Properties

Label 900.2.o.d
Level $900$
Weight $2$
Character orbit 900.o
Analytic conductor $7.187$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(299,900)]
 
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([3, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.299");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.o (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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 6 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 6 q^{6} + 4 q^{9} - 30 q^{14} - 8 q^{21} - 4 q^{24} + 24 q^{29} + 12 q^{34} - 18 q^{36} + 60 q^{41} - 24 q^{46} - 48 q^{49} + 28 q^{54} - 84 q^{56} - 12 q^{64} - 106 q^{66} + 24 q^{69} + 84 q^{74} + 12 q^{76} + 60 q^{81} + 146 q^{84} - 108 q^{86} + 18 q^{94} - 108 q^{96}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1 −1.41402 + 0.0231729i 1.38433 + 1.04098i 1.99893 0.0655341i 0 −1.98160 1.43989i 2.09698 3.63208i −2.82501 + 0.138988i 0.832737 + 2.88211i 0
299.2 −1.41203 + 0.0785846i −0.883270 1.48991i 1.98765 0.221927i 0 1.36429 + 2.03438i −0.753289 + 1.30473i −2.78918 + 0.469566i −1.43967 + 2.63199i 0
299.3 −1.40223 + 0.183692i −0.138348 + 1.72652i 1.93251 0.515159i 0 −0.123151 2.44639i 0.472281 0.818015i −2.61520 + 1.07736i −2.96172 0.477722i 0
299.4 −1.38473 0.287286i 1.71411 + 0.248628i 1.83493 + 0.795625i 0 −2.30215 0.836722i −0.859188 + 1.48816i −2.31231 1.62887i 2.87637 + 0.852352i 0
299.5 −1.30270 + 0.550438i −1.28923 + 1.15667i 1.39404 1.43411i 0 1.04280 2.21643i 0.00927831 0.0160705i −1.02662 + 2.63554i 0.324236 2.98243i 0
299.6 −1.29375 0.571155i 0.626740 1.61468i 1.34756 + 1.47786i 0 −1.73308 + 1.73102i 1.75781 3.04461i −0.899318 2.68165i −2.21439 2.02397i 0
299.7 −1.28841 0.583101i −1.73149 + 0.0441645i 1.31999 + 1.50254i 0 2.25661 + 0.952730i −0.535425 + 0.927384i −0.824545 2.70557i 2.99610 0.152941i 0
299.8 −1.26730 + 0.627655i −1.19759 1.25131i 1.21210 1.59085i 0 2.30310 + 0.834117i 2.43888 4.22426i −0.537583 + 2.77687i −0.131568 + 2.99711i 0
299.9 −1.22744 + 0.702415i 0.688876 1.58917i 1.01323 1.72435i 0 0.270699 + 2.43449i −2.03748 + 3.52902i −0.0324711 + 2.82824i −2.05090 2.18948i 0
299.10 −1.14918 0.824243i −1.73149 + 0.0441645i 0.641247 + 1.89441i 0 2.02620 + 1.37641i −0.535425 + 0.927384i 0.824545 2.70557i 2.99610 0.152941i 0
299.11 −1.14151 0.834840i 0.626740 1.61468i 0.606083 + 1.90595i 0 −2.06343 + 1.31994i 1.75781 3.04461i 0.899318 2.68165i −2.21439 2.02397i 0
299.12 −1.04909 + 0.948372i 1.73081 + 0.0655074i 0.201183 1.98986i 0 −1.87790 + 1.57273i −1.64491 + 2.84907i 1.67606 + 2.27834i 2.99142 + 0.226762i 0
299.13 −0.941160 1.05557i 1.71411 + 0.248628i −0.228435 + 1.98691i 0 −1.35081 2.04336i −0.859188 + 1.48816i 2.31231 1.62887i 2.87637 + 0.852352i 0
