character orbit 252.be Analytic conductor $2.012$ Analytic rank $0$ Dimension $32$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 252.be (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 16q^{13} + 4q^{16} - 32q^{22} + 24q^{25} - 44q^{28} - 16q^{34} + 8q^{37} - 52q^{40} - 24q^{46} - 16q^{49} - 52q^{52} - 12q^{58} - 16q^{61} + 120q^{64} + 60q^{70} - 8q^{73} + 72q^{76} + 68q^{82} - 32q^{85} + 44q^{88} + 60q^{94} - 176q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −1.41312 0.0556843i 0 1.99380 + 0.157377i −1.80224 1.04052i 0 −1.89429 + 1.84707i −2.80871 0.333415i 0 2.48883 + 1.57073i
107.2 −1.40272 0.179921i 0 1.93526 + 0.504757i −0.604694 0.349120i 0 −1.16254 2.37666i −2.62381 1.05623i 0 0.785403 + 0.598515i
107.3 −1.30138 + 0.553555i 0 1.38715 1.44076i 3.35279 + 1.93573i 0 1.03277 2.43585i −1.00767 + 2.64284i 0 −5.43477 0.663164i
107.4 −0.857177 1.12483i 0 −0.530496 + 1.92836i −0.604694 0.349120i 0 1.16254 + 2.37666i 2.62381 1.05623i 0 0.125628 + 0.979437i
107.5 −0.783636 + 1.17725i 0 −0.771830 1.84507i 2.15525 + 1.24433i 0 −2.64453 + 0.0803545i 2.77694 + 0.537226i 0 −3.15382 + 1.56216i
107.6 −0.754782 1.19595i 0 −0.860607 + 1.80537i −1.80224 1.04052i 0 1.89429 1.84707i 2.80871 0.333415i 0 0.115881 + 2.94076i
107.7 −0.627710 + 1.26727i 0 −1.21196 1.59096i −2.15525 1.24433i 0 2.64453 0.0803545i 2.77694 0.537226i 0 2.92978 1.95021i
107.8 −0.171295 1.40380i 0 −1.94132 + 0.480929i 3.35279 + 1.93573i 0 −1.03277 + 2.43585i 1.00767 + 2.64284i 0 2.14307 5.03823i
107.9 0.171295 + 1.40380i 0 −1.94132 + 0.480929i −3.35279 1.93573i 0 −1.03277 + 2.43585i −1.00767 2.64284i 0 2.14307 5.03823i
107.10 0.627710 1.26727i 0 −1.21196 1.59096i 2.15525 + 1.24433i 0 2.64453 0.0803545i −2.77694 + 0.537226i 0 2.92978 1.95021i
107.11 0.754782 + 1.19595i 0 −0.860607 + 1.80537i 1.80224 + 1.04052i 0 1.89429 1.84707i −2.80871 + 0.333415i 0 0.115881 + 2.94076i
107.12 0.783636 1.17725i 0 −0.771830 1.84507i −2.15525 1.24433i 0 −2.64453 + 0.0803545i −2.77694 0.537226i 0 −3.15382 + 1.56216i
107.13 0.857177 + 1.12483i 0 −0.530496 + 1.92836i 0.604694 + 0.349120i 0 1.16254 + 2.37666i −2.62381 + 1.05623i 0 0.125628 + 0.979437i
107.14 1.30138 0.553555i 0 1.38715 1.44076i −3.35279 1.93573i 0 1.03277 2.43585i 1.00767 2.64284i 0 −5.43477 0.663164i
107.15 1.40272 + 0.179921i 0 1.93526 + 0.504757i 0.604694 + 0.349120i 0 −1.16254 2.37666i 2.62381 + 1.05623i 0 0.785403 + 0.598515i
107.16 1.41312 + 0.0556843i 0 1.99380 + 0.157377i 1.80224 + 1.04052i 0 −1.89429 + 1.84707i 2.80871 + 0.333415i 0 2.48883 + 1.57073i
179.1 −1.41312 + 0.0556843i 0 1.99380 0.157377i −1.80224 + 1.04052i 0 −1.89429 1.84707i −2.80871 + 0.333415i 0 2.48883 1.57073i
179.2 −1.40272 + 0.179921i 0 1.93526 0.504757i −0.604694 + 0.349120i 0 −1.16254 + 2.37666i −2.62381 + 1.05623i 0 0.785403 0.598515i
179.3 −1.30138 0.553555i 0 1.38715 + 1.44076i 3.35279 1.93573i 0 1.03277 + 2.43585i −1.00767 2.64284i 0 −5.43477 + 0.663164i
179.4 −0.857177 + 1.12483i 0 −0.530496 1.92836i −0.604694 + 0.349120i 0 1.16254 2.37666i 2.62381 + 1.05623i 0 0.125628 0.979437i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.16 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
7.c even 3 1 inner
12.b even 2 1 inner
21.h odd 6 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.be.a 32
3.b odd 2 1 inner 252.2.be.a 32
4.b odd 2 1 inner 252.2.be.a 32
7.c even 3 1 inner 252.2.be.a 32
7.c even 3 1 1764.2.e.i 16
7.d odd 6 1 1764.2.e.h 16
12.b even 2 1 inner 252.2.be.a 32
21.g even 6 1 1764.2.e.h 16
21.h odd 6 1 inner 252.2.be.a 32
21.h odd 6 1 1764.2.e.i 16
28.f even 6 1 1764.2.e.h 16
28.g odd 6 1 inner 252.2.be.a 32
28.g odd 6 1 1764.2.e.i 16
84.j odd 6 1 1764.2.e.h 16
84.n even 6 1 inner 252.2.be.a 32
84.n even 6 1 1764.2.e.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.be.a 32 1.a even 1 1 trivial
252.2.be.a 32 3.b odd 2 1 inner
252.2.be.a 32 4.b odd 2 1 inner
252.2.be.a 32 7.c even 3 1 inner
252.2.be.a 32 12.b even 2 1 inner
252.2.be.a 32 21.h odd 6 1 inner
252.2.be.a 32 28.g odd 6 1 inner
252.2.be.a 32 84.n even 6 1 inner
1764.2.e.h 16 7.d odd 6 1
1764.2.e.h 16 21.g even 6 1
1764.2.e.h 16 28.f even 6 1
1764.2.e.h 16 84.j odd 6 1
1764.2.e.i 16 7.c even 3 1
1764.2.e.i 16 21.h odd 6 1
1764.2.e.i 16 28.g odd 6 1
1764.2.e.i 16 84.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database