# Properties

 Label 4012.2.b.b Level 4012 Weight 2 Character orbit 4012.b Analytic conductor 32.036 Analytic rank 0 Dimension 46 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: $$46$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$46q - 54q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100})$$

## 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 3.24132i 0 0.276658i 0 5.07344i 0 −7.50616 0
237.2 0 3.20235i 0 1.49915i 0 2.89750i 0 −7.25507 0
237.3 0 3.18403i 0 1.45906i 0 1.23253i 0 −7.13805 0
237.4 0 3.13247i 0 3.91603i 0 2.98385i 0 −6.81236 0
237.5 0 2.81557i 0 1.77357i 0 1.01302i 0 −4.92746 0
237.6 0 2.74502i 0 1.20235i 0 3.93843i 0 −4.53516 0
237.7 0 2.57223i 0 2.31913i 0 3.71324i 0 −3.61638 0
237.8 0 2.34087i 0 2.91532i 0 3.43927i 0 −2.47969 0
237.9 0 2.27623i 0 2.55864i 0 1.00518i 0 −2.18122 0
237.10 0 1.82354i 0 0.284793i 0 0.645277i 0 −0.325298 0
237.11 0 1.79340i 0 1.19915i 0 1.20458i 0 −0.216279 0
237.12 0 1.77972i 0 1.91545i 0 2.45686i 0 −0.167415 0
237.13 0 1.58213i 0 3.58424i 0 1.89303i 0 0.496852 0
237.14 0 1.50352i 0 4.31120i 0 4.93216i 0 0.739426 0
237.15 0 1.45665i 0 3.69601i 0 3.33013i 0 0.878183 0
237.16 0 1.43806i 0 2.66437i 0 3.93282i 0 0.931993 0
237.17 0 1.38906i 0 2.35139i 0 0.244801i 0 1.07052 0
237.18 0 1.13278i 0 2.09684i 0 2.71039i 0 1.71682 0
237.19 0 0.593296i 0 3.87533i 0 2.92807i 0 2.64800 0
237.20 0 0.405081i 0 1.20499i 0 0.122460i 0 2.83591 0
See all 46 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 237.46 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.b 46
17.b even 2 1 inner 4012.2.b.b 46

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4012.2.b.b 46 1.a even 1 1 trivial
4012.2.b.b 46 17.b even 2 1 inner

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database