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Properties

Label 900.2.r.e
Level $900$
Weight $2$
Character orbit 900.r
Analytic conductor $7.187$
Analytic rank $0$
Dimension $48$
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] = [900,2,Mod(551,900)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.551");
 
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.r (of order \(6\), degree \(2\), 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: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{6} - 2 q^{9} + 7 q^{12} + 15 q^{14} + 7 q^{18} - 4 q^{21} + 2 q^{24} - 12 q^{29} - 16 q^{33} - 6 q^{34} - 9 q^{36} + 30 q^{38} + 30 q^{41} + 21 q^{42} - 12 q^{46} + 60 q^{48} + 24 q^{49} - 9 q^{52} - 14 q^{54} - 42 q^{56} + 24 q^{57} + 18 q^{58} + 6 q^{64} - 53 q^{66} + 69 q^{68} - 12 q^{69} - 9 q^{72} - 12 q^{73} - 42 q^{74} + 6 q^{76} + 24 q^{77} + 61 q^{78} + 30 q^{81} - 18 q^{82} - 73 q^{84} - 54 q^{86} - 60 q^{92} - 40 q^{93} - 9 q^{94} - 54 q^{96} + 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1 −1.41420 + 0.00541195i 1.58917 + 0.688876i 1.99994 − 0.0153072i 0 −2.25113 − 0.965610i 3.52902 + 2.03748i −2.82824 + 0.0324711i 2.05090 + 2.18948i 0
551.2 −1.41134 + 0.0900845i 1.25131 − 1.19759i 1.98377 − 0.254280i 0 −1.65815 + 1.80293i −4.22426 − 2.43888i −2.77687 + 0.537583i 0.131568 − 2.99711i 0
551.3 −1.33030 − 0.479908i −0.730690 − 1.57038i 1.53938 + 1.27684i 0 0.218396 + 2.43973i 2.37870 + 1.37334i −1.43506 − 2.43734i −1.93218 + 2.29492i 0
551.4 −1.30622 + 0.542034i −1.72652 − 0.138348i 1.41240 − 1.41603i 0 2.33019 − 0.755118i −0.818015 − 0.472281i −1.07736 + 2.61520i 2.96172 + 0.477722i 0
551.5 −1.08076 − 0.912116i 0.730690 + 1.57038i 0.336088 + 1.97156i 0 0.642667 − 2.36368i −2.37870 − 1.37334i 1.43506 − 2.43734i −1.93218 + 2.29492i 0
551.6 −1.05557 + 0.941160i −0.248628 + 1.71411i 0.228435 − 1.98691i 0 −1.35081 − 2.04336i 1.48816 + 0.859188i 1.62887 + 2.31231i −2.87637 − 0.852352i 0
551.7 −0.834840 + 1.14151i 1.61468 + 0.626740i −0.606083 − 1.90595i 0 −2.06343 + 1.31994i −3.04461 − 1.75781i 2.68165 + 0.899318i 2.21439 + 2.02397i 0
551.8 −0.824243 + 1.14918i −0.0441645 − 1.73149i −0.641247 − 1.89441i 0 2.02620 + 1.37641i 0.927384 + 0.535425i 2.70557 + 0.824545i −2.99610 + 0.152941i 0
551.9 −0.702415 − 1.22744i −1.58917 − 0.688876i −1.01323 + 1.72435i 0 0.270699 + 2.43449i −3.52902 − 2.03748i 2.82824 + 0.0324711i 2.05090 + 2.18948i 0
551.10 −0.627655 − 1.26730i −1.25131 + 1.19759i −1.21210 + 1.59085i 0 2.30310 + 0.834117i 4.22426 + 2.43888i 2.77687 + 0.537583i 0.131568 − 2.99711i 0
551.11 −0.183692 − 1.40223i 1.72652 + 0.138348i −1.93251 + 0.515159i 0 −0.123151 − 2.44639i 0.818015 + 0.472281i 1.07736 + 2.61520i 2.96172 + 0.477722i 0
551.12 0.0231729 + 1.41402i −1.04098 + 1.38433i −1.99893 + 0.0655341i 0 −1.98160 − 1.43989i −3.63208 − 2.09698i −0.138988 − 2.82501i −0.832737 − 2.88211i 0
551.13 0.0785846 + 1.41203i 1.48991 − 0.883270i −1.98765 + 0.221927i 0 1.36429 + 2.03438i 1.30473 + 0.753289i −0.469566 − 2.78918i 1.43967 − 2.63199i 0
551.14 0.287286 − 1.38473i 0.248628 − 1.71411i −1.83493 − 0.795625i 0 −2.30215 − 0.836722i −1.48816 − 0.859188i −1.62887 + 2.31231i −2.87637 − 0.852352i 0
551.15 0.550438 + 1.30270i −1.15667 − 1.28923i −1.39404 + 1.43411i 0 1.04280 − 2.21643i −0.0160705 − 0.00927831i −2.63554 − 1.02662i −0.324236 + 2.98243i 0
551.16 0.571155 − 1.29375i −1.61468 − 0.626740i −1.34756 − 1.47786i 0 −1.73308 + 1.73102i 3.04461 + 1.75781i −2.68165 + 0.899318i 2.21439 + 2.02397i 0
551.17 0.583101 − 1.28841i 0.0441645 + 1.73149i −1.31999 − 1.50254i 0 2.25661 + 0.952730i −0.927384 − 0.535425i −2.70557 + 0.824545i −2.99610 + 0.152941i 0
551.18 0.948372 + 1.04909i −0.0655074 + 1.73081i −0.201183 + 1.98986i 0 −1.87790 + 1.57273i 2.84907 + 1.64491i −2.27834 + 1.67606i −2.99142 − 0.226762i 0
551.19 1.17812 + 0.782320i 1.68149 − 0.415454i 0.775952 + 1.84334i 0 2.30602 + 0.826004i −0.134598 − 0.0777105i −0.527913 + 2.77872i 2.65480 − 1.39716i 0
551.20 1.23617 − 0.686944i 1.04098 − 1.38433i 1.05622 − 1.69835i 0 0.335864 − 2.42635i 3.63208 + 2.09698i 0.138988 − 2.82501i −0.832737 − 2.88211i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.24
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{7}^{48} - 96 T_{7}^{46} + 5355 T_{7}^{44} - 201092 T_{7}^{42} + 5657979 T_{7}^{40} - 123073686 T_{7}^{38} + \cdots + 160000 \) Copy content Toggle raw display
\( T_{13}^{24} + 84 T_{13}^{22} - 68 T_{13}^{21} + 4740 T_{13}^{20} - 4128 T_{13}^{19} + 148442 T_{13}^{18} + \cdots + 40000 \) Copy content Toggle raw display