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.
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 |
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 \(S_{2}^{\mathrm{new}}(300, [\chi])\).