Properties

Label 900.3.ba.a
Level $900$
Weight $3$
Character orbit 900.ba
Analytic conductor $24.523$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.ba (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80q + 16q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80q + 16q^{7} - 8q^{13} + 60q^{19} - 120q^{25} + 120q^{31} + 116q^{37} - 80q^{43} + 440q^{49} + 120q^{55} + 80q^{61} + 24q^{67} + 128q^{73} + 40q^{79} + 40q^{85} - 140q^{91} + 384q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 0 0 0 −4.89744 + 1.00750i 0 0.208186 0 0 0
161.2 0 0 0 −4.41393 + 2.34888i 0 −11.9145 0 0 0
161.3 0 0 0 −4.19489 + 2.72083i 0 5.95985 0 0 0
161.4 0 0 0 −4.15240 2.78525i 0 −1.13968 0 0 0
161.5 0 0 0 −3.66894 3.39689i 0 −6.21282 0 0 0
161.6 0 0 0 −3.45827 + 3.61115i 0 7.44862 0 0 0
161.7 0 0 0 −2.65177 4.23888i 0 6.40455 0 0 0
161.8 0 0 0 −1.42516 4.79259i 0 −1.71989 0 0 0
161.9 0 0 0 −0.922718 + 4.91412i 0 9.63452 0 0 0
161.10 0 0 0 −0.709639 + 4.94939i 0 −11.1410 0 0 0
161.11 0 0 0 0.709639 4.94939i 0 −11.1410 0 0 0
161.12 0 0 0 0.922718 4.91412i 0 9.63452 0 0 0
161.13 0 0 0 1.42516 + 4.79259i 0 −1.71989 0 0 0
161.14 0 0 0 2.65177 + 4.23888i 0 6.40455 0 0 0
161.15 0 0 0 3.45827 3.61115i 0 7.44862 0 0 0
161.16 0 0 0 3.66894 + 3.39689i 0 −6.21282 0 0 0
161.17 0 0 0 4.15240 + 2.78525i 0 −1.13968 0 0 0
161.18 0 0 0 4.19489 2.72083i 0 5.95985 0 0 0
161.19 0 0 0 4.41393 2.34888i 0 −11.9145 0 0 0
161.20 0 0 0 4.89744 1.00750i 0 0.208186 0 0 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.ba.a 80
3.b odd 2 1 inner 900.3.ba.a 80
25.d even 5 1 inner 900.3.ba.a 80
75.j odd 10 1 inner 900.3.ba.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.ba.a 80 1.a even 1 1 trivial
900.3.ba.a 80 3.b odd 2 1 inner
900.3.ba.a 80 25.d even 5 1 inner
900.3.ba.a 80 75.j odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(900, [\chi])\).