# Properties

 Label 4014.2.d.a Level 4014 Weight 2 Character orbit 4014.d Analytic conductor 32.052 Analytic rank 0 Dimension 72 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4014 = 2 \cdot 3^{2} \cdot 223$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4014.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.0519513713$$ Analytic rank: $$0$$ Dimension: $$72$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q - 72q^{4} + 16q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q - 72q^{4} + 16q^{7} + 72q^{16} - 40q^{19} + 96q^{25} - 16q^{28} - 24q^{37} - 8q^{43} + 56q^{49} + 40q^{58} - 72q^{64} - 32q^{73} + 40q^{76} + 16q^{82} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4013.1 1.00000i 0 −1.00000 3.51050 0 3.72470 1.00000i 0 3.51050i
4013.2 1.00000i 0 −1.00000 3.51050 0 3.72470 1.00000i 0 3.51050i
4013.3 1.00000i 0 −1.00000 −0.969046 0 −3.46872 1.00000i 0 0.969046i
4013.4 1.00000i 0 −1.00000 −0.969046 0 −3.46872 1.00000i 0 0.969046i
4013.5 1.00000i 0 −1.00000 −2.31709 0 −2.47546 1.00000i 0 2.31709i
4013.6 1.00000i 0 −1.00000 −2.31709 0 −2.47546 1.00000i 0 2.31709i
4013.7 1.00000i 0 −1.00000 0.125404 0 3.10184 1.00000i 0 0.125404i
4013.8 1.00000i 0 −1.00000 0.125404 0 3.10184 1.00000i 0 0.125404i
4013.9 1.00000i 0 −1.00000 3.59603 0 4.66381 1.00000i 0 3.59603i
4013.10 1.00000i 0 −1.00000 3.59603 0 4.66381 1.00000i 0 3.59603i
4013.11 1.00000i 0 −1.00000 −2.58876 0 −1.62417 1.00000i 0 2.58876i
4013.12 1.00000i 0 −1.00000 −2.58876 0 −1.62417 1.00000i 0 2.58876i
4013.13 1.00000i 0 −1.00000 2.29842 0 2.05558 1.00000i 0 2.29842i
4013.14 1.00000i 0 −1.00000 2.29842 0 2.05558 1.00000i 0 2.29842i
4013.15 1.00000i 0 −1.00000 −2.37383 0 3.81286 1.00000i 0 2.37383i
4013.16 1.00000i 0 −1.00000 −2.37383 0 3.81286 1.00000i 0 2.37383i
4013.17 1.00000i 0 −1.00000 0.707934 0 0.916361 1.00000i 0 0.707934i
4013.18 1.00000i 0 −1.00000 0.707934 0 0.916361 1.00000i 0 0.707934i
4013.19 1.00000i 0 −1.00000 −1.71459 0 2.72316 1.00000i 0 1.71459i
4013.20 1.00000i 0 −1.00000 −1.71459 0 2.72316 1.00000i 0 1.71459i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4013.72 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
223.b odd 2 1 inner
669.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4014.2.d.a 72
3.b odd 2 1 inner 4014.2.d.a 72
223.b odd 2 1 inner 4014.2.d.a 72
669.c even 2 1 inner 4014.2.d.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4014.2.d.a 72 1.a even 1 1 trivial
4014.2.d.a 72 3.b odd 2 1 inner
4014.2.d.a 72 223.b odd 2 1 inner
4014.2.d.a 72 669.c even 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(4014, [\chi])$$.