Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [435,2,Mod(16,435)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(435, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("435.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 435 = 3 \cdot 5 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 435.u (of order \(7\), degree \(6\), 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: | \(3.47349248793\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{7})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −1.70259 | − | 0.819922i | 0.623490 | − | 0.781831i | 0.979547 | + | 1.22831i | 0.900969 | + | 0.433884i | −1.70259 | + | 0.819922i | −0.471718 | + | 0.591515i | 0.180366 | + | 0.790235i | −0.222521 | − | 0.974928i | −1.17823 | − | 1.47745i |
16.2 | −1.57475 | − | 0.758361i | 0.623490 | − | 0.781831i | 0.657755 | + | 0.824799i | 0.900969 | + | 0.433884i | −1.57475 | + | 0.758361i | −2.68227 | + | 3.36346i | 0.367557 | + | 1.61037i | −0.222521 | − | 0.974928i | −1.08976 | − | 1.36652i |
16.3 | −0.212418 | − | 0.102295i | 0.623490 | − | 0.781831i | −1.21232 | − | 1.52020i | 0.900969 | + | 0.433884i | −0.212418 | + | 0.102295i | 2.04210 | − | 2.56071i | 0.206936 | + | 0.906644i | −0.222521 | − | 0.974928i | −0.146998 | − | 0.184330i |
16.4 | 1.52210 | + | 0.733005i | 0.623490 | − | 0.781831i | 0.532514 | + | 0.667752i | 0.900969 | + | 0.433884i | 1.52210 | − | 0.733005i | 1.30031 | − | 1.63054i | −0.430781 | − | 1.88737i | −0.222521 | − | 0.974928i | 1.05333 | + | 1.32083i |
16.5 | 2.19018 | + | 1.05473i | 0.623490 | − | 0.781831i | 2.43743 | + | 3.05645i | 0.900969 | + | 0.433884i | 2.19018 | − | 1.05473i | −1.71288 | + | 2.14789i | 1.03282 | + | 4.52507i | −0.222521 | − | 0.974928i | 1.51565 | + | 1.90056i |
136.1 | −1.70259 | + | 0.819922i | 0.623490 | + | 0.781831i | 0.979547 | − | 1.22831i | 0.900969 | − | 0.433884i | −1.70259 | − | 0.819922i | −0.471718 | − | 0.591515i | 0.180366 | − | 0.790235i | −0.222521 | + | 0.974928i | −1.17823 | + | 1.47745i |
136.2 | −1.57475 | + | 0.758361i | 0.623490 | + | 0.781831i | 0.657755 | − | 0.824799i | 0.900969 | − | 0.433884i | −1.57475 | − | 0.758361i | −2.68227 | − | 3.36346i | 0.367557 | − | 1.61037i | −0.222521 | + | 0.974928i | −1.08976 | + | 1.36652i |
136.3 | −0.212418 | + | 0.102295i | 0.623490 | + | 0.781831i | −1.21232 | + | 1.52020i | 0.900969 | − | 0.433884i | −0.212418 | − | 0.102295i | 2.04210 | + | 2.56071i | 0.206936 | − | 0.906644i | −0.222521 | + | 0.974928i | −0.146998 | + | 0.184330i |
136.4 | 1.52210 | − | 0.733005i | 0.623490 | + | 0.781831i | 0.532514 | − | 0.667752i | 0.900969 | − | 0.433884i | 1.52210 | + | 0.733005i | 1.30031 | + | 1.63054i | −0.430781 | + | 1.88737i | −0.222521 | + | 0.974928i | 1.05333 | − | 1.32083i |
136.5 | 2.19018 | − | 1.05473i | 0.623490 | + | 0.781831i | 2.43743 | − | 3.05645i | 0.900969 | − | 0.433884i | 2.19018 | + | 1.05473i | −1.71288 | − | 2.14789i | 1.03282 | − | 4.52507i | −0.222521 | + | 0.974928i | 1.51565 | − | 1.90056i |
