Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [234,2,Mod(25,234)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(234, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("234.25");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 234 = 2 \cdot 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 234.t (of order \(6\), degree \(2\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(1.86849940730\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 | −0.866025 | − | 0.500000i | −1.62009 | − | 0.612619i | 0.500000 | + | 0.866025i | 0.548308 | − | 0.316566i | 1.09673 | + | 1.34059i | −2.15366 | − | 1.24342i | − | 1.00000i | 2.24940 | + | 1.98500i | −0.633132 | |||
25.2 | −0.866025 | − | 0.500000i | −0.987257 | + | 1.42314i | 0.500000 | + | 0.866025i | 2.53992 | − | 1.46642i | 1.56656 | − | 0.738845i | −1.90832 | − | 1.10177i | − | 1.00000i | −1.05065 | − | 2.81001i | −2.93284 | |||
25.3 | −0.866025 | − | 0.500000i | −0.579434 | − | 1.63225i | 0.500000 | + | 0.866025i | −3.32246 | + | 1.91822i | −0.314323 | + | 1.70329i | 2.91802 | + | 1.68472i | − | 1.00000i | −2.32851 | + | 1.89157i | 3.83645 | |||
25.4 | −0.866025 | − | 0.500000i | −0.215223 | + | 1.71863i | 0.500000 | + | 0.866025i | −2.40542 | + | 1.38877i | 1.04570 | − | 1.38076i | −0.759011 | − | 0.438215i | − | 1.00000i | −2.90736 | − | 0.739776i | 2.77754 | |||
25.5 | −0.866025 | − | 0.500000i | 0.523143 | − | 1.65116i | 0.500000 | + | 0.866025i | 0.419378 | − | 0.242128i | −1.27863 | + | 1.16837i | −4.37722 | − | 2.52719i | − | 1.00000i | −2.45264 | − | 1.72758i | −0.484256 | |||
25.6 | −0.866025 | − | 0.500000i | 1.21476 | + | 1.23465i | 0.500000 | + | 0.866025i | 2.73536 | − | 1.57926i | −0.434692 | − | 1.67662i | 1.36392 | + | 0.787458i | − | 1.00000i | −0.0487053 | + | 2.99960i | −3.15852 | |||
25.7 | −0.866025 | − | 0.500000i | 1.66410 | − | 0.480381i | 0.500000 | + | 0.866025i | −0.515076 | + | 0.297379i | −1.68134 | − | 0.416029i | 1.45217 | + | 0.838409i | − | 1.00000i | 2.53847 | − | 1.59880i | 0.594758 | |||
25.8 | 0.866025 | + | 0.500000i | −1.62009 | − | 0.612619i | 0.500000 | + | 0.866025i | −0.548308 | + | 0.316566i | −1.09673 | − | 1.34059i | 2.15366 | + | 1.24342i | 1.00000i | 2.24940 | + | 1.98500i | −0.633132 | ||||
25.9 | 0.866025 | + | 0.500000i | −0.987257 | + | 1.42314i | 0.500000 | + | 0.866025i | −2.53992 | + | 1.46642i | −1.56656 | + | 0.738845i | 1.90832 | + | 1.10177i | 1.00000i | −1.05065 | − | 2.81001i | −2.93284 | ||||
25.10 | 0.866025 | + | 0.500000i | −0.579434 | − | 1.63225i | 0.500000 | + | 0.866025i | 3.32246 | − | 1.91822i | 0.314323 | − | 1.70329i | −2.91802 | − | 1.68472i | 1.00000i | −2.32851 | + | 1.89157i | 3.83645 | ||||
25.11 | 0.866025 | + | 0.500000i | −0.215223 | + | 1.71863i | 0.500000 | + | 0.866025i | 2.40542 | − | 1.38877i | −1.04570 | + | 1.38076i | 0.759011 | + | 0.438215i | 1.00000i | −2.90736 | − | 0.739776i | 2.77754 | ||||
25.12 | 0.866025 | + | 0.500000i | 0.523143 | − | 1.65116i | 0.500000 | + | 0.866025i | −0.419378 | + | 0.242128i | 1.27863 | − | 1.16837i | 4.37722 | + | 2.52719i | 1.00000i | −2.45264 | − | 1.72758i | −0.484256 | ||||
