↑

Properties

Label 405.5.d.a
Level $405$
Weight $5$
Character orbit 405.d
Analytic conductor $41.865$
Analytic rank $0$
Dimension $44$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [405,5,Mod(404,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.404"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 405.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(41.8648350490\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 324 q^{4} + 28 q^{10} + 2116 q^{16} - 8 q^{19} + 296 q^{25} + 2224 q^{31} + 872 q^{34} + 1700 q^{40} - 5668 q^{46} - 10792 q^{49} - 3072 q^{55} - 5564 q^{61} + 8348 q^{64} - 9564 q^{70} + 3552 q^{76}+ \cdots + 37652 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
404.1 −7.50729 0 40.3595 −20.1247 + 14.8322i 0 − 69.8446i −182.874 0 151.082 − 111.350i
404.2 −7.50729 0 40.3595 −20.1247 − 14.8322i 0 69.8446i −182.874 0 151.082 + 111.350i
404.3 −7.28232 0 37.0321 21.3751 − 12.9656i 0 − 44.4060i −153.163 0 −155.660 + 94.4195i
404.4 −7.28232 0 37.0321 21.3751 + 12.9656i 0 44.4060i −153.163 0 −155.660 − 94.4195i
404.5 −6.24699 0 23.0249 13.1570 − 21.2578i 0 − 11.6987i −43.8846 0 −82.1915 + 132.797i
404.6 −6.24699 0 23.0249 13.1570 + 21.2578i 0 11.6987i −43.8846 0 −82.1915 − 132.797i
404.7 −5.74722 0 17.0305 −17.1405 − 18.1989i 0 − 47.8220i −5.92246 0 98.5104 + 104.593i
404.8 −5.74722 0 17.0305 −17.1405 + 18.1989i 0 47.8220i −5.92246 0 98.5104 − 104.593i
404.9 −5.23381 0 11.3928 3.38042 + 24.7704i 0 − 57.8140i 24.1135 0 −17.6925 − 129.644i
404.10 −5.23381 0 11.3928 3.38042 − 24.7704i 0 57.8140i 24.1135 0 −17.6925 + 129.644i
404.11 −4.61842 0 5.32983 −22.1666 + 11.5603i 0 52.6736i 49.2794 0 102.375 − 53.3902i
404.12 −4.61842 0 5.32983 −22.1666 − 11.5603i 0 − 52.6736i 49.2794 0 102.375 + 53.3902i
404.13 −3.29622 0 −5.13494 24.8093 + 3.08189i 0 − 74.7640i 69.6654 0 −81.7769 − 10.1586i
404.14 −3.29622 0 −5.13494 24.8093 − 3.08189i 0 74.7640i 69.6654 0 −81.7769 + 10.1586i
404.15 −3.19213 0 −5.81034 17.8434 + 17.5104i 0 24.3518i 69.6213 0 −56.9583 − 55.8953i
404.16 −3.19213 0 −5.81034 17.8434 − 17.5104i 0 − 24.3518i 69.6213 0 −56.9583 + 55.8953i
404.17 −2.04378 0 −11.8230 −9.29647 + 23.2072i 0 − 2.42178i 56.8640 0 19.0000 − 47.4305i
404.18 −2.04378 0 −11.8230 −9.29647 − 23.2072i 0 2.42178i 56.8640 0 19.0000 + 47.4305i
404.19 −1.09451 0 −14.8020 −24.2165 − 6.20970i 0 − 19.5383i 33.7133 0 26.5053 + 6.79660i
404.20 −1.09451 0 −14.8020 −24.2165 + 6.20970i 0 19.5383i 33.7133 0 26.5053 − 6.79660i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 404.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.5.d.a 44
3.b odd 2 1 inner 405.5.d.a 44
5.b even 2 1 inner 405.5.d.a 44
9.c even 3 1 45.5.h.a ✓ 44
9.c even 3 1 135.5.h.a 44
9.d odd 6 1 45.5.h.a ✓ 44
9.d odd 6 1 135.5.h.a 44
15.d odd 2 1 inner 405.5.d.a 44
45.h odd 6 1 45.5.h.a ✓ 44
45.h odd 6 1 135.5.h.a 44
45.j even 6 1 45.5.h.a ✓ 44
45.j even 6 1 135.5.h.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.5.h.a ✓ 44 9.c even 3 1
45.5.h.a ✓ 44 9.d odd 6 1
45.5.h.a ✓ 44 45.h odd 6 1
45.5.h.a ✓ 44 45.j even 6 1
135.5.h.a 44 9.c even 3 1
135.5.h.a 44 9.d odd 6 1
135.5.h.a 44 45.h odd 6 1
135.5.h.a 44 45.j even 6 1
405.5.d.a 44 1.a even 1 1 trivial
405.5.d.a 44 3.b odd 2 1 inner
405.5.d.a 44 5.b even 2 1 inner
405.5.d.a 44 15.d odd 2 1 inner