Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [471,4,Mod(313,471)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(471, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("471.313");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 471 = 3 \cdot 157 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 471.b (of order \(2\), degree \(1\), 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: | \(27.7898996127\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
313.1 | − | 5.39623i | −3.00000 | −21.1193 | − | 16.5163i | 16.1887i | − | 25.8596i | 70.7949i | 9.00000 | −89.1257 | |||||||||||||||
313.2 | − | 5.31164i | −3.00000 | −20.2135 | 13.7454i | 15.9349i | 18.9295i | 64.8739i | 9.00000 | 73.0104 | |||||||||||||||||
313.3 | − | 5.19343i | −3.00000 | −18.9718 | − | 15.3454i | 15.5803i | 23.0710i | 56.9811i | 9.00000 | −79.6953 | ||||||||||||||||
313.4 | − | 5.06578i | −3.00000 | −17.6621 | 10.9807i | 15.1973i | − | 24.6884i | 48.9461i | 9.00000 | 55.6257 | ||||||||||||||||
313.5 | − | 4.67506i | −3.00000 | −13.8562 | − | 4.61864i | 14.0252i | − | 11.3073i | 27.3780i | 9.00000 | −21.5924 | |||||||||||||||
313.6 | − | 4.30375i | −3.00000 | −10.5223 | − | 4.32730i | 12.9113i | 21.9616i | 10.8553i | 9.00000 | −18.6236 | ||||||||||||||||
313.7 | − | 4.28965i | −3.00000 | −10.4011 | − | 1.35162i | 12.8690i | 9.03182i | 10.3000i | 9.00000 | −5.79797 | ||||||||||||||||
313.8 | − | 3.78788i | −3.00000 | −6.34802 | − | 9.26420i | 11.3636i | − | 4.30898i | − | 6.25748i | 9.00000 | −35.0916 | ||||||||||||||
313.9 | − | 3.68048i | −3.00000 | −5.54594 | 17.9346i | 11.0414i | − | 17.3085i | − | 9.03212i | 9.00000 | 66.0080 | |||||||||||||||
313.10 | − | 3.30421i | −3.00000 | −2.91779 | 13.7388i | 9.91262i | 33.0358i | − | 16.7927i | 9.00000 | 45.3958 | ||||||||||||||||
313.11 | − | 3.18058i | −3.00000 | −2.11606 | − | 2.38612i | 9.54173i | − | 14.3272i | − | 18.7143i | 9.00000 | −7.58923 | ||||||||||||||
313.12 | − | 2.64007i | −3.00000 | 1.03005 | − | 13.5204i | 7.92020i | − | 34.8670i | − | 23.8399i | 9.00000 | −35.6946 | ||||||||||||||
313.13 | − | 2.63265i | −3.00000 | 1.06918 | − | 20.9406i | 7.89794i | 20.4039i | − | 23.8759i | 9.00000 | −55.1291 | |||||||||||||||
313.14 | − | 2.09701i | −3.00000 | 3.60255 | 1.98891i | 6.29103i | 9.73610i | − | 24.3307i | 9.00000 | 4.17077 | ||||||||||||||||
313.15 | − | 1.46936i | −3.00000 | 5.84098 | − | 4.94652i | 4.40809i | 12.8754i | − | 20.3374i | 9.00000 | −7.26823 | |||||||||||||||
313.16 | − | 1.38921i | −3.00000 | 6.07011 | 4.30021i | 4.16762i | − | 36.7534i | − | 19.5463i | 9.00000 | 5.97388 | |||||||||||||||
313.17 | − | 0.999086i | −3.00000 | 7.00183 | 10.0353i | 2.99726i | 4.69571i | − | 14.9881i | 9.00000 | 10.0261 | ||||||||||||||||
313.18 | − | 0.783695i | −3.00000 | 7.38582 | 11.1397i | 2.35108i | 5.77258i | − | 12.0578i | 9.00000 | 8.73009 | ||||||||||||||||
313.19 | − | 0.500423i | −3.00000 | 7.74958 | 10.0481i | 1.50127i | − | 27.3201i | − | 7.88145i | 9.00000 | 5.02829 | |||||||||||||||
313.20 | − | 0.275605i | −3.00000 | 7.92404 | − | 19.4532i | 0.826816i | − | 15.2471i | − | 4.38875i | 9.00000 | −5.36141 | ||||||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
157.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 471.4.b.a | ✓ | 40 |
157.b | even | 2 | 1 | inner | 471.4.b.a | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
471.4.b.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
471.4.b.a | ✓ | 40 | 157.b | even | 2 | 1 | inner |