# Properties

 Label 4012.2.b.a Level 4012 Weight 2 Character orbit 4012.b Analytic conductor 32.036 Analytic rank 0 Dimension 40 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4012 = 2^{2} \cdot 17 \cdot 59$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4012.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.0359812909$$ Analytic rank: $$0$$ Dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 36q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 36q^{9} + 8q^{13} + 20q^{15} - 8q^{17} + 20q^{19} + 6q^{21} - 24q^{25} - 14q^{33} - 52q^{35} + 22q^{43} - 10q^{47} + 8q^{49} - 6q^{51} - 2q^{53} - 12q^{55} - 40q^{59} - 24q^{67} - 36q^{69} - 38q^{77} + 16q^{81} + 32q^{83} - 22q^{85} - 18q^{87} + 40q^{89} + 22q^{93} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
237.1 0 3.27953i 0 4.21610i 0 0.547734i 0 −7.75529 0
237.2 0 3.07288i 0 1.34148i 0 0.0167420i 0 −6.44259 0
237.3 0 2.97688i 0 2.24434i 0 1.80545i 0 −5.86181 0
237.4 0 2.95725i 0 2.24490i 0 1.49294i 0 −5.74532 0
237.5 0 2.44713i 0 1.66009i 0 3.65944i 0 −2.98842 0
237.6 0 2.32869i 0 4.13581i 0 2.86231i 0 −2.42278 0
237.7 0 2.23721i 0 0.538464i 0 0.0679637i 0 −2.00513 0
237.8 0 2.21522i 0 2.69018i 0 2.88510i 0 −1.90721 0
237.9 0 2.18094i 0 0.351694i 0 3.15514i 0 −1.75651 0
237.10 0 1.86007i 0 2.89135i 0 5.10759i 0 −0.459864 0
237.11 0 1.67902i 0 0.714826i 0 1.81204i 0 0.180890 0
237.12 0 1.51601i 0 1.74989i 0 0.991299i 0 0.701704 0
237.13 0 1.46162i 0 3.03678i 0 1.90612i 0 0.863667 0
237.14 0 1.18944i 0 0.609716i 0 4.31103i 0 1.58523 0
237.15 0 1.02370i 0 3.54148i 0 0.166941i 0 1.95203 0
237.16 0 0.622620i 0 0.657734i 0 2.30157i 0 2.61234 0
237.17 0 0.510359i 0 1.04821i 0 1.04964i 0 2.73953 0
237.18 0 0.409767i 0 0.188112i 0 3.66007i 0 2.83209 0
237.19 0 0.346659i 0 2.94947i 0 4.18117i 0 2.87983 0
237.20 0 0.0489925i 0 3.24065i 0 0.683113i 0 2.99760 0
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 237.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4012.2.b.a 40
17.b even 2 1 inner 4012.2.b.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.b.a 40 1.a even 1 1 trivial
4012.2.b.a 40 17.b even 2 1 inner