# Properties

 Label 300.2.x.a Level $300$ Weight $2$ Character orbit 300.x Analytic conductor $2.396$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.x (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$10$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 2q^{3} + 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 2q^{3} + 4q^{7} + 12q^{13} + 10q^{15} + 20q^{19} + 40q^{25} - 14q^{27} - 20q^{33} + 12q^{37} - 40q^{39} + 12q^{43} - 60q^{45} - 76q^{57} - 98q^{63} - 36q^{67} - 70q^{69} - 44q^{73} - 90q^{75} - 40q^{79} + 20q^{81} - 100q^{85} - 70q^{87} - 18q^{93} - 56q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −1.73066 0.0694365i 0 0.354250 + 2.20783i 0 −1.67035 1.67035i 0 2.99036 + 0.240342i 0
17.2 0 −1.58605 + 0.696021i 0 −1.99640 1.00718i 0 0.814380 + 0.814380i 0 2.03111 2.20785i 0
17.3 0 −1.13130 + 1.31155i 0 2.14441 0.633635i 0 0.907947 + 0.907947i 0 −0.440321 2.96751i 0
17.4 0 −0.719024 1.57576i 0 −1.13602 + 1.92599i 0 3.00936 + 3.00936i 0 −1.96601 + 2.26601i 0
17.5 0 −0.351313 1.69605i 0 2.22846 0.184289i 0 −2.84010 2.84010i 0 −2.75316 + 1.19169i 0
17.6 0 0.670639 + 1.59695i 0 −2.14441 + 0.633635i 0 0.907947 + 0.907947i 0 −2.10049 + 2.14195i 0
17.7 0 0.858227 1.50448i 0 −2.22846 + 0.184289i 0 −2.84010 2.84010i 0 −1.52689 2.58236i 0
17.8 0 1.17077 1.27644i 0 1.13602 1.92599i 0 3.00936 + 3.00936i 0 −0.258608 2.98883i 0
17.9 0 1.29334 + 1.15207i 0 1.99640 + 1.00718i 0 0.814380 + 0.814380i 0 0.345460 + 2.98004i 0
17.10 0 1.66741 + 0.468765i 0 −0.354250 2.20783i 0 −1.67035 1.67035i 0 2.56052 + 1.56325i 0
53.1 0 −1.73066 + 0.0694365i 0 0.354250 2.20783i 0 −1.67035 + 1.67035i 0 2.99036 0.240342i 0
53.2 0 −1.58605 0.696021i 0 −1.99640 + 1.00718i 0 0.814380 0.814380i 0 2.03111 + 2.20785i 0
53.3 0 −1.13130 1.31155i 0 2.14441 + 0.633635i 0 0.907947 0.907947i 0 −0.440321 + 2.96751i 0
53.4 0 −0.719024 + 1.57576i 0 −1.13602 1.92599i 0 3.00936 3.00936i 0 −1.96601 2.26601i 0
53.5 0 −0.351313 + 1.69605i 0 2.22846 + 0.184289i 0 −2.84010 + 2.84010i 0 −2.75316 1.19169i 0
53.6 0 0.670639 1.59695i 0 −2.14441 0.633635i 0 0.907947 0.907947i 0 −2.10049 2.14195i 0
53.7 0 0.858227 + 1.50448i 0 −2.22846 0.184289i 0 −2.84010 + 2.84010i 0 −1.52689 + 2.58236i 0
53.8 0 1.17077 + 1.27644i 0 1.13602 + 1.92599i 0 3.00936 3.00936i 0 −0.258608 + 2.98883i 0
53.9 0 1.29334 1.15207i 0 1.99640 1.00718i 0 0.814380 0.814380i 0 0.345460 2.98004i 0
53.10 0 1.66741 0.468765i 0 −0.354250 + 2.20783i 0 −1.67035 + 1.67035i 0 2.56052 1.56325i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 233.10 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
25.f odd 20 1 inner
75.l even 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.x.a 80
3.b odd 2 1 inner 300.2.x.a 80
25.f odd 20 1 inner 300.2.x.a 80
75.l even 20 1 inner 300.2.x.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.x.a 80 1.a even 1 1 trivial
300.2.x.a 80 3.b odd 2 1 inner
300.2.x.a 80 25.f odd 20 1 inner
300.2.x.a 80 75.l even 20 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(300, [\chi])$$.