# Properties

 Label 4004.2.m.b Level 4004 Weight 2 Character orbit 4004.m Analytic conductor 31.972 Analytic rank 0 Dimension 30 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4004.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.9721009693$$ Analytic rank: $$0$$ Dimension: $$30$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 18q^{9} + 4q^{13} + 12q^{17} + 6q^{23} - 6q^{29} - 2q^{35} + 8q^{39} - 22q^{43} - 30q^{49} + 60q^{51} - 38q^{53} - 2q^{55} - 36q^{61} + 10q^{65} + 36q^{69} - 20q^{75} - 30q^{77} - 10q^{79} - 42q^{81} + 36q^{87} + 6q^{91} - 2q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2157.1 0 −2.58184 0 1.48218i 0 1.00000i 0 3.66591 0
2157.2 0 −2.58184 0 1.48218i 0 1.00000i 0 3.66591 0
2157.3 0 −2.50189 0 2.36927i 0 1.00000i 0 3.25946 0
2157.4 0 −2.50189 0 2.36927i 0 1.00000i 0 3.25946 0
2157.5 0 −2.42937 0 1.20006i 0 1.00000i 0 2.90185 0
2157.6 0 −2.42937 0 1.20006i 0 1.00000i 0 2.90185 0
2157.7 0 −2.41023 0 3.42759i 0 1.00000i 0 2.80920 0
2157.8 0 −2.41023 0 3.42759i 0 1.00000i 0 2.80920 0
2157.9 0 −1.49114 0 1.37882i 0 1.00000i 0 −0.776492 0
2157.10 0 −1.49114 0 1.37882i 0 1.00000i 0 −0.776492 0
2157.11 0 −0.822838 0 0.417533i 0 1.00000i 0 −2.32294 0
2157.12 0 −0.822838 0 0.417533i 0 1.00000i 0 −2.32294 0
2157.13 0 −0.0797053 0 0.579061i 0 1.00000i 0 −2.99365 0
2157.14 0 −0.0797053 0 0.579061i 0 1.00000i 0 −2.99365 0
2157.15 0 0.0840880 0 3.49341i 0 1.00000i 0 −2.99293 0
2157.16 0 0.0840880 0 3.49341i 0 1.00000i 0 −2.99293 0
2157.17 0 0.705440 0 2.65594i 0 1.00000i 0 −2.50236 0
2157.18 0 0.705440 0 2.65594i 0 1.00000i 0 −2.50236 0
2157.19 0 0.726270 0 2.96350i 0 1.00000i 0 −2.47253 0
2157.20 0 0.726270 0 2.96350i 0 1.00000i 0 −2.47253 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2157.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4004.2.m.b 30
13.b even 2 1 inner 4004.2.m.b 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.m.b 30 1.a even 1 1 trivial
4004.2.m.b 30 13.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{15} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(4004, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database