# Properties

 Label 4140.3.c.a Level $4140$ Weight $3$ Character orbit 4140.c Analytic conductor $112.807$ Analytic rank $0$ Dimension $88$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 4140.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$112.806829445$$ Analytic rank: $$0$$ Dimension: $$88$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$88 q+O(q^{10})$$ 88 * q $$\operatorname{Tr}(f)(q) =$$ $$88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100})$$ 88 * q - 16 * q^19 - 48 * q^25 + 272 * q^31 - 600 * q^49 + 112 * q^55 + 448 * q^61 - 32 * q^79 - 264 * q^85 - 16 * q^91

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4049.1 0 0 0 −4.99275 0.269219i 0 13.6272i 0 0 0
4049.2 0 0 0 −4.99275 + 0.269219i 0 13.6272i 0 0 0
4049.3 0 0 0 −4.95880 0.640555i 0 5.90914i 0 0 0
4049.4 0 0 0 −4.95880 + 0.640555i 0 5.90914i 0 0 0
4049.5 0 0 0 −4.81534 1.34629i 0 4.38152i 0 0 0
4049.6 0 0 0 −4.81534 + 1.34629i 0 4.38152i 0 0 0
4049.7 0 0 0 −4.79317 1.42321i 0 6.33514i 0 0 0
4049.8 0 0 0 −4.79317 + 1.42321i 0 6.33514i 0 0 0
4049.9 0 0 0 −4.77264 1.49061i 0 2.56481i 0 0 0
4049.10 0 0 0 −4.77264 + 1.49061i 0 2.56481i 0 0 0
4049.11 0 0 0 −4.60834 1.93989i 0 1.03288i 0 0 0
4049.12 0 0 0 −4.60834 + 1.93989i 0 1.03288i 0 0 0
4049.13 0 0 0 −4.49878 2.18197i 0 10.0913i 0 0 0
4049.14 0 0 0 −4.49878 + 2.18197i 0 10.0913i 0 0 0
4049.15 0 0 0 −4.41871 2.33986i 0 11.6062i 0 0 0
4049.16 0 0 0 −4.41871 + 2.33986i 0 11.6062i 0 0 0
4049.17 0 0 0 −3.90943 3.11711i 0 7.62373i 0 0 0
4049.18 0 0 0 −3.90943 + 3.11711i 0 7.62373i 0 0 0
4049.19 0 0 0 −3.78150 3.27113i 0 2.52965i 0 0 0
4049.20 0 0 0 −3.78150 + 3.27113i 0 2.52965i 0 0 0
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4049.88 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
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.3.c.a 88
3.b odd 2 1 inner 4140.3.c.a 88
5.b even 2 1 inner 4140.3.c.a 88
15.d odd 2 1 inner 4140.3.c.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.3.c.a 88 1.a even 1 1 trivial
4140.3.c.a 88 3.b odd 2 1 inner
4140.3.c.a 88 5.b even 2 1 inner
4140.3.c.a 88 15.d odd 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(4140, [\chi])$$.