Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [475,2,Mod(32,475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([9, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("475.32");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 475.bb (of order \(36\), degree \(12\), 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: | \(3.79289409601\) |
Analytic rank: | \(0\) |
Dimension: | \(168\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
32.1 | −0.219903 | + | 2.51350i | −0.420327 | − | 0.901394i | −4.29970 | − | 0.758153i | 0 | 2.35808 | − | 0.858272i | 0.630014 | − | 0.168812i | 1.54508 | − | 5.76632i | 1.29253 | − | 1.54037i | 0 | ||||
32.2 | −0.205646 | + | 2.35054i | 0.0966746 | + | 0.207319i | −3.51313 | − | 0.619460i | 0 | −0.507193 | + | 0.184603i | −4.84313 | + | 1.29771i | 0.957149 | − | 3.57213i | 1.89473 | − | 2.25805i | 0 | ||||
32.3 | −0.158607 | + | 1.81289i | 1.18655 | + | 2.54457i | −1.29180 | − | 0.227779i | 0 | −4.80122 | + | 1.74750i | 4.44244 | − | 1.19035i | −0.324178 | + | 1.20985i | −3.13855 | + | 3.74038i | 0 | ||||
32.4 | −0.122983 | + | 1.40570i | 0.364684 | + | 0.782068i | 0.00874799 | + | 0.00154251i | 0 | −1.14420 | + | 0.416456i | −1.95768 | + | 0.524559i | −0.733667 | + | 2.73808i | 1.44973 | − | 1.72772i | 0 | ||||
32.5 | −0.0696587 | + | 0.796203i | −1.25628 | − | 2.69410i | 1.34053 | + | 0.236371i | 0 | 2.23256 | − | 0.812585i | 4.61801 | − | 1.23739i | −0.695298 | + | 2.59489i | −3.75157 | + | 4.47094i | 0 | ||||
32.6 | −0.0525703 | + | 0.600882i | 0.0661813 | + | 0.141926i | 1.61132 | + | 0.284119i | 0 | −0.0887601 | + | 0.0323060i | 0.465075 | − | 0.124617i | −0.567657 | + | 2.11852i | 1.91260 | − | 2.27935i | 0 | ||||
32.7 | −0.00427917 | + | 0.0489112i | 1.29767 | + | 2.78287i | 1.96724 | + | 0.346878i | 0 | −0.141666 | + | 0.0515624i | 1.74113 | − | 0.466534i | −0.0507994 | + | 0.189586i | −4.13205 | + | 4.92438i | 0 | ||||
32.8 | 0.00427917 | − | 0.0489112i | −1.29767 | − | 2.78287i | 1.96724 | + | 0.346878i | 0 | −0.141666 | + | 0.0515624i | −1.74113 | + | 0.466534i | 0.0507994 | − | 0.189586i | −4.13205 | + | 4.92438i | 0 | ||||
32.9 | 0.0525703 | − | 0.600882i | −0.0661813 | − | 0.141926i | 1.61132 | + | 0.284119i | 0 | −0.0887601 | + | 0.0323060i | −0.465075 | + | 0.124617i | 0.567657 | − | 2.11852i | 1.91260 | − | 2.27935i | 0 | ||||
32.10 | 0.0696587 | − | 0.796203i | 1.25628 | + | 2.69410i | 1.34053 | + | 0.236371i | 0 | 2.23256 | − | 0.812585i | −4.61801 | + | 1.23739i | 0.695298 | − | 2.59489i | −3.75157 | + | 4.47094i | 0 | ||||
32.11 | 0.122983 | − | 1.40570i | −0.364684 | − | 0.782068i | 0.00874799 | + | 0.00154251i | 0 | −1.14420 | + | 0.416456i | 1.95768 | − | 0.524559i | 0.733667 | − | 2.73808i | 1.44973 | − | 1.72772i | 0 | ||||
32.12 | 0.158607 | − | 1.81289i | −1.18655 | − | 2.54457i | −1.29180 | − | 0.227779i | 0 | −4.80122 | + | 1.74750i | −4.44244 | + | 1.19035i | 0.324178 | − | 1.20985i | −3.13855 | + | 3.74038i | 0 | ||||