181.1 | −0.525740 | − | 2.30342i | −0.900969 | − | 0.433884i | −3.22739 | + | 1.55423i | 0.222521 | + | 0.974928i | −0.525740 | + | 2.30342i | 3.87321 | + | 1.86524i | 2.33062 | + | 2.92251i | 0.623490 | + | 0.781831i | 2.12868 | − | 1.02512i |
181.2 | −0.410184 | − | 1.79714i | −0.900969 | − | 0.433884i | −1.25951 | + | 0.606547i | 0.222521 | + | 0.974928i | −0.410184 | + | 1.79714i | −3.46070 | − | 1.66659i | −0.691946 | − | 0.867673i | 0.623490 | + | 0.781831i | 1.66080 | − | 0.799801i |
181.3 | −0.224650 | − | 0.984254i | −0.900969 | − | 0.433884i | 0.883648 | − | 0.425543i | 0.222521 | + | 0.974928i | −0.224650 | + | 0.984254i | 2.01432 | + | 0.970046i | −1.87626 | − | 2.35276i | 0.623490 | + | 0.781831i | 0.909588 | − | 0.438034i |
181.4 | 0.124254 | + | 0.544394i | −0.900969 | − | 0.433884i | 1.52101 | − | 0.732481i | 0.222521 | + | 0.974928i | 0.124254 | − | 0.544394i | 1.04157 | + | 0.501593i | 1.28406 | + | 1.61016i | 0.623490 | + | 0.781831i | −0.503096 | + | 0.242278i |
181.5 | 0.412830 | + | 1.80872i | −0.900969 | − | 0.433884i | −1.29912 | + | 0.625623i | 0.222521 | + | 0.974928i | 0.412830 | − | 1.80872i | −2.78995 | − | 1.34357i | 0.645551 | + | 0.809496i | 0.623490 | + | 0.781831i | −1.67151 | + | 0.804958i |
226.1 | −1.44601 | + | 1.81324i | −0.222521 | + | 0.974928i | −0.751852 | − | 3.29408i | −0.623490 | + | 0.781831i | −1.44601 | − | 1.81324i | 0.715709 | − | 3.13572i | 2.88105 | + | 1.38744i | −0.900969 | − | 0.433884i | −0.516076 | − | 2.26107i |
226.2 | −0.703739 | + | 0.882461i | −0.222521 | + | 0.974928i | 0.161554 | + | 0.707812i | −0.623490 | + | 0.781831i | −0.703739 | − | 0.882461i | 0.123775 | − | 0.542294i | −2.77217 | − | 1.33501i | −0.900969 | − | 0.433884i | −0.251161 | − | 1.10041i |
226.3 | 0.320185 | − | 0.401499i | −0.222521 | + | 0.974928i | 0.386359 | + | 1.69275i | −0.623490 | + | 0.781831i | 0.320185 | + | 0.401499i | −0.169813 | + | 0.744000i | 1.72870 | + | 0.832500i | −0.900969 | − | 0.433884i | 0.114273 | + | 0.500661i |
226.4 | 1.12553 | − | 1.41136i | −0.222521 | + | 0.974928i | −0.280099 | − | 1.22719i | −0.623490 | + | 0.781831i | 1.12553 | + | 1.41136i | −0.863563 | + | 3.78351i | 1.20558 | + | 0.580579i | −0.900969 | − | 0.433884i | 0.401695 | + | 1.75994i |
226.5 | 1.60501 | − | 2.01262i | −0.222521 | + | 0.974928i | −1.02953 | − | 4.51069i | −0.623490 | + | 0.781831i | 1.60501 | + | 2.01262i | 1.03990 | − | 4.55611i | −6.09209 | − | 2.93379i | −0.900969 | − | 0.433884i | 0.572821 | + | 2.50969i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.d | even | 7 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 435.2.u.a | ✓ | 30 |
29.d | even | 7 | 1 | inner | 435.2.u.a | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
435.2.u.a | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
435.2.u.a | ✓ | 30 | 29.d | even | 7 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{30} - T_{2}^{29} + 7 T_{2}^{28} - 8 T_{2}^{27} + 54 T_{2}^{26} - 13 T_{2}^{25} + 227 T_{2}^{24} + \cdots + 8281 \) acting on \(S_{2}^{\mathrm{new}}(435, [\chi])\).