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Properties

Label 900.2.r.d
Level $900$
Weight $2$
Character orbit 900.r
Analytic conductor $7.187$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,2,Mod(551,900)] 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("900.551"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0,-3,0,0,-2,0,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 3 q^{6} - 2 q^{9} - 7 q^{12} + 15 q^{14} - 7 q^{18} - 4 q^{21} + 2 q^{24} - 12 q^{29} + 16 q^{33} - 6 q^{34} - 9 q^{36} - 30 q^{38} + 30 q^{41} - 21 q^{42} - 12 q^{46} - 60 q^{48} + 24 q^{49} + 9 q^{52}+ \cdots - 12 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1 −1.40339 + 0.174655i −1.15667 − 1.28923i 1.93899 − 0.490217i 0 1.84842 + 1.60727i −0.0160705 − 0.00927831i −2.63554 + 1.02662i −0.324236 + 2.98243i 0
551.2 −1.38273 − 0.296768i −0.0655074 + 1.73081i 1.82386 + 0.820698i 0 0.604229 − 2.37380i 2.84907 + 1.64491i −2.27834 − 1.67606i −2.99142 − 0.226762i 0
551.3 −1.26657 − 0.629125i 1.68149 − 0.415454i 1.20840 + 1.59366i 0 −2.39109 − 0.531664i −0.134598 − 0.0777105i −0.527913 − 2.77872i 2.65480 − 1.39716i 0
551.4 −1.26214 + 0.637958i 1.48991 − 0.883270i 1.18602 − 1.61039i 0 −1.31699 + 2.06532i 1.30473 + 0.753289i −0.469566 + 2.78918i 1.43967 − 2.63199i 0
551.5 −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
551.6 −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
551.7 −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
551.8 −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
551.9 −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
551.10 −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
551.11 −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
551.12 −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
551.13 −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
551.14 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
551.15 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
551.16 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
551.17 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
551.18 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
551.19 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
551.20 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
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 551.24
Significant digits:
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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.d ✓ 48
4.b odd 2 1 inner 900.2.r.d ✓ 48
5.b even 2 1 900.2.r.e yes 48
5.c odd 4 2 900.2.o.d 96
9.d odd 6 1 inner 900.2.r.d ✓ 48
20.d odd 2 1 900.2.r.e yes 48
20.e even 4 2 900.2.o.d 96
36.h even 6 1 inner 900.2.r.d ✓ 48
45.h odd 6 1 900.2.r.e yes 48
45.l even 12 2 900.2.o.d 96
180.n even 6 1 900.2.r.e yes 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 1.a even 1 1 trivial
900.2.r.d ✓ 48 4.b odd 2 1 inner
900.2.r.d ✓ 48 9.d odd 6 1 inner
900.2.r.d ✓ 48 36.h even 6 1 inner
900.2.r.e yes 48 5.b even 2 1
900.2.r.e yes 48 20.d odd 2 1
900.2.r.e yes 48 45.h odd 6 1
900.2.r.e yes 48 180.n even 6 1

Hecke kernels

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

\( T_{7}^{48} - 96 T_{7}^{46} + 5355 T_{7}^{44} - 201092 T_{7}^{42} + 5657979 T_{7}^{40} - 123073686 T_{7}^{38} + \cdots + 160000 \) 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