32.13 | 0.205646 | − | 2.35054i | −0.0966746 | − | 0.207319i | −3.51313 | − | 0.619460i | 0 | −0.507193 | + | 0.184603i | 4.84313 | − | 1.29771i | −0.957149 | + | 3.57213i | 1.89473 | − | 2.25805i | 0 | ||||
32.14 | 0.219903 | − | 2.51350i | 0.420327 | + | 0.901394i | −4.29970 | − | 0.758153i | 0 | 2.35808 | − | 0.858272i | −0.630014 | + | 0.168812i | −1.54508 | + | 5.76632i | 1.29253 | − | 1.54037i | 0 | ||||
143.1 | −1.56389 | + | 2.23346i | 0.0995607 | − | 1.13798i | −1.85858 | − | 5.10639i | 0 | 2.38595 | + | 2.00205i | 0.344190 | + | 1.28454i | 9.04425 | + | 2.42340i | 1.66933 | + | 0.294348i | 0 | ||||
143.2 | −1.51810 | + | 2.16807i | −0.234015 | + | 2.67481i | −1.71186 | − | 4.70330i | 0 | −5.44391 | − | 4.56798i | −1.16345 | − | 4.34206i | 7.68278 | + | 2.05859i | −4.14540 | − | 0.730946i | 0 | ||||
143.3 | −1.14859 | + | 1.64036i | 0.200764 | − | 2.29474i | −0.687469 | − | 1.88881i | 0 | 3.53360 | + | 2.96504i | −0.402975 | − | 1.50392i | 0.0193927 | + | 0.00519625i | −2.27111 | − | 0.400458i | 0 | ||||
143.4 | −1.12765 | + | 1.61045i | −0.263219 | + | 3.00861i | −0.637911 | − | 1.75265i | 0 | −4.54838 | − | 3.81655i | 0.583268 | + | 2.17679i | −0.256124 | − | 0.0686282i | −6.02800 | − | 1.06290i | 0 | ||||
143.5 | −0.751242 | + | 1.07289i | −0.0801886 | + | 0.916560i | 0.0973225 | + | 0.267391i | 0 | −0.923122 | − | 0.774592i | 0.629559 | + | 2.34955i | −2.89024 | − | 0.774437i | 2.12077 | + | 0.373949i | 0 | ||||
143.6 | −0.558716 | + | 0.797930i | 0.285949 | − | 3.26841i | 0.359512 | + | 0.987752i | 0 | 2.44820 | + | 2.05428i | −0.476978 | − | 1.78010i | −2.87082 | − | 0.769235i | −7.64630 | − | 1.34825i | 0 | ||||
See next 80 embeddings (of 168 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
19.f | odd | 18 | 1 | inner |
95.o | odd | 18 | 1 | inner |
95.r | even | 36 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 475.2.bb.c | ✓ | 168 |
5.b | even | 2 | 1 | inner | 475.2.bb.c | ✓ | 168 |
5.c | odd | 4 | 2 | inner | 475.2.bb.c | ✓ | 168 |
19.f | odd | 18 | 1 | inner | 475.2.bb.c | ✓ | 168 |
95.o | odd | 18 | 1 | inner | 475.2.bb.c | ✓ | 168 |
95.r | even | 36 | 2 | inner | 475.2.bb.c | ✓ | 168 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
475.2.bb.c | ✓ | 168 | 1.a | even | 1 | 1 | trivial |
475.2.bb.c | ✓ | 168 | 5.b | even | 2 | 1 | inner |
475.2.bb.c | ✓ | 168 | 5.c | odd | 4 | 2 | inner |
475.2.bb.c | ✓ | 168 | 19.f | odd | 18 | 1 | inner |
475.2.bb.c | ✓ | 168 | 95.o | odd | 18 | 1 | inner |
475.2.bb.c | ✓ | 168 | 95.r | even | 36 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{168} + 90 T_{2}^{164} + 4437 T_{2}^{160} - 19482 T_{2}^{156} - 6597594 T_{2}^{152} - 468213210 T_{2}^{148} + 529796541 T_{2}^{144} + 462484115838 T_{2}^{140} + 37018558161000 T_{2}^{136} + \cdots + 282429536481 \)
acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).