299.14 −0.912116 + 1.08076i −1.57038 + 0.730690i −0.336088 1.97156i 0 0.642667 2.36368i −1.37334 + 2.37870i 2.43734 + 1.43506i 1.93218 2.29492i 0
299.15 −0.782320 + 1.17812i −0.415454 1.68149i −0.775952 1.84334i 0 2.30602 + 0.826004i 0.0777105 0.134598i 2.77872 + 0.527913i −2.65480 + 1.39716i 0
299.16 −0.686944 1.23617i 1.38433 + 1.04098i −1.05622 + 1.69835i 0 0.335864 2.42635i 2.09698 3.63208i 2.82501 + 0.138988i 0.832737 + 2.88211i 0
299.17 −0.637958 1.26214i −0.883270 1.48991i −1.18602 + 1.61039i 0 −1.31699 + 2.06532i −0.753289 + 1.30473i 2.78918 + 0.469566i −1.43967 + 2.63199i 0
299.18 −0.629125 + 1.26657i 0.415454 + 1.68149i −1.20840 1.59366i 0 −2.39109 0.531664i −0.0777105 + 0.134598i 2.77872 0.527913i −2.65480 + 1.39716i 0
299.19 −0.542034 1.30622i −0.138348 + 1.72652i −1.41240 + 1.41603i 0 2.33019 0.755118i 0.472281 0.818015i 2.61520 + 1.07736i −2.96172 0.477722i 0
299.20 −0.479908 + 1.33030i 1.57038 0.730690i −1.53938 1.27684i 0 0.218396 + 2.43973i 1.37334 2.37870i 2.43734 1.43506i 1.93218 2.29492i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.48
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.o.d 96
4.b odd 2 1 inner 900.2.o.d 96
5.b even 2 1 inner 900.2.o.d 96
5.c odd 4 1 900.2.r.d 48
5.c odd 4 1 900.2.r.e yes 48
9.d odd 6 1 inner 900.2.o.d 96
20.d odd 2 1 inner 900.2.o.d 96
20.e even 4 1 900.2.r.d 48
20.e even 4 1 900.2.r.e yes 48
36.h even 6 1 inner 900.2.o.d 96
45.h odd 6 1 inner 900.2.o.d 96
45.l even 12 1 900.2.r.d 48
45.l even 12 1 900.2.r.e yes 48
180.n even 6 1 inner 900.2.o.d 96
180.v odd 12 1 900.2.r.d 48
180.v odd 12 1 900.2.r.e yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.o.d 96 1.a even 1 1 trivial
900.2.o.d 96 4.b odd 2 1 inner
900.2.o.d 96 5.b even 2 1 inner
900.2.o.d 96 9.d odd 6 1 inner
900.2.o.d 96 20.d odd 2 1 inner
900.2.o.d 96 36.h even 6 1 inner
900.2.o.d 96 45.h odd 6 1 inner
900.2.o.d 96 180.n even 6 1 inner
900.2.r.d 48 5.c odd 4 1
900.2.r.d 48 20.e even 4 1
900.2.r.d 48 45.l even 12 1
900.2.r.d 48 180.v odd 12 1
900.2.r.e yes 48 5.c odd 4 1
900.2.r.e yes 48 20.e even 4 1
900.2.r.e yes 48 45.l even 12 1
900.2.r.e yes 48 180.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{48} + 96 T_{7}^{46} + 5355 T_{7}^{44} + 201092 T_{7}^{42} + 5657979 T_{7}^{40} + 123073686 T_{7}^{38} + 2133295130 T_{7}^{36} + 29672232036 T_{7}^{34} + 334230570957 T_{7}^{32} + \cdots + 160000 \) Copy content Toggle raw display
\( T_{13}^{48} - 168 T_{13}^{46} + 16536 T_{13}^{44} - 1088580 T_{13}^{42} + 53568642 T_{13}^{40} - 2029962192 T_{13}^{38} + 61142871718 T_{13}^{36} - 1471147003398 T_{13}^{34} + \cdots + 1600000000 \) Copy content Toggle raw display