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Properties

Label 900.3.u.d
Level $900$
Weight $3$
Character orbit 900.u
Analytic conductor $24.523$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,3,Mod(149,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.149");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.u (of order \(6\), degree \(2\), not 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: \(24.5232237924\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 28 q^{9} - 4 q^{19} + 2 q^{21} - 18 q^{29} + 16 q^{31} - 38 q^{39} + 108 q^{41} + 90 q^{49} + 180 q^{51} - 18 q^{59} - 110 q^{61} + 294 q^{69} - 22 q^{79} - 260 q^{81} - 268 q^{91} - 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 0 −2.83603 − 0.978225i 0 0 0 4.69840 − 2.71262i 0 7.08615 + 5.54856i 0
149.2 0 −2.70777 + 1.29149i 0 0 0 3.00081 − 1.73252i 0 5.66408 − 6.99415i 0
149.3 0 −2.56994 − 1.54771i 0 0 0 −1.25581 + 0.725042i 0 4.20921 + 7.95503i 0
149.4 0 −2.37034 + 1.83888i 0 0 0 −7.15437 + 4.13058i 0 2.23701 − 8.71756i 0
149.5 0 −1.60876 + 2.53217i 0 0 0 8.13249 − 4.69530i 0 −3.82376 − 8.14732i 0
149.6 0 −1.28947 − 2.70874i 0 0 0 −9.36178 + 5.40503i 0 −5.67451 + 6.98569i 0
149.7 0 −0.708005 + 2.91526i 0 0 0 −8.59526 + 4.96248i 0 −7.99746 − 4.12804i 0
149.8 0 −0.386835 − 2.97496i 0 0 0 4.03103 − 2.32731i 0 −8.70072 + 2.30163i 0
149.9 0 0.386835 + 2.97496i 0 0 0 −4.03103 + 2.32731i 0 −8.70072 + 2.30163i 0
149.10 0 0.708005 − 2.91526i 0 0 0 8.59526 − 4.96248i 0 −7.99746 − 4.12804i 0
149.11 0 1.28947 + 2.70874i 0 0 0 9.36178 − 5.40503i 0 −5.67451 + 6.98569i 0
149.12 0 1.60876 − 2.53217i 0 0 0 −8.13249 + 4.69530i 0 −3.82376 − 8.14732i 0
149.13 0 2.37034 − 1.83888i 0 0 0 7.15437 − 4.13058i 0 2.23701 − 8.71756i 0
149.14 0 2.56994 + 1.54771i 0 0 0 1.25581 − 0.725042i 0 4.20921 + 7.95503i 0
149.15 0 2.70777 − 1.29149i 0 0 0 −3.00081 + 1.73252i 0 5.66408 − 6.99415i 0
149.16 0 2.83603 + 0.978225i 0 0 0 −4.69840 + 2.71262i 0 7.08615 + 5.54856i 0
749.1 0 −2.83603 + 0.978225i 0 0 0 4.69840 + 2.71262i 0 7.08615 − 5.54856i 0
749.2 0 −2.70777 − 1.29149i 0 0 0 3.00081 + 1.73252i 0 5.66408 + 6.99415i 0
749.3 0 −2.56994 + 1.54771i 0 0 0 −1.25581 − 0.725042i 0 4.20921 − 7.95503i 0
749.4 0 −2.37034 − 1.83888i 0 0 0 −7.15437 − 4.13058i 0 2.23701 + 8.71756i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.u.d 32
3.b odd 2 1 2700.3.u.d 32
5.b even 2 1 inner 900.3.u.d 32
5.c odd 4 1 900.3.p.d ✓ 16
5.c odd 4 1 900.3.p.e yes 16
9.c even 3 1 2700.3.u.d 32
9.d odd 6 1 inner 900.3.u.d 32
15.d odd 2 1 2700.3.u.d 32
15.e even 4 1 2700.3.p.d 16
15.e even 4 1 2700.3.p.e 16
45.h odd 6 1 inner 900.3.u.d 32
45.j even 6 1 2700.3.u.d 32
45.k odd 12 1 2700.3.p.d 16
45.k odd 12 1 2700.3.p.e 16
45.l even 12 1 900.3.p.d ✓ 16
45.l even 12 1 900.3.p.e yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.p.d ✓ 16 5.c odd 4 1
900.3.p.d ✓ 16 45.l even 12 1
900.3.p.e yes 16 5.c odd 4 1
900.3.p.e yes 16 45.l even 12 1
900.3.u.d 32 1.a even 1 1 trivial
900.3.u.d 32 5.b even 2 1 inner
900.3.u.d 32 9.d odd 6 1 inner
900.3.u.d 32 45.h odd 6 1 inner
2700.3.p.d 16 15.e even 4 1
2700.3.p.d 16 45.k odd 12 1
2700.3.p.e 16 15.e even 4 1
2700.3.p.e 16 45.k odd 12 1
2700.3.u.d 32 3.b odd 2 1
2700.3.u.d 32 9.c even 3 1
2700.3.u.d 32 15.d odd 2 1
2700.3.u.d 32 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{32} - 437 T_{7}^{30} + 114123 T_{7}^{28} - 19658878 T_{7}^{26} + 2517239723 T_{7}^{24} + \cdots + 12\!\cdots\!76 \) acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display