25.13 | 0.866025 | + | 0.500000i | 1.21476 | + | 1.23465i | 0.500000 | + | 0.866025i | −2.73536 | + | 1.57926i | 0.434692 | + | 1.67662i | −1.36392 | − | 0.787458i | 1.00000i | −0.0487053 | + | 2.99960i | −3.15852 | ||||
25.14 | 0.866025 | + | 0.500000i | 1.66410 | − | 0.480381i | 0.500000 | + | 0.866025i | 0.515076 | − | 0.297379i | 1.68134 | + | 0.416029i | −1.45217 | − | 0.838409i | 1.00000i | 2.53847 | − | 1.59880i | 0.594758 | ||||
103.1 | −0.866025 | + | 0.500000i | −1.62009 | + | 0.612619i | 0.500000 | − | 0.866025i | 0.548308 | + | 0.316566i | 1.09673 | − | 1.34059i | −2.15366 | + | 1.24342i | 1.00000i | 2.24940 | − | 1.98500i | −0.633132 | ||||
103.2 | −0.866025 | + | 0.500000i | −0.987257 | − | 1.42314i | 0.500000 | − | 0.866025i | 2.53992 | + | 1.46642i | 1.56656 | + | 0.738845i | −1.90832 | + | 1.10177i | 1.00000i | −1.05065 | + | 2.81001i | −2.93284 | ||||
103.3 | −0.866025 | + | 0.500000i | −0.579434 | + | 1.63225i | 0.500000 | − | 0.866025i | −3.32246 | − | 1.91822i | −0.314323 | − | 1.70329i | 2.91802 | − | 1.68472i | 1.00000i | −2.32851 | − | 1.89157i | 3.83645 | ||||
103.4 | −0.866025 | + | 0.500000i | −0.215223 | − | 1.71863i | 0.500000 | − | 0.866025i | −2.40542 | − | 1.38877i | 1.04570 | + | 1.38076i | −0.759011 | + | 0.438215i | 1.00000i | −2.90736 | + | 0.739776i | 2.77754 | ||||
103.5 | −0.866025 | + | 0.500000i | 0.523143 | + | 1.65116i | 0.500000 | − | 0.866025i | 0.419378 | + | 0.242128i | −1.27863 | − | 1.16837i | −4.37722 | + | 2.52719i | 1.00000i | −2.45264 | + | 1.72758i | −0.484256 | ||||
103.6 | −0.866025 | + | 0.500000i | 1.21476 | − | 1.23465i | 0.500000 | − | 0.866025i | 2.73536 | + | 1.57926i | −0.434692 | + | 1.67662i | 1.36392 | − | 0.787458i | 1.00000i | −0.0487053 | − | 2.99960i | −3.15852 | ||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
13.b | even | 2 | 1 | inner |
117.t | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 234.2.t.a | ✓ | 28 |
3.b | odd | 2 | 1 | 702.2.t.a | 28 | ||
9.c | even | 3 | 1 | inner | 234.2.t.a | ✓ | 28 |
9.c | even | 3 | 1 | 2106.2.b.c | 14 | ||
9.d | odd | 6 | 1 | 702.2.t.a | 28 | ||
9.d | odd | 6 | 1 | 2106.2.b.d | 14 | ||
13.b | even | 2 | 1 | inner | 234.2.t.a | ✓ | 28 |
39.d | odd | 2 | 1 | 702.2.t.a | 28 | ||
117.n | odd | 6 | 1 | 702.2.t.a | 28 | ||
117.n | odd | 6 | 1 | 2106.2.b.d | 14 | ||
117.t | even | 6 | 1 | inner | 234.2.t.a | ✓ | 28 |
117.t | even | 6 | 1 | 2106.2.b.c | 14 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
234.2.t.a | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
234.2.t.a | ✓ | 28 | 9.c | even | 3 | 1 | inner |
234.2.t.a | ✓ | 28 | 13.b | even | 2 | 1 | inner |
234.2.t.a | ✓ | 28 | 117.t | even | 6 | 1 | inner |
702.2.t.a | 28 | 3.b | odd | 2 | 1 | ||
702.2.t.a | 28 | 9.d | odd | 6 | 1 | ||
702.2.t.a | 28 | 39.d | odd | 2 | 1 | ||
702.2.t.a | 28 | 117.n | odd | 6 | 1 | ||
2106.2.b.c | 14 | 9.c | even | 3 | 1 | ||
2106.2.b.c | 14 | 117.t | even | 6 | 1 | ||
2106.2.b.d | 14 | 9.d | odd | 6 | 1 | ||
2106.2.b.d | 14 | 117.n | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(234, [\chi])\).